LMFDB - Embedded newform 8450.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8450,2,Mod(1,8450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8450, 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("8450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.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:	$67.4735897080$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 50)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{11} +1.00000 q^{12} -2.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} -5.00000 q^{19} -2.00000 q^{21} +3.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} -5.00000 q^{27} -2.00000 q^{28} -2.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} -3.00000 q^{34} -2.00000 q^{36} -2.00000 q^{37} -5.00000 q^{38} +3.00000 q^{41} -2.00000 q^{42} -4.00000 q^{43} +3.00000 q^{44} +6.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -3.00000 q^{51} +6.00000 q^{53} -5.00000 q^{54} -2.00000 q^{56} -5.00000 q^{57} +2.00000 q^{61} -2.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} +13.0000 q^{67} -3.00000 q^{68} +6.00000 q^{69} -12.0000 q^{71} -2.00000 q^{72} -11.0000 q^{73} -2.00000 q^{74} -5.00000 q^{76} -6.00000 q^{77} -10.0000 q^{79} +1.00000 q^{81} +3.00000 q^{82} +9.00000 q^{83} -2.00000 q^{84} -4.00000 q^{86} +3.00000 q^{88} -15.0000 q^{89} +6.00000 q^{92} -2.00000 q^{93} -12.0000 q^{94} +1.00000 q^{96} -2.00000 q^{97} -3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−2.00000	−0.471405
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	3.00000	0.639602
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	0	0
$27$	−5.00000	−0.962250
$28$	−2.00000	−0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	3.00000	0.522233
$34$	−3.00000	−0.514496
$35$	0	0
$36$	−2.00000	−0.333333
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−5.00000	−0.811107
$39$	0	0
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	−2.00000	−0.308607
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	6.00000	0.884652
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	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
$56$	−2.00000	−0.267261
$57$	−5.00000	−0.662266
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−2.00000	−0.254000
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	13.0000	1.58820	0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000	1.58820	0.794101	+	0.607785i	$0.207942\pi$
$68$	−3.00000	−0.363803
$69$	6.00000	0.722315
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−2.00000	−0.235702
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−5.00000	−0.573539
$77$	−6.00000	−0.683763
$78$	0	0
$79$	−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
$81$	1.00000	0.111111
$82$	3.00000	0.331295
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	3.00000	0.319801
$89$	−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
$92$	6.00000	0.625543
$93$	−2.00000	−0.207390
$94$	−12.0000	−1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−3.00000	−0.303046
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	−3.00000	−0.297044
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	6.00000	0.582772
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	−5.00000	−0.481125
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	−2.00000	−0.188982
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	−5.00000	−0.468293
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	2.00000	0.181071
$123$	3.00000	0.270501
$124$	−2.00000	−0.179605
$125$	0	0
$126$	4.00000	0.356348
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	3.00000	0.261116
$133$	10.0000	0.867110
$134$	13.0000	1.12303
$135$	0	0
$136$	−3.00000	−0.257248
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	6.00000	0.510754
