LMFDB - Embedded newform 6003.2.a.o.1.7

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 11 x^{11} + 58 x^{10} + 24 x^{9} - 298 x^{8} + 97 x^{7} + 641 x^{6} - 402 x^{5} + \cdots - 40 $ x^13 - 4x^12 - 11x^11 + 58x^10 + 24x^9 - 298x^8 + 97x^7 + 641x^6 - 402x^5 - 547x^4 + 352x^3 + 219x^2 - 88x - 40
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.775068$ 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)

$f(q)$	$=$	$q-0.775068 q^{2} -1.39927 q^{4} -3.48646 q^{5} +0.0624539 q^{7} +2.63467 q^{8} +O(q^{10})$	\(q-0.775068 q^{2} -1.39927 q^{4} -3.48646 q^{5} +0.0624539 q^{7} +2.63467 q^{8} +2.70224 q^{10} -3.55627 q^{11} -0.927177 q^{13} -0.0484060 q^{14} +0.756494 q^{16} +3.32813 q^{17} -5.02979 q^{19} +4.87850 q^{20} +2.75635 q^{22} +1.00000 q^{23} +7.15541 q^{25} +0.718625 q^{26} -0.0873898 q^{28} -1.00000 q^{29} +4.47664 q^{31} -5.85566 q^{32} -2.57953 q^{34} -0.217743 q^{35} -1.59284 q^{37} +3.89843 q^{38} -9.18566 q^{40} -1.57116 q^{41} -2.30580 q^{43} +4.97618 q^{44} -0.775068 q^{46} +5.34319 q^{47} -6.99610 q^{49} -5.54593 q^{50} +1.29737 q^{52} +9.94550 q^{53} +12.3988 q^{55} +0.164545 q^{56} +0.775068 q^{58} -7.26432 q^{59} +9.22630 q^{61} -3.46970 q^{62} +3.02555 q^{64} +3.23257 q^{65} +14.2216 q^{67} -4.65695 q^{68} +0.168766 q^{70} +7.60080 q^{71} +6.51425 q^{73} +1.23456 q^{74} +7.03803 q^{76} -0.222103 q^{77} -15.6281 q^{79} -2.63749 q^{80} +1.21776 q^{82} +8.68598 q^{83} -11.6034 q^{85} +1.78715 q^{86} -9.36957 q^{88} +15.0849 q^{89} -0.0579058 q^{91} -1.39927 q^{92} -4.14134 q^{94} +17.5362 q^{95} +11.7883 q^{97} +5.42245 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8} + 10 q^{10} - 10 q^{11} + 7 q^{13} + 12 q^{14} + 2 q^{16} - 26 q^{17} - 25 q^{20} - 15 q^{22} + 13 q^{23} + 19 q^{25} + 15 q^{26} + 5 q^{28} - 13 q^{29} - 6 q^{31} - 16 q^{32} + 11 q^{34} - q^{35} + 15 q^{37} - 8 q^{38} + 14 q^{40} - 9 q^{41} + q^{43} - 29 q^{44} - 4 q^{46} - 15 q^{47} + 4 q^{49} - 31 q^{50} - 8 q^{52} - 43 q^{53} - 3 q^{55} + 5 q^{56} + 4 q^{58} + 9 q^{59} + 20 q^{61} - 11 q^{62} - 16 q^{64} + 25 q^{65} + q^{67} - 21 q^{68} - 2 q^{70} - 17 q^{71} + 26 q^{73} - 11 q^{74} + 8 q^{76} - 17 q^{77} + 5 q^{79} - 10 q^{80} - 25 q^{82} - 4 q^{83} + 20 q^{85} + 13 q^{86} - 32 q^{88} - 48 q^{89} - 9 q^{91} + 12 q^{92} - 65 q^{94} - 8 q^{95} + 30 q^{97} - 2 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8 + 10 * q^10 - 10 * q^11 + 7 * q^13 + 12 * q^14 + 2 * q^16 - 26 * q^17 - 25 * q^20 - 15 * q^22 + 13 * q^23 + 19 * q^25 + 15 * q^26 + 5 * q^28 - 13 * q^29 - 6 * q^31 - 16 * q^32 + 11 * q^34 - q^35 + 15 * q^37 - 8 * q^38 + 14 * q^40 - 9 * q^41 + q^43 - 29 * q^44 - 4 * q^46 - 15 * q^47 + 4 * q^49 - 31 * q^50 - 8 * q^52 - 43 * q^53 - 3 * q^55 + 5 * q^56 + 4 * q^58 + 9 * q^59 + 20 * q^61 - 11 * q^62 - 16 * q^64 + 25 * q^65 + q^67 - 21 * q^68 - 2 * q^70 - 17 * q^71 + 26 * q^73 - 11 * q^74 + 8 * q^76 - 17 * q^77 + 5 * q^79 - 10 * q^80 - 25 * q^82 - 4 * q^83 + 20 * q^85 + 13 * q^86 - 32 * q^88 - 48 * q^89 - 9 * q^91 + 12 * q^92 - 65 * q^94 - 8 * q^95 + 30 * q^97 - 2 * 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 $

$2$	−0.775068	−0.548056	−0.274028	−	0.961722i	$-0.588356\pi$
$2$	−0.775068	−0.548056	−0.274028	+	0.961722i	$0.588356\pi$
$3$	0	0
$3$	0	0
$4$	−1.39927	−0.699635
$5$	−3.48646	−1.55919	−0.779596	−	0.626282i	$-0.784576\pi$
$5$	−3.48646	−1.55919	−0.779596	+	0.626282i	$0.784576\pi$
$6$	0	0
$7$	0.0624539	0.0236054	0.0118027	−	0.999930i	$-0.496243\pi$
$7$	0.0624539	0.0236054	0.0118027	+	0.999930i	$0.496243\pi$
$8$	2.63467	0.931495
$9$	0	0
$10$	2.70224	0.854525
$11$	−3.55627	−1.07225	−0.536127	−	0.844137i	$-0.680113\pi$
$11$	−3.55627	−1.07225	−0.536127	+	0.844137i	$0.680113\pi$
$12$	0	0
$13$	−0.927177	−0.257153	−0.128576	−	0.991700i	$-0.541041\pi$
$13$	−0.927177	−0.257153	−0.128576	+	0.991700i	$0.541041\pi$
$14$	−0.0484060	−0.0129371
$15$	0	0
$16$	0.756494	0.189124
$17$	3.32813	0.807190	0.403595	−	0.914938i	$-0.367760\pi$
$17$	3.32813	0.807190	0.403595	+	0.914938i	$0.367760\pi$
$18$	0	0
$19$	−5.02979	−1.15391	−0.576956	−	0.816775i	$-0.695760\pi$
$19$	−5.02979	−1.15391	−0.576956	+	0.816775i	$0.695760\pi$
$20$	4.87850	1.09087
$21$	0	0
$22$	2.75635	0.587656
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	7.15541	1.43108
$26$	0.718625	0.140934
$27$	0	0
$28$	−0.0873898	−0.0165151
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.47664	0.804027	0.402014	−	0.915634i	$-0.368310\pi$
$31$	4.47664	0.804027	0.402014	+	0.915634i	$0.368310\pi$
$32$	−5.85566	−1.03515
$33$	0	0
$34$	−2.57953	−0.442385
$35$	−0.217743	−0.0368053
$36$	0	0
$37$	−1.59284	−0.261861	−0.130930	−	0.991392i	$-0.541796\pi$
$37$	−1.59284	−0.261861	−0.130930	+	0.991392i	$0.541796\pi$
$38$	3.89843	0.632408
$39$	0	0
$40$	−9.18566	−1.45238
$41$	−1.57116	−0.245374	−0.122687	−	0.992445i	$-0.539151\pi$
$41$	−1.57116	−0.245374	−0.122687	+	0.992445i	$0.539151\pi$
$42$	0	0
$43$	−2.30580	−0.351631	−0.175815	−	0.984423i	$-0.556256\pi$
$43$	−2.30580	−0.351631	−0.175815	+	0.984423i	$0.556256\pi$
$44$	4.97618	0.750187
$45$	0	0
$46$	−0.775068	−0.114278
$47$	5.34319	0.779385	0.389693	−	0.920945i	$-0.372581\pi$
$47$	5.34319	0.779385	0.389693	+	0.920945i	$0.372581\pi$
$48$	0	0
$49$	−6.99610	−0.999443
