LMFDB - Embedded newform 6003.2.a.q.1.14

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 22 x^{14} + 68 x^{13} + 187 x^{12} - 597 x^{11} - 795 x^{10} + 2592 x^{9} + \cdots + 64 $ x^16 - 3x^15 - 22x^14 + 68x^13 + 187x^12 - 597x^11 - 795x^10 + 2592x^9 + 1860x^8 - 5877x^7 - 2496x^6 + 6612x^5 + 1842x^4 - 3011x^3 - 505x^2 + 336*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-1.80038$ 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)

$f(q)$	$=$	$q+1.80038 q^{2} +1.24138 q^{4} -2.67126 q^{5} +1.66993 q^{7} -1.36580 q^{8} +O(q^{10})$	\(q+1.80038 q^{2} +1.24138 q^{4} -2.67126 q^{5} +1.66993 q^{7} -1.36580 q^{8} -4.80929 q^{10} +2.09236 q^{11} +2.30892 q^{13} +3.00652 q^{14} -4.94173 q^{16} -3.16946 q^{17} +0.855823 q^{19} -3.31605 q^{20} +3.76705 q^{22} -1.00000 q^{23} +2.13561 q^{25} +4.15695 q^{26} +2.07303 q^{28} +1.00000 q^{29} -7.33040 q^{31} -6.16542 q^{32} -5.70624 q^{34} -4.46082 q^{35} +6.32439 q^{37} +1.54081 q^{38} +3.64840 q^{40} +0.237540 q^{41} +9.69447 q^{43} +2.59742 q^{44} -1.80038 q^{46} -6.83751 q^{47} -4.21132 q^{49} +3.84491 q^{50} +2.86626 q^{52} -2.54640 q^{53} -5.58923 q^{55} -2.28080 q^{56} +1.80038 q^{58} -9.30330 q^{59} +5.36437 q^{61} -13.1975 q^{62} -1.21665 q^{64} -6.16772 q^{65} -4.05341 q^{67} -3.93451 q^{68} -8.03120 q^{70} +9.77123 q^{71} -14.4287 q^{73} +11.3863 q^{74} +1.06241 q^{76} +3.49410 q^{77} -5.24323 q^{79} +13.2006 q^{80} +0.427664 q^{82} -14.5348 q^{83} +8.46643 q^{85} +17.4538 q^{86} -2.85775 q^{88} -4.00108 q^{89} +3.85575 q^{91} -1.24138 q^{92} -12.3101 q^{94} -2.28612 q^{95} -18.5247 q^{97} -7.58199 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) $ 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8	$ 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8} - 14 q^{10} - 4 q^{11} + 15 q^{13} - 8 q^{14} + 23 q^{16} - 20 q^{17} - 4 q^{19} - 25 q^{20} + 13 q^{22} - 16 q^{23} + 30 q^{25} - 25 q^{26} - 13 q^{28} + 16 q^{29} + 19 q^{32} - 23 q^{34} - 5 q^{35} + 5 q^{37} - 38 q^{38} - 20 q^{40} - 7 q^{41} - 17 q^{43} + 21 q^{44} + 3 q^{46} - 29 q^{47} + 31 q^{49} + 44 q^{50} + 20 q^{52} - 63 q^{53} + q^{55} + 19 q^{56} - 3 q^{58} - 11 q^{59} - 33 q^{62} + 29 q^{64} - 53 q^{65} - 13 q^{67} - 63 q^{68} - 46 q^{70} + 23 q^{71} - 38 q^{73} + 47 q^{74} - 56 q^{76} - 97 q^{77} - 27 q^{79} - 8 q^{80} + 9 q^{82} - 36 q^{83} + 6 q^{85} + 11 q^{86} - 24 q^{88} + 16 q^{89} - 47 q^{91} - 21 q^{92} + 37 q^{94} + 12 q^{95} - 30 q^{97} + 27 q^{98}+O(q^{100}) $ 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8 - 14 * q^10 - 4 * q^11 + 15 * q^13 - 8 * q^14 + 23 * q^16 - 20 * q^17 - 4 * q^19 - 25 * q^20 + 13 * q^22 - 16 * q^23 + 30 * q^25 - 25 * q^26 - 13 * q^28 + 16 * q^29 + 19 * q^32 - 23 * q^34 - 5 * q^35 + 5 * q^37 - 38 * q^38 - 20 * q^40 - 7 * q^41 - 17 * q^43 + 21 * q^44 + 3 * q^46 - 29 * q^47 + 31 * q^49 + 44 * q^50 + 20 * q^52 - 63 * q^53 + q^55 + 19 * q^56 - 3 * q^58 - 11 * q^59 - 33 * q^62 + 29 * q^64 - 53 * q^65 - 13 * q^67 - 63 * q^68 - 46 * q^70 + 23 * q^71 - 38 * q^73 + 47 * q^74 - 56 * q^76 - 97 * q^77 - 27 * q^79 - 8 * q^80 + 9 * q^82 - 36 * q^83 + 6 * q^85 + 11 * q^86 - 24 * q^88 + 16 * q^89 - 47 * q^91 - 21 * q^92 + 37 * q^94 + 12 * q^95 - 30 * q^97 + 27 * 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 $

$2$	1.80038	1.27306	0.636532	−	0.771250i	$-0.280368\pi$
$2$	1.80038	1.27306	0.636532	+	0.771250i	$0.280368\pi$
$3$	0	0
$3$	0	0
$4$	1.24138	0.620692
$5$	−2.67126	−1.19462	−0.597311	−	0.802010i	$-0.703764\pi$
$5$	−2.67126	−1.19462	−0.597311	+	0.802010i	$0.703764\pi$
$6$	0	0
$7$	1.66993	0.631176	0.315588	−	0.948896i	$-0.397798\pi$
$7$	1.66993	0.631176	0.315588	+	0.948896i	$0.397798\pi$
$8$	−1.36580	−0.482884
$9$	0	0
$10$	−4.80929	−1.52083
$11$	2.09236	0.630870	0.315435	−	0.948947i	$-0.397850\pi$
$11$	2.09236	0.630870	0.315435	+	0.948947i	$0.397850\pi$
$12$	0	0
$13$	2.30892	0.640380	0.320190	−	0.947353i	$-0.396253\pi$
$13$	2.30892	0.640380	0.320190	+	0.947353i	$0.396253\pi$
$14$	3.00652	0.803527
$15$	0	0
$16$	−4.94173	−1.23543
$17$	−3.16946	−0.768706	−0.384353	−	0.923186i	$-0.625575\pi$
$17$	−3.16946	−0.768706	−0.384353	+	0.923186i	$0.625575\pi$
$18$	0	0
$19$	0.855823	0.196339	0.0981697	−	0.995170i	$-0.468701\pi$
$19$	0.855823	0.196339	0.0981697	+	0.995170i	$0.468701\pi$
$20$	−3.31605	−0.741492
$21$	0	0
$22$	3.76705	0.803138
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	2.13561	0.427122
$26$	4.15695	0.815244
$27$	0	0
$28$	2.07303	0.391766
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−7.33040	−1.31658	−0.658289	−	0.752765i	$-0.728720\pi$
$31$	−7.33040	−1.31658	−0.658289	+	0.752765i	$0.728720\pi$
$32$	−6.16542	−1.08990
$33$	0	0
$34$	−5.70624	−0.978612
$35$	−4.46082	−0.754017
$36$	0	0
$37$	6.32439	1.03972	0.519862	−	0.854251i	$-0.325984\pi$
$37$	6.32439	1.03972	0.519862	+	0.854251i	$0.325984\pi$
$38$	1.54081	0.249953
$39$	0	0
$40$	3.64840	0.576863
$41$	0.237540	0.0370976	0.0185488	−	0.999828i	$-0.494095\pi$
$41$	0.237540	0.0370976	0.0185488	+	0.999828i	$0.494095\pi$
$42$	0	0
$43$	9.69447	1.47839	0.739196	−	0.673490i	$-0.235206\pi$
$43$	9.69447	1.47839	0.739196	+	0.673490i	$0.235206\pi$
$44$	2.59742	0.391576
$45$	0	0
$46$	−1.80038	−0.265452
$47$	−6.83751	−0.997353	−0.498677	−	0.866788i	$-0.666181\pi$
