LMFDB - Embedded newform 5200.2.a.cf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5200,2,Mod(1,5200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5200, 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("5200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5200 = 2^{4} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5200.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:	$41.5222090511$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.940.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 4 $ x^3 - 7*x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.602705$ of defining polynomial
Character	$\chi$	$=$	5200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.602705 q^{3} -3.63675 q^{7} -2.63675 q^{9} -2.00000 q^{11} +1.00000 q^{13} -6.67079 q^{17} -8.06808 q^{19} +2.19189 q^{21} -0.794590 q^{23} +3.39730 q^{27} -8.06808 q^{29} -5.20541 q^{31} +1.20541 q^{33} +0.431337 q^{37} -0.602705 q^{39} +6.86267 q^{41} +5.80811 q^{43} +7.63675 q^{47} +6.22593 q^{49} +4.02052 q^{51} +0.794590 q^{53} +4.86267 q^{57} -8.06808 q^{59} -2.86267 q^{61} +9.58918 q^{63} +5.20541 q^{67} +0.478903 q^{69} +9.46538 q^{71} -6.00000 q^{73} +7.27349 q^{77} -1.93192 q^{79} +5.86267 q^{81} -2.06808 q^{83} +4.86267 q^{87} -8.06808 q^{89} -3.63675 q^{91} +3.13733 q^{93} -13.6573 q^{97} +5.27349 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{7} + 5 q^{9} - 6 q^{11} + 3 q^{13} + 4 q^{17} - 2 q^{19} + 12 q^{21} - 6 q^{23} + 12 q^{27} - 2 q^{29} - 12 q^{31} - 8 q^{37} + 2 q^{41} + 12 q^{43} + 10 q^{47} + 13 q^{49} + 10 q^{51} + 6 q^{53}+ \cdots - 10 q^{99}+O(q^{100}) $ 3 * q + 2 * q^7 + 5 * q^9 - 6 * q^11 + 3 * q^13 + 4 * q^17 - 2 * q^19 + 12 * q^21 - 6 * q^23 + 12 * q^27 - 2 * q^29 - 12 * q^31 - 8 * q^37 + 2 * q^41 + 12 * q^43 + 10 * q^47 + 13 * q^49 + 10 * q^51 + 6 * q^53 - 4 * q^57 - 2 * q^59 + 10 * q^61 + 36 * q^63 + 12 * q^67 - 28 * q^69 + 8 * q^71 - 18 * q^73 - 4 * q^77 - 28 * q^79 - q^81 + 16 * q^83 - 4 * q^87 - 2 * q^89 + 2 * q^91 + 28 * q^93 - 26 * q^97 - 10 * 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
$3$	−0.602705	−0.347972	−0.173986	−	0.984748i	$-0.555665\pi$
$3$	−0.602705	−0.347972	−0.173986	+	0.984748i	$0.555665\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.63675	−1.37456	−0.687281	−	0.726392i	$-0.741196\pi$
$7$	−3.63675	−1.37456	−0.687281	+	0.726392i	$0.741196\pi$
$8$	0	0
$9$	−2.63675	−0.878916
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.67079	−1.61790	−0.808952	−	0.587875i	$-0.799965\pi$
$17$	−6.67079	−1.61790	−0.808952	+	0.587875i	$0.799965\pi$
$18$	0	0
$19$	−8.06808	−1.85095	−0.925473	−	0.378814i	$-0.876332\pi$
$19$	−8.06808	−1.85095	−0.925473	+	0.378814i	$0.876332\pi$
$20$	0	0
$21$	2.19189	0.478309
$22$	0	0
$23$	−0.794590	−0.165683	−0.0828417	−	0.996563i	$-0.526400\pi$
$23$	−0.794590	−0.165683	−0.0828417	+	0.996563i	$0.526400\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.39730	0.653810
$28$	0	0
$29$	−8.06808	−1.49821	−0.749103	−	0.662454i	$-0.769515\pi$
$29$	−8.06808	−1.49821	−0.749103	+	0.662454i	$0.769515\pi$
$30$	0	0
$31$	−5.20541	−0.934919	−0.467460	−	0.884014i	$-0.654831\pi$
$31$	−5.20541	−0.934919	−0.467460	+	0.884014i	$0.654831\pi$
$32$	0	0
$33$	1.20541	0.209835
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.431337	0.0709114	0.0354557	−	0.999371i	$-0.488712\pi$
$37$	0.431337	0.0709114	0.0354557	+	0.999371i	$0.488712\pi$
$38$	0	0
$39$	−0.602705	−0.0965100
$40$	0	0
$41$	6.86267	1.07177	0.535885	−	0.844291i	$-0.319978\pi$
$41$	6.86267	1.07177	0.535885	+	0.844291i	$0.319978\pi$
$42$	0	0
$43$	5.80811	0.885729	0.442865	−	0.896589i	$-0.353962\pi$
$43$	5.80811	0.885729	0.442865	+	0.896589i	$0.353962\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.63675	1.11393	0.556967	−	0.830535i	$-0.311965\pi$
$47$	7.63675	1.11393	0.556967	+	0.830535i	$0.311965\pi$
$48$	0	0
$49$	6.22593	0.889418
$50$	0	0
$51$	4.02052	0.562985
$52$	0	0
$53$	0.794590	0.109145	0.0545727	−	0.998510i	$-0.482620\pi$
$53$	0.794590	0.109145	0.0545727	+	0.998510i	$0.482620\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.86267	0.644077
$58$	0	0
$59$	−8.06808	−1.05038	−0.525188	−	0.850987i	$-0.676005\pi$
$59$	−8.06808	−1.05038	−0.525188	+	0.850987i	$0.676005\pi$
$60$	0	0
$61$	−2.86267	−0.366528	−0.183264	−	0.983064i	$-0.558666\pi$
$61$	−2.86267	−0.366528	−0.183264	+	0.983064i	$0.558666\pi$
$62$	0	0
$63$	9.58918	1.20812
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.20541	0.635942	0.317971	−	0.948100i	$-0.396999\pi$
$67$	5.20541	0.635942	0.317971	+	0.948100i	$0.396999\pi$
$68$	0	0
$69$	0.478903	0.0576532
$70$	0	0
$71$	9.46538	1.12333	0.561667	−	0.827363i	$-0.310160\pi$
