LMFDB - Embedded newform 2205.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,2,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, 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("2205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2205.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:	$17.6070136457$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} +O(q^{10})$

$q+2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} +2.00000 q^{10} -1.00000 q^{11} +3.00000 q^{13} -4.00000 q^{16} +3.00000 q^{17} +6.00000 q^{19} +2.00000 q^{20} -2.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{29} +6.00000 q^{31} -8.00000 q^{32} +6.00000 q^{34} +12.0000 q^{38} -6.00000 q^{41} -6.00000 q^{43} -2.00000 q^{44} +8.00000 q^{46} +9.00000 q^{47} +2.00000 q^{50} +6.00000 q^{52} +10.0000 q^{53} -1.00000 q^{55} +2.00000 q^{58} +6.00000 q^{59} +12.0000 q^{62} -8.00000 q^{64} +3.00000 q^{65} -14.0000 q^{67} +6.00000 q^{68} +8.00000 q^{71} +6.00000 q^{73} +12.0000 q^{76} -1.00000 q^{79} -4.00000 q^{80} -12.0000 q^{82} -12.0000 q^{83} +3.00000 q^{85} -12.0000 q^{86} -12.0000 q^{89} +8.00000 q^{92} +18.0000 q^{94} +6.00000 q^{95} -15.0000 q^{97} +O(q^{100})$

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	2.00000	0.632456
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	−2.00000	−0.426401
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.00000	1.17670
$27$	0	0
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	12.0000	1.94666
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	8.00000	1.17954
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	0	0
$50$	2.00000	0.282843
$51$	0	0
$52$	6.00000	0.832050
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	2.00000	0.262613
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	12.0000	1.52400
$63$	0	0
$64$	−8.00000	−1.00000
$65$	3.00000	0.372104
$66$	0	0
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	12.0000	1.37649
$77$	0	0
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	−12.0000	−1.32518
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	−12.0000	−1.29399
$87$	0	0
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	0	0
$92$	8.00000	0.834058
$93$	0	0
$94$	18.0000	1.85656
$95$	6.00000	0.615587
$96$	0	0
$97$	−15.0000	−1.52302	−0.761510	−	0.648154i	$-0.775541\pi$
$97$	−15.0000	−1.52302	−0.761510	+	0.648154i	$0.775541\pi$
$98$	0	0
$99$	0	0
$100$	2.00000	0.200000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	−9.00000	−0.886796	−0.443398	−	0.896325i	$-0.646227\pi$
$103$	−9.00000	−0.886796	−0.443398	+	0.896325i	$0.646227\pi$
$104$	0	0
$105$	0	0
$106$	20.0000	1.94257
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	0	0
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	0	0
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	2.00000	0.185695
$117$	0	0
$118$	12.0000	1.10469
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0	0
$124$	12.0000	1.07763
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	0	0
$130$	6.00000	0.526235
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	−28.0000	−2.41883
$135$	0	0
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	16.0000	1.34269
