LMFDB - Embedded newform 6003.2.a.g.1.1

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:	$4$
Coefficient field:	4.4.5744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.291367$ 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)

$f(q)$	$=$	$q-2.00000 q^{4} -2.14073 q^{5} +2.72347 q^{7} +O(q^{10})$	$q-2.00000 q^{4} -2.14073 q^{5} +2.72347 q^{7} +0.451253 q^{11} -5.41294 q^{13} +4.00000 q^{16} +3.24046 q^{17} -3.89595 q^{19} +4.28146 q^{20} -1.00000 q^{23} -0.417266 q^{25} -5.44693 q^{28} -1.00000 q^{29} -1.55800 q^{31} -5.83021 q^{35} -4.21572 q^{37} -3.10674 q^{41} +10.6287 q^{43} -0.902507 q^{44} +2.14073 q^{47} +0.417266 q^{49} +10.8259 q^{52} +0.834532 q^{53} -0.966013 q^{55} +14.0049 q^{59} +6.69441 q^{61} -8.00000 q^{64} +11.5877 q^{65} -5.44693 q^{67} -6.48092 q^{68} +1.82528 q^{71} -8.72347 q^{73} +7.79190 q^{76} +1.22897 q^{77} -8.73048 q^{79} -8.56293 q^{80} -0.403693 q^{83} -6.93696 q^{85} -4.22498 q^{89} -14.7420 q^{91} +2.00000 q^{92} +8.34019 q^{95} -1.86852 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 4 * q - 8 * q^4 + 2 * q^5 - 2 * q^7	$ 4 q - 8 q^{4} + 2 q^{5} - 2 q^{7} + 16 q^{16} - 2 q^{17} + 4 q^{19} - 4 q^{20} - 4 q^{23} - 4 q^{25} + 4 q^{28} - 4 q^{29} + 2 q^{31} - 4 q^{35} + 4 q^{37} - 6 q^{41} - 2 q^{47} + 4 q^{49} + 8 q^{53} - 8 q^{55} + 22 q^{59} - 16 q^{61} - 32 q^{64} + 10 q^{65} + 4 q^{67} + 4 q^{68} + 22 q^{71} - 22 q^{73} - 8 q^{76} + 8 q^{77} - 20 q^{79} + 8 q^{80} + 10 q^{83} - 2 q^{85} + 14 q^{89} - 26 q^{91} + 8 q^{92} + 14 q^{95} - 8 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 + 2 * q^5 - 2 * q^7 + 16 * q^16 - 2 * q^17 + 4 * q^19 - 4 * q^20 - 4 * q^23 - 4 * q^25 + 4 * q^28 - 4 * q^29 + 2 * q^31 - 4 * q^35 + 4 * q^37 - 6 * q^41 - 2 * q^47 + 4 * q^49 + 8 * q^53 - 8 * q^55 + 22 * q^59 - 16 * q^61 - 32 * q^64 + 10 * q^65 + 4 * q^67 + 4 * q^68 + 22 * q^71 - 22 * q^73 - 8 * q^76 + 8 * q^77 - 20 * q^79 + 8 * q^80 + 10 * q^83 - 2 * q^85 + 14 * q^89 - 26 * q^91 + 8 * q^92 + 14 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	−2.14073	−0.957364	−0.478682	−	0.877988i	$-0.658885\pi$
$5$	−2.14073	−0.957364	−0.478682	+	0.877988i	$0.658885\pi$
$6$	0	0
$7$	2.72347	1.02937	0.514687	−	0.857378i	$-0.327908\pi$
$7$	2.72347	1.02937	0.514687	+	0.857378i	$0.327908\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.451253	0.136058	0.0680290	−	0.997683i	$-0.478329\pi$
$11$	0.451253	0.136058	0.0680290	+	0.997683i	$0.478329\pi$
$12$	0	0
$13$	−5.41294	−1.50128	−0.750640	−	0.660711i	$-0.770255\pi$
$13$	−5.41294	−1.50128	−0.750640	+	0.660711i	$0.770255\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	3.24046	0.785927	0.392963	−	0.919554i	$-0.371450\pi$
$17$	3.24046	0.785927	0.392963	+	0.919554i	$0.371450\pi$
$18$	0	0
$19$	−3.89595	−0.893792	−0.446896	−	0.894586i	$-0.647471\pi$
$19$	−3.89595	−0.893792	−0.446896	+	0.894586i	$0.647471\pi$
$20$	4.28146	0.957364
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−0.417266	−0.0834532
$26$	0	0
$27$	0	0
$28$	−5.44693	−1.02937
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−1.55800	−0.279825	−0.139912	−	0.990164i	$-0.544682\pi$
$31$	−1.55800	−0.279825	−0.139912	+	0.990164i	$0.544682\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.83021	−0.985485
$36$	0	0
$37$	−4.21572	−0.693061	−0.346530	−	0.938039i	$-0.612640\pi$
$37$	−4.21572	−0.693061	−0.346530	+	0.938039i	$0.612640\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.10674	−0.485192	−0.242596	−	0.970127i	$-0.577999\pi$
$41$	−3.10674	−0.485192	−0.242596	+	0.970127i	$0.577999\pi$
$42$	0	0
$43$	10.6287	1.62086	0.810428	−	0.585838i	$-0.199234\pi$
$43$	10.6287	1.62086	0.810428	+	0.585838i	$0.199234\pi$
$44$	−0.902507	−0.136058
$45$	0	0
$46$	0	0
$47$	2.14073	0.312258	0.156129	−	0.987737i	$-0.450098\pi$
$47$	2.14073	0.312258	0.156129	+	0.987737i	$0.450098\pi$
$48$	0	0
$49$	0.417266	0.0596095
$50$	0	0
$51$	0	0
$52$	10.8259	1.50128
$53$	0.834532	0.114632	0.0573159	−	0.998356i	$-0.481746\pi$
$53$	0.834532	0.114632	0.0573159	+	0.998356i	$0.481746\pi$
$54$	0	0
$55$	−0.966013	−0.130257
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.0049	1.82329	0.911643	−	0.410982i	$-0.134814\pi$
$59$	14.0049	1.82329	0.911643	+	0.410982i	$0.134814\pi$
$60$	0	0
$61$	6.69441	0.857131	0.428566	−	0.903511i	$-0.359019\pi$
$61$	6.69441	0.857131	0.428566	+	0.903511i	$0.359019\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	11.5877	1.43727
$66$	0	0
$67$	−5.44693	−0.665449	−0.332724	−	0.943024i	$-0.607968\pi$
$67$	−5.44693	−0.665449	−0.332724	+	0.943024i	$0.607968\pi$
$68$	−6.48092	−0.785927
$69$	0	0
$70$	0	0
