LMFDB - Embedded newform 115.7.c.c.114.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,7,Mod(114,115)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 7, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("115.114");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	115.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$26.4562196163$
Analytic rank:	$0$
Dimension:	$68$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			114.2
Character	$\chi$	$=$	115.114
Dual form			115.7.c.c.114.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.94808i q^{2} -27.5624i q^{3} +39.5165 q^{4} +(31.3018 + 121.017i) q^{5} +136.381 q^{6} -630.980 q^{7} +512.208i q^{8} -30.6878 q^{9} +O(q^{10})$	\(q+4.94808i q^{2} -27.5624i q^{3} +39.5165 q^{4} +(31.3018 + 121.017i) q^{5} +136.381 q^{6} -630.980 q^{7} +512.208i q^{8} -30.6878 q^{9} +(-598.804 + 154.884i) q^{10} -1384.76i q^{11} -1089.17i q^{12} -362.389i q^{13} -3122.14i q^{14} +(3335.53 - 862.753i) q^{15} -5.39320 q^{16} -4368.74 q^{17} -151.846i q^{18} -5766.01i q^{19} +(1236.94 + 4782.18i) q^{20} +17391.3i q^{21} +6851.92 q^{22} +(-7067.26 - 9904.02i) q^{23} +14117.7 q^{24} +(-13665.4 + 7576.11i) q^{25} +1793.13 q^{26} -19247.2i q^{27} -24934.1 q^{28} -9650.46 q^{29} +(4268.97 + 16504.5i) q^{30} +9775.73 q^{31} +32754.6i q^{32} -38167.4 q^{33} -21616.9i q^{34} +(-19750.8 - 76359.5i) q^{35} -1212.67 q^{36} -34964.1 q^{37} +28530.7 q^{38} -9988.32 q^{39} +(-61986.1 + 16033.0i) q^{40} -111106. q^{41} -86053.8 q^{42} +4839.94 q^{43} -54720.9i q^{44} +(-960.581 - 3713.75i) q^{45} +(49005.9 - 34969.4i) q^{46} -112432. i q^{47} +148.650i q^{48} +280487. q^{49} +(-37487.2 - 67617.5i) q^{50} +120413. i q^{51} -14320.3i q^{52} -86977.0 q^{53} +95236.7 q^{54} +(167580. - 43345.5i) q^{55} -323193. i q^{56} -158925. q^{57} -47751.3i q^{58} +81280.7 q^{59} +(131809. - 34092.9i) q^{60} -216368. i q^{61} +48371.1i q^{62} +19363.4 q^{63} -162418. q^{64} +(43855.4 - 11343.4i) q^{65} -188856. i q^{66} -6297.15 q^{67} -172637. q^{68} +(-272979. + 194791. i) q^{69} +(377833. - 97728.5i) q^{70} -592554. q^{71} -15718.5i q^{72} -166216. i q^{73} -173005. i q^{74} +(208816. + 376652. i) q^{75} -227852. i q^{76} +873758. i q^{77} -49423.0i q^{78} +931872. i q^{79} +(-168.817 - 652.671i) q^{80} -552871. q^{81} -549760. i q^{82} -88821.9 q^{83} +687245. i q^{84} +(-136749. - 528694. i) q^{85} +23948.4i q^{86} +265990. i q^{87} +709287. q^{88} +811194. i q^{89} +(18375.9 - 4753.03i) q^{90} +228660. i q^{91} +(-279273. - 391372. i) q^{92} -269443. i q^{93} +556324. q^{94} +(697787. - 180486. i) q^{95} +902797. q^{96} +979818. q^{97} +1.38787e6i q^{98} +42495.3i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 68 q - 2440 q^{4} + 352 q^{6} - 16908 q^{9}+O(q^{10}) $ 68 * q - 2440 * q^4 + 352 * q^6 - 16908 * q^9	$ 68 q - 2440 q^{4} + 352 q^{6} - 16908 q^{9} + 66968 q^{16} - 30916 q^{24} + 32588 q^{25} - 22072 q^{26} + 103360 q^{29} - 17256 q^{31} - 358168 q^{35} + 451984 q^{36} + 192432 q^{39} - 183552 q^{41} - 397956 q^{46} + 806756 q^{49} - 749960 q^{50} - 1638436 q^{54} - 1752 q^{55} - 505552 q^{59} - 4095100 q^{64} + 1354876 q^{69} + 1196604 q^{70} + 493688 q^{71} + 3178568 q^{75} + 2473820 q^{81} + 3306336 q^{85} - 3770196 q^{94} + 896144 q^{95} + 16928136 q^{96}+O(q^{100}) $ 68 * q - 2440 * q^4 + 352 * q^6 - 16908 * q^9 + 66968 * q^16 - 30916 * q^24 + 32588 * q^25 - 22072 * q^26 + 103360 * q^29 - 17256 * q^31 - 358168 * q^35 + 451984 * q^36 + 192432 * q^39 - 183552 * q^41 - 397956 * q^46 + 806756 * q^49 - 749960 * q^50 - 1638436 * q^54 - 1752 * q^55 - 505552 * q^59 - 4095100 * q^64 + 1354876 * q^69 + 1196604 * q^70 + 493688 * q^71 + 3178568 * q^75 + 2473820 * q^81 + 3306336 * q^85 - 3770196 * q^94 + 896144 * q^95 + 16928136 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/115\mathbb{Z}\right)^\times$.

$n$	$47$	$51$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			4.94808i			0.618510i	0.950979	+	0.309255i	$0.100080\pi$
$2$			4.94808i			0.618510i	−0.950979	+	0.309255i	$0.899920\pi$
$3$		−	27.5624i		−	1.02083i	−0.859928	−	0.510415i	$-0.829492\pi$
$3$		−	27.5624i		−	1.02083i	0.859928	−	0.510415i	$-0.170508\pi$
$4$	39.5165			0.617445
$5$	31.3018	+	121.017i	0.250414	+	0.968139i
$5$	31.3018	+	121.017i	0.250414	+	0.968139i
$6$	136.381			0.631394
$7$	−630.980			−1.83959			−0.919796	−	0.392397i	$-0.871646\pi$
$7$	−630.980			−1.83959			−0.919796	+	0.392397i	$0.871646\pi$
$8$			512.208i			1.00041i
$9$	−30.6878			−0.0420957
$10$	−598.804	+	154.884i	−0.598804	+	0.154884i
$11$		−	1384.76i		−	1.04039i	−0.854047	−	0.520196i	$-0.825859\pi$
$11$		−	1384.76i		−	1.04039i	0.854047	−	0.520196i	$-0.174141\pi$
$12$		−	1089.17i		−	0.630307i
$13$		−	362.389i		−	0.164947i	−0.996593	−	0.0824736i	$-0.973718\pi$
$13$		−	362.389i		−	0.164947i	0.996593	−	0.0824736i	$-0.0262820\pi$
$14$		−	3122.14i		−	1.13781i
$15$	3335.53	−	862.753i	0.988306	−	0.255630i
$16$	−5.39320			−0.00131670
$17$	−4368.74			−0.889221			−0.444611	−	0.895724i	$-0.646658\pi$
$17$	−4368.74			−0.889221			−0.444611	+	0.895724i	$0.646658\pi$
$18$		−	151.846i		−	0.0260366i
$19$		−	5766.01i		−	0.840649i	−0.907374	−	0.420324i	$-0.861916\pi$
$19$		−	5766.01i		−	0.840649i	0.907374	−	0.420324i	$-0.138084\pi$
$20$	1236.94	+	4782.18i	0.154617	+	0.597772i
$21$			17391.3i			1.87791i
$22$	6851.92			0.643494
$23$	−7067.26	−	9904.02i	−0.580855	−	0.814007i
$23$	−7067.26	−	9904.02i	−0.580855	−	0.814007i
$24$	14117.7			1.02125
$25$	−13665.4	+	7576.11i	−0.874586	+	0.484871i
$26$	1793.13			0.102022
$27$		−	19247.2i		−	0.977858i
$28$	−24934.1			−1.13585
$29$	−9650.46			−0.395689			−0.197845	−	0.980233i	$-0.563394\pi$
$29$	−9650.46			−0.395689			−0.197845	+	0.980233i	$0.563394\pi$
$30$	4268.97	+	16504.5i	0.158110	+	0.611277i
$31$	9775.73			0.328144			0.164072	−	0.986448i	$-0.447537\pi$
$31$	9775.73			0.328144			0.164072	+	0.986448i	$0.447537\pi$
$32$			32754.6i			0.999592i
$33$	−38167.4			−1.06206
$34$		−	21616.9i		−	0.549993i
$35$	−19750.8	−	76359.5i	−0.460660	−	1.78098i
$36$	−1212.67			−0.0259918
$37$	−34964.1			−0.690266			−0.345133	−	0.938554i	$-0.612166\pi$
$37$	−34964.1			−0.690266			−0.345133	+	0.938554i	$0.612166\pi$
$38$	28530.7			0.519950
$39$	−9988.32			−0.168383
$40$	−61986.1	+	16033.0i	−0.968532	+	0.250516i
$41$	−111106.			−1.61207			−0.806037	−	0.591865i	$-0.798392\pi$
$41$	−111106.			−1.61207			−0.806037	+	0.591865i	$0.798392\pi$
$42$	−86053.8			−1.16151
$43$	4839.94			0.0608743			0.0304372	−	0.999537i	$-0.490310\pi$
$43$	4839.94			0.0608743			0.0304372	+	0.999537i	$0.490310\pi$
$44$		−	54720.9i		−	0.642385i
$45$	−960.581	−	3713.75i	−0.0105414	−	0.0407545i
$46$	49005.9	−	34969.4i	0.503472	−	0.359265i
$47$		−	112432.i		−	1.08292i	−0.840726	−	0.541461i	$-0.817871\pi$
$47$		−	112432.i		−	1.08292i	0.840726	−	0.541461i	$-0.182129\pi$
$48$			148.650i			0.00134413i
$49$	280487.			2.38410
$50$	−37487.2	−	67617.5i	−0.299898	−	0.540940i
$51$			120413.i			0.907745i
$52$		−	14320.3i		−	0.101846i
$53$	−86977.0			−0.584221			−0.292110	−	0.956385i	$-0.594357\pi$
$53$	−86977.0			−0.584221			−0.292110	+	0.956385i	$0.594357\pi$
$54$	95236.7			0.604815
$55$	167580.	−	43345.5i	1.00724	−	0.260529i
$56$		−	323193.i		−	1.84034i
$57$	−158925.			−0.858160
$58$		−	47751.3i		−	0.244738i
$59$	81280.7			0.395760			0.197880	−	0.980226i	$-0.436594\pi$
$59$	81280.7			0.395760			0.197880	+	0.980226i	$0.436594\pi$
$60$	131809.	−	34092.9i	0.610225	−	0.157838i
$61$		−	216368.i		−	0.953241i	−0.879109	−	0.476621i	$-0.841862\pi$
