LMFDB - Embedded newform 7616.2.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7616, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.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:	$60.8140661794$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.13448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.546295$ of defining polynomial
Character	$\chi$	$=$	7616.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.40815 q^{3} -1.29341 q^{5} -1.00000 q^{7} +2.79917 q^{9} -1.09259 q^{11} -2.00000 q^{13} +3.11473 q^{15} +1.00000 q^{17} +2.40815 q^{21} +4.81630 q^{23} -3.32708 q^{25} +0.483617 q^{27} -9.32206 q^{29} +2.97283 q^{31} +2.63112 q^{33} +1.29341 q^{35} -0.907411 q^{37} +4.81630 q^{39} +5.90239 q^{41} +8.31703 q^{43} -3.62049 q^{45} -7.64264 q^{47} +1.00000 q^{49} -2.40815 q^{51} -5.21235 q^{53} +1.41317 q^{55} +2.18518 q^{59} -4.40815 q^{61} -2.79917 q^{63} +2.58683 q^{65} +3.01712 q^{67} -11.5983 q^{69} +5.01152 q^{71} +5.54502 q^{73} +8.01210 q^{75} +1.09259 q^{77} +7.64264 q^{79} -9.56214 q^{81} +17.2752 q^{83} -1.29341 q^{85} +22.4489 q^{87} +15.2309 q^{89} +2.00000 q^{91} -7.15902 q^{93} +7.60395 q^{97} -3.05835 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9} - 8 q^{13} + 4 q^{17} - 3 q^{21} - 6 q^{23} + 3 q^{25} + 6 q^{27} - 8 q^{29} + 7 q^{31} - 6 q^{33} + 5 q^{35} - 8 q^{37} - 6 q^{39} + 15 q^{41} - 9 q^{43} + 2 q^{45}+ \cdots - 50 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - 5 * q^5 - 4 * q^7 + 7 * q^9 - 8 * q^13 + 4 * q^17 - 3 * q^21 - 6 * q^23 + 3 * q^25 + 6 * q^27 - 8 * q^29 + 7 * q^31 - 6 * q^33 + 5 * q^35 - 8 * q^37 - 6 * q^39 + 15 * q^41 - 9 * q^43 + 2 * q^45 - 6 * q^47 + 4 * q^49 + 3 * q^51 - 17 * q^53 + 6 * q^55 - 5 * q^61 - 7 * q^63 + 10 * q^65 - 9 * q^67 - 38 * q^69 + 12 * q^71 - 11 * q^73 - 2 * q^75 + 6 * q^79 + 16 * q^81 - 6 * q^83 - 5 * q^85 + 14 * q^87 + 2 * q^89 + 8 * q^91 + 9 * q^97 - 50 * q^99

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
$3$	−2.40815	−1.39034	−0.695172	−	0.718843i	$-0.744672\pi$
$3$	−2.40815	−1.39034	−0.695172	+	0.718843i	$0.744672\pi$
$4$	0	0
$5$	−1.29341	−0.578433	−0.289216	−	0.957264i	$-0.593395\pi$
$5$	−1.29341	−0.578433	−0.289216	+	0.957264i	$0.593395\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.79917	0.933058
$10$	0	0
$11$	−1.09259	−0.329428	−0.164714	−	0.986341i	$-0.552670\pi$
$11$	−1.09259	−0.329428	−0.164714	+	0.986341i	$0.552670\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	3.11473	0.804221
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	2.40815	0.525501
$22$	0	0
$23$	4.81630	1.00427	0.502133	−	0.864790i	$-0.332549\pi$
$23$	4.81630	1.00427	0.502133	+	0.864790i	$0.332549\pi$
$24$	0	0
$25$	−3.32708	−0.665416
$26$	0	0
$27$	0.483617	0.0930721
$28$	0	0
$29$	−9.32206	−1.73106	−0.865531	−	0.500855i	$-0.833019\pi$
$29$	−9.32206	−1.73106	−0.865531	+	0.500855i	$0.833019\pi$
$30$	0	0
$31$	2.97283	0.533937	0.266968	−	0.963705i	$-0.413978\pi$
$31$	2.97283	0.533937	0.266968	+	0.963705i	$0.413978\pi$
$32$	0	0
$33$	2.63112	0.458019
$34$	0	0
$35$	1.29341	0.218627
$36$	0	0
$37$	−0.907411	−0.149177	−0.0745887	−	0.997214i	$-0.523764\pi$
$37$	−0.907411	−0.149177	−0.0745887	+	0.997214i	$0.523764\pi$
$38$	0	0
$39$	4.81630	0.771224
$40$	0	0
$41$	5.90239	0.921798	0.460899	−	0.887453i	$-0.347527\pi$
$41$	5.90239	0.921798	0.460899	+	0.887453i	$0.347527\pi$
$42$	0	0
$43$	8.31703	1.26834	0.634168	−	0.773195i	$-0.281343\pi$
$43$	8.31703	1.26834	0.634168	+	0.773195i	$0.281343\pi$
$44$	0	0
$45$	−3.62049	−0.539711
$46$	0	0
$47$	−7.64264	−1.11479	−0.557397	−	0.830246i	$-0.688200\pi$
$47$	−7.64264	−1.11479	−0.557397	+	0.830246i	$0.688200\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.40815	−0.337208
$52$	0	0
$53$	−5.21235	−0.715971	−0.357985	−	0.933727i	$-0.616536\pi$
$53$	−5.21235	−0.715971	−0.357985	+	0.933727i	$0.616536\pi$
$54$	0	0
$55$	1.41317	0.190552
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.18518	0.284486	0.142243	−	0.989832i	$-0.454569\pi$
$59$	2.18518	0.284486	0.142243	+	0.989832i	$0.454569\pi$
$60$	0	0
$61$	−4.40815	−0.564405	−0.282203	−	0.959355i	$-0.591065\pi$
$61$	−4.40815	−0.564405	−0.282203	+	0.959355i	$0.591065\pi$
$62$	0	0
$63$	−2.79917	−0.352663
$64$	0	0
$65$	2.58683	0.320857
$66$	0	0
$67$	3.01712	0.368600	0.184300	−	0.982870i	$-0.440998\pi$
$67$	3.01712	0.368600	0.184300	+	0.982870i	$0.440998\pi$
$68$	0	0
$69$	−11.5983	−1.39628
$70$	0	0
$71$	5.01152	0.594758	0.297379	−	0.954760i	$-0.403888\pi$
$71$	5.01152	0.594758	0.297379	+	0.954760i	$0.403888\pi$
$72$	0	0
$73$	5.54502	0.648996	0.324498	−	0.945886i	$-0.394805\pi$
