LMFDB - Embedded newform 8046.2.a.j.1.12

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			1.12
Root			$-2.89416$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.89416 q^{5} +0.647278 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.89416 q^{5} +0.647278 q^{7} -1.00000 q^{8} -2.89416 q^{10} -3.72846 q^{11} +0.715732 q^{13} -0.647278 q^{14} +1.00000 q^{16} -0.587143 q^{17} +6.18007 q^{19} +2.89416 q^{20} +3.72846 q^{22} -7.30961 q^{23} +3.37615 q^{25} -0.715732 q^{26} +0.647278 q^{28} -0.536329 q^{29} -8.61374 q^{31} -1.00000 q^{32} +0.587143 q^{34} +1.87333 q^{35} -2.58020 q^{37} -6.18007 q^{38} -2.89416 q^{40} +6.06139 q^{41} +0.0254145 q^{43} -3.72846 q^{44} +7.30961 q^{46} -3.03761 q^{47} -6.58103 q^{49} -3.37615 q^{50} +0.715732 q^{52} -13.3216 q^{53} -10.7908 q^{55} -0.647278 q^{56} +0.536329 q^{58} +8.44682 q^{59} -1.72772 q^{61} +8.61374 q^{62} +1.00000 q^{64} +2.07144 q^{65} -1.14341 q^{67} -0.587143 q^{68} -1.87333 q^{70} -6.82877 q^{71} +7.14332 q^{73} +2.58020 q^{74} +6.18007 q^{76} -2.41335 q^{77} +3.34399 q^{79} +2.89416 q^{80} -6.06139 q^{82} +12.5992 q^{83} -1.69928 q^{85} -0.0254145 q^{86} +3.72846 q^{88} -12.7234 q^{89} +0.463278 q^{91} -7.30961 q^{92} +3.03761 q^{94} +17.8861 q^{95} -15.1212 q^{97} +6.58103 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.89416	1.29431	0.647153	−	0.762360i	$-0.275959\pi$
$5$	2.89416	1.29431	0.647153	+	0.762360i	$0.275959\pi$
$6$	0	0
$7$	0.647278	0.244648	0.122324	−	0.992490i	$-0.460965\pi$
$7$	0.647278	0.244648	0.122324	+	0.992490i	$0.460965\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.89416	−0.915213
$11$	−3.72846	−1.12417	−0.562086	−	0.827078i	$-0.690001\pi$
$11$	−3.72846	−1.12417	−0.562086	+	0.827078i	$0.690001\pi$
$12$	0	0
$13$	0.715732	0.198508	0.0992542	−	0.995062i	$-0.468354\pi$
$13$	0.715732	0.198508	0.0992542	+	0.995062i	$0.468354\pi$
$14$	−0.647278	−0.172992
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.587143	−0.142403	−0.0712015	−	0.997462i	$-0.522683\pi$
$17$	−0.587143	−0.142403	−0.0712015	+	0.997462i	$0.522683\pi$
$18$	0	0
$19$	6.18007	1.41781	0.708903	−	0.705306i	$-0.249190\pi$
$19$	6.18007	1.41781	0.708903	+	0.705306i	$0.249190\pi$
$20$	2.89416	0.647153
$21$	0	0
$22$	3.72846	0.794910
$23$	−7.30961	−1.52416	−0.762079	−	0.647484i	$-0.775821\pi$
$23$	−7.30961	−1.52416	−0.762079	+	0.647484i	$0.775821\pi$
$24$	0	0
$25$	3.37615	0.675230
$26$	−0.715732	−0.140367
$27$	0	0
$28$	0.647278	0.122324
$29$	−0.536329	−0.0995937	−0.0497969	−	0.998759i	$-0.515857\pi$
$29$	−0.536329	−0.0995937	−0.0497969	+	0.998759i	$0.515857\pi$
$30$	0	0
$31$	−8.61374	−1.54707	−0.773536	−	0.633752i	$-0.781514\pi$
$31$	−8.61374	−1.54707	−0.773536	+	0.633752i	$0.781514\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.587143	0.100694
$35$	1.87333	0.316650
$36$	0	0
$37$	−2.58020	−0.424183	−0.212091	−	0.977250i	$-0.568027\pi$
$37$	−2.58020	−0.424183	−0.212091	+	0.977250i	$0.568027\pi$
$38$	−6.18007	−1.00254
$39$	0	0
$40$	−2.89416	−0.457607
$41$	6.06139	0.946630	0.473315	−	0.880893i	$-0.343057\pi$
$41$	6.06139	0.946630	0.473315	+	0.880893i	$0.343057\pi$
$42$	0	0
$43$	0.0254145	0.00387567	0.00193784	−	0.999998i	$-0.499383\pi$
$43$	0.0254145	0.00387567	0.00193784	+	0.999998i	$0.499383\pi$
$44$	−3.72846	−0.562086
$45$	0	0
$46$	7.30961	1.07774
$47$	−3.03761	−0.443081	−0.221540	−	0.975151i	$-0.571108\pi$
$47$	−3.03761	−0.443081	−0.221540	+	0.975151i	$0.571108\pi$
$48$	0	0
$49$	−6.58103	−0.940147
$50$	−3.37615	−0.477460
$51$	0	0
$52$	0.715732	0.0992542
$53$	−13.3216	−1.82987	−0.914933	−	0.403607i	$-0.867756\pi$
$53$	−13.3216	−1.82987	−0.914933	+	0.403607i	$0.867756\pi$
$54$	0	0
$55$	−10.7908	−1.45502
$56$	−0.647278	−0.0864962
$57$	0	0
$58$	0.536329	0.0704234
$59$	8.44682	1.09968	0.549841	−	0.835269i	$-0.314688\pi$
$59$	8.44682	1.09968	0.549841	+	0.835269i	$0.314688\pi$
$60$	0	0
$61$	−1.72772	−0.221212	−0.110606	−	0.993864i	$-0.535279\pi$
$61$	−1.72772	−0.221212	−0.110606	+	0.993864i	$0.535279\pi$
$62$	8.61374	1.09395
$63$	0	0
$64$	1.00000	0.125000
$65$	2.07144	0.256931
$66$	0	0
$67$	−1.14341	−0.139689	−0.0698447	−	0.997558i	$-0.522250\pi$
$67$	−1.14341	−0.139689	−0.0698447	+	0.997558i	$0.522250\pi$
$68$	−0.587143	−0.0712015
$69$	0	0
$70$	−1.87333	−0.223905
$71$	−6.82877	−0.810425	−0.405213	−	0.914222i	$-0.632803\pi$
$71$	−6.82877	−0.810425	−0.405213	+	0.914222i	$0.632803\pi$
$72$	0	0
$73$	7.14332	0.836063	0.418031	−	0.908433i	$-0.362720\pi$
$73$	7.14332	0.836063	0.418031	+	0.908433i	$0.362720\pi$
$74$	2.58020	0.299942
$75$	0	0
$76$	6.18007	0.708903
$77$	−2.41335	−0.275027
$78$	0	0
$79$	3.34399	0.376228	0.188114	−	0.982147i	$-0.439763\pi$
$79$	3.34399	0.376228	0.188114	+	0.982147i	$0.439763\pi$
$80$	2.89416	0.323577
$81$	0	0
$82$	−6.06139	−0.669369
$83$	12.5992	1.38294	0.691472	−	0.722404i	$-0.256963\pi$
