LMFDB - Embedded newform 5070.2.b.d.1351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5070.b (of order $2$ , degree $1$ , not minimal)

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:	no
Analytic conductor:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.d.1351.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.00000i q^{11} +1.00000 q^{12} +1.00000i q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000i q^{18} -4.00000i q^{19} +1.00000i q^{20} +4.00000 q^{22} -8.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +6.00000 q^{29} +1.00000 q^{30} +8.00000i q^{31} -1.00000i q^{32} -4.00000i q^{33} -6.00000i q^{34} -1.00000 q^{36} -10.0000i q^{37} -4.00000 q^{38} +1.00000 q^{40} +6.00000i q^{41} -4.00000 q^{43} -4.00000i q^{44} -1.00000i q^{45} +8.00000i q^{46} -1.00000 q^{48} +7.00000 q^{49} +1.00000i q^{50} -6.00000 q^{51} -10.0000 q^{53} +1.00000i q^{54} +4.00000 q^{55} +4.00000i q^{57} -6.00000i q^{58} +4.00000i q^{59} -1.00000i q^{60} -2.00000 q^{61} +8.00000 q^{62} -1.00000 q^{64} -4.00000 q^{66} +12.0000i q^{67} -6.00000 q^{68} +8.00000 q^{69} -16.0000i q^{71} +1.00000i q^{72} +2.00000i q^{73} -10.0000 q^{74} +1.00000 q^{75} +4.00000i q^{76} -16.0000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} +12.0000i q^{83} -6.00000i q^{85} +4.00000i q^{86} -6.00000 q^{87} -4.00000 q^{88} +10.0000i q^{89} -1.00000 q^{90} +8.00000 q^{92} -8.00000i q^{93} -4.00000 q^{95} +1.00000i q^{96} +6.00000i q^{97} -7.00000i q^{98} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} + 2 q^{16} + 12 q^{17} + 8 q^{22} - 16 q^{23} - 2 q^{25} - 2 q^{27} + 12 q^{29} + 2 q^{30} - 2 q^{36} - 8 q^{38} + 2 q^{40} - 8 q^{43} - 2 q^{48}+ \cdots - 8 q^{95}+O(q^{100})$ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^12 + 2 * q^16 + 12 * q^17 + 8 * q^22 - 16 * q^23 - 2 * q^25 - 2 * q^27 + 12 * q^29 + 2 * q^30 - 2 * q^36 - 8 * q^38 + 2 * q^40 - 8 * q^43 - 2 * q^48 + 14 * q^49 - 12 * q^51 - 20 * q^53 + 8 * q^55 - 4 * q^61 + 16 * q^62 - 2 * q^64 - 8 * q^66 - 12 * q^68 + 16 * q^69 - 20 * q^74 + 2 * q^75 - 32 * q^79 + 2 * q^81 + 12 * q^82 - 12 * q^87 - 8 * q^88 - 2 * q^90 + 16 * q^92 - 8 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$ .

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	1.00000i	0.408248i
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	− 1.00000i	− 0.235702i
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	1.00000i	0.223607i
$21$	0	0
$22$	4.00000	0.852803
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	1.00000	0.182574
$31$	8.00000i	1.43684i	0.695608	+	0.718421i	$0.255135\pi$
$31$	8.00000i	1.43684i	−0.695608	+	0.718421i	$0.744865\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.00000i	− 0.696311i
$34$	− 6.00000i	− 1.02899i
$35$	0	0
$36$	−1.00000	−0.166667
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	1.00000	0.158114
$41$	6.00000i	0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$	6.00000i	0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	− 4.00000i	− 0.603023i
$45$	− 1.00000i	− 0.149071i
$46$	8.00000i	1.17954i
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−1.00000	−0.144338
$49$	7.00000	1.00000
$50$	1.00000i	0.141421i
$51$	−6.00000	−0.840168
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	1.00000i	0.136083i
$55$	4.00000	0.539360
$56$	0	0
$57$	4.00000i	0.529813i
$58$	− 6.00000i	− 0.787839i
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	− 1.00000i	− 0.129099i
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	−6.00000	−0.727607
$69$	8.00000	0.963087
$70$	0	0
$71$	− 16.0000i	− 1.89885i	−0.313993	−	0.949425i	$-0.601667\pi$
$71$	− 16.0000i	− 1.89885i	0.313993	−	0.949425i	$-0.398333\pi$
$72$	1.00000i	0.117851i
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	−10.0000	−1.16248
$75$	1.00000	0.115470
$76$	4.00000i	0.458831i
