LMFDB - Embedded newform 8470.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -6.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} +2.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -4.00000 q^{37} -2.00000 q^{38} +4.00000 q^{39} -1.00000 q^{40} -12.0000 q^{41} +2.00000 q^{42} +4.00000 q^{43} -1.00000 q^{45} -6.00000 q^{46} +12.0000 q^{47} -2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -12.0000 q^{51} -2.00000 q^{52} +4.00000 q^{54} -1.00000 q^{56} +4.00000 q^{57} +2.00000 q^{60} -2.00000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +8.00000 q^{67} +6.00000 q^{68} +12.0000 q^{69} +1.00000 q^{70} +12.0000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -4.00000 q^{74} -2.00000 q^{75} -2.00000 q^{76} +4.00000 q^{78} -14.0000 q^{79} -1.00000 q^{80} -11.0000 q^{81} -12.0000 q^{82} -12.0000 q^{83} +2.00000 q^{84} -6.00000 q^{85} +4.00000 q^{86} +6.00000 q^{89} -1.00000 q^{90} +2.00000 q^{91} -6.00000 q^{92} -16.0000 q^{93} +12.0000 q^{94} +2.00000 q^{95} -2.00000 q^{96} +8.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.00000	−0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	−2.00000	−0.577350
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	2.00000	0.516398
$16$	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
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−1.00000	−0.223607
$21$	2.00000	0.436436
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−2.00000	−0.408248
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	4.00000	0.769800
$28$	−1.00000	−0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	2.00000	0.365148
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−2.00000	−0.324443
$39$	4.00000	0.640513
$40$	−1.00000	−0.158114
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	2.00000	0.308607
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	−6.00000	−0.884652
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−2.00000	−0.288675
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−12.0000	−1.68034
$52$	−2.00000	−0.277350
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	−1.00000	−0.133631
$57$	4.00000	0.529813
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	2.00000	0.258199
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	8.00000	1.01600
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	6.00000	0.727607
$69$	12.0000	1.44463
$70$	1.00000	0.119523
$71$	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
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−4.00000	−0.464991
$75$	−2.00000	−0.230940
$76$	−2.00000	−0.229416
$77$	0	0
$78$	4.00000	0.452911
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	−1.00000	−0.111803
$81$	−11.0000	−1.22222
$82$	−12.0000	−1.32518
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	2.00000	0.218218
$85$	−6.00000	−0.650791
$86$	4.00000	0.431331
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−0.105409
$91$	2.00000	0.209657
$92$	−6.00000	−0.625543
$93$	−16.0000	−1.65912
$94$	12.0000	1.23771
$95$	2.00000	0.205196
$96$	−2.00000	−0.204124
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−12.0000	−1.18818
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−2.00000	−0.196116
$105$	−2.00000	−0.195180
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	4.00000	0.384900
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	−1.00000	−0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	4.00000	0.374634
$115$	6.00000	0.559503
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−6.00000	−0.550019
$120$	2.00000	0.182574
$121$	0	0
$122$	−2.00000	−0.181071
$123$	24.0000	2.16401
$124$	8.00000	0.718421
$125$	−1.00000	−0.0894427
$126$	−1.00000	−0.0890871
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	−8.00000	−0.704361
