LMFDB - Embedded newform 6003.2.a.o.1.2

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			1.2
Root			$2.43510$ 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)

$f(q)$	$=$	$q-2.43510 q^{2} +3.92972 q^{4} -3.49556 q^{5} -4.87125 q^{7} -4.69905 q^{8} +O(q^{10})$	\(q-2.43510 q^{2} +3.92972 q^{4} -3.49556 q^{5} -4.87125 q^{7} -4.69905 q^{8} +8.51205 q^{10} +1.63115 q^{11} -5.07695 q^{13} +11.8620 q^{14} +3.58324 q^{16} -7.41068 q^{17} +0.289958 q^{19} -13.7366 q^{20} -3.97201 q^{22} +1.00000 q^{23} +7.21896 q^{25} +12.3629 q^{26} -19.1426 q^{28} -1.00000 q^{29} -1.20829 q^{31} +0.672566 q^{32} +18.0458 q^{34} +17.0278 q^{35} +2.03076 q^{37} -0.706078 q^{38} +16.4258 q^{40} +10.9339 q^{41} +5.66129 q^{43} +6.40996 q^{44} -2.43510 q^{46} -5.12682 q^{47} +16.7291 q^{49} -17.5789 q^{50} -19.9510 q^{52} -7.16900 q^{53} -5.70179 q^{55} +22.8903 q^{56} +2.43510 q^{58} +0.758039 q^{59} +11.5695 q^{61} +2.94232 q^{62} -8.80424 q^{64} +17.7468 q^{65} +3.60778 q^{67} -29.1219 q^{68} -41.4644 q^{70} -8.89620 q^{71} -2.62827 q^{73} -4.94510 q^{74} +1.13945 q^{76} -7.94575 q^{77} -3.77211 q^{79} -12.5254 q^{80} -26.6251 q^{82} +10.7050 q^{83} +25.9045 q^{85} -13.7858 q^{86} -7.66486 q^{88} +4.54315 q^{89} +24.7311 q^{91} +3.92972 q^{92} +12.4843 q^{94} -1.01357 q^{95} -2.62406 q^{97} -40.7371 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.43510	−1.72188	−0.860938	−	0.508710i	$-0.830123\pi$
$2$	−2.43510	−1.72188	−0.860938	+	0.508710i	$0.830123\pi$
$3$	0	0
$3$	0	0
$4$	3.92972	1.96486
$5$	−3.49556	−1.56326	−0.781632	−	0.623740i	$-0.785612\pi$
$5$	−3.49556	−1.56326	−0.781632	+	0.623740i	$0.785612\pi$
$6$	0	0
$7$	−4.87125	−1.84116	−0.920581	−	0.390553i	$-0.872284\pi$
$7$	−4.87125	−1.84116	−0.920581	+	0.390553i	$0.872284\pi$
$8$	−4.69905	−1.66137
$9$	0	0
$10$	8.51205	2.69175
$11$	1.63115	0.491810	0.245905	−	0.969294i	$-0.420915\pi$
$11$	1.63115	0.491810	0.245905	+	0.969294i	$0.420915\pi$
$12$	0	0
$13$	−5.07695	−1.40809	−0.704046	−	0.710155i	$-0.748625\pi$
$13$	−5.07695	−1.40809	−0.704046	+	0.710155i	$0.748625\pi$
$14$	11.8620	3.17025
$15$	0	0
$16$	3.58324	0.895809
$17$	−7.41068	−1.79735	−0.898677	−	0.438611i	$-0.855471\pi$
$17$	−7.41068	−1.79735	−0.898677	+	0.438611i	$0.855471\pi$
$18$	0	0
$19$	0.289958	0.0665210	0.0332605	−	0.999447i	$-0.489411\pi$
$19$	0.289958	0.0665210	0.0332605	+	0.999447i	$0.489411\pi$
$20$	−13.7366	−3.07159
$21$	0	0
$22$	−3.97201	−0.846836
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	7.21896	1.44379
$26$	12.3629	2.42456
$27$	0	0
$28$	−19.1426	−3.61762
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−1.20829	−0.217016	−0.108508	−	0.994096i	$-0.534607\pi$
$31$	−1.20829	−0.217016	−0.108508	+	0.994096i	$0.534607\pi$
$32$	0.672566	0.118894
$33$	0	0
$34$	18.0458	3.09482
$35$	17.0278	2.87822
$36$	0	0
$37$	2.03076	0.333855	0.166927	−	0.985969i	$-0.446615\pi$
$37$	2.03076	0.333855	0.166927	+	0.985969i	$0.446615\pi$
$38$	−0.706078	−0.114541
$39$	0	0
$40$	16.4258	2.59715
$41$	10.9339	1.70758	0.853792	−	0.520615i	$-0.174297\pi$
$41$	10.9339	1.70758	0.853792	+	0.520615i	$0.174297\pi$
$42$	0	0
$43$	5.66129	0.863339	0.431670	−	0.902032i	$-0.357925\pi$
$43$	5.66129	0.863339	0.431670	+	0.902032i	$0.357925\pi$
$44$	6.40996	0.966337
$45$	0	0
$46$	−2.43510	−0.359036
$47$	−5.12682	−0.747823	−0.373912	−	0.927464i	$-0.621984\pi$
$47$	−5.12682	−0.747823	−0.373912	+	0.927464i	$0.621984\pi$
$48$	0	0
$49$	16.7291	2.38987
$50$	−17.5789	−2.48603
$51$	0	0
$52$	−19.9510	−2.76670
$53$	−7.16900	−0.984737	−0.492369	−	0.870387i	$-0.663869\pi$
$53$	−7.16900	−0.984737	−0.492369	+	0.870387i	$0.663869\pi$
$54$	0	0
$55$	−5.70179	−0.768829
$56$	22.8903	3.05884
$57$	0	0
$58$	2.43510	0.319744
$59$	0.758039	0.0986883	0.0493441	−	0.998782i	$-0.484287\pi$
$59$	0.758039	0.0986883	0.0493441	+	0.998782i	$0.484287\pi$
$60$	0	0
$61$	11.5695	1.48133	0.740663	−	0.671876i	$-0.234511\pi$
$61$	11.5695	1.48133	0.740663	+	0.671876i	$0.234511\pi$
$62$	2.94232	0.373675
$63$	0	0
$64$	−8.80424	−1.10053
$65$	17.7468	2.20122
$66$	0	0
$67$	3.60778	0.440761	0.220380	−	0.975414i	$-0.429270\pi$
$67$	3.60778	0.440761	0.220380	+	0.975414i	$0.429270\pi$
$68$	−29.1219	−3.53155
$69$	0	0
$70$	−41.4644	−4.95594
$71$	−8.89620	−1.05579	−0.527893	−	0.849311i	$-0.677018\pi$
$71$	−8.89620	−1.05579	−0.527893	+	0.849311i	$0.677018\pi$
$72$	0	0
$73$	−2.62827	−0.307616	−0.153808	−	0.988101i	$-0.549154\pi$
$73$	−2.62827	−0.307616	−0.153808	+	0.988101i	$0.549154\pi$
$74$	−4.94510	−0.574857
$75$	0	0
$76$	1.13945	0.130704
$77$	−7.94575	−0.905502
$78$	0	0
$79$	−3.77211	−0.424395	−0.212198	−	0.977227i	$-0.568062\pi$
$79$	−3.77211	−0.424395	−0.212198	+	0.977227i	$0.568062\pi$
$80$	−12.5254	−1.40039
$81$	0	0
$82$	−26.6251	−2.94025
$83$	10.7050	1.17502	0.587510	−	0.809217i	$-0.300108\pi$
