LMFDB - Embedded newform 8085.2.a.v.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:	yes
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)

$f(q)$

$=$

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

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

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	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	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
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	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
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−3.00000	−0.612372
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	1.00000	0.182574
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	5.00000	0.883883
$33$	−1.00000	−0.174078
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	4.00000	0.648886
$39$	−4.00000	−0.640513
$40$	−3.00000	−0.474342
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	1.00000	0.150756
$45$	1.00000	0.149071
$46$	4.00000	0.589768
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	1.00000	0.141421
$51$	−2.00000	−0.280056
$52$	4.00000	0.554700
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	1.00000	0.136083
$55$	−1.00000	−0.134840
$56$	0	0
$57$	4.00000	0.529813
$58$	−4.00000	−0.525226
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	−1.00000	−0.129099
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	−1.00000	−0.123091
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	2.00000	0.242536
$69$	4.00000	0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−3.00000	−0.353553
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	0	0
$78$	−4.00000	−0.452911
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	−8.00000	−0.862662
$87$	−4.00000	−0.428845
$88$	3.00000	0.319801
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−4.00000	−0.417029
$93$	−4.00000	−0.414781
$94$	12.0000	1.23771
$95$	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	$0.596490\pi$
$102$	−2.00000	−0.198030
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	12.0000	1.17670
$105$	0	0
$106$	12.0000	1.16554
$107$	20.0000	1.93347	0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000	1.93347	0.966736	+	0.255774i	$0.0823304\pi$
$108$	−1.00000	−0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−1.00000	−0.0953463
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	4.00000	0.374634
$115$	4.00000	0.373002
$116$	4.00000	0.371391
$117$	−4.00000	−0.369800
$118$	−6.00000	−0.552345
$119$	0	0
$120$	−3.00000	−0.273861
$121$	1.00000	0.0909091
$122$	6.00000	0.543214
$123$	10.0000	0.901670
$124$	4.00000	0.359211
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−3.00000	−0.265165
$129$	−8.00000	−0.704361
$130$	−4.00000	−0.350823
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−2.00000	−0.172774
$135$	1.00000	0.0860663
$136$	6.00000	0.514496
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	4.00000	0.340503
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	4.00000	0.334497
$144$	−1.00000	−0.0833333
$145$	−4.00000	−0.332182
$146$	8.00000	0.662085
$147$	0	0
$148$	2.00000	0.164399
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	1.00000	0.0816497
$151$	22.0000	1.79033	0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000	1.79033	0.895167	+	0.445730i	$0.147056\pi$
$152$	−12.0000	−0.973329
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−4.00000	−0.321288
$156$	4.00000	0.320256
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	10.0000	0.795557
$159$	12.0000	0.951662
$160$	5.00000	0.395285
$161$	0	0
$162$	1.00000	0.0785674
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	−10.0000	−0.780869
$165$	−1.00000	−0.0778499
$166$	10.0000	0.776151
