LMFDB - Embedded newform 8034.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +8.00000 q^{19} +3.00000 q^{21} +5.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -3.00000 q^{28} -4.00000 q^{31} -1.00000 q^{32} +5.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -7.00000 q^{37} -8.00000 q^{38} -1.00000 q^{39} +7.00000 q^{41} -3.00000 q^{42} -1.00000 q^{43} -5.00000 q^{44} -1.00000 q^{46} +3.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +5.00000 q^{50} -2.00000 q^{51} +1.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} +3.00000 q^{56} -8.00000 q^{57} +8.00000 q^{59} +1.00000 q^{61} +4.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} +8.00000 q^{67} +2.00000 q^{68} -1.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} -6.00000 q^{73} +7.00000 q^{74} +5.00000 q^{75} +8.00000 q^{76} +15.0000 q^{77} +1.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} -7.00000 q^{82} +8.00000 q^{83} +3.00000 q^{84} +1.00000 q^{86} +5.00000 q^{88} +12.0000 q^{89} -3.00000 q^{91} +1.00000 q^{92} +4.00000 q^{93} -3.00000 q^{94} +1.00000 q^{96} +8.00000 q^{97} -2.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.00000	0.408248
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	3.00000	0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−1.00000	−0.235702
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	5.00000	1.06600
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	−3.00000	−0.566947
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	5.00000	0.870388
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−8.00000	−1.29777
$39$	−1.00000	−0.160128
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	−3.00000	−0.462910
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	−1.00000	−0.147442
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	5.00000	0.707107
$51$	−2.00000	−0.280056
$52$	1.00000	0.138675
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	−8.00000	−1.05963
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	4.00000	0.508001
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.00000	−0.615457
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	2.00000	0.242536
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−1.00000	−0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	7.00000	0.813733
$75$	5.00000	0.577350
$76$	8.00000	0.917663
$77$	15.0000	1.70941
$78$	1.00000	0.113228
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−7.00000	−0.773021
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	1.00000	0.107833
$87$	0	0
$88$	5.00000	0.533002
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	1.00000	0.104257
$93$	4.00000	0.414781
$94$	−3.00000	−0.309426
$95$	0	0
$96$	1.00000	0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	−2.00000	−0.202031
$99$	−5.00000	−0.502519
$100$	−5.00000	−0.500000
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	2.00000	0.198030
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−2.00000	−0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	13.0000	1.24517	0.622587	−	0.782551i	$-0.286082\pi$
$109$	13.0000	1.24517	0.622587	+	0.782551i	$0.286082\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	−3.00000	−0.283473
$113$	17.0000	1.59923	0.799613	−	0.600516i	$-0.205038\pi$
$113$	17.0000	1.59923	0.799613	+	0.600516i	$0.205038\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	−8.00000	−0.736460
$119$	−6.00000	−0.550019
$120$	0	0
$121$	14.0000	1.27273
$122$	−1.00000	−0.0905357
$123$	−7.00000	−0.631169
$124$	−4.00000	−0.359211
$125$	0	0
$126$	3.00000	0.267261
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	−1.00000	−0.0883883
$129$	1.00000	0.0880451
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	5.00000	0.435194
$133$	−24.0000	−2.08106
