LMFDB - Embedded newform 2450.2.a.f.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} -2.00000 q^{12} +5.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -3.00000 q^{22} -3.00000 q^{23} +2.00000 q^{24} -5.00000 q^{26} +4.00000 q^{27} -6.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -11.0000 q^{37} -1.00000 q^{38} -10.0000 q^{39} -3.00000 q^{41} +10.0000 q^{43} +3.00000 q^{44} +3.00000 q^{46} +3.00000 q^{47} -2.00000 q^{48} -12.0000 q^{51} +5.00000 q^{52} -3.00000 q^{53} -4.00000 q^{54} -2.00000 q^{57} +6.00000 q^{58} +4.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +6.00000 q^{66} +4.00000 q^{67} +6.00000 q^{68} +6.00000 q^{69} +12.0000 q^{71} -1.00000 q^{72} -4.00000 q^{73} +11.0000 q^{74} +1.00000 q^{76} +10.0000 q^{78} -10.0000 q^{79} -11.0000 q^{81} +3.00000 q^{82} -12.0000 q^{83} -10.0000 q^{86} +12.0000 q^{87} -3.00000 q^{88} -6.00000 q^{89} -3.00000 q^{92} -8.00000 q^{93} -3.00000 q^{94} +2.00000 q^{96} +14.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)

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
$3$	−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
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−2.00000	−0.577350
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−1.00000	−0.235702
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	−3.00000	−0.639602
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	−5.00000	−0.980581
$27$	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
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	−6.00000	−1.04447
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	−1.00000	−0.162221
$39$	−10.0000	−1.60128
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	3.00000	0.442326
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−2.00000	−0.288675
$49$	0	0
$50$	0	0
$51$	−12.0000	−1.68034
$52$	5.00000	0.693375
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	0	0
$57$	−2.00000	−0.264906
$58$	6.00000	0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	6.00000	0.727607
$69$	6.00000	0.722315
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−1.00000	−0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	11.0000	1.27872
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	10.0000	1.13228
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	3.00000	0.331295
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	0	0
$86$	−10.0000	−1.07833
$87$	12.0000	1.28654
$88$	−3.00000	−0.319801
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	−3.00000	−0.312772
$93$	−8.00000	−0.829561
$94$	−3.00000	−0.309426
$95$	0	0
$96$	2.00000	0.204124
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	12.0000	1.18818
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−5.00000	−0.490290
$105$	0	0
$106$	3.00000	0.291386
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	4.00000	0.384900
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	22.0000	2.08815
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	−6.00000	−0.557086
$117$	5.00000	0.462250
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−4.00000	−0.362143
$123$	6.00000	0.541002
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	19.0000	1.68598	0.842989	−	0.537931i	$-0.180794\pi$
$127$	19.0000	1.68598	0.842989	+	0.537931i	$0.180794\pi$
$128$	−1.00000	−0.0883883
$129$	−20.0000	−1.76090
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	−6.00000	−0.510754
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	−12.0000	−1.00702
$143$	15.0000	1.25436
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	−11.0000	−0.904194
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	−1.00000	−0.0811107
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	−10.0000	−0.800641
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	10.0000	0.795557
