LMFDB - Embedded newform 6003.2.a.t.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-2.18238 q^{2} +2.76278 q^{4} +0.541552 q^{5} +0.936842 q^{7} -1.66467 q^{8} +O(q^{10})$	\(q-2.18238 q^{2} +2.76278 q^{4} +0.541552 q^{5} +0.936842 q^{7} -1.66467 q^{8} -1.18187 q^{10} -3.69864 q^{11} -4.89549 q^{13} -2.04454 q^{14} -1.89261 q^{16} +7.10752 q^{17} -1.09532 q^{19} +1.49619 q^{20} +8.07183 q^{22} +1.00000 q^{23} -4.70672 q^{25} +10.6838 q^{26} +2.58829 q^{28} +1.00000 q^{29} -5.91418 q^{31} +7.45974 q^{32} -15.5113 q^{34} +0.507349 q^{35} +10.3636 q^{37} +2.39040 q^{38} -0.901507 q^{40} -0.971796 q^{41} -1.94722 q^{43} -10.2185 q^{44} -2.18238 q^{46} +6.84150 q^{47} -6.12233 q^{49} +10.2718 q^{50} -13.5252 q^{52} +10.6007 q^{53} -2.00300 q^{55} -1.55953 q^{56} -2.18238 q^{58} -5.79346 q^{59} -3.13155 q^{61} +12.9070 q^{62} -12.4948 q^{64} -2.65117 q^{65} +12.7930 q^{67} +19.6365 q^{68} -1.10723 q^{70} +11.6056 q^{71} +9.09037 q^{73} -22.6173 q^{74} -3.02612 q^{76} -3.46504 q^{77} +12.3322 q^{79} -1.02495 q^{80} +2.12083 q^{82} +3.23404 q^{83} +3.84909 q^{85} +4.24957 q^{86} +6.15702 q^{88} -18.7802 q^{89} -4.58630 q^{91} +2.76278 q^{92} -14.9307 q^{94} -0.593172 q^{95} -13.0946 q^{97} +13.3612 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * 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 $

$2$	−2.18238	−1.54318	−0.771588	−	0.636123i	$-0.780537\pi$
$2$	−2.18238	−1.54318	−0.771588	+	0.636123i	$0.780537\pi$
$3$	0	0
$3$	0	0
$4$	2.76278	1.38139
$5$	0.541552	0.242190	0.121095	−	0.992641i	$-0.461360\pi$
$5$	0.541552	0.242190	0.121095	+	0.992641i	$0.461360\pi$
$6$	0	0
$7$	0.936842	0.354093	0.177046	−	0.984203i	$-0.443346\pi$
$7$	0.936842	0.354093	0.177046	+	0.984203i	$0.443346\pi$
$8$	−1.66467	−0.588550
$9$	0	0
$10$	−1.18187	−0.373741
$11$	−3.69864	−1.11518	−0.557590	−	0.830116i	$-0.688274\pi$
$11$	−3.69864	−1.11518	−0.557590	+	0.830116i	$0.688274\pi$
$12$	0	0
$13$	−4.89549	−1.35777	−0.678883	−	0.734247i	$-0.737536\pi$
$13$	−4.89549	−1.35777	−0.678883	+	0.734247i	$0.737536\pi$
$14$	−2.04454	−0.546427
$15$	0	0
$16$	−1.89261	−0.473153
$17$	7.10752	1.72383	0.861914	−	0.507055i	$-0.169266\pi$
$17$	7.10752	1.72383	0.861914	+	0.507055i	$0.169266\pi$
$18$	0	0
$19$	−1.09532	−0.251283	−0.125642	−	0.992076i	$-0.540099\pi$
$19$	−1.09532	−0.251283	−0.125642	+	0.992076i	$0.540099\pi$
$20$	1.49619	0.334558
$21$	0	0
$22$	8.07183	1.72092
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.70672	−0.941344
$26$	10.6838	2.09527
$27$	0	0
$28$	2.58829	0.489140
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−5.91418	−1.06222	−0.531109	−	0.847303i	$-0.678225\pi$
$31$	−5.91418	−1.06222	−0.531109	+	0.847303i	$0.678225\pi$
$32$	7.45974	1.31871
$33$	0	0
$34$	−15.5113	−2.66017
$35$	0.507349	0.0857576
$36$	0	0
$37$	10.3636	1.70377	0.851883	−	0.523732i	$-0.175461\pi$
$37$	10.3636	1.70377	0.851883	+	0.523732i	$0.175461\pi$
$38$	2.39040	0.387774
$39$	0	0
$40$	−0.901507	−0.142541
$41$	−0.971796	−0.151769	−0.0758845	−	0.997117i	$-0.524178\pi$
$41$	−0.971796	−0.151769	−0.0758845	+	0.997117i	$0.524178\pi$
$42$	0	0
$43$	−1.94722	−0.296948	−0.148474	−	0.988916i	$-0.547436\pi$
$43$	−1.94722	−0.296948	−0.148474	+	0.988916i	$0.547436\pi$
$44$	−10.2185	−1.54050
$45$	0	0
$46$	−2.18238	−0.321774
$47$	6.84150	0.997935	0.498968	−	0.866621i	$-0.333713\pi$
$47$	6.84150	0.997935	0.498968	+	0.866621i	$0.333713\pi$
$48$	0	0
$49$	−6.12233	−0.874618
$50$	10.2718	1.45266
$51$	0	0
$52$	−13.5252	−1.87560
$53$	10.6007	1.45612	0.728061	−	0.685513i	$-0.240422\pi$
$53$	10.6007	1.45612	0.728061	+	0.685513i	$0.240422\pi$
$54$	0	0
$55$	−2.00300	−0.270085
$56$	−1.55953	−0.208401
$57$	0	0
$58$	−2.18238	−0.286560
$59$	−5.79346	−0.754244	−0.377122	−	0.926164i	$-0.623086\pi$
$59$	−5.79346	−0.754244	−0.377122	+	0.926164i	$0.623086\pi$
$60$	0	0
$61$	−3.13155	−0.400954	−0.200477	−	0.979698i	$-0.564249\pi$
$61$	−3.13155	−0.400954	−0.200477	+	0.979698i	$0.564249\pi$
$62$	12.9070	1.63919
$63$	0	0
$64$	−12.4948	−1.56184
$65$	−2.65117	−0.328837
$66$	0	0
$67$	12.7930	1.56292	0.781459	−	0.623957i	$-0.214476\pi$
$67$	12.7930	1.56292	0.781459	+	0.623957i	$0.214476\pi$
$68$	19.6365	2.38128
$69$	0	0
$70$	−1.10723	−0.132339
$71$	11.6056	1.37733	0.688666	−	0.725079i	$-0.258197\pi$
$71$	11.6056	1.37733	0.688666	+	0.725079i	$0.258197\pi$
$72$	0	0
$73$	9.09037	1.06395	0.531974	−	0.846761i	$-0.321451\pi$
$73$	9.09037	1.06395	0.531974	+	0.846761i	$0.321451\pi$
$74$	−22.6173	−2.62921
$75$	0	0
$76$	−3.02612	−0.347120
$77$	−3.46504	−0.394877
$78$	0	0
$79$	12.3322	1.38748	0.693741	−	0.720224i	$-0.255961\pi$
$79$	12.3322	1.38748	0.693741	+	0.720224i	$0.255961\pi$
$80$	−1.02495	−0.114593
$81$	0	0
$82$	2.12083	0.234206
$83$	3.23404	0.354982	0.177491	−	0.984122i	$-0.443202\pi$
$83$	3.23404	0.354982	0.177491	+	0.984122i	$0.443202\pi$
