LMFDB - Embedded newform 8470.2.a.q.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:	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)

$f(q)$

$=$

$q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +1.00000 q^{10} -3.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} +6.00000 q^{18} -7.00000 q^{19} +1.00000 q^{20} -3.00000 q^{21} +1.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} -9.00000 q^{27} +1.00000 q^{28} +8.00000 q^{29} -3.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{35} +6.00000 q^{36} +2.00000 q^{37} -7.00000 q^{38} -3.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -3.00000 q^{42} -6.00000 q^{43} +6.00000 q^{45} +1.00000 q^{46} -12.0000 q^{47} -3.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} -12.0000 q^{53} -9.00000 q^{54} +1.00000 q^{56} +21.0000 q^{57} +8.00000 q^{58} +3.00000 q^{59} -3.00000 q^{60} -6.00000 q^{61} -4.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} +1.00000 q^{65} +8.00000 q^{67} -3.00000 q^{69} +1.00000 q^{70} -8.00000 q^{71} +6.00000 q^{72} +16.0000 q^{73} +2.00000 q^{74} -3.00000 q^{75} -7.00000 q^{76} -3.00000 q^{78} -9.00000 q^{79} +1.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} -13.0000 q^{83} -3.00000 q^{84} -6.00000 q^{86} -24.0000 q^{87} +6.00000 q^{89} +6.00000 q^{90} +1.00000 q^{91} +1.00000 q^{92} +12.0000 q^{93} -12.0000 q^{94} -7.00000 q^{95} -3.00000 q^{96} -8.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−3.00000	−1.22474
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	6.00000	2.00000
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−3.00000	−0.866025
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	1.00000	0.267261
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	6.00000	1.41421
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	1.00000	0.223607
$21$	−3.00000	−0.654654
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	−3.00000	−0.612372
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	−9.00000	−1.73205
$28$	1.00000	0.188982
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	−3.00000	−0.547723
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	6.00000	1.00000
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−7.00000	−1.13555
$39$	−3.00000	−0.480384
$40$	1.00000	0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−3.00000	−0.462910
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	6.00000	0.894427
$46$	1.00000	0.147442
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	−3.00000	−0.433013
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	1.00000	0.138675
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	−9.00000	−1.22474
$55$	0	0
$56$	1.00000	0.133631
$57$	21.0000	2.78152
$58$	8.00000	1.05045
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	−3.00000	−0.387298
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−4.00000	−0.508001
$63$	6.00000	0.755929
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	1.00000	0.119523
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	6.00000	0.707107
$73$	16.0000	1.87266	0.936329	−	0.351123i	$-0.114200\pi$
$73$	16.0000	1.87266	0.936329	+	0.351123i	$0.114200\pi$
$74$	2.00000	0.232495
$75$	−3.00000	−0.346410
$76$	−7.00000	−0.802955
$77$	0	0
$78$	−3.00000	−0.339683
$79$	−9.00000	−1.01258	−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000	−1.01258	−0.506290	+	0.862364i	$0.668983\pi$
$80$	1.00000	0.111803
$81$	9.00000	1.00000
$82$	−6.00000	−0.662589
$83$	−13.0000	−1.42694	−0.713468	−	0.700688i	$-0.752876\pi$
$83$	−13.0000	−1.42694	−0.713468	+	0.700688i	$0.752876\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	−6.00000	−0.646997
$87$	−24.0000	−2.57307
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	6.00000	0.632456
$91$	1.00000	0.104828
$92$	1.00000	0.104257
$93$	12.0000	1.24434
$94$	−12.0000	−1.23771
$95$	−7.00000	−0.718185
$96$	−3.00000	−0.306186
