LMFDB - Embedded newform 7600.2.a.ck.1.2

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.66064384.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 13x^{2} - 1 $ x^6 - 9x^4 + 13x^2 - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.68667$ 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)

$f(q)$	$=$	$q-2.31446 q^{3} -1.45033 q^{7} +2.35673 q^{9} +3.89655 q^{11} +3.05888 q^{13} +3.92301 q^{17} +1.00000 q^{19} +3.35673 q^{21} +5.37334 q^{23} +1.48883 q^{27} +6.00000 q^{29} +8.43637 q^{31} -9.01841 q^{33} -5.95953 q^{37} -7.07965 q^{39} +10.4364 q^{41} +1.45033 q^{43} -4.90686 q^{47} -4.89655 q^{49} -9.07965 q^{51} -4.23127 q^{53} -2.31446 q^{57} -3.35673 q^{59} +10.3329 q^{61} -3.41802 q^{63} +9.84404 q^{67} -12.4364 q^{69} -8.64327 q^{71} -2.43418 q^{73} -5.65127 q^{77} +12.4364 q^{79} -10.5160 q^{81} -12.6635 q^{83} -13.8868 q^{87} +12.3662 q^{89} -4.43637 q^{91} -19.5256 q^{93} +3.05888 q^{97} +9.18310 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 14 q^{9} - 2 q^{11} + 6 q^{19} + 20 q^{21} + 36 q^{29} + 8 q^{39} + 12 q^{41} - 4 q^{49} - 4 q^{51} - 20 q^{59} - 14 q^{61} - 24 q^{69} - 52 q^{71} + 24 q^{79} + 38 q^{81} + 24 q^{89} + 24 q^{91}+ \cdots + 30 q^{99}+O(q^{100}) $ 6 * q + 14 * q^9 - 2 * q^11 + 6 * q^19 + 20 * q^21 + 36 * q^29 + 8 * q^39 + 12 * q^41 - 4 * q^49 - 4 * q^51 - 20 * q^59 - 14 * q^61 - 24 * q^69 - 52 * q^71 + 24 * q^79 + 38 * q^81 + 24 * q^89 + 24 * q^91 + 30 * 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
$3$	−2.31446	−1.33625	−0.668127	−	0.744047i	$-0.732904\pi$
$3$	−2.31446	−1.33625	−0.668127	+	0.744047i	$0.732904\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.45033	−0.548172	−0.274086	−	0.961705i	$-0.588375\pi$
$7$	−1.45033	−0.548172	−0.274086	+	0.961705i	$0.588375\pi$
$8$	0	0
$9$	2.35673	0.785575
$10$	0	0
$11$	3.89655	1.17485	0.587427	−	0.809277i	$-0.300141\pi$
$11$	3.89655	1.17485	0.587427	+	0.809277i	$0.300141\pi$
$12$	0	0
$13$	3.05888	0.848380	0.424190	−	0.905573i	$-0.360559\pi$
$13$	3.05888	0.848380	0.424190	+	0.905573i	$0.360559\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.92301	0.951469	0.475735	−	0.879589i	$-0.342182\pi$
$17$	3.92301	0.951469	0.475735	+	0.879589i	$0.342182\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	3.35673	0.732498
$22$	0	0
$23$	5.37334	1.12042	0.560209	−	0.828351i	$-0.310721\pi$
$23$	5.37334	1.12042	0.560209	+	0.828351i	$0.310721\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.48883	0.286526
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	8.43637	1.51522	0.757609	−	0.652709i	$-0.226368\pi$
$31$	8.43637	1.51522	0.757609	+	0.652709i	$0.226368\pi$
$32$	0	0
$33$	−9.01841	−1.56990
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.95953	−0.979741	−0.489871	−	0.871795i	$-0.662956\pi$
$37$	−5.95953	−0.979741	−0.489871	+	0.871795i	$0.662956\pi$
$38$	0	0
$39$	−7.07965	−1.13365
$40$	0	0
$41$	10.4364	1.62989	0.814944	−	0.579540i	$-0.196768\pi$
$41$	10.4364	1.62989	0.814944	+	0.579540i	$0.196768\pi$
$42$	0	0
$43$	1.45033	0.221173	0.110586	−	0.993867i	$-0.464727\pi$
$43$	1.45033	0.221173	0.110586	+	0.993867i	$0.464727\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.90686	−0.715739	−0.357869	−	0.933772i	$-0.616497\pi$
$47$	−4.90686	−0.715739	−0.357869	+	0.933772i	$0.616497\pi$
$48$	0	0
$49$	−4.89655	−0.699507
$50$	0	0
$51$	−9.07965	−1.27140
$52$	0	0
$53$	−4.23127	−0.581209	−0.290605	−	0.956843i	$-0.593856\pi$
$53$	−4.23127	−0.581209	−0.290605	+	0.956843i	$0.593856\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.31446	−0.306558
$58$	0	0
$59$	−3.35673	−0.437008	−0.218504	−	0.975836i	$-0.570118\pi$
$59$	−3.35673	−0.437008	−0.218504	+	0.975836i	$0.570118\pi$
$60$	0	0
$61$	10.3329	1.32300	0.661498	−	0.749947i	$-0.269921\pi$
$61$	10.3329	1.32300	0.661498	+	0.749947i	$0.269921\pi$
$62$	0	0
$63$	−3.41802	−0.430631
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.84404	1.20264	0.601320	−	0.799008i	$-0.294642\pi$
$67$	9.84404	1.20264	0.601320	+	0.799008i	$0.294642\pi$
$68$	0	0
$69$	−12.4364	−1.49716
$70$	0	0
$71$	−8.64327	−1.02577	−0.512884	−	0.858458i	$-0.671423\pi$
$71$	−8.64327	−1.02577	−0.512884	+	0.858458i	$0.671423\pi$
$72$	0	0
$73$	−2.43418	−0.284899	−0.142449	−	0.989802i	$-0.545498\pi$
$73$	−2.43418	−0.284899	−0.142449	+	0.989802i	$0.545498\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.65127	−0.644022
$78$	0	0
$79$	12.4364	1.39920	0.699601	−	0.714534i	$-0.253361\pi$
$79$	12.4364	1.39920	0.699601	+	0.714534i	$0.253361\pi$
$80$	0	0
$81$	−10.5160	−1.16845
$82$	0	0
