LMFDB - Embedded newform 8085.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8085,2,Mod(1,8085)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8085, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8085.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8085.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:	$64.5590500342$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8085.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{15} -1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} +8.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +6.00000 q^{29} +1.00000 q^{30} -8.00000 q^{31} -5.00000 q^{32} +1.00000 q^{33} -6.00000 q^{34} -1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -3.00000 q^{40} +6.00000 q^{41} -4.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} -8.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} -1.00000 q^{55} +4.00000 q^{57} -6.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} +10.0000 q^{61} +8.00000 q^{62} +7.00000 q^{64} -2.00000 q^{65} -1.00000 q^{66} -12.0000 q^{67} -6.00000 q^{68} +8.00000 q^{69} -8.00000 q^{71} +3.00000 q^{72} +6.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} -6.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} +3.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} -8.00000 q^{92} -8.00000 q^{93} -8.00000 q^{94} -4.00000 q^{95} -5.00000 q^{96} +14.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	−1.00000	−0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−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	$0.348268\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	3.00000	0.612372
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	1.00000	0.182574
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−5.00000	−0.883883
$33$	1.00000	0.174078
$34$	−6.00000	−1.02899
$35$	0	0
$36$	−1.00000	−0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.00000	−0.648886
$39$	2.00000	0.320256
$40$	−3.00000	−0.474342
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−1.00000	−0.150756
$45$	−1.00000	−0.149071
$46$	−8.00000	−1.17954
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−1.00000	−0.141421
$51$	6.00000	0.840168
$52$	−2.00000	−0.277350
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	−1.00000	−0.136083
$55$	−1.00000	−0.134840
$56$	0	0
$57$	4.00000	0.529813
$58$	−6.00000	−0.787839
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	1.00000	0.129099
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	7.00000	0.875000
$65$	−2.00000	−0.248069
$66$	−1.00000	−0.123091
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−6.00000	−0.727607
$69$	8.00000	0.963087
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	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
$77$	0	0
$78$	−2.00000	−0.226455
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	4.00000	0.431331
$87$	6.00000	0.643268
$88$	3.00000	0.319801
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−8.00000	−0.834058
$93$	−8.00000	−0.829561
$94$	−8.00000	−0.825137
$95$	−4.00000	−0.410391
$96$	−5.00000	−0.510310
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	−1.00000	−0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	−6.00000	−0.594089
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	10.0000	0.971286
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	1.00000	0.0953463
$111$	6.00000	0.569495
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−4.00000	−0.374634
$115$	−8.00000	−0.746004
$116$	−6.00000	−0.557086
$117$	2.00000	0.184900
$118$	−12.0000	−1.10469
$119$	0	0
$120$	−3.00000	−0.273861
$121$	1.00000	0.0909091
$122$	−10.0000	−0.905357
$123$	6.00000	0.541002
$124$	8.00000	0.718421
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	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
$130$	2.00000	0.175412
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	12.0000	1.03664
$135$	−1.00000	−0.0860663
$136$	18.0000	1.54349
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	−8.00000	−0.681005
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	8.00000	0.671345
$143$	2.00000	0.167248
$144$	−1.00000	−0.0833333
$145$	−6.00000	−0.498273
$146$	−6.00000	−0.496564
$147$	0	0
