LMFDB - Embedded newform 2450.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.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:	$19.5633484952$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 350)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{12} +2.00000 q^{13} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} +9.00000 q^{23} -2.00000 q^{24} +2.00000 q^{26} +4.00000 q^{27} -6.00000 q^{29} -5.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} -8.00000 q^{38} -4.00000 q^{39} +3.00000 q^{41} +10.0000 q^{43} +9.00000 q^{46} -3.00000 q^{47} -2.00000 q^{48} +6.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +4.00000 q^{54} +16.0000 q^{57} -6.00000 q^{58} -12.0000 q^{59} +4.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} -2.00000 q^{67} -3.00000 q^{68} -18.0000 q^{69} -9.00000 q^{71} +1.00000 q^{72} -10.0000 q^{73} -8.00000 q^{74} -8.00000 q^{76} -4.00000 q^{78} +5.00000 q^{79} -11.0000 q^{81} +3.00000 q^{82} -6.00000 q^{83} +10.0000 q^{86} +12.0000 q^{87} -3.00000 q^{89} +9.00000 q^{92} +10.0000 q^{93} -3.00000 q^{94} -2.00000 q^{96} +5.00000 q^{97} +O(q^{100})$

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−2.00000	−0.577350
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	1.00000	0.235702
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000	1.87663	0.938315	+	0.345782i	$0.112386\pi$
$24$	−2.00000	−0.408248
$25$	0	0
$26$	2.00000	0.392232
$27$	4.00000	0.769800
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−8.00000	−1.29777
$39$	−4.00000	−0.640513
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	0	0
$46$	9.00000	1.32698
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−2.00000	−0.288675
$49$	0	0
$50$	0	0
$51$	6.00000	0.840168
$52$	2.00000	0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	0	0
$57$	16.0000	2.11925
$58$	−6.00000	−0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	−5.00000	−0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−3.00000	−0.363803
$69$	−18.0000	−2.16695
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	1.00000	0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	−8.00000	−0.917663
$77$	0	0
$78$	−4.00000	−0.452911
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	3.00000	0.331295
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	10.0000	1.07833
$87$	12.0000	1.28654
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	0	0
$92$	9.00000	0.938315
$93$	10.0000	1.03695
$94$	−3.00000	−0.309426
$95$	0	0
$96$	−2.00000	−0.204124
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	6.00000	0.594089
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	4.00000	0.384900
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	16.0000	1.51865
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	16.0000	1.49854
$115$	0	0
$116$	−6.00000	−0.557086
$117$	2.00000	0.184900
$118$	−12.0000	−1.10469
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	4.00000	0.362143
$123$	−6.00000	−0.541002
$124$	−5.00000	−0.449013
$125$	0	0
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	−20.0000	−1.76090
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	−2.00000	−0.172774
$135$	0	0
$136$	−3.00000	−0.257248
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	−18.0000	−1.53226
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	−9.00000	−0.755263
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−10.0000	−0.827606
$147$	0	0
$148$	−8.00000	−0.657596
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	−8.00000	−0.648886
$153$	−3.00000	−0.242536
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	5.00000	0.397779
