LMFDB - Embedded newform 8046.2.a.h.1.1

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 9x^{7} + 25x^{6} + 29x^{5} - 58x^{4} - 43x^{3} + 34x^{2} + 25x + 2 $ x^9 - 3x^8 - 9x^7 + 25x^6 + 29x^5 - 58x^4 - 43x^3 + 34x^2 + 25x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.05829$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.80777 q^{5} +0.515673 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.80777 q^{5} +0.515673 q^{7} +1.00000 q^{8} -3.80777 q^{10} -0.940941 q^{11} -2.83646 q^{13} +0.515673 q^{14} +1.00000 q^{16} +3.93453 q^{17} -3.57365 q^{19} -3.80777 q^{20} -0.940941 q^{22} +1.39239 q^{23} +9.49910 q^{25} -2.83646 q^{26} +0.515673 q^{28} +8.36889 q^{29} +2.17513 q^{31} +1.00000 q^{32} +3.93453 q^{34} -1.96357 q^{35} +3.67086 q^{37} -3.57365 q^{38} -3.80777 q^{40} +9.56071 q^{41} -9.00371 q^{43} -0.940941 q^{44} +1.39239 q^{46} -2.89701 q^{47} -6.73408 q^{49} +9.49910 q^{50} -2.83646 q^{52} -1.17075 q^{53} +3.58289 q^{55} +0.515673 q^{56} +8.36889 q^{58} -12.3747 q^{59} +2.12700 q^{61} +2.17513 q^{62} +1.00000 q^{64} +10.8006 q^{65} -10.2760 q^{67} +3.93453 q^{68} -1.96357 q^{70} -2.35223 q^{71} +10.1300 q^{73} +3.67086 q^{74} -3.57365 q^{76} -0.485218 q^{77} -5.14150 q^{79} -3.80777 q^{80} +9.56071 q^{82} +3.57873 q^{83} -14.9818 q^{85} -9.00371 q^{86} -0.940941 q^{88} +0.00223128 q^{89} -1.46269 q^{91} +1.39239 q^{92} -2.89701 q^{94} +13.6076 q^{95} +3.82873 q^{97} -6.73408 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) $ 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8	$ 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8} - 4 q^{10} - 4 q^{11} - 8 q^{13} - 4 q^{14} + 9 q^{16} - q^{17} - 10 q^{19} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 3 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{29} - 17 q^{31} + 9 q^{32} - q^{34} - 10 q^{35} - 11 q^{37} - 10 q^{38} - 4 q^{40} - 16 q^{43} - 4 q^{44} - 8 q^{46} - 7 q^{47} - 5 q^{49} - 3 q^{50} - 8 q^{52} - 12 q^{53} - 23 q^{55} - 4 q^{56} - 4 q^{58} - 6 q^{59} - 13 q^{61} - 17 q^{62} + 9 q^{64} + 24 q^{65} - 14 q^{67} - q^{68} - 10 q^{70} - 30 q^{71} - 12 q^{73} - 11 q^{74} - 10 q^{76} - 12 q^{77} - 35 q^{79} - 4 q^{80} - 5 q^{83} - 27 q^{85} - 16 q^{86} - 4 q^{88} - 23 q^{89} - 28 q^{91} - 8 q^{92} - 7 q^{94} - 32 q^{95} - 21 q^{97} - 5 q^{98}+O(q^{100}) $ 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8 - 4 * q^10 - 4 * q^11 - 8 * q^13 - 4 * q^14 + 9 * q^16 - q^17 - 10 * q^19 - 4 * q^20 - 4 * q^22 - 8 * q^23 - 3 * q^25 - 8 * q^26 - 4 * q^28 - 4 * q^29 - 17 * q^31 + 9 * q^32 - q^34 - 10 * q^35 - 11 * q^37 - 10 * q^38 - 4 * q^40 - 16 * q^43 - 4 * q^44 - 8 * q^46 - 7 * q^47 - 5 * q^49 - 3 * q^50 - 8 * q^52 - 12 * q^53 - 23 * q^55 - 4 * q^56 - 4 * q^58 - 6 * q^59 - 13 * q^61 - 17 * q^62 + 9 * q^64 + 24 * q^65 - 14 * q^67 - q^68 - 10 * q^70 - 30 * q^71 - 12 * q^73 - 11 * q^74 - 10 * q^76 - 12 * q^77 - 35 * q^79 - 4 * q^80 - 5 * q^83 - 27 * q^85 - 16 * q^86 - 4 * q^88 - 23 * q^89 - 28 * q^91 - 8 * q^92 - 7 * q^94 - 32 * q^95 - 21 * q^97 - 5 * 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
$5$	−3.80777	−1.70289	−0.851443	−	0.524447i	$-0.824272\pi$
$5$	−3.80777	−1.70289	−0.851443	+	0.524447i	$0.824272\pi$
$6$	0	0
$7$	0.515673	0.194906	0.0974531	−	0.995240i	$-0.468930\pi$
$7$	0.515673	0.194906	0.0974531	+	0.995240i	$0.468930\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.80777	−1.20412
$11$	−0.940941	−0.283704	−0.141852	−	0.989888i	$-0.545306\pi$
$11$	−0.940941	−0.283704	−0.141852	+	0.989888i	$0.545306\pi$
$12$	0	0
$13$	−2.83646	−0.786693	−0.393346	−	0.919390i	$-0.628683\pi$
$13$	−2.83646	−0.786693	−0.393346	+	0.919390i	$0.628683\pi$
$14$	0.515673	0.137820
$15$	0	0
$16$	1.00000	0.250000
$17$	3.93453	0.954264	0.477132	−	0.878832i	$-0.341676\pi$
$17$	3.93453	0.954264	0.477132	+	0.878832i	$0.341676\pi$
$18$	0	0
$19$	−3.57365	−0.819852	−0.409926	−	0.912119i	$-0.634446\pi$
$19$	−3.57365	−0.819852	−0.409926	+	0.912119i	$0.634446\pi$
$20$	−3.80777	−0.851443
$21$	0	0
$22$	−0.940941	−0.200609
$23$	1.39239	0.290333	0.145167	−	0.989407i	$-0.453628\pi$
$23$	1.39239	0.290333	0.145167	+	0.989407i	$0.453628\pi$
$24$	0	0
$25$	9.49910	1.89982
$26$	−2.83646	−0.556276
$27$	0	0
$28$	0.515673	0.0974531
$29$	8.36889	1.55406	0.777032	−	0.629461i	$-0.216724\pi$
$29$	8.36889	1.55406	0.777032	+	0.629461i	$0.216724\pi$
$30$	0	0
$31$	2.17513	0.390666	0.195333	−	0.980737i	$-0.437421\pi$
$31$	2.17513	0.390666	0.195333	+	0.980737i	$0.437421\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.93453	0.674766
$35$	−1.96357	−0.331903
$36$	0	0
$37$	3.67086	0.603486	0.301743	−	0.953389i	$-0.402432\pi$
$37$	3.67086	0.603486	0.301743	+	0.953389i	$0.402432\pi$
$38$	−3.57365	−0.579723
$39$	0	0
$40$	−3.80777	−0.602061
$41$	9.56071	1.49313	0.746566	−	0.665312i	$-0.231701\pi$
$41$	9.56071	1.49313	0.746566	+	0.665312i	$0.231701\pi$
$42$	0	0
$43$	−9.00371	−1.37305	−0.686526	−	0.727105i	$-0.740865\pi$
$43$	−9.00371	−1.37305	−0.686526	+	0.727105i	$0.740865\pi$
$44$	−0.940941	−0.141852
$45$	0	0
$46$	1.39239	0.205297
$47$	−2.89701	−0.422573	−0.211286	−	0.977424i	$-0.567765\pi$
$47$	−2.89701	−0.422573	−0.211286	+	0.977424i	$0.567765\pi$
$48$	0	0
$49$	−6.73408	−0.962012
