LMFDB - Embedded newform 6003.2.a.m.1.9

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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 18 x^{9} + 30 x^{8} + 124 x^{7} - 152 x^{6} - 408 x^{5} + 285 x^{4} + 634 x^{3} + \cdots - 108 $ x^11 - 2x^10 - 18x^9 + 30x^8 + 124x^7 - 152x^6 - 408x^5 + 285x^4 + 634x^3 - 93x^2 - 369x - 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-1.94502$ of defining polynomial
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)

$f(q)$	$=$	$q+1.94502 q^{2} +1.78310 q^{4} -0.890641 q^{5} -3.69089 q^{7} -0.421884 q^{8} +O(q^{10})$	\(q+1.94502 q^{2} +1.78310 q^{4} -0.890641 q^{5} -3.69089 q^{7} -0.421884 q^{8} -1.73231 q^{10} +4.50039 q^{11} -2.37833 q^{13} -7.17884 q^{14} -4.38676 q^{16} +8.08663 q^{17} +3.74291 q^{19} -1.58810 q^{20} +8.75335 q^{22} -1.00000 q^{23} -4.20676 q^{25} -4.62589 q^{26} -6.58120 q^{28} +1.00000 q^{29} +0.830492 q^{31} -7.68856 q^{32} +15.7286 q^{34} +3.28725 q^{35} -6.72421 q^{37} +7.28003 q^{38} +0.375747 q^{40} -5.54692 q^{41} -4.97595 q^{43} +8.02463 q^{44} -1.94502 q^{46} -11.9191 q^{47} +6.62264 q^{49} -8.18222 q^{50} -4.24078 q^{52} +5.72723 q^{53} -4.00823 q^{55} +1.55712 q^{56} +1.94502 q^{58} +4.08424 q^{59} -2.66889 q^{61} +1.61532 q^{62} -6.18087 q^{64} +2.11823 q^{65} -0.373417 q^{67} +14.4192 q^{68} +6.39377 q^{70} -11.6083 q^{71} -12.7226 q^{73} -13.0787 q^{74} +6.67397 q^{76} -16.6104 q^{77} -11.1606 q^{79} +3.90703 q^{80} -10.7889 q^{82} +0.182560 q^{83} -7.20228 q^{85} -9.67832 q^{86} -1.89864 q^{88} -0.217110 q^{89} +8.77813 q^{91} -1.78310 q^{92} -23.1828 q^{94} -3.33359 q^{95} +13.8496 q^{97} +12.8812 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) $ 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8	$ 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8} + 14 q^{10} - 11 q^{11} - 5 q^{13} - 17 q^{14} + 20 q^{16} - 15 q^{17} - 6 q^{19} - 21 q^{20} - 10 q^{22} - 11 q^{23} + 3 q^{25} + 5 q^{26} + 7 q^{28} + 11 q^{29} + 35 q^{31} - 28 q^{32} + 28 q^{34} - 15 q^{35} - 28 q^{37} + 2 q^{38} - q^{40} - 10 q^{41} - 6 q^{43} - 18 q^{44} + 2 q^{46} - 15 q^{47} + 22 q^{49} - 15 q^{50} - 36 q^{52} + 7 q^{53} - 12 q^{55} - 56 q^{56} - 2 q^{58} + 20 q^{59} - 20 q^{61} + 11 q^{62} + 36 q^{64} - 11 q^{65} - 39 q^{67} - 35 q^{68} + 38 q^{70} - 49 q^{71} - 3 q^{73} - 37 q^{74} - 18 q^{76} - 25 q^{77} + 41 q^{79} - 51 q^{80} - 19 q^{82} - 13 q^{83} - 62 q^{86} - 40 q^{88} - 34 q^{89} + 2 q^{91} - 18 q^{92} - 14 q^{94} - 25 q^{95} - 11 q^{97} - 53 q^{98}+O(q^{100}) $ 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8 + 14 * q^10 - 11 * q^11 - 5 * q^13 - 17 * q^14 + 20 * q^16 - 15 * q^17 - 6 * q^19 - 21 * q^20 - 10 * q^22 - 11 * q^23 + 3 * q^25 + 5 * q^26 + 7 * q^28 + 11 * q^29 + 35 * q^31 - 28 * q^32 + 28 * q^34 - 15 * q^35 - 28 * q^37 + 2 * q^38 - q^40 - 10 * q^41 - 6 * q^43 - 18 * q^44 + 2 * q^46 - 15 * q^47 + 22 * q^49 - 15 * q^50 - 36 * q^52 + 7 * q^53 - 12 * q^55 - 56 * q^56 - 2 * q^58 + 20 * q^59 - 20 * q^61 + 11 * q^62 + 36 * q^64 - 11 * q^65 - 39 * q^67 - 35 * q^68 + 38 * q^70 - 49 * q^71 - 3 * q^73 - 37 * q^74 - 18 * q^76 - 25 * q^77 + 41 * q^79 - 51 * q^80 - 19 * q^82 - 13 * q^83 - 62 * q^86 - 40 * q^88 - 34 * q^89 + 2 * q^91 - 18 * q^92 - 14 * q^94 - 25 * q^95 - 11 * q^97 - 53 * 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 $

$2$	1.94502	1.37534	0.687668	−	0.726026i	$-0.258635\pi$
$2$	1.94502	1.37534	0.687668	+	0.726026i	$0.258635\pi$
$3$	0	0
$3$	0	0
$4$	1.78310	0.891548
$5$	−0.890641	−0.398307	−0.199153	−	0.979968i	$-0.563819\pi$
$5$	−0.890641	−0.398307	−0.199153	+	0.979968i	$0.563819\pi$
$6$	0	0
$7$	−3.69089	−1.39502	−0.697512	−	0.716573i	$-0.745710\pi$
$7$	−3.69089	−1.39502	−0.697512	+	0.716573i	$0.745710\pi$
$8$	−0.421884	−0.149158
$9$	0	0
$10$	−1.73231	−0.547805
$11$	4.50039	1.35692	0.678460	−	0.734638i	$-0.262648\pi$
$11$	4.50039	1.35692	0.678460	+	0.734638i	$0.262648\pi$
$12$	0	0
$13$	−2.37833	−0.659629	−0.329815	−	0.944046i	$-0.606986\pi$
$13$	−2.37833	−0.659629	−0.329815	+	0.944046i	$0.606986\pi$
$14$	−7.17884	−1.91863
$15$	0	0
$16$	−4.38676	−1.09669
$17$	8.08663	1.96130	0.980648	−	0.195779i	$-0.0627236\pi$
$17$	8.08663	1.96130	0.980648	+	0.195779i	$0.0627236\pi$
$18$	0	0
$19$	3.74291	0.858683	0.429342	−	0.903142i	$-0.358746\pi$
$19$	3.74291	0.858683	0.429342	+	0.903142i	$0.358746\pi$
$20$	−1.58810	−0.355109
$21$	0	0
$22$	8.75335	1.86622
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.20676	−0.841352
$26$	−4.62589	−0.907211
$27$	0	0
$28$	−6.58120	−1.24373
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.830492	0.149161	0.0745804	−	0.997215i	$-0.476238\pi$
$31$	0.830492	0.149161	0.0745804	+	0.997215i	$0.476238\pi$
$32$	−7.68856	−1.35916
$33$	0	0
$34$	15.7286	2.69744
$35$	3.28725	0.555647
$36$	0	0
$37$	−6.72421	−1.10545	−0.552727	−	0.833363i	$-0.686413\pi$
$37$	−6.72421	−1.10545	−0.552727	+	0.833363i	$0.686413\pi$
$38$	7.28003	1.18098
$39$	0	0
$40$	0.375747	0.0594108
$41$	−5.54692	−0.866283	−0.433141	−	0.901326i	$-0.642595\pi$
$41$	−5.54692	−0.866283	−0.433141	+	0.901326i	$0.642595\pi$
$42$	0	0
$43$	−4.97595	−0.758826	−0.379413	−	0.925227i	$-0.623874\pi$
$43$	−4.97595	−0.758826	−0.379413	+	0.925227i	$0.623874\pi$
$44$	8.02463	1.20976
$45$	0	0
