LMFDB - Embedded newform 69.7.d.a.22.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,7,Mod(22,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, 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("69.22");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	69.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:	$15.8737317698$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.17
Character	$\chi$	$=$	69.22
Dual form			69.7.d.a.22.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.36312 q^{2} +15.5885 q^{3} -23.5107 q^{4} -60.2242i q^{5} +99.1913 q^{6} -233.613i q^{7} -556.841 q^{8} +243.000 q^{9} +O(q^{10})$	\(q+6.36312 q^{2} +15.5885 q^{3} -23.5107 q^{4} -60.2242i q^{5} +99.1913 q^{6} -233.613i q^{7} -556.841 q^{8} +243.000 q^{9} -383.214i q^{10} -1397.49i q^{11} -366.495 q^{12} -1285.15 q^{13} -1486.51i q^{14} -938.802i q^{15} -2038.57 q^{16} -2821.93i q^{17} +1546.24 q^{18} -10185.1i q^{19} +1415.91i q^{20} -3641.66i q^{21} -8892.40i q^{22} +(-3980.90 + 11497.3i) q^{23} -8680.29 q^{24} +11998.1 q^{25} -8177.54 q^{26} +3788.00 q^{27} +5492.39i q^{28} +5523.92 q^{29} -5973.71i q^{30} -27306.5 q^{31} +22666.2 q^{32} -21784.7i q^{33} -17956.3i q^{34} -14069.1 q^{35} -5713.09 q^{36} -39765.7i q^{37} -64809.3i q^{38} -20033.4 q^{39} +33535.3i q^{40} -18455.9 q^{41} -23172.4i q^{42} +30920.7i q^{43} +32855.9i q^{44} -14634.5i q^{45} +(-25330.9 + 73158.9i) q^{46} +62793.6 q^{47} -31778.1 q^{48} +63074.0 q^{49} +76345.1 q^{50} -43989.5i q^{51} +30214.6 q^{52} +56165.6i q^{53} +24103.5 q^{54} -84162.6 q^{55} +130085. i q^{56} -158771. i q^{57} +35149.4 q^{58} -159966. q^{59} +22071.8i q^{60} +333535. i q^{61} -173755. q^{62} -56767.9i q^{63} +274696. q^{64} +77396.8i q^{65} -138619. i q^{66} +244298. i q^{67} +66345.4i q^{68} +(-62056.1 + 179225. i) q^{69} -89523.7 q^{70} +437275. q^{71} -135312. q^{72} +494935. q^{73} -253034. i q^{74} +187031. q^{75} +239459. i q^{76} -326471. q^{77} -127475. q^{78} +571913. i q^{79} +122771. i q^{80} +59049.0 q^{81} -117437. q^{82} -720923. i q^{83} +85617.9i q^{84} -169948. q^{85} +196752. i q^{86} +86109.4 q^{87} +778179. i q^{88} -785492. i q^{89} -93120.9i q^{90} +300226. i q^{91} +(93593.5 - 270310. i) q^{92} -425666. q^{93} +399563. q^{94} -613392. q^{95} +353331. q^{96} +274434. i q^{97} +401348. q^{98} -339590. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) $ 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9	$ 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9} + 384 q^{13} + 29544 q^{16} - 4860 q^{18} + 29336 q^{23} - 39204 q^{24} - 61272 q^{25} + 10088 q^{26} + 64672 q^{29} + 9696 q^{31} - 319620 q^{32} - 225744 q^{35} + 198288 q^{36} - 11664 q^{39} + 135280 q^{41} + 233232 q^{46} - 74336 q^{47} + 552096 q^{48} - 722136 q^{49} + 619324 q^{50} + 1059720 q^{52} - 78732 q^{54} - 1019328 q^{55} - 694344 q^{58} + 1057648 q^{59} - 488776 q^{62} - 273888 q^{64} - 23328 q^{69} + 2785512 q^{70} - 255392 q^{71} - 228420 q^{72} - 322560 q^{73} - 365472 q^{75} - 1002960 q^{77} - 171072 q^{78} + 1417176 q^{81} - 5732712 q^{82} - 2704704 q^{85} + 611712 q^{87} - 1611444 q^{92} + 2484432 q^{93} - 147720 q^{94} - 1672656 q^{95} - 1818612 q^{96} + 9104212 q^{98}+O(q^{100}) $ 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9 + 384 * q^13 + 29544 * q^16 - 4860 * q^18 + 29336 * q^23 - 39204 * q^24 - 61272 * q^25 + 10088 * q^26 + 64672 * q^29 + 9696 * q^31 - 319620 * q^32 - 225744 * q^35 + 198288 * q^36 - 11664 * q^39 + 135280 * q^41 + 233232 * q^46 - 74336 * q^47 + 552096 * q^48 - 722136 * q^49 + 619324 * q^50 + 1059720 * q^52 - 78732 * q^54 - 1019328 * q^55 - 694344 * q^58 + 1057648 * q^59 - 488776 * q^62 - 273888 * q^64 - 23328 * q^69 + 2785512 * q^70 - 255392 * q^71 - 228420 * q^72 - 322560 * q^73 - 365472 * q^75 - 1002960 * q^77 - 171072 * q^78 + 1417176 * q^81 - 5732712 * q^82 - 2704704 * q^85 + 611712 * q^87 - 1611444 * q^92 + 2484432 * q^93 - 147720 * q^94 - 1672656 * q^95 - 1818612 * q^96 + 9104212 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$1$

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 $

$2$	6.36312			0.795390			0.397695	−	0.917518i	$-0.369810\pi$
$2$	6.36312			0.795390			0.397695	+	0.917518i	$0.369810\pi$
$3$	15.5885			0.577350
$3$	15.5885			0.577350
$4$	−23.5107			−0.367354
$5$		−	60.2242i		−	0.481793i	−0.970551	−	0.240897i	$-0.922559\pi$
$5$		−	60.2242i		−	0.481793i	0.970551	−	0.240897i	$-0.0774415\pi$
$6$	99.1913			0.459219
$7$		−	233.613i		−	0.681087i	−0.940229	−	0.340544i	$-0.889389\pi$
$7$		−	233.613i		−	0.681087i	0.940229	−	0.340544i	$-0.110611\pi$
$8$	−556.841			−1.08758
$9$	243.000			0.333333
$10$		−	383.214i		−	0.383214i
$11$		−	1397.49i		−	1.04995i	−0.851116	−	0.524977i	$-0.824074\pi$
$11$		−	1397.49i		−	1.04995i	0.851116	−	0.524977i	$-0.175926\pi$
$12$	−366.495			−0.212092
$13$	−1285.15			−0.584955			−0.292477	−	0.956272i	$-0.594480\pi$
$13$	−1285.15			−0.584955			−0.292477	+	0.956272i	$0.594480\pi$
$14$		−	1486.51i		−	0.541730i
$15$		−	938.802i		−	0.278163i
$16$	−2038.57			−0.497697
$17$		−	2821.93i		−	0.574379i	−0.957874	−	0.287190i	$-0.907279\pi$
$17$		−	2821.93i		−	0.574379i	0.957874	−	0.287190i	$-0.0927210\pi$
$18$	1546.24			0.265130
$19$		−	10185.1i		−	1.48493i	−0.669884	−	0.742466i	$-0.733656\pi$
$19$		−	10185.1i		−	1.48493i	0.669884	−	0.742466i	$-0.266344\pi$
$20$			1415.91i			0.176989i
$21$		−	3641.66i		−	0.393226i
$22$		−	8892.40i		−	0.835123i
$23$	−3980.90	+	11497.3i	−0.327188	+	0.944959i
$23$	−3980.90	+	11497.3i	−0.327188	+	0.944959i
$24$	−8680.29			−0.627915
$25$	11998.1			0.767875
$26$	−8177.54			−0.465267
$27$	3788.00			0.192450
$28$			5492.39i			0.250200i
$29$	5523.92			0.226492			0.113246	−	0.993567i	$-0.463875\pi$
$29$	5523.92			0.226492			0.113246	+	0.993567i	$0.463875\pi$
$30$		−	5973.71i		−	0.221249i
$31$	−27306.5			−0.916603			−0.458301	−	0.888797i	$-0.651542\pi$
$31$	−27306.5			−0.916603			−0.458301	+	0.888797i	$0.651542\pi$
$32$	22666.2			0.691717
$33$		−	21784.7i		−	0.606191i
$34$		−	17956.3i		−	0.456856i
$35$	−14069.1			−0.328143
$36$	−5713.09			−0.122451
$37$		−	39765.7i		−	0.785060i	−0.919739	−	0.392530i	$-0.871600\pi$
$37$		−	39765.7i		−	0.785060i	0.919739	−	0.392530i	$-0.128400\pi$
$38$		−	64809.3i		−	1.18110i
$39$	−20033.4			−0.337724
$40$			33535.3i			0.523989i
$41$	−18455.9			−0.267784			−0.133892	−	0.990996i	$-0.542748\pi$
$41$	−18455.9			−0.267784			−0.133892	+	0.990996i	$0.542748\pi$
$42$		−	23172.4i		−	0.312768i
$43$			30920.7i			0.388906i	0.980912	+	0.194453i	$0.0622931\pi$
$43$			30920.7i			0.388906i	−0.980912	+	0.194453i	$0.937707\pi$
$44$			32855.9i			0.385705i
$45$		−	14634.5i		−	0.160598i
$46$	−25330.9	+	73158.9i	−0.260242	+	0.751612i
$47$	62793.6			0.604814			0.302407	−	0.953179i	$-0.402210\pi$
$47$	62793.6			0.604814			0.302407	+	0.953179i	$0.402210\pi$
$48$	−31778.1			−0.287345
$49$	63074.0			0.536120
$50$	76345.1			0.610761
$51$		−	43989.5i		−	0.331618i
$52$	30214.6			0.214885
$53$			56165.6i			0.377262i	0.982048	+	0.188631i	$0.0604050\pi$
$53$			56165.6i			0.377262i	−0.982048	+	0.188631i	$0.939595\pi$