$148$	−6.00000	−0.493197
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	−1.00000	−0.0816497
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	12.0000	0.973329
$153$	6.00000	0.485071
$154$	0	0
$155$	8.00000	0.642575
$156$	−2.00000	−0.160128
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−8.00000	−0.636446
$159$	−10.0000	−0.793052
$160$	5.00000	0.395285
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−6.00000	−0.468521
$165$	−1.00000	−0.0778499
$166$	−4.00000	−0.310460
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	6.00000	0.460179
$171$	4.00000	0.305888
$172$	4.00000	0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	12.0000	0.901975
$178$	10.0000	0.749532
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	1.00000	0.0745356
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	24.0000	1.76930
$185$	−6.00000	−0.441129
$186$	8.00000	0.586588
$187$	6.00000	0.438763
$188$	−8.00000	−0.583460
$189$	0	0
$190$	4.00000	0.290191
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	7.00000	0.505181
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	−14.0000	−1.00514
$195$	−2.00000	−0.143223
$196$	0	0
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	−1.00000	−0.0710669
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	3.00000	0.212132
$201$	−12.0000	−0.846415
$202$	6.00000	0.422159
$203$	0	0
$204$	−6.00000	−0.420084
$205$	−6.00000	−0.419058
$206$	16.0000	1.11477
$207$	8.00000	0.556038
$208$	−2.00000	−0.138675
$209$	4.00000	0.276686
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	10.0000	0.686803
$213$	−8.00000	−0.548151
$214$	12.0000	0.820303
$215$	4.00000	0.272798
$216$	3.00000	0.204124
$217$	0	0
$218$	2.00000	0.135457
$219$	6.00000	0.405442
$220$	1.00000	0.0674200
$221$	12.0000	0.807207
$222$	−6.00000	−0.402694
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	14.0000	0.931266
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−4.00000	−0.264906
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	18.0000	1.18176
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	−2.00000	−0.130744
$235$	−8.00000	−0.521862
$236$	−12.0000	−0.781133
$237$	8.00000	0.519656
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	1.00000	0.0645497
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	0	0
$246$	−6.00000	−0.382546
$247$	8.00000	0.509028
$248$	−24.0000	−1.52400
$249$	4.00000	0.253490
$250$	1.00000	0.0632456
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	0	0
$255$	−6.00000	−0.375735
$256$	−17.0000	−1.06250
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	2.00000	0.124035
$261$	6.00000	0.371391
$262$	−12.0000	−0.741362
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	3.00000	0.184637
$265$	10.0000	0.614295
$266$	0	0
$267$	−10.0000	−0.611990
$268$	12.0000	0.733017
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	1.00000	0.0608581
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	22.0000	1.32907
$275$	1.00000	0.0603023
$276$	−8.00000	−0.481543
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	4.00000	0.239904
$279$	−8.00000	−0.478947
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	−8.00000	−0.476393
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	8.00000	0.474713
$285$	−4.00000	−0.236940
$286$	−2.00000	−0.118262
$287$	0	0
$288$	−5.00000	−0.294628
$289$	19.0000	1.11765
$290$	6.00000	0.352332
$291$	14.0000	0.820695
$292$	−6.00000	−0.351123
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	18.0000	1.04623
$297$	1.00000	0.0580259
$298$	18.0000	1.04271
$299$	16.0000	0.925304
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	−16.0000	−0.920697
$303$	−6.00000	−0.344691
$304$	−4.00000	−0.229416
$305$	−10.0000	−0.572598
$306$	−6.00000	−0.342997
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	−8.00000	−0.454369
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	6.00000	0.339683
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−34.0000	−1.90963	−0.954815	−	0.297200i	$-0.903947\pi$
$317$	−34.0000	−1.90963	−0.954815	+	0.297200i	$0.903947\pi$
$318$	10.0000	0.560772
$319$	6.00000	0.335936
$320$	−7.00000	−0.391312
$321$	−12.0000	−0.669775
$322$	0	0
$323$	24.0000	1.33540
$324$	−1.00000	−0.0555556
