LMFDB - Embedded newform 2450.2.a.w.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
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)

$f(q)$

$=$

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

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} -2.00000 q^{9} -6.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} +1.00000 q^{16} -2.00000 q^{18} +2.00000 q^{19} -6.00000 q^{22} +3.00000 q^{23} -1.00000 q^{24} +4.00000 q^{26} +5.00000 q^{27} -3.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} -2.00000 q^{36} +4.00000 q^{37} +2.00000 q^{38} -4.00000 q^{39} +9.00000 q^{41} +7.00000 q^{43} -6.00000 q^{44} +3.00000 q^{46} -1.00000 q^{48} +4.00000 q^{52} +6.00000 q^{53} +5.00000 q^{54} -2.00000 q^{57} -3.00000 q^{58} -6.00000 q^{59} +5.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} -5.00000 q^{67} -3.00000 q^{69} -6.00000 q^{71} -2.00000 q^{72} +16.0000 q^{73} +4.00000 q^{74} +2.00000 q^{76} -4.00000 q^{78} +2.00000 q^{79} +1.00000 q^{81} +9.00000 q^{82} -3.00000 q^{83} +7.00000 q^{86} +3.00000 q^{87} -6.00000 q^{88} -15.0000 q^{89} +3.00000 q^{92} -8.00000 q^{93} -1.00000 q^{96} -14.0000 q^{97} +12.0000 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	−1.00000	−0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−2.00000	−0.471405
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	−6.00000	−1.27920
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	4.00000	0.784465
$27$	5.00000	0.962250
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	6.00000	1.04447
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	2.00000	0.324443
$39$	−4.00000	−0.640513
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	3.00000	0.442326
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	0	0
$51$	0	0
$52$	4.00000	0.554700
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	0	0
$57$	−2.00000	−0.264906
$58$	−3.00000	−0.393919
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−2.00000	−0.235702
$73$	16.0000	1.87266	0.936329	−	0.351123i	$-0.114200\pi$
$73$	16.0000	1.87266	0.936329	+	0.351123i	$0.114200\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	−4.00000	−0.452911
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	9.00000	0.993884
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	0	0
$86$	7.00000	0.754829
$87$	3.00000	0.321634
$88$	−6.00000	−0.639602
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	0	0
$92$	3.00000	0.312772
$93$	−8.00000	−0.829561
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	12.0000	1.20605
$100$	0	0
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	6.00000	0.582772
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	5.00000	0.481125
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	−3.00000	−0.278543
$117$	−8.00000	−0.739600
$118$	−6.00000	−0.552345
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	5.00000	0.452679
$123$	−9.00000	−0.811503
$124$	8.00000	0.718421
$125$	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
$129$	−7.00000	−0.616316
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	6.00000	0.522233
$133$	0	0
$134$	−5.00000	−0.431934
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	−3.00000	−0.255377
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	0	0
$142$	−6.00000	−0.503509
$143$	−24.0000	−2.00698
$144$	−2.00000	−0.166667
$145$	0	0
$146$	16.0000	1.32417
$147$	0	0
$148$	4.00000	0.328798
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	2.00000	0.159111
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	−3.00000	−0.232845
