LMFDB - Embedded newform 6003.2.a.t.1.13

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:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
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)

$f(q)$	$=$	$q+0.183492 q^{2} -1.96633 q^{4} +0.926834 q^{5} -0.385808 q^{7} -0.727789 q^{8} +O(q^{10})$	\(q+0.183492 q^{2} -1.96633 q^{4} +0.926834 q^{5} -0.385808 q^{7} -0.727789 q^{8} +0.170066 q^{10} +0.749819 q^{11} +2.20193 q^{13} -0.0707926 q^{14} +3.79912 q^{16} -1.34638 q^{17} -3.59030 q^{19} -1.82246 q^{20} +0.137586 q^{22} +1.00000 q^{23} -4.14098 q^{25} +0.404037 q^{26} +0.758626 q^{28} +1.00000 q^{29} -1.24496 q^{31} +2.15269 q^{32} -0.247049 q^{34} -0.357580 q^{35} +3.97019 q^{37} -0.658790 q^{38} -0.674540 q^{40} -9.73001 q^{41} +4.45412 q^{43} -1.47439 q^{44} +0.183492 q^{46} -0.400723 q^{47} -6.85115 q^{49} -0.759836 q^{50} -4.32973 q^{52} -1.84679 q^{53} +0.694958 q^{55} +0.280787 q^{56} +0.183492 q^{58} -0.0802578 q^{59} +3.33682 q^{61} -0.228440 q^{62} -7.20324 q^{64} +2.04083 q^{65} +9.55846 q^{67} +2.64743 q^{68} -0.0656130 q^{70} +14.2456 q^{71} +7.66525 q^{73} +0.728497 q^{74} +7.05971 q^{76} -0.289286 q^{77} -12.3660 q^{79} +3.52115 q^{80} -1.78538 q^{82} -16.2698 q^{83} -1.24787 q^{85} +0.817295 q^{86} -0.545710 q^{88} -2.45409 q^{89} -0.849523 q^{91} -1.96633 q^{92} -0.0735294 q^{94} -3.32761 q^{95} -0.126390 q^{97} -1.25713 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * 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 $

$2$	0.183492	0.129748	0.0648741	−	0.997893i	$-0.479335\pi$
$2$	0.183492	0.129748	0.0648741	+	0.997893i	$0.479335\pi$
$3$	0	0
$3$	0	0
$4$	−1.96633	−0.983165
$5$	0.926834	0.414493	0.207246	−	0.978289i	$-0.433550\pi$
$5$	0.926834	0.414493	0.207246	+	0.978289i	$0.433550\pi$
$6$	0	0
$7$	−0.385808	−0.145822	−0.0729108	−	0.997338i	$-0.523229\pi$
$7$	−0.385808	−0.145822	−0.0729108	+	0.997338i	$0.523229\pi$
$8$	−0.727789	−0.257312
$9$	0	0
$10$	0.170066	0.0537797
$11$	0.749819	0.226079	0.113039	−	0.993590i	$-0.463941\pi$
$11$	0.749819	0.226079	0.113039	+	0.993590i	$0.463941\pi$
$12$	0	0
$13$	2.20193	0.610706	0.305353	−	0.952239i	$-0.401225\pi$
$13$	2.20193	0.610706	0.305353	+	0.952239i	$0.401225\pi$
$14$	−0.0707926	−0.0189201
$15$	0	0
$16$	3.79912	0.949780
$17$	−1.34638	−0.326545	−0.163272	−	0.986581i	$-0.552205\pi$
$17$	−1.34638	−0.326545	−0.163272	+	0.986581i	$0.552205\pi$
$18$	0	0
$19$	−3.59030	−0.823671	−0.411835	−	0.911258i	$-0.635112\pi$
$19$	−3.59030	−0.823671	−0.411835	+	0.911258i	$0.635112\pi$
$20$	−1.82246	−0.407515
$21$	0	0
$22$	0.137586	0.0293334
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.14098	−0.828196
$26$	0.404037	0.0792381
$27$	0	0
$28$	0.758626	0.143367
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−1.24496	−0.223602	−0.111801	−	0.993731i	$-0.535662\pi$
$31$	−1.24496	−0.223602	−0.111801	+	0.993731i	$0.535662\pi$
$32$	2.15269	0.380545
$33$	0	0
$34$	−0.247049	−0.0423686
$35$	−0.357580	−0.0604420
$36$	0	0
$37$	3.97019	0.652695	0.326348	−	0.945250i	$-0.394182\pi$
$37$	3.97019	0.652695	0.326348	+	0.945250i	$0.394182\pi$
$38$	−0.658790	−0.106870
$39$	0	0
$40$	−0.674540	−0.106654
$41$	−9.73001	−1.51957	−0.759786	−	0.650173i	$-0.774696\pi$
$41$	−9.73001	−1.51957	−0.759786	+	0.650173i	$0.774696\pi$
$42$	0	0
$43$	4.45412	0.679247	0.339623	−	0.940561i	$-0.389700\pi$
$43$	4.45412	0.679247	0.339623	+	0.940561i	$0.389700\pi$
$44$	−1.47439	−0.222273
$45$	0	0
$46$	0.183492	0.0270544
$47$	−0.400723	−0.0584515	−0.0292257	−	0.999573i	$-0.509304\pi$
$47$	−0.400723	−0.0584515	−0.0292257	+	0.999573i	$0.509304\pi$
$48$	0	0
$49$	−6.85115	−0.978736
$50$	−0.759836	−0.107457
$51$	0	0
$52$	−4.32973	−0.600425
$53$	−1.84679	−0.253676	−0.126838	−	0.991923i	$-0.540483\pi$
$53$	−1.84679	−0.253676	−0.126838	+	0.991923i	$0.540483\pi$
$54$	0	0
$55$	0.694958	0.0937081
$56$	0.280787	0.0375217
$57$	0	0
$58$	0.183492	0.0240937
$59$	−0.0802578	−0.0104487	−0.00522434	−	0.999986i	$-0.501663\pi$
$59$	−0.0802578	−0.0104487	−0.00522434	+	0.999986i	$0.501663\pi$
$60$	0	0
$61$	3.33682	0.427237	0.213618	−	0.976917i	$-0.431475\pi$
$61$	3.33682	0.427237	0.213618	+	0.976917i	$0.431475\pi$
$62$	−0.228440	−0.0290120
$63$	0	0
$64$	−7.20324	−0.900405
$65$	2.04083	0.253133
$66$	0	0
$67$	9.55846	1.16775	0.583876	−	0.811843i	$-0.301535\pi$
$67$	9.55846	1.16775	0.583876	+	0.811843i	$0.301535\pi$
$68$	2.64743	0.321048
$69$	0	0
$70$	−0.0656130	−0.00784225
$71$	14.2456	1.69064	0.845321	−	0.534259i	$-0.179409\pi$
$71$	14.2456	1.69064	0.845321	+	0.534259i	$0.179409\pi$
$72$	0	0
$73$	7.66525	0.897150	0.448575	−	0.893745i	$-0.351932\pi$
$73$	7.66525	0.897150	0.448575	+	0.893745i	$0.351932\pi$
$74$	0.728497	0.0846861
$75$	0	0
$76$	7.05971	0.809805
$77$	−0.289286	−0.0329672
$78$	0	0
$79$	−12.3660	−1.39128	−0.695641	−	0.718390i	$-0.744879\pi$
$79$	−12.3660	−1.39128	−0.695641	+	0.718390i	$0.744879\pi$
