LMFDB - Embedded newform 7616.2.a.bm.1.4

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 = [4,0,0,0,4,0,4,0,-2,0,-2] 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:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3808)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.90211$ 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)

$f(q)$	$=$	$q+1.90211 q^{3} -0.902113 q^{5} +1.00000 q^{7} +0.618034 q^{9} -0.442463 q^{11} +5.75621 q^{13} -1.71592 q^{15} +1.00000 q^{17} -6.42882 q^{19} +1.90211 q^{21} -0.273457 q^{23} -4.18619 q^{25} -4.53077 q^{27} -10.5018 q^{29} -7.29657 q^{31} -0.841616 q^{33} -0.902113 q^{35} +5.86067 q^{37} +10.9490 q^{39} -5.79766 q^{41} -1.94542 q^{43} -0.557537 q^{45} -2.10445 q^{47} +1.00000 q^{49} +1.90211 q^{51} +9.20524 q^{53} +0.399152 q^{55} -12.2284 q^{57} -4.79830 q^{59} -1.68980 q^{61} +0.618034 q^{63} -5.19276 q^{65} -10.3636 q^{67} -0.520147 q^{69} +5.11798 q^{71} -16.0872 q^{73} -7.96261 q^{75} -0.442463 q^{77} +1.36176 q^{79} -10.4721 q^{81} +7.10143 q^{83} -0.902113 q^{85} -19.9756 q^{87} +5.30011 q^{89} +5.75621 q^{91} -13.8789 q^{93} +5.79953 q^{95} -2.11321 q^{97} -0.273457 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{7} - 2 q^{9} - 2 q^{11} + 2 q^{13} - 10 q^{15} + 4 q^{17} - 4 q^{19} - 4 q^{23} - 6 q^{25} - 6 q^{29} - 16 q^{31} + 4 q^{35} + 8 q^{37} + 10 q^{39} - 10 q^{41} - 4 q^{43} - 2 q^{45}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^7 - 2 * q^9 - 2 * q^11 + 2 * q^13 - 10 * q^15 + 4 * q^17 - 4 * q^19 - 4 * q^23 - 6 * q^25 - 6 * q^29 - 16 * q^31 + 4 * q^35 + 8 * q^37 + 10 * q^39 - 10 * q^41 - 4 * q^43 - 2 * q^45 - 14 * q^47 + 4 * q^49 + 14 * q^53 - 2 * q^55 - 10 * q^57 - 12 * q^59 + 2 * q^61 - 2 * q^63 - 8 * q^65 - 22 * q^67 + 10 * q^69 - 16 * q^71 - 14 * q^73 - 20 * q^75 - 2 * q^77 - 10 * q^79 - 24 * q^81 + 6 * q^83 + 4 * q^85 - 22 * q^89 + 2 * q^91 + 6 * q^95 - 12 * q^97 - 4 * 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
$3$	1.90211	1.09819	0.549093	−	0.835761i	$-0.314973\pi$
$3$	1.90211	1.09819	0.549093	+	0.835761i	$0.314973\pi$
$4$	0	0
$5$	−0.902113	−0.403437	−0.201719	−	0.979444i	$-0.564653\pi$
$5$	−0.902113	−0.403437	−0.201719	+	0.979444i	$0.564653\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.618034	0.206011
$10$	0	0
$11$	−0.442463	−0.133408	−0.0667039	−	0.997773i	$-0.521248\pi$
$11$	−0.442463	−0.133408	−0.0667039	+	0.997773i	$0.521248\pi$
$12$	0	0
$13$	5.75621	1.59649	0.798243	−	0.602335i	$-0.205763\pi$
$13$	5.75621	1.59649	0.798243	+	0.602335i	$0.205763\pi$
$14$	0	0
$15$	−1.71592	−0.443049
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−6.42882	−1.47487	−0.737437	−	0.675416i	$-0.763964\pi$
$19$	−6.42882	−1.47487	−0.737437	+	0.675416i	$0.763964\pi$
$20$	0	0
$21$	1.90211	0.415075
$22$	0	0
$23$	−0.273457	−0.0570198	−0.0285099	−	0.999594i	$-0.509076\pi$
$23$	−0.273457	−0.0570198	−0.0285099	+	0.999594i	$0.509076\pi$
$24$	0	0
$25$	−4.18619	−0.837238
$26$	0	0
$27$	−4.53077	−0.871947
$28$	0	0
$29$	−10.5018	−1.95014	−0.975068	−	0.221904i	$-0.928773\pi$
$29$	−10.5018	−1.95014	−0.975068	+	0.221904i	$0.928773\pi$
$30$	0	0
$31$	−7.29657	−1.31050	−0.655251	−	0.755411i	$-0.727437\pi$
$31$	−7.29657	−1.31050	−0.655251	+	0.755411i	$0.727437\pi$
$32$	0	0
$33$	−0.841616	−0.146506
$34$	0	0
$35$	−0.902113	−0.152485
$36$	0	0
$37$	5.86067	0.963488	0.481744	−	0.876312i	$-0.340004\pi$
$37$	5.86067	0.963488	0.481744	+	0.876312i	$0.340004\pi$
$38$	0	0
$39$	10.9490	1.75324
$40$	0	0
$41$	−5.79766	−0.905443	−0.452721	−	0.891652i	$-0.649547\pi$
$41$	−5.79766	−0.905443	−0.452721	+	0.891652i	$0.649547\pi$
$42$	0	0
$43$	−1.94542	−0.296674	−0.148337	−	0.988937i	$-0.547392\pi$
$43$	−1.94542	−0.296674	−0.148337	+	0.988937i	$0.547392\pi$
$44$	0	0
$45$	−0.557537	−0.0831126
$46$	0	0
$47$	−2.10445	−0.306966	−0.153483	−	0.988151i	$-0.549049\pi$
$47$	−2.10445	−0.306966	−0.153483	+	0.988151i	$0.549049\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.90211	0.266349
$52$	0	0
$53$	9.20524	1.26444	0.632219	−	0.774790i	$-0.282144\pi$
$53$	9.20524	1.26444	0.632219	+	0.774790i	$0.282144\pi$
$54$	0	0
$55$	0.399152	0.0538217
$56$	0	0
$57$	−12.2284	−1.61968
$58$	0	0
$59$	−4.79830	−0.624686	−0.312343	−	0.949969i	$-0.601114\pi$
$59$	−4.79830	−0.624686	−0.312343	+	0.949969i	$0.601114\pi$
$60$	0	0
$61$	−1.68980	−0.216356	−0.108178	−	0.994132i	$-0.534502\pi$
$61$	−1.68980	−0.216356	−0.108178	+	0.994132i	$0.534502\pi$
$62$	0	0
$63$	0.618034	0.0778650
$64$	0	0
$65$	−5.19276	−0.644082
$66$	0	0
$67$	−10.3636	−1.26612	−0.633059	−	0.774103i	$-0.718201\pi$
$67$	−10.3636	−1.26612	−0.633059	+	0.774103i	$0.718201\pi$
$68$	0	0
$69$	−0.520147	−0.0626183
