LMFDB - Embedded newform 8034.2.a.f.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:	$0$
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)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} -5.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} -5.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -5.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{20} +5.00000 q^{21} +1.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} +1.00000 q^{26} -1.00000 q^{27} -5.00000 q^{28} -4.00000 q^{30} +1.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} -20.0000 q^{35} +1.00000 q^{36} -5.00000 q^{37} -1.00000 q^{39} +4.00000 q^{40} +9.00000 q^{41} +5.00000 q^{42} +3.00000 q^{43} +1.00000 q^{44} +4.00000 q^{45} +1.00000 q^{46} -3.00000 q^{47} -1.00000 q^{48} +18.0000 q^{49} +11.0000 q^{50} -2.00000 q^{51} +1.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -5.00000 q^{56} -4.00000 q^{60} +13.0000 q^{61} -5.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -1.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} -1.00000 q^{69} -20.0000 q^{70} +8.00000 q^{71} +1.00000 q^{72} -10.0000 q^{73} -5.00000 q^{74} -11.0000 q^{75} -5.00000 q^{77} -1.00000 q^{78} -8.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +9.00000 q^{82} +5.00000 q^{84} +8.00000 q^{85} +3.00000 q^{86} +1.00000 q^{88} +4.00000 q^{89} +4.00000 q^{90} -5.00000 q^{91} +1.00000 q^{92} -3.00000 q^{94} -1.00000 q^{96} -8.00000 q^{97} +18.0000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	−1.00000	−0.408248
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	4.00000	1.26491
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−5.00000	−1.33631
$15$	−4.00000	−1.03280
$16$	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
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	4.00000	0.894427
$21$	5.00000	1.09109
$22$	1.00000	0.213201
$23$	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
$25$	11.0000	2.20000
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	−5.00000	−0.944911
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−4.00000	−0.730297
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	2.00000	0.342997
$35$	−20.0000	−3.38062
$36$	1.00000	0.166667
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	4.00000	0.632456
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	5.00000	0.771517
$43$	3.00000	0.457496	0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000	0.457496	0.228748	+	0.973486i	$0.426537\pi$
$44$	1.00000	0.150756
$45$	4.00000	0.596285
$46$	1.00000	0.147442
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−1.00000	−0.144338
$49$	18.0000	2.57143
$50$	11.0000	1.55563
$51$	−2.00000	−0.280056
$52$	1.00000	0.138675
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	4.00000	0.539360
$56$	−5.00000	−0.668153
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−4.00000	−0.516398
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	0	0
$63$	−5.00000	−0.629941
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	−1.00000	−0.123091
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	2.00000	0.242536
$69$	−1.00000	−0.120386
$70$	−20.0000	−2.39046
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	1.00000	0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−5.00000	−0.581238
$75$	−11.0000	−1.27017
$76$	0	0
$77$	−5.00000	−0.569803
$78$	−1.00000	−0.113228
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	9.00000	0.993884
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	5.00000	0.545545
$85$	8.00000	0.867722
$86$	3.00000	0.323498
$87$	0	0
$88$	1.00000	0.106600
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	4.00000	0.421637
$91$	−5.00000	−0.524142
$92$	1.00000	0.104257
$93$	0	0
$94$	−3.00000	−0.309426
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	18.0000	1.81827
$99$	1.00000	0.100504
$100$	11.0000	1.10000
