LMFDB - Embedded newform 6003.2.a.o.1.4

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			1.4
Root			$1.46878$ 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)

$f(q)$	$=$	$q-1.46878 q^{2} +0.157302 q^{4} -4.31834 q^{5} +1.97692 q^{7} +2.70651 q^{8} +O(q^{10})$	\(q-1.46878 q^{2} +0.157302 q^{4} -4.31834 q^{5} +1.97692 q^{7} +2.70651 q^{8} +6.34267 q^{10} -1.83320 q^{11} -3.87390 q^{13} -2.90366 q^{14} -4.28986 q^{16} -4.20465 q^{17} +5.07651 q^{19} -0.679284 q^{20} +2.69255 q^{22} +1.00000 q^{23} +13.6481 q^{25} +5.68989 q^{26} +0.310974 q^{28} -1.00000 q^{29} -5.47778 q^{31} +0.887822 q^{32} +6.17568 q^{34} -8.53703 q^{35} +9.06579 q^{37} -7.45626 q^{38} -11.6876 q^{40} -10.5865 q^{41} +4.73958 q^{43} -0.288365 q^{44} -1.46878 q^{46} +6.05915 q^{47} -3.09177 q^{49} -20.0459 q^{50} -0.609372 q^{52} -2.15287 q^{53} +7.91636 q^{55} +5.35057 q^{56} +1.46878 q^{58} +13.0731 q^{59} -11.1000 q^{61} +8.04563 q^{62} +7.27571 q^{64} +16.7288 q^{65} -9.49675 q^{67} -0.661399 q^{68} +12.5390 q^{70} +8.07109 q^{71} +16.6448 q^{73} -13.3156 q^{74} +0.798545 q^{76} -3.62409 q^{77} -2.02071 q^{79} +18.5251 q^{80} +15.5491 q^{82} +6.08722 q^{83} +18.1571 q^{85} -6.96138 q^{86} -4.96156 q^{88} +2.99601 q^{89} -7.65840 q^{91} +0.157302 q^{92} -8.89953 q^{94} -21.9221 q^{95} -2.79853 q^{97} +4.54112 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.46878	−1.03858	−0.519291	−	0.854598i	$-0.673804\pi$
$2$	−1.46878	−1.03858	−0.519291	+	0.854598i	$0.673804\pi$
$3$	0	0
$3$	0	0
$4$	0.157302	0.0786510
$5$	−4.31834	−1.93122	−0.965610	−	0.259994i	$-0.916279\pi$
$5$	−4.31834	−1.93122	−0.965610	+	0.259994i	$0.916279\pi$
$6$	0	0
$7$	1.97692	0.747207	0.373604	−	0.927588i	$-0.378122\pi$
$7$	1.97692	0.747207	0.373604	+	0.927588i	$0.378122\pi$
$8$	2.70651	0.956896
$9$	0	0
$10$	6.34267	2.00573
$11$	−1.83320	−0.552729	−0.276365	−	0.961053i	$-0.589130\pi$
$11$	−1.83320	−0.552729	−0.276365	+	0.961053i	$0.589130\pi$
$12$	0	0
$13$	−3.87390	−1.07443	−0.537213	−	0.843447i	$-0.680523\pi$
$13$	−3.87390	−1.07443	−0.537213	+	0.843447i	$0.680523\pi$
$14$	−2.90366	−0.776035
$15$	0	0
$16$	−4.28986	−1.07247
$17$	−4.20465	−1.01978	−0.509888	−	0.860241i	$-0.670313\pi$
$17$	−4.20465	−1.01978	−0.509888	+	0.860241i	$0.670313\pi$
$18$	0	0
$19$	5.07651	1.16463	0.582316	−	0.812963i	$-0.302147\pi$
$19$	5.07651	1.16463	0.582316	+	0.812963i	$0.302147\pi$
$20$	−0.679284	−0.151892
$21$	0	0
$22$	2.69255	0.574054
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	13.6481	2.72961
$26$	5.68989	1.11588
$27$	0	0
$28$	0.310974	0.0587686
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−5.47778	−0.983838	−0.491919	−	0.870641i	$-0.663704\pi$
$31$	−5.47778	−0.983838	−0.491919	+	0.870641i	$0.663704\pi$
$32$	0.887822	0.156946
$33$	0	0
$34$	6.17568	1.05912
$35$	−8.53703	−1.44302
$36$	0	0
$37$	9.06579	1.49041	0.745203	−	0.666838i	$-0.232353\pi$
$37$	9.06579	1.49041	0.745203	+	0.666838i	$0.232353\pi$
$38$	−7.45626	−1.20956
$39$	0	0
$40$	−11.6876	−1.84798
$41$	−10.5865	−1.65333	−0.826664	−	0.562696i	$-0.809764\pi$
$41$	−10.5865	−1.65333	−0.826664	+	0.562696i	$0.809764\pi$
$42$	0	0
$43$	4.73958	0.722779	0.361389	−	0.932415i	$-0.382303\pi$
$43$	4.73958	0.722779	0.361389	+	0.932415i	$0.382303\pi$
$44$	−0.288365	−0.0434727
$45$	0	0
$46$	−1.46878	−0.216559
$47$	6.05915	0.883817	0.441909	−	0.897060i	$-0.354302\pi$
$47$	6.05915	0.883817	0.441909	+	0.897060i	$0.354302\pi$
$48$	0	0
$49$	−3.09177	−0.441681
$50$	−20.0459	−2.83492
$51$	0	0
$52$	−0.609372	−0.0845047
$53$	−2.15287	−0.295719	−0.147859	−	0.989008i	$-0.547238\pi$
$53$	−2.15287	−0.295719	−0.147859	+	0.989008i	$0.547238\pi$
$54$	0	0
$55$	7.91636	1.06744
$56$	5.35057	0.714999
$57$	0	0
$58$	1.46878	0.192860
$59$	13.0731	1.70198	0.850988	−	0.525185i	$-0.176004\pi$
$59$	13.0731	1.70198	0.850988	+	0.525185i	$0.176004\pi$
$60$	0	0
$61$	−11.1000	−1.42121	−0.710607	−	0.703590i	$-0.751579\pi$
$61$	−11.1000	−1.42121	−0.710607	+	0.703590i	$0.751579\pi$
$62$	8.04563	1.02180
$63$	0	0
$64$	7.27571	0.909464
$65$	16.7288	2.07495
$66$	0	0
$67$	−9.49675	−1.16021	−0.580106	−	0.814541i	$-0.696989\pi$
$67$	−9.49675	−1.16021	−0.580106	+	0.814541i	$0.696989\pi$
$68$	−0.661399	−0.0802064
$69$	0	0
$70$	12.5390	1.49870
$71$	8.07109	0.957863	0.478931	−	0.877852i	$-0.341024\pi$
$71$	8.07109	0.957863	0.478931	+	0.877852i	$0.341024\pi$
$72$	0	0
$73$	16.6448	1.94812	0.974062	−	0.226279i	$-0.0726563\pi$
$73$	16.6448	1.94812	0.974062	+	0.226279i	$0.0726563\pi$
$74$	−13.3156	−1.54791
$75$	0	0
$76$	0.798545	0.0915994
$77$	−3.62409	−0.413003
$78$	0	0
$79$	−2.02071	−0.227347	−0.113674	−	0.993518i	$-0.536262\pi$
$79$	−2.02071	−0.227347	−0.113674	+	0.993518i	$0.536262\pi$
$80$	18.5251	2.07117
$81$	0	0
$82$	15.5491	1.71712
$83$	6.08722	0.668160	0.334080	−	0.942545i	$-0.391574\pi$
