LMFDB - Embedded newform 8046.2.a.j.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 29 x^{10} + 76 x^{9} + 320 x^{8} - 724 x^{7} - 1643 x^{6} + 3265 x^{5} + 3921 x^{4} + \cdots + 423 $ x^12 - 3x^11 - 29x^10 + 76x^9 + 320x^8 - 724x^7 - 1643x^6 + 3265x^5 + 3921x^4 - 6927x^3 - 3639x^2 + 5508*x + 423
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.41916$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.41916 q^{5} +0.521062 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.41916 q^{5} +0.521062 q^{7} -1.00000 q^{8} -2.41916 q^{10} +4.11026 q^{11} -6.34437 q^{13} -0.521062 q^{14} +1.00000 q^{16} +1.64302 q^{17} -6.78386 q^{19} +2.41916 q^{20} -4.11026 q^{22} -6.27696 q^{23} +0.852348 q^{25} +6.34437 q^{26} +0.521062 q^{28} +0.453542 q^{29} -2.03791 q^{31} -1.00000 q^{32} -1.64302 q^{34} +1.26053 q^{35} +7.37496 q^{37} +6.78386 q^{38} -2.41916 q^{40} -0.207953 q^{41} +11.8979 q^{43} +4.11026 q^{44} +6.27696 q^{46} +6.60210 q^{47} -6.72849 q^{49} -0.852348 q^{50} -6.34437 q^{52} -8.35293 q^{53} +9.94338 q^{55} -0.521062 q^{56} -0.453542 q^{58} -4.41959 q^{59} +2.95418 q^{61} +2.03791 q^{62} +1.00000 q^{64} -15.3481 q^{65} +5.02285 q^{67} +1.64302 q^{68} -1.26053 q^{70} -3.57557 q^{71} -0.738246 q^{73} -7.37496 q^{74} -6.78386 q^{76} +2.14170 q^{77} -14.0719 q^{79} +2.41916 q^{80} +0.207953 q^{82} -13.4234 q^{83} +3.97473 q^{85} -11.8979 q^{86} -4.11026 q^{88} -4.74356 q^{89} -3.30581 q^{91} -6.27696 q^{92} -6.60210 q^{94} -16.4113 q^{95} +0.702134 q^{97} +6.72849 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^4 - 3 * q^5 + 6 * q^7 - 12 * q^8	$ 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8} + 3 q^{10} - 10 q^{11} + 5 q^{13} - 6 q^{14} + 12 q^{16} - 8 q^{17} + 2 q^{19} - 3 q^{20} + 10 q^{22} - 9 q^{23} + 7 q^{25} - 5 q^{26} + 6 q^{28} - 19 q^{29} + 10 q^{31} - 12 q^{32} + 8 q^{34} - 20 q^{35} + 11 q^{37} - 2 q^{38} + 3 q^{40} - 8 q^{41} + 13 q^{43} - 10 q^{44} + 9 q^{46} - 11 q^{47} + 2 q^{49} - 7 q^{50} + 5 q^{52} - 24 q^{53} + 3 q^{55} - 6 q^{56} + 19 q^{58} - 10 q^{59} - 10 q^{62} + 12 q^{64} - 28 q^{65} + 21 q^{67} - 8 q^{68} + 20 q^{70} - 37 q^{71} - 2 q^{73} - 11 q^{74} + 2 q^{76} - 2 q^{77} + 7 q^{79} - 3 q^{80} + 8 q^{82} - 22 q^{83} + 15 q^{85} - 13 q^{86} + 10 q^{88} - 40 q^{89} + q^{91} - 9 q^{92} + 11 q^{94} - 11 q^{95} + 7 q^{97} - 2 q^{98}+O(q^{100}) $ 12 * q - 12 * q^2 + 12 * q^4 - 3 * q^5 + 6 * q^7 - 12 * q^8 + 3 * q^10 - 10 * q^11 + 5 * q^13 - 6 * q^14 + 12 * q^16 - 8 * q^17 + 2 * q^19 - 3 * q^20 + 10 * q^22 - 9 * q^23 + 7 * q^25 - 5 * q^26 + 6 * q^28 - 19 * q^29 + 10 * q^31 - 12 * q^32 + 8 * q^34 - 20 * q^35 + 11 * q^37 - 2 * q^38 + 3 * q^40 - 8 * q^41 + 13 * q^43 - 10 * q^44 + 9 * q^46 - 11 * q^47 + 2 * q^49 - 7 * q^50 + 5 * q^52 - 24 * q^53 + 3 * q^55 - 6 * q^56 + 19 * q^58 - 10 * q^59 - 10 * q^62 + 12 * q^64 - 28 * q^65 + 21 * q^67 - 8 * q^68 + 20 * q^70 - 37 * q^71 - 2 * q^73 - 11 * q^74 + 2 * q^76 - 2 * q^77 + 7 * q^79 - 3 * q^80 + 8 * q^82 - 22 * q^83 + 15 * q^85 - 13 * q^86 + 10 * q^88 - 40 * q^89 + q^91 - 9 * q^92 + 11 * q^94 - 11 * q^95 + 7 * q^97 - 2 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.41916	1.08188	0.540941	−	0.841060i	$-0.318068\pi$
$5$	2.41916	1.08188	0.540941	+	0.841060i	$0.318068\pi$
$6$	0	0
$7$	0.521062	0.196943	0.0984715	−	0.995140i	$-0.468605\pi$
$7$	0.521062	0.196943	0.0984715	+	0.995140i	$0.468605\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.41916	−0.765006
$11$	4.11026	1.23929	0.619644	−	0.784883i	$-0.287277\pi$
$11$	4.11026	1.23929	0.619644	+	0.784883i	$0.287277\pi$
$12$	0	0
$13$	−6.34437	−1.75961	−0.879806	−	0.475333i	$-0.842328\pi$
$13$	−6.34437	−1.75961	−0.879806	+	0.475333i	$0.842328\pi$
$14$	−0.521062	−0.139260
$15$	0	0
$16$	1.00000	0.250000
$17$	1.64302	0.398491	0.199245	−	0.979950i	$-0.436151\pi$
$17$	1.64302	0.398491	0.199245	+	0.979950i	$0.436151\pi$
$18$	0	0
$19$	−6.78386	−1.55632	−0.778162	−	0.628063i	$-0.783848\pi$
$19$	−6.78386	−1.55632	−0.778162	+	0.628063i	$0.783848\pi$
$20$	2.41916	0.540941
$21$	0	0
$22$	−4.11026	−0.876309
$23$	−6.27696	−1.30884	−0.654419	−	0.756132i	$-0.727087\pi$
$23$	−6.27696	−1.30884	−0.654419	+	0.756132i	$0.727087\pi$
$24$	0	0
$25$	0.852348	0.170470
$26$	6.34437	1.24423
$27$	0	0
$28$	0.521062	0.0984715
$29$	0.453542	0.0842207	0.0421104	−	0.999113i	$-0.486592\pi$
$29$	0.453542	0.0842207	0.0421104	+	0.999113i	$0.486592\pi$
$30$	0	0
$31$	−2.03791	−0.366020	−0.183010	−	0.983111i	$-0.558584\pi$
$31$	−2.03791	−0.366020	−0.183010	+	0.983111i	$0.558584\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.64302	−0.281775
$35$	1.26053	0.213069
$36$	0	0
$37$	7.37496	1.21244	0.606218	−	0.795299i	$-0.292686\pi$
$37$	7.37496	1.21244	0.606218	+	0.795299i	$0.292686\pi$
$38$	6.78386	1.10049
$39$	0	0
$40$	−2.41916	−0.382503
$41$	−0.207953	−0.0324767	−0.0162384	−	0.999868i	$-0.505169\pi$
$41$	−0.207953	−0.0324767	−0.0162384	+	0.999868i	$0.505169\pi$
$42$	0	0
$43$	11.8979	1.81441	0.907207	−	0.420685i	$-0.138210\pi$
$43$	11.8979	1.81441	0.907207	+	0.420685i	$0.138210\pi$
$44$	4.11026	0.619644
$45$	0	0
$46$	6.27696	0.925488
$47$	6.60210	0.963015	0.481507	−	0.876442i	$-0.340089\pi$
$47$	6.60210	0.963015	0.481507	+	0.876442i	$0.340089\pi$
$48$	0	0
