LMFDB - Embedded newform 6003.2.a.v.1.11

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			1.11
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)

$f(q)$	$=$	$q-0.959410 q^{2} -1.07953 q^{4} +2.76105 q^{5} +2.95320 q^{7} +2.95453 q^{8} +O(q^{10})$	\(q-0.959410 q^{2} -1.07953 q^{4} +2.76105 q^{5} +2.95320 q^{7} +2.95453 q^{8} -2.64898 q^{10} -1.28197 q^{11} +5.35813 q^{13} -2.83333 q^{14} -0.675546 q^{16} +5.11908 q^{17} -7.15882 q^{19} -2.98064 q^{20} +1.22993 q^{22} +1.00000 q^{23} +2.62340 q^{25} -5.14064 q^{26} -3.18807 q^{28} -1.00000 q^{29} +6.82622 q^{31} -5.26094 q^{32} -4.91130 q^{34} +8.15392 q^{35} +1.42848 q^{37} +6.86825 q^{38} +8.15762 q^{40} +0.739636 q^{41} +8.76755 q^{43} +1.38393 q^{44} -0.959410 q^{46} -4.04273 q^{47} +1.72136 q^{49} -2.51691 q^{50} -5.78427 q^{52} +11.3634 q^{53} -3.53958 q^{55} +8.72532 q^{56} +0.959410 q^{58} +1.57727 q^{59} -10.5564 q^{61} -6.54915 q^{62} +6.39849 q^{64} +14.7941 q^{65} +8.17702 q^{67} -5.52622 q^{68} -7.82295 q^{70} +8.05660 q^{71} -15.1877 q^{73} -1.37050 q^{74} +7.72818 q^{76} -3.78591 q^{77} +11.2549 q^{79} -1.86521 q^{80} -0.709614 q^{82} +10.6759 q^{83} +14.1340 q^{85} -8.41168 q^{86} -3.78762 q^{88} -4.10964 q^{89} +15.8236 q^{91} -1.07953 q^{92} +3.87864 q^{94} -19.7659 q^{95} +18.6565 q^{97} -1.65149 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.959410	−0.678405	−0.339203	−	0.940713i	$-0.610157\pi$
$2$	−0.959410	−0.678405	−0.339203	+	0.940713i	$0.610157\pi$
$3$	0	0
$3$	0	0
$4$	−1.07953	−0.539766
$5$	2.76105	1.23478	0.617389	−	0.786658i	$-0.288190\pi$
$5$	2.76105	1.23478	0.617389	+	0.786658i	$0.288190\pi$
$6$	0	0
$7$	2.95320	1.11620	0.558101	−	0.829773i	$-0.311530\pi$
$7$	2.95320	1.11620	0.558101	+	0.829773i	$0.311530\pi$
$8$	2.95453	1.04459
$9$	0	0
$10$	−2.64898	−0.837681
$11$	−1.28197	−0.386528	−0.193264	−	0.981147i	$-0.561907\pi$
$11$	−1.28197	−0.386528	−0.193264	+	0.981147i	$0.561907\pi$
$12$	0	0
$13$	5.35813	1.48608	0.743039	−	0.669248i	$-0.233384\pi$
$13$	5.35813	1.48608	0.743039	+	0.669248i	$0.233384\pi$
$14$	−2.83333	−0.757238
$15$	0	0
$16$	−0.675546	−0.168886
$17$	5.11908	1.24156	0.620780	−	0.783985i	$-0.286816\pi$
$17$	5.11908	1.24156	0.620780	+	0.783985i	$0.286816\pi$
$18$	0	0
$19$	−7.15882	−1.64235	−0.821173	−	0.570679i	$-0.806680\pi$
$19$	−7.15882	−1.64235	−0.821173	+	0.570679i	$0.806680\pi$
$20$	−2.98064	−0.666492
$21$	0	0
$22$	1.22993	0.262223
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	2.62340	0.524679
$26$	−5.14064	−1.00816
$27$	0	0
$28$	−3.18807	−0.602489
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	6.82622	1.22603	0.613013	−	0.790073i	$-0.289957\pi$
$31$	6.82622	1.22603	0.613013	+	0.790073i	$0.289957\pi$
$32$	−5.26094	−0.930012
$33$	0	0
$34$	−4.91130	−0.842281
$35$	8.15392	1.37826
$36$	0	0
$37$	1.42848	0.234840	0.117420	−	0.993082i	$-0.462538\pi$
$37$	1.42848	0.234840	0.117420	+	0.993082i	$0.462538\pi$
$38$	6.86825	1.11418
$39$	0	0
$40$	8.15762	1.28983
$41$	0.739636	0.115512	0.0577559	−	0.998331i	$-0.481606\pi$
$41$	0.739636	0.115512	0.0577559	+	0.998331i	$0.481606\pi$
$42$	0	0
$43$	8.76755	1.33704	0.668519	−	0.743695i	$-0.266928\pi$
$43$	8.76755	1.33704	0.668519	+	0.743695i	$0.266928\pi$
$44$	1.38393	0.208635
$45$	0	0
$46$	−0.959410	−0.141457
$47$	−4.04273	−0.589693	−0.294847	−	0.955545i	$-0.595269\pi$
$47$	−4.04273	−0.589693	−0.294847	+	0.955545i	$0.595269\pi$
$48$	0	0
$49$	1.72136	0.245909
$50$	−2.51691	−0.355945
$51$	0	0
$52$	−5.78427	−0.802134
$53$	11.3634	1.56088	0.780440	−	0.625231i	$-0.214995\pi$
$53$	11.3634	1.56088	0.780440	+	0.625231i	$0.214995\pi$
$54$	0	0
$55$	−3.53958	−0.477277
$56$	8.72532	1.16597
$57$	0	0
$58$	0.959410	0.125977
$59$	1.57727	0.205343	0.102672	−	0.994715i	$-0.467261\pi$
$59$	1.57727	0.205343	0.102672	+	0.994715i	$0.467261\pi$
$60$	0	0
$61$	−10.5564	−1.35161	−0.675803	−	0.737083i	$-0.736203\pi$
$61$	−10.5564	−1.35161	−0.675803	+	0.737083i	$0.736203\pi$
$62$	−6.54915	−0.831743
$63$	0	0
$64$	6.39849	0.799812
$65$	14.7941	1.83498
$66$	0	0
$67$	8.17702	0.998982	0.499491	−	0.866319i	$-0.333520\pi$
$67$	8.17702	0.998982	0.499491	+	0.866319i	$0.333520\pi$
$68$	−5.52622	−0.670152
$69$	0	0
$70$	−7.82295	−0.935022
$71$	8.05660	0.956142	0.478071	−	0.878321i	$-0.341336\pi$
$71$	8.05660	0.956142	0.478071	+	0.878321i	$0.341336\pi$
$72$	0	0
$73$	−15.1877	−1.77759	−0.888794	−	0.458306i	$-0.848456\pi$
$73$	−15.1877	−1.77759	−0.888794	+	0.458306i	$0.848456\pi$
$74$	−1.37050	−0.159317
$75$	0	0
$76$	7.72818	0.886483
$77$	−3.78591	−0.431444
$78$	0	0
$79$	11.2549	1.26627	0.633136	−	0.774040i	$-0.281767\pi$
$79$	11.2549	1.26627	0.633136	+	0.774040i	$0.281767\pi$
$80$	−1.86521	−0.208537
$81$	0	0
$82$	−0.709614	−0.0783638
$83$	10.6759	1.17183	0.585917	−	0.810371i	$-0.300735\pi$
$83$	10.6759	1.17183	0.585917	+	0.810371i	$0.300735\pi$
