LMFDB - Embedded newform 32.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,8,Mod(1,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("32.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	32.a (trivial)

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:	yes
Analytic conductor:	$9.99632081549$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{15}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 15 $ x^2 - 15
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.87298$ of defining polynomial
Character	$\chi$	$=$	32.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-61.9677 q^{3} +70.0000 q^{5} -1115.42 q^{7} +1653.00 q^{9} +O(q^{10})$	$q-61.9677 q^{3} +70.0000 q^{5} -1115.42 q^{7} +1653.00 q^{9} +7250.22 q^{11} +13758.0 q^{13} -4337.74 q^{15} +16994.0 q^{17} -34020.3 q^{19} +69120.0 q^{21} +32347.2 q^{23} -73225.0 q^{25} +33090.8 q^{27} +34190.0 q^{29} +120465. q^{31} -449280. q^{33} -78079.3 q^{35} +35206.0 q^{37} -852552. q^{39} -484550. q^{41} +672040. q^{43} +115710. q^{45} +1.20688e6 q^{47} +420617. q^{49} -1.05308e6 q^{51} +851702. q^{53} +507516. q^{55} +2.10816e6 q^{57} -695464. q^{59} +71630.0 q^{61} -1.84379e6 q^{63} +963060. q^{65} +307298. q^{67} -2.00448e6 q^{69} +757370. q^{71} +3.91204e6 q^{73} +4.53759e6 q^{75} -8.08704e6 q^{77} -314548. q^{79} -5.66567e6 q^{81} +1.53649e6 q^{83} +1.18958e6 q^{85} -2.11868e6 q^{87} -2.51063e6 q^{89} -1.53459e7 q^{91} -7.46496e6 q^{93} -2.38142e6 q^{95} -50094.0 q^{97} +1.19846e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 140 q^{5} + 3306 q^{9}+O(q^{10}) $ 2 * q + 140 * q^5 + 3306 * q^9	$ 2 q + 140 q^{5} + 3306 q^{9} + 27516 q^{13} + 33988 q^{17} + 138240 q^{21} - 146450 q^{25} + 68380 q^{29} - 898560 q^{33} + 70412 q^{37} - 969100 q^{41} + 231420 q^{45} + 841234 q^{49} + 1703404 q^{53} + 4216320 q^{57} + 143260 q^{61} + 1926120 q^{65} - 4008960 q^{69} + 7824084 q^{73} - 16174080 q^{77} - 11331342 q^{81} + 2379160 q^{85} - 5021260 q^{89} - 14929920 q^{93} - 100188 q^{97}+O(q^{100}) $ 2 * q + 140 * q^5 + 3306 * q^9 + 27516 * q^13 + 33988 * q^17 + 138240 * q^21 - 146450 * q^25 + 68380 * q^29 - 898560 * q^33 + 70412 * q^37 - 969100 * q^41 + 231420 * q^45 + 841234 * q^49 + 1703404 * q^53 + 4216320 * q^57 + 143260 * q^61 + 1926120 * q^65 - 4008960 * q^69 + 7824084 * q^73 - 16174080 * q^77 - 11331342 * q^81 + 2379160 * q^85 - 5021260 * q^89 - 14929920 * q^93 - 100188 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−61.9677	−1.32508	−0.662539	−	0.749028i	$-0.730521\pi$
$3$	−61.9677	−1.32508	−0.662539	+	0.749028i	$0.730521\pi$
$4$	0	0
$5$	70.0000	0.250440	0.125220	−	0.992129i	$-0.460036\pi$
$5$	70.0000	0.250440	0.125220	+	0.992129i	$0.460036\pi$
$6$	0	0
$7$	−1115.42	−1.22912	−0.614561	−	0.788869i	$-0.710667\pi$
$7$	−1115.42	−1.22912	−0.614561	+	0.788869i	$0.710667\pi$
$8$	0	0
$9$	1653.00	0.755830
$10$	0	0
$11$	7250.22	1.64239	0.821196	−	0.570646i	$-0.193307\pi$
$11$	7250.22	1.64239	0.821196	+	0.570646i	$0.193307\pi$
$12$	0	0
$13$	13758.0	1.73682	0.868408	−	0.495851i	$-0.165144\pi$
$13$	13758.0	1.73682	0.868408	+	0.495851i	$0.165144\pi$
$14$	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
$19$	−34020.3	−1.13789	−0.568945	−	0.822376i	$-0.692648\pi$
$19$	−34020.3	−1.13789	−0.568945	+	0.822376i	$0.692648\pi$
$20$	0	0
$21$	69120.0	1.62868
$22$	0	0
$23$	32347.2	0.554356	0.277178	−	0.960819i	$-0.410601\pi$
$23$	32347.2	0.554356	0.277178	+	0.960819i	$0.410601\pi$
$24$	0	0
$25$	−73225.0	−0.937280
$26$	0	0
$27$	33090.8	0.323544
$28$	0	0
$29$	34190.0	0.260319	0.130160	−	0.991493i	$-0.458451\pi$
$29$	34190.0	0.260319	0.130160	+	0.991493i	$0.458451\pi$
$30$	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
$35$	−78079.3	−0.307821
$36$	0	0
$37$	35206.0	0.114264	0.0571322	−	0.998367i	$-0.481804\pi$
$37$	35206.0	0.114264	0.0571322	+	0.998367i	$0.481804\pi$
$38$	0	0
$39$	−852552.	−2.30141
$40$	0	0
$41$	−484550.	−1.09798	−0.548991	−	0.835828i	$-0.684988\pi$
$41$	−484550.	−1.09798	−0.548991	+	0.835828i	$0.684988\pi$
$42$	0	0
$43$	672040.	1.28901	0.644504	−	0.764601i	$-0.277064\pi$
$43$	672040.	1.28901	0.644504	+	0.764601i	$0.277064\pi$
$44$	0	0
$45$	115710.	0.189290
$46$	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
$51$	−1.05308e6	−1.11164
$52$	0	0
$53$	851702.	0.785818	0.392909	−	0.919577i	$-0.371469\pi$
$53$	851702.	0.785818	0.392909	+	0.919577i	$0.371469\pi$