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−4.00000	−0.305888
$172$	7.00000	0.533745
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	−6.00000	−0.452267
$177$	6.00000	0.450988
$178$	−15.0000	−1.12430
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	0	0
$183$	−5.00000	−0.369611
$184$	3.00000	0.221163
$185$	0	0
$186$	−8.00000	−0.586588
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	−1.00000	−0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	12.0000	0.852803
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	5.00000	0.352673
$202$	15.0000	1.05540
$203$	0	0
$204$	0	0
$205$	0	0
$206$	1.00000	0.0696733
$207$	−6.00000	−0.417029
$208$	4.00000	0.277350
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	6.00000	0.412082
$213$	6.00000	0.411113
$214$	15.0000	1.02538
$215$	0	0
$216$	5.00000	0.340207
$217$	0	0
$218$	11.0000	0.745014
$219$	−16.0000	−1.08118
$220$	0	0
$221$	0	0
$222$	−4.00000	−0.268462
$223$	28.0000	1.87502	0.937509	−	0.347960i	$-0.113126\pi$
$223$	28.0000	1.87502	0.937509	+	0.347960i	$0.113126\pi$
$224$	0	0
$225$	0	0
$226$	−6.00000	−0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−2.00000	−0.132453
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	−3.00000	−0.196960
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	−8.00000	−0.522976
$235$	0	0
$236$	−6.00000	−0.390567
$237$	−2.00000	−0.129914
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	25.0000	1.60706
$243$	−16.0000	−1.02640
$244$	5.00000	0.320092
$245$	0	0
$246$	−9.00000	−0.573819
$247$	8.00000	0.509028
$248$	8.00000	0.508001
$249$	3.00000	0.190117
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−18.0000	−1.13165
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	−7.00000	−0.435801
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	0	0
$267$	15.0000	0.917985
$268$	−5.00000	−0.305424
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	−3.00000	−0.180579
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	−10.0000	−0.599760
$279$	−16.0000	−0.957895
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−24.0000	−1.41915
$287$	0	0
$288$	−2.00000	−0.117851
$289$	−17.0000	−1.00000
$290$	0	0
$291$	14.0000	0.820695
$292$	16.0000	0.936329
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	0	0
$296$	4.00000	0.232495
$297$	−30.0000	−1.74078
$298$	15.0000	0.868927
$299$	12.0000	0.693978
$300$	0	0
$301$	0	0
$302$	−4.00000	−0.230174
$303$	−15.0000	−0.861727
$304$	2.00000	0.114708
$305$	0	0
$306$	0	0
$307$	−5.00000	−0.285365	−0.142683	−	0.989769i	$-0.545573\pi$
$307$	−5.00000	−0.285365	−0.142683	+	0.989769i	$0.545573\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	−4.00000	−0.226455
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	2.00000	0.112509
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	−6.00000	−0.336463
$319$	18.0000	1.00781
$320$	0	0
$321$	−15.0000	−0.837218
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	−11.0000	−0.608301
$328$	9.00000	0.496942
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−3.00000	−0.164646
$333$	−8.00000	−0.438397
$334$	3.00000	0.164153
$335$	0	0
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	3.00000	0.163178
$339$	6.00000	0.325875
$340$	0	0
$341$	−48.0000	−2.59935
$342$	−4.00000	−0.216295
$343$	0	0
$344$	7.00000	0.377415
$345$	0	0
$346$	0	0
$347$	−9.00000	−0.483145	−0.241573	−	0.970383i	$-0.577663\pi$
$347$	−9.00000	−0.483145	−0.241573	+	0.970383i	$0.577663\pi$
$348$	3.00000	0.160817
