LMFDB - Embedded newform 8470.2.a.u.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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -6.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -2.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +1.00000 q^{38} +4.00000 q^{39} +1.00000 q^{40} +2.00000 q^{42} +7.00000 q^{43} +1.00000 q^{45} -6.00000 q^{46} -2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} +9.00000 q^{53} +4.00000 q^{54} -1.00000 q^{56} -2.00000 q^{57} -6.00000 q^{58} +3.00000 q^{59} -2.00000 q^{60} +7.00000 q^{61} -4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +11.0000 q^{67} -3.00000 q^{68} +12.0000 q^{69} -1.00000 q^{70} -3.00000 q^{71} +1.00000 q^{72} -11.0000 q^{73} +2.00000 q^{74} -2.00000 q^{75} +1.00000 q^{76} +4.00000 q^{78} -11.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +12.0000 q^{83} +2.00000 q^{84} -3.00000 q^{85} +7.00000 q^{86} +12.0000 q^{87} +1.00000 q^{90} +2.00000 q^{91} -6.00000 q^{92} +8.00000 q^{93} +1.00000 q^{95} -2.00000 q^{96} -1.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	1.00000	0.235702
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	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
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−2.00000	−0.365148
$31$	−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
$34$	−3.00000	−0.514496
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	1.00000	0.162221
$39$	4.00000	0.640513
$40$	1.00000	0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	2.00000	0.308607
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	−6.00000	−0.884652
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−2.00000	−0.288675
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	6.00000	0.840168
$52$	−2.00000	−0.277350
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−2.00000	−0.264906
$58$	−6.00000	−0.787839
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	−2.00000	−0.258199
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	−4.00000	−0.508001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	11.0000	1.34386	0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.0000	1.34386	0.671932	+	0.740613i	$0.265465\pi$
$68$	−3.00000	−0.363803
$69$	12.0000	1.44463
$70$	−1.00000	−0.119523
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	1.00000	0.117851
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	2.00000	0.232495
$75$	−2.00000	−0.230940
$76$	1.00000	0.114708
$77$	0	0
$78$	4.00000	0.452911
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	2.00000	0.218218
$85$	−3.00000	−0.325396
$86$	7.00000	0.754829
$87$	12.0000	1.28654
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	1.00000	0.105409
$91$	2.00000	0.209657
$92$	−6.00000	−0.625543
$93$	8.00000	0.829561
$94$	0	0
$95$	1.00000	0.102598
$96$	−2.00000	−0.204124
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	6.00000	0.594089
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	−2.00000	−0.196116
$105$	2.00000	0.195180
$106$	9.00000	0.874157
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	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
$111$	−4.00000	−0.379663
$112$	−1.00000	−0.0944911
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	−2.00000	−0.187317
$115$	−6.00000	−0.559503
$116$	−6.00000	−0.557086
$117$	−2.00000	−0.184900
$118$	3.00000	0.276172
$119$	3.00000	0.275010
$120$	−2.00000	−0.182574
$121$	0	0
$122$	7.00000	0.633750
$123$	0	0
$124$	−4.00000	−0.359211
$125$	1.00000	0.0894427
$126$	−1.00000	−0.0890871
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	1.00000	0.0883883