$77$	0	0
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	− 6.00000i	− 0.650791i
$86$	4.00000i	0.431331i
$87$	−6.00000	−0.643268
$88$	−4.00000	−0.426401
$89$	10.0000i	1.06000i	0.847998	+	0.529999i	$0.177808\pi$
$89$	10.0000i	1.06000i	−0.847998	+	0.529999i	$0.822192\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	8.00000	0.834058
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	−4.00000	−0.410391
$96$	1.00000i	0.102062i
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	− 7.00000i	− 0.707107i
$99$	4.00000i	0.402015i
$100$	1.00000	0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	6.00000i	0.594089i
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	10.0000i	0.971286i
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	10.0000i	0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$	10.0000i	0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$	− 4.00000i	− 0.381385i
$111$	10.0000i	0.949158i
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	4.00000	0.374634
$115$	8.00000i	0.746004i
$116$	−6.00000	−0.557086
$117$	0	0
$118$	4.00000	0.368230
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−5.00000	−0.454545
$122$	2.00000i	0.181071i
$123$	− 6.00000i	− 0.541002i
$124$	− 8.00000i	− 0.718421i
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000i	0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	4.00000i	0.348155i
$133$	0	0
$134$	12.0000	1.03664
$135$	1.00000i	0.0860663i
$136$	6.00000i	0.514496i
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	− 8.00000i	− 0.681005i
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	−16.0000	−1.34269
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 6.00000i	− 0.498273i
$146$	2.00000	0.165521
$147$	−7.00000	−0.577350
$148$	10.0000i	0.821995i
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	4.00000	0.324443
$153$	6.00000	0.485071
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	16.0000i	1.27289i
$159$	10.0000	0.793052
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	− 6.00000i	− 0.468521i
$165$	−4.00000	−0.311400
$166$	12.0000	0.931381
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	0	0
$169$	0	0
$170$	−6.00000	−0.460179
$171$	− 4.00000i	− 0.305888i
$172$	4.00000	0.304997
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	6.00000i	0.454859i
$175$	0	0
$176$	4.00000i	0.301511i
$177$	− 4.00000i	− 0.300658i
$178$	10.0000	0.749532
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	1.00000i	0.0745356i
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	− 8.00000i	− 0.589768i
$185$	−10.0000	−0.735215
$186$	−8.00000	−0.586588
$187$	24.0000i	1.75505i
$188$	0	0
$189$	0	0
$190$	4.00000i	0.290191i
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	4.00000	0.284268
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	− 12.0000i	− 0.846415i
$202$	− 2.00000i	− 0.140720i
$203$	0	0
$204$	6.00000	0.420084
$205$	6.00000	0.419058
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	10.0000	0.686803
$213$	16.0000i	1.09630i
$214$	12.0000i	0.820303i
$215$	4.00000i	0.272798i
$216$	− 1.00000i	− 0.0680414i
$217$	0	0
$218$	10.0000	0.677285
$219$	− 2.00000i	− 0.135147i
$220$	−4.00000	−0.269680
$221$	0	0
$222$	10.0000	0.671156
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	− 10.0000i	− 0.665190i
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	− 4.00000i	− 0.264906i
$229$	14.0000i	0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$	14.0000i	0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	6.00000i	0.393919i
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	0	0
$236$	− 4.00000i	− 0.260378i
$237$	16.0000	1.03931
$238$	0	0
$239$	8.00000i	0.517477i	0.965947	+	0.258738i	$0.0833068\pi$
$239$	8.00000i	0.517477i	−0.965947	+	0.258738i	$0.916693\pi$
$240$	1.00000i	0.0645497i
$241$	18.0000i	1.15948i	0.814801	+	0.579741i	$0.196846\pi$