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−2.00000	−0.153393
$171$	4.00000	0.305888
$172$	8.00000	0.609994
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	1.00000	0.0753778
$177$	−6.00000	−0.450988
$178$	−6.00000	−0.449719
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−1.00000	−0.0745356
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	−12.0000	−0.884652
$185$	−2.00000	−0.147043
$186$	−4.00000	−0.293294
$187$	2.00000	0.146254
$188$	−12.0000	−0.875190
$189$	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
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	14.0000	1.00514
$195$	−4.00000	−0.286446
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−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
$201$	−2.00000	−0.141069
$202$	−6.00000	−0.422159
$203$	0	0
$204$	2.00000	0.140028
$205$	10.0000	0.698430
$206$	−8.00000	−0.557386
$207$	4.00000	0.278019
$208$	4.00000	0.277350
$209$	−4.00000	−0.276686
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	20.0000	1.36717
$215$	−8.00000	−0.545595
$216$	−3.00000	−0.204124
$217$	0	0
$218$	10.0000	0.677285
$219$	8.00000	0.540590
$220$	1.00000	0.0674200
$221$	8.00000	0.538138
$222$	−2.00000	−0.134231
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−4.00000	−0.266076
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	−4.00000	−0.264906
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	4.00000	0.263752
$231$	0	0
$232$	12.0000	0.787839
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	−4.00000	−0.261488
$235$	12.0000	0.782794
$236$	6.00000	0.390567
$237$	10.0000	0.649570
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	−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
$244$	−6.00000	−0.384111
$245$	0	0
$246$	10.0000	0.637577
$247$	−16.0000	−1.01806
$248$	12.0000	0.762001
$249$	10.0000	0.633724
$250$	1.00000	0.0632456
$251$	22.0000	1.38863	0.694314	−	0.719672i	$-0.255708\pi$
$251$	22.0000	1.38863	0.694314	+	0.719672i	$0.255708\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−8.00000	−0.501965
$255$	−2.00000	−0.125245
$256$	−17.0000	−1.06250
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	−8.00000	−0.498058
$259$	0	0
$260$	4.00000	0.248069
$261$	−4.00000	−0.247594
$262$	−12.0000	−0.741362
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	3.00000	0.184637
$265$	12.0000	0.737154
$266$	0	0
$267$	−6.00000	−0.367194
$268$	2.00000	0.122169
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	1.00000	0.0608581
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−12.0000	−0.724947
$275$	−1.00000	−0.0603023
$276$	−4.00000	−0.240772
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	4.00000	0.239904
$279$	−4.00000	−0.239474
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	12.0000	0.714590
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	4.00000	0.236525
$287$	0	0
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	14.0000	0.820695
$292$	−8.00000	−0.468165
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	6.00000	0.348743
$297$	−1.00000	−0.0580259
$298$	−4.00000	−0.231714
$299$	−16.0000	−0.925304
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	22.0000	1.26596
$303$	−6.00000	−0.344691
$304$	−4.00000	−0.229416
$305$	6.00000	0.343559
$306$	−2.00000	−0.114332
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	−4.00000	−0.227185
$311$	−14.0000	−0.793867	−0.396934	−	0.917847i	$-0.629926\pi$
$311$	−14.0000	−0.793867	−0.396934	+	0.917847i	$0.629926\pi$
$312$	12.0000	0.679366
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	−10.0000	−0.562544
$317$	−8.00000	−0.449325	−0.224662	−	0.974437i	$-0.572128\pi$
$317$	−8.00000	−0.449325	−0.224662	+	0.974437i	$0.572128\pi$
$318$	12.0000	0.672927
$319$	4.00000	0.223957
$320$	7.00000	0.391312
$321$	20.0000	1.11629
$322$	0	0