$46$	−1.94502	−0.286777
$47$	−11.9191	−1.73857	−0.869286	−	0.494309i	$-0.835421\pi$
$47$	−11.9191	−1.73857	−0.869286	+	0.494309i	$0.835421\pi$
$48$	0	0
$49$	6.62264	0.946091
$50$	−8.18222	−1.15714
$51$	0	0
$52$	−4.24078	−0.588091
$53$	5.72723	0.786696	0.393348	−	0.919390i	$-0.371317\pi$
$53$	5.72723	0.786696	0.393348	+	0.919390i	$0.371317\pi$
$54$	0	0
$55$	−4.00823	−0.540470
$56$	1.55712	0.208080
$57$	0	0
$58$	1.94502	0.255393
$59$	4.08424	0.531723	0.265861	−	0.964011i	$-0.414344\pi$
$59$	4.08424	0.531723	0.265861	+	0.964011i	$0.414344\pi$
$60$	0	0
$61$	−2.66889	−0.341717	−0.170858	−	0.985296i	$-0.554654\pi$
$61$	−2.66889	−0.341717	−0.170858	+	0.985296i	$0.554654\pi$
$62$	1.61532	0.205146
$63$	0	0
$64$	−6.18087	−0.772609
$65$	2.11823	0.262735
$66$	0	0
$67$	−0.373417	−0.0456202	−0.0228101	−	0.999740i	$-0.507261\pi$
$67$	−0.373417	−0.0456202	−0.0228101	+	0.999740i	$0.507261\pi$
$68$	14.4192	1.74859
$69$	0	0
$70$	6.39377	0.764201
$71$	−11.6083	−1.37766	−0.688828	−	0.724925i	$-0.741875\pi$
$71$	−11.6083	−1.37766	−0.688828	+	0.724925i	$0.741875\pi$
$72$	0	0
$73$	−12.7226	−1.48907	−0.744536	−	0.667582i	$-0.767329\pi$
$73$	−12.7226	−1.48907	−0.744536	+	0.667582i	$0.767329\pi$
$74$	−13.0787	−1.52037
$75$	0	0
$76$	6.67397	0.765557
$77$	−16.6104	−1.89294
$78$	0	0
$79$	−11.1606	−1.25566	−0.627830	−	0.778350i	$-0.716057\pi$
$79$	−11.1606	−1.25566	−0.627830	+	0.778350i	$0.716057\pi$
$80$	3.90703	0.436819
$81$	0	0
$82$	−10.7889	−1.19143
$83$	0.182560	0.0200385	0.0100193	−	0.999950i	$-0.496811\pi$
$83$	0.182560	0.0200385	0.0100193	+	0.999950i	$0.496811\pi$
$84$	0	0
$85$	−7.20228	−0.781197
$86$	−9.67832	−1.04364
$87$	0	0
$88$	−1.89864	−0.202396
$89$	−0.217110	−0.0230136	−0.0115068	−	0.999934i	$-0.503663\pi$
$89$	−0.217110	−0.0230136	−0.0115068	+	0.999934i	$0.503663\pi$
$90$	0	0
$91$	8.77813	0.920198
$92$	−1.78310	−0.185901
$93$	0	0
$94$	−23.1828	−2.39112
$95$	−3.33359	−0.342019
$96$	0	0
$97$	13.8496	1.40621	0.703105	−	0.711086i	$-0.251797\pi$
$97$	13.8496	1.40621	0.703105	+	0.711086i	$0.251797\pi$
$98$	12.8812	1.30119
$99$	0	0
$100$	−7.50105	−0.750105
$101$	−2.55168	−0.253902	−0.126951	−	0.991909i	$-0.540519\pi$
$101$	−2.55168	−0.253902	−0.126951	+	0.991909i	$0.540519\pi$
$102$	0	0
$103$	−12.9829	−1.27925	−0.639624	−	0.768688i	$-0.720910\pi$
$103$	−12.9829	−1.27925	−0.639624	+	0.768688i	$0.720910\pi$
$104$	1.00338	0.0983892
$105$	0	0
$106$	11.1396	1.08197
$107$	15.1805	1.46755	0.733775	−	0.679392i	$-0.237757\pi$
$107$	15.1805	1.46755	0.733775	+	0.679392i	$0.237757\pi$
$108$	0	0
$109$	−5.87116	−0.562355	−0.281178	−	0.959656i	$-0.590725\pi$
$109$	−5.87116	−0.562355	−0.281178	+	0.959656i	$0.590725\pi$
$110$	−7.79609	−0.743328
$111$	0	0
$112$	16.1910	1.52991
$113$	−17.5604	−1.65195	−0.825974	−	0.563708i	$-0.809374\pi$
$113$	−17.5604	−1.65195	−0.825974	+	0.563708i	$0.809374\pi$
$114$	0	0
$115$	0.890641	0.0830527
$116$	1.78310	0.165556
$117$	0	0
$118$	7.94392	0.731297
$119$	−29.8468	−2.73605
$120$	0	0
$121$	9.25354	0.841231
$122$	−5.19105	−0.469975
$123$	0	0
$124$	1.48085	0.132984
$125$	8.19991	0.733423
$126$	0	0
$127$	6.07663	0.539214	0.269607	−	0.962970i	$-0.413106\pi$
$127$	6.07663	0.539214	0.269607	+	0.962970i	$0.413106\pi$
$128$	3.35522	0.296562
$129$	0	0
$130$	4.12000	0.361348
$131$	−9.14993	−0.799433	−0.399717	−	0.916639i	$-0.630891\pi$
$131$	−9.14993	−0.799433	−0.399717	+	0.916639i	$0.630891\pi$
$132$	0	0
$133$	−13.8147	−1.19788
$134$	−0.726303	−0.0627430
$135$	0	0
$136$	−3.41162	−0.292544
$137$	−13.4962	−1.15306	−0.576529	−	0.817076i	$-0.695593\pi$
$137$	−13.4962	−1.15306	−0.576529	+	0.817076i	$0.695593\pi$
$138$	0	0
$139$	13.1825	1.11812	0.559062	−	0.829126i	$-0.311161\pi$
$139$	13.1825	1.11812	0.559062	+	0.829126i	$0.311161\pi$
$140$	5.86149	0.495386
$141$	0	0
$142$	−22.5784	−1.89474
$143$	−10.7034	−0.895064
$144$	0	0
$145$	−0.890641	−0.0739637
$146$	−24.7458	−2.04797
$147$	0	0
$148$	−11.9899	−0.985564
$149$	6.62847	0.543025	0.271513	−	0.962435i	$-0.412476\pi$
$149$	6.62847	0.543025	0.271513	+	0.962435i	$0.412476\pi$
$150$	0	0
$151$	4.85570	0.395151	0.197576	−	0.980288i	$-0.436693\pi$
$151$	4.85570	0.395151	0.197576	+	0.980288i	$0.436693\pi$
$152$	−1.57907	−0.128080
$153$	0	0
$154$	−32.3076	−2.60342
$155$	−0.739670	−0.0594117
$156$	0	0
$157$	−11.5387	−0.920890	−0.460445	−	0.887688i	$-0.652310\pi$
$157$	−11.5387	−0.920890	−0.460445	+	0.887688i	$0.652310\pi$
$158$	−21.7075	−1.72695
$159$	0	0
$160$	6.84775	0.541362
$161$	3.69089	0.290883
$162$	0	0
$163$	12.6736	0.992677	0.496338	−	0.868129i	$-0.334678\pi$
$163$	12.6736	0.992677	0.496338	+	0.868129i	$0.334678\pi$
$164$	−9.89068	−0.772333
$165$	0	0
$166$	0.355082	0.0275597
$167$	−10.0950	−0.781176	−0.390588	−	0.920566i	$-0.627728\pi$
$167$	−10.0950	−0.781176	−0.390588	+	0.920566i	$0.627728\pi$
$168$	0	0
$169$	−7.34356	−0.564889
$170$	−14.0086	−1.07441
$171$	0	0
$172$	−8.87260	−0.676529
$173$	−8.38539	−0.637529	−0.318765	−	0.947834i	$-0.603268\pi$
$173$	−8.38539	−0.637529	−0.318765	+	0.947834i	$0.603268\pi$
$174$	0	0
$175$	15.5267	1.17371
$176$	−19.7422	−1.48812
$177$	0	0
$178$	−0.422283	−0.0316515
$179$	0.135648	0.0101388	0.00506941	−	0.999987i	$-0.498386\pi$
$179$	0.135648	0.0101388	0.00506941	+	0.999987i	$0.498386\pi$
$180$	0	0