$50$	9.49910	1.34338
$51$	0	0
$52$	−2.83646	−0.393346
$53$	−1.17075	−0.160815	−0.0804076	−	0.996762i	$-0.525622\pi$
$53$	−1.17075	−0.160815	−0.0804076	+	0.996762i	$0.525622\pi$
$54$	0	0
$55$	3.58289	0.483116
$56$	0.515673	0.0689098
$57$	0	0
$58$	8.36889	1.09889
$59$	−12.3747	−1.61105	−0.805523	−	0.592564i	$-0.798116\pi$
$59$	−12.3747	−1.61105	−0.805523	+	0.592564i	$0.798116\pi$
$60$	0	0
$61$	2.12700	0.272334	0.136167	−	0.990686i	$-0.456522\pi$
$61$	2.12700	0.272334	0.136167	+	0.990686i	$0.456522\pi$
$62$	2.17513	0.276242
$63$	0	0
$64$	1.00000	0.125000
$65$	10.8006	1.33965
$66$	0	0
$67$	−10.2760	−1.25542	−0.627709	−	0.778448i	$-0.716007\pi$
$67$	−10.2760	−1.25542	−0.627709	+	0.778448i	$0.716007\pi$
$68$	3.93453	0.477132
$69$	0	0
$70$	−1.96357	−0.234691
$71$	−2.35223	−0.279158	−0.139579	−	0.990211i	$-0.544575\pi$
$71$	−2.35223	−0.279158	−0.139579	+	0.990211i	$0.544575\pi$
$72$	0	0
$73$	10.1300	1.18562	0.592812	−	0.805341i	$-0.298018\pi$
$73$	10.1300	1.18562	0.592812	+	0.805341i	$0.298018\pi$
$74$	3.67086	0.426729
$75$	0	0
$76$	−3.57365	−0.409926
$77$	−0.485218	−0.0552958
$78$	0	0
$79$	−5.14150	−0.578464	−0.289232	−	0.957259i	$-0.593400\pi$
$79$	−5.14150	−0.578464	−0.289232	+	0.957259i	$0.593400\pi$
$80$	−3.80777	−0.425721
$81$	0	0
$82$	9.56071	1.05580
$83$	3.57873	0.392816	0.196408	−	0.980522i	$-0.437072\pi$
$83$	3.57873	0.392816	0.196408	+	0.980522i	$0.437072\pi$
$84$	0	0
$85$	−14.9818	−1.62500
$86$	−9.00371	−0.970895
$87$	0	0
$88$	−0.940941	−0.100305
$89$	0.00223128	0.000236515 0	0.000118258	−	1.00000i	$-0.499962\pi$
$89$	0.00223128	0.000236515 0	0.000118258		1.00000i	$0.499962\pi$
$90$	0	0
$91$	−1.46269	−0.153331
$92$	1.39239	0.145167
$93$	0	0
$94$	−2.89701	−0.298804
$95$	13.6076	1.39611
$96$	0	0
$97$	3.82873	0.388749	0.194374	−	0.980927i	$-0.437732\pi$
$97$	3.82873	0.388749	0.194374	+	0.980927i	$0.437732\pi$
$98$	−6.73408	−0.680245
$99$	0	0
$100$	9.49910	0.949910
$101$	2.22598	0.221494	0.110747	−	0.993849i	$-0.464676\pi$
$101$	2.22598	0.221494	0.110747	+	0.993849i	$0.464676\pi$
$102$	0	0
$103$	−13.3755	−1.31793	−0.658966	−	0.752173i	$-0.729006\pi$
$103$	−13.3755	−1.31793	−0.658966	+	0.752173i	$0.729006\pi$
$104$	−2.83646	−0.278138
$105$	0	0
$106$	−1.17075	−0.113714
$107$	−4.96212	−0.479707	−0.239853	−	0.970809i	$-0.577099\pi$
$107$	−4.96212	−0.479707	−0.239853	+	0.970809i	$0.577099\pi$
$108$	0	0
$109$	−2.01553	−0.193053	−0.0965264	−	0.995330i	$-0.530773\pi$
$109$	−2.01553	−0.193053	−0.0965264	+	0.995330i	$0.530773\pi$
$110$	3.58289	0.341615
$111$	0	0
$112$	0.515673	0.0487266
$113$	−7.87785	−0.741086	−0.370543	−	0.928815i	$-0.620828\pi$
$113$	−7.87785	−0.741086	−0.370543	+	0.928815i	$0.620828\pi$
$114$	0	0
$115$	−5.30190	−0.494404
$116$	8.36889	0.777032
$117$	0	0
$118$	−12.3747	−1.13918
$119$	2.02893	0.185992
$120$	0	0
$121$	−10.1146	−0.919512
$122$	2.12700	0.192569
$123$	0	0
$124$	2.17513	0.195333
$125$	−17.1315	−1.53229
$126$	0	0
$127$	−19.7227	−1.75011	−0.875054	−	0.484025i	$-0.839174\pi$
$127$	−19.7227	−1.75011	−0.875054	+	0.484025i	$0.839174\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	10.8006	0.947274
$131$	10.9019	0.952504	0.476252	−	0.879309i	$-0.341995\pi$
$131$	10.9019	0.952504	0.476252	+	0.879309i	$0.341995\pi$
$132$	0	0
$133$	−1.84284	−0.159794
$134$	−10.2760	−0.887715
$135$	0	0
$136$	3.93453	0.337383
$137$	6.40820	0.547489	0.273745	−	0.961802i	$-0.411738\pi$
$137$	6.40820	0.547489	0.273745	+	0.961802i	$0.411738\pi$
$138$	0	0
$139$	−17.0897	−1.44953	−0.724763	−	0.688998i	$-0.758051\pi$
$139$	−17.0897	−1.44953	−0.724763	+	0.688998i	$0.758051\pi$
$140$	−1.96357	−0.165952
$141$	0	0
$142$	−2.35223	−0.197394
$143$	2.66894	0.223188
$144$	0	0
$145$	−31.8668	−2.64639
$146$	10.1300	0.838363
$147$	0	0
$148$	3.67086	0.301743
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−13.2487	−1.07816	−0.539081	−	0.842254i	$-0.681228\pi$
$151$	−13.2487	−1.07816	−0.539081	+	0.842254i	$0.681228\pi$
$152$	−3.57365	−0.289862
$153$	0	0
$154$	−0.485218	−0.0391000
$155$	−8.28241	−0.665259
$156$	0	0
$157$	10.5613	0.842883	0.421442	−	0.906855i	$-0.361524\pi$
$157$	10.5613	0.842883	0.421442	+	0.906855i	$0.361524\pi$
$158$	−5.14150	−0.409036
$159$	0	0
$160$	−3.80777	−0.301030
$161$	0.718018	0.0565878
$162$	0	0
$163$	−12.9514	−1.01443	−0.507216	−	0.861819i	$-0.669325\pi$
$163$	−12.9514	−1.01443	−0.507216	+	0.861819i	$0.669325\pi$
$164$	9.56071	0.746566
$165$	0	0
$166$	3.57873	0.277763
$167$	−11.4775	−0.888159	−0.444079	−	0.895987i	$-0.646469\pi$
$167$	−11.4775	−0.888159	−0.444079	+	0.895987i	$0.646469\pi$
$168$	0	0
$169$	−4.95449	−0.381114
$170$	−14.9818	−1.14905
$171$	0	0
$172$	−9.00371	−0.686526
$173$	24.8186	1.88692	0.943461	−	0.331485i	$-0.107550\pi$
$173$	24.8186	1.88692	0.943461	+	0.331485i	$0.107550\pi$
$174$	0	0
$175$	4.89843	0.370287
$176$	−0.940941	−0.0709261
$177$	0	0
$178$	0.00223128	0.000167242 0
$179$	−9.86161	−0.737091	−0.368546	−	0.929610i	$-0.620144\pi$
$179$	−9.86161	−0.737091	−0.368546	+	0.929610i	$0.620144\pi$
$180$	0	0
$181$	−15.6306	−1.16182	−0.580908	−	0.813969i	$-0.697302\pi$
$181$	−15.6306	−1.16182	−0.580908	+	0.813969i	$0.697302\pi$
$182$	−1.46269	−0.108422
$183$	0	0
$184$	1.39239	0.102648
$185$	−13.9778	−1.02767
$186$	0	0
$187$	−3.70216	−0.270729