$80$	3.52115	0.393677
$81$	0	0
$82$	−1.78538	−0.197162
$83$	−16.2698	−1.78584	−0.892920	−	0.450215i	$-0.851347\pi$
$83$	−16.2698	−1.78584	−0.892920	+	0.450215i	$0.851347\pi$
$84$	0	0
$85$	−1.24787	−0.135350
$86$	0.817295	0.0881311
$87$	0	0
$88$	−0.545710	−0.0581729
$89$	−2.45409	−0.260133	−0.130066	−	0.991505i	$-0.541519\pi$
$89$	−2.45409	−0.260133	−0.130066	+	0.991505i	$0.541519\pi$
$90$	0	0
$91$	−0.849523	−0.0890542
$92$	−1.96633	−0.205004
$93$	0	0
$94$	−0.0735294	−0.00758398
$95$	−3.32761	−0.341406
$96$	0	0
$97$	−0.126390	−0.0128329	−0.00641646	−	0.999979i	$-0.502042\pi$
$97$	−0.126390	−0.0128329	−0.00641646	+	0.999979i	$0.502042\pi$
$98$	−1.25713	−0.126989
$99$	0	0
$100$	8.14253	0.814253
$101$	13.4109	1.33444	0.667219	−	0.744862i	$-0.267485\pi$
$101$	13.4109	1.33444	0.667219	+	0.744862i	$0.267485\pi$
$102$	0	0
$103$	10.7952	1.06368	0.531839	−	0.846845i	$-0.321501\pi$
$103$	10.7952	1.06368	0.531839	+	0.846845i	$0.321501\pi$
$104$	−1.60254	−0.157142
$105$	0	0
$106$	−0.338871	−0.0329140
$107$	−16.7814	−1.62232	−0.811161	−	0.584823i	$-0.801164\pi$
$107$	−16.7814	−1.62232	−0.811161	+	0.584823i	$0.801164\pi$
$108$	0	0
$109$	−12.4718	−1.19458	−0.597292	−	0.802024i	$-0.703757\pi$
$109$	−12.4718	−1.19458	−0.597292	+	0.802024i	$0.703757\pi$
$110$	0.127519	0.0121585
$111$	0	0
$112$	−1.46573	−0.138498
$113$	14.9119	1.40279	0.701397	−	0.712770i	$-0.252560\pi$
$113$	14.9119	1.40279	0.701397	+	0.712770i	$0.252560\pi$
$114$	0	0
$115$	0.926834	0.0864277
$116$	−1.96633	−0.182569
$117$	0	0
$118$	−0.0147266	−0.00135570
$119$	0.519443	0.0476173
$120$	0	0
$121$	−10.4378	−0.948888
$122$	0.612280	0.0554332
$123$	0	0
$124$	2.44801	0.219838
$125$	−8.47217	−0.757774
$126$	0	0
$127$	−10.5731	−0.938214	−0.469107	−	0.883141i	$-0.655424\pi$
$127$	−10.5731	−0.938214	−0.469107	+	0.883141i	$0.655424\pi$
$128$	−5.62711	−0.497371
$129$	0	0
$130$	0.374475	0.0328436
$131$	−13.0008	−1.13588	−0.567941	−	0.823069i	$-0.692260\pi$
$131$	−13.0008	−1.13588	−0.567941	+	0.823069i	$0.692260\pi$
$132$	0	0
$133$	1.38517	0.120109
$134$	1.75390	0.151514
$135$	0	0
$136$	0.979880	0.0840240
$137$	0.664712	0.0567902	0.0283951	−	0.999597i	$-0.490960\pi$
$137$	0.664712	0.0567902	0.0283951	+	0.999597i	$0.490960\pi$
$138$	0	0
$139$	2.31036	0.195962	0.0979810	−	0.995188i	$-0.468762\pi$
$139$	2.31036	0.195962	0.0979810	+	0.995188i	$0.468762\pi$
$140$	0.703120	0.0594245
$141$	0	0
$142$	2.61395	0.219358
$143$	1.65105	0.138068
$144$	0	0
$145$	0.926834	0.0769694
$146$	1.40651	0.116404
$147$	0	0
$148$	−7.80671	−0.641707
$149$	−9.64076	−0.789801	−0.394901	−	0.918724i	$-0.629221\pi$
$149$	−9.64076	−0.789801	−0.394901	+	0.918724i	$0.629221\pi$
$150$	0	0
$151$	13.3668	1.08778	0.543889	−	0.839157i	$-0.316951\pi$
$151$	13.3668	1.08778	0.543889	+	0.839157i	$0.316951\pi$
$152$	2.61298	0.211941
$153$	0	0
$154$	−0.0530816	−0.00427744
$155$	−1.15387	−0.0926814
$156$	0	0
$157$	7.62314	0.608393	0.304197	−	0.952609i	$-0.401612\pi$
$157$	7.62314	0.608393	0.304197	+	0.952609i	$0.401612\pi$
$158$	−2.26906	−0.180516
$159$	0	0
$160$	1.99518	0.157733
$161$	−0.385808	−0.0304059
$162$	0	0
$163$	−13.0638	−1.02324	−0.511619	−	0.859212i	$-0.670954\pi$
$163$	−13.0638	−1.02324	−0.511619	+	0.859212i	$0.670954\pi$
$164$	19.1324	1.49399
$165$	0	0
$166$	−2.98537	−0.231710
$167$	−9.82954	−0.760632	−0.380316	−	0.924857i	$-0.624185\pi$
$167$	−9.82954	−0.760632	−0.380316	+	0.924857i	$0.624185\pi$
$168$	0	0
$169$	−8.15149	−0.627038
$170$	−0.228974	−0.0175615
$171$	0	0
$172$	−8.75827	−0.667812
$173$	−0.210024	−0.0159678	−0.00798390	−	0.999968i	$-0.502541\pi$
$173$	−0.210024	−0.0159678	−0.00798390	+	0.999968i	$0.502541\pi$
$174$	0	0
$175$	1.59762	0.120769
$176$	2.84865	0.214725
$177$	0	0
$178$	−0.450305	−0.0337518
$179$	12.3911	0.926155	0.463078	−	0.886318i	$-0.346745\pi$
$179$	12.3911	0.926155	0.463078	+	0.886318i	$0.346745\pi$
$180$	0	0
$181$	−11.5594	−0.859202	−0.429601	−	0.903019i	$-0.641346\pi$
$181$	−11.5594	−0.859202	−0.429601	+	0.903019i	$0.641346\pi$
$182$	−0.155880	−0.0115546
$183$	0	0
$184$	−0.727789	−0.0536533
$185$	3.67971	0.270537
$186$	0	0
$187$	−1.00954	−0.0738249
$188$	0.787954	0.0574675
$189$	0	0
$190$	−0.610589	−0.0442968
$191$	6.13869	0.444180	0.222090	−	0.975026i	$-0.428712\pi$
$191$	6.13869	0.444180	0.222090	+	0.975026i	$0.428712\pi$
$192$	0	0
$193$	−17.8749	−1.28667	−0.643333	−	0.765586i	$-0.722449\pi$
$193$	−17.8749	−1.28667	−0.643333	+	0.765586i	$0.722449\pi$
$194$	−0.0231915	−0.00166505
$195$	0	0
$196$	13.4716	0.962259
$197$	12.6179	0.898986	0.449493	−	0.893284i	$-0.351605\pi$
$197$	12.6179	0.898986	0.449493	+	0.893284i	$0.351605\pi$
$198$	0	0
$199$	−26.6935	−1.89225	−0.946126	−	0.323800i	$-0.895040\pi$
$199$	−26.6935	−1.89225	−0.946126	+	0.323800i	$0.895040\pi$
$200$	3.01376	0.213105
$201$	0	0
$202$	2.46080	0.173141
$203$	−0.385808	−0.0270784
$204$	0	0
$205$	−9.01811	−0.629852
