LMFDB - Embedded newform 6036.2.a.i.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.38993 q^{5} +0.588107 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.38993 q^{5} +0.588107 q^{7} +1.00000 q^{9} -6.40011 q^{11} -3.59160 q^{13} +2.38993 q^{15} -3.38286 q^{17} +7.29913 q^{19} -0.588107 q^{21} +0.312022 q^{23} +0.711742 q^{25} -1.00000 q^{27} -5.17403 q^{29} -0.272176 q^{31} +6.40011 q^{33} -1.40553 q^{35} -2.33784 q^{37} +3.59160 q^{39} -1.99835 q^{41} -4.73040 q^{43} -2.38993 q^{45} -9.43664 q^{47} -6.65413 q^{49} +3.38286 q^{51} -12.3484 q^{53} +15.2958 q^{55} -7.29913 q^{57} +6.36719 q^{59} +2.40187 q^{61} +0.588107 q^{63} +8.58365 q^{65} -9.64729 q^{67} -0.312022 q^{69} +14.8072 q^{71} +2.81016 q^{73} -0.711742 q^{75} -3.76395 q^{77} +2.41978 q^{79} +1.00000 q^{81} -13.8550 q^{83} +8.08479 q^{85} +5.17403 q^{87} -9.03526 q^{89} -2.11225 q^{91} +0.272176 q^{93} -17.4444 q^{95} +15.1732 q^{97} -6.40011 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.38993	−1.06881	−0.534404	−	0.845229i	$-0.679464\pi$
$5$	−2.38993	−1.06881	−0.534404	+	0.845229i	$0.679464\pi$
$6$	0	0
$7$	0.588107	0.222284	0.111142	−	0.993805i	$-0.464549\pi$
$7$	0.588107	0.222284	0.111142	+	0.993805i	$0.464549\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.40011	−1.92970	−0.964852	−	0.262793i	$-0.915356\pi$
$11$	−6.40011	−1.92970	−0.964852	+	0.262793i	$0.915356\pi$
$12$	0	0
$13$	−3.59160	−0.996130	−0.498065	−	0.867140i	$-0.665956\pi$
$13$	−3.59160	−0.996130	−0.498065	+	0.867140i	$0.665956\pi$
$14$	0	0
$15$	2.38993	0.617076
$16$	0	0
$17$	−3.38286	−0.820465	−0.410232	−	0.911981i	$-0.634552\pi$
$17$	−3.38286	−0.820465	−0.410232	+	0.911981i	$0.634552\pi$
$18$	0	0
$19$	7.29913	1.67454	0.837268	−	0.546793i	$-0.184151\pi$
$19$	7.29913	1.67454	0.837268	+	0.546793i	$0.184151\pi$
$20$	0	0
$21$	−0.588107	−0.128336
$22$	0	0
$23$	0.312022	0.0650611	0.0325306	−	0.999471i	$-0.489643\pi$
$23$	0.312022	0.0650611	0.0325306	+	0.999471i	$0.489643\pi$
$24$	0	0
$25$	0.711742	0.142348
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.17403	−0.960793	−0.480397	−	0.877051i	$-0.659507\pi$
$29$	−5.17403	−0.960793	−0.480397	+	0.877051i	$0.659507\pi$
$30$	0	0
$31$	−0.272176	−0.0488842	−0.0244421	−	0.999701i	$-0.507781\pi$
$31$	−0.272176	−0.0488842	−0.0244421	+	0.999701i	$0.507781\pi$
$32$	0	0
$33$	6.40011	1.11412
$34$	0	0
$35$	−1.40553	−0.237578
$36$	0	0
$37$	−2.33784	−0.384339	−0.192169	−	0.981362i	$-0.561552\pi$
$37$	−2.33784	−0.384339	−0.192169	+	0.981362i	$0.561552\pi$
$38$	0	0
$39$	3.59160	0.575116
$40$	0	0
$41$	−1.99835	−0.312089	−0.156045	−	0.987750i	$-0.549874\pi$
$41$	−1.99835	−0.312089	−0.156045	+	0.987750i	$0.549874\pi$
$42$	0	0
$43$	−4.73040	−0.721379	−0.360689	−	0.932686i	$-0.617459\pi$
$43$	−4.73040	−0.721379	−0.360689	+	0.932686i	$0.617459\pi$
$44$	0	0
$45$	−2.38993	−0.356269
$46$	0	0
$47$	−9.43664	−1.37647	−0.688237	−	0.725486i	$-0.741615\pi$
$47$	−9.43664	−1.37647	−0.688237	+	0.725486i	$0.741615\pi$
$48$	0	0
$49$	−6.65413	−0.950590
$50$	0	0
$51$	3.38286	0.473695
$52$	0	0
$53$	−12.3484	−1.69618	−0.848091	−	0.529850i	$-0.822248\pi$
$53$	−12.3484	−1.69618	−0.848091	+	0.529850i	$0.822248\pi$
$54$	0	0
$55$	15.2958	2.06248
$56$	0	0
$57$	−7.29913	−0.966794
$58$	0	0
$59$	6.36719	0.828937	0.414469	−	0.910064i	$-0.363968\pi$
$59$	6.36719	0.828937	0.414469	+	0.910064i	$0.363968\pi$
$60$	0	0
$61$	2.40187	0.307528	0.153764	−	0.988108i	$-0.450860\pi$
$61$	2.40187	0.307528	0.153764	+	0.988108i	$0.450860\pi$
$62$	0	0
$63$	0.588107	0.0740946
$64$	0	0
$65$	8.58365	1.06467
$66$	0	0
$67$	−9.64729	−1.17860	−0.589302	−	0.807913i	$-0.700597\pi$
$67$	−9.64729	−1.17860	−0.589302	+	0.807913i	$0.700597\pi$
$68$	0	0
$69$	−0.312022	−0.0375631
$70$	0	0
$71$	14.8072	1.75729	0.878647	−	0.477472i	$-0.158447\pi$
$71$	14.8072	1.75729	0.878647	+	0.477472i	$0.158447\pi$
$72$	0	0
$73$	2.81016	0.328904	0.164452	−	0.986385i	$-0.447414\pi$
$73$	2.81016	0.328904	0.164452	+	0.986385i	$0.447414\pi$
$74$	0	0
$75$	−0.711742	−0.0821849
$76$	0	0
$77$	−3.76395	−0.428942
$78$	0	0
$79$	2.41978	0.272247	0.136123	−	0.990692i	$-0.456536\pi$
$79$	2.41978	0.272247	0.136123	+	0.990692i	$0.456536\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.8550	−1.52079	−0.760393	−	0.649463i	$-0.774994\pi$
$83$	−13.8550	−1.52079	−0.760393	+	0.649463i	$0.774994\pi$
$84$	0	0
$85$	8.08479	0.876918
$86$	0	0
$87$	5.17403	0.554714
$88$	0	0