$323$	−8.00000	−0.445132
$324$	−1.00000	−0.0555556
$325$	−4.00000	−0.221880
$326$	−6.00000	−0.332309
$327$	10.0000	0.553001
$328$	−30.0000	−1.65647
$329$	0	0
$330$	−1.00000	−0.0550482
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−10.0000	−0.548821
$333$	−2.00000	−0.109599
$334$	−2.00000	−0.109435
$335$	−2.00000	−0.109272
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	3.00000	0.163178
$339$	−4.00000	−0.217250
$340$	2.00000	0.108465
$341$	4.00000	0.216612
$342$	4.00000	0.216295
$343$	0	0
$344$	24.0000	1.29399
$345$	4.00000	0.215353
$346$	−14.0000	−0.752645
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	4.00000	0.214423
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	−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	$0.859994\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	20.0000	1.05703
$359$	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
$362$	−12.0000	−0.630706
$363$	1.00000	0.0524864
$364$	0	0
$365$	8.00000	0.418739
$366$	6.00000	0.313625
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−4.00000	−0.208514
$369$	10.0000	0.520579
$370$	−2.00000	−0.103975
$371$	0	0
$372$	4.00000	0.207390
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	2.00000	0.103418
$375$	1.00000	0.0516398
$376$	−36.0000	−1.85656
$377$	16.0000	0.824042
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	−4.00000	−0.205196
$381$	−8.00000	−0.409852
$382$	16.0000	0.818631
$383$	−32.0000	−1.63512	−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000	−1.63512	−0.817562	+	0.575841i	$0.804675\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	18.0000	0.916176
$387$	−8.00000	−0.406663
$388$	−14.0000	−0.710742
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	−4.00000	−0.202548
$391$	−8.00000	−0.404577
$392$	0	0
$393$	−12.0000	−0.605320
$394$	18.0000	0.906827
$395$	10.0000	0.503155
$396$	1.00000	0.0502519
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	−2.00000	−0.0997509
$403$	16.0000	0.797017
$404$	6.00000	0.298511
$405$	1.00000	0.0496904
$406$	0	0
$407$	2.00000	0.0991363
$408$	6.00000	0.297044
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	10.0000	0.493865
$411$	−12.0000	−0.591916
$412$	8.00000	0.394132
$413$	0	0
$414$	4.00000	0.196589
$415$	10.0000	0.490881
$416$	−20.0000	−0.980581
$417$	4.00000	0.195881
$418$	−4.00000	−0.195646
$419$	2.00000	0.0977064	0.0488532	−	0.998806i	$-0.484443\pi$
$419$	2.00000	0.0977064	0.0488532	+	0.998806i	$0.484443\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	2.00000	0.0973585
$423$	12.0000	0.583460
$424$	−36.0000	−1.74831
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	0	0
$428$	−20.0000	−0.966736
$429$	4.00000	0.193122
$430$	−8.00000	−0.385794
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	−1.00000	−0.0481125
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	−10.0000	−0.478913
$437$	16.0000	0.765384
$438$	8.00000	0.382255
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	3.00000	0.143019
$441$	0	0
$442$	8.00000	0.380521
$443$	−32.0000	−1.52037	−0.760183	−	0.649709i	$-0.774891\pi$
$443$	−32.0000	−1.52037	−0.760183	+	0.649709i	$0.774891\pi$
$444$	2.00000	0.0949158
$445$	−6.00000	−0.284427
$446$	−16.0000	−0.757622
$447$	−4.00000	−0.189194
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	1.00000	0.0471405
$451$	−10.0000	−0.470882
$452$	4.00000	0.188144
$453$	22.0000	1.03365
$454$	6.00000	0.281594
$455$	0	0
$456$	−12.0000	−0.561951
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	8.00000	0.373815
$459$	−2.00000	−0.0933520
$460$	−4.00000	−0.186501
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	18.0000	0.836531	0.418265	−	0.908325i	$-0.362638\pi$
$463$	18.0000	0.836531	0.418265	+	0.908325i	$0.362638\pi$
$464$	4.00000	0.185695
$465$	−4.00000	−0.185496
