LMFDB - Embedded newform 6003.2.a.r.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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + \cdots - 192 $ x^16 - x^15 - 28x^14 + 27x^13 + 316x^12 - 295x^11 - 1835x^10 + 1665x^9 + 5778x^8 - 5124x^7 - 9405x^6 + 8288x^5 + 6405x^4 - 6032x^3 - 400x^2 + 1088x - 192
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.386616$ 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)

$f(q)$	$=$	$q+0.386616 q^{2} -1.85053 q^{4} -0.287925 q^{5} -2.57107 q^{7} -1.48868 q^{8} +O(q^{10})$	\(q+0.386616 q^{2} -1.85053 q^{4} -0.287925 q^{5} -2.57107 q^{7} -1.48868 q^{8} -0.111317 q^{10} +4.70956 q^{11} -0.906534 q^{13} -0.994018 q^{14} +3.12551 q^{16} -4.50938 q^{17} +3.83061 q^{19} +0.532813 q^{20} +1.82080 q^{22} +1.00000 q^{23} -4.91710 q^{25} -0.350481 q^{26} +4.75784 q^{28} +1.00000 q^{29} -0.585232 q^{31} +4.18573 q^{32} -1.74340 q^{34} +0.740275 q^{35} -3.38692 q^{37} +1.48098 q^{38} +0.428627 q^{40} -11.5065 q^{41} +5.57988 q^{43} -8.71518 q^{44} +0.386616 q^{46} +0.820738 q^{47} -0.389600 q^{49} -1.90103 q^{50} +1.67757 q^{52} -6.67652 q^{53} -1.35600 q^{55} +3.82749 q^{56} +0.386616 q^{58} -10.9989 q^{59} +8.33911 q^{61} -0.226260 q^{62} -4.63275 q^{64} +0.261014 q^{65} +15.2588 q^{67} +8.34473 q^{68} +0.286203 q^{70} -13.9732 q^{71} -8.09288 q^{73} -1.30944 q^{74} -7.08865 q^{76} -12.1086 q^{77} +8.56579 q^{79} -0.899912 q^{80} -4.44859 q^{82} +9.72061 q^{83} +1.29836 q^{85} +2.15728 q^{86} -7.01102 q^{88} +15.0639 q^{89} +2.33076 q^{91} -1.85053 q^{92} +0.317311 q^{94} -1.10293 q^{95} -2.21161 q^{97} -0.150626 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.386616	0.273379	0.136690	−	0.990614i	$-0.456354\pi$
$2$	0.386616	0.273379	0.136690	+	0.990614i	$0.456354\pi$
$3$	0	0
$3$	0	0
$4$	−1.85053	−0.925264
$5$	−0.287925	−0.128764	−0.0643820	−	0.997925i	$-0.520508\pi$
$5$	−0.287925	−0.128764	−0.0643820	+	0.997925i	$0.520508\pi$
$6$	0	0
$7$	−2.57107	−0.971773	−0.485887	−	0.874022i	$-0.661503\pi$
$7$	−2.57107	−0.971773	−0.485887	+	0.874022i	$0.661503\pi$
$8$	−1.48868	−0.526327
$9$	0	0
$10$	−0.111317	−0.0352014
$11$	4.70956	1.41999	0.709994	−	0.704208i	$-0.248698\pi$
$11$	4.70956	1.41999	0.709994	+	0.704208i	$0.248698\pi$
$12$	0	0
$13$	−0.906534	−0.251427	−0.125714	−	0.992067i	$-0.540122\pi$
$13$	−0.906534	−0.251427	−0.125714	+	0.992067i	$0.540122\pi$
$14$	−0.994018	−0.265662
$15$	0	0
$16$	3.12551	0.781377
$17$	−4.50938	−1.09369	−0.546843	−	0.837235i	$-0.684170\pi$
$17$	−4.50938	−1.09369	−0.546843	+	0.837235i	$0.684170\pi$
$18$	0	0
$19$	3.83061	0.878802	0.439401	−	0.898291i	$-0.355191\pi$
$19$	3.83061	0.878802	0.439401	+	0.898291i	$0.355191\pi$
$20$	0.532813	0.119141
$21$	0	0
$22$	1.82080	0.388195
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.91710	−0.983420
$26$	−0.350481	−0.0687350
$27$	0	0
$28$	4.75784	0.899146
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−0.585232	−0.105111	−0.0525554	−	0.998618i	$-0.516737\pi$
$31$	−0.585232	−0.105111	−0.0525554	+	0.998618i	$0.516737\pi$
$32$	4.18573	0.739939
$33$	0	0
$34$	−1.74340	−0.298991
$35$	0.740275	0.125129
$36$	0	0
$37$	−3.38692	−0.556806	−0.278403	−	0.960464i	$-0.589805\pi$
$37$	−3.38692	−0.556806	−0.278403	+	0.960464i	$0.589805\pi$
$38$	1.48098	0.240246
$39$	0	0
$40$	0.428627	0.0677720
$41$	−11.5065	−1.79701	−0.898504	−	0.438965i	$-0.855345\pi$
$41$	−11.5065	−1.79701	−0.898504	+	0.438965i	$0.855345\pi$
$42$	0	0
$43$	5.57988	0.850924	0.425462	−	0.904976i	$-0.360112\pi$
$43$	5.57988	0.850924	0.425462	+	0.904976i	$0.360112\pi$
$44$	−8.71518	−1.31386
$45$	0	0
$46$	0.386616	0.0570035
$47$	0.820738	0.119717	0.0598585	−	0.998207i	$-0.480935\pi$
$47$	0.820738	0.119717	0.0598585	+	0.998207i	$0.480935\pi$
$48$	0	0
$49$	−0.389600	−0.0556572
$50$	−1.90103	−0.268846
$51$	0	0
$52$	1.67757	0.232637
$53$	−6.67652	−0.917091	−0.458545	−	0.888671i	$-0.651629\pi$
$53$	−6.67652	−0.917091	−0.458545	+	0.888671i	$0.651629\pi$
$54$	0	0
$55$	−1.35600	−0.182843
$56$	3.82749	0.511470
$57$	0	0
$58$	0.386616	0.0507652
$59$	−10.9989	−1.43193	−0.715965	−	0.698136i	$-0.754013\pi$
$59$	−10.9989	−1.43193	−0.715965	+	0.698136i	$0.754013\pi$
$60$	0	0
$61$	8.33911	1.06771	0.533857	−	0.845575i	$-0.320742\pi$
$61$	8.33911	1.06771	0.533857	+	0.845575i	$0.320742\pi$
$62$	−0.226260	−0.0287351
$63$	0	0
$64$	−4.63275	−0.579093
$65$	0.261014	0.0323748
$66$	0	0
$67$	15.2588	1.86416	0.932081	−	0.362250i	$-0.117991\pi$
$67$	15.2588	1.86416	0.932081	+	0.362250i	$0.117991\pi$
$68$	8.34473	1.01195
$69$	0	0
$70$	0.286203	0.0342078
$71$	−13.9732	−1.65831	−0.829155	−	0.559018i	$-0.811178\pi$
$71$	−13.9732	−1.65831	−0.829155	+	0.559018i	$0.811178\pi$
$72$	0	0
$73$	−8.09288	−0.947200	−0.473600	−	0.880740i	$-0.657046\pi$
$73$	−8.09288	−0.947200	−0.473600	+	0.880740i	$0.657046\pi$
$74$	−1.30944	−0.152219
$75$	0	0
$76$	−7.08865	−0.813124
