LMFDB - Embedded newform 6003.2.a.o.1.13

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			1.13
Root			$-2.26622$ 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)

$f(q)$	$=$	$q+2.26622 q^{2} +3.13576 q^{4} -0.126764 q^{5} +2.95229 q^{7} +2.57389 q^{8} +O(q^{10})$	\(q+2.26622 q^{2} +3.13576 q^{4} -0.126764 q^{5} +2.95229 q^{7} +2.57389 q^{8} -0.287275 q^{10} -5.41609 q^{11} -0.978947 q^{13} +6.69055 q^{14} -0.438517 q^{16} -5.88336 q^{17} -3.80767 q^{19} -0.397501 q^{20} -12.2741 q^{22} +1.00000 q^{23} -4.98393 q^{25} -2.21851 q^{26} +9.25768 q^{28} -1.00000 q^{29} -3.86785 q^{31} -6.14156 q^{32} -13.3330 q^{34} -0.374243 q^{35} -8.52106 q^{37} -8.62902 q^{38} -0.326276 q^{40} -1.33604 q^{41} +12.3792 q^{43} -16.9836 q^{44} +2.26622 q^{46} -3.56605 q^{47} +1.71602 q^{49} -11.2947 q^{50} -3.06975 q^{52} -9.81101 q^{53} +0.686563 q^{55} +7.59887 q^{56} -2.26622 q^{58} +12.6643 q^{59} +14.6345 q^{61} -8.76540 q^{62} -13.0411 q^{64} +0.124095 q^{65} +1.26829 q^{67} -18.4488 q^{68} -0.848119 q^{70} -15.1978 q^{71} +12.1315 q^{73} -19.3106 q^{74} -11.9399 q^{76} -15.9899 q^{77} -11.3081 q^{79} +0.0555880 q^{80} -3.02776 q^{82} -8.69580 q^{83} +0.745796 q^{85} +28.0539 q^{86} -13.9404 q^{88} +3.58092 q^{89} -2.89014 q^{91} +3.13576 q^{92} -8.08147 q^{94} +0.482674 q^{95} +16.5153 q^{97} +3.88888 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.26622	1.60246	0.801231	−	0.598356i	$-0.204179\pi$
$2$	2.26622	1.60246	0.801231	+	0.598356i	$0.204179\pi$
$3$	0	0
$3$	0	0
$4$	3.13576	1.56788
$5$	−0.126764	−0.0566905	−0.0283452	−	0.999598i	$-0.509024\pi$
$5$	−0.126764	−0.0566905	−0.0283452	+	0.999598i	$0.509024\pi$
$6$	0	0
$7$	2.95229	1.11586	0.557930	−	0.829888i	$-0.311596\pi$
$7$	2.95229	1.11586	0.557930	+	0.829888i	$0.311596\pi$
$8$	2.57389	0.910008
$9$	0	0
$10$	−0.287275	−0.0908443
$11$	−5.41609	−1.63301	−0.816506	−	0.577337i	$-0.804092\pi$
$11$	−5.41609	−1.63301	−0.816506	+	0.577337i	$0.804092\pi$
$12$	0	0
$13$	−0.978947	−0.271511	−0.135756	−	0.990742i	$-0.543346\pi$
$13$	−0.978947	−0.271511	−0.135756	+	0.990742i	$0.543346\pi$
$14$	6.69055	1.78812
$15$	0	0
$16$	−0.438517	−0.109629
$17$	−5.88336	−1.42692	−0.713462	−	0.700694i	$-0.752874\pi$
$17$	−5.88336	−1.42692	−0.713462	+	0.700694i	$0.752874\pi$
$18$	0	0
$19$	−3.80767	−0.873538	−0.436769	−	0.899574i	$-0.643877\pi$
$19$	−3.80767	−0.873538	−0.436769	+	0.899574i	$0.643877\pi$
$20$	−0.397501	−0.0888839
$21$	0	0
$22$	−12.2741	−2.61684
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.98393	−0.996786
$26$	−2.21851	−0.435086
$27$	0	0
$28$	9.25768	1.74954
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−3.86785	−0.694686	−0.347343	−	0.937738i	$-0.612916\pi$
$31$	−3.86785	−0.694686	−0.347343	+	0.937738i	$0.612916\pi$
$32$	−6.14156	−1.08568
$33$	0	0
$34$	−13.3330	−2.28659
$35$	−0.374243	−0.0632587
$36$	0	0
$37$	−8.52106	−1.40085	−0.700427	−	0.713724i	$-0.747007\pi$
$37$	−8.52106	−1.40085	−0.700427	+	0.713724i	$0.747007\pi$
$38$	−8.62902	−1.39981
$39$	0	0
$40$	−0.326276	−0.0515888
$41$	−1.33604	−0.208654	−0.104327	−	0.994543i	$-0.533269\pi$
$41$	−1.33604	−0.208654	−0.104327	+	0.994543i	$0.533269\pi$
$42$	0	0
$43$	12.3792	1.88780	0.943902	−	0.330226i	$-0.107125\pi$
$43$	12.3792	1.88780	0.943902	+	0.330226i	$0.107125\pi$
$44$	−16.9836	−2.56037
$45$	0	0
$46$	2.26622	0.334136
$47$	−3.56605	−0.520162	−0.260081	−	0.965587i	$-0.583749\pi$
$47$	−3.56605	−0.520162	−0.260081	+	0.965587i	$0.583749\pi$
$48$	0	0
$49$	1.71602	0.245146
$50$	−11.2947	−1.59731
$51$	0	0
$52$	−3.06975	−0.425697
$53$	−9.81101	−1.34765	−0.673823	−	0.738893i	$-0.735349\pi$
$53$	−9.81101	−1.34765	−0.673823	+	0.738893i	$0.735349\pi$
$54$	0	0
$55$	0.686563	0.0925762
$56$	7.59887	1.01544
$57$	0	0
$58$	−2.26622	−0.297570
$59$	12.6643	1.64875	0.824375	−	0.566045i	$-0.191527\pi$
$59$	12.6643	1.64875	0.824375	+	0.566045i	$0.191527\pi$
$60$	0	0
$61$	14.6345	1.87375	0.936875	−	0.349663i	$-0.113704\pi$
$61$	14.6345	1.87375	0.936875	+	0.349663i	$0.113704\pi$
$62$	−8.76540	−1.11321
$63$	0	0
$64$	−13.0411	−1.63014
$65$	0.124095	0.0153921
$66$	0	0
$67$	1.26829	0.154947	0.0774733	−	0.996994i	$-0.475315\pi$
$67$	1.26829	0.154947	0.0774733	+	0.996994i	$0.475315\pi$
$68$	−18.4488	−2.23725
$69$	0	0
$70$	−0.848119	−0.101370
$71$	−15.1978	−1.80365	−0.901825	−	0.432100i	$-0.857773\pi$
$71$	−15.1978	−1.80365	−0.901825	+	0.432100i	$0.857773\pi$
$72$	0	0
$73$	12.1315	1.41988	0.709942	−	0.704260i	$-0.248721\pi$
$73$	12.1315	1.41988	0.709942	+	0.704260i	$0.248721\pi$
$74$	−19.3106	−2.24481
$75$	0	0
$76$	−11.9399	−1.36960
$77$	−15.9899	−1.82221
$78$	0	0
$79$	−11.3081	−1.27227	−0.636133	−	0.771580i	$-0.719467\pi$
$79$	−11.3081	−1.27227	−0.636133	+	0.771580i	$0.719467\pi$
$80$	0.0555880	0.00621493
$81$	0	0
$82$	−3.02776	−0.334360
$83$	−8.69580	−0.954488	−0.477244	−	0.878771i	$-0.658364\pi$
