LMFDB - Embedded newform 8046.2.a.j.1.3

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			1.3
Root			$2.15566$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.15566 q^{5} -3.96511 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -2.15566 q^{5} -3.96511 q^{7} -1.00000 q^{8} +2.15566 q^{10} +1.12390 q^{11} +3.24395 q^{13} +3.96511 q^{14} +1.00000 q^{16} +0.0965560 q^{17} -2.93839 q^{19} -2.15566 q^{20} -1.12390 q^{22} -6.32576 q^{23} -0.353119 q^{25} -3.24395 q^{26} -3.96511 q^{28} +3.71653 q^{29} +7.97900 q^{31} -1.00000 q^{32} -0.0965560 q^{34} +8.54743 q^{35} -1.52602 q^{37} +2.93839 q^{38} +2.15566 q^{40} -4.49469 q^{41} +7.35768 q^{43} +1.12390 q^{44} +6.32576 q^{46} +0.857841 q^{47} +8.72206 q^{49} +0.353119 q^{50} +3.24395 q^{52} +6.36279 q^{53} -2.42275 q^{55} +3.96511 q^{56} -3.71653 q^{58} +1.21347 q^{59} -3.66519 q^{61} -7.97900 q^{62} +1.00000 q^{64} -6.99286 q^{65} +9.51851 q^{67} +0.0965560 q^{68} -8.54743 q^{70} -8.51164 q^{71} +4.49546 q^{73} +1.52602 q^{74} -2.93839 q^{76} -4.45639 q^{77} +5.19969 q^{79} -2.15566 q^{80} +4.49469 q^{82} -2.22728 q^{83} -0.208142 q^{85} -7.35768 q^{86} -1.12390 q^{88} -2.10826 q^{89} -12.8626 q^{91} -6.32576 q^{92} -0.857841 q^{94} +6.33417 q^{95} +1.31013 q^{97} -8.72206 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.15566	−0.964042	−0.482021	−	0.876160i	$-0.660097\pi$
$5$	−2.15566	−0.964042	−0.482021	+	0.876160i	$0.660097\pi$
$6$	0	0
$7$	−3.96511	−1.49867	−0.749334	−	0.662192i	$-0.769626\pi$
$7$	−3.96511	−1.49867	−0.749334	+	0.662192i	$0.769626\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.15566	0.681680
$11$	1.12390	0.338869	0.169435	−	0.985541i	$-0.445806\pi$
$11$	1.12390	0.338869	0.169435	+	0.985541i	$0.445806\pi$
$12$	0	0
$13$	3.24395	0.899710	0.449855	−	0.893102i	$-0.351476\pi$
$13$	3.24395	0.899710	0.449855	+	0.893102i	$0.351476\pi$
$14$	3.96511	1.05972
$15$	0	0
$16$	1.00000	0.250000
$17$	0.0965560	0.0234183	0.0117091	−	0.999931i	$-0.496273\pi$
$17$	0.0965560	0.0234183	0.0117091	+	0.999931i	$0.496273\pi$
$18$	0	0
$19$	−2.93839	−0.674113	−0.337056	−	0.941485i	$-0.609431\pi$
$19$	−2.93839	−0.674113	−0.337056	+	0.941485i	$0.609431\pi$
$20$	−2.15566	−0.482021
$21$	0	0
$22$	−1.12390	−0.239617
$23$	−6.32576	−1.31901	−0.659506	−	0.751699i	$-0.729235\pi$
$23$	−6.32576	−1.31901	−0.659506	+	0.751699i	$0.729235\pi$
$24$	0	0
$25$	−0.353119	−0.0706237
$26$	−3.24395	−0.636191
$27$	0	0
$28$	−3.96511	−0.749334
$29$	3.71653	0.690142	0.345071	−	0.938577i	$-0.387855\pi$
$29$	3.71653	0.690142	0.345071	+	0.938577i	$0.387855\pi$
$30$	0	0
$31$	7.97900	1.43307	0.716536	−	0.697550i	$-0.245727\pi$
$31$	7.97900	1.43307	0.716536	+	0.697550i	$0.245727\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.0965560	−0.0165592
$35$	8.54743	1.44478
$36$	0	0
$37$	−1.52602	−0.250876	−0.125438	−	0.992101i	$-0.540034\pi$
$37$	−1.52602	−0.250876	−0.125438	+	0.992101i	$0.540034\pi$
$38$	2.93839	0.476670
$39$	0	0
$40$	2.15566	0.340840
$41$	−4.49469	−0.701952	−0.350976	−	0.936384i	$-0.614150\pi$
$41$	−4.49469	−0.701952	−0.350976	+	0.936384i	$0.614150\pi$
$42$	0	0
$43$	7.35768	1.12204	0.561018	−	0.827804i	$-0.310410\pi$
$43$	7.35768	1.12204	0.561018	+	0.827804i	$0.310410\pi$
$44$	1.12390	0.169435
$45$	0	0
$46$	6.32576	0.932683
$47$	0.857841	0.125129	0.0625645	−	0.998041i	$-0.480072\pi$
$47$	0.857841	0.125129	0.0625645	+	0.998041i	$0.480072\pi$
$48$	0	0
$49$	8.72206	1.24601
$50$	0.353119	0.0499385
$51$	0	0
$52$	3.24395	0.449855
$53$	6.36279	0.873997	0.436998	−	0.899462i	$-0.356042\pi$
$53$	6.36279	0.873997	0.436998	+	0.899462i	$0.356042\pi$
$54$	0	0
$55$	−2.42275	−0.326684
$56$	3.96511	0.529859
$57$	0	0
$58$	−3.71653	−0.488004
$59$	1.21347	0.157981	0.0789905	−	0.996875i	$-0.474830\pi$
$59$	1.21347	0.157981	0.0789905	+	0.996875i	$0.474830\pi$
$60$	0	0
$61$	−3.66519	−0.469279	−0.234640	−	0.972082i	$-0.575391\pi$
$61$	−3.66519	−0.469279	−0.234640	+	0.972082i	$0.575391\pi$
$62$	−7.97900	−1.01333
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.99286	−0.867358
$66$	0	0
$67$	9.51851	1.16287	0.581436	−	0.813592i	$-0.302491\pi$
$67$	9.51851	1.16287	0.581436	+	0.813592i	$0.302491\pi$
$68$	0.0965560	0.0117091
$69$	0	0
$70$	−8.54743	−1.02161
$71$	−8.51164	−1.01015	−0.505073	−	0.863077i	$-0.668534\pi$
$71$	−8.51164	−1.01015	−0.505073	+	0.863077i	$0.668534\pi$
$72$	0	0
$73$	4.49546	0.526154	0.263077	−	0.964775i	$-0.415263\pi$
$73$	4.49546	0.526154	0.263077	+	0.964775i	$0.415263\pi$
$74$	1.52602	0.177396
$75$	0	0
$76$	−2.93839	−0.337056
$77$	−4.45639	−0.507852
$78$	0	0
$79$	5.19969	0.585011	0.292505	−	0.956264i	$-0.405511\pi$
$79$	5.19969	0.585011	0.292505	+	0.956264i	$0.405511\pi$
$80$	−2.15566	−0.241010
$81$	0	0
$82$	4.49469	0.496355
$83$	−2.22728	−0.244475	−0.122238	−	0.992501i	$-0.539007\pi$
