LMFDB - Embedded newform 8470.2.a.t.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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} -7.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} +6.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -7.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} -2.00000 q^{42} -1.00000 q^{43} -1.00000 q^{45} +6.00000 q^{46} -2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} +3.00000 q^{53} +4.00000 q^{54} +1.00000 q^{56} +14.0000 q^{57} -6.00000 q^{58} -3.00000 q^{59} +2.00000 q^{60} +11.0000 q^{61} +8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +5.00000 q^{67} +3.00000 q^{68} -12.0000 q^{69} -1.00000 q^{70} -15.0000 q^{71} +1.00000 q^{72} +11.0000 q^{73} -10.0000 q^{74} -2.00000 q^{75} -7.00000 q^{76} -4.00000 q^{78} -1.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} -2.00000 q^{84} -3.00000 q^{85} -1.00000 q^{86} +12.0000 q^{87} +12.0000 q^{89} -1.00000 q^{90} +2.00000 q^{91} +6.00000 q^{92} -16.0000 q^{93} +7.00000 q^{95} -2.00000 q^{96} -1.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	1.00000	0.235702
$19$	−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
$21$	−2.00000	−0.436436
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	−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
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.00000	0.514496
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−7.00000	−1.13555
$39$	−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
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	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
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	1.00000	0.133631
$57$	14.0000	1.85435
$58$	−6.00000	−0.787839
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	2.00000	0.258199
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	8.00000	1.01600
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	3.00000	0.363803
$69$	−12.0000	−1.44463
$70$	−1.00000	−0.119523
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	1.00000	0.117851
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	−10.0000	−1.16248
$75$	−2.00000	−0.230940
$76$	−7.00000	−0.802955
$77$	0	0
$78$	−4.00000	−0.452911
$79$	−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
$81$	−11.0000	−1.22222
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−2.00000	−0.218218
$85$	−3.00000	−0.325396
$86$	−1.00000	−0.107833
$87$	12.0000	1.28654
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	−1.00000	−0.105409
$91$	2.00000	0.209657
$92$	6.00000	0.625543
$93$	−16.0000	−1.65912
$94$	0	0
$95$	7.00000	0.718185
$96$	−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
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	2.00000	0.196116
$105$	2.00000	0.195180
$106$	3.00000	0.291386
$107$	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
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	20.0000	1.89832
$112$	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	$0.309095\pi$
$114$	14.0000	1.31122
$115$	−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
$122$	11.0000	0.995893
$123$	0	0
$124$	8.00000	0.718421
$125$	−1.00000	−0.0894427
$126$	1.00000	0.0890871
$127$	−22.0000	−1.95218	−0.976092	−	0.217357i	$-0.930256\pi$
$127$	−22.0000	−1.95218	−0.976092	+	0.217357i	$0.930256\pi$
$128$	1.00000	0.0883883
$129$	2.00000	0.176090
$130$	−2.00000	−0.175412
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	−7.00000	−0.606977