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	−12.0000	−1.00702
$143$	0	0
$144$	−2.00000	−0.166667
$145$	0	0
$146$	−11.0000	−0.910366
$147$	−3.00000	−0.247436
$148$	−2.00000	−0.164399
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	−5.00000	−0.405554
$153$	6.00000	0.485071
$154$	−6.00000	−0.483494
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−10.0000	−0.795557
$159$	6.00000	0.475831
$160$	0	0
$161$	−12.0000	−0.945732
$162$	1.00000	0.0785674
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	9.00000	0.698535
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−2.00000	−0.154303
$169$	0	0
$170$	0	0
$171$	10.0000	0.764719
$172$	−4.00000	−0.304997
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	−15.0000	−1.12430
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	6.00000	0.442326
$185$	0	0
$186$	−2.00000	−0.146647
$187$	−9.00000	−0.658145
$188$	−12.0000	−0.875190
$189$	10.0000	0.727393
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	1.00000	0.0721688
$193$	19.0000	1.36765	0.683825	−	0.729646i	$-0.260315\pi$
$193$	19.0000	1.36765	0.683825	+	0.729646i	$0.260315\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−6.00000	−0.426401
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	13.0000	0.916949
$202$	−18.0000	−1.26648
$203$	0	0
$204$	−3.00000	−0.210042
$205$	0	0
$206$	−4.00000	−0.278693
$207$	−12.0000	−0.834058
$208$	0	0
$209$	−15.0000	−1.03757
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	6.00000	0.412082
$213$	−12.0000	−0.822226
$214$	−3.00000	−0.205076
$215$	0	0
$216$	−5.00000	−0.340207
$217$	4.00000	0.271538
$218$	10.0000	0.677285
$219$	−11.0000	−0.743311
$220$	0	0
$221$	0	0
$222$	−2.00000	−0.134231
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−9.00000	−0.598671
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−5.00000	−0.331133
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10.0000	−0.649570
$238$	6.00000	0.388922
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−17.0000	−1.09507	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000	−1.09507	−0.547533	+	0.836784i	$0.684433\pi$
$242$	−2.00000	−0.128565
$243$	16.0000	1.02640
$244$	2.00000	0.128037
$245$	0	0
$246$	3.00000	0.191273
$247$	0	0
$248$	−2.00000	−0.127000
$249$	9.00000	0.570352
$250$	0	0
$251$	27.0000	1.70422	0.852112	−	0.523359i	$-0.175321\pi$
$251$	27.0000	1.70422	0.852112	+	0.523359i	$0.175321\pi$
$252$	4.00000	0.251976
$253$	18.0000	1.13165
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	−4.00000	−0.249029
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	12.0000	0.741362
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	10.0000	0.613139
$267$	−15.0000	−0.917985
$268$	13.0000	0.794101
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	3.00000	0.181237
$275$	0	0
$276$	6.00000	0.361158
$277$	32.0000	1.92269	0.961347	−	0.275340i	$-0.0887905\pi$
$277$	32.0000	1.92269	0.961347	+	0.275340i	$0.0887905\pi$
$278$	5.00000	0.299880
$279$	4.00000	0.239474
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	−12.0000	−0.714590
$283$	11.0000	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$283$	11.0000	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	−6.00000	−0.354169
$288$	−2.00000	−0.117851
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−2.00000	−0.117242
$292$	−11.0000	−0.643726
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−2.00000	−0.116248
$297$	−15.0000	−0.870388
$298$	0	0
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	−2.00000	−0.115087
$303$	−18.0000	−1.03407
$304$	−5.00000	−0.286770
$305$	0	0
$306$	6.00000	0.342997
$307$	−17.0000	−0.970241	−0.485121	−	0.874447i	$-0.661224\pi$
$307$	−17.0000	−0.970241	−0.485121	+	0.874447i	$0.661224\pi$
$308$	−6.00000	−0.341882
