LMFDB - Embedded newform 2790.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2790,2,Mod(1,2790)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2790, 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("2790.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2790.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:	$22.2782621639$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2790.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -5.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{19} +1.00000 q^{20} -5.00000 q^{22} +5.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} -4.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} +1.00000 q^{35} +12.0000 q^{37} +1.00000 q^{38} +1.00000 q^{40} -4.00000 q^{41} +11.0000 q^{43} -5.00000 q^{44} +5.00000 q^{46} +10.0000 q^{47} -6.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} -9.00000 q^{53} -5.00000 q^{55} +1.00000 q^{56} -4.00000 q^{58} +10.0000 q^{59} +10.0000 q^{61} -1.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +6.00000 q^{67} +4.00000 q^{68} +1.00000 q^{70} -15.0000 q^{71} -13.0000 q^{73} +12.0000 q^{74} +1.00000 q^{76} -5.00000 q^{77} +13.0000 q^{79} +1.00000 q^{80} -4.00000 q^{82} +8.00000 q^{83} +4.00000 q^{85} +11.0000 q^{86} -5.00000 q^{88} -3.00000 q^{89} +2.00000 q^{91} +5.00000 q^{92} +10.0000 q^{94} +1.00000 q^{95} -18.0000 q^{97} -6.00000 q^{98} +O(q^{100})\)

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
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−5.00000	−1.06600
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	1.00000	0.188982
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	0	0
$34$	4.00000	0.685994
$35$	1.00000	0.169031
$36$	0	0
$37$	12.0000	1.97279	0.986394	−	0.164399i	$-0.0525685\pi$
$37$	12.0000	1.97279	0.986394	+	0.164399i	$0.0525685\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	1.00000	0.158114
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	5.00000	0.737210
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	1.00000	0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0	0
$55$	−5.00000	−0.674200
$56$	1.00000	0.133631
$57$	0	0
$58$	−4.00000	−0.525226
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	1.00000	0.119523
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	0	0
$73$	−13.0000	−1.52153	−0.760767	−	0.649025i	$-0.775177\pi$
$73$	−13.0000	−1.52153	−0.760767	+	0.649025i	$0.775177\pi$
$74$	12.0000	1.39497
$75$	0	0
$76$	1.00000	0.114708
$77$	−5.00000	−0.569803
$78$	0	0
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−4.00000	−0.441726
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	11.0000	1.18616
$87$	0	0
$88$	−5.00000	−0.533002
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	5.00000	0.521286
$93$	0	0
$94$	10.0000	1.03142
$95$	1.00000	0.102598
$96$	0	0
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	−6.00000	−0.606092
$99$	0	0
$100$	1.00000	0.100000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−9.00000	−0.874157
$107$	7.00000	0.676716	0.338358	−	0.941018i	$-0.390129\pi$
$107$	7.00000	0.676716	0.338358	+	0.941018i	$0.390129\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	−5.00000	−0.476731
$111$	0	0
$112$	1.00000	0.0944911
$113$	3.00000	0.282216	0.141108	−	0.989994i	$-0.454933\pi$
$113$	3.00000	0.282216	0.141108	+	0.989994i	$0.454933\pi$
$114$	0	0
$115$	5.00000	0.466252
$116$	−4.00000	−0.371391
$117$	0	0
$118$	10.0000	0.920575
$119$	4.00000	0.366679
$120$	0	0