$71$	9.46538	1.12333	0.561667	+	0.827363i	$0.310160\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.27349	0.828892
$78$	0	0
$79$	−1.93192	−0.217358	−0.108679	−	0.994077i	$-0.534662\pi$
$79$	−1.93192	−0.217358	−0.108679	+	0.994077i	$0.534662\pi$
$80$	0	0
$81$	5.86267	0.651408
$82$	0	0
$83$	−2.06808	−0.227002	−0.113501	−	0.993538i	$-0.536206\pi$
$83$	−2.06808	−0.227002	−0.113501	+	0.993538i	$0.536206\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.86267	0.521333
$88$	0	0
$89$	−8.06808	−0.855215	−0.427608	−	0.903964i	$-0.640643\pi$
$89$	−8.06808	−0.855215	−0.427608	+	0.903964i	$0.640643\pi$
$90$	0	0
$91$	−3.63675	−0.381235
$92$	0	0
$93$	3.13733	0.325326
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.6573	−1.38669	−0.693343	−	0.720608i	$-0.743863\pi$
$97$	−13.6573	−1.38669	−0.693343	+	0.720608i	$0.743863\pi$
$98$	0	0
$99$	5.27349	0.530006
$100$	0	0
$101$	−15.6843	−1.56065	−0.780324	−	0.625376i	$-0.784946\pi$
$101$	−15.6843	−1.56065	−0.780324	+	0.625376i	$0.784946\pi$
$102$	0	0
$103$	15.3416	1.51165	0.755825	−	0.654773i	$-0.227236\pi$
$103$	15.3416	1.51165	0.755825	+	0.654773i	$0.227236\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.20541	−0.116531	−0.0582657	−	0.998301i	$-0.518557\pi$
$107$	−1.20541	−0.116531	−0.0582657	+	0.998301i	$0.518557\pi$
$108$	0	0
$109$	9.80811	0.939447	0.469724	−	0.882814i	$-0.344354\pi$
$109$	9.80811	0.939447	0.469724	+	0.882814i	$0.344354\pi$
$110$	0	0
$111$	−0.259969	−0.0246752
$112$	0	0
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.63675	−0.243767
$118$	0	0
$119$	24.2600	2.22391
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−4.13617	−0.372946
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.79459	−0.425451	−0.212726	−	0.977112i	$-0.568234\pi$
$127$	−4.79459	−0.425451	−0.212726	+	0.977112i	$0.568234\pi$
$128$	0	0
$129$	−3.50058	−0.308209
$130$	0	0
$131$	4.02052	0.351274	0.175637	−	0.984455i	$-0.443801\pi$
$131$	4.02052	0.351274	0.175637	+	0.984455i	$0.443801\pi$
$132$	0	0
$133$	29.3416	2.54424
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.2054	0.957343	0.478671	−	0.877994i	$-0.341119\pi$
$137$	11.2054	0.957343	0.478671	+	0.877994i	$0.341119\pi$
$138$	0	0
$139$	−8.84216	−0.749982	−0.374991	−	0.927028i	$-0.622354\pi$
$139$	−8.84216	−0.749982	−0.374991	+	0.927028i	$0.622354\pi$
$140$	0	0
$141$	−4.60270	−0.387618
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.75240	−0.309492
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−7.05456	−0.574092	−0.287046	−	0.957917i	$-0.592673\pi$
$151$	−7.05456	−0.574092	−0.287046	+	0.957917i	$0.592673\pi$
$152$	0	0
$153$	17.5892	1.42200
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.0680837	−0.00543367	−0.00271683	−	0.999996i	$-0.500865\pi$
$157$	−0.0680837	−0.00543367	−0.00271683	+	0.999996i	$0.500865\pi$
$158$	0	0
$159$	−0.478903	−0.0379795
$160$	0	0
$161$	2.88972	0.227742
$162$	0	0
$163$	7.65726	0.599763	0.299882	−	0.953976i	$-0.403053\pi$
$163$	7.65726	0.599763	0.299882	+	0.953976i	$0.403053\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.82164	0.373110	0.186555	−	0.982445i	$-0.440268\pi$
$167$	4.82164	0.373110	0.186555	+	0.982445i	$0.440268\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	21.2735	1.62682
$172$	0	0
$173$	−12.7946	−0.972755	−0.486377	−	0.873749i	$-0.661682\pi$
$173$	−12.7946	−0.972755	−0.486377	+	0.873749i	$0.661682\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.86267	0.365501
$178$	0	0
$179$	8.84216	0.660894	0.330447	−	0.943825i	$-0.392801\pi$
$179$	8.84216	0.660894	0.330447	+	0.943825i	$0.392801\pi$
$180$	0	0
$181$	1.61623	0.120133	0.0600667	−	0.998194i	$-0.480869\pi$
$181$	1.61623	0.120133	0.0600667	+	0.998194i	$0.480869\pi$
$182$	0	0
$183$	1.72535	0.127541
$184$	0	0
$185$	0	0
$186$	0	0
$187$	13.3416	0.975633
$188$	0	0
$189$	−12.3551	−0.898701
$190$	0	0
$191$	−21.3416	−1.54422	−0.772111	−	0.635487i	$-0.780799\pi$
$191$	−21.3416	−1.54422	−0.772111	+	0.635487i	$0.780799\pi$
$192$	0	0
$193$	−5.61623	−0.404265	−0.202133	−	0.979358i	$-0.564787\pi$
$193$	−5.61623	−0.404265	−0.202133	+	0.979358i	$0.564787\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.0205	0.998920	0.499460	−	0.866337i	$-0.333532\pi$
$197$	14.0205	0.998920	0.499460	+	0.866337i	$0.333532\pi$
$198$	0	0
$199$	15.7524	1.11666	0.558329	−	0.829620i	$-0.311443\pi$
$199$	15.7524	1.11666	0.558329	+	0.829620i	$0.311443\pi$
$200$	0	0
$201$	−3.13733	−0.221290
$202$	0	0
$203$	29.3416	2.05937
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.09513	0.145622
$208$	0	0
$209$	16.1362	1.11616
$210$	0	0
$211$	1.70483	0.117365	0.0586827	−	0.998277i	$-0.481310\pi$
$211$	1.70483	0.117365	0.0586827	+	0.998277i	$0.481310\pi$