$143$	−3.00000	−0.250873
$144$	0	0
$145$	1.00000	0.0830455
$146$	12.0000	0.993127
$147$	0	0
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	15.0000	1.22068	0.610341	−	0.792139i	$-0.291032\pi$
$151$	15.0000	1.22068	0.610341	+	0.792139i	$0.291032\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	−2.00000	−0.159111
$159$	0	0
$160$	−8.00000	−0.632456
$161$	0	0
$162$	0	0
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	−24.0000	−1.86276
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	6.00000	0.460179
$171$	0	0
$172$	−12.0000	−0.914991
$173$	3.00000	0.228086	0.114043	−	0.993476i	$-0.463620\pi$
$173$	3.00000	0.228086	0.114043	+	0.993476i	$0.463620\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	−24.0000	−1.79888
$179$	−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
$181$	−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
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.00000	−0.219382
$188$	18.0000	1.31278
$189$	0	0
$190$	12.0000	0.870572
$191$	−17.0000	−1.23008	−0.615038	−	0.788497i	$-0.710860\pi$
$191$	−17.0000	−1.23008	−0.615038	+	0.788497i	$0.710860\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	−30.0000	−2.15387
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	0	0
$201$	0	0
$202$	36.0000	2.53295
$203$	0	0
$204$	0	0
$205$	−6.00000	−0.419058
$206$	−18.0000	−1.25412
$207$	0	0
$208$	−12.0000	−0.832050
$209$	−6.00000	−0.415029
$210$	0	0
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	20.0000	1.37361
$213$	0	0
$214$	4.00000	0.273434
$215$	−6.00000	−0.409197
$216$	0	0
$217$	0	0
$218$	−30.0000	−2.03186
$219$	0	0
$220$	−2.00000	−0.134840
$221$	9.00000	0.605406
$222$	0	0
$223$	3.00000	0.200895	0.100447	−	0.994942i	$-0.467973\pi$
$223$	3.00000	0.200895	0.100447	+	0.994942i	$0.467973\pi$
$224$	0	0
$225$	0	0
$226$	−16.0000	−1.06430
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	0	0
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	12.0000	0.781133
$237$	0	0
$238$	0	0
$239$	−7.00000	−0.452792	−0.226396	−	0.974035i	$-0.572694\pi$
$239$	−7.00000	−0.452792	−0.226396	+	0.974035i	$0.572694\pi$
$240$	0	0
$241$	−24.0000	−1.54598	−0.772988	−	0.634421i	$-0.781239\pi$
$241$	−24.0000	−1.54598	−0.772988	+	0.634421i	$0.781239\pi$
$242$	−20.0000	−1.28565
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.0000	1.14531
$248$	0	0
$249$	0	0
$250$	2.00000	0.126491
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−4.00000	−0.250982
$255$	0	0
$256$	16.0000	1.00000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	0	0
$260$	6.00000	0.372104
$261$	0	0
$262$	−24.0000	−1.48272
$263$	10.0000	0.616626	0.308313	−	0.951285i	$-0.400236\pi$
$263$	10.0000	0.616626	0.308313	+	0.951285i	$0.400236\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	0	0
$268$	−28.0000	−1.71037
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	−12.0000	−0.727607
$273$	0	0
$274$	−16.0000	−0.966595
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.00000	0.298275	0.149137	−	0.988816i	$-0.452350\pi$
$281$	5.00000	0.298275	0.149137	+	0.988816i	$0.452350\pi$
$282$	0	0
$283$	21.0000	1.24832	0.624160	−	0.781296i	$-0.285441\pi$