$71$	1.82528	0.216621	0.108310	−	0.994117i	$-0.465456\pi$
$71$	1.82528	0.216621	0.108310	+	0.994117i	$0.465456\pi$
$72$	0	0
$73$	−8.72347	−1.02100	−0.510502	−	0.859876i	$-0.670540\pi$
$73$	−8.72347	−1.02100	−0.510502	+	0.859876i	$0.670540\pi$
$74$	0	0
$75$	0	0
$76$	7.79190	0.893792
$77$	1.22897	0.140055
$78$	0	0
$79$	−8.73048	−0.982256	−0.491128	−	0.871087i	$-0.663415\pi$
$79$	−8.73048	−0.982256	−0.491128	+	0.871087i	$0.663415\pi$
$80$	−8.56293	−0.957364
$81$	0	0
$82$	0	0
$83$	−0.403693	−0.0443110	−0.0221555	−	0.999755i	$-0.507053\pi$
$83$	−0.403693	−0.0443110	−0.0221555	+	0.999755i	$0.507053\pi$
$84$	0	0
$85$	−6.93696	−0.752418
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.22498	−0.447847	−0.223923	−	0.974607i	$-0.571887\pi$
$89$	−4.22498	−0.447847	−0.223923	+	0.974607i	$0.571887\pi$
$90$	0	0
$91$	−14.7420	−1.54538
$92$	2.00000	0.208514
$93$	0	0
$94$	0	0
$95$	8.34019	0.855685
$96$	0	0
$97$	−1.86852	−0.189719	−0.0948597	−	0.995491i	$-0.530240\pi$
$97$	−1.86852	−0.189719	−0.0948597	+	0.995491i	$0.530240\pi$
$98$	0	0
$99$	0	0
$100$	0.834532	0.0834532
$101$	−3.17472	−0.315896	−0.157948	−	0.987447i	$-0.550488\pi$
$101$	−3.17472	−0.315896	−0.157948	+	0.987447i	$0.550488\pi$
$102$	0	0
$103$	−7.66042	−0.754804	−0.377402	−	0.926050i	$-0.623182\pi$
$103$	−7.66042	−0.754804	−0.377402	+	0.926050i	$0.623182\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.15924	−0.402088	−0.201044	−	0.979582i	$-0.564433\pi$
$107$	−4.15924	−0.402088	−0.201044	+	0.979582i	$0.564433\pi$
$108$	0	0
$109$	−14.4334	−1.38246	−0.691232	−	0.722632i	$-0.742932\pi$
$109$	−14.4334	−1.38246	−0.691232	+	0.722632i	$0.742932\pi$
$110$	0	0
$111$	0	0
$112$	10.8939	1.02937
$113$	3.03399	0.285414	0.142707	−	0.989765i	$-0.454419\pi$
$113$	3.03399	0.285414	0.142707	+	0.989765i	$0.454419\pi$
$114$	0	0
$115$	2.14073	0.199624
$116$	2.00000	0.185695
$117$	0	0
$118$	0	0
$119$	8.82528	0.809012
$120$	0	0
$121$	−10.7964	−0.981488
$122$	0	0
$123$	0	0
$124$	3.11600	0.279825
$125$	11.5969	1.03726
$126$	0	0
$127$	−7.47599	−0.663387	−0.331693	−	0.943387i	$-0.607620\pi$
$127$	−7.47599	−0.663387	−0.331693	+	0.943387i	$0.607620\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.0185035	0.00161666	0.000808329	−	1.00000i	$-0.499743\pi$
$131$	0.0185035	0.00161666	0.000808329		1.00000i	$0.499743\pi$
$132$	0	0
$133$	−10.6105	−0.920046
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.61942	0.394663	0.197332	−	0.980337i	$-0.436772\pi$
$137$	4.61942	0.394663	0.197332	+	0.980337i	$0.436772\pi$
$138$	0	0
$139$	2.51476	0.213299	0.106650	−	0.994297i	$-0.465988\pi$
$139$	2.51476	0.213299	0.106650	+	0.994297i	$0.465988\pi$
$140$	11.6604	0.985485
$141$	0	0
$142$	0	0
$143$	−2.44261	−0.204261
$144$	0	0
$145$	2.14073	0.177778
$146$	0	0
$147$	0	0
$148$	8.43145	0.693061
$149$	13.2475	1.08528	0.542638	−	0.839967i	$-0.317426\pi$
$149$	13.2475	1.08528	0.542638	+	0.839967i	$0.317426\pi$
$150$	0	0
$151$	−0.267282	−0.0217511	−0.0108756	−	0.999941i	$-0.503462\pi$
$151$	−0.267282	−0.0217511	−0.0108756	+	0.999941i	$0.503462\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.33526	0.267894
$156$	0	0
$157$	16.4152	1.31007	0.655037	−	0.755597i	$-0.272653\pi$
$157$	16.4152	1.31007	0.655037	+	0.755597i	$0.272653\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.72347	−0.214639
$162$	0	0
$163$	13.0438	1.02167	0.510836	−	0.859678i	$-0.329336\pi$
$163$	13.0438	1.02167	0.510836	+	0.859678i	$0.329336\pi$
$164$	6.21349	0.485192
$165$	0	0
$166$	0	0
$167$	24.0531	1.86128	0.930642	−	0.365930i	$-0.119249\pi$
$167$	24.0531	1.86128	0.930642	+	0.365930i	$0.119249\pi$
$168$	0	0
$169$	16.3000	1.25384
$170$	0	0
$171$	0	0
$172$	−21.2573	−1.62086
$173$	−15.7968	−1.20101	−0.600505	−	0.799621i	$-0.705034\pi$
$173$	−15.7968	−1.20101	−0.600505	+	0.799621i	$0.705034\pi$
$174$	0	0
$175$	−1.13641	−0.0859045
$176$	1.80501	0.136058
$177$	0	0
$178$	0	0
$179$	25.3204	1.89253	0.946267	−	0.323386i	$-0.104821\pi$
$179$	25.3204	1.89253	0.946267	+	0.323386i	$0.104821\pi$
$180$	0	0
$181$	15.7284	1.16908	0.584541	−	0.811364i	$-0.301275\pi$
$181$	15.7284	1.16908	0.584541	+	0.811364i	$0.301275\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.02474	0.663512
$186$	0	0
$187$	1.46227	0.106932
$188$	−4.28146	−0.312258
$189$	0	0
$190$	0	0
$191$	8.87122	0.641899	0.320949	−	0.947096i	$-0.395998\pi$
$191$	8.87122	0.641899	0.320949	+	0.947096i	$0.395998\pi$
$192$	0	0
$193$	−0.330935	−0.0238212	−0.0119106	−	0.999929i	$-0.503791\pi$
$193$	−0.330935	−0.0238212	−0.0119106	+	0.999929i	$0.503791\pi$
$194$	0	0
$195$	0	0
$196$	−0.834532	−0.0596095
$197$	15.9414	1.13578	0.567890	−	0.823105i	$-0.307760\pi$
