LMFDB - Embedded newform 115.7.d.a.91.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,7,Mod(91,115)]

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([0, 1]))

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

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

chi := DirichletCharacter("115.91");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	115.d (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:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.7
Character	$\chi$	$=$	115.91
Dual form			115.7.d.a.91.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.5973 q^{2} -47.4272 q^{3} +94.6924 q^{4} +55.9017i q^{5} +597.456 q^{6} -458.638i q^{7} -386.642 q^{8} +1520.34 q^{9} +O(q^{10})$	\(q-12.5973 q^{2} -47.4272 q^{3} +94.6924 q^{4} +55.9017i q^{5} +597.456 q^{6} -458.638i q^{7} -386.642 q^{8} +1520.34 q^{9} -704.211i q^{10} +570.910i q^{11} -4491.00 q^{12} -1734.09 q^{13} +5777.60i q^{14} -2651.26i q^{15} -1189.66 q^{16} -2306.80i q^{17} -19152.2 q^{18} +6682.88i q^{19} +5293.47i q^{20} +21751.9i q^{21} -7191.93i q^{22} +(-6911.01 + 10013.7i) q^{23} +18337.4 q^{24} -3125.00 q^{25} +21844.9 q^{26} -37531.1 q^{27} -43429.5i q^{28} -15193.0 q^{29} +33398.8i q^{30} -33284.0 q^{31} +39731.6 q^{32} -27076.7i q^{33} +29059.4i q^{34} +25638.6 q^{35} +143965. q^{36} -80251.4i q^{37} -84186.3i q^{38} +82243.0 q^{39} -21614.0i q^{40} +53792.6 q^{41} -274016. i q^{42} +113442. i q^{43} +54060.8i q^{44} +84989.6i q^{45} +(87060.2 - 126146. i) q^{46} -80821.9 q^{47} +56422.3 q^{48} -92699.4 q^{49} +39366.6 q^{50} +109405. i q^{51} -164205. q^{52} +191262. i q^{53} +472791. q^{54} -31914.8 q^{55} +177329. i q^{56} -316950. i q^{57} +191391. q^{58} +239927. q^{59} -251054. i q^{60} -35571.9i q^{61} +419289. q^{62} -697285. i q^{63} -424374. q^{64} -96938.5i q^{65} +341093. i q^{66} +482146. i q^{67} -218436. i q^{68} +(327770. - 474921. i) q^{69} -322978. q^{70} -362192. q^{71} -587828. q^{72} -520561. q^{73} +1.01095e6i q^{74} +148210. q^{75} +632818. i q^{76} +261841. q^{77} -1.03604e6 q^{78} +201840. i q^{79} -66504.1i q^{80} +671667. q^{81} -677642. q^{82} -1.04230e6i q^{83} +2.05974e6i q^{84} +128954. q^{85} -1.42907e6i q^{86} +720563. q^{87} -220738. i q^{88} -132496. i q^{89} -1.07064e6i q^{90} +795318. i q^{91} +(-654420. + 948220. i) q^{92} +1.57857e6 q^{93} +1.01814e6 q^{94} -373584. q^{95} -1.88436e6 q^{96} -185139. i q^{97} +1.16776e6 q^{98} +867978. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 20 q^{2} + 1584 q^{4} + 410 q^{6} - 1210 q^{8} + 12896 q^{9}+O(q^{10}) $ 48 * q - 20 * q^2 + 1584 * q^4 + 410 * q^6 - 1210 * q^8 + 12896 * q^9	$ 48 q - 20 q^{2} + 1584 q^{4} + 410 q^{6} - 1210 q^{8} + 12896 q^{9} + 4290 q^{12} - 1440 q^{13} + 65400 q^{16} + 4610 q^{18} + 26600 q^{23} + 14940 q^{24} - 150000 q^{25} + 47594 q^{26} + 16080 q^{27} + 131800 q^{29} - 1392 q^{31} - 225040 q^{32} + 5000 q^{35} + 658786 q^{36} - 236320 q^{39} - 351496 q^{41} + 382692 q^{46} + 395680 q^{47} + 1042550 q^{48} - 637848 q^{49} + 62500 q^{50} + 523890 q^{52} - 241250 q^{54} - 402000 q^{55} - 479130 q^{58} - 466312 q^{59} - 1124330 q^{62} + 837582 q^{64} + 1021060 q^{69} - 396000 q^{70} - 114336 q^{71} - 1960750 q^{72} - 498720 q^{73} + 3610400 q^{77} - 1104610 q^{78} + 972888 q^{81} + 124950 q^{82} - 246000 q^{85} - 2090960 q^{87} + 4913480 q^{92} + 3234320 q^{93} - 5550378 q^{94} - 1664000 q^{95} - 776990 q^{96} + 9993220 q^{98}+O(q^{100}) $ 48 * q - 20 * q^2 + 1584 * q^4 + 410 * q^6 - 1210 * q^8 + 12896 * q^9 + 4290 * q^12 - 1440 * q^13 + 65400 * q^16 + 4610 * q^18 + 26600 * q^23 + 14940 * q^24 - 150000 * q^25 + 47594 * q^26 + 16080 * q^27 + 131800 * q^29 - 1392 * q^31 - 225040 * q^32 + 5000 * q^35 + 658786 * q^36 - 236320 * q^39 - 351496 * q^41 + 382692 * q^46 + 395680 * q^47 + 1042550 * q^48 - 637848 * q^49 + 62500 * q^50 + 523890 * q^52 - 241250 * q^54 - 402000 * q^55 - 479130 * q^58 - 466312 * q^59 - 1124330 * q^62 + 837582 * q^64 + 1021060 * q^69 - 396000 * q^70 - 114336 * q^71 - 1960750 * q^72 - 498720 * q^73 + 3610400 * q^77 - 1104610 * q^78 + 972888 * q^81 + 124950 * q^82 - 246000 * q^85 - 2090960 * q^87 + 4913480 * q^92 + 3234320 * q^93 - 5550378 * q^94 - 1664000 * q^95 - 776990 * q^96 + 9993220 * q^98

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 $

$2$	−12.5973			−1.57466			−0.787332	−	0.616529i	$-0.788538\pi$
$2$	−12.5973			−1.57466			−0.787332	+	0.616529i	$0.788538\pi$
$3$	−47.4272			−1.75656			−0.878282	−	0.478143i	$-0.841310\pi$
$3$	−47.4272			−1.75656			−0.878282	+	0.478143i	$0.841310\pi$
$4$	94.6924			1.47957
$5$			55.9017i			0.447214i
$5$			55.9017i			0.447214i
$6$	597.456			2.76600
$7$		−	458.638i		−	1.33714i	−0.743651	−	0.668568i	$-0.766908\pi$
$7$		−	458.638i		−	1.33714i	0.743651	−	0.668568i	$-0.233092\pi$
$8$	−386.642			−0.755160
$9$	1520.34			2.08552
$10$		−	704.211i		−	0.704211i
$11$			570.910i			0.428933i	0.976731	+	0.214467i	$0.0688013\pi$
$11$			570.910i			0.428933i	−0.976731	+	0.214467i	$0.931199\pi$
$12$	−4491.00			−2.59896
$13$	−1734.09			−0.789299			−0.394649	−	0.918832i	$-0.629134\pi$
$13$	−1734.09			−0.789299			−0.394649	+	0.918832i	$0.629134\pi$
$14$			5777.60i			2.10554i
$15$		−	2651.26i		−	0.785559i
$16$	−1189.66			−0.290444
$17$		−	2306.80i		−	0.469529i	−0.972052	−	0.234764i	$-0.924568\pi$
$17$		−	2306.80i		−	0.469529i	0.972052	−	0.234764i	$-0.0754319\pi$
$18$	−19152.2			−3.28399
$19$			6682.88i			0.974322i	0.873312	+	0.487161i	$0.161968\pi$
$19$			6682.88i			0.974322i	−0.873312	+	0.487161i	$0.838032\pi$
$20$			5293.47i			0.661683i
$21$			21751.9i			2.34876i
$22$		−	7191.93i		−	0.675426i
$23$	−6911.01	+	10013.7i	−0.568013	+	0.823020i
$23$	−6911.01	+	10013.7i	−0.568013	+	0.823020i
$24$	18337.4			1.32649
$25$	−3125.00			−0.200000
$26$	21844.9			1.24288
$27$	−37531.1			−1.90678
$28$		−	43429.5i		−	1.97838i
$29$	−15193.0			−0.622946			−0.311473	−	0.950255i	$-0.600822\pi$
$29$	−15193.0			−0.622946			−0.311473	+	0.950255i	$0.600822\pi$
$30$			33398.8i			1.23699i
$31$	−33284.0			−1.11725			−0.558625	−	0.829420i	$-0.688671\pi$
$31$	−33284.0			−1.11725			−0.558625	+	0.829420i	$0.688671\pi$
$32$	39731.6			1.21251
$33$		−	27076.7i		−	0.753448i
$34$			29059.4i			0.739351i
$35$	25638.6			0.597985
$36$	143965.			3.08566
$37$		−	80251.4i		−	1.58434i	−0.610303	−	0.792168i	$-0.708952\pi$
$37$		−	80251.4i		−	1.58434i	0.610303	−	0.792168i	$-0.291048\pi$
$38$		−	84186.3i		−	1.53423i
$39$	82243.0			1.38645
$40$		−	21614.0i		−	0.337718i
$41$	53792.6			0.780496			0.390248	−	0.920710i	$-0.372389\pi$
$41$	53792.6			0.780496			0.390248	+	0.920710i	$0.372389\pi$
$42$		−	274016.i		−	3.69852i
$43$			113442.i			1.42682i	0.700747	+	0.713410i	$0.252850\pi$
$43$			113442.i			1.42682i	−0.700747	+	0.713410i	$0.747150\pi$
$44$			54060.8i			0.634636i
$45$			84989.6i			0.932671i
$46$	87060.2	−	126146.i	0.894430	−	1.29598i
$47$	−80821.9			−0.778458			−0.389229	−	0.921141i	$-0.627259\pi$
$47$	−80821.9			−0.778458			−0.389229	+	0.921141i	$0.627259\pi$
$48$	56422.3			0.510184
$49$	−92699.4			−0.787932