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−2.00000	−0.171499
$137$	17.0000	1.45241	0.726204	−	0.687479i	$-0.241283\pi$
$137$	17.0000	1.45241	0.726204	+	0.687479i	$0.241283\pi$
$138$	1.00000	0.0851257
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	8.00000	0.671345
$143$	−5.00000	−0.418121
$144$	1.00000	0.0833333
$145$	0	0
$146$	6.00000	0.496564
$147$	−2.00000	−0.164957
$148$	−7.00000	−0.575396
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	−5.00000	−0.408248
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−8.00000	−0.648886
$153$	2.00000	0.161690
$154$	−15.0000	−1.20873
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	16.0000	1.27289
$159$	−2.00000	−0.158610
$160$	0	0
$161$	−3.00000	−0.236433
$162$	−1.00000	−0.0785674
$163$	−9.00000	−0.704934	−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000	−0.704934	−0.352467	+	0.935824i	$0.614657\pi$
$164$	7.00000	0.546608
$165$	0	0
$166$	−8.00000	−0.620920
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	−3.00000	−0.231455
$169$	1.00000	0.0769231
$170$	0	0
$171$	8.00000	0.611775
$172$	−1.00000	−0.0762493
$173$	−7.00000	−0.532200	−0.266100	−	0.963945i	$-0.585735\pi$
$173$	−7.00000	−0.532200	−0.266100	+	0.963945i	$0.585735\pi$
$174$	0	0
$175$	15.0000	1.13389
$176$	−5.00000	−0.376889
$177$	−8.00000	−0.601317
$178$	−12.0000	−0.899438
$179$	5.00000	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	3.00000	0.222375
$183$	−1.00000	−0.0739221
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	−4.00000	−0.293294
$187$	−10.0000	−0.731272
$188$	3.00000	0.218797
$189$	3.00000	0.218218
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−1.00000	−0.0721688
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	2.00000	0.142857
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	5.00000	0.355335
$199$	−21.0000	−1.48865	−0.744325	−	0.667817i	$-0.767229\pi$
$199$	−21.0000	−1.48865	−0.744325	+	0.667817i	$0.767229\pi$
$200$	5.00000	0.353553
$201$	−8.00000	−0.564276
$202$	9.00000	0.633238
$203$	0	0
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−1.00000	−0.0696733
$207$	1.00000	0.0695048
$208$	1.00000	0.0693375
$209$	−40.0000	−2.76686
$210$	0	0
$211$	−11.0000	−0.757271	−0.378636	−	0.925546i	$-0.623607\pi$
$211$	−11.0000	−0.757271	−0.378636	+	0.925546i	$0.623607\pi$
$212$	2.00000	0.137361
$213$	8.00000	0.548151
$214$	−12.0000	−0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	12.0000	0.814613
$218$	−13.0000	−0.880471
$219$	6.00000	0.405442
$220$	0	0
$221$	2.00000	0.134535
$222$	−7.00000	−0.469809
$223$	−21.0000	−1.40626	−0.703132	−	0.711059i	$-0.748216\pi$
$223$	−21.0000	−1.40626	−0.703132	+	0.711059i	$0.748216\pi$
$224$	3.00000	0.200446
$225$	−5.00000	−0.333333
$226$	−17.0000	−1.13082
$227$	−28.0000	−1.85843	−0.929213	−	0.369546i	$-0.879513\pi$
$227$	−28.0000	−1.85843	−0.929213	+	0.369546i	$0.879513\pi$
$228$	−8.00000	−0.529813
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	−15.0000	−0.986928
$232$	0	0
$233$	17.0000	1.11371	0.556854	−	0.830611i	$-0.312008\pi$
$233$	17.0000	1.11371	0.556854	+	0.830611i	$0.312008\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	8.00000	0.520756
$237$	16.0000	1.03931
$238$	6.00000	0.388922
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	−14.0000	−0.899954
$243$	−1.00000	−0.0641500
$244$	1.00000	0.0640184
$245$	0	0
$246$	7.00000	0.446304
$247$	8.00000	0.509028
$248$	4.00000	0.254000
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−3.00000	−0.188982
$253$	−5.00000	−0.314347
$254$	−7.00000	−0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−1.00000	−0.0622573
$259$	21.0000	1.30488
$260$	0	0
$261$	0	0
$262$	20.0000	1.23560
$263$	−14.0000	−0.863277	−0.431638	−	0.902047i	$-0.642064\pi$
$263$	−14.0000	−0.863277	−0.431638	+	0.902047i	$0.642064\pi$