$197$	15.9414	1.13578	0.567890	+	0.823105i	$0.307760\pi$
$198$	0	0
$199$	2.51537	0.178310	0.0891548	−	0.996018i	$-0.471583\pi$
$199$	2.51537	0.178310	0.0891548	+	0.996018i	$0.471583\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.72347	−0.191150
$204$	0	0
$205$	6.65071	0.464506
$206$	0	0
$207$	0	0
$208$	−21.6518	−1.50128
$209$	−1.75806	−0.121608
$210$	0	0
$211$	5.33587	0.367336	0.183668	−	0.982988i	$-0.441203\pi$
$211$	5.33587	0.367336	0.183668	+	0.982988i	$0.441203\pi$
$212$	−1.66906	−0.114632
$213$	0	0
$214$	0	0
$215$	−22.7531	−1.55175
$216$	0	0
$217$	−4.24316	−0.288044
$218$	0	0
$219$	0	0
$220$	1.93203	0.130257
$221$	−17.5404	−1.17990
$222$	0	0
$223$	10.1272	0.678165	0.339082	−	0.940757i	$-0.389883\pi$
$223$	10.1272	0.678165	0.339082	+	0.940757i	$0.389883\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.16849	0.542162	0.271081	−	0.962557i	$-0.412619\pi$
$227$	8.16849	0.542162	0.271081	+	0.962557i	$0.412619\pi$
$228$	0	0
$229$	28.5084	1.88388	0.941942	−	0.335774i	$-0.108998\pi$
$229$	28.5084	1.88388	0.941942	+	0.335774i	$0.108998\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.07723	−0.0705714	−0.0352857	−	0.999377i	$-0.511234\pi$
$233$	−1.07723	−0.0705714	−0.0352857	+	0.999377i	$0.511234\pi$
$234$	0	0
$235$	−4.58273	−0.298945
$236$	−28.0099	−1.82329
$237$	0	0
$238$	0	0
$239$	28.8197	1.86419	0.932094	−	0.362216i	$-0.117980\pi$
$239$	28.8197	1.86419	0.932094	+	0.362216i	$0.117980\pi$
$240$	0	0
$241$	−17.2388	−1.11045	−0.555225	−	0.831700i	$-0.687368\pi$
$241$	−17.2388	−1.11045	−0.555225	+	0.831700i	$0.687368\pi$
$242$	0	0
$243$	0	0
$244$	−13.3888	−0.857131
$245$	−0.893255	−0.0570680
$246$	0	0
$247$	21.0886	1.34183
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11.7552	0.741983	0.370991	−	0.928636i	$-0.379018\pi$
$251$	11.7552	0.741983	0.370991	+	0.928636i	$0.379018\pi$
$252$	0	0
$253$	−0.451253	−0.0283701
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	−9.34558	−0.582961	−0.291481	−	0.956577i	$-0.594148\pi$
$257$	−9.34558	−0.582961	−0.291481	+	0.956577i	$0.594148\pi$
$258$	0	0
$259$	−11.4814	−0.713418
$260$	−23.1753	−1.43727
$261$	0	0
$262$	0	0
$263$	21.2956	1.31315	0.656573	−	0.754263i	$-0.272005\pi$
$263$	21.2956	1.31315	0.656573	+	0.754263i	$0.272005\pi$
$264$	0	0
$265$	−1.78651	−0.109744
$266$	0	0
$267$	0	0
$268$	10.8939	0.665449
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	24.1783	1.46873	0.734365	−	0.678755i	$-0.237480\pi$
$271$	24.1783	1.46873	0.734365	+	0.678755i	$0.237480\pi$
$272$	12.9618	0.785927
$273$	0	0
$274$	0	0
$275$	−0.188293	−0.0113545
$276$	0	0
$277$	17.6507	1.06053	0.530264	−	0.847832i	$-0.322093\pi$
$277$	17.6507	1.06053	0.530264	+	0.847832i	$0.322093\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.8784	−0.947225	−0.473612	−	0.880733i	$-0.657050\pi$
$281$	−15.8784	−0.947225	−0.473612	+	0.880733i	$0.657050\pi$
$282$	0	0
$283$	−9.88893	−0.587836	−0.293918	−	0.955831i	$-0.594959\pi$
$283$	−9.88893	−0.587836	−0.293918	+	0.955831i	$0.594959\pi$
$284$	−3.65056	−0.216621
$285$	0	0
$286$	0	0
$287$	−8.46111	−0.499444
$288$	0	0
$289$	−6.49942	−0.382319
$290$	0	0
$291$	0	0
$292$	17.4469	1.02100
$293$	−11.5899	−0.677089	−0.338545	−	0.940950i	$-0.609935\pi$
$293$	−11.5899	−0.677089	−0.338545	+	0.940950i	$0.609935\pi$
$294$	0	0
$295$	−29.9808	−1.74555
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.41294	0.313039
$300$	0	0
$301$	28.9468	1.66847
$302$	0	0
$303$	0	0
$304$	−15.5838	−0.893792
$305$	−14.3309	−0.820587
$306$	0	0
$307$	30.4617	1.73854	0.869271	−	0.494336i	$-0.164589\pi$
$307$	30.4617	1.73854	0.869271	+	0.494336i	$0.164589\pi$
$308$	−2.45795	−0.140055
$309$	0	0
$310$	0	0
$311$	−14.4314	−0.818332	−0.409166	−	0.912460i	$-0.634180\pi$
$311$	−14.4314	−0.818332	−0.409166	+	0.912460i	$0.634180\pi$
$312$	0	0
$313$	24.1844	1.36698	0.683492	−	0.729958i	$-0.260460\pi$
$313$	24.1844	1.36698	0.683492	+	0.729958i	$0.260460\pi$
$314$	0	0
$315$	0	0
$316$	17.4610	0.982256
$317$	20.1452	1.13147	0.565734	−	0.824588i	$-0.308593\pi$
$317$	20.1452	1.13147	0.565734	+	0.824588i	$0.308593\pi$
$318$	0	0
$319$	−0.451253	−0.0252653
$320$	17.1259	0.957364
$321$	0	0
$322$	0	0
$323$	−12.6247	−0.702455
$324$	0	0
$325$	2.25864	0.125287
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.83021	0.321430
$330$	0	0
$331$	22.9618	1.26210	0.631048	−	0.775743i	$-0.282625\pi$
$331$	22.9618	1.26210	0.631048	+	0.775743i	$0.282625\pi$
$332$	0.807385	0.0443110
$333$	0	0
$334$	0	0
$335$	11.6604	0.637077
$336$	0	0
$337$	−8.62420	−0.469790	−0.234895	−	0.972021i	$-0.575475\pi$
$337$	−8.62420	−0.469790	−0.234895	+	0.972021i	$0.575475\pi$
$338$	0	0