$47$	−6.83751	−0.997353	−0.498677	+	0.866788i	$0.666181\pi$
$48$	0	0
$49$	−4.21132	−0.601617
$50$	3.84491	0.543753
$51$	0	0
$52$	2.86626	0.397478
$53$	−2.54640	−0.349775	−0.174888	−	0.984588i	$-0.555956\pi$
$53$	−2.54640	−0.349775	−0.174888	+	0.984588i	$0.555956\pi$
$54$	0	0
$55$	−5.58923	−0.753651
$56$	−2.28080	−0.304785
$57$	0	0
$58$	1.80038	0.236402
$59$	−9.30330	−1.21119	−0.605593	−	0.795774i	$-0.707064\pi$
$59$	−9.30330	−1.21119	−0.605593	+	0.795774i	$0.707064\pi$
$60$	0	0
$61$	5.36437	0.686837	0.343419	−	0.939182i	$-0.388415\pi$
$61$	5.36437	0.686837	0.343419	+	0.939182i	$0.388415\pi$
$62$	−13.1975	−1.67609
$63$	0	0
$64$	−1.21665	−0.152082
$65$	−6.16772	−0.765011
$66$	0	0
$67$	−4.05341	−0.495202	−0.247601	−	0.968862i	$-0.579642\pi$
$67$	−4.05341	−0.495202	−0.247601	+	0.968862i	$0.579642\pi$
$68$	−3.93451	−0.477129
$69$	0	0
$70$	−8.03120	−0.959911
$71$	9.77123	1.15963	0.579816	−	0.814747i	$-0.303124\pi$
$71$	9.77123	1.15963	0.579816	+	0.814747i	$0.303124\pi$
$72$	0	0
$73$	−14.4287	−1.68875	−0.844375	−	0.535753i	$-0.820028\pi$
$73$	−14.4287	−1.68875	−0.844375	+	0.535753i	$0.820028\pi$
$74$	11.3863	1.32363
$75$	0	0
$76$	1.06241	0.121866
$77$	3.49410	0.398190
$78$	0	0
$79$	−5.24323	−0.589909	−0.294955	−	0.955511i	$-0.595305\pi$
$79$	−5.24323	−0.589909	−0.294955	+	0.955511i	$0.595305\pi$
$80$	13.2006	1.47588
$81$	0	0
$82$	0.427664	0.0472276
$83$	−14.5348	−1.59540	−0.797700	−	0.603054i	$-0.793950\pi$
$83$	−14.5348	−1.59540	−0.797700	+	0.603054i	$0.793950\pi$
$84$	0	0
$85$	8.46643	0.918313
$86$	17.4538	1.88209
$87$	0	0
$88$	−2.85775	−0.304637
$89$	−4.00108	−0.424113	−0.212057	−	0.977257i	$-0.568016\pi$
$89$	−4.00108	−0.424113	−0.212057	+	0.977257i	$0.568016\pi$
$90$	0	0
$91$	3.85575	0.404192
$92$	−1.24138	−0.129423
$93$	0	0
$94$	−12.3101	−1.26969
$95$	−2.28612	−0.234551
$96$	0	0
$97$	−18.5247	−1.88090	−0.940448	−	0.339938i	$-0.889594\pi$
$97$	−18.5247	−1.88090	−0.940448	+	0.339938i	$0.889594\pi$
$98$	−7.58199	−0.765897
$99$	0	0
$100$	2.65111	0.265111
$101$	−9.44544	−0.939856	−0.469928	−	0.882705i	$-0.655720\pi$
$101$	−9.44544	−0.939856	−0.469928	+	0.882705i	$0.655720\pi$
$102$	0	0
$103$	10.1223	0.997377	0.498688	−	0.866781i	$-0.333815\pi$
$103$	10.1223	0.997377	0.498688	+	0.866781i	$0.333815\pi$
$104$	−3.15353	−0.309229
$105$	0	0
$106$	−4.58450	−0.445286
$107$	−6.37046	−0.615856	−0.307928	−	0.951410i	$-0.599636\pi$
$107$	−6.37046	−0.615856	−0.307928	+	0.951410i	$0.599636\pi$
$108$	0	0
$109$	−0.657653	−0.0629917	−0.0314958	−	0.999504i	$-0.510027\pi$
$109$	−0.657653	−0.0629917	−0.0314958	+	0.999504i	$0.510027\pi$
$110$	−10.0628	−0.959446
$111$	0	0
$112$	−8.25237	−0.779776
$113$	−2.91640	−0.274351	−0.137176	−	0.990547i	$-0.543802\pi$
$113$	−2.91640	−0.274351	−0.137176	+	0.990547i	$0.543802\pi$
$114$	0	0
$115$	2.67126	0.249096
$116$	1.24138	0.115260
$117$	0	0
$118$	−16.7495	−1.54192
$119$	−5.29278	−0.485189
$120$	0	0
$121$	−6.62203	−0.602003
$122$	9.65793	0.874388
$123$	0	0
$124$	−9.09983	−0.817189
$125$	7.65152	0.684373
$126$	0	0
$127$	−1.13798	−0.100979	−0.0504895	−	0.998725i	$-0.516078\pi$
$127$	−1.13798	−0.100979	−0.0504895	+	0.998725i	$0.516078\pi$
$128$	10.1404	0.896293
$129$	0	0
$130$	−11.1043	−0.973909
$131$	−2.97772	−0.260165	−0.130082	−	0.991503i	$-0.541524\pi$
$131$	−2.97772	−0.260165	−0.130082	+	0.991503i	$0.541524\pi$
$132$	0	0
$133$	1.42917	0.123925
$134$	−7.29769	−0.630424
$135$	0	0
$136$	4.32885	0.371196
$137$	−22.1228	−1.89008	−0.945040	−	0.326954i	$-0.893978\pi$
$137$	−22.1228	−1.89008	−0.945040	+	0.326954i	$0.893978\pi$
$138$	0	0
$139$	14.5315	1.23255	0.616275	−	0.787531i	$-0.288641\pi$
$139$	14.5315	1.23255	0.616275	+	0.787531i	$0.288641\pi$
$140$	−5.53759	−0.468012
$141$	0	0
$142$	17.5920	1.47629
$143$	4.83109	0.403996
$144$	0	0
$145$	−2.67126	−0.221836
$146$	−25.9772	−2.14989
$147$	0	0
$148$	7.85099	0.645348
$149$	−9.69697	−0.794407	−0.397203	−	0.917731i	$-0.630019\pi$
$149$	−9.69697	−0.794407	−0.397203	+	0.917731i	$0.630019\pi$
$150$	0	0
$151$	18.7262	1.52392	0.761958	−	0.647627i	$-0.224238\pi$
$151$	18.7262	1.52392	0.761958	+	0.647627i	$0.224238\pi$
$152$	−1.16888	−0.0948091
$153$	0	0
$154$	6.29073	0.506922
$155$	19.5814	1.57281
$156$	0	0
$157$	16.7286	1.33508	0.667542	−	0.744572i	$-0.267346\pi$
$157$	16.7286	1.33508	0.667542	+	0.744572i	$0.267346\pi$
$158$	−9.43983	−0.750992
$159$	0	0
$160$	16.4694	1.30202
$161$	−1.66993	−0.131609
$162$	0	0
$163$	−15.6039	−1.22219	−0.611095	−	0.791557i	$-0.709271\pi$
$163$	−15.6039	−1.22219	−0.611095	+	0.791557i	$0.709271\pi$
$164$	0.294879	0.0230261
$165$	0	0
$166$	−26.1682	−2.03105
$167$	−12.0575	−0.933041	−0.466521	−	0.884510i	$-0.654493\pi$
$167$	−12.0575	−0.933041	−0.466521	+	0.884510i	$0.654493\pi$
$168$	0	0
$169$	−7.66888	−0.589914
$170$	15.2428	1.16907
$171$	0	0
$172$	12.0346	0.917626
$173$	−18.7908	−1.42863	−0.714317	−	0.699822i	$-0.753263\pi$
$173$	−18.7908	−1.42863	−0.714317	+	0.699822i	$0.753263\pi$
$174$	0	0
$175$	3.56633	0.269589
$176$	−10.3399	−0.779398
$177$	0	0
$178$	−7.20348	−0.539923
$179$	11.4827	0.858257	0.429129	−	0.903243i	$-0.358821\pi$
$179$	11.4827	0.858257	0.429129	+	0.903243i	$0.358821\pi$
$180$	0	0
$181$	5.39334	0.400884	0.200442	−	0.979706i	$-0.435762\pi$
