LMFDB - Embedded newform 2704.2.f.o.337.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2704,2,Mod(337,2704)] 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("2704.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2704, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,4,0,0,0,0,0,-6,0,0,0,0,0,0,0,4,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$21.5915487066$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 169)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$1.24698i$ of defining polynomial
Character	$\chi$	$=$	2704.337
Dual form			2704.2.f.o.337.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.801938 q^{3} -2.80194i q^{5} -2.69202i q^{7} -2.35690 q^{9} -1.19806i q^{11} +2.24698i q^{15} -1.13706 q^{17} -1.93900i q^{19} +2.15883i q^{21} -4.60388 q^{23} -2.85086 q^{25} +4.29590 q^{27} -7.89977 q^{29} -5.89977i q^{31} +0.960771i q^{33} -7.54288 q^{35} -0.951083i q^{37} +3.31767i q^{41} +7.15883 q^{43} +6.60388i q^{45} -7.69202i q^{47} -0.246980 q^{49} +0.911854 q^{51} +5.87263 q^{53} -3.35690 q^{55} +1.55496i q^{57} -0.0120816i q^{59} -8.03684 q^{61} +6.34481i q^{63} +9.25667i q^{67} +3.69202 q^{69} +13.7409i q^{71} -12.8170i q^{73} +2.28621 q^{75} -3.22521 q^{77} -0.807315 q^{79} +3.62565 q^{81} +16.3327i q^{83} +3.18598i q^{85} +6.33513 q^{87} +14.7289i q^{89} +4.73125i q^{93} -5.43296 q^{95} +3.13169i q^{97} +2.82371i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 6 q^{9} + 4 q^{17} - 10 q^{23} + 10 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{35} + 26 q^{43} + 8 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} + 8 q^{61} + 12 q^{69} + 30 q^{75} - 16 q^{77} + 10 q^{79}+ \cdots + 6 q^{95}+O(q^{100}) $ 6 * q + 4 * q^3 - 6 * q^9 + 4 * q^17 - 10 * q^23 + 10 * q^25 - 2 * q^27 - 2 * q^29 - 8 * q^35 + 26 * q^43 + 8 * q^49 - 2 * q^51 + 2 * q^53 - 12 * q^55 + 8 * q^61 + 12 * q^69 + 30 * q^75 - 16 * q^77 + 10 * q^79 - 2 * q^81 + 36 * q^87 + 6 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times$.

$n$	$677$	$1185$	$2367$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	−0.801938	−0.462999	−0.231499	−	0.972835i	$-0.574363\pi$
$3$	−0.801938	−0.462999	−0.231499	+	0.972835i	$0.574363\pi$
$4$	0	0
$5$	− 2.80194i	− 1.25306i	−0.779395	−	0.626532i	$-0.784474\pi$
$5$	− 2.80194i	− 1.25306i	0.779395	−	0.626532i	$-0.215526\pi$
$6$	0	0
$7$	− 2.69202i	− 1.01749i	−0.860918	−	0.508744i	$-0.830110\pi$
$7$	− 2.69202i	− 1.01749i	0.860918	−	0.508744i	$-0.169890\pi$
$8$	0	0
$9$	−2.35690	−0.785632
$10$	0	0
$11$	− 1.19806i	− 0.361229i	−0.983554	−	0.180615i	$-0.942191\pi$
$11$	− 1.19806i	− 0.361229i	0.983554	−	0.180615i	$-0.0578087\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.24698i	0.580168i
$16$	0	0
$17$	−1.13706	−0.275778	−0.137889	−	0.990448i	$-0.544032\pi$
$17$	−1.13706	−0.275778	−0.137889	+	0.990448i	$0.544032\pi$
$18$	0	0
$19$	− 1.93900i	− 0.444837i	−0.974951	−	0.222419i	$-0.928605\pi$
$19$	− 1.93900i	− 0.444837i	0.974951	−	0.222419i	$-0.0713952\pi$
$20$	0	0
$21$	2.15883i	0.471096i
$22$	0	0
$23$	−4.60388	−0.959974	−0.479987	−	0.877275i	$-0.659359\pi$
$23$	−4.60388	−0.959974	−0.479987	+	0.877275i	$0.659359\pi$
$24$	0	0
$25$	−2.85086	−0.570171
$26$	0	0
$27$	4.29590	0.826746
$28$	0	0
$29$	−7.89977	−1.46695	−0.733475	−	0.679716i	$-0.762103\pi$
$29$	−7.89977	−1.46695	−0.733475	+	0.679716i	$0.762103\pi$
$30$	0	0
$31$	− 5.89977i	− 1.05963i	−0.848113	−	0.529815i	$-0.822261\pi$
$31$	− 5.89977i	− 1.05963i	0.848113	−	0.529815i	$-0.177739\pi$
$32$	0	0
$33$	0.960771i	0.167249i
$34$	0	0
$35$	−7.54288	−1.27498
$36$	0	0
$37$	− 0.951083i	− 0.156357i	−0.996939	−	0.0781785i	$-0.975090\pi$
$37$	− 0.951083i	− 0.156357i	0.996939	−	0.0781785i	$-0.0249104\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.31767i	0.518133i	0.965860	+	0.259066i	$0.0834148\pi$
$41$	3.31767i	0.518133i	−0.965860	+	0.259066i	$0.916585\pi$
$42$	0	0
$43$	7.15883	1.09171	0.545856	−	0.837879i	$-0.316205\pi$
$43$	7.15883	1.09171	0.545856	+	0.837879i	$0.316205\pi$
$44$	0	0
$45$	6.60388i	0.984448i
$46$	0	0
$47$	− 7.69202i	− 1.12200i	−0.827817	−	0.560998i	$-0.810417\pi$
$47$	− 7.69202i	− 1.12200i	0.827817	−	0.560998i	$-0.189583\pi$
$48$	0	0
$49$	−0.246980	−0.0352828
$50$	0	0
$51$	0.911854	0.127685
$52$	0	0
$53$	5.87263	0.806667	0.403334	−	0.915053i	$-0.367851\pi$
$53$	5.87263	0.806667	0.403334	+	0.915053i	$0.367851\pi$
$54$	0	0
$55$	−3.35690	−0.452644
$56$	0	0
$57$	1.55496i	0.205959i
$58$	0	0
$59$	− 0.0120816i	− 0.00157289i	−1.00000		0.000786444i	$-0.999750\pi$
$59$	− 0.0120816i	− 0.00157289i	1.00000		0.000786444i	$-0.000250333\pi$
$60$	0	0
$61$	−8.03684	−1.02901	−0.514506	−	0.857487i	$-0.672025\pi$
$61$	−8.03684	−1.02901	−0.514506	+	0.857487i	$0.672025\pi$
$62$	0	0
$63$	6.34481i	0.799371i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.25667i	1.13088i	0.824789	+	0.565441i	$0.191294\pi$
$67$	9.25667i	1.13088i	−0.824789	+	0.565441i	$0.808706\pi$
$68$	0	0
$69$	3.69202	0.444467
$70$	0	0
