LMFDB - Embedded newform 6003.2.a.w.1.12

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:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.619961 q^{2} -1.61565 q^{4} -0.522930 q^{5} -1.16804 q^{7} +2.24156 q^{8} +O(q^{10})$	\(q-0.619961 q^{2} -1.61565 q^{4} -0.522930 q^{5} -1.16804 q^{7} +2.24156 q^{8} +0.324196 q^{10} +4.66114 q^{11} -1.29242 q^{13} +0.724142 q^{14} +1.84161 q^{16} +5.95028 q^{17} +7.77455 q^{19} +0.844871 q^{20} -2.88973 q^{22} -1.00000 q^{23} -4.72654 q^{25} +0.801249 q^{26} +1.88715 q^{28} +1.00000 q^{29} -5.43440 q^{31} -5.62485 q^{32} -3.68894 q^{34} +0.610806 q^{35} +7.45997 q^{37} -4.81992 q^{38} -1.17218 q^{40} +1.08633 q^{41} +5.14797 q^{43} -7.53077 q^{44} +0.619961 q^{46} +8.76267 q^{47} -5.63567 q^{49} +2.93027 q^{50} +2.08809 q^{52} +7.42594 q^{53} -2.43745 q^{55} -2.61824 q^{56} -0.619961 q^{58} +6.63579 q^{59} +1.32461 q^{61} +3.36912 q^{62} -0.196039 q^{64} +0.675844 q^{65} -12.1634 q^{67} -9.61356 q^{68} -0.378676 q^{70} -2.41579 q^{71} +3.43816 q^{73} -4.62489 q^{74} -12.5609 q^{76} -5.44442 q^{77} +16.4017 q^{79} -0.963036 q^{80} -0.673484 q^{82} -14.8863 q^{83} -3.11158 q^{85} -3.19154 q^{86} +10.4482 q^{88} -15.4180 q^{89} +1.50960 q^{91} +1.61565 q^{92} -5.43251 q^{94} -4.06555 q^{95} +3.33142 q^{97} +3.49390 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) $ 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8	$ 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8} + 8 q^{10} + 36 q^{13} + 7 q^{14} + 47 q^{16} + 18 q^{17} + 16 q^{19} + 25 q^{22} - 30 q^{23} + 56 q^{25} + 11 q^{26} + 27 q^{28} + 30 q^{29} + 14 q^{31} - 7 q^{32} + 3 q^{34} - 22 q^{35} + 40 q^{37} + 6 q^{38} + 30 q^{40} + 14 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} + 74 q^{49} - 21 q^{50} + 71 q^{52} + 16 q^{53} + 22 q^{55} + 14 q^{56} + q^{58} - 32 q^{59} + 46 q^{61} + 20 q^{62} + 68 q^{64} + 12 q^{65} + 14 q^{67} + 27 q^{68} - 32 q^{71} + 50 q^{73} - 26 q^{74} + 56 q^{76} + 34 q^{77} + 16 q^{79} + 2 q^{80} + 38 q^{82} - 14 q^{83} + 38 q^{85} + 10 q^{86} + 40 q^{88} - 2 q^{89} + 32 q^{91} - 37 q^{92} + 29 q^{94} - 28 q^{95} + 56 q^{97} + 8 q^{98}+O(q^{100}) $ 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8 + 8 * q^10 + 36 * q^13 + 7 * q^14 + 47 * q^16 + 18 * q^17 + 16 * q^19 + 25 * q^22 - 30 * q^23 + 56 * q^25 + 11 * q^26 + 27 * q^28 + 30 * q^29 + 14 * q^31 - 7 * q^32 + 3 * q^34 - 22 * q^35 + 40 * q^37 + 6 * q^38 + 30 * q^40 + 14 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 2 * q^47 + 74 * q^49 - 21 * q^50 + 71 * q^52 + 16 * q^53 + 22 * q^55 + 14 * q^56 + q^58 - 32 * q^59 + 46 * q^61 + 20 * q^62 + 68 * q^64 + 12 * q^65 + 14 * q^67 + 27 * q^68 - 32 * q^71 + 50 * q^73 - 26 * q^74 + 56 * q^76 + 34 * q^77 + 16 * q^79 + 2 * q^80 + 38 * q^82 - 14 * q^83 + 38 * q^85 + 10 * q^86 + 40 * q^88 - 2 * q^89 + 32 * q^91 - 37 * q^92 + 29 * q^94 - 28 * q^95 + 56 * q^97 + 8 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.619961	−0.438379	−0.219189	−	0.975682i	$-0.570341\pi$
$2$	−0.619961	−0.438379	−0.219189	+	0.975682i	$0.570341\pi$
$3$	0	0
$3$	0	0
$4$	−1.61565	−0.807824
$5$	−0.522930	−0.233861	−0.116931	−	0.993140i	$-0.537306\pi$
$5$	−0.522930	−0.233861	−0.116931	+	0.993140i	$0.537306\pi$
$6$	0	0
$7$	−1.16804	−0.441479	−0.220740	−	0.975333i	$-0.570847\pi$
$7$	−1.16804	−0.441479	−0.220740	+	0.975333i	$0.570847\pi$
$8$	2.24156	0.792512
$9$	0	0
$10$	0.324196	0.102520
$11$	4.66114	1.40539	0.702694	−	0.711492i	$-0.251980\pi$
$11$	4.66114	1.40539	0.702694	+	0.711492i	$0.251980\pi$
$12$	0	0
$13$	−1.29242	−0.358452	−0.179226	−	0.983808i	$-0.557359\pi$
$13$	−1.29242	−0.358452	−0.179226	+	0.983808i	$0.557359\pi$
$14$	0.724142	0.193535
$15$	0	0
$16$	1.84161	0.460404
$17$	5.95028	1.44316	0.721578	−	0.692334i	$-0.243417\pi$
$17$	5.95028	1.44316	0.721578	+	0.692334i	$0.243417\pi$
$18$	0	0
$19$	7.77455	1.78361	0.891803	−	0.452425i	$-0.149441\pi$
$19$	7.77455	1.78361	0.891803	+	0.452425i	$0.149441\pi$
$20$	0.844871	0.188919
$21$	0	0
$22$	−2.88973	−0.616092
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.72654	−0.945309
$26$	0.801249	0.157138
$27$	0	0
$28$	1.88715	0.356638
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−5.43440	−0.976047	−0.488023	−	0.872831i	$-0.662282\pi$
$31$	−5.43440	−0.976047	−0.488023	+	0.872831i	$0.662282\pi$
$32$	−5.62485	−0.994343
$33$	0	0
$34$	−3.68894	−0.632649
$35$	0.610806	0.103245
$36$	0	0
$37$	7.45997	1.22641	0.613206	−	0.789923i	$-0.289880\pi$
$37$	7.45997	1.22641	0.613206	+	0.789923i	$0.289880\pi$
$38$	−4.81992	−0.781895
$39$	0	0
$40$	−1.17218	−0.185338
$41$	1.08633	0.169657	0.0848283	−	0.996396i	$-0.472966\pi$
$41$	1.08633	0.169657	0.0848283	+	0.996396i	$0.472966\pi$
$42$	0	0
$43$	5.14797	0.785059	0.392529	−	0.919739i	$-0.371600\pi$
$43$	5.14797	0.785059	0.392529	+	0.919739i	$0.371600\pi$
$44$	−7.53077	−1.13531
$45$	0	0
$46$	0.619961	0.0914083
$47$	8.76267	1.27817	0.639083	−	0.769138i	$-0.279314\pi$
$47$	8.76267	1.27817	0.639083	+	0.769138i	$0.279314\pi$
$48$	0	0
$49$	−5.63567	−0.805096
$50$	2.93027	0.414403
$51$	0	0
$52$	2.08809	0.289566
$53$	7.42594	1.02003	0.510016	−	0.860165i	$-0.329640\pi$
$53$	7.42594	1.02003	0.510016	+	0.860165i	$0.329640\pi$