$130$	2.00000	0.175412
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	8.00000	0.691095
$135$	−4.00000	−0.344265
$136$	6.00000	0.514496
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	12.0000	1.02151
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	1.00000	0.0845154
$141$	−24.0000	−2.02116
$142$	12.0000	1.00702
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	−2.00000	−0.164957
$148$	−4.00000	−0.328798
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	−2.00000	−0.163299
$151$	−14.0000	−1.13930	−0.569652	−	0.821886i	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	−2.00000	−0.162221
$153$	6.00000	0.485071
$154$	0	0
$155$	−8.00000	−0.642575
$156$	4.00000	0.320256
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−14.0000	−1.11378
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	6.00000	0.472866
$162$	−11.0000	−0.864242
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	−12.0000	−0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	2.00000	0.154303
$169$	−9.00000	−0.692308
$170$	−6.00000	−0.460179
$171$	−2.00000	−0.152944
$172$	4.00000	0.304997
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	−1.00000	−0.0745356
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	2.00000	0.148250
$183$	4.00000	0.295689
$184$	−6.00000	−0.442326
$185$	4.00000	0.294086
$186$	−16.0000	−1.17318
$187$	0	0
$188$	12.0000	0.875190
$189$	−4.00000	−0.290957
$190$	2.00000	0.145095
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	−2.00000	−0.144338
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	8.00000	0.574367
$195$	−4.00000	−0.286446
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	1.00000	0.0707107
$201$	−16.0000	−1.12855
$202$	6.00000	0.422159
$203$	0	0
$204$	−12.0000	−0.840168
$205$	12.0000	0.838116
$206$	−16.0000	−1.11477
$207$	−6.00000	−0.417029
$208$	−2.00000	−0.138675
$209$	0	0
$210$	−2.00000	−0.138013
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	−24.0000	−1.64445
$214$	−12.0000	−0.820303
$215$	−4.00000	−0.272798
$216$	4.00000	0.272166
$217$	−8.00000	−0.543075
$218$	16.0000	1.08366
$219$	4.00000	0.270295
$220$	0	0
$221$	−12.0000	−0.807207
$222$	8.00000	0.536925
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−1.00000	−0.0668153
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	4.00000	0.264906
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−2.00000	−0.130744
$235$	−12.0000	−0.782794
$236$	0	0
$237$	28.0000	1.81880
$238$	−6.00000	−0.388922
$239$	−18.0000	−1.16432	−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000	−1.16432	−0.582162	+	0.813073i	$0.697793\pi$
$240$	2.00000	0.129099
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	−2.00000	−0.128037
$245$	−1.00000	−0.0638877
$246$	24.0000	1.53018
$247$	4.00000	0.254514
$248$	8.00000	0.508001
$249$	24.0000	1.52094
$250$	−1.00000	−0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	−8.00000	−0.501965
$255$	12.0000	0.751469
$256$	1.00000	0.0625000
$257$	24.0000	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$257$	24.0000	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$258$	−8.00000	−0.498058
$259$	4.00000	0.248548
$260$	2.00000	0.124035
$261$	0	0
$262$	18.0000	1.11204
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	0	0
$266$	2.00000	0.122628
$267$	−12.0000	−0.734388
$268$	8.00000	0.488678
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	−4.00000	−0.243432
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	6.00000	0.363803
$273$	−4.00000	−0.242091
$274$	−6.00000	−0.362473
$275$	0	0
$276$	12.0000	0.722315
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−14.0000	−0.839664
$279$	8.00000	0.478947
$280$	1.00000	0.0597614
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−24.0000	−1.42918