$71$	13.7409i	1.63075i	0.578934	+	0.815375i	$0.303469\pi$
$71$	13.7409i	1.63075i	−0.578934	+	0.815375i	$0.696531\pi$
$72$	0	0
$73$	− 12.8170i	− 1.50012i	−0.661372	−	0.750058i	$-0.730025\pi$
$73$	− 12.8170i	− 1.50012i	0.661372	−	0.750058i	$-0.269975\pi$
$74$	0	0
$75$	2.28621	0.263989
$76$	0	0
$77$	−3.22521	−0.367547
$78$	0	0
$79$	−0.807315	−0.0908300	−0.0454150	−	0.998968i	$-0.514461\pi$
$79$	−0.807315	−0.0908300	−0.0454150	+	0.998968i	$0.514461\pi$
$80$	0	0
$81$	3.62565	0.402850
$82$	0	0
$83$	16.3327i	1.79275i	0.443296	+	0.896375i	$0.353809\pi$
$83$	16.3327i	1.79275i	−0.443296	+	0.896375i	$0.646191\pi$
$84$	0	0
$85$	3.18598i	0.345568i
$86$	0	0
$87$	6.33513	0.679197
$88$	0	0
$89$	14.7289i	1.56126i	0.624996	+	0.780628i	$0.285101\pi$
$89$	14.7289i	1.56126i	−0.624996	+	0.780628i	$0.714899\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.73125i	0.490608i
$94$	0	0
$95$	−5.43296	−0.557410
$96$	0	0
$97$	3.13169i	0.317975i	0.987281	+	0.158987i	$0.0508229\pi$
$97$	3.13169i	0.317975i	−0.987281	+	0.158987i	$0.949177\pi$
$98$	0	0
$99$	2.82371i	0.283793i
$100$	0	0
$101$	−5.29052	−0.526426	−0.263213	−	0.964738i	$-0.584782\pi$
$101$	−5.29052	−0.526426	−0.263213	+	0.964738i	$0.584782\pi$
$102$	0	0
$103$	−13.5308	−1.33323	−0.666614	−	0.745403i	$-0.732257\pi$
$103$	−13.5308	−1.33323	−0.666614	+	0.745403i	$0.732257\pi$
$104$	0	0
$105$	6.04892	0.590314
$106$	0	0
$107$	−5.63102	−0.544371	−0.272186	−	0.962245i	$-0.587747\pi$
$107$	−5.63102	−0.544371	−0.272186	+	0.962245i	$0.587747\pi$
$108$	0	0
$109$	4.17629i	0.400016i	0.979794	+	0.200008i	$0.0640969\pi$
$109$	4.17629i	0.400016i	−0.979794	+	0.200008i	$0.935903\pi$
$110$	0	0
$111$	0.762709i	0.0723931i
$112$	0	0
$113$	7.64310	0.719003	0.359501	−	0.933145i	$-0.382947\pi$
$113$	7.64310	0.719003	0.359501	+	0.933145i	$0.382947\pi$
$114$	0	0
$115$	12.8998i	1.20291i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.06100i	0.280601i
$120$	0	0
$121$	9.56465	0.869513
$122$	0	0
$123$	− 2.66056i	− 0.239895i
$124$	0	0
$125$	− 6.02177i	− 0.538604i
$126$	0	0
$127$	6.77777	0.601430	0.300715	−	0.953714i	$-0.402775\pi$
$127$	6.77777	0.601430	0.300715	+	0.953714i	$0.402775\pi$
$128$	0	0
$129$	−5.74094	−0.505461
$130$	0	0
$131$	13.6799	1.19522	0.597611	−	0.801786i	$-0.296117\pi$
$131$	13.6799	1.19522	0.597611	+	0.801786i	$0.296117\pi$
$132$	0	0
$133$	−5.21983	−0.452617
$134$	0	0
$135$	− 12.0368i	− 1.03597i
$136$	0	0
$137$	− 12.9879i	− 1.10963i	−0.831973	−	0.554816i	$-0.812789\pi$
$137$	− 12.9879i	− 1.10963i	0.831973	−	0.554816i	$-0.187211\pi$
$138$	0	0
$139$	−12.0465	−1.02177	−0.510886	−	0.859648i	$-0.670683\pi$
$139$	−12.0465	−1.02177	−0.510886	+	0.859648i	$0.670683\pi$
$140$	0	0
$141$	6.16852i	0.519483i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	22.1347i	1.83818i
$146$	0	0
$147$	0.198062	0.0163359
$148$	0	0
$149$	− 0.740939i	− 0.0607001i	−0.999539	−	0.0303500i	$-0.990338\pi$
$149$	− 0.740939i	− 0.0607001i	0.999539	−	0.0303500i	$-0.00966220\pi$
$150$	0	0
$151$	− 19.0737i	− 1.55219i	−0.630614	−	0.776097i	$-0.717197\pi$
$151$	− 19.0737i	− 1.55219i	0.630614	−	0.776097i	$-0.282803\pi$
$152$	0	0
$153$	2.67994	0.216660
$154$	0	0
$155$	−16.5308	−1.32779
$156$	0	0
$157$	−4.02177	−0.320972	−0.160486	−	0.987038i	$-0.551306\pi$
$157$	−4.02177	−0.320972	−0.160486	+	0.987038i	$0.551306\pi$
$158$	0	0
$159$	−4.70948	−0.373486
$160$	0	0
$161$	12.3937i	0.976763i
$162$	0	0
$163$	15.1371i	1.18563i	0.805340	+	0.592813i	$0.201983\pi$
$163$	15.1371i	1.18563i	−0.805340	+	0.592813i	$0.798017\pi$
$164$	0	0
$165$	2.69202	0.209574
$166$	0	0
$167$	− 6.26337i	− 0.484674i	−0.970192	−	0.242337i	$-0.922086\pi$
$167$	− 6.26337i	− 0.484674i	0.970192	−	0.242337i	$-0.0779140\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	4.57002i	0.349478i
$172$	0	0
$173$	−16.3913	−1.24621	−0.623105	−	0.782138i	$-0.714129\pi$
$173$	−16.3913	−1.24621	−0.623105	+	0.782138i	$0.714129\pi$
$174$	0	0
$175$	7.67456i	0.580142i
$176$	0	0
$177$	0.00968868i	0 0.000728246i
$178$	0	0
$179$	−2.45473	−0.183475	−0.0917376	−	0.995783i	$-0.529242\pi$
$179$	−2.45473	−0.183475	−0.0917376	+	0.995783i	$0.529242\pi$
$180$	0	0
$181$	−11.8073	−0.877631	−0.438815	−	0.898577i	$-0.644602\pi$
$181$	−11.8073	−0.877631	−0.438815	+	0.898577i	$0.644602\pi$
$182$	0	0
$183$	6.44504	0.476431
$184$	0	0
$185$	−2.66487	−0.195925
$186$	0	0
$187$	1.36227i	0.0996192i
$188$	0	0
$189$	− 11.5646i	− 0.841204i
$190$	0	0
$191$	8.99330	0.650732	0.325366	−	0.945588i	$-0.394512\pi$
$191$	8.99330	0.650732	0.325366	+	0.945588i	$0.394512\pi$
$192$	0	0
$193$	− 13.5254i	− 0.973581i	−0.873519	−	0.486790i	$-0.838168\pi$
$193$	− 13.5254i	− 0.973581i	0.873519	−	0.486790i	$-0.161832\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.9758i	0.924490i	0.886752	+	0.462245i	$0.152956\pi$
$197$	12.9758i	0.924490i	−0.886752	+	0.462245i	$0.847044\pi$
$198$	0	0
$199$	−13.5864	−0.963116	−0.481558	−	0.876414i	$-0.659929\pi$
$199$	−13.5864	−0.963116	−0.481558	+	0.876414i	$0.659929\pi$