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	11.0000	0.864242
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	12.0000	0.931381
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	1.00000	0.0764719
$172$	10.0000	0.762493
$173$	3.00000	0.228086	0.114043	−	0.993476i	$-0.463620\pi$
$173$	3.00000	0.228086	0.114043	+	0.993476i	$0.463620\pi$
$174$	−12.0000	−0.909718
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	6.00000	0.449719
$179$	−3.00000	−0.224231	−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000	−0.224231	−0.112115	+	0.993695i	$0.535763\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	3.00000	0.221163
$185$	0	0
$186$	8.00000	0.586588
$187$	18.0000	1.31629
$188$	3.00000	0.218797
$189$	0	0
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−2.00000	−0.144338
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	0	0
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	−3.00000	−0.213201
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	−12.0000	−0.844317
$203$	0	0
$204$	−12.0000	−0.840168
$205$	0	0
$206$	4.00000	0.278693
$207$	−3.00000	−0.208514
$208$	5.00000	0.346688
$209$	3.00000	0.207514
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	−3.00000	−0.206041
$213$	−24.0000	−1.64445
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−4.00000	−0.272166
$217$	0	0
$218$	4.00000	0.270914
$219$	8.00000	0.540590
$220$	0	0
$221$	30.0000	2.01802
$222$	−22.0000	−1.47654
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	0	0
$226$	12.0000	0.798228
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	−2.00000	−0.132453
$229$	28.0000	1.85029	0.925146	−	0.379611i	$-0.123942\pi$
$229$	28.0000	1.85029	0.925146	+	0.379611i	$0.123942\pi$
$230$	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	$0.437031\pi$
$234$	−5.00000	−0.326860
$235$	0	0
$236$	0	0
$237$	20.0000	1.29914
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	25.0000	1.61039	0.805196	−	0.593009i	$-0.202060\pi$
$241$	25.0000	1.61039	0.805196	+	0.593009i	$0.202060\pi$
$242$	2.00000	0.128565
$243$	10.0000	0.641500
$244$	4.00000	0.256074
$245$	0	0
$246$	−6.00000	−0.382546
$247$	5.00000	0.318142
$248$	−4.00000	−0.254000
$249$	24.0000	1.52094
$250$	0	0
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	0	0
$253$	−9.00000	−0.565825
$254$	−19.0000	−1.19217
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	20.0000	1.24515
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	3.00000	0.185341
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	0	0
$267$	12.0000	0.734388
$268$	4.00000	0.244339
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	6.00000	0.361158
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−4.00000	−0.239904
$279$	4.00000	0.239474
$280$	0	0
$281$	−3.00000	−0.178965	−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000	−0.178965	−0.0894825	+	0.995988i	$0.528521\pi$
$282$	6.00000	0.357295
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−15.0000	−0.886969
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	−28.0000	−1.64139
$292$	−4.00000	−0.234082
$293$	−27.0000	−1.57736	−0.788678	−	0.614806i	$-0.789234\pi$
$293$	−27.0000	−1.57736	−0.788678	+	0.614806i	$0.789234\pi$
$294$	0	0
$295$	0	0
$296$	11.0000	0.639362
$297$	12.0000	0.696311
$298$	−18.0000	−1.04271
$299$	−15.0000	−0.867472
$300$	0	0
$301$	0	0
$302$	−14.0000	−0.805609
$303$	−24.0000	−1.37876
$304$	1.00000	0.0573539
$305$	0	0
$306$	−6.00000	−0.342997
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	10.0000	0.566139
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	−5.00000	−0.282166
$315$	0	0
$316$	−10.0000	−0.562544
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−6.00000	−0.336463
$319$	−18.0000	−1.00781
$320$	0	0
$321$	−24.0000	−1.33955
$322$	0	0
$323$	6.00000	0.333849