$101$	19.0000	1.89057	0.945285	−	0.326245i	$-0.105783\pi$
$101$	19.0000	1.89057	0.945285	+	0.326245i	$0.105783\pi$
$102$	−2.00000	−0.198030
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	1.00000	0.0980581
$105$	20.0000	1.95180
$106$	−6.00000	−0.582772
$107$	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
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	4.00000	0.381385
$111$	5.00000	0.474579
$112$	−5.00000	−0.472456
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−10.0000	−0.916698
$120$	−4.00000	−0.365148
$121$	−10.0000	−0.909091
$122$	13.0000	1.17696
$123$	−9.00000	−0.811503
$124$	0	0
$125$	24.0000	2.14663
$126$	−5.00000	−0.445435
$127$	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
$129$	−3.00000	−0.264135
$130$	4.00000	0.350823
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	−4.00000	−0.345547
$135$	−4.00000	−0.344265
$136$	2.00000	0.171499
$137$	15.0000	1.28154	0.640768	−	0.767734i	$-0.278616\pi$
$137$	15.0000	1.28154	0.640768	+	0.767734i	$0.278616\pi$
$138$	−1.00000	−0.0851257
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	−20.0000	−1.69031
$141$	3.00000	0.252646
$142$	8.00000	0.671345
$143$	1.00000	0.0836242
$144$	1.00000	0.0833333
$145$	0	0
$146$	−10.0000	−0.827606
$147$	−18.0000	−1.48461
$148$	−5.00000	−0.410997
$149$	11.0000	0.901155	0.450578	−	0.892737i	$-0.351218\pi$
$149$	11.0000	0.901155	0.450578	+	0.892737i	$0.351218\pi$
$150$	−11.0000	−0.898146
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	−5.00000	−0.402911
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	−8.00000	−0.636446
$159$	6.00000	0.475831
$160$	4.00000	0.316228
$161$	−5.00000	−0.394055
$162$	1.00000	0.0785674
$163$	5.00000	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$163$	5.00000	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$164$	9.00000	0.702782
$165$	−4.00000	−0.311400
$166$	0	0
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	5.00000	0.385758
$169$	1.00000	0.0769231
$170$	8.00000	0.613572
$171$	0	0
$172$	3.00000	0.228748
$173$	13.0000	0.988372	0.494186	−	0.869356i	$-0.335466\pi$
$173$	13.0000	0.988372	0.494186	+	0.869356i	$0.335466\pi$
$174$	0	0
$175$	−55.0000	−4.15761
$176$	1.00000	0.0753778
$177$	0	0
$178$	4.00000	0.299813
$179$	17.0000	1.27064	0.635320	−	0.772249i	$-0.280868\pi$
$179$	17.0000	1.27064	0.635320	+	0.772249i	$0.280868\pi$
$180$	4.00000	0.298142
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	−5.00000	−0.370625
$183$	−13.0000	−0.960988
$184$	1.00000	0.0737210
$185$	−20.0000	−1.47043
$186$	0	0
$187$	2.00000	0.146254
$188$	−3.00000	−0.218797
$189$	5.00000	0.363696
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	−1.00000	−0.0721688
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	−8.00000	−0.574367
$195$	−4.00000	−0.286446
$196$	18.0000	1.28571
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	1.00000	0.0710669
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	11.0000	0.777817
$201$	4.00000	0.282138
$202$	19.0000	1.33684
$203$	0	0
$204$	−2.00000	−0.140028
$205$	36.0000	2.51435
$206$	1.00000	0.0696733
$207$	1.00000	0.0695048
$208$	1.00000	0.0693375
$209$	0	0
$210$	20.0000	1.38013
$211$	1.00000	0.0688428	0.0344214	−	0.999407i	$-0.489041\pi$
$211$	1.00000	0.0688428	0.0344214	+	0.999407i	$0.489041\pi$
$212$	−6.00000	−0.412082
$213$	−8.00000	−0.548151
$214$	12.0000	0.820303
$215$	12.0000	0.818393
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−1.00000	−0.0677285
$219$	10.0000	0.675737
$220$	4.00000	0.269680
$221$	2.00000	0.134535
$222$	5.00000	0.335578
$223$	13.0000	0.870544	0.435272	−	0.900299i	$-0.356652\pi$
$223$	13.0000	0.870544	0.435272	+	0.900299i	$0.356652\pi$
$224$	−5.00000	−0.334077
$225$	11.0000	0.733333
$226$	1.00000	0.0665190
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	4.00000	0.263752
$231$	5.00000	0.328976
$232$	0	0