$77$	−12.1086	−1.37991
$78$	0	0
$79$	8.56579	0.963727	0.481864	−	0.876246i	$-0.339960\pi$
$79$	8.56579	0.963727	0.481864	+	0.876246i	$0.339960\pi$
$80$	−0.899912	−0.100613
$81$	0	0
$82$	−4.44859	−0.491265
$83$	9.72061	1.06698	0.533488	−	0.845808i	$-0.320881\pi$
$83$	9.72061	1.06698	0.533488	+	0.845808i	$0.320881\pi$
$84$	0	0
$85$	1.29836	0.140827
$86$	2.15728	0.232625
$87$	0	0
$88$	−7.01102	−0.747377
$89$	15.0639	1.59677	0.798387	−	0.602145i	$-0.205687\pi$
$89$	15.0639	1.59677	0.798387	+	0.602145i	$0.205687\pi$
$90$	0	0
$91$	2.33076	0.244330
$92$	−1.85053	−0.192931
$93$	0	0
$94$	0.317311	0.0327281
$95$	−1.10293	−0.113158
$96$	0	0
$97$	−2.21161	−0.224555	−0.112278	−	0.993677i	$-0.535815\pi$
$97$	−2.21161	−0.224555	−0.112278	+	0.993677i	$0.535815\pi$
$98$	−0.150626	−0.0152155
$99$	0	0
$100$	9.09923	0.909923
$101$	−1.72771	−0.171914	−0.0859570	−	0.996299i	$-0.527395\pi$
$101$	−1.72771	−0.171914	−0.0859570	+	0.996299i	$0.527395\pi$
$102$	0	0
$103$	14.5991	1.43850	0.719248	−	0.694754i	$-0.244487\pi$
$103$	14.5991	1.43850	0.719248	+	0.694754i	$0.244487\pi$
$104$	1.34954	0.132333
$105$	0	0
$106$	−2.58125	−0.250713
$107$	7.84440	0.758347	0.379173	−	0.925326i	$-0.376208\pi$
$107$	7.84440	0.758347	0.379173	+	0.925326i	$0.376208\pi$
$108$	0	0
$109$	−5.62566	−0.538840	−0.269420	−	0.963023i	$-0.586832\pi$
$109$	−5.62566	−0.538840	−0.269420	+	0.963023i	$0.586832\pi$
$110$	−0.524253	−0.0499855
$111$	0	0
$112$	−8.03590	−0.759321
$113$	−11.1189	−1.04597	−0.522987	−	0.852341i	$-0.675182\pi$
$113$	−11.1189	−1.04597	−0.522987	+	0.852341i	$0.675182\pi$
$114$	0	0
$115$	−0.287925	−0.0268491
$116$	−1.85053	−0.171817
$117$	0	0
$118$	−4.25234	−0.391460
$119$	11.5939	1.06281
$120$	0	0
$121$	11.1800	1.01636
$122$	3.22404	0.291891
$123$	0	0
$124$	1.08299	0.0972551
$125$	2.85538	0.255393
$126$	0	0
$127$	19.0377	1.68933	0.844663	−	0.535298i	$-0.179801\pi$
$127$	19.0377	1.68933	0.844663	+	0.535298i	$0.179801\pi$
$128$	−10.1626	−0.898251
$129$	0	0
$130$	0.100912	0.00885059
$131$	−3.00962	−0.262951	−0.131476	−	0.991319i	$-0.541972\pi$
$131$	−3.00962	−0.262951	−0.131476	+	0.991319i	$0.541972\pi$
$132$	0	0
$133$	−9.84876	−0.853996
$134$	5.89931	0.509623
$135$	0	0
$136$	6.71301	0.575636
$137$	21.3718	1.82591	0.912957	−	0.408057i	$-0.133793\pi$
$137$	21.3718	1.82591	0.912957	+	0.408057i	$0.133793\pi$
$138$	0	0
$139$	16.6957	1.41611	0.708057	−	0.706156i	$-0.249572\pi$
$139$	16.6957	1.41611	0.708057	+	0.706156i	$0.249572\pi$
$140$	−1.36990	−0.115778
$141$	0	0
$142$	−5.40226	−0.453348
$143$	−4.26938	−0.357024
$144$	0	0
$145$	−0.287925	−0.0239109
$146$	−3.12884	−0.258945
$147$	0	0
$148$	6.26758	0.515192
$149$	0.179232	0.0146832	0.00734162	−	0.999973i	$-0.497663\pi$
$149$	0.179232	0.0146832	0.00734162	+	0.999973i	$0.497663\pi$
$150$	0	0
$151$	−1.37683	−0.112045	−0.0560223	−	0.998430i	$-0.517842\pi$
$151$	−1.37683	−0.112045	−0.0560223	+	0.998430i	$0.517842\pi$
$152$	−5.70254	−0.462537
$153$	0	0
$154$	−4.68139	−0.377237
$155$	0.168503	0.0135345
$156$	0	0
$157$	17.6294	1.40698	0.703488	−	0.710707i	$-0.251625\pi$
$157$	17.6294	1.40698	0.703488	+	0.710707i	$0.251625\pi$
$158$	3.31168	0.263463
$159$	0	0
$160$	−1.20518	−0.0952775
$161$	−2.57107	−0.202629
$162$	0	0
$163$	−1.94175	−0.152090	−0.0760449	−	0.997104i	$-0.524229\pi$
$163$	−1.94175	−0.152090	−0.0760449	+	0.997104i	$0.524229\pi$
$164$	21.2930	1.66271
$165$	0	0
$166$	3.75815	0.291689
$167$	−7.84563	−0.607113	−0.303557	−	0.952813i	$-0.598174\pi$
$167$	−7.84563	−0.607113	−0.303557	+	0.952813i	$0.598174\pi$
$168$	0	0
$169$	−12.1782	−0.936784
$170$	0.501969	0.0384992
$171$	0	0
$172$	−10.3257	−0.787330
$173$	−3.11956	−0.237176	−0.118588	−	0.992944i	$-0.537837\pi$
$173$	−3.11956	−0.237176	−0.118588	+	0.992944i	$0.537837\pi$
$174$	0	0
$175$	12.6422	0.955661
$176$	14.7198	1.10955
$177$	0	0
$178$	5.82396	0.436524
$179$	21.4735	1.60500	0.802502	−	0.596649i	$-0.203502\pi$
$179$	21.4735	1.60500	0.802502	+	0.596649i	$0.203502\pi$
$180$	0	0
$181$	14.0756	1.04623	0.523117	−	0.852261i	$-0.324769\pi$
$181$	14.0756	1.04623	0.523117	+	0.852261i	$0.324769\pi$
$182$	0.901111	0.0667948
$183$	0	0
$184$	−1.48868	−0.109747
$185$	0.975178	0.0716965
$186$	0	0
$187$	−21.2372	−1.55302
$188$	−1.51880	−0.110770
$189$	0	0
$190$	−0.426410	−0.0309351
$191$	−4.16450	−0.301333	−0.150666	−	0.988585i	$-0.548142\pi$
$191$	−4.16450	−0.301333	−0.150666	+	0.988585i	$0.548142\pi$
$192$	0	0
$193$	−4.67082	−0.336213	−0.168107	−	0.985769i	$-0.553765\pi$
$193$	−4.67082	−0.336213	−0.168107	+	0.985769i	$0.553765\pi$
$194$	−0.855046	−0.0613887
$195$	0	0
$196$	0.720966	0.0514976
$197$	−13.0423	−0.929222	−0.464611	−	0.885515i	$-0.653806\pi$
$197$	−13.0423	−0.929222	−0.464611	+	0.885515i	$0.653806\pi$
$198$	0	0
$199$	23.2352	1.64710	0.823551	−	0.567242i	$-0.191990\pi$
$199$	23.2352	1.64710	0.823551	+	0.567242i	$0.191990\pi$
$200$	7.31997	0.517600
