LMFDB - Embedded newform 8046.2.a.n.1.11

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			1.11
Root			$-3.43402$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.43402 q^{5} -1.90397 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.43402 q^{5} -1.90397 q^{7} +1.00000 q^{8} +3.43402 q^{10} -5.05887 q^{11} -4.50984 q^{13} -1.90397 q^{14} +1.00000 q^{16} +3.54115 q^{17} -4.96850 q^{19} +3.43402 q^{20} -5.05887 q^{22} +7.75578 q^{23} +6.79252 q^{25} -4.50984 q^{26} -1.90397 q^{28} -8.52570 q^{29} -8.93157 q^{31} +1.00000 q^{32} +3.54115 q^{34} -6.53829 q^{35} +2.08597 q^{37} -4.96850 q^{38} +3.43402 q^{40} +5.04962 q^{41} -4.60761 q^{43} -5.05887 q^{44} +7.75578 q^{46} +5.86559 q^{47} -3.37488 q^{49} +6.79252 q^{50} -4.50984 q^{52} -2.76603 q^{53} -17.3723 q^{55} -1.90397 q^{56} -8.52570 q^{58} -5.93786 q^{59} +5.94479 q^{61} -8.93157 q^{62} +1.00000 q^{64} -15.4869 q^{65} -6.16643 q^{67} +3.54115 q^{68} -6.53829 q^{70} +12.1240 q^{71} -11.5197 q^{73} +2.08597 q^{74} -4.96850 q^{76} +9.63197 q^{77} +11.4246 q^{79} +3.43402 q^{80} +5.04962 q^{82} +6.74370 q^{83} +12.1604 q^{85} -4.60761 q^{86} -5.05887 q^{88} -13.9218 q^{89} +8.58661 q^{91} +7.75578 q^{92} +5.86559 q^{94} -17.0619 q^{95} -18.4392 q^{97} -3.37488 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.43402	1.53574	0.767871	−	0.640605i	$-0.221316\pi$
$5$	3.43402	1.53574	0.767871	+	0.640605i	$0.221316\pi$
$6$	0	0
$7$	−1.90397	−0.719635	−0.359817	−	0.933023i	$-0.617161\pi$
$7$	−1.90397	−0.719635	−0.359817	+	0.933023i	$0.617161\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.43402	1.08593
$11$	−5.05887	−1.52531	−0.762654	−	0.646807i	$-0.776104\pi$
$11$	−5.05887	−1.52531	−0.762654	+	0.646807i	$0.776104\pi$
$12$	0	0
$13$	−4.50984	−1.25080	−0.625402	−	0.780303i	$-0.715065\pi$
$13$	−4.50984	−1.25080	−0.625402	+	0.780303i	$0.715065\pi$
$14$	−1.90397	−0.508859
$15$	0	0
$16$	1.00000	0.250000
$17$	3.54115	0.858855	0.429427	−	0.903101i	$-0.358715\pi$
$17$	3.54115	0.858855	0.429427	+	0.903101i	$0.358715\pi$
$18$	0	0
$19$	−4.96850	−1.13985	−0.569926	−	0.821696i	$-0.693028\pi$
$19$	−4.96850	−1.13985	−0.569926	+	0.821696i	$0.693028\pi$
$20$	3.43402	0.767871
$21$	0	0
$22$	−5.05887	−1.07856
$23$	7.75578	1.61719	0.808596	−	0.588364i	$-0.200228\pi$
$23$	7.75578	1.61719	0.808596	+	0.588364i	$0.200228\pi$
$24$	0	0
$25$	6.79252	1.35850
$26$	−4.50984	−0.884452
$27$	0	0
$28$	−1.90397	−0.359817
$29$	−8.52570	−1.58318	−0.791591	−	0.611051i	$-0.790747\pi$
$29$	−8.52570	−1.58318	−0.791591	+	0.611051i	$0.790747\pi$
$30$	0	0
$31$	−8.93157	−1.60416	−0.802079	−	0.597218i	$-0.796273\pi$
$31$	−8.93157	−1.60416	−0.802079	+	0.597218i	$0.796273\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.54115	0.607302
$35$	−6.53829	−1.10517
$36$	0	0
$37$	2.08597	0.342931	0.171466	−	0.985190i	$-0.445150\pi$
$37$	2.08597	0.342931	0.171466	+	0.985190i	$0.445150\pi$
$38$	−4.96850	−0.805997
$39$	0	0
$40$	3.43402	0.542967
$41$	5.04962	0.788619	0.394309	−	0.918978i	$-0.370984\pi$
$41$	5.04962	0.788619	0.394309	+	0.918978i	$0.370984\pi$
$42$	0	0
$43$	−4.60761	−0.702654	−0.351327	−	0.936253i	$-0.614270\pi$
$43$	−4.60761	−0.702654	−0.351327	+	0.936253i	$0.614270\pi$
$44$	−5.05887	−0.762654
$45$	0	0
$46$	7.75578	1.14353
$47$	5.86559	0.855584	0.427792	−	0.903877i	$-0.359292\pi$
$47$	5.86559	0.855584	0.427792	+	0.903877i	$0.359292\pi$
$48$	0	0
$49$	−3.37488	−0.482126
$50$	6.79252	0.960607
$51$	0	0
$52$	−4.50984	−0.625402
$53$	−2.76603	−0.379944	−0.189972	−	0.981789i	$-0.560840\pi$
$53$	−2.76603	−0.379944	−0.189972	+	0.981789i	$0.560840\pi$
$54$	0	0
$55$	−17.3723	−2.34248
$56$	−1.90397	−0.254429
$57$	0	0
$58$	−8.52570	−1.11948
$59$	−5.93786	−0.773044	−0.386522	−	0.922280i	$-0.626324\pi$
$59$	−5.93786	−0.773044	−0.386522	+	0.922280i	$0.626324\pi$
$60$	0	0
$61$	5.94479	0.761152	0.380576	−	0.924750i	$-0.375726\pi$
$61$	5.94479	0.761152	0.380576	+	0.924750i	$0.375726\pi$
$62$	−8.93157	−1.13431
$63$	0	0
$64$	1.00000	0.125000
$65$	−15.4869	−1.92091
$66$	0	0
$67$	−6.16643	−0.753349	−0.376674	−	0.926346i	$-0.622932\pi$
$67$	−6.16643	−0.753349	−0.376674	+	0.926346i	$0.622932\pi$
$68$	3.54115	0.429427
$69$	0	0
$70$	−6.53829	−0.781475
$71$	12.1240	1.43886	0.719429	−	0.694566i	$-0.244403\pi$
$71$	12.1240	1.43886	0.719429	+	0.694566i	$0.244403\pi$
$72$	0	0
$73$	−11.5197	−1.34828	−0.674140	−	0.738604i	$-0.735486\pi$
$73$	−11.5197	−1.34828	−0.674140	+	0.738604i	$0.735486\pi$
$74$	2.08597	0.242489
$75$	0	0
$76$	−4.96850	−0.569926
$77$	9.63197	1.09766
$78$	0	0
$79$	11.4246	1.28537	0.642685	−	0.766131i	$-0.277821\pi$
$79$	11.4246	1.28537	0.642685	+	0.766131i	$0.277821\pi$
$80$	3.43402	0.383935
$81$	0	0
$82$	5.04962	0.557638
$83$	6.74370	0.740218	0.370109	−	0.928988i	$-0.379320\pi$