$50$	−5.54593	−0.784313
$51$	0	0
$52$	1.29737	0.179913
$53$	9.94550	1.36612	0.683060	−	0.730362i	$-0.260649\pi$
$53$	9.94550	1.36612	0.683060	+	0.730362i	$0.260649\pi$
$54$	0	0
$55$	12.3988	1.67185
$56$	0.164545	0.0219883
$57$	0	0
$58$	0.775068	0.101771
$59$	−7.26432	−0.945734	−0.472867	−	0.881134i	$-0.656781\pi$
$59$	−7.26432	−0.945734	−0.472867	+	0.881134i	$0.656781\pi$
$60$	0	0
$61$	9.22630	1.18131	0.590653	−	0.806925i	$-0.298870\pi$
$61$	9.22630	1.18131	0.590653	+	0.806925i	$0.298870\pi$
$62$	−3.46970	−0.440652
$63$	0	0
$64$	3.02555	0.378194
$65$	3.23257	0.400950
$66$	0	0
$67$	14.2216	1.73744	0.868720	−	0.495303i	$-0.164943\pi$
$67$	14.2216	1.73744	0.868720	+	0.495303i	$0.164943\pi$
$68$	−4.65695	−0.564738
$69$	0	0
$70$	0.168766	0.0201714
$71$	7.60080	0.902049	0.451024	−	0.892512i	$-0.351059\pi$
$71$	7.60080	0.902049	0.451024	+	0.892512i	$0.351059\pi$
$72$	0	0
$73$	6.51425	0.762436	0.381218	−	0.924485i	$-0.375505\pi$
$73$	6.51425	0.762436	0.381218	+	0.924485i	$0.375505\pi$
$74$	1.23456	0.143514
$75$	0	0
$76$	7.03803	0.807317
$77$	−0.222103	−0.0253110
$78$	0	0
$79$	−15.6281	−1.75830	−0.879149	−	0.476547i	$-0.841888\pi$
$79$	−15.6281	−1.75830	−0.879149	+	0.476547i	$0.841888\pi$
$80$	−2.63749	−0.294880
$81$	0	0
$82$	1.21776	0.134479
$83$	8.68598	0.953410	0.476705	−	0.879063i	$-0.341831\pi$
$83$	8.68598	0.953410	0.476705	+	0.879063i	$0.341831\pi$
$84$	0	0
$85$	−11.6034	−1.25856
$86$	1.78715	0.192713
$87$	0	0
$88$	−9.36957	−0.998800
$89$	15.0849	1.59900	0.799500	−	0.600666i	$-0.205098\pi$
$89$	15.0849	1.59900	0.799500	+	0.600666i	$0.205098\pi$
$90$	0	0
$91$	−0.0579058	−0.00607018
$92$	−1.39927	−0.145884
$93$	0	0
$94$	−4.14134	−0.427147
$95$	17.5362	1.79917
$96$	0	0
$97$	11.7883	1.19692	0.598462	−	0.801151i	$-0.295779\pi$
$97$	11.7883	1.19692	0.598462	+	0.801151i	$0.295779\pi$
$98$	5.42245	0.547750
$99$	0	0
$100$	−10.0124	−1.00124
$101$	2.43366	0.242159	0.121079	−	0.992643i	$-0.461364\pi$
$101$	2.43366	0.242159	0.121079	+	0.992643i	$0.461364\pi$
$102$	0	0
$103$	7.20441	0.709871	0.354936	−	0.934891i	$-0.384503\pi$
$103$	7.20441	0.709871	0.354936	+	0.934891i	$0.384503\pi$
$104$	−2.44280	−0.239536
$105$	0	0
$106$	−7.70844	−0.748710
$107$	1.19795	0.115810	0.0579051	−	0.998322i	$-0.481558\pi$
$107$	1.19795	0.115810	0.0579051	+	0.998322i	$0.481558\pi$
$108$	0	0
$109$	1.74496	0.167137	0.0835684	−	0.996502i	$-0.473368\pi$
$109$	1.74496	0.167137	0.0835684	+	0.996502i	$0.473368\pi$
$110$	−9.60990	−0.916268
$111$	0	0
$112$	0.0472460	0.00446433
$113$	−18.5427	−1.74435	−0.872175	−	0.489195i	$-0.837291\pi$
$113$	−18.5427	−1.74435	−0.872175	+	0.489195i	$0.837291\pi$
$114$	0	0
$115$	−3.48646	−0.325114
$116$	1.39927	0.129919
$117$	0	0
$118$	5.63034	0.518315
$119$	0.207855	0.0190540
$120$	0	0
$121$	1.64704	0.149731
$122$	−7.15101	−0.647422
$123$	0	0
$124$	−6.26402	−0.562525
$125$	−7.51476	−0.672141
$126$	0	0
$127$	−1.31494	−0.116682	−0.0583412	−	0.998297i	$-0.518581\pi$
$127$	−1.31494	−0.116682	−0.0583412	+	0.998297i	$0.518581\pi$
$128$	9.36632	0.827874
$129$	0	0
$130$	−2.50546	−0.219743
$131$	3.07885	0.269000	0.134500	−	0.990914i	$-0.457057\pi$
$131$	3.07885	0.269000	0.134500	+	0.990914i	$0.457057\pi$
$132$	0	0
$133$	−0.314130	−0.0272385
$134$	−11.0227	−0.952214
$135$	0	0
$136$	8.76851	0.751893
$137$	−5.56864	−0.475761	−0.237880	−	0.971294i	$-0.576453\pi$
$137$	−5.56864	−0.475761	−0.237880	+	0.971294i	$0.576453\pi$
$138$	0	0
$139$	−12.6474	−1.07273	−0.536367	−	0.843985i	$-0.680204\pi$
$139$	−12.6474	−1.07273	−0.536367	+	0.843985i	$0.680204\pi$
$140$	0.304681	0.0257503
$141$	0	0
$142$	−5.89114	−0.494373
$143$	3.29729	0.275733
$144$	0	0
$145$	3.48646	0.289535
$146$	−5.04899	−0.417857
$147$	0	0
$148$	2.22881	0.183207
$149$	−18.4512	−1.51158	−0.755791	−	0.654813i	$-0.772747\pi$
$149$	−18.4512	−1.51158	−0.755791	+	0.654813i	$0.772747\pi$
$150$	0	0
$151$	−8.10008	−0.659175	−0.329588	−	0.944125i	$-0.606910\pi$
$151$	−8.10008	−0.659175	−0.329588	+	0.944125i	$0.606910\pi$
$152$	−13.2518	−1.07486
$153$	0	0
$154$	0.172145	0.0138718
$155$	−15.6076	−1.25363
$156$	0	0
$157$	5.60973	0.447706	0.223853	−	0.974623i	$-0.428137\pi$
$157$	5.60973	0.447706	0.223853	+	0.974623i	$0.428137\pi$
$158$	12.1128	0.963646
$159$	0	0
$160$	20.4155	1.61399
$161$	0.0624539	0.00492206
$162$	0	0
$163$	1.87570	0.146916	0.0734580	−	0.997298i	$-0.476597\pi$
$163$	1.87570	0.146916	0.0734580	+	0.997298i	$0.476597\pi$
$164$	2.19848	0.171672
$165$	0	0
$166$	−6.73223	−0.522522
$167$	11.4370	0.885020	0.442510	−	0.896764i	$-0.354088\pi$
$167$	11.4370	0.885020	0.442510	+	0.896764i	$0.354088\pi$
$168$	0	0
$169$	−12.1403	−0.933873
$170$	8.99342	0.689764
$171$	0	0
$172$	3.22643	0.246013
$173$	0.252942	0.0192308	0.00961542	−	0.999954i	$-0.496939\pi$
$173$	0.252942	0.0192308	0.00961542	+	0.999954i	$0.496939\pi$
$174$	0	0
$175$	0.446883	0.0337812
$176$	−2.69030	−0.202789
$177$	0	0
$178$	−11.6919	−0.876341
$179$	6.75538	0.504921	0.252461	−	0.967607i	$-0.418760\pi$
$179$	6.75538	0.504921	0.252461	+	0.967607i	$0.418760\pi$
$180$	0	0
$181$	−19.7057	−1.46471	−0.732356	−	0.680922i	$-0.761579\pi$
$181$	−19.7057	−1.46471	−0.732356	+	0.680922i	$0.761579\pi$
$182$	0.0448809	0.00332680
$183$	0	0
$184$	2.63467	0.194230
$185$	5.55337	0.408292