$61$		−	216368.i		−	0.953241i	0.879109	−	0.476621i	$-0.158138\pi$
$62$			48371.1i			0.202960i
$63$	19363.4			0.0774389
$64$	−162418.			−0.619575
$65$	43855.4	−	11343.4i	0.159692	−	0.0413051i
$66$		−	188856.i		−	0.656898i
$67$	−6297.15			−0.0209372			−0.0104686	−	0.999945i	$-0.503332\pi$
$67$	−6297.15			−0.0209372			−0.0104686	+	0.999945i	$0.503332\pi$
$68$	−172637.			−0.549045
$69$	−272979.	+	194791.i	−0.830964	+	0.592955i
$70$	377833.	−	97728.5i	1.10155	−	0.284923i
$71$	−592554.			−1.65559			−0.827796	−	0.561030i	$-0.810405\pi$
$71$	−592554.			−1.65559			−0.827796	+	0.561030i	$0.810405\pi$
$72$		−	15718.5i		−	0.0421128i
$73$		−	166216.i		−	0.427273i	−0.976913	−	0.213637i	$-0.931469\pi$
$73$		−	166216.i		−	0.427273i	0.976913	−	0.213637i	$-0.0685308\pi$
$74$		−	173005.i		−	0.426937i
$75$	208816.	+	376652.i	0.494971	+	0.892804i
$76$		−	227852.i		−	0.519054i
$77$			873758.i			1.91390i
$78$		−	49423.0i		−	0.104147i
$79$			931872.i			1.89006i	0.326986	+	0.945029i	$0.393967\pi$
$79$			931872.i			1.89006i	−0.326986	+	0.945029i	$0.606033\pi$
$80$	−168.817	−	652.671i	−0.000329720	−	0.00127475i
$81$	−552871.			−1.04032
$82$		−	549760.i		−	0.997084i
$83$	−88821.9			−0.155341			−0.0776704	−	0.996979i	$-0.524748\pi$
$83$	−88821.9			−0.155341			−0.0776704	+	0.996979i	$0.524748\pi$
$84$			687245.i			1.15951i
$85$	−136749.	−	528694.i	−0.222673	−	0.860890i
$86$			23948.4i			0.0376514i
$87$			265990.i			0.403932i
$88$	709287.			1.04082
$89$			811194.i			1.15068i	0.817914	+	0.575341i	$0.195130\pi$
$89$			811194.i			1.15068i	−0.817914	+	0.575341i	$0.804870\pi$
$90$	18375.9	−	4753.03i	0.0252071	−	0.00651993i
$91$			228660.i			0.303436i
$92$	−279273.	−	391372.i	−0.358646	−	0.502605i
$93$		−	269443.i		−	0.334979i
$94$	556324.			0.669799
$95$	697787.	−	180486.i	0.813865	−	0.210510i
$96$	902797.			1.02041
$97$	979818.			1.07357			0.536785	−	0.843719i	$-0.319639\pi$
$97$	979818.			1.07357			0.536785	+	0.843719i	$0.319639\pi$
$98$			1.38787e6i			1.47459i
$99$			42495.3i			0.0437960i
$100$	−540009.	+	299381.i	−0.540009	+	0.299381i
$101$	−1.70141e6			−1.65137			−0.825684	−	0.564133i	$-0.809210\pi$
$101$	−1.70141e6			−1.65137			−0.825684	+	0.564133i	$0.809210\pi$
$102$	−595815.			−0.561449
$103$	−994277.			−0.909904			−0.454952	−	0.890516i	$-0.650344\pi$
$103$	−994277.			−0.909904			−0.454952	+	0.890516i	$0.650344\pi$
$104$	185619.			0.165014
$105$	−2.10465e6	+	544380.i	−1.81808	+	0.470256i
$106$		−	430369.i		−	0.361346i
$107$	2.22904e6			1.81956			0.909782	−	0.415086i	$-0.136249\pi$
$107$	2.22904e6			1.81956			0.909782	+	0.415086i	$0.136249\pi$
$108$		−	760581.i		−	0.603774i
$109$		−	1.68041e6i		−	1.29758i	−0.760966	−	0.648791i	$-0.775275\pi$
$109$		−	1.68041e6i		−	1.29758i	0.760966	−	0.648791i	$-0.224725\pi$
$110$	214477.	+	829201.i	0.161140	+	0.622991i
$111$			963695.i			0.704645i
$112$	3403.00			0.00242219
$113$	1.94401e6			1.34730			0.673648	−	0.739052i	$-0.264726\pi$
$113$	1.94401e6			1.34730			0.673648	+	0.739052i	$0.264726\pi$
$114$		−	786375.i		−	0.530781i
$115$	977341.	−	1.16527e6i	0.642618	−	0.766187i
$116$	−381352.			−0.244316
$117$			11120.9i			0.00694356i
$118$			402184.i			0.244781i
$119$	2.75659e6			1.63580
$120$	441909.	+	1.70849e6i	0.255734	+	0.988708i
$121$	−146007.			−0.0824169
$122$	1.07060e6			0.589589
$123$			3.06234e6i			1.64565i
$124$	386302.			0.202611
$125$	−1.34459e6	−	1.41660e6i	−0.688431	−	0.725302i
$126$			95811.5i			0.0478968i
$127$			1.98190e6i			0.967542i	0.875195	+	0.483771i	$0.160733\pi$
$127$			1.98190e6i			0.967542i	−0.875195	+	0.483771i	$0.839267\pi$
$128$			1.29264e6i			0.616379i
$129$		−	133400.i		−	0.0621424i
$130$	56128.1	+	217000.i	0.0255476	+	0.0987710i
$131$	3.64622e6			1.62192			0.810959	−	0.585103i	$-0.198946\pi$
$131$	3.64622e6			1.62192			0.810959	+	0.585103i	$0.198946\pi$
$132$	−1.50824e6			−0.655767
$133$			3.63824e6i			1.54645i
$134$		−	31158.8i		−	0.0129499i
$135$	2.32924e6	−	602471.i	0.946703	−	0.244869i
$136$		−	2.23771e6i		−	0.889583i
$137$	522797.			0.203316			0.101658	−	0.994819i	$-0.467585\pi$
$137$	522797.			0.203316			0.101658	+	0.994819i	$0.467585\pi$
$138$	−963842.	−	1.35072e6i	−0.366749	−	0.513960i
$139$	3.68041e6			1.37042			0.685208	−	0.728348i	$-0.259711\pi$
$139$	3.68041e6			1.37042			0.685208	+	0.728348i	$0.259711\pi$
$140$	−780482.	−	3.01746e6i	−0.284432	−	1.09966i
$141$	−3.09891e6			−1.10548
$142$		−	2.93201e6i		−	1.02400i
$143$	−501823.			−0.171610
$144$	165.505			5.54274e−5
$145$	−302076.	−	1.16787e6i	−0.0990861	−	0.383082i
$146$	822453.			0.264273
$147$		−	7.73090e6i		−	2.43376i
$148$	−1.38166e6			−0.426202
$149$		−	482807.i		−	0.145954i	−0.997334	−	0.0729768i	$-0.976750\pi$
$149$		−	482807.i		−	0.145954i	0.997334	−	0.0729768i	$-0.0232499\pi$
$150$	−1.86370e6	+	1.03324e6i	−0.552208	+	0.306145i
$151$	−2.50950e6			−0.728881			−0.364440	−	0.931227i	$-0.618740\pi$
$151$	−2.50950e6			−0.728881			−0.364440	+	0.931227i	$0.618740\pi$
$152$	2.95340e6			0.840991
$153$	134067.			0.0374324
$154$	−4.32343e6			−1.18377
$155$	305997.	+	1.18303e6i	0.0821718	+	0.317689i
$156$	−394703.			−0.103967
$157$	−2.30357e6			−0.595253			−0.297627	−	0.954682i	$-0.596195\pi$
$157$	−2.30357e6			−0.595253			−0.297627	+	0.954682i	$0.596195\pi$
$158$	−4.61098e6			−1.16902
$159$			2.39730e6i			0.596390i
$160$	−3.96388e6	+	1.02528e6i	−0.967744	+	0.250312i
$161$	4.45930e6	+	6.24924e6i	1.06854	+	1.49744i
$162$		−	2.73565e6i		−	0.643451i
$163$		−	3.46570e6i		−	0.800255i	−0.916460	−	0.400127i	$-0.868966\pi$
$163$		−	3.46570e6i		−	0.800255i	0.916460	−	0.400127i	$-0.131034\pi$
$164$	−4.39051e6			−0.995367
$165$	−1.19471e6	−	4.61892e6i	−0.265956	−	1.02823i
$166$		−	439498.i		−	0.0960799i
$167$			5.29143e6i			1.13612i	0.822988	+	0.568059i	$0.192306\pi$
$167$			5.29143e6i			1.13612i	−0.822988	+	0.568059i	$0.807694\pi$
$168$	−8.90799e6			−1.87868
$169$	4.69548e6			0.972792
$170$	2.61602e6	−	676647.i	0.532469	−	0.137726i
$171$			176946.i			0.0353877i
$172$	191257.			0.0375866
$173$			9.18832e6i			1.77459i	0.461203	+	0.887295i	$0.347418\pi$
$173$			9.18832e6i			1.77459i	−0.461203	+	0.887295i	$0.652582\pi$
$174$	−1.31614e6			−0.249836
$175$	8.62260e6	−	4.78038e6i	1.60888	−	0.891965i
$176$			7468.31i			0.00136989i
$177$		−	2.24029e6i		−	0.404004i
$178$	−4.01386e6			−0.711708
$179$	−6.18359e6			−1.07816			−0.539078	−	0.842256i	$-0.681227\pi$
$179$	−6.18359e6			−1.07816			−0.539078	+	0.842256i	$0.681227\pi$
$180$	−37958.8	−	146754.i	−0.00650870	−	0.0251636i
$181$			163276.i			0.0275351i	0.999905	+	0.0137675i	$0.00438248\pi$
$181$			163276.i			0.0275351i	−0.999905	+	0.0137675i	$0.995618\pi$
$182$	−1.13143e6			−0.187678
$183$	−5.96362e6			−0.973098
$184$	5.07292e6	−	3.61991e6i	0.814338	−	0.581091i
$185$	−1.09444e6	−	4.23126e6i	−0.172852	−	0.668274i
$186$	1.33323e6			0.207188
$187$			6.04967e6i			0.925139i
$188$		−	4.44293e6i		−	0.668645i
$189$			1.21446e7i			1.79886i
$190$	893061.	+	3.45271e6i	0.130203	+	0.503384i
$191$		−	4.36527e6i		−	0.626486i	−0.949673	−	0.313243i	$-0.898585\pi$
$191$		−	4.36527e6i		−	0.626486i	0.949673	−	0.313243i	$-0.101415\pi$
$192$			4.47663e6i			0.632481i
$193$			1.36522e6i			0.189902i	0.995482	+	0.0949512i	$0.0302695\pi$
$193$			1.36522e6i			0.189902i	−0.995482	+	0.0949512i	$0.969730\pi$
$194$			4.84822e6i			0.664014i
$195$	−312652.	−	1.20876e6i	−0.0421655	−	0.163018i
$196$	1.10839e7			1.47205
$197$		−	9.32488e6i		−	1.21968i	−0.792526	−	0.609838i	$-0.791235\pi$
$197$		−	9.32488e6i		−	1.21968i	0.792526	−	0.609838i	$-0.208765\pi$
$198$	−210270.			−0.0270883