$83$	−12.6635	−1.39000	−0.694999	−	0.719011i	$-0.744595\pi$
$83$	−12.6635	−1.39000	−0.694999	+	0.719011i	$0.744595\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−13.8868	−1.48882
$88$	0	0
$89$	12.3662	1.31081	0.655407	−	0.755276i	$-0.272497\pi$
$89$	12.3662	1.31081	0.655407	+	0.755276i	$0.272497\pi$
$90$	0	0
$91$	−4.43637	−0.465058
$92$	0	0
$93$	−19.5256	−2.02472
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.05888	0.310582	0.155291	−	0.987869i	$-0.450369\pi$
$97$	3.05888	0.310582	0.155291	+	0.987869i	$0.450369\pi$
$98$	0	0
$99$	9.18310	0.922936
$100$	0	0
$101$	3.35673	0.334007	0.167003	−	0.985956i	$-0.446591\pi$
$101$	3.35673	0.334007	0.167003	+	0.985956i	$0.446591\pi$
$102$	0	0
$103$	−13.0611	−1.28695	−0.643476	−	0.765466i	$-0.722508\pi$
$103$	−13.0611	−1.28695	−0.643476	+	0.765466i	$0.722508\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.77099	0.557903	0.278951	−	0.960305i	$-0.410013\pi$
$107$	5.77099	0.557903	0.278951	+	0.960305i	$0.410013\pi$
$108$	0	0
$109$	6.64327	0.636310	0.318155	−	0.948039i	$-0.396937\pi$
$109$	6.64327	0.636310	0.318155	+	0.948039i	$0.396937\pi$
$110$	0	0
$111$	13.7931	1.30918
$112$	0	0
$113$	−9.41606	−0.885789	−0.442894	−	0.896574i	$-0.646048\pi$
$113$	−9.41606	−0.885789	−0.442894	+	0.896574i	$0.646048\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.20893	0.666466
$118$	0	0
$119$	−5.68965	−0.521569
$120$	0	0
$121$	4.18310	0.380282
$122$	0	0
$123$	−24.1546	−2.17794
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.0934	−0.984383	−0.492192	−	0.870487i	$-0.663804\pi$
$127$	−11.0934	−0.984383	−0.492192	+	0.870487i	$0.663804\pi$
$128$	0	0
$129$	−3.35673	−0.295543
$130$	0	0
$131$	−4.61000	−0.402778	−0.201389	−	0.979511i	$-0.564545\pi$
$131$	−4.61000	−0.402778	−0.201389	+	0.979511i	$0.564545\pi$
$132$	0	0
$133$	−1.45033	−0.125759
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.1808	−1.12612	−0.563058	−	0.826417i	$-0.690375\pi$
$137$	−13.1808	−1.12612	−0.563058	+	0.826417i	$0.690375\pi$
$138$	0	0
$139$	−1.18310	−0.100349	−0.0501745	−	0.998740i	$-0.515978\pi$
$139$	−1.18310	−0.100349	−0.0501745	+	0.998740i	$0.515978\pi$
$140$	0	0
$141$	11.3567	0.956409
$142$	0	0
$143$	11.9191	0.996722
$144$	0	0
$145$	0	0
$146$	0	0
$147$	11.3329	0.934719
$148$	0	0
$149$	5.46018	0.447315	0.223658	−	0.974668i	$-0.428200\pi$
$149$	5.46018	0.447315	0.223658	+	0.974668i	$0.428200\pi$
$150$	0	0
$151$	5.07965	0.413376	0.206688	−	0.978407i	$-0.433732\pi$
$151$	5.07965	0.413376	0.206688	+	0.978407i	$0.433732\pi$
$152$	0	0
$153$	9.24546	0.747451
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.11775	0.488250	0.244125	−	0.969744i	$-0.421499\pi$
$157$	6.11775	0.488250	0.244125	+	0.969744i	$0.421499\pi$
$158$	0	0
$159$	9.79310	0.776643
$160$	0	0
$161$	−7.79310	−0.614182
$162$	0	0
$163$	16.4365	1.28740	0.643701	−	0.765277i	$-0.277398\pi$
$163$	16.4365	1.28740	0.643701	+	0.765277i	$0.277398\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.80329	0.294308	0.147154	−	0.989114i	$-0.452989\pi$
$167$	3.80329	0.294308	0.147154	+	0.989114i	$0.452989\pi$
$168$	0	0
$169$	−3.64327	−0.280252
$170$	0	0
$171$	2.35673	0.180223
$172$	0	0
$173$	−11.3838	−0.865491	−0.432746	−	0.901516i	$-0.642455\pi$
$173$	−11.3838	−0.865491	−0.432746	+	0.901516i	$0.642455\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.76901	0.583954
$178$	0	0
$179$	−10.0702	−0.752680	−0.376340	−	0.926482i	$-0.622818\pi$
$179$	−10.0702	−0.752680	−0.376340	+	0.926482i	$0.622818\pi$
$180$	0	0
$181$	0.573097	0.0425980	0.0212990	−	0.999773i	$-0.493220\pi$
$181$	0.573097	0.0425980	0.0212990	+	0.999773i	$0.493220\pi$
$182$	0	0
$183$	−23.9151	−1.76786
$184$	0	0
$185$	0	0
$186$	0	0
$187$	15.2862	1.11784
$188$	0	0
$189$	−2.15930	−0.157066
$190$	0	0
$191$	3.18310	0.230321	0.115160	−	0.993347i	$-0.463262\pi$
$191$	3.18310	0.230321	0.115160	+	0.993347i	$0.463262\pi$
$192$	0	0
$193$	3.05888	0.220183	0.110091	−	0.993921i	$-0.464886\pi$
$193$	3.05888	0.220183	0.110091	+	0.993921i	$0.464886\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.4933	1.53134	0.765669	−	0.643235i	$-0.222408\pi$
$197$	21.4933	1.53134	0.765669	+	0.643235i	$0.222408\pi$
$198$	0	0
$199$	4.81690	0.341461	0.170731	−	0.985318i	$-0.445387\pi$
$199$	4.81690	0.341461	0.170731	+	0.985318i	$0.445387\pi$
$200$	0	0
$201$	−22.7836	−1.60703
$202$	0	0
$203$	−8.70197	−0.610758
$204$	0	0
$205$	0	0
$206$	0	0
$207$	12.6635	0.880173
$208$	0	0
$209$	3.89655	0.269530
$210$	0	0
$211$	−10.5066	−0.723301	−0.361650	−	0.932314i	$-0.617787\pi$
$211$	−10.5066	−0.723301	−0.361650	+	0.932314i	$0.617787\pi$