$54$	0	0
$55$	507516.	0.411320
$56$	0	0
$57$	2.10816e6	1.50779
$58$	0	0
$59$	−695464.	−0.440852	−0.220426	−	0.975404i	$-0.570745\pi$
$59$	−695464.	−0.440852	−0.220426	+	0.975404i	$0.570745\pi$
$60$	0	0
$61$	71630.0	0.0404055	0.0202028	−	0.999796i	$-0.493569\pi$
$61$	71630.0	0.0404055	0.0202028	+	0.999796i	$0.493569\pi$
$62$	0	0
$63$	−1.84379e6	−0.929007
$64$	0	0
$65$	963060.	0.434967
$66$	0	0
$67$	307298.	0.124824	0.0624120	−	0.998050i	$-0.480121\pi$
$67$	307298.	0.124824	0.0624120	+	0.998050i	$0.480121\pi$
$68$	0	0
$69$	−2.00448e6	−0.734564
$70$	0	0
$71$	757370.	0.251133	0.125566	−	0.992085i	$-0.459925\pi$
$71$	757370.	0.251133	0.125566	+	0.992085i	$0.459925\pi$
$72$	0	0
$73$	3.91204e6	1.17699	0.588496	−	0.808500i	$-0.299720\pi$
$73$	3.91204e6	1.17699	0.588496	+	0.808500i	$0.299720\pi$
$74$	0	0
$75$	4.53759e6	1.24197
$76$	0	0
$77$	−8.08704e6	−2.01870
$78$	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
$83$	1.53649e6	0.294955	0.147478	−	0.989065i	$-0.452885\pi$
$83$	1.53649e6	0.294955	0.147478	+	0.989065i	$0.452885\pi$
$84$	0	0
$85$	1.18958e6	0.210101
$86$	0	0
$87$	−2.11868e6	−0.344943
$88$	0	0
$89$	−2.51063e6	−0.377501	−0.188750	−	0.982025i	$-0.560444\pi$
$89$	−2.51063e6	−0.377501	−0.188750	+	0.982025i	$0.560444\pi$
$90$	0	0
$91$	−1.53459e7	−2.13476
$92$	0	0
$93$	−7.46496e6	−0.962359
$94$	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
$99$	1.19846e7	1.24137
$100$	0	0
$101$	1.50354e7	1.45208	0.726039	−	0.687653i	$-0.241359\pi$
$101$	1.50354e7	1.45208	0.726039	+	0.687653i	$0.241359\pi$
$102$	0	0
$103$	1.99917e7	1.80268	0.901340	−	0.433113i	$-0.142585\pi$
$103$	1.99917e7	1.80268	0.901340	+	0.433113i	$0.142585\pi$
$104$	0	0
$105$	4.83840e6	0.407886
$106$	0	0
$107$	−8.75102e6	−0.690582	−0.345291	−	0.938496i	$-0.612220\pi$
$107$	−8.75102e6	−0.690582	−0.345291	+	0.938496i	$0.612220\pi$
$108$	0	0
$109$	1.78070e7	1.31704	0.658518	−	0.752565i	$-0.271184\pi$
$109$	1.78070e7	1.31704	0.658518	+	0.752565i	$0.271184\pi$
$110$	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
$115$	2.26430e6	0.138833
$116$	0	0
$117$	2.27420e7	1.31274
$118$	0	0
$119$	−1.89554e7	−1.03114
$120$	0	0
$121$	3.30786e7	1.69745
$122$	0	0
$123$	3.00265e7	1.45491
$124$	0	0
$125$	−1.05945e7	−0.485172
$126$	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
$131$	−2.63468e7	−1.02395	−0.511974	−	0.859001i	$-0.671085\pi$
$131$	−2.63468e7	−1.02395	−0.511974	+	0.859001i	$0.671085\pi$
$132$	0	0
$133$	3.79469e7	1.39861
$134$	0	0
$135$	2.31635e6	0.0810283
$136$	0	0
$137$	−1.88745e7	−0.627125	−0.313563	−	0.949568i	$-0.601523\pi$
$137$	−1.88745e7	−0.627125	−0.313563	+	0.949568i	$0.601523\pi$
$138$	0	0
$139$	1.08848e7	0.343771	0.171886	−	0.985117i	$-0.445014\pi$
$139$	1.08848e7	0.343771	0.171886	+	0.985117i	$0.445014\pi$
$140$	0	0
$141$	−7.47878e7	−2.24680
$142$	0	0
$143$	9.97486e7	2.85253
$144$	0	0
$145$	2.39330e6	0.0651942
$146$	0	0
$147$	−2.60647e7	−0.676771
$148$	0	0
$149$	−3.44462e7	−0.853078	−0.426539	−	0.904469i	$-0.640267\pi$
$149$	−3.44462e7	−0.853078	−0.426539	+	0.904469i	$0.640267\pi$
$150$	0	0
$151$	−1.30493e7	−0.308438	−0.154219	−	0.988037i	$-0.549286\pi$
$151$	−1.30493e7	−0.308438	−0.154219	+	0.988037i	$0.549286\pi$
$152$	0	0
$153$	2.80911e7	0.634086
$154$	0	0
$155$	8.43257e6	0.181886
$156$	0	0
$157$	−2.04587e7	−0.421919	−0.210960	−	0.977495i	$-0.567659\pi$
$157$	−2.04587e7	−0.421919	−0.210960	+	0.977495i	$0.567659\pi$
$158$	0	0
$159$	−5.27780e7	−1.04127
$160$	0	0
$161$	−3.60806e7	−0.681371
$162$	0	0
$163$	−1.07978e7	−0.195290	−0.0976448	−	0.995221i	$-0.531131\pi$
$163$	−1.07978e7	−0.195290	−0.0976448	+	0.995221i	$0.531131\pi$
$164$	0	0
$165$	−3.14496e7	−0.545031
$166$	0	0
$167$	−5.58122e7	−0.927303	−0.463652	−	0.886018i	$-0.653461\pi$
$167$	−5.58122e7	−0.927303	−0.463652	+	0.886018i	$0.653461\pi$
$168$	0	0
$169$	1.26534e8	2.01653
$170$	0	0
$171$	−5.62355e7	−0.860051
$172$	0	0
$173$	−8.94250e6	−0.131310	−0.0656550	−	0.997842i	$-0.520914\pi$
$173$	−8.94250e6	−0.131310	−0.0656550	+	0.997842i	$0.520914\pi$
$174$	0	0
$175$	8.16766e7	1.15203
$176$	0	0
$177$	4.30963e7	0.584163
$178$	0	0
$179$	9.39334e7	1.22415	0.612074	−	0.790800i	$-0.290335\pi$
$179$	9.39334e7	1.22415	0.612074	+	0.790800i	$0.290335\pi$
$180$	0	0
$181$	9.62269e7	1.20621	0.603103	−	0.797663i	$-0.293931\pi$
$181$	9.62269e7	1.20621	0.603103	+	0.797663i	$0.293931\pi$
$182$	0	0
