LMFDB - Embedded newform 2475.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -7.00000 q^{4} -36.0000 q^{7} -15.0000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} -7.00000 q^{4} -36.0000 q^{7} -15.0000 q^{8} -11.0000 q^{11} -2.00000 q^{13} -36.0000 q^{14} +41.0000 q^{16} +66.0000 q^{17} +140.000 q^{19} -11.0000 q^{22} -68.0000 q^{23} -2.00000 q^{26} +252.000 q^{28} -150.000 q^{29} -128.000 q^{31} +161.000 q^{32} +66.0000 q^{34} +314.000 q^{37} +140.000 q^{38} +118.000 q^{41} -172.000 q^{43} +77.0000 q^{44} -68.0000 q^{46} -324.000 q^{47} +953.000 q^{49} +14.0000 q^{52} +82.0000 q^{53} +540.000 q^{56} -150.000 q^{58} +740.000 q^{59} +122.000 q^{61} -128.000 q^{62} -167.000 q^{64} +124.000 q^{67} -462.000 q^{68} +988.000 q^{71} -2.00000 q^{73} +314.000 q^{74} -980.000 q^{76} +396.000 q^{77} +1100.00 q^{79} +118.000 q^{82} -868.000 q^{83} -172.000 q^{86} +165.000 q^{88} +470.000 q^{89} +72.0000 q^{91} +476.000 q^{92} -324.000 q^{94} -1186.00 q^{97} +953.000 q^{98} +O(q^{100})$

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	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	1.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−0.875000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−36.0000	−1.94382	−0.971909	−	0.235358i	$-0.924374\pi$
$7$	−36.0000	−1.94382	−0.971909	+	0.235358i	$0.924374\pi$
$8$	−15.0000	−0.662913
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−2.00000	−0.0426692	−0.0213346	−	0.999772i	$-0.506792\pi$
$13$	−2.00000	−0.0426692	−0.0213346	+	0.999772i	$0.506792\pi$
$14$	−36.0000	−0.687243
$15$	0	0
$16$	41.0000	0.640625
$17$	66.0000	0.941609	0.470804	−	0.882238i	$-0.343964\pi$
$17$	66.0000	0.941609	0.470804	+	0.882238i	$0.343964\pi$
$18$	0	0
$19$	140.000	1.69043	0.845216	−	0.534425i	$-0.179472\pi$
$19$	140.000	1.69043	0.845216	+	0.534425i	$0.179472\pi$
$20$	0	0
$21$	0	0
$22$	−11.0000	−0.106600
$23$	−68.0000	−0.616477	−0.308239	−	0.951309i	$-0.599740\pi$
$23$	−68.0000	−0.616477	−0.308239	+	0.951309i	$0.599740\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.0150859
$27$	0	0
$28$	252.000	1.70084
$29$	−150.000	−0.960493	−0.480247	−	0.877134i	$-0.659453\pi$
$29$	−150.000	−0.960493	−0.480247	+	0.877134i	$0.659453\pi$
$30$	0	0
$31$	−128.000	−0.741596	−0.370798	−	0.928714i	$-0.620916\pi$
$31$	−128.000	−0.741596	−0.370798	+	0.928714i	$0.620916\pi$
$32$	161.000	0.889408
$33$	0	0
$34$	66.0000	0.332909
$35$	0	0
$36$	0	0
$37$	314.000	1.39517	0.697585	−	0.716502i	$-0.254258\pi$
$37$	314.000	1.39517	0.697585	+	0.716502i	$0.254258\pi$
$38$	140.000	0.597658
$39$	0	0
$40$	0	0
$41$	118.000	0.449476	0.224738	−	0.974419i	$-0.427847\pi$
$41$	118.000	0.449476	0.224738	+	0.974419i	$0.427847\pi$
$42$	0	0
$43$	−172.000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−172.000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	77.0000	0.263822
$45$	0	0
$46$	−68.0000	−0.217958
$47$	−324.000	−1.00554	−0.502769	−	0.864421i	$-0.667685\pi$
$47$	−324.000	−1.00554	−0.502769	+	0.864421i	$0.667685\pi$
$48$	0	0
$49$	953.000	2.77843
$50$	0	0
$51$	0	0
$52$	14.0000	0.0373356
$53$	82.0000	0.212520	0.106260	−	0.994338i	$-0.466112\pi$
$53$	82.0000	0.212520	0.106260	+	0.994338i	$0.466112\pi$
$54$	0	0
$55$	0	0
$56$	540.000	1.28858
$57$	0	0
$58$	−150.000	−0.339586
$59$	740.000	1.63288	0.816439	−	0.577432i	$-0.195945\pi$
$59$	740.000	1.63288	0.816439	+	0.577432i	$0.195945\pi$
$60$	0	0
$61$	122.000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	122.000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−128.000	−0.262194
$63$	0	0
$64$	−167.000	−0.326172
$65$	0	0
$66$	0	0
$67$	124.000	0.226105	0.113052	−	0.993589i	$-0.463937\pi$
$67$	124.000	0.226105	0.113052	+	0.993589i	$0.463937\pi$
$68$	−462.000	−0.823908
$69$	0	0
$70$	0	0
$71$	988.000	1.65147	0.825733	−	0.564062i	$-0.190762\pi$
$71$	988.000	1.65147	0.825733	+	0.564062i	$0.190762\pi$
$72$	0	0
$73$	−2.00000	−0.00320661	−0.00160330	−	0.999999i	$-0.500510\pi$
$73$	−2.00000	−0.00320661	−0.00160330	+	0.999999i	$0.500510\pi$
$74$	314.000	0.493267
$75$	0	0
$76$	−980.000	−1.47913
$77$	396.000	0.586083
$78$	0	0
$79$	1100.00	1.56658	0.783289	−	0.621658i	$-0.213540\pi$
$79$	1100.00	1.56658	0.783289	+	0.621658i	$0.213540\pi$
$80$	0	0
$81$	0	0
$82$	118.000	0.158914
$83$	−868.000	−1.14790	−0.573948	−	0.818892i	$-0.694589\pi$
$83$	−868.000	−1.14790	−0.573948	+	0.818892i	$0.694589\pi$
$84$	0	0
$85$	0	0
$86$	−172.000	−0.215666
