LMFDB - Embedded newform 6480.2.a.by.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6480,2,Mod(1,6480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6480, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6480.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6480 = 2^{4} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6480.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:	$51.7430605098$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.62352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 11x^{2} + 12x + 24 $ x^4 - 2x^3 - 11x^2 + 12*x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3240)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.61675$ of defining polynomial
Character	$\chi$	$=$	6480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +4.34880 q^{7} +2.70115 q^{11} -6.26439 q^{13} +5.37969 q^{17} +6.23349 q^{19} +5.61675 q^{23} +1.00000 q^{25} -0.762947 q^{29} -6.61675 q^{31} -4.34880 q^{35} +11.1491 q^{37} -9.02733 q^{41} -9.68498 q^{43} +5.22705 q^{47} +11.9120 q^{49} +10.6294 q^{53} -2.70115 q^{55} +1.73205 q^{59} +1.63292 q^{61} +6.26439 q^{65} -9.52589 q^{67} +14.3017 q^{71} -10.6132 q^{73} +11.7468 q^{77} -0.535898 q^{79} +0.146201 q^{83} -5.37969 q^{85} +7.85997 q^{89} -27.2425 q^{91} -6.23349 q^{95} -0.389697 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 2 q^{7} + 4 q^{11} + 2 q^{17} + 10 q^{23} + 4 q^{25} + 4 q^{29} - 14 q^{31} + 2 q^{35} + 14 q^{37} - 4 q^{41} - 22 q^{43} + 10 q^{49} + 8 q^{53} - 4 q^{55} + 4 q^{61} - 24 q^{67} + 28 q^{71}+ \cdots - 10 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 2 * q^7 + 4 * q^11 + 2 * q^17 + 10 * q^23 + 4 * q^25 + 4 * q^29 - 14 * q^31 + 2 * q^35 + 14 * q^37 - 4 * q^41 - 22 * q^43 + 10 * q^49 + 8 * q^53 - 4 * q^55 + 4 * q^61 - 24 * q^67 + 28 * q^71 + 2 * q^73 + 30 * q^77 - 16 * q^79 + 6 * q^83 - 2 * q^85 + 8 * q^89 - 30 * q^91 - 10 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.34880	1.64369	0.821845	−	0.569711i	$-0.192945\pi$
$7$	4.34880	1.64369	0.821845	+	0.569711i	$0.192945\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.70115	0.814429	0.407214	−	0.913333i	$-0.366500\pi$
$11$	2.70115	0.814429	0.407214	+	0.913333i	$0.366500\pi$
$12$	0	0
$13$	−6.26439	−1.73743	−0.868714	−	0.495314i	$-0.835053\pi$
$13$	−6.26439	−1.73743	−0.868714	+	0.495314i	$0.835053\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.37969	1.30477	0.652384	−	0.757889i	$-0.273769\pi$
$17$	5.37969	1.30477	0.652384	+	0.757889i	$0.273769\pi$
$18$	0	0
$19$	6.23349	1.43006	0.715030	−	0.699093i	$-0.246413\pi$
$19$	6.23349	1.43006	0.715030	+	0.699093i	$0.246413\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.61675	1.17117	0.585586	−	0.810610i	$-0.300864\pi$
$23$	5.61675	1.17117	0.585586	+	0.810610i	$0.300864\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.762947	−0.141676	−0.0708378	−	0.997488i	$-0.522567\pi$
$29$	−0.762947	−0.141676	−0.0708378	+	0.997488i	$0.522567\pi$
$30$	0	0
$31$	−6.61675	−1.18840	−0.594201	−	0.804316i	$-0.702532\pi$
$31$	−6.61675	−1.18840	−0.594201	+	0.804316i	$0.702532\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.34880	−0.735081
$36$	0	0
$37$	11.1491	1.83290	0.916449	−	0.400152i	$-0.131043\pi$
$37$	11.1491	1.83290	0.916449	+	0.400152i	$0.131043\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.02733	−1.40983	−0.704916	−	0.709290i	$-0.749015\pi$
$41$	−9.02733	−1.40983	−0.704916	+	0.709290i	$0.749015\pi$
$42$	0	0
$43$	−9.68498	−1.47695	−0.738473	−	0.674283i	$-0.764453\pi$
$43$	−9.68498	−1.47695	−0.738473	+	0.674283i	$0.764453\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.22705	0.762443	0.381222	−	0.924484i	$-0.375503\pi$
$47$	5.22705	0.762443	0.381222	+	0.924484i	$0.375503\pi$
$48$	0	0
$49$	11.9120	1.70172
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.6294	1.46005	0.730027	−	0.683418i	$-0.239507\pi$
$53$	10.6294	1.46005	0.730027	+	0.683418i	$0.239507\pi$
$54$	0	0
$55$	−2.70115	−0.364224
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.73205	0.225494	0.112747	−	0.993624i	$-0.464035\pi$
$59$	1.73205	0.225494	0.112747	+	0.993624i	$0.464035\pi$
$60$	0	0
$61$	1.63292	0.209074	0.104537	−	0.994521i	$-0.466664\pi$
$61$	1.63292	0.209074	0.104537	+	0.994521i	$0.466664\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.26439	0.777002
$66$	0	0
$67$	−9.52589	−1.16377	−0.581887	−	0.813270i	$-0.697685\pi$
$67$	−9.52589	−1.16377	−0.581887	+	0.813270i	$0.697685\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.3017	1.69730	0.848651	−	0.528953i	$-0.177415\pi$