$50$	39366.6			0.314933
$51$			109405.i			0.824757i
$52$	−164205.			−1.16782
$53$			191262.i			1.28470i	0.766412	+	0.642349i	$0.222040\pi$
$53$			191262.i			1.28470i	−0.766412	+	0.642349i	$0.777960\pi$
$54$	472791.			3.00253
$55$	−31914.8			−0.191825
$56$			177329.i			1.00975i
$57$		−	316950.i		−	1.71146i
$58$	191391.			0.980931
$59$	239927.			1.16822			0.584109	−	0.811675i	$-0.301444\pi$
$59$	239927.			1.16822			0.584109	+	0.811675i	$0.301444\pi$
$60$		−	251054.i		−	1.16229i
$61$		−	35571.9i		−	0.156717i	−0.996925	−	0.0783587i	$-0.975032\pi$
$61$		−	35571.9i		−	0.156717i	0.996925	−	0.0783587i	$-0.0249679\pi$
$62$	419289.			1.75930
$63$		−	697285.i		−	2.78862i
$64$	−424374.			−1.61886
$65$		−	96938.5i		−	0.352985i
$66$			341093.i			1.18643i
$67$			482146.i			1.60307i	0.597945	+	0.801537i	$0.295984\pi$
$67$			482146.i			1.60307i	−0.597945	+	0.801537i	$0.704016\pi$
$68$		−	218436.i		−	0.694700i
$69$	327770.	−	474921.i	0.997751	−	1.44569i
$70$	−322978.			−0.941626
$71$	−362192.			−1.01196			−0.505981	−	0.862545i	$-0.668869\pi$
$71$	−362192.			−1.01196			−0.505981	+	0.862545i	$0.668869\pi$
$72$	−587828.			−1.57490
$73$	−520561.			−1.33814			−0.669072	−	0.743198i	$-0.733308\pi$
$73$	−520561.			−1.33814			−0.669072	+	0.743198i	$0.733308\pi$
$74$			1.01095e6i			2.49480i
$75$	148210.			0.351313
$76$			632818.i			1.44158i
$77$	261841.			0.573542
$78$	−1.03604e6			−2.18320
$79$			201840.i			0.409379i	0.978827	+	0.204690i	$0.0656185\pi$
$79$			201840.i			0.409379i	−0.978827	+	0.204690i	$0.934382\pi$
$80$		−	66504.1i		−	0.129891i
$81$	671667.			1.26386
$82$	−677642.			−1.22902
$83$		−	1.04230e6i		−	1.82289i	−0.411422	−	0.911445i	$-0.634968\pi$
$83$		−	1.04230e6i		−	1.82289i	0.411422	−	0.911445i	$-0.365032\pi$
$84$			2.05974e6i			3.47516i
$85$	128954.			0.209980
$86$		−	1.42907e6i		−	2.24676i
$87$	720563.			1.09424
$88$		−	220738.i		−	0.323913i
$89$		−	132496.i		−	0.187946i	−0.995575	−	0.0939730i	$-0.970043\pi$
$89$		−	132496.i		−	0.187946i	0.995575	−	0.0939730i	$-0.0299568\pi$
$90$		−	1.07064e6i		−	1.46864i
$91$			795318.i			1.05540i
$92$	−654420.	+	948220.i	−0.840414	+	1.21771i
$93$	1.57857e6			1.96252
$94$	1.01814e6			1.22581
$95$	−373584.			−0.435730
$96$	−1.88436e6			−2.12986
$97$		−	185139.i		−	0.202853i	−0.994843	−	0.101427i	$-0.967659\pi$
$97$		−	185139.i		−	0.202853i	0.994843	−	0.101427i	$-0.0323407\pi$
$98$	1.16776e6			1.24073
$99$			867978.i			0.894547i
$100$	−295914.			−0.295914
$101$	1.29836e6			1.26018			0.630090	−	0.776523i	$-0.283018\pi$
$101$	1.29836e6			1.26018			0.630090	+	0.776523i	$0.283018\pi$
$102$		−	1.37821e6i		−	1.29872i
$103$			1.61286e6i			1.47600i	0.674802	+	0.737999i	$0.264229\pi$
$103$			1.61286e6i			1.47600i	−0.674802	+	0.737999i	$0.735771\pi$
$104$	670472.			0.596047
$105$	−1.21597e6			−1.05040
$106$		−	2.40939e6i		−	2.02297i
$107$		−	680207.i		−	0.555252i	−0.960689	−	0.277626i	$-0.910452\pi$
$107$		−	680207.i		−	0.555252i	0.960689	−	0.277626i	$-0.0895476\pi$
$108$	−3.55391e6			−2.82121
$109$		−	1.53267e6i		−	1.18350i	−0.806122	−	0.591749i	$-0.798438\pi$
$109$		−	1.53267e6i		−	1.18350i	0.806122	−	0.591749i	$-0.201562\pi$
$110$	402041.			0.302060
$111$			3.80610e6i			2.78299i
$112$			545623.i			0.388364i
$113$		−	1.38275e6i		−	0.958316i	−0.877729	−	0.479158i	$-0.840942\pi$
$113$		−	1.38275e6i		−	0.958316i	0.877729	−	0.479158i	$-0.159058\pi$
$114$			3.99272e6i			2.69497i
$115$	−559782.	−	386337.i	−0.368066	−	0.254023i
$116$	−1.43866e6			−0.921692
$117$	−2.63641e6			−1.64609
$118$	−3.02244e6			−1.83955
$119$	−1.05798e6			−0.627824
$120$			1.02509e6i			0.593223i
$121$	1.44562e6			0.816016
$122$			448110.i			0.246777i
$123$	−2.55123e6			−1.37099
$124$	−3.15174e6			−1.65305
$125$		−	174693.i		−	0.0894427i
$126$			8.78393e6i			4.39114i
$127$	218138.			0.106493			0.0532464	−	0.998581i	$-0.483043\pi$
$127$	218138.			0.106493			0.0532464	+	0.998581i	$0.483043\pi$
$128$	2.80315e6			1.33664
$129$		−	5.38025e6i		−	2.50630i
$130$			1.22117e6i			0.555833i
$131$	3.98378e6			1.77207			0.886037	−	0.463615i	$-0.153448\pi$
$131$	3.98378e6			1.77207			0.886037	+	0.463615i	$0.153448\pi$
$132$		−	2.56396e6i		−	1.11478i
$133$	3.06502e6			1.30280
$134$		−	6.07374e6i		−	2.52431i
$135$		−	2.09805e6i		−	0.852737i
$136$			891904.i			0.354570i
$137$			2.35277e6i			0.914992i	0.889212	+	0.457496i	$0.151254\pi$
$137$			2.35277e6i			0.914992i	−0.889212	+	0.457496i	$0.848746\pi$
$138$	−4.12902e6	+	5.98273e6i	−1.57112	+	2.27647i
$139$	384148.			0.143039			0.0715195	−	0.997439i	$-0.477215\pi$
$139$	384148.			0.143039			0.0715195	+	0.997439i	$0.477215\pi$
$140$	2.42778e6			0.884760
$141$	3.83316e6			1.36741
$142$	4.56265e6			1.59350
$143$		−	990009.i		−	0.338556i
$144$	−1.80869e6			−0.605727
$145$		−	849316.i		−	0.278590i
$146$	6.55767e6			2.10713
$147$	4.39647e6			1.38405
$148$		−	7.59920e6i		−	2.34413i
$149$		−	5.52454e6i		−	1.67008i	−0.550189	−	0.835040i	$-0.685444\pi$
$149$		−	5.52454e6i		−	1.67008i	0.550189	−	0.835040i	$-0.314556\pi$
$150$	−1.86705e6			−0.553200
$151$	952505.			0.276654			0.138327	−	0.990387i	$-0.455828\pi$
$151$	952505.			0.276654			0.138327	+	0.990387i	$0.455828\pi$
$152$		−	2.58388e6i		−	0.735770i
$153$		−	3.50712e6i		−	0.979210i
$154$	−3.29849e6			−0.903136
$155$		−	1.86063e6i		−	0.499650i
$156$	7.78779e6			2.05135
$157$		−	1.91846e6i		−	0.495739i	−0.968793	−	0.247869i	$-0.920270\pi$
$157$		−	1.91846e6i		−	0.495739i	0.968793	−	0.247869i	$-0.0797304\pi$
$158$		−	2.54264e6i		−	0.644635i
$159$		−	9.07102e6i		−	2.25665i
$160$			2.22107e6i			0.542252i
$161$	4.59265e6	+	3.16965e6i	1.10049	+	0.759510i
$162$	−8.46120e6			−1.99016
$163$	4.40503e6			1.01715			0.508576	−	0.861017i	$-0.330172\pi$
$163$	4.40503e6			1.01715			0.508576	+	0.861017i	$0.330172\pi$
$164$	5.09375e6			1.15480
$165$	1.51363e6			0.336952
$166$			1.31302e7i			2.87044i
$167$	4.09675e6			0.879610			0.439805	−	0.898093i	$-0.355048\pi$
$167$	4.09675e6			0.879610			0.439805	+	0.898093i	$0.355048\pi$
$168$		−	8.41020e6i		−	1.77369i
$169$	−1.81974e6			−0.377008
$170$	−1.62447e6			−0.330648
$171$			1.01603e7i			2.03196i
$172$			1.07421e7i			2.11108i
$173$	1.32236e6			0.255394			0.127697	−	0.991813i	$-0.459241\pi$
$173$	1.32236e6			0.255394			0.127697	+	0.991813i	$0.459241\pi$
$174$	−9.07716e6			−1.72307
$175$			1.43324e6i			0.267427i
$176$		−	679189.i		−	0.124581i
$177$	−1.13791e7			−2.05205
$178$			1.66910e6i			0.295952i
$179$	−6.56690e6			−1.14499			−0.572494	−	0.819909i	$-0.694024\pi$
$179$	−6.56690e6			−1.14499			−0.572494	+	0.819909i	$0.694024\pi$
$180$			8.04787e6i			1.37995i
$181$		−	238891.i		−	0.0402869i	−0.999797	−	0.0201435i	$-0.993588\pi$
$181$		−	238891.i		−	0.0402869i	0.999797	−	0.0201435i	$-0.00641229\pi$
$182$		−	1.00189e7i		−	1.66190i
$183$			1.68707e6i			0.275284i
$184$	2.67209e6	−	3.87171e6i	0.428941	−	0.621512i
$185$	4.48619e6			0.708537
$186$	−1.98857e7			−3.09031
$187$	1.31697e6			0.201396
$188$	−7.65322e6			−1.15178
$189$			1.72132e7i			2.54962i
$190$	4.70616e6			0.686129
$191$		−	6.09102e6i		−	0.874158i	−0.899423	−	0.437079i	$-0.856013\pi$
