LMFDB - Embedded newform 2548.2.a.q.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2548,2,Mod(1,2548)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2548.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2548, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2548 = 2^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2548.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,3,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$20.3458824350$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.29268.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 5x + 16 $ x^4 - x^3 - 9x^2 + 5x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 364)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.43292$ of defining polynomial
Character	$\chi$	$=$	2548.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.43292 q^{3} +0.919111 q^{5} +8.78496 q^{9} +1.60425 q^{11} -1.00000 q^{13} +3.15524 q^{15} -2.72231 q^{17} -2.11806 q^{19} +5.43292 q^{23} -4.15524 q^{25} +19.8593 q^{27} -6.50727 q^{29} +5.66905 q^{31} +5.50727 q^{33} +9.43292 q^{37} -3.43292 q^{39} -2.35203 q^{41} -0.513812 q^{43} +8.07435 q^{45} +4.63187 q^{47} -9.34549 q^{51} +5.62972 q^{53} +1.47448 q^{55} -7.27114 q^{57} +7.33594 q^{59} -0.606405 q^{61} -0.919111 q^{65} -11.4518 q^{67} +18.6508 q^{69} -15.3476 q^{71} -10.1829 q^{73} -14.2646 q^{75} +8.52552 q^{79} +41.8206 q^{81} -13.0561 q^{83} -2.50211 q^{85} -22.3389 q^{87} -12.5306 q^{89} +19.4614 q^{93} -1.94673 q^{95} -7.21788 q^{97} +14.0933 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - q^{5} + 9 q^{9} + 4 q^{11} - 4 q^{13} - 9 q^{15} + 11 q^{23} + 5 q^{25} + 27 q^{27} + 11 q^{29} - 5 q^{31} - 15 q^{33} + 27 q^{37} - 3 q^{39} + 6 q^{41} + 4 q^{43} + 6 q^{45} + 4 q^{47}+ \cdots - 3 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - q^5 + 9 * q^9 + 4 * q^11 - 4 * q^13 - 9 * q^15 + 11 * q^23 + 5 * q^25 + 27 * q^27 + 11 * q^29 - 5 * q^31 - 15 * q^33 + 27 * q^37 - 3 * q^39 + 6 * q^41 + 4 * q^43 + 6 * q^45 + 4 * q^47 + 9 * q^51 + 18 * q^53 + 23 * q^55 - 9 * q^57 - 16 * q^59 + 10 * q^61 + q^65 - 2 * q^67 + 27 * q^69 - 5 * q^71 - 7 * q^73 - 6 * q^75 + 17 * q^79 + 24 * q^81 - 6 * q^83 - 15 * q^85 - 12 * q^87 - 21 * q^89 + 33 * q^93 + 9 * q^95 + 8 * q^97 - 3 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.43292	1.98200	0.990999	−	0.133867i	$-0.0427395\pi$
$3$	3.43292	1.98200	0.990999	+	0.133867i	$0.0427395\pi$
$4$	0	0
$5$	0.919111	0.411039	0.205519	−	0.978653i	$-0.434112\pi$
$5$	0.919111	0.411039	0.205519	+	0.978653i	$0.434112\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	8.78496	2.92832
$10$	0	0
$11$	1.60425	0.483700	0.241850	−	0.970314i	$-0.422246\pi$
$11$	1.60425	0.483700	0.241850	+	0.970314i	$0.422246\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	3.15524	0.814678
$16$	0	0
$17$	−2.72231	−0.660258	−0.330129	−	0.943936i	$-0.607092\pi$
$17$	−2.72231	−0.660258	−0.330129	+	0.943936i	$0.607092\pi$
$18$	0	0
$19$	−2.11806	−0.485917	−0.242958	−	0.970037i	$-0.578118\pi$
$19$	−2.11806	−0.485917	−0.242958	+	0.970037i	$0.578118\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.43292	1.13284	0.566421	−	0.824116i	$-0.308328\pi$
$23$	5.43292	1.13284	0.566421	+	0.824116i	$0.308328\pi$
$24$	0	0
$25$	−4.15524	−0.831047
$26$	0	0
$27$	19.8593	3.82192
$28$	0	0
$29$	−6.50727	−1.20837	−0.604185	−	0.796844i	$-0.706501\pi$
$29$	−6.50727	−1.20837	−0.604185	+	0.796844i	$0.706501\pi$
$30$	0	0
$31$	5.66905	1.01819	0.509095	−	0.860710i	$-0.329980\pi$
$31$	5.66905	1.01819	0.509095	+	0.860710i	$0.329980\pi$
$32$	0	0
$33$	5.50727	0.958692
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.43292	1.55076	0.775381	−	0.631493i	$-0.217558\pi$
$37$	9.43292	1.55076	0.775381	+	0.631493i	$0.217558\pi$
$38$	0	0
$39$	−3.43292	−0.549708
$40$	0	0
$41$	−2.35203	−0.367326	−0.183663	−	0.982989i	$-0.558795\pi$
$41$	−2.35203	−0.367326	−0.183663	+	0.982989i	$0.558795\pi$
$42$	0	0
$43$	−0.513812	−0.0783555	−0.0391778	−	0.999232i	$-0.512474\pi$
$43$	−0.513812	−0.0783555	−0.0391778	+	0.999232i	$0.512474\pi$
$44$	0	0
$45$	8.07435	1.20365
$46$	0	0
$47$	4.63187	0.675628	0.337814	−	0.941213i	$-0.390312\pi$
$47$	4.63187	0.675628	0.337814	+	0.941213i	$0.390312\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−9.34549	−1.30863
$52$	0	0
$53$	5.62972	0.773301	0.386651	−	0.922226i	$-0.373632\pi$
$53$	5.62972	0.773301	0.386651	+	0.922226i	$0.373632\pi$
$54$	0	0
$55$	1.47448	0.198819
$56$	0	0
$57$	−7.27114	−0.963087
$58$	0	0
$59$	7.33594	0.955058	0.477529	−	0.878616i	$-0.341533\pi$
$59$	7.33594	0.955058	0.477529	+	0.878616i	$0.341533\pi$
$60$	0	0
$61$	−0.606405	−0.0776422	−0.0388211	−	0.999246i	$-0.512360\pi$