$49$	−6.72849	−0.961213
$50$	−0.852348	−0.120540
$51$	0	0
$52$	−6.34437	−0.879806
$53$	−8.35293	−1.14736	−0.573681	−	0.819079i	$-0.694485\pi$
$53$	−8.35293	−1.14736	−0.573681	+	0.819079i	$0.694485\pi$
$54$	0	0
$55$	9.94338	1.34076
$56$	−0.521062	−0.0696299
$57$	0	0
$58$	−0.453542	−0.0595530
$59$	−4.41959	−0.575382	−0.287691	−	0.957723i	$-0.592888\pi$
$59$	−4.41959	−0.575382	−0.287691	+	0.957723i	$0.592888\pi$
$60$	0	0
$61$	2.95418	0.378245	0.189122	−	0.981954i	$-0.439436\pi$
$61$	2.95418	0.378245	0.189122	+	0.981954i	$0.439436\pi$
$62$	2.03791	0.258815
$63$	0	0
$64$	1.00000	0.125000
$65$	−15.3481	−1.90369
$66$	0	0
$67$	5.02285	0.613639	0.306820	−	0.951768i	$-0.400735\pi$
$67$	5.02285	0.613639	0.306820	+	0.951768i	$0.400735\pi$
$68$	1.64302	0.199245
$69$	0	0
$70$	−1.26053	−0.150663
$71$	−3.57557	−0.424342	−0.212171	−	0.977233i	$-0.568053\pi$
$71$	−3.57557	−0.424342	−0.212171	+	0.977233i	$0.568053\pi$
$72$	0	0
$73$	−0.738246	−0.0864051	−0.0432026	−	0.999066i	$-0.513756\pi$
$73$	−0.738246	−0.0864051	−0.0432026	+	0.999066i	$0.513756\pi$
$74$	−7.37496	−0.857322
$75$	0	0
$76$	−6.78386	−0.778162
$77$	2.14170	0.244069
$78$	0	0
$79$	−14.0719	−1.58322	−0.791608	−	0.611030i	$-0.790756\pi$
$79$	−14.0719	−1.58322	−0.791608	+	0.611030i	$0.790756\pi$
$80$	2.41916	0.270471
$81$	0	0
$82$	0.207953	0.0229645
$83$	−13.4234	−1.47341	−0.736704	−	0.676216i	$-0.763619\pi$
$83$	−13.4234	−1.47341	−0.736704	+	0.676216i	$0.763619\pi$
$84$	0	0
$85$	3.97473	0.431120
$86$	−11.8979	−1.28298
$87$	0	0
$88$	−4.11026	−0.438155
$89$	−4.74356	−0.502817	−0.251408	−	0.967881i	$-0.580894\pi$
$89$	−4.74356	−0.502817	−0.251408	+	0.967881i	$0.580894\pi$
$90$	0	0
$91$	−3.30581	−0.346543
$92$	−6.27696	−0.654419
$93$	0	0
$94$	−6.60210	−0.680954
$95$	−16.4113	−1.68376
$96$	0	0
$97$	0.702134	0.0712909	0.0356455	−	0.999364i	$-0.488651\pi$
$97$	0.702134	0.0712909	0.0356455	+	0.999364i	$0.488651\pi$
$98$	6.72849	0.679681
$99$	0	0
$100$	0.852348	0.0852348
$101$	−6.49645	−0.646421	−0.323210	−	0.946327i	$-0.604762\pi$
$101$	−6.49645	−0.646421	−0.323210	+	0.946327i	$0.604762\pi$
$102$	0	0
$103$	5.63092	0.554831	0.277415	−	0.960750i	$-0.410522\pi$
$103$	5.63092	0.554831	0.277415	+	0.960750i	$0.410522\pi$
$104$	6.34437	0.622117
$105$	0	0
$106$	8.35293	0.811308
$107$	−12.9948	−1.25626	−0.628130	−	0.778109i	$-0.716179\pi$
$107$	−12.9948	−1.25626	−0.628130	+	0.778109i	$0.716179\pi$
$108$	0	0
$109$	9.86881	0.945261	0.472630	−	0.881261i	$-0.343305\pi$
$109$	9.86881	0.945261	0.472630	+	0.881261i	$0.343305\pi$
$110$	−9.94338	−0.948064
$111$	0	0
$112$	0.521062	0.0492358
$113$	−8.11349	−0.763253	−0.381626	−	0.924317i	$-0.624636\pi$
$113$	−8.11349	−0.763253	−0.381626	+	0.924317i	$0.624636\pi$
$114$	0	0
$115$	−15.1850	−1.41601
$116$	0.453542	0.0421104
$117$	0	0
$118$	4.41959	0.406857
$119$	0.856116	0.0784800
$120$	0	0
$121$	5.89420	0.535836
$122$	−2.95418	−0.267459
$123$	0	0
$124$	−2.03791	−0.183010
$125$	−10.0338	−0.897454
$126$	0	0
$127$	−10.3044	−0.914365	−0.457182	−	0.889373i	$-0.651141\pi$
$127$	−10.3044	−0.914365	−0.457182	+	0.889373i	$0.651141\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	15.3481	1.34611
$131$	11.4643	1.00164	0.500821	−	0.865551i	$-0.333032\pi$
$131$	11.4643	1.00164	0.500821	+	0.865551i	$0.333032\pi$
$132$	0	0
$133$	−3.53482	−0.306507
$134$	−5.02285	−0.433909
$135$	0	0
$136$	−1.64302	−0.140888
$137$	5.21205	0.445295	0.222648	−	0.974899i	$-0.428530\pi$
$137$	5.21205	0.445295	0.222648	+	0.974899i	$0.428530\pi$
$138$	0	0
$139$	−2.36544	−0.200634	−0.100317	−	0.994956i	$-0.531986\pi$
$139$	−2.36544	−0.200634	−0.100317	+	0.994956i	$0.531986\pi$
$140$	1.26053	0.106535
$141$	0	0
$142$	3.57557	0.300055
$143$	−26.0770	−2.18067
$144$	0	0
$145$	1.09719	0.0911169
$146$	0.738246	0.0610977
$147$	0	0
$148$	7.37496	0.606218
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	3.82307	0.311117	0.155559	−	0.987827i	$-0.450282\pi$
$151$	3.82307	0.311117	0.155559	+	0.987827i	$0.450282\pi$
$152$	6.78386	0.550244
$153$	0	0
$154$	−2.14170	−0.172583
$155$	−4.93005	−0.395991
$156$	0	0
$157$	−5.23107	−0.417485	−0.208742	−	0.977971i	$-0.566937\pi$
$157$	−5.23107	−0.417485	−0.208742	+	0.977971i	$0.566937\pi$
$158$	14.0719	1.11950
$159$	0	0
$160$	−2.41916	−0.191252
$161$	−3.27069	−0.257767
$162$	0	0
$163$	−5.70738	−0.447037	−0.223518	−	0.974700i	$-0.571754\pi$
$163$	−5.70738	−0.447037	−0.223518	+	0.974700i	$0.571754\pi$
$164$	−0.207953	−0.0162384
$165$	0	0
$166$	13.4234	1.04186
$167$	13.9386	1.07860	0.539300	−	0.842114i	$-0.318689\pi$
$167$	13.9386	1.07860	0.539300	+	0.842114i	$0.318689\pi$
$168$	0	0
$169$	27.2511	2.09624
$170$	−3.97473	−0.304848
$171$	0	0
$172$	11.8979	0.907207
$173$	−18.6275	−1.41622	−0.708110	−	0.706102i	$-0.750452\pi$
$173$	−18.6275	−1.41622	−0.708110	+	0.706102i	$0.750452\pi$
$174$	0	0
$175$	0.444127	0.0335728
$176$	4.11026	0.309822
$177$	0	0
$178$	4.74356	0.355545
$179$	−19.2947	−1.44215	−0.721076	−	0.692856i	$-0.756352\pi$
$179$	−19.2947	−1.44215	−0.721076	+	0.692856i	$0.756352\pi$
$180$	0	0
$181$	−5.48036	−0.407352	−0.203676	−	0.979038i	$-0.565289\pi$
$181$	−5.48036	−0.407352	−0.203676	+	0.979038i	$0.565289\pi$
$182$	3.30581	0.245043
$183$	0	0
$184$	6.27696	0.462744
$185$	17.8412	1.31171