$188$	−2.89701	−0.211286
$189$	0	0
$190$	13.6076	0.987202
$191$	−8.22930	−0.595451	−0.297726	−	0.954651i	$-0.596228\pi$
$191$	−8.22930	−0.595451	−0.297726	+	0.954651i	$0.596228\pi$
$192$	0	0
$193$	−3.29466	−0.237155	−0.118577	−	0.992945i	$-0.537833\pi$
$193$	−3.29466	−0.237155	−0.118577	+	0.992945i	$0.537833\pi$
$194$	3.82873	0.274887
$195$	0	0
$196$	−6.73408	−0.481006
$197$	−13.7739	−0.981348	−0.490674	−	0.871343i	$-0.663249\pi$
$197$	−13.7739	−0.981348	−0.490674	+	0.871343i	$0.663249\pi$
$198$	0	0
$199$	2.93461	0.208029	0.104014	−	0.994576i	$-0.466831\pi$
$199$	2.93461	0.208029	0.104014	+	0.994576i	$0.466831\pi$
$200$	9.49910	0.671688
$201$	0	0
$202$	2.22598	0.156620
$203$	4.31562	0.302897
$204$	0	0
$205$	−36.4050	−2.54263
$206$	−13.3755	−0.931918
$207$	0	0
$208$	−2.83646	−0.196673
$209$	3.36260	0.232596
$210$	0	0
$211$	13.5697	0.934175	0.467088	−	0.884211i	$-0.345303\pi$
$211$	13.5697	0.934175	0.467088	+	0.884211i	$0.345303\pi$
$212$	−1.17075	−0.0804076
$213$	0	0
$214$	−4.96212	−0.339204
$215$	34.2840	2.33815
$216$	0	0
$217$	1.12166	0.0761432
$218$	−2.01553	−0.136509
$219$	0	0
$220$	3.58289	0.241558
$221$	−11.1601	−0.750712
$222$	0	0
$223$	0.372556	0.0249482	0.0124741	−	0.999922i	$-0.496029\pi$
$223$	0.372556	0.0249482	0.0124741	+	0.999922i	$0.496029\pi$
$224$	0.515673	0.0344549
$225$	0	0
$226$	−7.87785	−0.524027
$227$	12.9343	0.858478	0.429239	−	0.903191i	$-0.358782\pi$
$227$	12.9343	0.858478	0.429239	+	0.903191i	$0.358782\pi$
$228$	0	0
$229$	−6.44573	−0.425946	−0.212973	−	0.977058i	$-0.568315\pi$
$229$	−6.44573	−0.425946	−0.212973	+	0.977058i	$0.568315\pi$
$230$	−5.30190	−0.349597
$231$	0	0
$232$	8.36889	0.549445
$233$	−7.75366	−0.507959	−0.253980	−	0.967210i	$-0.581740\pi$
$233$	−7.75366	−0.507959	−0.253980	+	0.967210i	$0.581740\pi$
$234$	0	0
$235$	11.0311	0.719593
$236$	−12.3747	−0.805523
$237$	0	0
$238$	2.02893	0.131516
$239$	−14.3567	−0.928660	−0.464330	−	0.885662i	$-0.653705\pi$
$239$	−14.3567	−0.928660	−0.464330	+	0.885662i	$0.653705\pi$
$240$	0	0
$241$	12.1139	0.780323	0.390162	−	0.920746i	$-0.372419\pi$
$241$	12.1139	0.780323	0.390162	+	0.920746i	$0.372419\pi$
$242$	−10.1146	−0.650193
$243$	0	0
$244$	2.12700	0.136167
$245$	25.6418	1.63820
$246$	0	0
$247$	10.1365	0.644972
$248$	2.17513	0.138121
$249$	0	0
$250$	−17.1315	−1.08349
$251$	−16.3635	−1.03286	−0.516428	−	0.856331i	$-0.672739\pi$
$251$	−16.3635	−1.03286	−0.516428	+	0.856331i	$0.672739\pi$
$252$	0	0
$253$	−1.31016	−0.0823688
$254$	−19.7227	−1.23751
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.104679	−0.00652969	−0.00326484	−	0.999995i	$-0.501039\pi$
$257$	−0.104679	−0.00652969	−0.00326484	+	0.999995i	$0.501039\pi$
$258$	0	0
$259$	1.89297	0.117623
$260$	10.8006	0.669824
$261$	0	0
$262$	10.9019	0.673522
$263$	−21.3035	−1.31363	−0.656815	−	0.754052i	$-0.728097\pi$
$263$	−21.3035	−1.31363	−0.656815	+	0.754052i	$0.728097\pi$
$264$	0	0
$265$	4.45795	0.273850
$266$	−1.84284	−0.112992
$267$	0	0
$268$	−10.2760	−0.627709
$269$	9.61799	0.586419	0.293210	−	0.956048i	$-0.405277\pi$
$269$	9.61799	0.586419	0.293210	+	0.956048i	$0.405277\pi$
$270$	0	0
$271$	14.1277	0.858195	0.429097	−	0.903258i	$-0.358832\pi$
$271$	14.1277	0.858195	0.429097	+	0.903258i	$0.358832\pi$
$272$	3.93453	0.238566
$273$	0	0
$274$	6.40820	0.387133
$275$	−8.93809	−0.538987
$276$	0	0
$277$	−20.2247	−1.21519	−0.607593	−	0.794248i	$-0.707865\pi$
$277$	−20.2247	−1.21519	−0.607593	+	0.794248i	$0.707865\pi$
$278$	−17.0897	−1.02497
$279$	0	0
$280$	−1.96357	−0.117345
$281$	19.8872	1.18637	0.593184	−	0.805067i	$-0.297871\pi$
$281$	19.8872	1.18637	0.593184	+	0.805067i	$0.297871\pi$
$282$	0	0
$283$	17.9462	1.06679	0.533395	−	0.845866i	$-0.320916\pi$
$283$	17.9462	1.06679	0.533395	+	0.845866i	$0.320916\pi$
$284$	−2.35223	−0.139579
$285$	0	0
$286$	2.66894	0.157818
$287$	4.93020	0.291021
$288$	0	0
$289$	−1.51947	−0.0893807
$290$	−31.8668	−1.87128
$291$	0	0
$292$	10.1300	0.592812
$293$	−3.06625	−0.179132	−0.0895661	−	0.995981i	$-0.528548\pi$
$293$	−3.06625	−0.179132	−0.0895661	+	0.995981i	$0.528548\pi$
$294$	0	0
$295$	47.1199	2.74343
$296$	3.67086	0.213365
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−3.94946	−0.228403
$300$	0	0
$301$	−4.64297	−0.267617
$302$	−13.2487	−0.762375
$303$	0	0
$304$	−3.57365	−0.204963
$305$	−8.09911	−0.463754
$306$	0	0
$307$	6.82680	0.389626	0.194813	−	0.980840i	$-0.437590\pi$
$307$	6.82680	0.389626	0.194813	+	0.980840i	$0.437590\pi$
$308$	−0.485218	−0.0276479
$309$	0	0
$310$	−8.28241	−0.470409
$311$	11.5998	0.657765	0.328882	−	0.944371i	$-0.393328\pi$
$311$	11.5998	0.657765	0.328882	+	0.944371i	$0.393328\pi$
$312$	0	0
$313$	16.4325	0.928818	0.464409	−	0.885621i	$-0.346267\pi$
$313$	16.4325	0.928818	0.464409	+	0.885621i	$0.346267\pi$
$314$	10.5613	0.596009
$315$	0	0
$316$	−5.14150	−0.289232
$317$	−12.2930	−0.690445	−0.345222	−	0.938521i	$-0.612197\pi$
$317$	−12.2930	−0.690445	−0.345222	+	0.938521i	$0.612197\pi$
$318$	0	0
$319$	−7.87463	−0.440895
$320$	−3.80777	−0.212861
$321$	0	0
$322$	0.718018	0.0400136
$323$	−14.0606	−0.782355
$324$	0	0
$325$	−26.9438	−1.49457
$326$	−12.9514	−0.717312
$327$	0	0
$328$	9.56071	0.527902
$329$	−1.49391	−0.0823620
$330$	0	0