$134$	5.00000	0.431934
$135$	−4.00000	−0.344265
$136$	3.00000	0.257248
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	−12.0000	−1.02151
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−15.0000	−1.25877
$143$	0	0
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	11.0000	0.910366
$147$	−2.00000	−0.164957
$148$	−10.0000	−0.821995
$149$	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	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−7.00000	−0.567775
$153$	3.00000	0.242536
$154$	0	0
$155$	−8.00000	−0.642575
$156$	−4.00000	−0.320256
$157$	−16.0000	−1.27694	−0.638470	−	0.769647i	$-0.720432\pi$
$157$	−16.0000	−1.27694	−0.638470	+	0.769647i	$0.720432\pi$
$158$	−1.00000	−0.0795557
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	6.00000	0.472866
$162$	−11.0000	−0.864242
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.0000	1.62503	0.812514	−	0.582941i	$-0.198098\pi$
$167$	21.0000	1.62503	0.812514	+	0.582941i	$0.198098\pi$
$168$	−2.00000	−0.154303
$169$	−9.00000	−0.692308
$170$	−3.00000	−0.230089
$171$	−7.00000	−0.535303
$172$	−1.00000	−0.0762493
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	12.0000	0.909718
$175$	1.00000	0.0755929
$176$	0	0
$177$	6.00000	0.450988
$178$	12.0000	0.899438
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−1.00000	−0.0745356
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	2.00000	0.148250
$183$	−22.0000	−1.62629
$184$	6.00000	0.442326
$185$	10.0000	0.735215
$186$	−16.0000	−1.17318
$187$	0	0
$188$	0	0
$189$	4.00000	0.290957
$190$	7.00000	0.507833
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	−2.00000	−0.144338
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	−1.00000	−0.0717958
$195$	4.00000	0.286446
$196$	1.00000	0.0714286
$197$	−27.0000	−1.92367	−0.961835	−	0.273629i	$-0.911776\pi$
$197$	−27.0000	−1.92367	−0.961835	+	0.273629i	$0.911776\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	1.00000	0.0707107
$201$	−10.0000	−0.705346
$202$	−6.00000	−0.422159
$203$	−6.00000	−0.421117
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−13.0000	−0.905753
$207$	6.00000	0.417029
$208$	2.00000	0.138675
$209$	0	0
$210$	2.00000	0.138013
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	3.00000	0.206041
$213$	30.0000	2.05557
$214$	9.00000	0.615227
$215$	1.00000	0.0681994
$216$	4.00000	0.272166
$217$	8.00000	0.543075
$218$	−4.00000	−0.270914
$219$	−22.0000	−1.48662
$220$	0	0
$221$	6.00000	0.403604
$222$	20.0000	1.34231
$223$	23.0000	1.54019	0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000	1.54019	0.770097	+	0.637927i	$0.220208\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	12.0000	0.798228
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	14.0000	0.927173
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	−6.00000	−0.393919
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−3.00000	−0.195283
$237$	2.00000	0.129914
$238$	3.00000	0.194461
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	2.00000	0.129099
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	11.0000	0.704203
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−14.0000	−0.890799
$248$	8.00000	0.508001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	−22.0000	−1.38040
$255$	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	$0.272680\pi$
$258$	2.00000	0.124515
$259$	−10.0000	−0.621370
$260$	−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
$265$	−3.00000	−0.184289
