LMFDB - Embedded newform 6003.2.a.t.1.20

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			1.20
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)

$f(q)$	$=$	$q+2.08661 q^{2} +2.35393 q^{4} +1.43041 q^{5} -0.274571 q^{7} +0.738503 q^{8} +O(q^{10})$	\(q+2.08661 q^{2} +2.35393 q^{4} +1.43041 q^{5} -0.274571 q^{7} +0.738503 q^{8} +2.98470 q^{10} -6.10520 q^{11} +0.188265 q^{13} -0.572921 q^{14} -3.16689 q^{16} -2.24526 q^{17} +3.65245 q^{19} +3.36708 q^{20} -12.7391 q^{22} +1.00000 q^{23} -2.95393 q^{25} +0.392834 q^{26} -0.646319 q^{28} +1.00000 q^{29} -4.05707 q^{31} -8.08505 q^{32} -4.68496 q^{34} -0.392749 q^{35} -4.16777 q^{37} +7.62122 q^{38} +1.05636 q^{40} -3.24174 q^{41} -4.69667 q^{43} -14.3712 q^{44} +2.08661 q^{46} +6.10569 q^{47} -6.92461 q^{49} -6.16369 q^{50} +0.443161 q^{52} +3.87730 q^{53} -8.73294 q^{55} -0.202771 q^{56} +2.08661 q^{58} +2.91676 q^{59} -2.77700 q^{61} -8.46551 q^{62} -10.5365 q^{64} +0.269296 q^{65} +7.02715 q^{67} -5.28516 q^{68} -0.819512 q^{70} -3.49275 q^{71} -12.1979 q^{73} -8.69650 q^{74} +8.59759 q^{76} +1.67631 q^{77} -8.81856 q^{79} -4.52994 q^{80} -6.76424 q^{82} -2.42899 q^{83} -3.21163 q^{85} -9.80011 q^{86} -4.50871 q^{88} -1.75647 q^{89} -0.0516920 q^{91} +2.35393 q^{92} +12.7402 q^{94} +5.22450 q^{95} -5.73211 q^{97} -14.4489 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.08661	1.47545	0.737727	−	0.675099i	$-0.235899\pi$
$2$	2.08661	1.47545	0.737727	+	0.675099i	$0.235899\pi$
$3$	0	0
$3$	0	0
$4$	2.35393	1.17696
$5$	1.43041	0.639698	0.319849	−	0.947468i	$-0.396368\pi$
$5$	1.43041	0.639698	0.319849	+	0.947468i	$0.396368\pi$
$6$	0	0
$7$	−0.274571	−0.103778	−0.0518890	−	0.998653i	$-0.516524\pi$
$7$	−0.274571	−0.103778	−0.0518890	+	0.998653i	$0.516524\pi$
$8$	0.738503	0.261100
$9$	0	0
$10$	2.98470	0.943845
$11$	−6.10520	−1.84079	−0.920394	−	0.390993i	$-0.872132\pi$
$11$	−6.10520	−1.84079	−0.920394	+	0.390993i	$0.872132\pi$
$12$	0	0
$13$	0.188265	0.0522152	0.0261076	−	0.999659i	$-0.491689\pi$
$13$	0.188265	0.0522152	0.0261076	+	0.999659i	$0.491689\pi$
$14$	−0.572921	−0.153120
$15$	0	0
$16$	−3.16689	−0.791721
$17$	−2.24526	−0.544554	−0.272277	−	0.962219i	$-0.587777\pi$
$17$	−2.24526	−0.544554	−0.272277	+	0.962219i	$0.587777\pi$
$18$	0	0
$19$	3.65245	0.837929	0.418965	−	0.908003i	$-0.362393\pi$
$19$	3.65245	0.837929	0.418965	+	0.908003i	$0.362393\pi$
$20$	3.36708	0.752901
$21$	0	0
$22$	−12.7391	−2.71600
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.95393	−0.590786
$26$	0.392834	0.0770412
$27$	0	0
$28$	−0.646319	−0.122143
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.05707	−0.728671	−0.364336	−	0.931268i	$-0.618704\pi$
$31$	−4.05707	−0.728671	−0.364336	+	0.931268i	$0.618704\pi$
$32$	−8.08505	−1.42925
$33$	0	0
$34$	−4.68496	−0.803465
$35$	−0.392749	−0.0663866
$36$	0	0
$37$	−4.16777	−0.685178	−0.342589	−	0.939485i	$-0.611304\pi$
$37$	−4.16777	−0.685178	−0.342589	+	0.939485i	$0.611304\pi$
$38$	7.62122	1.23633
$39$	0	0
$40$	1.05636	0.167025
$41$	−3.24174	−0.506275	−0.253138	−	0.967430i	$-0.581463\pi$
$41$	−3.24174	−0.506275	−0.253138	+	0.967430i	$0.581463\pi$
$42$	0	0
$43$	−4.69667	−0.716236	−0.358118	−	0.933676i	$-0.616581\pi$
$43$	−4.69667	−0.716236	−0.358118	+	0.933676i	$0.616581\pi$
$44$	−14.3712	−2.16654
$45$	0	0
$46$	2.08661	0.307653
$47$	6.10569	0.890606	0.445303	−	0.895380i	$-0.353096\pi$
$47$	6.10569	0.890606	0.445303	+	0.895380i	$0.353096\pi$
$48$	0	0
$49$	−6.92461	−0.989230
$50$	−6.16369	−0.871677
$51$	0	0
$52$	0.443161	0.0614554
$53$	3.87730	0.532588	0.266294	−	0.963892i	$-0.414201\pi$
$53$	3.87730	0.532588	0.266294	+	0.963892i	$0.414201\pi$
$54$	0	0
$55$	−8.73294	−1.17755
$56$	−0.202771	−0.0270965
$57$	0	0
$58$	2.08661	0.273985
$59$	2.91676	0.379730	0.189865	−	0.981810i	$-0.439195\pi$
$59$	2.91676	0.379730	0.189865	+	0.981810i	$0.439195\pi$
$60$	0	0
$61$	−2.77700	−0.355559	−0.177779	−	0.984070i	$-0.556891\pi$
$61$	−2.77700	−0.355559	−0.177779	+	0.984070i	$0.556891\pi$
$62$	−8.46551	−1.07512
$63$	0	0
$64$	−10.5365	−1.31707
$65$	0.269296	0.0334020
$66$	0	0
$67$	7.02715	0.858502	0.429251	−	0.903185i	$-0.358778\pi$
$67$	7.02715	0.858502	0.429251	+	0.903185i	$0.358778\pi$
$68$	−5.28516	−0.640920
$69$	0	0
$70$	−0.819512	−0.0979504
$71$	−3.49275	−0.414514	−0.207257	−	0.978287i	$-0.566454\pi$
$71$	−3.49275	−0.414514	−0.207257	+	0.978287i	$0.566454\pi$
$72$	0	0
$73$	−12.1979	−1.42766	−0.713830	−	0.700319i	$-0.753041\pi$
$73$	−12.1979	−1.42766	−0.713830	+	0.700319i	$0.753041\pi$
$74$	−8.69650	−1.01095
$75$	0	0
$76$	8.59759	0.986211
$77$	1.67631	0.191033
$78$	0	0
$79$	−8.81856	−0.992165	−0.496083	−	0.868275i	$-0.665229\pi$
$79$	−8.81856	−0.992165	−0.496083	+	0.868275i	$0.665229\pi$
$80$	−4.52994	−0.506463
$81$	0	0
$82$	−6.76424	−0.746986
$83$	−2.42899	−0.266616	−0.133308	−	0.991075i	$-0.542560\pi$