$325$	2.00000	0.110940
$326$	−20.0000	−1.10770
$327$	−2.00000	−0.110600
$328$	18.0000	0.993884
$329$	0	0
$330$	1.00000	0.0550482
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	−4.00000	−0.219529
$333$	6.00000	0.328798
$334$	−16.0000	−0.875481
$335$	12.0000	0.655630
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	9.00000	0.489535
$339$	−14.0000	−0.760376
$340$	6.00000	0.325396
$341$	−8.00000	−0.433224
$342$	−4.00000	−0.216295
$343$	0	0
$344$	−12.0000	−0.646997
$345$	−8.00000	−0.430706
$346$	6.00000	0.322562
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	−6.00000	−0.321634
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	−5.00000	−0.266501
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	−12.0000	−0.637793
$355$	8.00000	0.424596
$356$	10.0000	0.529999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	−3.00000	−0.158114
$361$	−3.00000	−0.157895
$362$	6.00000	0.315353
$363$	1.00000	0.0524864
$364$	0	0
$365$	−6.00000	−0.314054
$366$	−10.0000	−0.522708
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−8.00000	−0.417029
$369$	6.00000	0.312348
$370$	6.00000	0.311925
$371$	0	0
$372$	8.00000	0.414781
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−6.00000	−0.310253
$375$	−1.00000	−0.0516398
$376$	24.0000	1.23771
$377$	12.0000	0.618031
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	−16.0000	−0.818631
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	22.0000	1.11977
$387$	−4.00000	−0.203331
$388$	−14.0000	−0.710742
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	2.00000	0.101274
$391$	48.0000	2.42746
$392$	0	0
$393$	12.0000	0.605320
$394$	−22.0000	−1.10834
$395$	−8.00000	−0.402524
$396$	−1.00000	−0.0502519
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	12.0000	0.598506
$403$	−16.0000	−0.797017
$404$	6.00000	0.298511
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	6.00000	0.297409
$408$	18.0000	0.891133
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	6.00000	0.296319
$411$	−22.0000	−1.08518
$412$	16.0000	0.788263
$413$	0	0
$414$	−8.00000	−0.393179
$415$	−4.00000	−0.196352
$416$	−10.0000	−0.490290
$417$	−4.00000	−0.195881
$418$	−4.00000	−0.195646
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	4.00000	0.194717
$423$	8.00000	0.388973
$424$	−30.0000	−1.45693
$425$	6.00000	0.291043
$426$	8.00000	0.387601
$427$	0	0
$428$	12.0000	0.580042
$429$	2.00000	0.0965609
$430$	−4.00000	−0.192897
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	−1.00000	−0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	2.00000	0.0957826
$437$	32.0000	1.53077
$438$	−6.00000	−0.286691
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	−3.00000	−0.143019
$441$	0	0
$442$	−12.0000	−0.570782
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−6.00000	−0.284747
$445$	10.0000	0.474045
$446$	−8.00000	−0.378811
$447$	−18.0000	−0.851371
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	−1.00000	−0.0471405
$451$	6.00000	0.282529
$452$	14.0000	0.658505
$453$	16.0000	0.751746
$454$	12.0000	0.563188
$455$	0	0
$456$	12.0000	0.561951
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−10.0000	−0.467269
$459$	6.00000	0.280056
$460$	8.00000	0.373002
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−6.00000	−0.278543
$465$	8.00000	0.370991
$466$	22.0000	1.01913
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	8.00000	0.369012
$471$	2.00000	0.0921551
$472$	36.0000	1.65703
$473$	−4.00000	−0.183920
$474$	−8.00000	−0.367452
$475$	4.00000	0.183533
$476$	0	0
$477$	−10.0000	−0.457869
$478$	16.0000	0.731823
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	5.00000	0.228218
$481$	12.0000	0.547153
$482$	−22.0000	−1.00207
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	−14.0000	−0.635707
$486$	−1.00000	−0.0453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	30.0000	1.35804
$489$	20.0000	0.904431
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	−6.00000	−0.270501
$493$	36.0000	1.62136
$494$	−8.00000	−0.359937
$495$	−1.00000	−0.0449467
$496$	8.00000	0.359211
$497$	0	0
$498$	−4.00000	−0.179244
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	1.00000	0.0447214
$501$	16.0000	0.714827
$502$	20.0000	0.892644