$181$	5.39334	0.400884	0.200442	+	0.979706i	$0.435762\pi$
$182$	6.94183	0.514563
$183$	0	0
$184$	1.36580	0.100688
$185$	−16.8941	−1.24208
$186$	0	0
$187$	−6.63164	−0.484954
$188$	−8.48797	−0.619049
$189$	0	0
$190$	−4.11590	−0.298599
$191$	−22.0948	−1.59873	−0.799363	−	0.600848i	$-0.794830\pi$
$191$	−22.0948	−1.59873	−0.799363	+	0.600848i	$0.794830\pi$
$192$	0	0
$193$	16.9225	1.21811	0.609053	−	0.793130i	$-0.291550\pi$
$193$	16.9225	1.21811	0.609053	+	0.793130i	$0.291550\pi$
$194$	−33.3515	−2.39450
$195$	0	0
$196$	−5.22786	−0.373419
$197$	9.01852	0.642543	0.321272	−	0.946987i	$-0.395890\pi$
$197$	9.01852	0.642543	0.321272	+	0.946987i	$0.395890\pi$
$198$	0	0
$199$	−19.6161	−1.39055	−0.695274	−	0.718745i	$-0.744717\pi$
$199$	−19.6161	−1.39055	−0.695274	+	0.718745i	$0.744717\pi$
$200$	−2.91682	−0.206250
$201$	0	0
$202$	−17.0054	−1.19650
$203$	1.66993	0.117206
$204$	0	0
$205$	−0.634531	−0.0443176
$206$	18.2240	1.26972
$207$	0	0
$208$	−11.4101	−0.791146
$209$	1.79069	0.123865
$210$	0	0
$211$	17.7261	1.22032	0.610159	−	0.792279i	$-0.291106\pi$
$211$	17.7261	1.22032	0.610159	+	0.792279i	$0.291106\pi$
$212$	−3.16106	−0.217103
$213$	0	0
$214$	−11.4693	−0.784023
$215$	−25.8964	−1.76612
$216$	0	0
$217$	−12.2413	−0.830993
$218$	−1.18403	−0.0801925
$219$	0	0
$220$	−6.93838	−0.467785
$221$	−7.31802	−0.492264
$222$	0	0
$223$	−9.50918	−0.636782	−0.318391	−	0.947959i	$-0.603142\pi$
$223$	−9.50918	−0.636782	−0.318391	+	0.947959i	$0.603142\pi$
$224$	−10.2958	−0.687920
$225$	0	0
$226$	−5.25063	−0.349267
$227$	9.15505	0.607642	0.303821	−	0.952729i	$-0.401737\pi$
$227$	9.15505	0.607642	0.303821	+	0.952729i	$0.401737\pi$
$228$	0	0
$229$	16.5585	1.09422	0.547108	−	0.837062i	$-0.315729\pi$
$229$	16.5585	1.09422	0.547108	+	0.837062i	$0.315729\pi$
$230$	4.80929	0.317115
$231$	0	0
$232$	−1.36580	−0.0896692
$233$	−5.52661	−0.362060	−0.181030	−	0.983478i	$-0.557943\pi$
$233$	−5.52661	−0.362060	−0.181030	+	0.983478i	$0.557943\pi$
$234$	0	0
$235$	18.2647	1.19146
$236$	−11.5490	−0.751773
$237$	0	0
$238$	−9.52905	−0.617676
$239$	19.4467	1.25790	0.628950	−	0.777446i	$-0.283485\pi$
$239$	19.4467	1.25790	0.628950	+	0.777446i	$0.283485\pi$
$240$	0	0
$241$	−18.1428	−1.16868	−0.584341	−	0.811508i	$-0.698647\pi$
$241$	−18.1428	−1.16868	−0.584341	+	0.811508i	$0.698647\pi$
$242$	−11.9222	−0.766388
$243$	0	0
$244$	6.65924	0.426314
$245$	11.2495	0.718705
$246$	0	0
$247$	1.97603	0.125732
$248$	10.0119	0.635754
$249$	0	0
$250$	13.7757	0.871251
$251$	−27.9363	−1.76332	−0.881661	−	0.471883i	$-0.843575\pi$
$251$	−27.9363	−1.76332	−0.881661	+	0.471883i	$0.843575\pi$
$252$	0	0
$253$	−2.09236	−0.131546
$254$	−2.04879	−0.128553
$255$	0	0
$256$	20.6899	1.29312
$257$	24.5000	1.52827	0.764133	−	0.645059i	$-0.223167\pi$
$257$	24.5000	1.52827	0.764133	+	0.645059i	$0.223167\pi$
$258$	0	0
$259$	10.5613	0.656248
$260$	−7.65651	−0.474836
$261$	0	0
$262$	−5.36104	−0.331206
$263$	−26.5801	−1.63900	−0.819500	−	0.573079i	$-0.805749\pi$
$263$	−26.5801	−1.63900	−0.819500	+	0.573079i	$0.805749\pi$
$264$	0	0
$265$	6.80209	0.417849
$266$	2.57305	0.157764
$267$	0	0
$268$	−5.03183	−0.307368
$269$	11.6472	0.710145	0.355072	−	0.934839i	$-0.384456\pi$
$269$	11.6472	0.710145	0.355072	+	0.934839i	$0.384456\pi$
$270$	0	0
$271$	3.26194	0.198149	0.0990744	−	0.995080i	$-0.468412\pi$
$271$	3.26194	0.198149	0.0990744	+	0.995080i	$0.468412\pi$
$272$	15.6626	0.949685
$273$	0	0
$274$	−39.8296	−2.40619
$275$	4.46846	0.269458
$276$	0	0
$277$	−30.3851	−1.82567	−0.912833	−	0.408333i	$-0.866110\pi$
$277$	−30.3851	−1.82567	−0.912833	+	0.408333i	$0.866110\pi$
$278$	26.1624	1.56911
$279$	0	0
$280$	6.09260	0.364102
$281$	−2.54033	−0.151544	−0.0757718	−	0.997125i	$-0.524142\pi$
$281$	−2.54033	−0.151544	−0.0757718	+	0.997125i	$0.524142\pi$
$282$	0	0
$283$	−7.86054	−0.467260	−0.233630	−	0.972326i	$-0.575061\pi$
$283$	−7.86054	−0.467260	−0.233630	+	0.972326i	$0.575061\pi$
$284$	12.1298	0.719774
$285$	0	0
$286$	8.69783	0.514313
$287$	0.396677	0.0234151
$288$	0	0
$289$	−6.95455	−0.409091
$290$	−4.80929	−0.282411
$291$	0	0
$292$	−17.9115	−1.04819
$293$	21.4677	1.25416	0.627079	−	0.778955i	$-0.284250\pi$
$293$	21.4677	1.25416	0.627079	+	0.778955i	$0.284250\pi$
$294$	0	0
$295$	24.8515	1.44691
$296$	−8.63786	−0.502065
$297$	0	0
$298$	−17.4583	−1.01133
$299$	−2.30892	−0.133528
$300$	0	0
$301$	16.1891	0.933126
$302$	33.7143	1.94004
$303$	0	0
$304$	−4.22925	−0.242564
$305$	−14.3296	−0.820511
$306$	0	0
$307$	−10.7356	−0.612715	−0.306358	−	0.951916i	$-0.599110\pi$
$307$	−10.7356	−0.612715	−0.306358	+	0.951916i	$0.599110\pi$
$308$	4.33752	0.247153
$309$	0	0
$310$	35.2540	2.00229
$311$	−2.06047	−0.116838	−0.0584192	−	0.998292i	$-0.518606\pi$
$311$	−2.06047	−0.116838	−0.0584192	+	0.998292i	$0.518606\pi$
$312$	0	0
$313$	21.2563	1.20148	0.600738	−	0.799446i	$-0.294873\pi$
$313$	21.2563	1.20148	0.600738	+	0.799446i	$0.294873\pi$
$314$	30.1178	1.69965
$315$	0	0
$316$	−6.50886	−0.366152
$317$	−2.75770	−0.154888	−0.0774439	−	0.996997i	$-0.524676\pi$
$317$	−2.75770	−0.154888	−0.0774439	+	0.996997i	$0.524676\pi$
$318$	0	0
$319$	2.09236	0.117150
$320$	3.24999	0.181680
$321$	0	0
$322$	−3.00652	−0.167547
$323$	−2.71249	−0.150927
$324$	0	0
$325$	4.93095	0.273520
$326$	−28.0930	−1.55593
$327$	0	0
$328$	−0.324433	−0.0179138
