LMFDB - Embedded newform 4410.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(1,4410)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4410, 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("4410.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4410.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:	$35.2140272914$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4410.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{13} +1.00000 q^{16} +3.00000 q^{19} -1.00000 q^{20} +1.00000 q^{22} -7.00000 q^{23} +1.00000 q^{25} -1.00000 q^{26} +8.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} +11.0000 q^{37} +3.00000 q^{38} -1.00000 q^{40} -11.0000 q^{41} +8.00000 q^{43} +1.00000 q^{44} -7.00000 q^{46} -5.00000 q^{47} +1.00000 q^{50} -1.00000 q^{52} +11.0000 q^{53} -1.00000 q^{55} +8.00000 q^{58} +4.00000 q^{59} +2.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +6.00000 q^{71} +6.00000 q^{73} +11.0000 q^{74} +3.00000 q^{76} -8.00000 q^{79} -1.00000 q^{80} -11.0000 q^{82} +8.00000 q^{83} +8.00000 q^{86} +1.00000 q^{88} -10.0000 q^{89} -7.00000 q^{92} -5.00000 q^{94} -3.00000 q^{95} +16.0000 q^{97} +O(q^{100})$

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 $

$2$	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
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.0000	1.80839	0.904194	−	0.427121i	$-0.140472\pi$
$37$	11.0000	1.80839	0.904194	+	0.427121i	$0.140472\pi$
$38$	3.00000	0.486664
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−11.0000	−1.71791	−0.858956	−	0.512050i	$-0.828886\pi$
$41$	−11.0000	−1.71791	−0.858956	+	0.512050i	$0.828886\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−7.00000	−1.03209
$47$	−5.00000	−0.729325	−0.364662	−	0.931140i	$-0.618816\pi$
$47$	−5.00000	−0.729325	−0.364662	+	0.931140i	$0.618816\pi$
$48$	0	0
$49$	0	0
$50$	1.00000	0.141421
$51$	0	0
$52$	−1.00000	−0.138675
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	8.00000	1.05045
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	11.0000	1.27872
$75$	0	0
$76$	3.00000	0.344124
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−11.0000	−1.21475
$83$	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
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	1.00000	0.106600
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	−7.00000	−0.729800
$93$	0	0
$94$	−5.00000	−0.515711
$95$	−3.00000	−0.307794
$96$	0	0
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	11.0000	1.06841
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	7.00000	0.652753
$116$	8.00000	0.742781
$117$	0	0
$118$	4.00000	0.368230
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0	0
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.0000	−1.50851	−0.754253	−	0.656584i	$-0.772001\pi$
$127$	−17.0000	−1.50851	−0.754253	+	0.656584i	$0.772001\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.00000	0.0877058
$131$	−5.00000	−0.436852	−0.218426	−	0.975854i	$-0.570092\pi$
$131$	−5.00000	−0.436852	−0.218426	+	0.975854i	$0.570092\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	0	0
$142$	6.00000	0.503509
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−8.00000	−0.664364
$146$	6.00000	0.496564
$147$	0	0
$148$	11.0000	0.904194
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	3.00000	0.243332
$153$	0	0
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	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	$0.284443\pi$
$164$	−11.0000	−0.858956
$165$	0	0
$166$	8.00000	0.620920
$167$	−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
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	8.00000	0.609994