$181$	13.4258	0.997929	0.498965	−	0.866622i	$-0.333714\pi$
$181$	13.4258	0.997929	0.498965	+	0.866622i	$0.333714\pi$
$182$	17.0736	1.26558
$183$	0	0
$184$	0.421884	0.0311017
$185$	5.98885	0.440309
$186$	0	0
$187$	36.3930	2.66132
$188$	−21.2528	−1.55002
$189$	0	0
$190$	−6.48389	−0.470391
$191$	−14.9265	−1.08004	−0.540021	−	0.841651i	$-0.681584\pi$
$191$	−14.9265	−1.08004	−0.540021	+	0.841651i	$0.681584\pi$
$192$	0	0
$193$	−20.1714	−1.45197	−0.725985	−	0.687710i	$-0.758616\pi$
$193$	−20.1714	−1.45197	−0.725985	+	0.687710i	$0.758616\pi$
$194$	26.9376	1.93401
$195$	0	0
$196$	11.8088	0.843485
$197$	−3.29903	−0.235046	−0.117523	−	0.993070i	$-0.537495\pi$
$197$	−3.29903	−0.235046	−0.117523	+	0.993070i	$0.537495\pi$
$198$	0	0
$199$	−18.8208	−1.33417	−0.667087	−	0.744980i	$-0.732459\pi$
$199$	−18.8208	−1.33417	−0.667087	+	0.744980i	$0.732459\pi$
$200$	1.77476	0.125495
$201$	0	0
$202$	−4.96307	−0.349200
$203$	−3.69089	−0.259049
$204$	0	0
$205$	4.94031	0.345046
$206$	−25.2521	−1.75939
$207$	0	0
$208$	10.4332	0.723409
$209$	16.8446	1.16516
$210$	0	0
$211$	4.25252	0.292756	0.146378	−	0.989229i	$-0.453238\pi$
$211$	4.25252	0.292756	0.146378	+	0.989229i	$0.453238\pi$
$212$	10.2122	0.701377
$213$	0	0
$214$	29.5263	2.01837
$215$	4.43179	0.302245
$216$	0	0
$217$	−3.06525	−0.208083
$218$	−11.4195	−0.773427
$219$	0	0
$220$	−7.14706	−0.481855
$221$	−19.2326	−1.29373
$222$	0	0
$223$	−2.65381	−0.177712	−0.0888560	−	0.996044i	$-0.528321\pi$
$223$	−2.65381	−0.177712	−0.0888560	+	0.996044i	$0.528321\pi$
$224$	28.3776	1.89606
$225$	0	0
$226$	−34.1554	−2.27198
$227$	−23.1112	−1.53394	−0.766972	−	0.641681i	$-0.778237\pi$
$227$	−23.1112	−1.53394	−0.766972	+	0.641681i	$0.778237\pi$
$228$	0	0
$229$	17.6282	1.16490	0.582452	−	0.812865i	$-0.302093\pi$
$229$	17.6282	1.16490	0.582452	+	0.812865i	$0.302093\pi$
$230$	1.73231	0.114225
$231$	0	0
$232$	−0.421884	−0.0276980
$233$	3.16486	0.207337	0.103668	−	0.994612i	$-0.466942\pi$
$233$	3.16486	0.207337	0.103668	+	0.994612i	$0.466942\pi$
$234$	0	0
$235$	10.6156	0.692485
$236$	7.28259	0.474056
$237$	0	0
$238$	−58.0526	−3.76299
$239$	−12.0664	−0.780509	−0.390255	−	0.920707i	$-0.627613\pi$
$239$	−12.0664	−0.780509	−0.390255	+	0.920707i	$0.627613\pi$
$240$	0	0
$241$	15.9750	1.02904	0.514519	−	0.857479i	$-0.327971\pi$
$241$	15.9750	1.02904	0.514519	+	0.857479i	$0.327971\pi$
$242$	17.9983	1.15697
$243$	0	0
$244$	−4.75889	−0.304657
$245$	−5.89839	−0.376834
$246$	0	0
$247$	−8.90187	−0.566413
$248$	−0.350371	−0.0222486
$249$	0	0
$250$	15.9490	1.00870
$251$	−19.0291	−1.20111	−0.600555	−	0.799584i	$-0.705053\pi$
$251$	−19.0291	−1.20111	−0.600555	+	0.799584i	$0.705053\pi$
$252$	0	0
$253$	−4.50039	−0.282937
$254$	11.8192	0.741600
$255$	0	0
$256$	18.8877	1.18048
$257$	−3.28939	−0.205186	−0.102593	−	0.994723i	$-0.532714\pi$
$257$	−3.28939	−0.205186	−0.102593	+	0.994723i	$0.532714\pi$
$258$	0	0
$259$	24.8183	1.54213
$260$	3.77701	0.234240
$261$	0	0
$262$	−17.7968	−1.09949
$263$	6.60009	0.406979	0.203489	−	0.979077i	$-0.434772\pi$
$263$	6.60009	0.406979	0.203489	+	0.979077i	$0.434772\pi$
$264$	0	0
$265$	−5.10091	−0.313346
$266$	−26.8698	−1.64749
$267$	0	0
$268$	−0.665838	−0.0406725
$269$	3.23293	0.197115	0.0985575	−	0.995131i	$-0.468577\pi$
$269$	3.23293	0.197115	0.0985575	+	0.995131i	$0.468577\pi$
$270$	0	0
$271$	−2.16831	−0.131715	−0.0658576	−	0.997829i	$-0.520978\pi$
$271$	−2.16831	−0.131715	−0.0658576	+	0.997829i	$0.520978\pi$
$272$	−35.4741	−2.15093
$273$	0	0
$274$	−26.2504	−1.58584
$275$	−18.9321	−1.14165
$276$	0	0
$277$	9.47700	0.569418	0.284709	−	0.958614i	$-0.408103\pi$
$277$	9.47700	0.569418	0.284709	+	0.958614i	$0.408103\pi$
$278$	25.6402	1.53779
$279$	0	0
$280$	−1.38684	−0.0828794
$281$	9.93844	0.592878	0.296439	−	0.955052i	$-0.404201\pi$
$281$	9.93844	0.592878	0.296439	+	0.955052i	$0.404201\pi$
$282$	0	0
$283$	30.9884	1.84207	0.921033	−	0.389484i	$-0.127347\pi$
$283$	30.9884	1.84207	0.921033	+	0.389484i	$0.127347\pi$
$284$	−20.6988	−1.22825
$285$	0	0
$286$	−20.8183	−1.23101
$287$	20.4730	1.20849
$288$	0	0
$289$	48.3936	2.84668
$290$	−1.73231	−0.101725
$291$	0	0
$292$	−22.6857	−1.32758
$293$	−10.0605	−0.587741	−0.293871	−	0.955845i	$-0.594943\pi$
$293$	−10.0605	−0.587741	−0.293871	+	0.955845i	$0.594943\pi$
$294$	0	0
$295$	−3.63759	−0.211789
$296$	2.83683	0.164888
$297$	0	0
$298$	12.8925	0.746842
$299$	2.37833	0.137542
$300$	0	0
$301$	18.3657	1.05858
$302$	9.44442	0.543465
$303$	0	0
$304$	−16.4193	−0.941710
$305$	2.37703	0.136108
$306$	0	0
$307$	8.96988	0.511938	0.255969	−	0.966685i	$-0.417605\pi$
$307$	8.96988	0.511938	0.255969	+	0.966685i	$0.417605\pi$
$308$	−29.6180	−1.68764
$309$	0	0
$310$	−1.43867	−0.0817110
$311$	−7.22707	−0.409810	−0.204905	−	0.978782i	$-0.565688\pi$
$311$	−7.22707	−0.409810	−0.204905	+	0.978782i	$0.565688\pi$
$312$	0	0
$313$	−16.3038	−0.921543	−0.460772	−	0.887519i	$-0.652427\pi$
$313$	−16.3038	−0.921543	−0.460772	+	0.887519i	$0.652427\pi$
$314$	−22.4430	−1.26653
$315$	0	0
$316$	−19.9003	−1.11948
$317$	14.1617	0.795399	0.397700	−	0.917516i	$-0.369809\pi$
$317$	14.1617	0.795399	0.397700	+	0.917516i	$0.369809\pi$
$318$	0	0
$319$	4.50039	0.251974
$320$	5.50494	0.307735
$321$	0	0
$322$	7.17884	0.400061
$323$	30.2676	1.68413
$324$	0	0
$325$	10.0050	0.554980