$71$	14.3017	1.69730	0.848651	+	0.528953i	$0.177415\pi$
$72$	0	0
$73$	−10.6132	−1.24218	−0.621090	−	0.783740i	$-0.713310\pi$
$73$	−10.6132	−1.24218	−0.621090	+	0.783740i	$0.713310\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.7468	1.33867
$78$	0	0
$79$	−0.535898	−0.0602933	−0.0301466	−	0.999545i	$-0.509597\pi$
$79$	−0.535898	−0.0602933	−0.0301466	+	0.999545i	$0.509597\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.146201	0.0160477	0.00802385	−	0.999968i	$-0.497446\pi$
$83$	0.146201	0.0160477	0.00802385	+	0.999968i	$0.497446\pi$
$84$	0	0
$85$	−5.37969	−0.583510
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.85997	0.833155	0.416578	−	0.909100i	$-0.363229\pi$
$89$	7.85997	0.833155	0.416578	+	0.909100i	$0.363229\pi$
$90$	0	0
$91$	−27.2425	−2.85579
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.23349	−0.639543
$96$	0	0
$97$	−0.389697	−0.0395677	−0.0197839	−	0.999804i	$-0.506298\pi$
$97$	−0.389697	−0.0395677	−0.0197839	+	0.999804i	$0.506298\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.8628	−1.27990	−0.639951	−	0.768416i	$-0.721045\pi$
$101$	−12.8628	−1.27990	−0.639951	+	0.768416i	$0.721045\pi$
$102$	0	0
$103$	−5.89731	−0.581079	−0.290539	−	0.956863i	$-0.593835\pi$
$103$	−5.89731	−0.581079	−0.290539	+	0.956863i	$0.593835\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.3923	−1.39136	−0.695678	−	0.718353i	$-0.744896\pi$
$107$	−14.3923	−1.39136	−0.695678	+	0.718353i	$0.744896\pi$
$108$	0	0
$109$	5.15264	0.493534	0.246767	−	0.969075i	$-0.420632\pi$
$109$	5.15264	0.493534	0.246767	+	0.969075i	$0.420632\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.3826	1.72929	0.864643	−	0.502386i	$-0.167544\pi$
$113$	18.3826	1.72929	0.864643	+	0.502386i	$0.167544\pi$
$114$	0	0
$115$	−5.61675	−0.523764
$116$	0	0
$117$	0	0
$118$	0	0
$119$	23.3952	2.14463
$120$	0	0
$121$	−3.70376	−0.336706
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	7.27700	0.645729	0.322865	−	0.946445i	$-0.395354\pi$
$127$	7.27700	0.645729	0.322865	+	0.946445i	$0.395354\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.36497	0.468740	0.234370	−	0.972148i	$-0.424697\pi$
$131$	5.36497	0.468740	0.234370	+	0.972148i	$0.424697\pi$
$132$	0	0
$133$	27.1082	2.35058
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.69759	−0.401342	−0.200671	−	0.979659i	$-0.564312\pi$
$137$	−4.69759	−0.401342	−0.200671	+	0.979659i	$0.564312\pi$
$138$	0	0
$139$	0.662362	0.0561808	0.0280904	−	0.999605i	$-0.491057\pi$
$139$	0.662362	0.0561808	0.0280904	+	0.999605i	$0.491057\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−16.9211	−1.41501
$144$	0	0
$145$	0.762947	0.0633593
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.16169	−0.177093	−0.0885464	−	0.996072i	$-0.528222\pi$
$149$	−2.16169	−0.177093	−0.0885464	+	0.996072i	$0.528222\pi$
$150$	0	0
$151$	8.44793	0.687483	0.343741	−	0.939064i	$-0.388306\pi$
$151$	8.44793	0.687483	0.343741	+	0.939064i	$0.388306\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.61675	0.531470
$156$	0	0
$157$	5.75111	0.458988	0.229494	−	0.973310i	$-0.426293\pi$
$157$	5.75111	0.458988	0.229494	+	0.973310i	$0.426293\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	24.4261	1.92504
$162$	0	0
$163$	−12.8438	−1.00600	−0.503002	−	0.864285i	$-0.667771\pi$
$163$	−12.8438	−1.00600	−0.503002	+	0.864285i	$0.667771\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.9085	1.69533	0.847664	−	0.530533i	$-0.178008\pi$
$167$	21.9085	1.69533	0.847664	+	0.530533i	$0.178008\pi$
$168$	0	0
$169$	26.2425	2.01866
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.68854	0.204406	0.102203	−	0.994764i	$-0.467411\pi$
$173$	2.68854	0.204406	0.102203	+	0.994764i	$0.467411\pi$
$174$	0	0
$175$	4.34880	0.328738
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.02733	−0.674735	−0.337367	−	0.941373i	$-0.609536\pi$
$179$	−9.02733	−0.674735	−0.337367	+	0.941373i	$0.609536\pi$
$180$	0	0
$181$	−5.85024	−0.434845	−0.217422	−	0.976078i	$-0.569765\pi$
$181$	−5.85024	−0.434845	−0.217422	+	0.976078i	$0.569765\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−11.1491	−0.819697
$186$	0	0
$187$	14.5314	1.06264
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.9936	0.940181	0.470090	−	0.882618i	$-0.344221\pi$
$191$	12.9936	0.940181	0.470090	+	0.882618i	$0.344221\pi$
$192$	0	0
$193$	4.51328	0.324873	0.162437	−	0.986719i	$-0.448065\pi$
$193$	4.51328	0.324873	0.162437	+	0.986719i	$0.448065\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.8431	−1.20002	−0.600011	−	0.799992i	$-0.704837\pi$
$197$	−16.8431	−1.20002	−0.600011	+	0.799992i	$0.704837\pi$