$83$	−2.22728	−0.244475	−0.122238	+	0.992501i	$0.539007\pi$
$84$	0	0
$85$	−0.208142	−0.0225762
$86$	−7.35768	−0.793399
$87$	0	0
$88$	−1.12390	−0.119808
$89$	−2.10826	−0.223475	−0.111738	−	0.993738i	$-0.535642\pi$
$89$	−2.10826	−0.223475	−0.111738	+	0.993738i	$0.535642\pi$
$90$	0	0
$91$	−12.8626	−1.34837
$92$	−6.32576	−0.659506
$93$	0	0
$94$	−0.857841	−0.0884795
$95$	6.33417	0.649873
$96$	0	0
$97$	1.31013	0.133024	0.0665119	−	0.997786i	$-0.478813\pi$
$97$	1.31013	0.133024	0.0665119	+	0.997786i	$0.478813\pi$
$98$	−8.72206	−0.881061
$99$	0	0
$100$	−0.353119	−0.0353119
$101$	−2.81738	−0.280340	−0.140170	−	0.990127i	$-0.544765\pi$
$101$	−2.81738	−0.280340	−0.140170	+	0.990127i	$0.544765\pi$
$102$	0	0
$103$	14.7823	1.45654	0.728270	−	0.685291i	$-0.240325\pi$
$103$	14.7823	1.45654	0.728270	+	0.685291i	$0.240325\pi$
$104$	−3.24395	−0.318095
$105$	0	0
$106$	−6.36279	−0.618009
$107$	−9.77733	−0.945210	−0.472605	−	0.881274i	$-0.656686\pi$
$107$	−9.77733	−0.945210	−0.472605	+	0.881274i	$0.656686\pi$
$108$	0	0
$109$	−7.16298	−0.686089	−0.343044	−	0.939319i	$-0.611458\pi$
$109$	−7.16298	−0.686089	−0.343044	+	0.939319i	$0.611458\pi$
$110$	2.42275	0.231000
$111$	0	0
$112$	−3.96511	−0.374667
$113$	16.4629	1.54870	0.774350	−	0.632757i	$-0.218077\pi$
$113$	16.4629	1.54870	0.774350	+	0.632757i	$0.218077\pi$
$114$	0	0
$115$	13.6362	1.27158
$116$	3.71653	0.345071
$117$	0	0
$118$	−1.21347	−0.111709
$119$	−0.382854	−0.0350962
$120$	0	0
$121$	−9.73685	−0.885168
$122$	3.66519	0.331831
$123$	0	0
$124$	7.97900	0.716536
$125$	11.5395	1.03213
$126$	0	0
$127$	−4.84668	−0.430073	−0.215036	−	0.976606i	$-0.568987\pi$
$127$	−4.84668	−0.430073	−0.215036	+	0.976606i	$0.568987\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	6.99286	0.613315
$131$	−14.9599	−1.30705	−0.653526	−	0.756904i	$-0.726711\pi$
$131$	−14.9599	−1.30705	−0.653526	+	0.756904i	$0.726711\pi$
$132$	0	0
$133$	11.6510	1.01027
$134$	−9.51851	−0.822274
$135$	0	0
$136$	−0.0965560	−0.00827960
$137$	−5.99201	−0.511932	−0.255966	−	0.966686i	$-0.582394\pi$
$137$	−5.99201	−0.511932	−0.255966	+	0.966686i	$0.582394\pi$
$138$	0	0
$139$	15.3257	1.29991	0.649955	−	0.759972i	$-0.274788\pi$
$139$	15.3257	1.29991	0.649955	+	0.759972i	$0.274788\pi$
$140$	8.54743	0.722390
$141$	0	0
$142$	8.51164	0.714281
$143$	3.64588	0.304884
$144$	0	0
$145$	−8.01158	−0.665326
$146$	−4.49546	−0.372047
$147$	0	0
$148$	−1.52602	−0.125438
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	13.2736	1.08019	0.540094	−	0.841605i	$-0.318389\pi$
$151$	13.2736	1.08019	0.540094	+	0.841605i	$0.318389\pi$
$152$	2.93839	0.238335
$153$	0	0
$154$	4.45639	0.359106
$155$	−17.2000	−1.38154
$156$	0	0
$157$	15.5100	1.23783	0.618917	−	0.785456i	$-0.287572\pi$
$157$	15.5100	1.23783	0.618917	+	0.785456i	$0.287572\pi$
$158$	−5.19969	−0.413665
$159$	0	0
$160$	2.15566	0.170420
$161$	25.0823	1.97676
$162$	0	0
$163$	10.0464	0.786897	0.393449	−	0.919347i	$-0.371282\pi$
$163$	10.0464	0.786897	0.393449	+	0.919347i	$0.371282\pi$
$164$	−4.49469	−0.350976
$165$	0	0
$166$	2.22728	0.172870
$167$	5.45389	0.422034	0.211017	−	0.977482i	$-0.432322\pi$
$167$	5.45389	0.422034	0.211017	+	0.977482i	$0.432322\pi$
$168$	0	0
$169$	−2.47679	−0.190522
$170$	0.208142	0.0159638
$171$	0	0
$172$	7.35768	0.561018
$173$	−18.8846	−1.43577	−0.717885	−	0.696162i	$-0.754890\pi$
$173$	−18.8846	−1.43577	−0.717885	+	0.696162i	$0.754890\pi$
$174$	0	0
$175$	1.40015	0.105842
$176$	1.12390	0.0847173
$177$	0	0
$178$	2.10826	0.158021
$179$	−1.75154	−0.130916	−0.0654579	−	0.997855i	$-0.520851\pi$
$179$	−1.75154	−0.130916	−0.0654579	+	0.997855i	$0.520851\pi$
$180$	0	0
$181$	−21.2991	−1.58315	−0.791576	−	0.611070i	$-0.790739\pi$
$181$	−21.2991	−1.58315	−0.791576	+	0.611070i	$0.790739\pi$
$182$	12.8626	0.953440
$183$	0	0
$184$	6.32576	0.466341
$185$	3.28958	0.241855
$186$	0	0
$187$	0.108519	0.00793572
$188$	0.857841	0.0625645
$189$	0	0
$190$	−6.33417	−0.459529
$191$	14.5648	1.05387	0.526937	−	0.849904i	$-0.323340\pi$
$191$	14.5648	1.05387	0.526937	+	0.849904i	$0.323340\pi$
$192$	0	0
$193$	12.5204	0.901238	0.450619	−	0.892716i	$-0.351203\pi$
$193$	12.5204	0.901238	0.450619	+	0.892716i	$0.351203\pi$
$194$	−1.31013	−0.0940620
$195$	0	0
$196$	8.72206	0.623004
$197$	−12.2475	−0.872598	−0.436299	−	0.899802i	$-0.643711\pi$
$197$	−12.2475	−0.872598	−0.436299	+	0.899802i	$0.643711\pi$
$198$	0	0
$199$	−1.23542	−0.0875766	−0.0437883	−	0.999041i	$-0.513943\pi$
$199$	−1.23542	−0.0875766	−0.0437883	+	0.999041i	$0.513943\pi$
$200$	0.353119	0.0249693
$201$	0	0
$202$	2.81738	0.198230
$203$	−14.7364	−1.03429
$204$	0	0
$205$	9.68902	0.676711
$206$	−14.7823	−1.02993
$207$	0	0
$208$	3.24395	0.224927
$209$	−3.30246	−0.228436
$210$	0	0
$211$	−4.32672	−0.297864	−0.148932	−	0.988847i	$-0.547584\pi$
$211$	−4.32672	−0.297864	−0.148932	+	0.988847i	$0.547584\pi$