$54$	0	0
$55$	−2.43745	−0.328666
$56$	−2.61824	−0.349878
$57$	0	0
$58$	−0.619961	−0.0814049
$59$	6.63579	0.863906	0.431953	−	0.901896i	$-0.357825\pi$
$59$	6.63579	0.863906	0.431953	+	0.901896i	$0.357825\pi$
$60$	0	0
$61$	1.32461	0.169599	0.0847994	−	0.996398i	$-0.472975\pi$
$61$	1.32461	0.169599	0.0847994	+	0.996398i	$0.472975\pi$
$62$	3.36912	0.427878
$63$	0	0
$64$	−0.196039	−0.0245049
$65$	0.675844	0.0838282
$66$	0	0
$67$	−12.1634	−1.48600	−0.742998	−	0.669293i	$-0.766597\pi$
$67$	−12.1634	−1.48600	−0.742998	+	0.669293i	$0.766597\pi$
$68$	−9.61356	−1.16582
$69$	0	0
$70$	−0.378676	−0.0452604
$71$	−2.41579	−0.286702	−0.143351	−	0.989672i	$-0.545788\pi$
$71$	−2.41579	−0.286702	−0.143351	+	0.989672i	$0.545788\pi$
$72$	0	0
$73$	3.43816	0.402406	0.201203	−	0.979550i	$-0.435515\pi$
$73$	3.43816	0.402406	0.201203	+	0.979550i	$0.435515\pi$
$74$	−4.62489	−0.537633
$75$	0	0
$76$	−12.5609	−1.44084
$77$	−5.44442	−0.620450
$78$	0	0
$79$	16.4017	1.84533	0.922666	−	0.385600i	$-0.126006\pi$
$79$	16.4017	1.84533	0.922666	+	0.385600i	$0.126006\pi$
$80$	−0.963036	−0.107671
$81$	0	0
$82$	−0.673484	−0.0743739
$83$	−14.8863	−1.63398	−0.816991	−	0.576650i	$-0.804360\pi$
$83$	−14.8863	−1.63398	−0.816991	+	0.576650i	$0.804360\pi$
$84$	0	0
$85$	−3.11158	−0.337498
$86$	−3.19154	−0.344153
$87$	0	0
$88$	10.4482	1.11379
$89$	−15.4180	−1.63430	−0.817151	−	0.576424i	$-0.804448\pi$
$89$	−15.4180	−1.63430	−0.817151	+	0.576424i	$0.804448\pi$
$90$	0	0
$91$	1.50960	0.158249
$92$	1.61565	0.168443
$93$	0	0
$94$	−5.43251	−0.560321
$95$	−4.06555	−0.417116
$96$	0	0
$97$	3.33142	0.338255	0.169127	−	0.985594i	$-0.445905\pi$
$97$	3.33142	0.338255	0.169127	+	0.985594i	$0.445905\pi$
$98$	3.49390	0.352937
$99$	0	0
$100$	7.63643	0.763643
$101$	−8.38172	−0.834012	−0.417006	−	0.908904i	$-0.636921\pi$
$101$	−8.38172	−0.834012	−0.417006	+	0.908904i	$0.636921\pi$
$102$	0	0
$103$	1.53965	0.151706	0.0758529	−	0.997119i	$-0.475832\pi$
$103$	1.53965	0.151706	0.0758529	+	0.997119i	$0.475832\pi$
$104$	−2.89704	−0.284078
$105$	0	0
$106$	−4.60379	−0.447160
$107$	−7.31749	−0.707408	−0.353704	−	0.935357i	$-0.615078\pi$
$107$	−7.31749	−0.707408	−0.353704	+	0.935357i	$0.615078\pi$
$108$	0	0
$109$	1.19333	0.114301	0.0571504	−	0.998366i	$-0.481799\pi$
$109$	1.19333	0.114301	0.0571504	+	0.998366i	$0.481799\pi$
$110$	1.51113	0.144080
$111$	0	0
$112$	−2.15109	−0.203259
$113$	−19.6435	−1.84790	−0.923951	−	0.382511i	$-0.875059\pi$
$113$	−19.6435	−1.84790	−0.923951	+	0.382511i	$0.875059\pi$
$114$	0	0
$115$	0.522930	0.0487635
$116$	−1.61565	−0.150009
$117$	0	0
$118$	−4.11393	−0.378718
$119$	−6.95019	−0.637123
$120$	0	0
$121$	10.7263	0.975114
$122$	−0.821207	−0.0743485
$123$	0	0
$124$	8.78007	0.788474
$125$	5.08630	0.454933
$126$	0	0
$127$	−16.6735	−1.47954	−0.739769	−	0.672861i	$-0.765065\pi$
$127$	−16.6735	−1.47954	−0.739769	+	0.672861i	$0.765065\pi$
$128$	11.3712	1.00509
$129$	0	0
$130$	−0.418997	−0.0367485
$131$	−4.36055	−0.380983	−0.190492	−	0.981689i	$-0.561008\pi$
$131$	−4.36055	−0.380983	−0.190492	+	0.981689i	$0.561008\pi$
$132$	0	0
$133$	−9.08103	−0.787425
$134$	7.54084	0.651429
$135$	0	0
$136$	13.3379	1.14372
$137$	−1.64320	−0.140388	−0.0701939	−	0.997533i	$-0.522362\pi$
$137$	−1.64320	−0.140388	−0.0701939	+	0.997533i	$0.522362\pi$
$138$	0	0
$139$	−7.01349	−0.594877	−0.297438	−	0.954741i	$-0.596132\pi$
$139$	−7.01349	−0.594877	−0.297438	+	0.954741i	$0.596132\pi$
$140$	−0.986847	−0.0834038
$141$	0	0
$142$	1.49770	0.125684
$143$	−6.02415	−0.503764
$144$	0	0
$145$	−0.522930	−0.0434270
$146$	−2.13152	−0.176406
$147$	0	0
$148$	−12.0527	−0.990725
$149$	4.40334	0.360736	0.180368	−	0.983599i	$-0.442271\pi$
$149$	4.40334	0.360736	0.180368	+	0.983599i	$0.442271\pi$
$150$	0	0
$151$	21.6844	1.76465	0.882327	−	0.470637i	$-0.155976\pi$
$151$	21.6844	1.76465	0.882327	+	0.470637i	$0.155976\pi$
$152$	17.4271	1.41353
$153$	0	0
$154$	3.37533	0.271992
$155$	2.84181	0.228260
$156$	0	0
$157$	17.5877	1.40365	0.701825	−	0.712350i	$-0.252369\pi$
$157$	17.5877	1.40365	0.701825	+	0.712350i	$0.252369\pi$
$158$	−10.1684	−0.808954
$159$	0	0
$160$	2.94140	0.232538
$161$	1.16804	0.0920548
$162$	0	0
$163$	12.1625	0.952640	0.476320	−	0.879272i	$-0.341970\pi$
$163$	12.1625	0.952640	0.476320	+	0.879272i	$0.341970\pi$
$164$	−1.75513	−0.137053
$165$	0	0
$166$	9.22892	0.716303
$167$	11.4129	0.883157	0.441579	−	0.897223i	$-0.354419\pi$
$167$	11.4129	0.883157	0.441579	+	0.897223i	$0.354419\pi$
$168$	0	0
$169$	−11.3297	−0.871512
$170$	1.92906	0.147952
$171$	0	0
$172$	−8.31731	−0.634189
$173$	0.478838	0.0364054	0.0182027	−	0.999834i	$-0.494206\pi$
$173$	0.478838	0.0364054	0.0182027	+	0.999834i	$0.494206\pi$
$174$	0	0
$175$	5.52082	0.417334
$176$	8.58403	0.647046
$177$	0	0
$178$	9.55854	0.716443
$179$	25.2534	1.88753	0.943764	−	0.330621i	$-0.107258\pi$
$179$	25.2534	1.88753	0.943764	+	0.330621i	$0.107258\pi$
$180$	0	0
$181$	−2.11624	−0.157299	−0.0786495	−	0.996902i	$-0.525061\pi$
$181$	−2.11624	−0.157299	−0.0786495	+	0.996902i	$0.525061\pi$
$182$	−0.935895	−0.0693731
$183$	0	0
$184$	−2.24156	−0.165250
$185$	−3.90104	−0.286810
$186$	0	0
$187$	27.7351	2.02819
$188$	−14.1574	−1.03253
$189$	0	0
$190$	2.52048	0.182855