$200$	0	0
$201$	− 7.42327i	− 0.523597i
$202$	0	0
$203$	21.2664i	1.49261i
$204$	0	0
$205$	9.29590	0.649254
$206$	0	0
$207$	10.8509	0.754187
$208$	0	0
$209$	−2.32304	−0.160688
$210$	0	0
$211$	−10.4601	−0.720103	−0.360052	−	0.932932i	$-0.617241\pi$
$211$	−10.4601	−0.720103	−0.360052	+	0.932932i	$0.617241\pi$
$212$	0	0
$213$	− 11.0194i	− 0.755035i
$214$	0	0
$215$	− 20.0586i	− 1.36799i
$216$	0	0
$217$	−15.8823	−1.07816
$218$	0	0
$219$	10.2784i	0.694553i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	11.4058i	0.763790i	0.924206	+	0.381895i	$0.124728\pi$
$223$	11.4058i	0.763790i	−0.924206	+	0.381895i	$0.875272\pi$
$224$	0	0
$225$	6.71917	0.447945
$226$	0	0
$227$	− 10.6407i	− 0.706249i	−0.935576	−	0.353124i	$-0.885119\pi$
$227$	− 10.6407i	− 0.706249i	0.935576	−	0.353124i	$-0.114881\pi$
$228$	0	0
$229$	1.13946i	0.0752974i	0.999291	+	0.0376487i	$0.0119868\pi$
$229$	1.13946i	0.0752974i	−0.999291	+	0.0376487i	$0.988013\pi$
$230$	0	0
$231$	2.58642	0.170174
$232$	0	0
$233$	10.8509	0.710863	0.355432	−	0.934702i	$-0.384334\pi$
$233$	10.8509	0.710863	0.355432	+	0.934702i	$0.384334\pi$
$234$	0	0
$235$	−21.5526	−1.40593
$236$	0	0
$237$	0.647416	0.0420542
$238$	0	0
$239$	11.9293i	0.771643i	0.922573	+	0.385822i	$0.126082\pi$
$239$	11.9293i	0.771643i	−0.922573	+	0.385822i	$0.873918\pi$
$240$	0	0
$241$	3.64848i	0.235019i	0.993072	+	0.117510i	$0.0374911\pi$
$241$	3.64848i	0.235019i	−0.993072	+	0.117510i	$0.962509\pi$
$242$	0	0
$243$	−15.7952	−1.01326
$244$	0	0
$245$	0.692021i	0.0442116i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 13.0978i	− 0.830042i
$250$	0	0
$251$	1.37329	0.0866813	0.0433406	−	0.999060i	$-0.486200\pi$
$251$	1.37329	0.0866813	0.0433406	+	0.999060i	$0.486200\pi$
$252$	0	0
$253$	5.51573i	0.346771i
$254$	0	0
$255$	− 2.55496i	− 0.159998i
$256$	0	0
$257$	−29.4359	−1.83616	−0.918082	−	0.396391i	$-0.870263\pi$
$257$	−29.4359	−1.83616	−0.918082	+	0.396391i	$0.870263\pi$
$258$	0	0
$259$	−2.56033	−0.159091
$260$	0	0
$261$	18.6189	1.15248
$262$	0	0
$263$	−10.6963	−0.659564	−0.329782	−	0.944057i	$-0.606975\pi$
$263$	−10.6963	−0.659564	−0.329782	+	0.944057i	$0.606975\pi$
$264$	0	0
$265$	− 16.4547i	− 1.01081i
$266$	0	0
$267$	− 11.8116i	− 0.722860i
$268$	0	0
$269$	−10.1860	−0.621050	−0.310525	−	0.950565i	$-0.600505\pi$
$269$	−10.1860	−0.621050	−0.310525	+	0.950565i	$0.600505\pi$
$270$	0	0
$271$	− 29.4523i	− 1.78910i	−0.446966	−	0.894551i	$-0.647495\pi$
$271$	− 29.4523i	− 1.78910i	0.446966	−	0.894551i	$-0.352505\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.41550i	0.205963i
$276$	0	0
$277$	10.2446	0.615538	0.307769	−	0.951461i	$-0.400418\pi$
$277$	10.2446	0.615538	0.307769	+	0.951461i	$0.400418\pi$
$278$	0	0
$279$	13.9051i	0.832480i
$280$	0	0
$281$	11.5646i	0.689889i	0.938623	+	0.344944i	$0.112102\pi$
$281$	11.5646i	0.689889i	−0.938623	+	0.344944i	$0.887898\pi$
$282$	0	0
$283$	−30.7090	−1.82546	−0.912730	−	0.408562i	$-0.866030\pi$
$283$	−30.7090	−1.82546	−0.912730	+	0.408562i	$0.866030\pi$
$284$	0	0
$285$	4.35690	0.258080
$286$	0	0
$287$	8.93123	0.527194
$288$	0	0
$289$	−15.7071	−0.923946
$290$	0	0
$291$	− 2.51142i	− 0.147222i
$292$	0	0
$293$	18.6082i	1.08710i	0.839376	+	0.543551i	$0.182921\pi$
$293$	18.6082i	1.08710i	−0.839376	+	0.543551i	$0.817079\pi$
$294$	0	0
$295$	−0.0338518	−0.00197093
$296$	0	0
$297$	− 5.14675i	− 0.298645i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 19.2717i	− 1.11080i
$302$	0	0
$303$	4.24267	0.243735
$304$	0	0
$305$	22.5187i	1.28942i
$306$	0	0
$307$	8.94438i	0.510483i	0.966877	+	0.255241i	$0.0821549\pi$
$307$	8.94438i	0.510483i	−0.966877	+	0.255241i	$0.917845\pi$
$308$	0	0
$309$	10.8509	0.617284
$310$	0	0
$311$	21.0398	1.19306	0.596529	−	0.802591i	$-0.296546\pi$
$311$	21.0398	1.19306	0.596529	+	0.802591i	$0.296546\pi$
$312$	0	0
$313$	−7.12737	−0.402863	−0.201432	−	0.979503i	$-0.564559\pi$
$313$	−7.12737	−0.402863	−0.201432	+	0.979503i	$0.564559\pi$
$314$	0	0
$315$	17.7778	1.00166
$316$	0	0
$317$	23.9651i	1.34601i	0.739636	+	0.673007i	$0.234997\pi$
$317$	23.9651i	1.34601i	−0.739636	+	0.673007i	$0.765003\pi$
$318$	0	0
$319$	9.46442i	0.529906i
$320$	0	0
$321$	4.51573	0.252043
$322$	0	0
$323$	2.20477i	0.122677i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 3.34913i	− 0.185207i
$328$	0	0
$329$	−20.7071	−1.14162
$330$	0	0
$331$	− 2.89546i	− 0.159149i	−0.996829	−	0.0795745i	$-0.974644\pi$
$331$	− 2.89546i	− 0.159149i	0.996829	−	0.0795745i	$-0.0253561\pi$
$332$	0	0
$333$	2.24160i	0.122839i
$334$	0	0
$335$	25.9366	1.41707
$336$	0	0
$337$	3.10560	0.169173	0.0845865	−	0.996416i	$-0.473043\pi$
$337$	3.10560	0.169173	0.0845865	+	0.996416i	$0.473043\pi$
$338$	0	0
$339$	−6.12929	−0.332898
$340$	0	0
$341$	−7.06829	−0.382770
$342$	0	0
$343$	− 18.1793i	− 0.981589i
$344$	0	0
$345$	− 10.3448i	− 0.556946i
$346$	0	0
$347$	11.3787	0.610839	0.305419	−	0.952218i	$-0.401203\pi$
$347$	11.3787	0.610839	0.305419	+	0.952218i	$0.401203\pi$
$348$	0	0