$212$	0	0
$213$	−5.70483	−0.390889
$214$	0	0
$215$	0	0
$216$	0	0
$217$	18.9308	1.28510
$218$	0	0
$219$	3.61623	0.244362
$220$	0	0
$221$	−6.67079	−0.448726
$222$	0	0
$223$	−22.9513	−1.53693	−0.768466	−	0.639891i	$-0.778979\pi$
$223$	−22.9513	−1.53693	−0.768466	+	0.639891i	$0.778979\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.5470	−0.700028	−0.350014	−	0.936744i	$-0.613823\pi$
$227$	−10.5470	−0.700028	−0.350014	+	0.936744i	$0.613823\pi$
$228$	0	0
$229$	25.9443	1.71445	0.857223	−	0.514945i	$-0.172188\pi$
$229$	25.9443	1.71445	0.857223	+	0.514945i	$0.172188\pi$
$230$	0	0
$231$	−4.38377	−0.288431
$232$	0	0
$233$	11.4924	0.752894	0.376447	−	0.926438i	$-0.377146\pi$
$233$	11.4924	0.752894	0.376447	+	0.926438i	$0.377146\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.16438	0.0756343
$238$	0	0
$239$	−14.6708	−0.948974	−0.474487	−	0.880262i	$-0.657366\pi$
$239$	−14.6708	−0.948974	−0.474487	+	0.880262i	$0.657366\pi$
$240$	0	0
$241$	−18.6151	−1.19910	−0.599551	−	0.800337i	$-0.704654\pi$
$241$	−18.6151	−1.19910	−0.599551	+	0.800337i	$0.704654\pi$
$242$	0	0
$243$	−13.7253	−0.880481
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.06808	−0.513360
$248$	0	0
$249$	1.24644	0.0789902
$250$	0	0
$251$	14.4108	0.909603	0.454801	−	0.890593i	$-0.349710\pi$
$251$	14.4108	0.909603	0.454801	+	0.890593i	$0.349710\pi$
$252$	0	0
$253$	1.58918	0.0999109
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.123802	0.00772253	0.00386126	−	0.999993i	$-0.498771\pi$
$257$	0.123802	0.00772253	0.00386126	+	0.999993i	$0.498771\pi$
$258$	0	0
$259$	−1.56866	−0.0974720
$260$	0	0
$261$	21.2735	1.31680
$262$	0	0
$263$	−5.27349	−0.325178	−0.162589	−	0.986694i	$-0.551984\pi$
$263$	−5.27349	−0.325178	−0.162589	+	0.986694i	$0.551984\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.86267	0.297591
$268$	0	0
$269$	1.27349	0.0776463	0.0388231	−	0.999246i	$-0.487639\pi$
$269$	1.27349	0.0776463	0.0388231	+	0.999246i	$0.487639\pi$
$270$	0	0
$271$	−6.28702	−0.381909	−0.190955	−	0.981599i	$-0.561158\pi$
$271$	−6.28702	−0.381909	−0.190955	+	0.981599i	$0.561158\pi$
$272$	0	0
$273$	2.19189	0.132659
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.9988	−1.38187	−0.690933	−	0.722919i	$-0.742800\pi$
$277$	−22.9988	−1.38187	−0.690933	+	0.722919i	$0.742800\pi$
$278$	0	0
$279$	13.7253	0.821715
$280$	0	0
$281$	10.3427	0.616996	0.308498	−	0.951225i	$-0.400174\pi$
$281$	10.3427	0.616996	0.308498	+	0.951225i	$0.400174\pi$
$282$	0	0
$283$	−10.0270	−0.596046	−0.298023	−	0.954559i	$-0.596327\pi$
$283$	−10.0270	−0.596046	−0.298023	+	0.954559i	$0.596327\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−24.9578	−1.47321
$288$	0	0
$289$	27.4994	1.61761
$290$	0	0
$291$	8.23130	0.482527
$292$	0	0
$293$	−6.11565	−0.357280	−0.178640	−	0.983914i	$-0.557170\pi$
$293$	−6.11565	−0.357280	−0.178640	+	0.983914i	$0.557170\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−6.79459	−0.394262
$298$	0	0
$299$	−0.794590	−0.0459523
$300$	0	0
$301$	−21.1226	−1.21749
$302$	0	0
$303$	9.45301	0.543061
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.8627	0.962404	0.481202	−	0.876610i	$-0.340200\pi$
$307$	16.8627	0.962404	0.481202	+	0.876610i	$0.340200\pi$
$308$	0	0
$309$	−9.24644	−0.526012
$310$	0	0
$311$	−32.9988	−1.87119	−0.935596	−	0.353072i	$-0.885137\pi$
$311$	−32.9988	−1.87119	−0.935596	+	0.353072i	$0.885137\pi$
$312$	0	0
$313$	25.4654	1.43939	0.719694	−	0.694291i	$-0.244282\pi$
$313$	25.4654	1.43939	0.719694	+	0.694291i	$0.244282\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.1362	0.793966	0.396983	−	0.917826i	$-0.370057\pi$
$317$	14.1362	0.793966	0.396983	+	0.917826i	$0.370057\pi$
$318$	0	0
$319$	16.1362	0.903452
$320$	0	0
$321$	0.726506	0.0405496
$322$	0	0
$323$	53.8205	2.99465
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.91140	−0.326901
$328$	0	0
$329$	−27.7729	−1.53117
$330$	0	0
$331$	−18.5199	−1.01795	−0.508974	−	0.860782i	$-0.669975\pi$
$331$	−18.5199	−1.01795	−0.508974	+	0.860782i	$0.669975\pi$
$332$	0	0
$333$	−1.13733	−0.0623251
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.2870	1.21405	0.607026	−	0.794682i	$-0.292363\pi$
$337$	22.2870	1.21405	0.607026	+	0.794682i	$0.292363\pi$
$338$	0	0
$339$	−2.41082	−0.130938
$340$	0	0
$341$	10.4108	0.563777
$342$	0	0
$343$	2.81511	0.152002
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.5335	1.69280	0.846402	−	0.532544i	$-0.178764\pi$
$347$	31.5335	1.69280	0.846402	+	0.532544i	$0.178764\pi$
$348$	0	0
$349$	32.3551	1.73193	0.865964	−	0.500106i	$-0.166705\pi$
$349$	32.3551	1.73193	0.865964	+	0.500106i	$0.166705\pi$
$350$	0	0
$351$	3.39730	0.181334
$352$	0	0
$353$	20.0270	1.06593	0.532966	−	0.846137i	$-0.321077\pi$