$183$	−4.43875e6	−0.0535404
$184$	0	0
$185$	2.46442e6	0.0286163
$186$	0	0
$187$	1.23210e8	1.37785
$188$	0	0
$189$	−3.69101e7	−0.397675
$190$	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
$195$	−5.96786e7	−0.576365
$196$	0	0
$197$	9.36611e7	0.872826	0.436413	−	0.899747i	$-0.356249\pi$
$197$	9.36611e7	0.872826	0.436413	+	0.899747i	$0.356249\pi$
$198$	0	0
$199$	5.37621e7	0.483605	0.241802	−	0.970326i	$-0.422261\pi$
$199$	5.37621e7	0.483605	0.241802	+	0.970326i	$0.422261\pi$
$200$	0	0
$201$	−1.90426e7	−0.165401
$202$	0	0
$203$	−3.81362e7	−0.319964
$204$	0	0
$205$	−3.39185e7	−0.274978
$206$	0	0
$207$	5.34699e7	0.418999
$208$	0	0
$209$	−2.46655e8	−1.86886
$210$	0	0
$211$	−1.26710e8	−0.928586	−0.464293	−	0.885682i	$-0.653692\pi$
$211$	−1.26710e8	−0.928586	−0.464293	+	0.885682i	$0.653692\pi$
$212$	0	0
$213$	−4.69325e7	−0.332771
$214$	0	0
$215$	4.70428e7	0.322819
$216$	0	0
$217$	−1.34369e8	−0.892670
$218$	0	0
$219$	−2.42420e8	−1.55961
$220$	0	0
$221$	2.33803e8	1.45706
$222$	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
$227$	2.46632e8	1.39946	0.699728	−	0.714410i	$-0.253305\pi$
$227$	2.46632e8	1.39946	0.699728	+	0.714410i	$0.253305\pi$
$228$	0	0
$229$	−2.24690e8	−1.23640	−0.618201	−	0.786020i	$-0.712138\pi$
$229$	−2.24690e8	−1.23640	−0.618201	+	0.786020i	$0.712138\pi$
$230$	0	0
$231$	5.01136e8	2.67494
$232$	0	0
$233$	−2.49045e8	−1.28983	−0.644914	−	0.764256i	$-0.723107\pi$
$233$	−2.49045e8	−1.28983	−0.644914	+	0.764256i	$0.723107\pi$
$234$	0	0
$235$	8.44819e7	0.424645
$236$	0	0
$237$	1.94918e7	0.0951116
$238$	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
$243$	2.78719e8	1.24608
$244$	0	0
$245$	2.94432e7	0.127910
$246$	0	0
$247$	−4.68051e8	−1.97630
$248$	0	0
$249$	−9.52128e7	−0.390839
$250$	0	0
$251$	−1.60093e8	−0.639021	−0.319511	−	0.947583i	$-0.603518\pi$
$251$	−1.60093e8	−0.639021	−0.319511	+	0.947583i	$0.603518\pi$
$252$	0	0
$253$	2.34524e8	0.910470
$254$	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
$259$	−3.92694e7	−0.140445
$260$	0	0
$261$	5.65161e7	0.196757
$262$	0	0
$263$	−3.49243e8	−1.18381	−0.591906	−	0.806007i	$-0.701624\pi$
$263$	−3.49243e8	−1.18381	−0.591906	+	0.806007i	$0.701624\pi$
$264$	0	0
$265$	5.96191e7	0.196800
$266$	0	0
$267$	1.55578e8	0.500218
$268$	0	0
$269$	1.91355e8	0.599387	0.299693	−	0.954036i	$-0.403116\pi$
$269$	1.91355e8	0.599387	0.299693	+	0.954036i	$0.403116\pi$
$270$	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
$275$	−5.30898e8	−1.53938
$276$	0	0
$277$	−5.25734e8	−1.48623	−0.743116	−	0.669162i	$-0.766653\pi$
$277$	−5.25734e8	−1.48623	−0.743116	+	0.669162i	$0.766653\pi$
$278$	0	0
$279$	1.99129e8	0.548934
$280$	0	0
$281$	−4.24101e7	−0.114024	−0.0570122	−	0.998373i	$-0.518157\pi$
$281$	−4.24101e7	−0.114024	−0.0570122	+	0.998373i	$0.518157\pi$
$282$	0	0
$283$	9.44052e7	0.247596	0.123798	−	0.992307i	$-0.460493\pi$
$283$	9.44052e7	0.247596	0.123798	+	0.992307i	$0.460493\pi$
$284$	0	0
$285$	1.47571e8	0.377611
$286$	0	0
$287$	5.40476e8	1.34955
$288$	0	0
$289$	−1.21543e8	−0.296201
$290$	0	0
$291$	3.10421e6	0.00738458
$292$	0	0
$293$	4.01296e8	0.932027	0.466014	−	0.884778i	$-0.345690\pi$
$293$	4.01296e8	0.932027	0.466014	+	0.884778i	$0.345690\pi$
$294$	0	0
$295$	−4.86825e7	−0.110407
$296$	0	0
$297$	2.39916e8	0.531387
$298$	0	0
$299$	4.45032e8	0.962814
$300$	0	0
$301$	−7.49606e8	−1.58435
$302$	0	0
$303$	−9.31710e8	−1.92412
$304$	0	0
$305$	5.01410e6	0.0101191
$306$	0	0
$307$	−2.71076e8	−0.534697	−0.267348	−	0.963600i	$-0.586147\pi$
$307$	−2.71076e8	−0.534697	−0.267348	+	0.963600i	$0.586147\pi$
$308$	0	0
$309$	−1.23884e9	−2.38869
$310$	0	0
$311$	4.12911e8	0.778387	0.389193	−	0.921156i	$-0.372754\pi$
$311$	4.12911e8	0.778387	0.389193	+	0.921156i	$0.372754\pi$
$312$	0	0
$313$	−1.81101e8	−0.333823	−0.166911	−	0.985972i	$-0.553379\pi$
$313$	−1.81101e8	−0.333823	−0.166911	+	0.985972i	$0.553379\pi$
$314$	0	0
$315$	−1.29065e8	−0.232660
$316$	0	0
$317$	7.48850e8	1.32034	0.660172	−	0.751114i	$-0.270483\pi$
$317$	7.48850e8	1.32034	0.660172	+	0.751114i	$0.270483\pi$
$318$	0	0
$319$	2.47885e8	0.427546
$320$	0	0
$321$	5.42281e8	0.915075
$322$	0	0
$323$	−5.78141e8	−0.954607
$324$	0	0
$325$	−1.00743e9	−1.62788
$326$	0	0
$327$	−1.10346e9	−1.74517
$328$	0	0
$329$	−1.34618e9	−2.08410
$330$	0	0
$331$	1.59530e8	0.241793	0.120897	−	0.992665i	$-0.461423\pi$
$331$	1.59530e8	0.241793	0.120897	+	0.992665i	$0.461423\pi$