$206$	1.98082	0.138010
$207$	0	0
$208$	8.36540	0.580036
$209$	−2.69207	−0.186215
$210$	0	0
$211$	−14.2682	−0.982266	−0.491133	−	0.871084i	$-0.663417\pi$
$211$	−14.2682	−0.982266	−0.491133	+	0.871084i	$0.663417\pi$
$212$	3.63140	0.249405
$213$	0	0
$214$	−3.07926	−0.210494
$215$	4.12823	0.281543
$216$	0	0
$217$	0.480316	0.0326060
$218$	−2.28848	−0.154995
$219$	0	0
$220$	−1.36652	−0.0921306
$221$	−2.96464	−0.199423
$222$	0	0
$223$	−7.94405	−0.531973	−0.265986	−	0.963977i	$-0.585698\pi$
$223$	−7.94405	−0.531973	−0.265986	+	0.963977i	$0.585698\pi$
$224$	−0.830523	−0.0554916
$225$	0	0
$226$	2.73621	0.182010
$227$	10.5170	0.698038	0.349019	−	0.937116i	$-0.386515\pi$
$227$	10.5170	0.698038	0.349019	+	0.937116i	$0.386515\pi$
$228$	0	0
$229$	−27.2804	−1.80274	−0.901368	−	0.433053i	$-0.857436\pi$
$229$	−27.2804	−1.80274	−0.901368	+	0.433053i	$0.857436\pi$
$230$	0.170066	0.0112139
$231$	0	0
$232$	−0.727789	−0.0477817
$233$	21.6476	1.41818	0.709092	−	0.705116i	$-0.249105\pi$
$233$	21.6476	1.41818	0.709092	+	0.705116i	$0.249105\pi$
$234$	0	0
$235$	−0.371404	−0.0242277
$236$	0.157813	0.0102728
$237$	0	0
$238$	0.0953136	0.00617826
$239$	−0.517454	−0.0334713	−0.0167356	−	0.999860i	$-0.505327\pi$
$239$	−0.517454	−0.0334713	−0.0167356	+	0.999860i	$0.505327\pi$
$240$	0	0
$241$	−26.4992	−1.70696	−0.853481	−	0.521124i	$-0.825513\pi$
$241$	−26.4992	−1.70696	−0.853481	+	0.521124i	$0.825513\pi$
$242$	−1.91525	−0.123117
$243$	0	0
$244$	−6.56130	−0.420044
$245$	−6.34988	−0.405679
$246$	0	0
$247$	−7.90560	−0.503021
$248$	0.906070	0.0575355
$249$	0	0
$250$	−1.55457	−0.0983199
$251$	−26.2662	−1.65791	−0.828955	−	0.559315i	$-0.811064\pi$
$251$	−26.2662	−1.65791	−0.828955	+	0.559315i	$0.811064\pi$
$252$	0	0
$253$	0.749819	0.0471407
$254$	−1.94008	−0.121732
$255$	0	0
$256$	13.3739	0.835872
$257$	−10.5961	−0.660965	−0.330483	−	0.943812i	$-0.607212\pi$
$257$	−10.5961	−0.660965	−0.330483	+	0.943812i	$0.607212\pi$
$258$	0	0
$259$	−1.53173	−0.0951771
$260$	−4.01294	−0.248872
$261$	0	0
$262$	−2.38553	−0.147379
$263$	2.52748	0.155851	0.0779256	−	0.996959i	$-0.475170\pi$
$263$	2.52748	0.155851	0.0779256	+	0.996959i	$0.475170\pi$
$264$	0	0
$265$	−1.71167	−0.105147
$266$	0.254166	0.0155839
$267$	0	0
$268$	−18.7951	−1.14809
$269$	−9.80921	−0.598078	−0.299039	−	0.954241i	$-0.596666\pi$
$269$	−9.80921	−0.598078	−0.299039	+	0.954241i	$0.596666\pi$
$270$	0	0
$271$	−7.40116	−0.449588	−0.224794	−	0.974406i	$-0.572171\pi$
$271$	−7.40116	−0.449588	−0.224794	+	0.974406i	$0.572171\pi$
$272$	−5.11505	−0.310146
$273$	0	0
$274$	0.121969	0.00736843
$275$	−3.10498	−0.187238
$276$	0	0
$277$	−19.8615	−1.19336	−0.596681	−	0.802479i	$-0.703514\pi$
$277$	−19.8615	−1.19336	−0.596681	+	0.802479i	$0.703514\pi$
$278$	0.423932	0.0254257
$279$	0	0
$280$	0.260243	0.0155525
$281$	−0.951509	−0.0567623	−0.0283811	−	0.999597i	$-0.509035\pi$
$281$	−0.951509	−0.0567623	−0.0283811	+	0.999597i	$0.509035\pi$
$282$	0	0
$283$	−2.52964	−0.150371	−0.0751857	−	0.997170i	$-0.523955\pi$
$283$	−2.52964	−0.150371	−0.0751857	+	0.997170i	$0.523955\pi$
$284$	−28.0116	−1.66218
$285$	0	0
$286$	0.302954	0.0179141
$287$	3.75391	0.221587
$288$	0	0
$289$	−15.1873	−0.893369
$290$	0.170066	0.00998665
$291$	0	0
$292$	−15.0724	−0.882047
$293$	24.6757	1.44157	0.720784	−	0.693159i	$-0.243782\pi$
$293$	24.6757	1.44157	0.720784	+	0.693159i	$0.243782\pi$
$294$	0	0
$295$	−0.0743857	−0.00433090
$296$	−2.88946	−0.167946
$297$	0	0
$298$	−1.76900	−0.102475
$299$	2.20193	0.127341
$300$	0	0
$301$	−1.71843	−0.0990489
$302$	2.45270	0.141137
$303$	0	0
$304$	−13.6400	−0.782306
$305$	3.09268	0.177086
$306$	0	0
$307$	−7.53667	−0.430140	−0.215070	−	0.976599i	$-0.568998\pi$
$307$	−7.53667	−0.430140	−0.215070	+	0.976599i	$0.568998\pi$
$308$	0.568832	0.0324122
$309$	0	0
$310$	−0.211726	−0.0120252
$311$	−29.6639	−1.68209	−0.841043	−	0.540968i	$-0.818058\pi$
$311$	−29.6639	−1.68209	−0.841043	+	0.540968i	$0.818058\pi$
$312$	0	0
$313$	3.52220	0.199087	0.0995433	−	0.995033i	$-0.468262\pi$
$313$	3.52220	0.199087	0.0995433	+	0.995033i	$0.468262\pi$
$314$	1.39878	0.0789380
$315$	0	0
$316$	24.3156	1.36786
$317$	11.3919	0.639834	0.319917	−	0.947446i	$-0.396345\pi$
$317$	11.3919	0.639834	0.319917	+	0.947446i	$0.396345\pi$
$318$	0	0
$319$	0.749819	0.0419818
$320$	−6.67621	−0.373211
$321$	0	0
$322$	−0.0707926	−0.00394512
$323$	4.83390	0.268965
$324$	0	0
$325$	−9.11816	−0.505784
$326$	−2.39711	−0.132763
$327$	0	0
$328$	7.08140	0.391005
$329$	0.154602	0.00852349
$330$	0	0
$331$	−2.32690	−0.127898	−0.0639490	−	0.997953i	$-0.520370\pi$
$331$	−2.32690	−0.127898	−0.0639490	+	0.997953i	$0.520370\pi$
$332$	31.9918	1.75578
$333$	0	0
$334$	−1.80364	−0.0986907
$335$	8.85911	0.484025
$336$	0	0
$337$	7.86996	0.428704	0.214352	−	0.976756i	$-0.431236\pi$
$337$	7.86996	0.428704	0.214352	+	0.976756i	$0.431236\pi$
$338$	−1.49573	−0.0813571
$339$	0	0
$340$	2.45372	0.133072
