LMFDB - Embedded newform 6003.2.a.i.1.6

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 9x^{5} + 10x^{4} + 19x^{3} - 20x^{2} - 5x + 1 $ x^7 - x^6 - 9x^5 + 10x^4 + 19x^3 - 20x^2 - 5*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.13025$ 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)

$f(q)$	$=$	$q+1.55072 q^{2} +0.404722 q^{4} +0.920116 q^{5} -2.53796 q^{7} -2.47382 q^{8} +O(q^{10})$	\(q+1.55072 q^{2} +0.404722 q^{4} +0.920116 q^{5} -2.53796 q^{7} -2.47382 q^{8} +1.42684 q^{10} +0.294618 q^{11} +5.74171 q^{13} -3.93566 q^{14} -4.64564 q^{16} +1.70697 q^{17} -4.65004 q^{19} +0.372391 q^{20} +0.456868 q^{22} +1.00000 q^{23} -4.15339 q^{25} +8.90377 q^{26} -1.02717 q^{28} -1.00000 q^{29} +4.26756 q^{31} -2.25643 q^{32} +2.64703 q^{34} -2.33522 q^{35} -8.67527 q^{37} -7.21090 q^{38} -2.27621 q^{40} -6.05542 q^{41} +10.0457 q^{43} +0.119238 q^{44} +1.55072 q^{46} -1.68275 q^{47} -0.558745 q^{49} -6.44072 q^{50} +2.32380 q^{52} -3.05393 q^{53} +0.271082 q^{55} +6.27847 q^{56} -1.55072 q^{58} -3.88052 q^{59} -1.83997 q^{61} +6.61778 q^{62} +5.79221 q^{64} +5.28305 q^{65} -7.06076 q^{67} +0.690848 q^{68} -3.62127 q^{70} -0.554787 q^{71} -13.8773 q^{73} -13.4529 q^{74} -1.88197 q^{76} -0.747728 q^{77} -11.6642 q^{79} -4.27453 q^{80} -9.39024 q^{82} +9.84078 q^{83} +1.57061 q^{85} +15.5780 q^{86} -0.728832 q^{88} +0.967352 q^{89} -14.5723 q^{91} +0.404722 q^{92} -2.60947 q^{94} -4.27858 q^{95} +3.92233 q^{97} -0.866456 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8}+O(q^{10}) $ 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8	$ 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8} - 11 q^{10} + 12 q^{11} - 13 q^{13} + 3 q^{14} - 13 q^{16} + 12 q^{17} - 5 q^{19} + 8 q^{20} - q^{22} + 7 q^{23} - 4 q^{25} - 2 q^{26} - 21 q^{28} - 7 q^{29} - 8 q^{31} + 5 q^{32} - 28 q^{34} - 5 q^{35} - 24 q^{37} + 6 q^{38} - 20 q^{40} - 9 q^{41} - q^{43} + 23 q^{44} + q^{46} - 27 q^{47} - 14 q^{49} - 7 q^{50} - 9 q^{52} + q^{53} - 11 q^{55} + 20 q^{56} - q^{58} - 8 q^{59} + q^{61} + 3 q^{64} - 12 q^{65} - 16 q^{67} - 15 q^{68} + 40 q^{70} + 13 q^{71} - 23 q^{73} + 8 q^{74} - 2 q^{76} - 13 q^{77} - 44 q^{79} - 30 q^{80} - 10 q^{82} - 21 q^{83} - 6 q^{86} + 21 q^{88} + 5 q^{89} - 18 q^{91} + 7 q^{92} + 28 q^{94} - 9 q^{95} - 55 q^{97} - 36 q^{98}+O(q^{100}) $ 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8 - 11 * q^10 + 12 * q^11 - 13 * q^13 + 3 * q^14 - 13 * q^16 + 12 * q^17 - 5 * q^19 + 8 * q^20 - q^22 + 7 * q^23 - 4 * q^25 - 2 * q^26 - 21 * q^28 - 7 * q^29 - 8 * q^31 + 5 * q^32 - 28 * q^34 - 5 * q^35 - 24 * q^37 + 6 * q^38 - 20 * q^40 - 9 * q^41 - q^43 + 23 * q^44 + q^46 - 27 * q^47 - 14 * q^49 - 7 * q^50 - 9 * q^52 + q^53 - 11 * q^55 + 20 * q^56 - q^58 - 8 * q^59 + q^61 + 3 * q^64 - 12 * q^65 - 16 * q^67 - 15 * q^68 + 40 * q^70 + 13 * q^71 - 23 * q^73 + 8 * q^74 - 2 * q^76 - 13 * q^77 - 44 * q^79 - 30 * q^80 - 10 * q^82 - 21 * q^83 - 6 * q^86 + 21 * q^88 + 5 * q^89 - 18 * q^91 + 7 * q^92 + 28 * q^94 - 9 * q^95 - 55 * q^97 - 36 * 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 $

$2$	1.55072	1.09652	0.548261	−	0.836307i	$-0.315290\pi$
$2$	1.55072	1.09652	0.548261	+	0.836307i	$0.315290\pi$
$3$	0	0
$3$	0	0
$4$	0.404722	0.202361
$5$	0.920116	0.411489	0.205744	−	0.978606i	$-0.434038\pi$
$5$	0.920116	0.411489	0.205744	+	0.978606i	$0.434038\pi$
$6$	0	0
$7$	−2.53796	−0.959260	−0.479630	−	0.877471i	$-0.659229\pi$
$7$	−2.53796	−0.959260	−0.479630	+	0.877471i	$0.659229\pi$
$8$	−2.47382	−0.874629
$9$	0	0
$10$	1.42684	0.451206
$11$	0.294618	0.0888305	0.0444153	−	0.999013i	$-0.485858\pi$
$11$	0.294618	0.0888305	0.0444153	+	0.999013i	$0.485858\pi$
$12$	0	0
$13$	5.74171	1.59247	0.796233	−	0.604991i	$-0.206823\pi$
$13$	5.74171	1.59247	0.796233	+	0.604991i	$0.206823\pi$
$14$	−3.93566	−1.05185
$15$	0	0
$16$	−4.64564	−1.16141
$17$	1.70697	0.414001	0.207000	−	0.978341i	$-0.433630\pi$
$17$	1.70697	0.414001	0.207000	+	0.978341i	$0.433630\pi$
$18$	0	0
$19$	−4.65004	−1.06679	−0.533397	−	0.845865i	$-0.679085\pi$
$19$	−4.65004	−1.06679	−0.533397	+	0.845865i	$0.679085\pi$
$20$	0.372391	0.0832692
$21$	0	0
$22$	0.456868	0.0974046
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.15339	−0.830677
$26$	8.90377	1.74617
$27$	0	0
$28$	−1.02717	−0.194117
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.26756	0.766477	0.383238	−	0.923650i	$-0.374809\pi$
$31$	4.26756	0.766477	0.383238	+	0.923650i	$0.374809\pi$
$32$	−2.25643	−0.398884
$33$	0	0
$34$	2.64703	0.453961
$35$	−2.33522	−0.394724
$36$	0	0
$37$	−8.67527	−1.42621	−0.713103	−	0.701059i	$-0.752711\pi$
$37$	−8.67527	−1.42621	−0.713103	+	0.701059i	$0.752711\pi$
$38$	−7.21090	−1.16976
$39$	0	0
$40$	−2.27621	−0.359900
$41$	−6.05542	−0.945698	−0.472849	−	0.881144i	$-0.656774\pi$
$41$	−6.05542	−0.945698	−0.472849	+	0.881144i	$0.656774\pi$
$42$	0	0
$43$	10.0457	1.53195	0.765977	−	0.642868i	$-0.222256\pi$
$43$	10.0457	1.53195	0.765977	+	0.642868i	$0.222256\pi$
$44$	0.119238	0.0179758
$45$	0	0
$46$	1.55072	0.228641
$47$	−1.68275	−0.245455	−0.122727	−	0.992440i	$-0.539164\pi$
$47$	−1.68275	−0.245455	−0.122727	+	0.992440i	$0.539164\pi$
$48$	0	0
$49$	−0.558745	−0.0798208
$50$	−6.44072	−0.910856
$51$	0	0
$52$	2.32380	0.322253