$70$	0	0
$71$	5.11798	0.607392	0.303696	−	0.952769i	$-0.401779\pi$
$71$	5.11798	0.607392	0.303696	+	0.952769i	$0.401779\pi$
$72$	0	0
$73$	−16.0872	−1.88286	−0.941429	−	0.337210i	$-0.890517\pi$
$73$	−16.0872	−1.88286	−0.941429	+	0.337210i	$0.890517\pi$
$74$	0	0
$75$	−7.96261	−0.919443
$76$	0	0
$77$	−0.442463	−0.0504234
$78$	0	0
$79$	1.36176	0.153210	0.0766051	−	0.997062i	$-0.475592\pi$
$79$	1.36176	0.153210	0.0766051	+	0.997062i	$0.475592\pi$
$80$	0	0
$81$	−10.4721	−1.16357
$82$	0	0
$83$	7.10143	0.779484	0.389742	−	0.920924i	$-0.372564\pi$
$83$	7.10143	0.779484	0.389742	+	0.920924i	$0.372564\pi$
$84$	0	0
$85$	−0.902113	−0.0978479
$86$	0	0
$87$	−19.9756	−2.14161
$88$	0	0
$89$	5.30011	0.561811	0.280905	−	0.959735i	$-0.409365\pi$
$89$	5.30011	0.561811	0.280905	+	0.959735i	$0.409365\pi$
$90$	0	0
$91$	5.75621	0.603415
$92$	0	0
$93$	−13.8789	−1.43917
$94$	0	0
$95$	5.79953	0.595019
$96$	0	0
$97$	−2.11321	−0.214564	−0.107282	−	0.994229i	$-0.534215\pi$
$97$	−2.11321	−0.214564	−0.107282	+	0.994229i	$0.534215\pi$
$98$	0	0
$99$	−0.273457	−0.0274835
$100$	0	0
$101$	−14.6304	−1.45578	−0.727890	−	0.685694i	$-0.759499\pi$
$101$	−14.6304	−1.45578	−0.727890	+	0.685694i	$0.759499\pi$
$102$	0	0
$103$	1.30532	0.128617	0.0643086	−	0.997930i	$-0.479516\pi$
$103$	1.30532	0.128617	0.0643086	+	0.997930i	$0.479516\pi$
$104$	0	0
$105$	−1.71592	−0.167457
$106$	0	0
$107$	−0.0267679	−0.00258775	−0.00129388	−	0.999999i	$-0.500412\pi$
$107$	−0.0267679	−0.00258775	−0.00129388	+	0.999999i	$0.500412\pi$
$108$	0	0
$109$	11.8386	1.13393	0.566966	−	0.823741i	$-0.308117\pi$
$109$	11.8386	1.13393	0.566966	+	0.823741i	$0.308117\pi$
$110$	0	0
$111$	11.1477	1.05809
$112$	0	0
$113$	−4.45559	−0.419147	−0.209573	−	0.977793i	$-0.567208\pi$
$113$	−4.45559	−0.419147	−0.209573	+	0.977793i	$0.567208\pi$
$114$	0	0
$115$	0.246690	0.0230039
$116$	0	0
$117$	3.55754	0.328894
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−10.8042	−0.982202
$122$	0	0
$123$	−11.0278	−0.994344
$124$	0	0
$125$	8.28698	0.741210
$126$	0	0
$127$	13.2425	1.17508	0.587542	−	0.809194i	$-0.300096\pi$
$127$	13.2425	1.17508	0.587542	+	0.809194i	$0.300096\pi$
$128$	0	0
$129$	−3.70042	−0.325804
$130$	0	0
$131$	20.3357	1.77674	0.888369	−	0.459129i	$-0.151839\pi$
$131$	20.3357	1.77674	0.888369	+	0.459129i	$0.151839\pi$
$132$	0	0
$133$	−6.42882	−0.557450
$134$	0	0
$135$	4.08727	0.351776
$136$	0	0
$137$	10.2858	0.878778	0.439389	−	0.898297i	$-0.355195\pi$
$137$	10.2858	0.878778	0.439389	+	0.898297i	$0.355195\pi$
$138$	0	0
$139$	−7.56108	−0.641323	−0.320661	−	0.947194i	$-0.603905\pi$
$139$	−7.56108	−0.641323	−0.320661	+	0.947194i	$0.603905\pi$
$140$	0	0
$141$	−4.00290	−0.337105
$142$	0	0
$143$	−2.54691	−0.212984
$144$	0	0
$145$	9.47382	0.786758
$146$	0	0
$147$	1.90211	0.156884
$148$	0	0
$149$	−13.4221	−1.09959	−0.549793	−	0.835301i	$-0.685293\pi$
$149$	−13.4221	−1.09959	−0.549793	+	0.835301i	$0.685293\pi$
$150$	0	0
$151$	−23.4518	−1.90848	−0.954241	−	0.299039i	$-0.903334\pi$
$151$	−23.4518	−1.90848	−0.954241	+	0.299039i	$0.903334\pi$
$152$	0	0
$153$	0.618034	0.0499651
$154$	0	0
$155$	6.58233	0.528705
$156$	0	0
$157$	22.7876	1.81865	0.909323	−	0.416091i	$-0.136600\pi$
$157$	22.7876	1.81865	0.909323	+	0.416091i	$0.136600\pi$
$158$	0	0
$159$	17.5094	1.38859
$160$	0	0
$161$	−0.273457	−0.0215515
$162$	0	0
$163$	4.65427	0.364551	0.182275	−	0.983248i	$-0.441654\pi$
$163$	4.65427	0.364551	0.182275	+	0.983248i	$0.441654\pi$
$164$	0	0
$165$	0.759232	0.0591062
$166$	0	0
$167$	−2.41093	−0.186563	−0.0932815	−	0.995640i	$-0.529736\pi$
$167$	−2.41093	−0.186563	−0.0932815	+	0.995640i	$0.529736\pi$
$168$	0	0
$169$	20.1340	1.54877
$170$	0	0
$171$	−3.97323	−0.303841
$172$	0	0
$173$	4.51911	0.343581	0.171791	−	0.985133i	$-0.445045\pi$
$173$	4.51911	0.343581	0.171791	+	0.985133i	$0.445045\pi$
$174$	0	0
$175$	−4.18619	−0.316446
$176$	0	0
$177$	−9.12692	−0.686021
$178$	0	0
$179$	11.3962	0.851792	0.425896	−	0.904772i	$-0.359959\pi$
$179$	11.3962	0.851792	0.425896	+	0.904772i	$0.359959\pi$
$180$	0	0
$181$	11.2169	0.833746	0.416873	−	0.908965i	$-0.363126\pi$
$181$	11.2169	0.833746	0.416873	+	0.908965i	$0.363126\pi$
$182$	0	0
$183$	−3.21418	−0.237599
$184$	0	0
$185$	−5.28698	−0.388707
$186$	0	0
$187$	−0.442463	−0.0323561
$188$	0	0
$189$	−4.53077	−0.329565
$190$	0	0
$191$	−3.34085	−0.241735	−0.120868	−	0.992669i	$-0.538568\pi$
$191$	−3.34085	−0.241735	−0.120868	+	0.992669i	$0.538568\pi$
$192$	0	0
$193$	−10.2284	−0.736253	−0.368126	−	0.929776i	$-0.620001\pi$
$193$	−10.2284	−0.736253	−0.368126	+	0.929776i	$0.620001\pi$
$194$	0	0
$195$	−9.87721	−0.707322
$196$	0	0
$197$	13.9651	0.994973	0.497487	−	0.867472i	$-0.334256\pi$