$283$	21.0000	1.24832	0.624160	+	0.781296i	$0.285441\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	−6.00000	−0.354787
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	2.00000	0.117444
$291$	0	0
$292$	12.0000	0.702247
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	0	0
$297$	0	0
$298$	−20.0000	−1.15857
$299$	12.0000	0.693978
$300$	0	0
$301$	0	0
$302$	30.0000	1.72631
$303$	0	0
$304$	−24.0000	−1.37649
$305$	0	0
$306$	0	0
$307$	−3.00000	−0.171219	−0.0856095	−	0.996329i	$-0.527284\pi$
$307$	−3.00000	−0.171219	−0.0856095	+	0.996329i	$0.527284\pi$
$308$	0	0
$309$	0	0
$310$	12.0000	0.681554
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	3.00000	0.169570	0.0847850	−	0.996399i	$-0.472980\pi$
$313$	3.00000	0.169570	0.0847850	+	0.996399i	$0.472980\pi$
$314$	36.0000	2.03160
$315$	0	0
$316$	−2.00000	−0.112509
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	−1.00000	−0.0559893
$320$	−8.00000	−0.447214
$321$	0	0
$322$	0	0
$323$	18.0000	1.00155
$324$	0	0
$325$	3.00000	0.166410
$326$	32.0000	1.77232
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	−24.0000	−1.31717
$333$	0	0
$334$	−6.00000	−0.328305
$335$	−14.0000	−0.764902
$336$	0	0
$337$	−24.0000	−1.30736	−0.653682	−	0.756770i	$-0.726776\pi$
$337$	−24.0000	−1.30736	−0.653682	+	0.756770i	$0.726776\pi$
$338$	−8.00000	−0.435143
$339$	0	0
$340$	6.00000	0.325396
$341$	−6.00000	−0.324918
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	6.00000	0.322562
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	0	0
$352$	8.00000	0.426401
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	−24.0000	−1.27200
$357$	0	0
$358$	−8.00000	−0.422813
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−12.0000	−0.630706
$363$	0	0
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	−33.0000	−1.72259	−0.861293	−	0.508109i	$-0.830345\pi$
$367$	−33.0000	−1.72259	−0.861293	+	0.508109i	$0.830345\pi$
$368$	−16.0000	−0.834058
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	0	0
$377$	3.00000	0.154508
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	12.0000	0.615587
$381$	0	0
$382$	−34.0000	−1.73959
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	24.0000	1.22157
$387$	0	0
$388$	−30.0000	−1.52302
$389$	−13.0000	−0.659126	−0.329563	−	0.944134i	$-0.606901\pi$
$389$	−13.0000	−0.659126	−0.329563	+	0.944134i	$0.606901\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	0	0
$394$	4.00000	0.201517
$395$	−1.00000	−0.0503155
$396$	0	0
$397$	9.00000	0.451697	0.225849	−	0.974162i	$-0.427485\pi$
$397$	9.00000	0.451697	0.225849	+	0.974162i	$0.427485\pi$
$398$	12.0000	0.601506
$399$	0	0
$400$	−4.00000	−0.200000
$401$	19.0000	0.948815	0.474407	−	0.880305i	$-0.342662\pi$
$401$	19.0000	0.948815	0.474407	+	0.880305i	$0.342662\pi$
$402$	0	0
$403$	18.0000	0.896644
$404$	36.0000	1.79107
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	−12.0000	−0.592638
$411$	0	0
$412$	−18.0000	−0.886796
$413$	0	0
$414$	0	0
$415$	−12.0000	−0.589057
$416$	−24.0000	−1.17670
$417$	0	0
$418$	−12.0000	−0.586939
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	30.0000	1.46038
$423$	0	0
$424$	0	0
$425$	3.00000	0.145521
