LMFDB - Embedded newform 8470.2.a.bd.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +5.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -2.00000 q^{18} +5.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -3.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +5.00000 q^{26} -5.00000 q^{27} +1.00000 q^{28} -1.00000 q^{30} -10.0000 q^{31} +1.00000 q^{32} +6.00000 q^{34} -1.00000 q^{35} -2.00000 q^{36} +2.00000 q^{37} +5.00000 q^{38} +5.00000 q^{39} -1.00000 q^{40} +1.00000 q^{42} +8.00000 q^{43} +2.00000 q^{45} -3.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +5.00000 q^{52} +6.00000 q^{53} -5.00000 q^{54} +1.00000 q^{56} +5.00000 q^{57} +9.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} -10.0000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -5.00000 q^{65} +14.0000 q^{67} +6.00000 q^{68} -3.00000 q^{69} -1.00000 q^{70} -2.00000 q^{72} +2.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +5.00000 q^{76} +5.00000 q^{78} -1.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +9.00000 q^{83} +1.00000 q^{84} -6.00000 q^{85} +8.00000 q^{86} +2.00000 q^{90} +5.00000 q^{91} -3.00000 q^{92} -10.0000 q^{93} -6.00000 q^{94} -5.00000 q^{95} +1.00000 q^{96} +8.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
$3$	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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	1.00000	0.288675
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	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	$0.240633\pi$
$18$	−2.00000	−0.471405
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	−1.00000	−0.223607
$21$	1.00000	0.218218
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	5.00000	0.980581
$27$	−5.00000	−0.962250
$28$	1.00000	0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−1.00000	−0.182574
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	−1.00000	−0.169031
$36$	−2.00000	−0.333333
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	5.00000	0.811107
$39$	5.00000	0.800641
$40$	−1.00000	−0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	1.00000	0.154303
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	−3.00000	−0.442326
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	6.00000	0.840168
$52$	5.00000	0.693375
$53$	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
$56$	1.00000	0.133631
$57$	5.00000	0.662266
$58$	0	0
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	−1.00000	−0.129099
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−10.0000	−1.27000
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	−5.00000	−0.620174
$66$	0	0
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	6.00000	0.727607
$69$	−3.00000	−0.361158
$70$	−1.00000	−0.119523
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−2.00000	−0.235702
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	5.00000	0.573539
$77$	0	0
$78$	5.00000	0.566139
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	0	0
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	1.00000	0.109109
$85$	−6.00000	−0.650791
$86$	8.00000	0.862662
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	2.00000	0.210819
$91$	5.00000	0.524142
$92$	−3.00000	−0.312772
$93$	−10.0000	−1.03695
$94$	−6.00000	−0.618853
$95$	−5.00000	−0.512989
$96$	1.00000	0.102062
$97$	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
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	6.00000	0.594089
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	5.00000	0.490290
$105$	−1.00000	−0.0975900
$106$	6.00000	0.582772
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	−5.00000	−0.481125