$466$	14.0000	0.648537
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	4.00000	0.184900
$469$	0	0
$470$	12.0000	0.553519
$471$	6.00000	0.276465
$472$	18.0000	0.828517
$473$	8.00000	0.367840
$474$	10.0000	0.459315
$475$	4.00000	0.183533
$476$	0	0
$477$	12.0000	0.549442
$478$	24.0000	1.09773
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	5.00000	0.228218
$481$	8.00000	0.364769
$482$	22.0000	1.00207
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	14.0000	0.635707
$486$	1.00000	0.0453609
$487$	−18.0000	−0.815658	−0.407829	−	0.913058i	$-0.633714\pi$
$487$	−18.0000	−0.815658	−0.407829	+	0.913058i	$0.633714\pi$
$488$	−18.0000	−0.814822
$489$	−6.00000	−0.271329
$490$	0	0
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	−10.0000	−0.450835
$493$	8.00000	0.360302
$494$	−16.0000	−0.719874
$495$	−1.00000	−0.0449467
$496$	4.00000	0.179605
$497$	0	0
$498$	10.0000	0.448111
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−1.00000	−0.0447214
$501$	−2.00000	−0.0893534
$502$	22.0000	0.981908
$503$	−30.0000	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	−4.00000	−0.177822
$507$	3.00000	0.133235
$508$	8.00000	0.354943
$509$	22.0000	0.975133	0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000	0.975133	0.487566	+	0.873086i	$0.337885\pi$
$510$	−2.00000	−0.0885615
$511$	0	0
$512$	−11.0000	−0.486136
$513$	4.00000	0.176604
$514$	6.00000	0.264649
$515$	−8.00000	−0.352522
$516$	8.00000	0.352180
$517$	−12.0000	−0.527759
$518$	0	0
$519$	−14.0000	−0.614532
$520$	12.0000	0.526235
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	−4.00000	−0.175075
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	8.00000	0.348817
$527$	8.00000	0.348485
$528$	1.00000	0.0435194
$529$	−7.00000	−0.304348
$530$	12.0000	0.521247
$531$	−6.00000	−0.260378
$532$	0	0
$533$	−40.0000	−1.73259
$534$	−6.00000	−0.259645
$535$	20.0000	0.864675
$536$	6.00000	0.259161
$537$	20.0000	0.863064
$538$	−2.00000	−0.0862261
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	8.00000	0.343629
$543$	−12.0000	−0.514969
$544$	−10.0000	−0.428746
$545$	10.0000	0.428353
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	12.0000	0.512615
$549$	6.00000	0.256074
$550$	−1.00000	−0.0426401
$551$	−16.0000	−0.681623
$552$	−12.0000	−0.510754
$553$	0	0
$554$	2.00000	0.0849719
$555$	−2.00000	−0.0848953
$556$	−4.00000	−0.169638
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	−4.00000	−0.169334
$559$	32.0000	1.35346
$560$	0	0
$561$	2.00000	0.0844401
$562$	4.00000	0.168730
$563$	14.0000	0.590030	0.295015	−	0.955493i	$-0.404675\pi$
$563$	14.0000	0.590030	0.295015	+	0.955493i	$0.404675\pi$
$564$	−12.0000	−0.505291
$565$	−4.00000	−0.168281
$566$	−4.00000	−0.168133
$567$	0	0
$568$	0	0
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	4.00000	0.167542
$571$	30.0000	1.25546	0.627730	−	0.778431i	$-0.283984\pi$
$571$	30.0000	1.25546	0.627730	+	0.778431i	$0.283984\pi$
$572$	−4.00000	−0.167248
$573$	16.0000	0.668410
$574$	0	0
$575$	4.00000	0.166812
$576$	7.00000	0.291667
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	−13.0000	−0.540729
$579$	18.0000	0.748054
$580$	4.00000	0.166091
$581$	0	0
$582$	14.0000	0.580319
$583$	−12.0000	−0.496989
$584$	−24.0000	−0.993127
$585$	−4.00000	−0.165380
$586$	2.00000	0.0826192
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	−6.00000	−0.247016
$591$	18.0000	0.740421
$592$	2.00000	0.0821995
$593$	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
$596$	4.00000	0.163846
$597$	−16.0000	−0.654836
$598$	−16.0000	−0.654289
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	−3.00000	−0.122474
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	−22.0000	−0.895167
$605$	1.00000	0.0406558
$606$	−6.00000	−0.243733
