LMFDB - Embedded newform 7616.2.a.ba.1.3

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,-1,0,3,0,2,0,-6] 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:	$3$
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			1.3
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)

$f(q)$	$=$	$q+1.11491 q^{3} +1.47283 q^{5} +1.00000 q^{7} -1.75698 q^{9} -2.00000 q^{11} -0.715853 q^{13} +1.64207 q^{15} -1.00000 q^{17} -1.28415 q^{19} +1.11491 q^{21} -2.22982 q^{23} -2.83076 q^{25} -5.30359 q^{27} +9.17548 q^{29} +6.75698 q^{31} -2.22982 q^{33} +1.47283 q^{35} -8.45963 q^{37} -0.798110 q^{39} -6.77643 q^{41} -4.39905 q^{43} -2.58774 q^{45} -2.22982 q^{47} +1.00000 q^{49} -1.11491 q^{51} -6.64832 q^{53} -2.94567 q^{55} -1.43171 q^{57} -5.28415 q^{59} -0.885092 q^{61} -1.75698 q^{63} -1.05433 q^{65} -4.90454 q^{67} -2.48604 q^{69} +16.4596 q^{71} +11.2361 q^{73} -3.15604 q^{75} -2.00000 q^{77} -10.6894 q^{79} -0.642074 q^{81} -2.37737 q^{83} -1.47283 q^{85} +10.2298 q^{87} +7.51396 q^{89} -0.715853 q^{91} +7.53341 q^{93} -1.89134 q^{95} +5.85021 q^{97} +3.51396 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - q^{5} + 3 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{13} + 4 q^{15} - 3 q^{17} - 2 q^{19} - 3 q^{21} + 6 q^{23} - 4 q^{25} - 6 q^{27} + 4 q^{29} + 13 q^{31} + 6 q^{33} - q^{35} + 14 q^{39} - 5 q^{41}+ \cdots - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - q^5 + 3 * q^7 + 2 * q^9 - 6 * q^11 - 4 * q^13 + 4 * q^15 - 3 * q^17 - 2 * q^19 - 3 * q^21 + 6 * q^23 - 4 * q^25 - 6 * q^27 + 4 * q^29 + 13 * q^31 + 6 * q^33 - q^35 + 14 * q^39 - 5 * q^41 - 5 * q^43 + 4 * q^45 + 6 * q^47 + 3 * q^49 + 3 * q^51 + 9 * q^53 + 2 * q^55 - 8 * q^57 - 14 * q^59 - 9 * q^61 + 2 * q^63 - 14 * q^65 - 13 * q^67 - 22 * q^69 + 24 * q^71 - 7 * q^73 + 6 * q^75 - 6 * q^77 + 6 * q^79 - q^81 + q^85 + 18 * q^87 + 8 * q^89 - 4 * q^91 + 16 * q^95 + 5 * 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
$3$	1.11491	0.643692	0.321846	−	0.946792i	$-0.395697\pi$
$3$	1.11491	0.643692	0.321846	+	0.946792i	$0.395697\pi$
$4$	0	0
$5$	1.47283	0.658671	0.329336	−	0.944213i	$-0.393175\pi$
$5$	1.47283	0.658671	0.329336	+	0.944213i	$0.393175\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.75698	−0.585660
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−0.715853	−0.198542	−0.0992709	−	0.995060i	$-0.531651\pi$
$13$	−0.715853	−0.198542	−0.0992709	+	0.995060i	$0.531651\pi$
$14$	0	0
$15$	1.64207	0.423982
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−1.28415	−0.294604	−0.147302	−	0.989092i	$-0.547059\pi$
$19$	−1.28415	−0.294604	−0.147302	+	0.989092i	$0.547059\pi$
$20$	0	0
$21$	1.11491	0.243293
$22$	0	0
$23$	−2.22982	−0.464949	−0.232474	−	0.972603i	$-0.574682\pi$
$23$	−2.22982	−0.464949	−0.232474	+	0.972603i	$0.574682\pi$
$24$	0	0
$25$	−2.83076	−0.566152
$26$	0	0
$27$	−5.30359	−1.02068
$28$	0	0
$29$	9.17548	1.70384	0.851922	−	0.523668i	$-0.175437\pi$
$29$	9.17548	1.70384	0.851922	+	0.523668i	$0.175437\pi$
$30$	0	0
$31$	6.75698	1.21359	0.606795	−	0.794859i	$-0.292455\pi$
$31$	6.75698	1.21359	0.606795	+	0.794859i	$0.292455\pi$
$32$	0	0
$33$	−2.22982	−0.388161
$34$	0	0
$35$	1.47283	0.248954
$36$	0	0
$37$	−8.45963	−1.39075	−0.695377	−	0.718645i	$-0.744763\pi$
$37$	−8.45963	−1.39075	−0.695377	+	0.718645i	$0.744763\pi$
$38$	0	0
$39$	−0.798110	−0.127800
$40$	0	0
$41$	−6.77643	−1.05830	−0.529150	−	0.848528i	$-0.677489\pi$
$41$	−6.77643	−1.05830	−0.529150	+	0.848528i	$0.677489\pi$
$42$	0	0
$43$	−4.39905	−0.670850	−0.335425	−	0.942067i	$-0.608880\pi$
$43$	−4.39905	−0.670850	−0.335425	+	0.942067i	$0.608880\pi$
$44$	0	0
$45$	−2.58774	−0.385758
$46$	0	0
$47$	−2.22982	−0.325252	−0.162626	−	0.986688i	$-0.551996\pi$
$47$	−2.22982	−0.325252	−0.162626	+	0.986688i	$0.551996\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.11491	−0.156118
$52$	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
$55$	−2.94567	−0.397194
$56$	0	0
$57$	−1.43171	−0.189634
$58$	0	0
$59$	−5.28415	−0.687937	−0.343969	−	0.938981i	$-0.611771\pi$
$59$	−5.28415	−0.687937	−0.343969	+	0.938981i	$0.611771\pi$
$60$	0	0
$61$	−0.885092	−0.113324	−0.0566622	−	0.998393i	$-0.518046\pi$
$61$	−0.885092	−0.113324	−0.0566622	+	0.998393i	$0.518046\pi$
$62$	0	0
$63$	−1.75698	−0.221359
$64$	0	0
$65$	−1.05433	−0.130774
$66$	0	0
$67$	−4.90454	−0.599185	−0.299592	−	0.954067i	$-0.596851\pi$
$67$	−4.90454	−0.599185	−0.299592	+	0.954067i	$0.596851\pi$
$68$	0	0
$69$	−2.48604	−0.299284
$70$	0	0
$71$	16.4596	1.95340	0.976699	−	0.214612i	$-0.0688487\pi$
$71$	16.4596	1.95340	0.976699	+	0.214612i	$0.0688487\pi$
$72$	0	0
$73$	11.2361	1.31508	0.657541	−	0.753419i	$-0.271597\pi$
