LMFDB - Embedded newform 7600.2.a.bj.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1900)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.19869$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.19869 q^{3} +1.19869 q^{7} -1.56314 q^{9} +O(q^{10})$	$q+1.19869 q^{3} +1.19869 q^{7} -1.56314 q^{9} +5.86718 q^{11} +0.364448 q^{13} -1.19869 q^{17} +1.00000 q^{19} +1.43686 q^{21} -8.23163 q^{23} -5.46980 q^{27} -7.86718 q^{29} -7.30404 q^{31} +7.03293 q^{33} -7.13828 q^{37} +0.436861 q^{39} +2.43686 q^{41} -7.39738 q^{43} -13.7014 q^{47} -5.56314 q^{49} -1.43686 q^{51} +7.39738 q^{53} +1.19869 q^{57} -12.8606 q^{59} -1.30404 q^{61} -1.87372 q^{63} +11.9330 q^{67} -9.86718 q^{69} -2.12628 q^{71} -2.50273 q^{73} +7.03293 q^{77} +7.74090 q^{79} -1.86718 q^{81} -3.02093 q^{83} -9.43032 q^{87} -5.68942 q^{89} +0.436861 q^{91} -8.75529 q^{93} -1.09334 q^{97} -9.17122 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - 2 q^{7} + q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 - 2 * q^7 + q^9	$ 3 q - 2 q^{3} - 2 q^{7} + q^{9} + q^{11} + q^{13} + 2 q^{17} + 3 q^{19} + 10 q^{21} - 8 q^{23} - 11 q^{27} - 7 q^{29} - 11 q^{31} + 10 q^{33} - 5 q^{37} + 7 q^{39} + 13 q^{41} - 11 q^{43} - 19 q^{47} - 11 q^{49} - 10 q^{51} + 11 q^{53} - 2 q^{57} + 6 q^{59} + 7 q^{61} - 17 q^{63} - 3 q^{67} - 13 q^{69} + 5 q^{71} + 9 q^{73} + 10 q^{77} + 18 q^{79} + 11 q^{81} - 3 q^{83} - 6 q^{87} + 7 q^{91} + 13 q^{93} - 3 q^{97}+O(q^{100}) $ 3 * q - 2 * q^3 - 2 * q^7 + q^9 + q^11 + q^13 + 2 * q^17 + 3 * q^19 + 10 * q^21 - 8 * q^23 - 11 * q^27 - 7 * q^29 - 11 * q^31 + 10 * q^33 - 5 * q^37 + 7 * q^39 + 13 * q^41 - 11 * q^43 - 19 * q^47 - 11 * q^49 - 10 * q^51 + 11 * q^53 - 2 * q^57 + 6 * q^59 + 7 * q^61 - 17 * q^63 - 3 * q^67 - 13 * q^69 + 5 * q^71 + 9 * q^73 + 10 * q^77 + 18 * q^79 + 11 * q^81 - 3 * q^83 - 6 * q^87 + 7 * q^91 + 13 * q^93 - 3 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.19869	0.692065	0.346032	−	0.938223i	$-0.387529\pi$
$3$	1.19869	0.692065	0.346032	+	0.938223i	$0.387529\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.19869	0.453063	0.226531	−	0.974004i	$-0.427261\pi$
$7$	1.19869	0.453063	0.226531	+	0.974004i	$0.427261\pi$
$8$	0	0
$9$	−1.56314	−0.521046
$10$	0	0
$11$	5.86718	1.76902	0.884510	−	0.466521i	$-0.154493\pi$
$11$	5.86718	1.76902	0.884510	+	0.466521i	$0.154493\pi$
$12$	0	0
$13$	0.364448	0.101080	0.0505399	−	0.998722i	$-0.483906\pi$
$13$	0.364448	0.101080	0.0505399	+	0.998722i	$0.483906\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.19869	−0.290725	−0.145363	−	0.989378i	$-0.546435\pi$
$17$	−1.19869	−0.290725	−0.145363	+	0.989378i	$0.546435\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.43686	0.313549
$22$	0	0
$23$	−8.23163	−1.71641	−0.858206	−	0.513305i	$-0.828421\pi$
$23$	−8.23163	−1.71641	−0.858206	+	0.513305i	$0.828421\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.46980	−1.05266
$28$	0	0
$29$	−7.86718	−1.46090	−0.730449	−	0.682967i	$-0.760689\pi$
$29$	−7.86718	−1.46090	−0.730449	+	0.682967i	$0.760689\pi$
$30$	0	0
$31$	−7.30404	−1.31184	−0.655922	−	0.754829i	$-0.727720\pi$
$31$	−7.30404	−1.31184	−0.655922	+	0.754829i	$0.727720\pi$
$32$	0	0
$33$	7.03293	1.22428
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.13828	−1.17353	−0.586763	−	0.809759i	$-0.699598\pi$
$37$	−7.13828	−1.17353	−0.586763	+	0.809759i	$0.699598\pi$
$38$	0	0
$39$	0.436861	0.0699537
$40$	0	0
$41$	2.43686	0.380574	0.190287	−	0.981729i	$-0.439058\pi$
$41$	2.43686	0.380574	0.190287	+	0.981729i	$0.439058\pi$
$42$	0	0
$43$	−7.39738	−1.12809	−0.564045	−	0.825744i	$-0.690756\pi$
$43$	−7.39738	−1.12809	−0.564045	+	0.825744i	$0.690756\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−13.7014	−1.99856	−0.999279	−	0.0379717i	$-0.987910\pi$
$47$	−13.7014	−1.99856	−0.999279	+	0.0379717i	$0.987910\pi$
$48$	0	0
$49$	−5.56314	−0.794734
$50$	0	0
$51$	−1.43686	−0.201201
$52$	0	0
$53$	7.39738	1.01611	0.508054	−	0.861325i	$-0.330365\pi$
$53$	7.39738	1.01611	0.508054	+	0.861325i	$0.330365\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.19869	0.158771
$58$	0	0
$59$	−12.8606	−1.67431	−0.837156	−	0.546964i	$-0.815783\pi$
$59$	−12.8606	−1.67431	−0.837156	+	0.546964i	$0.815783\pi$
$60$	0	0
$61$	−1.30404	−0.166965	−0.0834825	−	0.996509i	$-0.526604\pi$
$61$	−1.30404	−0.166965	−0.0834825	+	0.996509i	$0.526604\pi$
$62$	0	0
$63$	−1.87372	−0.236067
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.9330	1.45785	0.728927	−	0.684592i	$-0.240019\pi$
$67$	11.9330	1.45785	0.728927	+	0.684592i	$0.240019\pi$
$68$	0	0
$69$	−9.86718	−1.18787
$70$	0	0
$71$	−2.12628	−0.252343	−0.126171	−	0.992008i	$-0.540269\pi$