$331$	−18.3673	−1.00956	−0.504779	−	0.863248i	$-0.668426\pi$
$331$	−18.3673	−1.00956	−0.504779	+	0.863248i	$0.668426\pi$
$332$	3.57873	0.196408
$333$	0	0
$334$	−11.4775	−0.628023
$335$	39.1288	2.13783
$336$	0	0
$337$	26.3649	1.43619	0.718094	−	0.695946i	$-0.245015\pi$
$337$	26.3649	1.43619	0.718094	+	0.695946i	$0.245015\pi$
$338$	−4.95449	−0.269489
$339$	0	0
$340$	−14.9818	−0.812501
$341$	−2.04667	−0.110834
$342$	0	0
$343$	−7.08230	−0.382408
$344$	−9.00371	−0.485448
$345$	0	0
$346$	24.8186	1.33425
$347$	−31.6944	−1.70144	−0.850721	−	0.525617i	$-0.823835\pi$
$347$	−31.6944	−1.70144	−0.850721	+	0.525617i	$0.823835\pi$
$348$	0	0
$349$	−13.0175	−0.696809	−0.348404	−	0.937344i	$-0.613276\pi$
$349$	−13.0175	−0.696809	−0.348404	+	0.937344i	$0.613276\pi$
$350$	4.89843	0.261832
$351$	0	0
$352$	−0.940941	−0.0501523
$353$	24.8615	1.32324	0.661622	−	0.749837i	$-0.269868\pi$
$353$	24.8615	1.32324	0.661622	+	0.749837i	$0.269868\pi$
$354$	0	0
$355$	8.95673	0.475374
$356$	0.00223128	0.000118258 0
$357$	0	0
$358$	−9.86161	−0.521202
$359$	8.53896	0.450669	0.225335	−	0.974281i	$-0.427653\pi$
$359$	8.53896	0.450669	0.225335	+	0.974281i	$0.427653\pi$
$360$	0	0
$361$	−6.22900	−0.327842
$362$	−15.6306	−0.821527
$363$	0	0
$364$	−1.46269	−0.0766657
$365$	−38.5726	−2.01898
$366$	0	0
$367$	22.8141	1.19089	0.595443	−	0.803397i	$-0.296976\pi$
$367$	22.8141	1.19089	0.595443	+	0.803397i	$0.296976\pi$
$368$	1.39239	0.0725833
$369$	0	0
$370$	−13.9778	−0.726671
$371$	−0.603726	−0.0313439
$372$	0	0
$373$	−34.0227	−1.76163	−0.880813	−	0.473464i	$-0.843003\pi$
$373$	−34.0227	−1.76163	−0.880813	+	0.473464i	$0.843003\pi$
$374$	−3.70216	−0.191434
$375$	0	0
$376$	−2.89701	−0.149402
$377$	−23.7380	−1.22257
$378$	0	0
$379$	3.03351	0.155821	0.0779105	−	0.996960i	$-0.475175\pi$
$379$	3.03351	0.155821	0.0779105	+	0.996960i	$0.475175\pi$
$380$	13.6076	0.698057
$381$	0	0
$382$	−8.22930	−0.421048
$383$	22.2167	1.13522	0.567609	−	0.823298i	$-0.307869\pi$
$383$	22.2167	1.13522	0.567609	+	0.823298i	$0.307869\pi$
$384$	0	0
$385$	1.84760	0.0941624
$386$	−3.29466	−0.167694
$387$	0	0
$388$	3.82873	0.194374
$389$	−34.5062	−1.74954	−0.874768	−	0.484543i	$-0.838986\pi$
$389$	−34.5062	−1.74954	−0.874768	+	0.484543i	$0.838986\pi$
$390$	0	0
$391$	5.47840	0.277055
$392$	−6.73408	−0.340122
$393$	0	0
$394$	−13.7739	−0.693918
$395$	19.5776	0.985058
$396$	0	0
$397$	−29.6718	−1.48918	−0.744592	−	0.667520i	$-0.767356\pi$
$397$	−29.6718	−1.48918	−0.744592	+	0.667520i	$0.767356\pi$
$398$	2.93461	0.147099
$399$	0	0
$400$	9.49910	0.474955
$401$	18.1554	0.906637	0.453318	−	0.891349i	$-0.350240\pi$
$401$	18.1554	0.906637	0.453318	+	0.891349i	$0.350240\pi$
$402$	0	0
$403$	−6.16968	−0.307334
$404$	2.22598	0.110747
$405$	0	0
$406$	4.31562	0.214180
$407$	−3.45407	−0.171212
$408$	0	0
$409$	−25.9816	−1.28471	−0.642355	−	0.766408i	$-0.722042\pi$
$409$	−25.9816	−1.28471	−0.642355	+	0.766408i	$0.722042\pi$
$410$	−36.4050	−1.79791
$411$	0	0
$412$	−13.3755	−0.658966
$413$	−6.38130	−0.314003
$414$	0	0
$415$	−13.6270	−0.668921
$416$	−2.83646	−0.139069
$417$	0	0
$418$	3.36260	0.164470
$419$	−30.8938	−1.50926	−0.754629	−	0.656151i	$-0.772183\pi$
$419$	−30.8938	−1.50926	−0.754629	+	0.656151i	$0.772183\pi$
$420$	0	0
$421$	17.9253	0.873625	0.436812	−	0.899553i	$-0.356107\pi$
$421$	17.9253	0.873625	0.436812	+	0.899553i	$0.356107\pi$
$422$	13.5697	0.660562
$423$	0	0
$424$	−1.17075	−0.0568568
$425$	37.3745	1.81293
$426$	0	0
$427$	1.09684	0.0530796
$428$	−4.96212	−0.239853
$429$	0	0
$430$	34.2840	1.65332
$431$	−9.24825	−0.445472	−0.222736	−	0.974879i	$-0.571499\pi$
$431$	−9.24825	−0.445472	−0.222736	+	0.974879i	$0.571499\pi$
$432$	0	0
$433$	−9.35001	−0.449333	−0.224666	−	0.974436i	$-0.572129\pi$
$433$	−9.35001	−0.449333	−0.224666	+	0.974436i	$0.572129\pi$
$434$	1.12166	0.0538413
$435$	0	0
$436$	−2.01553	−0.0965264
$437$	−4.97592	−0.238030
$438$	0	0
$439$	0.0383341	0.00182959	0.000914793	−	1.00000i	$-0.499709\pi$
$439$	0.0383341	0.00182959	0.000914793		1.00000i	$0.499709\pi$
$440$	3.58289	0.170807
$441$	0	0
$442$	−11.1601	−0.530834
$443$	7.37697	0.350490	0.175245	−	0.984525i	$-0.443928\pi$
$443$	7.37697	0.350490	0.175245	+	0.984525i	$0.443928\pi$
$444$	0	0
$445$	−0.00849621	−0.000402759 0
$446$	0.372556	0.0176411
$447$	0	0
$448$	0.515673	0.0243633
$449$	14.1872	0.669534	0.334767	−	0.942301i	$-0.391342\pi$
$449$	14.1872	0.669534	0.334767	+	0.942301i	$0.391342\pi$
$450$	0	0
$451$	−8.99606	−0.423608
$452$	−7.87785	−0.370543
$453$	0	0
$454$	12.9343	0.607035
$455$	5.56958	0.261106
$456$	0	0
$457$	9.03586	0.422680	0.211340	−	0.977413i	$-0.432217\pi$
$457$	9.03586	0.422680	0.211340	+	0.977413i	$0.432217\pi$
$458$	−6.44573	−0.301189
$459$	0	0
$460$	−5.30190	−0.247202
$461$	9.78904	0.455921	0.227961	−	0.973670i	$-0.426794\pi$
$461$	9.78904	0.455921	0.227961	+	0.973670i	$0.426794\pi$
$462$	0	0
$463$	11.6439	0.541136	0.270568	−	0.962701i	$-0.412788\pi$
$463$	11.6439	0.541136	0.270568	+	0.962701i	$0.412788\pi$
$464$	8.36889	0.388516
$465$	0	0
$466$	−7.75366	−0.359181
$467$	−9.99358	−0.462448	−0.231224	−	0.972901i	$-0.574273\pi$
$467$	−9.99358	−0.462448	−0.231224	+	0.972901i	$0.574273\pi$
$468$	0	0
$469$	−5.29908	−0.244689
$470$	11.0311	0.508829
$471$	0	0
$472$	−12.3747	−0.569591