$121$	14.0000	1.27273
$122$	10.0000	0.905357
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	2.00000	0.175412
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	6.00000	0.518321
$135$	0	0
$136$	4.00000	0.342997
$137$	−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
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−15.0000	−1.25877
$143$	−10.0000	−0.836242
$144$	0	0
$145$	−4.00000	−0.332182
$146$	−13.0000	−1.07589
$147$	0	0
$148$	12.0000	0.986394
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−5.00000	−0.402911
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	13.0000	1.03422
$159$	0	0
$160$	1.00000	0.0790569
$161$	5.00000	0.394055
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	8.00000	0.620920
$167$	5.00000	0.386912	0.193456	−	0.981109i	$-0.438030\pi$
$167$	5.00000	0.386912	0.193456	+	0.981109i	$0.438030\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	4.00000	0.306786
$171$	0	0
$172$	11.0000	0.838742
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−5.00000	−0.376889
$177$	0	0
$178$	−3.00000	−0.224860
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−21.0000	−1.56092	−0.780459	−	0.625207i	$-0.785014\pi$
$181$	−21.0000	−1.56092	−0.780459	+	0.625207i	$0.785014\pi$
$182$	2.00000	0.148250
$183$	0	0
$184$	5.00000	0.368605
$185$	12.0000	0.882258
$186$	0	0
$187$	−20.0000	−1.46254
$188$	10.0000	0.729325
$189$	0	0
$190$	1.00000	0.0725476
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	−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
$195$	0	0
$196$	−6.00000	−0.428571
$197$	26.0000	1.85242	0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000	1.85242	0.926212	+	0.377004i	$0.123046\pi$
$198$	0	0
$199$	13.0000	0.921546	0.460773	−	0.887518i	$-0.347572\pi$
$199$	13.0000	0.921546	0.460773	+	0.887518i	$0.347572\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−3.00000	−0.211079
$203$	−4.00000	−0.280745
$204$	0	0
$205$	−4.00000	−0.279372
$206$	16.0000	1.11477
$207$	0	0
$208$	2.00000	0.138675
$209$	−5.00000	−0.345857
$210$	0	0
$211$	−15.0000	−1.03264	−0.516321	−	0.856395i	$-0.672699\pi$
$211$	−15.0000	−1.03264	−0.516321	+	0.856395i	$0.672699\pi$
$212$	−9.00000	−0.618123
$213$	0	0
$214$	7.00000	0.478510
$215$	11.0000	0.750194
$216$	0	0
$217$	−1.00000	−0.0678844
$218$	0	0
$219$	0	0
$220$	−5.00000	−0.337100
$221$	8.00000	0.538138
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	3.00000	0.199557
$227$	−13.0000	−0.862840	−0.431420	−	0.902151i	$-0.641987\pi$
$227$	−13.0000	−0.862840	−0.431420	+	0.902151i	$0.641987\pi$
$228$	0	0
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	5.00000	0.329690
$231$	0	0
$232$	−4.00000	−0.262613
$233$	13.0000	0.851658	0.425829	−	0.904804i	$-0.359982\pi$
$233$	13.0000	0.851658	0.425829	+	0.904804i	$0.359982\pi$
$234$	0	0
$235$	10.0000	0.652328
$236$	10.0000	0.650945
$237$	0	0
$238$	4.00000	0.259281
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	14.0000	0.899954
$243$	0	0
$244$	10.0000	0.640184
$245$	−6.00000	−0.383326
$246$	0	0
$247$	2.00000	0.127257
$248$	−1.00000	−0.0635001
$249$	0	0
$250$	1.00000	0.0632456
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	−25.0000	−1.57174
$254$	−20.0000	−1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	2.00000	0.124035
$261$	0	0
$262$	−10.0000	−0.617802
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	1.00000	0.0613139
$267$	0	0
$268$	6.00000	0.366508
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−1.00000	−0.0607457	−0.0303728	−	0.999539i	$-0.509669\pi$