$89$	−9.03526	−0.957735	−0.478868	−	0.877887i	$-0.658953\pi$
$89$	−9.03526	−0.957735	−0.478868	+	0.877887i	$0.658953\pi$
$90$	0	0
$91$	−2.11225	−0.221423
$92$	0	0
$93$	0.272176	0.0282233
$94$	0	0
$95$	−17.4444	−1.78976
$96$	0	0
$97$	15.1732	1.54060	0.770301	−	0.637681i	$-0.220106\pi$
$97$	15.1732	1.54060	0.770301	+	0.637681i	$0.220106\pi$
$98$	0	0
$99$	−6.40011	−0.643235
$100$	0	0
$101$	−11.9767	−1.19172	−0.595862	−	0.803087i	$-0.703189\pi$
$101$	−11.9767	−1.19172	−0.595862	+	0.803087i	$0.703189\pi$
$102$	0	0
$103$	0.697308	0.0687078	0.0343539	−	0.999410i	$-0.489063\pi$
$103$	0.697308	0.0687078	0.0343539	+	0.999410i	$0.489063\pi$
$104$	0	0
$105$	1.40553	0.137166
$106$	0	0
$107$	−5.38252	−0.520348	−0.260174	−	0.965562i	$-0.583780\pi$
$107$	−5.38252	−0.520348	−0.260174	+	0.965562i	$0.583780\pi$
$108$	0	0
$109$	11.3581	1.08791	0.543953	−	0.839116i	$-0.316927\pi$
$109$	11.3581	1.08791	0.543953	+	0.839116i	$0.316927\pi$
$110$	0	0
$111$	2.33784	0.221898
$112$	0	0
$113$	18.3709	1.72819	0.864094	−	0.503330i	$-0.167892\pi$
$113$	18.3709	1.72819	0.864094	+	0.503330i	$0.167892\pi$
$114$	0	0
$115$	−0.745710	−0.0695378
$116$	0	0
$117$	−3.59160	−0.332043
$118$	0	0
$119$	−1.98949	−0.182376
$120$	0	0
$121$	29.9613	2.72376
$122$	0	0
$123$	1.99835	0.180185
$124$	0	0
$125$	10.2486	0.916664
$126$	0	0
$127$	13.7453	1.21970	0.609848	−	0.792518i	$-0.291230\pi$
$127$	13.7453	1.21970	0.609848	+	0.792518i	$0.291230\pi$
$128$	0	0
$129$	4.73040	0.416488
$130$	0	0
$131$	4.90929	0.428926	0.214463	−	0.976732i	$-0.431200\pi$
$131$	4.90929	0.428926	0.214463	+	0.976732i	$0.431200\pi$
$132$	0	0
$133$	4.29267	0.372222
$134$	0	0
$135$	2.38993	0.205692
$136$	0	0
$137$	−0.345498	−0.0295179	−0.0147589	−	0.999891i	$-0.504698\pi$
$137$	−0.345498	−0.0295179	−0.0147589	+	0.999891i	$0.504698\pi$
$138$	0	0
$139$	−20.9006	−1.77276	−0.886381	−	0.462956i	$-0.846789\pi$
$139$	−20.9006	−1.77276	−0.886381	+	0.462956i	$0.846789\pi$
$140$	0	0
$141$	9.43664	0.794708
$142$	0	0
$143$	22.9866	1.92224
$144$	0	0
$145$	12.3655	1.02690
$146$	0	0
$147$	6.65413	0.548823
$148$	0	0
$149$	16.3697	1.34106	0.670528	−	0.741884i	$-0.266068\pi$
$149$	16.3697	1.34106	0.670528	+	0.741884i	$0.266068\pi$
$150$	0	0
$151$	−17.0850	−1.39036	−0.695180	−	0.718836i	$-0.744675\pi$
$151$	−17.0850	−1.39036	−0.695180	+	0.718836i	$0.744675\pi$
$152$	0	0
$153$	−3.38286	−0.273488
$154$	0	0
$155$	0.650480	0.0522478
$156$	0	0
$157$	2.07454	0.165566	0.0827830	−	0.996568i	$-0.473619\pi$
$157$	2.07454	0.165566	0.0827830	+	0.996568i	$0.473619\pi$
$158$	0	0
$159$	12.3484	0.979292
$160$	0	0
$161$	0.183503	0.0144620
$162$	0	0
$163$	−4.37975	−0.343049	−0.171524	−	0.985180i	$-0.554869\pi$
$163$	−4.37975	−0.343049	−0.171524	+	0.985180i	$0.554869\pi$
$164$	0	0
$165$	−15.2958	−1.19077
$166$	0	0
$167$	−5.72958	−0.443368	−0.221684	−	0.975119i	$-0.571155\pi$
$167$	−5.72958	−0.443368	−0.221684	+	0.975119i	$0.571155\pi$
$168$	0	0
$169$	−0.100422	−0.00772473
$170$	0	0
$171$	7.29913	0.558179
$172$	0	0
$173$	−5.49514	−0.417788	−0.208894	−	0.977938i	$-0.566986\pi$
$173$	−5.49514	−0.417788	−0.208894	+	0.977938i	$0.566986\pi$
$174$	0	0
$175$	0.418581	0.0316417
$176$	0	0
$177$	−6.36719	−0.478587
$178$	0	0
$179$	13.4407	1.00461	0.502304	−	0.864691i	$-0.332486\pi$
$179$	13.4407	1.00461	0.502304	+	0.864691i	$0.332486\pi$
$180$	0	0
$181$	−19.2230	−1.42884	−0.714418	−	0.699719i	$-0.753309\pi$
$181$	−19.2230	−1.42884	−0.714418	+	0.699719i	$0.753309\pi$
$182$	0	0
$183$	−2.40187	−0.177552
$184$	0	0
$185$	5.58726	0.410784
$186$	0	0
$187$	21.6507	1.58325
$188$	0	0
$189$	−0.588107	−0.0427785
$190$	0	0
$191$	14.5416	1.05219	0.526097	−	0.850425i	$-0.323655\pi$
$191$	14.5416	1.05219	0.526097	+	0.850425i	$0.323655\pi$
$192$	0	0
$193$	−26.4953	−1.90718	−0.953588	−	0.301114i	$-0.902642\pi$
$193$	−26.4953	−1.90718	−0.953588	+	0.301114i	$0.902642\pi$
$194$	0	0
$195$	−8.58365	−0.614688
$196$	0	0
$197$	1.72972	0.123238	0.0616188	−	0.998100i	$-0.480374\pi$
$197$	1.72972	0.123238	0.0616188	+	0.998100i	$0.480374\pi$
$198$	0	0
$199$	−22.1817	−1.57242	−0.786211	−	0.617958i	$-0.787960\pi$
$199$	−22.1817	−1.57242	−0.786211	+	0.617958i	$0.787960\pi$
$200$	0	0
$201$	9.64729	0.680467
$202$	0	0
$203$	−3.04288	−0.213569
$204$	0	0
$205$	4.77590	0.333563
$206$	0	0
$207$	0.312022	0.0216870
$208$	0	0
$209$	−46.7152	−3.23136
$210$	0	0
$211$	28.3317	1.95044	0.975218	−	0.221245i	$-0.0710120\pi$
$211$	28.3317	1.95044	0.975218	+	0.221245i	$0.0710120\pi$