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	1.00000	0.0980581
$105$	−3.00000	−0.292770
$106$	−12.0000	−1.16554
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	−9.00000	−0.866025
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	1.00000	0.0944911
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	21.0000	1.96683
$115$	1.00000	0.0932505
$116$	8.00000	0.742781
$117$	6.00000	0.554700
$118$	3.00000	0.276172
$119$	0	0
$120$	−3.00000	−0.273861
$121$	0	0
$122$	−6.00000	−0.543214
$123$	18.0000	1.62301
$124$	−4.00000	−0.359211
$125$	1.00000	0.0894427
$126$	6.00000	0.534522
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	1.00000	0.0883883
$129$	18.0000	1.58481
$130$	1.00000	0.0877058
$131$	−11.0000	−0.961074	−0.480537	−	0.876974i	$-0.659558\pi$
$131$	−11.0000	−0.961074	−0.480537	+	0.876974i	$0.659558\pi$
$132$	0	0
$133$	−7.00000	−0.606977
$134$	8.00000	0.691095
$135$	−9.00000	−0.774597
$136$	0	0
$137$	−5.00000	−0.427179	−0.213589	−	0.976924i	$-0.568515\pi$
$137$	−5.00000	−0.427179	−0.213589	+	0.976924i	$0.568515\pi$
$138$	−3.00000	−0.255377
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	1.00000	0.0845154
$141$	36.0000	3.03175
$142$	−8.00000	−0.671345
$143$	0	0
$144$	6.00000	0.500000
$145$	8.00000	0.664364
$146$	16.0000	1.32417
$147$	−3.00000	−0.247436
$148$	2.00000	0.164399
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−3.00000	−0.244949
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	−7.00000	−0.567775
$153$	0	0
$154$	0	0
$155$	−4.00000	−0.321288
$156$	−3.00000	−0.240192
$157$	−21.0000	−1.67598	−0.837991	−	0.545684i	$-0.816270\pi$
$157$	−21.0000	−1.67598	−0.837991	+	0.545684i	$0.816270\pi$
$158$	−9.00000	−0.716002
$159$	36.0000	2.85499
$160$	1.00000	0.0790569
$161$	1.00000	0.0788110
$162$	9.00000	0.707107
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−13.0000	−1.00900
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	−3.00000	−0.231455
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−42.0000	−3.21182
$172$	−6.00000	−0.457496
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	−24.0000	−1.81944
$175$	1.00000	0.0755929
$176$	0	0
$177$	−9.00000	−0.676481
$178$	6.00000	0.449719
$179$	22.0000	1.64436	0.822179	−	0.569230i	$-0.192758\pi$
$179$	22.0000	1.64436	0.822179	+	0.569230i	$0.192758\pi$
$180$	6.00000	0.447214
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	1.00000	0.0741249
$183$	18.0000	1.33060
$184$	1.00000	0.0737210
$185$	2.00000	0.147043
$186$	12.0000	0.879883
$187$	0	0
$188$	−12.0000	−0.875190
$189$	−9.00000	−0.654654
$190$	−7.00000	−0.507833
$191$	9.00000	0.651217	0.325609	−	0.945505i	$-0.394431\pi$
$191$	9.00000	0.651217	0.325609	+	0.945505i	$0.394431\pi$
$192$	−3.00000	−0.216506
$193$	−13.0000	−0.935760	−0.467880	−	0.883792i	$-0.654982\pi$
$193$	−13.0000	−0.935760	−0.467880	+	0.883792i	$0.654982\pi$
$194$	−8.00000	−0.574367
$195$	−3.00000	−0.214834
$196$	1.00000	0.0714286
$197$	−4.00000	−0.284988	−0.142494	−	0.989796i	$-0.545512\pi$
$197$	−4.00000	−0.284988	−0.142494	+	0.989796i	$0.545512\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	1.00000	0.0707107
$201$	−24.0000	−1.69283
$202$	15.0000	1.05540
$203$	8.00000	0.561490
$204$	0	0
$205$	−6.00000	−0.419058
$206$	6.00000	0.418040
$207$	6.00000	0.417029
$208$	1.00000	0.0693375
$209$	0	0
$210$	−3.00000	−0.207020
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	−12.0000	−0.824163
$213$	24.0000	1.64445
$214$	2.00000	0.136717
$215$	−6.00000	−0.409197
$216$	−9.00000	−0.612372
$217$	−4.00000	−0.271538
$218$	0	0
$219$	−48.0000	−3.24354
$220$	0	0
$221$	0	0
$222$	−6.00000	−0.402694
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	1.00000	0.0668153
$225$	6.00000	0.400000
$226$	−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	$0.630377\pi$
$228$	21.0000	1.39076
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	1.00000	0.0659380
$231$	0	0
$232$	8.00000	0.525226