$71$	−2.12628	−0.252343	−0.126171	+	0.992008i	$0.540269\pi$
$72$	0	0
$73$	−2.50273	−0.292922	−0.146461	−	0.989216i	$-0.546788\pi$
$73$	−2.50273	−0.292922	−0.146461	+	0.989216i	$0.546788\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.03293	0.801477
$78$	0	0
$79$	7.74090	0.870919	0.435460	−	0.900208i	$-0.356586\pi$
$79$	7.74090	0.870919	0.435460	+	0.900208i	$0.356586\pi$
$80$	0	0
$81$	−1.86718	−0.207464
$82$	0	0
$83$	−3.02093	−0.331590	−0.165795	−	0.986160i	$-0.553019\pi$
$83$	−3.02093	−0.331590	−0.165795	+	0.986160i	$0.553019\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−9.43032	−1.01104
$88$	0	0
$89$	−5.68942	−0.603077	−0.301539	−	0.953454i	$-0.597500\pi$
$89$	−5.68942	−0.603077	−0.301539	+	0.953454i	$0.597500\pi$
$90$	0	0
$91$	0.436861	0.0457954
$92$	0	0
$93$	−8.75529	−0.907881
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.09334	−0.111012	−0.0555061	−	0.998458i	$-0.517677\pi$
$97$	−1.09334	−0.111012	−0.0555061	+	0.998458i	$0.517677\pi$
$98$	0	0
$99$	−9.17122	−0.921742
$100$	0	0
$101$	10.8672	1.08132	0.540662	−	0.841240i	$-0.318174\pi$
$101$	10.8672	1.08132	0.540662	+	0.841240i	$0.318174\pi$
$102$	0	0
$103$	−9.74090	−0.959799	−0.479900	−	0.877323i	$-0.659327\pi$
$103$	−9.74090	−0.959799	−0.479900	+	0.877323i	$0.659327\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.3634	1.58191	0.790953	−	0.611877i	$-0.209585\pi$
$107$	16.3634	1.58191	0.790953	+	0.611877i	$0.209585\pi$
$108$	0	0
$109$	9.29749	0.890538	0.445269	−	0.895397i	$-0.353108\pi$
$109$	9.29749	0.890538	0.445269	+	0.895397i	$0.353108\pi$
$110$	0	0
$111$	−8.55660	−0.812156
$112$	0	0
$113$	2.96707	0.279118	0.139559	−	0.990214i	$-0.455432\pi$
$113$	2.96707	0.279118	0.139559	+	0.990214i	$0.455432\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.569683	−0.0526672
$118$	0	0
$119$	−1.43686	−0.131717
$120$	0	0
$121$	23.4238	2.12943
$122$	0	0
$123$	2.92104	0.263382
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.3579	−1.00785	−0.503926	−	0.863747i	$-0.668111\pi$
$127$	−11.3579	−1.00785	−0.503926	+	0.863747i	$0.668111\pi$
$128$	0	0
$129$	−8.86718	−0.780711
$130$	0	0
$131$	1.56314	0.136572	0.0682861	−	0.997666i	$-0.478247\pi$
$131$	1.56314	0.136572	0.0682861	+	0.997666i	$0.478247\pi$
$132$	0	0
$133$	1.19869	0.103940
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.8397	1.35328	0.676639	−	0.736315i	$-0.263436\pi$
$137$	15.8397	1.35328	0.676639	+	0.736315i	$0.263436\pi$
$138$	0	0
$139$	19.8606	1.68456	0.842278	−	0.539043i	$-0.181214\pi$
$139$	19.8606	1.68456	0.842278	+	0.539043i	$0.181214\pi$
$140$	0	0
$141$	−16.4238	−1.38313
$142$	0	0
$143$	2.13828	0.178812
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.66849	−0.550007
$148$	0	0
$149$	2.68942	0.220326	0.110163	−	0.993914i	$-0.464863\pi$
$149$	2.68942	0.220326	0.110163	+	0.993914i	$0.464863\pi$
$150$	0	0
$151$	−17.1647	−1.39684	−0.698421	−	0.715688i	$-0.746113\pi$
$151$	−17.1647	−1.39684	−0.698421	+	0.715688i	$0.746113\pi$
$152$	0	0
$153$	1.87372	0.151481
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.3634	1.30594	0.652969	−	0.757384i	$-0.273523\pi$
$157$	16.3634	1.30594	0.652969	+	0.757384i	$0.273523\pi$
$158$	0	0
$159$	8.86718	0.703213
$160$	0	0
$161$	−9.86718	−0.777643
$162$	0	0
$163$	−8.28549	−0.648970	−0.324485	−	0.945891i	$-0.605191\pi$
$163$	−8.28549	−0.648970	−0.324485	+	0.945891i	$0.605191\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.49727	−0.580156	−0.290078	−	0.957003i	$-0.593681\pi$
$167$	−7.49727	−0.580156	−0.290078	+	0.957003i	$0.593681\pi$
$168$	0	0
$169$	−12.8672	−0.989783
$170$	0	0
$171$	−1.56314	−0.119536
$172$	0	0
$173$	14.7464	1.12114	0.560572	−	0.828105i	$-0.310581\pi$
$173$	14.7464	1.12114	0.560572	+	0.828105i	$0.310581\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−15.4159	−1.15873
$178$	0	0
$179$	−13.2526	−0.990543	−0.495271	−	0.868738i	$-0.664931\pi$
$179$	−13.2526	−0.990543	−0.495271	+	0.868738i	$0.664931\pi$
$180$	0	0
$181$	16.7344	1.24385	0.621927	−	0.783075i	$-0.286350\pi$
$181$	16.7344	1.24385	0.621927	+	0.783075i	$0.286350\pi$
$182$	0	0
$183$	−1.56314	−0.115551
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−7.03293	−0.514299
$188$	0	0
$189$	−6.55660	−0.476922
$190$	0	0
$191$	4.13282	0.299041	0.149520	−	0.988759i	$-0.452227\pi$
$191$	4.13282	0.299041	0.149520	+	0.988759i	$0.452227\pi$
$192$	0	0
$193$	−7.08680	−0.510119	−0.255060	−	0.966925i	$-0.582095\pi$
$193$	−7.08680	−0.510119	−0.255060	+	0.966925i	$0.582095\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.4753	−1.45880	−0.729401	−	0.684087i	$-0.760201\pi$
$197$	−20.4753	−1.45880	−0.729401	+	0.684087i	$0.760201\pi$
$198$	0	0