$426$	0	0
$427$	0	0
$428$	4.00000	0.193347
$429$	0	0
$430$	−12.0000	−0.578691
$431$	−5.00000	−0.240842	−0.120421	−	0.992723i	$-0.538424\pi$
$431$	−5.00000	−0.240842	−0.120421	+	0.992723i	$0.538424\pi$
$432$	0	0
$433$	−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
$435$	0	0
$436$	−30.0000	−1.43674
$437$	24.0000	1.14808
$438$	0	0
$439$	12.0000	0.572729	0.286364	−	0.958121i	$-0.407553\pi$
$439$	12.0000	0.572729	0.286364	+	0.958121i	$0.407553\pi$
$440$	0	0
$441$	0	0
$442$	18.0000	0.856173
$443$	−34.0000	−1.61539	−0.807694	−	0.589601i	$-0.799285\pi$
$443$	−34.0000	−1.61539	−0.807694	+	0.589601i	$0.799285\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	6.00000	0.284108
$447$	0	0
$448$	0	0
$449$	−23.0000	−1.08544	−0.542719	−	0.839915i	$-0.682605\pi$
$449$	−23.0000	−1.08544	−0.542719	+	0.839915i	$0.682605\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	−16.0000	−0.752577
$453$	0	0
$454$	−6.00000	−0.281594
$455$	0	0
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	−12.0000	−0.560723
$459$	0	0
$460$	8.00000	0.373002
$461$	−36.0000	−1.67669	−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000	−1.67669	−0.838344	+	0.545142i	$0.816476\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	16.0000	0.741186
$467$	15.0000	0.694117	0.347059	−	0.937843i	$-0.387180\pi$
$467$	15.0000	0.694117	0.347059	+	0.937843i	$0.387180\pi$
$468$	0	0
$469$	0	0
$470$	18.0000	0.830278
$471$	0	0
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	0	0
$478$	−14.0000	−0.640345
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	0	0
$482$	−48.0000	−2.18634
$483$	0	0
$484$	−20.0000	−0.909091
$485$	−15.0000	−0.681115
$486$	0	0
$487$	36.0000	1.63132	0.815658	−	0.578535i	$-0.196375\pi$
$487$	36.0000	1.63132	0.815658	+	0.578535i	$0.196375\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−23.0000	−1.03798	−0.518988	−	0.854782i	$-0.673691\pi$
$491$	−23.0000	−1.03798	−0.518988	+	0.854782i	$0.673691\pi$
$492$	0	0
$493$	3.00000	0.135113
$494$	36.0000	1.61972
$495$	0	0
$496$	−24.0000	−1.07763
$497$	0	0
$498$	0	0
$499$	−27.0000	−1.20869	−0.604343	−	0.796724i	$-0.706564\pi$
$499$	−27.0000	−1.20869	−0.604343	+	0.796724i	$0.706564\pi$
$500$	2.00000	0.0894427
$501$	0	0
$502$	24.0000	1.07117
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	−8.00000	−0.355643
$507$	0	0
$508$	−4.00000	−0.177471
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	0	0
$512$	32.0000	1.41421
$513$	0	0
$514$	−36.0000	−1.58789
$515$	−9.00000	−0.396587
$516$	0	0
$517$	−9.00000	−0.395820
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	−24.0000	−1.04844
$525$	0	0
$526$	20.0000	0.872041
$527$	18.0000	0.784092
$528$	0	0
$529$	−7.00000	−0.304348
$530$	20.0000	0.868744
$531$	0	0
$532$	0	0
$533$	−18.0000	−0.779667
$534$	0	0
$535$	2.00000	0.0864675
$536$	0	0
$537$	0	0
$538$	−48.0000	−2.06943
$539$	0	0
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	48.0000	2.06178
$543$	0	0
$544$	−24.0000	−1.02899
$545$	−15.0000	−0.642529
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−16.0000	−0.683486
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	6.00000	0.255609
$552$	0	0
$553$	0	0