$324$	−11.0000	−0.611111
$325$	0	0
$326$	−4.00000	−0.221540
$327$	8.00000	0.442401
$328$	3.00000	0.165647
$329$	0	0
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	−12.0000	−0.658586
$333$	−11.0000	−0.602796
$334$	−9.00000	−0.492458
$335$	0	0
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	−12.0000	−0.652714
$339$	24.0000	1.30350
$340$	0	0
$341$	12.0000	0.649836
$342$	−1.00000	−0.0540738
$343$	0	0
$344$	−10.0000	−0.539164
$345$	0	0
$346$	−3.00000	−0.161281
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	12.0000	0.643268
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	20.0000	1.06752
$352$	−3.00000	−0.159901
$353$	12.0000	0.638696	0.319348	−	0.947638i	$-0.396536\pi$
$353$	12.0000	0.638696	0.319348	+	0.947638i	$0.396536\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	3.00000	0.158555
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	2.00000	0.105118
$363$	4.00000	0.209946
$364$	0	0
$365$	0	0
$366$	8.00000	0.418167
$367$	−1.00000	−0.0521996	−0.0260998	−	0.999659i	$-0.508309\pi$
$367$	−1.00000	−0.0521996	−0.0260998	+	0.999659i	$0.508309\pi$
$368$	−3.00000	−0.156386
$369$	−3.00000	−0.156174
$370$	0	0
$371$	0	0
$372$	−8.00000	−0.414781
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	−3.00000	−0.154713
$377$	−30.0000	−1.54508
$378$	0	0
$379$	−25.0000	−1.28416	−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000	−1.28416	−0.642082	+	0.766636i	$0.721929\pi$
$380$	0	0
$381$	−38.0000	−1.94680
$382$	−12.0000	−0.613973
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	−4.00000	−0.203595
$387$	10.0000	0.508329
$388$	14.0000	0.710742
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	6.00000	0.302660
$394$	−3.00000	−0.151138
$395$	0	0
$396$	3.00000	0.150756
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	0	0
$401$	21.0000	1.04869	0.524345	−	0.851506i	$-0.324310\pi$
$401$	21.0000	1.04869	0.524345	+	0.851506i	$0.324310\pi$
$402$	8.00000	0.399004
$403$	20.0000	0.996271
$404$	12.0000	0.597022
$405$	0	0
$406$	0	0
$407$	−33.0000	−1.63575
$408$	12.0000	0.594089
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	24.0000	1.18383
$412$	−4.00000	−0.197066
$413$	0	0
$414$	3.00000	0.147442
$415$	0	0
$416$	−5.00000	−0.245145
$417$	−8.00000	−0.391762
$418$	−3.00000	−0.146735
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	1.00000	0.0486792
$423$	3.00000	0.145865
$424$	3.00000	0.145693
$425$	0	0
$426$	24.0000	1.16280
$427$	0	0
$428$	12.0000	0.580042
$429$	−30.0000	−1.44841
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	4.00000	0.192450
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	0	0
$436$	−4.00000	−0.191565
$437$	−3.00000	−0.143509
$438$	−8.00000	−0.382255
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	0	0
$442$	−30.0000	−1.42695
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	22.0000	1.04407
$445$	0	0
$446$	−8.00000	−0.378811
$447$	−36.0000	−1.70274
$448$	0	0
$449$	−3.00000	−0.141579	−0.0707894	−	0.997491i	$-0.522552\pi$
$449$	−3.00000	−0.141579	−0.0707894	+	0.997491i	$0.522552\pi$
$450$	0	0
$451$	−9.00000	−0.423793
$452$	−12.0000	−0.564433
$453$	−28.0000	−1.31555
$454$	24.0000	1.12638
$455$	0	0
$456$	2.00000	0.0936586
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−28.0000	−1.30835
$459$	24.0000	1.12022
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	19.0000	0.883005	0.441502	−	0.897260i	$-0.354446\pi$
$463$	19.0000	0.883005	0.441502	+	0.897260i	$0.354446\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	5.00000	0.231125
$469$	0	0
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	30.0000	1.37940
$474$	−20.0000	−0.918630
$475$	0	0
$476$	0	0
$477$	−3.00000	−0.137361
$478$	−6.00000	−0.274434
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−55.0000	−2.50778
$482$	−25.0000	−1.13872