$191$	1.05527	0.0763568	0.0381784	−	0.999271i	$-0.487844\pi$
$191$	1.05527	0.0763568	0.0381784	+	0.999271i	$0.487844\pi$
$192$	0	0
$193$	24.6011	1.77082	0.885412	−	0.464807i	$-0.153876\pi$
$193$	24.6011	1.77082	0.885412	+	0.464807i	$0.153876\pi$
$194$	−2.06535	−0.148284
$195$	0	0
$196$	9.10526	0.650376
$197$	−5.05313	−0.360020	−0.180010	−	0.983665i	$-0.557613\pi$
$197$	−5.05313	−0.360020	−0.180010	+	0.983665i	$0.557613\pi$
$198$	0	0
$199$	7.50670	0.532136	0.266068	−	0.963954i	$-0.414275\pi$
$199$	7.50670	0.532136	0.266068	+	0.963954i	$0.414275\pi$
$200$	−10.5948	−0.749168
$201$	0	0
$202$	5.19634	0.365613
$203$	−1.16804	−0.0819807
$204$	0	0
$205$	−0.568076	−0.0396762
$206$	−0.954521	−0.0665046
$207$	0	0
$208$	−2.38014	−0.165033
$209$	36.2383	2.50666
$210$	0	0
$211$	−3.97387	−0.273573	−0.136786	−	0.990601i	$-0.543677\pi$
$211$	−3.97387	−0.273573	−0.136786	+	0.990601i	$0.543677\pi$
$212$	−11.9977	−0.824006
$213$	0	0
$214$	4.53656	0.310113
$215$	−2.69203	−0.183595
$216$	0	0
$217$	6.34762	0.430904
$218$	−0.739821	−0.0501070
$219$	0	0
$220$	3.93806	0.265504
$221$	−7.69025	−0.517302
$222$	0	0
$223$	26.7519	1.79144	0.895720	−	0.444618i	$-0.146660\pi$
$223$	26.7519	1.79144	0.895720	+	0.444618i	$0.146660\pi$
$224$	6.57008	0.438982
$225$	0	0
$226$	12.1782	0.810081
$227$	−14.6624	−0.973178	−0.486589	−	0.873631i	$-0.661759\pi$
$227$	−14.6624	−0.973178	−0.486589	+	0.873631i	$0.661759\pi$
$228$	0	0
$229$	1.71940	0.113621	0.0568105	−	0.998385i	$-0.481907\pi$
$229$	1.71940	0.113621	0.0568105	+	0.998385i	$0.481907\pi$
$230$	−0.324196	−0.0213769
$231$	0	0
$232$	2.24156	0.147166
$233$	18.3953	1.20512	0.602559	−	0.798075i	$-0.294148\pi$
$233$	18.3953	1.20512	0.602559	+	0.798075i	$0.294148\pi$
$234$	0	0
$235$	−4.58226	−0.298914
$236$	−10.7211	−0.697884
$237$	0	0
$238$	4.30885	0.279301
$239$	−10.6606	−0.689579	−0.344790	−	0.938680i	$-0.612050\pi$
$239$	−10.6606	−0.689579	−0.344790	+	0.938680i	$0.612050\pi$
$240$	0	0
$241$	−1.84055	−0.118560	−0.0592800	−	0.998241i	$-0.518880\pi$
$241$	−1.84055	−0.118560	−0.0592800	+	0.998241i	$0.518880\pi$
$242$	−6.64986	−0.427469
$243$	0	0
$244$	−2.14010	−0.137006
$245$	2.94706	0.188281
$246$	0	0
$247$	−10.0480	−0.639337
$248$	−12.1815	−0.773528
$249$	0	0
$250$	−3.15331	−0.199433
$251$	−13.2693	−0.837551	−0.418776	−	0.908090i	$-0.637541\pi$
$251$	−13.2693	−0.837551	−0.418776	+	0.908090i	$0.637541\pi$
$252$	0	0
$253$	−4.66114	−0.293044
$254$	10.3369	0.648598
$255$	0	0
$256$	−6.65765	−0.416103
$257$	17.9447	1.11936	0.559680	−	0.828708i	$-0.310924\pi$
$257$	17.9447	1.11936	0.559680	+	0.828708i	$0.310924\pi$
$258$	0	0
$259$	−8.71358	−0.541436
$260$	−1.09193	−0.0677184
$261$	0	0
$262$	2.70337	0.167015
$263$	3.22549	0.198892	0.0994461	−	0.995043i	$-0.468293\pi$
$263$	3.22549	0.198892	0.0994461	+	0.995043i	$0.468293\pi$
$264$	0	0
$265$	−3.88325	−0.238546
$266$	5.62989	0.345190
$267$	0	0
$268$	19.6518	1.20042
$269$	−10.4987	−0.640115	−0.320057	−	0.947398i	$-0.603702\pi$
$269$	−10.4987	−0.640115	−0.320057	+	0.947398i	$0.603702\pi$
$270$	0	0
$271$	19.2008	1.16637	0.583184	−	0.812340i	$-0.301807\pi$
$271$	19.2008	1.16637	0.583184	+	0.812340i	$0.301807\pi$
$272$	10.9581	0.664434
$273$	0	0
$274$	1.01872	0.0615430
$275$	−22.0311	−1.32853
$276$	0	0
$277$	−18.4356	−1.10769	−0.553843	−	0.832621i	$-0.686839\pi$
$277$	−18.4356	−1.10769	−0.553843	+	0.832621i	$0.686839\pi$
$278$	4.34809	0.260781
$279$	0	0
$280$	1.36916	0.0818229
$281$	6.72161	0.400978	0.200489	−	0.979696i	$-0.435747\pi$
$281$	6.72161	0.400978	0.200489	+	0.979696i	$0.435747\pi$
$282$	0	0
$283$	−6.56626	−0.390324	−0.195162	−	0.980771i	$-0.562523\pi$
$283$	−6.56626	−0.390324	−0.195162	+	0.980771i	$0.562523\pi$
$284$	3.90307	0.231605
$285$	0	0
$286$	3.73474	0.220840
$287$	−1.26889	−0.0748999
$288$	0	0
$289$	18.4058	1.08270
$290$	0.324196	0.0190375
$291$	0	0
$292$	−5.55485	−0.325073
$293$	21.2397	1.24084	0.620418	−	0.784272i	$-0.286963\pi$
$293$	21.2397	1.24084	0.620418	+	0.784272i	$0.286963\pi$
$294$	0	0
$295$	−3.47005	−0.202034
$296$	16.7220	0.971946
$297$	0	0
$298$	−2.72990	−0.158139
$299$	1.29242	0.0747425
$300$	0	0
$301$	−6.01306	−0.346587
$302$	−13.4435	−0.773587
$303$	0	0
$304$	14.3177	0.821178
$305$	−0.692678	−0.0396626
$306$	0	0
$307$	28.1167	1.60470	0.802352	−	0.596852i	$-0.203582\pi$
$307$	28.1167	1.60470	0.802352	+	0.596852i	$0.203582\pi$
$308$	8.79627	0.501214
$309$	0	0
$310$	−1.76181	−0.100064
$311$	10.4325	0.591570	0.295785	−	0.955254i	$-0.404419\pi$
$311$	10.4325	0.591570	0.295785	+	0.955254i	$0.404419\pi$
$312$	0	0
$313$	−22.8946	−1.29408	−0.647040	−	0.762456i	$-0.723993\pi$
$313$	−22.8946	−1.29408	−0.647040	+	0.762456i	$0.723993\pi$
$314$	−10.9037	−0.615330
$315$	0	0
$316$	−26.4993	−1.49070
$317$	−24.2993	−1.36479	−0.682393	−	0.730985i	$-0.739061\pi$
$317$	−24.2993	−1.36479	−0.682393	+	0.730985i	$0.739061\pi$
$318$	0	0
$319$	4.66114	0.260974
$320$	0.102515	0.00573074
$321$	0	0
$322$	−0.724142	−0.0403549
$323$	46.2608	2.57402
$324$	0	0
$325$	6.10867	0.338848
$326$	−7.54027	−0.417617
$327$	0	0
$328$	2.43508	0.134455
$329$	−10.2352	−0.564284
$330$	0	0
$331$	−20.4102	−1.12185	−0.560924	−	0.827867i	$-0.689554\pi$
$331$	−20.4102	−1.12185	−0.560924	+	0.827867i	$0.689554\pi$
