LMFDB - Embedded newform 8046.2.a.t.1.9

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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 46 x^{14} + 192 x^{13} + 752 x^{12} - 3378 x^{11} - 5277 x^{10} + 27132 x^{9} + \cdots - 4260 $ x^16 - 4x^15 - 46x^14 + 192x^13 + 752x^12 - 3378x^11 - 5277x^10 + 27132x^9 + 15173x^8 - 102466x^7 - 16612x^6 + 186120x^5 + 117x^4 - 156722x^3 + 10545x^2 + 48580*x - 4260
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			1.9
Root			$0.939377$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.939377 q^{5} +3.42185 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +0.939377 q^{5} +3.42185 q^{7} +1.00000 q^{8} +0.939377 q^{10} +0.393747 q^{11} +1.28986 q^{13} +3.42185 q^{14} +1.00000 q^{16} +7.81081 q^{17} +0.720806 q^{19} +0.939377 q^{20} +0.393747 q^{22} +0.995353 q^{23} -4.11757 q^{25} +1.28986 q^{26} +3.42185 q^{28} +0.330862 q^{29} -5.48169 q^{31} +1.00000 q^{32} +7.81081 q^{34} +3.21440 q^{35} -1.27008 q^{37} +0.720806 q^{38} +0.939377 q^{40} +12.4125 q^{41} +5.23973 q^{43} +0.393747 q^{44} +0.995353 q^{46} -0.953016 q^{47} +4.70904 q^{49} -4.11757 q^{50} +1.28986 q^{52} -11.7544 q^{53} +0.369877 q^{55} +3.42185 q^{56} +0.330862 q^{58} -7.48878 q^{59} +8.28297 q^{61} -5.48169 q^{62} +1.00000 q^{64} +1.21166 q^{65} +12.0766 q^{67} +7.81081 q^{68} +3.21440 q^{70} -4.57951 q^{71} -0.609608 q^{73} -1.27008 q^{74} +0.720806 q^{76} +1.34734 q^{77} +11.6536 q^{79} +0.939377 q^{80} +12.4125 q^{82} -4.32050 q^{83} +7.33730 q^{85} +5.23973 q^{86} +0.393747 q^{88} +8.05460 q^{89} +4.41369 q^{91} +0.995353 q^{92} -0.953016 q^{94} +0.677109 q^{95} -9.94473 q^{97} +4.70904 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8	$ 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8} + 4 q^{10} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 10 q^{19} + 4 q^{20} + 6 q^{22} + 10 q^{23} + 28 q^{25} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 21 q^{31} + 16 q^{32} + q^{34} + 16 q^{35} + 17 q^{37} + 10 q^{38} + 4 q^{40} - 4 q^{41} + 16 q^{43} + 6 q^{44} + 10 q^{46} + 25 q^{47} + 36 q^{49} + 28 q^{50} + 6 q^{52} + 14 q^{53} + 19 q^{55} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 23 q^{61} + 21 q^{62} + 16 q^{64} + 20 q^{65} + 22 q^{67} + q^{68} + 16 q^{70} + 10 q^{71} + 16 q^{73} + 17 q^{74} + 10 q^{76} - 2 q^{77} + 37 q^{79} + 4 q^{80} - 4 q^{82} + 33 q^{83} + 43 q^{85} + 16 q^{86} + 6 q^{88} - 3 q^{89} + 28 q^{91} + 10 q^{92} + 25 q^{94} + 14 q^{95} - 3 q^{97} + 36 q^{98}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8 + 4 * q^10 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 10 * q^19 + 4 * q^20 + 6 * q^22 + 10 * q^23 + 28 * q^25 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 21 * q^31 + 16 * q^32 + q^34 + 16 * q^35 + 17 * q^37 + 10 * q^38 + 4 * q^40 - 4 * q^41 + 16 * q^43 + 6 * q^44 + 10 * q^46 + 25 * q^47 + 36 * q^49 + 28 * q^50 + 6 * q^52 + 14 * q^53 + 19 * q^55 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 23 * q^61 + 21 * q^62 + 16 * q^64 + 20 * q^65 + 22 * q^67 + q^68 + 16 * q^70 + 10 * q^71 + 16 * q^73 + 17 * q^74 + 10 * q^76 - 2 * q^77 + 37 * q^79 + 4 * q^80 - 4 * q^82 + 33 * q^83 + 43 * q^85 + 16 * q^86 + 6 * q^88 - 3 * q^89 + 28 * q^91 + 10 * q^92 + 25 * q^94 + 14 * q^95 - 3 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.939377	0.420102	0.210051	−	0.977690i	$-0.432637\pi$
$5$	0.939377	0.420102	0.210051	+	0.977690i	$0.432637\pi$
$6$	0	0
$7$	3.42185	1.29334	0.646668	−	0.762771i	$-0.276162\pi$
$7$	3.42185	1.29334	0.646668	+	0.762771i	$0.276162\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0.939377	0.297057
$11$	0.393747	0.118719	0.0593596	−	0.998237i	$-0.481094\pi$
$11$	0.393747	0.118719	0.0593596	+	0.998237i	$0.481094\pi$
$12$	0	0
$13$	1.28986	0.357742	0.178871	−	0.983873i	$-0.442756\pi$
$13$	1.28986	0.357742	0.178871	+	0.983873i	$0.442756\pi$
$14$	3.42185	0.914527
$15$	0	0
$16$	1.00000	0.250000
$17$	7.81081	1.89440	0.947200	−	0.320643i	$-0.103899\pi$
$17$	7.81081	1.89440	0.947200	+	0.320643i	$0.103899\pi$
$18$	0	0
$19$	0.720806	0.165364	0.0826821	−	0.996576i	$-0.473651\pi$
$19$	0.720806	0.165364	0.0826821	+	0.996576i	$0.473651\pi$
$20$	0.939377	0.210051
$21$	0	0
$22$	0.393747	0.0839471
$23$	0.995353	0.207546	0.103773	−	0.994601i	$-0.466909\pi$
$23$	0.995353	0.207546	0.103773	+	0.994601i	$0.466909\pi$
$24$	0	0
$25$	−4.11757	−0.823514
$26$	1.28986	0.252961
$27$	0	0
$28$	3.42185	0.646668
$29$	0.330862	0.0614395	0.0307198	−	0.999528i	$-0.490220\pi$
$29$	0.330862	0.0614395	0.0307198	+	0.999528i	$0.490220\pi$
$30$	0	0
$31$	−5.48169	−0.984541	−0.492271	−	0.870442i	$-0.663833\pi$
$31$	−5.48169	−0.984541	−0.492271	+	0.870442i	$0.663833\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.81081	1.33954
$35$	3.21440	0.543334
$36$	0	0
$37$	−1.27008	−0.208801	−0.104400	−	0.994535i	$-0.533292\pi$
$37$	−1.27008	−0.208801	−0.104400	+	0.994535i	$0.533292\pi$
$38$	0.720806	0.116930
$39$	0	0
$40$	0.939377	0.148529
$41$	12.4125	1.93851	0.969256	−	0.246056i	$-0.0791346\pi$
$41$	12.4125	1.93851	0.969256	+	0.246056i	$0.0791346\pi$
$42$	0	0
$43$	5.23973	0.799052	0.399526	−	0.916722i	$-0.369175\pi$
$43$	5.23973	0.799052	0.399526	+	0.916722i	$0.369175\pi$
$44$	0.393747	0.0593596
$45$	0	0