$129$	−14.0000	−1.23263
$130$	−2.00000	−0.175412
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	11.0000	0.950255
$135$	4.00000	0.344265
$136$	−3.00000	−0.257248
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	12.0000	1.02151
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−3.00000	−0.251754
$143$	0	0
$144$	1.00000	0.0833333
$145$	−6.00000	−0.498273
$146$	−11.0000	−0.910366
$147$	−2.00000	−0.164957
$148$	2.00000	0.164399
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	−2.00000	−0.163299
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	1.00000	0.0811107
$153$	−3.00000	−0.242536
$154$	0	0
$155$	−4.00000	−0.321288
$156$	4.00000	0.320256
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	−11.0000	−0.875113
$159$	−18.0000	−1.42749
$160$	1.00000	0.0790569
$161$	6.00000	0.472866
$162$	−11.0000	−0.864242
$163$	5.00000	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$163$	5.00000	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$164$	0	0
$165$	0	0
$166$	12.0000	0.931381
$167$	−9.00000	−0.696441	−0.348220	−	0.937413i	$-0.613214\pi$
$167$	−9.00000	−0.696441	−0.348220	+	0.937413i	$0.613214\pi$
$168$	2.00000	0.154303
$169$	−9.00000	−0.692308
$170$	−3.00000	−0.230089
$171$	1.00000	0.0764719
$172$	7.00000	0.533745
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	12.0000	0.909718
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	1.00000	0.0745356
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	2.00000	0.148250
$183$	−14.0000	−1.03491
$184$	−6.00000	−0.442326
$185$	2.00000	0.147043
$186$	8.00000	0.586588
$187$	0	0
$188$	0	0
$189$	−4.00000	−0.290957
$190$	1.00000	0.0725476
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−2.00000	−0.144338
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−1.00000	−0.0717958
$195$	4.00000	0.286446
$196$	1.00000	0.0714286
$197$	9.00000	0.641223	0.320612	−	0.947211i	$-0.396112\pi$
$197$	9.00000	0.641223	0.320612	+	0.947211i	$0.396112\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	1.00000	0.0707107
$201$	−22.0000	−1.55176
$202$	−6.00000	−0.422159
$203$	6.00000	0.421117
$204$	6.00000	0.420084
$205$	0	0
$206$	−1.00000	−0.0696733
$207$	−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	$0.456034\pi$
$212$	9.00000	0.618123
$213$	6.00000	0.411113
$214$	9.00000	0.615227
$215$	7.00000	0.477396
$216$	4.00000	0.272166
$217$	4.00000	0.271538
$218$	16.0000	1.08366
$219$	22.0000	1.48662
$220$	0	0
$221$	6.00000	0.403604
$222$	−4.00000	−0.268462
$223$	11.0000	0.736614	0.368307	−	0.929704i	$-0.379937\pi$
$223$	11.0000	0.736614	0.368307	+	0.929704i	$0.379937\pi$
$224$	−1.00000	−0.0668153
$225$	1.00000	0.0666667
$226$	12.0000	0.798228
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	−2.00000	−0.132453
$229$	−25.0000	−1.65205	−0.826023	−	0.563636i	$-0.809402\pi$
$229$	−25.0000	−1.65205	−0.826023	+	0.563636i	$0.809402\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	3.00000	0.195283
$237$	22.0000	1.42905
$238$	3.00000	0.194461
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	−2.00000	−0.129099
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	7.00000	0.448129
$245$	1.00000	0.0638877
$246$	0	0
$247$	−2.00000	−0.127257
$248$	−4.00000	−0.254000
$249$	−24.0000	−1.52094
$250$	1.00000	0.0632456
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	−2.00000	−0.125491
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	21.0000	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$258$	−14.0000	−0.871602
$259$	−2.00000	−0.124274
$260$	−2.00000	−0.124035
$261$	−6.00000	−0.371391