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	1.00000	0.0944911
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	5.00000	0.468293
$115$	3.00000	0.279751
$116$	0	0
$117$	−10.0000	−0.924500
$118$	9.00000	0.828517
$119$	6.00000	0.550019
$120$	−1.00000	−0.0912871
$121$	0	0
$122$	−10.0000	−0.905357
$123$	0	0
$124$	−10.0000	−0.898027
$125$	−1.00000	−0.0894427
$126$	−2.00000	−0.178174
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	1.00000	0.0883883
$129$	8.00000	0.704361
$130$	−5.00000	−0.438529
$131$	21.0000	1.83478	0.917389	−	0.397991i	$-0.130293\pi$
$131$	21.0000	1.83478	0.917389	+	0.397991i	$0.130293\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	14.0000	1.20942
$135$	5.00000	0.430331
$136$	6.00000	0.514496
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	−3.00000	−0.255377
$139$	−19.0000	−1.61156	−0.805779	−	0.592216i	$-0.798253\pi$
$139$	−19.0000	−1.61156	−0.805779	+	0.592216i	$0.798253\pi$
$140$	−1.00000	−0.0845154
$141$	−6.00000	−0.505291
$142$	0	0
$143$	0	0
$144$	−2.00000	−0.166667
$145$	0	0
$146$	2.00000	0.165521
$147$	1.00000	0.0824786
$148$	2.00000	0.164399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	1.00000	0.0816497
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	5.00000	0.405554
$153$	−12.0000	−0.970143
$154$	0	0
$155$	10.0000	0.803219
$156$	5.00000	0.400320
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	−1.00000	−0.0795557
$159$	6.00000	0.475831
$160$	−1.00000	−0.0790569
$161$	−3.00000	−0.236433
$162$	1.00000	0.0785674
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	0	0
$165$	0	0
$166$	9.00000	0.698535
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	1.00000	0.0771517
$169$	12.0000	0.923077
$170$	−6.00000	−0.460179
$171$	−10.0000	−0.764719
$172$	8.00000	0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	9.00000	0.676481
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	2.00000	0.149071
$181$	−25.0000	−1.85824	−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000	−1.85824	−0.929118	+	0.369784i	$0.879432\pi$
$182$	5.00000	0.370625
$183$	−10.0000	−0.739221
$184$	−3.00000	−0.221163
$185$	−2.00000	−0.147043
$186$	−10.0000	−0.733236
$187$	0	0
$188$	−6.00000	−0.437595
$189$	−5.00000	−0.363696
$190$	−5.00000	−0.362738
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	1.00000	0.0721688
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	8.00000	0.574367
$195$	−5.00000	−0.358057
$196$	1.00000	0.0714286
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	1.00000	0.0707107
$201$	14.0000	0.987484
$202$	−3.00000	−0.211079
$203$	0	0
$204$	6.00000	0.420084
$205$	0	0
$206$	−4.00000	−0.278693
$207$	6.00000	0.417029
$208$	5.00000	0.346688
$209$	0	0
$210$	−1.00000	−0.0690066
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	6.00000	0.410152
$215$	−8.00000	−0.545595
$216$	−5.00000	−0.340207
$217$	−10.0000	−0.678844
$218$	−10.0000	−0.677285
$219$	2.00000	0.135147
$220$	0	0
$221$	30.0000	2.01802
$222$	2.00000	0.134231
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	1.00000	0.0668153
$225$	−2.00000	−0.133333
$226$	9.00000	0.598671
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	5.00000	0.331133
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	3.00000	0.197814
$231$	0	0
$232$	0	0
$233$	−21.0000	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$234$	−10.0000	−0.653720
$235$	6.00000	0.391397
$236$	9.00000	0.585850
$237$	−1.00000	−0.0649570
$238$	6.00000	0.388922
$239$	−3.00000	−0.194054	−0.0970269	−	0.995282i	$-0.530933\pi$
$239$	−3.00000	−0.194054	−0.0970269	+	0.995282i	$0.530933\pi$