$46$	0.995353	0.146757
$47$	−0.953016	−0.139012	−0.0695059	−	0.997582i	$-0.522142\pi$
$47$	−0.953016	−0.139012	−0.0695059	+	0.997582i	$0.522142\pi$
$48$	0	0
$49$	4.70904	0.672720
$50$	−4.11757	−0.582312
$51$	0	0
$52$	1.28986	0.178871
$53$	−11.7544	−1.61459	−0.807295	−	0.590148i	$-0.799069\pi$
$53$	−11.7544	−1.61459	−0.807295	+	0.590148i	$0.799069\pi$
$54$	0	0
$55$	0.369877	0.0498742
$56$	3.42185	0.457264
$57$	0	0
$58$	0.330862	0.0434443
$59$	−7.48878	−0.974956	−0.487478	−	0.873135i	$-0.662083\pi$
$59$	−7.48878	−0.974956	−0.487478	+	0.873135i	$0.662083\pi$
$60$	0	0
$61$	8.28297	1.06053	0.530263	−	0.847833i	$-0.322093\pi$
$61$	8.28297	1.06053	0.530263	+	0.847833i	$0.322093\pi$
$62$	−5.48169	−0.696176
$63$	0	0
$64$	1.00000	0.125000
$65$	1.21166	0.150288
$66$	0	0
$67$	12.0766	1.47539	0.737695	−	0.675134i	$-0.235914\pi$
$67$	12.0766	1.47539	0.737695	+	0.675134i	$0.235914\pi$
$68$	7.81081	0.947200
$69$	0	0
$70$	3.21440	0.384195
$71$	−4.57951	−0.543488	−0.271744	−	0.962370i	$-0.587600\pi$
$71$	−4.57951	−0.543488	−0.271744	+	0.962370i	$0.587600\pi$
$72$	0	0
$73$	−0.609608	−0.0713493	−0.0356746	−	0.999363i	$-0.511358\pi$
$73$	−0.609608	−0.0713493	−0.0356746	+	0.999363i	$0.511358\pi$
$74$	−1.27008	−0.147644
$75$	0	0
$76$	0.720806	0.0826821
$77$	1.34734	0.153544
$78$	0	0
$79$	11.6536	1.31113	0.655567	−	0.755137i	$-0.272430\pi$
$79$	11.6536	1.31113	0.655567	+	0.755137i	$0.272430\pi$
$80$	0.939377	0.105026
$81$	0	0
$82$	12.4125	1.37073
$83$	−4.32050	−0.474236	−0.237118	−	0.971481i	$-0.576203\pi$
$83$	−4.32050	−0.474236	−0.237118	+	0.971481i	$0.576203\pi$
$84$	0	0
$85$	7.33730	0.795841
$86$	5.23973	0.565015
$87$	0	0
$88$	0.393747	0.0419736
$89$	8.05460	0.853786	0.426893	−	0.904302i	$-0.359608\pi$
$89$	8.05460	0.853786	0.426893	+	0.904302i	$0.359608\pi$
$90$	0	0
$91$	4.41369	0.462680
$92$	0.995353	0.103773
$93$	0	0
$94$	−0.953016	−0.0982961
$95$	0.677109	0.0694699
$96$	0	0
$97$	−9.94473	−1.00973	−0.504867	−	0.863197i	$-0.668459\pi$
$97$	−9.94473	−1.00973	−0.504867	+	0.863197i	$0.668459\pi$
$98$	4.70904	0.475685
$99$	0	0
$100$	−4.11757	−0.411757
$101$	−11.4797	−1.14228	−0.571138	−	0.820854i	$-0.693498\pi$
$101$	−11.4797	−1.14228	−0.571138	+	0.820854i	$0.693498\pi$
$102$	0	0
$103$	−2.52637	−0.248931	−0.124466	−	0.992224i	$-0.539722\pi$
$103$	−2.52637	−0.248931	−0.124466	+	0.992224i	$0.539722\pi$
$104$	1.28986	0.126481
$105$	0	0
$106$	−11.7544	−1.14169
$107$	−13.2399	−1.27995	−0.639976	−	0.768395i	$-0.721056\pi$
$107$	−13.2399	−1.27995	−0.639976	+	0.768395i	$0.721056\pi$
$108$	0	0
$109$	−13.9327	−1.33451	−0.667255	−	0.744829i	$-0.732531\pi$
$109$	−13.9327	−1.33451	−0.667255	+	0.744829i	$0.732531\pi$
$110$	0.369877	0.0352664
$111$	0	0
$112$	3.42185	0.323334
$113$	−18.4863	−1.73905	−0.869523	−	0.493893i	$-0.835573\pi$
$113$	−18.4863	−1.73905	−0.869523	+	0.493893i	$0.835573\pi$
$114$	0	0
$115$	0.935012	0.0871903
$116$	0.330862	0.0307198
$117$	0	0
$118$	−7.48878	−0.689398
$119$	26.7274	2.45010
$120$	0	0
$121$	−10.8450	−0.985906
$122$	8.28297	0.749905
$123$	0	0
$124$	−5.48169	−0.492271
$125$	−8.56484	−0.766062
$126$	0	0
$127$	14.1202	1.25297	0.626483	−	0.779435i	$-0.284494\pi$
$127$	14.1202	1.25297	0.626483	+	0.779435i	$0.284494\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.21166	0.106270
$131$	10.1080	0.883144	0.441572	−	0.897226i	$-0.354421\pi$
$131$	10.1080	0.883144	0.441572	+	0.897226i	$0.354421\pi$
$132$	0	0
$133$	2.46649	0.213872
$134$	12.0766	1.04326
$135$	0	0
$136$	7.81081	0.669772
$137$	−8.61167	−0.735745	−0.367872	−	0.929876i	$-0.619914\pi$
$137$	−8.61167	−0.735745	−0.367872	+	0.929876i	$0.619914\pi$
$138$	0	0
$139$	17.7270	1.50359	0.751794	−	0.659398i	$-0.229189\pi$
$139$	17.7270	1.50359	0.751794	+	0.659398i	$0.229189\pi$
$140$	3.21440	0.271667
$141$	0	0
$142$	−4.57951	−0.384304
$143$	0.507876	0.0424708
$144$	0	0
$145$	0.310804	0.0258109
$146$	−0.609608	−0.0504516
$147$	0	0
$148$	−1.27008	−0.104400
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	18.3530	1.49354	0.746772	−	0.665080i	$-0.231603\pi$
$151$	18.3530	1.49354	0.746772	+	0.665080i	$0.231603\pi$
$152$	0.720806	0.0584651
$153$	0	0
$154$	1.34734	0.108572
$155$	−5.14938	−0.413608
$156$	0	0
$157$	17.5723	1.40242	0.701211	−	0.712954i	$-0.252643\pi$
$157$	17.5723	1.40242	0.701211	+	0.712954i	$0.252643\pi$
$158$	11.6536	0.927111
$159$	0	0
$160$	0.939377	0.0742643
$161$	3.40595	0.268426
$162$	0	0
$163$	−10.1444	−0.794569	−0.397284	−	0.917696i	$-0.630047\pi$
$163$	−10.1444	−0.794569	−0.397284	+	0.917696i	$0.630047\pi$
$164$	12.4125	0.969256
$165$	0	0
$166$	−4.32050	−0.335336
$167$	−23.2455	−1.79879	−0.899394	−	0.437139i	$-0.855992\pi$
$167$	−23.2455	−1.79879	−0.899394	+	0.437139i	$0.855992\pi$
$168$	0	0
$169$	−11.3363	−0.872021
$170$	7.33730	0.562745
$171$	0	0
$172$	5.23973	0.399526
$173$	−0.0813046	−0.00618148	−0.00309074	−	0.999995i	$-0.500984\pi$
$173$	−0.0813046	−0.00618148	−0.00309074	+	0.999995i	$0.500984\pi$
$174$	0	0
$175$	−14.0897	−1.06508
$176$	0.393747	0.0296798
$177$	0	0
$178$	8.05460	0.603718
$179$	4.57555	0.341993	0.170996	−	0.985272i	$-0.445301\pi$
$179$	4.57555	0.341993	0.170996	+	0.985272i	$0.445301\pi$
$180$	0	0
$181$	23.2674	1.72945	0.864726	−	0.502244i	$-0.167492\pi$
$181$	23.2674	1.72945	0.864726	+	0.502244i	$0.167492\pi$