$83$	12.5992	1.38294	0.691472	+	0.722404i	$0.256963\pi$
$84$	0	0
$85$	−1.69928	−0.184313
$86$	−0.0254145	−0.00274051
$87$	0	0
$88$	3.72846	0.397455
$89$	−12.7234	−1.34868	−0.674341	−	0.738420i	$-0.735572\pi$
$89$	−12.7234	−1.34868	−0.674341	+	0.738420i	$0.735572\pi$
$90$	0	0
$91$	0.463278	0.0485647
$92$	−7.30961	−0.762079
$93$	0	0
$94$	3.03761	0.313305
$95$	17.8861	1.83508
$96$	0	0
$97$	−15.1212	−1.53533	−0.767663	−	0.640853i	$-0.778581\pi$
$97$	−15.1212	−1.53533	−0.767663	+	0.640853i	$0.778581\pi$
$98$	6.58103	0.664785
$99$	0	0
$100$	3.37615	0.337615
$101$	−4.60974	−0.458687	−0.229343	−	0.973346i	$-0.573658\pi$
$101$	−4.60974	−0.458687	−0.229343	+	0.973346i	$0.573658\pi$
$102$	0	0
$103$	19.7923	1.95019	0.975095	−	0.221788i	$-0.0711892\pi$
$103$	19.7923	1.95019	0.975095	+	0.221788i	$0.0711892\pi$
$104$	−0.715732	−0.0701833
$105$	0	0
$106$	13.3216	1.29391
$107$	0.116319	0.0112449	0.00562247	−	0.999984i	$-0.498210\pi$
$107$	0.116319	0.0112449	0.00562247	+	0.999984i	$0.498210\pi$
$108$	0	0
$109$	−17.4189	−1.66843	−0.834216	−	0.551438i	$-0.814079\pi$
$109$	−17.4189	−1.66843	−0.834216	+	0.551438i	$0.814079\pi$
$110$	10.7908	1.02886
$111$	0	0
$112$	0.647278	0.0611620
$113$	−14.7319	−1.38586	−0.692928	−	0.721007i	$-0.743680\pi$
$113$	−14.7319	−1.38586	−0.692928	+	0.721007i	$0.743680\pi$
$114$	0	0
$115$	−21.1552	−1.97273
$116$	−0.536329	−0.0497969
$117$	0	0
$118$	−8.44682	−0.777593
$119$	−0.380045	−0.0348386
$120$	0	0
$121$	2.90141	0.263765
$122$	1.72772	0.156421
$123$	0	0
$124$	−8.61374	−0.773536
$125$	−4.69968	−0.420352
$126$	0	0
$127$	16.5659	1.46999	0.734993	−	0.678074i	$-0.237185\pi$
$127$	16.5659	1.46999	0.734993	+	0.678074i	$0.237185\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.07144	−0.181677
$131$	−3.97969	−0.347707	−0.173853	−	0.984772i	$-0.555622\pi$
$131$	−3.97969	−0.347707	−0.173853	+	0.984772i	$0.555622\pi$
$132$	0	0
$133$	4.00023	0.346864
$134$	1.14341	0.0987753
$135$	0	0
$136$	0.587143	0.0503471
$137$	−17.7470	−1.51622	−0.758112	−	0.652124i	$-0.773878\pi$
$137$	−17.7470	−1.51622	−0.758112	+	0.652124i	$0.773878\pi$
$138$	0	0
$139$	−3.40466	−0.288779	−0.144390	−	0.989521i	$-0.546122\pi$
$139$	−3.40466	−0.288779	−0.144390	+	0.989521i	$0.546122\pi$
$140$	1.87333	0.158325
$141$	0	0
$142$	6.82877	0.573057
$143$	−2.66858	−0.223158
$144$	0	0
$145$	−1.55222	−0.128905
$146$	−7.14332	−0.591186
$147$	0	0
$148$	−2.58020	−0.212091
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−10.6585	−0.867373	−0.433687	−	0.901064i	$-0.642787\pi$
$151$	−10.6585	−0.867373	−0.433687	+	0.901064i	$0.642787\pi$
$152$	−6.18007	−0.501270
$153$	0	0
$154$	2.41335	0.194473
$155$	−24.9295	−2.00239
$156$	0	0
$157$	10.8183	0.863397	0.431699	−	0.902018i	$-0.357914\pi$
$157$	10.8183	0.863397	0.431699	+	0.902018i	$0.357914\pi$
$158$	−3.34399	−0.266034
$159$	0	0
$160$	−2.89416	−0.228803
$161$	−4.73135	−0.372883
$162$	0	0
$163$	6.33240	0.495992	0.247996	−	0.968761i	$-0.420228\pi$
$163$	6.33240	0.495992	0.247996	+	0.968761i	$0.420228\pi$
$164$	6.06139	0.473315
$165$	0	0
$166$	−12.5992	−0.977888
$167$	20.0433	1.55099	0.775497	−	0.631351i	$-0.217499\pi$
$167$	20.0433	1.55099	0.775497	+	0.631351i	$0.217499\pi$
$168$	0	0
$169$	−12.4877	−0.960594
$170$	1.69928	0.130329
$171$	0	0
$172$	0.0254145	0.00193784
$173$	−19.7813	−1.50395	−0.751974	−	0.659193i	$-0.770898\pi$
$173$	−19.7813	−1.50395	−0.751974	+	0.659193i	$0.770898\pi$
$174$	0	0
$175$	2.18531	0.165194
$176$	−3.72846	−0.281043
$177$	0	0
$178$	12.7234	0.953662
$179$	−8.80643	−0.658224	−0.329112	−	0.944291i	$-0.606749\pi$
$179$	−8.80643	−0.658224	−0.329112	+	0.944291i	$0.606749\pi$
$180$	0	0
$181$	−2.53023	−0.188070	−0.0940352	−	0.995569i	$-0.529977\pi$
$181$	−2.53023	−0.188070	−0.0940352	+	0.995569i	$0.529977\pi$
$182$	−0.463278	−0.0343404
$183$	0	0
$184$	7.30961	0.538872
$185$	−7.46751	−0.549022
$186$	0	0
$187$	2.18914	0.160086
$188$	−3.03761	−0.221540
$189$	0	0
$190$	−17.8861	−1.29759
$191$	−11.0216	−0.797497	−0.398748	−	0.917060i	$-0.630555\pi$
$191$	−11.0216	−0.797497	−0.398748	+	0.917060i	$0.630555\pi$
$192$	0	0
$193$	13.0378	0.938481	0.469240	−	0.883070i	$-0.344528\pi$
$193$	13.0378	0.938481	0.469240	+	0.883070i	$0.344528\pi$
$194$	15.1212	1.08564
$195$	0	0
$196$	−6.58103	−0.470074
$197$	−6.51445	−0.464136	−0.232068	−	0.972700i	$-0.574549\pi$
$197$	−6.51445	−0.464136	−0.232068	+	0.972700i	$0.574549\pi$
$198$	0	0
$199$	8.24335	0.584356	0.292178	−	0.956364i	$-0.405620\pi$
$199$	8.24335	0.584356	0.292178	+	0.956364i	$0.405620\pi$
$200$	−3.37615	−0.238730
$201$	0	0
$202$	4.60974	0.324340
$203$	−0.347154	−0.0243654
$204$	0	0
$205$	17.5426	1.22523
$206$	−19.7923	−1.37899
$207$	0	0
$208$	0.715732	0.0496271
$209$	−23.0422	−1.59386
$210$	0	0
$211$	14.4493	0.994729	0.497364	−	0.867542i	$-0.334301\pi$
$211$	14.4493	0.994729	0.497364	+	0.867542i	$0.334301\pi$
