LMFDB - Embedded newform 6069.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6069,2,Mod(1,6069)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6069, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6069.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6069 = 3 \cdot 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6069.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:	$48.4612089867$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6069.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} -1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} +4.00000 q^{22} -3.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} +2.00000 q^{30} -5.00000 q^{32} +4.00000 q^{33} +2.00000 q^{35} -1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} +6.00000 q^{40} -2.00000 q^{41} +1.00000 q^{42} -4.00000 q^{43} +4.00000 q^{44} +2.00000 q^{45} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -8.00000 q^{55} +3.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} +2.00000 q^{60} +2.00000 q^{61} +1.00000 q^{63} +7.00000 q^{64} -4.00000 q^{65} -4.00000 q^{66} +4.00000 q^{67} -2.00000 q^{70} +3.00000 q^{72} +6.00000 q^{73} +6.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -4.00000 q^{77} -2.00000 q^{78} +16.0000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -12.0000 q^{83} +1.00000 q^{84} +4.00000 q^{86} -2.00000 q^{87} -12.0000 q^{88} -14.0000 q^{89} -2.00000 q^{90} -2.00000 q^{91} +8.00000 q^{95} +5.00000 q^{96} -18.0000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	1.00000	0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	−2.00000	−0.516398
$16$	−1.00000	−0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−2.00000	−0.447214
$21$	−1.00000	−0.218218
$22$	4.00000	0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−3.00000	−0.612372
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	2.00000	0.365148
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−5.00000	−0.883883
$33$	4.00000	0.696311
$34$	0	0
$35$	2.00000	0.338062
$36$	−1.00000	−0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−4.00000	−0.648886
$39$	2.00000	0.320256
$40$	6.00000	0.948683
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	1.00000	0.154303
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.00000	0.603023
$45$	2.00000	0.298142
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	−8.00000	−1.07872
$56$	3.00000	0.400892
$57$	−4.00000	−0.529813
$58$	−2.00000	−0.262613
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	2.00000	0.258199
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	−4.00000	−0.492366
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	−2.00000	−0.239046
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	3.00000	0.353553
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	6.00000	0.697486
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	−4.00000	−0.455842
$78$	−2.00000	−0.226455
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	4.00000	0.431331
$87$	−2.00000	−0.214423
$88$	−12.0000	−1.27920
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	−2.00000	−0.210819
$91$	−2.00000	−0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.00000	0.820783
$96$	5.00000	0.510310
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	−1.00000	−0.101015
$99$	−4.00000	−0.402015
$100$	1.00000	0.100000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−6.00000	−0.588348
$105$	−2.00000	−0.195180
$106$	−6.00000	−0.582772
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	1.00000	0.0962250
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	8.00000	0.762770
$111$	6.00000	0.569495
$112$	−1.00000	−0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−2.00000	−0.185695
$117$	−2.00000	−0.184900
$118$	−12.0000	−1.10469
$119$	0	0
$120$	−6.00000	−0.547723
$121$	5.00000	0.454545
$122$	−2.00000	−0.181071
$123$	2.00000	0.180334
$124$	0	0
$125$	−12.0000	−1.07331
$126$	−1.00000	−0.0890871
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	3.00000	0.265165
$129$	4.00000	0.352180
$130$	4.00000	0.350823