$332$	0	0
$333$	5.81955e7	0.0863644
$334$	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
$339$	1.16809e9	1.62846
$340$	0	0
$341$	8.73400e8	1.19281
$342$	0	0
$343$	4.49431e8	0.601359
$344$	0	0
$345$	−1.40314e8	−0.183964
$346$	0	0
$347$	1.20856e9	1.55279	0.776397	−	0.630244i	$-0.217045\pi$
$347$	1.20856e9	1.55279	0.776397	+	0.630244i	$0.217045\pi$
$348$	0	0
$349$	1.10448e8	0.139081	0.0695404	−	0.997579i	$-0.477847\pi$
$349$	1.10448e8	0.139081	0.0695404	+	0.997579i	$0.477847\pi$
$350$	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
$355$	5.30159e7	0.0628936
$356$	0	0
$357$	1.17463e9	1.36635
$358$	0	0
$359$	−1.97988e8	−0.225844	−0.112922	−	0.993604i	$-0.536021\pi$
$359$	−1.97988e8	−0.225844	−0.112922	+	0.993604i	$0.536021\pi$
$360$	0	0
$361$	2.63508e8	0.294794
$362$	0	0
$363$	−2.04981e9	−2.24926
$364$	0	0
$365$	2.73843e8	0.294765
$366$	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
$371$	−9.50005e8	−0.965866
$372$	0	0
$373$	4.72832e8	0.471765	0.235883	−	0.971782i	$-0.424202\pi$
$373$	4.72832e8	0.471765	0.235883	+	0.971782i	$0.424202\pi$
$374$	0	0
$375$	6.56517e8	0.642890
$376$	0	0
$377$	4.70386e8	0.452126
$378$	0	0
$379$	−1.27673e9	−1.20465	−0.602324	−	0.798251i	$-0.705759\pi$
$379$	−1.27673e9	−1.20465	−0.602324	+	0.798251i	$0.705759\pi$
$380$	0	0
$381$	2.05259e9	1.90136
$382$	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
$387$	1.11088e9	0.974271
$388$	0	0
$389$	5.29163e8	0.455791	0.227896	−	0.973686i	$-0.426815\pi$
$389$	5.29163e8	0.455791	0.227896	+	0.973686i	$0.426815\pi$
$390$	0	0
$391$	5.49708e8	0.465064
$392$	0	0
$393$	1.63265e9	1.35681
$394$	0	0
$395$	−2.20184e7	−0.0179761
$396$	0	0
$397$	−1.59191e9	−1.27688	−0.638442	−	0.769670i	$-0.720421\pi$
$397$	−1.59191e9	−1.27688	−0.638442	+	0.769670i	$0.720421\pi$
$398$	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
$403$	1.65736e9	1.26139
$404$	0	0
$405$	−3.96597e8	−0.296659
$406$	0	0
$407$	2.55251e8	0.187667
$408$	0	0
$409$	−5.08439e8	−0.367458	−0.183729	−	0.982977i	$-0.558817\pi$
$409$	−5.08439e8	−0.367458	−0.183729	+	0.982977i	$0.558817\pi$
$410$	0	0
$411$	1.16961e9	0.830989
$412$	0	0
$413$	7.75734e8	0.541861
$414$	0	0
$415$	1.07554e8	0.0738685
$416$	0	0
$417$	−6.74508e8	−0.455523
$418$	0	0
$419$	2.96853e8	0.197148	0.0985739	−	0.995130i	$-0.468572\pi$
$419$	2.96853e8	0.197148	0.0985739	+	0.995130i	$0.468572\pi$
$420$	0	0
$421$	8.73091e8	0.570259	0.285130	−	0.958489i	$-0.407963\pi$
$421$	8.73091e8	0.570259	0.285130	+	0.958489i	$0.407963\pi$
$422$	0	0
$423$	1.99498e9	1.28158
$424$	0	0
$425$	−1.24439e9	−0.786310
$426$	0	0
$427$	−7.98975e7	−0.0496633
$428$	0	0
$429$	−6.18119e9	−3.77983
$430$	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
$435$	−1.48307e8	−0.0863874
$436$	0	0
$437$	−1.10046e9	−0.630796
$438$	0	0
$439$	−1.88969e9	−1.06602	−0.533009	−	0.846110i	$-0.678939\pi$
$439$	−1.88969e9	−1.06602	−0.533009	+	0.846110i	$0.678939\pi$
$440$	0	0
$441$	6.95280e8	0.386033
$442$	0	0
$443$	−2.14187e8	−0.117052	−0.0585261	−	0.998286i	$-0.518640\pi$
$443$	−2.14187e8	−0.117052	−0.0585261	+	0.998286i	$0.518640\pi$
$444$	0	0
$445$	−1.75744e8	−0.0945411
$446$	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
$451$	−3.51310e9	−1.80332
$452$	0	0
$453$	8.08635e8	0.408704
$454$	0	0
$455$	−1.07422e9	−0.534628
$456$	0	0
$457$	2.18777e9	1.07225	0.536123	−	0.844140i	$-0.319888\pi$
$457$	2.18777e9	1.07225	0.536123	+	0.844140i	$0.319888\pi$
$458$	0	0
$459$	5.62345e8	0.271430
$460$	0	0
$461$	−1.48625e9	−0.706541	−0.353271	−	0.935521i	$-0.614930\pi$
$461$	−1.48625e9	−0.706541	−0.353271	+	0.935521i	$0.614930\pi$
$462$	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
$467$	−1.84718e9	−0.839266	−0.419633	−	0.907694i	$-0.637841\pi$
$467$	−1.84718e9	−0.839266	−0.419633	+	0.907694i	$0.637841\pi$
$468$	0	0
$469$	−3.42766e8	−0.153424
$470$	0	0
$471$	1.26778e9	0.559076
$472$	0	0
$473$	4.87244e9	2.11706
$474$	0	0
$475$	2.49114e9	1.06652
$476$	0	0
$477$	1.40786e9	0.593945
$478$	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
$483$	2.23584e9	0.902869
$484$	0	0
$485$	−3.50658e6	−0.00139569
$486$	0	0
$487$	−3.63187e9	−1.42488	−0.712442	−	0.701731i	$-0.752411\pi$
$487$	−3.63187e9	−1.42488	−0.712442	+	0.701731i	$0.752411\pi$
$488$	0	0
$489$	6.69116e8	0.258774
$490$	0	0
$491$	1.32674e9	0.505827	0.252914	−	0.967489i	$-0.418611\pi$
