LMFDB - Embedded newform 6003.2.a.s.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + \cdots - 256 $ x^20 - 2x^19 - 33x^18 + 64x^17 + 453x^16 - 846x^15 - 3353x^14 + 5985x^13 + 14484x^12 - 24566x^11 - 36791x^10 + 59410x^9 + 52109x^8 - 82362x^7 - 34967x^6 + 60661x^5 + 5201x^4 - 19624x^3 + 2768x^2 + 1408*x - 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.74461$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.74461 q^{2} +1.04367 q^{4} -0.892571 q^{5} +3.89655 q^{7} +1.66842 q^{8} +O(q^{10})$	\(q-1.74461 q^{2} +1.04367 q^{4} -0.892571 q^{5} +3.89655 q^{7} +1.66842 q^{8} +1.55719 q^{10} -4.47217 q^{11} +2.21315 q^{13} -6.79797 q^{14} -4.99809 q^{16} +5.25735 q^{17} +0.817209 q^{19} -0.931553 q^{20} +7.80221 q^{22} -1.00000 q^{23} -4.20332 q^{25} -3.86109 q^{26} +4.06673 q^{28} -1.00000 q^{29} -4.19787 q^{31} +5.38290 q^{32} -9.17204 q^{34} -3.47795 q^{35} +8.35747 q^{37} -1.42571 q^{38} -1.48918 q^{40} +1.95546 q^{41} -1.55005 q^{43} -4.66749 q^{44} +1.74461 q^{46} +1.09928 q^{47} +8.18310 q^{49} +7.33316 q^{50} +2.30981 q^{52} +0.517604 q^{53} +3.99173 q^{55} +6.50108 q^{56} +1.74461 q^{58} +6.99779 q^{59} -0.394355 q^{61} +7.32367 q^{62} +0.605114 q^{64} -1.97539 q^{65} +7.01521 q^{67} +5.48696 q^{68} +6.06767 q^{70} +4.42865 q^{71} -0.726062 q^{73} -14.5806 q^{74} +0.852899 q^{76} -17.4260 q^{77} -7.50686 q^{79} +4.46115 q^{80} -3.41153 q^{82} -8.09267 q^{83} -4.69256 q^{85} +2.70424 q^{86} -7.46146 q^{88} +10.4230 q^{89} +8.62365 q^{91} -1.04367 q^{92} -1.91783 q^{94} -0.729417 q^{95} +12.3546 q^{97} -14.2763 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * q^98

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 $

$2$	−1.74461	−1.23363	−0.616814	−	0.787109i	$-0.711577\pi$
$2$	−1.74461	−1.23363	−0.616814	+	0.787109i	$0.711577\pi$
$3$	0	0
$3$	0	0
$4$	1.04367	0.521837
$5$	−0.892571	−0.399170	−0.199585	−	0.979881i	$-0.563959\pi$
$5$	−0.892571	−0.399170	−0.199585	+	0.979881i	$0.563959\pi$
$6$	0	0
$7$	3.89655	1.47276	0.736379	−	0.676570i	$-0.236534\pi$
$7$	3.89655	1.47276	0.736379	+	0.676570i	$0.236534\pi$
$8$	1.66842	0.589875
$9$	0	0
$10$	1.55719	0.492427
$11$	−4.47217	−1.34841	−0.674205	−	0.738544i	$-0.735514\pi$
$11$	−4.47217	−1.34841	−0.674205	+	0.738544i	$0.735514\pi$
$12$	0	0
$13$	2.21315	0.613817	0.306909	−	0.951739i	$-0.400705\pi$
$13$	2.21315	0.613817	0.306909	+	0.951739i	$0.400705\pi$
$14$	−6.79797	−1.81683
$15$	0	0
$16$	−4.99809	−1.24952
$17$	5.25735	1.27510	0.637548	−	0.770411i	$-0.279949\pi$
$17$	5.25735	1.27510	0.637548	+	0.770411i	$0.279949\pi$
$18$	0	0
$19$	0.817209	0.187481	0.0937403	−	0.995597i	$-0.470118\pi$
$19$	0.817209	0.187481	0.0937403	+	0.995597i	$0.470118\pi$
$20$	−0.931553	−0.208302
$21$	0	0
$22$	7.80221	1.66344
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.20332	−0.840663
$26$	−3.86109	−0.757222
$27$	0	0
$28$	4.06673	0.768539
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−4.19787	−0.753961	−0.376980	−	0.926221i	$-0.623038\pi$
$31$	−4.19787	−0.753961	−0.376980	+	0.926221i	$0.623038\pi$
$32$	5.38290	0.951571
$33$	0	0
$34$	−9.17204	−1.57299
$35$	−3.47795	−0.587880
$36$	0	0
$37$	8.35747	1.37396	0.686980	−	0.726676i	$-0.258936\pi$
$37$	8.35747	1.37396	0.686980	+	0.726676i	$0.258936\pi$
$38$	−1.42571	−0.231281
$39$	0	0
$40$	−1.48918	−0.235460
$41$	1.95546	0.305392	0.152696	−	0.988273i	$-0.451204\pi$
$41$	1.95546	0.305392	0.152696	+	0.988273i	$0.451204\pi$
$42$	0	0
$43$	−1.55005	−0.236381	−0.118190	−	0.992991i	$-0.537709\pi$
$43$	−1.55005	−0.236381	−0.118190	+	0.992991i	$0.537709\pi$
$44$	−4.66749	−0.703650
$45$	0	0
$46$	1.74461	0.257229
$47$	1.09928	0.160347	0.0801735	−	0.996781i	$-0.474453\pi$
$47$	1.09928	0.160347	0.0801735	+	0.996781i	$0.474453\pi$
$48$	0	0
$49$	8.18310	1.16901
$50$	7.33316	1.03707
$51$	0	0
$52$	2.30981	0.320312
$53$	0.517604	0.0710983	0.0355492	−	0.999368i	$-0.488682\pi$
$53$	0.517604	0.0710983	0.0355492	+	0.999368i	$0.488682\pi$
$54$	0	0
$55$	3.99173	0.538245
$56$	6.50108	0.868743
$57$	0	0
$58$	1.74461	0.229079
$59$	6.99779	0.911035	0.455518	−	0.890227i	$-0.349454\pi$
$59$	6.99779	0.911035	0.455518	+	0.890227i	$0.349454\pi$
$60$	0	0
$61$	−0.394355	−0.0504919	−0.0252460	−	0.999681i	$-0.508037\pi$
$61$	−0.394355	−0.0504919	−0.0252460	+	0.999681i	$0.508037\pi$
$62$	7.32367	0.930106
$63$	0	0
$64$	0.605114	0.0756392
$65$	−1.97539	−0.245017
$66$	0	0
$67$	7.01521	0.857044	0.428522	−	0.903531i	$-0.359034\pi$
$67$	7.01521	0.857044	0.428522	+	0.903531i	$0.359034\pi$
$68$	5.48696	0.665392
$69$	0	0
$70$	6.06767	0.725225
$71$	4.42865	0.525585	0.262792	−	0.964852i	$-0.415357\pi$
$71$	4.42865	0.525585	0.262792	+	0.964852i	$0.415357\pi$
$72$	0	0
$73$	−0.726062	−0.0849791	−0.0424896	−	0.999097i	$-0.513529\pi$
$73$	−0.726062	−0.0849791	−0.0424896	+	0.999097i	$0.513529\pi$
$74$	−14.5806	−1.69495
$75$	0	0