$240$	−1.00000	−0.0645497
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	−10.0000	−0.640184
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	25.0000	1.59071
$248$	−10.0000	−0.635001
$249$	9.00000	0.570352
$250$	−1.00000	−0.0632456
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	−7.00000	−0.439219
$255$	−6.00000	−0.375735
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	8.00000	0.498058
$259$	2.00000	0.124274
$260$	−5.00000	−0.310087
$261$	0	0
$262$	21.0000	1.29738
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	5.00000	0.306570
$267$	0	0
$268$	14.0000	0.855186
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	5.00000	0.304290
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	6.00000	0.363803
$273$	5.00000	0.302614
$274$	−3.00000	−0.181237
$275$	0	0
$276$	−3.00000	−0.180579
$277$	32.0000	1.92269	0.961347	−	0.275340i	$-0.0887905\pi$
$277$	32.0000	1.92269	0.961347	+	0.275340i	$0.0887905\pi$
$278$	−19.0000	−1.13954
$279$	20.0000	1.19737
$280$	−1.00000	−0.0597614
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	−6.00000	−0.357295
$283$	29.0000	1.72387	0.861936	−	0.507018i	$-0.169252\pi$
$283$	29.0000	1.72387	0.861936	+	0.507018i	$0.169252\pi$
$284$	0	0
$285$	−5.00000	−0.296174
$286$	0	0
$287$	0	0
$288$	−2.00000	−0.117851
$289$	19.0000	1.11765
$290$	0	0
$291$	8.00000	0.468968
$292$	2.00000	0.117041
$293$	27.0000	1.57736	0.788678	−	0.614806i	$-0.210766\pi$
$293$	27.0000	1.57736	0.788678	+	0.614806i	$0.210766\pi$
$294$	1.00000	0.0583212
$295$	−9.00000	−0.524000
$296$	2.00000	0.116248
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−15.0000	−0.867472
$300$	1.00000	0.0577350
$301$	8.00000	0.461112
$302$	−7.00000	−0.402805
$303$	−3.00000	−0.172345
$304$	5.00000	0.286770
$305$	10.0000	0.572598
$306$	−12.0000	−0.685994
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	10.0000	0.567962
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	5.00000	0.283069
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	5.00000	0.282166
$315$	2.00000	0.112687
$316$	−1.00000	−0.0562544
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	6.00000	0.336463
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	6.00000	0.334887
$322$	−3.00000	−0.167183
$323$	30.0000	1.66924
$324$	1.00000	0.0555556
$325$	5.00000	0.277350
$326$	2.00000	0.110770
$327$	−10.0000	−0.553001
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	9.00000	0.493939
$333$	−4.00000	−0.219199
$334$	0	0
$335$	−14.0000	−0.764902
$336$	1.00000	0.0545545
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	12.0000	0.652714
$339$	9.00000	0.488813
$340$	−6.00000	−0.325396
$341$	0	0
$342$	−10.0000	−0.540738
$343$	1.00000	0.0539949
$344$	8.00000	0.431331
$345$	3.00000	0.161515
$346$	6.00000	0.322562
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	11.0000	0.588817	0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000	0.588817	0.294408	+	0.955680i	$0.404877\pi$
$350$	1.00000	0.0534522
$351$	−25.0000	−1.33440
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	9.00000	0.478345
$355$	0	0
$356$	0	0
$357$	6.00000	0.317554
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	2.00000	0.105409
$361$	6.00000	0.315789
$362$	−25.0000	−1.31397
$363$	0	0
$364$	5.00000	0.262071
$365$	−2.00000	−0.104685
$366$	−10.0000	−0.522708
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	−2.00000	−0.103975
$371$	6.00000	0.311504
$372$	−10.0000	−0.518476
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	−6.00000	−0.309426
$377$	0	0
$378$	−5.00000	−0.257172
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	−5.00000	−0.256495