$491$	1.32674e9	0.505827	0.252914	+	0.967489i	$0.418611\pi$
$492$	0	0
$493$	5.81025e8	0.218389
$494$	0	0
$495$	8.38924e8	0.310888
$496$	0	0
$497$	−8.44785e8	−0.308673
$498$	0	0
$499$	−2.32010e8	−0.0835901	−0.0417950	−	0.999126i	$-0.513308\pi$
$499$	−2.32010e8	−0.0835901	−0.0417950	+	0.999126i	$0.513308\pi$
$500$	0	0
$501$	3.45856e9	1.22875
$502$	0	0
$503$	−1.22500e9	−0.429187	−0.214594	−	0.976703i	$-0.568843\pi$
$503$	−1.22500e9	−0.429187	−0.214594	+	0.976703i	$0.568843\pi$
$504$	0	0
$505$	1.05248e9	0.363658
$506$	0	0
$507$	−7.84103e9	−2.67205
$508$	0	0
$509$	6.54747e8	0.220070	0.110035	−	0.993928i	$-0.464904\pi$
$509$	6.54747e8	0.220070	0.110035	+	0.993928i	$0.464904\pi$
$510$	0	0
$511$	−4.36357e9	−1.44667
$512$	0	0
$513$	−1.12576e9	−0.368158
$514$	0	0
$515$	1.39942e9	0.451462
$516$	0	0
$517$	8.75018e9	2.78484
$518$	0	0
$519$	5.54146e8	0.173996
$520$	0	0
$521$	−2.39015e8	−0.0740445	−0.0370222	−	0.999314i	$-0.511787\pi$
$521$	−2.39015e8	−0.0740445	−0.0370222	+	0.999314i	$0.511787\pi$
$522$	0	0
$523$	5.76480e9	1.76209	0.881045	−	0.473032i	$-0.156841\pi$
$523$	5.76480e9	1.76209	0.881045	+	0.473032i	$0.156841\pi$
$524$	0	0
$525$	−5.06131e9	−1.52653
$526$	0	0
$527$	2.04719e9	0.609285
$528$	0	0
$529$	−2.35849e9	−0.692690
$530$	0	0
$531$	−1.14960e9	−0.333209
$532$	0	0
$533$	−6.66644e9	−1.90699
$534$	0	0
$535$	−6.12571e8	−0.172949
$536$	0	0
$537$	−5.82084e9	−1.62209
$538$	0	0
$539$	3.04957e9	0.838837
$540$	0	0
$541$	5.89419e9	1.60042	0.800210	−	0.599720i	$-0.204722\pi$
$541$	5.89419e9	1.60042	0.800210	+	0.599720i	$0.204722\pi$
$542$	0	0
$543$	−5.96296e9	−1.59832
$544$	0	0
$545$	1.24649e9	0.329838
$546$	0	0
$547$	1.11664e9	0.291714	0.145857	−	0.989306i	$-0.453406\pi$
$547$	1.11664e9	0.291714	0.145857	+	0.989306i	$0.453406\pi$
$548$	0	0
$549$	1.18404e8	0.0305397
$550$	0	0
$551$	−1.16315e9	−0.296215
$552$	0	0
$553$	3.50853e8	0.0882241
$554$	0	0
$555$	−1.52715e8	−0.0379188
$556$	0	0
$557$	−2.94322e9	−0.721656	−0.360828	−	0.932632i	$-0.617506\pi$
$557$	−2.94322e9	−0.721656	−0.360828	+	0.932632i	$0.617506\pi$
$558$	0	0
$559$	9.24593e9	2.23877
$560$	0	0
$561$	−7.63506e9	−1.82576
$562$	0	0
$563$	−5.84572e8	−0.138057	−0.0690285	−	0.997615i	$-0.521990\pi$
$563$	−5.84572e8	−0.138057	−0.0690285	+	0.997615i	$0.521990\pi$
$564$	0	0
$565$	−1.31949e9	−0.307778
$566$	0	0
$567$	6.31960e9	1.45596
$568$	0	0
$569$	−7.40531e9	−1.68519	−0.842597	−	0.538544i	$-0.818975\pi$
$569$	−7.40531e9	−1.68519	−0.842597	+	0.538544i	$0.818975\pi$
$570$	0	0
$571$	−3.06829e8	−0.0689715	−0.0344858	−	0.999405i	$-0.510979\pi$
$571$	−3.06829e8	−0.0689715	−0.0344858	+	0.999405i	$0.510979\pi$
$572$	0	0
$573$	6.62280e9	1.47062
$574$	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
$579$	−8.34662e9	−1.78705
$580$	0	0
$581$	−1.71383e9	−0.362536
$582$	0	0
$583$	6.17503e9	1.29062
$584$	0	0
$585$	1.59194e9	0.328761
$586$	0	0
$587$	−6.35297e9	−1.29641	−0.648206	−	0.761465i	$-0.724480\pi$
$587$	−6.35297e9	−1.29641	−0.648206	+	0.761465i	$0.724480\pi$
$588$	0	0
$589$	−4.09826e9	−0.826411
$590$	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
$595$	−1.32688e9	−0.258239
$596$	0	0
$597$	−3.33151e9	−0.640813
$598$	0	0
$599$	−5.27230e9	−1.00232	−0.501160	−	0.865355i	$-0.667093\pi$
$599$	−5.27230e9	−1.00232	−0.501160	+	0.865355i	$0.667093\pi$
$600$	0	0
$601$	1.96322e9	0.368899	0.184449	−	0.982842i	$-0.440950\pi$
$601$	1.96322e9	0.368899	0.184449	+	0.982842i	$0.440950\pi$
$602$	0	0
$603$	5.07964e8	0.0943457
$604$	0	0
$605$	2.31550e9	0.425110
$606$	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
$611$	1.66043e10	2.94494
$612$	0	0
$613$	1.02824e9	0.180295	0.0901476	−	0.995928i	$-0.471266\pi$
$613$	1.02824e9	0.180295	0.0901476	+	0.995928i	$0.471266\pi$
$614$	0	0
$615$	2.10185e9	0.364367
$616$	0	0
$617$	1.41933e8	0.0243267	0.0121634	−	0.999926i	$-0.496128\pi$
$617$	1.41933e8	0.0243267	0.0121634	+	0.999926i	$0.496128\pi$
$618$	0	0
$619$	−7.00946e9	−1.18787	−0.593933	−	0.804515i	$-0.702426\pi$
$619$	−7.00946e9	−1.18787	−0.593933	+	0.804515i	$0.702426\pi$
$620$	0	0
$621$	1.07039e9	0.179359
$622$	0	0
$623$	2.80040e9	0.463994
$624$	0	0
$625$	4.97909e9	0.815774
$626$	0	0
$627$	1.52846e10	2.47639
$628$	0	0
$629$	5.98291e8	0.0958595
$630$	0	0
$631$	−6.69697e8	−0.106115	−0.0530573	−	0.998591i	$-0.516897\pi$
$631$	−6.69697e8	−0.106115	−0.0530573	+	0.998591i	$0.516897\pi$
$632$	0	0
$633$	7.85193e9	1.23045
$634$	0	0