$212$	−13.3216	−0.914933
$213$	0	0
$214$	−0.116319	−0.00795137
$215$	0.0735535	0.00501631
$216$	0	0
$217$	−5.57548	−0.378489
$218$	17.4189	1.17976
$219$	0	0
$220$	−10.7908	−0.727512
$221$	−0.420237	−0.0282682
$222$	0	0
$223$	−14.0558	−0.941249	−0.470624	−	0.882334i	$-0.655971\pi$
$223$	−14.0558	−0.941249	−0.470624	+	0.882334i	$0.655971\pi$
$224$	−0.647278	−0.0432481
$225$	0	0
$226$	14.7319	0.979948
$227$	−17.2851	−1.14725	−0.573626	−	0.819118i	$-0.694464\pi$
$227$	−17.2851	−1.14725	−0.573626	+	0.819118i	$0.694464\pi$
$228$	0	0
$229$	−9.37902	−0.619783	−0.309892	−	0.950772i	$-0.600293\pi$
$229$	−9.37902	−0.619783	−0.309892	+	0.950772i	$0.600293\pi$
$230$	21.1552	1.39493
$231$	0	0
$232$	0.536329	0.0352117
$233$	−17.5605	−1.15043	−0.575213	−	0.818003i	$-0.695081\pi$
$233$	−17.5605	−1.15043	−0.575213	+	0.818003i	$0.695081\pi$
$234$	0	0
$235$	−8.79132	−0.573482
$236$	8.44682	0.549841
$237$	0	0
$238$	0.380045	0.0246346
$239$	28.9074	1.86987	0.934933	−	0.354823i	$-0.115459\pi$
$239$	28.9074	1.86987	0.934933	+	0.354823i	$0.115459\pi$
$240$	0	0
$241$	22.3729	1.44117	0.720583	−	0.693369i	$-0.243874\pi$
$241$	22.3729	1.44117	0.720583	+	0.693369i	$0.243874\pi$
$242$	−2.90141	−0.186510
$243$	0	0
$244$	−1.72772	−0.110606
$245$	−19.0465	−1.21684
$246$	0	0
$247$	4.42328	0.281446
$248$	8.61374	0.546973
$249$	0	0
$250$	4.69968	0.297234
$251$	−14.0275	−0.885410	−0.442705	−	0.896667i	$-0.645981\pi$
$251$	−14.0275	−0.885410	−0.442705	+	0.896667i	$0.645981\pi$
$252$	0	0
$253$	27.2536	1.71342
$254$	−16.5659	−1.03944
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.63634	−0.601099	−0.300549	−	0.953766i	$-0.597170\pi$
$257$	−9.63634	−0.601099	−0.300549	+	0.953766i	$0.597170\pi$
$258$	0	0
$259$	−1.67011	−0.103776
$260$	2.07144	0.128465
$261$	0	0
$262$	3.97969	0.245866
$263$	21.5566	1.32924	0.664620	−	0.747182i	$-0.268594\pi$
$263$	21.5566	1.32924	0.664620	+	0.747182i	$0.268594\pi$
$264$	0	0
$265$	−38.5549	−2.36841
$266$	−4.00023	−0.245270
$267$	0	0
$268$	−1.14341	−0.0698447
$269$	−26.4569	−1.61311	−0.806554	−	0.591161i	$-0.798670\pi$
$269$	−26.4569	−1.61311	−0.806554	+	0.591161i	$0.798670\pi$
$270$	0	0
$271$	11.9830	0.727915	0.363958	−	0.931415i	$-0.381425\pi$
$271$	11.9830	0.727915	0.363958	+	0.931415i	$0.381425\pi$
$272$	−0.587143	−0.0356008
$273$	0	0
$274$	17.7470	1.07213
$275$	−12.5878	−0.759075
$276$	0	0
$277$	23.0482	1.38483	0.692416	−	0.721499i	$-0.256546\pi$
$277$	23.0482	1.38483	0.692416	+	0.721499i	$0.256546\pi$
$278$	3.40466	0.204198
$279$	0	0
$280$	−1.87333	−0.111953
$281$	−16.6551	−0.993561	−0.496781	−	0.867876i	$-0.665485\pi$
$281$	−16.6551	−0.993561	−0.496781	+	0.867876i	$0.665485\pi$
$282$	0	0
$283$	−4.34695	−0.258399	−0.129200	−	0.991619i	$-0.541241\pi$
$283$	−4.34695	−0.258399	−0.129200	+	0.991619i	$0.541241\pi$
$284$	−6.82877	−0.405213
$285$	0	0
$286$	2.66858	0.157796
$287$	3.92341	0.231591
$288$	0	0
$289$	−16.6553	−0.979721
$290$	1.55222	0.0911495
$291$	0	0
$292$	7.14332	0.418031
$293$	17.5782	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$293$	17.5782	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$294$	0	0
$295$	24.4464	1.42333
$296$	2.58020	0.149971
$297$	0	0
$298$	1.00000	0.0579284
$299$	−5.23172	−0.302558
$300$	0	0
$301$	0.0164502	0.000948176 0
$302$	10.6585	0.613325
$303$	0	0
$304$	6.18007	0.354452
$305$	−5.00030	−0.286316
$306$	0	0
$307$	−13.8702	−0.791612	−0.395806	−	0.918334i	$-0.629535\pi$
$307$	−13.8702	−0.791612	−0.395806	+	0.918334i	$0.629535\pi$
$308$	−2.41335	−0.137513
$309$	0	0
$310$	24.9295	1.41590
$311$	9.97478	0.565618	0.282809	−	0.959176i	$-0.408734\pi$
$311$	9.97478	0.565618	0.282809	+	0.959176i	$0.408734\pi$
$312$	0	0
$313$	22.0525	1.24648	0.623240	−	0.782031i	$-0.285816\pi$
$313$	22.0525	1.24648	0.623240	+	0.782031i	$0.285816\pi$
$314$	−10.8183	−0.610514
$315$	0	0
$316$	3.34399	0.188114
$317$	−2.36150	−0.132635	−0.0663176	−	0.997799i	$-0.521125\pi$
$317$	−2.36150	−0.132635	−0.0663176	+	0.997799i	$0.521125\pi$
$318$	0	0
$319$	1.99968	0.111961
$320$	2.89416	0.161788
$321$	0	0
$322$	4.73135	0.263668
$323$	−3.62859	−0.201900
$324$	0	0
$325$	2.41642	0.134039
$326$	−6.33240	−0.350719
$327$	0	0
$328$	−6.06139	−0.334684
$329$	−1.96618	−0.108399
$330$	0	0
$331$	−24.6246	−1.35349	−0.676745	−	0.736217i	$-0.736610\pi$
$331$	−24.6246	−1.35349	−0.676745	+	0.736217i	$0.736610\pi$
$332$	12.5992	0.691472
$333$	0	0
$334$	−20.0433	−1.09672
$335$	−3.30920	−0.180801
$336$	0	0
$337$	4.12734	0.224831	0.112415	−	0.993661i	$-0.464141\pi$
$337$	4.12734	0.224831	0.112415	+	0.993661i	$0.464141\pi$
$338$	12.4877	0.679243
$339$	0	0
$340$	−1.69928	−0.0921566
$341$	32.1160	1.73918
$342$	0	0
$343$	−8.79070	−0.474653
$344$	−0.0254145	−0.00137026
$345$	0	0
$346$	19.7813	1.06345
$347$	−1.66183	−0.0892120	−0.0446060	−	0.999005i	$-0.514203\pi$
$347$	−1.66183	−0.0892120	−0.0446060	+	0.999005i	$0.514203\pi$