$83$	−2.42899	−0.266616	−0.133308	+	0.991075i	$0.542560\pi$
$84$	0	0
$85$	−3.21163	−0.348351
$86$	−9.80011	−1.05677
$87$	0	0
$88$	−4.50871	−0.480630
$89$	−1.75647	−0.186185	−0.0930926	−	0.995657i	$-0.529675\pi$
$89$	−1.75647	−0.186185	−0.0930926	+	0.995657i	$0.529675\pi$
$90$	0	0
$91$	−0.0516920	−0.00541879
$92$	2.35393	0.245414
$93$	0	0
$94$	12.7402	1.31405
$95$	5.22450	0.536022
$96$	0	0
$97$	−5.73211	−0.582008	−0.291004	−	0.956722i	$-0.593989\pi$
$97$	−5.73211	−0.582008	−0.291004	+	0.956722i	$0.593989\pi$
$98$	−14.4489	−1.45956
$99$	0	0
$100$	−6.95333	−0.695333
$101$	9.23902	0.919317	0.459659	−	0.888096i	$-0.347972\pi$
$101$	9.23902	0.919317	0.459659	+	0.888096i	$0.347972\pi$
$102$	0	0
$103$	−7.23178	−0.712568	−0.356284	−	0.934378i	$-0.615957\pi$
$103$	−7.23178	−0.712568	−0.356284	+	0.934378i	$0.615957\pi$
$104$	0.139034	0.0136334
$105$	0	0
$106$	8.09040	0.785809
$107$	−3.16077	−0.305563	−0.152781	−	0.988260i	$-0.548823\pi$
$107$	−3.16077	−0.305563	−0.152781	+	0.988260i	$0.548823\pi$
$108$	0	0
$109$	9.56401	0.916066	0.458033	−	0.888935i	$-0.348554\pi$
$109$	9.56401	0.916066	0.458033	+	0.888935i	$0.348554\pi$
$110$	−18.2222	−1.73742
$111$	0	0
$112$	0.869534	0.0821633
$113$	−19.3625	−1.82147	−0.910734	−	0.412994i	$-0.864483\pi$
$113$	−19.3625	−1.82147	−0.910734	+	0.412994i	$0.864483\pi$
$114$	0	0
$115$	1.43041	0.133386
$116$	2.35393	0.218556
$117$	0	0
$118$	6.08613	0.560274
$119$	0.616482	0.0565128
$120$	0	0
$121$	26.2735	2.38850
$122$	−5.79451	−0.524610
$123$	0	0
$124$	−9.55004	−0.857619
$125$	−11.3774	−1.01762
$126$	0	0
$127$	−12.6479	−1.12232	−0.561161	−	0.827706i	$-0.689645\pi$
$127$	−12.6479	−1.12232	−0.561161	+	0.827706i	$0.689645\pi$
$128$	−5.81552	−0.514024
$129$	0	0
$130$	0.561914	0.0492831
$131$	7.81787	0.683050	0.341525	−	0.939873i	$-0.389057\pi$
$131$	7.81787	0.683050	0.341525	+	0.939873i	$0.389057\pi$
$132$	0	0
$133$	−1.00286	−0.0869586
$134$	14.6629	1.26668
$135$	0	0
$136$	−1.65813	−0.142183
$137$	−2.22227	−0.189861	−0.0949305	−	0.995484i	$-0.530263\pi$
$137$	−2.22227	−0.189861	−0.0949305	+	0.995484i	$0.530263\pi$
$138$	0	0
$139$	13.7510	1.16635	0.583174	−	0.812347i	$-0.301811\pi$
$139$	13.7510	1.16635	0.583174	+	0.812347i	$0.301811\pi$
$140$	−0.924501	−0.0781346
$141$	0	0
$142$	−7.28800	−0.611595
$143$	−1.14939	−0.0961171
$144$	0	0
$145$	1.43041	0.118789
$146$	−25.4523	−2.10645
$147$	0	0
$148$	−9.81063	−0.806428
$149$	6.77408	0.554954	0.277477	−	0.960732i	$-0.410502\pi$
$149$	6.77408	0.554954	0.277477	+	0.960732i	$0.410502\pi$
$150$	0	0
$151$	−10.6096	−0.863395	−0.431698	−	0.902018i	$-0.642085\pi$
$151$	−10.6096	−0.863395	−0.431698	+	0.902018i	$0.642085\pi$
$152$	2.69734	0.218784
$153$	0	0
$154$	3.49780	0.281861
$155$	−5.80327	−0.466130
$156$	0	0
$157$	17.0603	1.36156	0.680778	−	0.732489i	$-0.261642\pi$
$157$	17.0603	1.36156	0.680778	+	0.732489i	$0.261642\pi$
$158$	−18.4009	−1.46389
$159$	0	0
$160$	−11.5649	−0.914288
$161$	−0.274571	−0.0216392
$162$	0	0
$163$	−24.0671	−1.88508	−0.942539	−	0.334096i	$-0.891569\pi$
$163$	−24.0671	−1.88508	−0.942539	+	0.334096i	$0.891569\pi$
$164$	−7.63082	−0.595867
$165$	0	0
$166$	−5.06835	−0.393380
$167$	4.02327	0.311330	0.155665	−	0.987810i	$-0.450248\pi$
$167$	4.02327	0.311330	0.155665	+	0.987810i	$0.450248\pi$
$168$	0	0
$169$	−12.9646	−0.997274
$170$	−6.70141	−0.513975
$171$	0	0
$172$	−11.0556	−0.842983
$173$	−0.0249861	−0.00189966	−0.000949830	−	1.00000i	$-0.500302\pi$
$173$	−0.0249861	−0.00189966	−0.000949830		1.00000i	$0.500302\pi$
$174$	0	0
$175$	0.811063	0.0613106
$176$	19.3345	1.45739
$177$	0	0
$178$	−3.66506	−0.274707
$179$	−1.00959	−0.0754606	−0.0377303	−	0.999288i	$-0.512013\pi$
$179$	−1.00959	−0.0754606	−0.0377303	+	0.999288i	$0.512013\pi$
$180$	0	0
$181$	22.8161	1.69591	0.847955	−	0.530069i	$-0.177834\pi$
$181$	22.8161	1.69591	0.847955	+	0.530069i	$0.177834\pi$
$182$	−0.107861	−0.00799518
$183$	0	0
$184$	0.738503	0.0544432
$185$	−5.96162	−0.438307
$186$	0	0
$187$	13.7077	1.00241
$188$	14.3723	1.04821
$189$	0	0
$190$	10.9015	0.790875
$191$	2.91312	0.210786	0.105393	−	0.994431i	$-0.466390\pi$
$191$	2.91312	0.210786	0.105393	+	0.994431i	$0.466390\pi$
$192$	0	0
$193$	−3.51894	−0.253299	−0.126650	−	0.991948i	$-0.540422\pi$
$193$	−3.51894	−0.253299	−0.126650	+	0.991948i	$0.540422\pi$
$194$	−11.9607	−0.858725
$195$	0	0
$196$	−16.3000	−1.16429
$197$	14.7731	1.05254	0.526272	−	0.850317i	$-0.323590\pi$
$197$	14.7731	1.05254	0.526272	+	0.850317i	$0.323590\pi$
$198$	0	0
$199$	23.7774	1.68553	0.842767	−	0.538278i	$-0.180925\pi$
$199$	23.7774	1.68553	0.842767	+	0.538278i	$0.180925\pi$
$200$	−2.18149	−0.154254
$201$	0	0
$202$	19.2782	1.35641
$203$	−0.274571	−0.0192711
$204$	0	0
$205$	−4.63702	−0.323864
$206$	−15.0899	−1.05136
$207$	0	0
$208$	−0.596213	−0.0413399
$209$	−22.2989	−1.54245
$210$	0	0