$212$	0	0
$213$	20.0045	1.37069
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−12.2355	−0.830600
$218$	0	0
$219$	5.63380	0.380697
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	16.8947	1.13136	0.565678	−	0.824626i	$-0.308615\pi$
$223$	16.8947	1.13136	0.565678	+	0.824626i	$0.308615\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.1342	−1.13724	−0.568618	−	0.822602i	$-0.692522\pi$
$227$	−17.1342	−1.13724	−0.568618	+	0.822602i	$0.692522\pi$
$228$	0	0
$229$	25.0464	1.65511	0.827555	−	0.561384i	$-0.189731\pi$
$229$	25.0464	1.65511	0.827555	+	0.561384i	$0.189731\pi$
$230$	0	0
$231$	13.0796	0.860578
$232$	0	0
$233$	19.2986	1.26429	0.632147	−	0.774849i	$-0.282174\pi$
$233$	19.2986	1.26429	0.632147	+	0.774849i	$0.282174\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−28.7835	−1.86969
$238$	0	0
$239$	−18.7693	−1.21408	−0.607042	−	0.794669i	$-0.707644\pi$
$239$	−18.7693	−1.21408	−0.607042	+	0.794669i	$0.707644\pi$
$240$	0	0
$241$	−14.4364	−0.929929	−0.464964	−	0.885329i	$-0.653933\pi$
$241$	−14.4364	−0.929929	−0.464964	+	0.885329i	$0.653933\pi$
$242$	0	0
$243$	19.8724	1.27482
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.05888	0.194632
$248$	0	0
$249$	29.3091	1.85739
$250$	0	0
$251$	10.9762	0.692811	0.346406	−	0.938085i	$-0.387402\pi$
$251$	10.9762	0.692811	0.346406	+	0.938085i	$0.387402\pi$
$252$	0	0
$253$	20.9375	1.31633
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.6392	−1.10030	−0.550150	−	0.835066i	$-0.685430\pi$
$257$	−17.6392	−1.10030	−0.550150	+	0.835066i	$0.685430\pi$
$258$	0	0
$259$	8.64327	0.537067
$260$	0	0
$261$	14.1404	0.875266
$262$	0	0
$263$	1.68976	0.104195	0.0520975	−	0.998642i	$-0.483409\pi$
$263$	1.68976	0.104195	0.0520975	+	0.998642i	$0.483409\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−28.6211	−1.75158
$268$	0	0
$269$	−27.1022	−1.65245	−0.826226	−	0.563339i	$-0.809516\pi$
$269$	−27.1022	−1.65245	−0.826226	+	0.563339i	$0.809516\pi$
$270$	0	0
$271$	−23.9524	−1.45500	−0.727502	−	0.686105i	$-0.759319\pi$
$271$	−23.9524	−1.45500	−0.727502	+	0.686105i	$0.759319\pi$
$272$	0	0
$273$	10.2678	0.621436
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.23549	0.494822	0.247411	−	0.968911i	$-0.420420\pi$
$277$	8.23549	0.494822	0.247411	+	0.968911i	$0.420420\pi$
$278$	0	0
$279$	19.8822	1.19032
$280$	0	0
$281$	−10.4364	−0.622582	−0.311291	−	0.950315i	$-0.600761\pi$
$281$	−10.4364	−0.622582	−0.311291	+	0.950315i	$0.600761\pi$
$282$	0	0
$283$	10.4687	0.622302	0.311151	−	0.950361i	$-0.399286\pi$
$283$	10.4687	0.622302	0.311151	+	0.950361i	$0.399286\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−15.1362	−0.893459
$288$	0	0
$289$	−1.61000	−0.0947059
$290$	0	0
$291$	−7.07965	−0.415016
$292$	0	0
$293$	−16.4668	−0.961999	−0.481000	−	0.876721i	$-0.659726\pi$
$293$	−16.4668	−0.961999	−0.481000	+	0.876721i	$0.659726\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.80131	0.336626
$298$	0	0
$299$	16.4364	0.950540
$300$	0	0
$301$	−2.10345	−0.121241
$302$	0	0
$303$	−7.76901	−0.446318
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7.87634	−0.449526	−0.224763	−	0.974413i	$-0.572161\pi$
$307$	−7.87634	−0.449526	−0.224763	+	0.974413i	$0.572161\pi$
$308$	0	0
$309$	30.2295	1.71969
$310$	0	0
$311$	−3.89655	−0.220953	−0.110477	−	0.993879i	$-0.535238\pi$
$311$	−3.89655	−0.220953	−0.110477	+	0.993879i	$0.535238\pi$
$312$	0	0
$313$	7.76901	0.439130	0.219565	−	0.975598i	$-0.429536\pi$
$313$	7.76901	0.439130	0.219565	+	0.975598i	$0.429536\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.0510	1.07001	0.535005	−	0.844849i	$-0.320310\pi$
$317$	19.0510	1.07001	0.535005	+	0.844849i	$0.320310\pi$
$318$	0	0
$319$	23.3793	1.30899
$320$	0	0
$321$	−13.3567	−0.745500
$322$	0	0
$323$	3.92301	0.218282
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−15.3756	−0.850272
$328$	0	0
$329$	7.11655	0.392348
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	0	0
$333$	−14.0450	−0.769660
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−6.89249	−0.375458	−0.187729	−	0.982221i	$-0.560113\pi$
$337$	−6.89249	−0.375458	−0.187729	+	0.982221i	$0.560113\pi$
$338$	0	0
$339$	21.7931	1.18364
$340$	0	0
$341$	32.8727	1.78016
$342$	0	0
$343$	17.2539	0.931623
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.5503	1.64002	0.820012	−	0.572346i	$-0.193967\pi$
$347$	30.5503	1.64002	0.820012	+	0.572346i	$0.193967\pi$
$348$	0	0
$349$	16.7693	0.897640	0.448820	−	0.893622i	$-0.351844\pi$
$349$	16.7693	0.897640	0.448820	+	0.893622i	$0.351844\pi$
$350$	0	0
$351$	4.55416	0.243083
$352$	0	0
