LMFDB - Embedded newform 8046.2.a.n.1.5

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} - 31 x^{10} + 82 x^{9} + 334 x^{8} - 684 x^{7} - 1561 x^{6} + 1551 x^{5} + 3573 x^{4} + \cdots - 3 $ x^12 - 3x^11 - 31x^10 + 82x^9 + 334x^8 - 684x^7 - 1561x^6 + 1551x^5 + 3573x^4 + 345x^3 - 1607x^2 - 594*x - 3
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			1.5
Root			$0.761360$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.761360 q^{5} +1.23946 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -0.761360 q^{5} +1.23946 q^{7} +1.00000 q^{8} -0.761360 q^{10} -4.16498 q^{11} +2.92984 q^{13} +1.23946 q^{14} +1.00000 q^{16} -2.05222 q^{17} +0.666318 q^{19} -0.761360 q^{20} -4.16498 q^{22} -8.48075 q^{23} -4.42033 q^{25} +2.92984 q^{26} +1.23946 q^{28} +1.77307 q^{29} +3.84602 q^{31} +1.00000 q^{32} -2.05222 q^{34} -0.943679 q^{35} +4.44887 q^{37} +0.666318 q^{38} -0.761360 q^{40} +2.52486 q^{41} +5.24402 q^{43} -4.16498 q^{44} -8.48075 q^{46} +3.71322 q^{47} -5.46373 q^{49} -4.42033 q^{50} +2.92984 q^{52} -8.26592 q^{53} +3.17105 q^{55} +1.23946 q^{56} +1.77307 q^{58} -9.37120 q^{59} +10.0229 q^{61} +3.84602 q^{62} +1.00000 q^{64} -2.23066 q^{65} -1.45888 q^{67} -2.05222 q^{68} -0.943679 q^{70} -6.19297 q^{71} -10.9177 q^{73} +4.44887 q^{74} +0.666318 q^{76} -5.16234 q^{77} -9.54970 q^{79} -0.761360 q^{80} +2.52486 q^{82} -5.69162 q^{83} +1.56248 q^{85} +5.24402 q^{86} -4.16498 q^{88} -7.72882 q^{89} +3.63143 q^{91} -8.48075 q^{92} +3.71322 q^{94} -0.507308 q^{95} -18.6599 q^{97} -5.46373 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 14 q^{11} - 3 q^{13} - 6 q^{14} + 12 q^{16} - 8 q^{17} - 4 q^{19} - 3 q^{20} - 14 q^{22} - 13 q^{23} + 11 q^{25} - 3 q^{26} - 6 q^{28} - 23 q^{29} - 14 q^{31} + 12 q^{32} - 8 q^{34} - 32 q^{35} - 19 q^{37} - 4 q^{38} - 3 q^{40} - 30 q^{41} - 15 q^{43} - 14 q^{44} - 13 q^{46} + q^{47} + 14 q^{49} + 11 q^{50} - 3 q^{52} - 16 q^{53} - 7 q^{55} - 6 q^{56} - 23 q^{58} - 26 q^{59} - 16 q^{61} - 14 q^{62} + 12 q^{64} - 8 q^{65} - 39 q^{67} - 8 q^{68} - 32 q^{70} - 15 q^{71} - 2 q^{73} - 19 q^{74} - 4 q^{76} - 34 q^{77} - 13 q^{79} - 3 q^{80} - 30 q^{82} - 6 q^{83} - 11 q^{85} - 15 q^{86} - 14 q^{88} - 18 q^{89} - 35 q^{91} - 13 q^{92} + q^{94} - 51 q^{95} + 19 q^{97} + 14 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 - 14 * q^11 - 3 * q^13 - 6 * q^14 + 12 * q^16 - 8 * q^17 - 4 * q^19 - 3 * q^20 - 14 * q^22 - 13 * q^23 + 11 * q^25 - 3 * q^26 - 6 * q^28 - 23 * q^29 - 14 * q^31 + 12 * q^32 - 8 * q^34 - 32 * q^35 - 19 * q^37 - 4 * q^38 - 3 * q^40 - 30 * q^41 - 15 * q^43 - 14 * q^44 - 13 * q^46 + q^47 + 14 * q^49 + 11 * q^50 - 3 * q^52 - 16 * q^53 - 7 * q^55 - 6 * q^56 - 23 * q^58 - 26 * q^59 - 16 * q^61 - 14 * q^62 + 12 * q^64 - 8 * q^65 - 39 * q^67 - 8 * q^68 - 32 * q^70 - 15 * q^71 - 2 * q^73 - 19 * q^74 - 4 * q^76 - 34 * q^77 - 13 * q^79 - 3 * q^80 - 30 * q^82 - 6 * q^83 - 11 * q^85 - 15 * q^86 - 14 * q^88 - 18 * q^89 - 35 * q^91 - 13 * q^92 + q^94 - 51 * q^95 + 19 * q^97 + 14 * 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
$5$	−0.761360	−0.340491	−0.170245	−	0.985402i	$-0.554456\pi$
$5$	−0.761360	−0.340491	−0.170245	+	0.985402i	$0.554456\pi$
$6$	0	0
$7$	1.23946	0.468474	0.234237	−	0.972180i	$-0.424741\pi$
$7$	1.23946	0.468474	0.234237	+	0.972180i	$0.424741\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.761360	−0.240763
$11$	−4.16498	−1.25579	−0.627894	−	0.778299i	$-0.716083\pi$
$11$	−4.16498	−1.25579	−0.627894	+	0.778299i	$0.716083\pi$
$12$	0	0
$13$	2.92984	0.812591	0.406296	−	0.913742i	$-0.366820\pi$
$13$	2.92984	0.812591	0.406296	+	0.913742i	$0.366820\pi$
$14$	1.23946	0.331261
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.05222	−0.497737	−0.248868	−	0.968537i	$-0.580059\pi$
$17$	−2.05222	−0.497737	−0.248868	+	0.968537i	$0.580059\pi$
$18$	0	0
$19$	0.666318	0.152864	0.0764319	−	0.997075i	$-0.475647\pi$
$19$	0.666318	0.152864	0.0764319	+	0.997075i	$0.475647\pi$
$20$	−0.761360	−0.170245
$21$	0	0
$22$	−4.16498	−0.887976
$23$	−8.48075	−1.76836	−0.884179	−	0.467148i	$-0.845281\pi$
$23$	−8.48075	−1.76836	−0.884179	+	0.467148i	$0.845281\pi$
$24$	0	0
$25$	−4.42033	−0.884066
$26$	2.92984	0.574589
$27$	0	0
$28$	1.23946	0.234237
$29$	1.77307	0.329250	0.164625	−	0.986356i	$-0.447359\pi$
$29$	1.77307	0.329250	0.164625	+	0.986356i	$0.447359\pi$
$30$	0	0
$31$	3.84602	0.690765	0.345382	−	0.938462i	$-0.387749\pi$
$31$	3.84602	0.690765	0.345382	+	0.938462i	$0.387749\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.05222	−0.351953
$35$	−0.943679	−0.159511
$36$	0	0
$37$	4.44887	0.731390	0.365695	−	0.930735i	$-0.380831\pi$
$37$	4.44887	0.731390	0.365695	+	0.930735i	$0.380831\pi$
$38$	0.666318	0.108091
$39$	0	0
$40$	−0.761360	−0.120382
$41$	2.52486	0.394317	0.197158	−	0.980372i	$-0.436829\pi$
$41$	2.52486	0.394317	0.197158	+	0.980372i	$0.436829\pi$
$42$	0	0
$43$	5.24402	0.799706	0.399853	−	0.916579i	$-0.369061\pi$
$43$	5.24402	0.799706	0.399853	+	0.916579i	$0.369061\pi$
$44$	−4.16498	−0.627894
$45$	0	0
$46$	−8.48075	−1.25042
$47$	3.71322	0.541630	0.270815	−	0.962631i	$-0.412707\pi$
$47$	3.71322	0.541630	0.270815	+	0.962631i	$0.412707\pi$
$48$	0	0
$49$	−5.46373	−0.780533