$473$	8.47196	0.389541
$474$	0	0
$475$	−33.9465	−1.55757
$476$	2.02893	0.0929960
$477$	0	0
$478$	−14.3567	−0.656662
$479$	22.4494	1.02574	0.512871	−	0.858466i	$-0.328582\pi$
$479$	22.4494	1.02574	0.512871	+	0.858466i	$0.328582\pi$
$480$	0	0
$481$	−10.4123	−0.474758
$482$	12.1139	0.551772
$483$	0	0
$484$	−10.1146	−0.459756
$485$	−14.5789	−0.661995
$486$	0	0
$487$	9.74065	0.441391	0.220695	−	0.975343i	$-0.429167\pi$
$487$	9.74065	0.441391	0.220695	+	0.975343i	$0.429167\pi$
$488$	2.12700	0.0962847
$489$	0	0
$490$	25.6418	1.15838
$491$	−31.6763	−1.42953	−0.714766	−	0.699364i	$-0.753467\pi$
$491$	−31.6763	−1.42953	−0.714766	+	0.699364i	$0.753467\pi$
$492$	0	0
$493$	32.9277	1.48299
$494$	10.1365	0.456064
$495$	0	0
$496$	2.17513	0.0976664
$497$	−1.21298	−0.0544096
$498$	0	0
$499$	6.13481	0.274632	0.137316	−	0.990527i	$-0.456152\pi$
$499$	6.13481	0.274632	0.137316	+	0.990527i	$0.456152\pi$
$500$	−17.1315	−0.766145
$501$	0	0
$502$	−16.3635	−0.730339
$503$	5.23723	0.233517	0.116758	−	0.993160i	$-0.462750\pi$
$503$	5.23723	0.233517	0.116758	+	0.993160i	$0.462750\pi$
$504$	0	0
$505$	−8.47603	−0.377178
$506$	−1.31016	−0.0582436
$507$	0	0
$508$	−19.7227	−0.875054
$509$	−31.0989	−1.37844	−0.689218	−	0.724554i	$-0.742046\pi$
$509$	−31.0989	−1.37844	−0.689218	+	0.724554i	$0.742046\pi$
$510$	0	0
$511$	5.22376	0.231086
$512$	1.00000	0.0441942
$513$	0	0
$514$	−0.104679	−0.00461719
$515$	50.9310	2.24429
$516$	0	0
$517$	2.72592	0.119886
$518$	1.89297	0.0831722
$519$	0	0
$520$	10.8006	0.473637
$521$	−8.77765	−0.384556	−0.192278	−	0.981341i	$-0.561588\pi$
$521$	−8.77765	−0.384556	−0.192278	+	0.981341i	$0.561588\pi$
$522$	0	0
$523$	33.8192	1.47881	0.739406	−	0.673260i	$-0.235106\pi$
$523$	33.8192	1.47881	0.739406	+	0.673260i	$0.235106\pi$
$524$	10.9019	0.476252
$525$	0	0
$526$	−21.3035	−0.928876
$527$	8.55813	0.372798
$528$	0	0
$529$	−21.0613	−0.915707
$530$	4.45795	0.193641
$531$	0	0
$532$	−1.84284	−0.0798972
$533$	−27.1186	−1.17464
$534$	0	0
$535$	18.8946	0.816886
$536$	−10.2760	−0.443857
$537$	0	0
$538$	9.61799	0.414661
$539$	6.33637	0.272927
$540$	0	0
$541$	28.0587	1.20634	0.603170	−	0.797613i	$-0.293904\pi$
$541$	28.0587	1.20634	0.603170	+	0.797613i	$0.293904\pi$
$542$	14.1277	0.606835
$543$	0	0
$544$	3.93453	0.168692
$545$	7.67468	0.328747
$546$	0	0
$547$	13.7186	0.586565	0.293282	−	0.956026i	$-0.405252\pi$
$547$	13.7186	0.586565	0.293282	+	0.956026i	$0.405252\pi$
$548$	6.40820	0.273745
$549$	0	0
$550$	−8.93809	−0.381121
$551$	−29.9075	−1.27410
$552$	0	0
$553$	−2.65134	−0.112746
$554$	−20.2247	−0.859267
$555$	0	0
$556$	−17.0897	−0.724763
$557$	−38.7258	−1.64087	−0.820433	−	0.571743i	$-0.806267\pi$
$557$	−38.7258	−1.64087	−0.820433	+	0.571743i	$0.806267\pi$
$558$	0	0
$559$	25.5387	1.08017
$560$	−1.96357	−0.0829758
$561$	0	0
$562$	19.8872	0.838889
$563$	27.0005	1.13793	0.568967	−	0.822360i	$-0.307343\pi$
$563$	27.0005	1.13793	0.568967	+	0.822360i	$0.307343\pi$
$564$	0	0
$565$	29.9970	1.26198
$566$	17.9462	0.754335
$567$	0	0
$568$	−2.35223	−0.0986972
$569$	44.4013	1.86140	0.930700	−	0.365784i	$-0.119199\pi$
$569$	44.4013	1.86140	0.930700	+	0.365784i	$0.119199\pi$
$570$	0	0
$571$	16.1826	0.677220	0.338610	−	0.940927i	$-0.390043\pi$
$571$	16.1826	0.677220	0.338610	+	0.940927i	$0.390043\pi$
$572$	2.66894	0.111594
$573$	0	0
$574$	4.93020	0.205783
$575$	13.2264	0.551581
$576$	0	0
$577$	−38.4291	−1.59983	−0.799913	−	0.600116i	$-0.795121\pi$
$577$	−38.4291	−1.59983	−0.799913	+	0.600116i	$0.795121\pi$
$578$	−1.51947	−0.0632017
$579$	0	0
$580$	−31.8668	−1.32320
$581$	1.84545	0.0765623
$582$	0	0
$583$	1.10161	0.0456240
$584$	10.1300	0.419182
$585$	0	0
$586$	−3.06625	−0.126666
$587$	−45.0593	−1.85980	−0.929898	−	0.367816i	$-0.880105\pi$
$587$	−45.0593	−1.85980	−0.929898	+	0.367816i	$0.880105\pi$
$588$	0	0
$589$	−7.77318	−0.320288
$590$	47.1199	1.93990
$591$	0	0
$592$	3.67086	0.150872
$593$	6.82079	0.280096	0.140048	−	0.990145i	$-0.455274\pi$
$593$	6.82079	0.280096	0.140048	+	0.990145i	$0.455274\pi$
$594$	0	0
$595$	−7.72571	−0.316723
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−3.94946	−0.161505
$599$	−34.9813	−1.42930	−0.714648	−	0.699485i	$-0.753413\pi$
$599$	−34.9813	−1.42930	−0.714648	+	0.699485i	$0.753413\pi$
$600$	0	0
$601$	−7.13369	−0.290989	−0.145495	−	0.989359i	$-0.546477\pi$
$601$	−7.13369	−0.290989	−0.145495	+	0.989359i	$0.546477\pi$
$602$	−4.64297	−0.189234
$603$	0	0
$604$	−13.2487	−0.539081
$605$	38.5142	1.56582
$606$	0	0
$607$	−15.1999	−0.616946	−0.308473	−	0.951233i	$-0.599818\pi$
$607$	−15.1999	−0.616946	−0.308473	+	0.951233i	$0.599818\pi$
$608$	−3.57365	−0.144931
$609$	0	0
$610$	−8.09911	−0.327924
$611$	8.21726	0.332435
$612$	0	0
$613$	−2.50818	−0.101304	−0.0506521	−	0.998716i	$-0.516130\pi$
$613$	−2.50818	−0.101304	−0.0506521	+	0.998716i	$0.516130\pi$
$614$	6.82680	0.275507
$615$	0	0
$616$	−0.485218	−0.0195500
$617$	−21.9733	−0.884612	−0.442306	−	0.896864i	$-0.645839\pi$
$617$	−21.9733	−0.884612	−0.442306	+	0.896864i	$0.645839\pi$
$618$	0	0
$619$	−19.0935	−0.767432	−0.383716	−	0.923451i	$-0.625356\pi$
$619$	−19.0935	−0.767432	−0.383716	+	0.923451i	$0.625356\pi$
$620$	−8.28241	−0.332629
$621$	0	0
$622$	11.5998	0.465110
$623$	0.00115061	4.60983e−5 0