$84$	0	0
$85$	3.84909	0.417493
$86$	4.24957	0.458243
$87$	0	0
$88$	6.15702	0.656340
$89$	−18.7802	−1.99070	−0.995350	−	0.0963207i	$-0.969293\pi$
$89$	−18.7802	−1.99070	−0.995350	+	0.0963207i	$0.969293\pi$
$90$	0	0
$91$	−4.58630	−0.480775
$92$	2.76278	0.288040
$93$	0	0
$94$	−14.9307	−1.53999
$95$	−0.593172	−0.0608582
$96$	0	0
$97$	−13.0946	−1.32956	−0.664778	−	0.747041i	$-0.731474\pi$
$97$	−13.0946	−1.32956	−0.664778	+	0.747041i	$0.731474\pi$
$98$	13.3612	1.34969
$99$	0	0
$100$	−13.0036	−1.30036
$101$	−10.0431	−0.999323	−0.499661	−	0.866221i	$-0.666542\pi$
$101$	−10.0431	−0.999323	−0.499661	+	0.866221i	$0.666542\pi$
$102$	0	0
$103$	−20.0279	−1.97341	−0.986705	−	0.162522i	$-0.948037\pi$
$103$	−20.0279	−1.97341	−0.986705	+	0.162522i	$0.948037\pi$
$104$	8.14939	0.799114
$105$	0	0
$106$	−23.1348	−2.24705
$107$	−5.51804	−0.533449	−0.266724	−	0.963773i	$-0.585941\pi$
$107$	−5.51804	−0.533449	−0.266724	+	0.963773i	$0.585941\pi$
$108$	0	0
$109$	−2.72927	−0.261416	−0.130708	−	0.991421i	$-0.541725\pi$
$109$	−2.72927	−0.261416	−0.130708	+	0.991421i	$0.541725\pi$
$110$	4.37132	0.416789
$111$	0	0
$112$	−1.77308	−0.167540
$113$	10.7166	1.00814	0.504068	−	0.863664i	$-0.331836\pi$
$113$	10.7166	1.00814	0.504068	+	0.863664i	$0.331836\pi$
$114$	0	0
$115$	0.541552	0.0505000
$116$	2.76278	0.256518
$117$	0	0
$118$	12.6435	1.16393
$119$	6.65862	0.610395
$120$	0	0
$121$	2.67991	0.243628
$122$	6.83424	0.618743
$123$	0	0
$124$	−16.3396	−1.46734
$125$	−5.25670	−0.470173
$126$	0	0
$127$	−13.5491	−1.20229	−0.601144	−	0.799141i	$-0.705288\pi$
$127$	−13.5491	−1.20229	−0.601144	+	0.799141i	$0.705288\pi$
$128$	12.3488	1.09149
$129$	0	0
$130$	5.78585	0.507452
$131$	−5.07093	−0.443049	−0.221525	−	0.975155i	$-0.571103\pi$
$131$	−5.07093	−0.443049	−0.221525	+	0.975155i	$0.571103\pi$
$132$	0	0
$133$	−1.02614	−0.0889776
$134$	−27.9192	−2.41186
$135$	0	0
$136$	−11.8317	−1.01456
$137$	−14.4059	−1.23078	−0.615388	−	0.788224i	$-0.711001\pi$
$137$	−14.4059	−1.23078	−0.615388	+	0.788224i	$0.711001\pi$
$138$	0	0
$139$	−11.2391	−0.953289	−0.476645	−	0.879096i	$-0.658147\pi$
$139$	−11.2391	−0.953289	−0.476645	+	0.879096i	$0.658147\pi$
$140$	1.40169	0.118465
$141$	0	0
$142$	−25.3278	−2.12546
$143$	18.1066	1.51415
$144$	0	0
$145$	0.541552	0.0449735
$146$	−19.8386	−1.64186
$147$	0	0
$148$	28.6323	2.35356
$149$	−8.89024	−0.728317	−0.364159	−	0.931337i	$-0.618643\pi$
$149$	−8.89024	−0.728317	−0.364159	+	0.931337i	$0.618643\pi$
$150$	0	0
$151$	8.93585	0.727189	0.363594	−	0.931557i	$-0.381549\pi$
$151$	8.93585	0.727189	0.363594	+	0.931557i	$0.381549\pi$
$152$	1.82335	0.147893
$153$	0	0
$154$	7.56202	0.609365
$155$	−3.20284	−0.257258
$156$	0	0
$157$	−9.64721	−0.769931	−0.384966	−	0.922931i	$-0.625787\pi$
$157$	−9.64721	−0.769931	−0.384966	+	0.922931i	$0.625787\pi$
$158$	−26.9136	−2.14113
$159$	0	0
$160$	4.03984	0.319377
$161$	0.936842	0.0738335
$162$	0	0
$163$	7.00313	0.548528	0.274264	−	0.961654i	$-0.411566\pi$
$163$	7.00313	0.548528	0.274264	+	0.961654i	$0.411566\pi$
$164$	−2.68486	−0.209652
$165$	0	0
$166$	−7.05790	−0.547799
$167$	3.28587	0.254268	0.127134	−	0.991886i	$-0.459422\pi$
$167$	3.28587	0.254268	0.127134	+	0.991886i	$0.459422\pi$
$168$	0	0
$169$	10.9659	0.843527
$170$	−8.40018	−0.644265
$171$	0	0
$172$	−5.37974	−0.410201
$173$	13.3537	1.01526	0.507630	−	0.861575i	$-0.330522\pi$
$173$	13.3537	1.01526	0.507630	+	0.861575i	$0.330522\pi$
$174$	0	0
$175$	−4.40945	−0.333323
$176$	7.00008	0.527651
$177$	0	0
$178$	40.9856	3.07200
$179$	−21.6038	−1.61474	−0.807371	−	0.590044i	$-0.799110\pi$
$179$	−21.6038	−1.61474	−0.807371	+	0.590044i	$0.799110\pi$
$180$	0	0
$181$	14.9136	1.10852	0.554258	−	0.832345i	$-0.313002\pi$
$181$	14.9136	1.10852	0.554258	+	0.832345i	$0.313002\pi$
$182$	10.0090	0.741920
$183$	0	0
$184$	−1.66467	−0.122721
$185$	5.61243	0.412634
$186$	0	0
$187$	−26.2881	−1.92238
$188$	18.9016	1.37854
$189$	0	0
$190$	1.29453	0.0939149
$191$	22.8259	1.65162	0.825811	−	0.563947i	$-0.190718\pi$
$191$	22.8259	1.65162	0.825811	+	0.563947i	$0.190718\pi$
$192$	0	0
$193$	−22.4698	−1.61741	−0.808707	−	0.588212i	$-0.799832\pi$
$193$	−22.4698	−1.61741	−0.808707	+	0.588212i	$0.799832\pi$
$194$	28.5774	2.05174
$195$	0	0
$196$	−16.9146	−1.20819
$197$	−3.21964	−0.229390	−0.114695	−	0.993401i	$-0.536589\pi$
$197$	−3.21964	−0.229390	−0.114695	+	0.993401i	$0.536589\pi$
$198$	0	0
$199$	4.22747	0.299677	0.149839	−	0.988710i	$-0.452125\pi$
$199$	4.22747	0.299677	0.149839	+	0.988710i	$0.452125\pi$
$200$	7.83515	0.554029
$201$	0	0
$202$	21.9178	1.54213
$203$	0.936842	0.0657534
$204$	0	0
$205$	−0.526278	−0.0367569
$206$	43.7085	3.04532
$207$	0	0
$208$	9.26527	0.642431
$209$	4.05119	0.280226
$210$	0	0
$211$	15.8411	1.09054	0.545272	−	0.838259i	$-0.316426\pi$