$73$	5.54502	0.648996	0.324498	+	0.945886i	$0.394805\pi$
$74$	0	0
$75$	8.01210	0.925157
$76$	0	0
$77$	1.09259	0.124512
$78$	0	0
$79$	7.64264	0.859864	0.429932	−	0.902861i	$-0.358538\pi$
$79$	7.64264	0.859864	0.429932	+	0.902861i	$0.358538\pi$
$80$	0	0
$81$	−9.56214	−1.06246
$82$	0	0
$83$	17.2752	1.89620	0.948101	−	0.317968i	$-0.103001\pi$
$83$	17.2752	1.89620	0.948101	+	0.317968i	$0.103001\pi$
$84$	0	0
$85$	−1.29341	−0.140291
$86$	0	0
$87$	22.4489	2.40677
$88$	0	0
$89$	15.2309	1.61448	0.807238	−	0.590226i	$-0.200961\pi$
$89$	15.2309	1.61448	0.807238	+	0.590226i	$0.200961\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	−7.15902	−0.742356
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.60395	0.772064	0.386032	−	0.922485i	$-0.373845\pi$
$97$	7.60395	0.772064	0.386032	+	0.922485i	$0.373845\pi$
$98$	0	0
$99$	−3.05835	−0.307376
$100$	0	0
$101$	4.22947	0.420848	0.210424	−	0.977610i	$-0.432516\pi$
$101$	4.22947	0.420848	0.210424	+	0.977610i	$0.432516\pi$
$102$	0	0
$103$	−12.1751	−1.19965	−0.599826	−	0.800131i	$-0.704763\pi$
$103$	−12.1751	−1.19965	−0.599826	+	0.800131i	$0.704763\pi$
$104$	0	0
$105$	−3.11473	−0.303967
$106$	0	0
$107$	3.91893	0.378857	0.189429	−	0.981894i	$-0.439336\pi$
$107$	3.91893	0.378857	0.189429	+	0.981894i	$0.439336\pi$
$108$	0	0
$109$	3.13688	0.300458	0.150229	−	0.988651i	$-0.451999\pi$
$109$	3.13688	0.300458	0.150229	+	0.988651i	$0.451999\pi$
$110$	0	0
$111$	2.18518	0.207408
$112$	0	0
$113$	−14.8606	−1.39797	−0.698983	−	0.715138i	$-0.746364\pi$
$113$	−14.8606	−1.39797	−0.698983	+	0.715138i	$0.746364\pi$
$114$	0	0
$115$	−6.22947	−0.580901
$116$	0	0
$117$	−5.59835	−0.517568
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−9.80625	−0.891477
$122$	0	0
$123$	−14.2138	−1.28162
$124$	0	0
$125$	10.7704	0.963331
$126$	0	0
$127$	0.210872	0.0187119	0.00935593	−	0.999956i	$-0.497022\pi$
$127$	0.210872	0.0187119	0.00935593	+	0.999956i	$0.497022\pi$
$128$	0	0
$129$	−20.0286	−1.76342
$130$	0	0
$131$	6.31053	0.551354	0.275677	−	0.961250i	$-0.411098\pi$
$131$	6.31053	0.551354	0.275677	+	0.961250i	$0.411098\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.625517	−0.0538359
$136$	0	0
$137$	−2.67440	−0.228489	−0.114244	−	0.993453i	$-0.536445\pi$
$137$	−2.67440	−0.228489	−0.114244	+	0.993453i	$0.536445\pi$
$138$	0	0
$139$	−1.83596	−0.155724	−0.0778619	−	0.996964i	$-0.524809\pi$
$139$	−1.83596	−0.155724	−0.0778619	+	0.996964i	$0.524809\pi$
$140$	0	0
$141$	18.4046	1.54995
$142$	0	0
$143$	2.18518	0.182734
$144$	0	0
$145$	12.0573	1.00130
$146$	0	0
$147$	−2.40815	−0.198621
$148$	0	0
$149$	−1.32560	−0.108598	−0.0542989	−	0.998525i	$-0.517292\pi$
$149$	−1.32560	−0.108598	−0.0542989	+	0.998525i	$0.517292\pi$
$150$	0	0
$151$	−11.5137	−0.936974	−0.468487	−	0.883470i	$-0.655201\pi$
$151$	−11.5137	−0.936974	−0.468487	+	0.883470i	$0.655201\pi$
$152$	0	0
$153$	2.79917	0.226300
$154$	0	0
$155$	−3.84511	−0.308846
$156$	0	0
$157$	−2.41464	−0.192710	−0.0963548	−	0.995347i	$-0.530718\pi$
$157$	−2.41464	−0.192710	−0.0963548	+	0.995347i	$0.530718\pi$
$158$	0	0
$159$	12.5521	0.995446
$160$	0	0
$161$	−4.81630	−0.379577
$162$	0	0
$163$	−7.96322	−0.623727	−0.311864	−	0.950127i	$-0.600953\pi$
$163$	−7.96322	−0.623727	−0.311864	+	0.950127i	$0.600953\pi$
$164$	0	0
$165$	−3.40312	−0.264933
$166$	0	0
$167$	−3.09909	−0.239815	−0.119907	−	0.992785i	$-0.538260\pi$
$167$	−3.09909	−0.239815	−0.119907	+	0.992785i	$0.538260\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.603373	0.0458736	0.0229368	−	0.999737i	$-0.492698\pi$
$173$	0.603373	0.0458736	0.0229368	+	0.999737i	$0.492698\pi$
$174$	0	0
$175$	3.32708	0.251504
$176$	0	0
$177$	−5.26223	−0.395534
$178$	0	0
$179$	3.50074	0.261657	0.130829	−	0.991405i	$-0.458236\pi$
$179$	3.50074	0.261657	0.130829	+	0.991405i	$0.458236\pi$
$180$	0	0
$181$	16.1283	1.19881	0.599404	−	0.800447i	$-0.295404\pi$
$181$	16.1283	1.19881	0.599404	+	0.800447i	$0.295404\pi$
$182$	0	0
$183$	10.6155	0.784718
$184$	0	0
$185$	1.17366	0.0862891
$186$	0	0
$187$	−1.09259	−0.0798980
$188$	0	0
$189$	−0.483617	−0.0351779
$190$	0	0
$191$	12.8006	0.926222	0.463111	−	0.886300i	$-0.346733\pi$
$191$	12.8006	0.926222	0.463111	+	0.886300i	$0.346733\pi$
$192$	0	0
$193$	−10.6211	−0.764521	−0.382261	−	0.924055i	$-0.624854\pi$
$193$	−10.6211	−0.764521	−0.382261	+	0.924055i	$0.624854\pi$
$194$	0	0
$195$	−6.22947	−0.446101
$196$	0	0
$197$	16.7252	1.19162	0.595810	−	0.803126i	$-0.296831\pi$
$197$	16.7252	1.19162	0.595810	+	0.803126i	$0.296831\pi$
$198$	0	0
$199$	16.7187	1.18516	0.592578	−	0.805513i	$-0.298110\pi$
$199$	16.7187	1.18516	0.592578	+	0.805513i	$0.298110\pi$
$200$	0	0