$212$	0	0
$213$	−14.8072	−1.01457
$214$	0	0
$215$	11.3053	0.771015
$216$	0	0
$217$	−0.160069	−0.0108662
$218$	0	0
$219$	−2.81016	−0.189893
$220$	0	0
$221$	12.1499	0.817290
$222$	0	0
$223$	14.6429	0.980561	0.490281	−	0.871565i	$-0.336894\pi$
$223$	14.6429	0.980561	0.490281	+	0.871565i	$0.336894\pi$
$224$	0	0
$225$	0.711742	0.0474495
$226$	0	0
$227$	−16.9117	−1.12247	−0.561236	−	0.827656i	$-0.689674\pi$
$227$	−16.9117	−1.12247	−0.561236	+	0.827656i	$0.689674\pi$
$228$	0	0
$229$	28.6468	1.89303	0.946516	−	0.322656i	$-0.104576\pi$
$229$	28.6468	1.89303	0.946516	+	0.322656i	$0.104576\pi$
$230$	0	0
$231$	3.76395	0.247650
$232$	0	0
$233$	−11.9058	−0.779976	−0.389988	−	0.920820i	$-0.627521\pi$
$233$	−11.9058	−0.779976	−0.389988	+	0.920820i	$0.627521\pi$
$234$	0	0
$235$	22.5529	1.47119
$236$	0	0
$237$	−2.41978	−0.157182
$238$	0	0
$239$	2.67752	0.173195	0.0865973	−	0.996243i	$-0.472401\pi$
$239$	2.67752	0.173195	0.0865973	+	0.996243i	$0.472401\pi$
$240$	0	0
$241$	16.2107	1.04422	0.522110	−	0.852878i	$-0.325145\pi$
$241$	16.2107	1.04422	0.522110	+	0.852878i	$0.325145\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	15.9029	1.01600
$246$	0	0
$247$	−26.2156	−1.66806
$248$	0	0
$249$	13.8550	0.878026
$250$	0	0
$251$	−6.74504	−0.425743	−0.212872	−	0.977080i	$-0.568282\pi$
$251$	−6.74504	−0.425743	−0.212872	+	0.977080i	$0.568282\pi$
$252$	0	0
$253$	−1.99698	−0.125549
$254$	0	0
$255$	−8.08479	−0.506289
$256$	0	0
$257$	22.9551	1.43190	0.715950	−	0.698151i	$-0.245994\pi$
$257$	22.9551	1.43190	0.715950	+	0.698151i	$0.245994\pi$
$258$	0	0
$259$	−1.37490	−0.0854322
$260$	0	0
$261$	−5.17403	−0.320264
$262$	0	0
$263$	−21.3876	−1.31881	−0.659407	−	0.751786i	$-0.729193\pi$
$263$	−21.3876	−1.31881	−0.659407	+	0.751786i	$0.729193\pi$
$264$	0	0
$265$	29.5117	1.81289
$266$	0	0
$267$	9.03526	0.552949
$268$	0	0
$269$	25.0659	1.52829	0.764146	−	0.645043i	$-0.223160\pi$
$269$	25.0659	1.52829	0.764146	+	0.645043i	$0.223160\pi$
$270$	0	0
$271$	16.4961	1.00206	0.501032	−	0.865429i	$-0.332954\pi$
$271$	16.4961	1.00206	0.501032	+	0.865429i	$0.332954\pi$
$272$	0	0
$273$	2.11225	0.127839
$274$	0	0
$275$	−4.55523	−0.274690
$276$	0	0
$277$	19.0521	1.14473	0.572365	−	0.819999i	$-0.306026\pi$
$277$	19.0521	1.14473	0.572365	+	0.819999i	$0.306026\pi$
$278$	0	0
$279$	−0.272176	−0.0162947
$280$	0	0
$281$	−20.5003	−1.22294	−0.611472	−	0.791266i	$-0.709422\pi$
$281$	−20.5003	−1.22294	−0.611472	+	0.791266i	$0.709422\pi$
$282$	0	0
$283$	−22.1389	−1.31602	−0.658011	−	0.753009i	$-0.728602\pi$
$283$	−22.1389	−1.31602	−0.658011	+	0.753009i	$0.728602\pi$
$284$	0	0
$285$	17.4444	1.03332
$286$	0	0
$287$	−1.17524	−0.0693723
$288$	0	0
$289$	−5.55624	−0.326838
$290$	0	0
$291$	−15.1732	−0.889467
$292$	0	0
$293$	−5.77896	−0.337611	−0.168805	−	0.985649i	$-0.553991\pi$
$293$	−5.77896	−0.337611	−0.168805	+	0.985649i	$0.553991\pi$
$294$	0	0
$295$	−15.2171	−0.885974
$296$	0	0
$297$	6.40011	0.371372
$298$	0	0
$299$	−1.12066	−0.0648094
$300$	0	0
$301$	−2.78198	−0.160351
$302$	0	0
$303$	11.9767	0.688042
$304$	0	0
$305$	−5.74030	−0.328689
$306$	0	0
$307$	12.3201	0.703146	0.351573	−	0.936160i	$-0.385647\pi$
$307$	12.3201	0.703146	0.351573	+	0.936160i	$0.385647\pi$
$308$	0	0
$309$	−0.697308	−0.0396685
$310$	0	0
$311$	−3.99551	−0.226565	−0.113282	−	0.993563i	$-0.536136\pi$
$311$	−3.99551	−0.226565	−0.113282	+	0.993563i	$0.536136\pi$
$312$	0	0
$313$	−10.3980	−0.587731	−0.293865	−	0.955847i	$-0.594942\pi$
$313$	−10.3980	−0.587731	−0.293865	+	0.955847i	$0.594942\pi$
$314$	0	0
$315$	−1.40553	−0.0791928
$316$	0	0
$317$	18.9689	1.06540	0.532700	−	0.846304i	$-0.321177\pi$
$317$	18.9689	1.06540	0.532700	+	0.846304i	$0.321177\pi$
$318$	0	0
$319$	33.1143	1.85405
$320$	0	0
$321$	5.38252	0.300423
$322$	0	0
$323$	−24.6920	−1.37390
$324$	0	0
$325$	−2.55629	−0.141798
$326$	0	0
$327$	−11.3581	−0.628103
$328$	0	0
$329$	−5.54975	−0.305968
$330$	0	0
$331$	−7.95346	−0.437162	−0.218581	−	0.975819i	$-0.570143\pi$
$331$	−7.95346	−0.437162	−0.218581	+	0.975819i	$0.570143\pi$
$332$	0	0
$333$	−2.33784	−0.128113
$334$	0	0
$335$	23.0563	1.25970
$336$	0	0
$337$	−9.48324	−0.516585	−0.258292	−	0.966067i	$-0.583160\pi$
$337$	−9.48324	−0.516585	−0.258292	+	0.966067i	$0.583160\pi$
$338$	0	0
$339$	−18.3709	−0.997770
$340$	0	0
$341$	1.74195	0.0943322
$342$	0	0
$343$	−8.03009	−0.433584
$344$	0	0
$345$	0.745710	0.0401477
$346$	0	0