$266$	−7.00000	−0.429198
$267$	−24.0000	−1.46878
$268$	5.00000	0.305424
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	−4.00000	−0.243432
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	3.00000	0.181902
$273$	−4.00000	−0.242091
$274$	0	0
$275$	0	0
$276$	−12.0000	−0.722315
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	11.0000	0.659736
$279$	8.00000	0.478947
$280$	−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	$0.471479\pi$
$282$	0	0
$283$	32.0000	1.90220	0.951101	−	0.308879i	$-0.0999539\pi$
$283$	32.0000	1.90220	0.951101	+	0.308879i	$0.0999539\pi$
$284$	−15.0000	−0.890086
$285$	−14.0000	−0.829288
$286$	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
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	−2.00000	−0.116642
$295$	3.00000	0.174667
$296$	−10.0000	−0.581238
$297$	0	0
$298$	0	0
$299$	12.0000	0.693978
$300$	−2.00000	−0.115470
$301$	−1.00000	−0.0576390
$302$	8.00000	0.460348
$303$	12.0000	0.689382
$304$	−7.00000	−0.401478
$305$	−11.0000	−0.629858
$306$	3.00000	0.171499
$307$	14.0000	0.799022	0.399511	−	0.916728i	$-0.369180\pi$
$307$	14.0000	0.799022	0.399511	+	0.916728i	$0.369180\pi$
$308$	0	0
$309$	26.0000	1.47909
$310$	−8.00000	−0.454369
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	−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	$0.481998\pi$
$314$	−16.0000	−0.902932
$315$	−1.00000	−0.0563436
$316$	−1.00000	−0.0562544
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−18.0000	−1.00466
$322$	6.00000	0.334367
$323$	−21.0000	−1.16847
$324$	−11.0000	−0.611111
$325$	2.00000	0.110940
$326$	11.0000	0.609234
$327$	8.00000	0.442401
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	0	0
$333$	−10.0000	−0.547997
$334$	21.0000	1.14907
$335$	−5.00000	−0.273179
$336$	−2.00000	−0.109109
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−9.00000	−0.489535
$339$	−24.0000	−1.30350
$340$	−3.00000	−0.162698
$341$	0	0
$342$	−7.00000	−0.378517
$343$	1.00000	0.0539949
$344$	−1.00000	−0.0539164
$345$	12.0000	0.646058
$346$	12.0000	0.645124
$347$	−27.0000	−1.44944	−0.724718	−	0.689046i	$-0.758030\pi$
$347$	−27.0000	−1.44944	−0.724718	+	0.689046i	$0.758030\pi$
$348$	12.0000	0.643268
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	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
$355$	15.0000	0.796117
$356$	12.0000	0.635999
$357$	−6.00000	−0.317554
$358$	0	0
$359$	9.00000	0.475002	0.237501	−	0.971387i	$-0.423672\pi$
$359$	9.00000	0.475002	0.237501	+	0.971387i	$0.423672\pi$
$360$	−1.00000	−0.0527046
$361$	30.0000	1.57895
$362$	11.0000	0.578147
$363$	0	0
$364$	2.00000	0.104828
$365$	−11.0000	−0.575766
$366$	−22.0000	−1.14996
$367$	−7.00000	−0.365397	−0.182699	−	0.983169i	$-0.558483\pi$
$367$	−7.00000	−0.365397	−0.182699	+	0.983169i	$0.558483\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	10.0000	0.519875
$371$	3.00000	0.155752
$372$	−16.0000	−0.829561
$373$	5.00000	0.258890	0.129445	−	0.991587i	$-0.458680\pi$
$373$	5.00000	0.258890	0.129445	+	0.991587i	$0.458680\pi$
$374$	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	$0.434129\pi$
$380$	7.00000	0.359092
$381$	44.0000	2.25419
$382$	24.0000	1.22795
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	20.0000	1.01797
$387$	−1.00000	−0.0508329
$388$	−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
$394$	−27.0000	−1.36024
$395$	1.00000	0.0503155
$396$	0	0
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	14.0000	0.701757
$399$	14.0000	0.700877