$87$	0	0
$88$	165.000	0.199876
$89$	470.000	0.559774	0.279887	−	0.960033i	$-0.409703\pi$
$89$	470.000	0.559774	0.279887	+	0.960033i	$0.409703\pi$
$90$	0	0
$91$	72.0000	0.0829412
$92$	476.000	0.539418
$93$	0	0
$94$	−324.000	−0.355511
$95$	0	0
$96$	0	0
$97$	−1186.00	−1.24144	−0.620722	−	0.784031i	$-0.713160\pi$
$97$	−1186.00	−1.24144	−0.620722	+	0.784031i	$0.713160\pi$
$98$	953.000	0.982322
$99$	0	0
$100$	0	0
$101$	−1502.00	−1.47975	−0.739874	−	0.672745i	$-0.765115\pi$
$101$	−1502.00	−1.47975	−0.739874	+	0.672745i	$0.765115\pi$
$102$	0	0
$103$	−32.0000	−0.0306122	−0.0153061	−	0.999883i	$-0.504872\pi$
$103$	−32.0000	−0.0306122	−0.0153061	+	0.999883i	$0.504872\pi$
$104$	30.0000	0.0282860
$105$	0	0
$106$	82.0000	0.0751372
$107$	1116.00	1.00830	0.504149	−	0.863617i	$-0.331806\pi$
$107$	1116.00	1.00830	0.504149	+	0.863617i	$0.331806\pi$
$108$	0	0
$109$	−2190.00	−1.92444	−0.962220	−	0.272273i	$-0.912225\pi$
$109$	−2190.00	−1.92444	−0.962220	+	0.272273i	$0.912225\pi$
$110$	0	0
$111$	0	0
$112$	−1476.00	−1.24526
$113$	−1398.00	−1.16383	−0.581915	−	0.813250i	$-0.697696\pi$
$113$	−1398.00	−1.16383	−0.581915	+	0.813250i	$0.697696\pi$
$114$	0	0
$115$	0	0
$116$	1050.00	0.840431
$117$	0	0
$118$	740.000	0.577310
$119$	−2376.00	−1.83032
$120$	0	0
$121$	121.000	0.0909091
$122$	122.000	0.0905357
$123$	0	0
$124$	896.000	0.648897
$125$	0	0
$126$	0	0
$127$	44.0000	0.0307431	0.0153715	−	0.999882i	$-0.495107\pi$
$127$	44.0000	0.0307431	0.0153715	+	0.999882i	$0.495107\pi$
$128$	−1455.00	−1.00473
$129$	0	0
$130$	0	0
$131$	1308.00	0.872370	0.436185	−	0.899857i	$-0.356329\pi$
$131$	1308.00	0.872370	0.436185	+	0.899857i	$0.356329\pi$
$132$	0	0
$133$	−5040.00	−3.28589
$134$	124.000	0.0799401
$135$	0	0
$136$	−990.000	−0.624204
$137$	1626.00	1.01400	0.507002	−	0.861945i	$-0.330754\pi$
$137$	1626.00	1.01400	0.507002	+	0.861945i	$0.330754\pi$
$138$	0	0
$139$	180.000	0.109837	0.0549187	−	0.998491i	$-0.482510\pi$
$139$	180.000	0.109837	0.0549187	+	0.998491i	$0.482510\pi$
$140$	0	0
$141$	0	0
$142$	988.000	0.583881
$143$	22.0000	0.0128653
$144$	0	0
$145$	0	0
$146$	−2.00000	−0.00113371
$147$	0	0
$148$	−2198.00	−1.22077
$149$	−1430.00	−0.786243	−0.393121	−	0.919487i	$-0.628605\pi$
$149$	−1430.00	−0.786243	−0.393121	+	0.919487i	$0.628605\pi$
$150$	0	0
$151$	−1948.00	−1.04984	−0.524921	−	0.851151i	$-0.675905\pi$
$151$	−1948.00	−1.04984	−0.524921	+	0.851151i	$0.675905\pi$
$152$	−2100.00	−1.12061
$153$	0	0
$154$	396.000	0.207212
$155$	0	0
$156$	0	0
$157$	−646.000	−0.328385	−0.164192	−	0.986428i	$-0.552502\pi$
$157$	−646.000	−0.328385	−0.164192	+	0.986428i	$0.552502\pi$
$158$	1100.00	0.553869
$159$	0	0
$160$	0	0
$161$	2448.00	1.19832
$162$	0	0
$163$	−3052.00	−1.46657	−0.733286	−	0.679921i	$-0.762014\pi$
$163$	−3052.00	−1.46657	−0.733286	+	0.679921i	$0.762014\pi$
$164$	−826.000	−0.393291
$165$	0	0
$166$	−868.000	−0.405843
$167$	1216.00	0.563455	0.281727	−	0.959495i	$-0.409093\pi$
$167$	1216.00	0.563455	0.281727	+	0.959495i	$0.409093\pi$
$168$	0	0
$169$	−2193.00	−0.998179
$170$	0	0
$171$	0	0
$172$	1204.00	0.533745
$173$	−3858.00	−1.69548	−0.847741	−	0.530411i	$-0.822038\pi$
$173$	−3858.00	−1.69548	−0.847741	+	0.530411i	$0.822038\pi$
$174$	0	0
$175$	0	0
$176$	−451.000	−0.193156
$177$	0	0
$178$	470.000	0.197910
$179$	380.000	0.158673	0.0793367	−	0.996848i	$-0.474720\pi$
$179$	380.000	0.158673	0.0793367	+	0.996848i	$0.474720\pi$
$180$	0	0
$181$	−538.000	−0.220935	−0.110467	−	0.993880i	$-0.535235\pi$
$181$	−538.000	−0.220935	−0.110467	+	0.993880i	$0.535235\pi$
$182$	72.0000	0.0293241
$183$	0	0
$184$	1020.00	0.408671
$185$	0	0
$186$	0	0
$187$	−726.000	−0.283906
$188$	2268.00	0.879845
$189$	0	0
$190$	0	0
$191$	−1412.00	−0.534915	−0.267457	−	0.963570i	$-0.586183\pi$
$191$	−1412.00	−0.534915	−0.267457	+	0.963570i	$0.586183\pi$
$192$	0	0
$193$	638.000	0.237949	0.118975	−	0.992897i	$-0.462039\pi$
$193$	638.000	0.237949	0.118975	+	0.992897i	$0.462039\pi$
$194$	−1186.00	−0.438917
$195$	0	0
$196$	−6671.00	−2.43112
$197$	3686.00	1.33308	0.666540	−	0.745470i	$-0.267775\pi$
$197$	3686.00	1.33308	0.666540	+	0.745470i	$0.267775\pi$
$198$	0	0
$199$	−240.000	−0.0854932	−0.0427466	−	0.999086i	$-0.513611\pi$
$199$	−240.000	−0.0854932	−0.0427466	+	0.999086i	$0.513611\pi$
$200$	0	0
$201$	0	0
$202$	−1502.00	−0.523170
$203$	5400.00	1.86702
$204$	0	0
$205$	0	0
$206$	−32.0000	−0.0108230
$207$	0	0
$208$	−82.0000	−0.0273350
$209$	−1540.00	−0.509684