$83$	10.7050	1.17502	0.587510	+	0.809217i	$0.300108\pi$
$84$	0	0
$85$	25.9045	2.80974
$86$	−13.7858	−1.48656
$87$	0	0
$88$	−7.66486	−0.817077
$89$	4.54315	0.481573	0.240786	−	0.970578i	$-0.422595\pi$
$89$	4.54315	0.481573	0.240786	+	0.970578i	$0.422595\pi$
$90$	0	0
$91$	24.7311	2.59252
$92$	3.92972	0.409701
$93$	0	0
$94$	12.4843	1.28766
$95$	−1.01357	−0.103990
$96$	0	0
$97$	−2.62406	−0.266433	−0.133217	−	0.991087i	$-0.542531\pi$
$97$	−2.62406	−0.266433	−0.133217	+	0.991087i	$0.542531\pi$
$98$	−40.7371	−4.11507
$99$	0	0
$100$	28.3685	2.83685
$101$	−18.5441	−1.84521	−0.922604	−	0.385748i	$-0.873943\pi$
$101$	−18.5441	−1.84521	−0.922604	+	0.385748i	$0.873943\pi$
$102$	0	0
$103$	6.66490	0.656712	0.328356	−	0.944554i	$-0.393505\pi$
$103$	6.66490	0.656712	0.328356	+	0.944554i	$0.393505\pi$
$104$	23.8568	2.33936
$105$	0	0
$106$	17.4572	1.69560
$107$	13.0781	1.26431	0.632156	−	0.774841i	$-0.282170\pi$
$107$	13.0781	1.26431	0.632156	+	0.774841i	$0.282170\pi$
$108$	0	0
$109$	13.3526	1.27895	0.639476	−	0.768811i	$-0.279152\pi$
$109$	13.3526	1.27895	0.639476	+	0.768811i	$0.279152\pi$
$110$	13.8844	1.32383
$111$	0	0
$112$	−17.4549	−1.64933
$113$	−4.16536	−0.391844	−0.195922	−	0.980619i	$-0.562770\pi$
$113$	−4.16536	−0.391844	−0.195922	+	0.980619i	$0.562770\pi$
$114$	0	0
$115$	−3.49556	−0.325963
$116$	−3.92972	−0.364865
$117$	0	0
$118$	−1.84590	−0.169929
$119$	36.0993	3.30922
$120$	0	0
$121$	−8.33935	−0.758123
$122$	−28.1730	−2.55066
$123$	0	0
$124$	−4.74825	−0.426405
$125$	−7.75651	−0.693763
$126$	0	0
$127$	−10.0964	−0.895909	−0.447954	−	0.894056i	$-0.647847\pi$
$127$	−10.0964	−0.895909	−0.447954	+	0.894056i	$0.647847\pi$
$128$	20.0941	1.77608
$129$	0	0
$130$	−43.2152	−3.79023
$131$	20.4223	1.78430	0.892151	−	0.451736i	$-0.149195\pi$
$131$	20.4223	1.78430	0.892151	+	0.451736i	$0.149195\pi$
$132$	0	0
$133$	−1.41246	−0.122476
$134$	−8.78531	−0.758935
$135$	0	0
$136$	34.8232	2.98606
$137$	10.9157	0.932591	0.466296	−	0.884629i	$-0.345588\pi$
$137$	10.9157	0.932591	0.466296	+	0.884629i	$0.345588\pi$
$138$	0	0
$139$	−9.60677	−0.814836	−0.407418	−	0.913242i	$-0.633571\pi$
$139$	−9.60677	−0.814836	−0.407418	+	0.913242i	$0.633571\pi$
$140$	66.9143	5.65529
$141$	0	0
$142$	21.6632	1.81793
$143$	−8.28126	−0.692514
$144$	0	0
$145$	3.49556	0.290291
$146$	6.40011	0.529677
$147$	0	0
$148$	7.98031	0.655977
$149$	3.92991	0.321950	0.160975	−	0.986958i	$-0.448536\pi$
$149$	3.92991	0.321950	0.160975	+	0.986958i	$0.448536\pi$
$150$	0	0
$151$	9.79351	0.796985	0.398492	−	0.917172i	$-0.369534\pi$
$151$	9.79351	0.796985	0.398492	+	0.917172i	$0.369534\pi$
$152$	−1.36253	−0.110516
$153$	0	0
$154$	19.3487	1.55916
$155$	4.22367	0.339253
$156$	0	0
$157$	−1.60539	−0.128124	−0.0640621	−	0.997946i	$-0.520406\pi$
$157$	−1.60539	−0.128124	−0.0640621	+	0.997946i	$0.520406\pi$
$158$	9.18546	0.730756
$159$	0	0
$160$	−2.35100	−0.185863
$161$	−4.87125	−0.383909
$162$	0	0
$163$	−15.5495	−1.21793	−0.608964	−	0.793198i	$-0.708414\pi$
$163$	−15.5495	−1.21793	−0.608964	+	0.793198i	$0.708414\pi$
$164$	42.9670	3.35516
$165$	0	0
$166$	−26.0676	−2.02324
$167$	−0.764987	−0.0591965	−0.0295982	−	0.999562i	$-0.509423\pi$
$167$	−0.764987	−0.0591965	−0.0295982	+	0.999562i	$0.509423\pi$
$168$	0	0
$169$	12.7754	0.982722
$170$	−63.0801	−4.83802
$171$	0	0
$172$	22.2473	1.69634
$173$	−2.00062	−0.152104	−0.0760521	−	0.997104i	$-0.524232\pi$
$173$	−2.00062	−0.152104	−0.0760521	+	0.997104i	$0.524232\pi$
$174$	0	0
$175$	−35.1654	−2.65825
$176$	5.84480	0.440568
$177$	0	0
$178$	−11.0630	−0.829209
$179$	−20.3309	−1.51961	−0.759803	−	0.650154i	$-0.774704\pi$
$179$	−20.3309	−1.51961	−0.759803	+	0.650154i	$0.774704\pi$
$180$	0	0
$181$	5.46867	0.406483	0.203242	−	0.979129i	$-0.434852\pi$
$181$	5.46867	0.406483	0.203242	+	0.979129i	$0.434852\pi$
$182$	−60.2227	−4.46401
$183$	0	0
$184$	−4.69905	−0.346419
$185$	−7.09865	−0.521903
$186$	0	0
$187$	−12.0879	−0.883957
$188$	−20.1469	−1.46937
$189$	0	0
$190$	2.46814	0.179058
$191$	−6.82852	−0.494095	−0.247047	−	0.969003i	$-0.579460\pi$
$191$	−6.82852	−0.494095	−0.247047	+	0.969003i	$0.579460\pi$
$192$	0	0
$193$	19.2138	1.38304	0.691518	−	0.722359i	$-0.256942\pi$
$193$	19.2138	1.38304	0.691518	+	0.722359i	$0.256942\pi$
$194$	6.38986	0.458765
$195$	0	0
$196$	65.7407	4.69576
$197$	10.5750	0.753440	0.376720	−	0.926327i	$-0.377052\pi$
$197$	10.5750	0.753440	0.376720	+	0.926327i	$0.377052\pi$
$198$	0	0
$199$	−0.0141811	−0.00100527	−0.000502634	−	1.00000i	$-0.500160\pi$
$199$	−0.0141811	−0.00100527	−0.000502634		1.00000i	$0.500160\pi$
$200$	−33.9223	−2.39867
$201$	0	0
$202$	45.1568	3.17722
$203$	4.87125	0.341895
$204$	0	0
$205$	−38.2200	−2.66940
$206$	−16.2297	−1.13078
$207$	0	0
$208$	−18.1919	−1.26138
$209$	0.472966	0.0327157
$210$	0	0
$211$	−8.93797	−0.615316	−0.307658	−	0.951497i	$-0.599545\pi$