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	12.0000	0.712069
$285$	−4.00000	−0.236940
$286$	0	0
$287$	12.0000	0.708338
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	−16.0000	−0.937937
$292$	−2.00000	−0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	−4.00000	−0.232495
$297$	0	0
$298$	0	0
$299$	12.0000	0.693978
$300$	−2.00000	−0.115470
$301$	−4.00000	−0.230556
$302$	−14.0000	−0.805609
$303$	−12.0000	−0.689382
$304$	−2.00000	−0.114708
$305$	2.00000	0.114520
$306$	6.00000	0.342997
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	32.0000	1.82042
$310$	−8.00000	−0.454369
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	4.00000	0.226455
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	14.0000	0.790066
$315$	1.00000	0.0563436
$316$	−14.0000	−0.787562
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	24.0000	1.33955
$322$	6.00000	0.334367
$323$	−12.0000	−0.667698
$324$	−11.0000	−0.611111
$325$	−2.00000	−0.110940
$326$	8.00000	0.443079
$327$	−32.0000	−1.76960
$328$	−12.0000	−0.662589
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−12.0000	−0.658586
$333$	−4.00000	−0.219199
$334$	0	0
$335$	−8.00000	−0.437087
$336$	2.00000	0.109109
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	−9.00000	−0.489535
$339$	12.0000	0.651751
$340$	−6.00000	−0.325396
$341$	0	0
$342$	−2.00000	−0.108148
$343$	−1.00000	−0.0539949
$344$	4.00000	0.215666
$345$	−12.0000	−0.646058
$346$	18.0000	0.967686
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	−1.00000	−0.0534522
$351$	−8.00000	−0.427008
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	6.00000	0.317999
$357$	12.0000	0.635107
$358$	−12.0000	−0.634220
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	−1.00000	−0.0527046
$361$	−15.0000	−0.789474
$362$	2.00000	0.105118
$363$	0	0
$364$	2.00000	0.104828
$365$	2.00000	0.104685
$366$	4.00000	0.209083
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−6.00000	−0.312772
$369$	−12.0000	−0.624695
$370$	4.00000	0.207950
$371$	0	0
$372$	−16.0000	−0.829561
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	2.00000	0.103280
$376$	12.0000	0.618853
$377$	0	0
$378$	−4.00000	−0.205738
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	2.00000	0.102598
$381$	16.0000	0.819705
$382$	−24.0000	−1.22795
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	−2.00000	−0.101797
$387$	4.00000	0.203331
$388$	8.00000	0.406138
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−4.00000	−0.202548
$391$	−36.0000	−1.82060
$392$	1.00000	0.0505076
$393$	−36.0000	−1.81596
$394$	−6.00000	−0.302276
$395$	14.0000	0.704416
$396$	0	0
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	−16.0000	−0.802008
$399$	−4.00000	−0.200250
$400$	1.00000	0.0500000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	−16.0000	−0.798007
$403$	−16.0000	−0.797017
$404$	6.00000	0.298511
$405$	11.0000	0.546594
$406$	0	0
$407$	0	0
$408$	−12.0000	−0.594089
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	12.0000	0.592638
$411$	12.0000	0.591916
$412$	−16.0000	−0.788263
$413$	0	0
$414$	−6.00000	−0.294884
$415$	12.0000	0.589057
$416$	−2.00000	−0.0980581
$417$	28.0000	1.37117
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	−2.00000	−0.0975900
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	4.00000	0.194717
$423$	12.0000	0.583460
$424$	0	0
$425$	6.00000	0.291043
$426$	−24.0000	−1.16280
$427$	2.00000	0.0967868
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−4.00000	−0.192897
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	4.00000	0.192450
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	16.0000	0.766261
$437$	12.0000	0.574038
$438$	4.00000	0.191127
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−12.0000	−0.570782