$624$	0	0
$625$	17.7374	0.709494
$626$	16.4325	0.656773
$627$	0	0
$628$	10.5613	0.421442
$629$	14.4431	0.575885
$630$	0	0
$631$	−42.9622	−1.71030	−0.855149	−	0.518382i	$-0.826534\pi$
$631$	−42.9622	−1.71030	−0.855149	+	0.518382i	$0.826534\pi$
$632$	−5.14150	−0.204518
$633$	0	0
$634$	−12.2930	−0.488218
$635$	75.0996	2.98023
$636$	0	0
$637$	19.1010	0.756808
$638$	−7.87463	−0.311760
$639$	0	0
$640$	−3.80777	−0.150515
$641$	−2.70665	−0.106906	−0.0534532	−	0.998570i	$-0.517023\pi$
$641$	−2.70665	−0.106906	−0.0534532	+	0.998570i	$0.517023\pi$
$642$	0	0
$643$	14.2882	0.563472	0.281736	−	0.959492i	$-0.409090\pi$
$643$	14.2882	0.563472	0.281736	+	0.959492i	$0.409090\pi$
$644$	0.718018	0.0282939
$645$	0	0
$646$	−14.0606	−0.553209
$647$	−17.7134	−0.696387	−0.348194	−	0.937423i	$-0.613205\pi$
$647$	−17.7134	−0.696387	−0.348194	+	0.937423i	$0.613205\pi$
$648$	0	0
$649$	11.6438	0.457061
$650$	−26.9438	−1.05682
$651$	0	0
$652$	−12.9514	−0.507216
$653$	13.0562	0.510927	0.255463	−	0.966819i	$-0.417772\pi$
$653$	13.0562	0.510927	0.255463	+	0.966819i	$0.417772\pi$
$654$	0	0
$655$	−41.5119	−1.62200
$656$	9.56071	0.373283
$657$	0	0
$658$	−1.49391	−0.0582387
$659$	5.96324	0.232295	0.116147	−	0.993232i	$-0.462945\pi$
$659$	5.96324	0.232295	0.116147	+	0.993232i	$0.462945\pi$
$660$	0	0
$661$	−15.6385	−0.608267	−0.304133	−	0.952629i	$-0.598367\pi$
$661$	−15.6385	−0.608267	−0.304133	+	0.952629i	$0.598367\pi$
$662$	−18.3673	−0.713866
$663$	0	0
$664$	3.57873	0.138881
$665$	7.01710	0.272112
$666$	0	0
$667$	11.6528	0.451197
$668$	−11.4775	−0.444079
$669$	0	0
$670$	39.1288	1.51168
$671$	−2.00138	−0.0772624
$672$	0	0
$673$	13.3684	0.515315	0.257657	−	0.966236i	$-0.417049\pi$
$673$	13.3684	0.515315	0.257657	+	0.966236i	$0.417049\pi$
$674$	26.3649	1.01554
$675$	0	0
$676$	−4.95449	−0.190557
$677$	−0.501351	−0.0192685	−0.00963424	−	0.999954i	$-0.503067\pi$
$677$	−0.501351	−0.0192685	−0.00963424	+	0.999954i	$0.503067\pi$
$678$	0	0
$679$	1.97438	0.0757696
$680$	−14.9818	−0.574525
$681$	0	0
$682$	−2.04667	−0.0783711
$683$	−38.7401	−1.48235	−0.741174	−	0.671313i	$-0.765731\pi$
$683$	−38.7401	−1.48235	−0.741174	+	0.671313i	$0.765731\pi$
$684$	0	0
$685$	−24.4009	−0.932311
$686$	−7.08230	−0.270404
$687$	0	0
$688$	−9.00371	−0.343263
$689$	3.32079	0.126512
$690$	0	0
$691$	17.5941	0.669310	0.334655	−	0.942341i	$-0.391380\pi$
$691$	17.5941	0.669310	0.334655	+	0.942341i	$0.391380\pi$
$692$	24.8186	0.943461
$693$	0	0
$694$	−31.6944	−1.20310
$695$	65.0735	2.46838
$696$	0	0
$697$	37.6169	1.42484
$698$	−13.0175	−0.492718
$699$	0	0
$700$	4.89843	0.185143
$701$	47.1097	1.77931	0.889655	−	0.456633i	$-0.150945\pi$
$701$	47.1097	1.77931	0.889655	+	0.456633i	$0.150945\pi$
$702$	0	0
$703$	−13.1184	−0.494769
$704$	−0.940941	−0.0354630
$705$	0	0
$706$	24.8615	0.935675
$707$	1.14788	0.0431705
$708$	0	0
$709$	−32.4176	−1.21747	−0.608734	−	0.793374i	$-0.708322\pi$
$709$	−32.4176	−1.21747	−0.608734	+	0.793374i	$0.708322\pi$
$710$	8.95673	0.336140
$711$	0	0
$712$	0.00223128	8.36209e−5 0
$713$	3.02863	0.113423
$714$	0	0
$715$	−10.1627	−0.380064
$716$	−9.86161	−0.368546
$717$	0	0
$718$	8.53896	0.318671
$719$	8.71247	0.324920	0.162460	−	0.986715i	$-0.448057\pi$
$719$	8.71247	0.324920	0.162460	+	0.986715i	$0.448057\pi$
$720$	0	0
$721$	−6.89741	−0.256873
$722$	−6.22900	−0.231819
$723$	0	0
$724$	−15.6306	−0.580908
$725$	79.4969	2.95244
$726$	0	0
$727$	−26.3912	−0.978795	−0.489398	−	0.872061i	$-0.662783\pi$
$727$	−26.3912	−0.978795	−0.489398	+	0.872061i	$0.662783\pi$
$728$	−1.46269	−0.0542108
$729$	0	0
$730$	−38.5726	−1.42764
$731$	−35.4254	−1.31025
$732$	0	0
$733$	9.01175	0.332856	0.166428	−	0.986054i	$-0.446777\pi$
$733$	9.01175	0.332856	0.166428	+	0.986054i	$0.446777\pi$
$734$	22.8141	0.842084
$735$	0	0
$736$	1.39239	0.0513242
$737$	9.66915	0.356168
$738$	0	0
$739$	22.9160	0.842980	0.421490	−	0.906833i	$-0.361507\pi$
$739$	22.9160	0.842980	0.421490	+	0.906833i	$0.361507\pi$
$740$	−13.9778	−0.513834
$741$	0	0
$742$	−0.603726	−0.0221635
$743$	7.25666	0.266221	0.133111	−	0.991101i	$-0.457503\pi$
$743$	7.25666	0.266221	0.133111	+	0.991101i	$0.457503\pi$
$744$	0	0
$745$	3.80777	0.139506
$746$	−34.0227	−1.24566
$747$	0	0
$748$	−3.70216	−0.135364
$749$	−2.55884	−0.0934978
$750$	0	0
$751$	−26.0409	−0.950244	−0.475122	−	0.879920i	$-0.657596\pi$
$751$	−26.0409	−0.950244	−0.475122	+	0.879920i	$0.657596\pi$
$752$	−2.89701	−0.105643
$753$	0	0
$754$	−23.7380	−0.864488
$755$	50.4479	1.83599
$756$	0	0
$757$	−23.4919	−0.853829	−0.426915	−	0.904292i	$-0.640400\pi$
$757$	−23.4919	−0.853829	−0.426915	+	0.904292i	$0.640400\pi$
$758$	3.03351	0.110182
$759$	0	0
$760$	13.6076	0.493601
$761$	42.5099	1.54098	0.770491	−	0.637451i	$-0.220011\pi$
$761$	42.5099	1.54098	0.770491	+	0.637451i	$0.220011\pi$
$762$	0	0
$763$	−1.03936	−0.0376272
$764$	−8.22930	−0.297726
$765$	0	0
$766$	22.2167	0.802720
$767$	35.1003	1.26740
$768$	0	0
$769$	−0.651559	−0.0234958	−0.0117479	−	0.999931i	$-0.503740\pi$
$769$	−0.651559	−0.0234958	−0.0117479	+	0.999931i	$0.503740\pi$
$770$	1.84760	0.0665828
$771$	0	0
$772$	−3.29466	−0.118577
$773$	−2.37546	−0.0854393	−0.0427196	−	0.999087i	$-0.513602\pi$
$773$	−2.37546	−0.0854393	−0.0427196	+	0.999087i	$0.513602\pi$
$774$	0	0