$50$	−4.42033	−0.625129
$51$	0	0
$52$	2.92984	0.406296
$53$	−8.26592	−1.13541	−0.567706	−	0.823231i	$-0.692169\pi$
$53$	−8.26592	−1.13541	−0.567706	+	0.823231i	$0.692169\pi$
$54$	0	0
$55$	3.17105	0.427584
$56$	1.23946	0.165630
$57$	0	0
$58$	1.77307	0.232815
$59$	−9.37120	−1.22003	−0.610013	−	0.792391i	$-0.708836\pi$
$59$	−9.37120	−1.22003	−0.610013	+	0.792391i	$0.708836\pi$
$60$	0	0
$61$	10.0229	1.28330	0.641649	−	0.766998i	$-0.278251\pi$
$61$	10.0229	1.28330	0.641649	+	0.766998i	$0.278251\pi$
$62$	3.84602	0.488445
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.23066	−0.276680
$66$	0	0
$67$	−1.45888	−0.178231	−0.0891155	−	0.996021i	$-0.528404\pi$
$67$	−1.45888	−0.178231	−0.0891155	+	0.996021i	$0.528404\pi$
$68$	−2.05222	−0.248868
$69$	0	0
$70$	−0.943679	−0.112791
$71$	−6.19297	−0.734970	−0.367485	−	0.930029i	$-0.619781\pi$
$71$	−6.19297	−0.734970	−0.367485	+	0.930029i	$0.619781\pi$
$72$	0	0
$73$	−10.9177	−1.27782	−0.638910	−	0.769281i	$-0.720614\pi$
$73$	−10.9177	−1.27782	−0.638910	+	0.769281i	$0.720614\pi$
$74$	4.44887	0.517171
$75$	0	0
$76$	0.666318	0.0764319
$77$	−5.16234	−0.588304
$78$	0	0
$79$	−9.54970	−1.07443	−0.537213	−	0.843447i	$-0.680523\pi$
$79$	−9.54970	−1.07443	−0.537213	+	0.843447i	$0.680523\pi$
$80$	−0.761360	−0.0851227
$81$	0	0
$82$	2.52486	0.278824
$83$	−5.69162	−0.624737	−0.312368	−	0.949961i	$-0.601122\pi$
$83$	−5.69162	−0.624737	−0.312368	+	0.949961i	$0.601122\pi$
$84$	0	0
$85$	1.56248	0.169475
$86$	5.24402	0.565477
$87$	0	0
$88$	−4.16498	−0.443988
$89$	−7.72882	−0.819253	−0.409627	−	0.912253i	$-0.634341\pi$
$89$	−7.72882	−0.819253	−0.409627	+	0.912253i	$0.634341\pi$
$90$	0	0
$91$	3.63143	0.380678
$92$	−8.48075	−0.884179
$93$	0	0
$94$	3.71322	0.382990
$95$	−0.507308	−0.0520487
$96$	0	0
$97$	−18.6599	−1.89463	−0.947315	−	0.320304i	$-0.896215\pi$
$97$	−18.6599	−1.89463	−0.947315	+	0.320304i	$0.896215\pi$
$98$	−5.46373	−0.551920
$99$	0	0
$100$	−4.42033	−0.442033
$101$	−5.89482	−0.586557	−0.293278	−	0.956027i	$-0.594746\pi$
$101$	−5.89482	−0.586557	−0.293278	+	0.956027i	$0.594746\pi$
$102$	0	0
$103$	−3.77939	−0.372395	−0.186197	−	0.982512i	$-0.559616\pi$
$103$	−3.77939	−0.372395	−0.186197	+	0.982512i	$0.559616\pi$
$104$	2.92984	0.287294
$105$	0	0
$106$	−8.26592	−0.802858
$107$	−7.59349	−0.734090	−0.367045	−	0.930203i	$-0.619631\pi$
$107$	−7.59349	−0.734090	−0.367045	+	0.930203i	$0.619631\pi$
$108$	0	0
$109$	1.05733	0.101274	0.0506371	−	0.998717i	$-0.483875\pi$
$109$	1.05733	0.101274	0.0506371	+	0.998717i	$0.483875\pi$
$110$	3.17105	0.302348
$111$	0	0
$112$	1.23946	0.117118
$113$	9.52892	0.896405	0.448203	−	0.893932i	$-0.352064\pi$
$113$	9.52892	0.896405	0.448203	+	0.893932i	$0.352064\pi$
$114$	0	0
$115$	6.45691	0.602110
$116$	1.77307	0.164625
$117$	0	0
$118$	−9.37120	−0.862689
$119$	−2.54366	−0.233177
$120$	0	0
$121$	6.34704	0.577004
$122$	10.0229	0.907429
$123$	0	0
$124$	3.84602	0.345382
$125$	7.17227	0.641507
$126$	0	0
$127$	−15.5448	−1.37938	−0.689689	−	0.724106i	$-0.742253\pi$
$127$	−15.5448	−1.37938	−0.689689	+	0.724106i	$0.742253\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.23066	−0.195642
$131$	−5.51152	−0.481544	−0.240772	−	0.970582i	$-0.577401\pi$
$131$	−5.51152	−0.481544	−0.240772	+	0.970582i	$0.577401\pi$
$132$	0	0
$133$	0.825877	0.0716127
$134$	−1.45888	−0.126028
$135$	0	0
$136$	−2.05222	−0.175977
$137$	10.2474	0.875495	0.437747	−	0.899098i	$-0.355776\pi$
$137$	10.2474	0.875495	0.437747	+	0.899098i	$0.355776\pi$
$138$	0	0
$139$	−9.23075	−0.782942	−0.391471	−	0.920190i	$-0.628034\pi$
$139$	−9.23075	−0.782942	−0.391471	+	0.920190i	$0.628034\pi$
$140$	−0.943679	−0.0797555
$141$	0	0
$142$	−6.19297	−0.519702
$143$	−12.2027	−1.02044
$144$	0	0
$145$	−1.34994	−0.112107
$146$	−10.9177	−0.903555
$147$	0	0
$148$	4.44887	0.365695
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	8.58020	0.698247	0.349124	−	0.937077i	$-0.386479\pi$
$151$	8.58020	0.698247	0.349124	+	0.937077i	$0.386479\pi$
$152$	0.666318	0.0540455
$153$	0	0
$154$	−5.16234	−0.415993
$155$	−2.92821	−0.235199
$156$	0	0
$157$	9.08853	0.725344	0.362672	−	0.931917i	$-0.381865\pi$
$157$	9.08853	0.725344	0.362672	+	0.931917i	$0.381865\pi$
$158$	−9.54970	−0.759733
$159$	0	0
$160$	−0.761360	−0.0601908
$161$	−10.5116	−0.828429
$162$	0	0
$163$	13.3116	1.04264	0.521322	−	0.853360i	$-0.325439\pi$
$163$	13.3116	1.04264	0.521322	+	0.853360i	$0.325439\pi$
$164$	2.52486	0.197158
$165$	0	0
$166$	−5.69162	−0.441756
$167$	6.38977	0.494455	0.247228	−	0.968957i	$-0.420480\pi$
$167$	6.38977	0.494455	0.247228	+	0.968957i	$0.420480\pi$
$168$	0	0
$169$	−4.41604	−0.339695
$170$	1.56248	0.119837
$171$	0	0
$172$	5.24402	0.399853
$173$	−1.23167	−0.0936421	−0.0468211	−	0.998903i	$-0.514909\pi$
$173$	−1.23167	−0.0936421	−0.0468211	+	0.998903i	$0.514909\pi$
$174$	0	0
$175$	−5.47884	−0.414162
$176$	−4.16498	−0.313947
$177$	0	0
$178$	−7.72882	−0.579299
$179$	10.4034	0.777585	0.388792	−	0.921325i	$-0.372892\pi$
$179$	10.4034	0.777585	0.388792	+	0.921325i	$0.372892\pi$
$180$	0	0
$181$	5.03575	0.374304	0.187152	−	0.982331i	$-0.440074\pi$
$181$	5.03575	0.374304	0.187152	+	0.982331i	$0.440074\pi$
$182$	3.63143	0.269180
$183$	0	0
$184$	−8.48075	−0.625209
$185$	−3.38720	−0.249032