$212$	6.36279	0.436998
$213$	0	0
$214$	9.77733	0.668365
$215$	−15.8607	−1.08169
$216$	0	0
$217$	−31.6376	−2.14770
$218$	7.16298	0.485138
$219$	0	0
$220$	−2.42275	−0.163342
$221$	0.313223	0.0210696
$222$	0	0
$223$	12.7666	0.854913	0.427457	−	0.904036i	$-0.359410\pi$
$223$	12.7666	0.854913	0.427457	+	0.904036i	$0.359410\pi$
$224$	3.96511	0.264930
$225$	0	0
$226$	−16.4629	−1.09510
$227$	22.7865	1.51239	0.756197	−	0.654344i	$-0.227055\pi$
$227$	22.7865	1.51239	0.756197	+	0.654344i	$0.227055\pi$
$228$	0	0
$229$	10.6307	0.702494	0.351247	−	0.936283i	$-0.385758\pi$
$229$	10.6307	0.702494	0.351247	+	0.936283i	$0.385758\pi$
$230$	−13.6362	−0.899145
$231$	0	0
$232$	−3.71653	−0.244002
$233$	−25.2615	−1.65493	−0.827467	−	0.561515i	$-0.810219\pi$
$233$	−25.2615	−1.65493	−0.827467	+	0.561515i	$0.810219\pi$
$234$	0	0
$235$	−1.84922	−0.120630
$236$	1.21347	0.0789905
$237$	0	0
$238$	0.382854	0.0248168
$239$	4.22269	0.273143	0.136571	−	0.990630i	$-0.456392\pi$
$239$	4.22269	0.273143	0.136571	+	0.990630i	$0.456392\pi$
$240$	0	0
$241$	−11.9213	−0.767916	−0.383958	−	0.923351i	$-0.625439\pi$
$241$	−11.9213	−0.767916	−0.383958	+	0.923351i	$0.625439\pi$
$242$	9.73685	0.625908
$243$	0	0
$244$	−3.66519	−0.234640
$245$	−18.8018	−1.20120
$246$	0	0
$247$	−9.53199	−0.606506
$248$	−7.97900	−0.506667
$249$	0	0
$250$	−11.5395	−0.729823
$251$	−17.8818	−1.12869	−0.564345	−	0.825539i	$-0.690871\pi$
$251$	−17.8818	−1.12869	−0.564345	+	0.825539i	$0.690871\pi$
$252$	0	0
$253$	−7.10953	−0.446972
$254$	4.84668	0.304107
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.387919	−0.0241977	−0.0120989	−	0.999927i	$-0.503851\pi$
$257$	−0.387919	−0.0241977	−0.0120989	+	0.999927i	$0.503851\pi$
$258$	0	0
$259$	6.05083	0.375980
$260$	−6.99286	−0.433679
$261$	0	0
$262$	14.9599	0.924226
$263$	4.34002	0.267617	0.133809	−	0.991007i	$-0.457279\pi$
$263$	4.34002	0.267617	0.133809	+	0.991007i	$0.457279\pi$
$264$	0	0
$265$	−13.7160	−0.842569
$266$	−11.6510	−0.714370
$267$	0	0
$268$	9.51851	0.581436
$269$	−28.7658	−1.75388	−0.876941	−	0.480599i	$-0.840419\pi$
$269$	−28.7658	−1.75388	−0.876941	+	0.480599i	$0.840419\pi$
$270$	0	0
$271$	19.2160	1.16729	0.583644	−	0.812010i	$-0.301627\pi$
$271$	19.2160	1.16729	0.583644	+	0.812010i	$0.301627\pi$
$272$	0.0965560	0.00585456
$273$	0	0
$274$	5.99201	0.361991
$275$	−0.396871	−0.0239322
$276$	0	0
$277$	13.5206	0.812376	0.406188	−	0.913789i	$-0.366858\pi$
$277$	13.5206	0.812376	0.406188	+	0.913789i	$0.366858\pi$
$278$	−15.3257	−0.919176
$279$	0	0
$280$	−8.54743	−0.510807
$281$	−23.6498	−1.41083	−0.705415	−	0.708794i	$-0.749240\pi$
$281$	−23.6498	−1.41083	−0.705415	+	0.708794i	$0.749240\pi$
$282$	0	0
$283$	−27.5799	−1.63945	−0.819726	−	0.572756i	$-0.805874\pi$
$283$	−27.5799	−1.63945	−0.819726	+	0.572756i	$0.805874\pi$
$284$	−8.51164	−0.505073
$285$	0	0
$286$	−3.64588	−0.215585
$287$	17.8219	1.05199
$288$	0	0
$289$	−16.9907	−0.999452
$290$	8.01158	0.470456
$291$	0	0
$292$	4.49546	0.263077
$293$	−20.8470	−1.21789	−0.608946	−	0.793211i	$-0.708408\pi$
$293$	−20.8470	−1.21789	−0.608946	+	0.793211i	$0.708408\pi$
$294$	0	0
$295$	−2.61584	−0.152300
$296$	1.52602	0.0886981
$297$	0	0
$298$	1.00000	0.0579284
$299$	−20.5205	−1.18673
$300$	0	0
$301$	−29.1740	−1.68156
$302$	−13.2736	−0.763809
$303$	0	0
$304$	−2.93839	−0.168528
$305$	7.90091	0.452405
$306$	0	0
$307$	−1.77284	−0.101181	−0.0505906	−	0.998719i	$-0.516110\pi$
$307$	−1.77284	−0.101181	−0.0505906	+	0.998719i	$0.516110\pi$
$308$	−4.45639	−0.253926
$309$	0	0
$310$	17.2000	0.976897
$311$	−15.0758	−0.854867	−0.427434	−	0.904047i	$-0.640582\pi$
$311$	−15.0758	−0.854867	−0.427434	+	0.904047i	$0.640582\pi$
$312$	0	0
$313$	−7.21243	−0.407670	−0.203835	−	0.979005i	$-0.565341\pi$
$313$	−7.21243	−0.407670	−0.203835	+	0.979005i	$0.565341\pi$
$314$	−15.5100	−0.875281
$315$	0	0
$316$	5.19969	0.292505
$317$	21.3367	1.19839	0.599193	−	0.800604i	$-0.295488\pi$
$317$	21.3367	1.19839	0.599193	+	0.800604i	$0.295488\pi$
$318$	0	0
$319$	4.17701	0.233868
$320$	−2.15566	−0.120505
$321$	0	0
$322$	−25.0823	−1.39778
$323$	−0.283719	−0.0157865
$324$	0	0
$325$	−1.14550	−0.0635409
$326$	−10.0464	−0.556420
$327$	0	0
$328$	4.49469	0.248177
$329$	−3.40143	−0.187527
$330$	0	0
$331$	−6.74660	−0.370827	−0.185413	−	0.982661i	$-0.559362\pi$
$331$	−6.74660	−0.370827	−0.185413	+	0.982661i	$0.559362\pi$
$332$	−2.22728	−0.122238
$333$	0	0
$334$	−5.45389	−0.298423
$335$	−20.5187	−1.12106
$336$	0	0
$337$	−31.3557	−1.70806	−0.854028	−	0.520227i	$-0.825847\pi$
$337$	−31.3557	−1.70806	−0.854028	+	0.520227i	$0.825847\pi$
$338$	2.47679	0.134719
$339$	0	0
$340$	−0.208142	−0.0112881
$341$	8.96761	0.485623
$342$	0	0
$343$	−6.82814	−0.368685
$344$	−7.35768	−0.396699
$345$	0	0
$346$	18.8846	1.01524
$347$	−11.2623	−0.604591	−0.302296	−	0.953214i	$-0.597753\pi$