$341$	−0.933497	−0.0505517
$342$	0	0
$343$	5.34388	0.288543
$344$	−3.24166	−0.174779
$345$	0	0
$346$	−0.0385376	−0.00207179
$347$	4.33984	0.232975	0.116487	−	0.993192i	$-0.462837\pi$
$347$	4.33984	0.232975	0.116487	+	0.993192i	$0.462837\pi$
$348$	0	0
$349$	3.80390	0.203618	0.101809	−	0.994804i	$-0.467537\pi$
$349$	3.80390	0.203618	0.101809	+	0.994804i	$0.467537\pi$
$350$	0.293150	0.0156696
$351$	0	0
$352$	1.61412	0.0860331
$353$	−35.7730	−1.90400	−0.952002	−	0.306093i	$-0.900978\pi$
$353$	−35.7730	−1.90400	−0.952002	+	0.306093i	$0.900978\pi$
$354$	0	0
$355$	13.2033	0.700759
$356$	4.82555	0.255754
$357$	0	0
$358$	2.27367	0.120167
$359$	31.8517	1.68107	0.840534	−	0.541759i	$-0.182241\pi$
$359$	31.8517	1.68107	0.840534	+	0.541759i	$0.182241\pi$
$360$	0	0
$361$	−6.10976	−0.321566
$362$	−2.12105	−0.111480
$363$	0	0
$364$	1.67044	0.0875550
$365$	7.10442	0.371862
$366$	0	0
$367$	−9.65204	−0.503832	−0.251916	−	0.967749i	$-0.581061\pi$
$367$	−9.65204	−0.503832	−0.251916	+	0.967749i	$0.581061\pi$
$368$	3.79912	0.198043
$369$	0	0
$370$	0.675196	0.0351018
$371$	0.712505	0.0369914
$372$	0	0
$373$	33.8951	1.75502	0.877511	−	0.479556i	$-0.159202\pi$
$373$	33.8951	1.75502	0.877511	+	0.479556i	$0.159202\pi$
$374$	−0.185242	−0.00957866
$375$	0	0
$376$	0.291642	0.0150403
$377$	2.20193	0.113405
$378$	0	0
$379$	10.2771	0.527898	0.263949	−	0.964537i	$-0.414975\pi$
$379$	10.2771	0.527898	0.263949	+	0.964537i	$0.414975\pi$
$380$	6.54318	0.335658
$381$	0	0
$382$	1.12640	0.0576316
$383$	−6.26703	−0.320230	−0.160115	−	0.987098i	$-0.551187\pi$
$383$	−6.26703	−0.320230	−0.160115	+	0.987098i	$0.551187\pi$
$384$	0	0
$385$	−0.268120	−0.0136647
$386$	−3.27991	−0.166943
$387$	0	0
$388$	0.248524	0.0126169
$389$	1.50479	0.0762960	0.0381480	−	0.999272i	$-0.487854\pi$
$389$	1.50479	0.0762960	0.0381480	+	0.999272i	$0.487854\pi$
$390$	0	0
$391$	−1.34638	−0.0680893
$392$	4.98619	0.251841
$393$	0	0
$394$	2.31527	0.116642
$395$	−11.4612	−0.576676
$396$	0	0
$397$	18.0745	0.907131	0.453566	−	0.891223i	$-0.350152\pi$
$397$	18.0745	0.907131	0.453566	+	0.891223i	$0.350152\pi$
$398$	−4.89804	−0.245516
$399$	0	0
$400$	−15.7321	−0.786603
$401$	−16.4662	−0.822282	−0.411141	−	0.911572i	$-0.634870\pi$
$401$	−16.4662	−0.822282	−0.411141	+	0.911572i	$0.634870\pi$
$402$	0	0
$403$	−2.74132	−0.136555
$404$	−26.3703	−1.31197
$405$	0	0
$406$	−0.0707926	−0.00351338
$407$	2.97692	0.147561
$408$	0	0
$409$	−33.6007	−1.66145	−0.830724	−	0.556685i	$-0.812073\pi$
$409$	−33.6007	−1.66145	−0.830724	+	0.556685i	$0.812073\pi$
$410$	−1.65475	−0.0817222
$411$	0	0
$412$	−21.2268	−1.04577
$413$	0.0309641	0.00152364
$414$	0	0
$415$	−15.0794	−0.740218
$416$	4.74007	0.232401
$417$	0	0
$418$	−0.493974	−0.0241610
$419$	37.4551	1.82980	0.914902	−	0.403677i	$-0.132268\pi$
$419$	37.4551	1.82980	0.914902	+	0.403677i	$0.132268\pi$
$420$	0	0
$421$	−27.1254	−1.32201	−0.661005	−	0.750382i	$-0.729870\pi$
$421$	−27.1254	−1.32201	−0.661005	+	0.750382i	$0.729870\pi$
$422$	−2.61811	−0.127447
$423$	0	0
$424$	1.34407	0.0652739
$425$	5.57532	0.270443
$426$	0	0
$427$	−1.28737	−0.0623003
$428$	32.9978	1.59501
$429$	0	0
$430$	0.757497	0.0365297
$431$	32.4650	1.56378	0.781892	−	0.623413i	$-0.214255\pi$
$431$	32.4650	1.56378	0.781892	+	0.623413i	$0.214255\pi$
$432$	0	0
$433$	−39.8665	−1.91586	−0.957932	−	0.286997i	$-0.907343\pi$
$433$	−39.8665	−1.91586	−0.957932	+	0.286997i	$0.907343\pi$
$434$	0.0881341	0.00423057
$435$	0	0
$436$	24.5237	1.17447
$437$	−3.59030	−0.171747
$438$	0	0
$439$	25.2011	1.20278	0.601391	−	0.798955i	$-0.294613\pi$
$439$	25.2011	1.20278	0.601391	+	0.798955i	$0.294613\pi$
$440$	−0.505783	−0.0241123
$441$	0	0
$442$	−0.543986	−0.0258748
$443$	−21.7528	−1.03351	−0.516754	−	0.856134i	$-0.672860\pi$
$443$	−21.7528	−1.03351	−0.516754	+	0.856134i	$0.672860\pi$
$444$	0	0
$445$	−2.27453	−0.107823
$446$	−1.45767	−0.0690225
$447$	0	0
$448$	2.77906	0.131298
$449$	−34.1849	−1.61328	−0.806641	−	0.591041i	$-0.798717\pi$
$449$	−34.1849	−1.61328	−0.806641	+	0.591041i	$0.798717\pi$
$450$	0	0
$451$	−7.29575	−0.343543
$452$	−29.3218	−1.37918
$453$	0	0
$454$	1.92978	0.0905692
$455$	−0.787367	−0.0369123
$456$	0	0
$457$	−34.3663	−1.60759	−0.803793	−	0.594909i	$-0.797188\pi$
$457$	−34.3663	−1.60759	−0.803793	+	0.594909i	$0.797188\pi$
$458$	−5.00572	−0.233902
$459$	0	0
$460$	−1.82246	−0.0849728
$461$	12.5371	0.583912	0.291956	−	0.956432i	$-0.405694\pi$
$461$	12.5371	0.583912	0.291956	+	0.956432i	$0.405694\pi$
$462$	0	0
$463$	−3.04685	−0.141599	−0.0707996	−	0.997491i	$-0.522555\pi$
$463$	−3.04685	−0.141599	−0.0707996	+	0.997491i	$0.522555\pi$
$464$	3.79912	0.176370
$465$	0	0
$466$	3.97216	0.184007
$467$	−9.69440	−0.448603	−0.224302	−	0.974520i	$-0.572010\pi$
$467$	−9.69440	−0.448603	−0.224302	+	0.974520i	$0.572010\pi$
$468$	0	0
$469$	−3.68773	−0.170284
$470$	−0.0681496	−0.00314351
$471$	0	0
$472$	0.0584108	0.00268857