$400$	1.00000	0.0500000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	−10.0000	−0.498755
$403$	16.0000	0.797017
$404$	−6.00000	−0.298511
$405$	11.0000	0.546594
$406$	−6.00000	−0.297775
$407$	0	0
$408$	−6.00000	−0.297044
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	0	0
$412$	−13.0000	−0.640464
$413$	−3.00000	−0.147620
$414$	6.00000	0.294884
$415$	0	0
$416$	2.00000	0.0980581
$417$	−22.0000	−1.07734
$418$	0	0
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	2.00000	0.0975900
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	8.00000	0.389434
$423$	0	0
$424$	3.00000	0.145693
$425$	3.00000	0.145521
$426$	30.0000	1.45350
$427$	11.0000	0.532327
$428$	9.00000	0.435031
$429$	0	0
$430$	1.00000	0.0482243
$431$	−33.0000	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$432$	4.00000	0.192450
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	8.00000	0.384012
$435$	−12.0000	−0.575356
$436$	−4.00000	−0.191565
$437$	−42.0000	−2.00913
$438$	−22.0000	−1.05120
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	6.00000	0.285391
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	20.0000	0.949158
$445$	−12.0000	−0.568855
$446$	23.0000	1.08908
$447$	0	0
$448$	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	$0.522552\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	12.0000	0.564433
$453$	−16.0000	−0.751746
$454$	12.0000	0.563188
$455$	−2.00000	−0.0937614
$456$	14.0000	0.655610
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	5.00000	0.233635
$459$	12.0000	0.560112
$460$	−6.00000	−0.279751
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	0	0
$463$	26.0000	1.20832	0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000	1.20832	0.604161	+	0.796862i	$0.293508\pi$
$464$	−6.00000	−0.278543
$465$	16.0000	0.741982
$466$	12.0000	0.555889
$467$	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
$469$	5.00000	0.230879
$470$	0	0
$471$	32.0000	1.47448
$472$	−3.00000	−0.138086
$473$	0	0
$474$	2.00000	0.0918630
$475$	−7.00000	−0.321182
$476$	3.00000	0.137505
$477$	3.00000	0.137361
$478$	0	0
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	2.00000	0.0912871
$481$	−20.0000	−0.911922
$482$	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	$0.528888\pi$
$488$	11.0000	0.497947
$489$	−22.0000	−0.994874
$490$	−1.00000	−0.0451754
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	−18.0000	−0.810679
$494$	−14.0000	−0.629890
$495$	0	0
$496$	8.00000	0.359211
$497$	−15.0000	−0.672842
$498$	0	0
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	−1.00000	−0.0447214
$501$	−42.0000	−1.87642
$502$	15.0000	0.669483
$503$	33.0000	1.47140	0.735699	−	0.677309i	$-0.236854\pi$
$503$	33.0000	1.47140	0.735699	+	0.677309i	$0.236854\pi$
$504$	1.00000	0.0445435
$505$	6.00000	0.266996
$506$	0	0
$507$	18.0000	0.799408
$508$	−22.0000	−0.976092
$509$	33.0000	1.46270	0.731350	−	0.682003i	$-0.238891\pi$
$509$	33.0000	1.46270	0.731350	+	0.682003i	$0.238891\pi$
$510$	6.00000	0.265684
$511$	11.0000	0.486611
$512$	1.00000	0.0441942
$513$	−28.0000	−1.23623
$514$	21.0000	0.926270
$515$	13.0000	0.572848
$516$	2.00000	0.0880451
$517$	0	0
$518$	−10.0000	−0.439375
$519$	−24.0000	−1.05348
$520$	−2.00000	−0.0877058
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−6.00000	−0.262613
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	−12.0000	−0.524222
$525$	−2.00000	−0.0872872
$526$	18.0000	0.784837
$527$	24.0000	1.04546
$528$	0	0
$529$	13.0000	0.565217
$530$	−3.00000	−0.130312