$554$	−56.0000	−2.37921
$555$	0	0
$556$	0	0
$557$	−4.00000	−0.169485	−0.0847427	−	0.996403i	$-0.527007\pi$
$557$	−4.00000	−0.169485	−0.0847427	+	0.996403i	$0.527007\pi$
$558$	0	0
$559$	−18.0000	−0.761319
$560$	0	0
$561$	0	0
$562$	10.0000	0.421825
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	42.0000	1.76539
$567$	0	0
$568$	0	0
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−6.00000	−0.250873
$573$	0	0
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	15.0000	0.624458	0.312229	−	0.950007i	$-0.398924\pi$
$577$	15.0000	0.624458	0.312229	+	0.950007i	$0.398924\pi$
$578$	−16.0000	−0.665512
$579$	0	0
$580$	2.00000	0.0830455
$581$	0	0
$582$	0	0
$583$	−10.0000	−0.414158
$584$	0	0
$585$	0	0
$586$	−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
$589$	36.0000	1.48335
$590$	12.0000	0.494032
$591$	0	0
$592$	0	0
$593$	45.0000	1.84793	0.923964	−	0.382479i	$-0.124930\pi$
$593$	45.0000	1.84793	0.923964	+	0.382479i	$0.124930\pi$
$594$	0	0
$595$	0	0
$596$	−20.0000	−0.819232
$597$	0	0
$598$	24.0000	0.981433
$599$	−17.0000	−0.694601	−0.347301	−	0.937754i	$-0.612902\pi$
$599$	−17.0000	−0.694601	−0.347301	+	0.937754i	$0.612902\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	0	0
$604$	30.0000	1.22068
$605$	−10.0000	−0.406558
$606$	0	0
$607$	21.0000	0.852364	0.426182	−	0.904638i	$-0.359858\pi$
$607$	21.0000	0.852364	0.426182	+	0.904638i	$0.359858\pi$
$608$	−48.0000	−1.94666
$609$	0	0
$610$	0	0
$611$	27.0000	1.09230
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−6.00000	−0.242140
$615$	0	0
$616$	0	0
$617$	−4.00000	−0.161034	−0.0805170	−	0.996753i	$-0.525657\pi$
$617$	−4.00000	−0.161034	−0.0805170	+	0.996753i	$0.525657\pi$
$618$	0	0
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	12.0000	0.481932
$621$	0	0
$622$	12.0000	0.481156
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	0	0
$628$	36.0000	1.43656
$629$	0	0
$630$	0	0
$631$	−11.0000	−0.437903	−0.218952	−	0.975736i	$-0.570264\pi$
$631$	−11.0000	−0.437903	−0.218952	+	0.975736i	$0.570264\pi$
$632$	0	0
$633$	0	0
$634$	44.0000	1.74746
$635$	−2.00000	−0.0793676
$636$	0	0
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	0	0
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	27.0000	1.06478	0.532388	−	0.846500i	$-0.321295\pi$
$643$	27.0000	1.06478	0.532388	+	0.846500i	$0.321295\pi$
$644$	0	0
$645$	0	0
$646$	36.0000	1.41640
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	6.00000	0.235339
$651$	0	0
$652$	32.0000	1.25322
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	24.0000	0.937043
$657$	0	0
$658$	0	0
$659$	19.0000	0.740135	0.370067	−	0.929005i	$-0.379335\pi$
$659$	19.0000	0.740135	0.370067	+	0.929005i	$0.379335\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	24.0000	0.932786
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.00000	0.154881
$668$	−6.00000	−0.232147
$669$	0	0
$670$	−28.0000	−1.08173
$671$	0	0
$672$	0	0
$673$	12.0000	0.462566	0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000	0.462566	0.231283	+	0.972887i	$0.425708\pi$
$674$	−48.0000	−1.84889
$675$	0	0
$676$	−8.00000	−0.307692
$677$	9.00000	0.345898	0.172949	−	0.984931i	$-0.444670\pi$