$186$	0	0
$187$	8.54746	0.625052
$188$	3.71322	0.270815
$189$	0	0
$190$	−0.507308	−0.0368040
$191$	14.0973	1.02004	0.510022	−	0.860161i	$-0.329637\pi$
$191$	14.0973	1.02004	0.510022	+	0.860161i	$0.329637\pi$
$192$	0	0
$193$	7.50264	0.540052	0.270026	−	0.962853i	$-0.412968\pi$
$193$	7.50264	0.540052	0.270026	+	0.962853i	$0.412968\pi$
$194$	−18.6599	−1.33971
$195$	0	0
$196$	−5.46373	−0.390266
$197$	8.01296	0.570900	0.285450	−	0.958394i	$-0.407857\pi$
$197$	8.01296	0.570900	0.285450	+	0.958394i	$0.407857\pi$
$198$	0	0
$199$	−14.7857	−1.04813	−0.524064	−	0.851679i	$-0.675585\pi$
$199$	−14.7857	−1.04813	−0.524064	+	0.851679i	$0.675585\pi$
$200$	−4.42033	−0.312565
$201$	0	0
$202$	−5.89482	−0.414758
$203$	2.19765	0.154245
$204$	0	0
$205$	−1.92233	−0.134261
$206$	−3.77939	−0.263323
$207$	0	0
$208$	2.92984	0.203148
$209$	−2.77520	−0.191965
$210$	0	0
$211$	3.54316	0.243921	0.121961	−	0.992535i	$-0.461082\pi$
$211$	3.54316	0.243921	0.121961	+	0.992535i	$0.461082\pi$
$212$	−8.26592	−0.567706
$213$	0	0
$214$	−7.59349	−0.519080
$215$	−3.99259	−0.272292
$216$	0	0
$217$	4.76700	0.323605
$218$	1.05733	0.0716117
$219$	0	0
$220$	3.17105	0.213792
$221$	−6.01268	−0.404457
$222$	0	0
$223$	12.5260	0.838806	0.419403	−	0.907800i	$-0.362239\pi$
$223$	12.5260	0.838806	0.419403	+	0.907800i	$0.362239\pi$
$224$	1.23946	0.0828152
$225$	0	0
$226$	9.52892	0.633854
$227$	−14.2435	−0.945376	−0.472688	−	0.881230i	$-0.656716\pi$
$227$	−14.2435	−0.945376	−0.472688	+	0.881230i	$0.656716\pi$
$228$	0	0
$229$	−22.3396	−1.47624	−0.738120	−	0.674670i	$-0.764286\pi$
$229$	−22.3396	−1.47624	−0.738120	+	0.674670i	$0.764286\pi$
$230$	6.45691	0.425756
$231$	0	0
$232$	1.77307	0.116407
$233$	13.9412	0.913316	0.456658	−	0.889642i	$-0.349046\pi$
$233$	13.9412	0.913316	0.456658	+	0.889642i	$0.349046\pi$
$234$	0	0
$235$	−2.82710	−0.184420
$236$	−9.37120	−0.610013
$237$	0	0
$238$	−2.54366	−0.164881
$239$	−15.8580	−1.02577	−0.512883	−	0.858458i	$-0.671423\pi$
$239$	−15.8580	−1.02577	−0.512883	+	0.858458i	$0.671423\pi$
$240$	0	0
$241$	−13.2150	−0.851253	−0.425626	−	0.904899i	$-0.639946\pi$
$241$	−13.2150	−0.851253	−0.425626	+	0.904899i	$0.639946\pi$
$242$	6.34704	0.408003
$243$	0	0
$244$	10.0229	0.641649
$245$	4.15987	0.265764
$246$	0	0
$247$	1.95220	0.124216
$248$	3.84602	0.244222
$249$	0	0
$250$	7.17227	0.453614
$251$	−8.73531	−0.551368	−0.275684	−	0.961248i	$-0.588904\pi$
$251$	−8.73531	−0.551368	−0.275684	+	0.961248i	$0.588904\pi$
$252$	0	0
$253$	35.3221	2.22068
$254$	−15.5448	−0.975367
$255$	0	0
$256$	1.00000	0.0625000
$257$	−17.0182	−1.06157	−0.530785	−	0.847507i	$-0.678103\pi$
$257$	−17.0182	−1.06157	−0.530785	+	0.847507i	$0.678103\pi$
$258$	0	0
$259$	5.51422	0.342637
$260$	−2.23066	−0.138340
$261$	0	0
$262$	−5.51152	−0.340503
$263$	−1.87735	−0.115762	−0.0578811	−	0.998323i	$-0.518434\pi$
$263$	−1.87735	−0.115762	−0.0578811	+	0.998323i	$0.518434\pi$
$264$	0	0
$265$	6.29335	0.386597
$266$	0.825877	0.0506378
$267$	0	0
$268$	−1.45888	−0.0891155
$269$	17.0976	1.04246	0.521230	−	0.853416i	$-0.325473\pi$
$269$	17.0976	1.04246	0.521230	+	0.853416i	$0.325473\pi$
$270$	0	0
$271$	2.52171	0.153183	0.0765916	−	0.997063i	$-0.475596\pi$
$271$	2.52171	0.153183	0.0765916	+	0.997063i	$0.475596\pi$
$272$	−2.05222	−0.124434
$273$	0	0
$274$	10.2474	0.619068
$275$	18.4106	1.11020
$276$	0	0
$277$	20.3576	1.22317	0.611586	−	0.791178i	$-0.290532\pi$
$277$	20.3576	1.22317	0.611586	+	0.791178i	$0.290532\pi$
$278$	−9.23075	−0.553624
$279$	0	0
$280$	−0.943679	−0.0563956
$281$	26.7498	1.59576	0.797880	−	0.602816i	$-0.205955\pi$
$281$	26.7498	1.59576	0.797880	+	0.602816i	$0.205955\pi$
$282$	0	0
$283$	−29.9892	−1.78267	−0.891336	−	0.453344i	$-0.850231\pi$
$283$	−29.9892	−1.78267	−0.891336	+	0.453344i	$0.850231\pi$
$284$	−6.19297	−0.367485
$285$	0	0
$286$	−12.2027	−0.721562
$287$	3.12947	0.184727
$288$	0	0
$289$	−12.7884	−0.752258
$290$	−1.34994	−0.0792713
$291$	0	0
$292$	−10.9177	−0.638910
$293$	−5.78618	−0.338032	−0.169016	−	0.985613i	$-0.554059\pi$
$293$	−5.78618	−0.338032	−0.169016	+	0.985613i	$0.554059\pi$
$294$	0	0
$295$	7.13486	0.415408
$296$	4.44887	0.258586
$297$	0	0
$298$	1.00000	0.0579284
$299$	−24.8472	−1.43695
$300$	0	0
$301$	6.49978	0.374641
$302$	8.58020	0.493735
$303$	0	0
$304$	0.666318	0.0382160
$305$	−7.63102	−0.436951
$306$	0	0
$307$	−15.9386	−0.909665	−0.454832	−	0.890577i	$-0.650301\pi$
$307$	−15.9386	−0.909665	−0.454832	+	0.890577i	$0.650301\pi$
$308$	−5.16234	−0.294152
$309$	0	0
$310$	−2.92821	−0.166311
$311$	−18.5036	−1.04924	−0.524621	−	0.851336i	$-0.675793\pi$
$311$	−18.5036	−1.04924	−0.524621	+	0.851336i	$0.675793\pi$
$312$	0	0
$313$	−8.94571	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$313$	−8.94571	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$314$	9.08853	0.512896
$315$	0	0
$316$	−9.54970	−0.537213
$317$	−33.2027	−1.86485	−0.932424	−	0.361365i	$-0.882311\pi$
$317$	−33.2027	−1.86485	−0.932424	+	0.361365i	$0.882311\pi$
$318$	0	0
$319$	−7.38478	−0.413468
$320$	−0.761360	−0.0425613
$321$	0	0
$322$	−10.5116	−0.585788
$323$	−1.36743	−0.0760859
$324$	0	0
$325$	−12.9509	−0.718384
$326$	13.3116	0.737260
$327$	0	0
$328$	2.52486	0.139412
$329$	4.60241	0.253739