$211$	−8.93797	−0.615316	−0.307658	+	0.951497i	$0.599545\pi$
$212$	−28.1721	−1.93487
$213$	0	0
$214$	−31.8466	−2.17699
$215$	−19.7894	−1.34963
$216$	0	0
$217$	5.88591	0.399561
$218$	−32.5150	−2.20220
$219$	0	0
$220$	−22.4064	−1.51064
$221$	37.6236	2.53084
$222$	0	0
$223$	22.4912	1.50612	0.753062	−	0.657950i	$-0.228576\pi$
$223$	22.4912	1.50612	0.753062	+	0.657950i	$0.228576\pi$
$224$	−3.27624	−0.218903
$225$	0	0
$226$	10.1431	0.674707
$227$	−12.6937	−0.842510	−0.421255	−	0.906942i	$-0.638410\pi$
$227$	−12.6937	−0.842510	−0.421255	+	0.906942i	$0.638410\pi$
$228$	0	0
$229$	−19.9279	−1.31687	−0.658437	−	0.752636i	$-0.728782\pi$
$229$	−19.9279	−1.31687	−0.658437	+	0.752636i	$0.728782\pi$
$230$	8.51205	0.561268
$231$	0	0
$232$	4.69905	0.308508
$233$	−15.8089	−1.03568	−0.517838	−	0.855479i	$-0.673263\pi$
$233$	−15.8089	−1.03568	−0.517838	+	0.855479i	$0.673263\pi$
$234$	0	0
$235$	17.9211	1.16904
$236$	2.97888	0.193908
$237$	0	0
$238$	−87.9055	−5.69807
$239$	28.5189	1.84474	0.922368	−	0.386313i	$-0.126252\pi$
$239$	28.5189	1.84474	0.922368	+	0.386313i	$0.126252\pi$
$240$	0	0
$241$	−10.4957	−0.676087	−0.338044	−	0.941130i	$-0.609765\pi$
$241$	−10.4957	−0.676087	−0.338044	+	0.941130i	$0.609765\pi$
$242$	20.3072	1.30539
$243$	0	0
$244$	45.4650	2.91060
$245$	−58.4777	−3.73600
$246$	0	0
$247$	−1.47210	−0.0936677
$248$	5.67783	0.360543
$249$	0	0
$250$	18.8879	1.19457
$251$	−0.515831	−0.0325589	−0.0162795	−	0.999867i	$-0.505182\pi$
$251$	−0.515831	−0.0325589	−0.0162795	+	0.999867i	$0.505182\pi$
$252$	0	0
$253$	1.63115	0.102550
$254$	24.5857	1.54264
$255$	0	0
$256$	−31.3226	−1.95766
$257$	−11.5127	−0.718142	−0.359071	−	0.933310i	$-0.616906\pi$
$257$	−11.5127	−0.718142	−0.359071	+	0.933310i	$0.616906\pi$
$258$	0	0
$259$	−9.89234	−0.614680
$260$	69.7398	4.32508
$261$	0	0
$262$	−49.7303	−3.07235
$263$	3.77657	0.232873	0.116437	−	0.993198i	$-0.462853\pi$
$263$	3.77657	0.232873	0.116437	+	0.993198i	$0.462853\pi$
$264$	0	0
$265$	25.0597	1.53940
$266$	3.43949	0.210888
$267$	0	0
$268$	14.1776	0.866032
$269$	13.0119	0.793348	0.396674	−	0.917960i	$-0.370164\pi$
$269$	13.0119	0.793348	0.396674	+	0.917960i	$0.370164\pi$
$270$	0	0
$271$	−23.5566	−1.43096	−0.715480	−	0.698633i	$-0.753792\pi$
$271$	−23.5566	−1.43096	−0.715480	+	0.698633i	$0.753792\pi$
$272$	−26.5542	−1.61009
$273$	0	0
$274$	−26.5808	−1.60581
$275$	11.7752	0.710072
$276$	0	0
$277$	8.30189	0.498812	0.249406	−	0.968399i	$-0.419765\pi$
$277$	8.30189	0.498812	0.249406	+	0.968399i	$0.419765\pi$
$278$	23.3935	1.40305
$279$	0	0
$280$	−80.0144	−4.78178
$281$	24.5091	1.46209	0.731046	−	0.682328i	$-0.239032\pi$
$281$	24.5091	1.46209	0.731046	+	0.682328i	$0.239032\pi$
$282$	0	0
$283$	16.8683	1.00271	0.501357	−	0.865240i	$-0.332834\pi$
$283$	16.8683	1.00271	0.501357	+	0.865240i	$0.332834\pi$
$284$	−34.9596	−2.07447
$285$	0	0
$286$	20.1657	1.19242
$287$	−53.2617	−3.14394
$288$	0	0
$289$	37.9182	2.23048
$290$	−8.51205	−0.499845
$291$	0	0
$292$	−10.3284	−0.604422
$293$	−9.99230	−0.583756	−0.291878	−	0.956456i	$-0.594280\pi$
$293$	−9.99230	−0.583756	−0.291878	+	0.956456i	$0.594280\pi$
$294$	0	0
$295$	−2.64977	−0.154276
$296$	−9.54264	−0.554655
$297$	0	0
$298$	−9.56972	−0.554359
$299$	−5.07695	−0.293607
$300$	0	0
$301$	−27.5776	−1.58955
$302$	−23.8482	−1.37231
$303$	0	0
$304$	1.03899	0.0595901
$305$	−40.4420	−2.31570
$306$	0	0
$307$	−10.3287	−0.589488	−0.294744	−	0.955576i	$-0.595234\pi$
$307$	−10.3287	−0.589488	−0.294744	+	0.955576i	$0.595234\pi$
$308$	−31.2245	−1.77918
$309$	0	0
$310$	−10.2851	−0.584152
$311$	−21.5343	−1.22110	−0.610550	−	0.791978i	$-0.709051\pi$
$311$	−21.5343	−1.22110	−0.610550	+	0.791978i	$0.709051\pi$
$312$	0	0
$313$	−1.23924	−0.0700459	−0.0350229	−	0.999387i	$-0.511150\pi$
$313$	−1.23924	−0.0700459	−0.0350229	+	0.999387i	$0.511150\pi$
$314$	3.90929	0.220614
$315$	0	0
$316$	−14.8233	−0.833876
$317$	24.4749	1.37465	0.687325	−	0.726350i	$-0.258785\pi$
$317$	24.4749	1.37465	0.687325	+	0.726350i	$0.258785\pi$
$318$	0	0
$319$	−1.63115	−0.0913269
$320$	30.7758	1.72042
$321$	0	0
$322$	11.8620	0.661043
$323$	−2.14879	−0.119562
$324$	0	0
$325$	−36.6503	−2.03299
$326$	37.8645	2.09712
$327$	0	0
$328$	−51.3788	−2.83692
$329$	24.9740	1.37686
$330$	0	0
$331$	5.08020	0.279233	0.139616	−	0.990206i	$-0.455413\pi$
$331$	5.08020	0.279233	0.139616	+	0.990206i	$0.455413\pi$
$332$	42.0674	2.30875
$333$	0	0
$334$	1.86282	0.101929
$335$	−12.6112	−0.689025
$336$	0	0
$337$	−30.1891	−1.64450	−0.822252	−	0.569124i	$-0.807282\pi$
$337$	−30.1891	−1.64450	−0.822252	+	0.569124i	$0.807282\pi$
$338$	−31.1094	−1.69213
$339$	0	0
$340$	101.797	5.52074
$341$	−1.97091	−0.106731
$342$	0	0
$343$	−47.3930	−2.55898
$344$	−26.6027	−1.43432
$345$	0	0
$346$	4.87171	0.261905
$347$	15.1414	0.812831	0.406416	−	0.913688i	$-0.366779\pi$