$349$	3.34721i	0.179172i	0.995979	+	0.0895859i	$0.0285544\pi$
$349$	3.34721i	0.179172i	−0.995979	+	0.0895859i	$0.971446\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.637727i	0.0339428i	0.999856	+	0.0169714i	$0.00540242\pi$
$353$	0.637727i	0.0339428i	−0.999856	+	0.0169714i	$0.994598\pi$
$354$	0	0
$355$	38.5013	2.04343
$356$	0	0
$357$	− 2.45473i	− 0.129918i
$358$	0	0
$359$	− 21.4590i	− 1.13256i	−0.824211	−	0.566282i	$-0.808381\pi$
$359$	− 21.4590i	− 1.13256i	0.824211	−	0.566282i	$-0.191619\pi$
$360$	0	0
$361$	15.2403	0.802120
$362$	0	0
$363$	−7.67025	−0.402584
$364$	0	0
$365$	−35.9124	−1.87974
$366$	0	0
$367$	9.38703	0.489999	0.244999	−	0.969523i	$-0.421212\pi$
$367$	9.38703	0.489999	0.244999	+	0.969523i	$0.421212\pi$
$368$	0	0
$369$	− 7.81940i	− 0.407062i
$370$	0	0
$371$	− 15.8092i	− 0.820775i
$372$	0	0
$373$	27.7265	1.43562	0.717811	−	0.696238i	$-0.245144\pi$
$373$	27.7265	1.43562	0.717811	+	0.696238i	$0.245144\pi$
$374$	0	0
$375$	4.82908i	0.249373i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 35.8702i	− 1.84253i	−0.388935	−	0.921265i	$-0.627157\pi$
$379$	− 35.8702i	− 1.84253i	0.388935	−	0.921265i	$-0.372843\pi$
$380$	0	0
$381$	−5.43535	−0.278462
$382$	0	0
$383$	− 4.85517i	− 0.248087i	−0.992277	−	0.124044i	$-0.960414\pi$
$383$	− 4.85517i	− 0.248087i	0.992277	−	0.124044i	$-0.0395863\pi$
$384$	0	0
$385$	9.03684i	0.460560i
$386$	0	0
$387$	−16.8726	−0.857684
$388$	0	0
$389$	−2.38537	−0.120943	−0.0604716	−	0.998170i	$-0.519260\pi$
$389$	−2.38537	−0.120943	−0.0604716	+	0.998170i	$0.519260\pi$
$390$	0	0
$391$	5.23490	0.264740
$392$	0	0
$393$	−10.9705	−0.553387
$394$	0	0
$395$	2.26205i	0.113816i
$396$	0	0
$397$	15.2664i	0.766196i	0.923708	+	0.383098i	$0.125143\pi$
$397$	15.2664i	0.766196i	−0.923708	+	0.383098i	$0.874857\pi$
$398$	0	0
$399$	4.18598	0.209561
$400$	0	0
$401$	− 12.7584i	− 0.637124i	−0.947902	−	0.318562i	$-0.896800\pi$
$401$	− 12.7584i	− 0.637124i	0.947902	−	0.318562i	$-0.103200\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 10.1588i	− 0.504797i
$406$	0	0
$407$	−1.13946	−0.0564807
$408$	0	0
$409$	− 25.3588i	− 1.25391i	−0.779054	−	0.626956i	$-0.784300\pi$
$409$	− 25.3588i	− 1.25391i	0.779054	−	0.626956i	$-0.215700\pi$
$410$	0	0
$411$	10.4155i	0.513759i
$412$	0	0
$413$	−0.0325239	−0.00160040
$414$	0	0
$415$	45.7633	2.24643
$416$	0	0
$417$	9.66056	0.473080
$418$	0	0
$419$	11.6673	0.569983	0.284992	−	0.958530i	$-0.408009\pi$
$419$	11.6673	0.569983	0.284992	+	0.958530i	$0.408009\pi$
$420$	0	0
$421$	− 8.29291i	− 0.404172i	−0.979368	−	0.202086i	$-0.935228\pi$
$421$	− 8.29291i	− 0.404172i	0.979368	−	0.202086i	$-0.0647720\pi$
$422$	0	0
$423$	18.1293i	0.881476i
$424$	0	0
$425$	3.24160	0.157241
$426$	0	0
$427$	21.6353i	1.04701i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 0.932296i	− 0.0449071i	−0.999748	−	0.0224536i	$-0.992852\pi$
$431$	− 0.932296i	− 0.0449071i	0.999748	−	0.0224536i	$-0.00714779\pi$
$432$	0	0
$433$	13.3502	0.641569	0.320785	−	0.947152i	$-0.396053\pi$
$433$	13.3502	0.641569	0.320785	+	0.947152i	$0.396053\pi$
$434$	0	0
$435$	− 17.7506i	− 0.851077i
$436$	0	0
$437$	8.92692i	0.427032i
$438$	0	0
$439$	13.9922	0.667813	0.333906	−	0.942606i	$-0.391633\pi$
$439$	13.9922	0.667813	0.333906	+	0.942606i	$0.391633\pi$
$440$	0	0
$441$	0.582105	0.0277193
$442$	0	0
$443$	−23.7017	−1.12610	−0.563051	−	0.826422i	$-0.690373\pi$
$443$	−23.7017	−1.12610	−0.563051	+	0.826422i	$0.690373\pi$
$444$	0	0
$445$	41.2693	1.95635
$446$	0	0
$447$	0.594187i	0.0281041i
$448$	0	0
$449$	12.5864i	0.593990i	0.954879	+	0.296995i	$0.0959844\pi$
$449$	12.5864i	0.593990i	−0.954879	+	0.296995i	$0.904016\pi$
$450$	0	0
$451$	3.97477	0.187165
$452$	0	0
$453$	15.2959i	0.718664i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 33.6383i	− 1.57353i	−0.617250	−	0.786767i	$-0.711753\pi$
$457$	− 33.6383i	− 1.57353i	0.617250	−	0.786767i	$-0.288247\pi$
$458$	0	0
$459$	−4.88471	−0.227999
$460$	0	0
$461$	− 1.40283i	− 0.0653363i	−0.999466	−	0.0326681i	$-0.989600\pi$
$461$	− 1.40283i	− 0.0653363i	0.999466	−	0.0326681i	$-0.0104004\pi$
$462$	0	0
$463$	− 15.2010i	− 0.706453i	−0.935538	−	0.353226i	$-0.885085\pi$
$463$	− 15.2010i	− 0.706453i	0.935538	−	0.353226i	$-0.114915\pi$
$464$	0	0
$465$	13.2567	0.614763
$466$	0	0
$467$	−39.3414	−1.82050	−0.910250	−	0.414058i	$-0.864111\pi$
$467$	−39.3414	−1.82050	−0.910250	+	0.414058i	$0.864111\pi$
$468$	0	0
$469$	24.9191	1.15066
$470$	0	0
$471$	3.22521	0.148610
$472$	0	0
$473$	− 8.57673i	− 0.394358i
$474$	0	0
$475$	5.52781i	0.253633i
$476$	0	0
$477$	−13.8412	−0.633743
$478$	0	0
$479$	− 22.3690i	− 1.02206i	−0.859561	−	0.511032i	$-0.829263\pi$
$479$	− 22.3690i	− 1.02206i	0.859561	−	0.511032i	$-0.170737\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 9.93900i	− 0.452240i
$484$	0	0
$485$	8.77479	0.398443
$486$	0	0
$487$	− 22.9205i	− 1.03863i	−0.854584	−	0.519313i	$-0.826188\pi$
$487$	− 22.9205i	− 1.03863i	0.854584	−	0.519313i	$-0.173812\pi$
$488$	0	0
$489$	− 12.1390i	− 0.548944i