$330$	0	0
$331$	−26.3503	−1.44834	−0.724172	−	0.689620i	$-0.757778\pi$
$331$	−26.3503	−1.44834	−0.724172	+	0.689620i	$0.757778\pi$
$332$	−5.69162	−0.312368
$333$	0	0
$334$	6.38977	0.349633
$335$	1.11074	0.0606860
$336$	0	0
$337$	4.70544	0.256322	0.128161	−	0.991753i	$-0.459093\pi$
$337$	4.70544	0.256322	0.128161	+	0.991753i	$0.459093\pi$
$338$	−4.41604	−0.240201
$339$	0	0
$340$	1.56248	0.0847374
$341$	−16.0186	−0.867454
$342$	0	0
$343$	−15.4483	−0.834132
$344$	5.24402	0.282739
$345$	0	0
$346$	−1.23167	−0.0662150
$347$	−27.0511	−1.45218	−0.726091	−	0.687599i	$-0.758665\pi$
$347$	−27.0511	−1.45218	−0.726091	+	0.687599i	$0.758665\pi$
$348$	0	0
$349$	9.98625	0.534552	0.267276	−	0.963620i	$-0.413877\pi$
$349$	9.98625	0.534552	0.267276	+	0.963620i	$0.413877\pi$
$350$	−5.47884	−0.292856
$351$	0	0
$352$	−4.16498	−0.221994
$353$	−9.07685	−0.483112	−0.241556	−	0.970387i	$-0.577658\pi$
$353$	−9.07685	−0.483112	−0.241556	+	0.970387i	$0.577658\pi$
$354$	0	0
$355$	4.71508	0.250251
$356$	−7.72882	−0.409627
$357$	0	0
$358$	10.4034	0.549836
$359$	13.4684	0.710836	0.355418	−	0.934707i	$-0.384339\pi$
$359$	13.4684	0.710836	0.355418	+	0.934707i	$0.384339\pi$
$360$	0	0
$361$	−18.5560	−0.976633
$362$	5.03575	0.264673
$363$	0	0
$364$	3.63143	0.190339
$365$	8.31230	0.435086
$366$	0	0
$367$	18.1866	0.949331	0.474665	−	0.880166i	$-0.342569\pi$
$367$	18.1866	0.949331	0.474665	+	0.880166i	$0.342569\pi$
$368$	−8.48075	−0.442089
$369$	0	0
$370$	−3.38720	−0.176092
$371$	−10.2453	−0.531911
$372$	0	0
$373$	15.9422	0.825456	0.412728	−	0.910854i	$-0.364576\pi$
$373$	15.9422	0.825456	0.412728	+	0.910854i	$0.364576\pi$
$374$	8.54746	0.441978
$375$	0	0
$376$	3.71322	0.191495
$377$	5.19480	0.267546
$378$	0	0
$379$	−29.0734	−1.49340	−0.746701	−	0.665160i	$-0.768363\pi$
$379$	−29.0734	−1.49340	−0.746701	+	0.665160i	$0.768363\pi$
$380$	−0.507308	−0.0260244
$381$	0	0
$382$	14.0973	0.721280
$383$	−25.2617	−1.29081	−0.645407	−	0.763839i	$-0.723312\pi$
$383$	−25.2617	−1.29081	−0.645407	+	0.763839i	$0.723312\pi$
$384$	0	0
$385$	3.93040	0.200312
$386$	7.50264	0.381874
$387$	0	0
$388$	−18.6599	−0.947315
$389$	−1.32764	−0.0673138	−0.0336569	−	0.999433i	$-0.510715\pi$
$389$	−1.32764	−0.0673138	−0.0336569	+	0.999433i	$0.510715\pi$
$390$	0	0
$391$	17.4044	0.880177
$392$	−5.46373	−0.275960
$393$	0	0
$394$	8.01296	0.403687
$395$	7.27077	0.365832
$396$	0	0
$397$	−2.12240	−0.106520	−0.0532601	−	0.998581i	$-0.516961\pi$
$397$	−2.12240	−0.106520	−0.0532601	+	0.998581i	$0.516961\pi$
$398$	−14.7857	−0.741138
$399$	0	0
$400$	−4.42033	−0.221017
$401$	−26.4011	−1.31841	−0.659204	−	0.751964i	$-0.729107\pi$
$401$	−26.4011	−1.31841	−0.659204	+	0.751964i	$0.729107\pi$
$402$	0	0
$403$	11.2682	0.561310
$404$	−5.89482	−0.293278
$405$	0	0
$406$	2.19765	0.109068
$407$	−18.5295	−0.918471
$408$	0	0
$409$	−10.0056	−0.494744	−0.247372	−	0.968921i	$-0.579567\pi$
$409$	−10.0056	−0.494744	−0.247372	+	0.968921i	$0.579567\pi$
$410$	−1.92233	−0.0949370
$411$	0	0
$412$	−3.77939	−0.186197
$413$	−11.6153	−0.571550
$414$	0	0
$415$	4.33338	0.212717
$416$	2.92984	0.143647
$417$	0	0
$418$	−2.77520	−0.135739
$419$	20.0307	0.978562	0.489281	−	0.872126i	$-0.337259\pi$
$419$	20.0307	0.978562	0.489281	+	0.872126i	$0.337259\pi$
$420$	0	0
$421$	26.9452	1.31323	0.656614	−	0.754227i	$-0.271988\pi$
$421$	26.9452	1.31323	0.656614	+	0.754227i	$0.271988\pi$
$422$	3.54316	0.172478
$423$	0	0
$424$	−8.26592	−0.401429
$425$	9.07150	0.440032
$426$	0	0
$427$	12.4230	0.601191
$428$	−7.59349	−0.367045
$429$	0	0
$430$	−3.99259	−0.192540
$431$	16.9357	0.815763	0.407881	−	0.913035i	$-0.366268\pi$
$431$	16.9357	0.815763	0.407881	+	0.913035i	$0.366268\pi$
$432$	0	0
$433$	−10.1849	−0.489453	−0.244727	−	0.969592i	$-0.578698\pi$
$433$	−10.1849	−0.489453	−0.244727	+	0.969592i	$0.578698\pi$
$434$	4.76700	0.228823
$435$	0	0
$436$	1.05733	0.0506371
$437$	−5.65087	−0.270318
$438$	0	0
$439$	32.8283	1.56681	0.783405	−	0.621511i	$-0.213481\pi$
$439$	32.8283	1.56681	0.783405	+	0.621511i	$0.213481\pi$
$440$	3.17105	0.151174
$441$	0	0
$442$	−6.01268	−0.285994
$443$	13.1581	0.625162	0.312581	−	0.949891i	$-0.398806\pi$
$443$	13.1581	0.625162	0.312581	+	0.949891i	$0.398806\pi$
$444$	0	0
$445$	5.88442	0.278948
$446$	12.5260	0.593126
$447$	0	0
$448$	1.23946	0.0585592
$449$	−1.34071	−0.0632721	−0.0316361	−	0.999499i	$-0.510072\pi$
$449$	−1.34071	−0.0632721	−0.0316361	+	0.999499i	$0.510072\pi$
$450$	0	0
$451$	−10.5160	−0.495178
$452$	9.52892	0.448203
$453$	0	0
$454$	−14.2435	−0.668482
$455$	−2.76483	−0.129617
$456$	0	0
$457$	4.66820	0.218369	0.109185	−	0.994021i	$-0.465176\pi$
$457$	4.66820	0.218369	0.109185	+	0.994021i	$0.465176\pi$
$458$	−22.3396	−1.04386
$459$	0	0
$460$	6.45691	0.301055
$461$	7.86962	0.366525	0.183262	−	0.983064i	$-0.441334\pi$
$461$	7.86962	0.366525	0.183262	+	0.983064i	$0.441334\pi$
$462$	0	0
$463$	19.3249	0.898106	0.449053	−	0.893505i	$-0.351761\pi$
$463$	19.3249	0.898106	0.449053	+	0.893505i	$0.351761\pi$
$464$	1.77307	0.0823125
$465$	0	0
$466$	13.9412	0.645812
$467$	35.3122	1.63405	0.817026	−	0.576601i	$-0.195621\pi$
$467$	35.3122	1.63405	0.817026	+	0.576601i	$0.195621\pi$
$468$	0	0
$469$	−1.80824	−0.0834965
$470$	−2.82710	−0.130405