$159$	12.0000	0.951662
$160$	0	0
$161$	0	0
$162$	−11.0000	−0.864242
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	−6.00000	−0.465690
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−8.00000	−0.611775
$172$	10.0000	0.762493
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	12.0000	0.909718
$175$	0	0
$176$	0	0
$177$	24.0000	1.80395
$178$	−3.00000	−0.224860
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	9.00000	0.663489
$185$	0	0
$186$	10.0000	0.733236
$187$	0	0
$188$	−3.00000	−0.218797
$189$	0	0
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	−2.00000	−0.144338
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	5.00000	0.358979
$195$	0	0
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−17.0000	−1.20510	−0.602549	−	0.798082i	$-0.705848\pi$
$199$	−17.0000	−1.20510	−0.602549	+	0.798082i	$0.705848\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	0	0
$204$	6.00000	0.420084
$205$	0	0
$206$	11.0000	0.766406
$207$	9.00000	0.625543
$208$	2.00000	0.138675
$209$	0	0
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	−6.00000	−0.412082
$213$	18.0000	1.23334
$214$	−12.0000	−0.820303
$215$	0	0
$216$	4.00000	0.272166
$217$	0	0
$218$	−10.0000	−0.677285
$219$	20.0000	1.35147
$220$	0	0
$221$	−6.00000	−0.403604
$222$	16.0000	1.07385
$223$	23.0000	1.54019	0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000	1.54019	0.770097	+	0.637927i	$0.220208\pi$
$224$	0	0
$225$	0	0
$226$	−15.0000	−0.997785
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	16.0000	1.05963
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−12.0000	−0.781133
$237$	−10.0000	−0.649570
$238$	0	0
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	−11.0000	−0.707107
$243$	10.0000	0.641500
$244$	4.00000	0.256074
$245$	0	0
$246$	−6.00000	−0.382546
$247$	−16.0000	−1.01806
$248$	−5.00000	−0.317500
$249$	12.0000	0.760469
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−20.0000	−1.24515
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.00000	0.367194
$268$	−2.00000	−0.122169
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	−11.0000	−0.668202	−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000	−0.668202	−0.334101	+	0.942537i	$0.608433\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−9.00000	−0.543710
$275$	0	0
$276$	−18.0000	−1.08347
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	−2.00000	−0.119952
$279$	−5.00000	−0.299342
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	6.00000	0.357295
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	−9.00000	−0.534052
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−10.0000	−0.586210
$292$	−10.0000	−0.585206
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	0	0
$296$	−8.00000	−0.464991
$297$	0	0
$298$	0	0
$299$	18.0000	1.04097
$300$	0	0
$301$	0	0
$302$	−4.00000	−0.230174
$303$	0	0
$304$	−8.00000	−0.458831
$305$	0	0
$306$	−3.00000	−0.171499
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	−22.0000	−1.25154
$310$	0	0
$311$	−21.0000	−1.19080	−0.595400	−	0.803429i	$-0.703007\pi$
$311$	−21.0000	−1.19080	−0.595400	+	0.803429i	$0.703007\pi$
$312$	−4.00000	−0.226455
$313$	17.0000	0.960897	0.480448	−	0.877023i	$-0.340474\pi$
$313$	17.0000	0.960897	0.480448	+	0.877023i	$0.340474\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	5.00000	0.281272
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	12.0000	0.672927
$319$	0	0
$320$	0	0
$321$	24.0000	1.33955
$322$	0	0
$323$	24.0000	1.33540
$324$	−11.0000	−0.611111
$325$	0	0
$326$	−8.00000	−0.443079
$327$	20.0000	1.10600
$328$	3.00000	0.165647
$329$	0	0
$330$	0	0
