LMFDB - Embedded newform 7616.2.a.bb.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7616,2,Mod(1,7616)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [3,0,-3,0,5,0,-3,0,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$60.8140661794$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.11491$ 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)

$f(q)$	$=$	$q-3.11491 q^{3} +3.47283 q^{5} -1.00000 q^{7} +6.70265 q^{9} +4.22982 q^{11} +1.28415 q^{13} -10.8176 q^{15} -1.00000 q^{17} +2.94567 q^{19} +3.11491 q^{21} +0.715853 q^{23} +7.06058 q^{25} -11.5334 q^{27} +6.94567 q^{29} -8.98680 q^{31} -13.1755 q^{33} -3.47283 q^{35} +10.9457 q^{37} -4.00000 q^{39} +2.06058 q^{41} -0.399055 q^{43} +23.2772 q^{45} +12.4596 q^{47} +1.00000 q^{49} +3.11491 q^{51} -6.64832 q^{53} +14.6894 q^{55} -9.17548 q^{57} -1.43171 q^{59} +6.88509 q^{61} -6.70265 q^{63} +4.45963 q^{65} +10.4185 q^{67} -2.22982 q^{69} +13.9736 q^{71} -14.5202 q^{73} -21.9930 q^{75} -4.22982 q^{77} -4.00000 q^{79} +15.8176 q^{81} -1.05433 q^{83} -3.47283 q^{85} -21.6351 q^{87} +4.60719 q^{89} -1.28415 q^{91} +27.9930 q^{93} +10.2298 q^{95} +10.2709 q^{97} +28.3510 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 5 q^{5} - 3 q^{7} + 2 q^{9} + 2 q^{13} - 8 q^{15} - 3 q^{17} - 2 q^{19} + 3 q^{21} + 4 q^{23} + 4 q^{25} - 12 q^{27} + 10 q^{29} - 7 q^{31} - 16 q^{33} - 5 q^{35} + 22 q^{37} - 12 q^{39}+ \cdots + 38 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 5 * q^5 - 3 * q^7 + 2 * q^9 + 2 * q^13 - 8 * q^15 - 3 * q^17 - 2 * q^19 + 3 * q^21 + 4 * q^23 + 4 * q^25 - 12 * q^27 + 10 * q^29 - 7 * q^31 - 16 * q^33 - 5 * q^35 + 22 * q^37 - 12 * q^39 - 11 * q^41 + 7 * q^43 + 20 * q^45 + 12 * q^47 + 3 * q^49 + 3 * q^51 + 9 * q^53 + 6 * q^55 - 4 * q^57 - 8 * q^59 + 27 * q^61 - 2 * q^63 - 12 * q^65 + 15 * q^67 + 6 * q^69 + 2 * q^71 - q^73 - 18 * q^75 - 12 * q^79 + 23 * q^81 - 14 * q^83 - 5 * q^85 - 16 * q^87 - 6 * q^89 - 2 * q^91 + 36 * q^93 + 18 * q^95 + 9 * q^97 + 38 * 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
$3$	−3.11491	−1.79839	−0.899196	−	0.437545i	$-0.855848\pi$
$3$	−3.11491	−1.79839	−0.899196	+	0.437545i	$0.855848\pi$
$4$	0	0
$5$	3.47283	1.55310	0.776549	−	0.630057i	$-0.216968\pi$
$5$	3.47283	1.55310	0.776549	+	0.630057i	$0.216968\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	6.70265	2.23422
$10$	0	0
$11$	4.22982	1.27534	0.637669	−	0.770311i	$-0.279899\pi$
$11$	4.22982	1.27534	0.637669	+	0.770311i	$0.279899\pi$
$12$	0	0
$13$	1.28415	0.356158	0.178079	−	0.984016i	$-0.443012\pi$
$13$	1.28415	0.356158	0.178079	+	0.984016i	$0.443012\pi$
$14$	0	0
$15$	−10.8176	−2.79308
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.94567	0.675783	0.337891	−	0.941185i	$-0.390286\pi$
$19$	2.94567	0.675783	0.337891	+	0.941185i	$0.390286\pi$
$20$	0	0
$21$	3.11491	0.679729
$22$	0	0
$23$	0.715853	0.149266	0.0746328	−	0.997211i	$-0.476222\pi$
$23$	0.715853	0.149266	0.0746328	+	0.997211i	$0.476222\pi$
$24$	0	0
$25$	7.06058	1.41212
$26$	0	0
$27$	−11.5334	−2.21961
$28$	0	0
$29$	6.94567	1.28978	0.644889	−	0.764276i	$-0.276904\pi$
$29$	6.94567	1.28978	0.644889	+	0.764276i	$0.276904\pi$
$30$	0	0
$31$	−8.98680	−1.61408	−0.807038	−	0.590499i	$-0.798931\pi$
$31$	−8.98680	−1.61408	−0.807038	+	0.590499i	$0.798931\pi$
$32$	0	0
$33$	−13.1755	−2.29356
$34$	0	0
$35$	−3.47283	−0.587016
$36$	0	0
$37$	10.9457	1.79946	0.899728	−	0.436450i	$-0.143765\pi$
$37$	10.9457	1.79946	0.899728	+	0.436450i	$0.143765\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	2.06058	0.321808	0.160904	−	0.986970i	$-0.448559\pi$
$41$	2.06058	0.321808	0.160904	+	0.986970i	$0.448559\pi$
$42$	0	0
$43$	−0.399055	−0.0608553	−0.0304276	−	0.999537i	$-0.509687\pi$
$43$	−0.399055	−0.0608553	−0.0304276	+	0.999537i	$0.509687\pi$
$44$	0	0
$45$	23.2772	3.46996
$46$	0	0
$47$	12.4596	1.81742	0.908712	−	0.417424i	$-0.137067\pi$
$47$	12.4596	1.81742	0.908712	+	0.417424i	$0.137067\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.11491	0.436174
$52$	0	0
$53$	−6.64832	−0.913217	−0.456608	−	0.889668i	$-0.650936\pi$
$53$	−6.64832	−0.913217	−0.456608	+	0.889668i	$0.650936\pi$
$54$	0	0
$55$	14.6894	1.98072
$56$	0	0
$57$	−9.17548	−1.21532
$58$	0	0
$59$	−1.43171	−0.186392	−0.0931961	−	0.995648i	$-0.529708\pi$
$59$	−1.43171	−0.186392	−0.0931961	+	0.995648i	$0.529708\pi$
$60$	0	0
$61$	6.88509	0.881546	0.440773	−	0.897619i	$-0.354704\pi$
$61$	6.88509	0.881546	0.440773	+	0.897619i	$0.354704\pi$
$62$	0	0
$63$	−6.70265	−0.844454
$64$	0	0
$65$	4.45963	0.553149
$66$	0	0
$67$	10.4185	1.27282	0.636411	−	0.771350i	$-0.280418\pi$
$67$	10.4185	1.27282	0.636411	+	0.771350i	$0.280418\pi$
$68$	0	0