$182$	4.41369	0.327164
$183$	0	0
$184$	0.995353	0.0733784
$185$	−1.19309	−0.0877176
$186$	0	0
$187$	3.07548	0.224902
$188$	−0.953016	−0.0695059
$189$	0	0
$190$	0.677109	0.0491226
$191$	−5.68788	−0.411560	−0.205780	−	0.978598i	$-0.565973\pi$
$191$	−5.68788	−0.411560	−0.205780	+	0.978598i	$0.565973\pi$
$192$	0	0
$193$	−21.2253	−1.52783	−0.763915	−	0.645317i	$-0.776725\pi$
$193$	−21.2253	−1.52783	−0.763915	+	0.645317i	$0.776725\pi$
$194$	−9.94473	−0.713990
$195$	0	0
$196$	4.70904	0.336360
$197$	5.12277	0.364982	0.182491	−	0.983208i	$-0.441584\pi$
$197$	5.12277	0.364982	0.182491	+	0.983208i	$0.441584\pi$
$198$	0	0
$199$	0.652658	0.0462657	0.0231329	−	0.999732i	$-0.492636\pi$
$199$	0.652658	0.0462657	0.0231329	+	0.999732i	$0.492636\pi$
$200$	−4.11757	−0.291156
$201$	0	0
$202$	−11.4797	−0.807712
$203$	1.13216	0.0794620
$204$	0	0
$205$	11.6600	0.814373
$206$	−2.52637	−0.176021
$207$	0	0
$208$	1.28986	0.0894354
$209$	0.283815	0.0196319
$210$	0	0
$211$	−4.91269	−0.338203	−0.169102	−	0.985599i	$-0.554087\pi$
$211$	−4.91269	−0.338203	−0.169102	+	0.985599i	$0.554087\pi$
$212$	−11.7544	−0.807295
$213$	0	0
$214$	−13.2399	−0.905063
$215$	4.92209	0.335683
$216$	0	0
$217$	−18.7575	−1.27334
$218$	−13.9327	−0.943641
$219$	0	0
$220$	0.369877	0.0249371
$221$	10.0748	0.677706
$222$	0	0
$223$	−8.14971	−0.545745	−0.272873	−	0.962050i	$-0.587974\pi$
$223$	−8.14971	−0.545745	−0.272873	+	0.962050i	$0.587974\pi$
$224$	3.42185	0.228632
$225$	0	0
$226$	−18.4863	−1.22969
$227$	18.1369	1.20379	0.601896	−	0.798575i	$-0.294412\pi$
$227$	18.1369	1.20379	0.601896	+	0.798575i	$0.294412\pi$
$228$	0	0
$229$	−0.00980482	−0.000647921 0	−0.000323960	−	1.00000i	$-0.500103\pi$
$229$	−0.00980482	−0.000647921 0	−0.000323960		1.00000i	$0.500103\pi$
$230$	0.935012	0.0616529
$231$	0	0
$232$	0.330862	0.0217222
$233$	20.4897	1.34232	0.671161	−	0.741311i	$-0.265796\pi$
$233$	20.4897	1.34232	0.671161	+	0.741311i	$0.265796\pi$
$234$	0	0
$235$	−0.895241	−0.0583991
$236$	−7.48878	−0.487478
$237$	0	0
$238$	26.7274	1.73248
$239$	−21.0947	−1.36450	−0.682251	−	0.731118i	$-0.738999\pi$
$239$	−21.0947	−1.36450	−0.682251	+	0.731118i	$0.738999\pi$
$240$	0	0
$241$	8.35135	0.537958	0.268979	−	0.963146i	$-0.413314\pi$
$241$	8.35135	0.537958	0.268979	+	0.963146i	$0.413314\pi$
$242$	−10.8450	−0.697141
$243$	0	0
$244$	8.28297	0.530263
$245$	4.42356	0.282611
$246$	0	0
$247$	0.929736	0.0591577
$248$	−5.48169	−0.348088
$249$	0	0
$250$	−8.56484	−0.541688
$251$	−17.6456	−1.11378	−0.556891	−	0.830586i	$-0.688006\pi$
$251$	−17.6456	−1.11378	−0.556891	+	0.830586i	$0.688006\pi$
$252$	0	0
$253$	0.391917	0.0246396
$254$	14.1202	0.885981
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.87999	−0.242027	−0.121014	−	0.992651i	$-0.538615\pi$
$257$	−3.87999	−0.242027	−0.121014	+	0.992651i	$0.538615\pi$
$258$	0	0
$259$	−4.34604	−0.270050
$260$	1.21166	0.0751440
$261$	0	0
$262$	10.1080	0.624477
$263$	10.7807	0.664767	0.332383	−	0.943144i	$-0.392147\pi$
$263$	10.7807	0.664767	0.332383	+	0.943144i	$0.392147\pi$
$264$	0	0
$265$	−11.0418	−0.678293
$266$	2.46649	0.151230
$267$	0	0
$268$	12.0766	0.737695
$269$	15.6420	0.953708	0.476854	−	0.878982i	$-0.341777\pi$
$269$	15.6420	0.953708	0.476854	+	0.878982i	$0.341777\pi$
$270$	0	0
$271$	−11.6742	−0.709160	−0.354580	−	0.935026i	$-0.615376\pi$
$271$	−11.6742	−0.709160	−0.354580	+	0.935026i	$0.615376\pi$
$272$	7.81081	0.473600
$273$	0	0
$274$	−8.61167	−0.520250
$275$	−1.62128	−0.0977669
$276$	0	0
$277$	26.4429	1.58880	0.794399	−	0.607397i	$-0.207786\pi$
$277$	26.4429	1.58880	0.794399	+	0.607397i	$0.207786\pi$
$278$	17.7270	1.06320
$279$	0	0
$280$	3.21440	0.192097
$281$	1.18970	0.0709717	0.0354858	−	0.999370i	$-0.488702\pi$
$281$	1.18970	0.0709717	0.0354858	+	0.999370i	$0.488702\pi$
$282$	0	0
$283$	12.3812	0.735985	0.367992	−	0.929829i	$-0.380045\pi$
$283$	12.3812	0.735985	0.367992	+	0.929829i	$0.380045\pi$
$284$	−4.57951	−0.271744
$285$	0	0
$286$	0.507876	0.0300314
$287$	42.4738	2.50715
$288$	0	0
$289$	44.0088	2.58875
$290$	0.310804	0.0182511
$291$	0	0
$292$	−0.609608	−0.0356746
$293$	21.4962	1.25582	0.627911	−	0.778285i	$-0.283910\pi$
$293$	21.4962	1.25582	0.627911	+	0.778285i	$0.283910\pi$
$294$	0	0
$295$	−7.03479	−0.409581
$296$	−1.27008	−0.0738222
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	1.28386	0.0742477
$300$	0	0
$301$	17.9296	1.03344
$302$	18.3530	1.05610
$303$	0	0
$304$	0.720806	0.0413411
$305$	7.78083	0.445529
$306$	0	0
$307$	9.49869	0.542119	0.271059	−	0.962563i	$-0.412626\pi$
$307$	9.49869	0.542119	0.271059	+	0.962563i	$0.412626\pi$
$308$	1.34734	0.0767719
$309$	0	0
$310$	−5.14938	−0.292465
$311$	−31.3499	−1.77769	−0.888844	−	0.458211i	$-0.848490\pi$
$311$	−31.3499	−1.77769	−0.888844	+	0.458211i	$0.848490\pi$
$312$	0	0
$313$	−7.05477	−0.398759	−0.199380	−	0.979922i	$-0.563893\pi$
$313$	−7.05477	−0.398759	−0.199380	+	0.979922i	$0.563893\pi$
$314$	17.5723	0.991662
$315$	0	0
$316$	11.6536	0.655567
$317$	8.13314	0.456803	0.228401	−	0.973567i	$-0.426650\pi$
$317$	8.13314	0.456803	0.228401	+	0.973567i	$0.426650\pi$
$318$	0	0
$319$	0.130276	0.00729405
$320$	0.939377	0.0525128
$321$	0	0
$322$	3.40595	0.189806
$323$	5.63008	0.313266
$324$	0	0
$325$	−5.31107	−0.294605
$326$	−10.1444	−0.561845