$54$	24103.5			0.153073
$55$	−84162.6			−0.505861
$56$			130085.i			0.740737i
$57$		−	158771.i		−	0.857326i
$58$	35149.4			0.180150
$59$	−159966.			−0.778880			−0.389440	−	0.921052i	$-0.627331\pi$
$59$	−159966.			−0.778880			−0.389440	+	0.921052i	$0.627331\pi$
$60$			22071.8i			0.102184i
$61$			333535.i			1.46944i	0.678372	+	0.734719i	$0.262686\pi$
$61$			333535.i			1.46944i	−0.678372	+	0.734719i	$0.737314\pi$
$62$	−173755.			−0.729057
$63$		−	56767.9i		−	0.227029i
$64$	274696.			1.04788
$65$			77396.8i			0.281827i
$66$		−	138619.i		−	0.482159i
$67$			244298.i			0.812260i	0.913815	+	0.406130i	$0.133122\pi$
$67$			244298.i			0.812260i	−0.913815	+	0.406130i	$0.866878\pi$
$68$			66345.4i			0.211001i
$69$	−62056.1	+	179225.i	−0.188902	+	0.545572i
$70$	−89523.7			−0.261002
$71$	437275.			1.22174			0.610871	−	0.791731i	$-0.290820\pi$
$71$	437275.			1.22174			0.610871	+	0.791731i	$0.290820\pi$
$72$	−135312.			−0.362527
$73$	494935.			1.27227			0.636135	−	0.771578i	$-0.280532\pi$
$73$	494935.			1.27227			0.636135	+	0.771578i	$0.280532\pi$
$74$		−	253034.i		−	0.624429i
$75$	187031.			0.443333
$76$			239459.i			0.545496i
$77$	−326471.			−0.715110
$78$	−127475.			−0.268622
$79$			571913.i			1.15998i	0.814625	+	0.579988i	$0.196943\pi$
$79$			571913.i			1.15998i	−0.814625	+	0.579988i	$0.803057\pi$
$80$			122771.i			0.239787i
$81$	59049.0			0.111111
$82$	−117437.			−0.212993
$83$		−	720923.i		−	1.26082i	−0.776261	−	0.630412i	$-0.782886\pi$
$83$		−	720923.i		−	1.26082i	0.776261	−	0.630412i	$-0.217114\pi$
$84$			85617.9i			0.144453i
$85$	−169948.			−0.276732
$86$			196752.i			0.309332i
$87$	86109.4			0.130765
$88$			778179.i			1.14191i
$89$		−	785492.i		−	1.11422i	−0.830438	−	0.557111i	$-0.811910\pi$
$89$		−	785492.i		−	1.11422i	0.830438	−	0.557111i	$-0.188090\pi$
$90$		−	93120.9i		−	0.127738i
$91$			300226.i			0.398405i
$92$	93593.5	−	270310.i	0.120194	−	0.347135i
$93$	−425666.			−0.529201
$94$	399563.			0.481063
$95$	−613392.			−0.715430
$96$	353331.			0.399363
$97$			274434.i			0.300692i	0.988633	+	0.150346i	$0.0480388\pi$
$97$			274434.i			0.300692i	−0.988633	+	0.150346i	$0.951961\pi$
$98$	401348.			0.426425
$99$		−	339590.i		−	0.349985i
$100$	−282082.			−0.282082
$101$	−999436.			−0.970043			−0.485021	−	0.874502i	$-0.661188\pi$
$101$	−999436.			−0.970043			−0.485021	+	0.874502i	$0.661188\pi$
$102$		−	279910.i		−	0.263766i
$103$		−	359691.i		−	0.329168i	−0.986363	−	0.164584i	$-0.947372\pi$
$103$		−	359691.i		−	0.329168i	0.986363	−	0.164584i	$-0.0526282\pi$
$104$	715622.			0.636185
$105$	−219316.			−0.189454
$106$			357389.i			0.300070i
$107$		−	2.15412e6i		−	1.75841i	−0.476447	−	0.879203i	$-0.658076\pi$
$107$		−	2.15412e6i		−	1.75841i	0.476447	−	0.879203i	$-0.341924\pi$
$108$	−89058.3			−0.0706973
$109$		−	1.90465e6i		−	1.47074i	−0.677668	−	0.735368i	$-0.737009\pi$
$109$		−	1.90465e6i		−	1.47074i	0.677668	−	0.735368i	$-0.262991\pi$
$110$	−535537.			−0.402357
$111$		−	619885.i		−	0.453255i
$112$			476235.i			0.338975i
$113$		−	2.60669e6i		−	1.80657i	−0.429041	−	0.903285i	$-0.641149\pi$
$113$		−	2.60669e6i		−	1.80657i	0.429041	−	0.903285i	$-0.358851\pi$
$114$		−	1.01028e6i		−	0.681909i
$115$	692416.	+	239746.i	0.455275	+	0.157637i
$116$	−129871.			−0.0832028
$117$	−312290.			−0.194985
$118$	−1.01788e6			−0.619514
$119$	−659238.			−0.391202
$120$			522763.i			0.302525i
$121$	−181415.			−0.102404
$122$			2.12232e6i			1.16878i
$123$	−287700.			−0.154605
$124$	641994.			0.336718
$125$		−	1.66357e6i		−	0.851750i
$126$		−	361221.i		−	0.180577i
$127$	1.00672e6			0.491473			0.245736	−	0.969337i	$-0.420970\pi$
$127$	1.00672e6			0.491473			0.245736	+	0.969337i	$0.420970\pi$
$128$	297289.			0.141758
$129$			482007.i			0.224535i
$130$			492485.i			0.224163i
$131$	−1.48663e6			−0.661284			−0.330642	−	0.943756i	$-0.607265\pi$
$131$	−1.48663e6			−0.661284			−0.330642	+	0.943756i	$0.607265\pi$
$132$			512173.i			0.222687i
$133$	−2.37938e6			−1.01137
$134$			1.55450e6i			0.646064i
$135$		−	228129.i		−	0.0927212i
$136$			1.57136e6i			0.624684i
$137$			933505.i			0.363041i	0.983387	+	0.181520i	$0.0581018\pi$
$137$			933505.i			0.363041i	−0.983387	+	0.181520i	$0.941898\pi$
$138$	−394870.	+	1.14043e6i	−0.150251	+	0.433943i
$139$	2.57479e6			0.958734			0.479367	−	0.877615i	$-0.340866\pi$
$139$	2.57479e6			0.958734			0.479367	+	0.877615i	$0.340866\pi$
$140$	330775.			0.120545
$141$	978855.			0.349189
$142$	2.78243e6			0.971761
$143$			1.79598e6i			0.614176i
$144$	−495372.			−0.165899
$145$		−	332673.i		−	0.109122i
$146$	3.14933e6			1.01195
$147$	983227.			0.309529
$148$			934917.i			0.288395i
$149$			665234.i			0.201102i	0.994932	+	0.100551i	$0.0320605\pi$
$149$			665234.i			0.201102i	−0.994932	+	0.100551i	$0.967940\pi$
$150$	1.19010e6			0.352623
$151$	−385234.			−0.111891			−0.0559453	−	0.998434i	$-0.517817\pi$
$151$	−385234.			−0.111891			−0.0559453	+	0.998434i	$0.517817\pi$
$152$			5.67151e6i			1.61498i
$153$		−	685728.i		−	0.191460i
$154$	−2.07738e6			−0.568792
$155$			1.64451e6i			0.441613i
$156$	470999.			0.124064
$157$			2.55515e6i			0.660263i	0.943935	+	0.330132i	$0.107093\pi$
$157$			2.55515e6i			0.660263i	−0.943935	+	0.330132i	$0.892907\pi$
$158$			3.63916e6i			0.922634i
$159$			875535.i			0.217812i
$160$		−	1.36505e6i		−	0.333265i
$161$	2.68592e6	+	929989.i	0.643599	+	0.222844i
$162$	375736.			0.0883767
$163$	−4.76008e6			−1.09914			−0.549568	−	0.835449i	$-0.685208\pi$
$163$	−4.76008e6			−1.09914			−0.549568	+	0.835449i	$0.685208\pi$
$164$	433911.			0.0983715
$165$	−1.31197e6			−0.292059
$166$		−	4.58732e6i		−	1.00285i
$167$	3.47396e6			0.745892			0.372946	−	0.927853i	$-0.378348\pi$
$167$	3.47396e6			0.745892			0.372946	+	0.927853i	$0.378348\pi$
$168$			2.02783e6i			0.427665i
$169$	−3.17521e6			−0.657828
$170$	−1.08140e6			−0.220110
$171$		−	2.47499e6i		−	0.494977i
$172$		−	726967.i		−	0.142866i
$173$	3.58125e6			0.691666			0.345833	−	0.938296i	$-0.387596\pi$
$173$	3.58125e6			0.691666			0.345833	+	0.938296i	$0.387596\pi$
$174$	547925.			0.104009
$175$		−	2.80290e6i		−	0.522990i
$176$			2.84887e6i			0.522559i
$177$	−2.49362e6			−0.449687
$178$		−	4.99818e6i		−	0.886242i
$179$	−1.81514e6			−0.316483			−0.158241	−	0.987400i	$-0.550582\pi$
$179$	−1.81514e6			−0.316483			−0.158241	+	0.987400i	$0.550582\pi$
$180$			344066.i			0.0589962i
$181$		−	5.36160e6i		−	0.904188i	−0.891970	−	0.452094i	$-0.850677\pi$
$181$		−	5.36160e6i		−	0.904188i	0.891970	−	0.452094i	$-0.149323\pi$
$182$			1.91038e6i			0.316888i
$183$			5.19929e6i			0.848380i
$184$	2.21673e6	−	6.40218e6i	0.355843	−	1.02772i
$185$	−2.39485e6			−0.378237
$186$	−2.70857e6			−0.420921
$187$	−3.94361e6			−0.603072
$188$	−1.47632e6			−0.222181
$189$		−	884924.i		−	0.131075i
$190$	−3.90309e6			−0.569046
$191$		−	6.26268e6i		−	0.898794i	−0.893332	−	0.449397i	$-0.851639\pi$
$191$		−	6.26268e6i		−	0.898794i	0.893332	−	0.449397i	$-0.148361\pi$
$192$	4.28209e6			0.604995
$193$	1.28599e7			1.78882			0.894408	−	0.447252i	$-0.147597\pi$
$193$	1.28599e7			1.78882			0.894408	+	0.447252i	$0.147597\pi$