$211$	15.8411	1.09054	0.545272	+	0.838259i	$0.316426\pi$
$212$	29.2875	2.01147
$213$	0	0
$214$	12.0424	0.823205
$215$	−1.05452	−0.0719178
$216$	0	0
$217$	−5.54065	−0.376124
$218$	5.95629	0.403411
$219$	0	0
$220$	−5.53386	−0.373093
$221$	−34.7948	−2.34055
$222$	0	0
$223$	5.13806	0.344070	0.172035	−	0.985091i	$-0.444966\pi$
$223$	5.13806	0.344070	0.172035	+	0.985091i	$0.444966\pi$
$224$	6.98859	0.466945
$225$	0	0
$226$	−23.3878	−1.55573
$227$	−23.1797	−1.53849	−0.769246	−	0.638953i	$-0.779368\pi$
$227$	−23.1797	−1.53849	−0.769246	+	0.638953i	$0.779368\pi$
$228$	0	0
$229$	−27.6792	−1.82909	−0.914545	−	0.404484i	$-0.867451\pi$
$229$	−27.6792	−1.82909	−0.914545	+	0.404484i	$0.867451\pi$
$230$	−1.18187	−0.0779304
$231$	0	0
$232$	−1.66467	−0.109291
$233$	−5.79047	−0.379347	−0.189673	−	0.981847i	$-0.560743\pi$
$233$	−5.79047	−0.379347	−0.189673	+	0.981847i	$0.560743\pi$
$234$	0	0
$235$	3.70503	0.241690
$236$	−16.0060	−1.04190
$237$	0	0
$238$	−14.5316	−0.941946
$239$	23.4674	1.51798	0.758991	−	0.651101i	$-0.225693\pi$
$239$	23.4674	1.51798	0.758991	+	0.651101i	$0.225693\pi$
$240$	0	0
$241$	−3.03269	−0.195353	−0.0976765	−	0.995218i	$-0.531141\pi$
$241$	−3.03269	−0.195353	−0.0976765	+	0.995218i	$0.531141\pi$
$242$	−5.84858	−0.375961
$243$	0	0
$244$	−8.65179	−0.553874
$245$	−3.31556	−0.211823
$246$	0	0
$247$	5.36213	0.341184
$248$	9.84518	0.625169
$249$	0	0
$250$	11.4721	0.725560
$251$	29.3351	1.85161	0.925807	−	0.377996i	$-0.123387\pi$
$251$	29.3351	1.85161	0.925807	+	0.377996i	$0.123387\pi$
$252$	0	0
$253$	−3.69864	−0.232531
$254$	29.5693	1.85534
$255$	0	0
$256$	−1.96029	−0.122518
$257$	−18.4474	−1.15071	−0.575357	−	0.817902i	$-0.695137\pi$
$257$	−18.4474	−1.15071	−0.575357	+	0.817902i	$0.695137\pi$
$258$	0	0
$259$	9.70905	0.603291
$260$	−7.32458	−0.454251
$261$	0	0
$262$	11.0667	0.683702
$263$	−2.40266	−0.148155	−0.0740773	−	0.997253i	$-0.523601\pi$
$263$	−2.40266	−0.148155	−0.0740773	+	0.997253i	$0.523601\pi$
$264$	0	0
$265$	5.74085	0.352657
$266$	2.23943	0.137308
$267$	0	0
$268$	35.3443	2.15900
$269$	8.01329	0.488579	0.244290	−	0.969702i	$-0.421445\pi$
$269$	8.01329	0.488579	0.244290	+	0.969702i	$0.421445\pi$
$270$	0	0
$271$	−22.2186	−1.34969	−0.674843	−	0.737961i	$-0.735789\pi$
$271$	−22.2186	−1.34969	−0.674843	+	0.737961i	$0.735789\pi$
$272$	−13.4518	−0.815634
$273$	0	0
$274$	31.4391	1.89930
$275$	17.4084	1.04977
$276$	0	0
$277$	18.8850	1.13469	0.567344	−	0.823481i	$-0.307971\pi$
$277$	18.8850	1.13469	0.567344	+	0.823481i	$0.307971\pi$
$278$	24.5280	1.47109
$279$	0	0
$280$	−0.844569	−0.0504727
$281$	−22.4266	−1.33786	−0.668928	−	0.743327i	$-0.733247\pi$
$281$	−22.4266	−1.33786	−0.668928	+	0.743327i	$0.733247\pi$
$282$	0	0
$283$	−16.9952	−1.01026	−0.505129	−	0.863044i	$-0.668555\pi$
$283$	−16.9952	−1.01026	−0.505129	+	0.863044i	$0.668555\pi$
$284$	32.0637	1.90263
$285$	0	0
$286$	−39.5156	−2.33660
$287$	−0.910419	−0.0537403
$288$	0	0
$289$	33.5169	1.97158
$290$	−1.18187	−0.0694019
$291$	0	0
$292$	25.1147	1.46973
$293$	−22.1283	−1.29275	−0.646374	−	0.763021i	$-0.723715\pi$
$293$	−22.1283	−1.29275	−0.646374	+	0.763021i	$0.723715\pi$
$294$	0	0
$295$	−3.13746	−0.182670
$296$	−17.2520	−1.00275
$297$	0	0
$298$	19.4019	1.12392
$299$	−4.89549	−0.283114
$300$	0	0
$301$	−1.82424	−0.105147
$302$	−19.5014	−1.12218
$303$	0	0
$304$	2.07301	0.118895
$305$	−1.69590	−0.0971069
$306$	0	0
$307$	−9.55579	−0.545378	−0.272689	−	0.962102i	$-0.587913\pi$
$307$	−9.55579	−0.545378	−0.272689	+	0.962102i	$0.587913\pi$
$308$	−9.57313	−0.545480
$309$	0	0
$310$	6.98981	0.396995
$311$	2.74915	0.155890	0.0779451	−	0.996958i	$-0.475164\pi$
$311$	2.74915	0.155890	0.0779451	+	0.996958i	$0.475164\pi$
$312$	0	0
$313$	13.1079	0.740905	0.370452	−	0.928851i	$-0.379203\pi$
$313$	13.1079	0.740905	0.370452	+	0.928851i	$0.379203\pi$
$314$	21.0539	1.18814
$315$	0	0
$316$	34.0712	1.91665
$317$	18.2291	1.02385	0.511923	−	0.859031i	$-0.328933\pi$
$317$	18.2291	1.02385	0.511923	+	0.859031i	$0.328933\pi$
$318$	0	0
$319$	−3.69864	−0.207084
$320$	−6.76656	−0.378262
$321$	0	0
$322$	−2.04454	−0.113938
$323$	−7.78500	−0.433169
$324$	0	0
$325$	23.0417	1.27812
$326$	−15.2835	−0.846474
$327$	0	0
$328$	1.61772	0.0893237
$329$	6.40940	0.353362
$330$	0	0
$331$	15.7316	0.864686	0.432343	−	0.901709i	$-0.357687\pi$
$331$	15.7316	0.864686	0.432343	+	0.901709i	$0.357687\pi$
$332$	8.93493	0.490368
$333$	0	0
$334$	−7.17102	−0.392381
$335$	6.92810	0.378522
$336$	0	0
$337$	−26.2825	−1.43170	−0.715849	−	0.698255i	$-0.753960\pi$
$337$	−26.2825	−1.43170	−0.715849	+	0.698255i	$0.753960\pi$
$338$	−23.9317	−1.30171
$339$	0	0
$340$	10.6342	0.576720
$341$	21.8744	1.18457
$342$	0	0
$343$	−12.2935	−0.663789
$344$	3.24148	0.174769
$345$	0	0
$346$	−29.1428	−1.56673
$347$	8.60890	0.462150	0.231075	−	0.972936i	$-0.425776\pi$