$326$	24.6505	1.36526
$327$	0	0
$328$	2.34015	0.129213
$329$	43.9919	2.42535
$330$	0	0
$331$	5.91686	0.325220	0.162610	−	0.986690i	$-0.448009\pi$
$331$	5.91686	0.325220	0.162610	+	0.986690i	$0.448009\pi$
$332$	0.325522	0.0178653
$333$	0	0
$334$	−19.6350	−1.07438
$335$	0.332580	0.0181708
$336$	0	0
$337$	−27.4448	−1.49501	−0.747506	−	0.664256i	$-0.768749\pi$
$337$	−27.4448	−1.49501	−0.747506	+	0.664256i	$0.768749\pi$
$338$	−14.2834	−0.776912
$339$	0	0
$340$	−12.8424	−0.696474
$341$	3.73754	0.202399
$342$	0	0
$343$	1.39280	0.0752039
$344$	2.09927	0.113185
$345$	0	0
$346$	−16.3097	−0.876817
$347$	20.7086	1.11170	0.555849	−	0.831283i	$-0.312393\pi$
$347$	20.7086	1.11170	0.555849	+	0.831283i	$0.312393\pi$
$348$	0	0
$349$	−5.21746	−0.279284	−0.139642	−	0.990202i	$-0.544595\pi$
$349$	−5.21746	−0.279284	−0.139642	+	0.990202i	$0.544595\pi$
$350$	30.1997	1.61424
$351$	0	0
$352$	−34.6016	−1.84427
$353$	30.2413	1.60958	0.804791	−	0.593559i	$-0.202277\pi$
$353$	30.2413	1.60958	0.804791	+	0.593559i	$0.202277\pi$
$354$	0	0
$355$	10.3389	0.548730
$356$	−0.387128	−0.0205177
$357$	0	0
$358$	0.263838	0.0139443
$359$	3.46133	0.182682	0.0913411	−	0.995820i	$-0.470885\pi$
$359$	3.46133	0.182682	0.0913411	+	0.995820i	$0.470885\pi$
$360$	0	0
$361$	−4.99060	−0.262663
$362$	26.1134	1.37249
$363$	0	0
$364$	15.6522	0.820401
$365$	11.3313	0.593107
$366$	0	0
$367$	2.14876	0.112164	0.0560821	−	0.998426i	$-0.482139\pi$
$367$	2.14876	0.112164	0.0560821	+	0.998426i	$0.482139\pi$
$368$	4.38676	0.228676
$369$	0	0
$370$	11.6484	0.605573
$371$	−21.1386	−1.09746
$372$	0	0
$373$	−35.8019	−1.85375	−0.926877	−	0.375365i	$-0.877517\pi$
$373$	−35.8019	−1.85375	−0.926877	+	0.375365i	$0.877517\pi$
$374$	70.7851	3.66021
$375$	0	0
$376$	5.02845	0.259323
$377$	−2.37833	−0.122490
$378$	0	0
$379$	21.1506	1.08643	0.543216	−	0.839593i	$-0.317206\pi$
$379$	21.1506	1.08643	0.543216	+	0.839593i	$0.317206\pi$
$380$	−5.94411	−0.304926
$381$	0	0
$382$	−29.0323	−1.48542
$383$	12.5981	0.643732	0.321866	−	0.946785i	$-0.395690\pi$
$383$	12.5981	0.643732	0.321866	+	0.946785i	$0.395690\pi$
$384$	0	0
$385$	14.7939	0.753969
$386$	−39.2338	−1.99695
$387$	0	0
$388$	24.6951	1.25370
$389$	27.2193	1.38008	0.690038	−	0.723773i	$-0.257594\pi$
$389$	27.2193	1.38008	0.690038	+	0.723773i	$0.257594\pi$
$390$	0	0
$391$	−8.08663	−0.408958
$392$	−2.79398	−0.141117
$393$	0	0
$394$	−6.41668	−0.323268
$395$	9.94005	0.500138
$396$	0	0
$397$	1.53013	0.0767952	0.0383976	−	0.999263i	$-0.487775\pi$
$397$	1.53013	0.0767952	0.0383976	+	0.999263i	$0.487775\pi$
$398$	−36.6069	−1.83494
$399$	0	0
$400$	18.4541	0.922703
$401$	24.0237	1.19969	0.599843	−	0.800118i	$-0.295230\pi$
$401$	24.0237	1.19969	0.599843	+	0.800118i	$0.295230\pi$
$402$	0	0
$403$	−1.97518	−0.0983908
$404$	−4.54989	−0.226366
$405$	0	0
$406$	−7.17884	−0.356280
$407$	−30.2616	−1.50001
$408$	0	0
$409$	3.44189	0.170190	0.0850951	−	0.996373i	$-0.472881\pi$
$409$	3.44189	0.170190	0.0850951	+	0.996373i	$0.472881\pi$
$410$	9.60899	0.474554
$411$	0	0
$412$	−23.1498	−1.14051
$413$	−15.0745	−0.741766
$414$	0	0
$415$	−0.162595	−0.00798149
$416$	18.2859	0.896541
$417$	0	0
$418$	32.7630	1.60249
$419$	4.54765	0.222167	0.111084	−	0.993811i	$-0.464568\pi$
$419$	4.54765	0.222167	0.111084	+	0.993811i	$0.464568\pi$
$420$	0	0
$421$	−7.22743	−0.352243	−0.176122	−	0.984368i	$-0.556355\pi$
$421$	−7.22743	−0.352243	−0.176122	+	0.984368i	$0.556355\pi$
$422$	8.27123	0.402637
$423$	0	0
$424$	−2.41623	−0.117342
$425$	−34.0185	−1.65014
$426$	0	0
$427$	9.85059	0.476703
$428$	27.0682	1.30839
$429$	0	0
$430$	8.61991	0.415689
$431$	30.8300	1.48503	0.742515	−	0.669830i	$-0.233633\pi$
$431$	30.8300	1.48503	0.742515	+	0.669830i	$0.233633\pi$
$432$	0	0
$433$	6.41433	0.308253	0.154126	−	0.988051i	$-0.450744\pi$
$433$	6.41433	0.308253	0.154126	+	0.988051i	$0.450744\pi$
$434$	−5.96197	−0.286184
$435$	0	0
$436$	−10.4688	−0.501367
$437$	−3.74291	−0.179048
$438$	0	0
$439$	−5.00105	−0.238687	−0.119344	−	0.992853i	$-0.538079\pi$
$439$	−5.00105	−0.238687	−0.119344	+	0.992853i	$0.538079\pi$
$440$	1.69101	0.0806156
$441$	0	0
$442$	−37.4078	−1.77931
$443$	−32.0940	−1.52483	−0.762416	−	0.647087i	$-0.775987\pi$
$443$	−32.0940	−1.52483	−0.762416	+	0.647087i	$0.775987\pi$
$444$	0	0
$445$	0.193367	0.00916648
$446$	−5.16170	−0.244414
$447$	0	0
$448$	22.8129	1.07781
$449$	−4.71662	−0.222591	−0.111296	−	0.993787i	$-0.535500\pi$
$449$	−4.71662	−0.222591	−0.111296	+	0.993787i	$0.535500\pi$
$450$	0	0
$451$	−24.9633	−1.17548
$452$	−31.3119	−1.47279
$453$	0	0
$454$	−44.9517	−2.10969
$455$	−7.81816	−0.366521
$456$	0	0
$457$	−15.3936	−0.720083	−0.360042	−	0.932936i	$-0.617237\pi$
$457$	−15.3936	−0.720083	−0.360042	+	0.932936i	$0.617237\pi$
$458$	34.2871	1.60213
$459$	0	0
$460$	1.58810	0.0740454
$461$	−5.23341	−0.243744	−0.121872	−	0.992546i	$-0.538890\pi$
$461$	−5.23341	−0.243744	−0.121872	+	0.992546i	$0.538890\pi$
$462$	0	0
$463$	25.7280	1.19568	0.597841	−	0.801615i	$-0.296026\pi$
$463$	25.7280	1.19568	0.597841	+	0.801615i	$0.296026\pi$
$464$	−4.38676	−0.203650
$465$	0	0
$466$	6.15571	0.285158
$467$	−14.6282	−0.676914	−0.338457	−	0.940982i	$-0.609905\pi$
$467$	−14.6282	−0.676914	−0.338457	+	0.940982i	$0.609905\pi$
$468$	0	0
$469$	1.37824	0.0636412