$210$	0	0
$211$	5092.00	1.66136	0.830682	−	0.556747i	$-0.187951\pi$
$211$	5092.00	1.66136	0.830682	+	0.556747i	$0.187951\pi$
$212$	−574.000	−0.185955
$213$	0	0
$214$	1116.00	0.356487
$215$	0	0
$216$	0	0
$217$	4608.00	1.44153
$218$	−2190.00	−0.680392
$219$	0	0
$220$	0	0
$221$	−132.000	−0.0401777
$222$	0	0
$223$	3808.00	1.14351	0.571755	−	0.820425i	$-0.306263\pi$
$223$	3808.00	1.14351	0.571755	+	0.820425i	$0.306263\pi$
$224$	−5796.00	−1.72885
$225$	0	0
$226$	−1398.00	−0.411476
$227$	−44.0000	−0.0128651	−0.00643256	−	0.999979i	$-0.502048\pi$
$227$	−44.0000	−0.0128651	−0.00643256	+	0.999979i	$0.502048\pi$
$228$	0	0
$229$	−2650.00	−0.764703	−0.382351	−	0.924017i	$-0.624886\pi$
$229$	−2650.00	−0.764703	−0.382351	+	0.924017i	$0.624886\pi$
$230$	0	0
$231$	0	0
$232$	2250.00	0.636723
$233$	−5718.00	−1.60772	−0.803860	−	0.594819i	$-0.797224\pi$
$233$	−5718.00	−1.60772	−0.803860	+	0.594819i	$0.797224\pi$
$234$	0	0
$235$	0	0
$236$	−5180.00	−1.42877
$237$	0	0
$238$	−2376.00	−0.647114
$239$	6520.00	1.76462	0.882309	−	0.470670i	$-0.155988\pi$
$239$	6520.00	1.76462	0.882309	+	0.470670i	$0.155988\pi$
$240$	0	0
$241$	−2438.00	−0.651641	−0.325820	−	0.945432i	$-0.605640\pi$
$241$	−2438.00	−0.651641	−0.325820	+	0.945432i	$0.605640\pi$
$242$	121.000	0.0321412
$243$	0	0
$244$	−854.000	−0.224065
$245$	0	0
$246$	0	0
$247$	−280.000	−0.0721294
$248$	1920.00	0.491613
$249$	0	0
$250$	0	0
$251$	5268.00	1.32475	0.662377	−	0.749171i	$-0.269548\pi$
$251$	5268.00	1.32475	0.662377	+	0.749171i	$0.269548\pi$
$252$	0	0
$253$	748.000	0.185875
$254$	44.0000	0.0108693
$255$	0	0
$256$	−119.000	−0.0290527
$257$	−5574.00	−1.35290	−0.676452	−	0.736486i	$-0.736484\pi$
$257$	−5574.00	−1.35290	−0.676452	+	0.736486i	$0.736484\pi$
$258$	0	0
$259$	−11304.0	−2.71196
$260$	0	0
$261$	0	0
$262$	1308.00	0.308429
$263$	2472.00	0.579582	0.289791	−	0.957090i	$-0.406414\pi$
$263$	2472.00	0.579582	0.289791	+	0.957090i	$0.406414\pi$
$264$	0	0
$265$	0	0
$266$	−5040.00	−1.16174
$267$	0	0
$268$	−868.000	−0.197842
$269$	−4410.00	−0.999563	−0.499781	−	0.866152i	$-0.666586\pi$
$269$	−4410.00	−0.999563	−0.499781	+	0.866152i	$0.666586\pi$
$270$	0	0
$271$	−1308.00	−0.293193	−0.146597	−	0.989196i	$-0.546832\pi$
$271$	−1308.00	−0.293193	−0.146597	+	0.989196i	$0.546832\pi$
$272$	2706.00	0.603218
$273$	0	0
$274$	1626.00	0.358505
$275$	0	0
$276$	0	0
$277$	−226.000	−0.0490217	−0.0245109	−	0.999700i	$-0.507803\pi$
$277$	−226.000	−0.0490217	−0.0245109	+	0.999700i	$0.507803\pi$
$278$	180.000	0.0388334
$279$	0	0
$280$	0	0
$281$	−522.000	−0.110818	−0.0554091	−	0.998464i	$-0.517646\pi$
$281$	−522.000	−0.110818	−0.0554091	+	0.998464i	$0.517646\pi$
$282$	0	0
$283$	1508.00	0.316754	0.158377	−	0.987379i	$-0.449374\pi$
$283$	1508.00	0.316754	0.158377	+	0.987379i	$0.449374\pi$
$284$	−6916.00	−1.44503
$285$	0	0
$286$	22.0000	0.00454856
$287$	−4248.00	−0.873699
$288$	0	0
$289$	−557.000	−0.113373
$290$	0	0
$291$	0	0
$292$	14.0000	0.00280578
$293$	−4658.00	−0.928748	−0.464374	−	0.885639i	$-0.653721\pi$
$293$	−4658.00	−0.928748	−0.464374	+	0.885639i	$0.653721\pi$
$294$	0	0
$295$	0	0
$296$	−4710.00	−0.924876
$297$	0	0
$298$	−1430.00	−0.277979
$299$	136.000	0.0263046
$300$	0	0
$301$	6192.00	1.18572
$302$	−1948.00	−0.371175
$303$	0	0
$304$	5740.00	1.08293
$305$	0	0
$306$	0	0
$307$	−6476.00	−1.20392	−0.601962	−	0.798525i	$-0.705614\pi$
$307$	−6476.00	−1.20392	−0.601962	+	0.798525i	$0.705614\pi$
$308$	−2772.00	−0.512823
$309$	0	0
$310$	0	0
$311$	−5652.00	−1.03053	−0.515266	−	0.857030i	$-0.672307\pi$
$311$	−5652.00	−1.03053	−0.515266	+	0.857030i	$0.672307\pi$
$312$	0	0
$313$	5638.00	1.01814	0.509071	−	0.860724i	$-0.329989\pi$
$313$	5638.00	1.01814	0.509071	+	0.860724i	$0.329989\pi$
$314$	−646.000	−0.116102
$315$	0	0
$316$	−7700.00	−1.37076
$317$	1506.00	0.266831	0.133415	−	0.991060i	$-0.457406\pi$
$317$	1506.00	0.266831	0.133415	+	0.991060i	$0.457406\pi$
$318$	0	0
$319$	1650.00	0.289600
$320$	0	0
$321$	0	0
$322$	2448.00	0.423670
$323$	9240.00	1.59173
$324$	0	0
$325$	0	0
$326$	−3052.00	−0.518511
$327$	0	0
$328$	−1770.00	−0.297963
$329$	11664.0	1.95458
$330$	0	0
$331$	−7268.00	−1.20690	−0.603452	−	0.797399i	$-0.706209\pi$
$331$	−7268.00	−1.20690	−0.603452	+	0.797399i	$0.706209\pi$
$332$	6076.00	1.00441
$333$	0	0
$334$	1216.00	0.199211
$335$	0	0
$336$	0	0
$337$	7254.00	1.17255	0.586277	−	0.810111i	$-0.300593\pi$
$337$	7254.00	1.17255	0.586277	+	0.810111i	$0.300593\pi$