$775$	20.6618	0.742194
$776$	3.82873	0.137444
$777$	0	0
$778$	−34.5062	−1.23711
$779$	−34.1667	−1.22415
$780$	0	0
$781$	2.21331	0.0791983
$782$	5.47840	0.195907
$783$	0	0
$784$	−6.73408	−0.240503
$785$	−40.2150	−1.43533
$786$	0	0
$787$	5.98295	0.213269	0.106635	−	0.994298i	$-0.465993\pi$
$787$	5.98295	0.213269	0.106635	+	0.994298i	$0.465993\pi$
$788$	−13.7739	−0.490674
$789$	0	0
$790$	19.5776	0.696541
$791$	−4.06240	−0.144442
$792$	0	0
$793$	−6.03315	−0.214243
$794$	−29.6718	−1.05301
$795$	0	0
$796$	2.93461	0.104014
$797$	28.4827	1.00891	0.504455	−	0.863438i	$-0.331694\pi$
$797$	28.4827	1.00891	0.504455	+	0.863438i	$0.331694\pi$
$798$	0	0
$799$	−11.3984	−0.403246
$800$	9.49910	0.335844
$801$	0	0
$802$	18.1554	0.641089
$803$	−9.53171	−0.336367
$804$	0	0
$805$	−2.73405	−0.0963625
$806$	−6.16968	−0.217318
$807$	0	0
$808$	2.22598	0.0783098
$809$	−20.2110	−0.710582	−0.355291	−	0.934756i	$-0.615618\pi$
$809$	−20.2110	−0.710582	−0.355291	+	0.934756i	$0.615618\pi$
$810$	0	0
$811$	−9.44577	−0.331686	−0.165843	−	0.986152i	$-0.553034\pi$
$811$	−9.44577	−0.331686	−0.165843	+	0.986152i	$0.553034\pi$
$812$	4.31562	0.151448
$813$	0	0
$814$	−3.45407	−0.121065
$815$	49.3159	1.72746
$816$	0	0
$817$	32.1761	1.12570
$818$	−25.9816	−0.908426
$819$	0	0
$820$	−36.4050	−1.27132
$821$	3.74294	0.130629	0.0653147	−	0.997865i	$-0.479195\pi$
$821$	3.74294	0.130629	0.0653147	+	0.997865i	$0.479195\pi$
$822$	0	0
$823$	35.8071	1.24816	0.624078	−	0.781362i	$-0.285475\pi$
$823$	35.8071	1.24816	0.624078	+	0.781362i	$0.285475\pi$
$824$	−13.3755	−0.465959
$825$	0	0
$826$	−6.38130	−0.222034
$827$	18.6515	0.648577	0.324289	−	0.945958i	$-0.394875\pi$
$827$	18.6515	0.648577	0.324289	+	0.945958i	$0.394875\pi$
$828$	0	0
$829$	−46.9704	−1.63135	−0.815674	−	0.578512i	$-0.803634\pi$
$829$	−46.9704	−1.63135	−0.815674	+	0.578512i	$0.803634\pi$
$830$	−13.6270	−0.472999
$831$	0	0
$832$	−2.83646	−0.0983366
$833$	−26.4954	−0.918013
$834$	0	0
$835$	43.7038	1.51243
$836$	3.36260	0.116298
$837$	0	0
$838$	−30.8938	−1.06721
$839$	−35.3679	−1.22104	−0.610518	−	0.792002i	$-0.709039\pi$
$839$	−35.3679	−1.22104	−0.610518	+	0.792002i	$0.709039\pi$
$840$	0	0
$841$	41.0384	1.41512
$842$	17.9253	0.617746
$843$	0	0
$844$	13.5697	0.467088
$845$	18.8655	0.648994
$846$	0	0
$847$	−5.21585	−0.179219
$848$	−1.17075	−0.0402038
$849$	0	0
$850$	37.3745	1.28193
$851$	5.11127	0.175212
$852$	0	0
$853$	−36.2844	−1.24236	−0.621178	−	0.783670i	$-0.713345\pi$
$853$	−36.2844	−1.24236	−0.621178	+	0.783670i	$0.713345\pi$
$854$	1.09684	0.0375330
$855$	0	0
$856$	−4.96212	−0.169602
$857$	−41.3401	−1.41215	−0.706076	−	0.708136i	$-0.749536\pi$
$857$	−41.3401	−1.41215	−0.706076	+	0.708136i	$0.749536\pi$
$858$	0	0
$859$	12.1747	0.415395	0.207697	−	0.978193i	$-0.433403\pi$
$859$	12.1747	0.415395	0.207697	+	0.978193i	$0.433403\pi$
$860$	34.2840	1.16908
$861$	0	0
$862$	−9.24825	−0.314996
$863$	26.6836	0.908319	0.454160	−	0.890920i	$-0.349940\pi$
$863$	26.6836	0.908319	0.454160	+	0.890920i	$0.349940\pi$
$864$	0	0
$865$	−94.5034	−3.21321
$866$	−9.35001	−0.317726
$867$	0	0
$868$	1.12166	0.0380716
$869$	4.83785	0.164113
$870$	0	0
$871$	29.1476	0.987628
$872$	−2.01553	−0.0682545
$873$	0	0
$874$	−4.97592	−0.168313
$875$	−8.83427	−0.298653
$876$	0	0
$877$	−23.8159	−0.804205	−0.402103	−	0.915595i	$-0.631720\pi$
$877$	−23.8159	−0.804205	−0.402103	+	0.915595i	$0.631720\pi$
$878$	0.0383341	0.00129371
$879$	0	0
$880$	3.58289	0.120779
$881$	27.5213	0.927216	0.463608	−	0.886040i	$-0.346555\pi$
$881$	27.5213	0.927216	0.463608	+	0.886040i	$0.346555\pi$
$882$	0	0
$883$	33.0700	1.11289	0.556447	−	0.830883i	$-0.312164\pi$
$883$	33.0700	1.11289	0.556447	+	0.830883i	$0.312164\pi$
$884$	−11.1601	−0.375356
$885$	0	0
$886$	7.37697	0.247834
$887$	−23.4709	−0.788074	−0.394037	−	0.919095i	$-0.628922\pi$
$887$	−23.4709	−0.788074	−0.394037	+	0.919095i	$0.628922\pi$
$888$	0	0
$889$	−10.1705	−0.341107
$890$	−0.00849621	−0.000284793 0
$891$	0	0
$892$	0.372556	0.0124741
$893$	10.3529	0.346447
$894$	0	0
$895$	37.5507	1.25518
$896$	0.515673	0.0172274
$897$	0	0
$898$	14.1872	0.473432
$899$	18.2035	0.607119
$900$	0	0
$901$	−4.60636	−0.153460
$902$	−8.99606	−0.299536
$903$	0	0
$904$	−7.87785	−0.262014
$905$	59.5178	1.97844
$906$	0	0
$907$	−51.5516	−1.71174	−0.855871	−	0.517189i	$-0.826978\pi$
$907$	−51.5516	−1.71174	−0.855871	+	0.517189i	$0.826978\pi$
$908$	12.9343	0.429239
$909$	0	0
$910$	5.56958	0.184630
$911$	−44.9739	−1.49005	−0.745027	−	0.667035i	$-0.767563\pi$
$911$	−44.9739	−1.49005	−0.745027	+	0.667035i	$0.767563\pi$
$912$	0	0
$913$	−3.36737	−0.111444
$914$	9.03586	0.298880
$915$	0	0
$916$	−6.44573	−0.212973
$917$	5.62182	0.185649
$918$	0	0
$919$	31.1582	1.02781	0.513907	−	0.857846i	$-0.328198\pi$
$919$	31.1582	1.02781	0.513907	+	0.857846i	$0.328198\pi$
$920$	−5.30190	−0.174798
$921$	0	0
$922$	9.78904	0.322385
$923$	6.67200	0.219611
$924$	0	0
$925$	34.8699	1.14651
$926$	11.6439	0.382641
$927$	0	0
$928$	8.36889	0.274722
$929$	28.6933	0.941397	0.470699	−	0.882294i	$-0.344002\pi$
$929$	28.6933	0.941397	0.470699	+	0.882294i	$0.344002\pi$
$930$	0	0
$931$	24.0653	0.788707
$932$	−7.75366	−0.253980
$933$	0	0
$934$	−9.99358	−0.327000
$935$	14.0970	0.461020
$936$	0	0