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	−4.00000	−0.348155
$133$	4.00000	0.346844
$134$	−4.00000	−0.345547
$135$	−2.00000	−0.172133
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	0	0
$143$	8.00000	0.668994
$144$	−1.00000	−0.0833333
$145$	4.00000	0.332182
$146$	−6.00000	−0.496564
$147$	−1.00000	−0.0824786
$148$	6.00000	0.493197
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−1.00000	−0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	12.0000	0.973329
$153$	0	0
$154$	4.00000	0.322329
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−16.0000	−1.27289
$159$	−6.00000	−0.475831
$160$	−10.0000	−0.790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	2.00000	0.156174
$165$	8.00000	0.622799
$166$	12.0000	0.931381
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	−3.00000	−0.231455
$169$	−9.00000	−0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	4.00000	0.304997
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	2.00000	0.151620
$175$	−1.00000	−0.0755929
$176$	4.00000	0.301511
$177$	−12.0000	−0.901975
$178$	14.0000	1.04934
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	−2.00000	−0.149071
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	2.00000	0.148250
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	−8.00000	−0.580381
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−7.00000	−0.505181
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	18.0000	1.29232
$195$	4.00000	0.286446
$196$	−1.00000	−0.0714286
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	4.00000	0.284268
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	−3.00000	−0.212132
$201$	−4.00000	−0.282138
$202$	−14.0000	−0.985037
$203$	2.00000	0.140372
$204$	0	0
$205$	−4.00000	−0.279372
$206$	−8.00000	−0.557386
$207$	0	0
$208$	2.00000	0.138675
$209$	−16.0000	−1.10674
$210$	2.00000	0.138013
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	4.00000	0.273434
$215$	−8.00000	−0.545595
$216$	−3.00000	−0.204124
$217$	0	0
$218$	−18.0000	−1.21911
$219$	−6.00000	−0.405442
$220$	8.00000	0.539360
$221$	0	0
$222$	−6.00000	−0.402694
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−5.00000	−0.334077
$225$	−1.00000	−0.0666667
$226$	−14.0000	−0.931266
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	4.00000	0.264906
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	6.00000	0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−12.0000	−0.781133
$237$	−16.0000	−1.03931
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	2.00000	0.129099
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	2.00000	0.127775
$246$	−2.00000	−0.127515
$247$	−8.00000	−0.509028
$248$	0	0
$249$	12.0000	0.760469
$250$	12.0000	0.758947
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−17.0000	−1.06250
$257$	26.0000	1.62184	0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000	1.62184	0.810918	+	0.585160i	$0.198968\pi$
$258$	−4.00000	−0.249029
$259$	−6.00000	−0.372822
$260$	4.00000	0.248069
$261$	2.00000	0.123797
$262$	4.00000	0.247121
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	12.0000	0.738549
$265$	12.0000	0.737154
$266$	−4.00000	−0.245256
$267$	14.0000	0.856786
$268$	−4.00000	−0.244339
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	2.00000	0.121716
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	6.00000	0.362473
$275$	4.00000	0.241209
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	12.0000	0.719712
$279$	0	0
$280$	6.00000	0.358569
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	−8.00000	−0.473050
$287$	−2.00000	−0.118056
$288$	−5.00000	−0.294628
$289$	0	0
$290$	−4.00000	−0.234888
$291$	18.0000	1.05518
$292$	−6.00000	−0.351123
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	1.00000	0.0583212
$295$	24.0000	1.39733
$296$	−18.0000	−1.04623
$297$	4.00000	0.232104
$298$	−6.00000	−0.347571
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−4.00000	−0.230556
$302$	−8.00000	−0.460348
$303$	−14.0000	−0.804279
$304$	−4.00000	−0.229416
$305$	4.00000	0.229039
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	4.00000	0.227921
$309$	−8.00000	−0.455104