$61$	−0.606405	−0.0776422	−0.0388211	+	0.999246i	$0.512360\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.919111	−0.114002
$66$	0	0
$67$	−11.4518	−1.39907	−0.699533	−	0.714600i	$-0.746609\pi$
$67$	−11.4518	−1.39907	−0.699533	+	0.714600i	$0.746609\pi$
$68$	0	0
$69$	18.6508	2.24529
$70$	0	0
$71$	−15.3476	−1.82143	−0.910715	−	0.413035i	$-0.864469\pi$
$71$	−15.3476	−1.82143	−0.910715	+	0.413035i	$0.864469\pi$
$72$	0	0
$73$	−10.1829	−1.19181	−0.595907	−	0.803054i	$-0.703207\pi$
$73$	−10.1829	−1.19181	−0.595907	+	0.803054i	$0.703207\pi$
$74$	0	0
$75$	−14.2646	−1.64713
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.52552	0.959195	0.479598	−	0.877488i	$-0.340783\pi$
$79$	8.52552	0.959195	0.479598	+	0.877488i	$0.340783\pi$
$80$	0	0
$81$	41.8206	4.64673
$82$	0	0
$83$	−13.0561	−1.43309	−0.716547	−	0.697539i	$-0.754278\pi$
$83$	−13.0561	−1.43309	−0.716547	+	0.697539i	$0.754278\pi$
$84$	0	0
$85$	−2.50211	−0.271392
$86$	0	0
$87$	−22.3389	−2.39499
$88$	0	0
$89$	−12.5306	−1.32824	−0.664120	−	0.747626i	$-0.731193\pi$
$89$	−12.5306	−1.32824	−0.664120	+	0.747626i	$0.731193\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	19.4614	2.01805
$94$	0	0
$95$	−1.94673	−0.199731
$96$	0	0
$97$	−7.21788	−0.732864	−0.366432	−	0.930445i	$-0.619421\pi$
$97$	−7.21788	−0.732864	−0.366432	+	0.930445i	$0.619421\pi$
$98$	0	0
$99$	14.0933	1.41643
$100$	0	0
$101$	0.757332	0.0753574	0.0376787	−	0.999290i	$-0.488004\pi$
$101$	0.757332	0.0753574	0.0376787	+	0.999290i	$0.488004\pi$
$102$	0	0
$103$	−0.330953	−0.0326098	−0.0163049	−	0.999867i	$-0.505190\pi$
$103$	−0.330953	−0.0326098	−0.0163049	+	0.999867i	$0.505190\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.2529	−0.991185	−0.495592	−	0.868555i	$-0.665049\pi$
$107$	−10.2529	−0.991185	−0.495592	+	0.868555i	$0.665049\pi$
$108$	0	0
$109$	−7.97521	−0.763887	−0.381943	−	0.924186i	$-0.624745\pi$
$109$	−7.97521	−0.763887	−0.381943	+	0.924186i	$0.624745\pi$
$110$	0	0
$111$	32.3825	3.07361
$112$	0	0
$113$	14.8308	1.39517	0.697583	−	0.716504i	$-0.254259\pi$
$113$	14.8308	1.39517	0.697583	+	0.716504i	$0.254259\pi$
$114$	0	0
$115$	4.99346	0.465642
$116$	0	0
$117$	−8.78496	−0.812169
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.42638	−0.766034
$122$	0	0
$123$	−8.07435	−0.728039
$124$	0	0
$125$	−8.41467	−0.752631
$126$	0	0
$127$	19.0102	1.68688	0.843442	−	0.537219i	$-0.180525\pi$
$127$	19.0102	1.68688	0.843442	+	0.537219i	$0.180525\pi$
$128$	0	0
$129$	−1.76388	−0.155301
$130$	0	0
$131$	13.7850	1.20440	0.602199	−	0.798346i	$-0.294292\pi$
$131$	13.7850	1.20440	0.602199	+	0.798346i	$0.294292\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	18.2529	1.57096
$136$	0	0
$137$	6.68299	0.570966	0.285483	−	0.958384i	$-0.407846\pi$
$137$	6.68299	0.570966	0.285483	+	0.958384i	$0.407846\pi$
$138$	0	0
$139$	−1.25944	−0.106824	−0.0534121	−	0.998573i	$-0.517010\pi$
$139$	−1.25944	−0.106824	−0.0534121	+	0.998573i	$0.517010\pi$
$140$	0	0
$141$	15.9009	1.33909
$142$	0	0
$143$	−1.60425	−0.134154
$144$	0	0
$145$	−5.98090	−0.496687
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.26177	0.431061	0.215530	−	0.976497i	$-0.430852\pi$
$149$	5.26177	0.431061	0.215530	+	0.976497i	$0.430852\pi$
$150$	0	0
$151$	13.7754	1.12103	0.560513	−	0.828145i	$-0.310604\pi$
$151$	13.7754	1.12103	0.560513	+	0.828145i	$0.310604\pi$
$152$	0	0
$153$	−23.9154	−1.93345
$154$	0	0
$155$	5.21048	0.418516
$156$	0	0
$157$	1.22675	0.0979052	0.0489526	−	0.998801i	$-0.484412\pi$
$157$	1.22675	0.0979052	0.0489526	+	0.998801i	$0.484412\pi$
$158$	0	0
$159$	19.3264	1.53268
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−15.4562	−1.21062	−0.605310	−	0.795990i	$-0.706951\pi$
$163$	−15.4562	−1.21062	−0.605310	+	0.795990i	$0.706951\pi$
$164$	0	0
$165$	5.06179	0.394060
$166$	0	0
$167$	−23.7091	−1.83466	−0.917331	−	0.398126i	$-0.869661\pi$
$167$	−23.7091	−1.83466	−0.917331	+	0.398126i	$0.869661\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−18.6071	−1.42292
$172$	0	0
$173$	−14.1763	−1.07781	−0.538903	−	0.842368i	$-0.681161\pi$
$173$	−14.1763	−1.07781	−0.538903	+	0.842368i	$0.681161\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	25.1837	1.89292
$178$	0	0
$179$	−6.46055	−0.482884	−0.241442	−	0.970415i	$-0.577620\pi$
$179$	−6.46055	−0.482884	−0.241442	+	0.970415i	$0.577620\pi$
$180$	0	0
$181$	11.9476	0.888057	0.444029	−	0.896013i	$-0.353549\pi$
$181$	11.9476	0.888057	0.444029	+	0.896013i	$0.353549\pi$
$182$	0	0
$183$	−2.08174	−0.153887
$184$	0	0
$185$	8.66990	0.637424
$186$	0	0
$187$	−4.36727	−0.319367
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.17855	0.157634	0.0788172	−	0.996889i	$-0.474886\pi$
$191$	2.17855	0.157634	0.0788172	+	0.996889i	$0.474886\pi$
$192$	0	0
$193$	−20.2988	−1.46114	−0.730569	−	0.682839i	$-0.760745\pi$