$76$	0.852899	0.0978342
$77$	−17.4260	−1.98588
$78$	0	0
$79$	−7.50686	−0.844587	−0.422294	−	0.906459i	$-0.638775\pi$
$79$	−7.50686	−0.844587	−0.422294	+	0.906459i	$0.638775\pi$
$80$	4.46115	0.498772
$81$	0	0
$82$	−3.41153	−0.376740
$83$	−8.09267	−0.888286	−0.444143	−	0.895956i	$-0.646492\pi$
$83$	−8.09267	−0.888286	−0.444143	+	0.895956i	$0.646492\pi$
$84$	0	0
$85$	−4.69256	−0.508980
$86$	2.70424	0.291606
$87$	0	0
$88$	−7.46146	−0.795394
$89$	10.4230	1.10484	0.552418	−	0.833567i	$-0.313705\pi$
$89$	10.4230	1.10484	0.552418	+	0.833567i	$0.313705\pi$
$90$	0	0
$91$	8.62365	0.904004
$92$	−1.04367	−0.108810
$93$	0	0
$94$	−1.91783	−0.197809
$95$	−0.729417	−0.0748366
$96$	0	0
$97$	12.3546	1.25442	0.627209	−	0.778851i	$-0.284197\pi$
$97$	12.3546	1.25442	0.627209	+	0.778851i	$0.284197\pi$
$98$	−14.2763	−1.44213
$99$	0	0
$100$	−4.38689	−0.438689
$101$	2.45708	0.244489	0.122244	−	0.992500i	$-0.460991\pi$
$101$	2.45708	0.244489	0.122244	+	0.992500i	$0.460991\pi$
$102$	0	0
$103$	−9.41863	−0.928046	−0.464023	−	0.885823i	$-0.653594\pi$
$103$	−9.41863	−0.928046	−0.464023	+	0.885823i	$0.653594\pi$
$104$	3.69246	0.362076
$105$	0	0
$106$	−0.903018	−0.0877088
$107$	−7.89713	−0.763445	−0.381722	−	0.924277i	$-0.624669\pi$
$107$	−7.89713	−0.763445	−0.381722	+	0.924277i	$0.624669\pi$
$108$	0	0
$109$	15.7824	1.51168	0.755840	−	0.654756i	$-0.227229\pi$
$109$	15.7824	1.51168	0.755840	+	0.654756i	$0.227229\pi$
$110$	−6.96402	−0.663993
$111$	0	0
$112$	−19.4753	−1.84024
$113$	−3.23551	−0.304372	−0.152186	−	0.988352i	$-0.548631\pi$
$113$	−3.23551	−0.304372	−0.152186	+	0.988352i	$0.548631\pi$
$114$	0	0
$115$	0.892571	0.0832327
$116$	−1.04367	−0.0969027
$117$	0	0
$118$	−12.2084	−1.12388
$119$	20.4855	1.87791
$120$	0	0
$121$	9.00031	0.818210
$122$	0.687996	0.0622882
$123$	0	0
$124$	−4.38121	−0.393444
$125$	8.21461	0.734737
$126$	0	0
$127$	−2.01959	−0.179209	−0.0896046	−	0.995977i	$-0.528560\pi$
$127$	−2.01959	−0.179209	−0.0896046	+	0.995977i	$0.528560\pi$
$128$	−11.8215	−1.04488
$129$	0	0
$130$	3.44630	0.302260
$131$	9.92888	0.867491	0.433745	−	0.901036i	$-0.357192\pi$
$131$	9.92888	0.867491	0.433745	+	0.901036i	$0.357192\pi$
$132$	0	0
$133$	3.18429	0.276113
$134$	−12.2388	−1.05727
$135$	0	0
$136$	8.77147	0.752147
$137$	−6.37499	−0.544652	−0.272326	−	0.962205i	$-0.587793\pi$
$137$	−6.37499	−0.544652	−0.272326	+	0.962205i	$0.587793\pi$
$138$	0	0
$139$	10.8258	0.918236	0.459118	−	0.888375i	$-0.348165\pi$
$139$	10.8258	0.918236	0.459118	+	0.888375i	$0.348165\pi$
$140$	−3.62984	−0.306778
$141$	0	0
$142$	−7.72629	−0.648376
$143$	−9.89758	−0.827678
$144$	0	0
$145$	0.892571	0.0741240
$146$	1.26670	0.104833
$147$	0	0
$148$	8.72247	0.716983
$149$	13.0542	1.06944	0.534721	−	0.845029i	$-0.320417\pi$
$149$	13.0542	1.06944	0.534721	+	0.845029i	$0.320417\pi$
$150$	0	0
$151$	−11.5418	−0.939262	−0.469631	−	0.882863i	$-0.655613\pi$
$151$	−11.5418	−0.939262	−0.469631	+	0.882863i	$0.655613\pi$
$152$	1.36345	0.110590
$153$	0	0
$154$	30.4017	2.44984
$155$	3.74690	0.300958
$156$	0	0
$157$	−9.86331	−0.787178	−0.393589	−	0.919287i	$-0.628767\pi$
$157$	−9.86331	−0.787178	−0.393589	+	0.919287i	$0.628767\pi$
$158$	13.0966	1.04191
$159$	0	0
$160$	−4.80462	−0.379838
$161$	−3.89655	−0.307091
$162$	0	0
$163$	−17.8041	−1.39452	−0.697260	−	0.716818i	$-0.745598\pi$
$163$	−17.8041	−1.39452	−0.697260	+	0.716818i	$0.745598\pi$
$164$	2.04086	0.159365
$165$	0	0
$166$	14.1186	1.09581
$167$	18.7004	1.44708	0.723539	−	0.690283i	$-0.242514\pi$
$167$	18.7004	1.44708	0.723539	+	0.690283i	$0.242514\pi$
$168$	0	0
$169$	−8.10197	−0.623228
$170$	8.18670	0.627891
$171$	0	0
$172$	−1.61775	−0.123352
$173$	−4.95268	−0.376545	−0.188273	−	0.982117i	$-0.560289\pi$
$173$	−4.95268	−0.376545	−0.188273	+	0.982117i	$0.560289\pi$
$174$	0	0
$175$	−16.3784	−1.23809
$176$	22.3523	1.68487
$177$	0	0
$178$	−18.1841	−1.36296
$179$	7.53642	0.563298	0.281649	−	0.959517i	$-0.409119\pi$
$179$	7.53642	0.563298	0.281649	+	0.959517i	$0.409119\pi$
$180$	0	0
$181$	11.5859	0.861175	0.430588	−	0.902549i	$-0.358306\pi$
$181$	11.5859	0.861175	0.430588	+	0.902549i	$0.358306\pi$
$182$	−15.0449	−1.11520
$183$	0	0
$184$	−1.66842	−0.122997
$185$	−7.45964	−0.548443
$186$	0	0
$187$	−23.5118	−1.71935
$188$	1.14729	0.0836750
$189$	0	0
$190$	1.27255	0.0923205
$191$	15.3233	1.10875	0.554376	−	0.832266i	$-0.312957\pi$
$191$	15.3233	1.10875	0.554376	+	0.832266i	$0.312957\pi$
$192$	0	0
$193$	7.59424	0.546646	0.273323	−	0.961922i	$-0.411877\pi$
$193$	7.59424	0.546646	0.273323	+	0.961922i	$0.411877\pi$
$194$	−21.5540	−1.54748
$195$	0	0
$196$	8.54048	0.610034
$197$	2.72033	0.193816	0.0969078	−	0.995293i	$-0.469105\pi$
$197$	2.72033	0.193816	0.0969078	+	0.995293i	$0.469105\pi$
$198$	0	0
$199$	−11.5802	−0.820896	−0.410448	−	0.911884i	$-0.634628\pi$
$199$	−11.5802	−0.820896	−0.410448	+	0.911884i	$0.634628\pi$