$262$	12.0000	0.741362
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	9.00000	0.552866
$266$	−1.00000	−0.0613139
$267$	0	0
$268$	11.0000	0.671932
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	4.00000	0.243432
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	−3.00000	−0.181902
$273$	−4.00000	−0.242091
$274$	12.0000	0.724947
$275$	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	$0.270164\pi$
$278$	−5.00000	−0.299880
$279$	−4.00000	−0.239474
$280$	−1.00000	−0.0597614
$281$	21.0000	1.25275	0.626377	−	0.779520i	$-0.284537\pi$
$281$	21.0000	1.25275	0.626377	+	0.779520i	$0.284537\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	−3.00000	−0.178017
$285$	−2.00000	−0.118470
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	−6.00000	−0.352332
$291$	2.00000	0.117242
$292$	−11.0000	−0.643726
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	−2.00000	−0.116642
$295$	3.00000	0.174667
$296$	2.00000	0.116248
$297$	0	0
$298$	0	0
$299$	12.0000	0.693978
$300$	−2.00000	−0.115470
$301$	−7.00000	−0.403473
$302$	−8.00000	−0.460348
$303$	12.0000	0.689382
$304$	1.00000	0.0573539
$305$	7.00000	0.400819
$306$	−3.00000	−0.171499
$307$	34.0000	1.94048	0.970241	−	0.242140i	$-0.0778494\pi$
$307$	34.0000	1.94048	0.970241	+	0.242140i	$0.0778494\pi$
$308$	0	0
$309$	2.00000	0.113776
$310$	−4.00000	−0.227185
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	4.00000	0.226455
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	8.00000	0.451466
$315$	−1.00000	−0.0563436
$316$	−11.0000	−0.618798
$317$	33.0000	1.85346	0.926732	−	0.375722i	$-0.122605\pi$
$317$	33.0000	1.85346	0.926732	+	0.375722i	$0.122605\pi$
$318$	−18.0000	−1.00939
$319$	0	0
$320$	1.00000	0.0559017
$321$	−18.0000	−1.00466
$322$	6.00000	0.334367
$323$	−3.00000	−0.166924
$324$	−11.0000	−0.611111
$325$	−2.00000	−0.110940
$326$	5.00000	0.276924
$327$	−32.0000	−1.76960
$328$	0	0
$329$	0	0
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	12.0000	0.658586
$333$	2.00000	0.109599
$334$	−9.00000	−0.492458
$335$	11.0000	0.600994
$336$	2.00000	0.109109
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−9.00000	−0.489535
$339$	−24.0000	−1.30350
$340$	−3.00000	−0.162698
$341$	0	0
$342$	1.00000	0.0540738
$343$	−1.00000	−0.0539949
$344$	7.00000	0.377415
$345$	12.0000	0.646058
$346$	−12.0000	−0.645124
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	12.0000	0.643268
$349$	−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
$353$	21.0000	1.11772	0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000	1.11772	0.558859	+	0.829263i	$0.311239\pi$
$354$	−6.00000	−0.318896
$355$	−3.00000	−0.159223
$356$	0	0
$357$	−6.00000	−0.317554
$358$	24.0000	1.26844
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	1.00000	0.0527046
$361$	−18.0000	−0.947368
$362$	−7.00000	−0.367912
$363$	0	0
$364$	2.00000	0.104828
$365$	−11.0000	−0.575766
$366$	−14.0000	−0.731792
$367$	29.0000	1.51379	0.756894	−	0.653538i	$-0.226716\pi$
$367$	29.0000	1.51379	0.756894	+	0.653538i	$0.226716\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	2.00000	0.103975
$371$	−9.00000	−0.467257
$372$	8.00000	0.414781
$373$	−23.0000	−1.19089	−0.595447	−	0.803394i	$-0.703025\pi$
$373$	−23.0000	−1.19089	−0.595447	+	0.803394i	$0.703025\pi$
$374$	0	0
$375$	−2.00000	−0.103280
$376$	0	0
$377$	12.0000	0.618031
$378$	−4.00000	−0.205738
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	1.00000	0.0512989
$381$	4.00000	0.204926
$382$	0	0
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	4.00000	0.203595
$387$	7.00000	0.355830
$388$	−1.00000	−0.0507673