$271$	−1.00000	−0.0607457	−0.0303728	+	0.999539i	$0.509669\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−5.00000	−0.301511
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	−14.0000	−0.839664
$279$	0	0
$280$	1.00000	0.0597614
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	−15.0000	−0.890086
$285$	0	0
$286$	−10.0000	−0.591312
$287$	−4.00000	−0.236113
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	−4.00000	−0.234888
$291$	0	0
$292$	−13.0000	−0.760767
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	10.0000	0.582223
$296$	12.0000	0.697486
$297$	0	0
$298$	−9.00000	−0.521356
$299$	10.0000	0.578315
$300$	0	0
$301$	11.0000	0.634029
$302$	8.00000	0.460348
$303$	0	0
$304$	1.00000	0.0573539
$305$	10.0000	0.572598
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	−5.00000	−0.284901
$309$	0	0
$310$	−1.00000	−0.0567962
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	−13.0000	−0.733632
$315$	0	0
$316$	13.0000	0.731307
$317$	−8.00000	−0.449325	−0.224662	−	0.974437i	$-0.572128\pi$
$317$	−8.00000	−0.449325	−0.224662	+	0.974437i	$0.572128\pi$
$318$	0	0
$319$	20.0000	1.11979
$320$	1.00000	0.0559017
$321$	0	0
$322$	5.00000	0.278639
$323$	4.00000	0.222566
$324$	0	0
$325$	2.00000	0.110940
$326$	6.00000	0.332309
$327$	0	0
$328$	−4.00000	−0.220863
$329$	10.0000	0.551318
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	8.00000	0.439057
$333$	0	0
$334$	5.00000	0.273588
$335$	6.00000	0.327815
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	4.00000	0.216930
$341$	5.00000	0.270765
$342$	0	0
$343$	−13.0000	−0.701934
$344$	11.0000	0.593080
$345$	0	0
$346$	2.00000	0.107521
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	−5.00000	−0.266501
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	−15.0000	−0.796117
$356$	−3.00000	−0.159000
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−31.0000	−1.63612	−0.818059	−	0.575135i	$-0.804950\pi$
$359$	−31.0000	−1.63612	−0.818059	+	0.575135i	$0.804950\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−21.0000	−1.10374
$363$	0	0
$364$	2.00000	0.104828
$365$	−13.0000	−0.680451
$366$	0	0
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	5.00000	0.260643
$369$	0	0
$370$	12.0000	0.623850
$371$	−9.00000	−0.467257
$372$	0	0
$373$	31.0000	1.60512	0.802560	−	0.596572i	$-0.203471\pi$
$373$	31.0000	1.60512	0.802560	+	0.596572i	$0.203471\pi$
$374$	−20.0000	−1.03418
$375$	0	0
$376$	10.0000	0.515711
$377$	−8.00000	−0.412021
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	1.00000	0.0512989
$381$	0	0
$382$	24.0000	1.22795
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	0	0
$385$	−5.00000	−0.254824
$386$	−2.00000	−0.101797
$387$	0	0
$388$	−18.0000	−0.913812
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	−6.00000	−0.303046
$393$	0	0
$394$	26.0000	1.30986
$395$	13.0000	0.654101
$396$	0	0
$397$	−37.0000	−1.85698	−0.928488	−	0.371361i	$-0.878891\pi$
$397$	−37.0000	−1.85698	−0.928488	+	0.371361i	$0.878891\pi$
$398$	13.0000	0.651631
$399$	0	0
$400$	1.00000	0.0500000
$401$	−19.0000	−0.948815	−0.474407	−	0.880305i	$-0.657338\pi$
$401$	−19.0000	−0.948815	−0.474407	+	0.880305i	$0.657338\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	−3.00000	−0.149256
$405$	0	0
$406$	−4.00000	−0.198517
$407$	−60.0000	−2.97409
$408$	0	0
$409$	−24.0000	−1.18672	−0.593362	−	0.804936i	$-0.702200\pi$
$409$	−24.0000	−1.18672	−0.593362	+	0.804936i	$0.702200\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	16.0000	0.788263