$211$	0.588832	0.0405369	0.0202684	−	0.999795i	$-0.493548\pi$
$211$	0.588832	0.0405369	0.0202684	+	0.999795i	$0.493548\pi$
$212$	9.12688	0.626837
$213$	0	0
$214$	−6.59527	−0.450844
$215$	−6.71817	−0.458175
$216$	0	0
$217$	1.11395	0.0756201
$218$	19.9563	1.35161
$219$	0	0
$220$	−20.5567	−1.38593
$221$	−0.422702	−0.0284340
$222$	0	0
$223$	−17.5937	−1.17816	−0.589079	−	0.808075i	$-0.700509\pi$
$223$	−17.5937	−1.17816	−0.589079	+	0.808075i	$0.700509\pi$
$224$	2.21992	0.148325
$225$	0	0
$226$	−40.4018	−2.68749
$227$	27.3659	1.81634	0.908169	−	0.418603i	$-0.137480\pi$
$227$	27.3659	1.81634	0.908169	+	0.418603i	$0.137480\pi$
$228$	0	0
$229$	−20.2749	−1.33980	−0.669900	−	0.742451i	$-0.733663\pi$
$229$	−20.2749	−1.33980	−0.669900	+	0.742451i	$0.733663\pi$
$230$	2.98470	0.196805
$231$	0	0
$232$	0.738503	0.0484851
$233$	−3.03886	−0.199082	−0.0995412	−	0.995033i	$-0.531737\pi$
$233$	−3.03886	−0.199082	−0.0995412	+	0.995033i	$0.531737\pi$
$234$	0	0
$235$	8.73363	0.569719
$236$	6.86584	0.446928
$237$	0	0
$238$	1.28635	0.0833820
$239$	−1.74365	−0.112787	−0.0563936	−	0.998409i	$-0.517960\pi$
$239$	−1.74365	−0.112787	−0.0563936	+	0.998409i	$0.517960\pi$
$240$	0	0
$241$	3.99474	0.257324	0.128662	−	0.991689i	$-0.458932\pi$
$241$	3.99474	0.257324	0.128662	+	0.991689i	$0.458932\pi$
$242$	54.8224	3.52412
$243$	0	0
$244$	−6.53685	−0.418479
$245$	−9.90503	−0.632809
$246$	0	0
$247$	0.687627	0.0437527
$248$	−2.99616	−0.190256
$249$	0	0
$250$	−23.7401	−1.50146
$251$	−7.41675	−0.468141	−0.234070	−	0.972220i	$-0.575205\pi$
$251$	−7.41675	−0.468141	−0.234070	+	0.972220i	$0.575205\pi$
$252$	0	0
$253$	−6.10520	−0.383831
$254$	−26.3913	−1.65594
$255$	0	0
$256$	8.93839	0.558649
$257$	19.0615	1.18902	0.594512	−	0.804087i	$-0.297345\pi$
$257$	19.0615	1.18902	0.594512	+	0.804087i	$0.297345\pi$
$258$	0	0
$259$	1.14435	0.0711064
$260$	0.633902	0.0393129
$261$	0	0
$262$	16.3128	1.00781
$263$	2.53582	0.156365	0.0781827	−	0.996939i	$-0.475088\pi$
$263$	2.53582	0.156365	0.0781827	+	0.996939i	$0.475088\pi$
$264$	0	0
$265$	5.54613	0.340696
$266$	−2.09257	−0.128303
$267$	0	0
$268$	16.5414	1.01043
$269$	−4.13080	−0.251859	−0.125930	−	0.992039i	$-0.540191\pi$
$269$	−4.13080	−0.251859	−0.125930	+	0.992039i	$0.540191\pi$
$270$	0	0
$271$	−4.10014	−0.249066	−0.124533	−	0.992215i	$-0.539743\pi$
$271$	−4.10014	−0.249066	−0.124533	+	0.992215i	$0.539743\pi$
$272$	7.11047	0.431135
$273$	0	0
$274$	−4.63700	−0.280131
$275$	18.0343	1.08751
$276$	0	0
$277$	7.14419	0.429253	0.214626	−	0.976696i	$-0.431147\pi$
$277$	7.14419	0.429253	0.214626	+	0.976696i	$0.431147\pi$
$278$	28.6930	1.72089
$279$	0	0
$280$	−0.290046	−0.0173336
$281$	−15.6506	−0.933635	−0.466817	−	0.884354i	$-0.654599\pi$
$281$	−15.6506	−0.933635	−0.466817	+	0.884354i	$0.654599\pi$
$282$	0	0
$283$	8.24117	0.489887	0.244943	−	0.969537i	$-0.421231\pi$
$283$	8.24117	0.489887	0.244943	+	0.969537i	$0.421231\pi$
$284$	−8.22168	−0.487867
$285$	0	0
$286$	−2.39833	−0.141816
$287$	0.890088	0.0525403
$288$	0	0
$289$	−11.9588	−0.703461
$290$	2.98470	0.175268
$291$	0	0
$292$	−28.7130	−1.68030
$293$	20.9302	1.22275	0.611377	−	0.791340i	$-0.290616\pi$
$293$	20.9302	1.22275	0.611377	+	0.791340i	$0.290616\pi$
$294$	0	0
$295$	4.17216	0.242913
$296$	−3.07791	−0.178900
$297$	0	0
$298$	14.1348	0.818809
$299$	0.188265	0.0108876
$300$	0	0
$301$	1.28957	0.0743296
$302$	−22.1380	−1.27390
$303$	0	0
$304$	−11.5669	−0.663406
$305$	−3.97225	−0.227450
$306$	0	0
$307$	19.1315	1.09189	0.545947	−	0.837819i	$-0.316170\pi$
$307$	19.1315	1.09189	0.545947	+	0.837819i	$0.316170\pi$
$308$	3.94591	0.224839
$309$	0	0
$310$	−12.1091	−0.687753
$311$	30.8388	1.74871	0.874353	−	0.485291i	$-0.161286\pi$
$311$	30.8388	1.74871	0.874353	+	0.485291i	$0.161286\pi$
$312$	0	0
$313$	0.287106	0.0162282	0.00811409	−	0.999967i	$-0.497417\pi$
$313$	0.287106	0.0162282	0.00811409	+	0.999967i	$0.497417\pi$
$314$	35.5980	2.00891
$315$	0	0
$316$	−20.7582	−1.16774
$317$	4.79550	0.269342	0.134671	−	0.990890i	$-0.457002\pi$
$317$	4.79550	0.269342	0.134671	+	0.990890i	$0.457002\pi$
$318$	0	0
$319$	−6.10520	−0.341826
$320$	−15.0716	−0.842526
$321$	0	0
$322$	−0.572921	−0.0319276
$323$	−8.20068	−0.456298
$324$	0	0
$325$	−0.556121	−0.0308480
$326$	−50.2185	−2.78134
$327$	0	0
$328$	−2.39404	−0.132189
$329$	−1.67644	−0.0924253
$330$	0	0
$331$	−7.68460	−0.422384	−0.211192	−	0.977445i	$-0.567735\pi$
$331$	−7.68460	−0.422384	−0.211192	+	0.977445i	$0.567735\pi$
$332$	−5.71766	−0.313798
$333$	0	0
$334$	8.39498	0.459353
$335$	10.0517	0.549183
$336$	0	0
$337$	0.712372	0.0388054	0.0194027	−	0.999812i	$-0.493824\pi$
$337$	0.712372	0.0388054	0.0194027	+	0.999812i	$0.493824\pi$
$338$	−27.0519	−1.47143
$339$	0	0
$340$	−7.55995	−0.409996
$341$	24.7692	1.34133
$342$	0	0
$343$	3.82329	0.206438
$344$	−3.46851	−0.187009
$345$	0	0
$346$	−0.0521362	−0.00280286