$329$	−11.4182	−0.629506
$330$	0	0
$331$	−11.5054	−0.632394	−0.316197	−	0.948694i	$-0.602406\pi$
$331$	−11.5054	−0.632394	−0.316197	+	0.948694i	$0.602406\pi$
$332$	−18.0432	−0.990252
$333$	0	0
$334$	−21.7082	−1.18782
$335$	10.8277	0.591580
$336$	0	0
$337$	33.6508	1.83307	0.916537	−	0.399950i	$-0.130972\pi$
$337$	33.6508	1.83307	0.916537	+	0.399950i	$0.130972\pi$
$338$	−13.8069	−0.750998
$339$	0	0
$340$	10.5101	0.569989
$341$	−15.3378	−0.830590
$342$	0	0
$343$	−18.7222	−1.01090
$344$	−13.2407	−0.713892
$345$	0	0
$346$	−33.8306	−1.81874
$347$	−13.9776	−0.750355	−0.375178	−	0.926953i	$-0.622418\pi$
$347$	−13.9776	−0.750355	−0.375178	+	0.926953i	$0.622418\pi$
$348$	0	0
$349$	1.32730	0.0710485	0.0355243	−	0.999369i	$-0.488690\pi$
$349$	1.32730	0.0710485	0.0355243	+	0.999369i	$0.488690\pi$
$350$	6.42076	0.343204
$351$	0	0
$352$	−12.9003	−0.687587
$353$	1.02867	0.0547506	0.0273753	−	0.999625i	$-0.491285\pi$
$353$	1.02867	0.0547506	0.0273753	+	0.999625i	$0.491285\pi$
$354$	0	0
$355$	−26.1015	−1.38532
$356$	−4.96687	−0.263244
$357$	0	0
$358$	20.6733	1.09262
$359$	20.5700	1.08564	0.542822	−	0.839848i	$-0.317356\pi$
$359$	20.5700	1.08564	0.542822	+	0.839848i	$0.317356\pi$
$360$	0	0
$361$	−18.2676	−0.961451
$362$	9.71008	0.510350
$363$	0	0
$364$	4.78646	0.250879
$365$	38.5427	2.01742
$366$	0	0
$367$	−24.0628	−1.25607	−0.628033	−	0.778187i	$-0.716140\pi$
$367$	−24.0628	−1.25607	−0.628033	+	0.778187i	$0.716140\pi$
$368$	4.94173	0.257606
$369$	0	0
$370$	−30.4158	−1.58124
$371$	−4.25232	−0.220770
$372$	0	0
$373$	10.9453	0.566726	0.283363	−	0.959013i	$-0.408550\pi$
$373$	10.9453	0.566726	0.283363	+	0.959013i	$0.408550\pi$
$374$	−11.9395	−0.617377
$375$	0	0
$376$	9.33868	0.481606
$377$	2.30892	0.118916
$378$	0	0
$379$	−12.4616	−0.640110	−0.320055	−	0.947399i	$-0.603701\pi$
$379$	−12.4616	−0.640110	−0.320055	+	0.947399i	$0.603701\pi$
$380$	−2.83796	−0.145584
$381$	0	0
$382$	−39.7792	−2.03528
$383$	−0.0834058	−0.00426184	−0.00213092	−	0.999998i	$-0.500678\pi$
$383$	−0.0834058	−0.00426184	−0.00213092	+	0.999998i	$0.500678\pi$
$384$	0	0
$385$	−9.33365	−0.475687
$386$	30.4669	1.55073
$387$	0	0
$388$	−22.9962	−1.16746
$389$	13.6022	0.689659	0.344830	−	0.938665i	$-0.387937\pi$
$389$	13.6022	0.689659	0.344830	+	0.938665i	$0.387937\pi$
$390$	0	0
$391$	3.16946	0.160286
$392$	5.75182	0.290511
$393$	0	0
$394$	16.2368	0.817999
$395$	14.0060	0.704719
$396$	0	0
$397$	26.2001	1.31495	0.657474	−	0.753477i	$-0.271625\pi$
$397$	26.2001	1.31495	0.657474	+	0.753477i	$0.271625\pi$
$398$	−35.3165	−1.77026
$399$	0	0
$400$	−10.5536	−0.527680
$401$	−27.3144	−1.36402	−0.682009	−	0.731344i	$-0.738894\pi$
$401$	−27.3144	−1.36402	−0.682009	+	0.731344i	$0.738894\pi$
$402$	0	0
$403$	−16.9253	−0.843110
$404$	−11.7254	−0.583361
$405$	0	0
$406$	3.00652	0.149211
$407$	13.2329	0.655930
$408$	0	0
$409$	33.3682	1.64995	0.824975	−	0.565169i	$-0.191189\pi$
$409$	33.3682	1.64995	0.824975	+	0.565169i	$0.191189\pi$
$410$	−1.14240	−0.0564191
$411$	0	0
$412$	12.5656	0.619064
$413$	−15.5359	−0.764472
$414$	0	0
$415$	38.8261	1.90590
$416$	−14.2355	−0.697951
$417$	0	0
$418$	3.22393	0.157688
$419$	−21.6033	−1.05539	−0.527696	−	0.849433i	$-0.676944\pi$
$419$	−21.6033	−1.05539	−0.527696	+	0.849433i	$0.676944\pi$
$420$	0	0
$421$	−27.2802	−1.32956	−0.664779	−	0.747040i	$-0.731474\pi$
$421$	−27.2802	−1.32956	−0.664779	+	0.747040i	$0.731474\pi$
$422$	31.9139	1.55354
$423$	0	0
$424$	3.47788	0.168901
$425$	−6.76871	−0.328331
$426$	0	0
$427$	8.95815	0.433515
$428$	−7.90818	−0.382256
$429$	0	0
$430$	−46.6235	−2.24838
$431$	9.04837	0.435845	0.217922	−	0.975966i	$-0.430072\pi$
$431$	9.04837	0.435845	0.217922	+	0.975966i	$0.430072\pi$
$432$	0	0
$433$	16.4712	0.791554	0.395777	−	0.918347i	$-0.370475\pi$
$433$	16.4712	0.791554	0.395777	+	0.918347i	$0.370475\pi$
$434$	−22.0390	−1.05791
$435$	0	0
$436$	−0.816399	−0.0390984
$437$	−0.855823	−0.0409396
$438$	0	0
$439$	−18.3212	−0.874424	−0.437212	−	0.899359i	$-0.644034\pi$
$439$	−18.3212	−0.874424	−0.437212	+	0.899359i	$0.644034\pi$
$440$	7.63378	0.363926
$441$	0	0
$442$	−13.1753	−0.626683
$443$	32.7174	1.55445	0.777226	−	0.629221i	$-0.216626\pi$
$443$	32.7174	1.55445	0.777226	+	0.629221i	$0.216626\pi$
$444$	0	0
$445$	10.6879	0.506655
$446$	−17.1202	−0.810664
$447$	0	0
$448$	−2.03173	−0.0959903
$449$	39.3491	1.85700	0.928499	−	0.371336i	$-0.121100\pi$
$449$	39.3491	1.85700	0.928499	+	0.371336i	$0.121100\pi$
$450$	0	0
$451$	0.497020	0.0234037
$452$	−3.62036	−0.170288
$453$	0	0
$454$	16.4826	0.773567
$455$	−10.2997	−0.482857
$456$	0	0
$457$	34.7280	1.62451	0.812255	−	0.583303i	$-0.198240\pi$
$457$	34.7280	1.62451	0.812255	+	0.583303i	$0.198240\pi$
$458$	29.8117	1.39301
$459$	0	0
$460$	3.31605	0.154612
$461$	−1.30696	−0.0608714	−0.0304357	−	0.999537i	$-0.509689\pi$
$461$	−1.30696	−0.0608714	−0.0304357	+	0.999537i	$0.509689\pi$
$462$	0	0
$463$	28.5878	1.32859	0.664293	−	0.747472i	$-0.268733\pi$
$463$	28.5878	1.32859	0.664293	+	0.747472i	$0.268733\pi$
$464$	−4.94173	−0.229414
$465$	0	0
$466$	−9.95002	−0.460926
$467$	−6.56840	−0.303950	−0.151975	−	0.988384i	$-0.548563\pi$
$467$	−6.56840	−0.303950	−0.151975	+	0.988384i	$0.548563\pi$
$468$	0	0
$469$	−6.76892	−0.312560
$470$	32.8836	1.51681
$471$	0	0