$381$	−7.00000	−0.358621
$382$	15.0000	0.767467
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	11.0000	0.559885
$387$	−16.0000	−0.813326
$388$	8.00000	0.406138
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	−5.00000	−0.253185
$391$	−18.0000	−0.910299
$392$	1.00000	0.0505076
$393$	21.0000	1.05931
$394$	12.0000	0.604551
$395$	1.00000	0.0503155
$396$	0	0
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	8.00000	0.401004
$399$	5.00000	0.250313
$400$	1.00000	0.0500000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	14.0000	0.698257
$403$	−50.0000	−2.49068
$404$	−3.00000	−0.149256
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	6.00000	0.297044
$409$	−28.0000	−1.38451	−0.692255	−	0.721653i	$-0.743383\pi$
$409$	−28.0000	−1.38451	−0.692255	+	0.721653i	$0.743383\pi$
$410$	0	0
$411$	−3.00000	−0.147979
$412$	−4.00000	−0.197066
$413$	9.00000	0.442861
$414$	6.00000	0.294884
$415$	−9.00000	−0.441793
$416$	5.00000	0.245145
$417$	−19.0000	−0.930434
$418$	0	0
$419$	21.0000	1.02592	0.512959	−	0.858413i	$-0.328549\pi$
$419$	21.0000	1.02592	0.512959	+	0.858413i	$0.328549\pi$
$420$	−1.00000	−0.0487950
$421$	−40.0000	−1.94948	−0.974740	−	0.223341i	$-0.928304\pi$
$421$	−40.0000	−1.94948	−0.974740	+	0.223341i	$0.928304\pi$
$422$	20.0000	0.973585
$423$	12.0000	0.583460
$424$	6.00000	0.291386
$425$	6.00000	0.291043
$426$	0	0
$427$	−10.0000	−0.483934
$428$	6.00000	0.290021
$429$	0	0
$430$	−8.00000	−0.385794
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	−5.00000	−0.240563
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	−10.0000	−0.480015
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−15.0000	−0.717547
$438$	2.00000	0.0955637
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	30.0000	1.42695
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	−10.0000	−0.473514
$447$	−6.00000	−0.283790
$448$	1.00000	0.0472456
$449$	−3.00000	−0.141579	−0.0707894	−	0.997491i	$-0.522552\pi$
$449$	−3.00000	−0.141579	−0.0707894	+	0.997491i	$0.522552\pi$
$450$	−2.00000	−0.0942809
$451$	0	0
$452$	9.00000	0.423324
$453$	−7.00000	−0.328889
$454$	24.0000	1.12638
$455$	−5.00000	−0.234404
$456$	5.00000	0.234146
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	−22.0000	−1.02799
$459$	−30.0000	−1.40028
$460$	3.00000	0.139876
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	−7.00000	−0.325318	−0.162659	−	0.986682i	$-0.552007\pi$
$463$	−7.00000	−0.325318	−0.162659	+	0.986682i	$0.552007\pi$
$464$	0	0
$465$	10.0000	0.463739
$466$	−21.0000	−0.972806
$467$	−3.00000	−0.138823	−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000	−0.138823	−0.0694117	+	0.997588i	$0.522112\pi$
$468$	−10.0000	−0.462250
$469$	14.0000	0.646460
$470$	6.00000	0.276759
$471$	5.00000	0.230388
$472$	9.00000	0.414259
$473$	0	0
$474$	−1.00000	−0.0459315
$475$	5.00000	0.229416
$476$	6.00000	0.275010
$477$	−12.0000	−0.549442
$478$	−3.00000	−0.137217
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	−1.00000	−0.0456435
$481$	10.0000	0.455961
$482$	−4.00000	−0.182195
$483$	−3.00000	−0.136505
$484$	0	0
$485$	−8.00000	−0.363261
$486$	16.0000	0.725775
$487$	−13.0000	−0.589086	−0.294543	−	0.955638i	$-0.595167\pi$
$487$	−13.0000	−0.589086	−0.294543	+	0.955638i	$0.595167\pi$
$488$	−10.0000	−0.452679
$489$	2.00000	0.0904431
$490$	−1.00000	−0.0451754
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	0	0
$494$	25.0000	1.12480
$495$	0	0
$496$	−10.0000	−0.449013
$497$	0	0
$498$	9.00000	0.403300
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	24.0000	1.07117