$327$	0	0
$328$	12.4125	0.685367
$329$	−3.26108	−0.179789
$330$	0	0
$331$	24.5183	1.34765	0.673825	−	0.738891i	$-0.264650\pi$
$331$	24.5183	1.34765	0.673825	+	0.738891i	$0.264650\pi$
$332$	−4.32050	−0.237118
$333$	0	0
$334$	−23.2455	−1.27194
$335$	11.3445	0.619814
$336$	0	0
$337$	−1.16523	−0.0634740	−0.0317370	−	0.999496i	$-0.510104\pi$
$337$	−1.16523	−0.0634740	−0.0317370	+	0.999496i	$0.510104\pi$
$338$	−11.3363	−0.616612
$339$	0	0
$340$	7.33730	0.397921
$341$	−2.15840	−0.116884
$342$	0	0
$343$	−7.83931	−0.423283
$344$	5.23973	0.282508
$345$	0	0
$346$	−0.0813046	−0.00437096
$347$	−12.0680	−0.647846	−0.323923	−	0.946083i	$-0.605002\pi$
$347$	−12.0680	−0.647846	−0.323923	+	0.946083i	$0.605002\pi$
$348$	0	0
$349$	−34.7355	−1.85935	−0.929675	−	0.368380i	$-0.879913\pi$
$349$	−34.7355	−1.85935	−0.929675	+	0.368380i	$0.879913\pi$
$350$	−14.0897	−0.753126
$351$	0	0
$352$	0.393747	0.0209868
$353$	5.61228	0.298711	0.149356	−	0.988784i	$-0.452280\pi$
$353$	5.61228	0.298711	0.149356	+	0.988784i	$0.452280\pi$
$354$	0	0
$355$	−4.30188	−0.228320
$356$	8.05460	0.426893
$357$	0	0
$358$	4.57555	0.241826
$359$	−23.8300	−1.25770	−0.628850	−	0.777526i	$-0.716474\pi$
$359$	−23.8300	−1.25770	−0.628850	+	0.777526i	$0.716474\pi$
$360$	0	0
$361$	−18.4804	−0.972655
$362$	23.2674	1.22291
$363$	0	0
$364$	4.41369	0.231340
$365$	−0.572652	−0.0299740
$366$	0	0
$367$	30.1399	1.57329	0.786646	−	0.617405i	$-0.211816\pi$
$367$	30.1399	1.57329	0.786646	+	0.617405i	$0.211816\pi$
$368$	0.995353	0.0518864
$369$	0	0
$370$	−1.19309	−0.0620257
$371$	−40.2217	−2.08821
$372$	0	0
$373$	13.0572	0.676075	0.338038	−	0.941133i	$-0.390237\pi$
$373$	13.0572	0.676075	0.338038	+	0.941133i	$0.390237\pi$
$374$	3.07548	0.159029
$375$	0	0
$376$	−0.953016	−0.0491481
$377$	0.426764	0.0219795
$378$	0	0
$379$	−21.7852	−1.11903	−0.559516	−	0.828820i	$-0.689013\pi$
$379$	−21.7852	−1.11903	−0.559516	+	0.828820i	$0.689013\pi$
$380$	0.677109	0.0347349
$381$	0	0
$382$	−5.68788	−0.291017
$383$	20.9262	1.06928	0.534640	−	0.845080i	$-0.320447\pi$
$383$	20.9262	1.06928	0.534640	+	0.845080i	$0.320447\pi$
$384$	0	0
$385$	1.26566	0.0645041
$386$	−21.2253	−1.08034
$387$	0	0
$388$	−9.94473	−0.504867
$389$	−4.36864	−0.221499	−0.110749	−	0.993848i	$-0.535325\pi$
$389$	−4.36864	−0.221499	−0.110749	+	0.993848i	$0.535325\pi$
$390$	0	0
$391$	7.77452	0.393174
$392$	4.70904	0.237842
$393$	0	0
$394$	5.12277	0.258081
$395$	10.9471	0.550810
$396$	0	0
$397$	−13.0226	−0.653587	−0.326793	−	0.945096i	$-0.605968\pi$
$397$	−13.0226	−0.653587	−0.326793	+	0.945096i	$0.605968\pi$
$398$	0.652658	0.0327148
$399$	0	0
$400$	−4.11757	−0.205879
$401$	−21.3200	−1.06467	−0.532335	−	0.846534i	$-0.678685\pi$
$401$	−21.3200	−1.06467	−0.532335	+	0.846534i	$0.678685\pi$
$402$	0	0
$403$	−7.07059	−0.352211
$404$	−11.4797	−0.571138
$405$	0	0
$406$	1.13216	0.0561881
$407$	−0.500092	−0.0247886
$408$	0	0
$409$	19.4308	0.960793	0.480396	−	0.877051i	$-0.340493\pi$
$409$	19.4308	0.960793	0.480396	+	0.877051i	$0.340493\pi$
$410$	11.6600	0.575848
$411$	0	0
$412$	−2.52637	−0.124466
$413$	−25.6255	−1.26095
$414$	0	0
$415$	−4.05858	−0.199228
$416$	1.28986	0.0632404
$417$	0	0
$418$	0.283815	0.0138818
$419$	−7.29883	−0.356571	−0.178286	−	0.983979i	$-0.557055\pi$
$419$	−7.29883	−0.356571	−0.178286	+	0.983979i	$0.557055\pi$
$420$	0	0
$421$	20.8884	1.01804	0.509020	−	0.860754i	$-0.330008\pi$
$421$	20.8884	1.01804	0.509020	+	0.860754i	$0.330008\pi$
$422$	−4.91269	−0.239146
$423$	0	0
$424$	−11.7544	−0.570844
$425$	−32.1616	−1.56007
$426$	0	0
$427$	28.3431	1.37162
$428$	−13.2399	−0.639976
$429$	0	0
$430$	4.92209	0.237364
$431$	−12.1489	−0.585194	−0.292597	−	0.956236i	$-0.594519\pi$
$431$	−12.1489	−0.585194	−0.292597	+	0.956236i	$0.594519\pi$
$432$	0	0
$433$	−21.5369	−1.03500	−0.517498	−	0.855684i	$-0.673137\pi$
$433$	−21.5369	−1.03500	−0.517498	+	0.855684i	$0.673137\pi$
$434$	−18.7575	−0.900390
$435$	0	0
$436$	−13.9327	−0.667255
$437$	0.717457	0.0343206
$438$	0	0
$439$	2.11515	0.100951	0.0504753	−	0.998725i	$-0.483926\pi$
$439$	2.11515	0.100951	0.0504753	+	0.998725i	$0.483926\pi$
$440$	0.369877	0.0176332
$441$	0	0
$442$	10.0748	0.479210
$443$	4.94385	0.234889	0.117445	−	0.993079i	$-0.462530\pi$
$443$	4.94385	0.234889	0.117445	+	0.993079i	$0.462530\pi$
$444$	0	0
$445$	7.56631	0.358677
$446$	−8.14971	−0.385900
$447$	0	0
$448$	3.42185	0.161667
$449$	−14.5035	−0.684465	−0.342232	−	0.939615i	$-0.611183\pi$
$449$	−14.5035	−0.684465	−0.342232	+	0.939615i	$0.611183\pi$
$450$	0	0
$451$	4.88739	0.230138
$452$	−18.4863	−0.869523
$453$	0	0
$454$	18.1369	0.851209
$455$	4.14612	0.194373
$456$	0	0
$457$	−18.2987	−0.855979	−0.427989	−	0.903784i	$-0.640778\pi$
$457$	−18.2987	−0.855979	−0.427989	+	0.903784i	$0.640778\pi$
$458$	−0.00980482	−0.000458149 0
$459$	0	0
$460$	0.935012	0.0435952
$461$	11.3647	0.529307	0.264654	−	0.964344i	$-0.414742\pi$
$461$	11.3647	0.529307	0.264654	+	0.964344i	$0.414742\pi$
$462$	0	0
$463$	31.6110	1.46909	0.734543	−	0.678562i	$-0.237397\pi$
$463$	31.6110	1.46909	0.734543	+	0.678562i	$0.237397\pi$
$464$	0.330862	0.0153599
$465$	0	0
$466$	20.4897	0.949165
$467$	14.2382	0.658865	0.329432	−	0.944179i	$-0.393143\pi$
$467$	14.2382	0.658865	0.329432	+	0.944179i	$0.393143\pi$
$468$	0	0
$469$	41.3242	1.90818