$191$		−	6.09102e6i		−	0.874158i	0.899423	−	0.437079i	$-0.143987\pi$
$192$	2.01269e7			2.84363
$193$	−6.77680e6			−0.942655			−0.471328	−	0.881958i	$-0.656225\pi$
$193$	−6.77680e6			−0.942655			−0.471328	+	0.881958i	$0.656225\pi$
$194$			2.33225e6i			0.319426i
$195$			4.59752e6i			0.620041i
$196$	−8.77793e6			−1.16580
$197$	1.34373e7			1.75757			0.878784	−	0.477219i	$-0.158355\pi$
$197$	1.34373e7			1.75757			0.878784	+	0.477219i	$0.158355\pi$
$198$		−	1.09342e7i		−	1.40861i
$199$		−	1.20252e7i		−	1.52593i	−0.646442	−	0.762963i	$-0.723744\pi$
$199$		−	1.20252e7i		−	1.52593i	0.646442	−	0.762963i	$-0.276256\pi$
$200$	1.20826e6			0.151032
$201$		−	2.28668e7i		−	2.81590i
$202$	−1.63559e7			−1.98436
$203$			6.96809e6i			0.832963i
$204$			1.03598e7i			1.22029i
$205$			3.00710e6i			0.349048i
$206$		−	2.03177e7i		−	2.32420i
$207$	−1.05071e7	+	1.52242e7i	−1.18460	+	1.71642i
$208$	2.06298e6			0.229247
$209$	−3.81532e6			−0.417919
$210$	1.53179e7			1.65403
$211$	−1.55960e7			−1.66022			−0.830112	−	0.557596i	$-0.811724\pi$
$211$	−1.55960e7			−1.66022			−0.830112	+	0.557596i	$0.811724\pi$
$212$			1.81111e7i			1.90080i
$213$	1.71778e7			1.77758
$214$			8.56879e6i			0.874335i
$215$	−6.34161e6			−0.638093
$216$	1.45111e7			1.43992
$217$			1.52653e7i			1.49392i
$218$			1.93075e7i			1.86361i
$219$	2.46888e7			2.35054
$220$	−3.02209e6			−0.283818
$221$			4.00019e6i			0.370598i
$222$		−	4.79466e7i		−	4.38227i
$223$	−9.65080e6			−0.870260			−0.435130	−	0.900368i	$-0.643298\pi$
$223$	−9.65080e6			−0.870260			−0.435130	+	0.900368i	$0.643298\pi$
$224$		−	1.82224e7i		−	1.62129i
$225$	−4.75107e6			−0.417103
$226$			1.74189e7i			1.50903i
$227$			8.35833e6i			0.714565i	0.933996	+	0.357283i	$0.116297\pi$
$227$			8.35833e6i			0.714565i	−0.933996	+	0.357283i	$0.883703\pi$
$228$		−	3.00128e7i		−	2.53222i
$229$		−	1.05721e7i		−	0.880349i	−0.897912	−	0.440175i	$-0.854917\pi$
$229$		−	1.05721e7i		−	0.880349i	0.897912	−	0.440175i	$-0.145083\pi$
$230$	7.05175e6	+	4.86681e6i	0.579580	+	0.400001i
$231$	−1.24184e7			−1.00746
$232$	5.87427e6			0.470424
$233$	−1.08691e7			−0.859260			−0.429630	−	0.903005i	$-0.641356\pi$
$233$	−1.08691e7			−0.859260			−0.429630	+	0.903005i	$0.641356\pi$
$234$	3.32116e7			2.59205
$235$		−	4.51808e6i		−	0.348137i
$236$	2.27193e7			1.72846
$237$		−	9.57270e6i		−	0.719100i
$238$	1.33277e7			0.988612
$239$	−4.36401e6			−0.319663			−0.159831	−	0.987144i	$-0.551095\pi$
$239$	−4.36401e6			−0.319663			−0.159831	+	0.987144i	$0.551095\pi$
$240$			3.15410e6i			0.228161i
$241$		−	6.91012e6i		−	0.493667i	−0.969058	−	0.246834i	$-0.920610\pi$
$241$		−	6.91012e6i		−	0.493667i	0.969058	−	0.246834i	$-0.0793901\pi$
$242$	−1.82110e7			−1.28495
$243$	−4.49513e6			−0.313273
$244$		−	3.36838e6i		−	0.231874i
$245$		−	5.18205e6i		−	0.352374i
$246$	3.21387e7			2.15885
$247$		−	1.15887e7i		−	0.769031i
$248$	1.28690e7			0.843704
$249$			4.94336e7i			3.20202i
$250$			2.20066e6i			0.140842i
$251$			1.57550e7i			0.996313i	0.867087	+	0.498157i	$0.165990\pi$
$251$			1.57550e7i			0.996313i	−0.867087	+	0.498157i	$0.834010\pi$
$252$		−	6.60276e7i		−	4.12595i
$253$	−5.71691e6	−	3.94557e6i	−0.353020	−	0.243640i
$254$	−2.74795e6			−0.167690
$255$	−6.11592e6			−0.368843
$256$	−8.15221e6			−0.485909
$257$	2.58408e7			1.52232			0.761161	−	0.648563i	$-0.224630\pi$
$257$	2.58408e7			1.52232			0.761161	+	0.648563i	$0.224630\pi$
$258$			6.77767e7i			3.94658i
$259$	−3.68063e7			−2.11847
$260$		−	9.17934e6i		−	0.522266i
$261$	−2.30986e7			−1.29916
$262$	−5.01850e7			−2.79042
$263$		−	2.35926e6i		−	0.129691i	−0.997895	−	0.0648453i	$-0.979345\pi$
$263$		−	2.35926e6i		−	0.129691i	0.997895	−	0.0648453i	$-0.0206554\pi$
$264$			1.04690e7i			0.568974i
$265$	−1.06919e7			−0.574534
$266$	−3.86110e7			−2.05148
$267$			6.28392e6i			0.330139i
$268$			4.56555e7i			2.37186i
$269$	2.45449e7			1.26097			0.630484	−	0.776203i	$-0.282857\pi$
$269$	2.45449e7			1.26097			0.630484	+	0.776203i	$0.282857\pi$
$270$			2.64298e7i			1.34277i
$271$	2.89734e7			1.45577			0.727883	−	0.685701i	$-0.240504\pi$
$271$	2.89734e7			1.45577			0.727883	+	0.685701i	$0.240504\pi$
$272$			2.74430e6i			0.136372i
$273$		−	3.77197e7i		−	1.85388i
$274$		−	2.96386e7i		−	1.44081i
$275$		−	1.78409e6i		−	0.0857866i
$276$	3.10373e7	−	4.49714e7i	1.47624	−	2.13899i
$277$	2.71423e7			1.27705			0.638523	−	0.769603i	$-0.279546\pi$
$277$	2.71423e7			1.27705			0.638523	+	0.769603i	$0.279546\pi$
$278$	−4.83924e6			−0.225239
$279$	−5.06031e7			−2.33004
$280$	−9.91297e6			−0.451575
$281$			3.11635e7i			1.40452i	0.711921	+	0.702260i	$0.247825\pi$
$281$			3.11635e7i			1.40452i	−0.711921	+	0.702260i	$0.752175\pi$
$282$	−4.82875e7			−2.15321
$283$			1.01356e7i			0.447189i	0.974682	+	0.223595i	$0.0717792\pi$
$283$			1.01356e7i			0.447189i	−0.974682	+	0.223595i	$0.928221\pi$
$284$	−3.42969e7			−1.49727
$285$	1.77181e7			0.765388
$286$			1.24715e7i			0.533113i
$287$		−	2.46713e7i		−	1.04363i
$288$	6.04056e7			2.52872
$289$	1.88163e7			0.779543
$290$			1.06991e7i			0.438686i
$291$			8.78062e6i			0.356325i
$292$	−4.92932e7			−1.97988
$293$		−	1.47865e7i		−	0.587846i	−0.955829	−	0.293923i	$-0.905039\pi$
$293$		−	1.47865e7i		−	0.587846i	0.955829	−	0.293923i	$-0.0949609\pi$
$294$	−5.53838e7			−2.17942
$295$			1.34123e7i			0.522443i
$296$			3.10286e7i			1.19643i
$297$		−	2.14269e7i		−	0.817880i
$298$			6.95944e7i			2.62982i
$299$	1.19843e7	−	1.73646e7i	0.448332	−	0.649608i
$300$	1.40344e7			0.519791
$301$	5.20288e7			1.90785
$302$	−1.19990e7			−0.435637
$303$	−6.15778e7			−2.21358
$304$		−	7.95036e6i		−	0.282987i
$305$	1.98853e6			0.0700861
$306$			4.41802e7i			1.54193i
$307$	−1.57096e7			−0.542936			−0.271468	−	0.962447i	$-0.587509\pi$
$307$	−1.57096e7			−0.542936			−0.271468	+	0.962447i	$0.587509\pi$
$308$	2.47943e7			0.848595
$309$		−	7.64936e7i		−	2.59268i
$310$			2.34390e7i			0.786781i
$311$	−9.85973e6			−0.327781			−0.163891	−	0.986479i	$-0.552404\pi$
$311$	−9.85973e6			−0.327781			−0.163891	+	0.986479i	$0.552404\pi$
$312$	−3.17986e7			−1.04699
$313$			1.09613e7i			0.357463i	0.983898	+	0.178731i	$0.0571993\pi$
$313$			1.09613e7i			0.357463i	−0.983898	+	0.178731i	$0.942801\pi$
$314$			2.41674e7i			0.780622i
$315$	3.89794e7			1.24711
$316$			1.91127e7i			0.605705i
$317$	9.67366e6			0.303678			0.151839	−	0.988405i	$-0.451481\pi$
$317$	9.67366e6			0.303678			0.151839	+	0.988405i	$0.451481\pi$
$318$			1.14271e8i			3.55347i
$319$		−	8.67385e6i		−	0.267202i
$320$		−	2.37232e7i		−	0.723975i
$321$			3.22603e7i			0.975335i
$322$	−5.78551e7	−	3.99291e7i	−1.73290	−	1.19597i
$323$	1.54160e7			0.457473
$324$	6.36018e7			1.86997
$325$	5.41903e6			0.157860
$326$	−5.54915e7			−1.60167
$327$			7.26900e7i			2.07889i
$328$	−2.07985e7			−0.589400
$329$			3.70679e7i			1.04090i
$330$	−1.90677e7			−0.530587
$331$	−5.61646e7			−1.54874			−0.774370	−	0.632733i	$-0.781933\pi$
$331$	−5.61646e7			−1.54874			−0.774370	+	0.632733i	$0.781933\pi$
$332$		−	9.86984e7i		−	2.69709i
$333$		−	1.22009e8i		−	3.30416i
$334$	−5.16081e7			−1.38509
$335$	−2.69528e7			−0.716917