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	−6.00000	−0.329293
$333$	−8.00000	−0.438397
$334$	24.0000	1.31322
$335$	0	0
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	−9.00000	−0.489535
$339$	30.0000	1.62938
$340$	0	0
$341$	0	0
$342$	−8.00000	−0.432590
$343$	0	0
$344$	10.0000	0.539164
$345$	0	0
$346$	−12.0000	−0.645124
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	12.0000	0.643268
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	0	0
$353$	−9.00000	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$354$	24.0000	1.27559
$355$	0	0
$356$	−3.00000	−0.159000
$357$	0	0
$358$	24.0000	1.26844
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	−20.0000	−1.05118
$363$	22.0000	1.15470
$364$	0	0
$365$	0	0
$366$	−8.00000	−0.418167
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	9.00000	0.469157
$369$	3.00000	0.156174
$370$	0	0
$371$	0	0
$372$	10.0000	0.518476
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	0	0
$375$	0	0
$376$	−3.00000	−0.154713
$377$	−12.0000	−0.618031
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	15.0000	0.767467
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	−5.00000	−0.254493
$387$	10.0000	0.508329
$388$	5.00000	0.253837
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−27.0000	−1.36545
$392$	0	0
$393$	0	0
$394$	−18.0000	−0.906827
$395$	0	0
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−17.0000	−0.852133
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	4.00000	0.199502
$403$	−10.0000	−0.498135
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	6.00000	0.297044
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	11.0000	0.541931
$413$	0	0
$414$	9.00000	0.442326
$415$	0	0
$416$	2.00000	0.0980581
$417$	4.00000	0.195881
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	2.00000	0.0973585
$423$	−3.00000	−0.145865
$424$	−6.00000	−0.291386
$425$	0	0
$426$	18.0000	0.872103
$427$	0	0
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	4.00000	0.192450
$433$	29.0000	1.39365	0.696826	−	0.717241i	$-0.254595\pi$
$433$	29.0000	1.39365	0.696826	+	0.717241i	$0.254595\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−72.0000	−3.44423
$438$	20.0000	0.955637
$439$	−23.0000	−1.09773	−0.548865	−	0.835911i	$-0.684940\pi$
$439$	−23.0000	−1.09773	−0.548865	+	0.835911i	$0.684940\pi$
$440$	0	0
$441$	0	0
$442$	−6.00000	−0.285391
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	16.0000	0.759326
$445$	0	0
$446$	23.0000	1.08908
$447$	0	0
$448$	0	0
$449$	−27.0000	−1.27421	−0.637104	−	0.770778i	$-0.719868\pi$
$449$	−27.0000	−1.27421	−0.637104	+	0.770778i	$0.719868\pi$
$450$	0	0
$451$	0	0
$452$	−15.0000	−0.705541
$453$	8.00000	0.375873
$454$	18.0000	0.844782
$455$	0	0
$456$	16.0000	0.749269
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−20.0000	−0.934539
$459$	−12.0000	−0.560112
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	25.0000	1.16185	0.580924	−	0.813958i	$-0.302691\pi$
$463$	25.0000	1.16185	0.580924	+	0.813958i	$0.302691\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	2.00000	0.0924500
$469$	0	0
$470$	0	0
$471$	−4.00000	−0.184310
$472$	−12.0000	−0.552345
$473$	0	0
$474$	−10.0000	−0.459315
$475$	0	0
$476$	0	0
$477$	−6.00000	−0.274721
$478$	15.0000	0.686084
$479$	27.0000	1.23366	0.616831	−	0.787096i	$-0.288416\pi$
$479$	27.0000	1.23366	0.616831	+	0.787096i	$0.288416\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	22.0000	1.00207
$483$	0	0
$484$	−11.0000	−0.500000
$485$	0	0
$486$	10.0000	0.453609
$487$	−11.0000	−0.498458	−0.249229	−	0.968445i	$-0.580177\pi$
$487$	−11.0000	−0.498458	−0.249229	+	0.968445i	$0.580177\pi$