$83$	6.74370	0.740218	0.370109	+	0.928988i	$0.379320\pi$
$84$	0	0
$85$	12.1604	1.31898
$86$	−4.60761	−0.496852
$87$	0	0
$88$	−5.05887	−0.539278
$89$	−13.9218	−1.47571	−0.737853	−	0.674961i	$-0.764160\pi$
$89$	−13.9218	−1.47571	−0.737853	+	0.674961i	$0.764160\pi$
$90$	0	0
$91$	8.58661	0.900122
$92$	7.75578	0.808596
$93$	0	0
$94$	5.86559	0.604989
$95$	−17.0619	−1.75052
$96$	0	0
$97$	−18.4392	−1.87222	−0.936110	−	0.351708i	$-0.885601\pi$
$97$	−18.4392	−1.87222	−0.936110	+	0.351708i	$0.885601\pi$
$98$	−3.37488	−0.340915
$99$	0	0
$100$	6.79252	0.679252
$101$	14.9963	1.49219	0.746093	−	0.665842i	$-0.231928\pi$
$101$	14.9963	1.49219	0.746093	+	0.665842i	$0.231928\pi$
$102$	0	0
$103$	−7.81447	−0.769983	−0.384992	−	0.922920i	$-0.625796\pi$
$103$	−7.81447	−0.769983	−0.384992	+	0.922920i	$0.625796\pi$
$104$	−4.50984	−0.442226
$105$	0	0
$106$	−2.76603	−0.268661
$107$	−18.9522	−1.83218	−0.916088	−	0.400978i	$-0.868670\pi$
$107$	−18.9522	−1.83218	−0.916088	+	0.400978i	$0.868670\pi$
$108$	0	0
$109$	−5.75906	−0.551618	−0.275809	−	0.961212i	$-0.588946\pi$
$109$	−5.75906	−0.551618	−0.275809	+	0.961212i	$0.588946\pi$
$110$	−17.3723	−1.65638
$111$	0	0
$112$	−1.90397	−0.179909
$113$	−10.7590	−1.01212	−0.506062	−	0.862497i	$-0.668899\pi$
$113$	−10.7590	−1.01212	−0.506062	+	0.862497i	$0.668899\pi$
$114$	0	0
$115$	26.6335	2.48359
$116$	−8.52570	−0.791591
$117$	0	0
$118$	−5.93786	−0.546625
$119$	−6.74226	−0.618062
$120$	0	0
$121$	14.5922	1.32657
$122$	5.94479	0.538216
$123$	0	0
$124$	−8.93157	−0.802079
$125$	6.15555	0.550569
$126$	0	0
$127$	−22.1839	−1.96851	−0.984253	−	0.176766i	$-0.943436\pi$
$127$	−22.1839	−1.96851	−0.984253	+	0.176766i	$0.943436\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−15.4869	−1.35829
$131$	−13.6597	−1.19346	−0.596729	−	0.802443i	$-0.703533\pi$
$131$	−13.6597	−1.19346	−0.596729	+	0.802443i	$0.703533\pi$
$132$	0	0
$133$	9.45990	0.820277
$134$	−6.16643	−0.532698
$135$	0	0
$136$	3.54115	0.303651
$137$	9.17091	0.783524	0.391762	−	0.920067i	$-0.371866\pi$
$137$	9.17091	0.783524	0.391762	+	0.920067i	$0.371866\pi$
$138$	0	0
$139$	−16.4861	−1.39834	−0.699168	−	0.714957i	$-0.746446\pi$
$139$	−16.4861	−1.39834	−0.699168	+	0.714957i	$0.746446\pi$
$140$	−6.53829	−0.552587
$141$	0	0
$142$	12.1240	1.01743
$143$	22.8147	1.90786
$144$	0	0
$145$	−29.2774	−2.43136
$146$	−11.5197	−0.953378
$147$	0	0
$148$	2.08597	0.171466
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−17.5202	−1.42578	−0.712888	−	0.701278i	$-0.752613\pi$
$151$	−17.5202	−1.42578	−0.712888	+	0.701278i	$0.752613\pi$
$152$	−4.96850	−0.402999
$153$	0	0
$154$	9.63197	0.776166
$155$	−30.6712	−2.46357
$156$	0	0
$157$	24.5751	1.96130	0.980652	−	0.195760i	$-0.0627173\pi$
$157$	24.5751	1.96130	0.980652	+	0.195760i	$0.0627173\pi$
$158$	11.4246	0.908893
$159$	0	0
$160$	3.43402	0.271483
$161$	−14.7668	−1.16379
$162$	0	0
$163$	−24.7090	−1.93536	−0.967681	−	0.252177i	$-0.918853\pi$
$163$	−24.7090	−1.93536	−0.967681	+	0.252177i	$0.918853\pi$
$164$	5.04962	0.394309
$165$	0	0
$166$	6.74370	0.523413
$167$	−0.350459	−0.0271194	−0.0135597	−	0.999908i	$-0.504316\pi$
$167$	−0.350459	−0.0271194	−0.0135597	+	0.999908i	$0.504316\pi$
$168$	0	0
$169$	7.33863	0.564510
$170$	12.1604	0.932659
$171$	0	0
$172$	−4.60761	−0.351327
$173$	0.877157	0.0666890	0.0333445	−	0.999444i	$-0.489384\pi$
$173$	0.877157	0.0666890	0.0333445	+	0.999444i	$0.489384\pi$
$174$	0	0
$175$	−12.9328	−0.977626
$176$	−5.05887	−0.381327
$177$	0	0
$178$	−13.9218	−1.04348
$179$	1.12528	0.0841075	0.0420538	−	0.999115i	$-0.486610\pi$
$179$	1.12528	0.0841075	0.0420538	+	0.999115i	$0.486610\pi$
$180$	0	0
$181$	−3.57092	−0.265424	−0.132712	−	0.991155i	$-0.542369\pi$
$181$	−3.57092	−0.265424	−0.132712	+	0.991155i	$0.542369\pi$
$182$	8.58661	0.636482
$183$	0	0
$184$	7.75578	0.571764
$185$	7.16327	0.526654
$186$	0	0
$187$	−17.9142	−1.31002
$188$	5.86559	0.427792
$189$	0	0
$190$	−17.0619	−1.23780
$191$	13.6802	0.989862	0.494931	−	0.868932i	$-0.335193\pi$
$191$	13.6802	0.989862	0.494931	+	0.868932i	$0.335193\pi$
$192$	0	0
$193$	7.02643	0.505774	0.252887	−	0.967496i	$-0.418620\pi$
$193$	7.02643	0.505774	0.252887	+	0.967496i	$0.418620\pi$
$194$	−18.4392	−1.32386
$195$	0	0
$196$	−3.37488	−0.241063
$197$	5.83711	0.415877	0.207938	−	0.978142i	$-0.433325\pi$
$197$	5.83711	0.415877	0.207938	+	0.978142i	$0.433325\pi$
$198$	0	0
$199$	−8.28688	−0.587442	−0.293721	−	0.955891i	$-0.594894\pi$
$199$	−8.28688	−0.587442	−0.293721	+	0.955891i	$0.594894\pi$
$200$	6.79252	0.480304
$201$	0	0
$202$	14.9963	1.05513
$203$	16.2327	1.13931
$204$	0	0
$205$	17.3405	1.21111
$206$	−7.81447	−0.544460
$207$	0	0
$208$	−4.50984	−0.312701
$209$	25.1350	1.73863
$210$	0	0
$211$	8.91088	0.613451	0.306725	−	0.951798i	$-0.400767\pi$
$211$	8.91088	0.613451	0.306725	+	0.951798i	$0.400767\pi$