$470$	−0.895241	−0.0412944
$471$	0	0
$472$	−7.48878	−0.344699
$473$	2.06313	0.0948628
$474$	0	0
$475$	−2.96797	−0.136180
$476$	26.7274	1.22505
$477$	0	0
$478$	−21.0947	−0.964849
$479$	31.8865	1.45693	0.728466	−	0.685082i	$-0.240233\pi$
$479$	31.8865	1.45693	0.728466	+	0.685082i	$0.240233\pi$
$480$	0	0
$481$	−1.63823	−0.0746967
$482$	8.35135	0.380394
$483$	0	0
$484$	−10.8450	−0.492953
$485$	−9.34185	−0.424192
$486$	0	0
$487$	35.3344	1.60115	0.800577	−	0.599230i	$-0.204526\pi$
$487$	35.3344	1.60115	0.800577	+	0.599230i	$0.204526\pi$
$488$	8.28297	0.374953
$489$	0	0
$490$	4.42356	0.199836
$491$	−25.5224	−1.15181	−0.575905	−	0.817517i	$-0.695350\pi$
$491$	−25.5224	−1.15181	−0.575905	+	0.817517i	$0.695350\pi$
$492$	0	0
$493$	2.58430	0.116391
$494$	0.929736	0.0418308
$495$	0	0
$496$	−5.48169	−0.246135
$497$	−15.6704	−0.702913
$498$	0	0
$499$	−14.5862	−0.652970	−0.326485	−	0.945202i	$-0.605864\pi$
$499$	−14.5862	−0.652970	−0.326485	+	0.945202i	$0.605864\pi$
$500$	−8.56484	−0.383031
$501$	0	0
$502$	−17.6456	−0.787563
$503$	−24.4552	−1.09040	−0.545202	−	0.838305i	$-0.683547\pi$
$503$	−24.4552	−1.09040	−0.545202	+	0.838305i	$0.683547\pi$
$504$	0	0
$505$	−10.7838	−0.479873
$506$	0.391917	0.0174228
$507$	0	0
$508$	14.1202	0.626483
$509$	−0.601939	−0.0266805	−0.0133402	−	0.999911i	$-0.504246\pi$
$509$	−0.601939	−0.0266805	−0.0133402	+	0.999911i	$0.504246\pi$
$510$	0	0
$511$	−2.08599	−0.0922786
$512$	1.00000	0.0441942
$513$	0	0
$514$	−3.87999	−0.171139
$515$	−2.37322	−0.104576
$516$	0	0
$517$	−0.375247	−0.0165033
$518$	−4.34604	−0.190954
$519$	0	0
$520$	1.21166	0.0531348
$521$	33.2391	1.45623	0.728116	−	0.685454i	$-0.240396\pi$
$521$	33.2391	1.45623	0.728116	+	0.685454i	$0.240396\pi$
$522$	0	0
$523$	−8.58871	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$523$	−8.58871	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$524$	10.1080	0.441572
$525$	0	0
$526$	10.7807	0.470061
$527$	−42.8165	−1.86512
$528$	0	0
$529$	−22.0093	−0.956925
$530$	−11.0418	−0.479625
$531$	0	0
$532$	2.46649	0.106936
$533$	16.0104	0.693486
$534$	0	0
$535$	−12.4373	−0.537710
$536$	12.0766	0.521629
$537$	0	0
$538$	15.6420	0.674373
$539$	1.85417	0.0798647
$540$	0	0
$541$	−10.0031	−0.430068	−0.215034	−	0.976607i	$-0.568986\pi$
$541$	−10.0031	−0.430068	−0.215034	+	0.976607i	$0.568986\pi$
$542$	−11.6742	−0.501452
$543$	0	0
$544$	7.81081	0.334886
$545$	−13.0881	−0.560631
$546$	0	0
$547$	−31.8992	−1.36391	−0.681957	−	0.731393i	$-0.738871\pi$
$547$	−31.8992	−1.36391	−0.681957	+	0.731393i	$0.738871\pi$
$548$	−8.61167	−0.367872
$549$	0	0
$550$	−1.62128	−0.0691316
$551$	0.238487	0.0101599
$552$	0	0
$553$	39.8769	1.69574
$554$	26.4429	1.12345
$555$	0	0
$556$	17.7270	0.751794
$557$	−16.4363	−0.696430	−0.348215	−	0.937415i	$-0.613212\pi$
$557$	−16.4363	−0.696430	−0.348215	+	0.937415i	$0.613212\pi$
$558$	0	0
$559$	6.75850	0.285854
$560$	3.21440	0.135833
$561$	0	0
$562$	1.18970	0.0501845
$563$	−13.3298	−0.561783	−0.280891	−	0.959740i	$-0.590630\pi$
$563$	−13.3298	−0.561783	−0.280891	+	0.959740i	$0.590630\pi$
$564$	0	0
$565$	−17.3656	−0.730577
$566$	12.3812	0.520420
$567$	0	0
$568$	−4.57951	−0.192152
$569$	−22.5469	−0.945214	−0.472607	−	0.881273i	$-0.656687\pi$
$569$	−22.5469	−0.945214	−0.472607	+	0.881273i	$0.656687\pi$
$570$	0	0
$571$	33.3146	1.39417	0.697086	−	0.716987i	$-0.254479\pi$
$571$	33.3146	1.39417	0.697086	+	0.716987i	$0.254479\pi$
$572$	0.507876	0.0212354
$573$	0	0
$574$	42.4738	1.77282
$575$	−4.09844	−0.170917
$576$	0	0
$577$	−44.4772	−1.85161	−0.925805	−	0.378001i	$-0.876612\pi$
$577$	−44.4772	−1.85161	−0.925805	+	0.378001i	$0.876612\pi$
$578$	44.0088	1.83052
$579$	0	0
$580$	0.310804	0.0129054
$581$	−14.7841	−0.613347
$582$	0	0
$583$	−4.62825	−0.191683
$584$	−0.609608	−0.0252258
$585$	0	0
$586$	21.4962	0.888000
$587$	2.19028	0.0904024	0.0452012	−	0.998978i	$-0.485607\pi$
$587$	2.19028	0.0904024	0.0452012	+	0.998978i	$0.485607\pi$
$588$	0	0
$589$	−3.95124	−0.162808
$590$	−7.03479	−0.289618
$591$	0	0
$592$	−1.27008	−0.0522002
$593$	−10.9399	−0.449248	−0.224624	−	0.974445i	$-0.572115\pi$
$593$	−10.9399	−0.449248	−0.224624	+	0.974445i	$0.572115\pi$
$594$	0	0
$595$	25.1071	1.02929
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	1.28386	0.0525010
$599$	5.93249	0.242395	0.121197	−	0.992628i	$-0.461327\pi$
$599$	5.93249	0.242395	0.121197	+	0.992628i	$0.461327\pi$
$600$	0	0
$601$	−18.3719	−0.749405	−0.374703	−	0.927145i	$-0.622255\pi$
$601$	−18.3719	−0.749405	−0.374703	+	0.927145i	$0.622255\pi$
$602$	17.9296	0.730755
$603$	0	0
$604$	18.3530	0.746772
$605$	−10.1875	−0.414181
$606$	0	0
$607$	39.3237	1.59610	0.798050	−	0.602591i	$-0.205865\pi$
$607$	39.3237	1.59610	0.798050	+	0.602591i	$0.205865\pi$
$608$	0.720806	0.0292325
$609$	0	0
$610$	7.78083	0.315037
$611$	−1.22925	−0.0497303
$612$	0	0
$613$	−1.32943	−0.0536953	−0.0268476	−	0.999640i	$-0.508547\pi$
$613$	−1.32943	−0.0536953	−0.0268476	+	0.999640i	$0.508547\pi$
$614$	9.49869	0.383336
$615$	0	0
$616$	1.34734	0.0542859
$617$	−5.99566	−0.241376	−0.120688	−	0.992690i	$-0.538510\pi$
$617$	−5.99566	−0.241376	−0.120688	+	0.992690i	$0.538510\pi$
$618$	0	0
$619$	−5.05130	−0.203029	−0.101514	−	0.994834i	$-0.532369\pi$
$619$	−5.05130	−0.203029	−0.101514	+	0.994834i	$0.532369\pi$