$347$	17.4217	0.935246	0.467623	−	0.883928i	$-0.345110\pi$
$347$	17.4217	0.935246	0.467623	+	0.883928i	$0.345110\pi$
$348$	0	0
$349$	2.28231	0.122169	0.0610847	−	0.998133i	$-0.480544\pi$
$349$	2.28231	0.122169	0.0610847	+	0.998133i	$0.480544\pi$
$350$	0	0
$351$	3.59160	0.191705
$352$	0	0
$353$	32.4238	1.72574	0.862872	−	0.505422i	$-0.168663\pi$
$353$	32.4238	1.72574	0.862872	+	0.505422i	$0.168663\pi$
$354$	0	0
$355$	−35.3882	−1.87821
$356$	0	0
$357$	1.98949	0.105295
$358$	0	0
$359$	22.6046	1.19303	0.596513	−	0.802603i	$-0.296552\pi$
$359$	22.6046	1.19303	0.596513	+	0.802603i	$0.296552\pi$
$360$	0	0
$361$	34.2773	1.80407
$362$	0	0
$363$	−29.9613	−1.57256
$364$	0	0
$365$	−6.71607	−0.351535
$366$	0	0
$367$	11.4818	0.599347	0.299674	−	0.954042i	$-0.403122\pi$
$367$	11.4818	0.599347	0.299674	+	0.954042i	$0.403122\pi$
$368$	0	0
$369$	−1.99835	−0.104030
$370$	0	0
$371$	−7.26218	−0.377034
$372$	0	0
$373$	−7.20131	−0.372870	−0.186435	−	0.982467i	$-0.559693\pi$
$373$	−7.20131	−0.372870	−0.186435	+	0.982467i	$0.559693\pi$
$374$	0	0
$375$	−10.2486	−0.529236
$376$	0	0
$377$	18.5830	0.957075
$378$	0	0
$379$	30.5023	1.56680	0.783399	−	0.621519i	$-0.213484\pi$
$379$	30.5023	1.56680	0.783399	+	0.621519i	$0.213484\pi$
$380$	0	0
$381$	−13.7453	−0.704192
$382$	0	0
$383$	22.4939	1.14939	0.574693	−	0.818369i	$-0.305122\pi$
$383$	22.4939	1.14939	0.574693	+	0.818369i	$0.305122\pi$
$384$	0	0
$385$	8.99556	0.458456
$386$	0	0
$387$	−4.73040	−0.240460
$388$	0	0
$389$	−3.76896	−0.191094	−0.0955469	−	0.995425i	$-0.530460\pi$
$389$	−3.76896	−0.191094	−0.0955469	+	0.995425i	$0.530460\pi$
$390$	0	0
$391$	−1.05553	−0.0533804
$392$	0	0
$393$	−4.90929	−0.247641
$394$	0	0
$395$	−5.78309	−0.290979
$396$	0	0
$397$	−17.5566	−0.881141	−0.440571	−	0.897718i	$-0.645224\pi$
$397$	−17.5566	−0.881141	−0.440571	+	0.897718i	$0.645224\pi$
$398$	0	0
$399$	−4.29267	−0.214902
$400$	0	0
$401$	17.3227	0.865056	0.432528	−	0.901621i	$-0.357622\pi$
$401$	17.3227	0.865056	0.432528	+	0.901621i	$0.357622\pi$
$402$	0	0
$403$	0.977547	0.0486951
$404$	0	0
$405$	−2.38993	−0.118756
$406$	0	0
$407$	14.9624	0.741660
$408$	0	0
$409$	13.2762	0.656465	0.328232	−	0.944597i	$-0.393547\pi$
$409$	13.2762	0.656465	0.328232	+	0.944597i	$0.393547\pi$
$410$	0	0
$411$	0.345498	0.0170422
$412$	0	0
$413$	3.74459	0.184259
$414$	0	0
$415$	33.1125	1.62543
$416$	0	0
$417$	20.9006	1.02351
$418$	0	0
$419$	−8.62663	−0.421438	−0.210719	−	0.977547i	$-0.567581\pi$
$419$	−8.62663	−0.421438	−0.210719	+	0.977547i	$0.567581\pi$
$420$	0	0
$421$	−5.48658	−0.267400	−0.133700	−	0.991022i	$-0.542686\pi$
$421$	−5.48658	−0.267400	−0.133700	+	0.991022i	$0.542686\pi$
$422$	0	0
$423$	−9.43664	−0.458825
$424$	0	0
$425$	−2.40773	−0.116792
$426$	0	0
$427$	1.41256	0.0683585
$428$	0	0
$429$	−22.9866	−1.10980
$430$	0	0
$431$	−5.86411	−0.282464	−0.141232	−	0.989976i	$-0.545106\pi$
$431$	−5.86411	−0.282464	−0.141232	+	0.989976i	$0.545106\pi$
$432$	0	0
$433$	19.5766	0.940793	0.470396	−	0.882455i	$-0.344111\pi$
$433$	19.5766	0.940793	0.470396	+	0.882455i	$0.344111\pi$
$434$	0	0
$435$	−12.3655	−0.592882
$436$	0	0
$437$	2.27749	0.108947
$438$	0	0
$439$	8.97589	0.428396	0.214198	−	0.976790i	$-0.431286\pi$
$439$	8.97589	0.428396	0.214198	+	0.976790i	$0.431286\pi$
$440$	0	0
$441$	−6.65413	−0.316863
$442$	0	0
$443$	13.3745	0.635440	0.317720	−	0.948185i	$-0.397083\pi$
$443$	13.3745	0.635440	0.317720	+	0.948185i	$0.397083\pi$
$444$	0	0
$445$	21.5936	1.02363
$446$	0	0
$447$	−16.3697	−0.774259
$448$	0	0
$449$	21.6819	1.02323	0.511615	−	0.859215i	$-0.329047\pi$
$449$	21.6819	1.02323	0.511615	+	0.859215i	$0.329047\pi$
$450$	0	0
$451$	12.7896	0.602240
$452$	0	0
$453$	17.0850	0.802725
$454$	0	0
$455$	5.04811	0.236659
$456$	0	0
$457$	24.3867	1.14076	0.570380	−	0.821381i	$-0.306796\pi$
$457$	24.3867	1.14076	0.570380	+	0.821381i	$0.306796\pi$
$458$	0	0
$459$	3.38286	0.157898
$460$	0	0
$461$	35.6549	1.66061	0.830307	−	0.557306i	$-0.188165\pi$
$461$	35.6549	1.66061	0.830307	+	0.557306i	$0.188165\pi$
$462$	0	0
$463$	10.2832	0.477902	0.238951	−	0.971032i	$-0.423196\pi$
$463$	10.2832	0.477902	0.238951	+	0.971032i	$0.423196\pi$
$464$	0	0
$465$	−0.650480	−0.0301653
$466$	0	0
$467$	−3.45629	−0.159938	−0.0799691	−	0.996797i	$-0.525482\pi$
$467$	−3.45629	−0.159938	−0.0799691	+	0.996797i	$0.525482\pi$
$468$	0	0
$469$	−5.67364	−0.261984
$470$	0	0
$471$	−2.07454	−0.0955896
$472$	0	0
$473$	30.2750	1.39205
$474$	0	0