$198$	0	0
$199$	−11.6006	−0.822343	−0.411171	−	0.911558i	$-0.634880\pi$
$199$	−11.6006	−0.822343	−0.411171	+	0.911558i	$0.634880\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.31790	−0.232871
$204$	0	0
$205$	9.02733	0.630496
$206$	0	0
$207$	0	0
$208$	0	0
$209$	16.8376	1.16468
$210$	0	0
$211$	0.863531	0.0594480	0.0297240	−	0.999558i	$-0.490537\pi$
$211$	0.863531	0.0594480	0.0297240	+	0.999558i	$0.490537\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.68498	0.660510
$216$	0	0
$217$	−28.7749	−1.95337
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−33.7005	−2.26694
$222$	0	0
$223$	8.86641	0.593739	0.296869	−	0.954918i	$-0.404057\pi$
$223$	8.86641	0.593739	0.296869	+	0.954918i	$0.404057\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.6850	−0.908304	−0.454152	−	0.890924i	$-0.650058\pi$
$227$	−13.6850	−0.908304	−0.454152	+	0.890924i	$0.650058\pi$
$228$	0	0
$229$	−2.46698	−0.163023	−0.0815114	−	0.996672i	$-0.525975\pi$
$229$	−2.46698	−0.163023	−0.0815114	+	0.996672i	$0.525975\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.8438	0.841425	0.420712	−	0.907194i	$-0.361780\pi$
$233$	12.8438	0.841425	0.420712	+	0.907194i	$0.361780\pi$
$234$	0	0
$235$	−5.22705	−0.340975
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.53590	0.552141	0.276071	−	0.961137i	$-0.410968\pi$
$239$	8.53590	0.552141	0.276071	+	0.961137i	$0.410968\pi$
$240$	0	0
$241$	5.13976	0.331081	0.165540	−	0.986203i	$-0.447063\pi$
$241$	5.13976	0.331081	0.165540	+	0.986203i	$0.447063\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−11.9120	−0.761032
$246$	0	0
$247$	−39.0490	−2.48463
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.13080	−0.0713753	−0.0356877	−	0.999363i	$-0.511362\pi$
$251$	−1.13080	−0.0713753	−0.0356877	+	0.999363i	$0.511362\pi$
$252$	0	0
$253$	15.1717	0.953837
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.1688	0.634313	0.317157	−	0.948373i	$-0.397272\pi$
$257$	10.1688	0.634313	0.317157	+	0.948373i	$0.397272\pi$
$258$	0	0
$259$	48.4851	3.01272
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.20731	−0.506084	−0.253042	−	0.967455i	$-0.581431\pi$
$263$	−8.20731	−0.506084	−0.253042	+	0.967455i	$0.581431\pi$
$264$	0	0
$265$	−10.6294	−0.652956
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.9246	1.64162	0.820812	−	0.571198i	$-0.193521\pi$
$269$	26.9246	1.64162	0.820812	+	0.571198i	$0.193521\pi$
$270$	0	0
$271$	4.30529	0.261528	0.130764	−	0.991414i	$-0.458257\pi$
$271$	4.30529	0.261528	0.130764	+	0.991414i	$0.458257\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.70115	0.162886
$276$	0	0
$277$	20.2181	1.21479	0.607394	−	0.794401i	$-0.292215\pi$
$277$	20.2181	1.21479	0.607394	+	0.794401i	$0.292215\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0.140803	0.00839959	0.00419979	−	0.999991i	$-0.498663\pi$
$281$	0.140803	0.00839959	0.00419979	+	0.999991i	$0.498663\pi$
$282$	0	0
$283$	−0.230611	−0.0137084	−0.00685420	−	0.999977i	$-0.502182\pi$
$283$	−0.230611	−0.0137084	−0.00685420	+	0.999977i	$0.502182\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−39.2580	−2.31733
$288$	0	0
$289$	11.9411	0.702417
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8.12290	−0.474545	−0.237272	−	0.971443i	$-0.576253\pi$
$293$	−8.12290	−0.474545	−0.237272	+	0.971443i	$0.576253\pi$
$294$	0	0
$295$	−1.73205	−0.100844
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−35.1855	−2.03483
$300$	0	0
$301$	−42.1180	−2.42764
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.63292	−0.0935008
$306$	0	0
$307$	0.444364	0.0253612	0.0126806	−	0.999920i	$-0.495964\pi$
$307$	0.444364	0.0253612	0.0126806	+	0.999920i	$0.495964\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.52521	−0.370011	−0.185005	−	0.982738i	$-0.559230\pi$
$311$	−6.52521	−0.370011	−0.185005	+	0.982738i	$0.559230\pi$
$312$	0	0
$313$	−16.9576	−0.958503	−0.479251	−	0.877678i	$-0.659092\pi$
$313$	−16.9576	−0.958503	−0.479251	+	0.877678i	$0.659092\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.9936	0.785957	0.392978	−	0.919548i	$-0.371445\pi$
$317$	13.9936	0.785957	0.392978	+	0.919548i	$0.371445\pi$
$318$	0	0
$319$	−2.06084	−0.115385
$320$	0	0
$321$	0	0
$322$	0	0
$323$	33.5343	1.86590
$324$	0	0
$325$	−6.26439	−0.347486
$326$	0	0
$327$	0	0
$328$	0	0
$329$	22.7314	1.25322
$330$	0	0
$331$	−19.2820	−1.05983	−0.529917	−	0.848050i	$-0.677777\pi$
$331$	−19.2820	−1.05983	−0.529917	+	0.848050i	$0.677777\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.52589	0.520455
$336$	0	0
$337$	17.2587	0.940142	0.470071	−	0.882629i	$-0.344228\pi$