$264$	−5.00000	−0.307729
$265$	0	0
$266$	24.0000	1.47153
$267$	−12.0000	−0.734388
$268$	8.00000	0.488678
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	2.00000	0.121268
$273$	3.00000	0.181568
$274$	−17.0000	−1.02701
$275$	25.0000	1.50756
$276$	−1.00000	−0.0601929
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	−12.0000	−0.719712
$279$	−4.00000	−0.239474
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	3.00000	0.178647
$283$	−9.00000	−0.534994	−0.267497	−	0.963559i	$-0.586197\pi$
$283$	−9.00000	−0.534994	−0.267497	+	0.963559i	$0.586197\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	5.00000	0.295656
$287$	−21.0000	−1.23959
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−8.00000	−0.468968
$292$	−6.00000	−0.351123
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	7.00000	0.406867
$297$	5.00000	0.290129
$298$	15.0000	0.868927
$299$	1.00000	0.0578315
$300$	5.00000	0.288675
$301$	3.00000	0.172917
$302$	−4.00000	−0.230174
$303$	9.00000	0.517036
$304$	8.00000	0.458831
$305$	0	0
$306$	−2.00000	−0.114332
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	15.0000	0.854704
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	1.00000	0.0566139
$313$	−35.0000	−1.97832	−0.989158	−	0.146852i	$-0.953086\pi$
$313$	−35.0000	−1.97832	−0.989158	+	0.146852i	$0.953086\pi$
$314$	8.00000	0.451466
$315$	0	0
$316$	−16.0000	−0.900070
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	2.00000	0.112154
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	3.00000	0.167183
$323$	16.0000	0.890264
$324$	1.00000	0.0555556
$325$	−5.00000	−0.277350
$326$	9.00000	0.498464
$327$	−13.0000	−0.718902
$328$	−7.00000	−0.386510
$329$	−9.00000	−0.496186
$330$	0	0
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	8.00000	0.439057
$333$	−7.00000	−0.383598
$334$	−18.0000	−0.984916
$335$	0	0
$336$	3.00000	0.163663
$337$	21.0000	1.14394	0.571971	−	0.820274i	$-0.306179\pi$
$337$	21.0000	1.14394	0.571971	+	0.820274i	$0.306179\pi$
$338$	−1.00000	−0.0543928
$339$	−17.0000	−0.923313
$340$	0	0
$341$	20.0000	1.08306
$342$	−8.00000	−0.432590
$343$	15.0000	0.809924
$344$	1.00000	0.0539164
$345$	0	0
$346$	7.00000	0.376322
$347$	−7.00000	−0.375780	−0.187890	−	0.982190i	$-0.560165\pi$
$347$	−7.00000	−0.375780	−0.187890	+	0.982190i	$0.560165\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	−15.0000	−0.801784
$351$	−1.00000	−0.0533761
$352$	5.00000	0.266501
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	12.0000	0.635999
$357$	6.00000	0.317554
$358$	−5.00000	−0.264258
$359$	−14.0000	−0.738892	−0.369446	−	0.929252i	$-0.620452\pi$
$359$	−14.0000	−0.738892	−0.369446	+	0.929252i	$0.620452\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	2.00000	0.105118
$363$	−14.0000	−0.734809
$364$	−3.00000	−0.157243
$365$	0	0
$366$	1.00000	0.0522708
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	1.00000	0.0521286
$369$	7.00000	0.364405
$370$	0	0
$371$	−6.00000	−0.311504
$372$	4.00000	0.207390
$373$	−7.00000	−0.362446	−0.181223	−	0.983442i	$-0.558006\pi$
$373$	−7.00000	−0.362446	−0.181223	+	0.983442i	$0.558006\pi$
$374$	10.0000	0.517088
$375$	0	0
$376$	−3.00000	−0.154713
$377$	0	0
$378$	−3.00000	−0.154303
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	−7.00000	−0.358621
$382$	12.0000	0.613973
$383$	−29.0000	−1.48183	−0.740915	−	0.671598i	$-0.765608\pi$
$383$	−29.0000	−1.48183	−0.740915	+	0.671598i	$0.765608\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−11.0000	−0.559885
$387$	−1.00000	−0.0508329
$388$	8.00000	0.406138
$389$	9.00000	0.456318	0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000	0.456318	0.228159	+	0.973624i	$0.426729\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	−2.00000	−0.101015
$393$	20.0000	1.00887
$394$	12.0000	0.604551
$395$	0	0