$53$	−3.05393	−0.419489	−0.209744	−	0.977756i	$-0.567263\pi$
$53$	−3.05393	−0.419489	−0.209744	+	0.977756i	$0.567263\pi$
$54$	0	0
$55$	0.271082	0.0365527
$56$	6.27847	0.838996
$57$	0	0
$58$	−1.55072	−0.203619
$59$	−3.88052	−0.505201	−0.252601	−	0.967571i	$-0.581286\pi$
$59$	−3.88052	−0.505201	−0.252601	+	0.967571i	$0.581286\pi$
$60$	0	0
$61$	−1.83997	−0.235584	−0.117792	−	0.993038i	$-0.537582\pi$
$61$	−1.83997	−0.235584	−0.117792	+	0.993038i	$0.537582\pi$
$62$	6.61778	0.840459
$63$	0	0
$64$	5.79221	0.724026
$65$	5.28305	0.655281
$66$	0	0
$67$	−7.06076	−0.862609	−0.431305	−	0.902206i	$-0.641947\pi$
$67$	−7.06076	−0.862609	−0.431305	+	0.902206i	$0.641947\pi$
$68$	0.690848	0.0837776
$69$	0	0
$70$	−3.62127	−0.432824
$71$	−0.554787	−0.0658411	−0.0329206	−	0.999458i	$-0.510481\pi$
$71$	−0.554787	−0.0658411	−0.0329206	+	0.999458i	$0.510481\pi$
$72$	0	0
$73$	−13.8773	−1.62421	−0.812106	−	0.583509i	$-0.801679\pi$
$73$	−13.8773	−1.62421	−0.812106	+	0.583509i	$0.801679\pi$
$74$	−13.4529	−1.56387
$75$	0	0
$76$	−1.88197	−0.215877
$77$	−0.747728	−0.0852116
$78$	0	0
$79$	−11.6642	−1.31233	−0.656163	−	0.754620i	$-0.727821\pi$
$79$	−11.6642	−1.31233	−0.656163	+	0.754620i	$0.727821\pi$
$80$	−4.27453	−0.477907
$81$	0	0
$82$	−9.39024	−1.03698
$83$	9.84078	1.08017	0.540083	−	0.841612i	$-0.318393\pi$
$83$	9.84078	1.08017	0.540083	+	0.841612i	$0.318393\pi$
$84$	0	0
$85$	1.57061	0.170357
$86$	15.5780	1.67982
$87$	0	0
$88$	−0.728832	−0.0776938
$89$	0.967352	0.102539	0.0512696	−	0.998685i	$-0.483673\pi$
$89$	0.967352	0.102539	0.0512696	+	0.998685i	$0.483673\pi$
$90$	0	0
$91$	−14.5723	−1.52759
$92$	0.404722	0.0421952
$93$	0	0
$94$	−2.60947	−0.269147
$95$	−4.27858	−0.438973
$96$	0	0
$97$	3.92233	0.398252	0.199126	−	0.979974i	$-0.436190\pi$
$97$	3.92233	0.398252	0.199126	+	0.979974i	$0.436190\pi$
$98$	−0.866456	−0.0875252
$99$	0	0
$100$	−1.68097	−0.168097
$101$	3.92185	0.390238	0.195119	−	0.980780i	$-0.437491\pi$
$101$	3.92185	0.390238	0.195119	+	0.980780i	$0.437491\pi$
$102$	0	0
$103$	−6.54643	−0.645039	−0.322520	−	0.946563i	$-0.604530\pi$
$103$	−6.54643	−0.645039	−0.322520	+	0.946563i	$0.604530\pi$
$104$	−14.2040	−1.39282
$105$	0	0
$106$	−4.73577	−0.459979
$107$	11.9957	1.15966	0.579832	−	0.814736i	$-0.303118\pi$
$107$	11.9957	1.15966	0.579832	+	0.814736i	$0.303118\pi$
$108$	0	0
$109$	−4.83555	−0.463161	−0.231581	−	0.972816i	$-0.574390\pi$
$109$	−4.83555	−0.463161	−0.231581	+	0.972816i	$0.574390\pi$
$110$	0.420372	0.0400809
$111$	0	0
$112$	11.7905	1.11409
$113$	5.84726	0.550064	0.275032	−	0.961435i	$-0.411312\pi$
$113$	5.84726	0.550064	0.275032	+	0.961435i	$0.411312\pi$
$114$	0	0
$115$	0.920116	0.0858013
$116$	−0.404722	−0.0375775
$117$	0	0
$118$	−6.01759	−0.553964
$119$	−4.33223	−0.397134
$120$	0	0
$121$	−10.9132	−0.992109
$122$	−2.85327	−0.258323
$123$	0	0
$124$	1.72718	0.155105
$125$	−8.42218	−0.753303
$126$	0	0
$127$	−9.54852	−0.847294	−0.423647	−	0.905827i	$-0.639250\pi$
$127$	−9.54852	−0.847294	−0.423647	+	0.905827i	$0.639250\pi$
$128$	13.4949	1.19279
$129$	0	0
$130$	8.19251	0.718530
$131$	8.35446	0.729932	0.364966	−	0.931021i	$-0.381081\pi$
$131$	8.35446	0.729932	0.364966	+	0.931021i	$0.381081\pi$
$132$	0	0
$133$	11.8016	1.02333
$134$	−10.9492	−0.945870
$135$	0	0
$136$	−4.22274	−0.362097
$137$	−15.5216	−1.32610	−0.663050	−	0.748575i	$-0.730739\pi$
$137$	−15.5216	−1.32610	−0.663050	+	0.748575i	$0.730739\pi$
$138$	0	0
$139$	−12.4550	−1.05642	−0.528211	−	0.849113i	$-0.677137\pi$
$139$	−12.4550	−1.05642	−0.528211	+	0.849113i	$0.677137\pi$
$140$	−0.945115	−0.0798768
$141$	0	0
$142$	−0.860317	−0.0721962
$143$	1.69161	0.141460
$144$	0	0
$145$	−0.920116	−0.0764115
$146$	−21.5197	−1.78099
$147$	0	0
$148$	−3.51107	−0.288608
$149$	−10.7891	−0.883879	−0.441940	−	0.897045i	$-0.645710\pi$
$149$	−10.7891	−0.883879	−0.441940	+	0.897045i	$0.645710\pi$
$150$	0	0
$151$	−13.3481	−1.08625	−0.543126	−	0.839651i	$-0.682760\pi$
$151$	−13.3481	−1.08625	−0.543126	+	0.839651i	$0.682760\pi$
$152$	11.5034	0.933048
$153$	0	0
$154$	−1.15951	−0.0934364
$155$	3.92665	0.315396
$156$	0	0
$157$	−23.1432	−1.84703	−0.923515	−	0.383562i	$-0.874697\pi$
$157$	−23.1432	−1.84703	−0.923515	+	0.383562i	$0.874697\pi$
$158$	−18.0879	−1.43899
$159$	0	0
$160$	−2.07618	−0.164136
$161$	−2.53796	−0.200019
$162$	0	0
$163$	−14.1699	−1.10988	−0.554938	−	0.831892i	$-0.687258\pi$
$163$	−14.1699	−1.10988	−0.554938	+	0.831892i	$0.687258\pi$
$164$	−2.45076	−0.191372
$165$	0	0
$166$	15.2603	1.18443
$167$	−0.670813	−0.0519091	−0.0259545	−	0.999663i	$-0.508263\pi$
$167$	−0.670813	−0.0519091	−0.0259545	+	0.999663i	$0.508263\pi$
$168$	0	0
$169$	19.9673	1.53594
$170$	2.43557	0.186800
$171$	0	0
$172$	4.06571	0.310008
$173$	−5.72576	−0.435322	−0.217661	−	0.976024i	$-0.569843\pi$
$173$	−5.72576	−0.435322	−0.217661	+	0.976024i	$0.569843\pi$
$174$	0	0
$175$	10.5411	0.796835
$176$	−1.36869	−0.103169
$177$	0	0
$178$	1.50009	0.112436
$179$	−19.2578	−1.43940	−0.719698	−	0.694288i	$-0.755720\pi$
$179$	−19.2578	−1.43940	−0.719698	+	0.694288i	$0.755720\pi$
$180$	0	0
$181$	−0.611032	−0.0454177	−0.0227088	−	0.999742i	$-0.507229\pi$
$181$	−0.611032	−0.0454177	−0.0227088	+	0.999742i	$0.507229\pi$
$182$	−22.5974	−1.67503
$183$	0	0
$184$	−2.47382	−0.182373
$185$	−7.98226	−0.586868