$349$	17.0000	0.909989	0.454995	−	0.890494i	$-0.349641\pi$
$349$	17.0000	0.909989	0.454995	+	0.890494i	$0.349641\pi$
$350$	0	0
$351$	20.0000	1.06752
$352$	−6.00000	−0.319801
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	−15.0000	−0.794998
$357$	0	0
$358$	−24.0000	−1.26844
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	11.0000	0.578147
$363$	−25.0000	−1.31216
$364$	0	0
$365$	0	0
$366$	−5.00000	−0.261354
$367$	−35.0000	−1.82699	−0.913493	−	0.406855i	$-0.866625\pi$
$367$	−35.0000	−1.82699	−0.913493	+	0.406855i	$0.866625\pi$
$368$	3.00000	0.156386
$369$	−18.0000	−0.937043
$370$	0	0
$371$	0	0
$372$	−8.00000	−0.414781
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	−6.00000	−0.306987
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	−14.0000	−0.711660
$388$	−14.0000	−0.710742
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	6.00000	0.302276
$395$	0	0
$396$	12.0000	0.603023
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	0	0
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	5.00000	0.249377
$403$	32.0000	1.59403
$404$	15.0000	0.746278
$405$	0	0
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	−13.0000	−0.642809	−0.321404	−	0.946942i	$-0.604155\pi$
$409$	−13.0000	−0.642809	−0.321404	+	0.946942i	$0.604155\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	1.00000	0.0492665
$413$	0	0
$414$	−6.00000	−0.294884
$415$	0	0
$416$	4.00000	0.196116
$417$	10.0000	0.489702
$418$	−12.0000	−0.586939
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	−10.0000	−0.486792
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	6.00000	0.290701
$427$	0	0
$428$	15.0000	0.725052
$429$	24.0000	1.15873
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	5.00000	0.240563
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	0	0
$435$	0	0
$436$	11.0000	0.526804
$437$	6.00000	0.287019
$438$	−16.0000	−0.764510
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.0000	−0.997740	−0.498870	−	0.866677i	$-0.666252\pi$
$443$	−21.0000	−0.997740	−0.498870	+	0.866677i	$0.666252\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	28.0000	1.32584
$447$	−15.0000	−0.709476
$448$	0	0
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	0	0
$451$	−54.0000	−2.54276
$452$	−6.00000	−0.282216
$453$	4.00000	0.187936
$454$	12.0000	0.563188
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	13.0000	0.604161	0.302081	−	0.953282i	$-0.402319\pi$
$463$	13.0000	0.604161	0.302081	+	0.953282i	$0.402319\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	12.0000	0.555889
$467$	15.0000	0.694117	0.347059	−	0.937843i	$-0.387180\pi$
$467$	15.0000	0.694117	0.347059	+	0.937843i	$0.387180\pi$
$468$	−8.00000	−0.369800
$469$	0	0
$470$	0	0
$471$	−22.0000	−1.01371
$472$	−6.00000	−0.276172
$473$	−42.0000	−1.93116
$474$	−2.00000	−0.0918630
$475$	0	0
$476$	0	0
$477$	−12.0000	−0.549442
$478$	12.0000	0.548867
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	2.00000	0.0910975
$483$	0	0
$484$	25.0000	1.13636
$485$	0	0
$486$	−16.0000	−0.725775
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	5.00000	0.226339
$489$	−4.00000	−0.180886
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	−9.00000	−0.405751
$493$	0	0
$494$	8.00000	0.359937
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	3.00000	0.134433
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	−3.00000	−0.134030
$502$	−12.0000	−0.535586
$503$	−21.0000	−0.936344	−0.468172	−	0.883637i	$-0.655087\pi$
$503$	−21.0000	−0.936344	−0.468172	+	0.883637i	$0.655087\pi$
$504$	0	0
$505$	0	0
$506$	−18.0000	−0.800198