$233$	9.00000	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$234$	1.00000	0.0653720
$235$	−12.0000	−0.782794
$236$	0	0
$237$	8.00000	0.519656
$238$	−10.0000	−0.648204
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	−4.00000	−0.258199
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	−10.0000	−0.642824
$243$	−1.00000	−0.0641500
$244$	13.0000	0.832240
$245$	72.0000	4.59991
$246$	−9.00000	−0.573819
$247$	0	0
$248$	0	0
$249$	0	0
$250$	24.0000	1.51789
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	−5.00000	−0.314970
$253$	1.00000	0.0628695
$254$	7.00000	0.439219
$255$	−8.00000	−0.500979
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	−3.00000	−0.186772
$259$	25.0000	1.55342
$260$	4.00000	0.248069
$261$	0	0
$262$	4.00000	0.247121
$263$	−14.0000	−0.863277	−0.431638	−	0.902047i	$-0.642064\pi$
$263$	−14.0000	−0.863277	−0.431638	+	0.902047i	$0.642064\pi$
$264$	−1.00000	−0.0615457
$265$	−24.0000	−1.47431
$266$	0	0
$267$	−4.00000	−0.244796
$268$	−4.00000	−0.244339
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	−4.00000	−0.243432
$271$	18.0000	1.09342	0.546711	−	0.837321i	$-0.315880\pi$
$271$	18.0000	1.09342	0.546711	+	0.837321i	$0.315880\pi$
$272$	2.00000	0.121268
$273$	5.00000	0.302614
$274$	15.0000	0.906183
$275$	11.0000	0.663325
$276$	−1.00000	−0.0601929
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−16.0000	−0.959616
$279$	0	0
$280$	−20.0000	−1.19523
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	3.00000	0.178647
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	1.00000	0.0591312
$287$	−45.0000	−2.65627
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	8.00000	0.468968
$292$	−10.0000	−0.585206
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	−18.0000	−1.04978
$295$	0	0
$296$	−5.00000	−0.290619
$297$	−1.00000	−0.0580259
$298$	11.0000	0.637213
$299$	1.00000	0.0578315
$300$	−11.0000	−0.635085
$301$	−15.0000	−0.864586
$302$	−12.0000	−0.690522
$303$	−19.0000	−1.09152
$304$	0	0
$305$	52.0000	2.97751
$306$	2.00000	0.114332
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	−5.00000	−0.284901
$309$	−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
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	8.00000	0.451466
$315$	−20.0000	−1.12687
$316$	−8.00000	−0.450035
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	6.00000	0.336463
$319$	0	0
$320$	4.00000	0.223607
$321$	−12.0000	−0.669775
$322$	−5.00000	−0.278639
$323$	0	0
$324$	1.00000	0.0555556
$325$	11.0000	0.610170
$326$	5.00000	0.276924
$327$	1.00000	0.0553001
$328$	9.00000	0.496942
$329$	15.0000	0.826977
$330$	−4.00000	−0.220193
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	0	0
$333$	−5.00000	−0.273998
$334$	18.0000	0.984916
$335$	−16.0000	−0.874173
$336$	5.00000	0.272772
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	1.00000	0.0543928
$339$	−1.00000	−0.0543125
$340$	8.00000	0.433861
$341$	0	0
$342$	0	0
$343$	−55.0000	−2.96972
$344$	3.00000	0.161749
$345$	−4.00000	−0.215353
$346$	13.0000	0.698884
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	−55.0000	−2.93987
$351$	−1.00000	−0.0533761
$352$	1.00000	0.0533002
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	32.0000	1.69838
$356$	4.00000	0.212000
$357$	10.0000	0.529256
$358$	17.0000	0.898478
$359$	−22.0000	−1.16112	−0.580558	−	0.814219i	$-0.697165\pi$
$359$	−22.0000	−1.16112	−0.580558	+	0.814219i	$0.697165\pi$
$360$	4.00000	0.210819
$361$	−19.0000	−1.00000
$362$	−2.00000	−0.105118
$363$	10.0000	0.524864
$364$	−5.00000	−0.262071
$365$	−40.0000	−2.09370
$366$	−13.0000	−0.679521
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	1.00000	0.0521286
$369$	9.00000	0.468521
$370$	−20.0000	−1.03975
$371$	30.0000	1.55752
$372$	0	0
$373$	−35.0000	−1.81223	−0.906116	−	0.423030i	$-0.860966\pi$
$373$	−35.0000	−1.81223	−0.906116	+	0.423030i	$0.860966\pi$
$374$	2.00000	0.103418