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	4.00000	0.202548
$391$	18.0000	0.910299
$392$	1.00000	0.0505076
$393$	−24.0000	−1.21064
$394$	9.00000	0.453413
$395$	−11.0000	−0.553470
$396$	0	0
$397$	−16.0000	−0.803017	−0.401508	−	0.915855i	$-0.631514\pi$
$397$	−16.0000	−0.803017	−0.401508	+	0.915855i	$0.631514\pi$
$398$	2.00000	0.100251
$399$	2.00000	0.100125
$400$	1.00000	0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−22.0000	−1.09726
$403$	8.00000	0.398508
$404$	−6.00000	−0.298511
$405$	−11.0000	−0.546594
$406$	6.00000	0.297775
$407$	0	0
$408$	6.00000	0.297044
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	−24.0000	−1.18383
$412$	−1.00000	−0.0492665
$413$	−3.00000	−0.147620
$414$	−6.00000	−0.294884
$415$	12.0000	0.589057
$416$	−2.00000	−0.0980581
$417$	10.0000	0.489702
$418$	0	0
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	2.00000	0.0975900
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	9.00000	0.437079
$425$	−3.00000	−0.145521
$426$	6.00000	0.290701
$427$	−7.00000	−0.338754
$428$	9.00000	0.435031
$429$	0	0
$430$	7.00000	0.337570
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	4.00000	0.192450
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	4.00000	0.192006
$435$	12.0000	0.575356
$436$	16.0000	0.766261
$437$	−6.00000	−0.287019
$438$	22.0000	1.05120
$439$	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
$441$	1.00000	0.0476190
$442$	6.00000	0.285391
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	11.0000	0.520865
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	21.0000	0.991051	0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000	0.991051	0.495526	+	0.868593i	$0.334975\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	12.0000	0.564433
$453$	16.0000	0.751746
$454$	0	0
$455$	2.00000	0.0937614
$456$	−2.00000	−0.0936586
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	−25.0000	−1.16817
$459$	−12.0000	−0.560112
$460$	−6.00000	−0.279751
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	−6.00000	−0.278543
$465$	8.00000	0.370991
$466$	−24.0000	−1.11178
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−2.00000	−0.0924500
$469$	−11.0000	−0.507933
$470$	0	0
$471$	−16.0000	−0.737241
$472$	3.00000	0.138086
$473$	0	0
$474$	22.0000	1.01049
$475$	1.00000	0.0458831
$476$	3.00000	0.137505
$477$	9.00000	0.412082
$478$	24.0000	1.09773
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	−2.00000	−0.0912871
$481$	−4.00000	−0.182384
$482$	−8.00000	−0.364390
$483$	−12.0000	−0.546019
$484$	0	0
$485$	−1.00000	−0.0454077
$486$	10.0000	0.453609
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	7.00000	0.316875
$489$	−10.0000	−0.452216
$490$	1.00000	0.0451754
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	18.0000	0.810679
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	−4.00000	−0.179605
$497$	3.00000	0.134568
$498$	−24.0000	−1.07547
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	1.00000	0.0447214
$501$	18.0000	0.804181
$502$	−15.0000	−0.669483
$503$	−21.0000	−0.936344	−0.468172	−	0.883637i	$-0.655087\pi$
$503$	−21.0000	−0.936344	−0.468172	+	0.883637i	$0.655087\pi$
$504$	−1.00000	−0.0445435
$505$	−6.00000	−0.266996
$506$	0	0
$507$	18.0000	0.799408
$508$	−2.00000	−0.0887357
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	6.00000	0.265684
$511$	11.0000	0.486611
$512$	1.00000	0.0441942
$513$	4.00000	0.176604
$514$	21.0000	0.926270
$515$	−1.00000	−0.0440653
$516$	−14.0000	−0.616316
$517$	0	0
$518$	−2.00000	−0.0878750
$519$	24.0000	1.05348
$520$	−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	$0.271757\pi$