$73$	11.2361	1.31508	0.657541	+	0.753419i	$0.271597\pi$
$74$	0	0
$75$	−3.15604	−0.364428
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−10.6894	−1.20266	−0.601328	−	0.799002i	$-0.705362\pi$
$79$	−10.6894	−1.20266	−0.601328	+	0.799002i	$0.705362\pi$
$80$	0	0
$81$	−0.642074	−0.0713415
$82$	0	0
$83$	−2.37737	−0.260951	−0.130475	−	0.991452i	$-0.541650\pi$
$83$	−2.37737	−0.260951	−0.130475	+	0.991452i	$0.541650\pi$
$84$	0	0
$85$	−1.47283	−0.159751
$86$	0	0
$87$	10.2298	1.09675
$88$	0	0
$89$	7.51396	0.796478	0.398239	−	0.917282i	$-0.369621\pi$
$89$	7.51396	0.796478	0.398239	+	0.917282i	$0.369621\pi$
$90$	0	0
$91$	−0.715853	−0.0750418
$92$	0	0
$93$	7.53341	0.781178
$94$	0	0
$95$	−1.89134	−0.194047
$96$	0	0
$97$	5.85021	0.593999	0.296999	−	0.954878i	$-0.404014\pi$
$97$	5.85021	0.593999	0.296999	+	0.954878i	$0.404014\pi$
$98$	0	0
$99$	3.51396	0.353167
$100$	0	0
$101$	−18.3510	−1.82599	−0.912995	−	0.407971i	$-0.866236\pi$
$101$	−18.3510	−1.82599	−0.912995	+	0.407971i	$0.866236\pi$
$102$	0	0
$103$	−0.338479	−0.0333514	−0.0166757	−	0.999861i	$-0.505308\pi$
$103$	−0.338479	−0.0333514	−0.0166757	+	0.999861i	$0.505308\pi$
$104$	0	0
$105$	1.64207	0.160250
$106$	0	0
$107$	−0.568295	−0.0549391	−0.0274696	−	0.999623i	$-0.508745\pi$
$107$	−0.568295	−0.0549391	−0.0274696	+	0.999623i	$0.508745\pi$
$108$	0	0
$109$	−4.71585	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$109$	−4.71585	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$110$	0	0
$111$	−9.43171	−0.895218
$112$	0	0
$113$	−12.9457	−1.21783	−0.608913	−	0.793237i	$-0.708394\pi$
$113$	−12.9457	−1.21783	−0.608913	+	0.793237i	$0.708394\pi$
$114$	0	0
$115$	−3.28415	−0.306248
$116$	0	0
$117$	1.25774	0.116278
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−7.55509	−0.681220
$124$	0	0
$125$	−11.5334	−1.03158
$126$	0	0
$127$	5.95887	0.528764	0.264382	−	0.964418i	$-0.414832\pi$
$127$	5.95887	0.528764	0.264382	+	0.964418i	$0.414832\pi$
$128$	0	0
$129$	−4.90454	−0.431821
$130$	0	0
$131$	−18.8106	−1.64349	−0.821745	−	0.569856i	$-0.806999\pi$
$131$	−18.8106	−1.64349	−0.821745	+	0.569856i	$0.806999\pi$
$132$	0	0
$133$	−1.28415	−0.111350
$134$	0	0
$135$	−7.81131	−0.672291
$136$	0	0
$137$	−14.1817	−1.21163	−0.605813	−	0.795607i	$-0.707152\pi$
$137$	−14.1817	−1.21163	−0.605813	+	0.795607i	$0.707152\pi$
$138$	0	0
$139$	−0.188687	−0.0160042	−0.00800210	−	0.999968i	$-0.502547\pi$
$139$	−0.188687	−0.0160042	−0.00800210	+	0.999968i	$0.502547\pi$
$140$	0	0
$141$	−2.48604	−0.209362
$142$	0	0
$143$	1.43171	0.119725
$144$	0	0
$145$	13.5140	1.12227
$146$	0	0
$147$	1.11491	0.0919560
$148$	0	0
$149$	−13.0885	−1.07225	−0.536126	−	0.844138i	$-0.680113\pi$
$149$	−13.0885	−1.07225	−0.536126	+	0.844138i	$0.680113\pi$
$150$	0	0
$151$	−5.49228	−0.446955	−0.223478	−	0.974709i	$-0.571741\pi$
$151$	−5.49228	−0.446955	−0.223478	+	0.974709i	$0.571741\pi$
$152$	0	0
$153$	1.75698	0.142044
$154$	0	0
$155$	9.95191	0.799357
$156$	0	0
$157$	−11.0279	−0.880124	−0.440062	−	0.897967i	$-0.645044\pi$
$157$	−11.0279	−0.880124	−0.440062	+	0.897967i	$0.645044\pi$
$158$	0	0
$159$	−7.41226	−0.587830
$160$	0	0
$161$	−2.22982	−0.175734
$162$	0	0
$163$	6.71585	0.526026	0.263013	−	0.964792i	$-0.415284\pi$
$163$	6.71585	0.526026	0.263013	+	0.964792i	$0.415284\pi$
$164$	0	0
$165$	−3.28415	−0.255671
$166$	0	0
$167$	11.8044	0.913448	0.456724	−	0.889608i	$-0.349023\pi$
$167$	11.8044	0.913448	0.456724	+	0.889608i	$0.349023\pi$
$168$	0	0
$169$	−12.4876	−0.960581
$170$	0	0
$171$	2.25622	0.172538
$172$	0	0
$173$	−13.3447	−1.01458	−0.507290	−	0.861775i	$-0.669353\pi$
$173$	−13.3447	−1.01458	−0.507290	+	0.861775i	$0.669353\pi$
$174$	0	0
$175$	−2.83076	−0.213985
$176$	0	0
$177$	−5.89134	−0.442820
$178$	0	0
$179$	1.37113	0.102483	0.0512415	−	0.998686i	$-0.483682\pi$
$179$	1.37113	0.102483	0.0512415	+	0.998686i	$0.483682\pi$
$180$	0	0
$181$	−7.13659	−0.530458	−0.265229	−	0.964185i	$-0.585448\pi$
$181$	−7.13659	−0.530458	−0.265229	+	0.964185i	$0.585448\pi$
$182$	0	0
$183$	−0.986796	−0.0729461
$184$	0	0
$185$	−12.4596	−0.916050
$186$	0	0
$187$	2.00000	0.146254
$188$	0	0
$189$	−5.30359	−0.385780
$190$	0	0
$191$	6.50076	0.470378	0.235189	−	0.971950i	$-0.424429\pi$
$191$	6.50076	0.470378	0.235189	+	0.971950i	$0.424429\pi$
$192$	0	0
$193$	16.3510	1.17697	0.588484	−	0.808509i	$-0.299725\pi$
$193$	16.3510	1.17697	0.588484	+	0.808509i	$0.299725\pi$
$194$	0	0
$195$	−1.17548	−0.0841781
$196$	0	0
$197$	20.4596	1.45769	0.728844	−	0.684680i	$-0.240058\pi$
$197$	20.4596	1.45769	0.728844	+	0.684680i	$0.240058\pi$
$198$	0	0
$199$	17.6957	1.25441	0.627207	−	0.778853i	$-0.284198\pi$