$337$	17.2587	0.940142	0.470071	+	0.882629i	$0.344228\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−17.8729	−0.967869
$342$	0	0
$343$	21.3614	1.15341
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.6032	0.730257	0.365128	−	0.930957i	$-0.381025\pi$
$347$	13.6032	0.730257	0.365128	+	0.930957i	$0.381025\pi$
$348$	0	0
$349$	−1.61963	−0.0866965	−0.0433483	−	0.999060i	$-0.513803\pi$
$349$	−1.61963	−0.0866965	−0.0433483	+	0.999060i	$0.513803\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−29.0802	−1.54778	−0.773890	−	0.633319i	$-0.781692\pi$
$353$	−29.0802	−1.54778	−0.773890	+	0.633319i	$0.781692\pi$
$354$	0	0
$355$	−14.3017	−0.759057
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13.9218	0.734762	0.367381	−	0.930070i	$-0.380254\pi$
$359$	13.9218	0.734762	0.367381	+	0.930070i	$0.380254\pi$
$360$	0	0
$361$	19.8564	1.04507
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.6132	0.555519
$366$	0	0
$367$	−18.6652	−0.974318	−0.487159	−	0.873313i	$-0.661967\pi$
$367$	−18.6652	−0.974318	−0.487159	+	0.873313i	$0.661967\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	46.2249	2.39988
$372$	0	0
$373$	19.1872	0.993475	0.496738	−	0.867901i	$-0.334531\pi$
$373$	19.1872	0.993475	0.496738	+	0.867901i	$0.334531\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.77939	0.246151
$378$	0	0
$379$	1.82213	0.0935966	0.0467983	−	0.998904i	$-0.485098\pi$
$379$	1.82213	0.0935966	0.0467983	+	0.998904i	$0.485098\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.95438	0.202060	0.101030	−	0.994883i	$-0.467786\pi$
$383$	3.95438	0.202060	0.101030	+	0.994883i	$0.467786\pi$
$384$	0	0
$385$	−11.7468	−0.598671
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−25.1646	−1.27589	−0.637947	−	0.770080i	$-0.720216\pi$
$389$	−25.1646	−1.27589	−0.637947	+	0.770080i	$0.720216\pi$
$390$	0	0
$391$	30.2164	1.52811
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.535898	0.0269640
$396$	0	0
$397$	−13.8635	−0.695791	−0.347895	−	0.937533i	$-0.613104\pi$
$397$	−13.8635	−0.695791	−0.347895	+	0.937533i	$0.613104\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.4190	1.81868	0.909338	−	0.416058i	$-0.136589\pi$
$401$	36.4190	1.81868	0.909338	+	0.416058i	$0.136589\pi$
$402$	0	0
$403$	41.4499	2.06476
$404$	0	0
$405$	0	0
$406$	0	0
$407$	30.1154	1.49276
$408$	0	0
$409$	30.3524	1.50083	0.750416	−	0.660966i	$-0.229853\pi$
$409$	30.3524	1.50083	0.750416	+	0.660966i	$0.229853\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.53234	0.370642
$414$	0	0
$415$	−0.146201	−0.00717675
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10.2306	−0.499798	−0.249899	−	0.968272i	$-0.580397\pi$
$419$	−10.2306	−0.499798	−0.249899	+	0.968272i	$0.580397\pi$
$420$	0	0
$421$	10.3043	0.502202	0.251101	−	0.967961i	$-0.419207\pi$
$421$	10.3043	0.502202	0.251101	+	0.967961i	$0.419207\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.37969	0.260953
$426$	0	0
$427$	7.10124	0.343653
$428$	0	0
$429$	0	0
$430$	0	0
$431$	41.0317	1.97643	0.988213	−	0.153086i	$-0.0489210\pi$
$431$	41.0317	1.97643	0.988213	+	0.153086i	$0.0489210\pi$
$432$	0	0
$433$	16.7270	0.803850	0.401925	−	0.915673i	$-0.368341\pi$
$433$	16.7270	0.803850	0.401925	+	0.915673i	$0.368341\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	35.0119	1.67485
$438$	0	0
$439$	−12.4860	−0.595926	−0.297963	−	0.954577i	$-0.596307\pi$
$439$	−12.4860	−0.595926	−0.297963	+	0.954577i	$0.596307\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.61606	0.361850	0.180925	−	0.983497i	$-0.442091\pi$
$443$	7.61606	0.361850	0.180925	+	0.983497i	$0.442091\pi$
$444$	0	0
$445$	−7.85997	−0.372598
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12.8585	−0.606831	−0.303415	−	0.952858i	$-0.598127\pi$
$449$	−12.8585	−0.606831	−0.303415	+	0.952858i	$0.598127\pi$
$450$	0	0
$451$	−24.3842	−1.14821
$452$	0	0
$453$	0	0
$454$	0	0
$455$	27.2425	1.27715
$456$	0	0
$457$	12.6132	0.590020	0.295010	−	0.955494i	$-0.404677\pi$
$457$	12.6132	0.590020	0.295010	+	0.955494i	$0.404677\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	31.2157	1.45386	0.726930	−	0.686712i	$-0.240947\pi$
$461$	31.2157	1.45386	0.726930	+	0.686712i	$0.240947\pi$
$462$	0	0
$463$	−3.95910	−0.183995	−0.0919975	−	0.995759i	$-0.529325\pi$
$463$	−3.95910	−0.183995	−0.0919975	+	0.995759i	$0.529325\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.7260	0.959083	0.479542	−	0.877519i	$-0.340803\pi$
$467$	20.7260	0.959083	0.479542	+	0.877519i	$0.340803\pi$
$468$	0	0
$469$	−41.4262	−1.91288
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−26.1606	−1.20287