$200$	−7.01290	−0.495887
$201$	0	0
$202$	−4.28666	−0.301608
$203$	−3.89655	−0.273484
$204$	0	0
$205$	−1.74539	−0.121903
$206$	16.4319	1.14486
$207$	0	0
$208$	−11.0615	−0.766979
$209$	−3.65470	−0.252801
$210$	0	0
$211$	−2.39087	−0.164594	−0.0822970	−	0.996608i	$-0.526226\pi$
$211$	−2.39087	−0.164594	−0.0822970	+	0.996608i	$0.526226\pi$
$212$	0.540209	0.0371017
$213$	0	0
$214$	13.7774	0.941806
$215$	1.38353	0.0943560
$216$	0	0
$217$	−16.3572	−1.11040
$218$	−27.5342	−1.86485
$219$	0	0
$220$	4.16606	0.280876
$221$	11.6353	0.782676
$222$	0	0
$223$	4.99410	0.334430	0.167215	−	0.985920i	$-0.446523\pi$
$223$	4.99410	0.334430	0.167215	+	0.985920i	$0.446523\pi$
$224$	20.9747	1.40143
$225$	0	0
$226$	5.64472	0.375481
$227$	−9.07560	−0.602369	−0.301184	−	0.953566i	$-0.597382\pi$
$227$	−9.07560	−0.602369	−0.301184	+	0.953566i	$0.597382\pi$
$228$	0	0
$229$	−10.2011	−0.674106	−0.337053	−	0.941486i	$-0.609430\pi$
$229$	−10.2011	−0.674106	−0.337053	+	0.941486i	$0.609430\pi$
$230$	−1.55719	−0.102678
$231$	0	0
$232$	−1.66842	−0.109537
$233$	−17.5259	−1.14816	−0.574080	−	0.818799i	$-0.694640\pi$
$233$	−17.5259	−1.14816	−0.574080	+	0.818799i	$0.694640\pi$
$234$	0	0
$235$	−0.981189	−0.0640057
$236$	7.30341	0.475412
$237$	0	0
$238$	−35.7393	−2.31664
$239$	−18.0140	−1.16523	−0.582614	−	0.812749i	$-0.697970\pi$
$239$	−18.0140	−1.16523	−0.582614	+	0.812749i	$0.697970\pi$
$240$	0	0
$241$	−10.3862	−0.669037	−0.334519	−	0.942389i	$-0.608574\pi$
$241$	−10.3862	−0.669037	−0.334519	+	0.942389i	$0.608574\pi$
$242$	−15.7021	−1.00937
$243$	0	0
$244$	−0.411577	−0.0263485
$245$	−7.30399	−0.466635
$246$	0	0
$247$	1.80861	0.115079
$248$	−7.00381	−0.444743
$249$	0	0
$250$	−14.3313	−0.906392
$251$	−29.6513	−1.87158	−0.935788	−	0.352564i	$-0.885310\pi$
$251$	−29.6513	−1.87158	−0.935788	+	0.352564i	$0.885310\pi$
$252$	0	0
$253$	4.47217	0.281163
$254$	3.52339	0.221077
$255$	0	0
$256$	19.4137	1.21336
$257$	21.8985	1.36599	0.682996	−	0.730422i	$-0.260677\pi$
$257$	21.8985	1.36599	0.682996	+	0.730422i	$0.260677\pi$
$258$	0	0
$259$	32.5653	2.02351
$260$	−2.06167	−0.127859
$261$	0	0
$262$	−17.3221	−1.07016
$263$	7.83697	0.483248	0.241624	−	0.970370i	$-0.422320\pi$
$263$	7.83697	0.483248	0.241624	+	0.970370i	$0.422320\pi$
$264$	0	0
$265$	−0.461998	−0.0283803
$266$	−5.55536	−0.340621
$267$	0	0
$268$	7.32159	0.447237
$269$	−6.81186	−0.415326	−0.207663	−	0.978200i	$-0.566586\pi$
$269$	−6.81186	−0.415326	−0.207663	+	0.978200i	$0.566586\pi$
$270$	0	0
$271$	−20.9427	−1.27218	−0.636090	−	0.771615i	$-0.719449\pi$
$271$	−20.9427	−1.27218	−0.636090	+	0.771615i	$0.719449\pi$
$272$	−26.2767	−1.59326
$273$	0	0
$274$	11.1219	0.671898
$275$	18.7980	1.13356
$276$	0	0
$277$	30.7080	1.84506	0.922531	−	0.385922i	$-0.126117\pi$
$277$	30.7080	1.84506	0.922531	+	0.385922i	$0.126117\pi$
$278$	−18.8869	−1.13276
$279$	0	0
$280$	−5.80267	−0.346776
$281$	29.8255	1.77924	0.889620	−	0.456701i	$-0.150969\pi$
$281$	29.8255	1.77924	0.889620	+	0.456701i	$0.150969\pi$
$282$	0	0
$283$	−17.8218	−1.05940	−0.529699	−	0.848185i	$-0.677695\pi$
$283$	−17.8218	−1.05940	−0.529699	+	0.848185i	$0.677695\pi$
$284$	4.62207	0.274269
$285$	0	0
$286$	17.2675	1.02105
$287$	7.61956	0.449768
$288$	0	0
$289$	10.6397	0.625868
$290$	−1.55719	−0.0914414
$291$	0	0
$292$	−0.757772	−0.0443452
$293$	27.0066	1.57774	0.788871	−	0.614559i	$-0.210666\pi$
$293$	27.0066	1.57774	0.788871	+	0.614559i	$0.210666\pi$
$294$	0	0
$295$	−6.24603	−0.363658
$296$	13.9438	0.810465
$297$	0	0
$298$	−22.7745	−1.31929
$299$	−2.21315	−0.127990
$300$	0	0
$301$	−6.03985	−0.348131
$302$	20.1360	1.15870
$303$	0	0
$304$	−4.08448	−0.234261
$305$	0.351989	0.0201549
$306$	0	0
$307$	12.7269	0.726365	0.363183	−	0.931718i	$-0.381690\pi$
$307$	12.7269	0.726365	0.363183	+	0.931718i	$0.381690\pi$
$308$	−18.1871	−1.03631
$309$	0	0
$310$	−6.53689	−0.371270
$311$	17.6022	0.998131	0.499066	−	0.866564i	$-0.333677\pi$
$311$	17.6022	0.998131	0.499066	+	0.866564i	$0.333677\pi$
$312$	0	0
$313$	−10.2142	−0.577341	−0.288671	−	0.957428i	$-0.593213\pi$
$313$	−10.2142	−0.577341	−0.288671	+	0.957428i	$0.593213\pi$
$314$	17.2077	0.971084
$315$	0	0
$316$	−7.83471	−0.440737
$317$	16.0811	0.903207	0.451603	−	0.892219i	$-0.350852\pi$
$317$	16.0811	0.903207	0.451603	+	0.892219i	$0.350852\pi$
$318$	0	0
$319$	4.47217	0.250393
$320$	−0.540107	−0.0301929
$321$	0	0
$322$	6.79797	0.378836
$323$	4.29635	0.239056
$324$	0	0
$325$	−9.30257	−0.516014
$326$	31.0612	1.72032
$327$	0	0
$328$	3.26253	0.180143
$329$	4.28342	0.236152
$330$	0	0
$331$	31.5407	1.73364	0.866818	−	0.498625i	$-0.166161\pi$
$331$	31.5407	1.73364	0.866818	+	0.498625i	$0.166161\pi$
$332$	−8.44611	−0.463540
$333$	0	0
$334$	−32.6249	−1.78516
$335$	−6.26157	−0.342106
$336$	0	0
$337$	−10.3837	−0.565635	−0.282817	−	0.959174i	$-0.591269\pi$
$337$	−10.3837	−0.565635	−0.282817	+	0.959174i	$0.591269\pi$