$199$	17.6957	1.25441	0.627207	+	0.778853i	$0.284198\pi$
$200$	0	0
$201$	−5.46811	−0.385691
$202$	0	0
$203$	9.17548	0.643993
$204$	0	0
$205$	−9.98055	−0.697072
$206$	0	0
$207$	3.91774	0.272302
$208$	0	0
$209$	2.56829	0.177653
$210$	0	0
$211$	−24.7283	−1.70237	−0.851185	−	0.524867i	$-0.824115\pi$
$211$	−24.7283	−1.70237	−0.851185	+	0.524867i	$0.824115\pi$
$212$	0	0
$213$	18.3510	1.25739
$214$	0	0
$215$	−6.47908	−0.441869
$216$	0	0
$217$	6.75698	0.458694
$218$	0	0
$219$	12.5272	0.846507
$220$	0	0
$221$	0.715853	0.0481535
$222$	0	0
$223$	−22.3510	−1.49673	−0.748366	−	0.663286i	$-0.769161\pi$
$223$	−22.3510	−1.49673	−0.748366	+	0.663286i	$0.769161\pi$
$224$	0	0
$225$	4.97359	0.331573
$226$	0	0
$227$	19.2989	1.28091	0.640455	−	0.767995i	$-0.278746\pi$
$227$	19.2989	1.28091	0.640455	+	0.767995i	$0.278746\pi$
$228$	0	0
$229$	−5.39281	−0.356367	−0.178184	−	0.983997i	$-0.557022\pi$
$229$	−5.39281	−0.356367	−0.178184	+	0.983997i	$0.557022\pi$
$230$	0	0
$231$	−2.22982	−0.146711
$232$	0	0
$233$	−8.10866	−0.531216	−0.265608	−	0.964081i	$-0.585573\pi$
$233$	−8.10866	−0.531216	−0.265608	+	0.964081i	$0.585573\pi$
$234$	0	0
$235$	−3.28415	−0.214234
$236$	0	0
$237$	−11.9177	−0.774141
$238$	0	0
$239$	−22.4115	−1.44968	−0.724841	−	0.688916i	$-0.758087\pi$
$239$	−22.4115	−1.44968	−0.724841	+	0.688916i	$0.758087\pi$
$240$	0	0
$241$	6.98680	0.450059	0.225030	−	0.974352i	$-0.427752\pi$
$241$	6.98680	0.450059	0.225030	+	0.974352i	$0.427752\pi$
$242$	0	0
$243$	15.1949	0.974755
$244$	0	0
$245$	1.47283	0.0940959
$246$	0	0
$247$	0.919260	0.0584911
$248$	0	0
$249$	−2.65055	−0.167972
$250$	0	0
$251$	15.6351	0.986880	0.493440	−	0.869780i	$-0.335739\pi$
$251$	15.6351	0.986880	0.493440	+	0.869780i	$0.335739\pi$
$252$	0	0
$253$	4.45963	0.280375
$254$	0	0
$255$	−1.64207	−0.102831
$256$	0	0
$257$	0.689445	0.0430064	0.0215032	−	0.999769i	$-0.493155\pi$
$257$	0.689445	0.0430064	0.0215032	+	0.999769i	$0.493155\pi$
$258$	0	0
$259$	−8.45963	−0.525656
$260$	0	0
$261$	−16.1212	−0.997874
$262$	0	0
$263$	17.4317	1.07489	0.537443	−	0.843300i	$-0.319391\pi$
$263$	17.4317	1.07489	0.537443	+	0.843300i	$0.319391\pi$
$264$	0	0
$265$	−9.79187	−0.601510
$266$	0	0
$267$	8.37737	0.512687
$268$	0	0
$269$	3.43171	0.209235	0.104617	−	0.994513i	$-0.466638\pi$
$269$	3.43171	0.209235	0.104617	+	0.994513i	$0.466638\pi$
$270$	0	0
$271$	2.68945	0.163372	0.0816861	−	0.996658i	$-0.473970\pi$
$271$	2.68945	0.163372	0.0816861	+	0.996658i	$0.473970\pi$
$272$	0	0
$273$	−0.798110	−0.0483038
$274$	0	0
$275$	5.66152	0.341403
$276$	0	0
$277$	−15.6615	−0.941010	−0.470505	−	0.882397i	$-0.655928\pi$
$277$	−15.6615	−0.941010	−0.470505	+	0.882397i	$0.655928\pi$
$278$	0	0
$279$	−11.8719	−0.710751
$280$	0	0
$281$	−5.72906	−0.341767	−0.170883	−	0.985291i	$-0.554662\pi$
$281$	−5.72906	−0.341767	−0.170883	+	0.985291i	$0.554662\pi$
$282$	0	0
$283$	−17.2430	−1.02499	−0.512496	−	0.858690i	$-0.671279\pi$
$283$	−17.2430	−1.02499	−0.512496	+	0.858690i	$0.671279\pi$
$284$	0	0
$285$	−2.10866	−0.124906
$286$	0	0
$287$	−6.77643	−0.400000
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	6.52244	0.382352
$292$	0	0
$293$	24.8370	1.45099	0.725497	−	0.688226i	$-0.241610\pi$
$293$	24.8370	1.45099	0.725497	+	0.688226i	$0.241610\pi$
$294$	0	0
$295$	−7.78267	−0.453125
$296$	0	0
$297$	10.6072	0.615491
$298$	0	0
$299$	1.59622	0.0923117
$300$	0	0
$301$	−4.39905	−0.253557
$302$	0	0
$303$	−20.4596	−1.17538
$304$	0	0
$305$	−1.30359	−0.0746436
$306$	0	0
$307$	24.6894	1.40910	0.704551	−	0.709654i	$-0.251149\pi$
$307$	24.6894	1.40910	0.704551	+	0.709654i	$0.251149\pi$
$308$	0	0
$309$	−0.377373	−0.0214680
$310$	0	0
$311$	12.2709	0.695821	0.347911	−	0.937528i	$-0.386891\pi$
$311$	12.2709	0.695821	0.347911	+	0.937528i	$0.386891\pi$
$312$	0	0
$313$	8.87813	0.501822	0.250911	−	0.968010i	$-0.419270\pi$
$313$	8.87813	0.501822	0.250911	+	0.968010i	$0.419270\pi$
$314$	0	0
$315$	−2.58774	−0.145803
$316$	0	0
$317$	24.5419	1.37841	0.689205	−	0.724567i	$-0.257960\pi$
$317$	24.5419	1.37841	0.689205	+	0.724567i	$0.257960\pi$
$318$	0	0
$319$	−18.3510	−1.02746
$320$	0	0
$321$	−0.633596	−0.0353639
$322$	0	0
$323$	1.28415	0.0714519
$324$	0	0
$325$	2.02641	0.112405
$326$	0	0
$327$	−5.25774	−0.290754
$328$	0	0
$329$	−2.22982	−0.122934
$330$	0	0
$331$	7.47283	0.410744	0.205372	−	0.978684i	$-0.434160\pi$
$331$	7.47283	0.410744	0.205372	+	0.978684i	$0.434160\pi$
$332$	0	0
$333$	14.8634	0.814510
$334$	0	0
$335$	−7.22357	−0.394666
$336$	0	0
$337$	−3.39281	−0.184818	−0.0924091	−	0.995721i	$-0.529457\pi$
$337$	−3.39281	−0.184818	−0.0924091	+	0.995721i	$0.529457\pi$
$338$	0	0
$339$	−14.4332	−0.783905
$340$	0	0
$341$	−13.5140	−0.731822