$84$	0	0
$85$	14.1340	1.53305
$86$	−8.41168	−0.907054
$87$	0	0
$88$	−3.78762	−0.403762
$89$	−4.10964	−0.435621	−0.217811	−	0.975991i	$-0.569891\pi$
$89$	−4.10964	−0.435621	−0.217811	+	0.975991i	$0.569891\pi$
$90$	0	0
$91$	15.8236	1.65876
$92$	−1.07953	−0.112549
$93$	0	0
$94$	3.87864	0.400051
$95$	−19.7659	−2.02794
$96$	0	0
$97$	18.6565	1.89428	0.947140	−	0.320820i	$-0.103959\pi$
$97$	18.6565	1.89428	0.947140	+	0.320820i	$0.103959\pi$
$98$	−1.65149	−0.166826
$99$	0	0
$100$	−2.83204	−0.283204
$101$	−17.6859	−1.75981	−0.879906	−	0.475148i	$-0.842394\pi$
$101$	−17.6859	−1.75981	−0.879906	+	0.475148i	$0.842394\pi$
$102$	0	0
$103$	9.81993	0.967587	0.483793	−	0.875182i	$-0.339259\pi$
$103$	9.81993	0.967587	0.483793	+	0.875182i	$0.339259\pi$
$104$	15.8308	1.55234
$105$	0	0
$106$	−10.9021	−1.05891
$107$	−19.9704	−1.93061	−0.965306	−	0.261121i	$-0.915908\pi$
$107$	−19.9704	−1.93061	−0.965306	+	0.261121i	$0.915908\pi$
$108$	0	0
$109$	10.5223	1.00785	0.503924	−	0.863748i	$-0.331889\pi$
$109$	10.5223	1.00785	0.503924	+	0.863748i	$0.331889\pi$
$110$	3.39591	0.323787
$111$	0	0
$112$	−1.99502	−0.188511
$113$	−6.11131	−0.574903	−0.287452	−	0.957795i	$-0.592808\pi$
$113$	−6.11131	−0.574903	−0.287452	+	0.957795i	$0.592808\pi$
$114$	0	0
$115$	2.76105	0.257469
$116$	1.07953	0.100232
$117$	0	0
$118$	−1.51325	−0.139306
$119$	15.1177	1.38583
$120$	0	0
$121$	−9.35655	−0.850596
$122$	10.1279	0.916936
$123$	0	0
$124$	−7.36913	−0.661767
$125$	−6.56192	−0.586916
$126$	0	0
$127$	−18.2628	−1.62056	−0.810279	−	0.586045i	$-0.800684\pi$
$127$	−18.2628	−1.62056	−0.810279	+	0.586045i	$0.800684\pi$
$128$	4.38311	0.387416
$129$	0	0
$130$	−14.1936	−1.24486
$131$	−10.2408	−0.894739	−0.447369	−	0.894349i	$-0.647639\pi$
$131$	−10.2408	−0.894739	−0.447369	+	0.894349i	$0.647639\pi$
$132$	0	0
$133$	−21.1414	−1.83319
$134$	−7.84512	−0.677715
$135$	0	0
$136$	15.1245	1.29692
$137$	1.12502	0.0961170	0.0480585	−	0.998845i	$-0.484697\pi$
$137$	1.12502	0.0961170	0.0480585	+	0.998845i	$0.484697\pi$
$138$	0	0
$139$	0.321885	0.0273020	0.0136510	−	0.999907i	$-0.495655\pi$
$139$	0.321885	0.0273020	0.0136510	+	0.999907i	$0.495655\pi$
$140$	−8.80242	−0.743940
$141$	0	0
$142$	−7.72958	−0.648652
$143$	−6.86896	−0.574411
$144$	0	0
$145$	−2.76105	−0.229293
$146$	14.5713	1.20593
$147$	0	0
$148$	−1.54209	−0.126759
$149$	12.8633	1.05380	0.526902	−	0.849926i	$-0.323354\pi$
$149$	12.8633	1.05380	0.526902	+	0.849926i	$0.323354\pi$
$150$	0	0
$151$	2.52063	0.205126	0.102563	−	0.994727i	$-0.467296\pi$
$151$	2.52063	0.205126	0.102563	+	0.994727i	$0.467296\pi$
$152$	−21.1510	−1.71557
$153$	0	0
$154$	3.63224	0.292694
$155$	18.8475	1.51387
$156$	0	0
$157$	17.1186	1.36621	0.683105	−	0.730320i	$-0.260629\pi$
$157$	17.1186	1.36621	0.683105	+	0.730320i	$0.260629\pi$
$158$	−10.7980	−0.859046
$159$	0	0
$160$	−14.5257	−1.14836
$161$	2.95320	0.232744
$162$	0	0
$163$	−1.88078	−0.147314	−0.0736570	−	0.997284i	$-0.523467\pi$
$163$	−1.88078	−0.147314	−0.0736570	+	0.997284i	$0.523467\pi$
$164$	−0.798461	−0.0623493
$165$	0	0
$166$	−10.2426	−0.794978
$167$	−7.68009	−0.594303	−0.297152	−	0.954830i	$-0.596037\pi$
$167$	−7.68009	−0.594303	−0.297152	+	0.954830i	$0.596037\pi$
$168$	0	0
$169$	15.7095	1.20843
$170$	−13.5603	−1.04003
$171$	0	0
$172$	−9.46485	−0.721688
$173$	−11.1878	−0.850590	−0.425295	−	0.905055i	$-0.639830\pi$
$173$	−11.1878	−0.850590	−0.425295	+	0.905055i	$0.639830\pi$
$174$	0	0
$175$	7.74740	0.585648
$176$	0.866029	0.0652794
$177$	0	0
$178$	3.94283	0.295528
$179$	−13.0249	−0.973530	−0.486765	−	0.873533i	$-0.661823\pi$
$179$	−13.0249	−0.973530	−0.486765	+	0.873533i	$0.661823\pi$
$180$	0	0
$181$	4.55216	0.338360	0.169180	−	0.985585i	$-0.445888\pi$
$181$	4.55216	0.338360	0.169180	+	0.985585i	$0.445888\pi$
$182$	−15.1813	−1.12531
$183$	0	0
$184$	2.95453	0.217811
$185$	3.94410	0.289976
$186$	0	0
$187$	−6.56251	−0.479898
$188$	4.36426	0.318296
$189$	0	0
$190$	18.9636	1.37576
$191$	−10.6762	−0.772506	−0.386253	−	0.922393i	$-0.626231\pi$
$191$	−10.6762	−0.772506	−0.386253	+	0.922393i	$0.626231\pi$
$192$	0	0
$193$	12.9053	0.928944	0.464472	−	0.885588i	$-0.346244\pi$
$193$	12.9053	0.928944	0.464472	+	0.885588i	$0.346244\pi$
$194$	−17.8992	−1.28509
$195$	0	0
$196$	−1.85827	−0.132733
$197$	14.1547	1.00848	0.504239	−	0.863564i	$-0.331773\pi$
$197$	14.1547	1.00848	0.504239	+	0.863564i	$0.331773\pi$
$198$	0	0
$199$	15.0600	1.06758	0.533788	−	0.845619i	$-0.320768\pi$
$199$	15.0600	1.06758	0.533788	+	0.845619i	$0.320768\pi$
$200$	7.75091	0.548072
$201$	0	0
$202$	16.9680	1.19387
$203$	−2.95320	−0.207274
$204$	0	0
$205$	2.04217	0.142631
$206$	−9.42134	−0.656416
$207$	0	0
$208$	−3.61966	−0.250978
$209$	9.17739	0.634814
$210$	0	0
$211$	−20.8024	−1.43210	−0.716049	−	0.698050i	$-0.754051\pi$