$490$	0	0
$491$	1.84356	0.0831987	0.0415993	−	0.999134i	$-0.486755\pi$
$491$	1.84356	0.0831987	0.0415993	+	0.999134i	$0.486755\pi$
$492$	0	0
$493$	8.98254	0.404553
$494$	0	0
$495$	7.91185	0.355611
$496$	0	0
$497$	36.9909	1.65927
$498$	0	0
$499$	12.0344i	0.538736i	0.963037	+	0.269368i	$0.0868147\pi$
$499$	12.0344i	0.538736i	−0.963037	+	0.269368i	$0.913185\pi$
$500$	0	0
$501$	5.02284i	0.224404i
$502$	0	0
$503$	30.5056	1.36018	0.680088	−	0.733130i	$-0.261942\pi$
$503$	30.5056	1.36018	0.680088	+	0.733130i	$0.261942\pi$
$504$	0	0
$505$	14.8237i	0.659646i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 1.51142i	− 0.0669924i	−0.999439	−	0.0334962i	$-0.989336\pi$
$509$	− 1.51142i	− 0.0669924i	0.999439	−	0.0334962i	$-0.0106642\pi$
$510$	0	0
$511$	−34.5036	−1.52635
$512$	0	0
$513$	− 8.32975i	− 0.367767i
$514$	0	0
$515$	37.9124i	1.67062i
$516$	0	0
$517$	−9.21552	−0.405298
$518$	0	0
$519$	13.1448	0.576994
$520$	0	0
$521$	−5.64012	−0.247098	−0.123549	−	0.992338i	$-0.539428\pi$
$521$	−5.64012	−0.247098	−0.123549	+	0.992338i	$0.539428\pi$
$522$	0	0
$523$	31.7506	1.38836	0.694179	−	0.719802i	$-0.255768\pi$
$523$	31.7506	1.38836	0.694179	+	0.719802i	$0.255768\pi$
$524$	0	0
$525$	− 6.15452i	− 0.268605i
$526$	0	0
$527$	6.70841i	0.292223i
$528$	0	0
$529$	−1.80433	−0.0784492
$530$	0	0
$531$	0.0284750i	0.00123571i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	15.7778i	0.682133i
$536$	0	0
$537$	1.96854	0.0849488
$538$	0	0
$539$	0.295897i	0.0127452i
$540$	0	0
$541$	− 24.3297i	− 1.04602i	−0.852327	−	0.523009i	$-0.824809\pi$
$541$	− 24.3297i	− 1.04602i	0.852327	−	0.523009i	$-0.175191\pi$
$542$	0	0
$543$	9.46873	0.406342
$544$	0	0
$545$	11.7017	0.501246
$546$	0	0
$547$	8.18896	0.350135	0.175067	−	0.984556i	$-0.443986\pi$
$547$	8.18896	0.350135	0.175067	+	0.984556i	$0.443986\pi$
$548$	0	0
$549$	18.9420	0.808424
$550$	0	0
$551$	15.3177i	0.652555i
$552$	0	0
$553$	2.17331i	0.0924185i
$554$	0	0
$555$	2.13706	0.0907133
$556$	0	0
$557$	− 25.3327i	− 1.07338i	−0.843779	−	0.536691i	$-0.819674\pi$
$557$	− 25.3327i	− 1.07338i	0.843779	−	0.536691i	$-0.180326\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 1.09246i	− 0.0461236i
$562$	0	0
$563$	−25.3937	−1.07022	−0.535109	−	0.844783i	$-0.679730\pi$
$563$	−25.3937	−1.07022	−0.535109	+	0.844783i	$0.679730\pi$
$564$	0	0
$565$	− 21.4155i	− 0.900957i
$566$	0	0
$567$	− 9.76032i	− 0.409895i
$568$	0	0
$569$	31.1347	1.30523	0.652617	−	0.757688i	$-0.273671\pi$
$569$	31.1347	1.30523	0.652617	+	0.757688i	$0.273671\pi$
$570$	0	0
$571$	−20.5090	−0.858276	−0.429138	−	0.903239i	$-0.641183\pi$
$571$	−20.5090	−0.858276	−0.429138	+	0.903239i	$0.641183\pi$
$572$	0	0
$573$	−7.21206	−0.301288
$574$	0	0
$575$	13.1250	0.547350
$576$	0	0
$577$	15.6890i	0.653143i	0.945172	+	0.326572i	$0.105893\pi$
$577$	15.6890i	0.653143i	−0.945172	+	0.326572i	$0.894107\pi$
$578$	0	0
$579$	10.8465i	0.450767i
$580$	0	0
$581$	43.9681	1.82410
$582$	0	0
$583$	− 7.03577i	− 0.291392i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 30.5687i	− 1.26171i	−0.775903	−	0.630853i	$-0.782705\pi$
$587$	− 30.5687i	− 1.26171i	0.775903	−	0.630853i	$-0.217295\pi$
$588$	0	0
$589$	−11.4397	−0.471363
$590$	0	0
$591$	− 10.4058i	− 0.428038i
$592$	0	0
$593$	29.6883i	1.21915i	0.792727	+	0.609576i	$0.208660\pi$
$593$	29.6883i	1.21915i	−0.792727	+	0.609576i	$0.791340\pi$
$594$	0	0
$595$	8.57673	0.351612
$596$	0	0
$597$	10.8955	0.445922
$598$	0	0
$599$	−24.2325	−0.990113	−0.495057	−	0.868861i	$-0.664853\pi$
$599$	−24.2325	−0.990113	−0.495057	+	0.868861i	$0.664853\pi$
$600$	0	0
$601$	16.4819	0.672310	0.336155	−	0.941807i	$-0.390873\pi$
$601$	16.4819	0.672310	0.336155	+	0.941807i	$0.390873\pi$
$602$	0	0
$603$	− 21.8170i	− 0.888457i
$604$	0	0
$605$	− 26.7995i	− 1.08956i
$606$	0	0
$607$	−1.43190	−0.0581188	−0.0290594	−	0.999578i	$-0.509251\pi$
$607$	−1.43190	−0.0581188	−0.0290594	+	0.999578i	$0.509251\pi$
$608$	0	0
$609$	− 17.0543i	− 0.691075i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	3.84846i	0.155438i	0.996975	+	0.0777190i	$0.0247637\pi$
$613$	3.84846i	0.155438i	−0.996975	+	0.0777190i	$0.975236\pi$
$614$	0	0
$615$	−7.45473	−0.300604
$616$	0	0
$617$	− 15.0388i	− 0.605437i	−0.953080	−	0.302719i	$-0.902106\pi$
$617$	− 15.0388i	− 0.605437i	0.953080	−	0.302719i	$-0.0978943\pi$
$618$	0	0
$619$	12.8170i	0.515159i	0.966257	+	0.257579i	$0.0829249\pi$
$619$	12.8170i	0.515159i	−0.966257	+	0.257579i	$0.917075\pi$
$620$	0	0
$621$	−19.7778	−0.793655
$622$	0	0
$623$	39.6504	1.58856
$624$	0	0
$625$	−31.1269	−1.24508
$626$	0	0
$627$	1.86294	0.0743985
$628$	0	0
$629$	1.08144i	0.0431199i
$630$	0	0
$631$	25.7517i	1.02516i	0.858640	+	0.512579i	$0.171310\pi$
$631$	25.7517i	1.02516i	−0.858640	+	0.512579i	$0.828690\pi$
$632$	0	0
$633$	8.38835	0.333407
$634$	0	0
$635$	− 18.9909i	− 0.753631i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 32.3860i	− 1.28117i
$640$	0	0
$641$	−24.4571	−0.965998	−0.482999	−	0.875621i	$-0.660453\pi$