$413$	10.0000	0.492068
$414$	0	0
$415$	8.00000	0.392705
$416$	2.00000	0.0980581
$417$	0	0
$418$	−5.00000	−0.244558
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−15.0000	−0.730189
$423$	0	0
$424$	−9.00000	−0.437079
$425$	4.00000	0.194029
$426$	0	0
$427$	10.0000	0.483934
$428$	7.00000	0.338358
$429$	0	0
$430$	11.0000	0.530467
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	1.00000	0.0480569	0.0240285	−	0.999711i	$-0.492351\pi$
$433$	1.00000	0.0480569	0.0240285	+	0.999711i	$0.492351\pi$
$434$	−1.00000	−0.0480015
$435$	0	0
$436$	0	0
$437$	5.00000	0.239182
$438$	0	0
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	−5.00000	−0.238366
$441$	0	0
$442$	8.00000	0.380521
$443$	−7.00000	−0.332580	−0.166290	−	0.986077i	$-0.553179\pi$
$443$	−7.00000	−0.332580	−0.166290	+	0.986077i	$0.553179\pi$
$444$	0	0
$445$	−3.00000	−0.142214
$446$	10.0000	0.473514
$447$	0	0
$448$	1.00000	0.0472456
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	3.00000	0.141108
$453$	0	0
$454$	−13.0000	−0.610120
$455$	2.00000	0.0937614
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	−5.00000	−0.233635
$459$	0	0
$460$	5.00000	0.233126
$461$	−20.0000	−0.931493	−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000	−0.931493	−0.465746	+	0.884918i	$0.654214\pi$
$462$	0	0
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	13.0000	0.602213
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	6.00000	0.277054
$470$	10.0000	0.461266
$471$	0	0
$472$	10.0000	0.460287
$473$	−55.0000	−2.52890
$474$	0	0
$475$	1.00000	0.0458831
$476$	4.00000	0.183340
$477$	0	0
$478$	16.0000	0.731823
$479$	−11.0000	−0.502603	−0.251301	−	0.967909i	$-0.580859\pi$
$479$	−11.0000	−0.502603	−0.251301	+	0.967909i	$0.580859\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	22.0000	1.00207
$483$	0	0
$484$	14.0000	0.636364
$485$	−18.0000	−0.817338
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	10.0000	0.452679
$489$	0	0
$490$	−6.00000	−0.271052
$491$	11.0000	0.496423	0.248212	−	0.968706i	$-0.420157\pi$
$491$	11.0000	0.496423	0.248212	+	0.968706i	$0.420157\pi$
$492$	0	0
$493$	−16.0000	−0.720604
$494$	2.00000	0.0899843
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	−15.0000	−0.672842
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	4.00000	0.178529
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	−3.00000	−0.133498
$506$	−25.0000	−1.11139
$507$	0	0
$508$	−20.0000	−0.887357
$509$	−4.00000	−0.177297	−0.0886484	−	0.996063i	$-0.528255\pi$
$509$	−4.00000	−0.177297	−0.0886484	+	0.996063i	$0.528255\pi$
$510$	0	0
$511$	−13.0000	−0.575086
$512$	1.00000	0.0441942
$513$	0	0
$514$	3.00000	0.132324
$515$	16.0000	0.705044
$516$	0	0
$517$	−50.0000	−2.19900
$518$	12.0000	0.527250
$519$	0	0
$520$	2.00000	0.0877058
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	15.0000	0.655904	0.327952	−	0.944694i	$-0.393642\pi$
$523$	15.0000	0.655904	0.327952	+	0.944694i	$0.393642\pi$
$524$	−10.0000	−0.436852
$525$	0	0
$526$	0	0
$527$	−4.00000	−0.174243
$528$	0	0
$529$	2.00000	0.0869565
$530$	−9.00000	−0.390935
$531$	0	0
$532$	1.00000	0.0433555
$533$	−8.00000	−0.346518
$534$	0	0
$535$	7.00000	0.302636
$536$	6.00000	0.259161
$537$	0	0
$538$	6.00000	0.258678
$539$	30.0000	1.29219
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−1.00000	−0.0429537
$543$	0	0
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	−5.00000	−0.213201
$551$	−4.00000	−0.170406
$552$	0	0