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	8.00000	0.379663
$445$	−6.00000	−0.284427
$446$	8.00000	0.378811
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−6.00000	−0.282216
$453$	28.0000	1.31555
$454$	−24.0000	−1.12638
$455$	−2.00000	−0.0937614
$456$	4.00000	0.187317
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−10.0000	−0.467269
$459$	24.0000	1.12022
$460$	6.00000	0.279751
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	0	0
$465$	16.0000	0.741982
$466$	−6.00000	−0.277945
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	−2.00000	−0.0924500
$469$	−8.00000	−0.369406
$470$	−12.0000	−0.553519
$471$	−28.0000	−1.29017
$472$	0	0
$473$	0	0
$474$	28.0000	1.28608
$475$	−2.00000	−0.0917663
$476$	−6.00000	−0.275010
$477$	0	0
$478$	−18.0000	−0.823301
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	2.00000	0.0912871
$481$	8.00000	0.364769
$482$	−20.0000	−0.910975
$483$	−12.0000	−0.546019
$484$	0	0
$485$	−8.00000	−0.363261
$486$	10.0000	0.453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−2.00000	−0.0905357
$489$	−16.0000	−0.723545
$490$	−1.00000	−0.0451754
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	24.0000	1.08200
$493$	0	0
$494$	4.00000	0.179969
$495$	0	0
$496$	8.00000	0.359211
$497$	−12.0000	−0.538274
$498$	24.0000	1.07547
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	−1.00000	−0.0445435
$505$	−6.00000	−0.266996
$506$	0	0
$507$	18.0000	0.799408
$508$	−8.00000	−0.354943
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	12.0000	0.531369
$511$	2.00000	0.0884748
$512$	1.00000	0.0441942
$513$	−8.00000	−0.353209
$514$	24.0000	1.05859
$515$	16.0000	0.705044
$516$	−8.00000	−0.352180
$517$	0	0
$518$	4.00000	0.175750
$519$	−36.0000	−1.58022
$520$	2.00000	0.0877058
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	−44.0000	−1.92399	−0.961993	−	0.273075i	$-0.911959\pi$
$523$	−44.0000	−1.92399	−0.961993	+	0.273075i	$0.911959\pi$
$524$	18.0000	0.786334
$525$	2.00000	0.0872872
$526$	−24.0000	−1.04645
$527$	48.0000	2.09091
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	2.00000	0.0867110
$533$	24.0000	1.03956
$534$	−12.0000	−0.519291
$535$	12.0000	0.518805
$536$	8.00000	0.345547
$537$	24.0000	1.03568
$538$	−18.0000	−0.776035
$539$	0	0
$540$	−4.00000	−0.172133
$541$	−20.0000	−0.859867	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000	−0.859867	−0.429934	+	0.902861i	$0.641463\pi$
$542$	−20.0000	−0.859074
$543$	−4.00000	−0.171656
$544$	6.00000	0.257248
$545$	−16.0000	−0.685365
$546$	−4.00000	−0.171184
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−6.00000	−0.256307
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	0	0
$552$	12.0000	0.510754
$553$	14.0000	0.595341
$554$	22.0000	0.934690
$555$	−8.00000	−0.339581
$556$	−14.0000	−0.593732
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	8.00000	0.338667
$559$	−8.00000	−0.338364
$560$	1.00000	0.0422577
$561$	0	0
$562$	−6.00000	−0.253095
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	−24.0000	−1.01058
$565$	6.00000	0.252422
$566$	16.0000	0.672530
$567$	11.0000	0.461957
$568$	12.0000	0.503509
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	−4.00000	−0.167542
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	48.0000	2.00523
$574$	12.0000	0.500870
$575$	−6.00000	−0.250217
$576$	1.00000	0.0416667
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	19.0000	0.790296
$579$	4.00000	0.166234
$580$	0	0
$581$	12.0000	0.497844
$582$	−16.0000	−0.663221
$583$	0	0
$584$	−2.00000	−0.0827606
$585$	2.00000	0.0826898
$586$	6.00000	0.247858
$587$	−30.0000	−1.23823	−0.619116	−	0.785299i	$-0.712509\pi$
$587$	−30.0000	−1.23823	−0.619116	+	0.785299i	$0.712509\pi$
$588$	−2.00000	−0.0824786
$589$	−16.0000	−0.659269
$590$	0	0