$199$		−	1.17378e7i		−	1.48945i	−0.667372	−	0.744725i	$-0.732581\pi$
$199$		−	1.17378e7i		−	1.48945i	0.667372	−	0.744725i	$-0.267419\pi$
$200$	−3.88055e6	−	6.99953e6i	−0.485068	−	0.874941i
$201$			173565.i			0.0213734i
$202$		−	8.41870e6i		−	1.02139i
$203$	6.08925e6			0.727907
$204$			4.75831e6i			0.560482i
$205$	−3.47780e6	−	1.34457e7i	−0.403686	−	1.56071i
$206$		−	4.91977e6i		−	0.562785i
$207$	216878.	+	303932.i	0.0244515	+	0.0342662i
$208$			1954.44i			0.000217186i
$209$	−7.98456e6			−0.874605
$210$	−2.69364e6	−	1.04140e7i	−0.290858	−	1.12450i
$211$	−1.00928e7			−1.07440			−0.537198	−	0.843456i	$-0.680517\pi$
$211$	−1.00928e7			−1.07440			−0.537198	+	0.843456i	$0.680517\pi$
$212$	−3.43702e6			−0.360724
$213$			1.63322e7i			1.69008i
$214$			1.10295e7i			1.12542i
$215$	151498.	+	585716.i	0.0152438	+	0.0589348i
$216$	9.85856e6			0.978256
$217$	−6.16829e6			−0.603651
$218$	8.31479e6			0.802568
$219$	−4.58133e6			−0.436173
$220$	6.62218e6	−	1.71286e6i	0.621918	−	0.160862i
$221$			1.58318e6i			0.146675i
$222$	−4.76844e6			−0.435830
$223$			2.55039e6i			0.229981i	0.993367	+	0.114990i	$0.0366837\pi$
$223$			2.55039e6i			0.229981i	−0.993367	+	0.114990i	$0.963316\pi$
$224$		−	2.06675e7i		−	1.83884i
$225$	419360.	−	232494.i	0.0368163	−	0.0204110i
$226$			9.61912e6i			0.833317i
$227$	−5.04778e6			−0.431542			−0.215771	−	0.976444i	$-0.569226\pi$
$227$	−5.04778e6			−0.431542			−0.215771	+	0.976444i	$0.569226\pi$
$228$	−6.28017e6			−0.529867
$229$		−	9.83048e6i		−	0.818594i	−0.912401	−	0.409297i	$-0.865774\pi$
$229$		−	9.83048e6i		−	0.818594i	0.912401	−	0.409297i	$-0.134226\pi$
$230$	5.76587e6	+	4.83596e6i	0.473895	+	0.397466i
$231$	2.40829e7			1.95377
$232$		−	4.94304e6i		−	0.395850i
$233$		−	361324.i		−	0.0285646i	−0.999898	−	0.0142823i	$-0.995454\pi$
$233$		−	361324.i		−	0.0285646i	0.999898	−	0.0142823i	$-0.00454636\pi$
$234$	−55027.1			−0.00429467
$235$	1.36063e7	−	3.51933e6i	1.04842	−	0.271179i
$236$	3.21193e6			0.244360
$237$	2.56847e7			1.92943
$238$			1.36398e7i			1.01176i
$239$	−1.81352e6			−0.132840			−0.0664200	−	0.997792i	$-0.521158\pi$
$239$	−1.81352e6			−0.132840			−0.0664200	+	0.997792i	$0.521158\pi$
$240$	−17989.2	+	4653.00i	−0.00130130	+	0.000336589i
$241$			2.05476e7i			1.46795i	0.679178	+	0.733973i	$0.262336\pi$
$241$			2.05476e7i			1.46795i	−0.679178	+	0.733973i	$0.737664\pi$
$242$		−	722453.i		−	0.0509757i
$243$			1.20726e6i			0.0841362i
$244$		−	8.55009e6i		−	0.588574i
$245$	8.77973e6	+	3.39438e7i	0.597012	+	2.30814i
$246$	−1.51527e7			−1.01785
$247$	−2.08954e6			−0.138663
$248$			5.00721e6i			0.328277i
$249$			2.44815e6i			0.158577i
$250$	7.00948e6	−	6.65315e6i	0.448607	−	0.425802i
$251$		−	1.08771e7i		−	0.687844i	−0.938998	−	0.343922i	$-0.888244\pi$
$251$		−	1.08771e7i		−	0.687844i	0.938998	−	0.343922i	$-0.111756\pi$
$252$	765172.			0.0478143
$253$	−1.37147e7	+	9.78648e6i	−0.846887	+	0.604317i
$254$	−9.80659e6			−0.598435
$255$	−1.45721e7	+	3.76915e6i	−0.878823	+	0.227312i
$256$	−1.67908e7			−1.00081
$257$			2.63417e7i			1.55183i	0.630838	+	0.775915i	$0.282711\pi$
$257$			2.63417e7i			1.55183i	−0.630838	+	0.775915i	$0.717289\pi$
$258$	660076.			0.0384357
$259$	2.20616e7			1.26981
$260$	1.73301e6	−	448252.i	0.0986009	−	0.0255036i
$261$	296151.			0.0166568
$262$			1.80418e7i			1.00317i
$263$	−7.74203e6			−0.425586			−0.212793	−	0.977097i	$-0.568256\pi$
$263$	−7.74203e6			−0.425586			−0.212793	+	0.977097i	$0.568256\pi$
$264$		−	1.95497e7i		−	1.06250i
$265$	−2.72253e6	−	1.05257e7i	−0.146297	−	0.565607i
$266$	−1.80023e7			−0.956496
$267$	2.23585e7			1.17465
$268$	−248841.			−0.0129276
$269$	−698740.			−0.0358971			−0.0179485	−	0.999839i	$-0.505714\pi$
$269$	−698740.			−0.0358971			−0.0179485	+	0.999839i	$0.505714\pi$
$270$	2.98107e6	+	1.15253e7i	0.151454	+	0.585545i
$271$	−1.01642e7			−0.510700			−0.255350	−	0.966849i	$-0.582191\pi$
$271$	−1.01642e7			−0.510700			−0.255350	+	0.966849i	$0.582191\pi$
$272$	23561.5			0.00117084
$273$	6.30243e6			0.309756
$274$			2.58684e6i			0.125753i
$275$	1.04911e7	+	1.89233e7i	0.504456	+	0.909912i
$276$	−1.07872e7	+	7.69745e6i	−0.513074	+	0.366117i
$277$		−	1.81417e6i		−	0.0853570i	−0.999089	−	0.0426785i	$-0.986411\pi$
$277$		−	1.81417e6i		−	0.0853570i	0.999089	−	0.0426785i	$-0.0135891\pi$
$278$			1.82110e7i			0.847616i
$279$	−299995.			−0.0138134
$280$	3.91120e7	−	1.01165e7i	1.78170	−	0.460847i
$281$			4.80363e6i			0.216496i	0.994124	+	0.108248i	$0.0345241\pi$
$281$			4.80363e6i			0.216496i	−0.994124	+	0.108248i	$0.965476\pi$
$282$		−	1.53336e7i		−	0.683751i
$283$	−2.52736e6			−0.111508			−0.0557542	−	0.998445i	$-0.517756\pi$
$283$	−2.52736e6			−0.111508			−0.0557542	+	0.998445i	$0.517756\pi$
$284$	−2.34157e7			−1.02224
$285$	−4.97464e6	−	1.92327e7i	−0.214895	−	0.830818i
$286$		−	2.48306e6i		−	0.106142i
$287$	7.01055e7			2.96556
$288$		−	1.00517e6i		−	0.0420785i
$289$	−5.05164e6			−0.209286
$290$	5.77873e6	−	1.49470e6i	0.236940	−	0.0612858i
$291$		−	2.70062e7i		−	1.09593i
$292$		−	6.56829e6i		−	0.263818i
$293$	−4.31842e7			−1.71681			−0.858405	−	0.512972i	$-0.828544\pi$
$293$	−4.31842e7			−1.71681			−0.858405	+	0.512972i	$0.828544\pi$
$294$	3.82531e7			1.50531
$295$	2.54423e6	+	9.83637e6i	0.0991037	+	0.383150i
$296$		−	1.79089e7i		−	0.690547i
$297$	−2.66528e7			−1.01736
$298$	2.38897e6			0.0902738
$299$	−3.58911e6	+	2.56110e6i	−0.134268	+	0.0958104i
$300$	8.25168e6	+	1.48839e7i	0.305618	+	0.551257i
$301$	−3.05390e6			−0.111984
$302$		−	1.24172e7i		−	0.450820i
$303$			4.68949e7i			1.68577i
$304$			31097.3i			0.00110688i
$305$	2.61842e7	−	6.77269e6i	0.922870	−	0.238705i
$306$			663374.i			0.0231523i
$307$		−	5.01133e7i		−	1.73196i	−0.500078	−	0.865980i	$-0.666695\pi$
$307$		−	5.01133e7i		−	1.73196i	0.500078	−	0.865980i	$-0.333305\pi$
$308$			3.45278e7i			1.18173i
$309$			2.74047e7i			0.928859i
$310$	−5.85374e6	+	1.51410e6i	−0.196494	+	0.0508241i
$311$	1.24312e7			0.413267			0.206634	−	0.978418i	$-0.433749\pi$
$311$	1.24312e7			0.413267			0.206634	+	0.978418i	$0.433749\pi$
$312$		−	5.11610e6i		−	0.168452i
$313$	4.42286e7			1.44235			0.721174	−	0.692754i	$-0.243603\pi$
$313$	4.42286e7			1.44235			0.721174	+	0.692754i	$0.243603\pi$
$314$		−	1.13982e7i		−	0.368170i
$315$	606107.	+	2.34330e6i	0.0193918	+	0.0749716i
$316$			3.68243e7i			1.16701i
$317$		−	1.90846e7i		−	0.599108i	−0.954079	−	0.299554i	$-0.903162\pi$
$317$		−	1.90846e7i		−	0.599108i	0.954079	−	0.299554i	$-0.0968378\pi$
$318$	−1.18620e7			−0.368874
$319$			1.33636e7i			0.411672i
$320$	−5.08396e6	−	1.96554e7i	−0.155150	−	0.599834i
$321$		−	6.14379e7i		−	1.85747i
$322$	−3.09218e7	+	2.20650e7i	−0.926183	+	0.660901i
$323$			2.51902e7i			0.747523i
$324$	−2.18475e7			−0.642343
$325$	2.74550e6	+	4.95219e6i	0.0799781	+	0.144260i
$326$	1.71486e7			0.494966
$327$	−4.63161e7			−1.32461
$328$		−	5.69093e7i		−	1.61273i
$329$			7.09425e7i			1.99214i
$330$	2.28548e7	−	5.91151e6i	0.635969	−	0.164497i
$331$	−4.07176e7			−1.12279			−0.561394	−	0.827549i	$-0.689735\pi$
$331$	−4.07176e7			−1.12279			−0.561394	+	0.827549i	$0.689735\pi$
$332$	−3.50993e6			−0.0959145
$333$	1.07297e6			0.0290572
$334$	−2.61824e7			−0.702701
$335$	−197112.	−	762064.i	−0.00524298	−	0.0202701i
$336$		−	93795.1i		−	0.00247265i
$337$	−2.40667e6			−0.0628822			−0.0314411	−	0.999506i	$-0.510010\pi$
$337$	−2.40667e6			−0.0628822			−0.0314411	+	0.999506i	$0.510010\pi$
$338$			2.32336e7i			0.601682i
$339$		−	5.35817e7i		−	1.37536i
$340$	−5.40385e6	−	2.08921e7i	−0.137489	−	0.531552i
$341$		−	1.35371e7i		−	0.341398i
$342$	−875543.			−0.0218877