$472$	12.7065	0.584862
$473$	20.2843	0.932674
$474$	0	0
$475$	1.82770	0.0838608
$476$	−6.57037	−0.301153
$477$	0	0
$478$	35.0115	1.60139
$479$	−31.5136	−1.43989	−0.719947	−	0.694029i	$-0.755834\pi$
$479$	−31.5136	−1.43989	−0.719947	+	0.694029i	$0.755834\pi$
$480$	0	0
$481$	14.6025	0.665817
$482$	−32.6641	−1.48781
$483$	0	0
$484$	−8.22048	−0.373658
$485$	49.4842	2.24696
$486$	0	0
$487$	−8.38768	−0.380082	−0.190041	−	0.981776i	$-0.560862\pi$
$487$	−8.38768	−0.380082	−0.190041	+	0.981776i	$0.560862\pi$
$488$	−7.32666	−0.331662
$489$	0	0
$490$	20.2534	0.914957
$491$	10.0413	0.453155	0.226578	−	0.973993i	$-0.427246\pi$
$491$	10.0413	0.453155	0.226578	+	0.973993i	$0.427246\pi$
$492$	0	0
$493$	−3.16946	−0.142745
$494$	3.55761	0.160065
$495$	0	0
$496$	36.2249	1.62654
$497$	16.3173	0.731932
$498$	0	0
$499$	11.9821	0.536392	0.268196	−	0.963364i	$-0.413573\pi$
$499$	11.9821	0.536392	0.268196	+	0.963364i	$0.413573\pi$
$500$	9.49848	0.424785
$501$	0	0
$502$	−50.2961	−2.24482
$503$	6.68248	0.297957	0.148979	−	0.988840i	$-0.452401\pi$
$503$	6.68248	0.297957	0.148979	+	0.988840i	$0.452401\pi$
$504$	0	0
$505$	25.2312	1.12277
$506$	−3.76705	−0.167466
$507$	0	0
$508$	−1.41266	−0.0626768
$509$	20.9085	0.926753	0.463377	−	0.886161i	$-0.346638\pi$
$509$	20.9085	0.926753	0.463377	+	0.886161i	$0.346638\pi$
$510$	0	0
$511$	−24.0950	−1.06590
$512$	16.9690	0.749931
$513$	0	0
$514$	44.1094	1.94558
$515$	−27.0392	−1.19149
$516$	0	0
$517$	−14.3065	−0.629201
$518$	19.0144	0.835446
$519$	0	0
$520$	8.42388	0.369412
$521$	−13.9056	−0.609216	−0.304608	−	0.952478i	$-0.598526\pi$
$521$	−13.9056	−0.609216	−0.304608	+	0.952478i	$0.598526\pi$
$522$	0	0
$523$	−10.5453	−0.461115	−0.230557	−	0.973059i	$-0.574055\pi$
$523$	−10.5453	−0.461115	−0.230557	+	0.973059i	$0.574055\pi$
$524$	−3.69649	−0.161482
$525$	0	0
$526$	−47.8544	−2.08655
$527$	23.2334	1.01206
$528$	0	0
$529$	1.00000	0.0434783
$530$	12.2464	0.531948
$531$	0	0
$532$	1.77415	0.0769190
$533$	0.548462	0.0237565
$534$	0	0
$535$	17.0171	0.735715
$536$	5.53615	0.239125
$537$	0	0
$538$	20.9695	0.904060
$539$	−8.81159	−0.379542
$540$	0	0
$541$	−34.2114	−1.47086	−0.735431	−	0.677600i	$-0.763020\pi$
$541$	−34.2114	−1.47086	−0.735431	+	0.677600i	$0.763020\pi$
$542$	5.87275	0.252256
$543$	0	0
$544$	19.5410	0.837814
$545$	1.75676	0.0752513
$546$	0	0
$547$	6.74782	0.288516	0.144258	−	0.989540i	$-0.453920\pi$
$547$	6.74782	0.288516	0.144258	+	0.989540i	$0.453920\pi$
$548$	−27.4629	−1.17316
$549$	0	0
$550$	8.04494	0.343038
$551$	0.855823	0.0364593
$552$	0	0
$553$	−8.75585	−0.372337
$554$	−54.7049	−2.32419
$555$	0	0
$556$	18.0392	0.765033
$557$	19.2080	0.813870	0.406935	−	0.913457i	$-0.366597\pi$
$557$	19.2080	0.813870	0.406935	+	0.913457i	$0.366597\pi$
$558$	0	0
$559$	22.3838	0.946732
$560$	22.0442	0.931538
$561$	0	0
$562$	−4.57358	−0.192925
$563$	6.69581	0.282195	0.141097	−	0.989996i	$-0.454937\pi$
$563$	6.69581	0.282195	0.141097	+	0.989996i	$0.454937\pi$
$564$	0	0
$565$	7.79044	0.327746
$566$	−14.1520	−0.594852
$567$	0	0
$568$	−13.3456	−0.559967
$569$	30.4923	1.27830	0.639151	−	0.769081i	$-0.279286\pi$
$569$	30.4923	1.27830	0.639151	+	0.769081i	$0.279286\pi$
$570$	0	0
$571$	−4.84450	−0.202736	−0.101368	−	0.994849i	$-0.532322\pi$
$571$	−4.84450	−0.202736	−0.101368	+	0.994849i	$0.532322\pi$
$572$	5.99724	0.250757
$573$	0	0
$574$	0.714171	0.0298089
$575$	−2.13561	−0.0890610
$576$	0	0
$577$	18.2641	0.760346	0.380173	−	0.924915i	$-0.375865\pi$
$577$	18.2641	0.760346	0.380173	+	0.924915i	$0.375865\pi$
$578$	−12.5209	−0.520799
$579$	0	0
$580$	−3.31605	−0.137692
$581$	−24.2722	−1.00698
$582$	0	0
$583$	−5.32799	−0.220663
$584$	19.7067	0.815470
$585$	0	0
$586$	38.6502	1.59662
$587$	19.6519	0.811121	0.405560	−	0.914068i	$-0.367076\pi$
$587$	19.6519	0.811121	0.405560	+	0.914068i	$0.367076\pi$
$588$	0	0
$589$	−6.27353	−0.258496
$590$	44.7422	1.84201
$591$	0	0
$592$	−31.2534	−1.28451
$593$	28.3554	1.16442	0.582208	−	0.813040i	$-0.302189\pi$
$593$	28.3554	1.16442	0.582208	+	0.813040i	$0.302189\pi$
$594$	0	0
$595$	14.1384	0.579617
$596$	−12.0377	−0.493082
$597$	0	0
$598$	−4.15695	−0.169990
$599$	13.6995	0.559745	0.279873	−	0.960037i	$-0.409708\pi$
$599$	13.6995	0.559745	0.279873	+	0.960037i	$0.409708\pi$
$600$	0	0
$601$	32.9585	1.34440	0.672202	−	0.740367i	$-0.265348\pi$
$601$	32.9585	1.34440	0.672202	+	0.740367i	$0.265348\pi$
$602$	29.1467	1.18793
$603$	0	0
$604$	23.2464	0.945882
$605$	17.6891	0.719166
$606$	0	0
$607$	−31.6252	−1.28363	−0.641815	−	0.766860i	$-0.721818\pi$
$607$	−31.6252	−1.28363	−0.641815	+	0.766860i	$0.721818\pi$
$608$	−5.27651	−0.213991
$609$	0	0
$610$	−25.7988	−1.04456
$611$	−15.7873	−0.638685
$612$	0	0
$613$	−23.3793	−0.944281	−0.472141	−	0.881523i	$-0.656519\pi$
$613$	−23.3793	−0.944281	−0.472141	+	0.881523i	$0.656519\pi$
$614$	−19.3283	−0.780026
$615$	0	0
$616$	−4.77225	−0.192280
$617$	−2.70486	−0.108894	−0.0544469	−	0.998517i	$-0.517340\pi$
$617$	−2.70486	−0.108894	−0.0544469	+	0.998517i	$0.517340\pi$
$618$	0	0
$619$	−6.20806	−0.249523	−0.124762	−	0.992187i	$-0.539817\pi$
$619$	−6.20806	−0.249523	−0.124762	+	0.992187i	$0.539817\pi$
$620$	24.3080	0.976232
$621$	0	0
$622$	−3.70964	−0.148743
$623$	−6.68154	−0.267690
$624$	0	0
$625$	−31.1172	−1.24469
$626$	38.2695	1.52956
