LMFDB - Embedded newform 6003.2.a.t.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6003,2,Mod(1,6003)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [22,-3,0,17,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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			1.9
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)

$f(q)$	$=$	\(q-1.06922 q^{2} -0.856774 q^{4} +0.420836 q^{5} +1.13415 q^{7} +3.05451 q^{8} -0.449966 q^{10} -2.91779 q^{11} -1.70064 q^{13} -1.21266 q^{14} -1.55239 q^{16} +0.536110 q^{17} +0.585904 q^{19} -0.360562 q^{20} +3.11976 q^{22} +1.00000 q^{23} -4.82290 q^{25} +1.81835 q^{26} -0.971713 q^{28} +1.00000 q^{29} +7.49962 q^{31} -4.44918 q^{32} -0.573218 q^{34} +0.477293 q^{35} -1.22414 q^{37} -0.626459 q^{38} +1.28545 q^{40} +12.2002 q^{41} -2.78460 q^{43} +2.49989 q^{44} -1.06922 q^{46} +0.180494 q^{47} -5.71370 q^{49} +5.15673 q^{50} +1.45706 q^{52} -11.0675 q^{53} -1.22791 q^{55} +3.46429 q^{56} -1.06922 q^{58} -5.86452 q^{59} +12.1083 q^{61} -8.01872 q^{62} +7.86192 q^{64} -0.715690 q^{65} -14.3374 q^{67} -0.459325 q^{68} -0.510330 q^{70} +13.2457 q^{71} -4.56789 q^{73} +1.30887 q^{74} -0.501988 q^{76} -3.30923 q^{77} -6.89701 q^{79} -0.653302 q^{80} -13.0447 q^{82} -8.57178 q^{83} +0.225614 q^{85} +2.97734 q^{86} -8.91244 q^{88} +2.75169 q^{89} -1.92878 q^{91} -0.856774 q^{92} -0.192988 q^{94} +0.246570 q^{95} -7.28509 q^{97} +6.10918 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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 $

$2$	−1.06922	−0.756051	−0.378025	−	0.925795i	$-0.623397\pi$
$2$	−1.06922	−0.756051	−0.378025	+	0.925795i	$0.623397\pi$
$3$	0	0
$3$	0	0
$4$	−0.856774	−0.428387
$5$	0.420836	0.188204	0.0941019	−	0.995563i	$-0.470002\pi$
$5$	0.420836	0.188204	0.0941019	+	0.995563i	$0.470002\pi$
$6$	0	0
$7$	1.13415	0.428670	0.214335	−	0.976760i	$-0.431242\pi$
$7$	1.13415	0.428670	0.214335	+	0.976760i	$0.431242\pi$
$8$	3.05451	1.07993
$9$	0	0
$10$	−0.449966	−0.142292
$11$	−2.91779	−0.879748	−0.439874	−	0.898059i	$-0.644977\pi$
$11$	−2.91779	−0.879748	−0.439874	+	0.898059i	$0.644977\pi$
$12$	0	0
$13$	−1.70064	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$13$	−1.70064	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$14$	−1.21266	−0.324096
$15$	0	0
$16$	−1.55239	−0.388097
$17$	0.536110	0.130026	0.0650128	−	0.997884i	$-0.479291\pi$
$17$	0.536110	0.130026	0.0650128	+	0.997884i	$0.479291\pi$
$18$	0	0
$19$	0.585904	0.134416	0.0672078	−	0.997739i	$-0.478591\pi$
$19$	0.585904	0.134416	0.0672078	+	0.997739i	$0.478591\pi$
$20$	−0.360562	−0.0806240
$21$	0	0
$22$	3.11976	0.665134
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.82290	−0.964579
$26$	1.81835	0.356608
$27$	0	0
$28$	−0.971713	−0.183637
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	7.49962	1.34697	0.673486	−	0.739200i	$-0.264796\pi$
$31$	7.49962	1.34697	0.673486	+	0.739200i	$0.264796\pi$
$32$	−4.44918	−0.786512
$33$	0	0
$34$	−0.573218	−0.0983060
$35$	0.477293	0.0806772
$36$	0	0
$37$	−1.22414	−0.201247	−0.100623	−	0.994925i	$-0.532084\pi$
$37$	−1.22414	−0.201247	−0.100623	+	0.994925i	$0.532084\pi$
$38$	−0.626459	−0.101625
$39$	0	0
$40$	1.28545	0.203247
$41$	12.2002	1.90535	0.952676	−	0.303987i	$-0.0983179\pi$
$41$	12.2002	1.90535	0.952676	+	0.303987i	$0.0983179\pi$
$42$	0	0
$43$	−2.78460	−0.424647	−0.212324	−	0.977199i	$-0.568103\pi$
$43$	−2.78460	−0.424647	−0.212324	+	0.977199i	$0.568103\pi$
$44$	2.49989	0.376873
$45$	0	0
$46$	−1.06922	−0.157648
$47$	0.180494	0.0263278	0.0131639	−	0.999913i	$-0.495810\pi$
$47$	0.180494	0.0263278	0.0131639	+	0.999913i	$0.495810\pi$
$48$	0	0
$49$	−5.71370	−0.816242
$50$	5.15673	0.729271
$51$	0	0
$52$	1.45706	0.202058
$53$	−11.0675	−1.52023	−0.760116	−	0.649787i	$-0.774858\pi$
$53$	−11.0675	−1.52023	−0.760116	+	0.649787i	$0.774858\pi$
$54$	0	0
$55$	−1.22791	−0.165572
$56$	3.46429	0.462935
$57$	0	0
$58$	−1.06922	−0.140395
$59$	−5.86452	−0.763496	−0.381748	−	0.924266i	$-0.624678\pi$
$59$	−5.86452	−0.763496	−0.381748	+	0.924266i	$0.624678\pi$
$60$	0	0
$61$	12.1083	1.55031	0.775153	−	0.631773i	$-0.217673\pi$
$61$	12.1083	1.55031	0.775153	+	0.631773i	$0.217673\pi$
$62$	−8.01872	−1.01838
$63$	0	0
$64$	7.86192	0.982740
$65$	−0.715690	−0.0887705
$66$	0	0
$67$	−14.3374	−1.75159	−0.875794	−	0.482686i	$-0.839661\pi$
$67$	−14.3374	−1.75159	−0.875794	+	0.482686i	$0.839661\pi$
$68$	−0.459325	−0.0557013
$69$	0	0
$70$	−0.510330	−0.0609961
$71$	13.2457	1.57198	0.785988	−	0.618242i	$-0.212155\pi$
$71$	13.2457	1.57198	0.785988	+	0.618242i	$0.212155\pi$
$72$	0	0
$73$	−4.56789	−0.534632	−0.267316	−	0.963609i	$-0.586137\pi$
$73$	−4.56789	−0.534632	−0.267316	+	0.963609i	$0.586137\pi$
$74$	1.30887	0.152153
$75$	0	0
$76$	−0.501988	−0.0575819
$77$	−3.30923	−0.377121
$78$	0	0
$79$	−6.89701	−0.775974	−0.387987	−	0.921665i	$-0.626829\pi$