$197$	13.9651	0.994973	0.497487	+	0.867472i	$0.334256\pi$
$198$	0	0
$199$	−0.762067	−0.0540215	−0.0270108	−	0.999635i	$-0.508599\pi$
$199$	−0.762067	−0.0540215	−0.0270108	+	0.999635i	$0.508599\pi$
$200$	0	0
$201$	−19.7128	−1.39043
$202$	0	0
$203$	−10.5018	−0.737082
$204$	0	0
$205$	5.23015	0.365289
$206$	0	0
$207$	−0.169006	−0.0117467
$208$	0	0
$209$	2.84452	0.196760
$210$	0	0
$211$	−23.1340	−1.59261	−0.796306	−	0.604895i	$-0.793215\pi$
$211$	−23.1340	−1.59261	−0.796306	+	0.604895i	$0.793215\pi$
$212$	0	0
$213$	9.73497	0.667029
$214$	0	0
$215$	1.75499	0.119690
$216$	0	0
$217$	−7.29657	−0.495323
$218$	0	0
$219$	−30.5996	−2.06773
$220$	0	0
$221$	5.75621	0.387205
$222$	0	0
$223$	−27.0776	−1.81325	−0.906624	−	0.421939i	$-0.861350\pi$
$223$	−27.0776	−1.81325	−0.906624	+	0.421939i	$0.861350\pi$
$224$	0	0
$225$	−2.58721	−0.172481
$226$	0	0
$227$	13.0260	0.864567	0.432283	−	0.901738i	$-0.357708\pi$
$227$	13.0260	0.864567	0.432283	+	0.901738i	$0.357708\pi$
$228$	0	0
$229$	−18.0971	−1.19589	−0.597946	−	0.801536i	$-0.704016\pi$
$229$	−18.0971	−1.19589	−0.597946	+	0.801536i	$0.704016\pi$
$230$	0	0
$231$	−0.841616	−0.0553742
$232$	0	0
$233$	−4.04331	−0.264886	−0.132443	−	0.991191i	$-0.542282\pi$
$233$	−4.04331	−0.264886	−0.132443	+	0.991191i	$0.542282\pi$
$234$	0	0
$235$	1.89845	0.123841
$236$	0	0
$237$	2.59023	0.168253
$238$	0	0
$239$	−14.7225	−0.952318	−0.476159	−	0.879359i	$-0.657971\pi$
$239$	−14.7225	−0.952318	−0.476159	+	0.879359i	$0.657971\pi$
$240$	0	0
$241$	15.2047	0.979423	0.489711	−	0.871885i	$-0.337102\pi$
$241$	15.2047	0.979423	0.489711	+	0.871885i	$0.337102\pi$
$242$	0	0
$243$	−6.32688	−0.405870
$244$	0	0
$245$	−0.902113	−0.0576339
$246$	0	0
$247$	−37.0057	−2.35462
$248$	0	0
$249$	13.5077	0.856018
$250$	0	0
$251$	13.3350	0.841697	0.420849	−	0.907131i	$-0.361732\pi$
$251$	13.3350	0.841697	0.420849	+	0.907131i	$0.361732\pi$
$252$	0	0
$253$	0.120995	0.00760689
$254$	0	0
$255$	−1.71592	−0.107455
$256$	0	0
$257$	−16.1774	−1.00912	−0.504560	−	0.863376i	$-0.668345\pi$
$257$	−16.1774	−1.00912	−0.504560	+	0.863376i	$0.668345\pi$
$258$	0	0
$259$	5.86067	0.364164
$260$	0	0
$261$	−6.49047	−0.401750
$262$	0	0
$263$	−20.6868	−1.27561	−0.637803	−	0.770200i	$-0.720157\pi$
$263$	−20.6868	−1.27561	−0.637803	+	0.770200i	$0.720157\pi$
$264$	0	0
$265$	−8.30417	−0.510121
$266$	0	0
$267$	10.0814	0.616972
$268$	0	0
$269$	17.8161	1.08626	0.543132	−	0.839647i	$-0.317238\pi$
$269$	17.8161	1.08626	0.543132	+	0.839647i	$0.317238\pi$
$270$	0	0
$271$	−26.1964	−1.59132	−0.795658	−	0.605746i	$-0.792875\pi$
$271$	−26.1964	−1.59132	−0.795658	+	0.605746i	$0.792875\pi$
$272$	0	0
$273$	10.9490	0.662662
$274$	0	0
$275$	1.85224	0.111694
$276$	0	0
$277$	−9.96982	−0.599028	−0.299514	−	0.954092i	$-0.596825\pi$
$277$	−9.96982	−0.599028	−0.299514	+	0.954092i	$0.596825\pi$
$278$	0	0
$279$	−4.50953	−0.269978
$280$	0	0
$281$	−24.6442	−1.47015	−0.735074	−	0.677987i	$-0.762853\pi$
$281$	−24.6442	−1.47015	−0.735074	+	0.677987i	$0.762853\pi$
$282$	0	0
$283$	−17.1601	−1.02006	−0.510032	−	0.860155i	$-0.670366\pi$
$283$	−17.1601	−1.02006	−0.510032	+	0.860155i	$0.670366\pi$
$284$	0	0
$285$	11.0314	0.653441
$286$	0	0
$287$	−5.79766	−0.342225
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−4.01956	−0.235631
$292$	0	0
$293$	24.9222	1.45597	0.727985	−	0.685593i	$-0.240457\pi$
$293$	24.9222	1.45597	0.727985	+	0.685593i	$0.240457\pi$
$294$	0	0
$295$	4.32861	0.252022
$296$	0	0
$297$	2.00470	0.116324
$298$	0	0
$299$	−1.57408	−0.0910314
$300$	0	0
$301$	−1.94542	−0.112132
$302$	0	0
$303$	−27.8287	−1.59872
$304$	0	0
$305$	1.52439	0.0872861
$306$	0	0
$307$	24.7099	1.41027	0.705134	−	0.709074i	$-0.250887\pi$
$307$	24.7099	1.41027	0.705134	+	0.709074i	$0.250887\pi$
$308$	0	0
$309$	2.48287	0.141246
$310$	0	0
$311$	−22.7305	−1.28893	−0.644464	−	0.764635i	$-0.722919\pi$
$311$	−22.7305	−1.28893	−0.644464	+	0.764635i	$0.722919\pi$
$312$	0	0
$313$	−27.9495	−1.57980	−0.789900	−	0.613236i	$-0.789867\pi$
$313$	−27.9495	−1.57980	−0.789900	+	0.613236i	$0.789867\pi$
$314$	0	0
$315$	−0.557537	−0.0314136
$316$	0	0
$317$	−13.2096	−0.741926	−0.370963	−	0.928648i	$-0.620972\pi$
$317$	−13.2096	−0.741926	−0.370963	+	0.928648i	$0.620972\pi$
$318$	0	0
$319$	4.64667	0.260163
$320$	0	0
$321$	−0.0509156	−0.00284183
$322$	0	0
$323$	−6.42882	−0.357709
$324$	0	0
$325$	−24.0966	−1.33664
$326$	0	0
$327$	22.5184	1.24527
$328$	0	0
$329$	−2.10445	−0.116022
$330$	0	0
$331$	−16.2164	−0.891333	−0.445666	−	0.895199i	$-0.647033\pi$
$331$	−16.2164	−0.891333	−0.445666	+	0.895199i	$0.647033\pi$
$332$	0	0
$333$	3.62209	0.198489
$334$	0	0
$335$	9.34916	0.510799
$336$	0	0