$336$		−	2.58774e7i		−	0.682186i
$337$			2.97857e6i			0.0778249i	0.999243	+	0.0389125i	$0.0123893\pi$
$337$			2.97857e6i			0.0778249i	−0.999243	+	0.0389125i	$0.987611\pi$
$338$	2.29239e7			0.593661
$339$			6.55800e7i			1.68334i
$340$	1.22109e7			0.310679
$341$		−	1.90022e7i		−	0.479226i
$342$		−	1.27992e8i		−	3.19966i
$343$		−	1.14428e7i		−	0.283564i
$344$		−	4.38615e7i		−	1.07748i
$345$	2.65489e7	+	1.83229e7i	0.646531	+	0.446208i
$346$	−1.66582e7			−0.402160
$347$	1.93249e7			0.462518			0.231259	−	0.972892i	$-0.425716\pi$
$347$	1.93249e7			0.462518			0.231259	+	0.972892i	$0.425716\pi$
$348$	6.82319e7			1.61901
$349$	6.19555e7			1.45748			0.728742	−	0.684789i	$-0.240105\pi$
$349$	6.19555e7			1.45748			0.728742	+	0.684789i	$0.240105\pi$
$350$		−	1.80550e7i		−	0.421108i
$351$	6.50823e7			1.50502
$352$			2.26832e7i			0.520087i
$353$	6.55488e7			1.49019			0.745093	−	0.666961i	$-0.232405\pi$
$353$	6.55488e7			1.49019			0.745093	+	0.666961i	$0.232405\pi$
$354$	1.43346e8			3.23129
$355$		−	2.02472e7i		−	0.452563i
$356$		−	1.25464e7i		−	0.278079i
$357$	5.01772e7			1.10281
$358$	8.27253e7			1.80297
$359$		−	3.36104e7i		−	0.726424i	−0.931707	−	0.363212i	$-0.881680\pi$
$359$		−	3.36104e7i		−	0.726424i	0.931707	−	0.363212i	$-0.118320\pi$
$360$		−	3.28606e7i		−	0.704316i
$361$	2.38502e6			0.0506957
$362$			3.00938e6i			0.0634384i
$363$	−6.85619e7			−1.43338
$364$			7.53106e7i			1.56154i
$365$		−	2.91002e7i		−	0.598436i
$366$		−	2.12526e7i		−	0.433480i
$367$		−	8.96769e7i		−	1.81419i	−0.420927	−	0.907095i	$-0.638295\pi$
$367$		−	8.96769e7i		−	1.81419i	0.420927	−	0.907095i	$-0.361705\pi$
$368$	8.22176e6	−	1.19129e7i	0.164976	−	0.239042i
$369$	8.17830e7			1.62774
$370$	−5.65139e7			−1.11571
$371$	8.77199e7			1.71782
$372$	1.49478e8			2.90369
$373$		−	1.73005e7i		−	0.333374i	−0.986010	−	0.166687i	$-0.946693\pi$
$373$		−	1.73005e7i		−	0.333374i	0.986010	−	0.166687i	$-0.0533070\pi$
$374$	−1.65903e7			−0.317132
$375$			8.28519e6i			0.157112i
$376$	3.12491e7			0.587861
$377$	2.63461e7			0.491690
$378$		−	2.16840e8i		−	4.01480i
$379$		−	8.15972e7i		−	1.49885i	−0.662090	−	0.749424i	$-0.730330\pi$
$379$		−	8.15972e7i		−	1.49885i	0.662090	−	0.749424i	$-0.269670\pi$
$380$	−3.53756e7			−0.644693
$381$	−1.03457e7			−0.187061
$382$			7.67305e7i			1.37651i
$383$			4.98344e7i			0.887019i	0.896270	+	0.443510i	$0.146267\pi$
$383$			4.98344e7i			0.887019i	−0.896270	+	0.443510i	$0.853733\pi$
$384$	−1.32945e8			−2.34790
$385$			1.46373e7i			0.256496i
$386$	8.53695e7			1.48437
$387$			1.72471e8i			2.97566i
$388$		−	1.75312e7i		−	0.300136i
$389$			7.35439e7i			1.24939i	0.780869	+	0.624694i	$0.214776\pi$
$389$			7.35439e7i			1.24939i	−0.780869	+	0.624694i	$0.785224\pi$
$390$		−	5.79165e7i		−	0.976356i
$391$	2.30995e7	+	1.59423e7i	0.386432	+	0.266698i
$392$	3.58415e7			0.595015
$393$	−1.88940e8			−3.11276
$394$	−1.69274e8			−2.76758
$395$	−1.12832e7			−0.183080
$396$			8.21909e7i			1.32354i
$397$	−4.36155e7			−0.697059			−0.348530	−	0.937298i	$-0.613319\pi$
$397$	−4.36155e7			−0.697059			−0.348530	+	0.937298i	$0.613319\pi$
$398$			1.51485e8i			2.40282i
$399$	−1.45365e8			−2.28845
$400$	3.71769e6			0.0580889
$401$		−	9.79005e7i		−	1.51828i	−0.650927	−	0.759140i	$-0.725620\pi$
$401$		−	9.79005e7i		−	1.51828i	0.650927	−	0.759140i	$-0.274380\pi$
$402$			2.88061e8i			4.43410i
$403$	5.77174e7			0.881844
$404$	1.22945e8			1.86452
$405$			3.75473e7i			0.565215i
$406$		−	8.77793e7i		−	1.31164i
$407$	4.58163e7			0.679574
$408$		−	4.23005e7i		−	0.622824i
$409$	−3.23089e7			−0.472229			−0.236114	−	0.971725i	$-0.575874\pi$
$409$	−3.23089e7			−0.472229			−0.236114	+	0.971725i	$0.575874\pi$
$410$		−	3.78813e7i		−	0.549634i
$411$		−	1.11585e8i		−	1.60724i
$412$			1.52726e8i			2.18384i
$413$		−	1.10040e8i		−	1.56207i
$414$	1.32361e8	−	1.91784e8i	1.86535	−	2.70279i
$415$	5.82666e7			0.815221
$416$	−6.88982e7			−0.957035
$417$	−1.82191e7			−0.251257
$418$	4.80628e7			0.658083
$419$			7.11233e7i			0.966874i	0.875379	+	0.483437i	$0.160612\pi$
$419$			7.11233e7i			0.966874i	−0.875379	+	0.483437i	$0.839388\pi$
$420$	−1.15143e8			−1.55414
$421$			1.27848e8i			1.71336i	0.515846	+	0.856681i	$0.327478\pi$
$421$			1.27848e8i			1.71336i	−0.515846	+	0.856681i	$0.672522\pi$
$422$	1.96468e8			2.61430
$423$	−1.22877e8			−1.62349
$424$		−	7.39499e7i		−	0.970153i
$425$			7.20874e6i			0.0939058i
$426$	−2.16394e8			−2.79909
$427$	−1.63146e7			−0.209552
$428$		−	6.44105e7i		−	0.821533i
$429$			4.69534e7i			0.594696i
$430$	7.98873e7			1.00478
$431$		−	4.65134e7i		−	0.580960i	−0.956881	−	0.290480i	$-0.906185\pi$
$431$		−	4.65134e7i		−	0.580960i	0.956881	−	0.290480i	$-0.0938149\pi$
$432$	4.46493e7			0.553813
$433$		−	9.79366e7i		−	1.20637i	−0.797600	−	0.603186i	$-0.793898\pi$
$433$		−	9.79366e7i		−	1.20637i	0.797600	−	0.603186i	$-0.206102\pi$
$434$		−	1.92302e8i		−	2.35242i
$435$			4.02807e7i			0.489361i
$436$		−	1.45132e8i		−	1.75107i
$437$	−6.69202e7	−	4.61854e7i	−0.801887	−	0.553428i
$438$	−3.11012e8			−3.70130
$439$	6.08173e7			0.718842			0.359421	−	0.933176i	$-0.382974\pi$
$439$	6.08173e7			0.718842			0.359421	+	0.933176i	$0.382974\pi$
$440$	1.23396e7			0.144858
$441$	−1.40935e8			−1.64324
$442$		−	5.03916e7i		−	0.583568i
$443$	1.16367e8			1.33850			0.669248	−	0.743039i	$-0.266616\pi$
$443$	1.16367e8			1.33850			0.669248	+	0.743039i	$0.266616\pi$
$444$			3.60409e8i			4.11762i
$445$	7.40676e6			0.0840520
$446$	1.21574e8			1.37037
$447$			2.62014e8i			2.93360i
$448$			1.94634e8i			2.16463i
$449$	−1.64305e8			−1.81515			−0.907574	−	0.419893i	$-0.862068\pi$
$449$	−1.64305e8			−1.81515			−0.907574	+	0.419893i	$0.862068\pi$
$450$	5.98507e7			0.656798
$451$			3.07107e7i			0.334781i
$452$		−	1.30936e8i		−	1.41789i
$453$	−4.51747e7			−0.485960
$454$		−	1.05293e8i		−	1.12520i
$455$	−4.44596e7			−0.471989
$456$			1.22546e8i			1.29243i
$457$		−	1.14443e8i		−	1.19906i	−0.800351	−	0.599532i	$-0.795354\pi$
$457$		−	1.14443e8i		−	1.19906i	0.800351	−	0.599532i	$-0.204646\pi$
$458$			1.33180e8i			1.38626i
$459$			8.65766e7i			0.895287i
$460$	−5.30071e7	−	3.65832e7i	−0.544578	−	0.375845i
$461$	−1.02863e8			−1.04992			−0.524958	−	0.851128i	$-0.675919\pi$
$461$	−1.02863e8			−1.04992			−0.524958	+	0.851128i	$0.675919\pi$
$462$	1.56438e8			1.58642
$463$	5.02198e7			0.505978			0.252989	−	0.967469i	$-0.418586\pi$
$463$	5.02198e7			0.505978			0.252989	+	0.967469i	$0.418586\pi$
$464$	1.80745e7			0.180931
$465$			8.82447e7i			0.877667i
$466$	1.36921e8			1.35305
$467$		−	1.16664e8i		−	1.14548i	−0.819738	−	0.572738i	$-0.805881\pi$
$467$		−	1.16664e8i		−	1.14548i	0.819738	−	0.572738i	$-0.194119\pi$
$468$	−2.49648e8			−2.43551
$469$	2.21130e8			2.14353
$470$			5.69157e7i			0.548199i
$471$			9.09870e7i			0.870797i
$472$	−9.27660e7			−0.882192
$473$	−6.47653e7			−0.612010
$474$			1.20590e8i			1.13234i
$475$		−	2.08840e7i		−	0.194864i
$476$	−1.00183e8			−0.928909
$477$			2.90783e8i			2.67926i
$478$	5.49748e7			0.503362
$479$		−	1.35591e8i		−	1.23374i	−0.787064	−	0.616871i	$-0.788400\pi$