$522$	−6.00000	−0.262613
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	12.0000	0.524222
$525$	2.00000	0.0872872
$526$	18.0000	0.784837
$527$	12.0000	0.522728
$528$	0	0
$529$	13.0000	0.565217
$530$	9.00000	0.390935
$531$	3.00000	0.130189
$532$	−1.00000	−0.0433555
$533$	0	0
$534$	0	0
$535$	9.00000	0.389104
$536$	11.0000	0.475128
$537$	−48.0000	−2.07135
$538$	−15.0000	−0.646696
$539$	0	0
$540$	4.00000	0.172133
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	−2.00000	−0.0859074
$543$	14.0000	0.600798
$544$	−3.00000	−0.128624
$545$	16.0000	0.685365
$546$	−4.00000	−0.171184
$547$	−35.0000	−1.49649	−0.748246	−	0.663421i	$-0.769104\pi$
$547$	−35.0000	−1.49649	−0.748246	+	0.663421i	$0.769104\pi$
$548$	12.0000	0.512615
$549$	7.00000	0.298753
$550$	0	0
$551$	−6.00000	−0.255609
$552$	12.0000	0.510754
$553$	11.0000	0.467768
$554$	22.0000	0.934690
$555$	−4.00000	−0.169791
$556$	−5.00000	−0.212047
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	−4.00000	−0.169334
$559$	−14.0000	−0.592137
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	21.0000	0.885832
$563$	−30.0000	−1.26435	−0.632175	−	0.774826i	$-0.717837\pi$
$563$	−30.0000	−1.26435	−0.632175	+	0.774826i	$0.717837\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	16.0000	0.672530
$567$	11.0000	0.461957
$568$	−3.00000	−0.125877
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	−2.00000	−0.0837708
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.00000	−0.250217
$576$	1.00000	0.0416667
$577$	11.0000	0.457936	0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000	0.457936	0.228968	+	0.973434i	$0.426465\pi$
$578$	−8.00000	−0.332756
$579$	−8.00000	−0.332469
$580$	−6.00000	−0.249136
$581$	−12.0000	−0.497844
$582$	2.00000	0.0829027
$583$	0	0
$584$	−11.0000	−0.455183
$585$	−2.00000	−0.0826898
$586$	−12.0000	−0.495715
$587$	30.0000	1.23823	0.619116	−	0.785299i	$-0.287491\pi$
$587$	30.0000	1.23823	0.619116	+	0.785299i	$0.287491\pi$
$588$	−2.00000	−0.0824786
$589$	−4.00000	−0.164817
$590$	3.00000	0.123508
$591$	−18.0000	−0.740421
$592$	2.00000	0.0821995
$593$	−39.0000	−1.60154	−0.800769	−	0.598973i	$-0.795576\pi$
$593$	−39.0000	−1.60154	−0.800769	+	0.598973i	$0.795576\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	0	0
$597$	−4.00000	−0.163709
$598$	12.0000	0.490716
$599$	27.0000	1.10319	0.551595	−	0.834112i	$-0.314019\pi$
$599$	27.0000	1.10319	0.551595	+	0.834112i	$0.314019\pi$
$600$	−2.00000	−0.0816497
$601$	40.0000	1.63163	0.815817	−	0.578310i	$-0.196288\pi$
$601$	40.0000	1.63163	0.815817	+	0.578310i	$0.196288\pi$
$602$	−7.00000	−0.285299
$603$	11.0000	0.447955
$604$	−8.00000	−0.325515
$605$	0	0
$606$	12.0000	0.487467
$607$	−5.00000	−0.202944	−0.101472	−	0.994838i	$-0.532355\pi$
$607$	−5.00000	−0.202944	−0.101472	+	0.994838i	$0.532355\pi$
$608$	1.00000	0.0405554
$609$	−12.0000	−0.486265
$610$	7.00000	0.283422
$611$	0	0
$612$	−3.00000	−0.121268
$613$	1.00000	0.0403896	0.0201948	−	0.999796i	$-0.493571\pi$
$613$	1.00000	0.0403896	0.0201948	+	0.999796i	$0.493571\pi$
$614$	34.0000	1.37213
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	2.00000	0.0804518
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	−4.00000	−0.160644
$621$	−24.0000	−0.963087
$622$	0	0
$623$	0	0
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	2.00000	0.0799361
$627$	0	0
$628$	8.00000	0.319235
$629$	−6.00000	−0.239236
$630$	−1.00000	−0.0398410
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	−11.0000	−0.437557
$633$	−8.00000	−0.317971
$634$	33.0000	1.31060
$635$	−2.00000	−0.0793676