$199$	−8.74090	−0.619626	−0.309813	−	0.950798i	$-0.600266\pi$
$199$	−8.74090	−0.619626	−0.309813	+	0.950798i	$0.600266\pi$
$200$	0	0
$201$	14.3040	1.00893
$202$	0	0
$203$	−9.43032	−0.661878
$204$	0	0
$205$	0	0
$206$	0	0
$207$	12.8672	0.894331
$208$	0	0
$209$	5.86718	0.405841
$210$	0	0
$211$	18.7344	1.28973	0.644863	−	0.764298i	$-0.276914\pi$
$211$	18.7344	1.28973	0.644863	+	0.764298i	$0.276914\pi$
$212$	0	0
$213$	−2.54875	−0.174638
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−8.75529	−0.594348
$218$	0	0
$219$	−3.00000	−0.202721
$220$	0	0
$221$	−0.436861	−0.0293864
$222$	0	0
$223$	4.27110	0.286014	0.143007	−	0.989722i	$-0.454323\pi$
$223$	4.27110	0.286014	0.143007	+	0.989722i	$0.454323\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.5631	−1.43120	−0.715598	−	0.698512i	$-0.753846\pi$
$227$	−21.5631	−1.43120	−0.715598	+	0.698512i	$0.753846\pi$
$228$	0	0
$229$	27.4172	1.81178	0.905891	−	0.423511i	$-0.139203\pi$
$229$	27.4172	1.81178	0.905891	+	0.423511i	$0.139203\pi$
$230$	0	0
$231$	8.43032	0.554674
$232$	0	0
$233$	−7.86718	−0.515396	−0.257698	−	0.966226i	$-0.582964\pi$
$233$	−7.86718	−0.515396	−0.257698	+	0.966226i	$0.582964\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.27895	0.602732
$238$	0	0
$239$	−9.30404	−0.601828	−0.300914	−	0.953651i	$-0.597292\pi$
$239$	−9.30404	−0.601828	−0.300914	+	0.953651i	$0.597292\pi$
$240$	0	0
$241$	−21.9056	−1.41106	−0.705531	−	0.708679i	$-0.749291\pi$
$241$	−21.9056	−1.41106	−0.705531	+	0.708679i	$0.749291\pi$
$242$	0	0
$243$	14.1712	0.909084
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.364448	0.0231893
$248$	0	0
$249$	−3.62116	−0.229482
$250$	0	0
$251$	7.43032	0.468997	0.234499	−	0.972116i	$-0.424655\pi$
$251$	7.43032	0.468997	0.234499	+	0.972116i	$0.424655\pi$
$252$	0	0
$253$	−48.2964	−3.03637
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.05387	−0.315251	−0.157626	−	0.987499i	$-0.550384\pi$
$257$	−5.05387	−0.315251	−0.157626	+	0.987499i	$0.550384\pi$
$258$	0	0
$259$	−8.55660	−0.531681
$260$	0	0
$261$	12.2975	0.761196
$262$	0	0
$263$	−17.7673	−1.09558	−0.547789	−	0.836617i	$-0.684530\pi$
$263$	−17.7673	−1.09558	−0.547789	+	0.836617i	$0.684530\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.81986	−0.417368
$268$	0	0
$269$	9.17776	0.559578	0.279789	−	0.960062i	$-0.409736\pi$
$269$	9.17776	0.559578	0.279789	+	0.960062i	$0.409736\pi$
$270$	0	0
$271$	20.7278	1.25912	0.629562	−	0.776950i	$-0.283234\pi$
$271$	20.7278	1.25912	0.629562	+	0.776950i	$0.283234\pi$
$272$	0	0
$273$	0.523661	0.0316934
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−24.5895	−1.47744	−0.738721	−	0.674012i	$-0.764570\pi$
$277$	−24.5895	−1.47744	−0.738721	+	0.674012i	$0.764570\pi$
$278$	0	0
$279$	11.4172	0.683532
$280$	0	0
$281$	3.73436	0.222773	0.111386	−	0.993777i	$-0.464471\pi$
$281$	3.73436	0.222773	0.111386	+	0.993777i	$0.464471\pi$
$282$	0	0
$283$	−13.0318	−0.774663	−0.387332	−	0.921941i	$-0.626603\pi$
$283$	−13.0318	−0.774663	−0.387332	+	0.921941i	$0.626603\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.92104	0.172424
$288$	0	0
$289$	−15.5631	−0.915479
$290$	0	0
$291$	−1.31058	−0.0768277
$292$	0	0
$293$	−19.9001	−1.16258	−0.581288	−	0.813698i	$-0.697451\pi$
$293$	−19.9001	−1.16258	−0.581288	+	0.813698i	$0.697451\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−32.0923	−1.86218
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	−8.86718	−0.511096
$302$	0	0
$303$	13.0264	0.748347
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.02093	0.457779	0.228889	−	0.973452i	$-0.426491\pi$
$307$	8.02093	0.457779	0.228889	+	0.973452i	$0.426491\pi$
$308$	0	0
$309$	−11.6763	−0.664243
$310$	0	0
$311$	16.1778	0.917357	0.458678	−	0.888602i	$-0.348323\pi$
$311$	16.1778	0.917357	0.458678	+	0.888602i	$0.348323\pi$
$312$	0	0
$313$	7.13828	0.403480	0.201740	−	0.979439i	$-0.435340\pi$
$313$	7.13828	0.403480	0.201740	+	0.979439i	$0.435340\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.96052	−0.222445	−0.111223	−	0.993796i	$-0.535477\pi$
$317$	−3.96052	−0.222445	−0.111223	+	0.993796i	$0.535477\pi$
$318$	0	0
$319$	−46.1581	−2.58436
$320$	0	0
$321$	19.6146	1.09478
$322$	0	0
$323$	−1.19869	−0.0666970
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.1448	0.616310
$328$	0	0
$329$	−16.4238	−0.905472
$330$	0	0
$331$	5.94852	0.326960	0.163480	−	0.986547i	$-0.447728\pi$
$331$	5.94852	0.326960	0.163480	+	0.986547i	$0.447728\pi$
$332$	0	0
$333$	11.1581	0.611462
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−8.90666	−0.485176	−0.242588	−	0.970129i	$-0.577996\pi$
$337$	−8.90666	−0.485176	−0.242588	+	0.970129i	$0.577996\pi$
$338$	0	0
$339$	3.55660	0.193168
$340$	0	0
$341$	−42.8541	−2.32068