$347$	−20.4402	−1.09729	−0.548645	−	0.836056i	$-0.684856\pi$
$347$	−20.4402	−1.09729	−0.548645	+	0.836056i	$0.684856\pi$
$348$	0	0
$349$	3.84665	0.205907	0.102953	−	0.994686i	$-0.467171\pi$
$349$	3.84665	0.205907	0.102953	+	0.994686i	$0.467171\pi$
$350$	1.69237	0.0904609
$351$	0	0
$352$	49.3608	2.63094
$353$	6.68615	0.355868	0.177934	−	0.984042i	$-0.443059\pi$
$353$	6.68615	0.355868	0.177934	+	0.984042i	$0.443059\pi$
$354$	0	0
$355$	−4.99607	−0.265164
$356$	−4.13459	−0.219133
$357$	0	0
$358$	−2.10662	−0.111339
$359$	−25.6460	−1.35355	−0.676773	−	0.736191i	$-0.736622\pi$
$359$	−25.6460	−1.35355	−0.676773	+	0.736191i	$0.736622\pi$
$360$	0	0
$361$	−5.65962	−0.297875
$362$	47.6083	2.50223
$363$	0	0
$364$	−0.121679	−0.00637772
$365$	−17.4480	−0.913272
$366$	0	0
$367$	12.3016	0.642141	0.321070	−	0.947055i	$-0.395957\pi$
$367$	12.3016	0.642141	0.321070	+	0.947055i	$0.395957\pi$
$368$	−3.16689	−0.165085
$369$	0	0
$370$	−12.4396	−0.646702
$371$	−1.06459	−0.0552710
$372$	0	0
$373$	−22.4264	−1.16120	−0.580598	−	0.814190i	$-0.697181\pi$
$373$	−22.4264	−1.16120	−0.580598	+	0.814190i	$0.697181\pi$
$374$	28.6026	1.47901
$375$	0	0
$376$	4.50907	0.232537
$377$	0.188265	0.00969613
$378$	0	0
$379$	33.7025	1.73118	0.865590	−	0.500754i	$-0.166944\pi$
$379$	33.7025	1.73118	0.865590	+	0.500754i	$0.166944\pi$
$380$	12.2981	0.630878
$381$	0	0
$382$	6.07853	0.311004
$383$	21.0513	1.07567	0.537836	−	0.843050i	$-0.319242\pi$
$383$	21.0513	1.07567	0.537836	+	0.843050i	$0.319242\pi$
$384$	0	0
$385$	2.39781	0.122204
$386$	−7.34265	−0.373731
$387$	0	0
$388$	−13.4930	−0.685001
$389$	−34.5723	−1.75288	−0.876442	−	0.481507i	$-0.840090\pi$
$389$	−34.5723	−1.75288	−0.876442	+	0.481507i	$0.840090\pi$
$390$	0	0
$391$	−2.24526	−0.113547
$392$	−5.11385	−0.258288
$393$	0	0
$394$	30.8257	1.55298
$395$	−12.6141	−0.634686
$396$	0	0
$397$	12.2736	0.615995	0.307997	−	0.951387i	$-0.400341\pi$
$397$	12.2736	0.615995	0.307997	+	0.951387i	$0.400341\pi$
$398$	49.6140	2.48693
$399$	0	0
$400$	9.35476	0.467738
$401$	−9.95916	−0.497337	−0.248668	−	0.968589i	$-0.579993\pi$
$401$	−9.95916	−0.497337	−0.248668	+	0.968589i	$0.579993\pi$
$402$	0	0
$403$	−0.763803	−0.0380478
$404$	21.7480	1.08200
$405$	0	0
$406$	−0.572921	−0.0284336
$407$	25.4451	1.26127
$408$	0	0
$409$	−28.5666	−1.41253	−0.706264	−	0.707949i	$-0.749621\pi$
$409$	−28.5666	−1.41253	−0.706264	+	0.707949i	$0.749621\pi$
$410$	−9.67564	−0.477846
$411$	0	0
$412$	−17.0231	−0.838666
$413$	−0.800858	−0.0394076
$414$	0	0
$415$	−3.47445	−0.170554
$416$	−1.52213	−0.0746285
$417$	0	0
$418$	−46.5291	−2.27581
$419$	7.55392	0.369033	0.184517	−	0.982829i	$-0.440928\pi$
$419$	7.55392	0.369033	0.184517	+	0.982829i	$0.440928\pi$
$420$	0	0
$421$	−7.74446	−0.377442	−0.188721	−	0.982031i	$-0.560434\pi$
$421$	−7.74446	−0.377442	−0.188721	+	0.982031i	$0.560434\pi$
$422$	1.22866	0.0598103
$423$	0	0
$424$	2.86340	0.139059
$425$	6.63233	0.321715
$426$	0	0
$427$	0.762484	0.0368992
$428$	−7.44021	−0.359636
$429$	0	0
$430$	−14.0182	−0.676016
$431$	38.6217	1.86034	0.930172	−	0.367124i	$-0.119658\pi$
$431$	38.6217	1.86034	0.930172	+	0.367124i	$0.119658\pi$
$432$	0	0
$433$	−5.41271	−0.260118	−0.130059	−	0.991506i	$-0.541517\pi$
$433$	−5.41271	−0.260118	−0.130059	+	0.991506i	$0.541517\pi$
$434$	2.32438	0.111574
$435$	0	0
$436$	22.5130	1.07818
$437$	3.65245	0.174720
$438$	0	0
$439$	−12.2447	−0.584407	−0.292203	−	0.956356i	$-0.594388\pi$
$439$	−12.2447	−0.584407	−0.292203	+	0.956356i	$0.594388\pi$
$440$	−6.44930	−0.307458
$441$	0	0
$442$	−0.882013	−0.0419531
$443$	−11.2775	−0.535808	−0.267904	−	0.963446i	$-0.586331\pi$
$443$	−11.2775	−0.535808	−0.267904	+	0.963446i	$0.586331\pi$
$444$	0	0
$445$	−2.51247	−0.119102
$446$	−36.7110	−1.73832
$447$	0	0
$448$	2.89303	0.136683
$449$	−20.3814	−0.961858	−0.480929	−	0.876760i	$-0.659700\pi$
$449$	−20.3814	−0.961858	−0.480929	+	0.876760i	$0.659700\pi$
$450$	0	0
$451$	19.7915	0.931945
$452$	−45.5778	−2.14380
$453$	0	0
$454$	57.1019	2.67992
$455$	−0.0739407	−0.00346639
$456$	0	0
$457$	−14.2683	−0.667442	−0.333721	−	0.942672i	$-0.608304\pi$
$457$	−14.2683	−0.667442	−0.333721	+	0.942672i	$0.608304\pi$
$458$	−42.3057	−1.97681
$459$	0	0
$460$	3.36708	0.156991
$461$	−28.9224	−1.34705	−0.673525	−	0.739164i	$-0.735221\pi$
$461$	−28.9224	−1.34705	−0.673525	+	0.739164i	$0.735221\pi$
$462$	0	0
$463$	−10.8540	−0.504429	−0.252215	−	0.967671i	$-0.581159\pi$
$463$	−10.8540	−0.504429	−0.252215	+	0.967671i	$0.581159\pi$
$464$	−3.16689	−0.147019
$465$	0	0
$466$	−6.34090	−0.293737
$467$	−18.3096	−0.847267	−0.423633	−	0.905834i	$-0.639245\pi$
$467$	−18.3096	−0.847267	−0.423633	+	0.905834i	$0.639245\pi$
$468$	0	0
$469$	−1.92945	−0.0890937
$470$	18.2236	0.840594
$471$	0	0
$472$	2.15404	0.0991476
$473$	28.6741	1.31844
$474$	0	0
$475$	−10.7891	−0.495037
$476$	1.45115	0.0665134
$477$	0	0