$474$	0	0
$475$	6.23349	0.286012
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−29.7221	−1.35804	−0.679020	−	0.734120i	$-0.737595\pi$
$479$	−29.7221	−1.35804	−0.679020	+	0.734120i	$0.737595\pi$
$480$	0	0
$481$	−69.8422	−3.18453
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.389697	0.0176952
$486$	0	0
$487$	0.0532416	0.00241260	0.00120630	−	0.999999i	$-0.499616\pi$
$487$	0.0532416	0.00241260	0.00120630	+	0.999999i	$0.499616\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.8585	0.580297	0.290148	−	0.956982i	$-0.406295\pi$
$491$	12.8585	0.580297	0.290148	+	0.956982i	$0.406295\pi$
$492$	0	0
$493$	−4.10442	−0.184854
$494$	0	0
$495$	0	0
$496$	0	0
$497$	62.1953	2.78984
$498$	0	0
$499$	−38.2192	−1.71093	−0.855464	−	0.517862i	$-0.826728\pi$
$499$	−38.2192	−1.71093	−0.855464	+	0.517862i	$0.826728\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	31.0828	1.38591	0.692956	−	0.720980i	$-0.256308\pi$
$503$	31.0828	1.38591	0.692956	+	0.720980i	$0.256308\pi$
$504$	0	0
$505$	12.8628	0.572389
$506$	0	0
$507$	0	0
$508$	0	0
$509$	34.9340	1.54842	0.774210	−	0.632928i	$-0.218147\pi$
$509$	34.9340	1.54842	0.774210	+	0.632928i	$0.218147\pi$
$510$	0	0
$511$	−46.1546	−2.04176
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.89731	0.259866
$516$	0	0
$517$	14.1191	0.620956
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−16.3880	−0.717970	−0.358985	−	0.933343i	$-0.616877\pi$
$521$	−16.3880	−0.717970	−0.358985	+	0.933343i	$0.616877\pi$
$522$	0	0
$523$	−14.6682	−0.641393	−0.320697	−	0.947182i	$-0.603917\pi$
$523$	−14.6682	−0.641393	−0.320697	+	0.947182i	$0.603917\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−35.5961	−1.55059
$528$	0	0
$529$	8.54783	0.371645
$530$	0	0
$531$	0	0
$532$	0	0
$533$	56.5507	2.44948
$534$	0	0
$535$	14.3923	0.622234
$536$	0	0
$537$	0	0
$538$	0	0
$539$	32.1762	1.38593
$540$	0	0
$541$	29.9301	1.28680	0.643398	−	0.765532i	$-0.277524\pi$
$541$	29.9301	1.28680	0.643398	+	0.765532i	$0.277524\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.15264	−0.220715
$546$	0	0
$547$	40.4233	1.72838	0.864188	−	0.503170i	$-0.167833\pi$
$547$	40.4233	1.72838	0.864188	+	0.503170i	$0.167833\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.75582	−0.202605
$552$	0	0
$553$	−2.33051	−0.0991035
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.62909	−0.238512	−0.119256	−	0.992864i	$-0.538051\pi$
$557$	−5.62909	−0.238512	−0.119256	+	0.992864i	$0.538051\pi$
$558$	0	0
$559$	60.6705	2.56609
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.28555	0.0541795	0.0270897	−	0.999633i	$-0.491376\pi$
$563$	1.28555	0.0541795	0.0270897	+	0.999633i	$0.491376\pi$
$564$	0	0
$565$	−18.3826	−0.773361
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.40154	0.226444	0.113222	−	0.993570i	$-0.463883\pi$
$569$	5.40154	0.226444	0.113222	+	0.993570i	$0.463883\pi$
$570$	0	0
$571$	23.8008	0.996032	0.498016	−	0.867168i	$-0.334062\pi$
$571$	23.8008	0.996032	0.498016	+	0.867168i	$0.334062\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.61675	0.234234
$576$	0	0
$577$	−26.9926	−1.12372	−0.561858	−	0.827233i	$-0.689913\pi$
$577$	−26.9926	−1.12372	−0.561858	+	0.827233i	$0.689913\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.635800	0.0263774
$582$	0	0
$583$	28.7115	1.18911
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26.3208	−1.08637	−0.543187	−	0.839611i	$-0.682783\pi$
$587$	−26.3208	−1.08637	−0.543187	+	0.839611i	$0.682783\pi$
$588$	0	0
$589$	−41.2454	−1.69949
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−27.4951	−1.12909	−0.564544	−	0.825403i	$-0.690948\pi$
$593$	−27.4951	−1.12909	−0.564544	+	0.825403i	$0.690948\pi$
$594$	0	0
$595$	−23.3952	−0.959109
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.96409	0.407122	0.203561	−	0.979062i	$-0.434749\pi$
$599$	9.96409	0.407122	0.203561	+	0.979062i	$0.434749\pi$
$600$	0	0
$601$	31.2782	1.27586	0.637931	−	0.770093i	$-0.279790\pi$
$601$	31.2782	1.27586	0.637931	+	0.770093i	$0.279790\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.70376	0.150579
$606$	0	0
$607$	−26.3331	−1.06883	−0.534414	−	0.845223i	$-0.679468\pi$
$607$	−26.3331	−1.06883	−0.534414	+	0.845223i	$0.679468\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32.7443	−1.32469
$612$	0	0
$613$	−18.6020	−0.751329	−0.375664	−	0.926756i	$-0.622585\pi$
$613$	−18.6020	−0.751329	−0.375664	+	0.926756i	$0.622585\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17.3276	0.697584	0.348792	−	0.937200i	$-0.386592\pi$
$617$	17.3276	0.697584	0.348792	+	0.937200i	$0.386592\pi$
$618$	0	0