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−3.66152	−0.197130
$346$	0	0
$347$	−6.67696	−0.358438	−0.179219	−	0.983809i	$-0.557357\pi$
$347$	−6.67696	−0.358438	−0.179219	+	0.983809i	$0.557357\pi$
$348$	0	0
$349$	−13.0543	−0.698782	−0.349391	−	0.936977i	$-0.613612\pi$
$349$	−13.0543	−0.698782	−0.349391	+	0.936977i	$0.613612\pi$
$350$	0	0
$351$	3.79659	0.202647
$352$	0	0
$353$	−25.0279	−1.33210	−0.666051	−	0.745906i	$-0.732017\pi$
$353$	−25.0279	−1.33210	−0.666051	+	0.745906i	$0.732017\pi$
$354$	0	0
$355$	24.2423	1.28665
$356$	0	0
$357$	−1.11491	−0.0590072
$358$	0	0
$359$	11.1732	0.589701	0.294851	−	0.955543i	$-0.404730\pi$
$359$	11.1732	0.589701	0.294851	+	0.955543i	$0.404730\pi$
$360$	0	0
$361$	−17.3510	−0.913209
$362$	0	0
$363$	−7.80435	−0.409622
$364$	0	0
$365$	16.5488	0.866206
$366$	0	0
$367$	19.8044	1.03378	0.516889	−	0.856052i	$-0.327090\pi$
$367$	19.8044	1.03378	0.516889	+	0.856052i	$0.327090\pi$
$368$	0	0
$369$	11.9061	0.619805
$370$	0	0
$371$	−6.64832	−0.345163
$372$	0	0
$373$	−7.53564	−0.390181	−0.195090	−	0.980785i	$-0.562500\pi$
$373$	−7.53564	−0.390181	−0.195090	+	0.980785i	$0.562500\pi$
$374$	0	0
$375$	−12.8587	−0.664020
$376$	0	0
$377$	−6.56829	−0.338284
$378$	0	0
$379$	−2.75475	−0.141502	−0.0707509	−	0.997494i	$-0.522540\pi$
$379$	−2.75475	−0.141502	−0.0707509	+	0.997494i	$0.522540\pi$
$380$	0	0
$381$	6.64359	0.340361
$382$	0	0
$383$	0.0388938	0.00198738	0.000993691	−	1.00000i	$-0.499684\pi$
$383$	0.0388938	0.00198738	0.000993691		1.00000i	$0.499684\pi$
$384$	0	0
$385$	−2.94567	−0.150125
$386$	0	0
$387$	7.72906	0.392890
$388$	0	0
$389$	−5.05210	−0.256152	−0.128076	−	0.991764i	$-0.540880\pi$
$389$	−5.05210	−0.256152	−0.128076	+	0.991764i	$0.540880\pi$
$390$	0	0
$391$	2.22982	0.112767
$392$	0	0
$393$	−20.9721	−1.05790
$394$	0	0
$395$	−15.7438	−0.792155
$396$	0	0
$397$	−2.06754	−0.103767	−0.0518833	−	0.998653i	$-0.516522\pi$
$397$	−2.06754	−0.103767	−0.0518833	+	0.998653i	$0.516522\pi$
$398$	0	0
$399$	−1.43171	−0.0716749
$400$	0	0
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	0	0
$403$	−4.83700	−0.240948
$404$	0	0
$405$	−0.945668	−0.0469906
$406$	0	0
$407$	16.9193	0.838657
$408$	0	0
$409$	2.59470	0.128300	0.0641499	−	0.997940i	$-0.479566\pi$
$409$	2.59470	0.128300	0.0641499	+	0.997940i	$0.479566\pi$
$410$	0	0
$411$	−15.8113	−0.779915
$412$	0	0
$413$	−5.28415	−0.260016
$414$	0	0
$415$	−3.50148	−0.171881
$416$	0	0
$417$	−0.210368	−0.0103018
$418$	0	0
$419$	22.9993	1.12359	0.561794	−	0.827277i	$-0.310111\pi$
$419$	22.9993	1.12359	0.561794	+	0.827277i	$0.310111\pi$
$420$	0	0
$421$	−1.47979	−0.0721208	−0.0360604	−	0.999350i	$-0.511481\pi$
$421$	−1.47979	−0.0721208	−0.0360604	+	0.999350i	$0.511481\pi$
$422$	0	0
$423$	3.91774	0.190487
$424$	0	0
$425$	2.83076	0.137312
$426$	0	0
$427$	−0.885092	−0.0428326
$428$	0	0
$429$	1.59622	0.0770662
$430$	0	0
$431$	9.97359	0.480411	0.240206	−	0.970722i	$-0.422785\pi$
$431$	9.97359	0.480411	0.240206	+	0.970722i	$0.422785\pi$
$432$	0	0
$433$	−20.3943	−0.980089	−0.490044	−	0.871697i	$-0.663019\pi$
$433$	−20.3943	−0.980089	−0.490044	+	0.871697i	$0.663019\pi$
$434$	0	0
$435$	15.0668	0.722399
$436$	0	0
$437$	2.86341	0.136976
$438$	0	0
$439$	32.3851	1.54566	0.772829	−	0.634614i	$-0.218841\pi$
$439$	32.3851	1.54566	0.772829	+	0.634614i	$0.218841\pi$
$440$	0	0
$441$	−1.75698	−0.0836658
$442$	0	0
$443$	13.8913	0.659997	0.329999	−	0.943981i	$-0.392952\pi$
$443$	13.8913	0.659997	0.329999	+	0.943981i	$0.392952\pi$
$444$	0	0
$445$	11.0668	0.524618
$446$	0	0
$447$	−14.5925	−0.690200
$448$	0	0
$449$	−15.3928	−0.726432	−0.363216	−	0.931705i	$-0.618321\pi$
$449$	−15.3928	−0.726432	−0.363216	+	0.931705i	$0.618321\pi$
$450$	0	0
$451$	13.5529	0.638179
$452$	0	0
$453$	−6.12339	−0.287702
$454$	0	0
$455$	−1.05433	−0.0494279
$456$	0	0
$457$	−32.3223	−1.51197	−0.755987	−	0.654586i	$-0.772843\pi$
$457$	−32.3223	−1.51197	−0.755987	+	0.654586i	$0.772843\pi$
$458$	0	0
$459$	5.30359	0.247551
$460$	0	0
$461$	21.9736	1.02341	0.511706	−	0.859161i	$-0.329014\pi$
$461$	21.9736	1.02341	0.511706	+	0.859161i	$0.329014\pi$
$462$	0	0
$463$	25.8308	1.20046	0.600229	−	0.799828i	$-0.295076\pi$
$463$	25.8308	1.20046	0.600229	+	0.799828i	$0.295076\pi$
$464$	0	0
$465$	11.0955	0.514540
$466$	0	0
$467$	−27.2577	−1.26134	−0.630669	−	0.776052i	$-0.717219\pi$
$467$	−27.2577	−1.26134	−0.630669	+	0.776052i	$0.717219\pi$
$468$	0	0
$469$	−4.90454	−0.226471
$470$	0	0
$471$	−12.2951	−0.566529
$472$	0	0
$473$	8.79811	0.404538
$474$	0	0
$475$	3.63511	0.166790
$476$	0	0
$477$	11.6810	0.534835
$478$	0	0
$479$	26.4938	1.21053	0.605266	−	0.796023i	$-0.293067\pi$
