LMFDB - Embedded newform 6003.2.a.k.1.8

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 17x^{8} + 23x^{7} + 69x^{6} - 88x^{5} - 106x^{4} + 101x^{3} + 60x^{2} - 23x - 11 $ x^10 - x^9 - 17x^8 + 23x^7 + 69x^6 - 88x^5 - 106x^4 + 101x^3 + 60x^2 - 23x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.435319$ 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)

$f(q)$	$=$	$q+2.09108 q^{2} +2.37262 q^{4} -4.27000 q^{5} +3.25035 q^{7} +0.779182 q^{8} +O(q^{10})$	\(q+2.09108 q^{2} +2.37262 q^{4} -4.27000 q^{5} +3.25035 q^{7} +0.779182 q^{8} -8.92891 q^{10} +3.52823 q^{11} -5.02055 q^{13} +6.79676 q^{14} -3.11591 q^{16} -0.0201193 q^{17} -0.844205 q^{19} -10.1311 q^{20} +7.37782 q^{22} +1.00000 q^{23} +13.2329 q^{25} -10.4984 q^{26} +7.71186 q^{28} -1.00000 q^{29} +9.32330 q^{31} -8.07399 q^{32} -0.0420712 q^{34} -13.8790 q^{35} -8.07528 q^{37} -1.76530 q^{38} -3.32711 q^{40} -2.76429 q^{41} -0.721845 q^{43} +8.37116 q^{44} +2.09108 q^{46} -5.90133 q^{47} +3.56480 q^{49} +27.6710 q^{50} -11.9119 q^{52} -11.8408 q^{53} -15.0656 q^{55} +2.53262 q^{56} -2.09108 q^{58} -2.90270 q^{59} -13.8176 q^{61} +19.4958 q^{62} -10.6515 q^{64} +21.4377 q^{65} +11.5596 q^{67} -0.0477356 q^{68} -29.0221 q^{70} -11.2291 q^{71} +1.28040 q^{73} -16.8861 q^{74} -2.00298 q^{76} +11.4680 q^{77} +7.24288 q^{79} +13.3049 q^{80} -5.78036 q^{82} -6.40814 q^{83} +0.0859096 q^{85} -1.50944 q^{86} +2.74914 q^{88} -18.1777 q^{89} -16.3186 q^{91} +2.37262 q^{92} -12.3402 q^{94} +3.60475 q^{95} -5.01067 q^{97} +7.45429 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) $ 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8	$ 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8} - 4 q^{10} - 9 q^{11} - 16 q^{13} - 16 q^{14} + 27 q^{16} + q^{19} - 21 q^{20} + 17 q^{22} + 10 q^{23} - 4 q^{25} - 28 q^{26} - 14 q^{28} - 10 q^{29} + 17 q^{31} - 21 q^{32} - 3 q^{34} - 29 q^{35} + q^{37} - 32 q^{38} + 13 q^{40} - 5 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 31 q^{49} + 22 q^{50} - 21 q^{52} - 35 q^{53} - 20 q^{55} - 18 q^{56} + 3 q^{58} - 49 q^{59} + 8 q^{61} - 15 q^{62} + 12 q^{64} + 3 q^{65} + 35 q^{67} + 18 q^{68} - 16 q^{70} - 30 q^{71} - 15 q^{73} - 23 q^{74} + 10 q^{76} - 23 q^{77} + 24 q^{79} - 23 q^{80} - 5 q^{82} - q^{83} + 10 q^{86} + 18 q^{88} - 15 q^{89} + 26 q^{91} + 17 q^{92} + 3 q^{94} - 7 q^{95} - 35 q^{97} - 3 q^{98}+O(q^{100}) $ 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8 - 4 * q^10 - 9 * q^11 - 16 * q^13 - 16 * q^14 + 27 * q^16 + q^19 - 21 * q^20 + 17 * q^22 + 10 * q^23 - 4 * q^25 - 28 * q^26 - 14 * q^28 - 10 * q^29 + 17 * q^31 - 21 * q^32 - 3 * q^34 - 29 * q^35 + q^37 - 32 * q^38 + 13 * q^40 - 5 * q^43 - 33 * q^44 - 3 * q^46 - 15 * q^47 + 31 * q^49 + 22 * q^50 - 21 * q^52 - 35 * q^53 - 20 * q^55 - 18 * q^56 + 3 * q^58 - 49 * q^59 + 8 * q^61 - 15 * q^62 + 12 * q^64 + 3 * q^65 + 35 * q^67 + 18 * q^68 - 16 * q^70 - 30 * q^71 - 15 * q^73 - 23 * q^74 + 10 * q^76 - 23 * q^77 + 24 * q^79 - 23 * q^80 - 5 * q^82 - q^83 + 10 * q^86 + 18 * q^88 - 15 * q^89 + 26 * q^91 + 17 * q^92 + 3 * q^94 - 7 * q^95 - 35 * q^97 - 3 * 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 $

$2$	2.09108	1.47862	0.739309	−	0.673366i	$-0.235152\pi$
$2$	2.09108	1.47862	0.739309	+	0.673366i	$0.235152\pi$
$3$	0	0
$3$	0	0
$4$	2.37262	1.18631
$5$	−4.27000	−1.90960	−0.954801	−	0.297247i	$-0.903932\pi$
$5$	−4.27000	−1.90960	−0.954801	+	0.297247i	$0.903932\pi$
$6$	0	0
$7$	3.25035	1.22852	0.614259	−	0.789104i	$-0.289455\pi$
$7$	3.25035	1.22852	0.614259	+	0.789104i	$0.289455\pi$
$8$	0.779182	0.275482
$9$	0	0
$10$	−8.92891	−2.82357
$11$	3.52823	1.06380	0.531901	−	0.846806i	$-0.321478\pi$
$11$	3.52823	1.06380	0.531901	+	0.846806i	$0.321478\pi$
$12$	0	0
$13$	−5.02055	−1.39245	−0.696225	−	0.717824i	$-0.745138\pi$
$13$	−5.02055	−1.39245	−0.696225	+	0.717824i	$0.745138\pi$
$14$	6.79676	1.81651
$15$	0	0
$16$	−3.11591	−0.778978
$17$	−0.0201193	−0.00487966	−0.00243983	−	0.999997i	$-0.500777\pi$
$17$	−0.0201193	−0.00487966	−0.00243983	+	0.999997i	$0.500777\pi$
$18$	0	0
$19$	−0.844205	−0.193674	−0.0968369	−	0.995300i	$-0.530873\pi$
$19$	−0.844205	−0.193674	−0.0968369	+	0.995300i	$0.530873\pi$
$20$	−10.1311	−2.26538
$21$	0	0
$22$	7.37782	1.57296
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	13.2329	2.64658
$26$	−10.4984	−2.05890
$27$	0	0
$28$	7.71186	1.45740
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	9.32330	1.67451	0.837257	−	0.546809i	$-0.184158\pi$
$31$	9.32330	1.67451	0.837257	+	0.546809i	$0.184158\pi$
$32$	−8.07399	−1.42729
$33$	0	0
$34$	−0.0420712	−0.00721515
$35$	−13.8790	−2.34598
$36$	0	0
$37$	−8.07528	−1.32757	−0.663784	−	0.747925i	$-0.731050\pi$
$37$	−8.07528	−1.32757	−0.663784	+	0.747925i	$0.731050\pi$
$38$	−1.76530	−0.286370
$39$	0	0
$40$	−3.32711	−0.526062
$41$	−2.76429	−0.431710	−0.215855	−	0.976425i	$-0.569254\pi$
$41$	−2.76429	−0.431710	−0.215855	+	0.976425i	$0.569254\pi$
$42$	0	0
$43$	−0.721845	−0.110080	−0.0550402	−	0.998484i	$-0.517529\pi$
$43$	−0.721845	−0.110080	−0.0550402	+	0.998484i	$0.517529\pi$
$44$	8.37116	1.26200
$45$	0	0
$46$	2.09108	0.308313
$47$	−5.90133	−0.860798	−0.430399	−	0.902639i	$-0.641627\pi$
$47$	−5.90133	−0.860798	−0.430399	+	0.902639i	$0.641627\pi$