$233$	−29.0000	−1.89985	−0.949927	−	0.312473i	$-0.898843\pi$
$233$	−29.0000	−1.89985	−0.949927	+	0.312473i	$0.898843\pi$
$234$	6.00000	0.392232
$235$	−12.0000	−0.782794
$236$	3.00000	0.195283
$237$	27.0000	1.75384
$238$	0	0
$239$	17.0000	1.09964	0.549819	−	0.835284i	$-0.314697\pi$
$239$	17.0000	1.09964	0.549819	+	0.835284i	$0.314697\pi$
$240$	−3.00000	−0.193649
$241$	−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
$243$	0	0
$244$	−6.00000	−0.384111
$245$	1.00000	0.0638877
$246$	18.0000	1.14764
$247$	−7.00000	−0.445399
$248$	−4.00000	−0.254000
$249$	39.0000	2.47152
$250$	1.00000	0.0632456
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	6.00000	0.377964
$253$	0	0
$254$	7.00000	0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	18.0000	1.12063
$259$	2.00000	0.124274
$260$	1.00000	0.0620174
$261$	48.0000	2.97113
$262$	−11.0000	−0.679582
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	−7.00000	−0.429198
$267$	−18.0000	−1.10158
$268$	8.00000	0.488678
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	−9.00000	−0.547723
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	0	0
$273$	−3.00000	−0.181568
$274$	−5.00000	−0.302061
$275$	0	0
$276$	−3.00000	−0.180579
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	5.00000	0.299880
$279$	−24.0000	−1.43684
$280$	1.00000	0.0597614
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	36.0000	2.14377
$283$	7.00000	0.416107	0.208053	−	0.978117i	$-0.433287\pi$
$283$	7.00000	0.416107	0.208053	+	0.978117i	$0.433287\pi$
$284$	−8.00000	−0.474713
$285$	21.0000	1.24393
$286$	0	0
$287$	−6.00000	−0.354169
$288$	6.00000	0.353553
$289$	−17.0000	−1.00000
$290$	8.00000	0.469776
$291$	24.0000	1.40690
$292$	16.0000	0.936329
$293$	19.0000	1.10999	0.554996	−	0.831853i	$-0.312720\pi$
$293$	19.0000	1.10999	0.554996	+	0.831853i	$0.312720\pi$
$294$	−3.00000	−0.174964
$295$	3.00000	0.174667
$296$	2.00000	0.116248
$297$	0	0
$298$	18.0000	1.04271
$299$	1.00000	0.0578315
$300$	−3.00000	−0.173205
$301$	−6.00000	−0.345834
$302$	−7.00000	−0.402805
$303$	−45.0000	−2.58518
$304$	−7.00000	−0.401478
$305$	−6.00000	−0.343559
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	−18.0000	−1.02398
$310$	−4.00000	−0.227185
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	−3.00000	−0.169842
$313$	−18.0000	−1.01742	−0.508710	−	0.860938i	$-0.669877\pi$
$313$	−18.0000	−1.01742	−0.508710	+	0.860938i	$0.669877\pi$
$314$	−21.0000	−1.18510
$315$	6.00000	0.338062
$316$	−9.00000	−0.506290
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	36.0000	2.01878
$319$	0	0
$320$	1.00000	0.0559017
$321$	−6.00000	−0.334887
$322$	1.00000	0.0557278
$323$	0	0
$324$	9.00000	0.500000
$325$	1.00000	0.0554700
$326$	6.00000	0.332309
$327$	0	0
$328$	−6.00000	−0.331295
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−22.0000	−1.20923	−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000	−1.20923	−0.604615	+	0.796518i	$0.706673\pi$
$332$	−13.0000	−0.713468
$333$	12.0000	0.657596
$334$	−2.00000	−0.109435
$335$	8.00000	0.437087
$336$	−3.00000	−0.163663
$337$	−25.0000	−1.36184	−0.680918	−	0.732359i	$-0.738419\pi$
$337$	−25.0000	−1.36184	−0.680918	+	0.732359i	$0.738419\pi$
$338$	−12.0000	−0.652714
$339$	27.0000	1.46644
$340$	0	0
$341$	0	0
$342$	−42.0000	−2.27110
$343$	1.00000	0.0539949
$344$	−6.00000	−0.323498
$345$	−3.00000	−0.161515
$346$	−14.0000	−0.752645
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	−24.0000	−1.28654
$349$	−23.0000	−1.23116	−0.615581	−	0.788074i	$-0.711079\pi$
$349$	−23.0000	−1.23116	−0.615581	+	0.788074i	$0.711079\pi$
$350$	1.00000	0.0534522
$351$	−9.00000	−0.480384
$352$	0	0
$353$	36.0000	1.91609	0.958043	−	0.286623i	$-0.0925328\pi$
$353$	36.0000	1.91609	0.958043	+	0.286623i	$0.0925328\pi$
$354$	−9.00000	−0.478345
$355$	−8.00000	−0.424596
$356$	6.00000	0.317999
$357$	0	0
$358$	22.0000	1.16274
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	6.00000	0.316228