$83$	−8.69580	−0.954488	−0.477244	+	0.878771i	$0.658364\pi$
$84$	0	0
$85$	0.745796	0.0808930
$86$	28.0539	3.02513
$87$	0	0
$88$	−13.9404	−1.48605
$89$	3.58092	0.379577	0.189789	−	0.981825i	$-0.439220\pi$
$89$	3.58092	0.379577	0.189789	+	0.981825i	$0.439220\pi$
$90$	0	0
$91$	−2.89014	−0.302969
$92$	3.13576	0.326926
$93$	0	0
$94$	−8.08147	−0.833540
$95$	0.482674	0.0495213
$96$	0	0
$97$	16.5153	1.67688	0.838439	−	0.544995i	$-0.183469\pi$
$97$	16.5153	1.67688	0.838439	+	0.544995i	$0.183469\pi$
$98$	3.88888	0.392836
$99$	0	0
$100$	−15.6284	−1.56284
$101$	−1.36650	−0.135972	−0.0679858	−	0.997686i	$-0.521657\pi$
$101$	−1.36650	−0.135972	−0.0679858	+	0.997686i	$0.521657\pi$
$102$	0	0
$103$	−2.91398	−0.287123	−0.143562	−	0.989641i	$-0.545856\pi$
$103$	−2.91398	−0.287123	−0.143562	+	0.989641i	$0.545856\pi$
$104$	−2.51970	−0.247077
$105$	0	0
$106$	−22.2339	−2.15955
$107$	18.5607	1.79433	0.897167	−	0.441691i	$-0.145621\pi$
$107$	18.5607	1.79433	0.897167	+	0.441691i	$0.145621\pi$
$108$	0	0
$109$	8.98791	0.860886	0.430443	−	0.902618i	$-0.358357\pi$
$109$	8.98791	0.860886	0.430443	+	0.902618i	$0.358357\pi$
$110$	1.55591	0.148350
$111$	0	0
$112$	−1.29463	−0.122331
$113$	3.28188	0.308733	0.154367	−	0.988014i	$-0.450666\pi$
$113$	3.28188	0.308733	0.154367	+	0.988014i	$0.450666\pi$
$114$	0	0
$115$	−0.126764	−0.0118208
$116$	−3.13576	−0.291148
$117$	0	0
$118$	28.7001	2.64206
$119$	−17.3694	−1.59225
$120$	0	0
$121$	18.3340	1.66673
$122$	33.1649	3.00261
$123$	0	0
$124$	−12.1287	−1.08919
$125$	1.26560	0.113199
$126$	0	0
$127$	3.82898	0.339767	0.169883	−	0.985464i	$-0.445661\pi$
$127$	3.82898	0.339767	0.169883	+	0.985464i	$0.445661\pi$
$128$	−17.2709	−1.52655
$129$	0	0
$130$	0.281227	0.0246652
$131$	−5.77160	−0.504267	−0.252133	−	0.967693i	$-0.581132\pi$
$131$	−5.77160	−0.504267	−0.252133	+	0.967693i	$0.581132\pi$
$132$	0	0
$133$	−11.2413	−0.974747
$134$	2.87423	0.248296
$135$	0	0
$136$	−15.1431	−1.29851
$137$	22.6645	1.93636	0.968178	−	0.250261i	$-0.0805165\pi$
$137$	22.6645	1.93636	0.968178	+	0.250261i	$0.0805165\pi$
$138$	0	0
$139$	−4.23769	−0.359436	−0.179718	−	0.983718i	$-0.557519\pi$
$139$	−4.23769	−0.359436	−0.179718	+	0.983718i	$0.557519\pi$
$140$	−1.17354	−0.0991821
$141$	0	0
$142$	−34.4417	−2.89028
$143$	5.30206	0.443381
$144$	0	0
$145$	0.126764	0.0105272
$146$	27.4927	2.27531
$147$	0	0
$148$	−26.7200	−2.19637
$149$	−16.8929	−1.38392	−0.691961	−	0.721934i	$-0.743253\pi$
$149$	−16.8929	−1.38392	−0.691961	+	0.721934i	$0.743253\pi$
$150$	0	0
$151$	16.4671	1.34007	0.670036	−	0.742328i	$-0.266279\pi$
$151$	16.4671	1.34007	0.670036	+	0.742328i	$0.266279\pi$
$152$	−9.80052	−0.794927
$153$	0	0
$154$	−36.2366	−2.92003
$155$	0.490303	0.0393821
$156$	0	0
$157$	11.6243	0.927719	0.463860	−	0.885909i	$-0.346464\pi$
$157$	11.6243	0.927719	0.463860	+	0.885909i	$0.346464\pi$
$158$	−25.6268	−2.03876
$159$	0	0
$160$	0.778527	0.0615480
$161$	2.95229	0.232673
$162$	0	0
$163$	−6.00704	−0.470508	−0.235254	−	0.971934i	$-0.575592\pi$
$163$	−6.00704	−0.470508	−0.235254	+	0.971934i	$0.575592\pi$
$164$	−4.18950	−0.327145
$165$	0	0
$166$	−19.7066	−1.52953
$167$	3.94570	0.305327	0.152664	−	0.988278i	$-0.451215\pi$
$167$	3.94570	0.305327	0.152664	+	0.988278i	$0.451215\pi$
$168$	0	0
$169$	−12.0417	−0.926282
$170$	1.69014	0.129628
$171$	0	0
$172$	38.8181	2.95985
$173$	−7.49513	−0.569844	−0.284922	−	0.958551i	$-0.591968\pi$
$173$	−7.49513	−0.569844	−0.284922	+	0.958551i	$0.591968\pi$
$174$	0	0
$175$	−14.7140	−1.11227
$176$	2.37504	0.179026
$177$	0	0
$178$	8.11517	0.608258
$179$	−5.86398	−0.438295	−0.219147	−	0.975692i	$-0.570328\pi$
$179$	−5.86398	−0.438295	−0.219147	+	0.975692i	$0.570328\pi$
$180$	0	0
$181$	−17.1985	−1.27836	−0.639178	−	0.769059i	$-0.720725\pi$
$181$	−17.1985	−1.27836	−0.639178	+	0.769059i	$0.720725\pi$
$182$	−6.54969	−0.485496
$183$	0	0
$184$	2.57389	0.189750
$185$	1.08016	0.0794151
$186$	0	0
$187$	31.8648	2.33018
$188$	−11.1823	−0.815553
$189$	0	0
$190$	1.09385	0.0793559
$191$	2.72337	0.197056	0.0985279	−	0.995134i	$-0.468587\pi$
$191$	2.72337	0.197056	0.0985279	+	0.995134i	$0.468587\pi$
$192$	0	0
$193$	19.4222	1.39804	0.699020	−	0.715103i	$-0.253620\pi$
$193$	19.4222	1.39804	0.699020	+	0.715103i	$0.253620\pi$
$194$	37.4274	2.68713
$195$	0	0
$196$	5.38103	0.384359
$197$	−7.94311	−0.565923	−0.282961	−	0.959131i	$-0.591317\pi$
$197$	−7.94311	−0.565923	−0.282961	+	0.959131i	$0.591317\pi$
$198$	0	0
$199$	−19.5634	−1.38681	−0.693407	−	0.720546i	$-0.743891\pi$
$199$	−19.5634	−1.38681	−0.693407	+	0.720546i	$0.743891\pi$
$200$	−12.8281	−0.907083
$201$	0	0
$202$	−3.09679	−0.217889
$203$	−2.95229	−0.207210
$204$	0	0
$205$	0.169361	0.0118287
$206$	−6.60374	−0.460104
$207$	0	0
$208$	0.429285	0.0297655
$209$	20.6226	1.42650
$210$	0	0
$211$	0.726373	0.0500056	0.0250028	−	0.999687i	$-0.492041\pi$