$186$	0	0
$187$	0.502903	0.0367759
$188$	−0.681047	−0.0496705
$189$	0	0
$190$	−6.63487	−0.481344
$191$	10.8586	0.785699	0.392849	−	0.919603i	$-0.371489\pi$
$191$	10.8586	0.785699	0.392849	+	0.919603i	$0.371489\pi$
$192$	0	0
$193$	9.44410	0.679801	0.339900	−	0.940461i	$-0.389607\pi$
$193$	9.44410	0.679801	0.339900	+	0.940461i	$0.389607\pi$
$194$	6.08242	0.436692
$195$	0	0
$196$	−0.226136	−0.0161526
$197$	17.0771	1.21669	0.608347	−	0.793671i	$-0.291833\pi$
$197$	17.0771	1.21669	0.608347	+	0.793671i	$0.291833\pi$
$198$	0	0
$199$	−13.8211	−0.979751	−0.489876	−	0.871792i	$-0.662958\pi$
$199$	−13.8211	−0.979751	−0.489876	+	0.871792i	$0.662958\pi$
$200$	10.2747	0.726534
$201$	0	0
$202$	6.08167	0.427905
$203$	2.53796	0.178130
$204$	0	0
$205$	−5.57169	−0.389144
$206$	−10.1517	−0.707300
$207$	0	0
$208$	−26.6740	−1.84951
$209$	−1.36998	−0.0947638
$210$	0	0
$211$	−2.77270	−0.190880	−0.0954401	−	0.995435i	$-0.530426\pi$
$211$	−2.77270	−0.190880	−0.0954401	+	0.995435i	$0.530426\pi$
$212$	−1.23599	−0.0848882
$213$	0	0
$214$	18.6019	1.27160
$215$	9.24321	0.630382
$216$	0	0
$217$	−10.8309	−0.735250
$218$	−7.49856	−0.507867
$219$	0	0
$220$	0.109713	0.00739685
$221$	9.80093	0.659282
$222$	0	0
$223$	9.49038	0.635523	0.317762	−	0.948171i	$-0.397069\pi$
$223$	9.49038	0.635523	0.317762	+	0.948171i	$0.397069\pi$
$224$	5.72673	0.382633
$225$	0	0
$226$	9.06744	0.603157
$227$	19.2167	1.27546	0.637730	−	0.770260i	$-0.279874\pi$
$227$	19.2167	1.27546	0.637730	+	0.770260i	$0.279874\pi$
$228$	0	0
$229$	10.3141	0.681573	0.340787	−	0.940141i	$-0.389307\pi$
$229$	10.3141	0.681573	0.340787	+	0.940141i	$0.389307\pi$
$230$	1.42684	0.0940830
$231$	0	0
$232$	2.47382	0.162415
$233$	−15.4420	−1.01164	−0.505819	−	0.862640i	$-0.668810\pi$
$233$	−15.4420	−1.01164	−0.505819	+	0.862640i	$0.668810\pi$
$234$	0	0
$235$	−1.54833	−0.101002
$236$	−1.57053	−0.102233
$237$	0	0
$238$	−6.71805	−0.435467
$239$	0.257622	0.0166642	0.00833209	−	0.999965i	$-0.497348\pi$
$239$	0.257622	0.0166642	0.00833209	+	0.999965i	$0.497348\pi$
$240$	0	0
$241$	−4.32351	−0.278502	−0.139251	−	0.990257i	$-0.544469\pi$
$241$	−4.32351	−0.278502	−0.139251	+	0.990257i	$0.544469\pi$
$242$	−16.9233	−1.08787
$243$	0	0
$244$	−0.744677	−0.0476730
$245$	−0.514111	−0.0328453
$246$	0	0
$247$	−26.6992	−1.69883
$248$	−10.5572	−0.670383
$249$	0	0
$250$	−13.0604	−0.826013
$251$	18.9394	1.19544	0.597722	−	0.801704i	$-0.296073\pi$
$251$	18.9394	1.19544	0.597722	+	0.801704i	$0.296073\pi$
$252$	0	0
$253$	0.294618	0.0185224
$254$	−14.8070	−0.929077
$255$	0	0
$256$	9.34239	0.583900
$257$	18.9624	1.18284	0.591421	−	0.806363i	$-0.298567\pi$
$257$	18.9624	1.18284	0.591421	+	0.806363i	$0.298567\pi$
$258$	0	0
$259$	22.0175	1.36810
$260$	2.13816	0.132603
$261$	0	0
$262$	12.9554	0.800387
$263$	−9.48797	−0.585053	−0.292527	−	0.956257i	$-0.594496\pi$
$263$	−9.48797	−0.585053	−0.292527	+	0.956257i	$0.594496\pi$
$264$	0	0
$265$	−2.80997	−0.172615
$266$	18.3010	1.12211
$267$	0	0
$268$	−2.85764	−0.174558
$269$	−15.1348	−0.922787	−0.461394	−	0.887196i	$-0.652650\pi$
$269$	−15.1348	−0.922787	−0.461394	+	0.887196i	$0.652650\pi$
$270$	0	0
$271$	−5.65542	−0.343542	−0.171771	−	0.985137i	$-0.554949\pi$
$271$	−5.65542	−0.343542	−0.171771	+	0.985137i	$0.554949\pi$
$272$	−7.92997	−0.480825
$273$	0	0
$274$	−24.0696	−1.45410
$275$	−1.22366	−0.0737895
$276$	0	0
$277$	−12.4470	−0.747871	−0.373935	−	0.927455i	$-0.621992\pi$
$277$	−12.4470	−0.747871	−0.373935	+	0.927455i	$0.621992\pi$
$278$	−19.3142	−1.15839
$279$	0	0
$280$	5.77693	0.345237
$281$	−12.8006	−0.763618	−0.381809	−	0.924241i	$-0.624699\pi$
$281$	−12.8006	−0.763618	−0.381809	+	0.924241i	$0.624699\pi$
$282$	0	0
$283$	6.43854	0.382732	0.191366	−	0.981519i	$-0.438708\pi$
$283$	6.43854	0.382732	0.191366	+	0.981519i	$0.438708\pi$
$284$	−0.224534	−0.0133237
$285$	0	0
$286$	2.62321	0.155113
$287$	15.3684	0.907170
$288$	0	0
$289$	−14.0863	−0.828603
$290$	−1.42684	−0.0837869
$291$	0	0
$292$	−5.61644	−0.328677
$293$	5.55778	0.324689	0.162344	−	0.986734i	$-0.448094\pi$
$293$	5.55778	0.324689	0.162344	+	0.986734i	$0.448094\pi$
$294$	0	0
$295$	−3.57053	−0.207884
$296$	21.4611	1.24740
$297$	0	0
$298$	−16.7309	−0.969193
$299$	5.74171	0.332052
$300$	0	0
$301$	−25.4956	−1.46954
$302$	−20.6991	−1.19110
$303$	0	0
$304$	21.6024	1.23899
$305$	−1.69299	−0.0969402
$306$	0	0
$307$	−5.76163	−0.328833	−0.164417	−	0.986391i	$-0.552574\pi$
$307$	−5.76163	−0.328833	−0.164417	+	0.986391i	$0.552574\pi$
$308$	−0.302622	−0.0172435
$309$	0	0
$310$	6.08912	0.345839
$311$	23.2993	1.32118	0.660590	−	0.750747i	$-0.270306\pi$
$311$	23.2993	1.32118	0.660590	+	0.750747i	$0.270306\pi$
$312$	0	0
$313$	16.4820	0.931616	0.465808	−	0.884886i	$-0.345764\pi$
$313$	16.4820	0.931616	0.465808	+	0.884886i	$0.345764\pi$
$314$	−35.8886	−2.02531
$315$	0	0
$316$	−4.72076	−0.265563
$317$	−19.1446	−1.07527	−0.537633	−	0.843179i	$-0.680681\pi$
$317$	−19.1446	−1.07527	−0.537633	+	0.843179i	$0.680681\pi$
$318$	0	0
$319$	−0.294618	−0.0164954
$320$	5.32950	0.297928
$321$	0	0
$322$	−3.93566	−0.219326
$323$	−7.93748	−0.441653
$324$	0	0
$325$	−23.8476	−1.32282
$326$	−21.9736	−1.21700
$327$	0	0
$328$	14.9800	0.827135
$329$	4.27077	0.235455
$330$	0	0
$331$	16.3110	0.896533	0.448266	−	0.893900i	$-0.352042\pi$