$478$	−3.63831	−0.166412
$479$	29.2790	1.33779	0.668896	−	0.743356i	$-0.266767\pi$
$479$	29.2790	1.33779	0.668896	+	0.743356i	$0.266767\pi$
$480$	0	0
$481$	−0.784645	−0.0357767
$482$	8.33544	0.379669
$483$	0	0
$484$	61.8458	2.81117
$485$	−8.19926	−0.372309
$486$	0	0
$487$	2.51914	0.114153	0.0570766	−	0.998370i	$-0.481822\pi$
$487$	2.51914	0.114153	0.0570766	+	0.998370i	$0.481822\pi$
$488$	−2.05082	−0.0928365
$489$	0	0
$490$	−20.6679	−0.933680
$491$	−18.0791	−0.815896	−0.407948	−	0.913005i	$-0.633756\pi$
$491$	−18.0791	−0.815896	−0.407948	+	0.913005i	$0.633756\pi$
$492$	0	0
$493$	−2.24526	−0.101121
$494$	1.43481	0.0645550
$495$	0	0
$496$	12.8483	0.576905
$497$	0.959008	0.0430174
$498$	0	0
$499$	19.8082	0.886736	0.443368	−	0.896340i	$-0.353784\pi$
$499$	19.8082	0.886736	0.443368	+	0.896340i	$0.353784\pi$
$500$	−26.7815	−1.19770
$501$	0	0
$502$	−15.4758	−0.690720
$503$	12.8927	0.574855	0.287428	−	0.957802i	$-0.407200\pi$
$503$	12.8927	0.574855	0.287428	+	0.957802i	$0.407200\pi$
$504$	0	0
$505$	13.2156	0.588086
$506$	−12.7391	−0.566324
$507$	0	0
$508$	−29.7723	−1.32093
$509$	8.87677	0.393456	0.196728	−	0.980458i	$-0.436968\pi$
$509$	8.87677	0.393456	0.196728	+	0.980458i	$0.436968\pi$
$510$	0	0
$511$	3.34920	0.148160
$512$	30.2819	1.33829
$513$	0	0
$514$	39.7739	1.75435
$515$	−10.3444	−0.455829
$516$	0	0
$517$	−37.2764	−1.63942
$518$	2.38780	0.104914
$519$	0	0
$520$	0.198876	0.00872127
$521$	4.07848	0.178681	0.0893407	−	0.996001i	$-0.471524\pi$
$521$	4.07848	0.178681	0.0893407	+	0.996001i	$0.471524\pi$
$522$	0	0
$523$	−41.7215	−1.82436	−0.912178	−	0.409795i	$-0.865600\pi$
$523$	−41.7215	−1.82436	−0.912178	+	0.409795i	$0.865600\pi$
$524$	18.4027	0.803925
$525$	0	0
$526$	5.29126	0.230710
$527$	9.10916	0.396801
$528$	0	0
$529$	1.00000	0.0434783
$530$	11.5726	0.502681
$531$	0	0
$532$	−2.36065	−0.102347
$533$	−0.610306	−0.0264353
$534$	0	0
$535$	−4.52119	−0.195468
$536$	5.18957	0.224155
$537$	0	0
$538$	−8.61935	−0.371607
$539$	42.2761	1.82096
$540$	0	0
$541$	−20.0737	−0.863038	−0.431519	−	0.902104i	$-0.642022\pi$
$541$	−20.0737	−0.863038	−0.431519	+	0.902104i	$0.642022\pi$
$542$	−8.55538	−0.367485
$543$	0	0
$544$	18.1530	0.778303
$545$	13.6804	0.586006
$546$	0	0
$547$	−13.2512	−0.566583	−0.283291	−	0.959034i	$-0.591426\pi$
$547$	−13.2512	−0.566583	−0.283291	+	0.959034i	$0.591426\pi$
$548$	−5.23105	−0.223459
$549$	0	0
$550$	37.6305	1.60457
$551$	3.65245	0.155600
$552$	0	0
$553$	2.42132	0.102965
$554$	14.9071	0.633342
$555$	0	0
$556$	32.3689	1.37275
$557$	−29.8454	−1.26459	−0.632295	−	0.774727i	$-0.717887\pi$
$557$	−29.8454	−1.26459	−0.632295	+	0.774727i	$0.717887\pi$
$558$	0	0
$559$	−0.884218	−0.0373984
$560$	1.24379	0.0525597
$561$	0	0
$562$	−32.6566	−1.37753
$563$	−6.32526	−0.266578	−0.133289	−	0.991077i	$-0.542554\pi$
$563$	−6.32526	−0.266578	−0.133289	+	0.991077i	$0.542554\pi$
$564$	0	0
$565$	−27.6962	−1.16519
$566$	17.1961	0.722805
$567$	0	0
$568$	−2.57941	−0.108230
$569$	−13.4246	−0.562788	−0.281394	−	0.959592i	$-0.590797\pi$
$569$	−13.4246	−0.562788	−0.281394	+	0.959592i	$0.590797\pi$
$570$	0	0
$571$	−31.9335	−1.33638	−0.668188	−	0.743992i	$-0.732930\pi$
$571$	−31.9335	−1.33638	−0.668188	+	0.743992i	$0.732930\pi$
$572$	−2.70559	−0.113126
$573$	0	0
$574$	1.85726	0.0775207
$575$	−2.95393	−0.123187
$576$	0	0
$577$	12.4350	0.517677	0.258838	−	0.965921i	$-0.416660\pi$
$577$	12.4350	0.517677	0.258838	+	0.965921i	$0.416660\pi$
$578$	−24.9534	−1.03792
$579$	0	0
$580$	3.36708	0.139810
$581$	0.666930	0.0276689
$582$	0	0
$583$	−23.6717	−0.980382
$584$	−9.00821	−0.372762
$585$	0	0
$586$	43.6730	1.80412
$587$	24.0000	0.990586	0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000	0.990586	0.495293	+	0.868726i	$0.335061\pi$
$588$	0	0
$589$	−14.8182	−0.610575
$590$	8.70566	0.358406
$591$	0	0
$592$	13.1989	0.542470
$593$	−19.4433	−0.798441	−0.399221	−	0.916855i	$-0.630719\pi$
$593$	−19.4433	−0.798441	−0.399221	+	0.916855i	$0.630719\pi$
$594$	0	0
$595$	0.881821	0.0361511
$596$	15.9457	0.653161
$597$	0	0
$598$	0.392834	0.0160642
$599$	13.7703	0.562640	0.281320	−	0.959614i	$-0.409228\pi$
$599$	13.7703	0.562640	0.281320	+	0.959614i	$0.409228\pi$
$600$	0	0
$601$	−1.23499	−0.0503764	−0.0251882	−	0.999683i	$-0.508018\pi$
$601$	−1.23499	−0.0503764	−0.0251882	+	0.999683i	$0.508018\pi$
$602$	2.69082	0.109670
$603$	0	0
$604$	−24.9742	−1.01618
$605$	37.5818	1.52792
$606$	0	0
$607$	−29.5165	−1.19804	−0.599019	−	0.800735i	$-0.704443\pi$
$607$	−29.5165	−1.19804	−0.599019	+	0.800735i	$0.704443\pi$
$608$	−29.5302	−1.19761
$609$	0	0
$610$	−8.28852	−0.335592
$611$	1.14949	0.0465032
$612$	0	0
$613$	11.7152	0.473172	0.236586	−	0.971611i	$-0.423972\pi$
$613$	11.7152	0.473172	0.236586	+	0.971611i	$0.423972\pi$
$614$	39.9200	1.61104
$615$	0	0
$616$	1.23796	0.0498788
$617$	29.9849	1.20715	0.603573	−	0.797308i	$-0.293743\pi$