$48$	0	0
$49$	3.56480	0.509257
$50$	27.6710	3.91328
$51$	0	0
$52$	−11.9119	−1.65188
$53$	−11.8408	−1.62646	−0.813232	−	0.581940i	$-0.802294\pi$
$53$	−11.8408	−1.62646	−0.813232	+	0.581940i	$0.802294\pi$
$54$	0	0
$55$	−15.0656	−2.03144
$56$	2.53262	0.338435
$57$	0	0
$58$	−2.09108	−0.274572
$59$	−2.90270	−0.377899	−0.188949	−	0.981987i	$-0.560508\pi$
$59$	−2.90270	−0.377899	−0.188949	+	0.981987i	$0.560508\pi$
$60$	0	0
$61$	−13.8176	−1.76916	−0.884581	−	0.466386i	$-0.845556\pi$
$61$	−13.8176	−1.76916	−0.884581	+	0.466386i	$0.845556\pi$
$62$	19.4958	2.47597
$63$	0	0
$64$	−10.6515	−1.33144
$65$	21.4377	2.65902
$66$	0	0
$67$	11.5596	1.41223	0.706117	−	0.708095i	$-0.250445\pi$
$67$	11.5596	1.41223	0.706117	+	0.708095i	$0.250445\pi$
$68$	−0.0477356	−0.00578879
$69$	0	0
$70$	−29.0221	−3.46881
$71$	−11.2291	−1.33265	−0.666324	−	0.745662i	$-0.732133\pi$
$71$	−11.2291	−1.33265	−0.666324	+	0.745662i	$0.732133\pi$
$72$	0	0
$73$	1.28040	0.149860	0.0749300	−	0.997189i	$-0.476127\pi$
$73$	1.28040	0.149860	0.0749300	+	0.997189i	$0.476127\pi$
$74$	−16.8861	−1.96297
$75$	0	0
$76$	−2.00298	−0.229757
$77$	11.4680	1.30690
$78$	0	0
$79$	7.24288	0.814887	0.407444	−	0.913230i	$-0.366420\pi$
$79$	7.24288	0.814887	0.407444	+	0.913230i	$0.366420\pi$
$80$	13.3049	1.48754
$81$	0	0
$82$	−5.78036	−0.638334
$83$	−6.40814	−0.703385	−0.351692	−	0.936116i	$-0.614394\pi$
$83$	−6.40814	−0.703385	−0.351692	+	0.936116i	$0.614394\pi$
$84$	0	0
$85$	0.0859096	0.00931820
$86$	−1.50944	−0.162767
$87$	0	0
$88$	2.74914	0.293059
$89$	−18.1777	−1.92683	−0.963414	−	0.268018i	$-0.913631\pi$
$89$	−18.1777	−1.92683	−0.963414	+	0.268018i	$0.913631\pi$
$90$	0	0
$91$	−16.3186	−1.71065
$92$	2.37262	0.247363
$93$	0	0
$94$	−12.3402	−1.27279
$95$	3.60475	0.369840
$96$	0	0
$97$	−5.01067	−0.508757	−0.254378	−	0.967105i	$-0.581871\pi$
$97$	−5.01067	−0.508757	−0.254378	+	0.967105i	$0.581871\pi$
$98$	7.45429	0.752997
$99$	0	0
$100$	31.3966	3.13966
$101$	−2.10491	−0.209447	−0.104723	−	0.994501i	$-0.533396\pi$
$101$	−2.10491	−0.209447	−0.104723	+	0.994501i	$0.533396\pi$
$102$	0	0
$103$	15.6569	1.54272	0.771360	−	0.636400i	$-0.219577\pi$
$103$	15.6569	1.54272	0.771360	+	0.636400i	$0.219577\pi$
$104$	−3.91192	−0.383595
$105$	0	0
$106$	−24.7602	−2.40492
$107$	−5.88378	−0.568806	−0.284403	−	0.958705i	$-0.591795\pi$
$107$	−5.88378	−0.568806	−0.284403	+	0.958705i	$0.591795\pi$
$108$	0	0
$109$	1.60764	0.153984	0.0769921	−	0.997032i	$-0.475468\pi$
$109$	1.60764	0.153984	0.0769921	+	0.997032i	$0.475468\pi$
$110$	−31.5033	−3.00372
$111$	0	0
$112$	−10.1278	−0.956988
$113$	−1.85337	−0.174351	−0.0871753	−	0.996193i	$-0.527784\pi$
$113$	−1.85337	−0.174351	−0.0871753	+	0.996193i	$0.527784\pi$
$114$	0	0
$115$	−4.27000	−0.398179
$116$	−2.37262	−0.220292
$117$	0	0
$118$	−6.06977	−0.558768
$119$	−0.0653950	−0.00599475
$120$	0	0
$121$	1.44844	0.131676
$122$	−28.8937	−2.61592
$123$	0	0
$124$	22.1207	1.98649
$125$	−35.1544	−3.14431
$126$	0	0
$127$	12.0665	1.07073	0.535364	−	0.844621i	$-0.320174\pi$
$127$	12.0665	1.07073	0.535364	+	0.844621i	$0.320174\pi$
$128$	−6.12527	−0.541402
$129$	0	0
$130$	44.8280	3.93168
$131$	−3.85596	−0.336897	−0.168448	−	0.985710i	$-0.553876\pi$
$131$	−3.85596	−0.336897	−0.168448	+	0.985710i	$0.553876\pi$
$132$	0	0
$133$	−2.74396	−0.237932
$134$	24.1722	2.08816
$135$	0	0
$136$	−0.0156766	−0.00134426
$137$	15.6037	1.33311	0.666557	−	0.745454i	$-0.267767\pi$
$137$	15.6037	1.33311	0.666557	+	0.745454i	$0.267767\pi$
$138$	0	0
$139$	−6.24868	−0.530006	−0.265003	−	0.964248i	$-0.585373\pi$
$139$	−6.24868	−0.530006	−0.265003	+	0.964248i	$0.585373\pi$
$140$	−32.9296	−2.78306
$141$	0	0
$142$	−23.4809	−1.97048
$143$	−17.7137	−1.48129
$144$	0	0
$145$	4.27000	0.354604
$146$	2.67743	0.221586
$147$	0	0
$148$	−19.1596	−1.57491
$149$	−14.5432	−1.19142	−0.595712	−	0.803198i	$-0.703130\pi$
$149$	−14.5432	−1.19142	−0.595712	+	0.803198i	$0.703130\pi$
$150$	0	0
$151$	−10.6692	−0.868246	−0.434123	−	0.900854i	$-0.642942\pi$
$151$	−10.6692	−0.868246	−0.434123	+	0.900854i	$0.642942\pi$
$152$	−0.657789	−0.0533538
$153$	0	0
$154$	23.9805	1.93241
$155$	−39.8105	−3.19765
$156$	0	0
$157$	−15.7549	−1.25738	−0.628690	−	0.777656i	$-0.716409\pi$
$157$	−15.7549	−1.25738	−0.628690	+	0.777656i	$0.716409\pi$
$158$	15.1454	1.20491
$159$	0	0
$160$	34.4759	2.72556
$161$	3.25035	0.256164
$162$	0	0
$163$	−12.9934	−1.01772	−0.508859	−	0.860850i	$-0.669933\pi$
$163$	−12.9934	−1.01772	−0.508859	+	0.860850i	$0.669933\pi$
$164$	−6.55862	−0.512142
$165$	0	0
$166$	−13.3999	−1.04004
$167$	−6.37516	−0.493325	−0.246662	−	0.969101i	$-0.579334\pi$
$167$	−6.37516	−0.493325	−0.246662	+	0.969101i	$0.579334\pi$
$168$	0	0
$169$	12.2059	0.938915
$170$	0.179644	0.0137781
$171$	0	0
$172$	−1.71267	−0.130590
$173$	3.84645	0.292440	0.146220	−	0.989252i	$-0.453289\pi$
$173$	3.84645	0.292440	0.146220	+	0.989252i	$0.453289\pi$
$174$	0	0
$175$	43.0116	3.25137
$176$	−10.9937	−0.828678
$177$	0	0
$178$	−38.0110	−2.84904
$179$	−6.55212	−0.489728	−0.244864	−	0.969557i	$-0.578743\pi$
$179$	−6.55212	−0.489728	−0.244864	+	0.969557i	$0.578743\pi$
$180$	0	0
$181$	−18.2300	−1.35502	−0.677512	−	0.735511i	$-0.736942\pi$
$181$	−18.2300	−1.35502	−0.677512	+	0.735511i	$0.736942\pi$