$201$	−7.26567	−0.512481
$202$	0	0
$203$	9.32206	0.654280
$204$	0	0
$205$	−7.63423	−0.533198
$206$	0	0
$207$	13.4817	0.937040
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.13835	0.284896	0.142448	−	0.989802i	$-0.454503\pi$
$211$	4.13835	0.284896	0.142448	+	0.989802i	$0.454503\pi$
$212$	0	0
$213$	−12.0685	−0.826919
$214$	0	0
$215$	−10.7574	−0.733647
$216$	0	0
$217$	−2.97283	−0.201809
$218$	0	0
$219$	−13.3532	−0.902328
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	15.3588	1.02850	0.514252	−	0.857639i	$-0.328070\pi$
$223$	15.3588	1.02850	0.514252	+	0.857639i	$0.328070\pi$
$224$	0	0
$225$	−9.31308	−0.620872
$226$	0	0
$227$	−10.7538	−0.713756	−0.356878	−	0.934151i	$-0.616159\pi$
$227$	−10.7538	−0.713756	−0.356878	+	0.934151i	$0.616159\pi$
$228$	0	0
$229$	5.44741	0.359975	0.179988	−	0.983669i	$-0.442394\pi$
$229$	5.44741	0.359975	0.179988	+	0.983669i	$0.442394\pi$
$230$	0	0
$231$	−2.63112	−0.173115
$232$	0	0
$233$	26.4388	1.73207	0.866033	−	0.499987i	$-0.166662\pi$
$233$	26.4388	1.73207	0.866033	+	0.499987i	$0.166662\pi$
$234$	0	0
$235$	9.88510	0.644833
$236$	0	0
$237$	−18.4046	−1.19551
$238$	0	0
$239$	−19.7644	−1.27846	−0.639228	−	0.769017i	$-0.720746\pi$
$239$	−19.7644	−1.27846	−0.639228	+	0.769017i	$0.720746\pi$
$240$	0	0
$241$	−10.8319	−0.697747	−0.348873	−	0.937170i	$-0.613436\pi$
$241$	−10.8319	−0.697747	−0.348873	+	0.937170i	$0.613436\pi$
$242$	0	0
$243$	21.5762	1.38411
$244$	0	0
$245$	−1.29341	−0.0826332
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−41.6013	−2.63638
$250$	0	0
$251$	−5.10010	−0.321915	−0.160958	−	0.986961i	$-0.551458\pi$
$251$	−5.10010	−0.321915	−0.160958	+	0.986961i	$0.551458\pi$
$252$	0	0
$253$	−5.26223	−0.330834
$254$	0	0
$255$	3.11473	0.195052
$256$	0	0
$257$	−20.2867	−1.26545	−0.632726	−	0.774376i	$-0.718064\pi$
$257$	−20.2867	−1.26545	−0.632726	+	0.774376i	$0.718064\pi$
$258$	0	0
$259$	0.907411	0.0563838
$260$	0	0
$261$	−26.0941	−1.61518
$262$	0	0
$263$	−25.0800	−1.54650	−0.773250	−	0.634102i	$-0.781370\pi$
$263$	−25.0800	−1.54650	−0.773250	+	0.634102i	$0.781370\pi$
$264$	0	0
$265$	6.74172	0.414141
$266$	0	0
$267$	−36.6784	−2.24468
$268$	0	0
$269$	−22.1513	−1.35059	−0.675296	−	0.737547i	$-0.735984\pi$
$269$	−22.1513	−1.35059	−0.675296	+	0.737547i	$0.735984\pi$
$270$	0	0
$271$	−5.61960	−0.341366	−0.170683	−	0.985326i	$-0.554597\pi$
$271$	−5.61960	−0.341366	−0.170683	+	0.985326i	$0.554597\pi$
$272$	0	0
$273$	−4.81630	−0.291495
$274$	0	0
$275$	3.63513	0.219207
$276$	0	0
$277$	2.27629	0.136769	0.0683846	−	0.997659i	$-0.478215\pi$
$277$	2.27629	0.136769	0.0683846	+	0.997659i	$0.478215\pi$
$278$	0	0
$279$	8.32148	0.498194
$280$	0	0
$281$	13.8021	0.823366	0.411683	−	0.911327i	$-0.364941\pi$
$281$	13.8021	0.823366	0.411683	+	0.911327i	$0.364941\pi$
$282$	0	0
$283$	1.92453	0.114401	0.0572007	−	0.998363i	$-0.481782\pi$
$283$	1.92453	0.114401	0.0572007	+	0.998363i	$0.481782\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.90239	−0.348407
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−18.3114	−1.07344
$292$	0	0
$293$	−14.0130	−0.818648	−0.409324	−	0.912389i	$-0.634235\pi$
$293$	−14.0130	−0.818648	−0.409324	+	0.912389i	$0.634235\pi$
$294$	0	0
$295$	−2.82634	−0.164556
$296$	0	0
$297$	−0.528394	−0.0306606
$298$	0	0
$299$	−9.63259	−0.557067
$300$	0	0
$301$	−8.31703	−0.479386
$302$	0	0
$303$	−10.1852	−0.585123
$304$	0	0
$305$	5.70156	0.326470
$306$	0	0
$307$	16.8606	0.962284	0.481142	−	0.876643i	$-0.340222\pi$
$307$	16.8606	0.962284	0.481142	+	0.876643i	$0.340222\pi$
$308$	0	0
$309$	29.3195	1.66793
$310$	0	0
$311$	−21.7992	−1.23612	−0.618059	−	0.786132i	$-0.712081\pi$
$311$	−21.7992	−1.23612	−0.618059	+	0.786132i	$0.712081\pi$
$312$	0	0
$313$	6.80065	0.384395	0.192198	−	0.981356i	$-0.438439\pi$
$313$	6.80065	0.384395	0.192198	+	0.981356i	$0.438439\pi$
$314$	0	0
$315$	3.62049	0.203992
$316$	0	0
$317$	−8.96469	−0.503507	−0.251754	−	0.967791i	$-0.581007\pi$
$317$	−8.96469	−0.503507	−0.251754	+	0.967791i	$0.581007\pi$
$318$	0	0
$319$	10.1852	0.570261
$320$	0	0
$321$	−9.43736	−0.526742
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6.65416	0.369106
$326$	0	0
$327$	−7.55406	−0.417741
$328$	0	0
$329$	7.64264	0.421352
$330$	0	0
$331$	−11.0171	−0.605556	−0.302778	−	0.953061i	$-0.597914\pi$
$331$	−11.0171	−0.605556	−0.302778	+	0.953061i	$0.597914\pi$
$332$	0	0
$333$	−2.54000	−0.139191
$334$	0	0
$335$	−3.90239	−0.213210
$336$	0	0
$337$	−10.5969	−0.577249	−0.288624	−	0.957442i	$-0.593198\pi$
$337$	−10.5969	−0.577249	−0.288624	+	0.957442i	$0.593198\pi$
$338$	0	0
$339$	35.7865	1.94365
$340$	0	0
$341$	−3.24809	−0.175894