$338$	14.1348	0.768831
$339$	0	0
$340$	−4.89750	−0.265604
$341$	18.7736	1.01665
$342$	0	0
$343$	4.60999	0.248916
$344$	−2.58614	−0.139435
$345$	0	0
$346$	8.64050	0.464516
$347$	16.0250	0.860266	0.430133	−	0.902765i	$-0.358467\pi$
$347$	16.0250	0.860266	0.430133	+	0.902765i	$0.358467\pi$
$348$	0	0
$349$	36.3793	1.94734	0.973670	−	0.227962i	$-0.0732061\pi$
$349$	36.3793	1.94734	0.973670	+	0.227962i	$0.0732061\pi$
$350$	28.5740	1.52735
$351$	0	0
$352$	−24.0732	−1.28311
$353$	−26.5184	−1.41143	−0.705716	−	0.708495i	$-0.749374\pi$
$353$	−26.5184	−1.41143	−0.705716	+	0.708495i	$0.749374\pi$
$354$	0	0
$355$	−3.95289	−0.209797
$356$	10.8782	0.576544
$357$	0	0
$358$	−13.1481	−0.694900
$359$	−9.97949	−0.526697	−0.263349	−	0.964701i	$-0.584827\pi$
$359$	−9.97949	−0.526697	−0.263349	+	0.964701i	$0.584827\pi$
$360$	0	0
$361$	−18.3322	−0.964851
$362$	−20.2130	−1.06237
$363$	0	0
$364$	9.00027	0.471743
$365$	0.648062	0.0339211
$366$	0	0
$367$	−0.780644	−0.0407493	−0.0203746	−	0.999792i	$-0.506486\pi$
$367$	−0.780644	−0.0407493	−0.0203746	+	0.999792i	$0.506486\pi$
$368$	4.99809	0.260544
$369$	0	0
$370$	13.0142	0.676575
$371$	2.01687	0.104711
$372$	0	0
$373$	−19.3681	−1.00284	−0.501420	−	0.865204i	$-0.667189\pi$
$373$	−19.3681	−1.00284	−0.501420	+	0.865204i	$0.667189\pi$
$374$	41.0189	2.12104
$375$	0	0
$376$	1.83407	0.0945848
$377$	−2.21315	−0.113983
$378$	0	0
$379$	24.2021	1.24318	0.621590	−	0.783343i	$-0.286487\pi$
$379$	24.2021	1.24318	0.621590	+	0.783343i	$0.286487\pi$
$380$	−0.761273	−0.0390525
$381$	0	0
$382$	−26.7332	−1.36779
$383$	25.5806	1.30711	0.653553	−	0.756881i	$-0.273278\pi$
$383$	25.5806	1.30711	0.653553	+	0.756881i	$0.273278\pi$
$384$	0	0
$385$	15.5540	0.792704
$386$	−13.2490	−0.674357
$387$	0	0
$388$	12.8942	0.654601
$389$	−7.52819	−0.381694	−0.190847	−	0.981620i	$-0.561123\pi$
$389$	−7.52819	−0.381694	−0.190847	+	0.981620i	$0.561123\pi$
$390$	0	0
$391$	−5.25735	−0.265876
$392$	13.6528	0.689572
$393$	0	0
$394$	−4.74593	−0.239096
$395$	6.70040	0.337134
$396$	0	0
$397$	32.5670	1.63449	0.817245	−	0.576290i	$-0.195500\pi$
$397$	32.5670	1.63449	0.817245	+	0.576290i	$0.195500\pi$
$398$	20.2029	1.01268
$399$	0	0
$400$	21.0086	1.05043
$401$	13.5106	0.674689	0.337345	−	0.941381i	$-0.390471\pi$
$401$	13.5106	0.674689	0.337345	+	0.941381i	$0.390471\pi$
$402$	0	0
$403$	−9.29053	−0.462794
$404$	2.56439	0.127583
$405$	0	0
$406$	6.79797	0.337378
$407$	−37.3760	−1.85266
$408$	0	0
$409$	21.3259	1.05450	0.527248	−	0.849711i	$-0.323224\pi$
$409$	21.3259	1.05450	0.527248	+	0.849711i	$0.323224\pi$
$410$	3.04503	0.150383
$411$	0	0
$412$	−9.82998	−0.484288
$413$	27.2672	1.34173
$414$	0	0
$415$	7.22328	0.354577
$416$	11.9132	0.584091
$417$	0	0
$418$	6.37603	0.311862
$419$	3.88439	0.189765	0.0948825	−	0.995488i	$-0.469752\pi$
$419$	3.88439	0.189765	0.0948825	+	0.995488i	$0.469752\pi$
$420$	0	0
$421$	−0.915374	−0.0446126	−0.0223063	−	0.999751i	$-0.507101\pi$
$421$	−0.915374	−0.0446126	−0.0223063	+	0.999751i	$0.507101\pi$
$422$	4.17113	0.203048
$423$	0	0
$424$	0.863580	0.0419391
$425$	−22.0983	−1.07193
$426$	0	0
$427$	−1.53662	−0.0743623
$428$	−8.24203	−0.398393
$429$	0	0
$430$	−2.41373	−0.116400
$431$	3.51349	0.169239	0.0846195	−	0.996413i	$-0.473033\pi$
$431$	3.51349	0.169239	0.0846195	+	0.996413i	$0.473033\pi$
$432$	0	0
$433$	−15.7078	−0.754868	−0.377434	−	0.926036i	$-0.623194\pi$
$433$	−15.7078	−0.754868	−0.377434	+	0.926036i	$0.623194\pi$
$434$	28.5370	1.36982
$435$	0	0
$436$	16.4717	0.788850
$437$	−0.817209	−0.0390924
$438$	0	0
$439$	22.0238	1.05114	0.525569	−	0.850751i	$-0.323852\pi$
$439$	22.0238	1.05114	0.525569	+	0.850751i	$0.323852\pi$
$440$	6.65988	0.317497
$441$	0	0
$442$	−20.2991	−0.965530
$443$	−17.2084	−0.817595	−0.408798	−	0.912625i	$-0.634052\pi$
$443$	−17.2084	−0.817595	−0.408798	+	0.912625i	$0.634052\pi$
$444$	0	0
$445$	−9.30327	−0.441017
$446$	−8.71277	−0.412562
$447$	0	0
$448$	2.35786	0.111398
$449$	20.5864	0.971531	0.485766	−	0.874089i	$-0.338541\pi$
$449$	20.5864	0.971531	0.485766	+	0.874089i	$0.338541\pi$
$450$	0	0
$451$	−8.74516	−0.411794
$452$	−3.37682	−0.158832
$453$	0	0
$454$	15.8334	0.743099
$455$	−7.69722	−0.360851
$456$	0	0
$457$	6.35046	0.297062	0.148531	−	0.988908i	$-0.452545\pi$
$457$	6.35046	0.297062	0.148531	+	0.988908i	$0.452545\pi$
$458$	17.7969	0.831596
$459$	0	0
$460$	0.931553	0.0434339
$461$	−22.8900	−1.06609	−0.533047	−	0.846086i	$-0.678953\pi$
$461$	−22.8900	−1.06609	−0.533047	+	0.846086i	$0.678953\pi$
$462$	0	0
$463$	5.69016	0.264444	0.132222	−	0.991220i	$-0.457789\pi$
$463$	5.69016	0.264444	0.132222	+	0.991220i	$0.457789\pi$
$464$	4.99809	0.232031
$465$	0	0
$466$	30.5759	1.41640
$467$	−31.9619	−1.47902	−0.739510	−	0.673146i	$-0.764943\pi$
$467$	−31.9619	−1.47902	−0.739510	+	0.673146i	$0.764943\pi$
$468$	0	0
$469$	27.3351	1.26222
