LMFDB - Embedded newform 1240.2.a.m.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1240,2,Mod(1,1240)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1240.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1240 = 2^{3} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1240.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-1,0,6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.90144985064$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 19x^{4} + 15x^{3} + 98x^{2} - 44x - 108 $ x^6 - x^5 - 19x^4 + 15x^3 + 98x^2 - 44x - 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-3.31203$ of defining polynomial
Character	$\chi$	$=$	1240.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.31203 q^{3} +1.00000 q^{5} -1.94649 q^{7} +7.96954 q^{9} -2.17597 q^{11} +4.63020 q^{13} +3.31203 q^{15} +7.48800 q^{17} -3.83663 q^{19} -6.44682 q^{21} -5.53362 q^{23} +1.00000 q^{25} +16.4593 q^{27} -9.21290 q^{29} +1.00000 q^{31} -7.20688 q^{33} -1.94649 q^{35} +4.12561 q^{37} +15.3354 q^{39} +9.37025 q^{41} -12.1717 q^{43} +7.96954 q^{45} +8.39330 q^{47} -3.21119 q^{49} +24.8005 q^{51} -5.20500 q^{53} -2.17597 q^{55} -12.7070 q^{57} +7.43592 q^{59} -9.03566 q^{61} -15.5126 q^{63} +4.63020 q^{65} +10.1163 q^{67} -18.3275 q^{69} -6.46069 q^{71} +9.26466 q^{73} +3.31203 q^{75} +4.23550 q^{77} +3.71573 q^{79} +30.6049 q^{81} -11.1225 q^{83} +7.48800 q^{85} -30.5134 q^{87} -5.81048 q^{89} -9.01262 q^{91} +3.31203 q^{93} -3.83663 q^{95} -12.6991 q^{97} -17.3415 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 6 q^{5} - 2 q^{7} + 21 q^{9} + 8 q^{11} + 4 q^{13} - q^{15} + 3 q^{17} - 5 q^{19} + 8 q^{21} - 12 q^{23} + 6 q^{25} - 7 q^{27} - 4 q^{29} + 6 q^{31} + 14 q^{33} - 2 q^{35} + 3 q^{37} + 10 q^{39}+ \cdots - 40 q^{99}+O(q^{100}) $ 6 * q - q^3 + 6 * q^5 - 2 * q^7 + 21 * q^9 + 8 * q^11 + 4 * q^13 - q^15 + 3 * q^17 - 5 * q^19 + 8 * q^21 - 12 * q^23 + 6 * q^25 - 7 * q^27 - 4 * q^29 + 6 * q^31 + 14 * q^33 - 2 * q^35 + 3 * q^37 + 10 * q^39 + 17 * q^41 - 17 * q^43 + 21 * q^45 - 6 * q^47 + 34 * q^49 + 23 * q^51 + 9 * q^53 + 8 * q^55 - q^57 + 39 * q^59 + 2 * q^61 - 16 * q^63 + 4 * q^65 + 12 * q^67 - 8 * q^69 + 21 * q^71 + 43 * q^73 - q^75 + 20 * q^77 - 2 * q^79 + 50 * q^81 - 5 * q^83 + 3 * q^85 - 44 * q^87 + 6 * q^89 + 24 * q^91 - q^93 - 5 * q^95 - 12 * q^97 - 40 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.31203	1.91220	0.956101	−	0.293039i	$-0.0946665\pi$
$3$	3.31203	1.91220	0.956101	+	0.293039i	$0.0946665\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.94649	−0.735703	−0.367851	−	0.929885i	$-0.619906\pi$
$7$	−1.94649	−0.735703	−0.367851	+	0.929885i	$0.619906\pi$
$8$	0	0
$9$	7.96954	2.65651
$10$	0	0
$11$	−2.17597	−0.656080	−0.328040	−	0.944664i	$-0.606388\pi$
$11$	−2.17597	−0.656080	−0.328040	+	0.944664i	$0.606388\pi$
$12$	0	0
$13$	4.63020	1.28419	0.642093	−	0.766627i	$-0.278066\pi$
$13$	4.63020	1.28419	0.642093	+	0.766627i	$0.278066\pi$
$14$	0	0
$15$	3.31203	0.855162
$16$	0	0
$17$	7.48800	1.81611	0.908054	−	0.418854i	$-0.137568\pi$
$17$	7.48800	1.81611	0.908054	+	0.418854i	$0.137568\pi$
$18$	0	0
$19$	−3.83663	−0.880184	−0.440092	−	0.897953i	$-0.645054\pi$
$19$	−3.83663	−0.880184	−0.440092	+	0.897953i	$0.645054\pi$
$20$	0	0
$21$	−6.44682	−1.40681
$22$	0	0
$23$	−5.53362	−1.15384	−0.576920	−	0.816801i	$-0.695745\pi$
$23$	−5.53362	−1.15384	−0.576920	+	0.816801i	$0.695745\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	16.4593	3.16759
$28$	0	0
$29$	−9.21290	−1.71079	−0.855396	−	0.517974i	$-0.826686\pi$
$29$	−9.21290	−1.71079	−0.855396	+	0.517974i	$0.826686\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−7.20688	−1.25456
$34$	0	0
$35$	−1.94649	−0.329016
$36$	0	0
$37$	4.12561	0.678246	0.339123	−	0.940742i	$-0.389870\pi$
$37$	4.12561	0.678246	0.339123	+	0.940742i	$0.389870\pi$
$38$	0	0
$39$	15.3354	2.45562
$40$	0	0
$41$	9.37025	1.46339	0.731694	−	0.681634i	$-0.238730\pi$
$41$	9.37025	1.46339	0.731694	+	0.681634i	$0.238730\pi$
$42$	0	0
$43$	−12.1717	−1.85617	−0.928085	−	0.372369i	$-0.878545\pi$
$43$	−12.1717	−1.85617	−0.928085	+	0.372369i	$0.878545\pi$
$44$	0	0
$45$	7.96954	1.18803
$46$	0	0
$47$	8.39330	1.22429	0.612145	−	0.790746i	$-0.290307\pi$
$47$	8.39330	1.22429	0.612145	+	0.790746i	$0.290307\pi$
$48$	0	0
$49$	−3.21119	−0.458742
$50$	0	0
$51$	24.8005	3.47276
$52$	0	0
$53$	−5.20500	−0.714962	−0.357481	−	0.933920i	$-0.616364\pi$
$53$	−5.20500	−0.714962	−0.357481	+	0.933920i	$0.616364\pi$
$54$	0	0
$55$	−2.17597	−0.293408
$56$	0	0
$57$	−12.7070	−1.68309
$58$	0	0
$59$	7.43592	0.968074	0.484037	−	0.875047i	$-0.339170\pi$
$59$	7.43592	0.968074	0.484037	+	0.875047i	$0.339170\pi$
$60$	0	0
$61$	−9.03566	−1.15690	−0.578449	−	0.815719i	$-0.696342\pi$
