LMFDB - Embedded newform 6003.2.a.s.1.15

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			1.15
Root			$-1.41510$ 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)

$f(q)$	$=$	$q+1.41510 q^{2} +0.00250280 q^{4} -0.0661489 q^{5} +1.68271 q^{7} -2.82665 q^{8} +O(q^{10})$	\(q+1.41510 q^{2} +0.00250280 q^{4} -0.0661489 q^{5} +1.68271 q^{7} -2.82665 q^{8} -0.0936071 q^{10} +6.36138 q^{11} +1.48279 q^{13} +2.38120 q^{14} -4.00500 q^{16} +0.164936 q^{17} +4.96034 q^{19} -0.000165558 q^{20} +9.00198 q^{22} -1.00000 q^{23} -4.99562 q^{25} +2.09830 q^{26} +0.00421148 q^{28} -1.00000 q^{29} +3.38542 q^{31} -0.0141580 q^{32} +0.233401 q^{34} -0.111309 q^{35} +10.0009 q^{37} +7.01937 q^{38} +0.186980 q^{40} -7.66574 q^{41} +3.17946 q^{43} +0.0159213 q^{44} -1.41510 q^{46} -3.81508 q^{47} -4.16850 q^{49} -7.06930 q^{50} +0.00371114 q^{52} -1.23830 q^{53} -0.420798 q^{55} -4.75643 q^{56} -1.41510 q^{58} +7.72903 q^{59} -11.1893 q^{61} +4.79071 q^{62} +7.98996 q^{64} -0.0980852 q^{65} +0.107831 q^{67} +0.000412803 q^{68} -0.157513 q^{70} +5.18281 q^{71} +11.9211 q^{73} +14.1523 q^{74} +0.0124147 q^{76} +10.7043 q^{77} +2.20102 q^{79} +0.264926 q^{80} -10.8478 q^{82} -8.12880 q^{83} -0.0109103 q^{85} +4.49925 q^{86} -17.9814 q^{88} -12.9994 q^{89} +2.49511 q^{91} -0.00250280 q^{92} -5.39871 q^{94} -0.328121 q^{95} +15.0574 q^{97} -5.89883 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.41510	1.00063	0.500313	−	0.865845i	$-0.333218\pi$
$2$	1.41510	1.00063	0.500313	+	0.865845i	$0.333218\pi$
$3$	0	0
$3$	0	0
$4$	0.00250280	0.00125140
$5$	−0.0661489	−0.0295827	−0.0147913	−	0.999891i	$-0.504708\pi$
$5$	−0.0661489	−0.0295827	−0.0147913	+	0.999891i	$0.504708\pi$
$6$	0	0
$7$	1.68271	0.636004	0.318002	−	0.948090i	$-0.396988\pi$
$7$	1.68271	0.636004	0.318002	+	0.948090i	$0.396988\pi$
$8$	−2.82665	−0.999373
$9$	0	0
$10$	−0.0936071	−0.0296012
$11$	6.36138	1.91803	0.959014	−	0.283359i	$-0.0914487\pi$
$11$	6.36138	1.91803	0.959014	+	0.283359i	$0.0914487\pi$
$12$	0	0
$13$	1.48279	0.411253	0.205627	−	0.978631i	$-0.434077\pi$
$13$	1.48279	0.411253	0.205627	+	0.978631i	$0.434077\pi$
$14$	2.38120	0.636401
$15$	0	0
$16$	−4.00500	−1.00125
$17$	0.164936	0.0400029	0.0200015	−	0.999800i	$-0.493633\pi$
$17$	0.164936	0.0400029	0.0200015	+	0.999800i	$0.493633\pi$
$18$	0	0
$19$	4.96034	1.13798	0.568990	−	0.822345i	$-0.307334\pi$
$19$	4.96034	1.13798	0.568990	+	0.822345i	$0.307334\pi$
$20$	−0.000165558 0	−3.70198e−5 0
$21$	0	0
$22$	9.00198	1.91923
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.99562	−0.999125
$26$	2.09830	0.411510
$27$	0	0
$28$	0.00421148	0.000795896 0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	3.38542	0.608040	0.304020	−	0.952666i	$-0.401671\pi$
$31$	3.38542	0.608040	0.304020	+	0.952666i	$0.401671\pi$
$32$	−0.0141580	−0.00250280
$33$	0	0
$34$	0.233401	0.0400279
$35$	−0.111309	−0.0188147
$36$	0	0
$37$	10.0009	1.64414	0.822072	−	0.569383i	$-0.192818\pi$
$37$	10.0009	1.64414	0.822072	+	0.569383i	$0.192818\pi$
$38$	7.01937	1.13869
$39$	0	0
$40$	0.186980	0.0295641
$41$	−7.66574	−1.19719	−0.598594	−	0.801053i	$-0.704274\pi$
$41$	−7.66574	−1.19719	−0.598594	+	0.801053i	$0.704274\pi$
$42$	0	0
$43$	3.17946	0.484863	0.242431	−	0.970169i	$-0.422055\pi$
$43$	3.17946	0.484863	0.242431	+	0.970169i	$0.422055\pi$
$44$	0.0159213	0.00240022
$45$	0	0
$46$	−1.41510	−0.208645
$47$	−3.81508	−0.556486	−0.278243	−	0.960511i	$-0.589752\pi$
$47$	−3.81508	−0.556486	−0.278243	+	0.960511i	$0.589752\pi$
$48$	0	0
$49$	−4.16850	−0.595499
$50$	−7.06930	−0.999750
$51$	0	0
$52$	0.00371114	0.000514643 0
$53$	−1.23830	−0.170094	−0.0850470	−	0.996377i	$-0.527104\pi$
$53$	−1.23830	−0.170094	−0.0850470	+	0.996377i	$0.527104\pi$
$54$	0	0
$55$	−0.420798	−0.0567404
$56$	−4.75643	−0.635605
$57$	0	0
$58$	−1.41510	−0.185811
$59$	7.72903	1.00623	0.503117	−	0.864218i	$-0.332186\pi$
$59$	7.72903	1.00623	0.503117	+	0.864218i	$0.332186\pi$
$60$	0	0
$61$	−11.1893	−1.43264	−0.716321	−	0.697771i	$-0.754175\pi$
$61$	−11.1893	−1.43264	−0.716321	+	0.697771i	$0.754175\pi$
$62$	4.79071	0.608420
$63$	0	0
$64$	7.98996	0.998745
$65$	−0.0980852	−0.0121660
$66$	0	0
$67$	0.107831	0.0131736	0.00658681	−	0.999978i	$-0.497903\pi$
$67$	0.107831	0.0131736	0.00658681	+	0.999978i	$0.497903\pi$
$68$	0.000412803 0	5.00597e−5 0
$69$	0	0
$70$	−0.157513	−0.0188265
$71$	5.18281	0.615087	0.307543	−	0.951534i	$-0.400493\pi$
$71$	5.18281	0.615087	0.307543	+	0.951534i	$0.400493\pi$
$72$	0	0
$73$	11.9211	1.39526	0.697631	−	0.716458i	$-0.254238\pi$
$73$	11.9211	1.39526	0.697631	+	0.716458i	$0.254238\pi$
$74$	14.1523	1.64517
$75$	0	0
$76$	0.0124147	0.00142407
$77$	10.7043	1.21987
$78$	0	0
$79$	2.20102	0.247634	0.123817	−	0.992305i	$-0.460486\pi$
$79$	2.20102	0.247634	0.123817	+	0.992305i	$0.460486\pi$
$80$	0.264926	0.0296196
$81$	0	0
$82$	−10.8478	−1.19794
$83$	−8.12880	−0.892252	−0.446126	−	0.894970i	$-0.647197\pi$