$83$	6.08722	0.668160	0.334080	+	0.942545i	$0.391574\pi$
$84$	0	0
$85$	18.1571	1.96941
$86$	−6.96138	−0.750664
$87$	0	0
$88$	−4.96156	−0.528904
$89$	2.99601	0.317576	0.158788	−	0.987313i	$-0.449241\pi$
$89$	2.99601	0.317576	0.158788	+	0.987313i	$0.449241\pi$
$90$	0	0
$91$	−7.65840	−0.802819
$92$	0.157302	0.0163999
$93$	0	0
$94$	−8.89953	−0.917916
$95$	−21.9221	−2.24916
$96$	0	0
$97$	−2.79853	−0.284148	−0.142074	−	0.989856i	$-0.545377\pi$
$97$	−2.79853	−0.284148	−0.142074	+	0.989856i	$0.545377\pi$
$98$	4.54112	0.458722
$99$	0	0
$100$	2.14687	0.214687
$101$	−1.46716	−0.145988	−0.0729939	−	0.997332i	$-0.523255\pi$
$101$	−1.46716	−0.145988	−0.0729939	+	0.997332i	$0.523255\pi$
$102$	0	0
$103$	−2.23140	−0.219866	−0.109933	−	0.993939i	$-0.535064\pi$
$103$	−2.23140	−0.219866	−0.109933	+	0.993939i	$0.535064\pi$
$104$	−10.4847	−1.02811
$105$	0	0
$106$	3.16208	0.307128
$107$	−6.98440	−0.675208	−0.337604	−	0.941288i	$-0.609616\pi$
$107$	−6.98440	−0.675208	−0.337604	+	0.941288i	$0.609616\pi$
$108$	0	0
$109$	−0.478672	−0.0458485	−0.0229242	−	0.999737i	$-0.507298\pi$
$109$	−0.478672	−0.0458485	−0.0229242	+	0.999737i	$0.507298\pi$
$110$	−11.6274	−1.10863
$111$	0	0
$112$	−8.48073	−0.801354
$113$	19.1178	1.79845	0.899227	−	0.437482i	$-0.144130\pi$
$113$	19.1178	1.79845	0.899227	+	0.437482i	$0.144130\pi$
$114$	0	0
$115$	−4.31834	−0.402687
$116$	−0.157302	−0.0146051
$117$	0	0
$118$	−19.2015	−1.76764
$119$	−8.31227	−0.761984
$120$	0	0
$121$	−7.63939	−0.694490
$122$	16.3035	1.47605
$123$	0	0
$124$	−0.861665	−0.0773798
$125$	−37.3453	−3.34026
$126$	0	0
$127$	7.04182	0.624860	0.312430	−	0.949941i	$-0.398857\pi$
$127$	7.04182	0.624860	0.312430	+	0.949941i	$0.398857\pi$
$128$	−12.4620	−1.10150
$129$	0	0
$130$	−24.5709	−2.15501
$131$	10.0499	0.878062	0.439031	−	0.898472i	$-0.355322\pi$
$131$	10.0499	0.878062	0.439031	+	0.898472i	$0.355322\pi$
$132$	0	0
$133$	10.0359	0.870221
$134$	13.9486	1.20498
$135$	0	0
$136$	−11.3799	−0.975820
$137$	5.22983	0.446814	0.223407	−	0.974725i	$-0.428282\pi$
$137$	5.22983	0.446814	0.223407	+	0.974725i	$0.428282\pi$
$138$	0	0
$139$	6.50655	0.551879	0.275939	−	0.961175i	$-0.411011\pi$
$139$	6.50655	0.551879	0.275939	+	0.961175i	$0.411011\pi$
$140$	−1.34289	−0.113495
$141$	0	0
$142$	−11.8546	−0.994818
$143$	7.10161	0.593867
$144$	0	0
$145$	4.31834	0.358619
$146$	−24.4475	−2.02329
$147$	0	0
$148$	1.42607	0.117222
$149$	12.7045	1.04080	0.520398	−	0.853924i	$-0.325784\pi$
$149$	12.7045	1.04080	0.520398	+	0.853924i	$0.325784\pi$
$150$	0	0
$151$	−8.10479	−0.659558	−0.329779	−	0.944058i	$-0.606974\pi$
$151$	−8.10479	−0.659558	−0.329779	+	0.944058i	$0.606974\pi$
$152$	13.7396	1.11443
$153$	0	0
$154$	5.32297	0.428938
$155$	23.6549	1.90001
$156$	0	0
$157$	2.66910	0.213017	0.106509	−	0.994312i	$-0.466033\pi$
$157$	2.66910	0.213017	0.106509	+	0.994312i	$0.466033\pi$
$158$	2.96796	0.236119
$159$	0	0
$160$	−3.83392	−0.303098
$161$	1.97692	0.155803
$162$	0	0
$163$	−0.170762	−0.0133751	−0.00668755	−	0.999978i	$-0.502129\pi$
$163$	−0.170762	−0.0133751	−0.00668755	+	0.999978i	$0.502129\pi$
$164$	−1.66527	−0.130036
$165$	0	0
$166$	−8.94077	−0.693938
$167$	−7.64267	−0.591407	−0.295704	−	0.955280i	$-0.595554\pi$
$167$	−7.64267	−0.591407	−0.295704	+	0.955280i	$0.595554\pi$
$168$	0	0
$169$	2.00708	0.154391
$170$	−26.6687	−2.04540
$171$	0	0
$172$	0.745545	0.0568473
$173$	9.05474	0.688419	0.344209	−	0.938893i	$-0.388147\pi$
$173$	9.05474	0.688419	0.344209	+	0.938893i	$0.388147\pi$
$174$	0	0
$175$	26.9812	2.03959
$176$	7.86415	0.592783
$177$	0	0
$178$	−4.40047	−0.329829
$179$	13.8782	1.03730	0.518651	−	0.854986i	$-0.326434\pi$
$179$	13.8782	1.03730	0.518651	+	0.854986i	$0.326434\pi$
$180$	0	0
$181$	18.4738	1.37314	0.686572	−	0.727061i	$-0.259114\pi$
$181$	18.4738	1.37314	0.686572	+	0.727061i	$0.259114\pi$
$182$	11.2485	0.833792
$183$	0	0
$184$	2.70651	0.199527
$185$	−39.1491	−2.87830
$186$	0	0
$187$	7.70794	0.563660
$188$	0.953116	0.0695131
$189$	0	0
$190$	32.1986	2.33594
$191$	19.2748	1.39467	0.697337	−	0.716743i	$-0.254368\pi$
$191$	19.2748	1.39467	0.697337	+	0.716743i	$0.254368\pi$
$192$	0	0
$193$	−2.64644	−0.190495	−0.0952474	−	0.995454i	$-0.530364\pi$
$193$	−2.64644	−0.190495	−0.0952474	+	0.995454i	$0.530364\pi$
$194$	4.11041	0.295110
$195$	0	0
$196$	−0.486342	−0.0347387
$197$	−19.0630	−1.35818	−0.679091	−	0.734054i	$-0.737626\pi$
$197$	−19.0630	−1.35818	−0.679091	+	0.734054i	$0.737626\pi$
$198$	0	0
$199$	11.2745	0.799225	0.399613	−	0.916684i	$-0.369145\pi$
$199$	11.2745	0.799225	0.399613	+	0.916684i	$0.369145\pi$
$200$	36.9386	2.61195
$201$	0	0
$202$	2.15493	0.151620
$203$	−1.97692	−0.138753
$204$	0	0
$205$	45.7160	3.19294
$206$	3.27743	0.228349
$207$	0	0
$208$	16.6185	1.15228
$209$	−9.30624	−0.643726
$210$	0	0
$211$	−6.25288	−0.430466	−0.215233	−	0.976563i	$-0.569051\pi$