$617$	29.9849	1.20715	0.603573	+	0.797308i	$0.293743\pi$
$618$	0	0
$619$	7.17569	0.288415	0.144208	−	0.989547i	$-0.453937\pi$
$619$	7.17569	0.288415	0.144208	+	0.989547i	$0.453937\pi$
$620$	−13.6605	−0.548618
$621$	0	0
$622$	64.3484	2.58013
$623$	0.482275	0.0193219
$624$	0	0
$625$	−1.50465	−0.0601861
$626$	0.599077	0.0239439
$627$	0	0
$628$	40.1586	1.60250
$629$	9.35771	0.373116
$630$	0	0
$631$	−22.8076	−0.907955	−0.453977	−	0.891013i	$-0.649995\pi$
$631$	−22.8076	−0.907955	−0.453977	+	0.891013i	$0.649995\pi$
$632$	−6.51253	−0.259055
$633$	0	0
$634$	10.0063	0.397402
$635$	−18.0917	−0.717948
$636$	0	0
$637$	−1.30366	−0.0516529
$638$	−12.7391	−0.504348
$639$	0	0
$640$	−8.31857	−0.328820
$641$	2.68538	0.106066	0.0530331	−	0.998593i	$-0.483111\pi$
$641$	2.68538	0.106066	0.0530331	+	0.998593i	$0.483111\pi$
$642$	0	0
$643$	8.39727	0.331156	0.165578	−	0.986197i	$-0.447051\pi$
$643$	8.39727	0.331156	0.165578	+	0.986197i	$0.447051\pi$
$644$	−0.646319	−0.0254685
$645$	0	0
$646$	−17.1116	−0.673246
$647$	−13.2389	−0.520476	−0.260238	−	0.965545i	$-0.583801\pi$
$647$	−13.2389	−0.520476	−0.260238	+	0.965545i	$0.583801\pi$
$648$	0	0
$649$	−17.8074	−0.699002
$650$	−1.16040	−0.0455148
$651$	0	0
$652$	−56.6521	−2.21867
$653$	30.6244	1.19842	0.599212	−	0.800591i	$-0.295481\pi$
$653$	30.6244	1.19842	0.599212	+	0.800591i	$0.295481\pi$
$654$	0	0
$655$	11.1827	0.436946
$656$	10.2662	0.400829
$657$	0	0
$658$	−3.49808	−0.136369
$659$	33.9602	1.32290	0.661450	−	0.749989i	$-0.269941\pi$
$659$	33.9602	1.32290	0.661450	+	0.749989i	$0.269941\pi$
$660$	0	0
$661$	16.1979	0.630027	0.315013	−	0.949087i	$-0.397991\pi$
$661$	16.1979	0.630027	0.315013	+	0.949087i	$0.397991\pi$
$662$	−16.0347	−0.623208
$663$	0	0
$664$	−1.79382	−0.0696136
$665$	−1.43449	−0.0556273
$666$	0	0
$667$	1.00000	0.0387202
$668$	9.47047	0.366424
$669$	0	0
$670$	20.9739	0.810293
$671$	16.9542	0.654508
$672$	0	0
$673$	−38.7576	−1.49400	−0.746998	−	0.664827i	$-0.768505\pi$
$673$	−38.7576	−1.49400	−0.746998	+	0.664827i	$0.768505\pi$
$674$	1.48644	0.0572555
$675$	0	0
$676$	−30.5176	−1.17375
$677$	34.2894	1.31785	0.658925	−	0.752208i	$-0.271011\pi$
$677$	34.2894	1.31785	0.658925	+	0.752208i	$0.271011\pi$
$678$	0	0
$679$	1.57387	0.0603996
$680$	−2.37180	−0.0909544
$681$	0	0
$682$	51.6836	1.97907
$683$	7.15688	0.273851	0.136925	−	0.990581i	$-0.456278\pi$
$683$	7.15688	0.273851	0.136925	+	0.990581i	$0.456278\pi$
$684$	0	0
$685$	−3.17875	−0.121454
$686$	7.97770	0.304590
$687$	0	0
$688$	14.8738	0.567060
$689$	0.729959	0.0278092
$690$	0	0
$691$	−27.4957	−1.04599	−0.522993	−	0.852337i	$-0.675184\pi$
$691$	−27.4957	−1.04599	−0.522993	+	0.852337i	$0.675184\pi$
$692$	−0.0588155	−0.00223583
$693$	0	0
$694$	−42.6507	−1.61900
$695$	19.6696	0.746111
$696$	0	0
$697$	7.27854	0.275694
$698$	8.02645	0.303806
$699$	0	0
$700$	1.90918	0.0721603
$701$	−13.8109	−0.521630	−0.260815	−	0.965389i	$-0.583991\pi$
$701$	−13.8109	−0.521630	−0.260815	+	0.965389i	$0.583991\pi$
$702$	0	0
$703$	−15.2226	−0.574130
$704$	64.3277	2.42444
$705$	0	0
$706$	13.9514	0.525067
$707$	−2.53677	−0.0954049
$708$	0	0
$709$	−0.924764	−0.0347303	−0.0173651	−	0.999849i	$-0.505528\pi$
$709$	−0.924764	−0.0347303	−0.0173651	+	0.999849i	$0.505528\pi$
$710$	−10.4248	−0.391237
$711$	0	0
$712$	−1.29716	−0.0486130
$713$	−4.05707	−0.151938
$714$	0	0
$715$	−1.64410	−0.0614860
$716$	−2.37651	−0.0888143
$717$	0	0
$718$	−53.5132	−1.99709
$719$	20.1522	0.751552	0.375776	−	0.926711i	$-0.377376\pi$
$719$	20.1522	0.751552	0.375776	+	0.926711i	$0.377376\pi$
$720$	0	0
$721$	1.98564	0.0739489
$722$	−11.8094	−0.439500
$723$	0	0
$724$	53.7075	1.99602
$725$	−2.95393	−0.109706
$726$	0	0
$727$	2.08270	0.0772429	0.0386215	−	0.999254i	$-0.487703\pi$
$727$	2.08270	0.0772429	0.0386215	+	0.999254i	$0.487703\pi$
$728$	−0.0381747	−0.00141485
$729$	0	0
$730$	−36.4072	−1.34749
$731$	10.5452	0.390030
$732$	0	0
$733$	8.15634	0.301261	0.150631	−	0.988590i	$-0.451870\pi$
$733$	8.15634	0.301261	0.150631	+	0.988590i	$0.451870\pi$
$734$	25.6687	0.947449
$735$	0	0
$736$	−8.08505	−0.298019
$737$	−42.9021	−1.58032
$738$	0	0
$739$	−25.8699	−0.951638	−0.475819	−	0.879543i	$-0.657848\pi$
$739$	−25.8699	−0.951638	−0.475819	+	0.879543i	$0.657848\pi$
$740$	−14.0332	−0.515871
$741$	0	0
$742$	−2.22139	−0.0815497
$743$	34.9391	1.28179	0.640896	−	0.767628i	$-0.278563\pi$
$743$	34.9391	1.28179	0.640896	+	0.767628i	$0.278563\pi$
$744$	0	0
$745$	9.68971	0.355003
$746$	−46.7951	−1.71329
$747$	0	0
$748$	32.2670	1.17980
$749$	0.867854	0.0317107
$750$	0	0
$751$	10.4297	0.380585	0.190293	−	0.981727i	$-0.439056\pi$
$751$	10.4297	0.380585	0.190293	+	0.981727i	$0.439056\pi$
$752$	−19.3360	−0.705112
$753$	0	0
$754$	0.392834	0.0143062
$755$	−15.1760	−0.552313
$756$	0	0
$757$	46.0068	1.67215	0.836073	−	0.548619i	$-0.184846\pi$