$353$	29.0999	1.54883	0.774417	−	0.632676i	$-0.218044\pi$
$353$	29.0999	1.54883	0.774417	+	0.632676i	$0.218044\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	13.1685	0.696949
$358$	0	0
$359$	11.6896	0.616956	0.308478	−	0.951231i	$-0.400180\pi$
$359$	11.6896	0.616956	0.308478	+	0.951231i	$0.400180\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−9.68161	−0.508153
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.20095	0.219288	0.109644	−	0.993971i	$-0.465029\pi$
$367$	4.20095	0.219288	0.109644	+	0.993971i	$0.465029\pi$
$368$	0	0
$369$	24.5957	1.28040
$370$	0	0
$371$	6.13672	0.318603
$372$	0	0
$373$	−16.9456	−0.877412	−0.438706	−	0.898631i	$-0.644563\pi$
$373$	−16.9456	−0.877412	−0.438706	+	0.898631i	$0.644563\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.3533	0.945241
$378$	0	0
$379$	−10.3662	−0.532476	−0.266238	−	0.963907i	$-0.585781\pi$
$379$	−10.3662	−0.532476	−0.266238	+	0.963907i	$0.585781\pi$
$380$	0	0
$381$	25.6753	1.31539
$382$	0	0
$383$	−20.5907	−1.05214	−0.526068	−	0.850442i	$-0.676334\pi$
$383$	−20.5907	−1.05214	−0.526068	+	0.850442i	$0.676334\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.41802	0.173748
$388$	0	0
$389$	8.10345	0.410861	0.205431	−	0.978672i	$-0.434141\pi$
$389$	8.10345	0.410861	0.205431	+	0.978672i	$0.434141\pi$
$390$	0	0
$391$	21.0796	1.06604
$392$	0	0
$393$	10.6697	0.538213
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.46891	0.174100	0.0870499	−	0.996204i	$-0.472256\pi$
$397$	3.46891	0.174100	0.0870499	+	0.996204i	$0.472256\pi$
$398$	0	0
$399$	3.35673	0.168046
$400$	0	0
$401$	−15.9524	−0.796625	−0.398312	−	0.917250i	$-0.630404\pi$
$401$	−15.9524	−0.796625	−0.398312	+	0.917250i	$0.630404\pi$
$402$	0	0
$403$	25.8058	1.28548
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−23.2216	−1.15105
$408$	0	0
$409$	−3.92982	−0.194317	−0.0971586	−	0.995269i	$-0.530975\pi$
$409$	−3.92982	−0.194317	−0.0971586	+	0.995269i	$0.530975\pi$
$410$	0	0
$411$	30.5066	1.50478
$412$	0	0
$413$	4.86835	0.239556
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.73823	0.134092
$418$	0	0
$419$	34.2996	1.67565	0.837824	−	0.545941i	$-0.183828\pi$
$419$	34.2996	1.67565	0.837824	+	0.545941i	$0.183828\pi$
$420$	0	0
$421$	−26.0891	−1.27151	−0.635753	−	0.771893i	$-0.719310\pi$
$421$	−26.0891	−1.27151	−0.635753	+	0.771893i	$0.719310\pi$
$422$	0	0
$423$	−11.5641	−0.562267
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−14.9861	−0.725229
$428$	0	0
$429$	−27.5862	−1.33187
$430$	0	0
$431$	−10.2996	−0.496117	−0.248058	−	0.968745i	$-0.579792\pi$
$431$	−10.2996	−0.496117	−0.248058	+	0.968745i	$0.579792\pi$
$432$	0	0
$433$	−36.2319	−1.74119	−0.870596	−	0.491999i	$-0.836266\pi$
$433$	−36.2319	−1.74119	−0.870596	+	0.491999i	$0.836266\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.37334	0.257042
$438$	0	0
$439$	24.0891	1.14971	0.574855	−	0.818255i	$-0.305058\pi$
$439$	24.0891	1.14971	0.574855	+	0.818255i	$0.305058\pi$
$440$	0	0
$441$	−11.5398	−0.549515
$442$	0	0
$443$	5.52337	0.262423	0.131212	−	0.991354i	$-0.458113\pi$
$443$	5.52337	0.262423	0.131212	+	0.991354i	$0.458113\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−12.6374	−0.597727
$448$	0	0
$449$	7.92982	0.374231	0.187116	−	0.982338i	$-0.440086\pi$
$449$	7.92982	0.374231	0.187116	+	0.982338i	$0.440086\pi$
$450$	0	0
$451$	40.6658	1.91488
$452$	0	0
$453$	−11.7566	−0.552375
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.6534	−1.05968	−0.529840	−	0.848098i	$-0.677748\pi$
$457$	−22.6534	−1.05968	−0.529840	+	0.848098i	$0.677748\pi$
$458$	0	0
$459$	5.84070	0.272621
$460$	0	0
$461$	13.3900	0.623634	0.311817	−	0.950142i	$-0.399062\pi$
$461$	13.3900	0.623634	0.311817	+	0.950142i	$0.399062\pi$
$462$	0	0
$463$	16.9029	0.785546	0.392773	−	0.919635i	$-0.371516\pi$
$463$	16.9029	0.785546	0.392773	+	0.919635i	$0.371516\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.2501	1.02961	0.514807	−	0.857306i	$-0.327864\pi$
$467$	22.2501	1.02961	0.514807	+	0.857306i	$0.327864\pi$
$468$	0	0
$469$	−14.2771	−0.659254
$470$	0	0
$471$	−14.1593	−0.652426
$472$	0	0
$473$	5.65127	0.259846
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−9.97194	−0.456584
$478$	0	0
$479$	−0.366196	−0.0167319	−0.00836597	−	0.999965i	$-0.502663\pi$
$479$	−0.366196	−0.0167319	−0.00836597	+	0.999965i	$0.502663\pi$
$480$	0	0
$481$	−18.2295	−0.831192
$482$	0	0
$483$	18.0368	0.820704
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.8461	1.21651	0.608257	−	0.793740i	$-0.291869\pi$
$487$	26.8461	1.21651	0.608257	+	0.793740i	$0.291869\pi$
$488$	0	0
$489$	−38.0415	−1.72030