$635$	−2.31864e9	−0.359357
$636$	0	0
$637$	5.78685e9	0.887062
$638$	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
$643$	−7.39491e9	−1.09697	−0.548485	−	0.836160i	$-0.684795\pi$
$643$	−7.39491e9	−1.09697	−0.548485	+	0.836160i	$0.684795\pi$
$644$	0	0
$645$	−2.91514e9	−0.427760
$646$	0	0
$647$	7.79266e9	1.13115	0.565575	−	0.824697i	$-0.308654\pi$
$647$	7.79266e9	1.13115	0.565575	+	0.824697i	$0.308654\pi$
$648$	0	0
$649$	−5.04227e9	−0.724052
$650$	0	0
$651$	8.32656e9	1.18286
$652$	0	0
$653$	−1.23260e10	−1.73232	−0.866159	−	0.499768i	$-0.833419\pi$
$653$	−1.23260e10	−1.73232	−0.866159	+	0.499768i	$0.833419\pi$
$654$	0	0
$655$	−1.84427e9	−0.256437
$656$	0	0
$657$	6.46661e9	0.889606
$658$	0	0
$659$	−1.03521e10	−1.40906	−0.704532	−	0.709672i	$-0.748843\pi$
$659$	−1.03521e10	−1.40906	−0.704532	+	0.709672i	$0.748843\pi$
$660$	0	0
$661$	−2.58143e9	−0.347661	−0.173830	−	0.984776i	$-0.555614\pi$
$661$	−2.58143e9	−0.347661	−0.173830	+	0.984776i	$0.555614\pi$
$662$	0	0
$663$	−1.44883e10	−1.93072
$664$	0	0
$665$	2.65628e9	0.350266
$666$	0	0
$667$	1.10595e9	0.144309
$668$	0	0
$669$	1.24137e10	1.60291
$670$	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
$675$	−2.42307e9	−0.303252
$676$	0	0
$677$	−1.13228e10	−1.40247	−0.701233	−	0.712932i	$-0.747367\pi$
$677$	−1.13228e10	−1.40247	−0.701233	+	0.712932i	$0.747367\pi$
$678$	0	0
$679$	5.58758e7	0.00684983
$680$	0	0
$681$	−1.52832e10	−1.85439
$682$	0	0
$683$	−1.30342e10	−1.56535	−0.782676	−	0.622429i	$-0.786146\pi$
$683$	−1.30342e10	−1.56535	−0.782676	+	0.622429i	$0.786146\pi$
$684$	0	0
$685$	−1.32122e9	−0.157057
$686$	0	0
$687$	1.39235e10	1.63833
$688$	0	0
$689$	1.17177e10	1.36482
$690$	0	0
$691$	1.79137e9	0.206544	0.103272	−	0.994653i	$-0.467069\pi$
$691$	1.79137e9	0.206544	0.103272	+	0.994653i	$0.467069\pi$
$692$	0	0
$693$	−1.33679e10	−1.52579
$694$	0	0
$695$	7.61937e8	0.0860939
$696$	0	0
$697$	−8.23444e9	−0.921127
$698$	0	0
$699$	1.54327e10	1.70912
$700$	0	0
$701$	1.08521e10	1.18987	0.594937	−	0.803772i	$-0.297177\pi$
$701$	1.08521e10	1.18987	0.594937	+	0.803772i	$0.297177\pi$
$702$	0	0
$703$	−1.19772e9	−0.130020
$704$	0	0
$705$	−5.23515e9	−0.562687
$706$	0	0
$707$	−1.67708e10	−1.78478
$708$	0	0
$709$	−7.12248e9	−0.750532	−0.375266	−	0.926917i	$-0.622449\pi$
$709$	−7.12248e9	−0.750532	−0.375266	+	0.926917i	$0.622449\pi$
$710$	0	0
$711$	−5.19948e8	−0.0542521
$712$	0	0
$713$	3.89671e9	0.402610
$714$	0	0
$715$	6.98240e9	0.714387
$716$	0	0
$717$	6.49051e9	0.657600
$718$	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
$723$	−4.98574e9	−0.490621
$724$	0	0
$725$	−2.50356e9	−0.243992
$726$	0	0
$727$	7.41219e9	0.715445	0.357722	−	0.933828i	$-0.383553\pi$
$727$	7.41219e9	0.715445	0.357722	+	0.933828i	$0.383553\pi$
$728$	0	0
$729$	−4.88078e9	−0.466598
$730$	0	0
$731$	1.14206e10	1.08138
$732$	0	0
$733$	−1.23085e10	−1.15436	−0.577182	−	0.816615i	$-0.695848\pi$
$733$	−1.23085e10	−1.15436	−0.577182	+	0.816615i	$0.695848\pi$
$734$	0	0
$735$	−1.82453e9	−0.169490
$736$	0	0
$737$	2.22798e9	0.205010
$738$	0	0
$739$	1.11431e10	1.01567	0.507833	−	0.861455i	$-0.330447\pi$
$739$	1.11431e10	1.01567	0.507833	+	0.861455i	$0.330447\pi$
$740$	0	0
$741$	2.90041e10	2.61876
$742$	0	0
$743$	−5.48011e9	−0.490150	−0.245075	−	0.969504i	$-0.578813\pi$
$743$	−5.48011e9	−0.490150	−0.245075	+	0.969504i	$0.578813\pi$
$744$	0	0
$745$	−2.41123e9	−0.213645
$746$	0	0
$747$	2.53982e9	0.222936
$748$	0	0
$749$	9.76106e9	0.848810
$750$	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
$755$	−9.13450e8	−0.0772450
$756$	0	0
$757$	5.58618e9	0.468036	0.234018	−	0.972232i	$-0.424813\pi$
$757$	5.58618e9	0.468036	0.234018	+	0.972232i	$0.424813\pi$
$758$	0	0
$759$	−1.45329e10	−1.20644
$760$	0	0
$761$	−1.26117e10	−1.03736	−0.518679	−	0.854969i	$-0.673576\pi$
$761$	−1.26117e10	−1.03736	−0.518679	+	0.854969i	$0.673576\pi$
$762$	0	0
$763$	−1.98622e10	−1.61880
$764$	0	0
$765$	1.96638e9	0.158800
$766$	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
$771$	−1.09476e10	−0.860258
$772$	0	0
$773$	−1.75882e8	−0.0136959	−0.00684797	−	0.999977i	$-0.502180\pi$
$773$	−1.75882e8	−0.0136959	−0.00684797	+	0.999977i	$0.502180\pi$
$774$	0	0
$775$	−8.82107e9	−0.680715
$776$	0	0
$777$	2.43344e9	0.186100
$778$	0	0
$779$	1.64845e10	1.24938
$780$	0	0
$781$	5.49110e9	0.412459
$782$	0	0
$783$	1.13137e9	0.0842248
$784$	0	0
$785$	−1.43211e9	−0.105665