$211$	0.726373	0.0500056	0.0250028	+	0.999687i	$0.492041\pi$
$212$	−30.7650	−2.11295
$213$	0	0
$214$	42.0628	2.87535
$215$	−1.56923	−0.107020
$216$	0	0
$217$	−11.4190	−0.775173
$218$	20.3686	1.37954
$219$	0	0
$220$	2.15290	0.145149
$221$	5.75950	0.387426
$222$	0	0
$223$	−13.6808	−0.916137	−0.458069	−	0.888917i	$-0.651459\pi$
$223$	−13.6808	−0.916137	−0.458069	+	0.888917i	$0.651459\pi$
$224$	−18.1317	−1.21147
$225$	0	0
$226$	7.43746	0.494733
$227$	−14.4918	−0.961853	−0.480926	−	0.876761i	$-0.659700\pi$
$227$	−14.4918	−0.961853	−0.480926	+	0.876761i	$0.659700\pi$
$228$	0	0
$229$	13.2702	0.876917	0.438459	−	0.898751i	$-0.355525\pi$
$229$	13.2702	0.876917	0.438459	+	0.898751i	$0.355525\pi$
$230$	−0.287275	−0.0189423
$231$	0	0
$232$	−2.57389	−0.168984
$233$	−19.7193	−1.29185	−0.645926	−	0.763400i	$-0.723528\pi$
$233$	−19.7193	−1.29185	−0.645926	+	0.763400i	$0.723528\pi$
$234$	0	0
$235$	0.452046	0.0294882
$236$	39.7122	2.58504
$237$	0	0
$238$	−39.3629	−2.55152
$239$	3.21756	0.208127	0.104063	−	0.994571i	$-0.466816\pi$
$239$	3.21756	0.208127	0.104063	+	0.994571i	$0.466816\pi$
$240$	0	0
$241$	19.2388	1.23928	0.619639	−	0.784887i	$-0.287279\pi$
$241$	19.2388	1.23928	0.619639	+	0.784887i	$0.287279\pi$
$242$	41.5489	2.67086
$243$	0	0
$244$	45.8902	2.93782
$245$	−0.217529	−0.0138974
$246$	0	0
$247$	3.72750	0.237175
$248$	−9.95542	−0.632170
$249$	0	0
$250$	2.86813	0.181397
$251$	−17.6279	−1.11266	−0.556332	−	0.830960i	$-0.687792\pi$
$251$	−17.6279	−1.11266	−0.556332	+	0.830960i	$0.687792\pi$
$252$	0	0
$253$	−5.41609	−0.340506
$254$	8.67731	0.544463
$255$	0	0
$256$	−13.0575	−0.816096
$257$	−20.3622	−1.27016	−0.635081	−	0.772446i	$-0.719033\pi$
$257$	−20.3622	−1.27016	−0.635081	+	0.772446i	$0.719033\pi$
$258$	0	0
$259$	−25.1567	−1.56316
$260$	0.389133	0.0241330
$261$	0	0
$262$	−13.0797	−0.808068
$263$	2.97723	0.183584	0.0917920	−	0.995778i	$-0.470741\pi$
$263$	2.97723	0.183584	0.0917920	+	0.995778i	$0.470741\pi$
$264$	0	0
$265$	1.24368	0.0763987
$266$	−25.4754	−1.56199
$267$	0	0
$268$	3.97707	0.242938
$269$	−23.1228	−1.40982	−0.704910	−	0.709297i	$-0.749012\pi$
$269$	−23.1228	−1.40982	−0.704910	+	0.709297i	$0.749012\pi$
$270$	0	0
$271$	−25.0539	−1.52191	−0.760957	−	0.648802i	$-0.775270\pi$
$271$	−25.0539	−1.52191	−0.760957	+	0.648802i	$0.775270\pi$
$272$	2.57995	0.156432
$273$	0	0
$274$	51.3627	3.10294
$275$	26.9934	1.62776
$276$	0	0
$277$	5.42075	0.325701	0.162851	−	0.986651i	$-0.447931\pi$
$277$	5.42075	0.325701	0.162851	+	0.986651i	$0.447931\pi$
$278$	−9.60355	−0.575983
$279$	0	0
$280$	−0.963262	−0.0575659
$281$	28.2699	1.68644	0.843219	−	0.537570i	$-0.180658\pi$
$281$	28.2699	1.68644	0.843219	+	0.537570i	$0.180658\pi$
$282$	0	0
$283$	−8.59398	−0.510859	−0.255430	−	0.966828i	$-0.582217\pi$
$283$	−8.59398	−0.510859	−0.255430	+	0.966828i	$0.582217\pi$
$284$	−47.6568	−2.82791
$285$	0	0
$286$	12.0157	0.710501
$287$	−3.94437	−0.232829
$288$	0	0
$289$	17.6139	1.03611
$290$	0.287275	0.0168694
$291$	0	0
$292$	38.0415	2.22621
$293$	−19.5921	−1.14458	−0.572292	−	0.820050i	$-0.693946\pi$
$293$	−19.5921	−1.14458	−0.572292	+	0.820050i	$0.693946\pi$
$294$	0	0
$295$	−1.60537	−0.0934684
$296$	−21.9323	−1.27479
$297$	0	0
$298$	−38.2831	−2.21768
$299$	−0.978947	−0.0566140
$300$	0	0
$301$	36.5469	2.10653
$302$	37.3181	2.14741
$303$	0	0
$304$	1.66972	0.0957653
$305$	−1.85512	−0.106224
$306$	0	0
$307$	−0.549924	−0.0313858	−0.0156929	−	0.999877i	$-0.504995\pi$
$307$	−0.549924	−0.0313858	−0.0156929	+	0.999877i	$0.504995\pi$
$308$	−50.1404	−2.85702
$309$	0	0
$310$	1.11114	0.0631082
$311$	4.18523	0.237323	0.118661	−	0.992935i	$-0.462140\pi$
$311$	4.18523	0.237323	0.118661	+	0.992935i	$0.462140\pi$
$312$	0	0
$313$	−16.8240	−0.950948	−0.475474	−	0.879730i	$-0.657723\pi$
$313$	−16.8240	−0.950948	−0.475474	+	0.879730i	$0.657723\pi$
$314$	26.3432	1.48663
$315$	0	0
$316$	−35.4596	−1.99476
$317$	−24.6910	−1.38679	−0.693393	−	0.720559i	$-0.743885\pi$
$317$	−24.6910	−1.38679	−0.693393	+	0.720559i	$0.743885\pi$
$318$	0	0
$319$	5.41609	0.303243
$320$	1.65314	0.0924133
$321$	0	0
$322$	6.69055	0.372850
$323$	22.4019	1.24647
$324$	0	0
$325$	4.87901	0.270639
$326$	−13.6133	−0.753970
$327$	0	0
$328$	−3.43881	−0.189877
$329$	−10.5280	−0.580429
$330$	0	0
$331$	−9.55640	−0.525267	−0.262634	−	0.964896i	$-0.584591\pi$
$331$	−9.55640	−0.525267	−0.262634	+	0.964896i	$0.584591\pi$
$332$	−27.2680	−1.49652
$333$	0	0
$334$	8.94183	0.489275
$335$	−0.160774	−0.00878400
$336$	0	0
$337$	−6.91137	−0.376486	−0.188243	−	0.982122i	$-0.560279\pi$
$337$	−6.91137	−0.376486	−0.188243	+	0.982122i	$0.560279\pi$
$338$	−27.2891	−1.48433
$339$	0	0
$340$	2.33864	0.126831
$341$	20.9486	1.13443
$342$	0	0
$343$	−15.5998	−0.842313
$344$	31.8626	1.71792
$345$	0	0
$346$	−16.9856	−0.913153
$347$	−11.1356	−0.597791	−0.298896	−	0.954286i	$-0.596618\pi$