$490$	0	0
$491$	23.7266	1.07076	0.535382	−	0.844610i	$-0.320168\pi$
$491$	23.7266	1.07076	0.535382	+	0.844610i	$0.320168\pi$
$492$	0	0
$493$	23.5381	1.06010
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.5356	0.562298
$498$	0	0
$499$	−6.81690	−0.305166	−0.152583	−	0.988291i	$-0.548759\pi$
$499$	−6.81690	−0.305166	−0.152583	+	0.988291i	$0.548759\pi$
$500$	0	0
$501$	−8.80257	−0.393270
$502$	0	0
$503$	23.4102	1.04381	0.521904	−	0.853004i	$-0.325222\pi$
$503$	23.4102	1.04381	0.521904	+	0.853004i	$0.325222\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.43221	0.374488
$508$	0	0
$509$	16.9204	0.749981	0.374991	−	0.927029i	$-0.377646\pi$
$509$	16.9204	0.749981	0.374991	+	0.927029i	$0.377646\pi$
$510$	0	0
$511$	3.53035	0.156174
$512$	0	0
$513$	1.48883	0.0657336
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−19.1198	−0.840888
$518$	0	0
$519$	26.3473	1.15652
$520$	0	0
$521$	−3.49345	−0.153051	−0.0765254	−	0.997068i	$-0.524383\pi$
$521$	−3.49345	−0.153051	−0.0765254	+	0.997068i	$0.524383\pi$
$522$	0	0
$523$	14.9271	0.652714	0.326357	−	0.945247i	$-0.394179\pi$
$523$	14.9271	0.652714	0.326357	+	0.945247i	$0.394179\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	33.0960	1.44168
$528$	0	0
$529$	5.87275	0.255337
$530$	0	0
$531$	−7.91088	−0.343303
$532$	0	0
$533$	31.9236	1.38276
$534$	0	0
$535$	0	0
$536$	0	0
$537$	23.3070	1.00577
$538$	0	0
$539$	−19.0796	−0.821819
$540$	0	0
$541$	−12.6991	−0.545978	−0.272989	−	0.962017i	$-0.588012\pi$
$541$	−12.6991	−0.545978	−0.272989	+	0.962017i	$0.588012\pi$
$542$	0	0
$543$	−1.32641	−0.0569217
$544$	0	0
$545$	0	0
$546$	0	0
$547$	31.8162	1.36036	0.680182	−	0.733043i	$-0.261901\pi$
$547$	31.8162	1.36036	0.680182	+	0.733043i	$0.261901\pi$
$548$	0	0
$549$	24.3519	1.03931
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−18.0368	−0.767003
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.64731	−0.408770	−0.204385	−	0.978891i	$-0.565519\pi$
$557$	−9.64731	−0.408770	−0.204385	+	0.978891i	$0.565519\pi$
$558$	0	0
$559$	4.43637	0.187639
$560$	0	0
$561$	−35.3793	−1.49372
$562$	0	0
$563$	4.28216	0.180471	0.0902357	−	0.995920i	$-0.471238\pi$
$563$	4.28216	0.180471	0.0902357	+	0.995920i	$0.471238\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	15.2517	0.640510
$568$	0	0
$569$	42.2295	1.77035	0.885176	−	0.465257i	$-0.154038\pi$
$569$	42.2295	1.77035	0.885176	+	0.465257i	$0.154038\pi$
$570$	0	0
$571$	19.2200	0.804332	0.402166	−	0.915567i	$-0.368257\pi$
$571$	19.2200	0.804332	0.402166	+	0.915567i	$0.368257\pi$
$572$	0	0
$573$	−7.36715	−0.307767
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−40.7919	−1.69819	−0.849096	−	0.528239i	$-0.822852\pi$
$577$	−40.7919	−1.69819	−0.849096	+	0.528239i	$0.822852\pi$
$578$	0	0
$579$	−7.07965	−0.294220
$580$	0	0
$581$	18.3662	0.761958
$582$	0	0
$583$	−16.4873	−0.682836
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.8851	1.31604	0.658019	−	0.753001i	$-0.271395\pi$
$587$	31.8851	1.31604	0.658019	+	0.753001i	$0.271395\pi$
$588$	0	0
$589$	8.43637	0.347615
$590$	0	0
$591$	−49.7455	−2.04626
$592$	0	0
$593$	−38.8973	−1.59732	−0.798660	−	0.601783i	$-0.794457\pi$
$593$	−38.8973	−1.59732	−0.798660	+	0.601783i	$0.794457\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.1485	−0.456279
$598$	0	0
$599$	−28.1629	−1.15071	−0.575353	−	0.817905i	$-0.695135\pi$
$599$	−28.1629	−1.15071	−0.575353	+	0.817905i	$0.695135\pi$
$600$	0	0
$601$	−5.56363	−0.226945	−0.113473	−	0.993541i	$-0.536197\pi$
$601$	−5.56363	−0.226945	−0.113473	+	0.993541i	$0.536197\pi$
$602$	0	0
$603$	23.1997	0.944764
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−33.9986	−1.37996	−0.689980	−	0.723828i	$-0.742381\pi$
$607$	−33.9986	−1.37996	−0.689980	+	0.723828i	$0.742381\pi$
$608$	0	0
$609$	20.1404	0.816128
$610$	0	0
$611$	−15.0095	−0.607218
$612$	0	0
$613$	17.5703	0.709659	0.354830	−	0.934931i	$-0.384539\pi$
$613$	17.5703	0.709659	0.354830	+	0.934931i	$0.384539\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.0791	−0.526544	−0.263272	−	0.964722i	$-0.584802\pi$
$617$	−13.0791	−0.526544	−0.263272	+	0.964722i	$0.584802\pi$
$618$	0	0
$619$	18.9393	0.761234	0.380617	−	0.924733i	$-0.375712\pi$
$619$	18.9393	0.761234	0.380617	+	0.924733i	$0.375712\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	−17.9350	−0.718552
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−9.01841	−0.360161
$628$	0	0
$629$	−23.3793	−0.932194
$630$	0	0
$631$	−31.6896	−1.26154	−0.630772	−	0.775968i	$-0.717262\pi$
$631$	−31.6896	−1.26154	−0.630772	+	0.775968i	$0.717262\pi$
$632$	0	0