$182$	−34.1234	−2.52940
$183$	0	0
$184$	0.779182	0.0574421
$185$	34.4814	2.53512
$186$	0	0
$187$	−0.0709858	−0.00519099
$188$	−14.0016	−1.02117
$189$	0	0
$190$	7.53783	0.546852
$191$	−9.76931	−0.706882	−0.353441	−	0.935457i	$-0.614989\pi$
$191$	−9.76931	−0.706882	−0.353441	+	0.935457i	$0.614989\pi$
$192$	0	0
$193$	−1.59407	−0.114743	−0.0573717	−	0.998353i	$-0.518272\pi$
$193$	−1.59407	−0.114743	−0.0573717	+	0.998353i	$0.518272\pi$
$194$	−10.4777	−0.752257
$195$	0	0
$196$	8.45793	0.604138
$197$	17.4479	1.24311	0.621555	−	0.783370i	$-0.286501\pi$
$197$	17.4479	1.24311	0.621555	+	0.783370i	$0.286501\pi$
$198$	0	0
$199$	15.9600	1.13137	0.565686	−	0.824620i	$-0.308611\pi$
$199$	15.9600	1.13137	0.565686	+	0.824620i	$0.308611\pi$
$200$	10.3108	0.729086
$201$	0	0
$202$	−4.40155	−0.309692
$203$	−3.25035	−0.228130
$204$	0	0
$205$	11.8035	0.824394
$206$	32.7398	2.28109
$207$	0	0
$208$	15.6436	1.08469
$209$	−2.97855	−0.206031
$210$	0	0
$211$	−7.87384	−0.542057	−0.271029	−	0.962571i	$-0.587364\pi$
$211$	−7.87384	−0.542057	−0.271029	+	0.962571i	$0.587364\pi$
$212$	−28.0938	−1.92949
$213$	0	0
$214$	−12.3035	−0.841047
$215$	3.08228	0.210210
$216$	0	0
$217$	30.3040	2.05717
$218$	3.36171	0.227684
$219$	0	0
$220$	−35.7449	−2.40992
$221$	0.101010	0.00679468
$222$	0	0
$223$	−19.3035	−1.29266	−0.646330	−	0.763058i	$-0.723697\pi$
$223$	−19.3035	−1.29266	−0.646330	+	0.763058i	$0.723697\pi$
$224$	−26.2433	−1.75346
$225$	0	0
$226$	−3.87555	−0.257798
$227$	18.1220	1.20280	0.601401	−	0.798947i	$-0.294609\pi$
$227$	18.1220	1.20280	0.601401	+	0.798947i	$0.294609\pi$
$228$	0	0
$229$	5.94760	0.393029	0.196514	−	0.980501i	$-0.437038\pi$
$229$	5.94760	0.393029	0.196514	+	0.980501i	$0.437038\pi$
$230$	−8.92891	−0.588755
$231$	0	0
$232$	−0.779182	−0.0511558
$233$	7.09230	0.464632	0.232316	−	0.972640i	$-0.425370\pi$
$233$	7.09230	0.464632	0.232316	+	0.972640i	$0.425370\pi$
$234$	0	0
$235$	25.1987	1.64378
$236$	−6.88700	−0.448306
$237$	0	0
$238$	−0.136746	−0.00886394
$239$	−9.30675	−0.602004	−0.301002	−	0.953624i	$-0.597321\pi$
$239$	−9.30675	−0.602004	−0.301002	+	0.953624i	$0.597321\pi$
$240$	0	0
$241$	6.58703	0.424308	0.212154	−	0.977236i	$-0.431952\pi$
$241$	6.58703	0.424308	0.212154	+	0.977236i	$0.431952\pi$
$242$	3.02880	0.194698
$243$	0	0
$244$	−32.7839	−2.09878
$245$	−15.2217	−0.972479
$246$	0	0
$247$	4.23837	0.269681
$248$	7.26455	0.461299
$249$	0	0
$250$	−73.5107	−4.64923
$251$	10.0983	0.637397	0.318698	−	0.947856i	$-0.396754\pi$
$251$	10.0983	0.637397	0.318698	+	0.947856i	$0.396754\pi$
$252$	0	0
$253$	3.52823	0.221818
$254$	25.2320	1.58320
$255$	0	0
$256$	8.49465	0.530915
$257$	2.03420	0.126890	0.0634449	−	0.997985i	$-0.479791\pi$
$257$	2.03420	0.126890	0.0634449	+	0.997985i	$0.479791\pi$
$258$	0	0
$259$	−26.2475	−1.63094
$260$	50.8636	3.15443
$261$	0	0
$262$	−8.06312	−0.498142
$263$	31.4605	1.93994	0.969968	−	0.243232i	$-0.0782078\pi$
$263$	31.4605	1.93994	0.969968	+	0.243232i	$0.0782078\pi$
$264$	0	0
$265$	50.5603	3.10590
$266$	−5.73785	−0.351810
$267$	0	0
$268$	27.4267	1.67535
$269$	−3.12220	−0.190364	−0.0951820	−	0.995460i	$-0.530343\pi$
$269$	−3.12220	−0.190364	−0.0951820	+	0.995460i	$0.530343\pi$
$270$	0	0
$271$	29.0814	1.76657	0.883284	−	0.468838i	$-0.155327\pi$
$271$	29.0814	1.76657	0.883284	+	0.468838i	$0.155327\pi$
$272$	0.0626901	0.00380114
$273$	0	0
$274$	32.6286	1.97117
$275$	46.6887	2.81544
$276$	0	0
$277$	22.0686	1.32598	0.662988	−	0.748630i	$-0.269288\pi$
$277$	22.0686	1.32598	0.662988	+	0.748630i	$0.269288\pi$
$278$	−13.0665	−0.783676
$279$	0	0
$280$	−10.8143	−0.646276
$281$	−12.6153	−0.752568	−0.376284	−	0.926504i	$-0.622798\pi$
$281$	−12.6153	−0.752568	−0.376284	+	0.926504i	$0.622798\pi$
$282$	0	0
$283$	−19.5697	−1.16330	−0.581649	−	0.813440i	$-0.697592\pi$
$283$	−19.5697	−1.16330	−0.581649	+	0.813440i	$0.697592\pi$
$284$	−26.6424	−1.58093
$285$	0	0
$286$	−37.0407	−2.19026
$287$	−8.98493	−0.530364
$288$	0	0
$289$	−16.9996	−0.999976
$290$	8.92891	0.524324
$291$	0	0
$292$	3.03791	0.177781
$293$	−14.4860	−0.846281	−0.423141	−	0.906064i	$-0.639072\pi$
$293$	−14.4860	−0.846281	−0.423141	+	0.906064i	$0.639072\pi$
$294$	0	0
$295$	12.3945	0.721636
$296$	−6.29211	−0.365722
$297$	0	0
$298$	−30.4110	−1.76166
$299$	−5.02055	−0.290346
$300$	0	0
$301$	−2.34625	−0.135236
$302$	−22.3101	−1.28380
$303$	0	0
$304$	2.63047	0.150868
$305$	59.0011	3.37840
$306$	0	0
$307$	−27.0211	−1.54218	−0.771088	−	0.636729i	$-0.780287\pi$
$307$	−27.0211	−1.54218	−0.771088	+	0.636729i	$0.780287\pi$
$308$	27.2092	1.55039
$309$	0	0
$310$	−83.2470	−4.72811
$311$	−24.1529	−1.36959	−0.684793	−	0.728737i	$-0.740108\pi$
$311$	−24.1529	−1.36959	−0.684793	+	0.728737i	$0.740108\pi$
$312$	0	0
$313$	25.3428	1.43246	0.716229	−	0.697865i	$-0.245866\pi$
$313$	25.3428	1.43246	0.716229	+	0.697865i	$0.245866\pi$
$314$	−32.9448	−1.85918
$315$	0	0
$316$	17.1846	0.966710
$317$	−6.57718	−0.369411	−0.184706	−	0.982794i	$-0.559133\pi$
$317$	−6.57718	−0.369411	−0.184706	+	0.982794i	$0.559133\pi$
$318$	0	0
$319$	−3.52823	−0.197543
$320$	45.4821	2.54252
$321$	0	0
$322$	6.79676	0.378768
$323$	0.0169848	0.000945062 0
$324$	0	0
$325$	−66.4363	−3.68522
$326$	−27.1702	−1.50482
$327$	0	0