$339$	0	0
$340$	13.8739	0.752418
$341$	−0.703052	−0.0380724
$342$	0	0
$343$	−17.9279	−0.968013
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.276534	0.0148451	0.00742257	−	0.999972i	$-0.497637\pi$
$347$	0.276534	0.0148451	0.00742257	+	0.999972i	$0.497637\pi$
$348$	0	0
$349$	10.1753	0.544673	0.272336	−	0.962202i	$-0.412204\pi$
$349$	10.1753	0.544673	0.272336	+	0.962202i	$0.412204\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−11.8895	−0.632816	−0.316408	−	0.948623i	$-0.602477\pi$
$353$	−11.8895	−0.632816	−0.316408	+	0.948623i	$0.602477\pi$
$354$	0	0
$355$	−3.90744	−0.207385
$356$	8.44995	0.447847
$357$	0	0
$358$	0	0
$359$	11.1923	0.590706	0.295353	−	0.955388i	$-0.404563\pi$
$359$	11.1923	0.590706	0.295353	+	0.955388i	$0.404563\pi$
$360$	0	0
$361$	−3.82157	−0.201135
$362$	0	0
$363$	0	0
$364$	29.4839	1.54538
$365$	18.6746	0.977473
$366$	0	0
$367$	1.37896	0.0719810	0.0359905	−	0.999352i	$-0.488541\pi$
$367$	1.37896	0.0719810	0.0359905	+	0.999352i	$0.488541\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	0	0
$371$	2.27282	0.117999
$372$	0	0
$373$	−5.90744	−0.305875	−0.152938	−	0.988236i	$-0.548873\pi$
$373$	−5.90744	−0.305875	−0.152938	+	0.988236i	$0.548873\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.41294	0.278781
$378$	0	0
$379$	−27.2114	−1.39776	−0.698878	−	0.715241i	$-0.746317\pi$
$379$	−27.2114	−1.39776	−0.698878	+	0.715241i	$0.746317\pi$
$380$	−16.6804	−0.855685
$381$	0	0
$382$	0	0
$383$	12.5939	0.643518	0.321759	−	0.946822i	$-0.395726\pi$
$383$	12.5939	0.643518	0.321759	+	0.946822i	$0.395726\pi$
$384$	0	0
$385$	−2.63090	−0.134083
$386$	0	0
$387$	0	0
$388$	3.73704	0.189719
$389$	12.6491	0.641334	0.320667	−	0.947192i	$-0.396093\pi$
$389$	12.6491	0.641334	0.320667	+	0.947192i	$0.396093\pi$
$390$	0	0
$391$	−3.24046	−0.163877
$392$	0	0
$393$	0	0
$394$	0	0
$395$	18.6896	0.940377
$396$	0	0
$397$	17.4358	0.875076	0.437538	−	0.899200i	$-0.355851\pi$
$397$	17.4358	0.875076	0.437538	+	0.899200i	$0.355851\pi$
$398$	0	0
$399$	0	0
$400$	−1.66906	−0.0834532
$401$	8.47228	0.423085	0.211543	−	0.977369i	$-0.432151\pi$
$401$	8.47228	0.423085	0.211543	+	0.977369i	$0.432151\pi$
$402$	0	0
$403$	8.43336	0.420095
$404$	6.34944	0.315896
$405$	0	0
$406$	0	0
$407$	−1.90236	−0.0942965
$408$	0	0
$409$	0.768468	0.0379983	0.0189992	−	0.999819i	$-0.493952\pi$
$409$	0.768468	0.0379983	0.0189992	+	0.999819i	$0.493952\pi$
$410$	0	0
$411$	0	0
$412$	15.3208	0.754804
$413$	38.1419	1.87684
$414$	0	0
$415$	0.864198	0.0424218
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−15.2573	−0.745370	−0.372685	−	0.927958i	$-0.621563\pi$
$419$	−15.2573	−0.745370	−0.372685	+	0.927958i	$0.621563\pi$
$420$	0	0
$421$	1.23855	0.0603632	0.0301816	−	0.999544i	$-0.490391\pi$
$421$	1.23855	0.0603632	0.0301816	+	0.999544i	$0.490391\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.35213	−0.0655881
$426$	0	0
$427$	18.2320	0.882308
$428$	8.31847	0.402088
$429$	0	0
$430$	0	0
$431$	−11.8458	−0.570594	−0.285297	−	0.958439i	$-0.592092\pi$
$431$	−11.8458	−0.570594	−0.285297	+	0.958439i	$0.592092\pi$
$432$	0	0
$433$	−27.4131	−1.31739	−0.658695	−	0.752410i	$-0.728891\pi$
$433$	−27.4131	−1.31739	−0.658695	+	0.752410i	$0.728891\pi$
$434$	0	0
$435$	0	0
$436$	28.8667	1.38246
$437$	3.89595	0.186369
$438$	0	0
$439$	21.5246	1.02731	0.513657	−	0.857996i	$-0.328290\pi$
$439$	21.5246	1.02731	0.513657	+	0.857996i	$0.328290\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.671122	0.0318860	0.0159430	−	0.999873i	$-0.494925\pi$
$443$	0.671122	0.0318860	0.0159430	+	0.999873i	$0.494925\pi$
$444$	0	0
$445$	9.04454	0.428752
$446$	0	0
$447$	0	0
$448$	−21.7877	−1.02937
$449$	−21.9517	−1.03597	−0.517983	−	0.855391i	$-0.673317\pi$
$449$	−21.9517	−1.03597	−0.517983	+	0.855391i	$0.673317\pi$
$450$	0	0
$451$	−1.40193	−0.0660143
$452$	−6.06797	−0.285414
$453$	0	0
$454$	0	0
$455$	31.5586	1.47949
$456$	0	0
$457$	0.310521	0.0145256	0.00726279	−	0.999974i	$-0.497688\pi$
$457$	0.310521	0.0145256	0.00726279	+	0.999974i	$0.497688\pi$
$458$	0	0
$459$	0	0
$460$	−4.28146	−0.199624
$461$	−15.9604	−0.743349	−0.371675	−	0.928363i	$-0.621216\pi$
$461$	−15.9604	−0.743349	−0.371675	+	0.928363i	$0.621216\pi$
$462$	0	0
$463$	16.6451	0.773563	0.386781	−	0.922171i	$-0.373587\pi$
$463$	16.6451	0.773563	0.386781	+	0.922171i	$0.373587\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	0	0
$467$	12.7126	0.588268	0.294134	−	0.955764i	$-0.404969\pi$
$467$	12.7126	0.588268	0.294134	+	0.955764i	$0.404969\pi$
$468$	0	0
$469$	−14.8345	−0.684995
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.79622	0.220531
$474$	0	0
$475$	1.62565	0.0745899