$347$	15.1414	0.812831	0.406416	+	0.913688i	$0.366779\pi$
$348$	0	0
$349$	25.2696	1.35265	0.676325	−	0.736603i	$-0.263571\pi$
$349$	25.2696	1.35265	0.676325	+	0.736603i	$0.263571\pi$
$350$	85.6313	4.57718
$351$	0	0
$352$	1.09706	0.0584733
$353$	21.9355	1.16751	0.583755	−	0.811930i	$-0.301583\pi$
$353$	21.9355	1.16751	0.583755	+	0.811930i	$0.301583\pi$
$354$	0	0
$355$	31.0972	1.65047
$356$	17.8533	0.946222
$357$	0	0
$358$	49.5079	2.61657
$359$	11.5228	0.608148	0.304074	−	0.952648i	$-0.401653\pi$
$359$	11.5228	0.608148	0.304074	+	0.952648i	$0.401653\pi$
$360$	0	0
$361$	−18.9159	−0.995575
$362$	−13.3168	−0.699914
$363$	0	0
$364$	97.1862	5.09394
$365$	9.18729	0.480885
$366$	0	0
$367$	30.2955	1.58141	0.790707	−	0.612195i	$-0.209713\pi$
$367$	30.2955	1.58141	0.790707	+	0.612195i	$0.209713\pi$
$368$	3.58324	0.186789
$369$	0	0
$370$	17.2859	0.898652
$371$	34.9220	1.81306
$372$	0	0
$373$	12.2767	0.635664	0.317832	−	0.948147i	$-0.397045\pi$
$373$	12.2767	0.635664	0.317832	+	0.948147i	$0.397045\pi$
$374$	29.4353	1.52207
$375$	0	0
$376$	24.0912	1.24241
$377$	5.07695	0.261476
$378$	0	0
$379$	−10.3849	−0.533438	−0.266719	−	0.963774i	$-0.585940\pi$
$379$	−10.3849	−0.533438	−0.266719	+	0.963774i	$0.585940\pi$
$380$	−3.98303	−0.204325
$381$	0	0
$382$	16.6281	0.850770
$383$	−22.4826	−1.14880	−0.574402	−	0.818573i	$-0.694766\pi$
$383$	−22.4826	−1.14880	−0.574402	+	0.818573i	$0.694766\pi$
$384$	0	0
$385$	27.7749	1.41554
$386$	−46.7874	−2.38142
$387$	0	0
$388$	−10.3118	−0.523504
$389$	9.81702	0.497743	0.248871	−	0.968537i	$-0.419940\pi$
$389$	9.81702	0.497743	0.248871	+	0.968537i	$0.419940\pi$
$390$	0	0
$391$	−7.41068	−0.374774
$392$	−78.6110	−3.97046
$393$	0	0
$394$	−25.7513	−1.29733
$395$	13.1856	0.663441
$396$	0	0
$397$	1.40591	0.0705608	0.0352804	−	0.999377i	$-0.488768\pi$
$397$	1.40591	0.0705608	0.0352804	+	0.999377i	$0.488768\pi$
$398$	0.0345323	0.00173095
$399$	0	0
$400$	25.8672	1.29336
$401$	12.9350	0.645943	0.322971	−	0.946409i	$-0.395318\pi$
$401$	12.9350	0.645943	0.322971	+	0.946409i	$0.395318\pi$
$402$	0	0
$403$	6.13444	0.305578
$404$	−72.8731	−3.62557
$405$	0	0
$406$	−11.8620	−0.588701
$407$	3.31247	0.164193
$408$	0	0
$409$	25.6740	1.26950	0.634748	−	0.772720i	$-0.281104\pi$
$409$	25.6740	1.26950	0.634748	+	0.772720i	$0.281104\pi$
$410$	93.0696	4.59638
$411$	0	0
$412$	26.1912	1.29035
$413$	−3.69260	−0.181701
$414$	0	0
$415$	−37.4198	−1.83687
$416$	−3.41458	−0.167414
$417$	0	0
$418$	−1.15172	−0.0563324
$419$	−2.80144	−0.136859	−0.0684297	−	0.997656i	$-0.521799\pi$
$419$	−2.80144	−0.136859	−0.0684297	+	0.997656i	$0.521799\pi$
$420$	0	0
$421$	23.0333	1.12258	0.561288	−	0.827621i	$-0.310306\pi$
$421$	23.0333	1.12258	0.561288	+	0.827621i	$0.310306\pi$
$422$	21.7649	1.05950
$423$	0	0
$424$	33.6875	1.63601
$425$	−53.4974	−2.59501
$426$	0	0
$427$	−56.3581	−2.72736
$428$	51.3934	2.48419
$429$	0	0
$430$	48.1892	2.32389
$431$	−11.6913	−0.563150	−0.281575	−	0.959539i	$-0.590857\pi$
$431$	−11.6913	−0.563150	−0.281575	+	0.959539i	$0.590857\pi$
$432$	0	0
$433$	9.89512	0.475529	0.237765	−	0.971323i	$-0.423585\pi$
$433$	9.89512	0.475529	0.237765	+	0.971323i	$0.423585\pi$
$434$	−14.3328	−0.687995
$435$	0	0
$436$	52.4721	2.51296
$437$	0.289958	0.0138706
$438$	0	0
$439$	−25.3217	−1.20854	−0.604269	−	0.796780i	$-0.706535\pi$
$439$	−25.3217	−1.20854	−0.604269	+	0.796780i	$0.706535\pi$
$440$	26.7930	1.27731
$441$	0	0
$442$	−91.6174	−4.35779
$443$	−35.7692	−1.69944	−0.849722	−	0.527230i	$-0.823231\pi$
$443$	−35.7692	−1.69944	−0.849722	+	0.527230i	$0.823231\pi$
$444$	0	0
$445$	−15.8809	−0.752825
$446$	−54.7684	−2.59336
$447$	0	0
$448$	42.8877	2.02625
$449$	−22.9074	−1.08106	−0.540532	−	0.841323i	$-0.681777\pi$
$449$	−22.9074	−1.08106	−0.540532	+	0.841323i	$0.681777\pi$
$450$	0	0
$451$	17.8348	0.839807
$452$	−16.3687	−0.769918
$453$	0	0
$454$	30.9104	1.45070
$455$	−86.4491	−4.05280
$456$	0	0
$457$	16.8884	0.790005	0.395002	−	0.918680i	$-0.370744\pi$
$457$	16.8884	0.790005	0.395002	+	0.918680i	$0.370744\pi$
$458$	48.5265	2.26749
$459$	0	0
$460$	−13.7366	−0.640471
$461$	−28.9020	−1.34610	−0.673050	−	0.739597i	$-0.735016\pi$
$461$	−28.9020	−1.34610	−0.673050	+	0.739597i	$0.735016\pi$
$462$	0	0
$463$	−12.8535	−0.597351	−0.298675	−	0.954355i	$-0.596545\pi$
$463$	−12.8535	−0.597351	−0.298675	+	0.954355i	$0.596545\pi$
$464$	−3.58324	−0.166348
$465$	0	0
$466$	38.4963	1.78330
$467$	7.93973	0.367407	0.183703	−	0.982982i	$-0.441191\pi$
$467$	7.93973	0.367407	0.183703	+	0.982982i	$0.441191\pi$
$468$	0	0
$469$	−17.5744	−0.811511
$470$	−43.6397	−2.01295
$471$	0	0
$472$	−3.56207	−0.163957
$473$	9.23442	0.424599
$474$	0	0
$475$	2.09320	0.0960425
$476$	141.860	6.50215
$477$	0	0
$478$	−69.4464	−3.17641
$479$	−32.7143	−1.49475	−0.747377	−	0.664400i	$-0.768687\pi$