$620$	−5.14938	−0.206804
$621$	0	0
$622$	−31.3499	−1.25701
$623$	27.5616	1.10423
$624$	0	0
$625$	12.5422	0.501690
$626$	−7.05477	−0.281965
$627$	0	0
$628$	17.5723	0.701211
$629$	−9.92039	−0.395552
$630$	0	0
$631$	−4.94378	−0.196809	−0.0984044	−	0.995147i	$-0.531374\pi$
$631$	−4.94378	−0.196809	−0.0984044	+	0.995147i	$0.531374\pi$
$632$	11.6536	0.463556
$633$	0	0
$634$	8.13314	0.323008
$635$	13.2642	0.526374
$636$	0	0
$637$	6.07398	0.240660
$638$	0.130276	0.00515767
$639$	0	0
$640$	0.939377	0.0371321
$641$	40.2331	1.58911	0.794557	−	0.607190i	$-0.207703\pi$
$641$	40.2331	1.58911	0.794557	+	0.607190i	$0.207703\pi$
$642$	0	0
$643$	−6.43980	−0.253961	−0.126980	−	0.991905i	$-0.540529\pi$
$643$	−6.43980	−0.253961	−0.126980	+	0.991905i	$0.540529\pi$
$644$	3.40595	0.134213
$645$	0	0
$646$	5.63008	0.221513
$647$	44.9807	1.76838	0.884188	−	0.467132i	$-0.154713\pi$
$647$	44.9807	1.76838	0.884188	+	0.467132i	$0.154713\pi$
$648$	0	0
$649$	−2.94868	−0.115746
$650$	−5.31107	−0.208317
$651$	0	0
$652$	−10.1444	−0.397284
$653$	−15.0070	−0.587268	−0.293634	−	0.955918i	$-0.594865\pi$
$653$	−15.0070	−0.587268	−0.293634	+	0.955918i	$0.594865\pi$
$654$	0	0
$655$	9.49527	0.371011
$656$	12.4125	0.484628
$657$	0	0
$658$	−3.26108	−0.127130
$659$	32.0510	1.24853	0.624265	−	0.781213i	$-0.285399\pi$
$659$	32.0510	1.24853	0.624265	+	0.781213i	$0.285399\pi$
$660$	0	0
$661$	12.8228	0.498749	0.249374	−	0.968407i	$-0.419775\pi$
$661$	12.8228	0.498749	0.249374	+	0.968407i	$0.419775\pi$
$662$	24.5183	0.952932
$663$	0	0
$664$	−4.32050	−0.167668
$665$	2.31696	0.0898479
$666$	0	0
$667$	0.329325	0.0127515
$668$	−23.2455	−0.899394
$669$	0	0
$670$	11.3445	0.438275
$671$	3.26139	0.125905
$672$	0	0
$673$	19.7255	0.760362	0.380181	−	0.924912i	$-0.375862\pi$
$673$	19.7255	0.760362	0.380181	+	0.924912i	$0.375862\pi$
$674$	−1.16523	−0.0448829
$675$	0	0
$676$	−11.3363	−0.436011
$677$	38.4782	1.47884	0.739418	−	0.673246i	$-0.235101\pi$
$677$	38.4782	1.47884	0.739418	+	0.673246i	$0.235101\pi$
$678$	0	0
$679$	−34.0294	−1.30593
$680$	7.33730	0.281372
$681$	0	0
$682$	−2.15840	−0.0826494
$683$	4.38035	0.167609	0.0838047	−	0.996482i	$-0.473293\pi$
$683$	4.38035	0.167609	0.0838047	+	0.996482i	$0.473293\pi$
$684$	0	0
$685$	−8.08960	−0.309088
$686$	−7.83931	−0.299306
$687$	0	0
$688$	5.23973	0.199763
$689$	−15.1615	−0.577606
$690$	0	0
$691$	−51.7445	−1.96845	−0.984227	−	0.176911i	$-0.943390\pi$
$691$	−51.7445	−1.96845	−0.984227	+	0.176911i	$0.943390\pi$
$692$	−0.0813046	−0.00309074
$693$	0	0
$694$	−12.0680	−0.458097
$695$	16.6524	0.631661
$696$	0	0
$697$	96.9519	3.67232
$698$	−34.7355	−1.31476
$699$	0	0
$700$	−14.0897	−0.532541
$701$	−39.6526	−1.49766	−0.748830	−	0.662763i	$-0.769384\pi$
$701$	−39.6526	−1.49766	−0.748830	+	0.662763i	$0.769384\pi$
$702$	0	0
$703$	−0.915485	−0.0345282
$704$	0.393747	0.0148399
$705$	0	0
$706$	5.61228	0.211221
$707$	−39.2819	−1.47735
$708$	0	0
$709$	22.7458	0.854238	0.427119	−	0.904195i	$-0.359529\pi$
$709$	22.7458	0.854238	0.427119	+	0.904195i	$0.359529\pi$
$710$	−4.30188	−0.161447
$711$	0	0
$712$	8.05460	0.301859
$713$	−5.45622	−0.204337
$714$	0	0
$715$	0.477087	0.0178421
$716$	4.57555	0.170996
$717$	0	0
$718$	−23.8300	−0.889329
$719$	−33.6347	−1.25436	−0.627181	−	0.778873i	$-0.715791\pi$
$719$	−33.6347	−1.25436	−0.627181	+	0.778873i	$0.715791\pi$
$720$	0	0
$721$	−8.64487	−0.321952
$722$	−18.4804	−0.687771
$723$	0	0
$724$	23.2674	0.864726
$725$	−1.36235	−0.0505963
$726$	0	0
$727$	28.3010	1.04963	0.524813	−	0.851217i	$-0.324135\pi$
$727$	28.3010	1.04963	0.524813	+	0.851217i	$0.324135\pi$
$728$	4.41369	0.163582
$729$	0	0
$730$	−0.572652	−0.0211948
$731$	40.9266	1.51372
$732$	0	0
$733$	−0.501893	−0.0185378	−0.00926891	−	0.999957i	$-0.502950\pi$
$733$	−0.501893	−0.0185378	−0.00926891	+	0.999957i	$0.502950\pi$
$734$	30.1399	1.11248
$735$	0	0
$736$	0.995353	0.0366892
$737$	4.75512	0.175157
$738$	0	0
$739$	25.7247	0.946300	0.473150	−	0.880982i	$-0.343117\pi$
$739$	25.7247	0.946300	0.473150	+	0.880982i	$0.343117\pi$
$740$	−1.19309	−0.0438588
$741$	0	0
$742$	−40.2217	−1.47659
$743$	−45.6112	−1.67331	−0.836657	−	0.547727i	$-0.815493\pi$
$743$	−45.6112	−1.67331	−0.836657	+	0.547727i	$0.815493\pi$
$744$	0	0
$745$	−0.939377	−0.0344161
$746$	13.0572	0.478057
$747$	0	0
$748$	3.07548	0.112451
$749$	−45.3050	−1.65541
$750$	0	0
$751$	−7.22690	−0.263714	−0.131857	−	0.991269i	$-0.542094\pi$
$751$	−7.22690	−0.263714	−0.131857	+	0.991269i	$0.542094\pi$
$752$	−0.953016	−0.0347529
$753$	0	0
$754$	0.426764	0.0155418
$755$	17.2404	0.627441
$756$	0	0
$757$	33.7106	1.22523	0.612617	−	0.790380i	$-0.290117\pi$
$757$	33.7106	1.22523	0.612617	+	0.790380i	$0.290117\pi$
$758$	−21.7852	−0.791275
$759$	0	0
$760$	0.677109	0.0245613
$761$	28.8357	1.04529	0.522647	−	0.852549i	$-0.324945\pi$
$761$	28.8357	1.04529	0.522647	+	0.852549i	$0.324945\pi$
$762$	0	0
$763$	−47.6756	−1.72597
$764$	−5.68788	−0.205780
$765$	0	0
$766$	20.9262	0.756095
$767$	−9.65944	−0.348782
$768$	0	0
$769$	51.4902	1.85678	0.928392	−	0.371601i	$-0.121191\pi$
$769$	51.4902	1.85678	0.928392	+	0.371601i	$0.121191\pi$
$770$	1.26566	0.0456113
$771$	0	0
$772$	−21.2253	−0.763915
$773$	−29.3749	−1.05654	−0.528272	−	0.849075i	$-0.677160\pi$
$773$	−29.3749	−1.05654	−0.528272	+	0.849075i	$0.677160\pi$