$79$	−6.89701	−0.775974	−0.387987	+	0.921665i	$0.626829\pi$
$80$	−0.653302	−0.0730414
$81$	0	0
$82$	−13.0447	−1.44054
$83$	−8.57178	−0.940875	−0.470438	−	0.882433i	$-0.655904\pi$
$83$	−8.57178	−0.940875	−0.470438	+	0.882433i	$0.655904\pi$
$84$	0	0
$85$	0.225614	0.0244713
$86$	2.97734	0.321055
$87$	0	0
$88$	−8.91244	−0.950069
$89$	2.75169	0.291678	0.145839	−	0.989308i	$-0.453412\pi$
$89$	2.75169	0.291678	0.145839	+	0.989308i	$0.453412\pi$
$90$	0	0
$91$	−1.92878	−0.202192
$92$	−0.856774	−0.0893249
$93$	0	0
$94$	−0.192988	−0.0199052
$95$	0.246570	0.0252975
$96$	0	0
$97$	−7.28509	−0.739689	−0.369844	−	0.929094i	$-0.620589\pi$
$97$	−7.28509	−0.739689	−0.369844	+	0.929094i	$0.620589\pi$
$98$	6.10918	0.617121
$99$	0	0
$100$	4.13213	0.413213
$101$	−7.18183	−0.714618	−0.357309	−	0.933986i	$-0.616306\pi$
$101$	−7.18183	−0.714618	−0.357309	+	0.933986i	$0.616306\pi$
$102$	0	0
$103$	3.02471	0.298034	0.149017	−	0.988835i	$-0.452389\pi$
$103$	3.02471	0.298034	0.149017	+	0.988835i	$0.452389\pi$
$104$	−5.19462	−0.509374
$105$	0	0
$106$	11.8335	1.14937
$107$	9.61926	0.929929	0.464965	−	0.885329i	$-0.346067\pi$
$107$	9.61926	0.929929	0.464965	+	0.885329i	$0.346067\pi$
$108$	0	0
$109$	10.0908	0.966521	0.483261	−	0.875477i	$-0.339452\pi$
$109$	10.0908	0.966521	0.483261	+	0.875477i	$0.339452\pi$
$110$	1.31291	0.125181
$111$	0	0
$112$	−1.76065	−0.166366
$113$	6.82687	0.642218	0.321109	−	0.947042i	$-0.395944\pi$
$113$	6.82687	0.642218	0.321109	+	0.947042i	$0.395944\pi$
$114$	0	0
$115$	0.420836	0.0392432
$116$	−0.856774	−0.0795495
$117$	0	0
$118$	6.27045	0.577242
$119$	0.608031	0.0557381
$120$	0	0
$121$	−2.48647	−0.226043
$122$	−12.9464	−1.17211
$123$	0	0
$124$	−6.42548	−0.577025
$125$	−4.13383	−0.369741
$126$	0	0
$127$	−5.53291	−0.490967	−0.245483	−	0.969401i	$-0.578947\pi$
$127$	−5.53291	−0.490967	−0.245483	+	0.969401i	$0.578947\pi$
$128$	0.492261	0.0435101
$129$	0	0
$130$	0.765229	0.0671150
$131$	13.4974	1.17927	0.589637	−	0.807668i	$-0.299271\pi$
$131$	13.4974	1.17927	0.589637	+	0.807668i	$0.299271\pi$
$132$	0	0
$133$	0.664505	0.0576199
$134$	15.3298	1.32429
$135$	0	0
$136$	1.63755	0.140419
$137$	5.83987	0.498934	0.249467	−	0.968383i	$-0.419745\pi$
$137$	5.83987	0.498934	0.249467	+	0.968383i	$0.419745\pi$
$138$	0	0
$139$	−22.3348	−1.89442	−0.947208	−	0.320619i	$-0.896109\pi$
$139$	−22.3348	−1.89442	−0.947208	+	0.320619i	$0.896109\pi$
$140$	−0.408932	−0.0345611
$141$	0	0
$142$	−14.1625	−1.18849
$143$	4.96211	0.414953
$144$	0	0
$145$	0.420836	0.0349486
$146$	4.88407	0.404209
$147$	0	0
$148$	1.04881	0.0862116
$149$	−0.856186	−0.0701415	−0.0350707	−	0.999385i	$-0.511166\pi$
$149$	−0.856186	−0.0701415	−0.0350707	+	0.999385i	$0.511166\pi$
$150$	0	0
$151$	−5.82116	−0.473720	−0.236860	−	0.971544i	$-0.576118\pi$
$151$	−5.82116	−0.473720	−0.236860	+	0.971544i	$0.576118\pi$
$152$	1.78965	0.145160
$153$	0	0
$154$	3.53828	0.285123
$155$	3.15611	0.253505
$156$	0	0
$157$	−1.59058	−0.126942	−0.0634709	−	0.997984i	$-0.520217\pi$
$157$	−1.59058	−0.126942	−0.0634709	+	0.997984i	$0.520217\pi$
$158$	7.37440	0.586676
$159$	0	0
$160$	−1.87238	−0.148024
$161$	1.13415	0.0893838
$162$	0	0
$163$	−9.80469	−0.767963	−0.383981	−	0.923341i	$-0.625447\pi$
$163$	−9.80469	−0.767963	−0.383981	+	0.923341i	$0.625447\pi$
$164$	−10.4528	−0.816228
$165$	0	0
$166$	9.16510	0.711349
$167$	16.2385	1.25658	0.628288	−	0.777981i	$-0.283756\pi$
$167$	16.2385	1.25658	0.628288	+	0.777981i	$0.283756\pi$
$168$	0	0
$169$	−10.1078	−0.777525
$170$	−0.241231	−0.0185016
$171$	0	0
$172$	2.38577	0.181913
$173$	−14.2547	−1.08376	−0.541881	−	0.840455i	$-0.682288\pi$
$173$	−14.2547	−1.08376	−0.541881	+	0.840455i	$0.682288\pi$
$174$	0	0
$175$	−5.46991	−0.413486
$176$	4.52955	0.341428
$177$	0	0
$178$	−2.94215	−0.220524
$179$	0.143629	0.0107353	0.00536766	−	0.999986i	$-0.498291\pi$
$179$	0.143629	0.0107353	0.00536766	+	0.999986i	$0.498291\pi$
$180$	0	0
$181$	−11.6508	−0.865997	−0.432998	−	0.901395i	$-0.642544\pi$
$181$	−11.6508	−0.865997	−0.432998	+	0.901395i	$0.642544\pi$
$182$	2.06229	0.152867
$183$	0	0
$184$	3.05451	0.225182
$185$	−0.515161	−0.0378754
$186$	0	0
$187$	−1.56426	−0.114390
$188$	−0.154643	−0.0112785
$189$	0	0
$190$	−0.263637	−0.0191262
$191$	26.1784	1.89420	0.947100	−	0.320939i	$-0.103998\pi$
$191$	26.1784	1.89420	0.947100	+	0.320939i	$0.103998\pi$
$192$	0	0
$193$	7.02480	0.505656	0.252828	−	0.967511i	$-0.418639\pi$
$193$	7.02480	0.505656	0.252828	+	0.967511i	$0.418639\pi$
$194$	7.78934	0.559242
$195$	0	0
$196$	4.89535	0.349668
$197$	3.74205	0.266610	0.133305	−	0.991075i	$-0.457441\pi$
$197$	3.74205	0.266610	0.133305	+	0.991075i	$0.457441\pi$
$198$	0	0
$199$	13.8087	0.978875	0.489437	−	0.872038i	$-0.337202\pi$
$199$	13.8087	0.978875	0.489437	+	0.872038i	$0.337202\pi$
$200$	−14.7316	−1.04168
$201$	0	0
$202$	7.67893	0.540288
$203$	1.13415	0.0796020
$204$	0	0
$205$	5.13429	0.358594