$361$	30.0000	1.57895
$362$	11.0000	0.578147
$363$	0	0
$364$	1.00000	0.0524142
$365$	16.0000	0.837478
$366$	18.0000	0.940875
$367$	−30.0000	−1.56599	−0.782994	−	0.622030i	$-0.786308\pi$
$367$	−30.0000	−1.56599	−0.782994	+	0.622030i	$0.786308\pi$
$368$	1.00000	0.0521286
$369$	−36.0000	−1.87409
$370$	2.00000	0.103975
$371$	−12.0000	−0.623009
$372$	12.0000	0.622171
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	−12.0000	−0.618853
$377$	8.00000	0.412021
$378$	−9.00000	−0.462910
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	−7.00000	−0.359092
$381$	−21.0000	−1.07586
$382$	9.00000	0.460480
$383$	−38.0000	−1.94171	−0.970855	−	0.239669i	$-0.922961\pi$
$383$	−38.0000	−1.94171	−0.970855	+	0.239669i	$0.922961\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	−13.0000	−0.661683
$387$	−36.0000	−1.82998
$388$	−8.00000	−0.406138
$389$	−28.0000	−1.41966	−0.709828	−	0.704375i	$-0.751227\pi$
$389$	−28.0000	−1.41966	−0.709828	+	0.704375i	$0.751227\pi$
$390$	−3.00000	−0.151911
$391$	0	0
$392$	1.00000	0.0505076
$393$	33.0000	1.66463
$394$	−4.00000	−0.201517
$395$	−9.00000	−0.452839
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	4.00000	0.200502
$399$	21.0000	1.05131
$400$	1.00000	0.0500000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	−24.0000	−1.19701
$403$	−4.00000	−0.199254
$404$	15.0000	0.746278
$405$	9.00000	0.447214
$406$	8.00000	0.397033
$407$	0	0
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	−6.00000	−0.296319
$411$	15.0000	0.739895
$412$	6.00000	0.295599
$413$	3.00000	0.147620
$414$	6.00000	0.294884
$415$	−13.0000	−0.638145
$416$	1.00000	0.0490290
$417$	−15.0000	−0.734553
$418$	0	0
$419$	3.00000	0.146560	0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000	0.146560	0.0732798	+	0.997311i	$0.476653\pi$
$420$	−3.00000	−0.146385
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	16.0000	0.778868
$423$	−72.0000	−3.50076
$424$	−12.0000	−0.582772
$425$	0	0
$426$	24.0000	1.16280
$427$	−6.00000	−0.290360
$428$	2.00000	0.0966736
$429$	0	0
$430$	−6.00000	−0.289346
$431$	−7.00000	−0.337178	−0.168589	−	0.985686i	$-0.553921\pi$
$431$	−7.00000	−0.337178	−0.168589	+	0.985686i	$0.553921\pi$
$432$	−9.00000	−0.433013
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	−4.00000	−0.192006
$435$	−24.0000	−1.15071
$436$	0	0
$437$	−7.00000	−0.334855
$438$	−48.0000	−2.29353
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	−6.00000	−0.284747
$445$	6.00000	0.284427
$446$	0	0
$447$	−54.0000	−2.55411
$448$	1.00000	0.0472456
$449$	−7.00000	−0.330350	−0.165175	−	0.986264i	$-0.552819\pi$
$449$	−7.00000	−0.330350	−0.165175	+	0.986264i	$0.552819\pi$
$450$	6.00000	0.282843
$451$	0	0
$452$	−9.00000	−0.423324
$453$	21.0000	0.986666
$454$	−12.0000	−0.563188
$455$	1.00000	0.0468807
$456$	21.0000	0.983415
$457$	−25.0000	−1.16945	−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000	−1.16945	−0.584725	+	0.811231i	$0.698798\pi$
$458$	2.00000	0.0934539
$459$	0	0
$460$	1.00000	0.0466252
$461$	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
$463$	−23.0000	−1.06890	−0.534450	−	0.845200i	$-0.679481\pi$
$463$	−23.0000	−1.06890	−0.534450	+	0.845200i	$0.679481\pi$
$464$	8.00000	0.371391
$465$	12.0000	0.556487
$466$	−29.0000	−1.34340
$467$	−27.0000	−1.24941	−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000	−1.24941	−0.624705	+	0.780860i	$0.714781\pi$
$468$	6.00000	0.277350
$469$	8.00000	0.369406
$470$	−12.0000	−0.553519
$471$	63.0000	2.90289
$472$	3.00000	0.138086
$473$	0	0
$474$	27.0000	1.24015
$475$	−7.00000	−0.321182
$476$	0	0
$477$	−72.0000	−3.29665
$478$	17.0000	0.777562
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	−3.00000	−0.136931
$481$	2.00000	0.0911922
$482$	−20.0000	−0.910975
$483$	−3.00000	−0.136505
$484$	0	0
$485$	−8.00000	−0.363261
$486$	0	0
$487$	3.00000	0.135943	0.0679715	−	0.997687i	$-0.478347\pi$