$473$	3.33978	0.153563
$474$	0	0
$475$	14.8673	0.682161
$476$	−1.02140	−0.0468157
$477$	0	0
$478$	−0.0949485	−0.00434284
$479$	33.2429	1.51891	0.759453	−	0.650562i	$-0.225467\pi$
$479$	33.2429	1.51891	0.759453	+	0.650562i	$0.225467\pi$
$480$	0	0
$481$	8.74209	0.398605
$482$	−4.86238	−0.221475
$483$	0	0
$484$	20.5241	0.932914
$485$	−0.117142	−0.00531916
$486$	0	0
$487$	−20.1894	−0.914868	−0.457434	−	0.889244i	$-0.651231\pi$
$487$	−20.1894	−0.914868	−0.457434	+	0.889244i	$0.651231\pi$
$488$	−2.42850	−0.109933
$489$	0	0
$490$	−1.16515	−0.0526362
$491$	29.1577	1.31587	0.657935	−	0.753075i	$-0.271430\pi$
$491$	29.1577	1.31587	0.657935	+	0.753075i	$0.271430\pi$
$492$	0	0
$493$	−1.34638	−0.0606378
$494$	−1.45061	−0.0652661
$495$	0	0
$496$	−4.72976	−0.212372
$497$	−5.49606	−0.246532
$498$	0	0
$499$	12.6666	0.567035	0.283517	−	0.958967i	$-0.408499\pi$
$499$	12.6666	0.567035	0.283517	+	0.958967i	$0.408499\pi$
$500$	16.6591	0.745017
$501$	0	0
$502$	−4.81964	−0.215111
$503$	16.7240	0.745688	0.372844	−	0.927894i	$-0.378383\pi$
$503$	16.7240	0.745688	0.372844	+	0.927894i	$0.378383\pi$
$504$	0	0
$505$	12.4297	0.553115
$506$	0.137586	0.00611643
$507$	0	0
$508$	20.7903	0.922419
$509$	20.1057	0.891168	0.445584	−	0.895240i	$-0.352996\pi$
$509$	20.1057	0.891168	0.445584	+	0.895240i	$0.352996\pi$
$510$	0	0
$511$	−2.95731	−0.130824
$512$	13.7082	0.605823
$513$	0	0
$514$	−1.94429	−0.0857591
$515$	10.0053	0.440887
$516$	0	0
$517$	−0.300470	−0.0132147
$518$	−0.281060	−0.0123491
$519$	0	0
$520$	−1.48529	−0.0651344
$521$	−35.8407	−1.57021	−0.785104	−	0.619363i	$-0.787391\pi$
$521$	−35.8407	−1.57021	−0.785104	+	0.619363i	$0.787391\pi$
$522$	0	0
$523$	−34.1618	−1.49379	−0.746895	−	0.664942i	$-0.768456\pi$
$523$	−34.1618	−1.49379	−0.746895	+	0.664942i	$0.768456\pi$
$524$	25.5638	1.11676
$525$	0	0
$526$	0.463772	0.0202214
$527$	1.67619	0.0730160
$528$	0	0
$529$	1.00000	0.0434783
$530$	−0.314077	−0.0136426
$531$	0	0
$532$	−2.72369	−0.118087
$533$	−21.4248	−0.928012
$534$	0	0
$535$	−15.5536	−0.672441
$536$	−6.95655	−0.300477
$537$	0	0
$538$	−1.79991	−0.0775996
$539$	−5.13713	−0.221272
$540$	0	0
$541$	38.2085	1.64271	0.821357	−	0.570415i	$-0.193218\pi$
$541$	38.2085	1.64271	0.821357	+	0.570415i	$0.193218\pi$
$542$	−1.35805	−0.0583333
$543$	0	0
$544$	−2.89833	−0.124265
$545$	−11.5593	−0.495147
$546$	0	0
$547$	25.1716	1.07626	0.538131	−	0.842861i	$-0.319131\pi$
$547$	25.1716	1.07626	0.538131	+	0.842861i	$0.319131\pi$
$548$	−1.30704	−0.0558341
$549$	0	0
$550$	−0.569739	−0.0242938
$551$	−3.59030	−0.152952
$552$	0	0
$553$	4.77089	0.202879
$554$	−3.64442	−0.154837
$555$	0	0
$556$	−4.54293	−0.192663
$557$	12.9380	0.548202	0.274101	−	0.961701i	$-0.411620\pi$
$557$	12.9380	0.548202	0.274101	+	0.961701i	$0.411620\pi$
$558$	0	0
$559$	9.80768	0.414820
$560$	−1.35849	−0.0574066
$561$	0	0
$562$	−0.174594	−0.00736481
$563$	−33.6052	−1.41629	−0.708145	−	0.706067i	$-0.750468\pi$
$563$	−33.6052	−1.41629	−0.708145	+	0.706067i	$0.750468\pi$
$564$	0	0
$565$	13.8209	0.581448
$566$	−0.464168	−0.0195104
$567$	0	0
$568$	−10.3678	−0.435023
$569$	−7.99893	−0.335333	−0.167666	−	0.985844i	$-0.553623\pi$
$569$	−7.99893	−0.335333	−0.167666	+	0.985844i	$0.553623\pi$
$570$	0	0
$571$	19.1730	0.802367	0.401183	−	0.915998i	$-0.368599\pi$
$571$	19.1730	0.802367	0.401183	+	0.915998i	$0.368599\pi$
$572$	−3.24651	−0.135744
$573$	0	0
$574$	0.688812	0.0287505
$575$	−4.14098	−0.172691
$576$	0	0
$577$	−24.9987	−1.04071	−0.520354	−	0.853951i	$-0.674200\pi$
$577$	−24.9987	−1.04071	−0.520354	+	0.853951i	$0.674200\pi$
$578$	−2.78674	−0.115913
$579$	0	0
$580$	−1.82246	−0.0756736
$581$	6.27701	0.260414
$582$	0	0
$583$	−1.38476	−0.0573508
$584$	−5.57869	−0.230848
$585$	0	0
$586$	4.52779	0.187041
$587$	23.0785	0.952551	0.476275	−	0.879296i	$-0.341987\pi$
$587$	23.0785	0.952551	0.476275	+	0.879296i	$0.341987\pi$
$588$	0	0
$589$	4.46979	0.184174
$590$	−0.0136492	−0.000561927 0
$591$	0	0
$592$	15.0832	0.619916
$593$	−14.9820	−0.615236	−0.307618	−	0.951510i	$-0.599532\pi$
$593$	−14.9820	−0.615236	−0.307618	+	0.951510i	$0.599532\pi$
$594$	0	0
$595$	0.481438	0.0197370
$596$	18.9569	0.776505
$597$	0	0
$598$	0.404037	0.0165223
$599$	−32.0926	−1.31127	−0.655633	−	0.755080i	$-0.727598\pi$
$599$	−32.0926	−1.31127	−0.655633	+	0.755080i	$0.727598\pi$
$600$	0	0
$601$	−24.2848	−0.990598	−0.495299	−	0.868723i	$-0.664941\pi$
$601$	−24.2848	−0.990598	−0.495299	+	0.868723i	$0.664941\pi$
$602$	−0.315319	−0.0128514
$603$	0	0
$604$	−26.2836	−1.06947
$605$	−9.67408	−0.393307
$606$	0	0
$607$	45.9454	1.86486	0.932432	−	0.361345i	$-0.117682\pi$
$607$	45.9454	1.86486	0.932432	+	0.361345i	$0.117682\pi$
$608$	−7.72878	−0.313444
$609$	0	0
$610$	0.567482	0.0229767
$611$	−0.882366	−0.0356967
$612$	0	0
$613$	−28.4632	−1.14962	−0.574810	−	0.818287i	$-0.694924\pi$
$613$	−28.4632	−1.14962	−0.574810	+	0.818287i	$0.694924\pi$