$331$	16.3110	0.896533	0.448266	+	0.893900i	$0.352042\pi$
$332$	3.98278	0.218583
$333$	0	0
$334$	−1.04024	−0.0569195
$335$	−6.49672	−0.354954
$336$	0	0
$337$	−10.5552	−0.574979	−0.287490	−	0.957784i	$-0.592821\pi$
$337$	−10.5552	−0.574979	−0.287490	+	0.957784i	$0.592821\pi$
$338$	30.9636	1.68420
$339$	0	0
$340$	0.635660	0.0344735
$341$	1.25730	0.0680865
$342$	0	0
$343$	19.1838	1.03583
$344$	−24.8513	−1.33989
$345$	0	0
$346$	−8.87903	−0.477340
$347$	−31.6377	−1.69840	−0.849201	−	0.528070i	$-0.822916\pi$
$347$	−31.6377	−1.69840	−0.849201	+	0.528070i	$0.822916\pi$
$348$	0	0
$349$	6.92488	0.370681	0.185340	−	0.982674i	$-0.440661\pi$
$349$	6.92488	0.370681	0.185340	+	0.982674i	$0.440661\pi$
$350$	16.3463	0.873747
$351$	0	0
$352$	−0.664783	−0.0354331
$353$	−29.7641	−1.58418	−0.792092	−	0.610402i	$-0.791008\pi$
$353$	−29.7641	−1.58418	−0.792092	+	0.610402i	$0.791008\pi$
$354$	0	0
$355$	−0.510469	−0.0270929
$356$	0.391509	0.0207499
$357$	0	0
$358$	−29.8634	−1.57833
$359$	−3.16103	−0.166833	−0.0834165	−	0.996515i	$-0.526583\pi$
$359$	−3.16103	−0.166833	−0.0834165	+	0.996515i	$0.526583\pi$
$360$	0	0
$361$	2.62291	0.138048
$362$	−0.947538	−0.0498015
$363$	0	0
$364$	−5.89771	−0.309124
$365$	−12.7687	−0.668345
$366$	0	0
$367$	21.4343	1.11886	0.559431	−	0.828877i	$-0.311020\pi$
$367$	21.4343	1.11886	0.559431	+	0.828877i	$0.311020\pi$
$368$	−4.64564	−0.242171
$369$	0	0
$370$	−12.3782	−0.643513
$371$	7.75075	0.402399
$372$	0	0
$373$	2.05560	0.106435	0.0532174	−	0.998583i	$-0.483052\pi$
$373$	2.05560	0.106435	0.0532174	+	0.998583i	$0.483052\pi$
$374$	0.779860	0.0403256
$375$	0	0
$376$	4.16284	0.214682
$377$	−5.74171	−0.295713
$378$	0	0
$379$	−3.15343	−0.161981	−0.0809905	−	0.996715i	$-0.525808\pi$
$379$	−3.15343	−0.161981	−0.0809905	+	0.996715i	$0.525808\pi$
$380$	−1.73164	−0.0888310
$381$	0	0
$382$	16.8386	0.861536
$383$	35.3997	1.80884	0.904420	−	0.426643i	$-0.140304\pi$
$383$	35.3997	1.80884	0.904420	+	0.426643i	$0.140304\pi$
$384$	0	0
$385$	−0.687997	−0.0350636
$386$	14.6451	0.745417
$387$	0	0
$388$	1.58745	0.0805907
$389$	39.1672	1.98585	0.992927	−	0.118723i	$-0.0378801\pi$
$389$	39.1672	1.98585	0.992927	+	0.118723i	$0.0378801\pi$
$390$	0	0
$391$	1.70697	0.0863252
$392$	1.38224	0.0698136
$393$	0	0
$394$	26.4818	1.33413
$395$	−10.7324	−0.540007
$396$	0	0
$397$	17.3448	0.870512	0.435256	−	0.900307i	$-0.356658\pi$
$397$	17.3448	0.870512	0.435256	+	0.900307i	$0.356658\pi$
$398$	−21.4326	−1.07432
$399$	0	0
$400$	19.2952	0.964758
$401$	−23.1874	−1.15793	−0.578963	−	0.815354i	$-0.696542\pi$
$401$	−23.1874	−1.15793	−0.578963	+	0.815354i	$0.696542\pi$
$402$	0	0
$403$	24.5031	1.22059
$404$	1.58726	0.0789690
$405$	0	0
$406$	3.93566	0.195324
$407$	−2.55589	−0.126691
$408$	0	0
$409$	−17.5081	−0.865720	−0.432860	−	0.901461i	$-0.642496\pi$
$409$	−17.5081	−0.865720	−0.432860	+	0.901461i	$0.642496\pi$
$410$	−8.64011	−0.426705
$411$	0	0
$412$	−2.64948	−0.130531
$413$	9.84862	0.484619
$414$	0	0
$415$	9.05466	0.444476
$416$	−12.9558	−0.635209
$417$	0	0
$418$	−2.12446	−0.103911
$419$	38.1925	1.86582	0.932912	−	0.360103i	$-0.117259\pi$
$419$	38.1925	1.86582	0.932912	+	0.360103i	$0.117259\pi$
$420$	0	0
$421$	10.8222	0.527444	0.263722	−	0.964599i	$-0.415050\pi$
$421$	10.8222	0.527444	0.263722	+	0.964599i	$0.415050\pi$
$422$	−4.29966	−0.209304
$423$	0	0
$424$	7.55487	0.366897
$425$	−7.08970	−0.343901
$426$	0	0
$427$	4.66978	0.225986
$428$	4.85490	0.234671
$429$	0	0
$430$	14.3336	0.691227
$431$	24.8260	1.19583	0.597914	−	0.801561i	$-0.295997\pi$
$431$	24.8260	1.19583	0.597914	+	0.801561i	$0.295997\pi$
$432$	0	0
$433$	4.67395	0.224616	0.112308	−	0.993673i	$-0.464176\pi$
$433$	4.67395	0.224616	0.112308	+	0.993673i	$0.464176\pi$
$434$	−16.7957	−0.806218
$435$	0	0
$436$	−1.95705	−0.0937257
$437$	−4.65004	−0.222442
$438$	0	0
$439$	−7.10914	−0.339301	−0.169650	−	0.985504i	$-0.554264\pi$
$439$	−7.10914	−0.339301	−0.169650	+	0.985504i	$0.554264\pi$
$440$	−0.670610	−0.0319701
$441$	0	0
$442$	15.1985	0.722917
$443$	−4.20233	−0.199659	−0.0998293	−	0.995005i	$-0.531830\pi$
$443$	−4.20233	−0.199659	−0.0998293	+	0.995005i	$0.531830\pi$
$444$	0	0
$445$	0.890076	0.0421937
$446$	14.7169	0.696865
$447$	0	0
$448$	−14.7004	−0.694529
$449$	19.1700	0.904688	0.452344	−	0.891843i	$-0.350588\pi$
$449$	19.1700	0.904688	0.452344	+	0.891843i	$0.350588\pi$
$450$	0	0
$451$	−1.78403	−0.0840068
$452$	2.36651	0.111311
$453$	0	0
$454$	29.7997	1.39857
$455$	−13.4082	−0.628585
$456$	0	0
$457$	27.5937	1.29078	0.645389	−	0.763854i	$-0.276695\pi$
$457$	27.5937	1.29078	0.645389	+	0.763854i	$0.276695\pi$
$458$	15.9942	0.747360
$459$	0	0
$460$	0.372391	0.0173628
$461$	17.6892	0.823868	0.411934	−	0.911214i	$-0.364853\pi$
$461$	17.6892	0.823868	0.411934	+	0.911214i	$0.364853\pi$
$462$	0	0
$463$	22.2642	1.03470	0.517352	−	0.855773i	$-0.326918\pi$
$463$	22.2642	1.03470	0.517352	+	0.855773i	$0.326918\pi$
$464$	4.64564	0.215669
$465$	0	0
$466$	−23.9461	−1.10928
$467$	4.91837	0.227595	0.113797	−	0.993504i	$-0.463699\pi$
$467$	4.91837	0.227595	0.113797	+	0.993504i	$0.463699\pi$
$468$	0	0
$469$	17.9199	0.827466
$470$	−2.40102	−0.110751
$471$	0	0
$472$	9.59973	0.441863
$473$	2.95964	0.136084
$474$	0	0
$475$	19.3134	0.886161
$476$	−1.75335	−0.0803645