$212$	−2.76603	−0.189972
$213$	0	0
$214$	−18.9522	−1.29554
$215$	−15.8227	−1.07910
$216$	0	0
$217$	17.0055	1.15441
$218$	−5.75906	−0.390053
$219$	0	0
$220$	−17.3723	−1.17124
$221$	−15.9700	−1.07426
$222$	0	0
$223$	8.89504	0.595656	0.297828	−	0.954620i	$-0.403738\pi$
$223$	8.89504	0.595656	0.297828	+	0.954620i	$0.403738\pi$
$224$	−1.90397	−0.127215
$225$	0	0
$226$	−10.7590	−0.715680
$227$	−12.0793	−0.801731	−0.400866	−	0.916137i	$-0.631291\pi$
$227$	−12.0793	−0.801731	−0.400866	+	0.916137i	$0.631291\pi$
$228$	0	0
$229$	0.111965	0.00739886	0.00369943	−	0.999993i	$-0.498822\pi$
$229$	0.111965	0.00739886	0.00369943	+	0.999993i	$0.498822\pi$
$230$	26.6335	1.75616
$231$	0	0
$232$	−8.52570	−0.559739
$233$	−5.71066	−0.374118	−0.187059	−	0.982349i	$-0.559896\pi$
$233$	−5.71066	−0.374118	−0.187059	+	0.982349i	$0.559896\pi$
$234$	0	0
$235$	20.1426	1.31396
$236$	−5.93786	−0.386522
$237$	0	0
$238$	−6.74226	−0.437035
$239$	21.7102	1.40432	0.702158	−	0.712021i	$-0.252220\pi$
$239$	21.7102	1.40432	0.702158	+	0.712021i	$0.252220\pi$
$240$	0	0
$241$	14.0658	0.906060	0.453030	−	0.891495i	$-0.350343\pi$
$241$	14.0658	0.906060	0.453030	+	0.891495i	$0.350343\pi$
$242$	14.5922	0.938023
$243$	0	0
$244$	5.94479	0.380576
$245$	−11.5894	−0.740421
$246$	0	0
$247$	22.4071	1.42573
$248$	−8.93157	−0.567155
$249$	0	0
$250$	6.15555	0.389311
$251$	−16.2009	−1.02259	−0.511295	−	0.859406i	$-0.670834\pi$
$251$	−16.2009	−1.02259	−0.511295	+	0.859406i	$0.670834\pi$
$252$	0	0
$253$	−39.2355	−2.46672
$254$	−22.1839	−1.39194
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.0484	0.689179	0.344589	−	0.938754i	$-0.388018\pi$
$257$	11.0484	0.689179	0.344589	+	0.938754i	$0.388018\pi$
$258$	0	0
$259$	−3.97163	−0.246785
$260$	−15.4869	−0.960456
$261$	0	0
$262$	−13.6597	−0.843902
$263$	−19.4287	−1.19802	−0.599012	−	0.800740i	$-0.704440\pi$
$263$	−19.4287	−1.19802	−0.599012	+	0.800740i	$0.704440\pi$
$264$	0	0
$265$	−9.49863	−0.583496
$266$	9.45990	0.580024
$267$	0	0
$268$	−6.16643	−0.376674
$269$	17.8368	1.08753	0.543765	−	0.839238i	$-0.316998\pi$
$269$	17.8368	1.08753	0.543765	+	0.839238i	$0.316998\pi$
$270$	0	0
$271$	9.65417	0.586449	0.293225	−	0.956044i	$-0.405272\pi$
$271$	9.65417	0.586449	0.293225	+	0.956044i	$0.405272\pi$
$272$	3.54115	0.214714
$273$	0	0
$274$	9.17091	0.554035
$275$	−34.3625	−2.07214
$276$	0	0
$277$	−2.38956	−0.143575	−0.0717873	−	0.997420i	$-0.522870\pi$
$277$	−2.38956	−0.143575	−0.0717873	+	0.997420i	$0.522870\pi$
$278$	−16.4861	−0.988773
$279$	0	0
$280$	−6.53829	−0.390738
$281$	12.2560	0.731132	0.365566	−	0.930785i	$-0.380875\pi$
$281$	12.2560	0.731132	0.365566	+	0.930785i	$0.380875\pi$
$282$	0	0
$283$	−5.68135	−0.337721	−0.168861	−	0.985640i	$-0.554009\pi$
$283$	−5.68135	−0.337721	−0.168861	+	0.985640i	$0.554009\pi$
$284$	12.1240	0.719429
$285$	0	0
$286$	22.8147	1.34906
$287$	−9.61435	−0.567517
$288$	0	0
$289$	−4.46027	−0.262369
$290$	−29.2774	−1.71923
$291$	0	0
$292$	−11.5197	−0.674140
$293$	8.46595	0.494586	0.247293	−	0.968941i	$-0.420459\pi$
$293$	8.46595	0.494586	0.247293	+	0.968941i	$0.420459\pi$
$294$	0	0
$295$	−20.3908	−1.18720
$296$	2.08597	0.121244
$297$	0	0
$298$	1.00000	0.0579284
$299$	−34.9773	−2.02279
$300$	0	0
$301$	8.77278	0.505654
$302$	−17.5202	−1.00818
$303$	0	0
$304$	−4.96850	−0.284963
$305$	20.4146	1.16893
$306$	0	0
$307$	19.4134	1.10798	0.553989	−	0.832524i	$-0.313105\pi$
$307$	19.4134	1.10798	0.553989	+	0.832524i	$0.313105\pi$
$308$	9.63197	0.548832
$309$	0	0
$310$	−30.6712	−1.74201
$311$	−9.32581	−0.528818	−0.264409	−	0.964411i	$-0.585177\pi$
$311$	−9.32581	−0.528818	−0.264409	+	0.964411i	$0.585177\pi$
$312$	0	0
$313$	20.7473	1.17271	0.586355	−	0.810054i	$-0.300562\pi$
$313$	20.7473	1.17271	0.586355	+	0.810054i	$0.300562\pi$
$314$	24.5751	1.38685
$315$	0	0
$316$	11.4246	0.642685
$317$	19.6554	1.10396	0.551980	−	0.833857i	$-0.313872\pi$
$317$	19.6554	1.10396	0.551980	+	0.833857i	$0.313872\pi$
$318$	0	0
$319$	43.1304	2.41484
$320$	3.43402	0.191968
$321$	0	0
$322$	−14.7668	−0.822922
$323$	−17.5942	−0.978967
$324$	0	0
$325$	−30.6331	−1.69922
$326$	−24.7090	−1.36851
$327$	0	0
$328$	5.04962	0.278819
$329$	−11.1679	−0.615708
$330$	0	0
$331$	33.1029	1.81950	0.909749	−	0.415159i	$-0.136274\pi$
$331$	33.1029	1.81950	0.909749	+	0.415159i	$0.136274\pi$
$332$	6.74370	0.370109
$333$	0	0
$334$	−0.350459	−0.0191763
$335$	−21.1757	−1.15695
$336$	0	0
$337$	10.2651	0.559174	0.279587	−	0.960120i	$-0.409802\pi$
$337$	10.2651	0.559174	0.279587	+	0.960120i	$0.409802\pi$
$338$	7.33863	0.399169
$339$	0	0
$340$	12.1604	0.659490
$341$	45.1837	2.44683
$342$	0	0
$343$	19.7535	1.06659
$344$	−4.60761	−0.248426
$345$	0	0
$346$	0.877157	0.0471562
$347$	3.08394	0.165555	0.0827773	−	0.996568i	$-0.473621\pi$
$347$	3.08394	0.165555	0.0827773	+	0.996568i	$0.473621\pi$