$470$	20.6475	0.952399
$471$	0	0
$472$	−1.72307	−0.0793109
$473$	−22.3937	−1.02967
$474$	0	0
$475$	−15.7455	−0.722455
$476$	−53.2197	−2.43932
$477$	0	0
$478$	−23.4693	−1.07346
$479$	−26.9689	−1.23224	−0.616121	−	0.787652i	$-0.711297\pi$
$479$	−26.9689	−1.23224	−0.616121	+	0.787652i	$0.711297\pi$
$480$	0	0
$481$	15.9924	0.729189
$482$	31.0716	1.41527
$483$	0	0
$484$	16.4999	0.749997
$485$	−12.3350	−0.560102
$486$	0	0
$487$	−9.49480	−0.430251	−0.215125	−	0.976586i	$-0.569016\pi$
$487$	−9.49480	−0.430251	−0.215125	+	0.976586i	$0.569016\pi$
$488$	1.12596	0.0509700
$489$	0	0
$490$	−11.4725	−0.518274
$491$	−40.7189	−1.83762	−0.918810	−	0.394701i	$-0.870848\pi$
$491$	−40.7189	−1.83762	−0.918810	+	0.394701i	$0.870848\pi$
$492$	0	0
$493$	8.08663	0.364204
$494$	−17.3143	−0.779007
$495$	0	0
$496$	−3.64317	−0.163583
$497$	42.8451	1.92186
$498$	0	0
$499$	7.73103	0.346088	0.173044	−	0.984914i	$-0.444640\pi$
$499$	7.73103	0.346088	0.173044	+	0.984914i	$0.444640\pi$
$500$	14.6212	0.653881
$501$	0	0
$502$	−37.0120	−1.65193
$503$	17.0295	0.759309	0.379655	−	0.925128i	$-0.376043\pi$
$503$	17.0295	0.759309	0.379655	+	0.925128i	$0.376043\pi$
$504$	0	0
$505$	2.27263	0.101131
$506$	−8.75335	−0.389134
$507$	0	0
$508$	10.8352	0.480735
$509$	43.3883	1.92315	0.961576	−	0.274538i	$-0.0885250\pi$
$509$	43.3883	1.92315	0.961576	+	0.274538i	$0.0885250\pi$
$510$	0	0
$511$	46.9578	2.07729
$512$	30.0265	1.32700
$513$	0	0
$514$	−6.39792	−0.282200
$515$	11.5631	0.509533
$516$	0	0
$517$	−53.6404	−2.35910
$518$	48.2720	2.12095
$519$	0	0
$520$	−0.893648	−0.0391891
$521$	29.4799	1.29154	0.645769	−	0.763533i	$-0.276537\pi$
$521$	29.4799	1.29154	0.645769	+	0.763533i	$0.276537\pi$
$522$	0	0
$523$	−29.3504	−1.28340	−0.641702	−	0.766954i	$-0.721771\pi$
$523$	−29.3504	−1.28340	−0.641702	+	0.766954i	$0.721771\pi$
$524$	−16.3152	−0.712733
$525$	0	0
$526$	12.8373	0.559732
$527$	6.71588	0.292548
$528$	0	0
$529$	1.00000	0.0434783
$530$	−9.92136	−0.430956
$531$	0	0
$532$	−24.6329	−1.06797
$533$	13.1924	0.571425
$534$	0	0
$535$	−13.5203	−0.584535
$536$	0.157539	0.00680463
$537$	0	0
$538$	6.28810	0.271099
$539$	29.8045	1.28377
$540$	0	0
$541$	−13.9790	−0.601005	−0.300503	−	0.953781i	$-0.597154\pi$
$541$	−13.9790	−0.601005	−0.300503	+	0.953781i	$0.597154\pi$
$542$	−4.21739	−0.181153
$543$	0	0
$544$	−62.1746	−2.66571
$545$	5.22910	0.223990
$546$	0	0
$547$	−15.1676	−0.648522	−0.324261	−	0.945968i	$-0.605116\pi$
$547$	−15.1676	−0.648522	−0.324261	+	0.945968i	$0.605116\pi$
$548$	−24.0650	−1.02801
$549$	0	0
$550$	−36.8232	−1.57015
$551$	3.74291	0.159453
$552$	0	0
$553$	41.1923	1.75168
$554$	18.4329	0.783141
$555$	0	0
$556$	23.5056	0.996860
$557$	−16.5036	−0.699282	−0.349641	−	0.936884i	$-0.613696\pi$
$557$	−16.5036	−0.699282	−0.349641	+	0.936884i	$0.613696\pi$
$558$	0	0
$559$	11.8344	0.500544
$560$	−14.4204	−0.609373
$561$	0	0
$562$	19.3304	0.815406
$563$	37.6335	1.58606	0.793032	−	0.609181i	$-0.208502\pi$
$563$	37.6335	1.58606	0.793032	+	0.609181i	$0.208502\pi$
$564$	0	0
$565$	15.6400	0.657982
$566$	60.2729	2.53346
$567$	0	0
$568$	4.89737	0.205489
$569$	−27.3924	−1.14835	−0.574174	−	0.818733i	$-0.694677\pi$
$569$	−27.3924	−1.14835	−0.574174	+	0.818733i	$0.694677\pi$
$570$	0	0
$571$	37.6260	1.57460	0.787300	−	0.616570i	$-0.211478\pi$
$571$	37.6260	1.57460	0.787300	+	0.616570i	$0.211478\pi$
$572$	−19.0852	−0.797992
$573$	0	0
$574$	39.8204	1.66207
$575$	4.20676	0.175434
$576$	0	0
$577$	2.57125	0.107043	0.0535213	−	0.998567i	$-0.482956\pi$
$577$	2.57125	0.107043	0.0535213	+	0.998567i	$0.482956\pi$
$578$	94.1264	3.91514
$579$	0	0
$580$	−1.58810	−0.0659421
$581$	−0.673808	−0.0279543
$582$	0	0
$583$	25.7748	1.06748
$584$	5.36747	0.222108
$585$	0	0
$586$	−19.5679	−0.808341
$587$	31.4684	1.29884	0.649421	−	0.760429i	$-0.275011\pi$
$587$	31.4684	1.29884	0.649421	+	0.760429i	$0.275011\pi$
$588$	0	0
$589$	3.10846	0.128082
$590$	−7.07518	−0.291281
$591$	0	0
$592$	29.4975	1.21234
$593$	−17.9331	−0.736426	−0.368213	−	0.929742i	$-0.620030\pi$
$593$	−17.9331	−0.736426	−0.368213	+	0.929742i	$0.620030\pi$
$594$	0	0
$595$	26.5828	1.08979
$596$	11.8192	0.484133
$597$	0	0
$598$	4.62589	0.189167
$599$	−36.3152	−1.48380	−0.741900	−	0.670510i	$-0.766075\pi$
$599$	−36.3152	−1.48380	−0.741900	+	0.670510i	$0.766075\pi$
$600$	0	0
$601$	−20.3773	−0.831208	−0.415604	−	0.909546i	$-0.636430\pi$
$601$	−20.3773	−0.831208	−0.415604	+	0.909546i	$0.636430\pi$
$602$	35.7216	1.45590
$603$	0	0
$604$	8.65817	0.352296
$605$	−8.24158	−0.335068
$606$	0	0
$607$	33.7527	1.36998	0.684991	−	0.728552i	$-0.259806\pi$
$607$	33.7527	1.36998	0.684991	+	0.728552i	$0.259806\pi$
$608$	−28.7776	−1.16709
$609$	0	0
$610$	4.62336	0.187194
$611$	28.3474	1.14681
$612$	0	0
$613$	8.15566	0.329404	0.164702	−	0.986343i	$-0.447334\pi$
$613$	8.15566	0.329404	0.164702	+	0.986343i	$0.447334\pi$
$614$	17.4466	0.704087
$615$	0	0
$616$	7.00767	0.282347
$617$	−39.4382	−1.58772	−0.793861	−	0.608099i	$-0.791932\pi$
$617$	−39.4382	−1.58772	−0.793861	+	0.608099i	$0.791932\pi$
$618$	0	0
$619$	−22.8899	−0.920024	−0.460012	−	0.887913i	$-0.652155\pi$
$619$	−22.8899	−0.920024	−0.460012	+	0.887913i	$0.652155\pi$
$620$	−1.31890	−0.0529684
$621$	0	0
$622$	−14.0568	−0.563626
$623$	0.801329	0.0321046
$624$	0	0
$625$	13.7306	0.549225