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	−8.00000	−0.355643
$507$	−9.00000	−0.399704
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	6.00000	0.265684
$511$	0	0
$512$	11.0000	0.486136
$513$	4.00000	0.176604
$514$	−30.0000	−1.32324
$515$	16.0000	0.705044
$516$	4.00000	0.176090
$517$	8.00000	0.351840
$518$	0	0
$519$	−6.00000	−0.263371
$520$	−6.00000	−0.263117
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	−6.00000	−0.262613
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−16.0000	−0.697633
$527$	−48.0000	−2.09091
$528$	−1.00000	−0.0435194
$529$	41.0000	1.78261
$530$	−10.0000	−0.434372
$531$	12.0000	0.520756
$532$	0	0
$533$	12.0000	0.519778
$534$	10.0000	0.432742
$535$	12.0000	0.518805
$536$	−36.0000	−1.55496
$537$	4.00000	0.172613
$538$	14.0000	0.603583
$539$	0	0
$540$	1.00000	0.0430331
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	8.00000	0.343629
$543$	−6.00000	−0.257485
$544$	−30.0000	−1.28624
$545$	2.00000	0.0856706
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	22.0000	0.939793
$549$	10.0000	0.426790
$550$	−1.00000	−0.0426401
$551$	24.0000	1.02243
$552$	24.0000	1.02151
$553$	0	0
$554$	18.0000	0.764747
$555$	−6.00000	−0.254686
$556$	4.00000	0.169638
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	8.00000	0.338667
$559$	−8.00000	−0.338364
$560$	0	0
$561$	6.00000	0.253320
$562$	−18.0000	−0.759284
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	−8.00000	−0.336861
$565$	14.0000	0.588984
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−24.0000	−1.00702
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	4.00000	0.167542
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−2.00000	−0.0836242
$573$	16.0000	0.668410
$574$	0	0
$575$	8.00000	0.333623
$576$	7.00000	0.291667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	−19.0000	−0.790296
$579$	−22.0000	−0.914289
$580$	6.00000	0.249136
$581$	0	0
$582$	−14.0000	−0.580319
$583$	−10.0000	−0.414158
$584$	18.0000	0.744845
$585$	−2.00000	−0.0826898
$586$	−18.0000	−0.743573
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	12.0000	0.494032
$591$	22.0000	0.904959
$592$	−6.00000	−0.246598
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	18.0000	0.737309
$597$	−16.0000	−0.654836
$598$	−16.0000	−0.654289
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	3.00000	0.122474
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	−16.0000	−0.651031
$605$	−1.00000	−0.0406558
$606$	6.00000	0.243733
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	−20.0000	−0.811107
$609$	0	0
$610$	10.0000	0.404888
$611$	16.0000	0.647291
$612$	−6.00000	−0.242536
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	4.00000	0.161427
$615$	−6.00000	−0.241943
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	16.0000	0.643614
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	−8.00000	−0.321288
$621$	8.00000	0.321029
$622$	−16.0000	−0.641542
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	10.0000	0.399680
$627$	4.00000	0.159745
$628$	−2.00000	−0.0798087
$629$	36.0000	1.43541
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	24.0000	0.954669
$633$	−4.00000	−0.158986
$634$	34.0000	1.35031
$635$	0	0
$636$	10.0000	0.396526
$637$	0	0
$638$	−6.00000	−0.237542
$639$	−8.00000	−0.316475
$640$	−3.00000	−0.118585
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	12.0000	0.473602
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	−24.0000	−0.944267
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	3.00000	0.117851
$649$	12.0000	0.471041
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−20.0000	−0.783260
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	2.00000	0.0782062
$655$	−12.0000	−0.468879
$656$	−6.00000	−0.234261
$657$	6.00000	0.234082
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	1.00000	0.0389249
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	−28.0000	−1.08825
$663$	12.0000	0.466041
$664$	12.0000	0.465690
$665$	0	0
$666$	−6.00000	−0.232495
$667$	48.0000	1.85857
$668$	−16.0000	−0.619059
$669$	8.00000	0.309298
$670$	−12.0000	−0.463600
$671$	10.0000	0.386046
$672$	0	0
$673$	−38.0000	−1.46479	−0.732396	−	0.680879i	$-0.761598\pi$
$673$	−38.0000	−1.46479	−0.732396	+	0.680879i	$0.761598\pi$