$347$	−11.2623	−0.604591	−0.302296	+	0.953214i	$0.597753\pi$
$348$	0	0
$349$	−28.9957	−1.55210	−0.776051	−	0.630670i	$-0.782780\pi$
$349$	−28.9957	−1.55210	−0.776051	+	0.630670i	$0.782780\pi$
$350$	−1.40015	−0.0748413
$351$	0	0
$352$	−1.12390	−0.0599041
$353$	28.4759	1.51562	0.757808	−	0.652477i	$-0.226270\pi$
$353$	28.4759	1.51562	0.757808	+	0.652477i	$0.226270\pi$
$354$	0	0
$355$	18.3482	0.973823
$356$	−2.10826	−0.111738
$357$	0	0
$358$	1.75154	0.0925715
$359$	−6.75847	−0.356698	−0.178349	−	0.983967i	$-0.557076\pi$
$359$	−6.75847	−0.356698	−0.178349	+	0.983967i	$0.557076\pi$
$360$	0	0
$361$	−10.3659	−0.545572
$362$	21.2991	1.11946
$363$	0	0
$364$	−12.8626	−0.674184
$365$	−9.69070	−0.507234
$366$	0	0
$367$	−24.7374	−1.29128	−0.645642	−	0.763640i	$-0.723410\pi$
$367$	−24.7374	−1.29128	−0.645642	+	0.763640i	$0.723410\pi$
$368$	−6.32576	−0.329753
$369$	0	0
$370$	−3.28958	−0.171017
$371$	−25.2291	−1.30983
$372$	0	0
$373$	9.52425	0.493147	0.246573	−	0.969124i	$-0.420695\pi$
$373$	9.52425	0.493147	0.246573	+	0.969124i	$0.420695\pi$
$374$	−0.108519	−0.00561140
$375$	0	0
$376$	−0.857841	−0.0442398
$377$	12.0562	0.620928
$378$	0	0
$379$	−6.76673	−0.347583	−0.173792	−	0.984782i	$-0.555602\pi$
$379$	−6.76673	−0.347583	−0.173792	+	0.984782i	$0.555602\pi$
$380$	6.33417	0.324936
$381$	0	0
$382$	−14.5648	−0.745202
$383$	19.4512	0.993910	0.496955	−	0.867776i	$-0.334451\pi$
$383$	19.4512	0.993910	0.496955	+	0.867776i	$0.334451\pi$
$384$	0	0
$385$	9.60647	0.489591
$386$	−12.5204	−0.637271
$387$	0	0
$388$	1.31013	0.0665119
$389$	11.3920	0.577596	0.288798	−	0.957390i	$-0.406744\pi$
$389$	11.3920	0.577596	0.288798	+	0.957390i	$0.406744\pi$
$390$	0	0
$391$	−0.610790	−0.0308890
$392$	−8.72206	−0.440530
$393$	0	0
$394$	12.2475	0.617020
$395$	−11.2088	−0.563975
$396$	0	0
$397$	−21.0348	−1.05571	−0.527853	−	0.849336i	$-0.677003\pi$
$397$	−21.0348	−1.05571	−0.527853	+	0.849336i	$0.677003\pi$
$398$	1.23542	0.0619260
$399$	0	0
$400$	−0.353119	−0.0176559
$401$	1.42155	0.0709886	0.0354943	−	0.999370i	$-0.488699\pi$
$401$	1.42155	0.0709886	0.0354943	+	0.999370i	$0.488699\pi$
$402$	0	0
$403$	25.8835	1.28935
$404$	−2.81738	−0.140170
$405$	0	0
$406$	14.7364	0.731357
$407$	−1.71510	−0.0850141
$408$	0	0
$409$	−20.5108	−1.01419	−0.507096	−	0.861890i	$-0.669281\pi$
$409$	−20.5108	−1.01419	−0.507096	+	0.861890i	$0.669281\pi$
$410$	−9.68902	−0.478507
$411$	0	0
$412$	14.7823	0.728270
$413$	−4.81155	−0.236761
$414$	0	0
$415$	4.80126	0.235685
$416$	−3.24395	−0.159048
$417$	0	0
$418$	3.30246	0.161529
$419$	−22.5840	−1.10330	−0.551650	−	0.834075i	$-0.686002\pi$
$419$	−22.5840	−1.10330	−0.551650	+	0.834075i	$0.686002\pi$
$420$	0	0
$421$	−9.44910	−0.460521	−0.230260	−	0.973129i	$-0.573958\pi$
$421$	−9.44910	−0.460521	−0.230260	+	0.973129i	$0.573958\pi$
$422$	4.32672	0.210621
$423$	0	0
$424$	−6.36279	−0.309004
$425$	−0.0340957	−0.00165388
$426$	0	0
$427$	14.5329	0.703294
$428$	−9.77733	−0.472605
$429$	0	0
$430$	15.8607	0.764870
$431$	0.748219	0.0360404	0.0180202	−	0.999838i	$-0.494264\pi$
$431$	0.748219	0.0360404	0.0180202	+	0.999838i	$0.494264\pi$
$432$	0	0
$433$	7.18287	0.345187	0.172593	−	0.984993i	$-0.444785\pi$
$433$	7.18287	0.345187	0.172593	+	0.984993i	$0.444785\pi$
$434$	31.6376	1.51865
$435$	0	0
$436$	−7.16298	−0.343044
$437$	18.5875	0.889163
$438$	0	0
$439$	−3.06995	−0.146521	−0.0732604	−	0.997313i	$-0.523340\pi$
$439$	−3.06995	−0.146521	−0.0732604	+	0.997313i	$0.523340\pi$
$440$	2.42275	0.115500
$441$	0	0
$442$	−0.313223	−0.0148985
$443$	−11.1300	−0.528802	−0.264401	−	0.964413i	$-0.585174\pi$
$443$	−11.1300	−0.528802	−0.264401	+	0.964413i	$0.585174\pi$
$444$	0	0
$445$	4.54470	0.215440
$446$	−12.7666	−0.604515
$447$	0	0
$448$	−3.96511	−0.187334
$449$	−9.93204	−0.468722	−0.234361	−	0.972150i	$-0.575300\pi$
$449$	−9.93204	−0.468722	−0.234361	+	0.972150i	$0.575300\pi$
$450$	0	0
$451$	−5.05158	−0.237870
$452$	16.4629	0.774350
$453$	0	0
$454$	−22.7865	−1.06942
$455$	27.7274	1.29988
$456$	0	0
$457$	−14.3761	−0.672485	−0.336243	−	0.941775i	$-0.609156\pi$
$457$	−14.3761	−0.672485	−0.336243	+	0.941775i	$0.609156\pi$
$458$	−10.6307	−0.496738
$459$	0	0
$460$	13.6362	0.635791
$461$	3.63676	0.169381	0.0846905	−	0.996407i	$-0.473010\pi$
$461$	3.63676	0.169381	0.0846905	+	0.996407i	$0.473010\pi$
$462$	0	0
$463$	30.9033	1.43620	0.718098	−	0.695942i	$-0.245013\pi$
$463$	30.9033	1.43620	0.718098	+	0.695942i	$0.245013\pi$
$464$	3.71653	0.172536
$465$	0	0
$466$	25.2615	1.17021
$467$	−26.0997	−1.20775	−0.603875	−	0.797079i	$-0.706378\pi$
$467$	−26.0997	−1.20775	−0.603875	+	0.797079i	$0.706378\pi$
$468$	0	0
$469$	−37.7419	−1.74276
$470$	1.84922	0.0852980
$471$	0	0
$472$	−1.21347	−0.0558547
$473$	8.26930	0.380223
$474$	0	0
$475$	1.03760	0.0476083
$476$	−0.382854	−0.0175481
$477$	0	0