$641$	−24.4571	−0.965998	−0.482999	+	0.875621i	$0.660453\pi$
$642$	0	0
$643$	9.97344i	0.393314i	0.980472	+	0.196657i	$0.0630086\pi$
$643$	9.97344i	0.393314i	−0.980472	+	0.196657i	$0.936991\pi$
$644$	0	0
$645$	16.0858i	0.633376i
$646$	0	0
$647$	11.8431	0.465600	0.232800	−	0.972525i	$-0.425211\pi$
$647$	11.8431	0.465600	0.232800	+	0.972525i	$0.425211\pi$
$648$	0	0
$649$	−0.0144745	−0.000568173 0
$650$	0	0
$651$	12.7366	0.499188
$652$	0	0
$653$	−7.47411	−0.292484	−0.146242	−	0.989249i	$-0.546718\pi$
$653$	−7.47411	−0.292484	−0.146242	+	0.989249i	$0.546718\pi$
$654$	0	0
$655$	− 38.3303i	− 1.49769i
$656$	0	0
$657$	30.2083i	1.17854i
$658$	0	0
$659$	−34.1739	−1.33123	−0.665613	−	0.746297i	$-0.731830\pi$
$659$	−34.1739	−1.33123	−0.665613	+	0.746297i	$0.731830\pi$
$660$	0	0
$661$	− 33.6088i	− 1.30723i	−0.756827	−	0.653615i	$-0.773252\pi$
$661$	− 33.6088i	− 1.30723i	0.756827	−	0.653615i	$-0.226748\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	14.6256i	0.567158i
$666$	0	0
$667$	36.3696	1.40824
$668$	0	0
$669$	− 9.14675i	− 0.353634i
$670$	0	0
$671$	9.62863i	0.371709i
$672$	0	0
$673$	−48.0320	−1.85150	−0.925750	−	0.378137i	$-0.876565\pi$
$673$	−48.0320	−1.85150	−0.925750	+	0.378137i	$0.876565\pi$
$674$	0	0
$675$	−12.2470	−0.471386
$676$	0	0
$677$	−33.6582	−1.29359	−0.646794	−	0.762665i	$-0.723891\pi$
$677$	−33.6582	−1.29359	−0.646794	+	0.762665i	$0.723891\pi$
$678$	0	0
$679$	8.43057	0.323535
$680$	0	0
$681$	8.53319i	0.326992i
$682$	0	0
$683$	15.9041i	0.608553i	0.952584	+	0.304276i	$0.0984147\pi$
$683$	15.9041i	0.608553i	−0.952584	+	0.304276i	$0.901585\pi$
$684$	0	0
$685$	−36.3913	−1.39044
$686$	0	0
$687$	− 0.913773i	− 0.0348626i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 33.1903i	− 1.26262i	−0.775531	−	0.631309i	$-0.782518\pi$
$691$	− 33.1903i	− 1.26262i	0.775531	−	0.631309i	$-0.217482\pi$
$692$	0	0
$693$	7.60148	0.288756
$694$	0	0
$695$	33.7536i	1.28035i
$696$	0	0
$697$	− 3.77240i	− 0.142890i
$698$	0	0
$699$	−8.70171	−0.329129
$700$	0	0
$701$	14.9129	0.563253	0.281627	−	0.959524i	$-0.409126\pi$
$701$	14.9129	0.563253	0.281627	+	0.959524i	$0.409126\pi$
$702$	0	0
$703$	−1.84415	−0.0695534
$704$	0	0
$705$	17.2838	0.650946
$706$	0	0
$707$	14.2422i	0.535633i
$708$	0	0
$709$	− 38.4312i	− 1.44331i	−0.692252	−	0.721656i	$-0.743381\pi$
$709$	− 38.4312i	− 1.44331i	0.692252	−	0.721656i	$-0.256619\pi$
$710$	0	0
$711$	1.90276	0.0713589
$712$	0	0
$713$	27.1618i	1.01722i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 9.56657i	− 0.357270i
$718$	0	0
$719$	−11.4373	−0.426538	−0.213269	−	0.976993i	$-0.568411\pi$
$719$	−11.4373	−0.426538	−0.213269	+	0.976993i	$0.568411\pi$
$720$	0	0
$721$	36.4252i	1.35654i
$722$	0	0
$723$	− 2.92585i	− 0.108814i
$724$	0	0
$725$	22.5211	0.836413
$726$	0	0
$727$	3.63640	0.134867	0.0674333	−	0.997724i	$-0.478519\pi$
$727$	3.63640	0.134867	0.0674333	+	0.997724i	$0.478519\pi$
$728$	0	0
$729$	1.78986	0.0662910
$730$	0	0
$731$	−8.14005	−0.301071
$732$	0	0
$733$	− 3.52217i	− 0.130094i	−0.997882	−	0.0650472i	$-0.979280\pi$
$733$	− 3.52217i	− 0.130094i	0.997882	−	0.0650472i	$-0.0207198\pi$
$734$	0	0
$735$	− 0.554958i	− 0.0204699i
$736$	0	0
$737$	11.0901	0.408508
$738$	0	0
$739$	0.420288i	0.0154605i	0.999970	+	0.00773027i	$0.00246064\pi$
$739$	0.420288i	0.0154605i	−0.999970	+	0.00773027i	$0.997539\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	25.3623i	0.930452i	0.885192	+	0.465226i	$0.154027\pi$
$743$	25.3623i	0.930452i	−0.885192	+	0.465226i	$0.845973\pi$
$744$	0	0
$745$	−2.07606	−0.0760611
$746$	0	0
$747$	− 38.4946i	− 1.40844i
$748$	0	0
$749$	15.1588i	0.553892i
$750$	0	0
$751$	0.650874	0.0237507	0.0118754	−	0.999929i	$-0.496220\pi$
$751$	0.650874	0.0237507	0.0118754	+	0.999929i	$0.496220\pi$
$752$	0	0
$753$	−1.10129	−0.0401333
$754$	0	0
$755$	−53.4432	−1.94500
$756$	0	0
$757$	−16.7909	−0.610276	−0.305138	−	0.952308i	$-0.598703\pi$
$757$	−16.7909	−0.610276	−0.305138	+	0.952308i	$0.598703\pi$
$758$	0	0
$759$	− 4.42327i	− 0.160555i
$760$	0	0
$761$	− 30.9221i	− 1.12093i	−0.828179	−	0.560463i	$-0.810623\pi$
$761$	− 30.9221i	− 1.12093i	0.828179	−	0.560463i	$-0.189377\pi$
$762$	0	0
$763$	11.2427	0.407012
$764$	0	0
$765$	− 7.50902i	− 0.271489i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 43.7689i	− 1.57835i	−0.614169	−	0.789174i	$-0.710509\pi$
$769$	− 43.7689i	− 1.57835i	0.614169	−	0.789174i	$-0.289491\pi$
$770$	0	0
$771$	23.6058	0.850142
$772$	0	0
$773$	− 42.4209i	− 1.52577i	−0.646532	−	0.762886i	$-0.723782\pi$
$773$	− 42.4209i	− 1.52577i	0.646532	−	0.762886i	$-0.276218\pi$
$774$	0	0
$775$	16.8194i	0.604171i
$776$	0	0
$777$	2.05323	0.0736592
$778$	0	0
$779$	6.43296	0.230485
$780$	0	0
$781$	16.4625	0.589075
$782$	0	0
$783$	−33.9366	−1.21280
$784$	0	0
$785$	11.2687i	0.402199i
$786$	0	0
$787$	− 36.0116i	− 1.28368i	−0.766841	−	0.641838i	$-0.778172\pi$
$787$	− 36.0116i	− 1.28368i	0.766841	−	0.641838i	$-0.221828\pi$
$788$	0	0
$789$	8.57779	0.305378
$790$	0	0