$343$	−1.02748e8			−2.54618
$344$			2.47905e6i			0.0608991i
$345$	−3.21178e7	−	2.69379e7i	−0.782147	−	0.656004i
$346$	−4.54646e7			−1.09760
$347$			9.17607e6i			0.219618i	0.993953	+	0.109809i	$0.0350239\pi$
$347$			9.17607e6i			0.219618i	−0.993953	+	0.109809i	$0.964976\pi$
$348$			1.05110e7i			0.249406i
$349$	−1.38470e7			−0.325746			−0.162873	−	0.986647i	$-0.552076\pi$
$349$	−1.38470e7			−0.325746			−0.162873	+	0.986647i	$0.552076\pi$
$350$	2.36537e7	+	4.26653e7i	0.551690	+	0.995109i
$351$	−6.97497e6			−0.161295
$352$	4.53574e7			1.03997
$353$		−	2.53380e7i		−	0.576035i	−0.957625	−	0.288017i	$-0.907004\pi$
$353$		−	2.53380e7i		−	0.576035i	0.957625	−	0.288017i	$-0.0929961\pi$
$354$	1.10852e7			0.249880
$355$	−1.85480e7	−	7.17093e7i	−0.414583	−	1.60284i
$356$			3.20555e7i			0.710482i
$357$		−	7.59783e7i		−	1.66988i
$358$		−	3.05969e7i		−	0.666850i
$359$			4.04833e7i			0.874968i	0.899226	+	0.437484i	$0.144130\pi$
$359$			4.04833e7i			0.874968i	−0.899226	+	0.437484i	$0.855870\pi$
$360$	1.90221e6	−	492017.i	0.0407710	−	0.0105456i
$361$	1.37990e7			0.293309
$362$	−807903.			−0.0170307
$363$			4.02430e6i			0.0841337i
$364$			9.03585e6i			0.187355i
$365$	2.01151e7	−	5.20287e6i	0.413660	−	0.106995i
$366$		−	2.95085e7i		−	0.601871i
$367$	−2.75132e7			−0.556600			−0.278300	−	0.960494i	$-0.589771\pi$
$367$	−2.75132e7			−0.556600			−0.278300	+	0.960494i	$0.589771\pi$
$368$	38115.2	+	53414.4i	0.000764812	+	0.00107180i
$369$	3.40959e6			0.0678614
$370$	2.09366e7	−	5.41536e6i	0.413334	−	0.106911i
$371$	5.48808e7			1.07473
$372$		−	1.06474e7i		−	0.206831i
$373$	1.48874e7			0.286874			0.143437	−	0.989659i	$-0.454185\pi$
$373$	1.48874e7			0.286874			0.143437	+	0.989659i	$0.454185\pi$
$374$	−2.99343e7			−0.572208
$375$	−3.90451e7	+	3.70602e7i	−0.740410	+	0.702772i
$376$	5.75887e7			1.08336
$377$			3.49722e6i			0.0652678i
$378$	−6.00924e7			−1.11261
$379$		−	6.08204e7i		−	1.11720i	−0.829437	−	0.558601i	$-0.811338\pi$
$379$		−	6.08204e7i		−	1.11720i	0.829437	−	0.558601i	$-0.188662\pi$
$380$	2.75741e7	−	7.13218e6i	0.502517	−	0.129979i
$381$	5.46259e7			0.987697
$382$	2.15997e7			0.387488
$383$	6.47549e7			1.15259			0.576297	−	0.817240i	$-0.304497\pi$
$383$	6.47549e7			1.15259			0.576297	+	0.817240i	$0.304497\pi$
$384$	3.56283e7			0.629218
$385$	−1.05740e8	+	2.73502e7i	−1.85292	+	0.479267i
$386$	−6.75522e6			−0.117457
$387$	−148527.			−0.00256255
$388$	3.87190e7			0.662870
$389$		−	1.21683e7i		−	0.206720i	−0.994644	−	0.103360i	$-0.967041\pi$
$389$		−	1.21683e7i		−	0.206720i	0.994644	−	0.103360i	$-0.0329593\pi$
$390$	5.98105e6	−	1.54703e6i	0.100828	−	0.0260798i
$391$	3.08751e7	+	4.32681e7i	0.516508	+	0.723832i
$392$			1.43668e8i			2.38507i
$393$		−	1.00499e8i		−	1.65570i
$394$	4.61403e7			0.754382
$395$	−1.12773e8	+	2.91692e7i	−1.82984	+	0.473297i
$396$			1.67926e6i			0.0270416i
$397$		−	5.37026e7i		−	0.858270i	−0.903240	−	0.429135i	$-0.858818\pi$
$397$		−	5.37026e7i		−	0.858270i	0.903240	−	0.429135i	$-0.141182\pi$
$398$	5.80794e7			0.921240
$399$	1.00279e8			1.57867
$400$	73700.3	−	40859.5i	0.00115157	−	0.000638430i
$401$			8.01340e6i			0.124275i	0.998068	+	0.0621375i	$0.0197917\pi$
$401$			8.01340e6i			0.124275i	−0.998068	+	0.0621375i	$0.980208\pi$
$402$	−858812.			−0.0132197
$403$		−	3.54262e6i		−	0.0541264i
$404$	−6.72336e7			−1.01963
$405$	−1.73058e7	−	6.69069e7i	−0.260512	−	1.00718i
$406$			3.01301e7i			0.450218i
$407$			4.84169e7i			0.718148i
$408$	−6.16766e7			−0.908113
$409$	−6.77807e7			−0.990686			−0.495343	−	0.868698i	$-0.664958\pi$
$409$	−6.77807e7			−0.990686			−0.495343	+	0.868698i	$0.664958\pi$
$410$	6.65305e7	−	1.72085e7i	0.965316	−	0.249684i
$411$		−	1.44096e7i		−	0.207551i
$412$	−3.92903e7			−0.561816
$413$	−5.12865e7			−0.728036
$414$	−1.50388e6	+	1.07313e6i	−0.0211940	+	0.0151235i
$415$	−2.78028e6	−	1.07490e7i	−0.0388995	−	0.150392i
$416$	1.18699e7			0.164880
$417$		−	1.01441e8i		−	1.39896i
$418$		−	3.95082e7i		−	0.540952i
$419$		−	8.44414e7i		−	1.14792i	−0.818882	−	0.573962i	$-0.805406\pi$
$419$		−	8.44414e7i		−	1.14792i	0.818882	−	0.573962i	$-0.194594\pi$
$420$	−8.31686e7	+	2.15120e7i	−1.12256	+	0.290357i
$421$		−	1.14700e7i		−	0.153716i	−0.997042	−	0.0768578i	$-0.975511\pi$
$421$		−	1.14700e7i		−	0.153716i	0.997042	−	0.0768578i	$-0.0244887\pi$
$422$		−	4.99400e7i		−	0.664525i
$423$			3.45029e6i			0.0455864i
$424$		−	4.45503e7i		−	0.584458i
$425$	5.97006e7	−	3.30981e7i	0.777700	−	0.431158i
$426$	−8.08133e7			−1.04533
$427$			1.36524e8i			1.75358i
$428$	8.80840e7			1.12348
$429$			1.38315e7i			0.175185i
$430$	−2.89817e6	+	749627.i	−0.0364518	+	0.00942844i
$431$			7.26126e7i			0.906943i	0.891271	+	0.453472i	$0.149815\pi$
$431$			7.26126e7i			0.906943i	−0.891271	+	0.453472i	$0.850185\pi$
$432$			103804.i			0.00128755i
$433$	−3.28590e7			−0.404753			−0.202377	−	0.979308i	$-0.564867\pi$
$433$	−3.28590e7			−0.404753			−0.202377	+	0.979308i	$0.564867\pi$
$434$		−	3.05212e7i		−	0.373364i
$435$	−3.21894e7	+	8.32596e6i	−0.391062	+	0.101150i
$436$		−	6.64038e7i		−	0.801186i
$437$	−5.71067e7	+	4.07499e7i	−0.684294	+	0.488295i
$438$		−	2.26688e7i		−	0.269778i
$439$	1.51059e8			1.78547			0.892736	−	0.450580i	$-0.148783\pi$
$439$	1.51059e8			1.78547			0.892736	+	0.450580i	$0.148783\pi$
$440$	2.22019e7	+	8.58360e7i	0.260635	+	1.00765i
$441$	−8.60751e6			−0.100360
$442$	−7.83373e6			−0.0907197
$443$		−	5.31378e7i		−	0.611213i	−0.952158	−	0.305607i	$-0.901141\pi$
$443$		−	5.31378e7i		−	0.611213i	0.952158	−	0.305607i	$-0.0988592\pi$
$444$			3.80818e7i			0.435080i
$445$	−9.81686e7	+	2.53918e7i	−1.11402	+	0.288147i
$446$	−1.26195e7			−0.142245
$447$	−1.33073e7			−0.148994
$448$	1.02482e8			1.13976
$449$	1.04195e8			1.15109			0.575545	−	0.817770i	$-0.304790\pi$
$449$	1.04195e8			1.15109			0.575545	+	0.817770i	$0.304790\pi$
$450$	1.15040e6	+	2.07503e6i	0.0126244	+	0.0227712i
$451$			1.53855e8i			1.67719i
$452$	7.68204e7			0.831882
$453$			6.91679e7i			0.744064i
$454$		−	2.49768e7i		−	0.266913i
$455$	−2.76719e7	+	7.15747e6i	−0.293768	+	0.0759845i
$456$		−	8.14028e7i		−	0.858509i
$457$	−1.27252e8			−1.33326			−0.666631	−	0.745388i	$-0.732264\pi$
$457$	−1.27252e8			−1.33326			−0.666631	+	0.745388i	$0.732264\pi$
$458$	4.86420e7			0.506309
$459$			8.40860e7i			0.869532i
$460$	3.86211e7	−	4.60476e7i	0.396781	−	0.473078i
$461$	−6.52076e7			−0.665572			−0.332786	−	0.943002i	$-0.607989\pi$
$461$	−6.52076e7			−0.665572			−0.332786	+	0.943002i	$0.607989\pi$
$462$			1.19164e8i			1.20842i
$463$		−	2.30472e7i		−	0.232207i	−0.993237	−	0.116103i	$-0.962960\pi$
$463$		−	2.30472e7i		−	0.232207i	0.993237	−	0.116103i	$-0.0370404\pi$
$464$	52046.9			0.000521004
$465$	3.26073e7	−	8.43403e6i	0.324306	−	0.0838835i
$466$	1.78786e6			0.0176675
$467$	5.68246e7			0.557938			0.278969	−	0.960300i	$-0.410007\pi$
$467$	5.68246e7			0.557938			0.278969	+	0.960300i	$0.410007\pi$
$468$			439459.i			0.00428727i
$469$	3.97337e6			0.0385160
$470$	1.74139e7	+	6.73249e7i	0.167727	+	0.648458i
$471$			6.34919e7i			0.607653i
$472$			4.16326e7i			0.395920i
$473$		−	6.70216e6i		−	0.0633332i
$474$			1.27090e8i			1.19337i
$475$	4.36839e7	+	7.87948e7i	0.407606	+	0.735219i
$476$	1.08931e8			1.01002
$477$	2.66913e6			0.0245932
$478$		−	8.97345e6i		−	0.0821629i
$479$			1.11385e8i			1.01349i	0.862095	+	0.506747i	$0.169152\pi$
$479$			1.11385e8i			1.01349i	−0.862095	+	0.506747i	$0.830848\pi$
$480$	2.82591e7	+	1.09254e8i	0.255526	+	0.987903i
$481$			1.26706e7i			0.113858i
$482$	−1.01671e8			−0.907940
$483$	1.72244e8	−	1.22909e8i	1.52863	−	1.09079i