$375$	−24.0000	−1.23935
$376$	−3.00000	−0.154713
$377$	0	0
$378$	5.00000	0.257172
$379$	−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
$382$	16.0000	0.818631
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	−1.00000	−0.0510310
$385$	−20.0000	−1.01929
$386$	5.00000	0.254493
$387$	3.00000	0.152499
$388$	−8.00000	−0.406138
$389$	−3.00000	−0.152106	−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000	−0.152106	−0.0760530	+	0.997104i	$0.524232\pi$
$390$	−4.00000	−0.202548
$391$	2.00000	0.101144
$392$	18.0000	0.909137
$393$	−4.00000	−0.201773
$394$	−8.00000	−0.403034
$395$	−32.0000	−1.61009
$396$	1.00000	0.0502519
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	3.00000	0.150376
$399$	0	0
$400$	11.0000	0.550000
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	19.0000	0.945285
$405$	4.00000	0.198762
$406$	0	0
$407$	−5.00000	−0.247841
$408$	−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	$0.256617\pi$
$410$	36.0000	1.77791
$411$	−15.0000	−0.739895
$412$	1.00000	0.0492665
$413$	0	0
$414$	1.00000	0.0491473
$415$	0	0
$416$	1.00000	0.0490290
$417$	16.0000	0.783523
$418$	0	0
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	20.0000	0.975900
$421$	−24.0000	−1.16969	−0.584844	−	0.811146i	$-0.698844\pi$
$421$	−24.0000	−1.16969	−0.584844	+	0.811146i	$0.698844\pi$
$422$	1.00000	0.0486792
$423$	−3.00000	−0.145865
$424$	−6.00000	−0.291386
$425$	22.0000	1.06716
$426$	−8.00000	−0.387601
$427$	−65.0000	−3.14557
$428$	12.0000	0.580042
$429$	−1.00000	−0.0482805
$430$	12.0000	0.578691
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	−1.00000	−0.0481125
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	0	0
$435$	0	0
$436$	−1.00000	−0.0478913
$437$	0	0
$438$	10.0000	0.477818
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	4.00000	0.190693
$441$	18.0000	0.857143
$442$	2.00000	0.0951303
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	5.00000	0.237289
$445$	16.0000	0.758473
$446$	13.0000	0.615568
$447$	−11.0000	−0.520282
$448$	−5.00000	−0.236228
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	11.0000	0.518545
$451$	9.00000	0.423793
$452$	1.00000	0.0470360
$453$	12.0000	0.563809
$454$	12.0000	0.563188
$455$	−20.0000	−0.937614
$456$	0	0
$457$	−3.00000	−0.140334	−0.0701670	−	0.997535i	$-0.522353\pi$
$457$	−3.00000	−0.140334	−0.0701670	+	0.997535i	$0.522353\pi$
$458$	−8.00000	−0.373815
$459$	−2.00000	−0.0933520
$460$	4.00000	0.186501
$461$	9.00000	0.419172	0.209586	−	0.977790i	$-0.432788\pi$
$461$	9.00000	0.419172	0.209586	+	0.977790i	$0.432788\pi$
$462$	5.00000	0.232621
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	0	0
$465$	0	0
$466$	9.00000	0.416917
$467$	15.0000	0.694117	0.347059	−	0.937843i	$-0.387180\pi$
$467$	15.0000	0.694117	0.347059	+	0.937843i	$0.387180\pi$
$468$	1.00000	0.0462250
$469$	20.0000	0.923514
$470$	−12.0000	−0.553519
$471$	−8.00000	−0.368621
$472$	0	0
$473$	3.00000	0.137940
$474$	8.00000	0.367452
$475$	0	0
$476$	−10.0000	−0.458349
$477$	−6.00000	−0.274721
$478$	−20.0000	−0.914779
$479$	25.0000	1.14228	0.571140	−	0.820853i	$-0.306501\pi$
$479$	25.0000	1.14228	0.571140	+	0.820853i	$0.306501\pi$
$480$	−4.00000	−0.182574
$481$	−5.00000	−0.227980
$482$	7.00000	0.318841
$483$	5.00000	0.227508
$484$	−10.0000	−0.454545
$485$	−32.0000	−1.45305
$486$	−1.00000	−0.0453609
$487$	−14.0000	−0.634401	−0.317200	−	0.948359i	$-0.602743\pi$
$487$	−14.0000	−0.634401	−0.317200	+	0.948359i	$0.602743\pi$
$488$	13.0000	0.588482
$489$	−5.00000	−0.226108
$490$	72.0000	3.25263
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	−9.00000	−0.405751
$493$	0	0
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	−40.0000	−1.79425
$498$	0	0
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	24.0000	1.07331
$501$	−18.0000	−0.804181
$502$	28.0000	1.24970
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	−5.00000	−0.222718