$348$	0	0
$349$	9.27295	0.496370	0.248185	−	0.968713i	$-0.420166\pi$
$349$	9.27295	0.496370	0.248185	+	0.968713i	$0.420166\pi$
$350$	−12.9328	−0.691286
$351$	0	0
$352$	−5.05887	−0.269639
$353$	−2.41443	−0.128507	−0.0642537	−	0.997934i	$-0.520467\pi$
$353$	−2.41443	−0.128507	−0.0642537	+	0.997934i	$0.520467\pi$
$354$	0	0
$355$	41.6342	2.20972
$356$	−13.9218	−0.737853
$357$	0	0
$358$	1.12528	0.0594730
$359$	−33.4019	−1.76288	−0.881442	−	0.472293i	$-0.843426\pi$
$359$	−33.4019	−1.76288	−0.881442	+	0.472293i	$0.843426\pi$
$360$	0	0
$361$	5.68600	0.299263
$362$	−3.57092	−0.187683
$363$	0	0
$364$	8.58661	0.450061
$365$	−39.5590	−2.07061
$366$	0	0
$367$	30.9456	1.61535	0.807673	−	0.589630i	$-0.200727\pi$
$367$	30.9456	1.61535	0.807673	+	0.589630i	$0.200727\pi$
$368$	7.75578	0.404298
$369$	0	0
$370$	7.16327	0.372401
$371$	5.26646	0.273421
$372$	0	0
$373$	15.7547	0.815748	0.407874	−	0.913038i	$-0.366270\pi$
$373$	15.7547	0.815748	0.407874	+	0.913038i	$0.366270\pi$
$374$	−17.9142	−0.926323
$375$	0	0
$376$	5.86559	0.302495
$377$	38.4495	1.98025
$378$	0	0
$379$	18.0781	0.928610	0.464305	−	0.885675i	$-0.346304\pi$
$379$	18.0781	0.928610	0.464305	+	0.885675i	$0.346304\pi$
$380$	−17.0619	−0.875259
$381$	0	0
$382$	13.6802	0.699938
$383$	36.5542	1.86783	0.933915	−	0.357494i	$-0.116369\pi$
$383$	36.5542	1.86783	0.933915	+	0.357494i	$0.116369\pi$
$384$	0	0
$385$	33.0764	1.68573
$386$	7.02643	0.357636
$387$	0	0
$388$	−18.4392	−0.936110
$389$	−32.4824	−1.64692	−0.823462	−	0.567372i	$-0.807960\pi$
$389$	−32.4824	−1.64692	−0.823462	+	0.567372i	$0.807960\pi$
$390$	0	0
$391$	27.4644	1.38893
$392$	−3.37488	−0.170457
$393$	0	0
$394$	5.83711	0.294069
$395$	39.2324	1.97400
$396$	0	0
$397$	−26.4450	−1.32724	−0.663619	−	0.748071i	$-0.730980\pi$
$397$	−26.4450	−1.32724	−0.663619	+	0.748071i	$0.730980\pi$
$398$	−8.28688	−0.415384
$399$	0	0
$400$	6.79252	0.339626
$401$	−6.66723	−0.332946	−0.166473	−	0.986046i	$-0.553238\pi$
$401$	−6.66723	−0.332946	−0.166473	+	0.986046i	$0.553238\pi$
$402$	0	0
$403$	40.2799	2.00649
$404$	14.9963	0.746093
$405$	0	0
$406$	16.2327	0.805616
$407$	−10.5527	−0.523076
$408$	0	0
$409$	23.1492	1.14465	0.572327	−	0.820026i	$-0.306041\pi$
$409$	23.1492	1.14465	0.572327	+	0.820026i	$0.306041\pi$
$410$	17.3405	0.856388
$411$	0	0
$412$	−7.81447	−0.384992
$413$	11.3055	0.556309
$414$	0	0
$415$	23.1580	1.13678
$416$	−4.50984	−0.221113
$417$	0	0
$418$	25.1350	1.22939
$419$	5.17742	0.252934	0.126467	−	0.991971i	$-0.459636\pi$
$419$	5.17742	0.252934	0.126467	+	0.991971i	$0.459636\pi$
$420$	0	0
$421$	24.5799	1.19795	0.598977	−	0.800767i	$-0.295574\pi$
$421$	24.5799	1.19795	0.598977	+	0.800767i	$0.295574\pi$
$422$	8.91088	0.433775
$423$	0	0
$424$	−2.76603	−0.134331
$425$	24.0533	1.16676
$426$	0	0
$427$	−11.3187	−0.547752
$428$	−18.9522	−0.916088
$429$	0	0
$430$	−15.8227	−0.763036
$431$	−16.1595	−0.778374	−0.389187	−	0.921159i	$-0.627244\pi$
$431$	−16.1595	−0.778374	−0.389187	+	0.921159i	$0.627244\pi$
$432$	0	0
$433$	36.1699	1.73821	0.869107	−	0.494623i	$-0.164694\pi$
$433$	36.1699	1.73821	0.869107	+	0.494623i	$0.164694\pi$
$434$	17.0055	0.816289
$435$	0	0
$436$	−5.75906	−0.275809
$437$	−38.5346	−1.84336
$438$	0	0
$439$	−30.1285	−1.43796	−0.718978	−	0.695033i	$-0.755390\pi$
$439$	−30.1285	−1.43796	−0.718978	+	0.695033i	$0.755390\pi$
$440$	−17.3723	−0.828192
$441$	0	0
$442$	−15.9700	−0.759615
$443$	−22.6806	−1.07759	−0.538794	−	0.842438i	$-0.681120\pi$
$443$	−22.6806	−1.07759	−0.538794	+	0.842438i	$0.681120\pi$
$444$	0	0
$445$	−47.8077	−2.26630
$446$	8.89504	0.421193
$447$	0	0
$448$	−1.90397	−0.0899543
$449$	−19.5804	−0.924056	−0.462028	−	0.886865i	$-0.652878\pi$
$449$	−19.5804	−0.924056	−0.462028	+	0.886865i	$0.652878\pi$
$450$	0	0
$451$	−25.5454	−1.20289
$452$	−10.7590	−0.506062
$453$	0	0
$454$	−12.0793	−0.566910
$455$	29.4866	1.38235
$456$	0	0
$457$	−26.4538	−1.23746	−0.618729	−	0.785605i	$-0.712352\pi$
$457$	−26.4538	−1.23746	−0.618729	+	0.785605i	$0.712352\pi$
$458$	0.111965	0.00523179
$459$	0	0
$460$	26.6335	1.24180
$461$	37.1292	1.72928	0.864640	−	0.502392i	$-0.167547\pi$
$461$	37.1292	1.72928	0.864640	+	0.502392i	$0.167547\pi$
$462$	0	0
$463$	−1.88633	−0.0876653	−0.0438327	−	0.999039i	$-0.513957\pi$
$463$	−1.88633	−0.0876653	−0.0438327	+	0.999039i	$0.513957\pi$
$464$	−8.52570	−0.395796
$465$	0	0
$466$	−5.71066	−0.264541
$467$	−31.7341	−1.46848	−0.734238	−	0.678892i	$-0.762461\pi$
$467$	−31.7341	−1.46848	−0.734238	+	0.678892i	$0.762461\pi$
$468$	0	0
$469$	11.7407	0.542136
$470$	20.1426	0.929107
$471$	0	0
$472$	−5.93786	−0.273312
$473$	23.3093	1.07176
$474$	0	0
$475$	−33.7486	−1.54849
$476$	−6.74226	−0.309031
$477$	0	0
$478$	21.7102	0.993002
$479$	−0.498846	−0.0227929	−0.0113964	−	0.999935i	$-0.503628\pi$