$531$	−3.00000	−0.130189
$532$	−7.00000	−0.303488
$533$	0	0
$534$	−24.0000	−1.03858
$535$	−9.00000	−0.389104
$536$	5.00000	0.215967
$537$	0	0
$538$	−21.0000	−0.905374
$539$	0	0
$540$	−4.00000	−0.172133
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	2.00000	0.0859074
$543$	−22.0000	−0.944110
$544$	3.00000	0.128624
$545$	4.00000	0.171341
$546$	−4.00000	−0.171184
$547$	−19.0000	−0.812381	−0.406191	−	0.913788i	$-0.633143\pi$
$547$	−19.0000	−0.812381	−0.406191	+	0.913788i	$0.633143\pi$
$548$	0	0
$549$	11.0000	0.469469
$550$	0	0
$551$	42.0000	1.78926
$552$	−12.0000	−0.510754
$553$	−1.00000	−0.0425243
$554$	−10.0000	−0.424859
$555$	−20.0000	−0.848953
$556$	11.0000	0.466504
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	8.00000	0.338667
$559$	−2.00000	−0.0845910
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	3.00000	0.126547
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	32.0000	1.34506
$567$	−11.0000	−0.461957
$568$	−15.0000	−0.629386
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	−14.0000	−0.586395
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	−48.0000	−2.00523
$574$	0	0
$575$	6.00000	0.250217
$576$	1.00000	0.0416667
$577$	−37.0000	−1.54033	−0.770165	−	0.637845i	$-0.779826\pi$
$577$	−37.0000	−1.54033	−0.770165	+	0.637845i	$0.779826\pi$
$578$	−8.00000	−0.332756
$579$	−40.0000	−1.66234
$580$	6.00000	0.249136
$581$	0	0
$582$	2.00000	0.0829027
$583$	0	0
$584$	11.0000	0.455183
$585$	−2.00000	−0.0826898
$586$	−24.0000	−0.991431
$587$	−30.0000	−1.23823	−0.619116	−	0.785299i	$-0.712509\pi$
$587$	−30.0000	−1.23823	−0.619116	+	0.785299i	$0.712509\pi$
$588$	−2.00000	−0.0824786
$589$	−56.0000	−2.30744
$590$	3.00000	0.123508
$591$	54.0000	2.22126
$592$	−10.0000	−0.410997
$593$	−33.0000	−1.35515	−0.677574	−	0.735455i	$-0.736969\pi$
$593$	−33.0000	−1.35515	−0.677574	+	0.735455i	$0.736969\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	0	0
$597$	−28.0000	−1.14596
$598$	12.0000	0.490716
$599$	39.0000	1.59350	0.796748	−	0.604311i	$-0.206552\pi$
$599$	39.0000	1.59350	0.796748	+	0.604311i	$0.206552\pi$
$600$	−2.00000	−0.0816497
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	−1.00000	−0.0407570
$603$	5.00000	0.203616
$604$	8.00000	0.325515
$605$	0	0
$606$	12.0000	0.487467
$607$	17.0000	0.690009	0.345004	−	0.938601i	$-0.387877\pi$
$607$	17.0000	0.690009	0.345004	+	0.938601i	$0.387877\pi$
$608$	−7.00000	−0.283887
$609$	12.0000	0.486265
$610$	−11.0000	−0.445377
$611$	0	0
$612$	3.00000	0.121268
$613$	29.0000	1.17130	0.585649	−	0.810564i	$-0.300840\pi$
$613$	29.0000	1.17130	0.585649	+	0.810564i	$0.300840\pi$
$614$	14.0000	0.564994
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	26.0000	1.04587
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−8.00000	−0.321288
$621$	24.0000	0.963087
$622$	−12.0000	−0.481156
$623$	12.0000	0.480770
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	2.00000	0.0799361
$627$	0	0
$628$	−16.0000	−0.638470
$629$	−30.0000	−1.19618
$630$	−1.00000	−0.0398410
$631$	−7.00000	−0.278666	−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000	−0.278666	−0.139333	+	0.990246i	$0.544496\pi$
$632$	−1.00000	−0.0397779
$633$	−16.0000	−0.635943
$634$	3.00000	0.119145
$635$	22.0000	0.873043
$636$	−6.00000	−0.237915
$637$	2.00000	0.0792429
$638$	0	0
$639$	−15.0000	−0.593391
$640$	−1.00000	−0.0395285
$641$	−3.00000	−0.118493	−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000	−0.118493	−0.0592464	+	0.998243i	$0.518870\pi$