$479$		−	1.35591e8i		−	1.23374i	0.787064	−	0.616871i	$-0.211600\pi$
$480$		−	1.05339e8i		−	0.952501i
$481$			1.39163e8i			1.25051i
$482$			8.70490e7i			0.777360i
$483$	−2.17817e8	−	1.50328e8i	−1.93308	−	1.33413i
$484$	1.36890e8			1.20735
$485$	1.03496e7			0.0907188
$486$	5.66265e7			0.493300
$487$	−9.43277e7			−0.816681			−0.408340	−	0.912830i	$-0.633892\pi$
$487$	−9.43277e7			−0.816681			−0.408340	+	0.912830i	$0.633892\pi$
$488$			1.37536e7i			0.118347i
$489$	−2.08918e8			−1.78669
$490$			6.52800e7i			0.554871i
$491$	6.58373e7			0.556195			0.278098	−	0.960553i	$-0.410296\pi$
$491$	6.58373e7			0.556195			0.278098	+	0.960553i	$0.410296\pi$
$492$	−2.41582e8			−2.02848
$493$			3.50472e7i			0.292491i
$494$			1.45987e8i			1.21097i
$495$	−4.85214e7			−0.400053
$496$	3.95967e7			0.324499
$497$			1.66115e8i			1.35313i
$498$		−	6.22731e8i		−	5.04211i
$499$	−2.27978e7			−0.183481			−0.0917404	−	0.995783i	$-0.529243\pi$
$499$	−2.27978e7			−0.183481			−0.0917404	+	0.995783i	$0.529243\pi$
$500$		−	1.65421e7i		−	0.132337i
$501$	−1.94297e8			−1.54509
$502$		−	1.98470e8i		−	1.56886i
$503$		−	1.99337e7i		−	0.156633i	−0.996929	−	0.0783165i	$-0.975046\pi$
$503$		−	1.99337e7i		−	0.156633i	0.996929	−	0.0783165i	$-0.0249545\pi$
$504$			2.69600e8i			2.10585i
$505$			7.25807e7i			0.563569i
$506$	7.20177e7	+	4.97035e7i	0.555889	+	0.383651i
$507$	8.63054e7			0.662238
$508$	2.06560e7			0.157563
$509$	6.64780e7			0.504109			0.252055	−	0.967713i	$-0.418894\pi$
$509$	6.64780e7			0.504109			0.252055	+	0.967713i	$0.418894\pi$
$510$	7.70442e7			0.580804
$511$			2.38749e8i			1.78928i
$512$	−7.67054e7			−0.571500
$513$		−	2.50816e8i		−	1.85782i
$514$	−3.25525e8			−2.39715
$515$	−9.01618e7			−0.660086
$516$		−	5.09469e8i		−	3.70824i
$517$		−	4.61420e7i		−	0.333906i
$518$	4.63661e8			3.33588
$519$	−6.27157e7			−0.448616
$520$			3.74805e7i			0.266560i
$521$			6.00264e7i			0.424452i	0.977221	+	0.212226i	$0.0680714\pi$
$521$			6.00264e7i			0.424452i	−0.977221	+	0.212226i	$0.931929\pi$
$522$	2.90980e8			2.04575
$523$		−	1.12497e8i		−	0.786386i	−0.919456	−	0.393193i	$-0.871371\pi$
$523$		−	1.12497e8i		−	0.786386i	0.919456	−	0.393193i	$-0.128629\pi$
$524$	3.77234e8			2.62191
$525$		−	6.79747e7i		−	0.469753i
$526$			2.97203e7i			0.204219i
$527$			7.67794e7i			0.524582i
$528$			3.22120e7i			0.218835i
$529$	−5.25117e7	−	1.38409e8i	−0.354723	−	0.934971i
$530$	1.34689e8			0.904699
$531$	3.64771e8			2.43634
$532$	2.90234e8			1.92758
$533$	−9.32811e7			−0.616044
$534$		−	7.91606e7i		−	0.519859i
$535$	3.80247e7			0.248316
$536$		−	1.86418e8i		−	1.21058i
$537$	3.11450e8			2.01125
$538$	−3.09199e8			−1.98560
$539$		−	5.29230e7i		−	0.337970i
$540$		−	1.98670e8i		−	1.26168i
$541$	1.43170e8			0.904188			0.452094	−	0.891970i	$-0.350677\pi$
$541$	1.43170e8			0.904188			0.452094	+	0.891970i	$0.350677\pi$
$542$	−3.64987e8			−2.29234
$543$			1.13299e7i			0.0707665i
$544$		−	9.16527e7i		−	0.569310i
$545$	8.56786e7			0.529277
$546$			4.75167e8i			2.91923i
$547$	8.39494e6			0.0512927			0.0256464	−	0.999671i	$-0.491836\pi$
$547$	8.39494e6			0.0512927			0.0256464	+	0.999671i	$0.491836\pi$
$548$			2.22789e8i			1.35379i
$549$		−	5.40813e7i		−	0.326836i
$550$			2.24748e7i			0.135085i
$551$		−	1.01533e8i		−	0.606950i
$552$	−1.26730e8	+	1.83624e8i	−0.753462	+	1.09173i
$553$	9.25713e7			0.547395
$554$	−3.41920e8			−2.01092
$555$	−2.12767e8			−1.24459
$556$	3.63759e7			0.211636
$557$			2.09692e8i			1.21344i	0.794917	+	0.606718i	$0.207514\pi$
$557$			2.09692e8i			1.21344i	−0.794917	+	0.606718i	$0.792486\pi$
$558$	6.37463e8			3.66904
$559$		−	1.96719e8i		−	1.12619i
$560$	−3.05013e7			−0.173682
$561$	−6.24603e7			−0.353766
$562$		−	3.92577e8i		−	2.21165i
$563$			2.41541e8i			1.35353i	0.736201	+	0.676763i	$0.236618\pi$
$563$			2.41541e8i			1.35353i	−0.736201	+	0.676763i	$0.763382\pi$
$564$	3.62971e8			2.02318
$565$	7.72981e7			0.428572
$566$		−	1.27682e8i		−	0.704173i
$567$		−	3.08052e8i		−	1.68995i
$568$	1.40039e8			0.764194
$569$			2.84639e8i			1.54510i	0.634953	+	0.772550i	$0.281019\pi$
$569$			2.84639e8i			1.54510i	−0.634953	+	0.772550i	$0.718981\pi$
$570$	−2.23200e8			−1.20523
$571$		−	2.34627e8i		−	1.26029i	−0.776478	−	0.630144i	$-0.782996\pi$
$571$		−	2.34627e8i		−	1.26029i	0.776478	−	0.630144i	$-0.217004\pi$
$572$		−	9.37463e7i		−	0.500917i
$573$			2.88880e8i			1.53551i
$574$			3.10792e8i			1.64337i
$575$	2.15969e7	−	3.12928e7i	0.113603	−	0.164604i
$576$	−6.45193e8			−3.37615
$577$	6.92074e7			0.360268			0.180134	−	0.983642i	$-0.442347\pi$
$577$	6.92074e7			0.360268			0.180134	+	0.983642i	$0.442347\pi$
$578$	−2.37034e8			−1.22752
$579$	3.21405e8			1.65583
$580$		−	8.04238e7i		−	0.412193i
$581$	−4.78040e8			−2.43745
$582$		−	1.10612e8i		−	0.561092i
$583$	−1.09193e8			−0.551049
$584$	2.01271e8			1.01051
$585$		−	1.47380e8i		−	0.736156i
$586$			1.86271e8i			0.925660i
$587$	−2.46313e8			−1.21779			−0.608896	−	0.793250i	$-0.708387\pi$
$587$	−2.46313e8			−1.21779			−0.608896	+	0.793250i	$0.708387\pi$
$588$	4.16313e8			2.04780
$589$		−	2.22433e8i		−	1.08856i
$590$		−	1.68960e8i		−	0.822672i
$591$	−6.37292e8			−3.08728
$592$			9.54719e7i			0.460162i
$593$	3.18328e8			1.52655			0.763275	−	0.646074i	$-0.223590\pi$
$593$	3.18328e8			1.52655			0.763275	+	0.646074i	$0.223590\pi$
$594$			2.69921e8i			1.28789i
$595$		−	5.91430e7i		−	0.280771i
$596$		−	5.23132e8i		−	2.47100i
$597$			5.70323e8i			2.68039i
$598$	−1.50970e8	+	2.18748e8i	−0.705972	+	1.02292i
$599$	−1.82859e8			−0.850817			−0.425409	−	0.905001i	$-0.639870\pi$
$599$	−1.82859e8			−0.850817			−0.425409	+	0.905001i	$0.639870\pi$
$600$	−5.73043e7			−0.265297
$601$	−3.98700e7			−0.183663			−0.0918317	−	0.995775i	$-0.529272\pi$
$601$	−3.98700e7			−0.183663			−0.0918317	+	0.995775i	$0.529272\pi$
$602$	−6.55424e8			−3.00423
$603$			7.33026e8i			3.34324i
$604$	9.01950e7			0.409328
$605$			8.08128e7i			0.364934i
$606$	7.75715e8			3.48565
$607$	2.25489e8			1.00823			0.504114	−	0.863637i	$-0.331819\pi$
$607$	2.25489e8			1.00823			0.504114	+	0.863637i	$0.331819\pi$
$608$			2.65522e8i			1.18138i
$609$		−	3.30477e8i		−	1.46315i
$610$	−2.50501e7			−0.110362
$611$	1.40152e8			0.614436
$612$		−	3.32097e8i		−	1.44881i
$613$		−	1.28510e7i		−	0.0557901i	−0.999611	−	0.0278951i	$-0.991120\pi$
$613$		−	1.28510e7i		−	0.0557901i	0.999611	−	0.0278951i	$-0.00888042\pi$
$614$	1.97898e8			0.854943
$615$		−	1.42618e8i		−	0.613126i
$616$	−1.01239e8			−0.433116
$617$		−	1.51697e8i		−	0.645836i	−0.946427	−	0.322918i	$-0.895336\pi$
$617$		−	1.51697e8i		−	0.645836i	0.946427	−	0.322918i	$-0.104664\pi$
$618$			9.63614e8i			4.08261i
$619$		−	5.18711e7i		−	0.218702i	−0.994003	−	0.109351i	$-0.965123\pi$
$619$		−	5.18711e7i		−	0.218702i	0.994003	−	0.109351i	$-0.0348773\pi$
$620$		−	1.76188e8i		−	0.739266i
$621$	2.59378e8	−	3.75824e8i	1.08307	−	1.56932i
$622$	1.24206e8			0.516146
$623$	−6.07677e7			−0.251309
$624$	−9.78413e7			−0.402688
$625$	9.76562e6			0.0400000
$626$		−	1.38084e8i		−	0.562884i
$627$	1.80950e8			0.734102
$628$		−	1.81663e8i		−	0.733480i
$629$	−1.85124e8			−0.743892
$630$	−4.91036e8			−1.96378