$396$	−5.00000	−0.251259
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	21.0000	1.05263
$399$	24.0000	1.20150
$400$	−5.00000	−0.250000
$401$	−23.0000	−1.14857	−0.574283	−	0.818657i	$-0.694719\pi$
$401$	−23.0000	−1.14857	−0.574283	+	0.818657i	$0.694719\pi$
$402$	8.00000	0.399004
$403$	−4.00000	−0.199254
$404$	−9.00000	−0.447767
$405$	0	0
$406$	0	0
$407$	35.0000	1.73489
$408$	2.00000	0.0990148
$409$	28.0000	1.38451	0.692255	−	0.721653i	$-0.256617\pi$
$409$	28.0000	1.38451	0.692255	+	0.721653i	$0.256617\pi$
$410$	0	0
$411$	−17.0000	−0.838548
$412$	1.00000	0.0492665
$413$	−24.0000	−1.18096
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	−12.0000	−0.587643
$418$	40.0000	1.95646
$419$	−35.0000	−1.70986	−0.854931	−	0.518742i	$-0.826401\pi$
$419$	−35.0000	−1.70986	−0.854931	+	0.518742i	$0.826401\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	11.0000	0.535472
$423$	3.00000	0.145865
$424$	−2.00000	−0.0971286
$425$	−10.0000	−0.485071
$426$	−8.00000	−0.387601
$427$	−3.00000	−0.145180
$428$	12.0000	0.580042
$429$	5.00000	0.241402
$430$	0	0
$431$	14.0000	0.674356	0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000	0.674356	0.337178	+	0.941441i	$0.390528\pi$
$432$	−1.00000	−0.0481125
$433$	−8.00000	−0.384455	−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000	−0.384455	−0.192228	+	0.981350i	$0.561571\pi$
$434$	−12.0000	−0.576018
$435$	0	0
$436$	13.0000	0.622587
$437$	8.00000	0.382692
$438$	−6.00000	−0.286691
$439$	−23.0000	−1.09773	−0.548865	−	0.835911i	$-0.684940\pi$
$439$	−23.0000	−1.09773	−0.548865	+	0.835911i	$0.684940\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	−2.00000	−0.0951303
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	7.00000	0.332205
$445$	0	0
$446$	21.0000	0.994379
$447$	15.0000	0.709476
$448$	−3.00000	−0.141737
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	5.00000	0.235702
$451$	−35.0000	−1.64809
$452$	17.0000	0.799613
$453$	−4.00000	−0.187936
$454$	28.0000	1.31411
$455$	0	0
$456$	8.00000	0.374634
$457$	−21.0000	−0.982339	−0.491169	−	0.871064i	$-0.663430\pi$
$457$	−21.0000	−0.982339	−0.491169	+	0.871064i	$0.663430\pi$
$458$	20.0000	0.934539
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	15.0000	0.697863
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	0	0
$465$	0	0
$466$	−17.0000	−0.787510
$467$	−37.0000	−1.71216	−0.856078	−	0.516847i	$-0.827106\pi$
$467$	−37.0000	−1.71216	−0.856078	+	0.516847i	$0.827106\pi$
$468$	1.00000	0.0462250
$469$	−24.0000	−1.10822
$470$	0	0
$471$	8.00000	0.368621
$472$	−8.00000	−0.368230
$473$	5.00000	0.229900
$474$	−16.0000	−0.734904
$475$	−40.0000	−1.83533
$476$	−6.00000	−0.275010
$477$	2.00000	0.0915737
$478$	16.0000	0.731823
$479$	−9.00000	−0.411220	−0.205610	−	0.978634i	$-0.565918\pi$
$479$	−9.00000	−0.411220	−0.205610	+	0.978634i	$0.565918\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	7.00000	0.318841
$483$	3.00000	0.136505
$484$	14.0000	0.636364
$485$	0	0
$486$	1.00000	0.0453609
$487$	−30.0000	−1.35943	−0.679715	−	0.733476i	$-0.737896\pi$
$487$	−30.0000	−1.35943	−0.679715	+	0.733476i	$0.737896\pi$
$488$	−1.00000	−0.0452679
$489$	9.00000	0.406994
$490$	0	0
$491$	32.0000	1.44414	0.722070	−	0.691820i	$-0.243191\pi$
$491$	32.0000	1.44414	0.722070	+	0.691820i	$0.243191\pi$
$492$	−7.00000	−0.315584
$493$	0	0
$494$	−8.00000	−0.359937
$495$	0	0
$496$	−4.00000	−0.179605
$497$	24.0000	1.07655
$498$	8.00000	0.358489
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	12.0000	0.535586
$503$	−33.0000	−1.47140	−0.735699	−	0.677309i	$-0.763146\pi$
$503$	−33.0000	−1.47140	−0.735699	+	0.677309i	$0.763146\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	5.00000	0.222277
$507$	−1.00000	−0.0444116
$508$	7.00000	0.310575
$509$	1.00000	0.0443242	0.0221621	−	0.999754i	$-0.492945\pi$