$347$	8.60890	0.462150	0.231075	+	0.972936i	$0.425776\pi$
$348$	0	0
$349$	8.19419	0.438625	0.219312	−	0.975655i	$-0.429619\pi$
$349$	8.19419	0.438625	0.219312	+	0.975655i	$0.429619\pi$
$350$	9.62310	0.514376
$351$	0	0
$352$	−27.5909	−1.47060
$353$	11.6822	0.621780	0.310890	−	0.950446i	$-0.399373\pi$
$353$	11.6822	0.621780	0.310890	+	0.950446i	$0.399373\pi$
$354$	0	0
$355$	6.28504	0.333575
$356$	−51.8856	−2.74993
$357$	0	0
$358$	47.1476	2.49183
$359$	2.76963	0.146175	0.0730877	−	0.997326i	$-0.476715\pi$
$359$	2.76963	0.146175	0.0730877	+	0.997326i	$0.476715\pi$
$360$	0	0
$361$	−17.8003	−0.936857
$362$	−32.5471	−1.71064
$363$	0	0
$364$	−12.6709	−0.664137
$365$	4.92291	0.257677
$366$	0	0
$367$	−19.3751	−1.01137	−0.505687	−	0.862717i	$-0.668761\pi$
$367$	−19.3751	−1.01137	−0.505687	+	0.862717i	$0.668761\pi$
$368$	−1.89261	−0.0986592
$369$	0	0
$370$	−12.2485	−0.636767
$371$	9.93120	0.515602
$372$	0	0
$373$	−7.92717	−0.410453	−0.205227	−	0.978714i	$-0.565793\pi$
$373$	−7.92717	−0.410453	−0.205227	+	0.978714i	$0.565793\pi$
$374$	57.3707	2.96657
$375$	0	0
$376$	−11.3889	−0.587335
$377$	−4.89549	−0.252131
$378$	0	0
$379$	−22.6325	−1.16255	−0.581275	−	0.813707i	$-0.697446\pi$
$379$	−22.6325	−1.16255	−0.581275	+	0.813707i	$0.697446\pi$
$380$	−1.63880	−0.0840689
$381$	0	0
$382$	−49.8147	−2.54874
$383$	1.29715	0.0662813	0.0331406	−	0.999451i	$-0.489449\pi$
$383$	1.29715	0.0662813	0.0331406	+	0.999451i	$0.489449\pi$
$384$	0	0
$385$	−1.87650	−0.0956352
$386$	49.0377	2.49595
$387$	0	0
$388$	−36.1775	−1.83663
$389$	−12.9895	−0.658595	−0.329297	−	0.944226i	$-0.606812\pi$
$389$	−12.9895	−0.658595	−0.329297	+	0.944226i	$0.606812\pi$
$390$	0	0
$391$	7.10752	0.359443
$392$	10.1917	0.514757
$393$	0	0
$394$	7.02647	0.353988
$395$	6.67854	0.336034
$396$	0	0
$397$	−24.3012	−1.21964	−0.609822	−	0.792538i	$-0.708759\pi$
$397$	−24.3012	−1.21964	−0.609822	+	0.792538i	$0.708759\pi$
$398$	−9.22594	−0.462455
$399$	0	0
$400$	8.90800	0.445400
$401$	0.473592	0.0236500	0.0118250	−	0.999930i	$-0.496236\pi$
$401$	0.473592	0.0236500	0.0118250	+	0.999930i	$0.496236\pi$
$402$	0	0
$403$	28.9528	1.44224
$404$	−27.7468	−1.38045
$405$	0	0
$406$	−2.04454	−0.101469
$407$	−38.3312	−1.90001
$408$	0	0
$409$	−15.4034	−0.761649	−0.380824	−	0.924647i	$-0.624360\pi$
$409$	−15.4034	−0.761649	−0.380824	+	0.924647i	$0.624360\pi$
$410$	1.14854	0.0567223
$411$	0	0
$412$	−55.3327	−2.72605
$413$	−5.42755	−0.267072
$414$	0	0
$415$	1.75140	0.0859729
$416$	−36.5191	−1.79050
$417$	0	0
$418$	−8.84122	−0.432438
$419$	−7.23895	−0.353646	−0.176823	−	0.984243i	$-0.556582\pi$
$419$	−7.23895	−0.353646	−0.176823	+	0.984243i	$0.556582\pi$
$420$	0	0
$421$	18.7625	0.914430	0.457215	−	0.889356i	$-0.348847\pi$
$421$	18.7625	0.914430	0.457215	+	0.889356i	$0.348847\pi$
$422$	−34.5712	−1.68290
$423$	0	0
$424$	−17.6467	−0.857001
$425$	−33.4531	−1.62271
$426$	0	0
$427$	−2.93377	−0.141975
$428$	−15.2451	−0.736900
$429$	0	0
$430$	2.30137	0.110982
$431$	35.3900	1.70468	0.852338	−	0.522991i	$-0.175184\pi$
$431$	35.3900	1.70468	0.852338	+	0.522991i	$0.175184\pi$
$432$	0	0
$433$	−34.1692	−1.64207	−0.821034	−	0.570879i	$-0.806603\pi$
$433$	−34.1692	−1.64207	−0.821034	+	0.570879i	$0.806603\pi$
$434$	12.0918	0.580425
$435$	0	0
$436$	−7.54036	−0.361118
$437$	−1.09532	−0.0523962
$438$	0	0
$439$	−23.9651	−1.14379	−0.571897	−	0.820326i	$-0.693792\pi$
$439$	−23.9651	−1.14379	−0.571897	+	0.820326i	$0.693792\pi$
$440$	3.33435	0.158959
$441$	0	0
$442$	75.9355	3.61188
$443$	25.7391	1.22290	0.611450	−	0.791283i	$-0.290586\pi$
$443$	25.7391	1.22290	0.611450	+	0.791283i	$0.290586\pi$
$444$	0	0
$445$	−10.1705	−0.482127
$446$	−11.2132	−0.530960
$447$	0	0
$448$	−11.7056	−0.553038
$449$	−18.3083	−0.864023	−0.432011	−	0.901868i	$-0.642196\pi$
$449$	−18.3083	−0.864023	−0.432011	+	0.901868i	$0.642196\pi$
$450$	0	0
$451$	3.59432	0.169250
$452$	29.6077	1.39263
$453$	0	0
$454$	50.5869	2.37416
$455$	−2.48372	−0.116439
$456$	0	0
$457$	14.6410	0.684875	0.342437	−	0.939541i	$-0.388748\pi$
$457$	14.6410	0.684875	0.342437	+	0.939541i	$0.388748\pi$
$458$	60.4064	2.82261
$459$	0	0
$460$	1.49619	0.0697602
$461$	−27.0760	−1.26106	−0.630528	−	0.776167i	$-0.717162\pi$
$461$	−27.0760	−1.26106	−0.630528	+	0.776167i	$0.717162\pi$
$462$	0	0
$463$	33.9300	1.57686	0.788431	−	0.615123i	$-0.210894\pi$
$463$	33.9300	1.57686	0.788431	+	0.615123i	$0.210894\pi$
$464$	−1.89261	−0.0878623
$465$	0	0
$466$	12.6370	0.585398
$467$	−27.3244	−1.26442	−0.632210	−	0.774797i	$-0.717852\pi$
$467$	−27.3244	−1.26442	−0.632210	+	0.774797i	$0.717852\pi$
$468$	0	0
$469$	11.9850	0.553418
$470$	−8.08578	−0.372969
$471$	0	0
$472$	9.64421	0.443911
$473$	7.20206	0.331151
$474$	0	0
$475$	5.15536	0.236544
$476$	18.3963	0.843193
$477$	0	0
$478$	−51.2148	−2.34251
$479$	−11.4058	−0.521146	−0.260573	−	0.965454i	$-0.583911\pi$