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	−10.0000	−0.562544
$317$	−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
$319$	0	0
$320$	0	0
$321$	−3.00000	−0.167444
$322$	−12.0000	−0.668734
$323$	15.0000	0.834622
$324$	1.00000	0.0555556
$325$	0	0
$326$	−11.0000	−0.609234
$327$	10.0000	0.553001
$328$	3.00000	0.165647
$329$	24.0000	1.32316
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	9.00000	0.493939
$333$	4.00000	0.219199
$334$	−12.0000	−0.656611
$335$	0	0
$336$	−2.00000	−0.109109
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	0	0
$339$	−9.00000	−0.488813
$340$	0	0
$341$	−6.00000	−0.324918
$342$	10.0000	0.540738
$343$	20.0000	1.07990
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−24.0000	−1.29025
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	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
$351$	0	0
$352$	3.00000	0.159901
$353$	−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
$356$	−15.0000	−0.794998
$357$	6.00000	0.317554
$358$	−15.0000	−0.792775
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	2.00000	0.105118
$363$	−2.00000	−0.104973
$364$	0	0
$365$	0	0
$366$	2.00000	0.104542
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	6.00000	0.312772
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−12.0000	−0.623009
$372$	−2.00000	−0.103695
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	−9.00000	−0.465379
$375$	0	0
$376$	−12.0000	−0.618853
$377$	0	0
$378$	10.0000	0.514344
$379$	25.0000	1.28416	0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000	1.28416	0.642082	+	0.766636i	$0.278071\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	−18.0000	−0.920960
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	19.0000	0.967075
$387$	8.00000	0.406663
$388$	−2.00000	−0.101535
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	−3.00000	−0.151523
$393$	12.0000	0.605320
$394$	18.0000	0.906827
$395$	0	0
$396$	−6.00000	−0.301511
$397$	28.0000	1.40528	0.702640	−	0.711546i	$-0.252005\pi$
$397$	28.0000	1.40528	0.702640	+	0.711546i	$0.252005\pi$
$398$	20.0000	1.00251
$399$	10.0000	0.500626
$400$	0	0
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	13.0000	0.648381
$403$	0	0
$404$	−18.0000	−0.895533
$405$	0	0
$406$	0	0
$407$	−6.00000	−0.297409
$408$	−3.00000	−0.148522
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	3.00000	0.147979
$412$	−4.00000	−0.197066
$413$	0	0
$414$	−12.0000	−0.589768
$415$	0	0
$416$	0	0
$417$	5.00000	0.244851
$418$	−15.0000	−0.733674
$419$	−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
$421$	28.0000	1.36464	0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000	1.36464	0.682318	+	0.731055i	$0.260972\pi$
$422$	−13.0000	−0.632830
$423$	24.0000	1.16692
$424$	6.00000	0.291386
$425$	0	0
$426$	−12.0000	−0.581402
$427$	−4.00000	−0.193574
$428$	−3.00000	−0.145010
$429$	0	0
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	−5.00000	−0.240563
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	10.0000	0.478913
$437$	−30.0000	−1.43509
$438$	−11.0000	−0.525600
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	−9.00000	−0.427603	−0.213801	−	0.976877i	$-0.568585\pi$
$443$	−9.00000	−0.427603	−0.213801	+	0.976877i	$0.568585\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	4.00000	0.189405
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	−9.00000	−0.423324
$453$	−2.00000	−0.0939682
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−5.00000	−0.234146
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	−20.0000	−0.934539
$459$	15.0000	0.700140
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	−6.00000	−0.279145
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	0	0
$465$	0	0
$466$	6.00000	0.277945