$201$	0	0
$202$	−0.667963	−0.0469977
$203$	−2.57107	−0.180454
$204$	0	0
$205$	3.31300	0.231390
$206$	5.64426	0.393255
$207$	0	0
$208$	−2.83338	−0.196460
$209$	18.0405	1.24789
$210$	0	0
$211$	−7.35145	−0.506095	−0.253047	−	0.967454i	$-0.581433\pi$
$211$	−7.35145	−0.506095	−0.253047	+	0.967454i	$0.581433\pi$
$212$	12.3551	0.848551
$213$	0	0
$214$	3.03277	0.207316
$215$	−1.60659	−0.109568
$216$	0	0
$217$	1.50467	0.102144
$218$	−2.17497	−0.147308
$219$	0	0
$220$	2.50932	0.169178
$221$	4.08791	0.274982
$222$	0	0
$223$	17.2637	1.15606	0.578030	−	0.816016i	$-0.303822\pi$
$223$	17.2637	1.15606	0.578030	+	0.816016i	$0.303822\pi$
$224$	−10.7618	−0.719053
$225$	0	0
$226$	−4.29873	−0.285947
$227$	−24.4444	−1.62243	−0.811215	−	0.584747i	$-0.801194\pi$
$227$	−24.4444	−1.62243	−0.811215	+	0.584747i	$0.801194\pi$
$228$	0	0
$229$	6.54407	0.432444	0.216222	−	0.976344i	$-0.430626\pi$
$229$	6.54407	0.432444	0.216222	+	0.976344i	$0.430626\pi$
$230$	−0.111317	−0.00734000
$231$	0	0
$232$	−1.48868	−0.0977365
$233$	26.1002	1.70988	0.854939	−	0.518728i	$-0.173594\pi$
$233$	26.1002	1.70988	0.854939	+	0.518728i	$0.173594\pi$
$234$	0	0
$235$	−0.236311	−0.0154152
$236$	20.3537	1.32491
$237$	0	0
$238$	4.48240	0.290551
$239$	7.25068	0.469008	0.234504	−	0.972115i	$-0.424653\pi$
$239$	7.25068	0.469008	0.234504	+	0.972115i	$0.424653\pi$
$240$	0	0
$241$	−22.9939	−1.48117	−0.740583	−	0.671965i	$-0.765451\pi$
$241$	−22.9939	−1.48117	−0.740583	+	0.671965i	$0.765451\pi$
$242$	4.32237	0.277853
$243$	0	0
$244$	−15.4318	−0.987917
$245$	0.112176	0.00716664
$246$	0	0
$247$	−3.47258	−0.220955
$248$	0.871221	0.0553226
$249$	0	0
$250$	1.10394	0.0698191
$251$	−16.7032	−1.05430	−0.527148	−	0.849774i	$-0.676739\pi$
$251$	−16.7032	−1.05430	−0.527148	+	0.849774i	$0.676739\pi$
$252$	0	0
$253$	4.70956	0.296088
$254$	7.36031	0.461827
$255$	0	0
$256$	5.33648	0.333530
$257$	31.4979	1.96479	0.982393	−	0.186826i	$-0.0598201\pi$
$257$	31.4979	1.96479	0.982393	+	0.186826i	$0.0598201\pi$
$258$	0	0
$259$	8.70800	0.541089
$260$	−0.483013	−0.0299552
$261$	0	0
$262$	−1.16357	−0.0718854
$263$	28.5998	1.76354	0.881769	−	0.471681i	$-0.156353\pi$
$263$	28.5998	1.76354	0.881769	+	0.471681i	$0.156353\pi$
$264$	0	0
$265$	1.92234	0.118088
$266$	−3.80769	−0.233465
$267$	0	0
$268$	−28.2369	−1.72484
$269$	12.3966	0.755837	0.377919	−	0.925839i	$-0.376640\pi$
$269$	12.3966	0.755837	0.377919	+	0.925839i	$0.376640\pi$
$270$	0	0
$271$	9.61788	0.584244	0.292122	−	0.956381i	$-0.405639\pi$
$271$	9.61788	0.584244	0.292122	+	0.956381i	$0.405639\pi$
$272$	−14.0941	−0.854580
$273$	0	0
$274$	8.26268	0.499167
$275$	−23.1574	−1.39644
$276$	0	0
$277$	−5.69568	−0.342220	−0.171110	−	0.985252i	$-0.554735\pi$
$277$	−5.69568	−0.342220	−0.171110	+	0.985252i	$0.554735\pi$
$278$	6.45484	0.387136
$279$	0	0
$280$	−1.10203	−0.0658590
$281$	−6.76690	−0.403680	−0.201840	−	0.979419i	$-0.564692\pi$
$281$	−6.76690	−0.403680	−0.201840	+	0.979419i	$0.564692\pi$
$282$	0	0
$283$	12.1255	0.720789	0.360394	−	0.932800i	$-0.382642\pi$
$283$	12.1255	0.720789	0.360394	+	0.932800i	$0.382642\pi$
$284$	25.8577	1.53438
$285$	0	0
$286$	−1.65061	−0.0976028
$287$	29.5839	1.74628
$288$	0	0
$289$	3.33450	0.196147
$290$	−0.111317	−0.00653673
$291$	0	0
$292$	14.9761	0.876410
$293$	−1.15709	−0.0675979	−0.0337989	−	0.999429i	$-0.510761\pi$
$293$	−1.15709	−0.0675979	−0.0337989	+	0.999429i	$0.510761\pi$
$294$	0	0
$295$	3.16685	0.184381
$296$	5.04203	0.293062
$297$	0	0
$298$	0.0692939	0.00401409
$299$	−0.906534	−0.0524262
$300$	0	0
$301$	−14.3463	−0.826905
$302$	−0.532304	−0.0306307
$303$	0	0
$304$	11.9726	0.686676
$305$	−2.40104	−0.137483
$306$	0	0
$307$	−2.29266	−0.130849	−0.0654246	−	0.997858i	$-0.520840\pi$
$307$	−2.29266	−0.130849	−0.0654246	+	0.997858i	$0.520840\pi$
$308$	22.4073	1.27678
$309$	0	0
$310$	0.0651460	0.00370004
$311$	15.1271	0.857782	0.428891	−	0.903356i	$-0.358904\pi$
$311$	15.1271	0.857782	0.428891	+	0.903356i	$0.358904\pi$
$312$	0	0
$313$	6.96731	0.393816	0.196908	−	0.980422i	$-0.436910\pi$
$313$	6.96731	0.393816	0.196908	+	0.980422i	$0.436910\pi$
$314$	6.81580	0.384638
$315$	0	0
$316$	−15.8512	−0.891702
$317$	18.7165	1.05122	0.525611	−	0.850725i	$-0.323837\pi$
$317$	18.7165	1.05122	0.525611	+	0.850725i	$0.323837\pi$
$318$	0	0
$319$	4.70956	0.263685
$320$	1.33388	0.0745663
$321$	0	0
$322$	−0.994018	−0.0553944
$323$	−17.2737	−0.961133
$324$	0	0
$325$	4.45752	0.247259
$326$	−0.750714	−0.0415782
$327$	0	0
$328$	17.1294	0.945814
$329$	−2.11018	−0.116338
$330$	0	0
$331$	9.50628	0.522512	0.261256	−	0.965269i	$-0.415863\pi$
$331$	9.50628	0.522512	0.261256	+	0.965269i	$0.415863\pi$
$332$	−17.9883	−0.987234
$333$	0	0
$334$	−3.03325	−0.165972
$335$	−4.39340	−0.240037
$336$	0	0
$337$	20.8700	1.13686	0.568430	−	0.822732i	$-0.307551\pi$
$337$	20.8700	1.13686	0.568430	+	0.822732i	$0.307551\pi$
$338$	−4.70829	−0.256097
$339$	0	0