$206$	−3.23407	−0.225329
$207$	0	0
$208$	2.64005	0.183055
$209$	−1.70955	−0.118252
$210$	0	0
$211$	5.17516	0.356273	0.178136	−	0.984006i	$-0.442993\pi$
$211$	5.17516	0.356273	0.178136	+	0.984006i	$0.442993\pi$
$212$	9.48231	0.651248
$213$	0	0
$214$	−10.2851	−0.703074
$215$	−1.17186	−0.0799202
$216$	0	0
$217$	8.50572	0.577406
$218$	−10.7892	−0.730739
$219$	0	0
$220$	1.05205	0.0709289
$221$	−0.911728	−0.0613295
$222$	0	0
$223$	−12.8518	−0.860617	−0.430308	−	0.902682i	$-0.641595\pi$
$223$	−12.8518	−0.860617	−0.430308	+	0.902682i	$0.641595\pi$
$224$	−5.04606	−0.337154
$225$	0	0
$226$	−7.29941	−0.485550
$227$	−12.5847	−0.835278	−0.417639	−	0.908613i	$-0.637142\pi$
$227$	−12.5847	−0.835278	−0.417639	+	0.908613i	$0.637142\pi$
$228$	0	0
$229$	19.4879	1.28779	0.643897	−	0.765112i	$-0.277316\pi$
$229$	19.4879	1.28779	0.643897	+	0.765112i	$0.277316\pi$
$230$	−0.449966	−0.0296699
$231$	0	0
$232$	3.05451	0.200539
$233$	−11.0040	−0.720893	−0.360446	−	0.932780i	$-0.617376\pi$
$233$	−11.0040	−0.720893	−0.360446	+	0.932780i	$0.617376\pi$
$234$	0	0
$235$	0.0759586	0.00495499
$236$	5.02457	0.327072
$237$	0	0
$238$	−0.650117	−0.0421408
$239$	9.61008	0.621625	0.310812	−	0.950471i	$-0.399399\pi$
$239$	9.61008	0.621625	0.310812	+	0.950471i	$0.399399\pi$
$240$	0	0
$241$	−5.96508	−0.384244	−0.192122	−	0.981371i	$-0.561537\pi$
$241$	−5.96508	−0.384244	−0.192122	+	0.981371i	$0.561537\pi$
$242$	2.65858	0.170900
$243$	0	0
$244$	−10.3741	−0.664131
$245$	−2.40453	−0.153620
$246$	0	0
$247$	−0.996411	−0.0634001
$248$	22.9077	1.45464
$249$	0	0
$250$	4.41997	0.279543
$251$	−9.33642	−0.589310	−0.294655	−	0.955604i	$-0.595205\pi$
$251$	−9.33642	−0.589310	−0.294655	+	0.955604i	$0.595205\pi$
$252$	0	0
$253$	−2.91779	−0.183440
$254$	5.91589	0.371196
$255$	0	0
$256$	−16.2502	−1.01564
$257$	27.9492	1.74342	0.871711	−	0.490021i	$-0.163011\pi$
$257$	27.9492	1.74342	0.871711	+	0.490021i	$0.163011\pi$
$258$	0	0
$259$	−1.38836	−0.0862685
$260$	0.613185	0.0380281
$261$	0	0
$262$	−14.4317	−0.891591
$263$	−27.8535	−1.71752	−0.858760	−	0.512379i	$-0.828764\pi$
$263$	−27.8535	−1.71752	−0.858760	+	0.512379i	$0.828764\pi$
$264$	0	0
$265$	−4.65759	−0.286113
$266$	−0.710501	−0.0435636
$267$	0	0
$268$	12.2839	0.750357
$269$	8.33222	0.508025	0.254012	−	0.967201i	$-0.418250\pi$
$269$	8.33222	0.508025	0.254012	+	0.967201i	$0.418250\pi$
$270$	0	0
$271$	−25.0222	−1.51999	−0.759995	−	0.649929i	$-0.774799\pi$
$271$	−25.0222	−1.51999	−0.759995	+	0.649929i	$0.774799\pi$
$272$	−0.832251	−0.0504626
$273$	0	0
$274$	−6.24409	−0.377219
$275$	14.0722	0.848587
$276$	0	0
$277$	−17.5910	−1.05694	−0.528469	−	0.848952i	$-0.677234\pi$
$277$	−17.5910	−1.05694	−0.528469	+	0.848952i	$0.677234\pi$
$278$	23.8808	1.43228
$279$	0	0
$280$	1.45790	0.0871260
$281$	2.63805	0.157373	0.0786864	−	0.996899i	$-0.474927\pi$
$281$	2.63805	0.157373	0.0786864	+	0.996899i	$0.474927\pi$
$282$	0	0
$283$	−16.9340	−1.00662	−0.503312	−	0.864105i	$-0.667885\pi$
$283$	−16.9340	−1.00662	−0.503312	+	0.864105i	$0.667885\pi$
$284$	−11.3486	−0.673414
$285$	0	0
$286$	−5.30558	−0.313725
$287$	13.8369	0.816767
$288$	0	0
$289$	−16.7126	−0.983093
$290$	−0.449966	−0.0264229
$291$	0	0
$292$	3.91365	0.229029
$293$	4.73836	0.276818	0.138409	−	0.990375i	$-0.455801\pi$
$293$	4.73836	0.276818	0.138409	+	0.990375i	$0.455801\pi$
$294$	0	0
$295$	−2.46800	−0.143693
$296$	−3.73914	−0.217333
$297$	0	0
$298$	0.915449	0.0530305
$299$	−1.70064	−0.0983504
$300$	0	0
$301$	−3.15816	−0.182034
$302$	6.22409	0.358156
$303$	0	0
$304$	−0.909552	−0.0521664
$305$	5.09561	0.291774
$306$	0	0
$307$	−34.0679	−1.94436	−0.972178	−	0.234243i	$-0.924739\pi$
$307$	−34.0679	−1.94436	−0.972178	+	0.234243i	$0.924739\pi$
$308$	2.83526	0.161554
$309$	0	0
$310$	−3.37457	−0.191663
$311$	−3.97157	−0.225207	−0.112604	−	0.993640i	$-0.535919\pi$
$311$	−3.97157	−0.225207	−0.112604	+	0.993640i	$0.535919\pi$
$312$	0	0
$313$	21.4087	1.21009	0.605045	−	0.796192i	$-0.293155\pi$
$313$	21.4087	1.21009	0.605045	+	0.796192i	$0.293155\pi$
$314$	1.70067	0.0959744
$315$	0	0
$316$	5.90918	0.332417
$317$	−28.2785	−1.58828	−0.794140	−	0.607735i	$-0.792078\pi$
$317$	−28.2785	−1.58828	−0.794140	+	0.607735i	$0.792078\pi$
$318$	0	0
$319$	−2.91779	−0.163365
$320$	3.30858	0.184955
$321$	0	0
$322$	−1.21266	−0.0675787
$323$	0.314109	0.0174775
$324$	0	0
$325$	8.20200	0.454965
$326$	10.4833	0.580619
$327$	0	0
$328$	37.2657	2.05765
$329$	0.204708	0.0112859
$330$	0	0
$331$	−17.6922	−0.972450	−0.486225	−	0.873834i	$-0.661627\pi$
$331$	−17.6922	−0.972450	−0.486225	+	0.873834i	$0.661627\pi$
$332$	7.34408	0.403059
$333$	0	0
$334$	−17.3625	−0.950035
$335$	−6.03368	−0.329655
$336$	0	0
$337$	−19.4660	−1.06038	−0.530191	−	0.847878i	$-0.677880\pi$
$337$	−19.4660	−1.06038	−0.530191	+	0.847878i	$0.677880\pi$
$338$	10.8075	0.587849
$339$	0	0
$340$	−0.193301	−0.0104832
$341$	−21.8823	−1.18500
