LMFDB - Embedded newform 256.8.b.i.129.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,8,Mod(129,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	256.b (of order $2$, degree $1$, not 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:	$79.9705665239$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 16 $ x^4 - 7*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$1.93649 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.8.b.i.129.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-61.9677i q^{3} -70.0000i q^{5} +1115.42 q^{7} -1653.00 q^{9} +O(q^{10})$	$q-61.9677i q^{3} -70.0000i q^{5} +1115.42 q^{7} -1653.00 q^{9} -7250.22i q^{11} +13758.0i q^{13} -4337.74 q^{15} +16994.0 q^{17} -34020.3i q^{19} -69120.0i q^{21} -32347.2 q^{23} +73225.0 q^{25} -33090.8i q^{27} +34190.0i q^{29} +120465. q^{31} -449280. q^{33} -78079.3i q^{35} -35206.0i q^{37} +852552. q^{39} +484550. q^{41} -672040. i q^{43} +115710. i q^{45} +1.20688e6 q^{47} +420617. q^{49} -1.05308e6i q^{51} -851702. i q^{53} -507516. q^{55} -2.10816e6 q^{57} +695464. i q^{59} +71630.0i q^{61} -1.84379e6 q^{63} +963060. q^{65} +307298. i q^{67} +2.00448e6i q^{69} -757370. q^{71} -3.91204e6 q^{73} -4.53759e6i q^{75} -8.08704e6i q^{77} -314548. q^{79} -5.66567e6 q^{81} +1.53649e6i q^{83} -1.18958e6i q^{85} +2.11868e6 q^{87} +2.51063e6 q^{89} +1.53459e7i q^{91} -7.46496e6i q^{93} -2.38142e6 q^{95} -50094.0 q^{97} +1.19846e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6612 q^{9}+O(q^{10}) $ 4 * q - 6612 * q^9	$ 4 q - 6612 q^{9} + 67976 q^{17} + 292900 q^{25} - 1797120 q^{33} + 1938200 q^{41} + 1682468 q^{49} - 8432640 q^{57} + 3852240 q^{65} - 15648168 q^{73} - 22662684 q^{81} + 10042520 q^{89} - 200376 q^{97}+O(q^{100}) $ 4 * q - 6612 * q^9 + 67976 * q^17 + 292900 * q^25 - 1797120 * q^33 + 1938200 * q^41 + 1682468 * q^49 - 8432640 * q^57 + 3852240 * q^65 - 15648168 * q^73 - 22662684 * q^81 + 10042520 * q^89 - 200376 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 61.9677i	− 1.32508i	−0.749028	−	0.662539i	$-0.769479\pi$
$3$	− 61.9677i	− 1.32508i	0.749028	−	0.662539i	$-0.230521\pi$
$4$	0	0
$5$	− 70.0000i	− 0.250440i	−0.992129	−	0.125220i	$-0.960036\pi$
$5$	− 70.0000i	− 0.250440i	0.992129	−	0.125220i	$-0.0399636\pi$
$6$	0	0
$7$	1115.42	1.22912	0.614561	−	0.788869i	$-0.289333\pi$
$7$	1115.42	1.22912	0.614561	+	0.788869i	$0.289333\pi$
$8$	0	0
$9$	−1653.00	−0.755830
$10$	0	0
$11$	− 7250.22i	− 1.64239i	−0.570646	−	0.821196i	$-0.693307\pi$
$11$	− 7250.22i	− 1.64239i	0.570646	−	0.821196i	$-0.306693\pi$
$12$	0	0
$13$	13758.0i	1.73682i	0.495851	+	0.868408i	$0.334856\pi$
$13$	13758.0i	1.73682i	−0.495851	+	0.868408i	$0.665144\pi$
$14$	0	0
$15$	−4337.74	−0.331852
$16$	0	0
$17$	16994.0	0.838927	0.419464	−	0.907772i	$-0.362218\pi$
$17$	16994.0	0.838927	0.419464	+	0.907772i	$0.362218\pi$
$18$	0	0
$19$	− 34020.3i	− 1.13789i	−0.822376	−	0.568945i	$-0.807352\pi$
$19$	− 34020.3i	− 1.13789i	0.822376	−	0.568945i	$-0.192648\pi$
$20$	0	0
$21$	− 69120.0i	− 1.62868i
$22$	0	0
$23$	−32347.2	−0.554356	−0.277178	−	0.960819i	$-0.589399\pi$
$23$	−32347.2	−0.554356	−0.277178	+	0.960819i	$0.589399\pi$
$24$	0	0
$25$	73225.0	0.937280
$26$	0	0
$27$	− 33090.8i	− 0.323544i
$28$	0	0
$29$	34190.0i	0.260319i	0.991493	+	0.130160i	$0.0415490\pi$
$29$	34190.0i	0.260319i	−0.991493	+	0.130160i	$0.958451\pi$
$30$	0	0
$31$	120465.	0.726266	0.363133	−	0.931737i	$-0.381707\pi$
$31$	120465.	0.726266	0.363133	+	0.931737i	$0.381707\pi$
$32$	0	0
$33$	−449280.	−2.17630
$34$	0	0
$35$	− 78079.3i	− 0.307821i
$36$	0	0
$37$	− 35206.0i	− 0.114264i	−0.998367	−	0.0571322i	$-0.981804\pi$
$37$	− 35206.0i	− 0.114264i	0.998367	−	0.0571322i	$-0.0181956\pi$
$38$	0	0
$39$	852552.	2.30141
$40$	0	0
$41$	484550.	1.09798	0.548991	−	0.835828i	$-0.315012\pi$
$41$	484550.	1.09798	0.548991	+	0.835828i	$0.315012\pi$
$42$	0	0
$43$	− 672040.i	− 1.28901i	−0.764601	−	0.644504i	$-0.777064\pi$
$43$	− 672040.i	− 1.28901i	0.764601	−	0.644504i	$-0.222936\pi$
$44$	0	0
$45$	115710.i	0.189290i
$46$	0	0
$47$	1.20688e6	1.69560	0.847799	−	0.530318i	$-0.177927\pi$
$47$	1.20688e6	1.69560	0.847799	+	0.530318i	$0.177927\pi$
$48$	0	0
$49$	420617.	0.510741
$50$	0	0
$51$	− 1.05308e6i	− 1.11164i
$52$	0	0
$53$	− 851702.i	− 0.785818i	−0.919577	−	0.392909i	$-0.871469\pi$
$53$	− 851702.i	− 0.785818i	0.919577	−	0.392909i	$-0.128531\pi$
$54$	0	0
$55$	−507516.	−0.411320
$56$	0	0
$57$	−2.10816e6	−1.50779
$58$	0	0
$59$	695464.i	0.440852i	0.975404	+	0.220426i	$0.0707447\pi$
$59$	695464.i	0.440852i	−0.975404	+	0.220426i	$0.929255\pi$
$60$	0	0
$61$	71630.0i	0.0404055i	0.999796	+	0.0202028i	$0.00643117\pi$
$61$	71630.0i	0.0404055i	−0.999796	+	0.0202028i	$0.993569\pi$
$62$	0	0
$63$	−1.84379e6	−0.929007
$64$	0	0
$65$	963060.	0.434967
$66$	0	0
$67$	307298.i	0.124824i	0.998050	+	0.0624120i	$0.0198793\pi$
$67$	307298.i	0.124824i	−0.998050	+	0.0624120i	$0.980121\pi$
$68$	0	0
$69$	2.00448e6i	0.734564i
$70$	0	0
$71$	−757370.	−0.251133	−0.125566	−	0.992085i	$-0.540075\pi$
