LMFDB - Embedded newform 2760.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,2,Mod(1,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2760.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2760.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$22.0387109579$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	2760.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +1.39821 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +1.39821 q^{7} +1.00000 q^{9} +6.64681 q^{13} +1.00000 q^{15} +3.39821 q^{17} +1.39821 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.601793 q^{29} -6.04502 q^{31} +1.39821 q^{35} -8.69182 q^{37} +6.64681 q^{39} +5.24860 q^{41} -7.44322 q^{43} +1.00000 q^{45} +2.79641 q^{47} -5.04502 q^{49} +3.39821 q^{51} +9.24860 q^{53} -3.24860 q^{59} -9.44322 q^{61} +1.39821 q^{63} +6.64681 q^{65} +10.6918 q^{67} +1.00000 q^{69} -7.89541 q^{71} +4.79641 q^{73} +1.00000 q^{75} +5.85039 q^{79} +1.00000 q^{81} +11.8954 q^{83} +3.39821 q^{85} +0.601793 q^{87} -6.09003 q^{89} +9.29362 q^{91} -6.04502 q^{93} +9.44322 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 + q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} + q^{7} + 3 q^{9} + 4 q^{13} + 3 q^{15} + 7 q^{17} + q^{21} + 3 q^{23} + 3 q^{25} + 3 q^{27} + 5 q^{29} + q^{31} + q^{35} + 9 q^{37} + 4 q^{39} + 3 q^{41} + 3 q^{45} + 2 q^{47} + 4 q^{49} + 7 q^{51} + 15 q^{53} + 3 q^{59} - 6 q^{61} + q^{63} + 4 q^{65} - 3 q^{67} + 3 q^{69} + 5 q^{71} + 8 q^{73} + 3 q^{75} + 8 q^{79} + 3 q^{81} + 7 q^{83} + 7 q^{85} + 5 q^{87} + 20 q^{89} - 4 q^{91} + q^{93} + 6 q^{97}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 + q^7 + 3 * q^9 + 4 * q^13 + 3 * q^15 + 7 * q^17 + q^21 + 3 * q^23 + 3 * q^25 + 3 * q^27 + 5 * q^29 + q^31 + q^35 + 9 * q^37 + 4 * q^39 + 3 * q^41 + 3 * q^45 + 2 * q^47 + 4 * q^49 + 7 * q^51 + 15 * q^53 + 3 * q^59 - 6 * q^61 + q^63 + 4 * q^65 - 3 * q^67 + 3 * q^69 + 5 * q^71 + 8 * q^73 + 3 * q^75 + 8 * q^79 + 3 * q^81 + 7 * q^83 + 7 * q^85 + 5 * q^87 + 20 * q^89 - 4 * q^91 + q^93 + 6 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.39821	0.528473	0.264236	−	0.964458i	$-0.414880\pi$
$7$	1.39821	0.528473	0.264236	+	0.964458i	$0.414880\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	6.64681	1.84349	0.921747	−	0.387793i	$-0.126762\pi$
$13$	6.64681	1.84349	0.921747	+	0.387793i	$0.126762\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	3.39821	0.824186	0.412093	−	0.911142i	$-0.364798\pi$
$17$	3.39821	0.824186	0.412093	+	0.911142i	$0.364798\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	1.39821	0.305114
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.601793	0.111750	0.0558750	−	0.998438i	$-0.482205\pi$
$29$	0.601793	0.111750	0.0558750	+	0.998438i	$0.482205\pi$
$30$	0	0
$31$	−6.04502	−1.08572	−0.542858	−	0.839824i	$-0.682658\pi$
$31$	−6.04502	−1.08572	−0.542858	+	0.839824i	$0.682658\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.39821	0.236340
$36$	0	0
$37$	−8.69182	−1.42893	−0.714464	−	0.699673i	$-0.753329\pi$
$37$	−8.69182	−1.42893	−0.714464	+	0.699673i	$0.753329\pi$
$38$	0	0
$39$	6.64681	1.06434
$40$	0	0
$41$	5.24860	0.819694	0.409847	−	0.912154i	$-0.365582\pi$
$41$	5.24860	0.819694	0.409847	+	0.912154i	$0.365582\pi$
$42$	0	0
$43$	−7.44322	−1.13508	−0.567540	−	0.823346i	$-0.692105\pi$
$43$	−7.44322	−1.13508	−0.567540	+	0.823346i	$0.692105\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	2.79641	0.407899	0.203950	−	0.978981i	$-0.434622\pi$
$47$	2.79641	0.407899	0.203950	+	0.978981i	$0.434622\pi$
$48$	0	0
$49$	−5.04502	−0.720717
$50$	0	0
$51$	3.39821	0.475844
$52$	0	0
$53$	9.24860	1.27039	0.635197	−	0.772351i	$-0.280919\pi$
$53$	9.24860	1.27039	0.635197	+	0.772351i	$0.280919\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.24860	−0.422932	−0.211466	−	0.977385i	$-0.567824\pi$
$59$	−3.24860	−0.422932	−0.211466	+	0.977385i	$0.567824\pi$
$60$	0	0
$61$	−9.44322	−1.20908	−0.604540	−	0.796574i	$-0.706643\pi$
$61$	−9.44322	−1.20908	−0.604540	+	0.796574i	$0.706643\pi$
$62$	0	0
$63$	1.39821	0.176158
$64$	0	0
$65$	6.64681	0.824435
$66$	0	0
$67$	10.6918	1.30621	0.653107	−	0.757266i	$-0.273465\pi$
$67$	10.6918	1.30621	0.653107	+	0.757266i	$0.273465\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−7.89541	−0.937013	−0.468506	−	0.883460i	$-0.655208\pi$
$71$	−7.89541	−0.937013	−0.468506	+	0.883460i	$0.655208\pi$
$72$	0	0
$73$	4.79641	0.561378	0.280689	−	0.959799i	$-0.409437\pi$