$353$	20.0270	1.06593	0.532966	+	0.846137i	$0.321077\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−14.6216	−0.773857
$358$	0	0
$359$	−22.9308	−1.21024	−0.605120	−	0.796135i	$-0.706875\pi$
$359$	−22.9308	−1.21024	−0.605120	+	0.796135i	$0.706875\pi$
$360$	0	0
$361$	46.0940	2.42600
$362$	0	0
$363$	4.21893	0.221437
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−34.1362	−1.78189	−0.890947	−	0.454108i	$-0.849958\pi$
$367$	−34.1362	−1.78189	−0.890947	+	0.454108i	$0.849958\pi$
$368$	0	0
$369$	−18.0951	−0.941995
$370$	0	0
$371$	−2.88972	−0.150027
$372$	0	0
$373$	−26.9988	−1.39795	−0.698974	−	0.715148i	$-0.746359\pi$
$373$	−26.9988	−1.39795	−0.698974	+	0.715148i	$0.746359\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.06808	−0.415527
$378$	0	0
$379$	−12.9308	−0.664208	−0.332104	−	0.943243i	$-0.607759\pi$
$379$	−12.9308	−0.664208	−0.332104	+	0.943243i	$0.607759\pi$
$380$	0	0
$381$	2.88972	0.148045
$382$	0	0
$383$	21.3621	1.09155	0.545776	−	0.837931i	$-0.316235\pi$
$383$	21.3621	1.09155	0.545776	+	0.837931i	$0.316235\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−15.3145	−0.778481
$388$	0	0
$389$	−29.4507	−1.49321	−0.746605	−	0.665268i	$-0.768317\pi$
$389$	−29.4507	−1.49321	−0.746605	+	0.665268i	$0.768317\pi$
$390$	0	0
$391$	5.30054	0.268060
$392$	0	0
$393$	−2.42319	−0.122234
$394$	0	0
$395$	0	0
$396$	0	0
$397$	16.6832	0.837304	0.418652	−	0.908147i	$-0.362503\pi$
$397$	16.6832	0.837304	0.418652	+	0.908147i	$0.362503\pi$
$398$	0	0
$399$	−17.6843	−0.885323
$400$	0	0
$401$	−7.34158	−0.366621	−0.183310	−	0.983055i	$-0.558681\pi$
$401$	−7.34158	−0.366621	−0.183310	+	0.983055i	$0.558681\pi$
$402$	0	0
$403$	−5.20541	−0.259300
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.862674	−0.0427612
$408$	0	0
$409$	−35.4777	−1.75426	−0.877131	−	0.480252i	$-0.840545\pi$
$409$	−35.4777	−1.75426	−0.877131	+	0.480252i	$0.840545\pi$
$410$	0	0
$411$	−6.75356	−0.333128
$412$	0	0
$413$	29.3416	1.44380
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.32921	0.260973
$418$	0	0
$419$	−0.883191	−0.0431467	−0.0215734	−	0.999767i	$-0.506868\pi$
$419$	−0.883191	−0.0431467	−0.0215734	+	0.999767i	$0.506868\pi$
$420$	0	0
$421$	27.2859	1.32983	0.664916	−	0.746918i	$-0.268467\pi$
$421$	27.2859	1.32983	0.664916	+	0.746918i	$0.268467\pi$
$422$	0	0
$423$	−20.1362	−0.979054
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10.4108	0.503815
$428$	0	0
$429$	1.20541	0.0581977
$430$	0	0
$431$	−0.259969	−0.0125223	−0.00626113	−	0.999980i	$-0.501993\pi$
$431$	−0.259969	−0.0125223	−0.00626113	+	0.999980i	$0.501993\pi$
$432$	0	0
$433$	−17.4654	−0.839333	−0.419666	−	0.907678i	$-0.637853\pi$
$433$	−17.4654	−0.839333	−0.419666	+	0.907678i	$0.637853\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.41082	0.306671
$438$	0	0
$439$	−17.6843	−0.844026	−0.422013	−	0.906590i	$-0.638676\pi$
$439$	−17.6843	−0.844026	−0.422013	+	0.906590i	$0.638676\pi$
$440$	0	0
$441$	−16.4162	−0.781723
$442$	0	0
$443$	−32.4913	−1.54371	−0.771853	−	0.635801i	$-0.780670\pi$
$443$	−32.4913	−1.54371	−0.771853	+	0.635801i	$0.780670\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	35.0940	1.65619	0.828094	−	0.560590i	$-0.189426\pi$
$449$	35.0940	1.65619	0.828094	+	0.560590i	$0.189426\pi$
$450$	0	0
$451$	−13.7253	−0.646301
$452$	0	0
$453$	4.25182	0.199768
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.5470	0.961148	0.480574	−	0.876954i	$-0.340428\pi$
$457$	20.5470	0.961148	0.480574	+	0.876954i	$0.340428\pi$
$458$	0	0
$459$	−22.6626	−1.05780
$460$	0	0
$461$	−7.14969	−0.332994	−0.166497	−	0.986042i	$-0.553246\pi$
$461$	−7.14969	−0.332994	−0.166497	+	0.986042i	$0.553246\pi$
$462$	0	0
$463$	−22.6832	−1.05418	−0.527088	−	0.849811i	$-0.676716\pi$
$463$	−22.6832	−1.05418	−0.527088	+	0.849811i	$0.676716\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.2054	1.16637	0.583184	−	0.812340i	$-0.301807\pi$
$467$	25.2054	1.16637	0.583184	+	0.812340i	$0.301807\pi$
$468$	0	0
$469$	−18.9308	−0.874141
$470$	0	0
$471$	0.0410344	0.00189076
$472$	0	0
$473$	−11.6162	−0.534115
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.09513	−0.0959295
$478$	0	0
$479$	−4.01237	−0.183330	−0.0916648	−	0.995790i	$-0.529219\pi$
$479$	−4.01237	−0.183330	−0.0916648	+	0.995790i	$0.529219\pi$
$480$	0	0
$481$	0.431337	0.0196673
$482$	0	0
$483$	−1.74165	−0.0792478
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	−4.61507	−0.208701
$490$	0	0
$491$	8.88319	0.400893	0.200446	−	0.979705i	$-0.435761\pi$
$491$	8.88319	0.400893	0.200446	+	0.979705i	$0.435761\pi$
$492$	0	0
$493$	53.8205	2.42395
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−34.4232	−1.54409
$498$	0	0