$479$	−0.498846	−0.0227929	−0.0113964	+	0.999935i	$0.503628\pi$
$480$	0	0
$481$	−9.40738	−0.428940
$482$	14.0658	0.640681
$483$	0	0
$484$	14.5922	0.663283
$485$	−63.3207	−2.87525
$486$	0	0
$487$	−17.5179	−0.793812	−0.396906	−	0.917859i	$-0.629916\pi$
$487$	−17.5179	−0.793812	−0.396906	+	0.917859i	$0.629916\pi$
$488$	5.94479	0.269108
$489$	0	0
$490$	−11.5894	−0.523557
$491$	27.8433	1.25655	0.628275	−	0.777991i	$-0.283761\pi$
$491$	27.8433	1.25655	0.628275	+	0.777991i	$0.283761\pi$
$492$	0	0
$493$	−30.1908	−1.35972
$494$	22.4071	1.00814
$495$	0	0
$496$	−8.93157	−0.401039
$497$	−23.0839	−1.03545
$498$	0	0
$499$	28.5254	1.27697	0.638485	−	0.769634i	$-0.279561\pi$
$499$	28.5254	1.27697	0.638485	+	0.769634i	$0.279561\pi$
$500$	6.15555	0.275284
$501$	0	0
$502$	−16.2009	−0.723080
$503$	−19.9347	−0.888844	−0.444422	−	0.895818i	$-0.646591\pi$
$503$	−19.9347	−0.888844	−0.444422	+	0.895818i	$0.646591\pi$
$504$	0	0
$505$	51.4976	2.29161
$506$	−39.2355	−1.74423
$507$	0	0
$508$	−22.1839	−0.984253
$509$	−13.1218	−0.581615	−0.290808	−	0.956782i	$-0.593924\pi$
$509$	−13.1218	−0.581615	−0.290808	+	0.956782i	$0.593924\pi$
$510$	0	0
$511$	21.9332	0.970269
$512$	1.00000	0.0441942
$513$	0	0
$514$	11.0484	0.487323
$515$	−26.8351	−1.18250
$516$	0	0
$517$	−29.6733	−1.30503
$518$	−3.97163	−0.174503
$519$	0	0
$520$	−15.4869	−0.679145
$521$	21.1780	0.927824	0.463912	−	0.885881i	$-0.346445\pi$
$521$	21.1780	0.927824	0.463912	+	0.885881i	$0.346445\pi$
$522$	0	0
$523$	−15.9077	−0.695594	−0.347797	−	0.937570i	$-0.613070\pi$
$523$	−15.9077	−0.695594	−0.347797	+	0.937570i	$0.613070\pi$
$524$	−13.6597	−0.596729
$525$	0	0
$526$	−19.4287	−0.847131
$527$	−31.6280	−1.37774
$528$	0	0
$529$	37.1522	1.61531
$530$	−9.49863	−0.412594
$531$	0	0
$532$	9.45990	0.410139
$533$	−22.7730	−0.986407
$534$	0	0
$535$	−65.0822	−2.81375
$536$	−6.16643	−0.266349
$537$	0	0
$538$	17.8368	0.769000
$539$	17.0731	0.735391
$540$	0	0
$541$	−13.5612	−0.583041	−0.291521	−	0.956565i	$-0.594161\pi$
$541$	−13.5612	−0.583041	−0.291521	+	0.956565i	$0.594161\pi$
$542$	9.65417	0.414682
$543$	0	0
$544$	3.54115	0.151825
$545$	−19.7768	−0.847143
$546$	0	0
$547$	−26.0301	−1.11297	−0.556483	−	0.830859i	$-0.687849\pi$
$547$	−26.0301	−1.11297	−0.556483	+	0.830859i	$0.687849\pi$
$548$	9.17091	0.391762
$549$	0	0
$550$	−34.3625	−1.46522
$551$	42.3599	1.80459
$552$	0	0
$553$	−21.7522	−0.924996
$554$	−2.38956	−0.101523
$555$	0	0
$556$	−16.4861	−0.699168
$557$	5.05666	0.214258	0.107129	−	0.994245i	$-0.465834\pi$
$557$	5.05666	0.214258	0.107129	+	0.994245i	$0.465834\pi$
$558$	0	0
$559$	20.7796	0.878883
$560$	−6.53829	−0.276293
$561$	0	0
$562$	12.2560	0.516988
$563$	−38.1264	−1.60684	−0.803418	−	0.595415i	$-0.796988\pi$
$563$	−38.1264	−1.60684	−0.803418	+	0.595415i	$0.796988\pi$
$564$	0	0
$565$	−36.9468	−1.55436
$566$	−5.68135	−0.238805
$567$	0	0
$568$	12.1240	0.508713
$569$	15.1381	0.634624	0.317312	−	0.948321i	$-0.397220\pi$
$569$	15.1381	0.634624	0.317312	+	0.948321i	$0.397220\pi$
$570$	0	0
$571$	−21.2021	−0.887280	−0.443640	−	0.896205i	$-0.646313\pi$
$571$	−21.2021	−0.887280	−0.443640	+	0.896205i	$0.646313\pi$
$572$	22.8147	0.953931
$573$	0	0
$574$	−9.61435	−0.401295
$575$	52.6813	2.19696
$576$	0	0
$577$	25.7094	1.07030	0.535149	−	0.844758i	$-0.320255\pi$
$577$	25.7094	1.07030	0.535149	+	0.844758i	$0.320255\pi$
$578$	−4.46027	−0.185523
$579$	0	0
$580$	−29.2774	−1.21568
$581$	−12.8398	−0.532686
$582$	0	0
$583$	13.9930	0.579532
$584$	−11.5197	−0.476689
$585$	0	0
$586$	8.46595	0.349725
$587$	−27.3617	−1.12934	−0.564669	−	0.825317i	$-0.690996\pi$
$587$	−27.3617	−1.12934	−0.564669	+	0.825317i	$0.690996\pi$
$588$	0	0
$589$	44.3765	1.82850
$590$	−20.3908	−0.839474
$591$	0	0
$592$	2.08597	0.0857328
$593$	26.2310	1.07718	0.538589	−	0.842569i	$-0.318958\pi$
$593$	26.2310	1.07718	0.538589	+	0.842569i	$0.318958\pi$
$594$	0	0
$595$	−23.1531	−0.949183
$596$	1.00000	0.0409616
$597$	0	0
$598$	−34.9773	−1.43033
$599$	35.4640	1.44902	0.724509	−	0.689265i	$-0.242066\pi$
$599$	35.4640	1.44902	0.724509	+	0.689265i	$0.242066\pi$
$600$	0	0
$601$	−12.1645	−0.496200	−0.248100	−	0.968734i	$-0.579806\pi$
$601$	−12.1645	−0.496200	−0.248100	+	0.968734i	$0.579806\pi$
$602$	8.77278	0.357552
$603$	0	0
$604$	−17.5202	−0.712888
$605$	50.1100	2.03726
$606$	0	0
$607$	−18.1093	−0.735032	−0.367516	−	0.930017i	$-0.619792\pi$
$607$	−18.1093	−0.735032	−0.367516	+	0.930017i	$0.619792\pi$
$608$	−4.96850	−0.201499
$609$	0	0
$610$	20.4146	0.826561
$611$	−26.4528	−1.07017
$612$	0	0
$613$	19.3044	0.779697	0.389848	−	0.920879i	$-0.372527\pi$
$613$	19.3044	0.779697	0.389848	+	0.920879i	$0.372527\pi$
$614$	19.4134	0.783459
$615$	0	0
$616$	9.63197	0.388083
$617$	9.21303	0.370903	0.185451	−	0.982653i	$-0.440625\pi$
$617$	9.21303	0.370903	0.185451	+	0.982653i	$0.440625\pi$