$71$	−757370.	−0.251133	−0.125566	+	0.992085i	$0.540075\pi$
$72$	0	0
$73$	−3.91204e6	−1.17699	−0.588496	−	0.808500i	$-0.700280\pi$
$73$	−3.91204e6	−1.17699	−0.588496	+	0.808500i	$0.700280\pi$
$74$	0	0
$75$	− 4.53759e6i	− 1.24197i
$76$	0	0
$77$	− 8.08704e6i	− 2.01870i
$78$	0	0
$79$	−314548.	−0.0717782	−0.0358891	−	0.999356i	$-0.511426\pi$
$79$	−314548.	−0.0717782	−0.0358891	+	0.999356i	$0.511426\pi$
$80$	0	0
$81$	−5.66567e6	−1.18455
$82$	0	0
$83$	1.53649e6i	0.294955i	0.989065	+	0.147478i	$0.0471155\pi$
$83$	1.53649e6i	0.294955i	−0.989065	+	0.147478i	$0.952885\pi$
$84$	0	0
$85$	− 1.18958e6i	− 0.210101i
$86$	0	0
$87$	2.11868e6	0.344943
$88$	0	0
$89$	2.51063e6	0.377501	0.188750	−	0.982025i	$-0.439556\pi$
$89$	2.51063e6	0.377501	0.188750	+	0.982025i	$0.439556\pi$
$90$	0	0
$91$	1.53459e7i	2.13476i
$92$	0	0
$93$	− 7.46496e6i	− 0.962359i
$94$	0	0
$95$	−2.38142e6	−0.284973
$96$	0	0
$97$	−50094.0	−0.00557294	−0.00278647	−	0.999996i	$-0.500887\pi$
$97$	−50094.0	−0.00557294	−0.00278647	+	0.999996i	$0.500887\pi$
$98$	0	0
$99$	1.19846e7i	1.24137i
$100$	0	0
$101$	− 1.50354e7i	− 1.45208i	−0.687653	−	0.726039i	$-0.741359\pi$
$101$	− 1.50354e7i	− 1.45208i	0.687653	−	0.726039i	$-0.258641\pi$
$102$	0	0
$103$	−1.99917e7	−1.80268	−0.901340	−	0.433113i	$-0.857415\pi$
$103$	−1.99917e7	−1.80268	−0.901340	+	0.433113i	$0.857415\pi$
$104$	0	0
$105$	−4.83840e6	−0.407886
$106$	0	0
$107$	8.75102e6i	0.690582i	0.938496	+	0.345291i	$0.112220\pi$
$107$	8.75102e6i	0.690582i	−0.938496	+	0.345291i	$0.887780\pi$
$108$	0	0
$109$	1.78070e7i	1.31704i	0.752565	+	0.658518i	$0.228816\pi$
$109$	1.78070e7i	1.31704i	−0.752565	+	0.658518i	$0.771184\pi$
$110$	0	0
$111$	−2.18164e6	−0.151409
$112$	0	0
$113$	−1.88499e7	−1.22895	−0.614475	−	0.788936i	$-0.710632\pi$
$113$	−1.88499e7	−1.22895	−0.614475	+	0.788936i	$0.710632\pi$
$114$	0	0
$115$	2.26430e6i	0.138833i
$116$	0	0
$117$	− 2.27420e7i	− 1.31274i
$118$	0	0
$119$	1.89554e7	1.03114
$120$	0	0
$121$	−3.30786e7	−1.69745
$122$	0	0
$123$	− 3.00265e7i	− 1.45491i
$124$	0	0
$125$	− 1.05945e7i	− 0.485172i
$126$	0	0
$127$	−3.31235e7	−1.43490	−0.717452	−	0.696608i	$-0.754692\pi$
$127$	−3.31235e7	−1.43490	−0.717452	+	0.696608i	$0.754692\pi$
$128$	0	0
$129$	−4.16448e7	−1.70804
$130$	0	0
$131$	− 2.63468e7i	− 1.02395i	−0.859001	−	0.511974i	$-0.828915\pi$
$131$	− 2.63468e7i	− 1.02395i	0.859001	−	0.511974i	$-0.171085\pi$
$132$	0	0
$133$	− 3.79469e7i	− 1.39861i
$134$	0	0
$135$	−2.31635e6	−0.0810283
$136$	0	0
$137$	1.88745e7	0.627125	0.313563	−	0.949568i	$-0.398477\pi$
$137$	1.88745e7	0.627125	0.313563	+	0.949568i	$0.398477\pi$
$138$	0	0
$139$	− 1.08848e7i	− 0.343771i	−0.985117	−	0.171886i	$-0.945014\pi$
$139$	− 1.08848e7i	− 0.343771i	0.985117	−	0.171886i	$-0.0549859\pi$
$140$	0	0
$141$	− 7.47878e7i	− 2.24680i
$142$	0	0
$143$	9.97486e7	2.85253
$144$	0	0
$145$	2.39330e6	0.0651942
$146$	0	0
$147$	− 2.60647e7i	− 0.676771i
$148$	0	0
$149$	3.44462e7i	0.853078i	0.904469	+	0.426539i	$0.140267\pi$
$149$	3.44462e7i	0.853078i	−0.904469	+	0.426539i	$0.859733\pi$
$150$	0	0
$151$	1.30493e7	0.308438	0.154219	−	0.988037i	$-0.450714\pi$
$151$	1.30493e7	0.308438	0.154219	+	0.988037i	$0.450714\pi$
$152$	0	0
$153$	−2.80911e7	−0.634086
$154$	0	0
$155$	− 8.43257e6i	− 0.181886i
$156$	0	0
$157$	− 2.04587e7i	− 0.421919i	−0.977495	−	0.210960i	$-0.932341\pi$
$157$	− 2.04587e7i	− 0.421919i	0.977495	−	0.210960i	$-0.0676589\pi$
$158$	0	0
$159$	−5.27780e7	−1.04127
$160$	0	0
$161$	−3.60806e7	−0.681371
$162$	0	0
$163$	− 1.07978e7i	− 0.195290i	−0.995221	−	0.0976448i	$-0.968869\pi$
$163$	− 1.07978e7i	− 0.195290i	0.995221	−	0.0976448i	$-0.0311309\pi$
$164$	0	0
$165$	3.14496e7i	0.545031i
$166$	0	0
$167$	5.58122e7	0.927303	0.463652	−	0.886018i	$-0.346539\pi$
$167$	5.58122e7	0.927303	0.463652	+	0.886018i	$0.346539\pi$
$168$	0	0
$169$	−1.26534e8	−2.01653
$170$	0	0
$171$	5.62355e7i	0.860051i
$172$	0	0
$173$	− 8.94250e6i	− 0.131310i	−0.997842	−	0.0656550i	$-0.979086\pi$
$173$	− 8.94250e6i	− 0.131310i	0.997842	−	0.0656550i	$-0.0209137\pi$
$174$	0	0
$175$	8.16766e7	1.15203
$176$	0	0
$177$	4.30963e7	0.584163
$178$	0	0
$179$	9.39334e7i	1.22415i	0.790800	+	0.612074i	$0.209665\pi$
$179$	9.39334e7i	1.22415i	−0.790800	+	0.612074i	$0.790335\pi$
$180$	0	0
$181$	− 9.62269e7i	− 1.20621i	−0.797663	−	0.603103i	$-0.793931\pi$
$181$	− 9.62269e7i	− 1.20621i	0.797663	−	0.603103i	$-0.206069\pi$
$182$	0	0
$183$	4.43875e6	0.0535404
$184$	0	0
$185$	−2.46442e6	−0.0286163
$186$	0	0
$187$	− 1.23210e8i	− 1.37785i
$188$	0	0
$189$	− 3.69101e7i	− 0.397675i
$190$	0	0
$191$	−1.06875e8	−1.10984	−0.554919	−	0.831905i	$-0.687251\pi$
$191$	−1.06875e8	−1.10984	−0.554919	+	0.831905i	$0.687251\pi$
$192$	0	0
$193$	1.34693e8	1.34863	0.674317	−	0.738442i	$-0.264438\pi$
$193$	1.34693e8	1.34863	0.674317	+	0.738442i	$0.264438\pi$
$194$	0	0
$195$	− 5.96786e7i	− 0.576365i
$196$	0	0
$197$	− 9.36611e7i	− 0.872826i	−0.899747	−	0.436413i	$-0.856249\pi$
$197$	− 9.36611e7i	− 0.872826i	0.899747	−	0.436413i	$-0.143751\pi$
$198$	0	0
$199$	−5.37621e7	−0.483605	−0.241802	−	0.970326i	$-0.577739\pi$