$73$	4.79641	0.561378	0.280689	+	0.959799i	$0.409437\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.85039	0.658221	0.329110	−	0.944291i	$-0.393251\pi$
$79$	5.85039	0.658221	0.329110	+	0.944291i	$0.393251\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.8954	1.30569	0.652845	−	0.757491i	$-0.273575\pi$
$83$	11.8954	1.30569	0.652845	+	0.757491i	$0.273575\pi$
$84$	0	0
$85$	3.39821	0.368587
$86$	0	0
$87$	0.601793	0.0645189
$88$	0	0
$89$	−6.09003	−0.645542	−0.322771	−	0.946477i	$-0.604614\pi$
$89$	−6.09003	−0.645542	−0.322771	+	0.946477i	$0.604614\pi$
$90$	0	0
$91$	9.29362	0.974236
$92$	0	0
$93$	−6.04502	−0.626839
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.44322	0.958814	0.479407	−	0.877593i	$-0.340852\pi$
$97$	9.44322	0.958814	0.479407	+	0.877593i	$0.340852\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.601793	0.0598806	0.0299403	−	0.999552i	$-0.490468\pi$
$101$	0.601793	0.0598806	0.0299403	+	0.999552i	$0.490468\pi$
$102$	0	0
$103$	1.85039	0.182325	0.0911624	−	0.995836i	$-0.470942\pi$
$103$	1.85039	0.182325	0.0911624	+	0.995836i	$0.470942\pi$
$104$	0	0
$105$	1.39821	0.136451
$106$	0	0
$107$	−5.39821	−0.521864	−0.260932	−	0.965357i	$-0.584030\pi$
$107$	−5.39821	−0.521864	−0.260932	+	0.965357i	$0.584030\pi$
$108$	0	0
$109$	19.9404	1.90995	0.954973	−	0.296692i	$-0.0958835\pi$
$109$	19.9404	1.90995	0.954973	+	0.296692i	$0.0958835\pi$
$110$	0	0
$111$	−8.69182	−0.824991
$112$	0	0
$113$	−5.89541	−0.554593	−0.277297	−	0.960784i	$-0.589439\pi$
$113$	−5.89541	−0.554593	−0.277297	+	0.960784i	$0.589439\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	6.64681	0.614498
$118$	0	0
$119$	4.75140	0.435560
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	5.24860	0.473250
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.64681	−0.767280	−0.383640	−	0.923483i	$-0.625330\pi$
$127$	−8.64681	−0.767280	−0.383640	+	0.923483i	$0.625330\pi$
$128$	0	0
$129$	−7.44322	−0.655339
$130$	0	0
$131$	9.29362	0.811987	0.405994	−	0.913876i	$-0.366926\pi$
$131$	9.29362	0.811987	0.405994	+	0.913876i	$0.366926\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	9.44322	0.806789	0.403395	−	0.915026i	$-0.367830\pi$
$137$	9.44322	0.806789	0.403395	+	0.915026i	$0.367830\pi$
$138$	0	0
$139$	−12.8414	−1.08920	−0.544598	−	0.838697i	$-0.683318\pi$
$139$	−12.8414	−1.08920	−0.544598	+	0.838697i	$0.683318\pi$
$140$	0	0
$141$	2.79641	0.235501
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0.601793	0.0499762
$146$	0	0
$147$	−5.04502	−0.416106
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	3.39821	0.274729
$154$	0	0
$155$	−6.04502	−0.485547
$156$	0	0
$157$	−2.19462	−0.175150	−0.0875750	−	0.996158i	$-0.527912\pi$
$157$	−2.19462	−0.175150	−0.0875750	+	0.996158i	$0.527912\pi$
$158$	0	0
$159$	9.24860	0.733462
$160$	0	0
$161$	1.39821	0.110194
$162$	0	0
$163$	−18.4972	−1.44881	−0.724406	−	0.689373i	$-0.757886\pi$
$163$	−18.4972	−1.44881	−0.724406	+	0.689373i	$0.757886\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.20359	−0.402666	−0.201333	−	0.979523i	$-0.564527\pi$
$167$	−5.20359	−0.402666	−0.201333	+	0.979523i	$0.564527\pi$
$168$	0	0
$169$	31.1801	2.39847
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.20359	0.547678	0.273839	−	0.961775i	$-0.411706\pi$
$173$	7.20359	0.547678	0.273839	+	0.961775i	$0.411706\pi$
$174$	0	0
$175$	1.39821	0.105695
$176$	0	0
$177$	−3.24860	−0.244180
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−1.05398	−0.0783416	−0.0391708	−	0.999233i	$-0.512472\pi$
$181$	−1.05398	−0.0783416	−0.0391708	+	0.999233i	$0.512472\pi$
$182$	0	0
$183$	−9.44322	−0.698063
$184$	0	0
$185$	−8.69182	−0.639036
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.39821	0.101705
$190$	0	0
$191$	6.49720	0.470121	0.235061	−	0.971981i	$-0.424471\pi$
$191$	6.49720	0.470121	0.235061	+	0.971981i	$0.424471\pi$
$192$	0	0
$193$	−12.4972	−0.899568	−0.449784	−	0.893137i	$-0.648499\pi$
$193$	−12.4972	−0.899568	−0.449784	+	0.893137i	$0.648499\pi$
$194$	0	0
$195$	6.64681	0.475988
$196$	0	0
$197$	−1.70079	−0.121176	−0.0605880	−	0.998163i	$-0.519298\pi$
$197$	−1.70079	−0.121176	−0.0605880	+	0.998163i	$0.519298\pi$
$198$	0	0
$199$	8.64681	0.612956	0.306478	−	0.951878i	$-0.400849\pi$
$199$	8.64681	0.612956	0.306478	+	0.951878i	$0.400849\pi$
$200$	0	0
$201$	10.6918	0.754143
$202$	0	0
$203$	0.841431	0.0590569
$204$	0	0
$205$	5.24860	0.366578
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−4.45219	−0.306501	−0.153251	−	0.988187i	$-0.548974\pi$
