LMFDB - Embedded newform 3450.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(1,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.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:	$27.5483886973$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 690)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -4.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} -6.00000 q^{39} +10.0000 q^{41} -4.00000 q^{43} -4.00000 q^{44} -1.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} -6.00000 q^{51} +6.00000 q^{52} +14.0000 q^{53} -1.00000 q^{54} -4.00000 q^{57} -6.00000 q^{58} +10.0000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} +1.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -6.00000 q^{74} +4.00000 q^{76} -6.00000 q^{78} -12.0000 q^{79} +1.00000 q^{81} +10.0000 q^{82} +16.0000 q^{83} -4.00000 q^{86} +6.00000 q^{87} -4.00000 q^{88} -2.00000 q^{89} -1.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} -1.00000 q^{96} +14.0000 q^{97} -7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	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
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	−4.00000	−0.852803
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	0	0
$26$	6.00000	1.17670
$27$	−1.00000	−0.192450
$28$	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
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	4.00000	0.696311
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	4.00000	0.648886
$39$	−6.00000	−0.960769
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−1.00000	−0.147442
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−6.00000	−0.840168
$52$	6.00000	0.832050
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	−6.00000	−0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	6.00000	0.727607
$69$	1.00000	0.120386
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	1.00000	0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	−6.00000	−0.679366
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	6.00000	0.643268
$88$	−4.00000	−0.426401
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	0	0
$92$	−1.00000	−0.104257
$93$	8.00000	0.829561
$94$	8.00000	0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−7.00000	−0.707107
$99$	−4.00000	−0.402015
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	−6.00000	−0.594089
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	14.0000	1.35980
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	−1.00000	−0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−6.00000	−0.557086
$117$	6.00000	0.554700
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	10.0000	0.905357
$123$	−10.0000	−0.901670
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	1.00000	0.0851257
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	8.00000	0.671345
$143$	−24.0000	−2.00698
$144$	1.00000	0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	7.00000	0.577350
$148$	−6.00000	−0.493197
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	4.00000	0.324443
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	−6.00000	−0.480384
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−12.0000	−0.954669
$159$	−14.0000	−1.11027
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	16.0000	1.24184
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	4.00000	0.305888
$172$	−4.00000	−0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	−2.00000	−0.149906
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	8.00000	0.586588
$187$	−24.0000	−1.75505
$188$	8.00000	0.583460
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−1.00000	−0.0721688
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	−26.0000	−1.85242	−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000	−1.85242	−0.926212	+	0.377004i	$0.876954\pi$
$198$	−4.00000	−0.284268
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	18.0000	1.26648
$203$	0	0
$204$	−6.00000	−0.420084
$205$	0	0
$206$	16.0000	1.11477
$207$	−1.00000	−0.0695048
$208$	6.00000	0.416025
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	14.0000	0.961524
$213$	−8.00000	−0.548151
$214$	−8.00000	−0.546869
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	2.00000	0.135457
$219$	2.00000	0.135147
$220$	0	0
$221$	36.0000	2.42162
$222$	6.00000	0.402694
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	0	0
$225$	0	0
$226$	14.0000	0.931266
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	−4.00000	−0.264906
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	0	0
$237$	12.0000	0.779484
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	10.0000	0.640184
$245$	0	0
$246$	−10.0000	−0.637577
$247$	24.0000	1.52708
$248$	−8.00000	−0.508001
$249$	−16.0000	−1.01396
$250$	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
$253$	4.00000	0.251478
$254$	−12.0000	−0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	8.00000	0.494242
$263$	32.0000	1.97320	0.986602	−	0.163144i	$-0.0521635\pi$
$263$	32.0000	1.97320	0.986602	+	0.163144i	$0.0521635\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	0	0
$267$	2.00000	0.122398
$268$	−4.00000	−0.244339
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−2.00000	−0.120824
$275$	0	0
$276$	1.00000	0.0601929