$186$	0	0
$187$	−11.8357	−0.865513
$188$	−7.47657	−0.545285
$189$	0	0
$190$	−13.5917	−0.986047
$191$	−3.34437	−0.241990	−0.120995	−	0.992653i	$-0.538609\pi$
$191$	−3.34437	−0.241990	−0.120995	+	0.992653i	$0.538609\pi$
$192$	0	0
$193$	−8.02376	−0.577563	−0.288782	−	0.957395i	$-0.593250\pi$
$193$	−8.02376	−0.577563	−0.288782	+	0.957395i	$0.593250\pi$
$194$	−9.13676	−0.655981
$195$	0	0
$196$	9.78943	0.699245
$197$	−27.2979	−1.94489	−0.972447	−	0.233123i	$-0.925106\pi$
$197$	−27.2979	−1.94489	−0.972447	+	0.233123i	$0.925106\pi$
$198$	0	0
$199$	12.1839	0.863692	0.431846	−	0.901947i	$-0.357862\pi$
$199$	12.1839	0.863692	0.431846	+	0.901947i	$0.357862\pi$
$200$	18.8521	1.33305
$201$	0	0
$202$	−1.88626	−0.132716
$203$	−0.0624539	−0.00438340
$204$	0	0
$205$	5.47779	0.382585
$206$	−5.58391	−0.389049
$207$	0	0
$208$	−0.701404	−0.0486336
$209$	17.8873	1.23729
$210$	0	0
$211$	4.16412	0.286670	0.143335	−	0.989674i	$-0.454217\pi$
$211$	4.16412	0.286670	0.143335	+	0.989674i	$0.454217\pi$
$212$	−13.9164	−0.955785
$213$	0	0
$214$	−0.928493	−0.0634705
$215$	8.03907	0.548260
$216$	0	0
$217$	0.279583	0.0189794
$218$	−1.35246	−0.0916003
$219$	0	0
$220$	−17.3492	−1.16969
$221$	−3.08576	−0.207571
$222$	0	0
$223$	16.4546	1.10188	0.550942	−	0.834544i	$-0.314269\pi$
$223$	16.4546	1.10188	0.550942	+	0.834544i	$0.314269\pi$
$224$	−0.365709	−0.0244350
$225$	0	0
$226$	14.3718	0.956001
$227$	16.9207	1.12307	0.561534	−	0.827453i	$-0.310211\pi$
$227$	16.9207	1.12307	0.561534	+	0.827453i	$0.310211\pi$
$228$	0	0
$229$	18.3965	1.21567	0.607837	−	0.794062i	$-0.292037\pi$
$229$	18.3965	1.21567	0.607837	+	0.794062i	$0.292037\pi$
$230$	2.70224	0.178181
$231$	0	0
$232$	−2.63467	−0.172974
$233$	20.2748	1.32824	0.664122	−	0.747624i	$-0.268806\pi$
$233$	20.2748	1.32824	0.664122	+	0.747624i	$0.268806\pi$
$234$	0	0
$235$	−18.6288	−1.21521
$236$	10.1647	0.661668
$237$	0	0
$238$	−0.161102	−0.0104427
$239$	−15.2829	−0.988571	−0.494286	−	0.869300i	$-0.664570\pi$
$239$	−15.2829	−0.988571	−0.494286	+	0.869300i	$0.664570\pi$
$240$	0	0
$241$	−4.39354	−0.283013	−0.141506	−	0.989937i	$-0.545195\pi$
$241$	−4.39354	−0.283013	−0.141506	+	0.989937i	$0.545195\pi$
$242$	−1.27656	−0.0820607
$243$	0	0
$244$	−12.9101	−0.826483
$245$	24.3916	1.55832
$246$	0	0
$247$	4.66350	0.296731
$248$	11.7944	0.748947
$249$	0	0
$250$	5.82445	0.368371
$251$	−9.25415	−0.584117	−0.292058	−	0.956401i	$-0.594340\pi$
$251$	−9.25415	−0.584117	−0.292058	+	0.956401i	$0.594340\pi$
$252$	0	0
$253$	−3.55627	−0.223581
$254$	1.01917	0.0639485
$255$	0	0
$256$	−13.3106	−0.831915
$257$	0.576309	0.0359491	0.0179746	−	0.999838i	$-0.494278\pi$
$257$	0.576309	0.0359491	0.0179746	+	0.999838i	$0.494278\pi$
$258$	0	0
$259$	−0.0994789	−0.00618132
$260$	−4.52323	−0.280519
$261$	0	0
$262$	−2.38632	−0.147427
$263$	−8.16943	−0.503749	−0.251874	−	0.967760i	$-0.581047\pi$
$263$	−8.16943	−0.503749	−0.251874	+	0.967760i	$0.581047\pi$
$264$	0	0
$265$	−34.6746	−2.13004
$266$	0.243472	0.0149282
$267$	0	0
$268$	−19.8998	−1.21557
$269$	−25.1561	−1.53380	−0.766898	−	0.641769i	$-0.778201\pi$
$269$	−25.1561	−1.53380	−0.766898	+	0.641769i	$0.778201\pi$
$270$	0	0
$271$	23.1343	1.40531	0.702656	−	0.711530i	$-0.251997\pi$
$271$	23.1343	1.40531	0.702656	+	0.711530i	$0.251997\pi$
$272$	2.51771	0.152659
$273$	0	0
$274$	4.31607	0.260743
$275$	−25.4466	−1.53449
$276$	0	0
$277$	3.39955	0.204259	0.102130	−	0.994771i	$-0.467434\pi$
$277$	3.39955	0.204259	0.102130	+	0.994771i	$0.467434\pi$
$278$	9.80256	0.587919
$279$	0	0
$280$	−0.573680	−0.0342840
$281$	−28.4141	−1.69505	−0.847523	−	0.530759i	$-0.821907\pi$
$281$	−28.4141	−1.69505	−0.847523	+	0.530759i	$0.821907\pi$
$282$	0	0
$283$	−7.69721	−0.457552	−0.228776	−	0.973479i	$-0.573472\pi$
$283$	−7.69721	−0.457552	−0.228776	+	0.973479i	$0.573472\pi$
$284$	−10.6356	−0.631105
$285$	0	0
$286$	−2.55562	−0.151117
$287$	−0.0981251	−0.00579214
$288$	0	0
$289$	−5.92355	−0.348444
$290$	−2.70224	−0.158681
$291$	0	0
$292$	−9.11520	−0.533427
$293$	−11.1030	−0.648644	−0.324322	−	0.945947i	$-0.605136\pi$
$293$	−11.1030	−0.648644	−0.324322	+	0.945947i	$0.605136\pi$
$294$	0	0
$295$	25.3268	1.47458
$296$	−4.19659	−0.243922
$297$	0	0
$298$	14.3009	0.828431
$299$	−0.927177	−0.0536200
$300$	0	0
$301$	−0.144006	−0.00830037
$302$	6.27811	0.361265
$303$	0	0
$304$	−3.80500	−0.218232
$305$	−32.1671	−1.84188
$306$	0	0
$307$	−21.5009	−1.22712	−0.613561	−	0.789647i	$-0.710264\pi$
$307$	−21.5009	−1.22712	−0.613561	+	0.789647i	$0.710264\pi$
$308$	0.310782	0.0177084
$309$	0	0
$310$	12.0970	0.687061
$311$	−25.6647	−1.45531	−0.727655	−	0.685943i	$-0.759390\pi$
$311$	−25.6647	−1.45531	−0.727655	+	0.685943i	$0.759390\pi$
$312$	0	0
$313$	30.7981	1.74081	0.870406	−	0.492334i	$-0.163856\pi$
$313$	30.7981	1.74081	0.870406	+	0.492334i	$0.163856\pi$
$314$	−4.34793	−0.245368
$315$	0	0
$316$	21.8679	1.23017
$317$	3.16459	0.177741	0.0888706	−	0.996043i	$-0.471674\pi$
$317$	3.16459	0.177741	0.0888706	+	0.996043i	$0.471674\pi$
$318$	0	0
$319$	3.55627	0.199113
$320$	−10.5485	−0.589677
$321$	0	0
$322$	−0.0484060	−0.00269756
$323$	−16.7398	−0.931426
$324$	0	0
$325$	−6.63433	−0.368007
$326$	−1.45379	−0.0805181
$327$	0	0
$328$	−4.13948	−0.228565
$329$	0.333703	0.0183977
$330$	0	0