$342$	0	0
$343$	−15.0593	−0.813127
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.5961	−0.998290	−0.499145	−	0.866519i	$-0.666352\pi$
$347$	−18.5961	−0.998290	−0.499145	+	0.866519i	$0.666352\pi$
$348$	0	0
$349$	−14.3040	−0.765678	−0.382839	−	0.923815i	$-0.625054\pi$
$349$	−14.3040	−0.765678	−0.382839	+	0.923815i	$0.625054\pi$
$350$	0	0
$351$	−1.99346	−0.106403
$352$	0	0
$353$	12.7673	0.679534	0.339767	−	0.940510i	$-0.389652\pi$
$353$	12.7673	0.679534	0.339767	+	0.940510i	$0.389652\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.72235	−0.0911566
$358$	0	0
$359$	−2.69596	−0.142287	−0.0711437	−	0.997466i	$-0.522665\pi$
$359$	−2.69596	−0.142287	−0.0711437	+	0.997466i	$0.522665\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	28.0779	1.47371
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.43140	0.179118	0.0895589	−	0.995982i	$-0.471454\pi$
$367$	3.43140	0.179118	0.0895589	+	0.995982i	$0.471454\pi$
$368$	0	0
$369$	−3.80915	−0.198297
$370$	0	0
$371$	8.86718	0.460361
$372$	0	0
$373$	−15.5697	−0.806168	−0.403084	−	0.915163i	$-0.632062\pi$
$373$	−15.5697	−0.806168	−0.403084	+	0.915163i	$0.632062\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.86718	−0.147667
$378$	0	0
$379$	0.164672	0.00845864	0.00422932	−	0.999991i	$-0.498654\pi$
$379$	0.164672	0.00845864	0.00422932	+	0.999991i	$0.498654\pi$
$380$	0	0
$381$	−13.6146	−0.697498
$382$	0	0
$383$	22.3514	1.14210	0.571051	−	0.820915i	$-0.306536\pi$
$383$	22.3514	1.14210	0.571051	+	0.820915i	$0.306536\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.5631	0.587787
$388$	0	0
$389$	−9.34243	−0.473680	−0.236840	−	0.971549i	$-0.576112\pi$
$389$	−9.34243	−0.473680	−0.236840	+	0.971549i	$0.576112\pi$
$390$	0	0
$391$	9.86718	0.499005
$392$	0	0
$393$	1.87372	0.0945167
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−4.83533	−0.242678	−0.121339	−	0.992611i	$-0.538719\pi$
$397$	−4.83533	−0.242678	−0.121339	+	0.992611i	$0.538719\pi$
$398$	0	0
$399$	1.43686	0.0719330
$400$	0	0
$401$	37.4622	1.87077	0.935386	−	0.353629i	$-0.115053\pi$
$401$	37.4622	1.87077	0.935386	+	0.353629i	$0.115053\pi$
$402$	0	0
$403$	−2.66194	−0.132601
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−41.8816	−2.07599
$408$	0	0
$409$	9.42377	0.465976	0.232988	−	0.972480i	$-0.425150\pi$
$409$	9.42377	0.465976	0.232988	+	0.972480i	$0.425150\pi$
$410$	0	0
$411$	18.9869	0.936555
$412$	0	0
$413$	−15.4159	−0.758568
$414$	0	0
$415$	0	0
$416$	0	0
$417$	23.8068	1.16582
$418$	0	0
$419$	−27.4303	−1.34006	−0.670029	−	0.742335i	$-0.733718\pi$
$419$	−27.4303	−1.34006	−0.670029	+	0.742335i	$0.733718\pi$
$420$	0	0
$421$	18.4303	0.898239	0.449119	−	0.893472i	$-0.351738\pi$
$421$	18.4303	0.898239	0.449119	+	0.893472i	$0.351738\pi$
$422$	0	0
$423$	21.4172	1.04134
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.56314	−0.0756456
$428$	0	0
$429$	2.56314	0.123750
$430$	0	0
$431$	15.5117	0.747170	0.373585	−	0.927596i	$-0.378128\pi$
$431$	15.5117	0.747170	0.373585	+	0.927596i	$0.378128\pi$
$432$	0	0
$433$	31.8870	1.53239	0.766196	−	0.642607i	$-0.222147\pi$
$433$	31.8870	1.53239	0.766196	+	0.642607i	$0.222147\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.23163	−0.393772
$438$	0	0
$439$	−14.7278	−0.702920	−0.351460	−	0.936203i	$-0.614315\pi$
$439$	−14.7278	−0.702920	−0.351460	+	0.936203i	$0.614315\pi$
$440$	0	0
$441$	8.69596	0.414093
$442$	0	0
$443$	−6.46087	−0.306965	−0.153483	−	0.988151i	$-0.549049\pi$
$443$	−6.46087	−0.306965	−0.153483	+	0.988151i	$0.549049\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.22378	0.152480
$448$	0	0
$449$	21.2910	1.00478	0.502391	−	0.864641i	$-0.332454\pi$
$449$	21.2910	1.00478	0.502391	+	0.864641i	$0.332454\pi$
$450$	0	0
$451$	14.2975	0.673243
$452$	0	0
$453$	−20.5751	−0.966705
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.6465	−0.919023	−0.459512	−	0.888172i	$-0.651976\pi$
$457$	−19.6465	−0.919023	−0.459512	+	0.888172i	$0.651976\pi$
$458$	0	0
$459$	6.55660	0.306036
$460$	0	0
$461$	1.16467	0.0542442	0.0271221	−	0.999632i	$-0.491366\pi$
$461$	1.16467	0.0542442	0.0271221	+	0.999632i	$0.491366\pi$
$462$	0	0
$463$	5.15375	0.239515	0.119758	−	0.992803i	$-0.461788\pi$
$463$	5.15375	0.239515	0.119758	+	0.992803i	$0.461788\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.17122	0.193021	0.0965104	−	0.995332i	$-0.469232\pi$
$467$	4.17122	0.193021	0.0965104	+	0.995332i	$0.469232\pi$
$468$	0	0
$469$	14.3040	0.660499
$470$	0	0
$471$	19.6146	0.903794
$472$	0	0
$473$	−43.4018	−1.99561
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−11.5631	−0.529440
$478$	0	0
$479$	2.35552	0.107626	0.0538132	−	0.998551i	$-0.482862\pi$