$499$	10.9988	0.492376	0.246188	−	0.969222i	$-0.420822\pi$
$499$	10.9988	0.492376	0.246188	+	0.969222i	$0.420822\pi$
$500$	0	0
$501$	−2.90603	−0.129832
$502$	0	0
$503$	−5.65726	−0.252245	−0.126122	−	0.992015i	$-0.540253\pi$
$503$	−5.65726	−0.252245	−0.126122	+	0.992015i	$0.540253\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−0.602705	−0.0267671
$508$	0	0
$509$	−9.72535	−0.431068	−0.215534	−	0.976496i	$-0.569149\pi$
$509$	−9.72535	−0.431068	−0.215534	+	0.976496i	$0.569149\pi$
$510$	0	0
$511$	21.8205	0.965281
$512$	0	0
$513$	−27.4097	−1.21017
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−15.2735	−0.671727
$518$	0	0
$519$	7.71136	0.338491
$520$	0	0
$521$	−28.7741	−1.26062	−0.630308	−	0.776346i	$-0.717071\pi$
$521$	−28.7741	−1.26062	−0.630308	+	0.776346i	$0.717071\pi$
$522$	0	0
$523$	37.6139	1.64474	0.822371	−	0.568952i	$-0.192651\pi$
$523$	37.6139	1.64474	0.822371	+	0.568952i	$0.192651\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	34.7242	1.51261
$528$	0	0
$529$	−22.3686	−0.972549
$530$	0	0
$531$	21.2735	0.923191
$532$	0	0
$533$	6.86267	0.297255
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5.32921	−0.229972
$538$	0	0
$539$	−12.4519	−0.536339
$540$	0	0
$541$	21.8081	0.937604	0.468802	−	0.883303i	$-0.344686\pi$
$541$	21.8081	0.937604	0.468802	+	0.883303i	$0.344686\pi$
$542$	0	0
$543$	−0.974110	−0.0418030
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.73887	0.202620	0.101310	−	0.994855i	$-0.467697\pi$
$547$	4.73887	0.202620	0.101310	+	0.994855i	$0.467697\pi$
$548$	0	0
$549$	7.54815	0.322147
$550$	0	0
$551$	65.0940	2.77310
$552$	0	0
$553$	7.02589	0.298771
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.74702	−0.285881	−0.142940	−	0.989731i	$-0.545656\pi$
$557$	−6.74702	−0.285881	−0.142940	+	0.989731i	$0.545656\pi$
$558$	0	0
$559$	5.80811	0.245657
$560$	0	0
$561$	−8.04103	−0.339493
$562$	0	0
$563$	41.9443	1.76774	0.883870	−	0.467732i	$-0.154929\pi$
$563$	41.9443	1.76774	0.883870	+	0.467732i	$0.154929\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−21.3211	−0.895400
$568$	0	0
$569$	−7.95243	−0.333383	−0.166692	−	0.986009i	$-0.553308\pi$
$569$	−7.95243	−0.333383	−0.166692	+	0.986009i	$0.553308\pi$
$570$	0	0
$571$	14.5265	0.607914	0.303957	−	0.952686i	$-0.401692\pi$
$571$	14.5265	0.607914	0.303957	+	0.952686i	$0.401692\pi$
$572$	0	0
$573$	12.8627	0.537346
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.17836	0.215578	0.107789	−	0.994174i	$-0.465623\pi$
$577$	5.17836	0.215578	0.107789	+	0.994174i	$0.465623\pi$
$578$	0	0
$579$	3.38493	0.140673
$580$	0	0
$581$	7.52110	0.312028
$582$	0	0
$583$	−1.58918	−0.0658171
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.8897	0.944760	0.472380	−	0.881395i	$-0.343395\pi$
$587$	22.8897	0.944760	0.472380	+	0.881395i	$0.343395\pi$
$588$	0	0
$589$	41.9977	1.73048
$590$	0	0
$591$	−8.45023	−0.347596
$592$	0	0
$593$	−14.3427	−0.588986	−0.294493	−	0.955654i	$-0.595151\pi$
$593$	−14.3427	−0.588986	−0.294493	+	0.955654i	$0.595151\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−9.49405	−0.388565
$598$	0	0
$599$	−5.63021	−0.230044	−0.115022	−	0.993363i	$-0.536694\pi$
$599$	−5.63021	−0.230044	−0.115022	+	0.993363i	$0.536694\pi$
$600$	0	0
$601$	−11.3211	−0.461796	−0.230898	−	0.972978i	$-0.574166\pi$
$601$	−11.3211	−0.461796	−0.230898	+	0.972978i	$0.574166\pi$
$602$	0	0
$603$	−13.7253	−0.558939
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−25.0259	−1.01577	−0.507885	−	0.861425i	$-0.669572\pi$
$607$	−25.0259	−1.01577	−0.507885	+	0.861425i	$0.669572\pi$
$608$	0	0
$609$	−17.6843	−0.716605
$610$	0	0
$611$	7.63675	0.308950
$612$	0	0
$613$	−25.4507	−1.02794	−0.513972	−	0.857807i	$-0.671826\pi$
$613$	−25.4507	−1.02794	−0.513972	+	0.857807i	$0.671826\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.6843	0.631427	0.315713	−	0.948855i	$-0.397756\pi$
$617$	15.6843	0.631427	0.315713	+	0.948855i	$0.397756\pi$
$618$	0	0
$619$	44.5880	1.79214	0.896072	−	0.443909i	$-0.146409\pi$
$619$	44.5880	1.79214	0.896072	+	0.443909i	$0.146409\pi$
$620$	0	0
$621$	−2.69946	−0.108325
$622$	0	0
$623$	29.3416	1.17555
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−9.72535	−0.388393
$628$	0	0
$629$	−2.87736	−0.114728
$630$	0	0
$631$	38.8070	1.54488	0.772440	−	0.635087i	$-0.219036\pi$
$631$	38.8070	1.54488	0.772440	+	0.635087i	$0.219036\pi$
$632$	0	0
$633$	−1.02751	−0.0408398
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.22593	0.246680
$638$	0	0
$639$	−24.9578	−0.987316
$640$	0	0
$641$	−8.27465	−0.326829	−0.163415	−	0.986557i	$-0.552251\pi$
$641$	−8.27465	−0.326829	−0.163415	+	0.986557i	$0.552251\pi$
$642$	0	0
$643$	−6.16322	−0.243054	−0.121527	−	0.992588i	$-0.538779\pi$