$211$	−20.8024	−1.43210	−0.716049	+	0.698050i	$0.754051\pi$
$212$	−12.2671	−0.842510
$213$	0	0
$214$	19.1598	1.30974
$215$	24.2076	1.65095
$216$	0	0
$217$	20.1592	1.36849
$218$	−10.0952	−0.683730
$219$	0	0
$220$	3.82109	0.257618
$221$	27.4287	1.84506
$222$	0	0
$223$	−5.64221	−0.377830	−0.188915	−	0.981993i	$-0.560497\pi$
$223$	−5.64221	−0.377830	−0.188915	+	0.981993i	$0.560497\pi$
$224$	−15.5366	−1.03808
$225$	0	0
$226$	5.86325	0.390018
$227$	−18.0673	−1.19917	−0.599585	−	0.800311i	$-0.704668\pi$
$227$	−18.0673	−1.19917	−0.599585	+	0.800311i	$0.704668\pi$
$228$	0	0
$229$	27.9240	1.84527	0.922634	−	0.385676i	$-0.126032\pi$
$229$	27.9240	1.84527	0.922634	+	0.385676i	$0.126032\pi$
$230$	−2.64898	−0.174668
$231$	0	0
$232$	−2.95453	−0.193975
$233$	−9.37216	−0.613991	−0.306995	−	0.951711i	$-0.599324\pi$
$233$	−9.37216	−0.613991	−0.306995	+	0.951711i	$0.599324\pi$
$234$	0	0
$235$	−11.1622	−0.728141
$236$	−1.70272	−0.110837
$237$	0	0
$238$	−14.5040	−0.940157
$239$	4.28687	0.277295	0.138647	−	0.990342i	$-0.455725\pi$
$239$	4.28687	0.277295	0.138647	+	0.990342i	$0.455725\pi$
$240$	0	0
$241$	13.6885	0.881754	0.440877	−	0.897567i	$-0.354667\pi$
$241$	13.6885	0.881754	0.440877	+	0.897567i	$0.354667\pi$
$242$	8.97677	0.577049
$243$	0	0
$244$	11.3959	0.729551
$245$	4.75277	0.303643
$246$	0	0
$247$	−38.3579	−2.44065
$248$	20.1683	1.28069
$249$	0	0
$250$	6.29558	0.398167
$251$	−28.6639	−1.80925	−0.904624	−	0.426210i	$-0.859848\pi$
$251$	−28.6639	−1.80925	−0.904624	+	0.426210i	$0.859848\pi$
$252$	0	0
$253$	−1.28197	−0.0805967
$254$	17.5215	1.09939
$255$	0	0
$256$	−17.0022	−1.06264
$257$	−18.5325	−1.15602	−0.578012	−	0.816028i	$-0.696172\pi$
$257$	−18.5325	−1.15602	−0.578012	+	0.816028i	$0.696172\pi$
$258$	0	0
$259$	4.21857	0.262129
$260$	−15.9707	−0.990459
$261$	0	0
$262$	9.82508	0.606996
$263$	−23.4357	−1.44510	−0.722552	−	0.691316i	$-0.757031\pi$
$263$	−23.4357	−1.44510	−0.722552	+	0.691316i	$0.757031\pi$
$264$	0	0
$265$	31.3748	1.92734
$266$	20.2833	1.24365
$267$	0	0
$268$	−8.82736	−0.539217
$269$	25.8021	1.57318	0.786592	−	0.617474i	$-0.211844\pi$
$269$	25.8021	1.57318	0.786592	+	0.617474i	$0.211844\pi$
$270$	0	0
$271$	−5.69026	−0.345659	−0.172829	−	0.984952i	$-0.555291\pi$
$271$	−5.69026	−0.345659	−0.172829	+	0.984952i	$0.555291\pi$
$272$	−3.45818	−0.209683
$273$	0	0
$274$	−1.07936	−0.0652063
$275$	−3.36311	−0.202803
$276$	0	0
$277$	−13.7131	−0.823942	−0.411971	−	0.911197i	$-0.635160\pi$
$277$	−13.7131	−0.823942	−0.411971	+	0.911197i	$0.635160\pi$
$278$	−0.308820	−0.0185218
$279$	0	0
$280$	24.0910	1.43971
$281$	−20.8026	−1.24098	−0.620490	−	0.784214i	$-0.713066\pi$
$281$	−20.8026	−1.24098	−0.620490	+	0.784214i	$0.713066\pi$
$282$	0	0
$283$	21.0736	1.25269	0.626347	−	0.779544i	$-0.284549\pi$
$283$	21.0736	1.25269	0.626347	+	0.779544i	$0.284549\pi$
$284$	−8.69736	−0.516093
$285$	0	0
$286$	6.59015	0.389684
$287$	2.18429	0.128935
$288$	0	0
$289$	9.20503	0.541472
$290$	2.64898	0.155553
$291$	0	0
$292$	16.3956	0.959482
$293$	20.9524	1.22405	0.612025	−	0.790839i	$-0.290355\pi$
$293$	20.9524	1.22405	0.612025	+	0.790839i	$0.290355\pi$
$294$	0	0
$295$	4.35493	0.253554
$296$	4.22049	0.245311
$297$	0	0
$298$	−12.3412	−0.714906
$299$	5.35813	0.309869
$300$	0	0
$301$	25.8923	1.49241
$302$	−2.41832	−0.139159
$303$	0	0
$304$	4.83611	0.277370
$305$	−29.1467	−1.66893
$306$	0	0
$307$	11.2742	0.643455	0.321727	−	0.946832i	$-0.395737\pi$
$307$	11.2742	0.643455	0.321727	+	0.946832i	$0.395737\pi$
$308$	4.08701	0.232879
$309$	0	0
$310$	−18.0825	−1.02702
$311$	−14.6907	−0.833033	−0.416516	−	0.909128i	$-0.636749\pi$
$311$	−14.6907	−0.833033	−0.416516	+	0.909128i	$0.636749\pi$
$312$	0	0
$313$	16.5576	0.935889	0.467945	−	0.883758i	$-0.344995\pi$
$313$	16.5576	0.935889	0.467945	+	0.883758i	$0.344995\pi$
$314$	−16.4237	−0.926844
$315$	0	0
$316$	−12.1500	−0.683491
$317$	11.0272	0.619350	0.309675	−	0.950843i	$-0.399780\pi$
$317$	11.0272	0.619350	0.309675	+	0.950843i	$0.399780\pi$
$318$	0	0
$319$	1.28197	0.0717765
$320$	17.6666	0.987591
$321$	0	0
$322$	−2.83333	−0.157895
$323$	−36.6466	−2.03907
$324$	0	0
$325$	14.0565	0.779714
$326$	1.80444	0.0999387
$327$	0	0
$328$	2.18528	0.120662
$329$	−11.9390	−0.658217
$330$	0	0
$331$	−8.85210	−0.486555	−0.243278	−	0.969957i	$-0.578223\pi$
$331$	−8.85210	−0.486555	−0.243278	+	0.969957i	$0.578223\pi$
$332$	−11.5250	−0.632516
$333$	0	0
$334$	7.36835	0.403178
$335$	22.5772	1.23352
$336$	0	0
$337$	−22.3289	−1.21633	−0.608165	−	0.793810i	$-0.708094\pi$
$337$	−22.3289	−1.21633	−0.608165	+	0.793810i	$0.708094\pi$
$338$	−15.0719	−0.819803
$339$	0	0
$340$	−15.2582	−0.827490
$341$	−8.75101	−0.473894
$342$	0	0
$343$	−15.5888	−0.841719
$344$	25.9040	1.39665
$345$	0	0
$346$	10.7337	0.577045
$347$	−25.5751	−1.37294	−0.686471	−	0.727157i	$-0.740841\pi$
$347$	−25.5751	−1.37294	−0.686471	+	0.727157i	$0.740841\pi$