$484$	−5.76967e6			−0.0508879
$485$	3.06700e7	+	1.18575e8i	0.268837	+	1.03936i
$486$	−5.97363e6			−0.0520391
$487$		−	4.60533e7i		−	0.398725i	−0.979926	−	0.199363i	$-0.936113\pi$
$487$		−	4.60533e7i		−	0.398725i	0.979926	−	0.199363i	$-0.0638872\pi$
$488$	1.10825e8			0.953629
$489$	−9.55231e7			−0.816925
$490$	−1.67957e8	+	4.34428e7i	−1.42761	+	0.369258i
$491$	−2.00880e8			−1.69704			−0.848520	−	0.529163i	$-0.822506\pi$
$491$	−2.00880e8			−1.69704			−0.848520	+	0.529163i	$0.822506\pi$
$492$			1.21013e8i			1.01610i
$493$	4.21604e7			0.351855
$494$		−	1.03392e7i		−	0.0857643i
$495$	−5.14266e6	+	1.33018e6i	−0.0424006	+	0.0109671i
$496$	−52722.5			−0.000432067
$497$	3.73890e8			3.04561
$498$	−1.21136e7			−0.0980814
$499$	−1.82933e8			−1.47228			−0.736141	−	0.676828i	$-0.763354\pi$
$499$	−1.82933e8			−1.47228			−0.736141	+	0.676828i	$0.763354\pi$
$500$	−5.31335e7	−	5.59792e7i	−0.425068	−	0.447834i
$501$	1.45845e8			1.15978
$502$	5.38206e7			0.425439
$503$	1.68979e8			1.32779			0.663893	−	0.747828i	$-0.268903\pi$
$503$	1.68979e8			1.32779			0.663893	+	0.747828i	$0.268903\pi$
$504$			9.91807e6i			0.0774704i
$505$	−5.32570e7	−	2.05900e8i	−0.413526	−	1.59875i
$506$	−4.84243e7	−	6.78616e7i	−0.373776	−	0.523808i
$507$		−	1.29419e8i		−	0.993057i
$508$			7.83176e7i			0.597404i
$509$	−1.06204e8			−0.805357			−0.402679	−	0.915341i	$-0.631921\pi$
$509$	−1.06204e8			−0.805357			−0.402679	+	0.915341i	$0.631921\pi$
$510$	−1.86500e7	−	7.21039e7i	−0.140595	−	0.543561i
$511$			1.04879e8i			0.786008i
$512$		−	353449.i		−	0.00263340i
$513$	−1.10979e8			−0.822036
$514$	−1.30341e8			−0.959823
$515$	−3.11226e7	−	1.20325e8i	−0.227853	−	0.880914i
$516$		−	5.27151e6i		−	0.0383695i
$517$	−1.55692e8			−1.12666
$518$			1.09163e8i			0.785390i
$519$	2.53252e8			1.81156
$520$	5.81019e6	+	2.24631e7i	0.0413219	+	0.159757i
$521$		−	1.82832e8i		−	1.29282i	−0.762990	−	0.646410i	$-0.776270\pi$
$521$		−	1.82832e8i		−	1.29282i	0.762990	−	0.646410i	$-0.223730\pi$
$522$			1.46538e6i			0.0103024i
$523$	−4.84564e7			−0.338724			−0.169362	−	0.985554i	$-0.554171\pi$
$523$	−4.84564e7			−0.338724			−0.169362	+	0.985554i	$0.554171\pi$
$524$	1.44086e8			1.00145
$525$	−1.31759e8	−	2.37660e8i	−0.910546	−	1.64240i
$526$		−	3.83082e7i		−	0.263230i
$527$	−4.27077e7			−0.291792
$528$	205845.			0.00139842
$529$	−4.81435e7	+	1.39989e8i	−0.325215	+	0.945640i
$530$	5.20822e7	−	1.34713e7i	0.349833	−	0.0904862i
$531$	−2.49432e6			−0.0166598
$532$			1.43770e8i			0.954849i
$533$			4.02635e7i			0.265907i
$534$			1.10632e8i			0.726534i
$535$	6.97730e7	+	2.69753e8i	0.455644	+	1.76159i
$536$		−	3.22545e6i		−	0.0209457i
$537$			1.70435e8i			1.10061i
$538$		−	3.45742e6i		−	0.0222027i
$539$		−	3.88408e8i		−	2.48040i
$540$	9.20435e7	−	2.38075e7i	0.584537	−	0.151193i
$541$	5.54698e7			0.350320			0.175160	−	0.984540i	$-0.443956\pi$
$541$	5.54698e7			0.350320			0.175160	+	0.984540i	$0.443956\pi$
$542$		−	5.02933e7i		−	0.315873i
$543$	4.50028e6			0.0281087
$544$		−	1.43097e8i		−	0.888858i
$545$	2.03358e8	−	5.25997e7i	1.25624	−	0.324933i
$546$			3.11850e7i			0.191588i
$547$			1.18237e8i			0.722422i	0.932484	+	0.361211i	$0.117637\pi$
$547$			1.18237e8i			0.722422i	−0.932484	+	0.361211i	$0.882363\pi$
$548$	2.06591e7			0.125536
$549$			6.63984e6i			0.0401273i
$550$	−9.36342e7	+	5.19109e7i	−0.562790	+	0.312011i
$551$			5.56447e7i			0.332636i
$552$	−9.97735e7	−	1.39822e8i	−0.593196	−	0.831301i
$553$		−	5.87993e8i		−	3.47694i
$554$	8.97667e6			0.0527942
$555$	−1.16624e8	+	3.01653e7i	−0.682195	+	0.176453i
$556$	1.45437e8			0.846156
$557$	−3.43742e8			−1.98915			−0.994574	−	0.104033i	$-0.966825\pi$
$557$	−3.43742e8			−1.98915			−0.994574	+	0.104033i	$0.966825\pi$
$558$		−	1.48440e6i		−	0.00854375i
$559$		−	1.75394e6i		−	0.0100410i
$560$	106520.	+	411823.i	0.000606551	+	0.00234502i
$561$	1.66744e8			0.944411
$562$	−2.37688e7			−0.133905
$563$	−5.32172e7			−0.298213			−0.149107	−	0.988821i	$-0.547640\pi$
$563$	−5.32172e7			−0.298213			−0.149107	+	0.988821i	$0.547640\pi$
$564$	−1.22458e8			−0.682573
$565$	6.08509e7	+	2.35259e8i	0.337382	+	1.30437i
$566$		−	1.25056e7i		−	0.0689690i
$567$	3.48850e8			1.91377
$568$		−	3.03511e8i		−	1.65626i
$569$			2.11197e8i			1.14644i	0.819401	+	0.573221i	$0.194306\pi$
$569$			2.11197e8i			1.14644i	−0.819401	+	0.573221i	$0.805694\pi$
$570$	9.51651e7	−	2.46149e7i	0.513870	−	0.132915i
$571$		−	2.58655e8i		−	1.38935i	−0.719322	−	0.694677i	$-0.755547\pi$
$571$		−	2.58655e8i		−	1.38935i	0.719322	−	0.694677i	$-0.244453\pi$
$572$	−1.98303e7			−0.105960
$573$	−1.20318e8			−0.639536
$574$			3.46888e8i			1.83423i
$575$	1.71611e8	+	8.18001e7i	0.902696	+	0.430279i
$576$	4.98424e6			0.0260814
$577$		−	1.38259e8i		−	0.719722i	−0.933006	−	0.359861i	$-0.882824\pi$
$577$		−	1.38259e8i		−	0.719722i	0.933006	−	0.359861i	$-0.117176\pi$
$578$		−	2.49960e7i		−	0.129445i
$579$	3.76288e7			0.193858
$580$	−1.19370e7	−	4.61502e7i	−0.0611802	−	0.236532i
$581$	5.60449e7			0.285764
$582$	1.33629e8			0.677846
$583$			1.20442e8i			0.607819i
$584$	8.51374e7			0.427447
$585$	−1.34582e6	+	348104.i	−0.00672233	+	0.00173877i
$586$		−	2.13679e8i		−	1.06186i
$587$		−	9.52791e7i		−	0.471068i	−0.971866	−	0.235534i	$-0.924316\pi$
$587$		−	9.52791e7i		−	0.471068i	0.971866	−	0.235534i	$-0.0756838\pi$
$588$		−	3.05498e8i		−	1.50271i
$589$		−	5.63670e7i		−	0.275854i
$590$	−4.86712e7	+	1.25891e7i	−0.236982	+	0.0612967i
$591$	−2.57016e8			−1.24508
$592$	188568.			0.000908874
$593$			3.16820e8i			1.51932i	0.650322	+	0.759659i	$0.274634\pi$
$593$			3.16820e8i			1.51932i	−0.650322	+	0.759659i	$0.725366\pi$
$594$		−	1.31880e8i		−	0.629246i
$595$	8.62861e7	+	3.33595e8i	0.409628	+	1.58369i
$596$		−	1.90788e7i		−	0.0901183i
$597$	−3.23521e8			−1.52048
$598$	−1.26725e7	−	1.77592e7i	−0.0592597	−	0.0830463i
$599$	−2.73205e8			−1.27118			−0.635591	−	0.772026i	$-0.719244\pi$
$599$	−2.73205e8			−1.27118			−0.635591	+	0.772026i	$0.719244\pi$
$600$	−1.92924e8	+	1.06957e8i	−0.893167	+	0.495173i
$601$	3.04147e8			1.40107			0.700536	−	0.713617i	$-0.252944\pi$
$601$	3.04147e8			1.40107			0.700536	+	0.713617i	$0.252944\pi$
$602$		−	1.51110e7i		−	0.0692632i
$603$	193245.			0.000881367
$604$	−9.91666e7			−0.450044
$605$	−4.57026e6	−	1.76693e7i	−0.0206384	−	0.0797910i
$606$	−2.32040e8			−1.04266
$607$		−	3.74096e8i		−	1.67270i	−0.548199	−	0.836348i	$-0.684686\pi$
$607$		−	3.74096e8i		−	1.67270i	0.548199	−	0.836348i	$-0.315314\pi$
$608$	1.88864e8			0.840306
$609$		−	1.67835e8i		−	0.743070i
$610$	3.35118e7	+	1.29562e8i	0.147641	+	0.570804i
$611$	−4.07442e7			−0.178625
$612$	5.29785e6			0.0231124
$613$	1.84236e8			0.799822			0.399911	−	0.916554i	$-0.369041\pi$
$613$	1.84236e8			0.799822			0.399911	+	0.916554i	$0.369041\pi$
$614$	2.47965e8			1.07124
$615$	−3.70597e8	+	9.58568e7i	−1.59322	+	0.412095i
$616$	−4.47546e8			−1.91468
$617$	1.31512e7			0.0559898			0.0279949	−	0.999608i	$-0.491088\pi$
$617$	1.31512e7			0.0559898			0.0279949	+	0.999608i	$0.491088\pi$
$618$	−1.35601e8			−0.574509
$619$		−	2.72742e8i		−	1.14995i	−0.818170	−	0.574976i	$-0.805011\pi$
$619$		−	2.72742e8i		−	1.14995i	0.818170	−	0.574976i	$-0.194989\pi$
$620$	1.20919e7	+	4.67493e7i	0.0507366	+	0.196155i
$621$	−1.90625e8	+	1.36025e8i	−0.795984	+	0.567994i
$622$			6.15105e7i			0.255610i
$623$		−	5.11848e8i		−	2.11678i
$624$	53869.1			0.000221710
$625$	1.29346e8	−	2.07061e8i	0.529800	−	0.848123i
$626$			2.18847e8i			0.892108i
$627$			2.20074e8i			0.892824i
$628$	−9.10288e7			−0.367536
$629$	1.52749e8			0.613800
$630$	−1.15949e7	+	2.99907e6i	−0.0463707	+	0.0119940i