$328$	−2.15389	−0.118929
$329$	−19.1814	−1.05751
$330$	0	0
$331$	28.9988	1.59392	0.796958	−	0.604034i	$-0.206441\pi$
$331$	28.9988	1.59392	0.796958	+	0.604034i	$0.206441\pi$
$332$	−15.2041	−0.834433
$333$	0	0
$334$	−13.3310	−0.729439
$335$	−49.3597	−2.69681
$336$	0	0
$337$	14.4842	0.789003	0.394501	−	0.918895i	$-0.370917\pi$
$337$	14.4842	0.789003	0.394501	+	0.918895i	$0.370917\pi$
$338$	25.5235	1.38830
$339$	0	0
$340$	0.203831	0.0110543
$341$	32.8948	1.78135
$342$	0	0
$343$	−11.1656	−0.602886
$344$	−0.562449	−0.0303252
$345$	0	0
$346$	8.04324	0.432407
$347$	1.73755	0.0932764	0.0466382	−	0.998912i	$-0.485149\pi$
$347$	1.73755	0.0932764	0.0466382	+	0.998912i	$0.485149\pi$
$348$	0	0
$349$	9.73121	0.520900	0.260450	−	0.965487i	$-0.416129\pi$
$349$	9.73121	0.520900	0.260450	+	0.965487i	$0.416129\pi$
$350$	89.9407	4.80753
$351$	0	0
$352$	−28.4869	−1.51836
$353$	−3.57545	−0.190302	−0.0951511	−	0.995463i	$-0.530333\pi$
$353$	−3.57545	−0.190302	−0.0951511	+	0.995463i	$0.530333\pi$
$354$	0	0
$355$	47.9482	2.54483
$356$	−43.1287	−2.28582
$357$	0	0
$358$	−13.7010	−0.724121
$359$	17.1057	0.902803	0.451402	−	0.892321i	$-0.350924\pi$
$359$	17.1057	0.902803	0.451402	+	0.892321i	$0.350924\pi$
$360$	0	0
$361$	−18.2873	−0.962490
$362$	−38.1204	−2.00356
$363$	0	0
$364$	−38.7178	−2.02936
$365$	−5.46732	−0.286173
$366$	0	0
$367$	−25.8324	−1.34844	−0.674219	−	0.738532i	$-0.735520\pi$
$367$	−25.8324	−1.34844	−0.674219	+	0.738532i	$0.735520\pi$
$368$	−3.11591	−0.162428
$369$	0	0
$370$	72.1035	3.74848
$371$	−38.4869	−1.99814
$372$	0	0
$373$	0.929245	0.0481145	0.0240572	−	0.999711i	$-0.492342\pi$
$373$	0.929245	0.0481145	0.0240572	+	0.999711i	$0.492342\pi$
$374$	−0.148437	−0.00767549
$375$	0	0
$376$	−4.59821	−0.237135
$377$	5.02055	0.258571
$378$	0	0
$379$	34.0540	1.74924	0.874619	−	0.484810i	$-0.161111\pi$
$379$	34.0540	1.74924	0.874619	+	0.484810i	$0.161111\pi$
$380$	8.55272	0.438745
$381$	0	0
$382$	−20.4284	−1.04521
$383$	−11.4951	−0.587371	−0.293685	−	0.955902i	$-0.594882\pi$
$383$	−11.4951	−0.587371	−0.293685	+	0.955902i	$0.594882\pi$
$384$	0	0
$385$	−48.9684	−2.49566
$386$	−3.33332	−0.169662
$387$	0	0
$388$	−11.8884	−0.603544
$389$	37.5870	1.90574	0.952869	−	0.303382i	$-0.0981157\pi$
$389$	37.5870	1.90574	0.952869	+	0.303382i	$0.0981157\pi$
$390$	0	0
$391$	−0.0201193	−0.00101748
$392$	2.77763	0.140291
$393$	0	0
$394$	36.4850	1.83809
$395$	−30.9271	−1.55611
$396$	0	0
$397$	−25.4305	−1.27632	−0.638161	−	0.769903i	$-0.720305\pi$
$397$	−25.4305	−1.27632	−0.638161	+	0.769903i	$0.720305\pi$
$398$	33.3736	1.67287
$399$	0	0
$400$	−41.2325	−2.06162
$401$	35.8401	1.78977	0.894886	−	0.446296i	$-0.147257\pi$
$401$	35.8401	1.78977	0.894886	+	0.446296i	$0.147257\pi$
$402$	0	0
$403$	−46.8081	−2.33168
$404$	−4.99417	−0.248469
$405$	0	0
$406$	−6.79676	−0.337317
$407$	−28.4915	−1.41227
$408$	0	0
$409$	2.10354	0.104013	0.0520067	−	0.998647i	$-0.483438\pi$
$409$	2.10354	0.104013	0.0520067	+	0.998647i	$0.483438\pi$
$410$	24.6821	1.21896
$411$	0	0
$412$	37.1479	1.83014
$413$	−9.43479	−0.464256
$414$	0	0
$415$	27.3627	1.34318
$416$	40.5358	1.98743
$417$	0	0
$418$	−6.22840	−0.304641
$419$	−1.76510	−0.0862309	−0.0431154	−	0.999070i	$-0.513728\pi$
$419$	−1.76510	−0.0862309	−0.0431154	+	0.999070i	$0.513728\pi$
$420$	0	0
$421$	35.8642	1.74791	0.873957	−	0.486004i	$-0.161546\pi$
$421$	35.8642	1.74791	0.873957	+	0.486004i	$0.161546\pi$
$422$	−16.4648	−0.801495
$423$	0	0
$424$	−9.22617	−0.448062
$425$	−0.266237	−0.0129144
$426$	0	0
$427$	−44.9121	−2.17345
$428$	−13.9600	−0.674781
$429$	0	0
$430$	6.44529	0.310820
$431$	−23.2484	−1.11984	−0.559918	−	0.828548i	$-0.689167\pi$
$431$	−23.2484	−1.11984	−0.559918	+	0.828548i	$0.689167\pi$
$432$	0	0
$433$	6.84029	0.328724	0.164362	−	0.986400i	$-0.447444\pi$
$433$	6.84029	0.328724	0.164362	+	0.986400i	$0.447444\pi$
$434$	63.3682	3.04177
$435$	0	0
$436$	3.81433	0.182673
$437$	−0.844205	−0.0403838
$438$	0	0
$439$	−10.6962	−0.510503	−0.255251	−	0.966875i	$-0.582158\pi$
$439$	−10.6962	−0.510503	−0.255251	+	0.966875i	$0.582158\pi$
$440$	−11.7388	−0.559626
$441$	0	0
$442$	0.211220	0.0100467
$443$	−0.758143	−0.0360205	−0.0180102	−	0.999838i	$-0.505733\pi$
$443$	−0.758143	−0.0360205	−0.0180102	+	0.999838i	$0.505733\pi$
$444$	0	0
$445$	77.6186	3.67947
$446$	−40.3652	−1.91135
$447$	0	0
$448$	−34.6213	−1.63570
$449$	17.2635	0.814714	0.407357	−	0.913269i	$-0.366451\pi$
$449$	17.2635	0.814714	0.407357	+	0.913269i	$0.366451\pi$
$450$	0	0
$451$	−9.75308	−0.459254
$452$	−4.39735	−0.206834
$453$	0	0
$454$	37.8947	1.77849
$455$	69.6802	3.26666
$456$	0	0
$457$	0.915744	0.0428367	0.0214183	−	0.999771i	$-0.493182\pi$
$457$	0.915744	0.0428367	0.0214183	+	0.999771i	$0.493182\pi$
$458$	12.4369	0.581139
$459$	0	0
$460$	−10.1311	−0.472365
$461$	−21.5469	−1.00354	−0.501769	−	0.865002i	$-0.667317\pi$
$461$	−21.5469	−1.00354	−0.501769	+	0.865002i	$0.667317\pi$
$462$	0	0
$463$	−27.0273	−1.25607	−0.628034	−	0.778186i	$-0.716140\pi$
$463$	−27.0273	−1.25607	−0.628034	+	0.778186i	$0.716140\pi$
$464$	3.11591	0.144652
$465$	0	0
$466$	14.8306	0.687013
$467$	28.9856	1.34130	0.670648	−	0.741776i	$-0.266016\pi$
$467$	28.9856	1.34130	0.670648	+	0.741776i	$0.266016\pi$
$468$	0	0
$469$	37.5729	1.73496