$194$			1.74626e6i			0.239168i
$195$			1.20650e6i			0.162713i
$196$	−1.48291e6			−0.196946
$197$	9.53336e6			1.24694			0.623472	−	0.781845i	$-0.285721\pi$
$197$	9.53336e6			1.24694			0.623472	+	0.781845i	$0.285721\pi$
$198$		−	2.16085e6i		−	0.278374i
$199$			1.75998e6i			0.223331i	0.993746	+	0.111665i	$0.0356185\pi$
$199$			1.75998e6i			0.223331i	−0.993746	+	0.111665i	$0.964382\pi$
$200$	−6.68101e6			−0.835126
$201$			3.80823e6i			0.468959i
$202$	−6.35954e6			−0.771563
$203$		−	1.29046e6i		−	0.154261i
$204$			1.03422e6i			0.121821i
$205$			1.11149e6i			0.129017i
$206$		−	2.28876e6i		−	0.261817i
$207$	−967358.	+	2.79385e6i	−0.109063	+	0.314986i
$208$	2.61985e6			0.291130
$209$	−1.42336e7			−1.55911
$210$	−1.39554e6			−0.150690
$211$	−1.02042e6			−0.108626			−0.0543128	−	0.998524i	$-0.517297\pi$
$211$	−1.02042e6			−0.108626			−0.0543128	+	0.998524i	$0.517297\pi$
$212$		−	1.32049e6i		−	0.138589i
$213$	6.81644e6			0.705372
$214$		−	1.37069e7i		−	1.39862i
$215$	1.86218e6			0.187372
$216$	−2.10931e6			−0.209305
$217$			6.37915e6i			0.624286i
$218$		−	1.21195e7i		−	1.16981i
$219$	7.71527e6			0.734545
$220$	1.97872e6			0.185830
$221$			3.62659e6i			0.335986i
$222$		−	3.94441e6i		−	0.360515i
$223$	−5.16861e6			−0.466078			−0.233039	−	0.972467i	$-0.574867\pi$
$223$	−5.16861e6			−0.466078			−0.233039	+	0.972467i	$0.574867\pi$
$224$		−	5.29511e6i		−	0.471119i
$225$	2.91553e6			0.255958
$226$		−	1.65867e7i		−	1.43693i
$227$			7.01786e6i			0.599967i	0.953944	+	0.299983i	$0.0969812\pi$
$227$			7.01786e6i			0.599967i	−0.953944	+	0.299983i	$0.903019\pi$
$228$			3.73280e6i			0.314942i
$229$			1.15614e7i			0.962731i	0.876520	+	0.481366i	$0.159859\pi$
$229$			1.15614e7i			0.962731i	−0.876520	+	0.481366i	$0.840141\pi$
$230$	4.40593e6	+	1.52553e6i	0.362121	+	0.125383i
$231$	−5.08919e6			−0.412869
$232$	−3.07594e6			−0.246328
$233$	−1.34263e7			−1.06143			−0.530713	−	0.847552i	$-0.678076\pi$
$233$	−1.34263e7			−1.06143			−0.530713	+	0.847552i	$0.678076\pi$
$234$	−1.98714e6			−0.155089
$235$		−	3.78169e6i		−	0.291395i
$236$	3.76090e6			0.286125
$237$			8.91525e6i			0.669712i
$238$	−4.19481e6			−0.311159
$239$	7.65228e6			0.560528			0.280264	−	0.959923i	$-0.409578\pi$
$239$	7.65228e6			0.560528			0.280264	+	0.959923i	$0.409578\pi$
$240$			1.91381e6i			0.138441i
$241$			9.36770e6i			0.669240i	0.942353	+	0.334620i	$0.108608\pi$
$241$			9.36770e6i			0.669240i	−0.942353	+	0.334620i	$0.891392\pi$
$242$	−1.15436e6			−0.0814510
$243$	920483.			0.0641500
$244$		−	7.84162e6i		−	0.539804i
$245$		−	3.79858e6i		−	0.258299i
$246$	−1.83067e6			−0.122971
$247$			1.30894e7i			0.868618i
$248$	1.52054e7			0.996879
$249$		−	1.12381e7i		−	0.727937i
$250$		−	1.05855e7i		−	0.677474i
$251$			109973.i			0.00695448i	0.999994	+	0.00347724i	$0.00110684\pi$
$251$			109973.i			0.00695448i	−0.999994	+	0.00347724i	$0.998893\pi$
$252$			1.33465e6i			0.0834000i
$253$	1.60674e7	+	5.56326e6i	0.992164	+	0.343533i
$254$	6.40591e6			0.390913
$255$	−2.64923e6			−0.159771
$256$	−1.56889e7			−0.935129
$257$	5.24170e6			0.308797			0.154398	−	0.988009i	$-0.450656\pi$
$257$	5.24170e6			0.308797			0.154398	+	0.988009i	$0.450656\pi$
$258$			3.06707e6i			0.178593i
$259$	−9.28977e6			−0.534694
$260$		−	1.81965e6i		−	0.103530i
$261$	1.34231e6			0.0754974
$262$	−9.45959e6			−0.525979
$263$		−	2.01529e7i		−	1.10782i	−0.832576	−	0.553911i	$-0.813135\pi$
$263$		−	2.01529e7i		−	1.10782i	0.832576	−	0.553911i	$-0.186865\pi$
$264$			1.21306e7i			0.659282i
$265$	3.38253e6			0.181762
$266$	−1.51403e7			−0.804432
$267$		−	1.22446e7i		−	0.643297i
$268$		−	5.74360e6i		−	0.298387i
$269$	−3.23774e7			−1.66336			−0.831678	−	0.555258i	$-0.812620\pi$
$269$	−3.23774e7			−1.66336			−0.831678	+	0.555258i	$0.812620\pi$
$270$		−	1.45161e6i		−	0.0737495i
$271$	1.93637e7			0.972929			0.486465	−	0.873700i	$-0.338286\pi$
$271$	1.93637e7			0.972929			0.486465	+	0.873700i	$0.338286\pi$
$272$			5.75268e6i			0.285867i
$273$			4.68007e6i			0.230019i
$274$			5.94001e6i			0.288759i
$275$		−	1.67671e7i		−	0.806234i
$276$	1.45898e6	−	4.21371e6i	0.0693940	−	0.200418i
$277$	3.05629e7			1.43799			0.718994	−	0.695016i	$-0.244603\pi$
$277$	3.05629e7			1.43799			0.718994	+	0.695016i	$0.244603\pi$
$278$	1.63837e7			0.762568
$279$	−6.63548e6			−0.305534
$280$	7.83427e6			0.356882
$281$			1.20980e7i			0.545249i	0.962120	+	0.272625i	$0.0878917\pi$
$281$			1.20980e7i			0.545249i	−0.962120	+	0.272625i	$0.912108\pi$
$282$	6.22857e6			0.277742
$283$			3.45135e7i			1.52275i	0.648310	+	0.761377i	$0.275476\pi$
$283$			3.45135e7i			1.52275i	−0.648310	+	0.761377i	$0.724524\pi$
$284$	−1.02806e7			−0.448812
$285$	−9.56183e6			−0.413054
$286$			1.14280e7i			0.488509i
$287$			4.31154e6i			0.182384i
$288$	5.50788e6			0.230572
$289$	1.61743e7			0.670088
$290$		−	2.11684e6i		−	0.0867949i
$291$			4.27800e6i			0.173605i
$292$	−1.16362e7			−0.467373
$293$			3.33872e7i			1.32733i	0.748032	+	0.663663i	$0.230999\pi$
$293$			3.33872e7i			1.32733i	−0.748032	+	0.663663i	$0.769001\pi$
$294$	6.25639e6			0.246197
$295$			9.63379e6i			0.375259i
$296$			2.21432e7i			0.853816i
$297$		−	5.29368e6i		−	0.202064i
$298$			4.23296e6i			0.159954i
$299$	5.11603e6	−	1.47757e7i	0.191390	−	0.552758i
$300$	−4.39722e6			−0.162860
$301$	7.22348e6			0.264879
$302$	−2.45129e6			−0.0889968
$303$	−1.55797e7			−0.560055
$304$			2.07631e7i			0.739046i
$305$	2.00868e7			0.707965
$306$		−	4.36337e6i		−	0.152285i
$307$	−5.47778e7			−1.89317			−0.946585	−	0.322455i	$-0.895492\pi$
$307$	−5.47778e7			−1.89317			−0.946585	+	0.322455i	$0.895492\pi$
$308$	7.67556e6			0.262699
$309$		−	5.60703e6i		−	0.190045i
$310$			1.04642e7i			0.351255i
$311$	2.86189e7			0.951418			0.475709	−	0.879603i	$-0.342192\pi$
$311$	2.86189e7			0.951418			0.475709	+	0.879603i	$0.342192\pi$
$312$	1.11554e7			0.367302
$313$			3.59059e7i			1.17094i	0.810695	+	0.585468i	$0.199089\pi$
$313$			3.59059e7i			1.17094i	−0.810695	+	0.585468i	$0.800911\pi$
$314$			1.62587e7i			0.525167i
$315$	−3.41880e6			−0.109381
$316$		−	1.34461e7i		−	0.426122i
$317$	3.22286e7			1.01173			0.505865	−	0.862613i	$-0.331174\pi$
$317$	3.22286e7			1.01173			0.505865	+	0.862613i	$0.331174\pi$
$318$			5.57114e6i			0.173246i
$319$		−	7.71962e6i		−	0.237806i
$320$		−	1.65433e7i		−	0.504862i
$321$		−	3.35795e7i		−	1.01522i
$322$	1.70908e7	+	5.91763e6i	0.511913	+	0.177248i
$323$	−2.87417e7			−0.852914
$324$	−1.38828e6			−0.0408171
$325$	−1.54192e7			−0.449172
$326$	−3.02890e7			−0.874242
$327$		−	2.96905e7i		−	0.849130i
$328$	1.02770e7			0.291237
$329$		−	1.46694e7i		−	0.411931i
$330$	−8.34820e6			−0.232301
$331$	3.35391e7			0.924842			0.462421	−	0.886661i	$-0.346981\pi$
$331$	3.35391e7			0.924842			0.462421	+	0.886661i	$0.346981\pi$
$332$			1.69494e7i			0.463169i
$333$		−	9.66306e6i		−	0.261687i
$334$	2.21053e7			0.593275
$335$	1.47126e7			0.391341
$336$			7.42377e6i			0.195707i
$337$			5.76754e7i			1.50696i	0.657472	+	0.753479i	$0.271626\pi$
$337$			5.76754e7i			1.50696i	−0.657472	+	0.753479i	$0.728374\pi$
$338$	−2.02043e7			−0.523230
$339$		−	4.06343e7i		−	1.04302i
$340$	3.99559e6			0.101659