$674$	−26.0000	−1.00148
$675$	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
$679$	0	0
$680$	−18.0000	−0.690268
$681$	−12.0000	−0.459841
$682$	8.00000	0.306336
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−4.00000	−0.152944
$685$	22.0000	0.840577
$686$	0	0
$687$	10.0000	0.381524
$688$	4.00000	0.152499
$689$	−20.0000	−0.761939
$690$	8.00000	0.304555
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−20.0000	−0.759190
$695$	4.00000	0.151729
$696$	18.0000	0.682288
$697$	36.0000	1.36360
$698$	−10.0000	−0.378506
$699$	−22.0000	−0.832116
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	−2.00000	−0.0754851
$703$	24.0000	0.905177
$704$	7.00000	0.263822
$705$	−8.00000	−0.301297
$706$	34.0000	1.27961
$707$	0	0
$708$	−12.0000	−0.450988
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	−8.00000	−0.300235
$711$	8.00000	0.300023
$712$	−30.0000	−1.12430
$713$	−64.0000	−2.39682
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	−4.00000	−0.149487
$717$	−16.0000	−0.597531
$718$	−8.00000	−0.298557
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	3.00000	0.111648
$723$	22.0000	0.818189
$724$	6.00000	0.222988
$725$	6.00000	0.222834
$726$	−1.00000	−0.0371135
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	6.00000	0.222070
$731$	−24.0000	−0.887672
$732$	−10.0000	−0.369611
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	−40.0000	−1.47442
$737$	−12.0000	−0.442026
$738$	−6.00000	−0.220863
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	6.00000	0.220564
$741$	8.00000	0.293887
$742$	0	0
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	−24.0000	−0.879883
$745$	18.0000	0.659469
$746$	−14.0000	−0.512576
$747$	4.00000	0.146352
$748$	−6.00000	−0.219382
$749$	0	0
$750$	1.00000	0.0365148
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	−8.00000	−0.291730
$753$	−20.0000	−0.728841
$754$	−12.0000	−0.437014
$755$	−16.0000	−0.582300
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	20.0000	0.726433
$759$	8.00000	0.290382
$760$	−12.0000	−0.435286
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	0	0
$764$	−16.0000	−0.578860
$765$	−6.00000	−0.216930
$766$	24.0000	0.867155
$767$	24.0000	0.866590
$768$	−17.0000	−0.613435
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	22.0000	0.791797
$773$	10.0000	0.359675	0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000	0.359675	0.179838	+	0.983696i	$0.442443\pi$
$774$	4.00000	0.143777
$775$	−8.00000	−0.287368
$776$	42.0000	1.50771
$777$	0	0
$778$	10.0000	0.358517
$779$	24.0000	0.859889
$780$	2.00000	0.0716115
$781$	−8.00000	−0.286263
$782$	−48.0000	−1.71648
$783$	6.00000	0.214423
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	−12.0000	−0.428026
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−22.0000	−0.783718
$789$	16.0000	0.569615
$790$	8.00000	0.284627
$791$	0	0
$792$	3.00000	0.106600
$793$	20.0000	0.710221
$794$	−18.0000	−0.638796
$795$	10.0000	0.354663
$796$	16.0000	0.567105
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	−5.00000	−0.176777
$801$	−10.0000	−0.353333
$802$	−2.00000	−0.0706225
$803$	6.00000	0.211735
$804$	12.0000	0.423207
$805$	0	0
$806$	16.0000	0.563576
$807$	−14.0000	−0.492823
$808$	−18.0000	−0.633238
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	1.00000	0.0351364
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	−6.00000	−0.210300
$815$	−20.0000	−0.700569
$816$	−6.00000	−0.210042
$817$	−16.0000	−0.559769
$818$	−14.0000	−0.489499
$819$	0	0
$820$	6.00000	0.209529
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	22.0000	0.767338
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	−48.0000	−1.67216
$825$	1.00000	0.0348155
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−8.00000	−0.278019
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	4.00000	0.138842
$831$	−18.0000	−0.624413
$832$	14.0000	0.485363
$833$	0	0
$834$	4.00000	0.138509
$835$	−16.0000	−0.553703
$836$	−4.00000	−0.138343
$837$	−8.00000	−0.276520
$838$	−36.0000	−1.24360
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−6.00000	−0.206774
$843$	18.0000	0.619953
$844$	4.00000	0.137686
$845$	9.00000	0.309609
$846$	−8.00000	−0.275046
$847$	0	0
$848$	10.0000	0.343401