$614$	−1.38292	−0.0558100
$615$	0	0
$616$	0.210539	0.00848287
$617$	−17.7981	−0.716524	−0.358262	−	0.933621i	$-0.616631\pi$
$617$	−17.7981	−0.716524	−0.358262	+	0.933621i	$0.616631\pi$
$618$	0	0
$619$	−37.3427	−1.50093	−0.750465	−	0.660910i	$-0.770170\pi$
$619$	−37.3427	−1.50093	−0.750465	+	0.660910i	$0.770170\pi$
$620$	2.26890	0.0911211
$621$	0	0
$622$	−5.44309	−0.218248
$623$	0.946806	0.0379330
$624$	0	0
$625$	12.8526	0.514104
$626$	0.646295	0.0258311
$627$	0	0
$628$	−14.9896	−0.598151
$629$	−5.34538	−0.213134
$630$	0	0
$631$	−2.33993	−0.0931513	−0.0465756	−	0.998915i	$-0.514831\pi$
$631$	−2.33993	−0.0931513	−0.0465756	+	0.998915i	$0.514831\pi$
$632$	8.99983	0.357994
$633$	0	0
$634$	2.09032	0.0830174
$635$	−9.79954	−0.388883
$636$	0	0
$637$	−15.0858	−0.597720
$638$	0.137586	0.00544707
$639$	0	0
$640$	−5.21539	−0.206157
$641$	26.9889	1.06600	0.532998	−	0.846116i	$-0.321065\pi$
$641$	26.9889	1.06600	0.532998	+	0.846116i	$0.321065\pi$
$642$	0	0
$643$	25.0301	0.987090	0.493545	−	0.869720i	$-0.335701\pi$
$643$	25.0301	0.987090	0.493545	+	0.869720i	$0.335701\pi$
$644$	0.758626	0.0298940
$645$	0	0
$646$	0.886981	0.0348978
$647$	−28.4657	−1.11910	−0.559550	−	0.828796i	$-0.689026\pi$
$647$	−28.4657	−1.11910	−0.559550	+	0.828796i	$0.689026\pi$
$648$	0	0
$649$	−0.0601788	−0.00236223
$650$	−1.67311	−0.0656247
$651$	0	0
$652$	25.6878	1.00601
$653$	−5.11404	−0.200128	−0.100064	−	0.994981i	$-0.531905\pi$
$653$	−5.11404	−0.200128	−0.100064	+	0.994981i	$0.531905\pi$
$654$	0	0
$655$	−12.0496	−0.470815
$656$	−36.9655	−1.44326
$657$	0	0
$658$	0.0283682	0.00110591
$659$	13.4130	0.522496	0.261248	−	0.965272i	$-0.415866\pi$
$659$	13.4130	0.522496	0.261248	+	0.965272i	$0.415866\pi$
$660$	0	0
$661$	−6.72595	−0.261609	−0.130805	−	0.991408i	$-0.541756\pi$
$661$	−6.72595	−0.261609	−0.130805	+	0.991408i	$0.541756\pi$
$662$	−0.426967	−0.0165946
$663$	0	0
$664$	11.8410	0.459519
$665$	1.28382	0.0497843
$666$	0	0
$667$	1.00000	0.0387202
$668$	19.3281	0.747827
$669$	0	0
$670$	1.62557	0.0628014
$671$	2.50201	0.0965892
$672$	0	0
$673$	9.11955	0.351533	0.175766	−	0.984432i	$-0.443760\pi$
$673$	9.11955	0.351533	0.175766	+	0.984432i	$0.443760\pi$
$674$	1.44407	0.0556236
$675$	0	0
$676$	16.0285	0.616482
$677$	−24.5508	−0.943564	−0.471782	−	0.881715i	$-0.656389\pi$
$677$	−24.5508	−0.943564	−0.471782	+	0.881715i	$0.656389\pi$
$678$	0	0
$679$	0.0487621	0.00187132
$680$	0.908186	0.0348273
$681$	0	0
$682$	−0.171289	−0.00655899
$683$	43.5434	1.66614	0.833071	−	0.553166i	$-0.186580\pi$
$683$	43.5434	1.66614	0.833071	+	0.553166i	$0.186580\pi$
$684$	0	0
$685$	0.616078	0.0235391
$686$	0.980559	0.0374379
$687$	0	0
$688$	16.9217	0.645135
$689$	−4.06651	−0.154922
$690$	0	0
$691$	14.7682	0.561807	0.280904	−	0.959736i	$-0.409366\pi$
$691$	14.7682	0.561807	0.280904	+	0.959736i	$0.409366\pi$
$692$	0.412976	0.0156990
$693$	0	0
$694$	0.796326	0.0302281
$695$	2.14132	0.0812248
$696$	0	0
$697$	13.1003	0.496208
$698$	0.697984	0.0264191
$699$	0	0
$700$	−3.14145	−0.118736
$701$	35.2124	1.32995	0.664977	−	0.746864i	$-0.268441\pi$
$701$	35.2124	1.32995	0.664977	+	0.746864i	$0.268441\pi$
$702$	0	0
$703$	−14.2542	−0.537606
$704$	−5.40112	−0.203563
$705$	0	0
$706$	−6.56405	−0.247041
$707$	−5.17404	−0.194590
$708$	0	0
$709$	−29.0997	−1.09286	−0.546432	−	0.837504i	$-0.684014\pi$
$709$	−29.0997	−1.09286	−0.546432	+	0.837504i	$0.684014\pi$
$710$	2.42270	0.0909223
$711$	0	0
$712$	1.78606	0.0669354
$713$	−1.24496	−0.0466242
$714$	0	0
$715$	1.53025	0.0572281
$716$	−24.3650	−0.910564
$717$	0	0
$718$	5.84452	0.218116
$719$	−20.7747	−0.774766	−0.387383	−	0.921919i	$-0.626621\pi$
$719$	−20.7747	−0.774766	−0.387383	+	0.921919i	$0.626621\pi$
$720$	0	0
$721$	−4.16485	−0.155107
$722$	−1.12109	−0.0417227
$723$	0	0
$724$	22.7296	0.844738
$725$	−4.14098	−0.153792
$726$	0	0
$727$	−1.69278	−0.0627817	−0.0313908	−	0.999507i	$-0.509994\pi$
$727$	−1.69278	−0.0627817	−0.0313908	+	0.999507i	$0.509994\pi$
$728$	0.618274	0.0229147
$729$	0	0
$730$	1.30360	0.0482485
$731$	−5.99693	−0.221805
$732$	0	0
$733$	30.7492	1.13575	0.567874	−	0.823116i	$-0.307766\pi$
$733$	30.7492	1.13575	0.567874	+	0.823116i	$0.307766\pi$
$734$	−1.77107	−0.0653714
$735$	0	0
$736$	2.15269	0.0793490
$737$	7.16712	0.264004
$738$	0	0
$739$	−21.8356	−0.803237	−0.401619	−	0.915807i	$-0.631552\pi$
$739$	−21.8356	−0.803237	−0.401619	+	0.915807i	$0.631552\pi$
$740$	−7.23552	−0.265983
$741$	0	0
$742$	0.130739	0.00479958
$743$	−25.3649	−0.930548	−0.465274	−	0.885167i	$-0.654044\pi$
$743$	−25.3649	−0.930548	−0.465274	+	0.885167i	$0.654044\pi$
$744$	0	0
$745$	−8.93538	−0.327367
$746$	6.21948	0.227711
$747$	0	0
$748$	1.98509	0.0725821
$749$	6.47441	0.236570
$750$	0	0
$751$	39.4256	1.43866	0.719330	−	0.694669i	$-0.244449\pi$
$751$	39.4256	1.43866	0.719330	+	0.694669i	$0.244449\pi$
$752$	−1.52239	−0.0555160
$753$	0	0
$754$	0.404037	0.0147141
$755$	12.3888	0.450876