$470$	1.71180	0.0789592
$471$	0	0
$472$	11.6753	0.537397
$473$	6.93209	0.318738
$474$	0	0
$475$	−3.43499	−0.157608
$476$	21.3802	0.979960
$477$	0	0
$478$	31.4274	1.43746
$479$	−32.8406	−1.50053	−0.750263	−	0.661140i	$-0.770073\pi$
$479$	−32.8406	−1.50053	−0.750263	+	0.661140i	$0.770073\pi$
$480$	0	0
$481$	18.4963	0.843360
$482$	18.1200	0.825342
$483$	0	0
$484$	9.39338	0.426972
$485$	−11.0273	−0.500726
$486$	0	0
$487$	38.6272	1.75036	0.875182	−	0.483793i	$-0.160741\pi$
$487$	38.6272	1.75036	0.875182	+	0.483793i	$0.160741\pi$
$488$	−0.657949	−0.0297839
$489$	0	0
$490$	12.7426	0.575654
$491$	23.2977	1.05141	0.525704	−	0.850667i	$-0.323802\pi$
$491$	23.2977	1.05141	0.525704	+	0.850667i	$0.323802\pi$
$492$	0	0
$493$	−5.25735	−0.236779
$494$	−3.15532	−0.141964
$495$	0	0
$496$	20.9814	0.942091
$497$	17.2565	0.774058
$498$	0	0
$499$	14.8514	0.664841	0.332421	−	0.943131i	$-0.392135\pi$
$499$	14.8514	0.664841	0.332421	+	0.943131i	$0.392135\pi$
$500$	8.57337	0.383413
$501$	0	0
$502$	51.7301	2.30883
$503$	44.0329	1.96333	0.981666	−	0.190610i	$-0.0610466\pi$
$503$	44.0329	1.96333	0.981666	+	0.190610i	$0.0610466\pi$
$504$	0	0
$505$	−2.19312	−0.0975925
$506$	−7.80221	−0.346850
$507$	0	0
$508$	−2.10779	−0.0935179
$509$	−26.1277	−1.15809	−0.579044	−	0.815296i	$-0.696574\pi$
$509$	−26.1277	−1.15809	−0.579044	+	0.815296i	$0.696574\pi$
$510$	0	0
$511$	−2.82914	−0.125154
$512$	−10.2264	−0.451947
$513$	0	0
$514$	−38.2044	−1.68513
$515$	8.40680	0.370448
$516$	0	0
$517$	−4.91619	−0.216214
$518$	−56.8138	−2.49626
$519$	0	0
$520$	−3.29578	−0.144530
$521$	19.9360	0.873412	0.436706	−	0.899604i	$-0.356145\pi$
$521$	19.9360	0.873412	0.436706	+	0.899604i	$0.356145\pi$
$522$	0	0
$523$	31.3922	1.37268	0.686342	−	0.727279i	$-0.259215\pi$
$523$	31.3922	1.37268	0.686342	+	0.727279i	$0.259215\pi$
$524$	10.3625	0.452689
$525$	0	0
$526$	−13.6725	−0.596148
$527$	−22.0697	−0.961371
$528$	0	0
$529$	1.00000	0.0434783
$530$	0.806007	0.0350107
$531$	0	0
$532$	3.32336	0.144086
$533$	4.32773	0.187455
$534$	0	0
$535$	7.04875	0.304744
$536$	11.7043	0.505549
$537$	0	0
$538$	11.8841	0.512358
$539$	−36.5962	−1.57631
$540$	0	0
$541$	−7.55331	−0.324742	−0.162371	−	0.986730i	$-0.551914\pi$
$541$	−7.55331	−0.324742	−0.162371	+	0.986730i	$0.551914\pi$
$542$	36.5370	1.56940
$543$	0	0
$544$	28.2998	1.21334
$545$	−14.0869	−0.603417
$546$	0	0
$547$	−0.579523	−0.0247786	−0.0123893	−	0.999923i	$-0.503944\pi$
$547$	−0.579523	−0.0247786	−0.0123893	+	0.999923i	$0.503944\pi$
$548$	−6.65341	−0.284220
$549$	0	0
$550$	−32.7951	−1.39839
$551$	−0.817209	−0.0348143
$552$	0	0
$553$	−29.2508	−1.24387
$554$	−53.5735	−2.27612
$555$	0	0
$556$	11.2986	0.479169
$557$	10.3346	0.437891	0.218945	−	0.975737i	$-0.429738\pi$
$557$	10.3346	0.437891	0.218945	+	0.975737i	$0.429738\pi$
$558$	0	0
$559$	−3.43050	−0.145095
$560$	17.3831	0.734570
$561$	0	0
$562$	−52.0340	−2.19492
$563$	15.2717	0.643627	0.321814	−	0.946803i	$-0.395708\pi$
$563$	15.2717	0.643627	0.321814	+	0.946803i	$0.395708\pi$
$564$	0	0
$565$	2.88793	0.121496
$566$	31.0922	1.30690
$567$	0	0
$568$	7.38885	0.310029
$569$	−10.9113	−0.457424	−0.228712	−	0.973494i	$-0.573451\pi$
$569$	−10.9113	−0.457424	−0.228712	+	0.973494i	$0.573451\pi$
$570$	0	0
$571$	31.1468	1.30345	0.651726	−	0.758455i	$-0.274045\pi$
$571$	31.1468	1.30345	0.651726	+	0.758455i	$0.274045\pi$
$572$	−10.3298	−0.431913
$573$	0	0
$574$	−13.2932	−0.554846
$575$	4.20332	0.175290
$576$	0	0
$577$	20.3250	0.846141	0.423071	−	0.906097i	$-0.360952\pi$
$577$	20.3250	0.846141	0.423071	+	0.906097i	$0.360952\pi$
$578$	−18.5622	−0.772087
$579$	0	0
$580$	0.931553	0.0386806
$581$	−31.5335	−1.30823
$582$	0	0
$583$	−2.31481	−0.0958697
$584$	−1.21138	−0.0501271
$585$	0	0
$586$	−47.1161	−1.94635
$587$	25.9647	1.07168	0.535839	−	0.844320i	$-0.319996\pi$
$587$	25.9647	1.07168	0.535839	+	0.844320i	$0.319996\pi$
$588$	0	0
$589$	−3.43054	−0.141353
$590$	10.8969	0.448618
$591$	0	0
$592$	−41.7714	−1.71679
$593$	26.0859	1.07122	0.535609	−	0.844466i	$-0.320082\pi$
$593$	26.0859	1.07122	0.535609	+	0.844466i	$0.320082\pi$
$594$	0	0
$595$	−18.2848	−0.749603
$596$	13.6243	0.558074
$597$	0	0
$598$	3.86109	0.157892
$599$	−28.1346	−1.14955	−0.574774	−	0.818312i	$-0.694910\pi$
$599$	−28.1346	−1.14955	−0.574774	+	0.818312i	$0.694910\pi$
$600$	0	0
$601$	−16.9863	−0.692884	−0.346442	−	0.938071i	$-0.612610\pi$
$601$	−16.9863	−0.692884	−0.346442	+	0.938071i	$0.612610\pi$
$602$	10.5372	0.429464
$603$	0	0
$604$	−12.0459	−0.490141
$605$	−8.03341	−0.326605
$606$	0	0
$607$	7.50970	0.304809	0.152405	−	0.988318i	$-0.451298\pi$
$607$	7.50970	0.304809	0.152405	+	0.988318i	$0.451298\pi$
$608$	4.39895	0.178401
$609$	0	0
$610$	−0.614085	−0.0248636
$611$	2.43288	0.0984238
$612$	0	0
$613$	44.6186	1.80213	0.901065	−	0.433684i	$-0.142787\pi$
$613$	44.6186	1.80213	0.901065	+	0.433684i	$0.142787\pi$