$211$	−6.25288	−0.430466	−0.215233	+	0.976563i	$0.569051\pi$
$212$	−0.338650	−0.0232586
$213$	0	0
$214$	10.2585	0.701258
$215$	−20.4671	−1.39585
$216$	0	0
$217$	−10.8292	−0.735131
$218$	0.703062	0.0476174
$219$	0	0
$220$	1.24526	0.0839554
$221$	16.2884	1.09567
$222$	0	0
$223$	−18.9022	−1.26578	−0.632891	−	0.774241i	$-0.718132\pi$
$223$	−18.9022	−1.26578	−0.632891	+	0.774241i	$0.718132\pi$
$224$	1.75516	0.117271
$225$	0	0
$226$	−28.0798	−1.86784
$227$	−10.8786	−0.722036	−0.361018	−	0.932559i	$-0.617571\pi$
$227$	−10.8786	−0.722036	−0.361018	+	0.932559i	$0.617571\pi$
$228$	0	0
$229$	−17.1411	−1.13271	−0.566357	−	0.824160i	$-0.691648\pi$
$229$	−17.1411	−1.13271	−0.566357	+	0.824160i	$0.691648\pi$
$230$	6.34267	0.418223
$231$	0	0
$232$	−2.70651	−0.177691
$233$	4.32390	0.283268	0.141634	−	0.989919i	$-0.454764\pi$
$233$	4.32390	0.283268	0.141634	+	0.989919i	$0.454764\pi$
$234$	0	0
$235$	−26.1654	−1.70685
$236$	2.05643	0.133862
$237$	0	0
$238$	12.2089	0.791383
$239$	−8.89820	−0.575577	−0.287788	−	0.957694i	$-0.592920\pi$
$239$	−8.89820	−0.575577	−0.287788	+	0.957694i	$0.592920\pi$
$240$	0	0
$241$	−4.59323	−0.295876	−0.147938	−	0.988997i	$-0.547264\pi$
$241$	−4.59323	−0.295876	−0.147938	+	0.988997i	$0.547264\pi$
$242$	11.2206	0.721285
$243$	0	0
$244$	−1.74606	−0.111780
$245$	13.3513	0.852984
$246$	0	0
$247$	−19.6659	−1.25131
$248$	−14.8257	−0.941430
$249$	0	0
$250$	54.8518	3.46913
$251$	−19.9458	−1.25897	−0.629484	−	0.777013i	$-0.716734\pi$
$251$	−19.9458	−1.25897	−0.629484	+	0.777013i	$0.716734\pi$
$252$	0	0
$253$	−1.83320	−0.115252
$254$	−10.3429	−0.648968
$255$	0	0
$256$	3.75251	0.234532
$257$	23.1679	1.44518	0.722588	−	0.691279i	$-0.242952\pi$
$257$	23.1679	1.44518	0.722588	+	0.691279i	$0.242952\pi$
$258$	0	0
$259$	17.9224	1.11364
$260$	2.63147	0.163197
$261$	0	0
$262$	−14.7610	−0.911939
$263$	−0.500370	−0.0308542	−0.0154271	−	0.999881i	$-0.504911\pi$
$263$	−0.500370	−0.0308542	−0.0154271	+	0.999881i	$0.504911\pi$
$264$	0	0
$265$	9.29681	0.571098
$266$	−14.7405	−0.903795
$267$	0	0
$268$	−1.49386	−0.0912519
$269$	−6.17017	−0.376202	−0.188101	−	0.982150i	$-0.560233\pi$
$269$	−6.17017	−0.376202	−0.188101	+	0.982150i	$0.560233\pi$
$270$	0	0
$271$	16.3328	0.992146	0.496073	−	0.868281i	$-0.334775\pi$
$271$	16.3328	0.992146	0.496073	+	0.868281i	$0.334775\pi$
$272$	18.0373	1.09367
$273$	0	0
$274$	−7.68144	−0.464053
$275$	−25.0196	−1.50874
$276$	0	0
$277$	−1.81971	−0.109336	−0.0546679	−	0.998505i	$-0.517410\pi$
$277$	−1.81971	−0.109336	−0.0546679	+	0.998505i	$0.517410\pi$
$278$	−9.55667	−0.573171
$279$	0	0
$280$	−23.1056	−1.38082
$281$	−1.22881	−0.0733047	−0.0366523	−	0.999328i	$-0.511669\pi$
$281$	−1.22881	−0.0733047	−0.0366523	+	0.999328i	$0.511669\pi$
$282$	0	0
$283$	17.8740	1.06250	0.531249	−	0.847216i	$-0.321723\pi$
$283$	17.8740	1.06250	0.531249	+	0.847216i	$0.321723\pi$
$284$	1.26960	0.0753369
$285$	0	0
$286$	−10.4307	−0.616779
$287$	−20.9286	−1.23538
$288$	0	0
$289$	0.679044	0.0399438
$290$	−6.34267	−0.372455
$291$	0	0
$292$	2.61826	0.153222
$293$	−21.6627	−1.26555	−0.632775	−	0.774335i	$-0.718084\pi$
$293$	−21.6627	−1.26555	−0.632775	+	0.774335i	$0.718084\pi$
$294$	0	0
$295$	−56.4542	−3.28689
$296$	24.5366	1.42616
$297$	0	0
$298$	−18.6601	−1.08095
$299$	−3.87390	−0.224033
$300$	0	0
$301$	9.36979	0.540066
$302$	11.9041	0.685005
$303$	0	0
$304$	−21.7775	−1.24903
$305$	47.9337	2.74468
$306$	0	0
$307$	−19.9586	−1.13910	−0.569548	−	0.821958i	$-0.692882\pi$
$307$	−19.9586	−1.13910	−0.569548	+	0.821958i	$0.692882\pi$
$308$	−0.570077	−0.0324831
$309$	0	0
$310$	−34.7438	−1.97331
$311$	−30.1721	−1.71090	−0.855452	−	0.517882i	$-0.826721\pi$
$311$	−30.1721	−1.71090	−0.855452	+	0.517882i	$0.826721\pi$
$312$	0	0
$313$	7.10502	0.401600	0.200800	−	0.979632i	$-0.435646\pi$
$313$	7.10502	0.401600	0.200800	+	0.979632i	$0.435646\pi$
$314$	−3.92031	−0.221236
$315$	0	0
$316$	−0.317861	−0.0178811
$317$	−21.9769	−1.23435	−0.617173	−	0.786828i	$-0.711722\pi$
$317$	−21.9769	−1.23435	−0.617173	+	0.786828i	$0.711722\pi$
$318$	0	0
$319$	1.83320	0.102639
$320$	−31.4190	−1.75637
$321$	0	0
$322$	−2.90366	−0.161815
$323$	−21.3449	−1.18766
$324$	0	0
$325$	−52.8712	−2.93277
$326$	0.250811	0.0138911
$327$	0	0
$328$	−28.6524	−1.58206
$329$	11.9785	0.660395
$330$	0	0
$331$	−27.6093	−1.51754	−0.758771	−	0.651357i	$-0.774200\pi$
$331$	−27.6093	−1.51754	−0.758771	+	0.651357i	$0.774200\pi$
$332$	0.957533	0.0525514
$333$	0	0
$334$	11.2254	0.614224
$335$	41.0102	2.24063
$336$	0	0
$337$	4.74384	0.258414	0.129207	−	0.991618i	$-0.458757\pi$
$337$	4.74384	0.258414	0.129207	+	0.991618i	$0.458757\pi$
$338$	−2.94795	−0.160347
$339$	0	0
$340$	2.85615	0.154896
$341$	10.0418	0.543796
$342$	0	0
$343$	−19.9507	−1.07723
$344$	12.8277	0.691624
$345$	0	0
$346$	−13.2994	−0.714979
$347$	1.82991	0.0982348	0.0491174	−	0.998793i	$-0.484359\pi$