$479$	−32.7143	−1.49475	−0.747377	+	0.664400i	$0.768687\pi$
$480$	0	0
$481$	−10.3101	−0.470098
$482$	25.5581	1.16414
$483$	0	0
$484$	−32.7713	−1.48960
$485$	9.17258	0.416505
$486$	0	0
$487$	6.76700	0.306642	0.153321	−	0.988176i	$-0.451003\pi$
$487$	6.76700	0.306642	0.153321	+	0.988176i	$0.451003\pi$
$488$	−54.3658	−2.46103
$489$	0	0
$490$	142.399	6.43294
$491$	14.2398	0.642634	0.321317	−	0.946972i	$-0.395874\pi$
$491$	14.2398	0.642634	0.321317	+	0.946972i	$0.395874\pi$
$492$	0	0
$493$	7.41068	0.333760
$494$	3.58472	0.161284
$495$	0	0
$496$	−4.32960	−0.194405
$497$	43.3357	1.94387
$498$	0	0
$499$	−5.53838	−0.247932	−0.123966	−	0.992286i	$-0.539561\pi$
$499$	−5.53838	−0.247932	−0.123966	+	0.992286i	$0.539561\pi$
$500$	−30.4809	−1.36315
$501$	0	0
$502$	1.25610	0.0560625
$503$	−35.3182	−1.57476	−0.787381	−	0.616466i	$-0.788564\pi$
$503$	−35.3182	−1.57476	−0.787381	+	0.616466i	$0.788564\pi$
$504$	0	0
$505$	64.8221	2.88455
$506$	−3.97201	−0.176578
$507$	0	0
$508$	−39.6759	−1.76033
$509$	24.8206	1.10015	0.550076	−	0.835114i	$-0.314599\pi$
$509$	24.8206	1.10015	0.550076	+	0.835114i	$0.314599\pi$
$510$	0	0
$511$	12.8030	0.566371
$512$	36.0856	1.59477
$513$	0	0
$514$	28.0346	1.23655
$515$	−23.2976	−1.02661
$516$	0	0
$517$	−8.36261	−0.367787
$518$	24.0889	1.05840
$519$	0	0
$520$	−83.3931	−3.65703
$521$	−25.3463	−1.11044	−0.555221	−	0.831703i	$-0.687366\pi$
$521$	−25.3463	−1.11044	−0.555221	+	0.831703i	$0.687366\pi$
$522$	0	0
$523$	−1.12658	−0.0492617	−0.0246309	−	0.999697i	$-0.507841\pi$
$523$	−1.12658	−0.0492617	−0.0246309	+	0.999697i	$0.507841\pi$
$524$	80.2538	3.50590
$525$	0	0
$526$	−9.19633	−0.400979
$527$	8.95428	0.390054
$528$	0	0
$529$	1.00000	0.0434783
$530$	−61.0228	−2.65066
$531$	0	0
$532$	−5.55057	−0.240648
$533$	−55.5107	−2.40443
$534$	0	0
$535$	−45.7154	−1.97645
$536$	−16.9532	−0.732265
$537$	0	0
$538$	−31.6852	−1.36605
$539$	27.2877	1.17536
$540$	0	0
$541$	−37.1099	−1.59548	−0.797740	−	0.603001i	$-0.793972\pi$
$541$	−37.1099	−1.59548	−0.797740	+	0.603001i	$0.793972\pi$
$542$	57.3626	2.46394
$543$	0	0
$544$	−4.98417	−0.213695
$545$	−46.6750	−1.99934
$546$	0	0
$547$	4.86456	0.207993	0.103997	−	0.994578i	$-0.466837\pi$
$547$	4.86456	0.207993	0.103997	+	0.994578i	$0.466837\pi$
$548$	42.8956	1.83241
$549$	0	0
$550$	−28.6738	−1.22266
$551$	−0.289958	−0.0123526
$552$	0	0
$553$	18.3749	0.781380
$554$	−20.2159	−0.858893
$555$	0	0
$556$	−37.7519	−1.60104
$557$	23.4839	0.995043	0.497521	−	0.867452i	$-0.334244\pi$
$557$	23.4839	0.995043	0.497521	+	0.867452i	$0.334244\pi$
$558$	0	0
$559$	−28.7421	−1.21566
$560$	61.0145	2.57833
$561$	0	0
$562$	−59.6822	−2.51754
$563$	42.1697	1.77724	0.888620	−	0.458645i	$-0.151665\pi$
$563$	42.1697	1.77724	0.888620	+	0.458645i	$0.151665\pi$
$564$	0	0
$565$	14.5603	0.612555
$566$	−41.0759	−1.72655
$567$	0	0
$568$	41.8037	1.75405
$569$	2.21802	0.0929842	0.0464921	−	0.998919i	$-0.485196\pi$
$569$	2.21802	0.0929842	0.0464921	+	0.998919i	$0.485196\pi$
$570$	0	0
$571$	19.4117	0.812353	0.406177	−	0.913795i	$-0.366862\pi$
$571$	19.4117	0.812353	0.406177	+	0.913795i	$0.366862\pi$
$572$	−32.5430	−1.36069
$573$	0	0
$574$	129.698	5.41347
$575$	7.21896	0.301051
$576$	0	0
$577$	−1.65159	−0.0687565	−0.0343782	−	0.999409i	$-0.510945\pi$
$577$	−1.65159	−0.0687565	−0.0343782	+	0.999409i	$0.510945\pi$
$578$	−92.3346	−3.84061
$579$	0	0
$580$	13.7366	0.570380
$581$	−52.1465	−2.16340
$582$	0	0
$583$	−11.6937	−0.484304
$584$	12.3504	0.511063
$585$	0	0
$586$	24.3322	1.00516
$587$	17.7277	0.731698	0.365849	−	0.930674i	$-0.380779\pi$
$587$	17.7277	0.731698	0.365849	+	0.930674i	$0.380779\pi$
$588$	0	0
$589$	−0.350355	−0.0144361
$590$	6.45246	0.265644
$591$	0	0
$592$	7.27669	0.299070
$593$	1.20316	0.0494078	0.0247039	−	0.999695i	$-0.492136\pi$
$593$	1.20316	0.0494078	0.0247039	+	0.999695i	$0.492136\pi$
$594$	0	0
$595$	−126.187	−5.17318
$596$	15.4434	0.632587
$597$	0	0
$598$	12.3629	0.505556
$599$	36.3309	1.48444	0.742221	−	0.670155i	$-0.233773\pi$
$599$	36.3309	1.48444	0.742221	+	0.670155i	$0.233773\pi$
$600$	0	0
$601$	−2.62167	−0.106940	−0.0534701	−	0.998569i	$-0.517028\pi$
$601$	−2.62167	−0.106940	−0.0534701	+	0.998569i	$0.517028\pi$
$602$	67.1543	2.73700
$603$	0	0
$604$	38.4857	1.56596
$605$	29.1507	1.18515
$606$	0	0
$607$	−6.15983	−0.250020	−0.125010	−	0.992155i	$-0.539896\pi$
$607$	−6.15983	−0.250020	−0.125010	+	0.992155i	$0.539896\pi$
$608$	0.195016	0.00790895
$609$	0	0
$610$	98.4804	3.98735
$611$	26.0286	1.05300
$612$	0	0
$613$	18.2042	0.735259	0.367630	−	0.929972i	$-0.380169\pi$
$613$	18.2042	0.735259	0.367630	+	0.929972i	$0.380169\pi$
$614$	25.1513	1.01502
$615$	0	0
$616$	37.3375	1.50437
$617$	−7.55796	−0.304272	−0.152136	−	0.988360i	$-0.548615\pi$
$617$	−7.55796	−0.304272	−0.152136	+	0.988360i	$0.548615\pi$
$618$	0	0