$631$			7.07688e7i			0.281679i	0.990032	+	0.140839i	$0.0449801\pi$
$631$			7.07688e7i			0.281679i	−0.990032	+	0.140839i	$0.955020\pi$
$632$		−	7.80398e7i		−	0.309147i
$633$	7.39677e8			2.91629
$634$	−1.21862e8			−0.478191
$635$			1.21943e7i			0.0476250i
$636$		−	8.58957e8i		−	3.33887i
$637$	1.60749e8			0.621913
$638$			1.09267e8i			0.420754i
$639$	−5.50656e8			−2.11046
$640$			1.56701e8i			0.597765i
$641$		−	4.01021e8i		−	1.52262i	−0.648386	−	0.761312i	$-0.724556\pi$
$641$		−	4.01021e8i		−	1.52262i	0.648386	−	0.761312i	$-0.275444\pi$
$642$		−	4.06394e8i		−	1.53583i
$643$			3.96230e8i			1.49044i	0.666820	+	0.745219i	$0.267655\pi$
$643$			3.96230e8i			1.49044i	−0.666820	+	0.745219i	$0.732345\pi$
$644$	4.34889e8	+	3.00142e8i	1.62825	+	1.12375i
$645$	3.00765e8			1.12085
$646$	−1.94201e8			−0.720366
$647$	1.93291e8			0.713674			0.356837	−	0.934167i	$-0.383855\pi$
$647$	1.93291e8			0.713674			0.356837	+	0.934167i	$0.383855\pi$
$648$	−2.59695e8			−0.954417
$649$			1.36977e8i			0.501087i
$650$	−6.82652e7			−0.248576
$651$		−	7.23991e8i		−	2.62416i
$652$	4.17123e8			1.50495
$653$	7.41121e6			0.0266164			0.0133082	−	0.999911i	$-0.495764\pi$
$653$	7.41121e6			0.0266164			0.0133082	+	0.999911i	$0.495764\pi$
$654$		−	9.15699e8i		−	3.27356i
$655$			2.22700e8i			0.792495i
$656$	−6.39949e7			−0.226691
$657$	−7.91430e8			−2.79072
$658$		−	4.66957e8i		−	1.63908i
$659$			3.42078e8i			1.19528i	0.801765	+	0.597640i	$0.203895\pi$
$659$			3.42078e8i			1.19528i	−0.801765	+	0.597640i	$0.796105\pi$
$660$	1.43329e8			0.498544
$661$		−	2.61610e8i		−	0.905837i	−0.891552	−	0.452919i	$-0.850383\pi$
$661$		−	2.61610e8i		−	0.905837i	0.891552	−	0.452919i	$-0.149617\pi$
$662$	7.07523e8			2.43875
$663$		−	1.89718e8i		−	0.650980i
$664$			4.02999e8i			1.37657i
$665$			1.71340e8i			0.582630i
$666$			1.53699e9i			5.20294i
$667$	1.04999e8	−	1.52138e8i	0.353841	−	0.512697i
$668$	3.87931e8			1.30144
$669$	4.57711e8			1.52867
$670$	3.39532e8			1.12890
$671$	2.03083e7			0.0672212
$672$			8.64239e8i			2.84791i
$673$	−2.78529e8			−0.913744			−0.456872	−	0.889532i	$-0.651030\pi$
$673$	−2.78529e8			−0.913744			−0.456872	+	0.889532i	$0.651030\pi$
$674$		−	3.75220e7i		−	0.122548i
$675$	1.17285e8			0.381355
$676$	−1.72316e8			−0.557809
$677$		−	1.45331e8i		−	0.468374i	−0.972192	−	0.234187i	$-0.924757\pi$
$677$		−	1.45331e8i		−	0.468374i	0.972192	−	0.234187i	$-0.0752428\pi$
$678$		−	8.26132e8i		−	2.65070i
$679$	−8.49116e7			−0.271243
$680$	−4.98590e7			−0.158568
$681$		−	3.96412e8i		−	1.25518i
$682$			2.39376e8i			0.754620i
$683$	7.43319e7			0.233299			0.116650	−	0.993173i	$-0.462785\pi$
$683$	7.43319e7			0.233299			0.116650	+	0.993173i	$0.462785\pi$
$684$			9.62099e8i			3.00643i
$685$	−1.31524e8			−0.409197
$686$			1.44149e8i			0.446518i
$687$			5.01406e8i			1.54639i
$688$		−	1.34958e8i		−	0.414412i
$689$		−	3.31665e8i		−	1.01401i
$690$	−3.34445e8	−	2.30819e8i	−1.01807	−	0.702627i
$691$	−7.11639e7			−0.215688			−0.107844	−	0.994168i	$-0.534395\pi$
$691$	−7.11639e7			−0.215688			−0.107844	+	0.994168i	$0.534395\pi$
$692$	1.25217e8			0.377873
$693$	3.98087e8			1.19613
$694$	−2.43442e8			−0.728310
$695$			2.14746e7i			0.0639690i
$696$	−2.78600e8			−0.826330
$697$		−	1.24088e8i		−	0.366465i
$698$	−7.80473e8			−2.29505
$699$	5.15490e8			1.50934
$700$			1.35717e8i			0.395677i
$701$		−	3.97225e8i		−	1.15314i	−0.817048	−	0.576570i	$-0.804391\pi$
$701$		−	3.97225e8i		−	1.15314i	0.817048	−	0.576570i	$-0.195609\pi$
$702$	−8.19862e8			−2.36990
$703$	5.36310e8			1.54365
$704$		−	2.42279e8i		−	0.694381i
$705$			2.14280e8i			0.611525i
$706$	−8.25739e8			−2.34654
$707$		−	5.95478e8i		−	1.68503i
$708$	−1.07751e9			−3.03615
$709$		−	3.10310e8i		−	0.870677i	−0.900267	−	0.435338i	$-0.856629\pi$
$709$		−	3.10310e8i		−	0.870677i	0.900267	−	0.435338i	$-0.143371\pi$
$710$			2.55060e8i			0.712636i
$711$			3.06865e8i			0.853766i
$712$			5.12286e7i			0.141929i
$713$	2.30026e8	−	3.33296e8i	0.634613	−	0.919519i
$714$	−6.32098e8			−1.73656
$715$	5.53432e7			0.151407
$716$	−6.21835e8			−1.69409
$717$	2.06973e8			0.561508
$718$			4.23401e8i			1.14387i
$719$	6.90283e8			1.85712			0.928561	−	0.371181i	$-0.121047\pi$
$719$	6.90283e8			1.85712			0.928561	+	0.371181i	$0.121047\pi$
$720$		−	1.01109e8i		−	0.270889i
$721$	7.39719e8			1.97361
$722$	−3.00449e7			−0.0798287
$723$			3.27728e8i			0.867158i
$724$		−	2.26212e7i		−	0.0596073i
$725$	4.74782e7			0.124589
$726$	8.63696e8			2.25710
$727$		−	2.38219e8i		−	0.619972i	−0.950741	−	0.309986i	$-0.899676\pi$
$727$		−	2.38219e8i		−	0.619972i	0.950741	−	0.309986i	$-0.100324\pi$
$728$		−	3.07504e8i		−	0.796996i
$729$	−2.76454e8			−0.713576
$730$			3.66585e8i			0.942336i
$731$	2.61688e8			0.669933
$732$			1.59753e8i			0.407302i
$733$		−	1.37385e8i		−	0.348841i	−0.984671	−	0.174420i	$-0.944195\pi$
$733$		−	1.37385e8i		−	0.348841i	0.984671	−	0.174420i	$-0.0558052\pi$
$734$			1.12969e9i			2.85674i
$735$			2.45770e8i			0.618967i
$736$	−2.74586e8	+	3.97860e8i	−0.688723	+	0.997922i
$737$	−2.75262e8			−0.687612
$738$	−1.03025e9			−2.56314
$739$	1.83773e8			0.455354			0.227677	−	0.973737i	$-0.426887\pi$
$739$	1.83773e8			0.455354			0.227677	+	0.973737i	$0.426887\pi$
$740$	4.24808e8			1.04833
$741$			5.49620e8i			1.35085i
$742$	−1.10504e9			−2.70498
$743$			6.21626e7i			0.151552i	0.997125	+	0.0757762i	$0.0241435\pi$
$743$			6.21626e7i			0.151552i	−0.997125	+	0.0757762i	$0.975857\pi$
$744$	−6.10341e8			−1.48202
$745$	3.08831e8			0.746883
$746$			2.17940e8i			0.524953i
$747$		−	1.58466e9i		−	3.80167i
$748$	1.24707e8			0.297980
$749$	−3.11969e8			−0.742447
$750$		−	1.04371e8i		−	0.247398i
$751$			2.08916e8i			0.493234i	0.969113	+	0.246617i	$0.0793189\pi$
$751$			2.08916e8i			0.493234i	−0.969113	+	0.246617i	$0.920681\pi$
$752$	9.61506e7			0.226099
$753$		−	7.47213e8i		−	1.75009i
$754$	−3.31890e8			−0.774247
$755$			5.32467e7i			0.123723i
$756$			1.62996e9i			3.77234i
$757$			4.87075e8i			1.12281i	0.827540	+	0.561407i	$0.189740\pi$
$757$			4.87075e8i			1.12281i	−0.827540	+	0.561407i	$0.810260\pi$
$758$			1.02791e9i			2.36018i
$759$	2.71137e8	+	1.87127e8i	0.620103	+	0.427968i
$760$	1.44443e8			0.329046
$761$	8.34719e8			1.89403			0.947014	−	0.321193i	$-0.104084\pi$
$761$	8.34719e8			1.89403			0.947014	+	0.321193i	$0.104084\pi$
$762$	1.30328e8			0.294559
$763$	−7.02938e8			−1.58250
$764$		−	5.76773e8i		−	1.29338i
$765$	1.96054e8			0.437916
$766$		−	6.27780e8i		−	1.39676i
$767$	−4.16055e8			−0.922072
$768$	3.86636e8			0.853531
$769$			4.98614e8i			1.09644i	0.836334	+	0.548221i	$0.184695\pi$
$769$			4.98614e8i			1.09644i	−0.836334	+	0.548221i	$0.815305\pi$
$770$		−	1.84391e8i		−	0.403895i
$771$	−1.22556e9			−2.67405
$772$	−6.41712e8			−1.39472
$773$			1.11620e8i			0.241659i	0.992673	+	0.120830i	$0.0385555\pi$
$773$			1.11620e8i			0.241659i	−0.992673	+	0.120830i	$0.961444\pi$
$774$		−	2.17267e9i		−	4.68566i
$775$	1.04013e8			0.223450
$776$			7.15825e7i			0.153187i
$777$	1.74562e9			3.72123
$778$		−	9.26456e8i		−	1.96737i
$779$			3.59489e8i			0.760455i
$780$			4.35351e8i			0.917393i