$347$	1.82991	0.0982348	0.0491174	+	0.998793i	$0.484359\pi$
$348$	0	0
$349$	−4.01236	−0.214777	−0.107388	−	0.994217i	$-0.534249\pi$
$349$	−4.01236	−0.214777	−0.107388	+	0.994217i	$0.534249\pi$
$350$	−39.6293	−2.11828
$351$	0	0
$352$	−1.62755	−0.0867488
$353$	29.1168	1.54973	0.774865	−	0.632126i	$-0.217818\pi$
$353$	29.1168	1.54973	0.774865	+	0.632126i	$0.217818\pi$
$354$	0	0
$355$	−34.8537	−1.84984
$356$	0.471278	0.0249777
$357$	0	0
$358$	−20.3839	−1.07732
$359$	8.79160	0.464003	0.232002	−	0.972715i	$-0.425473\pi$
$359$	8.79160	0.464003	0.232002	+	0.972715i	$0.425473\pi$
$360$	0	0
$361$	6.77096	0.356366
$362$	−27.1338	−1.42612
$363$	0	0
$364$	−1.20468	−0.0631425
$365$	−71.8778	−3.76226
$366$	0	0
$367$	10.8159	0.564583	0.282292	−	0.959329i	$-0.408905\pi$
$367$	10.8159	0.564583	0.282292	+	0.959329i	$0.408905\pi$
$368$	−4.28986	−0.223624
$369$	0	0
$370$	57.5013	2.98935
$371$	−4.25605	−0.220963
$372$	0	0
$373$	−0.0211051	−0.00109278	−0.000546391	−	1.00000i	$-0.500174\pi$
$373$	−0.0211051	−0.00109278	−0.000546391		1.00000i	$0.500174\pi$
$374$	−11.3212	−0.585407
$375$	0	0
$376$	16.3991	0.845721
$377$	3.87390	0.199516
$378$	0	0
$379$	−28.6833	−1.47336	−0.736681	−	0.676240i	$-0.763608\pi$
$379$	−28.6833	−1.47336	−0.736681	+	0.676240i	$0.763608\pi$
$380$	−3.44839	−0.176899
$381$	0	0
$382$	−28.3103	−1.44848
$383$	10.9234	0.558160	0.279080	−	0.960268i	$-0.409970\pi$
$383$	10.9234	0.558160	0.279080	+	0.960268i	$0.409970\pi$
$384$	0	0
$385$	15.6501	0.797600
$386$	3.88703	0.197844
$387$	0	0
$388$	−0.440214	−0.0223485
$389$	−7.22681	−0.366414	−0.183207	−	0.983074i	$-0.558648\pi$
$389$	−7.22681	−0.366414	−0.183207	+	0.983074i	$0.558648\pi$
$390$	0	0
$391$	−4.20465	−0.212638
$392$	−8.36790	−0.422643
$393$	0	0
$394$	27.9993	1.41058
$395$	8.72610	0.439058
$396$	0	0
$397$	−7.14485	−0.358590	−0.179295	−	0.983795i	$-0.557382\pi$
$397$	−7.14485	−0.358590	−0.179295	+	0.983795i	$0.557382\pi$
$398$	−16.5596	−0.830060
$399$	0	0
$400$	−58.5483	−2.92741
$401$	−15.8074	−0.789386	−0.394693	−	0.918813i	$-0.629149\pi$
$401$	−15.8074	−0.789386	−0.394693	+	0.918813i	$0.629149\pi$
$402$	0	0
$403$	21.2203	1.05706
$404$	−0.230787	−0.0114821
$405$	0	0
$406$	2.90366	0.144106
$407$	−16.6194	−0.823791
$408$	0	0
$409$	19.7351	0.975839	0.487920	−	0.872889i	$-0.337756\pi$
$409$	19.7351	0.975839	0.487920	+	0.872889i	$0.337756\pi$
$410$	−67.1465	−3.31613
$411$	0	0
$412$	−0.351004	−0.0172927
$413$	25.8446	1.27173
$414$	0	0
$415$	−26.2867	−1.29036
$416$	−3.43933	−0.168627
$417$	0	0
$418$	13.6688	0.668562
$419$	−9.62363	−0.470145	−0.235073	−	0.971978i	$-0.575533\pi$
$419$	−9.62363	−0.470145	−0.235073	+	0.971978i	$0.575533\pi$
$420$	0	0
$421$	−34.0713	−1.66054	−0.830268	−	0.557365i	$-0.811812\pi$
$421$	−34.0713	−1.66054	−0.830268	+	0.557365i	$0.811812\pi$
$422$	9.18408	0.447074
$423$	0	0
$424$	−5.82675	−0.282972
$425$	−57.3853	−2.78359
$426$	0	0
$427$	−21.9439	−1.06194
$428$	−1.09866	−0.0531058
$429$	0	0
$430$	30.0616	1.44970
$431$	−5.47912	−0.263920	−0.131960	−	0.991255i	$-0.542127\pi$
$431$	−5.47912	−0.263920	−0.131960	+	0.991255i	$0.542127\pi$
$432$	0	0
$433$	−21.6270	−1.03933	−0.519664	−	0.854371i	$-0.673943\pi$
$433$	−21.6270	−1.03933	−0.519664	+	0.854371i	$0.673943\pi$
$434$	15.9056	0.763493
$435$	0	0
$436$	−0.0752961	−0.00360603
$437$	5.07651	0.242842
$438$	0	0
$439$	−15.5504	−0.742178	−0.371089	−	0.928597i	$-0.621016\pi$
$439$	−15.5504	−0.742178	−0.371089	+	0.928597i	$0.621016\pi$
$440$	21.4257	1.02143
$441$	0	0
$442$	−23.9240	−1.13795
$443$	−41.6611	−1.97938	−0.989690	−	0.143225i	$-0.954253\pi$
$443$	−41.6611	−1.97938	−0.989690	+	0.143225i	$0.954253\pi$
$444$	0	0
$445$	−12.9378	−0.613310
$446$	27.7630	1.31462
$447$	0	0
$448$	14.3835	0.679558
$449$	25.8510	1.21998	0.609992	−	0.792408i	$-0.291173\pi$
$449$	25.8510	1.21998	0.609992	+	0.792408i	$0.291173\pi$
$450$	0	0
$451$	19.4071	0.913843
$452$	3.00727	0.141450
$453$	0	0
$454$	15.9782	0.749893
$455$	33.0716	1.55042
$456$	0	0
$457$	−14.4237	−0.674713	−0.337357	−	0.941377i	$-0.609533\pi$
$457$	−14.4237	−0.674713	−0.337357	+	0.941377i	$0.609533\pi$
$458$	25.1764	1.17642
$459$	0	0
$460$	−0.679284	−0.0316718
$461$	12.6609	0.589676	0.294838	−	0.955547i	$-0.404734\pi$
$461$	12.6609	0.589676	0.294838	+	0.955547i	$0.404734\pi$
$462$	0	0
$463$	−0.406547	−0.0188938	−0.00944691	−	0.999955i	$-0.503007\pi$
$463$	−0.406547	−0.0188938	−0.00944691	+	0.999955i	$0.503007\pi$
$464$	4.28986	0.199152
$465$	0	0
$466$	−6.35084	−0.294197
$467$	−1.68222	−0.0778439	−0.0389220	−	0.999242i	$-0.512392\pi$
$467$	−1.68222	−0.0778439	−0.0389220	+	0.999242i	$0.512392\pi$
$468$	0	0
$469$	−18.7744	−0.866919
$470$	38.4312	1.77270
$471$	0	0
$472$	35.3825	1.62861
$473$	−8.68857	−0.399501
$474$	0	0
$475$	69.2845	3.17899
$476$	−1.30754	−0.0599308
$477$	0	0
$478$	13.0695	0.597783