$186$	0	0
$187$	6.75323	0.493845
$188$	6.60210	0.481507
$189$	0	0
$190$	16.4113	1.19060
$191$	11.9265	0.862971	0.431486	−	0.902120i	$-0.357990\pi$
$191$	11.9265	0.862971	0.431486	+	0.902120i	$0.357990\pi$
$192$	0	0
$193$	−11.2228	−0.807837	−0.403919	−	0.914795i	$-0.632352\pi$
$193$	−11.2228	−0.807837	−0.403919	+	0.914795i	$0.632352\pi$
$194$	−0.702134	−0.0504103
$195$	0	0
$196$	−6.72849	−0.480607
$197$	0.571373	0.0407086	0.0203543	−	0.999793i	$-0.493521\pi$
$197$	0.571373	0.0407086	0.0203543	+	0.999793i	$0.493521\pi$
$198$	0	0
$199$	−14.4380	−1.02348	−0.511742	−	0.859139i	$-0.671000\pi$
$199$	−14.4380	−1.02348	−0.511742	+	0.859139i	$0.671000\pi$
$200$	−0.852348	−0.0602701
$201$	0	0
$202$	6.49645	0.457088
$203$	0.236324	0.0165867
$204$	0	0
$205$	−0.503071	−0.0351360
$206$	−5.63092	−0.392325
$207$	0	0
$208$	−6.34437	−0.439903
$209$	−27.8834	−1.92874
$210$	0	0
$211$	1.26551	0.0871213	0.0435607	−	0.999051i	$-0.486130\pi$
$211$	1.26551	0.0871213	0.0435607	+	0.999051i	$0.486130\pi$
$212$	−8.35293	−0.573681
$213$	0	0
$214$	12.9948	0.888310
$215$	28.7830	1.96298
$216$	0	0
$217$	−1.06188	−0.0720851
$218$	−9.86881	−0.668400
$219$	0	0
$220$	9.94338	0.670382
$221$	−10.4239	−0.701189
$222$	0	0
$223$	6.56730	0.439779	0.219889	−	0.975525i	$-0.429430\pi$
$223$	6.56730	0.439779	0.219889	+	0.975525i	$0.429430\pi$
$224$	−0.521062	−0.0348149
$225$	0	0
$226$	8.11349	0.539701
$227$	−20.8482	−1.38374	−0.691872	−	0.722020i	$-0.743214\pi$
$227$	−20.8482	−1.38374	−0.691872	+	0.722020i	$0.743214\pi$
$228$	0	0
$229$	−21.5345	−1.42304	−0.711520	−	0.702666i	$-0.751993\pi$
$229$	−21.5345	−1.42304	−0.711520	+	0.702666i	$0.751993\pi$
$230$	15.1850	1.00127
$231$	0	0
$232$	−0.453542	−0.0297765
$233$	−6.04955	−0.396319	−0.198160	−	0.980170i	$-0.563496\pi$
$233$	−6.04955	−0.396319	−0.198160	+	0.980170i	$0.563496\pi$
$234$	0	0
$235$	15.9715	1.04187
$236$	−4.41959	−0.287691
$237$	0	0
$238$	−0.856116	−0.0554937
$239$	3.90903	0.252854	0.126427	−	0.991976i	$-0.459649\pi$
$239$	3.90903	0.252854	0.126427	+	0.991976i	$0.459649\pi$
$240$	0	0
$241$	−9.32692	−0.600800	−0.300400	−	0.953813i	$-0.597120\pi$
$241$	−9.32692	−0.600800	−0.300400	+	0.953813i	$0.597120\pi$
$242$	−5.89420	−0.378893
$243$	0	0
$244$	2.95418	0.189122
$245$	−16.2773	−1.03992
$246$	0	0
$247$	43.0393	2.73853
$248$	2.03791	0.129408
$249$	0	0
$250$	10.0338	0.634596
$251$	12.1524	0.767052	0.383526	−	0.923530i	$-0.374710\pi$
$251$	12.1524	0.767052	0.383526	+	0.923530i	$0.374710\pi$
$252$	0	0
$253$	−25.7999	−1.62203
$254$	10.3044	0.646553
$255$	0	0
$256$	1.00000	0.0625000
$257$	26.7763	1.67026	0.835131	−	0.550052i	$-0.185392\pi$
$257$	26.7763	1.67026	0.835131	+	0.550052i	$0.185392\pi$
$258$	0	0
$259$	3.84281	0.238781
$260$	−15.3481	−0.951847
$261$	0	0
$262$	−11.4643	−0.708267
$263$	−20.7166	−1.27744	−0.638721	−	0.769439i	$-0.720536\pi$
$263$	−20.7166	−1.27744	−0.638721	+	0.769439i	$0.720536\pi$
$264$	0	0
$265$	−20.2071	−1.24131
$266$	3.53482	0.216733
$267$	0	0
$268$	5.02285	0.306820
$269$	19.9625	1.21714	0.608568	−	0.793502i	$-0.291744\pi$
$269$	19.9625	1.21714	0.608568	+	0.793502i	$0.291744\pi$
$270$	0	0
$271$	−6.02115	−0.365759	−0.182880	−	0.983135i	$-0.558542\pi$
$271$	−6.02115	−0.365759	−0.182880	+	0.983135i	$0.558542\pi$
$272$	1.64302	0.0996227
$273$	0	0
$274$	−5.21205	−0.314871
$275$	3.50337	0.211261
$276$	0	0
$277$	−5.49723	−0.330296	−0.165148	−	0.986269i	$-0.552810\pi$
$277$	−5.49723	−0.330296	−0.165148	+	0.986269i	$0.552810\pi$
$278$	2.36544	0.141870
$279$	0	0
$280$	−1.26053	−0.0753314
$281$	−17.1246	−1.02157	−0.510784	−	0.859709i	$-0.670645\pi$
$281$	−17.1246	−1.02157	−0.510784	+	0.859709i	$0.670645\pi$
$282$	0	0
$283$	9.80402	0.582789	0.291394	−	0.956603i	$-0.405881\pi$
$283$	9.80402	0.582789	0.291394	+	0.956603i	$0.405881\pi$
$284$	−3.57557	−0.212171
$285$	0	0
$286$	26.0770	1.54196
$287$	−0.108356	−0.00639607
$288$	0	0
$289$	−14.3005	−0.841205
$290$	−1.09719	−0.0644294
$291$	0	0
$292$	−0.738246	−0.0432026
$293$	−16.0534	−0.937847	−0.468924	−	0.883239i	$-0.655358\pi$
$293$	−16.0534	−0.937847	−0.468924	+	0.883239i	$0.655358\pi$
$294$	0	0
$295$	−10.6917	−0.622496
$296$	−7.37496	−0.428661
$297$	0	0
$298$	1.00000	0.0579284
$299$	39.8234	2.30305
$300$	0	0
$301$	6.19955	0.357336
$302$	−3.82307	−0.219993
$303$	0	0
$304$	−6.78386	−0.389081
$305$	7.14665	0.409216
$306$	0	0
$307$	0.803629	0.0458655	0.0229328	−	0.999737i	$-0.492700\pi$
$307$	0.803629	0.0458655	0.0229328	+	0.999737i	$0.492700\pi$
$308$	2.14170	0.122035
$309$	0	0
$310$	4.93005	0.280008
$311$	26.2837	1.49041	0.745207	−	0.666833i	$-0.232351\pi$
$311$	26.2837	1.49041	0.745207	+	0.666833i	$0.232351\pi$
$312$	0	0
$313$	25.3338	1.43195	0.715974	−	0.698127i	$-0.245983\pi$
$313$	25.3338	1.43195	0.715974	+	0.698127i	$0.245983\pi$
$314$	5.23107	0.295206
$315$	0	0
$316$	−14.0719	−0.791608
$317$	−7.42192	−0.416857	−0.208428	−	0.978038i	$-0.566835\pi$
$317$	−7.42192	−0.416857	−0.208428	+	0.978038i	$0.566835\pi$
$318$	0	0
$319$	1.86418	0.104374
$320$	2.41916	0.135235
$321$	0	0
$322$	3.27069	0.182268
$323$	−11.1460	−0.620181
$324$	0	0
$325$	−5.40761	−0.299960
$326$	5.70738	0.316103
$327$	0	0
$328$	0.207953	0.0114823
$329$	3.44010	0.189659
$330$	0	0
$331$	−23.5355	−1.29363	−0.646813	−	0.762649i	$-0.723899\pi$