$338$	−2193.00	−0.352910
$339$	0	0
$340$	0	0
$341$	1408.00	0.223600
$342$	0	0
$343$	−21960.0	−3.45693
$344$	2580.00	0.404373
$345$	0	0
$346$	−3858.00	−0.599443
$347$	2276.00	0.352110	0.176055	−	0.984380i	$-0.443666\pi$
$347$	2276.00	0.352110	0.176055	+	0.984380i	$0.443666\pi$
$348$	0	0
$349$	4610.00	0.707071	0.353535	−	0.935421i	$-0.384979\pi$
$349$	4610.00	0.707071	0.353535	+	0.935421i	$0.384979\pi$
$350$	0	0
$351$	0	0
$352$	−1771.00	−0.268167
$353$	4602.00	0.693880	0.346940	−	0.937887i	$-0.387221\pi$
$353$	4602.00	0.693880	0.346940	+	0.937887i	$0.387221\pi$
$354$	0	0
$355$	0	0
$356$	−3290.00	−0.489802
$357$	0	0
$358$	380.000	0.0560995
$359$	4160.00	0.611578	0.305789	−	0.952099i	$-0.401080\pi$
$359$	4160.00	0.611578	0.305789	+	0.952099i	$0.401080\pi$
$360$	0	0
$361$	12741.0	1.85756
$362$	−538.000	−0.0781123
$363$	0	0
$364$	−504.000	−0.0725736
$365$	0	0
$366$	0	0
$367$	5024.00	0.714579	0.357290	−	0.933994i	$-0.383701\pi$
$367$	5024.00	0.714579	0.357290	+	0.933994i	$0.383701\pi$
$368$	−2788.00	−0.394931
$369$	0	0
$370$	0	0
$371$	−2952.00	−0.413100
$372$	0	0
$373$	−10842.0	−1.50503	−0.752517	−	0.658573i	$-0.771160\pi$
$373$	−10842.0	−1.50503	−0.752517	+	0.658573i	$0.771160\pi$
$374$	−726.000	−0.100376
$375$	0	0
$376$	4860.00	0.666583
$377$	300.000	0.0409835
$378$	0	0
$379$	−1540.00	−0.208719	−0.104359	−	0.994540i	$-0.533279\pi$
$379$	−1540.00	−0.208719	−0.104359	+	0.994540i	$0.533279\pi$
$380$	0	0
$381$	0	0
$382$	−1412.00	−0.189121
$383$	−9468.00	−1.26317	−0.631583	−	0.775309i	$-0.717594\pi$
$383$	−9468.00	−1.26317	−0.631583	+	0.775309i	$0.717594\pi$
$384$	0	0
$385$	0	0
$386$	638.000	0.0841278
$387$	0	0
$388$	8302.00	1.08626
$389$	10590.0	1.38029	0.690147	−	0.723669i	$-0.257546\pi$
$389$	10590.0	1.38029	0.690147	+	0.723669i	$0.257546\pi$
$390$	0	0
$391$	−4488.00	−0.580481
$392$	−14295.0	−1.84185
$393$	0	0
$394$	3686.00	0.471315
$395$	0	0
$396$	0	0
$397$	7434.00	0.939803	0.469901	−	0.882719i	$-0.344289\pi$
$397$	7434.00	0.939803	0.469901	+	0.882719i	$0.344289\pi$
$398$	−240.000	−0.0302264
$399$	0	0
$400$	0	0
$401$	−11402.0	−1.41992	−0.709961	−	0.704241i	$-0.751287\pi$
$401$	−11402.0	−1.41992	−0.709961	+	0.704241i	$0.751287\pi$
$402$	0	0
$403$	256.000	0.0316433
$404$	10514.0	1.29478
$405$	0	0
$406$	5400.00	0.660092
$407$	−3454.00	−0.420660
$408$	0	0
$409$	−510.000	−0.0616574	−0.0308287	−	0.999525i	$-0.509815\pi$
$409$	−510.000	−0.0616574	−0.0308287	+	0.999525i	$0.509815\pi$
$410$	0	0
$411$	0	0
$412$	224.000	0.0267857
$413$	−26640.0	−3.17402
$414$	0	0
$415$	0	0
$416$	−322.000	−0.0379504
$417$	0	0
$418$	−1540.00	−0.180201
$419$	−2420.00	−0.282159	−0.141080	−	0.989998i	$-0.545057\pi$
$419$	−2420.00	−0.282159	−0.141080	+	0.989998i	$0.545057\pi$
$420$	0	0
$421$	−8178.00	−0.946725	−0.473363	−	0.880868i	$-0.656960\pi$
$421$	−8178.00	−0.946725	−0.473363	+	0.880868i	$0.656960\pi$
$422$	5092.00	0.587381
$423$	0	0
$424$	−1230.00	−0.140882
$425$	0	0
$426$	0	0
$427$	−4392.00	−0.497761
$428$	−7812.00	−0.882260
$429$	0	0
$430$	0	0
$431$	6768.00	0.756388	0.378194	−	0.925726i	$-0.376545\pi$
$431$	6768.00	0.756388	0.378194	+	0.925726i	$0.376545\pi$
$432$	0	0
$433$	478.000	0.0530513	0.0265257	−	0.999648i	$-0.491556\pi$
$433$	478.000	0.0530513	0.0265257	+	0.999648i	$0.491556\pi$
$434$	4608.00	0.509657
$435$	0	0
$436$	15330.0	1.68388
$437$	−9520.00	−1.04211
$438$	0	0
$439$	−3260.00	−0.354422	−0.177211	−	0.984173i	$-0.556708\pi$
$439$	−3260.00	−0.354422	−0.177211	+	0.984173i	$0.556708\pi$
$440$	0	0
$441$	0	0
$442$	−132.000	−0.0142050
$443$	6812.00	0.730582	0.365291	−	0.930893i	$-0.380969\pi$
$443$	6812.00	0.730582	0.365291	+	0.930893i	$0.380969\pi$
$444$	0	0
$445$	0	0
$446$	3808.00	0.404292
$447$	0	0
$448$	6012.00	0.634019
$449$	−15290.0	−1.60708	−0.803541	−	0.595250i	$-0.797053\pi$
$449$	−15290.0	−1.60708	−0.803541	+	0.595250i	$0.797053\pi$
$450$	0	0
$451$	−1298.00	−0.135522
$452$	9786.00	1.01835
$453$	0	0
$454$	−44.0000	−0.00454851
$455$	0	0
$456$	0	0
$457$	12854.0	1.31572	0.657861	−	0.753140i	$-0.271462\pi$
$457$	12854.0	1.31572	0.657861	+	0.753140i	$0.271462\pi$
$458$	−2650.00	−0.270363
$459$	0	0
$460$	0	0
$461$	−6782.00	−0.685183	−0.342591	−	0.939485i	$-0.611305\pi$
$461$	−6782.00	−0.685183	−0.342591	+	0.939485i	$0.611305\pi$
$462$	0	0
$463$	6408.00	0.643207	0.321604	−	0.946874i	$-0.395778\pi$
$463$	6408.00	0.643207	0.321604	+	0.946874i	$0.395778\pi$
$464$	−6150.00	−0.615316
$465$	0	0