$340$	−2.40266	−0.130302
$341$	−2.75619	−0.149256
$342$	0	0
$343$	18.9992	1.02586
$344$	−8.30665	−0.447864
$345$	0	0
$346$	−1.20607	−0.0648389
$347$	6.79461	0.364754	0.182377	−	0.983229i	$-0.441621\pi$
$347$	6.79461	0.364754	0.182377	+	0.983229i	$0.441621\pi$
$348$	0	0
$349$	24.4527	1.30892	0.654462	−	0.756095i	$-0.272895\pi$
$349$	24.4527	1.30892	0.654462	+	0.756095i	$0.272895\pi$
$350$	4.88768	0.261258
$351$	0	0
$352$	19.7130	1.05070
$353$	−31.6663	−1.68543	−0.842714	−	0.538362i	$-0.819043\pi$
$353$	−31.6663	−1.68543	−0.842714	+	0.538362i	$0.819043\pi$
$354$	0	0
$355$	4.02323	0.213531
$356$	−27.8762	−1.47744
$357$	0	0
$358$	8.30200	0.438775
$359$	−24.1599	−1.27511	−0.637555	−	0.770404i	$-0.720054\pi$
$359$	−24.1599	−1.27511	−0.637555	+	0.770404i	$0.720054\pi$
$360$	0	0
$361$	−4.32643	−0.227707
$362$	5.44187	0.286019
$363$	0	0
$364$	−4.31314	−0.226070
$365$	2.33014	0.121965
$366$	0	0
$367$	28.3996	1.48245	0.741224	−	0.671258i	$-0.234246\pi$
$367$	28.3996	1.48245	0.741224	+	0.671258i	$0.234246\pi$
$368$	3.12551	0.162928
$369$	0	0
$370$	0.377020	0.0196003
$371$	17.1658	0.891204
$372$	0	0
$373$	−23.7148	−1.22791	−0.613953	−	0.789343i	$-0.710422\pi$
$373$	−23.7148	−1.22791	−0.613953	+	0.789343i	$0.710422\pi$
$374$	−8.21066	−0.424563
$375$	0	0
$376$	−1.22181	−0.0630103
$377$	−0.906534	−0.0466889
$378$	0	0
$379$	−34.7595	−1.78548	−0.892738	−	0.450575i	$-0.851219\pi$
$379$	−34.7595	−1.78548	−0.892738	+	0.450575i	$0.851219\pi$
$380$	2.04100	0.104701
$381$	0	0
$382$	−1.61007	−0.0823781
$383$	22.5166	1.15054	0.575271	−	0.817963i	$-0.304896\pi$
$383$	22.5166	1.15054	0.575271	+	0.817963i	$0.304896\pi$
$384$	0	0
$385$	3.48637	0.177682
$386$	−1.80582	−0.0919136
$387$	0	0
$388$	4.09265	0.207773
$389$	2.98565	0.151378	0.0756892	−	0.997131i	$-0.475884\pi$
$389$	2.98565	0.151378	0.0756892	+	0.997131i	$0.475884\pi$
$390$	0	0
$391$	−4.50938	−0.228049
$392$	0.579989	0.0292939
$393$	0	0
$394$	−5.04235	−0.254030
$395$	−2.46631	−0.124093
$396$	0	0
$397$	−29.6332	−1.48725	−0.743625	−	0.668597i	$-0.766895\pi$
$397$	−29.6332	−1.48725	−0.743625	+	0.668597i	$0.766895\pi$
$398$	8.98312	0.450283
$399$	0	0
$400$	−15.3684	−0.768422
$401$	−13.3800	−0.668167	−0.334083	−	0.942544i	$-0.608427\pi$
$401$	−13.3800	−0.668167	−0.334083	+	0.942544i	$0.608427\pi$
$402$	0	0
$403$	0.530533	0.0264277
$404$	3.19718	0.159066
$405$	0	0
$406$	−0.994018	−0.0493323
$407$	−15.9509	−0.790657
$408$	0	0
$409$	−15.2597	−0.754544	−0.377272	−	0.926102i	$-0.623138\pi$
$409$	−15.2597	−0.754544	−0.377272	+	0.926102i	$0.623138\pi$
$410$	1.28086	0.0632572
$411$	0	0
$412$	−27.0161	−1.33099
$413$	28.2789	1.39151
$414$	0	0
$415$	−2.79881	−0.137388
$416$	−3.79451	−0.186041
$417$	0	0
$418$	6.97476	0.341146
$419$	1.51590	0.0740567	0.0370283	−	0.999314i	$-0.488211\pi$
$419$	1.51590	0.0740567	0.0370283	+	0.999314i	$0.488211\pi$
$420$	0	0
$421$	−30.9888	−1.51030	−0.755151	−	0.655551i	$-0.772437\pi$
$421$	−30.9888	−1.51030	−0.755151	+	0.655551i	$0.772437\pi$
$422$	−2.84219	−0.138356
$423$	0	0
$424$	9.93918	0.482689
$425$	22.1731	1.07555
$426$	0	0
$427$	−21.4404	−1.03758
$428$	−14.5163	−0.701671
$429$	0	0
$430$	−0.621134	−0.0299537
$431$	−4.42544	−0.213166	−0.106583	−	0.994304i	$-0.533991\pi$
$431$	−4.42544	−0.213166	−0.106583	+	0.994304i	$0.533991\pi$
$432$	0	0
$433$	−1.75648	−0.0844108	−0.0422054	−	0.999109i	$-0.513438\pi$
$433$	−1.75648	−0.0844108	−0.0422054	+	0.999109i	$0.513438\pi$
$434$	0.581731	0.0279240
$435$	0	0
$436$	10.4104	0.498569
$437$	3.83061	0.183243
$438$	0	0
$439$	10.6827	0.509857	0.254929	−	0.966960i	$-0.417948\pi$
$439$	10.6827	0.509857	0.254929	+	0.966960i	$0.417948\pi$
$440$	2.01865	0.0962353
$441$	0	0
$442$	1.58045	0.0751744
$443$	18.5116	0.879512	0.439756	−	0.898117i	$-0.355065\pi$
$443$	18.5116	0.879512	0.439756	+	0.898117i	$0.355065\pi$
$444$	0	0
$445$	−4.33728	−0.205607
$446$	6.67441	0.316043
$447$	0	0
$448$	11.9111	0.562747
$449$	−6.30134	−0.297379	−0.148689	−	0.988884i	$-0.547505\pi$
$449$	−6.30134	−0.297379	−0.148689	+	0.988884i	$0.547505\pi$
$450$	0	0
$451$	−54.1905	−2.55173
$452$	20.5758	0.967802
$453$	0	0
$454$	−9.45060	−0.443539
$455$	−0.671085	−0.0314609
$456$	0	0
$457$	21.1799	0.990756	0.495378	−	0.868677i	$-0.335030\pi$
$457$	21.1799	0.990756	0.495378	+	0.868677i	$0.335030\pi$
$458$	2.53005	0.118221
$459$	0	0
$460$	0.532813	0.0248425
$461$	−1.47287	−0.0685984	−0.0342992	−	0.999412i	$-0.510920\pi$
$461$	−1.47287	−0.0685984	−0.0342992	+	0.999412i	$0.510920\pi$
$462$	0	0
$463$	27.3343	1.27033	0.635167	−	0.772375i	$-0.280931\pi$
$463$	27.3343	1.27033	0.635167	+	0.772375i	$0.280931\pi$
$464$	3.12551	0.145098
$465$	0	0
$466$	10.0908	0.467445
$467$	41.9669	1.94200	0.970998	−	0.239086i	$-0.0768477\pi$
$467$	41.9669	1.94200	0.970998	+	0.239086i	$0.0768477\pi$
$468$	0	0
$469$	−39.2315	−1.81154
$470$	−0.0913618	−0.00421420
$471$	0	0