$505$	76.0000	3.38196
$506$	1.00000	0.0444554
$507$	−1.00000	−0.0444116
$508$	7.00000	0.310575
$509$	−13.0000	−0.576215	−0.288107	−	0.957598i	$-0.593026\pi$
$509$	−13.0000	−0.576215	−0.288107	+	0.957598i	$0.593026\pi$
$510$	−8.00000	−0.354246
$511$	50.0000	2.21187
$512$	1.00000	0.0441942
$513$	0	0
$514$	−14.0000	−0.617514
$515$	4.00000	0.176261
$516$	−3.00000	−0.132068
$517$	−3.00000	−0.131940
$518$	25.0000	1.09844
$519$	−13.0000	−0.570637
$520$	4.00000	0.175412
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	0	0
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	4.00000	0.174741
$525$	55.0000	2.40040
$526$	−14.0000	−0.610429
$527$	0	0
$528$	−1.00000	−0.0435194
$529$	−22.0000	−0.956522
$530$	−24.0000	−1.04249
$531$	0	0
$532$	0	0
$533$	9.00000	0.389833
$534$	−4.00000	−0.173097
$535$	48.0000	2.07522
$536$	−4.00000	−0.172774
$537$	−17.0000	−0.733604
$538$	30.0000	1.29339
$539$	18.0000	0.775315
$540$	−4.00000	−0.172133
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	18.0000	0.773166
$543$	2.00000	0.0858282
$544$	2.00000	0.0857493
$545$	−4.00000	−0.171341
$546$	5.00000	0.213980
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	15.0000	0.640768
$549$	13.0000	0.554826
$550$	11.0000	0.469042
$551$	0	0
$552$	−1.00000	−0.0425628
$553$	40.0000	1.70097
$554$	8.00000	0.339887
$555$	20.0000	0.848953
$556$	−16.0000	−0.678551
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	3.00000	0.126886
$560$	−20.0000	−0.845154
$561$	−2.00000	−0.0844401
$562$	18.0000	0.759284
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	3.00000	0.126323
$565$	4.00000	0.168281
$566$	−13.0000	−0.546431
$567$	−5.00000	−0.209980
$568$	8.00000	0.335673
$569$	3.00000	0.125767	0.0628833	−	0.998021i	$-0.479970\pi$
$569$	3.00000	0.125767	0.0628833	+	0.998021i	$0.479970\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	1.00000	0.0418121
$573$	−16.0000	−0.668410
$574$	−45.0000	−1.87826
$575$	11.0000	0.458732
$576$	1.00000	0.0416667
$577$	23.0000	0.957503	0.478751	−	0.877951i	$-0.341090\pi$
$577$	23.0000	0.957503	0.478751	+	0.877951i	$0.341090\pi$
$578$	−13.0000	−0.540729
$579$	−5.00000	−0.207793
$580$	0	0
$581$	0	0
$582$	8.00000	0.331611
$583$	−6.00000	−0.248495
$584$	−10.0000	−0.413803
$585$	4.00000	0.165380
$586$	−30.0000	−1.23929
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	−18.0000	−0.742307
$589$	0	0
$590$	0	0
$591$	8.00000	0.329076
$592$	−5.00000	−0.205499
$593$	−10.0000	−0.410651	−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000	−0.410651	−0.205325	+	0.978694i	$0.565825\pi$
$594$	−1.00000	−0.0410305
$595$	−40.0000	−1.63984
$596$	11.0000	0.450578
$597$	−3.00000	−0.122782
$598$	1.00000	0.0408930
$599$	26.0000	1.06233	0.531166	−	0.847268i	$-0.321754\pi$
$599$	26.0000	1.06233	0.531166	+	0.847268i	$0.321754\pi$
$600$	−11.0000	−0.449073
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	−15.0000	−0.611354
$603$	−4.00000	−0.162893
$604$	−12.0000	−0.488273
$605$	−40.0000	−1.62623
$606$	−19.0000	−0.771822
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	0	0
$609$	0	0
$610$	52.0000	2.10542
$611$	−3.00000	−0.121367
$612$	2.00000	0.0808452
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	−16.0000	−0.645707
$615$	−36.0000	−1.45166
$616$	−5.00000	−0.201456
$617$	28.0000	1.12724	0.563619	−	0.826035i	$-0.309409\pi$
$617$	28.0000	1.12724	0.563619	+	0.826035i	$0.309409\pi$
$618$	−1.00000	−0.0402259
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	−15.0000	−0.601445
$623$	−20.0000	−0.801283
$624$	−1.00000	−0.0400320
$625$	41.0000	1.64000
$626$	−19.0000	−0.759393
$627$	0	0
$628$	8.00000	0.319235
$629$	−10.0000	−0.398726
$630$	−20.0000	−0.796819
$631$	−49.0000	−1.95066	−0.975330	−	0.220754i	$-0.929148\pi$
$631$	−49.0000	−1.95066	−0.975330	+	0.220754i	$0.929148\pi$
$632$	−8.00000	−0.318223
$633$	−1.00000	−0.0397464
$634$	−22.0000	−0.873732