$193$	−20.2988	−1.46114	−0.730569	+	0.682839i	$0.760745\pi$
$194$	0	0
$195$	−3.15524	−0.225951
$196$	0	0
$197$	−2.00938	−0.143162	−0.0715810	−	0.997435i	$-0.522804\pi$
$197$	−2.00938	−0.143162	−0.0715810	+	0.997435i	$0.522804\pi$
$198$	0	0
$199$	−2.64142	−0.187246	−0.0936228	−	0.995608i	$-0.529845\pi$
$199$	−2.64142	−0.187246	−0.0936228	+	0.995608i	$0.529845\pi$
$200$	0	0
$201$	−39.3133	−2.77295
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.16178	−0.150985
$206$	0	0
$207$	47.7280	3.31732
$208$	0	0
$209$	−3.39790	−0.235038
$210$	0	0
$211$	9.92194	0.683055	0.341527	−	0.939872i	$-0.389056\pi$
$211$	9.92194	0.683055	0.341527	+	0.939872i	$0.389056\pi$
$212$	0	0
$213$	−52.6873	−3.61007
$214$	0	0
$215$	−0.472250	−0.0322072
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−34.9570	−2.36217
$220$	0	0
$221$	2.72231	0.183123
$222$	0	0
$223$	−9.20402	−0.616347	−0.308173	−	0.951330i	$-0.599718\pi$
$223$	−9.20402	−0.616347	−0.308173	+	0.951330i	$0.599718\pi$
$224$	0	0
$225$	−36.5036	−2.43357
$226$	0	0
$227$	−20.5349	−1.36295	−0.681474	−	0.731842i	$-0.738661\pi$
$227$	−20.5349	−1.36295	−0.681474	+	0.731842i	$0.738661\pi$
$228$	0	0
$229$	1.79433	0.118573	0.0592864	−	0.998241i	$-0.481117\pi$
$229$	1.79433	0.118573	0.0592864	+	0.998241i	$0.481117\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.8702	1.43276	0.716381	−	0.697710i	$-0.245797\pi$
$233$	21.8702	1.43276	0.716381	+	0.697710i	$0.245797\pi$
$234$	0	0
$235$	4.25720	0.277709
$236$	0	0
$237$	29.2674	1.90112
$238$	0	0
$239$	12.3468	0.798648	0.399324	−	0.916810i	$-0.369245\pi$
$239$	12.3468	0.798648	0.399324	+	0.916810i	$0.369245\pi$
$240$	0	0
$241$	−13.9795	−0.900500	−0.450250	−	0.892903i	$-0.648665\pi$
$241$	−13.9795	−0.900500	−0.450250	+	0.892903i	$0.648665\pi$
$242$	0	0
$243$	83.9889	5.38789
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.11806	0.134769
$248$	0	0
$249$	−44.8206	−2.84039
$250$	0	0
$251$	−5.72231	−0.361189	−0.180595	−	0.983558i	$-0.557802\pi$
$251$	−5.72231	−0.361189	−0.180595	+	0.983558i	$0.557802\pi$
$252$	0	0
$253$	8.71577	0.547956
$254$	0	0
$255$	−8.58954	−0.537898
$256$	0	0
$257$	−8.88262	−0.554082	−0.277041	−	0.960858i	$-0.589354\pi$
$257$	−8.88262	−0.554082	−0.277041	+	0.960858i	$0.589354\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−57.1661	−3.53849
$262$	0	0
$263$	16.9242	1.04359	0.521795	−	0.853071i	$-0.325263\pi$
$263$	16.9242	1.04359	0.521795	+	0.853071i	$0.325263\pi$
$264$	0	0
$265$	5.17434	0.317857
$266$	0	0
$267$	−43.0165	−2.63257
$268$	0	0
$269$	−1.89432	−0.115499	−0.0577494	−	0.998331i	$-0.518392\pi$
$269$	−1.89432	−0.115499	−0.0577494	+	0.998331i	$0.518392\pi$
$270$	0	0
$271$	−10.4664	−0.635788	−0.317894	−	0.948126i	$-0.602976\pi$
$271$	−10.4664	−0.635788	−0.317894	+	0.948126i	$0.602976\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.66604	−0.401977
$276$	0	0
$277$	−11.2871	−0.678174	−0.339087	−	0.940755i	$-0.610118\pi$
$277$	−11.2871	−0.678174	−0.339087	+	0.940755i	$0.610118\pi$
$278$	0	0
$279$	49.8023	2.98159
$280$	0	0
$281$	14.5517	0.868079	0.434040	−	0.900894i	$-0.357088\pi$
$281$	14.5517	0.868079	0.434040	+	0.900894i	$0.357088\pi$
$282$	0	0
$283$	13.5357	0.804616	0.402308	−	0.915504i	$-0.368208\pi$
$283$	13.5357	0.804616	0.402308	+	0.915504i	$0.368208\pi$
$284$	0	0
$285$	−6.68299	−0.395866
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−9.58901	−0.564059
$290$	0	0
$291$	−24.7784	−1.45254
$292$	0	0
$293$	14.4816	0.846026	0.423013	−	0.906124i	$-0.360973\pi$
$293$	14.4816	0.846026	0.423013	+	0.906124i	$0.360973\pi$
$294$	0	0
$295$	6.74254	0.392566
$296$	0	0
$297$	31.8593	1.84866
$298$	0	0
$299$	−5.43292	−0.314194
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.59986	0.149358
$304$	0	0
$305$	−0.557353	−0.0319140
$306$	0	0
$307$	−14.6530	−0.836288	−0.418144	−	0.908381i	$-0.637319\pi$
$307$	−14.6530	−0.836288	−0.418144	+	0.908381i	$0.637319\pi$
$308$	0	0
$309$	−1.13614	−0.0646325
$310$	0	0
$311$	−13.8265	−0.784030	−0.392015	−	0.919959i	$-0.628222\pi$
$311$	−13.8265	−0.784030	−0.392015	+	0.919959i	$0.628222\pi$
$312$	0	0
$313$	17.9658	1.01549	0.507744	−	0.861508i	$-0.330479\pi$
$313$	17.9658	1.01549	0.507744	+	0.861508i	$0.330479\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.64057	0.204475	0.102237	−	0.994760i	$-0.467400\pi$
$317$	3.64057	0.204475	0.102237	+	0.994760i	$0.467400\pi$
$318$	0	0
$319$	−10.4393	−0.584488
$320$	0	0
$321$	−35.1974	−1.96453
$322$	0	0
$323$	5.76603	0.320830
$324$	0	0
$325$	4.15524	0.230491
$326$	0	0
$327$	−27.3783	−1.51402
$328$	0	0
$329$	0	0
$330$	0	0
$331$	13.5625	0.745463	0.372732	−	0.927939i	$-0.378421\pi$
$331$	13.5625	0.745463	0.372732	+	0.927939i	$0.378421\pi$
$332$	0	0