$618$	0	0
$619$	−46.2202	−1.85775	−0.928873	−	0.370399i	$-0.879221\pi$
$619$	−46.2202	−1.85775	−0.928873	+	0.370399i	$0.879221\pi$
$620$	−30.6712	−1.23179
$621$	0	0
$622$	−9.32581	−0.373931
$623$	26.5067	1.06197
$624$	0	0
$625$	−12.8243	−0.512972
$626$	20.7473	0.829231
$627$	0	0
$628$	24.5751	0.980652
$629$	7.38673	0.294528
$630$	0	0
$631$	39.3007	1.56454	0.782268	−	0.622942i	$-0.214063\pi$
$631$	39.3007	1.56454	0.782268	+	0.622942i	$0.214063\pi$
$632$	11.4246	0.454447
$633$	0	0
$634$	19.6554	0.780618
$635$	−76.1801	−3.02312
$636$	0	0
$637$	15.2202	0.603045
$638$	43.1304	1.70755
$639$	0	0
$640$	3.43402	0.135742
$641$	23.2483	0.918254	0.459127	−	0.888371i	$-0.348162\pi$
$641$	23.2483	0.918254	0.459127	+	0.888371i	$0.348162\pi$
$642$	0	0
$643$	14.3230	0.564845	0.282423	−	0.959290i	$-0.408862\pi$
$643$	14.3230	0.564845	0.282423	+	0.959290i	$0.408862\pi$
$644$	−14.7668	−0.581894
$645$	0	0
$646$	−17.5942	−0.692234
$647$	−22.2795	−0.875898	−0.437949	−	0.899000i	$-0.644295\pi$
$647$	−22.2795	−0.875898	−0.437949	+	0.899000i	$0.644295\pi$
$648$	0	0
$649$	30.0389	1.17913
$650$	−30.6331	−1.20153
$651$	0	0
$652$	−24.7090	−0.967681
$653$	−26.5884	−1.04049	−0.520243	−	0.854019i	$-0.674158\pi$
$653$	−26.5884	−1.04049	−0.520243	+	0.854019i	$0.674158\pi$
$654$	0	0
$655$	−46.9079	−1.83284
$656$	5.04962	0.197155
$657$	0	0
$658$	−11.1679	−0.435371
$659$	−6.00650	−0.233980	−0.116990	−	0.993133i	$-0.537325\pi$
$659$	−6.00650	−0.233980	−0.116990	+	0.993133i	$0.537325\pi$
$660$	0	0
$661$	−16.3433	−0.635680	−0.317840	−	0.948144i	$-0.602957\pi$
$661$	−16.3433	−0.635680	−0.317840	+	0.948144i	$0.602957\pi$
$662$	33.1029	1.28658
$663$	0	0
$664$	6.74370	0.261706
$665$	32.4855	1.25973
$666$	0	0
$667$	−66.1235	−2.56031
$668$	−0.350459	−0.0135597
$669$	0	0
$670$	−21.1757	−0.818087
$671$	−30.0740	−1.16099
$672$	0	0
$673$	5.80760	0.223867	0.111933	−	0.993716i	$-0.464296\pi$
$673$	5.80760	0.223867	0.111933	+	0.993716i	$0.464296\pi$
$674$	10.2651	0.395396
$675$	0	0
$676$	7.33863	0.282255
$677$	9.93795	0.381946	0.190973	−	0.981595i	$-0.438836\pi$
$677$	9.93795	0.381946	0.190973	+	0.981595i	$0.438836\pi$
$678$	0	0
$679$	35.1078	1.34731
$680$	12.1604	0.466330
$681$	0	0
$682$	45.1837	1.73017
$683$	27.1692	1.03960	0.519800	−	0.854288i	$-0.326007\pi$
$683$	27.1692	1.03960	0.519800	+	0.854288i	$0.326007\pi$
$684$	0	0
$685$	31.4931	1.20329
$686$	19.7535	0.754192
$687$	0	0
$688$	−4.60761	−0.175664
$689$	12.4744	0.475236
$690$	0	0
$691$	13.3016	0.506015	0.253007	−	0.967464i	$-0.418580\pi$
$691$	13.3016	0.506015	0.253007	+	0.967464i	$0.418580\pi$
$692$	0.877157	0.0333445
$693$	0	0
$694$	3.08394	0.117065
$695$	−56.6138	−2.14748
$696$	0	0
$697$	17.8815	0.677309
$698$	9.27295	0.350986
$699$	0	0
$700$	−12.9328	−0.488813
$701$	−21.6719	−0.818535	−0.409268	−	0.912414i	$-0.634216\pi$
$701$	−21.6719	−0.818535	−0.409268	+	0.912414i	$0.634216\pi$
$702$	0	0
$703$	−10.3641	−0.390891
$704$	−5.05887	−0.190664
$705$	0	0
$706$	−2.41443	−0.0908684
$707$	−28.5525	−1.07383
$708$	0	0
$709$	36.6117	1.37498	0.687491	−	0.726193i	$-0.258712\pi$
$709$	36.6117	1.37498	0.687491	+	0.726193i	$0.258712\pi$
$710$	41.6342	1.56250
$711$	0	0
$712$	−13.9218	−0.521741
$713$	−69.2713	−2.59423
$714$	0	0
$715$	78.3462	2.92998
$716$	1.12528	0.0420538
$717$	0	0
$718$	−33.4019	−1.24655
$719$	37.6629	1.40459	0.702294	−	0.711887i	$-0.252159\pi$
$719$	37.6629	1.40459	0.702294	+	0.711887i	$0.252159\pi$
$720$	0	0
$721$	14.8786	0.554106
$722$	5.68600	0.211611
$723$	0	0
$724$	−3.57092	−0.132712
$725$	−57.9109	−2.15076
$726$	0	0
$727$	−19.2420	−0.713647	−0.356824	−	0.934172i	$-0.616140\pi$
$727$	−19.2420	−0.713647	−0.356824	+	0.934172i	$0.616140\pi$
$728$	8.58661	0.318241
$729$	0	0
$730$	−39.5590	−1.46414
$731$	−16.3162	−0.603478
$732$	0	0
$733$	−2.80002	−0.103421	−0.0517105	−	0.998662i	$-0.516467\pi$
$733$	−2.80002	−0.103421	−0.0517105	+	0.998662i	$0.516467\pi$
$734$	30.9456	1.14222
$735$	0	0
$736$	7.75578	0.285882
$737$	31.1952	1.14909
$738$	0	0
$739$	−45.4814	−1.67306	−0.836529	−	0.547922i	$-0.815419\pi$
$739$	−45.4814	−1.67306	−0.836529	+	0.547922i	$0.815419\pi$
$740$	7.16327	0.263327
$741$	0	0
$742$	5.26646	0.193338
$743$	−12.3191	−0.451945	−0.225973	−	0.974134i	$-0.572556\pi$
$743$	−12.3191	−0.451945	−0.225973	+	0.974134i	$0.572556\pi$
$744$	0	0
$745$	3.43402	0.125813
$746$	15.7547	0.576821
$747$	0	0
$748$	−17.9142	−0.655009
$749$	36.0844	1.31850
$750$	0	0
$751$	28.6626	1.04591	0.522957	−	0.852359i	$-0.324829\pi$
$751$	28.6626	1.04591	0.522957	+	0.852359i	$0.324829\pi$
$752$	5.86559	0.213896
$753$	0	0
$754$	38.4495	1.40025
$755$	−60.1649	−2.18962
$756$	0	0
$757$	−33.0639	−1.20173	−0.600864	−	0.799351i	$-0.705177\pi$
$757$	−33.0639	−1.20173	−0.600864	+	0.799351i	$0.705177\pi$