$332$	24.0510	1.31997
$333$	0	0
$334$	−7.07556	−0.387157
$335$	6.36061	0.347517
$336$	0	0
$337$	14.2370	0.775540	0.387770	−	0.921756i	$-0.373246\pi$
$337$	14.2370	0.775540	0.387770	+	0.921756i	$0.373246\pi$
$338$	7.02395	0.382052
$339$	0	0
$340$	5.02722	0.272639
$341$	−25.3305	−1.37172
$342$	0	0
$343$	14.7590	0.796913
$344$	11.5395	0.622168
$345$	0	0
$346$	−0.296861	−0.0159593
$347$	15.6646	0.840920	0.420460	−	0.907311i	$-0.361869\pi$
$347$	15.6646	0.840920	0.420460	+	0.907311i	$0.361869\pi$
$348$	0	0
$349$	0.150742	0.00806905	0.00403452	−	0.999992i	$-0.498716\pi$
$349$	0.150742	0.00806905	0.00403452	+	0.999992i	$0.498716\pi$
$350$	−3.42269	−0.182951
$351$	0	0
$352$	−26.2182	−1.39744
$353$	−10.8442	−0.577177	−0.288588	−	0.957453i	$-0.593186\pi$
$353$	−10.8442	−0.577177	−0.288588	+	0.957453i	$0.593186\pi$
$354$	0	0
$355$	1.26329	0.0670485
$356$	24.9100	1.32023
$357$	0	0
$358$	−15.6561	−0.827452
$359$	4.28204	0.225997	0.112999	−	0.993595i	$-0.463954\pi$
$359$	4.28204	0.225997	0.112999	+	0.993595i	$0.463954\pi$
$360$	0	0
$361$	41.4437	2.18125
$362$	1.31199	0.0689565
$363$	0	0
$364$	−2.43899	−0.127838
$365$	−1.79792	−0.0941072
$366$	0	0
$367$	−1.85603	−0.0968839	−0.0484420	−	0.998826i	$-0.515426\pi$
$367$	−1.85603	−0.0968839	−0.0484420	+	0.998826i	$0.515426\pi$
$368$	−1.84161	−0.0960008
$369$	0	0
$370$	2.41850	0.125732
$371$	−8.67383	−0.450323
$372$	0	0
$373$	2.75413	0.142603	0.0713017	−	0.997455i	$-0.477285\pi$
$373$	2.75413	0.142603	0.0713017	+	0.997455i	$0.477285\pi$
$374$	−17.1947	−0.889117
$375$	0	0
$376$	19.6421	1.01296
$377$	−1.29242	−0.0665629
$378$	0	0
$379$	6.64276	0.341216	0.170608	−	0.985339i	$-0.445427\pi$
$379$	6.64276	0.341216	0.170608	+	0.985339i	$0.445427\pi$
$380$	6.56850	0.336957
$381$	0	0
$382$	−0.654228	−0.0334732
$383$	1.80391	0.0921754	0.0460877	−	0.998937i	$-0.485325\pi$
$383$	1.80391	0.0921754	0.0460877	+	0.998937i	$0.485325\pi$
$384$	0	0
$385$	2.84705	0.145099
$386$	−15.2517	−0.776292
$387$	0	0
$388$	−5.38241	−0.273250
$389$	3.18206	0.161337	0.0806684	−	0.996741i	$-0.474295\pi$
$389$	3.18206	0.161337	0.0806684	+	0.996741i	$0.474295\pi$
$390$	0	0
$391$	−5.95028	−0.300919
$392$	−12.6327	−0.638048
$393$	0	0
$394$	3.13274	0.157825
$395$	−8.57693	−0.431552
$396$	0	0
$397$	17.8739	0.897067	0.448533	−	0.893766i	$-0.351947\pi$
$397$	17.8739	0.897067	0.448533	+	0.893766i	$0.351947\pi$
$398$	−4.65386	−0.233277
$399$	0	0
$400$	−8.70447	−0.435224
$401$	1.69712	0.0847500	0.0423750	−	0.999102i	$-0.486508\pi$
$401$	1.69712	0.0847500	0.0423750	+	0.999102i	$0.486508\pi$
$402$	0	0
$403$	7.02351	0.349866
$404$	13.5419	0.673735
$405$	0	0
$406$	0.724142	0.0359386
$407$	34.7720	1.72358
$408$	0	0
$409$	37.4962	1.85407	0.927033	−	0.374979i	$-0.122350\pi$
$409$	37.4962	1.85407	0.927033	+	0.374979i	$0.122350\pi$
$410$	0.352185	0.0173932
$411$	0	0
$412$	−2.48753	−0.122552
$413$	−7.75089	−0.381397
$414$	0	0
$415$	7.78449	0.382125
$416$	7.26966	0.356425
$417$	0	0
$418$	−22.4664	−1.09887
$419$	5.08483	0.248410	0.124205	−	0.992257i	$-0.460362\pi$
$419$	5.08483	0.248410	0.124205	+	0.992257i	$0.460362\pi$
$420$	0	0
$421$	−17.0091	−0.828971	−0.414486	−	0.910056i	$-0.636039\pi$
$421$	−17.0091	−0.828971	−0.414486	+	0.910056i	$0.636039\pi$
$422$	2.46365	0.119928
$423$	0	0
$424$	16.6457	0.808387
$425$	−28.1243	−1.36423
$426$	0	0
$427$	−1.54720	−0.0748744
$428$	11.8225	0.571461
$429$	0	0
$430$	1.66895	0.0804841
$431$	−20.2048	−0.973231	−0.486616	−	0.873616i	$-0.661769\pi$
$431$	−20.2048	−0.973231	−0.486616	+	0.873616i	$0.661769\pi$
$432$	0	0
$433$	−6.36864	−0.306057	−0.153029	−	0.988222i	$-0.548903\pi$
$433$	−6.36864	−0.306057	−0.153029	+	0.988222i	$0.548903\pi$
$434$	−3.93528	−0.188899
$435$	0	0
$436$	−1.92801	−0.0923349
$437$	−7.77455	−0.371907
$438$	0	0
$439$	17.1869	0.820284	0.410142	−	0.912022i	$-0.365479\pi$
$439$	17.1869	0.820284	0.410142	+	0.912022i	$0.365479\pi$
$440$	−5.46370	−0.260472
$441$	0	0
$442$	4.76766	0.226774
$443$	−28.7695	−1.36688	−0.683440	−	0.730007i	$-0.739517\pi$
$443$	−28.7695	−1.36688	−0.683440	+	0.730007i	$0.739517\pi$
$444$	0	0
$445$	8.06252	0.382200
$446$	−16.5852	−0.785330
$447$	0	0
$448$	0.228982	0.0108184
$449$	−37.5665	−1.77287	−0.886436	−	0.462851i	$-0.846826\pi$
$449$	−37.5665	−1.77287	−0.886436	+	0.462851i	$0.846826\pi$
$450$	0	0
$451$	5.06355	0.238433
$452$	31.7369	1.49278
$453$	0	0
$454$	9.09012	0.426620
$455$	−0.789416	−0.0370084
$456$	0	0
$457$	12.8260	0.599976	0.299988	−	0.953943i	$-0.403017\pi$
$457$	12.8260	0.599976	0.299988	+	0.953943i	$0.403017\pi$
$458$	−1.06596	−0.0498090
$459$	0	0
$460$	−0.844871	−0.0393923
$461$	35.5910	1.65764	0.828819	−	0.559517i	$-0.189013\pi$
$461$	35.5910	1.65764	0.828819	+	0.559517i	$0.189013\pi$
$462$	0	0
$463$	18.0208	0.837498	0.418749	−	0.908102i	$-0.362469\pi$
$463$	18.0208	0.837498	0.418749	+	0.908102i	$0.362469\pi$
$464$	1.84161	0.0854948
$465$	0	0
$466$	−11.4044	−0.528298
$467$	22.8948	1.05945	0.529723	−	0.848171i	$-0.322296\pi$
$467$	22.8948	1.05945	0.529723	+	0.848171i	$0.322296\pi$
$468$	0	0
$469$	14.2074	0.656037
$470$	2.84082	0.131037
$471$	0	0
$472$	14.8745	0.684655
$473$	23.9954	1.10331
$474$	0	0
$475$	−36.7468	−1.68606
$476$	11.2291	0.514684