$509$	1.00000	0.0443242	0.0221621	+	0.999754i	$0.492945\pi$
$510$	0	0
$511$	18.0000	0.796273
$512$	−1.00000	−0.0441942
$513$	−8.00000	−0.353209
$514$	−18.0000	−0.793946
$515$	0	0
$516$	1.00000	0.0440225
$517$	−15.0000	−0.659699
$518$	−21.0000	−0.922687
$519$	7.00000	0.307266
$520$	0	0
$521$	5.00000	0.219054	0.109527	−	0.993984i	$-0.465066\pi$
$521$	5.00000	0.219054	0.109527	+	0.993984i	$0.465066\pi$
$522$	0	0
$523$	−30.0000	−1.31181	−0.655904	−	0.754844i	$-0.727712\pi$
$523$	−30.0000	−1.31181	−0.655904	+	0.754844i	$0.727712\pi$
$524$	−20.0000	−0.873704
$525$	−15.0000	−0.654654
$526$	14.0000	0.610429
$527$	−8.00000	−0.348485
$528$	5.00000	0.217597
$529$	−22.0000	−0.956522
$530$	0	0
$531$	8.00000	0.347170
$532$	−24.0000	−1.04053
$533$	7.00000	0.303204
$534$	12.0000	0.519291
$535$	0	0
$536$	−8.00000	−0.345547
$537$	−5.00000	−0.215766
$538$	10.0000	0.431131
$539$	−10.0000	−0.430730
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	2.00000	0.0859074
$543$	2.00000	0.0858282
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	−3.00000	−0.128388
$547$	−18.0000	−0.769624	−0.384812	−	0.922995i	$-0.625734\pi$
$547$	−18.0000	−0.769624	−0.384812	+	0.922995i	$0.625734\pi$
$548$	17.0000	0.726204
$549$	1.00000	0.0426790
$550$	−25.0000	−1.06600
$551$	0	0
$552$	1.00000	0.0425628
$553$	48.0000	2.04117
$554$	4.00000	0.169944
$555$	0	0
$556$	12.0000	0.508913
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	4.00000	0.169334
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	10.0000	0.422200
$562$	−30.0000	−1.26547
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	−3.00000	−0.126323
$565$	0	0
$566$	9.00000	0.378298
$567$	−3.00000	−0.125988
$568$	8.00000	0.335673
$569$	−37.0000	−1.55112	−0.775560	−	0.631273i	$-0.782533\pi$
$569$	−37.0000	−1.55112	−0.775560	+	0.631273i	$0.782533\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	−5.00000	−0.209061
$573$	12.0000	0.501307
$574$	21.0000	0.876523
$575$	−5.00000	−0.208514
$576$	1.00000	0.0416667
$577$	−39.0000	−1.62359	−0.811796	−	0.583942i	$-0.801510\pi$
$577$	−39.0000	−1.62359	−0.811796	+	0.583942i	$0.801510\pi$
$578$	13.0000	0.540729
$579$	−11.0000	−0.457144
$580$	0	0
$581$	−24.0000	−0.995688
$582$	8.00000	0.331611
$583$	−10.0000	−0.414158
$584$	6.00000	0.248282
$585$	0	0
$586$	−26.0000	−1.07405
$587$	40.0000	1.65098	0.825488	−	0.564419i	$-0.190900\pi$
$587$	40.0000	1.65098	0.825488	+	0.564419i	$0.190900\pi$
$588$	−2.00000	−0.0824786
$589$	−32.0000	−1.31854
$590$	0	0
$591$	12.0000	0.493614
$592$	−7.00000	−0.287698
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	−5.00000	−0.205152
$595$	0	0
$596$	−15.0000	−0.614424
$597$	21.0000	0.859473
$598$	−1.00000	−0.0408930
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	−5.00000	−0.204124
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	−3.00000	−0.122271
$603$	8.00000	0.325785
$604$	4.00000	0.162758
$605$	0	0
$606$	−9.00000	−0.365600
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	−8.00000	−0.324443
$609$	0	0
$610$	0	0
$611$	3.00000	0.121367
$612$	2.00000	0.0808452
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	−15.0000	−0.604367
$617$	40.0000	1.61034	0.805170	−	0.593045i	$-0.202074\pi$
$617$	40.0000	1.61034	0.805170	+	0.593045i	$0.202074\pi$
$618$	1.00000	0.0402259
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	15.0000	0.601445
$623$	−36.0000	−1.44231
$624$	−1.00000	−0.0400320
$625$	25.0000	1.00000
$626$	35.0000	1.39888
$627$	40.0000	1.59745
$628$	−8.00000	−0.319235
$629$	−14.0000	−0.558217
$630$	0	0
$631$	1.00000	0.0398094	0.0199047	−	0.999802i	$-0.493664\pi$
$631$	1.00000	0.0398094	0.0199047	+	0.999802i	$0.493664\pi$
$632$	16.0000	0.636446
$633$	11.0000	0.437211
$634$	−30.0000	−1.19145
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	2.00000	0.0792429