$614$	−22.2036	−0.896064
$615$	0	0
$616$	−29.0739	−1.17142
$617$	−9.13991	−0.367959	−0.183980	−	0.982930i	$-0.558898\pi$
$617$	−9.13991	−0.367959	−0.183980	+	0.982930i	$0.558898\pi$
$618$	0	0
$619$	−12.9320	−0.519779	−0.259890	−	0.965638i	$-0.583686\pi$
$619$	−12.9320	−0.519779	−0.259890	+	0.965638i	$0.583686\pi$
$620$	3.91054	0.157051
$621$	0	0
$622$	−30.7091	−1.23132
$623$	40.6138	1.62716
$624$	0	0
$625$	13.6845	0.547378
$626$	17.8198	0.712224
$627$	0	0
$628$	−10.2941	−0.410778
$629$	43.9382	1.75193
$630$	0	0
$631$	−10.7411	−0.427598	−0.213799	−	0.976878i	$-0.568584\pi$
$631$	−10.7411	−0.427598	−0.213799	+	0.976878i	$0.568584\pi$
$632$	−12.5246	−0.498201
$633$	0	0
$634$	−28.0554	−1.11422
$635$	1.80262	0.0715349
$636$	0	0
$637$	18.1104	0.717561
$638$	−7.80221	−0.308892
$639$	0	0
$640$	10.5515	0.417085
$641$	7.68854	0.303679	0.151839	−	0.988405i	$-0.451480\pi$
$641$	7.68854	0.303679	0.151839	+	0.988405i	$0.451480\pi$
$642$	0	0
$643$	17.2471	0.680161	0.340080	−	0.940396i	$-0.389546\pi$
$643$	17.2471	0.680161	0.340080	+	0.940396i	$0.389546\pi$
$644$	−4.06673	−0.160251
$645$	0	0
$646$	−7.49547	−0.294905
$647$	4.19619	0.164969	0.0824846	−	0.996592i	$-0.473714\pi$
$647$	4.19619	0.164969	0.0824846	+	0.996592i	$0.473714\pi$
$648$	0	0
$649$	−31.2953	−1.22845
$650$	16.2294	0.636569
$651$	0	0
$652$	−18.5816	−0.727712
$653$	6.76412	0.264701	0.132350	−	0.991203i	$-0.457748\pi$
$653$	6.76412	0.264701	0.132350	+	0.991203i	$0.457748\pi$
$654$	0	0
$655$	−8.86223	−0.346276
$656$	−9.77358	−0.381594
$657$	0	0
$658$	−7.47290	−0.291324
$659$	35.0908	1.36694	0.683471	−	0.729977i	$-0.260469\pi$
$659$	35.0908	1.36694	0.683471	+	0.729977i	$0.260469\pi$
$660$	0	0
$661$	18.0132	0.700632	0.350316	−	0.936632i	$-0.386074\pi$
$661$	18.0132	0.700632	0.350316	+	0.936632i	$0.386074\pi$
$662$	−55.0264	−2.13866
$663$	0	0
$664$	−13.5020	−0.523978
$665$	−2.84221	−0.110216
$666$	0	0
$667$	1.00000	0.0387202
$668$	19.5171	0.755139
$669$	0	0
$670$	10.9240	0.422032
$671$	1.76362	0.0680838
$672$	0	0
$673$	37.7043	1.45339	0.726697	−	0.686958i	$-0.241054\pi$
$673$	37.7043	1.45339	0.726697	+	0.686958i	$0.241054\pi$
$674$	18.1155	0.697782
$675$	0	0
$676$	−8.45581	−0.325223
$677$	−10.7829	−0.414421	−0.207211	−	0.978296i	$-0.566439\pi$
$677$	−10.7829	−0.414421	−0.207211	+	0.978296i	$0.566439\pi$
$678$	0	0
$679$	48.1402	1.84745
$680$	−7.82916	−0.300234
$681$	0	0
$682$	−32.7527	−1.25416
$683$	−17.3465	−0.663744	−0.331872	−	0.943324i	$-0.607680\pi$
$683$	−17.3465	−0.663744	−0.331872	+	0.943324i	$0.607680\pi$
$684$	0	0
$685$	5.69013	0.217409
$686$	−8.04266	−0.307070
$687$	0	0
$688$	7.74730	0.295363
$689$	1.14553	0.0436414
$690$	0	0
$691$	7.94187	0.302123	0.151061	−	0.988524i	$-0.451731\pi$
$691$	7.94187	0.302123	0.151061	+	0.988524i	$0.451731\pi$
$692$	−5.16898	−0.196495
$693$	0	0
$694$	−27.9574	−1.06125
$695$	−9.66283	−0.366532
$696$	0	0
$697$	10.2806	0.389404
$698$	−63.4678	−2.40229
$699$	0	0
$700$	−17.0937	−0.646083
$701$	−31.3499	−1.18407	−0.592035	−	0.805913i	$-0.701675\pi$
$701$	−31.3499	−1.18407	−0.592035	+	0.805913i	$0.701675\pi$
$702$	0	0
$703$	6.82980	0.257591
$704$	−2.70617	−0.101993
$705$	0	0
$706$	46.2643	1.74118
$707$	9.57414	0.360073
$708$	0	0
$709$	3.77036	0.141599	0.0707995	−	0.997491i	$-0.477445\pi$
$709$	3.77036	0.141599	0.0707995	+	0.997491i	$0.477445\pi$
$710$	6.89626	0.258812
$711$	0	0
$712$	17.3899	0.651716
$713$	4.19787	0.157212
$714$	0	0
$715$	8.83430	0.330384
$716$	7.86556	0.293950
$717$	0	0
$718$	17.4103	0.649748
$719$	−2.39941	−0.0894828	−0.0447414	−	0.998999i	$-0.514246\pi$
$719$	−2.39941	−0.0894828	−0.0447414	+	0.998999i	$0.514246\pi$
$720$	0	0
$721$	−36.7002	−1.36679
$722$	31.9825	1.19027
$723$	0	0
$724$	12.0919	0.449393
$725$	4.20332	0.156107
$726$	0	0
$727$	5.76498	0.213811	0.106906	−	0.994269i	$-0.465906\pi$
$727$	5.76498	0.213811	0.106906	+	0.994269i	$0.465906\pi$
$728$	14.3879	0.533250
$729$	0	0
$730$	−1.13062	−0.0418460
$731$	−8.14917	−0.301408
$732$	0	0
$733$	−9.54781	−0.352656	−0.176328	−	0.984331i	$-0.556422\pi$
$733$	−9.54781	−0.352656	−0.176328	+	0.984331i	$0.556422\pi$
$734$	1.36192	0.0502694
$735$	0	0
$736$	−5.38290	−0.198416
$737$	−31.3732	−1.15565
$738$	0	0
$739$	7.33328	0.269759	0.134879	−	0.990862i	$-0.456935\pi$
$739$	7.33328	0.269759	0.134879	+	0.990862i	$0.456935\pi$
$740$	−7.78542	−0.286198
$741$	0	0
$742$	−3.51865	−0.129174
$743$	−33.1647	−1.21670	−0.608348	−	0.793671i	$-0.708167\pi$
$743$	−33.1647	−1.21670	−0.608348	+	0.793671i	$0.708167\pi$
$744$	0	0
$745$	−11.6518	−0.426889
$746$	33.7898	1.23713
$747$	0	0
$748$	−24.5386	−0.897221
$749$	−30.7716	−1.12437
$750$	0	0
$751$	−18.2886	−0.667362	−0.333681	−	0.942686i	$-0.608291\pi$
$751$	−18.2886	−0.667362	−0.333681	+	0.942686i	$0.608291\pi$
$752$	−5.49432	−0.200357
$753$	0	0
$754$	3.86109	0.140613
$755$	10.3019	0.374925