$347$	−11.1356	−0.597791	−0.298896	+	0.954286i	$0.596618\pi$
$348$	0	0
$349$	−4.86757	−0.260555	−0.130278	−	0.991478i	$-0.541587\pi$
$349$	−4.86757	−0.260555	−0.130278	+	0.991478i	$0.541587\pi$
$350$	−33.3452	−1.78238
$351$	0	0
$352$	33.2632	1.77294
$353$	−0.322369	−0.0171579	−0.00857897	−	0.999963i	$-0.502731\pi$
$353$	−0.322369	−0.0171579	−0.00857897	+	0.999963i	$0.502731\pi$
$354$	0	0
$355$	1.92653	0.102250
$356$	11.2289	0.595132
$357$	0	0
$358$	−13.2891	−0.702350
$359$	3.05280	0.161121	0.0805603	−	0.996750i	$-0.474329\pi$
$359$	3.05280	0.161121	0.0805603	+	0.996750i	$0.474329\pi$
$360$	0	0
$361$	−4.50169	−0.236931
$362$	−38.9757	−2.04852
$363$	0	0
$364$	−9.06278	−0.475019
$365$	−1.53783	−0.0804939
$366$	0	0
$367$	11.6478	0.608012	0.304006	−	0.952670i	$-0.401676\pi$
$367$	11.6478	0.608012	0.304006	+	0.952670i	$0.401676\pi$
$368$	−0.438517	−0.0228593
$369$	0	0
$370$	2.44789	0.127260
$371$	−28.9650	−1.50379
$372$	0	0
$373$	−21.2921	−1.10246	−0.551232	−	0.834352i	$-0.685842\pi$
$373$	−21.2921	−1.10246	−0.551232	+	0.834352i	$0.685842\pi$
$374$	72.2126	3.73403
$375$	0	0
$376$	−9.17863	−0.473352
$377$	0.978947	0.0504184
$378$	0	0
$379$	2.98632	0.153397	0.0766984	−	0.997054i	$-0.475562\pi$
$379$	2.98632	0.153397	0.0766984	+	0.997054i	$0.475562\pi$
$380$	1.51355	0.0776435
$381$	0	0
$382$	6.17175	0.315774
$383$	3.76622	0.192445	0.0962224	−	0.995360i	$-0.469324\pi$
$383$	3.76622	0.192445	0.0962224	+	0.995360i	$0.469324\pi$
$384$	0	0
$385$	2.02693	0.103302
$386$	44.0150	2.24030
$387$	0	0
$388$	51.7882	2.62915
$389$	24.6945	1.25206	0.626031	−	0.779798i	$-0.284678\pi$
$389$	24.6945	1.25206	0.626031	+	0.779798i	$0.284678\pi$
$390$	0	0
$391$	−5.88336	−0.297534
$392$	4.41685	0.223084
$393$	0	0
$394$	−18.0008	−0.906869
$395$	1.43346	0.0721253
$396$	0	0
$397$	33.7378	1.69325	0.846627	−	0.532187i	$-0.178630\pi$
$397$	33.7378	1.69325	0.846627	+	0.532187i	$0.178630\pi$
$398$	−44.3351	−2.22232
$399$	0	0
$400$	2.18554	0.109277
$401$	−9.90120	−0.494442	−0.247221	−	0.968959i	$-0.579517\pi$
$401$	−9.90120	−0.494442	−0.247221	+	0.968959i	$0.579517\pi$
$402$	0	0
$403$	3.78642	0.188615
$404$	−4.28501	−0.213187
$405$	0	0
$406$	−6.69055	−0.332046
$407$	46.1508	2.28761
$408$	0	0
$409$	−38.4271	−1.90010	−0.950048	−	0.312102i	$-0.898967\pi$
$409$	−38.4271	−1.90010	−0.950048	+	0.312102i	$0.898967\pi$
$410$	0.383810	0.0189550
$411$	0	0
$412$	−9.13757	−0.450176
$413$	37.3886	1.83977
$414$	0	0
$415$	1.10231	0.0541104
$416$	6.01226	0.294775
$417$	0	0
$418$	46.7355	2.28591
$419$	−1.32133	−0.0645514	−0.0322757	−	0.999479i	$-0.510275\pi$
$419$	−1.32133	−0.0645514	−0.0322757	+	0.999479i	$0.510275\pi$
$420$	0	0
$421$	−25.2689	−1.23153	−0.615765	−	0.787930i	$-0.711153\pi$
$421$	−25.2689	−1.23153	−0.615765	+	0.787930i	$0.711153\pi$
$422$	1.64612	0.0801320
$423$	0	0
$424$	−25.2525	−1.22637
$425$	29.3222	1.42234
$426$	0	0
$427$	43.2052	2.09085
$428$	58.2021	2.81330
$429$	0	0
$430$	−3.55622	−0.171496
$431$	3.15084	0.151770	0.0758852	−	0.997117i	$-0.475822\pi$
$431$	3.15084	0.151770	0.0758852	+	0.997117i	$0.475822\pi$
$432$	0	0
$433$	−6.07570	−0.291980	−0.145990	−	0.989286i	$-0.546637\pi$
$433$	−6.07570	−0.291980	−0.145990	+	0.989286i	$0.546637\pi$
$434$	−25.8780	−1.24218
$435$	0	0
$436$	28.1840	1.34977
$437$	−3.80767	−0.182145
$438$	0	0
$439$	3.39072	0.161830	0.0809151	−	0.996721i	$-0.474216\pi$
$439$	3.39072	0.161830	0.0809151	+	0.996721i	$0.474216\pi$
$440$	1.76714	0.0842451
$441$	0	0
$442$	13.0523	0.620835
$443$	32.4820	1.54326	0.771632	−	0.636069i	$-0.219441\pi$
$443$	32.4820	1.54326	0.771632	+	0.636069i	$0.219441\pi$
$444$	0	0
$445$	−0.453931	−0.0215184
$446$	−31.0038	−1.46807
$447$	0	0
$448$	−38.5011	−1.81901
$449$	0.234196	0.0110524	0.00552620	−	0.999985i	$-0.498241\pi$
$449$	0.234196	0.0110524	0.00552620	+	0.999985i	$0.498241\pi$
$450$	0	0
$451$	7.23609	0.340734
$452$	10.2912	0.484057
$453$	0	0
$454$	−32.8416	−1.54133
$455$	0.366365	0.0171754
$456$	0	0
$457$	−30.6758	−1.43495	−0.717476	−	0.696583i	$-0.754703\pi$
$457$	−30.6758	−1.43495	−0.717476	+	0.696583i	$0.754703\pi$
$458$	30.0731	1.40523
$459$	0	0
$460$	−0.397501	−0.0185336
$461$	−3.71121	−0.172848	−0.0864241	−	0.996258i	$-0.527544\pi$
$461$	−3.71121	−0.172848	−0.0864241	+	0.996258i	$0.527544\pi$
$462$	0	0
$463$	−20.7959	−0.966469	−0.483235	−	0.875491i	$-0.660538\pi$
$463$	−20.7959	−0.966469	−0.483235	+	0.875491i	$0.660538\pi$
$464$	0.438517	0.0203576
$465$	0	0
$466$	−44.6882	−2.07014
$467$	7.59575	0.351489	0.175745	−	0.984436i	$-0.443767\pi$
$467$	7.59575	0.351489	0.175745	+	0.984436i	$0.443767\pi$
$468$	0	0
$469$	3.74437	0.172899
$470$	1.02444	0.0472538
$471$	0	0
$472$	32.5965	1.50037
$473$	−67.0466	−3.08281
$474$	0	0
$475$	18.9771	0.870731
$476$	−54.4663	−2.49646
$477$	0	0
$478$	7.29170	0.333515