$479$	−25.4029	−1.16069	−0.580343	−	0.814372i	$-0.697082\pi$
$479$	−25.4029	−1.16069	−0.580343	+	0.814372i	$0.697082\pi$
$480$	0	0
$481$	−35.1199	−1.60133
$482$	6.74642	0.307291
$483$	0	0
$484$	−1.20169	−0.0546224
$485$	12.0850	0.548751
$486$	0	0
$487$	34.2968	1.55414	0.777069	−	0.629415i	$-0.216706\pi$
$487$	34.2968	1.55414	0.777069	+	0.629415i	$0.216706\pi$
$488$	−30.0423	−1.35995
$489$	0	0
$490$	−19.6101	−0.885893
$491$	−28.8727	−1.30300	−0.651502	−	0.758647i	$-0.725861\pi$
$491$	−28.8727	−1.30300	−0.651502	+	0.758647i	$0.725861\pi$
$492$	0	0
$493$	4.20465	0.189368
$494$	28.8848	1.29959
$495$	0	0
$496$	23.4989	1.05513
$497$	15.9559	0.715722
$498$	0	0
$499$	−1.97330	−0.0883371	−0.0441685	−	0.999024i	$-0.514064\pi$
$499$	−1.97330	−0.0883371	−0.0441685	+	0.999024i	$0.514064\pi$
$500$	−5.87448	−0.262715
$501$	0	0
$502$	29.2959	1.30754
$503$	−24.9289	−1.11152	−0.555762	−	0.831342i	$-0.687573\pi$
$503$	−24.9289	−1.11152	−0.555762	+	0.831342i	$0.687573\pi$
$504$	0	0
$505$	6.33569	0.281935
$506$	2.69255	0.119699
$507$	0	0
$508$	1.10769	0.0491459
$509$	−37.1015	−1.64449	−0.822247	−	0.569131i	$-0.807280\pi$
$509$	−37.1015	−1.64449	−0.822247	+	0.569131i	$0.807280\pi$
$510$	0	0
$511$	32.9055	1.45565
$512$	19.4125	0.857918
$513$	0	0
$514$	−34.0285	−1.50093
$515$	9.63594	0.424610
$516$	0	0
$517$	−11.1076	−0.488512
$518$	−26.3239	−1.15661
$519$	0	0
$520$	45.2767	1.98551
$521$	−26.5629	−1.16374	−0.581870	−	0.813282i	$-0.697679\pi$
$521$	−26.5629	−1.16374	−0.581870	+	0.813282i	$0.697679\pi$
$522$	0	0
$523$	−23.3634	−1.02161	−0.510806	−	0.859696i	$-0.670653\pi$
$523$	−23.3634	−1.02161	−0.510806	+	0.859696i	$0.670653\pi$
$524$	1.58087	0.0690604
$525$	0	0
$526$	0.734932	0.0320445
$527$	23.0321	1.00329
$528$	0	0
$529$	1.00000	0.0434783
$530$	−13.6549	−0.593132
$531$	0	0
$532$	1.57866	0.0684438
$533$	41.0109	1.77638
$534$	0	0
$535$	30.1610	1.30397
$536$	−25.7031	−1.11020
$537$	0	0
$538$	9.06259	0.390716
$539$	5.66782	0.244130
$540$	0	0
$541$	2.48754	0.106948	0.0534738	−	0.998569i	$-0.482971\pi$
$541$	2.48754	0.106948	0.0534738	+	0.998569i	$0.482971\pi$
$542$	−23.9892	−1.03042
$543$	0	0
$544$	−3.73298	−0.160050
$545$	2.06707	0.0885435
$546$	0	0
$547$	−3.69422	−0.157953	−0.0789767	−	0.996876i	$-0.525165\pi$
$547$	−3.69422	−0.157953	−0.0789767	+	0.996876i	$0.525165\pi$
$548$	0.822662	0.0351424
$549$	0	0
$550$	36.7481	1.56695
$551$	−5.07651	−0.216267
$552$	0	0
$553$	−3.99478	−0.169875
$554$	2.67275	0.113554
$555$	0	0
$556$	1.02349	0.0434058
$557$	30.6634	1.29925	0.649626	−	0.760254i	$-0.274926\pi$
$557$	30.6634	1.29925	0.649626	+	0.760254i	$0.274926\pi$
$558$	0	0
$559$	−18.3606	−0.776572
$560$	36.6227	1.54759
$561$	0	0
$562$	1.80485	0.0761328
$563$	−15.8280	−0.667070	−0.333535	−	0.942738i	$-0.608242\pi$
$563$	−15.8280	−0.667070	−0.333535	+	0.942738i	$0.608242\pi$
$564$	0	0
$565$	−82.5573	−3.47321
$566$	−26.2529	−1.10349
$567$	0	0
$568$	21.8445	0.916575
$569$	25.2392	1.05808	0.529042	−	0.848596i	$-0.322552\pi$
$569$	25.2392	1.05808	0.529042	+	0.848596i	$0.322552\pi$
$570$	0	0
$571$	40.6814	1.70247	0.851233	−	0.524789i	$-0.175856\pi$
$571$	40.6814	1.70247	0.851233	+	0.524789i	$0.175856\pi$
$572$	1.11710	0.0467082
$573$	0	0
$574$	30.7395	1.28304
$575$	13.6481	0.569163
$576$	0	0
$577$	−29.9842	−1.24826	−0.624128	−	0.781322i	$-0.714546\pi$
$577$	−29.9842	−1.24826	−0.624128	+	0.781322i	$0.714546\pi$
$578$	−0.997363	−0.0414848
$579$	0	0
$580$	0.679284	0.0282057
$581$	12.0340	0.499254
$582$	0	0
$583$	3.94662	0.163452
$584$	45.0493	1.86415
$585$	0	0
$586$	31.8177	1.31438
$587$	−37.9283	−1.56547	−0.782734	−	0.622356i	$-0.786176\pi$
$587$	−37.9283	−1.56547	−0.782734	+	0.622356i	$0.786176\pi$
$588$	0	0
$589$	−27.8080	−1.14581
$590$	82.9185	3.41370
$591$	0	0
$592$	−38.8910	−1.59841
$593$	−17.6162	−0.723410	−0.361705	−	0.932293i	$-0.617805\pi$
$593$	−17.6162	−0.723410	−0.361705	+	0.932293i	$0.617805\pi$
$594$	0	0
$595$	35.8952	1.47156
$596$	1.99845	0.0818596
$597$	0	0
$598$	5.68989	0.232677
$599$	11.9876	0.489799	0.244900	−	0.969548i	$-0.421245\pi$
$599$	11.9876	0.489799	0.244900	+	0.969548i	$0.421245\pi$
$600$	0	0
$601$	36.1976	1.47653	0.738266	−	0.674510i	$-0.235645\pi$
$601$	36.1976	1.47653	0.738266	+	0.674510i	$0.235645\pi$
$602$	−13.7621	−0.560902
$603$	0	0
$604$	−1.27490	−0.0518749
$605$	32.9895	1.34121
$606$	0	0
$607$	9.60236	0.389748	0.194874	−	0.980828i	$-0.437570\pi$
$607$	9.60236	0.389748	0.194874	+	0.980828i	$0.437570\pi$
$608$	4.50704	0.182785
$609$	0	0
$610$	−70.4039	−2.85057
$611$	−23.4725	−0.949596
$612$	0	0
$613$	10.5423	0.425799	0.212900	−	0.977074i	$-0.431709\pi$
$613$	10.5423	0.425799	0.212900	+	0.977074i	$0.431709\pi$
$614$	29.3147	1.18304
$615$	0	0
$616$	−9.80864	−0.395201
$617$	44.5861	1.79497	0.897484	−	0.441048i	$-0.145393\pi$
$617$	44.5861	1.79497	0.897484	+	0.441048i	$0.145393\pi$
$618$	0	0