$466$	−5718.00	−0.568415
$467$	10476.0	1.03805	0.519027	−	0.854758i	$-0.326294\pi$
$467$	10476.0	1.03805	0.519027	+	0.854758i	$0.326294\pi$
$468$	0	0
$469$	−4464.00	−0.439506
$470$	0	0
$471$	0	0
$472$	−11100.0	−1.08246
$473$	1892.00	0.183920
$474$	0	0
$475$	0	0
$476$	16632.0	1.60153
$477$	0	0
$478$	6520.00	0.623887
$479$	−1920.00	−0.183146	−0.0915731	−	0.995798i	$-0.529190\pi$
$479$	−1920.00	−0.183146	−0.0915731	+	0.995798i	$0.529190\pi$
$480$	0	0
$481$	−628.000	−0.0595308
$482$	−2438.00	−0.230390
$483$	0	0
$484$	−847.000	−0.0795455
$485$	0	0
$486$	0	0
$487$	−8416.00	−0.783091	−0.391546	−	0.920159i	$-0.628059\pi$
$487$	−8416.00	−0.783091	−0.391546	+	0.920159i	$0.628059\pi$
$488$	−1830.00	−0.169755
$489$	0	0
$490$	0	0
$491$	−17732.0	−1.62980	−0.814902	−	0.579598i	$-0.803209\pi$
$491$	−17732.0	−1.62980	−0.814902	+	0.579598i	$0.803209\pi$
$492$	0	0
$493$	−9900.00	−0.904409
$494$	−280.000	−0.0255016
$495$	0	0
$496$	−5248.00	−0.475085
$497$	−35568.0	−3.21015
$498$	0	0
$499$	−5580.00	−0.500591	−0.250296	−	0.968169i	$-0.580528\pi$
$499$	−5580.00	−0.500591	−0.250296	+	0.968169i	$0.580528\pi$
$500$	0	0
$501$	0	0
$502$	5268.00	0.468371
$503$	12512.0	1.10911	0.554555	−	0.832147i	$-0.312888\pi$
$503$	12512.0	1.10911	0.554555	+	0.832147i	$0.312888\pi$
$504$	0	0
$505$	0	0
$506$	748.000	0.0657167
$507$	0	0
$508$	−308.000	−0.0269002
$509$	22390.0	1.94974	0.974872	−	0.222767i	$-0.0715091\pi$
$509$	22390.0	1.94974	0.974872	+	0.222767i	$0.0715091\pi$
$510$	0	0
$511$	72.0000	0.00623306
$512$	11521.0	0.994455
$513$	0	0
$514$	−5574.00	−0.478324
$515$	0	0
$516$	0	0
$517$	3564.00	0.303181
$518$	−11304.0	−0.958821
$519$	0	0
$520$	0	0
$521$	−882.000	−0.0741672	−0.0370836	−	0.999312i	$-0.511807\pi$
$521$	−882.000	−0.0741672	−0.0370836	+	0.999312i	$0.511807\pi$
$522$	0	0
$523$	−18172.0	−1.51932	−0.759662	−	0.650319i	$-0.774635\pi$
$523$	−18172.0	−1.51932	−0.759662	+	0.650319i	$0.774635\pi$
$524$	−9156.00	−0.763324
$525$	0	0
$526$	2472.00	0.204913
$527$	−8448.00	−0.698293
$528$	0	0
$529$	−7543.00	−0.619956
$530$	0	0
$531$	0	0
$532$	35280.0	2.87515
$533$	−236.000	−0.0191788
$534$	0	0
$535$	0	0
$536$	−1860.00	−0.149888
$537$	0	0
$538$	−4410.00	−0.353399
$539$	−10483.0	−0.837727
$540$	0	0
$541$	−17998.0	−1.43030	−0.715152	−	0.698969i	$-0.753643\pi$
$541$	−17998.0	−1.43030	−0.715152	+	0.698969i	$0.753643\pi$
$542$	−1308.00	−0.103659
$543$	0	0
$544$	10626.0	0.837474
$545$	0	0
$546$	0	0
$547$	4164.00	0.325484	0.162742	−	0.986669i	$-0.447966\pi$
$547$	4164.00	0.325484	0.162742	+	0.986669i	$0.447966\pi$
$548$	−11382.0	−0.887254
$549$	0	0
$550$	0	0
$551$	−21000.0	−1.62365
$552$	0	0
$553$	−39600.0	−3.04514
$554$	−226.000	−0.0173318
$555$	0	0
$556$	−1260.00	−0.0961077
$557$	13366.0	1.01676	0.508380	−	0.861133i	$-0.330244\pi$
$557$	13366.0	1.01676	0.508380	+	0.861133i	$0.330244\pi$
$558$	0	0
$559$	344.000	0.0260280
$560$	0	0
$561$	0	0
$562$	−522.000	−0.0391801
$563$	24612.0	1.84240	0.921201	−	0.389087i	$-0.127210\pi$
$563$	24612.0	1.84240	0.921201	+	0.389087i	$0.127210\pi$
$564$	0	0
$565$	0	0
$566$	1508.00	0.111989
$567$	0	0
$568$	−14820.0	−1.09478
$569$	70.0000	0.00515739	0.00257869	−	0.999997i	$-0.499179\pi$
$569$	70.0000	0.00515739	0.00257869	+	0.999997i	$0.499179\pi$
$570$	0	0
$571$	−12348.0	−0.904987	−0.452494	−	0.891768i	$-0.649465\pi$
$571$	−12348.0	−0.904987	−0.452494	+	0.891768i	$0.649465\pi$
$572$	−154.000	−0.0112571
$573$	0	0
$574$	−4248.00	−0.308899
$575$	0	0
$576$	0	0
$577$	−23426.0	−1.69019	−0.845093	−	0.534620i	$-0.820455\pi$
$577$	−23426.0	−1.69019	−0.845093	+	0.534620i	$0.820455\pi$
$578$	−557.000	−0.0400833
$579$	0	0
$580$	0	0
$581$	31248.0	2.23130
$582$	0	0
$583$	−902.000	−0.0640772
$584$	30.0000	0.00212570
$585$	0	0
$586$	−4658.00	−0.328362
$587$	21036.0	1.47913	0.739564	−	0.673086i	$-0.235032\pi$
$587$	21036.0	1.47913	0.739564	+	0.673086i	$0.235032\pi$
$588$	0	0
$589$	−17920.0	−1.25362
$590$	0	0
$591$	0	0
$592$	12874.0	0.893781
$593$	−2798.00	−0.193761	−0.0968803	−	0.995296i	$-0.530886\pi$
$593$	−2798.00	−0.193761	−0.0968803	+	0.995296i	$0.530886\pi$
$594$	0	0
$595$	0	0
$596$	10010.0	0.687962
$597$	0	0
$598$	136.000	0.00930009
$599$	11100.0	0.757151	0.378576	−	0.925570i	$-0.376414\pi$
$599$	11100.0	0.757151	0.378576	+	0.925570i	$0.376414\pi$
$600$	0	0
$601$	15242.0	1.03450	0.517250	−	0.855835i	$-0.326956\pi$
$601$	15242.0	1.03450	0.517250	+	0.855835i	$0.326956\pi$
$602$	6192.00	0.419214
$603$	0	0