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−16.0000	−0.631962	−0.315981	−	0.948766i	$-0.602334\pi$
$641$	−16.0000	−0.631962	−0.315981	+	0.948766i	$0.602334\pi$
$642$	12.0000	0.473602
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	−3.00000	−0.118217
$645$	0	0
$646$	−16.0000	−0.629512
$647$	13.0000	0.511083	0.255541	−	0.966798i	$-0.417746\pi$
$647$	13.0000	0.511083	0.255541	+	0.966798i	$0.417746\pi$
$648$	−1.00000	−0.0392837
$649$	−40.0000	−1.57014
$650$	5.00000	0.196116
$651$	−12.0000	−0.470317
$652$	−9.00000	−0.352467
$653$	−9.00000	−0.352197	−0.176099	−	0.984373i	$-0.556348\pi$
$653$	−9.00000	−0.352197	−0.176099	+	0.984373i	$0.556348\pi$
$654$	13.0000	0.508340
$655$	0	0
$656$	7.00000	0.273304
$657$	−6.00000	−0.234082
$658$	9.00000	0.350857
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	26.0000	1.01052
$663$	−2.00000	−0.0776736
$664$	−8.00000	−0.310460
$665$	0	0
$666$	7.00000	0.271244
$667$	0	0
$668$	18.0000	0.696441
$669$	21.0000	0.811907
$670$	0	0
$671$	−5.00000	−0.193023
$672$	−3.00000	−0.115728
$673$	−31.0000	−1.19496	−0.597481	−	0.801883i	$-0.703832\pi$
$673$	−31.0000	−1.19496	−0.597481	+	0.801883i	$0.703832\pi$
$674$	−21.0000	−0.808890
$675$	5.00000	0.192450
$676$	1.00000	0.0384615
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	17.0000	0.652881
$679$	−24.0000	−0.921035
$680$	0	0
$681$	28.0000	1.07296
$682$	−20.0000	−0.765840
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	8.00000	0.305888
$685$	0	0
$686$	−15.0000	−0.572703
$687$	20.0000	0.763048
$688$	−1.00000	−0.0381246
$689$	2.00000	0.0761939
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	−7.00000	−0.266100
$693$	15.0000	0.569803
$694$	7.00000	0.265716
$695$	0	0
$696$	0	0
$697$	14.0000	0.530288
$698$	−30.0000	−1.13552
$699$	−17.0000	−0.642999
$700$	15.0000	0.566947
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	1.00000	0.0377426
$703$	−56.0000	−2.11208
$704$	−5.00000	−0.188445
$705$	0	0
$706$	−18.0000	−0.677439
$707$	27.0000	1.01544
$708$	−8.00000	−0.300658
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	−12.0000	−0.449719
$713$	−4.00000	−0.149801
$714$	−6.00000	−0.224544
$715$	0	0
$716$	5.00000	0.186859
$717$	16.0000	0.597531
$718$	14.0000	0.522475
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	−3.00000	−0.111726
$722$	−45.0000	−1.67473
$723$	7.00000	0.260333
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	14.0000	0.519589
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	3.00000	0.111187
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.00000	−0.0739727
$732$	−1.00000	−0.0369611
$733$	−13.0000	−0.480166	−0.240083	−	0.970752i	$-0.577175\pi$
$733$	−13.0000	−0.480166	−0.240083	+	0.970752i	$0.577175\pi$
$734$	−22.0000	−0.812035
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−40.0000	−1.47342
$738$	−7.00000	−0.257674
$739$	5.00000	0.183928	0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000	0.183928	0.0919640	+	0.995762i	$0.470686\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	6.00000	0.220267
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	7.00000	0.256288
$747$	8.00000	0.292705
$748$	−10.0000	−0.365636
$749$	−36.0000	−1.31541
$750$	0	0
$751$	−30.0000	−1.09472	−0.547358	−	0.836899i	$-0.684366\pi$
$751$	−30.0000	−1.09472	−0.547358	+	0.836899i	$0.684366\pi$
$752$	3.00000	0.109399
$753$	12.0000	0.437304
$754$	0	0
$755$	0	0
$756$	3.00000	0.109109
$757$	−5.00000	−0.181728	−0.0908640	−	0.995863i	$-0.528963\pi$
$757$	−5.00000	−0.181728	−0.0908640	+	0.995863i	$0.528963\pi$
$758$	2.00000	0.0726433
$759$	5.00000	0.181489
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	7.00000	0.253583
$763$	−39.0000	−1.41189
$764$	−12.0000	−0.434145
$765$	0	0
$766$	29.0000	1.04781
$767$	8.00000	0.288863
$768$	−1.00000	−0.0360844