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	−26.0000	−1.20057
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	−12.0000	−0.551761
$474$	−10.0000	−0.459315
$475$	0	0
$476$	6.00000	0.275010
$477$	−12.0000	−0.549442
$478$	0	0
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	0	0
$482$	−17.0000	−0.774329
$483$	−12.0000	−0.546019
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	16.0000	0.725775
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	2.00000	0.0905357
$489$	−11.0000	−0.497437
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	3.00000	0.135250
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	24.0000	1.07655
$498$	9.00000	0.403300
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	27.0000	1.20507
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	18.0000	0.800198
$507$	0	0
$508$	2.00000	0.0887357
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	22.0000	0.973223
$512$	1.00000	0.0441942
$513$	25.0000	1.10378
$514$	−18.0000	−0.793946
$515$	0	0
$516$	−4.00000	−0.176090
$517$	−36.0000	−1.58328
$518$	4.00000	0.175750
$519$	−24.0000	−1.05348
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	0	0
$523$	41.0000	1.79280	0.896402	−	0.443241i	$-0.146171\pi$
$523$	41.0000	1.79280	0.896402	+	0.443241i	$0.146171\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	6.00000	0.261612
$527$	6.00000	0.261364
$528$	3.00000	0.130558
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	10.0000	0.433555
$533$	0	0
$534$	−15.0000	−0.649113
$535$	0	0
$536$	13.0000	0.561514
$537$	−15.0000	−0.647298
$538$	0	0
$539$	−9.00000	−0.387657
$540$	0	0
$541$	−32.0000	−1.37579	−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000	−1.37579	−0.687894	+	0.725811i	$0.741464\pi$
$542$	−2.00000	−0.0859074
$543$	2.00000	0.0858282
$544$	−3.00000	−0.128624
$545$	0	0
$546$	0	0
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	3.00000	0.128154
$549$	−4.00000	−0.170716
$550$	0	0
$551$	0	0
$552$	6.00000	0.255377
$553$	20.0000	0.850487
$554$	32.0000	1.35955
$555$	0	0
$556$	5.00000	0.212047
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	4.00000	0.169334
$559$	0	0
$560$	0	0
$561$	−9.00000	−0.379980
$562$	18.0000	0.759284
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	11.0000	0.462364
$567$	−2.00000	−0.0839921
$568$	−12.0000	−0.503509
$569$	−15.0000	−0.628833	−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000	−0.628833	−0.314416	+	0.949285i	$0.601809\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	−18.0000	−0.751961
$574$	−6.00000	−0.250435
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	13.0000	0.541197	0.270599	−	0.962692i	$-0.412778\pi$
$577$	13.0000	0.541197	0.270599	+	0.962692i	$0.412778\pi$
$578$	−8.00000	−0.332756
$579$	19.0000	0.789613
$580$	0	0
$581$	−18.0000	−0.746766
$582$	−2.00000	−0.0829027
$583$	18.0000	0.745484
$584$	−11.0000	−0.455183
$585$	0	0
$586$	−6.00000	−0.247858
$587$	33.0000	1.36206	0.681028	−	0.732257i	$-0.261533\pi$
$587$	33.0000	1.36206	0.681028	+	0.732257i	$0.261533\pi$
$588$	−3.00000	−0.123718
$589$	10.0000	0.412043
$590$	0	0
$591$	18.0000	0.740421
$592$	−2.00000	−0.0821995
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	−15.0000	−0.615457
$595$	0	0
$596$	0	0
$597$	20.0000	0.818546
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	8.00000	0.326056
$603$	−26.0000	−1.05880
$604$	−2.00000	−0.0813788
$605$	0	0
$606$	−18.0000	−0.731200
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	0	0
$611$	0	0
$612$	6.00000	0.242536
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	−17.0000	−0.686064
$615$	0	0
$616$	−6.00000	−0.241747
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	−4.00000	−0.160904
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	−30.0000	−1.20386
$622$	−18.0000	−0.721734
$623$	30.0000	1.20192