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	−2.00000	−0.0890871
$505$	3.00000	0.133498
$506$	0	0
$507$	12.0000	0.532939
$508$	−7.00000	−0.310575
$509$	3.00000	0.132973	0.0664863	−	0.997787i	$-0.478821\pi$
$509$	3.00000	0.132973	0.0664863	+	0.997787i	$0.478821\pi$
$510$	−6.00000	−0.265684
$511$	2.00000	0.0884748
$512$	1.00000	0.0441942
$513$	−25.0000	−1.10378
$514$	6.00000	0.264649
$515$	4.00000	0.176261
$516$	8.00000	0.352180
$517$	0	0
$518$	2.00000	0.0878750
$519$	6.00000	0.263371
$520$	−5.00000	−0.219265
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	0	0
$523$	17.0000	0.743358	0.371679	−	0.928361i	$-0.378782\pi$
$523$	17.0000	0.743358	0.371679	+	0.928361i	$0.378782\pi$
$524$	21.0000	0.917389
$525$	1.00000	0.0436436
$526$	−9.00000	−0.392419
$527$	−60.0000	−2.61364
$528$	0	0
$529$	−14.0000	−0.608696
$530$	−6.00000	−0.260623
$531$	−18.0000	−0.781133
$532$	5.00000	0.216777
$533$	0	0
$534$	0	0
$535$	−6.00000	−0.259403
$536$	14.0000	0.604708
$537$	0	0
$538$	−21.0000	−0.905374
$539$	0	0
$540$	5.00000	0.215166
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	20.0000	0.859074
$543$	−25.0000	−1.07285
$544$	6.00000	0.257248
$545$	10.0000	0.428353
$546$	5.00000	0.213980
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−3.00000	−0.128154
$549$	20.0000	0.853579
$550$	0	0
$551$	0	0
$552$	−3.00000	−0.127688
$553$	−1.00000	−0.0425243
$554$	32.0000	1.35955
$555$	−2.00000	−0.0848953
$556$	−19.0000	−0.805779
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	20.0000	0.846668
$559$	40.0000	1.69182
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−15.0000	−0.632737
$563$	9.00000	0.379305	0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000	0.379305	0.189652	+	0.981851i	$0.439264\pi$
$564$	−6.00000	−0.252646
$565$	−9.00000	−0.378633
$566$	29.0000	1.21896
$567$	1.00000	0.0419961
$568$	0	0
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	−5.00000	−0.209427
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	15.0000	0.626634
$574$	0	0
$575$	−3.00000	−0.125109
$576$	−2.00000	−0.0833333
$577$	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
$579$	11.0000	0.457144
$580$	0	0
$581$	9.00000	0.373383
$582$	8.00000	0.331611
$583$	0	0
$584$	2.00000	0.0827606
$585$	10.0000	0.413449
$586$	27.0000	1.11536
$587$	15.0000	0.619116	0.309558	−	0.950881i	$-0.399819\pi$
$587$	15.0000	0.619116	0.309558	+	0.950881i	$0.399819\pi$
$588$	1.00000	0.0412393
$589$	−50.0000	−2.06021
$590$	−9.00000	−0.370524
$591$	12.0000	0.493614
$592$	2.00000	0.0821995
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	−6.00000	−0.245770
$597$	8.00000	0.327418
$598$	−15.0000	−0.613396
$599$	3.00000	0.122577	0.0612883	−	0.998120i	$-0.480479\pi$
$599$	3.00000	0.122577	0.0612883	+	0.998120i	$0.480479\pi$
$600$	1.00000	0.0408248
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	8.00000	0.326056
$603$	−28.0000	−1.14025
$604$	−7.00000	−0.284826
$605$	0	0
$606$	−3.00000	−0.121867
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	10.0000	0.404888
$611$	−30.0000	−1.21367
$612$	−12.0000	−0.485071
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	0	0
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	−4.00000	−0.160904
$619$	23.0000	0.924448	0.462224	−	0.886763i	$-0.347052\pi$
$619$	23.0000	0.924448	0.462224	+	0.886763i	$0.347052\pi$
$620$	10.0000	0.401610
$621$	15.0000	0.601929
$622$	0	0
$623$	0	0
$624$	5.00000	0.200160
$625$	1.00000	0.0400000
$626$	−34.0000	−1.35891
$627$	0	0
$628$	5.00000	0.199522
$629$	12.0000	0.478471
$630$	2.00000	0.0796819
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	−1.00000	−0.0397779