$333$	82.8678	4.54113
$334$	0	0
$335$	−10.5255	−0.575070
$336$	0	0
$337$	−16.1880	−0.881818	−0.440909	−	0.897552i	$-0.645344\pi$
$337$	−16.1880	−0.881818	−0.440909	+	0.897552i	$0.645344\pi$
$338$	0	0
$339$	50.9131	2.76522
$340$	0	0
$341$	9.09457	0.492499
$342$	0	0
$343$	0	0
$344$	0	0
$345$	17.1421	0.922902
$346$	0	0
$347$	14.1922	0.761879	0.380940	−	0.924600i	$-0.375601\pi$
$347$	14.1922	0.761879	0.380940	+	0.924600i	$0.375601\pi$
$348$	0	0
$349$	−4.22675	−0.226253	−0.113126	−	0.993581i	$-0.536086\pi$
$349$	−4.22675	−0.226253	−0.113126	+	0.993581i	$0.536086\pi$
$350$	0	0
$351$	−19.8593	−1.06001
$352$	0	0
$353$	29.2390	1.55623	0.778116	−	0.628120i	$-0.216175\pi$
$353$	29.2390	1.55623	0.778116	+	0.628120i	$0.216175\pi$
$354$	0	0
$355$	−14.1062	−0.748679
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.29446	0.437765	0.218882	−	0.975751i	$-0.429759\pi$
$359$	8.29446	0.437765	0.218882	+	0.975751i	$0.429759\pi$
$360$	0	0
$361$	−14.5138	−0.763885
$362$	0	0
$363$	−28.9271	−1.51828
$364$	0	0
$365$	−9.35917	−0.489882
$366$	0	0
$367$	−24.7251	−1.29064	−0.645321	−	0.763911i	$-0.723276\pi$
$367$	−24.7251	−1.29064	−0.645321	+	0.763911i	$0.723276\pi$
$368$	0	0
$369$	−20.6625	−1.07565
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7.93082	−0.410642	−0.205321	−	0.978695i	$-0.565824\pi$
$373$	−7.93082	−0.410642	−0.205321	+	0.978695i	$0.565824\pi$
$374$	0	0
$375$	−28.8869	−1.49171
$376$	0	0
$377$	6.50727	0.335141
$378$	0	0
$379$	−34.5050	−1.77240	−0.886202	−	0.463299i	$-0.846666\pi$
$379$	−34.5050	−1.77240	−0.886202	+	0.463299i	$0.846666\pi$
$380$	0	0
$381$	65.2606	3.34340
$382$	0	0
$383$	15.9286	0.813912	0.406956	−	0.913448i	$-0.366590\pi$
$383$	15.9286	0.813912	0.406956	+	0.913448i	$0.366590\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.51381	−0.229450
$388$	0	0
$389$	8.09690	0.410529	0.205265	−	0.978707i	$-0.434194\pi$
$389$	8.09690	0.410529	0.205265	+	0.978707i	$0.434194\pi$
$390$	0	0
$391$	−14.7901	−0.747968
$392$	0	0
$393$	47.3227	2.38711
$394$	0	0
$395$	7.83589	0.394267
$396$	0	0
$397$	−31.5910	−1.58551	−0.792753	−	0.609543i	$-0.791353\pi$
$397$	−31.5910	−1.58551	−0.792753	+	0.609543i	$0.791353\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.41382	0.0706029	0.0353015	−	0.999377i	$-0.488761\pi$
$401$	1.41382	0.0706029	0.0353015	+	0.999377i	$0.488761\pi$
$402$	0	0
$403$	−5.66905	−0.282395
$404$	0	0
$405$	38.4377	1.90999
$406$	0	0
$407$	15.1328	0.750104
$408$	0	0
$409$	18.4699	0.913279	0.456639	−	0.889652i	$-0.349053\pi$
$409$	18.4699	0.913279	0.456639	+	0.889652i	$0.349053\pi$
$410$	0	0
$411$	22.9422	1.13165
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	−4.32356	−0.211726
$418$	0	0
$419$	−15.1529	−0.740268	−0.370134	−	0.928978i	$-0.620688\pi$
$419$	−15.1529	−0.740268	−0.370134	+	0.928978i	$0.620688\pi$
$420$	0	0
$421$	15.8804	0.773962	0.386981	−	0.922088i	$-0.373518\pi$
$421$	15.8804	0.773962	0.386981	+	0.922088i	$0.373518\pi$
$422$	0	0
$423$	40.6908	1.97845
$424$	0	0
$425$	11.3119	0.548705
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−5.50727	−0.265893
$430$	0	0
$431$	−15.0028	−0.722661	−0.361331	−	0.932438i	$-0.617677\pi$
$431$	−15.0028	−0.722661	−0.361331	+	0.932438i	$0.617677\pi$
$432$	0	0
$433$	9.81258	0.471562	0.235781	−	0.971806i	$-0.424235\pi$
$433$	9.81258	0.471562	0.235781	+	0.971806i	$0.424235\pi$
$434$	0	0
$435$	−20.5320	−0.984432
$436$	0	0
$437$	−11.5073	−0.550467
$438$	0	0
$439$	1.40445	0.0670306	0.0335153	−	0.999438i	$-0.489330\pi$
$439$	1.40445	0.0670306	0.0335153	+	0.999438i	$0.489330\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	29.2352	1.38901	0.694504	−	0.719489i	$-0.255624\pi$
$443$	29.2352	1.38901	0.694504	+	0.719489i	$0.255624\pi$
$444$	0	0
$445$	−11.5170	−0.545958
$446$	0	0
$447$	18.0632	0.854362
$448$	0	0
$449$	9.13416	0.431067	0.215534	−	0.976496i	$-0.430851\pi$
$449$	9.13416	0.431067	0.215534	+	0.976496i	$0.430851\pi$
$450$	0	0
$451$	−3.77325	−0.177675
$452$	0	0
$453$	47.2899	2.22187
$454$	0	0
$455$	0	0
$456$	0	0
$457$	40.9342	1.91482	0.957409	−	0.288734i	$-0.0932342\pi$
$457$	40.9342	1.91482	0.957409	+	0.288734i	$0.0932342\pi$
$458$	0	0
$459$	−54.0632	−2.52346
$460$	0	0
$461$	−33.4731	−1.55900	−0.779499	−	0.626404i	$-0.784526\pi$
$461$	−33.4731	−1.55900	−0.779499	+	0.626404i	$0.784526\pi$
$462$	0	0
$463$	−6.94390	−0.322710	−0.161355	−	0.986896i	$-0.551586\pi$
$463$	−6.94390	−0.322710	−0.161355	+	0.986896i	$0.551586\pi$
$464$	0	0
$465$	17.8872	0.829498
$466$	0	0
$467$	−12.1872	−0.563955	−0.281977	−	0.959421i	$-0.590990\pi$
$467$	−12.1872	−0.563955	−0.281977	+	0.959421i	$0.590990\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4.21133	0.194048
$472$	0	0
$473$	−0.824283	−0.0379006
$474$	0	0