$475$	5.19510	0.238368
$476$	0	0
$477$	−12.3484	−0.565394
$478$	0	0
$479$	−40.0140	−1.82829	−0.914143	−	0.405393i	$-0.867135\pi$
$479$	−40.0140	−1.82829	−0.914143	+	0.405393i	$0.867135\pi$
$480$	0	0
$481$	8.39658	0.382851
$482$	0	0
$483$	−0.183503	−0.00834966
$484$	0	0
$485$	−36.2627	−1.64661
$486$	0	0
$487$	15.2621	0.691593	0.345796	−	0.938310i	$-0.387609\pi$
$487$	15.2621	0.691593	0.345796	+	0.938310i	$0.387609\pi$
$488$	0	0
$489$	4.37975	0.198059
$490$	0	0
$491$	36.1991	1.63364	0.816821	−	0.576891i	$-0.195734\pi$
$491$	36.1991	1.63364	0.816821	+	0.576891i	$0.195734\pi$
$492$	0	0
$493$	17.5030	0.788297
$494$	0	0
$495$	15.2958	0.687494
$496$	0	0
$497$	8.70824	0.390618
$498$	0	0
$499$	−40.7341	−1.82351	−0.911754	−	0.410736i	$-0.865272\pi$
$499$	−40.7341	−1.82351	−0.911754	+	0.410736i	$0.865272\pi$
$500$	0	0
$501$	5.72958	0.255979
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	28.6234	1.27372
$506$	0	0
$507$	0.100422	0.00445988
$508$	0	0
$509$	−16.1247	−0.714713	−0.357356	−	0.933968i	$-0.616322\pi$
$509$	−16.1247	−0.714713	−0.357356	+	0.933968i	$0.616322\pi$
$510$	0	0
$511$	1.65268	0.0731101
$512$	0	0
$513$	−7.29913	−0.322265
$514$	0	0
$515$	−1.66651	−0.0734354
$516$	0	0
$517$	60.3955	2.65619
$518$	0	0
$519$	5.49514	0.241210
$520$	0	0
$521$	9.76844	0.427963	0.213982	−	0.976838i	$-0.431357\pi$
$521$	9.76844	0.427963	0.213982	+	0.976838i	$0.431357\pi$
$522$	0	0
$523$	24.3708	1.06566	0.532829	−	0.846223i	$-0.321129\pi$
$523$	24.3708	1.06566	0.532829	+	0.846223i	$0.321129\pi$
$524$	0	0
$525$	−0.418581	−0.0182684
$526$	0	0
$527$	0.920734	0.0401078
$528$	0	0
$529$	−22.9026	−0.995767
$530$	0	0
$531$	6.36719	0.276312
$532$	0	0
$533$	7.17726	0.310881
$534$	0	0
$535$	12.8638	0.556152
$536$	0	0
$537$	−13.4407	−0.580011
$538$	0	0
$539$	42.5871	1.83436
$540$	0	0
$541$	1.49822	0.0644135	0.0322067	−	0.999481i	$-0.489747\pi$
$541$	1.49822	0.0644135	0.0322067	+	0.999481i	$0.489747\pi$
$542$	0	0
$543$	19.2230	0.824939
$544$	0	0
$545$	−27.1450	−1.16276
$546$	0	0
$547$	−15.0557	−0.643736	−0.321868	−	0.946785i	$-0.604311\pi$
$547$	−15.0557	−0.643736	−0.321868	+	0.946785i	$0.604311\pi$
$548$	0	0
$549$	2.40187	0.102509
$550$	0	0
$551$	−37.7659	−1.60888
$552$	0	0
$553$	1.42309	0.0605159
$554$	0	0
$555$	−5.58726	−0.237166
$556$	0	0
$557$	−6.34801	−0.268974	−0.134487	−	0.990915i	$-0.542939\pi$
$557$	−6.34801	−0.268974	−0.134487	+	0.990915i	$0.542939\pi$
$558$	0	0
$559$	16.9897	0.718587
$560$	0	0
$561$	−21.6507	−0.914092
$562$	0	0
$563$	−24.4057	−1.02858	−0.514290	−	0.857617i	$-0.671944\pi$
$563$	−24.4057	−1.02858	−0.514290	+	0.857617i	$0.671944\pi$
$564$	0	0
$565$	−43.9051	−1.84710
$566$	0	0
$567$	0.588107	0.0246982
$568$	0	0
$569$	16.3905	0.687124	0.343562	−	0.939130i	$-0.388366\pi$
$569$	16.3905	0.687124	0.343562	+	0.939130i	$0.388366\pi$
$570$	0	0
$571$	−16.1798	−0.677103	−0.338552	−	0.940948i	$-0.609937\pi$
$571$	−16.1798	−0.677103	−0.338552	+	0.940948i	$0.609937\pi$
$572$	0	0
$573$	−14.5416	−0.607484
$574$	0	0
$575$	0.222079	0.00926135
$576$	0	0
$577$	19.2936	0.803204	0.401602	−	0.915814i	$-0.368454\pi$
$577$	19.2936	0.803204	0.401602	+	0.915814i	$0.368454\pi$
$578$	0	0
$579$	26.4953	1.10111
$580$	0	0
$581$	−8.14824	−0.338046
$582$	0	0
$583$	79.0310	3.27313
$584$	0	0
$585$	8.58365	0.354890
$586$	0	0
$587$	−15.8376	−0.653688	−0.326844	−	0.945078i	$-0.605985\pi$
$587$	−15.8376	−0.653688	−0.326844	+	0.945078i	$0.605985\pi$
$588$	0	0
$589$	−1.98665	−0.0818584
$590$	0	0
$591$	−1.72972	−0.0711513
$592$	0	0
$593$	−32.5864	−1.33816	−0.669082	−	0.743188i	$-0.733313\pi$
$593$	−32.5864	−1.33816	−0.669082	+	0.743188i	$0.733313\pi$
$594$	0	0
$595$	4.75472	0.194925
$596$	0	0
$597$	22.1817	0.907838
$598$	0	0
$599$	−39.8792	−1.62942	−0.814709	−	0.579869i	$-0.803104\pi$
$599$	−39.8792	−1.62942	−0.814709	+	0.579869i	$0.803104\pi$
$600$	0	0
$601$	4.11864	0.168003	0.0840015	−	0.996466i	$-0.473230\pi$
$601$	4.11864	0.168003	0.0840015	+	0.996466i	$0.473230\pi$
$602$	0	0
$603$	−9.64729	−0.392868
$604$	0	0
$605$	−71.6054	−2.91117
$606$	0	0
$607$	32.6159	1.32384	0.661920	−	0.749575i	$-0.269742\pi$
$607$	32.6159	1.32384	0.661920	+	0.749575i	$0.269742\pi$
$608$	0	0
$609$	3.04288	0.123304
$610$	0	0
$611$	33.8926	1.37115
$612$	0	0
$613$	−31.6708	−1.27917	−0.639587	−	0.768719i	$-0.720895\pi$
$613$	−31.6708	−1.27917	−0.639587	+	0.768719i	$0.720895\pi$
$614$	0	0
$615$	−4.77590	−0.192583
$616$	0	0