$769$	−33.0000	−1.19001	−0.595005	−	0.803722i	$-0.702850\pi$
$769$	−33.0000	−1.19001	−0.595005	+	0.803722i	$0.702850\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	11.0000	0.395899
$773$	21.0000	0.755318	0.377659	−	0.925945i	$-0.376729\pi$
$773$	21.0000	0.755318	0.377659	+	0.925945i	$0.376729\pi$
$774$	1.00000	0.0359443
$775$	20.0000	0.718421
$776$	−8.00000	−0.287183
$777$	−21.0000	−0.753371
$778$	−9.00000	−0.322666
$779$	56.0000	2.00641
$780$	0	0
$781$	40.0000	1.43131
$782$	−2.00000	−0.0715199
$783$	0	0
$784$	2.00000	0.0714286
$785$	0	0
$786$	−20.0000	−0.713376
$787$	25.0000	0.891154	0.445577	−	0.895244i	$-0.352999\pi$
$787$	25.0000	0.891154	0.445577	+	0.895244i	$0.352999\pi$
$788$	−12.0000	−0.427482
$789$	14.0000	0.498413
$790$	0	0
$791$	−51.0000	−1.81335
$792$	5.00000	0.177667
$793$	1.00000	0.0355110
$794$	14.0000	0.496841
$795$	0	0
$796$	−21.0000	−0.744325
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	−24.0000	−0.849591
$799$	6.00000	0.212265
$800$	5.00000	0.176777
$801$	12.0000	0.423999
$802$	23.0000	0.812158
$803$	30.0000	1.05868
$804$	−8.00000	−0.282138
$805$	0	0
$806$	4.00000	0.140894
$807$	10.0000	0.352017
$808$	9.00000	0.316619
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	0	0
$813$	2.00000	0.0701431
$814$	−35.0000	−1.22675
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	−8.00000	−0.279885
$818$	−28.0000	−0.978997
$819$	−3.00000	−0.104828
$820$	0	0
$821$	17.0000	0.593304	0.296652	−	0.954986i	$-0.404130\pi$
$821$	17.0000	0.593304	0.296652	+	0.954986i	$0.404130\pi$
$822$	17.0000	0.592943
$823$	37.0000	1.28974	0.644869	−	0.764293i	$-0.276912\pi$
$823$	37.0000	1.28974	0.644869	+	0.764293i	$0.276912\pi$
$824$	−1.00000	−0.0348367
$825$	−25.0000	−0.870388
$826$	24.0000	0.835067
$827$	5.00000	0.173867	0.0869335	−	0.996214i	$-0.472293\pi$
$827$	5.00000	0.173867	0.0869335	+	0.996214i	$0.472293\pi$
$828$	1.00000	0.0347524
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	4.00000	0.138758
$832$	1.00000	0.0346688
$833$	4.00000	0.138592
$834$	12.0000	0.415526
$835$	0	0
$836$	−40.0000	−1.38343
$837$	4.00000	0.138260
$838$	35.0000	1.20905
$839$	26.0000	0.897620	0.448810	−	0.893627i	$-0.351848\pi$
$839$	26.0000	0.897620	0.448810	+	0.893627i	$0.351848\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−8.00000	−0.275698
$843$	−30.0000	−1.03325
$844$	−11.0000	−0.378636
$845$	0	0
$846$	−3.00000	−0.103142
$847$	−42.0000	−1.44314
$848$	2.00000	0.0686803
$849$	9.00000	0.308879
$850$	10.0000	0.342997
$851$	−7.00000	−0.239957
$852$	8.00000	0.274075
$853$	−8.00000	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$853$	−8.00000	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$854$	3.00000	0.102658
$855$	0	0
$856$	−12.0000	−0.410152
$857$	20.0000	0.683187	0.341593	−	0.939848i	$-0.389033\pi$
$857$	20.0000	0.683187	0.341593	+	0.939848i	$0.389033\pi$
$858$	−5.00000	−0.170697
$859$	−41.0000	−1.39890	−0.699451	−	0.714681i	$-0.746572\pi$
$859$	−41.0000	−1.39890	−0.699451	+	0.714681i	$0.746572\pi$
$860$	0	0
$861$	21.0000	0.715678
$862$	−14.0000	−0.476842
$863$	−19.0000	−0.646768	−0.323384	−	0.946268i	$-0.604820\pi$
$863$	−19.0000	−0.646768	−0.323384	+	0.946268i	$0.604820\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	8.00000	0.271851
$867$	13.0000	0.441503
$868$	12.0000	0.407307
$869$	80.0000	2.71381
$870$	0	0
$871$	8.00000	0.271070
$872$	−13.0000	−0.440236
$873$	8.00000	0.270759
$874$	−8.00000	−0.270604
$875$	0	0
$876$	6.00000	0.202721
$877$	41.0000	1.38447	0.692236	−	0.721671i	$-0.256626\pi$
$877$	41.0000	1.38447	0.692236	+	0.721671i	$0.256626\pi$
$878$	23.0000	0.776212
$879$	−26.0000	−0.876958
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	−2.00000	−0.0673435