$61$	−9.03566	−1.15690	−0.578449	+	0.815719i	$0.696342\pi$
$62$	0	0
$63$	−15.5126	−1.95440
$64$	0	0
$65$	4.63020	0.574306
$66$	0	0
$67$	10.1163	1.23590	0.617952	−	0.786215i	$-0.287962\pi$
$67$	10.1163	1.23590	0.617952	+	0.786215i	$0.287962\pi$
$68$	0	0
$69$	−18.3275	−2.20637
$70$	0	0
$71$	−6.46069	−0.766743	−0.383372	−	0.923594i	$-0.625237\pi$
$71$	−6.46069	−0.766743	−0.383372	+	0.923594i	$0.625237\pi$
$72$	0	0
$73$	9.26466	1.08435	0.542173	−	0.840267i	$-0.317602\pi$
$73$	9.26466	1.08435	0.542173	+	0.840267i	$0.317602\pi$
$74$	0	0
$75$	3.31203	0.382440
$76$	0	0
$77$	4.23550	0.482680
$78$	0	0
$79$	3.71573	0.418053	0.209026	−	0.977910i	$-0.432971\pi$
$79$	3.71573	0.418053	0.209026	+	0.977910i	$0.432971\pi$
$80$	0	0
$81$	30.6049	3.40055
$82$	0	0
$83$	−11.1225	−1.22085	−0.610427	−	0.792073i	$-0.709002\pi$
$83$	−11.1225	−1.22085	−0.610427	+	0.792073i	$0.709002\pi$
$84$	0	0
$85$	7.48800	0.812188
$86$	0	0
$87$	−30.5134	−3.27138
$88$	0	0
$89$	−5.81048	−0.615910	−0.307955	−	0.951401i	$-0.599645\pi$
$89$	−5.81048	−0.615910	−0.307955	+	0.951401i	$0.599645\pi$
$90$	0	0
$91$	−9.01262	−0.944779
$92$	0	0
$93$	3.31203	0.343441
$94$	0	0
$95$	−3.83663	−0.393630
$96$	0	0
$97$	−12.6991	−1.28940	−0.644701	−	0.764435i	$-0.723018\pi$
$97$	−12.6991	−1.28940	−0.644701	+	0.764435i	$0.723018\pi$
$98$	0	0
$99$	−17.3415	−1.74289
$100$	0	0
$101$	−12.3537	−1.22924	−0.614618	−	0.788825i	$-0.710690\pi$
$101$	−12.3537	−1.22924	−0.614618	+	0.788825i	$0.710690\pi$
$102$	0	0
$103$	−4.21119	−0.414941	−0.207471	−	0.978241i	$-0.566523\pi$
$103$	−4.21119	−0.414941	−0.207471	+	0.978241i	$0.566523\pi$
$104$	0	0
$105$	−6.44682	−0.629145
$106$	0	0
$107$	2.16509	0.209307	0.104653	−	0.994509i	$-0.466627\pi$
$107$	2.16509	0.209307	0.104653	+	0.994509i	$0.466627\pi$
$108$	0	0
$109$	7.77571	0.744778	0.372389	−	0.928077i	$-0.378539\pi$
$109$	7.77571	0.744778	0.372389	+	0.928077i	$0.378539\pi$
$110$	0	0
$111$	13.6641	1.29694
$112$	0	0
$113$	5.76912	0.542713	0.271357	−	0.962479i	$-0.412528\pi$
$113$	5.76912	0.542713	0.271357	+	0.962479i	$0.412528\pi$
$114$	0	0
$115$	−5.53362	−0.516013
$116$	0	0
$117$	36.9006	3.41146
$118$	0	0
$119$	−14.5753	−1.33611
$120$	0	0
$121$	−6.26514	−0.569559
$122$	0	0
$123$	31.0346	2.79829
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−5.31015	−0.471199	−0.235600	−	0.971850i	$-0.575705\pi$
$127$	−5.31015	−0.471199	−0.235600	+	0.971850i	$0.575705\pi$
$128$	0	0
$129$	−40.3131	−3.54937
$130$	0	0
$131$	−11.1394	−0.973251	−0.486626	−	0.873611i	$-0.661772\pi$
$131$	−11.1394	−0.973251	−0.486626	+	0.873611i	$0.661772\pi$
$132$	0	0
$133$	7.46795	0.647554
$134$	0	0
$135$	16.4593	1.41659
$136$	0	0
$137$	−2.85179	−0.243645	−0.121822	−	0.992552i	$-0.538874\pi$
$137$	−2.85179	−0.243645	−0.121822	+	0.992552i	$0.538874\pi$
$138$	0	0
$139$	4.56453	0.387159	0.193579	−	0.981085i	$-0.437990\pi$
$139$	4.56453	0.387159	0.193579	+	0.981085i	$0.437990\pi$
$140$	0	0
$141$	27.7989	2.34109
$142$	0	0
$143$	−10.0752	−0.842529
$144$	0	0
$145$	−9.21290	−0.765090
$146$	0	0
$147$	−10.6356	−0.877206
$148$	0	0
$149$	−12.5032	−1.02430	−0.512149	−	0.858896i	$-0.671151\pi$
$149$	−12.5032	−1.02430	−0.512149	+	0.858896i	$0.671151\pi$
$150$	0	0
$151$	−12.7527	−1.03780	−0.518898	−	0.854836i	$-0.673658\pi$
$151$	−12.7527	−1.03780	−0.518898	+	0.854836i	$0.673658\pi$
$152$	0	0
$153$	59.6759	4.82451
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−5.16552	−0.412254	−0.206127	−	0.978525i	$-0.566086\pi$
$157$	−5.16552	−0.412254	−0.206127	+	0.978525i	$0.566086\pi$
$158$	0	0
$159$	−17.2391	−1.36715
$160$	0	0
$161$	10.7711	0.848883
$162$	0	0
$163$	−3.88356	−0.304184	−0.152092	−	0.988366i	$-0.548601\pi$
$163$	−3.88356	−0.304184	−0.152092	+	0.988366i	$0.548601\pi$
$164$	0	0
$165$	−7.20688	−0.561055
$166$	0	0
$167$	−3.40817	−0.263732	−0.131866	−	0.991268i	$-0.542097\pi$
$167$	−3.40817	−0.263732	−0.131866	+	0.991268i	$0.542097\pi$
$168$	0	0
$169$	8.43875	0.649134
$170$	0	0
$171$	−30.5762	−2.33822
$172$	0	0
$173$	−1.27255	−0.0967504	−0.0483752	−	0.998829i	$-0.515404\pi$
$173$	−1.27255	−0.0967504	−0.0483752	+	0.998829i	$0.515404\pi$
$174$	0	0
$175$	−1.94649	−0.147141
$176$	0	0
$177$	24.6280	1.85115
$178$	0	0
$179$	10.6012	0.792372	0.396186	−	0.918170i	$-0.370333\pi$
$179$	10.6012	0.792372	0.396186	+	0.918170i	$0.370333\pi$
$180$	0	0
$181$	−2.16382	−0.160835	−0.0804177	−	0.996761i	$-0.525625\pi$
$181$	−2.16382	−0.160835	−0.0804177	+	0.996761i	$0.525625\pi$
$182$	0	0
$183$	−29.9264	−2.21222
$184$	0	0
$185$	4.12561	0.303321
$186$	0	0
$187$	−16.2937	−1.19151
$188$	0	0
$189$	−32.0377	−2.33040
$190$	0	0
$191$	17.0720	1.23529	0.617643	−	0.786459i	$-0.288088\pi$
$191$	17.0720	1.23529	0.617643	+	0.786459i	$0.288088\pi$