$472$	16.3738	0.753664
$473$	26.2788	1.20830
$474$	0	0
$475$	−18.8355	−0.864231
$476$	−21.4549	−0.983383
$477$	0	0
$478$	2.80323	0.128217
$479$	17.5408	0.801460	0.400730	−	0.916196i	$-0.368757\pi$
$479$	17.5408	0.801460	0.400730	+	0.916196i	$0.368757\pi$
$480$	0	0
$481$	3.07036	0.139996
$482$	−8.88981	−0.404920
$483$	0	0
$484$	−20.6889	−0.940405
$485$	0.636779	0.0289146
$486$	0	0
$487$	−14.5887	−0.661079	−0.330539	−	0.943792i	$-0.607231\pi$
$487$	−14.5887	−0.661079	−0.330539	+	0.943792i	$0.607231\pi$
$488$	−12.4142	−0.561967
$489$	0	0
$490$	0.0433690	0.00195921
$491$	−5.38388	−0.242971	−0.121486	−	0.992593i	$-0.538766\pi$
$491$	−5.38388	−0.242971	−0.121486	+	0.992593i	$0.538766\pi$
$492$	0	0
$493$	−4.50938	−0.203092
$494$	−1.34256	−0.0604045
$495$	0	0
$496$	−1.82915	−0.0821311
$497$	35.9260	1.61150
$498$	0	0
$499$	4.25174	0.190334	0.0951669	−	0.995461i	$-0.469662\pi$
$499$	4.25174	0.190334	0.0951669	+	0.995461i	$0.469662\pi$
$500$	−5.28396	−0.236306
$501$	0	0
$502$	−6.45772	−0.288222
$503$	−27.5946	−1.23038	−0.615190	−	0.788379i	$-0.710921\pi$
$503$	−27.5946	−1.23038	−0.615190	+	0.788379i	$0.710921\pi$
$504$	0	0
$505$	0.497452	0.0221363
$506$	1.82080	0.0809442
$507$	0	0
$508$	−35.2299	−1.56307
$509$	13.8528	0.614013	0.307007	−	0.951707i	$-0.400673\pi$
$509$	13.8528	0.614013	0.307007	+	0.951707i	$0.400673\pi$
$510$	0	0
$511$	20.8074	0.920463
$512$	22.3883	0.989431
$513$	0	0
$514$	12.1776	0.537131
$515$	−4.20346	−0.185226
$516$	0	0
$517$	3.86532	0.169997
$518$	3.36666	0.147922
$519$	0	0
$520$	−0.388565	−0.0170397
$521$	−26.3988	−1.15655	−0.578277	−	0.815841i	$-0.696275\pi$
$521$	−26.3988	−1.15655	−0.578277	+	0.815841i	$0.696275\pi$
$522$	0	0
$523$	39.6615	1.73428	0.867139	−	0.498067i	$-0.165957\pi$
$523$	39.6615	1.73428	0.867139	+	0.498067i	$0.165957\pi$
$524$	5.56938	0.243299
$525$	0	0
$526$	11.0571	0.482115
$527$	2.63903	0.114958
$528$	0	0
$529$	1.00000	0.0434783
$530$	0.743207	0.0322829
$531$	0	0
$532$	18.2254	0.790172
$533$	10.4310	0.451817
$534$	0	0
$535$	−2.25860	−0.0976478
$536$	−22.7155	−0.981158
$537$	0	0
$538$	4.79275	0.206630
$539$	−1.83485	−0.0790325
$540$	0	0
$541$	−21.8378	−0.938880	−0.469440	−	0.882964i	$-0.655544\pi$
$541$	−21.8378	−0.938880	−0.469440	+	0.882964i	$0.655544\pi$
$542$	3.71843	0.159720
$543$	0	0
$544$	−18.8750	−0.809260
$545$	1.61977	0.0693832
$546$	0	0
$547$	−33.5249	−1.43342	−0.716712	−	0.697370i	$-0.754354\pi$
$547$	−33.5249	−1.43342	−0.716712	+	0.697370i	$0.754354\pi$
$548$	−39.5490	−1.68945
$549$	0	0
$550$	−8.95303	−0.381759
$551$	3.83061	0.163189
$552$	0	0
$553$	−22.0233	−0.936524
$554$	−2.20204	−0.0935558
$555$	0	0
$556$	−30.8959	−1.31028
$557$	12.8124	0.542878	0.271439	−	0.962456i	$-0.412500\pi$
$557$	12.8124	0.542878	0.271439	+	0.962456i	$0.412500\pi$
$558$	0	0
$559$	−5.05836	−0.213946
$560$	2.31374	0.0977732
$561$	0	0
$562$	−2.61620	−0.110358
$563$	−3.39408	−0.143043	−0.0715217	−	0.997439i	$-0.522786\pi$
$563$	−3.39408	−0.143043	−0.0715217	+	0.997439i	$0.522786\pi$
$564$	0	0
$565$	3.20140	0.134684
$566$	4.68793	0.197049
$567$	0	0
$568$	20.8016	0.872814
$569$	−23.4806	−0.984358	−0.492179	−	0.870494i	$-0.663799\pi$
$569$	−23.4806	−0.984358	−0.492179	+	0.870494i	$0.663799\pi$
$570$	0	0
$571$	−40.4939	−1.69462	−0.847309	−	0.531101i	$-0.821779\pi$
$571$	−40.4939	−1.69462	−0.847309	+	0.531101i	$0.821779\pi$
$572$	7.90061	0.330341
$573$	0	0
$574$	11.4376	0.477398
$575$	−4.91710	−0.205057
$576$	0	0
$577$	5.68234	0.236559	0.118279	−	0.992980i	$-0.462262\pi$
$577$	5.68234	0.236559	0.118279	+	0.992980i	$0.462262\pi$
$578$	1.28917	0.0536225
$579$	0	0
$580$	0.532813	0.0221239
$581$	−24.9924	−1.03686
$582$	0	0
$583$	−31.4435	−1.30226
$584$	12.0477	0.498537
$585$	0	0
$586$	−0.447350	−0.0184798
$587$	14.5024	0.598580	0.299290	−	0.954162i	$-0.403250\pi$
$587$	14.5024	0.598580	0.299290	+	0.954162i	$0.403250\pi$
$588$	0	0
$589$	−2.24179	−0.0923715
$590$	1.22436	0.0504059
$591$	0	0
$592$	−10.5858	−0.435075
$593$	−7.79222	−0.319988	−0.159994	−	0.987118i	$-0.551148\pi$
$593$	−7.79222	−0.319988	−0.159994	+	0.987118i	$0.551148\pi$
$594$	0	0
$595$	−3.33818	−0.136852
$596$	−0.331673	−0.0135859
$597$	0	0
$598$	−0.350481	−0.0143322
$599$	19.8445	0.810823	0.405412	−	0.914134i	$-0.367128\pi$
$599$	19.8445	0.810823	0.405412	+	0.914134i	$0.367128\pi$
$600$	0	0
$601$	44.8062	1.82768	0.913841	−	0.406072i	$-0.133102\pi$
$601$	44.8062	1.82768	0.913841	+	0.406072i	$0.133102\pi$
$602$	−5.54650	−0.226059
$603$	0	0
$604$	2.54786	0.103671
$605$	−3.21900	−0.130871
$606$	0	0
$607$	−24.1809	−0.981472	−0.490736	−	0.871308i	$-0.663272\pi$
$607$	−24.1809	−0.981472	−0.490736	+	0.871308i	$0.663272\pi$
$608$	16.0339	0.650260
$609$	0	0
$610$	−0.928281	−0.0375850
$611$	−0.744027	−0.0301001
$612$	0	0
$613$	35.9050	1.45019	0.725095	−	0.688648i	$-0.241796\pi$
$613$	35.9050	1.45019	0.725095	+	0.688648i	$0.241796\pi$