$642$	−18.0000	−0.710403
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	6.00000	0.236433
$645$	−2.00000	−0.0787499
$646$	−21.0000	−0.826234
$647$	9.00000	0.353827	0.176913	−	0.984226i	$-0.443389\pi$
$647$	9.00000	0.353827	0.176913	+	0.984226i	$0.443389\pi$
$648$	−11.0000	−0.432121
$649$	0	0
$650$	2.00000	0.0784465
$651$	−16.0000	−0.627089
$652$	11.0000	0.430793
$653$	15.0000	0.586995	0.293498	−	0.955960i	$-0.405181\pi$
$653$	15.0000	0.586995	0.293498	+	0.955960i	$0.405181\pi$
$654$	8.00000	0.312825
$655$	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
$661$	50.0000	1.94477	0.972387	−	0.233373i	$-0.0749763\pi$
$661$	50.0000	1.94477	0.972387	+	0.233373i	$0.0749763\pi$
$662$	−10.0000	−0.388661
$663$	−12.0000	−0.466041
$664$	0	0
$665$	7.00000	0.271448
$666$	−10.0000	−0.387492
$667$	−36.0000	−1.39393
$668$	21.0000	0.812514
$669$	−46.0000	−1.77846
$670$	−5.00000	−0.193167
$671$	0	0
$672$	−2.00000	−0.0771517
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	−22.0000	−0.847408
$675$	4.00000	0.153960
$676$	−9.00000	−0.346154
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	−24.0000	−0.921714
$679$	−1.00000	−0.0383765
$680$	−3.00000	−0.115045
$681$	−24.0000	−0.919682
$682$	0	0
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	−7.00000	−0.267652
$685$	0	0
$686$	1.00000	0.0381802
$687$	−10.0000	−0.381524
$688$	−1.00000	−0.0381246
$689$	6.00000	0.228582
$690$	12.0000	0.456832
$691$	35.0000	1.33146	0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000	1.33146	0.665731	+	0.746191i	$0.268120\pi$
$692$	12.0000	0.456172
$693$	0	0
$694$	−27.0000	−1.02491
$695$	−11.0000	−0.417254
$696$	12.0000	0.454859
$697$	0	0
$698$	2.00000	0.0757011
$699$	−24.0000	−0.907763
$700$	1.00000	0.0377964
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	8.00000	0.301941
$703$	70.0000	2.64010
$704$	0	0
$705$	0	0
$706$	21.0000	0.790345
$707$	−6.00000	−0.225653
$708$	6.00000	0.225494
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	15.0000	0.562940
$711$	−1.00000	−0.0375029
$712$	12.0000	0.449719
$713$	48.0000	1.79761
$714$	−6.00000	−0.224544
$715$	0	0
$716$	0	0
$717$	0	0
$718$	9.00000	0.335877
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	−1.00000	−0.0372678
$721$	−13.0000	−0.484145
$722$	30.0000	1.11648
$723$	−16.0000	−0.595046
$724$	11.0000	0.408812
$725$	−6.00000	−0.222834
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	2.00000	0.0741249
$729$	13.0000	0.481481
$730$	−11.0000	−0.407128
$731$	−3.00000	−0.110959
$732$	−22.0000	−0.813143
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	−7.00000	−0.258375
$735$	2.00000	0.0737711
$736$	6.00000	0.221163
$737$	0	0
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	10.0000	0.367607
$741$	28.0000	1.02861
$742$	3.00000	0.110133
$743$	−42.0000	−1.54083	−0.770415	−	0.637542i	$-0.779951\pi$
$743$	−42.0000	−1.54083	−0.770415	+	0.637542i	$0.779951\pi$
$744$	−16.0000	−0.586588
$745$	0	0
$746$	5.00000	0.183063
$747$	0	0
$748$	0	0
$749$	9.00000	0.328853
$750$	2.00000	0.0730297
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	−30.0000	−1.09326
$754$	−12.0000	−0.437014
$755$	−8.00000	−0.291150
$756$	4.00000	0.145479
$757$	−43.0000	−1.56286	−0.781431	−	0.623992i	$-0.785510\pi$
$757$	−43.0000	−1.56286	−0.781431	+	0.623992i	$0.785510\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	7.00000	0.253917
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	44.0000	1.59395
$763$	−4.00000	−0.144810