$635$	28.0000	1.11115
$636$	6.00000	0.237915
$637$	18.0000	0.713186
$638$	0	0
$639$	8.00000	0.316475
$640$	4.00000	0.158114
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	−12.0000	−0.473602
$643$	−33.0000	−1.30139	−0.650696	−	0.759338i	$-0.725523\pi$
$643$	−33.0000	−1.30139	−0.650696	+	0.759338i	$0.725523\pi$
$644$	−5.00000	−0.197028
$645$	−12.0000	−0.472500
$646$	0	0
$647$	−11.0000	−0.432455	−0.216227	−	0.976343i	$-0.569375\pi$
$647$	−11.0000	−0.432455	−0.216227	+	0.976343i	$0.569375\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	11.0000	0.431455
$651$	0	0
$652$	5.00000	0.195815
$653$	−5.00000	−0.195665	−0.0978326	−	0.995203i	$-0.531191\pi$
$653$	−5.00000	−0.195665	−0.0978326	+	0.995203i	$0.531191\pi$
$654$	1.00000	0.0391031
$655$	16.0000	0.625172
$656$	9.00000	0.351391
$657$	−10.0000	−0.390137
$658$	15.0000	0.584761
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	−4.00000	−0.155700
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	10.0000	0.388661
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	0	0
$666$	−5.00000	−0.193746
$667$	0	0
$668$	18.0000	0.696441
$669$	−13.0000	−0.502609
$670$	−16.0000	−0.618134
$671$	13.0000	0.501859
$672$	5.00000	0.192879
$673$	41.0000	1.58043	0.790217	−	0.612827i	$-0.209968\pi$
$673$	41.0000	1.58043	0.790217	+	0.612827i	$0.209968\pi$
$674$	13.0000	0.500741
$675$	−11.0000	−0.423390
$676$	1.00000	0.0384615
$677$	46.0000	1.76792	0.883962	−	0.467559i	$-0.154866\pi$
$677$	46.0000	1.76792	0.883962	+	0.467559i	$0.154866\pi$
$678$	−1.00000	−0.0384048
$679$	40.0000	1.53506
$680$	8.00000	0.306786
$681$	−12.0000	−0.459841
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	60.0000	2.29248
$686$	−55.0000	−2.09991
$687$	8.00000	0.305219
$688$	3.00000	0.114374
$689$	−6.00000	−0.228582
$690$	−4.00000	−0.152277
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	13.0000	0.494186
$693$	−5.00000	−0.189934
$694$	−3.00000	−0.113878
$695$	−64.0000	−2.42766
$696$	0	0
$697$	18.0000	0.681799
$698$	−14.0000	−0.529908
$699$	−9.00000	−0.340411
$700$	−55.0000	−2.07880
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	−1.00000	−0.0377426
$703$	0	0
$704$	1.00000	0.0376889
$705$	12.0000	0.451946
$706$	−30.0000	−1.12906
$707$	−95.0000	−3.57284
$708$	0	0
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	32.0000	1.20094
$711$	−8.00000	−0.300023
$712$	4.00000	0.149906
$713$	0	0
$714$	10.0000	0.374241
$715$	4.00000	0.149592
$716$	17.0000	0.635320
$717$	20.0000	0.746914
$718$	−22.0000	−0.821033
$719$	38.0000	1.41716	0.708580	−	0.705630i	$-0.249336\pi$
$719$	38.0000	1.41716	0.708580	+	0.705630i	$0.249336\pi$
$720$	4.00000	0.149071
$721$	−5.00000	−0.186210
$722$	−19.0000	−0.707107
$723$	−7.00000	−0.260333
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	10.0000	0.371135
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	−5.00000	−0.185312
$729$	1.00000	0.0370370
$730$	−40.0000	−1.48047
$731$	6.00000	0.221918
$732$	−13.0000	−0.480494
$733$	−7.00000	−0.258551	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	−7.00000	−0.258551	−0.129275	+	0.991609i	$0.541265\pi$
$734$	18.0000	0.664392
$735$	−72.0000	−2.65576
$736$	1.00000	0.0368605
$737$	−4.00000	−0.147342
$738$	9.00000	0.331295
$739$	−25.0000	−0.919640	−0.459820	−	0.888012i	$-0.652086\pi$
$739$	−25.0000	−0.919640	−0.459820	+	0.888012i	$0.652086\pi$
$740$	−20.0000	−0.735215
$741$	0	0
$742$	30.0000	1.10133
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	44.0000	1.61204
$746$	−35.0000	−1.28144
$747$	0	0
$748$	2.00000	0.0731272
$749$	−60.0000	−2.19235
$750$	−24.0000	−0.876356
$751$	−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
$753$	−28.0000	−1.02038
$754$	0	0
$755$	−48.0000	−1.74690
$756$	5.00000	0.181848
$757$	−9.00000	−0.327111	−0.163555	−	0.986534i	$-0.552296\pi$
$757$	−9.00000	−0.327111	−0.163555	+	0.986534i	$0.552296\pi$
$758$	−2.00000	−0.0726433