$643$	−6.16322	−0.243054	−0.121527	+	0.992588i	$0.538779\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.24644	0.284887	0.142444	−	0.989803i	$-0.454504\pi$
$647$	7.24644	0.284887	0.142444	+	0.989803i	$0.454504\pi$
$648$	0	0
$649$	16.1362	0.633400
$650$	0	0
$651$	−11.4097	−0.447180
$652$	0	0
$653$	5.27349	0.206368	0.103184	−	0.994662i	$-0.467097\pi$
$653$	5.27349	0.206368	0.103184	+	0.994662i	$0.467097\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	15.8205	0.617216
$658$	0	0
$659$	18.6832	0.727792	0.363896	−	0.931440i	$-0.381446\pi$
$659$	18.6832	0.727792	0.363896	+	0.931440i	$0.381446\pi$
$660$	0	0
$661$	15.3145	0.595666	0.297833	−	0.954618i	$-0.403736\pi$
$661$	15.3145	0.595666	0.297833	+	0.954618i	$0.403736\pi$
$662$	0	0
$663$	4.02052	0.156144
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.41082	0.248228
$668$	0	0
$669$	13.8328	0.534809
$670$	0	0
$671$	5.72535	0.221025
$672$	0	0
$673$	−10.2870	−0.396535	−0.198268	−	0.980148i	$-0.563532\pi$
$673$	−10.2870	−0.396535	−0.198268	+	0.980148i	$0.563532\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.88856	0.380048	0.190024	−	0.981779i	$-0.439143\pi$
$677$	9.88856	0.380048	0.190024	+	0.981779i	$0.439143\pi$
$678$	0	0
$679$	49.6680	1.90608
$680$	0	0
$681$	6.35672	0.243590
$682$	0	0
$683$	−2.06808	−0.0791330	−0.0395665	−	0.999217i	$-0.512598\pi$
$683$	−2.06808	−0.0791330	−0.0395665	+	0.999217i	$0.512598\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−15.6367	−0.596579
$688$	0	0
$689$	0.794590	0.0302715
$690$	0	0
$691$	−15.8205	−0.601839	−0.300920	−	0.953649i	$-0.597294\pi$
$691$	−15.8205	−0.601839	−0.300920	+	0.953649i	$0.597294\pi$
$692$	0	0
$693$	−19.1784	−0.728526
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−45.7794	−1.73402
$698$	0	0
$699$	−6.92654	−0.261986
$700$	0	0
$701$	14.6561	0.553553	0.276777	−	0.960934i	$-0.410734\pi$
$701$	14.6561	0.553553	0.276777	+	0.960934i	$0.410734\pi$
$702$	0	0
$703$	−3.48006	−0.131253
$704$	0	0
$705$	0	0
$706$	0	0
$707$	57.0399	2.14521
$708$	0	0
$709$	−23.3145	−0.875595	−0.437798	−	0.899073i	$-0.644241\pi$
$709$	−23.3145	−0.875595	−0.437798	+	0.899073i	$0.644241\pi$
$710$	0	0
$711$	5.09397	0.191039
$712$	0	0
$713$	4.13617	0.154901
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.84216	0.330216
$718$	0	0
$719$	29.3416	1.09426	0.547128	−	0.837049i	$-0.315721\pi$
$719$	29.3416	1.09426	0.547128	+	0.837049i	$0.315721\pi$
$720$	0	0
$721$	−55.7934	−2.07786
$722$	0	0
$723$	11.2194	0.417254
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−15.4530	−0.573121	−0.286560	−	0.958062i	$-0.592512\pi$
$727$	−15.4530	−0.573121	−0.286560	+	0.958062i	$0.592512\pi$
$728$	0	0
$729$	−9.31569	−0.345025
$730$	0	0
$731$	−38.7447	−1.43302
$732$	0	0
$733$	−30.2108	−1.11586	−0.557930	−	0.829888i	$-0.688404\pi$
$733$	−30.2108	−1.11586	−0.557930	+	0.829888i	$0.688404\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.4108	−0.383487
$738$	0	0
$739$	24.5880	0.904485	0.452242	−	0.891895i	$-0.350624\pi$
$739$	24.5880	0.904485	0.452242	+	0.891895i	$0.350624\pi$
$740$	0	0
$741$	4.86267	0.178635
$742$	0	0
$743$	−48.0452	−1.76261	−0.881305	−	0.472549i	$-0.843334\pi$
$743$	−48.0452	−1.76261	−0.881305	+	0.472549i	$0.843334\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5.45301	0.199515
$748$	0	0
$749$	4.38377	0.160179
$750$	0	0
$751$	4.90371	0.178939	0.0894694	−	0.995990i	$-0.471483\pi$
$751$	4.90371	0.178939	0.0894694	+	0.995990i	$0.471483\pi$
$752$	0	0
$753$	−8.68547	−0.316516
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−11.3416	−0.412217	−0.206108	−	0.978529i	$-0.566080\pi$
$757$	−11.3416	−0.412217	−0.206108	+	0.978529i	$0.566080\pi$
$758$	0	0
$759$	−0.957807	−0.0347662
$760$	0	0
$761$	10.3427	0.374924	0.187462	−	0.982272i	$-0.439974\pi$
$761$	10.3427	0.374924	0.187462	+	0.982272i	$0.439974\pi$
$762$	0	0
$763$	−35.6696	−1.29133
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.06808	−0.291322
$768$	0	0
$769$	27.6843	0.998322	0.499161	−	0.866509i	$-0.333642\pi$
$769$	27.6843	0.998322	0.499161	+	0.866509i	$0.333642\pi$
$770$	0	0
$771$	−0.0746158	−0.00268722
$772$	0	0
$773$	−12.2952	−0.442227	−0.221113	−	0.975248i	$-0.570969\pi$
$773$	−12.2952	−0.442227	−0.221113	+	0.975248i	$0.570969\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.945441	0.0339175
$778$	0	0
$779$	−55.3686	−1.98379
$780$	0	0
$781$	−18.9308	−0.677396
$782$	0	0
$783$	−27.4097	−0.979541
$784$	0	0
$785$	0	0
$786$	0	0
$787$	43.5048	1.55078	0.775389	−	0.631484i	$-0.217554\pi$
$787$	43.5048	1.55078	0.775389	+	0.631484i	$0.217554\pi$
$788$	0	0
$789$	3.17836	0.113153
$790$	0	0
$791$	−14.5470	−0.517231
$792$	0	0
$793$	−2.86267	−0.101657