$475$	8.80105	0.403820
$476$	0	0
$477$	49.4568	2.26447
$478$	0	0
$479$	8.08528	0.369426	0.184713	−	0.982793i	$-0.440864\pi$
$479$	8.08528	0.369426	0.184713	+	0.982793i	$0.440864\pi$
$480$	0	0
$481$	−9.43292	−0.430104
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.63403	−0.301236
$486$	0	0
$487$	−26.4205	−1.19723	−0.598614	−	0.801038i	$-0.704282\pi$
$487$	−26.4205	−1.19723	−0.598614	+	0.801038i	$0.704282\pi$
$488$	0	0
$489$	−53.0598	−2.39945
$490$	0	0
$491$	21.9504	0.990609	0.495304	−	0.868720i	$-0.335057\pi$
$491$	21.9504	0.990609	0.495304	+	0.868720i	$0.335057\pi$
$492$	0	0
$493$	17.7148	0.797836
$494$	0	0
$495$	12.9533	0.582207
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.76671	0.392452	0.196226	−	0.980559i	$-0.437131\pi$
$499$	8.76671	0.392452	0.196226	+	0.980559i	$0.437131\pi$
$500$	0	0
$501$	−81.3913	−3.63630
$502$	0	0
$503$	19.7135	0.878983	0.439492	−	0.898247i	$-0.355159\pi$
$503$	19.7135	0.878983	0.439492	+	0.898247i	$0.355159\pi$
$504$	0	0
$505$	0.696072	0.0309748
$506$	0	0
$507$	3.43292	0.152461
$508$	0	0
$509$	21.1094	0.935656	0.467828	−	0.883819i	$-0.345037\pi$
$509$	21.1094	0.935656	0.467828	+	0.883819i	$0.345037\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−42.0632	−1.85714
$514$	0	0
$515$	−0.304182	−0.0134039
$516$	0	0
$517$	7.43069	0.326801
$518$	0	0
$519$	−48.6662	−2.13621
$520$	0	0
$521$	−30.1481	−1.32081	−0.660405	−	0.750909i	$-0.729616\pi$
$521$	−30.1481	−1.32081	−0.660405	+	0.750909i	$0.729616\pi$
$522$	0	0
$523$	−0.280015	−0.0122442	−0.00612211	−	0.999981i	$-0.501949\pi$
$523$	−0.280015	−0.0122442	−0.00612211	+	0.999981i	$0.501949\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.4329	−0.672269
$528$	0	0
$529$	6.51664	0.283332
$530$	0	0
$531$	64.4459	2.79671
$532$	0	0
$533$	2.35203	0.101878
$534$	0	0
$535$	−9.42355	−0.407415
$536$	0	0
$537$	−22.1786	−0.957075
$538$	0	0
$539$	0	0
$540$	0	0
$541$	10.5677	0.454340	0.227170	−	0.973855i	$-0.427053\pi$
$541$	10.5677	0.454340	0.227170	+	0.973855i	$0.427053\pi$
$542$	0	0
$543$	41.0151	1.76013
$544$	0	0
$545$	−7.33010	−0.313987
$546$	0	0
$547$	−21.0940	−0.901912	−0.450956	−	0.892546i	$-0.648917\pi$
$547$	−21.0940	−0.901912	−0.450956	+	0.892546i	$0.648917\pi$
$548$	0	0
$549$	−5.32724	−0.227361
$550$	0	0
$551$	13.7828	0.587167
$552$	0	0
$553$	0	0
$554$	0	0
$555$	29.7631	1.26337
$556$	0	0
$557$	40.3694	1.71051	0.855253	−	0.518210i	$-0.173402\pi$
$557$	40.3694	1.71051	0.855253	+	0.518210i	$0.173402\pi$
$558$	0	0
$559$	0.513812	0.0217319
$560$	0	0
$561$	−14.9925	−0.632984
$562$	0	0
$563$	−30.8504	−1.30019	−0.650095	−	0.759853i	$-0.725271\pi$
$563$	−30.8504	−1.30019	−0.650095	+	0.759853i	$0.725271\pi$
$564$	0	0
$565$	13.6312	0.573468
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.42553	−0.353216	−0.176608	−	0.984281i	$-0.556513\pi$
$569$	−8.42553	−0.353216	−0.176608	+	0.984281i	$0.556513\pi$
$570$	0	0
$571$	3.16030	0.132255	0.0661273	−	0.997811i	$-0.478936\pi$
$571$	3.16030	0.132255	0.0661273	+	0.997811i	$0.478936\pi$
$572$	0	0
$573$	7.47879	0.312431
$574$	0	0
$575$	−22.5751	−0.941446
$576$	0	0
$577$	15.9286	0.663115	0.331558	−	0.943435i	$-0.392426\pi$
$577$	15.9286	0.663115	0.331558	+	0.943435i	$0.392426\pi$
$578$	0	0
$579$	−69.6841	−2.89597
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9.03148	0.374046
$584$	0	0
$585$	−8.07435	−0.333833
$586$	0	0
$587$	−25.0415	−1.03357	−0.516786	−	0.856114i	$-0.672872\pi$
$587$	−25.0415	−1.03357	−0.516786	+	0.856114i	$0.672872\pi$
$588$	0	0
$589$	−12.0074	−0.494756
$590$	0	0
$591$	−6.89803	−0.283747
$592$	0	0
$593$	22.1973	0.911534	0.455767	−	0.890099i	$-0.349365\pi$
$593$	22.1973	0.911534	0.455767	+	0.890099i	$0.349365\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−9.06780	−0.371120
$598$	0	0
$599$	−36.4394	−1.48888	−0.744438	−	0.667692i	$-0.767282\pi$
$599$	−36.4394	−1.48888	−0.744438	+	0.667692i	$0.767282\pi$
$600$	0	0
$601$	−36.3090	−1.48107	−0.740537	−	0.672015i	$-0.765429\pi$
$601$	−36.3090	−1.48107	−0.740537	+	0.672015i	$0.765429\pi$
$602$	0	0
$603$	−100.604	−4.09691
$604$	0	0
$605$	−7.74478	−0.314870
$606$	0	0
$607$	−8.85043	−0.359228	−0.179614	−	0.983737i	$-0.557485\pi$
$607$	−8.85043	−0.359228	−0.179614	+	0.983737i	$0.557485\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.63187	−0.187386
$612$	0	0
$613$	20.9761	0.847215	0.423608	−	0.905846i	$-0.360764\pi$
$613$	20.9761	0.847215	0.423608	+	0.905846i	$0.360764\pi$
$614$	0	0
$615$	−7.42122	−0.299252
$616$	0	0
$617$	−18.4816	−0.744042	−0.372021	−	0.928224i	$-0.621335\pi$
$617$	−18.4816	−0.744042	−0.372021	+	0.928224i	$0.621335\pi$
$618$	0	0
$619$	19.8134	0.796369	0.398185	−	0.917305i	$-0.369640\pi$
$619$	19.8134	0.796369	0.398185	+	0.917305i	$0.369640\pi$