$764$	24.0000	0.868290
$765$	−3.00000	−0.108465
$766$	9.00000	0.325183
$767$	−6.00000	−0.216647
$768$	−2.00000	−0.0721688
$769$	−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
$771$	−42.0000	−1.51259
$772$	20.0000	0.719816
$773$	−48.0000	−1.72644	−0.863220	−	0.504828i	$-0.831556\pi$
$773$	−48.0000	−1.72644	−0.863220	+	0.504828i	$0.831556\pi$
$774$	−1.00000	−0.0359443
$775$	8.00000	0.287368
$776$	−1.00000	−0.0358979
$777$	20.0000	0.717496
$778$	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
$785$	16.0000	0.571064
$786$	24.0000	0.856052
$787$	14.0000	0.499046	0.249523	−	0.968369i	$-0.419726\pi$
$787$	14.0000	0.499046	0.249523	+	0.968369i	$0.419726\pi$
$788$	−27.0000	−0.961835
$789$	−36.0000	−1.28163
$790$	1.00000	0.0355784
$791$	12.0000	0.426671
$792$	0	0
$793$	22.0000	0.781243
$794$	20.0000	0.709773
$795$	6.00000	0.212798
$796$	14.0000	0.496217
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	14.0000	0.495595
$799$	0	0
$800$	1.00000	0.0353553
$801$	12.0000	0.423999
$802$	30.0000	1.05934
$803$	0	0
$804$	−10.0000	−0.352673
$805$	−6.00000	−0.211472
$806$	16.0000	0.563576
$807$	42.0000	1.47847
$808$	−6.00000	−0.211079
$809$	27.0000	0.949269	0.474635	−	0.880183i	$-0.342580\pi$
$809$	27.0000	0.949269	0.474635	+	0.880183i	$0.342580\pi$
$810$	11.0000	0.386501
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	−6.00000	−0.210559
$813$	−4.00000	−0.140286
$814$	0	0
$815$	−11.0000	−0.385313
$816$	−6.00000	−0.210042
$817$	7.00000	0.244899
$818$	20.0000	0.699284
$819$	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	$0.362450\pi$
$822$	0	0
$823$	−46.0000	−1.60346	−0.801730	−	0.597687i	$-0.796087\pi$
$823$	−46.0000	−1.60346	−0.801730	+	0.597687i	$0.796087\pi$
$824$	−13.0000	−0.452876
$825$	0	0
$826$	−3.00000	−0.104383
$827$	3.00000	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$828$	6.00000	0.208514
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	20.0000	0.693792
$832$	2.00000	0.0693375
$833$	3.00000	0.103944
$834$	−22.0000	−0.761798
$835$	−21.0000	−0.726735
$836$	0	0
$837$	32.0000	1.10608
$838$	−9.00000	−0.310900
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	2.00000	0.0690066
$841$	7.00000	0.241379
$842$	20.0000	0.689246
$843$	−6.00000	−0.206651
$844$	8.00000	0.275371
$845$	9.00000	0.309609
$846$	0	0
$847$	0	0
$848$	3.00000	0.103020
$849$	−64.0000	−2.19647
$850$	3.00000	0.102899
$851$	−60.0000	−2.05677
$852$	30.0000	1.02778
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	11.0000	0.376412
$855$	7.00000	0.239395
$856$	9.00000	0.307614
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	17.0000	0.580033	0.290016	−	0.957022i	$-0.406339\pi$
$859$	17.0000	0.580033	0.290016	+	0.957022i	$0.406339\pi$
$860$	1.00000	0.0340997
$861$	0	0
$862$	−33.0000	−1.12398
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	4.00000	0.136083
$865$	−12.0000	−0.408012
$866$	−22.0000	−0.747590
$867$	16.0000	0.543388
$868$	8.00000	0.271538
$869$	0	0
$870$	−12.0000	−0.406838
$871$	10.0000	0.338837
$872$	−4.00000	−0.135457
$873$	−1.00000	−0.0338449
$874$	−42.0000	−1.42067
$875$	−1.00000	−0.0338062
$876$	−22.0000	−0.743311
$877$	−37.0000	−1.24940	−0.624701	−	0.780864i	$-0.714779\pi$
$877$	−37.0000	−1.24940	−0.624701	+	0.780864i	$0.714779\pi$
$878$	14.0000	0.472477
$879$	48.0000	1.61900
$880$	0	0
$881$	−48.0000	−1.61716	−0.808581	−	0.588386i	$-0.799764\pi$
$881$	−48.0000	−1.61716	−0.808581	+	0.588386i	$0.799764\pi$
$882$	1.00000	0.0336718