$241$	18.0000i	1.15948i	−0.814801	+	0.579741i	$0.803154\pi$
$242$	5.00000i	0.321412i
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	− 7.00000i	− 0.447214i
$246$	−6.00000	−0.382546
$247$	0	0
$248$	−8.00000	−0.508001
$249$	− 12.0000i	− 0.760469i
$250$	1.00000	0.0632456
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	0	0
$253$	− 32.0000i	− 2.01182i
$254$	8.00000i	0.501965i
$255$	6.00000i	0.375735i
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	− 4.00000i	− 0.249029i
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	− 12.0000i	− 0.741362i
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	4.00000	0.246183
$265$	10.0000i	0.614295i
$266$	0	0
$267$	− 10.0000i	− 0.611990i
$268$	− 12.0000i	− 0.733017i
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	1.00000	0.0608581
$271$	24.0000i	1.45790i	0.684569	+	0.728948i	$0.259990\pi$
$271$	24.0000i	1.45790i	−0.684569	+	0.728948i	$0.740010\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	10.0000	0.604122
$275$	− 4.00000i	− 0.241209i
$276$	−8.00000	−0.481543
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	− 12.0000i	− 0.719712i
$279$	8.00000i	0.478947i
$280$	0	0
$281$	10.0000i	0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$	10.0000i	0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	16.0000i	0.949425i
$285$	4.00000	0.236940
$286$	0	0
$287$	0	0
$288$	− 1.00000i	− 0.0589256i
$289$	19.0000	1.11765
$290$	−6.00000	−0.352332
$291$	− 6.00000i	− 0.351726i
$292$	− 2.00000i	− 0.117041i
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	7.00000i	0.408248i
$295$	4.00000	0.232889
$296$	10.0000	0.581238
$297$	− 4.00000i	− 0.232104i
$298$	−6.00000	−0.347571
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	0	0
$303$	−2.00000	−0.114897
$304$	− 4.00000i	− 0.229416i
$305$	2.00000i	0.114520i
$306$	− 6.00000i	− 0.342997i
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	0	0
$310$	− 8.00000i	− 0.454369i
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	2.00000i	0.112867i
$315$	0	0
$316$	16.0000	0.900070
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	− 10.0000i	− 0.560772i
$319$	24.0000i	1.34374i
$320$	1.00000i	0.0559017i
$321$	12.0000	0.669775
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	− 10.0000i	− 0.553001i
$328$	−6.00000	−0.331295
$329$	0	0
$330$	4.00000i	0.220193i
$331$	20.0000i	1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$	20.0000i	1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$	− 12.0000i	− 0.658586i
$333$	− 10.0000i	− 0.547997i
$334$	8.00000	0.437741
$335$	12.0000	0.655630
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	6.00000i	0.325396i
$341$	−32.0000	−1.73290
$342$	−4.00000	−0.216295
$343$	0	0
$344$	− 4.00000i	− 0.215666i
$345$	− 8.00000i	− 0.430706i
$346$	− 18.0000i	− 0.967686i
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	6.00000	0.321634
$349$	6.00000i	0.321173i	0.987022	+	0.160586i	$0.0513385\pi$
$349$	6.00000i	0.321173i	−0.987022	+	0.160586i	$0.948662\pi$
$350$	0	0
$351$	0	0
$352$	4.00000	0.213201
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	−4.00000	−0.212598
$355$	−16.0000	−0.849192
$356$	− 10.0000i	− 0.529999i
$357$	0	0
$358$	12.0000i	0.634220i
$359$	16.0000i	0.844448i	0.906492	+	0.422224i	$0.138750\pi$
$359$	16.0000i	0.844448i	−0.906492	+	0.422224i	$0.861250\pi$
$360$	1.00000	0.0527046
$361$	3.00000	0.157895
$362$	6.00000i	0.315353i
$363$	5.00000	0.262432
$364$	0	0
$365$	2.00000	0.104685
$366$	− 2.00000i	− 0.104542i
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	−8.00000	−0.417029
$369$	6.00000i	0.312348i
$370$	10.0000i	0.519875i
$371$	0	0
$372$	8.00000i	0.414781i
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	24.0000	1.24101