$173$	−15.0000	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	−10.0000	−0.749532
$179$	19.0000	1.42013	0.710063	−	0.704138i	$-0.248666\pi$
$179$	19.0000	1.42013	0.710063	+	0.704138i	$0.248666\pi$
$180$	0	0
$181$	24.0000	1.78391	0.891953	−	0.452128i	$-0.149335\pi$
$181$	24.0000	1.78391	0.891953	+	0.452128i	$0.149335\pi$
$182$	0	0
$183$	0	0
$184$	−7.00000	−0.516047
$185$	−11.0000	−0.808736
$186$	0	0
$187$	0	0
$188$	−5.00000	−0.364662
$189$	0	0
$190$	−3.00000	−0.217643
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	0	0
$197$	1.00000	0.0712470	0.0356235	−	0.999365i	$-0.488658\pi$
$197$	1.00000	0.0712470	0.0356235	+	0.999365i	$0.488658\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	11.0000	0.768273
$206$	16.0000	1.11477
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	3.00000	0.207514
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	11.0000	0.755483
$213$	0	0
$214$	10.0000	0.683586
$215$	−8.00000	−0.545595
$216$	0	0
$217$	0	0
$218$	6.00000	0.406371
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	0	0
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	0	0
$225$	0	0
$226$	−6.00000	−0.399114
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	7.00000	0.461566
$231$	0	0
$232$	8.00000	0.525226
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	5.00000	0.326164
$236$	4.00000	0.260378
$237$	0	0
$238$	0	0
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	0	0
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	−10.0000	−0.642824
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.00000	−0.190885
$248$	2.00000	0.127000
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	13.0000	0.820553	0.410276	−	0.911961i	$-0.365432\pi$
$251$	13.0000	0.820553	0.410276	+	0.911961i	$0.365432\pi$
$252$	0	0
$253$	−7.00000	−0.440086
$254$	−17.0000	−1.06667
$255$	0	0
$256$	1.00000	0.0625000
$257$	−20.0000	−1.24757	−0.623783	−	0.781598i	$-0.714405\pi$
$257$	−20.0000	−1.24757	−0.623783	+	0.781598i	$0.714405\pi$
$258$	0	0
$259$	0	0
$260$	1.00000	0.0620174
$261$	0	0
$262$	−5.00000	−0.308901
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−11.0000	−0.675725
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.0000	1.21942	0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000	1.21942	0.609711	+	0.792624i	$0.291286\pi$
$270$	0	0
$271$	−32.0000	−1.94386	−0.971931	−	0.235267i	$-0.924404\pi$
$271$	−32.0000	−1.94386	−0.971931	+	0.235267i	$0.924404\pi$
$272$	0	0
$273$	0	0
$274$	18.0000	1.08742
$275$	1.00000	0.0603023
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−20.0000	−1.19952
$279$	0	0
$280$	0	0
$281$	1.00000	0.0596550	0.0298275	−	0.999555i	$-0.490504\pi$
$281$	1.00000	0.0596550	0.0298275	+	0.999555i	$0.490504\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	−1.00000	−0.0591312
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−8.00000	−0.469776
$291$	0	0
$292$	6.00000	0.351123
$293$	27.0000	1.57736	0.788678	−	0.614806i	$-0.210766\pi$
$293$	27.0000	1.57736	0.788678	+	0.614806i	$0.210766\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	11.0000	0.639362
$297$	0	0
$298$	0	0
$299$	7.00000	0.404820
$300$	0	0
$301$	0	0
$302$	6.00000	0.345261
$303$	0	0
$304$	3.00000	0.172062
$305$	0	0
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	0	0
$310$	−2.00000	−0.113592
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	12.0000	0.678280	0.339140	−	0.940736i	$-0.389864\pi$
$313$	12.0000	0.678280	0.339140	+	0.940736i	$0.389864\pi$
$314$	7.00000	0.395033
$315$	0	0
$316$	−8.00000	−0.450035
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	16.0000	0.886158
$327$	0	0
$328$	−11.0000	−0.607373
$329$	0	0
$330$	0	0