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	6.00000	0.339683
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	2.00000	0.112867
$315$	2.00000	0.112687
$316$	−16.0000	−0.900070
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	6.00000	0.336463
$319$	−8.00000	−0.447914
$320$	14.0000	0.782624
$321$	4.00000	0.223258
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	2.00000	0.110940
$326$	4.00000	0.221540
$327$	−18.0000	−0.995402
$328$	−6.00000	−0.331295
$329$	0	0
$330$	−8.00000	−0.440386
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	12.0000	0.658586
$333$	−6.00000	−0.328798
$334$	−8.00000	−0.437741
$335$	8.00000	0.437087
$336$	1.00000	0.0545545
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	9.00000	0.489535
$339$	−14.0000	−0.760376
$340$	0	0
$341$	0	0
$342$	−4.00000	−0.216295
$343$	1.00000	0.0539949
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−10.0000	−0.537603
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	2.00000	0.107211
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	1.00000	0.0534522
$351$	2.00000	0.106752
$352$	20.0000	1.06600
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	14.0000	0.741999
$357$	0	0
$358$	4.00000	0.211407
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	6.00000	0.316228
$361$	−3.00000	−0.157895
$362$	−26.0000	−1.36653
$363$	−5.00000	−0.262432
$364$	2.00000	0.104828
$365$	12.0000	0.628109
$366$	2.00000	0.104542
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	12.0000	0.623850
$371$	6.00000	0.311504
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	−4.00000	−0.206010
$378$	1.00000	0.0514344
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	−8.00000	−0.410391
$381$	0	0
$382$	8.00000	0.409316
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−3.00000	−0.153093
$385$	−8.00000	−0.407718
$386$	2.00000	0.101797
$387$	−4.00000	−0.203331
$388$	18.0000	0.913812
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−4.00000	−0.202548
$391$	0	0
$392$	3.00000	0.151523
$393$	4.00000	0.201773
$394$	22.0000	1.10834
$395$	32.0000	1.61009
$396$	4.00000	0.201008
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	24.0000	1.20301
$399$	−4.00000	−0.200250
$400$	1.00000	0.0500000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	−14.0000	−0.696526
$405$	2.00000	0.0993808
$406$	−2.00000	−0.0992583
$407$	24.0000	1.18964
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	4.00000	0.197546
$411$	6.00000	0.295958
$412$	−8.00000	−0.394132
$413$	12.0000	0.590481
$414$	0	0
$415$	−24.0000	−1.17811
$416$	10.0000	0.490290
$417$	12.0000	0.587643
$418$	16.0000	0.782586
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	2.00000	0.0975900
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	18.0000	0.874157
$425$	0	0
$426$	0	0
$427$	2.00000	0.0967868
$428$	4.00000	0.193347
$429$	−8.00000	−0.386244
$430$	8.00000	0.385794
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	1.00000	0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	−18.0000	−0.862044
$437$	0	0
$438$	6.00000	0.286691
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	−24.0000	−1.14416
$441$	1.00000	0.0476190
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	−6.00000	−0.284747
$445$	−28.0000	−1.32733
$446$	−16.0000	−0.757622
$447$	−6.00000	−0.283790
$448$	7.00000	0.330719
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	1.00000	0.0471405
$451$	8.00000	0.376705
$452$	−14.0000	−0.658505
$453$	−8.00000	−0.375873
$454$	−12.0000	−0.563188
$455$	−4.00000	−0.187523
$456$	−12.0000	−0.561951
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	0	0
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	−4.00000	−0.186097
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−6.00000	−0.277945
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	2.00000	0.0924500
$469$	4.00000	0.184703
$470$	0	0
$471$	2.00000	0.0921551
$472$	36.0000	1.65703
$473$	16.0000	0.735681
$474$	16.0000	0.734904
$475$	−4.00000	−0.183533
$476$	0	0
$477$	6.00000	0.274721
$478$	−24.0000	−1.09773
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	10.0000	0.456435