$479$	−11.4058	−0.521146	−0.260573	+	0.965454i	$0.583911\pi$
$480$	0	0
$481$	−50.7350	−2.31331
$482$	6.61849	0.301464
$483$	0	0
$484$	7.40400	0.336545
$485$	−7.09141	−0.322005
$486$	0	0
$487$	15.5110	0.702871	0.351435	−	0.936212i	$-0.385694\pi$
$487$	15.5110	0.702871	0.351435	+	0.936212i	$0.385694\pi$
$488$	5.21301	0.235982
$489$	0	0
$490$	7.23581	0.326881
$491$	−1.80111	−0.0812829	−0.0406415	−	0.999174i	$-0.512940\pi$
$491$	−1.80111	−0.0812829	−0.0406415	+	0.999174i	$0.512940\pi$
$492$	0	0
$493$	7.10752	0.320107
$494$	−11.7022	−0.526507
$495$	0	0
$496$	11.1933	0.502592
$497$	10.8726	0.487703
$498$	0	0
$499$	−41.5009	−1.85783	−0.928917	−	0.370289i	$-0.879259\pi$
$499$	−41.5009	−1.85783	−0.928917	+	0.370289i	$0.879259\pi$
$500$	−14.5231	−0.649492
$501$	0	0
$502$	−64.0203	−2.85737
$503$	−29.8332	−1.33020	−0.665099	−	0.746755i	$-0.731611\pi$
$503$	−29.8332	−1.33020	−0.665099	+	0.746755i	$0.731611\pi$
$504$	0	0
$505$	−5.43885	−0.242025
$506$	8.07183	0.358836
$507$	0	0
$508$	−37.4332	−1.66083
$509$	−25.3981	−1.12575	−0.562875	−	0.826542i	$-0.690305\pi$
$509$	−25.3981	−1.12575	−0.562875	+	0.826542i	$0.690305\pi$
$510$	0	0
$511$	8.51623	0.376736
$512$	−20.4195	−0.902425
$513$	0	0
$514$	40.2591	1.77575
$515$	−10.8462	−0.477939
$516$	0	0
$517$	−25.3042	−1.11288
$518$	−21.1888	−0.930984
$519$	0	0
$520$	4.41332	0.193537
$521$	−25.0390	−1.09698	−0.548489	−	0.836158i	$-0.684797\pi$
$521$	−25.0390	−1.09698	−0.548489	+	0.836158i	$0.684797\pi$
$522$	0	0
$523$	−20.3520	−0.889930	−0.444965	−	0.895548i	$-0.646784\pi$
$523$	−20.3520	−0.889930	−0.444965	+	0.895548i	$0.646784\pi$
$524$	−14.0099	−0.612023
$525$	0	0
$526$	5.24352	0.228628
$527$	−42.0352	−1.83108
$528$	0	0
$529$	1.00000	0.0434783
$530$	−12.5287	−0.544212
$531$	0	0
$532$	−2.83500	−0.122913
$533$	4.75742	0.206067
$534$	0	0
$535$	−2.98830	−0.129196
$536$	−21.2962	−0.919856
$537$	0	0
$538$	−17.4880	−0.753963
$539$	22.6443	0.975358
$540$	0	0
$541$	14.7556	0.634391	0.317196	−	0.948360i	$-0.397259\pi$
$541$	14.7556	0.634391	0.317196	+	0.948360i	$0.397259\pi$
$542$	48.4895	2.08280
$543$	0	0
$544$	53.0203	2.27323
$545$	−1.47804	−0.0633123
$546$	0	0
$547$	31.0173	1.32620	0.663102	−	0.748529i	$-0.269240\pi$
$547$	31.0173	1.32620	0.663102	+	0.748529i	$0.269240\pi$
$548$	−39.8002	−1.70018
$549$	0	0
$550$	−37.9918	−1.61998
$551$	−1.09532	−0.0466622
$552$	0	0
$553$	11.5533	0.491297
$554$	−41.2142	−1.75102
$555$	0	0
$556$	−31.0512	−1.31686
$557$	−20.4619	−0.866997	−0.433498	−	0.901154i	$-0.642721\pi$
$557$	−20.4619	−0.866997	−0.433498	+	0.901154i	$0.642721\pi$
$558$	0	0
$559$	9.53261	0.403186
$560$	−0.960214	−0.0405764
$561$	0	0
$562$	48.9433	2.06455
$563$	8.08437	0.340716	0.170358	−	0.985382i	$-0.445508\pi$
$563$	8.08437	0.340716	0.170358	+	0.985382i	$0.445508\pi$
$564$	0	0
$565$	5.80362	0.244160
$566$	37.0899	1.55901
$567$	0	0
$568$	−19.3195	−0.810629
$569$	21.3865	0.896569	0.448284	−	0.893891i	$-0.352035\pi$
$569$	21.3865	0.896569	0.448284	+	0.893891i	$0.352035\pi$
$570$	0	0
$571$	17.9758	0.752262	0.376131	−	0.926567i	$-0.377254\pi$
$571$	17.9758	0.752262	0.376131	+	0.926567i	$0.377254\pi$
$572$	50.0247	2.09164
$573$	0	0
$574$	1.98688	0.0829307
$575$	−4.70672	−0.196284
$576$	0	0
$577$	44.1313	1.83721	0.918604	−	0.395178i	$-0.129317\pi$
$577$	44.1313	1.83721	0.918604	+	0.395178i	$0.129317\pi$
$578$	−73.1465	−3.04249
$579$	0	0
$580$	1.49619	0.0621259
$581$	3.02978	0.125696
$582$	0	0
$583$	−39.2082	−1.62384
$584$	−15.1325	−0.626187
$585$	0	0
$586$	48.2923	1.99494
$587$	−19.9086	−0.821717	−0.410858	−	0.911699i	$-0.634771\pi$
$587$	−19.9086	−0.821717	−0.410858	+	0.911699i	$0.634771\pi$
$588$	0	0
$589$	6.47792	0.266918
$590$	6.84713	0.281892
$591$	0	0
$592$	−19.6143	−0.806142
$593$	−4.05650	−0.166581	−0.0832903	−	0.996525i	$-0.526543\pi$
$593$	−4.05650	−0.166581	−0.0832903	+	0.996525i	$0.526543\pi$
$594$	0	0
$595$	3.60599	0.147831
$596$	−24.5618	−1.00609
$597$	0	0
$598$	10.6838	0.436894
$599$	−41.4655	−1.69423	−0.847117	−	0.531407i	$-0.821664\pi$
$599$	−41.4655	−1.69423	−0.847117	+	0.531407i	$0.821664\pi$
$600$	0	0
$601$	35.8970	1.46427	0.732134	−	0.681160i	$-0.238524\pi$
$601$	35.8970	1.46427	0.732134	+	0.681160i	$0.238524\pi$
$602$	3.98118	0.162261
$603$	0	0
$604$	24.6878	1.00453
$605$	1.45131	0.0590042
$606$	0	0
$607$	−20.4894	−0.831640	−0.415820	−	0.909447i	$-0.636505\pi$
$607$	−20.4894	−0.831640	−0.415820	+	0.909447i	$0.636505\pi$
$608$	−8.17079	−0.331369
$609$	0	0
$610$	3.70110	0.149853
$611$	−33.4925	−1.35496
$612$	0	0
$613$	34.3964	1.38926	0.694629	−	0.719368i	$-0.255568\pi$
$613$	34.3964	1.38926	0.694629	+	0.719368i	$0.255568\pi$
$614$	20.8544	0.841614
$615$	0	0
$616$	5.76815	0.232405
$617$	25.0198	1.00726	0.503629	−	0.863920i	$-0.331998\pi$
$617$	25.0198	1.00726	0.503629	+	0.863920i	$0.331998\pi$