$476$	−17.6506	−0.809012
$477$	0	0
$478$	0	0
$479$	33.1686	1.51551	0.757757	−	0.652537i	$-0.226295\pi$
$479$	33.1686	1.51551	0.757757	+	0.652537i	$0.226295\pi$
$480$	0	0
$481$	22.8195	1.04048
$482$	0	0
$483$	0	0
$484$	21.5927	0.981488
$485$	4.00000	0.181631
$486$	0	0
$487$	−23.8851	−1.08234	−0.541168	−	0.840915i	$-0.682018\pi$
$487$	−23.8851	−1.08234	−0.541168	+	0.840915i	$0.682018\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.290716	0.0131198	0.00655991	−	0.999978i	$-0.497912\pi$
$491$	0.290716	0.0131198	0.00655991	+	0.999978i	$0.497912\pi$
$492$	0	0
$493$	−3.24046	−0.145943
$494$	0	0
$495$	0	0
$496$	−6.23199	−0.279825
$497$	4.97109	0.222984
$498$	0	0
$499$	16.0735	0.719549	0.359775	−	0.933039i	$-0.382854\pi$
$499$	16.0735	0.719549	0.359775	+	0.933039i	$0.382854\pi$
$500$	−23.1938	−1.03726
$501$	0	0
$502$	0	0
$503$	6.77517	0.302090	0.151045	−	0.988527i	$-0.451736\pi$
$503$	6.77517	0.302090	0.151045	+	0.988527i	$0.451736\pi$
$504$	0	0
$505$	6.79622	0.302428
$506$	0	0
$507$	0	0
$508$	14.9520	0.663387
$509$	−38.8024	−1.71988	−0.859942	−	0.510391i	$-0.829501\pi$
$509$	−38.8024	−1.71988	−0.859942	+	0.510391i	$0.829501\pi$
$510$	0	0
$511$	−23.7581	−1.05099
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.3989	0.722622
$516$	0	0
$517$	0.966013	0.0424852
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.2961	1.06443	0.532216	−	0.846609i	$-0.321359\pi$
$521$	24.2961	1.06443	0.532216	+	0.846609i	$0.321359\pi$
$522$	0	0
$523$	−6.94682	−0.303763	−0.151882	−	0.988399i	$-0.548533\pi$
$523$	−6.94682	−0.303763	−0.151882	+	0.988399i	$0.548533\pi$
$524$	−0.0370070	−0.00161666
$525$	0	0
$526$	0	0
$527$	−5.04863	−0.219922
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	21.2210	0.920046
$533$	16.8166	0.728409
$534$	0	0
$535$	8.90381	0.384945
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.188293	0.00811035
$540$	0	0
$541$	10.0692	0.432908	0.216454	−	0.976293i	$-0.430551\pi$
$541$	10.0692	0.432908	0.216454	+	0.976293i	$0.430551\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	30.8980	1.32352
$546$	0	0
$547$	−3.63220	−0.155302	−0.0776509	−	0.996981i	$-0.524742\pi$
$547$	−3.63220	−0.155302	−0.0776509	+	0.996981i	$0.524742\pi$
$548$	−9.23883	−0.394663
$549$	0	0
$550$	0	0
$551$	3.89595	0.165973
$552$	0	0
$553$	−23.7772	−1.01111
$554$	0	0
$555$	0	0
$556$	−5.02952	−0.213299
$557$	−17.0260	−0.721413	−0.360706	−	0.932679i	$-0.617464\pi$
$557$	−17.0260	−0.721413	−0.360706	+	0.932679i	$0.617464\pi$
$558$	0	0
$559$	−57.5324	−2.43336
$560$	−23.3208	−0.985485
$561$	0	0
$562$	0	0
$563$	−5.80051	−0.244463	−0.122231	−	0.992502i	$-0.539005\pi$
$563$	−5.80051	−0.244463	−0.122231	+	0.992502i	$0.539005\pi$
$564$	0	0
$565$	−6.49495	−0.273245
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16.6197	−0.696736	−0.348368	−	0.937358i	$-0.613264\pi$
$569$	−16.6197	−0.696736	−0.348368	+	0.937358i	$0.613264\pi$
$570$	0	0
$571$	9.26789	0.387849	0.193925	−	0.981016i	$-0.437878\pi$
$571$	9.26789	0.387849	0.193925	+	0.981016i	$0.437878\pi$
$572$	4.88522	0.204261
$573$	0	0
$574$	0	0
$575$	0.417266	0.0174012
$576$	0	0
$577$	43.2098	1.79885	0.899423	−	0.437079i	$-0.143987\pi$
$577$	43.2098	1.79885	0.899423	+	0.437079i	$0.143987\pi$
$578$	0	0
$579$	0	0
$580$	−4.28146	−0.177778
$581$	−1.09944	−0.0456126
$582$	0	0
$583$	0.376586	0.0155966
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.2690	1.45571	0.727853	−	0.685733i	$-0.240518\pi$
$587$	35.2690	1.45571	0.727853	+	0.685733i	$0.240518\pi$
$588$	0	0
$589$	6.06988	0.250105
$590$	0	0
$591$	0	0
$592$	−16.8629	−0.693061
$593$	−15.8508	−0.650913	−0.325457	−	0.945557i	$-0.605518\pi$
$593$	−15.8508	−0.650913	−0.325457	+	0.945557i	$0.605518\pi$
$594$	0	0
$595$	−18.8926	−0.774520
$596$	−26.4950	−1.08528
$597$	0	0
$598$	0	0
$599$	36.6357	1.49689	0.748447	−	0.663195i	$-0.230800\pi$
$599$	36.6357	1.49689	0.748447	+	0.663195i	$0.230800\pi$
$600$	0	0
$601$	−9.28448	−0.378722	−0.189361	−	0.981908i	$-0.560642\pi$
$601$	−9.28448	−0.378722	−0.189361	+	0.981908i	$0.560642\pi$
$602$	0	0
$603$	0	0
$604$	0.534565	0.0217511
$605$	23.1121	0.939642
$606$	0	0
$607$	−9.75298	−0.395861	−0.197931	−	0.980216i	$-0.563422\pi$
$607$	−9.75298	−0.395861	−0.197931	+	0.980216i	$0.563422\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11.5877	−0.468787
$612$	0	0
$613$	−23.0624	−0.931480	−0.465740	−	0.884922i	$-0.654212\pi$
$613$	−23.0624	−0.931480	−0.465740	+	0.884922i	$0.654212\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−31.1963	−1.25592	−0.627959	−	0.778247i	$-0.716109\pi$
$617$	−31.1963	−1.25592	−0.627959	+	0.778247i	$0.716109\pi$
$618$	0	0