$478$	−4.22269	−0.193141
$479$	−0.185023	−0.00845390	−0.00422695	−	0.999991i	$-0.501345\pi$
$479$	−0.185023	−0.00845390	−0.00422695	+	0.999991i	$0.501345\pi$
$480$	0	0
$481$	−4.95033	−0.225716
$482$	11.9213	0.542999
$483$	0	0
$484$	−9.73685	−0.442584
$485$	−2.82420	−0.128240
$486$	0	0
$487$	12.9888	0.588581	0.294290	−	0.955716i	$-0.404917\pi$
$487$	12.9888	0.588581	0.294290	+	0.955716i	$0.404917\pi$
$488$	3.66519	0.165915
$489$	0	0
$490$	18.8018	0.849379
$491$	−10.9945	−0.496176	−0.248088	−	0.968738i	$-0.579802\pi$
$491$	−10.9945	−0.496176	−0.248088	+	0.968738i	$0.579802\pi$
$492$	0	0
$493$	0.358853	0.0161619
$494$	9.53199	0.428864
$495$	0	0
$496$	7.97900	0.358268
$497$	33.7496	1.51387
$498$	0	0
$499$	−22.7533	−1.01858	−0.509288	−	0.860596i	$-0.670091\pi$
$499$	−22.7533	−1.01858	−0.509288	+	0.860596i	$0.670091\pi$
$500$	11.5395	0.516063
$501$	0	0
$502$	17.8818	0.798104
$503$	14.6114	0.651489	0.325745	−	0.945458i	$-0.394385\pi$
$503$	14.6114	0.651489	0.325745	+	0.945458i	$0.394385\pi$
$504$	0	0
$505$	6.07333	0.270260
$506$	7.10953	0.316057
$507$	0	0
$508$	−4.84668	−0.215036
$509$	−1.36980	−0.0607152	−0.0303576	−	0.999539i	$-0.509665\pi$
$509$	−1.36980	−0.0607152	−0.0303576	+	0.999539i	$0.509665\pi$
$510$	0	0
$511$	−17.8250	−0.788530
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	0.387919	0.0171104
$515$	−31.8656	−1.40416
$516$	0	0
$517$	0.964129	0.0424023
$518$	−6.05083	−0.265858
$519$	0	0
$520$	6.99286	0.306657
$521$	−15.9176	−0.697361	−0.348681	−	0.937242i	$-0.613370\pi$
$521$	−15.9176	−0.697361	−0.348681	+	0.937242i	$0.613370\pi$
$522$	0	0
$523$	5.99278	0.262046	0.131023	−	0.991379i	$-0.458174\pi$
$523$	5.99278	0.262046	0.131023	+	0.991379i	$0.458174\pi$
$524$	−14.9599	−0.653526
$525$	0	0
$526$	−4.34002	−0.189234
$527$	0.770420	0.0335600
$528$	0	0
$529$	17.0153	0.739794
$530$	13.7160	0.595786
$531$	0	0
$532$	11.6510	0.505136
$533$	−14.5805	−0.631553
$534$	0	0
$535$	21.0766	0.911222
$536$	−9.51851	−0.411137
$537$	0	0
$538$	28.7658	1.24018
$539$	9.80273	0.422234
$540$	0	0
$541$	−8.03756	−0.345562	−0.172781	−	0.984960i	$-0.555275\pi$
$541$	−8.03756	−0.345562	−0.172781	+	0.984960i	$0.555275\pi$
$542$	−19.2160	−0.825397
$543$	0	0
$544$	−0.0965560	−0.00413980
$545$	15.4410	0.661418
$546$	0	0
$547$	7.19273	0.307539	0.153769	−	0.988107i	$-0.450859\pi$
$547$	7.19273	0.307539	0.153769	+	0.988107i	$0.450859\pi$
$548$	−5.99201	−0.255966
$549$	0	0
$550$	0.396871	0.0169226
$551$	−10.9206	−0.465233
$552$	0	0
$553$	−20.6173	−0.876737
$554$	−13.5206	−0.574437
$555$	0	0
$556$	15.3257	0.649955
$557$	−2.40926	−0.102084	−0.0510418	−	0.998697i	$-0.516254\pi$
$557$	−2.40926	−0.102084	−0.0510418	+	0.998697i	$0.516254\pi$
$558$	0	0
$559$	23.8679	1.00951
$560$	8.54743	0.361195
$561$	0	0
$562$	23.6498	0.997608
$563$	−3.49642	−0.147357	−0.0736783	−	0.997282i	$-0.523474\pi$
$563$	−3.49642	−0.147357	−0.0736783	+	0.997282i	$0.523474\pi$
$564$	0	0
$565$	−35.4885	−1.49301
$566$	27.5799	1.15927
$567$	0	0
$568$	8.51164	0.357141
$569$	−0.242490	−0.0101657	−0.00508285	−	0.999987i	$-0.501618\pi$
$569$	−0.242490	−0.0101657	−0.00508285	+	0.999987i	$0.501618\pi$
$570$	0	0
$571$	37.3859	1.56455	0.782275	−	0.622933i	$-0.214059\pi$
$571$	37.3859	1.56455	0.782275	+	0.622933i	$0.214059\pi$
$572$	3.64588	0.152442
$573$	0	0
$574$	−17.8219	−0.743872
$575$	2.23374	0.0931536
$576$	0	0
$577$	23.9303	0.996233	0.498116	−	0.867110i	$-0.334025\pi$
$577$	23.9303	0.996233	0.498116	+	0.867110i	$0.334025\pi$
$578$	16.9907	0.706719
$579$	0	0
$580$	−8.01158	−0.332663
$581$	8.83139	0.366388
$582$	0	0
$583$	7.15115	0.296170
$584$	−4.49546	−0.186023
$585$	0	0
$586$	20.8470	0.861180
$587$	14.4272	0.595474	0.297737	−	0.954648i	$-0.403768\pi$
$587$	14.4272	0.595474	0.297737	+	0.954648i	$0.403768\pi$
$588$	0	0
$589$	−23.4454	−0.966051
$590$	2.61584	0.107693
$591$	0	0
$592$	−1.52602	−0.0627190
$593$	−11.9471	−0.490607	−0.245303	−	0.969446i	$-0.578888\pi$
$593$	−11.9471	−0.490607	−0.245303	+	0.969446i	$0.578888\pi$
$594$	0	0
$595$	0.825305	0.0338342
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	20.5205	0.839144
$599$	23.7388	0.969943	0.484972	−	0.874530i	$-0.338830\pi$
$599$	23.7388	0.969943	0.484972	+	0.874530i	$0.338830\pi$
$600$	0	0
$601$	34.5580	1.40965	0.704825	−	0.709381i	$-0.251025\pi$
$601$	34.5580	1.40965	0.704825	+	0.709381i	$0.251025\pi$
$602$	29.1740	1.18904
$603$	0	0
$604$	13.2736	0.540094
$605$	20.9894	0.853339
$606$	0	0
$607$	−27.4660	−1.11481	−0.557406	−	0.830240i	$-0.688203\pi$
$607$	−27.4660	−1.11481	−0.557406	+	0.830240i	$0.688203\pi$
$608$	2.93839	0.119167
$609$	0	0
$610$	−7.90091	−0.319898
$611$	2.78279	0.112580
$612$	0	0
$613$	4.22930	0.170820	0.0854100	−	0.996346i	$-0.472780\pi$
$613$	4.22930	0.170820	0.0854100	+	0.996346i	$0.472780\pi$
$614$	1.77284	0.0715459
$615$	0	0