$479$	−28.2076	−1.28884	−0.644419	−	0.764672i	$-0.722901\pi$
$479$	−28.2076	−1.28884	−0.644419	+	0.764672i	$0.722901\pi$
$480$	0	0
$481$	8.34167	0.380347
$482$	43.5993	1.98590
$483$	0	0
$484$	57.4911	2.61323
$485$	−2.09355	−0.0950630
$486$	0	0
$487$	−31.6309	−1.43334	−0.716668	−	0.697415i	$-0.754333\pi$
$487$	−31.6309	−1.43334	−0.716668	+	0.697415i	$0.754333\pi$
$488$	37.6675	1.70513
$489$	0	0
$490$	−0.492969	−0.0222701
$491$	27.3455	1.23408	0.617042	−	0.786930i	$-0.288331\pi$
$491$	27.3455	1.23408	0.617042	+	0.786930i	$0.288331\pi$
$492$	0	0
$493$	5.88336	0.264973
$494$	8.44735	0.380064
$495$	0	0
$496$	1.69612	0.0761578
$497$	−44.8684	−2.01262
$498$	0	0
$499$	20.4725	0.916474	0.458237	−	0.888830i	$-0.348481\pi$
$499$	20.4725	0.916474	0.458237	+	0.888830i	$0.348481\pi$
$500$	3.96862	0.177482
$501$	0	0
$502$	−39.9488	−1.78300
$503$	21.9821	0.980133	0.490067	−	0.871685i	$-0.336972\pi$
$503$	21.9821	0.980133	0.490067	+	0.871685i	$0.336972\pi$
$504$	0	0
$505$	0.173222	0.00770829
$506$	−12.2741	−0.545648
$507$	0	0
$508$	12.0068	0.532714
$509$	−6.75255	−0.299301	−0.149651	−	0.988739i	$-0.547815\pi$
$509$	−6.75255	−0.299301	−0.149651	+	0.988739i	$0.547815\pi$
$510$	0	0
$511$	35.8157	1.58439
$512$	4.95056	0.218786
$513$	0	0
$514$	−46.1453	−2.03538
$515$	0.369388	0.0162772
$516$	0	0
$517$	19.3140	0.849431
$518$	−57.0106	−2.50490
$519$	0	0
$520$	0.319407	0.0140069
$521$	21.5031	0.942069	0.471034	−	0.882115i	$-0.343881\pi$
$521$	21.5031	0.942069	0.471034	+	0.882115i	$0.343881\pi$
$522$	0	0
$523$	1.46391	0.0640125	0.0320063	−	0.999488i	$-0.489810\pi$
$523$	1.46391	0.0640125	0.0320063	+	0.999488i	$0.489810\pi$
$524$	−18.0984	−0.790630
$525$	0	0
$526$	6.74707	0.294186
$527$	22.7559	0.991264
$528$	0	0
$529$	1.00000	0.0434783
$530$	2.81846	0.122426
$531$	0	0
$532$	−35.2502	−1.52829
$533$	1.30791	0.0566519
$534$	0	0
$535$	−2.35283	−0.101722
$536$	3.26445	0.141003
$537$	0	0
$538$	−52.4013	−2.25918
$539$	−9.29411	−0.400326
$540$	0	0
$541$	−23.8709	−1.02629	−0.513145	−	0.858302i	$-0.671520\pi$
$541$	−23.8709	−1.02629	−0.513145	+	0.858302i	$0.671520\pi$
$542$	−56.7776	−2.43881
$543$	0	0
$544$	36.1330	1.54919
$545$	−1.13934	−0.0488040
$546$	0	0
$547$	−18.8563	−0.806235	−0.403118	−	0.915148i	$-0.632073\pi$
$547$	−18.8563	−0.806235	−0.403118	+	0.915148i	$0.632073\pi$
$548$	71.0704	3.03598
$549$	0	0
$550$	61.1730	2.60843
$551$	3.80767	0.162212
$552$	0	0
$553$	−33.3849	−1.41967
$554$	12.2846	0.521923
$555$	0	0
$556$	−13.2884	−0.563554
$557$	15.6484	0.663044	0.331522	−	0.943448i	$-0.392438\pi$
$557$	15.6484	0.663044	0.331522	+	0.943448i	$0.392438\pi$
$558$	0	0
$559$	−12.1185	−0.512560
$560$	0.164112	0.00693500
$561$	0	0
$562$	64.0658	2.70245
$563$	−8.20131	−0.345644	−0.172822	−	0.984953i	$-0.555289\pi$
$563$	−8.20131	−0.345644	−0.172822	+	0.984953i	$0.555289\pi$
$564$	0	0
$565$	−0.416023	−0.0175022
$566$	−19.4759	−0.818632
$567$	0	0
$568$	−39.1176	−1.64134
$569$	−23.8115	−0.998228	−0.499114	−	0.866536i	$-0.666341\pi$
$569$	−23.8115	−0.998228	−0.499114	+	0.866536i	$0.666341\pi$
$570$	0	0
$571$	−9.06117	−0.379198	−0.189599	−	0.981862i	$-0.560719\pi$
$571$	−9.06117	−0.379198	−0.189599	+	0.981862i	$0.560719\pi$
$572$	16.6260	0.695169
$573$	0	0
$574$	−8.93882	−0.373099
$575$	−4.98393	−0.207844
$576$	0	0
$577$	−26.0777	−1.08563	−0.542814	−	0.839853i	$-0.682641\pi$
$577$	−26.0777	−1.08563	−0.542814	+	0.839853i	$0.682641\pi$
$578$	39.9170	1.66033
$579$	0	0
$580$	0.397501	0.0165053
$581$	−25.6725	−1.06508
$582$	0	0
$583$	53.1373	2.20072
$584$	31.2252	1.29211
$585$	0	0
$586$	−44.4001	−1.83415
$587$	−24.1351	−0.996160	−0.498080	−	0.867131i	$-0.665961\pi$
$587$	−24.1351	−0.996160	−0.498080	+	0.867131i	$0.665961\pi$
$588$	0	0
$589$	14.7275	0.606835
$590$	−3.63813	−0.149779
$591$	0	0
$592$	3.73663	0.153574
$593$	−3.81516	−0.156670	−0.0783349	−	0.996927i	$-0.524960\pi$
$593$	−3.81516	−0.156670	−0.0783349	+	0.996927i	$0.524960\pi$
$594$	0	0
$595$	2.20181	0.0902653
$596$	−52.9722	−2.16983
$597$	0	0
$598$	−2.21851	−0.0907217
$599$	15.5069	0.633593	0.316796	−	0.948494i	$-0.397393\pi$
$599$	15.5069	0.633593	0.316796	+	0.948494i	$0.397393\pi$
$600$	0	0
$601$	40.0434	1.63341	0.816703	−	0.577058i	$-0.195799\pi$
$601$	40.0434	1.63341	0.816703	+	0.577058i	$0.195799\pi$
$602$	82.8233	3.37563
$603$	0	0
$604$	51.6369	2.10107
$605$	−2.32409	−0.0944875
$606$	0	0
$607$	−6.91112	−0.280514	−0.140257	−	0.990115i	$-0.544793\pi$
$607$	−6.91112	−0.280514	−0.140257	+	0.990115i	$0.544793\pi$
$608$	23.3850	0.948387
$609$	0	0
$610$	−4.20411	−0.170220
$611$	3.49098	0.141230
$612$	0	0
$613$	−14.8180	−0.598492	−0.299246	−	0.954176i	$-0.596735\pi$
$613$	−14.8180	−0.598492	−0.299246	+	0.954176i	$0.596735\pi$
$614$	−1.24625	−0.0502946
$615$	0	0
$616$	−41.1562	−1.65823
$617$	21.6971	0.873491	0.436746	−	0.899585i	$-0.356131\pi$
$617$	21.6971	0.873491	0.436746	+	0.899585i	$0.356131\pi$