$619$	−7.87917	−0.316690	−0.158345	−	0.987384i	$-0.550616\pi$
$619$	−7.87917	−0.316690	−0.158345	+	0.987384i	$0.550616\pi$
$620$	3.72096	0.149438
$621$	0	0
$622$	44.3161	1.77691
$623$	5.92289	0.237295
$624$	0	0
$625$	93.0292	3.72117
$626$	−10.4357	−0.417094
$627$	0	0
$628$	0.419855	0.0167540
$629$	−38.1184	−1.51988
$630$	0	0
$631$	−30.3776	−1.20931	−0.604657	−	0.796486i	$-0.706690\pi$
$631$	−30.3776	−1.20931	−0.604657	+	0.796486i	$0.706690\pi$
$632$	−5.46906	−0.217548
$633$	0	0
$634$	32.2791	1.28197
$635$	−30.4090	−1.20674
$636$	0	0
$637$	11.9772	0.474554
$638$	−2.69255	−0.106599
$639$	0	0
$640$	53.8153	2.12724
$641$	13.3578	0.527600	0.263800	−	0.964577i	$-0.415024\pi$
$641$	13.3578	0.527600	0.263800	+	0.964577i	$0.415024\pi$
$642$	0	0
$643$	12.3603	0.487443	0.243722	−	0.969845i	$-0.421632\pi$
$643$	12.3603	0.487443	0.243722	+	0.969845i	$0.421632\pi$
$644$	0.310974	0.0122541
$645$	0	0
$646$	31.3509	1.23349
$647$	−37.8303	−1.48726	−0.743632	−	0.668589i	$-0.766898\pi$
$647$	−37.8303	−1.48726	−0.743632	+	0.668589i	$0.766898\pi$
$648$	0	0
$649$	−23.9656	−0.940732
$650$	77.6559	3.04591
$651$	0	0
$652$	−0.0268612	−0.00105196
$653$	−34.8560	−1.36402	−0.682011	−	0.731342i	$-0.738894\pi$
$653$	−34.8560	−1.36402	−0.682011	+	0.731342i	$0.738894\pi$
$654$	0	0
$655$	−43.3988	−1.69573
$656$	45.4145	1.77314
$657$	0	0
$658$	−17.5937	−0.685873
$659$	20.4878	0.798090	0.399045	−	0.916931i	$-0.369342\pi$
$659$	20.4878	0.798090	0.399045	+	0.916931i	$0.369342\pi$
$660$	0	0
$661$	27.2935	1.06160	0.530798	−	0.847498i	$-0.321892\pi$
$661$	27.2935	1.06160	0.530798	+	0.847498i	$0.321892\pi$
$662$	40.5518	1.57609
$663$	0	0
$664$	16.4751	0.639359
$665$	−43.3383	−1.68059
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−1.20221	−0.0465148
$669$	0	0
$670$	−60.2348	−2.32707
$671$	20.3485	0.785546
$672$	0	0
$673$	−9.50307	−0.366316	−0.183158	−	0.983083i	$-0.558632\pi$
$673$	−9.50307	−0.366316	−0.183158	+	0.983083i	$0.558632\pi$
$674$	−6.96764	−0.268383
$675$	0	0
$676$	0.315717	0.0121430
$677$	−17.6511	−0.678388	−0.339194	−	0.940717i	$-0.610154\pi$
$677$	−17.6511	−0.678388	−0.339194	+	0.940717i	$0.610154\pi$
$678$	0	0
$679$	−5.53248	−0.212317
$680$	49.1423	1.88452
$681$	0	0
$682$	−14.7492	−0.564776
$683$	−6.40203	−0.244967	−0.122483	−	0.992471i	$-0.539086\pi$
$683$	−6.40203	−0.244967	−0.122483	+	0.992471i	$0.539086\pi$
$684$	0	0
$685$	−22.5842	−0.862897
$686$	29.3031	1.11880
$687$	0	0
$688$	−20.3321	−0.775155
$689$	8.33998	0.317728
$690$	0	0
$691$	−50.4784	−1.92029	−0.960144	−	0.279507i	$-0.909829\pi$
$691$	−50.4784	−1.92029	−0.960144	+	0.279507i	$0.909829\pi$
$692$	1.42433	0.0541448
$693$	0	0
$694$	−2.68773	−0.102025
$695$	−28.0975	−1.06580
$696$	0	0
$697$	44.5123	1.68603
$698$	5.89325	0.223063
$699$	0	0
$700$	4.24419	0.160415
$701$	26.8061	1.01245	0.506227	−	0.862400i	$-0.331040\pi$
$701$	26.8061	1.01245	0.506227	+	0.862400i	$0.331040\pi$
$702$	0	0
$703$	46.0226	1.73577
$704$	−13.3378	−0.502687
$705$	0	0
$706$	−42.7660	−1.60952
$707$	−2.90046	−0.109083
$708$	0	0
$709$	−21.5712	−0.810122	−0.405061	−	0.914290i	$-0.632750\pi$
$709$	−21.5712	−0.810122	−0.405061	+	0.914290i	$0.632750\pi$
$710$	51.1923	1.92121
$711$	0	0
$712$	8.10873	0.303888
$713$	−5.47778	−0.205144
$714$	0	0
$715$	−30.6672	−1.14689
$716$	2.18306	0.0815848
$717$	0	0
$718$	−12.9129	−0.481905
$719$	−20.2309	−0.754486	−0.377243	−	0.926114i	$-0.623128\pi$
$719$	−20.2309	−0.754486	−0.377243	+	0.926114i	$0.623128\pi$
$720$	0	0
$721$	−4.41131	−0.164286
$722$	−9.94503	−0.370116
$723$	0	0
$724$	2.90596	0.107999
$725$	−13.6481	−0.506876
$726$	0	0
$727$	−9.12267	−0.338341	−0.169171	−	0.985587i	$-0.554109\pi$
$727$	−9.12267	−0.338341	−0.169171	+	0.985587i	$0.554109\pi$
$728$	−20.7275	−0.768214
$729$	0	0
$730$	105.572	3.90741
$731$	−19.9282	−0.737073
$732$	0	0
$733$	−3.48854	−0.128852	−0.0644262	−	0.997922i	$-0.520522\pi$
$733$	−3.48854	−0.128852	−0.0644262	+	0.997922i	$0.520522\pi$
$734$	−15.8861	−0.586365
$735$	0	0
$736$	0.887822	0.0327256
$737$	17.4094	0.641284
$738$	0	0
$739$	−16.9510	−0.623552	−0.311776	−	0.950156i	$-0.600924\pi$
$739$	−16.9510	−0.623552	−0.311776	+	0.950156i	$0.600924\pi$
$740$	−6.15824	−0.226381
$741$	0	0
$742$	6.25119	0.229488
$743$	35.5481	1.30413	0.652067	−	0.758162i	$-0.273902\pi$
$743$	35.5481	1.30413	0.652067	+	0.758162i	$0.273902\pi$
$744$	0	0
$745$	−54.8625	−2.01001
$746$	0.0309987	0.00113494
$747$	0	0
$748$	1.21247	0.0443324
$749$	−13.8076	−0.504520
$750$	0	0
$751$	7.34241	0.267929	0.133964	−	0.990986i	$-0.457229\pi$
$751$	7.34241	0.267929	0.133964	+	0.990986i	$0.457229\pi$
$752$	−25.9929	−0.947863
$753$	0	0
$754$	−5.68989	−0.207213
$755$	34.9992	1.27375
$756$	0	0
$757$	30.4932	1.10830	0.554148	−	0.832418i	$-0.313044\pi$
$757$	30.4932	1.10830	0.554148	+	0.832418i	$0.313044\pi$
$758$	42.1293	1.53021