$331$	−23.5355	−1.29363	−0.646813	+	0.762649i	$0.723899\pi$
$332$	−13.4234	−0.736704
$333$	0	0
$334$	−13.9386	−0.762686
$335$	12.1511	0.663886
$336$	0	0
$337$	−21.1836	−1.15394	−0.576972	−	0.816764i	$-0.695766\pi$
$337$	−21.1836	−1.15394	−0.576972	+	0.816764i	$0.695766\pi$
$338$	−27.2511	−1.48226
$339$	0	0
$340$	3.97473	0.215560
$341$	−8.37635	−0.453605
$342$	0	0
$343$	−7.15340	−0.386247
$344$	−11.8979	−0.641492
$345$	0	0
$346$	18.6275	1.00142
$347$	−7.79972	−0.418711	−0.209355	−	0.977840i	$-0.567137\pi$
$347$	−7.79972	−0.418711	−0.209355	+	0.977840i	$0.567137\pi$
$348$	0	0
$349$	−5.48920	−0.293830	−0.146915	−	0.989149i	$-0.546934\pi$
$349$	−5.48920	−0.293830	−0.146915	+	0.989149i	$0.546934\pi$
$350$	−0.444127	−0.0237396
$351$	0	0
$352$	−4.11026	−0.219077
$353$	2.46784	0.131350	0.0656749	−	0.997841i	$-0.479080\pi$
$353$	2.46784	0.131350	0.0656749	+	0.997841i	$0.479080\pi$
$354$	0	0
$355$	−8.64989	−0.459088
$356$	−4.74356	−0.251408
$357$	0	0
$358$	19.2947	1.01976
$359$	−23.2051	−1.22472	−0.612358	−	0.790581i	$-0.709779\pi$
$359$	−23.2051	−1.22472	−0.612358	+	0.790581i	$0.709779\pi$
$360$	0	0
$361$	27.0208	1.42215
$362$	5.48036	0.288041
$363$	0	0
$364$	−3.30581	−0.173272
$365$	−1.78594	−0.0934802
$366$	0	0
$367$	34.8189	1.81753	0.908765	−	0.417309i	$-0.137027\pi$
$367$	34.8189	1.81753	0.908765	+	0.417309i	$0.137027\pi$
$368$	−6.27696	−0.327209
$369$	0	0
$370$	−17.8412	−0.927521
$371$	−4.35240	−0.225965
$372$	0	0
$373$	21.1517	1.09519	0.547597	−	0.836742i	$-0.315543\pi$
$373$	21.1517	1.09519	0.547597	+	0.836742i	$0.315543\pi$
$374$	−6.75323	−0.349201
$375$	0	0
$376$	−6.60210	−0.340477
$377$	−2.87744	−0.148196
$378$	0	0
$379$	−16.3820	−0.841485	−0.420743	−	0.907180i	$-0.638230\pi$
$379$	−16.3820	−0.841485	−0.420743	+	0.907180i	$0.638230\pi$
$380$	−16.4113	−0.841880
$381$	0	0
$382$	−11.9265	−0.610213
$383$	12.6455	0.646156	0.323078	−	0.946372i	$-0.395282\pi$
$383$	12.6455	0.646156	0.323078	+	0.946372i	$0.395282\pi$
$384$	0	0
$385$	5.18112	0.264054
$386$	11.2228	0.571227
$387$	0	0
$388$	0.702134	0.0356455
$389$	−1.54826	−0.0785000	−0.0392500	−	0.999229i	$-0.512497\pi$
$389$	−1.54826	−0.0785000	−0.0392500	+	0.999229i	$0.512497\pi$
$390$	0	0
$391$	−10.3132	−0.521560
$392$	6.72849	0.339840
$393$	0	0
$394$	−0.571373	−0.0287853
$395$	−34.0423	−1.71285
$396$	0	0
$397$	36.3925	1.82649	0.913244	−	0.407413i	$-0.133569\pi$
$397$	36.3925	1.82649	0.913244	+	0.407413i	$0.133569\pi$
$398$	14.4380	0.723712
$399$	0	0
$400$	0.852348	0.0426174
$401$	3.19605	0.159603	0.0798014	−	0.996811i	$-0.474571\pi$
$401$	3.19605	0.159603	0.0798014	+	0.996811i	$0.474571\pi$
$402$	0	0
$403$	12.9293	0.644054
$404$	−6.49645	−0.323210
$405$	0	0
$406$	−0.236324	−0.0117286
$407$	30.3130	1.50256
$408$	0	0
$409$	18.1471	0.897316	0.448658	−	0.893704i	$-0.351902\pi$
$409$	18.1471	0.897316	0.448658	+	0.893704i	$0.351902\pi$
$410$	0.503071	0.0248449
$411$	0	0
$412$	5.63092	0.277415
$413$	−2.30288	−0.113318
$414$	0	0
$415$	−32.4733	−1.59405
$416$	6.34437	0.311058
$417$	0	0
$418$	27.8834	1.36382
$419$	−3.71552	−0.181515	−0.0907576	−	0.995873i	$-0.528929\pi$
$419$	−3.71552	−0.181515	−0.0907576	+	0.995873i	$0.528929\pi$
$420$	0	0
$421$	7.81057	0.380664	0.190332	−	0.981720i	$-0.439044\pi$
$421$	7.81057	0.380664	0.190332	+	0.981720i	$0.439044\pi$
$422$	−1.26551	−0.0616041
$423$	0	0
$424$	8.35293	0.405654
$425$	1.40042	0.0679306
$426$	0	0
$427$	1.53931	0.0744926
$428$	−12.9948	−0.628130
$429$	0	0
$430$	−28.7830	−1.38804
$431$	16.3875	0.789356	0.394678	−	0.918820i	$-0.370856\pi$
$431$	16.3875	0.789356	0.394678	+	0.918820i	$0.370856\pi$
$432$	0	0
$433$	−3.09999	−0.148976	−0.0744880	−	0.997222i	$-0.523732\pi$
$433$	−3.09999	−0.148976	−0.0744880	+	0.997222i	$0.523732\pi$
$434$	1.06188	0.0509719
$435$	0	0
$436$	9.86881	0.472630
$437$	42.5821	2.03698
$438$	0	0
$439$	−18.5942	−0.887453	−0.443727	−	0.896162i	$-0.646344\pi$
$439$	−18.5942	−0.887453	−0.443727	+	0.896162i	$0.646344\pi$
$440$	−9.94338	−0.474032
$441$	0	0
$442$	10.4239	0.495816
$443$	23.6084	1.12167	0.560835	−	0.827928i	$-0.310480\pi$
$443$	23.6084	1.12167	0.560835	+	0.827928i	$0.310480\pi$
$444$	0	0
$445$	−11.4755	−0.543989
$446$	−6.56730	−0.310970
$447$	0	0
$448$	0.521062	0.0246179
$449$	−41.1909	−1.94392	−0.971961	−	0.235144i	$-0.924444\pi$
$449$	−41.1909	−1.94392	−0.971961	+	0.235144i	$0.924444\pi$
$450$	0	0
$451$	−0.854738	−0.0402480
$452$	−8.11349	−0.381626
$453$	0	0
$454$	20.8482	0.978454
$455$	−7.99730	−0.374919
$456$	0	0
$457$	3.92311	0.183515	0.0917577	−	0.995781i	$-0.470751\pi$
$457$	3.92311	0.183515	0.0917577	+	0.995781i	$0.470751\pi$
$458$	21.5345	1.00624
$459$	0	0
$460$	−15.1850	−0.708004
$461$	−3.42177	−0.159368	−0.0796838	−	0.996820i	$-0.525391\pi$
$461$	−3.42177	−0.159368	−0.0796838	+	0.996820i	$0.525391\pi$
$462$	0	0
$463$	4.06287	0.188817	0.0944087	−	0.995534i	$-0.469904\pi$
$463$	4.06287	0.188817	0.0944087	+	0.995534i	$0.469904\pi$
$464$	0.453542	0.0210552
$465$	0	0
$466$	6.04955	0.280240
$467$	−27.2711	−1.26195	−0.630977	−	0.775801i	$-0.717346\pi$
$467$	−27.2711	−1.26195	−0.630977	+	0.775801i	$0.717346\pi$
$468$	0	0
$469$	2.61722	0.120852
$470$	−15.9715	−0.736712
$471$	0	0
$472$	4.41959	0.203428
$473$	48.9034	2.24858