$756$	0	0
$757$	−6.47394	−0.235299	−0.117650	−	0.993055i	$-0.537536\pi$
$757$	−6.47394	−0.235299	−0.117650	+	0.993055i	$0.537536\pi$
$758$	1.88576	0.0684939
$759$	0	0
$760$	2.42180	0.0878479
$761$	53.9004	1.95389	0.976945	−	0.213492i	$-0.0684838\pi$
$761$	53.9004	1.95389	0.976945	+	0.213492i	$0.0684838\pi$
$762$	0	0
$763$	4.81173	0.174196
$764$	−12.0707	−0.436702
$765$	0	0
$766$	−1.14995	−0.0415493
$767$	−0.176722	−0.00638107
$768$	0	0
$769$	11.0960	0.400133	0.200066	−	0.979782i	$-0.435884\pi$
$769$	11.0960	0.400133	0.200066	+	0.979782i	$0.435884\pi$
$770$	−0.0491979	−0.00177297
$771$	0	0
$772$	35.1480	1.26501
$773$	41.9688	1.50951	0.754756	−	0.656005i	$-0.227755\pi$
$773$	41.9688	1.50951	0.754756	+	0.656005i	$0.227755\pi$
$774$	0	0
$775$	5.15536	0.185186
$776$	0.0919850	0.00330207
$777$	0	0
$778$	0.276117	0.00989928
$779$	34.9336	1.25163
$780$	0	0
$781$	10.6816	0.382219
$782$	−0.247049	−0.00883447
$783$	0	0
$784$	−26.0283	−0.929583
$785$	7.06539	0.252175
$786$	0	0
$787$	−7.36397	−0.262497	−0.131249	−	0.991349i	$-0.541899\pi$
$787$	−7.36397	−0.262497	−0.131249	+	0.991349i	$0.541899\pi$
$788$	−24.8109	−0.883852
$789$	0	0
$790$	−2.10304	−0.0748228
$791$	−5.75313	−0.204558
$792$	0	0
$793$	7.34746	0.260916
$794$	3.31652	0.117699
$795$	0	0
$796$	52.4882	1.86040
$797$	35.5482	1.25918	0.629590	−	0.776927i	$-0.283223\pi$
$797$	35.5482	1.25918	0.629590	+	0.776927i	$0.283223\pi$
$798$	0	0
$799$	0.539525	0.0190870
$800$	−8.91422	−0.315165
$801$	0	0
$802$	−3.02141	−0.106690
$803$	5.74755	0.202827
$804$	0	0
$805$	−0.357580	−0.0126030
$806$	−0.503010	−0.0177178
$807$	0	0
$808$	−9.76033	−0.343367
$809$	−24.8265	−0.872852	−0.436426	−	0.899740i	$-0.643756\pi$
$809$	−24.8265	−0.872852	−0.436426	+	0.899740i	$0.643756\pi$
$810$	0	0
$811$	4.46243	0.156697	0.0783486	−	0.996926i	$-0.475035\pi$
$811$	4.46243	0.156697	0.0783486	+	0.996926i	$0.475035\pi$
$812$	0.758626	0.0266225
$813$	0	0
$814$	0.546241	0.0191457
$815$	−12.1080	−0.424125
$816$	0	0
$817$	−15.9916	−0.559476
$818$	−6.16545	−0.215570
$819$	0	0
$820$	17.7326	0.619249
$821$	−32.1527	−1.12214	−0.561069	−	0.827769i	$-0.689610\pi$
$821$	−32.1527	−1.12214	−0.561069	+	0.827769i	$0.689610\pi$
$822$	0	0
$823$	36.9464	1.28787	0.643936	−	0.765079i	$-0.277300\pi$
$823$	36.9464	1.28787	0.643936	+	0.765079i	$0.277300\pi$
$824$	−7.85660	−0.273697
$825$	0	0
$826$	0.00568166	0.000197690 0
$827$	−17.9151	−0.622969	−0.311485	−	0.950251i	$-0.600826\pi$
$827$	−17.9151	−0.622969	−0.311485	+	0.950251i	$0.600826\pi$
$828$	0	0
$829$	−0.432599	−0.0150248	−0.00751239	−	0.999972i	$-0.502391\pi$
$829$	−0.432599	−0.0150248	−0.00751239	+	0.999972i	$0.502391\pi$
$830$	−2.76694	−0.0960420
$831$	0	0
$832$	−15.8610	−0.549883
$833$	9.22424	0.319601
$834$	0	0
$835$	−9.11035	−0.315277
$836$	5.29351	0.183080
$837$	0	0
$838$	6.87271	0.237414
$839$	−31.6009	−1.09098	−0.545492	−	0.838116i	$-0.683657\pi$
$839$	−31.6009	−1.09098	−0.545492	+	0.838116i	$0.683657\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−4.97728	−0.171528
$843$	0	0
$844$	28.0561	0.965730
$845$	−7.55508	−0.259903
$846$	0	0
$847$	4.02697	0.138368
$848$	−7.01617	−0.240936
$849$	0	0
$850$	1.02303	0.0350895
$851$	3.97019	0.136096
$852$	0	0
$853$	−36.6916	−1.25630	−0.628148	−	0.778094i	$-0.716187\pi$
$853$	−36.6916	−1.25630	−0.628148	+	0.778094i	$0.716187\pi$
$854$	−0.236222	−0.00808336
$855$	0	0
$856$	12.2133	0.417444
$857$	14.6234	0.499527	0.249763	−	0.968307i	$-0.419647\pi$
$857$	14.6234	0.499527	0.249763	+	0.968307i	$0.419647\pi$
$858$	0	0
$859$	17.8847	0.610219	0.305110	−	0.952317i	$-0.401307\pi$
$859$	17.8847	0.610219	0.305110	+	0.952317i	$0.401307\pi$
$860$	−8.11747	−0.276803
$861$	0	0
$862$	5.95706	0.202898
$863$	−5.51718	−0.187807	−0.0939034	−	0.995581i	$-0.529934\pi$
$863$	−5.51718	−0.187807	−0.0939034	+	0.995581i	$0.529934\pi$
$864$	0	0
$865$	−0.194657	−0.00661854
$866$	−7.31518	−0.248580
$867$	0	0
$868$	−0.944460	−0.0320571
$869$	−9.27225	−0.314540
$870$	0	0
$871$	21.0471	0.713154
$872$	9.07686	0.307381
$873$	0	0
$874$	−0.658790	−0.0222839
$875$	3.26863	0.110500
$876$	0	0
$877$	7.12677	0.240654	0.120327	−	0.992734i	$-0.461606\pi$
$877$	7.12677	0.240654	0.120327	+	0.992734i	$0.461606\pi$
$878$	4.62419	0.156059
$879$	0	0
$880$	2.64023	0.0890021
$881$	20.5458	0.692206	0.346103	−	0.938197i	$-0.387505\pi$
$881$	20.5458	0.692206	0.346103	+	0.938197i	$0.387505\pi$
$882$	0	0
$883$	49.5380	1.66709	0.833543	−	0.552455i	$-0.186309\pi$
$883$	49.5380	1.66709	0.833543	+	0.552455i	$0.186309\pi$
$884$	5.82945	0.196066
$885$	0	0
$886$	−3.99147	−0.134096
$887$	−40.4280	−1.35744	−0.678720	−	0.734397i	$-0.737465\pi$
$887$	−40.4280	−1.35744	−0.678720	+	0.734397i	$0.737465\pi$
$888$	0	0
$889$	4.07920	0.136812
$890$	−0.417358	−0.0139899
$891$	0	0
$892$	15.6206	0.523017
$893$	1.43872	0.0481448
$894$	0	0
$895$	11.4845	0.383885
$896$	2.17098	0.0725274