$781$		−	2.06779e8i		−	0.434064i
$782$	−2.90992e8	−	2.00830e8i	−0.608500	−	0.419961i
$783$	5.70211e8			1.18782
$784$	1.10281e8			0.228850
$785$	1.07245e8			0.221701
$786$	2.38013e9			4.90155
$787$		−	1.14255e8i		−	0.234396i	−0.993109	−	0.117198i	$-0.962609\pi$
$787$		−	1.14255e8i		−	0.234396i	0.993109	−	0.117198i	$-0.0373913\pi$
$788$	1.27241e9			2.60044
$789$			1.11893e8i			0.227810i
$790$	1.42138e8			0.288289
$791$	−6.34181e8			−1.28140
$792$		−	3.35597e8i		−	0.675526i
$793$			6.16848e7i			0.123697i
$794$	5.49439e8			1.09763
$795$	5.07086e8			1.00921
$796$		−	1.13870e9i		−	2.25771i
$797$		−	3.61613e8i		−	0.714281i	−0.934051	−	0.357140i	$-0.883752\pi$
$797$		−	3.61613e8i		−	0.714281i	0.934051	−	0.357140i	$-0.116248\pi$
$798$	1.83121e9			3.60355
$799$			1.86439e8i			0.365509i
$800$	−1.24161e8			−0.242503
$801$		−	2.01439e8i		−	0.391965i
$802$			1.23328e9i			2.39078i
$803$		−	2.97193e8i		−	0.573974i
$804$		−	2.16531e9i		−	4.16632i
$805$	−1.77189e8	+	2.56737e8i	−0.339663	+	0.492154i
$806$	−7.27085e8			−1.38861
$807$	−1.16409e9			−2.21497
$808$	−5.02002e8			−0.951637
$809$	−1.85597e8			−0.350529			−0.175265	−	0.984521i	$-0.556078\pi$
$809$	−1.85597e8			−0.350529			−0.175265	+	0.984521i	$0.556078\pi$
$810$		−	4.72996e8i		−	0.890025i
$811$	4.14201e8			0.776513			0.388257	−	0.921551i	$-0.373077\pi$
$811$	4.14201e8			0.776513			0.388257	+	0.921551i	$0.373077\pi$
$812$			6.59826e8i			1.23243i
$813$	−1.37413e9			−2.55715
$814$	−5.77163e8			−1.07010
$815$			2.46248e8i			0.454884i
$816$		−	1.30155e8i		−	0.239546i
$817$	−7.58120e8			−1.39018
$818$	4.07006e8			0.743602
$819$			1.20915e9i			2.20105i
$820$			2.84749e8i			0.516441i
$821$	−4.08130e8			−0.737513			−0.368756	−	0.929526i	$-0.620216\pi$
$821$	−4.08130e8			−0.737513			−0.368756	+	0.929526i	$0.620216\pi$
$822$			1.40567e9i			2.53087i
$823$	−7.28113e8			−1.30617			−0.653085	−	0.757285i	$-0.726525\pi$
$823$	−7.28113e8			−1.30617			−0.653085	+	0.757285i	$0.726525\pi$
$824$		−	6.23601e8i		−	1.11462i
$825$			8.46146e7i			0.150690i
$826$			1.38621e9i			2.45973i
$827$			5.28485e8i			0.934364i	0.884161	+	0.467182i	$0.154731\pi$
$827$			5.28485e8i			0.934364i	−0.884161	+	0.467182i	$0.845269\pi$
$828$	−9.94942e8	+	1.44162e9i	−1.75270	+	2.53956i
$829$	−5.19526e8			−0.911893			−0.455946	−	0.890007i	$-0.650699\pi$
$829$	−5.19526e8			−0.911893			−0.455946	+	0.890007i	$0.650699\pi$
$830$	−7.34003e8			−1.28370
$831$	−1.28728e9			−2.24321
$832$	7.35902e8			1.27776
$833$			2.13839e8i			0.369957i
$834$	2.29512e8			0.395646
$835$			2.29015e8i			0.393373i
$836$	−3.61282e8			−0.618340
$837$	1.24919e9			2.13035
$838$		−	8.95963e8i		−	1.52250i
$839$			5.91530e8i			1.00159i	0.865566	+	0.500796i	$0.166959\pi$
$839$			5.91530e8i			1.00159i	−0.865566	+	0.500796i	$0.833041\pi$
$840$	4.70145e8			0.793220
$841$	−3.63995e8			−0.611938
$842$		−	1.61055e9i		−	2.69797i
$843$		−	1.47800e9i		−	2.46713i
$844$	−1.47683e9			−2.45642
$845$		−	1.01727e8i		−	0.168603i
$846$	1.54792e9			2.55645
$847$		−	6.63017e8i		−	1.09112i
$848$		−	2.27537e8i		−	0.373133i
$849$		−	4.80705e8i		−	0.785516i
$850$		−	9.08107e7i		−	0.147870i
$851$	8.03612e8	+	5.54618e8i	1.30394	+	0.899923i
$852$	1.62661e9			2.63005
$853$	−1.10069e9			−1.77344			−0.886722	−	0.462303i	$-0.847023\pi$
$853$	−1.10069e9			−1.77344			−0.886722	+	0.462303i	$0.847023\pi$
$854$	2.05520e8			0.329975
$855$	−5.67975e8			−0.908722
$856$			2.62997e8i			0.419304i
$857$	1.91360e8			0.304025			0.152012	−	0.988379i	$-0.451425\pi$
$857$	1.91360e8			0.304025			0.152012	+	0.988379i	$0.451425\pi$
$858$		−	5.91486e8i		−	0.936446i
$859$	3.60038e8			0.568027			0.284013	−	0.958820i	$-0.408334\pi$
$859$	3.60038e8			0.568027			0.284013	+	0.958820i	$0.408334\pi$
$860$	−6.00502e8			−0.944103
$861$			1.17009e9i			1.83320i
$862$			5.85944e8i			0.914817i
$863$	1.29193e8			0.201005			0.100503	−	0.994937i	$-0.467955\pi$
$863$	1.29193e8			0.201005			0.100503	+	0.994937i	$0.467955\pi$
$864$	−1.49117e9			−2.31199
$865$			7.39220e7i			0.114216i
$866$			1.23374e9i			1.89963i
$867$	−8.92403e8			−1.36932
$868$			1.44551e9i			2.21035i
$869$	−1.15232e8			−0.175596
$870$		−	5.07429e8i		−	0.770579i
$871$		−	8.36083e8i		−	1.26530i
$872$			5.92593e8i			0.893731i
$873$		−	2.81474e8i		−	0.423054i
$874$	8.43015e8	+	5.81813e8i	1.26270	+	0.871463i
$875$	−8.01207e7			−0.119597
$876$	2.33784e9			3.47778
$877$	2.35312e8			0.348856			0.174428	−	0.984670i	$-0.444192\pi$
$877$	2.35312e8			0.348856			0.174428	+	0.984670i	$0.444192\pi$
$878$	−7.66134e8			−1.13193
$879$			7.01284e8i			1.03259i
$880$	3.79678e7			0.0557144
$881$			9.96028e8i			1.45661i	0.685252	+	0.728306i	$0.259692\pi$
$881$			9.96028e8i			1.45661i	−0.685252	+	0.728306i	$0.740308\pi$
$882$	1.77540e9			2.58756
$883$	5.02627e8			0.730068			0.365034	−	0.930994i	$-0.381057\pi$
$883$	5.02627e8			0.730068			0.365034	+	0.930994i	$0.381057\pi$
$884$			3.78787e8i			0.548326i
$885$		−	6.36110e8i		−	0.917704i
$886$	−1.46591e9			−2.10768
$887$	−3.28373e8			−0.470541			−0.235270	−	0.971930i	$-0.575598\pi$
$887$	−3.28373e8			−0.470541			−0.235270	+	0.971930i	$0.575598\pi$
$888$		−	1.47160e9i		−	2.10160i
$889$		−	1.00046e8i		−	0.142395i
$890$	−9.33053e7			−0.132354
$891$			3.83461e8i			0.542111i
$892$	−9.13858e8			−1.28761
$893$		−	5.40123e8i		−	0.758469i
$894$		−	3.30067e9i		−	4.61944i
$895$		−	3.67101e8i		−	0.512055i
$896$		−	1.28563e9i		−	1.78727i
$897$	−5.68382e8	+	8.23555e8i	−0.787523	+	1.14108i
$898$	2.06980e9			2.85825
$899$	5.05685e8			0.695987
$900$	−4.49890e8			−0.617133
$901$	4.41202e8			0.603203
$902$		−	3.86873e8i		−	0.527167i
$903$	−2.46758e9			−3.35126
$904$			5.34630e8i			0.723682i
$905$	1.33544e7			0.0180169
$906$	5.69080e8			0.765224
$907$			9.65342e7i			0.129378i	0.997905	+	0.0646889i	$0.0206055\pi$
$907$			9.65342e7i			0.129378i	−0.997905	+	0.0646889i	$0.979395\pi$
$908$			7.91470e8i			1.05725i
$909$	1.97396e9			2.62812
$910$	5.60072e8			0.743224
$911$			6.38093e8i			0.843974i	0.906602	+	0.421987i	$0.138667\pi$
$911$			6.38093e8i			0.843974i	−0.906602	+	0.421987i	$0.861333\pi$
$912$			3.77063e8i			0.497084i
$913$	5.95062e8			0.781898
$914$			1.44168e9i			1.88812i
$915$	−9.43103e7			−0.123111
$916$		−	1.00110e9i		−	1.30254i
$917$		−	1.82711e9i		−	2.36950i
$918$		−	1.09063e9i		−	1.40978i
$919$			1.40141e9i			1.80558i	0.430077	+	0.902792i	$0.358486\pi$
$919$			1.40141e9i			1.80558i	−0.430077	+	0.902792i	$0.641514\pi$
$920$	2.16435e8	+	1.49374e8i	0.277949	+	0.191828i
$921$	7.45061e8			0.953702
$922$	1.29579e9			1.65327
$923$	6.28074e8			0.798740
$924$	−1.17593e9			−1.49061
$925$			2.50786e8i			0.316867i
$926$	−6.32635e8			−0.796746
$927$			2.45210e9i			3.07822i
$928$	−6.03644e8			−0.755330
$929$	5.36650e8			0.669335			0.334668	−	0.942336i	$-0.391376\pi$
$929$	5.36650e8			0.669335			0.334668	+	0.942336i	$0.391376\pi$
$930$		−	1.11165e9i		−	1.38203i
$931$		−	6.19499e8i		−	0.767700i
$932$	−1.02922e9			−1.27133
$933$	4.67620e8			0.575769
$934$			1.46965e9i			1.80374i
$935$			7.36210e7i			0.0900672i
$936$	1.01935e9			1.24307