$192$	0	0
$193$	−6.10703	−0.439593	−0.219797	−	0.975546i	$-0.570539\pi$
$193$	−6.10703	−0.439593	−0.219797	+	0.975546i	$0.570539\pi$
$194$	0	0
$195$	15.3354	1.09819
$196$	0	0
$197$	19.5904	1.39576	0.697879	−	0.716215i	$-0.254127\pi$
$197$	19.5904	1.39576	0.697879	+	0.716215i	$0.254127\pi$
$198$	0	0
$199$	−6.80130	−0.482131	−0.241066	−	0.970509i	$-0.577497\pi$
$199$	−6.80130	−0.482131	−0.241066	+	0.970509i	$0.577497\pi$
$200$	0	0
$201$	33.5055	2.36330
$202$	0	0
$203$	17.9328	1.25863
$204$	0	0
$205$	9.37025	0.654447
$206$	0	0
$207$	−44.1004	−3.06519
$208$	0	0
$209$	8.34841	0.577471
$210$	0	0
$211$	21.7165	1.49503	0.747513	−	0.664247i	$-0.231248\pi$
$211$	21.7165	1.49503	0.747513	+	0.664247i	$0.231248\pi$
$212$	0	0
$213$	−21.3980	−1.46617
$214$	0	0
$215$	−12.1717	−0.830104
$216$	0	0
$217$	−1.94649	−0.132136
$218$	0	0
$219$	30.6848	2.07349
$220$	0	0
$221$	34.6709	2.33222
$222$	0	0
$223$	−17.9929	−1.20489	−0.602446	−	0.798160i	$-0.705807\pi$
$223$	−17.9929	−1.20489	−0.602446	+	0.798160i	$0.705807\pi$
$224$	0	0
$225$	7.96954	0.531303
$226$	0	0
$227$	21.5018	1.42712	0.713562	−	0.700593i	$-0.247081\pi$
$227$	21.5018	1.42712	0.713562	+	0.700593i	$0.247081\pi$
$228$	0	0
$229$	9.56484	0.632063	0.316031	−	0.948749i	$-0.397650\pi$
$229$	9.56484	0.632063	0.316031	+	0.948749i	$0.397650\pi$
$230$	0	0
$231$	14.0281	0.922981
$232$	0	0
$233$	−13.4266	−0.879605	−0.439803	−	0.898094i	$-0.644952\pi$
$233$	−13.4266	−0.879605	−0.439803	+	0.898094i	$0.644952\pi$
$234$	0	0
$235$	8.39330	0.547519
$236$	0	0
$237$	12.3066	0.799400
$238$	0	0
$239$	12.9337	0.836609	0.418305	−	0.908307i	$-0.362624\pi$
$239$	12.9337	0.836609	0.418305	+	0.908307i	$0.362624\pi$
$240$	0	0
$241$	−2.62419	−0.169039	−0.0845193	−	0.996422i	$-0.526935\pi$
$241$	−2.62419	−0.169039	−0.0845193	+	0.996422i	$0.526935\pi$
$242$	0	0
$243$	51.9867	3.33495
$244$	0	0
$245$	−3.21119	−0.205156
$246$	0	0
$247$	−17.7644	−1.13032
$248$	0	0
$249$	−36.8381	−2.33452
$250$	0	0
$251$	−4.17961	−0.263815	−0.131907	−	0.991262i	$-0.542110\pi$
$251$	−4.17961	−0.263815	−0.131907	+	0.991262i	$0.542110\pi$
$252$	0	0
$253$	12.0410	0.757011
$254$	0	0
$255$	24.8005	1.55307
$256$	0	0
$257$	16.4746	1.02766	0.513828	−	0.857894i	$-0.328227\pi$
$257$	16.4746	1.02766	0.513828	+	0.857894i	$0.328227\pi$
$258$	0	0
$259$	−8.03044	−0.498987
$260$	0	0
$261$	−73.4225	−4.54474
$262$	0	0
$263$	14.0014	0.863363	0.431681	−	0.902026i	$-0.357920\pi$
$263$	14.0014	0.863363	0.431681	+	0.902026i	$0.357920\pi$
$264$	0	0
$265$	−5.20500	−0.319741
$266$	0	0
$267$	−19.2445	−1.17774
$268$	0	0
$269$	−5.09645	−0.310736	−0.155368	−	0.987857i	$-0.549656\pi$
$269$	−5.09645	−0.310736	−0.155368	+	0.987857i	$0.549656\pi$
$270$	0	0
$271$	−18.5718	−1.12816	−0.564078	−	0.825722i	$-0.690768\pi$
$271$	−18.5718	−1.12816	−0.564078	+	0.825722i	$0.690768\pi$
$272$	0	0
$273$	−29.8501	−1.80661
$274$	0	0
$275$	−2.17597	−0.131216
$276$	0	0
$277$	−15.0434	−0.903867	−0.451934	−	0.892052i	$-0.649266\pi$
$277$	−15.0434	−0.903867	−0.451934	+	0.892052i	$0.649266\pi$
$278$	0	0
$279$	7.96954	0.477124
$280$	0	0
$281$	4.35653	0.259889	0.129944	−	0.991521i	$-0.458520\pi$
$281$	4.35653	0.259889	0.129944	+	0.991521i	$0.458520\pi$
$282$	0	0
$283$	−26.3103	−1.56398	−0.781991	−	0.623290i	$-0.785796\pi$
$283$	−26.3103	−1.56398	−0.781991	+	0.623290i	$0.785796\pi$
$284$	0	0
$285$	−12.7070	−0.752700
$286$	0	0
$287$	−18.2391	−1.07662
$288$	0	0
$289$	39.0702	2.29825
$290$	0	0
$291$	−42.0599	−2.46560
$292$	0	0
$293$	−28.9092	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$293$	−28.9092	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$294$	0	0
$295$	7.43592	0.432936
$296$	0	0
$297$	−35.8149	−2.07819
$298$	0	0
$299$	−25.6218	−1.48174
$300$	0	0
$301$	23.6921	1.36559
$302$	0	0
$303$	−40.9157	−2.35055
$304$	0	0
$305$	−9.03566	−0.517380
$306$	0	0
$307$	−12.1650	−0.694291	−0.347145	−	0.937811i	$-0.612849\pi$
$307$	−12.1650	−0.694291	−0.347145	+	0.937811i	$0.612849\pi$
$308$	0	0
$309$	−13.9476	−0.793451
$310$	0	0
$311$	12.4058	0.703471	0.351735	−	0.936099i	$-0.385592\pi$
$311$	12.4058	0.703471	0.351735	+	0.936099i	$0.385592\pi$
$312$	0	0
$313$	−13.7236	−0.775705	−0.387853	−	0.921721i	$-0.626783\pi$
$313$	−13.7236	−0.775705	−0.387853	+	0.921721i	$0.626783\pi$
$314$	0	0
$315$	−15.5126	−0.874036
$316$	0	0
$317$	−13.4350	−0.754583	−0.377292	−	0.926094i	$-0.623145\pi$
$317$	−13.4350	−0.754583	−0.377292	+	0.926094i	$0.623145\pi$
$318$	0	0
$319$	20.0470	1.12242
$320$	0	0
$321$	7.17083	0.400237
$322$	0	0
$323$	−28.7287	−1.59851
$324$	0	0
$325$	4.63020	0.256837
$326$	0	0
$327$	25.7534	1.42417
$328$	0	0
$329$	−16.3375	−0.900713
$330$	0	0
$331$	4.52804	0.248884	0.124442	−	0.992227i	$-0.460286\pi$