$774$	0	0
$775$	22.5713	0.810784
$776$	−9.94473	−0.356995
$777$	0	0
$778$	−4.36864	−0.156623
$779$	8.94703	0.320560
$780$	0	0
$781$	−1.80317	−0.0645224
$782$	7.77452	0.278016
$783$	0	0
$784$	4.70904	0.168180
$785$	16.5070	0.589160
$786$	0	0
$787$	2.62467	0.0935594	0.0467797	−	0.998905i	$-0.485104\pi$
$787$	2.62467	0.0935594	0.0467797	+	0.998905i	$0.485104\pi$
$788$	5.12277	0.182491
$789$	0	0
$790$	10.9471	0.389481
$791$	−63.2573	−2.24917
$792$	0	0
$793$	10.6838	0.379394
$794$	−13.0226	−0.462156
$795$	0	0
$796$	0.652658	0.0231329
$797$	44.2023	1.56573	0.782863	−	0.622194i	$-0.213758\pi$
$797$	44.2023	1.56573	0.782863	+	0.622194i	$0.213758\pi$
$798$	0	0
$799$	−7.44383	−0.263344
$800$	−4.11757	−0.145578
$801$	0	0
$802$	−21.3200	−0.752836
$803$	−0.240031	−0.00847052
$804$	0	0
$805$	3.19947	0.112766
$806$	−7.07059	−0.249051
$807$	0	0
$808$	−11.4797	−0.403856
$809$	−6.99173	−0.245816	−0.122908	−	0.992418i	$-0.539222\pi$
$809$	−6.99173	−0.245816	−0.122908	+	0.992418i	$0.539222\pi$
$810$	0	0
$811$	−47.1383	−1.65525	−0.827624	−	0.561282i	$-0.810308\pi$
$811$	−47.1383	−1.65525	−0.827624	+	0.561282i	$0.810308\pi$
$812$	1.13216	0.0397310
$813$	0	0
$814$	−0.500092	−0.0175282
$815$	−9.52939	−0.333800
$816$	0	0
$817$	3.77683	0.132135
$818$	19.4308	0.679383
$819$	0	0
$820$	11.6600	0.407186
$821$	−36.7256	−1.28173	−0.640867	−	0.767652i	$-0.721425\pi$
$821$	−36.7256	−1.28173	−0.640867	+	0.767652i	$0.721425\pi$
$822$	0	0
$823$	10.1326	0.353200	0.176600	−	0.984283i	$-0.443490\pi$
$823$	10.1326	0.353200	0.176600	+	0.984283i	$0.443490\pi$
$824$	−2.52637	−0.0880104
$825$	0	0
$826$	−25.6255	−0.891624
$827$	29.8224	1.03703	0.518514	−	0.855069i	$-0.326485\pi$
$827$	29.8224	1.03703	0.518514	+	0.855069i	$0.326485\pi$
$828$	0	0
$829$	−25.8955	−0.899387	−0.449693	−	0.893183i	$-0.648467\pi$
$829$	−25.8955	−0.899387	−0.449693	+	0.893183i	$0.648467\pi$
$830$	−4.05858	−0.140875
$831$	0	0
$832$	1.28986	0.0447177
$833$	36.7814	1.27440
$834$	0	0
$835$	−21.8362	−0.755675
$836$	0.283815	0.00981595
$837$	0	0
$838$	−7.29883	−0.252134
$839$	−8.62657	−0.297822	−0.148911	−	0.988851i	$-0.547577\pi$
$839$	−8.62657	−0.297822	−0.148911	+	0.988851i	$0.547577\pi$
$840$	0	0
$841$	−28.8905	−0.996225
$842$	20.8884	0.719863
$843$	0	0
$844$	−4.91269	−0.169102
$845$	−10.6490	−0.366338
$846$	0	0
$847$	−37.1098	−1.27511
$848$	−11.7544	−0.403647
$849$	0	0
$850$	−32.1616	−1.10313
$851$	−1.26418	−0.0433356
$852$	0	0
$853$	49.7138	1.70217	0.851084	−	0.525029i	$-0.175946\pi$
$853$	49.7138	1.70217	0.851084	+	0.525029i	$0.175946\pi$
$854$	28.3431	0.969880
$855$	0	0
$856$	−13.2399	−0.452531
$857$	−26.3124	−0.898814	−0.449407	−	0.893327i	$-0.648365\pi$
$857$	−26.3124	−0.898814	−0.449407	+	0.893327i	$0.648365\pi$
$858$	0	0
$859$	43.5449	1.48573	0.742866	−	0.669440i	$-0.233466\pi$
$859$	43.5449	1.48573	0.742866	+	0.669440i	$0.233466\pi$
$860$	4.92209	0.167842
$861$	0	0
$862$	−12.1489	−0.413795
$863$	−50.2615	−1.71092	−0.855460	−	0.517869i	$-0.826725\pi$
$863$	−50.2615	−1.71092	−0.855460	+	0.517869i	$0.826725\pi$
$864$	0	0
$865$	−0.0763757	−0.00259685
$866$	−21.5369	−0.731853
$867$	0	0
$868$	−18.7575	−0.636672
$869$	4.58857	0.155657
$870$	0	0
$871$	15.5770	0.527808
$872$	−13.9327	−0.471821
$873$	0	0
$874$	0.717457	0.0242683
$875$	−29.3076	−0.990776
$876$	0	0
$877$	22.5390	0.761086	0.380543	−	0.924763i	$-0.375737\pi$
$877$	22.5390	0.761086	0.380543	+	0.924763i	$0.375737\pi$
$878$	2.11515	0.0713829
$879$	0	0
$880$	0.369877	0.0124685
$881$	14.5825	0.491295	0.245648	−	0.969359i	$-0.420999\pi$
$881$	14.5825	0.491295	0.245648	+	0.969359i	$0.420999\pi$
$882$	0	0
$883$	24.9262	0.838832	0.419416	−	0.907794i	$-0.362235\pi$
$883$	24.9262	0.838832	0.419416	+	0.907794i	$0.362235\pi$
$884$	10.0748	0.338853
$885$	0	0
$886$	4.94385	0.166092
$887$	53.4262	1.79388	0.896938	−	0.442156i	$-0.145786\pi$
$887$	53.4262	1.79388	0.896938	+	0.442156i	$0.145786\pi$
$888$	0	0
$889$	48.3172	1.62051
$890$	7.56631	0.253623
$891$	0	0
$892$	−8.14971	−0.272873
$893$	−0.686940	−0.0229876
$894$	0	0
$895$	4.29817	0.143672
$896$	3.42185	0.114316
$897$	0	0
$898$	−14.5035	−0.483990
$899$	−1.81368	−0.0604898
$900$	0	0
$901$	−91.8113	−3.05868
$902$	4.88739	0.162732
$903$	0	0
$904$	−18.4863	−0.614845
$905$	21.8568	0.726546
$906$	0	0
$907$	33.9444	1.12711	0.563553	−	0.826080i	$-0.309434\pi$
$907$	33.9444	1.12711	0.563553	+	0.826080i	$0.309434\pi$
$908$	18.1369	0.601896
$909$	0	0
$910$	4.14612	0.137442
$911$	28.7838	0.953651	0.476826	−	0.878998i	$-0.341787\pi$
$911$	28.7838	0.953651	0.476826	+	0.878998i	$0.341787\pi$
$912$	0	0
$913$	−1.70118	−0.0563009
$914$	−18.2987	−0.605268
$915$	0	0
$916$	−0.00980482	−0.000323960 0
$917$	34.5882	1.14220
$918$	0	0
$919$	0.500581	0.0165127	0.00825633	−	0.999966i	$-0.497372\pi$
$919$	0.500581	0.0165127	0.00825633	+	0.999966i	$0.497372\pi$
$920$	0.935012	0.0308264
$921$	0	0
$922$	11.3647	0.374277
$923$	−5.90690	−0.194428
$924$	0	0
$925$	5.22966	0.171950
$926$	31.6110	1.03880
$927$	0	0
$928$	0.330862	0.0108611
$929$	−19.0004	−0.623381	−0.311691	−	0.950184i	$-0.600895\pi$
$929$	−19.0004	−0.623381	−0.311691	+	0.950184i	$0.600895\pi$
$930$	0	0
$931$	3.39430	0.111244
$932$	20.4897	0.671161
$933$	0	0
$934$	14.2382	0.465888
$935$	2.88904	0.0944816