$211$	−4.45219	−0.306501	−0.153251	+	0.988187i	$0.548974\pi$
$212$	0	0
$213$	−7.89541	−0.540985
$214$	0	0
$215$	−7.44322	−0.507624
$216$	0	0
$217$	−8.45219	−0.573772
$218$	0	0
$219$	4.79641	0.324112
$220$	0	0
$221$	22.5872	1.51938
$222$	0	0
$223$	−25.9404	−1.73710	−0.868550	−	0.495602i	$-0.834947\pi$
$223$	−25.9404	−1.73710	−0.868550	+	0.495602i	$0.834947\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−17.0361	−1.13072	−0.565361	−	0.824843i	$-0.691263\pi$
$227$	−17.0361	−1.13072	−0.565361	+	0.824843i	$0.691263\pi$
$228$	0	0
$229$	−22.8269	−1.50844	−0.754221	−	0.656621i	$-0.771985\pi$
$229$	−22.8269	−1.50844	−0.754221	+	0.656621i	$0.771985\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0900	1.18512	0.592559	−	0.805527i	$-0.298118\pi$
$233$	18.0900	1.18512	0.592559	+	0.805527i	$0.298118\pi$
$234$	0	0
$235$	2.79641	0.182418
$236$	0	0
$237$	5.85039	0.380024
$238$	0	0
$239$	27.0811	1.75173	0.875864	−	0.482557i	$-0.160292\pi$
$239$	27.0811	1.75173	0.875864	+	0.482557i	$0.160292\pi$
$240$	0	0
$241$	6.09003	0.392293	0.196147	−	0.980575i	$-0.437157\pi$
$241$	6.09003	0.392293	0.196147	+	0.980575i	$0.437157\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−5.04502	−0.322314
$246$	0	0
$247$	0	0
$248$	0	0
$249$	11.8954	0.753841
$250$	0	0
$251$	−26.9765	−1.70274	−0.851370	−	0.524565i	$-0.824228\pi$
$251$	−26.9765	−1.70274	−0.851370	+	0.524565i	$0.824228\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	3.39821	0.212804
$256$	0	0
$257$	11.5928	0.723141	0.361570	−	0.932345i	$-0.382241\pi$
$257$	11.5928	0.723141	0.361570	+	0.932345i	$0.382241\pi$
$258$	0	0
$259$	−12.1530	−0.755149
$260$	0	0
$261$	0.601793	0.0372500
$262$	0	0
$263$	−14.0450	−0.866053	−0.433026	−	0.901381i	$-0.642554\pi$
$263$	−14.0450	−0.866053	−0.433026	+	0.901381i	$0.642554\pi$
$264$	0	0
$265$	9.24860	0.568137
$266$	0	0
$267$	−6.09003	−0.372704
$268$	0	0
$269$	13.8054	0.841729	0.420864	−	0.907124i	$-0.361727\pi$
$269$	13.8054	0.841729	0.420864	+	0.907124i	$0.361727\pi$
$270$	0	0
$271$	7.33863	0.445790	0.222895	−	0.974842i	$-0.428449\pi$
$271$	7.33863	0.445790	0.222895	+	0.974842i	$0.428449\pi$
$272$	0	0
$273$	9.29362	0.562475
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.74244	−0.104693	−0.0523464	−	0.998629i	$-0.516670\pi$
$277$	−1.74244	−0.104693	−0.0523464	+	0.998629i	$0.516670\pi$
$278$	0	0
$279$	−6.04502	−0.361906
$280$	0	0
$281$	−11.6829	−0.696941	−0.348471	−	0.937320i	$-0.613299\pi$
$281$	−11.6829	−0.696941	−0.348471	+	0.937320i	$0.613299\pi$
$282$	0	0
$283$	14.7819	0.878690	0.439345	−	0.898318i	$-0.355211\pi$
$283$	14.7819	0.878690	0.439345	+	0.898318i	$0.355211\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.33863	0.433186
$288$	0	0
$289$	−5.45219	−0.320717
$290$	0	0
$291$	9.44322	0.553572
$292$	0	0
$293$	8.34423	0.487475	0.243738	−	0.969841i	$-0.421626\pi$
$293$	8.34423	0.487475	0.243738	+	0.969841i	$0.421626\pi$
$294$	0	0
$295$	−3.24860	−0.189141
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.64681	0.384395
$300$	0	0
$301$	−10.4072	−0.599859
$302$	0	0
$303$	0.601793	0.0345721
$304$	0	0
$305$	−9.44322	−0.540717
$306$	0	0
$307$	5.29362	0.302123	0.151061	−	0.988524i	$-0.451731\pi$
$307$	5.29362	0.302123	0.151061	+	0.988524i	$0.451731\pi$
$308$	0	0
$309$	1.85039	0.105265
$310$	0	0
$311$	−13.0361	−0.739207	−0.369603	−	0.929190i	$-0.620506\pi$
$311$	−13.0361	−0.739207	−0.369603	+	0.929190i	$0.620506\pi$
$312$	0	0
$313$	−2.19462	−0.124047	−0.0620237	−	0.998075i	$-0.519755\pi$
$313$	−2.19462	−0.124047	−0.0620237	+	0.998075i	$0.519755\pi$
$314$	0	0
$315$	1.39821	0.0787801
$316$	0	0
$317$	22.9944	1.29149	0.645747	−	0.763551i	$-0.276546\pi$
$317$	22.9944	1.29149	0.645747	+	0.763551i	$0.276546\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−5.39821	−0.301299
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6.64681	0.368699
$326$	0	0
$327$	19.9404	1.10271
$328$	0	0
$329$	3.90997	0.215564
$330$	0	0
$331$	−15.6378	−0.859534	−0.429767	−	0.902940i	$-0.641404\pi$
$331$	−15.6378	−0.859534	−0.429767	+	0.902940i	$0.641404\pi$
$332$	0	0
$333$	−8.69182	−0.476309
$334$	0	0
$335$	10.6918	0.584157
$336$	0	0
$337$	−9.74244	−0.530704	−0.265352	−	0.964152i	$-0.585488\pi$
$337$	−9.74244	−0.530704	−0.265352	+	0.964152i	$0.585488\pi$
$338$	0	0
$339$	−5.89541	−0.320195
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.8414	−0.909352
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−2.45219	−0.131263	−0.0656313	−	0.997844i	$-0.520906\pi$
$349$	−2.45219	−0.131263	−0.0656313	+	0.997844i	$0.520906\pi$