$757$	46.0068	1.67215	0.836073	+	0.548619i	$0.184846\pi$
$758$	70.3238	2.55428
$759$	0	0
$760$	3.85831	0.139956
$761$	−33.1684	−1.20235	−0.601177	−	0.799116i	$-0.705301\pi$
$761$	−33.1684	−1.20235	−0.601177	+	0.799116i	$0.705301\pi$
$762$	0	0
$763$	−2.62600	−0.0950675
$764$	6.85726	0.248087
$765$	0	0
$766$	43.9258	1.58710
$767$	0.549123	0.0198277
$768$	0	0
$769$	−23.9544	−0.863816	−0.431908	−	0.901918i	$-0.642159\pi$
$769$	−23.9544	−0.863816	−0.431908	+	0.901918i	$0.642159\pi$
$770$	5.00328	0.180306
$771$	0	0
$772$	−8.28333	−0.298124
$773$	−37.4046	−1.34535	−0.672674	−	0.739939i	$-0.734854\pi$
$773$	−37.4046	−1.34535	−0.672674	+	0.739939i	$0.734854\pi$
$774$	0	0
$775$	11.9843	0.430489
$776$	−4.23318	−0.151962
$777$	0	0
$778$	−72.1387	−2.58630
$779$	−11.8403	−0.424223
$780$	0	0
$781$	21.3240	0.763031
$782$	−4.68496	−0.167534
$783$	0	0
$784$	21.9295	0.783195
$785$	24.4032	0.870986
$786$	0	0
$787$	4.05977	0.144715	0.0723576	−	0.997379i	$-0.476948\pi$
$787$	4.05977	0.144715	0.0723576	+	0.997379i	$0.476948\pi$
$788$	34.7749	1.23880
$789$	0	0
$790$	−26.3208	−0.936450
$791$	5.31637	0.189028
$792$	0	0
$793$	−0.522811	−0.0185656
$794$	25.6102	0.908871
$795$	0	0
$796$	55.9702	1.98381
$797$	−1.56174	−0.0553196	−0.0276598	−	0.999617i	$-0.508806\pi$
$797$	−1.56174	−0.0553196	−0.0276598	+	0.999617i	$0.508806\pi$
$798$	0	0
$799$	−13.7088	−0.484983
$800$	23.8827	0.844380
$801$	0	0
$802$	−20.7808	−0.733797
$803$	74.4708	2.62802
$804$	0	0
$805$	−0.392749	−0.0138426
$806$	−1.59376	−0.0561377
$807$	0	0
$808$	6.82305	0.240034
$809$	−9.09173	−0.319648	−0.159824	−	0.987146i	$-0.551093\pi$
$809$	−9.09173	−0.319648	−0.159824	+	0.987146i	$0.551093\pi$
$810$	0	0
$811$	6.07851	0.213445	0.106723	−	0.994289i	$-0.465964\pi$
$811$	6.07851	0.213445	0.106723	+	0.994289i	$0.465964\pi$
$812$	−0.646319	−0.0226814
$813$	0	0
$814$	53.0939	1.86094
$815$	−34.4257	−1.20588
$816$	0	0
$817$	−17.1544	−0.600155
$818$	−59.6073	−2.08412
$819$	0	0
$820$	−10.9152	−0.381175
$821$	29.7440	1.03807	0.519036	−	0.854752i	$-0.326291\pi$
$821$	29.7440	1.03807	0.519036	+	0.854752i	$0.326291\pi$
$822$	0	0
$823$	12.8740	0.448758	0.224379	−	0.974502i	$-0.427965\pi$
$823$	12.8740	0.448758	0.224379	+	0.974502i	$0.427965\pi$
$824$	−5.34069	−0.186052
$825$	0	0
$826$	−1.67107	−0.0581441
$827$	36.2049	1.25897	0.629484	−	0.777014i	$-0.283266\pi$
$827$	36.2049	1.25897	0.629484	+	0.777014i	$0.283266\pi$
$828$	0	0
$829$	−31.7283	−1.10197	−0.550985	−	0.834515i	$-0.685748\pi$
$829$	−31.7283	−1.10197	−0.550985	+	0.834515i	$0.685748\pi$
$830$	−7.24981	−0.251645
$831$	0	0
$832$	−1.98366	−0.0687710
$833$	15.5475	0.538690
$834$	0	0
$835$	5.75492	0.199157
$836$	−52.4900	−1.81541
$837$	0	0
$838$	15.7621	0.544491
$839$	−29.9699	−1.03468	−0.517338	−	0.855781i	$-0.673077\pi$
$839$	−29.9699	−1.03468	−0.517338	+	0.855781i	$0.673077\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−16.1596	−0.556898
$843$	0	0
$844$	1.38607	0.0477104
$845$	−18.5446	−0.637954
$846$	0	0
$847$	−7.21393	−0.247874
$848$	−12.2790	−0.421662
$849$	0	0
$850$	13.8391	0.474676
$851$	−4.16777	−0.142869
$852$	0	0
$853$	−43.0788	−1.47499	−0.737495	−	0.675352i	$-0.763992\pi$
$853$	−43.0788	−1.47499	−0.737495	+	0.675352i	$0.763992\pi$
$854$	1.59100	0.0544430
$855$	0	0
$856$	−2.33424	−0.0797825
$857$	−56.4937	−1.92979	−0.964895	−	0.262637i	$-0.915408\pi$
$857$	−56.4937	−1.92979	−0.964895	+	0.262637i	$0.915408\pi$
$858$	0	0
$859$	13.8657	0.473093	0.236547	−	0.971620i	$-0.423984\pi$
$859$	13.8657	0.473093	0.236547	+	0.971620i	$0.423984\pi$
$860$	−15.8141	−0.539255
$861$	0	0
$862$	80.5884	2.74485
$863$	12.0348	0.409669	0.204835	−	0.978797i	$-0.434334\pi$
$863$	12.0348	0.409669	0.204835	+	0.978797i	$0.434334\pi$
$864$	0	0
$865$	−0.0357404	−0.00121521
$866$	−11.2942	−0.383792
$867$	0	0
$868$	2.62216	0.0890020
$869$	53.8391	1.82636
$870$	0	0
$871$	1.32296	0.0448269
$872$	7.06305	0.239185
$873$	0	0
$874$	7.62122	0.257792
$875$	3.12389	0.105607
$876$	0	0
$877$	−26.0150	−0.878463	−0.439232	−	0.898374i	$-0.644749\pi$
$877$	−26.0150	−0.878463	−0.439232	+	0.898374i	$0.644749\pi$
$878$	−25.5498	−0.862265
$879$	0	0
$880$	27.6562	0.932291
$881$	18.7821	0.632785	0.316392	−	0.948628i	$-0.397528\pi$
$881$	18.7821	0.632785	0.316392	+	0.948628i	$0.397528\pi$
$882$	0	0
$883$	−8.62443	−0.290235	−0.145118	−	0.989414i	$-0.546356\pi$
$883$	−8.62443	−0.290235	−0.145118	+	0.989414i	$0.546356\pi$
$884$	−0.995010	−0.0334658
$885$	0	0
$886$	−23.5316	−0.790560
$887$	39.3782	1.32219	0.661095	−	0.750302i	$-0.270092\pi$
$887$	39.3782	1.32219	0.661095	+	0.750302i	$0.270092\pi$
$888$	0	0
$889$	3.47275	0.116472
$890$	−5.24253	−0.175730
$891$	0	0
$892$	−41.4142	−1.38665
$893$	22.3007	0.746265
$894$	0	0
$895$	−1.44413	−0.0482720
$896$	1.59677	0.0533444
$897$	0	0
$898$	−42.5280	−1.41918
$899$	−4.05707	−0.135311
$900$	0	0
$901$	−8.70553	−0.290023
