LMFDB - Embedded newform 5070.2.b.m.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.m.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} -2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -1.00000i q^{18} +2.00000i q^{19} -1.00000i q^{20} -2.00000i q^{21} +6.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +2.00000i q^{28} +1.00000 q^{30} +8.00000i q^{31} -1.00000i q^{32} +2.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} +2.00000 q^{38} -1.00000 q^{40} +6.00000i q^{41} -2.00000 q^{42} +4.00000 q^{43} +1.00000i q^{45} -6.00000i q^{46} +1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} -6.00000 q^{53} -1.00000i q^{54} +2.00000 q^{56} +2.00000i q^{57} -1.00000i q^{60} +14.0000 q^{61} +8.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} +6.00000 q^{69} -2.00000i q^{70} +1.00000i q^{72} +4.00000i q^{73} -2.00000 q^{74} -1.00000 q^{75} -2.00000i q^{76} -16.0000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} -12.0000i q^{83} +2.00000i q^{84} -4.00000i q^{86} +6.00000i q^{89} +1.00000 q^{90} -6.00000 q^{92} +8.00000i q^{93} -2.00000 q^{95} -1.00000i q^{96} -4.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} - 2 q^{12} - 4 q^{14} + 2 q^{16} + 12 q^{23} - 2 q^{25} + 2 q^{27} + 2 q^{30} + 4 q^{35} - 2 q^{36} + 4 q^{38} - 2 q^{40} - 4 q^{42} + 8 q^{43} + 2 q^{48} + 6 q^{49} - 12 q^{53} + 4 q^{56} + 28 q^{61} + 16 q^{62} - 2 q^{64} + 12 q^{69} - 4 q^{74} - 2 q^{75} - 32 q^{79} + 2 q^{81} + 12 q^{82} + 2 q^{90} - 12 q^{92} - 4 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 - 2 * q^12 - 4 * q^14 + 2 * q^16 + 12 * q^23 - 2 * q^25 + 2 * q^27 + 2 * q^30 + 4 * q^35 - 2 * q^36 + 4 * q^38 - 2 * q^40 - 4 * q^42 + 8 * q^43 + 2 * q^48 + 6 * q^49 - 12 * q^53 + 4 * q^56 + 28 * q^61 + 16 * q^62 - 2 * q^64 + 12 * q^69 - 4 * q^74 - 2 * q^75 - 32 * q^79 + 2 * q^81 + 12 * q^82 + 2 * q^90 - 12 * q^92 - 4 * q^95

Display 100 coefficients 10 coefficients

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
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−2.00000	−0.534522
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 1.00000i	− 0.235702i
$19$	2.00000i	0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$	2.00000i	0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 2.00000i	− 0.436436i
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	2.00000i	0.377964i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	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
$33$	0	0
$34$	0	0
$35$	2.00000	0.338062
$36$	−1.00000	−0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	2.00000	0.324443
$39$	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	$0.844787\pi$
$42$	−2.00000	−0.308607
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	1.00000i	0.149071i
$46$	− 6.00000i	− 0.884652i
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000	0.144338
$49$	3.00000	0.428571
$50$	1.00000i	0.141421i
$51$	0	0
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	− 1.00000i	− 0.136083i
$55$	0	0
$56$	2.00000	0.267261
$57$	2.00000i	0.264906i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	− 1.00000i	− 0.129099i
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	8.00000	1.01600
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	− 2.00000i	− 0.239046i
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.00000i	0.117851i
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−2.00000	−0.232495
$75$	−1.00000	−0.115470
$76$	− 2.00000i	− 0.229416i
$77$	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	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	2.00000i	0.218218i
$85$	0	0
$86$	− 4.00000i	− 0.431331i
$87$	0	0
$88$	0	0
$89$	6.00000i	0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$	6.00000i	0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−6.00000	−0.625543
$93$	8.00000i	0.829561i
$94$	0	0
$95$	−2.00000	−0.205196
$96$	− 1.00000i	− 0.102062i
$97$	− 4.00000i	− 0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$	− 4.00000i	− 0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	1.00000	0.100000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	6.00000i	0.582772i
$107$	−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
$109$	− 4.00000i	− 0.383131i	−0.981480	−	0.191565i	$-0.938644\pi$
$109$	− 4.00000i	− 0.383131i	0.981480	−	0.191565i	$-0.0613564\pi$
$110$	0	0
$111$	− 2.00000i	− 0.189832i
$112$	− 2.00000i	− 0.188982i
$113$	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
$115$	6.00000i	0.559503i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	11.0000	1.00000
$122$	− 14.0000i	− 1.26750i
$123$	6.00000i	0.541002i
$124$	− 8.00000i	− 0.718421i
$125$	− 1.00000i	− 0.0894427i
$126$	−2.00000	−0.178174
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000i	0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−4.00000	−0.345547
$135$	1.00000i	0.0860663i
$136$	0	0
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	− 6.00000i	− 0.510754i