$331$	4.52804	0.248884	0.124442	+	0.992227i	$0.460286\pi$
$332$	0	0
$333$	32.8792	1.80177
$334$	0	0
$335$	10.1163	0.552713
$336$	0	0
$337$	−20.2494	−1.10306	−0.551528	−	0.834157i	$-0.685955\pi$
$337$	−20.2494	−1.10306	−0.551528	+	0.834157i	$0.685955\pi$
$338$	0	0
$339$	19.1075	1.03778
$340$	0	0
$341$	−2.17597	−0.117836
$342$	0	0
$343$	19.8759	1.07320
$344$	0	0
$345$	−18.3275	−0.986720
$346$	0	0
$347$	24.9127	1.33738	0.668692	−	0.743539i	$-0.266854\pi$
$347$	24.9127	1.33738	0.668692	+	0.743539i	$0.266854\pi$
$348$	0	0
$349$	6.01134	0.321780	0.160890	−	0.986972i	$-0.448564\pi$
$349$	6.01134	0.321780	0.160890	+	0.986972i	$0.448564\pi$
$350$	0	0
$351$	76.2097	4.06777
$352$	0	0
$353$	20.1090	1.07030	0.535148	−	0.844758i	$-0.320256\pi$
$353$	20.1090	1.07030	0.535148	+	0.844758i	$0.320256\pi$
$354$	0	0
$355$	−6.46069	−0.342898
$356$	0	0
$357$	−48.2738	−2.55492
$358$	0	0
$359$	−9.47026	−0.499821	−0.249911	−	0.968269i	$-0.580401\pi$
$359$	−9.47026	−0.499821	−0.249911	+	0.968269i	$0.580401\pi$
$360$	0	0
$361$	−4.28025	−0.225276
$362$	0	0
$363$	−20.7503	−1.08911
$364$	0	0
$365$	9.26466	0.484934
$366$	0	0
$367$	17.2254	0.899159	0.449579	−	0.893240i	$-0.351574\pi$
$367$	17.2254	0.899159	0.449579	+	0.893240i	$0.351574\pi$
$368$	0	0
$369$	74.6766	3.88751
$370$	0	0
$371$	10.1315	0.525999
$372$	0	0
$373$	−18.4712	−0.956400	−0.478200	−	0.878251i	$-0.658711\pi$
$373$	−18.4712	−0.956400	−0.478200	+	0.878251i	$0.658711\pi$
$374$	0	0
$375$	3.31203	0.171032
$376$	0	0
$377$	−42.6576	−2.19698
$378$	0	0
$379$	−12.7605	−0.655461	−0.327730	−	0.944771i	$-0.606284\pi$
$379$	−12.7605	−0.655461	−0.327730	+	0.944771i	$0.606284\pi$
$380$	0	0
$381$	−17.5874	−0.901028
$382$	0	0
$383$	20.3657	1.04064	0.520320	−	0.853971i	$-0.325813\pi$
$383$	20.3657	1.04064	0.520320	+	0.853971i	$0.325813\pi$
$384$	0	0
$385$	4.23550	0.215861
$386$	0	0
$387$	−97.0030	−4.93094
$388$	0	0
$389$	−11.5556	−0.585890	−0.292945	−	0.956129i	$-0.594635\pi$
$389$	−11.5556	−0.585890	−0.292945	+	0.956129i	$0.594635\pi$
$390$	0	0
$391$	−41.4358	−2.09550
$392$	0	0
$393$	−36.8939	−1.86105
$394$	0	0
$395$	3.71573	0.186959
$396$	0	0
$397$	9.52621	0.478107	0.239053	−	0.971006i	$-0.423163\pi$
$397$	9.52621	0.478107	0.239053	+	0.971006i	$0.423163\pi$
$398$	0	0
$399$	24.7341	1.23825
$400$	0	0
$401$	18.2814	0.912930	0.456465	−	0.889741i	$-0.349115\pi$
$401$	18.2814	0.912930	0.456465	+	0.889741i	$0.349115\pi$
$402$	0	0
$403$	4.63020	0.230647
$404$	0	0
$405$	30.6049	1.52077
$406$	0	0
$407$	−8.97721	−0.444984
$408$	0	0
$409$	−17.4082	−0.860781	−0.430391	−	0.902643i	$-0.641624\pi$
$409$	−17.4082	−0.860781	−0.430391	+	0.902643i	$0.641624\pi$
$410$	0	0
$411$	−9.44521	−0.465898
$412$	0	0
$413$	−14.4739	−0.712215
$414$	0	0
$415$	−11.1225	−0.545982
$416$	0	0
$417$	15.1179	0.740325
$418$	0	0
$419$	31.2133	1.52487	0.762436	−	0.647064i	$-0.224003\pi$
$419$	31.2133	1.52487	0.762436	+	0.647064i	$0.224003\pi$
$420$	0	0
$421$	12.0417	0.586874	0.293437	−	0.955978i	$-0.405201\pi$
$421$	12.0417	0.586874	0.293437	+	0.955978i	$0.405201\pi$
$422$	0	0
$423$	66.8908	3.25234
$424$	0	0
$425$	7.48800	0.363221
$426$	0	0
$427$	17.5878	0.851132
$428$	0	0
$429$	−33.3693	−1.61109
$430$	0	0
$431$	37.4995	1.80629	0.903144	−	0.429338i	$-0.141253\pi$
$431$	37.4995	1.80629	0.903144	+	0.429338i	$0.141253\pi$
$432$	0	0
$433$	10.4844	0.503849	0.251924	−	0.967747i	$-0.418937\pi$
$433$	10.4844	0.503849	0.251924	+	0.967747i	$0.418937\pi$
$434$	0	0
$435$	−30.5134	−1.46301
$436$	0	0
$437$	21.2305	1.01559
$438$	0	0
$439$	12.0767	0.576388	0.288194	−	0.957572i	$-0.406945\pi$
$439$	12.0767	0.576388	0.288194	+	0.957572i	$0.406945\pi$
$440$	0	0
$441$	−25.5917	−1.21865
$442$	0	0
$443$	−16.5231	−0.785037	−0.392518	−	0.919744i	$-0.628396\pi$
$443$	−16.5231	−0.785037	−0.392518	+	0.919744i	$0.628396\pi$
$444$	0	0
$445$	−5.81048	−0.275443
$446$	0	0
$447$	−41.4108	−1.95866
$448$	0	0
$449$	20.9700	0.989637	0.494818	−	0.868996i	$-0.335235\pi$
$449$	20.9700	0.989637	0.494818	+	0.868996i	$0.335235\pi$
$450$	0	0
$451$	−20.3894	−0.960100
$452$	0	0
$453$	−42.2372	−1.98448
$454$	0	0
$455$	−9.01262	−0.422518
$456$	0	0
$457$	−4.01404	−0.187769	−0.0938843	−	0.995583i	$-0.529928\pi$
$457$	−4.01404	−0.187769	−0.0938843	+	0.995583i	$0.529928\pi$
$458$	0	0
$459$	123.247	5.75268
$460$	0	0
$461$	29.0945	1.35507	0.677533	−	0.735493i	$-0.263049\pi$
$461$	29.0945	1.35507	0.677533	+	0.735493i	$0.263049\pi$
$462$	0	0
$463$	12.4089	0.576690	0.288345	−	0.957527i	$-0.406895\pi$
$463$	12.4089	0.576690	0.288345	+	0.957527i	$0.406895\pi$
$464$	0	0
$465$	3.31203	0.153592
$466$	0	0
$467$	−4.09221	−0.189365	−0.0946825	−	0.995508i	$-0.530184\pi$
$467$	−4.09221	−0.189365	−0.0946825	+	0.995508i	$0.530184\pi$
$468$	0	0
$469$	−19.6913	−0.909258
$470$	0	0