$937$		−	4.46417e8i		−	0.542653i	−0.962487	−	0.271327i	$-0.912538\pi$
$937$		−	4.46417e8i		−	0.542653i	0.962487	−	0.271327i	$-0.0874624\pi$
$938$	−2.78565e9			−3.37534
$939$		−	5.19866e8i		−	0.627906i
$940$		−	4.27828e8i		−	0.515093i
$941$		−	4.34185e8i		−	0.521082i	−0.965463	−	0.260541i	$-0.916099\pi$
$941$		−	4.34185e8i		−	0.521082i	0.965463	−	0.260541i	$-0.0839010\pi$
$942$		−	1.14619e9i		−	1.37121i
$943$	−3.71761e8	+	5.38662e8i	−0.443332	+	0.642364i
$944$	−2.85432e8			−0.339302
$945$	−9.62245e8			−1.14022
$946$	8.15869e8			0.963711
$947$	−7.49595e8			−0.882627			−0.441313	−	0.897353i	$-0.645487\pi$
$947$	−7.49595e8			−0.882627			−0.441313	+	0.897353i	$0.645487\pi$
$948$		−	9.06462e8i		−	1.06396i
$949$	9.02699e8			1.05620
$950$			2.63082e8i			0.306846i
$951$	−4.58795e8			−0.533429
$952$	4.09061e8			0.474108
$953$		−	6.30317e8i		−	0.728250i	−0.931350	−	0.364125i	$-0.881368\pi$
$953$		−	6.30317e8i		−	0.728250i	0.931350	−	0.364125i	$-0.118632\pi$
$954$		−	3.66309e9i		−	4.21893i
$955$	3.40498e8			0.390935
$956$	−4.13239e8			−0.472963
$957$			4.11377e8i			0.469358i
$958$			1.70808e9i			1.94273i
$959$	1.07907e9			1.22347
$960$			1.12513e9i			1.27171i
$961$	2.20322e8			0.248249
$962$		−	1.75308e9i		−	1.96914i
$963$		−	1.03415e9i		−	1.15799i
$964$		−	6.54336e8i		−	0.730415i
$965$		−	3.78835e8i		−	0.421568i
$966$	2.74390e9	+	1.89373e9i	3.04395	+	2.10080i
$967$	8.40877e8			0.929936			0.464968	−	0.885327i	$-0.346066\pi$
$967$	8.40877e8			0.929936			0.464968	+	0.885327i	$0.346066\pi$
$968$	−5.58939e8			−0.616223
$969$	−7.31139e8			−0.803580
$970$	−1.30377e8			−0.142852
$971$		−	2.60827e8i		−	0.284902i	−0.989802	−	0.142451i	$-0.954502\pi$
$971$		−	2.60827e8i		−	0.284902i	0.989802	−	0.142451i	$-0.0454983\pi$
$972$	−4.25654e8			−0.463509
$973$		−	1.76185e8i		−	0.191263i
$974$	1.18828e9			1.28600
$975$	−2.57009e8			−0.277291
$976$			4.23184e7i			0.0455177i
$977$			3.06962e8i			0.329155i	0.986364	+	0.164578i	$0.0526261\pi$
$977$			3.06962e8i			0.329155i	−0.986364	+	0.164578i	$0.947374\pi$
$978$	2.63181e9			2.81344
$979$	7.56434e7			0.0806163
$980$		−	4.90701e8i		−	0.521361i
$981$		−	2.33017e9i		−	2.46820i
$982$	−8.29373e8			−0.875821
$983$			1.27968e9i			1.34723i	0.739085	+	0.673613i	$0.235258\pi$
$983$			1.27968e9i			1.34723i	−0.739085	+	0.673613i	$0.764742\pi$
$984$	9.86414e8			1.03532
$985$			7.51166e8i			0.786009i
$986$		−	4.41501e8i		−	0.460575i
$987$		−	1.75803e9i		−	1.82841i
$988$		−	1.09736e9i		−	1.13783i
$989$	−1.13597e9	−	7.84000e8i	−1.17430	−	0.810452i
$990$	6.11240e8			0.629950
$991$	9.44701e8			0.970674			0.485337	−	0.874327i	$-0.338697\pi$
$991$	9.44701e8			0.970674			0.485337	+	0.874327i	$0.338697\pi$
$992$	−1.32243e9			−1.35468
$993$	2.66373e9			2.72046
$994$		−	2.09260e9i		−	2.13073i
$995$	6.72230e8			0.682415
$996$			4.68099e9i			4.73761i
$997$	8.25895e8			0.833373			0.416686	−	0.909050i	$-0.363191\pi$
$997$	8.25895e8			0.833373			0.416686	+	0.909050i	$0.363191\pi$
$998$	2.87191e8			0.288921
$999$			3.01192e9i			3.02098i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.7.d.a.91.7	✓	48
23.22	odd	2	inner	115.7.d.a.91.8	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.7.d.a.91.7	✓	48	1.1	even	1	trivial
115.7.d.a.91.8	yes	48	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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-12.5973 q^{2} -47.4272 q^{3} +94.6924 q^{4} +55.9017i q^{5} +597.456 q^{6} -458.638i q^{7} -386.642 q^{8} +1520.34 q^{9} +O(q^{10})\)	\(q-12.5973 q^{2} -47.4272 q^{3} +94.6924 q^{4} +55.9017i q^{5} +597.456 q^{6} -458.638i q^{7} -386.642 q^{8} +1520.34 q^{9} -704.211i q^{10} +570.910i q^{11} -4491.00 q^{12} -1734.09 q^{13} +5777.60i q^{14} -2651.26i q^{15} -1189.66 q^{16} -2306.80i q^{17} -19152.2 q^{18} +6682.88i q^{19} +5293.47i q^{20} +21751.9i q^{21} -7191.93i q^{22} +(-6911.01 + 10013.7i) q^{23} +18337.4 q^{24} -3125.00 q^{25} +21844.9 q^{26} -37531.1 q^{27} -43429.5i q^{28} -15193.0 q^{29} +33398.8i q^{30} -33284.0 q^{31} +39731.6 q^{32} -27076.7i q^{33} +29059.4i q^{34} +25638.6 q^{35} +143965. q^{36} -80251.4i q^{37} -84186.3i q^{38} +82243.0 q^{39} -21614.0i q^{40} +53792.6 q^{41} -274016. i q^{42} +113442. i q^{43} +54060.8i q^{44} +84989.6i q^{45} +(87060.2 - 126146. i) q^{46} -80821.9 q^{47} +56422.3 q^{48} -92699.4 q^{49} +39366.6 q^{50} +109405. i q^{51} -164205. q^{52} +191262. i q^{53} +472791. q^{54} -31914.8 q^{55} +177329. i q^{56} -316950. i q^{57} +191391. q^{58} +239927. q^{59} -251054. i q^{60} -35571.9i q^{61} +419289. q^{62} -697285. i q^{63} -424374. q^{64} -96938.5i q^{65} +341093. i q^{66} +482146. i q^{67} -218436. i q^{68} +(327770. - 474921. i) q^{69} -322978. q^{70} -362192. q^{71} -587828. q^{72} -520561. q^{73} +1.01095e6i q^{74} +148210. q^{75} +632818. i q^{76} +261841. q^{77} -1.03604e6 q^{78} +201840. i q^{79} -66504.1i q^{80} +671667. q^{81} -677642. q^{82} -1.04230e6i q^{83} +2.05974e6i q^{84} +128954. q^{85} -1.42907e6i q^{86} +720563. q^{87} -220738. i q^{88} -132496. i q^{89} -1.07064e6i q^{90} +795318. i q^{91} +(-654420. + 948220. i) q^{92} +1.57857e6 q^{93} +1.01814e6 q^{94} -373584. q^{95} -1.88436e6 q^{96} -185139. i q^{97} +1.16776e6 q^{98} +867978. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 20 q^{2} + 1584 q^{4} + 410 q^{6} - 1210 q^{8} + 12896 q^{9}+O(q^{10}) \) 48 * q - 20 * q^2 + 1584 * q^4 + 410 * q^6 - 1210 * q^8 + 12896 * q^9	\( 48 q - 20 q^{2} + 1584 q^{4} + 410 q^{6} - 1210 q^{8} + 12896 q^{9} + 4290 q^{12} - 1440 q^{13} + 65400 q^{16} + 4610 q^{18} + 26600 q^{23} + 14940 q^{24} - 150000 q^{25} + 47594 q^{26} + 16080 q^{27} + 131800 q^{29} - 1392 q^{31} - 225040 q^{32} + 5000 q^{35} + 658786 q^{36} - 236320 q^{39} - 351496 q^{41} + 382692 q^{46} + 395680 q^{47} + 1042550 q^{48} - 637848 q^{49} + 62500 q^{50} + 523890 q^{52} - 241250 q^{54} - 402000 q^{55} - 479130 q^{58} - 466312 q^{59} - 1124330 q^{62} + 837582 q^{64} + 1021060 q^{69} - 396000 q^{70} - 114336 q^{71} - 1960750 q^{72} - 498720 q^{73} + 3610400 q^{77} - 1104610 q^{78} + 972888 q^{81} + 124950 q^{82} - 246000 q^{85} - 2090960 q^{87} + 4913480 q^{92} + 3234320 q^{93} - 5550378 q^{94} - 1664000 q^{95} - 776990 q^{96} + 9993220 q^{98}+O(q^{100}) \) 48 * q - 20 * q^2 + 1584 * q^4 + 410 * q^6 - 1210 * q^8 + 12896 * q^9 + 4290 * q^12 - 1440 * q^13 + 65400 * q^16 + 4610 * q^18 + 26600 * q^23 + 14940 * q^24 - 150000 * q^25 + 47594 * q^26 + 16080 * q^27 + 131800 * q^29 - 1392 * q^31 - 225040 * q^32 + 5000 * q^35 + 658786 * q^36 - 236320 * q^39 - 351496 * q^41 + 382692 * q^46 + 395680 * q^47 + 1042550 * q^48 - 637848 * q^49 + 62500 * q^50 + 523890 * q^52 - 241250 * q^54 - 402000 * q^55 - 479130 * q^58 - 466312 * q^59 - 1124330 * q^62 + 837582 * q^64 + 1021060 * q^69 - 396000 * q^70 - 114336 * q^71 - 1960750 * q^72 - 498720 * q^73 + 3610400 * q^77 - 1104610 * q^78 + 972888 * q^81 + 124950 * q^82 - 246000 * q^85 - 2090960 * q^87 + 4913480 * q^92 + 3234320 * q^93 - 5550378 * q^94 - 1664000 * q^95 - 776990 * q^96 + 9993220 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more