$481$	12.0000	0.547153
$482$	2.00000	0.0910975
$483$	0	0
$484$	−5.00000	−0.227273
$485$	−36.0000	−1.63468
$486$	1.00000	0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	6.00000	0.271607
$489$	4.00000	0.180886
$490$	−2.00000	−0.0903508
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−2.00000	−0.0901670
$493$	0	0
$494$	8.00000	0.359937
$495$	−8.00000	−0.359573
$496$	0	0
$497$	0	0
$498$	−12.0000	−0.537733
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	12.0000	0.536656
$501$	−8.00000	−0.357414
$502$	20.0000	0.892644
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	3.00000	0.133631
$505$	28.0000	1.24598
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	11.0000	0.486136
$513$	−4.00000	−0.176604
$514$	−26.0000	−1.14681
$515$	16.0000	0.705044
$516$	−4.00000	−0.176090
$517$	0	0
$518$	6.00000	0.263625
$519$	−10.0000	−0.438951
$520$	−12.0000	−0.526235
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	−2.00000	−0.0875376
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	4.00000	0.174741
$525$	1.00000	0.0436436
$526$	−16.0000	−0.697633
$527$	0	0
$528$	−4.00000	−0.174078
$529$	−23.0000	−1.00000
$530$	−12.0000	−0.521247
$531$	12.0000	0.520756
$532$	−4.00000	−0.173422
$533$	4.00000	0.173259
$534$	−14.0000	−0.605839
$535$	−8.00000	−0.345870
$536$	12.0000	0.518321
$537$	4.00000	0.172613
$538$	6.00000	0.258678
$539$	−4.00000	−0.172292
$540$	2.00000	0.0860663
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	−16.0000	−0.687259
$543$	−26.0000	−1.11577
$544$	0	0
$545$	36.0000	1.54207
$546$	−2.00000	−0.0855921
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	6.00000	0.256307
$549$	2.00000	0.0853579
$550$	−4.00000	−0.170561
$551$	8.00000	0.340811
$552$	0	0
$553$	16.0000	0.680389
$554$	22.0000	0.934690
$555$	12.0000	0.509372
$556$	12.0000	0.508913
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	22.0000	0.928014
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	28.0000	1.17797
$566$	−20.0000	−0.840663
$567$	1.00000	0.0419961
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	8.00000	0.335083
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−8.00000	−0.334497
$573$	8.00000	0.334205
$574$	2.00000	0.0834784
$575$	0	0
$576$	7.00000	0.291667
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	0	0
$579$	2.00000	0.0831172
$580$	−4.00000	−0.166091
$581$	−12.0000	−0.497844
$582$	−18.0000	−0.746124
$583$	−24.0000	−0.993978
$584$	18.0000	0.744845
$585$	−4.00000	−0.165380
$586$	−14.0000	−0.578335
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	1.00000	0.0412393
$589$	0	0
$590$	−24.0000	−0.988064
$591$	22.0000	0.904959
$592$	6.00000	0.246598
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−6.00000	−0.245770
$597$	24.0000	0.982255
$598$	0	0
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	3.00000	0.122474
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	4.00000	0.163028
$603$	4.00000	0.162893
$604$	−8.00000	−0.325515
$605$	10.0000	0.406558
$606$	14.0000	0.568711
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	−20.0000	−0.811107
$609$	−2.00000	−0.0810441
$610$	−4.00000	−0.161955
$611$	0	0
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	−4.00000	−0.161427
$615$	4.00000	0.161296
$616$	−12.0000	−0.483494
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	8.00000	0.321807
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	0	0
$622$	−24.0000	−0.962312
$623$	−14.0000	−0.560898
$624$	−2.00000	−0.0800641
$625$	−19.0000	−0.760000
$626$	26.0000	1.03917
$627$	16.0000	0.638978
$628$	2.00000	0.0798087
$629$	0	0
$630$	−2.00000	−0.0796819
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	48.0000	1.90934
$633$	4.00000	0.158986
$634$	−18.0000	−0.714871
$635$	0	0
$636$	6.00000	0.237915
$637$	−2.00000	−0.0792429
$638$	8.00000	0.316723
$639$	0	0
$640$	6.00000	0.237171
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	−4.00000	−0.157867
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−40.0000	−1.57256	−0.786281	−	0.617869i	$-0.787996\pi$
$647$	−40.0000	−1.57256	−0.786281	+	0.617869i	$0.787996\pi$