$341$			3.81606e7i			0.962391i
$342$		−	1.57487e7i		−	0.393700i
$343$		−	4.22192e7i		−	1.04623i
$344$		−	1.72179e7i		−	0.422966i
$345$	1.07937e7	+	3.73727e6i	0.262853	+	0.0910118i
$346$	2.27879e7			0.550144
$347$	−2.70139e7			−0.646545			−0.323272	−	0.946306i	$-0.604783\pi$
$347$	−2.70139e7			−0.646545			−0.323272	+	0.946306i	$0.604783\pi$
$348$	−2.02449e6			−0.0480372
$349$	3.28000e6			0.0771609			0.0385804	−	0.999255i	$-0.487716\pi$
$349$	3.28000e6			0.0771609			0.0385804	+	0.999255i	$0.487716\pi$
$350$		−	1.78352e7i		−	0.415981i
$351$	−4.86812e6			−0.112575
$352$		−	3.16757e7i		−	0.726271i
$353$	−4.41288e7			−1.00323			−0.501613	−	0.865092i	$-0.667260\pi$
$353$	−4.41288e7			−1.00323			−0.501613	+	0.865092i	$0.667260\pi$
$354$	−1.58672e7			−0.357676
$355$		−	2.63345e7i		−	0.588627i
$356$			1.84674e7i			0.409314i
$357$	−1.02765e7			−0.225861
$358$	−1.15499e7			−0.251728
$359$		−	3.84523e7i		−	0.831073i	−0.909576	−	0.415537i	$-0.863594\pi$
$359$		−	3.84523e7i		−	0.831073i	0.909576	−	0.415537i	$-0.136406\pi$
$360$			8.14907e6i			0.174663i
$361$	−5.66913e7			−1.20502
$362$		−	3.41165e7i		−	0.719182i
$363$	−2.82797e6			−0.0591229
$364$		−	7.05852e6i		−	0.146356i
$365$		−	2.98070e7i		−	0.612971i
$366$			3.30837e7i			0.674794i
$367$		−	1.90097e7i		−	0.384571i	−0.981339	−	0.192286i	$-0.938410\pi$
$367$		−	1.90097e7i		−	0.384571i	0.981339	−	0.192286i	$-0.0615900\pi$
$368$	8.11533e6	−	2.34380e7i	0.162841	−	0.470303i
$369$	−4.48479e6			−0.0892613
$370$	−1.52387e7			−0.300846
$371$	1.31210e7			0.256948
$372$	1.00077e7			0.194404
$373$			6.82339e7i			1.31484i	0.753523	+	0.657421i	$0.228353\pi$
$373$			6.82339e7i			1.31484i	−0.753523	+	0.657421i	$0.771647\pi$
$374$	−2.50937e7			−0.479678
$375$		−	2.59326e7i		−	0.491758i
$376$	−3.49660e7			−0.657783
$377$	−7.09904e6			−0.132488
$378$		−	5.63088e6i		−	0.104256i
$379$		−	5.18792e7i		−	0.952961i	−0.879185	−	0.476481i	$-0.841912\pi$
$379$		−	5.18792e7i		−	0.952961i	0.879185	−	0.476481i	$-0.158088\pi$
$380$	1.44212e7			0.262816
$381$	1.56933e7			0.283752
$382$		−	3.98502e7i		−	0.714892i
$383$		−	1.02080e8i		−	1.81696i	−0.417931	−	0.908479i	$-0.637245\pi$
$383$		−	1.02080e8i		−	1.81696i	0.417931	−	0.908479i	$-0.362755\pi$
$384$	4.63427e6			0.0818442
$385$			1.96615e7i			0.344535i
$386$	8.18291e7			1.42281
$387$			7.51374e6i			0.129635i
$388$		−	6.45212e6i		−	0.110461i
$389$		−	2.97488e7i		−	0.505383i	−0.967547	−	0.252691i	$-0.918684\pi$
$389$		−	2.97488e7i		−	0.505383i	0.967547	−	0.252691i	$-0.0813157\pi$
$390$			7.67709e6i			0.129420i
$391$	3.24446e7	+	1.12338e7i	0.542765	+	0.187930i
$392$	−3.51222e7			−0.583074
$393$	−2.31742e7			−0.381793
$394$	6.06619e7			0.991808
$395$	3.44430e7			0.558869
$396$			7.98398e6i			0.128568i
$397$	−5.08435e6			−0.0812576			−0.0406288	−	0.999174i	$-0.512936\pi$
$397$	−5.08435e6			−0.0812576			−0.0406288	+	0.999174i	$0.512936\pi$
$398$			1.11990e7i			0.177635i
$399$	−3.70909e7			−0.583913
$400$	−2.44588e7			−0.382169
$401$			8.46941e7i			1.31347i	0.754122	+	0.656735i	$0.228063\pi$
$401$			8.46941e7i			1.31347i	−0.754122	+	0.656735i	$0.771937\pi$
$402$			2.42322e7i			0.373005i
$403$	3.50928e7			0.536171
$404$	2.34974e7			0.356349
$405$		−	3.55618e6i		−	0.0535326i
$406$		−	8.21134e6i		−	0.122698i
$407$	−5.55721e7			−0.824277
$408$			2.44952e7i			0.360661i
$409$	1.01530e7			0.148397			0.0741985	−	0.997243i	$-0.476360\pi$
$409$	1.01530e7			0.148397			0.0741985	+	0.997243i	$0.476360\pi$
$410$			7.07257e6i			0.102618i
$411$			1.45519e7i			0.209602i
$412$			8.45657e6i			0.120921i
$413$			3.73700e7i			0.530485i
$414$	−6.15542e6	+	1.77776e7i	−0.0867474	+	0.250537i
$415$	−4.34170e7			−0.607456
$416$	−2.91293e7			−0.404623
$417$	4.01371e7			0.553525
$418$	−9.05703e7			−1.24010
$419$		−	2.98259e7i		−	0.405464i	−0.979234	−	0.202732i	$-0.935018\pi$
$419$		−	2.98259e7i		−	0.405464i	0.979234	−	0.202732i	$-0.0649820\pi$
$420$	5.15627e6			0.0695965
$421$			8.63872e7i			1.15772i	0.815427	+	0.578860i	$0.196502\pi$
$421$			8.63872e7i			1.15772i	−0.815427	+	0.578860i	$0.803498\pi$
$422$	−6.49306e6			−0.0863997
$423$	1.52588e7			0.201605
$424$		−	3.12753e7i		−	0.410303i
$425$		−	3.38576e7i		−	0.441052i
$426$	4.33738e7			0.561047
$427$	7.79179e7			1.00082
$428$			5.06449e7i			0.645958i
$429$			2.79965e7i			0.354594i
$430$	1.18492e7			0.149034
$431$			4.07286e7i			0.508707i	0.967111	+	0.254353i	$0.0818626\pi$
$431$			4.07286e7i			0.508707i	−0.967111	+	0.254353i	$0.918137\pi$
$432$	−7.72208e6			−0.0957818
$433$		−	1.05183e8i		−	1.29563i	−0.761798	−	0.647814i	$-0.775683\pi$
$433$		−	1.05183e8i		−	1.29563i	0.761798	−	0.647814i	$-0.224317\pi$
$434$			4.05913e7i			0.496551i
$435$		−	5.18586e6i		−	0.0630019i
$436$			4.47795e7i			0.540281i
$437$	1.17102e8	+	4.05460e7i	1.40320	+	0.485852i
$438$	4.90932e7			0.584250
$439$	−5.25897e7			−0.621594			−0.310797	−	0.950476i	$-0.600596\pi$
$439$	−5.25897e7			−0.621594			−0.310797	+	0.950476i	$0.600596\pi$
$440$	4.68652e7			0.550164
$441$	1.53270e7			0.178707
$442$			2.30764e7i			0.267240i
$443$	−1.46092e8			−1.68041			−0.840207	−	0.542266i	$-0.817566\pi$
$443$	−1.46092e8			−1.68041			−0.840207	+	0.542266i	$0.817566\pi$
$444$			1.45739e7i			0.166505i
$445$	−4.73056e7			−0.536825
$446$	−3.28885e7			−0.370714
$447$			1.03700e7i			0.116106i
$448$		−	6.41725e7i		−	0.713699i
$449$	3.09669e7			0.342104			0.171052	−	0.985262i	$-0.445283\pi$
$449$	3.09669e7			0.342104			0.171052	+	0.985262i	$0.445283\pi$
$450$	1.85519e7			0.203587
$451$			2.57920e7i			0.281161i
$452$			6.12851e7i			0.663651i
$453$	−6.00520e6			−0.0646001
$454$			4.46555e7i			0.477208i
$455$	1.80809e7			0.191949
$456$			8.84101e7i			0.932410i
$457$			3.64508e7i			0.381908i	0.981599	+	0.190954i	$0.0611581\pi$
$457$			3.64508e7i			0.381908i	−0.981599	+	0.190954i	$0.938842\pi$
$458$			7.35668e7i			0.765747i
$459$		−	1.06894e7i		−	0.110539i
$460$	−1.62792e7	−	5.63659e6i	−0.167247	−	0.0579086i
$461$	3.26518e7			0.333276			0.166638	−	0.986018i	$-0.446709\pi$
$461$	3.26518e7			0.333276			0.166638	+	0.986018i	$0.446709\pi$
$462$	−3.23831e7			−0.328392
$463$	1.67599e8			1.68860			0.844302	−	0.535868i	$-0.180016\pi$
$463$	1.67599e8			1.68860			0.844302	+	0.535868i	$0.180016\pi$
$464$	−1.12609e7			−0.112724
$465$			2.56354e7i			0.254965i
$466$	−8.54334e7			−0.844248
$467$			3.58074e7i			0.351578i	0.984428	+	0.175789i	$0.0562477\pi$
$467$			3.58074e7i			0.351578i	−0.984428	+	0.175789i	$0.943752\pi$
$468$	7.34215e6			0.0716285
$469$	5.70711e7			0.553220
$470$		−	2.40634e7i		−	0.231773i
$471$			3.98308e7i			0.381203i
$472$	8.90754e7			0.847094
$473$	4.32114e7			0.408333
$474$			5.67288e7i			0.532683i
$475$		−	1.22202e8i		−	1.14024i
$476$	1.54991e7			0.143710
$477$			1.36482e7i			0.125754i
$478$	4.86924e7			0.445838
$479$		−	4.01741e7i		−	0.365544i	−0.983155	−	0.182772i	$-0.941493\pi$
$479$		−	4.01741e7i		−	0.365544i	0.983155	−	0.182772i	$-0.0585070\pi$
$480$		−	2.12791e7i		−	0.192410i
$481$			5.11047e7i			0.459225i
$482$			5.96078e7i			0.532307i
$483$	4.18694e7	+	1.44971e7i	0.371582	+	0.128659i