$199$	−5.37621e7	−0.483605	−0.241802	+	0.970326i	$0.577739\pi$
$200$	0	0
$201$	1.90426e7	0.165401
$202$	0	0
$203$	3.81362e7i	0.319964i
$204$	0	0
$205$	− 3.39185e7i	− 0.274978i
$206$	0	0
$207$	5.34699e7	0.418999
$208$	0	0
$209$	−2.46655e8	−1.86886
$210$	0	0
$211$	− 1.26710e8i	− 0.928586i	−0.885682	−	0.464293i	$-0.846308\pi$
$211$	− 1.26710e8i	− 0.928586i	0.885682	−	0.464293i	$-0.153692\pi$
$212$	0	0
$213$	4.69325e7i	0.332771i
$214$	0	0
$215$	−4.70428e7	−0.322819
$216$	0	0
$217$	1.34369e8	0.892670
$218$	0	0
$219$	2.42420e8i	1.55961i
$220$	0	0
$221$	2.33803e8i	1.45706i
$222$	0	0
$223$	−2.00325e8	−1.20967	−0.604836	−	0.796350i	$-0.706761\pi$
$223$	−2.00325e8	−1.20967	−0.604836	+	0.796350i	$0.706761\pi$
$224$	0	0
$225$	−1.21041e8	−0.708424
$226$	0	0
$227$	2.46632e8i	1.39946i	0.714410	+	0.699728i	$0.246695\pi$
$227$	2.46632e8i	1.39946i	−0.714410	+	0.699728i	$0.753305\pi$
$228$	0	0
$229$	2.24690e8i	1.23640i	0.786020	+	0.618201i	$0.212138\pi$
$229$	2.24690e8i	1.23640i	−0.786020	+	0.618201i	$0.787862\pi$
$230$	0	0
$231$	−5.01136e8	−2.67494
$232$	0	0
$233$	2.49045e8	1.28983	0.644914	−	0.764256i	$-0.276893\pi$
$233$	2.49045e8	1.28983	0.644914	+	0.764256i	$0.276893\pi$
$234$	0	0
$235$	− 8.44819e7i	− 0.424645i
$236$	0	0
$237$	1.94918e7i	0.0951116i
$238$	0	0
$239$	−1.04740e8	−0.496273	−0.248136	−	0.968725i	$-0.579818\pi$
$239$	−1.04740e8	−0.496273	−0.248136	+	0.968725i	$0.579818\pi$
$240$	0	0
$241$	8.04571e7	0.370258	0.185129	−	0.982714i	$-0.440730\pi$
$241$	8.04571e7	0.370258	0.185129	+	0.982714i	$0.440730\pi$
$242$	0	0
$243$	2.78719e8i	1.24608i
$244$	0	0
$245$	− 2.94432e7i	− 0.127910i
$246$	0	0
$247$	4.68051e8	1.97630
$248$	0	0
$249$	9.52128e7	0.390839
$250$	0	0
$251$	1.60093e8i	0.639021i	0.947583	+	0.319511i	$0.103518\pi$
$251$	1.60093e8i	0.639021i	−0.947583	+	0.319511i	$0.896482\pi$
$252$	0	0
$253$	2.34524e8i	0.910470i
$254$	0	0
$255$	−7.37156e7	−0.278400
$256$	0	0
$257$	1.76666e8	0.649213	0.324607	−	0.945849i	$-0.394768\pi$
$257$	1.76666e8	0.649213	0.324607	+	0.945849i	$0.394768\pi$
$258$	0	0
$259$	− 3.92694e7i	− 0.140445i
$260$	0	0
$261$	− 5.65161e7i	− 0.196757i
$262$	0	0
$263$	3.49243e8	1.18381	0.591906	−	0.806007i	$-0.298376\pi$
$263$	3.49243e8	1.18381	0.591906	+	0.806007i	$0.298376\pi$
$264$	0	0
$265$	−5.96191e7	−0.196800
$266$	0	0
$267$	− 1.55578e8i	− 0.500218i
$268$	0	0
$269$	1.91355e8i	0.599387i	0.954036	+	0.299693i	$0.0968844\pi$
$269$	1.91355e8i	0.599387i	−0.954036	+	0.299693i	$0.903116\pi$
$270$	0	0
$271$	5.62669e8	1.71736	0.858678	−	0.512516i	$-0.171286\pi$
$271$	5.62669e8	1.71736	0.858678	+	0.512516i	$0.171286\pi$
$272$	0	0
$273$	9.50953e8	2.82872
$274$	0	0
$275$	− 5.30898e8i	− 1.53938i
$276$	0	0
$277$	5.25734e8i	1.48623i	0.669162	+	0.743116i	$0.266653\pi$
$277$	5.25734e8i	1.48623i	−0.669162	+	0.743116i	$0.733347\pi$
$278$	0	0
$279$	−1.99129e8	−0.548934
$280$	0	0
$281$	4.24101e7	0.114024	0.0570122	−	0.998373i	$-0.481843\pi$
$281$	4.24101e7	0.114024	0.0570122	+	0.998373i	$0.481843\pi$
$282$	0	0
$283$	− 9.44052e7i	− 0.247596i	−0.992307	−	0.123798i	$-0.960493\pi$
$283$	− 9.44052e7i	− 0.247596i	0.992307	−	0.123798i	$-0.0395075\pi$
$284$	0	0
$285$	1.47571e8i	0.377611i
$286$	0	0
$287$	5.40476e8	1.34955
$288$	0	0
$289$	−1.21543e8	−0.296201
$290$	0	0
$291$	3.10421e6i	0.00738458i
$292$	0	0
$293$	− 4.01296e8i	− 0.932027i	−0.884778	−	0.466014i	$-0.845690\pi$
$293$	− 4.01296e8i	− 0.932027i	0.884778	−	0.466014i	$-0.154310\pi$
$294$	0	0
$295$	4.86825e7	0.110407
$296$	0	0
$297$	−2.39916e8	−0.531387
$298$	0	0
$299$	− 4.45032e8i	− 0.962814i
$300$	0	0
$301$	− 7.49606e8i	− 1.58435i
$302$	0	0
$303$	−9.31710e8	−1.92412
$304$	0	0
$305$	5.01410e6	0.0101191
$306$	0	0
$307$	− 2.71076e8i	− 0.534697i	−0.963600	−	0.267348i	$-0.913853\pi$
$307$	− 2.71076e8i	− 0.534697i	0.963600	−	0.267348i	$-0.0861474\pi$
$308$	0	0
$309$	1.23884e9i	2.38869i
$310$	0	0
$311$	−4.12911e8	−0.778387	−0.389193	−	0.921156i	$-0.627246\pi$
$311$	−4.12911e8	−0.778387	−0.389193	+	0.921156i	$0.627246\pi$
$312$	0	0
$313$	1.81101e8	0.333823	0.166911	−	0.985972i	$-0.446621\pi$
$313$	1.81101e8	0.333823	0.166911	+	0.985972i	$0.446621\pi$
$314$	0	0
$315$	1.29065e8i	0.232660i
$316$	0	0
$317$	7.48850e8i	1.32034i	0.751114	+	0.660172i	$0.229517\pi$
$317$	7.48850e8i	1.32034i	−0.751114	+	0.660172i	$0.770483\pi$
$318$	0	0
$319$	2.47885e8	0.427546
$320$	0	0
$321$	5.42281e8	0.915075
$322$	0	0
$323$	− 5.78141e8i	− 0.954607i
$324$	0	0
$325$	1.00743e9i	1.62788i
$326$	0	0
$327$	1.10346e9	1.74517
$328$	0	0
$329$	1.34618e9	2.08410
$330$	0	0
$331$	− 1.59530e8i	− 0.241793i	−0.992665	−	0.120897i	$-0.961423\pi$
$331$	− 1.59530e8i	− 0.241793i	0.992665	−	0.120897i	$-0.0385770\pi$
$332$	0	0
$333$	5.81955e7i	0.0863644i
$334$	0	0
$335$	2.15109e7	0.0312609
$336$	0	0
$337$	−5.50578e8	−0.783636	−0.391818	−	0.920043i	$-0.628154\pi$
$337$	−5.50578e8	−0.783636	−0.391818	+	0.920043i	$0.628154\pi$
$338$	0	0
$339$	1.16809e9i	1.62846i
$340$	0	0
$341$	− 8.73400e8i	− 1.19281i
$342$	0	0
$343$	−4.49431e8	−0.601359
$344$	0	0
$345$	1.40314e8	0.183964
$346$	0	0