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	6.00000	0.201802
$885$	−6.00000	−0.201688
$886$	12.0000	0.403148
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	20.0000	0.671156
$889$	−22.0000	−0.737856
$890$	−12.0000	−0.402241
$891$	0	0
$892$	23.0000	0.770097
$893$	0	0
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	−24.0000	−0.801337
$898$	−3.00000	−0.100111
$899$	−48.0000	−1.60089
$900$	1.00000	0.0333333
$901$	9.00000	0.299833
$902$	0	0
$903$	2.00000	0.0665558
$904$	12.0000	0.399114
$905$	−11.0000	−0.365652
$906$	−16.0000	−0.531564
$907$	41.0000	1.36138	0.680691	−	0.732570i	$-0.261680\pi$
$907$	41.0000	1.36138	0.680691	+	0.732570i	$0.261680\pi$
$908$	12.0000	0.398234
$909$	−6.00000	−0.199007
$910$	−2.00000	−0.0662994
$911$	39.0000	1.29213	0.646064	−	0.763283i	$-0.276414\pi$
$911$	39.0000	1.29213	0.646064	+	0.763283i	$0.276414\pi$
$912$	14.0000	0.463586
$913$	0	0
$914$	−28.0000	−0.926158
$915$	22.0000	0.727298
$916$	5.00000	0.165205
$917$	−12.0000	−0.396275
$918$	12.0000	0.396059
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	−6.00000	−0.197814
$921$	−28.0000	−0.922631
$922$	3.00000	0.0987997
$923$	−30.0000	−0.987462
$924$	0	0
$925$	−10.0000	−0.328798
$926$	26.0000	0.854413
$927$	−13.0000	−0.426976
$928$	−6.00000	−0.196960
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	16.0000	0.524661
$931$	−7.00000	−0.229416
$932$	12.0000	0.393073
$933$	24.0000	0.785725
$934$	12.0000	0.392652
$935$	0	0
$936$	2.00000	0.0653720
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	5.00000	0.163256
$939$	−4.00000	−0.130535
$940$	0	0
$941$	21.0000	0.684580	0.342290	−	0.939594i	$-0.388797\pi$
$941$	21.0000	0.684580	0.342290	+	0.939594i	$0.388797\pi$
$942$	32.0000	1.04262
$943$	0	0
$944$	−3.00000	−0.0976417
$945$	−4.00000	−0.130120
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	2.00000	0.0649570
$949$	22.0000	0.714150
$950$	−7.00000	−0.227110
$951$	−6.00000	−0.194563
$952$	3.00000	0.0972306
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	3.00000	0.0971286
$955$	−24.0000	−0.776622
$956$	0	0
$957$	0	0
$958$	36.0000	1.16311
$959$	0	0
$960$	2.00000	0.0645497
$961$	33.0000	1.06452
$962$	−20.0000	−0.644826
$963$	9.00000	0.290021
$964$	8.00000	0.257663
$965$	−20.0000	−0.643823
$966$	−12.0000	−0.386094
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	0	0
$969$	42.0000	1.34923
$970$	1.00000	0.0321081
$971$	51.0000	1.63667	0.818334	−	0.574743i	$-0.194898\pi$
$971$	51.0000	1.63667	0.818334	+	0.574743i	$0.194898\pi$
$972$	10.0000	0.320750
$973$	11.0000	0.352644
$974$	−4.00000	−0.128168
$975$	−4.00000	−0.128103
$976$	11.0000	0.352101
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	−22.0000	−0.703482
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−4.00000	−0.127710
$982$	24.0000	0.765871
$983$	−60.0000	−1.91370	−0.956851	−	0.290578i	$-0.906153\pi$
$983$	−60.0000	−1.91370	−0.956851	+	0.290578i	$0.906153\pi$
$984$	0	0
$985$	27.0000	0.860292
$986$	−18.0000	−0.573237
$987$	0	0
$988$	−14.0000	−0.445399
$989$	−6.00000	−0.190789
$990$	0	0
$991$	−55.0000	−1.74713	−0.873566	−	0.486705i	$-0.838199\pi$
$991$	−55.0000	−1.74713	−0.873566	+	0.486705i	$0.838199\pi$
$992$	8.00000	0.254000
$993$	20.0000	0.634681
$994$	−15.0000	−0.475771
$995$	−14.0000	−0.443830
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	44.0000	1.39280
$999$	−40.0000	−1.26554

Twists

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

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