$759$	0	0
$760$	−59.3324	−2.15221
$761$	48.3414	1.75237	0.876187	−	0.481972i	$-0.160079\pi$
$761$	48.3414	1.75237	0.876187	+	0.481972i	$0.160079\pi$
$762$	0	0
$763$	−0.946299	−0.0342583
$764$	3.03196	0.109693
$765$	0	0
$766$	−16.0440	−0.579695
$767$	−50.6439	−1.82865
$768$	0	0
$769$	−32.9910	−1.18969	−0.594843	−	0.803842i	$-0.702786\pi$
$769$	−32.9910	−1.18969	−0.594843	+	0.803842i	$0.702786\pi$
$770$	−22.9864	−0.828373
$771$	0	0
$772$	−0.416290	−0.0149826
$773$	−27.2251	−0.979218	−0.489609	−	0.871942i	$-0.662860\pi$
$773$	−27.2251	−0.979218	−0.489609	+	0.871942i	$0.662860\pi$
$774$	0	0
$775$	−74.7610	−2.68550
$776$	−7.57425	−0.271900
$777$	0	0
$778$	10.6146	0.380551
$779$	−53.7423	−1.92552
$780$	0	0
$781$	−14.7959	−0.529439
$782$	6.17568	0.220842
$783$	0	0
$784$	13.2633	0.473688
$785$	−11.5261	−0.411384
$786$	0	0
$787$	−17.4869	−0.623341	−0.311671	−	0.950190i	$-0.600889\pi$
$787$	−17.4869	−0.623341	−0.311671	+	0.950190i	$0.600889\pi$
$788$	−2.99865	−0.106822
$789$	0	0
$790$	−12.8167	−0.455997
$791$	37.7945	1.34382
$792$	0	0
$793$	43.0004	1.52699
$794$	10.4942	0.372424
$795$	0	0
$796$	1.77349	0.0628599
$797$	−46.5153	−1.64766	−0.823829	−	0.566838i	$-0.808166\pi$
$797$	−46.5153	−1.64766	−0.823829	+	0.566838i	$0.808166\pi$
$798$	0	0
$799$	−25.4766	−0.901296
$800$	12.1170	0.428402
$801$	0	0
$802$	23.2176	0.819841
$803$	−30.5132	−1.07679
$804$	0	0
$805$	−8.53703	−0.300891
$806$	−31.1679	−1.09784
$807$	0	0
$808$	−3.97088	−0.139695
$809$	43.0308	1.51288	0.756441	−	0.654062i	$-0.226936\pi$
$809$	43.0308	1.51288	0.756441	+	0.654062i	$0.226936\pi$
$810$	0	0
$811$	13.3752	0.469667	0.234834	−	0.972036i	$-0.424545\pi$
$811$	13.3752	0.469667	0.234834	+	0.972036i	$0.424545\pi$
$812$	−0.310974	−0.0109131
$813$	0	0
$814$	24.4101	0.855574
$815$	0.737407	0.0258303
$816$	0	0
$817$	24.0605	0.841771
$818$	−28.9865	−1.01349
$819$	0	0
$820$	7.19121	0.251128
$821$	37.0332	1.29247	0.646235	−	0.763139i	$-0.276343\pi$
$821$	37.0332	1.29247	0.646235	+	0.763139i	$0.276343\pi$
$822$	0	0
$823$	20.7639	0.723786	0.361893	−	0.932220i	$-0.382131\pi$
$823$	20.7639	0.723786	0.361893	+	0.932220i	$0.382131\pi$
$824$	−6.03931	−0.210389
$825$	0	0
$826$	−37.9599	−1.32079
$827$	−35.9493	−1.25008	−0.625040	−	0.780593i	$-0.714917\pi$
$827$	−35.9493	−1.25008	−0.625040	+	0.780593i	$0.714917\pi$
$828$	0	0
$829$	−38.3647	−1.33246	−0.666231	−	0.745745i	$-0.732094\pi$
$829$	−38.3647	−1.33246	−0.666231	+	0.745745i	$0.732094\pi$
$830$	38.6093	1.34015
$831$	0	0
$832$	−28.1853	−0.977151
$833$	12.9998	0.450416
$834$	0	0
$835$	33.0036	1.14214
$836$	−1.46389	−0.0506297
$837$	0	0
$838$	14.1350	0.488284
$839$	−34.1180	−1.17789	−0.588943	−	0.808175i	$-0.700456\pi$
$839$	−34.1180	−1.17789	−0.588943	+	0.808175i	$0.700456\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	50.0431	1.72460
$843$	0	0
$844$	−0.983591	−0.0338566
$845$	−8.66724	−0.298162
$846$	0	0
$847$	−15.1025	−0.518928
$848$	9.23549	0.317148
$849$	0	0
$850$	84.2861	2.89099
$851$	9.06579	0.310771
$852$	0	0
$853$	17.8613	0.611560	0.305780	−	0.952102i	$-0.401083\pi$
$853$	17.8613	0.611560	0.305780	+	0.952102i	$0.401083\pi$
$854$	32.2307	1.10291
$855$	0	0
$856$	−18.9034	−0.646103
$857$	40.9341	1.39828	0.699142	−	0.714983i	$-0.253566\pi$
$857$	40.9341	1.39828	0.699142	+	0.714983i	$0.253566\pi$
$858$	0	0
$859$	−4.71520	−0.160881	−0.0804403	−	0.996759i	$-0.525633\pi$
$859$	−4.71520	−0.160881	−0.0804403	+	0.996759i	$0.525633\pi$
$860$	−3.21952	−0.109785
$861$	0	0
$862$	8.04759	0.274102
$863$	−25.6983	−0.874779	−0.437389	−	0.899272i	$-0.644097\pi$
$863$	−25.6983	−0.874779	−0.437389	+	0.899272i	$0.644097\pi$
$864$	0	0
$865$	−39.1014	−1.32949
$866$	31.7652	1.07943
$867$	0	0
$868$	−1.70345	−0.0578188
$869$	3.70435	0.125661
$870$	0	0
$871$	36.7894	1.24656
$872$	−1.29553	−0.0438722
$873$	0	0
$874$	−7.45626	−0.252212
$875$	−73.8288	−2.49587
$876$	0	0
$877$	46.7316	1.57801	0.789007	−	0.614385i	$-0.210596\pi$
$877$	46.7316	1.57801	0.789007	+	0.614385i	$0.210596\pi$
$878$	22.8400	0.770812
$879$	0	0
$880$	−33.9601	−1.14479
$881$	−12.0024	−0.404372	−0.202186	−	0.979347i	$-0.564804\pi$
$881$	−12.0024	−0.404372	−0.202186	+	0.979347i	$0.564804\pi$
$882$	0	0
$883$	19.3283	0.650448	0.325224	−	0.945637i	$-0.394560\pi$
$883$	19.3283	0.650448	0.325224	+	0.945637i	$0.394560\pi$
$884$	2.56219	0.0861758
$885$	0	0
$886$	61.1909	2.05575
$887$	−4.73456	−0.158971	−0.0794855	−	0.996836i	$-0.525328\pi$
$887$	−4.73456	−0.158971	−0.0794855	+	0.996836i	$0.525328\pi$
$888$	0	0
$889$	13.9211	0.466900
$890$	19.0027	0.636972
$891$	0	0
$892$	−2.97335	−0.0995550
$893$	30.7593	1.02932
$894$	0	0
$895$	−59.9306	−2.00326
$896$	−24.6365	−0.823047
$897$	0	0
$898$	−37.9693	−1.26705
$899$	5.47778	0.182694
$900$	0	0
$901$	9.05204	0.301567
$902$	−28.5046	−0.949100
$903$	0	0
$904$	51.7426	1.72093
$905$	−79.7760	−2.65185