$277$	30.0000	1.80253	0.901263	−	0.433273i	$-0.142641\pi$
$277$	30.0000	1.80253	0.901263	+	0.433273i	$0.142641\pi$
$278$	12.0000	0.719712
$279$	−8.00000	−0.478947
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	−8.00000	−0.476393
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−24.0000	−1.41915
$287$	0	0
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	−14.0000	−0.820695
$292$	−2.00000	−0.117041
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	−6.00000	−0.348743
$297$	4.00000	0.232104
$298$	10.0000	0.579284
$299$	−6.00000	−0.346989
$300$	0	0
$301$	0	0
$302$	16.0000	0.920697
$303$	−18.0000	−1.03407
$304$	4.00000	0.229416
$305$	0	0
$306$	6.00000	0.342997
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	−6.00000	−0.339683
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−14.0000	−0.785081
$319$	24.0000	1.34374
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	24.0000	1.33540
$324$	1.00000	0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	−2.00000	−0.110600
$328$	10.0000	0.552158
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	16.0000	0.878114
$333$	−6.00000	−0.328798
$334$	16.0000	0.875481
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	23.0000	1.25104
$339$	−14.0000	−0.760376
$340$	0	0
$341$	32.0000	1.73290
$342$	4.00000	0.216295
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−18.0000	−0.967686
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	6.00000	0.321634
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	−4.00000	−0.213201
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	0	0
$356$	−2.00000	−0.106000
$357$	0	0
$358$	0	0
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−6.00000	−0.315353
$363$	−5.00000	−0.262432
$364$	0	0
$365$	0	0
$366$	−10.0000	−0.522708
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	−1.00000	−0.0521286
$369$	10.0000	0.520579
$370$	0	0
$371$	0	0
$372$	8.00000	0.414781
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	−24.0000	−1.24101
$375$	0	0
$376$	8.00000	0.412568
$377$	−36.0000	−1.85409
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	−8.00000	−0.409316
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−10.0000	−0.508987
$387$	−4.00000	−0.203331
$388$	14.0000	0.710742
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	−7.00000	−0.353553
$393$	−8.00000	−0.403547
$394$	−26.0000	−1.30986
$395$	0	0
$396$	−4.00000	−0.201008
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−12.0000	−0.601506
$399$	0	0
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	4.00000	0.199502
$403$	−48.0000	−2.39105
$404$	18.0000	0.895533
$405$	0	0
$406$	0	0
$407$	24.0000	1.18964
$408$	−6.00000	−0.297044
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	16.0000	0.788263
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	6.00000	0.294174
$417$	−12.0000	−0.587643
$418$	−16.0000	−0.782586
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−38.0000	−1.85201	−0.926003	−	0.377515i	$-0.876779\pi$
$421$	−38.0000	−1.85201	−0.926003	+	0.377515i	$0.876779\pi$
$422$	−20.0000	−0.973585
$423$	8.00000	0.388973
$424$	14.0000	0.679900
$425$	0	0
$426$	−8.00000	−0.387601
$427$	0	0
$428$	−8.00000	−0.386695
$429$	24.0000	1.15873
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	−1.00000	−0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	−4.00000	−0.191346
$438$	2.00000	0.0955637
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	36.0000	1.71235
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	20.0000	0.947027
$447$	−10.0000	−0.472984
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	14.0000	0.658505
$453$	−16.0000	−0.751746
$454$	−8.00000	−0.375459
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−14.0000	−0.654177
$459$	−6.00000	−0.280056
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−22.0000	−1.01913
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	6.00000	0.277350
$469$	0	0
$470$	0	0
$471$	6.00000	0.276465
$472$	0	0
$473$	16.0000	0.735681
$474$	12.0000	0.551178
$475$	0	0
$476$	0	0
$477$	14.0000	0.641016
$478$	−8.00000	−0.365911
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−36.0000	−1.64146
$482$	2.00000	0.0910975
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	10.0000	0.452679
$489$	4.00000	0.180886
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	−10.0000	−0.450835
$493$	−36.0000	−1.62136
$494$	24.0000	1.07981
$495$	0	0
$496$	−8.00000	−0.359211
$497$	0	0
$498$	−16.0000	−0.716977
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	−12.0000	−0.535586
$503$	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
$506$	4.00000	0.177822
$507$	−23.0000	−1.02147
$508$	−12.0000	−0.532414
$509$	−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
$514$	−6.00000	−0.264649
$515$	0	0
$516$	4.00000	0.176090
$517$	−32.0000	−1.40736
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	−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	$0.355946\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	32.0000	1.39527
$527$	−48.0000	−2.09091
$528$	4.00000	0.174078
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	60.0000	2.59889
$534$	2.00000	0.0865485
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	10.0000	0.431131
$539$	28.0000	1.20605