$507$	−3.00000	−0.133235
$508$	−8.00000	−0.354943
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	10.0000	0.441511
$514$	0	0
$515$	0	0
$516$	−7.00000	−0.308158
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	6.00000	0.262613
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	0	0
$525$	0	0
$526$	21.0000	0.915644
$527$	0	0
$528$	6.00000	0.261116
$529$	−14.0000	−0.608696
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	36.0000	1.55933
$534$	15.0000	0.649113
$535$	0	0
$536$	−5.00000	−0.215967
$537$	24.0000	1.03568
$538$	15.0000	0.646696
$539$	0	0
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	2.00000	0.0859074
$543$	−11.0000	−0.472055
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19.0000	0.812381	0.406191	−	0.913788i	$-0.366857\pi$
$547$	19.0000	0.812381	0.406191	+	0.913788i	$0.366857\pi$
$548$	−12.0000	−0.512615
$549$	−10.0000	−0.426790
$550$	0	0
$551$	−6.00000	−0.255609
$552$	−3.00000	−0.127688
$553$	0	0
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−10.0000	−0.424094
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	−16.0000	−0.677334
$559$	28.0000	1.18427
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	−27.0000	−1.13791	−0.568957	−	0.822367i	$-0.692653\pi$
$563$	−27.0000	−1.13791	−0.568957	+	0.822367i	$0.692653\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	0	0
$568$	−6.00000	−0.251754
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	−24.0000	−1.00349
$573$	6.00000	0.250654
$574$	0	0
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	−17.0000	−0.707107
$579$	2.00000	0.0831172
$580$	0	0
$581$	0	0
$582$	14.0000	0.580319
$583$	−36.0000	−1.49097
$584$	16.0000	0.662085
$585$	0	0
$586$	−12.0000	−0.495715
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	0	0
$591$	−6.00000	−0.246807
$592$	4.00000	0.164399
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	−30.0000	−1.23091
$595$	0	0
$596$	15.0000	0.614424
$597$	4.00000	0.163709
$598$	12.0000	0.490716
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	0	0
$603$	10.0000	0.407231
$604$	−4.00000	−0.162758
$605$	0	0
$606$	−15.0000	−0.609333
$607$	−23.0000	−0.933541	−0.466771	−	0.884378i	$-0.654583\pi$
$607$	−23.0000	−0.933541	−0.466771	+	0.884378i	$0.654583\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	−5.00000	−0.201784
$615$	0	0
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	−1.00000	−0.0402259
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	0	0
$621$	15.0000	0.601929
$622$	−18.0000	−0.721734
$623$	0	0
$624$	−4.00000	−0.160128
$625$	0	0
$626$	−8.00000	−0.319744
$627$	12.0000	0.479234
$628$	22.0000	0.877896
$629$	0	0
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	2.00000	0.0795557
$633$	10.0000	0.397464
$634$	−12.0000	−0.476581
$635$	0	0
$636$	−6.00000	−0.237915
$637$	0	0
$638$	18.0000	0.712627
$639$	12.0000	0.474713
$640$	0	0
$641$	−3.00000	−0.118493	−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000	−0.118493	−0.0592464	+	0.998243i	$0.518870\pi$
$642$	−15.0000	−0.592003
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	1.00000	0.0392837
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	4.00000	0.156652
$653$	48.0000	1.87839	0.939193	−	0.343391i	$-0.111576\pi$
$653$	48.0000	1.87839	0.939193	+	0.343391i	$0.111576\pi$
$654$	−11.0000	−0.430134
$655$	0	0
$656$	9.00000	0.351391
$657$	−32.0000	−1.24844
$658$	0	0
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	41.0000	1.59472	0.797358	−	0.603507i	$-0.206231\pi$
$661$	41.0000	1.59472	0.797358	+	0.603507i	$0.206231\pi$