$633$	20.0000	0.794929
$634$	30.0000	1.19145
$635$	7.00000	0.277787
$636$	6.00000	0.237915
$637$	5.00000	0.198107
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−21.0000	−0.829450	−0.414725	−	0.909947i	$-0.636122\pi$
$641$	−21.0000	−0.829450	−0.414725	+	0.909947i	$0.636122\pi$
$642$	6.00000	0.236801
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	−3.00000	−0.118217
$645$	−8.00000	−0.315000
$646$	30.0000	1.18033
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	5.00000	0.196116
$651$	−10.0000	−0.391931
$652$	2.00000	0.0783260
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	−10.0000	−0.391031
$655$	−21.0000	−0.820538
$656$	0	0
$657$	−4.00000	−0.156055
$658$	−6.00000	−0.233904
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	5.00000	0.194477	0.0972387	−	0.995261i	$-0.468999\pi$
$661$	5.00000	0.194477	0.0972387	+	0.995261i	$0.468999\pi$
$662$	8.00000	0.310929
$663$	30.0000	1.16510
$664$	9.00000	0.349268
$665$	−5.00000	−0.193892
$666$	−4.00000	−0.154997
$667$	0	0
$668$	0	0
$669$	−10.0000	−0.386622
$670$	−14.0000	−0.540867
$671$	0	0
$672$	1.00000	0.0385758
$673$	17.0000	0.655302	0.327651	−	0.944799i	$-0.393743\pi$
$673$	17.0000	0.655302	0.327651	+	0.944799i	$0.393743\pi$
$674$	23.0000	0.885927
$675$	−5.00000	−0.192450
$676$	12.0000	0.461538
$677$	−3.00000	−0.115299	−0.0576497	−	0.998337i	$-0.518361\pi$
$677$	−3.00000	−0.115299	−0.0576497	+	0.998337i	$0.518361\pi$
$678$	9.00000	0.345643
$679$	8.00000	0.307012
$680$	−6.00000	−0.230089
$681$	24.0000	0.919682
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−10.0000	−0.382360
$685$	3.00000	0.114624
$686$	1.00000	0.0381802
$687$	−22.0000	−0.839352
$688$	8.00000	0.304997
$689$	30.0000	1.14291
$690$	3.00000	0.114208
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−6.00000	−0.227757
$695$	19.0000	0.720711
$696$	0	0
$697$	0	0
$698$	11.0000	0.416356
$699$	−21.0000	−0.794293
$700$	1.00000	0.0377964
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	−25.0000	−0.943564
$703$	10.0000	0.377157
$704$	0	0
$705$	6.00000	0.225973
$706$	30.0000	1.12906
$707$	−3.00000	−0.112827
$708$	9.00000	0.338241
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	0	0
$713$	30.0000	1.12351
$714$	6.00000	0.224544
$715$	0	0
$716$	0	0
$717$	−3.00000	−0.112037
$718$	−24.0000	−0.895672
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	2.00000	0.0745356
$721$	−4.00000	−0.148968
$722$	6.00000	0.223297
$723$	−4.00000	−0.148762
$724$	−25.0000	−0.929118
$725$	0	0
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	5.00000	0.185312
$729$	13.0000	0.481481
$730$	−2.00000	−0.0740233
$731$	48.0000	1.77534
$732$	−10.0000	−0.369611
$733$	−13.0000	−0.480166	−0.240083	−	0.970752i	$-0.577175\pi$
$733$	−13.0000	−0.480166	−0.240083	+	0.970752i	$0.577175\pi$
$734$	−16.0000	−0.590571
$735$	−1.00000	−0.0368856
$736$	−3.00000	−0.110581
$737$	0	0
$738$	0	0
$739$	−22.0000	−0.809283	−0.404642	−	0.914475i	$-0.632604\pi$
$739$	−22.0000	−0.809283	−0.404642	+	0.914475i	$0.632604\pi$
$740$	−2.00000	−0.0735215
$741$	25.0000	0.918398
$742$	6.00000	0.220267
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	−10.0000	−0.366618
$745$	6.00000	0.219823
$746$	−22.0000	−0.805477
$747$	−18.0000	−0.658586
$748$	0	0
$749$	6.00000	0.219235
$750$	−1.00000	−0.0365148
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	−6.00000	−0.218797
$753$	24.0000	0.874609
$754$	0	0
$755$	7.00000	0.254756
$756$	−5.00000	−0.181848
$757$	20.0000	0.726912	0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000	0.726912	0.363456	+	0.931611i	$0.381597\pi$
$758$	−34.0000	−1.23494
$759$	0	0