$619$	−3.36490	−0.135247	−0.0676233	−	0.997711i	$-0.521542\pi$
$619$	−3.36490	−0.135247	−0.0676233	+	0.997711i	$0.521542\pi$
$620$	16.5978	0.666584
$621$	0	0
$622$	52.4382	2.10258
$623$	−22.1308	−0.886653
$624$	0	0
$625$	−8.98142	−0.359257
$626$	3.01767	0.120610
$627$	0	0
$628$	−6.30873	−0.251746
$629$	−15.0493	−0.600055
$630$	0	0
$631$	4.41948	0.175937	0.0879684	−	0.996123i	$-0.471963\pi$
$631$	4.41948	0.175937	0.0879684	+	0.996123i	$0.471963\pi$
$632$	17.7253	0.705076
$633$	0	0
$634$	−59.5989	−2.36698
$635$	35.2925	1.40054
$636$	0	0
$637$	−84.9329	−3.36516
$638$	3.97201	0.157254
$639$	0	0
$640$	−70.2401	−2.77648
$641$	−27.1330	−1.07169	−0.535845	−	0.844316i	$-0.680007\pi$
$641$	−27.1330	−1.07169	−0.535845	+	0.844316i	$0.680007\pi$
$642$	0	0
$643$	18.4069	0.725896	0.362948	−	0.931809i	$-0.381770\pi$
$643$	18.4069	0.725896	0.362948	+	0.931809i	$0.381770\pi$
$644$	−19.1426	−0.754326
$645$	0	0
$646$	5.23252	0.205871
$647$	−31.4390	−1.23600	−0.617998	−	0.786180i	$-0.712056\pi$
$647$	−31.4390	−1.23600	−0.617998	+	0.786180i	$0.712056\pi$
$648$	0	0
$649$	1.23648	0.0485359
$650$	89.2471	3.50056
$651$	0	0
$652$	−61.1049	−2.39305
$653$	3.57315	0.139828	0.0699140	−	0.997553i	$-0.477728\pi$
$653$	3.57315	0.139828	0.0699140	+	0.997553i	$0.477728\pi$
$654$	0	0
$655$	−71.3874	−2.78934
$656$	39.1786	1.52967
$657$	0	0
$658$	−60.8143	−2.37079
$659$	−4.59543	−0.179012	−0.0895062	−	0.995986i	$-0.528529\pi$
$659$	−4.59543	−0.179012	−0.0895062	+	0.995986i	$0.528529\pi$
$660$	0	0
$661$	−29.4071	−1.14380	−0.571901	−	0.820322i	$-0.693794\pi$
$661$	−29.4071	−1.14380	−0.571901	+	0.820322i	$0.693794\pi$
$662$	−12.3708	−0.480804
$663$	0	0
$664$	−50.3031	−1.95214
$665$	4.93735	0.191462
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−3.00618	−0.116313
$669$	0	0
$670$	30.7096	1.18642
$671$	18.8716	0.728532
$672$	0	0
$673$	25.9468	1.00018	0.500088	−	0.865975i	$-0.333301\pi$
$673$	25.9468	1.00018	0.500088	+	0.865975i	$0.333301\pi$
$674$	73.5134	2.83163
$675$	0	0
$676$	50.2037	1.93091
$677$	−12.6758	−0.487170	−0.243585	−	0.969880i	$-0.578323\pi$
$677$	−12.6758	−0.487170	−0.243585	+	0.969880i	$0.578323\pi$
$678$	0	0
$679$	12.7825	0.490547
$680$	−121.727	−4.66800
$681$	0	0
$682$	4.79936	0.183777
$683$	−41.1437	−1.57432	−0.787159	−	0.616750i	$-0.788449\pi$
$683$	−41.1437	−1.57432	−0.787159	+	0.616750i	$0.788449\pi$
$684$	0	0
$685$	−38.1565	−1.45789
$686$	115.407	4.40625
$687$	0	0
$688$	20.2858	0.773387
$689$	36.3966	1.38660
$690$	0	0
$691$	−44.6369	−1.69807	−0.849035	−	0.528337i	$-0.822816\pi$
$691$	−44.6369	−1.69807	−0.849035	+	0.528337i	$0.822816\pi$
$692$	−7.86187	−0.298863
$693$	0	0
$694$	−36.8708	−1.39959
$695$	33.5811	1.27380
$696$	0	0
$697$	−81.0274	−3.06913
$698$	−61.5340	−2.32910
$699$	0	0
$700$	−138.190	−5.22309
$701$	18.9701	0.716489	0.358244	−	0.933628i	$-0.383375\pi$
$701$	18.9701	0.716489	0.358244	+	0.933628i	$0.383375\pi$
$702$	0	0
$703$	0.588836	0.0222084
$704$	−14.3610	−0.541252
$705$	0	0
$706$	−53.4152	−2.01031
$707$	90.3331	3.39733
$708$	0	0
$709$	−12.2432	−0.459802	−0.229901	−	0.973214i	$-0.573840\pi$
$709$	−12.2432	−0.459802	−0.229901	+	0.973214i	$0.573840\pi$
$710$	−75.7249	−2.84191
$711$	0	0
$712$	−21.3485	−0.800069
$713$	−1.20829	−0.0452509
$714$	0	0
$715$	28.9477	1.08258
$716$	−79.8948	−2.98581
$717$	0	0
$718$	−28.0591	−1.04716
$719$	−6.46404	−0.241068	−0.120534	−	0.992709i	$-0.538461\pi$
$719$	−6.46404	−0.241068	−0.120534	+	0.992709i	$0.538461\pi$
$720$	0	0
$721$	−32.4664	−1.20911
$722$	46.0622	1.71426
$723$	0	0
$724$	21.4903	0.798682
$725$	−7.21896	−0.268105
$726$	0	0
$727$	12.3230	0.457034	0.228517	−	0.973540i	$-0.426612\pi$
$727$	12.3230	0.457034	0.228517	+	0.973540i	$0.426612\pi$
$728$	−116.213	−4.30713
$729$	0	0
$730$	−22.3720	−0.828024
$731$	−41.9541	−1.55173
$732$	0	0
$733$	19.1603	0.707702	0.353851	−	0.935302i	$-0.384872\pi$
$733$	19.1603	0.707702	0.353851	+	0.935302i	$0.384872\pi$
$734$	−73.7726	−2.72300
$735$	0	0
$736$	0.672566	0.0247911
$737$	5.88483	0.216771
$738$	0	0
$739$	32.4622	1.19414	0.597071	−	0.802189i	$-0.296331\pi$
$739$	32.4622	1.19414	0.597071	+	0.802189i	$0.296331\pi$
$740$	−27.8957	−1.02546
$741$	0	0
$742$	−85.0386	−3.12187
$743$	13.5331	0.496481	0.248240	−	0.968698i	$-0.420148\pi$
$743$	13.5331	0.496481	0.248240	+	0.968698i	$0.420148\pi$
$744$	0	0
$745$	−13.7372	−0.503293
$746$	−29.8950	−1.09453
$747$	0	0
$748$	−47.5021	−1.73685
$749$	−63.7069	−2.32780
$750$	0	0
$751$	−27.8722	−1.01707	−0.508535	−	0.861041i	$-0.669813\pi$
$751$	−27.8722	−1.01707	−0.508535	+	0.861041i	$0.669813\pi$
$752$	−18.3706	−0.669907
$753$	0	0
$754$	−12.3629	−0.450229
$755$	−34.2338	−1.24590
$756$	0	0
$757$	−41.0638	−1.49249	−0.746245	−	0.665672i	$-0.768145\pi$
$757$	−41.0638	−1.49249	−0.746245	+	0.665672i	$0.768145\pi$
$758$	25.2883	0.918514
$759$	0	0
$760$	4.76281	0.172765