$662$	−28.0000	−1.08825
$663$	0	0
$664$	−3.00000	−0.116423
$665$	0	0
$666$	−8.00000	−0.309994
$667$	−9.00000	−0.348481
$668$	3.00000	0.116073
$669$	−28.0000	−1.08254
$670$	0	0
$671$	−30.0000	−1.15814
$672$	0	0
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	3.00000	0.115385
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	−48.0000	−1.83801
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	0	0
$687$	−14.0000	−0.534133
$688$	7.00000	0.266872
$689$	24.0000	0.914327
$690$	0	0
$691$	−22.0000	−0.836919	−0.418460	−	0.908235i	$-0.637430\pi$
$691$	−22.0000	−0.836919	−0.418460	+	0.908235i	$0.637430\pi$
$692$	0	0
$693$	0	0
$694$	−9.00000	−0.341635
$695$	0	0
$696$	3.00000	0.113715
$697$	0	0
$698$	17.0000	0.643459
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−3.00000	−0.113308	−0.0566542	−	0.998394i	$-0.518043\pi$
$701$	−3.00000	−0.113308	−0.0566542	+	0.998394i	$0.518043\pi$
$702$	20.0000	0.754851
$703$	8.00000	0.301726
$704$	−6.00000	−0.226134
$705$	0	0
$706$	6.00000	0.225813
$707$	0	0
$708$	6.00000	0.225494
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	−15.0000	−0.562149
$713$	24.0000	0.898807
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	−12.0000	−0.448148
$718$	24.0000	0.895672
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	0	0
$722$	−15.0000	−0.558242
$723$	−2.00000	−0.0743808
$724$	11.0000	0.408812
$725$	0	0
$726$	−25.0000	−0.927837
$727$	19.0000	0.704671	0.352335	−	0.935874i	$-0.385388\pi$
$727$	19.0000	0.704671	0.352335	+	0.935874i	$0.385388\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	−5.00000	−0.184805
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−35.0000	−1.29187
$735$	0	0
$736$	3.00000	0.110581
$737$	30.0000	1.10506
$738$	−18.0000	−0.662589
$739$	26.0000	0.956425	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956425	0.478213	+	0.878244i	$0.341285\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−39.0000	−1.43077	−0.715386	−	0.698730i	$-0.753749\pi$
$743$	−39.0000	−1.43077	−0.715386	+	0.698730i	$0.753749\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	4.00000	0.146450
$747$	6.00000	0.219529
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	28.0000	1.01768	0.508839	−	0.860862i	$-0.330075\pi$
$757$	28.0000	1.01768	0.508839	+	0.860862i	$0.330075\pi$
$758$	−34.0000	−1.23494
$759$	18.0000	0.653359
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	8.00000	0.289809
$763$	0	0
$764$	−6.00000	−0.217072
$765$	0	0
$766$	15.0000	0.541972
$767$	−24.0000	−0.866590
$768$	−1.00000	−0.0360844
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	12.0000	0.431610	0.215805	−	0.976436i	$-0.430762\pi$
$773$	12.0000	0.431610	0.215805	+	0.976436i	$0.430762\pi$
$774$	−14.0000	−0.503220
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	−30.0000	−1.07555
$779$	18.0000	0.644917
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	−15.0000	−0.536056
$784$	0	0
$785$	0	0
$786$	0	0
$787$	43.0000	1.53278	0.766392	−	0.642373i	$-0.222050\pi$
$787$	43.0000	1.53278	0.766392	+	0.642373i	$0.222050\pi$
$788$	6.00000	0.213741
$789$	−21.0000	−0.747620
$790$	0	0
$791$	0	0
$792$	12.0000	0.426401
$793$	20.0000	0.710221
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−4.00000	−0.141776
$797$	48.0000	1.70025	0.850124	−	0.526583i	$-0.176527\pi$
$797$	48.0000	1.70025	0.850124	+	0.526583i	$0.176527\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	30.0000	1.06000
$802$	15.0000	0.529668
$803$	−96.0000	−3.38777
$804$	5.00000	0.176336
$805$	0	0
$806$	32.0000	1.12715