$484$	4.26518e6			0.0376185
$485$	1.65275e7			0.144872
$486$	5.85715e6			0.0510243
$487$	−1.60417e8			−1.38888			−0.694439	−	0.719552i	$-0.744347\pi$
$487$	−1.60417e8			−1.38888			−0.694439	+	0.719552i	$0.744347\pi$
$488$		−	1.85726e8i		−	1.59813i
$489$	−7.42023e7			−0.634586
$490$		−	2.41708e7i		−	0.205449i
$491$	−7.40765e7			−0.625800			−0.312900	−	0.949786i	$-0.601301\pi$
$491$	−7.40765e7			−0.625800			−0.312900	+	0.949786i	$0.601301\pi$
$492$	6.76401e6			0.0567948
$493$		−	1.55881e7i		−	0.130092i
$494$			8.32894e7i			0.690890i
$495$	−2.04515e7			−0.168620
$496$	5.56661e7			0.456190
$497$		−	1.02153e8i		−	0.832112i
$498$		−	7.15092e7i		−	0.578994i
$499$	−1.97181e8			−1.58695			−0.793474	−	0.608604i	$-0.791730\pi$
$499$	−1.97181e8			−1.58695			−0.793474	+	0.608604i	$0.791730\pi$
$500$			3.91117e7i			0.312894i
$501$	5.41537e7			0.430641
$502$			699772.i			0.00553153i
$503$			6.73802e6i			0.0529454i	0.999650	+	0.0264727i	$0.00842750\pi$
$503$			6.73802e6i			0.0529454i	−0.999650	+	0.0264727i	$0.991572\pi$
$504$			3.16107e7i			0.246912i
$505$			6.01902e7i			0.467360i
$506$	1.02239e8	+	3.53997e7i	0.789158	+	0.273242i
$507$	−4.94966e7			−0.379797
$508$	−2.36688e7			−0.180544
$509$	2.39743e7			0.181799			0.0908997	−	0.995860i	$-0.471026\pi$
$509$	2.39743e7			0.181799			0.0908997	+	0.995860i	$0.471026\pi$
$510$	−1.68574e7			−0.127081
$511$		−	1.15623e8i		−	0.866526i
$512$	−1.18857e8			−0.885551
$513$		−	3.85813e7i		−	0.285775i
$514$	3.33536e7			0.245614
$515$	−2.16621e7			−0.158591
$516$		−	1.13323e7i		−	0.0824838i
$517$		−	8.77533e7i		−	0.635027i
$518$	−5.91119e7			−0.425291
$519$	5.58262e7			0.399334
$520$		−	4.30977e7i		−	0.306510i
$521$			1.14966e8i			0.812933i	0.913666	+	0.406467i	$0.133239\pi$
$521$			1.14966e8i			0.812933i	−0.913666	+	0.406467i	$0.866761\pi$
$522$	8.54130e6			0.0600499
$523$		−	2.70926e8i		−	1.89385i	−0.321456	−	0.946925i	$-0.604172\pi$
$523$		−	2.70926e8i		−	1.89385i	0.321456	−	0.946925i	$-0.395828\pi$
$524$	3.49516e7			0.242926
$525$		−	4.36929e7i		−	0.301948i
$526$		−	1.28235e8i		−	0.881152i
$527$			7.70570e7i			0.526478i
$528$			4.44096e7i			0.301700i
$529$	−1.16341e8	−	9.15393e7i	−0.785896	−	0.618359i
$530$	2.15234e7			0.144572
$531$	−3.88716e7			−0.259627
$532$	5.59408e7			0.371530
$533$	2.37186e7			0.156641
$534$		−	7.79140e7i		−	0.511672i
$535$	−1.29730e8			−0.847188
$536$		−	1.36035e8i		−	0.883398i
$537$	−2.82952e7			−0.182722
$538$	−2.06021e8			−1.32302
$539$		−	8.81453e7i		−	0.562902i
$540$			5.36346e6i			0.0340615i
$541$	1.43573e8			0.906739			0.453370	−	0.891323i	$-0.350222\pi$
$541$	1.43573e8			0.906739			0.453370	+	0.891323i	$0.350222\pi$
$542$	1.23214e8			0.773859
$543$		−	8.35790e7i		−	0.522033i
$544$		−	6.39623e7i		−	0.397308i
$545$	−1.14706e8			−0.708591
$546$			2.97798e7i			0.182955i
$547$	2.07872e8			1.27009			0.635045	−	0.772475i	$-0.280982\pi$
$547$	2.07872e8			1.27009			0.635045	+	0.772475i	$0.280982\pi$
$548$		−	2.19473e7i		−	0.133364i
$549$			8.10489e7i			0.489813i
$550$		−	1.06691e8i		−	0.641271i
$551$		−	5.62619e7i		−	0.336325i
$552$	3.45554e7	−	9.98001e7i	0.205446	−	0.593354i
$553$	1.33606e8			0.790044
$554$	1.94475e8			1.14376
$555$	−3.73321e7			−0.218375
$556$	−6.05351e7			−0.352195
$557$			2.03427e8i			1.17718i	0.808431	+	0.588591i	$0.200317\pi$
$557$			2.03427e8i			1.17718i	−0.808431	+	0.588591i	$0.799683\pi$
$558$	−4.22224e7			−0.243019
$559$		−	3.97376e7i		−	0.227492i
$560$	2.86809e7			0.163316
$561$	−6.14748e7			−0.348184
$562$			7.69812e7i			0.433686i
$563$			4.02092e7i			0.225320i	0.993634	+	0.112660i	$0.0359371\pi$
$563$			4.02092e7i			0.225320i	−0.993634	+	0.112660i	$0.964063\pi$
$564$	−2.30135e7			−0.128276
$565$	−1.56986e8			−0.870393
$566$			2.19614e8i			1.21118i
$567$		−	1.37946e7i		−	0.0756763i
$568$	−2.43492e8			−1.32874
$569$		−	2.17982e8i		−	1.18327i	−0.806205	−	0.591636i	$-0.798482\pi$
$569$		−	2.17982e8i		−	1.18327i	0.806205	−	0.591636i	$-0.201518\pi$
$570$	−6.08431e7			−0.328539
$571$			1.53635e8i			0.825243i	0.910903	+	0.412621i	$0.135387\pi$
$571$			1.53635e8i			0.825243i	−0.910903	+	0.412621i	$0.864613\pi$
$572$		−	4.22246e7i		−	0.225620i
$573$		−	9.76256e7i		−	0.518919i
$574$			2.74349e7i			0.145067i
$575$	−4.77630e7	+	1.37945e8i	−0.251240	+	0.725611i
$576$	6.67511e7			0.349294
$577$	4.00536e7			0.208504			0.104252	−	0.994551i	$-0.466755\pi$
$577$	4.00536e7			0.208504			0.104252	+	0.994551i	$0.466755\pi$
$578$	1.02919e8			0.532982
$579$	2.00466e8			1.03277
$580$			7.82137e6i			0.0400866i
$581$	−1.68417e8			−0.858731
$582$			2.72214e7i			0.138084i
$583$	7.84908e7			0.396108
$584$	−2.75600e8			−1.38370
$585$			1.88074e7i			0.0939424i
$586$			2.12447e8i			1.05574i
$587$	−1.79154e8			−0.885750			−0.442875	−	0.896583i	$-0.646042\pi$
$587$	−1.79154e8			−0.885750			−0.442875	+	0.896583i	$0.646042\pi$
$588$	−2.31163e7			−0.113707
$589$			2.78121e8i			1.36109i
$590$			6.13010e7i			0.298477i
$591$	1.48610e8			0.719924
$592$			8.10649e7i			0.390722i
$593$	3.94002e8			1.88944			0.944722	−	0.327872i	$-0.106331\pi$
$593$	3.94002e8			1.88944			0.944722	+	0.327872i	$0.106331\pi$
$594$		−	3.36843e7i		−	0.160720i
$595$			3.97021e7i			0.188479i
$596$		−	1.56401e7i		−	0.0738755i
$597$			2.74354e7i			0.128940i
$598$	3.25539e7	−	9.40198e7i	0.152230	−	0.439659i
$599$	3.26132e8			1.51745			0.758723	−	0.651413i	$-0.225823\pi$
$599$	3.26132e8			1.51745			0.758723	+	0.651413i	$0.225823\pi$
$600$	−1.04147e8			−0.482160
$601$	3.34444e8			1.54064			0.770318	−	0.637659i	$-0.220097\pi$
$601$	3.34444e8			1.54064			0.770318	+	0.637659i	$0.220097\pi$
$602$	4.59639e7			0.210682
$603$			5.93644e7i			0.270753i
$604$	9.05711e6			0.0411035
$605$			1.09255e7i			0.0493375i
$606$	−9.91353e7			−0.445462
$607$	−2.70163e8			−1.20798			−0.603991	−	0.796991i	$-0.706424\pi$
$607$	−2.70163e8			−1.20798			−0.603991	+	0.796991i	$0.706424\pi$
$608$		−	2.30858e8i		−	1.02715i
$609$		−	2.01163e7i		−	0.0890626i
$610$	1.27815e8			0.563109
$611$	−8.06989e7			−0.353789
$612$			1.61219e7i			0.0703335i
$613$			1.66910e8i			0.724606i	0.932060	+	0.362303i	$0.118009\pi$
$613$			1.66910e8i			0.724606i	−0.932060	+	0.362303i	$0.881991\pi$
$614$	−3.48558e8			−1.50581
$615$			1.73265e7i			0.0744877i
$616$	1.81793e8			0.777740
$617$			3.40980e6i			0.0145169i	0.999974	+	0.00725843i	$0.00231045\pi$
$617$			3.40980e6i			0.0145169i	−0.999974	+	0.00725843i	$0.997690\pi$
$618$		−	3.56782e7i		−	0.151160i
$619$			7.78890e7i			0.328401i	0.986427	+	0.164200i	$0.0525043\pi$
$619$			7.78890e7i			0.328401i	−0.986427	+	0.164200i	$0.947496\pi$
$620$		−	3.86636e7i		−	0.162228i
$621$	−1.50796e7	+	4.35518e7i	−0.0629674	+	0.181857i
$622$	1.82105e8			0.756748
$623$	−1.83501e8			−0.758882
$624$	4.08395e7			0.168084
$625$	8.72821e7			0.357508
$626$			2.28474e8i			0.931352i
$627$	−2.21880e8			−0.900153
$628$		−	6.00732e7i		−	0.242550i
$629$	−1.12216e8			−0.450923
$630$	−2.17542e7			−0.0870006
$631$			2.97365e8i			1.18359i	0.806089	+	0.591795i	$0.201580\pi$
$631$			2.97365e8i			1.18359i	−0.806089	+	0.591795i	$0.798420\pi$
$632$		−	3.18465e8i		−	1.26157i
$633$	−1.59068e7			−0.0627150