$902$	41.2971	1.37504
$903$	0	0
$904$	−14.2992	−0.475586
$905$	32.6364	1.08487
$906$	0	0
$907$	8.33324	0.276701	0.138350	−	0.990383i	$-0.455820\pi$
$907$	8.33324	0.276701	0.138350	+	0.990383i	$0.455820\pi$
$908$	64.4173	2.13776
$909$	0	0
$910$	−0.154285	−0.00511450
$911$	40.2510	1.33358	0.666788	−	0.745247i	$-0.267669\pi$
$911$	40.2510	1.33358	0.666788	+	0.745247i	$0.267669\pi$
$912$	0	0
$913$	14.8295	0.490784
$914$	−29.7723	−0.984780
$915$	0	0
$916$	−47.7255	−1.57690
$917$	−2.14656	−0.0708856
$918$	0	0
$919$	34.7016	1.14470	0.572351	−	0.820009i	$-0.306032\pi$
$919$	34.7016	1.14470	0.572351	+	0.820009i	$0.306032\pi$
$920$	1.05636	0.0348272
$921$	0	0
$922$	−60.3497	−1.98751
$923$	−0.657562	−0.0216439
$924$	0	0
$925$	12.3113	0.404793
$926$	−22.6481	−0.744262
$927$	0	0
$928$	−8.08505	−0.265405
$929$	25.0656	0.822376	0.411188	−	0.911551i	$-0.365114\pi$
$929$	25.0656	0.822376	0.411188	+	0.911551i	$0.365114\pi$
$930$	0	0
$931$	−25.2918	−0.828905
$932$	−7.15325	−0.234313
$933$	0	0
$934$	−38.2049	−1.25010
$935$	19.6077	0.641239
$936$	0	0
$937$	−48.2923	−1.57764	−0.788821	−	0.614623i	$-0.789308\pi$
$937$	−48.2923	−1.57764	−0.788821	+	0.614623i	$0.789308\pi$
$938$	−4.02600	−0.131454
$939$	0	0
$940$	20.5583	0.670538
$941$	16.0613	0.523582	0.261791	−	0.965125i	$-0.415687\pi$
$941$	16.0613	0.523582	0.261791	+	0.965125i	$0.415687\pi$
$942$	0	0
$943$	−3.24174	−0.105566
$944$	−9.23705	−0.300640
$945$	0	0
$946$	59.8316	1.94529
$947$	36.9475	1.20063	0.600316	−	0.799763i	$-0.295041\pi$
$947$	36.9475	1.20063	0.600316	+	0.799763i	$0.295041\pi$
$948$	0	0
$949$	−2.29644	−0.0745456
$950$	−22.5126	−0.730404
$951$	0	0
$952$	0.455274	0.0147555
$953$	31.1650	1.00953	0.504766	−	0.863256i	$-0.331579\pi$
$953$	31.1650	1.00953	0.504766	+	0.863256i	$0.331579\pi$
$954$	0	0
$955$	4.16695	0.134839
$956$	−4.10442	−0.132746
$957$	0	0
$958$	61.0938	1.97385
$959$	0.610170	0.0197034
$960$	0	0
$961$	−14.5402	−0.469038
$962$	−1.63724	−0.0527869
$963$	0	0
$964$	9.40331	0.302860
$965$	−5.03353	−0.162035
$966$	0	0
$967$	57.2912	1.84236	0.921181	−	0.389134i	$-0.127226\pi$
$967$	57.2912	1.84236	0.921181	+	0.389134i	$0.127226\pi$
$968$	19.4030	0.623637
$969$	0	0
$970$	−17.1086	−0.549325
$971$	−36.2034	−1.16182	−0.580912	−	0.813966i	$-0.697304\pi$
$971$	−36.2034	−1.16182	−0.580912	+	0.813966i	$0.697304\pi$
$972$	0	0
$973$	−3.77563	−0.121041
$974$	5.25646	0.168428
$975$	0	0
$976$	8.79445	0.281503
$977$	−36.0090	−1.15203	−0.576015	−	0.817439i	$-0.695393\pi$
$977$	−36.0090	−1.15203	−0.576015	+	0.817439i	$0.695393\pi$
$978$	0	0
$979$	10.7236	0.342727
$980$	−23.3157	−0.744793
$981$	0	0
$982$	−37.7239	−1.20382
$983$	14.3777	0.458576	0.229288	−	0.973359i	$-0.426360\pi$
$983$	14.3777	0.458576	0.229288	+	0.973359i	$0.426360\pi$
$984$	0	0
$985$	21.1316	0.673310
$986$	−4.68496	−0.149200
$987$	0	0
$988$	1.61862	0.0514953
$989$	−4.69667	−0.149346
$990$	0	0
$991$	5.18303	0.164644	0.0823221	−	0.996606i	$-0.473766\pi$
$991$	5.18303	0.164644	0.0823221	+	0.996606i	$0.473766\pi$
$992$	32.8016	1.04145
$993$	0	0
$994$	2.00107	0.0634702
$995$	34.0114	1.07823
$996$	0	0
$997$	−50.0974	−1.58660	−0.793301	−	0.608830i	$-0.791639\pi$
$997$	−50.0974	−1.58660	−0.793301	+	0.608830i	$0.791639\pi$
$998$	41.3319	1.30834
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.20	✓	22
3.2	odd	2		6003.2.a.u.1.3	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.20	✓	22	1.1	even	1	trivial
6003.2.a.u.1.3	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)

\(f(q)\)	\(=\)	\(q+2.08661 q^{2} +2.35393 q^{4} +1.43041 q^{5} -0.274571 q^{7} +0.738503 q^{8} +O(q^{10})\)	\(q+2.08661 q^{2} +2.35393 q^{4} +1.43041 q^{5} -0.274571 q^{7} +0.738503 q^{8} +2.98470 q^{10} -6.10520 q^{11} +0.188265 q^{13} -0.572921 q^{14} -3.16689 q^{16} -2.24526 q^{17} +3.65245 q^{19} +3.36708 q^{20} -12.7391 q^{22} +1.00000 q^{23} -2.95393 q^{25} +0.392834 q^{26} -0.646319 q^{28} +1.00000 q^{29} -4.05707 q^{31} -8.08505 q^{32} -4.68496 q^{34} -0.392749 q^{35} -4.16777 q^{37} +7.62122 q^{38} +1.05636 q^{40} -3.24174 q^{41} -4.69667 q^{43} -14.3712 q^{44} +2.08661 q^{46} +6.10569 q^{47} -6.92461 q^{49} -6.16369 q^{50} +0.443161 q^{52} +3.87730 q^{53} -8.73294 q^{55} -0.202771 q^{56} +2.08661 q^{58} +2.91676 q^{59} -2.77700 q^{61} -8.46551 q^{62} -10.5365 q^{64} +0.269296 q^{65} +7.02715 q^{67} -5.28516 q^{68} -0.819512 q^{70} -3.49275 q^{71} -12.1979 q^{73} -8.69650 q^{74} +8.59759 q^{76} +1.67631 q^{77} -8.81856 q^{79} -4.52994 q^{80} -6.76424 q^{82} -2.42899 q^{83} -3.21163 q^{85} -9.80011 q^{86} -4.50871 q^{88} -1.75647 q^{89} -0.0516920 q^{91} +2.35393 q^{92} +12.7402 q^{94} +5.22450 q^{95} -5.73211 q^{97} -14.4489 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 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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