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	15.0015	0.807652
$346$	0	0
$347$	9.09259	0.488116	0.244058	−	0.969761i	$-0.421521\pi$
$347$	9.09259	0.488116	0.244058	+	0.969761i	$0.421521\pi$
$348$	0	0
$349$	−25.0900	−1.34304	−0.671520	−	0.740987i	$-0.734358\pi$
$349$	−25.0900	−1.34304	−0.671520	+	0.740987i	$0.734358\pi$
$350$	0	0
$351$	−0.967233	−0.0516271
$352$	0	0
$353$	−9.97991	−0.531177	−0.265588	−	0.964087i	$-0.585566\pi$
$353$	−9.97991	−0.531177	−0.265588	+	0.964087i	$0.585566\pi$
$354$	0	0
$355$	−6.48197	−0.344027
$356$	0	0
$357$	2.40815	0.127453
$358$	0	0
$359$	25.2909	1.33480	0.667401	−	0.744699i	$-0.267407\pi$
$359$	25.2909	1.33480	0.667401	+	0.744699i	$0.267407\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	23.6149	1.23946
$364$	0	0
$365$	−7.17202	−0.375400
$366$	0	0
$367$	23.9154	1.24837	0.624186	−	0.781275i	$-0.285431\pi$
$367$	23.9154	1.24837	0.624186	+	0.781275i	$0.285431\pi$
$368$	0	0
$369$	16.5218	0.860091
$370$	0	0
$371$	5.21235	0.270612
$372$	0	0
$373$	−33.3185	−1.72517	−0.862583	−	0.505915i	$-0.831155\pi$
$373$	−33.3185	−1.72517	−0.862583	+	0.505915i	$0.831155\pi$
$374$	0	0
$375$	−25.9366	−1.33936
$376$	0	0
$377$	18.6441	0.960221
$378$	0	0
$379$	−31.9874	−1.64308	−0.821542	−	0.570149i	$-0.806886\pi$
$379$	−31.9874	−1.64308	−0.821542	+	0.570149i	$0.806886\pi$
$380$	0	0
$381$	−0.507811	−0.0260159
$382$	0	0
$383$	−17.7705	−0.908032	−0.454016	−	0.890994i	$-0.650009\pi$
$383$	−17.7705	−0.908032	−0.454016	+	0.890994i	$0.650009\pi$
$384$	0	0
$385$	−1.41317	−0.0720219
$386$	0	0
$387$	23.2808	1.18343
$388$	0	0
$389$	9.82222	0.498006	0.249003	−	0.968503i	$-0.419897\pi$
$389$	9.82222	0.498006	0.249003	+	0.968503i	$0.419897\pi$
$390$	0	0
$391$	4.81630	0.243571
$392$	0	0
$393$	−15.1967	−0.766572
$394$	0	0
$395$	−9.88510	−0.497373
$396$	0	0
$397$	14.9833	0.751990	0.375995	−	0.926622i	$-0.377301\pi$
$397$	14.9833	0.751990	0.375995	+	0.926622i	$0.377301\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.1881	−1.10802	−0.554011	−	0.832509i	$-0.686903\pi$
$401$	−22.1881	−1.10802	−0.554011	+	0.832509i	$0.686903\pi$
$402$	0	0
$403$	−5.94567	−0.296175
$404$	0	0
$405$	12.3678	0.614562
$406$	0	0
$407$	0.991427	0.0491432
$408$	0	0
$409$	−24.4046	−1.20673	−0.603365	−	0.797465i	$-0.706174\pi$
$409$	−24.4046	−1.20673	−0.603365	+	0.797465i	$0.706174\pi$
$410$	0	0
$411$	6.44034	0.317678
$412$	0	0
$413$	−2.18518	−0.107526
$414$	0	0
$415$	−22.3440	−1.09683
$416$	0	0
$417$	4.42125	0.216510
$418$	0	0
$419$	14.9361	0.729674	0.364837	−	0.931071i	$-0.381125\pi$
$419$	14.9361	0.729674	0.364837	+	0.931071i	$0.381125\pi$
$420$	0	0
$421$	25.5822	1.24680	0.623400	−	0.781903i	$-0.285751\pi$
$421$	25.5822	1.24680	0.623400	+	0.781903i	$0.285751\pi$
$422$	0	0
$423$	−21.3931	−1.04017
$424$	0	0
$425$	−3.32708	−0.161387
$426$	0	0
$427$	4.40815	0.213325
$428$	0	0
$429$	−5.26223	−0.254063
$430$	0	0
$431$	−19.7605	−0.951829	−0.475915	−	0.879492i	$-0.657883\pi$
$431$	−19.7605	−0.951829	−0.475915	+	0.879492i	$0.657883\pi$
$432$	0	0
$433$	−35.8963	−1.72507	−0.862533	−	0.506001i	$-0.831123\pi$
$433$	−35.8963	−1.72507	−0.862533	+	0.506001i	$0.831123\pi$
$434$	0	0
$435$	−29.0357	−1.39216
$436$	0	0
$437$	0	0
$438$	0	0
$439$	40.5494	1.93532	0.967660	−	0.252259i	$-0.0811735\pi$
$439$	40.5494	1.93532	0.967660	+	0.252259i	$0.0811735\pi$
$440$	0	0
$441$	2.79917	0.133294
$442$	0	0
$443$	6.98848	0.332033	0.166016	−	0.986123i	$-0.446910\pi$
$443$	6.98848	0.332033	0.166016	+	0.986123i	$0.446910\pi$
$444$	0	0
$445$	−19.6999	−0.933866
$446$	0	0
$447$	3.19225	0.150988
$448$	0	0
$449$	−28.4176	−1.34111	−0.670555	−	0.741860i	$-0.733944\pi$
$449$	−28.4176	−1.34111	−0.670555	+	0.741860i	$0.733944\pi$
$450$	0	0
$451$	−6.44889	−0.303666
$452$	0	0
$453$	27.7268	1.30272
$454$	0	0
$455$	−2.58683	−0.121272
$456$	0	0
$457$	25.9108	1.21206	0.606028	−	0.795443i	$-0.292762\pi$
$457$	25.9108	1.21206	0.606028	+	0.795443i	$0.292762\pi$
$458$	0	0
$459$	0.483617	0.0225733
$460$	0	0
$461$	6.63406	0.308979	0.154490	−	0.987994i	$-0.450627\pi$
$461$	6.63406	0.308979	0.154490	+	0.987994i	$0.450627\pi$
$462$	0	0
$463$	−37.1463	−1.72634	−0.863168	−	0.504917i	$-0.831523\pi$
$463$	−37.1463	−1.72634	−0.863168	+	0.504917i	$0.831523\pi$
$464$	0	0
$465$	9.25958	0.429403
$466$	0	0
$467$	−12.0372	−0.557015	−0.278507	−	0.960434i	$-0.589840\pi$
$467$	−12.0372	−0.557015	−0.278507	+	0.960434i	$0.589840\pi$
$468$	0	0
$469$	−3.01712	−0.139318
$470$	0	0
$471$	5.81482	0.267933
$472$	0	0
$473$	−9.08710	−0.417825
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−14.5903	−0.668042
$478$	0	0
$479$	−11.9355	−0.545346	−0.272673	−	0.962107i	$-0.587908\pi$