$634$	2.05075e8			0.804720
$635$		−	6.06291e7i		−	0.236788i
$636$		−	2.05844e7i		−	0.0800142i
$637$	−8.10593e7			−0.313606
$638$		−	4.91209e7i		−	0.189149i
$639$	1.06258e8			0.407247
$640$		−	1.79040e7i		−	0.0682982i
$641$		−	1.39938e8i		−	0.531328i	−0.964066	−	0.265664i	$-0.914409\pi$
$641$		−	1.39938e8i		−	0.531328i	0.964066	−	0.265664i	$-0.0855912\pi$
$642$		−	2.13670e8i		−	0.807493i
$643$		−	3.06153e8i		−	1.15161i	−0.817587	−	0.575805i	$-0.804689\pi$
$643$		−	3.06153e8i		−	1.15161i	0.817587	−	0.575805i	$-0.195311\pi$
$644$	−6.31478e7	−	2.18647e7i	−0.236429	−	0.0818625i
$645$	2.90284e7			0.108179
$646$	−1.82887e8			−0.678400
$647$	2.49111e8			0.919771			0.459885	−	0.887978i	$-0.347890\pi$
$647$	2.49111e8			0.919771			0.459885	+	0.887978i	$0.347890\pi$
$648$	−3.28809e7			−0.120842
$649$			2.23550e8i			0.817788i
$650$	−9.81145e7			−0.357267
$651$			9.94412e7i			0.360432i
$652$	1.11913e8			0.403772
$653$	−4.16924e8			−1.49733			−0.748665	−	0.662948i	$-0.769305\pi$
$653$	−4.16924e8			−1.49733			−0.748665	+	0.662948i	$0.769305\pi$
$654$		−	1.88924e8i		−	0.675390i
$655$			8.95309e7i			0.318602i
$656$	3.76237e7			0.133275
$657$	1.20269e8			0.424090
$658$		−	9.33431e7i		−	0.327646i
$659$		−	2.42010e8i		−	0.845622i	−0.906218	−	0.422811i	$-0.861043\pi$
$659$		−	2.42010e8i		−	0.845622i	0.906218	−	0.422811i	$-0.138957\pi$
$660$	3.08452e7			0.107289
$661$			4.85278e8i			1.68030i	0.542356	+	0.840149i	$0.317532\pi$
$661$			4.85278e8i			1.68030i	−0.542356	+	0.840149i	$0.682468\pi$
$662$	2.13413e8			0.735610
$663$			5.65329e7i			0.193982i
$664$			4.01439e8i			1.37125i
$665$			1.43296e8i			0.487270i
$666$		−	6.14872e7i		−	0.208143i
$667$	−2.19902e7	+	6.35102e7i	−0.0741056	+	0.214026i
$668$	−8.16752e7			−0.274006
$669$	−8.05706e7			−0.269090
$670$	9.36183e7			0.311269
$671$	4.66111e8			1.54284
$672$		−	8.25426e7i		−	0.272001i
$673$	−3.54234e8			−1.16211			−0.581053	−	0.813866i	$-0.697359\pi$
$673$	−3.54234e8			−1.16211			−0.581053	+	0.813866i	$0.697359\pi$
$674$			3.66996e8i			1.19862i
$675$	4.54486e7			0.147778
$676$	7.46513e7			0.241656
$677$		−	1.26295e8i		−	0.407023i	−0.979073	−	0.203512i	$-0.934765\pi$
$677$		−	1.26295e8i		−	0.407023i	0.979073	−	0.203512i	$-0.0652354\pi$
$678$		−	2.58561e8i		−	0.829611i
$679$	6.41113e7			0.204798
$680$	9.46341e7			0.300968
$681$			1.09398e8i			0.346391i
$682$			2.42820e8i			0.765477i
$683$	2.78165e7			0.0873054			0.0436527	−	0.999047i	$-0.486101\pi$
$683$	2.78165e7			0.0873054			0.0436527	+	0.999047i	$0.486101\pi$
$684$			5.81887e7i			0.181832i
$685$	5.62196e7			0.174910
$686$		−	2.68646e8i		−	0.832163i
$687$			1.80225e8i			0.555833i
$688$		−	6.30340e7i		−	0.193557i
$689$		−	7.21810e7i		−	0.220681i
$690$	6.86817e7	+	2.37807e7i	0.209071	+	0.0723899i
$691$	6.07780e8			1.84210			0.921048	−	0.389449i	$-0.127335\pi$
$691$	6.07780e8			1.84210			0.921048	+	0.389449i	$0.127335\pi$
$692$	−8.41976e7			−0.254086
$693$	−7.93325e7			−0.238370
$694$	−1.71893e8			−0.514256
$695$		−	1.55065e8i		−	0.461912i
$696$	−4.79492e7			−0.142218
$697$			5.20813e7i			0.153810i
$698$	2.08710e7			0.0613730
$699$	−2.09296e8			−0.612814
$700$			6.58980e7i			0.192122i
$701$			3.75291e8i			1.08947i	0.838609	+	0.544733i	$0.183369\pi$
$701$			3.75291e8i			1.08947i	−0.838609	+	0.544733i	$0.816631\pi$
$702$	−3.09765e7			−0.0895407
$703$	−4.05019e8			−1.16576
$704$		−	3.83885e8i		−	1.10023i
$705$		−	5.89507e7i		−	0.168237i
$706$	−2.80797e8			−0.797956
$707$			2.33481e8i			0.660684i
$708$	5.86266e7			0.165194
$709$			4.32348e8i			1.21309i	0.795048	+	0.606547i	$0.207446\pi$
$709$			4.32348e8i			1.21309i	−0.795048	+	0.606547i	$0.792554\pi$
$710$		−	1.67570e8i		−	0.468188i
$711$			1.38975e8i			0.386659i
$712$			4.37394e8i			1.21181i
$713$	1.08704e8	−	3.13952e8i	0.299902	−	0.866152i
$714$	−6.53907e7			−0.179648
$715$	1.08161e8			0.295906
$716$	4.26751e7			0.116261
$717$	1.19287e8			0.323621
$718$		−	2.44677e8i		−	0.661028i
$719$	−1.83199e8			−0.492875			−0.246437	−	0.969159i	$-0.579260\pi$
$719$	−1.83199e8			−0.492875			−0.246437	+	0.969159i	$0.579260\pi$
$720$			2.98333e7i			0.0799290i
$721$	−8.40284e7			−0.224192
$722$	−3.60734e8			−0.958462
$723$			1.46028e8i			0.386386i
$724$			1.26055e8i			0.332157i
$725$	6.62763e7			0.173918
$726$	−1.79947e7			−0.0470258
$727$		−	3.32462e8i		−	0.865244i	−0.901575	−	0.432622i	$-0.857588\pi$
$727$		−	3.32462e8i		−	0.865244i	0.901575	−	0.432622i	$-0.142412\pi$
$728$		−	1.67178e8i		−	0.433297i
$729$	1.43489e7			0.0370370
$730$		−	1.89666e8i		−	0.487551i
$731$	8.72560e7			0.223380
$732$		−	1.22239e8i		−	0.311656i
$733$		−	5.89347e8i		−	1.49644i	−0.663450	−	0.748220i	$-0.730909\pi$
$733$		−	5.89347e8i		−	1.49644i	0.663450	−	0.748220i	$-0.269091\pi$
$734$		−	1.20961e8i		−	0.305884i
$735$		−	5.92140e7i		−	0.149129i
$736$	−9.02318e7	+	2.60600e8i	−0.226322	+	0.653644i
$737$	3.41403e8			0.852836
$738$	−2.85373e7			−0.0709976
$739$	5.99727e8			1.48601			0.743003	−	0.669288i	$-0.233401\pi$
$739$	5.99727e8			1.48601			0.743003	+	0.669288i	$0.233401\pi$
$740$	5.63046e7			0.138947
$741$			2.04043e8i			0.501497i
$742$	8.34906e7			0.204374
$743$			6.45667e7i			0.157414i	0.996898	+	0.0787068i	$0.0250791\pi$
$743$			6.45667e7i			0.157414i	−0.996898	+	0.0787068i	$0.974921\pi$
$744$	2.37029e8			0.575549
$745$	4.00631e7			0.0968894
$746$			4.34181e8i			1.04581i
$747$		−	1.75184e8i		−	0.420275i
$748$	9.27169e7			0.221541
$749$	−5.03231e8			−1.19763
$750$		−	1.65012e8i		−	0.391140i
$751$		−	8.08683e7i		−	0.190923i	−0.995433	−	0.0954616i	$-0.969567\pi$
$751$		−	8.08683e7i		−	0.190923i	0.995433	−	0.0954616i	$-0.0304327\pi$
$752$	−1.28009e8			−0.301014
$753$			1.71431e6i			0.00401517i
$754$	−4.51721e7			−0.105379
$755$			2.32004e7i			0.0539082i
$756$			2.08052e7i			0.0481510i
$757$		−	6.29360e7i		−	0.145081i	−0.997365	−	0.0725406i	$-0.976889\pi$
$757$		−	6.29360e7i		−	0.145081i	0.997365	−	0.0725406i	$-0.0231107\pi$
$758$		−	3.30113e8i		−	0.757976i
$759$	2.50466e8	+	8.67227e7i	0.572826	+	0.198339i
$760$	3.41562e8			0.778088
$761$	−5.45904e8			−1.23869			−0.619344	−	0.785119i	$-0.712602\pi$
$761$	−5.45904e8			−1.23869			−0.619344	+	0.785119i	$0.712602\pi$
$762$	9.98583e7			0.225694
$763$	−4.44950e8			−1.00170
$764$			1.47240e8i			0.330176i
$765$	−4.12974e7			−0.0922441
$766$		−	6.49548e8i		−	1.44519i
$767$	2.05579e8			0.455609
$768$	−2.44565e8			−0.539897
$769$		−	1.19089e8i		−	0.261874i	−0.991391	−	0.130937i	$-0.958201\pi$
$769$		−	1.19089e8i		−	0.261874i	0.991391	−	0.130937i	$-0.0417985\pi$
$770$			1.25108e8i			0.274040i
$771$	8.17100e7			0.178284
$772$	−3.02345e8			−0.657129
$773$		−	3.69267e8i		−	0.799470i	−0.916631	−	0.399735i	$-0.869102\pi$
$773$		−	3.69267e8i		−	0.799470i	0.916631	−	0.399735i	$-0.130898\pi$
$774$			4.78108e7i			0.103111i
$775$	−3.27625e8			−0.703837
$776$		−	1.52816e8i		−	0.327027i
$777$	−1.44813e8			−0.308706
$778$		−	1.89295e8i		−	0.401977i
$779$			1.87976e8i			0.397641i
$780$		−	2.83655e7i		−	0.0597733i
$781$		−	6.11086e8i		−	1.28277i
$782$	2.06449e8	+	7.14821e7i	0.431710	+	0.149478i
$783$	2.09246e7			0.0435884