$627$	0	0
$628$	20.7666	0.828676
$629$	−20.0449	−0.799241
$630$	0	0
$631$	8.78362	0.349670	0.174835	−	0.984598i	$-0.444061\pi$
$631$	8.78362	0.349670	0.174835	+	0.984598i	$0.444061\pi$
$632$	7.16121	0.284858
$633$	0	0
$634$	−4.96492	−0.197182
$635$	3.03982	0.120632
$636$	0	0
$637$	−9.72360	−0.385263
$638$	3.76705	0.149139
$639$	0	0
$640$	−27.0876	−1.07073
$641$	−37.1787	−1.46847	−0.734235	−	0.678896i	$-0.762459\pi$
$641$	−37.1787	−1.46847	−0.734235	+	0.678896i	$0.762459\pi$
$642$	0	0
$643$	37.4545	1.47706	0.738531	−	0.674220i	$-0.235520\pi$
$643$	37.4545	1.47706	0.738531	+	0.674220i	$0.235520\pi$
$644$	−2.07303	−0.0816888
$645$	0	0
$646$	−4.88353	−0.192140
$647$	12.1024	0.475796	0.237898	−	0.971290i	$-0.423542\pi$
$647$	12.1024	0.475796	0.237898	+	0.971290i	$0.423542\pi$
$648$	0	0
$649$	−19.4659	−0.764102
$650$	8.87760	0.348208
$651$	0	0
$652$	−19.3704	−0.758603
$653$	27.0887	1.06006	0.530032	−	0.847978i	$-0.322180\pi$
$653$	27.0887	1.06006	0.530032	+	0.847978i	$0.322180\pi$
$654$	0	0
$655$	7.95426	0.310798
$656$	−1.17386	−0.0458316
$657$	0	0
$658$	−20.5571	−0.801401
$659$	33.9335	1.32186	0.660930	−	0.750448i	$-0.270162\pi$
$659$	33.9335	1.32186	0.660930	+	0.750448i	$0.270162\pi$
$660$	0	0
$661$	2.59247	0.100835	0.0504176	−	0.998728i	$-0.483945\pi$
$661$	2.59247	0.100835	0.0504176	+	0.998728i	$0.483945\pi$
$662$	−20.7141	−0.805078
$663$	0	0
$664$	19.8516	0.770393
$665$	−3.81768	−0.148043
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−14.9680	−0.579131
$669$	0	0
$670$	19.4940	0.753119
$671$	11.2242	0.433305
$672$	0	0
$673$	−36.9053	−1.42260	−0.711298	−	0.702891i	$-0.751892\pi$
$673$	−36.9053	−1.42260	−0.711298	+	0.702891i	$0.751892\pi$
$674$	60.5843	2.33362
$675$	0	0
$676$	−9.52002	−0.366155
$677$	−5.47441	−0.210399	−0.105199	−	0.994451i	$-0.533548\pi$
$677$	−5.47441	−0.210399	−0.105199	+	0.994451i	$0.533548\pi$
$678$	0	0
$679$	−30.9350	−1.18718
$680$	−11.5635	−0.443438
$681$	0	0
$682$	−27.6140	−1.05739
$683$	0.415708	0.0159066	0.00795331	−	0.999968i	$-0.497468\pi$
$683$	0.415708	0.0159066	0.00795331	+	0.999968i	$0.497468\pi$
$684$	0	0
$685$	59.0957	2.25793
$686$	−33.7071	−1.28694
$687$	0	0
$688$	−47.9075	−1.82646
$689$	−5.87944	−0.223989
$690$	0	0
$691$	29.9264	1.13845	0.569227	−	0.822180i	$-0.307243\pi$
$691$	29.9264	1.13845	0.569227	+	0.822180i	$0.307243\pi$
$692$	−23.3265	−0.886742
$693$	0	0
$694$	−25.1650	−0.955250
$695$	−38.8175	−1.47243
$696$	0	0
$697$	−0.752873	−0.0285171
$698$	2.38964	0.0904493
$699$	0	0
$700$	4.42718	0.167332
$701$	14.4942	0.547439	0.273720	−	0.961810i	$-0.411746\pi$
$701$	14.4942	0.547439	0.273720	+	0.961810i	$0.411746\pi$
$702$	0	0
$703$	5.41256	0.204139
$704$	−2.54568	−0.0959438
$705$	0	0
$706$	1.85200	0.0697010
$707$	−15.7733	−0.593215
$708$	0	0
$709$	−19.8992	−0.747332	−0.373666	−	0.927563i	$-0.621899\pi$
$709$	−19.8992	−0.747332	−0.373666	+	0.927563i	$0.621899\pi$
$710$	−46.9927	−1.76360
$711$	0	0
$712$	5.46468	0.204797
$713$	7.33040	0.274526
$714$	0	0
$715$	−12.9051	−0.482623
$716$	14.2544	0.532713
$717$	0	0
$718$	37.0340	1.38210
$719$	8.34648	0.311271	0.155636	−	0.987815i	$-0.450257\pi$
$719$	8.34648	0.311271	0.155636	+	0.987815i	$0.450257\pi$
$720$	0	0
$721$	16.9035	0.629520
$722$	−32.8886	−1.22399
$723$	0	0
$724$	6.69520	0.248825
$725$	2.13561	0.0793145
$726$	0	0
$727$	13.8747	0.514583	0.257292	−	0.966334i	$-0.417170\pi$
$727$	13.8747	0.514583	0.257292	+	0.966334i	$0.417170\pi$
$728$	−5.26619	−0.195178
$729$	0	0
$730$	69.3917	2.56830
$731$	−30.7262	−1.13645
$732$	0	0
$733$	22.9744	0.848577	0.424289	−	0.905527i	$-0.360524\pi$
$733$	22.9744	0.848577	0.424289	+	0.905527i	$0.360524\pi$
$734$	−43.3222	−1.59905
$735$	0	0
$736$	6.16542	0.227260
$737$	−8.48118	−0.312408
$738$	0	0
$739$	20.9037	0.768954	0.384477	−	0.923135i	$-0.374382\pi$
$739$	20.9037	0.768954	0.384477	+	0.923135i	$0.374382\pi$
$740$	−20.9720	−0.770946
$741$	0	0
$742$	−7.65582	−0.281054
$743$	−51.6364	−1.89436	−0.947178	−	0.320707i	$-0.896079\pi$
$743$	−51.6364	−1.89436	−0.947178	+	0.320707i	$0.896079\pi$
$744$	0	0
$745$	25.9031	0.949016
$746$	19.7057	0.721478
$747$	0	0
$748$	−8.23241	−0.301007
$749$	−10.6383	−0.388713
$750$	0	0
$751$	−44.0079	−1.60587	−0.802935	−	0.596067i	$-0.796729\pi$
$751$	−44.0079	−1.60587	−0.802935	+	0.596067i	$0.796729\pi$
$752$	33.7892	1.23216
$753$	0	0
$754$	4.15695	0.151387
$755$	−50.0224	−1.82050
$756$	0	0
$757$	−20.1006	−0.730569	−0.365284	−	0.930896i	$-0.619028\pi$
$757$	−20.1006	−0.730569	−0.365284	+	0.930896i	$0.619028\pi$
$758$	−22.4357	−0.814901
$759$	0	0
$760$	3.12239	0.113261
$761$	14.4611	0.524216	0.262108	−	0.965039i	$-0.415582\pi$
$761$	14.4611	0.524216	0.262108	+	0.965039i	$0.415582\pi$
$762$	0	0
$763$	−1.09824	−0.0397588
$764$	−27.4282	−0.992316
$765$	0	0
$766$	−0.150163	−0.00542559
$767$	−21.4806	−0.775619
$768$	0	0
$769$	−24.8366	−0.895629	−0.447815	−	0.894126i	$-0.647798\pi$
$769$	−24.8366	−0.895629	−0.447815	+	0.894126i	$0.647798\pi$
$770$	−16.8042	−0.605580
$771$	0	0
$772$	21.0073	0.756068
$773$	−8.57947	−0.308582	−0.154291	−	0.988025i	$-0.549309\pi$
$773$	−8.57947	−0.308582	−0.154291	+	0.988025i	$0.549309\pi$
$774$	0	0
$775$	−15.6549	−0.562339
$776$	25.3010	0.908254
$777$	0	0
$778$	24.4892	0.877980
$779$	0.203292	0.00728371