$375$	− 1.00000i	− 0.0516398i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	4.00000i	0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$	4.00000i	0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	4.00000	0.205196
$381$	8.00000	0.409852
$382$	0	0
$383$	− 32.0000i	− 1.63512i	−0.575841	−	0.817562i	$-0.695325\pi$
$383$	− 32.0000i	− 1.63512i	0.575841	−	0.817562i	$-0.304675\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	0	0
$386$	10.0000	0.508987
$387$	−4.00000	−0.203331
$388$	− 6.00000i	− 0.304604i
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	7.00000i	0.353553i
$393$	−12.0000	−0.605320
$394$	−6.00000	−0.302276
$395$	16.0000i	0.805047i
$396$	− 4.00000i	− 0.201008i
$397$	30.0000i	1.50566i	0.658217	+	0.752828i	$0.271311\pi$
$397$	30.0000i	1.50566i	−0.658217	+	0.752828i	$0.728689\pi$
$398$	− 8.00000i	− 0.401004i
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	34.0000i	1.69788i	0.528490	+	0.848939i	$0.322758\pi$
$401$	34.0000i	1.69788i	−0.528490	+	0.848939i	$0.677242\pi$
$402$	−12.0000	−0.598506
$403$	0	0
$404$	−2.00000	−0.0995037
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	40.0000	1.98273
$408$	− 6.00000i	− 0.297044i
$409$	22.0000i	1.08783i	0.839140	+	0.543915i	$0.183059\pi$
$409$	22.0000i	1.08783i	−0.839140	+	0.543915i	$0.816941\pi$
$410$	− 6.00000i	− 0.296319i
$411$	− 10.0000i	− 0.493264i
$412$	0	0
$413$	0	0
$414$	8.00000i	0.393179i
$415$	12.0000	0.589057
$416$	0	0
$417$	−12.0000	−0.587643
$418$	− 16.0000i	− 0.782586i
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	34.0000i	1.65706i	0.559946	+	0.828529i	$0.310822\pi$
$421$	34.0000i	1.65706i	−0.559946	+	0.828529i	$0.689178\pi$
$422$	12.0000i	0.584151i
$423$	0	0
$424$	− 10.0000i	− 0.485643i
$425$	−6.00000	−0.291043
$426$	16.0000	0.775203
$427$	0	0
$428$	12.0000	0.580042
$429$	0	0
$430$	4.00000	0.192897
$431$	24.0000i	1.15604i	0.816023	+	0.578020i	$0.196174\pi$
$431$	24.0000i	1.15604i	−0.816023	+	0.578020i	$0.803826\pi$
$432$	−1.00000	−0.0481125
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	6.00000i	0.287678i
$436$	− 10.0000i	− 0.478913i
$437$	32.0000i	1.53077i
$438$	−2.00000	−0.0955637
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	4.00000i	0.190693i
$441$	7.00000	0.333333
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	− 10.0000i	− 0.474579i
$445$	10.0000	0.474045
$446$	−8.00000	−0.378811
$447$	6.00000i	0.283790i
$448$	0	0
$449$	18.0000i	0.849473i	0.905317	+	0.424736i	$0.139633\pi$
$449$	18.0000i	0.849473i	−0.905317	+	0.424736i	$0.860367\pi$
$450$	1.00000i	0.0471405i
$451$	−24.0000	−1.13012
$452$	−10.0000	−0.470360
$453$	0	0
$454$	−4.00000	−0.187729
$455$	0	0
$456$	−4.00000	−0.187317
$457$	− 34.0000i	− 1.59045i	−0.606313	−	0.795226i	$-0.707352\pi$
$457$	− 34.0000i	− 1.59045i	0.606313	−	0.795226i	$-0.292648\pi$
$458$	14.0000	0.654177
$459$	−6.00000	−0.280056
$460$	− 8.00000i	− 0.373002i
$461$	18.0000i	0.838344i	0.907907	+	0.419172i	$0.137680\pi$
$461$	18.0000i	0.838344i	−0.907907	+	0.419172i	$0.862320\pi$
$462$	0	0
$463$	8.00000i	0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$	8.00000i	0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$	6.00000	0.278543
$465$	−8.00000	−0.370991
$466$	− 14.0000i	− 0.648537i
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2.00000	0.0921551
$472$	−4.00000	−0.184115
$473$	− 16.0000i	− 0.735681i
$474$	− 16.0000i	− 0.734904i
$475$	4.00000i	0.183533i
$476$	0	0
$477$	−10.0000	−0.457869
$478$	8.00000	0.365911
$479$	− 8.00000i	− 0.365529i	−0.983157	−	0.182765i	$-0.941495\pi$
$479$	− 8.00000i	− 0.365529i	0.983157	−	0.182765i	$-0.0585046\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	18.0000	0.819878
$483$	0	0
$484$	5.00000	0.227273
$485$	6.00000	0.272446
$486$	1.00000i	0.0453609i
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	− 4.00000i	− 0.180886i
$490$	−7.00000	−0.316228
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	6.00000i	0.270501i