$487$	3.00000	0.135943	0.0679715	+	0.997687i	$0.478347\pi$
$488$	−6.00000	−0.271607
$489$	−18.0000	−0.813988
$490$	1.00000	0.0451754
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	18.0000	0.811503
$493$	0	0
$494$	−7.00000	−0.314945
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−8.00000	−0.358849
$498$	39.0000	1.74763
$499$	−18.0000	−0.805791	−0.402895	−	0.915246i	$-0.631996\pi$
$499$	−18.0000	−0.805791	−0.402895	+	0.915246i	$0.631996\pi$
$500$	1.00000	0.0447214
$501$	6.00000	0.268060
$502$	−12.0000	−0.535586
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	6.00000	0.267261
$505$	15.0000	0.667491
$506$	0	0
$507$	36.0000	1.59882
$508$	7.00000	0.310575
$509$	−45.0000	−1.99459	−0.997295	−	0.0735034i	$-0.976582\pi$
$509$	−45.0000	−1.99459	−0.997295	+	0.0735034i	$0.976582\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	1.00000	0.0441942
$513$	63.0000	2.78152
$514$	12.0000	0.529297
$515$	6.00000	0.264392
$516$	18.0000	0.792406
$517$	0	0
$518$	2.00000	0.0878750
$519$	42.0000	1.84360
$520$	1.00000	0.0438529
$521$	40.0000	1.75243	0.876216	−	0.481919i	$-0.160060\pi$
$521$	40.0000	1.75243	0.876216	+	0.481919i	$0.160060\pi$
$522$	48.0000	2.10090
$523$	23.0000	1.00572	0.502860	−	0.864368i	$-0.332281\pi$
$523$	23.0000	1.00572	0.502860	+	0.864368i	$0.332281\pi$
$524$	−11.0000	−0.480537
$525$	−3.00000	−0.130931
$526$	9.00000	0.392419
$527$	0	0
$528$	0	0
$529$	−22.0000	−0.956522
$530$	−12.0000	−0.521247
$531$	18.0000	0.781133
$532$	−7.00000	−0.303488
$533$	−6.00000	−0.259889
$534$	−18.0000	−0.778936
$535$	2.00000	0.0864675
$536$	8.00000	0.345547
$537$	−66.0000	−2.84811
$538$	3.00000	0.129339
$539$	0	0
$540$	−9.00000	−0.387298
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	−24.0000	−1.03089
$543$	−33.0000	−1.41617
$544$	0	0
$545$	0	0
$546$	−3.00000	−0.128388
$547$	−42.0000	−1.79579	−0.897895	−	0.440209i	$-0.854904\pi$
$547$	−42.0000	−1.79579	−0.897895	+	0.440209i	$0.854904\pi$
$548$	−5.00000	−0.213589
$549$	−36.0000	−1.53644
$550$	0	0
$551$	−56.0000	−2.38568
$552$	−3.00000	−0.127688
$553$	−9.00000	−0.382719
$554$	2.00000	0.0849719
$555$	−6.00000	−0.254686
$556$	5.00000	0.212047
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	−24.0000	−1.01600
$559$	−6.00000	−0.253773
$560$	1.00000	0.0422577
$561$	0	0
$562$	3.00000	0.126547
$563$	7.00000	0.295015	0.147507	−	0.989061i	$-0.452875\pi$
$563$	7.00000	0.295015	0.147507	+	0.989061i	$0.452875\pi$
$564$	36.0000	1.51587
$565$	−9.00000	−0.378633
$566$	7.00000	0.294232
$567$	9.00000	0.377964
$568$	−8.00000	−0.335673
$569$	1.00000	0.0419222	0.0209611	−	0.999780i	$-0.493327\pi$
$569$	1.00000	0.0419222	0.0209611	+	0.999780i	$0.493327\pi$
$570$	21.0000	0.879593
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	−27.0000	−1.12794
$574$	−6.00000	−0.250435
$575$	1.00000	0.0417029
$576$	6.00000	0.250000
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−17.0000	−0.707107
$579$	39.0000	1.62078
$580$	8.00000	0.332182
$581$	−13.0000	−0.539331
$582$	24.0000	0.994832
$583$	0	0
$584$	16.0000	0.662085
$585$	6.00000	0.248069
$586$	19.0000	0.784883
$587$	−37.0000	−1.52715	−0.763577	−	0.645717i	$-0.776559\pi$
$587$	−37.0000	−1.52715	−0.763577	+	0.645717i	$0.776559\pi$
$588$	−3.00000	−0.123718
$589$	28.0000	1.15372
$590$	3.00000	0.123508
$591$	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
$595$	0	0
$596$	18.0000	0.737309
$597$	−12.0000	−0.491127
$598$	1.00000	0.0408930
$599$	−15.0000	−0.612883	−0.306442	−	0.951889i	$-0.599138\pi$
$599$	−15.0000	−0.612883	−0.306442	+	0.951889i	$0.599138\pi$
$600$	−3.00000	−0.122474
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	−6.00000	−0.244542
$603$	48.0000	1.95471
$604$	−7.00000	−0.284826
$605$	0	0
$606$	−45.0000	−1.82800
$607$	−2.00000	−0.0811775	−0.0405887	−	0.999176i	$-0.512923\pi$
$607$	−2.00000	−0.0811775	−0.0405887	+	0.999176i	$0.512923\pi$
$608$	−7.00000	−0.283887
$609$	−24.0000	−0.972529