$69$	−2.22982	−0.268438
$70$	0	0
$71$	13.9736	1.65836	0.829180	−	0.558981i	$-0.188808\pi$
$71$	13.9736	1.65836	0.829180	+	0.558981i	$0.188808\pi$
$72$	0	0
$73$	−14.5202	−1.69946	−0.849731	−	0.527217i	$-0.823236\pi$
$73$	−14.5202	−1.69946	−0.849731	+	0.527217i	$0.823236\pi$
$74$	0	0
$75$	−21.9930	−2.53954
$76$	0	0
$77$	−4.22982	−0.482032
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	15.8176	1.75751
$82$	0	0
$83$	−1.05433	−0.115728	−0.0578640	−	0.998324i	$-0.518429\pi$
$83$	−1.05433	−0.115728	−0.0578640	+	0.998324i	$0.518429\pi$
$84$	0	0
$85$	−3.47283	−0.376682
$86$	0	0
$87$	−21.6351	−2.31953
$88$	0	0
$89$	4.60719	0.488361	0.244180	−	0.969730i	$-0.421481\pi$
$89$	4.60719	0.488361	0.244180	+	0.969730i	$0.421481\pi$
$90$	0	0
$91$	−1.28415	−0.134615
$92$	0	0
$93$	27.9930	2.90274
$94$	0	0
$95$	10.2298	1.04956
$96$	0	0
$97$	10.2709	1.04286	0.521428	−	0.853295i	$-0.325399\pi$
$97$	10.2709	1.04286	0.521428	+	0.853295i	$0.325399\pi$
$98$	0	0
$99$	28.3510	2.84938
$100$	0	0
$101$	−2.79811	−0.278422	−0.139211	−	0.990263i	$-0.544457\pi$
$101$	−2.79811	−0.278422	−0.139211	+	0.990263i	$0.544457\pi$
$102$	0	0
$103$	−6.94567	−0.684377	−0.342188	−	0.939631i	$-0.611168\pi$
$103$	−6.94567	−0.684377	−0.342188	+	0.939631i	$0.611168\pi$
$104$	0	0
$105$	10.8176	1.05569
$106$	0	0
$107$	−1.74378	−0.168577	−0.0842887	−	0.996441i	$-0.526862\pi$
$107$	−1.74378	−0.168577	−0.0842887	+	0.996441i	$0.526862\pi$
$108$	0	0
$109$	10.6072	1.01598	0.507992	−	0.861362i	$-0.330388\pi$
$109$	10.6072	1.01598	0.507992	+	0.861362i	$0.330388\pi$
$110$	0	0
$111$	−34.0947	−3.23613
$112$	0	0
$113$	−18.2034	−1.71243	−0.856216	−	0.516618i	$-0.827191\pi$
$113$	−18.2034	−1.71243	−0.856216	+	0.516618i	$0.827191\pi$
$114$	0	0
$115$	2.48604	0.231824
$116$	0	0
$117$	8.60719	0.795735
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	6.89134	0.626485
$122$	0	0
$123$	−6.41850	−0.578737
$124$	0	0
$125$	7.15604	0.640055
$126$	0	0
$127$	−10.0411	−0.891006	−0.445503	−	0.895280i	$-0.646975\pi$
$127$	−10.0411	−0.891006	−0.445503	+	0.895280i	$0.646975\pi$
$128$	0	0
$129$	1.24302	0.109442
$130$	0	0
$131$	−0.824517	−0.0720384	−0.0360192	−	0.999351i	$-0.511468\pi$
$131$	−0.824517	−0.0720384	−0.0360192	+	0.999351i	$0.511468\pi$
$132$	0	0
$133$	−2.94567	−0.255422
$134$	0	0
$135$	−40.0536	−3.44727
$136$	0	0
$137$	−0.290390	−0.0248097	−0.0124049	−	0.999923i	$-0.503949\pi$
$137$	−0.290390	−0.0248097	−0.0124049	+	0.999923i	$0.503949\pi$
$138$	0	0
$139$	−20.5397	−1.74215	−0.871075	−	0.491150i	$-0.836577\pi$
$139$	−20.5397	−1.74215	−0.871075	+	0.491150i	$0.836577\pi$
$140$	0	0
$141$	−38.8106	−3.26844
$142$	0	0
$143$	5.43171	0.454222
$144$	0	0
$145$	24.1212	2.00315
$146$	0	0
$147$	−3.11491	−0.256913
$148$	0	0
$149$	16.2904	1.33456	0.667280	−	0.744807i	$-0.267458\pi$
$149$	16.2904	1.33456	0.667280	+	0.744807i	$0.267458\pi$
$150$	0	0
$151$	7.72210	0.628415	0.314208	−	0.949354i	$-0.398261\pi$
$151$	7.72210	0.628415	0.314208	+	0.949354i	$0.398261\pi$
$152$	0	0
$153$	−6.70265	−0.541877
$154$	0	0
$155$	−31.2097	−2.50682
$156$	0	0
$157$	−17.4053	−1.38909	−0.694547	−	0.719447i	$-0.744395\pi$
$157$	−17.4053	−1.38909	−0.694547	+	0.719447i	$0.744395\pi$
$158$	0	0
$159$	20.7089	1.64232
$160$	0	0
$161$	−0.715853	−0.0564171
$162$	0	0
$163$	3.89134	0.304793	0.152396	−	0.988319i	$-0.451301\pi$
$163$	3.89134	0.304793	0.152396	+	0.988319i	$0.451301\pi$
$164$	0	0
$165$	−45.7563	−3.56212
$166$	0	0
$167$	12.8587	0.995035	0.497517	−	0.867454i	$-0.334245\pi$
$167$	12.8587	0.995035	0.497517	+	0.867454i	$0.334245\pi$
$168$	0	0
$169$	−11.3510	−0.873151
$170$	0	0
$171$	19.7438	1.50984
$172$	0	0
$173$	15.3447	1.16664	0.583319	−	0.812243i	$-0.301754\pi$
$173$	15.3447	1.16664	0.583319	+	0.812243i	$0.301754\pi$
$174$	0	0
$175$	−7.06058	−0.533729
$176$	0	0
$177$	4.45963	0.335206
$178$	0	0
$179$	−26.4115	−1.97409	−0.987046	−	0.160440i	$-0.948709\pi$
$179$	−26.4115	−1.97409	−0.987046	+	0.160440i	$0.948709\pi$
$180$	0	0
$181$	−9.63511	−0.716172	−0.358086	−	0.933689i	$-0.616571\pi$
$181$	−9.63511	−0.716172	−0.358086	+	0.933689i	$0.616571\pi$
$182$	0	0
$183$	−21.4464	−1.58537
$184$	0	0
$185$	38.0125	2.79473
$186$	0	0
$187$	−4.22982	−0.309315
$188$	0	0
$189$	11.5334	0.838932
$190$	0	0
$191$	23.2555	1.68271	0.841355	−	0.540483i	$-0.181759\pi$
$191$	23.2555	1.68271	0.841355	+	0.540483i	$0.181759\pi$
$192$	0	0
$193$	−9.06682	−0.652644	−0.326322	−	0.945259i	$-0.605809\pi$
$193$	−9.06682	−0.652644	−0.326322	+	0.945259i	$0.605809\pi$
$194$	0	0
$195$	−13.8913	−0.994779
$196$	0	0
$197$	−16.2034	−1.15444	−0.577222	−	0.816587i	$-0.695863\pi$
$197$	−16.2034	−1.15444	−0.577222	+	0.816587i	$0.695863\pi$
$198$	0	0