$758$	18.0781	0.656627
$759$	0	0
$760$	−17.0619	−0.618902
$761$	−39.5260	−1.43282	−0.716408	−	0.697682i	$-0.754215\pi$
$761$	−39.5260	−1.43282	−0.716408	+	0.697682i	$0.754215\pi$
$762$	0	0
$763$	10.9651	0.396963
$764$	13.6802	0.494931
$765$	0	0
$766$	36.5542	1.32076
$767$	26.7788	0.966926
$768$	0	0
$769$	35.3194	1.27365	0.636825	−	0.771009i	$-0.280247\pi$
$769$	35.3194	1.27365	0.636825	+	0.771009i	$0.280247\pi$
$770$	33.0764	1.19199
$771$	0	0
$772$	7.02643	0.252887
$773$	−31.3199	−1.12650	−0.563249	−	0.826287i	$-0.690449\pi$
$773$	−31.3199	−1.12650	−0.563249	+	0.826287i	$0.690449\pi$
$774$	0	0
$775$	−60.6679	−2.17925
$776$	−18.4392	−0.661930
$777$	0	0
$778$	−32.4824	−1.16455
$779$	−25.0891	−0.898909
$780$	0	0
$781$	−61.3340	−2.19470
$782$	27.4644	0.982124
$783$	0	0
$784$	−3.37488	−0.120531
$785$	84.3914	3.01206
$786$	0	0
$787$	−38.4161	−1.36938	−0.684692	−	0.728832i	$-0.740063\pi$
$787$	−38.4161	−1.36938	−0.684692	+	0.728832i	$0.740063\pi$
$788$	5.83711	0.207938
$789$	0	0
$790$	39.2324	1.39583
$791$	20.4849	0.728360
$792$	0	0
$793$	−26.8100	−0.952052
$794$	−26.4450	−0.938498
$795$	0	0
$796$	−8.28688	−0.293721
$797$	36.8825	1.30644	0.653222	−	0.757166i	$-0.273417\pi$
$797$	36.8825	1.30644	0.653222	+	0.757166i	$0.273417\pi$
$798$	0	0
$799$	20.7709	0.734822
$800$	6.79252	0.240152
$801$	0	0
$802$	−6.66723	−0.235428
$803$	58.2768	2.05654
$804$	0	0
$805$	−50.7096	−1.78728
$806$	40.2799	1.41880
$807$	0	0
$808$	14.9963	0.527567
$809$	11.1962	0.393637	0.196819	−	0.980440i	$-0.436939\pi$
$809$	11.1962	0.393637	0.196819	+	0.980440i	$0.436939\pi$
$810$	0	0
$811$	5.32077	0.186838	0.0934188	−	0.995627i	$-0.470220\pi$
$811$	5.32077	0.186838	0.0934188	+	0.995627i	$0.470220\pi$
$812$	16.2327	0.569656
$813$	0	0
$814$	−10.5527	−0.369870
$815$	−84.8515	−2.97222
$816$	0	0
$817$	22.8929	0.800922
$818$	23.1492	0.809392
$819$	0	0
$820$	17.3405	0.605557
$821$	−12.0445	−0.420355	−0.210177	−	0.977663i	$-0.567404\pi$
$821$	−12.0445	−0.420355	−0.210177	+	0.977663i	$0.567404\pi$
$822$	0	0
$823$	−15.3907	−0.536486	−0.268243	−	0.963351i	$-0.586443\pi$
$823$	−15.3907	−0.536486	−0.268243	+	0.963351i	$0.586443\pi$
$824$	−7.81447	−0.272230
$825$	0	0
$826$	11.3055	0.393370
$827$	−0.661832	−0.0230142	−0.0115071	−	0.999934i	$-0.503663\pi$
$827$	−0.661832	−0.0230142	−0.0115071	+	0.999934i	$0.503663\pi$
$828$	0	0
$829$	−14.0032	−0.486350	−0.243175	−	0.969982i	$-0.578189\pi$
$829$	−14.0032	−0.486350	−0.243175	+	0.969982i	$0.578189\pi$
$830$	23.1580	0.803827
$831$	0	0
$832$	−4.50984	−0.156350
$833$	−11.9510	−0.414076
$834$	0	0
$835$	−1.20349	−0.0416483
$836$	25.1350	0.869313
$837$	0	0
$838$	5.17742	0.178851
$839$	31.7972	1.09776	0.548881	−	0.835900i	$-0.315054\pi$
$839$	31.7972	1.09776	0.548881	+	0.835900i	$0.315054\pi$
$840$	0	0
$841$	43.6875	1.50647
$842$	24.5799	0.847081
$843$	0	0
$844$	8.91088	0.306725
$845$	25.2010	0.866942
$846$	0	0
$847$	−27.7832	−0.954642
$848$	−2.76603	−0.0949860
$849$	0	0
$850$	24.0533	0.825022
$851$	16.1783	0.554586
$852$	0	0
$853$	17.6419	0.604047	0.302024	−	0.953300i	$-0.402338\pi$
$853$	17.6419	0.604047	0.302024	+	0.953300i	$0.402338\pi$
$854$	−11.3187	−0.387319
$855$	0	0
$856$	−18.9522	−0.647772
$857$	−12.6484	−0.432061	−0.216031	−	0.976387i	$-0.569311\pi$
$857$	−12.6484	−0.432061	−0.216031	+	0.976387i	$0.569311\pi$
$858$	0	0
$859$	19.2414	0.656507	0.328254	−	0.944590i	$-0.393540\pi$
$859$	19.2414	0.656507	0.328254	+	0.944590i	$0.393540\pi$
$860$	−15.8227	−0.539548
$861$	0	0
$862$	−16.1595	−0.550394
$863$	−6.61547	−0.225193	−0.112597	−	0.993641i	$-0.535917\pi$
$863$	−6.61547	−0.225193	−0.112597	+	0.993641i	$0.535917\pi$
$864$	0	0
$865$	3.01218	0.102417
$866$	36.1699	1.22910
$867$	0	0
$868$	17.0055	0.577204
$869$	−57.7957	−1.96058
$870$	0	0
$871$	27.8096	0.942292
$872$	−5.75906	−0.195026
$873$	0	0
$874$	−38.5346	−1.30345
$875$	−11.7200	−0.396208
$876$	0	0
$877$	−7.11458	−0.240242	−0.120121	−	0.992759i	$-0.538328\pi$
$877$	−7.11458	−0.240242	−0.120121	+	0.992759i	$0.538328\pi$
$878$	−30.1285	−1.01679
$879$	0	0
$880$	−17.3723	−0.585620
$881$	−53.6060	−1.80603	−0.903016	−	0.429606i	$-0.858652\pi$
$881$	−53.6060	−1.80603	−0.903016	+	0.429606i	$0.858652\pi$
$882$	0	0
$883$	13.5553	0.456173	0.228087	−	0.973641i	$-0.426753\pi$
$883$	13.5553	0.456173	0.228087	+	0.973641i	$0.426753\pi$
$884$	−15.9700	−0.537129
$885$	0	0
$886$	−22.6806	−0.761970
$887$	28.3142	0.950699	0.475349	−	0.879797i	$-0.342322\pi$
$887$	28.3142	0.950699	0.475349	+	0.879797i	$0.342322\pi$
$888$	0	0
$889$	42.2376	1.41660
$890$	−47.8077	−1.60252
$891$	0	0
$892$	8.89504	0.297828
$893$	−29.1432	−0.975239
$894$	0	0
$895$	3.86425	0.129167
$896$	−1.90397	−0.0636073
$897$	0	0
$898$	−19.5804	−0.653406
$899$	76.1479	2.53967
$900$	0	0
$901$	−9.79494	−0.326317
$902$	−25.5454	−0.850569