$474$	0	0
$475$	−5.78221	−0.265306
$476$	0.856116	0.0392400
$477$	0	0
$478$	−3.90903	−0.178795
$479$	−14.5587	−0.665204	−0.332602	−	0.943067i	$-0.607927\pi$
$479$	−14.5587	−0.665204	−0.332602	+	0.943067i	$0.607927\pi$
$480$	0	0
$481$	−46.7895	−2.13342
$482$	9.32692	0.424830
$483$	0	0
$484$	5.89420	0.267918
$485$	1.69858	0.0771284
$486$	0	0
$487$	23.3034	1.05598	0.527990	−	0.849251i	$-0.322946\pi$
$487$	23.3034	1.05598	0.527990	+	0.849251i	$0.322946\pi$
$488$	−2.95418	−0.133730
$489$	0	0
$490$	16.2773	0.735334
$491$	19.7732	0.892354	0.446177	−	0.894945i	$-0.352785\pi$
$491$	19.7732	0.892354	0.446177	+	0.894945i	$0.352785\pi$
$492$	0	0
$493$	0.745179	0.0335612
$494$	−43.0393	−1.93643
$495$	0	0
$496$	−2.03791	−0.0915051
$497$	−1.86310	−0.0835712
$498$	0	0
$499$	22.2601	0.996498	0.498249	−	0.867034i	$-0.333977\pi$
$499$	22.2601	0.996498	0.498249	+	0.867034i	$0.333977\pi$
$500$	−10.0338	−0.448727
$501$	0	0
$502$	−12.1524	−0.542387
$503$	−9.66562	−0.430968	−0.215484	−	0.976507i	$-0.569133\pi$
$503$	−9.66562	−0.430968	−0.215484	+	0.976507i	$0.569133\pi$
$504$	0	0
$505$	−15.7160	−0.699351
$506$	25.7999	1.14695
$507$	0	0
$508$	−10.3044	−0.457182
$509$	−27.5034	−1.21907	−0.609534	−	0.792760i	$-0.708643\pi$
$509$	−27.5034	−1.21907	−0.609534	+	0.792760i	$0.708643\pi$
$510$	0	0
$511$	−0.384672	−0.0170169
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−26.7763	−1.18105
$515$	13.6221	0.600262
$516$	0	0
$517$	27.1363	1.19345
$518$	−3.84281	−0.168844
$519$	0	0
$520$	15.3481	0.673057
$521$	−8.85903	−0.388121	−0.194060	−	0.980990i	$-0.562166\pi$
$521$	−8.85903	−0.388121	−0.194060	+	0.980990i	$0.562166\pi$
$522$	0	0
$523$	32.4818	1.42033	0.710165	−	0.704035i	$-0.248620\pi$
$523$	32.4818	1.42033	0.710165	+	0.704035i	$0.248620\pi$
$524$	11.4643	0.500821
$525$	0	0
$526$	20.7166	0.903288
$527$	−3.34833	−0.145856
$528$	0	0
$529$	16.4003	0.713056
$530$	20.2071	0.877740
$531$	0	0
$532$	−3.53482	−0.153254
$533$	1.31933	0.0571465
$534$	0	0
$535$	−31.4367	−1.35913
$536$	−5.02285	−0.216954
$537$	0	0
$538$	−19.9625	−0.860644
$539$	−27.6558	−1.19122
$540$	0	0
$541$	−17.9164	−0.770287	−0.385144	−	0.922857i	$-0.625848\pi$
$541$	−17.9164	−0.770287	−0.385144	+	0.922857i	$0.625848\pi$
$542$	6.02115	0.258631
$543$	0	0
$544$	−1.64302	−0.0704439
$545$	23.8743	1.02266
$546$	0	0
$547$	15.1104	0.646074	0.323037	−	0.946386i	$-0.395296\pi$
$547$	15.1104	0.646074	0.323037	+	0.946386i	$0.395296\pi$
$548$	5.21205	0.222648
$549$	0	0
$550$	−3.50337	−0.149384
$551$	−3.07677	−0.131075
$552$	0	0
$553$	−7.33235	−0.311803
$554$	5.49723	0.233555
$555$	0	0
$556$	−2.36544	−0.100317
$557$	−0.106658	−0.00451923	−0.00225962	−	0.999997i	$-0.500719\pi$
$557$	−0.106658	−0.00451923	−0.00225962	+	0.999997i	$0.500719\pi$
$558$	0	0
$559$	−75.4847	−3.19266
$560$	1.26053	0.0532673
$561$	0	0
$562$	17.1246	0.722358
$563$	4.17813	0.176087	0.0880436	−	0.996117i	$-0.471939\pi$
$563$	4.17813	0.176087	0.0880436	+	0.996117i	$0.471939\pi$
$564$	0	0
$565$	−19.6278	−0.825749
$566$	−9.80402	−0.412094
$567$	0	0
$568$	3.57557	0.150028
$569$	−15.8101	−0.662792	−0.331396	−	0.943492i	$-0.607520\pi$
$569$	−15.8101	−0.662792	−0.331396	+	0.943492i	$0.607520\pi$
$570$	0	0
$571$	21.1434	0.884824	0.442412	−	0.896812i	$-0.354123\pi$
$571$	21.1434	0.884824	0.442412	+	0.896812i	$0.354123\pi$
$572$	−26.0770	−1.09033
$573$	0	0
$574$	0.108356	0.00452270
$575$	−5.35016	−0.223117
$576$	0	0
$577$	−18.3013	−0.761892	−0.380946	−	0.924597i	$-0.624402\pi$
$577$	−18.3013	−0.761892	−0.380946	+	0.924597i	$0.624402\pi$
$578$	14.3005	0.594822
$579$	0	0
$580$	1.09719	0.0455585
$581$	−6.99442	−0.290177
$582$	0	0
$583$	−34.3327	−1.42191
$584$	0.738246	0.0305488
$585$	0	0
$586$	16.0534	0.663158
$587$	−36.8334	−1.52028	−0.760138	−	0.649762i	$-0.774869\pi$
$587$	−36.8334	−1.52028	−0.760138	+	0.649762i	$0.774869\pi$
$588$	0	0
$589$	13.8249	0.569646
$590$	10.6917	0.440171
$591$	0	0
$592$	7.37496	0.303109
$593$	−14.2026	−0.583232	−0.291616	−	0.956536i	$-0.594193\pi$
$593$	−14.2026	−0.583232	−0.291616	+	0.956536i	$0.594193\pi$
$594$	0	0
$595$	2.07108	0.0849061
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−39.8234	−1.62850
$599$	−34.5130	−1.41016	−0.705081	−	0.709127i	$-0.749089\pi$
$599$	−34.5130	−1.41016	−0.705081	+	0.709127i	$0.749089\pi$
$600$	0	0
$601$	25.0442	1.02157	0.510787	−	0.859707i	$-0.329354\pi$
$601$	25.0442	1.02157	0.510787	+	0.859707i	$0.329354\pi$
$602$	−6.19955	−0.252675
$603$	0	0
$604$	3.82307	0.155559
$605$	14.2590	0.579712
$606$	0	0
$607$	30.7189	1.24684	0.623421	−	0.781886i	$-0.285742\pi$
$607$	30.7189	1.24684	0.623421	+	0.781886i	$0.285742\pi$
$608$	6.78386	0.275122
$609$	0	0
$610$	−7.14665	−0.289359
$611$	−41.8862	−1.69453
$612$	0	0
$613$	33.3549	1.34719	0.673596	−	0.739100i	$-0.264749\pi$
$613$	33.3549	1.34719	0.673596	+	0.739100i	$0.264749\pi$
$614$	−0.803629	−0.0324318
$615$	0	0
$616$	−2.14170	−0.0862915
$617$	−18.4604	−0.743189	−0.371595	−	0.928395i	$-0.621189\pi$
$617$	−18.4604	−0.743189	−0.371595	+	0.928395i	$0.621189\pi$
$618$	0	0
$619$	−9.74307	−0.391607	−0.195804	−	0.980643i	$-0.562731\pi$
$619$	−9.74307	−0.391607	−0.195804	+	0.980643i	$0.562731\pi$
$620$	−4.93005	−0.197995
$621$	0	0
$622$	−26.2837	−1.05388