$761$	−38.0521	−1.37939	−0.689694	−	0.724101i	$-0.742255\pi$
$761$	−38.0521	−1.37939	−0.689694	+	0.724101i	$0.742255\pi$
$762$	0	0
$763$	−65.0441	−2.35476
$764$	−26.8342	−0.970826
$765$	0	0
$766$	54.7473	1.97810
$767$	−3.84852	−0.138962
$768$	0	0
$769$	22.2151	0.801097	0.400549	−	0.916276i	$-0.368820\pi$
$769$	22.2151	0.801097	0.400549	+	0.916276i	$0.368820\pi$
$770$	−67.6346	−2.43738
$771$	0	0
$772$	75.5046	2.71747
$773$	−42.9479	−1.54473	−0.772365	−	0.635179i	$-0.780926\pi$
$773$	−42.9479	−1.54473	−0.772365	+	0.635179i	$0.780926\pi$
$774$	0	0
$775$	−8.72262	−0.313326
$776$	12.3306	0.442643
$777$	0	0
$778$	−23.9054	−0.857051
$779$	3.17037	0.113590
$780$	0	0
$781$	−14.5110	−0.519246
$782$	18.0458	0.645315
$783$	0	0
$784$	59.9444	2.14087
$785$	5.61175	0.200292
$786$	0	0
$787$	31.1228	1.10941	0.554703	−	0.832048i	$-0.312832\pi$
$787$	31.1228	1.10941	0.554703	+	0.832048i	$0.312832\pi$
$788$	41.5569	1.48040
$789$	0	0
$790$	−32.1084	−1.14236
$791$	20.2905	0.721448
$792$	0	0
$793$	−58.7379	−2.08584
$794$	−3.42354	−0.121497
$795$	0	0
$796$	−0.0557275	−0.00197521
$797$	−24.1202	−0.854383	−0.427191	−	0.904161i	$-0.640497\pi$
$797$	−24.1202	−0.854383	−0.427191	+	0.904161i	$0.640497\pi$
$798$	0	0
$799$	37.9932	1.34410
$800$	4.85523	0.171658
$801$	0	0
$802$	−31.4980	−1.11223
$803$	−4.28711	−0.151289
$804$	0	0
$805$	17.0278	0.600150
$806$	−14.9380	−0.526168
$807$	0	0
$808$	87.1398	3.06557
$809$	22.2731	0.783081	0.391541	−	0.920161i	$-0.371942\pi$
$809$	22.2731	0.783081	0.391541	+	0.920161i	$0.371942\pi$
$810$	0	0
$811$	−38.2815	−1.34424	−0.672122	−	0.740440i	$-0.734617\pi$
$811$	−38.2815	−1.34424	−0.672122	+	0.740440i	$0.734617\pi$
$812$	19.1426	0.671775
$813$	0	0
$814$	−8.06621	−0.282720
$815$	54.3541	1.90394
$816$	0	0
$817$	1.64154	0.0574302
$818$	−62.5187	−2.18591
$819$	0	0
$820$	−150.194	−5.24500
$821$	−17.3465	−0.605395	−0.302698	−	0.953087i	$-0.597887\pi$
$821$	−17.3465	−0.605395	−0.302698	+	0.953087i	$0.597887\pi$
$822$	0	0
$823$	5.80452	0.202333	0.101166	−	0.994870i	$-0.467743\pi$
$823$	5.80452	0.202333	0.101166	+	0.994870i	$0.467743\pi$
$824$	−31.3187	−1.09104
$825$	0	0
$826$	8.99186	0.312867
$827$	27.5059	0.956475	0.478237	−	0.878231i	$-0.341276\pi$
$827$	27.5059	0.956475	0.478237	+	0.878231i	$0.341276\pi$
$828$	0	0
$829$	18.3056	0.635779	0.317890	−	0.948128i	$-0.397026\pi$
$829$	18.3056	0.635779	0.317890	+	0.948128i	$0.397026\pi$
$830$	91.1210	3.16286
$831$	0	0
$832$	44.6986	1.54965
$833$	−123.974	−4.29545
$834$	0	0
$835$	2.67406	0.0925397
$836$	1.85862	0.0642817
$837$	0	0
$838$	6.82179	0.235655
$839$	0.291572	0.0100662	0.00503308	−	0.999987i	$-0.498398\pi$
$839$	0.291572	0.0100662	0.00503308	+	0.999987i	$0.498398\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−56.0885	−1.93294
$843$	0	0
$844$	−35.1237	−1.20901
$845$	−44.6572	−1.53625
$846$	0	0
$847$	40.6231	1.39583
$848$	−25.6882	−0.882136
$849$	0	0
$850$	130.272	4.46828
$851$	2.03076	0.0696135
$852$	0	0
$853$	−25.6773	−0.879175	−0.439587	−	0.898200i	$-0.644875\pi$
$853$	−25.6773	−0.879175	−0.439587	+	0.898200i	$0.644875\pi$
$854$	137.238	4.69618
$855$	0	0
$856$	−61.4549	−2.10048
$857$	40.3915	1.37975	0.689874	−	0.723930i	$-0.257666\pi$
$857$	40.3915	1.37975	0.689874	+	0.723930i	$0.257666\pi$
$858$	0	0
$859$	−5.56391	−0.189838	−0.0949191	−	0.995485i	$-0.530259\pi$
$859$	−5.56391	−0.189838	−0.0949191	+	0.995485i	$0.530259\pi$
$860$	−77.7668	−2.65182
$861$	0	0
$862$	28.4695	0.969674
$863$	−42.7773	−1.45616	−0.728078	−	0.685494i	$-0.759586\pi$
$863$	−42.7773	−1.45616	−0.728078	+	0.685494i	$0.759586\pi$
$864$	0	0
$865$	6.99329	0.237779
$866$	−24.0956	−0.818802
$867$	0	0
$868$	23.1299	0.785081
$869$	−6.15287	−0.208722
$870$	0	0
$871$	−18.3165	−0.620631
$872$	−62.7448	−2.12481
$873$	0	0
$874$	−0.706078	−0.0238834
$875$	37.7839	1.27733
$876$	0	0
$877$	−44.3278	−1.49684	−0.748422	−	0.663223i	$-0.769188\pi$
$877$	−44.3278	−1.49684	−0.748422	+	0.663223i	$0.769188\pi$
$878$	61.6609	2.08095
$879$	0	0
$880$	−20.4308	−0.688724
$881$	30.0372	1.01198	0.505990	−	0.862539i	$-0.331127\pi$
$881$	30.0372	1.01198	0.505990	+	0.862539i	$0.331127\pi$
$882$	0	0
$883$	−29.4304	−0.990411	−0.495205	−	0.868776i	$-0.664907\pi$
$883$	−29.4304	−0.990411	−0.495205	+	0.868776i	$0.664907\pi$
$884$	147.850	4.97274
$885$	0	0
$886$	87.1016	2.92623
$887$	57.8556	1.94260	0.971301	−	0.237854i	$-0.0764439\pi$
$887$	57.8556	1.94260	0.971301	+	0.237854i	$0.0764439\pi$
$888$	0	0
$889$	49.1820	1.64951
$890$	38.6715	1.29627
$891$	0	0
$892$	88.3841	2.95932
$893$	−1.48656	−0.0497460
$894$	0	0
$895$	71.0681	2.37554
$896$	−97.8833	−3.27005
$897$	0	0
$898$	55.7817	1.86146
$899$	1.20829	0.0402988
$900$	0	0
$901$	53.1271	1.76992
$902$	−43.4295	−1.44604
$903$	0	0
$904$	19.5732	0.650996
$905$	−19.1161	−0.635440
$906$	0	0