$604$	13636.0	0.918611
$605$	0	0
$606$	0	0
$607$	−13876.0	−0.927857	−0.463929	−	0.885873i	$-0.653561\pi$
$607$	−13876.0	−0.927857	−0.463929	+	0.885873i	$0.653561\pi$
$608$	22540.0	1.50348
$609$	0	0
$610$	0	0
$611$	648.000	0.0429055
$612$	0	0
$613$	−8722.00	−0.574679	−0.287340	−	0.957829i	$-0.592771\pi$
$613$	−8722.00	−0.574679	−0.287340	+	0.957829i	$0.592771\pi$
$614$	−6476.00	−0.425652
$615$	0	0
$616$	−5940.00	−0.388522
$617$	−8014.00	−0.522904	−0.261452	−	0.965217i	$-0.584201\pi$
$617$	−8014.00	−0.522904	−0.261452	+	0.965217i	$0.584201\pi$
$618$	0	0
$619$	26020.0	1.68955	0.844776	−	0.535121i	$-0.179734\pi$
$619$	26020.0	1.68955	0.844776	+	0.535121i	$0.179734\pi$
$620$	0	0
$621$	0	0
$622$	−5652.00	−0.364348
$623$	−16920.0	−1.08810
$624$	0	0
$625$	0	0
$626$	5638.00	0.359968
$627$	0	0
$628$	4522.00	0.287337
$629$	20724.0	1.31370
$630$	0	0
$631$	−23528.0	−1.48437	−0.742183	−	0.670197i	$-0.766209\pi$
$631$	−23528.0	−1.48437	−0.742183	+	0.670197i	$0.766209\pi$
$632$	−16500.0	−1.03850
$633$	0	0
$634$	1506.00	0.0943390
$635$	0	0
$636$	0	0
$637$	−1906.00	−0.118553
$638$	1650.00	0.102389
$639$	0	0
$640$	0	0
$641$	18998.0	1.17063	0.585317	−	0.810805i	$-0.300970\pi$
$641$	18998.0	1.17063	0.585317	+	0.810805i	$0.300970\pi$
$642$	0	0
$643$	4348.00	0.266669	0.133335	−	0.991071i	$-0.457431\pi$
$643$	4348.00	0.266669	0.133335	+	0.991071i	$0.457431\pi$
$644$	−17136.0	−1.04853
$645$	0	0
$646$	9240.00	0.562760
$647$	31916.0	1.93933	0.969666	−	0.244435i	$-0.0786024\pi$
$647$	31916.0	1.93933	0.969666	+	0.244435i	$0.0786024\pi$
$648$	0	0
$649$	−8140.00	−0.492331
$650$	0	0
$651$	0	0
$652$	21364.0	1.28325
$653$	25682.0	1.53907	0.769536	−	0.638603i	$-0.220487\pi$
$653$	25682.0	1.53907	0.769536	+	0.638603i	$0.220487\pi$
$654$	0	0
$655$	0	0
$656$	4838.00	0.287945
$657$	0	0
$658$	11664.0	0.691049
$659$	−14940.0	−0.883126	−0.441563	−	0.897230i	$-0.645576\pi$
$659$	−14940.0	−0.883126	−0.441563	+	0.897230i	$0.645576\pi$
$660$	0	0
$661$	4502.00	0.264913	0.132457	−	0.991189i	$-0.457714\pi$
$661$	4502.00	0.264913	0.132457	+	0.991189i	$0.457714\pi$
$662$	−7268.00	−0.426705
$663$	0	0
$664$	13020.0	0.760955
$665$	0	0
$666$	0	0
$667$	10200.0	0.592122
$668$	−8512.00	−0.493023
$669$	0	0
$670$	0	0
$671$	−1342.00	−0.0772091
$672$	0	0
$673$	−5362.00	−0.307117	−0.153559	−	0.988140i	$-0.549073\pi$
$673$	−5362.00	−0.307117	−0.153559	+	0.988140i	$0.549073\pi$
$674$	7254.00	0.414560
$675$	0	0
$676$	15351.0	0.873407
$677$	14566.0	0.826908	0.413454	−	0.910525i	$-0.364322\pi$
$677$	14566.0	0.826908	0.413454	+	0.910525i	$0.364322\pi$
$678$	0	0
$679$	42696.0	2.41314
$680$	0	0
$681$	0	0
$682$	1408.00	0.0790544
$683$	11852.0	0.663989	0.331994	−	0.943281i	$-0.392279\pi$
$683$	11852.0	0.663989	0.331994	+	0.943281i	$0.392279\pi$
$684$	0	0
$685$	0	0
$686$	−21960.0	−1.22221
$687$	0	0
$688$	−7052.00	−0.390778
$689$	−164.000	−0.00906807
$690$	0	0
$691$	−1828.00	−0.100637	−0.0503187	−	0.998733i	$-0.516024\pi$
$691$	−1828.00	−0.100637	−0.0503187	+	0.998733i	$0.516024\pi$
$692$	27006.0	1.48355
$693$	0	0
$694$	2276.00	0.124490
$695$	0	0
$696$	0	0
$697$	7788.00	0.423230
$698$	4610.00	0.249987
$699$	0	0
$700$	0	0
$701$	−23182.0	−1.24903	−0.624516	−	0.781012i	$-0.714704\pi$
$701$	−23182.0	−1.24903	−0.624516	+	0.781012i	$0.714704\pi$
$702$	0	0
$703$	43960.0	2.35844
$704$	1837.00	0.0983445
$705$	0	0
$706$	4602.00	0.245324
$707$	54072.0	2.87636
$708$	0	0
$709$	−33850.0	−1.79304	−0.896519	−	0.443006i	$-0.853912\pi$
$709$	−33850.0	−1.79304	−0.896519	+	0.443006i	$0.853912\pi$
$710$	0	0
$711$	0	0
$712$	−7050.00	−0.371081
$713$	8704.00	0.457177
$714$	0	0
$715$	0	0
$716$	−2660.00	−0.138839
$717$	0	0
$718$	4160.00	0.216225
$719$	7140.00	0.370344	0.185172	−	0.982706i	$-0.440716\pi$
$719$	7140.00	0.370344	0.185172	+	0.982706i	$0.440716\pi$
$720$	0	0
$721$	1152.00	0.0595045
$722$	12741.0	0.656746
$723$	0	0
$724$	3766.00	0.193318
$725$	0	0
$726$	0	0
$727$	−8896.00	−0.453830	−0.226915	−	0.973915i	$-0.572864\pi$
$727$	−8896.00	−0.453830	−0.226915	+	0.973915i	$0.572864\pi$
$728$	−1080.00	−0.0549828
$729$	0	0
$730$	0	0
$731$	−11352.0	−0.574376
$732$	0	0
$733$	13038.0	0.656984	0.328492	−	0.944507i	$-0.393460\pi$
$733$	13038.0	0.656984	0.328492	+	0.944507i	$0.393460\pi$
$734$	5024.00	0.252642
$735$	0	0
$736$	−10948.0	−0.548300
$737$	−1364.00	−0.0681731
$738$	0	0
$739$	11620.0	0.578415	0.289207	−	0.957266i	$-0.406608\pi$
$739$	11620.0	0.578415	0.289207	+	0.957266i	$0.406608\pi$