$337$	26.6962	1.45424	0.727119	−	0.686512i	$-0.240859\pi$
$337$	26.6962	1.45424	0.727119	+	0.686512i	$0.240859\pi$
$338$	0	0
$339$	−8.47504	−0.460301
$340$	0	0
$341$	3.22846	0.174831
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.469231	0.0252626
$346$	0	0
$347$	−21.6085	−1.16000	−0.580001	−	0.814616i	$-0.696948\pi$
$347$	−21.6085	−1.16000	−0.580001	+	0.814616i	$0.696948\pi$
$348$	0	0
$349$	21.2114	1.13542	0.567710	−	0.823229i	$-0.307830\pi$
$349$	21.2114	1.13542	0.567710	+	0.823229i	$0.307830\pi$
$350$	0	0
$351$	−26.0801	−1.39205
$352$	0	0
$353$	9.71663	0.517164	0.258582	−	0.965989i	$-0.416745\pi$
$353$	9.71663	0.517164	0.258582	+	0.965989i	$0.416745\pi$
$354$	0	0
$355$	−4.61699	−0.245045
$356$	0	0
$357$	1.90211	0.100670
$358$	0	0
$359$	1.53224	0.0808683	0.0404342	−	0.999182i	$-0.487126\pi$
$359$	1.53224	0.0808683	0.0404342	+	0.999182i	$0.487126\pi$
$360$	0	0
$361$	22.3298	1.17525
$362$	0	0
$363$	−20.5509	−1.07864
$364$	0	0
$365$	14.5124	0.759615
$366$	0	0
$367$	5.77881	0.301652	0.150826	−	0.988560i	$-0.451807\pi$
$367$	5.77881	0.301652	0.150826	+	0.988560i	$0.451807\pi$
$368$	0	0
$369$	−3.58315	−0.186531
$370$	0	0
$371$	9.20524	0.477912
$372$	0	0
$373$	−20.4685	−1.05982	−0.529909	−	0.848055i	$-0.677774\pi$
$373$	−20.4685	−1.05982	−0.529909	+	0.848055i	$0.677774\pi$
$374$	0	0
$375$	15.7628	0.813986
$376$	0	0
$377$	−60.4507	−3.11337
$378$	0	0
$379$	29.5573	1.51826	0.759129	−	0.650941i	$-0.225625\pi$
$379$	29.5573	1.51826	0.759129	+	0.650941i	$0.225625\pi$
$380$	0	0
$381$	25.1888	1.29046
$382$	0	0
$383$	15.9586	0.815448	0.407724	−	0.913105i	$-0.366323\pi$
$383$	15.9586	0.815448	0.407724	+	0.913105i	$0.366323\pi$
$384$	0	0
$385$	0.399152	0.0203427
$386$	0	0
$387$	−1.20234	−0.0611183
$388$	0	0
$389$	37.7813	1.91559	0.957794	−	0.287456i	$-0.0928097\pi$
$389$	37.7813	1.91559	0.957794	+	0.287456i	$0.0928097\pi$
$390$	0	0
$391$	−0.273457	−0.0138293
$392$	0	0
$393$	38.6808	1.95119
$394$	0	0
$395$	−1.22846	−0.0618107
$396$	0	0
$397$	−14.9425	−0.749944	−0.374972	−	0.927036i	$-0.622348\pi$
$397$	−14.9425	−0.749944	−0.374972	+	0.927036i	$0.622348\pi$
$398$	0	0
$399$	−12.2284	−0.612183
$400$	0	0
$401$	28.5519	1.42581	0.712907	−	0.701259i	$-0.247378\pi$
$401$	28.5519	1.42581	0.712907	+	0.701259i	$0.247378\pi$
$402$	0	0
$403$	−42.0006	−2.09220
$404$	0	0
$405$	9.44705	0.469428
$406$	0	0
$407$	−2.59313	−0.128537
$408$	0	0
$409$	−13.7557	−0.680176	−0.340088	−	0.940394i	$-0.610457\pi$
$409$	−13.7557	−0.680176	−0.340088	+	0.940394i	$0.610457\pi$
$410$	0	0
$411$	19.5648	0.965061
$412$	0	0
$413$	−4.79830	−0.236109
$414$	0	0
$415$	−6.40630	−0.314473
$416$	0	0
$417$	−14.3820	−0.704291
$418$	0	0
$419$	15.8978	0.776659	0.388329	−	0.921521i	$-0.373052\pi$
$419$	15.8978	0.776659	0.388329	+	0.921521i	$0.373052\pi$
$420$	0	0
$421$	7.67146	0.373884	0.186942	−	0.982371i	$-0.440142\pi$
$421$	7.67146	0.373884	0.186942	+	0.982371i	$0.440142\pi$
$422$	0	0
$423$	−1.30062	−0.0632384
$424$	0	0
$425$	−4.18619	−0.203060
$426$	0	0
$427$	−1.68980	−0.0817749
$428$	0	0
$429$	−4.84452	−0.233896
$430$	0	0
$431$	11.7444	0.565706	0.282853	−	0.959163i	$-0.408719\pi$
$431$	11.7444	0.565706	0.282853	+	0.959163i	$0.408719\pi$
$432$	0	0
$433$	3.91679	0.188229	0.0941145	−	0.995561i	$-0.469998\pi$
$433$	3.91679	0.188229	0.0941145	+	0.995561i	$0.469998\pi$
$434$	0	0
$435$	18.0203	0.864006
$436$	0	0
$437$	1.75801	0.0840970
$438$	0	0
$439$	34.4182	1.64269	0.821345	−	0.570432i	$-0.193224\pi$
$439$	34.4182	1.64269	0.821345	+	0.570432i	$0.193224\pi$
$440$	0	0
$441$	0.618034	0.0294302
$442$	0	0
$443$	−27.0343	−1.28444	−0.642218	−	0.766522i	$-0.721986\pi$
$443$	−27.0343	−1.28444	−0.642218	+	0.766522i	$0.721986\pi$
$444$	0	0
$445$	−4.78130	−0.226655
$446$	0	0
$447$	−25.5304	−1.20755
$448$	0	0
$449$	24.2436	1.14412	0.572062	−	0.820210i	$-0.306144\pi$
$449$	24.2436	1.14412	0.572062	+	0.820210i	$0.306144\pi$
$450$	0	0
$451$	2.56525	0.120793
$452$	0	0
$453$	−44.6080	−2.09587
$454$	0	0
$455$	−5.19276	−0.243440
$456$	0	0
$457$	−3.25274	−0.152157	−0.0760785	−	0.997102i	$-0.524240\pi$
$457$	−3.25274	−0.152157	−0.0760785	+	0.997102i	$0.524240\pi$
$458$	0	0
$459$	−4.53077	−0.211478
$460$	0	0
$461$	−1.54814	−0.0721039	−0.0360520	−	0.999350i	$-0.511478\pi$
$461$	−1.54814	−0.0721039	−0.0360520	+	0.999350i	$0.511478\pi$
$462$	0	0
$463$	−22.7152	−1.05566	−0.527832	−	0.849349i	$-0.676995\pi$
$463$	−22.7152	−1.05566	−0.527832	+	0.849349i	$0.676995\pi$
$464$	0	0
$465$	12.5203	0.580616
$466$	0	0
$467$	−35.7356	−1.65365	−0.826823	−	0.562463i	$-0.809854\pi$
$467$	−35.7356	−1.65365	−0.826823	+	0.562463i	$0.809854\pi$
$468$	0	0
$469$	−10.3636	−0.478548
$470$	0	0
$471$	43.3445	1.99721
$472$	0	0