$617$	42.5002	1.71099	0.855497	−	0.517808i	$-0.173252\pi$
$617$	42.5002	1.71099	0.855497	+	0.517808i	$0.173252\pi$
$618$	0	0
$619$	−43.1901	−1.73596	−0.867979	−	0.496601i	$-0.834581\pi$
$619$	−43.1901	−1.73596	−0.867979	+	0.496601i	$0.834581\pi$
$620$	0	0
$621$	−0.312022	−0.0125210
$622$	0	0
$623$	−5.31370	−0.212889
$624$	0	0
$625$	−28.0521	−1.12209
$626$	0	0
$627$	46.7152	1.86563
$628$	0	0
$629$	7.90859	0.315336
$630$	0	0
$631$	−40.8635	−1.62675	−0.813374	−	0.581740i	$-0.802372\pi$
$631$	−40.8635	−1.62675	−0.813374	+	0.581740i	$0.802372\pi$
$632$	0	0
$633$	−28.3317	−1.12609
$634$	0	0
$635$	−32.8502	−1.30362
$636$	0	0
$637$	23.8990	0.946911
$638$	0	0
$639$	14.8072	0.585765
$640$	0	0
$641$	−45.9590	−1.81527	−0.907637	−	0.419757i	$-0.862115\pi$
$641$	−45.9590	−1.81527	−0.907637	+	0.419757i	$0.862115\pi$
$642$	0	0
$643$	−14.3778	−0.567004	−0.283502	−	0.958972i	$-0.591496\pi$
$643$	−14.3778	−0.567004	−0.283502	+	0.958972i	$0.591496\pi$
$644$	0	0
$645$	−11.3053	−0.445145
$646$	0	0
$647$	−38.4085	−1.50999	−0.754997	−	0.655729i	$-0.772362\pi$
$647$	−38.4085	−1.50999	−0.754997	+	0.655729i	$0.772362\pi$
$648$	0	0
$649$	−40.7507	−1.59960
$650$	0	0
$651$	0.160069	0.00627359
$652$	0	0
$653$	−15.6342	−0.611812	−0.305906	−	0.952062i	$-0.598959\pi$
$653$	−15.6342	−0.611812	−0.305906	+	0.952062i	$0.598959\pi$
$654$	0	0
$655$	−11.7328	−0.458440
$656$	0	0
$657$	2.81016	0.109635
$658$	0	0
$659$	18.7437	0.730151	0.365076	−	0.930978i	$-0.381043\pi$
$659$	18.7437	0.730151	0.365076	+	0.930978i	$0.381043\pi$
$660$	0	0
$661$	−21.6538	−0.842235	−0.421118	−	0.907006i	$-0.638362\pi$
$661$	−21.6538	−0.842235	−0.421118	+	0.907006i	$0.638362\pi$
$662$	0	0
$663$	−12.1499	−0.471862
$664$	0	0
$665$	−10.2592	−0.397833
$666$	0	0
$667$	−1.61441	−0.0625103
$668$	0	0
$669$	−14.6429	−0.566127
$670$	0	0
$671$	−15.3722	−0.593439
$672$	0	0
$673$	−18.7691	−0.723497	−0.361749	−	0.932276i	$-0.617820\pi$
$673$	−18.7691	−0.723497	−0.361749	+	0.932276i	$0.617820\pi$
$674$	0	0
$675$	−0.711742	−0.0273950
$676$	0	0
$677$	1.75391	0.0674081	0.0337040	−	0.999432i	$-0.489270\pi$
$677$	1.75391	0.0674081	0.0337040	+	0.999432i	$0.489270\pi$
$678$	0	0
$679$	8.92345	0.342451
$680$	0	0
$681$	16.9117	0.648059
$682$	0	0
$683$	−47.3013	−1.80993	−0.904967	−	0.425481i	$-0.860105\pi$
$683$	−47.3013	−1.80993	−0.904967	+	0.425481i	$0.860105\pi$
$684$	0	0
$685$	0.825714	0.0315489
$686$	0	0
$687$	−28.6468	−1.09294
$688$	0	0
$689$	44.3505	1.68962
$690$	0	0
$691$	20.8270	0.792295	0.396148	−	0.918187i	$-0.370347\pi$
$691$	20.8270	0.792295	0.396148	+	0.918187i	$0.370347\pi$
$692$	0	0
$693$	−3.76395	−0.142981
$694$	0	0
$695$	49.9508	1.89474
$696$	0	0
$697$	6.76013	0.256058
$698$	0	0
$699$	11.9058	0.450319
$700$	0	0
$701$	20.1425	0.760772	0.380386	−	0.924828i	$-0.375791\pi$
$701$	20.1425	0.760772	0.380386	+	0.924828i	$0.375791\pi$
$702$	0	0
$703$	−17.0642	−0.643589
$704$	0	0
$705$	−22.5529	−0.849390
$706$	0	0
$707$	−7.04357	−0.264901
$708$	0	0
$709$	−39.6246	−1.48813	−0.744067	−	0.668105i	$-0.767106\pi$
$709$	−39.6246	−1.48813	−0.744067	+	0.668105i	$0.767106\pi$
$710$	0	0
$711$	2.41978	0.0907488
$712$	0	0
$713$	−0.0849250	−0.00318047
$714$	0	0
$715$	−54.9363	−2.05450
$716$	0	0
$717$	−2.67752	−0.0999939
$718$	0	0
$719$	8.19406	0.305587	0.152793	−	0.988258i	$-0.451173\pi$
$719$	8.19406	0.305587	0.152793	+	0.988258i	$0.451173\pi$
$720$	0	0
$721$	0.410092	0.0152726
$722$	0	0
$723$	−16.2107	−0.602881
$724$	0	0
$725$	−3.68258	−0.136767
$726$	0	0
$727$	−7.36758	−0.273249	−0.136624	−	0.990623i	$-0.543625\pi$
$727$	−7.36758	−0.273249	−0.136624	+	0.990623i	$0.543625\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.0023	0.591866
$732$	0	0
$733$	16.3787	0.604962	0.302481	−	0.953155i	$-0.402185\pi$
$733$	16.3787	0.604962	0.302481	+	0.953155i	$0.402185\pi$
$734$	0	0
$735$	−15.9029	−0.586586
$736$	0	0
$737$	61.7436	2.27436
$738$	0	0
$739$	−8.99317	−0.330819	−0.165410	−	0.986225i	$-0.552895\pi$
$739$	−8.99317	−0.330819	−0.165410	+	0.986225i	$0.552895\pi$
$740$	0	0
$741$	26.2156	0.963052
$742$	0	0
$743$	−12.6592	−0.464420	−0.232210	−	0.972666i	$-0.574596\pi$
$743$	−12.6592	−0.464420	−0.232210	+	0.972666i	$0.574596\pi$
$744$	0	0
$745$	−39.1223	−1.43333
$746$	0	0
$747$	−13.8550	−0.506929
$748$	0	0
$749$	−3.16550	−0.115665
$750$	0	0
$751$	4.01514	0.146515	0.0732573	−	0.997313i	$-0.476661\pi$
$751$	4.01514	0.146515	0.0732573	+	0.997313i	$0.476661\pi$
$752$	0	0
$753$	6.74504	0.245803