$619$	−20.8368	−0.837500	−0.418750	−	0.908101i	$-0.637532\pi$
$619$	−20.8368	−0.837500	−0.418750	+	0.908101i	$0.637532\pi$
$620$	−6.67051	−0.267894
$621$	0	0
$622$	0	0
$623$	−11.5066	−0.461001
$624$	0	0
$625$	−22.7396	−0.909582
$626$	0	0
$627$	0	0
$628$	−32.8304	−1.31007
$629$	−13.6609	−0.544695
$630$	0	0
$631$	−4.18959	−0.166785	−0.0833926	−	0.996517i	$-0.526576\pi$
$631$	−4.18959	−0.166785	−0.0833926	+	0.996517i	$0.526576\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	16.0041	0.635103
$636$	0	0
$637$	−2.25864	−0.0894905
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24.9755	−0.986475	−0.493237	−	0.869895i	$-0.664187\pi$
$641$	−24.9755	−0.986475	−0.493237	+	0.869895i	$0.664187\pi$
$642$	0	0
$643$	12.4455	0.490802	0.245401	−	0.969422i	$-0.421080\pi$
$643$	12.4455	0.490802	0.245401	+	0.969422i	$0.421080\pi$
$644$	5.44693	0.214639
$645$	0	0
$646$	0	0
$647$	38.3894	1.50924	0.754622	−	0.656160i	$-0.227820\pi$
$647$	38.3894	1.50924	0.754622	+	0.656160i	$0.227820\pi$
$648$	0	0
$649$	6.31977	0.248073
$650$	0	0
$651$	0	0
$652$	−26.0877	−1.02167
$653$	−38.3414	−1.50042	−0.750208	−	0.661202i	$-0.770046\pi$
$653$	−38.3414	−1.50042	−0.750208	+	0.661202i	$0.770046\pi$
$654$	0	0
$655$	−0.0396110	−0.00154773
$656$	−12.4270	−0.485192
$657$	0	0
$658$	0	0
$659$	31.3052	1.21948	0.609738	−	0.792603i	$-0.291275\pi$
$659$	31.3052	1.21948	0.609738	+	0.792603i	$0.291275\pi$
$660$	0	0
$661$	−0.243763	−0.00948129	−0.00474065	−	0.999989i	$-0.501509\pi$
$661$	−0.243763	−0.00948129	−0.00474065	+	0.999989i	$0.501509\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	22.7142	0.880819
$666$	0	0
$667$	1.00000	0.0387202
$668$	−48.1062	−1.86128
$669$	0	0
$670$	0	0
$671$	3.02088	0.116620
$672$	0	0
$673$	−16.6506	−0.641832	−0.320916	−	0.947108i	$-0.603991\pi$
$673$	−16.6506	−0.641832	−0.320916	+	0.947108i	$0.603991\pi$
$674$	0	0
$675$	0	0
$676$	−32.5999	−1.25384
$677$	−11.8162	−0.454133	−0.227066	−	0.973879i	$-0.572913\pi$
$677$	−11.8162	−0.454133	−0.227066	+	0.973879i	$0.572913\pi$
$678$	0	0
$679$	−5.08885	−0.195292
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−37.5343	−1.43621	−0.718106	−	0.695934i	$-0.754991\pi$
$683$	−37.5343	−1.43621	−0.718106	+	0.695934i	$0.754991\pi$
$684$	0	0
$685$	−9.88893	−0.377837
$686$	0	0
$687$	0	0
$688$	42.5147	1.62086
$689$	−4.51728	−0.172095
$690$	0	0
$691$	−13.1048	−0.498531	−0.249266	−	0.968435i	$-0.580189\pi$
$691$	−13.1048	−0.498531	−0.249266	+	0.968435i	$0.580189\pi$
$692$	31.5937	1.20101
$693$	0	0
$694$	0	0
$695$	−5.38343	−0.204205
$696$	0	0
$697$	−10.0673	−0.381325
$698$	0	0
$699$	0	0
$700$	2.27282	0.0859045
$701$	−4.19113	−0.158297	−0.0791483	−	0.996863i	$-0.525220\pi$
$701$	−4.19113	−0.158297	−0.0791483	+	0.996863i	$0.525220\pi$
$702$	0	0
$703$	16.4243	0.619452
$704$	−3.61003	−0.136058
$705$	0	0
$706$	0	0
$707$	−8.64624	−0.325175
$708$	0	0
$709$	47.8449	1.79685	0.898426	−	0.439126i	$-0.144712\pi$
$709$	47.8449	1.79685	0.898426	+	0.439126i	$0.144712\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.55800	0.0583475
$714$	0	0
$715$	5.22897	0.195553
$716$	−50.6408	−1.89253
$717$	0	0
$718$	0	0
$719$	−49.5524	−1.84799	−0.923996	−	0.382402i	$-0.875097\pi$
$719$	−49.5524	−1.84799	−0.923996	+	0.382402i	$0.875097\pi$
$720$	0	0
$721$	−20.8629	−0.776975
$722$	0	0
$723$	0	0
$724$	−31.4568	−1.16908
$725$	0.417266	0.0154969
$726$	0	0
$727$	49.3100	1.82881	0.914403	−	0.404805i	$-0.132661\pi$
$727$	49.3100	1.82881	0.914403	+	0.404805i	$0.132661\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	34.4418	1.27387
$732$	0	0
$733$	−5.79190	−0.213929	−0.106964	−	0.994263i	$-0.534113\pi$
$733$	−5.79190	−0.213929	−0.106964	+	0.994263i	$0.534113\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.45795	−0.0905396
$738$	0	0
$739$	31.3623	1.15368	0.576841	−	0.816857i	$-0.304285\pi$
$739$	31.3623	1.15368	0.576841	+	0.816857i	$0.304285\pi$
$740$	−18.0495	−0.663512
$741$	0	0
$742$	0	0
$743$	−32.2217	−1.18210	−0.591050	−	0.806635i	$-0.701286\pi$
$743$	−32.2217	−1.18210	−0.591050	+	0.806635i	$0.701286\pi$
$744$	0	0
$745$	−28.3593	−1.03900
$746$	0	0
$747$	0	0
$748$	−2.92454	−0.106932
$749$	−11.3275	−0.413899
$750$	0	0
$751$	10.0644	0.367257	0.183628	−	0.982996i	$-0.441216\pi$
$751$	10.0644	0.367257	0.183628	+	0.982996i	$0.441216\pi$
$752$	8.56293	0.312258
$753$	0	0
$754$	0	0
$755$	0.572180	0.0208238
$756$	0	0
$757$	12.9551	0.470862	0.235431	−	0.971891i	$-0.424350\pi$
$757$	12.9551	0.470862	0.235431	+	0.971891i	$0.424350\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	26.0304	0.943602	0.471801	−	0.881705i	$-0.343604\pi$
$761$	26.0304	0.943602	0.471801	+	0.881705i	$0.343604\pi$