$331$	−13.0000	−0.714545	−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000	−0.714545	−0.357272	+	0.934000i	$0.616293\pi$
$332$	8.00000	0.439057
$333$	0	0
$334$	−3.00000	−0.164153
$335$	0	0
$336$	0	0
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	0	0
$344$	8.00000	0.431331
$345$	0	0
$346$	−15.0000	−0.806405
$347$	−14.0000	−0.751559	−0.375780	−	0.926709i	$-0.622625\pi$
$347$	−14.0000	−0.751559	−0.375780	+	0.926709i	$0.622625\pi$
$348$	0	0
$349$	−12.0000	−0.642345	−0.321173	−	0.947021i	$-0.604077\pi$
$349$	−12.0000	−0.642345	−0.321173	+	0.947021i	$0.604077\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	−10.0000	−0.529999
$357$	0	0
$358$	19.0000	1.00418
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	24.0000	1.26141
$363$	0	0
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−25.0000	−1.30499	−0.652495	−	0.757793i	$-0.726278\pi$
$367$	−25.0000	−1.30499	−0.652495	+	0.757793i	$0.726278\pi$
$368$	−7.00000	−0.364900
$369$	0	0
$370$	−11.0000	−0.571863
$371$	0	0
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	0	0
$376$	−5.00000	−0.257855
$377$	−8.00000	−0.412021
$378$	0	0
$379$	−19.0000	−0.975964	−0.487982	−	0.872854i	$-0.662267\pi$
$379$	−19.0000	−0.975964	−0.487982	+	0.872854i	$0.662267\pi$
$380$	−3.00000	−0.153897
$381$	0	0
$382$	6.00000	0.306987
$383$	35.0000	1.78842	0.894208	−	0.447651i	$-0.147739\pi$
$383$	35.0000	1.78842	0.894208	+	0.447651i	$0.147739\pi$
$384$	0	0
$385$	0	0
$386$	22.0000	1.11977
$387$	0	0
$388$	16.0000	0.812277
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	1.00000	0.0503793
$395$	8.00000	0.402524
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	24.0000	1.20301
$399$	0	0
$400$	1.00000	0.0500000
$401$	5.00000	0.249688	0.124844	−	0.992176i	$-0.460157\pi$
$401$	5.00000	0.249688	0.124844	+	0.992176i	$0.460157\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.0000	0.545250
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	11.0000	0.543251
$411$	0	0
$412$	16.0000	0.788263
$413$	0	0
$414$	0	0
$415$	−8.00000	−0.392705
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	3.00000	0.146735
$419$	−5.00000	−0.244266	−0.122133	−	0.992514i	$-0.538973\pi$
$419$	−5.00000	−0.244266	−0.122133	+	0.992514i	$0.538973\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	5.00000	0.243396
$423$	0	0
$424$	11.0000	0.534207
$425$	0	0
$426$	0	0
$427$	0	0
$428$	10.0000	0.483368
$429$	0	0
$430$	−8.00000	−0.385794
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	32.0000	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$433$	32.0000	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$434$	0	0
$435$	0	0
$436$	6.00000	0.287348
$437$	−21.0000	−1.00457
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	0	0
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	−12.0000	−0.568216
$447$	0	0
$448$	0	0
$449$	−11.0000	−0.519122	−0.259561	−	0.965727i	$-0.583578\pi$
$449$	−11.0000	−0.519122	−0.259561	+	0.965727i	$0.583578\pi$
$450$	0	0
$451$	−11.0000	−0.517970
$452$	−6.00000	−0.282216
$453$	0	0
$454$	8.00000	0.375459
$455$	0	0
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	−14.0000	−0.654177
$459$	0	0
$460$	7.00000	0.326377
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	−13.0000	−0.604161	−0.302081	−	0.953282i	$-0.597681\pi$
$463$	−13.0000	−0.604161	−0.302081	+	0.953282i	$0.597681\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	−18.0000	−0.833834
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	5.00000	0.230633
$471$	0	0
$472$	4.00000	0.184115
$473$	8.00000	0.367840
$474$	0	0
$475$	3.00000	0.137649
$476$	0	0
$477$	0	0
$478$	18.0000	0.823301