$479$	26.4938	1.21053	0.605266	+	0.796023i	$0.293067\pi$
$480$	0	0
$481$	6.05585	0.276123
$482$	0	0
$483$	−2.48604	−0.113119
$484$	0	0
$485$	8.61638	0.391250
$486$	0	0
$487$	17.9736	0.814461	0.407231	−	0.913325i	$-0.366495\pi$
$487$	17.9736	0.814461	0.407231	+	0.913325i	$0.366495\pi$
$488$	0	0
$489$	7.48755	0.338599
$490$	0	0
$491$	−17.1468	−0.773826	−0.386913	−	0.922116i	$-0.626459\pi$
$491$	−17.1468	−0.773826	−0.386913	+	0.922116i	$0.626459\pi$
$492$	0	0
$493$	−9.17548	−0.413243
$494$	0	0
$495$	5.17548	0.232621
$496$	0	0
$497$	16.4596	0.738315
$498$	0	0
$499$	20.2298	0.905611	0.452805	−	0.891609i	$-0.350423\pi$
$499$	20.2298	0.905611	0.452805	+	0.891609i	$0.350423\pi$
$500$	0	0
$501$	13.1608	0.587979
$502$	0	0
$503$	−2.16228	−0.0964113	−0.0482056	−	0.998837i	$-0.515350\pi$
$503$	−2.16228	−0.0964113	−0.0482056	+	0.998837i	$0.515350\pi$
$504$	0	0
$505$	−27.0279	−1.20273
$506$	0	0
$507$	−13.9225	−0.618319
$508$	0	0
$509$	−29.2144	−1.29490	−0.647452	−	0.762106i	$-0.724165\pi$
$509$	−29.2144	−1.29490	−0.647452	+	0.762106i	$0.724165\pi$
$510$	0	0
$511$	11.2361	0.497054
$512$	0	0
$513$	6.81060	0.300695
$514$	0	0
$515$	−0.498524	−0.0219676
$516$	0	0
$517$	4.45963	0.196134
$518$	0	0
$519$	−14.8781	−0.653078
$520$	0	0
$521$	−10.1428	−0.444366	−0.222183	−	0.975005i	$-0.571318\pi$
$521$	−10.1428	−0.444366	−0.222183	+	0.975005i	$0.571318\pi$
$522$	0	0
$523$	24.8929	1.08849	0.544244	−	0.838927i	$-0.316816\pi$
$523$	24.8929	1.08849	0.544244	+	0.838927i	$0.316816\pi$
$524$	0	0
$525$	−3.15604	−0.137741
$526$	0	0
$527$	−6.75698	−0.294339
$528$	0	0
$529$	−18.0279	−0.783823
$530$	0	0
$531$	9.28415	0.402898
$532$	0	0
$533$	4.85092	0.210117
$534$	0	0
$535$	−0.837003	−0.0361868
$536$	0	0
$537$	1.52868	0.0659676
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	15.3664	0.660653	0.330327	−	0.943867i	$-0.392841\pi$
$541$	15.3664	0.660653	0.330327	+	0.943867i	$0.392841\pi$
$542$	0	0
$543$	−7.95664	−0.341452
$544$	0	0
$545$	−6.94567	−0.297520
$546$	0	0
$547$	−4.31207	−0.184371	−0.0921854	−	0.995742i	$-0.529385\pi$
$547$	−4.31207	−0.184371	−0.0921854	+	0.995742i	$0.529385\pi$
$548$	0	0
$549$	1.55509	0.0663697
$550$	0	0
$551$	−11.7827	−0.501959
$552$	0	0
$553$	−10.6894	−0.454561
$554$	0	0
$555$	−13.8913	−0.589654
$556$	0	0
$557$	32.5933	1.38102	0.690511	−	0.723322i	$-0.257386\pi$
$557$	32.5933	1.38102	0.690511	+	0.723322i	$0.257386\pi$
$558$	0	0
$559$	3.14908	0.133192
$560$	0	0
$561$	2.22982	0.0941429
$562$	0	0
$563$	11.6740	0.492001	0.246000	−	0.969270i	$-0.420884\pi$
$563$	11.6740	0.492001	0.246000	+	0.969270i	$0.420884\pi$
$564$	0	0
$565$	−19.0668	−0.802147
$566$	0	0
$567$	−0.642074	−0.0269646
$568$	0	0
$569$	−23.6134	−0.989927	−0.494963	−	0.868914i	$-0.664818\pi$
$569$	−23.6134	−0.989927	−0.494963	+	0.868914i	$0.664818\pi$
$570$	0	0
$571$	−45.2313	−1.89287	−0.946436	−	0.322891i	$-0.895345\pi$
$571$	−45.2313	−1.89287	−0.946436	+	0.322891i	$0.895345\pi$
$572$	0	0
$573$	7.24774	0.302779
$574$	0	0
$575$	6.31207	0.263232
$576$	0	0
$577$	8.98903	0.374218	0.187109	−	0.982339i	$-0.440088\pi$
$577$	8.98903	0.374218	0.187109	+	0.982339i	$0.440088\pi$
$578$	0	0
$579$	18.2298	0.757605
$580$	0	0
$581$	−2.37737	−0.0986301
$582$	0	0
$583$	13.2966	0.550690
$584$	0	0
$585$	1.85244	0.0765890
$586$	0	0
$587$	−19.0404	−0.785882	−0.392941	−	0.919564i	$-0.628542\pi$
$587$	−19.0404	−0.785882	−0.392941	+	0.919564i	$0.628542\pi$
$588$	0	0
$589$	−8.67696	−0.357528
$590$	0	0
$591$	22.8106	0.938303
$592$	0	0
$593$	12.0947	0.496672	0.248336	−	0.968674i	$-0.420116\pi$
$593$	12.0947	0.496672	0.248336	+	0.968674i	$0.420116\pi$
$594$	0	0
$595$	−1.47283	−0.0603803
$596$	0	0
$597$	19.7291	0.807457
$598$	0	0
$599$	3.30583	0.135073	0.0675363	−	0.997717i	$-0.478486\pi$
$599$	3.30583	0.135073	0.0675363	+	0.997717i	$0.478486\pi$
$600$	0	0
$601$	25.4876	1.03966	0.519830	−	0.854270i	$-0.325995\pi$
$601$	25.4876	1.03966	0.519830	+	0.854270i	$0.325995\pi$
$602$	0	0
$603$	8.61718	0.350919
$604$	0	0
$605$	−10.3098	−0.419154
$606$	0	0
$607$	−30.9993	−1.25822	−0.629111	−	0.777315i	$-0.716581\pi$
$607$	−30.9993	−1.25822	−0.629111	+	0.777315i	$0.716581\pi$
$608$	0	0
$609$	10.2298	0.414533
$610$	0	0
$611$	1.59622	0.0645761
$612$	0	0
$613$	−17.5117	−0.707292	−0.353646	−	0.935379i	$-0.615058\pi$
$613$	−17.5117	−0.707292	−0.353646	+	0.935379i	$0.615058\pi$
$614$	0	0
$615$	−11.1274	−0.448700
$616$	0	0
$617$	23.2144	0.934576	0.467288	−	0.884105i	$-0.345231\pi$
$617$	23.2144	0.934576	0.467288	+	0.884105i	$0.345231\pi$
$618$	0	0
$619$	47.5125	1.90969	0.954845	−	0.297105i	$-0.0960211\pi$
$619$	47.5125	1.90969	0.954845	+	0.297105i	$0.0960211\pi$
$620$	0	0
$621$	11.8260	0.474562
$622$	0	0