$331$	−4.73849	−0.260451	−0.130226	−	0.991484i	$-0.541570\pi$
$331$	−4.73849	−0.260451	−0.130226	+	0.991484i	$0.541570\pi$
$332$	−12.1540	−0.667039
$333$	0	0
$334$	−8.86444	−0.485040
$335$	−49.5829	−2.70900
$336$	0	0
$337$	31.8821	1.73673	0.868364	−	0.495927i	$-0.165172\pi$
$337$	31.8821	1.73673	0.868364	+	0.495927i	$0.165172\pi$
$338$	9.40959	0.511814
$339$	0	0
$340$	16.2363	0.880536
$341$	−15.9201	−0.862122
$342$	0	0
$343$	−0.874111	−0.0471976
$344$	−6.07501	−0.327542
$345$	0	0
$346$	−0.196047	−0.0105396
$347$	2.76211	0.148278	0.0741390	−	0.997248i	$-0.476379\pi$
$347$	2.76211	0.148278	0.0741390	+	0.997248i	$0.476379\pi$
$348$	0	0
$349$	−21.7242	−1.16287	−0.581435	−	0.813593i	$-0.697508\pi$
$349$	−21.7242	−1.16287	−0.581435	+	0.813593i	$0.697508\pi$
$350$	−0.346365	−0.0185140
$351$	0	0
$352$	20.8243	1.10994
$353$	−33.1855	−1.76629	−0.883143	−	0.469105i	$-0.844577\pi$
$353$	−33.1855	−1.76629	−0.883143	+	0.469105i	$0.844577\pi$
$354$	0	0
$355$	−26.4999	−1.40647
$356$	−21.1079	−1.11872
$357$	0	0
$358$	−5.23588	−0.276725
$359$	9.46450	0.499517	0.249759	−	0.968308i	$-0.419649\pi$
$359$	9.46450	0.499517	0.249759	+	0.968308i	$0.419649\pi$
$360$	0	0
$361$	6.29875	0.331513
$362$	15.2732	0.802744
$363$	0	0
$364$	0.0810258	0.00424691
$365$	−22.7117	−1.18878
$366$	0	0
$367$	4.00106	0.208853	0.104427	−	0.994533i	$-0.466699\pi$
$367$	4.00106	0.208853	0.104427	+	0.994533i	$0.466699\pi$
$368$	0.756494	0.0394350
$369$	0	0
$370$	−4.30424	−0.223767
$371$	0.621135	0.0322477
$372$	0	0
$373$	−25.2640	−1.30812	−0.654061	−	0.756441i	$-0.726936\pi$
$373$	−25.2640	−1.30812	−0.654061	+	0.756441i	$0.726936\pi$
$374$	9.17349	0.474350
$375$	0	0
$376$	14.0775	0.725993
$377$	0.927177	0.0477520
$378$	0	0
$379$	−2.49577	−0.128199	−0.0640995	−	0.997944i	$-0.520418\pi$
$379$	−2.49577	−0.128199	−0.0640995	+	0.997944i	$0.520418\pi$
$380$	−24.5378	−1.25876
$381$	0	0
$382$	2.59212	0.132624
$383$	−34.1153	−1.74321	−0.871606	−	0.490207i	$-0.836921\pi$
$383$	−34.1153	−1.74321	−0.871606	+	0.490207i	$0.836921\pi$
$384$	0	0
$385$	0.774353	0.0394647
$386$	6.21896	0.316537
$387$	0	0
$388$	−16.4951	−0.837410
$389$	10.5613	0.535481	0.267740	−	0.963491i	$-0.413723\pi$
$389$	10.5613	0.535481	0.267740	+	0.963491i	$0.413723\pi$
$390$	0	0
$391$	3.32813	0.168311
$392$	−18.4324	−0.930976
$393$	0	0
$394$	21.1577	1.06591
$395$	54.4867	2.74153
$396$	0	0
$397$	17.5404	0.880325	0.440163	−	0.897918i	$-0.354921\pi$
$397$	17.5404	0.880325	0.440163	+	0.897918i	$0.354921\pi$
$398$	−9.44333	−0.473352
$399$	0	0
$400$	5.41303	0.270651
$401$	−22.2069	−1.10896	−0.554479	−	0.832198i	$-0.687082\pi$
$401$	−22.2069	−1.10896	−0.554479	+	0.832198i	$0.687082\pi$
$402$	0	0
$403$	−4.15063	−0.206758
$404$	−3.40535	−0.169423
$405$	0	0
$406$	0.0484060	0.00240235
$407$	5.66456	0.280782
$408$	0	0
$409$	−17.7893	−0.879627	−0.439813	−	0.898089i	$-0.644955\pi$
$409$	−17.7893	−0.879627	−0.439813	+	0.898089i	$0.644955\pi$
$410$	−4.24566	−0.209678
$411$	0	0
$412$	−10.0809	−0.496651
$413$	−0.453685	−0.0223244
$414$	0	0
$415$	−30.2833	−1.48655
$416$	5.42924	0.266190
$417$	0	0
$418$	−13.8638	−0.678103
$419$	−19.6622	−0.960562	−0.480281	−	0.877115i	$-0.659465\pi$
$419$	−19.6622	−0.960562	−0.480281	+	0.877115i	$0.659465\pi$
$420$	0	0
$421$	32.9612	1.60643	0.803216	−	0.595688i	$-0.203121\pi$
$421$	32.9612	1.60643	0.803216	+	0.595688i	$0.203121\pi$
$422$	−3.22748	−0.157111
$423$	0	0
$424$	26.2031	1.27253
$425$	23.8141	1.15516
$426$	0	0
$427$	0.576218	0.0278852
$428$	−1.67626	−0.0810249
$429$	0	0
$430$	−6.23083	−0.300477
$431$	−3.39015	−0.163298	−0.0816488	−	0.996661i	$-0.526019\pi$
$431$	−3.39015	−0.163298	−0.0816488	+	0.996661i	$0.526019\pi$
$432$	0	0
$433$	−19.7726	−0.950211	−0.475105	−	0.879929i	$-0.657590\pi$
$433$	−19.7726	−0.950211	−0.475105	+	0.879929i	$0.657590\pi$
$434$	−0.216696	−0.0104017
$435$	0	0
$436$	−2.44167	−0.116935
$437$	−5.02979	−0.240607
$438$	0	0
$439$	−17.6369	−0.841762	−0.420881	−	0.907116i	$-0.638279\pi$
$439$	−17.6369	−0.841762	−0.420881	+	0.907116i	$0.638279\pi$
$440$	32.6667	1.55732
$441$	0	0
$442$	2.39168	0.113760
$443$	17.5315	0.832947	0.416474	−	0.909148i	$-0.363266\pi$
$443$	17.5315	0.832947	0.416474	+	0.909148i	$0.363266\pi$
$444$	0	0
$445$	−52.5930	−2.49315
$446$	−12.7535	−0.603894
$447$	0	0
$448$	0.188957	0.00892740
$449$	−17.3047	−0.816660	−0.408330	−	0.912834i	$-0.633889\pi$
$449$	−17.3047	−0.816660	−0.408330	+	0.912834i	$0.633889\pi$
$450$	0	0
$451$	5.58747	0.263104
$452$	25.9462	1.22041
$453$	0	0
$454$	−13.1147	−0.615504
$455$	0.201886	0.00946458
$456$	0	0
$457$	35.6007	1.66533	0.832666	−	0.553776i	$-0.186814\pi$
$457$	35.6007	1.66533	0.832666	+	0.553776i	$0.186814\pi$
$458$	−14.2585	−0.666257
$459$	0	0
$460$	4.87850	0.227461
$461$	−28.9928	−1.35033	−0.675165	−	0.737667i	$-0.735927\pi$
$461$	−28.9928	−1.35033	−0.675165	+	0.737667i	$0.735927\pi$
$462$	0	0
$463$	14.8195	0.688720	0.344360	−	0.938838i	$-0.388096\pi$
$463$	14.8195	0.688720	0.344360	+	0.938838i	$0.388096\pi$
$464$	−0.756494	−0.0351194
$465$	0	0
$466$	−15.7143	−0.727952
$467$	−21.1430	−0.978382	−0.489191	−	0.872177i	$-0.662708\pi$
$467$	−21.1430	−0.978382	−0.489191	+	0.872177i	$0.662708\pi$
$468$	0	0
$469$	0.888192	0.0410129
$470$	14.4386	0.666004
$471$	0	0
$472$	−19.1391	−0.880946