$614$	−0.886381	−0.0357714
$615$	0	0
$616$	18.0258	0.726281
$617$	25.5271	1.02768	0.513841	−	0.857885i	$-0.328222\pi$
$617$	25.5271	1.02768	0.513841	+	0.857885i	$0.328222\pi$
$618$	0	0
$619$	1.27108	0.0510891	0.0255446	−	0.999674i	$-0.491868\pi$
$619$	1.27108	0.0510891	0.0255446	+	0.999674i	$0.491868\pi$
$620$	−0.311819	−0.0125230
$621$	0	0
$622$	5.84840	0.234500
$623$	−38.7304	−1.55170
$624$	0	0
$625$	23.7634	0.950534
$626$	2.69368	0.107661
$627$	0	0
$628$	−32.6236	−1.30182
$629$	15.2729	0.608970
$630$	0	0
$631$	−25.7750	−1.02609	−0.513044	−	0.858362i	$-0.671482\pi$
$631$	−25.7750	−1.02609	−0.513044	+	0.858362i	$0.671482\pi$
$632$	−12.7517	−0.507235
$633$	0	0
$634$	7.23610	0.287382
$635$	−5.48144	−0.217524
$636$	0	0
$637$	0.353186	0.0139937
$638$	1.82080	0.0720860
$639$	0	0
$640$	2.92605	0.115662
$641$	−13.5772	−0.536267	−0.268133	−	0.963382i	$-0.586407\pi$
$641$	−13.5772	−0.536267	−0.268133	+	0.963382i	$0.586407\pi$
$642$	0	0
$643$	3.72566	0.146926	0.0734629	−	0.997298i	$-0.476595\pi$
$643$	3.72566	0.146926	0.0734629	+	0.997298i	$0.476595\pi$
$644$	4.75784	0.187485
$645$	0	0
$646$	−6.67829	−0.262754
$647$	20.2185	0.794869	0.397435	−	0.917630i	$-0.369901\pi$
$647$	20.2185	0.794869	0.397435	+	0.917630i	$0.369901\pi$
$648$	0	0
$649$	−51.7999	−2.03332
$650$	1.72335	0.0675954
$651$	0	0
$652$	3.59327	0.140723
$653$	−41.9179	−1.64037	−0.820187	−	0.572096i	$-0.806131\pi$
$653$	−41.9179	−1.64037	−0.820187	+	0.572096i	$0.806131\pi$
$654$	0	0
$655$	0.866544	0.0338587
$656$	−35.9636	−1.40414
$657$	0	0
$658$	−0.815829	−0.0318043
$659$	20.1209	0.783800	0.391900	−	0.920008i	$-0.371818\pi$
$659$	20.1209	0.783800	0.391900	+	0.920008i	$0.371818\pi$
$660$	0	0
$661$	47.1610	1.83435	0.917176	−	0.398483i	$-0.130463\pi$
$661$	47.1610	1.83435	0.917176	+	0.398483i	$0.130463\pi$
$662$	3.67528	0.142844
$663$	0	0
$664$	−14.4709	−0.561578
$665$	2.83571	0.109964
$666$	0	0
$667$	1.00000	0.0387202
$668$	14.5186	0.561740
$669$	0	0
$670$	−1.69856	−0.0656211
$671$	39.2736	1.51614
$672$	0	0
$673$	44.6166	1.71984	0.859922	−	0.510425i	$-0.170512\pi$
$673$	44.6166	1.71984	0.859922	+	0.510425i	$0.170512\pi$
$674$	8.06867	0.310794
$675$	0	0
$676$	22.5361	0.866773
$677$	−15.4385	−0.593349	−0.296675	−	0.954979i	$-0.595878\pi$
$677$	−15.4385	−0.593349	−0.296675	+	0.954979i	$0.595878\pi$
$678$	0	0
$679$	5.68621	0.218217
$680$	−1.93284	−0.0741212
$681$	0	0
$682$	−1.06559	−0.0408034
$683$	−42.7476	−1.63569	−0.817846	−	0.575437i	$-0.804832\pi$
$683$	−42.7476	−1.63569	−0.817846	+	0.575437i	$0.804832\pi$
$684$	0	0
$685$	−6.15347	−0.235112
$686$	7.34539	0.280448
$687$	0	0
$688$	17.4400	0.664893
$689$	6.05249	0.230582
$690$	0	0
$691$	8.76609	0.333478	0.166739	−	0.986001i	$-0.446676\pi$
$691$	8.76609	0.333478	0.166739	+	0.986001i	$0.446676\pi$
$692$	5.77283	0.219450
$693$	0	0
$694$	2.62691	0.0997160
$695$	−4.80712	−0.182344
$696$	0	0
$697$	51.8870	1.96536
$698$	9.45382	0.357833
$699$	0	0
$700$	−23.3948	−0.884238
$701$	−22.6787	−0.856564	−0.428282	−	0.903645i	$-0.640881\pi$
$701$	−22.6787	−0.856564	−0.428282	+	0.903645i	$0.640881\pi$
$702$	0	0
$703$	−12.9740	−0.489322
$704$	−21.8182	−0.822305
$705$	0	0
$706$	−12.2427	−0.460761
$707$	4.44208	0.167061
$708$	0	0
$709$	−19.5032	−0.732460	−0.366230	−	0.930524i	$-0.619352\pi$
$709$	−19.5032	−0.732460	−0.366230	+	0.930524i	$0.619352\pi$
$710$	1.55545	0.0583748
$711$	0	0
$712$	−22.4253	−0.840425
$713$	−0.585232	−0.0219171
$714$	0	0
$715$	1.22926	0.0459718
$716$	−39.7373	−1.48505
$717$	0	0
$718$	−9.34062	−0.348589
$719$	−11.9661	−0.446261	−0.223130	−	0.974789i	$-0.571628\pi$
$719$	−11.9661	−0.446261	−0.223130	+	0.974789i	$0.571628\pi$
$720$	0	0
$721$	−37.5354	−1.39789
$722$	−1.67267	−0.0622503
$723$	0	0
$724$	−26.0474	−0.968042
$725$	−4.91710	−0.182616
$726$	0	0
$727$	−23.6984	−0.878923	−0.439462	−	0.898261i	$-0.644831\pi$
$727$	−23.6984	−0.878923	−0.439462	+	0.898261i	$0.644831\pi$
$728$	−3.46975	−0.128598
$729$	0	0
$730$	0.900871	0.0333427
$731$	−25.1618	−0.930643
$732$	0	0
$733$	51.4839	1.90160	0.950801	−	0.309802i	$-0.100263\pi$
$733$	51.4839	1.90160	0.950801	+	0.309802i	$0.100263\pi$
$734$	10.9798	0.405270
$735$	0	0
$736$	4.18573	0.154288
$737$	71.8624	2.64709
$738$	0	0
$739$	−47.9688	−1.76456	−0.882280	−	0.470725i	$-0.843992\pi$
$739$	−47.9688	−1.76456	−0.882280	+	0.470725i	$0.843992\pi$
$740$	−1.80459	−0.0663382
$741$	0	0
$742$	6.63658	0.243637
$743$	11.4140	0.418740	0.209370	−	0.977837i	$-0.432859\pi$
$743$	11.4140	0.418740	0.209370	+	0.977837i	$0.432859\pi$
$744$	0	0
$745$	−0.0516053	−0.00189067
$746$	−9.16853	−0.335684
$747$	0	0
$748$	39.3000	1.43695
$749$	−20.1685	−0.736941
$750$	0	0
$751$	38.4086	1.40155	0.700776	−	0.713382i	$-0.252837\pi$
$751$	38.4086	1.40155	0.700776	+	0.713382i	$0.252837\pi$
$752$	2.56522	0.0935441
$753$	0	0
$754$	−0.350481	−0.0127638
$755$	0.396423	0.0144273
$756$	0	0