$791$	− 20.5754i	− 0.731577i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	13.1957i	0.468002i
$796$	0	0
$797$	31.7101	1.12323	0.561614	−	0.827399i	$-0.310181\pi$
$797$	31.7101	1.12323	0.561614	+	0.827399i	$0.310181\pi$
$798$	0	0
$799$	8.74632i	0.309422i
$800$	0	0
$801$	− 34.7144i	− 1.22657i
$802$	0	0
$803$	−15.3556	−0.541886
$804$	0	0
$805$	34.7265	1.22395
$806$	0	0
$807$	8.16852	0.287546
$808$	0	0
$809$	−45.2814	−1.59201	−0.796005	−	0.605290i	$-0.793057\pi$
$809$	−45.2814	−1.59201	−0.796005	+	0.605290i	$0.793057\pi$
$810$	0	0
$811$	42.8635i	1.50514i	0.658511	+	0.752571i	$0.271187\pi$
$811$	42.8635i	1.50514i	−0.658511	+	0.752571i	$0.728813\pi$
$812$	0	0
$813$	23.6189i	0.828352i
$814$	0	0
$815$	42.4131	1.48567
$816$	0	0
$817$	− 13.8810i	− 0.485634i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 7.82776i	− 0.273191i	−0.990627	−	0.136595i	$-0.956384\pi$
$821$	− 7.82776i	− 0.273191i	0.990627	−	0.136595i	$-0.0436160\pi$
$822$	0	0
$823$	−36.7754	−1.28191	−0.640955	−	0.767579i	$-0.721461\pi$
$823$	−36.7754	−1.28191	−0.640955	+	0.767579i	$0.721461\pi$
$824$	0	0
$825$	− 2.73902i	− 0.0953604i
$826$	0	0
$827$	47.3293i	1.64580i	0.568186	+	0.822900i	$0.307645\pi$
$827$	47.3293i	1.64580i	−0.568186	+	0.822900i	$0.692355\pi$
$828$	0	0
$829$	−25.2687	−0.877620	−0.438810	−	0.898580i	$-0.644600\pi$
$829$	−25.2687	−0.877620	−0.438810	+	0.898580i	$0.644600\pi$
$830$	0	0
$831$	−8.21552	−0.284993
$832$	0	0
$833$	0.280831	0.00973023
$834$	0	0
$835$	−17.5496	−0.607328
$836$	0	0
$837$	− 25.3448i	− 0.876045i
$838$	0	0
$839$	− 37.6883i	− 1.30114i	−0.759444	−	0.650572i	$-0.774529\pi$
$839$	− 37.6883i	− 1.30114i	0.759444	−	0.650572i	$-0.225471\pi$
$840$	0	0
$841$	33.4064	1.15194
$842$	0	0
$843$	− 9.27413i	− 0.319418i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 25.7482i	− 0.884720i
$848$	0	0
$849$	24.6267	0.845187
$850$	0	0
$851$	4.37867i	0.150099i
$852$	0	0
$853$	31.0121i	1.06183i	0.847424	+	0.530917i	$0.178152\pi$
$853$	31.0121i	1.06183i	−0.847424	+	0.530917i	$0.821848\pi$
$854$	0	0
$855$	12.8049	0.437919
$856$	0	0
$857$	−12.4692	−0.425940	−0.212970	−	0.977059i	$-0.568314\pi$
$857$	−12.4692	−0.425940	−0.212970	+	0.977059i	$0.568314\pi$
$858$	0	0
$859$	17.3163	0.590826	0.295413	−	0.955370i	$-0.404543\pi$
$859$	17.3163	0.590826	0.295413	+	0.955370i	$0.404543\pi$
$860$	0	0
$861$	−7.16229	−0.244090
$862$	0	0
$863$	3.46383i	0.117910i	0.998261	+	0.0589550i	$0.0187769\pi$
$863$	3.46383i	0.117910i	−0.998261	+	0.0589550i	$0.981223\pi$
$864$	0	0
$865$	45.9275i	1.56158i
$866$	0	0
$867$	12.5961	0.427786
$868$	0	0
$869$	0.967213i	0.0328105i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 7.38106i	− 0.249811i
$874$	0	0
$875$	−16.2107	−0.548023
$876$	0	0
$877$	57.2549i	1.93336i	0.255989	+	0.966680i	$0.417599\pi$
$877$	57.2549i	1.93336i	−0.255989	+	0.966680i	$0.582401\pi$
$878$	0	0
$879$	− 14.9226i	− 0.503327i
$880$	0	0
$881$	43.1782	1.45471	0.727355	−	0.686261i	$-0.240749\pi$
$881$	43.1782	1.45471	0.727355	+	0.686261i	$0.240749\pi$
$882$	0	0
$883$	49.9560	1.68115	0.840576	−	0.541693i	$-0.182216\pi$
$883$	49.9560	1.68115	0.840576	+	0.541693i	$0.182216\pi$
$884$	0	0
$885$	0.0271471	0.000912539 0
$886$	0	0
$887$	−17.6746	−0.593454	−0.296727	−	0.954962i	$-0.595895\pi$
$887$	−17.6746	−0.593454	−0.296727	+	0.954962i	$0.595895\pi$
$888$	0	0
$889$	− 18.2459i	− 0.611948i
$890$	0	0
$891$	− 4.34375i	− 0.145521i
$892$	0	0
$893$	−14.9148	−0.499106
$894$	0	0
$895$	6.87800i	0.229906i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	46.6069i	1.55443i
$900$	0	0
$901$	−6.67755	−0.222461
$902$	0	0
$903$	15.4547i	0.514301i
$904$	0	0
$905$	33.0834i	1.09973i
$906$	0	0
$907$	7.73423	0.256811	0.128406	−	0.991722i	$-0.459014\pi$
$907$	7.73423	0.256811	0.128406	+	0.991722i	$0.459014\pi$
$908$	0	0
$909$	12.4692	0.413577
$910$	0	0
$911$	−39.6179	−1.31260	−0.656299	−	0.754501i	$-0.727879\pi$
$911$	−39.6179	−1.31260	−0.656299	+	0.754501i	$0.727879\pi$
$912$	0	0
$913$	19.5676	0.647594
$914$	0	0
$915$	− 18.0586i	− 0.596999i
$916$	0	0
$917$	− 36.8267i	− 1.21612i
$918$	0	0
$919$	−14.6213	−0.482313	−0.241157	−	0.970486i	$-0.577527\pi$
$919$	−14.6213	−0.482313	−0.241157	+	0.970486i	$0.577527\pi$
$920$	0	0
$921$	− 7.17283i	− 0.236353i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	2.71140i	0.0891502i
$926$	0	0
$927$	31.8907	1.04743
$928$	0	0
$929$	3.55735i	0.116713i	0.998296	+	0.0583565i	$0.0185860\pi$
$929$	3.55735i	0.116713i	−0.998296	+	0.0583565i	$0.981414\pi$
$930$	0	0
$931$	0.478894i	0.0156951i
$932$	0	0
$933$	−16.8726	−0.552385
$934$	0	0
$935$	3.81700	0.124829
$936$	0	0
$937$	34.5526	1.12878	0.564392	−	0.825507i	$-0.309111\pi$
$937$	34.5526	1.12878	0.564392	+	0.825507i	$0.309111\pi$
$938$	0	0
$939$	5.71571	0.186525
$940$	0	0
$941$	20.6233i	0.672299i	0.941809	+	0.336149i	$0.109125\pi$
$941$	20.6233i	0.672299i	−0.941809	+	0.336149i	$0.890875\pi$
$942$	0	0
$943$	− 15.2741i	− 0.497394i
$944$	0	0
$945$	−32.4034	−1.05408
$946$	0	0