$347$	− 1.20856e9i	− 1.55279i	−0.630244	−	0.776397i	$-0.717045\pi$
$347$	− 1.20856e9i	− 1.55279i	0.630244	−	0.776397i	$-0.282955\pi$
$348$	0	0
$349$	1.10448e8i	0.139081i	0.997579	+	0.0695404i	$0.0221533\pi$
$349$	1.10448e8i	0.139081i	−0.997579	+	0.0695404i	$0.977847\pi$
$350$	0	0
$351$	4.55263e8	0.561937
$352$	0	0
$353$	6.42950e8	0.777975	0.388988	−	0.921243i	$-0.372825\pi$
$353$	6.42950e8	0.777975	0.388988	+	0.921243i	$0.372825\pi$
$354$	0	0
$355$	5.30159e7i	0.0628936i
$356$	0	0
$357$	− 1.17463e9i	− 1.36635i
$358$	0	0
$359$	1.97988e8	0.225844	0.112922	−	0.993604i	$-0.463979\pi$
$359$	1.97988e8	0.225844	0.112922	+	0.993604i	$0.463979\pi$
$360$	0	0
$361$	−2.63508e8	−0.294794
$362$	0	0
$363$	2.04981e9i	2.24926i
$364$	0	0
$365$	2.73843e8i	0.294765i
$366$	0	0
$367$	−2.68287e8	−0.283315	−0.141657	−	0.989916i	$-0.545243\pi$
$367$	−2.68287e8	−0.283315	−0.141657	+	0.989916i	$0.545243\pi$
$368$	0	0
$369$	−8.00961e8	−0.829887
$370$	0	0
$371$	− 9.50005e8i	− 0.965866i
$372$	0	0
$373$	− 4.72832e8i	− 0.471765i	−0.971782	−	0.235883i	$-0.924202\pi$
$373$	− 4.72832e8i	− 0.471765i	0.971782	−	0.235883i	$-0.0757981\pi$
$374$	0	0
$375$	−6.56517e8	−0.642890
$376$	0	0
$377$	−4.70386e8	−0.452126
$378$	0	0
$379$	1.27673e9i	1.20465i	0.798251	+	0.602324i	$0.205759\pi$
$379$	1.27673e9i	1.20465i	−0.798251	+	0.602324i	$0.794241\pi$
$380$	0	0
$381$	2.05259e9i	1.90136i
$382$	0	0
$383$	7.29377e8	0.663371	0.331685	−	0.943390i	$-0.392383\pi$
$383$	7.29377e8	0.663371	0.331685	+	0.943390i	$0.392383\pi$
$384$	0	0
$385$	−5.66093e8	−0.505563
$386$	0	0
$387$	1.11088e9i	0.974271i
$388$	0	0
$389$	− 5.29163e8i	− 0.455791i	−0.973686	−	0.227896i	$-0.926815\pi$
$389$	− 5.29163e8i	− 0.455791i	0.973686	−	0.227896i	$-0.0731845\pi$
$390$	0	0
$391$	−5.49708e8	−0.465064
$392$	0	0
$393$	−1.63265e9	−1.35681
$394$	0	0
$395$	2.20184e7i	0.0179761i
$396$	0	0
$397$	− 1.59191e9i	− 1.27688i	−0.769670	−	0.638442i	$-0.779579\pi$
$397$	− 1.59191e9i	− 1.27688i	0.769670	−	0.638442i	$-0.220421\pi$
$398$	0	0
$399$	−2.35148e9	−1.85326
$400$	0	0
$401$	−1.03748e9	−0.803480	−0.401740	−	0.915754i	$-0.631594\pi$
$401$	−1.03748e9	−0.803480	−0.401740	+	0.915754i	$0.631594\pi$
$402$	0	0
$403$	1.65736e9i	1.26139i
$404$	0	0
$405$	3.96597e8i	0.296659i
$406$	0	0
$407$	−2.55251e8	−0.187667
$408$	0	0
$409$	5.08439e8	0.367458	0.183729	−	0.982977i	$-0.441183\pi$
$409$	5.08439e8	0.367458	0.183729	+	0.982977i	$0.441183\pi$
$410$	0	0
$411$	− 1.16961e9i	− 0.830989i
$412$	0	0
$413$	7.75734e8i	0.541861i
$414$	0	0
$415$	1.07554e8	0.0738685
$416$	0	0
$417$	−6.74508e8	−0.455523
$418$	0	0
$419$	2.96853e8i	0.197148i	0.995130	+	0.0985739i	$0.0314281\pi$
$419$	2.96853e8i	0.197148i	−0.995130	+	0.0985739i	$0.968572\pi$
$420$	0	0
$421$	− 8.73091e8i	− 0.570259i	−0.958489	−	0.285130i	$-0.907963\pi$
$421$	− 8.73091e8i	− 0.570259i	0.958489	−	0.285130i	$-0.0920366\pi$
$422$	0	0
$423$	−1.99498e9	−1.28158
$424$	0	0
$425$	1.24439e9	0.786310
$426$	0	0
$427$	7.98975e7i	0.0496633i
$428$	0	0
$429$	− 6.18119e9i	− 3.77983i
$430$	0	0
$431$	−2.90031e9	−1.74491	−0.872457	−	0.488691i	$-0.837475\pi$
$431$	−2.90031e9	−1.74491	−0.872457	+	0.488691i	$0.837475\pi$
$432$	0	0
$433$	2.46197e9	1.45739	0.728695	−	0.684838i	$-0.240127\pi$
$433$	2.46197e9	1.45739	0.728695	+	0.684838i	$0.240127\pi$
$434$	0	0
$435$	− 1.48307e8i	− 0.0863874i
$436$	0	0
$437$	1.10046e9i	0.630796i
$438$	0	0
$439$	1.88969e9	1.06602	0.533009	−	0.846110i	$-0.321061\pi$
$439$	1.88969e9	1.06602	0.533009	+	0.846110i	$0.321061\pi$
$440$	0	0
$441$	−6.95280e8	−0.386033
$442$	0	0
$443$	2.14187e8i	0.117052i	0.998286	+	0.0585261i	$0.0186401\pi$
$443$	2.14187e8i	0.117052i	−0.998286	+	0.0585261i	$0.981360\pi$
$444$	0	0
$445$	− 1.75744e8i	− 0.0945411i
$446$	0	0
$447$	2.13455e9	1.13039
$448$	0	0
$449$	−2.48824e9	−1.29727	−0.648635	−	0.761100i	$-0.724660\pi$
$449$	−2.48824e9	−1.29727	−0.648635	+	0.761100i	$0.724660\pi$
$450$	0	0
$451$	− 3.51310e9i	− 1.80332i
$452$	0	0
$453$	− 8.08635e8i	− 0.408704i
$454$	0	0
$455$	1.07422e9	0.534628
$456$	0	0
$457$	−2.18777e9	−1.07225	−0.536123	−	0.844140i	$-0.680112\pi$
$457$	−2.18777e9	−1.07225	−0.536123	+	0.844140i	$0.680112\pi$
$458$	0	0
$459$	− 5.62345e8i	− 0.271430i
$460$	0	0
$461$	− 1.48625e9i	− 0.706541i	−0.935521	−	0.353271i	$-0.885070\pi$
$461$	− 1.48625e9i	− 0.706541i	0.935521	−	0.353271i	$-0.114930\pi$
$462$	0	0
$463$	2.59340e9	1.21433	0.607164	−	0.794576i	$-0.292307\pi$
$463$	2.59340e9	1.21433	0.607164	+	0.794576i	$0.292307\pi$
$464$	0	0
$465$	−5.22547e8	−0.241013
$466$	0	0
$467$	− 1.84718e9i	− 0.839266i	−0.907694	−	0.419633i	$-0.862159\pi$
$467$	− 1.84718e9i	− 0.839266i	0.907694	−	0.419633i	$-0.137841\pi$
$468$	0	0
$469$	3.42766e8i	0.153424i
$470$	0	0
$471$	−1.26778e9	−0.559076
$472$	0	0
$473$	−4.87244e9	−2.11706
$474$	0	0
$475$	− 2.49114e9i	− 1.06652i
$476$	0	0
$477$	1.40786e9i	0.593945i
$478$	0	0
$479$	3.45543e9	1.43657	0.718287	−	0.695747i	$-0.244927\pi$
$479$	3.45543e9	1.43657	0.718287	+	0.695747i	$0.244927\pi$
$480$	0	0
$481$	4.84364e8	0.198456
$482$	0	0
$483$	2.23584e9i	0.902869i