$624$	0	0
$625$	0	0
$626$	26.0000	1.03917
$627$	−15.0000	−0.599042
$628$	2.00000	0.0798087
$629$	6.00000	0.239236
$630$	0	0
$631$	−2.00000	−0.0796187	−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000	−0.0796187	−0.0398094	+	0.999207i	$0.512675\pi$
$632$	−10.0000	−0.397779
$633$	−13.0000	−0.516704
$634$	−12.0000	−0.476581
$635$	0	0
$636$	6.00000	0.237915
$637$	0	0
$638$	0	0
$639$	24.0000	0.949425
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	−3.00000	−0.118401
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	15.0000	0.590167
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	4.00000	0.156772
$652$	−11.0000	−0.430793
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	3.00000	0.117130
$657$	22.0000	0.858302
$658$	24.0000	0.935617
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	−17.0000	−0.660724
$663$	0	0
$664$	9.00000	0.349268
$665$	0	0
$666$	4.00000	0.154997
$667$	0	0
$668$	−12.0000	−0.464294
$669$	4.00000	0.154649
$670$	0	0
$671$	6.00000	0.231627
$672$	−2.00000	−0.0771517
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	0	0
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	−9.00000	−0.345643
$679$	4.00000	0.153506
$680$	0	0
$681$	−12.0000	−0.459841
$682$	−6.00000	−0.229752
$683$	39.0000	1.49229	0.746147	−	0.665782i	$-0.231902\pi$
$683$	39.0000	1.49229	0.746147	+	0.665782i	$0.231902\pi$
$684$	10.0000	0.382360
$685$	0	0
$686$	20.0000	0.763604
$687$	−20.0000	−0.763048
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	−17.0000	−0.646710	−0.323355	−	0.946278i	$-0.604811\pi$
$691$	−17.0000	−0.646710	−0.323355	+	0.946278i	$0.604811\pi$
$692$	−24.0000	−0.912343
$693$	12.0000	0.455842
$694$	−3.00000	−0.113878
$695$	0	0
$696$	0	0
$697$	−9.00000	−0.340899
$698$	10.0000	0.378506
$699$	6.00000	0.226941
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	3.00000	0.113067
$705$	0	0
$706$	−6.00000	−0.225813
$707$	36.0000	1.35392
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	−15.0000	−0.562149
$713$	−12.0000	−0.449404
$714$	6.00000	0.224544
$715$	0	0
$716$	−15.0000	−0.560576
$717$	0	0
$718$	−30.0000	−1.11959
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	6.00000	0.223297
$723$	−17.0000	−0.632237
$724$	2.00000	0.0743294
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	12.0000	0.443836
$732$	2.00000	0.0739221
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	6.00000	0.221163
$737$	39.0000	1.43658
$738$	−6.00000	−0.220863
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	−12.0000	−0.440534
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	−2.00000	−0.0733236
$745$	0	0
$746$	26.0000	0.951928
$747$	−18.0000	−0.658586
$748$	−9.00000	−0.329073
$749$	6.00000	0.219235
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	−12.0000	−0.437595
$753$	27.0000	0.983935
$754$	0	0
$755$	0	0
$756$	10.0000	0.363696
$757$	32.0000	1.16306	0.581530	−	0.813525i	$-0.302454\pi$
$757$	32.0000	1.16306	0.581530	+	0.813525i	$0.302454\pi$
$758$	25.0000	0.908041
$759$	18.0000	0.653359
$760$	0	0
$761$	3.00000	0.108750	0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000	0.108750	0.0543750	+	0.998521i	$0.482683\pi$
$762$	2.00000	0.0724524
$763$	−20.0000	−0.724049
$764$	−18.0000	−0.651217
$765$	0	0
$766$	−6.00000	−0.216789
$767$	0	0
$768$	1.00000	0.0360844
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	19.0000	0.683825
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	4.00000	0.143499
$778$	30.0000	1.07555
$779$	−15.0000	−0.537431
$780$	0	0
$781$	−36.0000	−1.28818
$782$	−18.0000	−0.643679
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	12.0000	0.428026
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	18.0000	0.641223
$789$	6.00000	0.213606
$790$	0	0
$791$	18.0000	0.640006
$792$	−6.00000	−0.213201