$350$	0	0
$351$	6.64681	0.354780
$352$	0	0
$353$	9.70079	0.516321	0.258160	−	0.966102i	$-0.416884\pi$
$353$	9.70079	0.516321	0.258160	+	0.966102i	$0.416884\pi$
$354$	0	0
$355$	−7.89541	−0.419045
$356$	0	0
$357$	4.75140	0.251471
$358$	0	0
$359$	22.4972	1.18736	0.593678	−	0.804702i	$-0.297675\pi$
$359$	22.4972	1.18736	0.593678	+	0.804702i	$0.297675\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−11.0000	−0.577350
$364$	0	0
$365$	4.79641	0.251056
$366$	0	0
$367$	4.19462	0.218958	0.109479	−	0.993989i	$-0.465082\pi$
$367$	4.19462	0.218958	0.109479	+	0.993989i	$0.465082\pi$
$368$	0	0
$369$	5.24860	0.273231
$370$	0	0
$371$	12.9315	0.671368
$372$	0	0
$373$	28.0305	1.45136	0.725681	−	0.688031i	$-0.241525\pi$
$373$	28.0305	1.45136	0.725681	+	0.688031i	$0.241525\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	−30.8864	−1.58653	−0.793265	−	0.608876i	$-0.791621\pi$
$379$	−30.8864	−1.58653	−0.793265	+	0.608876i	$0.791621\pi$
$380$	0	0
$381$	−8.64681	−0.442989
$382$	0	0
$383$	−14.0450	−0.717667	−0.358833	−	0.933402i	$-0.616825\pi$
$383$	−14.0450	−0.717667	−0.358833	+	0.933402i	$0.616825\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.44322	−0.378360
$388$	0	0
$389$	23.2936	1.18103	0.590517	−	0.807025i	$-0.298924\pi$
$389$	23.2936	1.18103	0.590517	+	0.807025i	$0.298924\pi$
$390$	0	0
$391$	3.39821	0.171855
$392$	0	0
$393$	9.29362	0.468801
$394$	0	0
$395$	5.85039	0.294365
$396$	0	0
$397$	−19.9404	−1.00078	−0.500391	−	0.865800i	$-0.666810\pi$
$397$	−19.9404	−1.00078	−0.500391	+	0.865800i	$0.666810\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−32.2880	−1.61239	−0.806193	−	0.591652i	$-0.798476\pi$
$401$	−32.2880	−1.61239	−0.806193	+	0.591652i	$0.798476\pi$
$402$	0	0
$403$	−40.1801	−2.00151
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−10.5422	−0.521279	−0.260640	−	0.965436i	$-0.583933\pi$
$409$	−10.5422	−0.521279	−0.260640	+	0.965436i	$0.583933\pi$
$410$	0	0
$411$	9.44322	0.465800
$412$	0	0
$413$	−4.54222	−0.223508
$414$	0	0
$415$	11.8954	0.583923
$416$	0	0
$417$	−12.8414	−0.628848
$418$	0	0
$419$	−3.70079	−0.180795	−0.0903976	−	0.995906i	$-0.528814\pi$
$419$	−3.70079	−0.180795	−0.0903976	+	0.995906i	$0.528814\pi$
$420$	0	0
$421$	−18.3476	−0.894207	−0.447104	−	0.894482i	$-0.647544\pi$
$421$	−18.3476	−0.894207	−0.447104	+	0.894482i	$0.647544\pi$
$422$	0	0
$423$	2.79641	0.135966
$424$	0	0
$425$	3.39821	0.164837
$426$	0	0
$427$	−13.2036	−0.638966
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.8864	0.717055	0.358527	−	0.933519i	$-0.383279\pi$
$431$	14.8864	0.717055	0.358527	+	0.933519i	$0.383279\pi$
$432$	0	0
$433$	−31.5783	−1.51755	−0.758777	−	0.651350i	$-0.774203\pi$
$433$	−31.5783	−1.51755	−0.758777	+	0.651350i	$0.774203\pi$
$434$	0	0
$435$	0.601793	0.0288537
$436$	0	0
$437$	0	0
$438$	0	0
$439$	33.4737	1.59761	0.798806	−	0.601589i	$-0.205465\pi$
$439$	33.4737	1.59761	0.798806	+	0.601589i	$0.205465\pi$
$440$	0	0
$441$	−5.04502	−0.240239
$442$	0	0
$443$	−22.7964	−1.08309	−0.541545	−	0.840672i	$-0.682160\pi$
$443$	−22.7964	−1.08309	−0.541545	+	0.840672i	$0.682160\pi$
$444$	0	0
$445$	−6.09003	−0.288695
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	33.3386	1.57335	0.786674	−	0.617369i	$-0.211801\pi$
$449$	33.3386	1.57335	0.786674	+	0.617369i	$0.211801\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9.29362	0.435691
$456$	0	0
$457$	21.9854	1.02844	0.514218	−	0.857660i	$-0.328082\pi$
$457$	21.9854	1.02844	0.514218	+	0.857660i	$0.328082\pi$
$458$	0	0
$459$	3.39821	0.158615
$460$	0	0
$461$	−27.0361	−1.25919	−0.629597	−	0.776922i	$-0.716780\pi$
$461$	−27.0361	−1.25919	−0.629597	+	0.776922i	$0.716780\pi$
$462$	0	0
$463$	−18.1496	−0.843484	−0.421742	−	0.906716i	$-0.638581\pi$
$463$	−18.1496	−0.843484	−0.421742	+	0.906716i	$0.638581\pi$
$464$	0	0
$465$	−6.04502	−0.280331
$466$	0	0
$467$	−39.9854	−1.85031	−0.925153	−	0.379595i	$-0.876063\pi$
$467$	−39.9854	−1.85031	−0.925153	+	0.379595i	$0.876063\pi$
$468$	0	0
$469$	14.9494	0.690299
$470$	0	0
$471$	−2.19462	−0.101123
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.24860	0.423464
$478$	0	0
$479$	7.09563	0.324207	0.162104	−	0.986774i	$-0.448172\pi$
$479$	7.09563	0.324207	0.162104	+	0.986774i	$0.448172\pi$
$480$	0	0
$481$	−57.7729	−2.63422
$482$	0	0
$483$	1.39821	0.0636206
$484$	0	0
$485$	9.44322	0.428795
$486$	0	0
$487$	−24.8269	−1.12501	−0.562506	−	0.826793i	$-0.690163\pi$
$487$	−24.8269	−1.12501	−0.562506	+	0.826793i	$0.690163\pi$