$616$	4.45639	0.179553
$617$	26.3829	1.06214	0.531068	−	0.847329i	$-0.321791\pi$
$617$	26.3829	1.06214	0.531068	+	0.847329i	$0.321791\pi$
$618$	0	0
$619$	2.92730	0.117658	0.0588291	−	0.998268i	$-0.481263\pi$
$619$	2.92730	0.117658	0.0588291	+	0.998268i	$0.481263\pi$
$620$	−17.2000	−0.690770
$621$	0	0
$622$	15.0758	0.604483
$623$	8.35949	0.334916
$624$	0	0
$625$	−23.1097	−0.924389
$626$	7.21243	0.288267
$627$	0	0
$628$	15.5100	0.618917
$629$	−0.147346	−0.00587508
$630$	0	0
$631$	−29.4077	−1.17070	−0.585351	−	0.810780i	$-0.699043\pi$
$631$	−29.4077	−1.17070	−0.585351	+	0.810780i	$0.699043\pi$
$632$	−5.19969	−0.206833
$633$	0	0
$634$	−21.3367	−0.847387
$635$	10.4478	0.414608
$636$	0	0
$637$	28.2939	1.12105
$638$	−4.17701	−0.165369
$639$	0	0
$640$	2.15566	0.0852100
$641$	−12.4149	−0.490359	−0.245180	−	0.969478i	$-0.578847\pi$
$641$	−12.4149	−0.490359	−0.245180	+	0.969478i	$0.578847\pi$
$642$	0	0
$643$	−34.5602	−1.36292	−0.681460	−	0.731855i	$-0.738655\pi$
$643$	−34.5602	−1.36292	−0.681460	+	0.731855i	$0.738655\pi$
$644$	25.0823	0.988381
$645$	0	0
$646$	0.283719	0.0111628
$647$	30.5414	1.20071	0.600353	−	0.799735i	$-0.295027\pi$
$647$	30.5414	1.20071	0.600353	+	0.799735i	$0.295027\pi$
$648$	0	0
$649$	1.36383	0.0535349
$650$	1.14550	0.0449302
$651$	0	0
$652$	10.0464	0.393449
$653$	21.3686	0.836219	0.418109	−	0.908397i	$-0.362693\pi$
$653$	21.3686	0.836219	0.418109	+	0.908397i	$0.362693\pi$
$654$	0	0
$655$	32.2485	1.26005
$656$	−4.49469	−0.175488
$657$	0	0
$658$	3.40143	0.132602
$659$	−14.2126	−0.553645	−0.276823	−	0.960921i	$-0.589281\pi$
$659$	−14.2126	−0.553645	−0.276823	+	0.960921i	$0.589281\pi$
$660$	0	0
$661$	−20.0789	−0.780980	−0.390490	−	0.920607i	$-0.627694\pi$
$661$	−20.0789	−0.780980	−0.390490	+	0.920607i	$0.627694\pi$
$662$	6.74660	0.262214
$663$	0	0
$664$	2.22728	0.0864351
$665$	−25.1157	−0.973944
$666$	0	0
$667$	−23.5099	−0.910306
$668$	5.45389	0.211017
$669$	0	0
$670$	20.5187	0.792707
$671$	−4.11931	−0.159024
$672$	0	0
$673$	−25.4010	−0.979138	−0.489569	−	0.871964i	$-0.662846\pi$
$673$	−25.4010	−0.979138	−0.489569	+	0.871964i	$0.662846\pi$
$674$	31.3557	1.20778
$675$	0	0
$676$	−2.47679	−0.0952611
$677$	31.5694	1.21331	0.606654	−	0.794966i	$-0.292511\pi$
$677$	31.5694	1.21331	0.606654	+	0.794966i	$0.292511\pi$
$678$	0	0
$679$	−5.19481	−0.199358
$680$	0.208142	0.00798188
$681$	0	0
$682$	−8.96761	−0.343388
$683$	−16.5155	−0.631949	−0.315975	−	0.948768i	$-0.602331\pi$
$683$	−16.5155	−0.631949	−0.315975	+	0.948768i	$0.602331\pi$
$684$	0	0
$685$	12.9168	0.493524
$686$	6.82814	0.260700
$687$	0	0
$688$	7.35768	0.280509
$689$	20.6406	0.786343
$690$	0	0
$691$	−4.08327	−0.155335	−0.0776675	−	0.996979i	$-0.524747\pi$
$691$	−4.08327	−0.155335	−0.0776675	+	0.996979i	$0.524747\pi$
$692$	−18.8846	−0.717885
$693$	0	0
$694$	11.2623	0.427511
$695$	−33.0371	−1.25317
$696$	0	0
$697$	−0.433989	−0.0164385
$698$	28.9957	1.09750
$699$	0	0
$700$	1.40015	0.0529208
$701$	16.7314	0.631935	0.315967	−	0.948770i	$-0.397671\pi$
$701$	16.7314	0.631935	0.315967	+	0.948770i	$0.397671\pi$
$702$	0	0
$703$	4.48404	0.169119
$704$	1.12390	0.0423586
$705$	0	0
$706$	−28.4759	−1.07170
$707$	11.1712	0.420137
$708$	0	0
$709$	−31.1120	−1.16844	−0.584218	−	0.811597i	$-0.698599\pi$
$709$	−31.1120	−1.16844	−0.584218	+	0.811597i	$0.698599\pi$
$710$	−18.3482	−0.688597
$711$	0	0
$712$	2.10826	0.0790105
$713$	−50.4733	−1.89024
$714$	0	0
$715$	−7.85929	−0.293921
$716$	−1.75154	−0.0654579
$717$	0	0
$718$	6.75847	0.252224
$719$	−21.1950	−0.790439	−0.395220	−	0.918587i	$-0.629332\pi$
$719$	−21.1950	−0.790439	−0.395220	+	0.918587i	$0.629332\pi$
$720$	0	0
$721$	−58.6132	−2.18287
$722$	10.3659	0.385778
$723$	0	0
$724$	−21.2991	−0.791576
$725$	−1.31238	−0.0487404
$726$	0	0
$727$	−41.3725	−1.53442	−0.767211	−	0.641394i	$-0.778356\pi$
$727$	−41.3725	−1.53442	−0.767211	+	0.641394i	$0.778356\pi$
$728$	12.8626	0.476720
$729$	0	0
$730$	9.69070	0.358669
$731$	0.710428	0.0262761
$732$	0	0
$733$	−9.61119	−0.354997	−0.177499	−	0.984121i	$-0.556801\pi$
$733$	−9.61119	−0.354997	−0.177499	+	0.984121i	$0.556801\pi$
$734$	24.7374	0.913076
$735$	0	0
$736$	6.32576	0.233171
$737$	10.6979	0.394061
$738$	0	0
$739$	−40.1643	−1.47747	−0.738733	−	0.673998i	$-0.764576\pi$
$739$	−40.1643	−1.47747	−0.738733	+	0.673998i	$0.764576\pi$
$740$	3.28958	0.120927
$741$	0	0
$742$	25.2291	0.926191
$743$	−23.2162	−0.851719	−0.425860	−	0.904789i	$-0.640028\pi$
$743$	−23.2162	−0.851719	−0.425860	+	0.904789i	$0.640028\pi$
$744$	0	0
$745$	2.15566	0.0789774
$746$	−9.52425	−0.348707
$747$	0	0
$748$	0.108519	0.00396786
$749$	38.7681	1.41656
$750$	0	0
$751$	−27.6393	−1.00857	−0.504286	−	0.863537i	$-0.668244\pi$
$751$	−27.6393	−1.00857	−0.504286	+	0.863537i	$0.668244\pi$
$752$	0.857841	0.0312822
$753$	0	0
$754$	−12.0562	−0.439062
$755$	−28.6134	−1.04135