$479$	2.35552	0.107626	0.0538132	+	0.998551i	$0.482862\pi$
$480$	0	0
$481$	−2.60153	−0.118620
$482$	0	0
$483$	−11.8277	−0.538179
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−19.9056	−0.902008	−0.451004	−	0.892522i	$-0.648934\pi$
$487$	−19.9056	−0.902008	−0.451004	+	0.892522i	$0.648934\pi$
$488$	0	0
$489$	−9.93175	−0.449129
$490$	0	0
$491$	−7.90557	−0.356773	−0.178387	−	0.983960i	$-0.557088\pi$
$491$	−7.90557	−0.356773	−0.178387	+	0.983960i	$0.557088\pi$
$492$	0	0
$493$	9.43032	0.424720
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.54875	−0.114327
$498$	0	0
$499$	35.7147	1.59881	0.799405	−	0.600792i	$-0.205148\pi$
$499$	35.7147	1.59881	0.799405	+	0.600792i	$0.205148\pi$
$500$	0	0
$501$	−8.98691	−0.401506
$502$	0	0
$503$	30.7991	1.37327	0.686633	−	0.727004i	$-0.259088\pi$
$503$	30.7991	1.37327	0.686633	+	0.727004i	$0.259088\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−15.4238	−0.684994
$508$	0	0
$509$	17.5631	0.778472	0.389236	−	0.921138i	$-0.372739\pi$
$509$	17.5631	0.778472	0.389236	+	0.921138i	$0.372739\pi$
$510$	0	0
$511$	−3.00000	−0.132712
$512$	0	0
$513$	−5.46980	−0.241497
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−80.3887	−3.53549
$518$	0	0
$519$	17.6763	0.775905
$520$	0	0
$521$	−24.6081	−1.07810	−0.539050	−	0.842274i	$-0.681217\pi$
$521$	−24.6081	−1.07810	−0.539050	+	0.842274i	$0.681217\pi$
$522$	0	0
$523$	43.7147	1.91151	0.955756	−	0.294162i	$-0.0950404\pi$
$523$	43.7147	1.91151	0.955756	+	0.294162i	$0.0950404\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.75529	0.381386
$528$	0	0
$529$	44.7597	1.94607
$530$	0	0
$531$	20.1030	0.872394
$532$	0	0
$533$	0.888109	0.0384683
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−15.8857	−0.685520
$538$	0	0
$539$	−32.6399	−1.40590
$540$	0	0
$541$	−8.38538	−0.360516	−0.180258	−	0.983619i	$-0.557693\pi$
$541$	−8.38538	−0.360516	−0.180258	+	0.983619i	$0.557693\pi$
$542$	0	0
$543$	20.0593	0.860828
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−26.3040	−1.12468	−0.562340	−	0.826906i	$-0.690099\pi$
$547$	−26.3040	−1.12468	−0.562340	+	0.826906i	$0.690099\pi$
$548$	0	0
$549$	2.03839	0.0869965
$550$	0	0
$551$	−7.86718	−0.335153
$552$	0	0
$553$	9.27895	0.394581
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.66740	0.113021	0.0565107	−	0.998402i	$-0.482002\pi$
$557$	2.66740	0.113021	0.0565107	+	0.998402i	$0.482002\pi$
$558$	0	0
$559$	−2.69596	−0.114027
$560$	0	0
$561$	−8.43032	−0.355928
$562$	0	0
$563$	−37.7751	−1.59203	−0.796016	−	0.605276i	$-0.793063\pi$
$563$	−37.7751	−1.59203	−0.796016	+	0.605276i	$0.793063\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.23817	−0.0939943
$568$	0	0
$569$	38.8606	1.62912	0.814561	−	0.580078i	$-0.196978\pi$
$569$	38.8606	1.62912	0.814561	+	0.580078i	$0.196978\pi$
$570$	0	0
$571$	−4.17122	−0.174560	−0.0872800	−	0.996184i	$-0.527817\pi$
$571$	−4.17122	−0.174560	−0.0872800	+	0.996184i	$0.527817\pi$
$572$	0	0
$573$	4.95398	0.206955
$574$	0	0
$575$	0	0
$576$	0	0
$577$	39.3413	1.63780	0.818901	−	0.573935i	$-0.194584\pi$
$577$	39.3413	1.63780	0.818901	+	0.573935i	$0.194584\pi$
$578$	0	0
$579$	−8.49489	−0.353035
$580$	0	0
$581$	−3.62116	−0.150231
$582$	0	0
$583$	43.4018	1.79752
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.8672	0.737457	0.368729	−	0.929537i	$-0.379793\pi$
$587$	17.8672	0.737457	0.368729	+	0.929537i	$0.379793\pi$
$588$	0	0
$589$	−7.30404	−0.300958
$590$	0	0
$591$	−24.5435	−1.00959
$592$	0	0
$593$	−13.0803	−0.537142	−0.268571	−	0.963260i	$-0.586551\pi$
$593$	−13.0803	−0.537142	−0.268571	+	0.963260i	$0.586551\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10.4776	−0.428821
$598$	0	0
$599$	−10.7278	−0.438326	−0.219163	−	0.975688i	$-0.570333\pi$
$599$	−10.7278	−0.438326	−0.219163	+	0.975688i	$0.570333\pi$
$600$	0	0
$601$	39.8026	1.62358	0.811791	−	0.583948i	$-0.198493\pi$
$601$	39.8026	1.62358	0.811791	+	0.583948i	$0.198493\pi$
$602$	0	0
$603$	−18.6530	−0.759609
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−43.0318	−1.74661	−0.873304	−	0.487175i	$-0.838027\pi$
$607$	−43.0318	−1.74661	−0.873304	+	0.487175i	$0.838027\pi$
$608$	0	0
$609$	−11.3040	−0.458063
$610$	0	0
$611$	−4.99346	−0.202014
$612$	0	0
$613$	−15.4722	−0.624915	−0.312458	−	0.949932i	$-0.601152\pi$
$613$	−15.4722	−0.624915	−0.312458	+	0.949932i	$0.601152\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.5237	1.43013	0.715064	−	0.699059i	$-0.246397\pi$
$617$	35.5237	1.43013	0.715064	+	0.699059i	$0.246397\pi$
$618$	0	0
$619$	15.0449	0.604707	0.302354	−	0.953196i	$-0.402228\pi$
$619$	15.0449	0.604707	0.302354	+	0.953196i	$0.402228\pi$
$620$	0	0
$621$	45.0253	1.80680
$622$	0	0
$623$	−6.81986	−0.273232