$754$	0	0
$755$	40.8319	1.48603
$756$	0	0
$757$	22.0647	0.801954	0.400977	−	0.916088i	$-0.368671\pi$
$757$	22.0647	0.801954	0.400977	+	0.916088i	$0.368671\pi$
$758$	0	0
$759$	1.99698	0.0724856
$760$	0	0
$761$	−2.68911	−0.0974801	−0.0487400	−	0.998811i	$-0.515521\pi$
$761$	−2.68911	−0.0974801	−0.0487400	+	0.998811i	$0.515521\pi$
$762$	0	0
$763$	6.67977	0.241824
$764$	0	0
$765$	8.08479	0.292306
$766$	0	0
$767$	−22.8684	−0.825729
$768$	0	0
$769$	26.1831	0.944187	0.472094	−	0.881548i	$-0.343498\pi$
$769$	26.1831	0.944187	0.472094	+	0.881548i	$0.343498\pi$
$770$	0	0
$771$	−22.9551	−0.826708
$772$	0	0
$773$	−28.9270	−1.04043	−0.520216	−	0.854035i	$-0.674149\pi$
$773$	−28.9270	−1.04043	−0.520216	+	0.854035i	$0.674149\pi$
$774$	0	0
$775$	−0.193719	−0.00695860
$776$	0	0
$777$	1.37490	0.0493243
$778$	0	0
$779$	−14.5862	−0.522605
$780$	0	0
$781$	−94.7678	−3.39106
$782$	0	0
$783$	5.17403	0.184905
$784$	0	0
$785$	−4.95799	−0.176958
$786$	0	0
$787$	17.9161	0.638641	0.319321	−	0.947647i	$-0.396545\pi$
$787$	17.9161	0.638641	0.319321	+	0.947647i	$0.396545\pi$
$788$	0	0
$789$	21.3876	0.761418
$790$	0	0
$791$	10.8041	0.384148
$792$	0	0
$793$	−8.62657	−0.306338
$794$	0	0
$795$	−29.5117	−1.04667
$796$	0	0
$797$	33.9111	1.20119	0.600597	−	0.799552i	$-0.294930\pi$
$797$	33.9111	1.20119	0.600597	+	0.799552i	$0.294930\pi$
$798$	0	0
$799$	31.9228	1.12935
$800$	0	0
$801$	−9.03526	−0.319245
$802$	0	0
$803$	−17.9853	−0.634688
$804$	0	0
$805$	−0.438557	−0.0154571
$806$	0	0
$807$	−25.0659	−0.882360
$808$	0	0
$809$	−51.9598	−1.82681	−0.913405	−	0.407051i	$-0.866557\pi$
$809$	−51.9598	−1.82681	−0.913405	+	0.407051i	$0.866557\pi$
$810$	0	0
$811$	−14.1320	−0.496240	−0.248120	−	0.968729i	$-0.579813\pi$
$811$	−14.1320	−0.496240	−0.248120	+	0.968729i	$0.579813\pi$
$812$	0	0
$813$	−16.4961	−0.578542
$814$	0	0
$815$	10.4673	0.366653
$816$	0	0
$817$	−34.5278	−1.20797
$818$	0	0
$819$	−2.11225	−0.0738078
$820$	0	0
$821$	−38.4230	−1.34097	−0.670485	−	0.741923i	$-0.733914\pi$
$821$	−38.4230	−1.34097	−0.670485	+	0.741923i	$0.733914\pi$
$822$	0	0
$823$	21.1853	0.738474	0.369237	−	0.929335i	$-0.379619\pi$
$823$	21.1853	0.738474	0.369237	+	0.929335i	$0.379619\pi$
$824$	0	0
$825$	4.55523	0.158593
$826$	0	0
$827$	−29.2491	−1.01709	−0.508546	−	0.861035i	$-0.669817\pi$
$827$	−29.2491	−1.01709	−0.508546	+	0.861035i	$0.669817\pi$
$828$	0	0
$829$	3.15645	0.109628	0.0548140	−	0.998497i	$-0.482543\pi$
$829$	3.15645	0.109628	0.0548140	+	0.998497i	$0.482543\pi$
$830$	0	0
$831$	−19.0521	−0.660910
$832$	0	0
$833$	22.5100	0.779925
$834$	0	0
$835$	13.6933	0.473875
$836$	0	0
$837$	0.272176	0.00940778
$838$	0	0
$839$	−17.8210	−0.615250	−0.307625	−	0.951508i	$-0.599534\pi$
$839$	−17.8210	−0.615250	−0.307625	+	0.951508i	$0.599534\pi$
$840$	0	0
$841$	−2.22941	−0.0768763
$842$	0	0
$843$	20.5003	0.706067
$844$	0	0
$845$	0.240000	0.00825625
$846$	0	0
$847$	17.6205	0.605447
$848$	0	0
$849$	22.1389	0.759805
$850$	0	0
$851$	−0.729458	−0.0250055
$852$	0	0
$853$	−24.0437	−0.823242	−0.411621	−	0.911355i	$-0.635037\pi$
$853$	−24.0437	−0.823242	−0.411621	+	0.911355i	$0.635037\pi$
$854$	0	0
$855$	−17.4444	−0.596585
$856$	0	0
$857$	35.2784	1.20509	0.602544	−	0.798086i	$-0.294154\pi$
$857$	35.2784	1.20509	0.602544	+	0.798086i	$0.294154\pi$
$858$	0	0
$859$	−44.0022	−1.50134	−0.750668	−	0.660680i	$-0.770268\pi$
$859$	−44.0022	−1.50134	−0.750668	+	0.660680i	$0.770268\pi$
$860$	0	0
$861$	1.17524	0.0400521
$862$	0	0
$863$	31.6997	1.07907	0.539535	−	0.841963i	$-0.318600\pi$
$863$	31.6997	1.07907	0.539535	+	0.841963i	$0.318600\pi$
$864$	0	0
$865$	13.1330	0.446535
$866$	0	0
$867$	5.55624	0.188700
$868$	0	0
$869$	−15.4868	−0.525355
$870$	0	0
$871$	34.6492	1.17404
$872$	0	0
$873$	15.1732	0.513534
$874$	0	0
$875$	6.02729	0.203759
$876$	0	0
$877$	19.2543	0.650171	0.325086	−	0.945685i	$-0.394607\pi$
$877$	19.2543	0.650171	0.325086	+	0.945685i	$0.394607\pi$
$878$	0	0
$879$	5.77896	0.194920
$880$	0	0
$881$	46.2169	1.55709	0.778543	−	0.627591i	$-0.215959\pi$
$881$	46.2169	1.55709	0.778543	+	0.627591i	$0.215959\pi$
$882$	0	0
$883$	23.0857	0.776896	0.388448	−	0.921471i	$-0.373011\pi$
$883$	23.0857	0.776896	0.388448	+	0.921471i	$0.373011\pi$
$884$	0	0
$885$	15.2171	0.511517
$886$	0	0
$887$	−30.7720	−1.03322	−0.516611	−	0.856220i	$-0.672807\pi$
$887$	−30.7720	−1.03322	−0.516611	+	0.856220i	$0.672807\pi$
$888$	0	0
$889$	8.08370	0.271119
$890$	0	0