$471$	−17.1084	−0.788312
$472$	0	0
$473$	26.4853	1.21780
$474$	0	0
$475$	−3.83663	−0.176037
$476$	0	0
$477$	−41.4815	−1.89931
$478$	0	0
$479$	1.96969	0.0899973	0.0449986	−	0.998987i	$-0.485672\pi$
$479$	1.96969	0.0899973	0.0449986	+	0.998987i	$0.485672\pi$
$480$	0	0
$481$	19.1024	0.870994
$482$	0	0
$483$	35.6742	1.62323
$484$	0	0
$485$	−12.6991	−0.576638
$486$	0	0
$487$	−5.13960	−0.232897	−0.116449	−	0.993197i	$-0.537151\pi$
$487$	−5.13960	−0.232897	−0.116449	+	0.993197i	$0.537151\pi$
$488$	0	0
$489$	−12.8624	−0.581660
$490$	0	0
$491$	−21.6597	−0.977489	−0.488745	−	0.872427i	$-0.662545\pi$
$491$	−21.6597	−0.977489	−0.488745	+	0.872427i	$0.662545\pi$
$492$	0	0
$493$	−68.9862	−3.10698
$494$	0	0
$495$	−17.3415	−0.779442
$496$	0	0
$497$	12.5756	0.564095
$498$	0	0
$499$	23.4911	1.05161	0.525803	−	0.850606i	$-0.323765\pi$
$499$	23.4911	1.05161	0.525803	+	0.850606i	$0.323765\pi$
$500$	0	0
$501$	−11.2880	−0.504309
$502$	0	0
$503$	34.6823	1.54641	0.773204	−	0.634157i	$-0.218653\pi$
$503$	34.6823	1.54641	0.773204	+	0.634157i	$0.218653\pi$
$504$	0	0
$505$	−12.3537	−0.549731
$506$	0	0
$507$	27.9494	1.24128
$508$	0	0
$509$	37.8462	1.67751	0.838753	−	0.544513i	$-0.183285\pi$
$509$	37.8462	1.67751	0.838753	+	0.544513i	$0.183285\pi$
$510$	0	0
$511$	−18.0335	−0.797756
$512$	0	0
$513$	−63.1481	−2.78806
$514$	0	0
$515$	−4.21119	−0.185567
$516$	0	0
$517$	−18.2636	−0.803232
$518$	0	0
$519$	−4.21473	−0.185006
$520$	0	0
$521$	8.80322	0.385676	0.192838	−	0.981231i	$-0.438231\pi$
$521$	8.80322	0.385676	0.192838	+	0.981231i	$0.438231\pi$
$522$	0	0
$523$	−19.9176	−0.870937	−0.435469	−	0.900204i	$-0.643417\pi$
$523$	−19.9176	−0.870937	−0.435469	+	0.900204i	$0.643417\pi$
$524$	0	0
$525$	−6.44682	−0.281362
$526$	0	0
$527$	7.48800	0.326182
$528$	0	0
$529$	7.62094	0.331345
$530$	0	0
$531$	59.2609	2.57170
$532$	0	0
$533$	43.3861	1.87926
$534$	0	0
$535$	2.16509	0.0936048
$536$	0	0
$537$	35.1116	1.51518
$538$	0	0
$539$	6.98747	0.300971
$540$	0	0
$541$	33.7566	1.45131	0.725656	−	0.688058i	$-0.241537\pi$
$541$	33.7566	1.45131	0.725656	+	0.688058i	$0.241537\pi$
$542$	0	0
$543$	−7.16663	−0.307550
$544$	0	0
$545$	7.77571	0.333075
$546$	0	0
$547$	−29.3038	−1.25294	−0.626469	−	0.779446i	$-0.715501\pi$
$547$	−29.3038	−1.25294	−0.626469	+	0.779446i	$0.715501\pi$
$548$	0	0
$549$	−72.0100	−3.07331
$550$	0	0
$551$	35.3465	1.50581
$552$	0	0
$553$	−7.23262	−0.307562
$554$	0	0
$555$	13.6641	0.580010
$556$	0	0
$557$	1.43423	0.0607705	0.0303852	−	0.999538i	$-0.490327\pi$
$557$	1.43423	0.0607705	0.0303852	+	0.999538i	$0.490327\pi$
$558$	0	0
$559$	−56.3575	−2.38367
$560$	0	0
$561$	−53.9652	−2.27841
$562$	0	0
$563$	−19.3861	−0.817028	−0.408514	−	0.912752i	$-0.633953\pi$
$563$	−19.3861	−0.817028	−0.408514	+	0.912752i	$0.633953\pi$
$564$	0	0
$565$	5.76912	0.242709
$566$	0	0
$567$	−59.5721	−2.50179
$568$	0	0
$569$	−35.7835	−1.50012	−0.750061	−	0.661368i	$-0.769976\pi$
$569$	−35.7835	−1.50012	−0.750061	+	0.661368i	$0.769976\pi$
$570$	0	0
$571$	44.0490	1.84339	0.921696	−	0.387914i	$-0.126804\pi$
$571$	44.0490	1.84339	0.921696	+	0.387914i	$0.126804\pi$
$572$	0	0
$573$	56.5429	2.36211
$574$	0	0
$575$	−5.53362	−0.230768
$576$	0	0
$577$	30.0326	1.25027	0.625137	−	0.780515i	$-0.285043\pi$
$577$	30.0326	1.25027	0.625137	+	0.780515i	$0.285043\pi$
$578$	0	0
$579$	−20.2267	−0.840591
$580$	0	0
$581$	21.6498	0.898185
$582$	0	0
$583$	11.3259	0.469072
$584$	0	0
$585$	36.9006	1.52565
$586$	0	0
$587$	0.699636	0.0288771	0.0144385	−	0.999896i	$-0.495404\pi$
$587$	0.699636	0.0288771	0.0144385	+	0.999896i	$0.495404\pi$
$588$	0	0
$589$	−3.83663	−0.158086
$590$	0	0
$591$	64.8840	2.66897
$592$	0	0
$593$	14.4909	0.595071	0.297535	−	0.954711i	$-0.403835\pi$
$593$	14.4909	0.595071	0.297535	+	0.954711i	$0.403835\pi$
$594$	0	0
$595$	−14.5753	−0.597529
$596$	0	0
$597$	−22.5261	−0.921932
$598$	0	0
$599$	14.5490	0.594455	0.297227	−	0.954807i	$-0.403938\pi$
$599$	14.5490	0.594455	0.297227	+	0.954807i	$0.403938\pi$
$600$	0	0
$601$	7.09797	0.289532	0.144766	−	0.989466i	$-0.453757\pi$
$601$	7.09797	0.289532	0.144766	+	0.989466i	$0.453757\pi$
$602$	0	0
$603$	80.6224	3.28320
$604$	0	0
$605$	−6.26514	−0.254714
$606$	0	0
$607$	−27.6801	−1.12350	−0.561751	−	0.827307i	$-0.689872\pi$
$607$	−27.6801	−1.12350	−0.561751	+	0.827307i	$0.689872\pi$
$608$	0	0
$609$	59.3939	2.40676
$610$	0	0
$611$	38.8627	1.57222
$612$	0	0
$613$	0.697261	0.0281621	0.0140810	−	0.999901i	$-0.495518\pi$
$613$	0.697261	0.0281621	0.0140810	+	0.999901i	$0.495518\pi$
$614$	0	0
$615$	31.0346	1.25143
$616$	0	0
$617$	−22.2947	−0.897549	−0.448775	−	0.893645i	$-0.648139\pi$
$617$	−22.2947	−0.897549	−0.448775	+	0.893645i	$0.648139\pi$
$618$	0	0
$619$	−32.8750	−1.32136	−0.660679	−	0.750669i	$-0.729731\pi$