$780$	0	0
$781$	20.4449	0.731577
$782$	5.70624	0.204055
$783$	0	0
$784$	20.8112	0.743258
$785$	−44.6863	−1.59492
$786$	0	0
$787$	8.68947	0.309746	0.154873	−	0.987934i	$-0.450503\pi$
$787$	8.68947	0.309746	0.154873	+	0.987934i	$0.450503\pi$
$788$	11.1954	0.398821
$789$	0	0
$790$	25.2162	0.897152
$791$	−4.87019	−0.173164
$792$	0	0
$793$	12.3859	0.439837
$794$	47.1703	1.67401
$795$	0	0
$796$	−24.3511	−0.863102
$797$	−40.9587	−1.45083	−0.725417	−	0.688310i	$-0.758353\pi$
$797$	−40.9587	−1.45083	−0.725417	+	0.688310i	$0.758353\pi$
$798$	0	0
$799$	21.6712	0.766671
$800$	−13.1669	−0.465521
$801$	0	0
$802$	−49.1765	−1.73648
$803$	−30.1900	−1.06538
$804$	0	0
$805$	4.46082	0.157223
$806$	−30.4721	−1.07333
$807$	0	0
$808$	12.9006	0.453841
$809$	29.3151	1.03066	0.515332	−	0.856990i	$-0.327669\pi$
$809$	29.3151	1.03066	0.515332	+	0.856990i	$0.327669\pi$
$810$	0	0
$811$	47.2670	1.65977	0.829884	−	0.557936i	$-0.188407\pi$
$811$	47.2670	1.65977	0.829884	+	0.557936i	$0.188407\pi$
$812$	2.07303	0.0727491
$813$	0	0
$814$	23.8243	0.835041
$815$	41.6820	1.46006
$816$	0	0
$817$	8.29675	0.290267
$818$	60.0756	2.10049
$819$	0	0
$820$	−0.787696	−0.0275075
$821$	4.71487	0.164550	0.0822750	−	0.996610i	$-0.473781\pi$
$821$	4.71487	0.164550	0.0822750	+	0.996610i	$0.473781\pi$
$822$	0	0
$823$	20.6481	0.719747	0.359873	−	0.933001i	$-0.382820\pi$
$823$	20.6481	0.719747	0.359873	+	0.933001i	$0.382820\pi$
$824$	−13.8250	−0.481617
$825$	0	0
$826$	−27.9706	−0.973222
$827$	25.6122	0.890624	0.445312	−	0.895375i	$-0.353093\pi$
$827$	25.6122	0.890624	0.445312	+	0.895375i	$0.353093\pi$
$828$	0	0
$829$	2.63616	0.0915576	0.0457788	−	0.998952i	$-0.485423\pi$
$829$	2.63616	0.0915576	0.0457788	+	0.998952i	$0.485423\pi$
$830$	69.9020	2.42633
$831$	0	0
$832$	−2.80916	−0.0973900
$833$	13.3476	0.462466
$834$	0	0
$835$	32.2088	1.11463
$836$	2.22293	0.0768818
$837$	0	0
$838$	−38.8943	−1.34358
$839$	−17.0389	−0.588247	−0.294124	−	0.955767i	$-0.595028\pi$
$839$	−17.0389	−0.588247	−0.294124	+	0.955767i	$0.595028\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−49.1149	−1.69261
$843$	0	0
$844$	22.0049	0.757441
$845$	20.4855	0.704724
$846$	0	0
$847$	−11.0584	−0.379970
$848$	12.5836	0.432124
$849$	0	0
$850$	−12.1863	−0.417986
$851$	−6.32439	−0.216797
$852$	0	0
$853$	0.598601	0.0204957	0.0102479	−	0.999947i	$-0.496738\pi$
$853$	0.598601	0.0204957	0.0102479	+	0.999947i	$0.496738\pi$
$854$	16.1281	0.551893
$855$	0	0
$856$	8.70078	0.297387
$857$	−8.72118	−0.297910	−0.148955	−	0.988844i	$-0.547591\pi$
$857$	−8.72118	−0.297910	−0.148955	+	0.988844i	$0.547591\pi$
$858$	0	0
$859$	20.4118	0.696443	0.348221	−	0.937412i	$-0.386786\pi$
$859$	20.4118	0.696443	0.348221	+	0.937412i	$0.386786\pi$
$860$	−32.1474	−1.09622
$861$	0	0
$862$	16.2905	0.554858
$863$	4.12094	0.140278	0.0701391	−	0.997537i	$-0.477656\pi$
$863$	4.12094	0.140278	0.0701391	+	0.997537i	$0.477656\pi$
$864$	0	0
$865$	50.1949	1.70668
$866$	29.6545	1.00770
$867$	0	0
$868$	−15.1961	−0.515790
$869$	−10.9707	−0.372156
$870$	0	0
$871$	−9.35900	−0.317117
$872$	0.898223	0.0304177
$873$	0	0
$874$	−1.54081	−0.0521187
$875$	12.7775	0.431960
$876$	0	0
$877$	4.80288	0.162182	0.0810909	−	0.996707i	$-0.474160\pi$
$877$	4.80288	0.162182	0.0810909	+	0.996707i	$0.474160\pi$
$878$	−32.9852	−1.11320
$879$	0	0
$880$	27.6205	0.931086
$881$	−4.55946	−0.153612	−0.0768060	−	0.997046i	$-0.524472\pi$
$881$	−4.55946	−0.153612	−0.0768060	+	0.997046i	$0.524472\pi$
$882$	0	0
$883$	−22.6282	−0.761500	−0.380750	−	0.924678i	$-0.624334\pi$
$883$	−22.6282	−0.761500	−0.380750	+	0.924678i	$0.624334\pi$
$884$	−9.08447	−0.305544
$885$	0	0
$886$	58.9040	1.97892
$887$	−4.58047	−0.153797	−0.0768985	−	0.997039i	$-0.524502\pi$
$887$	−4.58047	−0.153797	−0.0768985	+	0.997039i	$0.524502\pi$
$888$	0	0
$889$	−1.90035	−0.0637355
$890$	19.2423	0.645004
$891$	0	0
$892$	−11.8045	−0.395245
$893$	−5.85170	−0.195820
$894$	0	0
$895$	−30.6732	−1.02529
$896$	16.9338	0.565718
$897$	0	0
$898$	70.8434	2.36408
$899$	−7.33040	−0.244482
$900$	0	0
$901$	8.07071	0.268874
$902$	0.894826	0.0297945
$903$	0	0
$904$	3.98322	0.132480
$905$	−14.4070	−0.478904
$906$	0	0
$907$	35.3993	1.17542	0.587708	−	0.809073i	$-0.300031\pi$
$907$	35.3993	1.17542	0.587708	+	0.809073i	$0.300031\pi$
$908$	11.3649	0.377158
$909$	0	0
$910$	−18.5434	−0.614708
$911$	25.8326	0.855872	0.427936	−	0.903809i	$-0.359241\pi$
$911$	25.8326	0.855872	0.427936	+	0.903809i	$0.359241\pi$
$912$	0	0
$913$	−30.4120	−1.00649
$914$	62.5238	2.06810
$915$	0	0
$916$	20.5555	0.679171
$917$	−4.97260	−0.164210
$918$	0	0
$919$	19.5672	0.645464	0.322732	−	0.946490i	$-0.395399\pi$
$919$	19.5672	0.645464	0.322732	+	0.946490i	$0.395399\pi$
$920$	−3.64840	−0.120284
$921$	0	0
$922$	−2.35304	−0.0774931
$923$	22.5610	0.742605
$924$	0	0
$925$	13.5064	0.444088
$926$	51.4689	1.69137
$927$	0	0
$928$	−6.16542	−0.202390
$929$	52.3694	1.71818	0.859092	−	0.511820i	$-0.171029\pi$
$929$	52.3694	1.71818	0.859092	+	0.511820i	$0.171029\pi$
$930$	0	0
$931$	−3.60414	−0.118121
$932$	−6.86064	−0.224728
$933$	0	0
$934$	−11.8257	−0.386947
$935$	17.7148	0.579336
$936$	0	0
$937$	−18.8943	−0.617251	−0.308625	−	0.951184i	$-0.599869\pi$
$937$	−18.8943	−0.617251	−0.308625	+	0.951184i	$0.599869\pi$
$938$	−12.1867	−0.397909