$762$	0	0
$763$	−39.3088	−1.42307
$764$	−17.7424	−0.641899
$765$	0	0
$766$	0	0
$767$	−75.8079	−2.73726
$768$	0	0
$769$	5.79831	0.209092	0.104546	−	0.994520i	$-0.466661\pi$
$769$	5.79831	0.209092	0.104546	+	0.994520i	$0.466661\pi$
$770$	0	0
$771$	0	0
$772$	0.661870	0.0238212
$773$	−44.6714	−1.60672	−0.803358	−	0.595496i	$-0.796956\pi$
$773$	−44.6714	−1.60672	−0.803358	+	0.595496i	$0.796956\pi$
$774$	0	0
$775$	0.650100	0.0233523
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.1037	0.433661
$780$	0	0
$781$	0.823664	0.0294730
$782$	0	0
$783$	0	0
$784$	1.66906	0.0596095
$785$	−35.1405	−1.25422
$786$	0	0
$787$	16.7171	0.595900	0.297950	−	0.954582i	$-0.403697\pi$
$787$	16.7171	0.595900	0.297950	+	0.954582i	$0.403697\pi$
$788$	−31.8828	−1.13578
$789$	0	0
$790$	0	0
$791$	8.26296	0.293797
$792$	0	0
$793$	−36.2365	−1.28679
$794$	0	0
$795$	0	0
$796$	−5.03074	−0.178310
$797$	31.3789	1.11150	0.555748	−	0.831351i	$-0.312432\pi$
$797$	31.3789	1.11150	0.555748	+	0.831351i	$0.312432\pi$
$798$	0	0
$799$	6.93696	0.245412
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.93649	−0.138916
$804$	0	0
$805$	5.83021	0.205488
$806$	0	0
$807$	0	0
$808$	0	0
$809$	10.4174	0.366257	0.183128	−	0.983089i	$-0.441378\pi$
$809$	10.4174	0.366257	0.183128	+	0.983089i	$0.441378\pi$
$810$	0	0
$811$	−40.6886	−1.42877	−0.714386	−	0.699752i	$-0.753294\pi$
$811$	−40.6886	−1.42877	−0.714386	+	0.699752i	$0.753294\pi$
$812$	5.44693	0.191150
$813$	0	0
$814$	0	0
$815$	−27.9234	−0.978113
$816$	0	0
$817$	−41.4088	−1.44871
$818$	0	0
$819$	0	0
$820$	−13.3014	−0.464506
$821$	−9.34688	−0.326208	−0.163104	−	0.986609i	$-0.552151\pi$
$821$	−9.34688	−0.326208	−0.163104	+	0.986609i	$0.552151\pi$
$822$	0	0
$823$	−43.0610	−1.50101	−0.750506	−	0.660863i	$-0.770190\pi$
$823$	−43.0610	−1.50101	−0.750506	+	0.660863i	$0.770190\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−52.3168	−1.81923	−0.909617	−	0.415447i	$-0.863625\pi$
$827$	−52.3168	−1.81923	−0.909617	+	0.415447i	$0.863625\pi$
$828$	0	0
$829$	26.5597	0.922456	0.461228	−	0.887282i	$-0.347409\pi$
$829$	26.5597	0.922456	0.461228	+	0.887282i	$0.347409\pi$
$830$	0	0
$831$	0	0
$832$	43.3036	1.50128
$833$	1.35213	0.0468487
$834$	0	0
$835$	−51.4912	−1.78193
$836$	3.51612	0.121608
$837$	0	0
$838$	0	0
$839$	38.9318	1.34407	0.672037	−	0.740517i	$-0.265419\pi$
$839$	38.9318	1.34407	0.672037	+	0.740517i	$0.265419\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	−10.6717	−0.367336
$845$	−34.8939	−1.20039
$846$	0	0
$847$	−29.4035	−1.01032
$848$	3.33813	0.114632
$849$	0	0
$850$	0	0
$851$	4.21572	0.144513
$852$	0	0
$853$	33.8334	1.15843	0.579216	−	0.815174i	$-0.303359\pi$
$853$	33.8334	1.15843	0.579216	+	0.815174i	$0.303359\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.9332	1.09082	0.545409	−	0.838170i	$-0.316374\pi$
$857$	31.9332	1.09082	0.545409	+	0.838170i	$0.316374\pi$
$858$	0	0
$859$	3.13194	0.106860	0.0534302	−	0.998572i	$-0.482985\pi$
$859$	3.13194	0.106860	0.0534302	+	0.998572i	$0.482985\pi$
$860$	45.5063	1.55175
$861$	0	0
$862$	0	0
$863$	34.4962	1.17426	0.587132	−	0.809491i	$-0.300257\pi$
$863$	34.4962	1.17426	0.587132	+	0.809491i	$0.300257\pi$
$864$	0	0
$865$	33.8168	1.14981
$866$	0	0
$867$	0	0
$868$	8.48631	0.288044
$869$	−3.93966	−0.133644
$870$	0	0
$871$	29.4839	0.999025
$872$	0	0
$873$	0	0
$874$	0	0
$875$	31.5838	1.06773
$876$	0	0
$877$	−10.6747	−0.360461	−0.180230	−	0.983624i	$-0.557684\pi$
$877$	−10.6747	−0.360461	−0.180230	+	0.983624i	$0.557684\pi$
$878$	0	0
$879$	0	0
$880$	−3.86405	−0.130257
$881$	18.4669	0.622166	0.311083	−	0.950383i	$-0.399308\pi$
$881$	18.4669	0.622166	0.311083	+	0.950383i	$0.399308\pi$
$882$	0	0
$883$	26.3253	0.885917	0.442959	−	0.896542i	$-0.353929\pi$
$883$	26.3253	0.885917	0.442959	+	0.896542i	$0.353929\pi$
$884$	35.0809	1.17990
$885$	0	0
$886$	0	0
$887$	−25.0617	−0.841491	−0.420745	−	0.907179i	$-0.638231\pi$
$887$	−25.0617	−0.841491	−0.420745	+	0.907179i	$0.638231\pi$
$888$	0	0
$889$	−20.3606	−0.682873
$890$	0	0
$891$	0	0
$892$	−20.2543	−0.678165
$893$	−8.34019	−0.279094
$894$	0	0
$895$	−54.2042	−1.81184
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.55800	0.0519622
$900$	0	0
$901$	2.70427	0.0900922
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−33.6703	−1.11924
$906$	0	0
$907$	−34.3691	−1.14121	−0.570604	−	0.821225i	$-0.693291\pi$
$907$	−34.3691	−1.14121	−0.570604	+	0.821225i	$0.693291\pi$
$908$	−16.3370	−0.542162
$909$	0	0
$910$	0	0
$911$	−22.5447	−0.746941	−0.373470	−	0.927642i	$-0.621832\pi$
$911$	−22.5447	−0.746941	−0.373470	+	0.927642i	$0.621832\pi$
$912$	0	0
$913$	−0.182168	−0.00602887