$342$	0	0
$343$	−14.4193	−0.778568
$344$	−8.50559	−0.458591
$345$	0	0
$346$	15.2413	0.819379
$347$	−7.18999	−0.385979	−0.192989	−	0.981201i	$-0.561818\pi$
$347$	−7.18999	−0.385979	−0.192989	+	0.981201i	$0.561818\pi$
$348$	0	0
$349$	19.0083	1.01749	0.508746	−	0.860917i	$-0.330109\pi$
$349$	19.0083	1.01749	0.508746	+	0.860917i	$0.330109\pi$
$350$	5.84852	0.312616
$351$	0	0
$352$	12.9818	0.691932
$353$	−3.18574	−0.169560	−0.0847799	−	0.996400i	$-0.527019\pi$
$353$	−3.18574	−0.169560	−0.0847799	+	0.996400i	$0.527019\pi$
$354$	0	0
$355$	5.57427	0.295852
$356$	−2.35758	−0.124951
$357$	0	0
$358$	−0.153571	−0.00811645
$359$	10.3826	0.547973	0.273986	−	0.961734i	$-0.411658\pi$
$359$	10.3826	0.547973	0.273986	+	0.961734i	$0.411658\pi$
$360$	0	0
$361$	−18.6567	−0.981932
$362$	12.4572	0.654737
$363$	0	0
$364$	1.65253	0.0866163
$365$	−1.92234	−0.100620
$366$	0	0
$367$	2.15378	0.112426	0.0562132	−	0.998419i	$-0.482097\pi$
$367$	2.15378	0.112426	0.0562132	+	0.998419i	$0.482097\pi$
$368$	−1.55239	−0.0809239
$369$	0	0
$370$	0.550820	0.0286357
$371$	−12.5522	−0.651678
$372$	0	0
$373$	10.6902	0.553517	0.276758	−	0.960940i	$-0.410740\pi$
$373$	10.6902	0.553517	0.276758	+	0.960940i	$0.410740\pi$
$374$	1.67253	0.0864846
$375$	0	0
$376$	0.551323	0.0284323
$377$	−1.70064	−0.0875873
$378$	0	0
$379$	9.12577	0.468759	0.234380	−	0.972145i	$-0.424694\pi$
$379$	9.12577	0.468759	0.234380	+	0.972145i	$0.424694\pi$
$380$	−0.211255	−0.0108371
$381$	0	0
$382$	−27.9904	−1.43211
$383$	−11.0898	−0.566663	−0.283332	−	0.959022i	$-0.591440\pi$
$383$	−11.0898	−0.566663	−0.283332	+	0.959022i	$0.591440\pi$
$384$	0	0
$385$	−1.39264	−0.0709757
$386$	−7.51104	−0.382302
$387$	0	0
$388$	6.24168	0.316873
$389$	−7.52137	−0.381349	−0.190674	−	0.981653i	$-0.561067\pi$
$389$	−7.52137	−0.381349	−0.190674	+	0.981653i	$0.561067\pi$
$390$	0	0
$391$	0.536110	0.0271122
$392$	−17.4526	−0.881487
$393$	0	0
$394$	−4.00106	−0.201571
$395$	−2.90251	−0.146041
$396$	0	0
$397$	7.78635	0.390786	0.195393	−	0.980725i	$-0.437402\pi$
$397$	7.78635	0.390786	0.195393	+	0.980725i	$0.437402\pi$
$398$	−14.7645	−0.740079
$399$	0	0
$400$	7.48702	0.374351
$401$	−29.2302	−1.45968	−0.729842	−	0.683616i	$-0.760406\pi$
$401$	−29.2302	−1.45968	−0.729842	+	0.683616i	$0.760406\pi$
$402$	0	0
$403$	−12.7541	−0.635329
$404$	6.15320	0.306133
$405$	0	0
$406$	−1.21266	−0.0601831
$407$	3.57178	0.177047
$408$	0	0
$409$	−13.3398	−0.659611	−0.329806	−	0.944049i	$-0.606983\pi$
$409$	−13.3398	−0.659611	−0.329806	+	0.944049i	$0.606983\pi$
$410$	−5.48967	−0.271116
$411$	0	0
$412$	−2.59149	−0.127674
$413$	−6.65127	−0.327288
$414$	0	0
$415$	−3.60732	−0.177076
$416$	7.56645	0.370976
$417$	0	0
$418$	1.82788	0.0894045
$419$	−28.6408	−1.39919	−0.699597	−	0.714537i	$-0.746637\pi$
$419$	−28.6408	−1.39919	−0.699597	+	0.714537i	$0.746637\pi$
$420$	0	0
$421$	−35.9225	−1.75075	−0.875377	−	0.483441i	$-0.839387\pi$
$421$	−35.9225	−1.75075	−0.875377	+	0.483441i	$0.839387\pi$
$422$	−5.53337	−0.269360
$423$	0	0
$424$	−33.8057	−1.64175
$425$	−2.58560	−0.125420
$426$	0	0
$427$	13.7327	0.664570
$428$	−8.24154	−0.398370
$429$	0	0
$430$	1.25297	0.0604238
$431$	−27.2374	−1.31198	−0.655989	−	0.754770i	$-0.727748\pi$
$431$	−27.2374	−1.31198	−0.655989	+	0.754770i	$0.727748\pi$
$432$	0	0
$433$	−23.7658	−1.14211	−0.571057	−	0.820910i	$-0.693466\pi$
$433$	−23.7658	−1.14211	−0.571057	+	0.820910i	$0.693466\pi$
$434$	−9.09446	−0.436548
$435$	0	0
$436$	−8.64552	−0.414045
$437$	0.585904	0.0280276
$438$	0	0
$439$	27.4214	1.30875	0.654377	−	0.756169i	$-0.272931\pi$
$439$	27.4214	1.30875	0.654377	+	0.756169i	$0.272931\pi$
$440$	−3.75068	−0.178807
$441$	0	0
$442$	0.974836	0.0463682
$443$	−24.9966	−1.18763	−0.593813	−	0.804603i	$-0.702378\pi$
$443$	−24.9966	−1.18763	−0.593813	+	0.804603i	$0.702378\pi$
$444$	0	0
$445$	1.15801	0.0548950
$446$	13.7413	0.650670
$447$	0	0
$448$	8.91663	0.421271
$449$	−19.5495	−0.922598	−0.461299	−	0.887245i	$-0.652616\pi$
$449$	−19.5495	−0.922598	−0.461299	+	0.887245i	$0.652616\pi$
$450$	0	0
$451$	−35.5977	−1.67623
$452$	−5.84909	−0.275118
$453$	0	0
$454$	13.4558	0.631512
$455$	−0.811703	−0.0380532
$456$	0	0
$457$	−11.7789	−0.550995	−0.275498	−	0.961302i	$-0.588843\pi$
$457$	−11.7789	−0.550995	−0.275498	+	0.961302i	$0.588843\pi$
$458$	−20.8368	−0.973638
$459$	0	0
$460$	−0.360562	−0.0168113
$461$	2.35402	0.109638	0.0548189	−	0.998496i	$-0.482542\pi$
$461$	2.35402	0.109638	0.0548189	+	0.998496i	$0.482542\pi$
$462$	0	0
$463$	33.5600	1.55967	0.779833	−	0.625988i	$-0.215304\pi$
$463$	33.5600	1.55967	0.779833	+	0.625988i	$0.215304\pi$
$464$	−1.55239	−0.0720679
$465$	0	0
$466$	11.7656	0.545032
$467$	17.2653	0.798945	0.399472	−	0.916745i	$-0.369193\pi$
$467$	17.2653	0.798945	0.399472	+	0.916745i	$0.369193\pi$
$468$	0	0
$469$	−16.2608	−0.750852
$470$	−0.0812163	−0.00374623
$471$	0	0
$472$	−17.9133	−0.824525