$784$	−1.28581e8			−0.266826
$785$	1.53882e8			0.318110
$786$	−1.47460e8			−0.303674
$787$		−	9.26119e8i		−	1.89995i	−0.312324	−	0.949976i	$-0.601107\pi$
$787$		−	9.26119e8i		−	1.89995i	0.312324	−	0.949976i	$-0.398893\pi$
$788$	−2.24136e8			−0.458070
$789$		−	3.14153e8i		−	0.639602i
$790$	2.19165e8			0.444519
$791$	−6.08957e8			−1.23043
$792$			1.89098e8i			0.380637i
$793$		−	4.28640e8i		−	0.859555i
$794$	−3.23523e7			−0.0646315
$795$	5.27284e7			0.104940
$796$		−	4.13783e7i		−	0.0820415i
$797$			7.70727e8i			1.52239i	0.648523	+	0.761195i	$0.275387\pi$
$797$			7.70727e8i			1.52239i	−0.648523	+	0.761195i	$0.724613\pi$
$798$	−2.36014e8			−0.464439
$799$		−	1.77199e8i		−	0.347393i
$800$	2.71950e8			0.531152
$801$		−	1.90875e8i		−	0.371407i
$802$			5.38919e8i			1.04472i
$803$		−	6.91666e8i		−	1.33582i
$804$		−	8.95339e7i		−	0.172274i
$805$	5.60078e7	−	1.61757e8i	0.107365	−	0.310082i
$806$	2.23300e8			0.426465
$807$	−5.04714e8			−0.960339
$808$	5.56527e8			1.05500
$809$	5.09249e8			0.961800			0.480900	−	0.876775i	$-0.340310\pi$
$809$	5.09249e8			0.961800			0.480900	+	0.876775i	$0.340310\pi$
$810$		−	2.26284e7i		−	0.0425793i
$811$	−9.15685e8			−1.71666			−0.858328	−	0.513101i	$-0.828497\pi$
$811$	−9.15685e8			−1.71666			−0.858328	+	0.513101i	$0.828497\pi$
$812$			3.03395e7i			0.0566684i
$813$	3.01851e8			0.561721
$814$	−3.53612e8			−0.655622
$815$			2.86672e8i			0.529556i
$816$			8.96755e7i			0.165045i
$817$	3.14932e8			0.577498
$818$	6.46049e7			0.118034
$819$			7.29550e7i			0.132802i
$820$		−	2.61319e7i		−	0.0473947i
$821$	4.22090e8			0.762738			0.381369	−	0.924423i	$-0.375453\pi$
$821$	4.22090e8			0.762738			0.381369	+	0.924423i	$0.375453\pi$
$822$			9.25956e7i			0.166715i
$823$	−8.84464e8			−1.58665			−0.793324	−	0.608800i	$-0.791651\pi$
$823$	−8.84464e8			−1.58665			−0.793324	+	0.608800i	$0.791651\pi$
$824$			2.00291e8i			0.357997i
$825$		−	2.61374e8i		−	0.465479i
$826$			2.37790e8i			0.421943i
$827$			1.06398e9i			1.88111i	0.339636	+	0.940557i	$0.389696\pi$
$827$			1.06398e9i			1.88111i	−0.339636	+	0.940557i	$0.610304\pi$
$828$	2.27432e7	−	6.56852e7i	0.0400646	−	0.115712i
$829$	8.34964e8			1.46556			0.732781	−	0.680465i	$-0.238222\pi$
$829$	8.34964e8			1.46556			0.732781	+	0.680465i	$0.238222\pi$
$830$	−2.76267e8			−0.483165
$831$	4.76428e8			0.830223
$832$	−3.53024e8			−0.612963
$833$		−	1.77990e8i		−	0.307937i
$834$	2.55397e8			0.440269
$835$		−	2.09217e8i		−	0.359366i
$836$	3.34642e8			0.572745
$837$	−1.03437e8			−0.176400
$838$		−	1.89786e8i		−	0.322502i
$839$			3.90151e8i			0.660613i	0.943874	+	0.330307i	$0.107152\pi$
$839$			3.90151e8i			0.660613i	−0.943874	+	0.330307i	$0.892848\pi$
$840$	1.22124e8			0.206046
$841$	−5.64310e8			−0.948701
$842$			5.49693e8i			0.920839i
$843$			1.88589e8i			0.314800i
$844$	2.39908e7			0.0399040
$845$			1.91224e8i			0.316937i
$846$	9.70939e7			0.160354
$847$			4.23808e7i			0.0697459i
$848$		−	1.14497e8i		−	0.187762i
$849$			5.38012e8i			0.879162i
$850$		−	2.15440e8i		−	0.350808i
$851$	4.57198e8	+	1.58303e8i	0.741850	+	0.256862i
$852$	−1.60259e8			−0.259121
$853$	−1.58229e8			−0.254940			−0.127470	−	0.991842i	$-0.540686\pi$
$853$	−1.58229e8			−0.254940			−0.127470	+	0.991842i	$0.540686\pi$
$854$	4.95802e8			0.796039
$855$	−1.49054e8			−0.238477
$856$			1.19950e9i			1.91241i
$857$	−3.52002e8			−0.559245			−0.279623	−	0.960110i	$-0.590209\pi$
$857$	−3.52002e8			−0.559245			−0.279623	+	0.960110i	$0.590209\pi$
$858$			1.78145e8i			0.282041i
$859$	−5.14413e7			−0.0811582			−0.0405791	−	0.999176i	$-0.512920\pi$
$859$	−5.14413e7			−0.0811582			−0.0405791	+	0.999176i	$0.512920\pi$
$860$	−4.37810e7			−0.0688319
$861$			6.72103e7i			0.105300i
$862$			2.59161e8i			0.404620i
$863$	4.96102e8			0.771861			0.385930	−	0.922528i	$-0.373881\pi$
$863$	4.96102e8			0.771861			0.385930	+	0.922528i	$0.373881\pi$
$864$	8.58594e7			0.133121
$865$		−	2.15678e8i		−	0.333240i
$866$		−	6.69290e8i		−	1.03053i
$867$	2.52132e8			0.386876
$868$		−	1.49978e8i		−	0.229334i
$869$	7.99243e8			1.21792
$870$		−	3.29983e7i		−	0.0501111i
$871$		−	3.13958e8i		−	0.475135i
$872$			1.06059e9i			1.59954i
$873$			6.66874e7i			0.100231i
$874$	7.45134e8	+	2.57999e8i	1.11609	+	0.386442i
$875$	−3.88632e8			−0.580116
$876$	−1.81391e8			−0.269838
$877$	4.89169e8			0.725203			0.362602	−	0.931944i	$-0.381889\pi$
$877$	4.89169e8			0.725203			0.362602	+	0.931944i	$0.381889\pi$
$878$	−3.34635e8			−0.494410
$879$			5.20455e8i			0.766332i
$880$	1.71571e8			0.251765
$881$			3.68496e8i			0.538896i	0.963015	+	0.269448i	$0.0868412\pi$
$881$			3.68496e8i			0.538896i	−0.963015	+	0.269448i	$0.913159\pi$
$882$	9.75275e7			0.142142
$883$	7.59012e8			1.10247			0.551235	−	0.834350i	$-0.314157\pi$
$883$	7.59012e8			1.10247			0.551235	+	0.834350i	$0.314157\pi$
$884$		−	8.52634e7i		−	0.123426i
$885$			1.50176e8i			0.216656i
$886$	−9.29603e8			−1.33658
$887$	−2.70519e8			−0.387638			−0.193819	−	0.981037i	$-0.562087\pi$
$887$	−2.70519e8			−0.387638			−0.193819	+	0.981037i	$0.562087\pi$
$888$			3.45178e8i			0.492951i
$889$		−	2.35184e8i		−	0.334736i
$890$	−3.01011e8			−0.426985
$891$		−	8.25203e7i		−	0.116662i
$892$	1.21517e8			0.171216
$893$		−	6.39562e8i		−	0.898107i
$894$			6.59854e7i			0.0923496i
$895$			1.09315e8i			0.152479i
$896$		−	6.94504e7i		−	0.0965497i
$897$	7.97511e7	−	2.30331e8i	0.110499	−	0.319135i
$898$	1.97046e8			0.272107
$899$	−1.50839e8			−0.207603
$900$	−6.85460e7			−0.0940274
$901$	1.58495e8			0.216691
$902$			1.64117e8i			0.223633i
$903$	1.12603e8			0.152928
$904$			1.45151e9i			1.96479i
$905$	−3.22898e8			−0.435631
$906$	−3.82119e7			−0.0513823
$907$		−	1.23189e9i		−	1.65101i	−0.564394	−	0.825505i	$-0.690890\pi$
$907$		−	1.23189e9i		−	1.65101i	0.564394	−	0.825505i	$-0.309110\pi$
$908$		−	1.64995e8i		−	0.220400i
$909$	−2.42863e8			−0.323348
$910$	1.15051e8			0.152674
$911$			1.21327e9i			1.60473i	0.596832	+	0.802366i	$0.296426\pi$
$911$			1.21327e9i			1.60473i	−0.596832	+	0.802366i	$0.703574\pi$
$912$			3.23665e8i			0.426688i
$913$	−1.00748e9			−1.32381
$914$			2.31941e8i			0.303766i
$915$	3.13123e8			0.408744
$916$		−	2.71817e8i		−	0.353663i
$917$			3.47295e8i			0.450392i
$918$		−	6.80182e7i		−	0.0879220i
$919$			6.64269e8i			0.855850i	0.903814	+	0.427925i	$0.140755\pi$
$919$			6.64269e8i			0.855850i	−0.903814	+	0.427925i	$0.859245\pi$
$920$	−3.85566e8	−	1.33501e8i	−0.495148	−	0.171443i
$921$	−8.53902e8			−1.09302
$922$	2.07767e8			0.265084
$923$	−5.61961e8			−0.714663
$924$	1.19650e8			0.151669
$925$		−	4.77110e8i		−	0.602828i
$926$	1.06645e9			1.34310
$927$		−	8.74049e7i		−	0.109723i
$928$	1.25206e8			0.156669
$929$	−9.67171e8			−1.20630			−0.603151	−	0.797627i	$-0.706088\pi$
$929$	−9.67171e8			−1.20630			−0.603151	+	0.797627i	$0.706088\pi$
$930$			1.63121e8i			0.202797i
$931$		−	6.42418e8i		−	0.796102i
$932$	3.15662e8			0.389919
$933$	4.46124e8			0.549301
$934$			2.27847e8i			0.279642i
$935$			2.37501e8i			0.290556i
$936$	1.73896e8			0.212062
$937$			2.93220e8i			0.356430i	0.983992	+	0.178215i	$0.0570323\pi$
$937$			2.93220e8i			0.356430i	−0.983992	+	0.178215i	$0.942968\pi$