$907$	−27.2768	−0.905710	−0.452855	−	0.891584i	$-0.649595\pi$
$907$	−27.2768	−0.905710	−0.452855	+	0.891584i	$0.649595\pi$
$908$	−49.8826	−1.65541
$909$	0	0
$910$	210.512	6.97842
$911$	24.0599	0.797139	0.398569	−	0.917138i	$-0.369507\pi$
$911$	24.0599	0.797139	0.398569	+	0.917138i	$0.369507\pi$
$912$	0	0
$913$	17.4614	0.577887
$914$	−41.1249	−1.36029
$915$	0	0
$916$	−78.3110	−2.58747
$917$	−99.4821	−3.28519
$918$	0	0
$919$	−38.2959	−1.26326	−0.631632	−	0.775269i	$-0.717615\pi$
$919$	−38.2959	−1.26326	−0.631632	+	0.775269i	$0.717615\pi$
$920$	16.4258	0.541544
$921$	0	0
$922$	70.3792	2.31782
$923$	45.1656	1.48664
$924$	0	0
$925$	14.6600	0.482017
$926$	31.2995	1.02856
$927$	0	0
$928$	−0.672566	−0.0220781
$929$	−16.9016	−0.554523	−0.277262	−	0.960794i	$-0.589427\pi$
$929$	−16.9016	−0.554523	−0.277262	+	0.960794i	$0.589427\pi$
$930$	0	0
$931$	4.85075	0.158977
$932$	−62.1245	−2.03495
$933$	0	0
$934$	−19.3340	−0.632629
$935$	42.2541	1.38186
$936$	0	0
$937$	0.00392697	0.000128289 0	6.41443e−5	−	1.00000i	$-0.499980\pi$
$937$	0.00392697	0.000128289 0	6.41443e−5		1.00000i	$0.499980\pi$
$938$	42.7955	1.39732
$939$	0	0
$940$	70.4249	2.29701
$941$	25.9434	0.845731	0.422865	−	0.906193i	$-0.361024\pi$
$941$	25.9434	0.845731	0.422865	+	0.906193i	$0.361024\pi$
$942$	0	0
$943$	10.9339	0.356056
$944$	2.71623	0.0884058
$945$	0	0
$946$	−22.4867	−0.731107
$947$	−17.3620	−0.564188	−0.282094	−	0.959387i	$-0.591029\pi$
$947$	−17.3620	−0.564188	−0.282094	+	0.959387i	$0.591029\pi$
$948$	0	0
$949$	13.3436	0.433151
$950$	−5.09715	−0.165373
$951$	0	0
$952$	−169.633	−5.49782
$953$	−28.5774	−0.925713	−0.462857	−	0.886433i	$-0.653175\pi$
$953$	−28.5774	−0.925713	−0.462857	+	0.886433i	$0.653175\pi$
$954$	0	0
$955$	23.8695	0.772400
$956$	112.071	3.62464
$957$	0	0
$958$	79.6626	2.57378
$959$	−53.1732	−1.71705
$960$	0	0
$961$	−29.5400	−0.952904
$962$	25.1060	0.809451
$963$	0	0
$964$	−41.2451	−1.32842
$965$	−67.1629	−2.16205
$966$	0	0
$967$	−3.08127	−0.0990869	−0.0495435	−	0.998772i	$-0.515777\pi$
$967$	−3.08127	−0.0990869	−0.0495435	+	0.998772i	$0.515777\pi$
$968$	39.1870	1.25952
$969$	0	0
$970$	−22.3362	−0.717171
$971$	10.1788	0.326652	0.163326	−	0.986572i	$-0.447778\pi$
$971$	10.1788	0.326652	0.163326	+	0.986572i	$0.447778\pi$
$972$	0	0
$973$	46.7970	1.50024
$974$	−16.4783	−0.528000
$975$	0	0
$976$	41.4563	1.32699
$977$	51.5899	1.65051	0.825253	−	0.564763i	$-0.191032\pi$
$977$	51.5899	1.65051	0.825253	+	0.564763i	$0.191032\pi$
$978$	0	0
$979$	7.41056	0.236843
$980$	−229.801	−7.34072
$981$	0	0
$982$	−34.6754	−1.10654
$983$	40.0166	1.27633	0.638166	−	0.769899i	$-0.279693\pi$
$983$	40.0166	1.27633	0.638166	+	0.769899i	$0.279693\pi$
$984$	0	0
$985$	−36.9657	−1.17783
$986$	−18.0458	−0.574694
$987$	0	0
$988$	−5.78495	−0.184044
$989$	5.66129	0.180019
$990$	0	0
$991$	−16.9438	−0.538238	−0.269119	−	0.963107i	$-0.586733\pi$
$991$	−16.9438	−0.538238	−0.269119	+	0.963107i	$0.586733\pi$
$992$	−0.812657	−0.0258019
$993$	0	0
$994$	−105.527	−3.34710
$995$	0.0495708	0.00157150
$996$	0	0
$997$	15.0505	0.476655	0.238327	−	0.971185i	$-0.423401\pi$
$997$	15.0505	0.476655	0.238327	+	0.971185i	$0.423401\pi$
$998$	13.4865	0.426908
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.o.1.2		13
3.2	odd	2		667.2.a.c.1.12	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.c.1.12	✓	13	3.2	odd	2
6003.2.a.o.1.2		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)

\(f(q)\)	\(=\)	\(q-2.43510 q^{2} +3.92972 q^{4} -3.49556 q^{5} -4.87125 q^{7} -4.69905 q^{8} +O(q^{10})\)	\(q-2.43510 q^{2} +3.92972 q^{4} -3.49556 q^{5} -4.87125 q^{7} -4.69905 q^{8} +8.51205 q^{10} +1.63115 q^{11} -5.07695 q^{13} +11.8620 q^{14} +3.58324 q^{16} -7.41068 q^{17} +0.289958 q^{19} -13.7366 q^{20} -3.97201 q^{22} +1.00000 q^{23} +7.21896 q^{25} +12.3629 q^{26} -19.1426 q^{28} -1.00000 q^{29} -1.20829 q^{31} +0.672566 q^{32} +18.0458 q^{34} +17.0278 q^{35} +2.03076 q^{37} -0.706078 q^{38} +16.4258 q^{40} +10.9339 q^{41} +5.66129 q^{43} +6.40996 q^{44} -2.43510 q^{46} -5.12682 q^{47} +16.7291 q^{49} -17.5789 q^{50} -19.9510 q^{52} -7.16900 q^{53} -5.70179 q^{55} +22.8903 q^{56} +2.43510 q^{58} +0.758039 q^{59} +11.5695 q^{61} +2.94232 q^{62} -8.80424 q^{64} +17.7468 q^{65} +3.60778 q^{67} -29.1219 q^{68} -41.4644 q^{70} -8.89620 q^{71} -2.62827 q^{73} -4.94510 q^{74} +1.13945 q^{76} -7.94575 q^{77} -3.77211 q^{79} -12.5254 q^{80} -26.6251 q^{82} +10.7050 q^{83} +25.9045 q^{85} -13.7858 q^{86} -7.66486 q^{88} +4.54315 q^{89} +24.7311 q^{91} +3.92972 q^{92} +12.4843 q^{94} -1.01357 q^{95} -2.62406 q^{97} -40.7371 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) 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

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Database

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