$479$	22.0000	1.00521	0.502603	−	0.864517i	$-0.332376\pi$
$479$	22.0000	1.00521	0.502603	+	0.864517i	$0.332376\pi$
$480$	0	0
$481$	−11.0000	−0.501557
$482$	−7.00000	−0.318841
$483$	0	0
$484$	−10.0000	−0.454545
$485$	−16.0000	−0.726523
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	0	0
$494$	−3.00000	−0.134976
$495$	0	0
$496$	2.00000	0.0898027
$497$	0	0
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	13.0000	0.580218
$503$	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
$505$	0	0
$506$	−7.00000	−0.311188
$507$	0	0
$508$	−17.0000	−0.754253
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−20.0000	−0.882162
$515$	−16.0000	−0.705044
$516$	0	0
$517$	−5.00000	−0.219900
$518$	0	0
$519$	0	0
$520$	1.00000	0.0438529
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	0	0
$523$	−2.00000	−0.0874539	−0.0437269	−	0.999044i	$-0.513923\pi$
$523$	−2.00000	−0.0874539	−0.0437269	+	0.999044i	$0.513923\pi$
$524$	−5.00000	−0.218426
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	0	0
$529$	26.0000	1.13043
$530$	−11.0000	−0.477809
$531$	0	0
$532$	0	0
$533$	11.0000	0.476463
$534$	0	0
$535$	−10.0000	−0.432338
$536$	0	0
$537$	0	0
$538$	20.0000	0.862261
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−32.0000	−1.37452
$543$	0	0
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	1.00000	0.0426401
$551$	24.0000	1.02243
$552$	0	0
$553$	0	0
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−20.0000	−0.848189
$557$	33.0000	1.39825	0.699127	−	0.714997i	$-0.253572\pi$
$557$	33.0000	1.39825	0.699127	+	0.714997i	$0.253572\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	1.00000	0.0421825
$563$	−38.0000	−1.60151	−0.800755	−	0.598993i	$-0.795568\pi$
$563$	−38.0000	−1.60151	−0.800755	+	0.598993i	$0.795568\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	14.0000	0.588464
$567$	0	0
$568$	6.00000	0.251754
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	−1.00000	−0.0418121
$573$	0	0
$574$	0	0
$575$	−7.00000	−0.291920
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	−8.00000	−0.332182
$581$	0	0
$582$	0	0
$583$	11.0000	0.455573
$584$	6.00000	0.248282
$585$	0	0
$586$	27.0000	1.11536
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	−4.00000	−0.164677
$591$	0	0
$592$	11.0000	0.452097
$593$	16.0000	0.657041	0.328521	−	0.944497i	$-0.393450\pi$
$593$	16.0000	0.657041	0.328521	+	0.944497i	$0.393450\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	7.00000	0.286251
$599$	−2.00000	−0.0817178	−0.0408589	−	0.999165i	$-0.513009\pi$
$599$	−2.00000	−0.0817178	−0.0408589	+	0.999165i	$0.513009\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	0	0
$604$	6.00000	0.244137
$605$	10.0000	0.406558
$606$	0	0
$607$	37.0000	1.50178	0.750892	−	0.660425i	$-0.229624\pi$
$607$	37.0000	1.50178	0.750892	+	0.660425i	$0.229624\pi$
$608$	3.00000	0.121666
$609$	0	0
$610$	0	0
$611$	5.00000	0.202278
$612$	0	0
$613$	−41.0000	−1.65597	−0.827987	−	0.560747i	$-0.810514\pi$
$613$	−41.0000	−1.65597	−0.827987	+	0.560747i	$0.810514\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	0	0
$617$	−28.0000	−1.12724	−0.563619	−	0.826035i	$-0.690591\pi$
$617$	−28.0000	−1.12724	−0.563619	+	0.826035i	$0.690591\pi$
$618$	0	0
$619$	−29.0000	−1.16561	−0.582804	−	0.812613i	$-0.698045\pi$
$619$	−29.0000	−1.16561	−0.582804	+	0.812613i	$0.698045\pi$
$620$	−2.00000	−0.0803219
$621$	0	0
$622$	12.0000	0.481156
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	12.0000	0.479616
$627$	0	0
$628$	7.00000	0.279330
$629$	0	0
$630$	0	0
$631$	−26.0000	−1.03504	−0.517522	−	0.855670i	$-0.673145\pi$
$631$	−26.0000	−1.03504	−0.517522	+	0.855670i	$0.673145\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	18.0000	0.714871