$619$	−32.8750	−1.32136	−0.660679	+	0.750669i	$0.729731\pi$
$620$	0	0
$621$	−91.0793	−3.65489
$622$	0	0
$623$	11.3100	0.453126
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	27.6502	1.10424
$628$	0	0
$629$	30.8926	1.23177
$630$	0	0
$631$	29.1361	1.15989	0.579945	−	0.814656i	$-0.303074\pi$
$631$	29.1361	1.15989	0.579945	+	0.814656i	$0.303074\pi$
$632$	0	0
$633$	71.9257	2.85879
$634$	0	0
$635$	−5.31015	−0.210727
$636$	0	0
$637$	−14.8685	−0.589110
$638$	0	0
$639$	−51.4887	−2.03686
$640$	0	0
$641$	−7.95110	−0.314050	−0.157025	−	0.987595i	$-0.550190\pi$
$641$	−7.95110	−0.314050	−0.157025	+	0.987595i	$0.550190\pi$
$642$	0	0
$643$	−28.9037	−1.13985	−0.569925	−	0.821697i	$-0.693028\pi$
$643$	−28.9037	−1.13985	−0.569925	+	0.821697i	$0.693028\pi$
$644$	0	0
$645$	−40.3131	−1.58733
$646$	0	0
$647$	14.0622	0.552842	0.276421	−	0.961037i	$-0.410852\pi$
$647$	14.0622	0.552842	0.276421	+	0.961037i	$0.410852\pi$
$648$	0	0
$649$	−16.1804	−0.635135
$650$	0	0
$651$	−6.44682	−0.252671
$652$	0	0
$653$	40.1717	1.57204	0.786020	−	0.618201i	$-0.212138\pi$
$653$	40.1717	1.57204	0.786020	+	0.618201i	$0.212138\pi$
$654$	0	0
$655$	−11.1394	−0.435251
$656$	0	0
$657$	73.8350	2.88058
$658$	0	0
$659$	−4.87356	−0.189847	−0.0949234	−	0.995485i	$-0.530261\pi$
$659$	−4.87356	−0.189847	−0.0949234	+	0.995485i	$0.530261\pi$
$660$	0	0
$661$	−8.88808	−0.345706	−0.172853	−	0.984948i	$-0.555299\pi$
$661$	−8.88808	−0.345706	−0.172853	+	0.984948i	$0.555299\pi$
$662$	0	0
$663$	114.831	4.45967
$664$	0	0
$665$	7.46795	0.289595
$666$	0	0
$667$	50.9807	1.97398
$668$	0	0
$669$	−59.5929	−2.30400
$670$	0	0
$671$	19.6613	0.759018
$672$	0	0
$673$	16.9899	0.654912	0.327456	−	0.944866i	$-0.393809\pi$
$673$	16.9899	0.654912	0.327456	+	0.944866i	$0.393809\pi$
$674$	0	0
$675$	16.4593	0.633517
$676$	0	0
$677$	24.8453	0.954882	0.477441	−	0.878664i	$-0.341564\pi$
$677$	24.8453	0.954882	0.477441	+	0.878664i	$0.341564\pi$
$678$	0	0
$679$	24.7187	0.948617
$680$	0	0
$681$	71.2145	2.72895
$682$	0	0
$683$	30.0980	1.15167	0.575834	−	0.817567i	$-0.304678\pi$
$683$	30.0980	1.15167	0.575834	+	0.817567i	$0.304678\pi$
$684$	0	0
$685$	−2.85179	−0.108961
$686$	0	0
$687$	31.6790	1.20863
$688$	0	0
$689$	−24.1002	−0.918144
$690$	0	0
$691$	−20.1124	−0.765111	−0.382556	−	0.923932i	$-0.624956\pi$
$691$	−20.1124	−0.765111	−0.382556	+	0.923932i	$0.624956\pi$
$692$	0	0
$693$	33.7550	1.28225
$694$	0	0
$695$	4.56453	0.173143
$696$	0	0
$697$	70.1645	2.65767
$698$	0	0
$699$	−44.4693	−1.68198
$700$	0	0
$701$	−2.23289	−0.0843351	−0.0421675	−	0.999111i	$-0.513426\pi$
$701$	−2.23289	−0.0843351	−0.0421675	+	0.999111i	$0.513426\pi$
$702$	0	0
$703$	−15.8284	−0.596981
$704$	0	0
$705$	27.7989	1.04697
$706$	0	0
$707$	24.0462	0.904352
$708$	0	0
$709$	8.90342	0.334375	0.167187	−	0.985925i	$-0.446531\pi$
$709$	8.90342	0.334375	0.167187	+	0.985925i	$0.446531\pi$
$710$	0	0
$711$	29.6127	1.11056
$712$	0	0
$713$	−5.53362	−0.207236
$714$	0	0
$715$	−10.0752	−0.376791
$716$	0	0
$717$	42.8367	1.59976
$718$	0	0
$719$	40.1789	1.49842	0.749210	−	0.662333i	$-0.230433\pi$
$719$	40.1789	1.49842	0.749210	+	0.662333i	$0.230433\pi$
$720$	0	0
$721$	8.19703	0.305273
$722$	0	0
$723$	−8.69138	−0.323236
$724$	0	0
$725$	−9.21290	−0.342158
$726$	0	0
$727$	−10.6515	−0.395042	−0.197521	−	0.980299i	$-0.563289\pi$
$727$	−10.6515	−0.395042	−0.197521	+	0.980299i	$0.563289\pi$
$728$	0	0
$729$	80.3666	2.97654
$730$	0	0
$731$	−91.1418	−3.37100
$732$	0	0
$733$	7.11588	0.262831	0.131416	−	0.991327i	$-0.458048\pi$
$733$	7.11588	0.262831	0.131416	+	0.991327i	$0.458048\pi$
$734$	0	0
$735$	−10.6356	−0.392299
$736$	0	0
$737$	−22.0128	−0.810853
$738$	0	0
$739$	−12.2746	−0.451529	−0.225764	−	0.974182i	$-0.572488\pi$
$739$	−12.2746	−0.451529	−0.225764	+	0.974182i	$0.572488\pi$
$740$	0	0
$741$	−58.8361	−2.16140
$742$	0	0
$743$	−14.3267	−0.525594	−0.262797	−	0.964851i	$-0.584645\pi$
$743$	−14.3267	−0.525594	−0.262797	+	0.964851i	$0.584645\pi$
$744$	0	0
$745$	−12.5032	−0.458080
$746$	0	0
$747$	−88.6413	−3.24321
$748$	0	0
$749$	−4.21431	−0.153988
$750$	0	0
$751$	35.4641	1.29410	0.647051	−	0.762447i	$-0.276002\pi$
$751$	35.4641	1.29410	0.647051	+	0.762447i	$0.276002\pi$
$752$	0	0
$753$	−13.8430	−0.504467
$754$	0	0
$755$	−12.7527	−0.464117
$756$	0	0
$757$	−3.47744	−0.126390	−0.0631949	−	0.998001i	$-0.520129\pi$
$757$	−3.47744	−0.126390	−0.0631949	+	0.998001i	$0.520129\pi$
$758$	0	0
$759$	39.8802	1.44756
$760$	0	0
$761$	−31.9706	−1.15893	−0.579466	−	0.814996i	$-0.696739\pi$
$761$	−31.9706	−1.15893	−0.579466	+	0.814996i	$0.696739\pi$
$762$	0	0
$763$	−15.1353	−0.547935
$764$	0	0
$765$	59.6759	2.15759
$766$	0	0
$767$	34.4298	1.24319
$768$	0	0
$769$	46.0338	1.66002	0.830010	−	0.557748i	$-0.188334\pi$