$610$	−6.00000	−0.242933
$611$	−12.0000	−0.485468
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	−28.0000	−1.12999
$615$	18.0000	0.725830
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−18.0000	−0.724066
$619$	1.00000	0.0401934	0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000	0.0401934	0.0200967	+	0.999798i	$0.493603\pi$
$620$	−4.00000	−0.160644
$621$	−9.00000	−0.361158
$622$	30.0000	1.20289
$623$	6.00000	0.240385
$624$	−3.00000	−0.120096
$625$	1.00000	0.0400000
$626$	−18.0000	−0.719425
$627$	0	0
$628$	−21.0000	−0.837991
$629$	0	0
$630$	6.00000	0.239046
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	−9.00000	−0.358001
$633$	−48.0000	−1.90783
$634$	−12.0000	−0.476581
$635$	7.00000	0.277787
$636$	36.0000	1.42749
$637$	1.00000	0.0396214
$638$	0	0
$639$	−48.0000	−1.89885
$640$	1.00000	0.0395285
$641$	−41.0000	−1.61940	−0.809701	−	0.586842i	$-0.800371\pi$
$641$	−41.0000	−1.61940	−0.809701	+	0.586842i	$0.800371\pi$
$642$	−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	$0.789257\pi$
$644$	1.00000	0.0394055
$645$	18.0000	0.708749
$646$	0	0
$647$	2.00000	0.0786281	0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000	0.0786281	0.0393141	+	0.999227i	$0.487483\pi$
$648$	9.00000	0.353553
$649$	0	0
$650$	1.00000	0.0392232
$651$	12.0000	0.470317
$652$	6.00000	0.234978
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	0	0
$655$	−11.0000	−0.429806
$656$	−6.00000	−0.234261
$657$	96.0000	3.74532
$658$	−12.0000	−0.467809
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	29.0000	1.12797	0.563985	−	0.825785i	$-0.309268\pi$
$661$	29.0000	1.12797	0.563985	+	0.825785i	$0.309268\pi$
$662$	−22.0000	−0.855054
$663$	0	0
$664$	−13.0000	−0.504498
$665$	−7.00000	−0.271448
$666$	12.0000	0.464991
$667$	8.00000	0.309761
$668$	−2.00000	−0.0773823
$669$	0	0
$670$	8.00000	0.309067
$671$	0	0
$672$	−3.00000	−0.115728
$673$	17.0000	0.655302	0.327651	−	0.944799i	$-0.393743\pi$
$673$	17.0000	0.655302	0.327651	+	0.944799i	$0.393743\pi$
$674$	−25.0000	−0.962964
$675$	−9.00000	−0.346410
$676$	−12.0000	−0.461538
$677$	−23.0000	−0.883962	−0.441981	−	0.897024i	$-0.645724\pi$
$677$	−23.0000	−0.883962	−0.441981	+	0.897024i	$0.645724\pi$
$678$	27.0000	1.03693
$679$	−8.00000	−0.307012
$680$	0	0
$681$	36.0000	1.37952
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	−42.0000	−1.60591
$685$	−5.00000	−0.191040
$686$	1.00000	0.0381802
$687$	−6.00000	−0.228914
$688$	−6.00000	−0.228748
$689$	−12.0000	−0.457164
$690$	−3.00000	−0.114208
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	18.0000	0.683271
$695$	5.00000	0.189661
$696$	−24.0000	−0.909718
$697$	0	0
$698$	−23.0000	−0.870563
$699$	87.0000	3.29064
$700$	1.00000	0.0377964
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	−9.00000	−0.339683
$703$	−14.0000	−0.528020
$704$	0	0
$705$	36.0000	1.35584
$706$	36.0000	1.35488
$707$	15.0000	0.564133
$708$	−9.00000	−0.338241
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	−8.00000	−0.300235
$711$	−54.0000	−2.02516
$712$	6.00000	0.224860
$713$	−4.00000	−0.149801
$714$	0	0
$715$	0	0
$716$	22.0000	0.822179
$717$	−51.0000	−1.90463
$718$	−24.0000	−0.895672
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	6.00000	0.223607
$721$	6.00000	0.223452
$722$	30.0000	1.11648
$723$	60.0000	2.23142
$724$	11.0000	0.408812
$725$	8.00000	0.297113
$726$	0	0
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	1.00000	0.0370625
$729$	−27.0000	−1.00000
$730$	16.0000	0.592187
$731$	0	0
$732$	18.0000	0.665299
$733$	−17.0000	−0.627909	−0.313955	−	0.949438i	$-0.601654\pi$
$733$	−17.0000	−0.627909	−0.313955	+	0.949438i	$0.601654\pi$
$734$	−30.0000	−1.10732
$735$	−3.00000	−0.110657
$736$	1.00000	0.0368605
$737$	0	0
$738$	−36.0000	−1.32518
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	2.00000	0.0735215
$741$	21.0000	0.771454
$742$	−12.0000	−0.440534
$743$	20.0000	0.733729	0.366864	−	0.930274i	$-0.380431\pi$