$591$	12.0000	0.493614
$592$	−4.00000	−0.164399
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	0	0
$597$	32.0000	1.30967
$598$	12.0000	0.490716
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	−2.00000	−0.0816497
$601$	−32.0000	−1.30531	−0.652654	−	0.757656i	$-0.726344\pi$
$601$	−32.0000	−1.30531	−0.652654	+	0.757656i	$0.726344\pi$
$602$	−4.00000	−0.163028
$603$	8.00000	0.325785
$604$	−14.0000	−0.569652
$605$	0	0
$606$	−12.0000	−0.487467
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	2.00000	0.0809776
$611$	−24.0000	−0.970936
$612$	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	$0.591239\pi$
$614$	4.00000	0.161427
$615$	−24.0000	−0.967773
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	32.0000	1.28723
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	−8.00000	−0.321288
$621$	−24.0000	−0.963087
$622$	24.0000	0.962312
$623$	−6.00000	−0.240385
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	8.00000	0.319744
$627$	0	0
$628$	14.0000	0.558661
$629$	−24.0000	−0.956943
$630$	1.00000	0.0398410
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	−14.0000	−0.556890
$633$	−8.00000	−0.317971
$634$	−12.0000	−0.476581
$635$	8.00000	0.317470
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	12.0000	0.474713
$640$	−1.00000	−0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	24.0000	0.947204
$643$	−34.0000	−1.34083	−0.670415	−	0.741987i	$-0.733884\pi$
$643$	−34.0000	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$644$	6.00000	0.236433
$645$	8.00000	0.315000
$646$	−12.0000	−0.472134
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	−11.0000	−0.432121
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	16.0000	0.627089
$652$	8.00000	0.313304
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	−32.0000	−1.25130
$655$	−18.0000	−0.703318
$656$	−12.0000	−0.468521
$657$	−2.00000	−0.0780274
$658$	−12.0000	−0.467809
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	−28.0000	−1.08825
$663$	24.0000	0.932083
$664$	−12.0000	−0.465690
$665$	−2.00000	−0.0775567
$666$	−4.00000	−0.154997
$667$	0	0
$668$	0	0
$669$	−16.0000	−0.618596
$670$	−8.00000	−0.309067
$671$	0	0
$672$	2.00000	0.0771517
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	−14.0000	−0.539260
$675$	4.00000	0.153960
$676$	−9.00000	−0.346154
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	12.0000	0.460857
$679$	−8.00000	−0.307012
$680$	−6.00000	−0.230089
$681$	48.0000	1.83936
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	−2.00000	−0.0764719
$685$	6.00000	0.229248
$686$	−1.00000	−0.0381802
$687$	20.0000	0.763048
$688$	4.00000	0.152499
$689$	0	0
$690$	−12.0000	−0.456832
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−12.0000	−0.455514
$695$	14.0000	0.531050
$696$	0	0
$697$	−72.0000	−2.72719
$698$	−14.0000	−0.529908
$699$	12.0000	0.453882
$700$	−1.00000	−0.0377964
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	−8.00000	−0.301941
$703$	8.00000	0.301726
$704$	0	0
$705$	24.0000	0.903892
$706$	−24.0000	−0.903252
$707$	−6.00000	−0.225653
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	−12.0000	−0.450352
$711$	−14.0000	−0.525041
$712$	6.00000	0.224860
$713$	−48.0000	−1.79761
$714$	12.0000	0.449089
$715$	0	0
$716$	−12.0000	−0.448461
$717$	36.0000	1.34444
$718$	−30.0000	−1.11959
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	−1.00000	−0.0372678
$721$	16.0000	0.595871
$722$	−15.0000	−0.558242
$723$	40.0000	1.48762
$724$	2.00000	0.0743294
$725$	0	0
$726$	0	0
$727$	20.0000	0.741759	0.370879	−	0.928681i	$-0.379056\pi$
$727$	20.0000	0.741759	0.370879	+	0.928681i	$0.379056\pi$
$728$	2.00000	0.0741249
$729$	13.0000	0.481481
$730$	2.00000	0.0740233
$731$	24.0000	0.887672
$732$	4.00000	0.147844