$633$	24.3170	0.966514
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−14.9779	−0.593448
$638$	0	0
$639$	−20.3698	−0.805818
$640$	0	0
$641$	47.9750	1.89490	0.947449	−	0.319908i	$-0.103652\pi$
$641$	47.9750	1.89490	0.947449	+	0.319908i	$0.103652\pi$
$642$	0	0
$643$	0.200927	0.00792378	0.00396189	−	0.999992i	$-0.498739\pi$
$643$	0.200927	0.00792378	0.00396189	+	0.999992i	$0.498739\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.58798	−0.0624299	−0.0312150	−	0.999513i	$-0.509938\pi$
$647$	−1.58798	−0.0624299	−0.0312150	+	0.999513i	$0.509938\pi$
$648$	0	0
$649$	−13.0796	−0.513421
$650$	0	0
$651$	28.3186	1.10989
$652$	0	0
$653$	−33.5624	−1.31340	−0.656700	−	0.754152i	$-0.728048\pi$
$653$	−33.5624	−1.31340	−0.656700	+	0.754152i	$0.728048\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5.73669	−0.223809
$658$	0	0
$659$	−15.3567	−0.598213	−0.299107	−	0.954220i	$-0.596689\pi$
$659$	−15.3567	−0.598213	−0.299107	+	0.954220i	$0.596689\pi$
$660$	0	0
$661$	12.8026	0.497962	0.248981	−	0.968508i	$-0.419904\pi$
$661$	12.8026	0.497962	0.248981	+	0.968508i	$0.419904\pi$
$662$	0	0
$663$	−27.7735	−1.07863
$664$	0	0
$665$	0	0
$666$	0	0
$667$	32.2400	1.24834
$668$	0	0
$669$	−39.1022	−1.51178
$670$	0	0
$671$	40.2627	1.55433
$672$	0	0
$673$	7.82545	0.301649	0.150824	−	0.988561i	$-0.451807\pi$
$673$	7.82545	0.301649	0.150824	+	0.988561i	$0.451807\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.6516	0.832137	0.416069	−	0.909333i	$-0.363408\pi$
$677$	21.6516	0.832137	0.416069	+	0.909333i	$0.363408\pi$
$678$	0	0
$679$	−4.43637	−0.170252
$680$	0	0
$681$	39.6564	1.51964
$682$	0	0
$683$	−6.38751	−0.244411	−0.122206	−	0.992505i	$-0.538997\pi$
$683$	−6.38751	−0.244411	−0.122206	+	0.992505i	$0.538997\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−57.9688	−2.21165
$688$	0	0
$689$	−12.9429	−0.493086
$690$	0	0
$691$	−44.4958	−1.69270	−0.846351	−	0.532626i	$-0.821205\pi$
$691$	−44.4958	−1.69270	−0.846351	+	0.532626i	$0.821205\pi$
$692$	0	0
$693$	−13.3185	−0.505928
$694$	0	0
$695$	0	0
$696$	0	0
$697$	40.9420	1.55079
$698$	0	0
$699$	−44.6658	−1.68942
$700$	0	0
$701$	17.5160	0.661571	0.330785	−	0.943706i	$-0.392686\pi$
$701$	17.5160	0.661571	0.330785	+	0.943706i	$0.392686\pi$
$702$	0	0
$703$	−5.95953	−0.224768
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.86835	−0.183093
$708$	0	0
$709$	11.4269	0.429146	0.214573	−	0.976708i	$-0.431164\pi$
$709$	11.4269	0.429146	0.214573	+	0.976708i	$0.431164\pi$
$710$	0	0
$711$	29.3091	1.09918
$712$	0	0
$713$	45.3315	1.69768
$714$	0	0
$715$	0	0
$716$	0	0
$717$	43.4408	1.62233
$718$	0	0
$719$	−15.8965	−0.592841	−0.296421	−	0.955057i	$-0.595793\pi$
$719$	−15.8965	−0.592841	−0.296421	+	0.955057i	$0.595793\pi$
$720$	0	0
$721$	18.9429	0.705471
$722$	0	0
$723$	33.4124	1.24262
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−41.9905	−1.55734	−0.778670	−	0.627434i	$-0.784105\pi$
$727$	−41.9905	−1.55734	−0.778670	+	0.627434i	$0.784105\pi$
$728$	0	0
$729$	−14.4458	−0.535031
$730$	0	0
$731$	5.68965	0.210439
$732$	0	0
$733$	−0.632884	−0.0233761	−0.0116881	−	0.999932i	$-0.503721\pi$
$733$	−0.632884	−0.0233761	−0.0116881	+	0.999932i	$0.503721\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38.3578	1.41293
$738$	0	0
$739$	29.5493	1.08699	0.543494	−	0.839413i	$-0.317101\pi$
$739$	29.5493	1.08699	0.543494	+	0.839413i	$0.317101\pi$
$740$	0	0
$741$	−7.07965	−0.260077
$742$	0	0
$743$	31.3374	1.14966	0.574829	−	0.818274i	$-0.305069\pi$
$743$	31.3374	1.14966	0.574829	+	0.818274i	$0.305069\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−29.8443	−1.09195
$748$	0	0
$749$	−8.36983	−0.305827
$750$	0	0
$751$	−25.0131	−0.912741	−0.456371	−	0.889790i	$-0.650851\pi$
$751$	−25.0131	−0.912741	−0.456371	+	0.889790i	$0.650851\pi$
$752$	0	0
$753$	−25.4040	−0.925772
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−32.0900	−1.16633	−0.583165	−	0.812354i	$-0.698186\pi$
$757$	−32.0900	−1.16633	−0.583165	+	0.812354i	$0.698186\pi$
$758$	0	0
$759$	−48.4589	−1.75895
$760$	0	0
$761$	−40.4922	−1.46784	−0.733921	−	0.679235i	$-0.762312\pi$
$761$	−40.4922	−1.46784	−0.733921	+	0.679235i	$0.762312\pi$
$762$	0	0
$763$	−9.63492	−0.348808
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.2678	−0.370749
$768$	0	0
$769$	3.09398	0.111572	0.0557859	−	0.998443i	$-0.482234\pi$
$769$	3.09398	0.111572	0.0557859	+	0.998443i	$0.482234\pi$
$770$	0	0
$771$	40.8251	1.47028
$772$	0	0
$773$	−1.96350	−0.0706220	−0.0353110	−	0.999376i	$-0.511242\pi$
$773$	−1.96350	−0.0706220	−0.0353110	+	0.999376i	$0.511242\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−20.0045	−0.717658