$938$	3.63150e8			0.440026
$939$			5.59718e8i			0.676040i
$940$			8.89100e7i			0.107045i
$941$		−	1.21758e9i		−	1.46127i	−0.682769	−	0.730635i	$-0.739224\pi$
$941$		−	1.21758e9i		−	1.46127i	0.682769	−	0.730635i	$-0.260776\pi$
$942$			2.53448e8i			0.303205i
$943$	7.34712e7	−	2.12194e8i	0.0876157	−	0.253045i
$944$	3.26100e8			0.387646
$945$	−5.32938e7			−0.0631512
$946$	2.74959e8			0.324784
$947$	8.42419e8			0.991924			0.495962	−	0.868344i	$-0.334816\pi$
$947$	8.42419e8			0.991924			0.495962	+	0.868344i	$0.334816\pi$
$948$		−	2.09603e8i		−	0.246022i
$949$	−6.36063e8			−0.744220
$950$		−	7.77586e8i		−	0.906938i
$951$	5.02395e8			0.584122
$952$	3.67091e8			0.425464
$953$		−	1.79752e8i		−	0.207680i	−0.994594	−	0.103840i	$-0.966887\pi$
$953$		−	1.79752e8i		−	0.207680i	0.994594	−	0.103840i	$-0.0331129\pi$
$954$			8.68455e7i			0.100023i
$955$	−3.77165e8			−0.433033
$956$	−1.79910e8			−0.205912
$957$		−	1.20337e8i		−	0.137298i
$958$		−	2.55633e8i		−	0.290750i
$959$	2.18079e8			0.247262
$960$		−	2.57885e8i		−	0.291482i
$961$	−1.41858e8			−0.159839
$962$			3.25185e8i			0.365263i
$963$		−	5.23452e8i		−	0.586135i
$964$		−	2.20241e8i		−	0.245848i
$965$		−	7.74477e8i		−	0.861839i
$966$	2.66420e8	+	9.22468e7i	0.295553	+	0.102334i
$967$	1.27323e9			1.40808			0.704039	−	0.710161i	$-0.251378\pi$
$967$	1.27323e9			1.40808			0.704039	+	0.710161i	$0.251378\pi$
$968$	1.01019e8			0.111372
$969$	−4.48039e8			−0.492430
$970$	1.05167e8			0.115229
$971$			3.28170e7i			0.0358461i	0.999839	+	0.0179230i	$0.00570538\pi$
$971$			3.28170e7i			0.0358461i	−0.999839	+	0.0179230i	$0.994295\pi$
$972$	−2.16412e7			−0.0235658
$973$		−	6.01505e8i		−	0.652981i
$974$	−1.02075e9			−1.10470
$975$	−2.40362e8			−0.259330
$976$		−	6.79932e8i		−	0.731335i
$977$		−	7.22268e8i		−	0.774488i	−0.921977	−	0.387244i	$-0.873427\pi$
$977$		−	7.22268e8i		−	0.774488i	0.921977	−	0.387244i	$-0.126573\pi$
$978$	−4.72158e8			−0.504744
$979$	−1.09772e9			−1.16988
$980$			8.93071e7i			0.0948873i
$981$		−	4.62829e8i		−	0.490246i
$982$	−4.71358e8			−0.497756
$983$		−	1.40301e9i		−	1.47706i	−0.674219	−	0.738532i	$-0.735519\pi$
$983$		−	1.40301e9i		−	1.47706i	0.674219	−	0.738532i	$-0.264481\pi$
$984$	1.60203e8			0.168146
$985$		−	5.74138e8i		−	0.600770i
$986$		−	9.91889e7i		−	0.103474i
$987$		−	2.28673e8i		−	0.237828i
$988$		−	3.07740e8i		−	0.319090i
$989$	−3.55506e8	−	1.23092e8i	−0.367500	−	0.127245i
$990$	−1.30135e8			−0.134119
$991$	1.77438e8			0.182316			0.0911581	−	0.995836i	$-0.470943\pi$
$991$	1.77438e8			0.182316			0.0911581	+	0.995836i	$0.470943\pi$
$992$	−6.18934e8			−0.634030
$993$	5.22823e8			0.533958
$994$		−	6.50012e8i		−	0.661854i
$995$	1.05993e8			0.107599
$996$			2.64215e8i			0.267411i
$997$	−1.51194e9			−1.52563			−0.762813	−	0.646619i	$-0.776182\pi$
$997$	−1.51194e9			−1.52563			−0.762813	+	0.646619i	$0.776182\pi$
$998$	−1.25469e9			−1.26224
$999$		−	1.50632e8i		−	0.151085i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.d.a.22.17	✓	24
3.2	odd	2		207.7.d.e.91.8		24
23.22	odd	2	inner	69.7.d.a.22.18	yes	24
69.68	even	2		207.7.d.e.91.7		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.d.a.22.17	✓	24	1.1	even	1	trivial
69.7.d.a.22.18	yes	24	23.22	odd	2	inner
207.7.d.e.91.7		24	69.68	even	2
207.7.d.e.91.8		24	3.2	odd	2

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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	69.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.36312 q^{2} +15.5885 q^{3} -23.5107 q^{4} -60.2242i q^{5} +99.1913 q^{6} -233.613i q^{7} -556.841 q^{8} +243.000 q^{9} +O(q^{10})\)	\(q+6.36312 q^{2} +15.5885 q^{3} -23.5107 q^{4} -60.2242i q^{5} +99.1913 q^{6} -233.613i q^{7} -556.841 q^{8} +243.000 q^{9} -383.214i q^{10} -1397.49i q^{11} -366.495 q^{12} -1285.15 q^{13} -1486.51i q^{14} -938.802i q^{15} -2038.57 q^{16} -2821.93i q^{17} +1546.24 q^{18} -10185.1i q^{19} +1415.91i q^{20} -3641.66i q^{21} -8892.40i q^{22} +(-3980.90 + 11497.3i) q^{23} -8680.29 q^{24} +11998.1 q^{25} -8177.54 q^{26} +3788.00 q^{27} +5492.39i q^{28} +5523.92 q^{29} -5973.71i q^{30} -27306.5 q^{31} +22666.2 q^{32} -21784.7i q^{33} -17956.3i q^{34} -14069.1 q^{35} -5713.09 q^{36} -39765.7i q^{37} -64809.3i q^{38} -20033.4 q^{39} +33535.3i q^{40} -18455.9 q^{41} -23172.4i q^{42} +30920.7i q^{43} +32855.9i q^{44} -14634.5i q^{45} +(-25330.9 + 73158.9i) q^{46} +62793.6 q^{47} -31778.1 q^{48} +63074.0 q^{49} +76345.1 q^{50} -43989.5i q^{51} +30214.6 q^{52} +56165.6i q^{53} +24103.5 q^{54} -84162.6 q^{55} +130085. i q^{56} -158771. i q^{57} +35149.4 q^{58} -159966. q^{59} +22071.8i q^{60} +333535. i q^{61} -173755. q^{62} -56767.9i q^{63} +274696. q^{64} +77396.8i q^{65} -138619. i q^{66} +244298. i q^{67} +66345.4i q^{68} +(-62056.1 + 179225. i) q^{69} -89523.7 q^{70} +437275. q^{71} -135312. q^{72} +494935. q^{73} -253034. i q^{74} +187031. q^{75} +239459. i q^{76} -326471. q^{77} -127475. q^{78} +571913. i q^{79} +122771. i q^{80} +59049.0 q^{81} -117437. q^{82} -720923. i q^{83} +85617.9i q^{84} -169948. q^{85} +196752. i q^{86} +86109.4 q^{87} +778179. i q^{88} -785492. i q^{89} -93120.9i q^{90} +300226. i q^{91} +(93593.5 - 270310. i) q^{92} -425666. q^{93} +399563. q^{94} -613392. q^{95} +353331. q^{96} +274434. i q^{97} +401348. q^{98} -339590. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) \) 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9} + 384 q^{13} + 29544 q^{16} - 4860 q^{18} + 29336 q^{23} - 39204 q^{24} - 61272 q^{25} + 10088 q^{26} + 64672 q^{29} + 9696 q^{31} - 319620 q^{32} - 225744 q^{35} + 198288 q^{36} - 11664 q^{39} + 135280 q^{41} + 233232 q^{46} - 74336 q^{47} + 552096 q^{48} - 722136 q^{49} + 619324 q^{50} + 1059720 q^{52} - 78732 q^{54} - 1019328 q^{55} - 694344 q^{58} + 1057648 q^{59} - 488776 q^{62} - 273888 q^{64} - 23328 q^{69} + 2785512 q^{70} - 255392 q^{71} - 228420 q^{72} - 322560 q^{73} - 365472 q^{75} - 1002960 q^{77} - 171072 q^{78} + 1417176 q^{81} - 5732712 q^{82} - 2704704 q^{85} + 611712 q^{87} - 1611444 q^{92} + 2484432 q^{93} - 147720 q^{94} - 1672656 q^{95} - 1818612 q^{96} + 9104212 q^{98}+O(q^{100}) \) 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9 + 384 * q^13 + 29544 * q^16 - 4860 * q^18 + 29336 * q^23 - 39204 * q^24 - 61272 * q^25 + 10088 * q^26 + 64672 * q^29 + 9696 * q^31 - 319620 * q^32 - 225744 * q^35 + 198288 * q^36 - 11664 * q^39 + 135280 * q^41 + 233232 * q^46 - 74336 * q^47 + 552096 * q^48 - 722136 * q^49 + 619324 * q^50 + 1059720 * q^52 - 78732 * q^54 - 1019328 * q^55 - 694344 * q^58 + 1057648 * q^59 - 488776 * q^62 - 273888 * q^64 - 23328 * q^69 + 2785512 * q^70 - 255392 * q^71 - 228420 * q^72 - 322560 * q^73 - 365472 * q^75 - 1002960 * q^77 - 171072 * q^78 + 1417176 * q^81 - 5732712 * q^82 - 2704704 * q^85 + 611712 * q^87 - 1611444 * q^92 + 2484432 * q^93 - 147720 * q^94 - 1672656 * q^95 - 1818612 * q^96 + 9104212 * q^98

Introduction

L-functions

Modular forms

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Database

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