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	−8.00000	−0.343629
$543$	6.00000	0.257485
$544$	6.00000	0.257248
$545$	0	0
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−2.00000	−0.0854358
$549$	10.0000	0.426790
$550$	0	0
$551$	−24.0000	−1.02243
$552$	1.00000	0.0425628
$553$	0	0
$554$	30.0000	1.27458
$555$	0	0
$556$	12.0000	0.508913
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	−8.00000	−0.338667
$559$	−24.0000	−1.01509
$560$	0	0
$561$	24.0000	1.01328
$562$	30.0000	1.26547
$563$	8.00000	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$563$	8.00000	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	8.00000	0.335673
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	−24.0000	−1.00349
$573$	8.00000	0.334205
$574$	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	$0.750276\pi$
$578$	19.0000	0.790296
$579$	10.0000	0.415586
$580$	0	0
$581$	0	0
$582$	−14.0000	−0.580319
$583$	−56.0000	−2.31928
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	14.0000	0.578335
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	7.00000	0.288675
$589$	−32.0000	−1.31854
$590$	0	0
$591$	26.0000	1.06950
$592$	−6.00000	−0.246598
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	10.0000	0.409616
$597$	12.0000	0.491127
$598$	−6.00000	−0.245358
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	16.0000	0.651031
$605$	0	0
$606$	−18.0000	−0.731200
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	0	0
$611$	48.0000	1.94187
$612$	6.00000	0.242536
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−16.0000	−0.643614
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	−8.00000	−0.320771
$623$	0	0
$624$	−6.00000	−0.240192
$625$	0	0
$626$	−10.0000	−0.399680
$627$	16.0000	0.638978
$628$	−6.00000	−0.239426
$629$	−36.0000	−1.43541
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	−12.0000	−0.477334
$633$	20.0000	0.794929
$634$	−18.0000	−0.714871
$635$	0	0
$636$	−14.0000	−0.555136
$637$	−42.0000	−1.66410
$638$	24.0000	0.950169
$639$	8.00000	0.316475
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	8.00000	0.315735
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	0	0
$645$	0	0
$646$	24.0000	0.944267
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−4.00000	−0.156652
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	10.0000	0.390434
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	34.0000	1.32245	0.661223	−	0.750189i	$-0.270038\pi$
$661$	34.0000	1.32245	0.661223	+	0.750189i	$0.270038\pi$
$662$	−20.0000	−0.777322
$663$	−36.0000	−1.39812
$664$	16.0000	0.620920
$665$	0	0
$666$	−6.00000	−0.232495
$667$	6.00000	0.232321
$668$	16.0000	0.619059
$669$	−20.0000	−0.773245
$670$	0	0
$671$	−40.0000	−1.54418
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	23.0000	0.884615
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	−14.0000	−0.537667
$679$	0	0
$680$	0	0
$681$	8.00000	0.306561
$682$	32.0000	1.22534
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	0	0
$687$	14.0000	0.534133
$688$	−4.00000	−0.152499
$689$	84.0000	3.20015
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	28.0000	1.06287
$695$	0	0
$696$	6.00000	0.227429
$697$	60.0000	2.27266
$698$	−34.0000	−1.28692
$699$	22.0000	0.832116
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	−6.00000	−0.226455
$703$	−24.0000	−0.905177
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−6.00000	−0.225813
$707$	0	0
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	−2.00000	−0.0749532
$713$	8.00000	0.299602
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.00000	0.298765
$718$	−8.00000	−0.298557
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	0	0
$721$	0	0
$722$	−3.00000	−0.111648
$723$	−2.00000	−0.0743808
$724$	−6.00000	−0.222988
$725$	0	0
$726$	−5.00000	−0.185567
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	−10.0000	−0.369611
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	32.0000	1.18114
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	16.0000	0.589368
$738$	10.0000	0.368105
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	−24.0000	−0.881662
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	18.0000	0.659027
$747$	16.0000	0.585409
$748$	−24.0000	−0.877527
$749$	0	0
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	8.00000	0.291730
$753$	12.0000	0.437304
$754$	−36.0000	−1.31104
$755$	0	0
$756$	0	0
$757$	−54.0000	−1.96266	−0.981332	−	0.192323i	$-0.938398\pi$
$757$	−54.0000	−1.96266	−0.981332	+	0.192323i	$0.938398\pi$
$758$	4.00000	0.145287
$759$	−4.00000	−0.145191
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	12.0000	0.434714
$763$	0	0
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−16.0000	−0.578103
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−10.0000	−0.359908
$773$	−2.00000	−0.0719350	−0.0359675	−	0.999353i	$-0.511451\pi$
$773$	−2.00000	−0.0719350	−0.0359675	+	0.999353i	$0.511451\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	−6.00000	−0.215110
$779$	40.0000	1.43315
$780$	0	0
$781$	−32.0000	−1.14505
$782$	−6.00000	−0.214560
$783$	6.00000	0.214423
$784$	−7.00000	−0.250000
$785$	0	0
$786$	−8.00000	−0.285351
$787$	−36.0000	−1.28326	−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000	−1.28326	−0.641631	+	0.767014i	$0.721742\pi$
$788$	−26.0000	−0.926212