$484$	0	0
$485$	3.50658e6i	0.00139569i
$486$	0	0
$487$	3.63187e9	1.42488	0.712442	−	0.701731i	$-0.247589\pi$
$487$	3.63187e9	1.42488	0.712442	+	0.701731i	$0.247589\pi$
$488$	0	0
$489$	−6.69116e8	−0.258774
$490$	0	0
$491$	− 1.32674e9i	− 0.505827i	−0.967489	−	0.252914i	$-0.918611\pi$
$491$	− 1.32674e9i	− 0.505827i	0.967489	−	0.252914i	$-0.0813888\pi$
$492$	0	0
$493$	5.81025e8i	0.218389i
$494$	0	0
$495$	8.38924e8	0.310888
$496$	0	0
$497$	−8.44785e8	−0.308673
$498$	0	0
$499$	− 2.32010e8i	− 0.0835901i	−0.999126	−	0.0417950i	$-0.986692\pi$
$499$	− 2.32010e8i	− 0.0835901i	0.999126	−	0.0417950i	$-0.0133077\pi$
$500$	0	0
$501$	− 3.45856e9i	− 1.22875i
$502$	0	0
$503$	1.22500e9	0.429187	0.214594	−	0.976703i	$-0.431157\pi$
$503$	1.22500e9	0.429187	0.214594	+	0.976703i	$0.431157\pi$
$504$	0	0
$505$	−1.05248e9	−0.363658
$506$	0	0
$507$	7.84103e9i	2.67205i
$508$	0	0
$509$	6.54747e8i	0.220070i	0.993928	+	0.110035i	$0.0350964\pi$
$509$	6.54747e8i	0.220070i	−0.993928	+	0.110035i	$0.964904\pi$
$510$	0	0
$511$	−4.36357e9	−1.44667
$512$	0	0
$513$	−1.12576e9	−0.368158
$514$	0	0
$515$	1.39942e9i	0.451462i
$516$	0	0
$517$	− 8.75018e9i	− 2.78484i
$518$	0	0
$519$	−5.54146e8	−0.173996
$520$	0	0
$521$	2.39015e8	0.0740445	0.0370222	−	0.999314i	$-0.488213\pi$
$521$	2.39015e8	0.0740445	0.0370222	+	0.999314i	$0.488213\pi$
$522$	0	0
$523$	− 5.76480e9i	− 1.76209i	−0.473032	−	0.881045i	$-0.656841\pi$
$523$	− 5.76480e9i	− 1.76209i	0.473032	−	0.881045i	$-0.343159\pi$
$524$	0	0
$525$	− 5.06131e9i	− 1.52653i
$526$	0	0
$527$	2.04719e9	0.609285
$528$	0	0
$529$	−2.35849e9	−0.692690
$530$	0	0
$531$	− 1.14960e9i	− 0.333209i
$532$	0	0
$533$	6.66644e9i	1.90699i
$534$	0	0
$535$	6.12571e8	0.172949
$536$	0	0
$537$	5.82084e9	1.62209
$538$	0	0
$539$	− 3.04957e9i	− 0.838837i
$540$	0	0
$541$	5.89419e9i	1.60042i	0.599720	+	0.800210i	$0.295278\pi$
$541$	5.89419e9i	1.60042i	−0.599720	+	0.800210i	$0.704722\pi$
$542$	0	0
$543$	−5.96296e9	−1.59832
$544$	0	0
$545$	1.24649e9	0.329838
$546$	0	0
$547$	1.11664e9i	0.291714i	0.989306	+	0.145857i	$0.0465940\pi$
$547$	1.11664e9i	0.291714i	−0.989306	+	0.145857i	$0.953406\pi$
$548$	0	0
$549$	− 1.18404e8i	− 0.0305397i
$550$	0	0
$551$	1.16315e9	0.296215
$552$	0	0
$553$	−3.50853e8	−0.0882241
$554$	0	0
$555$	1.52715e8i	0.0379188i
$556$	0	0
$557$	− 2.94322e9i	− 0.721656i	−0.932632	−	0.360828i	$-0.882494\pi$
$557$	− 2.94322e9i	− 0.721656i	0.932632	−	0.360828i	$-0.117506\pi$
$558$	0	0
$559$	9.24593e9	2.23877
$560$	0	0
$561$	−7.63506e9	−1.82576
$562$	0	0
$563$	− 5.84572e8i	− 0.138057i	−0.997615	−	0.0690285i	$-0.978010\pi$
$563$	− 5.84572e8i	− 0.138057i	0.997615	−	0.0690285i	$-0.0219899\pi$
$564$	0	0
$565$	1.31949e9i	0.307778i
$566$	0	0
$567$	−6.31960e9	−1.45596
$568$	0	0
$569$	7.40531e9	1.68519	0.842597	−	0.538544i	$-0.181025\pi$
$569$	7.40531e9	1.68519	0.842597	+	0.538544i	$0.181025\pi$
$570$	0	0
$571$	3.06829e8i	0.0689715i	0.999405	+	0.0344858i	$0.0109793\pi$
$571$	3.06829e8i	0.0689715i	−0.999405	+	0.0344858i	$0.989021\pi$
$572$	0	0
$573$	6.62280e9i	1.47062i
$574$	0	0
$575$	−2.36862e9	−0.519587
$576$	0	0
$577$	−3.60748e8	−0.0781786	−0.0390893	−	0.999236i	$-0.512446\pi$
$577$	−3.60748e8	−0.0781786	−0.0390893	+	0.999236i	$0.512446\pi$
$578$	0	0
$579$	− 8.34662e9i	− 1.78705i
$580$	0	0
$581$	1.71383e9i	0.362536i
$582$	0	0
$583$	−6.17503e9	−1.29062
$584$	0	0
$585$	−1.59194e9	−0.328761
$586$	0	0
$587$	6.35297e9i	1.29641i	0.761465	+	0.648206i	$0.224480\pi$
$587$	6.35297e9i	1.29641i	−0.761465	+	0.648206i	$0.775520\pi$
$588$	0	0
$589$	− 4.09826e9i	− 0.826411i
$590$	0	0
$591$	−5.80397e9	−1.15656
$592$	0	0
$593$	8.89548e9	1.75177	0.875886	−	0.482517i	$-0.160277\pi$
$593$	8.89548e9	1.75177	0.875886	+	0.482517i	$0.160277\pi$
$594$	0	0
$595$	− 1.32688e9i	− 0.258239i
$596$	0	0
$597$	3.33151e9i	0.640813i
$598$	0	0
$599$	5.27230e9	1.00232	0.501160	−	0.865355i	$-0.332907\pi$
$599$	5.27230e9	1.00232	0.501160	+	0.865355i	$0.332907\pi$
$600$	0	0
$601$	−1.96322e9	−0.368899	−0.184449	−	0.982842i	$-0.559050\pi$
$601$	−1.96322e9	−0.368899	−0.184449	+	0.982842i	$0.559050\pi$
$602$	0	0
$603$	− 5.07964e8i	− 0.0943457i
$604$	0	0
$605$	2.31550e9i	0.425110i
$606$	0	0
$607$	−6.98364e9	−1.26742	−0.633712	−	0.773569i	$-0.718469\pi$
$607$	−6.98364e9	−1.26742	−0.633712	+	0.773569i	$0.718469\pi$
$608$	0	0
$609$	2.36321e9	0.423977
$610$	0	0
$611$	1.66043e10i	2.94494i
$612$	0	0
$613$	− 1.02824e9i	− 0.180295i	−0.995928	−	0.0901476i	$-0.971266\pi$
$613$	− 1.02824e9i	− 0.180295i	0.995928	−	0.0901476i	$-0.0287339\pi$
$614$	0	0
$615$	−2.10185e9	−0.364367
$616$	0	0
$617$	−1.41933e8	−0.0243267	−0.0121634	−	0.999926i	$-0.503872\pi$
$617$	−1.41933e8	−0.0243267	−0.0121634	+	0.999926i	$0.503872\pi$
$618$	0	0
$619$	7.00946e9i	1.18787i	0.804515	+	0.593933i	$0.202426\pi$
$619$	7.00946e9i	1.18787i	−0.804515	+	0.593933i	$0.797574\pi$
$620$	0	0
$621$	1.07039e9i	0.179359i
$622$	0	0
$623$	2.80040e9	0.463994
$624$	0	0
$625$	4.97909e9	0.815774
$626$	0	0
$627$	1.52846e10i	2.47639i
$628$	0	0
$629$	− 5.98291e8i	− 0.0958595i
$630$	0	0