$807$	−15.0000	−0.528025
$808$	15.0000	0.527698
$809$	21.0000	0.738321	0.369160	−	0.929366i	$-0.379645\pi$
$809$	21.0000	0.738321	0.369160	+	0.929366i	$0.379645\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	0	0
$813$	−2.00000	−0.0701431
$814$	−24.0000	−0.841200
$815$	0	0
$816$	0	0
$817$	14.0000	0.489798
$818$	−13.0000	−0.454534
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	12.0000	0.418548
$823$	19.0000	0.662298	0.331149	−	0.943578i	$-0.392564\pi$
$823$	19.0000	0.662298	0.331149	+	0.943578i	$0.392564\pi$
$824$	1.00000	0.0348367
$825$	0	0
$826$	0	0
$827$	−15.0000	−0.521601	−0.260801	−	0.965393i	$-0.583986\pi$
$827$	−15.0000	−0.521601	−0.260801	+	0.965393i	$0.583986\pi$
$828$	−6.00000	−0.208514
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	4.00000	0.138675
$833$	0	0
$834$	10.0000	0.346272
$835$	0	0
$836$	−12.0000	−0.415029
$837$	40.0000	1.38260
$838$	0	0
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	17.0000	0.585859
$843$	6.00000	0.206651
$844$	−10.0000	−0.344214
$845$	0	0
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	−4.00000	−0.137280
$850$	0	0
$851$	12.0000	0.411355
$852$	6.00000	0.205557
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	0	0
$855$	0	0
$856$	15.0000	0.512689
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	24.0000	0.819346
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	0	0
$861$	0	0
$862$	30.0000	1.02180
$863$	−27.0000	−0.919091	−0.459545	−	0.888154i	$-0.651988\pi$
$863$	−27.0000	−0.919091	−0.459545	+	0.888154i	$0.651988\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	22.0000	0.747590
$867$	17.0000	0.577350
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	−20.0000	−0.677674
$872$	11.0000	0.372507
$873$	28.0000	0.947656
$874$	6.00000	0.202953
$875$	0	0
$876$	−16.0000	−0.540590
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−28.0000	−0.944954
$879$	12.0000	0.404750
$880$	0	0
$881$	57.0000	1.92038	0.960189	−	0.279350i	$-0.0901189\pi$
$881$	57.0000	1.92038	0.960189	+	0.279350i	$0.0901189\pi$
$882$	0	0
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	0	0
$885$	0	0
$886$	−21.0000	−0.705509
$887$	21.0000	0.705111	0.352555	−	0.935791i	$-0.385313\pi$
$887$	21.0000	0.705111	0.352555	+	0.935791i	$0.385313\pi$
$888$	−4.00000	−0.134231
$889$	0	0
$890$	0	0
$891$	−6.00000	−0.201008
$892$	28.0000	0.937509
$893$	0	0
$894$	−15.0000	−0.501675
$895$	0	0
$896$	0	0
$897$	−12.0000	−0.400668
$898$	−9.00000	−0.300334
$899$	−24.0000	−0.800445
$900$	0	0
$901$	0	0
$902$	−54.0000	−1.79800
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	4.00000	0.132891
$907$	25.0000	0.830111	0.415056	−	0.909796i	$-0.363762\pi$
$907$	25.0000	0.830111	0.415056	+	0.909796i	$0.363762\pi$
$908$	12.0000	0.398234
$909$	−30.0000	−0.995037
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	−2.00000	−0.0662266
$913$	18.0000	0.595713
$914$	−32.0000	−1.05847
$915$	0	0
$916$	14.0000	0.462573
$917$	0	0
$918$	0	0
$919$	14.0000	0.461817	0.230909	−	0.972975i	$-0.425830\pi$
$919$	14.0000	0.461817	0.230909	+	0.972975i	$0.425830\pi$
$920$	0	0
$921$	5.00000	0.164756
$922$	18.0000	0.592798
$923$	−24.0000	−0.789970
$924$	0	0
$925$	0	0
$926$	13.0000	0.427207
$927$	−2.00000	−0.0656886
$928$	−3.00000	−0.0984798
$929$	−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
$932$	12.0000	0.393073
$933$	18.0000	0.589294
$934$	15.0000	0.490815
$935$	0	0
$936$	−8.00000	−0.261488
$937$	28.0000	0.914720	0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000	0.914720	0.457360	+	0.889282i	$0.348795\pi$