$778$	0	0
$779$	10.4364	0.373922
$780$	0	0
$781$	−33.6789	−1.20513
$782$	0	0
$783$	8.93300	0.319239
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0.107331	0.00382595	0.00191297	−	0.999998i	$-0.499391\pi$
$787$	0.107331	0.00382595	0.00191297	+	0.999998i	$0.499391\pi$
$788$	0	0
$789$	−3.91088	−0.139231
$790$	0	0
$791$	13.6564	0.485565
$792$	0	0
$793$	31.6071	1.12240
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−8.32068	−0.294734	−0.147367	−	0.989082i	$-0.547080\pi$
$797$	−8.32068	−0.294734	−0.147367	+	0.989082i	$0.547080\pi$
$798$	0	0
$799$	−19.2496	−0.681003
$800$	0	0
$801$	29.1437	1.02974
$802$	0	0
$803$	−9.48489	−0.334714
$804$	0	0
$805$	0	0
$806$	0	0
$807$	62.7270	2.20810
$808$	0	0
$809$	−27.6231	−0.971177	−0.485588	−	0.874188i	$-0.661395\pi$
$809$	−27.6231	−0.971177	−0.485588	+	0.874188i	$0.661395\pi$
$810$	0	0
$811$	−23.0095	−0.807972	−0.403986	−	0.914765i	$-0.632376\pi$
$811$	−23.0095	−0.807972	−0.403986	+	0.914765i	$0.632376\pi$
$812$	0	0
$813$	55.4369	1.94426
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.45033	0.0507405
$818$	0	0
$819$	−10.4553	−0.365338
$820$	0	0
$821$	31.1355	1.08664	0.543318	−	0.839527i	$-0.317168\pi$
$821$	31.1355	1.08664	0.543318	+	0.839527i	$0.317168\pi$
$822$	0	0
$823$	−20.4201	−0.711800	−0.355900	−	0.934524i	$-0.615826\pi$
$823$	−20.4201	−0.711800	−0.355900	+	0.934524i	$0.615826\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0.902638	0.0313878	0.0156939	−	0.999877i	$-0.495004\pi$
$827$	0.902638	0.0313878	0.0156939	+	0.999877i	$0.495004\pi$
$828$	0	0
$829$	−13.4971	−0.468773	−0.234386	−	0.972143i	$-0.575308\pi$
$829$	−13.4971	−0.468773	−0.234386	+	0.972143i	$0.575308\pi$
$830$	0	0
$831$	−19.0607	−0.661209
$832$	0	0
$833$	−19.2092	−0.665560
$834$	0	0
$835$	0	0
$836$	0	0
$837$	12.5603	0.434149
$838$	0	0
$839$	−33.1022	−1.14282	−0.571408	−	0.820666i	$-0.693603\pi$
$839$	−33.1022	−1.14282	−0.571408	+	0.820666i	$0.693603\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	24.1546	0.831928
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.06686	−0.208460
$848$	0	0
$849$	−24.2295	−0.831553
$850$	0	0
$851$	−32.0226	−1.09772
$852$	0	0
$853$	50.9097	1.74312	0.871558	−	0.490293i	$-0.163110\pi$
$853$	50.9097	1.74312	0.871558	+	0.490293i	$0.163110\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.2333	0.725317	0.362659	−	0.931922i	$-0.381869\pi$
$857$	21.2333	0.725317	0.362659	+	0.931922i	$0.381869\pi$
$858$	0	0
$859$	−29.4827	−1.00594	−0.502969	−	0.864304i	$-0.667759\pi$
$859$	−29.4827	−1.00594	−0.502969	+	0.864304i	$0.667759\pi$
$860$	0	0
$861$	35.0320	1.19389
$862$	0	0
$863$	−25.7755	−0.877408	−0.438704	−	0.898632i	$-0.644562\pi$
$863$	−25.7755	−0.877408	−0.438704	+	0.898632i	$0.644562\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.72628	0.126551
$868$	0	0
$869$	48.4589	1.64386
$870$	0	0
$871$	30.1117	1.02030
$872$	0	0
$873$	7.20893	0.243985
$874$	0	0
$875$	0	0
$876$	0	0
$877$	54.2687	1.83252	0.916261	−	0.400581i	$-0.131192\pi$
$877$	54.2687	1.83252	0.916261	+	0.400581i	$0.131192\pi$
$878$	0	0
$879$	38.1117	1.28548
$880$	0	0
$881$	−28.4922	−0.959927	−0.479964	−	0.877288i	$-0.659350\pi$
$881$	−28.4922	−0.959927	−0.479964	+	0.877288i	$0.659350\pi$
$882$	0	0
$883$	−40.5264	−1.36382	−0.681911	−	0.731435i	$-0.738851\pi$
$883$	−40.5264	−1.36382	−0.681911	+	0.731435i	$0.738851\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.4705	0.385142	0.192571	−	0.981283i	$-0.438317\pi$
$887$	11.4705	0.385142	0.192571	+	0.981283i	$0.438317\pi$
$888$	0	0
$889$	16.0891	0.539612
$890$	0	0
$891$	−40.9762	−1.37275
$892$	0	0
$893$	−4.90686	−0.164202
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−38.0413	−1.27016
$898$	0	0
$899$	50.6182	1.68821
$900$	0	0
$901$	−16.5993	−0.553003
$902$	0	0
$903$	4.86835	0.162009
$904$	0	0
$905$	0	0
$906$	0	0
$907$	48.1000	1.59714	0.798568	−	0.601905i	$-0.205591\pi$
$907$	48.1000	1.59714	0.798568	+	0.601905i	$0.205591\pi$
$908$	0	0
$909$	7.91088	0.262387
$910$	0	0
$911$	12.8062	0.424288	0.212144	−	0.977238i	$-0.431955\pi$
$911$	12.8062	0.424288	0.212144	+	0.977238i	$0.431955\pi$
$912$	0	0
$913$	−49.3439	−1.63304
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.68601	0.220792
$918$	0	0
$919$	38.7135	1.27704	0.638519	−	0.769606i	$-0.279547\pi$
$919$	38.7135	1.27704	0.638519	+	0.769606i	$0.279547\pi$
$920$	0	0
$921$	18.2295	0.600682
$922$	0	0
$923$	−26.4387	−0.870241
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−30.7815	−1.01100
$928$	0	0