$348$	0	0
$349$	−9.67092	−0.517673	−0.258836	−	0.965921i	$-0.583339\pi$
$349$	−9.67092	−0.517673	−0.258836	+	0.965921i	$0.583339\pi$
$350$	−2.18531	−0.116810
$351$	0	0
$352$	3.72846	0.198728
$353$	28.8909	1.53771	0.768853	−	0.639426i	$-0.220828\pi$
$353$	28.8909	1.53771	0.768853	+	0.639426i	$0.220828\pi$
$354$	0	0
$355$	−19.7635	−1.04894
$356$	−12.7234	−0.674341
$357$	0	0
$358$	8.80643	0.465434
$359$	−6.28476	−0.331697	−0.165849	−	0.986151i	$-0.553036\pi$
$359$	−6.28476	−0.331697	−0.165849	+	0.986151i	$0.553036\pi$
$360$	0	0
$361$	19.1933	1.01017
$362$	2.53023	0.132986
$363$	0	0
$364$	0.463278	0.0242823
$365$	20.6739	1.08212
$366$	0	0
$367$	−23.5043	−1.22691	−0.613457	−	0.789728i	$-0.710221\pi$
$367$	−23.5043	−1.22691	−0.613457	+	0.789728i	$0.710221\pi$
$368$	−7.30961	−0.381040
$369$	0	0
$370$	7.46751	0.388218
$371$	−8.62279	−0.447673
$372$	0	0
$373$	8.22391	0.425818	0.212909	−	0.977072i	$-0.431706\pi$
$373$	8.22391	0.425818	0.212909	+	0.977072i	$0.431706\pi$
$374$	−2.18914	−0.113198
$375$	0	0
$376$	3.03761	0.156653
$377$	−0.383868	−0.0197702
$378$	0	0
$379$	17.0746	0.877065	0.438533	−	0.898715i	$-0.355498\pi$
$379$	17.0746	0.877065	0.438533	+	0.898715i	$0.355498\pi$
$380$	17.8861	0.917538
$381$	0	0
$382$	11.0216	0.563915
$383$	9.91645	0.506707	0.253354	−	0.967374i	$-0.418466\pi$
$383$	9.91645	0.506707	0.253354	+	0.967374i	$0.418466\pi$
$384$	0	0
$385$	−6.98462	−0.355969
$386$	−13.0378	−0.663606
$387$	0	0
$388$	−15.1212	−0.767663
$389$	20.6837	1.04871	0.524353	−	0.851501i	$-0.324307\pi$
$389$	20.6837	1.04871	0.524353	+	0.851501i	$0.324307\pi$
$390$	0	0
$391$	4.29178	0.217045
$392$	6.58103	0.332392
$393$	0	0
$394$	6.51445	0.328193
$395$	9.67804	0.486955
$396$	0	0
$397$	−10.8701	−0.545557	−0.272778	−	0.962077i	$-0.587943\pi$
$397$	−10.8701	−0.545557	−0.272778	+	0.962077i	$0.587943\pi$
$398$	−8.24335	−0.413202
$399$	0	0
$400$	3.37615	0.168807
$401$	27.8798	1.39225	0.696125	−	0.717921i	$-0.254906\pi$
$401$	27.8798	1.39225	0.696125	+	0.717921i	$0.254906\pi$
$402$	0	0
$403$	−6.16513	−0.307107
$404$	−4.60974	−0.229343
$405$	0	0
$406$	0.347154	0.0172290
$407$	9.62018	0.476855
$408$	0	0
$409$	−21.3723	−1.05679	−0.528397	−	0.848997i	$-0.677207\pi$
$409$	−21.3723	−1.05679	−0.528397	+	0.848997i	$0.677207\pi$
$410$	−17.5426	−0.866368
$411$	0	0
$412$	19.7923	0.975095
$413$	5.46744	0.269035
$414$	0	0
$415$	36.4641	1.78995
$416$	−0.715732	−0.0350916
$417$	0	0
$418$	23.0422	1.12703
$419$	−0.754815	−0.0368751	−0.0184376	−	0.999830i	$-0.505869\pi$
$419$	−0.754815	−0.0368751	−0.0184376	+	0.999830i	$0.505869\pi$
$420$	0	0
$421$	24.0721	1.17320	0.586601	−	0.809876i	$-0.300466\pi$
$421$	24.0721	1.17320	0.586601	+	0.809876i	$0.300466\pi$
$422$	−14.4493	−0.703379
$423$	0	0
$424$	13.3216	0.646955
$425$	−1.98228	−0.0961548
$426$	0	0
$427$	−1.11832	−0.0541191
$428$	0.116319	0.00562247
$429$	0	0
$430$	−0.0735535	−0.00354707
$431$	−10.4667	−0.504162	−0.252081	−	0.967706i	$-0.581115\pi$
$431$	−10.4667	−0.504162	−0.252081	+	0.967706i	$0.581115\pi$
$432$	0	0
$433$	2.47142	0.118769	0.0593843	−	0.998235i	$-0.481086\pi$
$433$	2.47142	0.118769	0.0593843	+	0.998235i	$0.481086\pi$
$434$	5.57548	0.267632
$435$	0	0
$436$	−17.4189	−0.834216
$437$	−45.1739	−2.16096
$438$	0	0
$439$	37.7653	1.80244	0.901221	−	0.433361i	$-0.142672\pi$
$439$	37.7653	1.80244	0.901221	+	0.433361i	$0.142672\pi$
$440$	10.7908	0.514429
$441$	0	0
$442$	0.420237	0.0199886
$443$	−10.3172	−0.490184	−0.245092	−	0.969500i	$-0.578818\pi$
$443$	−10.3172	−0.490184	−0.245092	+	0.969500i	$0.578818\pi$
$444$	0	0
$445$	−36.8237	−1.74561
$446$	14.0558	0.665563
$447$	0	0
$448$	0.647278	0.0305810
$449$	18.1795	0.857946	0.428973	−	0.903317i	$-0.358876\pi$
$449$	18.1795	0.857946	0.428973	+	0.903317i	$0.358876\pi$
$450$	0	0
$451$	−22.5996	−1.06418
$452$	−14.7319	−0.692928
$453$	0	0
$454$	17.2851	0.811229
$455$	1.34080	0.0628576
$456$	0	0
$457$	34.7422	1.62517	0.812587	−	0.582841i	$-0.198059\pi$
$457$	34.7422	1.62517	0.812587	+	0.582841i	$0.198059\pi$
$458$	9.37902	0.438253
$459$	0	0
$460$	−21.1552	−0.986365
$461$	28.6089	1.33245	0.666224	−	0.745752i	$-0.267909\pi$
$461$	28.6089	1.33245	0.666224	+	0.745752i	$0.267909\pi$
$462$	0	0
$463$	19.6770	0.914470	0.457235	−	0.889346i	$-0.348840\pi$
$463$	19.6770	0.914470	0.457235	+	0.889346i	$0.348840\pi$
$464$	−0.536329	−0.0248984
$465$	0	0
$466$	17.5605	0.813475
$467$	35.9243	1.66238	0.831188	−	0.555991i	$-0.187661\pi$
$467$	35.9243	1.66238	0.831188	+	0.555991i	$0.187661\pi$
$468$	0	0
$469$	−0.740102	−0.0341747
$470$	8.79132	0.405513
$471$	0	0
$472$	−8.44682	−0.388796
$473$	−0.0947569	−0.00435693
$474$	0	0
$475$	20.8648	0.957345
$476$	−0.380045	−0.0174193
$477$	0	0
$478$	−28.9074	−1.32220
$479$	−23.7403	−1.08472	−0.542360	−	0.840146i	$-0.682469\pi$
$479$	−23.7403	−1.08472	−0.542360	+	0.840146i	$0.682469\pi$