$757$	−3.25821	−0.118421	−0.0592107	−	0.998246i	$-0.518858\pi$
$757$	−3.25821	−0.118421	−0.0592107	+	0.998246i	$0.518858\pi$
$758$	−13.4386	−0.488112
$759$	0	0
$760$	1.64190	0.0595581
$761$	−9.73174	−0.352775	−0.176388	−	0.984321i	$-0.556441\pi$
$761$	−9.73174	−0.352775	−0.176388	+	0.984321i	$0.556441\pi$
$762$	0	0
$763$	14.4640	0.523630
$764$	7.70653	0.278812
$765$	0	0
$766$	8.70527	0.314534
$767$	9.97085	0.360026
$768$	0	0
$769$	30.7353	1.10834	0.554171	−	0.832403i	$-0.313035\pi$
$769$	30.7353	1.10834	0.554171	+	0.832403i	$0.313035\pi$
$770$	1.34789	0.0485746
$771$	0	0
$772$	8.64348	0.311086
$773$	−9.05798	−0.325793	−0.162896	−	0.986643i	$-0.552084\pi$
$773$	−9.05798	−0.325793	−0.162896	+	0.986643i	$0.552084\pi$
$774$	0	0
$775$	2.87764	0.103368
$776$	3.29238	0.118189
$777$	0	0
$778$	1.15430	0.0413837
$779$	−44.0768	−1.57921
$780$	0	0
$781$	−65.8076	−2.35478
$782$	−1.74340	−0.0623439
$783$	0	0
$784$	−1.21770	−0.0434893
$785$	−5.07593	−0.181168
$786$	0	0
$787$	9.41379	0.335565	0.167783	−	0.985824i	$-0.446339\pi$
$787$	9.41379	0.335565	0.167783	+	0.985824i	$0.446339\pi$
$788$	24.1351	0.859776
$789$	0	0
$790$	−0.953515	−0.0339245
$791$	28.5874	1.01645
$792$	0	0
$793$	−7.55969	−0.268452
$794$	−11.4567	−0.406583
$795$	0	0
$796$	−42.9974	−1.52400
$797$	43.4980	1.54078	0.770389	−	0.637574i	$-0.220062\pi$
$797$	43.4980	1.54078	0.770389	+	0.637574i	$0.220062\pi$
$798$	0	0
$799$	−3.70102	−0.130933
$800$	−20.5816	−0.727671
$801$	0	0
$802$	−5.17294	−0.182663
$803$	−38.1139	−1.34501
$804$	0	0
$805$	0.740275	0.0260913
$806$	0.205113	0.00722478
$807$	0	0
$808$	2.57201	0.0904830
$809$	11.9829	0.421296	0.210648	−	0.977562i	$-0.432443\pi$
$809$	11.9829	0.421296	0.210648	+	0.977562i	$0.432443\pi$
$810$	0	0
$811$	30.1902	1.06012	0.530061	−	0.847960i	$-0.322169\pi$
$811$	30.1902	1.06012	0.530061	+	0.847960i	$0.322169\pi$
$812$	4.75784	0.166967
$813$	0	0
$814$	−6.16688	−0.216149
$815$	0.559079	0.0195837
$816$	0	0
$817$	21.3744	0.747794
$818$	−5.89965	−0.206277
$819$	0	0
$820$	−6.13080	−0.214097
$821$	31.9957	1.11666	0.558328	−	0.829620i	$-0.311443\pi$
$821$	31.9957	1.11666	0.558328	+	0.829620i	$0.311443\pi$
$822$	0	0
$823$	−28.1400	−0.980898	−0.490449	−	0.871470i	$-0.663167\pi$
$823$	−28.1400	−0.980898	−0.490449	+	0.871470i	$0.663167\pi$
$824$	−21.7334	−0.757119
$825$	0	0
$826$	10.9331	0.380410
$827$	−12.4134	−0.431656	−0.215828	−	0.976431i	$-0.569245\pi$
$827$	−12.4134	−0.431656	−0.215828	+	0.976431i	$0.569245\pi$
$828$	0	0
$829$	16.7653	0.582284	0.291142	−	0.956680i	$-0.405965\pi$
$829$	16.7653	0.582284	0.291142	+	0.956680i	$0.405965\pi$
$830$	−1.08207	−0.0375590
$831$	0	0
$832$	4.19974	0.145600
$833$	1.75686	0.0608714
$834$	0	0
$835$	2.25895	0.0781743
$836$	−33.3845	−1.15463
$837$	0	0
$838$	0.586073	0.0202455
$839$	−8.12999	−0.280679	−0.140339	−	0.990103i	$-0.544819\pi$
$839$	−8.12999	−0.280679	−0.140339	+	0.990103i	$0.544819\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−11.9808	−0.412885
$843$	0	0
$844$	13.6041	0.468271
$845$	3.50641	0.120624
$846$	0	0
$847$	−28.7446	−0.987675
$848$	−20.8675	−0.716594
$849$	0	0
$850$	8.57247	0.294033
$851$	−3.38692	−0.116102
$852$	0	0
$853$	1.26296	0.0432428	0.0216214	−	0.999766i	$-0.493117\pi$
$853$	1.26296	0.0432428	0.0216214	+	0.999766i	$0.493117\pi$
$854$	−8.28923	−0.283651
$855$	0	0
$856$	−11.6778	−0.399138
$857$	0.254072	0.00867895	0.00433947	−	0.999991i	$-0.498619\pi$
$857$	0.254072	0.00867895	0.00433947	+	0.999991i	$0.498619\pi$
$858$	0	0
$859$	41.0688	1.40125	0.700625	−	0.713530i	$-0.252905\pi$
$859$	41.0688	1.40125	0.700625	+	0.713530i	$0.252905\pi$
$860$	2.97304	0.101380
$861$	0	0
$862$	−1.71095	−0.0582751
$863$	14.8984	0.507147	0.253574	−	0.967316i	$-0.418394\pi$
$863$	14.8984	0.507147	0.253574	+	0.967316i	$0.418394\pi$
$864$	0	0
$865$	0.898199	0.0305397
$866$	−0.679082	−0.0230762
$867$	0	0
$868$	−2.78444	−0.0945099
$869$	40.3412	1.36848
$870$	0	0
$871$	−13.8326	−0.468701
$872$	8.37479	0.283606
$873$	0	0
$874$	1.48098	0.0500948
$875$	−7.34138	−0.248184
$876$	0	0
$877$	−4.72399	−0.159518	−0.0797589	−	0.996814i	$-0.525415\pi$
$877$	−4.72399	−0.159518	−0.0797589	+	0.996814i	$0.525415\pi$
$878$	4.13011	0.139384
$879$	0	0
$880$	−4.23819	−0.142869
$881$	−4.39995	−0.148238	−0.0741190	−	0.997249i	$-0.523614\pi$
$881$	−4.39995	−0.148238	−0.0741190	+	0.997249i	$0.523614\pi$
$882$	0	0
$883$	7.42849	0.249988	0.124994	−	0.992157i	$-0.460109\pi$
$883$	7.42849	0.249988	0.124994	+	0.992157i	$0.460109\pi$
$884$	−7.56478	−0.254431
$885$	0	0
$886$	7.15688	0.240440
$887$	−8.17408	−0.274459	−0.137229	−	0.990539i	$-0.543820\pi$
$887$	−8.17408	−0.274459	−0.137229	+	0.990539i	$0.543820\pi$
$888$	0	0
$889$	−48.9474	−1.64164
$890$	−1.67686	−0.0562086
$891$	0	0
$892$	−31.9469	−1.06966
$893$	3.14393	0.105208
$894$	0	0
$895$	−6.18276	−0.206667
$896$	26.1286	0.872896
$897$	0	0
$898$	−2.43620	−0.0812971