$473$	8.12489	0.373583
$474$	0	0
$475$	−2.82576	−0.129655
$476$	−0.520945	−0.0238775
$477$	0	0
$478$	−10.2753	−0.469980
$479$	−31.1558	−1.42354	−0.711771	−	0.702411i	$-0.752107\pi$
$479$	−31.1558	−1.42354	−0.711771	+	0.702411i	$0.752107\pi$
$480$	0	0
$481$	2.08181	0.0949226
$482$	6.37796	0.290508
$483$	0	0
$484$	2.13035	0.0968339
$485$	−3.06583	−0.139212
$486$	0	0
$487$	13.4246	0.608326	0.304163	−	0.952620i	$-0.401623\pi$
$487$	13.4246	0.608326	0.304163	+	0.952620i	$0.401623\pi$
$488$	36.9849	1.67423
$489$	0	0
$490$	2.57097	0.116144
$491$	−4.34995	−0.196310	−0.0981552	−	0.995171i	$-0.531294\pi$
$491$	−4.34995	−0.196310	−0.0981552	+	0.995171i	$0.531294\pi$
$492$	0	0
$493$	0.536110	0.0241452
$494$	1.06538	0.0479337
$495$	0	0
$496$	−11.6423	−0.522756
$497$	15.0227	0.673858
$498$	0	0
$499$	−34.3267	−1.53668	−0.768338	−	0.640044i	$-0.778916\pi$
$499$	−34.3267	−1.53668	−0.768338	+	0.640044i	$0.778916\pi$
$500$	3.54176	0.158392
$501$	0	0
$502$	9.98266	0.445548
$503$	35.4539	1.58081	0.790406	−	0.612584i	$-0.209870\pi$
$503$	35.4539	1.58081	0.790406	+	0.612584i	$0.209870\pi$
$504$	0	0
$505$	−3.02237	−0.134494
$506$	3.11976	0.138690
$507$	0	0
$508$	4.74046	0.210324
$509$	−11.3342	−0.502378	−0.251189	−	0.967938i	$-0.580822\pi$
$509$	−11.3342	−0.502378	−0.251189	+	0.967938i	$0.580822\pi$
$510$	0	0
$511$	−5.18069	−0.229180
$512$	16.3905	0.724363
$513$	0	0
$514$	−29.8837	−1.31812
$515$	1.27291	0.0560910
$516$	0	0
$517$	−0.526646	−0.0231619
$518$	1.48446	0.0652234
$519$	0	0
$520$	−2.18609	−0.0958662
$521$	13.6651	0.598679	0.299339	−	0.954147i	$-0.403234\pi$
$521$	13.6651	0.598679	0.299339	+	0.954147i	$0.403234\pi$
$522$	0	0
$523$	4.43578	0.193963	0.0969816	−	0.995286i	$-0.469081\pi$
$523$	4.43578	0.193963	0.0969816	+	0.995286i	$0.469081\pi$
$524$	−11.5642	−0.505186
$525$	0	0
$526$	29.7814	1.29853
$527$	4.02062	0.175141
$528$	0	0
$529$	1.00000	0.0434783
$530$	4.97997	0.216316
$531$	0	0
$532$	−0.569331	−0.0246836
$533$	−20.7481	−0.898702
$534$	0	0
$535$	4.04814	0.175016
$536$	−43.7936	−1.89160
$537$	0	0
$538$	−8.90896	−0.384092
$539$	16.6714	0.718088
$540$	0	0
$541$	−43.6790	−1.87791	−0.938954	−	0.344044i	$-0.888203\pi$
$541$	−43.6790	−1.87791	−0.938954	+	0.344044i	$0.888203\pi$
$542$	26.7542	1.14919
$543$	0	0
$544$	−2.38525	−0.102267
$545$	4.24657	0.181903
$546$	0	0
$547$	−22.1181	−0.945702	−0.472851	−	0.881142i	$-0.656775\pi$
$547$	−22.1181	−0.945702	−0.472851	+	0.881142i	$0.656775\pi$
$548$	−5.00345	−0.213737
$549$	0	0
$550$	−15.0463	−0.641575
$551$	0.585904	0.0249604
$552$	0	0
$553$	−7.82226	−0.332637
$554$	18.8086	0.799099
$555$	0	0
$556$	19.1359	0.811544
$557$	4.37611	0.185422	0.0927108	−	0.995693i	$-0.470447\pi$
$557$	4.37611	0.185422	0.0927108	+	0.995693i	$0.470447\pi$
$558$	0	0
$559$	4.73560	0.200294
$560$	−0.740945	−0.0313106
$561$	0	0
$562$	−2.82065	−0.118982
$563$	−11.8239	−0.498318	−0.249159	−	0.968463i	$-0.580154\pi$
$563$	−11.8239	−0.498318	−0.249159	+	0.968463i	$0.580154\pi$
$564$	0	0
$565$	2.87300	0.120868
$566$	18.1062	0.761058
$567$	0	0
$568$	40.4592	1.69763
$569$	−38.5706	−1.61696	−0.808482	−	0.588521i	$-0.799710\pi$
$569$	−38.5706	−1.61696	−0.808482	+	0.588521i	$0.799710\pi$
$570$	0	0
$571$	23.5913	0.987264	0.493632	−	0.869671i	$-0.335669\pi$
$571$	23.5913	0.987264	0.493632	+	0.869671i	$0.335669\pi$
$572$	−4.25141	−0.177760
$573$	0	0
$574$	−14.7947	−0.617517
$575$	−4.82290	−0.201129
$576$	0	0
$577$	1.28918	0.0536692	0.0268346	−	0.999640i	$-0.491457\pi$
$577$	1.28918	0.0536692	0.0268346	+	0.999640i	$0.491457\pi$
$578$	17.8694	0.743269
$579$	0	0
$580$	−0.360562	−0.0149715
$581$	−9.72172	−0.403325
$582$	0	0
$583$	32.2926	1.33742
$584$	−13.9527	−0.577366
$585$	0	0
$586$	−5.06634	−0.209288
$587$	7.18244	0.296451	0.148226	−	0.988954i	$-0.452644\pi$
$587$	7.18244	0.296451	0.148226	+	0.988954i	$0.452644\pi$
$588$	0	0
$589$	4.39406	0.181054
$590$	2.63883	0.108639
$591$	0	0
$592$	1.90034	0.0781034
$593$	−29.1976	−1.19900	−0.599502	−	0.800373i	$-0.704635\pi$
$593$	−29.1976	−1.19900	−0.599502	+	0.800373i	$0.704635\pi$
$594$	0	0
$595$	0.255881	0.0104901
$596$	0.733558	0.0300477
$597$	0	0
$598$	1.81835	0.0743579
$599$	4.66540	0.190623	0.0953115	−	0.995448i	$-0.469615\pi$
$599$	4.66540	0.190623	0.0953115	+	0.995448i	$0.469615\pi$
$600$	0	0
$601$	25.9769	1.05962	0.529810	−	0.848117i	$-0.322263\pi$
$601$	25.9769	1.05962	0.529810	+	0.848117i	$0.322263\pi$
$602$	3.37676	0.137627
$603$	0	0
$604$	4.98742	0.202935
$605$	−1.04640	−0.0425422
$606$	0	0
$607$	−14.6632	−0.595162	−0.297581	−	0.954697i	$-0.596180\pi$
$607$	−14.6632	−0.595162	−0.297581	+	0.954697i	$0.596180\pi$
$608$	−2.60680	−0.105720
$609$	0	0
$610$	−5.44831	−0.220596
$611$	−0.306956	−0.0124181
$612$	0	0
$613$	−15.1212	−0.610741	−0.305371	−	0.952234i	$-0.598780\pi$
$613$	−15.1212	−0.610741	−0.305371	+	0.952234i	$0.598780\pi$
$614$	36.4260	1.47003