$677$	9.00000	0.345898	0.172949	+	0.984931i	$0.444670\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	−12.0000	−0.459504
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	0	0
$685$	−8.00000	−0.305664
$686$	0	0
$687$	0	0
$688$	24.0000	0.914991
$689$	30.0000	1.14291
$690$	0	0
$691$	36.0000	1.36950	0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000	1.36950	0.684752	+	0.728776i	$0.259910\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−32.0000	−1.21470
$695$	0	0
$696$	0	0
$697$	−18.0000	−0.681799
$698$	−60.0000	−2.27103
$699$	0	0
$700$	0	0
$701$	−47.0000	−1.77517	−0.887583	−	0.460648i	$-0.847617\pi$
$701$	−47.0000	−1.77517	−0.887583	+	0.460648i	$0.847617\pi$
$702$	0	0
$703$	0	0
$704$	8.00000	0.301511
$705$	0	0
$706$	6.00000	0.225813
$707$	0	0
$708$	0	0
$709$	−33.0000	−1.23934	−0.619671	−	0.784862i	$-0.712734\pi$
$709$	−33.0000	−1.23934	−0.619671	+	0.784862i	$0.712734\pi$
$710$	16.0000	0.600469
$711$	0	0
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	−3.00000	−0.112194
$716$	−8.00000	−0.298974
$717$	0	0
$718$	−64.0000	−2.38846
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	0	0
$722$	34.0000	1.26535
$723$	0	0
$724$	−12.0000	−0.445976
$725$	1.00000	0.0371391
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	0	0
$730$	12.0000	0.444140
$731$	−18.0000	−0.665754
$732$	0	0
$733$	15.0000	0.554038	0.277019	−	0.960864i	$-0.410654\pi$
$733$	15.0000	0.554038	0.277019	+	0.960864i	$0.410654\pi$
$734$	−66.0000	−2.43610
$735$	0	0
$736$	−32.0000	−1.17954
$737$	14.0000	0.515697
$738$	0	0
$739$	−43.0000	−1.58178	−0.790890	−	0.611958i	$-0.790382\pi$
$739$	−43.0000	−1.58178	−0.790890	+	0.611958i	$0.790382\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	22.0000	0.807102	0.403551	−	0.914957i	$-0.367776\pi$
$743$	22.0000	0.807102	0.403551	+	0.914957i	$0.367776\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	8.00000	0.292901
$747$	0	0
$748$	−6.00000	−0.219382
$749$	0	0
$750$	0	0
$751$	15.0000	0.547358	0.273679	−	0.961821i	$-0.411759\pi$
$751$	15.0000	0.547358	0.273679	+	0.961821i	$0.411759\pi$
$752$	−36.0000	−1.31278
$753$	0	0
$754$	6.00000	0.218507
$755$	15.0000	0.545906
$756$	0	0
$757$	48.0000	1.74459	0.872295	−	0.488980i	$-0.162631\pi$
$757$	48.0000	1.74459	0.872295	+	0.488980i	$0.162631\pi$
$758$	−48.0000	−1.74344
$759$	0	0
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	0	0
$764$	−34.0000	−1.23008
$765$	0	0
$766$	0	0
$767$	18.0000	0.649942
$768$	0	0
$769$	42.0000	1.51456	0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000	1.51456	0.757279	+	0.653091i	$0.226528\pi$
$770$	0	0
$771$	0	0
$772$	24.0000	0.863779
$773$	−3.00000	−0.107903	−0.0539513	−	0.998544i	$-0.517182\pi$
$773$	−3.00000	−0.107903	−0.0539513	+	0.998544i	$0.517182\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	0	0
$777$	0	0
$778$	−26.0000	−0.932145
$779$	−36.0000	−1.28983
$780$	0	0
$781$	−8.00000	−0.286263
$782$	24.0000	0.858238
$783$	0	0
$784$	0	0
$785$	18.0000	0.642448
$786$	0	0
$787$	39.0000	1.39020	0.695100	−	0.718913i	$-0.255360\pi$
$787$	39.0000	1.39020	0.695100	+	0.718913i	$0.255360\pi$
$788$	4.00000	0.142494
$789$	0	0
$790$	−2.00000	−0.0711568
$791$	0	0