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	20.0000	0.811107
$609$	0	0
$610$	6.00000	0.242933
$611$	−48.0000	−1.94187
$612$	2.00000	0.0808452
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	−16.0000	−0.645707
$615$	10.0000	0.403239
$616$	0	0
$617$	8.00000	0.322068	0.161034	−	0.986949i	$-0.448517\pi$
$617$	8.00000	0.322068	0.161034	+	0.986949i	$0.448517\pi$
$618$	−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	$0.368338\pi$
$620$	4.00000	0.160644
$621$	4.00000	0.160514
$622$	−14.0000	−0.561349
$623$	0	0
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	−2.00000	−0.0799361
$627$	−4.00000	−0.159745
$628$	−6.00000	−0.239426
$629$	4.00000	0.159490
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	−30.0000	−1.19334
$633$	2.00000	0.0794929
$634$	−8.00000	−0.317721
$635$	−8.00000	−0.317470
$636$	−12.0000	−0.475831
$637$	0	0
$638$	4.00000	0.158362
$639$	0	0
$640$	−3.00000	−0.118585
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	20.0000	0.789337
$643$	−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
$646$	−8.00000	−0.314756
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	−3.00000	−0.117851
$649$	6.00000	0.235521
$650$	−4.00000	−0.156893
$651$	0	0
$652$	6.00000	0.234978
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	10.0000	0.391031
$655$	−12.0000	−0.468879
$656$	−10.0000	−0.390434
$657$	8.00000	0.312110
$658$	0	0
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	1.00000	0.0389249
$661$	8.00000	0.311164	0.155582	−	0.987823i	$-0.450275\pi$
$661$	8.00000	0.311164	0.155582	+	0.987823i	$0.450275\pi$
$662$	−20.0000	−0.777322
$663$	8.00000	0.310694
$664$	−30.0000	−1.16423
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	−16.0000	−0.619522
$668$	2.00000	0.0773823
$669$	−16.0000	−0.618596
$670$	−2.00000	−0.0772667
$671$	−6.00000	−0.231627
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	−26.0000	−1.00148
$675$	1.00000	0.0384900
$676$	−3.00000	−0.115385
$677$	10.0000	0.384331	0.192166	−	0.981363i	$-0.438449\pi$
$677$	10.0000	0.384331	0.192166	+	0.981363i	$0.438449\pi$
$678$	−4.00000	−0.153619
$679$	0	0
$680$	6.00000	0.230089
$681$	6.00000	0.229920
$682$	4.00000	0.153168
$683$	−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
$685$	−12.0000	−0.458496
$686$	0	0
$687$	8.00000	0.305219
$688$	8.00000	0.304997
$689$	−48.0000	−1.82865
$690$	4.00000	0.152277
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	20.0000	0.759190
$695$	4.00000	0.151729
$696$	12.0000	0.454859
$697$	−20.0000	−0.757554
$698$	−2.00000	−0.0757011
$699$	14.0000	0.529529
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	−4.00000	−0.150970
$703$	−8.00000	−0.301726
$704$	−7.00000	−0.263822
$705$	12.0000	0.451946
$706$	−34.0000	−1.27961
$707$	0	0
$708$	6.00000	0.225494
$709$	−42.0000	−1.57734	−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000	−1.57734	−0.788672	+	0.614815i	$0.789231\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	18.0000	0.674579
$713$	−16.0000	−0.599205
$714$	0	0
$715$	4.00000	0.149592
$716$	−20.0000	−0.747435
$717$	24.0000	0.896296
$718$	8.00000	0.298557
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	−3.00000	−0.111648
$723$	22.0000	0.818189
$724$	12.0000	0.445976
$725$	−4.00000	−0.148556
$726$	1.00000	0.0371135
$727$	−24.0000	−0.890111	−0.445055	−	0.895503i	$-0.646816\pi$
$727$	−24.0000	−0.890111	−0.445055	+	0.895503i	$0.646816\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	8.00000	0.296093
$731$	16.0000	0.591781
$732$	−6.00000	−0.221766
$733$	−16.0000	−0.590973	−0.295487	−	0.955347i	$-0.595482\pi$
$733$	−16.0000	−0.590973	−0.295487	+	0.955347i	$0.595482\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	20.0000	0.737210
$737$	2.00000	0.0736709
$738$	10.0000	0.368105
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	2.00000	0.0735215