$648$	3.00000	0.117851
$649$	−48.0000	−1.88416
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	4.00000	0.156652
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	18.0000	0.703856
$655$	−8.00000	−0.312586
$656$	2.00000	0.0780869
$657$	6.00000	0.234082
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	−8.00000	−0.311400
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−36.0000	−1.39707
$665$	8.00000	0.310227
$666$	6.00000	0.232495
$667$	0	0
$668$	−8.00000	−0.309529
$669$	−16.0000	−0.618596
$670$	−8.00000	−0.309067
$671$	−8.00000	−0.308837
$672$	5.00000	0.192879
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	−14.0000	−0.539260
$675$	1.00000	0.0384900
$676$	9.00000	0.346154
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	14.0000	0.537667
$679$	−18.0000	−0.690777
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	−4.00000	−0.152944
$685$	−12.0000	−0.458496
$686$	−1.00000	−0.0381802
$687$	10.0000	0.381524
$688$	4.00000	0.152499
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−10.0000	−0.380143
$693$	−4.00000	−0.151947
$694$	−28.0000	−1.06287
$695$	−24.0000	−0.910372
$696$	−6.00000	−0.227429
$697$	0	0
$698$	2.00000	0.0757011
$699$	−6.00000	−0.226941
$700$	1.00000	0.0377964
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	−2.00000	−0.0754851
$703$	−24.0000	−0.905177
$704$	−28.0000	−1.05529
$705$	0	0
$706$	−10.0000	−0.376355
$707$	14.0000	0.526524
$708$	12.0000	0.450988
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	−42.0000	−1.57402
$713$	0	0
$714$	0	0
$715$	16.0000	0.598366
$716$	4.00000	0.149487
$717$	−24.0000	−0.896296
$718$	−32.0000	−1.19423
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	−2.00000	−0.0745356
$721$	8.00000	0.297936
$722$	3.00000	0.111648
$723$	2.00000	0.0743808
$724$	−26.0000	−0.966282
$725$	−2.00000	−0.0742781
$726$	5.00000	0.185567
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	−12.0000	−0.444140
$731$	0	0
$732$	2.00000	0.0739221
$733$	−18.0000	−0.664845	−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000	−0.664845	−0.332423	+	0.943131i	$0.607866\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	−16.0000	−0.589368
$738$	2.00000	0.0736210
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	12.0000	0.441129
$741$	8.00000	0.293887
$742$	−6.00000	−0.220267
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	10.0000	0.366126
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−4.00000	−0.146157
$750$	−12.0000	−0.438178
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	4.00000	0.145671
$755$	16.0000	0.582300
$756$	1.00000	0.0363696
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	12.0000	0.435860
$759$	0	0
$760$	24.0000	0.870572
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	18.0000	0.651644
$764$	8.00000	0.289430
$765$	0	0
$766$	0	0
$767$	−24.0000	−0.866590
$768$	17.0000	0.613435
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	8.00000	0.288300
$771$	−26.0000	−0.936367
$772$	2.00000	0.0719816
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−54.0000	−1.93849
$777$	6.00000	0.215249
$778$	−6.00000	−0.215110
$779$	−8.00000	−0.286630
$780$	−4.00000	−0.143223
$781$	0	0
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	−1.00000	−0.0357143
$785$	−4.00000	−0.142766
$786$	−4.00000	−0.142675
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	22.0000	0.783718
$789$	−16.0000	−0.569615
$790$	−32.0000	−1.13851
$791$	14.0000	0.497783
$792$	−12.0000	−0.426401
$793$	−4.00000	−0.142044
$794$	−18.0000	−0.638796
$795$	−12.0000	−0.425596
$796$	24.0000	0.850657
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	4.00000	0.141598
$799$	0	0
$800$	5.00000	0.176777
$801$	−14.0000	−0.494666
$802$	−30.0000	−1.05934
$803$	−24.0000	−0.846942
$804$	4.00000	0.141069
$805$	0	0
$806$	0	0
$807$	6.00000	0.211210
$808$	42.0000	1.47755
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	−2.00000	−0.0702728
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	−2.00000	−0.0701862
$813$	−16.0000	−0.561144
$814$	−24.0000	−0.841200
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−16.0000	−0.559769