$348$	0	0
$349$	−0.279761	−0.0149753	−0.00748764	−	0.999972i	$-0.502383\pi$
$349$	−0.279761	−0.0149753	−0.00748764	+	0.999972i	$0.502383\pi$
$350$	−7.43293	−0.397307
$351$	0	0
$352$	6.74437	0.359476
$353$	7.86461	0.418591	0.209295	−	0.977852i	$-0.432883\pi$
$353$	7.86461	0.418591	0.209295	+	0.977852i	$0.432883\pi$
$354$	0	0
$355$	22.2447	1.18062
$356$	4.43649	0.235134
$357$	0	0
$358$	12.4963	0.660448
$359$	22.3968	1.18206	0.591029	−	0.806651i	$-0.298722\pi$
$359$	22.3968	1.18206	0.591029	+	0.806651i	$0.298722\pi$
$360$	0	0
$361$	32.2488	1.69730
$362$	−4.36739	−0.229545
$363$	0	0
$364$	−17.0821	−0.895345
$365$	−41.9341	−2.19493
$366$	0	0
$367$	7.70626	0.402264	0.201132	−	0.979564i	$-0.435538\pi$
$367$	7.70626	0.402264	0.201132	+	0.979564i	$0.435538\pi$
$368$	−0.675546	−0.0352152
$369$	0	0
$370$	−3.78401	−0.196721
$371$	33.5583	1.74226
$372$	0	0
$373$	−21.7661	−1.12701	−0.563503	−	0.826114i	$-0.690547\pi$
$373$	−21.7661	−1.12701	−0.563503	+	0.826114i	$0.690547\pi$
$374$	6.29614	0.325566
$375$	0	0
$376$	−11.9444	−0.615985
$377$	−5.35813	−0.275958
$378$	0	0
$379$	29.8556	1.53358	0.766791	−	0.641897i	$-0.221852\pi$
$379$	29.8556	1.53358	0.766791	+	0.641897i	$0.221852\pi$
$380$	21.3379	1.09461
$381$	0	0
$382$	10.2429	0.524072
$383$	−1.48267	−0.0757608	−0.0378804	−	0.999282i	$-0.512061\pi$
$383$	−1.48267	−0.0757608	−0.0378804	+	0.999282i	$0.512061\pi$
$384$	0	0
$385$	−10.4531	−0.532738
$386$	−12.3815	−0.630201
$387$	0	0
$388$	−20.1403	−1.02247
$389$	9.27829	0.470428	0.235214	−	0.971944i	$-0.424421\pi$
$389$	9.27829	0.470428	0.235214	+	0.971944i	$0.424421\pi$
$390$	0	0
$391$	5.11908	0.258883
$392$	5.08582	0.256873
$393$	0	0
$394$	−13.5801	−0.684156
$395$	31.0753	1.56357
$396$	0	0
$397$	4.87160	0.244499	0.122249	−	0.992499i	$-0.460989\pi$
$397$	4.87160	0.244499	0.122249	+	0.992499i	$0.460989\pi$
$398$	−14.4487	−0.724249
$399$	0	0
$400$	−1.77222	−0.0886112
$401$	28.7127	1.43384	0.716921	−	0.697154i	$-0.245551\pi$
$401$	28.7127	1.43384	0.716921	+	0.697154i	$0.245551\pi$
$402$	0	0
$403$	36.5758	1.82197
$404$	19.0925	0.949887
$405$	0	0
$406$	2.83333	0.140616
$407$	−1.83126	−0.0907724
$408$	0	0
$409$	17.6645	0.873455	0.436727	−	0.899594i	$-0.356137\pi$
$409$	17.6645	0.873455	0.436727	+	0.899594i	$0.356137\pi$
$410$	−1.95928	−0.0967619
$411$	0	0
$412$	−10.6009	−0.522271
$413$	4.65799	0.229205
$414$	0	0
$415$	29.4767	1.44695
$416$	−28.1888	−1.38207
$417$	0	0
$418$	−8.80488	−0.430661
$419$	−14.4940	−0.708077	−0.354039	−	0.935231i	$-0.615192\pi$
$419$	−14.4940	−0.708077	−0.354039	+	0.935231i	$0.615192\pi$
$420$	0	0
$421$	25.2467	1.23045	0.615224	−	0.788352i	$-0.289065\pi$
$421$	25.2467	1.23045	0.615224	+	0.788352i	$0.289065\pi$
$422$	19.9581	0.971543
$423$	0	0
$424$	33.5735	1.63047
$425$	13.4294	0.651421
$426$	0	0
$427$	−31.1750	−1.50867
$428$	21.5587	1.04208
$429$	0	0
$430$	−23.2251	−1.12001
$431$	11.7015	0.563639	0.281820	−	0.959467i	$-0.409062\pi$
$431$	11.7015	0.563639	0.281820	+	0.959467i	$0.409062\pi$
$432$	0	0
$433$	−17.3514	−0.833857	−0.416928	−	0.908939i	$-0.636893\pi$
$433$	−17.3514	−0.833857	−0.416928	+	0.908939i	$0.636893\pi$
$434$	−19.3409	−0.928394
$435$	0	0
$436$	−11.3591	−0.544003
$437$	−7.15882	−0.342453
$438$	0	0
$439$	29.9501	1.42944	0.714719	−	0.699411i	$-0.246554\pi$
$439$	29.9501	1.42944	0.714719	+	0.699411i	$0.246554\pi$
$440$	−10.4578	−0.498557
$441$	0	0
$442$	−26.3154	−1.25170
$443$	6.01086	0.285585	0.142792	−	0.989753i	$-0.454392\pi$
$443$	6.01086	0.285585	0.142792	+	0.989753i	$0.454392\pi$
$444$	0	0
$445$	−11.3469	−0.537896
$446$	5.41319	0.256322
$447$	0	0
$448$	18.8960	0.892752
$449$	1.07827	0.0508869	0.0254435	−	0.999676i	$-0.491900\pi$
$449$	1.07827	0.0508869	0.0254435	+	0.999676i	$0.491900\pi$
$450$	0	0
$451$	−0.948191	−0.0446486
$452$	6.59735	0.310313
$453$	0	0
$454$	17.3340	0.813524
$455$	43.6897	2.04821
$456$	0	0
$457$	16.7408	0.783100	0.391550	−	0.920157i	$-0.371939\pi$
$457$	16.7408	0.783100	0.391550	+	0.920157i	$0.371939\pi$
$458$	−26.7905	−1.25184
$459$	0	0
$460$	−2.98064	−0.138973
$461$	31.5762	1.47065	0.735326	−	0.677713i	$-0.237029\pi$
$461$	31.5762	1.47065	0.735326	+	0.677713i	$0.237029\pi$
$462$	0	0
$463$	−7.96330	−0.370086	−0.185043	−	0.982730i	$-0.559242\pi$
$463$	−7.96330	−0.370086	−0.185043	+	0.982730i	$0.559242\pi$
$464$	0.675546	0.0313614
$465$	0	0
$466$	8.99175	0.416535
$467$	26.5092	1.22670	0.613349	−	0.789812i	$-0.289822\pi$
$467$	26.5092	1.22670	0.613349	+	0.789812i	$0.289822\pi$
$468$	0	0
$469$	24.1483	1.11507
$470$	10.7091	0.493975
$471$	0	0
$472$	4.66010	0.214499
$473$	−11.2397	−0.516803
$474$	0	0
$475$	−18.7804	−0.861705
$476$	−16.3200	−0.748026
$477$	0	0
$478$	−4.11287	−0.188118
$479$	−35.2409	−1.61020	−0.805099	−	0.593140i	$-0.797888\pi$
$479$	−35.2409	−1.61020	−0.805099	+	0.593140i	$0.797888\pi$