$626$	−31.7111	−1.26743
$627$	0	0
$628$	−20.5746	−0.821017
$629$	−54.3762	−2.16812
$630$	0	0
$631$	10.4183	0.414747	0.207374	−	0.978262i	$-0.433508\pi$
$631$	10.4183	0.414747	0.207374	+	0.978262i	$0.433508\pi$
$632$	4.70846	0.187292
$633$	0	0
$634$	27.5447	1.09394
$635$	−5.41210	−0.214773
$636$	0	0
$637$	−15.7508	−0.624069
$638$	8.75335	0.346548
$639$	0	0
$640$	−2.98830	−0.118123
$641$	−34.5328	−1.36396	−0.681982	−	0.731369i	$-0.738882\pi$
$641$	−34.5328	−1.36396	−0.681982	+	0.731369i	$0.738882\pi$
$642$	0	0
$643$	1.08338	0.0427243	0.0213621	−	0.999772i	$-0.493200\pi$
$643$	1.08338	0.0427243	0.0213621	+	0.999772i	$0.493200\pi$
$644$	6.58120	0.259336
$645$	0	0
$646$	58.8710	2.31625
$647$	37.1584	1.46085	0.730424	−	0.682994i	$-0.239322\pi$
$647$	37.1584	1.46085	0.730424	+	0.682994i	$0.239322\pi$
$648$	0	0
$649$	18.3807	0.721505
$650$	19.4600	0.763284
$651$	0	0
$652$	22.5983	0.885019
$653$	23.3144	0.912362	0.456181	−	0.889887i	$-0.349217\pi$
$653$	23.3144	0.912362	0.456181	+	0.889887i	$0.349217\pi$
$654$	0	0
$655$	8.14930	0.318419
$656$	24.3330	0.950044
$657$	0	0
$658$	85.5650	3.33567
$659$	48.4850	1.88871	0.944355	−	0.328929i	$-0.106688\pi$
$659$	48.4850	1.88871	0.944355	+	0.328929i	$0.106688\pi$
$660$	0	0
$661$	−2.79308	−0.108638	−0.0543191	−	0.998524i	$-0.517299\pi$
$661$	−2.79308	−0.108638	−0.0543191	+	0.998524i	$0.517299\pi$
$662$	11.5084	0.447287
$663$	0	0
$664$	−0.0770190	−0.00298892
$665$	12.3039	0.477125
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−18.0004	−0.696456
$669$	0	0
$670$	0.646875	0.0249910
$671$	−12.0111	−0.463682
$672$	0	0
$673$	13.8118	0.532406	0.266203	−	0.963917i	$-0.414231\pi$
$673$	13.8118	0.532406	0.266203	+	0.963917i	$0.414231\pi$
$674$	−53.3805	−2.05614
$675$	0	0
$676$	−13.0943	−0.503626
$677$	−28.5318	−1.09657	−0.548284	−	0.836292i	$-0.684719\pi$
$677$	−28.5318	−1.09657	−0.548284	+	0.836292i	$0.684719\pi$
$678$	0	0
$679$	−51.1171	−1.96170
$680$	3.03852	0.116522
$681$	0	0
$682$	7.26958	0.278367
$683$	6.40227	0.244976	0.122488	−	0.992470i	$-0.460913\pi$
$683$	6.40227	0.244976	0.122488	+	0.992470i	$0.460913\pi$
$684$	0	0
$685$	12.0203	0.459271
$686$	2.70901	0.103431
$687$	0	0
$688$	21.8283	0.832197
$689$	−13.6212	−0.518928
$690$	0	0
$691$	−13.6105	−0.517767	−0.258883	−	0.965909i	$-0.583355\pi$
$691$	−13.6105	−0.517767	−0.258883	+	0.965909i	$0.583355\pi$
$692$	−14.9519	−0.568388
$693$	0	0
$694$	40.2787	1.52896
$695$	−11.7409	−0.445356
$696$	0	0
$697$	−44.8559	−1.69904
$698$	−10.1481	−0.384110
$699$	0	0
$700$	27.6855	1.04641
$701$	34.8262	1.31537	0.657683	−	0.753295i	$-0.271537\pi$
$701$	34.8262	1.31537	0.657683	+	0.753295i	$0.271537\pi$
$702$	0	0
$703$	−25.1681	−0.949234
$704$	−27.8164	−1.04837
$705$	0	0
$706$	58.8199	2.21371
$707$	9.41797	0.354199
$708$	0	0
$709$	20.7395	0.778887	0.389444	−	0.921050i	$-0.372667\pi$
$709$	20.7395	0.778887	0.389444	+	0.921050i	$0.372667\pi$
$710$	20.1093	0.754687
$711$	0	0
$712$	0.0915952	0.00343268
$713$	−0.830492	−0.0311022
$714$	0	0
$715$	9.53289	0.356510
$716$	0.241874	0.00903924
$717$	0	0
$718$	6.73235	0.251249
$719$	−34.7376	−1.29549	−0.647747	−	0.761856i	$-0.724289\pi$
$719$	−34.7376	−1.29549	−0.647747	+	0.761856i	$0.724289\pi$
$720$	0	0
$721$	47.9186	1.78458
$722$	−9.70680	−0.361250
$723$	0	0
$724$	23.9394	0.889702
$725$	−4.20676	−0.156235
$726$	0	0
$727$	−48.7362	−1.80753	−0.903763	−	0.428033i	$-0.859207\pi$
$727$	−48.7362	−1.80753	−0.903763	+	0.428033i	$0.859207\pi$
$728$	−3.70335	−0.137255
$729$	0	0
$730$	22.0396	0.815722
$731$	−40.2387	−1.48828
$732$	0	0
$733$	34.5425	1.27586	0.637929	−	0.770095i	$-0.279791\pi$
$733$	34.5425	1.27586	0.637929	+	0.770095i	$0.279791\pi$
$734$	4.17937	0.154264
$735$	0	0
$736$	7.68856	0.283404
$737$	−1.68052	−0.0619029
$738$	0	0
$739$	0.863005	0.0317462	0.0158731	−	0.999874i	$-0.494947\pi$
$739$	0.863005	0.0317462	0.0158731	+	0.999874i	$0.494947\pi$
$740$	10.6787	0.392557
$741$	0	0
$742$	−41.1149	−1.50938
$743$	−18.8605	−0.691924	−0.345962	−	0.938249i	$-0.612447\pi$
$743$	−18.8605	−0.691924	−0.345962	+	0.938249i	$0.612447\pi$
$744$	0	0
$745$	−5.90358	−0.216290
$746$	−69.6354	−2.54953
$747$	0	0
$748$	64.8922	2.37269
$749$	−56.0293	−2.04727
$750$	0	0
$751$	35.9838	1.31307	0.656534	−	0.754297i	$-0.272022\pi$
$751$	35.9838	1.31307	0.656534	+	0.754297i	$0.272022\pi$
$752$	52.2860	1.90668
$753$	0	0
$754$	−4.62589	−0.168465
$755$	−4.32468	−0.157391
$756$	0	0
$757$	−29.0366	−1.05535	−0.527676	−	0.849446i	$-0.676936\pi$
$757$	−29.0366	−1.05535	−0.527676	+	0.849446i	$0.676936\pi$
$758$	41.1383	1.49421
$759$	0	0
$760$	1.40639	0.0510150
$761$	23.6765	0.858274	0.429137	−	0.903239i	$-0.358818\pi$
$761$	23.6765	0.858274	0.429137	+	0.903239i	$0.358818\pi$
$762$	0	0
$763$	21.6698	0.784499
$764$	−26.6153	−0.962909
$765$	0	0
$766$	24.5035	0.885347
$767$	−9.71366	−0.350740
$768$	0	0
$769$	−25.5332	−0.920752	−0.460376	−	0.887724i	$-0.652285\pi$
$769$	−25.5332	−0.920752	−0.460376	+	0.887724i	$0.652285\pi$
$770$	28.7745	1.03696
$771$	0	0
$772$	−35.9676	−1.29450
$773$	−11.8275	−0.425407	−0.212703	−	0.977117i	$-0.568227\pi$
$773$	−11.8275	−0.425407	−0.212703	+	0.977117i	$0.568227\pi$
$774$	0	0
$775$	−3.49368	−0.125497
$776$	−5.84290	−0.209748
$777$	0	0
$778$	52.9421	1.89807