$760$	−5.00000	−0.181369
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	−7.00000	−0.253583
$763$	−10.0000	−0.362024
$764$	15.0000	0.542681
$765$	12.0000	0.433861
$766$	12.0000	0.433578
$767$	45.0000	1.62486
$768$	1.00000	0.0360844
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	11.0000	0.395899
$773$	39.0000	1.40273	0.701366	−	0.712801i	$-0.252574\pi$
$773$	39.0000	1.40273	0.701366	+	0.712801i	$0.252574\pi$
$774$	−16.0000	−0.575108
$775$	−10.0000	−0.359211
$776$	8.00000	0.287183
$777$	2.00000	0.0717496
$778$	30.0000	1.07555
$779$	0	0
$780$	−5.00000	−0.179029
$781$	0	0
$782$	−18.0000	−0.643679
$783$	0	0
$784$	1.00000	0.0357143
$785$	−5.00000	−0.178458
$786$	21.0000	0.749045
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	12.0000	0.427482
$789$	−9.00000	−0.320408
$790$	1.00000	0.0355784
$791$	9.00000	0.320003
$792$	0	0
$793$	−50.0000	−1.77555
$794$	38.0000	1.34857
$795$	−6.00000	−0.212798
$796$	8.00000	0.283552
$797$	33.0000	1.16892	0.584460	−	0.811423i	$-0.301306\pi$
$797$	33.0000	1.16892	0.584460	+	0.811423i	$0.301306\pi$
$798$	5.00000	0.176998
$799$	−36.0000	−1.27359
$800$	1.00000	0.0353553
$801$	0	0
$802$	−30.0000	−1.05934
$803$	0	0
$804$	14.0000	0.493742
$805$	3.00000	0.105736
$806$	−50.0000	−1.76117
$807$	−21.0000	−0.739235
$808$	−3.00000	−0.105540
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	−1.00000	−0.0351364
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	6.00000	0.210042
$817$	40.0000	1.39942
$818$	−28.0000	−0.978997
$819$	−10.0000	−0.349428
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	−3.00000	−0.104637
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	9.00000	0.313150
$827$	−30.0000	−1.04320	−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000	−1.04320	−0.521601	+	0.853189i	$0.674665\pi$
$828$	6.00000	0.208514
$829$	17.0000	0.590434	0.295217	−	0.955430i	$-0.404608\pi$
$829$	17.0000	0.590434	0.295217	+	0.955430i	$0.404608\pi$
$830$	−9.00000	−0.312395
$831$	32.0000	1.11007
$832$	5.00000	0.173344
$833$	6.00000	0.207888
$834$	−19.0000	−0.657916
$835$	0	0
$836$	0	0
$837$	50.0000	1.72825
$838$	21.0000	0.725433
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	−1.00000	−0.0345033
$841$	−29.0000	−1.00000
$842$	−40.0000	−1.37849
$843$	−15.0000	−0.516627
$844$	20.0000	0.688428
$845$	−12.0000	−0.412813
$846$	12.0000	0.412568
$847$	0	0
$848$	6.00000	0.206041
$849$	29.0000	0.995277
$850$	6.00000	0.205798
$851$	−6.00000	−0.205677
$852$	0	0
$853$	−19.0000	−0.650548	−0.325274	−	0.945620i	$-0.605456\pi$
$853$	−19.0000	−0.650548	−0.325274	+	0.945620i	$0.605456\pi$
$854$	−10.0000	−0.342193
$855$	10.0000	0.341993
$856$	6.00000	0.205076
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−3.00000	−0.102180
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	−5.00000	−0.170103
$865$	−6.00000	−0.204006
$866$	−4.00000	−0.135926
$867$	19.0000	0.645274
$868$	−10.0000	−0.339422
$869$	0	0
$870$	0	0
$871$	70.0000	2.37186
$872$	−10.0000	−0.338643
$873$	−16.0000	−0.541518
$874$	−15.0000	−0.507383
$875$	−1.00000	−0.0338062
$876$	2.00000	0.0675737
$877$	−40.0000	−1.35070	−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000	−1.35070	−0.675352	+	0.737496i	$0.736008\pi$
$878$	−16.0000	−0.539974
$879$	27.0000	0.910687
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	−2.00000	−0.0673435
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	30.0000	1.00901
$885$	−9.00000	−0.302532
$886$	−24.0000	−0.806296
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	2.00000	0.0671156
$889$	−7.00000	−0.234772
$890$	0	0
$891$	0	0
$892$	−10.0000	−0.334825
$893$	−30.0000	−1.00391