$473$	8.20003	0.377038
$474$	0	0
$475$	−35.9902	−1.65134
$476$	−0.290845	−0.0133308
$477$	0	0
$478$	11.8453	0.541792
$479$	27.3454	1.24944	0.624721	−	0.780848i	$-0.285213\pi$
$479$	27.3454	1.24944	0.624721	+	0.780848i	$0.285213\pi$
$480$	0	0
$481$	1.47684	0.0673382
$482$	3.40529	0.155107
$483$	0	0
$484$	−2.30465	−0.104757
$485$	−41.0996	−1.86624
$486$	0	0
$487$	−14.2658	−0.646446	−0.323223	−	0.946323i	$-0.604766\pi$
$487$	−14.2658	−0.646446	−0.323223	+	0.946323i	$0.604766\pi$
$488$	24.3082	1.10038
$489$	0	0
$490$	−18.9052	−0.854049
$491$	35.2425	1.59047	0.795236	−	0.606300i	$-0.207347\pi$
$491$	35.2425	1.59047	0.795236	+	0.606300i	$0.207347\pi$
$492$	0	0
$493$	−3.32813	−0.149891
$494$	−3.61453	−0.162625
$495$	0	0
$496$	3.38655	0.152061
$497$	0.474700	0.0212932
$498$	0	0
$499$	1.01926	0.0456283	0.0228142	−	0.999740i	$-0.492737\pi$
$499$	1.01926	0.0456283	0.0228142	+	0.999740i	$0.492737\pi$
$500$	10.5152	0.470253
$501$	0	0
$502$	7.17259	0.320129
$503$	13.4878	0.601392	0.300696	−	0.953720i	$-0.402781\pi$
$503$	13.4878	0.601392	0.300696	+	0.953720i	$0.402781\pi$
$504$	0	0
$505$	−8.48488	−0.377572
$506$	2.75635	0.122535
$507$	0	0
$508$	1.83996	0.0816351
$509$	−5.55764	−0.246338	−0.123169	−	0.992386i	$-0.539306\pi$
$509$	−5.55764	−0.246338	−0.123169	+	0.992386i	$0.539306\pi$
$510$	0	0
$511$	0.406841	0.0179976
$512$	−8.41599	−0.371938
$513$	0	0
$514$	−0.446678	−0.0197021
$515$	−25.1179	−1.10683
$516$	0	0
$517$	−19.0018	−0.835699
$518$	0.0771029	0.00338771
$519$	0	0
$520$	8.51673	0.373483
$521$	−0.731714	−0.0320570	−0.0160285	−	0.999872i	$-0.505102\pi$
$521$	−0.731714	−0.0320570	−0.0160285	+	0.999872i	$0.505102\pi$
$522$	0	0
$523$	23.3248	1.01992	0.509960	−	0.860198i	$-0.329660\pi$
$523$	23.3248	1.01992	0.509960	+	0.860198i	$0.329660\pi$
$524$	−4.30814	−0.188202
$525$	0	0
$526$	6.33186	0.276082
$527$	14.8988	0.649003
$528$	0	0
$529$	1.00000	0.0434783
$530$	26.8752	1.16738
$531$	0	0
$532$	0.439552	0.0190570
$533$	1.45674	0.0630986
$534$	0	0
$535$	−4.17661	−0.180570
$536$	37.4691	1.61842
$537$	0	0
$538$	19.4977	0.840606
$539$	24.8800	1.07166
$540$	0	0
$541$	−27.8806	−1.19868	−0.599341	−	0.800494i	$-0.704571\pi$
$541$	−27.8806	−1.19868	−0.599341	+	0.800494i	$0.704571\pi$
$542$	−17.9307	−0.770189
$543$	0	0
$544$	−19.4884	−0.835559
$545$	−6.08373	−0.260598
$546$	0	0
$547$	11.0330	0.471736	0.235868	−	0.971785i	$-0.424207\pi$
$547$	11.0330	0.471736	0.235868	+	0.971785i	$0.424207\pi$
$548$	7.79202	0.332859
$549$	0	0
$550$	19.7228	0.840984
$551$	5.02979	0.214276
$552$	0	0
$553$	−0.976036	−0.0415053
$554$	−2.63488	−0.111946
$555$	0	0
$556$	17.6971	0.750523
$557$	18.1806	0.770338	0.385169	−	0.922846i	$-0.374143\pi$
$557$	18.1806	0.770338	0.385169	+	0.922846i	$0.374143\pi$
$558$	0	0
$559$	2.13788	0.0904228
$560$	−0.164721	−0.00696075
$561$	0	0
$562$	22.0229	0.928980
$563$	−6.60912	−0.278541	−0.139271	−	0.990254i	$-0.544476\pi$
$563$	−6.60912	−0.278541	−0.139271	+	0.990254i	$0.544476\pi$
$564$	0	0
$565$	64.6484	2.71978
$566$	5.96586	0.250764
$567$	0	0
$568$	20.0256	0.840254
$569$	32.1341	1.34713	0.673566	−	0.739127i	$-0.264762\pi$
$569$	32.1341	1.34713	0.673566	+	0.739127i	$0.264762\pi$
$570$	0	0
$571$	−32.8338	−1.37405	−0.687025	−	0.726633i	$-0.741084\pi$
$571$	−32.8338	−1.37405	−0.687025	+	0.726633i	$0.741084\pi$
$572$	−4.61379	−0.192912
$573$	0	0
$574$	0.0760536	0.00317442
$575$	7.15541	0.298401
$576$	0	0
$577$	−8.09372	−0.336946	−0.168473	−	0.985706i	$-0.553884\pi$
$577$	−8.09372	−0.336946	−0.168473	+	0.985706i	$0.553884\pi$
$578$	4.59116	0.190967
$579$	0	0
$580$	−4.87850	−0.202569
$581$	0.542473	0.0225056
$582$	0	0
$583$	−35.3689	−1.46483
$584$	17.1629	0.710205
$585$	0	0
$586$	8.60558	0.355493
$587$	−6.27695	−0.259078	−0.129539	−	0.991574i	$-0.541350\pi$
$587$	−6.27695	−0.259078	−0.129539	+	0.991574i	$0.541350\pi$
$588$	0	0
$589$	−22.5165	−0.927777
$590$	−19.6300	−0.808153
$591$	0	0
$592$	−1.20497	−0.0495241
$593$	12.9248	0.530758	0.265379	−	0.964144i	$-0.414503\pi$
$593$	12.9248	0.530758	0.265379	+	0.964144i	$0.414503\pi$
$594$	0	0
$595$	−0.724677	−0.0297089
$596$	25.8182	1.05756
$597$	0	0
$598$	0.718625	0.0293868
$599$	−13.1317	−0.536546	−0.268273	−	0.963343i	$-0.586453\pi$
$599$	−13.1317	−0.536546	−0.268273	+	0.963343i	$0.586453\pi$
$600$	0	0
$601$	2.15214	0.0877877	0.0438938	−	0.999036i	$-0.486024\pi$
$601$	2.15214	0.0877877	0.0438938	+	0.999036i	$0.486024\pi$
$602$	0.111615	0.00454907
$603$	0	0
$604$	11.3342	0.461182
$605$	−5.74233	−0.233459
$606$	0	0
$607$	−22.1876	−0.900566	−0.450283	−	0.892886i	$-0.648677\pi$
$607$	−22.1876	−0.900566	−0.450283	+	0.892886i	$0.648677\pi$
$608$	29.4527	1.19447
$609$	0	0
$610$	24.9317	1.00946
$611$	−4.95409	−0.200421
$612$	0	0
$613$	47.7378	1.92811	0.964057	−	0.265696i	$-0.0856019\pi$
$613$	47.7378	1.92811	0.964057	+	0.265696i	$0.0856019\pi$
$614$	16.6647	0.672532
$615$	0	0
$616$	−0.585166	−0.0235770
$617$	12.8265	0.516376	0.258188	−	0.966095i	$-0.416875\pi$
$617$	12.8265	0.516376	0.258188	+	0.966095i	$0.416875\pi$
$618$	0	0
$619$	−42.7985	−1.72022	−0.860108	−	0.510113i	$-0.829604\pi$
$619$	−42.7985	−1.72022	−0.860108	+	0.510113i	$0.829604\pi$
$620$	21.8393	0.877086
$621$	0	0
$622$	19.8919	0.797591
$623$	0.942113	0.0377450