$471$	0	0
$472$	−9.37120	−0.431345
$473$	−21.8412	−1.00426
$474$	0	0
$475$	−2.94535	−0.135142
$476$	−2.54366	−0.116588
$477$	0	0
$478$	−15.8580	−0.725326
$479$	5.70156	0.260511	0.130255	−	0.991480i	$-0.458420\pi$
$479$	5.70156	0.260511	0.130255	+	0.991480i	$0.458420\pi$
$480$	0	0
$481$	13.0345	0.594322
$482$	−13.2150	−0.601927
$483$	0	0
$484$	6.34704	0.288502
$485$	14.2069	0.645104
$486$	0	0
$487$	−24.4051	−1.10590	−0.552950	−	0.833214i	$-0.686498\pi$
$487$	−24.4051	−1.10590	−0.552950	+	0.833214i	$0.686498\pi$
$488$	10.0229	0.453714
$489$	0	0
$490$	4.15987	0.187924
$491$	−14.7538	−0.665828	−0.332914	−	0.942957i	$-0.608032\pi$
$491$	−14.7538	−0.665828	−0.332914	+	0.942957i	$0.608032\pi$
$492$	0	0
$493$	−3.63872	−0.163880
$494$	1.95220	0.0878338
$495$	0	0
$496$	3.84602	0.172691
$497$	−7.67597	−0.344314
$498$	0	0
$499$	3.13631	0.140400	0.0702002	−	0.997533i	$-0.477636\pi$
$499$	3.13631	0.140400	0.0702002	+	0.997533i	$0.477636\pi$
$500$	7.17227	0.320754
$501$	0	0
$502$	−8.73531	−0.389876
$503$	32.1683	1.43431	0.717157	−	0.696912i	$-0.245443\pi$
$503$	32.1683	1.43431	0.717157	+	0.696912i	$0.245443\pi$
$504$	0	0
$505$	4.48808	0.199717
$506$	35.3221	1.57026
$507$	0	0
$508$	−15.5448	−0.689689
$509$	−12.2885	−0.544677	−0.272339	−	0.962202i	$-0.587797\pi$
$509$	−12.2885	−0.544677	−0.272339	+	0.962202i	$0.587797\pi$
$510$	0	0
$511$	−13.5321	−0.598625
$512$	1.00000	0.0441942
$513$	0	0
$514$	−17.0182	−0.750643
$515$	2.87748	0.126797
$516$	0	0
$517$	−15.4655	−0.680172
$518$	5.51422	0.242281
$519$	0	0
$520$	−2.23066	−0.0978211
$521$	−24.8807	−1.09004	−0.545022	−	0.838421i	$-0.683479\pi$
$521$	−24.8807	−1.09004	−0.545022	+	0.838421i	$0.683479\pi$
$522$	0	0
$523$	5.95819	0.260534	0.130267	−	0.991479i	$-0.458417\pi$
$523$	5.95819	0.260534	0.130267	+	0.991479i	$0.458417\pi$
$524$	−5.51152	−0.240772
$525$	0	0
$526$	−1.87735	−0.0818562
$527$	−7.89288	−0.343819
$528$	0	0
$529$	48.9231	2.12709
$530$	6.29335	0.273366
$531$	0	0
$532$	0.825877	0.0358063
$533$	7.39743	0.320418
$534$	0	0
$535$	5.78138	0.249951
$536$	−1.45888	−0.0630142
$537$	0	0
$538$	17.0976	0.737130
$539$	22.7563	0.980183
$540$	0	0
$541$	19.2533	0.827762	0.413881	−	0.910331i	$-0.364173\pi$
$541$	19.2533	0.827762	0.413881	+	0.910331i	$0.364173\pi$
$542$	2.52171	0.108317
$543$	0	0
$544$	−2.05222	−0.0879883
$545$	−0.805012	−0.0344829
$546$	0	0
$547$	−21.0064	−0.898171	−0.449085	−	0.893489i	$-0.648250\pi$
$547$	−21.0064	−0.898171	−0.449085	+	0.893489i	$0.648250\pi$
$548$	10.2474	0.437747
$549$	0	0
$550$	18.4106	0.785030
$551$	1.18143	0.0503304
$552$	0	0
$553$	−11.8365	−0.503340
$554$	20.3576	0.864913
$555$	0	0
$556$	−9.23075	−0.391471
$557$	6.48989	0.274986	0.137493	−	0.990503i	$-0.456096\pi$
$557$	6.48989	0.274986	0.137493	+	0.990503i	$0.456096\pi$
$558$	0	0
$559$	15.3641	0.649834
$560$	−0.943679	−0.0398777
$561$	0	0
$562$	26.7498	1.12837
$563$	−44.9514	−1.89447	−0.947237	−	0.320533i	$-0.896138\pi$
$563$	−44.9514	−1.89447	−0.947237	+	0.320533i	$0.896138\pi$
$564$	0	0
$565$	−7.25494	−0.305218
$566$	−29.9892	−1.26054
$567$	0	0
$568$	−6.19297	−0.259851
$569$	18.5364	0.777085	0.388542	−	0.921431i	$-0.372979\pi$
$569$	18.5364	0.777085	0.388542	+	0.921431i	$0.372979\pi$
$570$	0	0
$571$	11.1531	0.466741	0.233370	−	0.972388i	$-0.425025\pi$
$571$	11.1531	0.466741	0.233370	+	0.972388i	$0.425025\pi$
$572$	−12.2027	−0.510221
$573$	0	0
$574$	3.12947	0.130622
$575$	37.4877	1.56335
$576$	0	0
$577$	4.89010	0.203578	0.101789	−	0.994806i	$-0.467543\pi$
$577$	4.89010	0.203578	0.101789	+	0.994806i	$0.467543\pi$
$578$	−12.7884	−0.531927
$579$	0	0
$580$	−1.34994	−0.0560533
$581$	−7.05456	−0.292673
$582$	0	0
$583$	34.4274	1.42584
$584$	−10.9177	−0.451778
$585$	0	0
$586$	−5.78618	−0.239025
$587$	37.4879	1.54729	0.773646	−	0.633618i	$-0.218431\pi$
$587$	37.4879	1.54729	0.773646	+	0.633618i	$0.218431\pi$
$588$	0	0
$589$	2.56267	0.105593
$590$	7.13486	0.293738
$591$	0	0
$592$	4.44887	0.182848
$593$	11.4627	0.470719	0.235359	−	0.971908i	$-0.424373\pi$
$593$	11.4627	0.470719	0.235359	+	0.971908i	$0.424373\pi$
$594$	0	0
$595$	1.93664	0.0793944
$596$	1.00000	0.0409616
$597$	0	0
$598$	−24.8472	−1.01608
$599$	4.91928	0.200996	0.100498	−	0.994937i	$-0.467956\pi$
$599$	4.91928	0.200996	0.100498	+	0.994937i	$0.467956\pi$
$600$	0	0
$601$	3.31114	0.135064	0.0675321	−	0.997717i	$-0.478487\pi$
$601$	3.31114	0.135064	0.0675321	+	0.997717i	$0.478487\pi$
$602$	6.49978	0.264911
$603$	0	0
$604$	8.58020	0.349124
$605$	−4.83239	−0.196464
$606$	0	0
$607$	−36.6659	−1.48822	−0.744112	−	0.668055i	$-0.767127\pi$
$607$	−36.6659	−1.48822	−0.744112	+	0.668055i	$0.767127\pi$
$608$	0.666318	0.0270228
$609$	0	0
$610$	−7.63102	−0.308971
$611$	10.8792	0.440123
$612$	0	0
$613$	23.5176	0.949865	0.474932	−	0.880022i	$-0.342472\pi$
$613$	23.5176	0.949865	0.474932	+	0.880022i	$0.342472\pi$
$614$	−15.9386	−0.643230
$615$	0	0
$616$	−5.16234	−0.207997
$617$	−9.64068	−0.388119	−0.194060	−	0.980990i	$-0.562166\pi$
$617$	−9.64068	−0.388119	−0.194060	+	0.980990i	$0.562166\pi$
$618$	0	0
$619$	11.9779	0.481431	0.240715	−	0.970596i	$-0.422618\pi$
$619$	11.9779	0.481431	0.240715	+	0.970596i	$0.422618\pi$
$620$	−2.92821	−0.117600
$621$	0	0
$622$	−18.5036	−0.741927