$83$	−8.12880	−0.892252	−0.446126	+	0.894970i	$0.647197\pi$
$84$	0	0
$85$	−0.0109103	−0.00118339
$86$	4.49925	0.485166
$87$	0	0
$88$	−17.9814	−1.91683
$89$	−12.9994	−1.37794	−0.688969	−	0.724791i	$-0.741936\pi$
$89$	−12.9994	−1.37794	−0.688969	+	0.724791i	$0.741936\pi$
$90$	0	0
$91$	2.49511	0.261559
$92$	−0.00250280	−0.000260935 0
$93$	0	0
$94$	−5.39871	−0.556834
$95$	−0.328121	−0.0336645
$96$	0	0
$97$	15.0574	1.52884	0.764422	−	0.644717i	$-0.223025\pi$
$97$	15.0574	1.52884	0.764422	+	0.644717i	$0.223025\pi$
$98$	−5.89883	−0.595872
$99$	0	0
$100$	−0.0125031	−0.00125031
$101$	17.6465	1.75589	0.877945	−	0.478762i	$-0.158914\pi$
$101$	17.6465	1.75589	0.877945	+	0.478762i	$0.158914\pi$
$102$	0	0
$103$	9.78080	0.963731	0.481866	−	0.876245i	$-0.339959\pi$
$103$	9.78080	0.963731	0.481866	+	0.876245i	$0.339959\pi$
$104$	−4.19135	−0.410995
$105$	0	0
$106$	−1.75232	−0.170200
$107$	−12.4134	−1.20005	−0.600023	−	0.799983i	$-0.704842\pi$
$107$	−12.4134	−1.20005	−0.600023	+	0.799983i	$0.704842\pi$
$108$	0	0
$109$	−9.55772	−0.915464	−0.457732	−	0.889090i	$-0.651338\pi$
$109$	−9.55772	−0.915464	−0.457732	+	0.889090i	$0.651338\pi$
$110$	−0.595470	−0.0567759
$111$	0	0
$112$	−6.73924	−0.636799
$113$	20.7986	1.95656	0.978282	−	0.207279i	$-0.0664609\pi$
$113$	20.7986	1.95656	0.978282	+	0.207279i	$0.0664609\pi$
$114$	0	0
$115$	0.0661489	0.00616841
$116$	−0.00250280	−0.000232379 0
$117$	0	0
$118$	10.9373	1.00686
$119$	0.277539	0.0254420
$120$	0	0
$121$	29.4671	2.67883
$122$	−15.8339	−1.43354
$123$	0	0
$124$	0.00847304	0.000760902 0
$125$	0.661199	0.0591395
$126$	0	0
$127$	15.0834	1.33844	0.669220	−	0.743065i	$-0.266628\pi$
$127$	15.0834	1.33844	0.669220	+	0.743065i	$0.266628\pi$
$128$	11.3349	1.00187
$129$	0	0
$130$	−0.138800	−0.0121736
$131$	8.14086	0.711270	0.355635	−	0.934625i	$-0.384265\pi$
$131$	8.14086	0.711270	0.355635	+	0.934625i	$0.384265\pi$
$132$	0	0
$133$	8.34680	0.723759
$134$	0.152591	0.0131819
$135$	0	0
$136$	−0.466218	−0.0399778
$137$	−16.0272	−1.36930	−0.684649	−	0.728873i	$-0.740045\pi$
$137$	−16.0272	−1.36930	−0.684649	+	0.728873i	$0.740045\pi$
$138$	0	0
$139$	12.8642	1.09113	0.545565	−	0.838068i	$-0.316315\pi$
$139$	12.8642	1.09113	0.545565	+	0.838068i	$0.316315\pi$
$140$	−0.000278585 0	−2.35447e−5 0
$141$	0	0
$142$	7.33419	0.615471
$143$	9.43262	0.788795
$144$	0	0
$145$	0.0661489	0.00549336
$146$	16.8696	1.39613
$147$	0	0
$148$	0.0250304	0.00205748
$149$	−15.7895	−1.29353	−0.646764	−	0.762690i	$-0.723878\pi$
$149$	−15.7895	−1.29353	−0.646764	+	0.762690i	$0.723878\pi$
$150$	0	0
$151$	15.5878	1.26852	0.634258	−	0.773122i	$-0.281306\pi$
$151$	15.5878	1.26852	0.634258	+	0.773122i	$0.281306\pi$
$152$	−14.0212	−1.13727
$153$	0	0
$154$	15.1477	1.22064
$155$	−0.223942	−0.0179874
$156$	0	0
$157$	1.86070	0.148500	0.0742500	−	0.997240i	$-0.476344\pi$
$157$	1.86070	0.148500	0.0742500	+	0.997240i	$0.476344\pi$
$158$	3.11466	0.247789
$159$	0	0
$160$	0.000936534 0	7.40395e−5 0
$161$	−1.68271	−0.132616
$162$	0	0
$163$	8.92976	0.699433	0.349717	−	0.936856i	$-0.386278\pi$
$163$	8.92976	0.699433	0.349717	+	0.936856i	$0.386278\pi$
$164$	−0.0191858	−0.00149816
$165$	0	0
$166$	−11.5031	−0.892810
$167$	−5.48419	−0.424379	−0.212190	−	0.977228i	$-0.568059\pi$
$167$	−5.48419	−0.424379	−0.212190	+	0.977228i	$0.568059\pi$
$168$	0	0
$169$	−10.8013	−0.830871
$170$	−0.0154392	−0.00118413
$171$	0	0
$172$	0.00795755	0.000606758 0
$173$	−9.90355	−0.752953	−0.376476	−	0.926426i	$-0.622864\pi$
$173$	−9.90355	−0.752953	−0.376476	+	0.926426i	$0.622864\pi$
$174$	0	0
$175$	−8.40617	−0.635447
$176$	−25.4773	−1.92043
$177$	0	0
$178$	−18.3955	−1.37880
$179$	0.604420	0.0451765	0.0225882	−	0.999745i	$-0.492809\pi$
$179$	0.604420	0.0451765	0.0225882	+	0.999745i	$0.492809\pi$
$180$	0	0
$181$	−2.15864	−0.160451	−0.0802254	−	0.996777i	$-0.525564\pi$
$181$	−2.15864	−0.160451	−0.0802254	+	0.996777i	$0.525564\pi$
$182$	3.53082	0.261722
$183$	0	0
$184$	2.82665	0.208384
$185$	−0.661551	−0.0486382
$186$	0	0
$187$	1.04922	0.0767267
$188$	−0.00954839	−0.000696388 0
$189$	0	0
$190$	−0.464323	−0.0336855
$191$	20.4433	1.47922	0.739611	−	0.673034i	$-0.235009\pi$
$191$	20.4433	1.47922	0.739611	+	0.673034i	$0.235009\pi$
$192$	0	0
$193$	−27.6836	−1.99271	−0.996353	−	0.0853263i	$-0.972807\pi$
$193$	−27.6836	−1.99271	−0.996353	+	0.0853263i	$0.972807\pi$
$194$	21.3076	1.52980
$195$	0	0
$196$	−0.0104329	−0.000745209 0
$197$	−12.8501	−0.915534	−0.457767	−	0.889072i	$-0.651351\pi$
$197$	−12.8501	−0.915534	−0.457767	+	0.889072i	$0.651351\pi$
$198$	0	0
$199$	6.46992	0.458641	0.229320	−	0.973351i	$-0.426350\pi$
$199$	6.46992	0.458641	0.229320	+	0.973351i	$0.426350\pi$
$200$	14.1209	0.998499
$201$	0	0
$202$	24.9715	1.75699
$203$	−1.68271	−0.118103
$204$	0	0
$205$	0.507080	0.0354160
$206$	13.8408	0.964334
$207$	0	0
$208$	−5.93859	−0.411767
$209$	31.5546	2.18268
$210$	0	0
$211$	18.1502	1.24951	0.624755	−	0.780821i	$-0.285199\pi$