$786$	0	0
$787$	1.27829e10	0.934802	0.467401	−	0.884046i	$-0.345191\pi$
$787$	1.27829e10	0.934802	0.467401	+	0.884046i	$0.345191\pi$
$788$	0	0
$789$	2.16418e10	1.56864
$790$	0	0
$791$	2.10255e10	1.51053
$792$	0	0
$793$	9.85486e8	0.0701769
$794$	0	0
$795$	−3.69446e9	−0.260775
$796$	0	0
$797$	2.22622e10	1.55763	0.778814	−	0.627255i	$-0.215822\pi$
$797$	2.22622e10	1.55763	0.778814	+	0.627255i	$0.215822\pi$
$798$	0	0
$799$	2.05098e10	1.42248
$800$	0	0
$801$	−4.15007e9	−0.285326
$802$	0	0
$803$	2.83632e10	1.93308
$804$	0	0
$805$	−2.52564e9	−0.170642
$806$	0	0
$807$	−1.18578e10	−0.794234
$808$	0	0
$809$	2.37057e10	1.57410	0.787050	−	0.616889i	$-0.211607\pi$
$809$	2.37057e10	1.57410	0.787050	+	0.616889i	$0.211607\pi$
$810$	0	0
$811$	−2.02345e10	−1.33205	−0.666025	−	0.745930i	$-0.732005\pi$
$811$	−2.02345e10	−1.33205	−0.666025	+	0.745930i	$0.732005\pi$
$812$	0	0
$813$	−3.48673e10	−2.27563
$814$	0	0
$815$	−7.55847e8	−0.0489083
$816$	0	0
$817$	−2.28630e10	−1.46675
$818$	0	0
$819$	−2.53668e10	−1.61351
$820$	0	0
$821$	−2.64348e10	−1.66715	−0.833575	−	0.552407i	$-0.813710\pi$
$821$	−2.64348e10	−1.66715	−0.833575	+	0.552407i	$0.813710\pi$
$822$	0	0
$823$	2.73175e10	1.70821	0.854104	−	0.520102i	$-0.174106\pi$
$823$	2.73175e10	1.70821	0.854104	+	0.520102i	$0.174106\pi$
$824$	0	0
$825$	3.28985e10	2.03980
$826$	0	0
$827$	−1.63558e10	−1.00555	−0.502773	−	0.864418i	$-0.667687\pi$
$827$	−1.63558e10	−1.00555	−0.502773	+	0.864418i	$0.667687\pi$
$828$	0	0
$829$	−6.74626e9	−0.411266	−0.205633	−	0.978629i	$-0.565925\pi$
$829$	−6.74626e9	−0.411266	−0.205633	+	0.978629i	$0.565925\pi$
$830$	0	0
$831$	3.25785e10	1.96937
$832$	0	0
$833$	7.14797e9	0.428474
$834$	0	0
$835$	−3.90686e9	−0.232233
$836$	0	0
$837$	3.98629e9	0.234979
$838$	0	0
$839$	−1.37908e10	−0.806161	−0.403080	−	0.915165i	$-0.632060\pi$
$839$	−1.37908e10	−0.806161	−0.403080	+	0.915165i	$0.632060\pi$
$840$	0	0
$841$	−1.60809e10	−0.932234
$842$	0	0
$843$	2.62806e9	0.151091
$844$	0	0
$845$	8.85738e9	0.505018
$846$	0	0
$847$	−3.68965e10	−2.08638
$848$	0	0
$849$	−5.85007e9	−0.328084
$850$	0	0
$851$	1.13881e9	0.0633431
$852$	0	0
$853$	−7.09615e9	−0.391472	−0.195736	−	0.980657i	$-0.562710\pi$
$853$	−7.09615e9	−0.391472	−0.195736	+	0.980657i	$0.562710\pi$
$854$	0	0
$855$	−3.93649e9	−0.215391
$856$	0	0
$857$	−2.09501e10	−1.13698	−0.568490	−	0.822690i	$-0.692472\pi$
$857$	−2.09501e10	−1.13698	−0.568490	+	0.822690i	$0.692472\pi$
$858$	0	0
$859$	1.36920e10	0.737040	0.368520	−	0.929620i	$-0.379865\pi$
$859$	1.36920e10	0.737040	0.368520	+	0.929620i	$0.379865\pi$
$860$	0	0
$861$	−3.34921e10	−1.78826
$862$	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
$867$	7.53172e9	0.392489
$868$	0	0
$869$	−2.28055e9	−0.117888
$870$	0	0
$871$	4.22781e9	0.216796
$872$	0	0
$873$	−8.28054e7	−0.00421220
$874$	0	0
$875$	1.18173e10	0.596335
$876$	0	0
$877$	−2.71632e10	−1.35982	−0.679912	−	0.733294i	$-0.737982\pi$
$877$	−2.71632e10	−1.35982	−0.679912	+	0.733294i	$0.737982\pi$
$878$	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
$883$	3.30370e10	1.61487	0.807437	−	0.589954i	$-0.200854\pi$
$883$	3.30370e10	1.61487	0.807437	+	0.589954i	$0.200854\pi$
$884$	0	0
$885$	3.01674e9	0.146297
$886$	0	0
$887$	1.66589e10	0.801521	0.400760	−	0.916183i	$-0.368746\pi$
$887$	1.66589e10	0.801521	0.400760	+	0.916183i	$0.368746\pi$
$888$	0	0
$889$	3.69466e10	1.76367
$890$	0	0
$891$	−4.10774e10	−1.94550
$892$	0	0
$893$	−4.10585e10	−1.92940
$894$	0	0
$895$	6.57533e9	0.306575
$896$	0	0
$897$	−2.75776e10	−1.27580
$898$	0	0
$899$	4.11871e9	0.189061
$900$	0	0
$901$	1.44738e10	0.659244
$902$	0	0
$903$	4.64514e10	2.09938
$904$	0	0
$905$	6.73588e9	0.302082
$906$	0	0
$907$	1.81507e10	0.807731	0.403865	−	0.914818i	$-0.367666\pi$
$907$	1.81507e10	0.807731	0.403865	+	0.914818i	$0.367666\pi$
$908$	0	0
$909$	2.48535e10	1.09752
$910$	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
$915$	−3.10712e8	−0.0134086
$916$	0	0
$917$	2.93877e10	1.25856
$918$	0	0
$919$	−1.94899e10	−0.828333	−0.414166	−	0.910201i	$-0.635927\pi$
$919$	−1.94899e10	−0.828333	−0.414166	+	0.910201i	$0.635927\pi$
$920$	0	0
$921$	1.67980e10	0.708514
$922$	0	0
$923$	1.04199e10	0.436171
$924$	0	0
$925$	−2.57796e9	−0.107098
$926$	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
$931$	−1.43095e10	−0.581167
$932$	0	0
$933$	−2.55872e10	−1.03142
$934$	0	0
$935$	8.62472e9	0.345068
$936$	0	0