$473$	0.860779	0.0395787
$474$	0	0
$475$	26.9123	1.23482
$476$	0	0
$477$	5.68915	0.260488
$478$	0	0
$479$	−5.63888	−0.257647	−0.128824	−	0.991668i	$-0.541120\pi$
$479$	−5.63888	−0.257647	−0.128824	+	0.991668i	$0.541120\pi$
$480$	0	0
$481$	33.7353	1.53820
$482$	0	0
$483$	−0.520147	−0.0236675
$484$	0	0
$485$	1.90635	0.0865630
$486$	0	0
$487$	0.915425	0.0414818	0.0207409	−	0.999785i	$-0.493397\pi$
$487$	0.915425	0.0414818	0.0207409	+	0.999785i	$0.493397\pi$
$488$	0	0
$489$	8.85295	0.400344
$490$	0	0
$491$	−25.1596	−1.13544	−0.567719	−	0.823222i	$-0.692174\pi$
$491$	−25.1596	−1.13544	−0.567719	+	0.823222i	$0.692174\pi$
$492$	0	0
$493$	−10.5018	−0.472978
$494$	0	0
$495$	0.246690	0.0110879
$496$	0	0
$497$	5.11798	0.229573
$498$	0	0
$499$	−21.5898	−0.966492	−0.483246	−	0.875485i	$-0.660542\pi$
$499$	−21.5898	−0.966492	−0.483246	+	0.875485i	$0.660542\pi$
$500$	0	0
$501$	−4.58585	−0.204881
$502$	0	0
$503$	−41.5302	−1.85174	−0.925870	−	0.377841i	$-0.876667\pi$
$503$	−41.5302	−1.85174	−0.925870	+	0.377841i	$0.876667\pi$
$504$	0	0
$505$	13.1983	0.587316
$506$	0	0
$507$	38.2972	1.70084
$508$	0	0
$509$	−7.26284	−0.321920	−0.160960	−	0.986961i	$-0.551459\pi$
$509$	−7.26284	−0.321920	−0.160960	+	0.986961i	$0.551459\pi$
$510$	0	0
$511$	−16.0872	−0.711654
$512$	0	0
$513$	29.1275	1.28601
$514$	0	0
$515$	−1.17755	−0.0518890
$516$	0	0
$517$	0.931143	0.0409516
$518$	0	0
$519$	8.59585	0.377316
$520$	0	0
$521$	10.1370	0.444108	0.222054	−	0.975034i	$-0.428724\pi$
$521$	10.1370	0.444108	0.222054	+	0.975034i	$0.428724\pi$
$522$	0	0
$523$	18.0468	0.789132	0.394566	−	0.918868i	$-0.370895\pi$
$523$	18.0468	0.789132	0.394566	+	0.918868i	$0.370895\pi$
$524$	0	0
$525$	−7.96261	−0.347517
$526$	0	0
$527$	−7.29657	−0.317843
$528$	0	0
$529$	−22.9252	−0.996749
$530$	0	0
$531$	−2.96552	−0.128692
$532$	0	0
$533$	−33.3726	−1.44553
$534$	0	0
$535$	0.0241477	0.00104400
$536$	0	0
$537$	21.6769	0.935426
$538$	0	0
$539$	−0.442463	−0.0190583
$540$	0	0
$541$	−6.71561	−0.288726	−0.144363	−	0.989525i	$-0.546113\pi$
$541$	−6.71561	−0.288726	−0.144363	+	0.989525i	$0.546113\pi$
$542$	0	0
$543$	21.3358	0.915608
$544$	0	0
$545$	−10.6798	−0.457470
$546$	0	0
$547$	18.9621	0.810761	0.405380	−	0.914148i	$-0.367139\pi$
$547$	18.9621	0.810761	0.405380	+	0.914148i	$0.367139\pi$
$548$	0	0
$549$	−1.04435	−0.0445718
$550$	0	0
$551$	67.5143	2.87621
$552$	0	0
$553$	1.36176	0.0579080
$554$	0	0
$555$	−10.0564	−0.426872
$556$	0	0
$557$	11.0426	0.467890	0.233945	−	0.972250i	$-0.424836\pi$
$557$	11.0426	0.467890	0.233945	+	0.972250i	$0.424836\pi$
$558$	0	0
$559$	−11.1983	−0.473637
$560$	0	0
$561$	−0.841616	−0.0355330
$562$	0	0
$563$	−14.8742	−0.626872	−0.313436	−	0.949609i	$-0.601480\pi$
$563$	−14.8742	−0.626872	−0.313436	+	0.949609i	$0.601480\pi$
$564$	0	0
$565$	4.01945	0.169099
$566$	0	0
$567$	−10.4721	−0.439788
$568$	0	0
$569$	4.00574	0.167929	0.0839647	−	0.996469i	$-0.473242\pi$
$569$	4.00574	0.167929	0.0839647	+	0.996469i	$0.473242\pi$
$570$	0	0
$571$	−36.6992	−1.53581	−0.767906	−	0.640563i	$-0.778701\pi$
$571$	−36.6992	−1.53581	−0.767906	+	0.640563i	$0.778701\pi$
$572$	0	0
$573$	−6.35467	−0.265470
$574$	0	0
$575$	1.14475	0.0477392
$576$	0	0
$577$	−37.3126	−1.55334	−0.776671	−	0.629906i	$-0.783093\pi$
$577$	−37.3126	−1.55334	−0.776671	+	0.629906i	$0.783093\pi$
$578$	0	0
$579$	−19.4555	−0.808542
$580$	0	0
$581$	7.10143	0.294617
$582$	0	0
$583$	−4.07298	−0.168686
$584$	0	0
$585$	−3.20930	−0.132688
$586$	0	0
$587$	28.8015	1.18876	0.594382	−	0.804183i	$-0.297397\pi$
$587$	28.8015	1.18876	0.594382	+	0.804183i	$0.297397\pi$
$588$	0	0
$589$	46.9083	1.93282
$590$	0	0
$591$	26.5632	1.09267
$592$	0	0
$593$	1.10613	0.0454235	0.0227117	−	0.999742i	$-0.492770\pi$
$593$	1.10613	0.0454235	0.0227117	+	0.999742i	$0.492770\pi$
$594$	0	0
$595$	−0.902113	−0.0369830
$596$	0	0
$597$	−1.44954	−0.0593256
$598$	0	0
$599$	−19.6645	−0.803468	−0.401734	−	0.915756i	$-0.631592\pi$
$599$	−19.6645	−0.803468	−0.401734	+	0.915756i	$0.631592\pi$
$600$	0	0
$601$	1.48538	0.0605898	0.0302949	−	0.999541i	$-0.490355\pi$
$601$	1.48538	0.0605898	0.0302949	+	0.999541i	$0.490355\pi$
$602$	0	0
$603$	−6.40507	−0.260835
$604$	0	0
$605$	9.74663	0.396257
$606$	0	0
$607$	−14.7167	−0.597333	−0.298666	−	0.954358i	$-0.596542\pi$
$607$	−14.7167	−0.597333	−0.298666	+	0.954358i	$0.596542\pi$
$608$	0	0
$609$	−19.9756	−0.809453
$610$	0	0
$611$	−12.1137	−0.490067
$612$	0	0
$613$	13.2646	0.535752	0.267876	−	0.963453i	$-0.413678\pi$
$613$	13.2646	0.535752	0.267876	+	0.963453i	$0.413678\pi$
$614$	0	0
$615$	9.94833	0.401155
$616$	0	0
$617$	6.48495	0.261074	0.130537	−	0.991443i	$-0.458330\pi$
$617$	6.48495	0.261074	0.130537	+	0.991443i	$0.458330\pi$