$493$	36.0000	1.62136
$494$	0	0
$495$	4.00000	0.179787
$496$	8.00000i	0.359211i
$497$	0	0
$498$	−12.0000	−0.537733
$499$	− 20.0000i	− 0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$	− 20.0000i	− 0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 8.00000i	− 0.357414i
$502$	− 28.0000i	− 1.24970i
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	0	0
$505$	− 2.00000i	− 0.0889988i
$506$	−32.0000	−1.42257
$507$	0	0
$508$	8.00000	0.354943
$509$	− 30.0000i	− 1.32973i	−0.746965	−	0.664863i	$-0.768490\pi$
$509$	− 30.0000i	− 1.32973i	0.746965	−	0.664863i	$-0.231510\pi$
$510$	6.00000	0.265684
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	4.00000i	0.176604i
$514$	− 6.00000i	− 0.264649i
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	− 6.00000i	− 0.262613i
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	− 24.0000i	− 1.04645i
$527$	48.0000i	2.09091i
$528$	− 4.00000i	− 0.174078i
$529$	41.0000	1.78261
$530$	10.0000	0.434372
$531$	4.00000i	0.173585i
$532$	0	0
$533$	0	0
$534$	−10.0000	−0.432742
$535$	12.0000i	0.518805i
$536$	−12.0000	−0.518321
$537$	12.0000	0.517838
$538$	26.0000i	1.12094i
$539$	28.0000i	1.20605i
$540$	− 1.00000i	− 0.0430331i
$541$	22.0000i	0.945854i	0.881102	+	0.472927i	$0.156803\pi$
$541$	22.0000i	0.945854i	−0.881102	+	0.472927i	$0.843197\pi$
$542$	24.0000	1.03089
$543$	6.00000	0.257485
$544$	− 6.00000i	− 0.257248i
$545$	10.0000	0.428353
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	− 10.0000i	− 0.427179i
$549$	−2.00000	−0.0853579
$550$	−4.00000	−0.170561
$551$	− 24.0000i	− 1.02243i
$552$	8.00000i	0.340503i
$553$	0	0
$554$	− 10.0000i	− 0.424859i
$555$	10.0000	0.424476
$556$	−12.0000	−0.508913
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	8.00000	0.338667
$559$	0	0
$560$	0	0
$561$	− 24.0000i	− 1.01328i
$562$	10.0000	0.421825
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	− 10.0000i	− 0.420703i
$566$	20.0000i	0.840663i
$567$	0	0
$568$	16.0000	0.671345
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	− 4.00000i	− 0.167542i
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.00000	0.333623
$576$	−1.00000	−0.0416667
$577$	− 42.0000i	− 1.74848i	−0.485491	−	0.874241i	$-0.661359\pi$
$577$	− 42.0000i	− 1.74848i	0.485491	−	0.874241i	$-0.338641\pi$
$578$	− 19.0000i	− 0.790296i
$579$	− 10.0000i	− 0.415586i
$580$	6.00000i	0.249136i
$581$	0	0
$582$	−6.00000	−0.248708
$583$	− 40.0000i	− 1.65663i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−26.0000	−1.07405
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	7.00000	0.288675
$589$	32.0000	1.31854
$590$	− 4.00000i	− 0.164677i
$591$	6.00000i	0.246807i
$592$	− 10.0000i	− 0.410997i
$593$	34.0000i	1.39621i	0.715994	+	0.698106i	$0.245974\pi$
$593$	34.0000i	1.39621i	−0.715994	+	0.698106i	$0.754026\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	6.00000i	0.245770i
$597$	−8.00000	−0.327418
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	1.00000i	0.0408248i
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	0	0
$603$	12.0000i	0.488678i
$604$	0	0
$605$	5.00000i	0.203279i
$606$	2.00000i	0.0812444i
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	2.00000	0.0809776
$611$	0	0
$612$	−6.00000	−0.242536
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	−12.0000	−0.484281
$615$	−6.00000	−0.241943
$616$	0	0
$617$	− 42.0000i	− 1.69086i	−0.534089	−	0.845428i	$-0.679345\pi$
$617$	− 42.0000i	− 1.69086i	0.534089	−	0.845428i	$-0.320655\pi$
$618$	0	0
$619$	− 4.00000i	− 0.160774i	−0.996764	−	0.0803868i	$-0.974384\pi$
$619$	− 4.00000i	− 0.160774i	0.996764	−	0.0803868i	$-0.0256155\pi$
$620$	−8.00000	−0.321288
$621$	8.00000	0.321029
$622$	− 24.0000i	− 0.962312i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	22.0000i	0.879297i
$627$	−16.0000	−0.638978
$628$	2.00000	0.0798087
$629$	− 60.0000i	− 2.39236i