$793$	0	0
$794$	28.0000	0.993683
$795$	0	0
$796$	20.0000	0.708881
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	10.0000	0.353996
$799$	36.0000	1.27359
$800$	0	0
$801$	30.0000	1.06000
$802$	3.00000	0.105934
$803$	−33.0000	−1.16454
$804$	13.0000	0.458475
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−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
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	−2.00000	−0.0701431
$814$	−6.00000	−0.210300
$815$	0	0
$816$	−3.00000	−0.105021
$817$	20.0000	0.699711
$818$	−5.00000	−0.174821
$819$	0	0
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	3.00000	0.104637
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	0	0
$827$	33.0000	1.14752	0.573761	−	0.819023i	$-0.305484\pi$
$827$	33.0000	1.14752	0.573761	+	0.819023i	$0.305484\pi$
$828$	−12.0000	−0.417029
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	32.0000	1.11007
$832$	0	0
$833$	9.00000	0.311832
$834$	5.00000	0.173136
$835$	0	0
$836$	−15.0000	−0.518786
$837$	10.0000	0.345651
$838$	−15.0000	−0.518166
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	28.0000	0.964944
$843$	18.0000	0.619953
$844$	−13.0000	−0.447478
$845$	0	0
$846$	24.0000	0.825137
$847$	4.00000	0.137442
$848$	6.00000	0.206041
$849$	11.0000	0.377519
$850$	0	0
$851$	−12.0000	−0.411355
$852$	−12.0000	−0.411113
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	−3.00000	−0.102538
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	5.00000	0.170598	0.0852989	−	0.996355i	$-0.472815\pi$
$859$	5.00000	0.170598	0.0852989	+	0.996355i	$0.472815\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	18.0000	0.613082
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	−19.0000	−0.645646
$867$	−8.00000	−0.271694
$868$	4.00000	0.135769
$869$	−30.0000	−1.01768
$870$	0	0
$871$	0	0
$872$	10.0000	0.338643
$873$	4.00000	0.135379
$874$	−30.0000	−1.01477
$875$	0	0
$876$	−11.0000	−0.371656
$877$	28.0000	0.945493	0.472746	−	0.881199i	$-0.343263\pi$
$877$	28.0000	0.945493	0.472746	+	0.881199i	$0.343263\pi$
$878$	−40.0000	−1.34993
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	6.00000	0.202031
$883$	41.0000	1.37976	0.689880	−	0.723924i	$-0.257663\pi$
$883$	41.0000	1.37976	0.689880	+	0.723924i	$0.257663\pi$
$884$	0	0
$885$	0	0
$886$	−9.00000	−0.302361
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	−2.00000	−0.0671156
$889$	−4.00000	−0.134156
$890$	0	0
$891$	3.00000	0.100504
$892$	4.00000	0.133930
$893$	60.0000	2.00782
$894$	0	0
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	15.0000	0.500556
$899$	0	0
$900$	0	0
$901$	−18.0000	−0.599667
$902$	9.00000	0.299667
$903$	8.00000	0.266223
$904$	−9.00000	−0.299336
$905$	0	0
$906$	−2.00000	−0.0664455
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−12.0000	−0.398234
$909$	36.0000	1.19404
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	−5.00000	−0.165567
$913$	27.0000	0.893570
$914$	−17.0000	−0.562310
$915$	0	0
$916$	−20.0000	−0.660819
$917$	−24.0000	−0.792550
$918$	15.0000	0.495074
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	0	0
$921$	−17.0000	−0.560169
$922$	−12.0000	−0.395199
$923$	0	0
$924$	−6.00000	−0.197386
$925$	0	0
$926$	4.00000	0.131448
$927$	8.00000	0.262754
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	15.0000	0.491605
$932$	6.00000	0.196537
$933$	−18.0000	−0.589294
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	−13.0000	−0.424691	−0.212346	−	0.977195i	$-0.568110\pi$
$937$	−13.0000	−0.424691	−0.212346	+	0.977195i	$0.568110\pi$
$938$	−26.0000	−0.848930
$939$	26.0000	0.848478
$940$	0	0
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	2.00000	0.0651635
$943$	18.0000	0.586161
$944$	0	0
$945$	0	0
$946$	−12.0000	−0.390154
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	−10.0000	−0.324785