$488$	4.00000	0.181071
$489$	16.0000	0.723545
$490$	0	0
$491$	42.0000	1.89543	0.947717	−	0.319113i	$-0.103385\pi$
$491$	42.0000	1.89543	0.947717	+	0.319113i	$0.103385\pi$
$492$	−6.00000	−0.270501
$493$	18.0000	0.810679
$494$	−16.0000	−0.719874
$495$	0	0
$496$	−5.00000	−0.224507
$497$	0	0
$498$	12.0000	0.537733
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	0	0
$501$	−48.0000	−2.14448
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	18.0000	0.799408
$508$	−8.00000	−0.354943
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	−32.0000	−1.41283
$514$	18.0000	0.793946
$515$	0	0
$516$	−20.0000	−0.880451
$517$	0	0
$518$	0	0
$519$	24.0000	1.05348
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	−6.00000	−0.262613
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	0	0
$525$	0	0
$526$	−9.00000	−0.392419
$527$	15.0000	0.653410
$528$	0	0
$529$	58.0000	2.52174
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	6.00000	0.259889
$534$	6.00000	0.259645
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	−48.0000	−2.07135
$538$	30.0000	1.29339
$539$	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	$0.568964\pi$
$542$	−11.0000	−0.472490
$543$	40.0000	1.71656
$544$	−3.00000	−0.128624
$545$	0	0
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−9.00000	−0.384461
$549$	4.00000	0.170716
$550$	0	0
$551$	48.0000	2.04487
$552$	−18.0000	−0.766131
$553$	0	0
$554$	4.00000	0.169944
$555$	0	0
$556$	−2.00000	−0.0848189
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	−5.00000	−0.211667
$559$	20.0000	0.845910
$560$	0	0
$561$	0	0
$562$	3.00000	0.126547
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	2.00000	0.0840663
$567$	0	0
$568$	−9.00000	−0.377632
$569$	−3.00000	−0.125767	−0.0628833	−	0.998021i	$-0.520030\pi$
$569$	−3.00000	−0.125767	−0.0628833	+	0.998021i	$0.520030\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	−30.0000	−1.25327
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	−8.00000	−0.332756
$579$	10.0000	0.415586
$580$	0	0
$581$	0	0
$582$	−10.0000	−0.414513
$583$	0	0
$584$	−10.0000	−0.413803
$585$	0	0
$586$	6.00000	0.247858
$587$	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
$589$	40.0000	1.64817
$590$	0	0
$591$	36.0000	1.48084
$592$	−8.00000	−0.328798
$593$	3.00000	0.123195	0.0615976	−	0.998101i	$-0.480380\pi$
$593$	3.00000	0.123195	0.0615976	+	0.998101i	$0.480380\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	34.0000	1.39153
$598$	18.0000	0.736075
$599$	21.0000	0.858037	0.429018	−	0.903296i	$-0.358860\pi$
$599$	21.0000	0.858037	0.429018	+	0.903296i	$0.358860\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	−4.00000	−0.162758
$605$	0	0
$606$	0	0
$607$	−25.0000	−1.01472	−0.507359	−	0.861735i	$-0.669378\pi$
$607$	−25.0000	−1.01472	−0.507359	+	0.861735i	$0.669378\pi$
$608$	−8.00000	−0.324443
$609$	0	0
$610$	0	0
$611$	−6.00000	−0.242734
$612$	−3.00000	−0.121268
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	0	0
$617$	−15.0000	−0.603877	−0.301939	−	0.953327i	$-0.597634\pi$
$617$	−15.0000	−0.603877	−0.301939	+	0.953327i	$0.597634\pi$
$618$	−22.0000	−0.884970
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	36.0000	1.44463
$622$	−21.0000	−0.842023
$623$	0	0
$624$	−4.00000	−0.160128
$625$	0	0
$626$	17.0000	0.679457
$627$	0	0
$628$	2.00000	0.0798087
$629$	24.0000	0.956943
$630$	0	0
$631$	−19.0000	−0.756378	−0.378189	−	0.925728i	$-0.623453\pi$
$631$	−19.0000	−0.756378	−0.378189	+	0.925728i	$0.623453\pi$
$632$	5.00000	0.198889
$633$	−4.00000	−0.158986
$634$	12.0000	0.476581
$635$	0	0
$636$	12.0000	0.475831
$637$	0	0
$638$	0	0