$623$	−9.57960	−0.383798
$624$	0	0
$625$	16.6410	0.665639
$626$	−8.94571	−0.357542
$627$	0	0
$628$	9.08853	0.362672
$629$	−9.13007	−0.364040
$630$	0	0
$631$	−26.2849	−1.04638	−0.523192	−	0.852215i	$-0.675259\pi$
$631$	−26.2849	−1.04638	−0.523192	+	0.852215i	$0.675259\pi$
$632$	−9.54970	−0.379867
$633$	0	0
$634$	−33.2027	−1.31865
$635$	11.8352	0.469665
$636$	0	0
$637$	−16.0078	−0.634254
$638$	−7.38478	−0.292366
$639$	0	0
$640$	−0.761360	−0.0300954
$641$	3.50080	0.138273	0.0691366	−	0.997607i	$-0.477976\pi$
$641$	3.50080	0.138273	0.0691366	+	0.997607i	$0.477976\pi$
$642$	0	0
$643$	4.61292	0.181916	0.0909578	−	0.995855i	$-0.471007\pi$
$643$	4.61292	0.181916	0.0909578	+	0.995855i	$0.471007\pi$
$644$	−10.5116	−0.414214
$645$	0	0
$646$	−1.36743	−0.0538009
$647$	17.2644	0.678735	0.339368	−	0.940654i	$-0.389787\pi$
$647$	17.2644	0.678735	0.339368	+	0.940654i	$0.389787\pi$
$648$	0	0
$649$	39.0309	1.53210
$650$	−12.9509	−0.507974
$651$	0	0
$652$	13.3116	0.521322
$653$	−7.86506	−0.307784	−0.153892	−	0.988088i	$-0.549181\pi$
$653$	−7.86506	−0.307784	−0.153892	+	0.988088i	$0.549181\pi$
$654$	0	0
$655$	4.19625	0.163961
$656$	2.52486	0.0985792
$657$	0	0
$658$	4.60241	0.179421
$659$	−25.2923	−0.985247	−0.492623	−	0.870243i	$-0.663962\pi$
$659$	−25.2923	−0.985247	−0.492623	+	0.870243i	$0.663962\pi$
$660$	0	0
$661$	18.4738	0.718546	0.359273	−	0.933233i	$-0.383025\pi$
$661$	18.4738	0.718546	0.359273	+	0.933233i	$0.383025\pi$
$662$	−26.3503	−1.02413
$663$	0	0
$664$	−5.69162	−0.220878
$665$	−0.628790	−0.0243834
$666$	0	0
$667$	−15.0369	−0.582232
$668$	6.38977	0.247228
$669$	0	0
$670$	1.11074	0.0429115
$671$	−41.7451	−1.61155
$672$	0	0
$673$	38.6055	1.48813	0.744067	−	0.668105i	$-0.232895\pi$
$673$	38.6055	1.48813	0.744067	+	0.668105i	$0.232895\pi$
$674$	4.70544	0.181247
$675$	0	0
$676$	−4.41604	−0.169848
$677$	−8.94930	−0.343950	−0.171975	−	0.985101i	$-0.555015\pi$
$677$	−8.94930	−0.343950	−0.171975	+	0.985101i	$0.555015\pi$
$678$	0	0
$679$	−23.1283	−0.887584
$680$	1.56248	0.0599184
$681$	0	0
$682$	−16.0186	−0.613383
$683$	−24.7539	−0.947182	−0.473591	−	0.880745i	$-0.657042\pi$
$683$	−24.7539	−0.947182	−0.473591	+	0.880745i	$0.657042\pi$
$684$	0	0
$685$	−7.80197	−0.298098
$686$	−15.4483	−0.589821
$687$	0	0
$688$	5.24402	0.199926
$689$	−24.2178	−0.922626
$690$	0	0
$691$	17.9206	0.681733	0.340867	−	0.940112i	$-0.389280\pi$
$691$	17.9206	0.681733	0.340867	+	0.940112i	$0.389280\pi$
$692$	−1.23167	−0.0468211
$693$	0	0
$694$	−27.0511	−1.02685
$695$	7.02793	0.266585
$696$	0	0
$697$	−5.18157	−0.196266
$698$	9.98625	0.377985
$699$	0	0
$700$	−5.47884	−0.207081
$701$	−29.8120	−1.12598	−0.562992	−	0.826463i	$-0.690350\pi$
$701$	−29.8120	−1.12598	−0.562992	+	0.826463i	$0.690350\pi$
$702$	0	0
$703$	2.96436	0.111803
$704$	−4.16498	−0.156974
$705$	0	0
$706$	−9.07685	−0.341612
$707$	−7.30642	−0.274786
$708$	0	0
$709$	17.8874	0.671776	0.335888	−	0.941902i	$-0.390964\pi$
$709$	17.8874	0.671776	0.335888	+	0.941902i	$0.390964\pi$
$710$	4.71508	0.176954
$711$	0	0
$712$	−7.72882	−0.289650
$713$	−32.6171	−1.22152
$714$	0	0
$715$	9.29067	0.347451
$716$	10.4034	0.388792
$717$	0	0
$718$	13.4684	0.502637
$719$	−31.1012	−1.15988	−0.579939	−	0.814660i	$-0.696924\pi$
$719$	−31.1012	−1.15988	−0.579939	+	0.814660i	$0.696924\pi$
$720$	0	0
$721$	−4.68442	−0.174457
$722$	−18.5560	−0.690584
$723$	0	0
$724$	5.03575	0.187152
$725$	−7.83754	−0.291079
$726$	0	0
$727$	39.7921	1.47581	0.737904	−	0.674906i	$-0.235816\pi$
$727$	39.7921	1.47581	0.737904	+	0.674906i	$0.235816\pi$
$728$	3.63143	0.134590
$729$	0	0
$730$	8.31230	0.307652
$731$	−10.7619	−0.398043
$732$	0	0
$733$	−28.9878	−1.07069	−0.535344	−	0.844634i	$-0.679818\pi$
$733$	−28.9878	−1.07069	−0.535344	+	0.844634i	$0.679818\pi$
$734$	18.1866	0.671278
$735$	0	0
$736$	−8.48075	−0.312604
$737$	6.07622	0.223820
$738$	0	0
$739$	−9.23444	−0.339694	−0.169847	−	0.985470i	$-0.554327\pi$
$739$	−9.23444	−0.339694	−0.169847	+	0.985470i	$0.554327\pi$
$740$	−3.38720	−0.124516
$741$	0	0
$742$	−10.2453	−0.376118
$743$	−4.29294	−0.157493	−0.0787463	−	0.996895i	$-0.525092\pi$
$743$	−4.29294	−0.157493	−0.0787463	+	0.996895i	$0.525092\pi$
$744$	0	0
$745$	−0.761360	−0.0278941
$746$	15.9422	0.583685
$747$	0	0
$748$	8.54746	0.312526
$749$	−9.41186	−0.343902
$750$	0	0
$751$	33.4787	1.22165	0.610827	−	0.791764i	$-0.290837\pi$
$751$	33.4787	1.22165	0.610827	+	0.791764i	$0.290837\pi$
$752$	3.71322	0.135407
$753$	0	0
$754$	5.19480	0.189183
$755$	−6.53263	−0.237747
$756$	0	0
$757$	−40.1690	−1.45997	−0.729984	−	0.683464i	$-0.760473\pi$
$757$	−40.1690	−1.45997	−0.729984	+	0.683464i	$0.760473\pi$
$758$	−29.0734	−1.05599
$759$	0	0
$760$	−0.507308	−0.0184020
$761$	27.1823	0.985356	0.492678	−	0.870212i	$-0.336018\pi$
$761$	27.1823	0.985356	0.492678	+	0.870212i	$0.336018\pi$
$762$	0	0
$763$	1.31053	0.0474443
$764$	14.0973	0.510022
$765$	0	0
$766$	−25.2617	−0.912744
$767$	−27.4561	−0.991383
$768$	0	0
$769$	27.1059	0.977464	0.488732	−	0.872434i	$-0.337460\pi$
$769$	27.1059	0.977464	0.488732	+	0.872434i	$0.337460\pi$
$770$	3.93040	0.141642
$771$	0	0
$772$	7.50264	0.270026
$773$	30.4154	1.09397	0.546984	−	0.837143i	$-0.315776\pi$
$773$	30.4154	1.09397	0.546984	+	0.837143i	$0.315776\pi$