$470$	52.6925	2.43052
$471$	0	0
$472$	−2.26173	−0.104105
$473$	−2.54684	−0.117104
$474$	0	0
$475$	−11.1713	−0.512573
$476$	−0.155158	−0.00711164
$477$	0	0
$478$	−19.4612	−0.890134
$479$	−5.03847	−0.230214	−0.115107	−	0.993353i	$-0.536721\pi$
$479$	−5.03847	−0.230214	−0.115107	+	0.993353i	$0.536721\pi$
$480$	0	0
$481$	40.5423	1.84857
$482$	13.7740	0.627390
$483$	0	0
$484$	3.43659	0.156209
$485$	21.3956	0.971523
$486$	0	0
$487$	−23.8712	−1.08171	−0.540854	−	0.841117i	$-0.681899\pi$
$487$	−23.8712	−1.08171	−0.540854	+	0.841117i	$0.681899\pi$
$488$	−10.7664	−0.487373
$489$	0	0
$490$	−31.8298	−1.43792
$491$	−28.4182	−1.28249	−0.641247	−	0.767335i	$-0.721583\pi$
$491$	−28.4182	−1.28249	−0.641247	+	0.767335i	$0.721583\pi$
$492$	0	0
$493$	0.0201193	0.000906130 0
$494$	8.86278	0.398755
$495$	0	0
$496$	−29.0506	−1.30441
$497$	−36.4985	−1.63718
$498$	0	0
$499$	29.2166	1.30792	0.653958	−	0.756531i	$-0.273107\pi$
$499$	29.2166	1.30792	0.653958	+	0.756531i	$0.273107\pi$
$500$	−83.4081	−3.73012
$501$	0	0
$502$	21.1163	0.942466
$503$	−7.41728	−0.330720	−0.165360	−	0.986233i	$-0.552879\pi$
$503$	−7.41728	−0.330720	−0.165360	+	0.986233i	$0.552879\pi$
$504$	0	0
$505$	8.98798	0.399960
$506$	7.37782	0.327984
$507$	0	0
$508$	28.6292	1.27022
$509$	34.3566	1.52283	0.761415	−	0.648264i	$-0.224505\pi$
$509$	34.3566	1.52283	0.761415	+	0.648264i	$0.224505\pi$
$510$	0	0
$511$	4.16177	0.184106
$512$	30.0135	1.32642
$513$	0	0
$514$	4.25367	0.187621
$515$	−66.8549	−2.94598
$516$	0	0
$517$	−20.8213	−0.915719
$518$	−54.8857	−2.41154
$519$	0	0
$520$	16.7039	0.732514
$521$	8.80762	0.385869	0.192935	−	0.981212i	$-0.438200\pi$
$521$	8.80762	0.385869	0.192935	+	0.981212i	$0.438200\pi$
$522$	0	0
$523$	8.70954	0.380842	0.190421	−	0.981703i	$-0.439015\pi$
$523$	8.70954	0.380842	0.190421	+	0.981703i	$0.439015\pi$
$524$	−9.14873	−0.399664
$525$	0	0
$526$	65.7864	2.86842
$527$	−0.187579	−0.00817106
$528$	0	0
$529$	1.00000	0.0434783
$530$	105.726	4.59244
$531$	0	0
$532$	−6.51039	−0.282261
$533$	13.8783	0.601134
$534$	0	0
$535$	25.1237	1.08619
$536$	9.00706	0.389046
$537$	0	0
$538$	−6.52878	−0.281476
$539$	12.5775	0.541749
$540$	0	0
$541$	−45.0504	−1.93687	−0.968434	−	0.249271i	$-0.919809\pi$
$541$	−45.0504	−1.93687	−0.968434	+	0.249271i	$0.919809\pi$
$542$	60.8116	2.61208
$543$	0	0
$544$	0.162443	0.00696470
$545$	−6.86463	−0.294048
$546$	0	0
$547$	−22.2007	−0.949234	−0.474617	−	0.880192i	$-0.657413\pi$
$547$	−22.2007	−0.949234	−0.474617	+	0.880192i	$0.657413\pi$
$548$	37.0217	1.58149
$549$	0	0
$550$	97.6299	4.16295
$551$	0.844205	0.0359643
$552$	0	0
$553$	23.5419	1.00110
$554$	46.1473	1.96061
$555$	0	0
$556$	−14.8257	−0.628752
$557$	36.5137	1.54714	0.773568	−	0.633713i	$-0.218470\pi$
$557$	36.5137	1.54714	0.773568	+	0.633713i	$0.218470\pi$
$558$	0	0
$559$	3.62406	0.153281
$560$	43.2457	1.82747
$561$	0	0
$562$	−26.3797	−1.11276
$563$	27.9546	1.17814	0.589072	−	0.808080i	$-0.299493\pi$
$563$	27.9546	1.17814	0.589072	+	0.808080i	$0.299493\pi$
$564$	0	0
$565$	7.91390	0.332940
$566$	−40.9218	−1.72007
$567$	0	0
$568$	−8.74951	−0.367121
$569$	−15.4258	−0.646684	−0.323342	−	0.946282i	$-0.604806\pi$
$569$	−15.4258	−0.646684	−0.323342	+	0.946282i	$0.604806\pi$
$570$	0	0
$571$	−9.08243	−0.380088	−0.190044	−	0.981776i	$-0.560863\pi$
$571$	−9.08243	−0.380088	−0.190044	+	0.981776i	$0.560863\pi$
$572$	−42.0278	−1.75727
$573$	0	0
$574$	−18.7882	−0.784205
$575$	13.2329	0.551850
$576$	0	0
$577$	10.7508	0.447563	0.223781	−	0.974639i	$-0.428160\pi$
$577$	10.7508	0.447563	0.223781	+	0.974639i	$0.428160\pi$
$578$	−35.5475	−1.47858
$579$	0	0
$580$	10.1311	0.420671
$581$	−20.8287	−0.864121
$582$	0	0
$583$	−41.7772	−1.73024
$584$	0.997668	0.0412838
$585$	0	0
$586$	−30.2914	−1.25133
$587$	3.81891	0.157623	0.0788117	−	0.996890i	$-0.474887\pi$
$587$	3.81891	0.157623	0.0788117	+	0.996890i	$0.474887\pi$
$588$	0	0
$589$	−7.87078	−0.324310
$590$	25.9179	1.06702
$591$	0	0
$592$	25.1618	1.03415
$593$	27.3374	1.12261	0.561306	−	0.827608i	$-0.310299\pi$
$593$	27.3374	1.12261	0.561306	+	0.827608i	$0.310299\pi$
$594$	0	0
$595$	0.279237	0.0114476
$596$	−34.5054	−1.41340
$597$	0	0
$598$	−10.4984	−0.429310
$599$	15.7743	0.644519	0.322259	−	0.946651i	$-0.395558\pi$
$599$	15.7743	0.644519	0.322259	+	0.946651i	$0.395558\pi$
$600$	0	0
$601$	−20.0160	−0.816468	−0.408234	−	0.912877i	$-0.633855\pi$
$601$	−20.0160	−0.816468	−0.408234	+	0.912877i	$0.633855\pi$
$602$	−4.90621	−0.199962
$603$	0	0
$604$	−25.3139	−1.03001
$605$	−6.18482	−0.251449
$606$	0	0
$607$	−27.9764	−1.13553	−0.567763	−	0.823192i	$-0.692191\pi$
$607$	−27.9764	−1.13553	−0.567763	+	0.823192i	$0.692191\pi$
$608$	6.81610	0.276429
$609$	0	0
$610$	123.376	4.99536
$611$	29.6279	1.19862
$612$	0	0
$613$	17.7179	0.715620	0.357810	−	0.933794i	$-0.383524\pi$
$613$	17.7179	0.715620	0.357810	+	0.933794i	$0.383524\pi$
$614$	−56.5033	−2.28029
$615$	0	0
$616$	8.93567	0.360028
$617$	−36.5526	−1.47155	−0.735777	−	0.677224i	$-0.763183\pi$
$617$	−36.5526	−1.47155	−0.735777	+	0.677224i	$0.763183\pi$
$618$	0	0
$619$	1.85013	0.0743631	0.0371815	−	0.999309i	$-0.488162\pi$
$619$	1.85013	0.0743631	0.0371815	+	0.999309i	$0.488162\pi$
$620$	−94.4552	−3.79341
$621$	0	0
$622$	−50.5057	−2.02510