$949$	0	0
$950$	0	0
$951$	−12.0000	−0.389127
$952$	6.00000	0.194461
$953$	−39.0000	−1.26333	−0.631667	−	0.775240i	$-0.717629\pi$
$953$	−39.0000	−1.26333	−0.631667	+	0.775240i	$0.717629\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	0	0
$957$	0	0
$958$	−30.0000	−0.969256
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	6.00000	0.193347
$964$	−17.0000	−0.547533
$965$	0	0
$966$	−12.0000	−0.386094
$967$	−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
$969$	15.0000	0.481869
$970$	0	0
$971$	−3.00000	−0.0962746	−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000	−0.0962746	−0.0481373	+	0.998841i	$0.515328\pi$
$972$	16.0000	0.513200
$973$	−10.0000	−0.320585
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	2.00000	0.0640184
$977$	−57.0000	−1.82359	−0.911796	−	0.410644i	$-0.865304\pi$
$977$	−57.0000	−1.82359	−0.911796	+	0.410644i	$0.865304\pi$
$978$	−11.0000	−0.351741
$979$	−45.0000	−1.43821
$980$	0	0
$981$	−20.0000	−0.638551
$982$	12.0000	0.382935
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	3.00000	0.0956365
$985$	0	0
$986$	0	0
$987$	24.0000	0.763928
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−58.0000	−1.84243	−0.921215	−	0.389053i	$-0.872802\pi$
$991$	−58.0000	−1.84243	−0.921215	+	0.389053i	$0.872802\pi$
$992$	−2.00000	−0.0635001
$993$	−17.0000	−0.539479
$994$	24.0000	0.761234
$995$	0	0
$996$	9.00000	0.285176
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	−20.0000	−0.633089
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.v.1.1		1
5.4	even	2		8450.2.a.d.1.1		1
13.12	even	2		50.2.a.a.1.1	✓	1
39.38	odd	2		450.2.a.g.1.1		1
52.51	odd	2		400.2.a.d.1.1		1
65.12	odd	4		50.2.b.a.49.1		2
65.38	odd	4		50.2.b.a.49.2		2
65.64	even	2		50.2.a.b.1.1	yes	1
91.90	odd	2		2450.2.a.g.1.1		1
104.51	odd	2		1600.2.a.p.1.1		1
104.77	even	2		1600.2.a.j.1.1		1
143.142	odd	2		6050.2.a.bi.1.1		1
156.155	even	2		3600.2.a.l.1.1		1
195.38	even	4		450.2.c.c.199.1		2
195.77	even	4		450.2.c.c.199.2		2
195.194	odd	2		450.2.a.c.1.1		1
260.103	even	4		400.2.c.c.49.1		2
260.207	even	4		400.2.c.c.49.2		2
260.259	odd	2		400.2.a.f.1.1		1
455.272	even	4		2450.2.c.m.99.1		2
455.363	even	4		2450.2.c.m.99.2		2
455.454	odd	2		2450.2.a.bd.1.1		1
520.77	odd	4		1600.2.c.i.449.2		2
520.259	odd	2		1600.2.a.i.1.1		1
520.363	even	4		1600.2.c.h.449.2		2
520.389	even	2		1600.2.a.q.1.1		1
520.467	even	4		1600.2.c.h.449.1		2
520.493	odd	4		1600.2.c.i.449.1		2
715.714	odd	2		6050.2.a.h.1.1		1
780.467	odd	4		3600.2.f.f.2449.1		2
780.623	odd	4		3600.2.f.f.2449.2		2
780.779	even	2		3600.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.2.a.a.1.1	✓	1	13.12	even	2
50.2.a.b.1.1	yes	1	65.64	even	2
50.2.b.a.49.1		2	65.12	odd	4
50.2.b.a.49.2		2	65.38	odd	4
400.2.a.d.1.1		1	52.51	odd	2
400.2.a.f.1.1		1	260.259	odd	2
400.2.c.c.49.1		2	260.103	even	4
400.2.c.c.49.2		2	260.207	even	4
450.2.a.c.1.1		1	195.194	odd	2
450.2.a.g.1.1		1	39.38	odd	2
450.2.c.c.199.1		2	195.38	even	4
450.2.c.c.199.2		2	195.77	even	4
1600.2.a.i.1.1		1	520.259	odd	2
1600.2.a.j.1.1		1	104.77	even	2
1600.2.a.p.1.1		1	104.51	odd	2
1600.2.a.q.1.1		1	520.389	even	2
1600.2.c.h.449.1		2	520.467	even	4
1600.2.c.h.449.2		2	520.363	even	4
1600.2.c.i.449.1		2	520.493	odd	4
1600.2.c.i.449.2		2	520.77	odd	4
2450.2.a.g.1.1		1	91.90	odd	2
2450.2.a.bd.1.1		1	455.454	odd	2
2450.2.c.m.99.1		2	455.272	even	4
2450.2.c.m.99.2		2	455.363	even	4
3600.2.a.l.1.1		1	156.155	even	2
3600.2.a.bc.1.1		1	780.779	even	2
3600.2.f.f.2449.1		2	780.467	odd	4
3600.2.f.f.2449.2		2	780.623	odd	4
6050.2.a.h.1.1		1	715.714	odd	2
6050.2.a.bi.1.1		1	143.142	odd	2
8450.2.a.d.1.1		1	5.4	even	2
8450.2.a.v.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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