$553$	13.0000	0.552816
$554$	−28.0000	−1.18961
$555$	0	0
$556$	−14.0000	−0.593732
$557$	27.0000	1.14403	0.572013	−	0.820244i	$-0.306163\pi$
$557$	27.0000	1.14403	0.572013	+	0.820244i	$0.306163\pi$
$558$	0	0
$559$	22.0000	0.930501
$560$	1.00000	0.0422577
$561$	0	0
$562$	12.0000	0.506189
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	3.00000	0.126211
$566$	−28.0000	−1.17693
$567$	0	0
$568$	−15.0000	−0.629386
$569$	37.0000	1.55112	0.775560	−	0.631273i	$-0.217467\pi$
$569$	37.0000	1.55112	0.775560	+	0.631273i	$0.217467\pi$
$570$	0	0
$571$	−30.0000	−1.25546	−0.627730	−	0.778431i	$-0.716016\pi$
$571$	−30.0000	−1.25546	−0.627730	+	0.778431i	$0.716016\pi$
$572$	−10.0000	−0.418121
$573$	0	0
$574$	−4.00000	−0.166957
$575$	5.00000	0.208514
$576$	0	0
$577$	12.0000	0.499567	0.249783	−	0.968302i	$-0.419641\pi$
$577$	12.0000	0.499567	0.249783	+	0.968302i	$0.419641\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	−4.00000	−0.166091
$581$	8.00000	0.331896
$582$	0	0
$583$	45.0000	1.86371
$584$	−13.0000	−0.537944
$585$	0	0
$586$	6.00000	0.247858
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	−1.00000	−0.0412043
$590$	10.0000	0.411693
$591$	0	0
$592$	12.0000	0.493197
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	−9.00000	−0.368654
$597$	0	0
$598$	10.0000	0.408930
$599$	−21.0000	−0.858037	−0.429018	−	0.903296i	$-0.641140\pi$
$599$	−21.0000	−0.858037	−0.429018	+	0.903296i	$0.641140\pi$
$600$	0	0
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	11.0000	0.448327
$603$	0	0
$604$	8.00000	0.325515
$605$	14.0000	0.569181
$606$	0	0
$607$	1.00000	0.0405887	0.0202944	−	0.999794i	$-0.493540\pi$
$607$	1.00000	0.0405887	0.0202944	+	0.999794i	$0.493540\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	10.0000	0.404888
$611$	20.0000	0.809113
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	−5.00000	−0.201456
$617$	15.0000	0.603877	0.301939	−	0.953327i	$-0.402366\pi$
$617$	15.0000	0.603877	0.301939	+	0.953327i	$0.402366\pi$
$618$	0	0
$619$	34.0000	1.36658	0.683288	−	0.730149i	$-0.260549\pi$
$619$	34.0000	1.36658	0.683288	+	0.730149i	$0.260549\pi$
$620$	−1.00000	−0.0401610
$621$	0	0
$622$	0	0
$623$	−3.00000	−0.120192
$624$	0	0
$625$	1.00000	0.0400000
$626$	2.00000	0.0799361
$627$	0	0
$628$	−13.0000	−0.518756
$629$	48.0000	1.91389
$630$	0	0
$631$	−13.0000	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$632$	13.0000	0.517112
$633$	0	0
$634$	−8.00000	−0.317721
$635$	−20.0000	−0.793676
$636$	0	0
$637$	−12.0000	−0.475457
$638$	20.0000	0.791808
$639$	0	0
$640$	1.00000	0.0395285
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	1.00000	0.0394362	0.0197181	−	0.999806i	$-0.493723\pi$
$643$	1.00000	0.0394362	0.0197181	+	0.999806i	$0.493723\pi$
$644$	5.00000	0.197028
$645$	0	0
$646$	4.00000	0.157378
$647$	−29.0000	−1.14011	−0.570054	−	0.821607i	$-0.693078\pi$
$647$	−29.0000	−1.14011	−0.570054	+	0.821607i	$0.693078\pi$
$648$	0	0
$649$	−50.0000	−1.96267
$650$	2.00000	0.0784465
$651$	0	0
$652$	6.00000	0.234978
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	−10.0000	−0.390732
$656$	−4.00000	−0.156174
$657$	0	0
$658$	10.0000	0.389841
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	0	0
$661$	34.0000	1.32245	0.661223	−	0.750189i	$-0.270038\pi$
$661$	34.0000	1.32245	0.661223	+	0.750189i	$0.270038\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	8.00000	0.310460
$665$	1.00000	0.0387783
$666$	0	0
$667$	−20.0000	−0.774403
$668$	5.00000	0.193456
$669$	0	0