$624$	0	0
$625$	−9.57713	−0.383085
$626$	−23.8706	−0.954063
$627$	0	0
$628$	−7.84953	−0.313230
$629$	−5.30117	−0.211371
$630$	0	0
$631$	−2.68116	−0.106735	−0.0533676	−	0.998575i	$-0.516996\pi$
$631$	−2.68116	−0.106735	−0.0533676	+	0.998575i	$0.516996\pi$
$632$	−41.1748	−1.63785
$633$	0	0
$634$	−2.45277	−0.0974121
$635$	4.58450	0.181931
$636$	0	0
$637$	6.48662	0.257009
$638$	−2.75635	−0.109125
$639$	0	0
$640$	−32.6553	−1.29081
$641$	−18.3001	−0.722812	−0.361406	−	0.932409i	$-0.617703\pi$
$641$	−18.3001	−0.722812	−0.361406	+	0.932409i	$0.617703\pi$
$642$	0	0
$643$	−44.1602	−1.74151	−0.870754	−	0.491719i	$-0.836369\pi$
$643$	−44.1602	−1.74151	−0.870754	+	0.491719i	$0.836369\pi$
$644$	−0.0873898	−0.00344364
$645$	0	0
$646$	12.9745	0.510474
$647$	46.8890	1.84340	0.921699	−	0.387906i	$-0.126801\pi$
$647$	46.8890	1.84340	0.921699	+	0.387906i	$0.126801\pi$
$648$	0	0
$649$	25.8339	1.01407
$650$	5.14206	0.201688
$651$	0	0
$652$	−2.62461	−0.102787
$653$	−42.8826	−1.67813	−0.839063	−	0.544035i	$-0.816896\pi$
$653$	−42.8826	−1.67813	−0.839063	+	0.544035i	$0.816896\pi$
$654$	0	0
$655$	−10.7343	−0.419424
$656$	−1.18857	−0.0464060
$657$	0	0
$658$	−0.258643	−0.0100829
$659$	4.08065	0.158960	0.0794799	−	0.996836i	$-0.474674\pi$
$659$	4.08065	0.158960	0.0794799	+	0.996836i	$0.474674\pi$
$660$	0	0
$661$	−31.6491	−1.23101	−0.615503	−	0.788134i	$-0.711047\pi$
$661$	−31.6491	−1.23101	−0.615503	+	0.788134i	$0.711047\pi$
$662$	3.67265	0.142742
$663$	0	0
$664$	22.8846	0.888097
$665$	1.09520	0.0424701
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−16.0034	−0.619191
$669$	0	0
$670$	38.4301	1.48469
$671$	−32.8112	−1.26666
$672$	0	0
$673$	2.89214	0.111484	0.0557418	−	0.998445i	$-0.482248\pi$
$673$	2.89214	0.111484	0.0557418	+	0.998445i	$0.482248\pi$
$674$	−24.7108	−0.951824
$675$	0	0
$676$	16.9876	0.653370
$677$	6.83199	0.262575	0.131287	−	0.991344i	$-0.458089\pi$
$677$	6.83199	0.262575	0.131287	+	0.991344i	$0.458089\pi$
$678$	0	0
$679$	0.736227	0.0282538
$680$	−30.5711	−1.17235
$681$	0	0
$682$	12.3392	0.472491
$683$	37.3148	1.42781	0.713906	−	0.700241i	$-0.246924\pi$
$683$	37.3148	1.42781	0.713906	+	0.700241i	$0.246924\pi$
$684$	0	0
$685$	19.4148	0.741803
$686$	0.677496	0.0258669
$687$	0	0
$688$	−1.74432	−0.0665017
$689$	−9.22124	−0.351301
$690$	0	0
$691$	−26.5406	−1.00965	−0.504826	−	0.863221i	$-0.668443\pi$
$691$	−26.5406	−1.00965	−0.504826	+	0.863221i	$0.668443\pi$
$692$	−0.353934	−0.0134546
$693$	0	0
$694$	−2.14082	−0.0812646
$695$	44.0945	1.67260
$696$	0	0
$697$	−5.22903	−0.198063
$698$	16.8377	0.637317
$699$	0	0
$700$	−0.625310	−0.0236345
$701$	15.9071	0.600804	0.300402	−	0.953813i	$-0.402879\pi$
$701$	15.9071	0.600804	0.300402	+	0.953813i	$0.402879\pi$
$702$	0	0
$703$	8.01163	0.302164
$704$	−10.7597	−0.405520
$705$	0	0
$706$	25.7210	0.968023
$707$	0.151992	0.00571624
$708$	0	0
$709$	44.3680	1.66628	0.833138	−	0.553065i	$-0.186542\pi$
$709$	44.3680	1.66628	0.833138	+	0.553065i	$0.186542\pi$
$710$	20.5392	0.770823
$711$	0	0
$712$	39.7438	1.48946
$713$	4.47664	0.167651
$714$	0	0
$715$	−11.4959	−0.429921
$716$	−9.45260	−0.353260
$717$	0	0
$718$	−7.33563	−0.273763
$719$	34.4101	1.28328	0.641641	−	0.767005i	$-0.278254\pi$
$719$	34.4101	1.28328	0.641641	+	0.767005i	$0.278254\pi$
$720$	0	0
$721$	0.449943	0.0167568
$722$	−4.88196	−0.181688
$723$	0	0
$724$	27.5736	1.02476
$725$	−7.15541	−0.265745
$726$	0	0
$727$	−8.54732	−0.317003	−0.158501	−	0.987359i	$-0.550666\pi$
$727$	−8.54732	−0.317003	−0.158501	+	0.987359i	$0.550666\pi$
$728$	−0.152562	−0.00565434
$729$	0	0
$730$	17.6031	0.651520
$731$	−7.67399	−0.283833
$732$	0	0
$733$	13.3991	0.494905	0.247453	−	0.968900i	$-0.420407\pi$
$733$	13.3991	0.494905	0.247453	+	0.968900i	$0.420407\pi$
$734$	−3.10109	−0.114463
$735$	0	0
$736$	−5.85566	−0.215843
$737$	−50.5757	−1.86298
$738$	0	0
$739$	−44.3998	−1.63327	−0.816637	−	0.577152i	$-0.804164\pi$
$739$	−44.3998	−1.63327	−0.816637	+	0.577152i	$0.804164\pi$
$740$	−7.77066	−0.285655
$741$	0	0
$742$	−0.481422	−0.0176736
$743$	35.2336	1.29259	0.646297	−	0.763086i	$-0.276317\pi$
$743$	35.2336	1.29259	0.646297	+	0.763086i	$0.276317\pi$
$744$	0	0
$745$	64.3294	2.35685
$746$	19.5814	0.716924
$747$	0	0
$748$	16.5614	0.605543
$749$	0.0748167	0.00273374
$750$	0	0
$751$	12.5784	0.458994	0.229497	−	0.973309i	$-0.426292\pi$
$751$	12.5784	0.458994	0.229497	+	0.973309i	$0.426292\pi$
$752$	4.04210	0.147400
$753$	0	0
$754$	−0.718625	−0.0261708
$755$	28.2406	1.02778
$756$	0	0
$757$	−37.9618	−1.37975	−0.689873	−	0.723931i	$-0.742334\pi$
$757$	−37.9618	−1.37975	−0.689873	+	0.723931i	$0.742334\pi$
$758$	1.93439	0.0702602
$759$	0	0
$760$	46.2019	1.67592
$761$	15.3541	0.556584	0.278292	−	0.960496i	$-0.410232\pi$
$761$	15.3541	0.556584	0.278292	+	0.960496i	$0.410232\pi$
$762$	0	0
$763$	0.108979	0.00394532
$764$	4.67968	0.169305
$765$	0	0
$766$	26.4417	0.955378
$767$	6.73531	0.243198
$768$	0	0
$769$	−7.50788	−0.270741	−0.135371	−	0.990795i	$-0.543222\pi$
$769$	−7.50788	−0.270741	−0.135371	+	0.990795i	$0.543222\pi$
$770$	−0.600176	−0.0216288
$771$	0	0
$772$	11.2274	0.404083
$773$	−24.2141	−0.870919	−0.435459	−	0.900208i	$-0.643414\pi$
$773$	−24.2141	−0.870919	−0.435459	+	0.900208i	$0.643414\pi$
$774$	0	0
$775$	32.0322	1.15063