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	8.00000	0.295285
$735$	2.00000	0.0737711
$736$	−6.00000	−0.221163
$737$	0	0
$738$	−12.0000	−0.441726
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	4.00000	0.147043
$741$	−8.00000	−0.293887
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	−16.0000	−0.586588
$745$	0	0
$746$	−14.0000	−0.512576
$747$	−12.0000	−0.439057
$748$	0	0
$749$	12.0000	0.438470
$750$	2.00000	0.0730297
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	0	0
$755$	14.0000	0.509512
$756$	−4.00000	−0.145479
$757$	−16.0000	−0.581530	−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000	−0.581530	−0.290765	+	0.956795i	$0.593910\pi$
$758$	−28.0000	−1.01701
$759$	0	0
$760$	2.00000	0.0725476
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	16.0000	0.579619
$763$	−16.0000	−0.579239
$764$	−24.0000	−0.868290
$765$	−6.00000	−0.216930
$766$	−24.0000	−0.867155
$767$	0	0
$768$	−2.00000	−0.0721688
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	−48.0000	−1.72868
$772$	−2.00000	−0.0719816
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	4.00000	0.143777
$775$	8.00000	0.287368
$776$	8.00000	0.287183
$777$	−8.00000	−0.286998
$778$	−6.00000	−0.215110
$779$	24.0000	0.859889
$780$	−4.00000	−0.143223
$781$	0	0
$782$	−36.0000	−1.28736
$783$	0	0
$784$	1.00000	0.0357143
$785$	−14.0000	−0.499681
$786$	−36.0000	−1.28408
$787$	16.0000	0.570338	0.285169	−	0.958477i	$-0.407950\pi$
$787$	16.0000	0.570338	0.285169	+	0.958477i	$0.407950\pi$
$788$	−6.00000	−0.213741
$789$	48.0000	1.70885
$790$	14.0000	0.498098
$791$	6.00000	0.213335
$792$	0	0
$793$	4.00000	0.142044
$794$	38.0000	1.34857
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	−4.00000	−0.141598
$799$	72.0000	2.54718
$800$	1.00000	0.0353553
$801$	6.00000	0.212000
$802$	30.0000	1.05934
$803$	0	0
$804$	−16.0000	−0.564276
$805$	−6.00000	−0.211472
$806$	−16.0000	−0.563576
$807$	36.0000	1.26726
$808$	6.00000	0.211079
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	11.0000	0.386501
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	0	0
$813$	40.0000	1.40286
$814$	0	0
$815$	−8.00000	−0.280228
$816$	−12.0000	−0.420084
$817$	−8.00000	−0.279885
$818$	4.00000	0.139857
$819$	2.00000	0.0698857
$820$	12.0000	0.419058
$821$	−12.0000	−0.418803	−0.209401	−	0.977830i	$-0.567152\pi$
$821$	−12.0000	−0.418803	−0.209401	+	0.977830i	$0.567152\pi$
$822$	12.0000	0.418548
$823$	50.0000	1.74289	0.871445	−	0.490493i	$-0.163183\pi$
$823$	50.0000	1.74289	0.871445	+	0.490493i	$0.163183\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−6.00000	−0.208514
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	12.0000	0.416526
$831$	−44.0000	−1.52634
$832$	−2.00000	−0.0693375
$833$	6.00000	0.207888
$834$	28.0000	0.969561
$835$	0	0
$836$	0	0
$837$	32.0000	1.10608
$838$	0	0
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	−2.00000	−0.0690066
$841$	−29.0000	−1.00000
$842$	−10.0000	−0.344623
$843$	12.0000	0.413302
$844$	4.00000	0.137686
$845$	9.00000	0.309609
$846$	12.0000	0.412568
$847$	0	0
$848$	0	0
$849$	−32.0000	−1.09824
$850$	6.00000	0.205798
$851$	24.0000	0.822709
$852$	−24.0000	−0.822226
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	2.00000	0.0684386
$855$	2.00000	0.0683986
$856$	−12.0000	−0.410152
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	−16.0000	−0.545913	−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	−4.00000	−0.136399
$861$	−24.0000	−0.817918
$862$	−30.0000	−1.02180
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	4.00000	0.136083
$865$	−18.0000	−0.612018
$866$	−4.00000	−0.135926
$867$	−38.0000	−1.29055
$868$	−8.00000	−0.271538
$869$	0	0
$870$	0	0
$871$	−16.0000	−0.542139
$872$	16.0000	0.541828
$873$	8.00000	0.270759
$874$	12.0000	0.405906
$875$	1.00000	0.0338062