$488$	0	0
$489$	−18.4972	−0.836472
$490$	0	0
$491$	−23.3386	−1.05326	−0.526629	−	0.850095i	$-0.676544\pi$
$491$	−23.3386	−1.05326	−0.526629	+	0.850095i	$0.676544\pi$
$492$	0	0
$493$	2.04502	0.0921029
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−11.0394	−0.495186
$498$	0	0
$499$	32.3330	1.44743	0.723713	−	0.690101i	$-0.242434\pi$
$499$	32.3330	1.44743	0.723713	+	0.690101i	$0.242434\pi$
$500$	0	0
$501$	−5.20359	−0.232479
$502$	0	0
$503$	−4.75140	−0.211854	−0.105927	−	0.994374i	$-0.533781\pi$
$503$	−4.75140	−0.211854	−0.105927	+	0.994374i	$0.533781\pi$
$504$	0	0
$505$	0.601793	0.0267794
$506$	0	0
$507$	31.1801	1.38476
$508$	0	0
$509$	16.3297	0.723800	0.361900	−	0.932217i	$-0.382128\pi$
$509$	16.3297	0.723800	0.361900	+	0.932217i	$0.382128\pi$
$510$	0	0
$511$	6.70638	0.296673
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.85039	0.0815381
$516$	0	0
$517$	0	0
$518$	0	0
$519$	7.20359	0.316202
$520$	0	0
$521$	3.20359	0.140352	0.0701758	−	0.997535i	$-0.477644\pi$
$521$	3.20359	0.140352	0.0701758	+	0.997535i	$0.477644\pi$
$522$	0	0
$523$	−8.55678	−0.374162	−0.187081	−	0.982345i	$-0.559903\pi$
$523$	−8.55678	−0.374162	−0.187081	+	0.982345i	$0.559903\pi$
$524$	0	0
$525$	1.39821	0.0610228
$526$	0	0
$527$	−20.5422	−0.894833
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.24860	−0.140977
$532$	0	0
$533$	34.8864	1.51110
$534$	0	0
$535$	−5.39821	−0.233385
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	35.7729	1.53800	0.768998	−	0.639251i	$-0.220755\pi$
$541$	35.7729	1.53800	0.768998	+	0.639251i	$0.220755\pi$
$542$	0	0
$543$	−1.05398	−0.0452306
$544$	0	0
$545$	19.9404	0.854154
$546$	0	0
$547$	−22.5872	−0.965760	−0.482880	−	0.875686i	$-0.660409\pi$
$547$	−22.5872	−0.965760	−0.482880	+	0.875686i	$0.660409\pi$
$548$	0	0
$549$	−9.44322	−0.403027
$550$	0	0
$551$	0	0
$552$	0	0
$553$	8.18006	0.347852
$554$	0	0
$555$	−8.69182	−0.368947
$556$	0	0
$557$	−2.45219	−0.103902	−0.0519512	−	0.998650i	$-0.516544\pi$
$557$	−2.45219	−0.103902	−0.0519512	+	0.998650i	$0.516544\pi$
$558$	0	0
$559$	−49.4737	−2.09251
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−26.6018	−1.12113	−0.560566	−	0.828110i	$-0.689416\pi$
$563$	−26.6018	−1.12113	−0.560566	+	0.828110i	$0.689416\pi$
$564$	0	0
$565$	−5.89541	−0.248022
$566$	0	0
$567$	1.39821	0.0587192
$568$	0	0
$569$	44.8864	1.88174	0.940869	−	0.338771i	$-0.110011\pi$
$569$	44.8864	1.88174	0.940869	+	0.338771i	$0.110011\pi$
$570$	0	0
$571$	41.8629	1.75191	0.875954	−	0.482394i	$-0.160233\pi$
$571$	41.8629	1.75191	0.875954	+	0.482394i	$0.160233\pi$
$572$	0	0
$573$	6.49720	0.271425
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−22.3892	−0.932076	−0.466038	−	0.884765i	$-0.654319\pi$
$577$	−22.3892	−0.932076	−0.466038	+	0.884765i	$0.654319\pi$
$578$	0	0
$579$	−12.4972	−0.519366
$580$	0	0
$581$	16.6323	0.690022
$582$	0	0
$583$	0	0
$584$	0	0
$585$	6.64681	0.274812
$586$	0	0
$587$	−11.0956	−0.457966	−0.228983	−	0.973430i	$-0.573540\pi$
$587$	−11.0956	−0.457966	−0.228983	+	0.973430i	$0.573540\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−1.70079	−0.0699610
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	4.75140	0.194788
$596$	0	0
$597$	8.64681	0.353890
$598$	0	0
$599$	−10.0305	−0.409833	−0.204917	−	0.978779i	$-0.565692\pi$
$599$	−10.0305	−0.409833	−0.204917	+	0.978779i	$0.565692\pi$
$600$	0	0
$601$	−41.6378	−1.69844	−0.849222	−	0.528037i	$-0.822928\pi$
$601$	−41.6378	−1.69844	−0.849222	+	0.528037i	$0.822928\pi$
$602$	0	0
$603$	10.6918	0.435405
$604$	0	0
$605$	−11.0000	−0.447214
$606$	0	0
$607$	0.646809	0.0262531	0.0131266	−	0.999914i	$-0.495822\pi$
$607$	0.646809	0.0262531	0.0131266	+	0.999914i	$0.495822\pi$
$608$	0	0
$609$	0.841431	0.0340965
$610$	0	0
$611$	18.5872	0.751959
$612$	0	0
$613$	25.6233	1.03491	0.517457	−	0.855709i	$-0.326879\pi$
$613$	25.6233	1.03491	0.517457	+	0.855709i	$0.326879\pi$
$614$	0	0
$615$	5.24860	0.211644
$616$	0	0
$617$	−11.6974	−0.470920	−0.235460	−	0.971884i	$-0.575660\pi$
$617$	−11.6974	−0.470920	−0.235460	+	0.971884i	$0.575660\pi$
$618$	0	0
$619$	−13.5928	−0.546342	−0.273171	−	0.961965i	$-0.588072\pi$
$619$	−13.5928	−0.546342	−0.273171	+	0.961965i	$0.588072\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−8.51513	−0.341151
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−29.5366	−1.17770
$630$	0	0
$631$	35.2340	1.40265	0.701323	−	0.712844i	$-0.252593\pi$
$631$	35.2340	1.40265	0.701323	+	0.712844i	$0.252593\pi$
$632$	0	0
$633$	−4.45219	−0.176959
$634$	0	0