$636$	−18.0000	−0.713746
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	−3.00000	−0.118678
$640$	1.00000	0.0395285
$641$	21.0000	0.829450	0.414725	−	0.909947i	$-0.363878\pi$
$641$	21.0000	0.829450	0.414725	+	0.909947i	$0.363878\pi$
$642$	−18.0000	−0.710403
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	6.00000	0.236433
$645$	−14.0000	−0.551249
$646$	−3.00000	−0.118033
$647$	45.0000	1.76913	0.884566	−	0.466415i	$-0.154454\pi$
$647$	45.0000	1.76913	0.884566	+	0.466415i	$0.154454\pi$
$648$	−11.0000	−0.432121
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	−8.00000	−0.313545
$652$	5.00000	0.195815
$653$	21.0000	0.821794	0.410897	−	0.911682i	$-0.365216\pi$
$653$	21.0000	0.821794	0.410897	+	0.911682i	$0.365216\pi$
$654$	−32.0000	−1.25130
$655$	12.0000	0.468879
$656$	0	0
$657$	−11.0000	−0.429151
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	26.0000	1.01052
$663$	−12.0000	−0.466041
$664$	12.0000	0.465690
$665$	−1.00000	−0.0387783
$666$	2.00000	0.0774984
$667$	36.0000	1.39393
$668$	−9.00000	−0.348220
$669$	−22.0000	−0.850569
$670$	11.0000	0.424967
$671$	0	0
$672$	2.00000	0.0771517
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	22.0000	0.847408
$675$	4.00000	0.153960
$676$	−9.00000	−0.346154
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	−24.0000	−0.921714
$679$	1.00000	0.0383765
$680$	−3.00000	−0.115045
$681$	0	0
$682$	0	0
$683$	15.0000	0.573959	0.286980	−	0.957937i	$-0.407349\pi$
$683$	15.0000	0.573959	0.286980	+	0.957937i	$0.407349\pi$
$684$	1.00000	0.0382360
$685$	12.0000	0.458496
$686$	−1.00000	−0.0381802
$687$	50.0000	1.90762
$688$	7.00000	0.266872
$689$	−18.0000	−0.685745
$690$	12.0000	0.456832
$691$	−43.0000	−1.63580	−0.817899	−	0.575362i	$-0.804861\pi$
$691$	−43.0000	−1.63580	−0.817899	+	0.575362i	$0.804861\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	−3.00000	−0.113878
$695$	−5.00000	−0.189661
$696$	12.0000	0.454859
$697$	0	0
$698$	−14.0000	−0.529908
$699$	48.0000	1.81553
$700$	−1.00000	−0.0377964
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	−8.00000	−0.301941
$703$	2.00000	0.0754314
$704$	0	0
$705$	0	0
$706$	21.0000	0.790345
$707$	6.00000	0.225653
$708$	−6.00000	−0.225494
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	−3.00000	−0.112588
$711$	−11.0000	−0.412532
$712$	0	0
$713$	24.0000	0.898807
$714$	−6.00000	−0.224544
$715$	0	0
$716$	24.0000	0.896922
$717$	−48.0000	−1.79259
$718$	3.00000	0.111959
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	1.00000	0.0372678
$721$	1.00000	0.0372419
$722$	−18.0000	−0.669891
$723$	16.0000	0.595046
$724$	−7.00000	−0.260153
$725$	−6.00000	−0.222834
$726$	0	0
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	2.00000	0.0741249
$729$	13.0000	0.481481
$730$	−11.0000	−0.407128
$731$	−21.0000	−0.776713
$732$	−14.0000	−0.517455
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	29.0000	1.07041
$735$	−2.00000	−0.0737711
$736$	−6.00000	−0.221163
$737$	0	0
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	2.00000	0.0735215
$741$	4.00000	0.146944
$742$	−9.00000	−0.330400
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	−23.0000	−0.842090
$747$	12.0000	0.439057
$748$	0	0
$749$	−9.00000	−0.328853
$750$	−2.00000	−0.0730297
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	30.0000	1.09326
$754$	12.0000	0.437014
$755$	−8.00000	−0.291150
$756$	−4.00000	−0.145479
$757$	47.0000	1.70824	0.854122	−	0.520073i	$-0.174095\pi$
$757$	47.0000	1.70824	0.854122	+	0.520073i	$0.174095\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	1.00000	0.0362738