$480$	0	0
$481$	−1.84673	−0.0842038
$482$	−22.3729	−1.01906
$483$	0	0
$484$	2.90141	0.131882
$485$	−43.7632	−1.98718
$486$	0	0
$487$	−17.2993	−0.783906	−0.391953	−	0.919985i	$-0.628200\pi$
$487$	−17.2993	−0.783906	−0.391953	+	0.919985i	$0.628200\pi$
$488$	1.72772	0.0782103
$489$	0	0
$490$	19.0465	0.860435
$491$	−0.651841	−0.0294172	−0.0147086	−	0.999892i	$-0.504682\pi$
$491$	−0.651841	−0.0294172	−0.0147086	+	0.999892i	$0.504682\pi$
$492$	0	0
$493$	0.314902	0.0141825
$494$	−4.42328	−0.199013
$495$	0	0
$496$	−8.61374	−0.386768
$497$	−4.42011	−0.198269
$498$	0	0
$499$	−15.6414	−0.700203	−0.350102	−	0.936712i	$-0.613853\pi$
$499$	−15.6414	−0.700203	−0.350102	+	0.936712i	$0.613853\pi$
$500$	−4.69968	−0.210176
$501$	0	0
$502$	14.0275	0.626079
$503$	−9.68777	−0.431956	−0.215978	−	0.976398i	$-0.569294\pi$
$503$	−9.68777	−0.431956	−0.215978	+	0.976398i	$0.569294\pi$
$504$	0	0
$505$	−13.3413	−0.593681
$506$	−27.2536	−1.21157
$507$	0	0
$508$	16.5659	0.734993
$509$	4.48613	0.198844	0.0994222	−	0.995045i	$-0.468301\pi$
$509$	4.48613	0.198844	0.0994222	+	0.995045i	$0.468301\pi$
$510$	0	0
$511$	4.62372	0.204541
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	9.63634	0.425041
$515$	57.2819	2.52414
$516$	0	0
$517$	11.3256	0.498099
$518$	1.67011	0.0733804
$519$	0	0
$520$	−2.07144	−0.0908387
$521$	−29.3370	−1.28528	−0.642638	−	0.766170i	$-0.722160\pi$
$521$	−29.3370	−1.28528	−0.642638	+	0.766170i	$0.722160\pi$
$522$	0	0
$523$	−1.24942	−0.0546335	−0.0273167	−	0.999627i	$-0.508696\pi$
$523$	−1.24942	−0.0546335	−0.0273167	+	0.999627i	$0.508696\pi$
$524$	−3.97969	−0.173853
$525$	0	0
$526$	−21.5566	−0.939914
$527$	5.05749	0.220308
$528$	0	0
$529$	30.4304	1.32306
$530$	38.5549	1.67472
$531$	0	0
$532$	4.00023	0.173432
$533$	4.33833	0.187914
$534$	0	0
$535$	0.336644	0.0145544
$536$	1.14341	0.0493877
$537$	0	0
$538$	26.4569	1.14064
$539$	24.5371	1.05689
$540$	0	0
$541$	23.0580	0.991341	0.495670	−	0.868511i	$-0.334922\pi$
$541$	23.0580	0.991341	0.495670	+	0.868511i	$0.334922\pi$
$542$	−11.9830	−0.514714
$543$	0	0
$544$	0.587143	0.0251735
$545$	−50.4131	−2.15946
$546$	0	0
$547$	−45.8753	−1.96149	−0.980744	−	0.195298i	$-0.937433\pi$
$547$	−45.8753	−1.96149	−0.980744	+	0.195298i	$0.937433\pi$
$548$	−17.7470	−0.758112
$549$	0	0
$550$	12.5878	0.536747
$551$	−3.31455	−0.141205
$552$	0	0
$553$	2.16449	0.0920436
$554$	−23.0482	−0.979224
$555$	0	0
$556$	−3.40466	−0.144390
$557$	−40.9294	−1.73424	−0.867118	−	0.498103i	$-0.834030\pi$
$557$	−40.9294	−1.73424	−0.867118	+	0.498103i	$0.834030\pi$
$558$	0	0
$559$	0.0181900	0.000769353 0
$560$	1.87333	0.0791624
$561$	0	0
$562$	16.6551	0.702554
$563$	−41.8178	−1.76241	−0.881205	−	0.472734i	$-0.843267\pi$
$563$	−41.8178	−1.76241	−0.881205	+	0.472734i	$0.843267\pi$
$564$	0	0
$565$	−42.6363	−1.79372
$566$	4.34695	0.182716
$567$	0	0
$568$	6.82877	0.286529
$569$	−12.9590	−0.543271	−0.271635	−	0.962400i	$-0.587564\pi$
$569$	−12.9590	−0.543271	−0.271635	+	0.962400i	$0.587564\pi$
$570$	0	0
$571$	2.26520	0.0947956	0.0473978	−	0.998876i	$-0.484907\pi$
$571$	2.26520	0.0947956	0.0473978	+	0.998876i	$0.484907\pi$
$572$	−2.66858	−0.111579
$573$	0	0
$574$	−3.92341	−0.163760
$575$	−24.6783	−1.02916
$576$	0	0
$577$	−45.8681	−1.90951	−0.954757	−	0.297388i	$-0.903884\pi$
$577$	−45.8681	−1.90951	−0.954757	+	0.297388i	$0.903884\pi$
$578$	16.6553	0.692768
$579$	0	0
$580$	−1.55222	−0.0644524
$581$	8.15519	0.338334
$582$	0	0
$583$	49.6691	2.05708
$584$	−7.14332	−0.295593
$585$	0	0
$586$	−17.5782	−0.726147
$587$	−24.5809	−1.01456	−0.507282	−	0.861780i	$-0.669350\pi$
$587$	−24.5809	−1.01456	−0.507282	+	0.861780i	$0.669350\pi$
$588$	0	0
$589$	−53.2335	−2.19345
$590$	−24.4464	−1.00644
$591$	0	0
$592$	−2.58020	−0.106046
$593$	34.9322	1.43449	0.717246	−	0.696820i	$-0.245402\pi$
$593$	34.9322	1.43449	0.717246	+	0.696820i	$0.245402\pi$
$594$	0	0
$595$	−1.09991	−0.0450919
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	5.23172	0.213941
$599$	−24.6003	−1.00514	−0.502571	−	0.864536i	$-0.667613\pi$
$599$	−24.6003	−1.00514	−0.502571	+	0.864536i	$0.667613\pi$
$600$	0	0
$601$	−27.8018	−1.13406	−0.567029	−	0.823698i	$-0.691907\pi$
$601$	−27.8018	−1.13406	−0.567029	+	0.823698i	$0.691907\pi$
$602$	−0.0164502	−0.000670462 0
$603$	0	0
$604$	−10.6585	−0.433687
$605$	8.39715	0.341393
$606$	0	0
$607$	19.6853	0.799000	0.399500	−	0.916733i	$-0.369184\pi$
$607$	19.6853	0.799000	0.399500	+	0.916733i	$0.369184\pi$
$608$	−6.18007	−0.250635
$609$	0	0
$610$	5.00030	0.202456
$611$	−2.17411	−0.0879552
$612$	0	0
$613$	−41.9191	−1.69310	−0.846548	−	0.532313i	$-0.821323\pi$
$613$	−41.9191	−1.69310	−0.846548	+	0.532313i	$0.821323\pi$
$614$	13.8702	0.559754
$615$	0	0
$616$	2.41335	0.0972367
$617$	−47.4315	−1.90952	−0.954760	−	0.297378i	$-0.903888\pi$
$617$	−47.4315	−1.90952	−0.954760	+	0.297378i	$0.903888\pi$