$635$	−8.64681	−0.343138
$636$	0	0
$637$	−33.5333	−1.32864
$638$	0	0
$639$	−7.89541	−0.312338
$640$	0	0
$641$	15.6829	0.619436	0.309718	−	0.950828i	$-0.399765\pi$
$641$	15.6829	0.619436	0.309718	+	0.950828i	$0.399765\pi$
$642$	0	0
$643$	17.5783	0.693219	0.346610	−	0.938009i	$-0.387333\pi$
$643$	17.5783	0.693219	0.346610	+	0.938009i	$0.387333\pi$
$644$	0	0
$645$	−7.44322	−0.293077
$646$	0	0
$647$	−26.5872	−1.04525	−0.522626	−	0.852562i	$-0.675048\pi$
$647$	−26.5872	−1.04525	−0.522626	+	0.852562i	$0.675048\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−8.45219	−0.331267
$652$	0	0
$653$	−50.2701	−1.96722	−0.983610	−	0.180307i	$-0.942291\pi$
$653$	−50.2701	−1.96722	−0.983610	+	0.180307i	$0.942291\pi$
$654$	0	0
$655$	9.29362	0.363132
$656$	0	0
$657$	4.79641	0.187126
$658$	0	0
$659$	−19.1857	−0.747367	−0.373684	−	0.927556i	$-0.621905\pi$
$659$	−19.1857	−0.747367	−0.373684	+	0.927556i	$0.621905\pi$
$660$	0	0
$661$	−25.6233	−0.996630	−0.498315	−	0.866996i	$-0.666048\pi$
$661$	−25.6233	−0.996630	−0.498315	+	0.866996i	$0.666048\pi$
$662$	0	0
$663$	22.5872	0.877215
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.601793	0.0233015
$668$	0	0
$669$	−25.9404	−1.00291
$670$	0	0
$671$	0	0
$672$	0	0
$673$	11.4737	0.442278	0.221139	−	0.975242i	$-0.429023\pi$
$673$	11.4737	0.442278	0.221139	+	0.975242i	$0.429023\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	6.63225	0.254898	0.127449	−	0.991845i	$-0.459321\pi$
$677$	6.63225	0.254898	0.127449	+	0.991845i	$0.459321\pi$
$678$	0	0
$679$	13.2036	0.506707
$680$	0	0
$681$	−17.0361	−0.652823
$682$	0	0
$683$	−45.6829	−1.74801	−0.874003	−	0.485920i	$-0.838484\pi$
$683$	−45.6829	−1.74801	−0.874003	+	0.485920i	$0.838484\pi$
$684$	0	0
$685$	9.44322	0.360807
$686$	0	0
$687$	−22.8269	−0.870900
$688$	0	0
$689$	61.4737	2.34196
$690$	0	0
$691$	−27.0665	−1.02966	−0.514829	−	0.857293i	$-0.672145\pi$
$691$	−27.0665	−1.02966	−0.514829	+	0.857293i	$0.672145\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.8414	−0.487103
$696$	0	0
$697$	17.8358	0.675580
$698$	0	0
$699$	18.0900	0.684228
$700$	0	0
$701$	−22.9944	−0.868487	−0.434243	−	0.900796i	$-0.642984\pi$
$701$	−22.9944	−0.868487	−0.434243	+	0.900796i	$0.642984\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	2.79641	0.105319
$706$	0	0
$707$	0.841431	0.0316453
$708$	0	0
$709$	−1.44322	−0.0542014	−0.0271007	−	0.999633i	$-0.508627\pi$
$709$	−1.44322	−0.0542014	−0.0271007	+	0.999633i	$0.508627\pi$
$710$	0	0
$711$	5.85039	0.219407
$712$	0	0
$713$	−6.04502	−0.226388
$714$	0	0
$715$	0	0
$716$	0	0
$717$	27.0811	1.01136
$718$	0	0
$719$	44.8898	1.67411	0.837054	−	0.547121i	$-0.184276\pi$
$719$	44.8898	1.67411	0.837054	+	0.547121i	$0.184276\pi$
$720$	0	0
$721$	2.58723	0.0963536
$722$	0	0
$723$	6.09003	0.226491
$724$	0	0
$725$	0.601793	0.0223500
$726$	0	0
$727$	41.1890	1.52762	0.763808	−	0.645443i	$-0.223327\pi$
$727$	41.1890	1.52762	0.763808	+	0.645443i	$0.223327\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−25.2936	−0.935518
$732$	0	0
$733$	−33.9854	−1.25528	−0.627640	−	0.778503i	$-0.715979\pi$
$733$	−33.9854	−1.25528	−0.627640	+	0.778503i	$0.715979\pi$
$734$	0	0
$735$	−5.04502	−0.186088
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−22.3442	−0.821946	−0.410973	−	0.911648i	$-0.634811\pi$
$739$	−22.3442	−0.821946	−0.410973	+	0.911648i	$0.634811\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.5872	0.388408	0.194204	−	0.980961i	$-0.437788\pi$
$743$	10.5872	0.388408	0.194204	+	0.980961i	$0.437788\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	11.8954	0.435230
$748$	0	0
$749$	−7.54781	−0.275791
$750$	0	0
$751$	−8.43763	−0.307893	−0.153947	−	0.988079i	$-0.549198\pi$
$751$	−8.43763	−0.307893	−0.153947	+	0.988079i	$0.549198\pi$
$752$	0	0
$753$	−26.9765	−0.983078
$754$	0	0
$755$	0	0
$756$	0	0
$757$	23.2791	0.846092	0.423046	−	0.906108i	$-0.360961\pi$
$757$	23.2791	0.846092	0.423046	+	0.906108i	$0.360961\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−10.7514	−0.389738	−0.194869	−	0.980829i	$-0.562428\pi$
$761$	−10.7514	−0.389738	−0.194869	+	0.980829i	$0.562428\pi$
$762$	0	0
$763$	27.8809	1.00935
$764$	0	0
$765$	3.39821	0.122862
$766$	0	0
$767$	−21.5928	−0.779672
$768$	0	0
$769$	−21.4016	−0.771761	−0.385880	−	0.922549i	$-0.626102\pi$
$769$	−21.4016	−0.771761	−0.385880	+	0.922549i	$0.626102\pi$
$770$	0	0
$771$	11.5928	0.417506
$772$	0	0
$773$	25.7008	0.924393	0.462197	−	0.886778i	$-0.347061\pi$
$773$	25.7008	0.924393	0.462197	+	0.886778i	$0.347061\pi$