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.03293	0.280868
$628$	0	0
$629$	8.55660	0.341174
$630$	0	0
$631$	−35.6399	−1.41880	−0.709402	−	0.704805i	$-0.751035\pi$
$631$	−35.6399	−1.41880	−0.709402	+	0.704805i	$0.751035\pi$
$632$	0	0
$633$	22.4567	0.892574
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.02748	−0.0803315
$638$	0	0
$639$	3.32367	0.131482
$640$	0	0
$641$	6.98691	0.275966	0.137983	−	0.990435i	$-0.455938\pi$
$641$	6.98691	0.275966	0.137983	+	0.990435i	$0.455938\pi$
$642$	0	0
$643$	−3.74744	−0.147785	−0.0738924	−	0.997266i	$-0.523542\pi$
$643$	−3.74744	−0.147785	−0.0738924	+	0.997266i	$0.523542\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−34.1658	−1.34319	−0.671597	−	0.740916i	$-0.734391\pi$
$647$	−34.1658	−1.34319	−0.671597	+	0.740916i	$0.734391\pi$
$648$	0	0
$649$	−75.4556	−2.96189
$650$	0	0
$651$	−10.4949	−0.411327
$652$	0	0
$653$	−13.5446	−0.530041	−0.265020	−	0.964243i	$-0.585379\pi$
$653$	−13.5446	−0.530041	−0.265020	+	0.964243i	$0.585379\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.91211	0.152626
$658$	0	0
$659$	22.6334	0.881671	0.440836	−	0.897588i	$-0.354682\pi$
$659$	22.6334	0.881671	0.440836	+	0.897588i	$0.354682\pi$
$660$	0	0
$661$	−3.96161	−0.154089	−0.0770443	−	0.997028i	$-0.524548\pi$
$661$	−3.96161	−0.154089	−0.0770443	+	0.997028i	$0.524548\pi$
$662$	0	0
$663$	−0.523661	−0.0203373
$664$	0	0
$665$	0	0
$666$	0	0
$667$	64.7597	2.50750
$668$	0	0
$669$	5.11973	0.197940
$670$	0	0
$671$	−7.65102	−0.295365
$672$	0	0
$673$	28.9605	1.11635	0.558173	−	0.829725i	$-0.311503\pi$
$673$	28.9605	1.11635	0.558173	+	0.829725i	$0.311503\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.7518	0.951290	0.475645	−	0.879637i	$-0.342215\pi$
$677$	24.7518	0.951290	0.475645	+	0.879637i	$0.342215\pi$
$678$	0	0
$679$	−1.31058	−0.0502955
$680$	0	0
$681$	−25.8475	−0.990480
$682$	0	0
$683$	−40.0593	−1.53283	−0.766414	−	0.642347i	$-0.777961\pi$
$683$	−40.0593	−1.53283	−0.766414	+	0.642347i	$0.777961\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	32.8648	1.25387
$688$	0	0
$689$	2.69596	0.102708
$690$	0	0
$691$	−35.2162	−1.33969	−0.669843	−	0.742503i	$-0.733639\pi$
$691$	−35.2162	−1.33969	−0.669843	+	0.742503i	$0.733639\pi$
$692$	0	0
$693$	−10.9935	−0.417607
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2.92104	−0.110642
$698$	0	0
$699$	−9.43032	−0.356687
$700$	0	0
$701$	−38.1283	−1.44008	−0.720042	−	0.693930i	$-0.755878\pi$
$701$	−38.1283	−1.44008	−0.720042	+	0.693930i	$0.755878\pi$
$702$	0	0
$703$	−7.13828	−0.269225
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.0264	0.489908
$708$	0	0
$709$	−6.08789	−0.228635	−0.114318	−	0.993444i	$-0.536468\pi$
$709$	−6.08789	−0.228635	−0.114318	+	0.993444i	$0.536468\pi$
$710$	0	0
$711$	−12.1001	−0.453789
$712$	0	0
$713$	60.1241	2.25167
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.1527	−0.416504
$718$	0	0
$719$	12.1647	0.453666	0.226833	−	0.973934i	$-0.427163\pi$
$719$	12.1647	0.453666	0.226833	+	0.973934i	$0.427163\pi$
$720$	0	0
$721$	−11.6763	−0.434849
$722$	0	0
$723$	−26.2580	−0.976546
$724$	0	0
$725$	0	0
$726$	0	0
$727$	25.6859	0.952639	0.476320	−	0.879272i	$-0.341971\pi$
$727$	25.6859	0.952639	0.476320	+	0.879272i	$0.341971\pi$
$728$	0	0
$729$	22.5884	0.836609
$730$	0	0
$731$	8.86718	0.327964
$732$	0	0
$733$	−43.4502	−1.60487	−0.802434	−	0.596741i	$-0.796462\pi$
$733$	−43.4502	−1.60487	−0.802434	+	0.596741i	$0.796462\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	70.0133	2.57897
$738$	0	0
$739$	43.4172	1.59713	0.798564	−	0.601910i	$-0.205593\pi$
$739$	43.4172	1.59713	0.798564	+	0.601910i	$0.205593\pi$
$740$	0	0
$741$	0.436861	0.0160485
$742$	0	0
$743$	−26.6709	−0.978459	−0.489230	−	0.872155i	$-0.662722\pi$
$743$	−26.6709	−0.978459	−0.489230	+	0.872155i	$0.662722\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.72214	0.172774
$748$	0	0
$749$	19.6146	0.716703
$750$	0	0
$751$	1.52674	0.0557114	0.0278557	−	0.999612i	$-0.491132\pi$
$751$	1.52674	0.0557114	0.0278557	+	0.999612i	$0.491132\pi$
$752$	0	0
$753$	8.90666	0.324577
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−9.38191	−0.340991	−0.170496	−	0.985358i	$-0.554537\pi$
$757$	−9.38191	−0.340991	−0.170496	+	0.985358i	$0.554537\pi$
$758$	0	0
$759$	−57.8925	−2.10136
$760$	0	0
$761$	−6.70251	−0.242966	−0.121483	−	0.992594i	$-0.538765\pi$
$761$	−6.70251	−0.242966	−0.121483	+	0.992594i	$0.538765\pi$
$762$	0	0
$763$	11.1448	0.403470
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.68703	−0.169239
$768$	0	0
$769$	−36.0637	−1.30049	−0.650245	−	0.759724i	$-0.725334\pi$
$769$	−36.0637	−1.30049	−0.650245	+	0.759724i	$0.725334\pi$
$770$	0	0