$792$	0	0
$793$	0	0
$794$	18.0000	0.638796
$795$	0	0
$796$	12.0000	0.425329
$797$	27.0000	0.956389	0.478195	−	0.878254i	$-0.341291\pi$
$797$	27.0000	0.956389	0.478195	+	0.878254i	$0.341291\pi$
$798$	0	0
$799$	27.0000	0.955191
$800$	−8.00000	−0.282843
$801$	0	0
$802$	38.0000	1.34183
$803$	−6.00000	−0.211735
$804$	0	0
$805$	0	0
$806$	36.0000	1.26805
$807$	0	0
$808$	0	0
$809$	35.0000	1.23053	0.615267	−	0.788319i	$-0.289048\pi$
$809$	35.0000	1.23053	0.615267	+	0.788319i	$0.289048\pi$
$810$	0	0
$811$	48.0000	1.68551	0.842754	−	0.538299i	$-0.180933\pi$
$811$	48.0000	1.68551	0.842754	+	0.538299i	$0.180933\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	16.0000	0.560456
$816$	0	0
$817$	−36.0000	−1.25948
$818$	60.0000	2.09785
$819$	0	0
$820$	−12.0000	−0.419058
$821$	23.0000	0.802706	0.401353	−	0.915924i	$-0.368540\pi$
$821$	23.0000	0.802706	0.401353	+	0.915924i	$0.368540\pi$
$822$	0	0
$823$	50.0000	1.74289	0.871445	−	0.490493i	$-0.163183\pi$
$823$	50.0000	1.74289	0.871445	+	0.490493i	$0.163183\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.0000	0.556375	0.278187	−	0.960527i	$-0.410266\pi$
$827$	16.0000	0.556375	0.278187	+	0.960527i	$0.410266\pi$
$828$	0	0
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	−24.0000	−0.833052
$831$	0	0
$832$	−24.0000	−0.832050
$833$	0	0
$834$	0	0
$835$	−3.00000	−0.103819
$836$	−12.0000	−0.415029
$837$	0	0
$838$	12.0000	0.414533
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	2.00000	0.0689246
$843$	0	0
$844$	30.0000	1.03264
$845$	−4.00000	−0.137604
$846$	0	0
$847$	0	0
$848$	−40.0000	−1.37361
$849$	0	0
$850$	6.00000	0.205798
$851$	0	0
$852$	0	0
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	−12.0000	−0.409197
$861$	0	0
$862$	−10.0000	−0.340601
$863$	34.0000	1.15737	0.578687	−	0.815550i	$-0.303565\pi$
$863$	34.0000	1.15737	0.578687	+	0.815550i	$0.303565\pi$
$864$	0	0
$865$	3.00000	0.102003
$866$	−60.0000	−2.03888
$867$	0	0
$868$	0	0
$869$	1.00000	0.0339227
$870$	0	0
$871$	−42.0000	−1.42312
$872$	0	0
$873$	0	0
$874$	48.0000	1.62362
$875$	0	0
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	24.0000	0.809961
$879$	0	0
$880$	4.00000	0.134840
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	18.0000	0.605406
$885$	0	0
$886$	−68.0000	−2.28450
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	0	0
$890$	−24.0000	−0.804482
$891$	0	0
$892$	6.00000	0.200895
$893$	54.0000	1.80704
$894$	0	0
$895$	−4.00000	−0.133705
$896$	0	0
$897$	0	0
$898$	−46.0000	−1.53504
$899$	6.00000	0.200111
$900$	0	0
$901$	30.0000	0.999445
$902$	12.0000	0.399556
$903$	0	0
$904$	0	0
$905$	−6.00000	−0.199447
$906$	0	0
$907$	24.0000	0.796907	0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000	0.796907	0.398453	+	0.917189i	$0.369547\pi$
$908$	−6.00000	−0.199117
$909$	0	0
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	12.0000	0.396925
$915$	0	0
$916$	−12.0000	−0.396491
$917$	0	0
$918$	0	0
$919$	−9.00000	−0.296883	−0.148441	−	0.988921i	$-0.547426\pi$
$919$	−9.00000	−0.296883	−0.148441	+	0.988921i	$0.547426\pi$
$920$	0	0
$921$	0	0
$922$	−72.0000	−2.37119
$923$	24.0000	0.789970
$924$	0	0
$925$	0	0
$926$	−8.00000	−0.262896
$927$	0	0