$635$	17.0000	0.674624
$636$	0	0
$637$	0	0
$638$	8.00000	0.316723
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	0	0
$643$	10.0000	0.394362	0.197181	−	0.980367i	$-0.436821\pi$
$643$	10.0000	0.394362	0.197181	+	0.980367i	$0.436821\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17.0000	−0.668339	−0.334169	−	0.942513i	$-0.608456\pi$
$647$	−17.0000	−0.668339	−0.334169	+	0.942513i	$0.608456\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	16.0000	0.626608
$653$	−7.00000	−0.273931	−0.136966	−	0.990576i	$-0.543735\pi$
$653$	−7.00000	−0.273931	−0.136966	+	0.990576i	$0.543735\pi$
$654$	0	0
$655$	5.00000	0.195366
$656$	−11.0000	−0.429478
$657$	0	0
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	24.0000	0.933492	0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000	0.933492	0.466746	+	0.884391i	$0.345426\pi$
$662$	−13.0000	−0.505259
$663$	0	0
$664$	8.00000	0.310460
$665$	0	0
$666$	0	0
$667$	−56.0000	−2.16833
$668$	−3.00000	−0.116073
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	−12.0000	−0.462223
$675$	0	0
$676$	−12.0000	−0.461538
$677$	−13.0000	−0.499631	−0.249815	−	0.968294i	$-0.580370\pi$
$677$	−13.0000	−0.499631	−0.249815	+	0.968294i	$0.580370\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	2.00000	0.0765840
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	0	0
$688$	8.00000	0.304997
$689$	−11.0000	−0.419067
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	−15.0000	−0.570214
$693$	0	0
$694$	−14.0000	−0.531433
$695$	20.0000	0.758643
$696$	0	0
$697$	0	0
$698$	−12.0000	−0.454207
$699$	0	0
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	33.0000	1.24462
$704$	1.00000	0.0376889
$705$	0	0
$706$	24.0000	0.903252
$707$	0	0
$708$	0	0
$709$	44.0000	1.65245	0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000	1.65245	0.826227	+	0.563337i	$0.190483\pi$
$710$	−6.00000	−0.225176
$711$	0	0
$712$	−10.0000	−0.374766
$713$	−14.0000	−0.524304
$714$	0	0
$715$	1.00000	0.0373979
$716$	19.0000	0.710063
$717$	0	0
$718$	−4.00000	−0.149279
$719$	2.00000	0.0745874	0.0372937	−	0.999304i	$-0.488126\pi$
$719$	2.00000	0.0745874	0.0372937	+	0.999304i	$0.488126\pi$
$720$	0	0
$721$	0	0
$722$	−10.0000	−0.372161
$723$	0	0
$724$	24.0000	0.891953
$725$	8.00000	0.297113
$726$	0	0
$727$	−11.0000	−0.407967	−0.203984	−	0.978974i	$-0.565389\pi$
$727$	−11.0000	−0.407967	−0.203984	+	0.978974i	$0.565389\pi$
$728$	0	0
$729$	0	0
$730$	−6.00000	−0.222070
$731$	0	0
$732$	0	0
$733$	−11.0000	−0.406294	−0.203147	−	0.979148i	$-0.565117\pi$
$733$	−11.0000	−0.406294	−0.203147	+	0.979148i	$0.565117\pi$
$734$	−25.0000	−0.922767
$735$	0	0
$736$	−7.00000	−0.258023
$737$	0	0
$738$	0	0
$739$	−19.0000	−0.698926	−0.349463	−	0.936950i	$-0.613636\pi$
$739$	−19.0000	−0.698926	−0.349463	+	0.936950i	$0.613636\pi$
$740$	−11.0000	−0.404368
$741$	0	0
$742$	0	0
$743$	−49.0000	−1.79764	−0.898818	−	0.438322i	$-0.855573\pi$
$743$	−49.0000	−1.79764	−0.898818	+	0.438322i	$0.855573\pi$
$744$	0	0
$745$	0	0
$746$	6.00000	0.219676
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	−5.00000	−0.182331
$753$	0	0
$754$	−8.00000	−0.291343
$755$	−6.00000	−0.218362
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−19.0000	−0.690111
$759$	0	0
$760$	−3.00000	−0.108821
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	0	0
$764$	6.00000	0.217072
$765$	0	0
$766$	35.0000	1.26460
$767$	−4.00000	−0.144432
$768$	0	0
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	0	0
$772$	22.0000	0.791797
$773$	33.0000	1.18693	0.593464	−	0.804861i	$-0.297760\pi$
$773$	33.0000	1.18693	0.593464	+	0.804861i	$0.297760\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	16.0000	0.574367