$618$	0	0
$619$	−31.4016	−1.26214	−0.631069	−	0.775727i	$-0.717384\pi$
$619$	−31.4016	−1.26214	−0.631069	+	0.775727i	$0.717384\pi$
$620$	0	0
$621$	1.23897	0.0497183
$622$	0	0
$623$	5.30011	0.212345
$624$	0	0
$625$	13.4552	0.538207
$626$	0	0
$627$	5.41060	0.216079
$628$	0	0
$629$	5.86067	0.233680
$630$	0	0
$631$	7.05780	0.280966	0.140483	−	0.990083i	$-0.455134\pi$
$631$	7.05780	0.280966	0.140483	+	0.990083i	$0.455134\pi$
$632$	0	0
$633$	−44.0035	−1.74898
$634$	0	0
$635$	−11.9462	−0.474072
$636$	0	0
$637$	5.75621	0.228070
$638$	0	0
$639$	3.16308	0.125130
$640$	0	0
$641$	10.6849	0.422029	0.211014	−	0.977483i	$-0.432323\pi$
$641$	10.6849	0.422029	0.211014	+	0.977483i	$0.432323\pi$
$642$	0	0
$643$	−27.7576	−1.09465	−0.547326	−	0.836920i	$-0.684354\pi$
$643$	−27.7576	−1.09465	−0.547326	+	0.836920i	$0.684354\pi$
$644$	0	0
$645$	3.33819	0.131441
$646$	0	0
$647$	21.9893	0.864487	0.432244	−	0.901757i	$-0.357722\pi$
$647$	21.9893	0.864487	0.432244	+	0.901757i	$0.357722\pi$
$648$	0	0
$649$	2.12307	0.0833380
$650$	0	0
$651$	−13.8789	−0.543957
$652$	0	0
$653$	48.4601	1.89639	0.948194	−	0.317690i	$-0.102907\pi$
$653$	48.4601	1.89639	0.948194	+	0.317690i	$0.102907\pi$
$654$	0	0
$655$	−18.3451	−0.716803
$656$	0	0
$657$	−9.94241	−0.387890
$658$	0	0
$659$	42.3384	1.64927	0.824636	−	0.565664i	$-0.191380\pi$
$659$	42.3384	1.64927	0.824636	+	0.565664i	$0.191380\pi$
$660$	0	0
$661$	−35.2521	−1.37115	−0.685574	−	0.728003i	$-0.740449\pi$
$661$	−35.2521	−1.37115	−0.685574	+	0.728003i	$0.740449\pi$
$662$	0	0
$663$	10.9490	0.425223
$664$	0	0
$665$	5.79953	0.224896
$666$	0	0
$667$	2.87180	0.111196
$668$	0	0
$669$	−51.5046	−1.99128
$670$	0	0
$671$	0.747673	0.0288636
$672$	0	0
$673$	22.7138	0.875553	0.437777	−	0.899084i	$-0.355766\pi$
$673$	22.7138	0.875553	0.437777	+	0.899084i	$0.355766\pi$
$674$	0	0
$675$	18.9667	0.730027
$676$	0	0
$677$	7.77860	0.298956	0.149478	−	0.988765i	$-0.452241\pi$
$677$	7.77860	0.298956	0.149478	+	0.988765i	$0.452241\pi$
$678$	0	0
$679$	−2.11321	−0.0810975
$680$	0	0
$681$	24.7769	0.949455
$682$	0	0
$683$	−40.8852	−1.56443	−0.782215	−	0.623009i	$-0.785910\pi$
$683$	−40.8852	−1.56443	−0.782215	+	0.623009i	$0.785910\pi$
$684$	0	0
$685$	−9.27898	−0.354532
$686$	0	0
$687$	−34.4228	−1.31331
$688$	0	0
$689$	52.9874	2.01866
$690$	0	0
$691$	37.3658	1.42146	0.710731	−	0.703464i	$-0.248364\pi$
$691$	37.3658	1.42146	0.710731	+	0.703464i	$0.248364\pi$
$692$	0	0
$693$	−0.273457	−0.0103878
$694$	0	0
$695$	6.82095	0.258733
$696$	0	0
$697$	−5.79766	−0.219602
$698$	0	0
$699$	−7.69084	−0.290894
$700$	0	0
$701$	43.7365	1.65191	0.825953	−	0.563739i	$-0.190638\pi$
$701$	43.7365	1.65191	0.825953	+	0.563739i	$0.190638\pi$
$702$	0	0
$703$	−37.6772	−1.42102
$704$	0	0
$705$	3.61107	0.136001
$706$	0	0
$707$	−14.6304	−0.550233
$708$	0	0
$709$	11.4513	0.430062	0.215031	−	0.976607i	$-0.431015\pi$
$709$	11.4513	0.430062	0.215031	+	0.976607i	$0.431015\pi$
$710$	0	0
$711$	0.841616	0.0315630
$712$	0	0
$713$	1.99530	0.0747246
$714$	0	0
$715$	2.29761	0.0859256
$716$	0	0
$717$	−28.0038	−1.04582
$718$	0	0
$719$	−18.7789	−0.700336	−0.350168	−	0.936687i	$-0.613875\pi$
$719$	−18.7789	−0.700336	−0.350168	+	0.936687i	$0.613875\pi$
$720$	0	0
$721$	1.30532	0.0486127
$722$	0	0
$723$	28.9211	1.07559
$724$	0	0
$725$	43.9626	1.63273
$726$	0	0
$727$	24.7173	0.916716	0.458358	−	0.888768i	$-0.348438\pi$
$727$	24.7173	0.916716	0.458358	+	0.888768i	$0.348438\pi$
$728$	0	0
$729$	19.3820	0.717851
$730$	0	0
$731$	−1.94542	−0.0719541
$732$	0	0
$733$	−14.1197	−0.521522	−0.260761	−	0.965403i	$-0.583973\pi$
$733$	−14.1197	−0.521522	−0.260761	+	0.965403i	$0.583973\pi$
$734$	0	0
$735$	−1.71592	−0.0632927
$736$	0	0
$737$	4.58553	0.168910
$738$	0	0
$739$	−33.4979	−1.23224	−0.616120	−	0.787652i	$-0.711296\pi$
$739$	−33.4979	−1.23224	−0.616120	+	0.787652i	$0.711296\pi$
$740$	0	0
$741$	−70.3890	−2.58581
$742$	0	0
$743$	20.4149	0.748949	0.374474	−	0.927237i	$-0.377823\pi$
$743$	20.4149	0.748949	0.374474	+	0.927237i	$0.377823\pi$
$744$	0	0
$745$	12.1083	0.443614
$746$	0	0
$747$	4.38893	0.160582
$748$	0	0
$749$	−0.0267679	−0.000978079 0
$750$	0	0
$751$	27.5534	1.00544	0.502720	−	0.864450i	$-0.332333\pi$
$751$	27.5534	1.00544	0.502720	+	0.864450i	$0.332333\pi$
$752$	0	0
$753$	25.3647	0.924340
$754$	0	0
$755$	21.1562	0.769953
$756$	0	0
$757$	13.8291	0.502628	0.251314	−	0.967906i	$-0.419137\pi$
$757$	13.8291	0.502628	0.251314	+	0.967906i	$0.419137\pi$
$758$	0	0
$759$	0.230146	0.00835377
$760$	0	0
$761$	31.4676	1.14070	0.570351	−	0.821401i	$-0.306807\pi$
$761$	31.4676	1.14070	0.570351	+	0.821401i	$0.306807\pi$
$762$	0	0
$763$	11.8386	0.428586
$764$	0	0