$743$	20.0000	0.733729	0.366864	+	0.930274i	$0.380431\pi$
$744$	12.0000	0.439941
$745$	18.0000	0.659469
$746$	18.0000	0.659027
$747$	−78.0000	−2.85387
$748$	0	0
$749$	2.00000	0.0730784
$750$	−3.00000	−0.109545
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	−12.0000	−0.437595
$753$	36.0000	1.31191
$754$	8.00000	0.291343
$755$	−7.00000	−0.254756
$756$	−9.00000	−0.327327
$757$	−14.0000	−0.508839	−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000	−0.508839	−0.254419	+	0.967094i	$0.581884\pi$
$758$	−14.0000	−0.508503
$759$	0	0
$760$	−7.00000	−0.253917
$761$	52.0000	1.88500	0.942499	−	0.334208i	$-0.108469\pi$
$761$	52.0000	1.88500	0.942499	+	0.334208i	$0.108469\pi$
$762$	−21.0000	−0.760750
$763$	0	0
$764$	9.00000	0.325609
$765$	0	0
$766$	−38.0000	−1.37300
$767$	3.00000	0.108324
$768$	−3.00000	−0.108253
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	−36.0000	−1.29651
$772$	−13.0000	−0.467880
$773$	−39.0000	−1.40273	−0.701366	−	0.712801i	$-0.747426\pi$
$773$	−39.0000	−1.40273	−0.701366	+	0.712801i	$0.747426\pi$
$774$	−36.0000	−1.29399
$775$	−4.00000	−0.143684
$776$	−8.00000	−0.287183
$777$	−6.00000	−0.215249
$778$	−28.0000	−1.00385
$779$	42.0000	1.50481
$780$	−3.00000	−0.107417
$781$	0	0
$782$	0	0
$783$	−72.0000	−2.57307
$784$	1.00000	0.0357143
$785$	−21.0000	−0.749522
$786$	33.0000	1.17707
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−4.00000	−0.142494
$789$	−27.0000	−0.961225
$790$	−9.00000	−0.320206
$791$	−9.00000	−0.320003
$792$	0	0
$793$	−6.00000	−0.213066
$794$	−22.0000	−0.780751
$795$	36.0000	1.27679
$796$	4.00000	0.141776
$797$	23.0000	0.814702	0.407351	−	0.913272i	$-0.366453\pi$
$797$	23.0000	0.814702	0.407351	+	0.913272i	$0.366453\pi$
$798$	21.0000	0.743392
$799$	0	0
$800$	1.00000	0.0353553
$801$	36.0000	1.27200
$802$	−10.0000	−0.353112
$803$	0	0
$804$	−24.0000	−0.846415
$805$	1.00000	0.0352454
$806$	−4.00000	−0.140894
$807$	−9.00000	−0.316815
$808$	15.0000	0.527698
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	9.00000	0.316228
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	8.00000	0.280745
$813$	72.0000	2.52515
$814$	0	0
$815$	6.00000	0.210171
$816$	0	0
$817$	42.0000	1.46939
$818$	−4.00000	−0.139857
$819$	6.00000	0.209657
$820$	−6.00000	−0.209529
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	15.0000	0.523185
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	3.00000	0.104383
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	6.00000	0.208514
$829$	33.0000	1.14614	0.573069	−	0.819507i	$-0.305753\pi$
$829$	33.0000	1.14614	0.573069	+	0.819507i	$0.305753\pi$
$830$	−13.0000	−0.451237
$831$	−6.00000	−0.208138
$832$	1.00000	0.0346688
$833$	0	0
$834$	−15.0000	−0.519408
$835$	−2.00000	−0.0692129
$836$	0	0
$837$	36.0000	1.24434
$838$	3.00000	0.103633
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	−3.00000	−0.103510
$841$	35.0000	1.20690
$842$	32.0000	1.10279
$843$	−9.00000	−0.309976
$844$	16.0000	0.550743
$845$	−12.0000	−0.412813
$846$	−72.0000	−2.47541
$847$	0	0
$848$	−12.0000	−0.412082
$849$	−21.0000	−0.720718
$850$	0	0
$851$	2.00000	0.0685591
$852$	24.0000	0.822226
$853$	49.0000	1.67773	0.838864	−	0.544341i	$-0.183220\pi$
$853$	49.0000	1.67773	0.838864	+	0.544341i	$0.183220\pi$
$854$	−6.00000	−0.205316
$855$	−42.0000	−1.43637
$856$	2.00000	0.0683586
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	−6.00000	−0.204598
$861$	18.0000	0.613438
$862$	−7.00000	−0.238421
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	−9.00000	−0.306186
$865$	−14.0000	−0.476014
$866$	14.0000	0.475739
$867$	51.0000	1.73205
$868$	−4.00000	−0.135769
$869$	0	0
$870$	−24.0000	−0.813676
$871$	8.00000	0.271070
$872$	0	0
$873$	−48.0000	−1.62455
$874$	−7.00000	−0.236779
$875$	1.00000	0.0338062
$876$	−48.0000	−1.62177
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	10.0000	0.337484
$879$	−57.0000	−1.92256
$880$	0	0