$619$	−21.5155	−0.864780	−0.432390	−	0.901687i	$-0.642330\pi$
$619$	−21.5155	−0.864780	−0.432390	+	0.901687i	$0.642330\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	34.1814	1.36945
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	59.9786	2.39150
$630$	0	0
$631$	20.9696	0.834786	0.417393	−	0.908726i	$-0.362944\pi$
$631$	20.9696	0.834786	0.417393	+	0.908726i	$0.362944\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−7.27700	−0.288779
$636$	0	0
$637$	−74.6216	−2.95661
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−15.8405	−0.625662	−0.312831	−	0.949809i	$-0.601277\pi$
$641$	−15.8405	−0.625662	−0.312831	+	0.949809i	$0.601277\pi$
$642$	0	0
$643$	16.9508	0.668475	0.334238	−	0.942489i	$-0.391521\pi$
$643$	16.9508	0.668475	0.334238	+	0.942489i	$0.391521\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−41.0504	−1.61386	−0.806929	−	0.590648i	$-0.798872\pi$
$647$	−41.0504	−1.61386	−0.806929	+	0.590648i	$0.798872\pi$
$648$	0	0
$649$	4.67854	0.183649
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.7270	−1.20244	−0.601221	−	0.799083i	$-0.705319\pi$
$653$	−30.7270	−1.20244	−0.601221	+	0.799083i	$0.705319\pi$
$654$	0	0
$655$	−5.36497	−0.209627
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.2349	−0.944059	−0.472030	−	0.881583i	$-0.656479\pi$
$659$	−24.2349	−0.944059	−0.472030	+	0.881583i	$0.656479\pi$
$660$	0	0
$661$	36.2849	1.41132	0.705659	−	0.708552i	$-0.250651\pi$
$661$	36.2849	1.41132	0.705659	+	0.708552i	$0.250651\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−27.1082	−1.05121
$666$	0	0
$667$	−4.28528	−0.165927
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.41077	0.170276
$672$	0	0
$673$	8.41231	0.324271	0.162135	−	0.986769i	$-0.448162\pi$
$673$	8.41231	0.324271	0.162135	+	0.986769i	$0.448162\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.1090	1.61838	0.809189	−	0.587548i	$-0.199907\pi$
$677$	42.1090	1.61838	0.809189	+	0.587548i	$0.199907\pi$
$678$	0	0
$679$	−1.69471	−0.0650371
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−23.2140	−0.888260	−0.444130	−	0.895962i	$-0.646487\pi$
$683$	−23.2140	−0.888260	−0.444130	+	0.895962i	$0.646487\pi$
$684$	0	0
$685$	4.69759	0.179486
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−66.5864	−2.53674
$690$	0	0
$691$	30.7432	1.16953	0.584763	−	0.811204i	$-0.301187\pi$
$691$	30.7432	1.16953	0.584763	+	0.811204i	$0.301187\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.662362	−0.0251248
$696$	0	0
$697$	−48.5643	−1.83950
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−15.1624	−0.572675	−0.286338	−	0.958129i	$-0.592438\pi$
$701$	−15.1624	−0.572675	−0.286338	+	0.958129i	$0.592438\pi$
$702$	0	0
$703$	69.4977	2.62116
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−55.9379	−2.10376
$708$	0	0
$709$	−39.6115	−1.48764	−0.743821	−	0.668378i	$-0.766989\pi$
$709$	−39.6115	−1.48764	−0.743821	+	0.668378i	$0.766989\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−37.1646	−1.39182
$714$	0	0
$715$	16.9211	0.632812
$716$	0	0
$717$	0	0
$718$	0	0
$719$	35.7093	1.33173	0.665865	−	0.746072i	$-0.268063\pi$
$719$	35.7093	1.33173	0.665865	+	0.746072i	$0.268063\pi$
$720$	0	0
$721$	−25.6462	−0.955114
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.762947	−0.0283351
$726$	0	0
$727$	−25.6538	−0.951447	−0.475724	−	0.879595i	$-0.657814\pi$
$727$	−25.6538	−0.951447	−0.475724	+	0.879595i	$0.657814\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−52.1022	−1.92707
$732$	0	0
$733$	2.30529	0.0851477	0.0425739	−	0.999093i	$-0.486444\pi$
$733$	2.30529	0.0851477	0.0425739	+	0.999093i	$0.486444\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25.7309	−0.947810
$738$	0	0
$739$	−7.86679	−0.289385	−0.144692	−	0.989477i	$-0.546219\pi$
$739$	−7.86679	−0.289385	−0.144692	+	0.989477i	$0.546219\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.56507	0.277535	0.138768	−	0.990325i	$-0.455686\pi$
$743$	7.56507	0.277535	0.138768	+	0.990325i	$0.455686\pi$
$744$	0	0
$745$	2.16169	0.0791983
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−62.5892	−2.28696
$750$	0	0
$751$	50.6087	1.84674	0.923368	−	0.383916i	$-0.125425\pi$
$751$	50.6087	1.84674	0.923368	+	0.383916i	$0.125425\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.44793	−0.307452
$756$	0	0
$757$	−54.7114	−1.98852	−0.994259	−	0.106999i	$-0.965876\pi$
$757$	−54.7114	−1.98852	−0.994259	+	0.106999i	$0.965876\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35.9555	−1.30339	−0.651694	−	0.758482i	$-0.725941\pi$
$761$	−35.9555	−1.30339	−0.651694	+	0.758482i	$0.725941\pi$