$211$	18.1502	1.24951	0.624755	+	0.780821i	$0.285199\pi$
$212$	−0.00309923	−0.000212856 0
$213$	0	0
$214$	−17.5661	−1.20080
$215$	−0.210318	−0.0143435
$216$	0	0
$217$	5.69668	0.386716
$218$	−13.5251	−0.916036
$219$	0	0
$220$	−0.00105317	−7.10050e−5 0
$221$	0.244566	0.0164513
$222$	0	0
$223$	17.8721	1.19681	0.598403	−	0.801195i	$-0.295802\pi$
$223$	17.8721	1.19681	0.598403	+	0.801195i	$0.295802\pi$
$224$	−0.0238237	−0.00159179
$225$	0	0
$226$	29.4320	1.95779
$227$	14.8714	0.987048	0.493524	−	0.869732i	$-0.335709\pi$
$227$	14.8714	0.987048	0.493524	+	0.869732i	$0.335709\pi$
$228$	0	0
$229$	−19.2578	−1.27259	−0.636294	−	0.771446i	$-0.719534\pi$
$229$	−19.2578	−1.27259	−0.636294	+	0.771446i	$0.719534\pi$
$230$	0.0936071	0.00617227
$231$	0	0
$232$	2.82665	0.185579
$233$	−26.9074	−1.76276	−0.881380	−	0.472408i	$-0.843385\pi$
$233$	−26.9074	−1.76276	−0.881380	+	0.472408i	$0.843385\pi$
$234$	0	0
$235$	0.252363	0.0164624
$236$	0.0193442	0.00125920
$237$	0	0
$238$	0.392745	0.0254579
$239$	4.47083	0.289194	0.144597	−	0.989491i	$-0.453811\pi$
$239$	4.47083	0.289194	0.144597	+	0.989491i	$0.453811\pi$
$240$	0	0
$241$	16.5316	1.06489	0.532446	−	0.846464i	$-0.321273\pi$
$241$	16.5316	1.06489	0.532446	+	0.846464i	$0.321273\pi$
$242$	41.6989	2.68051
$243$	0	0
$244$	−0.0280046	−0.00179281
$245$	0.275741	0.0176165
$246$	0	0
$247$	7.35516	0.467998
$248$	−9.56942	−0.607659
$249$	0	0
$250$	0.935662	0.0591764
$251$	13.0096	0.821157	0.410579	−	0.911825i	$-0.365327\pi$
$251$	13.0096	0.821157	0.410579	+	0.911825i	$0.365327\pi$
$252$	0	0
$253$	−6.36138	−0.399936
$254$	21.3446	1.33928
$255$	0	0
$256$	0.0600672	0.00375420
$257$	−0.836189	−0.0521601	−0.0260800	−	0.999660i	$-0.508302\pi$
$257$	−0.836189	−0.0521601	−0.0260800	+	0.999660i	$0.508302\pi$
$258$	0	0
$259$	16.8287	1.04568
$260$	−0.000245488 0	−1.52245e−5 0
$261$	0	0
$262$	11.5201	0.711715
$263$	5.86335	0.361550	0.180775	−	0.983525i	$-0.442139\pi$
$263$	5.86335	0.361550	0.180775	+	0.983525i	$0.442139\pi$
$264$	0	0
$265$	0.0819123	0.00503183
$266$	11.8115	0.724212
$267$	0	0
$268$	0.000269879 0	1.64855e−5 0
$269$	9.35063	0.570118	0.285059	−	0.958510i	$-0.407987\pi$
$269$	9.35063	0.570118	0.285059	+	0.958510i	$0.407987\pi$
$270$	0	0
$271$	0.200145	0.0121579	0.00607897	−	0.999982i	$-0.498065\pi$
$271$	0.200145	0.0121579	0.00607897	+	0.999982i	$0.498065\pi$
$272$	−0.660569	−0.0400529
$273$	0	0
$274$	−22.6801	−1.37016
$275$	−31.7791	−1.91635
$276$	0	0
$277$	28.2932	1.69998	0.849988	−	0.526802i	$-0.176609\pi$
$277$	28.2932	1.69998	0.849988	+	0.526802i	$0.176609\pi$
$278$	18.2042	1.09181
$279$	0	0
$280$	0.314633	0.0188029
$281$	7.56733	0.451429	0.225715	−	0.974193i	$-0.427528\pi$
$281$	7.56733	0.451429	0.225715	+	0.974193i	$0.427528\pi$
$282$	0	0
$283$	25.8217	1.53494	0.767470	−	0.641085i	$-0.221516\pi$
$283$	25.8217	1.53494	0.767470	+	0.641085i	$0.221516\pi$
$284$	0.0129716	0.000769720 0
$285$	0	0
$286$	13.3481	0.789288
$287$	−12.8992	−0.761416
$288$	0	0
$289$	−16.9728	−0.998400
$290$	0.0936071	0.00549680
$291$	0	0
$292$	0.0298362	0.00174603
$293$	−29.0271	−1.69578	−0.847892	−	0.530170i	$-0.822128\pi$
$293$	−29.0271	−1.69578	−0.847892	+	0.530170i	$0.822128\pi$
$294$	0	0
$295$	−0.511267	−0.0297671
$296$	−28.2692	−1.64311
$297$	0	0
$298$	−22.3437	−1.29434
$299$	−1.48279	−0.0857522
$300$	0	0
$301$	5.35010	0.308374
$302$	22.0582	1.26931
$303$	0	0
$304$	−19.8662	−1.13940
$305$	0.740159	0.0423814
$306$	0	0
$307$	24.0803	1.37433	0.687166	−	0.726500i	$-0.258854\pi$
$307$	24.0803	1.37433	0.687166	+	0.726500i	$0.258854\pi$
$308$	0.0267908	0.00152655
$309$	0	0
$310$	−0.316900	−0.0179987
$311$	−6.39445	−0.362596	−0.181298	−	0.983428i	$-0.558030\pi$
$311$	−6.39445	−0.362596	−0.181298	+	0.983428i	$0.558030\pi$
$312$	0	0
$313$	9.97215	0.563659	0.281830	−	0.959464i	$-0.409059\pi$
$313$	9.97215	0.563659	0.281830	+	0.959464i	$0.409059\pi$
$314$	2.63307	0.148593
$315$	0	0
$316$	0.00550872	0.000309890 0
$317$	29.6240	1.66385	0.831925	−	0.554888i	$-0.187239\pi$
$317$	29.6240	1.66385	0.831925	+	0.554888i	$0.187239\pi$
$318$	0	0
$319$	−6.36138	−0.356169
$320$	−0.528527	−0.0295456
$321$	0	0
$322$	−2.38120	−0.132699
$323$	0.818139	0.0455225
$324$	0	0
$325$	−7.40748	−0.410893
$326$	12.6365	0.699871
$327$	0	0
$328$	21.6684	1.19644
$329$	−6.41966	−0.353927
$330$	0	0
$331$	19.4206	1.06745	0.533725	−	0.845658i	$-0.320792\pi$
$331$	19.4206	1.06745	0.533725	+	0.845658i	$0.320792\pi$
$332$	−0.0203448	−0.00111656
$333$	0	0
$334$	−7.76067	−0.424645
$335$	−0.00713288	−0.000389711 0
$336$	0	0
$337$	−26.9097	−1.46587	−0.732933	−	0.680301i	$-0.761849\pi$
$337$	−26.9097	−1.46587	−0.732933	+	0.680301i	$0.761849\pi$
$338$	−15.2849	−0.831391
$339$	0	0
$340$	−2.73064e−5 0	−1.48090e−6 0
$341$	21.5360	1.16624
$342$	0	0
$343$	−18.7933	−1.01474
$344$	−8.98723	−0.484559
$345$	0	0
$346$	−14.0145	−0.753424
$347$	−15.4245	−0.828029	−0.414015	−	0.910270i	$-0.635874\pi$