$881$	38.0000	1.28025	0.640126	−	0.768270i	$-0.278882\pi$
$881$	38.0000	1.28025	0.640126	+	0.768270i	$0.278882\pi$
$882$	6.00000	0.202031
$883$	54.0000	1.81724	0.908622	−	0.417619i	$-0.137135\pi$
$883$	54.0000	1.81724	0.908622	+	0.417619i	$0.137135\pi$
$884$	0	0
$885$	−9.00000	−0.302532
$886$	−28.0000	−0.940678
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	−6.00000	−0.201347
$889$	7.00000	0.234772
$890$	6.00000	0.201120
$891$	0	0
$892$	0	0
$893$	84.0000	2.81095
$894$	−54.0000	−1.80603
$895$	22.0000	0.735379
$896$	1.00000	0.0334077
$897$	−3.00000	−0.100167
$898$	−7.00000	−0.233593
$899$	−32.0000	−1.06726
$900$	6.00000	0.200000
$901$	0	0
$902$	0	0
$903$	18.0000	0.599002
$904$	−9.00000	−0.299336
$905$	11.0000	0.365652
$906$	21.0000	0.697678
$907$	40.0000	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$908$	−12.0000	−0.398234
$909$	90.0000	2.98511
$910$	1.00000	0.0331497
$911$	−21.0000	−0.695761	−0.347881	−	0.937539i	$-0.613099\pi$
$911$	−21.0000	−0.695761	−0.347881	+	0.937539i	$0.613099\pi$
$912$	21.0000	0.695379
$913$	0	0
$914$	−25.0000	−0.826927
$915$	18.0000	0.595062
$916$	2.00000	0.0660819
$917$	−11.0000	−0.363252
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	1.00000	0.0329690
$921$	84.0000	2.76789
$922$	30.0000	0.987997
$923$	−8.00000	−0.263323
$924$	0	0
$925$	2.00000	0.0657596
$926$	−23.0000	−0.755827
$927$	36.0000	1.18240
$928$	8.00000	0.262613
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	12.0000	0.393496
$931$	−7.00000	−0.229416
$932$	−29.0000	−0.949927
$933$	−90.0000	−2.94647
$934$	−27.0000	−0.883467
$935$	0	0
$936$	6.00000	0.196116
$937$	−28.0000	−0.914720	−0.457360	−	0.889282i	$-0.651205\pi$
$937$	−28.0000	−0.914720	−0.457360	+	0.889282i	$0.651205\pi$
$938$	8.00000	0.261209
$939$	54.0000	1.76222
$940$	−12.0000	−0.391397
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	63.0000	2.05265
$943$	−6.00000	−0.195387
$944$	3.00000	0.0976417
$945$	−9.00000	−0.292770
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	27.0000	0.876919
$949$	16.0000	0.519382
$950$	−7.00000	−0.227110
$951$	36.0000	1.16738
$952$	0	0
$953$	−41.0000	−1.32812	−0.664060	−	0.747679i	$-0.731168\pi$
$953$	−41.0000	−1.32812	−0.664060	+	0.747679i	$0.731168\pi$
$954$	−72.0000	−2.33109
$955$	9.00000	0.291233
$956$	17.0000	0.549819
$957$	0	0
$958$	−18.0000	−0.581554
$959$	−5.00000	−0.161458
$960$	−3.00000	−0.0968246
$961$	−15.0000	−0.483871
$962$	2.00000	0.0644826
$963$	12.0000	0.386695
$964$	−20.0000	−0.644157
$965$	−13.0000	−0.418485
$966$	−3.00000	−0.0965234
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	0	0
$970$	−8.00000	−0.256865
$971$	−39.0000	−1.25157	−0.625785	−	0.779996i	$-0.715221\pi$
$971$	−39.0000	−1.25157	−0.625785	+	0.779996i	$0.715221\pi$
$972$	0	0
$973$	5.00000	0.160293
$974$	3.00000	0.0961262
$975$	−3.00000	−0.0960769
$976$	−6.00000	−0.192055
$977$	−57.0000	−1.82359	−0.911796	−	0.410644i	$-0.865304\pi$
$977$	−57.0000	−1.82359	−0.911796	+	0.410644i	$0.865304\pi$
$978$	−18.0000	−0.575577
$979$	0	0
$980$	1.00000	0.0319438
$981$	0	0
$982$	8.00000	0.255290
$983$	30.0000	0.956851	0.478426	−	0.878128i	$-0.341208\pi$
$983$	30.0000	0.956851	0.478426	+	0.878128i	$0.341208\pi$
$984$	18.0000	0.573819
$985$	−4.00000	−0.127451
$986$	0	0
$987$	36.0000	1.14589
$988$	−7.00000	−0.222700
$989$	−6.00000	−0.190789
$990$	0	0
$991$	9.00000	0.285894	0.142947	−	0.989730i	$-0.454342\pi$
$991$	9.00000	0.285894	0.142947	+	0.989730i	$0.454342\pi$
$992$	−4.00000	−0.127000
$993$	66.0000	2.09445
$994$	−8.00000	−0.253745
$995$	4.00000	0.126809
$996$	39.0000	1.23576
$997$	−57.0000	−1.80521	−0.902604	−	0.430472i	$-0.858347\pi$
$997$	−57.0000	−1.80521	−0.902604	+	0.430472i	$0.858347\pi$
$998$	−18.0000	−0.569780
$999$	−18.0000	−0.569495

Twists

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

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

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