$779$	−20.7616	−0.743863
$780$	0	0
$781$	−52.2421	−1.86937
$782$	−15.7286	−0.562455
$783$	0	0
$784$	−29.0519	−1.03757
$785$	10.2768	0.366796
$786$	0	0
$787$	−25.5112	−0.909374	−0.454687	−	0.890651i	$-0.650249\pi$
$787$	−25.5112	−0.909374	−0.454687	+	0.890651i	$0.650249\pi$
$788$	−5.88249	−0.209555
$789$	0	0
$790$	19.3336	0.687857
$791$	64.8136	2.30451
$792$	0	0
$793$	6.34750	0.225406
$794$	2.97614	0.105619
$795$	0	0
$796$	−33.5594	−1.18948
$797$	−47.4936	−1.68231	−0.841154	−	0.540795i	$-0.818124\pi$
$797$	−47.4936	−1.68231	−0.841154	+	0.540795i	$0.818124\pi$
$798$	0	0
$799$	−96.3850	−3.40985
$800$	32.3439	1.14353
$801$	0	0
$802$	46.7265	1.64997
$803$	−57.2569	−2.02055
$804$	0	0
$805$	−3.28725	−0.115860
$806$	−3.84176	−0.135320
$807$	0	0
$808$	1.07651	0.0378716
$809$	−11.2763	−0.396453	−0.198226	−	0.980156i	$-0.563518\pi$
$809$	−11.2763	−0.396453	−0.198226	+	0.980156i	$0.563518\pi$
$810$	0	0
$811$	−42.6721	−1.49842	−0.749210	−	0.662333i	$-0.769567\pi$
$811$	−42.6721	−1.49842	−0.749210	+	0.662333i	$0.769567\pi$
$812$	−6.58120	−0.230955
$813$	0	0
$814$	−58.8593	−2.06302
$815$	−11.2877	−0.395390
$816$	0	0
$817$	−18.6246	−0.651591
$818$	6.69453	0.234069
$819$	0	0
$820$	8.80904	0.307625
$821$	−14.8975	−0.519925	−0.259963	−	0.965619i	$-0.583710\pi$
$821$	−14.8975	−0.519925	−0.259963	+	0.965619i	$0.583710\pi$
$822$	0	0
$823$	49.7000	1.73243	0.866216	−	0.499669i	$-0.166545\pi$
$823$	49.7000	1.73243	0.866216	+	0.499669i	$0.166545\pi$
$824$	5.47729	0.190810
$825$	0	0
$826$	−29.3201	−1.02018
$827$	−3.92447	−0.136467	−0.0682336	−	0.997669i	$-0.521736\pi$
$827$	−3.92447	−0.136467	−0.0682336	+	0.997669i	$0.521736\pi$
$828$	0	0
$829$	40.1066	1.39296	0.696480	−	0.717576i	$-0.254749\pi$
$829$	40.1066	1.39296	0.696480	+	0.717576i	$0.254749\pi$
$830$	−0.316251	−0.0109772
$831$	0	0
$832$	14.7001	0.509635
$833$	53.5548	1.85557
$834$	0	0
$835$	8.99104	0.311148
$836$	30.0355	1.03880
$837$	0	0
$838$	8.84526	0.305554
$839$	−10.9698	−0.378719	−0.189360	−	0.981908i	$-0.560641\pi$
$839$	−10.9698	−0.378719	−0.189360	+	0.981908i	$0.560641\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−14.0575	−0.484453
$843$	0	0
$844$	7.58265	0.261006
$845$	6.54048	0.224999
$846$	0	0
$847$	−34.1538	−1.17354
$848$	−25.1240	−0.862762
$849$	0	0
$850$	−66.1666	−2.26950
$851$	6.72421	0.230503
$852$	0	0
$853$	−46.2251	−1.58272	−0.791359	−	0.611351i	$-0.790626\pi$
$853$	−46.2251	−1.58272	−0.791359	+	0.611351i	$0.790626\pi$
$854$	19.1596	0.655627
$855$	0	0
$856$	−6.40439	−0.218897
$857$	19.2667	0.658137	0.329069	−	0.944306i	$-0.393265\pi$
$857$	19.2667	0.658137	0.329069	+	0.944306i	$0.393265\pi$
$858$	0	0
$859$	−1.44604	−0.0493382	−0.0246691	−	0.999696i	$-0.507853\pi$
$859$	−1.44604	−0.0493382	−0.0246691	+	0.999696i	$0.507853\pi$
$860$	7.90230	0.269466
$861$	0	0
$862$	59.9649	2.04241
$863$	−5.63105	−0.191683	−0.0958416	−	0.995397i	$-0.530554\pi$
$863$	−5.63105	−0.191683	−0.0958416	+	0.995397i	$0.530554\pi$
$864$	0	0
$865$	7.46837	0.253932
$866$	12.4760	0.423951
$867$	0	0
$868$	−5.46563	−0.185516
$869$	−50.2269	−1.70383
$870$	0	0
$871$	0.888108	0.0300924
$872$	2.47695	0.0838800
$873$	0	0
$874$	−7.28003	−0.246251
$875$	−30.2649	−1.02314
$876$	0	0
$877$	4.44244	0.150011	0.0750053	−	0.997183i	$-0.476103\pi$
$877$	4.44244	0.150011	0.0750053	+	0.997183i	$0.476103\pi$
$878$	−9.72714	−0.328275
$879$	0	0
$880$	17.5832	0.592728
$881$	−35.8832	−1.20894	−0.604468	−	0.796630i	$-0.706614\pi$
$881$	−35.8832	−1.20894	−0.604468	+	0.796630i	$0.706614\pi$
$882$	0	0
$883$	−9.76965	−0.328775	−0.164387	−	0.986396i	$-0.552565\pi$
$883$	−9.76965	−0.328775	−0.164387	+	0.986396i	$0.552565\pi$
$884$	−34.2936	−1.15342
$885$	0	0
$886$	−62.4234	−2.09716
$887$	11.7923	0.395948	0.197974	−	0.980207i	$-0.436564\pi$
$887$	11.7923	0.395948	0.197974	+	0.980207i	$0.436564\pi$
$888$	0	0
$889$	−22.4282	−0.752216
$890$	0.376102	0.0126070
$891$	0	0
$892$	−4.73199	−0.158439
$893$	−44.6120	−1.49288
$894$	0	0
$895$	−0.120814	−0.00403836
$896$	−12.3837	−0.413712
$897$	0	0
$898$	−9.17392	−0.306138
$899$	0.830492	0.0276985
$900$	0	0
$901$	46.3140	1.54294
$902$	−48.5541	−1.61667
$903$	0	0
$904$	7.40847	0.246402
$905$	−11.9575	−0.397482
$906$	0	0
$907$	19.0093	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$907$	19.0093	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$908$	−41.2094	−1.36758
$909$	0	0
$910$	−15.2065	−0.504089
$911$	45.1868	1.49711	0.748553	−	0.663075i	$-0.230749\pi$
$911$	45.1868	1.49711	0.748553	+	0.663075i	$0.230749\pi$
$912$	0	0
$913$	0.821591	0.0271907
$914$	−29.9409	−0.990356
$915$	0	0
$916$	31.4327	1.03857
$917$	33.7713	1.11523
$918$	0	0
$919$	10.6947	0.352785	0.176392	−	0.984320i	$-0.443557\pi$
$919$	10.6947	0.352785	0.176392	+	0.984320i	$0.443557\pi$
$920$	−0.375747	−0.0123880
$921$	0	0
$922$	−10.1791	−0.335230
$923$	27.6084	0.908742
$924$	0	0
$925$	28.2871	0.930075
$926$	50.0414	1.64446
$927$	0	0
$928$	−7.68856	−0.252389
$929$	−43.2595	−1.41930	−0.709649	−	0.704556i	$-0.751146\pi$
$929$	−43.2595	−1.41930	−0.709649	+	0.704556i	$0.751146\pi$
$930$	0	0
$931$	24.7880	0.812393
$932$	5.64325	0.184851
$933$	0	0
$934$	−28.4522	−0.930984
$935$	−32.4131	−1.06002
$936$	0	0
$937$	−15.7181	−0.513487	−0.256744	−	0.966480i	$-0.582650\pi$