$899$	−0.585232	−0.0195186
$900$	0	0
$901$	30.1070	1.00301
$902$	−20.9509	−0.697589
$903$	0	0
$904$	16.5524	0.550524
$905$	−4.05273	−0.134717
$906$	0	0
$907$	8.71889	0.289506	0.144753	−	0.989468i	$-0.453761\pi$
$907$	8.71889	0.289506	0.144753	+	0.989468i	$0.453761\pi$
$908$	45.2350	1.50118
$909$	0	0
$910$	−0.259452	−0.00860077
$911$	−29.8552	−0.989146	−0.494573	−	0.869136i	$-0.664676\pi$
$911$	−29.8552	−0.989146	−0.494573	+	0.869136i	$0.664676\pi$
$912$	0	0
$913$	45.7799	1.51509
$914$	8.18852	0.270852
$915$	0	0
$916$	−12.1100	−0.400125
$917$	7.73793	0.255529
$918$	0	0
$919$	49.5220	1.63358	0.816791	−	0.576934i	$-0.195751\pi$
$919$	49.5220	1.63358	0.816791	+	0.576934i	$0.195751\pi$
$920$	0.428627	0.0141314
$921$	0	0
$922$	−0.569436	−0.0187534
$923$	12.6672	0.416945
$924$	0	0
$925$	16.6538	0.547574
$926$	10.5679	0.347283
$927$	0	0
$928$	4.18573	0.137403
$929$	−27.6463	−0.907045	−0.453523	−	0.891245i	$-0.649833\pi$
$929$	−27.6463	−0.907045	−0.453523	+	0.891245i	$0.649833\pi$
$930$	0	0
$931$	−1.49241	−0.0489117
$932$	−48.2991	−1.58209
$933$	0	0
$934$	16.2251	0.530901
$935$	6.11472	0.199973
$936$	0	0
$937$	58.7856	1.92044	0.960220	−	0.279244i	$-0.0900838\pi$
$937$	58.7856	1.92044	0.960220	+	0.279244i	$0.0900838\pi$
$938$	−15.1675	−0.495238
$939$	0	0
$940$	0.437300	0.0142632
$941$	37.0088	1.20645	0.603226	−	0.797570i	$-0.293882\pi$
$941$	37.0088	1.20645	0.603226	+	0.797570i	$0.293882\pi$
$942$	0	0
$943$	−11.5065	−0.374702
$944$	−34.3770	−1.11888
$945$	0	0
$946$	10.1598	0.330324
$947$	38.6118	1.25472	0.627358	−	0.778731i	$-0.284136\pi$
$947$	38.6118	1.25472	0.627358	+	0.778731i	$0.284136\pi$
$948$	0	0
$949$	7.33647	0.238152
$950$	−7.28211	−0.236263
$951$	0	0
$952$	−17.2596	−0.559387
$953$	24.2350	0.785049	0.392524	−	0.919742i	$-0.371602\pi$
$953$	24.2350	0.785049	0.392524	+	0.919742i	$0.371602\pi$
$954$	0	0
$955$	1.19907	0.0388008
$956$	−13.4176	−0.433956
$957$	0	0
$958$	6.78157	0.219102
$959$	−54.9483	−1.77437
$960$	0	0
$961$	−30.6575	−0.988952
$962$	1.18705	0.0382720
$963$	0	0
$964$	42.5508	1.37047
$965$	1.34485	0.0432921
$966$	0	0
$967$	−20.4310	−0.657018	−0.328509	−	0.944501i	$-0.606546\pi$
$967$	−20.4310	−0.657018	−0.328509	+	0.944501i	$0.606546\pi$
$968$	−16.6434	−0.534940
$969$	0	0
$970$	0.246189	0.00790466
$971$	−4.62601	−0.148456	−0.0742278	−	0.997241i	$-0.523649\pi$
$971$	−4.62601	−0.148456	−0.0742278	+	0.997241i	$0.523649\pi$
$972$	0	0
$973$	−42.9259	−1.37614
$974$	−5.64025	−0.180725
$975$	0	0
$976$	26.0640	0.834287
$977$	27.8443	0.890819	0.445409	−	0.895327i	$-0.353058\pi$
$977$	27.8443	0.890819	0.445409	+	0.895327i	$0.353058\pi$
$978$	0	0
$979$	70.9445	2.26740
$980$	−0.207584	−0.00663103
$981$	0	0
$982$	−2.08150	−0.0664232
$983$	−33.0220	−1.05324	−0.526619	−	0.850102i	$-0.676540\pi$
$983$	−33.0220	−1.05324	−0.526619	+	0.850102i	$0.676540\pi$
$984$	0	0
$985$	3.75519	0.119650
$986$	−1.74340	−0.0555212
$987$	0	0
$988$	6.42610	0.204442
$989$	5.57988	0.177430
$990$	0	0
$991$	−32.3931	−1.02900	−0.514500	−	0.857490i	$-0.672022\pi$
$991$	−32.3931	−1.02900	−0.514500	+	0.857490i	$0.672022\pi$
$992$	−2.44962	−0.0777755
$993$	0	0
$994$	13.8896	0.440551
$995$	−6.69000	−0.212087
$996$	0	0
$997$	−61.5979	−1.95083	−0.975413	−	0.220386i	$-0.929268\pi$
$997$	−61.5979	−1.95083	−0.975413	+	0.220386i	$0.929268\pi$
$998$	1.64379	0.0520333
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.r.1.9		16
3.2	odd	2		2001.2.a.n.1.8	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.n.1.8	✓	16	3.2	odd	2
6003.2.a.r.1.9		16	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)

\(f(q)\)	\(=\)	\(q+0.386616 q^{2} -1.85053 q^{4} -0.287925 q^{5} -2.57107 q^{7} -1.48868 q^{8} +O(q^{10})\)	\(q+0.386616 q^{2} -1.85053 q^{4} -0.287925 q^{5} -2.57107 q^{7} -1.48868 q^{8} -0.111317 q^{10} +4.70956 q^{11} -0.906534 q^{13} -0.994018 q^{14} +3.12551 q^{16} -4.50938 q^{17} +3.83061 q^{19} +0.532813 q^{20} +1.82080 q^{22} +1.00000 q^{23} -4.91710 q^{25} -0.350481 q^{26} +4.75784 q^{28} +1.00000 q^{29} -0.585232 q^{31} +4.18573 q^{32} -1.74340 q^{34} +0.740275 q^{35} -3.38692 q^{37} +1.48098 q^{38} +0.428627 q^{40} -11.5065 q^{41} +5.57988 q^{43} -8.71518 q^{44} +0.386616 q^{46} +0.820738 q^{47} -0.389600 q^{49} -1.90103 q^{50} +1.67757 q^{52} -6.67652 q^{53} -1.35600 q^{55} +3.82749 q^{56} +0.386616 q^{58} -10.9989 q^{59} +8.33911 q^{61} -0.226260 q^{62} -4.63275 q^{64} +0.261014 q^{65} +15.2588 q^{67} +8.34473 q^{68} +0.286203 q^{70} -13.9732 q^{71} -8.09288 q^{73} -1.30944 q^{74} -7.08865 q^{76} -12.1086 q^{77} +8.56579 q^{79} -0.899912 q^{80} -4.44859 q^{82} +9.72061 q^{83} +1.29836 q^{85} +2.15728 q^{86} -7.01102 q^{88} +15.0639 q^{89} +2.33076 q^{91} -1.85053 q^{92} +0.317311 q^{94} -1.10293 q^{95} -2.21161 q^{97} -0.150626 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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