$914$	0	0
$915$	0	0
$916$	−57.0167	−1.88388
$917$	0.0503936	0.00166414
$918$	0	0
$919$	−17.3790	−0.573279	−0.286639	−	0.958039i	$-0.592538\pi$
$919$	−17.3790	−0.573279	−0.286639	+	0.958039i	$0.592538\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9.88014	−0.325209
$924$	0	0
$925$	1.75908	0.0578382
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−44.4583	−1.45863	−0.729315	−	0.684178i	$-0.760161\pi$
$929$	−44.4583	−1.45863	−0.729315	+	0.684178i	$0.760161\pi$
$930$	0	0
$931$	−1.62565	−0.0532785
$932$	2.15445	0.0705714
$933$	0	0
$934$	0	0
$935$	−3.13033	−0.102373
$936$	0	0
$937$	44.0111	1.43778	0.718890	−	0.695124i	$-0.244651\pi$
$937$	44.0111	1.43778	0.718890	+	0.695124i	$0.244651\pi$
$938$	0	0
$939$	0	0
$940$	9.16547	0.298945
$941$	−43.8742	−1.43026	−0.715129	−	0.698992i	$-0.753632\pi$
$941$	−43.8742	−1.43026	−0.715129	+	0.698992i	$0.753632\pi$
$942$	0	0
$943$	3.10674	0.101170
$944$	56.0197	1.82329
$945$	0	0
$946$	0	0
$947$	53.9967	1.75466	0.877329	−	0.479889i	$-0.159323\pi$
$947$	53.9967	1.75466	0.877329	+	0.479889i	$0.159323\pi$
$948$	0	0
$949$	47.2196	1.53281
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.62727	−0.0527126	−0.0263563	−	0.999653i	$-0.508390\pi$
$953$	−1.62727	−0.0527126	−0.0263563	+	0.999653i	$0.508390\pi$
$954$	0	0
$955$	−18.9909	−0.614531
$956$	−57.6393	−1.86419
$957$	0	0
$958$	0	0
$959$	12.5808	0.406256
$960$	0	0
$961$	−28.5726	−0.921698
$962$	0	0
$963$	0	0
$964$	34.4777	1.11045
$965$	0.708443	0.0228056
$966$	0	0
$967$	22.3204	0.717775	0.358888	−	0.933381i	$-0.383156\pi$
$967$	22.3204	0.717775	0.358888	+	0.933381i	$0.383156\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.5661	1.17346	0.586731	−	0.809782i	$-0.300415\pi$
$971$	36.5661	1.17346	0.586731	+	0.809782i	$0.300415\pi$
$972$	0	0
$973$	6.84886	0.219564
$974$	0	0
$975$	0	0
$976$	26.7776	0.857131
$977$	14.7024	0.470373	0.235186	−	0.971950i	$-0.424430\pi$
$977$	14.7024	0.470373	0.235186	+	0.971950i	$0.424430\pi$
$978$	0	0
$979$	−1.90653	−0.0609331
$980$	1.78651	0.0570680
$981$	0	0
$982$	0	0
$983$	9.76221	0.311366	0.155683	−	0.987807i	$-0.450242\pi$
$983$	9.76221	0.311366	0.155683	+	0.987807i	$0.450242\pi$
$984$	0	0
$985$	−34.1263	−1.08735
$986$	0	0
$987$	0	0
$988$	−42.1771	−1.34183
$989$	−10.6287	−0.337972
$990$	0	0
$991$	12.6899	0.403109	0.201555	−	0.979477i	$-0.435401\pi$
$991$	12.6899	0.403109	0.201555	+	0.979477i	$0.435401\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.38473	−0.170707
$996$	0	0
$997$	18.8584	0.597252	0.298626	−	0.954370i	$-0.403472\pi$
$997$	18.8584	0.597252	0.298626	+	0.954370i	$0.403472\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.g.1.1		4
3.2	odd	2		2001.2.a.g.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.g.1.4	✓	4	3.2	odd	2
6003.2.a.g.1.1		4	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} -2.14073 q^{5} +2.72347 q^{7} +O(q^{10})\)	\(q-2.00000 q^{4} -2.14073 q^{5} +2.72347 q^{7} +0.451253 q^{11} -5.41294 q^{13} +4.00000 q^{16} +3.24046 q^{17} -3.89595 q^{19} +4.28146 q^{20} -1.00000 q^{23} -0.417266 q^{25} -5.44693 q^{28} -1.00000 q^{29} -1.55800 q^{31} -5.83021 q^{35} -4.21572 q^{37} -3.10674 q^{41} +10.6287 q^{43} -0.902507 q^{44} +2.14073 q^{47} +0.417266 q^{49} +10.8259 q^{52} +0.834532 q^{53} -0.966013 q^{55} +14.0049 q^{59} +6.69441 q^{61} -8.00000 q^{64} +11.5877 q^{65} -5.44693 q^{67} -6.48092 q^{68} +1.82528 q^{71} -8.72347 q^{73} +7.79190 q^{76} +1.22897 q^{77} -8.73048 q^{79} -8.56293 q^{80} -0.403693 q^{83} -6.93696 q^{85} -4.22498 q^{89} -14.7420 q^{91} +2.00000 q^{92} +8.34019 q^{95} -1.86852 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 4 * q - 8 * q^4 + 2 * q^5 - 2 * q^7	\( 4 q - 8 q^{4} + 2 q^{5} - 2 q^{7} + 16 q^{16} - 2 q^{17} + 4 q^{19} - 4 q^{20} - 4 q^{23} - 4 q^{25} + 4 q^{28} - 4 q^{29} + 2 q^{31} - 4 q^{35} + 4 q^{37} - 6 q^{41} - 2 q^{47} + 4 q^{49} + 8 q^{53} - 8 q^{55} + 22 q^{59} - 16 q^{61} - 32 q^{64} + 10 q^{65} + 4 q^{67} + 4 q^{68} + 22 q^{71} - 22 q^{73} - 8 q^{76} + 8 q^{77} - 20 q^{79} + 8 q^{80} + 10 q^{83} - 2 q^{85} + 14 q^{89} - 26 q^{91} + 8 q^{92} + 14 q^{95} - 8 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 + 2 * q^5 - 2 * q^7 + 16 * q^16 - 2 * q^17 + 4 * q^19 - 4 * q^20 - 4 * q^23 - 4 * q^25 + 4 * q^28 - 4 * q^29 + 2 * q^31 - 4 * q^35 + 4 * q^37 - 6 * q^41 - 2 * q^47 + 4 * q^49 + 8 * q^53 - 8 * q^55 + 22 * q^59 - 16 * q^61 - 32 * q^64 + 10 * q^65 + 4 * q^67 + 4 * q^68 + 22 * q^71 - 22 * q^73 - 8 * q^76 + 8 * q^77 - 20 * q^79 + 8 * q^80 + 10 * q^83 - 2 * q^85 + 14 * q^89 - 26 * q^91 + 8 * q^92 + 14 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more