$477$	0	0
$478$	0.399499	0.0182726
$479$	−7.39386	−0.337834	−0.168917	−	0.985630i	$-0.554027\pi$
$479$	−7.39386	−0.337834	−0.168917	+	0.985630i	$0.554027\pi$
$480$	0	0
$481$	−49.8109	−2.27118
$482$	−6.70453	−0.305383
$483$	0	0
$484$	−4.41681	−0.200764
$485$	3.60900	0.163876
$486$	0	0
$487$	12.8052	0.580260	0.290130	−	0.956987i	$-0.406301\pi$
$487$	12.8052	0.580260	0.290130	+	0.956987i	$0.406301\pi$
$488$	4.55177	0.206049
$489$	0	0
$490$	−0.797240	−0.0360156
$491$	−8.90802	−0.402014	−0.201007	−	0.979590i	$-0.564421\pi$
$491$	−8.90802	−0.402014	−0.201007	+	0.979590i	$0.564421\pi$
$492$	0	0
$493$	−1.70697	−0.0768780
$494$	−41.4029	−1.86281
$495$	0	0
$496$	−19.8256	−0.890194
$497$	1.40803	0.0631587
$498$	0	0
$499$	−23.5854	−1.05583	−0.527914	−	0.849298i	$-0.677026\pi$
$499$	−23.5854	−1.05583	−0.527914	+	0.849298i	$0.677026\pi$
$500$	−3.40864	−0.152439
$501$	0	0
$502$	29.3696	1.31083
$503$	−1.13086	−0.0504227	−0.0252113	−	0.999682i	$-0.508026\pi$
$503$	−1.13086	−0.0504227	−0.0252113	+	0.999682i	$0.508026\pi$
$504$	0	0
$505$	3.60855	0.160579
$506$	0.456868	0.0203103
$507$	0	0
$508$	−3.86449	−0.171459
$509$	−7.10342	−0.314853	−0.157427	−	0.987531i	$-0.550320\pi$
$509$	−7.10342	−0.314853	−0.157427	+	0.987531i	$0.550320\pi$
$510$	0	0
$511$	35.2200	1.55804
$512$	−12.5025	−0.552535
$513$	0	0
$514$	29.4053	1.29701
$515$	−6.02348	−0.265426
$516$	0	0
$517$	−0.495769	−0.0218039
$518$	34.1429	1.50015
$519$	0	0
$520$	−13.0693	−0.573128
$521$	−11.1529	−0.488616	−0.244308	−	0.969698i	$-0.578561\pi$
$521$	−11.1529	−0.488616	−0.244308	+	0.969698i	$0.578561\pi$
$522$	0	0
$523$	43.1168	1.88537	0.942683	−	0.333690i	$-0.108294\pi$
$523$	43.1168	1.88537	0.942683	+	0.333690i	$0.108294\pi$
$524$	3.38123	0.147710
$525$	0	0
$526$	−14.7131	−0.641524
$527$	7.28460	0.317322
$528$	0	0
$529$	1.00000	0.0434783
$530$	−4.35746	−0.189276
$531$	0	0
$532$	4.77638	0.207082
$533$	−34.7685	−1.50599
$534$	0	0
$535$	11.0374	0.477188
$536$	17.4671	0.754463
$537$	0	0
$538$	−23.4698	−1.01186
$539$	−0.164616	−0.00709052
$540$	0	0
$541$	31.8465	1.36919	0.684594	−	0.728925i	$-0.259980\pi$
$541$	31.8465	1.36919	0.684594	+	0.728925i	$0.259980\pi$
$542$	−8.76996	−0.376702
$543$	0	0
$544$	−3.85165	−0.165138
$545$	−4.44926	−0.190586
$546$	0	0
$547$	36.8946	1.57750	0.788750	−	0.614714i	$-0.210728\pi$
$547$	36.8946	1.57750	0.788750	+	0.614714i	$0.210728\pi$
$548$	−6.28194	−0.268351
$549$	0	0
$550$	−1.89755	−0.0809118
$551$	4.65004	0.198099
$552$	0	0
$553$	29.6033	1.25886
$554$	−19.3018	−0.820057
$555$	0	0
$556$	−5.04082	−0.213779
$557$	6.48485	0.274772	0.137386	−	0.990518i	$-0.456130\pi$
$557$	6.48485	0.274772	0.137386	+	0.990518i	$0.456130\pi$
$558$	0	0
$559$	57.6795	2.43958
$560$	10.8486	0.458437
$561$	0	0
$562$	−19.8501	−0.837324
$563$	10.5825	0.445998	0.222999	−	0.974819i	$-0.428415\pi$
$563$	10.5825	0.445998	0.222999	+	0.974819i	$0.428415\pi$
$564$	0	0
$565$	5.38016	0.226345
$566$	9.98436	0.419674
$567$	0	0
$568$	1.37245	0.0575865
$569$	−25.5371	−1.07057	−0.535286	−	0.844671i	$-0.679796\pi$
$569$	−25.5371	−1.07057	−0.535286	+	0.844671i	$0.679796\pi$
$570$	0	0
$571$	7.97917	0.333918	0.166959	−	0.985964i	$-0.446605\pi$
$571$	7.97917	0.333918	0.166959	+	0.985964i	$0.446605\pi$
$572$	0.684631	0.0286259
$573$	0	0
$574$	23.8321	0.994732
$575$	−4.15339	−0.173208
$576$	0	0
$577$	36.2622	1.50961	0.754807	−	0.655947i	$-0.227731\pi$
$577$	36.2622	1.50961	0.754807	+	0.655947i	$0.227731\pi$
$578$	−21.8438	−0.908582
$579$	0	0
$580$	−0.372391	−0.0154627
$581$	−24.9755	−1.03616
$582$	0	0
$583$	−0.899740	−0.0372634
$584$	34.3300	1.42058
$585$	0	0
$586$	8.61854	0.356029
$587$	8.65368	0.357175	0.178588	−	0.983924i	$-0.442847\pi$
$587$	8.65368	0.357175	0.178588	+	0.983924i	$0.442847\pi$
$588$	0	0
$589$	−19.8443	−0.817672
$590$	−5.53688	−0.227950
$591$	0	0
$592$	40.3022	1.65641
$593$	−14.8793	−0.611021	−0.305510	−	0.952189i	$-0.598827\pi$
$593$	−14.8793	−0.611021	−0.305510	+	0.952189i	$0.598827\pi$
$594$	0	0
$595$	−3.98615	−0.163416
$596$	−4.36659	−0.178863
$597$	0	0
$598$	8.90377	0.364102
$599$	−21.1474	−0.864060	−0.432030	−	0.901859i	$-0.642202\pi$
$599$	−21.1474	−0.864060	−0.432030	+	0.901859i	$0.642202\pi$
$600$	0	0
$601$	−2.18931	−0.0893037	−0.0446518	−	0.999003i	$-0.514218\pi$
$601$	−2.18931	−0.0893037	−0.0446518	+	0.999003i	$0.514218\pi$
$602$	−39.5365	−1.61139
$603$	0	0
$604$	−5.40227	−0.219815
$605$	−10.0414	−0.408242
$606$	0	0
$607$	10.7539	0.436486	0.218243	−	0.975894i	$-0.429968\pi$
$607$	10.7539	0.436486	0.218243	+	0.975894i	$0.429968\pi$
$608$	10.4925	0.425527
$609$	0	0
$610$	−2.62534	−0.106297
$611$	−9.66189	−0.390878
$612$	0	0
$613$	−29.9979	−1.21160	−0.605801	−	0.795616i	$-0.707147\pi$
$613$	−29.9979	−1.21160	−0.605801	+	0.795616i	$0.707147\pi$
$614$	−8.93465	−0.360573
$615$	0	0
$616$	1.84975	0.0745285
$617$	−41.8380	−1.68433	−0.842167	−	0.539217i	$-0.818720\pi$
$617$	−41.8380	−1.68433	−0.842167	+	0.539217i	$0.818720\pi$
$618$	0	0
$619$	−17.2717	−0.694209	−0.347104	−	0.937826i	$-0.612835\pi$
$619$	−17.2717	−0.694209	−0.347104	+	0.937826i	$0.612835\pi$
$620$	1.58920	0.0638239
$621$	0	0
$622$	36.1306	1.44870
$623$	−2.45510	−0.0983616
$624$	0	0
$625$	13.0175	0.520702
$626$	25.5589	1.02154
$627$	0	0
$628$	−9.36657	−0.373767