$199$	−9.49228	−0.672890	−0.336445	−	0.941703i	$-0.609225\pi$
$199$	−9.49228	−0.672890	−0.336445	+	0.941703i	$0.609225\pi$
$200$	0	0
$201$	−32.4527	−2.28903
$202$	0	0
$203$	−6.94567	−0.487490
$204$	0	0
$205$	7.15604	0.499799
$206$	0	0
$207$	4.79811	0.333492
$208$	0	0
$209$	12.4596	0.861851
$210$	0	0
$211$	−1.40530	−0.0967447	−0.0483724	−	0.998829i	$-0.515403\pi$
$211$	−1.40530	−0.0967447	−0.0483724	+	0.998829i	$0.515403\pi$
$212$	0	0
$213$	−43.5264	−2.98238
$214$	0	0
$215$	−1.38585	−0.0945143
$216$	0	0
$217$	8.98680	0.610063
$218$	0	0
$219$	45.2291	3.05630
$220$	0	0
$221$	−1.28415	−0.0863811
$222$	0	0
$223$	21.5140	1.44068	0.720341	−	0.693620i	$-0.243985\pi$
$223$	21.5140	1.44068	0.720341	+	0.693620i	$0.243985\pi$
$224$	0	0
$225$	47.3246	3.15497
$226$	0	0
$227$	19.0257	1.26278	0.631390	−	0.775466i	$-0.282485\pi$
$227$	19.0257	1.26278	0.631390	+	0.775466i	$0.282485\pi$
$228$	0	0
$229$	12.2298	0.808169	0.404084	−	0.914722i	$-0.367590\pi$
$229$	12.2298	0.808169	0.404084	+	0.914722i	$0.367590\pi$
$230$	0	0
$231$	13.1755	0.866883
$232$	0	0
$233$	−12.2687	−0.803750	−0.401875	−	0.915695i	$-0.631641\pi$
$233$	−12.2687	−0.803750	−0.401875	+	0.915695i	$0.631641\pi$
$234$	0	0
$235$	43.2702	2.82264
$236$	0	0
$237$	12.4596	0.809340
$238$	0	0
$239$	−18.6289	−1.20500	−0.602501	−	0.798118i	$-0.705829\pi$
$239$	−18.6289	−1.20500	−0.602501	+	0.798118i	$0.705829\pi$
$240$	0	0
$241$	22.7306	1.46420	0.732102	−	0.681195i	$-0.238539\pi$
$241$	22.7306	1.46420	0.732102	+	0.681195i	$0.238539\pi$
$242$	0	0
$243$	−14.6700	−0.941081
$244$	0	0
$245$	3.47283	0.221871
$246$	0	0
$247$	3.78267	0.240686
$248$	0	0
$249$	3.28415	0.208124
$250$	0	0
$251$	6.94567	0.438407	0.219203	−	0.975679i	$-0.429654\pi$
$251$	6.94567	0.438407	0.219203	+	0.975679i	$0.429654\pi$
$252$	0	0
$253$	3.02792	0.190364
$254$	0	0
$255$	10.8176	0.677422
$256$	0	0
$257$	−26.6630	−1.66319	−0.831597	−	0.555379i	$-0.812573\pi$
$257$	−26.6630	−1.66319	−0.831597	+	0.555379i	$0.812573\pi$
$258$	0	0
$259$	−10.9457	−0.680131
$260$	0	0
$261$	46.5544	2.88164
$262$	0	0
$263$	13.1366	0.810037	0.405018	−	0.914309i	$-0.367265\pi$
$263$	13.1366	0.810037	0.405018	+	0.914309i	$0.367265\pi$
$264$	0	0
$265$	−23.0885	−1.41832
$266$	0	0
$267$	−14.3510	−0.878265
$268$	0	0
$269$	−6.14756	−0.374823	−0.187412	−	0.982281i	$-0.560010\pi$
$269$	−6.14756	−0.374823	−0.187412	+	0.982281i	$0.560010\pi$
$270$	0	0
$271$	−23.7438	−1.44233	−0.721166	−	0.692762i	$-0.756393\pi$
$271$	−23.7438	−1.44233	−0.721166	+	0.692762i	$0.756393\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	29.8649	1.80092
$276$	0	0
$277$	−16.4985	−0.991300	−0.495650	−	0.868522i	$-0.665070\pi$
$277$	−16.4985	−0.991300	−0.495650	+	0.868522i	$0.665070\pi$
$278$	0	0
$279$	−60.2353	−3.60620
$280$	0	0
$281$	18.2709	1.08995	0.544977	−	0.838451i	$-0.316539\pi$
$281$	18.2709	1.08995	0.544977	+	0.838451i	$0.316539\pi$
$282$	0	0
$283$	−20.6219	−1.22585	−0.612923	−	0.790143i	$-0.710006\pi$
$283$	−20.6219	−1.22585	−0.612923	+	0.790143i	$0.710006\pi$
$284$	0	0
$285$	−31.8649	−1.88752
$286$	0	0
$287$	−2.06058	−0.121632
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−31.9930	−1.87547
$292$	0	0
$293$	0.824517	0.0481688	0.0240844	−	0.999710i	$-0.492333\pi$
$293$	0.824517	0.0481688	0.0240844	+	0.999710i	$0.492333\pi$
$294$	0	0
$295$	−4.97208	−0.289485
$296$	0	0
$297$	−48.7842	−2.83075
$298$	0	0
$299$	0.919260	0.0531622
$300$	0	0
$301$	0.399055	0.0230011
$302$	0	0
$303$	8.71585	0.500713
$304$	0	0
$305$	23.9108	1.36913
$306$	0	0
$307$	−31.9038	−1.82085	−0.910424	−	0.413677i	$-0.864244\pi$
$307$	−31.9038	−1.82085	−0.910424	+	0.413677i	$0.864244\pi$
$308$	0	0
$309$	21.6351	1.23078
$310$	0	0
$311$	16.3098	0.924846	0.462423	−	0.886659i	$-0.346980\pi$
$311$	16.3098	0.924846	0.462423	+	0.886659i	$0.346980\pi$
$312$	0	0
$313$	−28.5397	−1.61316	−0.806578	−	0.591127i	$-0.798683\pi$
$313$	−28.5397	−1.61316	−0.806578	+	0.591127i	$0.798683\pi$
$314$	0	0
$315$	−23.2772	−1.31152
$316$	0	0
$317$	18.5683	1.04290	0.521450	−	0.853282i	$-0.325391\pi$
$317$	18.5683	1.04290	0.521450	+	0.853282i	$0.325391\pi$
$318$	0	0
$319$	29.3789	1.64490
$320$	0	0
$321$	5.43171	0.303168
$322$	0	0
$323$	−2.94567	−0.163901
$324$	0	0
$325$	9.06682	0.502937
$326$	0	0
$327$	−33.0404	−1.82714
$328$	0	0
$329$	−12.4596	−0.686922
$330$	0	0
$331$	2.79588	0.153675	0.0768376	−	0.997044i	$-0.475518\pi$
$331$	2.79588	0.153675	0.0768376	+	0.997044i	$0.475518\pi$
$332$	0	0
$333$	73.3650	4.02038
$334$	0	0
$335$	36.1817	1.97682
$336$	0	0
$337$	−5.57926	−0.303922	−0.151961	−	0.988387i	$-0.548559\pi$
$337$	−5.57926	−0.303922	−0.151961	+	0.988387i	$0.548559\pi$
$338$	0	0
$339$	56.7019	3.07963
$340$	0	0
$341$	−38.0125	−2.05849