$947$	29.4999i	0.958619i	0.877646	+	0.479309i	$0.159113\pi$
$947$	29.4999i	0.958619i	−0.877646	+	0.479309i	$0.840887\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 19.2185i	− 0.623203i
$952$	0	0
$953$	−26.2389	−0.849963	−0.424981	−	0.905202i	$-0.639719\pi$
$953$	−26.2389	−0.849963	−0.424981	+	0.905202i	$0.639719\pi$
$954$	0	0
$955$	− 25.1987i	− 0.815409i
$956$	0	0
$957$	− 7.58987i	− 0.245346i
$958$	0	0
$959$	−34.9638	−1.12904
$960$	0	0
$961$	−3.80731	−0.122817
$962$	0	0
$963$	13.2717	0.427676
$964$	0	0
$965$	−37.8974	−1.21996
$966$	0	0
$967$	− 17.5176i	− 0.563330i	−0.959513	−	0.281665i	$-0.909113\pi$
$967$	− 17.5176i	− 0.563330i	0.959513	−	0.281665i	$-0.0908866\pi$
$968$	0	0
$969$	− 1.76809i	− 0.0567991i
$970$	0	0
$971$	−20.5120	−0.658262	−0.329131	−	0.944284i	$-0.606756\pi$
$971$	−20.5120	−0.658262	−0.329131	+	0.944284i	$0.606756\pi$
$972$	0	0
$973$	32.4295i	1.03964i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.4450i	0.814059i	0.913415	+	0.407030i	$0.133435\pi$
$977$	25.4450i	0.814059i	−0.913415	+	0.407030i	$0.866565\pi$
$978$	0	0
$979$	17.6461	0.563971
$980$	0	0
$981$	− 9.84309i	− 0.314266i
$982$	0	0
$983$	− 39.5244i	− 1.26063i	−0.776339	−	0.630316i	$-0.782926\pi$
$983$	− 39.5244i	− 1.26063i	0.776339	−	0.630316i	$-0.217074\pi$
$984$	0	0
$985$	36.3575	1.15845
$986$	0	0
$987$	16.6058	0.528568
$988$	0	0
$989$	−32.9584	−1.04802
$990$	0	0
$991$	29.8377	0.947826	0.473913	−	0.880572i	$-0.342841\pi$
$991$	29.8377	0.947826	0.473913	+	0.880572i	$0.342841\pi$
$992$	0	0
$993$	2.32198i	0.0736858i
$994$	0	0
$995$	38.0683i	1.20685i
$996$	0	0
$997$	−4.93123	−0.156174	−0.0780868	−	0.996947i	$-0.524881\pi$
$997$	−4.93123	−0.156174	−0.0780868	+	0.996947i	$0.524881\pi$
$998$	0	0
$999$	− 4.08575i	− 0.129268i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.f.o.337.1		6
4.3	odd	2		169.2.b.b.168.3		6
12.11	even	2		1521.2.b.l.1351.4		6
13.5	odd	4		2704.2.a.z.1.1		3
13.8	odd	4		2704.2.a.ba.1.1		3
13.12	even	2	inner	2704.2.f.o.337.2		6
52.3	odd	6		169.2.e.b.147.4		12
52.7	even	12		169.2.c.b.146.2		6
52.11	even	12		169.2.c.b.22.2		6
52.15	even	12		169.2.c.c.22.2		6
52.19	even	12		169.2.c.c.146.2		6
52.23	odd	6		169.2.e.b.147.3		12
52.31	even	4		169.2.a.b.1.2	✓	3
52.35	odd	6		169.2.e.b.23.3		12
52.43	odd	6		169.2.e.b.23.4		12
52.47	even	4		169.2.a.c.1.2	yes	3
52.51	odd	2		169.2.b.b.168.4		6
156.47	odd	4		1521.2.a.o.1.2		3
156.83	odd	4		1521.2.a.r.1.2		3
156.155	even	2		1521.2.b.l.1351.3		6
260.99	even	4		4225.2.a.bb.1.2		3
260.239	even	4		4225.2.a.bg.1.2		3
364.83	odd	4		8281.2.a.bf.1.2		3
364.307	odd	4		8281.2.a.bj.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.2	✓	3	52.31	even	4
169.2.a.c.1.2	yes	3	52.47	even	4
169.2.b.b.168.3		6	4.3	odd	2
169.2.b.b.168.4		6	52.51	odd	2
169.2.c.b.22.2		6	52.11	even	12
169.2.c.b.146.2		6	52.7	even	12
169.2.c.c.22.2		6	52.15	even	12
169.2.c.c.146.2		6	52.19	even	12
169.2.e.b.23.3		12	52.35	odd	6
169.2.e.b.23.4		12	52.43	odd	6
169.2.e.b.147.3		12	52.23	odd	6
169.2.e.b.147.4		12	52.3	odd	6
1521.2.a.o.1.2		3	156.47	odd	4
1521.2.a.r.1.2		3	156.83	odd	4
1521.2.b.l.1351.3		6	156.155	even	2
1521.2.b.l.1351.4		6	12.11	even	2
2704.2.a.z.1.1		3	13.5	odd	4
2704.2.a.ba.1.1		3	13.8	odd	4
2704.2.f.o.337.1		6	1.1	even	1	trivial
2704.2.f.o.337.2		6	13.12	even	2	inner
4225.2.a.bb.1.2		3	260.99	even	4
4225.2.a.bg.1.2		3	260.239	even	4
8281.2.a.bf.1.2		3	364.83	odd	4
8281.2.a.bj.1.2		3	364.307	odd	4

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:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.801938 q^{3} -2.80194i q^{5} -2.69202i q^{7} -2.35690 q^{9} -1.19806i q^{11} +2.24698i q^{15} -1.13706 q^{17} -1.93900i q^{19} +2.15883i q^{21} -4.60388 q^{23} -2.85086 q^{25} +4.29590 q^{27} -7.89977 q^{29} -5.89977i q^{31} +0.960771i q^{33} -7.54288 q^{35} -0.951083i q^{37} +3.31767i q^{41} +7.15883 q^{43} +6.60388i q^{45} -7.69202i q^{47} -0.246980 q^{49} +0.911854 q^{51} +5.87263 q^{53} -3.35690 q^{55} +1.55496i q^{57} -0.0120816i q^{59} -8.03684 q^{61} +6.34481i q^{63} +9.25667i q^{67} +3.69202 q^{69} +13.7409i q^{71} -12.8170i q^{73} +2.28621 q^{75} -3.22521 q^{77} -0.807315 q^{79} +3.62565 q^{81} +16.3327i q^{83} +3.18598i q^{85} +6.33513 q^{87} +14.7289i q^{89} +4.73125i q^{93} -5.43296 q^{95} +3.13169i q^{97} +2.82371i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 6 q^{9} + 4 q^{17} - 10 q^{23} + 10 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{35} + 26 q^{43} + 8 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} + 8 q^{61} + 12 q^{69} + 30 q^{75} - 16 q^{77} + 10 q^{79}+ \cdots + 6 q^{95}+O(q^{100}) \) 6 * q + 4 * q^3 - 6 * q^9 + 4 * q^17 - 10 * q^23 + 10 * q^25 - 2 * q^27 - 2 * q^29 - 8 * q^35 + 26 * q^43 + 8 * q^49 - 2 * q^51 + 2 * q^53 - 12 * q^55 + 8 * q^61 + 12 * q^69 + 30 * q^75 - 16 * q^77 + 10 * q^79 - 2 * q^81 + 36 * q^87 + 6 * q^95

Introduction

L-functions

Modular forms

Varieties

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