$618$	0	0
$619$	−32.0158	−1.28682	−0.643412	−	0.765520i	$-0.722482\pi$
$619$	−32.0158	−1.28682	−0.643412	+	0.765520i	$0.722482\pi$
$620$	−24.9295	−1.00119
$621$	0	0
$622$	−9.97478	−0.399952
$623$	−8.23561	−0.329953
$624$	0	0
$625$	−30.4824	−1.21929
$626$	−22.0525	−0.881394
$627$	0	0
$628$	10.8183	0.431699
$629$	1.51495	0.0604049
$630$	0	0
$631$	−11.8268	−0.470817	−0.235408	−	0.971897i	$-0.575643\pi$
$631$	−11.8268	−0.470817	−0.235408	+	0.971897i	$0.575643\pi$
$632$	−3.34399	−0.133017
$633$	0	0
$634$	2.36150	0.0937873
$635$	47.9443	1.90261
$636$	0	0
$637$	−4.71025	−0.186627
$638$	−1.99968	−0.0791681
$639$	0	0
$640$	−2.89416	−0.114402
$641$	26.3481	1.04069	0.520344	−	0.853957i	$-0.325804\pi$
$641$	26.3481	1.04069	0.520344	+	0.853957i	$0.325804\pi$
$642$	0	0
$643$	−1.08450	−0.0427687	−0.0213843	−	0.999771i	$-0.506807\pi$
$643$	−1.08450	−0.0427687	−0.0213843	+	0.999771i	$0.506807\pi$
$644$	−4.73135	−0.186441
$645$	0	0
$646$	3.62859	0.142765
$647$	26.9245	1.05851	0.529256	−	0.848462i	$-0.322471\pi$
$647$	26.9245	1.05851	0.529256	+	0.848462i	$0.322471\pi$
$648$	0	0
$649$	−31.4936	−1.23623
$650$	−2.41642	−0.0947797
$651$	0	0
$652$	6.33240	0.247996
$653$	7.23572	0.283156	0.141578	−	0.989927i	$-0.454782\pi$
$653$	7.23572	0.283156	0.141578	+	0.989927i	$0.454782\pi$
$654$	0	0
$655$	−11.5178	−0.450039
$656$	6.06139	0.236658
$657$	0	0
$658$	1.96618	0.0766496
$659$	−21.4670	−0.836237	−0.418119	−	0.908392i	$-0.637310\pi$
$659$	−21.4670	−0.836237	−0.418119	+	0.908392i	$0.637310\pi$
$660$	0	0
$661$	3.65121	0.142016	0.0710078	−	0.997476i	$-0.477378\pi$
$661$	3.65121	0.142016	0.0710078	+	0.997476i	$0.477378\pi$
$662$	24.6246	0.957063
$663$	0	0
$664$	−12.5992	−0.488944
$665$	11.5773	0.448948
$666$	0	0
$667$	3.92035	0.151797
$668$	20.0433	0.775497
$669$	0	0
$670$	3.30920	0.127846
$671$	6.44174	0.248681
$672$	0	0
$673$	−12.0356	−0.463939	−0.231970	−	0.972723i	$-0.574517\pi$
$673$	−12.0356	−0.463939	−0.231970	+	0.972723i	$0.574517\pi$
$674$	−4.12734	−0.158979
$675$	0	0
$676$	−12.4877	−0.480297
$677$	−42.7141	−1.64163	−0.820817	−	0.571191i	$-0.806482\pi$
$677$	−42.7141	−1.64163	−0.820817	+	0.571191i	$0.806482\pi$
$678$	0	0
$679$	−9.78763	−0.375615
$680$	1.69928	0.0651646
$681$	0	0
$682$	−32.1160	−1.22978
$683$	33.1663	1.26907	0.634537	−	0.772892i	$-0.281191\pi$
$683$	33.1663	1.26907	0.634537	+	0.772892i	$0.281191\pi$
$684$	0	0
$685$	−51.3625	−1.96246
$686$	8.79070	0.335631
$687$	0	0
$688$	0.0254145	0.000968918 0
$689$	−9.53471	−0.363243
$690$	0	0
$691$	9.66686	0.367745	0.183872	−	0.982950i	$-0.441137\pi$
$691$	9.66686	0.367745	0.183872	+	0.982950i	$0.441137\pi$
$692$	−19.7813	−0.751974
$693$	0	0
$694$	1.66183	0.0630824
$695$	−9.85362	−0.373769
$696$	0	0
$697$	−3.55890	−0.134803
$698$	9.67092	0.366050
$699$	0	0
$700$	2.18531	0.0825969
$701$	−13.0861	−0.494256	−0.247128	−	0.968983i	$-0.579487\pi$
$701$	−13.0861	−0.494256	−0.247128	+	0.968983i	$0.579487\pi$
$702$	0	0
$703$	−15.9458	−0.601409
$704$	−3.72846	−0.140522
$705$	0	0
$706$	−28.8909	−1.08732
$707$	−2.98379	−0.112217
$708$	0	0
$709$	23.8607	0.896108	0.448054	−	0.894006i	$-0.352117\pi$
$709$	23.8607	0.896108	0.448054	+	0.894006i	$0.352117\pi$
$710$	19.7635	0.741712
$711$	0	0
$712$	12.7234	0.476831
$713$	62.9631	2.35798
$714$	0	0
$715$	−7.72329	−0.288835
$716$	−8.80643	−0.329112
$717$	0	0
$718$	6.28476	0.234545
$719$	2.06440	0.0769893	0.0384946	−	0.999259i	$-0.487744\pi$
$719$	2.06440	0.0769893	0.0384946	+	0.999259i	$0.487744\pi$
$720$	0	0
$721$	12.8111	0.477110
$722$	−19.1933	−0.714301
$723$	0	0
$724$	−2.53023	−0.0940352
$725$	−1.81073	−0.0672487
$726$	0	0
$727$	13.8825	0.514875	0.257437	−	0.966295i	$-0.417122\pi$
$727$	13.8825	0.514875	0.257437	+	0.966295i	$0.417122\pi$
$728$	−0.463278	−0.0171702
$729$	0	0
$730$	−20.6739	−0.765176
$731$	−0.0149219	−0.000551908 0
$732$	0	0
$733$	34.2164	1.26381	0.631906	−	0.775045i	$-0.282273\pi$
$733$	34.2164	1.26381	0.631906	+	0.775045i	$0.282273\pi$
$734$	23.5043	0.867559
$735$	0	0
$736$	7.30961	0.269436
$737$	4.26315	0.157035
$738$	0	0
$739$	−19.0265	−0.699901	−0.349950	−	0.936768i	$-0.613802\pi$
$739$	−19.0265	−0.699901	−0.349950	+	0.936768i	$0.613802\pi$
$740$	−7.46751	−0.274511
$741$	0	0
$742$	8.62279	0.316553
$743$	−6.44555	−0.236464	−0.118232	−	0.992986i	$-0.537723\pi$
$743$	−6.44555	−0.236464	−0.118232	+	0.992986i	$0.537723\pi$
$744$	0	0
$745$	−2.89416	−0.106034
$746$	−8.22391	−0.301099
$747$	0	0
$748$	2.18914	0.0800428
$749$	0.0752904	0.00275105
$750$	0	0
$751$	−17.8606	−0.651741	−0.325870	−	0.945414i	$-0.605657\pi$
$751$	−17.8606	−0.651741	−0.325870	+	0.945414i	$0.605657\pi$
$752$	−3.03761	−0.110770
$753$	0	0
$754$	0.383868	0.0139796
$755$	−30.8473	−1.12265
$756$	0	0
$757$	−4.27617	−0.155420	−0.0777099	−	0.996976i	$-0.524761\pi$
$757$	−4.27617	−0.155420	−0.0777099	+	0.996976i	$0.524761\pi$