$776$	31.0583	1.11493
$777$	0	0
$778$	−8.18575	−0.293473
$779$	7.90260	0.283140
$780$	0	0
$781$	−27.0305	−0.967226
$782$	−2.57953	−0.0922437
$783$	0	0
$784$	−5.29251	−0.189018
$785$	−19.5581	−0.698059
$786$	0	0
$787$	22.7577	0.811225	0.405613	−	0.914045i	$-0.367058\pi$
$787$	22.7577	0.811225	0.405613	+	0.914045i	$0.367058\pi$
$788$	38.1971	1.36072
$789$	0	0
$790$	−42.2309	−1.50251
$791$	−1.15806	−0.0411760
$792$	0	0
$793$	−8.55441	−0.303776
$794$	−13.5950	−0.482467
$795$	0	0
$796$	−17.0485	−0.604269
$797$	4.06083	0.143842	0.0719210	−	0.997410i	$-0.477087\pi$
$797$	4.06083	0.143842	0.0719210	+	0.997410i	$0.477087\pi$
$798$	0	0
$799$	17.7828	0.629112
$800$	−41.8997	−1.48138
$801$	0	0
$802$	17.2118	0.607771
$803$	−23.1664	−0.817525
$804$	0	0
$805$	−0.217743	−0.00767444
$806$	3.21702	0.113315
$807$	0	0
$808$	6.41189	0.225570
$809$	−48.2669	−1.69697	−0.848487	−	0.529217i	$-0.822486\pi$
$809$	−48.2669	−1.69697	−0.848487	+	0.529217i	$0.822486\pi$
$810$	0	0
$811$	−31.6174	−1.11024	−0.555118	−	0.831771i	$-0.687327\pi$
$811$	−31.6174	−1.11024	−0.555118	+	0.831771i	$0.687327\pi$
$812$	0.0873898	0.00306678
$813$	0	0
$814$	−4.39042	−0.153884
$815$	−6.53955	−0.229070
$816$	0	0
$817$	11.5977	0.405751
$818$	13.7880	0.482085
$819$	0	0
$820$	−7.66490	−0.267670
$821$	34.0753	1.18924	0.594618	−	0.804009i	$-0.297303\pi$
$821$	34.0753	1.18924	0.594618	+	0.804009i	$0.297303\pi$
$822$	0	0
$823$	48.4018	1.68718	0.843591	−	0.536987i	$-0.180438\pi$
$823$	48.4018	1.68718	0.843591	+	0.536987i	$0.180438\pi$
$824$	18.9812	0.661241
$825$	0	0
$826$	0.351637	0.0122350
$827$	3.72177	0.129419	0.0647094	−	0.997904i	$-0.479388\pi$
$827$	3.72177	0.129419	0.0647094	+	0.997904i	$0.479388\pi$
$828$	0	0
$829$	−23.5721	−0.818692	−0.409346	−	0.912379i	$-0.634243\pi$
$829$	−23.5721	−0.818692	−0.409346	+	0.912379i	$0.634243\pi$
$830$	23.4716	0.814713
$831$	0	0
$832$	−2.80522	−0.0972535
$833$	−23.2839	−0.806740
$834$	0	0
$835$	−39.8746	−1.37992
$836$	−25.0291	−0.865650
$837$	0	0
$838$	15.2395	0.526442
$839$	−45.6828	−1.57714	−0.788572	−	0.614942i	$-0.789179\pi$
$839$	−45.6828	−1.57714	−0.788572	+	0.614942i	$0.789179\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−25.5472	−0.880414
$843$	0	0
$844$	−5.82673	−0.200564
$845$	42.3268	1.45609
$846$	0	0
$847$	0.102864	0.00353444
$848$	7.52371	0.258365
$849$	0	0
$850$	−18.4576	−0.633090
$851$	−1.59284	−0.0546018
$852$	0	0
$853$	24.4605	0.837510	0.418755	−	0.908099i	$-0.362467\pi$
$853$	24.4605	0.837510	0.418755	+	0.908099i	$0.362467\pi$
$854$	−0.446609	−0.0152826
$855$	0	0
$856$	3.15620	0.107877
$857$	5.28505	0.180534	0.0902670	−	0.995918i	$-0.471228\pi$
$857$	5.28505	0.180534	0.0902670	+	0.995918i	$0.471228\pi$
$858$	0	0
$859$	−46.2825	−1.57914	−0.789570	−	0.613661i	$-0.789696\pi$
$859$	−46.2825	−1.57914	−0.789570	+	0.613661i	$0.789696\pi$
$860$	−11.2488	−0.383582
$861$	0	0
$862$	2.62759	0.0894962
$863$	−33.5105	−1.14071	−0.570355	−	0.821398i	$-0.693194\pi$
$863$	−33.5105	−1.14071	−0.570355	+	0.821398i	$0.693194\pi$
$864$	0	0
$865$	−0.881874	−0.0299846
$866$	15.3251	0.520769
$867$	0	0
$868$	−0.391212	−0.0132786
$869$	55.5777	1.88534
$870$	0	0
$871$	−13.1859	−0.446787
$872$	4.59738	0.155687
$873$	0	0
$874$	3.89843	0.131866
$875$	−0.469326	−0.0158661
$876$	0	0
$877$	20.1578	0.680680	0.340340	−	0.940302i	$-0.389458\pi$
$877$	20.1578	0.680680	0.340340	+	0.940302i	$0.389458\pi$
$878$	13.6698	0.461333
$879$	0	0
$880$	9.37961	0.316187
$881$	37.7877	1.27310	0.636549	−	0.771236i	$-0.280361\pi$
$881$	37.7877	1.27310	0.636549	+	0.771236i	$0.280361\pi$
$882$	0	0
$883$	−25.0568	−0.843228	−0.421614	−	0.906775i	$-0.638536\pi$
$883$	−25.0568	−0.843228	−0.421614	+	0.906775i	$0.638536\pi$
$884$	4.31782	0.145224
$885$	0	0
$886$	−13.5881	−0.456502
$887$	54.0378	1.81441	0.907206	−	0.420688i	$-0.138211\pi$
$887$	54.0378	1.81441	0.907206	+	0.420688i	$0.138211\pi$
$888$	0	0
$889$	−0.0821234	−0.00275433
$890$	40.7632	1.36639
$891$	0	0
$892$	−23.0245	−0.770916
$893$	−26.8751	−0.899342
$894$	0	0
$895$	−23.5524	−0.787269
$896$	0.584963	0.0195423
$897$	0	0
$898$	13.4123	0.447575
$899$	−4.47664	−0.149304
$900$	0	0
$901$	33.0999	1.10272
$902$	−4.33067	−0.144195
$903$	0	0
$904$	−48.8538	−1.62485
$905$	68.7031	2.28377
$906$	0	0
$907$	13.0995	0.434961	0.217481	−	0.976065i	$-0.430216\pi$
$907$	13.0995	0.434961	0.217481	+	0.976065i	$0.430216\pi$
$908$	−23.6767	−0.785738
$909$	0	0
$910$	−0.156476	−0.00518712
$911$	7.72405	0.255909	0.127955	−	0.991780i	$-0.459159\pi$
$911$	7.72405	0.255909	0.127955	+	0.991780i	$0.459159\pi$
$912$	0	0
$913$	−30.8897	−1.02230
$914$	−27.5930	−0.912695
$915$	0	0
$916$	−25.7416	−0.850528
$917$	0.192286	0.00634985
$918$	0	0
$919$	−9.57286	−0.315779	−0.157890	−	0.987457i	$-0.550469\pi$
$919$	−9.57286	−0.315779	−0.157890	+	0.987457i	$0.550469\pi$
$920$	−9.18566	−0.302842
$921$	0	0
$922$	22.4714	0.740056
$923$	−7.04728	−0.231964
$924$	0	0
$925$	−11.3974	−0.374745
$926$	−11.4861	−0.377457
$927$	0	0
$928$	5.85566	0.192222
$929$	−16.9036	−0.554590	−0.277295	−	0.960785i	$-0.589438\pi$
$929$	−16.9036	−0.554590	−0.277295	+	0.960785i	$0.589438\pi$
$930$	0	0
$931$	35.1889	1.15327
$932$	−28.3699	−0.929286
$933$	0	0
$934$	16.3873	0.536208
$935$	41.2648	1.34950
$936$	0	0