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−7.74378	−0.416911
$346$	0	0
$347$	25.9472	1.39292	0.696459	−	0.717597i	$-0.254758\pi$
$347$	25.9472	1.39292	0.696459	+	0.717597i	$0.254758\pi$
$348$	0	0
$349$	18.8370	1.00832	0.504161	−	0.863610i	$-0.331802\pi$
$349$	18.8370	1.00832	0.504161	+	0.863610i	$0.331802\pi$
$350$	0	0
$351$	−14.8106	−0.790531
$352$	0	0
$353$	−31.5000	−1.67658	−0.838289	−	0.545226i	$-0.816444\pi$
$353$	−31.5000	−1.67658	−0.838289	+	0.545226i	$0.816444\pi$
$354$	0	0
$355$	48.5280	2.57560
$356$	0	0
$357$	−3.11491	−0.164858
$358$	0	0
$359$	−10.7959	−0.569784	−0.284892	−	0.958560i	$-0.591958\pi$
$359$	−10.7959	−0.569784	−0.284892	+	0.958560i	$0.591958\pi$
$360$	0	0
$361$	−10.3230	−0.543318
$362$	0	0
$363$	−21.4659	−1.12667
$364$	0	0
$365$	−50.4263	−2.63943
$366$	0	0
$367$	3.47979	0.181644	0.0908219	−	0.995867i	$-0.471051\pi$
$367$	3.47979	0.181644	0.0908219	+	0.995867i	$0.471051\pi$
$368$	0	0
$369$	13.8113	0.718988
$370$	0	0
$371$	6.64832	0.345163
$372$	0	0
$373$	−35.5606	−1.84126	−0.920629	−	0.390437i	$-0.872324\pi$
$373$	−35.5606	−1.84126	−0.920629	+	0.390437i	$0.872324\pi$
$374$	0	0
$375$	−22.2904	−1.15107
$376$	0	0
$377$	8.91926	0.459365
$378$	0	0
$379$	12.9068	0.662976	0.331488	−	0.943459i	$-0.392449\pi$
$379$	12.9068	0.662976	0.331488	+	0.943459i	$0.392449\pi$
$380$	0	0
$381$	31.2772	1.60238
$382$	0	0
$383$	0.135072	0.00690185	0.00345092	−	0.999994i	$-0.498902\pi$
$383$	0.135072	0.00690185	0.00345092	+	0.999994i	$0.498902\pi$
$384$	0	0
$385$	−14.6894	−0.748643
$386$	0	0
$387$	−2.67472	−0.135964
$388$	0	0
$389$	20.9170	1.06054	0.530268	−	0.847830i	$-0.322091\pi$
$389$	20.9170	1.06054	0.530268	+	0.847830i	$0.322091\pi$
$390$	0	0
$391$	−0.715853	−0.0362022
$392$	0	0
$393$	2.56829	0.129553
$394$	0	0
$395$	−13.8913	−0.698949
$396$	0	0
$397$	−14.4185	−0.723644	−0.361822	−	0.932247i	$-0.617845\pi$
$397$	−14.4185	−0.723644	−0.361822	+	0.932247i	$0.617845\pi$
$398$	0	0
$399$	9.17548	0.459349
$400$	0	0
$401$	20.4332	1.02039	0.510193	−	0.860060i	$-0.329574\pi$
$401$	20.4332	1.02039	0.510193	+	0.860060i	$0.329574\pi$
$402$	0	0
$403$	−11.5404	−0.574867
$404$	0	0
$405$	54.9317	2.72958
$406$	0	0
$407$	46.2982	2.29491
$408$	0	0
$409$	−3.29663	−0.163008	−0.0815040	−	0.996673i	$-0.525972\pi$
$409$	−3.29663	−0.163008	−0.0815040	+	0.996673i	$0.525972\pi$
$410$	0	0
$411$	0.904539	0.0446176
$412$	0	0
$413$	1.43171	0.0704496
$414$	0	0
$415$	−3.66152	−0.179737
$416$	0	0
$417$	63.9791	3.13307
$418$	0	0
$419$	26.8906	1.31369	0.656846	−	0.754024i	$-0.271890\pi$
$419$	26.8906	1.31369	0.656846	+	0.754024i	$0.271890\pi$
$420$	0	0
$421$	26.2251	1.27813	0.639066	−	0.769152i	$-0.279321\pi$
$421$	26.2251	1.27813	0.639066	+	0.769152i	$0.279321\pi$
$422$	0	0
$423$	83.5125	4.06052
$424$	0	0
$425$	−7.06058	−0.342488
$426$	0	0
$427$	−6.88509	−0.333193
$428$	0	0
$429$	−16.9193	−0.816870
$430$	0	0
$431$	−1.09323	−0.0526588	−0.0263294	−	0.999653i	$-0.508382\pi$
$431$	−1.09323	−0.0526588	−0.0263294	+	0.999653i	$0.508382\pi$
$432$	0	0
$433$	15.3789	0.739062	0.369531	−	0.929218i	$-0.379518\pi$
$433$	15.3789	0.739062	0.369531	+	0.929218i	$0.379518\pi$
$434$	0	0
$435$	−75.1352	−3.60245
$436$	0	0
$437$	2.10866	0.100871
$438$	0	0
$439$	29.4395	1.40507	0.702535	−	0.711650i	$-0.252052\pi$
$439$	29.4395	1.40507	0.702535	+	0.711650i	$0.252052\pi$
$440$	0	0
$441$	6.70265	0.319174
$442$	0	0
$443$	−1.43171	−0.0680224	−0.0340112	−	0.999421i	$-0.510828\pi$
$443$	−1.43171	−0.0680224	−0.0340112	+	0.999421i	$0.510828\pi$
$444$	0	0
$445$	16.0000	0.758473
$446$	0	0
$447$	−50.7431	−2.40006
$448$	0	0
$449$	14.7547	0.696320	0.348160	−	0.937435i	$-0.386807\pi$
$449$	14.7547	0.696320	0.348160	+	0.937435i	$0.386807\pi$
$450$	0	0
$451$	8.71585	0.410413
$452$	0	0
$453$	−24.0536	−1.13014
$454$	0	0
$455$	−4.45963	−0.209071
$456$	0	0
$457$	−1.89357	−0.0885775	−0.0442887	−	0.999019i	$-0.514102\pi$
$457$	−1.89357	−0.0885775	−0.0442887	+	0.999019i	$0.514102\pi$
$458$	0	0
$459$	11.5334	0.538333
$460$	0	0
$461$	14.7547	0.687197	0.343599	−	0.939117i	$-0.388354\pi$
$461$	14.7547	0.687197	0.343599	+	0.939117i	$0.388354\pi$
$462$	0	0
$463$	14.9673	0.695592	0.347796	−	0.937570i	$-0.386930\pi$
$463$	14.9673	0.695592	0.347796	+	0.937570i	$0.386930\pi$
$464$	0	0
$465$	97.2152	4.50825
$466$	0	0
$467$	−0.473551	−0.0219133	−0.0109567	−	0.999940i	$-0.503488\pi$
$467$	−0.473551	−0.0219133	−0.0109567	+	0.999940i	$0.503488\pi$
$468$	0	0
$469$	−10.4185	−0.481082
$470$	0	0
$471$	54.2159	2.49814
$472$	0	0
$473$	−1.68793	−0.0776110
$474$	0	0
$475$	20.7981	0.954283
$476$	0	0
$477$	−44.5613	−2.04032
$478$	0	0
$479$	4.27790	0.195462	0.0977312	−	0.995213i	$-0.468841\pi$