$894$	−6.00000	−0.200670
$895$	0	0
$896$	1.00000	0.0334077
$897$	−15.0000	−0.500835
$898$	−3.00000	−0.100111
$899$	0	0
$900$	−2.00000	−0.0666667
$901$	36.0000	1.19933
$902$	0	0
$903$	8.00000	0.266223
$904$	9.00000	0.299336
$905$	25.0000	0.831028
$906$	−7.00000	−0.232559
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	24.0000	0.796468
$909$	6.00000	0.199007
$910$	−5.00000	−0.165748
$911$	45.0000	1.49092	0.745458	−	0.666552i	$-0.232231\pi$
$911$	45.0000	1.49092	0.745458	+	0.666552i	$0.232231\pi$
$912$	5.00000	0.165567
$913$	0	0
$914$	−1.00000	−0.0330771
$915$	10.0000	0.330590
$916$	−22.0000	−0.726900
$917$	21.0000	0.693481
$918$	−30.0000	−0.990148
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	3.00000	0.0989071
$921$	−4.00000	−0.131804
$922$	30.0000	0.987997
$923$	0	0
$924$	0	0
$925$	2.00000	0.0657596
$926$	−7.00000	−0.230034
$927$	8.00000	0.262754
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	10.0000	0.327913
$931$	5.00000	0.163868
$932$	−21.0000	−0.687878
$933$	0	0
$934$	−3.00000	−0.0981630
$935$	0	0
$936$	−10.0000	−0.326860
$937$	−52.0000	−1.69877	−0.849383	−	0.527777i	$-0.823026\pi$
$937$	−52.0000	−1.69877	−0.849383	+	0.527777i	$0.823026\pi$
$938$	14.0000	0.457116
$939$	−34.0000	−1.10955
$940$	6.00000	0.195698
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	5.00000	0.162909
$943$	0	0
$944$	9.00000	0.292925
$945$	5.00000	0.162650
$946$	0	0
$947$	6.00000	0.194974	0.0974869	−	0.995237i	$-0.468920\pi$
$947$	6.00000	0.194974	0.0974869	+	0.995237i	$0.468920\pi$
$948$	−1.00000	−0.0324785
$949$	10.0000	0.324614
$950$	5.00000	0.162221
$951$	30.0000	0.972817
$952$	6.00000	0.194461
$953$	−57.0000	−1.84641	−0.923206	−	0.384307i	$-0.874441\pi$
$953$	−57.0000	−1.84641	−0.923206	+	0.384307i	$0.874441\pi$
$954$	−12.0000	−0.388514
$955$	−15.0000	−0.485389
$956$	−3.00000	−0.0970269
$957$	0	0
$958$	−36.0000	−1.16311
$959$	−3.00000	−0.0968751
$960$	−1.00000	−0.0322749
$961$	69.0000	2.22581
$962$	10.0000	0.322413
$963$	−12.0000	−0.386695
$964$	−4.00000	−0.128831
$965$	−11.0000	−0.354103
$966$	−3.00000	−0.0965234
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	30.0000	0.963739
$970$	−8.00000	−0.256865
$971$	−9.00000	−0.288824	−0.144412	−	0.989518i	$-0.546129\pi$
$971$	−9.00000	−0.288824	−0.144412	+	0.989518i	$0.546129\pi$
$972$	16.0000	0.513200
$973$	−19.0000	−0.609112
$974$	−13.0000	−0.416547
$975$	5.00000	0.160128
$976$	−10.0000	−0.320092
$977$	−27.0000	−0.863807	−0.431903	−	0.901920i	$-0.642158\pi$
$977$	−27.0000	−0.863807	−0.431903	+	0.901920i	$0.642158\pi$
$978$	2.00000	0.0639529
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	20.0000	0.638551
$982$	−6.00000	−0.191468
$983$	−42.0000	−1.33959	−0.669796	−	0.742545i	$-0.733618\pi$
$983$	−42.0000	−1.33959	−0.669796	+	0.742545i	$0.733618\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	0	0
$987$	−6.00000	−0.190982
$988$	25.0000	0.795356
$989$	−24.0000	−0.763156
$990$	0	0
$991$	35.0000	1.11181	0.555906	−	0.831245i	$-0.312372\pi$
$991$	35.0000	1.11181	0.555906	+	0.831245i	$0.312372\pi$
$992$	−10.0000	−0.317500
$993$	8.00000	0.253872
$994$	0	0
$995$	−8.00000	−0.253617
$996$	9.00000	0.285176
$997$	−25.0000	−0.791758	−0.395879	−	0.918303i	$-0.629560\pi$
$997$	−25.0000	−0.791758	−0.395879	+	0.918303i	$0.629560\pi$
$998$	−34.0000	−1.07625
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.bd.1.1	yes	1
11.10	odd	2		8470.2.a.k.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.k.1.1	✓	1	11.10	odd	2
8470.2.a.bd.1.1	yes	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

Related objects

Downloads

Learn more