$818$	22.0000	0.769212
$819$	−2.00000	−0.0698857
$820$	4.00000	0.139686
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	−6.00000	−0.209274
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	24.0000	0.836080
$825$	−4.00000	−0.139262
$826$	−12.0000	−0.417533
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	24.0000	0.833052
$831$	22.0000	0.763172
$832$	−14.0000	−0.485363
$833$	0	0
$834$	−12.0000	−0.415526
$835$	16.0000	0.553703
$836$	16.0000	0.553372
$837$	0	0
$838$	−12.0000	−0.414533
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	−6.00000	−0.207020
$841$	−25.0000	−0.862069
$842$	−38.0000	−1.30957
$843$	22.0000	0.757720
$844$	4.00000	0.137686
$845$	−18.0000	−0.619219
$846$	0	0
$847$	5.00000	0.171802
$848$	−6.00000	−0.206041
$849$	−20.0000	−0.686398
$850$	0	0
$851$	0	0
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	−2.00000	−0.0684386
$855$	8.00000	0.273594
$856$	−12.0000	−0.410152
$857$	14.0000	0.478231	0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000	0.478231	0.239115	+	0.970991i	$0.423143\pi$
$858$	8.00000	0.273115
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	8.00000	0.272798
$861$	2.00000	0.0681598
$862$	−24.0000	−0.817443
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	5.00000	0.170103
$865$	20.0000	0.680020
$866$	14.0000	0.475739
$867$	0	0
$868$	0	0
$869$	−64.0000	−2.17105
$870$	4.00000	0.135613
$871$	−8.00000	−0.271070
$872$	54.0000	1.82867
$873$	−18.0000	−0.609208
$874$	0	0
$875$	−12.0000	−0.405674
$876$	6.00000	0.202721
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	−24.0000	−0.809961
$879$	−14.0000	−0.472208
$880$	8.00000	0.269680
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	−1.00000	−0.0336718
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	0	0
$885$	−24.0000	−0.806751
$886$	−36.0000	−1.20944
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	18.0000	0.604040
$889$	0	0
$890$	28.0000	0.938562
$891$	−4.00000	−0.134005
$892$	−16.0000	−0.535720
$893$	0	0
$894$	6.00000	0.200670
$895$	−8.00000	−0.267411
$896$	3.00000	0.100223
$897$	0	0
$898$	−30.0000	−1.00111
$899$	0	0
$900$	1.00000	0.0333333
$901$	0	0
$902$	−8.00000	−0.266371
$903$	4.00000	0.133112
$904$	42.0000	1.39690
$905$	52.0000	1.72854
$906$	8.00000	0.265782
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	−12.0000	−0.398234
$909$	14.0000	0.464351
$910$	4.00000	0.132599
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	4.00000	0.132453
$913$	48.0000	1.58857
$914$	−10.0000	−0.330771
$915$	−4.00000	−0.132236
$916$	10.0000	0.330409
$917$	−4.00000	−0.132092
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	10.0000	0.329332
$923$	0	0
$924$	−4.00000	−0.131590
$925$	6.00000	0.197279
$926$	−16.0000	−0.525793
$927$	8.00000	0.262754
$928$	−10.0000	−0.328266
$929$	−26.0000	−0.853032	−0.426516	−	0.904480i	$-0.640259\pi$
$929$	−26.0000	−0.853032	−0.426516	+	0.904480i	$0.640259\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	−6.00000	−0.196537
$933$	−24.0000	−0.785725
$934$	−36.0000	−1.17796
$935$	0	0
$936$	−6.00000	−0.196116
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	−4.00000	−0.130605
$939$	26.0000	0.848478
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	−2.00000	−0.0651635
$943$	0	0
$944$	−12.0000	−0.390567
$945$	−2.00000	−0.0650600
$946$	−16.0000	−0.520205
$947$	−44.0000	−1.42981	−0.714904	−	0.699223i	$-0.753530\pi$
$947$	−44.0000	−1.42981	−0.714904	+	0.699223i	$0.753530\pi$
$948$	16.0000	0.519656
$949$	−12.0000	−0.389536
$950$	4.00000	0.129777
$951$	−18.0000	−0.583690
$952$	0	0
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	−6.00000	−0.194257
$955$	−16.0000	−0.517748
$956$	−24.0000	−0.776215
$957$	8.00000	0.258603
$958$	−16.0000	−0.516937
$959$	−6.00000	−0.193750
$960$	−14.0000	−0.451848
$961$	−31.0000	−1.00000
$962$	−12.0000	−0.386896
$963$	−4.00000	−0.128898
$964$	2.00000	0.0644157
$965$	−4.00000	−0.128765
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	15.0000	0.482118
$969$	0	0
$970$	36.0000	1.15589
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000	0.0320750
$973$	−12.0000	−0.384702