$623$	−59.0838	−2.36714
$624$	0	0
$625$	83.9449	3.35779
$626$	52.9938	2.11806
$627$	0	0
$628$	−37.3805	−1.49164
$629$	0.162469	0.00647808
$630$	0	0
$631$	−4.10725	−0.163507	−0.0817535	−	0.996653i	$-0.526052\pi$
$631$	−4.10725	−0.163507	−0.0817535	+	0.996653i	$0.526052\pi$
$632$	5.64352	0.224487
$633$	0	0
$634$	−13.7534	−0.546218
$635$	−51.5239	−2.04466
$636$	0	0
$637$	−17.8973	−0.709115
$638$	−7.37782	−0.292091
$639$	0	0
$640$	26.1549	1.03386
$641$	10.0818	0.398209	0.199104	−	0.979978i	$-0.436197\pi$
$641$	10.0818	0.398209	0.199104	+	0.979978i	$0.436197\pi$
$642$	0	0
$643$	13.9525	0.550233	0.275117	−	0.961411i	$-0.411284\pi$
$643$	13.9525	0.550233	0.275117	+	0.961411i	$0.411284\pi$
$644$	7.71186	0.303890
$645$	0	0
$646$	0.0355167	0.00139739
$647$	5.56296	0.218703	0.109351	−	0.994003i	$-0.465123\pi$
$647$	5.56296	0.218703	0.109351	+	0.994003i	$0.465123\pi$
$648$	0	0
$649$	−10.2414	−0.402010
$650$	−138.924	−5.44904
$651$	0	0
$652$	−30.8283	−1.20733
$653$	−16.9574	−0.663593	−0.331796	−	0.943351i	$-0.607655\pi$
$653$	−16.9574	−0.663593	−0.331796	+	0.943351i	$0.607655\pi$
$654$	0	0
$655$	16.4649	0.643338
$656$	8.61329	0.336293
$657$	0	0
$658$	−40.1099	−1.56365
$659$	21.7981	0.849134	0.424567	−	0.905396i	$-0.360426\pi$
$659$	21.7981	0.849134	0.424567	+	0.905396i	$0.360426\pi$
$660$	0	0
$661$	−47.7860	−1.85866	−0.929330	−	0.369249i	$-0.879615\pi$
$661$	−47.7860	−1.85866	−0.929330	+	0.369249i	$0.879615\pi$
$662$	60.6388	2.35679
$663$	0	0
$664$	−4.99311	−0.193770
$665$	11.7167	0.454355
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−15.1258	−0.585237
$669$	0	0
$670$	−103.215	−3.98755
$671$	−48.7517	−1.88204
$672$	0	0
$673$	−26.7260	−1.03021	−0.515107	−	0.857126i	$-0.672248\pi$
$673$	−26.7260	−1.03021	−0.515107	+	0.857126i	$0.672248\pi$
$674$	30.2876	1.16663
$675$	0	0
$676$	28.9600	1.11384
$677$	−11.9940	−0.460965	−0.230483	−	0.973076i	$-0.574030\pi$
$677$	−11.9940	−0.460965	−0.230483	+	0.973076i	$0.574030\pi$
$678$	0	0
$679$	−16.2865	−0.625017
$680$	0.0669392	0.00256700
$681$	0	0
$682$	68.7857	2.63394
$683$	9.29936	0.355830	0.177915	−	0.984046i	$-0.443065\pi$
$683$	9.29936	0.355830	0.177915	+	0.984046i	$0.443065\pi$
$684$	0	0
$685$	−66.6278	−2.54572
$686$	−23.3482	−0.891438
$687$	0	0
$688$	2.24920	0.0857501
$689$	59.4475	2.26477
$690$	0	0
$691$	15.0512	0.572575	0.286288	−	0.958144i	$-0.407579\pi$
$691$	15.0512	0.572575	0.286288	+	0.958144i	$0.407579\pi$
$692$	9.12617	0.346925
$693$	0	0
$694$	3.63335	0.137920
$695$	26.6818	1.01210
$696$	0	0
$697$	0.0556158	0.00210660
$698$	20.3487	0.770211
$699$	0	0
$700$	102.050	3.85713
$701$	29.8218	1.12635	0.563176	−	0.826337i	$-0.309579\pi$
$701$	29.8218	1.12635	0.563176	+	0.826337i	$0.309579\pi$
$702$	0	0
$703$	6.81719	0.257115
$704$	−37.5811	−1.41639
$705$	0	0
$706$	−7.47656	−0.281384
$707$	−6.84172	−0.257309
$708$	0	0
$709$	34.5946	1.29923	0.649614	−	0.760264i	$-0.274931\pi$
$709$	34.5946	1.29923	0.649614	+	0.760264i	$0.274931\pi$
$710$	100.264	3.76283
$711$	0	0
$712$	−14.1637	−0.530807
$713$	9.32330	0.349160
$714$	0	0
$715$	75.6373	2.82868
$716$	−15.5457	−0.580970
$717$	0	0
$718$	35.7694	1.33490
$719$	−9.33825	−0.348258	−0.174129	−	0.984723i	$-0.555711\pi$
$719$	−9.33825	−0.348258	−0.174129	+	0.984723i	$0.555711\pi$
$720$	0	0
$721$	50.8904	1.89526
$722$	−38.2403	−1.42316
$723$	0	0
$724$	−43.2529	−1.60748
$725$	−13.2329	−0.491457
$726$	0	0
$727$	−5.85769	−0.217250	−0.108625	−	0.994083i	$-0.534645\pi$
$727$	−5.85769	−0.217250	−0.108625	+	0.994083i	$0.534645\pi$
$728$	−12.7151	−0.471254
$729$	0	0
$730$	−11.4326	−0.423140
$731$	0.0145231	0.000537155 0
$732$	0	0
$733$	−9.06161	−0.334698	−0.167349	−	0.985898i	$-0.553521\pi$
$733$	−9.06161	−0.334698	−0.167349	+	0.985898i	$0.553521\pi$
$734$	−54.0176	−1.99382
$735$	0	0
$736$	−8.07399	−0.297611
$737$	40.7851	1.50234
$738$	0	0
$739$	25.4207	0.935115	0.467557	−	0.883963i	$-0.345134\pi$
$739$	25.4207	0.935115	0.467557	+	0.883963i	$0.345134\pi$
$740$	81.8114	3.00745
$741$	0	0
$742$	−80.4793	−2.95449
$743$	−34.5910	−1.26902	−0.634510	−	0.772914i	$-0.718798\pi$
$743$	−34.5910	−1.26902	−0.634510	+	0.772914i	$0.718798\pi$
$744$	0	0
$745$	62.0993	2.27514
$746$	1.94313	0.0711429
$747$	0	0
$748$	−0.168422	−0.00615813
$749$	−19.1244	−0.698789
$750$	0	0
$751$	13.9340	0.508458	0.254229	−	0.967144i	$-0.418178\pi$
$751$	13.9340	0.508458	0.254229	+	0.967144i	$0.418178\pi$
$752$	18.3880	0.670542
$753$	0	0
$754$	10.4984	0.382328
$755$	45.5574	1.65800
$756$	0	0
$757$	−31.3210	−1.13838	−0.569191	−	0.822206i	$-0.692743\pi$
$757$	−31.3210	−1.13838	−0.569191	+	0.822206i	$0.692743\pi$
$758$	71.2098	2.58646
$759$	0	0
$760$	2.80876	0.101884
$761$	−35.8476	−1.29947	−0.649737	−	0.760159i	$-0.725121\pi$
$761$	−35.8476	−1.29947	−0.649737	+	0.760159i	$0.725121\pi$
$762$	0	0
$763$	5.22541	0.189172
$764$	−23.1789	−0.838582
$765$	0	0
$766$	−24.0371	−0.868497
$767$	14.5731	0.526205
$768$	0	0
$769$	35.7151	1.28792	0.643959	−	0.765060i	$-0.277291\pi$
$769$	35.7151	1.28792	0.643959	+	0.765060i	$0.277291\pi$
$770$	−102.397	−3.69013
$771$	0	0
$772$	−3.78211	−0.136121
$773$	43.2256	1.55472	0.777358	−	0.629058i	$-0.216559\pi$
$773$	43.2256	1.55472	0.777358	+	0.629058i	$0.216559\pi$
$774$	0	0
$775$	123.374	4.43173