$480$	0	0
$481$	7.65397	0.348991
$482$	−13.1329	−0.598187
$483$	0	0
$484$	10.1007	0.459123
$485$	51.5115	2.33902
$486$	0	0
$487$	−15.1791	−0.687829	−0.343914	−	0.939001i	$-0.611753\pi$
$487$	−15.1791	−0.687829	−0.343914	+	0.939001i	$0.611753\pi$
$488$	−31.1892	−1.41187
$489$	0	0
$490$	−4.55985	−0.205993
$491$	−6.82981	−0.308225	−0.154113	−	0.988053i	$-0.549252\pi$
$491$	−6.82981	−0.308225	−0.154113	+	0.988053i	$0.549252\pi$
$492$	0	0
$493$	−5.11908	−0.230552
$494$	36.8010	1.65575
$495$	0	0
$496$	−4.61143	−0.207059
$497$	23.7927	1.06725
$498$	0	0
$499$	−21.9579	−0.982973	−0.491486	−	0.870885i	$-0.663546\pi$
$499$	−21.9579	−0.982973	−0.491486	+	0.870885i	$0.663546\pi$
$500$	7.08381	0.316798
$501$	0	0
$502$	27.5004	1.22740
$503$	9.92840	0.442685	0.221343	−	0.975196i	$-0.428956\pi$
$503$	9.92840	0.442685	0.221343	+	0.975196i	$0.428956\pi$
$504$	0	0
$505$	−48.8316	−2.17298
$506$	1.22993	0.0546773
$507$	0	0
$508$	19.7152	0.874722
$509$	29.5479	1.30969	0.654844	−	0.755764i	$-0.272734\pi$
$509$	29.5479	1.30969	0.654844	+	0.755764i	$0.272734\pi$
$510$	0	0
$511$	−44.8523	−1.98415
$512$	7.54585	0.333483
$513$	0	0
$514$	17.7803	0.784254
$515$	27.1133	1.19476
$516$	0	0
$517$	5.18266	0.227933
$518$	−4.04734	−0.177830
$519$	0	0
$520$	43.7096	1.91679
$521$	39.8014	1.74373	0.871866	−	0.489744i	$-0.162910\pi$
$521$	39.8014	1.74373	0.871866	+	0.489744i	$0.162910\pi$
$522$	0	0
$523$	−40.0579	−1.75161	−0.875805	−	0.482665i	$-0.839669\pi$
$523$	−40.0579	−1.75161	−0.875805	+	0.482665i	$0.839669\pi$
$524$	11.0552	0.482950
$525$	0	0
$526$	22.4844	0.980367
$527$	34.9440	1.52219
$528$	0	0
$529$	1.00000	0.0434783
$530$	−30.1013	−1.30752
$531$	0	0
$532$	22.8228	0.989495
$533$	3.96306	0.171659
$534$	0	0
$535$	−55.1393	−2.38388
$536$	24.1593	1.04352
$537$	0	0
$538$	−24.7548	−1.06726
$539$	−2.20673	−0.0950508
$540$	0	0
$541$	−13.7571	−0.591462	−0.295731	−	0.955271i	$-0.595563\pi$
$541$	−13.7571	−0.591462	−0.295731	+	0.955271i	$0.595563\pi$
$542$	5.45929	0.234497
$543$	0	0
$544$	−26.9312	−1.15467
$545$	29.0525	1.24447
$546$	0	0
$547$	22.2272	0.950365	0.475183	−	0.879887i	$-0.342382\pi$
$547$	22.2272	0.950365	0.475183	+	0.879887i	$0.342382\pi$
$548$	−1.21450	−0.0518807
$549$	0	0
$550$	3.22660	0.137583
$551$	7.15882	0.304976
$552$	0	0
$553$	33.2378	1.41342
$554$	13.1565	0.558967
$555$	0	0
$556$	−0.347486	−0.0147367
$557$	−23.3162	−0.987939	−0.493969	−	0.869479i	$-0.664454\pi$
$557$	−23.3162	−0.987939	−0.493969	+	0.869479i	$0.664454\pi$
$558$	0	0
$559$	46.9777	1.98694
$560$	−5.50834	−0.232770
$561$	0	0
$562$	19.9582	0.841887
$563$	−5.44073	−0.229299	−0.114650	−	0.993406i	$-0.536575\pi$
$563$	−5.44073	−0.229299	−0.114650	+	0.993406i	$0.536575\pi$
$564$	0	0
$565$	−16.8736	−0.709879
$566$	−20.2182	−0.849834
$567$	0	0
$568$	23.8035	0.998772
$569$	12.9644	0.543496	0.271748	−	0.962369i	$-0.412398\pi$
$569$	12.9644	0.543496	0.271748	+	0.962369i	$0.412398\pi$
$570$	0	0
$571$	2.96652	0.124145	0.0620726	−	0.998072i	$-0.480229\pi$
$571$	2.96652	0.124145	0.0620726	+	0.998072i	$0.480229\pi$
$572$	7.41526	0.310048
$573$	0	0
$574$	−2.09563	−0.0874699
$575$	2.62340	0.109403
$576$	0	0
$577$	7.79048	0.324322	0.162161	−	0.986764i	$-0.448154\pi$
$577$	7.79048	0.324322	0.162161	+	0.986764i	$0.448154\pi$
$578$	−8.83140	−0.367338
$579$	0	0
$580$	2.98064	0.123764
$581$	31.5280	1.30800
$582$	0	0
$583$	−14.5675	−0.603324
$584$	−44.8727	−1.85684
$585$	0	0
$586$	−20.1019	−0.830402
$587$	24.6587	1.01777	0.508887	−	0.860833i	$-0.330057\pi$
$587$	24.6587	1.01777	0.508887	+	0.860833i	$0.330057\pi$
$588$	0	0
$589$	−48.8677	−2.01356
$590$	−4.17816	−0.172012
$591$	0	0
$592$	−0.965002	−0.0396613
$593$	43.3443	1.77994	0.889969	−	0.456021i	$-0.150726\pi$
$593$	43.3443	1.77994	0.889969	+	0.456021i	$0.150726\pi$
$594$	0	0
$595$	41.7406	1.71120
$596$	−13.8864	−0.568807
$597$	0	0
$598$	−5.14064	−0.210217
$599$	−34.3519	−1.40358	−0.701789	−	0.712384i	$-0.747615\pi$
$599$	−34.3519	−1.40358	−0.701789	+	0.712384i	$0.747615\pi$
$600$	0	0
$601$	2.75619	0.112427	0.0562137	−	0.998419i	$-0.482097\pi$
$601$	2.75619	0.112427	0.0562137	+	0.998419i	$0.482097\pi$
$602$	−24.8413	−1.01246
$603$	0	0
$604$	−2.72110	−0.110720
$605$	−25.8339	−1.05030
$606$	0	0
$607$	−41.4287	−1.68154	−0.840770	−	0.541393i	$-0.817897\pi$
$607$	−41.4287	−1.68154	−0.840770	+	0.541393i	$0.817897\pi$
$608$	37.6622	1.52740
$609$	0	0
$610$	27.9636	1.13221
$611$	−21.6615	−0.876330
$612$	0	0
$613$	19.7257	0.796713	0.398357	−	0.917231i	$-0.369581\pi$
$613$	19.7257	0.796713	0.398357	+	0.917231i	$0.369581\pi$
$614$	−10.8166	−0.436523
$615$	0	0
$616$	−11.1856	−0.450680
$617$	−16.9025	−0.680469	−0.340234	−	0.940341i	$-0.610506\pi$
$617$	−16.9025	−0.680469	−0.340234	+	0.940341i	$0.610506\pi$
$618$	0	0
$619$	5.74579	0.230943	0.115471	−	0.993311i	$-0.463162\pi$