$615$	0	0
$616$	−10.1081	−0.407266
$617$	8.37624	0.337215	0.168607	−	0.985683i	$-0.446073\pi$
$617$	8.37624	0.337215	0.168607	+	0.985683i	$0.446073\pi$
$618$	0	0
$619$	24.4794	0.983912	0.491956	−	0.870620i	$-0.336282\pi$
$619$	24.4794	0.983912	0.491956	+	0.870620i	$0.336282\pi$
$620$	−2.70408	−0.108598
$621$	0	0
$622$	4.24648	0.170268
$623$	3.12084	0.125034
$624$	0	0
$625$	22.3748	0.894993
$626$	−22.8905	−0.914889
$627$	0	0
$628$	1.36276	0.0543802
$629$	−0.656272	−0.0261673
$630$	0	0
$631$	−19.8813	−0.791463	−0.395731	−	0.918366i	$-0.629509\pi$
$631$	−19.8813	−0.791463	−0.395731	+	0.918366i	$0.629509\pi$
$632$	−21.0670	−0.838000
$633$	0	0
$634$	30.2359	1.20082
$635$	−2.32845	−0.0924018
$636$	0	0
$637$	9.71693	0.384999
$638$	3.11976	0.123512
$639$	0	0
$640$	0.207161	0.00818876
$641$	−30.4724	−1.20359	−0.601793	−	0.798652i	$-0.705547\pi$
$641$	−30.4724	−1.20359	−0.601793	+	0.798652i	$0.705547\pi$
$642$	0	0
$643$	16.0989	0.634879	0.317440	−	0.948278i	$-0.397177\pi$
$643$	16.0989	0.634879	0.317440	+	0.948278i	$0.397177\pi$
$644$	−0.971713	−0.0382909
$645$	0	0
$646$	−0.335851	−0.0132139
$647$	26.4180	1.03860	0.519299	−	0.854593i	$-0.326193\pi$
$647$	26.4180	1.03860	0.519299	+	0.854593i	$0.326193\pi$
$648$	0	0
$649$	17.1115	0.671684
$650$	−8.76972	−0.343977
$651$	0	0
$652$	8.40041	0.328985
$653$	−29.0738	−1.13775	−0.568873	−	0.822425i	$-0.692621\pi$
$653$	−29.0738	−1.13775	−0.568873	+	0.822425i	$0.692621\pi$
$654$	0	0
$655$	5.68020	0.221944
$656$	−18.9395	−0.739462
$657$	0	0
$658$	−0.218878	−0.00853275
$659$	−22.7601	−0.886607	−0.443303	−	0.896372i	$-0.646194\pi$
$659$	−22.7601	−0.886607	−0.443303	+	0.896372i	$0.646194\pi$
$660$	0	0
$661$	35.5742	1.38368	0.691838	−	0.722053i	$-0.256801\pi$
$661$	35.5742	1.38368	0.691838	+	0.722053i	$0.256801\pi$
$662$	18.9168	0.735222
$663$	0	0
$664$	−26.1826	−1.01608
$665$	0.279648	0.0108443
$666$	0	0
$667$	1.00000	0.0387202
$668$	−13.9128	−0.538301
$669$	0	0
$670$	6.45132	0.249236
$671$	−35.3295	−1.36388
$672$	0	0
$673$	−45.9525	−1.77134	−0.885669	−	0.464318i	$-0.846300\pi$
$673$	−45.9525	−1.77134	−0.885669	+	0.464318i	$0.846300\pi$
$674$	20.8134	0.801703
$675$	0	0
$676$	8.66013	0.333082
$677$	48.5051	1.86420	0.932102	−	0.362196i	$-0.117973\pi$
$677$	48.5051	1.86420	0.932102	+	0.362196i	$0.117973\pi$
$678$	0	0
$679$	−8.26241	−0.317082
$680$	0.689142	0.0264274
$681$	0	0
$682$	23.3970	0.895917
$683$	−1.90270	−0.0728048	−0.0364024	−	0.999337i	$-0.511590\pi$
$683$	−1.90270	−0.0728048	−0.0364024	+	0.999337i	$0.511590\pi$
$684$	0	0
$685$	2.45763	0.0939012
$686$	15.4173	0.588637
$687$	0	0
$688$	4.32278	0.164805
$689$	18.8217	0.717051
$690$	0	0
$691$	4.71656	0.179426	0.0897132	−	0.995968i	$-0.471405\pi$
$691$	4.71656	0.179426	0.0897132	+	0.995968i	$0.471405\pi$
$692$	12.2130	0.464270
$693$	0	0
$694$	7.68766	0.291820
$695$	−9.39931	−0.356536
$696$	0	0
$697$	6.54065	0.247745
$698$	−20.3240	−0.769276
$699$	0	0
$700$	4.68647	0.177132
$701$	−21.6293	−0.816926	−0.408463	−	0.912775i	$-0.633935\pi$
$701$	−21.6293	−0.816926	−0.408463	+	0.912775i	$0.633935\pi$
$702$	0	0
$703$	−0.717227	−0.0270507
$704$	−22.9395	−0.864564
$705$	0	0
$706$	3.40625	0.128196
$707$	−8.14529	−0.306335
$708$	0	0
$709$	−9.74667	−0.366044	−0.183022	−	0.983109i	$-0.558588\pi$
$709$	−9.74667	−0.366044	−0.183022	+	0.983109i	$0.558588\pi$
$710$	−5.96011	−0.223679
$711$	0	0
$712$	8.40507	0.314993
$713$	7.49962	0.280863
$714$	0	0
$715$	2.08824	0.0780957
$716$	−0.123058	−0.00459888
$717$	0	0
$718$	−11.1013	−0.414295
$719$	24.3097	0.906599	0.453300	−	0.891358i	$-0.350247\pi$
$719$	24.3097	0.906599	0.453300	+	0.891358i	$0.350247\pi$
$720$	0	0
$721$	3.43049	0.127758
$722$	19.9481	0.742391
$723$	0	0
$724$	9.98210	0.370982
$725$	−4.82290	−0.179118
$726$	0	0
$727$	1.84325	0.0683625	0.0341812	−	0.999416i	$-0.489118\pi$
$727$	1.84325	0.0683625	0.0341812	+	0.999416i	$0.489118\pi$
$728$	−5.89150	−0.218353
$729$	0	0
$730$	2.05540	0.0760736
$731$	−1.49285	−0.0552151
$732$	0	0
$733$	−16.7848	−0.619961	−0.309981	−	0.950743i	$-0.600323\pi$
$733$	−16.7848	−0.619961	−0.309981	+	0.950743i	$0.600323\pi$
$734$	−2.30286	−0.0850001
$735$	0	0
$736$	−4.44918	−0.163999
$737$	41.8335	1.54096
$738$	0	0
$739$	43.0787	1.58467	0.792337	−	0.610083i	$-0.208864\pi$
$739$	43.0787	1.58467	0.792337	+	0.610083i	$0.208864\pi$
$740$	0.441377	0.0162253
$741$	0	0
$742$	13.4210	0.492701
$743$	0.629086	0.0230789	0.0115395	−	0.999933i	$-0.496327\pi$
$743$	0.629086	0.0230789	0.0115395	+	0.999933i	$0.496327\pi$
$744$	0	0
$745$	−0.360314	−0.0132009
$746$	−11.4301	−0.418487
$747$	0	0
$748$	1.34022	0.0490031
$749$	10.9097	0.398633
$750$	0	0
$751$	32.3302	1.17975	0.589873	−	0.807496i	$-0.299178\pi$
$751$	32.3302	1.17975	0.589873	+	0.807496i	$0.299178\pi$
$752$	−0.280198	−0.0102178
$753$	0	0
$754$	1.81835	0.0662205
$755$	−2.44976	−0.0891558
$756$	0	0