$479$	4.27790	0.195462	0.0977312	+	0.995213i	$0.468841\pi$
$480$	0	0
$481$	14.0558	0.640892
$482$	0	0
$483$	2.22982	0.101460
$484$	0	0
$485$	35.6693	1.61966
$486$	0	0
$487$	−4.04336	−0.183222	−0.0916111	−	0.995795i	$-0.529202\pi$
$487$	−4.04336	−0.183222	−0.0916111	+	0.995795i	$0.529202\pi$
$488$	0	0
$489$	−12.1212	−0.548137
$490$	0	0
$491$	4.98680	0.225051	0.112525	−	0.993649i	$-0.464106\pi$
$491$	4.98680	0.225051	0.112525	+	0.993649i	$0.464106\pi$
$492$	0	0
$493$	−6.94567	−0.312817
$494$	0	0
$495$	98.4582	4.42537
$496$	0	0
$497$	−13.9736	−0.626801
$498$	0	0
$499$	17.6226	0.788897	0.394449	−	0.918918i	$-0.370936\pi$
$499$	17.6226	0.788897	0.394449	+	0.918918i	$0.370936\pi$
$500$	0	0
$501$	−40.0536	−1.78946
$502$	0	0
$503$	−3.55509	−0.158514	−0.0792568	−	0.996854i	$-0.525255\pi$
$503$	−3.55509	−0.158514	−0.0792568	+	0.996854i	$0.525255\pi$
$504$	0	0
$505$	−9.71737	−0.432417
$506$	0	0
$507$	35.3572	1.57027
$508$	0	0
$509$	7.89134	0.349777	0.174889	−	0.984588i	$-0.444043\pi$
$509$	7.89134	0.349777	0.174889	+	0.984588i	$0.444043\pi$
$510$	0	0
$511$	14.5202	0.642336
$512$	0	0
$513$	−33.9736	−1.49997
$514$	0	0
$515$	−24.1212	−1.06290
$516$	0	0
$517$	52.7019	2.31783
$518$	0	0
$519$	−47.7974	−2.09807
$520$	0	0
$521$	13.2097	0.578725	0.289363	−	0.957220i	$-0.406557\pi$
$521$	13.2097	0.578725	0.289363	+	0.957220i	$0.406557\pi$
$522$	0	0
$523$	38.5544	1.68587	0.842933	−	0.538019i	$-0.180827\pi$
$523$	38.5544	1.68587	0.842933	+	0.538019i	$0.180827\pi$
$524$	0	0
$525$	21.9930	0.959855
$526$	0	0
$527$	8.98680	0.391471
$528$	0	0
$529$	−22.4876	−0.977720
$530$	0	0
$531$	−9.59622	−0.416440
$532$	0	0
$533$	2.64608	0.114615
$534$	0	0
$535$	−6.05585	−0.261817
$536$	0	0
$537$	82.2695	3.55019
$538$	0	0
$539$	4.22982	0.182191
$540$	0	0
$541$	6.36489	0.273648	0.136824	−	0.990595i	$-0.456311\pi$
$541$	6.36489	0.273648	0.136824	+	0.990595i	$0.456311\pi$
$542$	0	0
$543$	30.0125	1.28796
$544$	0	0
$545$	36.8370	1.57792
$546$	0	0
$547$	43.0015	1.83861	0.919306	−	0.393543i	$-0.128751\pi$
$547$	43.0015	1.83861	0.919306	+	0.393543i	$0.128751\pi$
$548$	0	0
$549$	46.1484	1.96956
$550$	0	0
$551$	20.4596	0.871610
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	−118.405	−5.02603
$556$	0	0
$557$	41.9472	1.77736	0.888680	−	0.458529i	$-0.151623\pi$
$557$	41.9472	1.77736	0.888680	+	0.458529i	$0.151623\pi$
$558$	0	0
$559$	−0.512445	−0.0216741
$560$	0	0
$561$	13.1755	0.556269
$562$	0	0
$563$	20.3246	0.856578	0.428289	−	0.903642i	$-0.359117\pi$
$563$	20.3246	0.856578	0.428289	+	0.903642i	$0.359117\pi$
$564$	0	0
$565$	−63.2174	−2.65958
$566$	0	0
$567$	−15.8176	−0.664275
$568$	0	0
$569$	16.4115	0.688008	0.344004	−	0.938968i	$-0.388217\pi$
$569$	16.4115	0.688008	0.344004	+	0.938968i	$0.388217\pi$
$570$	0	0
$571$	4.52493	0.189362	0.0946812	−	0.995508i	$-0.469817\pi$
$571$	4.52493	0.189362	0.0946812	+	0.995508i	$0.469817\pi$
$572$	0	0
$573$	−72.4387	−3.02617
$574$	0	0
$575$	5.05433	0.210780
$576$	0	0
$577$	−4.35097	−0.181133	−0.0905665	−	0.995890i	$-0.528868\pi$
$577$	−4.35097	−0.181133	−0.0905665	+	0.995890i	$0.528868\pi$
$578$	0	0
$579$	28.2423	1.17371
$580$	0	0
$581$	1.05433	0.0437411
$582$	0	0
$583$	−28.1212	−1.16466
$584$	0	0
$585$	29.8913	1.23585
$586$	0	0
$587$	11.0668	0.456776	0.228388	−	0.973570i	$-0.426654\pi$
$587$	11.0668	0.456776	0.228388	+	0.973570i	$0.426654\pi$
$588$	0	0
$589$	−26.4721	−1.09076
$590$	0	0
$591$	50.4721	2.07615
$592$	0	0
$593$	−15.5529	−0.638679	−0.319340	−	0.947640i	$-0.603461\pi$
$593$	−15.5529	−0.638679	−0.319340	+	0.947640i	$0.603461\pi$
$594$	0	0
$595$	3.47283	0.142372
$596$	0	0
$597$	29.5676	1.21012
$598$	0	0
$599$	15.0885	0.616499	0.308250	−	0.951305i	$-0.400257\pi$
$599$	15.0885	0.616499	0.308250	+	0.951305i	$0.400257\pi$
$600$	0	0
$601$	6.45963	0.263494	0.131747	−	0.991283i	$-0.457941\pi$
$601$	6.45963	0.263494	0.131747	+	0.991283i	$0.457941\pi$
$602$	0	0
$603$	69.8316	2.84376
$604$	0	0
$605$	23.9325	0.972993
$606$	0	0
$607$	12.0147	0.487662	0.243831	−	0.969818i	$-0.421596\pi$
$607$	12.0147	0.487662	0.243831	+	0.969818i	$0.421596\pi$
$608$	0	0
$609$	21.6351	0.876699
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	9.29887	0.375578	0.187789	−	0.982209i	$-0.439868\pi$
$613$	9.29887	0.375578	0.187789	+	0.982209i	$0.439868\pi$
$614$	0	0
$615$	−22.2904	−0.898835
$616$	0	0
$617$	−3.01544	−0.121397	−0.0606985	−	0.998156i	$-0.519333\pi$
$617$	−3.01544	−0.121397	−0.0606985	+	0.998156i	$0.519333\pi$
$618$	0	0
$619$	0.607188	0.0244050	0.0122025	−	0.999926i	$-0.496116\pi$
$619$	0.607188	0.0244050	0.0122025	+	0.999926i	$0.496116\pi$
$620$	0	0
$621$	−8.25622	−0.331311
$622$	0	0
$623$	−4.60719	−0.184583
$624$	0	0