$740$	0	0
$741$	0	0
$742$	−2952.00	−0.146053
$743$	−1328.00	−0.0655715	−0.0327857	−	0.999462i	$-0.510438\pi$
$743$	−1328.00	−0.0655715	−0.0327857	+	0.999462i	$0.510438\pi$
$744$	0	0
$745$	0	0
$746$	−10842.0	−0.532110
$747$	0	0
$748$	5082.00	0.248418
$749$	−40176.0	−1.95995
$750$	0	0
$751$	−37808.0	−1.83706	−0.918531	−	0.395349i	$-0.870624\pi$
$751$	−37808.0	−1.83706	−0.918531	+	0.395349i	$0.870624\pi$
$752$	−13284.0	−0.644172
$753$	0	0
$754$	300.000	0.0144899
$755$	0	0
$756$	0	0
$757$	−34326.0	−1.64808	−0.824042	−	0.566529i	$-0.808286\pi$
$757$	−34326.0	−1.64808	−0.824042	+	0.566529i	$0.808286\pi$
$758$	−1540.00	−0.0737933
$759$	0	0
$760$	0	0
$761$	−4282.00	−0.203972	−0.101986	−	0.994786i	$-0.532520\pi$
$761$	−4282.00	−0.203972	−0.101986	+	0.994786i	$0.532520\pi$
$762$	0	0
$763$	78840.0	3.74076
$764$	9884.00	0.468050
$765$	0	0
$766$	−9468.00	−0.446596
$767$	−1480.00	−0.0696737
$768$	0	0
$769$	24410.0	1.14466	0.572332	−	0.820022i	$-0.306039\pi$
$769$	24410.0	1.14466	0.572332	+	0.820022i	$0.306039\pi$
$770$	0	0
$771$	0	0
$772$	−4466.00	−0.208206
$773$	8162.00	0.379776	0.189888	−	0.981806i	$-0.439188\pi$
$773$	8162.00	0.379776	0.189888	+	0.981806i	$0.439188\pi$
$774$	0	0
$775$	0	0
$776$	17790.0	0.822969
$777$	0	0
$778$	10590.0	0.488008
$779$	16520.0	0.759808
$780$	0	0
$781$	−10868.0	−0.497935
$782$	−4488.00	−0.205231
$783$	0	0
$784$	39073.0	1.77993
$785$	0	0
$786$	0	0
$787$	−26596.0	−1.20463	−0.602316	−	0.798258i	$-0.705755\pi$
$787$	−26596.0	−1.20463	−0.602316	+	0.798258i	$0.705755\pi$
$788$	−25802.0	−1.16644
$789$	0	0
$790$	0	0
$791$	50328.0	2.26227
$792$	0	0
$793$	−244.000	−0.0109265
$794$	7434.00	0.332271
$795$	0	0
$796$	1680.00	0.0748066
$797$	−24614.0	−1.09394	−0.546972	−	0.837151i	$-0.684219\pi$
$797$	−24614.0	−1.09394	−0.546972	+	0.837151i	$0.684219\pi$
$798$	0	0
$799$	−21384.0	−0.946823
$800$	0	0
$801$	0	0
$802$	−11402.0	−0.502018
$803$	22.0000	0.000966828 0
$804$	0	0
$805$	0	0
$806$	256.000	0.0111876
$807$	0	0
$808$	22530.0	0.980944
$809$	−39930.0	−1.73531	−0.867654	−	0.497169i	$-0.834373\pi$
$809$	−39930.0	−1.73531	−0.867654	+	0.497169i	$0.834373\pi$
$810$	0	0
$811$	2412.00	0.104435	0.0522175	−	0.998636i	$-0.483371\pi$
$811$	2412.00	0.104435	0.0522175	+	0.998636i	$0.483371\pi$
$812$	−37800.0	−1.63365
$813$	0	0
$814$	−3454.00	−0.148726
$815$	0	0
$816$	0	0
$817$	−24080.0	−1.03115
$818$	−510.000	−0.0217992
$819$	0	0
$820$	0	0
$821$	6018.00	0.255822	0.127911	−	0.991786i	$-0.459173\pi$
$821$	6018.00	0.255822	0.127911	+	0.991786i	$0.459173\pi$
$822$	0	0
$823$	−34632.0	−1.46682	−0.733412	−	0.679785i	$-0.762073\pi$
$823$	−34632.0	−1.46682	−0.733412	+	0.679785i	$0.762073\pi$
$824$	480.000	0.0202932
$825$	0	0
$826$	−26640.0	−1.12218
$827$	−40044.0	−1.68376	−0.841878	−	0.539668i	$-0.818550\pi$
$827$	−40044.0	−1.68376	−0.841878	+	0.539668i	$0.818550\pi$
$828$	0	0
$829$	−44090.0	−1.84718	−0.923588	−	0.383386i	$-0.874758\pi$
$829$	−44090.0	−1.84718	−0.923588	+	0.383386i	$0.874758\pi$
$830$	0	0
$831$	0	0
$832$	334.000	0.0139175
$833$	62898.0	2.61619
$834$	0	0
$835$	0	0
$836$	10780.0	0.445974
$837$	0	0
$838$	−2420.00	−0.0997584
$839$	23780.0	0.978518	0.489259	−	0.872138i	$-0.337267\pi$
$839$	23780.0	0.978518	0.489259	+	0.872138i	$0.337267\pi$
$840$	0	0
$841$	−1889.00	−0.0774530
$842$	−8178.00	−0.334718
$843$	0	0
$844$	−35644.0	−1.45369
$845$	0	0
$846$	0	0
$847$	−4356.00	−0.176711
$848$	3362.00	0.136146
$849$	0	0
$850$	0	0
$851$	−21352.0	−0.860091
$852$	0	0
$853$	4078.00	0.163691	0.0818453	−	0.996645i	$-0.473919\pi$
$853$	4078.00	0.163691	0.0818453	+	0.996645i	$0.473919\pi$
$854$	−4392.00	−0.175985
$855$	0	0
$856$	−16740.0	−0.668413
$857$	30666.0	1.22232	0.611161	−	0.791506i	$-0.290703\pi$
$857$	30666.0	1.22232	0.611161	+	0.791506i	$0.290703\pi$
$858$	0	0
$859$	4780.00	0.189862	0.0949310	−	0.995484i	$-0.469737\pi$
$859$	4780.00	0.189862	0.0949310	+	0.995484i	$0.469737\pi$
$860$	0	0
$861$	0	0
$862$	6768.00	0.267423
$863$	−9428.00	−0.371880	−0.185940	−	0.982561i	$-0.559533\pi$
$863$	−9428.00	−0.371880	−0.185940	+	0.982561i	$0.559533\pi$
$864$	0	0
$865$	0	0
$866$	478.000	0.0187565
$867$	0	0
$868$	−32256.0	−1.26134
$869$	−12100.0	−0.472341
$870$	0	0
$871$	−248.000	−0.00964771
$872$	32850.0	1.27574
$873$	0	0
$874$	−9520.00	−0.368443
$875$	0	0
$876$	0	0
$877$	−15266.0	−0.587795	−0.293897	−	0.955837i	$-0.594952\pi$
$877$	−15266.0	−0.587795	−0.293897	+	0.955837i	$0.594952\pi$