$891$	−6.40011	−0.214412
$892$	0	0
$893$	−68.8793	−2.30496
$894$	0	0
$895$	−32.1224	−1.07373
$896$	0	0
$897$	1.12066	0.0374177
$898$	0	0
$899$	1.40825	0.0469677
$900$	0	0
$901$	41.7729	1.39166
$902$	0	0
$903$	2.78198	0.0925785
$904$	0	0
$905$	45.9416	1.52715
$906$	0	0
$907$	0.611367	0.0203001	0.0101501	−	0.999948i	$-0.496769\pi$
$907$	0.611367	0.0203001	0.0101501	+	0.999948i	$0.496769\pi$
$908$	0	0
$909$	−11.9767	−0.397241
$910$	0	0
$911$	−5.11765	−0.169555	−0.0847776	−	0.996400i	$-0.527018\pi$
$911$	−5.11765	−0.169555	−0.0847776	+	0.996400i	$0.527018\pi$
$912$	0	0
$913$	88.6736	2.93467
$914$	0	0
$915$	5.74030	0.189768
$916$	0	0
$917$	2.88719	0.0953433
$918$	0	0
$919$	−32.2959	−1.06534	−0.532672	−	0.846322i	$-0.678812\pi$
$919$	−32.2959	−1.06534	−0.532672	+	0.846322i	$0.678812\pi$
$920$	0	0
$921$	−12.3201	−0.405962
$922$	0	0
$923$	−53.1816	−1.75049
$924$	0	0
$925$	−1.66394	−0.0547100
$926$	0	0
$927$	0.697308	0.0229026
$928$	0	0
$929$	30.9312	1.01482	0.507410	−	0.861704i	$-0.330603\pi$
$929$	30.9312	1.01482	0.507410	+	0.861704i	$0.330603\pi$
$930$	0	0
$931$	−48.5694	−1.59180
$932$	0	0
$933$	3.99551	0.130807
$934$	0	0
$935$	−51.7435	−1.69219
$936$	0	0
$937$	−35.0285	−1.14433	−0.572165	−	0.820138i	$-0.693896\pi$
$937$	−35.0285	−1.14433	−0.572165	+	0.820138i	$0.693896\pi$
$938$	0	0
$939$	10.3980	0.339327
$940$	0	0
$941$	−0.459980	−0.0149949	−0.00749747	−	0.999972i	$-0.502387\pi$
$941$	−0.459980	−0.0149949	−0.00749747	+	0.999972i	$0.502387\pi$
$942$	0	0
$943$	−0.623528	−0.0203049
$944$	0	0
$945$	1.40553	0.0457220
$946$	0	0
$947$	−8.99124	−0.292176	−0.146088	−	0.989272i	$-0.546668\pi$
$947$	−8.99124	−0.292176	−0.146088	+	0.989272i	$0.546668\pi$
$948$	0	0
$949$	−10.0930	−0.327631
$950$	0	0
$951$	−18.9689	−0.615109
$952$	0	0
$953$	−20.6118	−0.667683	−0.333841	−	0.942629i	$-0.608345\pi$
$953$	−20.6118	−0.667683	−0.333841	+	0.942629i	$0.608345\pi$
$954$	0	0
$955$	−34.7533	−1.12459
$956$	0	0
$957$	−33.1143	−1.07043
$958$	0	0
$959$	−0.203190	−0.00656134
$960$	0	0
$961$	−30.9259	−0.997610
$962$	0	0
$963$	−5.38252	−0.173449
$964$	0	0
$965$	63.3219	2.03840
$966$	0	0
$967$	−20.7297	−0.666621	−0.333311	−	0.942817i	$-0.608166\pi$
$967$	−20.7297	−0.666621	−0.333311	+	0.942817i	$0.608166\pi$
$968$	0	0
$969$	24.6920	0.793220
$970$	0	0
$971$	−22.5434	−0.723451	−0.361726	−	0.932285i	$-0.617812\pi$
$971$	−22.5434	−0.723451	−0.361726	+	0.932285i	$0.617812\pi$
$972$	0	0
$973$	−12.2918	−0.394056
$974$	0	0
$975$	2.55629	0.0818669
$976$	0	0
$977$	−42.8482	−1.37084	−0.685418	−	0.728150i	$-0.740381\pi$
$977$	−42.8482	−1.37084	−0.685418	+	0.728150i	$0.740381\pi$
$978$	0	0
$979$	57.8266	1.84815
$980$	0	0
$981$	11.3581	0.362635
$982$	0	0
$983$	−9.34154	−0.297949	−0.148974	−	0.988841i	$-0.547597\pi$
$983$	−9.34154	−0.297949	−0.148974	+	0.988841i	$0.547597\pi$
$984$	0	0
$985$	−4.13391	−0.131717
$986$	0	0
$987$	5.54975	0.176651
$988$	0	0
$989$	−1.47599	−0.0469337
$990$	0	0
$991$	21.0204	0.667737	0.333868	−	0.942620i	$-0.391646\pi$
$991$	21.0204	0.667737	0.333868	+	0.942620i	$0.391646\pi$
$992$	0	0
$993$	7.95346	0.252396
$994$	0	0
$995$	53.0127	1.68062
$996$	0	0
$997$	26.9967	0.854994	0.427497	−	0.904017i	$-0.359395\pi$
$997$	26.9967	0.854994	0.427497	+	0.904017i	$0.359395\pi$
$998$	0	0
$999$	2.33784	0.0739660

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.7	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.7	✓	26	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.38993 q^{5} +0.588107 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.38993 q^{5} +0.588107 q^{7} +1.00000 q^{9} -6.40011 q^{11} -3.59160 q^{13} +2.38993 q^{15} -3.38286 q^{17} +7.29913 q^{19} -0.588107 q^{21} +0.312022 q^{23} +0.711742 q^{25} -1.00000 q^{27} -5.17403 q^{29} -0.272176 q^{31} +6.40011 q^{33} -1.40553 q^{35} -2.33784 q^{37} +3.59160 q^{39} -1.99835 q^{41} -4.73040 q^{43} -2.38993 q^{45} -9.43664 q^{47} -6.65413 q^{49} +3.38286 q^{51} -12.3484 q^{53} +15.2958 q^{55} -7.29913 q^{57} +6.36719 q^{59} +2.40187 q^{61} +0.588107 q^{63} +8.58365 q^{65} -9.64729 q^{67} -0.312022 q^{69} +14.8072 q^{71} +2.81016 q^{73} -0.711742 q^{75} -3.76395 q^{77} +2.41978 q^{79} +1.00000 q^{81} -13.8550 q^{83} +8.08479 q^{85} +5.17403 q^{87} -9.03526 q^{89} -2.11225 q^{91} +0.272176 q^{93} -17.4444 q^{95} +15.1732 q^{97} -6.40011 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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