$629$	−14.8084	−0.590451
$630$	0	0
$631$	−6.56178	−0.261220	−0.130610	−	0.991434i	$-0.541694\pi$
$631$	−6.56178	−0.261220	−0.130610	+	0.991434i	$0.541694\pi$
$632$	28.8552	1.14780
$633$	0	0
$634$	−29.6878	−1.17905
$635$	−8.78575	−0.348652
$636$	0	0
$637$	−3.20816	−0.127112
$638$	−0.456868	−0.0180876
$639$	0	0
$640$	12.4169	0.490821
$641$	−7.14245	−0.282110	−0.141055	−	0.990002i	$-0.545049\pi$
$641$	−7.14245	−0.282110	−0.141055	+	0.990002i	$0.545049\pi$
$642$	0	0
$643$	−3.73453	−0.147276	−0.0736378	−	0.997285i	$-0.523461\pi$
$643$	−3.73453	−0.147276	−0.0736378	+	0.997285i	$0.523461\pi$
$644$	−1.02717	−0.0404761
$645$	0	0
$646$	−12.3088	−0.484283
$647$	28.3980	1.11644	0.558221	−	0.829692i	$-0.311484\pi$
$647$	28.3980	1.11644	0.558221	+	0.829692i	$0.311484\pi$
$648$	0	0
$649$	−1.14327	−0.0448773
$650$	−36.9808	−1.45051
$651$	0	0
$652$	−5.73488	−0.224595
$653$	35.4484	1.38720	0.693601	−	0.720359i	$-0.256023\pi$
$653$	35.4484	1.38720	0.693601	+	0.720359i	$0.256023\pi$
$654$	0	0
$655$	7.68707	0.300359
$656$	28.1313	1.09834
$657$	0	0
$658$	6.62275	0.258182
$659$	−4.93354	−0.192183	−0.0960916	−	0.995372i	$-0.530634\pi$
$659$	−4.93354	−0.192183	−0.0960916	+	0.995372i	$0.530634\pi$
$660$	0	0
$661$	−24.0960	−0.937227	−0.468614	−	0.883403i	$-0.655246\pi$
$661$	−24.0960	−0.937227	−0.468614	+	0.883403i	$0.655246\pi$
$662$	25.2937	0.983068
$663$	0	0
$664$	−24.3444	−0.944744
$665$	10.8589	0.421089
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−0.271493	−0.0105044
$669$	0	0
$670$	−10.0746	−0.389215
$671$	−0.542088	−0.0209271
$672$	0	0
$673$	−41.4624	−1.59826	−0.799129	−	0.601159i	$-0.794706\pi$
$673$	−41.4624	−1.59826	−0.799129	+	0.601159i	$0.794706\pi$
$674$	−16.3682	−0.630478
$675$	0	0
$676$	8.08120	0.310815
$677$	−8.35422	−0.321079	−0.160539	−	0.987029i	$-0.551323\pi$
$677$	−8.35422	−0.321079	−0.160539	+	0.987029i	$0.551323\pi$
$678$	0	0
$679$	−9.95473	−0.382027
$680$	−3.88541	−0.148999
$681$	0	0
$682$	1.94971	0.0746584
$683$	47.0105	1.79881	0.899403	−	0.437121i	$-0.144002\pi$
$683$	47.0105	1.79881	0.899403	+	0.437121i	$0.144002\pi$
$684$	0	0
$685$	−14.2817	−0.545675
$686$	29.7487	1.13581
$687$	0	0
$688$	−46.6687	−1.77923
$689$	−17.5348	−0.668021
$690$	0	0
$691$	−7.55835	−0.287533	−0.143767	−	0.989612i	$-0.545921\pi$
$691$	−7.55835	−0.287533	−0.143767	+	0.989612i	$0.545921\pi$
$692$	−2.31734	−0.0880921
$693$	0	0
$694$	−49.0611	−1.86234
$695$	−11.4601	−0.434705
$696$	0	0
$697$	−10.3364	−0.391520
$698$	10.7385	0.406459
$699$	0	0
$700$	4.26623	0.161248
$701$	−47.4732	−1.79304	−0.896518	−	0.443007i	$-0.853912\pi$
$701$	−47.4732	−1.79304	−0.896518	+	0.443007i	$0.853912\pi$
$702$	0	0
$703$	40.3404	1.52147
$704$	1.70649	0.0643156
$705$	0	0
$706$	−46.1557	−1.73709
$707$	−9.95350	−0.374340
$708$	0	0
$709$	−13.4626	−0.505600	−0.252800	−	0.967519i	$-0.581351\pi$
$709$	−13.4626	−0.505600	−0.252800	+	0.967519i	$0.581351\pi$
$710$	−0.791592	−0.0297079
$711$	0	0
$712$	−2.39306	−0.0896837
$713$	4.26756	0.159821
$714$	0	0
$715$	1.55648	0.0582090
$716$	−7.79405	−0.291277
$717$	0	0
$718$	−4.90187	−0.182936
$719$	0.150106	0.00559802	0.00279901	−	0.999996i	$-0.499109\pi$
$719$	0.150106	0.00559802	0.00279901	+	0.999996i	$0.499109\pi$
$720$	0	0
$721$	16.6146	0.618760
$722$	4.06739	0.151372
$723$	0	0
$724$	−0.247298	−0.00919076
$725$	4.15339	0.154253
$726$	0	0
$727$	−22.0694	−0.818508	−0.409254	−	0.912421i	$-0.634211\pi$
$727$	−22.0694	−0.818508	−0.409254	+	0.912421i	$0.634211\pi$
$728$	36.0492	1.33607
$729$	0	0
$730$	−19.8007	−0.732855
$731$	17.1477	0.634231
$732$	0	0
$733$	41.5903	1.53617	0.768087	−	0.640345i	$-0.221209\pi$
$733$	41.5903	1.53617	0.768087	+	0.640345i	$0.221209\pi$
$734$	33.2385	1.22686
$735$	0	0
$736$	−2.25643	−0.0831731
$737$	−2.08022	−0.0766260
$738$	0	0
$739$	−45.3933	−1.66982	−0.834910	−	0.550387i	$-0.814480\pi$
$739$	−45.3933	−1.66982	−0.834910	+	0.550387i	$0.814480\pi$
$740$	−3.23060	−0.118759
$741$	0	0
$742$	12.0192	0.441239
$743$	−0.403704	−0.0148105	−0.00740524	−	0.999973i	$-0.502357\pi$
$743$	−0.403704	−0.0148105	−0.00740524	+	0.999973i	$0.502357\pi$
$744$	0	0
$745$	−9.92725	−0.363706
$746$	3.18765	0.116708
$747$	0	0
$748$	0.203536	0.00744201
$749$	−30.4445	−1.11242
$750$	0	0
$751$	3.41838	0.124738	0.0623692	−	0.998053i	$-0.480134\pi$
$751$	3.41838	0.124738	0.0623692	+	0.998053i	$0.480134\pi$
$752$	7.81748	0.285074
$753$	0	0
$754$	−8.90377	−0.324256
$755$	−12.2818	−0.446980
$756$	0	0
$757$	−32.6212	−1.18564	−0.592819	−	0.805335i	$-0.701985\pi$
$757$	−32.6212	−1.18564	−0.592819	+	0.805335i	$0.701985\pi$
$758$	−4.89008	−0.177616
$759$	0	0
$760$	10.5845	0.383939
$761$	−35.3195	−1.28033	−0.640166	−	0.768237i	$-0.721134\pi$
$761$	−35.3195	−1.28033	−0.640166	+	0.768237i	$0.721134\pi$
$762$	0	0
$763$	12.2724	0.444292
$764$	4.39470	0.158995
$765$	0	0
$766$	54.8949	1.98343
$767$	−22.2809	−0.804515
$768$	0	0
$769$	−9.57460	−0.345269	−0.172634	−	0.984986i	$-0.555228\pi$
$769$	−9.57460	−0.345269	−0.172634	+	0.984986i	$0.555228\pi$
$770$	−1.06689	−0.0384480
$771$	0	0
$772$	3.82223	0.137565
$773$	−3.28418	−0.118124	−0.0590619	−	0.998254i	$-0.518811\pi$
$773$	−3.28418	−0.118124	−0.0590619	+	0.998254i	$0.518811\pi$
$774$	0	0
$775$	−17.7248	−0.636695
$776$	−9.70315	−0.348323