$771$	−6.05802	−0.218174
$772$	0	0
$773$	−11.6763	−0.419968	−0.209984	−	0.977705i	$-0.567341\pi$
$773$	−11.6763	−0.419968	−0.209984	+	0.977705i	$0.567341\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−10.2567	−0.367958
$778$	0	0
$779$	2.43686	0.0873096
$780$	0	0
$781$	−12.4753	−0.446400
$782$	0	0
$783$	43.0318	1.53783
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−43.1880	−1.53949	−0.769743	−	0.638354i	$-0.779616\pi$
$787$	−43.1880	−1.53949	−0.769743	+	0.638354i	$0.779616\pi$
$788$	0	0
$789$	−21.2975	−0.758211
$790$	0	0
$791$	3.55660	0.126458
$792$	0	0
$793$	−0.475254	−0.0168768
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.1658	1.03310	0.516552	−	0.856256i	$-0.327215\pi$
$797$	29.1658	1.03310	0.516552	+	0.856256i	$0.327215\pi$
$798$	0	0
$799$	16.4238	0.581031
$800$	0	0
$801$	8.89335	0.314231
$802$	0	0
$803$	−14.6840	−0.518186
$804$	0	0
$805$	0	0
$806$	0	0
$807$	11.0013	0.387264
$808$	0	0
$809$	2.43032	0.0854454	0.0427227	−	0.999087i	$-0.486397\pi$
$809$	2.43032	0.0854454	0.0427227	+	0.999087i	$0.486397\pi$
$810$	0	0
$811$	42.2779	1.48458	0.742288	−	0.670081i	$-0.233741\pi$
$811$	42.2779	1.48458	0.742288	+	0.670081i	$0.233741\pi$
$812$	0	0
$813$	24.8462	0.871396
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−7.39738	−0.258802
$818$	0	0
$819$	−0.682874	−0.0238616
$820$	0	0
$821$	52.5136	1.83274	0.916369	−	0.400334i	$-0.131106\pi$
$821$	52.5136	1.83274	0.916369	+	0.400334i	$0.131106\pi$
$822$	0	0
$823$	36.3150	1.26586	0.632930	−	0.774209i	$-0.281852\pi$
$823$	36.3150	1.26586	0.632930	+	0.774209i	$0.281852\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−46.3150	−1.61053	−0.805264	−	0.592916i	$-0.797977\pi$
$827$	−46.3150	−1.61053	−0.805264	+	0.592916i	$0.797977\pi$
$828$	0	0
$829$	20.0899	0.697750	0.348875	−	0.937169i	$-0.386564\pi$
$829$	20.0899	0.697750	0.348875	+	0.937169i	$0.386564\pi$
$830$	0	0
$831$	−29.4753	−1.02249
$832$	0	0
$833$	6.66849	0.231049
$834$	0	0
$835$	0	0
$836$	0	0
$837$	39.9516	1.38093
$838$	0	0
$839$	10.5930	0.365711	0.182855	−	0.983140i	$-0.441466\pi$
$839$	10.5930	0.365711	0.182855	+	0.983140i	$0.441466\pi$
$840$	0	0
$841$	32.8925	1.13422
$842$	0	0
$843$	4.47634	0.154173
$844$	0	0
$845$	0	0
$846$	0	0
$847$	28.0779	0.964767
$848$	0	0
$849$	−15.6212	−0.536117
$850$	0	0
$851$	58.7597	2.01426
$852$	0	0
$853$	40.0648	1.37179	0.685896	−	0.727700i	$-0.259410\pi$
$853$	40.0648	1.37179	0.685896	+	0.727700i	$0.259410\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38.3019	−1.30837	−0.654183	−	0.756336i	$-0.726988\pi$
$857$	−38.3019	−1.30837	−0.654183	+	0.756336i	$0.726988\pi$
$858$	0	0
$859$	39.9056	1.36156	0.680780	−	0.732488i	$-0.261641\pi$
$859$	39.9056	1.36156	0.680780	+	0.732488i	$0.261641\pi$
$860$	0	0
$861$	3.50143	0.119328
$862$	0	0
$863$	−23.3315	−0.794214	−0.397107	−	0.917772i	$-0.629986\pi$
$863$	−23.3315	−0.794214	−0.397107	+	0.917772i	$0.629986\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−18.6554	−0.633571
$868$	0	0
$869$	45.4172	1.54067
$870$	0	0
$871$	4.34898	0.147359
$872$	0	0
$873$	1.70905	0.0578426
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−10.4105	−0.351537	−0.175768	−	0.984432i	$-0.556241\pi$
$877$	−10.4105	−0.351537	−0.175768	+	0.984432i	$0.556241\pi$
$878$	0	0
$879$	−23.8541	−0.804578
$880$	0	0
$881$	−23.7662	−0.800704	−0.400352	−	0.916361i	$-0.631112\pi$
$881$	−23.7662	−0.800704	−0.400352	+	0.916361i	$0.631112\pi$
$882$	0	0
$883$	−28.3908	−0.955428	−0.477714	−	0.878515i	$-0.658534\pi$
$883$	−28.3908	−0.955428	−0.477714	+	0.878515i	$0.658534\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32.5644	−1.09341	−0.546703	−	0.837326i	$-0.684117\pi$
$887$	−32.5644	−1.09341	−0.546703	+	0.837326i	$0.684117\pi$
$888$	0	0
$889$	−13.6146	−0.456620
$890$	0	0
$891$	−10.9551	−0.367008
$892$	0	0
$893$	−13.7014	−0.458501
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.59607	−0.120069
$898$	0	0
$899$	57.4622	1.91647
$900$	0	0
$901$	−8.86718	−0.295409
$902$	0	0
$903$	−10.6290	−0.353711
$904$	0	0
$905$	0	0
$906$	0	0
$907$	57.2109	1.89966	0.949829	−	0.312771i	$-0.101257\pi$
$907$	57.2109	1.89966	0.949829	+	0.312771i	$0.101257\pi$
$908$	0	0
$909$	−16.9869	−0.563420
$910$	0	0
$911$	42.0617	1.39357	0.696783	−	0.717282i	$-0.254614\pi$
$911$	42.0617	1.39357	0.696783	+	0.717282i	$0.254614\pi$
$912$	0	0
$913$	−17.7243	−0.586590
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.87372	0.0618757
$918$	0	0
$919$	−5.68287	−0.187461	−0.0937304	−	0.995598i	$-0.529879\pi$
$919$	−5.68287	−0.187461	−0.0937304	+	0.995598i	$0.529879\pi$
$920$	0	0
$921$	9.61462	0.316813
$922$	0	0
$923$	−0.774918	−0.0255067
$924$	0	0
$925$	0	0
$926$	0	0