$631$	6.69697e8	0.106115	0.0530573	−	0.998591i	$-0.483103\pi$
$631$	6.69697e8	0.106115	0.0530573	+	0.998591i	$0.483103\pi$
$632$	0	0
$633$	−7.85193e9	−1.23045
$634$	0	0
$635$	2.31864e9i	0.359357i
$636$	0	0
$637$	5.78685e9i	0.887062i
$638$	0	0
$639$	1.25193e9	0.189814
$640$	0	0
$641$	1.05455e10	1.58149	0.790743	−	0.612148i	$-0.209694\pi$
$641$	1.05455e10	1.58149	0.790743	+	0.612148i	$0.209694\pi$
$642$	0	0
$643$	− 7.39491e9i	− 1.09697i	−0.836160	−	0.548485i	$-0.815205\pi$
$643$	− 7.39491e9i	− 1.09697i	0.836160	−	0.548485i	$-0.184795\pi$
$644$	0	0
$645$	2.91514e9i	0.427760i
$646$	0	0
$647$	−7.79266e9	−1.13115	−0.565575	−	0.824697i	$-0.691346\pi$
$647$	−7.79266e9	−1.13115	−0.565575	+	0.824697i	$0.691346\pi$
$648$	0	0
$649$	5.04227e9	0.724052
$650$	0	0
$651$	− 8.32656e9i	− 1.18286i
$652$	0	0
$653$	− 1.23260e10i	− 1.73232i	−0.499768	−	0.866159i	$-0.666581\pi$
$653$	− 1.23260e10i	− 1.73232i	0.499768	−	0.866159i	$-0.333419\pi$
$654$	0	0
$655$	−1.84427e9	−0.256437
$656$	0	0
$657$	6.46661e9	0.889606
$658$	0	0
$659$	− 1.03521e10i	− 1.40906i	−0.709672	−	0.704532i	$-0.751157\pi$
$659$	− 1.03521e10i	− 1.40906i	0.709672	−	0.704532i	$-0.248843\pi$
$660$	0	0
$661$	2.58143e9i	0.347661i	0.984776	+	0.173830i	$0.0556145\pi$
$661$	2.58143e9i	0.347661i	−0.984776	+	0.173830i	$0.944386\pi$
$662$	0	0
$663$	1.44883e10	1.93072
$664$	0	0
$665$	−2.65628e9	−0.350266
$666$	0	0
$667$	− 1.10595e9i	− 0.144309i
$668$	0	0
$669$	1.24137e10i	1.60291i
$670$	0	0
$671$	5.19334e8	0.0663617
$672$	0	0
$673$	−6.53171e9	−0.825989	−0.412995	−	0.910734i	$-0.635517\pi$
$673$	−6.53171e9	−0.825989	−0.412995	+	0.910734i	$0.635517\pi$
$674$	0	0
$675$	− 2.42307e9i	− 0.303252i
$676$	0	0
$677$	1.13228e10i	1.40247i	0.712932	+	0.701233i	$0.247367\pi$
$677$	1.13228e10i	1.40247i	−0.712932	+	0.701233i	$0.752633\pi$
$678$	0	0
$679$	−5.58758e7	−0.00684983
$680$	0	0
$681$	1.52832e10	1.85439
$682$	0	0
$683$	1.30342e10i	1.56535i	0.622429	+	0.782676i	$0.286146\pi$
$683$	1.30342e10i	1.56535i	−0.622429	+	0.782676i	$0.713854\pi$
$684$	0	0
$685$	− 1.32122e9i	− 0.157057i
$686$	0	0
$687$	1.39235e10	1.63833
$688$	0	0
$689$	1.17177e10	1.36482
$690$	0	0
$691$	1.79137e9i	0.206544i	0.994653	+	0.103272i	$0.0329312\pi$
$691$	1.79137e9i	0.206544i	−0.994653	+	0.103272i	$0.967069\pi$
$692$	0	0
$693$	1.33679e10i	1.52579i
$694$	0	0
$695$	−7.61937e8	−0.0860939
$696$	0	0
$697$	8.23444e9	0.921127
$698$	0	0
$699$	− 1.54327e10i	− 1.70912i
$700$	0	0
$701$	1.08521e10i	1.18987i	0.803772	+	0.594937i	$0.202823\pi$
$701$	1.08521e10i	1.18987i	−0.803772	+	0.594937i	$0.797177\pi$
$702$	0	0
$703$	−1.19772e9	−0.130020
$704$	0	0
$705$	−5.23515e9	−0.562687
$706$	0	0
$707$	− 1.67708e10i	− 1.78478i
$708$	0	0
$709$	7.12248e9i	0.750532i	0.926917	+	0.375266i	$0.122449\pi$
$709$	7.12248e9i	0.750532i	−0.926917	+	0.375266i	$0.877551\pi$
$710$	0	0
$711$	5.19948e8	0.0542521
$712$	0	0
$713$	−3.89671e9	−0.402610
$714$	0	0
$715$	− 6.98240e9i	− 0.714387i
$716$	0	0
$717$	6.49051e9i	0.657600i
$718$	0	0
$719$	5.34850e9	0.536638	0.268319	−	0.963330i	$-0.413532\pi$
$719$	5.34850e9	0.536638	0.268319	+	0.963330i	$0.413532\pi$
$720$	0	0
$721$	−2.22991e10	−2.21571
$722$	0	0
$723$	− 4.98574e9i	− 0.490621i
$724$	0	0
$725$	2.50356e9i	0.243992i
$726$	0	0
$727$	−7.41219e9	−0.715445	−0.357722	−	0.933828i	$-0.616447\pi$
$727$	−7.41219e9	−0.715445	−0.357722	+	0.933828i	$0.616447\pi$
$728$	0	0
$729$	4.88078e9	0.466598
$730$	0	0
$731$	− 1.14206e10i	− 1.08138i
$732$	0	0
$733$	− 1.23085e10i	− 1.15436i	−0.816615	−	0.577182i	$-0.804152\pi$
$733$	− 1.23085e10i	− 1.15436i	0.816615	−	0.577182i	$-0.195848\pi$
$734$	0	0
$735$	−1.82453e9	−0.169490
$736$	0	0
$737$	2.22798e9	0.205010
$738$	0	0
$739$	1.11431e10i	1.01567i	0.861455	+	0.507833i	$0.169553\pi$
$739$	1.11431e10i	1.01567i	−0.861455	+	0.507833i	$0.830447\pi$
$740$	0	0
$741$	− 2.90041e10i	− 2.61876i
$742$	0	0
$743$	5.48011e9	0.490150	0.245075	−	0.969504i	$-0.421187\pi$
$743$	5.48011e9	0.490150	0.245075	+	0.969504i	$0.421187\pi$
$744$	0	0
$745$	2.41123e9	0.213645
$746$	0	0
$747$	− 2.53982e9i	− 0.222936i
$748$	0	0
$749$	9.76106e9i	0.848810i
$750$	0	0
$751$	−7.76351e9	−0.668834	−0.334417	−	0.942425i	$-0.608539\pi$
$751$	−7.76351e9	−0.668834	−0.334417	+	0.942425i	$0.608539\pi$
$752$	0	0
$753$	9.92062e9	0.846752
$754$	0	0
$755$	− 9.13450e8i	− 0.0772450i
$756$	0	0
$757$	− 5.58618e9i	− 0.468036i	−0.972232	−	0.234018i	$-0.924813\pi$
$757$	− 5.58618e9i	− 0.468036i	0.972232	−	0.234018i	$-0.0751874\pi$
$758$	0	0
$759$	1.45329e10	1.20644
$760$	0	0
$761$	1.26117e10	1.03736	0.518679	−	0.854969i	$-0.326424\pi$
$761$	1.26117e10	1.03736	0.518679	+	0.854969i	$0.326424\pi$
$762$	0	0
$763$	1.98622e10i	1.61880i
$764$	0	0
$765$	1.96638e9i	0.158800i
$766$	0	0
$767$	−9.56819e9	−0.765678
$768$	0	0
$769$	1.33475e10	1.05842	0.529208	−	0.848492i	$-0.322489\pi$
$769$	1.33475e10	1.05842	0.529208	+	0.848492i	$0.322489\pi$
$770$	0	0
$771$	− 1.09476e10i	− 0.860258i
$772$	0	0
$773$	1.75882e8i	0.0136959i	0.999977	+	0.00684797i	$0.00217979\pi$
$773$	1.75882e8i	0.0136959i	−0.999977	+	0.00684797i	$0.997820\pi$