$903$	0	0
$904$	−10.7590	−0.357840
$905$	−12.2626	−0.407623
$906$	0	0
$907$	38.6910	1.28471	0.642357	−	0.766406i	$-0.277957\pi$
$907$	38.6910	1.28471	0.642357	+	0.766406i	$0.277957\pi$
$908$	−12.0793	−0.400866
$909$	0	0
$910$	29.4866	0.977472
$911$	−19.6986	−0.652644	−0.326322	−	0.945259i	$-0.605809\pi$
$911$	−19.6986	−0.652644	−0.326322	+	0.945259i	$0.605809\pi$
$912$	0	0
$913$	−34.1155	−1.12906
$914$	−26.4538	−0.875015
$915$	0	0
$916$	0.111965	0.00369943
$917$	26.0078	0.858853
$918$	0	0
$919$	14.2356	0.469589	0.234795	−	0.972045i	$-0.424558\pi$
$919$	14.2356	0.469589	0.234795	+	0.972045i	$0.424558\pi$
$920$	26.6335	0.878082
$921$	0	0
$922$	37.1292	1.22279
$923$	−54.6774	−1.79973
$924$	0	0
$925$	14.1690	0.465873
$926$	−1.88633	−0.0619888
$927$	0	0
$928$	−8.52570	−0.279870
$929$	7.98727	0.262054	0.131027	−	0.991379i	$-0.458173\pi$
$929$	7.98727	0.262054	0.131027	+	0.991379i	$0.458173\pi$
$930$	0	0
$931$	16.7681	0.549552
$932$	−5.71066	−0.187059
$933$	0	0
$934$	−31.7341	−1.03837
$935$	−61.5179	−2.01185
$936$	0	0
$937$	−19.8309	−0.647847	−0.323924	−	0.946083i	$-0.605002\pi$
$937$	−19.8309	−0.647847	−0.323924	+	0.946083i	$0.605002\pi$
$938$	11.7407	0.383348
$939$	0	0
$940$	20.1426	0.656978
$941$	24.3712	0.794477	0.397239	−	0.917715i	$-0.369969\pi$
$941$	24.3712	0.794477	0.397239	+	0.917715i	$0.369969\pi$
$942$	0	0
$943$	39.1638	1.27535
$944$	−5.93786	−0.193261
$945$	0	0
$946$	23.3093	0.757852
$947$	−8.25146	−0.268136	−0.134068	−	0.990972i	$-0.542804\pi$
$947$	−8.25146	−0.268136	−0.134068	+	0.990972i	$0.542804\pi$
$948$	0	0
$949$	51.9520	1.68643
$950$	−33.7486	−1.09495
$951$	0	0
$952$	−6.74226	−0.218518
$953$	−14.4807	−0.469076	−0.234538	−	0.972107i	$-0.575358\pi$
$953$	−14.4807	−0.469076	−0.234538	+	0.972107i	$0.575358\pi$
$954$	0	0
$955$	46.9780	1.52017
$956$	21.7102	0.702158
$957$	0	0
$958$	−0.498846	−0.0161170
$959$	−17.4612	−0.563851
$960$	0	0
$961$	48.7730	1.57332
$962$	−9.40738	−0.303306
$963$	0	0
$964$	14.0658	0.453030
$965$	24.1289	0.776738
$966$	0	0
$967$	−45.9995	−1.47924	−0.739622	−	0.673023i	$-0.764996\pi$
$967$	−45.9995	−1.47924	−0.739622	+	0.673023i	$0.764996\pi$
$968$	14.5922	0.469012
$969$	0	0
$970$	−63.3207	−2.03311
$971$	−60.5426	−1.94290	−0.971452	−	0.237236i	$-0.923758\pi$
$971$	−60.5426	−1.94290	−0.971452	+	0.237236i	$0.923758\pi$
$972$	0	0
$973$	31.3892	1.00629
$974$	−17.5179	−0.561310
$975$	0	0
$976$	5.94479	0.190288
$977$	12.4286	0.397626	0.198813	−	0.980037i	$-0.436291\pi$
$977$	12.4286	0.397626	0.198813	+	0.980037i	$0.436291\pi$
$978$	0	0
$979$	70.4286	2.25091
$980$	−11.5894	−0.370211
$981$	0	0
$982$	27.8433	0.888516
$983$	17.1806	0.547977	0.273988	−	0.961733i	$-0.411657\pi$
$983$	17.1806	0.547977	0.273988	+	0.961733i	$0.411657\pi$
$984$	0	0
$985$	20.0448	0.638679
$986$	−30.1908	−0.961470
$987$	0	0
$988$	22.4071	0.712866
$989$	−35.7357	−1.13633
$990$	0	0
$991$	−20.5098	−0.651514	−0.325757	−	0.945453i	$-0.605619\pi$
$991$	−20.5098	−0.651514	−0.325757	+	0.945453i	$0.605619\pi$
$992$	−8.93157	−0.283578
$993$	0	0
$994$	−23.0839	−0.732175
$995$	−28.4574	−0.902159
$996$	0	0
$997$	21.7656	0.689323	0.344661	−	0.938727i	$-0.387994\pi$
$997$	21.7656	0.689323	0.344661	+	0.938727i	$0.387994\pi$
$998$	28.5254	0.902955
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.n.1.11	yes	12
3.2	odd	2		8046.2.a.k.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.k.1.2	✓	12	3.2	odd	2
8046.2.a.n.1.11	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.43402 q^{5} -1.90397 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.43402 q^{5} -1.90397 q^{7} +1.00000 q^{8} +3.43402 q^{10} -5.05887 q^{11} -4.50984 q^{13} -1.90397 q^{14} +1.00000 q^{16} +3.54115 q^{17} -4.96850 q^{19} +3.43402 q^{20} -5.05887 q^{22} +7.75578 q^{23} +6.79252 q^{25} -4.50984 q^{26} -1.90397 q^{28} -8.52570 q^{29} -8.93157 q^{31} +1.00000 q^{32} +3.54115 q^{34} -6.53829 q^{35} +2.08597 q^{37} -4.96850 q^{38} +3.43402 q^{40} +5.04962 q^{41} -4.60761 q^{43} -5.05887 q^{44} +7.75578 q^{46} +5.86559 q^{47} -3.37488 q^{49} +6.79252 q^{50} -4.50984 q^{52} -2.76603 q^{53} -17.3723 q^{55} -1.90397 q^{56} -8.52570 q^{58} -5.93786 q^{59} +5.94479 q^{61} -8.93157 q^{62} +1.00000 q^{64} -15.4869 q^{65} -6.16643 q^{67} +3.54115 q^{68} -6.53829 q^{70} +12.1240 q^{71} -11.5197 q^{73} +2.08597 q^{74} -4.96850 q^{76} +9.63197 q^{77} +11.4246 q^{79} +3.43402 q^{80} +5.04962 q^{82} +6.74370 q^{83} +12.1604 q^{85} -4.60761 q^{86} -5.05887 q^{88} -13.9218 q^{89} +8.58661 q^{91} +7.75578 q^{92} +5.86559 q^{94} -17.0619 q^{95} -18.4392 q^{97} -3.37488 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 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

Representations

Groups

Database

Properties

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