$631$		−	2.37815e7i		−	0.0946568i	−0.998879	−	0.0473284i	$-0.984929\pi$
$631$		−	2.37815e7i		−	0.0946568i	0.998879	−	0.0473284i	$-0.0150707\pi$
$632$	−4.77313e8			−1.89083
$633$			2.78182e8i			1.09678i
$634$	9.44321e7			0.370554
$635$	−2.39844e8	+	6.20368e7i	−0.936715	+	0.242286i
$636$			9.47328e7i			0.368238i
$637$		−	1.01645e8i		−	0.393251i
$638$	−6.61242e7			−0.254623
$639$	1.81842e7			0.0696932
$640$	−1.56432e8	+	4.04619e7i	−0.596740	+	0.154350i
$641$			3.01947e8i			1.14645i	0.819397	+	0.573227i	$0.194309\pi$
$641$			3.01947e8i			1.14645i	−0.819397	+	0.573227i	$0.805691\pi$
$642$	3.04000e8			1.14886
$643$	3.01122e8			1.13269			0.566343	−	0.824170i	$-0.308358\pi$
$643$	3.01122e8			1.13269			0.566343	+	0.824170i	$0.308358\pi$
$644$	1.76216e8	+	2.46948e8i	0.659762	+	0.924588i
$645$	1.61438e7	−	4.17567e6i	0.0601625	−	0.0155613i
$646$	−1.24643e8			−0.462351
$647$			2.63862e8i			0.974234i	0.873337	+	0.487117i	$0.161952\pi$
$647$			2.63862e8i			0.974234i	−0.873337	+	0.487117i	$0.838048\pi$
$648$		−	2.83185e8i		−	1.04075i
$649$		−	1.12554e8i		−	0.411745i
$650$	−2.45038e7	+	1.35850e7i	−0.0892266	+	0.0494673i
$651$			1.70013e8i			0.616225i
$652$		−	1.36952e8i		−	0.494113i
$653$		−	5.51764e7i		−	0.198159i	−0.995080	−	0.0990796i	$-0.968410\pi$
$653$		−	5.51764e7i		−	0.198159i	0.995080	−	0.0990796i	$-0.0315898\pi$
$654$		−	2.29176e8i		−	0.819287i
$655$	1.14133e8	+	4.41256e8i	0.406151	+	1.57024i
$656$	599216.			0.00212262
$657$			5.10081e6i			0.0179864i
$658$	−3.51029e8			−1.23216
$659$		−	1.79422e8i		−	0.626930i	−0.949600	−	0.313465i	$-0.898510\pi$
$659$		−	1.79422e8i		−	0.626930i	0.949600	−	0.313465i	$-0.101490\pi$
$660$	−4.72106e7	−	1.82524e8i	−0.164213	−	0.634873i
$661$		−	2.95049e8i		−	1.02162i	−0.859693	−	0.510811i	$-0.829345\pi$
$661$		−	2.95049e8i		−	1.02162i	0.859693	−	0.510811i	$-0.170655\pi$
$662$		−	2.01474e8i		−	0.694456i
$663$	4.36364e7			0.149730
$664$		−	4.54953e7i		−	0.155404i
$665$	−4.40290e8	+	1.13883e8i	−1.49718	+	0.387253i
$666$			5.30914e6i			0.0179722i
$667$	6.82023e7	+	9.55784e7i	0.229838	+	0.322094i
$668$			2.09099e8i			0.701491i
$669$	7.02949e7			0.234771
$670$	3.77075e6	−	975325.i	0.0125373	−	0.00324284i
$671$	−2.99618e8			−0.991745
$672$	−5.69647e8			−1.87715
$673$		−	1.41855e8i		−	0.465371i	−0.972552	−	0.232686i	$-0.925249\pi$
$673$		−	1.41855e8i		−	0.465371i	0.972552	−	0.232686i	$-0.0747513\pi$
$674$		−	1.19084e7i		−	0.0388933i
$675$	1.45819e8	+	2.63020e8i	0.474135	+	0.855221i
$676$	1.85549e8			0.600646
$677$	6.54322e7			0.210875			0.105438	−	0.994426i	$-0.466376\pi$
$677$	6.54322e7			0.210875			0.105438	+	0.994426i	$0.466376\pi$
$678$	2.65126e8			0.850676
$679$	−6.18246e8			−1.97493
$680$	2.70801e8	−	7.00441e7i	0.861239	−	0.222764i
$681$			1.39129e8i			0.440531i
$682$	6.69825e7			0.211158
$683$			4.15958e8i			1.30553i	0.757560	+	0.652765i	$0.226391\pi$
$683$			4.15958e8i			1.30553i	−0.757560	+	0.652765i	$0.773609\pi$
$684$			6.99228e6i			0.0218500i
$685$	1.63645e7	+	6.32675e7i	0.0509132	+	0.196838i
$686$		−	5.08403e8i		−	1.57484i
$687$	−2.70952e8			−0.835646
$688$	−26102.8			−8.01532e−5
$689$			3.15195e7i			0.0963655i
$690$	1.33291e8	−	1.58922e8i	0.405745	−	0.483766i
$691$	3.15030e8			0.954811			0.477406	−	0.878683i	$-0.341577\pi$
$691$	3.15030e8			0.954811			0.477406	+	0.878683i	$0.341577\pi$
$692$			3.63090e8i			1.09571i
$693$		−	2.68137e7i		−	0.0805669i
$694$	−4.54039e7			−0.135836
$695$	1.15203e8	+	4.45394e8i	0.343171	+	1.32675i
$696$	−1.36242e8			−0.404096
$697$	4.85393e8			1.43349
$698$		−	6.85160e7i		−	0.201477i
$699$	−9.95896e6			−0.0291596
$700$	3.40735e8	−	1.88904e8i	0.993395	−	0.550739i
$701$		−	2.49999e8i		−	0.725745i	−0.931839	−	0.362873i	$-0.881796\pi$
$701$		−	2.49999e8i		−	0.725745i	0.931839	−	0.362873i	$-0.118204\pi$
$702$		−	3.45127e7i		−	0.0997626i
$703$			2.01603e8i			0.580272i
$704$			2.24910e8i			0.644601i
$705$	−9.70012e7	−	3.75021e8i	−0.276828	−	1.07026i
$706$	1.25375e8			0.356283
$707$	1.07355e9			3.03784
$708$		−	8.85285e7i		−	0.249450i
$709$		−	3.29160e8i		−	0.923567i	−0.886993	−	0.461784i	$-0.847210\pi$
$709$		−	3.29160e8i		−	0.923567i	0.886993	−	0.461784i	$-0.152790\pi$
$710$	3.54824e8	−	9.17770e7i	0.991374	−	0.256424i
$711$		−	2.85971e7i		−	0.0795633i
$712$	−4.15500e8			−1.15115
$713$	−6.90876e7	−	9.68190e7i	−0.190604	−	0.267111i
$714$	3.75947e8			1.03284
$715$	−1.57079e7	−	6.07293e7i	−0.0429735	−	0.166142i
$716$	−2.44354e8			−0.665702
$717$			4.99850e7i			0.135607i
$718$	−2.00315e8			−0.541177
$719$	1.28832e8			0.346605			0.173303	−	0.984869i	$-0.444556\pi$
$719$	1.28832e8			0.346605			0.173303	+	0.984869i	$0.444556\pi$
$720$	5180.61	+	20029.0i	1.38798e−5	+	5.36614e-5i
$721$	6.27369e8			1.67385
$722$			6.82786e7i			0.181415i
$723$	5.66342e8			1.49853
$724$			6.45209e6i			0.0170014i
$725$	1.31877e8	−	7.31130e7i	0.346064	−	0.191858i
$726$	−1.99126e7			−0.0520376
$727$	4.12634e8			1.07390			0.536948	−	0.843616i	$-0.319577\pi$
$727$	4.12634e8			1.07390			0.536948	+	0.843616i	$0.319577\pi$
$728$	−1.17122e8			−0.303559
$729$	−3.69768e8			−0.954435
$730$	2.57442e7	+	9.95311e7i	0.0661776	+	0.255853i
$731$	−2.11444e7			−0.0541308
$732$	−2.35661e8			−0.600834
$733$	3.67306e8			0.932645			0.466323	−	0.884615i	$-0.345579\pi$
$733$	3.67306e8			0.932645			0.466323	+	0.884615i	$0.345579\pi$
$734$		−	1.36138e8i		−	0.344263i
$735$	9.35573e8	−	2.41991e8i	2.35622	−	0.609448i
$736$	3.24403e8	−	2.31486e8i	0.813675	−	0.580618i
$737$			8.72005e6i			0.0217829i
$738$			1.68709e7i			0.0419729i
$739$	9.59330e7			0.237703			0.118851	−	0.992912i	$-0.462079\pi$
$739$	9.59330e7			0.237703			0.118851	+	0.992912i	$0.462079\pi$
$740$	−4.32483e7	−	1.67204e8i	−0.106727	−	0.412622i
$741$			5.75928e7i			0.141551i
$742$			2.71555e8i			0.664730i
$743$	−2.89146e8			−0.704939			−0.352469	−	0.935823i	$-0.614658\pi$
$743$	−2.89146e8			−0.704939			−0.352469	+	0.935823i	$0.614658\pi$
$744$	1.38011e8			0.335115
$745$	5.84280e7	−	1.51127e7i	0.141303	−	0.0365488i
$746$			7.36639e7i			0.177434i
$747$	2.72574e6			0.00653918
$748$			2.39062e8i			0.571223i
$749$	−1.40648e9			−3.34726
$750$	−1.83377e8	−	1.93198e8i	−0.434672	−	0.457951i
$751$		−	1.42211e8i		−	0.335749i	−0.985808	−	0.167874i	$-0.946310\pi$
$751$		−	1.42211e8i		−	0.335749i	0.985808	−	0.167874i	$-0.0536903\pi$
$752$			606370.i			0.00142588i
$753$	−2.99798e8			−0.702173
$754$	−1.73045e7			−0.0403688
$755$	−7.85518e7	−	3.03693e8i	−0.182522	−	0.705658i
$756$			4.79911e8i			1.11070i
$757$	3.82776e8			0.882383			0.441192	−	0.897413i	$-0.354556\pi$
$757$	3.82776e8			0.882383			0.441192	+	0.897413i	$0.354556\pi$
$758$	3.00944e8			0.691000
$759$	2.69739e8	+	3.78011e8i	0.616906	+	0.864528i
$760$	9.24465e7	+	3.57412e8i	0.210596	+	0.814196i
$761$	2.43883e8			0.553385			0.276693	−	0.960958i	$-0.410762\pi$
$761$	2.43883e8			0.553385			0.276693	+	0.960958i	$0.410762\pi$
$762$			2.70293e8i			0.610900i
$763$			1.06030e9i			2.38702i
$764$		−	1.72500e8i		−	0.386821i
$765$	4.19653e6	+	1.62244e7i	0.00937359	+	0.0362397i
$766$			3.20413e8i			0.712891i
$767$		−	2.94552e7i		−	0.0652794i
$768$			4.62796e8i			1.02166i
$769$			7.34206e7i			0.161450i	0.996736	+	0.0807252i	$0.0257236\pi$
$769$			7.34206e7i			0.161450i	−0.996736	+	0.0807252i	$0.974276\pi$
$770$	−1.35331e8	−	5.23209e8i	−0.296432	−	1.14605i
$771$	7.26041e8			1.58416
$772$			5.39487e7i			0.117254i
$773$	−3.53874e8			−0.766144			−0.383072	−	0.923719i	$-0.625134\pi$
$773$	−3.53874e8			−0.766144			−0.383072	+	0.923719i	$0.625134\pi$
$774$		−	734923.i		−	0.00158496i