$758$	−17.0746	−0.620179
$759$	0	0
$760$	−17.8861	−0.648797
$761$	8.11840	0.294292	0.147146	−	0.989115i	$-0.452991\pi$
$761$	8.11840	0.294292	0.147146	+	0.989115i	$0.452991\pi$
$762$	0	0
$763$	−11.2749	−0.408179
$764$	−11.0216	−0.398748
$765$	0	0
$766$	−9.91645	−0.358296
$767$	6.04566	0.218296
$768$	0	0
$769$	15.0788	0.543755	0.271878	−	0.962332i	$-0.412355\pi$
$769$	15.0788	0.543755	0.271878	+	0.962332i	$0.412355\pi$
$770$	6.98462	0.251708
$771$	0	0
$772$	13.0378	0.469240
$773$	−15.9177	−0.572520	−0.286260	−	0.958152i	$-0.592412\pi$
$773$	−15.9177	−0.572520	−0.286260	+	0.958152i	$0.592412\pi$
$774$	0	0
$775$	−29.0813	−1.04463
$776$	15.1212	0.542820
$777$	0	0
$778$	−20.6837	−0.741548
$779$	37.4598	1.34214
$780$	0	0
$781$	25.4608	0.911058
$782$	−4.29178	−0.153474
$783$	0	0
$784$	−6.58103	−0.235037
$785$	31.3100	1.11750
$786$	0	0
$787$	−51.4233	−1.83304	−0.916521	−	0.399986i	$-0.869015\pi$
$787$	−51.4233	−1.83304	−0.916521	+	0.399986i	$0.869015\pi$
$788$	−6.51445	−0.232068
$789$	0	0
$790$	−9.67804	−0.344329
$791$	−9.53561	−0.339047
$792$	0	0
$793$	−1.23659	−0.0439125
$794$	10.8701	0.385767
$795$	0	0
$796$	8.24335	0.292178
$797$	37.6677	1.33426	0.667129	−	0.744942i	$-0.267523\pi$
$797$	37.6677	1.33426	0.667129	+	0.744942i	$0.267523\pi$
$798$	0	0
$799$	1.78351	0.0630960
$800$	−3.37615	−0.119365
$801$	0	0
$802$	−27.8798	−0.984469
$803$	−26.6336	−0.939879
$804$	0	0
$805$	−13.6933	−0.482624
$806$	6.16513	0.217157
$807$	0	0
$808$	4.60974	0.162170
$809$	1.00886	0.0354696	0.0177348	−	0.999843i	$-0.494355\pi$
$809$	1.00886	0.0354696	0.0177348	+	0.999843i	$0.494355\pi$
$810$	0	0
$811$	−12.6227	−0.443244	−0.221622	−	0.975133i	$-0.571135\pi$
$811$	−12.6227	−0.443244	−0.221622	+	0.975133i	$0.571135\pi$
$812$	−0.347154	−0.0121827
$813$	0	0
$814$	−9.62018	−0.337187
$815$	18.3270	0.641965
$816$	0	0
$817$	0.157063	0.00549495
$818$	21.3723	0.747267
$819$	0	0
$820$	17.5426	0.612615
$821$	−31.5916	−1.10256	−0.551278	−	0.834322i	$-0.685860\pi$
$821$	−31.5916	−1.10256	−0.551278	+	0.834322i	$0.685860\pi$
$822$	0	0
$823$	31.7590	1.10705	0.553525	−	0.832832i	$-0.313282\pi$
$823$	31.7590	1.10705	0.553525	+	0.832832i	$0.313282\pi$
$824$	−19.7923	−0.689496
$825$	0	0
$826$	−5.46744	−0.190237
$827$	−0.680815	−0.0236743	−0.0118371	−	0.999930i	$-0.503768\pi$
$827$	−0.680815	−0.0236743	−0.0118371	+	0.999930i	$0.503768\pi$
$828$	0	0
$829$	18.9640	0.658648	0.329324	−	0.944217i	$-0.393179\pi$
$829$	18.9640	0.658648	0.329324	+	0.944217i	$0.393179\pi$
$830$	−36.4641	−1.26569
$831$	0	0
$832$	0.715732	0.0248135
$833$	3.86401	0.133880
$834$	0	0
$835$	58.0084	2.00746
$836$	−23.0422	−0.796930
$837$	0	0
$838$	0.754815	0.0260747
$839$	−19.3566	−0.668263	−0.334132	−	0.942526i	$-0.608443\pi$
$839$	−19.3566	−0.668263	−0.334132	+	0.942526i	$0.608443\pi$
$840$	0	0
$841$	−28.7124	−0.990081
$842$	−24.0721	−0.829579
$843$	0	0
$844$	14.4493	0.497364
$845$	−36.1415	−1.24330
$846$	0	0
$847$	1.87802	0.0645296
$848$	−13.3216	−0.457466
$849$	0	0
$850$	1.98228	0.0679917
$851$	18.8603	0.646522
$852$	0	0
$853$	−42.7432	−1.46350	−0.731750	−	0.681573i	$-0.761296\pi$
$853$	−42.7432	−1.46350	−0.731750	+	0.681573i	$0.761296\pi$
$854$	1.11832	0.0382680
$855$	0	0
$856$	−0.116319	−0.00397568
$857$	8.25018	0.281821	0.140910	−	0.990022i	$-0.454997\pi$
$857$	8.25018	0.281821	0.140910	+	0.990022i	$0.454997\pi$
$858$	0	0
$859$	44.8257	1.52943	0.764716	−	0.644368i	$-0.222879\pi$
$859$	44.8257	1.52943	0.764716	+	0.644368i	$0.222879\pi$
$860$	0.0735535	0.00250815
$861$	0	0
$862$	10.4667	0.356496
$863$	−11.2994	−0.384635	−0.192317	−	0.981333i	$-0.561600\pi$
$863$	−11.2994	−0.384635	−0.192317	+	0.981333i	$0.561600\pi$
$864$	0	0
$865$	−57.2503	−1.94657
$866$	−2.47142	−0.0839821
$867$	0	0
$868$	−5.57548	−0.189244
$869$	−12.4679	−0.422946
$870$	0	0
$871$	−0.818373	−0.0277295
$872$	17.4189	0.589880
$873$	0	0
$874$	45.1739	1.52803
$875$	−3.04200	−0.102838
$876$	0	0
$877$	−34.5953	−1.16820	−0.584101	−	0.811681i	$-0.698553\pi$
$877$	−34.5953	−1.16820	−0.584101	+	0.811681i	$0.698553\pi$
$878$	−37.7653	−1.27452
$879$	0	0
$880$	−10.7908	−0.363756
$881$	−18.6458	−0.628192	−0.314096	−	0.949391i	$-0.601701\pi$
$881$	−18.6458	−0.628192	−0.314096	+	0.949391i	$0.601701\pi$
$882$	0	0
$883$	−19.6055	−0.659777	−0.329889	−	0.944020i	$-0.607011\pi$
$883$	−19.6055	−0.659777	−0.329889	+	0.944020i	$0.607011\pi$
$884$	−0.420237	−0.0141341
$885$	0	0
$886$	10.3172	0.346613
$887$	14.3840	0.482969	0.241484	−	0.970405i	$-0.422366\pi$
$887$	14.3840	0.482969	0.241484	+	0.970405i	$0.422366\pi$
$888$	0	0
$889$	10.7227	0.359629
$890$	36.8237	1.23433
$891$	0	0
$892$	−14.0558	−0.470624
$893$	−18.7726	−0.628202
$894$	0	0
$895$	−25.4872	−0.851943
$896$	−0.647278	−0.0216240
$897$	0	0
$898$	−18.1795	−0.606659
$899$	4.61979	0.154079
$900$	0	0
$901$	7.82169	0.260578
$902$	22.5996	0.752486