$774$	0	0
$775$	8.82107e9	0.680715
$776$	0	0
$777$	−2.43344e9	−0.186100
$778$	0	0
$779$	− 1.64845e10i	− 1.24938i
$780$	0	0
$781$	5.49110e9i	0.412459i
$782$	0	0
$783$	1.13137e9	0.0842248
$784$	0	0
$785$	−1.43211e9	−0.105665
$786$	0	0
$787$	1.27829e10i	0.934802i	0.884046	+	0.467401i	$0.154809\pi$
$787$	1.27829e10i	0.934802i	−0.884046	+	0.467401i	$0.845191\pi$
$788$	0	0
$789$	− 2.16418e10i	− 1.56864i
$790$	0	0
$791$	−2.10255e10	−1.51053
$792$	0	0
$793$	−9.85486e8	−0.0701769
$794$	0	0
$795$	3.69446e9i	0.260775i
$796$	0	0
$797$	2.22622e10i	1.55763i	0.627255	+	0.778814i	$0.284178\pi$
$797$	2.22622e10i	1.55763i	−0.627255	+	0.778814i	$0.715822\pi$
$798$	0	0
$799$	2.05098e10	1.42248
$800$	0	0
$801$	−4.15007e9	−0.285326
$802$	0	0
$803$	2.83632e10i	1.93308i
$804$	0	0
$805$	2.52564e9i	0.170642i
$806$	0	0
$807$	1.18578e10	0.794234
$808$	0	0
$809$	−2.37057e10	−1.57410	−0.787050	−	0.616889i	$-0.788393\pi$
$809$	−2.37057e10	−1.57410	−0.787050	+	0.616889i	$0.788393\pi$
$810$	0	0
$811$	2.02345e10i	1.33205i	0.745930	+	0.666025i	$0.232005\pi$
$811$	2.02345e10i	1.33205i	−0.745930	+	0.666025i	$0.767995\pi$
$812$	0	0
$813$	− 3.48673e10i	− 2.27563i
$814$	0	0
$815$	−7.55847e8	−0.0489083
$816$	0	0
$817$	−2.28630e10	−1.46675
$818$	0	0
$819$	− 2.53668e10i	− 1.61351i
$820$	0	0
$821$	2.64348e10i	1.66715i	0.552407	+	0.833575i	$0.313710\pi$
$821$	2.64348e10i	1.66715i	−0.552407	+	0.833575i	$0.686290\pi$
$822$	0	0
$823$	−2.73175e10	−1.70821	−0.854104	−	0.520102i	$-0.825894\pi$
$823$	−2.73175e10	−1.70821	−0.854104	+	0.520102i	$0.825894\pi$
$824$	0	0
$825$	−3.28985e10	−2.03980
$826$	0	0
$827$	1.63558e10i	1.00555i	0.864418	+	0.502773i	$0.167687\pi$
$827$	1.63558e10i	1.00555i	−0.864418	+	0.502773i	$0.832313\pi$
$828$	0	0
$829$	− 6.74626e9i	− 0.411266i	−0.978629	−	0.205633i	$-0.934075\pi$
$829$	− 6.74626e9i	− 0.411266i	0.978629	−	0.205633i	$-0.0659253\pi$
$830$	0	0
$831$	3.25785e10	1.96937
$832$	0	0
$833$	7.14797e9	0.428474
$834$	0	0
$835$	− 3.90686e9i	− 0.232233i
$836$	0	0
$837$	− 3.98629e9i	− 0.234979i
$838$	0	0
$839$	1.37908e10	0.806161	0.403080	−	0.915165i	$-0.367940\pi$
$839$	1.37908e10	0.806161	0.403080	+	0.915165i	$0.367940\pi$
$840$	0	0
$841$	1.60809e10	0.932234
$842$	0	0
$843$	− 2.62806e9i	− 0.151091i
$844$	0	0
$845$	8.85738e9i	0.505018i
$846$	0	0
$847$	−3.68965e10	−2.08638
$848$	0	0
$849$	−5.85007e9	−0.328084
$850$	0	0
$851$	1.13881e9i	0.0633431i
$852$	0	0
$853$	7.09615e9i	0.391472i	0.980657	+	0.195736i	$0.0627096\pi$
$853$	7.09615e9i	0.391472i	−0.980657	+	0.195736i	$0.937290\pi$
$854$	0	0
$855$	3.93649e9	0.215391
$856$	0	0
$857$	2.09501e10	1.13698	0.568490	−	0.822690i	$-0.307528\pi$
$857$	2.09501e10	1.13698	0.568490	+	0.822690i	$0.307528\pi$
$858$	0	0
$859$	− 1.36920e10i	− 0.737040i	−0.929620	−	0.368520i	$-0.879865\pi$
$859$	− 1.36920e10i	− 0.737040i	0.929620	−	0.368520i	$-0.120135\pi$
$860$	0	0
$861$	− 3.34921e10i	− 1.78826i
$862$	0	0
$863$	−2.83553e10	−1.50174	−0.750872	−	0.660447i	$-0.770367\pi$
$863$	−2.83553e10	−1.50174	−0.750872	+	0.660447i	$0.770367\pi$
$864$	0	0
$865$	−6.25975e8	−0.0328852
$866$	0	0
$867$	7.53172e9i	0.392489i
$868$	0	0
$869$	2.28055e9i	0.117888i
$870$	0	0
$871$	−4.22781e9	−0.216796
$872$	0	0
$873$	8.28054e7	0.00421220
$874$	0	0
$875$	− 1.18173e10i	− 0.596335i
$876$	0	0
$877$	− 2.71632e10i	− 1.35982i	−0.733294	−	0.679912i	$-0.762018\pi$
$877$	− 2.71632e10i	− 1.35982i	0.733294	−	0.679912i	$-0.237982\pi$
$878$	0	0
$879$	−2.48674e10	−1.23501
$880$	0	0
$881$	1.39431e10	0.686978	0.343489	−	0.939157i	$-0.388391\pi$
$881$	1.39431e10	0.686978	0.343489	+	0.939157i	$0.388391\pi$
$882$	0	0
$883$	3.30370e10i	1.61487i	0.589954	+	0.807437i	$0.299146\pi$
$883$	3.30370e10i	1.61487i	−0.589954	+	0.807437i	$0.700854\pi$
$884$	0	0
$885$	− 3.01674e9i	− 0.146297i
$886$	0	0
$887$	−1.66589e10	−0.801521	−0.400760	−	0.916183i	$-0.631254\pi$
$887$	−1.66589e10	−0.801521	−0.400760	+	0.916183i	$0.631254\pi$
$888$	0	0
$889$	−3.69466e10	−1.76367
$890$	0	0
$891$	4.10774e10i	1.94550i
$892$	0	0
$893$	− 4.10585e10i	− 1.92940i
$894$	0	0
$895$	6.57533e9	0.306575
$896$	0	0
$897$	−2.75776e10	−1.27580
$898$	0	0
$899$	4.11871e9i	0.189061i
$900$	0	0
$901$	− 1.44738e10i	− 0.659244i
$902$	0	0
$903$	−4.64514e10	−2.09938
$904$	0	0
$905$	−6.73588e9	−0.302082
$906$	0	0
$907$	− 1.81507e10i	− 0.807731i	−0.914818	−	0.403865i	$-0.867666\pi$
$907$	− 1.81507e10i	− 0.807731i	0.914818	−	0.403865i	$-0.132334\pi$
$908$	0	0
$909$	2.48535e10i	1.09752i
$910$	0	0
$911$	−1.78139e10	−0.780630	−0.390315	−	0.920681i	$-0.627634\pi$
$911$	−1.78139e10	−0.780630	−0.390315	+	0.920681i	$0.627634\pi$
$912$	0	0
$913$	1.11399e10	0.484433
$914$	0	0
$915$	− 3.10712e8i	− 0.0134086i
$916$	0	0
$917$	− 2.93877e10i	− 1.25856i
$918$	0	0
$919$	1.94899e10	0.828333	0.414166	−	0.910201i	$-0.364073\pi$
$919$	1.94899e10	0.828333	0.414166	+	0.910201i	$0.364073\pi$
$920$	0	0
$921$	−1.67980e10	−0.708514
$922$	0	0
$923$	− 1.04199e10i	− 0.436171i
$924$	0	0
$925$	− 2.57796e9i	− 0.107098i
$926$	0	0
$927$	3.30462e10	1.36252
$928$	0	0