$670$	6.00000	0.231800
$671$	−50.0000	−1.93023
$672$	0	0
$673$	−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
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−9.00000	−0.345898	−0.172949	−	0.984931i	$-0.555330\pi$
$677$	−9.00000	−0.345898	−0.172949	+	0.984931i	$0.555330\pi$
$678$	0	0
$679$	−18.0000	−0.690777
$680$	4.00000	0.153393
$681$	0	0
$682$	5.00000	0.191460
$683$	−13.0000	−0.497431	−0.248716	−	0.968577i	$-0.580008\pi$
$683$	−13.0000	−0.497431	−0.248716	+	0.968577i	$0.580008\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	−13.0000	−0.496342
$687$	0	0
$688$	11.0000	0.419371
$689$	−18.0000	−0.685745
$690$	0	0
$691$	−11.0000	−0.418460	−0.209230	−	0.977866i	$-0.567096\pi$
$691$	−11.0000	−0.418460	−0.209230	+	0.977866i	$0.567096\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	−32.0000	−1.21470
$695$	−14.0000	−0.531050
$696$	0	0
$697$	−16.0000	−0.606043
$698$	0	0
$699$	0	0
$700$	1.00000	0.0377964
$701$	1.00000	0.0377695	0.0188847	−	0.999822i	$-0.493988\pi$
$701$	1.00000	0.0377695	0.0188847	+	0.999822i	$0.493988\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	−5.00000	−0.188445
$705$	0	0
$706$	14.0000	0.526897
$707$	−3.00000	−0.112827
$708$	0	0
$709$	33.0000	1.23934	0.619671	−	0.784862i	$-0.287266\pi$
$709$	33.0000	1.23934	0.619671	+	0.784862i	$0.287266\pi$
$710$	−15.0000	−0.562940
$711$	0	0
$712$	−3.00000	−0.112430
$713$	−5.00000	−0.187251
$714$	0	0
$715$	−10.0000	−0.373979
$716$	−4.00000	−0.149487
$717$	0	0
$718$	−31.0000	−1.15691
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	−18.0000	−0.669891
$723$	0	0
$724$	−21.0000	−0.780459
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−23.0000	−0.853023	−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000	−0.853023	−0.426511	+	0.904482i	$0.640258\pi$
$728$	2.00000	0.0741249
$729$	0	0
$730$	−13.0000	−0.481152
$731$	44.0000	1.62740
$732$	0	0
$733$	−42.0000	−1.55131	−0.775653	−	0.631160i	$-0.782579\pi$
$733$	−42.0000	−1.55131	−0.775653	+	0.631160i	$0.782579\pi$
$734$	22.0000	0.812035
$735$	0	0
$736$	5.00000	0.184302
$737$	−30.0000	−1.10506
$738$	0	0
$739$	−52.0000	−1.91285	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000	−1.91285	−0.956425	+	0.291977i	$0.905687\pi$
$740$	12.0000	0.441129
$741$	0	0
$742$	−9.00000	−0.330400
$743$	19.0000	0.697042	0.348521	−	0.937301i	$-0.386684\pi$
$743$	19.0000	0.697042	0.348521	+	0.937301i	$0.386684\pi$
$744$	0	0
$745$	−9.00000	−0.329734
$746$	31.0000	1.13499
$747$	0	0
$748$	−20.0000	−0.731272
$749$	7.00000	0.255774
$750$	0	0
$751$	10.0000	0.364905	0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000	0.364905	0.182453	+	0.983215i	$0.441596\pi$
$752$	10.0000	0.364662
$753$	0	0
$754$	−8.00000	−0.291343
$755$	8.00000	0.291150
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	−29.0000	−1.05333
$759$	0	0
$760$	1.00000	0.0362738
$761$	17.0000	0.616250	0.308125	−	0.951346i	$-0.400299\pi$
$761$	17.0000	0.616250	0.308125	+	0.951346i	$0.400299\pi$
$762$	0	0
$763$	0	0
$764$	24.0000	0.868290
$765$	0	0
$766$	−20.0000	−0.722629
$767$	20.0000	0.722158
$768$	0	0
$769$	17.0000	0.613036	0.306518	−	0.951865i	$-0.400836\pi$
$769$	17.0000	0.613036	0.306518	+	0.951865i	$0.400836\pi$
$770$	−5.00000	−0.180187
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	13.0000	0.467578	0.233789	−	0.972287i	$-0.424888\pi$
$773$	13.0000	0.467578	0.233789	+	0.972287i	$0.424888\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	−18.0000	−0.646162
$777$	0	0
$778$	−14.0000	−0.501924