$777$	0	0
$778$	60.7372	2.17753
$779$	28.1580	1.00886
$780$	0	0
$781$	−0.163450	−0.00584870
$782$	2.64703	0.0946575
$783$	0	0
$784$	2.59573	0.0927047
$785$	−21.2945	−0.760032
$786$	0	0
$787$	−22.7463	−0.810817	−0.405408	−	0.914136i	$-0.632871\pi$
$787$	−22.7463	−0.810817	−0.405408	+	0.914136i	$0.632871\pi$
$788$	6.91148	0.246211
$789$	0	0
$790$	−16.6429	−0.592129
$791$	−14.8401	−0.527654
$792$	0	0
$793$	−10.5646	−0.375160
$794$	26.8969	0.954536
$795$	0	0
$796$	−5.59370	−0.198263
$797$	18.0025	0.637680	0.318840	−	0.947809i	$-0.396707\pi$
$797$	18.0025	0.637680	0.318840	+	0.947809i	$0.396707\pi$
$798$	0	0
$799$	−2.87241	−0.101619
$800$	9.37182	0.331344
$801$	0	0
$802$	−35.9571	−1.26969
$803$	−4.08849	−0.144280
$804$	0	0
$805$	−2.33522	−0.0823057
$806$	37.9974	1.33840
$807$	0	0
$808$	−9.70196	−0.341314
$809$	−13.8919	−0.488414	−0.244207	−	0.969723i	$-0.578528\pi$
$809$	−13.8919	−0.488414	−0.244207	+	0.969723i	$0.578528\pi$
$810$	0	0
$811$	23.9714	0.841748	0.420874	−	0.907119i	$-0.361723\pi$
$811$	23.9714	0.841748	0.420874	+	0.907119i	$0.361723\pi$
$812$	1.02717	0.0360466
$813$	0	0
$814$	−3.96346	−0.138919
$815$	−13.0380	−0.456701
$816$	0	0
$817$	−46.7129	−1.63428
$818$	−27.1501	−0.949282
$819$	0	0
$820$	−2.25498	−0.0787475
$821$	36.3813	1.26972	0.634858	−	0.772629i	$-0.281058\pi$
$821$	36.3813	1.26972	0.634858	+	0.772629i	$0.281058\pi$
$822$	0	0
$823$	18.7577	0.653852	0.326926	−	0.945050i	$-0.393987\pi$
$823$	18.7577	0.653852	0.326926	+	0.945050i	$0.393987\pi$
$824$	16.1947	0.564170
$825$	0	0
$826$	15.2724	0.531396
$827$	15.4047	0.535675	0.267837	−	0.963464i	$-0.413691\pi$
$827$	15.4047	0.535675	0.267837	+	0.963464i	$0.413691\pi$
$828$	0	0
$829$	53.0291	1.84178	0.920889	−	0.389824i	$-0.127464\pi$
$829$	53.0291	1.84178	0.920889	+	0.389824i	$0.127464\pi$
$830$	14.0412	0.487377
$831$	0	0
$832$	33.2572	1.15299
$833$	−0.953761	−0.0330459
$834$	0	0
$835$	−0.617226	−0.0213600
$836$	−0.554463	−0.0191765
$837$	0	0
$838$	59.2257	2.04592
$839$	1.52069	0.0525001	0.0262500	−	0.999655i	$-0.491643\pi$
$839$	1.52069	0.0525001	0.0262500	+	0.999655i	$0.491643\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	16.7822	0.578354
$843$	0	0
$844$	−1.12217	−0.0386267
$845$	18.3722	0.632024
$846$	0	0
$847$	27.6973	0.951690
$848$	14.1875	0.487199
$849$	0	0
$850$	−10.9941	−0.377095
$851$	−8.67527	−0.297385
$852$	0	0
$853$	−38.0427	−1.30256	−0.651279	−	0.758839i	$-0.725767\pi$
$853$	−38.0427	−1.30256	−0.651279	+	0.758839i	$0.725767\pi$
$854$	7.24150	0.247799
$855$	0	0
$856$	−29.6751	−1.01428
$857$	−0.572918	−0.0195705	−0.00978525	−	0.999952i	$-0.503115\pi$
$857$	−0.572918	−0.0195705	−0.00978525	+	0.999952i	$0.503115\pi$
$858$	0	0
$859$	20.5437	0.700941	0.350471	−	0.936574i	$-0.386022\pi$
$859$	20.5437	0.700941	0.350471	+	0.936574i	$0.386022\pi$
$860$	3.74093	0.127565
$861$	0	0
$862$	38.4981	1.31125
$863$	32.0064	1.08951	0.544755	−	0.838595i	$-0.316623\pi$
$863$	32.0064	1.08951	0.544755	+	0.838595i	$0.316623\pi$
$864$	0	0
$865$	−5.26837	−0.179130
$866$	7.24798	0.246296
$867$	0	0
$868$	−4.38351	−0.148786
$869$	−3.43648	−0.116575
$870$	0	0
$871$	−40.5409	−1.37368
$872$	11.9623	0.405094
$873$	0	0
$874$	−7.21090	−0.243912
$875$	21.3752	0.722613
$876$	0	0
$877$	21.9419	0.740924	0.370462	−	0.928848i	$-0.379199\pi$
$877$	21.9419	0.740924	0.370462	+	0.928848i	$0.379199\pi$
$878$	−11.0243	−0.372051
$879$	0	0
$880$	−1.25935	−0.0424528
$881$	56.8734	1.91612	0.958058	−	0.286576i	$-0.0925171\pi$
$881$	56.8734	1.91612	0.958058	+	0.286576i	$0.0925171\pi$
$882$	0	0
$883$	−44.5561	−1.49943	−0.749717	−	0.661759i	$-0.769810\pi$
$883$	−44.5561	−1.49943	−0.749717	+	0.661759i	$0.769810\pi$
$884$	3.96665	0.133413
$885$	0	0
$886$	−6.51662	−0.218930
$887$	−37.8541	−1.27102	−0.635508	−	0.772095i	$-0.719209\pi$
$887$	−37.8541	−1.27102	−0.635508	+	0.772095i	$0.719209\pi$
$888$	0	0
$889$	24.2338	0.812775
$890$	1.38026	0.0462663
$891$	0	0
$892$	3.84097	0.128605
$893$	7.82488	0.261850
$894$	0	0
$895$	−17.7194	−0.592295
$896$	−34.2496	−1.14420
$897$	0	0
$898$	29.7272	0.992011
$899$	−4.26756	−0.142331
$900$	0	0
$901$	−5.21296	−0.173669
$902$	−2.76653	−0.0921154
$903$	0	0
$904$	−14.4651	−0.481102
$905$	−0.562221	−0.0186889
$906$	0	0
$907$	−6.38111	−0.211881	−0.105941	−	0.994372i	$-0.533785\pi$
$907$	−6.38111	−0.211881	−0.105941	+	0.994372i	$0.533785\pi$
$908$	7.77743	0.258103
$909$	0	0
$910$	−20.7923	−0.689257
$911$	−17.7182	−0.587029	−0.293515	−	0.955955i	$-0.594825\pi$
$911$	−17.7182	−0.587029	−0.293515	+	0.955955i	$0.594825\pi$
$912$	0	0
$913$	2.89927	0.0959517
$914$	42.7900	1.41537
$915$	0	0
$916$	4.17433	0.137924
$917$	−21.2033	−0.700195
$918$	0	0
$919$	−18.1235	−0.597840	−0.298920	−	0.954278i	$-0.596626\pi$
$919$	−18.1235	−0.597840	−0.298920	+	0.954278i	$0.596626\pi$
$920$	−2.27621	−0.0750443
$921$	0	0
$922$	27.4309	0.903390
$923$	−3.18543	−0.104850
$924$	0	0
$925$	36.0318	1.18472
$926$	34.5254	1.13457
$927$	0	0
$928$	2.25643	0.0740709
$929$	30.2422	0.992215	0.496108	−	0.868261i	$-0.334762\pi$
$929$	30.2422	0.992215	0.496108	+	0.868261i	$0.334762\pi$
$930$	0	0
$931$	2.59819	0.0851523
$932$	−6.24970	−0.204716
$933$	0	0
$934$	7.62700	0.249563
$935$	0.462729	0.0151329
$936$	0	0
$937$	−21.6319	−0.706684	−0.353342	−	0.935494i	$-0.614955\pi$