$757$	41.8728	1.52189	0.760946	−	0.648815i	$-0.224735\pi$
$757$	41.8728	1.52189	0.760946	+	0.648815i	$0.224735\pi$
$758$	−9.75744	−0.354406
$759$	0	0
$760$	0.753151	0.0273196
$761$	22.7021	0.822950	0.411475	−	0.911421i	$-0.365014\pi$
$761$	22.7021	0.822950	0.411475	+	0.911421i	$0.365014\pi$
$762$	0	0
$763$	11.4445	0.414318
$764$	−22.4289	−0.811451
$765$	0	0
$766$	11.8574	0.428426
$767$	9.97343	0.360120
$768$	0	0
$769$	−39.9953	−1.44227	−0.721134	−	0.692795i	$-0.756379\pi$
$769$	−39.9953	−1.44227	−0.721134	+	0.692795i	$0.756379\pi$
$770$	1.48904	0.0536612
$771$	0	0
$772$	−6.01867	−0.216617
$773$	6.67792	0.240188	0.120094	−	0.992763i	$-0.461680\pi$
$773$	6.67792	0.240188	0.120094	+	0.992763i	$0.461680\pi$
$774$	0	0
$775$	−36.1699	−1.29926
$776$	−22.2524	−0.798815
$777$	0	0
$778$	8.04198	0.288319
$779$	7.14815	0.256109
$780$	0	0
$781$	−38.6482	−1.38294
$782$	−0.573218	−0.0204982
$783$	0	0
$784$	8.86988	0.316782
$785$	−0.669372	−0.0238909
$786$	0	0
$787$	26.5400	0.946048	0.473024	−	0.881050i	$-0.343162\pi$
$787$	26.5400	0.946048	0.473024	+	0.881050i	$0.343162\pi$
$788$	−3.20609	−0.114212
$789$	0	0
$790$	3.10342	0.110415
$791$	7.74272	0.275299
$792$	0	0
$793$	−20.5918	−0.731237
$794$	−8.32530	−0.295454
$795$	0	0
$796$	−11.8310	−0.419337
$797$	31.8075	1.12668	0.563340	−	0.826225i	$-0.309516\pi$
$797$	31.8075	1.12668	0.563340	+	0.826225i	$0.309516\pi$
$798$	0	0
$799$	0.0967648	0.00342329
$800$	21.4580	0.758653
$801$	0	0
$802$	31.2534	1.10360
$803$	13.3282	0.470341
$804$	0	0
$805$	0.477293	0.0168224
$806$	13.6369	0.480341
$807$	0	0
$808$	−21.9370	−0.771740
$809$	−17.4024	−0.611836	−0.305918	−	0.952058i	$-0.598963\pi$
$809$	−17.4024	−0.611836	−0.305918	+	0.952058i	$0.598963\pi$
$810$	0	0
$811$	48.5297	1.70411	0.852055	−	0.523452i	$-0.175356\pi$
$811$	48.5297	1.70411	0.852055	+	0.523452i	$0.175356\pi$
$812$	−0.971713	−0.0341005
$813$	0	0
$814$	−3.81901	−0.133856
$815$	−4.12617	−0.144533
$816$	0	0
$817$	−1.63151	−0.0570793
$818$	14.2632	0.498700
$819$	0	0
$820$	−4.39893	−0.153617
$821$	−21.8969	−0.764207	−0.382103	−	0.924120i	$-0.624800\pi$
$821$	−21.8969	−0.764207	−0.382103	+	0.924120i	$0.624800\pi$
$822$	0	0
$823$	−12.2048	−0.425433	−0.212717	−	0.977114i	$-0.568231\pi$
$823$	−12.2048	−0.425433	−0.212717	+	0.977114i	$0.568231\pi$
$824$	9.23902	0.321856
$825$	0	0
$826$	7.11165	0.247446
$827$	−36.2068	−1.25903	−0.629516	−	0.776987i	$-0.716747\pi$
$827$	−36.2068	−1.25903	−0.629516	+	0.776987i	$0.716747\pi$
$828$	0	0
$829$	−2.38205	−0.0827320	−0.0413660	−	0.999144i	$-0.513171\pi$
$829$	−2.38205	−0.0827320	−0.0413660	+	0.999144i	$0.513171\pi$
$830$	3.85701	0.133879
$831$	0	0
$832$	−13.3703	−0.463531
$833$	−3.06317	−0.106132
$834$	0	0
$835$	6.83377	0.236492
$836$	1.46470	0.0506576
$837$	0	0
$838$	30.6232	1.05786
$839$	−52.3933	−1.80882	−0.904408	−	0.426668i	$-0.859687\pi$
$839$	−52.3933	−1.80882	−0.904408	+	0.426668i	$0.859687\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	38.4089	1.32366
$843$	0	0
$844$	−4.43394	−0.152623
$845$	−4.25374	−0.146333
$846$	0	0
$847$	−2.82004	−0.0968978
$848$	17.1810	0.589998
$849$	0	0
$850$	2.76457	0.0948240
$851$	−1.22414	−0.0419629
$852$	0	0
$853$	−12.3989	−0.424529	−0.212265	−	0.977212i	$-0.568084\pi$
$853$	−12.3989	−0.424529	−0.212265	+	0.977212i	$0.568084\pi$
$854$	−14.6832	−0.502448
$855$	0	0
$856$	29.3822	1.00426
$857$	−11.3105	−0.386359	−0.193179	−	0.981163i	$-0.561880\pi$
$857$	−11.3105	−0.386359	−0.193179	+	0.981163i	$0.561880\pi$
$858$	0	0
$859$	−27.2778	−0.930706	−0.465353	−	0.885125i	$-0.654073\pi$
$859$	−27.2778	−0.930706	−0.465353	+	0.885125i	$0.654073\pi$
$860$	1.00402	0.0342368
$861$	0	0
$862$	29.1227	0.991923
$863$	26.0532	0.886863	0.443431	−	0.896308i	$-0.353761\pi$
$863$	26.0532	0.886863	0.443431	+	0.896308i	$0.353761\pi$
$864$	0	0
$865$	−5.99888	−0.203968
$866$	25.4109	0.863496
$867$	0	0
$868$	−7.28748	−0.247353
$869$	20.1240	0.682662
$870$	0	0
$871$	24.3827	0.826175
$872$	30.8224	1.04378
$873$	0	0
$874$	−0.626459	−0.0211903
$875$	−4.68840	−0.158497
$876$	0	0
$877$	−53.0592	−1.79168	−0.895841	−	0.444375i	$-0.853426\pi$
$877$	−53.0592	−1.79168	−0.895841	+	0.444375i	$0.853426\pi$
$878$	−29.3195	−0.989484
$879$	0	0
$880$	1.90620	0.0642580
$881$	40.1909	1.35406	0.677032	−	0.735953i	$-0.263266\pi$
$881$	40.1909	1.35406	0.677032	+	0.735953i	$0.263266\pi$
$882$	0	0
$883$	20.1420	0.677833	0.338916	−	0.940816i	$-0.389940\pi$
$883$	20.1420	0.677833	0.338916	+	0.940816i	$0.389940\pi$
$884$	0.781145	0.0262728
$885$	0	0
$886$	26.7268	0.897906
$887$	−28.3924	−0.953325	−0.476662	−	0.879086i	$-0.658154\pi$
$887$	−28.3924	−0.953325	−0.476662	+	0.879086i	$0.658154\pi$
$888$	0	0
$889$	−6.27517	−0.210463
$890$	−1.23816	−0.0415034
$891$	0	0
$892$	11.0110	0.368677
$893$	0.105752	0.00353887
$894$	0	0
$895$	0.0604443	0.00202043
$896$	0.558299	0.0186515
$897$	0	0
$898$	20.9027	0.697531