$479$	−11.9355	−0.545346	−0.272673	+	0.962107i	$0.587908\pi$
$480$	0	0
$481$	1.81482	0.0827487
$482$	0	0
$483$	11.5983	0.527743
$484$	0	0
$485$	−9.83506	−0.446587
$486$	0	0
$487$	−2.56674	−0.116310	−0.0581549	−	0.998308i	$-0.518522\pi$
$487$	−2.56674	−0.116310	−0.0581549	+	0.998308i	$0.518522\pi$
$488$	0	0
$489$	19.1766	0.867196
$490$	0	0
$491$	−10.4073	−0.469673	−0.234836	−	0.972035i	$-0.575455\pi$
$491$	−10.4073	−0.469673	−0.234836	+	0.972035i	$0.575455\pi$
$492$	0	0
$493$	−9.32206	−0.419844
$494$	0	0
$495$	3.95571	0.177796
$496$	0	0
$497$	−5.01152	−0.224797
$498$	0	0
$499$	−15.3793	−0.688474	−0.344237	−	0.938883i	$-0.611862\pi$
$499$	−15.3793	−0.688474	−0.344237	+	0.938883i	$0.611862\pi$
$500$	0	0
$501$	7.46306	0.333425
$502$	0	0
$503$	−8.36591	−0.373018	−0.186509	−	0.982453i	$-0.559717\pi$
$503$	−8.36591	−0.373018	−0.186509	+	0.982453i	$0.559717\pi$
$504$	0	0
$505$	−5.47045	−0.243432
$506$	0	0
$507$	21.6733	0.962546
$508$	0	0
$509$	6.50322	0.288250	0.144125	−	0.989559i	$-0.453963\pi$
$509$	6.50322	0.288250	0.144125	+	0.989559i	$0.453963\pi$
$510$	0	0
$511$	−5.54502	−0.245297
$512$	0	0
$513$	0	0
$514$	0	0
$515$	15.7475	0.693917
$516$	0	0
$517$	8.35026	0.367244
$518$	0	0
$519$	−1.45301	−0.0637801
$520$	0	0
$521$	−33.6265	−1.47320	−0.736602	−	0.676327i	$-0.763571\pi$
$521$	−33.6265	−1.47320	−0.736602	+	0.676327i	$0.763571\pi$
$522$	0	0
$523$	18.4690	0.807592	0.403796	−	0.914849i	$-0.367691\pi$
$523$	18.4690	0.807592	0.403796	+	0.914849i	$0.367691\pi$
$524$	0	0
$525$	−8.01210	−0.349677
$526$	0	0
$527$	2.97283	0.129499
$528$	0	0
$529$	0.196699	0.00855215
$530$	0	0
$531$	6.11670	0.265442
$532$	0	0
$533$	−11.8048	−0.511322
$534$	0	0
$535$	−5.06880	−0.219143
$536$	0	0
$537$	−8.43029	−0.363794
$538$	0	0
$539$	−1.09259	−0.0470612
$540$	0	0
$541$	−26.2562	−1.12884	−0.564421	−	0.825487i	$-0.690901\pi$
$541$	−26.2562	−1.12884	−0.564421	+	0.825487i	$0.690901\pi$
$542$	0	0
$543$	−38.8393	−1.66676
$544$	0	0
$545$	−4.05728	−0.173795
$546$	0	0
$547$	−16.3135	−0.697514	−0.348757	−	0.937213i	$-0.613396\pi$
$547$	−16.3135	−0.697514	−0.348757	+	0.937213i	$0.613396\pi$
$548$	0	0
$549$	−12.3392	−0.526623
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−7.64264	−0.324998
$554$	0	0
$555$	−2.82634	−0.119972
$556$	0	0
$557$	−5.35884	−0.227061	−0.113531	−	0.993534i	$-0.536216\pi$
$557$	−5.35884	−0.227061	−0.113531	+	0.993534i	$0.536216\pi$
$558$	0	0
$559$	−16.6341	−0.703546
$560$	0	0
$561$	2.63112	0.111086
$562$	0	0
$563$	43.6798	1.84089	0.920443	−	0.390878i	$-0.127828\pi$
$563$	43.6798	1.84089	0.920443	+	0.390878i	$0.127828\pi$
$564$	0	0
$565$	19.2209	0.808629
$566$	0	0
$567$	9.56214	0.401572
$568$	0	0
$569$	−39.1333	−1.64055	−0.820277	−	0.571966i	$-0.806181\pi$
$569$	−39.1333	−1.64055	−0.820277	+	0.571966i	$0.806181\pi$
$570$	0	0
$571$	−35.9431	−1.50417	−0.752086	−	0.659064i	$-0.770952\pi$
$571$	−35.9431	−1.50417	−0.752086	+	0.659064i	$0.770952\pi$
$572$	0	0
$573$	−30.8259	−1.28777
$574$	0	0
$575$	−16.0242	−0.668255
$576$	0	0
$577$	−35.8650	−1.49308	−0.746540	−	0.665341i	$-0.768286\pi$
$577$	−35.8650	−1.49308	−0.746540	+	0.665341i	$0.768286\pi$
$578$	0	0
$579$	25.5771	1.06295
$580$	0	0
$581$	−17.2752	−0.716697
$582$	0	0
$583$	5.69495	0.235861
$584$	0	0
$585$	7.24099	0.299378
$586$	0	0
$587$	−20.5162	−0.846795	−0.423397	−	0.905944i	$-0.639163\pi$
$587$	−20.5162	−0.846795	−0.423397	+	0.905944i	$0.639163\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−40.2767	−1.65676
$592$	0	0
$593$	38.2324	1.57002	0.785009	−	0.619485i	$-0.212659\pi$
$593$	38.2324	1.57002	0.785009	+	0.619485i	$0.212659\pi$
$594$	0	0
$595$	1.29341	0.0530248
$596$	0	0
$597$	−40.2611	−1.64778
$598$	0	0
$599$	−39.7774	−1.62526	−0.812631	−	0.582779i	$-0.801966\pi$
$599$	−39.7774	−1.62526	−0.812631	+	0.582779i	$0.801966\pi$
$600$	0	0
$601$	1.00857	0.0411405	0.0205703	−	0.999788i	$-0.493452\pi$
$601$	1.00857	0.0411405	0.0205703	+	0.999788i	$0.493452\pi$
$602$	0	0
$603$	8.44545	0.343925
$604$	0	0
$605$	12.6835	0.515659
$606$	0	0
$607$	30.6598	1.24444	0.622221	−	0.782842i	$-0.286231\pi$
$607$	30.6598	1.24444	0.622221	+	0.782842i	$0.286231\pi$
$608$	0	0
$609$	−22.4489	−0.909675
$610$	0	0
$611$	15.2853	0.618376
$612$	0	0
$613$	−25.8334	−1.04340	−0.521701	−	0.853128i	$-0.674702\pi$
$613$	−25.8334	−1.04340	−0.521701	+	0.853128i	$0.674702\pi$
$614$	0	0
$615$	18.3844	0.741329
$616$	0	0
$617$	43.9829	1.77068	0.885342	−	0.464941i	$-0.153925\pi$
$617$	43.9829	1.77068	0.885342	+	0.464941i	$0.153925\pi$
$618$	0	0
$619$	−32.5843	−1.30967	−0.654837	−	0.755771i	$-0.727263\pi$
$619$	−32.5843	−1.30967	−0.654837	+	0.755771i	$0.727263\pi$