$630$	0	0
$631$	− 16.0000i	− 0.636950i	−0.947931	−	0.318475i	$-0.896829\pi$
$631$	− 16.0000i	− 0.636950i	0.947931	−	0.318475i	$-0.103171\pi$
$632$	− 16.0000i	− 0.636446i
$633$	12.0000	0.476957
$634$	2.00000	0.0794301
$635$	8.00000i	0.317470i
$636$	−10.0000	−0.396526
$637$	0	0
$638$	24.0000	0.950169
$639$	− 16.0000i	− 0.632950i
$640$	1.00000	0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	− 12.0000i	− 0.473602i
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	0	0
$645$	− 4.00000i	− 0.157500i
$646$	−24.0000	−0.944267
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	1.00000i	0.0392837i
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	−10.0000	−0.391031
$655$	− 12.0000i	− 0.468879i
$656$	6.00000i	0.234261i
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	4.00000	0.155700
$661$	14.0000i	0.544537i	0.962221	+	0.272268i	$0.0877739\pi$
$661$	14.0000i	0.544537i	−0.962221	+	0.272268i	$0.912226\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−10.0000	−0.387492
$667$	−48.0000	−1.85857
$668$	− 8.00000i	− 0.309529i
$669$	8.00000i	0.309298i
$670$	− 12.0000i	− 0.463600i
$671$	− 8.00000i	− 0.308837i
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	2.00000i	0.0770371i
$675$	1.00000	0.0384900
$676$	0	0
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	10.0000i	0.384048i
$679$	0	0
$680$	6.00000	0.230089
$681$	4.00000i	0.153280i
$682$	32.0000i	1.22534i
$683$	28.0000i	1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$	28.0000i	1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$	4.00000i	0.152944i
$685$	10.0000	0.382080
$686$	0	0
$687$	− 14.0000i	− 0.534133i
$688$	−4.00000	−0.152499
$689$	0	0
$690$	−8.00000	−0.304555
$691$	12.0000i	0.456502i	0.973602	+	0.228251i	$0.0733006\pi$
$691$	12.0000i	0.456502i	−0.973602	+	0.228251i	$0.926699\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	− 20.0000i	− 0.759190i
$695$	− 12.0000i	− 0.455186i
$696$	− 6.00000i	− 0.227429i
$697$	36.0000i	1.36360i
$698$	6.00000	0.227103
$699$	−14.0000	−0.529529
$700$	0	0
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	0	0
$703$	−40.0000	−1.50863
$704$	− 4.00000i	− 0.150756i
$705$	0	0
$706$	30.0000	1.12906
$707$	0	0
$708$	4.00000i	0.150329i
$709$	− 50.0000i	− 1.87779i	−0.344204	−	0.938895i	$-0.611851\pi$
$709$	− 50.0000i	− 1.87779i	0.344204	−	0.938895i	$-0.388149\pi$
$710$	16.0000i	0.600469i
$711$	−16.0000	−0.600047
$712$	−10.0000	−0.374766
$713$	− 64.0000i	− 2.39682i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	− 8.00000i	− 0.298765i
$718$	16.0000	0.597115
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	0	0
$722$	− 3.00000i	− 0.111648i
$723$	− 18.0000i	− 0.669427i
$724$	6.00000	0.222988
$725$	−6.00000	−0.222834
$726$	− 5.00000i	− 0.185567i
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 2.00000i	− 0.0740233i
$731$	−24.0000	−0.887672
$732$	−2.00000	−0.0739221
$733$	50.0000i	1.84679i	0.383849	+	0.923396i	$0.374598\pi$
$733$	50.0000i	1.84679i	−0.383849	+	0.923396i	$0.625402\pi$
$734$	24.0000i	0.885856i
$735$	7.00000i	0.258199i
$736$	8.00000i	0.294884i
$737$	−48.0000	−1.76810
$738$	6.00000	0.220863
$739$	20.0000i	0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$	20.0000i	0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$	10.0000	0.367607
$741$	0	0
$742$	0	0
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	8.00000	0.293294
$745$	−6.00000	−0.219823
$746$	− 22.0000i	− 0.805477i
$747$	12.0000i	0.439057i
$748$	− 24.0000i	− 0.877527i
$749$	0	0
$750$	−1.00000	−0.0365148
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	−28.0000	−1.02038
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	4.00000	0.145287
$759$	32.0000i	1.16153i
$760$	− 4.00000i	− 0.145095i
$761$	42.0000i	1.52250i	0.648459	+	0.761249i	$0.275414\pi$
$761$	42.0000i	1.52250i	−0.648459	+	0.761249i	$0.724586\pi$