$774$	0	0
$775$	−17.0007	−0.610682
$776$	−18.6599	−0.669853
$777$	0	0
$778$	−1.32764	−0.0475980
$779$	1.68236	0.0602768
$780$	0	0
$781$	25.7936	0.922967
$782$	17.4044	0.622379
$783$	0	0
$784$	−5.46373	−0.195133
$785$	−6.91965	−0.246973
$786$	0	0
$787$	−17.2434	−0.614660	−0.307330	−	0.951603i	$-0.599436\pi$
$787$	−17.2434	−0.614660	−0.307330	+	0.951603i	$0.599436\pi$
$788$	8.01296	0.285450
$789$	0	0
$790$	7.27077	0.258682
$791$	11.8108	0.419942
$792$	0	0
$793$	29.3654	1.04280
$794$	−2.12240	−0.0753211
$795$	0	0
$796$	−14.7857	−0.524064
$797$	53.3978	1.89145	0.945724	−	0.324971i	$-0.105355\pi$
$797$	53.3978	1.89145	0.945724	+	0.324971i	$0.105355\pi$
$798$	0	0
$799$	−7.62036	−0.269589
$800$	−4.42033	−0.156282
$801$	0	0
$802$	−26.4011	−0.932256
$803$	45.4720	1.60467
$804$	0	0
$805$	8.00311	0.282072
$806$	11.2682	0.396906
$807$	0	0
$808$	−5.89482	−0.207379
$809$	−33.8973	−1.19177	−0.595883	−	0.803071i	$-0.703198\pi$
$809$	−33.8973	−1.19177	−0.595883	+	0.803071i	$0.703198\pi$
$810$	0	0
$811$	8.67149	0.304497	0.152249	−	0.988342i	$-0.451349\pi$
$811$	8.67149	0.304497	0.152249	+	0.988342i	$0.451349\pi$
$812$	2.19765	0.0771225
$813$	0	0
$814$	−18.5295	−0.649457
$815$	−10.1349	−0.355010
$816$	0	0
$817$	3.49418	0.122246
$818$	−10.0056	−0.349837
$819$	0	0
$820$	−1.92233	−0.0671306
$821$	12.9393	0.451585	0.225793	−	0.974175i	$-0.427503\pi$
$821$	12.9393	0.451585	0.225793	+	0.974175i	$0.427503\pi$
$822$	0	0
$823$	2.76391	0.0963437	0.0481719	−	0.998839i	$-0.484660\pi$
$823$	2.76391	0.0963437	0.0481719	+	0.998839i	$0.484660\pi$
$824$	−3.77939	−0.131661
$825$	0	0
$826$	−11.6153	−0.404147
$827$	10.9901	0.382163	0.191082	−	0.981574i	$-0.438800\pi$
$827$	10.9901	0.382163	0.191082	+	0.981574i	$0.438800\pi$
$828$	0	0
$829$	−9.82219	−0.341139	−0.170569	−	0.985346i	$-0.554561\pi$
$829$	−9.82219	−0.341139	−0.170569	+	0.985346i	$0.554561\pi$
$830$	4.33338	0.150414
$831$	0	0
$832$	2.92984	0.101574
$833$	11.2128	0.388500
$834$	0	0
$835$	−4.86492	−0.168357
$836$	−2.77520	−0.0959823
$837$	0	0
$838$	20.0307	0.691948
$839$	−4.03589	−0.139334	−0.0696671	−	0.997570i	$-0.522194\pi$
$839$	−4.03589	−0.139334	−0.0696671	+	0.997570i	$0.522194\pi$
$840$	0	0
$841$	−25.8562	−0.891594
$842$	26.9452	0.928592
$843$	0	0
$844$	3.54316	0.121961
$845$	3.36220	0.115663
$846$	0	0
$847$	7.86693	0.270311
$848$	−8.26592	−0.283853
$849$	0	0
$850$	9.07150	0.311150
$851$	−37.7298	−1.29336
$852$	0	0
$853$	11.3792	0.389616	0.194808	−	0.980841i	$-0.437592\pi$
$853$	11.3792	0.389616	0.194808	+	0.980841i	$0.437592\pi$
$854$	12.4230	0.425106
$855$	0	0
$856$	−7.59349	−0.259540
$857$	−28.3238	−0.967522	−0.483761	−	0.875200i	$-0.660730\pi$
$857$	−28.3238	−0.967522	−0.483761	+	0.875200i	$0.660730\pi$
$858$	0	0
$859$	52.8590	1.80352	0.901762	−	0.432233i	$-0.142274\pi$
$859$	52.8590	1.80352	0.901762	+	0.432233i	$0.142274\pi$
$860$	−3.99259	−0.136146
$861$	0	0
$862$	16.9357	0.576831
$863$	19.7991	0.673968	0.336984	−	0.941510i	$-0.390593\pi$
$863$	19.7991	0.673968	0.336984	+	0.941510i	$0.390593\pi$
$864$	0	0
$865$	0.937745	0.0318843
$866$	−10.1849	−0.346096
$867$	0	0
$868$	4.76700	0.161803
$869$	39.7743	1.34925
$870$	0	0
$871$	−4.27430	−0.144829
$872$	1.05733	0.0358059
$873$	0	0
$874$	−5.65087	−0.191144
$875$	8.88977	0.300529
$876$	0	0
$877$	−13.9418	−0.470781	−0.235391	−	0.971901i	$-0.575637\pi$
$877$	−13.9418	−0.470781	−0.235391	+	0.971901i	$0.575637\pi$
$878$	32.8283	1.10790
$879$	0	0
$880$	3.17105	0.106896
$881$	−35.1454	−1.18408	−0.592039	−	0.805910i	$-0.701677\pi$
$881$	−35.1454	−1.18408	−0.592039	+	0.805910i	$0.701677\pi$
$882$	0	0
$883$	22.9553	0.772506	0.386253	−	0.922393i	$-0.373769\pi$
$883$	22.9553	0.772506	0.386253	+	0.922393i	$0.373769\pi$
$884$	−6.01268	−0.202228
$885$	0	0
$886$	13.1581	0.442056
$887$	4.29374	0.144170	0.0720849	−	0.997398i	$-0.477035\pi$
$887$	4.29374	0.144170	0.0720849	+	0.997398i	$0.477035\pi$
$888$	0	0
$889$	−19.2672	−0.646202
$890$	5.88442	0.197246
$891$	0	0
$892$	12.5260	0.419403
$893$	2.47419	0.0827956
$894$	0	0
$895$	−7.92072	−0.264760
$896$	1.23946	0.0414076
$897$	0	0
$898$	−1.34071	−0.0447401
$899$	6.81924	0.227434
$900$	0	0
$901$	16.9635	0.565136
$902$	−10.5160	−0.350144
$903$	0	0
$904$	9.52892	0.316927
$905$	−3.83402	−0.127447
$906$	0	0
$907$	−16.0955	−0.534442	−0.267221	−	0.963635i	$-0.586105\pi$
$907$	−16.0955	−0.534442	−0.267221	+	0.963635i	$0.586105\pi$
$908$	−14.2435	−0.472688
$909$	0	0
$910$	−2.76483	−0.0916532
$911$	−29.8829	−0.990063	−0.495032	−	0.868875i	$-0.664843\pi$
$911$	−29.8829	−0.990063	−0.495032	+	0.868875i	$0.664843\pi$
$912$	0	0
$913$	23.7055	0.784537
$914$	4.66820	0.154410
$915$	0	0
$916$	−22.3396	−0.738120
$917$	−6.83134	−0.225591
$918$	0	0
$919$	32.2582	1.06410	0.532050	−	0.846713i	$-0.321422\pi$
$919$	32.2582	1.06410	0.532050	+	0.846713i	$0.321422\pi$
$920$	6.45691	0.212878
$921$	0	0
$922$	7.86962	0.259172
$923$	−18.1444	−0.597230
$924$	0	0
$925$	−19.6655	−0.646597
$926$	19.3249	0.635057
$927$	0	0
$928$	1.77307	0.0582037
$929$	−3.07707	−0.100955	−0.0504777	−	0.998725i	$-0.516074\pi$
$929$	−3.07707	−0.100955	−0.0504777	+	0.998725i	$0.516074\pi$
$930$	0	0
$931$	−3.64058	−0.119315
$932$	13.9412	0.456658
$933$	0	0