$938$	0	0
$939$	8.00000	0.261070
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	−22.0000	−0.716799
$943$	27.0000	0.879241
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−42.0000	−1.36554
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	−2.00000	−0.0649570
$949$	64.0000	2.07753
$950$	0	0
$951$	12.0000	0.389127
$952$	0	0
$953$	−60.0000	−1.94359	−0.971795	−	0.235826i	$-0.924220\pi$
$953$	−60.0000	−1.94359	−0.971795	+	0.235826i	$0.924220\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	12.0000	0.388108
$957$	−18.0000	−0.581857
$958$	12.0000	0.387702
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	16.0000	0.515861
$963$	−30.0000	−0.966736
$964$	2.00000	0.0644157
$965$	0	0
$966$	0	0
$967$	−35.0000	−1.12552	−0.562762	−	0.826619i	$-0.690261\pi$
$967$	−35.0000	−1.12552	−0.562762	+	0.826619i	$0.690261\pi$
$968$	25.0000	0.803530
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	−16.0000	−0.513200
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	5.00000	0.160046
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	−4.00000	−0.127906
$979$	90.0000	2.87641
$980$	0	0
$981$	−22.0000	−0.702406
$982$	0	0
$983$	−39.0000	−1.24391	−0.621953	−	0.783054i	$-0.713661\pi$
$983$	−39.0000	−1.24391	−0.621953	+	0.783054i	$0.713661\pi$
$984$	−9.00000	−0.286910
$985$	0	0
$986$	0	0
$987$	0	0
$988$	8.00000	0.254514
$989$	21.0000	0.667761
$990$	0	0
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	8.00000	0.254000
$993$	28.0000	0.888553
$994$	0	0
$995$	0	0
$996$	3.00000	0.0950586
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	−22.0000	−0.696398
$999$	20.0000	0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.a.w.1.1		1
5.2	odd	4		2450.2.c.g.99.2		2
5.3	odd	4		2450.2.c.g.99.1		2
5.4	even	2		490.2.a.c.1.1		1
7.2	even	3		350.2.e.e.151.1		2
7.4	even	3		350.2.e.e.51.1		2
7.6	odd	2		2450.2.a.bc.1.1		1
15.14	odd	2		4410.2.a.bm.1.1		1
20.19	odd	2		3920.2.a.p.1.1		1
35.2	odd	12		350.2.j.b.249.2		4
35.4	even	6		70.2.e.c.51.1	yes	2
35.9	even	6		70.2.e.c.11.1	✓	2
35.13	even	4		2450.2.c.l.99.1		2
35.18	odd	12		350.2.j.b.149.2		4
35.19	odd	6		490.2.e.h.361.1		2
35.23	odd	12		350.2.j.b.249.1		4
35.24	odd	6		490.2.e.h.471.1		2
35.27	even	4		2450.2.c.l.99.2		2
35.32	odd	12		350.2.j.b.149.1		4
35.34	odd	2		490.2.a.b.1.1		1
105.44	odd	6		630.2.k.b.361.1		2
105.74	odd	6		630.2.k.b.541.1		2
105.104	even	2		4410.2.a.bd.1.1		1
140.39	odd	6		560.2.q.g.401.1		2
140.79	odd	6		560.2.q.g.81.1		2
140.139	even	2		3920.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.e.c.11.1	✓	2	35.9	even	6
70.2.e.c.51.1	yes	2	35.4	even	6
350.2.e.e.51.1		2	7.4	even	3
350.2.e.e.151.1		2	7.2	even	3
350.2.j.b.149.1		4	35.32	odd	12
350.2.j.b.149.2		4	35.18	odd	12
350.2.j.b.249.1		4	35.23	odd	12
350.2.j.b.249.2		4	35.2	odd	12
490.2.a.b.1.1		1	35.34	odd	2
490.2.a.c.1.1		1	5.4	even	2
490.2.e.h.361.1		2	35.19	odd	6
490.2.e.h.471.1		2	35.24	odd	6
560.2.q.g.81.1		2	140.79	odd	6
560.2.q.g.401.1		2	140.39	odd	6
630.2.k.b.361.1		2	105.44	odd	6
630.2.k.b.541.1		2	105.74	odd	6
2450.2.a.w.1.1		1	1.1	even	1	trivial
2450.2.a.bc.1.1		1	7.6	odd	2
2450.2.c.g.99.1		2	5.3	odd	4
2450.2.c.g.99.2		2	5.2	odd	4
2450.2.c.l.99.1		2	35.13	even	4
2450.2.c.l.99.2		2	35.27	even	4
3920.2.a.p.1.1		1	20.19	odd	2
3920.2.a.bc.1.1		1	140.139	even	2
4410.2.a.bd.1.1		1	105.104	even	2
4410.2.a.bm.1.1		1	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

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Representations

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