$623$	−2.47169	−0.0990263
$624$	0	0
$625$	−28.5352	−1.14141
$626$	−25.3338	−1.01254
$627$	0	0
$628$	−5.23107	−0.208742
$629$	12.1172	0.483144
$630$	0	0
$631$	28.7610	1.14496	0.572478	−	0.819920i	$-0.305982\pi$
$631$	28.7610	1.14496	0.572478	+	0.819920i	$0.305982\pi$
$632$	14.0719	0.559751
$633$	0	0
$634$	7.42192	0.294762
$635$	−24.9279	−0.989235
$636$	0	0
$637$	42.6881	1.69136
$638$	−1.86418	−0.0738034
$639$	0	0
$640$	−2.41916	−0.0956258
$641$	−30.8974	−1.22037	−0.610187	−	0.792257i	$-0.708906\pi$
$641$	−30.8974	−1.22037	−0.610187	+	0.792257i	$0.708906\pi$
$642$	0	0
$643$	−41.3216	−1.62957	−0.814783	−	0.579767i	$-0.803144\pi$
$643$	−41.3216	−1.62957	−0.814783	+	0.579767i	$0.803144\pi$
$644$	−3.27069	−0.128883
$645$	0	0
$646$	11.1460	0.438534
$647$	31.2228	1.22750	0.613748	−	0.789502i	$-0.289661\pi$
$647$	31.2228	1.22750	0.613748	+	0.789502i	$0.289661\pi$
$648$	0	0
$649$	−18.1657	−0.713065
$650$	5.40761	0.212104
$651$	0	0
$652$	−5.70738	−0.223518
$653$	−16.8610	−0.659821	−0.329911	−	0.944012i	$-0.607019\pi$
$653$	−16.8610	−0.659821	−0.329911	+	0.944012i	$0.607019\pi$
$654$	0	0
$655$	27.7340	1.08366
$656$	−0.207953	−0.00811918
$657$	0	0
$658$	−3.44010	−0.134109
$659$	23.3382	0.909129	0.454565	−	0.890714i	$-0.349795\pi$
$659$	23.3382	0.909129	0.454565	+	0.890714i	$0.349795\pi$
$660$	0	0
$661$	10.6973	0.416078	0.208039	−	0.978121i	$-0.433292\pi$
$661$	10.6973	0.416078	0.208039	+	0.978121i	$0.433292\pi$
$662$	23.5355	0.914732
$663$	0	0
$664$	13.4234	0.520928
$665$	−8.55129	−0.331605
$666$	0	0
$667$	−2.84687	−0.110231
$668$	13.9386	0.539300
$669$	0	0
$670$	−12.1511	−0.469438
$671$	12.1425	0.468754
$672$	0	0
$673$	−20.7774	−0.800910	−0.400455	−	0.916316i	$-0.631148\pi$
$673$	−20.7774	−0.800910	−0.400455	+	0.916316i	$0.631148\pi$
$674$	21.1836	0.815962
$675$	0	0
$676$	27.2511	1.04812
$677$	−36.6517	−1.40864	−0.704319	−	0.709884i	$-0.748747\pi$
$677$	−36.6517	−1.40864	−0.704319	+	0.709884i	$0.748747\pi$
$678$	0	0
$679$	0.365856	0.0140403
$680$	−3.97473	−0.152424
$681$	0	0
$682$	8.37635	0.320747
$683$	−26.0672	−0.997434	−0.498717	−	0.866765i	$-0.666195\pi$
$683$	−26.0672	−0.997434	−0.498717	+	0.866765i	$0.666195\pi$
$684$	0	0
$685$	12.6088	0.481757
$686$	7.15340	0.273118
$687$	0	0
$688$	11.8979	0.453603
$689$	52.9941	2.01891
$690$	0	0
$691$	−9.22720	−0.351019	−0.175510	−	0.984478i	$-0.556157\pi$
$691$	−9.22720	−0.351019	−0.175510	+	0.984478i	$0.556157\pi$
$692$	−18.6275	−0.708110
$693$	0	0
$694$	7.79972	0.296073
$695$	−5.72239	−0.217062
$696$	0	0
$697$	−0.341670	−0.0129417
$698$	5.48920	0.207769
$699$	0	0
$700$	0.444127	0.0167864
$701$	29.4693	1.11304	0.556520	−	0.830834i	$-0.312136\pi$
$701$	29.4693	1.11304	0.556520	+	0.830834i	$0.312136\pi$
$702$	0	0
$703$	−50.0307	−1.88694
$704$	4.11026	0.154911
$705$	0	0
$706$	−2.46784	−0.0928783
$707$	−3.38505	−0.127308
$708$	0	0
$709$	−16.1810	−0.607690	−0.303845	−	0.952721i	$-0.598270\pi$
$709$	−16.1810	−0.607690	−0.303845	+	0.952721i	$0.598270\pi$
$710$	8.64989	0.324624
$711$	0	0
$712$	4.74356	0.177773
$713$	12.7919	0.479061
$714$	0	0
$715$	−63.0845	−2.35923
$716$	−19.2947	−0.721076
$717$	0	0
$718$	23.2051	0.866005
$719$	−35.6905	−1.33103	−0.665516	−	0.746384i	$-0.731788\pi$
$719$	−35.6905	−1.33103	−0.665516	+	0.746384i	$0.731788\pi$
$720$	0	0
$721$	2.93406	0.109270
$722$	−27.0208	−1.00561
$723$	0	0
$724$	−5.48036	−0.203676
$725$	0.386576	0.0143571
$726$	0	0
$727$	39.0585	1.44860	0.724299	−	0.689486i	$-0.242164\pi$
$727$	39.0585	1.44860	0.724299	+	0.689486i	$0.242164\pi$
$728$	3.30581	0.122522
$729$	0	0
$730$	1.78594	0.0661005
$731$	19.5485	0.723027
$732$	0	0
$733$	−36.6626	−1.35417	−0.677083	−	0.735907i	$-0.736756\pi$
$733$	−36.6626	−1.35417	−0.677083	+	0.735907i	$0.736756\pi$
$734$	−34.8189	−1.28519
$735$	0	0
$736$	6.27696	0.231372
$737$	20.6452	0.760476
$738$	0	0
$739$	−33.6966	−1.23955	−0.619774	−	0.784780i	$-0.712776\pi$
$739$	−33.6966	−1.23955	−0.619774	+	0.784780i	$0.712776\pi$
$740$	17.8412	0.655856
$741$	0	0
$742$	4.35240	0.159782
$743$	28.9057	1.06045	0.530224	−	0.847858i	$-0.322108\pi$
$743$	28.9057	1.06045	0.530224	+	0.847858i	$0.322108\pi$
$744$	0	0
$745$	−2.41916	−0.0886313
$746$	−21.1517	−0.774419
$747$	0	0
$748$	6.75323	0.246923
$749$	−6.77113	−0.247412
$750$	0	0
$751$	−30.2122	−1.10246	−0.551229	−	0.834354i	$-0.685841\pi$
$751$	−30.2122	−1.10246	−0.551229	+	0.834354i	$0.685841\pi$
$752$	6.60210	0.240754
$753$	0	0
$754$	2.87744	0.104790
$755$	9.24863	0.336592
$756$	0	0
$757$	39.6556	1.44131	0.720654	−	0.693295i	$-0.243842\pi$
$757$	39.6556	1.44131	0.720654	+	0.693295i	$0.243842\pi$
$758$	16.3820	0.595020
$759$	0	0
$760$	16.4113	0.595299
$761$	−18.5261	−0.671569	−0.335785	−	0.941939i	$-0.609001\pi$
$761$	−18.5261	−0.671569	−0.335785	+	0.941939i	$0.609001\pi$
$762$	0	0
$763$	5.14227	0.186163
$764$	11.9265	0.431486
$765$	0	0
$766$	−12.6455	−0.456901
$767$	28.0396	1.01245
$768$	0	0
$769$	9.05426	0.326505	0.163253	−	0.986584i	$-0.447801\pi$
$769$	9.05426	0.326505	0.163253	+	0.986584i	$0.447801\pi$
$770$	−5.18112	−0.186715
$771$	0	0
$772$	−11.2228	−0.403919
$773$	41.8973	1.50694	0.753470	−	0.657482i	$-0.228378\pi$
$773$	41.8973	1.50694	0.753470	+	0.657482i	$0.228378\pi$
$774$	0	0
$775$	−1.73701	−0.0623953