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	−10.0000	−0.453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	−4.00000	−0.181071
$489$	−8.00000	−0.361773
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	6.00000	0.270501
$493$	−36.0000	−1.62136
$494$	−5.00000	−0.224961
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	−24.0000	−1.07547
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	−15.0000	−0.669483
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	0	0
$506$	9.00000	0.400099
$507$	−24.0000	−1.06588
$508$	19.0000	0.842989
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	12.0000	0.529297
$515$	0	0
$516$	−20.0000	−0.880451
$517$	9.00000	0.395820
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	6.00000	0.262613
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−3.00000	−0.131056
$525$	0	0
$526$	0	0
$527$	24.0000	1.04546
$528$	−6.00000	−0.261116
$529$	−14.0000	−0.608696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−15.0000	−0.649722
$534$	−12.0000	−0.519291
$535$	0	0
$536$	−4.00000	−0.172774
$537$	6.00000	0.258919
$538$	−12.0000	−0.517357
$539$	0	0
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	−16.0000	−0.687259
$543$	4.00000	0.171656
$544$	−6.00000	−0.257248
$545$	0	0
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−12.0000	−0.512615
$549$	4.00000	0.170716
$550$	0	0
$551$	−6.00000	−0.255609
$552$	−6.00000	−0.255377
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	4.00000	0.169638
$557$	27.0000	1.14403	0.572013	−	0.820244i	$-0.306163\pi$
$557$	27.0000	1.14403	0.572013	+	0.820244i	$0.306163\pi$
$558$	−4.00000	−0.169334
$559$	50.0000	2.11477
$560$	0	0
$561$	−36.0000	−1.51992
$562$	3.00000	0.126547
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	−26.0000	−1.09286
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−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	$0.362561\pi$
$572$	15.0000	0.627182
$573$	−24.0000	−1.00261
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	20.0000	0.832611	0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000	0.832611	0.416305	+	0.909225i	$0.363325\pi$
$578$	−19.0000	−0.790296
$579$	−8.00000	−0.332469
$580$	0	0
$581$	0	0
$582$	28.0000	1.16064
$583$	−9.00000	−0.372742
$584$	4.00000	0.165521
$585$	0	0
$586$	27.0000	1.11536
$587$	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
$589$	4.00000	0.164817
$590$	0	0
$591$	−6.00000	−0.246807
$592$	−11.0000	−0.452097
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	−12.0000	−0.492366
$595$	0	0
$596$	18.0000	0.737309
$597$	−8.00000	−0.327418
$598$	15.0000	0.613396
$599$	42.0000	1.71607	0.858037	−	0.513588i	$-0.171684\pi$
$599$	42.0000	1.71607	0.858037	+	0.513588i	$0.171684\pi$
$600$	0	0
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	14.0000	0.569652
$605$	0	0
$606$	24.0000	0.974933
$607$	−19.0000	−0.771186	−0.385593	−	0.922669i	$-0.626003\pi$
$607$	−19.0000	−0.771186	−0.385593	+	0.922669i	$0.626003\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	15.0000	0.606835
$612$	6.00000	0.242536
$613$	−47.0000	−1.89831	−0.949156	−	0.314806i	$-0.898061\pi$
$613$	−47.0000	−1.89831	−0.949156	+	0.314806i	$0.898061\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−8.00000	−0.321807
$619$	1.00000	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$620$	0	0
$621$	−12.0000	−0.481543
$622$	12.0000	0.481156
$623$	0	0
$624$	−10.0000	−0.400320
$625$	0	0
$626$	−8.00000	−0.319744
$627$	−6.00000	−0.239617
$628$	5.00000	0.199522
$629$	−66.0000	−2.63159
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	10.0000	0.397779
$633$	2.00000	0.0794929
$634$	18.0000	0.714871
$635$	0	0
$636$	6.00000	0.237915
$637$	0	0
$638$	18.0000	0.712627
$639$	12.0000	0.474713
$640$	0	0
$641$	−45.0000	−1.77739	−0.888697	−	0.458496i	$-0.848388\pi$