$777$	0	0
$778$	−10.0000	−0.358517
$779$	−33.0000	−1.18235
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.00000	−0.249841
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	1.00000	0.0356235
$789$	0	0
$790$	8.00000	0.284627
$791$	0	0
$792$	0	0
$793$	0	0
$794$	−14.0000	−0.496841
$795$	0	0
$796$	24.0000	0.850657
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	0	0
$802$	5.00000	0.176556
$803$	6.00000	0.211735
$804$	0	0
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	0	0
$808$	0	0
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	−9.00000	−0.316033	−0.158016	−	0.987436i	$-0.550510\pi$
$811$	−9.00000	−0.316033	−0.158016	+	0.987436i	$0.550510\pi$
$812$	0	0
$813$	0	0
$814$	11.0000	0.385550
$815$	−16.0000	−0.560456
$816$	0	0
$817$	24.0000	0.839654
$818$	−2.00000	−0.0699284
$819$	0	0
$820$	11.0000	0.384137
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	0	0
$827$	10.0000	0.347734	0.173867	−	0.984769i	$-0.444374\pi$
$827$	10.0000	0.347734	0.173867	+	0.984769i	$0.444374\pi$
$828$	0	0
$829$	8.00000	0.277851	0.138926	−	0.990303i	$-0.455635\pi$
$829$	8.00000	0.277851	0.138926	+	0.990303i	$0.455635\pi$
$830$	−8.00000	−0.277684
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	0	0
$835$	3.00000	0.103819
$836$	3.00000	0.103757
$837$	0	0
$838$	−5.00000	−0.172722
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	22.0000	0.758170
$843$	0	0
$844$	5.00000	0.172107
$845$	12.0000	0.412813
$846$	0	0
$847$	0	0
$848$	11.0000	0.377742
$849$	0	0
$850$	0	0
$851$	−77.0000	−2.63953
$852$	0	0
$853$	41.0000	1.40381	0.701907	−	0.712269i	$-0.252332\pi$
$853$	41.0000	1.40381	0.701907	+	0.712269i	$0.252332\pi$
$854$	0	0
$855$	0	0
$856$	10.0000	0.341793
$857$	−14.0000	−0.478231	−0.239115	−	0.970991i	$-0.576857\pi$
$857$	−14.0000	−0.478231	−0.239115	+	0.970991i	$0.576857\pi$
$858$	0	0
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	0	0
$863$	53.0000	1.80414	0.902070	−	0.431589i	$-0.142047\pi$
$863$	53.0000	1.80414	0.902070	+	0.431589i	$0.142047\pi$
$864$	0	0
$865$	15.0000	0.510015
$866$	32.0000	1.08740
$867$	0	0
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	0	0
$872$	6.00000	0.203186
$873$	0	0
$874$	−21.0000	−0.710336
$875$	0	0
$876$	0	0
$877$	17.0000	0.574049	0.287025	−	0.957923i	$-0.407334\pi$
$877$	17.0000	0.574049	0.287025	+	0.957923i	$0.407334\pi$
$878$	−32.0000	−1.07995
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	−53.0000	−1.78562	−0.892808	−	0.450438i	$-0.851268\pi$
$881$	−53.0000	−1.78562	−0.892808	+	0.450438i	$0.851268\pi$
$882$	0	0
$883$	−42.0000	−1.41341	−0.706706	−	0.707507i	$-0.749820\pi$
$883$	−42.0000	−1.41341	−0.706706	+	0.707507i	$0.749820\pi$
$884$	0	0
$885$	0	0
$886$	−16.0000	−0.537531
$887$	44.0000	1.47738	0.738688	−	0.674048i	$-0.235446\pi$
$887$	44.0000	1.47738	0.738688	+	0.674048i	$0.235446\pi$
$888$	0	0
$889$	0	0
$890$	10.0000	0.335201
$891$	0	0
$892$	−12.0000	−0.401790
$893$	−15.0000	−0.501956
$894$	0	0
$895$	−19.0000	−0.635100
$896$	0	0
$897$	0	0
$898$	−11.0000	−0.367075
$899$	16.0000	0.533630
$900$	0	0
$901$	0	0
$902$	−11.0000	−0.366260
$903$	0	0
$904$	−6.00000	−0.199557
$905$	−24.0000	−0.797787
$906$	0	0
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	8.00000	0.265489
$909$	0	0
$910$	0	0
$911$	2.00000	0.0662630	0.0331315	−	0.999451i	$-0.489452\pi$
$911$	2.00000	0.0662630	0.0331315	+	0.999451i	$0.489452\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−14.0000	−0.462573
$917$	0	0
$918$	0	0
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	7.00000	0.230783
$921$	0	0
$922$	−12.0000	−0.395199
$923$	−6.00000	−0.197492
$924$	0	0
$925$	11.0000	0.361678