$619$	5.74579	0.230943	0.115471	+	0.993311i	$0.463162\pi$
$620$	−20.3465	−0.817136
$621$	0	0
$622$	14.0944	0.565134
$623$	−12.1366	−0.486242
$624$	0	0
$625$	−31.2348	−1.24939
$626$	−15.8855	−0.634912
$627$	0	0
$628$	−18.4800	−0.737434
$629$	7.31250	0.291568
$630$	0	0
$631$	4.48369	0.178493	0.0892464	−	0.996010i	$-0.471554\pi$
$631$	4.48369	0.178493	0.0892464	+	0.996010i	$0.471554\pi$
$632$	33.2529	1.32273
$633$	0	0
$634$	−10.5796	−0.420170
$635$	−50.4244	−2.00103
$636$	0	0
$637$	9.22328	0.365440
$638$	−1.22993	−0.0486936
$639$	0	0
$640$	12.1020	0.478373
$641$	29.1938	1.15309	0.576543	−	0.817067i	$-0.304401\pi$
$641$	29.1938	1.15309	0.576543	+	0.817067i	$0.304401\pi$
$642$	0	0
$643$	−7.62070	−0.300531	−0.150266	−	0.988646i	$-0.548013\pi$
$643$	−7.62070	−0.300531	−0.150266	+	0.988646i	$0.548013\pi$
$644$	−3.18807	−0.125628
$645$	0	0
$646$	35.1591	1.38332
$647$	−31.9294	−1.25527	−0.627637	−	0.778506i	$-0.715978\pi$
$647$	−31.9294	−1.25527	−0.627637	+	0.778506i	$0.715978\pi$
$648$	0	0
$649$	−2.02201	−0.0793710
$650$	−13.4859	−0.528962
$651$	0	0
$652$	2.03036	0.0795152
$653$	−21.4471	−0.839292	−0.419646	−	0.907688i	$-0.637846\pi$
$653$	−21.4471	−0.839292	−0.419646	+	0.907688i	$0.637846\pi$
$654$	0	0
$655$	−28.2752	−1.10480
$656$	−0.499658	−0.0195084
$657$	0	0
$658$	11.4544	0.446538
$659$	−0.822737	−0.0320493	−0.0160246	−	0.999872i	$-0.505101\pi$
$659$	−0.822737	−0.0320493	−0.0160246	+	0.999872i	$0.505101\pi$
$660$	0	0
$661$	0.481496	0.0187280	0.00936401	−	0.999956i	$-0.497019\pi$
$661$	0.481496	0.0187280	0.00936401	+	0.999956i	$0.497019\pi$
$662$	8.49280	0.330082
$663$	0	0
$664$	31.5423	1.22408
$665$	−58.3725	−2.26359
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	8.29090	0.320785
$669$	0	0
$670$	−21.6608	−0.836828
$671$	13.5329	0.522434
$672$	0	0
$673$	22.7597	0.877321	0.438661	−	0.898653i	$-0.355453\pi$
$673$	22.7597	0.877321	0.438661	+	0.898653i	$0.355453\pi$
$674$	21.4225	0.825165
$675$	0	0
$676$	−16.9590	−0.652268
$677$	−19.0280	−0.731307	−0.365654	−	0.930751i	$-0.619155\pi$
$677$	−19.0280	−0.731307	−0.365654	+	0.930751i	$0.619155\pi$
$678$	0	0
$679$	55.0963	2.11440
$680$	41.7595	1.60140
$681$	0	0
$682$	8.39581	0.321492
$683$	29.0022	1.10974	0.554868	−	0.831938i	$-0.312769\pi$
$683$	29.0022	1.10974	0.554868	+	0.831938i	$0.312769\pi$
$684$	0	0
$685$	3.10624	0.118683
$686$	14.9561	0.571026
$687$	0	0
$688$	−5.92288	−0.225808
$689$	60.8864	2.31959
$690$	0	0
$691$	−3.59144	−0.136625	−0.0683124	−	0.997664i	$-0.521761\pi$
$691$	−3.59144	−0.136625	−0.0683124	+	0.997664i	$0.521761\pi$
$692$	12.0776	0.459120
$693$	0	0
$694$	24.5370	0.931411
$695$	0.888741	0.0337119
$696$	0	0
$697$	3.78626	0.143415
$698$	0.268406	0.0101593
$699$	0	0
$700$	−8.36357	−0.316113
$701$	−12.4149	−0.468904	−0.234452	−	0.972128i	$-0.575330\pi$
$701$	−12.4149	−0.468904	−0.234452	+	0.972128i	$0.575330\pi$
$702$	0	0
$703$	−10.2262	−0.385689
$704$	−8.20267	−0.309150
$705$	0	0
$706$	−7.54538	−0.283974
$707$	−52.2299	−1.96431
$708$	0	0
$709$	32.5094	1.22091	0.610457	−	0.792049i	$-0.290986\pi$
$709$	32.5094	1.22091	0.610457	+	0.792049i	$0.290986\pi$
$710$	−21.3418	−0.800942
$711$	0	0
$712$	−12.1421	−0.455044
$713$	6.82622	0.255644
$714$	0	0
$715$	−18.9655	−0.709271
$716$	14.0608	0.525478
$717$	0	0
$718$	−21.4877	−0.801914
$719$	−29.8279	−1.11239	−0.556197	−	0.831051i	$-0.687740\pi$
$719$	−29.8279	−1.11239	−0.556197	+	0.831051i	$0.687740\pi$
$720$	0	0
$721$	29.0002	1.08002
$722$	−30.9398	−1.15146
$723$	0	0
$724$	−4.91421	−0.182635
$725$	−2.62340	−0.0974305
$726$	0	0
$727$	34.7574	1.28908	0.644540	−	0.764570i	$-0.277049\pi$
$727$	34.7574	1.28908	0.644540	+	0.764570i	$0.277049\pi$
$728$	46.7514	1.73272
$729$	0	0
$730$	40.2320	1.48905
$731$	44.8818	1.66001
$732$	0	0
$733$	−41.0741	−1.51711	−0.758554	−	0.651610i	$-0.774094\pi$
$733$	−41.0741	−1.51711	−0.758554	+	0.651610i	$0.774094\pi$
$734$	−7.39347	−0.272898
$735$	0	0
$736$	−5.26094	−0.193921
$737$	−10.4827	−0.386135
$738$	0	0
$739$	41.3570	1.52134	0.760671	−	0.649137i	$-0.224870\pi$
$739$	41.3570	1.52134	0.760671	+	0.649137i	$0.224870\pi$
$740$	−4.25778	−0.156519
$741$	0	0
$742$	−32.1961	−1.18196
$743$	26.1440	0.959131	0.479565	−	0.877506i	$-0.340794\pi$
$743$	26.1440	0.959131	0.479565	+	0.877506i	$0.340794\pi$
$744$	0	0
$745$	35.5162	1.30121
$746$	20.8826	0.764566
$747$	0	0
$748$	7.08444	0.259033
$749$	−58.9765	−2.15495
$750$	0	0
$751$	−3.66473	−0.133728	−0.0668640	−	0.997762i	$-0.521299\pi$
$751$	−3.66473	−0.133728	−0.0668640	+	0.997762i	$0.521299\pi$
$752$	2.73105	0.0995912
$753$	0	0
$754$	5.14064	0.187211
$755$	6.95958	0.253285
$756$	0	0
$757$	19.7859	0.719130	0.359565	−	0.933120i	$-0.382925\pi$
$757$	19.7859	0.719130	0.359565	+	0.933120i	$0.382925\pi$
$758$	−28.6438	−1.04039
$759$	0	0
$760$	−58.3989	−2.11835