$929$	−36.0189	−1.18174	−0.590872	−	0.806766i	$-0.701216\pi$
$929$	−36.0189	−1.18174	−0.590872	+	0.806766i	$0.701216\pi$
$930$	0	0
$931$	−4.89655	−0.160478
$932$	0	0
$933$	9.01841	0.295249
$934$	0	0
$935$	0	0
$936$	0	0
$937$	45.2421	1.47799	0.738997	−	0.673709i	$-0.235300\pi$
$937$	45.2421	1.47799	0.738997	+	0.673709i	$0.235300\pi$
$938$	0	0
$939$	−17.9811	−0.586790
$940$	0	0
$941$	−11.7455	−0.382892	−0.191446	−	0.981503i	$-0.561318\pi$
$941$	−11.7455	−0.382892	−0.191446	+	0.981503i	$0.561318\pi$
$942$	0	0
$943$	56.0781	1.82616
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13.7752	−0.447635	−0.223817	−	0.974631i	$-0.571852\pi$
$947$	−13.7752	−0.447635	−0.223817	+	0.974631i	$0.571852\pi$
$948$	0	0
$949$	−7.44584	−0.241702
$950$	0	0
$951$	−44.0927	−1.42981
$952$	0	0
$953$	−9.01421	−0.291999	−0.145999	−	0.989285i	$-0.546640\pi$
$953$	−9.01421	−0.291999	−0.145999	+	0.989285i	$0.546640\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−54.1105	−1.74914
$958$	0	0
$959$	19.1166	0.617306
$960$	0	0
$961$	40.1724	1.29588
$962$	0	0
$963$	13.6006	0.438274
$964$	0	0
$965$	0	0
$966$	0	0
$967$	30.3232	0.975129	0.487564	−	0.873087i	$-0.337885\pi$
$967$	30.3232	0.975129	0.487564	+	0.873087i	$0.337885\pi$
$968$	0	0
$969$	−9.07965	−0.291680
$970$	0	0
$971$	31.5636	1.01292	0.506462	−	0.862262i	$-0.330953\pi$
$971$	31.5636	1.01292	0.506462	+	0.862262i	$0.330953\pi$
$972$	0	0
$973$	1.71588	0.0550086
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.1285	−1.37980	−0.689902	−	0.723903i	$-0.742346\pi$
$977$	−43.1285	−1.37980	−0.689902	+	0.723903i	$0.742346\pi$
$978$	0	0
$979$	48.1855	1.54002
$980$	0	0
$981$	15.6564	0.499870
$982$	0	0
$983$	7.81570	0.249282	0.124641	−	0.992202i	$-0.460222\pi$
$983$	7.81570	0.249282	0.124641	+	0.992202i	$0.460222\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−16.4710	−0.524277
$988$	0	0
$989$	7.79310	0.247806
$990$	0	0
$991$	23.5197	0.747126	0.373563	−	0.927605i	$-0.378136\pi$
$991$	23.5197	0.747126	0.373563	+	0.927605i	$0.378136\pi$
$992$	0	0
$993$	18.5157	0.587577
$994$	0	0
$995$	0	0
$996$	0	0
$997$	33.6395	1.06537	0.532686	−	0.846313i	$-0.321183\pi$
$997$	33.6395	1.06537	0.532686	+	0.846313i	$0.321183\pi$
$998$	0	0
$999$	−8.87275	−0.280721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.ck.1.2		6
4.3	odd	2		475.2.a.j.1.5		6
5.2	odd	4		1520.2.d.h.609.5		6
5.3	odd	4		1520.2.d.h.609.2		6
5.4	even	2	inner	7600.2.a.ck.1.5		6
12.11	even	2		4275.2.a.br.1.2		6
20.3	even	4		95.2.b.b.39.2	✓	6
20.7	even	4		95.2.b.b.39.5	yes	6
20.19	odd	2		475.2.a.j.1.2		6
60.23	odd	4		855.2.c.d.514.5		6
60.47	odd	4		855.2.c.d.514.2		6
60.59	even	2		4275.2.a.br.1.5		6
76.75	even	2		9025.2.a.bx.1.2		6
380.227	odd	4		1805.2.b.e.1084.2		6
380.303	odd	4		1805.2.b.e.1084.5		6
380.379	even	2		9025.2.a.bx.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.b.b.39.2	✓	6	20.3	even	4
95.2.b.b.39.5	yes	6	20.7	even	4
475.2.a.j.1.2		6	20.19	odd	2
475.2.a.j.1.5		6	4.3	odd	2
855.2.c.d.514.2		6	60.47	odd	4
855.2.c.d.514.5		6	60.23	odd	4
1520.2.d.h.609.2		6	5.3	odd	4
1520.2.d.h.609.5		6	5.2	odd	4
1805.2.b.e.1084.2		6	380.227	odd	4
1805.2.b.e.1084.5		6	380.303	odd	4
4275.2.a.br.1.2		6	12.11	even	2
4275.2.a.br.1.5		6	60.59	even	2
7600.2.a.ck.1.2		6	1.1	even	1	trivial
7600.2.a.ck.1.5		6	5.4	even	2	inner
9025.2.a.bx.1.2		6	76.75	even	2
9025.2.a.bx.1.5		6	380.379	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q-2.31446 q^{3} -1.45033 q^{7} +2.35673 q^{9} +3.89655 q^{11} +3.05888 q^{13} +3.92301 q^{17} +1.00000 q^{19} +3.35673 q^{21} +5.37334 q^{23} +1.48883 q^{27} +6.00000 q^{29} +8.43637 q^{31} -9.01841 q^{33} -5.95953 q^{37} -7.07965 q^{39} +10.4364 q^{41} +1.45033 q^{43} -4.90686 q^{47} -4.89655 q^{49} -9.07965 q^{51} -4.23127 q^{53} -2.31446 q^{57} -3.35673 q^{59} +10.3329 q^{61} -3.41802 q^{63} +9.84404 q^{67} -12.4364 q^{69} -8.64327 q^{71} -2.43418 q^{73} -5.65127 q^{77} +12.4364 q^{79} -10.5160 q^{81} -12.6635 q^{83} -13.8868 q^{87} +12.3662 q^{89} -4.43637 q^{91} -19.5256 q^{93} +3.05888 q^{97} +9.18310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 14 q^{9} - 2 q^{11} + 6 q^{19} + 20 q^{21} + 36 q^{29} + 8 q^{39} + 12 q^{41} - 4 q^{49} - 4 q^{51} - 20 q^{59} - 14 q^{61} - 24 q^{69} - 52 q^{71} + 24 q^{79} + 38 q^{81} + 24 q^{89} + 24 q^{91}+ \cdots + 30 q^{99}+O(q^{100}) \) 6 * q + 14 * q^9 - 2 * q^11 + 6 * q^19 + 20 * q^21 + 36 * q^29 + 8 * q^39 + 12 * q^41 - 4 * q^49 - 4 * q^51 - 20 * q^59 - 14 * q^61 - 24 * q^69 - 52 * q^71 + 24 * q^79 + 38 * q^81 + 24 * q^89 + 24 * q^91 + 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more