$618$	0	0
$619$	−12.5357	−0.503852	−0.251926	−	0.967747i	$-0.581064\pi$
$619$	−12.5357	−0.503852	−0.251926	+	0.967747i	$0.581064\pi$
$620$	1.53747	0.0617464
$621$	0	0
$622$	9.48467	0.380301
$623$	10.5719	0.423555
$624$	0	0
$625$	24.7592	0.990369
$626$	−38.1269	−1.52386
$627$	0	0
$628$	36.4510	1.45455
$629$	50.1325	1.99891
$630$	0	0
$631$	6.78244	0.270005	0.135002	−	0.990845i	$-0.456896\pi$
$631$	6.78244	0.270005	0.135002	+	0.990845i	$0.456896\pi$
$632$	−29.1059	−1.15777
$633$	0	0
$634$	−55.9554	−2.22227
$635$	−0.485376	−0.0192615
$636$	0	0
$637$	−1.67989	−0.0665598
$638$	12.2741	0.485935
$639$	0	0
$640$	2.18933	0.0865407
$641$	10.6789	0.421790	0.210895	−	0.977509i	$-0.432362\pi$
$641$	10.6789	0.421790	0.210895	+	0.977509i	$0.432362\pi$
$642$	0	0
$643$	12.2323	0.482394	0.241197	−	0.970476i	$-0.422460\pi$
$643$	12.2323	0.482394	0.241197	+	0.970476i	$0.422460\pi$
$644$	9.25768	0.364804
$645$	0	0
$646$	50.7676	1.99742
$647$	15.3358	0.602911	0.301456	−	0.953480i	$-0.402527\pi$
$647$	15.3358	0.602911	0.301456	+	0.953480i	$0.402527\pi$
$648$	0	0
$649$	−68.5908	−2.69243
$650$	11.0569	0.433688
$651$	0	0
$652$	−18.8367	−0.737700
$653$	6.16737	0.241348	0.120674	−	0.992692i	$-0.461494\pi$
$653$	6.16737	0.241348	0.120674	+	0.992692i	$0.461494\pi$
$654$	0	0
$655$	0.731629	0.0285871
$656$	0.585875	0.0228746
$657$	0	0
$658$	−23.8588	−0.930114
$659$	9.26404	0.360876	0.180438	−	0.983586i	$-0.442249\pi$
$659$	9.26404	0.360876	0.180438	+	0.983586i	$0.442249\pi$
$660$	0	0
$661$	−23.7166	−0.922468	−0.461234	−	0.887279i	$-0.652593\pi$
$661$	−23.7166	−0.922468	−0.461234	+	0.887279i	$0.652593\pi$
$662$	−21.6569	−0.841720
$663$	0	0
$664$	−22.3820	−0.868591
$665$	1.42499	0.0552589
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	12.3728	0.478717
$669$	0	0
$670$	−0.364349	−0.0140760
$671$	−79.2615	−3.05986
$672$	0	0
$673$	14.1811	0.546643	0.273322	−	0.961923i	$-0.411878\pi$
$673$	14.1811	0.546643	0.273322	+	0.961923i	$0.411878\pi$
$674$	−15.6627	−0.603304
$675$	0	0
$676$	−37.7598	−1.45230
$677$	−37.7188	−1.44965	−0.724825	−	0.688933i	$-0.758079\pi$
$677$	−37.7188	−1.44965	−0.724825	+	0.688933i	$0.758079\pi$
$678$	0	0
$679$	48.7581	1.87116
$680$	1.91960	0.0736132
$681$	0	0
$682$	47.4742	1.81788
$683$	23.4359	0.896749	0.448374	−	0.893846i	$-0.352003\pi$
$683$	23.4359	0.896749	0.448374	+	0.893846i	$0.352003\pi$
$684$	0	0
$685$	−2.87303	−0.109773
$686$	−35.3527	−1.34977
$687$	0	0
$688$	−5.42847	−0.206958
$689$	9.60446	0.365901
$690$	0	0
$691$	3.58106	0.136230	0.0681150	−	0.997677i	$-0.478302\pi$
$691$	3.58106	0.136230	0.0681150	+	0.997677i	$0.478302\pi$
$692$	−23.5030	−0.893448
$693$	0	0
$694$	−25.2358	−0.957937
$695$	0.537186	0.0203766
$696$	0	0
$697$	7.86038	0.297733
$698$	−11.0310	−0.417529
$699$	0	0
$700$	−46.1397	−1.74391
$701$	−30.5445	−1.15365	−0.576824	−	0.816868i	$-0.695708\pi$
$701$	−30.5445	−1.15365	−0.576824	+	0.816868i	$0.695708\pi$
$702$	0	0
$703$	32.4454	1.22370
$704$	70.6317	2.66203
$705$	0	0
$706$	−0.730559	−0.0274949
$707$	−4.03430	−0.151725
$708$	0	0
$709$	−21.3010	−0.799974	−0.399987	−	0.916521i	$-0.630985\pi$
$709$	−21.3010	−0.799974	−0.399987	+	0.916521i	$0.630985\pi$
$710$	4.36595	0.163851
$711$	0	0
$712$	9.21691	0.345418
$713$	−3.86785	−0.144852
$714$	0	0
$715$	−0.672109	−0.0251355
$716$	−18.3881	−0.687194
$717$	0	0
$718$	6.91832	0.258190
$719$	5.97508	0.222833	0.111416	−	0.993774i	$-0.464461\pi$
$719$	5.97508	0.222833	0.111416	+	0.993774i	$0.464461\pi$
$720$	0	0
$721$	−8.60293	−0.320390
$722$	−10.2018	−0.379672
$723$	0	0
$724$	−53.9305	−2.00431
$725$	4.98393	0.185099
$726$	0	0
$727$	0.900942	0.0334141	0.0167070	−	0.999860i	$-0.494682\pi$
$727$	0.900942	0.0334141	0.0167070	+	0.999860i	$0.494682\pi$
$728$	−7.43890	−0.275704
$729$	0	0
$730$	−3.48507	−0.128988
$731$	−72.8310	−2.69375
$732$	0	0
$733$	−45.4372	−1.67826	−0.839131	−	0.543930i	$-0.816936\pi$
$733$	−45.4372	−1.67826	−0.839131	+	0.543930i	$0.816936\pi$
$734$	26.3966	0.974316
$735$	0	0
$736$	−6.14156	−0.226381
$737$	−6.86918	−0.253030
$738$	0	0
$739$	−14.7734	−0.543449	−0.271724	−	0.962375i	$-0.587594\pi$
$739$	−14.7734	−0.543449	−0.271724	+	0.962375i	$0.587594\pi$
$740$	3.38713	0.124513
$741$	0	0
$742$	−65.6410	−2.40976
$743$	30.7951	1.12976	0.564881	−	0.825172i	$-0.308922\pi$
$743$	30.7951	1.12976	0.564881	+	0.825172i	$0.308922\pi$
$744$	0	0
$745$	2.14141	0.0784552
$746$	−48.2527	−1.76666
$747$	0	0
$748$	99.9204	3.65345
$749$	54.7967	2.00223
$750$	0	0
$751$	27.8492	1.01623	0.508116	−	0.861289i	$-0.330342\pi$
$751$	27.8492	1.01623	0.508116	+	0.861289i	$0.330342\pi$
$752$	1.56377	0.0570249
$753$	0	0
$754$	2.21851	0.0807934
$755$	−2.08743	−0.0759693
$756$	0	0
$757$	17.6755	0.642425	0.321213	−	0.947007i	$-0.395910\pi$
$757$	17.6755	0.642425	0.321213	+	0.947007i	$0.395910\pi$
$758$	6.76766	0.245812