$776$	−0.702134	−0.0252051
$777$	0	0
$778$	1.54826	0.0555079
$779$	1.41072	0.0505443
$780$	0	0
$781$	−14.6965	−0.525882
$782$	10.3132	0.368798
$783$	0	0
$784$	−6.72849	−0.240303
$785$	−12.6548	−0.451670
$786$	0	0
$787$	13.5823	0.484157	0.242079	−	0.970257i	$-0.422171\pi$
$787$	13.5823	0.484157	0.242079	+	0.970257i	$0.422171\pi$
$788$	0.571373	0.0203543
$789$	0	0
$790$	34.0423	1.21117
$791$	−4.22763	−0.150317
$792$	0	0
$793$	−18.7424	−0.665564
$794$	−36.3925	−1.29152
$795$	0	0
$796$	−14.4380	−0.511742
$797$	−15.3911	−0.545179	−0.272590	−	0.962130i	$-0.587880\pi$
$797$	−15.3911	−0.545179	−0.272590	+	0.962130i	$0.587880\pi$
$798$	0	0
$799$	10.8474	0.383752
$800$	−0.852348	−0.0301351
$801$	0	0
$802$	−3.19605	−0.112856
$803$	−3.03438	−0.107081
$804$	0	0
$805$	−7.91233	−0.278873
$806$	−12.9293	−0.455415
$807$	0	0
$808$	6.49645	0.228544
$809$	−36.2060	−1.27294	−0.636468	−	0.771303i	$-0.719605\pi$
$809$	−36.2060	−1.27294	−0.636468	+	0.771303i	$0.719605\pi$
$810$	0	0
$811$	−34.9293	−1.22653	−0.613267	−	0.789875i	$-0.710145\pi$
$811$	−34.9293	−1.22653	−0.613267	+	0.789875i	$0.710145\pi$
$812$	0.236324	0.00829334
$813$	0	0
$814$	−30.3130	−1.06247
$815$	−13.8071	−0.483641
$816$	0	0
$817$	−80.7137	−2.82382
$818$	−18.1471	−0.634498
$819$	0	0
$820$	−0.503071	−0.0175680
$821$	16.5873	0.578901	0.289450	−	0.957193i	$-0.406528\pi$
$821$	16.5873	0.578901	0.289450	+	0.957193i	$0.406528\pi$
$822$	0	0
$823$	13.3263	0.464525	0.232263	−	0.972653i	$-0.425387\pi$
$823$	13.3263	0.464525	0.232263	+	0.972653i	$0.425387\pi$
$824$	−5.63092	−0.196162
$825$	0	0
$826$	2.30288	0.0801276
$827$	25.0104	0.869697	0.434849	−	0.900504i	$-0.356802\pi$
$827$	25.0104	0.869697	0.434849	+	0.900504i	$0.356802\pi$
$828$	0	0
$829$	4.99747	0.173569	0.0867847	−	0.996227i	$-0.472341\pi$
$829$	4.99747	0.173569	0.0867847	+	0.996227i	$0.472341\pi$
$830$	32.4733	1.12717
$831$	0	0
$832$	−6.34437	−0.219952
$833$	−11.0550	−0.383035
$834$	0	0
$835$	33.7197	1.16692
$836$	−27.8834	−0.964368
$837$	0	0
$838$	3.71552	0.128351
$839$	25.9337	0.895330	0.447665	−	0.894201i	$-0.352256\pi$
$839$	25.9337	0.895330	0.447665	+	0.894201i	$0.352256\pi$
$840$	0	0
$841$	−28.7943	−0.992907
$842$	−7.81057	−0.269170
$843$	0	0
$844$	1.26551	0.0435607
$845$	65.9247	2.26788
$846$	0	0
$847$	3.07124	0.105529
$848$	−8.35293	−0.286841
$849$	0	0
$850$	−1.40042	−0.0480342
$851$	−46.2924	−1.58688
$852$	0	0
$853$	40.9561	1.40231	0.701155	−	0.713009i	$-0.252668\pi$
$853$	40.9561	1.40231	0.701155	+	0.713009i	$0.252668\pi$
$854$	−1.53931	−0.0526743
$855$	0	0
$856$	12.9948	0.444155
$857$	6.73633	0.230109	0.115054	−	0.993359i	$-0.463296\pi$
$857$	6.73633	0.230109	0.115054	+	0.993359i	$0.463296\pi$
$858$	0	0
$859$	−27.9828	−0.954761	−0.477380	−	0.878697i	$-0.658414\pi$
$859$	−27.9828	−0.954761	−0.477380	+	0.878697i	$0.658414\pi$
$860$	28.7830	0.981491
$861$	0	0
$862$	−16.3875	−0.558159
$863$	8.17113	0.278149	0.139074	−	0.990282i	$-0.455587\pi$
$863$	8.17113	0.278149	0.139074	+	0.990282i	$0.455587\pi$
$864$	0	0
$865$	−45.0629	−1.53218
$866$	3.09999	0.105342
$867$	0	0
$868$	−1.06188	−0.0360426
$869$	−57.8392	−1.96206
$870$	0	0
$871$	−31.8669	−1.07977
$872$	−9.86881	−0.334200
$873$	0	0
$874$	−42.5821	−1.44036
$875$	−5.22826	−0.176747
$876$	0	0
$877$	−20.5345	−0.693400	−0.346700	−	0.937976i	$-0.612698\pi$
$877$	−20.5345	−0.693400	−0.346700	+	0.937976i	$0.612698\pi$
$878$	18.5942	0.627524
$879$	0	0
$880$	9.94338	0.335191
$881$	−15.4319	−0.519913	−0.259957	−	0.965620i	$-0.583708\pi$
$881$	−15.4319	−0.519913	−0.259957	+	0.965620i	$0.583708\pi$
$882$	0	0
$883$	33.4539	1.12581	0.562907	−	0.826520i	$-0.309683\pi$
$883$	33.4539	1.12581	0.562907	+	0.826520i	$0.309683\pi$
$884$	−10.4239	−0.350595
$885$	0	0
$886$	−23.6084	−0.793140
$887$	−6.24655	−0.209739	−0.104869	−	0.994486i	$-0.533442\pi$
$887$	−6.24655	−0.209739	−0.104869	+	0.994486i	$0.533442\pi$
$888$	0	0
$889$	−5.36922	−0.180078
$890$	11.4755	0.384658
$891$	0	0
$892$	6.56730	0.219889
$893$	−44.7877	−1.49876
$894$	0	0
$895$	−46.6770	−1.56024
$896$	−0.521062	−0.0174075
$897$	0	0
$898$	41.1909	1.37456
$899$	−0.924281	−0.0308265
$900$	0	0
$901$	−13.7240	−0.457213
$902$	0.854738	0.0284597
$903$	0	0
$904$	8.11349	0.269851
$905$	−13.2579	−0.440707
$906$	0	0
$907$	1.30100	0.0431989	0.0215995	−	0.999767i	$-0.493124\pi$
$907$	1.30100	0.0431989	0.0215995	+	0.999767i	$0.493124\pi$
$908$	−20.8482	−0.691872
$909$	0	0
$910$	7.99730	0.265108
$911$	−5.63570	−0.186719	−0.0933596	−	0.995632i	$-0.529761\pi$
$911$	−5.63570	−0.186719	−0.0933596	+	0.995632i	$0.529761\pi$
$912$	0	0
$913$	−55.1735	−1.82598
$914$	−3.92311	−0.129765
$915$	0	0
$916$	−21.5345	−0.711520
$917$	5.97362	0.197266
$918$	0	0
$919$	−22.4251	−0.739737	−0.369869	−	0.929084i	$-0.620597\pi$
$919$	−22.4251	−0.739737	−0.369869	+	0.929084i	$0.620597\pi$
$920$	15.1850	0.500635
$921$	0	0
$922$	3.42177	0.112690
$923$	22.6847	0.746678
$924$	0	0
$925$	6.28603	0.206684
$926$	−4.06287	−0.133514
$927$	0	0
$928$	−0.453542	−0.0148883
$929$	33.5383	1.10036	0.550178	−	0.835047i	$-0.314560\pi$
$929$	33.5383	1.10036	0.550178	+	0.835047i	$0.314560\pi$
$930$	0	0
$931$	45.6452	1.49596
$932$	−6.04955	−0.198160
$933$	0	0
$934$	27.2711	0.892336
$935$	16.3372	0.534282
$936$	0	0