$620$	0	0
$621$	107.894	4.32964
$622$	0	0
$623$	0	0
$624$	0	0
$625$	13.0422	0.521686
$626$	0	0
$627$	−11.6647	−0.465845
$628$	0	0
$629$	−25.6794	−1.02390
$630$	0	0
$631$	8.40898	0.334756	0.167378	−	0.985893i	$-0.446470\pi$
$631$	8.40898	0.334756	0.167378	+	0.985893i	$0.446470\pi$
$632$	0	0
$633$	34.0613	1.35381
$634$	0	0
$635$	17.4725	0.693375
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−134.828	−5.33373
$640$	0	0
$641$	−30.0267	−1.18599	−0.592993	−	0.805208i	$-0.702054\pi$
$641$	−30.0267	−1.18599	−0.592993	+	0.805208i	$0.702054\pi$
$642$	0	0
$643$	−11.1349	−0.439119	−0.219559	−	0.975599i	$-0.570462\pi$
$643$	−11.1349	−0.439119	−0.219559	+	0.975599i	$0.570462\pi$
$644$	0	0
$645$	−1.62120	−0.0638346
$646$	0	0
$647$	−28.4525	−1.11858	−0.559292	−	0.828971i	$-0.688927\pi$
$647$	−28.4525	−1.11858	−0.559292	+	0.828971i	$0.688927\pi$
$648$	0	0
$649$	11.7687	0.461961
$650$	0	0
$651$	0	0
$652$	0	0
$653$	33.6610	1.31726	0.658629	−	0.752468i	$-0.271137\pi$
$653$	33.6610	1.31726	0.658629	+	0.752468i	$0.271137\pi$
$654$	0	0
$655$	12.6699	0.495054
$656$	0	0
$657$	−89.4560	−3.49001
$658$	0	0
$659$	17.2594	0.672332	0.336166	−	0.941803i	$-0.390870\pi$
$659$	17.2594	0.672332	0.336166	+	0.941803i	$0.390870\pi$
$660$	0	0
$661$	−24.8280	−0.965696	−0.482848	−	0.875704i	$-0.660398\pi$
$661$	−24.8280	−0.965696	−0.482848	+	0.875704i	$0.660398\pi$
$662$	0	0
$663$	9.34549	0.362949
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−35.3535	−1.36889
$668$	0	0
$669$	−31.5967	−1.22160
$670$	0	0
$671$	−0.972826	−0.0375555
$672$	0	0
$673$	−12.7567	−0.491735	−0.245867	−	0.969303i	$-0.579073\pi$
$673$	−12.7567	−0.491735	−0.245867	+	0.969303i	$0.579073\pi$
$674$	0	0
$675$	−82.5201	−3.17620
$676$	0	0
$677$	−13.3292	−0.512284	−0.256142	−	0.966639i	$-0.582451\pi$
$677$	−13.3292	−0.512284	−0.256142	+	0.966639i	$0.582451\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−70.4947	−2.70136
$682$	0	0
$683$	−34.3175	−1.31312	−0.656562	−	0.754272i	$-0.727990\pi$
$683$	−34.3175	−1.31312	−0.656562	+	0.754272i	$0.727990\pi$
$684$	0	0
$685$	6.14240	0.234689
$686$	0	0
$687$	6.15980	0.235011
$688$	0	0
$689$	−5.62972	−0.214475
$690$	0	0
$691$	−7.65088	−0.291053	−0.145527	−	0.989354i	$-0.546488\pi$
$691$	−7.65088	−0.291053	−0.145527	+	0.989354i	$0.546488\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.15756	−0.0439089
$696$	0	0
$697$	6.40297	0.242530
$698$	0	0
$699$	75.0785	2.83973
$700$	0	0
$701$	32.7331	1.23631	0.618157	−	0.786055i	$-0.287880\pi$
$701$	32.7331	1.23631	0.618157	+	0.786055i	$0.287880\pi$
$702$	0	0
$703$	−19.9795	−0.753542
$704$	0	0
$705$	14.6147	0.550420
$706$	0	0
$707$	0	0
$708$	0	0
$709$	3.67877	0.138159	0.0690796	−	0.997611i	$-0.477994\pi$
$709$	3.67877	0.138159	0.0690796	+	0.997611i	$0.477994\pi$
$710$	0	0
$711$	74.8963	2.80883
$712$	0	0
$713$	30.7995	1.15345
$714$	0	0
$715$	−1.47448	−0.0551426
$716$	0	0
$717$	42.3856	1.58292
$718$	0	0
$719$	11.3777	0.424316	0.212158	−	0.977235i	$-0.431951\pi$
$719$	11.3777	0.424316	0.212158	+	0.977235i	$0.431951\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−47.9906	−1.78479
$724$	0	0
$725$	27.0392	1.00421
$726$	0	0
$727$	−42.4642	−1.57491	−0.787456	−	0.616371i	$-0.788602\pi$
$727$	−42.4642	−1.57491	−0.787456	+	0.616371i	$0.788602\pi$
$728$	0	0
$729$	162.866	6.03206
$730$	0	0
$731$	1.39876	0.0517349
$732$	0	0
$733$	−1.37768	−0.0508856	−0.0254428	−	0.999676i	$-0.508100\pi$
$733$	−1.37768	−0.0508856	−0.0254428	+	0.999676i	$0.508100\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18.3716	−0.676728
$738$	0	0
$739$	8.66018	0.318570	0.159285	−	0.987233i	$-0.449081\pi$
$739$	8.66018	0.318570	0.159285	+	0.987233i	$0.449081\pi$
$740$	0	0
$741$	7.27114	0.267112
$742$	0	0
$743$	47.1952	1.73143	0.865713	−	0.500541i	$-0.166866\pi$
$743$	47.1952	1.73143	0.865713	+	0.500541i	$0.166866\pi$
$744$	0	0
$745$	4.83615	0.177183
$746$	0	0
$747$	−114.697	−4.19655
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16.4518	0.600334	0.300167	−	0.953887i	$-0.402958\pi$
$751$	16.4518	0.600334	0.300167	+	0.953887i	$0.402958\pi$
$752$	0	0
$753$	−19.6443	−0.715877
$754$	0	0
$755$	12.6611	0.460785
$756$	0	0
$757$	−11.4147	−0.414873	−0.207437	−	0.978248i	$-0.566512\pi$
$757$	−11.4147	−0.414873	−0.207437	+	0.978248i	$0.566512\pi$
$758$	0	0
$759$	29.9206	1.08605
$760$	0	0
$761$	−10.3038	−0.373514	−0.186757	−	0.982406i	$-0.559798\pi$
$761$	−10.3038	−0.373514	−0.186757	+	0.982406i	$0.559798\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−21.9809	−0.794721
$766$	0	0
$767$	−7.33594	−0.264885
$768$	0	0
$769$	37.3716	1.34766	0.673828	−	0.738888i	$-0.264649\pi$
$769$	37.3716	1.34766	0.673828	+	0.738888i	$0.264649\pi$
$770$	0	0
$771$	−30.4933	−1.09819