$623$	7.51396	0.301041
$624$	0	0
$625$	−2.83299	−0.113320
$626$	0	0
$627$	2.86341	0.114354
$628$	0	0
$629$	8.45963	0.337308
$630$	0	0
$631$	12.3921	0.493321	0.246661	−	0.969102i	$-0.420667\pi$
$631$	12.3921	0.493321	0.246661	+	0.969102i	$0.420667\pi$
$632$	0	0
$633$	−27.5698	−1.09580
$634$	0	0
$635$	8.77643	0.348282
$636$	0	0
$637$	−0.715853	−0.0283631
$638$	0	0
$639$	−28.9193	−1.14403
$640$	0	0
$641$	−20.2298	−0.799030	−0.399515	−	0.916727i	$-0.630821\pi$
$641$	−20.2298	−0.799030	−0.399515	+	0.916727i	$0.630821\pi$
$642$	0	0
$643$	−26.5466	−1.04690	−0.523448	−	0.852058i	$-0.675355\pi$
$643$	−26.5466	−1.04690	−0.523448	+	0.852058i	$0.675355\pi$
$644$	0	0
$645$	−7.22357	−0.284428
$646$	0	0
$647$	−31.0668	−1.22136	−0.610681	−	0.791876i	$-0.709104\pi$
$647$	−31.0668	−1.22136	−0.610681	+	0.791876i	$0.709104\pi$
$648$	0	0
$649$	10.5683	0.414842
$650$	0	0
$651$	7.53341	0.295258
$652$	0	0
$653$	2.36489	0.0925452	0.0462726	−	0.998929i	$-0.485266\pi$
$653$	2.36489	0.0925452	0.0462726	+	0.998929i	$0.485266\pi$
$654$	0	0
$655$	−27.7049	−1.08252
$656$	0	0
$657$	−19.7415	−0.770191
$658$	0	0
$659$	22.6358	0.881767	0.440883	−	0.897564i	$-0.354665\pi$
$659$	22.6358	0.881767	0.440883	+	0.897564i	$0.354665\pi$
$660$	0	0
$661$	−17.9302	−0.697405	−0.348703	−	0.937233i	$-0.613378\pi$
$661$	−17.9302	−0.697405	−0.348703	+	0.937233i	$0.613378\pi$
$662$	0	0
$663$	0.798110	0.0309960
$664$	0	0
$665$	−1.89134	−0.0733429
$666$	0	0
$667$	−20.4596	−0.792200
$668$	0	0
$669$	−24.9193	−0.963434
$670$	0	0
$671$	1.77018	0.0683372
$672$	0	0
$673$	−23.0932	−0.890178	−0.445089	−	0.895486i	$-0.646828\pi$
$673$	−23.0932	−0.890178	−0.445089	+	0.895486i	$0.646828\pi$
$674$	0	0
$675$	15.0132	0.577858
$676$	0	0
$677$	36.2982	1.39505	0.697526	−	0.716560i	$-0.254284\pi$
$677$	36.2982	1.39505	0.697526	+	0.716560i	$0.254284\pi$
$678$	0	0
$679$	5.85021	0.224510
$680$	0	0
$681$	21.5165	0.824512
$682$	0	0
$683$	−12.1476	−0.464813	−0.232407	−	0.972619i	$-0.574660\pi$
$683$	−12.1476	−0.464813	−0.232407	+	0.972619i	$0.574660\pi$
$684$	0	0
$685$	−20.8873	−0.798064
$686$	0	0
$687$	−6.01249	−0.229391
$688$	0	0
$689$	4.75922	0.181312
$690$	0	0
$691$	−49.6957	−1.89051	−0.945257	−	0.326328i	$-0.894189\pi$
$691$	−49.6957	−1.89051	−0.945257	+	0.326328i	$0.894189\pi$
$692$	0	0
$693$	3.51396	0.133484
$694$	0	0
$695$	−0.277904	−0.0105415
$696$	0	0
$697$	6.77643	0.256676
$698$	0	0
$699$	−9.04041	−0.341940
$700$	0	0
$701$	−38.4068	−1.45061	−0.725303	−	0.688430i	$-0.758300\pi$
$701$	−38.4068	−1.45061	−0.725303	+	0.688430i	$0.758300\pi$
$702$	0	0
$703$	10.8634	0.409721
$704$	0	0
$705$	−3.66152	−0.137901
$706$	0	0
$707$	−18.3510	−0.690159
$708$	0	0
$709$	−23.9861	−0.900816	−0.450408	−	0.892823i	$-0.648721\pi$
$709$	−23.9861	−0.900816	−0.450408	+	0.892823i	$0.648721\pi$
$710$	0	0
$711$	18.7812	0.704348
$712$	0	0
$713$	−15.0668	−0.564257
$714$	0	0
$715$	2.10866	0.0788596
$716$	0	0
$717$	−24.9868	−0.933149
$718$	0	0
$719$	−19.6832	−0.734060	−0.367030	−	0.930209i	$-0.619625\pi$
$719$	−19.6832	−0.734060	−0.367030	+	0.930209i	$0.619625\pi$
$720$	0	0
$721$	−0.338479	−0.0126056
$722$	0	0
$723$	7.78963	0.289700
$724$	0	0
$725$	−25.9736	−0.964635
$726$	0	0
$727$	−9.97359	−0.369900	−0.184950	−	0.982748i	$-0.559212\pi$
$727$	−9.97359	−0.369900	−0.184950	+	0.982748i	$0.559212\pi$
$728$	0	0
$729$	18.8672	0.698784
$730$	0	0
$731$	4.39905	0.162705
$732$	0	0
$733$	48.0294	1.77401	0.887004	−	0.461762i	$-0.152783\pi$
$733$	48.0294	1.77401	0.887004	+	0.461762i	$0.152783\pi$
$734$	0	0
$735$	1.64207	0.0605688
$736$	0	0
$737$	9.80908	0.361322
$738$	0	0
$739$	18.7500	0.689731	0.344865	−	0.938652i	$-0.387925\pi$
$739$	18.7500	0.689731	0.344865	+	0.938652i	$0.387925\pi$
$740$	0	0
$741$	1.02489	0.0376503
$742$	0	0
$743$	48.5669	1.78175	0.890873	−	0.454253i	$-0.150094\pi$
$743$	48.5669	1.78175	0.890873	+	0.454253i	$0.150094\pi$
$744$	0	0
$745$	−19.2772	−0.706261
$746$	0	0
$747$	4.17700	0.152828
$748$	0	0
$749$	−0.568295	−0.0207650
$750$	0	0
$751$	22.1336	0.807668	0.403834	−	0.914832i	$-0.367677\pi$
$751$	22.1336	0.807668	0.403834	+	0.914832i	$0.367677\pi$
$752$	0	0
$753$	17.4317	0.635247
$754$	0	0
$755$	−8.08922	−0.294397
$756$	0	0
$757$	−47.7779	−1.73652	−0.868259	−	0.496110i	$-0.834761\pi$
$757$	−47.7779	−1.73652	−0.868259	+	0.496110i	$0.834761\pi$
$758$	0	0
$759$	4.97208	0.180475
$760$	0	0
$761$	29.6087	1.07331	0.536657	−	0.843800i	$-0.319687\pi$
$761$	29.6087	1.07331	0.536657	+	0.843800i	$0.319687\pi$
$762$	0	0
$763$	−4.71585	−0.170725
$764$	0	0
$765$	2.58774	0.0935600
$766$	0	0
$767$	3.78267	0.136584
$768$	0	0
$769$	4.14756	0.149565	0.0747824	−	0.997200i	$-0.476174\pi$