$347$	−15.4245	−0.828029	−0.414015	+	0.910270i	$0.635874\pi$
$348$	0	0
$349$	−34.4497	−1.84405	−0.922025	−	0.387131i	$-0.873466\pi$
$349$	−34.4497	−1.84405	−0.922025	+	0.387131i	$0.873466\pi$
$350$	−11.8956	−0.635845
$351$	0	0
$352$	−0.0900643	−0.00480044
$353$	−33.1664	−1.76527	−0.882636	−	0.470057i	$-0.844233\pi$
$353$	−33.1664	−1.76527	−0.882636	+	0.470057i	$0.844233\pi$
$354$	0	0
$355$	−0.342837	−0.0181959
$356$	−0.0325350	−0.00172435
$357$	0	0
$358$	0.855314	0.0452047
$359$	21.6502	1.14265	0.571327	−	0.820723i	$-0.306429\pi$
$359$	21.6502	1.14265	0.571327	+	0.820723i	$0.306429\pi$
$360$	0	0
$361$	5.60496	0.294998
$362$	−3.05469	−0.160551
$363$	0	0
$364$	0.00624476	0.000327315 0
$365$	−0.788568	−0.0412756
$366$	0	0
$367$	−24.8289	−1.29606	−0.648029	−	0.761615i	$-0.724407\pi$
$367$	−24.8289	−1.29606	−0.648029	+	0.761615i	$0.724407\pi$
$368$	4.00500	0.208775
$369$	0	0
$370$	−0.936159	−0.0486686
$371$	−2.08370	−0.108180
$372$	0	0
$373$	−19.8447	−1.02752	−0.513760	−	0.857934i	$-0.671748\pi$
$373$	−19.8447	−1.02752	−0.513760	+	0.857934i	$0.671748\pi$
$374$	1.48475	0.0767747
$375$	0	0
$376$	10.7839	0.556138
$377$	−1.48279	−0.0763678
$378$	0	0
$379$	−34.7997	−1.78754	−0.893770	−	0.448527i	$-0.851949\pi$
$379$	−34.7997	−1.78754	−0.893770	+	0.448527i	$0.851949\pi$
$380$	−0.000821221 0	−4.21278e−5 0
$381$	0	0
$382$	28.9292	1.48015
$383$	2.84300	0.145270	0.0726352	−	0.997359i	$-0.476859\pi$
$383$	2.84300	0.145270	0.0726352	+	0.997359i	$0.476859\pi$
$384$	0	0
$385$	−0.708080	−0.0360871
$386$	−39.1750	−1.99395
$387$	0	0
$388$	0.0376856	0.00191320
$389$	34.6012	1.75435	0.877176	−	0.480169i	$-0.159425\pi$
$389$	34.6012	1.75435	0.877176	+	0.480169i	$0.159425\pi$
$390$	0	0
$391$	−0.164936	−0.00834118
$392$	11.7829	0.595126
$393$	0	0
$394$	−18.1842	−0.916107
$395$	−0.145595	−0.00732568
$396$	0	0
$397$	8.09800	0.406427	0.203213	−	0.979134i	$-0.434861\pi$
$397$	8.09800	0.406427	0.203213	+	0.979134i	$0.434861\pi$
$398$	9.15558	0.458928
$399$	0	0
$400$	20.0075	1.00037
$401$	11.9067	0.594594	0.297297	−	0.954785i	$-0.403915\pi$
$401$	11.9067	0.594594	0.297297	+	0.954785i	$0.403915\pi$
$402$	0	0
$403$	5.01989	0.250058
$404$	0.0441656	0.00219732
$405$	0	0
$406$	−2.38120	−0.118177
$407$	63.6198	3.15351
$408$	0	0
$409$	18.3346	0.906590	0.453295	−	0.891361i	$-0.350248\pi$
$409$	18.3346	0.906590	0.453295	+	0.891361i	$0.350248\pi$
$410$	0.717568	0.0354382
$411$	0	0
$412$	0.0244794	0.00120601
$413$	13.0057	0.639969
$414$	0	0
$415$	0.537711	0.0263952
$416$	−0.0209934	−0.00102928
$417$	0	0
$418$	44.6529	2.18404
$419$	−26.3913	−1.28930	−0.644651	−	0.764477i	$-0.722997\pi$
$419$	−26.3913	−1.28930	−0.644651	+	0.764477i	$0.722997\pi$
$420$	0	0
$421$	−3.46444	−0.168846	−0.0844232	−	0.996430i	$-0.526905\pi$
$421$	−3.46444	−0.168846	−0.0844232	+	0.996430i	$0.526905\pi$
$422$	25.6843	1.25029
$423$	0	0
$424$	3.50025	0.169987
$425$	−0.823959	−0.0399679
$426$	0	0
$427$	−18.8283	−0.911165
$428$	−0.0310682	−0.00150174
$429$	0	0
$430$	−0.297620	−0.0143525
$431$	−7.11092	−0.342521	−0.171260	−	0.985226i	$-0.554784\pi$
$431$	−7.11092	−0.342521	−0.171260	+	0.985226i	$0.554784\pi$
$432$	0	0
$433$	−2.39345	−0.115022	−0.0575108	−	0.998345i	$-0.518316\pi$
$433$	−2.39345	−0.115022	−0.0575108	+	0.998345i	$0.518316\pi$
$434$	8.06136	0.386957
$435$	0	0
$436$	−0.0239211	−0.00114561
$437$	−4.96034	−0.237285
$438$	0	0
$439$	4.77158	0.227735	0.113867	−	0.993496i	$-0.463676\pi$
$439$	4.77158	0.227735	0.113867	+	0.993496i	$0.463676\pi$
$440$	1.18945	0.0567048
$441$	0	0
$442$	0.346086	0.0164616
$443$	5.00340	0.237719	0.118859	−	0.992911i	$-0.462076\pi$
$443$	5.00340	0.237719	0.118859	+	0.992911i	$0.462076\pi$
$444$	0	0
$445$	0.859898	0.0407631
$446$	25.2908	1.19755
$447$	0	0
$448$	13.4448	0.635206
$449$	15.7800	0.744704	0.372352	−	0.928092i	$-0.378551\pi$
$449$	15.7800	0.744704	0.372352	+	0.928092i	$0.378551\pi$
$450$	0	0
$451$	−48.7647	−2.29624
$452$	0.0520547	0.00244845
$453$	0	0
$454$	21.0445	0.987665
$455$	−0.165049	−0.00773760
$456$	0	0
$457$	−9.39664	−0.439556	−0.219778	−	0.975550i	$-0.570533\pi$
$457$	−9.39664	−0.439556	−0.219778	+	0.975550i	$0.570533\pi$
$458$	−27.2516	−1.27338
$459$	0	0
$460$	0.000165558 0	7.71916e−6 0
$461$	−27.7472	−1.29232	−0.646159	−	0.763203i	$-0.723626\pi$
$461$	−27.7472	−1.29232	−0.646159	+	0.763203i	$0.723626\pi$
$462$	0	0
$463$	−32.2822	−1.50028	−0.750140	−	0.661279i	$-0.770014\pi$
$463$	−32.2822	−1.50028	−0.750140	+	0.661279i	$0.770014\pi$
$464$	4.00500	0.185927
$465$	0	0
$466$	−38.0766	−1.76386
$467$	9.90787	0.458482	0.229241	−	0.973370i	$-0.426376\pi$
$467$	9.90787	0.458482	0.229241	+	0.973370i	$0.426376\pi$
$468$	0	0
$469$	0.181448	0.00837847
$470$	0.357119	0.0164727
$471$	0	0
$472$	−21.8473	−1.00560
$473$	20.2257	0.929980
$474$	0	0
$475$	−24.7800	−1.13698
$476$	0.000694626 0	3.18381e−5 0
$477$	0	0
$478$	6.32666	0.289375