$774$	0	0
$775$	−6.04502	−0.217143
$776$	0	0
$777$	−12.1530	−0.435986
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0.601793	0.0215063
$784$	0	0
$785$	−2.19462	−0.0783294
$786$	0	0
$787$	26.1767	0.933098	0.466549	−	0.884495i	$-0.345497\pi$
$787$	26.1767	0.933098	0.466549	+	0.884495i	$0.345497\pi$
$788$	0	0
$789$	−14.0450	−0.500016
$790$	0	0
$791$	−8.24301	−0.293088
$792$	0	0
$793$	−62.7673	−2.22893
$794$	0	0
$795$	9.24860	0.328014
$796$	0	0
$797$	−46.5422	−1.64861	−0.824305	−	0.566146i	$-0.808434\pi$
$797$	−46.5422	−1.64861	−0.824305	+	0.566146i	$0.808434\pi$
$798$	0	0
$799$	9.50280	0.336185
$800$	0	0
$801$	−6.09003	−0.215181
$802$	0	0
$803$	0	0
$804$	0	0
$805$	1.39821	0.0492803
$806$	0	0
$807$	13.8054	0.485972
$808$	0	0
$809$	17.1586	0.603263	0.301632	−	0.953425i	$-0.402469\pi$
$809$	17.1586	0.603263	0.301632	+	0.953425i	$0.402469\pi$
$810$	0	0
$811$	−38.5243	−1.35277	−0.676385	−	0.736548i	$-0.736455\pi$
$811$	−38.5243	−1.35277	−0.676385	+	0.736548i	$0.736455\pi$
$812$	0	0
$813$	7.33863	0.257377
$814$	0	0
$815$	−18.4972	−0.647929
$816$	0	0
$817$	0	0
$818$	0	0
$819$	9.29362	0.324745
$820$	0	0
$821$	−35.7312	−1.24703	−0.623515	−	0.781812i	$-0.714296\pi$
$821$	−35.7312	−1.24703	−0.623515	+	0.781812i	$0.714296\pi$
$822$	0	0
$823$	−30.0305	−1.04680	−0.523398	−	0.852088i	$-0.675336\pi$
$823$	−30.0305	−1.04680	−0.523398	+	0.852088i	$0.675336\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.58387	0.298490	0.149245	−	0.988800i	$-0.452316\pi$
$827$	8.58387	0.298490	0.149245	+	0.988800i	$0.452316\pi$
$828$	0	0
$829$	20.8235	0.723230	0.361615	−	0.932327i	$-0.382225\pi$
$829$	20.8235	0.723230	0.361615	+	0.932327i	$0.382225\pi$
$830$	0	0
$831$	−1.74244	−0.0604444
$832$	0	0
$833$	−17.1440	−0.594005
$834$	0	0
$835$	−5.20359	−0.180077
$836$	0	0
$837$	−6.04502	−0.208946
$838$	0	0
$839$	11.3115	0.390518	0.195259	−	0.980752i	$-0.437445\pi$
$839$	11.3115	0.390518	0.195259	+	0.980752i	$0.437445\pi$
$840$	0	0
$841$	−28.6378	−0.987512
$842$	0	0
$843$	−11.6829	−0.402379
$844$	0	0
$845$	31.1801	1.07263
$846$	0	0
$847$	−15.3803	−0.528473
$848$	0	0
$849$	14.7819	0.507312
$850$	0	0
$851$	−8.69182	−0.297952
$852$	0	0
$853$	−51.2161	−1.75361	−0.876803	−	0.480849i	$-0.840328\pi$
$853$	−51.2161	−1.75361	−0.876803	+	0.480849i	$0.840328\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−44.5872	−1.52307	−0.761535	−	0.648123i	$-0.775554\pi$
$857$	−44.5872	−1.52307	−0.761535	+	0.648123i	$0.775554\pi$
$858$	0	0
$859$	−32.3330	−1.10319	−0.551595	−	0.834112i	$-0.685980\pi$
$859$	−32.3330	−1.10319	−0.551595	+	0.834112i	$0.685980\pi$
$860$	0	0
$861$	7.33863	0.250100
$862$	0	0
$863$	−14.4972	−0.493491	−0.246745	−	0.969080i	$-0.579361\pi$
$863$	−14.4972	−0.493491	−0.246745	+	0.969080i	$0.579361\pi$
$864$	0	0
$865$	7.20359	0.244929
$866$	0	0
$867$	−5.45219	−0.185166
$868$	0	0
$869$	0	0
$870$	0	0
$871$	71.0665	2.40800
$872$	0	0
$873$	9.44322	0.319605
$874$	0	0
$875$	1.39821	0.0472680
$876$	0	0
$877$	−49.3241	−1.66556	−0.832778	−	0.553607i	$-0.813251\pi$
$877$	−49.3241	−1.66556	−0.832778	+	0.553607i	$0.813251\pi$
$878$	0	0
$879$	8.34423	0.281444
$880$	0	0
$881$	39.9888	1.34726	0.673629	−	0.739070i	$-0.264735\pi$
$881$	39.9888	1.34726	0.673629	+	0.739070i	$0.264735\pi$
$882$	0	0
$883$	−14.7964	−0.497939	−0.248970	−	0.968511i	$-0.580092\pi$
$883$	−14.7964	−0.497939	−0.248970	+	0.968511i	$0.580092\pi$
$884$	0	0
$885$	−3.24860	−0.109201
$886$	0	0
$887$	29.5637	0.992652	0.496326	−	0.868136i	$-0.334682\pi$
$887$	29.5637	0.992652	0.496326	+	0.868136i	$0.334682\pi$
$888$	0	0
$889$	−12.0900	−0.405487
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.64681	0.221930
$898$	0	0
$899$	−3.63785	−0.121329
$900$	0	0
$901$	31.4287	1.04704
$902$	0	0
$903$	−10.4072	−0.346329
$904$	0	0
$905$	−1.05398	−0.0350354
$906$	0	0
$907$	30.3926	1.00917	0.504585	−	0.863362i	$-0.331645\pi$
$907$	30.3926	1.00917	0.504585	+	0.863362i	$0.331645\pi$
$908$	0	0
$909$	0.601793	0.0199602
$910$	0	0
$911$	26.4072	0.874909	0.437454	−	0.899241i	$-0.355880\pi$
$911$	26.4072	0.874909	0.437454	+	0.899241i	$0.355880\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−9.44322	−0.312183
$916$	0	0
$917$	12.9944	0.429113
$918$	0	0
$919$	14.9639	0.493615	0.246808	−	0.969065i	$-0.420618\pi$
$919$	14.9639	0.493615	0.246808	+	0.969065i	$0.420618\pi$
$920$	0	0
$921$	5.29362	0.174431
$922$	0	0
$923$	−52.4793	−1.72738
$924$	0	0
$925$	−8.69182	−0.285785
$926$	0	0
$927$	1.85039	0.0607749
$928$	0	0