$897$	0	0
$898$	−6.27264	−0.209321
$899$	−1.24496	−0.0415218
$900$	0	0
$901$	2.48648	0.0828366
$902$	−1.33871	−0.0445742
$903$	0	0
$904$	−10.8527	−0.360956
$905$	−10.7136	−0.356133
$906$	0	0
$907$	−31.7948	−1.05573	−0.527864	−	0.849329i	$-0.677007\pi$
$907$	−31.7948	−1.05573	−0.527864	+	0.849329i	$0.677007\pi$
$908$	−20.6799	−0.686287
$909$	0	0
$910$	−0.144475	−0.00478931
$911$	51.3127	1.70007	0.850034	−	0.526729i	$-0.176582\pi$
$911$	51.3127	1.70007	0.850034	+	0.526729i	$0.176582\pi$
$912$	0	0
$913$	−12.1994	−0.403741
$914$	−6.30593	−0.208582
$915$	0	0
$916$	53.6422	1.77239
$917$	5.01580	0.165636
$918$	0	0
$919$	−41.3474	−1.36393	−0.681963	−	0.731387i	$-0.738874\pi$
$919$	−41.3474	−1.36393	−0.681963	+	0.731387i	$0.738874\pi$
$920$	−0.674540	−0.0222389
$921$	0	0
$922$	2.30046	0.0757616
$923$	31.3679	1.03249
$924$	0	0
$925$	−16.4405	−0.540559
$926$	−0.559072	−0.0183723
$927$	0	0
$928$	2.15269	0.0706654
$929$	0.914960	0.0300189	0.0150094	−	0.999887i	$-0.495222\pi$
$929$	0.914960	0.0300189	0.0150094	+	0.999887i	$0.495222\pi$
$930$	0	0
$931$	24.5977	0.806156
$932$	−42.5664	−1.39431
$933$	0	0
$934$	−1.77884	−0.0582055
$935$	−0.935677	−0.0305999
$936$	0	0
$937$	48.1226	1.57210	0.786048	−	0.618165i	$-0.212124\pi$
$937$	48.1226	1.57210	0.786048	+	0.618165i	$0.212124\pi$
$938$	−0.676668	−0.0220940
$939$	0	0
$940$	0.730303	0.0238199
$941$	14.3235	0.466934	0.233467	−	0.972365i	$-0.424993\pi$
$941$	14.3235	0.466934	0.233467	+	0.972365i	$0.424993\pi$
$942$	0	0
$943$	−9.73001	−0.316853
$944$	−0.304909	−0.00992394
$945$	0	0
$946$	0.612823	0.0199246
$947$	−11.7482	−0.381767	−0.190883	−	0.981613i	$-0.561135\pi$
$947$	−11.7482	−0.381767	−0.190883	+	0.981613i	$0.561135\pi$
$948$	0	0
$949$	16.8784	0.547895
$950$	2.72804	0.0885092
$951$	0	0
$952$	−0.378045	−0.0122525
$953$	−49.0159	−1.58778	−0.793890	−	0.608061i	$-0.791947\pi$
$953$	−49.0159	−1.58778	−0.793890	+	0.608061i	$0.791947\pi$
$954$	0	0
$955$	5.68955	0.184109
$956$	1.01748	0.0329078
$957$	0	0
$958$	6.09980	0.197075
$959$	−0.256451	−0.00828124
$960$	0	0
$961$	−29.4501	−0.950002
$962$	1.60410	0.0517183
$963$	0	0
$964$	52.1062	1.67823
$965$	−16.5671	−0.533314
$966$	0	0
$967$	44.3688	1.42681	0.713403	−	0.700754i	$-0.247153\pi$
$967$	44.3688	1.42681	0.713403	+	0.700754i	$0.247153\pi$
$968$	7.59650	0.244161
$969$	0	0
$970$	−0.0214946	−0.000690152 0
$971$	59.4564	1.90805	0.954024	−	0.299731i	$-0.0968969\pi$
$971$	59.4564	1.90805	0.954024	+	0.299731i	$0.0968969\pi$
$972$	0	0
$973$	−0.891354	−0.0285755
$974$	−3.70458	−0.118703
$975$	0	0
$976$	12.6770	0.405780
$977$	20.5068	0.656069	0.328035	−	0.944666i	$-0.393614\pi$
$977$	20.5068	0.656069	0.328035	+	0.944666i	$0.393614\pi$
$978$	0	0
$979$	−1.84012	−0.0588106
$980$	12.4860	0.398850
$981$	0	0
$982$	5.35020	0.170732
$983$	−36.1826	−1.15405	−0.577023	−	0.816728i	$-0.695786\pi$
$983$	−36.1826	−1.15405	−0.577023	+	0.816728i	$0.695786\pi$
$984$	0	0
$985$	11.6947	0.372623
$986$	−0.247049	−0.00786766
$987$	0	0
$988$	15.5450	0.494553
$989$	4.45412	0.141633
$990$	0	0
$991$	39.6822	1.26055	0.630274	−	0.776373i	$-0.282943\pi$
$991$	39.6822	1.26055	0.630274	+	0.776373i	$0.282943\pi$
$992$	−2.68001	−0.0850905
$993$	0	0
$994$	−1.00848	−0.0319871
$995$	−24.7404	−0.784325
$996$	0	0
$997$	32.3711	1.02520	0.512601	−	0.858627i	$-0.328682\pi$
$997$	32.3711	1.02520	0.512601	+	0.858627i	$0.328682\pi$
$998$	2.32422	0.0735718
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.13	✓	22
3.2	odd	2		6003.2.a.u.1.10	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.13	✓	22	1.1	even	1	trivial
6003.2.a.u.1.10	yes	22	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+0.183492 q^{2} -1.96633 q^{4} +0.926834 q^{5} -0.385808 q^{7} -0.727789 q^{8} +O(q^{10})\)	\(q+0.183492 q^{2} -1.96633 q^{4} +0.926834 q^{5} -0.385808 q^{7} -0.727789 q^{8} +0.170066 q^{10} +0.749819 q^{11} +2.20193 q^{13} -0.0707926 q^{14} +3.79912 q^{16} -1.34638 q^{17} -3.59030 q^{19} -1.82246 q^{20} +0.137586 q^{22} +1.00000 q^{23} -4.14098 q^{25} +0.404037 q^{26} +0.758626 q^{28} +1.00000 q^{29} -1.24496 q^{31} +2.15269 q^{32} -0.247049 q^{34} -0.357580 q^{35} +3.97019 q^{37} -0.658790 q^{38} -0.674540 q^{40} -9.73001 q^{41} +4.45412 q^{43} -1.47439 q^{44} +0.183492 q^{46} -0.400723 q^{47} -6.85115 q^{49} -0.759836 q^{50} -4.32973 q^{52} -1.84679 q^{53} +0.694958 q^{55} +0.280787 q^{56} +0.183492 q^{58} -0.0802578 q^{59} +3.33682 q^{61} -0.228440 q^{62} -7.20324 q^{64} +2.04083 q^{65} +9.55846 q^{67} +2.64743 q^{68} -0.0656130 q^{70} +14.2456 q^{71} +7.66525 q^{73} +0.728497 q^{74} +7.05971 q^{76} -0.289286 q^{77} -12.3660 q^{79} +3.52115 q^{80} -1.78538 q^{82} -16.2698 q^{83} -1.24787 q^{85} +0.817295 q^{86} -0.545710 q^{88} -2.45409 q^{89} -0.849523 q^{91} -1.96633 q^{92} -0.0735294 q^{94} -3.32761 q^{95} -0.126390 q^{97} -1.25713 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more