$789$	−32.0000	−1.13923
$790$	0	0
$791$	0	0
$792$	−4.00000	−0.142134
$793$	60.0000	2.13066
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−12.0000	−0.425329
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	6.00000	0.211867
$803$	8.00000	0.282314
$804$	4.00000	0.141069
$805$	0	0
$806$	−48.0000	−1.69073
$807$	−10.0000	−0.352017
$808$	18.0000	0.633238
$809$	−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
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	24.0000	0.841200
$815$	0	0
$816$	−6.00000	−0.210042
$817$	−16.0000	−0.559769
$818$	10.0000	0.349642
$819$	0	0
$820$	0	0
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	2.00000	0.0697580
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	0	0
$827$	8.00000	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$828$	−1.00000	−0.0347524
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	0	0
$831$	−30.0000	−1.04069
$832$	6.00000	0.208013
$833$	−42.0000	−1.45521
$834$	−12.0000	−0.415526
$835$	0	0
$836$	−16.0000	−0.553372
$837$	8.00000	0.276520
$838$	−12.0000	−0.414533
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−38.0000	−1.30957
$843$	−30.0000	−1.03325
$844$	−20.0000	−0.688428
$845$	0	0
$846$	8.00000	0.275046
$847$	0	0
$848$	14.0000	0.480762
$849$	4.00000	0.137280
$850$	0	0
$851$	6.00000	0.205677
$852$	−8.00000	−0.274075
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	0	0
$856$	−8.00000	−0.273434
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	24.0000	0.819346
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	0	0
$862$	8.00000	0.272481
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	−19.0000	−0.645274
$868$	0	0
$869$	48.0000	1.62829
$870$	0	0
$871$	−24.0000	−0.813209
$872$	2.00000	0.0677285
$873$	14.0000	0.473828
$874$	−4.00000	−0.135302
$875$	0	0
$876$	2.00000	0.0675737
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−24.0000	−0.809961
$879$	−14.0000	−0.472208
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	−7.00000	−0.235702
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	36.0000	1.21081
$885$	0	0
$886$	−12.0000	−0.403148
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	6.00000	0.201347
$889$	0	0
$890$	0	0
$891$	−4.00000	−0.134005
$892$	20.0000	0.669650
$893$	32.0000	1.07084
$894$	−10.0000	−0.334450
$895$	0	0
$896$	0	0
$897$	6.00000	0.200334
$898$	10.0000	0.333704
$899$	48.0000	1.60089
$900$	0	0
$901$	84.0000	2.79845
$902$	−40.0000	−1.33185
$903$	0	0
$904$	14.0000	0.465633
$905$	0	0
$906$	−16.0000	−0.531564
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	−8.00000	−0.265489
$909$	18.0000	0.597022
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	−4.00000	−0.132453
$913$	−64.0000	−2.11809
$914$	−10.0000	−0.330771
$915$	0	0
$916$	−14.0000	−0.462573
$917$	0	0
$918$	−6.00000	−0.198030
$919$	12.0000	0.395843	0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000	0.395843	0.197922	+	0.980218i	$0.436581\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	2.00000	0.0658665
$923$	48.0000	1.57994
$924$	0	0
$925$	0	0
$926$	20.0000	0.657241
$927$	16.0000	0.525509
$928$	−6.00000	−0.196960
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	−22.0000	−0.720634
$933$	8.00000	0.261908
$934$	−8.00000	−0.261768
$935$	0	0
$936$	6.00000	0.196116
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	6.00000	0.195491
$943$	−10.0000	−0.325645
$944$	0	0
$945$	0	0
$946$	16.0000	0.520205
$947$	20.0000	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$947$	20.0000	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$948$	12.0000	0.389742
$949$	−12.0000	−0.389536
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	14.0000	0.453267
$955$	0	0
$956$	−8.00000	−0.258738
$957$	−24.0000	−0.775810
$958$	24.0000	0.775405
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	−36.0000	−1.16069
$963$	−8.00000	−0.257796
$964$	2.00000	0.0644157
$965$	0	0
$966$	0	0
$967$	−28.0000	−0.900419	−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000	−0.900419	−0.450210	+	0.892923i	$0.648651\pi$
$968$	5.00000	0.160706
$969$	−24.0000	−0.770991
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−20.0000	−0.640841
$975$	0	0
$976$	10.0000	0.320092
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	4.00000	0.127906
$979$	8.00000	0.255681
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	−10.0000	−0.318788
$985$	0	0
$986$	−36.0000	−1.14647
$987$	0	0
$988$	24.0000	0.763542
$989$	4.00000	0.127193
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	−8.00000	−0.254000
$993$	20.0000	0.634681
$994$	0	0
$995$	0	0
$996$	−16.0000	−0.506979
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	12.0000	0.379853
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.p.1.1		1
5.2	odd	4		3450.2.d.a.2899.2		2
5.3	odd	4		3450.2.d.a.2899.1		2
5.4	even	2		690.2.a.e.1.1	✓	1
15.14	odd	2		2070.2.a.r.1.1		1
20.19	odd	2		5520.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.e.1.1	✓	1	5.4	even	2
2070.2.a.r.1.1		1	15.14	odd	2
3450.2.a.p.1.1		1	1.1	even	1	trivial
3450.2.d.a.2899.1		2	5.3	odd	4
3450.2.d.a.2899.2		2	5.2	odd	4
5520.2.a.f.1.1		1	20.19	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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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