$756$	0	0
$757$	−16.9297	−0.615320	−0.307660	−	0.951496i	$-0.599546\pi$
$757$	−16.9297	−0.615320	−0.307660	+	0.951496i	$0.599546\pi$
$758$	−42.2234	−1.53362
$759$	0	0
$760$	−1.21697	−0.0441442
$761$	14.6565	0.531297	0.265649	−	0.964070i	$-0.414414\pi$
$761$	14.6565	0.531297	0.265649	+	0.964070i	$0.414414\pi$
$762$	0	0
$763$	61.4969	2.22634
$764$	15.9925	0.578588
$765$	0	0
$766$	−44.6282	−1.61248
$767$	15.4872	0.559209
$768$	0	0
$769$	−15.9253	−0.574282	−0.287141	−	0.957888i	$-0.592705\pi$
$769$	−15.9253	−0.574282	−0.287141	+	0.957888i	$0.592705\pi$
$770$	−27.1357	−0.977901
$771$	0	0
$772$	7.92591	0.285260
$773$	−29.0904	−1.04631	−0.523154	−	0.852238i	$-0.675245\pi$
$773$	−29.0904	−1.04631	−0.523154	+	0.852238i	$0.675245\pi$
$774$	0	0
$775$	17.6450	0.633827
$776$	20.6126	0.739950
$777$	0	0
$778$	13.1338	0.470869
$779$	1.59802	0.0572551
$780$	0	0
$781$	−19.8057	−0.708704
$782$	9.17204	0.327992
$783$	0	0
$784$	−40.8999	−1.46071
$785$	8.80370	0.314218
$786$	0	0
$787$	4.37350	0.155898	0.0779492	−	0.996957i	$-0.475163\pi$
$787$	4.37350	0.155898	0.0779492	+	0.996957i	$0.475163\pi$
$788$	2.83914	0.101140
$789$	0	0
$790$	−11.6896	−0.415898
$791$	−12.6073	−0.448266
$792$	0	0
$793$	−0.872766	−0.0309928
$794$	−56.8168	−2.01635
$795$	0	0
$796$	−12.0859	−0.428374
$797$	3.40045	0.120450	0.0602251	−	0.998185i	$-0.480818\pi$
$797$	3.40045	0.120450	0.0602251	+	0.998185i	$0.480818\pi$
$798$	0	0
$799$	5.77932	0.204458
$800$	−22.6260	−0.799951
$801$	0	0
$802$	−23.5708	−0.832315
$803$	3.24707	0.114587
$804$	0	0
$805$	3.47795	0.122582
$806$	16.2084	0.570916
$807$	0	0
$808$	4.09944	0.144218
$809$	30.2188	1.06244	0.531218	−	0.847235i	$-0.321734\pi$
$809$	30.2188	1.06244	0.531218	+	0.847235i	$0.321734\pi$
$810$	0	0
$811$	35.8361	1.25838	0.629188	−	0.777253i	$-0.283388\pi$
$811$	35.8361	1.25838	0.629188	+	0.777253i	$0.283388\pi$
$812$	−4.06673	−0.142714
$813$	0	0
$814$	65.2067	2.28549
$815$	15.8914	0.556651
$816$	0	0
$817$	−1.26672	−0.0443168
$818$	−37.2054	−1.30086
$819$	0	0
$820$	−1.82162	−0.0636136
$821$	35.2255	1.22938	0.614689	−	0.788770i	$-0.289282\pi$
$821$	35.2255	1.22938	0.614689	+	0.788770i	$0.289282\pi$
$822$	0	0
$823$	−31.0109	−1.08097	−0.540485	−	0.841353i	$-0.681759\pi$
$823$	−31.0109	−1.08097	−0.540485	+	0.841353i	$0.681759\pi$
$824$	−15.7142	−0.547431
$825$	0	0
$826$	−47.5708	−1.65520
$827$	6.42598	0.223453	0.111727	−	0.993739i	$-0.464362\pi$
$827$	6.42598	0.223453	0.111727	+	0.993739i	$0.464362\pi$
$828$	0	0
$829$	−9.29528	−0.322838	−0.161419	−	0.986886i	$-0.551607\pi$
$829$	−9.29528	−0.322838	−0.161419	+	0.986886i	$0.551607\pi$
$830$	−12.6018	−0.437416
$831$	0	0
$832$	1.33921	0.0464287
$833$	43.0214	1.49060
$834$	0	0
$835$	−16.6914	−0.577630
$836$	−3.81431	−0.131921
$837$	0	0
$838$	−6.77676	−0.234099
$839$	−19.1022	−0.659481	−0.329740	−	0.944072i	$-0.606961\pi$
$839$	−19.1022	−0.659481	−0.329740	+	0.944072i	$0.606961\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	1.59697	0.0550353
$843$	0	0
$844$	−2.49528	−0.0858912
$845$	7.23158	0.248774
$846$	0	0
$847$	35.0701	1.20502
$848$	−2.58703	−0.0888390
$849$	0	0
$850$	38.5530	1.32236
$851$	−8.35747	−0.286490
$852$	0	0
$853$	−25.2028	−0.862929	−0.431465	−	0.902130i	$-0.642003\pi$
$853$	−25.2028	−0.862929	−0.431465	+	0.902130i	$0.642003\pi$
$854$	2.68081	0.0917354
$855$	0	0
$856$	−13.1757	−0.450337
$857$	−13.0587	−0.446075	−0.223038	−	0.974810i	$-0.571597\pi$
$857$	−13.0587	−0.446075	−0.223038	+	0.974810i	$0.571597\pi$
$858$	0	0
$859$	10.3899	0.354498	0.177249	−	0.984166i	$-0.443280\pi$
$859$	10.3899	0.354498	0.177249	+	0.984166i	$0.443280\pi$
$860$	1.44395	0.0492384
$861$	0	0
$862$	−6.12968	−0.208778
$863$	4.29004	0.146035	0.0730173	−	0.997331i	$-0.476737\pi$
$863$	4.29004	0.146035	0.0730173	+	0.997331i	$0.476737\pi$
$864$	0	0
$865$	4.42061	0.150305
$866$	27.4040	0.931226
$867$	0	0
$868$	−17.0716	−0.579448
$869$	33.5719	1.13885
$870$	0	0
$871$	15.5257	0.526069
$872$	26.3317	0.891703
$873$	0	0
$874$	1.42571	0.0482255
$875$	32.0086	1.08209
$876$	0	0
$877$	−45.0380	−1.52082	−0.760412	−	0.649441i	$-0.775003\pi$
$877$	−45.0380	−1.52082	−0.760412	+	0.649441i	$0.775003\pi$
$878$	−38.4230	−1.29671
$879$	0	0
$880$	−19.9510	−0.672549
$881$	57.5805	1.93994	0.969969	−	0.243229i	$-0.0782067\pi$
$881$	57.5805	1.93994	0.969969	+	0.243229i	$0.0782067\pi$
$882$	0	0
$883$	30.9156	1.04039	0.520197	−	0.854046i	$-0.325859\pi$
$883$	30.9156	1.04039	0.520197	+	0.854046i	$0.325859\pi$
$884$	12.1435	0.408429
$885$	0	0
$886$	30.0220	1.00861
$887$	45.0355	1.51214	0.756072	−	0.654488i	$-0.227116\pi$
$887$	45.0355	1.51214	0.756072	+	0.654488i	$0.227116\pi$
$888$	0	0
$889$	−7.86941	−0.263932
$890$	16.2306	0.544051
$891$	0	0
$892$	5.21221	0.174518
$893$	0.898345	0.0300620
$894$	0	0
$895$	−6.72679	−0.224852
$896$	−46.0630	−1.53886
$897$	0	0
$898$	−35.9153	−1.19851