$769$	46.0338	1.66002	0.830010	+	0.557748i	$0.188334\pi$
$770$	0	0
$771$	54.5642	1.96508
$772$	0	0
$773$	40.7466	1.46555	0.732777	−	0.680468i	$-0.238224\pi$
$773$	40.7466	1.46555	0.732777	+	0.680468i	$0.238224\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−26.5971	−0.954164
$778$	0	0
$779$	−35.9502	−1.28805
$780$	0	0
$781$	14.0583	0.503045
$782$	0	0
$783$	−151.637	−5.41908
$784$	0	0
$785$	−5.16552	−0.184365
$786$	0	0
$787$	−9.21677	−0.328542	−0.164271	−	0.986415i	$-0.552527\pi$
$787$	−9.21677	−0.328542	−0.164271	+	0.986415i	$0.552527\pi$
$788$	0	0
$789$	46.3730	1.65092
$790$	0	0
$791$	−11.2295	−0.399275
$792$	0	0
$793$	−41.8369	−1.48567
$794$	0	0
$795$	−17.2391	−0.611409
$796$	0	0
$797$	−24.0806	−0.852978	−0.426489	−	0.904493i	$-0.640250\pi$
$797$	−24.0806	−0.852978	−0.426489	+	0.904493i	$0.640250\pi$
$798$	0	0
$799$	62.8491	2.22344
$800$	0	0
$801$	−46.3068	−1.63617
$802$	0	0
$803$	−20.1596	−0.711418
$804$	0	0
$805$	10.7711	0.379632
$806$	0	0
$807$	−16.8796	−0.594190
$808$	0	0
$809$	32.3319	1.13673	0.568365	−	0.822777i	$-0.307576\pi$
$809$	32.3319	1.13673	0.568365	+	0.822777i	$0.307576\pi$
$810$	0	0
$811$	−14.3530	−0.504002	−0.252001	−	0.967727i	$-0.581089\pi$
$811$	−14.3530	−0.504002	−0.252001	+	0.967727i	$0.581089\pi$
$812$	0	0
$813$	−61.5103	−2.15726
$814$	0	0
$815$	−3.88356	−0.136035
$816$	0	0
$817$	46.6984	1.63377
$818$	0	0
$819$	−71.8264	−2.50982
$820$	0	0
$821$	1.84557	0.0644109	0.0322055	−	0.999481i	$-0.489747\pi$
$821$	1.84557	0.0644109	0.0322055	+	0.999481i	$0.489747\pi$
$822$	0	0
$823$	−46.0156	−1.60400	−0.802001	−	0.597323i	$-0.796231\pi$
$823$	−46.0156	−1.60400	−0.802001	+	0.597323i	$0.796231\pi$
$824$	0	0
$825$	−7.20688	−0.250912
$826$	0	0
$827$	−16.3369	−0.568091	−0.284045	−	0.958811i	$-0.591677\pi$
$827$	−16.3369	−0.568091	−0.284045	+	0.958811i	$0.591677\pi$
$828$	0	0
$829$	47.6449	1.65478	0.827388	−	0.561631i	$-0.189826\pi$
$829$	47.6449	1.65478	0.827388	+	0.561631i	$0.189826\pi$
$830$	0	0
$831$	−49.8240	−1.72838
$832$	0	0
$833$	−24.0454	−0.833124
$834$	0	0
$835$	−3.40817	−0.117945
$836$	0	0
$837$	16.4593	0.568915
$838$	0	0
$839$	−12.7204	−0.439158	−0.219579	−	0.975595i	$-0.570468\pi$
$839$	−12.7204	−0.439158	−0.219579	+	0.975595i	$0.570468\pi$
$840$	0	0
$841$	55.8775	1.92681
$842$	0	0
$843$	14.4289	0.496959
$844$	0	0
$845$	8.43875	0.290302
$846$	0	0
$847$	12.1950	0.419026
$848$	0	0
$849$	−87.1403	−2.99065
$850$	0	0
$851$	−22.8295	−0.782587
$852$	0	0
$853$	−8.65778	−0.296437	−0.148218	−	0.988955i	$-0.547354\pi$
$853$	−8.65778	−0.296437	−0.148218	+	0.988955i	$0.547354\pi$
$854$	0	0
$855$	−30.5762	−1.04568
$856$	0	0
$857$	33.9276	1.15894	0.579472	−	0.814992i	$-0.303259\pi$
$857$	33.9276	1.15894	0.579472	+	0.814992i	$0.303259\pi$
$858$	0	0
$859$	−21.3257	−0.727623	−0.363811	−	0.931473i	$-0.618525\pi$
$859$	−21.3257	−0.727623	−0.363811	+	0.931473i	$0.618525\pi$
$860$	0	0
$861$	−60.4083	−2.05871
$862$	0	0
$863$	22.4943	0.765716	0.382858	−	0.923807i	$-0.374940\pi$
$863$	22.4943	0.765716	0.382858	+	0.923807i	$0.374940\pi$
$864$	0	0
$865$	−1.27255	−0.0432681
$866$	0	0
$867$	129.402	4.39471
$868$	0	0
$869$	−8.08533	−0.274276
$870$	0	0
$871$	46.8406	1.58713
$872$	0	0
$873$	−101.206	−3.42531
$874$	0	0
$875$	−1.94649	−0.0658032
$876$	0	0
$877$	−19.7833	−0.668034	−0.334017	−	0.942567i	$-0.608404\pi$
$877$	−19.7833	−0.668034	−0.334017	+	0.942567i	$0.608404\pi$
$878$	0	0
$879$	−95.7482	−3.22951
$880$	0	0
$881$	−23.8028	−0.801938	−0.400969	−	0.916092i	$-0.631327\pi$
$881$	−23.8028	−0.801938	−0.400969	+	0.916092i	$0.631327\pi$
$882$	0	0
$883$	8.23309	0.277066	0.138533	−	0.990358i	$-0.455761\pi$
$883$	8.23309	0.277066	0.138533	+	0.990358i	$0.455761\pi$
$884$	0	0
$885$	24.6280	0.827861
$886$	0	0
$887$	4.30132	0.144424	0.0722121	−	0.997389i	$-0.476994\pi$
$887$	4.30132	0.144424	0.0722121	+	0.997389i	$0.476994\pi$
$888$	0	0
$889$	10.3361	0.346663
$890$	0	0
$891$	−66.5955	−2.23103
$892$	0	0
$893$	−32.2020	−1.07760
$894$	0	0
$895$	10.6012	0.354360
$896$	0	0
$897$	−84.8600	−2.83339
$898$	0	0
$899$	−9.21290	−0.307267
$900$	0	0
$901$	−38.9751	−1.29845
$902$	0	0
$903$	78.4688	2.61128
$904$	0	0
$905$	−2.16382	−0.0719278
$906$	0	0
$907$	16.0044	0.531418	0.265709	−	0.964053i	$-0.414394\pi$
$907$	16.0044	0.531418	0.265709	+	0.964053i	$0.414394\pi$
$908$	0	0
$909$	−98.4530	−3.26548
$910$	0	0
$911$	−3.67017	−0.121598	−0.0607991	−	0.998150i	$-0.519365\pi$
$911$	−3.67017	−0.121598	−0.0607991	+	0.998150i	$0.519365\pi$
$912$	0	0
$913$	24.2023	0.800978
$914$	0	0
$915$	−29.9264	−0.989335
$916$	0	0
$917$	21.6826	0.716024
$918$	0	0
$919$	9.58343	0.316128	0.158064	−	0.987429i	$-0.449475\pi$
$919$	9.58343	0.316128	0.158064	+	0.987429i	$0.449475\pi$
$920$	0	0