$937$	−15.7181	−0.513487	−0.256744	+	0.966480i	$0.582650\pi$
$938$	2.68070	0.0875280
$939$	0	0
$940$	18.9286	0.617383
$941$	−27.5774	−0.898997	−0.449499	−	0.893281i	$-0.648397\pi$
$941$	−27.5774	−0.898997	−0.449499	+	0.893281i	$0.648397\pi$
$942$	0	0
$943$	5.54692	0.180632
$944$	−17.9166	−0.583135
$945$	0	0
$946$	−43.5562	−1.41614
$947$	11.1517	0.362383	0.181191	−	0.983448i	$-0.442005\pi$
$947$	11.1517	0.362383	0.181191	+	0.983448i	$0.442005\pi$
$948$	0	0
$949$	30.2586	0.982236
$950$	−30.6254	−0.993618
$951$	0	0
$952$	12.5919	0.408106
$953$	8.64031	0.279887	0.139944	−	0.990159i	$-0.455308\pi$
$953$	8.64031	0.279887	0.139944	+	0.990159i	$0.455308\pi$
$954$	0	0
$955$	13.2941	0.430188
$956$	−21.5155	−0.695861
$957$	0	0
$958$	−52.4550	−1.69475
$959$	49.8130	1.60854
$960$	0	0
$961$	−30.3103	−0.977751
$962$	31.1054	1.00288
$963$	0	0
$964$	28.4849	0.917436
$965$	17.9655	0.578330
$966$	0	0
$967$	−38.8650	−1.24981	−0.624907	−	0.780699i	$-0.714863\pi$
$967$	−38.8650	−1.24981	−0.624907	+	0.780699i	$0.714863\pi$
$968$	−3.90392	−0.125477
$969$	0	0
$970$	−23.9918	−0.770329
$971$	18.1124	0.581254	0.290627	−	0.956836i	$-0.406136\pi$
$971$	18.1124	0.581254	0.290627	+	0.956836i	$0.406136\pi$
$972$	0	0
$973$	−48.6550	−1.55981
$974$	−18.4676	−0.591739
$975$	0	0
$976$	11.7078	0.374758
$977$	−39.3962	−1.26040	−0.630198	−	0.776435i	$-0.717026\pi$
$977$	−39.3962	−1.26040	−0.630198	+	0.776435i	$0.717026\pi$
$978$	0	0
$979$	−0.977081	−0.0312276
$980$	−10.5174	−0.335966
$981$	0	0
$982$	−79.1990	−2.52734
$983$	−11.8203	−0.377010	−0.188505	−	0.982072i	$-0.560364\pi$
$983$	−11.8203	−0.377010	−0.188505	+	0.982072i	$0.560364\pi$
$984$	0	0
$985$	2.93825	0.0936205
$986$	15.7286	0.500902
$987$	0	0
$988$	−15.8729	−0.504984
$989$	4.97595	0.158226
$990$	0	0
$991$	48.6281	1.54472	0.772361	−	0.635183i	$-0.219076\pi$
$991$	48.6281	1.54472	0.772361	+	0.635183i	$0.219076\pi$
$992$	−6.38529	−0.202733
$993$	0	0
$994$	83.3344	2.64321
$995$	16.7626	0.531410
$996$	0	0
$997$	−14.8680	−0.470874	−0.235437	−	0.971890i	$-0.575652\pi$
$997$	−14.8680	−0.470874	−0.235437	+	0.971890i	$0.575652\pi$
$998$	15.0370	0.475988
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.m.1.9		11
3.2	odd	2		2001.2.a.l.1.3	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.l.1.3	✓	11	3.2	odd	2
6003.2.a.m.1.9		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.94502 q^{2} +1.78310 q^{4} -0.890641 q^{5} -3.69089 q^{7} -0.421884 q^{8} +O(q^{10})\)	\(q+1.94502 q^{2} +1.78310 q^{4} -0.890641 q^{5} -3.69089 q^{7} -0.421884 q^{8} -1.73231 q^{10} +4.50039 q^{11} -2.37833 q^{13} -7.17884 q^{14} -4.38676 q^{16} +8.08663 q^{17} +3.74291 q^{19} -1.58810 q^{20} +8.75335 q^{22} -1.00000 q^{23} -4.20676 q^{25} -4.62589 q^{26} -6.58120 q^{28} +1.00000 q^{29} +0.830492 q^{31} -7.68856 q^{32} +15.7286 q^{34} +3.28725 q^{35} -6.72421 q^{37} +7.28003 q^{38} +0.375747 q^{40} -5.54692 q^{41} -4.97595 q^{43} +8.02463 q^{44} -1.94502 q^{46} -11.9191 q^{47} +6.62264 q^{49} -8.18222 q^{50} -4.24078 q^{52} +5.72723 q^{53} -4.00823 q^{55} +1.55712 q^{56} +1.94502 q^{58} +4.08424 q^{59} -2.66889 q^{61} +1.61532 q^{62} -6.18087 q^{64} +2.11823 q^{65} -0.373417 q^{67} +14.4192 q^{68} +6.39377 q^{70} -11.6083 q^{71} -12.7226 q^{73} -13.0787 q^{74} +6.67397 q^{76} -16.6104 q^{77} -11.1606 q^{79} +3.90703 q^{80} -10.7889 q^{82} +0.182560 q^{83} -7.20228 q^{85} -9.67832 q^{86} -1.89864 q^{88} -0.217110 q^{89} +8.77813 q^{91} -1.78310 q^{92} -23.1828 q^{94} -3.33359 q^{95} +13.8496 q^{97} +12.8812 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) \) 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8	\( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8} + 14 q^{10} - 11 q^{11} - 5 q^{13} - 17 q^{14} + 20 q^{16} - 15 q^{17} - 6 q^{19} - 21 q^{20} - 10 q^{22} - 11 q^{23} + 3 q^{25} + 5 q^{26} + 7 q^{28} + 11 q^{29} + 35 q^{31} - 28 q^{32} + 28 q^{34} - 15 q^{35} - 28 q^{37} + 2 q^{38} - q^{40} - 10 q^{41} - 6 q^{43} - 18 q^{44} + 2 q^{46} - 15 q^{47} + 22 q^{49} - 15 q^{50} - 36 q^{52} + 7 q^{53} - 12 q^{55} - 56 q^{56} - 2 q^{58} + 20 q^{59} - 20 q^{61} + 11 q^{62} + 36 q^{64} - 11 q^{65} - 39 q^{67} - 35 q^{68} + 38 q^{70} - 49 q^{71} - 3 q^{73} - 37 q^{74} - 18 q^{76} - 25 q^{77} + 41 q^{79} - 51 q^{80} - 19 q^{82} - 13 q^{83} - 62 q^{86} - 40 q^{88} - 34 q^{89} + 2 q^{91} - 18 q^{92} - 14 q^{94} - 25 q^{95} - 11 q^{97} - 53 q^{98}+O(q^{100}) \) 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8 + 14 * q^10 - 11 * q^11 - 5 * q^13 - 17 * q^14 + 20 * q^16 - 15 * q^17 - 6 * q^19 - 21 * q^20 - 10 * q^22 - 11 * q^23 + 3 * q^25 + 5 * q^26 + 7 * q^28 + 11 * q^29 + 35 * q^31 - 28 * q^32 + 28 * q^34 - 15 * q^35 - 28 * q^37 + 2 * q^38 - q^40 - 10 * q^41 - 6 * q^43 - 18 * q^44 + 2 * q^46 - 15 * q^47 + 22 * q^49 - 15 * q^50 - 36 * q^52 + 7 * q^53 - 12 * q^55 - 56 * q^56 - 2 * q^58 + 20 * q^59 - 20 * q^61 + 11 * q^62 + 36 * q^64 - 11 * q^65 - 39 * q^67 - 35 * q^68 + 38 * q^70 - 49 * q^71 - 3 * q^73 - 37 * q^74 - 18 * q^76 - 25 * q^77 + 41 * q^79 - 51 * q^80 - 19 * q^82 - 13 * q^83 - 62 * q^86 - 40 * q^88 - 34 * q^89 + 2 * q^91 - 18 * q^92 - 14 * q^94 - 25 * q^95 - 11 * q^97 - 53 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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