$776$	−3.90423	−0.140154
$777$	0	0
$778$	78.5975	2.81786
$779$	2.33363	0.0836110
$780$	0	0
$781$	−39.6189	−1.41767
$782$	−0.0420712	−0.00150446
$783$	0	0
$784$	−11.1076	−0.396700
$785$	67.2735	2.40109
$786$	0	0
$787$	17.2847	0.616133	0.308066	−	0.951365i	$-0.400318\pi$
$787$	17.2847	0.616133	0.308066	+	0.951365i	$0.400318\pi$
$788$	41.3972	1.47472
$789$	0	0
$790$	−64.6710	−2.30089
$791$	−6.02412	−0.214193
$792$	0	0
$793$	69.3719	2.46347
$794$	−53.1773	−1.88719
$795$	0	0
$796$	37.8670	1.34216
$797$	4.41343	0.156332	0.0781659	−	0.996940i	$-0.475094\pi$
$797$	4.41343	0.156332	0.0781659	+	0.996940i	$0.475094\pi$
$798$	0	0
$799$	0.118731	0.00420040
$800$	−106.842	−3.77744
$801$	0	0
$802$	74.9446	2.64639
$803$	4.51757	0.159421
$804$	0	0
$805$	−13.8790	−0.489171
$806$	−97.8795	−3.44766
$807$	0	0
$808$	−1.64011	−0.0576989
$809$	7.24744	0.254807	0.127403	−	0.991851i	$-0.459336\pi$
$809$	7.24744	0.254807	0.127403	+	0.991851i	$0.459336\pi$
$810$	0	0
$811$	42.3629	1.48756	0.743782	−	0.668422i	$-0.233030\pi$
$811$	42.3629	1.48756	0.743782	+	0.668422i	$0.233030\pi$
$812$	−7.71186	−0.270633
$813$	0	0
$814$	−59.5780	−2.08821
$815$	55.4816	1.94344
$816$	0	0
$817$	0.609385	0.0213197
$818$	4.39868	0.153796
$819$	0	0
$820$	28.0053	0.977988
$821$	−18.6464	−0.650763	−0.325382	−	0.945583i	$-0.605493\pi$
$821$	−18.6464	−0.650763	−0.325382	+	0.945583i	$0.605493\pi$
$822$	0	0
$823$	−20.6409	−0.719497	−0.359749	−	0.933049i	$-0.617138\pi$
$823$	−20.6409	−0.719497	−0.359749	+	0.933049i	$0.617138\pi$
$824$	12.1996	0.424992
$825$	0	0
$826$	−19.7289	−0.686457
$827$	10.0136	0.348208	0.174104	−	0.984727i	$-0.444297\pi$
$827$	10.0136	0.348208	0.174104	+	0.984727i	$0.444297\pi$
$828$	0	0
$829$	−8.49741	−0.295127	−0.147564	−	0.989053i	$-0.547143\pi$
$829$	−8.49741	−0.295127	−0.147564	+	0.989053i	$0.547143\pi$
$830$	57.2177	1.98606
$831$	0	0
$832$	53.4766	1.85397
$833$	−0.0717215	−0.00248500
$834$	0	0
$835$	27.2219	0.942054
$836$	−7.06698	−0.244417
$837$	0	0
$838$	−3.69097	−0.127503
$839$	−23.9144	−0.825616	−0.412808	−	0.910818i	$-0.635452\pi$
$839$	−23.9144	−0.825616	−0.412808	+	0.910818i	$0.635452\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	74.9949	2.58450
$843$	0	0
$844$	−18.6816	−0.643048
$845$	−52.1191	−1.79295
$846$	0	0
$847$	4.70793	0.161766
$848$	36.8950	1.26698
$849$	0	0
$850$	−0.556723	−0.0190955
$851$	−8.07528	−0.276817
$852$	0	0
$853$	48.7794	1.67018	0.835088	−	0.550116i	$-0.185417\pi$
$853$	48.7794	1.67018	0.835088	+	0.550116i	$0.185417\pi$
$854$	−93.9149	−3.21370
$855$	0	0
$856$	−4.58453	−0.156696
$857$	27.7788	0.948906	0.474453	−	0.880281i	$-0.342646\pi$
$857$	27.7788	0.948906	0.474453	+	0.880281i	$0.342646\pi$
$858$	0	0
$859$	6.01242	0.205141	0.102571	−	0.994726i	$-0.467293\pi$
$859$	6.01242	0.205141	0.102571	+	0.994726i	$0.467293\pi$
$860$	7.31308	0.249374
$861$	0	0
$862$	−48.6143	−1.65581
$863$	28.4007	0.966771	0.483385	−	0.875408i	$-0.339407\pi$
$863$	28.4007	0.966771	0.483385	+	0.875408i	$0.339407\pi$
$864$	0	0
$865$	−16.4243	−0.558444
$866$	14.3036	0.486056
$867$	0	0
$868$	71.9000	2.44044
$869$	25.5546	0.866879
$870$	0	0
$871$	−58.0357	−1.96647
$872$	1.25265	0.0424199
$873$	0	0
$874$	−1.76530	−0.0597122
$875$	−114.264	−3.86284
$876$	0	0
$877$	25.2774	0.853558	0.426779	−	0.904356i	$-0.359648\pi$
$877$	25.2774	0.853558	0.426779	+	0.904356i	$0.359648\pi$
$878$	−22.3667	−0.754839
$879$	0	0
$880$	46.9429	1.58245
$881$	21.9829	0.740624	0.370312	−	0.928907i	$-0.379251\pi$
$881$	21.9829	0.740624	0.370312	+	0.928907i	$0.379251\pi$
$882$	0	0
$883$	−21.2605	−0.715472	−0.357736	−	0.933823i	$-0.616451\pi$
$883$	−21.2605	−0.715472	−0.357736	+	0.933823i	$0.616451\pi$
$884$	0.239659	0.00806060
$885$	0	0
$886$	−1.58534	−0.0532605
$887$	23.2355	0.780173	0.390087	−	0.920778i	$-0.372445\pi$
$887$	23.2355	0.780173	0.390087	+	0.920778i	$0.372445\pi$
$888$	0	0
$889$	39.2204	1.31541
$890$	162.307	5.44053
$891$	0	0
$892$	−45.7999	−1.53350
$893$	4.98193	0.166714
$894$	0	0
$895$	27.9775	0.935186
$896$	−19.9093	−0.665123
$897$	0	0
$898$	36.0993	1.20465
$899$	−9.32330	−0.310949
$900$	0	0
$901$	0.238230	0.00793659
$902$	−20.3945	−0.679062
$903$	0	0
$904$	−1.44411	−0.0480305
$905$	77.8420	2.58756
$906$	0	0
$907$	40.7049	1.35158	0.675791	−	0.737093i	$-0.263802\pi$
$907$	40.7049	1.35158	0.675791	+	0.737093i	$0.263802\pi$
$908$	42.9968	1.42690
$909$	0	0
$910$	145.707	4.83014
$911$	11.8285	0.391896	0.195948	−	0.980614i	$-0.437222\pi$
$911$	11.8285	0.391896	0.195948	+	0.980614i	$0.437222\pi$
$912$	0	0
$913$	−22.6094	−0.748262
$914$	1.91489	0.0633391
$915$	0	0
$916$	14.1114	0.466254
$917$	−12.5332	−0.413884
$918$	0	0
$919$	10.3685	0.342025	0.171012	−	0.985269i	$-0.445296\pi$
$919$	10.3685	0.342025	0.171012	+	0.985269i	$0.445296\pi$
$920$	−3.32711	−0.109691
$921$	0	0
$922$	−45.0563	−1.48385
$923$	56.3762	1.85564
$924$	0	0
$925$	−106.859	−3.51351
$926$	−56.5164	−1.85724
$927$	0	0
$928$	8.07399	0.265042
$929$	4.53697	0.148853	0.0744266	−	0.997226i	$-0.476287\pi$
$929$	4.53697	0.148853	0.0744266	+	0.997226i	$0.476287\pi$
$930$	0	0
$931$	−3.00942	−0.0986299
$932$	16.8273	0.551198
$933$	0	0
$934$	60.6113	1.98326
$935$	0.303109	0.00991273
$936$	0	0
$937$	−54.6753	−1.78616	−0.893082	−	0.449893i	$-0.851462\pi$