$625$	−10.4512	−0.418046
$626$	0	0
$627$	−38.8106	−1.54995
$628$	0	0
$629$	−10.9457	−0.436432
$630$	0	0
$631$	−3.33776	−0.132874	−0.0664371	−	0.997791i	$-0.521163\pi$
$631$	−3.33776	−0.132874	−0.0664371	+	0.997791i	$0.521163\pi$
$632$	0	0
$633$	4.37737	0.173985
$634$	0	0
$635$	−34.8712	−1.38382
$636$	0	0
$637$	1.28415	0.0508798
$638$	0	0
$639$	93.6601	3.70514
$640$	0	0
$641$	7.05433	0.278629	0.139315	−	0.990248i	$-0.455510\pi$
$641$	7.05433	0.278629	0.139315	+	0.990248i	$0.455510\pi$
$642$	0	0
$643$	−45.3447	−1.78822	−0.894111	−	0.447846i	$-0.852191\pi$
$643$	−45.3447	−1.78822	−0.894111	+	0.447846i	$0.852191\pi$
$644$	0	0
$645$	4.31680	0.169974
$646$	0	0
$647$	−37.7563	−1.48435	−0.742176	−	0.670205i	$-0.766206\pi$
$647$	−37.7563	−1.48435	−0.742176	+	0.670205i	$0.766206\pi$
$648$	0	0
$649$	−6.05585	−0.237713
$650$	0	0
$651$	−27.9930	−1.09713
$652$	0	0
$653$	14.9890	0.586566	0.293283	−	0.956026i	$-0.405252\pi$
$653$	14.9890	0.586566	0.293283	+	0.956026i	$0.405252\pi$
$654$	0	0
$655$	−2.86341	−0.111883
$656$	0	0
$657$	−97.3238	−3.79696
$658$	0	0
$659$	−2.33624	−0.0910072	−0.0455036	−	0.998964i	$-0.514489\pi$
$659$	−2.33624	−0.0910072	−0.0455036	+	0.998964i	$0.514489\pi$
$660$	0	0
$661$	−15.6785	−0.609822	−0.304911	−	0.952381i	$-0.598627\pi$
$661$	−15.6785	−0.609822	−0.304911	+	0.952381i	$0.598627\pi$
$662$	0	0
$663$	4.00000	0.155347
$664$	0	0
$665$	−10.2298	−0.396695
$666$	0	0
$667$	4.97208	0.192520
$668$	0	0
$669$	−67.0140	−2.59091
$670$	0	0
$671$	29.1227	1.12427
$672$	0	0
$673$	27.2144	1.04904	0.524519	−	0.851399i	$-0.324245\pi$
$673$	27.2144	1.04904	0.524519	+	0.851399i	$0.324245\pi$
$674$	0	0
$675$	−81.4325	−3.13434
$676$	0	0
$677$	−14.7717	−0.567723	−0.283861	−	0.958865i	$-0.591616\pi$
$677$	−14.7717	−0.567723	−0.283861	+	0.958865i	$0.591616\pi$
$678$	0	0
$679$	−10.2709	−0.394163
$680$	0	0
$681$	−59.2633	−2.27097
$682$	0	0
$683$	−3.13659	−0.120018	−0.0600091	−	0.998198i	$-0.519113\pi$
$683$	−3.13659	−0.120018	−0.0600091	+	0.998198i	$0.519113\pi$
$684$	0	0
$685$	−1.00848	−0.0385320
$686$	0	0
$687$	−38.0947	−1.45341
$688$	0	0
$689$	−8.53742	−0.325250
$690$	0	0
$691$	−14.8976	−0.566731	−0.283365	−	0.959012i	$-0.591451\pi$
$691$	−14.8976	−0.566731	−0.283365	+	0.959012i	$0.591451\pi$
$692$	0	0
$693$	−28.3510	−1.07696
$694$	0	0
$695$	−71.3308	−2.70573
$696$	0	0
$697$	−2.06058	−0.0780499
$698$	0	0
$699$	38.2159	1.44546
$700$	0	0
$701$	44.8106	1.69247	0.846236	−	0.532808i	$-0.178863\pi$
$701$	44.8106	1.69247	0.846236	+	0.532808i	$0.178863\pi$
$702$	0	0
$703$	32.2423	1.21604
$704$	0	0
$705$	−134.783	−5.07621
$706$	0	0
$707$	2.79811	0.105234
$708$	0	0
$709$	−27.2702	−1.02415	−0.512077	−	0.858939i	$-0.671124\pi$
$709$	−27.2702	−1.02415	−0.512077	+	0.858939i	$0.671124\pi$
$710$	0	0
$711$	−26.8106	−1.00548
$712$	0	0
$713$	−6.43322	−0.240926
$714$	0	0
$715$	18.8634	0.705452
$716$	0	0
$717$	58.0272	2.16707
$718$	0	0
$719$	6.75002	0.251733	0.125867	−	0.992047i	$-0.459829\pi$
$719$	6.75002	0.251733	0.125867	+	0.992047i	$0.459829\pi$
$720$	0	0
$721$	6.94567	0.258670
$722$	0	0
$723$	−70.8036	−2.63322
$724$	0	0
$725$	49.0404	1.82132
$726$	0	0
$727$	22.0125	0.816398	0.408199	−	0.912893i	$-0.366157\pi$
$727$	22.0125	0.816398	0.408199	+	0.912893i	$0.366157\pi$
$728$	0	0
$729$	−1.75698	−0.0650734
$730$	0	0
$731$	0.399055	0.0147596
$732$	0	0
$733$	18.4596	0.681822	0.340911	−	0.940096i	$-0.389265\pi$
$733$	18.4596	0.681822	0.340911	+	0.940096i	$0.389265\pi$
$734$	0	0
$735$	−10.8176	−0.399012
$736$	0	0
$737$	44.0683	1.62328
$738$	0	0
$739$	−25.2221	−0.927811	−0.463906	−	0.885885i	$-0.653552\pi$
$739$	−25.2221	−0.927811	−0.463906	+	0.885885i	$0.653552\pi$
$740$	0	0
$741$	−11.7827	−0.432847
$742$	0	0
$743$	9.05433	0.332171	0.166086	−	0.986111i	$-0.446887\pi$
$743$	9.05433	0.332171	0.166086	+	0.986111i	$0.446887\pi$
$744$	0	0
$745$	56.5738	2.07270
$746$	0	0
$747$	−7.06682	−0.258561
$748$	0	0
$749$	1.74378	0.0637162
$750$	0	0
$751$	−39.4053	−1.43792	−0.718960	−	0.695052i	$-0.755382\pi$
$751$	−39.4053	−1.43792	−0.718960	+	0.695052i	$0.755382\pi$
$752$	0	0
$753$	−21.6351	−0.788427
$754$	0	0
$755$	26.8176	0.975991
$756$	0	0
$757$	−12.9673	−0.471306	−0.235653	−	0.971837i	$-0.575723\pi$
$757$	−12.9673	−0.471306	−0.235653	+	0.971837i	$0.575723\pi$
$758$	0	0
$759$	−9.43171	−0.342349
$760$	0	0
$761$	5.20189	0.188568	0.0942842	−	0.995545i	$-0.469944\pi$
$761$	5.20189	0.188568	0.0942842	+	0.995545i	$0.469944\pi$
$762$	0	0
$763$	−10.6072	−0.384006
$764$	0	0
$765$	−23.2772	−0.841588
$766$	0	0
$767$	−1.83852	−0.0663851
$768$	0	0
$769$	−17.6226	−0.635488	−0.317744	−	0.948177i	$-0.602925\pi$
$769$	−17.6226	−0.635488	−0.317744	+	0.948177i	$0.602925\pi$
$770$	0	0