$765$	−0.557537	−0.0201578
$766$	0	0
$767$	−27.6201	−0.997303
$768$	0	0
$769$	32.3806	1.16767	0.583837	−	0.811871i	$-0.301551\pi$
$769$	32.3806	1.16767	0.583837	+	0.811871i	$0.301551\pi$
$770$	0	0
$771$	−30.7713	−1.10820
$772$	0	0
$773$	−11.5896	−0.416849	−0.208425	−	0.978038i	$-0.566834\pi$
$773$	−11.5896	−0.416849	−0.208425	+	0.978038i	$0.566834\pi$
$774$	0	0
$775$	30.5448	1.09720
$776$	0	0
$777$	11.1477	0.399920
$778$	0	0
$779$	37.2721	1.33541
$780$	0	0
$781$	−2.26452	−0.0810308
$782$	0	0
$783$	47.5813	1.70042
$784$	0	0
$785$	−20.5570	−0.733709
$786$	0	0
$787$	6.55258	0.233574	0.116787	−	0.993157i	$-0.462740\pi$
$787$	6.55258	0.233574	0.116787	+	0.993157i	$0.462740\pi$
$788$	0	0
$789$	−39.3487	−1.40085
$790$	0	0
$791$	−4.45559	−0.158423
$792$	0	0
$793$	−9.72683	−0.345410
$794$	0	0
$795$	−15.7955	−0.560208
$796$	0	0
$797$	−31.2555	−1.10713	−0.553564	−	0.832807i	$-0.686732\pi$
$797$	−31.2555	−1.10713	−0.553564	+	0.832807i	$0.686732\pi$
$798$	0	0
$799$	−2.10445	−0.0744501
$800$	0	0
$801$	3.27565	0.115739
$802$	0	0
$803$	7.11798	0.251188
$804$	0	0
$805$	0.246690	0.00869466
$806$	0	0
$807$	33.8882	1.19292
$808$	0	0
$809$	0.576782	0.0202786	0.0101393	−	0.999949i	$-0.496773\pi$
$809$	0.576782	0.0202786	0.0101393	+	0.999949i	$0.496773\pi$
$810$	0	0
$811$	−26.2284	−0.921004	−0.460502	−	0.887659i	$-0.652331\pi$
$811$	−26.2284	−0.921004	−0.460502	+	0.887659i	$0.652331\pi$
$812$	0	0
$813$	−49.8285	−1.74756
$814$	0	0
$815$	−4.19868	−0.147073
$816$	0	0
$817$	12.5068	0.437557
$818$	0	0
$819$	3.55754	0.124310
$820$	0	0
$821$	−7.51764	−0.262367	−0.131184	−	0.991358i	$-0.541878\pi$
$821$	−7.51764	−0.262367	−0.131184	+	0.991358i	$0.541878\pi$
$822$	0	0
$823$	41.6605	1.45219	0.726096	−	0.687593i	$-0.241333\pi$
$823$	41.6605	1.45219	0.726096	+	0.687593i	$0.241333\pi$
$824$	0	0
$825$	3.52316	0.122661
$826$	0	0
$827$	13.8522	0.481689	0.240845	−	0.970564i	$-0.422576\pi$
$827$	13.8522	0.481689	0.240845	+	0.970564i	$0.422576\pi$
$828$	0	0
$829$	53.5164	1.85870	0.929351	−	0.369196i	$-0.120367\pi$
$829$	53.5164	1.85870	0.929351	+	0.369196i	$0.120367\pi$
$830$	0	0
$831$	−18.9637	−0.657844
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	2.17493	0.0752665
$836$	0	0
$837$	33.0590	1.14269
$838$	0	0
$839$	22.9726	0.793103	0.396551	−	0.918013i	$-0.370207\pi$
$839$	22.9726	0.793103	0.396551	+	0.918013i	$0.370207\pi$
$840$	0	0
$841$	81.2880	2.80303
$842$	0	0
$843$	−46.8760	−1.61450
$844$	0	0
$845$	−18.1632	−0.624831
$846$	0	0
$847$	−10.8042	−0.371238
$848$	0	0
$849$	−32.6405	−1.12022
$850$	0	0
$851$	−1.60264	−0.0549379
$852$	0	0
$853$	−31.7484	−1.08704	−0.543522	−	0.839395i	$-0.682910\pi$
$853$	−31.7484	−1.08704	−0.543522	+	0.839395i	$0.682910\pi$
$854$	0	0
$855$	3.58430	0.122581
$856$	0	0
$857$	18.4564	0.630458	0.315229	−	0.949016i	$-0.397919\pi$
$857$	18.4564	0.630458	0.315229	+	0.949016i	$0.397919\pi$
$858$	0	0
$859$	−46.5780	−1.58922	−0.794611	−	0.607119i	$-0.792325\pi$
$859$	−46.5780	−1.58922	−0.794611	+	0.607119i	$0.792325\pi$
$860$	0	0
$861$	−11.0278	−0.375827
$862$	0	0
$863$	16.3424	0.556301	0.278151	−	0.960537i	$-0.410279\pi$
$863$	16.3424	0.556301	0.278151	+	0.960537i	$0.410279\pi$
$864$	0	0
$865$	−4.07675	−0.138614
$866$	0	0
$867$	1.90211	0.0645991
$868$	0	0
$869$	−0.602530	−0.0204394
$870$	0	0
$871$	−59.6553	−2.02134
$872$	0	0
$873$	−1.30603	−0.0442026
$874$	0	0
$875$	8.28698	0.280151
$876$	0	0
$877$	−30.1892	−1.01942	−0.509709	−	0.860347i	$-0.670247\pi$
$877$	−30.1892	−1.01942	−0.509709	+	0.860347i	$0.670247\pi$
$878$	0	0
$879$	47.4048	1.59893
$880$	0	0
$881$	−46.1013	−1.55319	−0.776596	−	0.629999i	$-0.783055\pi$
$881$	−46.1013	−1.55319	−0.776596	+	0.629999i	$0.783055\pi$
$882$	0	0
$883$	32.5463	1.09527	0.547635	−	0.836718i	$-0.315528\pi$
$883$	32.5463	1.09527	0.547635	+	0.836718i	$0.315528\pi$
$884$	0	0
$885$	8.23351	0.276766
$886$	0	0
$887$	−37.1849	−1.24855	−0.624273	−	0.781206i	$-0.714605\pi$
$887$	−37.1849	−1.24855	−0.624273	+	0.781206i	$0.714605\pi$
$888$	0	0
$889$	13.2425	0.444140
$890$	0	0
$891$	4.63354	0.155229
$892$	0	0
$893$	13.5291	0.452736
$894$	0	0
$895$	−10.2807	−0.343645
$896$	0	0
$897$	−2.99408	−0.0999694
$898$	0	0
$899$	76.6271	2.55566
$900$	0	0
$901$	9.20524	0.306671
$902$	0	0
$903$	−3.70042	−0.123142
$904$	0	0
$905$	−10.1189	−0.336364
$906$	0	0
$907$	20.6502	0.685678	0.342839	−	0.939394i	$-0.388612\pi$
$907$	20.6502	0.685678	0.342839	+	0.939394i	$0.388612\pi$
$908$	0	0
$909$	−9.04209	−0.299907
$910$	0	0
$911$	37.6232	1.24651	0.623256	−	0.782018i	$-0.285809\pi$
$911$	37.6232	1.24651	0.623256	+	0.782018i	$0.285809\pi$
$912$	0	0
$913$	−3.14213	−0.103989
$914$	0	0
$915$	2.89956	0.0958564
$916$	0	0