$639$	−9.00000	−0.356034
$640$	0	0
$641$	45.0000	1.77739	0.888697	−	0.458496i	$-0.151612\pi$
$641$	45.0000	1.77739	0.888697	+	0.458496i	$0.151612\pi$
$642$	24.0000	0.947204
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	0	0
$645$	0	0
$646$	24.0000	0.944267
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	−11.0000	−0.432121
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	20.0000	0.782062
$655$	0	0
$656$	3.00000	0.117130
$657$	−10.0000	−0.390137
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	16.0000	0.622328	0.311164	−	0.950356i	$-0.399281\pi$
$661$	16.0000	0.622328	0.311164	+	0.950356i	$0.399281\pi$
$662$	14.0000	0.544125
$663$	12.0000	0.466041
$664$	−6.00000	−0.232845
$665$	0	0
$666$	−8.00000	−0.309994
$667$	−54.0000	−2.09089
$668$	24.0000	0.928588
$669$	−46.0000	−1.77846
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−5.00000	−0.192736	−0.0963679	−	0.995346i	$-0.530723\pi$
$673$	−5.00000	−0.192736	−0.0963679	+	0.995346i	$0.530723\pi$
$674$	13.0000	0.500741
$675$	0	0
$676$	−9.00000	−0.346154
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	30.0000	1.15214
$679$	0	0
$680$	0	0
$681$	−36.0000	−1.37952
$682$	0	0
$683$	30.0000	1.14792	0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000	1.14792	0.573959	+	0.818884i	$0.305407\pi$
$684$	−8.00000	−0.305888
$685$	0	0
$686$	0	0
$687$	40.0000	1.52610
$688$	10.0000	0.381246
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−50.0000	−1.90209	−0.951045	−	0.309053i	$-0.899988\pi$
$691$	−50.0000	−1.90209	−0.951045	+	0.309053i	$0.899988\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	6.00000	0.227757
$695$	0	0
$696$	12.0000	0.454859
$697$	−9.00000	−0.340899
$698$	−26.0000	−0.984115
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	8.00000	0.301941
$703$	64.0000	2.41381
$704$	0	0
$705$	0	0
$706$	−9.00000	−0.338719
$707$	0	0
$708$	24.0000	0.901975
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	0	0
$711$	5.00000	0.187515
$712$	−3.00000	−0.112430
$713$	−45.0000	−1.68526
$714$	0	0
$715$	0	0
$716$	24.0000	0.896922
$717$	−30.0000	−1.12037
$718$	−24.0000	−0.895672
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	0	0
$722$	45.0000	1.67473
$723$	−44.0000	−1.63638
$724$	−20.0000	−0.743294
$725$	0	0
$726$	22.0000	0.816497
$727$	−31.0000	−1.14973	−0.574863	−	0.818250i	$-0.694945\pi$
$727$	−31.0000	−1.14973	−0.574863	+	0.818250i	$0.694945\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−30.0000	−1.10959
$732$	−8.00000	−0.295689
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	9.00000	0.331744
$737$	0	0
$738$	3.00000	0.110432
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	0	0
$741$	32.0000	1.17555
$742$	0	0
$743$	−33.0000	−1.21065	−0.605326	−	0.795977i	$-0.706957\pi$
$743$	−33.0000	−1.21065	−0.605326	+	0.795977i	$0.706957\pi$
$744$	10.0000	0.366618
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	−6.00000	−0.219529
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	−3.00000	−0.109399
$753$	0	0
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	0	0
$761$	−45.0000	−1.63125	−0.815624	−	0.578582i	$-0.803606\pi$
$761$	−45.0000	−1.63125	−0.815624	+	0.578582i	$0.803606\pi$
$762$	16.0000	0.579619
$763$	0	0
$764$	15.0000	0.542681
$765$	0	0
$766$	9.00000	0.325183
$767$	−24.0000	−0.866590
$768$	−2.00000	−0.0721688
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−36.0000	−1.29651
$772$	−5.00000	−0.179954
$773$	−12.0000	−0.431610	−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000	−0.431610	−0.215805	+	0.976436i	$0.569238\pi$
$774$	10.0000	0.359443
$775$	0	0
$776$	5.00000	0.179490
$777$	0	0