$620$	0	0
$621$	2.32924	0.0934692
$622$	0	0
$623$	−15.2309	−0.610215
$624$	0	0
$625$	2.70485	0.108194
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.907411	−0.0361808
$630$	0	0
$631$	29.2797	1.16561	0.582803	−	0.812614i	$-0.301956\pi$
$631$	29.2797	1.16561	0.582803	+	0.812614i	$0.301956\pi$
$632$	0	0
$633$	−9.96576	−0.396103
$634$	0	0
$635$	−0.272745	−0.0108235
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	14.0281	0.554944
$640$	0	0
$641$	−12.4489	−0.491701	−0.245851	−	0.969308i	$-0.579067\pi$
$641$	−12.4489	−0.491701	−0.245851	+	0.969308i	$0.579067\pi$
$642$	0	0
$643$	−3.08240	−0.121558	−0.0607790	−	0.998151i	$-0.519358\pi$
$643$	−3.08240	−0.121558	−0.0607790	+	0.998151i	$0.519358\pi$
$644$	0	0
$645$	25.9053	1.02002
$646$	0	0
$647$	−23.3095	−0.916390	−0.458195	−	0.888852i	$-0.651504\pi$
$647$	−23.3095	−0.916390	−0.458195	+	0.888852i	$0.651504\pi$
$648$	0	0
$649$	−2.38750	−0.0937177
$650$	0	0
$651$	7.15902	0.280584
$652$	0	0
$653$	40.6608	1.59118	0.795590	−	0.605835i	$-0.207161\pi$
$653$	40.6608	1.59118	0.795590	+	0.605835i	$0.207161\pi$
$654$	0	0
$655$	−8.16214	−0.318921
$656$	0	0
$657$	15.5215	0.605551
$658$	0	0
$659$	−31.6069	−1.23123	−0.615615	−	0.788047i	$-0.711092\pi$
$659$	−31.6069	−1.23123	−0.615615	+	0.788047i	$0.711092\pi$
$660$	0	0
$661$	1.26518	0.0492098	0.0246049	−	0.999697i	$-0.492167\pi$
$661$	1.26518	0.0492098	0.0246049	+	0.999697i	$0.492167\pi$
$662$	0	0
$663$	4.81630	0.187049
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−44.8978	−1.73845
$668$	0	0
$669$	−36.9863	−1.42997
$670$	0	0
$671$	4.81630	0.185931
$672$	0	0
$673$	26.4719	1.02042	0.510209	−	0.860051i	$-0.329568\pi$
$673$	26.4719	1.02042	0.510209	+	0.860051i	$0.329568\pi$
$674$	0	0
$675$	−1.60903	−0.0619316
$676$	0	0
$677$	50.5016	1.94093	0.970467	−	0.241233i	$-0.0775518\pi$
$677$	50.5016	1.94093	0.970467	+	0.241233i	$0.0775518\pi$
$678$	0	0
$679$	−7.60395	−0.291813
$680$	0	0
$681$	25.8968	0.992367
$682$	0	0
$683$	28.7464	1.09995	0.549976	−	0.835181i	$-0.314637\pi$
$683$	28.7464	1.09995	0.549976	+	0.835181i	$0.314637\pi$
$684$	0	0
$685$	3.45910	0.132165
$686$	0	0
$687$	−13.1182	−0.500489
$688$	0	0
$689$	10.4247	0.397149
$690$	0	0
$691$	23.9652	0.911678	0.455839	−	0.890062i	$-0.349339\pi$
$691$	23.9652	0.911678	0.455839	+	0.890062i	$0.349339\pi$
$692$	0	0
$693$	3.05835	0.116177
$694$	0	0
$695$	2.37465	0.0900757
$696$	0	0
$697$	5.90239	0.223569
$698$	0	0
$699$	−63.6686	−2.40817
$700$	0	0
$701$	−29.7977	−1.12544	−0.562721	−	0.826647i	$-0.690246\pi$
$701$	−29.7977	−1.12544	−0.562721	+	0.826647i	$0.690246\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−23.8048	−0.896540
$706$	0	0
$707$	−4.22947	−0.159065
$708$	0	0
$709$	−34.1513	−1.28258	−0.641290	−	0.767298i	$-0.721601\pi$
$709$	−34.1513	−1.28258	−0.641290	+	0.767298i	$0.721601\pi$
$710$	0	0
$711$	21.3931	0.802303
$712$	0	0
$713$	14.3180	0.536215
$714$	0	0
$715$	−2.82634	−0.105699
$716$	0	0
$717$	47.5957	1.77749
$718$	0	0
$719$	−31.6316	−1.17966	−0.589829	−	0.807528i	$-0.700805\pi$
$719$	−31.6316	−1.17966	−0.589829	+	0.807528i	$0.700805\pi$
$720$	0	0
$721$	12.1751	0.453426
$722$	0	0
$723$	26.0849	0.970108
$724$	0	0
$725$	31.0152	1.15188
$726$	0	0
$727$	−47.0517	−1.74505	−0.872525	−	0.488570i	$-0.837519\pi$
$727$	−47.0517	−1.74505	−0.872525	+	0.488570i	$0.837519\pi$
$728$	0	0
$729$	−23.2723	−0.861935
$730$	0	0
$731$	8.31703	0.307617
$732$	0	0
$733$	−8.17218	−0.301847	−0.150923	−	0.988545i	$-0.548225\pi$
$733$	−8.17218	−0.301847	−0.150923	+	0.988545i	$0.548225\pi$
$734$	0	0
$735$	3.11473	0.114889
$736$	0	0
$737$	−3.29647	−0.121427
$738$	0	0
$739$	−17.6729	−0.650109	−0.325054	−	0.945695i	$-0.605383\pi$
$739$	−17.6729	−0.650109	−0.325054	+	0.945695i	$0.605383\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	46.5876	1.70913	0.854567	−	0.519340i	$-0.173822\pi$
$743$	46.5876	1.70913	0.854567	+	0.519340i	$0.173822\pi$
$744$	0	0
$745$	1.71456	0.0628165
$746$	0	0
$747$	48.3564	1.76927
$748$	0	0
$749$	−3.91893	−0.143195
$750$	0	0
$751$	−28.6471	−1.04535	−0.522673	−	0.852533i	$-0.675065\pi$
$751$	−28.6471	−1.04535	−0.522673	+	0.852533i	$0.675065\pi$
$752$	0	0
$753$	12.2818	0.447573
$754$	0	0
$755$	14.8920	0.541976
$756$	0	0
$757$	−33.9597	−1.23429	−0.617143	−	0.786851i	$-0.711710\pi$
$757$	−33.9597	−1.23429	−0.617143	+	0.786851i	$0.711710\pi$
$758$	0	0
$759$	12.6722	0.459973
$760$	0	0
$761$	24.6110	0.892149	0.446074	−	0.894996i	$-0.352822\pi$
$761$	24.6110	0.892149	0.446074	+	0.894996i	$0.352822\pi$
$762$	0	0
$763$	−3.13688	−0.113563
$764$	0	0
$765$	−3.62049	−0.130899
$766$	0	0
$767$	−4.37036	−0.157804