$883$	−10.0000	−0.336527	−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000	−0.336527	−0.168263	+	0.985742i	$0.553816\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	0	0
$887$	−53.0000	−1.77957	−0.889783	−	0.456384i	$-0.849144\pi$
$887$	−53.0000	−1.77957	−0.889783	+	0.456384i	$0.849144\pi$
$888$	−7.00000	−0.234905
$889$	−21.0000	−0.704317
$890$	0	0
$891$	−5.00000	−0.167506
$892$	−21.0000	−0.703132
$893$	24.0000	0.803129
$894$	−15.0000	−0.501675
$895$	0	0
$896$	3.00000	0.100223
$897$	−1.00000	−0.0333890
$898$	−6.00000	−0.200223
$899$	0	0
$900$	−5.00000	−0.166667
$901$	4.00000	0.133259
$902$	35.0000	1.16537
$903$	−3.00000	−0.0998337
$904$	−17.0000	−0.565412
$905$	0	0
$906$	4.00000	0.132891
$907$	46.0000	1.52740	0.763702	−	0.645568i	$-0.223379\pi$
$907$	46.0000	1.52740	0.763702	+	0.645568i	$0.223379\pi$
$908$	−28.0000	−0.929213
$909$	−9.00000	−0.298511
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	−8.00000	−0.264906
$913$	−40.0000	−1.32381
$914$	21.0000	0.694618
$915$	0	0
$916$	−20.0000	−0.660819
$917$	60.0000	1.98137
$918$	2.00000	0.0660098
$919$	27.0000	0.890648	0.445324	−	0.895370i	$-0.353089\pi$
$919$	27.0000	0.890648	0.445324	+	0.895370i	$0.353089\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	−3.00000	−0.0987997
$923$	−8.00000	−0.263323
$924$	−15.0000	−0.493464
$925$	35.0000	1.15079
$926$	−4.00000	−0.131448
$927$	1.00000	0.0328443
$928$	0	0
$929$	−27.0000	−0.885841	−0.442921	−	0.896561i	$-0.646058\pi$
$929$	−27.0000	−0.885841	−0.442921	+	0.896561i	$0.646058\pi$
$930$	0	0
$931$	16.0000	0.524379
$932$	17.0000	0.556854
$933$	15.0000	0.491078
$934$	37.0000	1.21068
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	28.0000	0.914720	0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000	0.914720	0.457360	+	0.889282i	$0.348795\pi$
$938$	24.0000	0.783628
$939$	35.0000	1.14218
$940$	0	0
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	−8.00000	−0.260654
$943$	7.00000	0.227951
$944$	8.00000	0.260378
$945$	0	0
$946$	−5.00000	−0.162564
$947$	−17.0000	−0.552426	−0.276213	−	0.961096i	$-0.589079\pi$
$947$	−17.0000	−0.552426	−0.276213	+	0.961096i	$0.589079\pi$
$948$	16.0000	0.519656
$949$	−6.00000	−0.194768
$950$	40.0000	1.29777
$951$	−30.0000	−0.972817
$952$	6.00000	0.194461
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	−16.0000	−0.517477
$957$	0	0
$958$	9.00000	0.290777
$959$	−51.0000	−1.64688
$960$	0	0
$961$	−15.0000	−0.483871
$962$	7.00000	0.225689
$963$	12.0000	0.386695
$964$	−7.00000	−0.225455
$965$	0	0
$966$	−3.00000	−0.0965234
$967$	−12.0000	−0.385894	−0.192947	−	0.981209i	$-0.561805\pi$
$967$	−12.0000	−0.385894	−0.192947	+	0.981209i	$0.561805\pi$
$968$	−14.0000	−0.449977
$969$	−16.0000	−0.513994
$970$	0	0
$971$	32.0000	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$972$	−1.00000	−0.0320750
$973$	−36.0000	−1.15411
$974$	30.0000	0.961262
$975$	5.00000	0.160128
$976$	1.00000	0.0320092
$977$	1.00000	0.0319928	0.0159964	−	0.999872i	$-0.494908\pi$
$977$	1.00000	0.0319928	0.0159964	+	0.999872i	$0.494908\pi$
$978$	−9.00000	−0.287788
$979$	−60.0000	−1.91761
$980$	0	0
$981$	13.0000	0.415058
$982$	−32.0000	−1.02116
$983$	−34.0000	−1.08443	−0.542216	−	0.840239i	$-0.682414\pi$
$983$	−34.0000	−1.08443	−0.542216	+	0.840239i	$0.682414\pi$
$984$	7.00000	0.223152
$985$	0	0
$986$	0	0
$987$	9.00000	0.286473
$988$	8.00000	0.254514
$989$	−1.00000	−0.0317982
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	4.00000	0.127000
$993$	26.0000	0.825085
$994$	−24.0000	−0.761234
$995$	0	0
$996$	−8.00000	−0.253490
$997$	50.0000	1.58352	0.791758	−	0.610835i	$-0.209166\pi$
$997$	50.0000	1.58352	0.791758	+	0.610835i	$0.209166\pi$
$998$	−22.0000	−0.696398
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.b.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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