$762$	− 8.00000i	− 0.289809i
$763$	0	0
$764$	0	0
$765$	− 6.00000i	− 0.216930i
$766$	−32.0000	−1.15621
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 34.0000i	− 1.22607i	−0.790055	−	0.613036i	$-0.789948\pi$
$769$	− 34.0000i	− 1.22607i	0.790055	−	0.613036i	$-0.210052\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	− 10.0000i	− 0.359908i
$773$	− 38.0000i	− 1.36677i	−0.730061	−	0.683383i	$-0.760508\pi$
$773$	− 38.0000i	− 1.36677i	0.730061	−	0.683383i	$-0.239492\pi$
$774$	4.00000i	0.143777i
$775$	− 8.00000i	− 0.287368i
$776$	−6.00000	−0.215387
$777$	0	0
$778$	− 34.0000i	− 1.21896i
$779$	24.0000	0.859889
$780$	0	0
$781$	64.0000	2.29010
$782$	48.0000i	1.71648i
$783$	−6.00000	−0.214423
$784$	7.00000	0.250000
$785$	2.00000i	0.0713831i
$786$	12.0000i	0.428026i
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	6.00000i	0.213741i
$789$	−24.0000	−0.854423
$790$	16.0000	0.569254
$791$	0	0
$792$	−4.00000	−0.142134
$793$	0	0
$794$	30.0000	1.06466
$795$	− 10.0000i	− 0.354663i
$796$	−8.00000	−0.283552
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	0	0
$799$	0	0
$800$	1.00000i	0.0353553i
$801$	10.0000i	0.353333i
$802$	34.0000	1.20058
$803$	−8.00000	−0.282314
$804$	12.0000i	0.423207i
$805$	0	0
$806$	0	0
$807$	26.0000	0.915243
$808$	2.00000i	0.0703598i
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	−1.00000	−0.0351364
$811$	− 12.0000i	− 0.421377i	−0.977553	−	0.210688i	$-0.932429\pi$
$811$	− 12.0000i	− 0.421377i	0.977553	−	0.210688i	$-0.0675706\pi$
$812$	0	0
$813$	− 24.0000i	− 0.841717i
$814$	− 40.0000i	− 1.40200i
$815$	4.00000	0.140114
$816$	−6.00000	−0.210042
$817$	16.0000i	0.559769i
$818$	22.0000	0.769212
$819$	0	0
$820$	−6.00000	−0.209529
$821$	− 6.00000i	− 0.209401i	−0.994504	−	0.104701i	$-0.966612\pi$
$821$	− 6.00000i	− 0.209401i	0.994504	−	0.104701i	$-0.0333885\pi$
$822$	−10.0000	−0.348790
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	4.00000i	0.139262i
$826$	0	0
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	8.00000	0.278019
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	− 12.0000i	− 0.416526i
$831$	−10.0000	−0.346896
$832$	0	0
$833$	42.0000	1.45521
$834$	12.0000i	0.415526i
$835$	8.00000	0.276851
$836$	−16.0000	−0.553372
$837$	− 8.00000i	− 0.276520i
$838$	4.00000i	0.138178i
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	34.0000	1.17172
$843$	− 10.0000i	− 0.344418i
$844$	12.0000	0.413057
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−10.0000	−0.343401
$849$	20.0000	0.686398
$850$	6.00000i	0.205798i
$851$	80.0000i	2.74236i
$852$	− 16.0000i	− 0.548151i
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	− 12.0000i	− 0.410152i
$857$	−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
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	− 4.00000i	− 0.136399i
$861$	0	0
$862$	24.0000	0.817443
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	1.00000i	0.0340207i
$865$	− 18.0000i	− 0.612018i
$866$	34.0000i	1.15537i
$867$	−19.0000	−0.645274
$868$	0	0
$869$	− 64.0000i	− 2.17105i
$870$	6.00000	0.203419
$871$	0	0
$872$	−10.0000	−0.338643
$873$	6.00000i	0.203069i
$874$	32.0000	1.08242
$875$	0	0
$876$	2.00000i	0.0675737i
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	− 40.0000i	− 1.34993i
$879$	26.0000i	0.876958i
$880$	4.00000	0.134840
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	− 7.00000i	− 0.235702i
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	− 36.0000i	− 1.20944i
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	−10.0000	−0.335578
$889$	0	0
$890$	− 10.0000i	− 0.335201i
$891$	4.00000i	0.134005i
$892$	8.00000i	0.267860i
$893$	0	0
$894$	6.00000	0.200670
$895$	12.0000i	0.401116i
$896$	0	0
$897$	0	0
$898$	18.0000	0.600668
$899$	48.0000i	1.60089i
$900$	1.00000	0.0333333
$901$	−60.0000	−1.99889
$902$	24.0000i	0.799113i
$903$	0	0