$876$	4.00000	0.135147
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	28.0000	0.944954
$879$	−12.0000	−0.404750
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	1.00000	0.0336718
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	24.0000	0.806296
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	8.00000	0.268462
$889$	8.00000	0.268311
$890$	−6.00000	−0.201120
$891$	0	0
$892$	8.00000	0.267860
$893$	−24.0000	−0.803129
$894$	0	0
$895$	12.0000	0.401116
$896$	−1.00000	−0.0334077
$897$	−24.0000	−0.801337
$898$	−18.0000	−0.600668
$899$	0	0
$900$	1.00000	0.0333333
$901$	0	0
$902$	0	0
$903$	8.00000	0.266223
$904$	−6.00000	−0.199557
$905$	−2.00000	−0.0664822
$906$	28.0000	0.930238
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	−24.0000	−0.796468
$909$	6.00000	0.199007
$910$	−2.00000	−0.0662994
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	10.0000	0.330771
$915$	−4.00000	−0.132236
$916$	−10.0000	−0.330409
$917$	−18.0000	−0.594412
$918$	24.0000	0.792118
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	6.00000	0.197814
$921$	−8.00000	−0.263609
$922$	18.0000	0.592798
$923$	−24.0000	−0.789970
$924$	0	0
$925$	−4.00000	−0.131519
$926$	−22.0000	−0.722965
$927$	−16.0000	−0.525509
$928$	0	0
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	16.0000	0.524661
$931$	−2.00000	−0.0655474
$932$	−6.00000	−0.196537
$933$	−48.0000	−1.57145
$934$	−6.00000	−0.196326
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	−8.00000	−0.261209
$939$	−16.0000	−0.522140
$940$	−12.0000	−0.391397
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	−28.0000	−0.912289
$943$	72.0000	2.34464
$944$	0	0
$945$	4.00000	0.130120
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	28.0000	0.909398
$949$	4.00000	0.129845
$950$	−2.00000	−0.0648886
$951$	24.0000	0.778253
$952$	−6.00000	−0.194461
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	0	0
$955$	24.0000	0.776622
$956$	−18.0000	−0.582162
$957$	0	0
$958$	−24.0000	−0.775405
$959$	6.00000	0.193750
$960$	2.00000	0.0645497
$961$	33.0000	1.06452
$962$	8.00000	0.257930
$963$	−12.0000	−0.386695
$964$	−20.0000	−0.644157
$965$	2.00000	0.0643823
$966$	−12.0000	−0.386094
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	−8.00000	−0.256865
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	10.0000	0.320750
$973$	14.0000	0.448819
$974$	2.00000	0.0640841
$975$	4.00000	0.128103
$976$	−2.00000	−0.0640184
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	−16.0000	−0.511624
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	16.0000	0.510841
$982$	36.0000	1.14881
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	24.0000	0.765092
$985$	6.00000	0.191176
$986$	0	0
$987$	24.0000	0.763928
$988$	4.00000	0.127257
$989$	−24.0000	−0.763156
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	8.00000	0.254000
$993$	56.0000	1.77711
$994$	−12.0000	−0.380617
$995$	16.0000	0.507234
$996$	24.0000	0.760469
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	−4.00000	−0.126618
$999$	−16.0000	−0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.r.1.1		1
11.10	odd	2		770.2.a.a.1.1	✓	1
33.32	even	2		6930.2.a.bm.1.1		1
44.43	even	2		6160.2.a.k.1.1		1
55.32	even	4		3850.2.c.o.1849.1		2
55.43	even	4		3850.2.c.o.1849.2		2
55.54	odd	2		3850.2.a.ba.1.1		1
77.76	even	2		5390.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.a.1.1	✓	1	11.10	odd	2
3850.2.a.ba.1.1		1	55.54	odd	2
3850.2.c.o.1849.1		2	55.32	even	4
3850.2.c.o.1849.2		2	55.43	even	4
5390.2.a.r.1.1		1	77.76	even	2
6160.2.a.k.1.1		1	44.43	even	2
6930.2.a.bm.1.1		1	33.32	even	2
8470.2.a.r.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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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