$477$	0	0
$478$	6.60918	0.302297
$479$	−10.4510	−0.477520	−0.238760	−	0.971079i	$-0.576741\pi$
$479$	−10.4510	−0.477520	−0.238760	+	0.971079i	$0.576741\pi$
$480$	0	0
$481$	−9.64140	−0.439610
$482$	1.14107	0.0519742
$483$	0	0
$484$	−17.3299	−0.787721
$485$	−1.74210	−0.0791047
$486$	0	0
$487$	35.9399	1.62859	0.814297	−	0.580448i	$-0.197123\pi$
$487$	35.9399	1.62859	0.814297	+	0.580448i	$0.197123\pi$
$488$	2.96919	0.134409
$489$	0	0
$490$	−1.82706	−0.0825384
$491$	13.0961	0.591017	0.295508	−	0.955340i	$-0.404511\pi$
$491$	13.0961	0.591017	0.295508	+	0.955340i	$0.404511\pi$
$492$	0	0
$493$	5.95028	0.267987
$494$	6.22936	0.280272
$495$	0	0
$496$	−10.0081	−0.449375
$497$	2.82175	0.126573
$498$	0	0
$499$	−0.0114209	−0.000511271 0	−0.000255636	−	1.00000i	$-0.500081\pi$
$499$	−0.0114209	−0.000511271 0	−0.000255636		1.00000i	$0.500081\pi$
$500$	−8.21767	−0.367506
$501$	0	0
$502$	8.22646	0.367165
$503$	18.1517	0.809343	0.404672	−	0.914462i	$-0.367386\pi$
$503$	18.1517	0.809343	0.404672	+	0.914462i	$0.367386\pi$
$504$	0	0
$505$	4.38305	0.195043
$506$	2.88973	0.128464
$507$	0	0
$508$	26.9386	1.19521
$509$	11.1632	0.494802	0.247401	−	0.968913i	$-0.420424\pi$
$509$	11.1632	0.494802	0.247401	+	0.968913i	$0.420424\pi$
$510$	0	0
$511$	−4.01592	−0.177654
$512$	−18.6150	−0.822675
$513$	0	0
$514$	−11.1250	−0.490704
$515$	−0.805127	−0.0354781
$516$	0	0
$517$	40.8440	1.79632
$518$	5.40208	0.237354
$519$	0	0
$520$	1.51495	0.0664348
$521$	−25.6906	−1.12553	−0.562763	−	0.826619i	$-0.690261\pi$
$521$	−25.6906	−1.12553	−0.562763	+	0.826619i	$0.690261\pi$
$522$	0	0
$523$	−23.3590	−1.02142	−0.510710	−	0.859753i	$-0.670617\pi$
$523$	−23.3590	−1.02142	−0.510710	+	0.859753i	$0.670617\pi$
$524$	7.04512	0.307767
$525$	0	0
$526$	−1.99968	−0.0871901
$527$	−32.3362	−1.40859
$528$	0	0
$529$	1.00000	0.0434783
$530$	2.40746	0.104573
$531$	0	0
$532$	14.6717	0.636101
$533$	−1.40400	−0.0608138
$534$	0	0
$535$	3.82653	0.165436
$536$	−27.2650	−1.17767
$537$	0	0
$538$	6.50876	0.280613
$539$	−26.2687	−1.13147
$540$	0	0
$541$	−23.2096	−0.997860	−0.498930	−	0.866642i	$-0.666274\pi$
$541$	−23.2096	−0.997860	−0.498930	+	0.866642i	$0.666274\pi$
$542$	−11.9038	−0.511311
$543$	0	0
$544$	−33.4695	−1.43499
$545$	−0.624031	−0.0267305
$546$	0	0
$547$	40.4383	1.72901	0.864507	−	0.502620i	$-0.167631\pi$
$547$	40.4383	1.72901	0.864507	+	0.502620i	$0.167631\pi$
$548$	2.65483	0.113409
$549$	0	0
$550$	13.6584	0.582397
$551$	7.77455	0.331207
$552$	0	0
$553$	−19.1579	−0.814676
$554$	11.4293	0.485586
$555$	0	0
$556$	11.3313	0.480556
$557$	18.6845	0.791687	0.395844	−	0.918318i	$-0.370452\pi$
$557$	18.6845	0.791687	0.395844	+	0.918318i	$0.370452\pi$
$558$	0	0
$559$	−6.65333	−0.281406
$560$	1.12487	0.0475344
$561$	0	0
$562$	−4.16714	−0.175780
$563$	8.68708	0.366117	0.183058	−	0.983102i	$-0.441400\pi$
$563$	8.68708	0.366117	0.183058	+	0.983102i	$0.441400\pi$
$564$	0	0
$565$	10.2722	0.432153
$566$	4.07083	0.171110
$567$	0	0
$568$	−5.41515	−0.227215
$569$	36.5164	1.53085	0.765424	−	0.643527i	$-0.222529\pi$
$569$	36.5164	1.53085	0.765424	+	0.643527i	$0.222529\pi$
$570$	0	0
$571$	2.62277	0.109760	0.0548798	−	0.998493i	$-0.482522\pi$
$571$	2.62277	0.109760	0.0548798	+	0.998493i	$0.482522\pi$
$572$	9.73290	0.406953
$573$	0	0
$574$	0.786660	0.0328345
$575$	4.72654	0.197111
$576$	0	0
$577$	−15.5769	−0.648473	−0.324237	−	0.945976i	$-0.605107\pi$
$577$	−15.5769	−0.648473	−0.324237	+	0.945976i	$0.605107\pi$
$578$	−11.4109	−0.474631
$579$	0	0
$580$	0.844871	0.0350814
$581$	17.3878	0.721370
$582$	0	0
$583$	34.6134	1.43354
$584$	7.70684	0.318911
$585$	0	0
$586$	−13.1678	−0.543956
$587$	−37.2568	−1.53775	−0.768877	−	0.639396i	$-0.779184\pi$
$587$	−37.2568	−1.53775	−0.768877	+	0.639396i	$0.779184\pi$
$588$	0	0
$589$	−42.2500	−1.74088
$590$	2.15130	0.0885675
$591$	0	0
$592$	13.7384	0.564645
$593$	−24.2523	−0.995924	−0.497962	−	0.867199i	$-0.665918\pi$
$593$	−24.2523	−0.995924	−0.497962	+	0.867199i	$0.665918\pi$
$594$	0	0
$595$	3.63447	0.148999
$596$	−7.11425	−0.291411
$597$	0	0
$598$	−0.801249	−0.0327655
$599$	1.80563	0.0737759	0.0368880	−	0.999319i	$-0.488256\pi$
$599$	1.80563	0.0737759	0.0368880	+	0.999319i	$0.488256\pi$
$600$	0	0
$601$	2.51487	0.102584	0.0512918	−	0.998684i	$-0.483666\pi$
$601$	2.51487	0.102584	0.0512918	+	0.998684i	$0.483666\pi$
$602$	3.72787	0.151936
$603$	0	0
$604$	−35.0344	−1.42553
$605$	−5.60908	−0.228042
$606$	0	0
$607$	−30.1182	−1.22246	−0.611230	−	0.791453i	$-0.709325\pi$
$607$	−30.1182	−1.22246	−0.611230	+	0.791453i	$0.709325\pi$
$608$	−43.7307	−1.77352
$609$	0	0
$610$	0.429434	0.0173873
$611$	−11.3250	−0.458162
$612$	0	0
$613$	11.7839	0.475946	0.237973	−	0.971272i	$-0.423517\pi$
$613$	11.7839	0.475946	0.237973	+	0.971272i	$0.423517\pi$
$614$	−17.4312	−0.703468
$615$	0	0
$616$	−12.2040	−0.491714
$617$	39.8299	1.60349	0.801745	−	0.597666i	$-0.203905\pi$
$617$	39.8299	1.60349	0.801745	+	0.597666i	$0.203905\pi$
$618$	0	0
$619$	−31.2512	−1.25609	−0.628046	−	0.778176i	$-0.716145\pi$
$619$	−31.2512	−1.25609	−0.628046	+	0.778176i	$0.716145\pi$
$620$	−4.59136	−0.184394
$621$	0	0
$622$	−6.46772	−0.259332
$623$	18.0089	0.721511
$624$	0	0
$625$	20.9729	0.838918
$626$	14.1938	0.567297
$627$	0	0