$479$	−22.7189	−1.03805	−0.519026	−	0.854758i	$-0.673705\pi$
$479$	−22.7189	−1.03805	−0.519026	+	0.854758i	$0.673705\pi$
$480$	0	0
$481$	14.8293	0.676160
$482$	23.3938	1.06556
$483$	0	0
$484$	0.0737504	0.00335229
$485$	−0.996027	−0.0452273
$486$	0	0
$487$	−22.5117	−1.02010	−0.510051	−	0.860144i	$-0.670373\pi$
$487$	−22.5117	−1.02010	−0.510051	+	0.860144i	$0.670373\pi$
$488$	31.6283	1.43174
$489$	0	0
$490$	0.390201	0.0176275
$491$	4.13066	0.186414	0.0932070	−	0.995647i	$-0.470288\pi$
$491$	4.13066	0.186414	0.0932070	+	0.995647i	$0.470288\pi$
$492$	0	0
$493$	−0.164936	−0.00742835
$494$	10.4083	0.468291
$495$	0	0
$496$	−13.5586	−0.608800
$497$	8.72116	0.391197
$498$	0	0
$499$	−17.2172	−0.770748	−0.385374	−	0.922760i	$-0.625928\pi$
$499$	−17.2172	−0.770748	−0.385374	+	0.922760i	$0.625928\pi$
$500$	0.00165485	7.40072e−5 0
$501$	0	0
$502$	18.4098	0.821671
$503$	20.6576	0.921076	0.460538	−	0.887640i	$-0.347657\pi$
$503$	20.6576	0.921076	0.460538	+	0.887640i	$0.347657\pi$
$504$	0	0
$505$	−1.16729	−0.0519439
$506$	−9.00198	−0.400187
$507$	0	0
$508$	0.0377509	0.00167492
$509$	−24.2073	−1.07297	−0.536485	−	0.843910i	$-0.680248\pi$
$509$	−24.2073	−1.07297	−0.536485	+	0.843910i	$0.680248\pi$
$510$	0	0
$511$	20.0598	0.887391
$512$	−22.5848	−0.998116
$513$	0	0
$514$	−1.18329	−0.0521927
$515$	−0.646989	−0.0285097
$516$	0	0
$517$	−24.2692	−1.06736
$518$	23.8142	1.04634
$519$	0	0
$520$	0.277253	0.0121583
$521$	2.66812	0.116892	0.0584462	−	0.998291i	$-0.481385\pi$
$521$	2.66812	0.116892	0.0584462	+	0.998291i	$0.481385\pi$
$522$	0	0
$523$	−29.8778	−1.30646	−0.653232	−	0.757158i	$-0.726587\pi$
$523$	−29.8778	−1.30646	−0.653232	+	0.757158i	$0.726587\pi$
$524$	0.0203750	0.000890084 0
$525$	0	0
$526$	8.29722	0.361776
$527$	0.558379	0.0243234
$528$	0	0
$529$	1.00000	0.0434783
$530$	0.115914	0.00503498
$531$	0	0
$532$	0.0208904	0.000905713 0
$533$	−11.3667	−0.492347
$534$	0	0
$535$	0.821131	0.0355006
$536$	−0.304800	−0.0131654
$537$	0	0
$538$	13.2321	0.570474
$539$	−26.5174	−1.14218
$540$	0	0
$541$	−32.0957	−1.37990	−0.689951	−	0.723856i	$-0.742368\pi$
$541$	−32.0957	−1.37990	−0.689951	+	0.723856i	$0.742368\pi$
$542$	0.283225	0.0121655
$543$	0	0
$544$	−0.00233516	−0.000100119 0
$545$	0.632233	0.0270819
$546$	0	0
$547$	−12.7510	−0.545194	−0.272597	−	0.962128i	$-0.587883\pi$
$547$	−12.7510	−0.545194	−0.272597	+	0.962128i	$0.587883\pi$
$548$	−0.0401130	−0.00171354
$549$	0	0
$550$	−44.9705	−1.91755
$551$	−4.96034	−0.211318
$552$	0	0
$553$	3.70368	0.157496
$554$	40.0377	1.70104
$555$	0	0
$556$	0.0321966	0.00136544
$557$	−3.10403	−0.131522	−0.0657609	−	0.997835i	$-0.520947\pi$
$557$	−3.10403	−0.131522	−0.0657609	+	0.997835i	$0.520947\pi$
$558$	0	0
$559$	4.71448	0.199401
$560$	0.445793	0.0188382
$561$	0	0
$562$	10.7085	0.451712
$563$	24.8014	1.04525	0.522627	−	0.852561i	$-0.324952\pi$
$563$	24.8014	1.04525	0.522627	+	0.852561i	$0.324952\pi$
$564$	0	0
$565$	−1.37580	−0.0578804
$566$	36.5402	1.53590
$567$	0	0
$568$	−14.6500	−0.614701
$569$	42.8490	1.79633	0.898163	−	0.439663i	$-0.144902\pi$
$569$	42.8490	1.79633	0.898163	+	0.439663i	$0.144902\pi$
$570$	0	0
$571$	−34.5336	−1.44519	−0.722594	−	0.691273i	$-0.757050\pi$
$571$	−34.5336	−1.44519	−0.722594	+	0.691273i	$0.757050\pi$
$572$	0.0236080	0.000987099 0
$573$	0	0
$574$	−18.2536	−0.761892
$575$	4.99562	0.208332
$576$	0	0
$577$	−31.9373	−1.32957	−0.664784	−	0.747036i	$-0.731476\pi$
$577$	−31.9373	−1.32957	−0.664784	+	0.747036i	$0.731476\pi$
$578$	−24.0182	−0.999024
$579$	0	0
$580$	0.000165558 0	6.87440e−6 0
$581$	−13.6784	−0.567475
$582$	0	0
$583$	−7.87731	−0.326245
$584$	−33.6969	−1.39439
$585$	0	0
$586$	−41.0762	−1.69684
$587$	15.2752	0.630473	0.315236	−	0.949013i	$-0.397916\pi$
$587$	15.2752	0.630473	0.315236	+	0.949013i	$0.397916\pi$
$588$	0	0
$589$	16.7928	0.691937
$590$	−0.723492	−0.0297857
$591$	0	0
$592$	−40.0538	−1.64620
$593$	−20.4058	−0.837964	−0.418982	−	0.907995i	$-0.637613\pi$
$593$	−20.4058	−0.837964	−0.418982	+	0.907995i	$0.637613\pi$
$594$	0	0
$595$	−0.0183589	−0.000752642 0
$596$	−0.0395180	−0.00161872
$597$	0	0
$598$	−2.09830	−0.0858059
$599$	5.56455	0.227361	0.113681	−	0.993517i	$-0.463736\pi$
$599$	5.56455	0.227361	0.113681	+	0.993517i	$0.463736\pi$
$600$	0	0
$601$	14.9528	0.609937	0.304969	−	0.952362i	$-0.401354\pi$
$601$	14.9528	0.609937	0.304969	+	0.952362i	$0.401354\pi$
$602$	7.57091	0.308567
$603$	0	0
$604$	0.0390131	0.00158742
$605$	−1.94922	−0.0792470
$606$	0	0
$607$	−17.1691	−0.696872	−0.348436	−	0.937333i	$-0.613287\pi$
$607$	−17.1691	−0.696872	−0.348436	+	0.937333i	$0.613287\pi$
$608$	−0.0702284	−0.00284814
$609$	0	0
$610$	1.04740	0.0424079
$611$	−5.65698	−0.228857
$612$	0	0
$613$	−48.0492	−1.94069	−0.970344	−	0.241730i	$-0.922285\pi$
$613$	−48.0492	−1.94069	−0.970344	+	0.241730i	$0.922285\pi$
$614$	34.0759	1.37519
$615$	0	0
$616$	−30.2575	−1.21911
$617$	36.7302	1.47870	0.739351	−	0.673320i	$-0.235132\pi$