$771$	83.0529	2.99108
$772$	0	0
$773$	−34.4457	−1.23893	−0.619463	−	0.785026i	$-0.712650\pi$
$773$	−34.4457	−1.23893	−0.619463	+	0.785026i	$0.712650\pi$
$774$	0	0
$775$	−63.4520	−2.27926
$776$	0	0
$777$	34.0947	1.22314
$778$	0	0
$779$	6.06977	0.217472
$780$	0	0
$781$	59.1057	2.11497
$782$	0	0
$783$	−80.1072	−2.86280
$784$	0	0
$785$	−60.4457	−2.15740
$786$	0	0
$787$	9.28415	0.330944	0.165472	−	0.986214i	$-0.447085\pi$
$787$	9.28415	0.330944	0.165472	+	0.986214i	$0.447085\pi$
$788$	0	0
$789$	−40.9193	−1.45676
$790$	0	0
$791$	18.2034	0.647239
$792$	0	0
$793$	8.84147	0.313970
$794$	0	0
$795$	71.9185	2.55069
$796$	0	0
$797$	−51.7952	−1.83468	−0.917339	−	0.398106i	$-0.869668\pi$
$797$	−51.7952	−1.83468	−0.917339	+	0.398106i	$0.869668\pi$
$798$	0	0
$799$	−12.4596	−0.440790
$800$	0	0
$801$	30.8804	1.09110
$802$	0	0
$803$	−61.4178	−2.16739
$804$	0	0
$805$	−2.48604	−0.0876213
$806$	0	0
$807$	19.1491	0.674079
$808$	0	0
$809$	−13.7827	−0.484573	−0.242286	−	0.970205i	$-0.577897\pi$
$809$	−13.7827	−0.484573	−0.242286	+	0.970205i	$0.577897\pi$
$810$	0	0
$811$	31.3572	1.10110	0.550550	−	0.834802i	$-0.314418\pi$
$811$	31.3572	1.10110	0.550550	+	0.834802i	$0.314418\pi$
$812$	0	0
$813$	73.9597	2.59388
$814$	0	0
$815$	13.5140	0.473373
$816$	0	0
$817$	−1.17548	−0.0411249
$818$	0	0
$819$	−8.60719	−0.300760
$820$	0	0
$821$	5.29663	0.184854	0.0924269	−	0.995719i	$-0.470538\pi$
$821$	5.29663	0.184854	0.0924269	+	0.995719i	$0.470538\pi$
$822$	0	0
$823$	−24.1600	−0.842166	−0.421083	−	0.907022i	$-0.638350\pi$
$823$	−24.1600	−0.842166	−0.421083	+	0.907022i	$0.638350\pi$
$824$	0	0
$825$	−93.0265	−3.23877
$826$	0	0
$827$	32.9457	1.14563	0.572817	−	0.819684i	$-0.305851\pi$
$827$	32.9457	1.14563	0.572817	+	0.819684i	$0.305851\pi$
$828$	0	0
$829$	−15.6879	−0.544864	−0.272432	−	0.962175i	$-0.587828\pi$
$829$	−15.6879	−0.544864	−0.272432	+	0.962175i	$0.587828\pi$
$830$	0	0
$831$	51.3914	1.78275
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	44.6561	1.54539
$836$	0	0
$837$	103.648	3.58261
$838$	0	0
$839$	12.7019	0.438519	0.219260	−	0.975667i	$-0.429636\pi$
$839$	12.7019	0.438519	0.219260	+	0.975667i	$0.429636\pi$
$840$	0	0
$841$	19.2423	0.663528
$842$	0	0
$843$	−56.9123	−1.96016
$844$	0	0
$845$	−39.4200	−1.35609
$846$	0	0
$847$	−6.89134	−0.236789
$848$	0	0
$849$	64.2353	2.20455
$850$	0	0
$851$	7.83549	0.268597
$852$	0	0
$853$	−21.6351	−0.740772	−0.370386	−	0.928878i	$-0.620775\pi$
$853$	−21.6351	−0.740772	−0.370386	+	0.928878i	$0.620775\pi$
$854$	0	0
$855$	68.5669	2.34494
$856$	0	0
$857$	26.3782	0.901061	0.450531	−	0.892761i	$-0.351235\pi$
$857$	26.3782	0.901061	0.450531	+	0.892761i	$0.351235\pi$
$858$	0	0
$859$	−25.8385	−0.881599	−0.440799	−	0.897606i	$-0.645305\pi$
$859$	−25.8385	−0.881599	−0.440799	+	0.897606i	$0.645305\pi$
$860$	0	0
$861$	6.41850	0.218742
$862$	0	0
$863$	−45.8238	−1.55986	−0.779930	−	0.625867i	$-0.784745\pi$
$863$	−45.8238	−1.55986	−0.779930	+	0.625867i	$0.784745\pi$
$864$	0	0
$865$	53.2897	1.81190
$866$	0	0
$867$	−3.11491	−0.105788
$868$	0	0
$869$	−16.9193	−0.573947
$870$	0	0
$871$	13.3789	0.453326
$872$	0	0
$873$	68.8425	2.32997
$874$	0	0
$875$	−7.15604	−0.241918
$876$	0	0
$877$	47.0668	1.58933	0.794667	−	0.607046i	$-0.207646\pi$
$877$	47.0668	1.58933	0.794667	+	0.607046i	$0.207646\pi$
$878$	0	0
$879$	−2.56829	−0.0866264
$880$	0	0
$881$	−19.7026	−0.663799	−0.331900	−	0.943315i	$-0.607690\pi$
$881$	−19.7026	−0.663799	−0.331900	+	0.943315i	$0.607690\pi$
$882$	0	0
$883$	2.55357	0.0859346	0.0429673	−	0.999076i	$-0.486319\pi$
$883$	2.55357	0.0859346	0.0429673	+	0.999076i	$0.486319\pi$
$884$	0	0
$885$	15.4876	0.520608
$886$	0	0
$887$	−10.8003	−0.362640	−0.181320	−	0.983424i	$-0.558037\pi$
$887$	−10.8003	−0.362640	−0.181320	+	0.983424i	$0.558037\pi$
$888$	0	0
$889$	10.0411	0.336769
$890$	0	0
$891$	66.9053	2.24141
$892$	0	0
$893$	36.7019	1.22818
$894$	0	0
$895$	−91.7229	−3.06596
$896$	0	0
$897$	−2.86341	−0.0956065
$898$	0	0
$899$	−62.4193	−2.08180
$900$	0	0
$901$	6.64832	0.221488
$902$	0	0
$903$	−1.24302	−0.0413651
$904$	0	0
$905$	−33.4611	−1.11229
$906$	0	0
$907$	43.5962	1.44759	0.723794	−	0.690016i	$-0.242396\pi$
$907$	43.5962	1.44759	0.723794	+	0.690016i	$0.242396\pi$
$908$	0	0
$909$	−18.7547	−0.622056
$910$	0	0
$911$	−2.47212	−0.0819049	−0.0409524	−	0.999161i	$-0.513039\pi$
$911$	−2.47212	−0.0819049	−0.0409524	+	0.999161i	$0.513039\pi$
$912$	0	0
$913$	−4.45963	−0.147592
$914$	0	0
$915$	−74.4799	−2.46223
$916$	0	0
$917$	0.824517	0.0272280
$918$	0	0
$919$	−39.8866	−1.31574	−0.657869	−	0.753132i	$-0.728542\pi$
$919$	−39.8866	−1.31574	−0.657869	+	0.753132i	$0.728542\pi$
$920$	0	0
$921$	99.3775	3.27460
$922$	0	0
$923$	17.9442	0.590639