$775$	−1.33589e8	+	7.40620e7i	−0.286990	+	0.159107i
$776$			5.01871e8i			1.07401i
$777$		−	6.08072e8i		−	1.29626i
$778$	6.02099e7			0.127858
$779$			6.40637e8i			1.35519i
$780$	−1.23549e7	−	4.77659e7i	−0.0260349	−	0.100655i
$781$			8.20547e8i			1.72246i
$782$	−2.14094e8	+	1.52772e8i	−0.447698	+	0.319466i
$783$			1.85744e8i			0.386928i
$784$	−1.51272e6			−0.00313914
$785$	−7.21057e7	−	2.78771e8i	−0.149060	−	0.576288i
$786$	4.97276e8			1.02407
$787$	−7.80581e8			−1.60138			−0.800689	−	0.599080i	$-0.795533\pi$
$787$	−7.80581e8			−1.60138			−0.800689	+	0.599080i	$0.795533\pi$
$788$		−	3.68486e8i		−	0.753083i
$789$			2.13389e8i			0.434452i
$790$	−1.44332e8	−	5.58009e8i	−0.292739	−	1.13177i
$791$	−1.22663e9			−2.47848
$792$	−2.17664e7			−0.0438138
$793$	−7.84092e7			−0.157234
$794$	2.65725e8			0.530849
$795$	−2.90115e8	+	7.50396e7i	−0.577389	+	0.149345i
$796$		−	4.63835e8i		−	0.919653i
$797$	4.34566e8			0.858381			0.429191	−	0.903214i	$-0.358799\pi$
$797$	4.34566e8			0.858381			0.429191	+	0.903214i	$0.358799\pi$
$798$			4.96187e8i			0.976421i
$799$			4.91188e8i			0.962958i
$800$	−2.48153e8	−	4.47605e8i	−0.484673	−	0.874229i
$801$		−	2.48937e7i		−	0.0484387i
$802$	−3.96510e7			−0.0768654
$803$	−2.30170e8			−0.444532
$804$			6.85866e6i			0.0131969i
$805$	−6.16683e8	+	7.35265e8i	−1.18215	+	1.40947i
$806$	1.75292e7			0.0334777
$807$			1.92590e7i			0.0366448i
$808$		−	8.71474e8i		−	1.65204i
$809$	−3.29883e8			−0.623038			−0.311519	−	0.950240i	$-0.600838\pi$
$809$	−3.29883e8			−0.623038			−0.311519	+	0.950240i	$0.600838\pi$
$810$	3.31061e8	−	8.56306e7i	0.622950	−	0.161129i
$811$	3.55550e8			0.666559			0.333279	−	0.942828i	$-0.391845\pi$
$811$	3.55550e8			0.666559			0.333279	+	0.942828i	$0.391845\pi$
$812$	2.40626e8			0.449442
$813$			2.80150e8i			0.521338i
$814$	−2.39571e8			−0.444182
$815$	4.19410e8	−	1.08482e8i	0.774757	−	0.200395i
$816$		−	649413.i		−	0.00119523i
$817$		−	2.79071e7i		−	0.0511739i
$818$		−	3.35384e8i		−	0.612749i
$819$		−	7.01707e6i		−	0.0127733i
$820$	−1.37431e8	−	5.31328e8i	−0.249254	−	0.963653i
$821$	6.32869e8			1.14363			0.571813	−	0.820384i	$-0.306240\pi$
$821$	6.32869e8			1.14363			0.571813	+	0.820384i	$0.306240\pi$
$822$	7.12997e7			0.128373
$823$			7.34479e8i			1.31759i	0.752323	+	0.658795i	$0.228933\pi$
$823$			7.34479e8i			1.31759i	−0.752323	+	0.658795i	$0.771067\pi$
$824$		−	5.09277e8i		−	0.910274i
$825$	5.21573e8	−	2.89161e8i	0.928867	−	0.514965i
$826$		−	2.53770e8i		−	0.450298i
$827$	−1.87511e8			−0.331520			−0.165760	−	0.986166i	$-0.553008\pi$
$827$	−1.87511e8			−0.331520			−0.165760	+	0.986166i	$0.553008\pi$
$828$	8.57027e6	+	1.20103e7i	0.0150974	+	0.0211575i
$829$	−2.87056e8			−0.503853			−0.251926	−	0.967746i	$-0.581064\pi$
$829$	−2.87056e8			−0.503853			−0.251926	+	0.967746i	$0.581064\pi$
$830$	5.31869e7	−	1.37571e7i	0.0930187	−	0.0240598i
$831$	−5.00030e7			−0.0871351
$832$			5.88584e7i			0.102197i
$833$	−1.22538e9			−2.11999
$834$	5.01939e8			0.865273
$835$	−6.40355e8	+	1.65631e8i	−1.09992	+	0.284500i
$836$	−3.15522e8			−0.540020
$837$		−	1.88155e8i		−	0.320878i
$838$	4.17823e8			0.710003
$839$		−	3.28187e8i		−	0.555694i	−0.960625	−	0.277847i	$-0.910379\pi$
$839$		−	3.28187e8i		−	0.555694i	0.960625	−	0.277847i	$-0.0896208\pi$
$840$	−2.78836e8	−	1.07802e9i	−0.470447	−	1.81882i
$841$	−5.01692e8			−0.843430
$842$	5.67546e7			0.0950747
$843$	1.32400e8			0.221006
$844$	−3.98832e8			−0.663381
$845$	1.46977e8	+	5.68235e8i	0.243601	+	0.941798i
$846$	−1.70723e7			−0.0281956
$847$	9.21273e7			0.151614
$848$	469085.			0.000769243
$849$			6.96601e7i			0.113831i
$850$	1.63772e8	+	2.95404e8i	0.266675	+	0.481016i
$851$	2.47100e8	+	3.46285e8i	0.400945	+	0.561882i
$852$			6.45393e8i			1.04353i
$853$		−	2.74791e8i		−	0.442748i	−0.975189	−	0.221374i	$-0.928946\pi$
$853$		−	2.74791e8i		−	0.442748i	0.975189	−	0.221374i	$-0.0710541\pi$
$854$	−6.75530e8			−1.08460
$855$	−2.14135e7	+	5.53872e6i	−0.0342602	+	0.00886158i
$856$			1.14173e9i			1.82030i
$857$		−	9.15147e8i		−	1.45395i	−0.686665	−	0.726974i	$-0.740926\pi$
$857$		−	9.15147e8i		−	1.45395i	0.686665	−	0.726974i	$-0.259074\pi$
$858$	−6.84392e7			−0.108353
$859$	8.01648e8			1.26475			0.632375	−	0.774663i	$-0.282080\pi$
$859$	8.01648e8			1.26475			0.632375	+	0.774663i	$0.282080\pi$
$860$	5.98669e6	+	2.31454e7i	0.00941220	+	0.0363890i
$861$		−	1.93228e9i		−	3.02733i
$862$	−3.59293e8			−0.560954
$863$		−	6.03222e8i		−	0.938523i	−0.883059	−	0.469261i	$-0.844520\pi$
$863$		−	6.03222e8i		−	0.938523i	0.883059	−	0.469261i	$-0.155480\pi$
$864$	6.30434e8			0.977459
$865$	−1.11195e9	+	2.87611e8i	−1.71805	+	0.444382i
$866$		−	1.62589e8i		−	0.250344i
$867$			1.39236e8i			0.213645i
$868$	−2.43749e8			−0.372721
$869$	1.29042e9			1.96640
$870$	−4.11975e7	−	1.59276e8i	−0.0625624	−	0.241876i
$871$			2.28202e6i			0.00345354i
$872$	8.60718e8			1.29811
$873$	−3.00684e7			−0.0451927
$874$	−2.01634e8	−	2.82569e8i	−0.302016	−	0.423243i
$875$	8.48411e8	+	8.93850e8i	1.26643	+	1.33426i
$876$	−1.81038e8			−0.269313
$877$		−	1.24145e9i		−	1.84048i	−0.391354	−	0.920240i	$-0.627993\pi$
$877$		−	1.24145e9i		−	1.84048i	0.391354	−	0.920240i	$-0.372007\pi$
$878$			7.47452e8i			1.10433i
$879$			1.19026e9i			1.75257i
$880$	−903795.	+	233771.i	−0.00132624	+	0.000343039i
$881$			1.33935e9i			1.95870i	0.202180	+	0.979348i	$0.435197\pi$
$881$			1.33935e9i			1.95870i	−0.202180	+	0.979348i	$0.564803\pi$
$882$		−	4.25907e7i		−	0.0620739i
$883$			7.21605e8i			1.04814i	0.851677	+	0.524068i	$0.175586\pi$
$883$			7.21605e8i			1.04814i	−0.851677	+	0.524068i	$0.824414\pi$
$884$			6.25619e7i			0.0905635i
$885$	2.71114e8	−	7.01251e7i	0.391132	−	0.101168i
$886$	2.62930e8			0.378042
$887$		−	5.72574e8i		−	0.820466i	−0.911981	−	0.410233i	$-0.865447\pi$
$887$		−	5.72574e8i		−	0.820466i	0.911981	−	0.410233i	$-0.134553\pi$
$888$	−4.93612e8			−0.704932
$889$		−	1.25054e9i		−	1.77988i
$890$	−1.25641e8	−	4.85746e8i	−0.178222	−	0.689032i
$891$			7.65595e8i			1.08235i
$892$			1.00782e8i			0.142000i
$893$	−6.48286e8			−0.910358
$894$		−	6.58458e7i		−	0.0921543i
$895$	−1.93557e8	−	7.48321e8i	−0.269985	−	1.04380i
$896$		−	8.15630e8i		−	1.13389i
$897$	7.05901e7	+	9.89246e7i	0.0978062	+	0.137065i
$898$			5.15567e8i			0.711961i
$899$	−9.43403e7			−0.129843
$900$	1.65716e7	−	9.18734e6i	0.0227320	−	0.0126027i
$901$	3.79980e8			0.519501
$902$	−7.61288e8			−1.03736
$903$			8.41730e7i			0.114317i
$904$			9.95738e8i			1.34784i
$905$	−1.97592e7	+	5.11082e6i	−0.0266578	+	0.00689517i
$906$	−3.42249e8			−0.460211
$907$	1.10967e9			1.48721			0.743606	−	0.668618i	$-0.233114\pi$
$907$	1.10967e9			1.48721			0.743606	+	0.668618i	$0.233114\pi$
$908$	−1.99470e8			−0.266453
$909$	5.22123e7			0.0695154
$910$	−3.54157e7	−	1.36923e8i	−0.0469972	−	0.181698i
$911$		−	4.63479e8i		−	0.613020i	−0.951867	−	0.306510i	$-0.900839\pi$
$911$		−	4.63479e8i		−	0.613020i	0.951867	−	0.306510i	$-0.0991613\pi$
$912$	857117.			0.00112994
$913$			1.22997e8i			0.161616i
$914$		−	6.29653e8i		−	0.824637i
$915$	−1.86672e8	−	7.21701e8i	−0.243677	−	0.942094i
$916$		−	3.88466e8i		−	0.505437i
$917$	−2.30069e9			−2.98367
$918$	−4.16065e8			−0.537815
$919$		−	1.53307e9i		−	1.97522i	−0.156923	−	0.987611i	$-0.550158\pi$
$919$		−	1.53307e9i		−	1.97522i	0.156923	−	0.987611i	$-0.449842\pi$
$920$	5.96863e8	+	5.00602e8i	0.766498	+	0.642879i
$921$	−1.38125e9			−1.76804
$922$		−	3.22652e8i		−	0.411663i
$923$			2.14735e8i			0.273085i
$924$	9.51671e8			1.20634
$925$	4.77798e8	−	2.64892e8i	0.603697	−	0.334690i
$926$	1.14039e8			0.143622
$927$	3.05121e7			0.0383031