$974$	−8.00000	−0.256337
$975$	−2.00000	−0.0640513
$976$	−2.00000	−0.0640184
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−4.00000	−0.127906
$979$	56.0000	1.78977
$980$	−2.00000	−0.0638877
$981$	18.0000	0.574696
$982$	−20.0000	−0.638226
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	6.00000	0.191273
$985$	−44.0000	−1.40196
$986$	0	0
$987$	0	0
$988$	8.00000	0.254514
$989$	0	0
$990$	8.00000	0.254257
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	4.00000	0.126936
$994$	0	0
$995$	−48.0000	−1.52170
$996$	−12.0000	−0.380235
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	4.00000	0.126618
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6069.2.a.b.1.1		1
17.16	even	2		21.2.a.a.1.1	✓	1
51.50	odd	2		63.2.a.a.1.1		1
68.67	odd	2		336.2.a.a.1.1		1
85.33	odd	4		525.2.d.a.274.2		2
85.67	odd	4		525.2.d.a.274.1		2
85.84	even	2		525.2.a.d.1.1		1
119.16	even	6		147.2.e.b.67.1		2
119.33	odd	6		147.2.e.c.67.1		2
119.67	even	6		147.2.e.b.79.1		2
119.101	odd	6		147.2.e.c.79.1		2
119.118	odd	2		147.2.a.a.1.1		1
136.67	odd	2		1344.2.a.s.1.1		1
136.101	even	2		1344.2.a.g.1.1		1
153.16	even	6		567.2.f.g.190.1		2
153.50	odd	6		567.2.f.b.379.1		2
153.67	even	6		567.2.f.g.379.1		2
153.101	odd	6		567.2.f.b.190.1		2
187.186	odd	2		2541.2.a.j.1.1		1
204.203	even	2		1008.2.a.l.1.1		1
221.220	even	2		3549.2.a.c.1.1		1
255.152	even	4		1575.2.d.a.1324.2		2
255.203	even	4		1575.2.d.a.1324.1		2
255.254	odd	2		1575.2.a.c.1.1		1
272.67	odd	4		5376.2.c.l.2689.1		2
272.101	even	4		5376.2.c.r.2689.1		2
272.203	odd	4		5376.2.c.l.2689.2		2
272.237	even	4		5376.2.c.r.2689.2		2
323.322	odd	2		7581.2.a.d.1.1		1
340.339	odd	2		8400.2.a.bn.1.1		1
357.101	even	6		441.2.e.b.226.1		2
357.152	even	6		441.2.e.b.361.1		2
357.254	odd	6		441.2.e.a.361.1		2
357.305	odd	6		441.2.e.a.226.1		2
357.356	even	2		441.2.a.f.1.1		1
408.101	odd	2		4032.2.a.h.1.1		1
408.203	even	2		4032.2.a.k.1.1		1
476.67	odd	6		2352.2.q.x.961.1		2
476.135	odd	6		2352.2.q.x.1537.1		2
476.271	even	6		2352.2.q.e.1537.1		2
476.339	even	6		2352.2.q.e.961.1		2
476.475	even	2		2352.2.a.v.1.1		1
561.560	even	2		7623.2.a.g.1.1		1
595.594	odd	2		3675.2.a.n.1.1		1
952.237	odd	2		9408.2.a.bv.1.1		1
952.475	even	2		9408.2.a.m.1.1		1
1428.1427	odd	2		7056.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.2.a.a.1.1	✓	1	17.16	even	2
63.2.a.a.1.1		1	51.50	odd	2
147.2.a.a.1.1		1	119.118	odd	2
147.2.e.b.67.1		2	119.16	even	6
147.2.e.b.79.1		2	119.67	even	6
147.2.e.c.67.1		2	119.33	odd	6
147.2.e.c.79.1		2	119.101	odd	6
336.2.a.a.1.1		1	68.67	odd	2
441.2.a.f.1.1		1	357.356	even	2
441.2.e.a.226.1		2	357.305	odd	6
441.2.e.a.361.1		2	357.254	odd	6
441.2.e.b.226.1		2	357.101	even	6
441.2.e.b.361.1		2	357.152	even	6
525.2.a.d.1.1		1	85.84	even	2
525.2.d.a.274.1		2	85.67	odd	4
525.2.d.a.274.2		2	85.33	odd	4
567.2.f.b.190.1		2	153.101	odd	6
567.2.f.b.379.1		2	153.50	odd	6
567.2.f.g.190.1		2	153.16	even	6
567.2.f.g.379.1		2	153.67	even	6
1008.2.a.l.1.1		1	204.203	even	2
1344.2.a.g.1.1		1	136.101	even	2
1344.2.a.s.1.1		1	136.67	odd	2
1575.2.a.c.1.1		1	255.254	odd	2
1575.2.d.a.1324.1		2	255.203	even	4
1575.2.d.a.1324.2		2	255.152	even	4
2352.2.a.v.1.1		1	476.475	even	2
2352.2.q.e.961.1		2	476.339	even	6
2352.2.q.e.1537.1		2	476.271	even	6
2352.2.q.x.961.1		2	476.67	odd	6
2352.2.q.x.1537.1		2	476.135	odd	6
2541.2.a.j.1.1		1	187.186	odd	2
3549.2.a.c.1.1		1	221.220	even	2
3675.2.a.n.1.1		1	595.594	odd	2
4032.2.a.h.1.1		1	408.101	odd	2
4032.2.a.k.1.1		1	408.203	even	2
5376.2.c.l.2689.1		2	272.67	odd	4
5376.2.c.l.2689.2		2	272.203	odd	4
5376.2.c.r.2689.1		2	272.101	even	4
5376.2.c.r.2689.2		2	272.237	even	4
6069.2.a.b.1.1		1	1.1	even	1	trivial
7056.2.a.p.1.1		1	1428.1427	odd	2
7581.2.a.d.1.1		1	323.322	odd	2
7623.2.a.g.1.1		1	561.560	even	2
8400.2.a.bn.1.1		1	340.339	odd	2
9408.2.a.m.1.1		1	952.475	even	2
9408.2.a.bv.1.1		1	952.237	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6069 = 3 \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6069.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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