$849$	4.00000	0.137280
$850$	−6.00000	−0.205798
$851$	48.0000	1.64542
$852$	8.00000	0.274075
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	−36.0000	−1.23045
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	−2.00000	−0.0682789
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	16.0000	0.544962
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	−5.00000	−0.170103
$865$	6.00000	0.204006
$866$	2.00000	0.0679628
$867$	19.0000	0.645274
$868$	0	0
$869$	8.00000	0.271381
$870$	6.00000	0.203419
$871$	−24.0000	−0.813209
$872$	−6.00000	−0.203186
$873$	14.0000	0.473828
$874$	−32.0000	−1.08242
$875$	0	0
$876$	−6.00000	−0.202721
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	32.0000	1.07995
$879$	18.0000	0.607125
$880$	1.00000	0.0337100
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	−12.0000	−0.403604
$885$	−12.0000	−0.403376
$886$	−12.0000	−0.403148
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	18.0000	0.604040
$889$	0	0
$890$	−10.0000	−0.335201
$891$	1.00000	0.0335013
$892$	−8.00000	−0.267860
$893$	32.0000	1.07084
$894$	18.0000	0.602010
$895$	−4.00000	−0.133705
$896$	0	0
$897$	16.0000	0.534224
$898$	−2.00000	−0.0667409
$899$	−48.0000	−1.60089
$900$	−1.00000	−0.0333333
$901$	−60.0000	−1.99889
$902$	−6.00000	−0.199778
$903$	0	0
$904$	−42.0000	−1.39690
$905$	6.00000	0.199447
$906$	−16.0000	−0.531564
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	12.0000	0.398234
$909$	−6.00000	−0.199007
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−4.00000	−0.132453
$913$	4.00000	0.132381
$914$	−18.0000	−0.595387
$915$	−10.0000	−0.330590
$916$	−10.0000	−0.330409
$917$	0	0
$918$	−6.00000	−0.198030
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	−24.0000	−0.791257
$921$	−4.00000	−0.131804
$922$	14.0000	0.461065
$923$	−16.0000	−0.526646
$924$	0	0
$925$	6.00000	0.197279
$926$	−16.0000	−0.525793
$927$	−16.0000	−0.525509
$928$	−30.0000	−0.984798
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	−8.00000	−0.262330
$931$	0	0
$932$	22.0000	0.720634
$933$	16.0000	0.523816
$934$	28.0000	0.916188
$935$	−6.00000	−0.196221
$936$	6.00000	0.196116
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	8.00000	0.260931
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	−2.00000	−0.0651635
$943$	48.0000	1.56310
$944$	−12.0000	−0.390567
$945$	0	0
$946$	4.00000	0.130051
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	−8.00000	−0.259828
$949$	12.0000	0.389536
$950$	−4.00000	−0.129777
$951$	−34.0000	−1.10253
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	10.0000	0.323762
$955$	−16.0000	−0.517748
$956$	16.0000	0.517477
$957$	6.00000	0.193952
$958$	−16.0000	−0.516937
$959$	0	0
$960$	−7.00000	−0.225924
$961$	33.0000	1.06452
$962$	−12.0000	−0.386896
$963$	−12.0000	−0.386695
$964$	−22.0000	−0.708572
$965$	22.0000	0.708205
$966$	0	0
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	3.00000	0.0964237
$969$	24.0000	0.770991
$970$	14.0000	0.449513
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−24.0000	−0.769010
$975$	2.00000	0.0640513
$976$	−10.0000	−0.320092
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−20.0000	−0.639529
$979$	−10.0000	−0.319601
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	20.0000	0.638226
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	18.0000	0.573819
$985$	−22.0000	−0.700978
$986$	−36.0000	−1.14647
$987$	0	0
$988$	−8.00000	−0.254514
$989$	−32.0000	−1.01754
$990$	1.00000	0.0317821
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	40.0000	1.27000
$993$	28.0000	0.888553
$994$	0	0
$995$	16.0000	0.507234
$996$	−4.00000	−0.126745
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	12.0000	0.379853
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8085.2.a.g.1.1		1
7.6	odd	2		1155.2.a.e.1.1	✓	1
21.20	even	2		3465.2.a.l.1.1		1
35.34	odd	2		5775.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.e.1.1	✓	1	7.6	odd	2
3465.2.a.l.1.1		1	21.20	even	2
5775.2.a.v.1.1		1	35.34	odd	2
8085.2.a.g.1.1		1	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:	\( N \)	\(=\)	\( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8085.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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