$761$	−7.76993	−0.281660	−0.140830	−	0.990034i	$-0.544977\pi$
$761$	−7.76993	−0.281660	−0.140830	+	0.990034i	$0.544977\pi$
$762$	0	0
$763$	31.0743	1.12496
$764$	11.5254	0.416973
$765$	0	0
$766$	1.42249	0.0513965
$767$	8.45122	0.305156
$768$	0	0
$769$	−37.2684	−1.34393	−0.671966	−	0.740582i	$-0.734550\pi$
$769$	−37.2684	−1.34393	−0.671966	+	0.740582i	$0.734550\pi$
$770$	10.0288	0.361412
$771$	0	0
$772$	−13.9317	−0.501413
$773$	−28.9149	−1.04000	−0.519998	−	0.854168i	$-0.674067\pi$
$773$	−28.9149	−1.04000	−0.519998	+	0.854168i	$0.674067\pi$
$774$	0	0
$775$	17.9079	0.643270
$776$	55.1213	1.97874
$777$	0	0
$778$	−8.90168	−0.319141
$779$	−5.29492	−0.189710
$780$	0	0
$781$	−10.3283	−0.369576
$782$	−4.91130	−0.175628
$783$	0	0
$784$	−1.16286	−0.0415307
$785$	47.2652	1.68697
$786$	0	0
$787$	46.7401	1.66610	0.833052	−	0.553195i	$-0.186592\pi$
$787$	46.7401	1.66610	0.833052	+	0.553195i	$0.186592\pi$
$788$	−15.2804	−0.544342
$789$	0	0
$790$	−29.8139	−1.06073
$791$	−18.0479	−0.641709
$792$	0	0
$793$	−56.5624	−2.00859
$794$	−4.67386	−0.165869
$795$	0	0
$796$	−16.2578	−0.576241
$797$	53.5902	1.89826	0.949131	−	0.314881i	$-0.101965\pi$
$797$	53.5902	1.89826	0.949131	+	0.314881i	$0.101965\pi$
$798$	0	0
$799$	−20.6951	−0.732140
$800$	−13.8015	−0.487958
$801$	0	0
$802$	−27.5472	−0.972727
$803$	19.4702	0.687088
$804$	0	0
$805$	8.15392	0.287388
$806$	−35.0912	−1.23603
$807$	0	0
$808$	−52.2536	−1.83827
$809$	8.25180	0.290118	0.145059	−	0.989423i	$-0.453663\pi$
$809$	8.25180	0.290118	0.145059	+	0.989423i	$0.453663\pi$
$810$	0	0
$811$	−10.1269	−0.355602	−0.177801	−	0.984066i	$-0.556898\pi$
$811$	−10.1269	−0.355602	−0.177801	+	0.984066i	$0.556898\pi$
$812$	3.18807	0.111879
$813$	0	0
$814$	1.75693	0.0615805
$815$	−5.19293	−0.181900
$816$	0	0
$817$	−62.7653	−2.19588
$818$	−16.9475	−0.592556
$819$	0	0
$820$	−2.20459	−0.0769876
$821$	34.0029	1.18671	0.593355	−	0.804941i	$-0.297803\pi$
$821$	34.0029	1.18671	0.593355	+	0.804941i	$0.297803\pi$
$822$	0	0
$823$	−22.7510	−0.793049	−0.396525	−	0.918024i	$-0.629784\pi$
$823$	−22.7510	−0.793049	−0.396525	+	0.918024i	$0.629784\pi$
$824$	29.0133	1.01073
$825$	0	0
$826$	−4.46892	−0.155494
$827$	−30.4412	−1.05854	−0.529272	−	0.848452i	$-0.677535\pi$
$827$	−30.4412	−1.05854	−0.529272	+	0.848452i	$0.677535\pi$
$828$	0	0
$829$	−40.3935	−1.40292	−0.701462	−	0.712707i	$-0.747469\pi$
$829$	−40.3935	−1.40292	−0.701462	+	0.712707i	$0.747469\pi$
$830$	−28.2803	−0.981622
$831$	0	0
$832$	34.2840	1.18858
$833$	8.81180	0.305311
$834$	0	0
$835$	−21.2051	−0.733833
$836$	−9.90729	−0.342651
$837$	0	0
$838$	13.9057	0.480363
$839$	−5.81493	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$839$	−5.81493	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−24.2219	−0.834743
$843$	0	0
$844$	22.4569	0.772998
$845$	43.3748	1.49214
$846$	0	0
$847$	−27.6317	−0.949438
$848$	−7.67648	−0.263611
$849$	0	0
$850$	−12.8843	−0.441927
$851$	1.42848	0.0489676
$852$	0	0
$853$	−47.4043	−1.62309	−0.811545	−	0.584289i	$-0.801373\pi$
$853$	−47.4043	−1.62309	−0.811545	+	0.584289i	$0.801373\pi$
$854$	29.9096	1.02349
$855$	0	0
$856$	−59.0032	−2.01669
$857$	17.3158	0.591497	0.295748	−	0.955266i	$-0.404431\pi$
$857$	17.3158	0.591497	0.295748	+	0.955266i	$0.404431\pi$
$858$	0	0
$859$	15.0403	0.513167	0.256583	−	0.966522i	$-0.417403\pi$
$859$	15.0403	0.513167	0.256583	+	0.966522i	$0.417403\pi$
$860$	−26.1329	−0.891126
$861$	0	0
$862$	−11.2265	−0.382376
$863$	−36.2271	−1.23318	−0.616592	−	0.787282i	$-0.711487\pi$
$863$	−36.2271	−1.23318	−0.616592	+	0.787282i	$0.711487\pi$
$864$	0	0
$865$	−30.8900	−1.05029
$866$	16.6471	0.565693
$867$	0	0
$868$	−21.7625	−0.738667
$869$	−14.4284	−0.489450
$870$	0	0
$871$	43.8135	1.48457
$872$	31.0884	1.05278
$873$	0	0
$874$	6.86825	0.232322
$875$	−19.3786	−0.655118
$876$	0	0
$877$	9.59571	0.324024	0.162012	−	0.986789i	$-0.448202\pi$
$877$	9.59571	0.324024	0.162012	+	0.986789i	$0.448202\pi$
$878$	−28.7344	−0.969739
$879$	0	0
$880$	2.39115	0.0806056
$881$	12.0891	0.407292	0.203646	−	0.979045i	$-0.434721\pi$
$881$	12.0891	0.407292	0.203646	+	0.979045i	$0.434721\pi$
$882$	0	0
$883$	26.7425	0.899957	0.449979	−	0.893039i	$-0.351432\pi$
$883$	26.7425	0.899957	0.449979	+	0.893039i	$0.351432\pi$
$884$	−29.6102	−0.995898
$885$	0	0
$886$	−5.76688	−0.193742
$887$	38.2725	1.28507	0.642533	−	0.766258i	$-0.277884\pi$
$887$	38.2725	1.28507	0.642533	+	0.766258i	$0.277884\pi$
$888$	0	0
$889$	−53.9335	−1.80887
$890$	10.8864	0.364911
$891$	0	0
$892$	6.09095	0.203940
$893$	28.9412	0.968481
$894$	0	0
$895$	−35.9625	−1.20209
$896$	12.9442	0.432434
$897$	0	0
$898$	−1.03451	−0.0345220
$899$	−6.82622	−0.227667
$900$	0	0
$901$	58.1701	1.93793
$902$	0.909704	0.0302898
$903$	0	0
$904$	−18.0561	−0.600536
$905$	12.5688	0.417799
$906$	0	0