$628$	−28.4155	−1.13390
$629$	44.3889	1.76990
$630$	0	0
$631$	−39.0864	−1.55600	−0.778002	−	0.628262i	$-0.783766\pi$
$631$	−39.0864	−1.55600	−0.778002	+	0.628262i	$0.783766\pi$
$632$	36.7654	1.46245
$633$	0	0
$634$	15.0646	0.598293
$635$	8.71909	0.346007
$636$	0	0
$637$	7.28364	0.288588
$638$	−2.88973	−0.114405
$639$	0	0
$640$	−5.94636	−0.235051
$641$	−9.16902	−0.362154	−0.181077	−	0.983469i	$-0.557958\pi$
$641$	−9.16902	−0.362154	−0.181077	+	0.983469i	$0.557958\pi$
$642$	0	0
$643$	−7.36714	−0.290532	−0.145266	−	0.989393i	$-0.546404\pi$
$643$	−7.36714	−0.290532	−0.145266	+	0.989393i	$0.546404\pi$
$644$	−1.88715	−0.0743641
$645$	0	0
$646$	−28.6799	−1.12840
$647$	5.65797	0.222438	0.111219	−	0.993796i	$-0.464525\pi$
$647$	5.65797	0.222438	0.111219	+	0.993796i	$0.464525\pi$
$648$	0	0
$649$	30.9303	1.21412
$650$	−3.78714	−0.148544
$651$	0	0
$652$	−19.6503	−0.769565
$653$	1.18538	0.0463876	0.0231938	−	0.999731i	$-0.492617\pi$
$653$	1.18538	0.0463876	0.0231938	+	0.999731i	$0.492617\pi$
$654$	0	0
$655$	2.28026	0.0890973
$656$	2.00061	0.0781106
$657$	0	0
$658$	6.34542	0.247370
$659$	7.07467	0.275590	0.137795	−	0.990461i	$-0.455998\pi$
$659$	7.07467	0.275590	0.137795	+	0.990461i	$0.455998\pi$
$660$	0	0
$661$	−39.8552	−1.55019	−0.775094	−	0.631846i	$-0.782298\pi$
$661$	−39.8552	−1.55019	−0.775094	+	0.631846i	$0.782298\pi$
$662$	12.6536	0.491795
$663$	0	0
$664$	−33.3685	−1.29495
$665$	4.74874	0.184148
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−18.4392	−0.713436
$669$	0	0
$670$	−3.94333	−0.152344
$671$	6.17420	0.238352
$672$	0	0
$673$	9.88954	0.381214	0.190607	−	0.981666i	$-0.438954\pi$
$673$	9.88954	0.381214	0.190607	+	0.981666i	$0.438954\pi$
$674$	−8.82640	−0.339980
$675$	0	0
$676$	18.3047	0.704028
$677$	−49.6623	−1.90868	−0.954339	−	0.298727i	$-0.903438\pi$
$677$	−49.6623	−1.90868	−0.954339	+	0.298727i	$0.903438\pi$
$678$	0	0
$679$	−3.89125	−0.149333
$680$	−6.97480	−0.267471
$681$	0	0
$682$	15.7039	0.601335
$683$	11.0820	0.424042	0.212021	−	0.977265i	$-0.431995\pi$
$683$	11.0820	0.424042	0.212021	+	0.977265i	$0.431995\pi$
$684$	0	0
$685$	0.859277	0.0328313
$686$	−9.15003	−0.349350
$687$	0	0
$688$	9.48058	0.361444
$689$	−9.59742	−0.365633
$690$	0	0
$691$	−37.2853	−1.41840	−0.709200	−	0.705008i	$-0.750944\pi$
$691$	−37.2853	−1.41840	−0.709200	+	0.705008i	$0.750944\pi$
$692$	−0.773634	−0.0294091
$693$	0	0
$694$	−9.71144	−0.368641
$695$	3.66757	0.139119
$696$	0	0
$697$	6.46398	0.244841
$698$	−0.0934544	−0.00353730
$699$	0	0
$700$	−8.91969	−0.337133
$701$	−30.3587	−1.14663	−0.573316	−	0.819334i	$-0.694343\pi$
$701$	−30.3587	−1.14663	−0.573316	+	0.819334i	$0.694343\pi$
$702$	0	0
$703$	57.9980	2.18743
$704$	−0.913765	−0.0344388
$705$	0	0
$706$	6.72297	0.253022
$707$	9.79022	0.368199
$708$	0	0
$709$	38.8755	1.46000	0.729999	−	0.683448i	$-0.239520\pi$
$709$	38.8755	1.46000	0.729999	+	0.683448i	$0.239520\pi$
$710$	−0.783191	−0.0293927
$711$	0	0
$712$	−34.5603	−1.29520
$713$	5.43440	0.203520
$714$	0	0
$715$	3.15021	0.117811
$716$	−40.8006	−1.52479
$717$	0	0
$718$	−2.65470	−0.0990725
$719$	25.8495	0.964025	0.482012	−	0.876164i	$-0.339906\pi$
$719$	25.8495	0.964025	0.482012	+	0.876164i	$0.339906\pi$
$720$	0	0
$721$	−1.79837	−0.0669750
$722$	−25.6935	−0.956213
$723$	0	0
$724$	3.41910	0.127070
$725$	−4.72654	−0.175539
$726$	0	0
$727$	−2.32175	−0.0861088	−0.0430544	−	0.999073i	$-0.513709\pi$
$727$	−2.32175	−0.0861088	−0.0430544	+	0.999073i	$0.513709\pi$
$728$	3.38387	0.125414
$729$	0	0
$730$	1.11464	0.0412546
$731$	30.6319	1.13296
$732$	0	0
$733$	28.4122	1.04943	0.524714	−	0.851279i	$-0.324172\pi$
$733$	28.4122	1.04943	0.524714	+	0.851279i	$0.324172\pi$
$734$	1.15067	0.0424719
$735$	0	0
$736$	5.62485	0.207335
$737$	−56.6954	−2.08840
$738$	0	0
$739$	−42.3458	−1.55772	−0.778858	−	0.627201i	$-0.784201\pi$
$739$	−42.3458	−1.55772	−0.778858	+	0.627201i	$0.784201\pi$
$740$	6.30271	0.231692
$741$	0	0
$742$	5.37744	0.197412
$743$	11.5174	0.422534	0.211267	−	0.977428i	$-0.432241\pi$
$743$	11.5174	0.422534	0.211267	+	0.977428i	$0.432241\pi$
$744$	0	0
$745$	−2.30264	−0.0843622
$746$	−1.70745	−0.0625143
$747$	0	0
$748$	−44.8102	−1.63842
$749$	8.54715	0.312306
$750$	0	0
$751$	−21.1382	−0.771343	−0.385672	−	0.922636i	$-0.626030\pi$
$751$	−21.1382	−0.771343	−0.385672	+	0.922636i	$0.626030\pi$
$752$	16.1375	0.588472
$753$	0	0
$754$	0.801249	0.0291798
$755$	−11.3394	−0.412685
$756$	0	0
$757$	1.23574	0.0449139	0.0224569	−	0.999748i	$-0.492851\pi$
$757$	1.23574	0.0449139	0.0224569	+	0.999748i	$0.492851\pi$
$758$	−4.11826	−0.149582
$759$	0	0
$760$	−9.11318	−0.330570
$761$	−11.5975	−0.420408	−0.210204	−	0.977658i	$-0.567413\pi$
$761$	−11.5975	−0.420408	−0.210204	+	0.977658i	$0.567413\pi$
$762$	0	0
$763$	−1.39387	−0.0504614
$764$	−1.70495	−0.0616828
$765$	0	0
$766$	−1.11835	−0.0404077
$767$	−8.57621	−0.309669
$768$	0	0
$769$	−17.5135	−0.631554	−0.315777	−	0.948833i	$-0.602265\pi$
$769$	−17.5135	−0.631554	−0.315777	+	0.948833i	$0.602265\pi$
$770$	−1.76506	−0.0636084
$771$	0	0
$772$	−39.7467	−1.43051
$773$	25.0144	0.899707	0.449854	−	0.893102i	$-0.351476\pi$
$773$	25.0144	0.899707	0.449854	+	0.893102i	$0.351476\pi$
$774$	0	0
$775$	25.6859	0.922665