$756$	0	0
$757$	−30.9761	−1.12585	−0.562923	−	0.826510i	$-0.690323\pi$
$757$	−30.9761	−1.12585	−0.562923	+	0.826510i	$0.690323\pi$
$758$	6.76673	0.245779
$759$	0	0
$760$	−6.33417	−0.229765
$761$	36.0461	1.30667	0.653336	−	0.757068i	$-0.273369\pi$
$761$	36.0461	1.30667	0.653336	+	0.757068i	$0.273369\pi$
$762$	0	0
$763$	28.4020	1.02822
$764$	14.5648	0.526937
$765$	0	0
$766$	−19.4512	−0.702800
$767$	3.93645	0.142137
$768$	0	0
$769$	53.8427	1.94162	0.970810	−	0.239852i	$-0.0770988\pi$
$769$	53.8427	1.94162	0.970810	+	0.239852i	$0.0770988\pi$
$770$	−9.60647	−0.346193
$771$	0	0
$772$	12.5204	0.450619
$773$	−33.1205	−1.19126	−0.595631	−	0.803259i	$-0.703098\pi$
$773$	−33.1205	−1.19126	−0.595631	+	0.803259i	$0.703098\pi$
$774$	0	0
$775$	−2.81753	−0.101209
$776$	−1.31013	−0.0470310
$777$	0	0
$778$	−11.3920	−0.408422
$779$	13.2071	0.473195
$780$	0	0
$781$	−9.56625	−0.342307
$782$	0.610790	0.0218418
$783$	0	0
$784$	8.72206	0.311502
$785$	−33.4344	−1.19332
$786$	0	0
$787$	−6.52820	−0.232705	−0.116353	−	0.993208i	$-0.537120\pi$
$787$	−6.52820	−0.232705	−0.116353	+	0.993208i	$0.537120\pi$
$788$	−12.2475	−0.436299
$789$	0	0
$790$	11.2088	0.398790
$791$	−65.2772	−2.32099
$792$	0	0
$793$	−11.8897	−0.422215
$794$	21.0348	0.746497
$795$	0	0
$796$	−1.23542	−0.0437883
$797$	5.21140	0.184597	0.0922987	−	0.995731i	$-0.470579\pi$
$797$	5.21140	0.184597	0.0922987	+	0.995731i	$0.470579\pi$
$798$	0	0
$799$	0.0828297	0.00293030
$800$	0.353119	0.0124846
$801$	0	0
$802$	−1.42155	−0.0501965
$803$	5.05245	0.178297
$804$	0	0
$805$	−54.0690	−1.90568
$806$	−25.8835	−0.911707
$807$	0	0
$808$	2.81738	0.0991152
$809$	−43.9427	−1.54494	−0.772471	−	0.635050i	$-0.780979\pi$
$809$	−43.9427	−1.54494	−0.772471	+	0.635050i	$0.780979\pi$
$810$	0	0
$811$	22.1938	0.779331	0.389666	−	0.920956i	$-0.372591\pi$
$811$	22.1938	0.779331	0.389666	+	0.920956i	$0.372591\pi$
$812$	−14.7364	−0.517147
$813$	0	0
$814$	1.71510	0.0601141
$815$	−21.6567	−0.758602
$816$	0	0
$817$	−21.6197	−0.756378
$818$	20.5108	0.717142
$819$	0	0
$820$	9.68902	0.338355
$821$	−17.1951	−0.600114	−0.300057	−	0.953921i	$-0.597006\pi$
$821$	−17.1951	−0.600114	−0.300057	+	0.953921i	$0.597006\pi$
$822$	0	0
$823$	−56.2378	−1.96033	−0.980163	−	0.198194i	$-0.936492\pi$
$823$	−56.2378	−1.96033	−0.980163	+	0.198194i	$0.936492\pi$
$824$	−14.7823	−0.514964
$825$	0	0
$826$	4.81155	0.167415
$827$	15.4087	0.535812	0.267906	−	0.963445i	$-0.413668\pi$
$827$	15.4087	0.535812	0.267906	+	0.963445i	$0.413668\pi$
$828$	0	0
$829$	34.0979	1.18427	0.592134	−	0.805839i	$-0.298286\pi$
$829$	34.0979	1.18427	0.592134	+	0.805839i	$0.298286\pi$
$830$	−4.80126	−0.166654
$831$	0	0
$832$	3.24395	0.112464
$833$	0.842167	0.0291793
$834$	0	0
$835$	−11.7567	−0.406859
$836$	−3.30246	−0.114218
$837$	0	0
$838$	22.5840	0.780151
$839$	39.6777	1.36983	0.684913	−	0.728625i	$-0.259840\pi$
$839$	39.6777	1.36983	0.684913	+	0.728625i	$0.259840\pi$
$840$	0	0
$841$	−15.1874	−0.523704
$842$	9.44910	0.325637
$843$	0	0
$844$	−4.32672	−0.148932
$845$	5.33912	0.183671
$846$	0	0
$847$	38.6076	1.32657
$848$	6.36279	0.218499
$849$	0	0
$850$	0.0340957	0.00116947
$851$	9.65323	0.330909
$852$	0	0
$853$	18.6846	0.639749	0.319874	−	0.947460i	$-0.396359\pi$
$853$	18.6846	0.639749	0.319874	+	0.947460i	$0.396359\pi$
$854$	−14.5329	−0.497304
$855$	0	0
$856$	9.77733	0.334182
$857$	−29.4136	−1.00475	−0.502375	−	0.864650i	$-0.667540\pi$
$857$	−29.4136	−1.00475	−0.502375	+	0.864650i	$0.667540\pi$
$858$	0	0
$859$	−16.3033	−0.556262	−0.278131	−	0.960543i	$-0.589715\pi$
$859$	−16.3033	−0.556262	−0.278131	+	0.960543i	$0.589715\pi$
$860$	−15.8607	−0.540844
$861$	0	0
$862$	−0.748219	−0.0254844
$863$	−33.2394	−1.13148	−0.565741	−	0.824583i	$-0.691410\pi$
$863$	−33.2394	−1.13148	−0.565741	+	0.824583i	$0.691410\pi$
$864$	0	0
$865$	40.7088	1.38414
$866$	−7.18287	−0.244084
$867$	0	0
$868$	−31.6376	−1.07385
$869$	5.84394	0.198242
$870$	0	0
$871$	30.8776	1.04625
$872$	7.16298	0.242569
$873$	0	0
$874$	−18.5875	−0.628733
$875$	−45.7554	−1.54681
$876$	0	0
$877$	46.3313	1.56450	0.782248	−	0.622967i	$-0.214073\pi$
$877$	46.3313	1.56450	0.782248	+	0.622967i	$0.214073\pi$
$878$	3.06995	0.103606
$879$	0	0
$880$	−2.42275	−0.0816710
$881$	−49.6464	−1.67263	−0.836314	−	0.548250i	$-0.815294\pi$
$881$	−49.6464	−1.67263	−0.836314	+	0.548250i	$0.815294\pi$
$882$	0	0
$883$	0.0836456	0.00281490	0.00140745	−	0.999999i	$-0.499552\pi$
$883$	0.0836456	0.00281490	0.00140745	+	0.999999i	$0.499552\pi$
$884$	0.313223	0.0105348
$885$	0	0
$886$	11.1300	0.373920
$887$	−56.8653	−1.90935	−0.954674	−	0.297653i	$-0.903796\pi$
$887$	−56.8653	−1.90935	−0.954674	+	0.297653i	$0.903796\pi$
$888$	0	0
$889$	19.2176	0.644537
$890$	−4.54470	−0.152339
$891$	0	0
$892$	12.7666	0.427457
$893$	−2.52067	−0.0843510
$894$	0	0
$895$	3.77572	0.126208