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.9718	0.459484	0.229742	−	0.973252i	$-0.426212\pi$
$797$	12.9718	0.459484	0.229742	+	0.973252i	$0.426212\pi$
$798$	0	0
$799$	−50.9431	−1.80224
$800$	0	0
$801$	21.2735	0.751662
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.767541	−0.0270187
$808$	0	0
$809$	8.73304	0.307037	0.153519	−	0.988146i	$-0.450939\pi$
$809$	8.73304	0.307037	0.153519	+	0.988146i	$0.450939\pi$
$810$	0	0
$811$	−46.3134	−1.62628	−0.813141	−	0.582067i	$-0.802244\pi$
$811$	−46.3134	−1.62628	−0.813141	+	0.582067i	$0.802244\pi$
$812$	0	0
$813$	3.78922	0.132894
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−46.8604	−1.63944
$818$	0	0
$819$	9.58918	0.335073
$820$	0	0
$821$	31.9172	1.11392	0.556960	−	0.830540i	$-0.311968\pi$
$821$	31.9172	1.11392	0.556960	+	0.830540i	$0.311968\pi$
$822$	0	0
$823$	−6.42480	−0.223955	−0.111977	−	0.993711i	$-0.535718\pi$
$823$	−6.42480	−0.223955	−0.111977	+	0.993711i	$0.535718\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−38.7242	−1.34657	−0.673286	−	0.739382i	$-0.735118\pi$
$827$	−38.7242	−1.34657	−0.673286	+	0.739382i	$0.735118\pi$
$828$	0	0
$829$	−18.3427	−0.637070	−0.318535	−	0.947911i	$-0.603191\pi$
$829$	−18.3427	−0.637070	−0.318535	+	0.947911i	$0.603191\pi$
$830$	0	0
$831$	13.8615	0.480851
$832$	0	0
$833$	−41.5318	−1.43899
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−17.6843	−0.611259
$838$	0	0
$839$	12.6561	0.436937	0.218469	−	0.975844i	$-0.429894\pi$
$839$	12.6561	0.436937	0.218469	+	0.975844i	$0.429894\pi$
$840$	0	0
$841$	36.0940	1.24462
$842$	0	0
$843$	−6.23362	−0.214697
$844$	0	0
$845$	0	0
$846$	0	0
$847$	25.4572	0.874721
$848$	0	0
$849$	6.04335	0.207407
$850$	0	0
$851$	−0.342736	−0.0117488
$852$	0	0
$853$	−44.7988	−1.53388	−0.766941	−	0.641718i	$-0.778222\pi$
$853$	−44.7988	−1.53388	−0.766941	+	0.641718i	$0.778222\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−46.8193	−1.59932	−0.799659	−	0.600455i	$-0.794986\pi$
$857$	−46.8193	−1.59932	−0.799659	+	0.600455i	$0.794986\pi$
$858$	0	0
$859$	23.8638	0.814223	0.407112	−	0.913378i	$-0.366536\pi$
$859$	23.8638	0.814223	0.407112	+	0.913378i	$0.366536\pi$
$860$	0	0
$861$	15.0422	0.512637
$862$	0	0
$863$	−12.2271	−0.416215	−0.208107	−	0.978106i	$-0.566730\pi$
$863$	−12.2271	−0.416215	−0.208107	+	0.978106i	$0.566730\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.5740	−0.562884
$868$	0	0
$869$	3.86383	0.131072
$870$	0	0
$871$	5.20541	0.176379
$872$	0	0
$873$	36.0107	1.21878
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−0.336204	−0.0113528	−0.00567640	−	0.999984i	$-0.501807\pi$
$877$	−0.336204	−0.0113528	−0.00567640	+	0.999984i	$0.501807\pi$
$878$	0	0
$879$	3.68593	0.124323
$880$	0	0
$881$	−41.0464	−1.38289	−0.691444	−	0.722430i	$-0.743025\pi$
$881$	−41.0464	−1.38289	−0.691444	+	0.722430i	$0.743025\pi$
$882$	0	0
$883$	−40.0534	−1.34790	−0.673952	−	0.738775i	$-0.735405\pi$
$883$	−40.0534	−1.34790	−0.673952	+	0.738775i	$0.735405\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28.7242	−0.964464	−0.482232	−	0.876044i	$-0.660174\pi$
$887$	−28.7242	−0.964464	−0.482232	+	0.876044i	$0.660174\pi$
$888$	0	0
$889$	17.4367	0.584808
$890$	0	0
$891$	−11.7253	−0.392814
$892$	0	0
$893$	−61.6139	−2.06183
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.478903	0.0159901
$898$	0	0
$899$	41.9977	1.40070
$900$	0	0
$901$	−5.30054	−0.176587
$902$	0	0
$903$	12.7307	0.423652
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−48.7636	−1.61917	−0.809584	−	0.587003i	$-0.800308\pi$
$907$	−48.7636	−1.61917	−0.809584	+	0.587003i	$0.800308\pi$
$908$	0	0
$909$	41.3556	1.37168
$910$	0	0
$911$	−15.4801	−0.512877	−0.256439	−	0.966561i	$-0.582549\pi$
$911$	−15.4801	−0.512877	−0.256439	+	0.966561i	$0.582549\pi$
$912$	0	0
$913$	4.13617	0.136887
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14.6216	−0.482848
$918$	0	0
$919$	−45.8615	−1.51283	−0.756416	−	0.654091i	$-0.773051\pi$
$919$	−45.8615	−1.51283	−0.756416	+	0.654091i	$0.773051\pi$
$920$	0	0
$921$	−10.1632	−0.334889
$922$	0	0
$923$	9.46538	0.311557
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−40.4519	−1.32861
$928$	0	0
$929$	−26.9168	−0.883111	−0.441555	−	0.897234i	$-0.645573\pi$
$929$	−26.9168	−0.883111	−0.441555	+	0.897234i	$0.645573\pi$
$930$	0	0
$931$	−50.2313	−1.64626
$932$	0	0
$933$	19.8886	0.651122
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.90371	0.160197	0.0800986	−	0.996787i	$-0.474476\pi$
$937$	4.90371	0.160197	0.0800986	+	0.996787i	$0.474476\pi$
$938$	0	0
$939$	−15.3481	−0.500867
$940$	0	0
$941$	−46.6004	−1.51913	−0.759565	−	0.650432i	$-0.774588\pi$
$941$	−46.6004	−1.51913	−0.759565	+	0.650432i	$0.774588\pi$
$942$	0	0