$934$	35.3122	1.15545
$935$	−6.50770	−0.212824
$936$	0	0
$937$	22.4776	0.734311	0.367156	−	0.930160i	$-0.380332\pi$
$937$	22.4776	0.734311	0.367156	+	0.930160i	$0.380332\pi$
$938$	−1.80824	−0.0590410
$939$	0	0
$940$	−2.82710	−0.0922099
$941$	−22.0470	−0.718712	−0.359356	−	0.933201i	$-0.617004\pi$
$941$	−22.0470	−0.718712	−0.359356	+	0.933201i	$0.617004\pi$
$942$	0	0
$943$	−21.4127	−0.697293
$944$	−9.37120	−0.305007
$945$	0	0
$946$	−21.8412	−0.710120
$947$	5.17219	0.168073	0.0840367	−	0.996463i	$-0.473219\pi$
$947$	5.17219	0.168073	0.0840367	+	0.996463i	$0.473219\pi$
$948$	0	0
$949$	−31.9871	−1.03835
$950$	−2.94535	−0.0955596
$951$	0	0
$952$	−2.54366	−0.0824403
$953$	32.0348	1.03771	0.518855	−	0.854862i	$-0.326358\pi$
$953$	32.0348	1.03771	0.518855	+	0.854862i	$0.326358\pi$
$954$	0	0
$955$	−10.7331	−0.347316
$956$	−15.8580	−0.512883
$957$	0	0
$958$	5.70156	0.184209
$959$	12.7013	0.410146
$960$	0	0
$961$	−16.2082	−0.522844
$962$	13.0345	0.420249
$963$	0	0
$964$	−13.2150	−0.425626
$965$	−5.71221	−0.183883
$966$	0	0
$967$	−10.9279	−0.351419	−0.175709	−	0.984442i	$-0.556222\pi$
$967$	−10.9279	−0.351419	−0.175709	+	0.984442i	$0.556222\pi$
$968$	6.34704	0.204002
$969$	0	0
$970$	14.2069	0.456157
$971$	1.27203	0.0408214	0.0204107	−	0.999792i	$-0.493503\pi$
$971$	1.27203	0.0408214	0.0204107	+	0.999792i	$0.493503\pi$
$972$	0	0
$973$	−11.4412	−0.366788
$974$	−24.4051	−0.781990
$975$	0	0
$976$	10.0229	0.320825
$977$	−59.6830	−1.90943	−0.954714	−	0.297526i	$-0.903838\pi$
$977$	−59.6830	−1.90943	−0.954714	+	0.297526i	$0.903838\pi$
$978$	0	0
$979$	32.1904	1.02881
$980$	4.15987	0.132882
$981$	0	0
$982$	−14.7538	−0.470811
$983$	39.9669	1.27475	0.637374	−	0.770555i	$-0.280021\pi$
$983$	39.9669	1.27475	0.637374	+	0.770555i	$0.280021\pi$
$984$	0	0
$985$	−6.10075	−0.194386
$986$	−3.63872	−0.115881
$987$	0	0
$988$	1.95220	0.0621079
$989$	−44.4732	−1.41417
$990$	0	0
$991$	−18.7853	−0.596735	−0.298368	−	0.954451i	$-0.596442\pi$
$991$	−18.7853	−0.596735	−0.298368	+	0.954451i	$0.596442\pi$
$992$	3.84602	0.122111
$993$	0	0
$994$	−7.67597	−0.243467
$995$	11.2572	0.356878
$996$	0	0
$997$	42.9444	1.36006	0.680032	−	0.733183i	$-0.261966\pi$
$997$	42.9444	1.36006	0.680032	+	0.733183i	$0.261966\pi$
$998$	3.13631	0.0992781
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.n.1.5	yes	12
3.2	odd	2		8046.2.a.k.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.k.1.8	✓	12	3.2	odd	2
8046.2.a.n.1.5	yes	12	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.761360 q^{5} +1.23946 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.761360 q^{5} +1.23946 q^{7} +1.00000 q^{8} -0.761360 q^{10} -4.16498 q^{11} +2.92984 q^{13} +1.23946 q^{14} +1.00000 q^{16} -2.05222 q^{17} +0.666318 q^{19} -0.761360 q^{20} -4.16498 q^{22} -8.48075 q^{23} -4.42033 q^{25} +2.92984 q^{26} +1.23946 q^{28} +1.77307 q^{29} +3.84602 q^{31} +1.00000 q^{32} -2.05222 q^{34} -0.943679 q^{35} +4.44887 q^{37} +0.666318 q^{38} -0.761360 q^{40} +2.52486 q^{41} +5.24402 q^{43} -4.16498 q^{44} -8.48075 q^{46} +3.71322 q^{47} -5.46373 q^{49} -4.42033 q^{50} +2.92984 q^{52} -8.26592 q^{53} +3.17105 q^{55} +1.23946 q^{56} +1.77307 q^{58} -9.37120 q^{59} +10.0229 q^{61} +3.84602 q^{62} +1.00000 q^{64} -2.23066 q^{65} -1.45888 q^{67} -2.05222 q^{68} -0.943679 q^{70} -6.19297 q^{71} -10.9177 q^{73} +4.44887 q^{74} +0.666318 q^{76} -5.16234 q^{77} -9.54970 q^{79} -0.761360 q^{80} +2.52486 q^{82} -5.69162 q^{83} +1.56248 q^{85} +5.24402 q^{86} -4.16498 q^{88} -7.72882 q^{89} +3.63143 q^{91} -8.48075 q^{92} +3.71322 q^{94} -0.507308 q^{95} -18.6599 q^{97} -5.46373 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8} - 3 q^{10} - 14 q^{11} - 3 q^{13} - 6 q^{14} + 12 q^{16} - 8 q^{17} - 4 q^{19} - 3 q^{20} - 14 q^{22} - 13 q^{23} + 11 q^{25} - 3 q^{26} - 6 q^{28} - 23 q^{29} - 14 q^{31} + 12 q^{32} - 8 q^{34} - 32 q^{35} - 19 q^{37} - 4 q^{38} - 3 q^{40} - 30 q^{41} - 15 q^{43} - 14 q^{44} - 13 q^{46} + q^{47} + 14 q^{49} + 11 q^{50} - 3 q^{52} - 16 q^{53} - 7 q^{55} - 6 q^{56} - 23 q^{58} - 26 q^{59} - 16 q^{61} - 14 q^{62} + 12 q^{64} - 8 q^{65} - 39 q^{67} - 8 q^{68} - 32 q^{70} - 15 q^{71} - 2 q^{73} - 19 q^{74} - 4 q^{76} - 34 q^{77} - 13 q^{79} - 3 q^{80} - 30 q^{82} - 6 q^{83} - 11 q^{85} - 15 q^{86} - 14 q^{88} - 18 q^{89} - 35 q^{91} - 13 q^{92} + q^{94} - 51 q^{95} + 19 q^{97} + 14 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8 - 3 * q^10 - 14 * q^11 - 3 * q^13 - 6 * q^14 + 12 * q^16 - 8 * q^17 - 4 * q^19 - 3 * q^20 - 14 * q^22 - 13 * q^23 + 11 * q^25 - 3 * q^26 - 6 * q^28 - 23 * q^29 - 14 * q^31 + 12 * q^32 - 8 * q^34 - 32 * q^35 - 19 * q^37 - 4 * q^38 - 3 * q^40 - 30 * q^41 - 15 * q^43 - 14 * q^44 - 13 * q^46 + q^47 + 14 * q^49 + 11 * q^50 - 3 * q^52 - 16 * q^53 - 7 * q^55 - 6 * q^56 - 23 * q^58 - 26 * q^59 - 16 * q^61 - 14 * q^62 + 12 * q^64 - 8 * q^65 - 39 * q^67 - 8 * q^68 - 32 * q^70 - 15 * q^71 - 2 * q^73 - 19 * q^74 - 4 * q^76 - 34 * q^77 - 13 * q^79 - 3 * q^80 - 30 * q^82 - 6 * q^83 - 11 * q^85 - 15 * q^86 - 14 * q^88 - 18 * q^89 - 35 * q^91 - 13 * q^92 + q^94 - 51 * q^95 + 19 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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