$759$	−1.00000	−0.0362977
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	−7.00000	−0.253583
$763$	5.00000	0.181012
$764$	16.0000	0.578860
$765$	8.00000	0.289241
$766$	21.0000	0.758761
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−31.0000	−1.11789	−0.558944	−	0.829205i	$-0.688793\pi$
$769$	−31.0000	−1.11789	−0.558944	+	0.829205i	$0.688793\pi$
$770$	−20.0000	−0.720750
$771$	14.0000	0.504198
$772$	5.00000	0.179954
$773$	−33.0000	−1.18693	−0.593464	−	0.804861i	$-0.702240\pi$
$773$	−33.0000	−1.18693	−0.593464	+	0.804861i	$0.702240\pi$
$774$	3.00000	0.107833
$775$	0	0
$776$	−8.00000	−0.287183
$777$	−25.0000	−0.896870
$778$	−3.00000	−0.107555
$779$	0	0
$780$	−4.00000	−0.143223
$781$	8.00000	0.286263
$782$	2.00000	0.0715199
$783$	0	0
$784$	18.0000	0.642857
$785$	32.0000	1.14213
$786$	−4.00000	−0.142675
$787$	−13.0000	−0.463400	−0.231700	−	0.972787i	$-0.574429\pi$
$787$	−13.0000	−0.463400	−0.231700	+	0.972787i	$0.574429\pi$
$788$	−8.00000	−0.284988
$789$	14.0000	0.498413
$790$	−32.0000	−1.13851
$791$	−5.00000	−0.177780
$792$	1.00000	0.0355335
$793$	13.0000	0.461644
$794$	−2.00000	−0.0709773
$795$	24.0000	0.851192
$796$	3.00000	0.106332
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	11.0000	0.388909
$801$	4.00000	0.141333
$802$	15.0000	0.529668
$803$	−10.0000	−0.352892
$804$	4.00000	0.141069
$805$	−20.0000	−0.704907
$806$	0	0
$807$	−30.0000	−1.05605
$808$	19.0000	0.668418
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	4.00000	0.140546
$811$	−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
$813$	−18.0000	−0.631288
$814$	−5.00000	−0.175250
$815$	20.0000	0.700569
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	28.0000	0.978997
$819$	−5.00000	−0.174714
$820$	36.0000	1.25717
$821$	−37.0000	−1.29131	−0.645654	−	0.763630i	$-0.723415\pi$
$821$	−37.0000	−1.29131	−0.645654	+	0.763630i	$0.723415\pi$
$822$	−15.0000	−0.523185
$823$	−3.00000	−0.104573	−0.0522867	−	0.998632i	$-0.516651\pi$
$823$	−3.00000	−0.104573	−0.0522867	+	0.998632i	$0.516651\pi$
$824$	1.00000	0.0348367
$825$	−11.0000	−0.382971
$826$	0	0
$827$	39.0000	1.35616	0.678081	−	0.734987i	$-0.262812\pi$
$827$	39.0000	1.35616	0.678081	+	0.734987i	$0.262812\pi$
$828$	1.00000	0.0347524
$829$	−28.0000	−0.972480	−0.486240	−	0.873825i	$-0.661632\pi$
$829$	−28.0000	−0.972480	−0.486240	+	0.873825i	$0.661632\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	1.00000	0.0346688
$833$	36.0000	1.24733
$834$	16.0000	0.554035
$835$	72.0000	2.49166
$836$	0	0
$837$	0	0
$838$	9.00000	0.310900
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	20.0000	0.690066
$841$	−29.0000	−1.00000
$842$	−24.0000	−0.827095
$843$	−18.0000	−0.619953
$844$	1.00000	0.0344214
$845$	4.00000	0.137604
$846$	−3.00000	−0.103142
$847$	50.0000	1.71802
$848$	−6.00000	−0.206041
$849$	13.0000	0.446159
$850$	22.0000	0.754594
$851$	−5.00000	−0.171398
$852$	−8.00000	−0.274075
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	−65.0000	−2.22425
$855$	0	0
$856$	12.0000	0.410152
$857$	−28.0000	−0.956462	−0.478231	−	0.878234i	$-0.658722\pi$
$857$	−28.0000	−0.956462	−0.478231	+	0.878234i	$0.658722\pi$
$858$	−1.00000	−0.0341394
$859$	19.0000	0.648272	0.324136	−	0.946011i	$-0.394927\pi$
$859$	19.0000	0.648272	0.324136	+	0.946011i	$0.394927\pi$
$860$	12.0000	0.409197
$861$	45.0000	1.53360
$862$	10.0000	0.340601
$863$	3.00000	0.102121	0.0510606	−	0.998696i	$-0.483740\pi$
$863$	3.00000	0.102121	0.0510606	+	0.998696i	$0.483740\pi$
$864$	−1.00000	−0.0340207
$865$	52.0000	1.76805
$866$	8.00000	0.271851
$867$	13.0000	0.441503
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−4.00000	−0.135535
$872$	−1.00000	−0.0338643
$873$	−8.00000	−0.270759
$874$	0	0
$875$	−120.000	−4.05674
$876$	10.0000	0.337869
$877$	51.0000	1.72215	0.861074	−	0.508480i	$-0.169792\pi$
$877$	51.0000	1.72215	0.861074	+	0.508480i	$0.169792\pi$