$943$	−5.45301	−0.177575
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.7524	0.901832	0.450916	−	0.892566i	$-0.351097\pi$
$947$	27.7524	0.901832	0.450916	+	0.892566i	$0.351097\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	−8.51994	−0.276278
$952$	0	0
$953$	14.9725	0.485007	0.242503	−	0.970151i	$-0.422031\pi$
$953$	14.9725	0.485007	0.242503	+	0.970151i	$0.422031\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−9.72535	−0.314376
$958$	0	0
$959$	−40.7512	−1.31593
$960$	0	0
$961$	−3.90371	−0.125926
$962$	0	0
$963$	3.17836	0.102421
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−52.5946	−1.69133	−0.845663	−	0.533717i	$-0.820795\pi$
$967$	−52.5946	−1.69133	−0.845663	+	0.533717i	$0.820795\pi$
$968$	0	0
$969$	−32.4379	−1.04205
$970$	0	0
$971$	15.5253	0.498231	0.249115	−	0.968474i	$-0.419860\pi$
$971$	15.5253	0.498231	0.249115	+	0.968474i	$0.419860\pi$
$972$	0	0
$973$	32.1567	1.03090
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.5729	1.52199	0.760996	−	0.648757i	$-0.224711\pi$
$977$	47.5729	1.52199	0.760996	+	0.648757i	$0.224711\pi$
$978$	0	0
$979$	16.1362	0.515714
$980$	0	0
$981$	−25.8615	−0.825695
$982$	0	0
$983$	−37.3211	−1.19036	−0.595178	−	0.803594i	$-0.702919\pi$
$983$	−37.3211	−1.19036	−0.595178	+	0.803594i	$0.702919\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	16.7389	0.532804
$988$	0	0
$989$	−4.61507	−0.146751
$990$	0	0
$991$	−22.5060	−0.714925	−0.357463	−	0.933927i	$-0.616358\pi$
$991$	−22.5060	−0.714925	−0.357463	+	0.933927i	$0.616358\pi$
$992$	0	0
$993$	11.1621	0.354217
$994$	0	0
$995$	0	0
$996$	0	0
$997$	24.4108	0.773098	0.386549	−	0.922269i	$-0.373667\pi$
$997$	24.4108	0.773098	0.386549	+	0.922269i	$0.373667\pi$
$998$	0	0
$999$	1.46538	0.0463625

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5200.2.a.cf.1.2		3
4.3	odd	2		650.2.a.o.1.2		3
5.2	odd	4		1040.2.d.b.209.4		6
5.3	odd	4		1040.2.d.b.209.3		6
5.4	even	2		5200.2.a.ce.1.2		3
12.11	even	2		5850.2.a.cp.1.3		3
20.3	even	4		130.2.b.a.79.2	✓	6
20.7	even	4		130.2.b.a.79.5	yes	6
20.19	odd	2		650.2.a.n.1.2		3
52.51	odd	2		8450.2.a.bs.1.2		3
60.23	odd	4		1170.2.e.f.469.5		6
60.47	odd	4		1170.2.e.f.469.2		6
60.59	even	2		5850.2.a.cs.1.1		3
260.47	odd	4		1690.2.c.a.1689.3		6
260.83	odd	4		1690.2.c.a.1689.4		6
260.103	even	4		1690.2.b.a.339.5		6
260.187	odd	4		1690.2.c.d.1689.3		6
260.203	odd	4		1690.2.c.d.1689.4		6
260.207	even	4		1690.2.b.a.339.2		6
260.259	odd	2		8450.2.a.cc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.b.a.79.2	✓	6	20.3	even	4
130.2.b.a.79.5	yes	6	20.7	even	4
650.2.a.n.1.2		3	20.19	odd	2
650.2.a.o.1.2		3	4.3	odd	2
1040.2.d.b.209.3		6	5.3	odd	4
1040.2.d.b.209.4		6	5.2	odd	4
1170.2.e.f.469.2		6	60.47	odd	4
1170.2.e.f.469.5		6	60.23	odd	4
1690.2.b.a.339.2		6	260.207	even	4
1690.2.b.a.339.5		6	260.103	even	4
1690.2.c.a.1689.3		6	260.47	odd	4
1690.2.c.a.1689.4		6	260.83	odd	4
1690.2.c.d.1689.3		6	260.187	odd	4
1690.2.c.d.1689.4		6	260.203	odd	4
5200.2.a.ce.1.2		3	5.4	even	2
5200.2.a.cf.1.2		3	1.1	even	1	trivial
5850.2.a.cp.1.3		3	12.11	even	2
5850.2.a.cs.1.1		3	60.59	even	2
8450.2.a.bs.1.2		3	52.51	odd	2
8450.2.a.cc.1.2		3	260.259	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:	\( N \)	\(=\)	\( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5200.a (trivial)

\(f(q)\)	\(=\)	\(q-0.602705 q^{3} -3.63675 q^{7} -2.63675 q^{9} -2.00000 q^{11} +1.00000 q^{13} -6.67079 q^{17} -8.06808 q^{19} +2.19189 q^{21} -0.794590 q^{23} +3.39730 q^{27} -8.06808 q^{29} -5.20541 q^{31} +1.20541 q^{33} +0.431337 q^{37} -0.602705 q^{39} +6.86267 q^{41} +5.80811 q^{43} +7.63675 q^{47} +6.22593 q^{49} +4.02052 q^{51} +0.794590 q^{53} +4.86267 q^{57} -8.06808 q^{59} -2.86267 q^{61} +9.58918 q^{63} +5.20541 q^{67} +0.478903 q^{69} +9.46538 q^{71} -6.00000 q^{73} +7.27349 q^{77} -1.93192 q^{79} +5.86267 q^{81} -2.06808 q^{83} +4.86267 q^{87} -8.06808 q^{89} -3.63675 q^{91} +3.13733 q^{93} -13.6573 q^{97} +5.27349 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{7} + 5 q^{9} - 6 q^{11} + 3 q^{13} + 4 q^{17} - 2 q^{19} + 12 q^{21} - 6 q^{23} + 12 q^{27} - 2 q^{29} - 12 q^{31} - 8 q^{37} + 2 q^{41} + 12 q^{43} + 10 q^{47} + 13 q^{49} + 10 q^{51} + 6 q^{53}+ \cdots - 10 q^{99}+O(q^{100}) \) 3 * q + 2 * q^7 + 5 * q^9 - 6 * q^11 + 3 * q^13 + 4 * q^17 - 2 * q^19 + 12 * q^21 - 6 * q^23 + 12 * q^27 - 2 * q^29 - 12 * q^31 - 8 * q^37 + 2 * q^41 + 12 * q^43 + 10 * q^47 + 13 * q^49 + 10 * q^51 + 6 * q^53 - 4 * q^57 - 2 * q^59 + 10 * q^61 + 36 * q^63 + 12 * q^67 - 28 * q^69 + 8 * q^71 - 18 * q^73 - 4 * q^77 - 28 * q^79 - q^81 + 16 * q^83 - 4 * q^87 - 2 * q^89 + 2 * q^91 + 28 * q^93 - 26 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more