$776$	7.46759	0.268071
$777$	0	0
$778$	−1.97275	−0.0707267
$779$	8.44575	0.302600
$780$	0	0
$781$	−11.2604	−0.402927
$782$	3.68894	0.131916
$783$	0	0
$784$	−10.3787	−0.370669
$785$	−9.19713	−0.328260
$786$	0	0
$787$	−4.02641	−0.143526	−0.0717630	−	0.997422i	$-0.522863\pi$
$787$	−4.02641	−0.143526	−0.0717630	+	0.997422i	$0.522863\pi$
$788$	8.16407	0.290833
$789$	0	0
$790$	5.31736	0.189183
$791$	22.9444	0.815810
$792$	0	0
$793$	−1.71195	−0.0607931
$794$	−11.0811	−0.393255
$795$	0	0
$796$	−12.1282	−0.429872
$797$	6.15335	0.217963	0.108981	−	0.994044i	$-0.465241\pi$
$797$	6.15335	0.217963	0.108981	+	0.994044i	$0.465241\pi$
$798$	0	0
$799$	52.1403	1.84459
$800$	26.5861	0.939961
$801$	0	0
$802$	−1.05215	−0.0371526
$803$	16.0257	0.565536
$804$	0	0
$805$	−0.610806	−0.0215281
$806$	−4.35431	−0.153374
$807$	0	0
$808$	−18.7881	−0.660965
$809$	8.34032	0.293230	0.146615	−	0.989194i	$-0.453162\pi$
$809$	8.34032	0.293230	0.146615	+	0.989194i	$0.453162\pi$
$810$	0	0
$811$	−19.9063	−0.699006	−0.349503	−	0.936935i	$-0.613650\pi$
$811$	−19.9063	−0.699006	−0.349503	+	0.936935i	$0.613650\pi$
$812$	1.88715	0.0662260
$813$	0	0
$814$	−21.5573	−0.755583
$815$	−6.36013	−0.222786
$816$	0	0
$817$	40.0232	1.40023
$818$	−23.2462	−0.812784
$819$	0	0
$820$	0.917811	0.0320513
$821$	18.2575	0.637192	0.318596	−	0.947891i	$-0.396789\pi$
$821$	18.2575	0.637192	0.318596	+	0.947891i	$0.396789\pi$
$822$	0	0
$823$	−29.5490	−1.03001	−0.515007	−	0.857186i	$-0.672211\pi$
$823$	−29.5490	−1.03001	−0.515007	+	0.857186i	$0.672211\pi$
$824$	3.45121	0.120229
$825$	0	0
$826$	4.80525	0.167196
$827$	9.59382	0.333610	0.166805	−	0.985990i	$-0.446655\pi$
$827$	9.59382	0.333610	0.166805	+	0.985990i	$0.446655\pi$
$828$	0	0
$829$	−29.8738	−1.03756	−0.518779	−	0.854908i	$-0.673613\pi$
$829$	−29.8738	−1.03756	−0.518779	+	0.854908i	$0.673613\pi$
$830$	−4.82608	−0.167516
$831$	0	0
$832$	0.253364	0.00878382
$833$	−33.5338	−1.16188
$834$	0	0
$835$	−5.96815	−0.206536
$836$	−58.5484	−2.02494
$837$	0	0
$838$	−3.15240	−0.108898
$839$	17.4134	0.601178	0.300589	−	0.953754i	$-0.402817\pi$
$839$	17.4134	0.601178	0.300589	+	0.953754i	$0.402817\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	10.5450	0.363403
$843$	0	0
$844$	6.42038	0.220998
$845$	5.92462	0.203813
$846$	0	0
$847$	−12.5287	−0.430493
$848$	13.6757	0.469626
$849$	0	0
$850$	17.4360	0.598048
$851$	−7.45997	−0.255725
$852$	0	0
$853$	20.0068	0.685019	0.342509	−	0.939514i	$-0.388723\pi$
$853$	20.0068	0.685019	0.342509	+	0.939514i	$0.388723\pi$
$854$	0.959206	0.0328234
$855$	0	0
$856$	−16.4026	−0.560629
$857$	−23.2668	−0.794780	−0.397390	−	0.917650i	$-0.630084\pi$
$857$	−23.2668	−0.794780	−0.397390	+	0.917650i	$0.630084\pi$
$858$	0	0
$859$	12.6024	0.429988	0.214994	−	0.976615i	$-0.431027\pi$
$859$	12.6024	0.429988	0.214994	+	0.976615i	$0.431027\pi$
$860$	4.34937	0.148312
$861$	0	0
$862$	12.5262	0.426644
$863$	10.9736	0.373545	0.186773	−	0.982403i	$-0.440197\pi$
$863$	10.9736	0.373545	0.186773	+	0.982403i	$0.440197\pi$
$864$	0	0
$865$	−0.250399	−0.00851382
$866$	3.94831	0.134169
$867$	0	0
$868$	−10.2555	−0.348095
$869$	76.4505	2.59341
$870$	0	0
$871$	15.7202	0.532659
$872$	2.67493	0.0905847
$873$	0	0
$874$	4.81992	0.163036
$875$	−5.94103	−0.200843
$876$	0	0
$877$	19.5883	0.661449	0.330725	−	0.943727i	$-0.392707\pi$
$877$	19.5883	0.661449	0.330725	+	0.943727i	$0.392707\pi$
$878$	−10.6552	−0.359595
$879$	0	0
$880$	−4.48885	−0.151319
$881$	−45.4696	−1.53191	−0.765956	−	0.642894i	$-0.777734\pi$
$881$	−45.4696	−1.53191	−0.765956	+	0.642894i	$0.777734\pi$
$882$	0	0
$883$	24.0833	0.810468	0.405234	−	0.914213i	$-0.367190\pi$
$883$	24.0833	0.810468	0.405234	+	0.914213i	$0.367190\pi$
$884$	12.4247	0.417889
$885$	0	0
$886$	17.8360	0.599211
$887$	24.9015	0.836110	0.418055	−	0.908422i	$-0.362712\pi$
$887$	24.9015	0.836110	0.418055	+	0.908422i	$0.362712\pi$
$888$	0	0
$889$	19.4754	0.653185
$890$	−4.99845	−0.167548
$891$	0	0
$892$	−43.2217	−1.44717
$893$	68.1258	2.27974
$894$	0	0
$895$	−13.2058	−0.441420
$896$	−13.2821	−0.443724
$897$	0	0
$898$	23.2898	0.777189
$899$	−5.43440	−0.181247
$900$	0	0
$901$	44.1864	1.47206
$902$	−3.13921	−0.104524
$903$	0	0
$904$	−44.0320	−1.46448
$905$	1.10665	0.0367862
$906$	0	0
$907$	−11.6579	−0.387095	−0.193548	−	0.981091i	$-0.561999\pi$
$907$	−11.6579	−0.387095	−0.193548	+	0.981091i	$0.561999\pi$
$908$	23.6893	0.786156
$909$	0	0
$910$	0.489408	0.0162237
$911$	51.8747	1.71869	0.859343	−	0.511399i	$-0.170873\pi$
$911$	51.8747	1.71869	0.859343	+	0.511399i	$0.170873\pi$
$912$	0	0
$913$	−69.3871	−2.29638
$914$	−7.95164	−0.263017
$915$	0	0
$916$	−2.77794	−0.0917857
$917$	5.09332	0.168196
$918$	0	0
$919$	30.8363	1.01720	0.508598	−	0.861004i	$-0.330164\pi$
$919$	30.8363	1.01720	0.508598	+	0.861004i	$0.330164\pi$
$920$	1.17218	0.0386456
$921$	0	0
$922$	−22.0650	−0.726673
$923$	3.12222	0.102769
$924$	0	0
$925$	−35.2599	−1.15934
$926$	−11.1722	−0.367141
$927$	0	0
$928$	−5.62485	−0.184645
$929$	−49.8315	−1.63492	−0.817459	−	0.575987i	$-0.804618\pi$
$929$	−49.8315	−1.63492	−0.817459	+	0.575987i	$0.804618\pi$
$930$	0	0
$931$	−43.8148	−1.43597
$932$	−29.7204	−0.973523
$933$	0	0
$934$	−14.1939	−0.464439
$935$	−14.5035	−0.474316
$936$	0	0