$618$	0	0
$619$	−46.5200	−1.86980	−0.934899	−	0.354913i	$-0.884510\pi$
$619$	−46.5200	−1.86980	−0.934899	+	0.354913i	$0.884510\pi$
$620$	−8.84874	−0.355374
$621$	0	0
$622$	−5.99969	−0.240566
$623$	−17.5941	−0.704893
$624$	0	0
$625$	20.6868	0.827473
$626$	−28.6065	−1.14335
$627$	0	0
$628$	−26.6531	−1.06357
$629$	73.6595	2.93700
$630$	0	0
$631$	3.86585	0.153897	0.0769485	−	0.997035i	$-0.475482\pi$
$631$	3.86585	0.153897	0.0769485	+	0.997035i	$0.475482\pi$
$632$	−20.5291	−0.816603
$633$	0	0
$634$	−39.7827	−1.57997
$635$	−7.33754	−0.291182
$636$	0	0
$637$	29.9718	1.18753
$638$	8.07183	0.319567
$639$	0	0
$640$	6.68753	0.264348
$641$	2.36295	0.0933310	0.0466655	−	0.998911i	$-0.485141\pi$
$641$	2.36295	0.0933310	0.0466655	+	0.998911i	$0.485141\pi$
$642$	0	0
$643$	−0.885320	−0.0349136	−0.0174568	−	0.999848i	$-0.505557\pi$
$643$	−0.885320	−0.0349136	−0.0174568	+	0.999848i	$0.505557\pi$
$644$	2.58829	0.101993
$645$	0	0
$646$	16.9898	0.668456
$647$	7.00425	0.275366	0.137683	−	0.990476i	$-0.456035\pi$
$647$	7.00425	0.275366	0.137683	+	0.990476i	$0.456035\pi$
$648$	0	0
$649$	21.4279	0.841118
$650$	−50.2858	−1.97237
$651$	0	0
$652$	19.3481	0.757730
$653$	−35.3399	−1.38296	−0.691480	−	0.722396i	$-0.743041\pi$
$653$	−35.3399	−1.38296	−0.691480	+	0.722396i	$0.743041\pi$
$654$	0	0
$655$	−2.74617	−0.107302
$656$	1.83923	0.0718100
$657$	0	0
$658$	−13.9877	−0.545299
$659$	−14.6525	−0.570779	−0.285390	−	0.958412i	$-0.592123\pi$
$659$	−14.6525	−0.570779	−0.285390	+	0.958412i	$0.592123\pi$
$660$	0	0
$661$	−1.85751	−0.0722486	−0.0361243	−	0.999347i	$-0.511501\pi$
$661$	−1.85751	−0.0722486	−0.0361243	+	0.999347i	$0.511501\pi$
$662$	−34.3323	−1.33436
$663$	0	0
$664$	−5.38361	−0.208925
$665$	−0.555709	−0.0215495
$666$	0	0
$667$	1.00000	0.0387202
$668$	9.07813	0.351244
$669$	0	0
$670$	−15.1197	−0.584126
$671$	11.5825	0.447136
$672$	0	0
$673$	−1.59371	−0.0614330	−0.0307165	−	0.999528i	$-0.509779\pi$
$673$	−1.59371	−0.0614330	−0.0307165	+	0.999528i	$0.509779\pi$
$674$	57.3583	2.20936
$675$	0	0
$676$	30.2962	1.16524
$677$	−10.4718	−0.402466	−0.201233	−	0.979543i	$-0.564495\pi$
$677$	−10.4718	−0.402466	−0.201233	+	0.979543i	$0.564495\pi$
$678$	0	0
$679$	−12.2676	−0.470786
$680$	−6.40748	−0.245716
$681$	0	0
$682$	−47.7383	−1.82799
$683$	−14.6358	−0.560025	−0.280013	−	0.959996i	$-0.590339\pi$
$683$	−14.6358	−0.560025	−0.280013	+	0.959996i	$0.590339\pi$
$684$	0	0
$685$	−7.80153	−0.298081
$686$	26.8292	1.02434
$687$	0	0
$688$	3.68533	0.140502
$689$	−51.8958	−1.97707
$690$	0	0
$691$	−6.81240	−0.259156	−0.129578	−	0.991569i	$-0.541362\pi$
$691$	−6.81240	−0.259156	−0.129578	+	0.991569i	$0.541362\pi$
$692$	36.8932	1.40247
$693$	0	0
$694$	−18.7879	−0.713178
$695$	−6.08657	−0.230877
$696$	0	0
$697$	−6.90706	−0.261624
$698$	−17.8828	−0.676875
$699$	0	0
$700$	−12.1823	−0.460449
$701$	−24.0612	−0.908781	−0.454390	−	0.890803i	$-0.650143\pi$
$701$	−24.0612	−0.908781	−0.454390	+	0.890803i	$0.650143\pi$
$702$	0	0
$703$	−11.3515	−0.428128
$704$	46.2136	1.74174
$705$	0	0
$706$	−25.4949	−0.959515
$707$	−9.40876	−0.353853
$708$	0	0
$709$	7.05658	0.265016	0.132508	−	0.991182i	$-0.457697\pi$
$709$	7.05658	0.265016	0.132508	+	0.991182i	$0.457697\pi$
$710$	−13.7163	−0.514765
$711$	0	0
$712$	31.2629	1.17163
$713$	−5.91418	−0.221488
$714$	0	0
$715$	9.80570	0.366712
$716$	−59.6864	−2.23059
$717$	0	0
$718$	−6.04438	−0.225574
$719$	−1.04045	−0.0388023	−0.0194011	−	0.999812i	$-0.506176\pi$
$719$	−1.04045	−0.0388023	−0.0194011	+	0.999812i	$0.506176\pi$
$720$	0	0
$721$	−18.7630	−0.698770
$722$	38.8470	1.44573
$723$	0	0
$724$	41.2029	1.53129
$725$	−4.70672	−0.174803
$726$	0	0
$727$	−33.9812	−1.26029	−0.630146	−	0.776477i	$-0.717005\pi$
$727$	−33.9812	−1.26029	−0.630146	+	0.776477i	$0.717005\pi$
$728$	7.63469	0.282960
$729$	0	0
$730$	−10.7437	−0.397640
$731$	−13.8399	−0.511888
$732$	0	0
$733$	4.19306	0.154874	0.0774370	−	0.996997i	$-0.475326\pi$
$733$	4.19306	0.154874	0.0774370	+	0.996997i	$0.475326\pi$
$734$	42.2839	1.56073
$735$	0	0
$736$	7.45974	0.274970
$737$	−47.3168	−1.74294
$738$	0	0
$739$	6.50021	0.239114	0.119557	−	0.992827i	$-0.461853\pi$
$739$	6.50021	0.239114	0.119557	+	0.992827i	$0.461853\pi$
$740$	15.5059	0.570009
$741$	0	0
$742$	−21.6736	−0.795664
$743$	32.0119	1.17440	0.587201	−	0.809441i	$-0.300230\pi$
$743$	32.0119	1.17440	0.587201	+	0.809441i	$0.300230\pi$
$744$	0	0
$745$	−4.81453	−0.176391
$746$	17.3001	0.633401
$747$	0	0
$748$	−72.6283	−2.65555
$749$	−5.16952	−0.188890
$750$	0	0
$751$	−42.8138	−1.56230	−0.781150	−	0.624344i	$-0.785366\pi$
$751$	−42.8138	−1.56230	−0.781150	+	0.624344i	$0.785366\pi$
$752$	−12.9483	−0.472176
$753$	0	0
$754$	10.6838	0.389082
$755$	4.83923	0.176118
$756$	0	0
$757$	−21.0059	−0.763474	−0.381737	−	0.924271i	$-0.624674\pi$