$617$	36.7302	1.47870	0.739351	+	0.673320i	$0.235132\pi$
$618$	0	0
$619$	7.40958	0.297816	0.148908	−	0.988851i	$-0.452424\pi$
$619$	7.40958	0.297816	0.148908	+	0.988851i	$0.452424\pi$
$620$	−0.000560482 0	−2.25095e−5 0
$621$	0	0
$622$	−9.04877	−0.362823
$623$	−21.8743	−0.876373
$624$	0	0
$625$	24.9344	0.997375
$626$	14.1116	0.564012
$627$	0	0
$628$	0.00465696	0.000185833 0
$629$	1.64952	0.0657705
$630$	0	0
$631$	29.6444	1.18013	0.590063	−	0.807357i	$-0.299103\pi$
$631$	29.6444	1.18013	0.590063	+	0.807357i	$0.299103\pi$
$632$	−6.22153	−0.247479
$633$	0	0
$634$	41.9209	1.66489
$635$	−0.997753	−0.0395946
$636$	0	0
$637$	−6.18102	−0.244901
$638$	−9.00198	−0.356392
$639$	0	0
$640$	−0.749791	−0.0296381
$641$	10.3064	0.407080	0.203540	−	0.979067i	$-0.434755\pi$
$641$	10.3064	0.407080	0.203540	+	0.979067i	$0.434755\pi$
$642$	0	0
$643$	16.9166	0.667125	0.333563	−	0.942728i	$-0.391749\pi$
$643$	16.9166	0.667125	0.333563	+	0.942728i	$0.391749\pi$
$644$	−0.00421148	−0.000165956 0
$645$	0	0
$646$	1.15775	0.0455510
$647$	−34.4281	−1.35351	−0.676754	−	0.736209i	$-0.736614\pi$
$647$	−34.4281	−1.35351	−0.676754	+	0.736209i	$0.736614\pi$
$648$	0	0
$649$	49.1673	1.92999
$650$	−10.4823	−0.411150
$651$	0	0
$652$	0.0223494	0.000875271 0
$653$	−30.4723	−1.19247	−0.596237	−	0.802808i	$-0.703338\pi$
$653$	−30.4723	−1.19247	−0.596237	+	0.802808i	$0.703338\pi$
$654$	0	0
$655$	−0.538509	−0.0210413
$656$	30.7013	1.19868
$657$	0	0
$658$	−9.08445	−0.354149
$659$	−17.7960	−0.693232	−0.346616	−	0.938007i	$-0.612669\pi$
$659$	−17.7960	−0.693232	−0.346616	+	0.938007i	$0.612669\pi$
$660$	0	0
$661$	−42.9860	−1.67196	−0.835981	−	0.548758i	$-0.815101\pi$
$661$	−42.9860	−1.67196	−0.835981	+	0.548758i	$0.815101\pi$
$662$	27.4820	1.06812
$663$	0	0
$664$	22.9773	0.891693
$665$	−0.552131	−0.0214107
$666$	0	0
$667$	1.00000	0.0387202
$668$	−0.0137258	−0.000531069 0
$669$	0	0
$670$	−0.0100937	−0.000389955 0
$671$	−71.1793	−2.74785
$672$	0	0
$673$	21.9950	0.847847	0.423923	−	0.905698i	$-0.360653\pi$
$673$	21.9950	0.847847	0.423923	+	0.905698i	$0.360653\pi$
$674$	−38.0799	−1.46678
$675$	0	0
$676$	−0.0270336	−0.00103975
$677$	1.17847	0.0452923	0.0226462	−	0.999744i	$-0.492791\pi$
$677$	1.17847	0.0452923	0.0226462	+	0.999744i	$0.492791\pi$
$678$	0	0
$679$	25.3371	0.972350
$680$	0.0308398	0.00118265
$681$	0	0
$682$	30.4755	1.16697
$683$	8.35251	0.319600	0.159800	−	0.987149i	$-0.448915\pi$
$683$	8.35251	0.319600	0.159800	+	0.987149i	$0.448915\pi$
$684$	0	0
$685$	1.06018	0.0405075
$686$	−26.5944	−1.01538
$687$	0	0
$688$	−12.7337	−0.485469
$689$	−1.83615	−0.0699517
$690$	0	0
$691$	−31.9426	−1.21515	−0.607576	−	0.794261i	$-0.707858\pi$
$691$	−31.9426	−1.21515	−0.607576	+	0.794261i	$0.707858\pi$
$692$	−0.0247866	−0.000942246 0
$693$	0	0
$694$	−21.8271	−0.828547
$695$	−0.850955	−0.0322785
$696$	0	0
$697$	−1.26436	−0.0478910
$698$	−48.7497	−1.84520
$699$	0	0
$700$	−0.0210390	−0.000795199 0
$701$	−31.0114	−1.17129	−0.585643	−	0.810569i	$-0.699158\pi$
$701$	−31.0114	−1.17129	−0.585643	+	0.810569i	$0.699158\pi$
$702$	0	0
$703$	49.6081	1.87100
$704$	50.8272	1.91562
$705$	0	0
$706$	−46.9338	−1.76638
$707$	29.6939	1.11675
$708$	0	0
$709$	10.7850	0.405038	0.202519	−	0.979278i	$-0.435087\pi$
$709$	10.7850	0.405038	0.202519	+	0.979278i	$0.435087\pi$
$710$	−0.485148	−0.0182073
$711$	0	0
$712$	36.7449	1.37707
$713$	−3.38542	−0.126785
$714$	0	0
$715$	−0.623957	−0.0233347
$716$	0.00151274	5.65339e−5 0
$717$	0	0
$718$	30.6371	1.14337
$719$	−31.1791	−1.16278	−0.581391	−	0.813624i	$-0.697491\pi$
$719$	−31.1791	−1.16278	−0.581391	+	0.813624i	$0.697491\pi$
$720$	0	0
$721$	16.4582	0.612936
$722$	7.93157	0.295183
$723$	0	0
$724$	−0.00540266	−0.000200788 0
$725$	4.99562	0.185533
$726$	0	0
$727$	19.0809	0.707670	0.353835	−	0.935308i	$-0.384877\pi$
$727$	19.0809	0.707670	0.353835	+	0.935308i	$0.384877\pi$
$728$	−7.05281	−0.261395
$729$	0	0
$730$	−1.11590	−0.0413014
$731$	0.524408	0.0193959
$732$	0	0
$733$	−0.535265	−0.0197705	−0.00988523	−	0.999951i	$-0.503147\pi$
$733$	−0.535265	−0.0197705	−0.00988523	+	0.999951i	$0.503147\pi$
$734$	−35.1353	−1.29687
$735$	0	0
$736$	0.0141580	0.000521870 0
$737$	0.685952	0.0252674
$738$	0	0
$739$	−31.8288	−1.17084	−0.585420	−	0.810730i	$-0.699070\pi$
$739$	−31.8288	−1.17084	−0.585420	+	0.810730i	$0.699070\pi$
$740$	−0.00165573	−6.08659e−5 0
$741$	0	0
$742$	−2.94864	−0.108248
$743$	27.5922	1.01226	0.506129	−	0.862458i	$-0.331076\pi$
$743$	27.5922	1.01226	0.506129	+	0.862458i	$0.331076\pi$
$744$	0	0
$745$	1.04446	0.0382660
$746$	−28.0822	−1.02816
$747$	0	0
$748$	0.00262599	9.60159e−5 0
$749$	−20.8881	−0.763234
$750$	0	0
$751$	42.6275	1.55550	0.777751	−	0.628573i	$-0.216361\pi$
$751$	42.6275	1.55550	0.777751	+	0.628573i	$0.216361\pi$
$752$	15.2794	0.557182
$753$	0	0
$754$	−2.09830	−0.0764156
$755$	−1.03111	−0.0375261
$756$	0	0
$757$	15.7726	0.573264	0.286632	−	0.958041i	$-0.407464\pi$