$878$	−3260.00	−0.125307
$879$	0	0
$880$	0	0
$881$	5118.00	0.195721	0.0978603	−	0.995200i	$-0.468800\pi$
$881$	5118.00	0.195721	0.0978603	+	0.995200i	$0.468800\pi$
$882$	0	0
$883$	44188.0	1.68408	0.842041	−	0.539413i	$-0.181354\pi$
$883$	44188.0	1.68408	0.842041	+	0.539413i	$0.181354\pi$
$884$	924.000	0.0351555
$885$	0	0
$886$	6812.00	0.258300
$887$	−16864.0	−0.638374	−0.319187	−	0.947692i	$-0.603410\pi$
$887$	−16864.0	−0.638374	−0.319187	+	0.947692i	$0.603410\pi$
$888$	0	0
$889$	−1584.00	−0.0597589
$890$	0	0
$891$	0	0
$892$	−26656.0	−1.00057
$893$	−45360.0	−1.69979
$894$	0	0
$895$	0	0
$896$	52380.0	1.95301
$897$	0	0
$898$	−15290.0	−0.568189
$899$	19200.0	0.712298
$900$	0	0
$901$	5412.00	0.200111
$902$	−1298.00	−0.0479143
$903$	0	0
$904$	20970.0	0.771518
$905$	0	0
$906$	0	0
$907$	37924.0	1.38836	0.694182	−	0.719800i	$-0.255766\pi$
$907$	37924.0	1.38836	0.694182	+	0.719800i	$0.255766\pi$
$908$	308.000	0.0112570
$909$	0	0
$910$	0	0
$911$	36628.0	1.33210	0.666048	−	0.745909i	$-0.267985\pi$
$911$	36628.0	1.33210	0.666048	+	0.745909i	$0.267985\pi$
$912$	0	0
$913$	9548.00	0.346104
$914$	12854.0	0.465178
$915$	0	0
$916$	18550.0	0.669115
$917$	−47088.0	−1.69573
$918$	0	0
$919$	−21300.0	−0.764551	−0.382275	−	0.924048i	$-0.624859\pi$
$919$	−21300.0	−0.764551	−0.382275	+	0.924048i	$0.624859\pi$
$920$	0	0
$921$	0	0
$922$	−6782.00	−0.242249
$923$	−1976.00	−0.0704668
$924$	0	0
$925$	0	0
$926$	6408.00	0.227408
$927$	0	0
$928$	−24150.0	−0.854270
$929$	−31450.0	−1.11070	−0.555350	−	0.831616i	$-0.687416\pi$
$929$	−31450.0	−1.11070	−0.555350	+	0.831616i	$0.687416\pi$
$930$	0	0
$931$	133420.	4.69674
$932$	40026.0	1.40675
$933$	0	0
$934$	10476.0	0.367008
$935$	0	0
$936$	0	0
$937$	6174.00	0.215257	0.107628	−	0.994191i	$-0.465674\pi$
$937$	6174.00	0.215257	0.107628	+	0.994191i	$0.465674\pi$
$938$	−4464.00	−0.155389
$939$	0	0
$940$	0	0
$941$	−20422.0	−0.707479	−0.353740	−	0.935344i	$-0.615090\pi$
$941$	−20422.0	−0.707479	−0.353740	+	0.935344i	$0.615090\pi$
$942$	0	0
$943$	−8024.00	−0.277092
$944$	30340.0	1.04606
$945$	0	0
$946$	1892.00	0.0650256
$947$	12156.0	0.417125	0.208562	−	0.978009i	$-0.433122\pi$
$947$	12156.0	0.417125	0.208562	+	0.978009i	$0.433122\pi$
$948$	0	0
$949$	4.00000	0.000136823 0
$950$	0	0
$951$	0	0
$952$	35640.0	1.21334
$953$	14442.0	0.490894	0.245447	−	0.969410i	$-0.421065\pi$
$953$	14442.0	0.490894	0.245447	+	0.969410i	$0.421065\pi$
$954$	0	0
$955$	0	0
$956$	−45640.0	−1.54404
$957$	0	0
$958$	−1920.00	−0.0647520
$959$	−58536.0	−1.97104
$960$	0	0
$961$	−13407.0	−0.450035
$962$	−628.000	−0.0210473
$963$	0	0
$964$	17066.0	0.570186
$965$	0	0
$966$	0	0
$967$	−33356.0	−1.10926	−0.554631	−	0.832096i	$-0.687141\pi$
$967$	−33356.0	−1.10926	−0.554631	+	0.832096i	$0.687141\pi$
$968$	−1815.00	−0.0602648
$969$	0	0
$970$	0	0
$971$	−3852.00	−0.127309	−0.0636543	−	0.997972i	$-0.520275\pi$
$971$	−3852.00	−0.127309	−0.0636543	+	0.997972i	$0.520275\pi$
$972$	0	0
$973$	−6480.00	−0.213504
$974$	−8416.00	−0.276865
$975$	0	0
$976$	5002.00	0.164047
$977$	−37454.0	−1.22647	−0.613234	−	0.789901i	$-0.710132\pi$
$977$	−37454.0	−1.22647	−0.613234	+	0.789901i	$0.710132\pi$
$978$	0	0
$979$	−5170.00	−0.168778
$980$	0	0
$981$	0	0
$982$	−17732.0	−0.576223
$983$	−16228.0	−0.526544	−0.263272	−	0.964722i	$-0.584802\pi$
$983$	−16228.0	−0.526544	−0.263272	+	0.964722i	$0.584802\pi$
$984$	0	0
$985$	0	0
$986$	−9900.00	−0.319757
$987$	0	0
$988$	1960.00	0.0631133
$989$	11696.0	0.376048
$990$	0	0
$991$	−23728.0	−0.760590	−0.380295	−	0.924865i	$-0.624178\pi$
$991$	−23728.0	−0.760590	−0.380295	+	0.924865i	$0.624178\pi$
$992$	−20608.0	−0.659581
$993$	0	0
$994$	−35568.0	−1.13496
$995$	0	0
$996$	0	0
$997$	−41306.0	−1.31211	−0.656055	−	0.754713i	$-0.727776\pi$
$997$	−41306.0	−1.31211	−0.656055	+	0.754713i	$0.727776\pi$
$998$	−5580.00	−0.176986
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.g.1.1		1
3.2	odd	2		825.4.a.d.1.1		1
5.4	even	2		495.4.a.b.1.1		1
15.2	even	4		825.4.c.e.199.1		2
15.8	even	4		825.4.c.e.199.2		2
15.14	odd	2		165.4.a.b.1.1	✓	1
165.164	even	2		1815.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.b.1.1	✓	1	15.14	odd	2
495.4.a.b.1.1		1	5.4	even	2
825.4.a.d.1.1		1	3.2	odd	2
825.4.c.e.199.1		2	15.2	even	4
825.4.c.e.199.2		2	15.8	even	4
1815.4.a.c.1.1		1	165.164	even	2
2475.4.a.g.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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