$779$	−4.00000	−0.143315
$780$	0	0
$781$	75.0000	2.68371
$782$	20.0000	0.715199
$783$	0	0
$784$	−6.00000	−0.214286
$785$	−13.0000	−0.463990
$786$	0	0
$787$	31.0000	1.10503	0.552515	−	0.833503i	$-0.313668\pi$
$787$	31.0000	1.10503	0.552515	+	0.833503i	$0.313668\pi$
$788$	26.0000	0.926212
$789$	0	0
$790$	13.0000	0.462519
$791$	3.00000	0.106668
$792$	0	0
$793$	20.0000	0.710221
$794$	−37.0000	−1.31308
$795$	0	0
$796$	13.0000	0.460773
$797$	−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
$799$	40.0000	1.41510
$800$	1.00000	0.0353553
$801$	0	0
$802$	−19.0000	−0.670913
$803$	65.0000	2.29380
$804$	0	0
$805$	5.00000	0.176227
$806$	−2.00000	−0.0704470
$807$	0	0
$808$	−3.00000	−0.105540
$809$	29.0000	1.01959	0.509793	−	0.860297i	$-0.329722\pi$
$809$	29.0000	1.01959	0.509793	+	0.860297i	$0.329722\pi$
$810$	0	0
$811$	41.0000	1.43970	0.719852	−	0.694127i	$-0.244209\pi$
$811$	41.0000	1.43970	0.719852	+	0.694127i	$0.244209\pi$
$812$	−4.00000	−0.140372
$813$	0	0
$814$	−60.0000	−2.10300
$815$	6.00000	0.210171
$816$	0	0
$817$	11.0000	0.384841
$818$	−24.0000	−0.839140
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−8.00000	−0.279202	−0.139601	−	0.990208i	$-0.544582\pi$
$821$	−8.00000	−0.279202	−0.139601	+	0.990208i	$0.544582\pi$
$822$	0	0
$823$	10.0000	0.348578	0.174289	−	0.984695i	$-0.444237\pi$
$823$	10.0000	0.348578	0.174289	+	0.984695i	$0.444237\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	10.0000	0.347945
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−37.0000	−1.28506	−0.642532	−	0.766259i	$-0.722116\pi$
$829$	−37.0000	−1.28506	−0.642532	+	0.766259i	$0.722116\pi$
$830$	8.00000	0.277684
$831$	0	0
$832$	2.00000	0.0693375
$833$	−24.0000	−0.831551
$834$	0	0
$835$	5.00000	0.173032
$836$	−5.00000	−0.172929
$837$	0	0
$838$	−20.0000	−0.690889
$839$	21.0000	0.725001	0.362500	−	0.931984i	$-0.381923\pi$
$839$	21.0000	0.725001	0.362500	+	0.931984i	$0.381923\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−10.0000	−0.344623
$843$	0	0
$844$	−15.0000	−0.516321
$845$	−9.00000	−0.309609
$846$	0	0
$847$	14.0000	0.481046
$848$	−9.00000	−0.309061
$849$	0	0
$850$	4.00000	0.137199
$851$	60.0000	2.05677
$852$	0	0
$853$	−1.00000	−0.0342393	−0.0171197	−	0.999853i	$-0.505450\pi$
$853$	−1.00000	−0.0342393	−0.0171197	+	0.999853i	$0.505450\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	7.00000	0.239255
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	11.0000	0.375097
$861$	0	0
$862$	0	0
$863$	11.0000	0.374444	0.187222	−	0.982318i	$-0.440052\pi$
$863$	11.0000	0.374444	0.187222	+	0.982318i	$0.440052\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	1.00000	0.0339814
$867$	0	0
$868$	−1.00000	−0.0339422
$869$	−65.0000	−2.20497
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	0	0
$874$	5.00000	0.169128
$875$	1.00000	0.0338062
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	−22.0000	−0.742464
$879$	0	0
$880$	−5.00000	−0.168550
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	41.0000	1.37976	0.689880	−	0.723924i	$-0.257663\pi$
$883$	41.0000	1.37976	0.689880	+	0.723924i	$0.257663\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	−7.00000	−0.235170
$887$	−10.0000	−0.335767	−0.167884	−	0.985807i	$-0.553693\pi$
$887$	−10.0000	−0.335767	−0.167884	+	0.985807i	$0.553693\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	−3.00000	−0.100560
$891$	0	0
$892$	10.0000	0.334825
$893$	10.0000	0.334637
$894$	0	0
$895$	−4.00000	−0.133705
$896$	1.00000	0.0334077