$928$	−8.00000	−0.262613
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	0	0
$932$	16.0000	0.524097
$933$	0	0
$934$	30.0000	0.981630
$935$	−3.00000	−0.0981105
$936$	0	0
$937$	−27.0000	−0.882052	−0.441026	−	0.897494i	$-0.645385\pi$
$937$	−27.0000	−0.882052	−0.441026	+	0.897494i	$0.645385\pi$
$938$	0	0
$939$	0	0
$940$	18.0000	0.587095
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	−24.0000	−0.781133
$945$	0	0
$946$	12.0000	0.390154
$947$	2.00000	0.0649913	0.0324956	−	0.999472i	$-0.489654\pi$
$947$	2.00000	0.0649913	0.0324956	+	0.999472i	$0.489654\pi$
$948$	0	0
$949$	18.0000	0.584305
$950$	12.0000	0.389331
$951$	0	0
$952$	0	0
$953$	38.0000	1.23094	0.615470	−	0.788160i	$-0.288966\pi$
$953$	38.0000	1.23094	0.615470	+	0.788160i	$0.288966\pi$
$954$	0	0
$955$	−17.0000	−0.550107
$956$	−14.0000	−0.452792
$957$	0	0
$958$	48.0000	1.55081
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	0	0
$964$	−48.0000	−1.54598
$965$	12.0000	0.386294
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	0	0
$969$	0	0
$970$	−30.0000	−0.963242
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	0	0
$974$	72.0000	2.30703
$975$	0	0
$976$	0	0
$977$	−34.0000	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	0	0
$982$	−46.0000	−1.46792
$983$	−45.0000	−1.43528	−0.717639	−	0.696416i	$-0.754777\pi$
$983$	−45.0000	−1.43528	−0.717639	+	0.696416i	$0.754777\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	6.00000	0.191079
$987$	0	0
$988$	36.0000	1.14531
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−36.0000	−1.14358	−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000	−1.14358	−0.571789	+	0.820401i	$0.693750\pi$
$992$	−48.0000	−1.52400
$993$	0	0
$994$	0	0
$995$	6.00000	0.190213
$996$	0	0
$997$	−3.00000	−0.0950110	−0.0475055	−	0.998871i	$-0.515127\pi$
$997$	−3.00000	−0.0950110	−0.0475055	+	0.998871i	$0.515127\pi$
$998$	−54.0000	−1.70934
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.a.l.1.1		1
3.2	odd	2		245.2.a.b.1.1	yes	1
7.6	odd	2		2205.2.a.j.1.1		1
12.11	even	2		3920.2.a.a.1.1		1
15.2	even	4		1225.2.b.a.99.1		2
15.8	even	4		1225.2.b.a.99.2		2
15.14	odd	2		1225.2.a.h.1.1		1
21.2	odd	6		245.2.e.c.116.1		2
21.5	even	6		245.2.e.d.116.1		2
21.11	odd	6		245.2.e.c.226.1		2
21.17	even	6		245.2.e.d.226.1		2
21.20	even	2		245.2.a.a.1.1	✓	1
84.83	odd	2		3920.2.a.bj.1.1		1
105.62	odd	4		1225.2.b.b.99.1		2
105.83	odd	4		1225.2.b.b.99.2		2
105.104	even	2		1225.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.a.a.1.1	✓	1	21.20	even	2
245.2.a.b.1.1	yes	1	3.2	odd	2
245.2.e.c.116.1		2	21.2	odd	6
245.2.e.c.226.1		2	21.11	odd	6
245.2.e.d.116.1		2	21.5	even	6
245.2.e.d.226.1		2	21.17	even	6
1225.2.a.h.1.1		1	15.14	odd	2
1225.2.a.j.1.1		1	105.104	even	2
1225.2.b.a.99.1		2	15.2	even	4
1225.2.b.a.99.2		2	15.8	even	4
1225.2.b.b.99.1		2	105.62	odd	4
1225.2.b.b.99.2		2	105.83	odd	4
2205.2.a.j.1.1		1	7.6	odd	2
2205.2.a.l.1.1		1	1.1	even	1	trivial
3920.2.a.a.1.1		1	12.11	even	2
3920.2.a.bj.1.1		1	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

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L-functions

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