$778$	−18.0000	−0.645331
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	−27.0000	−0.965518
$783$	−24.0000	−0.857690
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−16.0000	−0.570338	−0.285169	−	0.958477i	$-0.592050\pi$
$787$	−16.0000	−0.570338	−0.285169	+	0.958477i	$0.592050\pi$
$788$	−18.0000	−0.641223
$789$	18.0000	0.640817
$790$	0	0
$791$	0	0
$792$	0	0
$793$	8.00000	0.284088
$794$	2.00000	0.0709773
$795$	0	0
$796$	−17.0000	−0.602549
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	9.00000	0.318397
$800$	0	0
$801$	−3.00000	−0.106000
$802$	18.0000	0.635602
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	−10.0000	−0.352235
$807$	−60.0000	−2.11210
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−26.0000	−0.912983	−0.456492	−	0.889728i	$-0.650894\pi$
$811$	−26.0000	−0.912983	−0.456492	+	0.889728i	$0.650894\pi$
$812$	0	0
$813$	22.0000	0.771574
$814$	0	0
$815$	0	0
$816$	6.00000	0.210042
$817$	−80.0000	−2.79885
$818$	7.00000	0.244749
$819$	0	0
$820$	0	0
$821$	−12.0000	−0.418803	−0.209401	−	0.977830i	$-0.567152\pi$
$821$	−12.0000	−0.418803	−0.209401	+	0.977830i	$0.567152\pi$
$822$	18.0000	0.627822
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	11.0000	0.383203
$825$	0	0
$826$	0	0
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	9.00000	0.312772
$829$	−44.0000	−1.52818	−0.764092	−	0.645108i	$-0.776812\pi$
$829$	−44.0000	−1.52818	−0.764092	+	0.645108i	$0.776812\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	2.00000	0.0693375
$833$	0	0
$834$	4.00000	0.138509
$835$	0	0
$836$	0	0
$837$	−20.0000	−0.691301
$838$	30.0000	1.03633
$839$	33.0000	1.13929	0.569643	−	0.821892i	$-0.307081\pi$
$839$	33.0000	1.13929	0.569643	+	0.821892i	$0.307081\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−4.00000	−0.137849
$843$	−6.00000	−0.206651
$844$	2.00000	0.0688428
$845$	0	0
$846$	−3.00000	−0.103142
$847$	0	0
$848$	−6.00000	−0.206041
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−72.0000	−2.46813
$852$	18.0000	0.616670
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	54.0000	1.84460	0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000	1.84460	0.922302	+	0.386469i	$0.126305\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	0	0
$862$	−3.00000	−0.102180
$863$	−21.0000	−0.714848	−0.357424	−	0.933942i	$-0.616345\pi$
$863$	−21.0000	−0.714848	−0.357424	+	0.933942i	$0.616345\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	29.0000	0.985460
$867$	16.0000	0.543388
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−4.00000	−0.135535
$872$	−10.0000	−0.338643
$873$	5.00000	0.169224
$874$	−72.0000	−2.43544
$875$	0	0
$876$	20.0000	0.675737
$877$	16.0000	0.540282	0.270141	−	0.962821i	$-0.412930\pi$
$877$	16.0000	0.540282	0.270141	+	0.962821i	$0.412930\pi$
$878$	−23.0000	−0.776212
$879$	−12.0000	−0.404750
$880$	0	0
$881$	−27.0000	−0.909653	−0.454827	−	0.890580i	$-0.650299\pi$
$881$	−27.0000	−0.909653	−0.454827	+	0.890580i	$0.650299\pi$
$882$	0	0
$883$	−26.0000	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	12.0000	0.403148
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	16.0000	0.536925
$889$	0	0
$890$	0	0
$891$	0	0
$892$	23.0000	0.770097
$893$	24.0000	0.803129
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−36.0000	−1.20201
$898$	−27.0000	−0.901002
$899$	30.0000	1.00056
$900$	0	0
$901$	18.0000	0.599667
$902$	0	0
$903$	0	0
$904$	−15.0000	−0.498893
$905$	0	0
$906$	8.00000	0.265782
$907$	40.0000	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$908$	18.0000	0.597351
$909$	0	0
$910$	0	0
$911$	3.00000	0.0993944	0.0496972	−	0.998764i	$-0.484174\pi$