$928$		−	3.16097e8i		−	0.395528i
$929$	1.25366e9			1.56363			0.781813	−	0.623512i	$-0.214295\pi$
$929$	1.25366e9			1.56363			0.781813	+	0.623512i	$0.214295\pi$
$930$	4.17323e7	+	1.61343e8i	0.0518828	+	0.200587i
$931$		−	1.61729e9i		−	2.00419i
$932$		−	1.42782e7i		−	0.0176371i
$933$		−	3.42633e8i		−	0.421876i
$934$			2.81173e8i			0.345090i
$935$	−7.32115e8	+	1.89365e8i	−0.895663	+	0.231668i
$936$	−5.69622e6			−0.00694639
$937$	1.19307e9			1.45026			0.725131	−	0.688611i	$-0.241779\pi$
$937$	1.19307e9			1.45026			0.725131	+	0.688611i	$0.241779\pi$
$938$			1.96606e7i			0.0238225i
$939$		−	1.21905e9i		−	1.47239i
$940$	5.37671e8	−	1.39071e8i	0.647341	−	0.167438i
$941$		−	1.64137e8i		−	0.196986i	−0.995138	−	0.0984932i	$-0.968598\pi$
$941$		−	1.64137e8i		−	0.196986i	0.995138	−	0.0984932i	$-0.0314023\pi$
$942$	−3.14163e8			−0.375840
$943$	7.85213e8	+	1.10039e9i	0.936381	+	1.31224i
$944$	−438363.			−0.000521097
$945$	−1.46971e9	+	3.80147e8i	−1.74155	+	0.450460i
$946$	3.31629e7			0.0391722
$947$		−	1.01072e9i		−	1.19009i	−0.803692	−	0.595045i	$-0.797134\pi$
$947$		−	1.01072e9i		−	1.19009i	0.803692	−	0.595045i	$-0.202866\pi$
$948$	1.01497e9			1.19132
$949$	−6.02350e7			−0.0704775
$950$	−3.89883e8	+	2.16152e8i	−0.454741	+	0.252109i
$951$	−5.26018e8			−0.611588
$952$			1.41195e9i			1.63647i
$953$	−4.80894e8			−0.555611			−0.277805	−	0.960637i	$-0.589607\pi$
$953$	−4.80894e8			−0.555611			−0.277805	+	0.960637i	$0.589607\pi$
$954$			1.32071e7i			0.0152111i
$955$	5.28274e8	−	1.36641e8i	0.606525	−	0.156881i
$956$	−7.16639e7			−0.0820214
$957$	3.68333e8			0.420248
$958$	−5.51143e8			−0.626856
$959$	−3.29875e8			−0.374019
$960$	−5.41750e8	+	1.40126e8i	−0.612329	+	0.158382i
$961$	−7.91939e8			−0.892322
$962$	−6.26951e7			−0.0704220
$963$	−6.84044e7			−0.0765958
$964$			8.11969e8i			0.906376i
$965$	−1.65215e8	+	4.27338e7i	−0.183852	+	0.0475543i
$966$	6.08165e8	+	8.52279e8i	0.674668	+	0.945476i
$967$		−	1.13517e9i		−	1.25540i	−0.778454	−	0.627702i	$-0.783996\pi$
$967$		−	1.13517e9i		−	1.25540i	0.778454	−	0.627702i	$-0.216004\pi$
$968$		−	7.47858e7i		−	0.0824504i
$969$	6.94304e8			0.763094
$970$	−5.86719e8	+	1.51758e8i	−0.642858	+	0.166278i
$971$		−	1.26549e9i		−	1.38229i	−0.722714	−	0.691147i	$-0.757106\pi$
$971$		−	1.26549e9i		−	1.38229i	0.722714	−	0.691147i	$-0.242894\pi$
$972$			4.77068e7i			0.0519495i
$973$	−2.32227e9			−2.52101
$974$	2.27876e8			0.246616
$975$	1.36494e8	−	7.56726e7i	0.147266	−	0.0816441i
$976$			1.16691e6i			0.00125513i
$977$	−8.01050e8			−0.858966			−0.429483	−	0.903075i	$-0.641304\pi$
$977$	−8.01050e8			−0.858966			−0.429483	+	0.903075i	$0.641304\pi$
$978$		−	4.72656e8i		−	0.505276i
$979$	1.12331e9			1.19716
$980$	3.46944e8	+	1.34134e9i	0.368622	+	1.42515i
$981$			5.15679e7i			0.0546226i
$982$		−	9.93971e8i		−	1.04964i
$983$	5.20105e7			0.0547558			0.0273779	−	0.999625i	$-0.491284\pi$
$983$	5.20105e7			0.0547558			0.0273779	+	0.999625i	$0.491284\pi$
$984$	−1.56856e9			−1.64632
$985$	1.12847e9	−	2.91885e8i	1.18082	−	0.305424i
$986$			2.08613e8i			0.217626i
$987$	1.95535e9			2.03363
$988$	−8.25712e7			−0.0856166
$989$	−3.42051e7	−	4.79348e7i	−0.0353592	−	0.0495521i
$990$	−6.58182e6	−	2.54463e7i	−0.00678329	−	0.0262252i
$991$	−1.31237e9			−1.34845			−0.674224	−	0.738527i	$-0.735522\pi$
$991$	−1.31237e9			−1.34845			−0.674224	+	0.738527i	$0.735522\pi$
$992$			3.20200e8i			0.328010i
$993$			1.12227e9i			1.14618i
$994$			1.85004e9i			1.88374i
$995$	1.42047e9	−	3.67412e8i	1.44199	−	0.372979i
$996$			9.67422e7i			0.0979124i
$997$		−	1.29114e9i		−	1.30283i	−0.758720	−	0.651416i	$-0.774175\pi$
$997$		−	1.29114e9i		−	1.30283i	0.758720	−	0.651416i	$-0.225825\pi$
$998$		−	9.05169e8i		−	0.910622i
$999$			6.72960e8i			0.674983i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.7.c.c.114.2	yes	68
5.4	even	2	inner	115.7.c.c.114.67	yes	68
23.22	odd	2	inner	115.7.c.c.114.68	yes	68
115.114	odd	2	inner	115.7.c.c.114.1	✓	68

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.7.c.c.114.1	✓	68	115.114	odd	2	inner
115.7.c.c.114.2	yes	68	1.1	even	1	trivial
115.7.c.c.114.67	yes	68	5.4	even	2	inner
115.7.c.c.114.68	yes	68	23.22	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	115.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.94808i q^{2} -27.5624i q^{3} +39.5165 q^{4} +(31.3018 + 121.017i) q^{5} +136.381 q^{6} -630.980 q^{7} +512.208i q^{8} -30.6878 q^{9} +O(q^{10})\)	\(q+4.94808i q^{2} -27.5624i q^{3} +39.5165 q^{4} +(31.3018 + 121.017i) q^{5} +136.381 q^{6} -630.980 q^{7} +512.208i q^{8} -30.6878 q^{9} +(-598.804 + 154.884i) q^{10} -1384.76i q^{11} -1089.17i q^{12} -362.389i q^{13} -3122.14i q^{14} +(3335.53 - 862.753i) q^{15} -5.39320 q^{16} -4368.74 q^{17} -151.846i q^{18} -5766.01i q^{19} +(1236.94 + 4782.18i) q^{20} +17391.3i q^{21} +6851.92 q^{22} +(-7067.26 - 9904.02i) q^{23} +14117.7 q^{24} +(-13665.4 + 7576.11i) q^{25} +1793.13 q^{26} -19247.2i q^{27} -24934.1 q^{28} -9650.46 q^{29} +(4268.97 + 16504.5i) q^{30} +9775.73 q^{31} +32754.6i q^{32} -38167.4 q^{33} -21616.9i q^{34} +(-19750.8 - 76359.5i) q^{35} -1212.67 q^{36} -34964.1 q^{37} +28530.7 q^{38} -9988.32 q^{39} +(-61986.1 + 16033.0i) q^{40} -111106. q^{41} -86053.8 q^{42} +4839.94 q^{43} -54720.9i q^{44} +(-960.581 - 3713.75i) q^{45} +(49005.9 - 34969.4i) q^{46} -112432. i q^{47} +148.650i q^{48} +280487. q^{49} +(-37487.2 - 67617.5i) q^{50} +120413. i q^{51} -14320.3i q^{52} -86977.0 q^{53} +95236.7 q^{54} +(167580. - 43345.5i) q^{55} -323193. i q^{56} -158925. q^{57} -47751.3i q^{58} +81280.7 q^{59} +(131809. - 34092.9i) q^{60} -216368. i q^{61} +48371.1i q^{62} +19363.4 q^{63} -162418. q^{64} +(43855.4 - 11343.4i) q^{65} -188856. i q^{66} -6297.15 q^{67} -172637. q^{68} +(-272979. + 194791. i) q^{69} +(377833. - 97728.5i) q^{70} -592554. q^{71} -15718.5i q^{72} -166216. i q^{73} -173005. i q^{74} +(208816. + 376652. i) q^{75} -227852. i q^{76} +873758. i q^{77} -49423.0i q^{78} +931872. i q^{79} +(-168.817 - 652.671i) q^{80} -552871. q^{81} -549760. i q^{82} -88821.9 q^{83} +687245. i q^{84} +(-136749. - 528694. i) q^{85} +23948.4i q^{86} +265990. i q^{87} +709287. q^{88} +811194. i q^{89} +(18375.9 - 4753.03i) q^{90} +228660. i q^{91} +(-279273. - 391372. i) q^{92} -269443. i q^{93} +556324. q^{94} +(697787. - 180486. i) q^{95} +902797. q^{96} +979818. q^{97} +1.38787e6i q^{98} +42495.3i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 2440 q^{4} + 352 q^{6} - 16908 q^{9}+O(q^{10}) \) 68 * q - 2440 * q^4 + 352 * q^6 - 16908 * q^9	\( 68 q - 2440 q^{4} + 352 q^{6} - 16908 q^{9} + 66968 q^{16} - 30916 q^{24} + 32588 q^{25} - 22072 q^{26} + 103360 q^{29} - 17256 q^{31} - 358168 q^{35} + 451984 q^{36} + 192432 q^{39} - 183552 q^{41} - 397956 q^{46} + 806756 q^{49} - 749960 q^{50} - 1638436 q^{54} - 1752 q^{55} - 505552 q^{59} - 4095100 q^{64} + 1354876 q^{69} + 1196604 q^{70} + 493688 q^{71} + 3178568 q^{75} + 2473820 q^{81} + 3306336 q^{85} - 3770196 q^{94} + 896144 q^{95} + 16928136 q^{96}+O(q^{100}) \) 68 * q - 2440 * q^4 + 352 * q^6 - 16908 * q^9 + 66968 * q^16 - 30916 * q^24 + 32588 * q^25 - 22072 * q^26 + 103360 * q^29 - 17256 * q^31 - 358168 * q^35 + 451984 * q^36 + 192432 * q^39 - 183552 * q^41 - 397956 * q^46 + 806756 * q^49 - 749960 * q^50 - 1638436 * q^54 - 1752 * q^55 - 505552 * q^59 - 4095100 * q^64 + 1354876 * q^69 + 1196604 * q^70 + 493688 * q^71 + 3178568 * q^75 + 2473820 * q^81 + 3306336 * q^85 - 3770196 * q^94 + 896144 * q^95 + 16928136 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more