$929$	11.7458	0.385367	0.192684	−	0.981261i	$-0.438281\pi$
$929$	11.7458	0.385367	0.192684	+	0.981261i	$0.438281\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−13.0361	−0.426781
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−27.7312	−0.905940	−0.452970	−	0.891526i	$-0.649636\pi$
$937$	−27.7312	−0.905940	−0.452970	+	0.891526i	$0.649636\pi$
$938$	0	0
$939$	−2.19462	−0.0716188
$940$	0	0
$941$	6.18006	0.201464	0.100732	−	0.994914i	$-0.467881\pi$
$941$	6.18006	0.201464	0.100732	+	0.994914i	$0.467881\pi$
$942$	0	0
$943$	5.24860	0.170918
$944$	0	0
$945$	1.39821	0.0454837
$946$	0	0
$947$	50.2880	1.63414	0.817071	−	0.576538i	$-0.195597\pi$
$947$	50.2880	1.63414	0.817071	+	0.576538i	$0.195597\pi$
$948$	0	0
$949$	31.8809	1.03490
$950$	0	0
$951$	22.9944	0.745645
$952$	0	0
$953$	21.1440	0.684922	0.342461	−	0.939532i	$-0.388740\pi$
$953$	21.1440	0.684922	0.342461	+	0.939532i	$0.388740\pi$
$954$	0	0
$955$	6.49720	0.210245
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13.2036	0.426366
$960$	0	0
$961$	5.54222	0.178781
$962$	0	0
$963$	−5.39821	−0.173955
$964$	0	0
$965$	−12.4972	−0.402299
$966$	0	0
$967$	36.9460	1.18810	0.594052	−	0.804427i	$-0.297527\pi$
$967$	36.9460	1.18810	0.594052	+	0.804427i	$0.297527\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	23.4016	0.750992	0.375496	−	0.926824i	$-0.377472\pi$
$971$	23.4016	0.750992	0.375496	+	0.926824i	$0.377472\pi$
$972$	0	0
$973$	−17.9550	−0.575610
$974$	0	0
$975$	6.64681	0.212868
$976$	0	0
$977$	−12.2125	−0.390714	−0.195357	−	0.980732i	$-0.562587\pi$
$977$	−12.2125	−0.390714	−0.195357	+	0.980732i	$0.562587\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	19.9404	0.636649
$982$	0	0
$983$	−43.9438	−1.40159	−0.700795	−	0.713363i	$-0.747171\pi$
$983$	−43.9438	−1.40159	−0.700795	+	0.713363i	$0.747171\pi$
$984$	0	0
$985$	−1.70079	−0.0541916
$986$	0	0
$987$	3.90997	0.124456
$988$	0	0
$989$	−7.44322	−0.236681
$990$	0	0
$991$	22.4343	0.712648	0.356324	−	0.934363i	$-0.384030\pi$
$991$	22.4343	0.712648	0.356324	+	0.934363i	$0.384030\pi$
$992$	0	0
$993$	−15.6378	−0.496252
$994$	0	0
$995$	8.64681	0.274122
$996$	0	0
$997$	16.5389	0.523791	0.261895	−	0.965096i	$-0.415652\pi$
$997$	16.5389	0.523791	0.261895	+	0.965096i	$0.415652\pi$
$998$	0	0
$999$	−8.69182	−0.274997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2760.2.a.t.1.2	✓	3
3.2	odd	2		8280.2.a.bh.1.2		3
4.3	odd	2		5520.2.a.ca.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.t.1.2	✓	3	1.1	even	1	trivial
5520.2.a.ca.1.2		3	4.3	odd	2
8280.2.a.bh.1.2		3	3.2	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2760.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.39821 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.39821 q^{7} +1.00000 q^{9} +6.64681 q^{13} +1.00000 q^{15} +3.39821 q^{17} +1.39821 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.601793 q^{29} -6.04502 q^{31} +1.39821 q^{35} -8.69182 q^{37} +6.64681 q^{39} +5.24860 q^{41} -7.44322 q^{43} +1.00000 q^{45} +2.79641 q^{47} -5.04502 q^{49} +3.39821 q^{51} +9.24860 q^{53} -3.24860 q^{59} -9.44322 q^{61} +1.39821 q^{63} +6.64681 q^{65} +10.6918 q^{67} +1.00000 q^{69} -7.89541 q^{71} +4.79641 q^{73} +1.00000 q^{75} +5.85039 q^{79} +1.00000 q^{81} +11.8954 q^{83} +3.39821 q^{85} +0.601793 q^{87} -6.09003 q^{89} +9.29362 q^{91} -6.04502 q^{93} +9.44322 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 + q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} + q^{7} + 3 q^{9} + 4 q^{13} + 3 q^{15} + 7 q^{17} + q^{21} + 3 q^{23} + 3 q^{25} + 3 q^{27} + 5 q^{29} + q^{31} + q^{35} + 9 q^{37} + 4 q^{39} + 3 q^{41} + 3 q^{45} + 2 q^{47} + 4 q^{49} + 7 q^{51} + 15 q^{53} + 3 q^{59} - 6 q^{61} + q^{63} + 4 q^{65} - 3 q^{67} + 3 q^{69} + 5 q^{71} + 8 q^{73} + 3 q^{75} + 8 q^{79} + 3 q^{81} + 7 q^{83} + 7 q^{85} + 5 q^{87} + 20 q^{89} - 4 q^{91} + q^{93} + 6 q^{97}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 + q^7 + 3 * q^9 + 4 * q^13 + 3 * q^15 + 7 * q^17 + q^21 + 3 * q^23 + 3 * q^25 + 3 * q^27 + 5 * q^29 + q^31 + q^35 + 9 * q^37 + 4 * q^39 + 3 * q^41 + 3 * q^45 + 2 * q^47 + 4 * q^49 + 7 * q^51 + 15 * q^53 + 3 * q^59 - 6 * q^61 + q^63 + 4 * q^65 - 3 * q^67 + 3 * q^69 + 5 * q^71 + 8 * q^73 + 3 * q^75 + 8 * q^79 + 3 * q^81 + 7 * q^83 + 7 * q^85 + 5 * q^87 + 20 * q^89 - 4 * q^91 + q^93 + 6 * q^97

Introduction

L-functions

Modular forms

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Database

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