$769$	4.14756	0.149565	0.0747824	+	0.997200i	$0.476174\pi$
$770$	0	0
$771$	0.768668	0.0276829
$772$	0	0
$773$	−26.9317	−0.968668	−0.484334	−	0.874883i	$-0.660938\pi$
$773$	−26.9317	−0.968668	−0.484334	+	0.874883i	$0.660938\pi$
$774$	0	0
$775$	−19.1274	−0.687076
$776$	0	0
$777$	−9.43171	−0.338361
$778$	0	0
$779$	8.70193	0.311779
$780$	0	0
$781$	−32.9193	−1.17794
$782$	0	0
$783$	−48.6630	−1.73908
$784$	0	0
$785$	−16.2423	−0.579713
$786$	0	0
$787$	−36.1645	−1.28913	−0.644563	−	0.764551i	$-0.722961\pi$
$787$	−36.1645	−1.28913	−0.644563	+	0.764551i	$0.722961\pi$
$788$	0	0
$789$	19.4347	0.691895
$790$	0	0
$791$	−12.9457	−0.460295
$792$	0	0
$793$	0.633596	0.0224996
$794$	0	0
$795$	−10.9170	−0.387187
$796$	0	0
$797$	15.8260	0.560587	0.280293	−	0.959914i	$-0.409568\pi$
$797$	15.8260	0.560587	0.280293	+	0.959914i	$0.409568\pi$
$798$	0	0
$799$	2.22982	0.0788852
$800$	0	0
$801$	−13.2019	−0.466466
$802$	0	0
$803$	−22.4721	−0.793024
$804$	0	0
$805$	−3.28415	−0.115751
$806$	0	0
$807$	3.82603	0.134683
$808$	0	0
$809$	−48.7967	−1.71560	−0.857800	−	0.513984i	$-0.828169\pi$
$809$	−48.7967	−1.71560	−0.857800	+	0.513984i	$0.828169\pi$
$810$	0	0
$811$	30.9101	1.08540	0.542700	−	0.839927i	$-0.317402\pi$
$811$	30.9101	1.08540	0.542700	+	0.839927i	$0.317402\pi$
$812$	0	0
$813$	2.99848	0.105161
$814$	0	0
$815$	9.89134	0.346478
$816$	0	0
$817$	5.64903	0.197635
$818$	0	0
$819$	1.25774	0.0439490
$820$	0	0
$821$	3.61816	0.126275	0.0631373	−	0.998005i	$-0.479889\pi$
$821$	3.61816	0.126275	0.0631373	+	0.998005i	$0.479889\pi$
$822$	0	0
$823$	37.9736	1.32368	0.661838	−	0.749647i	$-0.269777\pi$
$823$	37.9736	1.32368	0.661838	+	0.749647i	$0.269777\pi$
$824$	0	0
$825$	6.31207	0.219758
$826$	0	0
$827$	−6.79811	−0.236393	−0.118197	−	0.992990i	$-0.537711\pi$
$827$	−6.79811	−0.236393	−0.118197	+	0.992990i	$0.537711\pi$
$828$	0	0
$829$	−25.3928	−0.881929	−0.440964	−	0.897525i	$-0.645363\pi$
$829$	−25.3928	−0.881929	−0.440964	+	0.897525i	$0.645363\pi$
$830$	0	0
$831$	−17.4611	−0.605720
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	17.3859	0.601662
$836$	0	0
$837$	−35.8363	−1.23868
$838$	0	0
$839$	9.67401	0.333984	0.166992	−	0.985958i	$-0.446595\pi$
$839$	9.67401	0.333984	0.166992	+	0.985958i	$0.446595\pi$
$840$	0	0
$841$	55.1895	1.90309
$842$	0	0
$843$	−6.38737	−0.219993
$844$	0	0
$845$	−18.3921	−0.632707
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	−19.2244	−0.659779
$850$	0	0
$851$	18.8634	0.646629
$852$	0	0
$853$	−35.2144	−1.20572	−0.602859	−	0.797848i	$-0.705972\pi$
$853$	−35.2144	−1.20572	−0.602859	+	0.797848i	$0.705972\pi$
$854$	0	0
$855$	3.32304	0.113646
$856$	0	0
$857$	37.4728	1.28005	0.640024	−	0.768355i	$-0.278924\pi$
$857$	37.4728	1.28005	0.640024	+	0.768355i	$0.278924\pi$
$858$	0	0
$859$	−40.7283	−1.38963	−0.694816	−	0.719187i	$-0.744514\pi$
$859$	−40.7283	−1.38963	−0.694816	+	0.719187i	$0.744514\pi$
$860$	0	0
$861$	−7.55509	−0.257477
$862$	0	0
$863$	−39.4978	−1.34452	−0.672260	−	0.740315i	$-0.734677\pi$
$863$	−39.4978	−1.34452	−0.672260	+	0.740315i	$0.734677\pi$
$864$	0	0
$865$	−19.6546	−0.668275
$866$	0	0
$867$	1.11491	0.0378642
$868$	0	0
$869$	21.3789	0.725229
$870$	0	0
$871$	3.51093	0.118963
$872$	0	0
$873$	−10.2787	−0.347881
$874$	0	0
$875$	−11.5334	−0.389900
$876$	0	0
$877$	−14.6461	−0.494563	−0.247282	−	0.968944i	$-0.579537\pi$
$877$	−14.6461	−0.494563	−0.247282	+	0.968944i	$0.579537\pi$
$878$	0	0
$879$	27.6910	0.933993
$880$	0	0
$881$	41.2555	1.38993	0.694967	−	0.719042i	$-0.255419\pi$
$881$	41.2555	1.38993	0.694967	+	0.719042i	$0.255419\pi$
$882$	0	0
$883$	30.3362	1.02090	0.510448	−	0.859909i	$-0.329480\pi$
$883$	30.3362	1.02090	0.510448	+	0.859909i	$0.329480\pi$
$884$	0	0
$885$	−8.67696	−0.291673
$886$	0	0
$887$	24.1887	0.812177	0.406088	−	0.913834i	$-0.366893\pi$
$887$	24.1887	0.812177	0.406088	+	0.913834i	$0.366893\pi$
$888$	0	0
$889$	5.95887	0.199854
$890$	0	0
$891$	1.28415	0.0430206
$892$	0	0
$893$	2.86341	0.0958204
$894$	0	0
$895$	2.01945	0.0675027
$896$	0	0
$897$	1.77964	0.0594203
$898$	0	0
$899$	61.9986	2.06777
$900$	0	0
$901$	6.64832	0.221488
$902$	0	0
$903$	−4.90454	−0.163213
$904$	0	0
$905$	−10.5110	−0.349398
$906$	0	0
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	0	0
$909$	32.2423	1.06941
$910$	0	0
$911$	51.4736	1.70540	0.852699	−	0.522403i	$-0.174964\pi$
$911$	51.4736	1.70540	0.852699	+	0.522403i	$0.174964\pi$
$912$	0	0
$913$	4.75475	0.157359
$914$	0	0
$915$	−1.45339	−0.0480475
$916$	0	0
$917$	−18.8106	−0.621181
$918$	0	0
$919$	−47.9644	−1.58220	−0.791100	−	0.611687i	$-0.790491\pi$
$919$	−47.9644	−1.58220	−0.791100	+	0.611687i	$0.790491\pi$
$920$	0	0
$921$	27.5264	0.907027