$757$	15.7726	0.573264	0.286632	+	0.958041i	$0.407464\pi$
$758$	−49.2450	−1.78866
$759$	0	0
$760$	0.927484	0.0336434
$761$	−43.7021	−1.58420	−0.792099	−	0.610392i	$-0.791012\pi$
$761$	−43.7021	−1.58420	−0.792099	+	0.610392i	$0.791012\pi$
$762$	0	0
$763$	−16.0829	−0.582238
$764$	0.0511654	0.00185110
$765$	0	0
$766$	4.02312	0.145361
$767$	11.4606	0.413817
$768$	0	0
$769$	−26.1533	−0.943112	−0.471556	−	0.881836i	$-0.656308\pi$
$769$	−26.1533	−0.943112	−0.471556	+	0.881836i	$0.656308\pi$
$770$	−1.00200	−0.0361097
$771$	0	0
$772$	−0.0692865	−0.00249367
$773$	−9.28446	−0.333939	−0.166969	−	0.985962i	$-0.553398\pi$
$773$	−9.28446	−0.333939	−0.166969	+	0.985962i	$0.553398\pi$
$774$	0	0
$775$	−16.9123	−0.607508
$776$	−42.5620	−1.52789
$777$	0	0
$778$	48.9641	1.75545
$779$	−38.0247	−1.36238
$780$	0	0
$781$	32.9698	1.17975
$782$	−0.233401	−0.00834640
$783$	0	0
$784$	16.6948	0.596244
$785$	−0.123083	−0.00439302
$786$	0	0
$787$	14.8721	0.530133	0.265066	−	0.964230i	$-0.414606\pi$
$787$	14.8721	0.530133	0.265066	+	0.964230i	$0.414606\pi$
$788$	−0.0321613	−0.00114570
$789$	0	0
$790$	−0.206031	−0.00733027
$791$	34.9979	1.24438
$792$	0	0
$793$	−16.5914	−0.589179
$794$	11.4595	0.406681
$795$	0	0
$796$	0.0161929	0.000573943 0
$797$	30.0774	1.06540	0.532698	−	0.846305i	$-0.321178\pi$
$797$	30.0774	1.06540	0.532698	+	0.846305i	$0.321178\pi$
$798$	0	0
$799$	−0.629245	−0.0222611
$800$	0.0707279	0.00250061
$801$	0	0
$802$	16.8492	0.594966
$803$	75.8348	2.67615
$804$	0	0
$805$	0.111309	0.00392313
$806$	7.10363	0.250215
$807$	0	0
$808$	−49.8805	−1.75479
$809$	−5.34257	−0.187835	−0.0939173	−	0.995580i	$-0.529939\pi$
$809$	−5.34257	−0.187835	−0.0939173	+	0.995580i	$0.529939\pi$
$810$	0	0
$811$	−42.9156	−1.50697	−0.753486	−	0.657464i	$-0.771629\pi$
$811$	−42.9156	−1.50697	−0.753486	+	0.657464i	$0.771629\pi$
$812$	−0.00421148	−0.000147794 0
$813$	0	0
$814$	90.0282	3.15549
$815$	−0.590694	−0.0206911
$816$	0	0
$817$	15.7712	0.551764
$818$	25.9453	0.907157
$819$	0	0
$820$	0.00126912	4.43196e−5 0
$821$	11.7935	0.411596	0.205798	−	0.978594i	$-0.434021\pi$
$821$	11.7935	0.411596	0.205798	+	0.978594i	$0.434021\pi$
$822$	0	0
$823$	35.1864	1.22652	0.613260	−	0.789881i	$-0.289858\pi$
$823$	35.1864	1.22652	0.613260	+	0.789881i	$0.289858\pi$
$824$	−27.6469	−0.963127
$825$	0	0
$826$	18.4043	0.640369
$827$	0.424292	0.0147541	0.00737703	−	0.999973i	$-0.497652\pi$
$827$	0.424292	0.0147541	0.00737703	+	0.999973i	$0.497652\pi$
$828$	0	0
$829$	20.4576	0.710523	0.355262	−	0.934767i	$-0.384392\pi$
$829$	20.4576	0.710523	0.355262	+	0.934767i	$0.384392\pi$
$830$	0.760914	0.0264117
$831$	0	0
$832$	11.8475	0.410737
$833$	−0.687536	−0.0238217
$834$	0	0
$835$	0.362773	0.0125543
$836$	0.0789749	0.00273140
$837$	0	0
$838$	−37.3463	−1.29011
$839$	43.5410	1.50320	0.751602	−	0.659617i	$-0.229282\pi$
$839$	43.5410	1.50320	0.751602	+	0.659617i	$0.229282\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−4.90252	−0.168952
$843$	0	0
$844$	0.0454263	0.00156364
$845$	0.714495	0.0245794
$846$	0	0
$847$	49.5846	1.70375
$848$	4.95940	0.170307
$849$	0	0
$850$	−1.16598	−0.0399929
$851$	−10.0009	−0.342828
$852$	0	0
$853$	0.857018	0.0293437	0.0146719	−	0.999892i	$-0.495330\pi$
$853$	0.857018	0.0293437	0.0146719	+	0.999892i	$0.495330\pi$
$854$	−26.6439	−0.911735
$855$	0	0
$856$	35.0883	1.19929
$857$	18.3087	0.625412	0.312706	−	0.949850i	$-0.398765\pi$
$857$	18.3087	0.625412	0.312706	+	0.949850i	$0.398765\pi$
$858$	0	0
$859$	35.1332	1.19873	0.599364	−	0.800477i	$-0.295420\pi$
$859$	35.1332	1.19873	0.599364	+	0.800477i	$0.295420\pi$
$860$	−0.000526383 0	−1.79495e−5 0
$861$	0	0
$862$	−10.0626	−0.342735
$863$	15.0456	0.512157	0.256079	−	0.966656i	$-0.417569\pi$
$863$	15.0456	0.512157	0.256079	+	0.966656i	$0.417569\pi$
$864$	0	0
$865$	0.655108	0.0222744
$866$	−3.38696	−0.115094
$867$	0	0
$868$	0.0142577	0.000483936 0
$869$	14.0015	0.474970
$870$	0	0
$871$	0.159891	0.00541769
$872$	27.0164	0.914890
$873$	0	0
$874$	−7.01937	−0.237434
$875$	1.11260	0.0376129
$876$	0	0
$877$	−30.0634	−1.01517	−0.507584	−	0.861602i	$-0.669461\pi$
$877$	−30.0634	−1.01517	−0.507584	+	0.861602i	$0.669461\pi$
$878$	6.75225	0.227877
$879$	0	0
$880$	1.68530	0.0568113
$881$	−32.2354	−1.08604	−0.543019	−	0.839720i	$-0.682719\pi$
$881$	−32.2354	−1.08604	−0.543019	+	0.839720i	$0.682719\pi$
$882$	0	0
$883$	−15.1587	−0.510132	−0.255066	−	0.966924i	$-0.582097\pi$
$883$	−15.1587	−0.510132	−0.255066	+	0.966924i	$0.582097\pi$
$884$	0.000612101 0	2.05872e−5 0
$885$	0	0
$886$	7.08031	0.237868
$887$	48.0732	1.61414	0.807069	−	0.590457i	$-0.201052\pi$
$887$	48.0732	1.61414	0.807069	+	0.590457i	$0.201052\pi$
$888$	0	0
$889$	25.3810	0.851252
$890$	1.21684	0.0407886
$891$	0	0
$892$	0.0447304	0.00149768
$893$	−18.9241	−0.633270
$894$	0	0
$895$	−0.0399817	−0.00133644
$896$	19.0733	0.637195
$897$	0	0
$898$	22.3302	0.745170
$899$	−3.38542	−0.112910
$900$	0	0