$921$	−40.2907	−1.32762
$922$	0	0
$923$	−29.9143	−0.984641
$924$	0	0
$925$	4.12561	0.135649
$926$	0	0
$927$	−33.5613	−1.10230
$928$	0	0
$929$	51.4065	1.68659	0.843296	−	0.537450i	$-0.180612\pi$
$929$	51.4065	1.68659	0.843296	+	0.537450i	$0.180612\pi$
$930$	0	0
$931$	12.3202	0.403777
$932$	0	0
$933$	41.0885	1.34518
$934$	0	0
$935$	−16.2937	−0.532860
$936$	0	0
$937$	55.4143	1.81031	0.905153	−	0.425085i	$-0.139756\pi$
$937$	55.4143	1.81031	0.905153	+	0.425085i	$0.139756\pi$
$938$	0	0
$939$	−45.4531	−1.48330
$940$	0	0
$941$	−43.2999	−1.41154	−0.705768	−	0.708443i	$-0.749398\pi$
$941$	−43.2999	−1.41154	−0.705768	+	0.708443i	$0.749398\pi$
$942$	0	0
$943$	−51.8514	−1.68851
$944$	0	0
$945$	−32.0377	−1.04219
$946$	0	0
$947$	16.7956	0.545785	0.272893	−	0.962045i	$-0.412020\pi$
$947$	16.7956	0.545785	0.272893	+	0.962045i	$0.412020\pi$
$948$	0	0
$949$	42.8972	1.39250
$950$	0	0
$951$	−44.4970	−1.44292
$952$	0	0
$953$	−35.5097	−1.15027	−0.575136	−	0.818058i	$-0.695051\pi$
$953$	−35.5097	−1.15027	−0.575136	+	0.818058i	$0.695051\pi$
$954$	0	0
$955$	17.0720	0.552436
$956$	0	0
$957$	66.3963	2.14629
$958$	0	0
$959$	5.55097	0.179250
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	17.2547	0.556026
$964$	0	0
$965$	−6.10703	−0.196592
$966$	0	0
$967$	5.41717	0.174205	0.0871023	−	0.996199i	$-0.472239\pi$
$967$	5.41717	0.174205	0.0871023	+	0.996199i	$0.472239\pi$
$968$	0	0
$969$	−95.1503	−3.05667
$970$	0	0
$971$	−15.9776	−0.512746	−0.256373	−	0.966578i	$-0.582528\pi$
$971$	−15.9776	−0.512746	−0.256373	+	0.966578i	$0.582528\pi$
$972$	0	0
$973$	−8.88480	−0.284834
$974$	0	0
$975$	15.3354	0.491124
$976$	0	0
$977$	−27.3287	−0.874323	−0.437162	−	0.899383i	$-0.644016\pi$
$977$	−27.3287	−0.874323	−0.437162	+	0.899383i	$0.644016\pi$
$978$	0	0
$979$	12.6434	0.404086
$980$	0	0
$981$	61.9688	1.97851
$982$	0	0
$983$	15.7697	0.502974	0.251487	−	0.967861i	$-0.419080\pi$
$983$	15.7697	0.502974	0.251487	+	0.967861i	$0.419080\pi$
$984$	0	0
$985$	19.5904	0.624202
$986$	0	0
$987$	−54.1101	−1.72234
$988$	0	0
$989$	67.3536	2.14172
$990$	0	0
$991$	−52.1485	−1.65655	−0.828276	−	0.560320i	$-0.810678\pi$
$991$	−52.1485	−1.65655	−0.828276	+	0.560320i	$0.810678\pi$
$992$	0	0
$993$	14.9970	0.475916
$994$	0	0
$995$	−6.80130	−0.215616
$996$	0	0
$997$	−52.3565	−1.65815	−0.829073	−	0.559141i	$-0.811131\pi$
$997$	−52.3565	−1.65815	−0.829073	+	0.559141i	$0.811131\pi$
$998$	0	0
$999$	67.9045	2.14840

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1240.2.a.m.1.6	✓	6
4.3	odd	2		2480.2.a.ba.1.1		6
5.4	even	2		6200.2.a.w.1.1		6
8.3	odd	2		9920.2.a.co.1.6		6
8.5	even	2		9920.2.a.cp.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1240.2.a.m.1.6	✓	6	1.1	even	1	trivial
2480.2.a.ba.1.1		6	4.3	odd	2
6200.2.a.w.1.1		6	5.4	even	2
9920.2.a.co.1.6		6	8.3	odd	2
9920.2.a.cp.1.1		6	8.5	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:	\( N \)	\(=\)	\( 1240 = 2^{3} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1240.a (trivial)

\(f(q)\)	\(=\)	\(q+3.31203 q^{3} +1.00000 q^{5} -1.94649 q^{7} +7.96954 q^{9} -2.17597 q^{11} +4.63020 q^{13} +3.31203 q^{15} +7.48800 q^{17} -3.83663 q^{19} -6.44682 q^{21} -5.53362 q^{23} +1.00000 q^{25} +16.4593 q^{27} -9.21290 q^{29} +1.00000 q^{31} -7.20688 q^{33} -1.94649 q^{35} +4.12561 q^{37} +15.3354 q^{39} +9.37025 q^{41} -12.1717 q^{43} +7.96954 q^{45} +8.39330 q^{47} -3.21119 q^{49} +24.8005 q^{51} -5.20500 q^{53} -2.17597 q^{55} -12.7070 q^{57} +7.43592 q^{59} -9.03566 q^{61} -15.5126 q^{63} +4.63020 q^{65} +10.1163 q^{67} -18.3275 q^{69} -6.46069 q^{71} +9.26466 q^{73} +3.31203 q^{75} +4.23550 q^{77} +3.71573 q^{79} +30.6049 q^{81} -11.1225 q^{83} +7.48800 q^{85} -30.5134 q^{87} -5.81048 q^{89} -9.01262 q^{91} +3.31203 q^{93} -3.83663 q^{95} -12.6991 q^{97} -17.3415 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 6 q^{5} - 2 q^{7} + 21 q^{9} + 8 q^{11} + 4 q^{13} - q^{15} + 3 q^{17} - 5 q^{19} + 8 q^{21} - 12 q^{23} + 6 q^{25} - 7 q^{27} - 4 q^{29} + 6 q^{31} + 14 q^{33} - 2 q^{35} + 3 q^{37} + 10 q^{39}+ \cdots - 40 q^{99}+O(q^{100}) \) 6 * q - q^3 + 6 * q^5 - 2 * q^7 + 21 * q^9 + 8 * q^11 + 4 * q^13 - q^15 + 3 * q^17 - 5 * q^19 + 8 * q^21 - 12 * q^23 + 6 * q^25 - 7 * q^27 - 4 * q^29 + 6 * q^31 + 14 * q^33 - 2 * q^35 + 3 * q^37 + 10 * q^39 + 17 * q^41 - 17 * q^43 + 21 * q^45 - 6 * q^47 + 34 * q^49 + 23 * q^51 + 9 * q^53 + 8 * q^55 - q^57 + 39 * q^59 + 2 * q^61 - 16 * q^63 + 4 * q^65 + 12 * q^67 - 8 * q^69 + 21 * q^71 + 43 * q^73 - q^75 + 20 * q^77 - 2 * q^79 + 50 * q^81 - 5 * q^83 + 3 * q^85 - 44 * q^87 + 6 * q^89 + 24 * q^91 - q^93 - 5 * q^95 - 12 * q^97 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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