$929$	−3.36640e10	−1.37756	−0.688780	−	0.724971i	$-0.741853\pi$
$929$	−3.36640e10	−1.37756	−0.688780	+	0.724971i	$0.741853\pi$
$930$	0	0
$931$	− 1.43095e10i	− 0.581167i
$932$	0	0
$933$	2.55872e10i	1.03142i
$934$	0	0
$935$	−8.62472e9	−0.345068
$936$	0	0
$937$	−4.56214e10	−1.81167	−0.905837	−	0.423626i	$-0.860757\pi$
$937$	−4.56214e10	−1.81167	−0.905837	+	0.423626i	$0.860757\pi$
$938$	0	0
$939$	− 1.12224e10i	− 0.442341i
$940$	0	0
$941$	5.00373e10i	1.95763i	0.204752	+	0.978814i	$0.434361\pi$
$941$	5.00373e10i	1.95763i	−0.204752	+	0.978814i	$0.565639\pi$
$942$	0	0
$943$	−1.56738e10	−0.608673
$944$	0	0
$945$	−2.58371e9	−0.0995937
$946$	0	0
$947$	− 3.19677e10i	− 1.22317i	−0.791180	−	0.611584i	$-0.790533\pi$
$947$	− 3.19677e10i	− 1.22317i	0.791180	−	0.611584i	$-0.209467\pi$
$948$	0	0
$949$	− 5.38219e10i	− 2.04422i
$950$	0	0
$951$	4.64045e10	1.74956
$952$	0	0
$953$	3.10999e10	1.16395	0.581974	−	0.813208i	$-0.302281\pi$
$953$	3.10999e10	1.16395	0.581974	+	0.813208i	$0.302281\pi$
$954$	0	0
$955$	7.48125e9i	0.277947i
$956$	0	0
$957$	− 1.53609e10i	− 0.566532i
$958$	0	0
$959$	2.10530e10	0.770813
$960$	0	0
$961$	−1.30007e10	−0.472537
$962$	0	0
$963$	− 1.44654e10i	− 0.521963i
$964$	0	0
$965$	− 9.42851e9i	− 0.337752i
$966$	0	0
$967$	5.30176e10	1.88550	0.942751	−	0.333497i	$-0.108229\pi$
$967$	5.30176e10	1.88550	0.942751	+	0.333497i	$0.108229\pi$
$968$	0	0
$969$	−3.58261e10	−1.26493
$970$	0	0
$971$	4.84951e10i	1.69993i	0.526841	+	0.849964i	$0.323376\pi$
$971$	4.84951e10i	1.69993i	−0.526841	+	0.849964i	$0.676624\pi$
$972$	0	0
$973$	− 1.21411e10i	− 0.422537i
$974$	0	0
$975$	6.24281e10	2.15707
$976$	0	0
$977$	−2.30489e9	−0.0790714	−0.0395357	−	0.999218i	$-0.512588\pi$
$977$	−2.30489e9	−0.0790714	−0.0395357	+	0.999218i	$0.512588\pi$
$978$	0	0
$979$	− 1.82026e10i	− 0.620004i
$980$	0	0
$981$	− 2.94349e10i	− 0.995455i
$982$	0	0
$983$	−2.41653e10	−0.811438	−0.405719	−	0.913998i	$-0.632979\pi$
$983$	−2.41653e10	−0.811438	−0.405719	+	0.913998i	$0.632979\pi$
$984$	0	0
$985$	−6.55628e9	−0.218590
$986$	0	0
$987$	− 8.34198e10i	− 2.76159i
$988$	0	0
$989$	2.17386e10i	0.714569i
$990$	0	0
$991$	5.22086e10	1.70406	0.852029	−	0.523495i	$-0.175372\pi$
$991$	5.22086e10	1.70406	0.852029	+	0.523495i	$0.175372\pi$
$992$	0	0
$993$	−9.88572e9	−0.320395
$994$	0	0
$995$	3.76335e9i	0.121114i
$996$	0	0
$997$	− 7.83024e9i	− 0.250232i	−0.992142	−	0.125116i	$-0.960070\pi$
$997$	− 7.83024e9i	− 0.250232i	0.992142	−	0.125116i	$-0.0399303\pi$
$998$	0	0
$999$	−1.16499e9	−0.0369696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.8.b.i.129.1		4
4.3	odd	2	inner	256.8.b.i.129.3		4
8.3	odd	2	inner	256.8.b.i.129.2		4
8.5	even	2	inner	256.8.b.i.129.4		4
16.3	odd	4		64.8.a.i.1.1		2
16.5	even	4		32.8.a.c.1.1	✓	2
16.11	odd	4		32.8.a.c.1.2	yes	2
16.13	even	4		64.8.a.i.1.2		2
48.5	odd	4		288.8.a.k.1.1		2
48.11	even	4		288.8.a.k.1.2		2
48.29	odd	4		576.8.a.bk.1.1		2
48.35	even	4		576.8.a.bk.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.8.a.c.1.1	✓	2	16.5	even	4
32.8.a.c.1.2	yes	2	16.11	odd	4
64.8.a.i.1.1		2	16.3	odd	4
64.8.a.i.1.2		2	16.13	even	4
256.8.b.i.129.1		4	1.1	even	1	trivial
256.8.b.i.129.2		4	8.3	odd	2	inner
256.8.b.i.129.3		4	4.3	odd	2	inner
256.8.b.i.129.4		4	8.5	even	2	inner
288.8.a.k.1.1		2	48.5	odd	4
288.8.a.k.1.2		2	48.11	even	4
576.8.a.bk.1.1		2	48.29	odd	4
576.8.a.bk.1.2		2	48.35	even	4

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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-61.9677i q^{3} -70.0000i q^{5} +1115.42 q^{7} -1653.00 q^{9} +O(q^{10})\)	\(q-61.9677i q^{3} -70.0000i q^{5} +1115.42 q^{7} -1653.00 q^{9} -7250.22i q^{11} +13758.0i q^{13} -4337.74 q^{15} +16994.0 q^{17} -34020.3i q^{19} -69120.0i q^{21} -32347.2 q^{23} +73225.0 q^{25} -33090.8i q^{27} +34190.0i q^{29} +120465. q^{31} -449280. q^{33} -78079.3i q^{35} -35206.0i q^{37} +852552. q^{39} +484550. q^{41} -672040. i q^{43} +115710. i q^{45} +1.20688e6 q^{47} +420617. q^{49} -1.05308e6i q^{51} -851702. i q^{53} -507516. q^{55} -2.10816e6 q^{57} +695464. i q^{59} +71630.0i q^{61} -1.84379e6 q^{63} +963060. q^{65} +307298. i q^{67} +2.00448e6i q^{69} -757370. q^{71} -3.91204e6 q^{73} -4.53759e6i q^{75} -8.08704e6i q^{77} -314548. q^{79} -5.66567e6 q^{81} +1.53649e6i q^{83} -1.18958e6i q^{85} +2.11868e6 q^{87} +2.51063e6 q^{89} +1.53459e7i q^{91} -7.46496e6i q^{93} -2.38142e6 q^{95} -50094.0 q^{97} +1.19846e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6612 q^{9}+O(q^{10}) \) 4 * q - 6612 * q^9	\( 4 q - 6612 q^{9} + 67976 q^{17} + 292900 q^{25} - 1797120 q^{33} + 1938200 q^{41} + 1682468 q^{49} - 8432640 q^{57} + 3852240 q^{65} - 15648168 q^{73} - 22662684 q^{81} + 10042520 q^{89} - 200376 q^{97}+O(q^{100}) \) 4 * q - 6612 * q^9 + 67976 * q^17 + 292900 * q^25 - 1797120 * q^33 + 1938200 * q^41 + 1682468 * q^49 - 8432640 * q^57 + 3852240 * q^65 - 15648168 * q^73 - 22662684 * q^81 + 10042520 * q^89 - 200376 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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