$937$	−21.6319	−0.706684	−0.353342	+	0.935494i	$0.614955\pi$
$938$	27.7888	0.907335
$939$	0	0
$940$	−0.626643	−0.0204388
$941$	−37.5162	−1.22299	−0.611497	−	0.791246i	$-0.709432\pi$
$941$	−37.5162	−1.22299	−0.611497	+	0.791246i	$0.709432\pi$
$942$	0	0
$943$	−6.05542	−0.197192
$944$	18.0275	0.586746
$945$	0	0
$946$	4.58956	0.149219
$947$	−14.9169	−0.484735	−0.242367	−	0.970185i	$-0.577924\pi$
$947$	−14.9169	−0.484735	−0.242367	+	0.970185i	$0.577924\pi$
$948$	0	0
$949$	−79.6794	−2.58650
$950$	29.9497	0.971695
$951$	0	0
$952$	10.7172	0.347345
$953$	−51.2950	−1.66161	−0.830804	−	0.556565i	$-0.812119\pi$
$953$	−51.2950	−1.66161	−0.830804	+	0.556565i	$0.812119\pi$
$954$	0	0
$955$	9.99115	0.323306
$956$	0.104265	0.00337218
$957$	0	0
$958$	−11.4658	−0.370443
$959$	39.3933	1.27208
$960$	0	0
$961$	−12.7879	−0.412514
$962$	−77.2427	−2.49040
$963$	0	0
$964$	−1.74982	−0.0563578
$965$	8.68967	0.279730
$966$	0	0
$967$	24.8773	0.800000	0.400000	−	0.916515i	$-0.369010\pi$
$967$	24.8773	0.800000	0.400000	+	0.916515i	$0.369010\pi$
$968$	26.9973	0.867727
$969$	0	0
$970$	5.59654	0.179694
$971$	−39.7251	−1.27484	−0.637420	−	0.770516i	$-0.719998\pi$
$971$	−39.7251	−1.27484	−0.637420	+	0.770516i	$0.719998\pi$
$972$	0	0
$973$	31.6104	1.01338
$974$	19.8573	0.636268
$975$	0	0
$976$	8.54785	0.273610
$977$	−16.1055	−0.515260	−0.257630	−	0.966244i	$-0.582942\pi$
$977$	−16.1055	−0.515260	−0.257630	+	0.966244i	$0.582942\pi$
$978$	0	0
$979$	0.284999	0.00910860
$980$	−0.208072	−0.00664661
$981$	0	0
$982$	−13.8138	−0.440817
$983$	25.5195	0.813946	0.406973	−	0.913440i	$-0.366584\pi$
$983$	25.5195	0.813946	0.406973	+	0.913440i	$0.366584\pi$
$984$	0	0
$985$	15.7129	0.500656
$986$	−2.64703	−0.0842985
$987$	0	0
$988$	−10.8058	−0.343777
$989$	10.0457	0.319435
$990$	0	0
$991$	−48.8047	−1.55033	−0.775166	−	0.631757i	$-0.782334\pi$
$991$	−48.8047	−1.55033	−0.775166	+	0.631757i	$0.782334\pi$
$992$	−9.62945	−0.305735
$993$	0	0
$994$	2.18345	0.0692549
$995$	−12.7170	−0.403156
$996$	0	0
$997$	−29.6198	−0.938069	−0.469035	−	0.883180i	$-0.655398\pi$
$997$	−29.6198	−0.938069	−0.469035	+	0.883180i	$0.655398\pi$
$998$	−36.5743	−1.15774
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.i.1.6		7
3.2	odd	2		2001.2.a.j.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.j.1.2	✓	7	3.2	odd	2
6003.2.a.i.1.6		7	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)

\(f(q)\)	\(=\)	\(q+1.55072 q^{2} +0.404722 q^{4} +0.920116 q^{5} -2.53796 q^{7} -2.47382 q^{8} +O(q^{10})\)	\(q+1.55072 q^{2} +0.404722 q^{4} +0.920116 q^{5} -2.53796 q^{7} -2.47382 q^{8} +1.42684 q^{10} +0.294618 q^{11} +5.74171 q^{13} -3.93566 q^{14} -4.64564 q^{16} +1.70697 q^{17} -4.65004 q^{19} +0.372391 q^{20} +0.456868 q^{22} +1.00000 q^{23} -4.15339 q^{25} +8.90377 q^{26} -1.02717 q^{28} -1.00000 q^{29} +4.26756 q^{31} -2.25643 q^{32} +2.64703 q^{34} -2.33522 q^{35} -8.67527 q^{37} -7.21090 q^{38} -2.27621 q^{40} -6.05542 q^{41} +10.0457 q^{43} +0.119238 q^{44} +1.55072 q^{46} -1.68275 q^{47} -0.558745 q^{49} -6.44072 q^{50} +2.32380 q^{52} -3.05393 q^{53} +0.271082 q^{55} +6.27847 q^{56} -1.55072 q^{58} -3.88052 q^{59} -1.83997 q^{61} +6.61778 q^{62} +5.79221 q^{64} +5.28305 q^{65} -7.06076 q^{67} +0.690848 q^{68} -3.62127 q^{70} -0.554787 q^{71} -13.8773 q^{73} -13.4529 q^{74} -1.88197 q^{76} -0.747728 q^{77} -11.6642 q^{79} -4.27453 q^{80} -9.39024 q^{82} +9.84078 q^{83} +1.57061 q^{85} +15.5780 q^{86} -0.728832 q^{88} +0.967352 q^{89} -14.5723 q^{91} +0.404722 q^{92} -2.60947 q^{94} -4.27858 q^{95} +3.92233 q^{97} -0.866456 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8}+O(q^{10}) \) 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8	\( 7 q + q^{2} + 7 q^{4} + 5 q^{5} - 5 q^{7} - 3 q^{8} - 11 q^{10} + 12 q^{11} - 13 q^{13} + 3 q^{14} - 13 q^{16} + 12 q^{17} - 5 q^{19} + 8 q^{20} - q^{22} + 7 q^{23} - 4 q^{25} - 2 q^{26} - 21 q^{28} - 7 q^{29} - 8 q^{31} + 5 q^{32} - 28 q^{34} - 5 q^{35} - 24 q^{37} + 6 q^{38} - 20 q^{40} - 9 q^{41} - q^{43} + 23 q^{44} + q^{46} - 27 q^{47} - 14 q^{49} - 7 q^{50} - 9 q^{52} + q^{53} - 11 q^{55} + 20 q^{56} - q^{58} - 8 q^{59} + q^{61} + 3 q^{64} - 12 q^{65} - 16 q^{67} - 15 q^{68} + 40 q^{70} + 13 q^{71} - 23 q^{73} + 8 q^{74} - 2 q^{76} - 13 q^{77} - 44 q^{79} - 30 q^{80} - 10 q^{82} - 21 q^{83} - 6 q^{86} + 21 q^{88} + 5 q^{89} - 18 q^{91} + 7 q^{92} + 28 q^{94} - 9 q^{95} - 55 q^{97} - 36 q^{98}+O(q^{100}) \) 7 * q + q^2 + 7 * q^4 + 5 * q^5 - 5 * q^7 - 3 * q^8 - 11 * q^10 + 12 * q^11 - 13 * q^13 + 3 * q^14 - 13 * q^16 + 12 * q^17 - 5 * q^19 + 8 * q^20 - q^22 + 7 * q^23 - 4 * q^25 - 2 * q^26 - 21 * q^28 - 7 * q^29 - 8 * q^31 + 5 * q^32 - 28 * q^34 - 5 * q^35 - 24 * q^37 + 6 * q^38 - 20 * q^40 - 9 * q^41 - q^43 + 23 * q^44 + q^46 - 27 * q^47 - 14 * q^49 - 7 * q^50 - 9 * q^52 + q^53 - 11 * q^55 + 20 * q^56 - q^58 - 8 * q^59 + q^61 + 3 * q^64 - 12 * q^65 - 16 * q^67 - 15 * q^68 + 40 * q^70 + 13 * q^71 - 23 * q^73 + 8 * q^74 - 2 * q^76 - 13 * q^77 - 44 * q^79 - 30 * q^80 - 10 * q^82 - 21 * q^83 - 6 * q^86 + 21 * q^88 + 5 * q^89 - 18 * q^91 + 7 * q^92 + 28 * q^94 - 9 * q^95 - 55 * q^97 - 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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