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	4.00000	0.144905
$763$	−16.0000	−0.579239
$764$	0	0
$765$	−3.00000	−0.108465
$766$	21.0000	0.758761
$767$	−6.00000	−0.216647
$768$	−2.00000	−0.0721688
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	−42.0000	−1.51259
$772$	4.00000	0.143963
$773$	−12.0000	−0.431610	−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000	−0.431610	−0.215805	+	0.976436i	$0.569238\pi$
$774$	7.00000	0.251610
$775$	−4.00000	−0.143684
$776$	−1.00000	−0.0358979
$777$	4.00000	0.143499
$778$	12.0000	0.430221
$779$	0	0
$780$	4.00000	0.143223
$781$	0	0
$782$	18.0000	0.643679
$783$	−24.0000	−0.857690
$784$	1.00000	0.0357143
$785$	8.00000	0.285532
$786$	−24.0000	−0.856052
$787$	−50.0000	−1.78231	−0.891154	−	0.453701i	$-0.850103\pi$
$787$	−50.0000	−1.78231	−0.891154	+	0.453701i	$0.850103\pi$
$788$	9.00000	0.320612
$789$	−36.0000	−1.28163
$790$	−11.0000	−0.391362
$791$	−12.0000	−0.426671
$792$	0	0
$793$	−14.0000	−0.497155
$794$	−16.0000	−0.567819
$795$	−18.0000	−0.638394
$796$	2.00000	0.0708881
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	2.00000	0.0707992
$799$	0	0
$800$	1.00000	0.0353553
$801$	0	0
$802$	−18.0000	−0.635602
$803$	0	0
$804$	−22.0000	−0.775880
$805$	6.00000	0.211472
$806$	8.00000	0.281788
$807$	30.0000	1.05605
$808$	−6.00000	−0.211079
$809$	−3.00000	−0.105474	−0.0527372	−	0.998608i	$-0.516795\pi$
$809$	−3.00000	−0.105474	−0.0527372	+	0.998608i	$0.516795\pi$
$810$	−11.0000	−0.386501
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	6.00000	0.210559
$813$	4.00000	0.140286
$814$	0	0
$815$	5.00000	0.175142
$816$	6.00000	0.210042
$817$	7.00000	0.244899
$818$	4.00000	0.139857
$819$	2.00000	0.0698857
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	−24.0000	−0.837096
$823$	−22.0000	−0.766872	−0.383436	−	0.923567i	$-0.625259\pi$
$823$	−22.0000	−0.766872	−0.383436	+	0.923567i	$0.625259\pi$
$824$	−1.00000	−0.0348367
$825$	0	0
$826$	−3.00000	−0.104383
$827$	51.0000	1.77344	0.886722	−	0.462303i	$-0.152977\pi$
$827$	51.0000	1.77344	0.886722	+	0.462303i	$0.152977\pi$
$828$	−6.00000	−0.208514
$829$	50.0000	1.73657	0.868286	−	0.496064i	$-0.165222\pi$
$829$	50.0000	1.73657	0.868286	+	0.496064i	$0.165222\pi$
$830$	12.0000	0.416526
$831$	−44.0000	−1.52634
$832$	−2.00000	−0.0693375
$833$	−3.00000	−0.103944
$834$	10.0000	0.346272
$835$	−9.00000	−0.311458
$836$	0	0
$837$	−16.0000	−0.553041
$838$	9.00000	0.310900
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	2.00000	0.0690066
$841$	7.00000	0.241379
$842$	8.00000	0.275698
$843$	−42.0000	−1.44656
$844$	4.00000	0.137686
$845$	−9.00000	−0.309609
$846$	0	0
$847$	0	0
$848$	9.00000	0.309061
$849$	−32.0000	−1.09824
$850$	−3.00000	−0.102899
$851$	−12.0000	−0.411355
$852$	6.00000	0.205557
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	−7.00000	−0.239535
$855$	1.00000	0.0341993
$856$	9.00000	0.307614
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	47.0000	1.60362	0.801810	−	0.597580i	$-0.203871\pi$
$859$	47.0000	1.60362	0.801810	+	0.597580i	$0.203871\pi$
$860$	7.00000	0.238698
$861$	0	0
$862$	−3.00000	−0.102180
$863$	30.0000	1.02121	0.510606	−	0.859815i	$-0.329421\pi$
$863$	30.0000	1.02121	0.510606	+	0.859815i	$0.329421\pi$
$864$	4.00000	0.136083
$865$	−12.0000	−0.408012
$866$	2.00000	0.0679628
$867$	16.0000	0.543388
$868$	4.00000	0.135769
$869$	0	0
$870$	12.0000	0.406838
$871$	−22.0000	−0.745442
$872$	16.0000	0.541828
$873$	−1.00000	−0.0338449
$874$	−6.00000	−0.202953
$875$	−1.00000	−0.0338062
$876$	22.0000	0.743311
$877$	−17.0000	−0.574049	−0.287025	−	0.957923i	$-0.592666\pi$