$924$	0	0
$925$	77.2827	2.54104
$926$	0	0
$927$	−46.5544	−1.52905
$928$	0	0
$929$	17.9714	0.589621	0.294811	−	0.955556i	$-0.404743\pi$
$929$	17.9714	0.589621	0.294811	+	0.955556i	$0.404743\pi$
$930$	0	0
$931$	2.94567	0.0965404
$932$	0	0
$933$	−50.8036	−1.66324
$934$	0	0
$935$	−14.6894	−0.480396
$936$	0	0
$937$	3.47060	0.113380	0.0566898	−	0.998392i	$-0.481945\pi$
$937$	3.47060	0.113380	0.0566898	+	0.998392i	$0.481945\pi$
$938$	0	0
$939$	88.8984	2.90109
$940$	0	0
$941$	16.5591	0.539811	0.269906	−	0.962887i	$-0.413008\pi$
$941$	16.5591	0.539811	0.269906	+	0.962887i	$0.413008\pi$
$942$	0	0
$943$	1.47507	0.0480348
$944$	0	0
$945$	40.0536	1.30294
$946$	0	0
$947$	17.1491	0.557270	0.278635	−	0.960397i	$-0.410118\pi$
$947$	17.1491	0.557270	0.278635	+	0.960397i	$0.410118\pi$
$948$	0	0
$949$	−18.6461	−0.605277
$950$	0	0
$951$	−57.8385	−1.87554
$952$	0	0
$953$	16.7570	0.542812	0.271406	−	0.962465i	$-0.412511\pi$
$953$	16.7570	0.542812	0.271406	+	0.962465i	$0.412511\pi$
$954$	0	0
$955$	80.7625	2.61341
$956$	0	0
$957$	−91.5125	−2.95818
$958$	0	0
$959$	0.290390	0.00937720
$960$	0	0
$961$	49.7625	1.60524
$962$	0	0
$963$	−11.6879	−0.376638
$964$	0	0
$965$	−31.4876	−1.01362
$966$	0	0
$967$	−48.0855	−1.54633	−0.773163	−	0.634207i	$-0.781327\pi$
$967$	−48.0855	−1.54633	−0.773163	+	0.634207i	$0.781327\pi$
$968$	0	0
$969$	9.17548	0.294759
$970$	0	0
$971$	21.6785	0.695695	0.347848	−	0.937551i	$-0.386913\pi$
$971$	21.6785	0.695695	0.347848	+	0.937551i	$0.386913\pi$
$972$	0	0
$973$	20.5397	0.658471
$974$	0	0
$975$	−28.2423	−0.904478
$976$	0	0
$977$	−44.0800	−1.41024	−0.705122	−	0.709086i	$-0.749108\pi$
$977$	−44.0800	−1.41024	−0.705122	+	0.709086i	$0.749108\pi$
$978$	0	0
$979$	19.4876	0.622825
$980$	0	0
$981$	71.0963	2.26993
$982$	0	0
$983$	41.3114	1.31763	0.658814	−	0.752306i	$-0.271059\pi$
$983$	41.3114	1.31763	0.658814	+	0.752306i	$0.271059\pi$
$984$	0	0
$985$	−56.2717	−1.79297
$986$	0	0
$987$	38.8106	1.23535
$988$	0	0
$989$	−0.285664	−0.00908360
$990$	0	0
$991$	13.9302	0.442508	0.221254	−	0.975216i	$-0.428985\pi$
$991$	13.9302	0.442508	0.221254	+	0.975216i	$0.428985\pi$
$992$	0	0
$993$	−8.70889	−0.276368
$994$	0	0
$995$	−32.9651	−1.04506
$996$	0	0
$997$	−40.6803	−1.28836	−0.644178	−	0.764875i	$-0.722800\pi$
$997$	−40.6803	−1.28836	−0.644178	+	0.764875i	$0.722800\pi$
$998$	0	0
$999$	−126.241	−3.99408

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.bb.1.1		3
4.3	odd	2		7616.2.a.bh.1.3		3
8.3	odd	2		952.2.a.c.1.1	✓	3
8.5	even	2		1904.2.a.o.1.3		3
24.11	even	2		8568.2.a.be.1.3		3
56.27	even	2		6664.2.a.m.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.c.1.1	✓	3	8.3	odd	2
1904.2.a.o.1.3		3	8.5	even	2
6664.2.a.m.1.3		3	56.27	even	2
7616.2.a.bb.1.1		3	1.1	even	1	trivial
7616.2.a.bh.1.3		3	4.3	odd	2
8568.2.a.be.1.3		3	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)

\(f(q)\)	\(=\)	\(q-3.11491 q^{3} +3.47283 q^{5} -1.00000 q^{7} +6.70265 q^{9} +4.22982 q^{11} +1.28415 q^{13} -10.8176 q^{15} -1.00000 q^{17} +2.94567 q^{19} +3.11491 q^{21} +0.715853 q^{23} +7.06058 q^{25} -11.5334 q^{27} +6.94567 q^{29} -8.98680 q^{31} -13.1755 q^{33} -3.47283 q^{35} +10.9457 q^{37} -4.00000 q^{39} +2.06058 q^{41} -0.399055 q^{43} +23.2772 q^{45} +12.4596 q^{47} +1.00000 q^{49} +3.11491 q^{51} -6.64832 q^{53} +14.6894 q^{55} -9.17548 q^{57} -1.43171 q^{59} +6.88509 q^{61} -6.70265 q^{63} +4.45963 q^{65} +10.4185 q^{67} -2.22982 q^{69} +13.9736 q^{71} -14.5202 q^{73} -21.9930 q^{75} -4.22982 q^{77} -4.00000 q^{79} +15.8176 q^{81} -1.05433 q^{83} -3.47283 q^{85} -21.6351 q^{87} +4.60719 q^{89} -1.28415 q^{91} +27.9930 q^{93} +10.2298 q^{95} +10.2709 q^{97} +28.3510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 5 q^{5} - 3 q^{7} + 2 q^{9} + 2 q^{13} - 8 q^{15} - 3 q^{17} - 2 q^{19} + 3 q^{21} + 4 q^{23} + 4 q^{25} - 12 q^{27} + 10 q^{29} - 7 q^{31} - 16 q^{33} - 5 q^{35} + 22 q^{37} - 12 q^{39}+ \cdots + 38 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 5 * q^5 - 3 * q^7 + 2 * q^9 + 2 * q^13 - 8 * q^15 - 3 * q^17 - 2 * q^19 + 3 * q^21 + 4 * q^23 + 4 * q^25 - 12 * q^27 + 10 * q^29 - 7 * q^31 - 16 * q^33 - 5 * q^35 + 22 * q^37 - 12 * q^39 - 11 * q^41 + 7 * q^43 + 20 * q^45 + 12 * q^47 + 3 * q^49 + 3 * q^51 + 9 * q^53 + 6 * q^55 - 4 * q^57 - 8 * q^59 + 27 * q^61 - 2 * q^63 - 12 * q^65 + 15 * q^67 + 6 * q^69 + 2 * q^71 - q^73 - 18 * q^75 - 12 * q^79 + 23 * q^81 - 14 * q^83 - 5 * q^85 - 16 * q^87 - 6 * q^89 - 2 * q^91 + 36 * q^93 + 18 * q^95 + 9 * q^97 + 38 * q^99

Introduction

L-functions

Modular forms

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Database

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