$901$	−0.204241	−0.00680425
$902$	−69.0068	−2.29768
$903$	0	0
$904$	−58.7903	−1.95534
$905$	0.142792	0.00474656
$906$	0	0
$907$	−26.0297	−0.864301	−0.432150	−	0.901802i	$-0.642245\pi$
$907$	−26.0297	−0.864301	−0.432150	+	0.901802i	$0.642245\pi$
$908$	0.0372201	0.00123519
$909$	0	0
$910$	−0.233560	−0.00774244
$911$	7.69710	0.255016	0.127508	−	0.991838i	$-0.459302\pi$
$911$	7.69710	0.255016	0.127508	+	0.991838i	$0.459302\pi$
$912$	0	0
$913$	−51.7104	−1.71136
$914$	−13.2972	−0.439831
$915$	0	0
$916$	−0.0481984	−0.00159252
$917$	13.6987	0.452370
$918$	0	0
$919$	32.0462	1.05711	0.528554	−	0.848900i	$-0.322735\pi$
$919$	32.0462	1.05711	0.528554	+	0.848900i	$0.322735\pi$
$920$	−0.186980	−0.00616455
$921$	0	0
$922$	−39.2651	−1.29313
$923$	7.68504	0.252956
$924$	0	0
$925$	−49.9609	−1.64271
$926$	−45.6825	−1.50122
$927$	0	0
$928$	0.0141580	0.000464758 0
$929$	34.7566	1.14033	0.570164	−	0.821531i	$-0.306880\pi$
$929$	34.7566	1.14033	0.570164	+	0.821531i	$0.306880\pi$
$930$	0	0
$931$	−20.6772	−0.677666
$932$	−0.0673438	−0.00220592
$933$	0	0
$934$	14.0206	0.458768
$935$	−0.0694048	−0.00226978
$936$	0	0
$937$	−25.1222	−0.820707	−0.410353	−	0.911927i	$-0.634595\pi$
$937$	−25.1222	−0.820707	−0.410353	+	0.911927i	$0.634595\pi$
$938$	0.256766	0.00838371
$939$	0	0
$940$	0.000631615 0	2.06010e−5 0
$941$	−2.95650	−0.0963790	−0.0481895	−	0.998838i	$-0.515345\pi$
$941$	−2.95650	−0.0963790	−0.0481895	+	0.998838i	$0.515345\pi$
$942$	0	0
$943$	7.66574	0.249631
$944$	−30.9548	−1.00749
$945$	0	0
$946$	28.6214	0.930562
$947$	−21.4060	−0.695601	−0.347801	−	0.937569i	$-0.613071\pi$
$947$	−21.4060	−0.695601	−0.347801	+	0.937569i	$0.613071\pi$
$948$	0	0
$949$	17.6766	0.573806
$950$	−35.0661	−1.13770
$951$	0	0
$952$	−0.784508	−0.0254260
$953$	−11.7455	−0.380475	−0.190238	−	0.981738i	$-0.560926\pi$
$953$	−11.7455	−0.380475	−0.190238	+	0.981738i	$0.560926\pi$
$954$	0	0
$955$	−1.35230	−0.0437593
$956$	0.0111896	0.000361897 0
$957$	0	0
$958$	−32.1495	−1.03870
$959$	−26.9691	−0.870879
$960$	0	0
$961$	−19.5389	−0.630287
$962$	20.9850	0.676583
$963$	0	0
$964$	0.0413752	0.00133261
$965$	1.83124	0.0589496
$966$	0	0
$967$	51.2764	1.64894	0.824469	−	0.565907i	$-0.191474\pi$
$967$	51.2764	1.64894	0.824469	+	0.565907i	$0.191474\pi$
$968$	−83.2934	−2.67715
$969$	0	0
$970$	−1.40948	−0.0452556
$971$	−58.7158	−1.88428	−0.942140	−	0.335221i	$-0.891189\pi$
$971$	−58.7158	−1.88428	−0.942140	+	0.335221i	$0.891189\pi$
$972$	0	0
$973$	21.6467	0.693963
$974$	−31.8562	−1.02074
$975$	0	0
$976$	44.8131	1.43443
$977$	−24.4792	−0.783159	−0.391580	−	0.920144i	$-0.628071\pi$
$977$	−24.4792	−0.783159	−0.391580	+	0.920144i	$0.628071\pi$
$978$	0	0
$979$	−82.6944	−2.64292
$980$	0.000690126 0	2.20453e−5 0
$981$	0	0
$982$	5.84529	0.186531
$983$	−43.5416	−1.38876	−0.694380	−	0.719608i	$-0.744321\pi$
$983$	−43.5416	−1.38876	−0.694380	+	0.719608i	$0.744321\pi$
$984$	0	0
$985$	0.850022	0.0270840
$986$	−0.233401	−0.00743300
$987$	0	0
$988$	0.0184085	0.000585653 0
$989$	−3.17946	−0.101101
$990$	0	0
$991$	27.8175	0.883652	0.441826	−	0.897101i	$-0.354331\pi$
$991$	27.8175	0.883652	0.441826	+	0.897101i	$0.354331\pi$
$992$	−0.0479307	−0.00152180
$993$	0	0
$994$	12.3413	0.391442
$995$	−0.427978	−0.0135678
$996$	0	0
$997$	13.8723	0.439342	0.219671	−	0.975574i	$-0.429502\pi$
$997$	13.8723	0.439342	0.219671	+	0.975574i	$0.429502\pi$
$998$	−24.3640	−0.771230
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.6	✓	20	3.2	odd	2
6003.2.a.s.1.15		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)

\(f(q)\)	\(=\)	\(q+1.41510 q^{2} +0.00250280 q^{4} -0.0661489 q^{5} +1.68271 q^{7} -2.82665 q^{8} +O(q^{10})\)	\(q+1.41510 q^{2} +0.00250280 q^{4} -0.0661489 q^{5} +1.68271 q^{7} -2.82665 q^{8} -0.0936071 q^{10} +6.36138 q^{11} +1.48279 q^{13} +2.38120 q^{14} -4.00500 q^{16} +0.164936 q^{17} +4.96034 q^{19} -0.000165558 q^{20} +9.00198 q^{22} -1.00000 q^{23} -4.99562 q^{25} +2.09830 q^{26} +0.00421148 q^{28} -1.00000 q^{29} +3.38542 q^{31} -0.0141580 q^{32} +0.233401 q^{34} -0.111309 q^{35} +10.0009 q^{37} +7.01937 q^{38} +0.186980 q^{40} -7.66574 q^{41} +3.17946 q^{43} +0.0159213 q^{44} -1.41510 q^{46} -3.81508 q^{47} -4.16850 q^{49} -7.06930 q^{50} +0.00371114 q^{52} -1.23830 q^{53} -0.420798 q^{55} -4.75643 q^{56} -1.41510 q^{58} +7.72903 q^{59} -11.1893 q^{61} +4.79071 q^{62} +7.98996 q^{64} -0.0980852 q^{65} +0.107831 q^{67} +0.000412803 q^{68} -0.157513 q^{70} +5.18281 q^{71} +11.9211 q^{73} +14.1523 q^{74} +0.0124147 q^{76} +10.7043 q^{77} +2.20102 q^{79} +0.264926 q^{80} -10.8478 q^{82} -8.12880 q^{83} -0.0109103 q^{85} +4.49925 q^{86} -17.9814 q^{88} -12.9994 q^{89} +2.49511 q^{91} -0.00250280 q^{92} -5.39871 q^{94} -0.328121 q^{95} +15.0574 q^{97} -5.89883 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) 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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