$899$	7.49962	0.250126
$900$	0	0
$901$	−5.93337	−0.197669
$902$	38.0617	1.26732
$903$	0	0
$904$	20.8528	0.693553
$905$	−4.90308	−0.162984
$906$	0	0
$907$	−24.3598	−0.808854	−0.404427	−	0.914570i	$-0.632529\pi$
$907$	−24.3598	−0.808854	−0.404427	+	0.914570i	$0.632529\pi$
$908$	10.7823	0.357822
$909$	0	0
$910$	0.867887	0.0287702
$911$	−33.0626	−1.09541	−0.547707	−	0.836670i	$-0.684499\pi$
$911$	−33.0626	−1.09541	−0.547707	+	0.836670i	$0.684499\pi$
$912$	0	0
$913$	25.0107	0.827733
$914$	12.5942	0.416580
$915$	0	0
$916$	−16.6967	−0.551675
$917$	15.3081	0.505519
$918$	0	0
$919$	47.7236	1.57426	0.787128	−	0.616789i	$-0.211567\pi$
$919$	47.7236	1.57426	0.787128	+	0.616789i	$0.211567\pi$
$920$	1.28545	0.0423800
$921$	0	0
$922$	−2.51696	−0.0828918
$923$	−22.5261	−0.741457
$924$	0	0
$925$	5.90389	0.194119
$926$	−35.8829	−1.17919
$927$	0	0
$928$	−4.44918	−0.146052
$929$	−21.6699	−0.710968	−0.355484	−	0.934682i	$-0.615684\pi$
$929$	−21.6699	−0.710968	−0.355484	+	0.934682i	$0.615684\pi$
$930$	0	0
$931$	−3.34768	−0.109716
$932$	9.42790	0.308821
$933$	0	0
$934$	−18.4604	−0.604043
$935$	−0.658297	−0.0215286
$936$	0	0
$937$	11.3492	0.370764	0.185382	−	0.982667i	$-0.440648\pi$
$937$	11.3492	0.370764	0.185382	+	0.982667i	$0.440648\pi$
$938$	17.3863	0.567683
$939$	0	0
$940$	−0.0650794	−0.00212266
$941$	12.8107	0.417618	0.208809	−	0.977956i	$-0.433041\pi$
$941$	12.8107	0.417618	0.208809	+	0.977956i	$0.433041\pi$
$942$	0	0
$943$	12.2002	0.397293
$944$	9.10403	0.296311
$945$	0	0
$946$	−8.68727	−0.282448
$947$	−17.0001	−0.552428	−0.276214	−	0.961096i	$-0.589080\pi$
$947$	−17.0001	−0.552428	−0.276214	+	0.961096i	$0.589080\pi$
$948$	0	0
$949$	7.76834	0.252171
$950$	3.02135	0.0980255
$951$	0	0
$952$	1.85724	0.0601934
$953$	−36.4300	−1.18008	−0.590042	−	0.807373i	$-0.700889\pi$
$953$	−36.4300	−1.18008	−0.590042	+	0.807373i	$0.700889\pi$
$954$	0	0
$955$	11.0168	0.356495
$956$	−8.23367	−0.266296
$957$	0	0
$958$	33.3123	1.07627
$959$	6.62331	0.213878
$960$	0	0
$961$	25.2443	0.814332
$962$	−2.22591	−0.0717663
$963$	0	0
$964$	5.11072	0.164605
$965$	2.95629	0.0951664
$966$	0	0
$967$	−23.2204	−0.746719	−0.373359	−	0.927687i	$-0.621794\pi$
$967$	−23.2204	−0.746719	−0.373359	+	0.927687i	$0.621794\pi$
$968$	−7.59497	−0.244111
$969$	0	0
$970$	3.27804	0.105252
$971$	25.4051	0.815289	0.407644	−	0.913141i	$-0.366350\pi$
$971$	25.4051	0.815289	0.407644	+	0.913141i	$0.366350\pi$
$972$	0	0
$973$	−25.3311	−0.812079
$974$	−14.3538	−0.459926
$975$	0	0
$976$	−18.7968	−0.601670
$977$	−35.6930	−1.14192	−0.570961	−	0.820978i	$-0.693429\pi$
$977$	−35.6930	−1.14192	−0.570961	+	0.820978i	$0.693429\pi$
$978$	0	0
$979$	−8.02886	−0.256604
$980$	2.06014	0.0658088
$981$	0	0
$982$	4.65104	0.148421
$983$	30.1919	0.962973	0.481486	−	0.876454i	$-0.340097\pi$
$983$	30.1919	0.962973	0.481486	+	0.876454i	$0.340097\pi$
$984$	0	0
$985$	1.57479	0.0501770
$986$	−0.573218	−0.0182550
$987$	0	0
$988$	0.853699	0.0271598
$989$	−2.78460	−0.0885451
$990$	0	0
$991$	−18.5980	−0.590784	−0.295392	−	0.955376i	$-0.595450\pi$
$991$	−18.5980	−0.590784	−0.295392	+	0.955376i	$0.595450\pi$
$992$	−33.3672	−1.05941
$993$	0	0
$994$	−16.0625	−0.509471
$995$	5.81122	0.184228
$996$	0	0
$997$	−57.0452	−1.80664	−0.903319	−	0.428969i	$-0.858877\pi$
$997$	−57.0452	−1.80664	−0.903319	+	0.428969i	$0.858877\pi$
$998$	36.7027	1.16181
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.9	✓	22
3.2	odd	2		6003.2.a.u.1.14	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.9	✓	22	1.1	even	1	trivial
6003.2.a.u.1.14	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.06922 q^{2} -0.856774 q^{4} +0.420836 q^{5} +1.13415 q^{7} +3.05451 q^{8} -0.449966 q^{10} -2.91779 q^{11} -1.70064 q^{13} -1.21266 q^{14} -1.55239 q^{16} +0.536110 q^{17} +0.585904 q^{19} -0.360562 q^{20} +3.11976 q^{22} +1.00000 q^{23} -4.82290 q^{25} +1.81835 q^{26} -0.971713 q^{28} +1.00000 q^{29} +7.49962 q^{31} -4.44918 q^{32} -0.573218 q^{34} +0.477293 q^{35} -1.22414 q^{37} -0.626459 q^{38} +1.28545 q^{40} +12.2002 q^{41} -2.78460 q^{43} +2.49989 q^{44} -1.06922 q^{46} +0.180494 q^{47} -5.71370 q^{49} +5.15673 q^{50} +1.45706 q^{52} -11.0675 q^{53} -1.22791 q^{55} +3.46429 q^{56} -1.06922 q^{58} -5.86452 q^{59} +12.1083 q^{61} -8.01872 q^{62} +7.86192 q^{64} -0.715690 q^{65} -14.3374 q^{67} -0.459325 q^{68} -0.510330 q^{70} +13.2457 q^{71} -4.56789 q^{73} +1.30887 q^{74} -0.501988 q^{76} -3.30923 q^{77} -6.89701 q^{79} -0.653302 q^{80} -13.0447 q^{82} -8.57178 q^{83} +0.225614 q^{85} +2.97734 q^{86} -8.91244 q^{88} +2.75169 q^{89} -1.92878 q^{91} -0.856774 q^{92} -0.192988 q^{94} +0.246570 q^{95} -7.28509 q^{97} +6.10918 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34}+ \cdots - 28 q^{98}+O(q^{100}) \) 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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