$768$	0	0
$769$	−28.1409	−1.01479	−0.507393	−	0.861715i	$-0.669391\pi$
$769$	−28.1409	−1.01479	−0.507393	+	0.861715i	$0.669391\pi$
$770$	0	0
$771$	48.8535	1.75942
$772$	0	0
$773$	42.5898	1.53185	0.765924	−	0.642931i	$-0.222282\pi$
$773$	42.5898	1.53185	0.765924	+	0.642931i	$0.222282\pi$
$774$	0	0
$775$	−9.89085	−0.355290
$776$	0	0
$777$	−2.18518	−0.0783929
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−5.47553	−0.195930
$782$	0	0
$783$	−4.50830	−0.161114
$784$	0	0
$785$	3.12314	0.111470
$786$	0	0
$787$	−2.60733	−0.0929413	−0.0464706	−	0.998920i	$-0.514797\pi$
$787$	−2.60733	−0.0929413	−0.0464706	+	0.998920i	$0.514797\pi$
$788$	0	0
$789$	60.3963	2.15017
$790$	0	0
$791$	14.8606	0.528381
$792$	0	0
$793$	8.81630	0.313076
$794$	0	0
$795$	−16.2351	−0.575798
$796$	0	0
$797$	14.8606	0.526389	0.263194	−	0.964743i	$-0.415224\pi$
$797$	14.8606	0.526389	0.263194	+	0.964743i	$0.415224\pi$
$798$	0	0
$799$	−7.64264	−0.270377
$800$	0	0
$801$	42.6341	1.50640
$802$	0	0
$803$	−6.05843	−0.213798
$804$	0	0
$805$	6.22947	0.219560
$806$	0	0
$807$	53.3437	1.87779
$808$	0	0
$809$	14.7850	0.519813	0.259907	−	0.965634i	$-0.416308\pi$
$809$	14.7850	0.519813	0.259907	+	0.965634i	$0.416308\pi$
$810$	0	0
$811$	−23.3866	−0.821214	−0.410607	−	0.911812i	$-0.634683\pi$
$811$	−23.3866	−0.821214	−0.410607	+	0.911812i	$0.634683\pi$
$812$	0	0
$813$	13.5328	0.474617
$814$	0	0
$815$	10.2997	0.360784
$816$	0	0
$817$	0	0
$818$	0	0
$819$	5.59835	0.195622
$820$	0	0
$821$	14.6136	0.510017	0.255008	−	0.966939i	$-0.417922\pi$
$821$	14.6136	0.510017	0.255008	+	0.966939i	$0.417922\pi$
$822$	0	0
$823$	27.0718	0.943662	0.471831	−	0.881689i	$-0.343593\pi$
$823$	27.0718	0.943662	0.471831	+	0.881689i	$0.343593\pi$
$824$	0	0
$825$	−8.75393	−0.304773
$826$	0	0
$827$	−32.8974	−1.14395	−0.571977	−	0.820270i	$-0.693823\pi$
$827$	−32.8974	−1.14395	−0.571977	+	0.820270i	$0.693823\pi$
$828$	0	0
$829$	20.1651	0.700362	0.350181	−	0.936682i	$-0.386120\pi$
$829$	20.1651	0.700362	0.350181	+	0.936682i	$0.386120\pi$
$830$	0	0
$831$	−5.48165	−0.190156
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	4.00840	0.138717
$836$	0	0
$837$	1.43771	0.0496946
$838$	0	0
$839$	23.1967	0.800839	0.400419	−	0.916332i	$-0.368864\pi$
$839$	23.1967	0.800839	0.400419	+	0.916332i	$0.368864\pi$
$840$	0	0
$841$	57.9007	1.99658
$842$	0	0
$843$	−33.2375	−1.14476
$844$	0	0
$845$	11.6407	0.400453
$846$	0	0
$847$	9.80625	0.336947
$848$	0	0
$849$	−4.63455	−0.159058
$850$	0	0
$851$	−4.37036	−0.149814
$852$	0	0
$853$	10.4422	0.357536	0.178768	−	0.983891i	$-0.442789\pi$
$853$	10.4422	0.357536	0.178768	+	0.983891i	$0.442789\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.2135	−0.383046	−0.191523	−	0.981488i	$-0.561343\pi$
$857$	−11.2135	−0.383046	−0.191523	+	0.981488i	$0.561343\pi$
$858$	0	0
$859$	−40.9108	−1.39586	−0.697929	−	0.716167i	$-0.745895\pi$
$859$	−40.9108	−1.39586	−0.697929	+	0.716167i	$0.745895\pi$
$860$	0	0
$861$	14.2138	0.484406
$862$	0	0
$863$	−4.96279	−0.168935	−0.0844676	−	0.996426i	$-0.526919\pi$
$863$	−4.96279	−0.168935	−0.0844676	+	0.996426i	$0.526919\pi$
$864$	0	0
$865$	−0.780411	−0.0265348
$866$	0	0
$867$	−2.40815	−0.0817850
$868$	0	0
$869$	−8.35026	−0.283263
$870$	0	0
$871$	−6.03424	−0.204462
$872$	0	0
$873$	21.2848	0.720381
$874$	0	0
$875$	−10.7704	−0.364105
$876$	0	0
$877$	−29.8787	−1.00893	−0.504467	−	0.863431i	$-0.668311\pi$
$877$	−29.8787	−1.00893	−0.504467	+	0.863431i	$0.668311\pi$
$878$	0	0
$879$	33.7454	1.13820
$880$	0	0
$881$	−12.9858	−0.437504	−0.218752	−	0.975781i	$-0.570199\pi$
$881$	−12.9858	−0.437504	−0.218752	+	0.975781i	$0.570199\pi$
$882$	0	0
$883$	−41.1346	−1.38429	−0.692145	−	0.721758i	$-0.743334\pi$
$883$	−41.1346	−1.38429	−0.692145	+	0.721758i	$0.743334\pi$
$884$	0	0
$885$	6.80625	0.228790
$886$	0	0
$887$	15.1811	0.509730	0.254865	−	0.966977i	$-0.417969\pi$
$887$	15.1811	0.509730	0.254865	+	0.966977i	$0.417969\pi$
$888$	0	0
$889$	−0.210872	−0.00707242
$890$	0	0
$891$	10.4475	0.350004
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−4.52790	−0.151351
$896$	0	0
$897$	23.1967	0.774515
$898$	0	0
$899$	−27.7129	−0.924277
$900$	0	0
$901$	−5.21235	−0.173648
$902$	0	0
$903$	20.0286	0.666511
$904$	0	0
$905$	−20.8606	−0.693429
$906$	0	0
$907$	9.01903	0.299472	0.149736	−	0.988726i	$-0.452158\pi$
$907$	9.01903	0.299472	0.149736	+	0.988726i	$0.452158\pi$
$908$	0	0
$909$	11.8390	0.392675
$910$	0	0
$911$	5.42617	0.179777	0.0898884	−	0.995952i	$-0.471349\pi$
$911$	5.42617	0.179777	0.0898884	+	0.995952i	$0.471349\pi$
$912$	0	0
$913$	−18.8747	−0.624662
$914$	0	0
$915$	−13.7302	−0.453907
$916$	0	0