$911$	3.00000	0.0993944	0.0496972	+	0.998764i	$0.484174\pi$
$912$	16.0000	0.529813
$913$	0	0
$914$	22.0000	0.727695
$915$	0	0
$916$	−20.0000	−0.660819
$917$	0	0
$918$	−12.0000	−0.396059
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	0	0
$921$	32.0000	1.05444
$922$	24.0000	0.790398
$923$	−18.0000	−0.592477
$924$	0	0
$925$	0	0
$926$	25.0000	0.821551
$927$	11.0000	0.361287
$928$	−6.00000	−0.196960
$929$	−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
$932$	6.00000	0.196537
$933$	42.0000	1.37502
$934$	−6.00000	−0.196326
$935$	0	0
$936$	2.00000	0.0653720
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	−34.0000	−1.10955
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	−4.00000	−0.130327
$943$	27.0000	0.879241
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−10.0000	−0.324785
$949$	−20.0000	−0.649227
$950$	0	0
$951$	−24.0000	−0.778253
$952$	0	0
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	15.0000	0.485135
$957$	0	0
$958$	27.0000	0.872330
$959$	0	0
$960$	0	0
$961$	−6.00000	−0.193548
$962$	−16.0000	−0.515861
$963$	−12.0000	−0.386695
$964$	22.0000	0.708572
$965$	0	0
$966$	0	0
$967$	7.00000	0.225105	0.112552	−	0.993646i	$-0.464097\pi$
$967$	7.00000	0.225105	0.112552	+	0.993646i	$0.464097\pi$
$968$	−11.0000	−0.353553
$969$	−48.0000	−1.54198
$970$	0	0
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	10.0000	0.320750
$973$	0	0
$974$	−11.0000	−0.352463
$975$	0	0
$976$	4.00000	0.128037
$977$	51.0000	1.63163	0.815817	−	0.578310i	$-0.196287\pi$
$977$	51.0000	1.63163	0.815817	+	0.578310i	$0.196287\pi$
$978$	16.0000	0.511624
$979$	0	0
$980$	0	0
$981$	−10.0000	−0.319275
$982$	42.0000	1.34027
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	18.0000	0.573237
$987$	0	0
$988$	−16.0000	−0.509028
$989$	90.0000	2.86183
$990$	0	0
$991$	29.0000	0.921215	0.460608	−	0.887604i	$-0.347632\pi$
$991$	29.0000	0.921215	0.460608	+	0.887604i	$0.347632\pi$
$992$	−5.00000	−0.158750
$993$	−28.0000	−0.888553
$994$	0	0
$995$	0	0
$996$	12.0000	0.380235
$997$	−4.00000	−0.126681	−0.0633406	−	0.997992i	$-0.520175\pi$
$997$	−4.00000	−0.126681	−0.0633406	+	0.997992i	$0.520175\pi$
$998$	−34.0000	−1.07625
$999$	−32.0000	−1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.a.u.1.1		1
5.2	odd	4		2450.2.c.d.99.2		2
5.3	odd	4		2450.2.c.d.99.1		2
5.4	even	2		2450.2.a.o.1.1		1
7.3	odd	6		350.2.e.a.51.1	✓	2
7.5	odd	6		350.2.e.a.151.1	yes	2
7.6	odd	2		2450.2.a.be.1.1		1
35.3	even	12		350.2.j.e.149.2		4
35.12	even	12		350.2.j.e.249.2		4
35.13	even	4		2450.2.c.o.99.1		2
35.17	even	12		350.2.j.e.149.1		4
35.19	odd	6		350.2.e.k.151.1	yes	2
35.24	odd	6		350.2.e.k.51.1	yes	2
35.27	even	4		2450.2.c.o.99.2		2
35.33	even	12		350.2.j.e.249.1		4
35.34	odd	2		2450.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.e.a.51.1	✓	2	7.3	odd	6
350.2.e.a.151.1	yes	2	7.5	odd	6
350.2.e.k.51.1	yes	2	35.24	odd	6
350.2.e.k.151.1	yes	2	35.19	odd	6
350.2.j.e.149.1		4	35.17	even	12
350.2.j.e.149.2		4	35.3	even	12
350.2.j.e.249.1		4	35.33	even	12
350.2.j.e.249.2		4	35.12	even	12
2450.2.a.e.1.1		1	35.34	odd	2
2450.2.a.o.1.1		1	5.4	even	2
2450.2.a.u.1.1		1	1.1	even	1	trivial
2450.2.a.be.1.1		1	7.6	odd	2
2450.2.c.d.99.1		2	5.3	odd	4
2450.2.c.d.99.2		2	5.2	odd	4
2450.2.c.o.99.1		2	35.13	even	4
2450.2.c.o.99.2		2	35.27	even	4

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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

Introduction

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Modular forms

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Representations

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