$641$	−45.0000	−1.77739	−0.888697	+	0.458496i	$0.848388\pi$
$642$	24.0000	0.947204
$643$	38.0000	1.49857	0.749287	−	0.662246i	$-0.230396\pi$
$643$	38.0000	1.49857	0.749287	+	0.662246i	$0.230396\pi$
$644$	0	0
$645$	0	0
$646$	−6.00000	−0.236067
$647$	−21.0000	−0.825595	−0.412798	−	0.910823i	$-0.635448\pi$
$647$	−21.0000	−0.825595	−0.412798	+	0.910823i	$0.635448\pi$
$648$	11.0000	0.432121
$649$	0	0
$650$	0	0
$651$	0	0
$652$	4.00000	0.156652
$653$	−21.0000	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$653$	−21.0000	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$654$	−8.00000	−0.312825
$655$	0	0
$656$	−3.00000	−0.117130
$657$	−4.00000	−0.156055
$658$	0	0
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	−44.0000	−1.71140	−0.855701	−	0.517471i	$-0.826874\pi$
$661$	−44.0000	−1.71140	−0.855701	+	0.517471i	$0.826874\pi$
$662$	7.00000	0.272063
$663$	−60.0000	−2.33021
$664$	12.0000	0.465690
$665$	0	0
$666$	11.0000	0.426241
$667$	18.0000	0.696963
$668$	9.00000	0.348220
$669$	−16.0000	−0.618596
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	12.0000	0.461538
$677$	−3.00000	−0.115299	−0.0576497	−	0.998337i	$-0.518361\pi$
$677$	−3.00000	−0.115299	−0.0576497	+	0.998337i	$0.518361\pi$
$678$	−24.0000	−0.921714
$679$	0	0
$680$	0	0
$681$	48.0000	1.83936
$682$	−12.0000	−0.459504
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	0	0
$687$	−56.0000	−2.13653
$688$	10.0000	0.381246
$689$	−15.0000	−0.571454
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	3.00000	0.114043
$693$	0	0
$694$	24.0000	0.911028
$695$	0	0
$696$	−12.0000	−0.454859
$697$	−18.0000	−0.681799
$698$	−10.0000	−0.378506
$699$	−12.0000	−0.453882
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	−20.0000	−0.754851
$703$	−11.0000	−0.414873
$704$	3.00000	0.113067
$705$	0	0
$706$	−12.0000	−0.451626
$707$	0	0
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	6.00000	0.224860
$713$	−12.0000	−0.449404
$714$	0	0
$715$	0	0
$716$	−3.00000	−0.112115
$717$	−12.0000	−0.448148
$718$	−6.00000	−0.223918
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	0	0
$722$	18.0000	0.669891
$723$	−50.0000	−1.85952
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	−4.00000	−0.148454
$727$	29.0000	1.07555	0.537775	−	0.843088i	$-0.319265\pi$
$727$	29.0000	1.07555	0.537775	+	0.843088i	$0.319265\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	60.0000	2.21918
$732$	−8.00000	−0.295689
$733$	47.0000	1.73598	0.867992	−	0.496578i	$-0.165410\pi$
$733$	47.0000	1.73598	0.867992	+	0.496578i	$0.165410\pi$
$734$	1.00000	0.0369107
$735$	0	0
$736$	3.00000	0.110581
$737$	12.0000	0.442026
$738$	3.00000	0.110432
$739$	−37.0000	−1.36107	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−37.0000	−1.36107	−0.680534	+	0.732717i	$0.738252\pi$
$740$	0	0
$741$	−10.0000	−0.367359
$742$	0	0
$743$	−9.00000	−0.330178	−0.165089	−	0.986279i	$-0.552791\pi$
$743$	−9.00000	−0.330178	−0.165089	+	0.986279i	$0.552791\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	−34.0000	−1.24483
$747$	−12.0000	−0.439057
$748$	18.0000	0.658145
$749$	0	0
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	3.00000	0.109399
$753$	−30.0000	−1.09326
$754$	30.0000	1.09254
$755$	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	$0.656646\pi$
$758$	25.0000	0.908041
$759$	18.0000	0.653359
$760$	0	0
$761$	51.0000	1.84875	0.924374	−	0.381487i	$-0.124588\pi$
$761$	51.0000	1.84875	0.924374	+	0.381487i	$0.124588\pi$
$762$	38.0000	1.37659
$763$	0	0
$764$	12.0000	0.434145
$765$	0	0
$766$	−15.0000	−0.541972
$767$	0	0
$768$	−2.00000	−0.0721688
$769$	49.0000	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	4.00000	0.143963
$773$	39.0000	1.40273	0.701366	−	0.712801i	$-0.252574\pi$
$773$	39.0000	1.40273	0.701366	+	0.712801i	$0.252574\pi$
$774$	−10.0000	−0.359443