$937$	−48.2978	−1.57782	−0.788910	−	0.614509i	$-0.789354\pi$
$937$	−48.2978	−1.57782	−0.788910	+	0.614509i	$0.789354\pi$
$938$	−0.688409	−0.0224774
$939$	0	0
$940$	26.0668	0.850204
$941$	−18.2235	−0.594070	−0.297035	−	0.954867i	$-0.595998\pi$
$941$	−18.2235	−0.594070	−0.297035	+	0.954867i	$0.595998\pi$
$942$	0	0
$943$	−1.57116	−0.0511640
$944$	−5.49542	−0.178861
$945$	0	0
$946$	−6.35558	−0.206638
$947$	22.0443	0.716343	0.358172	−	0.933656i	$-0.383400\pi$
$947$	22.0443	0.716343	0.358172	+	0.933656i	$0.383400\pi$
$948$	0	0
$949$	−6.03986	−0.196062
$950$	27.8949	0.905029
$951$	0	0
$952$	0.547628	0.0177487
$953$	12.7950	0.414470	0.207235	−	0.978291i	$-0.433554\pi$
$953$	12.7950	0.414470	0.207235	+	0.978291i	$0.433554\pi$
$954$	0	0
$955$	11.6600	0.377310
$956$	21.3850	0.691639
$957$	0	0
$958$	−21.1945	−0.684764
$959$	−0.347783	−0.0112305
$960$	0	0
$961$	−10.9597	−0.353540
$962$	−1.14465	−0.0369051
$963$	0	0
$964$	6.14775	0.198006
$965$	27.9745	0.900532
$966$	0	0
$967$	−1.59257	−0.0512136	−0.0256068	−	0.999672i	$-0.508152\pi$
$967$	−1.59257	−0.0512136	−0.0256068	+	0.999672i	$0.508152\pi$
$968$	4.33939	0.139473
$969$	0	0
$970$	31.8550	1.02280
$971$	−1.41506	−0.0454115	−0.0227057	−	0.999742i	$-0.507228\pi$
$971$	−1.41506	−0.0454115	−0.0227057	+	0.999742i	$0.507228\pi$
$972$	0	0
$973$	−0.789877	−0.0253223
$974$	11.0570	0.354288
$975$	0	0
$976$	6.97964	0.223413
$977$	52.4116	1.67680	0.838398	−	0.545058i	$-0.183492\pi$
$977$	52.4116	1.67680	0.838398	+	0.545058i	$0.183492\pi$
$978$	0	0
$979$	−53.6461	−1.71454
$980$	−34.1305	−1.09026
$981$	0	0
$982$	−27.3154	−0.871668
$983$	−26.8515	−0.856430	−0.428215	−	0.903677i	$-0.640857\pi$
$983$	−26.8515	−0.856430	−0.428215	+	0.903677i	$0.640857\pi$
$984$	0	0
$985$	95.1731	3.03247
$986$	2.57953	0.0821489
$987$	0	0
$988$	−6.52550	−0.207604
$989$	−2.30580	−0.0733201
$990$	0	0
$991$	57.7403	1.83418	0.917090	−	0.398679i	$-0.130531\pi$
$991$	57.7403	1.83418	0.917090	+	0.398679i	$0.130531\pi$
$992$	−26.2137	−0.832285
$993$	0	0
$994$	−0.367924	−0.0116699
$995$	−42.4786	−1.34666
$996$	0	0
$997$	−1.93519	−0.0612882	−0.0306441	−	0.999530i	$-0.509756\pi$
$997$	−1.93519	−0.0612882	−0.0306441	+	0.999530i	$0.509756\pi$
$998$	−0.789995	−0.0250069
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.o.1.7		13
3.2	odd	2		667.2.a.c.1.7	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.c.1.7	✓	13	3.2	odd	2
6003.2.a.o.1.7		13	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)

\(f(q)\)	\(=\)	\(q-0.775068 q^{2} -1.39927 q^{4} -3.48646 q^{5} +0.0624539 q^{7} +2.63467 q^{8} +O(q^{10})\)	\(q-0.775068 q^{2} -1.39927 q^{4} -3.48646 q^{5} +0.0624539 q^{7} +2.63467 q^{8} +2.70224 q^{10} -3.55627 q^{11} -0.927177 q^{13} -0.0484060 q^{14} +0.756494 q^{16} +3.32813 q^{17} -5.02979 q^{19} +4.87850 q^{20} +2.75635 q^{22} +1.00000 q^{23} +7.15541 q^{25} +0.718625 q^{26} -0.0873898 q^{28} -1.00000 q^{29} +4.47664 q^{31} -5.85566 q^{32} -2.57953 q^{34} -0.217743 q^{35} -1.59284 q^{37} +3.89843 q^{38} -9.18566 q^{40} -1.57116 q^{41} -2.30580 q^{43} +4.97618 q^{44} -0.775068 q^{46} +5.34319 q^{47} -6.99610 q^{49} -5.54593 q^{50} +1.29737 q^{52} +9.94550 q^{53} +12.3988 q^{55} +0.164545 q^{56} +0.775068 q^{58} -7.26432 q^{59} +9.22630 q^{61} -3.46970 q^{62} +3.02555 q^{64} +3.23257 q^{65} +14.2216 q^{67} -4.65695 q^{68} +0.168766 q^{70} +7.60080 q^{71} +6.51425 q^{73} +1.23456 q^{74} +7.03803 q^{76} -0.222103 q^{77} -15.6281 q^{79} -2.63749 q^{80} +1.21776 q^{82} +8.68598 q^{83} -11.6034 q^{85} +1.78715 q^{86} -9.36957 q^{88} +15.0849 q^{89} -0.0579058 q^{91} -1.39927 q^{92} -4.14134 q^{94} +17.5362 q^{95} +11.7883 q^{97} +5.42245 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8} + 10 q^{10} - 10 q^{11} + 7 q^{13} + 12 q^{14} + 2 q^{16} - 26 q^{17} - 25 q^{20} - 15 q^{22} + 13 q^{23} + 19 q^{25} + 15 q^{26} + 5 q^{28} - 13 q^{29} - 6 q^{31} - 16 q^{32} + 11 q^{34} - q^{35} + 15 q^{37} - 8 q^{38} + 14 q^{40} - 9 q^{41} + q^{43} - 29 q^{44} - 4 q^{46} - 15 q^{47} + 4 q^{49} - 31 q^{50} - 8 q^{52} - 43 q^{53} - 3 q^{55} + 5 q^{56} + 4 q^{58} + 9 q^{59} + 20 q^{61} - 11 q^{62} - 16 q^{64} + 25 q^{65} + q^{67} - 21 q^{68} - 2 q^{70} - 17 q^{71} + 26 q^{73} - 11 q^{74} + 8 q^{76} - 17 q^{77} + 5 q^{79} - 10 q^{80} - 25 q^{82} - 4 q^{83} + 20 q^{85} + 13 q^{86} - 32 q^{88} - 48 q^{89} - 9 q^{91} + 12 q^{92} - 65 q^{94} - 8 q^{95} + 30 q^{97} - 2 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 16 * q^5 + q^7 - 6 * q^8 + 10 * q^10 - 10 * q^11 + 7 * q^13 + 12 * q^14 + 2 * q^16 - 26 * q^17 - 25 * q^20 - 15 * q^22 + 13 * q^23 + 19 * q^25 + 15 * q^26 + 5 * q^28 - 13 * q^29 - 6 * q^31 - 16 * q^32 + 11 * q^34 - q^35 + 15 * q^37 - 8 * q^38 + 14 * q^40 - 9 * q^41 + q^43 - 29 * q^44 - 4 * q^46 - 15 * q^47 + 4 * q^49 - 31 * q^50 - 8 * q^52 - 43 * q^53 - 3 * q^55 + 5 * q^56 + 4 * q^58 + 9 * q^59 + 20 * q^61 - 11 * q^62 - 16 * q^64 + 25 * q^65 + q^67 - 21 * q^68 - 2 * q^70 - 17 * q^71 + 26 * q^73 - 11 * q^74 + 8 * q^76 - 17 * q^77 + 5 * q^79 - 10 * q^80 - 25 * q^82 - 4 * q^83 + 20 * q^85 + 13 * q^86 - 32 * q^88 - 48 * q^89 - 9 * q^91 + 12 * q^92 - 65 * q^94 - 8 * q^95 + 30 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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