$917$	−6.31053	−0.208392
$918$	0	0
$919$	28.5677	0.942363	0.471181	−	0.882036i	$-0.343828\pi$
$919$	28.5677	0.942363	0.471181	+	0.882036i	$0.343828\pi$
$920$	0	0
$921$	−40.6028	−1.33791
$922$	0	0
$923$	−10.0230	−0.329912
$924$	0	0
$925$	3.01903	0.0992650
$926$	0	0
$927$	−34.0803	−1.11934
$928$	0	0
$929$	6.85793	0.225001	0.112501	−	0.993652i	$-0.464114\pi$
$929$	6.85793	0.225001	0.112501	+	0.993652i	$0.464114\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	52.4956	1.71863
$934$	0	0
$935$	1.41317	0.0462156
$936$	0	0
$937$	17.8621	0.583528	0.291764	−	0.956490i	$-0.405758\pi$
$937$	17.8621	0.583528	0.291764	+	0.956490i	$0.405758\pi$
$938$	0	0
$939$	−16.3770	−0.534442
$940$	0	0
$941$	−36.5227	−1.19061	−0.595303	−	0.803501i	$-0.702968\pi$
$941$	−36.5227	−1.19061	−0.595303	+	0.803501i	$0.702968\pi$
$942$	0	0
$943$	28.4276	0.925731
$944$	0	0
$945$	0.625517	0.0203481
$946$	0	0
$947$	−34.7402	−1.12890	−0.564452	−	0.825466i	$-0.690913\pi$
$947$	−34.7402	−1.12890	−0.564452	+	0.825466i	$0.690913\pi$
$948$	0	0
$949$	−11.0900	−0.359998
$950$	0	0
$951$	21.5883	0.700049
$952$	0	0
$953$	55.5546	1.79959	0.899795	−	0.436314i	$-0.143716\pi$
$953$	55.5546	1.79959	0.899795	+	0.436314i	$0.143716\pi$
$954$	0	0
$955$	−16.5565	−0.535757
$956$	0	0
$957$	−24.5274	−0.792859
$958$	0	0
$959$	2.67440	0.0863607
$960$	0	0
$961$	−22.1623	−0.714912
$962$	0	0
$963$	10.9698	0.353496
$964$	0	0
$965$	13.7374	0.442224
$966$	0	0
$967$	3.95967	0.127334	0.0636672	−	0.997971i	$-0.479720\pi$
$967$	3.95967	0.127334	0.0636672	+	0.997971i	$0.479720\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.13232	0.228887	0.114443	−	0.993430i	$-0.463492\pi$
$971$	7.13232	0.228887	0.114443	+	0.993430i	$0.463492\pi$
$972$	0	0
$973$	1.83596	0.0588581
$974$	0	0
$975$	−16.0242	−0.513185
$976$	0	0
$977$	7.88775	0.252351	0.126176	−	0.992008i	$-0.459730\pi$
$977$	7.88775	0.252351	0.126176	+	0.992008i	$0.459730\pi$
$978$	0	0
$979$	−16.6412	−0.531854
$980$	0	0
$981$	8.78067	0.280345
$982$	0	0
$983$	1.22649	0.0391191	0.0195595	−	0.999809i	$-0.493774\pi$
$983$	1.22649	0.0391191	0.0195595	+	0.999809i	$0.493774\pi$
$984$	0	0
$985$	−21.6326	−0.689272
$986$	0	0
$987$	−18.4046	−0.585825
$988$	0	0
$989$	40.0573	1.27375
$990$	0	0
$991$	26.4377	0.839821	0.419910	−	0.907566i	$-0.362061\pi$
$991$	26.4377	0.839821	0.419910	+	0.907566i	$0.362061\pi$
$992$	0	0
$993$	26.5309	0.841931
$994$	0	0
$995$	−21.6242	−0.685533
$996$	0	0
$997$	−12.4081	−0.392970	−0.196485	−	0.980507i	$-0.562953\pi$
$997$	−12.4081	−0.392970	−0.196485	+	0.980507i	$0.562953\pi$
$998$	0	0
$999$	−0.438839	−0.0138842

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.bo.1.1		4
4.3	odd	2		7616.2.a.bi.1.4		4
8.3	odd	2		952.2.a.h.1.1	✓	4
8.5	even	2		1904.2.a.r.1.4		4
24.11	even	2		8568.2.a.bg.1.2		4
56.27	even	2		6664.2.a.n.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.h.1.1	✓	4	8.3	odd	2
1904.2.a.r.1.4		4	8.5	even	2
6664.2.a.n.1.4		4	56.27	even	2
7616.2.a.bi.1.4		4	4.3	odd	2
7616.2.a.bo.1.1		4	1.1	even	1	trivial
8568.2.a.bg.1.2		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q-2.40815 q^{3} -1.29341 q^{5} -1.00000 q^{7} +2.79917 q^{9} -1.09259 q^{11} -2.00000 q^{13} +3.11473 q^{15} +1.00000 q^{17} +2.40815 q^{21} +4.81630 q^{23} -3.32708 q^{25} +0.483617 q^{27} -9.32206 q^{29} +2.97283 q^{31} +2.63112 q^{33} +1.29341 q^{35} -0.907411 q^{37} +4.81630 q^{39} +5.90239 q^{41} +8.31703 q^{43} -3.62049 q^{45} -7.64264 q^{47} +1.00000 q^{49} -2.40815 q^{51} -5.21235 q^{53} +1.41317 q^{55} +2.18518 q^{59} -4.40815 q^{61} -2.79917 q^{63} +2.58683 q^{65} +3.01712 q^{67} -11.5983 q^{69} +5.01152 q^{71} +5.54502 q^{73} +8.01210 q^{75} +1.09259 q^{77} +7.64264 q^{79} -9.56214 q^{81} +17.2752 q^{83} -1.29341 q^{85} +22.4489 q^{87} +15.2309 q^{89} +2.00000 q^{91} -7.15902 q^{93} +7.60395 q^{97} -3.05835 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9} - 8 q^{13} + 4 q^{17} - 3 q^{21} - 6 q^{23} + 3 q^{25} + 6 q^{27} - 8 q^{29} + 7 q^{31} - 6 q^{33} + 5 q^{35} - 8 q^{37} - 6 q^{39} + 15 q^{41} - 9 q^{43} + 2 q^{45}+ \cdots - 50 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - 5 * q^5 - 4 * q^7 + 7 * q^9 - 8 * q^13 + 4 * q^17 - 3 * q^21 - 6 * q^23 + 3 * q^25 + 6 * q^27 - 8 * q^29 + 7 * q^31 - 6 * q^33 + 5 * q^35 - 8 * q^37 - 6 * q^39 + 15 * q^41 - 9 * q^43 + 2 * q^45 - 6 * q^47 + 4 * q^49 + 3 * q^51 - 17 * q^53 + 6 * q^55 - 5 * q^61 - 7 * q^63 + 10 * q^65 - 9 * q^67 - 38 * q^69 + 12 * q^71 - 11 * q^73 - 2 * q^75 + 6 * q^79 + 16 * q^81 - 6 * q^83 - 5 * q^85 + 14 * q^87 + 2 * q^89 + 8 * q^91 + 9 * q^97 - 50 * q^99

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