$772$	0	0
$773$	16.2301	0.583756	0.291878	−	0.956456i	$-0.405720\pi$
$773$	16.2301	0.583756	0.291878	+	0.956456i	$0.405720\pi$
$774$	0	0
$775$	−23.5562	−0.846165
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.98175	0.178490
$780$	0	0
$781$	−24.6215	−0.881026
$782$	0	0
$783$	−129.230	−4.61830
$784$	0	0
$785$	1.12752	0.0402429
$786$	0	0
$787$	−2.13251	−0.0760156	−0.0380078	−	0.999277i	$-0.512101\pi$
$787$	−2.13251	−0.0760156	−0.0380078	+	0.999277i	$0.512101\pi$
$788$	0	0
$789$	58.0994	2.06839
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0.606405	0.0215341
$794$	0	0
$795$	17.7631	0.629992
$796$	0	0
$797$	38.0248	1.34691	0.673453	−	0.739230i	$-0.264810\pi$
$797$	38.0248	1.34691	0.673453	+	0.739230i	$0.264810\pi$
$798$	0	0
$799$	−12.6094	−0.446089
$800$	0	0
$801$	−110.081	−3.88951
$802$	0	0
$803$	−16.3359	−0.576480
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.50305	−0.228918
$808$	0	0
$809$	32.4774	1.14184	0.570922	−	0.821004i	$-0.306586\pi$
$809$	32.4774	1.14184	0.570922	+	0.821004i	$0.306586\pi$
$810$	0	0
$811$	12.0285	0.422377	0.211188	−	0.977445i	$-0.432267\pi$
$811$	12.0285	0.422377	0.211188	+	0.977445i	$0.432267\pi$
$812$	0	0
$813$	−35.9303	−1.26013
$814$	0	0
$815$	−14.2059	−0.497612
$816$	0	0
$817$	1.08828	0.0380743
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.8590	1.35619	0.678095	−	0.734974i	$-0.262806\pi$
$821$	38.8590	1.35619	0.678095	+	0.734974i	$0.262806\pi$
$822$	0	0
$823$	−17.5947	−0.613313	−0.306656	−	0.951820i	$-0.599210\pi$
$823$	−17.5947	−0.613313	−0.306656	+	0.951820i	$0.599210\pi$
$824$	0	0
$825$	−22.8840	−0.796719
$826$	0	0
$827$	30.5575	1.06259	0.531294	−	0.847187i	$-0.321706\pi$
$827$	30.5575	1.06259	0.531294	+	0.847187i	$0.321706\pi$
$828$	0	0
$829$	52.6164	1.82744	0.913721	−	0.406342i	$-0.133196\pi$
$829$	52.6164	1.82744	0.913721	+	0.406342i	$0.133196\pi$
$830$	0	0
$831$	−38.7476	−1.34414
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−21.7912	−0.754117
$836$	0	0
$837$	112.583	3.89145
$838$	0	0
$839$	−19.6341	−0.677845	−0.338922	−	0.940814i	$-0.610062\pi$
$839$	−19.6341	−0.677845	−0.338922	+	0.940814i	$0.610062\pi$
$840$	0	0
$841$	13.3445	0.460157
$842$	0	0
$843$	49.9547	1.72053
$844$	0	0
$845$	0.919111	0.0316184
$846$	0	0
$847$	0	0
$848$	0	0
$849$	46.4672	1.59475
$850$	0	0
$851$	51.2483	1.75677
$852$	0	0
$853$	0.00430866	0.000147526 0	7.37629e−5	−	1.00000i	$-0.499977\pi$
$853$	0.00430866	0.000147526 0	7.37629e−5		1.00000i	$0.499977\pi$
$854$	0	0
$855$	−17.1020	−0.584875
$856$	0	0
$857$	6.19972	0.211779	0.105889	−	0.994378i	$-0.466231\pi$
$857$	6.19972	0.211779	0.105889	+	0.994378i	$0.466231\pi$
$858$	0	0
$859$	−13.0504	−0.445274	−0.222637	−	0.974901i	$-0.571466\pi$
$859$	−13.0504	−0.445274	−0.222637	+	0.974901i	$0.571466\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0.284324	0.00967850	0.00483925	−	0.999988i	$-0.498460\pi$
$863$	0.284324	0.00967850	0.00483925	+	0.999988i	$0.498460\pi$
$864$	0	0
$865$	−13.0296	−0.443020
$866$	0	0
$867$	−32.9183	−1.11797
$868$	0	0
$869$	13.6771	0.463963
$870$	0	0
$871$	11.4518	0.388031
$872$	0	0
$873$	−63.4087	−2.14606
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27.9940	−0.945289	−0.472644	−	0.881253i	$-0.656701\pi$
$877$	−27.9940	−0.945289	−0.472644	+	0.881253i	$0.656701\pi$
$878$	0	0
$879$	49.7143	1.67682
$880$	0	0
$881$	3.18975	0.107465	0.0537327	−	0.998555i	$-0.482888\pi$
$881$	3.18975	0.107465	0.0537327	+	0.998555i	$0.482888\pi$
$882$	0	0
$883$	39.6084	1.33293	0.666464	−	0.745537i	$-0.267807\pi$
$883$	39.6084	1.33293	0.666464	+	0.745537i	$0.267807\pi$
$884$	0	0
$885$	23.1466	0.778065
$886$	0	0
$887$	7.37768	0.247718	0.123859	−	0.992300i	$-0.460473\pi$
$887$	7.37768	0.247718	0.123859	+	0.992300i	$0.460473\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	67.0907	2.24762
$892$	0	0
$893$	−9.81060	−0.328299
$894$	0	0
$895$	−5.93796	−0.198484
$896$	0	0
$897$	−18.6508	−0.622732
$898$	0	0
$899$	−36.8900	−1.23035
$900$	0	0
$901$	−15.3259	−0.510578
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.9812	0.365026
$906$	0	0
$907$	2.28473	0.0758633	0.0379317	−	0.999280i	$-0.487923\pi$
$907$	2.28473	0.0758633	0.0379317	+	0.999280i	$0.487923\pi$
$908$	0	0
$909$	6.65313	0.220670
$910$	0	0
$911$	−2.47732	−0.0820772	−0.0410386	−	0.999158i	$-0.513067\pi$
$911$	−2.47732	−0.0820772	−0.0410386	+	0.999158i	$0.513067\pi$
$912$	0	0
$913$	−20.9453	−0.693187
$914$	0	0
$915$	−1.91335	−0.0632534
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−38.2173	−1.26067	−0.630337	−	0.776322i	$-0.717083\pi$
$919$	−38.2173	−1.26067	−0.630337	+	0.776322i	$0.717083\pi$
$920$	0	0
$921$	−50.3025	−1.65752
$922$	0	0
$923$	15.3476	0.505174
$924$	0	0
$925$	−39.1960	−1.28876