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	24.0000	0.860442
$779$	−3.00000	−0.107486
$780$	0	0
$781$	36.0000	1.28818
$782$	18.0000	0.643679
$783$	−24.0000	−0.857690
$784$	0	0
$785$	0	0
$786$	−6.00000	−0.214013
$787$	−34.0000	−1.21197	−0.605985	−	0.795476i	$-0.707221\pi$
$787$	−34.0000	−1.21197	−0.605985	+	0.795476i	$0.707221\pi$
$788$	3.00000	0.106871
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−3.00000	−0.106600
$793$	20.0000	0.710221
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	4.00000	0.141776
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	0	0
$801$	−6.00000	−0.212000
$802$	−21.0000	−0.741536
$803$	−12.0000	−0.423471
$804$	−8.00000	−0.282138
$805$	0	0
$806$	−20.0000	−0.704470
$807$	−24.0000	−0.844840
$808$	−12.0000	−0.422159
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	−47.0000	−1.65039	−0.825197	−	0.564846i	$-0.808936\pi$
$811$	−47.0000	−1.65039	−0.825197	+	0.564846i	$0.808936\pi$
$812$	0	0
$813$	−32.0000	−1.12229
$814$	33.0000	1.15665
$815$	0	0
$816$	−12.0000	−0.420084
$817$	10.0000	0.349856
$818$	−22.0000	−0.769212
$819$	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	$0.601703\pi$
$822$	−24.0000	−0.837096
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	0	0
$827$	54.0000	1.87776	0.938882	−	0.344239i	$-0.111863\pi$
$827$	54.0000	1.87776	0.938882	+	0.344239i	$0.111863\pi$
$828$	−3.00000	−0.104257
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	4.00000	0.138758
$832$	5.00000	0.173344
$833$	0	0
$834$	8.00000	0.277017
$835$	0	0
$836$	3.00000	0.103757
$837$	16.0000	0.553041
$838$	15.0000	0.518166
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	34.0000	1.17172
$843$	6.00000	0.206651
$844$	−1.00000	−0.0344214
$845$	0	0
$846$	−3.00000	−0.103142
$847$	0	0
$848$	−3.00000	−0.103020
$849$	−52.0000	−1.78464
$850$	0	0
$851$	33.0000	1.13123
$852$	−24.0000	−0.822226
$853$	−1.00000	−0.0342393	−0.0171197	−	0.999853i	$-0.505450\pi$
$853$	−1.00000	−0.0342393	−0.0171197	+	0.999853i	$0.505450\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	30.0000	1.02418
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−39.0000	−1.32758	−0.663788	−	0.747921i	$-0.731052\pi$
$863$	−39.0000	−1.32758	−0.663788	+	0.747921i	$0.731052\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	16.0000	0.543702
$867$	−38.0000	−1.29055
$868$	0	0
$869$	−30.0000	−1.01768
$870$	0	0
$871$	20.0000	0.677674
$872$	4.00000	0.135457
$873$	14.0000	0.473828
$874$	3.00000	0.101477
$875$	0	0
$876$	8.00000	0.270295
$877$	7.00000	0.236373	0.118187	−	0.992991i	$-0.462292\pi$
$877$	7.00000	0.236373	0.118187	+	0.992991i	$0.462292\pi$
$878$	−10.0000	−0.337484
$879$	54.0000	1.82137
$880$	0	0
$881$	−33.0000	−1.11180	−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000	−1.11180	−0.555899	+	0.831250i	$0.687626\pi$
$882$	0	0
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	30.0000	1.00901
$885$	0	0
$886$	−24.0000	−0.806296
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	−22.0000	−0.738272
$889$	0	0
$890$	0	0
$891$	−33.0000	−1.10554
$892$	8.00000	0.267860
$893$	3.00000	0.100391
$894$	36.0000	1.20402
$895$	0	0
$896$	0	0
$897$	30.0000	1.00167
$898$	3.00000	0.100111
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−18.0000	−0.599667
$902$	9.00000	0.299667
$903$	0	0
$904$	12.0000	0.399114
$905$	0	0
$906$	28.0000	0.930238
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	−24.0000	−0.796468
$909$	12.0000	0.398015
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	−2.00000	−0.0662266
$913$	−36.0000	−1.19143
$914$	−22.0000	−0.727695
$915$	0	0
$916$	28.0000	0.925146
$917$	0	0
$918$	−24.0000	−0.792118
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	6.00000	0.197599
$923$	60.0000	1.97492
$924$	0	0
$925$	0	0
$926$	−19.0000	−0.624379
$927$	−4.00000	−0.131377
$928$	6.00000	0.196960