$878$	1.00000	0.0337484
$879$	30.0000	1.01187
$880$	4.00000	0.134840
$881$	50.0000	1.68454	0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000	1.68454	0.842271	+	0.539054i	$0.181218\pi$
$882$	18.0000	0.606092
$883$	−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
$886$	28.0000	0.940678
$887$	−37.0000	−1.24234	−0.621169	−	0.783676i	$-0.713342\pi$
$887$	−37.0000	−1.24234	−0.621169	+	0.783676i	$0.713342\pi$
$888$	5.00000	0.167789
$889$	−35.0000	−1.17386
$890$	16.0000	0.536321
$891$	1.00000	0.0335013
$892$	13.0000	0.435272
$893$	0	0
$894$	−11.0000	−0.367895
$895$	68.0000	2.27299
$896$	−5.00000	−0.167038
$897$	−1.00000	−0.0333890
$898$	2.00000	0.0667409
$899$	0	0
$900$	11.0000	0.366667
$901$	−12.0000	−0.399778
$902$	9.00000	0.299667
$903$	15.0000	0.499169
$904$	1.00000	0.0332595
$905$	−8.00000	−0.265929
$906$	12.0000	0.398673
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	12.0000	0.398234
$909$	19.0000	0.630190
$910$	−20.0000	−0.662994
$911$	42.0000	1.39152	0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000	1.39152	0.695761	+	0.718273i	$0.255067\pi$
$912$	0	0
$913$	0	0
$914$	−3.00000	−0.0992312
$915$	−52.0000	−1.71907
$916$	−8.00000	−0.264327
$917$	−20.0000	−0.660458
$918$	−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	$0.353089\pi$
$920$	4.00000	0.131876
$921$	16.0000	0.527218
$922$	9.00000	0.296399
$923$	8.00000	0.263323
$924$	5.00000	0.164488
$925$	−55.0000	−1.80839
$926$	−20.0000	−0.657241
$927$	1.00000	0.0328443
$928$	0	0
$929$	−29.0000	−0.951459	−0.475730	−	0.879592i	$-0.657816\pi$
$929$	−29.0000	−0.951459	−0.475730	+	0.879592i	$0.657816\pi$
$930$	0	0
$931$	0	0
$932$	9.00000	0.294805
$933$	15.0000	0.491078
$934$	15.0000	0.490815
$935$	8.00000	0.261628
$936$	1.00000	0.0326860
$937$	16.0000	0.522697	0.261349	−	0.965244i	$-0.415833\pi$
$937$	16.0000	0.522697	0.261349	+	0.965244i	$0.415833\pi$
$938$	20.0000	0.653023
$939$	19.0000	0.620042
$940$	−12.0000	−0.391397
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	−8.00000	−0.260654
$943$	9.00000	0.293080
$944$	0	0
$945$	20.0000	0.650600
$946$	3.00000	0.0975384
$947$	5.00000	0.162478	0.0812391	−	0.996695i	$-0.474112\pi$
$947$	5.00000	0.162478	0.0812391	+	0.996695i	$0.474112\pi$
$948$	8.00000	0.259828
$949$	−10.0000	−0.324614
$950$	0	0
$951$	22.0000	0.713399
$952$	−10.0000	−0.324102
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	−6.00000	−0.194257
$955$	64.0000	2.07099
$956$	−20.0000	−0.646846
$957$	0	0
$958$	25.0000	0.807713
$959$	−75.0000	−2.42188
$960$	−4.00000	−0.129099
$961$	−31.0000	−1.00000
$962$	−5.00000	−0.161206
$963$	12.0000	0.386695
$964$	7.00000	0.225455
$965$	20.0000	0.643823
$966$	5.00000	0.160872
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	−10.0000	−0.321412
$969$	0	0
$970$	−32.0000	−1.02746
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	−1.00000	−0.0320750
$973$	80.0000	2.56468
$974$	−14.0000	−0.448589
$975$	−11.0000	−0.352282
$976$	13.0000	0.416120
$977$	−1.00000	−0.0319928	−0.0159964	−	0.999872i	$-0.505092\pi$
$977$	−1.00000	−0.0319928	−0.0159964	+	0.999872i	$0.505092\pi$
$978$	−5.00000	−0.159882
$979$	4.00000	0.127841
$980$	72.0000	2.29996
$981$	−1.00000	−0.0319275
$982$	24.0000	0.765871
$983$	−14.0000	−0.446531	−0.223265	−	0.974758i	$-0.571672\pi$
$983$	−14.0000	−0.446531	−0.223265	+	0.974758i	$0.571672\pi$
$984$	−9.00000	−0.286910
$985$	−32.0000	−1.01960
$986$	0	0
$987$	−15.0000	−0.477455
$988$	0	0
$989$	3.00000	0.0953945
$990$	4.00000	0.127128
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	−10.0000	−0.317340
$994$	−40.0000	−1.26872
$995$	12.0000	0.380426
$996$	0	0
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	−14.0000	−0.443162
$999$	5.00000	0.158193

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.f.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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