$937$	28.3305	0.925519	0.462759	−	0.886484i	$-0.346859\pi$
$937$	28.3305	0.925519	0.462759	+	0.886484i	$0.346859\pi$
$938$	−8.80804	−0.287593
$939$	0	0
$940$	7.40332	0.241470
$941$	23.8194	0.776491	0.388246	−	0.921556i	$-0.373081\pi$
$941$	23.8194	0.776491	0.388246	+	0.921556i	$0.373081\pi$
$942$	0	0
$943$	−1.08633	−0.0353759
$944$	12.2206	0.397745
$945$	0	0
$946$	−14.8762	−0.483668
$947$	−22.3698	−0.726921	−0.363461	−	0.931610i	$-0.618405\pi$
$947$	−22.3698	−0.726921	−0.363461	+	0.931610i	$0.618405\pi$
$948$	0	0
$949$	−4.44354	−0.144243
$950$	22.7816	0.739132
$951$	0	0
$952$	−15.5793	−0.504928
$953$	−33.8114	−1.09526	−0.547629	−	0.836721i	$-0.684470\pi$
$953$	−33.8114	−1.09526	−0.547629	+	0.836721i	$0.684470\pi$
$954$	0	0
$955$	−0.551833	−0.0178569
$956$	17.2238	0.557059
$957$	0	0
$958$	6.47923	0.209335
$959$	1.91933	0.0619783
$960$	0	0
$961$	−1.46733	−0.0473331
$962$	5.97730	0.192716
$963$	0	0
$964$	2.97367	0.0957756
$965$	−12.8646	−0.414128
$966$	0	0
$967$	−35.6037	−1.14494	−0.572468	−	0.819927i	$-0.694014\pi$
$967$	−35.6037	−1.14494	−0.572468	+	0.819927i	$0.694014\pi$
$968$	24.0436	0.772790
$969$	0	0
$970$	1.08004	0.0346778
$971$	33.2022	1.06551	0.532755	−	0.846269i	$-0.321157\pi$
$971$	33.2022	1.06551	0.532755	+	0.846269i	$0.321157\pi$
$972$	0	0
$973$	8.19207	0.262626
$974$	−22.2814	−0.713941
$975$	0	0
$976$	2.43942	0.0780839
$977$	−62.0025	−1.98364	−0.991818	−	0.127663i	$-0.959252\pi$
$977$	−62.0025	−1.98364	−0.991818	+	0.127663i	$0.959252\pi$
$978$	0	0
$979$	−71.8654	−2.29683
$980$	−4.76142	−0.152098
$981$	0	0
$982$	−8.11904	−0.259089
$983$	29.5507	0.942520	0.471260	−	0.881994i	$-0.343799\pi$
$983$	29.5507	0.942520	0.471260	+	0.881994i	$0.343799\pi$
$984$	0	0
$985$	2.64243	0.0841949
$986$	−3.68894	−0.117480
$987$	0	0
$988$	16.2340	0.516472
$989$	−5.14797	−0.163696
$990$	0	0
$991$	42.2514	1.34216	0.671079	−	0.741385i	$-0.265831\pi$
$991$	42.2514	1.34216	0.671079	+	0.741385i	$0.265831\pi$
$992$	30.5677	0.970525
$993$	0	0
$994$	−1.74938	−0.0554869
$995$	−3.92548	−0.124446
$996$	0	0
$997$	17.7928	0.563503	0.281752	−	0.959487i	$-0.409085\pi$
$997$	17.7928	0.563503	0.281752	+	0.959487i	$0.409085\pi$
$998$	0.00708053	0.000224130 0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.w.1.12	yes	30
3.2	odd	2		6003.2.a.v.1.19	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.19	✓	30	3.2	odd	2
6003.2.a.w.1.12	yes	30	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-0.619961 q^{2} -1.61565 q^{4} -0.522930 q^{5} -1.16804 q^{7} +2.24156 q^{8} +O(q^{10})\)	\(q-0.619961 q^{2} -1.61565 q^{4} -0.522930 q^{5} -1.16804 q^{7} +2.24156 q^{8} +0.324196 q^{10} +4.66114 q^{11} -1.29242 q^{13} +0.724142 q^{14} +1.84161 q^{16} +5.95028 q^{17} +7.77455 q^{19} +0.844871 q^{20} -2.88973 q^{22} -1.00000 q^{23} -4.72654 q^{25} +0.801249 q^{26} +1.88715 q^{28} +1.00000 q^{29} -5.43440 q^{31} -5.62485 q^{32} -3.68894 q^{34} +0.610806 q^{35} +7.45997 q^{37} -4.81992 q^{38} -1.17218 q^{40} +1.08633 q^{41} +5.14797 q^{43} -7.53077 q^{44} +0.619961 q^{46} +8.76267 q^{47} -5.63567 q^{49} +2.93027 q^{50} +2.08809 q^{52} +7.42594 q^{53} -2.43745 q^{55} -2.61824 q^{56} -0.619961 q^{58} +6.63579 q^{59} +1.32461 q^{61} +3.36912 q^{62} -0.196039 q^{64} +0.675844 q^{65} -12.1634 q^{67} -9.61356 q^{68} -0.378676 q^{70} -2.41579 q^{71} +3.43816 q^{73} -4.62489 q^{74} -12.5609 q^{76} -5.44442 q^{77} +16.4017 q^{79} -0.963036 q^{80} -0.673484 q^{82} -14.8863 q^{83} -3.11158 q^{85} -3.19154 q^{86} +10.4482 q^{88} -15.4180 q^{89} +1.50960 q^{91} +1.61565 q^{92} -5.43251 q^{94} -4.06555 q^{95} +3.33142 q^{97} +3.49390 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8} + 8 q^{10} + 36 q^{13} + 7 q^{14} + 47 q^{16} + 18 q^{17} + 16 q^{19} + 25 q^{22} - 30 q^{23} + 56 q^{25} + 11 q^{26} + 27 q^{28} + 30 q^{29} + 14 q^{31} - 7 q^{32} + 3 q^{34} - 22 q^{35} + 40 q^{37} + 6 q^{38} + 30 q^{40} + 14 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} + 74 q^{49} - 21 q^{50} + 71 q^{52} + 16 q^{53} + 22 q^{55} + 14 q^{56} + q^{58} - 32 q^{59} + 46 q^{61} + 20 q^{62} + 68 q^{64} + 12 q^{65} + 14 q^{67} + 27 q^{68} - 32 q^{71} + 50 q^{73} - 26 q^{74} + 56 q^{76} + 34 q^{77} + 16 q^{79} + 2 q^{80} + 38 q^{82} - 14 q^{83} + 38 q^{85} + 10 q^{86} + 40 q^{88} - 2 q^{89} + 32 q^{91} - 37 q^{92} + 29 q^{94} - 28 q^{95} + 56 q^{97} + 8 q^{98}+O(q^{100}) \) 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8 + 8 * q^10 + 36 * q^13 + 7 * q^14 + 47 * q^16 + 18 * q^17 + 16 * q^19 + 25 * q^22 - 30 * q^23 + 56 * q^25 + 11 * q^26 + 27 * q^28 + 30 * q^29 + 14 * q^31 - 7 * q^32 + 3 * q^34 - 22 * q^35 + 40 * q^37 + 6 * q^38 + 30 * q^40 + 14 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 2 * q^47 + 74 * q^49 - 21 * q^50 + 71 * q^52 + 16 * q^53 + 22 * q^55 + 14 * q^56 + q^58 - 32 * q^59 + 46 * q^61 + 20 * q^62 + 68 * q^64 + 12 * q^65 + 14 * q^67 + 27 * q^68 - 32 * q^71 + 50 * q^73 - 26 * q^74 + 56 * q^76 + 34 * q^77 + 16 * q^79 + 2 * q^80 + 38 * q^82 - 14 * q^83 + 38 * q^85 + 10 * q^86 + 40 * q^88 - 2 * q^89 + 32 * q^91 - 37 * q^92 + 29 * q^94 - 28 * q^95 + 56 * q^97 + 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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