$904$	10.0000i	0.332595i
$905$	6.00000i	0.199447i
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	4.00000i	0.132745i
$909$	2.00000	0.0663358
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	4.00000i	0.132453i
$913$	−48.0000	−1.58857
$914$	−34.0000	−1.12462
$915$	− 2.00000i	− 0.0661180i
$916$	− 14.0000i	− 0.462573i
$917$	0	0
$918$	6.00000i	0.198030i
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	−8.00000	−0.263752
$921$	12.0000i	0.395413i
$922$	18.0000	0.592798
$923$	0	0
$924$	0	0
$925$	10.0000i	0.328798i
$926$	8.00000	0.262896
$927$	0	0
$928$	− 6.00000i	− 0.196960i
$929$	14.0000i	0.459325i	0.973270	+	0.229663i	$0.0737623\pi$
$929$	14.0000i	0.459325i	−0.973270	+	0.229663i	$0.926238\pi$
$930$	8.00000i	0.262330i
$931$	− 28.0000i	− 0.917663i
$932$	−14.0000	−0.458585
$933$	−24.0000	−0.785725
$934$	12.0000i	0.392652i
$935$	24.0000	0.784884
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	22.0000	0.717943
$940$	0	0
$941$	− 30.0000i	− 0.977972i	−0.872292	−	0.488986i	$-0.837367\pi$
$941$	− 30.0000i	− 0.977972i	0.872292	−	0.488986i	$-0.162633\pi$
$942$	− 2.00000i	− 0.0651635i
$943$	− 48.0000i	− 1.56310i
$944$	4.00000i	0.130189i
$945$	0	0
$946$	−16.0000	−0.520205
$947$	52.0000i	1.68977i	0.534946	+	0.844886i	$0.320332\pi$
$947$	52.0000i	1.68977i	−0.534946	+	0.844886i	$0.679668\pi$
$948$	−16.0000	−0.519656
$949$	0	0
$950$	4.00000	0.129777
$951$	− 2.00000i	− 0.0648544i
$952$	0	0
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	10.0000i	0.323762i
$955$	0	0
$956$	− 8.00000i	− 0.258738i
$957$	− 24.0000i	− 0.775810i
$958$	−8.00000	−0.258468
$959$	0	0
$960$	− 1.00000i	− 0.0322749i
$961$	−33.0000	−1.06452
$962$	0	0
$963$	−12.0000	−0.386695
$964$	− 18.0000i	− 0.579741i
$965$	10.0000	0.321911
$966$	0	0
$967$	− 16.0000i	− 0.514525i	−0.966342	−	0.257263i	$-0.917179\pi$
$967$	− 16.0000i	− 0.514525i	0.966342	−	0.257263i	$-0.0828206\pi$
$968$	− 5.00000i	− 0.160706i
$969$	24.0000i	0.770991i
$970$	− 6.00000i	− 0.192648i
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	−4.00000	−0.127906
$979$	−40.0000	−1.27841
$980$	7.00000i	0.223607i
$981$	10.0000i	0.319275i
$982$	4.00000i	0.127645i
$983$	8.00000i	0.255160i	0.991828	+	0.127580i	$0.0407210\pi$
$983$	8.00000i	0.255160i	−0.991828	+	0.127580i	$0.959279\pi$
$984$	6.00000	0.191273
$985$	−6.00000	−0.191176
$986$	− 36.0000i	− 1.14647i
$987$	0	0
$988$	0	0
$989$	32.0000	1.01754
$990$	− 4.00000i	− 0.127128i
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	8.00000	0.254000
$993$	− 20.0000i	− 0.634681i
$994$	0	0
$995$	− 8.00000i	− 0.253617i
$996$	12.0000i	0.380235i
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	−20.0000	−0.633089
$999$	10.0000i	0.316386i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.d.1351.1		2
13.5	odd	4		5070.2.a.a.1.1		1
13.8	odd	4		390.2.a.f.1.1	✓	1
13.12	even	2	inner	5070.2.b.d.1351.2		2
39.8	even	4		1170.2.a.a.1.1		1
52.47	even	4		3120.2.a.w.1.1		1
65.8	even	4		1950.2.e.g.1249.1		2
65.34	odd	4		1950.2.a.k.1.1		1
65.47	even	4		1950.2.e.g.1249.2		2
156.47	odd	4		9360.2.a.p.1.1		1
195.8	odd	4		5850.2.e.e.5149.2		2
195.47	odd	4		5850.2.e.e.5149.1		2
195.164	even	4		5850.2.a.bo.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.f.1.1	✓	1	13.8	odd	4
1170.2.a.a.1.1		1	39.8	even	4
1950.2.a.k.1.1		1	65.34	odd	4
1950.2.e.g.1249.1		2	65.8	even	4
1950.2.e.g.1249.2		2	65.47	even	4
3120.2.a.w.1.1		1	52.47	even	4
5070.2.a.a.1.1		1	13.5	odd	4
5070.2.b.d.1351.1		2	1.1	even	1	trivial
5070.2.b.d.1351.2		2	13.12	even	2	inner
5850.2.a.bo.1.1		1	195.164	even	4
5850.2.e.e.5149.1		2	195.47	odd	4
5850.2.e.e.5149.2		2	195.8	odd	4
9360.2.a.p.1.1		1	156.47	odd	4

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	5070.2.b.d.1351.1

Level	$5070$
Weight	$2$
Character	5070.1351
Analytic conductor	$40.484$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$