$929$	−33.0000	−1.08269	−0.541347	−	0.840799i	$-0.682086\pi$
$929$	−33.0000	−1.08269	−0.541347	+	0.840799i	$0.682086\pi$
$930$	0	0
$931$	0	0
$932$	6.00000	0.196537
$933$	24.0000	0.785725
$934$	18.0000	0.588978
$935$	0	0
$936$	−5.00000	−0.163430
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	−16.0000	−0.522140
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	10.0000	0.325818
$943$	9.00000	0.293080
$944$	0	0
$945$	0	0
$946$	−30.0000	−0.975384
$947$	−30.0000	−0.974869	−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000	−0.974869	−0.487435	+	0.873160i	$0.662067\pi$
$948$	20.0000	0.649570
$949$	−20.0000	−0.649227
$950$	0	0
$951$	36.0000	1.16738
$952$	0	0
$953$	12.0000	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$954$	3.00000	0.0971286
$955$	0	0
$956$	6.00000	0.194054
$957$	36.0000	1.16371
$958$	24.0000	0.775405
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	55.0000	1.77327
$963$	12.0000	0.386695
$964$	25.0000	0.805196
$965$	0	0
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	2.00000	0.0642824
$969$	−12.0000	−0.385496
$970$	0	0
$971$	−27.0000	−0.866471	−0.433236	−	0.901281i	$-0.642628\pi$
$971$	−27.0000	−0.866471	−0.433236	+	0.901281i	$0.642628\pi$
$972$	10.0000	0.320750
$973$	0	0
$974$	−16.0000	−0.512673
$975$	0	0
$976$	4.00000	0.128037
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	8.00000	0.255812
$979$	−18.0000	−0.575282
$980$	0	0
$981$	−4.00000	−0.127710
$982$	12.0000	0.382935
$983$	57.0000	1.81802	0.909009	−	0.416777i	$-0.136840\pi$
$983$	57.0000	1.81802	0.909009	+	0.416777i	$0.136840\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	36.0000	1.14647
$987$	0	0
$988$	5.00000	0.159071
$989$	−30.0000	−0.953945
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	−4.00000	−0.127000
$993$	14.0000	0.444277
$994$	0	0
$995$	0	0
$996$	24.0000	0.760469
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	28.0000	0.886325
$999$	−44.0000	−1.39210

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.a.f.1.1		1
5.2	odd	4		2450.2.c.p.99.1		2
5.3	odd	4		2450.2.c.p.99.2		2
5.4	even	2		490.2.a.j.1.1		1
7.3	odd	6		350.2.e.h.51.1		2
7.5	odd	6		350.2.e.h.151.1		2
7.6	odd	2		2450.2.a.p.1.1		1
15.14	odd	2		4410.2.a.c.1.1		1
20.19	odd	2		3920.2.a.g.1.1		1
35.3	even	12		350.2.j.a.149.1		4
35.4	even	6		490.2.e.a.471.1		2
35.9	even	6		490.2.e.a.361.1		2
35.12	even	12		350.2.j.a.249.1		4
35.13	even	4		2450.2.c.f.99.2		2
35.17	even	12		350.2.j.a.149.2		4
35.19	odd	6		70.2.e.b.11.1	✓	2
35.24	odd	6		70.2.e.b.51.1	yes	2
35.27	even	4		2450.2.c.f.99.1		2
35.33	even	12		350.2.j.a.249.2		4
35.34	odd	2		490.2.a.g.1.1		1
105.59	even	6		630.2.k.e.541.1		2
105.89	even	6		630.2.k.e.361.1		2
105.104	even	2		4410.2.a.m.1.1		1
140.19	even	6		560.2.q.d.81.1		2
140.59	even	6		560.2.q.d.401.1		2
140.139	even	2		3920.2.a.be.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.e.b.11.1	✓	2	35.19	odd	6
70.2.e.b.51.1	yes	2	35.24	odd	6
350.2.e.h.51.1		2	7.3	odd	6
350.2.e.h.151.1		2	7.5	odd	6
350.2.j.a.149.1		4	35.3	even	12
350.2.j.a.149.2		4	35.17	even	12
350.2.j.a.249.1		4	35.12	even	12
350.2.j.a.249.2		4	35.33	even	12
490.2.a.g.1.1		1	35.34	odd	2
490.2.a.j.1.1		1	5.4	even	2
490.2.e.a.361.1		2	35.9	even	6
490.2.e.a.471.1		2	35.4	even	6
560.2.q.d.81.1		2	140.19	even	6
560.2.q.d.401.1		2	140.59	even	6
630.2.k.e.361.1		2	105.89	even	6
630.2.k.e.541.1		2	105.59	even	6
2450.2.a.f.1.1		1	1.1	even	1	trivial
2450.2.a.p.1.1		1	7.6	odd	2
2450.2.c.f.99.1		2	35.27	even	4
2450.2.c.f.99.2		2	35.13	even	4
2450.2.c.p.99.1		2	5.2	odd	4
2450.2.c.p.99.2		2	5.3	odd	4
3920.2.a.g.1.1		1	20.19	odd	2
3920.2.a.be.1.1		1	140.139	even	2
4410.2.a.c.1.1		1	15.14	odd	2
4410.2.a.m.1.1		1	105.104	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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