$741$	−16.0000	−0.587775
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	12.0000	0.439941
$745$	−4.00000	−0.146549
$746$	10.0000	0.366126
$747$	10.0000	0.365881
$748$	−2.00000	−0.0731272
$749$	0	0
$750$	1.00000	0.0365148
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	−12.0000	−0.437595
$753$	22.0000	0.801725
$754$	16.0000	0.582686
$755$	22.0000	0.800662
$756$	0	0
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	32.0000	1.16229
$759$	−4.00000	−0.145191
$760$	−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	$0.465315\pi$
$762$	−8.00000	−0.289809
$763$	0	0
$764$	−16.0000	−0.578860
$765$	−2.00000	−0.0723102
$766$	−32.0000	−1.15621
$767$	24.0000	0.866590
$768$	−17.0000	−0.613435
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−18.0000	−0.647834
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	−8.00000	−0.287554
$775$	−4.00000	−0.143684
$776$	−42.0000	−1.50771
$777$	0	0
$778$	30.0000	1.07555
$779$	40.0000	1.43315
$780$	4.00000	0.143223
$781$	0	0
$782$	−8.00000	−0.286079
$783$	−4.00000	−0.142948
$784$	0	0
$785$	6.00000	0.214149
$786$	−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	$0.333684\pi$
$788$	−18.0000	−0.641223
$789$	8.00000	0.284808
$790$	10.0000	0.355784
$791$	0	0
$792$	3.00000	0.106600
$793$	−24.0000	−0.852265
$794$	2.00000	0.0709773
$795$	12.0000	0.425596
$796$	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
$799$	−24.0000	−0.849059
$800$	5.00000	0.176777
$801$	−6.00000	−0.212000
$802$	−22.0000	−0.776847
$803$	−8.00000	−0.282314
$804$	2.00000	0.0705346
$805$	0	0
$806$	16.0000	0.563576
$807$	−2.00000	−0.0704033
$808$	18.0000	0.633238
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	1.00000	0.0351364
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	2.00000	0.0701000
$815$	−6.00000	−0.210171
$816$	2.00000	0.0700140
$817$	−32.0000	−1.11954
$818$	14.0000	0.489499
$819$	0	0
$820$	−10.0000	−0.349215
$821$	−28.0000	−0.977207	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	−28.0000	−0.977207	−0.488603	+	0.872506i	$0.662493\pi$
$822$	−12.0000	−0.418548
$823$	−2.00000	−0.0697156	−0.0348578	−	0.999392i	$-0.511098\pi$
$823$	−2.00000	−0.0697156	−0.0348578	+	0.999392i	$0.511098\pi$
$824$	24.0000	0.836080
$825$	−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	$0.566904\pi$
$828$	−4.00000	−0.139010
$829$	−8.00000	−0.277851	−0.138926	−	0.990303i	$-0.544365\pi$
$829$	−8.00000	−0.277851	−0.138926	+	0.990303i	$0.544365\pi$
$830$	10.0000	0.347105
$831$	2.00000	0.0693792
$832$	−28.0000	−0.970725
$833$	0	0
$834$	4.00000	0.138509
$835$	−2.00000	−0.0692129
$836$	4.00000	0.138343
$837$	−4.00000	−0.138260
$838$	2.00000	0.0690889
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	2.00000	0.0689246
$843$	4.00000	0.137767
$844$	−2.00000	−0.0688428
$845$	3.00000	0.103203
$846$	12.0000	0.412568
$847$	0	0
$848$	−12.0000	−0.412082
$849$	−4.00000	−0.137280
$850$	−2.00000	−0.0685994
$851$	−8.00000	−0.274236
$852$	0	0
$853$	52.0000	1.78045	0.890223	−	0.455525i	$-0.150548\pi$
$853$	52.0000	1.78045	0.890223	+	0.455525i	$0.150548\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	−60.0000	−2.05076
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	4.00000	0.136558
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−36.0000	−1.22616
$863$	−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
$865$	−14.0000	−0.476014
$866$	−30.0000	−1.01944
$867$	−13.0000	−0.441503
$868$	0	0
$869$	−10.0000	−0.339227
$870$	−4.00000	−0.135613
$871$	8.00000	0.271070
$872$	−30.0000	−1.01593
$873$	14.0000	0.473828
$874$	16.0000	0.541208
$875$	0	0
$876$	−8.00000	−0.270295
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	16.0000	0.539974
$879$	2.00000	0.0674583
$880$	1.00000	0.0337100