$922$	0	0
$923$	−11.7827	−0.387831
$924$	0	0
$925$	23.9472	0.787379
$926$	0	0
$927$	0.594702	0.0195326
$928$	0	0
$929$	24.1189	0.791316	0.395658	−	0.918398i	$-0.370517\pi$
$929$	24.1189	0.791316	0.395658	+	0.918398i	$0.370517\pi$
$930$	0	0
$931$	−1.28415	−0.0420862
$932$	0	0
$933$	13.6810	0.447895
$934$	0	0
$935$	2.94567	0.0963336
$936$	0	0
$937$	−28.7967	−0.940746	−0.470373	−	0.882468i	$-0.655881\pi$
$937$	−28.7967	−0.940746	−0.470373	+	0.882468i	$0.655881\pi$
$938$	0	0
$939$	9.89830	0.323019
$940$	0	0
$941$	−10.0745	−0.328419	−0.164210	−	0.986425i	$-0.552507\pi$
$941$	−10.0745	−0.328419	−0.164210	+	0.986425i	$0.552507\pi$
$942$	0	0
$943$	15.1102	0.492055
$944$	0	0
$945$	−7.81131	−0.254102
$946$	0	0
$947$	30.8664	1.00302	0.501512	−	0.865150i	$-0.332777\pi$
$947$	30.8664	1.00302	0.501512	+	0.865150i	$0.332777\pi$
$948$	0	0
$949$	−8.04336	−0.261099
$950$	0	0
$951$	27.3619	0.887272
$952$	0	0
$953$	−0.0844915	−0.00273695	−0.00136847	−	0.999999i	$-0.500436\pi$
$953$	−0.0844915	−0.00273695	−0.00136847	+	0.999999i	$0.500436\pi$
$954$	0	0
$955$	9.57454	0.309825
$956$	0	0
$957$	−20.4596	−0.661366
$958$	0	0
$959$	−14.1817	−0.457952
$960$	0	0
$961$	14.6568	0.472800
$962$	0	0
$963$	0.998483	0.0321757
$964$	0	0
$965$	24.0823	0.775235
$966$	0	0
$967$	−14.9240	−0.479923	−0.239962	−	0.970782i	$-0.577135\pi$
$967$	−14.9240	−0.479923	−0.239962	+	0.970782i	$0.577135\pi$
$968$	0	0
$969$	1.43171	0.0459930
$970$	0	0
$971$	−16.8106	−0.539478	−0.269739	−	0.962934i	$-0.586937\pi$
$971$	−16.8106	−0.539478	−0.269739	+	0.962934i	$0.586937\pi$
$972$	0	0
$973$	−0.188687	−0.00604902
$974$	0	0
$975$	2.25926	0.0723541
$976$	0	0
$977$	33.3767	1.06781	0.533907	−	0.845543i	$-0.320723\pi$
$977$	33.3767	1.06781	0.533907	+	0.845543i	$0.320723\pi$
$978$	0	0
$979$	−15.0279	−0.480295
$980$	0	0
$981$	8.28566	0.264541
$982$	0	0
$983$	18.2974	0.583595	0.291797	−	0.956480i	$-0.405747\pi$
$983$	18.2974	0.583595	0.291797	+	0.956480i	$0.405747\pi$
$984$	0	0
$985$	30.1336	0.960138
$986$	0	0
$987$	−2.48604	−0.0791315
$988$	0	0
$989$	9.80908	0.311911
$990$	0	0
$991$	−0.958154	−0.0304368	−0.0152184	−	0.999884i	$-0.504844\pi$
$991$	−0.958154	−0.0304368	−0.0152184	+	0.999884i	$0.504844\pi$
$992$	0	0
$993$	8.33152	0.264393
$994$	0	0
$995$	26.0628	0.826247
$996$	0	0
$997$	56.5188	1.78997	0.894984	−	0.446099i	$-0.147187\pi$
$997$	56.5188	1.78997	0.894984	+	0.446099i	$0.147187\pi$
$998$	0	0
$999$	44.8664	1.41951

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.ba.1.3		3
4.3	odd	2		7616.2.a.bg.1.1		3
8.3	odd	2		952.2.a.d.1.3	✓	3
8.5	even	2		1904.2.a.p.1.1		3
24.11	even	2		8568.2.a.z.1.3		3
56.27	even	2		6664.2.a.l.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.d.1.3	✓	3	8.3	odd	2
1904.2.a.p.1.1		3	8.5	even	2
6664.2.a.l.1.1		3	56.27	even	2
7616.2.a.ba.1.3		3	1.1	even	1	trivial
7616.2.a.bg.1.1		3	4.3	odd	2
8568.2.a.z.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)

\(f(q)\)	\(=\)	\(q+1.11491 q^{3} +1.47283 q^{5} +1.00000 q^{7} -1.75698 q^{9} -2.00000 q^{11} -0.715853 q^{13} +1.64207 q^{15} -1.00000 q^{17} -1.28415 q^{19} +1.11491 q^{21} -2.22982 q^{23} -2.83076 q^{25} -5.30359 q^{27} +9.17548 q^{29} +6.75698 q^{31} -2.22982 q^{33} +1.47283 q^{35} -8.45963 q^{37} -0.798110 q^{39} -6.77643 q^{41} -4.39905 q^{43} -2.58774 q^{45} -2.22982 q^{47} +1.00000 q^{49} -1.11491 q^{51} -6.64832 q^{53} -2.94567 q^{55} -1.43171 q^{57} -5.28415 q^{59} -0.885092 q^{61} -1.75698 q^{63} -1.05433 q^{65} -4.90454 q^{67} -2.48604 q^{69} +16.4596 q^{71} +11.2361 q^{73} -3.15604 q^{75} -2.00000 q^{77} -10.6894 q^{79} -0.642074 q^{81} -2.37737 q^{83} -1.47283 q^{85} +10.2298 q^{87} +7.51396 q^{89} -0.715853 q^{91} +7.53341 q^{93} -1.89134 q^{95} +5.85021 q^{97} +3.51396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - q^{5} + 3 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{13} + 4 q^{15} - 3 q^{17} - 2 q^{19} - 3 q^{21} + 6 q^{23} - 4 q^{25} - 6 q^{27} + 4 q^{29} + 13 q^{31} + 6 q^{33} - q^{35} + 14 q^{39} - 5 q^{41}+ \cdots - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - q^5 + 3 * q^7 + 2 * q^9 - 6 * q^11 - 4 * q^13 + 4 * q^15 - 3 * q^17 - 2 * q^19 - 3 * q^21 + 6 * q^23 - 4 * q^25 - 6 * q^27 + 4 * q^29 + 13 * q^31 + 6 * q^33 - q^35 + 14 * q^39 - 5 * q^41 - 5 * q^43 + 4 * q^45 + 6 * q^47 + 3 * q^49 + 3 * q^51 + 9 * q^53 + 2 * q^55 - 8 * q^57 - 14 * q^59 - 9 * q^61 + 2 * q^63 - 14 * q^65 - 13 * q^67 - 22 * q^69 + 24 * q^71 - 7 * q^73 + 6 * q^75 - 6 * q^77 + 6 * q^79 - q^81 + q^85 + 18 * q^87 + 8 * q^89 - 4 * q^91 + 16 * q^95 + 5 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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