$926$	−13.0000	−0.427207
$927$	0	0
$928$	8.00000	0.262613
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	0	0
$931$	0	0
$932$	−18.0000	−0.589610
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−12.0000	−0.392023	−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000	−0.392023	−0.196011	+	0.980602i	$0.562799\pi$
$938$	0	0
$939$	0	0
$940$	5.00000	0.163082
$941$	54.0000	1.76035	0.880175	−	0.474650i	$-0.157425\pi$
$941$	54.0000	1.76035	0.880175	+	0.474650i	$0.157425\pi$
$942$	0	0
$943$	77.0000	2.50746
$944$	4.00000	0.130189
$945$	0	0
$946$	8.00000	0.260102
$947$	−22.0000	−0.714904	−0.357452	−	0.933932i	$-0.616354\pi$
$947$	−22.0000	−0.714904	−0.357452	+	0.933932i	$0.616354\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	3.00000	0.0973329
$951$	0	0
$952$	0	0
$953$	20.0000	0.647864	0.323932	−	0.946080i	$-0.394995\pi$
$953$	20.0000	0.647864	0.323932	+	0.946080i	$0.394995\pi$
$954$	0	0
$955$	−6.00000	−0.194155
$956$	18.0000	0.582162
$957$	0	0
$958$	22.0000	0.710788
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−11.0000	−0.354654
$963$	0	0
$964$	−7.00000	−0.225455
$965$	−22.0000	−0.708205
$966$	0	0
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	−10.0000	−0.321412
$969$	0	0
$970$	−16.0000	−0.513729
$971$	−45.0000	−1.44412	−0.722059	−	0.691831i	$-0.756804\pi$
$971$	−45.0000	−1.44412	−0.722059	+	0.691831i	$0.756804\pi$
$972$	0	0
$973$	0	0
$974$	−16.0000	−0.512673
$975$	0	0
$976$	0	0
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	0	0
$979$	−10.0000	−0.319601
$980$	0	0
$981$	0	0
$982$	−20.0000	−0.638226
$983$	49.0000	1.56286	0.781429	−	0.623995i	$-0.214491\pi$
$983$	49.0000	1.56286	0.781429	+	0.623995i	$0.214491\pi$
$984$	0	0
$985$	−1.00000	−0.0318626
$986$	0	0
$987$	0	0
$988$	−3.00000	−0.0954427
$989$	−56.0000	−1.78070
$990$	0	0
$991$	22.0000	0.698853	0.349427	−	0.936964i	$-0.386376\pi$
$991$	22.0000	0.698853	0.349427	+	0.936964i	$0.386376\pi$
$992$	2.00000	0.0635001
$993$	0	0
$994$	0	0
$995$	−24.0000	−0.760851
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	−32.0000	−1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.a.ba.1.1		1
3.2	odd	2		1470.2.a.h.1.1		1
7.3	odd	6		630.2.k.c.541.1		2
7.5	odd	6		630.2.k.c.361.1		2
7.6	odd	2		4410.2.a.bj.1.1		1
15.14	odd	2		7350.2.a.bu.1.1		1
21.2	odd	6		1470.2.i.m.361.1		2
21.5	even	6		210.2.i.d.151.1	yes	2
21.11	odd	6		1470.2.i.m.961.1		2
21.17	even	6		210.2.i.d.121.1	✓	2
21.20	even	2		1470.2.a.a.1.1		1
84.47	odd	6		1680.2.bg.g.1201.1		2
84.59	odd	6		1680.2.bg.g.961.1		2
105.17	odd	12		1050.2.o.i.499.2		4
105.38	odd	12		1050.2.o.i.499.1		4
105.47	odd	12		1050.2.o.i.949.1		4
105.59	even	6		1050.2.i.b.751.1		2
105.68	odd	12		1050.2.o.i.949.2		4
105.89	even	6		1050.2.i.b.151.1		2
105.104	even	2		7350.2.a.cp.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.i.d.121.1	✓	2	21.17	even	6
210.2.i.d.151.1	yes	2	21.5	even	6
630.2.k.c.361.1		2	7.5	odd	6
630.2.k.c.541.1		2	7.3	odd	6
1050.2.i.b.151.1		2	105.89	even	6
1050.2.i.b.751.1		2	105.59	even	6
1050.2.o.i.499.1		4	105.38	odd	12
1050.2.o.i.499.2		4	105.17	odd	12
1050.2.o.i.949.1		4	105.47	odd	12
1050.2.o.i.949.2		4	105.68	odd	12
1470.2.a.a.1.1		1	21.20	even	2
1470.2.a.h.1.1		1	3.2	odd	2
1470.2.i.m.361.1		2	21.2	odd	6
1470.2.i.m.961.1		2	21.11	odd	6
1680.2.bg.g.961.1		2	84.59	odd	6
1680.2.bg.g.1201.1		2	84.47	odd	6
4410.2.a.ba.1.1		1	1.1	even	1	trivial
4410.2.a.bj.1.1		1	7.6	odd	2
7350.2.a.bu.1.1		1	15.14	odd	2
7350.2.a.cp.1.1		1	105.104	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:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.a (trivial)

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