$881$	−34.0000	−1.14549	−0.572745	−	0.819734i	$-0.694121\pi$
$881$	−34.0000	−1.14549	−0.572745	+	0.819734i	$0.694121\pi$
$882$	0	0
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	−8.00000	−0.269069
$885$	−6.00000	−0.201688
$886$	−32.0000	−1.07506
$887$	−30.0000	−1.00730	−0.503651	−	0.863907i	$-0.668010\pi$
$887$	−30.0000	−1.00730	−0.503651	+	0.863907i	$0.668010\pi$
$888$	6.00000	0.201347
$889$	0	0
$890$	−6.00000	−0.201120
$891$	−1.00000	−0.0335013
$892$	16.0000	0.535720
$893$	48.0000	1.60626
$894$	−4.00000	−0.133780
$895$	20.0000	0.668526
$896$	0	0
$897$	−16.0000	−0.534224
$898$	18.0000	0.600668
$899$	16.0000	0.533630
$900$	−1.00000	−0.0333333
$901$	−24.0000	−0.799556
$902$	−10.0000	−0.332964
$903$	0	0
$904$	12.0000	0.399114
$905$	−12.0000	−0.398893
$906$	22.0000	0.730901
$907$	−2.00000	−0.0664089	−0.0332045	−	0.999449i	$-0.510571\pi$
$907$	−2.00000	−0.0664089	−0.0332045	+	0.999449i	$0.510571\pi$
$908$	−6.00000	−0.199117
$909$	−6.00000	−0.199007
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	−4.00000	−0.132453
$913$	−10.0000	−0.330952
$914$	22.0000	0.727695
$915$	6.00000	0.198354
$916$	−8.00000	−0.264327
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	−12.0000	−0.395628
$921$	−16.0000	−0.527218
$922$	6.00000	0.197599
$923$	0	0
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	18.0000	0.591517
$927$	−8.00000	−0.262754
$928$	−20.0000	−0.656532
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	−4.00000	−0.131165
$931$	0	0
$932$	−14.0000	−0.458585
$933$	−14.0000	−0.458339
$934$	−32.0000	−1.04707
$935$	2.00000	0.0654070
$936$	12.0000	0.392232
$937$	48.0000	1.56809	0.784046	−	0.620703i	$-0.213153\pi$
$937$	48.0000	1.56809	0.784046	+	0.620703i	$0.213153\pi$
$938$	0	0
$939$	−2.00000	−0.0652675
$940$	−12.0000	−0.391397
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	6.00000	0.195491
$943$	40.0000	1.30258
$944$	6.00000	0.195283
$945$	0	0
$946$	8.00000	0.260102
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	−10.0000	−0.324785
$949$	−32.0000	−1.03876
$950$	4.00000	0.129777
$951$	−8.00000	−0.259418
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	12.0000	0.388514
$955$	16.0000	0.517748
$956$	−24.0000	−0.776215
$957$	4.00000	0.129302
$958$	−36.0000	−1.16311
$959$	0	0
$960$	7.00000	0.225924
$961$	−15.0000	−0.483871
$962$	8.00000	0.257930
$963$	20.0000	0.644491
$964$	−22.0000	−0.708572
$965$	18.0000	0.579441
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	−3.00000	−0.0964237
$969$	−8.00000	−0.256997
$970$	14.0000	0.449513
$971$	−10.0000	−0.320915	−0.160458	−	0.987043i	$-0.551297\pi$
$971$	−10.0000	−0.320915	−0.160458	+	0.987043i	$0.551297\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−18.0000	−0.576757
$975$	−4.00000	−0.128103
$976$	−6.00000	−0.192055
$977$	24.0000	0.767828	0.383914	−	0.923369i	$-0.374576\pi$
$977$	24.0000	0.767828	0.383914	+	0.923369i	$0.374576\pi$
$978$	−6.00000	−0.191859
$979$	6.00000	0.191761
$980$	0	0
$981$	10.0000	0.319275
$982$	−24.0000	−0.765871
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−30.0000	−0.956365
$985$	18.0000	0.573528
$986$	8.00000	0.254772
$987$	0	0
$988$	16.0000	0.509028
$989$	−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	$0.418211\pi$
$992$	−20.0000	−0.635001
$993$	−20.0000	−0.634681
$994$	0	0
$995$	−16.0000	−0.507234
$996$	−10.0000	−0.316862
$997$	4.00000	0.126681	0.0633406	−	0.997992i	$-0.479825\pi$
$997$	4.00000	0.126681	0.0633406	+	0.997992i	$0.479825\pi$
$998$	20.0000	0.633089
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8085.2.a.v.1.1	yes	1
7.6	odd	2		8085.2.a.s.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8085.2.a.s.1.1	✓	1	7.6	odd	2
8085.2.a.v.1.1	yes	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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