$906$	0	0
$907$	46.6197	1.54798	0.773992	−	0.633196i	$-0.218257\pi$
$907$	46.6197	1.54798	0.773992	+	0.633196i	$0.218257\pi$
$908$	−1.71122	−0.0567889
$909$	0	0
$910$	−48.5747	−1.61024
$911$	−33.8874	−1.12274	−0.561371	−	0.827565i	$-0.689726\pi$
$911$	−33.8874	−1.12274	−0.561371	+	0.827565i	$0.689726\pi$
$912$	0	0
$913$	−11.1591	−0.369311
$914$	21.1852	0.700745
$915$	0	0
$916$	−2.69632	−0.0890891
$917$	19.8678	0.656094
$918$	0	0
$919$	−15.9739	−0.526931	−0.263465	−	0.964669i	$-0.584865\pi$
$919$	−15.9739	−0.526931	−0.263465	+	0.964669i	$0.584865\pi$
$920$	−11.6876	−0.385330
$921$	0	0
$922$	−18.5960	−0.612426
$923$	−31.2666	−1.02915
$924$	0	0
$925$	123.730	4.06823
$926$	0.597126	0.0196228
$927$	0	0
$928$	−0.887822	−0.0291442
$929$	6.83071	0.224108	0.112054	−	0.993702i	$-0.464257\pi$
$929$	6.83071	0.224108	0.112054	+	0.993702i	$0.464257\pi$
$930$	0	0
$931$	−15.6954	−0.514396
$932$	0.680158	0.0222793
$933$	0	0
$934$	2.47081	0.0808473
$935$	−33.2855	−1.08855
$936$	0	0
$937$	4.13943	0.135229	0.0676146	−	0.997712i	$-0.478461\pi$
$937$	4.13943	0.135229	0.0676146	+	0.997712i	$0.478461\pi$
$938$	27.5753	0.900366
$939$	0	0
$940$	−4.11588	−0.134245
$941$	33.2797	1.08489	0.542444	−	0.840092i	$-0.317499\pi$
$941$	33.2797	1.08489	0.542444	+	0.840092i	$0.317499\pi$
$942$	0	0
$943$	−10.5865	−0.344743
$944$	−56.0819	−1.82531
$945$	0	0
$946$	12.7616	0.414914
$947$	54.7802	1.78012	0.890058	−	0.455847i	$-0.150664\pi$
$947$	54.7802	1.78012	0.890058	+	0.455847i	$0.150664\pi$
$948$	0	0
$949$	−64.4802	−2.09312
$950$	−101.763	−3.30164
$951$	0	0
$952$	−22.4972	−0.729140
$953$	11.2383	0.364045	0.182023	−	0.983294i	$-0.441736\pi$
$953$	11.2383	0.364045	0.182023	+	0.983294i	$0.441736\pi$
$954$	0	0
$955$	−83.2351	−2.69342
$956$	−1.39970	−0.0452697
$957$	0	0
$958$	37.3111	1.20547
$959$	10.3390	0.333863
$960$	0	0
$961$	−0.993945	−0.0320627
$962$	51.5833	1.66311
$963$	0	0
$964$	−0.722524	−0.0232709
$965$	11.4282	0.367888
$966$	0	0
$967$	−40.6294	−1.30655	−0.653276	−	0.757120i	$-0.726606\pi$
$967$	−40.6294	−1.30655	−0.653276	+	0.757120i	$0.726606\pi$
$968$	−20.6761	−0.664555
$969$	0	0
$970$	−17.7502	−0.569923
$971$	13.1817	0.423022	0.211511	−	0.977376i	$-0.432162\pi$
$971$	13.1817	0.423022	0.211511	+	0.977376i	$0.432162\pi$
$972$	0	0
$973$	12.8630	0.412368
$974$	−50.3744	−1.61410
$975$	0	0
$976$	47.6176	1.52420
$977$	1.55708	0.0498154	0.0249077	−	0.999690i	$-0.492071\pi$
$977$	1.55708	0.0498154	0.0249077	+	0.999690i	$0.492071\pi$
$978$	0	0
$979$	−5.49227	−0.175534
$980$	2.10019	0.0670880
$981$	0	0
$982$	42.4075	1.35328
$983$	37.0826	1.18275	0.591376	−	0.806396i	$-0.298585\pi$
$983$	37.0826	1.18275	0.591376	+	0.806396i	$0.298585\pi$
$984$	0	0
$985$	82.3205	2.62295
$986$	−6.17568	−0.196674
$987$	0	0
$988$	−3.09348	−0.0984168
$989$	4.73958	0.150710
$990$	0	0
$991$	26.3834	0.838096	0.419048	−	0.907964i	$-0.362364\pi$
$991$	26.3834	0.838096	0.419048	+	0.907964i	$0.362364\pi$
$992$	−4.86329	−0.154410
$993$	0	0
$994$	−23.4357	−0.743335
$995$	−48.6869	−1.54348
$996$	0	0
$997$	−25.1891	−0.797747	−0.398873	−	0.917006i	$-0.630599\pi$
$997$	−25.1891	−0.797747	−0.398873	+	0.917006i	$0.630599\pi$
$998$	2.89834	0.0917452
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.o.1.4		13
3.2	odd	2		667.2.a.c.1.10	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.c.1.10	✓	13	3.2	odd	2
6003.2.a.o.1.4		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)

\(f(q)\)	\(=\)	\(q-1.46878 q^{2} +0.157302 q^{4} -4.31834 q^{5} +1.97692 q^{7} +2.70651 q^{8} +O(q^{10})\)	\(q-1.46878 q^{2} +0.157302 q^{4} -4.31834 q^{5} +1.97692 q^{7} +2.70651 q^{8} +6.34267 q^{10} -1.83320 q^{11} -3.87390 q^{13} -2.90366 q^{14} -4.28986 q^{16} -4.20465 q^{17} +5.07651 q^{19} -0.679284 q^{20} +2.69255 q^{22} +1.00000 q^{23} +13.6481 q^{25} +5.68989 q^{26} +0.310974 q^{28} -1.00000 q^{29} -5.47778 q^{31} +0.887822 q^{32} +6.17568 q^{34} -8.53703 q^{35} +9.06579 q^{37} -7.45626 q^{38} -11.6876 q^{40} -10.5865 q^{41} +4.73958 q^{43} -0.288365 q^{44} -1.46878 q^{46} +6.05915 q^{47} -3.09177 q^{49} -20.0459 q^{50} -0.609372 q^{52} -2.15287 q^{53} +7.91636 q^{55} +5.35057 q^{56} +1.46878 q^{58} +13.0731 q^{59} -11.1000 q^{61} +8.04563 q^{62} +7.27571 q^{64} +16.7288 q^{65} -9.49675 q^{67} -0.661399 q^{68} +12.5390 q^{70} +8.07109 q^{71} +16.6448 q^{73} -13.3156 q^{74} +0.798545 q^{76} -3.62409 q^{77} -2.02071 q^{79} +18.5251 q^{80} +15.5491 q^{82} +6.08722 q^{83} +18.1571 q^{85} -6.96138 q^{86} -4.96156 q^{88} +2.99601 q^{89} -7.65840 q^{91} +0.157302 q^{92} -8.89953 q^{94} -21.9221 q^{95} -2.79853 q^{97} +4.54112 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) 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

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Database

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