$899$	4.19787	0.140007
$900$	0	0
$901$	2.72122	0.0906571
$902$	15.2569	0.508000
$903$	0	0
$904$	−5.39820	−0.179541
$905$	−10.3413	−0.343755
$906$	0	0
$907$	39.2826	1.30436	0.652179	−	0.758065i	$-0.273855\pi$
$907$	39.2826	1.30436	0.652179	+	0.758065i	$0.273855\pi$
$908$	−9.47197	−0.314338
$909$	0	0
$910$	13.4287	0.445156
$911$	−15.1711	−0.502640	−0.251320	−	0.967904i	$-0.580865\pi$
$911$	−15.1711	−0.502640	−0.251320	+	0.967904i	$0.580865\pi$
$912$	0	0
$913$	36.1918	1.19777
$914$	−11.0791	−0.366464
$915$	0	0
$916$	−10.6466	−0.351774
$917$	38.6884	1.27760
$918$	0	0
$919$	34.8813	1.15063	0.575315	−	0.817932i	$-0.304880\pi$
$919$	34.8813	1.15063	0.575315	+	0.817932i	$0.304880\pi$
$920$	1.48918	0.0490969
$921$	0	0
$922$	39.9342	1.31516
$923$	9.80128	0.322613
$924$	0	0
$925$	−35.1291	−1.15504
$926$	−9.92713	−0.326226
$927$	0	0
$928$	−5.38290	−0.176702
$929$	−5.52578	−0.181295	−0.0906475	−	0.995883i	$-0.528894\pi$
$929$	−5.52578	−0.181295	−0.0906475	+	0.995883i	$0.528894\pi$
$930$	0	0
$931$	6.68730	0.219167
$932$	−18.2913	−0.599152
$933$	0	0
$934$	55.7611	1.82456
$935$	20.9859	0.686313
$936$	0	0
$937$	19.9977	0.653297	0.326649	−	0.945146i	$-0.394081\pi$
$937$	19.9977	0.653297	0.326649	+	0.945146i	$0.394081\pi$
$938$	−47.6892	−1.55711
$939$	0	0
$940$	−1.02404	−0.0334005
$941$	−12.5532	−0.409221	−0.204611	−	0.978843i	$-0.565593\pi$
$941$	−12.5532	−0.409221	−0.204611	+	0.978843i	$0.565593\pi$
$942$	0	0
$943$	−1.95546	−0.0636786
$944$	−34.9756	−1.13836
$945$	0	0
$946$	−12.0938	−0.393204
$947$	27.7156	0.900635	0.450317	−	0.892869i	$-0.351311\pi$
$947$	27.7156	0.900635	0.450317	+	0.892869i	$0.351311\pi$
$948$	0	0
$949$	−1.60688	−0.0521617
$950$	5.99272	0.194430
$951$	0	0
$952$	34.1785	1.10773
$953$	30.6448	0.992683	0.496342	−	0.868127i	$-0.334676\pi$
$953$	30.6448	0.992683	0.496342	+	0.868127i	$0.334676\pi$
$954$	0	0
$955$	−13.6771	−0.442580
$956$	−18.8007	−0.608059
$957$	0	0
$958$	57.2942	1.85109
$959$	−24.8405	−0.802141
$960$	0	0
$961$	−13.3778	−0.431544
$962$	−32.2689	−1.04039
$963$	0	0
$964$	−10.8399	−0.349128
$965$	−6.77840	−0.218204
$966$	0	0
$967$	53.6612	1.72563	0.862813	−	0.505522i	$-0.168700\pi$
$967$	53.6612	1.72563	0.862813	+	0.505522i	$0.168700\pi$
$968$	15.0163	0.482642
$969$	0	0
$970$	19.2384	0.617709
$971$	−22.8654	−0.733786	−0.366893	−	0.930263i	$-0.619578\pi$
$971$	−22.8654	−0.733786	−0.366893	+	0.930263i	$0.619578\pi$
$972$	0	0
$973$	42.1834	1.35234
$974$	−67.3895	−2.15930
$975$	0	0
$976$	1.97102	0.0630908
$977$	−40.5499	−1.29731	−0.648653	−	0.761084i	$-0.724667\pi$
$977$	−40.5499	−1.29731	−0.648653	+	0.761084i	$0.724667\pi$
$978$	0	0
$979$	−46.6135	−1.48977
$980$	−7.62299	−0.243507
$981$	0	0
$982$	−40.6454	−1.29705
$983$	−27.8600	−0.888595	−0.444297	−	0.895879i	$-0.646547\pi$
$983$	−27.8600	−0.888595	−0.444297	+	0.895879i	$0.646547\pi$
$984$	0	0
$985$	−2.42809	−0.0773654
$986$	9.17204	0.292097
$987$	0	0
$988$	1.88759	0.0600524
$989$	1.55005	0.0492888
$990$	0	0
$991$	11.8396	0.376096	0.188048	−	0.982160i	$-0.439784\pi$
$991$	11.8396	0.376096	0.188048	+	0.982160i	$0.439784\pi$
$992$	−22.5967	−0.717447
$993$	0	0
$994$	−30.1059	−0.954900
$995$	10.3361	0.327677
$996$	0	0
$997$	2.47262	0.0783087	0.0391544	−	0.999233i	$-0.487534\pi$
$997$	2.47262	0.0783087	0.0391544	+	0.999233i	$0.487534\pi$
$998$	−25.9100	−0.820167
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.s.1.6		20
3.2	odd	2		2001.2.a.o.1.15	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.15	✓	20	3.2	odd	2
6003.2.a.s.1.6		20	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-1.74461 q^{2} +1.04367 q^{4} -0.892571 q^{5} +3.89655 q^{7} +1.66842 q^{8} +O(q^{10})\)	\(q-1.74461 q^{2} +1.04367 q^{4} -0.892571 q^{5} +3.89655 q^{7} +1.66842 q^{8} +1.55719 q^{10} -4.47217 q^{11} +2.21315 q^{13} -6.79797 q^{14} -4.99809 q^{16} +5.25735 q^{17} +0.817209 q^{19} -0.931553 q^{20} +7.80221 q^{22} -1.00000 q^{23} -4.20332 q^{25} -3.86109 q^{26} +4.06673 q^{28} -1.00000 q^{29} -4.19787 q^{31} +5.38290 q^{32} -9.17204 q^{34} -3.47795 q^{35} +8.35747 q^{37} -1.42571 q^{38} -1.48918 q^{40} +1.95546 q^{41} -1.55005 q^{43} -4.66749 q^{44} +1.74461 q^{46} +1.09928 q^{47} +8.18310 q^{49} +7.33316 q^{50} +2.30981 q^{52} +0.517604 q^{53} +3.99173 q^{55} +6.50108 q^{56} +1.74461 q^{58} +6.99779 q^{59} -0.394355 q^{61} +7.32367 q^{62} +0.605114 q^{64} -1.97539 q^{65} +7.01521 q^{67} +5.48696 q^{68} +6.06767 q^{70} +4.42865 q^{71} -0.726062 q^{73} -14.5806 q^{74} +0.852899 q^{76} -17.4260 q^{77} -7.50686 q^{79} +4.46115 q^{80} -3.41153 q^{82} -8.09267 q^{83} -4.69256 q^{85} +2.70424 q^{86} -7.46146 q^{88} +10.4230 q^{89} +8.62365 q^{91} -1.04367 q^{92} -1.91783 q^{94} -0.729417 q^{95} +12.3546 q^{97} -14.2763 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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