$877$	−17.0000	−0.574049	−0.287025	+	0.957923i	$0.592666\pi$
$878$	10.0000	0.337484
$879$	24.0000	0.809500
$880$	0	0
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	1.00000	0.0336718
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	6.00000	0.201802
$885$	−6.00000	−0.201688
$886$	−12.0000	−0.403148
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−4.00000	−0.134231
$889$	2.00000	0.0670778
$890$	0	0
$891$	0	0
$892$	11.0000	0.368307
$893$	0	0
$894$	0	0
$895$	24.0000	0.802232
$896$	−1.00000	−0.0334077
$897$	−24.0000	−0.801337
$898$	21.0000	0.700779
$899$	24.0000	0.800445
$900$	1.00000	0.0333333
$901$	−27.0000	−0.899500
$902$	0	0
$903$	14.0000	0.465891
$904$	12.0000	0.399114
$905$	−7.00000	−0.232688
$906$	16.0000	0.531564
$907$	−1.00000	−0.0332045	−0.0166022	−	0.999862i	$-0.505285\pi$
$907$	−1.00000	−0.0332045	−0.0166022	+	0.999862i	$0.505285\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	2.00000	0.0662994
$911$	−21.0000	−0.695761	−0.347881	−	0.937539i	$-0.613099\pi$
$911$	−21.0000	−0.695761	−0.347881	+	0.937539i	$0.613099\pi$
$912$	−2.00000	−0.0662266
$913$	0	0
$914$	−8.00000	−0.264616
$915$	−14.0000	−0.462826
$916$	−25.0000	−0.826023
$917$	−12.0000	−0.396275
$918$	−12.0000	−0.396059
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	−6.00000	−0.197814
$921$	−68.0000	−2.24068
$922$	−9.00000	−0.296399
$923$	6.00000	0.197492
$924$	0	0
$925$	2.00000	0.0657596
$926$	−22.0000	−0.722965
$927$	−1.00000	−0.0328443
$928$	−6.00000	−0.196960
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	8.00000	0.262330
$931$	1.00000	0.0327737
$932$	−24.0000	−0.786146
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	−11.0000	−0.359163
$939$	−4.00000	−0.130535
$940$	0	0
$941$	−39.0000	−1.27136	−0.635682	−	0.771951i	$-0.719281\pi$
$941$	−39.0000	−1.27136	−0.635682	+	0.771951i	$0.719281\pi$
$942$	−16.0000	−0.521308
$943$	0	0
$944$	3.00000	0.0976417
$945$	−4.00000	−0.130120
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	22.0000	0.714527
$949$	22.0000	0.714150
$950$	1.00000	0.0324443
$951$	−66.0000	−2.14020
$952$	3.00000	0.0972306
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	9.00000	0.291386
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	−36.0000	−1.16311
$959$	−12.0000	−0.387500
$960$	−2.00000	−0.0645497
$961$	−15.0000	−0.483871
$962$	−4.00000	−0.128965
$963$	9.00000	0.290021
$964$	−8.00000	−0.257663
$965$	4.00000	0.128765
$966$	−12.0000	−0.386094
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	−1.00000	−0.0321081
$971$	21.0000	0.673922	0.336961	−	0.941519i	$-0.390601\pi$
$971$	21.0000	0.673922	0.336961	+	0.941519i	$0.390601\pi$
$972$	10.0000	0.320750
$973$	5.00000	0.160293
$974$	−4.00000	−0.128168
$975$	4.00000	0.128103
$976$	7.00000	0.224065
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	−10.0000	−0.319765
$979$	0	0
$980$	1.00000	0.0319438
$981$	16.0000	0.510841
$982$	12.0000	0.382935
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	9.00000	0.286764
$986$	18.0000	0.573237
$987$	0	0
$988$	−2.00000	−0.0636285
$989$	−42.0000	−1.33552
$990$	0	0
$991$	5.00000	0.158830	0.0794151	−	0.996842i	$-0.474695\pi$
$991$	5.00000	0.158830	0.0794151	+	0.996842i	$0.474695\pi$
$992$	−4.00000	−0.127000
$993$	−52.0000	−1.65017
$994$	3.00000	0.0951542
$995$	2.00000	0.0634043
$996$	−24.0000	−0.760469
$997$	−20.0000	−0.633406	−0.316703	−	0.948525i	$-0.602576\pi$
$997$	−20.0000	−0.633406	−0.316703	+	0.948525i	$0.602576\pi$
$998$	32.0000	1.01294
$999$	8.00000	0.253109

Twists

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

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