$897$	0	0
$898$	26.0000	0.867631
$899$	4.00000	0.133407
$900$	0	0
$901$	−36.0000	−1.19933
$902$	20.0000	0.665927
$903$	0	0
$904$	3.00000	0.0997785
$905$	−21.0000	−0.698064
$906$	0	0
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	−13.0000	−0.431420
$909$	0	0
$910$	2.00000	0.0662994
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	−40.0000	−1.32381
$914$	6.00000	0.198462
$915$	0	0
$916$	−5.00000	−0.165205
$917$	−10.0000	−0.330229
$918$	0	0
$919$	−26.0000	−0.857661	−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000	−0.857661	−0.428830	+	0.903385i	$0.641074\pi$
$920$	5.00000	0.164845
$921$	0	0
$922$	−20.0000	−0.658665
$923$	−30.0000	−0.987462
$924$	0	0
$925$	12.0000	0.394558
$926$	36.0000	1.18303
$927$	0	0
$928$	−4.00000	−0.131306
$929$	−29.0000	−0.951459	−0.475730	−	0.879592i	$-0.657816\pi$
$929$	−29.0000	−0.951459	−0.475730	+	0.879592i	$0.657816\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	13.0000	0.425829
$933$	0	0
$934$	−24.0000	−0.785304
$935$	−20.0000	−0.654070
$936$	0	0
$937$	16.0000	0.522697	0.261349	−	0.965244i	$-0.415833\pi$
$937$	16.0000	0.522697	0.261349	+	0.965244i	$0.415833\pi$
$938$	6.00000	0.195907
$939$	0	0
$940$	10.0000	0.326164
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	0	0
$943$	−20.0000	−0.651290
$944$	10.0000	0.325472
$945$	0	0
$946$	−55.0000	−1.78820
$947$	−46.0000	−1.49480	−0.747400	−	0.664375i	$-0.768698\pi$
$947$	−46.0000	−1.49480	−0.747400	+	0.664375i	$0.768698\pi$
$948$	0	0
$949$	−26.0000	−0.843996
$950$	1.00000	0.0324443
$951$	0	0
$952$	4.00000	0.129641
$953$	44.0000	1.42530	0.712650	−	0.701520i	$-0.247495\pi$
$953$	44.0000	1.42530	0.712650	+	0.701520i	$0.247495\pi$
$954$	0	0
$955$	24.0000	0.776622
$956$	16.0000	0.517477
$957$	0	0
$958$	−11.0000	−0.355394
$959$	−6.00000	−0.193750
$960$	0	0
$961$	1.00000	0.0322581
$962$	24.0000	0.773791
$963$	0	0
$964$	22.0000	0.708572
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	36.0000	1.15768	0.578841	−	0.815440i	$-0.303505\pi$
$967$	36.0000	1.15768	0.578841	+	0.815440i	$0.303505\pi$
$968$	14.0000	0.449977
$969$	0	0
$970$	−18.0000	−0.577945
$971$	26.0000	0.834380	0.417190	−	0.908819i	$-0.363015\pi$
$971$	26.0000	0.834380	0.417190	+	0.908819i	$0.363015\pi$
$972$	0	0
$973$	−14.0000	−0.448819
$974$	12.0000	0.384505
$975$	0	0
$976$	10.0000	0.320092
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	0	0
$979$	15.0000	0.479402
$980$	−6.00000	−0.191663
$981$	0	0
$982$	11.0000	0.351024
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	26.0000	0.828429
$986$	−16.0000	−0.509544
$987$	0	0
$988$	2.00000	0.0636285
$989$	55.0000	1.74890
$990$	0	0
$991$	−11.0000	−0.349427	−0.174713	−	0.984619i	$-0.555900\pi$
$991$	−11.0000	−0.349427	−0.174713	+	0.984619i	$0.555900\pi$
$992$	−1.00000	−0.0317500
$993$	0	0
$994$	−15.0000	−0.475771
$995$	13.0000	0.412128
$996$	0	0
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	−24.0000	−0.759707
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.bb.1.1		1
3.2	odd	2		930.2.a.f.1.1	✓	1
12.11	even	2		7440.2.a.c.1.1		1
15.2	even	4		4650.2.d.l.3349.1		2
15.8	even	4		4650.2.d.l.3349.2		2
15.14	odd	2		4650.2.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.f.1.1	✓	1	3.2	odd	2
2790.2.a.bb.1.1		1	1.1	even	1	trivial
4650.2.a.bb.1.1		1	15.14	odd	2
4650.2.d.l.3349.1		2	15.2	even	4
4650.2.d.l.3349.2		2	15.8	even	4
7440.2.a.c.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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