$937$	−54.6753	−1.78616	−0.893082	+	0.449893i	$0.851462\pi$
$938$	78.5681	2.56534
$939$	0	0
$940$	59.7869	1.95003
$941$	50.3720	1.64208	0.821040	−	0.570870i	$-0.193394\pi$
$941$	50.3720	1.64208	0.821040	+	0.570870i	$0.193394\pi$
$942$	0	0
$943$	−2.76429	−0.0900178
$944$	9.04454	0.294375
$945$	0	0
$946$	−5.32565	−0.173152
$947$	−40.8001	−1.32582	−0.662912	−	0.748697i	$-0.730680\pi$
$947$	−40.8001	−1.32582	−0.662912	+	0.748697i	$0.730680\pi$
$948$	0	0
$949$	−6.42833	−0.208672
$950$	−23.3600	−0.757899
$951$	0	0
$952$	−0.0509546	−0.00165145
$953$	−2.47746	−0.0802527	−0.0401263	−	0.999195i	$-0.512776\pi$
$953$	−2.47746	−0.0802527	−0.0401263	+	0.999195i	$0.512776\pi$
$954$	0	0
$955$	41.7149	1.34986
$956$	−22.0814	−0.714164
$957$	0	0
$958$	−10.5359	−0.340398
$959$	50.7175	1.63775
$960$	0	0
$961$	55.9239	1.80400
$962$	84.7773	2.73333
$963$	0	0
$964$	15.6285	0.503361
$965$	6.80666	0.219114
$966$	0	0
$967$	−41.8313	−1.34520	−0.672602	−	0.740004i	$-0.734823\pi$
$967$	−41.8313	−1.34520	−0.672602	+	0.740004i	$0.734823\pi$
$968$	1.12859	0.0362744
$969$	0	0
$970$	44.7399	1.43651
$971$	−56.6545	−1.81813	−0.909064	−	0.416657i	$-0.863202\pi$
$971$	−56.6545	−1.81813	−0.909064	+	0.416657i	$0.863202\pi$
$972$	0	0
$973$	−20.3104	−0.651122
$974$	−49.9166	−1.59943
$975$	0	0
$976$	43.0544	1.37814
$977$	30.0914	0.962708	0.481354	−	0.876526i	$-0.340145\pi$
$977$	30.0914	0.962708	0.481354	+	0.876526i	$0.340145\pi$
$978$	0	0
$979$	−64.1350	−2.04976
$980$	−36.1153	−1.15366
$981$	0	0
$982$	−59.4247	−1.89632
$983$	−56.8366	−1.81281	−0.906403	−	0.422414i	$-0.861183\pi$
$983$	−56.8366	−1.81281	−0.906403	+	0.422414i	$0.861183\pi$
$984$	0	0
$985$	−74.5025	−2.37385
$986$	0.0420712	0.00133982
$987$	0	0
$988$	10.0560	0.319926
$989$	−0.721845	−0.0229533
$990$	0	0
$991$	−44.2480	−1.40558	−0.702792	−	0.711396i	$-0.748063\pi$
$991$	−44.2480	−1.40558	−0.702792	+	0.711396i	$0.748063\pi$
$992$	−75.2762	−2.39002
$993$	0	0
$994$	−76.3214	−2.42077
$995$	−68.1491	−2.16047
$996$	0	0
$997$	−59.7747	−1.89308	−0.946541	−	0.322583i	$-0.895449\pi$
$997$	−59.7747	−1.89308	−0.946541	+	0.322583i	$0.895449\pi$
$998$	61.0944	1.93391
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.k.1.8		10
3.2	odd	2		2001.2.a.k.1.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.k.1.3	✓	10	3.2	odd	2
6003.2.a.k.1.8		10	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)

\(f(q)\)	\(=\)	\(q+2.09108 q^{2} +2.37262 q^{4} -4.27000 q^{5} +3.25035 q^{7} +0.779182 q^{8} +O(q^{10})\)	\(q+2.09108 q^{2} +2.37262 q^{4} -4.27000 q^{5} +3.25035 q^{7} +0.779182 q^{8} -8.92891 q^{10} +3.52823 q^{11} -5.02055 q^{13} +6.79676 q^{14} -3.11591 q^{16} -0.0201193 q^{17} -0.844205 q^{19} -10.1311 q^{20} +7.37782 q^{22} +1.00000 q^{23} +13.2329 q^{25} -10.4984 q^{26} +7.71186 q^{28} -1.00000 q^{29} +9.32330 q^{31} -8.07399 q^{32} -0.0420712 q^{34} -13.8790 q^{35} -8.07528 q^{37} -1.76530 q^{38} -3.32711 q^{40} -2.76429 q^{41} -0.721845 q^{43} +8.37116 q^{44} +2.09108 q^{46} -5.90133 q^{47} +3.56480 q^{49} +27.6710 q^{50} -11.9119 q^{52} -11.8408 q^{53} -15.0656 q^{55} +2.53262 q^{56} -2.09108 q^{58} -2.90270 q^{59} -13.8176 q^{61} +19.4958 q^{62} -10.6515 q^{64} +21.4377 q^{65} +11.5596 q^{67} -0.0477356 q^{68} -29.0221 q^{70} -11.2291 q^{71} +1.28040 q^{73} -16.8861 q^{74} -2.00298 q^{76} +11.4680 q^{77} +7.24288 q^{79} +13.3049 q^{80} -5.78036 q^{82} -6.40814 q^{83} +0.0859096 q^{85} -1.50944 q^{86} +2.74914 q^{88} -18.1777 q^{89} -16.3186 q^{91} +2.37262 q^{92} -12.3402 q^{94} +3.60475 q^{95} -5.01067 q^{97} +7.45429 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8	\( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8} - 4 q^{10} - 9 q^{11} - 16 q^{13} - 16 q^{14} + 27 q^{16} + q^{19} - 21 q^{20} + 17 q^{22} + 10 q^{23} - 4 q^{25} - 28 q^{26} - 14 q^{28} - 10 q^{29} + 17 q^{31} - 21 q^{32} - 3 q^{34} - 29 q^{35} + q^{37} - 32 q^{38} + 13 q^{40} - 5 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 31 q^{49} + 22 q^{50} - 21 q^{52} - 35 q^{53} - 20 q^{55} - 18 q^{56} + 3 q^{58} - 49 q^{59} + 8 q^{61} - 15 q^{62} + 12 q^{64} + 3 q^{65} + 35 q^{67} + 18 q^{68} - 16 q^{70} - 30 q^{71} - 15 q^{73} - 23 q^{74} + 10 q^{76} - 23 q^{77} + 24 q^{79} - 23 q^{80} - 5 q^{82} - q^{83} + 10 q^{86} + 18 q^{88} - 15 q^{89} + 26 q^{91} + 17 q^{92} + 3 q^{94} - 7 q^{95} - 35 q^{97} - 3 q^{98}+O(q^{100}) \) 10 * q - 3 * q^2 + 17 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8 - 4 * q^10 - 9 * q^11 - 16 * q^13 - 16 * q^14 + 27 * q^16 + q^19 - 21 * q^20 + 17 * q^22 + 10 * q^23 - 4 * q^25 - 28 * q^26 - 14 * q^28 - 10 * q^29 + 17 * q^31 - 21 * q^32 - 3 * q^34 - 29 * q^35 + q^37 - 32 * q^38 + 13 * q^40 - 5 * q^43 - 33 * q^44 - 3 * q^46 - 15 * q^47 + 31 * q^49 + 22 * q^50 - 21 * q^52 - 35 * q^53 - 20 * q^55 - 18 * q^56 + 3 * q^58 - 49 * q^59 + 8 * q^61 - 15 * q^62 + 12 * q^64 + 3 * q^65 + 35 * q^67 + 18 * q^68 - 16 * q^70 - 30 * q^71 - 15 * q^73 - 23 * q^74 + 10 * q^76 - 23 * q^77 + 24 * q^79 - 23 * q^80 - 5 * q^82 - q^83 + 10 * q^86 + 18 * q^88 - 15 * q^89 + 26 * q^91 + 17 * q^92 + 3 * q^94 - 7 * q^95 - 35 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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