$926$	0	0
$927$	−2.90741	−0.0954918
$928$	0	0
$929$	20.6428	0.677268	0.338634	−	0.940918i	$-0.390035\pi$
$929$	20.6428	0.677268	0.338634	+	0.940918i	$0.390035\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−47.4654	−1.55395
$934$	0	0
$935$	−4.01401	−0.131272
$936$	0	0
$937$	−46.3418	−1.51392	−0.756960	−	0.653461i	$-0.773316\pi$
$937$	−46.3418	−1.51392	−0.756960	+	0.653461i	$0.773316\pi$
$938$	0	0
$939$	61.6753	2.01270
$940$	0	0
$941$	32.6297	1.06370	0.531849	−	0.846839i	$-0.321497\pi$
$941$	32.6297	1.06370	0.531849	+	0.846839i	$0.321497\pi$
$942$	0	0
$943$	−12.7784	−0.416122
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17.2107	−0.559271	−0.279636	−	0.960106i	$-0.590214\pi$
$947$	−17.2107	−0.559271	−0.279636	+	0.960106i	$0.590214\pi$
$948$	0	0
$949$	10.1829	0.330550
$950$	0	0
$951$	12.4978	0.405269
$952$	0	0
$953$	50.0504	1.62129	0.810646	−	0.585537i	$-0.199116\pi$
$953$	50.0504	1.62129	0.810646	+	0.585537i	$0.199116\pi$
$954$	0	0
$955$	2.00233	0.0647938
$956$	0	0
$957$	−35.8373	−1.15845
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.13810	0.0367127
$962$	0	0
$963$	−90.0712	−2.90250
$964$	0	0
$965$	−18.6568	−0.600584
$966$	0	0
$967$	−15.9540	−0.513045	−0.256522	−	0.966538i	$-0.582577\pi$
$967$	−15.9540	−0.513045	−0.256522	+	0.966538i	$0.582577\pi$
$968$	0	0
$969$	19.7943	0.635886
$970$	0	0
$971$	5.09398	0.163473	0.0817367	−	0.996654i	$-0.473953\pi$
$971$	5.09398	0.163473	0.0817367	+	0.996654i	$0.473953\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	14.2646	0.456833
$976$	0	0
$977$	29.7850	0.952905	0.476453	−	0.879200i	$-0.341922\pi$
$977$	29.7850	0.952905	0.476453	+	0.879200i	$0.341922\pi$
$978$	0	0
$979$	−20.1022	−0.642469
$980$	0	0
$981$	−70.0619	−2.23690
$982$	0	0
$983$	−25.7771	−0.822162	−0.411081	−	0.911599i	$-0.634849\pi$
$983$	−25.7771	−0.822162	−0.411081	+	0.911599i	$0.634849\pi$
$984$	0	0
$985$	−1.84684	−0.0588452
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.79150	−0.0887645
$990$	0	0
$991$	−18.1153	−0.575451	−0.287725	−	0.957713i	$-0.592899\pi$
$991$	−18.1153	−0.575451	−0.287725	+	0.957713i	$0.592899\pi$
$992$	0	0
$993$	46.5591	1.47751
$994$	0	0
$995$	−2.42776	−0.0769652
$996$	0	0
$997$	14.6955	0.465413	0.232706	−	0.972547i	$-0.425242\pi$
$997$	14.6955	0.465413	0.232706	+	0.972547i	$0.425242\pi$
$998$	0	0
$999$	187.331	5.92690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2548.2.a.q.1.4		4
7.2	even	3		364.2.j.e.53.1	✓	8
7.3	odd	6		2548.2.j.q.1353.4		8
7.4	even	3		364.2.j.e.261.1	yes	8
7.5	odd	6		2548.2.j.q.1145.4		8
7.6	odd	2		2548.2.a.p.1.1		4
21.2	odd	6		3276.2.r.j.1873.3		8
21.11	odd	6		3276.2.r.j.2809.3		8
28.11	odd	6		1456.2.r.o.625.4		8
28.23	odd	6		1456.2.r.o.417.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.j.e.53.1	✓	8	7.2	even	3
364.2.j.e.261.1	yes	8	7.4	even	3
1456.2.r.o.417.4		8	28.23	odd	6
1456.2.r.o.625.4		8	28.11	odd	6
2548.2.a.p.1.1		4	7.6	odd	2
2548.2.a.q.1.4		4	1.1	even	1	trivial
2548.2.j.q.1145.4		8	7.5	odd	6
2548.2.j.q.1353.4		8	7.3	odd	6
3276.2.r.j.1873.3		8	21.2	odd	6
3276.2.r.j.2809.3		8	21.11	odd	6

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:	\( N \)	\(=\)	\( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2548.a (trivial)

\(f(q)\)	\(=\)	\(q+3.43292 q^{3} +0.919111 q^{5} +8.78496 q^{9} +1.60425 q^{11} -1.00000 q^{13} +3.15524 q^{15} -2.72231 q^{17} -2.11806 q^{19} +5.43292 q^{23} -4.15524 q^{25} +19.8593 q^{27} -6.50727 q^{29} +5.66905 q^{31} +5.50727 q^{33} +9.43292 q^{37} -3.43292 q^{39} -2.35203 q^{41} -0.513812 q^{43} +8.07435 q^{45} +4.63187 q^{47} -9.34549 q^{51} +5.62972 q^{53} +1.47448 q^{55} -7.27114 q^{57} +7.33594 q^{59} -0.606405 q^{61} -0.919111 q^{65} -11.4518 q^{67} +18.6508 q^{69} -15.3476 q^{71} -10.1829 q^{73} -14.2646 q^{75} +8.52552 q^{79} +41.8206 q^{81} -13.0561 q^{83} -2.50211 q^{85} -22.3389 q^{87} -12.5306 q^{89} +19.4614 q^{93} -1.94673 q^{95} -7.21788 q^{97} +14.0933 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - q^{5} + 9 q^{9} + 4 q^{11} - 4 q^{13} - 9 q^{15} + 11 q^{23} + 5 q^{25} + 27 q^{27} + 11 q^{29} - 5 q^{31} - 15 q^{33} + 27 q^{37} - 3 q^{39} + 6 q^{41} + 4 q^{43} + 6 q^{45} + 4 q^{47}+ \cdots - 3 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - q^5 + 9 * q^9 + 4 * q^11 - 4 * q^13 - 9 * q^15 + 11 * q^23 + 5 * q^25 + 27 * q^27 + 11 * q^29 - 5 * q^31 - 15 * q^33 + 27 * q^37 - 3 * q^39 + 6 * q^41 + 4 * q^43 + 6 * q^45 + 4 * q^47 + 9 * q^51 + 18 * q^53 + 23 * q^55 - 9 * q^57 - 16 * q^59 + 10 * q^61 + q^65 - 2 * q^67 + 27 * q^69 - 5 * q^71 - 7 * q^73 - 6 * q^75 + 17 * q^79 + 24 * q^81 - 6 * q^83 - 15 * q^85 - 12 * q^87 - 21 * q^89 + 33 * q^93 + 9 * q^95 + 8 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more