$937$	−47.8718	−1.56390	−0.781951	−	0.623339i	$-0.785776\pi$
$937$	−47.8718	−1.56390	−0.781951	+	0.623339i	$0.785776\pi$
$938$	−5.29908	−0.173021
$939$	0	0
$940$	11.0311	0.359796
$941$	−4.61707	−0.150512	−0.0752561	−	0.997164i	$-0.523977\pi$
$941$	−4.61707	−0.150512	−0.0752561	+	0.997164i	$0.523977\pi$
$942$	0	0
$943$	13.3122	0.433506
$944$	−12.3747	−0.402762
$945$	0	0
$946$	8.47196	0.275447
$947$	37.4692	1.21759	0.608793	−	0.793329i	$-0.291654\pi$
$947$	37.4692	1.21759	0.608793	+	0.793329i	$0.291654\pi$
$948$	0	0
$949$	−28.7333	−0.932722
$950$	−33.9465	−1.10137
$951$	0	0
$952$	2.02893	0.0657581
$953$	38.7731	1.25598	0.627992	−	0.778219i	$-0.283877\pi$
$953$	38.7731	1.25598	0.627992	+	0.778219i	$0.283877\pi$
$954$	0	0
$955$	31.3353	1.01399
$956$	−14.3567	−0.464330
$957$	0	0
$958$	22.4494	0.725308
$959$	3.30454	0.106709
$960$	0	0
$961$	−26.2688	−0.847380
$962$	−10.4123	−0.335705
$963$	0	0
$964$	12.1139	0.390162
$965$	12.5453	0.403848
$966$	0	0
$967$	23.4848	0.755221	0.377610	−	0.925965i	$-0.376746\pi$
$967$	23.4848	0.755221	0.377610	+	0.925965i	$0.376746\pi$
$968$	−10.1146	−0.325097
$969$	0	0
$970$	−14.5789	−0.468101
$971$	33.1490	1.06380	0.531901	−	0.846806i	$-0.321478\pi$
$971$	33.1490	1.06380	0.531901	+	0.846806i	$0.321478\pi$
$972$	0	0
$973$	−8.81269	−0.282522
$974$	9.74065	0.312111
$975$	0	0
$976$	2.12700	0.0680835
$977$	5.41485	0.173236	0.0866182	−	0.996242i	$-0.472394\pi$
$977$	5.41485	0.173236	0.0866182	+	0.996242i	$0.472394\pi$
$978$	0	0
$979$	−0.00209951	−6.71005e−5 0
$980$	25.6418	0.819098
$981$	0	0
$982$	−31.6763	−1.01083
$983$	43.8075	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$983$	43.8075	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$984$	0	0
$985$	52.4477	1.67112
$986$	32.9277	1.04863
$987$	0	0
$988$	10.1365	0.322486
$989$	−12.5367	−0.398643
$990$	0	0
$991$	7.59749	0.241342	0.120671	−	0.992693i	$-0.461495\pi$
$991$	7.59749	0.241342	0.120671	+	0.992693i	$0.461495\pi$
$992$	2.17513	0.0690606
$993$	0	0
$994$	−1.21298	−0.0384734
$995$	−11.1743	−0.354249
$996$	0	0
$997$	47.1875	1.49444	0.747221	−	0.664575i	$-0.231387\pi$
$997$	47.1875	1.49444	0.747221	+	0.664575i	$0.231387\pi$
$998$	6.13481	0.194194
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.h.1.1	yes	9
3.2	odd	2		8046.2.a.g.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.g.1.9	✓	9	3.2	odd	2
8046.2.a.h.1.1	yes	9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.80777 q^{5} +0.515673 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.80777 q^{5} +0.515673 q^{7} +1.00000 q^{8} -3.80777 q^{10} -0.940941 q^{11} -2.83646 q^{13} +0.515673 q^{14} +1.00000 q^{16} +3.93453 q^{17} -3.57365 q^{19} -3.80777 q^{20} -0.940941 q^{22} +1.39239 q^{23} +9.49910 q^{25} -2.83646 q^{26} +0.515673 q^{28} +8.36889 q^{29} +2.17513 q^{31} +1.00000 q^{32} +3.93453 q^{34} -1.96357 q^{35} +3.67086 q^{37} -3.57365 q^{38} -3.80777 q^{40} +9.56071 q^{41} -9.00371 q^{43} -0.940941 q^{44} +1.39239 q^{46} -2.89701 q^{47} -6.73408 q^{49} +9.49910 q^{50} -2.83646 q^{52} -1.17075 q^{53} +3.58289 q^{55} +0.515673 q^{56} +8.36889 q^{58} -12.3747 q^{59} +2.12700 q^{61} +2.17513 q^{62} +1.00000 q^{64} +10.8006 q^{65} -10.2760 q^{67} +3.93453 q^{68} -1.96357 q^{70} -2.35223 q^{71} +10.1300 q^{73} +3.67086 q^{74} -3.57365 q^{76} -0.485218 q^{77} -5.14150 q^{79} -3.80777 q^{80} +9.56071 q^{82} +3.57873 q^{83} -14.9818 q^{85} -9.00371 q^{86} -0.940941 q^{88} +0.00223128 q^{89} -1.46269 q^{91} +1.39239 q^{92} -2.89701 q^{94} +13.6076 q^{95} +3.82873 q^{97} -6.73408 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) \) 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8	\( 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8} - 4 q^{10} - 4 q^{11} - 8 q^{13} - 4 q^{14} + 9 q^{16} - q^{17} - 10 q^{19} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 3 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{29} - 17 q^{31} + 9 q^{32} - q^{34} - 10 q^{35} - 11 q^{37} - 10 q^{38} - 4 q^{40} - 16 q^{43} - 4 q^{44} - 8 q^{46} - 7 q^{47} - 5 q^{49} - 3 q^{50} - 8 q^{52} - 12 q^{53} - 23 q^{55} - 4 q^{56} - 4 q^{58} - 6 q^{59} - 13 q^{61} - 17 q^{62} + 9 q^{64} + 24 q^{65} - 14 q^{67} - q^{68} - 10 q^{70} - 30 q^{71} - 12 q^{73} - 11 q^{74} - 10 q^{76} - 12 q^{77} - 35 q^{79} - 4 q^{80} - 5 q^{83} - 27 q^{85} - 16 q^{86} - 4 q^{88} - 23 q^{89} - 28 q^{91} - 8 q^{92} - 7 q^{94} - 32 q^{95} - 21 q^{97} - 5 q^{98}+O(q^{100}) \) 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8 - 4 * q^10 - 4 * q^11 - 8 * q^13 - 4 * q^14 + 9 * q^16 - q^17 - 10 * q^19 - 4 * q^20 - 4 * q^22 - 8 * q^23 - 3 * q^25 - 8 * q^26 - 4 * q^28 - 4 * q^29 - 17 * q^31 + 9 * q^32 - q^34 - 10 * q^35 - 11 * q^37 - 10 * q^38 - 4 * q^40 - 16 * q^43 - 4 * q^44 - 8 * q^46 - 7 * q^47 - 5 * q^49 - 3 * q^50 - 8 * q^52 - 12 * q^53 - 23 * q^55 - 4 * q^56 - 4 * q^58 - 6 * q^59 - 13 * q^61 - 17 * q^62 + 9 * q^64 + 24 * q^65 - 14 * q^67 - q^68 - 10 * q^70 - 30 * q^71 - 12 * q^73 - 11 * q^74 - 10 * q^76 - 12 * q^77 - 35 * q^79 - 4 * q^80 - 5 * q^83 - 27 * q^85 - 16 * q^86 - 4 * q^88 - 23 * q^89 - 28 * q^91 - 8 * q^92 - 7 * q^94 - 32 * q^95 - 21 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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