$936$	0	0
$937$	−49.5244	−1.61789	−0.808946	−	0.587883i	$-0.799961\pi$
$937$	−49.5244	−1.61789	−0.808946	+	0.587883i	$0.799961\pi$
$938$	41.3242	1.34928
$939$	0	0
$940$	−0.895241	−0.0291996
$941$	−46.3405	−1.51066	−0.755329	−	0.655346i	$-0.772523\pi$
$941$	−46.3405	−1.51066	−0.755329	+	0.655346i	$0.772523\pi$
$942$	0	0
$943$	12.3549	0.402329
$944$	−7.48878	−0.243739
$945$	0	0
$946$	2.06313	0.0670781
$947$	21.4911	0.698368	0.349184	−	0.937054i	$-0.386459\pi$
$947$	21.4911	0.698368	0.349184	+	0.937054i	$0.386459\pi$
$948$	0	0
$949$	−0.786307	−0.0255246
$950$	−2.96797	−0.0962937
$951$	0	0
$952$	26.7274	0.866240
$953$	−17.4741	−0.566043	−0.283021	−	0.959114i	$-0.591337\pi$
$953$	−17.4741	−0.566043	−0.283021	+	0.959114i	$0.591337\pi$
$954$	0	0
$955$	−5.34306	−0.172897
$956$	−21.0947	−0.682251
$957$	0	0
$958$	31.8865	1.03021
$959$	−29.4678	−0.951565
$960$	0	0
$961$	−0.951028	−0.0306783
$962$	−1.63823	−0.0528185
$963$	0	0
$964$	8.35135	0.268979
$965$	−19.9385	−0.641844
$966$	0	0
$967$	−21.1917	−0.681480	−0.340740	−	0.940157i	$-0.610678\pi$
$967$	−21.1917	−0.681480	−0.340740	+	0.940157i	$0.610678\pi$
$968$	−10.8450	−0.348570
$969$	0	0
$970$	−9.34185	−0.299949
$971$	34.5381	1.10838	0.554190	−	0.832390i	$-0.313028\pi$
$971$	34.5381	1.10838	0.554190	+	0.832390i	$0.313028\pi$
$972$	0	0
$973$	60.6592	1.94465
$974$	35.3344	1.13219
$975$	0	0
$976$	8.28297	0.265132
$977$	−2.96196	−0.0947614	−0.0473807	−	0.998877i	$-0.515087\pi$
$977$	−2.96196	−0.0947614	−0.0473807	+	0.998877i	$0.515087\pi$
$978$	0	0
$979$	3.17147	0.101361
$980$	4.42356	0.141306
$981$	0	0
$982$	−25.5224	−0.814452
$983$	−40.0487	−1.27736	−0.638678	−	0.769474i	$-0.720518\pi$
$983$	−40.0487	−1.27736	−0.638678	+	0.769474i	$0.720518\pi$
$984$	0	0
$985$	4.81221	0.153330
$986$	2.58430	0.0823009
$987$	0	0
$988$	0.929736	0.0295788
$989$	5.21539	0.165840
$990$	0	0
$991$	6.36282	0.202122	0.101061	−	0.994880i	$-0.467776\pi$
$991$	6.36282	0.202122	0.101061	+	0.994880i	$0.467776\pi$
$992$	−5.48169	−0.174044
$993$	0	0
$994$	−15.6704	−0.497034
$995$	0.613092	0.0194363
$996$	0	0
$997$	1.79961	0.0569944	0.0284972	−	0.999594i	$-0.490928\pi$
$997$	1.79961	0.0569944	0.0284972	+	0.999594i	$0.490928\pi$
$998$	−14.5862	−0.461719
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.t.1.9	yes	16
3.2	odd	2		8046.2.a.s.1.8	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.s.1.8	✓	16	3.2	odd	2
8046.2.a.t.1.9	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.939377 q^{5} +3.42185 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.939377 q^{5} +3.42185 q^{7} +1.00000 q^{8} +0.939377 q^{10} +0.393747 q^{11} +1.28986 q^{13} +3.42185 q^{14} +1.00000 q^{16} +7.81081 q^{17} +0.720806 q^{19} +0.939377 q^{20} +0.393747 q^{22} +0.995353 q^{23} -4.11757 q^{25} +1.28986 q^{26} +3.42185 q^{28} +0.330862 q^{29} -5.48169 q^{31} +1.00000 q^{32} +7.81081 q^{34} +3.21440 q^{35} -1.27008 q^{37} +0.720806 q^{38} +0.939377 q^{40} +12.4125 q^{41} +5.23973 q^{43} +0.393747 q^{44} +0.995353 q^{46} -0.953016 q^{47} +4.70904 q^{49} -4.11757 q^{50} +1.28986 q^{52} -11.7544 q^{53} +0.369877 q^{55} +3.42185 q^{56} +0.330862 q^{58} -7.48878 q^{59} +8.28297 q^{61} -5.48169 q^{62} +1.00000 q^{64} +1.21166 q^{65} +12.0766 q^{67} +7.81081 q^{68} +3.21440 q^{70} -4.57951 q^{71} -0.609608 q^{73} -1.27008 q^{74} +0.720806 q^{76} +1.34734 q^{77} +11.6536 q^{79} +0.939377 q^{80} +12.4125 q^{82} -4.32050 q^{83} +7.33730 q^{85} +5.23973 q^{86} +0.393747 q^{88} +8.05460 q^{89} +4.41369 q^{91} +0.995353 q^{92} -0.953016 q^{94} +0.677109 q^{95} -9.94473 q^{97} +4.70904 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8} + 4 q^{10} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 10 q^{19} + 4 q^{20} + 6 q^{22} + 10 q^{23} + 28 q^{25} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 21 q^{31} + 16 q^{32} + q^{34} + 16 q^{35} + 17 q^{37} + 10 q^{38} + 4 q^{40} - 4 q^{41} + 16 q^{43} + 6 q^{44} + 10 q^{46} + 25 q^{47} + 36 q^{49} + 28 q^{50} + 6 q^{52} + 14 q^{53} + 19 q^{55} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 23 q^{61} + 21 q^{62} + 16 q^{64} + 20 q^{65} + 22 q^{67} + q^{68} + 16 q^{70} + 10 q^{71} + 16 q^{73} + 17 q^{74} + 10 q^{76} - 2 q^{77} + 37 q^{79} + 4 q^{80} - 4 q^{82} + 33 q^{83} + 43 q^{85} + 16 q^{86} + 6 q^{88} - 3 q^{89} + 28 q^{91} + 10 q^{92} + 25 q^{94} + 14 q^{95} - 3 q^{97} + 36 q^{98}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8 + 4 * q^10 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 10 * q^19 + 4 * q^20 + 6 * q^22 + 10 * q^23 + 28 * q^25 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 21 * q^31 + 16 * q^32 + q^34 + 16 * q^35 + 17 * q^37 + 10 * q^38 + 4 * q^40 - 4 * q^41 + 16 * q^43 + 6 * q^44 + 10 * q^46 + 25 * q^47 + 36 * q^49 + 28 * q^50 + 6 * q^52 + 14 * q^53 + 19 * q^55 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 23 * q^61 + 21 * q^62 + 16 * q^64 + 20 * q^65 + 22 * q^67 + q^68 + 16 * q^70 + 10 * q^71 + 16 * q^73 + 17 * q^74 + 10 * q^76 - 2 * q^77 + 37 * q^79 + 4 * q^80 - 4 * q^82 + 33 * q^83 + 43 * q^85 + 16 * q^86 + 6 * q^88 - 3 * q^89 + 28 * q^91 + 10 * q^92 + 25 * q^94 + 14 * q^95 - 3 * q^97 + 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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