$927$	15.2264	0.500100
$928$	0	0
$929$	−30.9289	−1.01474	−0.507372	−	0.861727i	$-0.669383\pi$
$929$	−30.9289	−1.01474	−0.507372	+	0.861727i	$0.669383\pi$
$930$	0	0
$931$	−5.56314	−0.182325
$932$	0	0
$933$	19.3921	0.634870
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−44.5357	−1.45492	−0.727458	−	0.686152i	$-0.759299\pi$
$937$	−44.5357	−1.45492	−0.727458	+	0.686152i	$0.759299\pi$
$938$	0	0
$939$	8.55660	0.279234
$940$	0	0
$941$	−2.08134	−0.0678498	−0.0339249	−	0.999424i	$-0.510801\pi$
$941$	−2.08134	−0.0678498	−0.0339249	+	0.999424i	$0.510801\pi$
$942$	0	0
$943$	−20.0593	−0.653221
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−52.7189	−1.71313	−0.856567	−	0.516036i	$-0.827407\pi$
$947$	−52.7189	−1.71313	−0.856567	+	0.516036i	$0.827407\pi$
$948$	0	0
$949$	−0.912115	−0.0296085
$950$	0	0
$951$	−4.74744	−0.153946
$952$	0	0
$953$	3.22071	0.104329	0.0521645	−	0.998639i	$-0.483388\pi$
$953$	3.22071	0.104329	0.0521645	+	0.998639i	$0.483388\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−55.3293	−1.78854
$958$	0	0
$959$	18.9869	0.613119
$960$	0	0
$961$	22.3490	0.720935
$962$	0	0
$963$	−25.5782	−0.824247
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−2.65956	−0.0855256	−0.0427628	−	0.999085i	$-0.513616\pi$
$967$	−2.65956	−0.0855256	−0.0427628	+	0.999085i	$0.513616\pi$
$968$	0	0
$969$	−1.43686	−0.0461586
$970$	0	0
$971$	−35.1263	−1.12726	−0.563628	−	0.826029i	$-0.690595\pi$
$971$	−35.1263	−1.12726	−0.563628	+	0.826029i	$0.690595\pi$
$972$	0	0
$973$	23.8068	0.763210
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.1668	0.581209	0.290604	−	0.956843i	$-0.406144\pi$
$977$	18.1668	0.581209	0.290604	+	0.956843i	$0.406144\pi$
$978$	0	0
$979$	−33.3808	−1.06686
$980$	0	0
$981$	−14.5333	−0.464012
$982$	0	0
$983$	−48.4556	−1.54549	−0.772747	−	0.634714i	$-0.781118\pi$
$983$	−48.4556	−1.54549	−0.772747	+	0.634714i	$0.781118\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−19.6870	−0.626645
$988$	0	0
$989$	60.8925	1.93627
$990$	0	0
$991$	−29.8157	−0.947127	−0.473563	−	0.880760i	$-0.657033\pi$
$991$	−29.8157	−0.947127	−0.473563	+	0.880760i	$0.657033\pi$
$992$	0	0
$993$	7.13044	0.226278
$994$	0	0
$995$	0	0
$996$	0	0
$997$	29.6894	0.940273	0.470137	−	0.882594i	$-0.344205\pi$
$997$	29.6894	0.940273	0.470137	+	0.882594i	$0.344205\pi$
$998$	0	0
$999$	39.0449	1.23533

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.bj.1.3		3
4.3	odd	2		1900.2.a.h.1.1	yes	3
5.4	even	2		7600.2.a.by.1.1		3
20.3	even	4		1900.2.c.g.1749.2		6
20.7	even	4		1900.2.c.g.1749.5		6
20.19	odd	2		1900.2.a.f.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.2.a.f.1.3	✓	3	20.19	odd	2
1900.2.a.h.1.1	yes	3	4.3	odd	2
1900.2.c.g.1749.2		6	20.3	even	4
1900.2.c.g.1749.5		6	20.7	even	4
7600.2.a.bj.1.3		3	1.1	even	1	trivial
7600.2.a.by.1.1		3	5.4	even	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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+1.19869 q^{3} +1.19869 q^{7} -1.56314 q^{9} +O(q^{10})\)	\(q+1.19869 q^{3} +1.19869 q^{7} -1.56314 q^{9} +5.86718 q^{11} +0.364448 q^{13} -1.19869 q^{17} +1.00000 q^{19} +1.43686 q^{21} -8.23163 q^{23} -5.46980 q^{27} -7.86718 q^{29} -7.30404 q^{31} +7.03293 q^{33} -7.13828 q^{37} +0.436861 q^{39} +2.43686 q^{41} -7.39738 q^{43} -13.7014 q^{47} -5.56314 q^{49} -1.43686 q^{51} +7.39738 q^{53} +1.19869 q^{57} -12.8606 q^{59} -1.30404 q^{61} -1.87372 q^{63} +11.9330 q^{67} -9.86718 q^{69} -2.12628 q^{71} -2.50273 q^{73} +7.03293 q^{77} +7.74090 q^{79} -1.86718 q^{81} -3.02093 q^{83} -9.43032 q^{87} -5.68942 q^{89} +0.436861 q^{91} -8.75529 q^{93} -1.09334 q^{97} -9.17122 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - 2 q^{7} + q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 - 2 * q^7 + q^9	\( 3 q - 2 q^{3} - 2 q^{7} + q^{9} + q^{11} + q^{13} + 2 q^{17} + 3 q^{19} + 10 q^{21} - 8 q^{23} - 11 q^{27} - 7 q^{29} - 11 q^{31} + 10 q^{33} - 5 q^{37} + 7 q^{39} + 13 q^{41} - 11 q^{43} - 19 q^{47} - 11 q^{49} - 10 q^{51} + 11 q^{53} - 2 q^{57} + 6 q^{59} + 7 q^{61} - 17 q^{63} - 3 q^{67} - 13 q^{69} + 5 q^{71} + 9 q^{73} + 10 q^{77} + 18 q^{79} + 11 q^{81} - 3 q^{83} - 6 q^{87} + 7 q^{91} + 13 q^{93} - 3 q^{97}+O(q^{100}) \) 3 * q - 2 * q^3 - 2 * q^7 + q^9 + q^11 + q^13 + 2 * q^17 + 3 * q^19 + 10 * q^21 - 8 * q^23 - 11 * q^27 - 7 * q^29 - 11 * q^31 + 10 * q^33 - 5 * q^37 + 7 * q^39 + 13 * q^41 - 11 * q^43 - 19 * q^47 - 11 * q^49 - 10 * q^51 + 11 * q^53 - 2 * q^57 + 6 * q^59 + 7 * q^61 - 17 * q^63 - 3 * q^67 - 13 * q^69 + 5 * q^71 + 9 * q^73 + 10 * q^77 + 18 * q^79 + 11 * q^81 - 3 * q^83 - 6 * q^87 + 7 * q^91 + 13 * q^93 - 3 * q^97

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