$757$	−21.0059	−0.763474	−0.381737	+	0.924271i	$0.624674\pi$
$758$	49.3926	1.79402
$759$	0	0
$760$	0.987438	0.0358181
$761$	42.5421	1.54215	0.771075	−	0.636744i	$-0.219719\pi$
$761$	42.5421	1.54215	0.771075	+	0.636744i	$0.219719\pi$
$762$	0	0
$763$	−2.55689	−0.0925656
$764$	63.0628	2.28153
$765$	0	0
$766$	−2.83087	−0.102284
$767$	28.3618	1.02409
$768$	0	0
$769$	−7.39163	−0.266549	−0.133274	−	0.991079i	$-0.542549\pi$
$769$	−7.39163	−0.266549	−0.133274	+	0.991079i	$0.542549\pi$
$770$	4.09523	0.147582
$771$	0	0
$772$	−62.0792	−2.23428
$773$	−44.3347	−1.59461	−0.797304	−	0.603578i	$-0.793741\pi$
$773$	−44.3347	−1.59461	−0.797304	+	0.603578i	$0.793741\pi$
$774$	0	0
$775$	27.8364	0.999914
$776$	21.7982	0.782511
$777$	0	0
$778$	28.3481	1.01633
$779$	1.06443	0.0381370
$780$	0	0
$781$	−42.9249	−1.53597
$782$	−15.5113	−0.554683
$783$	0	0
$784$	11.5872	0.413828
$785$	−5.22447	−0.186469
$786$	0	0
$787$	−6.28909	−0.224182	−0.112091	−	0.993698i	$-0.535755\pi$
$787$	−6.28909	−0.224182	−0.112091	+	0.993698i	$0.535755\pi$
$788$	−8.89515	−0.316876
$789$	0	0
$790$	−14.5751	−0.518559
$791$	10.0398	0.356974
$792$	0	0
$793$	15.3305	0.544402
$794$	53.0345	1.88212
$795$	0	0
$796$	11.6796	0.413971
$797$	14.1840	0.502422	0.251211	−	0.967932i	$-0.419171\pi$
$797$	14.1840	0.502422	0.251211	+	0.967932i	$0.419171\pi$
$798$	0	0
$799$	48.6261	1.72027
$800$	−35.1109	−1.24136
$801$	0	0
$802$	−1.03356	−0.0364961
$803$	−33.6220	−1.18649
$804$	0	0
$805$	0.507349	0.0178817
$806$	−63.1861	−2.22564
$807$	0	0
$808$	16.7184	0.588152
$809$	−51.7618	−1.81985	−0.909923	−	0.414777i	$-0.863860\pi$
$809$	−51.7618	−1.81985	−0.909923	+	0.414777i	$0.863860\pi$
$810$	0	0
$811$	−50.5755	−1.77595	−0.887973	−	0.459896i	$-0.847887\pi$
$811$	−50.5755	−1.77595	−0.887973	+	0.459896i	$0.847887\pi$
$812$	2.58829	0.0908310
$813$	0	0
$814$	83.6532	2.93204
$815$	3.79256	0.132848
$816$	0	0
$817$	2.13283	0.0746182
$818$	33.6160	1.17536
$819$	0	0
$820$	−1.45399	−0.0507756
$821$	−19.6402	−0.685447	−0.342724	−	0.939436i	$-0.611349\pi$
$821$	−19.6402	−0.685447	−0.342724	+	0.939436i	$0.611349\pi$
$822$	0	0
$823$	27.8234	0.969864	0.484932	−	0.874552i	$-0.338844\pi$
$823$	27.8234	0.969864	0.484932	+	0.874552i	$0.338844\pi$
$824$	33.3399	1.16145
$825$	0	0
$826$	11.8450	0.412139
$827$	39.4184	1.37071	0.685355	−	0.728209i	$-0.259647\pi$
$827$	39.4184	1.37071	0.685355	+	0.728209i	$0.259647\pi$
$828$	0	0
$829$	−21.9728	−0.763147	−0.381573	−	0.924339i	$-0.624618\pi$
$829$	−21.9728	−0.763147	−0.381573	+	0.924339i	$0.624618\pi$
$830$	−3.82222	−0.132671
$831$	0	0
$832$	61.1680	2.12062
$833$	−43.5146	−1.50769
$834$	0	0
$835$	1.77947	0.0615811
$836$	11.1925	0.387102
$837$	0	0
$838$	15.7981	0.545737
$839$	−27.8051	−0.959937	−0.479969	−	0.877286i	$-0.659352\pi$
$839$	−27.8051	−0.959937	−0.479969	+	0.877286i	$0.659352\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−40.9470	−1.41113
$843$	0	0
$844$	43.7654	1.50647
$845$	5.93858	0.204293
$846$	0	0
$847$	2.51065	0.0862670
$848$	−20.0631	−0.688968
$849$	0	0
$850$	73.0074	2.50413
$851$	10.3636	0.355260
$852$	0	0
$853$	−40.5376	−1.38798	−0.693990	−	0.719984i	$-0.744149\pi$
$853$	−40.5376	−1.38798	−0.693990	+	0.719984i	$0.744149\pi$
$854$	6.40260	0.219092
$855$	0	0
$856$	9.18572	0.313961
$857$	44.5279	1.52104	0.760522	−	0.649312i	$-0.224943\pi$
$857$	44.5279	1.52104	0.760522	+	0.649312i	$0.224943\pi$
$858$	0	0
$859$	50.4132	1.72008	0.860038	−	0.510230i	$-0.170440\pi$
$859$	50.4132	1.72008	0.860038	+	0.510230i	$0.170440\pi$
$860$	−2.91341	−0.0993465
$861$	0	0
$862$	−77.2344	−2.63061
$863$	19.2717	0.656016	0.328008	−	0.944675i	$-0.393623\pi$
$863$	19.2717	0.656016	0.328008	+	0.944675i	$0.393623\pi$
$864$	0	0
$865$	7.23171	0.245886
$866$	74.5702	2.53400
$867$	0	0
$868$	−15.3076	−0.519574
$869$	−45.6124	−1.54729
$870$	0	0
$871$	−62.6282	−2.12208
$872$	4.54333	0.153857
$873$	0	0
$874$	2.39040	0.0808565
$875$	−4.92469	−0.166485
$876$	0	0
$877$	−18.3077	−0.618208	−0.309104	−	0.951028i	$-0.600029\pi$
$877$	−18.3077	−0.618208	−0.309104	+	0.951028i	$0.600029\pi$
$878$	52.3010	1.76507
$879$	0	0
$880$	3.79091	0.127792
$881$	3.61086	0.121653	0.0608264	−	0.998148i	$-0.480626\pi$
$881$	3.61086	0.121653	0.0608264	+	0.998148i	$0.480626\pi$
$882$	0	0
$883$	−10.7008	−0.360111	−0.180056	−	0.983656i	$-0.557628\pi$
$883$	−10.7008	−0.360111	−0.180056	+	0.983656i	$0.557628\pi$
$884$	−96.1304	−3.23322
$885$	0	0
$886$	−56.1724	−1.88715
$887$	55.8629	1.87569	0.937846	−	0.347053i	$-0.112818\pi$
$887$	55.8629	1.87569	0.937846	+	0.347053i	$0.112818\pi$
$888$	0	0
$889$	−12.6934	−0.425721
$890$	22.1958	0.744006
$891$	0	0
$892$	14.1953	0.475295
$893$	−7.49362	−0.250765
$894$	0	0
$895$	−11.6996	−0.391074
$896$	11.5689	0.386489
$897$	0	0
$898$	39.9557	1.33334
$899$	−5.91418	−0.197249
$900$	0	0