$937$	4.56214e10	1.81167	0.905837	−	0.423626i	$-0.139243\pi$
$937$	4.56214e10	1.81167	0.905837	+	0.423626i	$0.139243\pi$
$938$	0	0
$939$	1.12224e10	0.442341
$940$	0	0
$941$	5.00373e10	1.95763	0.978814	−	0.204752i	$-0.0656388\pi$
$941$	5.00373e10	1.95763	0.978814	+	0.204752i	$0.0656388\pi$
$942$	0	0
$943$	−1.56738e10	−0.608673
$944$	0	0
$945$	−2.58371e9	−0.0995937
$946$	0	0
$947$	−3.19677e10	−1.22317	−0.611584	−	0.791180i	$-0.709467\pi$
$947$	−3.19677e10	−1.22317	−0.611584	+	0.791180i	$0.709467\pi$
$948$	0	0
$949$	5.38219e10	2.04422
$950$	0	0
$951$	−4.64045e10	−1.74956
$952$	0	0
$953$	−3.10999e10	−1.16395	−0.581974	−	0.813208i	$-0.697719\pi$
$953$	−3.10999e10	−1.16395	−0.581974	+	0.813208i	$0.697719\pi$
$954$	0	0
$955$	−7.48125e9	−0.277947
$956$	0	0
$957$	−1.53609e10	−0.566532
$958$	0	0
$959$	2.10530e10	0.770813
$960$	0	0
$961$	−1.30007e10	−0.472537
$962$	0	0
$963$	−1.44654e10	−0.521963
$964$	0	0
$965$	9.42851e9	0.337752
$966$	0	0
$967$	−5.30176e10	−1.88550	−0.942751	−	0.333497i	$-0.891771\pi$
$967$	−5.30176e10	−1.88550	−0.942751	+	0.333497i	$0.891771\pi$
$968$	0	0
$969$	3.58261e10	1.26493
$970$	0	0
$971$	−4.84951e10	−1.69993	−0.849964	−	0.526841i	$-0.823376\pi$
$971$	−4.84951e10	−1.69993	−0.849964	+	0.526841i	$0.823376\pi$
$972$	0	0
$973$	−1.21411e10	−0.422537
$974$	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
$979$	−1.82026e10	−0.620004
$980$	0	0
$981$	2.94349e10	0.995455
$982$	0	0
$983$	2.41653e10	0.811438	0.405719	−	0.913998i	$-0.367021\pi$
$983$	2.41653e10	0.811438	0.405719	+	0.913998i	$0.367021\pi$
$984$	0	0
$985$	6.55628e9	0.218590
$986$	0	0
$987$	8.34198e10	2.76159
$988$	0	0
$989$	2.17386e10	0.714569
$990$	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
$995$	3.76335e9	0.121114
$996$	0	0
$997$	7.83024e9	0.250232	0.125116	−	0.992142i	$-0.460070\pi$
$997$	7.83024e9	0.250232	0.125116	+	0.992142i	$0.460070\pi$
$998$	0	0
$999$	1.16499e9	0.0369696

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.8.a.c.1.1	✓	2	1.1	even	1	trivial
32.8.a.c.1.2	yes	2	4.3	odd	2	inner
64.8.a.i.1.1		2	8.3	odd	2
64.8.a.i.1.2		2	8.5	even	2
256.8.b.i.129.1		4	16.13	even	4
256.8.b.i.129.2		4	16.11	odd	4
256.8.b.i.129.3		4	16.3	odd	4
256.8.b.i.129.4		4	16.5	even	4
288.8.a.k.1.1		2	3.2	odd	2
288.8.a.k.1.2		2	12.11	even	2
576.8.a.bk.1.1		2	24.5	odd	2
576.8.a.bk.1.2		2	24.11	even	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:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	32.a (trivial)

\(f(q)\)	\(=\)	\(q-61.9677 q^{3} +70.0000 q^{5} -1115.42 q^{7} +1653.00 q^{9} +O(q^{10})\)	\(q-61.9677 q^{3} +70.0000 q^{5} -1115.42 q^{7} +1653.00 q^{9} +7250.22 q^{11} +13758.0 q^{13} -4337.74 q^{15} +16994.0 q^{17} -34020.3 q^{19} +69120.0 q^{21} +32347.2 q^{23} -73225.0 q^{25} +33090.8 q^{27} +34190.0 q^{29} +120465. q^{31} -449280. q^{33} -78079.3 q^{35} +35206.0 q^{37} -852552. q^{39} -484550. q^{41} +672040. q^{43} +115710. q^{45} +1.20688e6 q^{47} +420617. q^{49} -1.05308e6 q^{51} +851702. q^{53} +507516. q^{55} +2.10816e6 q^{57} -695464. q^{59} +71630.0 q^{61} -1.84379e6 q^{63} +963060. q^{65} +307298. q^{67} -2.00448e6 q^{69} +757370. q^{71} +3.91204e6 q^{73} +4.53759e6 q^{75} -8.08704e6 q^{77} -314548. q^{79} -5.66567e6 q^{81} +1.53649e6 q^{83} +1.18958e6 q^{85} -2.11868e6 q^{87} -2.51063e6 q^{89} -1.53459e7 q^{91} -7.46496e6 q^{93} -2.38142e6 q^{95} -50094.0 q^{97} +1.19846e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 140 q^{5} + 3306 q^{9}+O(q^{10}) \) 2 * q + 140 * q^5 + 3306 * q^9	\( 2 q + 140 q^{5} + 3306 q^{9} + 27516 q^{13} + 33988 q^{17} + 138240 q^{21} - 146450 q^{25} + 68380 q^{29} - 898560 q^{33} + 70412 q^{37} - 969100 q^{41} + 231420 q^{45} + 841234 q^{49} + 1703404 q^{53} + 4216320 q^{57} + 143260 q^{61} + 1926120 q^{65} - 4008960 q^{69} + 7824084 q^{73} - 16174080 q^{77} - 11331342 q^{81} + 2379160 q^{85} - 5021260 q^{89} - 14929920 q^{93} - 100188 q^{97}+O(q^{100}) \) 2 * q + 140 * q^5 + 3306 * q^9 + 27516 * q^13 + 33988 * q^17 + 138240 * q^21 - 146450 * q^25 + 68380 * q^29 - 898560 * q^33 + 70412 * q^37 - 969100 * q^41 + 231420 * q^45 + 841234 * q^49 + 1703404 * q^53 + 4216320 * q^57 + 143260 * q^61 + 1926120 * q^65 - 4008960 * q^69 + 7824084 * q^73 - 16174080 * q^77 - 11331342 * q^81 + 2379160 * q^85 - 5021260 * q^89 - 14929920 * q^93 - 100188 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more