$903$	0	0
$904$	14.7319	0.489974
$905$	−7.32288	−0.243421
$906$	0	0
$907$	49.7539	1.65205	0.826026	−	0.563632i	$-0.190596\pi$
$907$	49.7539	1.65205	0.826026	+	0.563632i	$0.190596\pi$
$908$	−17.2851	−0.573626
$909$	0	0
$910$	−1.34080	−0.0444470
$911$	59.1494	1.95971	0.979853	−	0.199721i	$-0.0640034\pi$
$911$	59.1494	1.95971	0.979853	+	0.199721i	$0.0640034\pi$
$912$	0	0
$913$	−46.9757	−1.55467
$914$	−34.7422	−1.14917
$915$	0	0
$916$	−9.37902	−0.309892
$917$	−2.57596	−0.0850658
$918$	0	0
$919$	3.31621	0.109392	0.0546959	−	0.998503i	$-0.482581\pi$
$919$	3.31621	0.109392	0.0546959	+	0.998503i	$0.482581\pi$
$920$	21.1552	0.697465
$921$	0	0
$922$	−28.6089	−0.942183
$923$	−4.88757	−0.160876
$924$	0	0
$925$	−8.71115	−0.286421
$926$	−19.6770	−0.646628
$927$	0	0
$928$	0.536329	0.0176059
$929$	5.62665	0.184604	0.0923021	−	0.995731i	$-0.470577\pi$
$929$	5.62665	0.184604	0.0923021	+	0.995731i	$0.470577\pi$
$930$	0	0
$931$	−40.6713	−1.33295
$932$	−17.5605	−0.575213
$933$	0	0
$934$	−35.9243	−1.17548
$935$	6.33571	0.207200
$936$	0	0
$937$	−35.2604	−1.15191	−0.575953	−	0.817483i	$-0.695369\pi$
$937$	−35.2604	−1.15191	−0.575953	+	0.817483i	$0.695369\pi$
$938$	0.740102	0.0241652
$939$	0	0
$940$	−8.79132	−0.286741
$941$	41.3278	1.34725	0.673624	−	0.739074i	$-0.264737\pi$
$941$	41.3278	1.34725	0.673624	+	0.739074i	$0.264737\pi$
$942$	0	0
$943$	−44.3064	−1.44281
$944$	8.44682	0.274921
$945$	0	0
$946$	0.0947569	0.00308081
$947$	23.3912	0.760112	0.380056	−	0.924963i	$-0.375905\pi$
$947$	23.3912	0.760112	0.380056	+	0.924963i	$0.375905\pi$
$948$	0	0
$949$	5.11270	0.165965
$950$	−20.8648	−0.676945
$951$	0	0
$952$	0.380045	0.0123173
$953$	4.73330	0.153327	0.0766633	−	0.997057i	$-0.475573\pi$
$953$	4.73330	0.153327	0.0766633	+	0.997057i	$0.475573\pi$
$954$	0	0
$955$	−31.8983	−1.03221
$956$	28.9074	0.934933
$957$	0	0
$958$	23.7403	0.767013
$959$	−11.4872	−0.370941
$960$	0	0
$961$	43.1965	1.39343
$962$	1.84673	0.0595411
$963$	0	0
$964$	22.3729	0.720583
$965$	37.7334	1.21468
$966$	0	0
$967$	43.4553	1.39743	0.698714	−	0.715401i	$-0.253756\pi$
$967$	43.4553	1.39743	0.698714	+	0.715401i	$0.253756\pi$
$968$	−2.90141	−0.0932550
$969$	0	0
$970$	43.7632	1.40515
$971$	37.4142	1.20068	0.600340	−	0.799745i	$-0.295032\pi$
$971$	37.4142	1.20068	0.600340	+	0.799745i	$0.295032\pi$
$972$	0	0
$973$	−2.20376	−0.0706493
$974$	17.2993	0.554305
$975$	0	0
$976$	−1.72772	−0.0553030
$977$	55.4340	1.77349	0.886746	−	0.462258i	$-0.152960\pi$
$977$	55.4340	1.77349	0.886746	+	0.462258i	$0.152960\pi$
$978$	0	0
$979$	47.4389	1.51615
$980$	−19.0465	−0.608419
$981$	0	0
$982$	0.651841	0.0208011
$983$	−37.5297	−1.19701	−0.598506	−	0.801118i	$-0.704239\pi$
$983$	−37.5297	−1.19701	−0.598506	+	0.801118i	$0.704239\pi$
$984$	0	0
$985$	−18.8539	−0.600734
$986$	−0.314902	−0.0100285
$987$	0	0
$988$	4.42328	0.140723
$989$	−0.185770	−0.00590714
$990$	0	0
$991$	−11.4603	−0.364049	−0.182025	−	0.983294i	$-0.558265\pi$
$991$	−11.4603	−0.364049	−0.182025	+	0.983294i	$0.558265\pi$
$992$	8.61374	0.273486
$993$	0	0
$994$	4.42011	0.140197
$995$	23.8576	0.756336
$996$	0	0
$997$	5.94633	0.188322	0.0941611	−	0.995557i	$-0.469983\pi$
$997$	5.94633	0.188322	0.0941611	+	0.995557i	$0.469983\pi$
$998$	15.6414	0.495118
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.j.1.12	✓	12
3.2	odd	2		8046.2.a.o.1.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.j.1.12	✓	12	1.1	even	1	trivial
8046.2.a.o.1.1	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.89416 q^{5} +0.647278 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.89416 q^{5} +0.647278 q^{7} -1.00000 q^{8} -2.89416 q^{10} -3.72846 q^{11} +0.715732 q^{13} -0.647278 q^{14} +1.00000 q^{16} -0.587143 q^{17} +6.18007 q^{19} +2.89416 q^{20} +3.72846 q^{22} -7.30961 q^{23} +3.37615 q^{25} -0.715732 q^{26} +0.647278 q^{28} -0.536329 q^{29} -8.61374 q^{31} -1.00000 q^{32} +0.587143 q^{34} +1.87333 q^{35} -2.58020 q^{37} -6.18007 q^{38} -2.89416 q^{40} +6.06139 q^{41} +0.0254145 q^{43} -3.72846 q^{44} +7.30961 q^{46} -3.03761 q^{47} -6.58103 q^{49} -3.37615 q^{50} +0.715732 q^{52} -13.3216 q^{53} -10.7908 q^{55} -0.647278 q^{56} +0.536329 q^{58} +8.44682 q^{59} -1.72772 q^{61} +8.61374 q^{62} +1.00000 q^{64} +2.07144 q^{65} -1.14341 q^{67} -0.587143 q^{68} -1.87333 q^{70} -6.82877 q^{71} +7.14332 q^{73} +2.58020 q^{74} +6.18007 q^{76} -2.41335 q^{77} +3.34399 q^{79} +2.89416 q^{80} -6.06139 q^{82} +12.5992 q^{83} -1.69928 q^{85} -0.0254145 q^{86} +3.72846 q^{88} -12.7234 q^{89} +0.463278 q^{91} -7.30961 q^{92} +3.03761 q^{94} +17.8861 q^{95} -15.1212 q^{97} +6.58103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 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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Representations

Groups

Database

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