$759$	0	0
$760$	1.24235	0.0450648
$761$	4.43186	0.160655	0.0803273	−	0.996769i	$-0.474403\pi$
$761$	4.43186	0.160655	0.0803273	+	0.996769i	$0.474403\pi$
$762$	0	0
$763$	26.5349	0.960629
$764$	8.53983	0.308960
$765$	0	0
$766$	8.53509	0.308385
$767$	−12.3977	−0.447654
$768$	0	0
$769$	44.3650	1.59984	0.799921	−	0.600105i	$-0.204874\pi$
$769$	44.3650	1.59984	0.799921	+	0.600105i	$0.204874\pi$
$770$	4.59348	0.165538
$771$	0	0
$772$	60.9033	2.19196
$773$	9.90517	0.356264	0.178132	−	0.984007i	$-0.442995\pi$
$773$	9.90517	0.356264	0.178132	+	0.984007i	$0.442995\pi$
$774$	0	0
$775$	19.2771	0.692453
$776$	42.5087	1.52597
$777$	0	0
$778$	55.9633	2.00638
$779$	5.08718	0.182267
$780$	0	0
$781$	82.3128	2.94538
$782$	−13.3330	−0.476787
$783$	0	0
$784$	−0.752503	−0.0268751
$785$	−1.47354	−0.0525929
$786$	0	0
$787$	−12.8948	−0.459649	−0.229824	−	0.973232i	$-0.573815\pi$
$787$	−12.8948	−0.459649	−0.229824	+	0.973232i	$0.573815\pi$
$788$	−24.9077	−0.887300
$789$	0	0
$790$	3.24854	0.115578
$791$	9.68906	0.344503
$792$	0	0
$793$	−14.3264	−0.508744
$794$	76.4574	2.71337
$795$	0	0
$796$	−61.3463	−2.17436
$797$	51.3846	1.82014	0.910068	−	0.414459i	$-0.136029\pi$
$797$	51.3846	1.82014	0.910068	+	0.414459i	$0.136029\pi$
$798$	0	0
$799$	20.9804	0.742232
$800$	30.6091	1.08220
$801$	0	0
$802$	−22.4383	−0.792324
$803$	−65.7053	−2.31869
$804$	0	0
$805$	−0.374243	−0.0131903
$806$	8.58087	0.302248
$807$	0	0
$808$	−3.51722	−0.123735
$809$	−33.2475	−1.16892	−0.584460	−	0.811423i	$-0.698694\pi$
$809$	−33.2475	−1.16892	−0.584460	+	0.811423i	$0.698694\pi$
$810$	0	0
$811$	−22.1107	−0.776411	−0.388206	−	0.921573i	$-0.626905\pi$
$811$	−22.1107	−0.776411	−0.388206	+	0.921573i	$0.626905\pi$
$812$	−9.25768	−0.324881
$813$	0	0
$814$	104.588	3.66581
$815$	0.761475	0.0266733
$816$	0	0
$817$	−47.1357	−1.64907
$818$	−87.0843	−3.04483
$819$	0	0
$820$	0.531076	0.0185460
$821$	−28.7126	−1.00208	−0.501038	−	0.865426i	$-0.667048\pi$
$821$	−28.7126	−1.00208	−0.501038	+	0.865426i	$0.667048\pi$
$822$	0	0
$823$	51.9609	1.81124	0.905621	−	0.424087i	$-0.139405\pi$
$823$	51.9609	1.81124	0.905621	+	0.424087i	$0.139405\pi$
$824$	−7.50028	−0.261285
$825$	0	0
$826$	84.7310	2.94817
$827$	5.48130	0.190604	0.0953018	−	0.995448i	$-0.469618\pi$
$827$	5.48130	0.190604	0.0953018	+	0.995448i	$0.469618\pi$
$828$	0	0
$829$	13.8666	0.481606	0.240803	−	0.970574i	$-0.422589\pi$
$829$	13.8666	0.481606	0.240803	+	0.970574i	$0.422589\pi$
$830$	2.49808	0.0867097
$831$	0	0
$832$	12.7666	0.442601
$833$	−10.0960	−0.349804
$834$	0	0
$835$	−0.500172	−0.0173092
$836$	64.6677	2.23658
$837$	0	0
$838$	−2.99444	−0.103441
$839$	27.0949	0.935419	0.467710	−	0.883882i	$-0.345079\pi$
$839$	27.0949	0.935419	0.467710	+	0.883882i	$0.345079\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−57.2649	−1.97348
$843$	0	0
$844$	2.27773	0.0784028
$845$	1.52645	0.0525113
$846$	0	0
$847$	54.1273	1.85984
$848$	4.30229	0.147741
$849$	0	0
$850$	66.4507	2.27924
$851$	−8.52106	−0.292098
$852$	0	0
$853$	23.2197	0.795028	0.397514	−	0.917596i	$-0.369873\pi$
$853$	23.2197	0.795028	0.397514	+	0.917596i	$0.369873\pi$
$854$	97.9125	3.35050
$855$	0	0
$856$	47.7733	1.63286
$857$	−19.7474	−0.674558	−0.337279	−	0.941405i	$-0.609507\pi$
$857$	−19.7474	−0.674558	−0.337279	+	0.941405i	$0.609507\pi$
$858$	0	0
$859$	−5.04653	−0.172185	−0.0860926	−	0.996287i	$-0.527438\pi$
$859$	−5.04653	−0.172185	−0.0860926	+	0.996287i	$0.527438\pi$
$860$	−4.92073	−0.167795
$861$	0	0
$862$	7.14050	0.243206
$863$	33.0579	1.12530	0.562651	−	0.826694i	$-0.309781\pi$
$863$	33.0579	1.12530	0.562651	+	0.826694i	$0.309781\pi$
$864$	0	0
$865$	0.950111	0.0323047
$866$	−13.7689	−0.467886
$867$	0	0
$868$	−35.8073	−1.21538
$869$	61.2459	2.07762
$870$	0	0
$871$	−1.24159	−0.0420697
$872$	23.1339	0.783413
$873$	0	0
$874$	−8.62902	−0.291881
$875$	3.73642	0.126314
$876$	0	0
$877$	24.3147	0.821050	0.410525	−	0.911849i	$-0.365345\pi$
$877$	24.3147	0.821050	0.410525	+	0.911849i	$0.365345\pi$
$878$	7.68412	0.259327
$879$	0	0
$880$	−0.301070	−0.0101491
$881$	45.4583	1.53153	0.765764	−	0.643121i	$-0.222361\pi$
$881$	45.4583	1.53153	0.765764	+	0.643121i	$0.222361\pi$
$882$	0	0
$883$	39.1644	1.31799	0.658993	−	0.752149i	$-0.270983\pi$
$883$	39.1644	1.31799	0.658993	+	0.752149i	$0.270983\pi$
$884$	18.0604	0.607438
$885$	0	0
$886$	73.6113	2.47302
$887$	−29.2143	−0.980920	−0.490460	−	0.871464i	$-0.663171\pi$
$887$	−29.2143	−0.980920	−0.490460	+	0.871464i	$0.663171\pi$
$888$	0	0
$889$	11.3043	0.379133
$890$	−1.02871	−0.0344824
$891$	0	0
$892$	−42.8999	−1.43639
$893$	13.5783	0.454382
$894$	0	0
$895$	0.743340	0.0248471
$896$	−50.9888	−1.70342
$897$	0	0
$898$	0.530741	0.0177110
$899$	3.86785	0.129000
$900$	0	0
$901$	57.7217	1.92299
$902$	16.3986	0.546014
$903$	0	0
$904$	8.44720	0.280950
$905$	2.18015	0.0724706
$906$	0	0