$937$	6.95266	0.227134	0.113567	−	0.993530i	$-0.463772\pi$
$937$	6.95266	0.227134	0.113567	+	0.993530i	$0.463772\pi$
$938$	−2.61722	−0.0854553
$939$	0	0
$940$	15.9715	0.520934
$941$	52.7987	1.72119	0.860595	−	0.509290i	$-0.170092\pi$
$941$	52.7987	1.72119	0.860595	+	0.509290i	$0.170092\pi$
$942$	0	0
$943$	1.30531	0.0425068
$944$	−4.41959	−0.143846
$945$	0	0
$946$	−48.9034	−1.58999
$947$	−25.4137	−0.825835	−0.412918	−	0.910768i	$-0.635490\pi$
$947$	−25.4137	−0.825835	−0.412918	+	0.910768i	$0.635490\pi$
$948$	0	0
$949$	4.68371	0.152040
$950$	5.78221	0.187600
$951$	0	0
$952$	−0.856116	−0.0277469
$953$	−9.81877	−0.318061	−0.159031	−	0.987274i	$-0.550837\pi$
$953$	−9.81877	−0.318061	−0.159031	+	0.987274i	$0.550837\pi$
$954$	0	0
$955$	28.8522	0.933634
$956$	3.90903	0.126427
$957$	0	0
$958$	14.5587	0.470371
$959$	2.71580	0.0876979
$960$	0	0
$961$	−26.8469	−0.866029
$962$	46.7895	1.50855
$963$	0	0
$964$	−9.32692	−0.300400
$965$	−27.1499	−0.873985
$966$	0	0
$967$	31.8175	1.02318	0.511590	−	0.859229i	$-0.329057\pi$
$967$	31.8175	1.02318	0.511590	+	0.859229i	$0.329057\pi$
$968$	−5.89420	−0.189447
$969$	0	0
$970$	−1.69858	−0.0545380
$971$	37.4558	1.20201	0.601007	−	0.799244i	$-0.294766\pi$
$971$	37.4558	1.20201	0.601007	+	0.799244i	$0.294766\pi$
$972$	0	0
$973$	−1.23254	−0.0395135
$974$	−23.3034	−0.746690
$975$	0	0
$976$	2.95418	0.0945611
$977$	48.4355	1.54959	0.774795	−	0.632213i	$-0.217853\pi$
$977$	48.4355	1.54959	0.774795	+	0.632213i	$0.217853\pi$
$978$	0	0
$979$	−19.4973	−0.623135
$980$	−16.2773	−0.519960
$981$	0	0
$982$	−19.7732	−0.630989
$983$	7.25425	0.231375	0.115687	−	0.993286i	$-0.463093\pi$
$983$	7.25425	0.231375	0.115687	+	0.993286i	$0.463093\pi$
$984$	0	0
$985$	1.38224	0.0440419
$986$	−0.745179	−0.0237313
$987$	0	0
$988$	43.0393	1.36926
$989$	−74.6827	−2.37477
$990$	0	0
$991$	−45.4233	−1.44292	−0.721460	−	0.692456i	$-0.756529\pi$
$991$	−45.4233	−1.44292	−0.721460	+	0.692456i	$0.756529\pi$
$992$	2.03791	0.0647038
$993$	0	0
$994$	1.86310	0.0590938
$995$	−34.9279	−1.10729
$996$	0	0
$997$	−54.0896	−1.71304	−0.856518	−	0.516118i	$-0.827377\pi$
$997$	−54.0896	−1.71304	−0.856518	+	0.516118i	$0.827377\pi$
$998$	−22.2601	−0.704630
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.j.1.10	✓	12
3.2	odd	2		8046.2.a.o.1.3	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.j.1.10	✓	12	1.1	even	1	trivial
8046.2.a.o.1.3	yes	12	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.41916 q^{5} +0.521062 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.41916 q^{5} +0.521062 q^{7} -1.00000 q^{8} -2.41916 q^{10} +4.11026 q^{11} -6.34437 q^{13} -0.521062 q^{14} +1.00000 q^{16} +1.64302 q^{17} -6.78386 q^{19} +2.41916 q^{20} -4.11026 q^{22} -6.27696 q^{23} +0.852348 q^{25} +6.34437 q^{26} +0.521062 q^{28} +0.453542 q^{29} -2.03791 q^{31} -1.00000 q^{32} -1.64302 q^{34} +1.26053 q^{35} +7.37496 q^{37} +6.78386 q^{38} -2.41916 q^{40} -0.207953 q^{41} +11.8979 q^{43} +4.11026 q^{44} +6.27696 q^{46} +6.60210 q^{47} -6.72849 q^{49} -0.852348 q^{50} -6.34437 q^{52} -8.35293 q^{53} +9.94338 q^{55} -0.521062 q^{56} -0.453542 q^{58} -4.41959 q^{59} +2.95418 q^{61} +2.03791 q^{62} +1.00000 q^{64} -15.3481 q^{65} +5.02285 q^{67} +1.64302 q^{68} -1.26053 q^{70} -3.57557 q^{71} -0.738246 q^{73} -7.37496 q^{74} -6.78386 q^{76} +2.14170 q^{77} -14.0719 q^{79} +2.41916 q^{80} +0.207953 q^{82} -13.4234 q^{83} +3.97473 q^{85} -11.8979 q^{86} -4.11026 q^{88} -4.74356 q^{89} -3.30581 q^{91} -6.27696 q^{92} -6.60210 q^{94} -16.4113 q^{95} +0.702134 q^{97} +6.72849 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^4 - 3 * q^5 + 6 * q^7 - 12 * q^8	\( 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8} + 3 q^{10} - 10 q^{11} + 5 q^{13} - 6 q^{14} + 12 q^{16} - 8 q^{17} + 2 q^{19} - 3 q^{20} + 10 q^{22} - 9 q^{23} + 7 q^{25} - 5 q^{26} + 6 q^{28} - 19 q^{29} + 10 q^{31} - 12 q^{32} + 8 q^{34} - 20 q^{35} + 11 q^{37} - 2 q^{38} + 3 q^{40} - 8 q^{41} + 13 q^{43} - 10 q^{44} + 9 q^{46} - 11 q^{47} + 2 q^{49} - 7 q^{50} + 5 q^{52} - 24 q^{53} + 3 q^{55} - 6 q^{56} + 19 q^{58} - 10 q^{59} - 10 q^{62} + 12 q^{64} - 28 q^{65} + 21 q^{67} - 8 q^{68} + 20 q^{70} - 37 q^{71} - 2 q^{73} - 11 q^{74} + 2 q^{76} - 2 q^{77} + 7 q^{79} - 3 q^{80} + 8 q^{82} - 22 q^{83} + 15 q^{85} - 13 q^{86} + 10 q^{88} - 40 q^{89} + q^{91} - 9 q^{92} + 11 q^{94} - 11 q^{95} + 7 q^{97} - 2 q^{98}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^4 - 3 * q^5 + 6 * q^7 - 12 * q^8 + 3 * q^10 - 10 * q^11 + 5 * q^13 - 6 * q^14 + 12 * q^16 - 8 * q^17 + 2 * q^19 - 3 * q^20 + 10 * q^22 - 9 * q^23 + 7 * q^25 - 5 * q^26 + 6 * q^28 - 19 * q^29 + 10 * q^31 - 12 * q^32 + 8 * q^34 - 20 * q^35 + 11 * q^37 - 2 * q^38 + 3 * q^40 - 8 * q^41 + 13 * q^43 - 10 * q^44 + 9 * q^46 - 11 * q^47 + 2 * q^49 - 7 * q^50 + 5 * q^52 - 24 * q^53 + 3 * q^55 - 6 * q^56 + 19 * q^58 - 10 * q^59 - 10 * q^62 + 12 * q^64 - 28 * q^65 + 21 * q^67 - 8 * q^68 + 20 * q^70 - 37 * q^71 - 2 * q^73 - 11 * q^74 + 2 * q^76 - 2 * q^77 + 7 * q^79 - 3 * q^80 + 8 * q^82 - 22 * q^83 + 15 * q^85 - 13 * q^86 + 10 * q^88 - 40 * q^89 + q^91 - 9 * q^92 + 11 * q^94 - 11 * q^95 + 7 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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