$762$	0	0
$763$	22.4078	0.811217
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.8502	−0.391779
$768$	0	0
$769$	−2.02327	−0.0729610	−0.0364805	−	0.999334i	$-0.511615\pi$
$769$	−2.02327	−0.0729610	−0.0364805	+	0.999334i	$0.511615\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−19.5708	−0.703914	−0.351957	−	0.936016i	$-0.614484\pi$
$773$	−19.5708	−0.703914	−0.351957	+	0.936016i	$0.614484\pi$
$774$	0	0
$775$	−6.61675	−0.237681
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−56.2718	−2.01615
$780$	0	0
$781$	38.6312	1.38233
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.75111	−0.205266
$786$	0	0
$787$	0.980262	0.0349426	0.0174713	−	0.999847i	$-0.494438\pi$
$787$	0.980262	0.0349426	0.0174713	+	0.999847i	$0.494438\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	79.9421	2.84241
$792$	0	0
$793$	−10.2292	−0.363251
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.1869	0.431683	0.215841	−	0.976428i	$-0.430751\pi$
$797$	12.1869	0.431683	0.215841	+	0.976428i	$0.430751\pi$
$798$	0	0
$799$	28.1199	0.994811
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−28.6678	−1.01167
$804$	0	0
$805$	−24.4261	−0.860906
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−13.6760	−0.480823	−0.240412	−	0.970671i	$-0.577282\pi$
$809$	−13.6760	−0.480823	−0.240412	+	0.970671i	$0.577282\pi$
$810$	0	0
$811$	13.4256	0.471436	0.235718	−	0.971822i	$-0.424256\pi$
$811$	13.4256	0.471436	0.235718	+	0.971822i	$0.424256\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12.8438	0.449898
$816$	0	0
$817$	−60.3712	−2.11212
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19.9346	−0.695724	−0.347862	−	0.937546i	$-0.613092\pi$
$821$	−19.9346	−0.695724	−0.347862	+	0.937546i	$0.613092\pi$
$822$	0	0
$823$	−52.5526	−1.83187	−0.915935	−	0.401327i	$-0.868549\pi$
$823$	−52.5526	−1.83187	−0.915935	+	0.401327i	$0.868549\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−50.1251	−1.74302	−0.871511	−	0.490376i	$-0.836859\pi$
$827$	−50.1251	−1.74302	−0.871511	+	0.490376i	$0.836859\pi$
$828$	0	0
$829$	−13.1526	−0.456810	−0.228405	−	0.973566i	$-0.573351\pi$
$829$	−13.1526	−0.456810	−0.228405	+	0.973566i	$0.573351\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	64.0830	2.22035
$834$	0	0
$835$	−21.9085	−0.758174
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.3593	−0.840976	−0.420488	−	0.907298i	$-0.638141\pi$
$839$	−24.3593	−0.840976	−0.420488	+	0.907298i	$0.638141\pi$
$840$	0	0
$841$	−28.4179	−0.979928
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−26.2425	−0.902771
$846$	0	0
$847$	−16.1069	−0.553440
$848$	0	0
$849$	0	0
$850$	0	0
$851$	62.6216	2.14664
$852$	0	0
$853$	5.57795	0.190985	0.0954927	−	0.995430i	$-0.469557\pi$
$853$	5.57795	0.190985	0.0954927	+	0.995430i	$0.469557\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.03235	0.137742	0.0688712	−	0.997626i	$-0.478060\pi$
$857$	4.03235	0.137742	0.0688712	+	0.997626i	$0.478060\pi$
$858$	0	0
$859$	38.5550	1.31548	0.657739	−	0.753246i	$-0.271513\pi$
$859$	38.5550	1.31548	0.657739	+	0.753246i	$0.271513\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.3959	1.06873	0.534364	−	0.845255i	$-0.320551\pi$
$863$	31.3959	1.06873	0.534364	+	0.845255i	$0.320551\pi$
$864$	0	0
$865$	−2.68854	−0.0914132
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.44754	−0.0491046
$870$	0	0
$871$	59.6739	2.02197
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.34880	−0.147016
$876$	0	0
$877$	24.1208	0.814501	0.407251	−	0.913316i	$-0.366488\pi$
$877$	24.1208	0.814501	0.407251	+	0.913316i	$0.366488\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42.3270	−1.42603	−0.713016	−	0.701148i	$-0.752671\pi$
$881$	−42.3270	−1.42603	−0.713016	+	0.701148i	$0.752671\pi$
$882$	0	0
$883$	−44.6242	−1.50172	−0.750861	−	0.660460i	$-0.770361\pi$
$883$	−44.6242	−1.50172	−0.750861	+	0.660460i	$0.770361\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.5957	0.859419	0.429709	−	0.902967i	$-0.358616\pi$
$887$	25.5957	0.859419	0.429709	+	0.902967i	$0.358616\pi$
$888$	0	0
$889$	31.6462	1.06138
$890$	0	0
$891$	0	0
$892$	0	0
$893$	32.5828	1.09034
$894$	0	0
$895$	9.02733	0.301750
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.04822	0.168368
$900$	0	0
$901$	57.1827	1.90503
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.85024	0.194468
$906$	0	0
$907$	−29.4009	−0.976242	−0.488121	−	0.872776i	$-0.662318\pi$
$907$	−29.4009	−0.976242	−0.488121	+	0.872776i	$0.662318\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−14.6869	−0.486599	−0.243299	−	0.969951i	$-0.578230\pi$