$901$	75.3449	2.51010
$902$	−7.84417	−0.261182
$903$	0	0
$904$	−17.8397	−0.593339
$905$	8.07648	0.268471
$906$	0	0
$907$	−28.0825	−0.932465	−0.466233	−	0.884662i	$-0.654389\pi$
$907$	−28.0825	−0.932465	−0.466233	+	0.884662i	$0.654389\pi$
$908$	−64.0404	−2.12526
$909$	0	0
$910$	5.42042	0.179685
$911$	−19.0498	−0.631147	−0.315574	−	0.948901i	$-0.602197\pi$
$911$	−19.0498	−0.631147	−0.315574	+	0.948901i	$0.602197\pi$
$912$	0	0
$913$	−11.9615	−0.395869
$914$	−31.9521	−1.05688
$915$	0	0
$916$	−76.4714	−2.52669
$917$	−4.75066	−0.156881
$918$	0	0
$919$	34.7771	1.14719	0.573595	−	0.819139i	$-0.305549\pi$
$919$	34.7771	1.14719	0.573595	+	0.819139i	$0.305549\pi$
$920$	−0.901507	−0.0297218
$921$	0	0
$922$	59.0901	1.94603
$923$	−56.8152	−1.87009
$924$	0	0
$925$	−48.7786	−1.60383
$926$	−74.0482	−2.43337
$927$	0	0
$928$	7.45974	0.244878
$929$	−19.4976	−0.639694	−0.319847	−	0.947469i	$-0.603632\pi$
$929$	−19.4976	−0.639694	−0.319847	+	0.947469i	$0.603632\pi$
$930$	0	0
$931$	6.70590	0.219777
$932$	−15.9978	−0.524025
$933$	0	0
$934$	59.6321	1.95122
$935$	−14.2364	−0.465580
$936$	0	0
$937$	−32.4024	−1.05854	−0.529270	−	0.848454i	$-0.677534\pi$
$937$	−32.4024	−1.05854	−0.529270	+	0.848454i	$0.677534\pi$
$938$	−26.1559	−0.854021
$939$	0	0
$940$	10.2362	0.333867
$941$	9.15194	0.298345	0.149172	−	0.988811i	$-0.452339\pi$
$941$	9.15194	0.298345	0.149172	+	0.988811i	$0.452339\pi$
$942$	0	0
$943$	−0.971796	−0.0316460
$944$	10.9648	0.356873
$945$	0	0
$946$	−15.7176	−0.511024
$947$	−42.1843	−1.37081	−0.685403	−	0.728164i	$-0.740374\pi$
$947$	−42.1843	−1.37081	−0.685403	+	0.728164i	$0.740374\pi$
$948$	0	0
$949$	−44.5018	−1.44459
$950$	−11.2510	−0.365029
$951$	0	0
$952$	−11.0844	−0.359248
$953$	29.2461	0.947376	0.473688	−	0.880693i	$-0.342923\pi$
$953$	29.2461	0.947376	0.473688	+	0.880693i	$0.342923\pi$
$954$	0	0
$955$	12.3614	0.400005
$956$	64.8353	2.09692
$957$	0	0
$958$	24.8918	0.804219
$959$	−13.4960	−0.435809
$960$	0	0
$961$	3.97758	0.128309
$962$	110.723	3.56985
$963$	0	0
$964$	−8.37866	−0.269859
$965$	−12.1686	−0.391721
$966$	0	0
$967$	2.39429	0.0769952	0.0384976	−	0.999259i	$-0.487743\pi$
$967$	2.39429	0.0769952	0.0384976	+	0.999259i	$0.487743\pi$
$968$	−4.46117	−0.143388
$969$	0	0
$970$	15.4762	0.496909
$971$	−16.9781	−0.544853	−0.272427	−	0.962177i	$-0.587826\pi$
$971$	−16.9781	−0.544853	−0.272427	+	0.962177i	$0.587826\pi$
$972$	0	0
$973$	−10.5293	−0.337553
$974$	−33.8509	−1.08465
$975$	0	0
$976$	5.92681	0.189713
$977$	46.4893	1.48733	0.743663	−	0.668555i	$-0.233087\pi$
$977$	46.4893	1.48733	0.743663	+	0.668555i	$0.233087\pi$
$978$	0	0
$979$	69.4613	2.21999
$980$	−9.16016	−0.292611
$981$	0	0
$982$	3.93070	0.125434
$983$	−49.4911	−1.57852	−0.789260	−	0.614059i	$-0.789536\pi$
$983$	−49.4911	−1.57852	−0.789260	+	0.614059i	$0.789536\pi$
$984$	0	0
$985$	−1.74360	−0.0555558
$986$	−15.5113	−0.493981
$987$	0	0
$988$	14.8144	0.471308
$989$	−1.94722	−0.0619180
$990$	0	0
$991$	21.7522	0.690983	0.345491	−	0.938422i	$-0.387712\pi$
$991$	21.7522	0.690983	0.345491	+	0.938422i	$0.387712\pi$
$992$	−44.1183	−1.40076
$993$	0	0
$994$	−23.7282	−0.752612
$995$	2.28940	0.0725787
$996$	0	0
$997$	−13.8349	−0.438157	−0.219079	−	0.975707i	$-0.570305\pi$
$997$	−13.8349	−0.438157	−0.219079	+	0.975707i	$0.570305\pi$
$998$	90.5706	2.86696
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.4	✓	22
3.2	odd	2		6003.2.a.u.1.19	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.4	✓	22	1.1	even	1	trivial
6003.2.a.u.1.19	yes	22	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-2.18238 q^{2} +2.76278 q^{4} +0.541552 q^{5} +0.936842 q^{7} -1.66467 q^{8} +O(q^{10})\)	\(q-2.18238 q^{2} +2.76278 q^{4} +0.541552 q^{5} +0.936842 q^{7} -1.66467 q^{8} -1.18187 q^{10} -3.69864 q^{11} -4.89549 q^{13} -2.04454 q^{14} -1.89261 q^{16} +7.10752 q^{17} -1.09532 q^{19} +1.49619 q^{20} +8.07183 q^{22} +1.00000 q^{23} -4.70672 q^{25} +10.6838 q^{26} +2.58829 q^{28} +1.00000 q^{29} -5.91418 q^{31} +7.45974 q^{32} -15.5113 q^{34} +0.507349 q^{35} +10.3636 q^{37} +2.39040 q^{38} -0.901507 q^{40} -0.971796 q^{41} -1.94722 q^{43} -10.2185 q^{44} -2.18238 q^{46} +6.84150 q^{47} -6.12233 q^{49} +10.2718 q^{50} -13.5252 q^{52} +10.6007 q^{53} -2.00300 q^{55} -1.55953 q^{56} -2.18238 q^{58} -5.79346 q^{59} -3.13155 q^{61} +12.9070 q^{62} -12.4948 q^{64} -2.65117 q^{65} +12.7930 q^{67} +19.6365 q^{68} -1.10723 q^{70} +11.6056 q^{71} +9.09037 q^{73} -22.6173 q^{74} -3.02612 q^{76} -3.46504 q^{77} +12.3322 q^{79} -1.02495 q^{80} +2.12083 q^{82} +3.23404 q^{83} +3.84909 q^{85} +4.24957 q^{86} +6.15702 q^{88} -18.7802 q^{89} -4.58630 q^{91} +2.76278 q^{92} -14.9307 q^{94} -0.593172 q^{95} -13.0946 q^{97} +13.3612 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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