$907$	25.0813	0.832812	0.416406	−	0.909179i	$-0.363289\pi$
$907$	25.0813	0.832812	0.416406	+	0.909179i	$0.363289\pi$
$908$	19.5043	0.647272
$909$	0	0
$910$	−41.9164	−1.38951
$911$	−4.29265	−0.142222	−0.0711109	−	0.997468i	$-0.522654\pi$
$911$	−4.29265	−0.142222	−0.0711109	+	0.997468i	$0.522654\pi$
$912$	0	0
$913$	−13.6862	−0.452947
$914$	−16.0613	−0.531259
$915$	0	0
$916$	−30.1448	−0.996013
$917$	−30.2429	−0.998710
$918$	0	0
$919$	−42.5174	−1.40252	−0.701260	−	0.712906i	$-0.747379\pi$
$919$	−42.5174	−1.40252	−0.701260	+	0.712906i	$0.747379\pi$
$920$	8.15762	0.268949
$921$	0	0
$922$	−30.2946	−0.997698
$923$	43.1683	1.42090
$924$	0	0
$925$	3.74746	0.123216
$926$	7.64007	0.251068
$927$	0	0
$928$	5.26094	0.172699
$929$	7.62768	0.250256	0.125128	−	0.992141i	$-0.460066\pi$
$929$	7.62768	0.250256	0.125128	+	0.992141i	$0.460066\pi$
$930$	0	0
$931$	−12.3229	−0.403868
$932$	10.1176	0.331411
$933$	0	0
$934$	−25.4332	−0.832198
$935$	−18.1194	−0.592568
$936$	0	0
$937$	−22.7696	−0.743849	−0.371924	−	0.928263i	$-0.621302\pi$
$937$	−22.7696	−0.743849	−0.371924	+	0.928263i	$0.621302\pi$
$938$	−23.1682	−0.756467
$939$	0	0
$940$	12.0499	0.393026
$941$	−45.6255	−1.48735	−0.743674	−	0.668542i	$-0.766919\pi$
$941$	−45.6255	−1.48735	−0.743674	+	0.668542i	$0.766919\pi$
$942$	0	0
$943$	0.739636	0.0240859
$944$	−1.06552	−0.0346797
$945$	0	0
$946$	10.7835	0.350602
$947$	−13.8796	−0.451025	−0.225512	−	0.974240i	$-0.572406\pi$
$947$	−13.8796	−0.451025	−0.225512	+	0.974240i	$0.572406\pi$
$948$	0	0
$949$	−81.3778	−2.64163
$950$	18.0181	0.584585
$951$	0	0
$952$	44.6656	1.44762
$953$	−37.2023	−1.20510	−0.602550	−	0.798081i	$-0.705849\pi$
$953$	−37.2023	−1.20510	−0.602550	+	0.798081i	$0.705849\pi$
$954$	0	0
$955$	−29.4777	−0.953874
$956$	−4.62782	−0.149674
$957$	0	0
$958$	33.8105	1.09237
$959$	3.32241	0.107286
$960$	0	0
$961$	15.5973	0.503140
$962$	−7.34329	−0.236757
$963$	0	0
$964$	−14.7772	−0.475941
$965$	35.6322	1.14704
$966$	0	0
$967$	−48.3743	−1.55561	−0.777806	−	0.628505i	$-0.783667\pi$
$967$	−48.3743	−1.55561	−0.777806	+	0.628505i	$0.783667\pi$
$968$	−27.6443	−0.888520
$969$	0	0
$970$	−49.4207	−1.58680
$971$	−53.0401	−1.70214	−0.851069	−	0.525054i	$-0.824045\pi$
$971$	−53.0401	−1.70214	−0.851069	+	0.525054i	$0.824045\pi$
$972$	0	0
$973$	0.950590	0.0304745
$974$	14.5629	0.466627
$975$	0	0
$976$	7.13131	0.228268
$977$	10.9606	0.350661	0.175331	−	0.984510i	$-0.443901\pi$
$977$	10.9606	0.350661	0.175331	+	0.984510i	$0.443901\pi$
$978$	0	0
$979$	5.26844	0.168380
$980$	−5.13076	−0.163896
$981$	0	0
$982$	6.55259	0.209102
$983$	−17.7844	−0.567235	−0.283617	−	0.958938i	$-0.591535\pi$
$983$	−17.7844	−0.567235	−0.283617	+	0.958938i	$0.591535\pi$
$984$	0	0
$985$	39.0817	1.24525
$986$	4.91130	0.156408
$987$	0	0
$988$	41.4086	1.31738
$989$	8.76755	0.278792
$990$	0	0
$991$	21.1068	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$991$	21.1068	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$992$	−35.9124	−1.14022
$993$	0	0
$994$	−22.8270	−0.724027
$995$	41.5814	1.31822
$996$	0	0
$997$	17.3564	0.549681	0.274841	−	0.961490i	$-0.411375\pi$
$997$	17.3564	0.549681	0.274841	+	0.961490i	$0.411375\pi$
$998$	21.0667	0.666854
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.v.1.11	✓	30
3.2	odd	2		6003.2.a.w.1.20	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.11	✓	30	1.1	even	1	trivial
6003.2.a.w.1.20	yes	30	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)

\(f(q)\)	\(=\)	\(q-0.959410 q^{2} -1.07953 q^{4} +2.76105 q^{5} +2.95320 q^{7} +2.95453 q^{8} +O(q^{10})\)	\(q-0.959410 q^{2} -1.07953 q^{4} +2.76105 q^{5} +2.95320 q^{7} +2.95453 q^{8} -2.64898 q^{10} -1.28197 q^{11} +5.35813 q^{13} -2.83333 q^{14} -0.675546 q^{16} +5.11908 q^{17} -7.15882 q^{19} -2.98064 q^{20} +1.22993 q^{22} +1.00000 q^{23} +2.62340 q^{25} -5.14064 q^{26} -3.18807 q^{28} -1.00000 q^{29} +6.82622 q^{31} -5.26094 q^{32} -4.91130 q^{34} +8.15392 q^{35} +1.42848 q^{37} +6.86825 q^{38} +8.15762 q^{40} +0.739636 q^{41} +8.76755 q^{43} +1.38393 q^{44} -0.959410 q^{46} -4.04273 q^{47} +1.72136 q^{49} -2.51691 q^{50} -5.78427 q^{52} +11.3634 q^{53} -3.53958 q^{55} +8.72532 q^{56} +0.959410 q^{58} +1.57727 q^{59} -10.5564 q^{61} -6.54915 q^{62} +6.39849 q^{64} +14.7941 q^{65} +8.17702 q^{67} -5.52622 q^{68} -7.82295 q^{70} +8.05660 q^{71} -15.1877 q^{73} -1.37050 q^{74} +7.72818 q^{76} -3.78591 q^{77} +11.2549 q^{79} -1.86521 q^{80} -0.709614 q^{82} +10.6759 q^{83} +14.1340 q^{85} -8.41168 q^{86} -3.78762 q^{88} -4.10964 q^{89} +15.8236 q^{91} -1.07953 q^{92} +3.87864 q^{94} -19.7659 q^{95} +18.6565 q^{97} -1.65149 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - q^{2} + 37 q^{4} + 10 q^{7} - 6 q^{8}+O(q^{10}) \) 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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