$896$	3.96511	0.132465
$897$	0	0
$898$	9.93204	0.331436
$899$	29.6542	0.989023
$900$	0	0
$901$	0.614365	0.0204675
$902$	5.05158	0.168199
$903$	0	0
$904$	−16.4629	−0.547548
$905$	45.9138	1.52623
$906$	0	0
$907$	−20.8216	−0.691369	−0.345684	−	0.938351i	$-0.612353\pi$
$907$	−20.8216	−0.691369	−0.345684	+	0.938351i	$0.612353\pi$
$908$	22.7865	0.756197
$909$	0	0
$910$	−27.7274	−0.919155
$911$	−4.26897	−0.141437	−0.0707187	−	0.997496i	$-0.522529\pi$
$911$	−4.26897	−0.141437	−0.0707187	+	0.997496i	$0.522529\pi$
$912$	0	0
$913$	−2.50324	−0.0828452
$914$	14.3761	0.475519
$915$	0	0
$916$	10.6307	0.351247
$917$	59.3176	1.95884
$918$	0	0
$919$	−9.02565	−0.297729	−0.148864	−	0.988858i	$-0.547562\pi$
$919$	−9.02565	−0.297729	−0.148864	+	0.988858i	$0.547562\pi$
$920$	−13.6362	−0.449572
$921$	0	0
$922$	−3.63676	−0.119770
$923$	−27.6113	−0.908838
$924$	0	0
$925$	0.538866	0.0177178
$926$	−30.9033	−1.01554
$927$	0	0
$928$	−3.71653	−0.122001
$929$	−53.2021	−1.74550	−0.872752	−	0.488164i	$-0.837667\pi$
$929$	−53.2021	−1.74550	−0.872752	+	0.488164i	$0.837667\pi$
$930$	0	0
$931$	−25.6288	−0.839950
$932$	−25.2615	−0.827467
$933$	0	0
$934$	26.0997	0.854009
$935$	−0.233931	−0.00765037
$936$	0	0
$937$	−17.0840	−0.558109	−0.279055	−	0.960275i	$-0.590021\pi$
$937$	−17.0840	−0.558109	−0.279055	+	0.960275i	$0.590021\pi$
$938$	37.7419	1.23232
$939$	0	0
$940$	−1.84922	−0.0603148
$941$	26.6781	0.869681	0.434841	−	0.900507i	$-0.356805\pi$
$941$	26.6781	0.869681	0.434841	+	0.900507i	$0.356805\pi$
$942$	0	0
$943$	28.4323	0.925883
$944$	1.21347	0.0394952
$945$	0	0
$946$	−8.26930	−0.268858
$947$	6.14436	0.199665	0.0998324	−	0.995004i	$-0.468169\pi$
$947$	6.14436	0.199665	0.0998324	+	0.995004i	$0.468169\pi$
$948$	0	0
$949$	14.5830	0.473386
$950$	−1.03760	−0.0336642
$951$	0	0
$952$	0.382854	0.0124084
$953$	−0.723083	−0.0234230	−0.0117115	−	0.999931i	$-0.503728\pi$
$953$	−0.723083	−0.0234230	−0.0117115	+	0.999931i	$0.503728\pi$
$954$	0	0
$955$	−31.3969	−1.01598
$956$	4.22269	0.136571
$957$	0	0
$958$	0.185023	0.00597781
$959$	23.7589	0.767216
$960$	0	0
$961$	32.6645	1.05369
$962$	4.95033	0.159605
$963$	0	0
$964$	−11.9213	−0.383958
$965$	−26.9897	−0.868831
$966$	0	0
$967$	33.6165	1.08103	0.540517	−	0.841333i	$-0.318229\pi$
$967$	33.6165	1.08103	0.540517	+	0.841333i	$0.318229\pi$
$968$	9.73685	0.312954
$969$	0	0
$970$	2.82420	0.0906796
$971$	24.5850	0.788969	0.394484	−	0.918903i	$-0.370923\pi$
$971$	24.5850	0.788969	0.394484	+	0.918903i	$0.370923\pi$
$972$	0	0
$973$	−60.7681	−1.94814
$974$	−12.9888	−0.416189
$975$	0	0
$976$	−3.66519	−0.117320
$977$	11.1003	0.355130	0.177565	−	0.984109i	$-0.443178\pi$
$977$	11.1003	0.355130	0.177565	+	0.984109i	$0.443178\pi$
$978$	0	0
$979$	−2.36948	−0.0757289
$980$	−18.8018	−0.600602
$981$	0	0
$982$	10.9945	0.350849
$983$	17.7913	0.567453	0.283726	−	0.958905i	$-0.408429\pi$
$983$	17.7913	0.567453	0.283726	+	0.958905i	$0.408429\pi$
$984$	0	0
$985$	26.4015	0.841221
$986$	−0.358853	−0.0114282
$987$	0	0
$988$	−9.53199	−0.303253
$989$	−46.5429	−1.47998
$990$	0	0
$991$	−4.64208	−0.147461	−0.0737303	−	0.997278i	$-0.523490\pi$
$991$	−4.64208	−0.147461	−0.0737303	+	0.997278i	$0.523490\pi$
$992$	−7.97900	−0.253334
$993$	0	0
$994$	−33.7496	−1.07047
$995$	2.66315	0.0844275
$996$	0	0
$997$	−24.0802	−0.762628	−0.381314	−	0.924446i	$-0.624528\pi$
$997$	−24.0802	−0.762628	−0.381314	+	0.924446i	$0.624528\pi$
$998$	22.7533	0.720242
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.15566 q^{5} -3.96511 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.15566 q^{5} -3.96511 q^{7} -1.00000 q^{8} +2.15566 q^{10} +1.12390 q^{11} +3.24395 q^{13} +3.96511 q^{14} +1.00000 q^{16} +0.0965560 q^{17} -2.93839 q^{19} -2.15566 q^{20} -1.12390 q^{22} -6.32576 q^{23} -0.353119 q^{25} -3.24395 q^{26} -3.96511 q^{28} +3.71653 q^{29} +7.97900 q^{31} -1.00000 q^{32} -0.0965560 q^{34} +8.54743 q^{35} -1.52602 q^{37} +2.93839 q^{38} +2.15566 q^{40} -4.49469 q^{41} +7.35768 q^{43} +1.12390 q^{44} +6.32576 q^{46} +0.857841 q^{47} +8.72206 q^{49} +0.353119 q^{50} +3.24395 q^{52} +6.36279 q^{53} -2.42275 q^{55} +3.96511 q^{56} -3.71653 q^{58} +1.21347 q^{59} -3.66519 q^{61} -7.97900 q^{62} +1.00000 q^{64} -6.99286 q^{65} +9.51851 q^{67} +0.0965560 q^{68} -8.54743 q^{70} -8.51164 q^{71} +4.49546 q^{73} +1.52602 q^{74} -2.93839 q^{76} -4.45639 q^{77} +5.19969 q^{79} -2.15566 q^{80} +4.49469 q^{82} -2.22728 q^{83} -0.208142 q^{85} -7.35768 q^{86} -1.12390 q^{88} -2.10826 q^{89} -12.8626 q^{91} -6.32576 q^{92} -0.857841 q^{94} +6.33417 q^{95} +1.31013 q^{97} -8.72206 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 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

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Database

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