$911$	−14.6869	−0.486599	−0.243299	+	0.969951i	$0.578230\pi$
$912$	0	0
$913$	0.394913	0.0130697
$914$	0	0
$915$	0	0
$916$	0	0
$917$	23.3312	0.770463
$918$	0	0
$919$	−0.353784	−0.0116703	−0.00583513	−	0.999983i	$-0.501857\pi$
$919$	−0.353784	−0.0116703	−0.00583513	+	0.999983i	$0.501857\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−89.5915	−2.94894
$924$	0	0
$925$	11.1491	0.366580
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27.3340	0.896801	0.448400	−	0.893833i	$-0.351994\pi$
$929$	27.3340	0.896801	0.448400	+	0.893833i	$0.351994\pi$
$930$	0	0
$931$	74.2535	2.43356
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−14.5314	−0.475227
$936$	0	0
$937$	−4.17594	−0.136422	−0.0682111	−	0.997671i	$-0.521729\pi$
$937$	−4.17594	−0.136422	−0.0682111	+	0.997671i	$0.521729\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−17.1969	−0.560604	−0.280302	−	0.959912i	$-0.590435\pi$
$941$	−17.1969	−0.560604	−0.280302	+	0.959912i	$0.590435\pi$
$942$	0	0
$943$	−50.7042	−1.65116
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.7086	1.06289	0.531443	−	0.847094i	$-0.321650\pi$
$947$	32.7086	1.06289	0.531443	+	0.847094i	$0.321650\pi$
$948$	0	0
$949$	66.4851	2.15820
$950$	0	0
$951$	0	0
$952$	0	0
$953$	33.1122	1.07261	0.536305	−	0.844024i	$-0.319820\pi$
$953$	33.1122	1.07261	0.536305	+	0.844024i	$0.319820\pi$
$954$	0	0
$955$	−12.9936	−0.420462
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−20.4289	−0.659683
$960$	0	0
$961$	12.7813	0.412301
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.51328	−0.145288
$966$	0	0
$967$	−49.5271	−1.59269	−0.796343	−	0.604846i	$-0.793235\pi$
$967$	−49.5271	−1.59269	−0.796343	+	0.604846i	$0.793235\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.82907	−0.154972	−0.0774862	−	0.996993i	$-0.524689\pi$
$971$	−4.82907	−0.154972	−0.0774862	+	0.996993i	$0.524689\pi$
$972$	0	0
$973$	2.88048	0.0923439
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14.9411	−0.478008	−0.239004	−	0.971019i	$-0.576821\pi$
$977$	−14.9411	−0.478008	−0.239004	+	0.971019i	$0.576821\pi$
$978$	0	0
$979$	21.2310	0.678545
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−50.5695	−1.61292	−0.806458	−	0.591291i	$-0.798618\pi$
$983$	−50.5695	−1.61292	−0.806458	+	0.591291i	$0.798618\pi$
$984$	0	0
$985$	16.8431	0.536666
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−54.3981	−1.72976
$990$	0	0
$991$	−37.4566	−1.18985	−0.594924	−	0.803782i	$-0.702818\pi$
$991$	−37.4566	−1.18985	−0.594924	+	0.803782i	$0.702818\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	11.6006	0.367763
$996$	0	0
$997$	−7.73822	−0.245072	−0.122536	−	0.992464i	$-0.539103\pi$
$997$	−7.73822	−0.245072	−0.122536	+	0.992464i	$0.539103\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.by.1.4		4
3.2	odd	2		6480.2.a.ca.1.4		4
4.3	odd	2		3240.2.a.t.1.1	✓	4
12.11	even	2		3240.2.a.v.1.1	yes	4
36.7	odd	6		3240.2.q.bh.1081.4		8
36.11	even	6		3240.2.q.bg.1081.4		8
36.23	even	6		3240.2.q.bg.2161.4		8
36.31	odd	6		3240.2.q.bh.2161.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3240.2.a.t.1.1	✓	4	4.3	odd	2
3240.2.a.v.1.1	yes	4	12.11	even	2
3240.2.q.bg.1081.4		8	36.11	even	6
3240.2.q.bg.2161.4		8	36.23	even	6
3240.2.q.bh.1081.4		8	36.7	odd	6
3240.2.q.bh.2161.4		8	36.31	odd	6
6480.2.a.by.1.4		4	1.1	even	1	trivial
6480.2.a.ca.1.4		4	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:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +4.34880 q^{7} +2.70115 q^{11} -6.26439 q^{13} +5.37969 q^{17} +6.23349 q^{19} +5.61675 q^{23} +1.00000 q^{25} -0.762947 q^{29} -6.61675 q^{31} -4.34880 q^{35} +11.1491 q^{37} -9.02733 q^{41} -9.68498 q^{43} +5.22705 q^{47} +11.9120 q^{49} +10.6294 q^{53} -2.70115 q^{55} +1.73205 q^{59} +1.63292 q^{61} +6.26439 q^{65} -9.52589 q^{67} +14.3017 q^{71} -10.6132 q^{73} +11.7468 q^{77} -0.535898 q^{79} +0.146201 q^{83} -5.37969 q^{85} +7.85997 q^{89} -27.2425 q^{91} -6.23349 q^{95} -0.389697 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 2 q^{7} + 4 q^{11} + 2 q^{17} + 10 q^{23} + 4 q^{25} + 4 q^{29} - 14 q^{31} + 2 q^{35} + 14 q^{37} - 4 q^{41} - 22 q^{43} + 10 q^{49} + 8 q^{53} - 4 q^{55} + 4 q^{61} - 24 q^{67} + 28 q^{71}+ \cdots - 10 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 2 * q^7 + 4 * q^11 + 2 * q^17 + 10 * q^23 + 4 * q^25 + 4 * q^29 - 14 * q^31 + 2 * q^35 + 14 * q^37 - 4 * q^41 - 22 * q^43 + 10 * q^49 + 8 * q^53 - 4 * q^55 + 4 * q^61 - 24 * q^67 + 28 * q^71 + 2 * q^73 + 30 * q^77 - 16 * q^79 + 6 * q^83 - 2 * q^85 + 8 * q^89 - 30 * q^91 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more