$939$	0	0
$940$	22.6735	0.739530
$941$	0.101917	0.00332241	0.00166120	−	0.999999i	$-0.499471\pi$
$941$	0.101917	0.00332241	0.00166120	+	0.999999i	$0.499471\pi$
$942$	0	0
$943$	−0.237540	−0.00773537
$944$	45.9744	1.49634
$945$	0	0
$946$	36.5196	1.18735
$947$	8.17439	0.265632	0.132816	−	0.991141i	$-0.457598\pi$
$947$	8.17439	0.265632	0.132816	+	0.991141i	$0.457598\pi$
$948$	0	0
$949$	−33.3147	−1.08144
$950$	3.29057	0.106760
$951$	0	0
$952$	7.22889	0.234290
$953$	34.2815	1.11049	0.555243	−	0.831688i	$-0.312625\pi$
$953$	34.2815	1.11049	0.555243	+	0.831688i	$0.312625\pi$
$954$	0	0
$955$	59.0210	1.90987
$956$	24.1408	0.780768
$957$	0	0
$958$	−56.7366	−1.83308
$959$	−36.9437	−1.19297
$960$	0	0
$961$	22.7347	0.733378
$962$	26.2901	0.847628
$963$	0	0
$964$	−22.5222	−0.725391
$965$	−45.2042	−1.45518
$966$	0	0
$967$	−25.8018	−0.829731	−0.414866	−	0.909883i	$-0.636171\pi$
$967$	−25.8018	−0.829731	−0.414866	+	0.909883i	$0.636171\pi$
$968$	9.04438	0.290697
$969$	0	0
$970$	89.0905	2.86052
$971$	−37.7530	−1.21155	−0.605776	−	0.795635i	$-0.707137\pi$
$971$	−37.7530	−1.21155	−0.605776	+	0.795635i	$0.707137\pi$
$972$	0	0
$973$	24.2667	0.777956
$974$	−15.1011	−0.483869
$975$	0	0
$976$	−26.5093	−0.848542
$977$	−5.08937	−0.162823	−0.0814116	−	0.996681i	$-0.525943\pi$
$977$	−5.08937	−0.162823	−0.0814116	+	0.996681i	$0.525943\pi$
$978$	0	0
$979$	−8.37169	−0.267561
$980$	13.9650	0.446094
$981$	0	0
$982$	18.0781	0.576896
$983$	1.25574	0.0400520	0.0200260	−	0.999799i	$-0.493625\pi$
$983$	1.25574	0.0400520	0.0200260	+	0.999799i	$0.493625\pi$
$984$	0	0
$985$	−24.0908	−0.767596
$986$	−5.70624	−0.181724
$987$	0	0
$988$	2.45301	0.0780406
$989$	−9.69447	−0.308266
$990$	0	0
$991$	−33.1445	−1.05287	−0.526435	−	0.850215i	$-0.676472\pi$
$991$	−33.1445	−1.05287	−0.526435	+	0.850215i	$0.676472\pi$
$992$	45.1950	1.43494
$993$	0	0
$994$	29.3775	0.931796
$995$	52.3996	1.66118
$996$	0	0
$997$	−22.4065	−0.709621	−0.354810	−	0.934938i	$-0.615455\pi$
$997$	−22.4065	−0.709621	−0.354810	+	0.934938i	$0.615455\pi$
$998$	21.5724	0.682861
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.q.1.14		16
3.2	odd	2		667.2.a.d.1.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.d.1.3	✓	16	3.2	odd	2
6003.2.a.q.1.14		16	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)

\(f(q)\)	\(=\)	\(q+1.80038 q^{2} +1.24138 q^{4} -2.67126 q^{5} +1.66993 q^{7} -1.36580 q^{8} +O(q^{10})\)	\(q+1.80038 q^{2} +1.24138 q^{4} -2.67126 q^{5} +1.66993 q^{7} -1.36580 q^{8} -4.80929 q^{10} +2.09236 q^{11} +2.30892 q^{13} +3.00652 q^{14} -4.94173 q^{16} -3.16946 q^{17} +0.855823 q^{19} -3.31605 q^{20} +3.76705 q^{22} -1.00000 q^{23} +2.13561 q^{25} +4.15695 q^{26} +2.07303 q^{28} +1.00000 q^{29} -7.33040 q^{31} -6.16542 q^{32} -5.70624 q^{34} -4.46082 q^{35} +6.32439 q^{37} +1.54081 q^{38} +3.64840 q^{40} +0.237540 q^{41} +9.69447 q^{43} +2.59742 q^{44} -1.80038 q^{46} -6.83751 q^{47} -4.21132 q^{49} +3.84491 q^{50} +2.86626 q^{52} -2.54640 q^{53} -5.58923 q^{55} -2.28080 q^{56} +1.80038 q^{58} -9.30330 q^{59} +5.36437 q^{61} -13.1975 q^{62} -1.21665 q^{64} -6.16772 q^{65} -4.05341 q^{67} -3.93451 q^{68} -8.03120 q^{70} +9.77123 q^{71} -14.4287 q^{73} +11.3863 q^{74} +1.06241 q^{76} +3.49410 q^{77} -5.24323 q^{79} +13.2006 q^{80} +0.427664 q^{82} -14.5348 q^{83} +8.46643 q^{85} +17.4538 q^{86} -2.85775 q^{88} -4.00108 q^{89} +3.85575 q^{91} -1.24138 q^{92} -12.3101 q^{94} -2.28612 q^{95} -18.5247 q^{97} -7.58199 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) \) 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8	\( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8} - 14 q^{10} - 4 q^{11} + 15 q^{13} - 8 q^{14} + 23 q^{16} - 20 q^{17} - 4 q^{19} - 25 q^{20} + 13 q^{22} - 16 q^{23} + 30 q^{25} - 25 q^{26} - 13 q^{28} + 16 q^{29} + 19 q^{32} - 23 q^{34} - 5 q^{35} + 5 q^{37} - 38 q^{38} - 20 q^{40} - 7 q^{41} - 17 q^{43} + 21 q^{44} + 3 q^{46} - 29 q^{47} + 31 q^{49} + 44 q^{50} + 20 q^{52} - 63 q^{53} + q^{55} + 19 q^{56} - 3 q^{58} - 11 q^{59} - 33 q^{62} + 29 q^{64} - 53 q^{65} - 13 q^{67} - 63 q^{68} - 46 q^{70} + 23 q^{71} - 38 q^{73} + 47 q^{74} - 56 q^{76} - 97 q^{77} - 27 q^{79} - 8 q^{80} + 9 q^{82} - 36 q^{83} + 6 q^{85} + 11 q^{86} - 24 q^{88} + 16 q^{89} - 47 q^{91} - 21 q^{92} + 37 q^{94} + 12 q^{95} - 30 q^{97} + 27 q^{98}+O(q^{100}) \) 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8 - 14 * q^10 - 4 * q^11 + 15 * q^13 - 8 * q^14 + 23 * q^16 - 20 * q^17 - 4 * q^19 - 25 * q^20 + 13 * q^22 - 16 * q^23 + 30 * q^25 - 25 * q^26 - 13 * q^28 + 16 * q^29 + 19 * q^32 - 23 * q^34 - 5 * q^35 + 5 * q^37 - 38 * q^38 - 20 * q^40 - 7 * q^41 - 17 * q^43 + 21 * q^44 + 3 * q^46 - 29 * q^47 + 31 * q^49 + 44 * q^50 + 20 * q^52 - 63 * q^53 + q^55 + 19 * q^56 - 3 * q^58 - 11 * q^59 - 33 * q^62 + 29 * q^64 - 53 * q^65 - 13 * q^67 - 63 * q^68 - 46 * q^70 + 23 * q^71 - 38 * q^73 + 47 * q^74 - 56 * q^76 - 97 * q^77 - 27 * q^79 - 8 * q^80 + 9 * q^82 - 36 * q^83 + 6 * q^85 + 11 * q^86 - 24 * q^88 + 16 * q^89 - 47 * q^91 - 21 * q^92 + 37 * q^94 + 12 * q^95 - 30 * q^97 + 27 * q^98

Introduction

L-functions

Modular forms

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Database

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