$907$	29.7948	0.989320	0.494660	−	0.869087i	$-0.335293\pi$
$907$	29.7948	0.989320	0.494660	+	0.869087i	$0.335293\pi$
$908$	−45.4428	−1.50807
$909$	0	0
$910$	0.830264	0.0275230
$911$	−13.6826	−0.453326	−0.226663	−	0.973973i	$-0.572782\pi$
$911$	−13.6826	−0.453326	−0.226663	+	0.973973i	$0.572782\pi$
$912$	0	0
$913$	47.0972	1.55869
$914$	−69.5182	−2.29946
$915$	0	0
$916$	41.6121	1.37490
$917$	−17.0394	−0.562691
$918$	0	0
$919$	−7.06537	−0.233065	−0.116533	−	0.993187i	$-0.537178\pi$
$919$	−7.06537	−0.233065	−0.116533	+	0.993187i	$0.537178\pi$
$920$	−0.326276	−0.0107570
$921$	0	0
$922$	−8.41043	−0.276983
$923$	14.8779	0.489711
$924$	0	0
$925$	42.4684	1.39635
$926$	−47.1282	−1.54873
$927$	0	0
$928$	6.14156	0.201607
$929$	9.05400	0.297052	0.148526	−	0.988908i	$-0.452547\pi$
$929$	9.05400	0.297052	0.148526	+	0.988908i	$0.452547\pi$
$930$	0	0
$931$	−6.53403	−0.214144
$932$	−61.8349	−2.02547
$933$	0	0
$934$	17.2137	0.563248
$935$	−4.03930	−0.132099
$936$	0	0
$937$	−0.0507149	−0.00165678	−0.000828392	−	1.00000i	$-0.500264\pi$
$937$	−0.0507149	−0.00165678	−0.000828392		1.00000i	$0.500264\pi$
$938$	8.48557	0.277064
$939$	0	0
$940$	1.41751	0.0462341
$941$	−47.4303	−1.54618	−0.773091	−	0.634295i	$-0.781291\pi$
$941$	−47.4303	−1.54618	−0.773091	+	0.634295i	$0.781291\pi$
$942$	0	0
$943$	−1.33604	−0.0435074
$944$	−5.55350	−0.180751
$945$	0	0
$946$	−151.942	−4.94008
$947$	−13.5491	−0.440286	−0.220143	−	0.975468i	$-0.570652\pi$
$947$	−13.5491	−0.440286	−0.220143	+	0.975468i	$0.570652\pi$
$948$	0	0
$949$	−11.8761	−0.385514
$950$	43.0064	1.39531
$951$	0	0
$952$	−44.7069	−1.44896
$953$	57.6425	1.86722	0.933612	−	0.358286i	$-0.116639\pi$
$953$	57.6425	1.86722	0.933612	+	0.358286i	$0.116639\pi$
$954$	0	0
$955$	−0.345224	−0.0111712
$956$	10.0895	0.326318
$957$	0	0
$958$	−63.9247	−2.06531
$959$	66.9121	2.16070
$960$	0	0
$961$	−16.0398	−0.517412
$962$	18.9041	0.609492
$963$	0	0
$964$	60.3283	1.94304
$965$	−2.46203	−0.0792555
$966$	0	0
$967$	−17.3136	−0.556769	−0.278385	−	0.960470i	$-0.589799\pi$
$967$	−17.3136	−0.556769	−0.278385	+	0.960470i	$0.589799\pi$
$968$	47.1897	1.51673
$969$	0	0
$970$	−4.74444	−0.152335
$971$	15.1688	0.486790	0.243395	−	0.969927i	$-0.421739\pi$
$971$	15.1688	0.486790	0.243395	+	0.969927i	$0.421739\pi$
$972$	0	0
$973$	−12.5109	−0.401081
$974$	−71.6828	−2.29686
$975$	0	0
$976$	−6.41746	−0.205418
$977$	25.5564	0.817621	0.408811	−	0.912619i	$-0.365944\pi$
$977$	25.5564	0.817621	0.408811	+	0.912619i	$0.365944\pi$
$978$	0	0
$979$	−19.3946	−0.619854
$980$	−0.682119	−0.0217895
$981$	0	0
$982$	61.9709	1.97757
$983$	−40.3341	−1.28646	−0.643230	−	0.765673i	$-0.722406\pi$
$983$	−40.3341	−1.28646	−0.643230	+	0.765673i	$0.722406\pi$
$984$	0	0
$985$	1.00690	0.0320824
$986$	13.3330	0.424609
$987$	0	0
$988$	11.6886	0.371863
$989$	12.3792	0.393634
$990$	0	0
$991$	−33.6326	−1.06837	−0.534187	−	0.845366i	$-0.679382\pi$
$991$	−33.6326	−1.06837	−0.534187	+	0.845366i	$0.679382\pi$
$992$	23.7546	0.754210
$993$	0	0
$994$	−101.682	−3.22515
$995$	2.47993	0.0786192
$996$	0	0
$997$	−24.1351	−0.764366	−0.382183	−	0.924087i	$-0.624828\pi$
$997$	−24.1351	−0.764366	−0.382183	+	0.924087i	$0.624828\pi$
$998$	46.3952	1.46861
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.o.1.13		13
3.2	odd	2		667.2.a.c.1.1	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.c.1.1	✓	13	3.2	odd	2
6003.2.a.o.1.13		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)

\(f(q)\)	\(=\)	\(q+2.26622 q^{2} +3.13576 q^{4} -0.126764 q^{5} +2.95229 q^{7} +2.57389 q^{8} +O(q^{10})\)	\(q+2.26622 q^{2} +3.13576 q^{4} -0.126764 q^{5} +2.95229 q^{7} +2.57389 q^{8} -0.287275 q^{10} -5.41609 q^{11} -0.978947 q^{13} +6.69055 q^{14} -0.438517 q^{16} -5.88336 q^{17} -3.80767 q^{19} -0.397501 q^{20} -12.2741 q^{22} +1.00000 q^{23} -4.98393 q^{25} -2.21851 q^{26} +9.25768 q^{28} -1.00000 q^{29} -3.86785 q^{31} -6.14156 q^{32} -13.3330 q^{34} -0.374243 q^{35} -8.52106 q^{37} -8.62902 q^{38} -0.326276 q^{40} -1.33604 q^{41} +12.3792 q^{43} -16.9836 q^{44} +2.26622 q^{46} -3.56605 q^{47} +1.71602 q^{49} -11.2947 q^{50} -3.06975 q^{52} -9.81101 q^{53} +0.686563 q^{55} +7.59887 q^{56} -2.26622 q^{58} +12.6643 q^{59} +14.6345 q^{61} -8.76540 q^{62} -13.0411 q^{64} +0.124095 q^{65} +1.26829 q^{67} -18.4488 q^{68} -0.848119 q^{70} -15.1978 q^{71} +12.1315 q^{73} -19.3106 q^{74} -11.9399 q^{76} -15.9899 q^{77} -11.3081 q^{79} +0.0555880 q^{80} -3.02776 q^{82} -8.69580 q^{83} +0.745796 q^{85} +28.0539 q^{86} -13.9404 q^{88} +3.58092 q^{89} -2.89014 q^{91} +3.13576 q^{92} -8.08147 q^{94} +0.482674 q^{95} +16.5153 q^{97} +3.88888 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 16 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) 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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