$917$	20.3357	0.671544
$918$	0	0
$919$	5.01508	0.165432	0.0827160	−	0.996573i	$-0.473641\pi$
$919$	5.01508	0.165432	0.0827160	+	0.996573i	$0.473641\pi$
$920$	0	0
$921$	47.0010	1.54874
$922$	0	0
$923$	29.4602	0.969694
$924$	0	0
$925$	−24.5339	−0.806669
$926$	0	0
$927$	0.806733	0.0264966
$928$	0	0
$929$	−2.83838	−0.0931243	−0.0465622	−	0.998915i	$-0.514827\pi$
$929$	−2.83838	−0.0931243	−0.0465622	+	0.998915i	$0.514827\pi$
$930$	0	0
$931$	−6.42882	−0.210696
$932$	0	0
$933$	−43.2360	−1.41548
$934$	0	0
$935$	0.399152	0.0130537
$936$	0	0
$937$	−8.32178	−0.271861	−0.135930	−	0.990718i	$-0.543402\pi$
$937$	−8.32178	−0.271861	−0.135930	+	0.990718i	$0.543402\pi$
$938$	0	0
$939$	−53.1631	−1.73491
$940$	0	0
$941$	27.1714	0.885761	0.442880	−	0.896581i	$-0.353957\pi$
$941$	27.1714	0.885761	0.442880	+	0.896581i	$0.353957\pi$
$942$	0	0
$943$	1.58541	0.0516282
$944$	0	0
$945$	4.08727	0.132959
$946$	0	0
$947$	−16.3843	−0.532419	−0.266209	−	0.963915i	$-0.585771\pi$
$947$	−16.3843	−0.532419	−0.266209	+	0.963915i	$0.585771\pi$
$948$	0	0
$949$	−92.6011	−3.00596
$950$	0	0
$951$	−25.1262	−0.814772
$952$	0	0
$953$	−11.0536	−0.358061	−0.179031	−	0.983843i	$-0.557296\pi$
$953$	−11.0536	−0.358061	−0.179031	+	0.983843i	$0.557296\pi$
$954$	0	0
$955$	3.01382	0.0975250
$956$	0	0
$957$	8.83849	0.285708
$958$	0	0
$959$	10.2858	0.332147
$960$	0	0
$961$	22.2399	0.717415
$962$	0	0
$963$	−0.0165435	−0.000533106 0
$964$	0	0
$965$	9.22713	0.297032
$966$	0	0
$967$	−4.07955	−0.131189	−0.0655947	−	0.997846i	$-0.520894\pi$
$967$	−4.07955	−0.131189	−0.0655947	+	0.997846i	$0.520894\pi$
$968$	0	0
$969$	−12.2284	−0.392831
$970$	0	0
$971$	27.1145	0.870147	0.435073	−	0.900395i	$-0.356722\pi$
$971$	27.1145	0.870147	0.435073	+	0.900395i	$0.356722\pi$
$972$	0	0
$973$	−7.56108	−0.242397
$974$	0	0
$975$	−45.8345	−1.46788
$976$	0	0
$977$	−7.97729	−0.255216	−0.127608	−	0.991825i	$-0.540730\pi$
$977$	−7.97729	−0.255216	−0.127608	+	0.991825i	$0.540730\pi$
$978$	0	0
$979$	−2.34511	−0.0749499
$980$	0	0
$981$	7.31666	0.233603
$982$	0	0
$983$	4.82005	0.153736	0.0768678	−	0.997041i	$-0.475508\pi$
$983$	4.82005	0.153736	0.0768678	+	0.997041i	$0.475508\pi$
$984$	0	0
$985$	−12.5981	−0.401409
$986$	0	0
$987$	−4.00290	−0.127414
$988$	0	0
$989$	0.531991	0.0169163
$990$	0	0
$991$	−4.10685	−0.130458	−0.0652291	−	0.997870i	$-0.520778\pi$
$991$	−4.10685	−0.130458	−0.0652291	+	0.997870i	$0.520778\pi$
$992$	0	0
$993$	−30.8454	−0.978848
$994$	0	0
$995$	0.687471	0.0217943
$996$	0	0
$997$	−3.43712	−0.108855	−0.0544274	−	0.998518i	$-0.517333\pi$
$997$	−3.43712	−0.108855	−0.0544274	+	0.998518i	$0.517333\pi$
$998$	0	0
$999$	−26.5533	−0.840110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.bm.1.4		4
4.3	odd	2		7616.2.a.bl.1.1		4
8.3	odd	2		3808.2.a.c.1.4	✓	4
8.5	even	2		3808.2.a.d.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.c.1.4	✓	4	8.3	odd	2
3808.2.a.d.1.1	yes	4	8.5	even	2
7616.2.a.bl.1.1		4	4.3	odd	2
7616.2.a.bm.1.4		4	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.90211 q^{3} -0.902113 q^{5} +1.00000 q^{7} +0.618034 q^{9} -0.442463 q^{11} +5.75621 q^{13} -1.71592 q^{15} +1.00000 q^{17} -6.42882 q^{19} +1.90211 q^{21} -0.273457 q^{23} -4.18619 q^{25} -4.53077 q^{27} -10.5018 q^{29} -7.29657 q^{31} -0.841616 q^{33} -0.902113 q^{35} +5.86067 q^{37} +10.9490 q^{39} -5.79766 q^{41} -1.94542 q^{43} -0.557537 q^{45} -2.10445 q^{47} +1.00000 q^{49} +1.90211 q^{51} +9.20524 q^{53} +0.399152 q^{55} -12.2284 q^{57} -4.79830 q^{59} -1.68980 q^{61} +0.618034 q^{63} -5.19276 q^{65} -10.3636 q^{67} -0.520147 q^{69} +5.11798 q^{71} -16.0872 q^{73} -7.96261 q^{75} -0.442463 q^{77} +1.36176 q^{79} -10.4721 q^{81} +7.10143 q^{83} -0.902113 q^{85} -19.9756 q^{87} +5.30011 q^{89} +5.75621 q^{91} -13.8789 q^{93} +5.79953 q^{95} -2.11321 q^{97} -0.273457 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{7} - 2 q^{9} - 2 q^{11} + 2 q^{13} - 10 q^{15} + 4 q^{17} - 4 q^{19} - 4 q^{23} - 6 q^{25} - 6 q^{29} - 16 q^{31} + 4 q^{35} + 8 q^{37} + 10 q^{39} - 10 q^{41} - 4 q^{43} - 2 q^{45}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^7 - 2 * q^9 - 2 * q^11 + 2 * q^13 - 10 * q^15 + 4 * q^17 - 4 * q^19 - 4 * q^23 - 6 * q^25 - 6 * q^29 - 16 * q^31 + 4 * q^35 + 8 * q^37 + 10 * q^39 - 10 * q^41 - 4 * q^43 - 2 * q^45 - 14 * q^47 + 4 * q^49 + 14 * q^53 - 2 * q^55 - 10 * q^57 - 12 * q^59 + 2 * q^61 - 2 * q^63 - 8 * q^65 - 22 * q^67 + 10 * q^69 - 16 * q^71 - 14 * q^73 - 20 * q^75 - 2 * q^77 - 10 * q^79 - 24 * q^81 + 6 * q^83 + 4 * q^85 - 22 * q^89 + 2 * q^91 + 6 * q^95 - 12 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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