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	3.00000	0.247436
$148$	2.00000i	0.164399i
$149$	18.0000i	1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$	18.0000i	1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$	1.00000i	0.0816497i
$151$	− 8.00000i	− 0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$	− 8.00000i	− 0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	16.0000i	1.27289i
$159$	−6.00000	−0.475831
$160$	1.00000	0.0790569
$161$	− 12.0000i	− 0.945732i
$162$	− 1.00000i	− 0.0785674i
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	− 6.00000i	− 0.468521i
$165$	0	0
$166$	−12.0000	−0.931381
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	2.00000	0.154303
$169$	0	0
$170$	0	0
$171$	2.00000i	0.152944i
$172$	−4.00000	−0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	2.00000i	0.151186i
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	6.00000i	0.442326i
$185$	2.00000	0.147043
$186$	8.00000	0.586588
$187$	0	0
$188$	0	0
$189$	− 2.00000i	− 0.145479i
$190$	2.00000i	0.145095i
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−1.00000	−0.0721688
$193$	− 8.00000i	− 0.575853i	−0.957653	−	0.287926i	$-0.907034\pi$
$193$	− 8.00000i	− 0.575853i	0.957653	−	0.287926i	$-0.0929658\pi$
$194$	−4.00000	−0.287183
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	− 4.00000i	− 0.282138i
$202$	− 12.0000i	− 0.844317i
$203$	0	0
$204$	0	0
$205$	−6.00000	−0.419058
$206$	− 16.0000i	− 1.11477i
$207$	6.00000	0.417029
$208$	0	0
$209$	0	0
$210$	− 2.00000i	− 0.138013i
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	12.0000i	0.820303i
$215$	4.00000i	0.272798i
$216$	1.00000i	0.0680414i
$217$	16.0000	1.08615
$218$	−4.00000	−0.270914
$219$	4.00000i	0.270295i
$220$	0	0
$221$	0	0
$222$	−2.00000	−0.134231
$223$	− 10.0000i	− 0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$	− 10.0000i	− 0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$	−2.00000	−0.133631
$225$	−1.00000	−0.0666667
$226$	− 12.0000i	− 0.798228i
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 2.00000i	− 0.132453i
$229$	4.00000i	0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$	4.00000i	0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	1.00000i	0.0645497i
$241$	− 26.0000i	− 1.67481i	−0.546585	−	0.837404i	$-0.684072\pi$
$241$	− 26.0000i	− 1.67481i	0.546585	−	0.837404i	$-0.315928\pi$
$242$	− 11.0000i	− 0.707107i
$243$	1.00000	0.0641500
$244$	−14.0000	−0.896258
$245$	3.00000i	0.191663i
$246$	6.00000	0.382546
$247$	0	0
$248$	−8.00000	−0.508001
$249$	− 12.0000i	− 0.760469i
$250$	−1.00000	−0.0632456
$251$	30.0000	1.89358	0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000	1.89358	0.946792	+	0.321847i	$0.104304\pi$
$252$	2.00000i	0.125988i
$253$	0	0
$254$	− 16.0000i	− 1.00393i
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	− 4.00000i	− 0.249029i
$259$	−4.00000	−0.248548
$260$	0	0
$261$	0	0
$262$	− 6.00000i	− 0.370681i
$263$	30.0000	1.84988	0.924940	−	0.380114i	$-0.124115\pi$
$263$	30.0000	1.84988	0.924940	+	0.380114i	$0.124115\pi$
$264$	0	0
$265$	− 6.00000i	− 0.368577i
$266$	− 4.00000i	− 0.245256i
$267$	6.00000i	0.367194i
$268$	4.00000i	0.244339i
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	1.00000	0.0608581
$271$	− 8.00000i	− 0.485965i	−0.970031	−	0.242983i	$-0.921874\pi$
$271$	− 8.00000i	− 0.485965i	0.970031	−	0.242983i	$-0.0781258\pi$
$272$	0	0
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	−6.00000	−0.361158
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	− 8.00000i	− 0.479808i
$279$	8.00000i	0.478947i
$280$	2.00000i	0.119523i
$281$	30.0000i	1.78965i	0.446417	+	0.894825i	$0.352700\pi$
$281$	30.0000i	1.78965i	−0.446417	+	0.894825i	$0.647300\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	12.0000	0.708338
$288$	− 1.00000i	− 0.0589256i
$289$	−17.0000	−1.00000
$290$	0	0
$291$	− 4.00000i	− 0.234484i
$292$	− 4.00000i	− 0.234082i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	− 3.00000i	− 0.174964i
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	18.0000	1.04271
$299$	0	0
$300$	1.00000	0.0577350
$301$	− 8.00000i	− 0.461112i
$302$	−8.00000	−0.460348
$303$	12.0000	0.689382
$304$	2.00000i	0.114708i
$305$	14.0000i	0.801638i
$306$	0	0
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	8.00000i	0.454369i
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	− 2.00000i	− 0.112867i
$315$	2.00000	0.112687
$316$	16.0000	0.900070
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	6.00000i	0.336463i
$319$	0	0
$320$	− 1.00000i	− 0.0559017i
$321$	−12.0000	−0.669775
$322$	−12.0000	−0.668734
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	16.0000	0.886158
$327$	− 4.00000i	− 0.221201i
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	2.00000i	0.109930i	0.998488	+	0.0549650i	$0.0175047\pi$
$331$	2.00000i	0.109930i	−0.998488	+	0.0549650i	$0.982495\pi$
$332$	12.0000i	0.658586i
$333$	− 2.00000i	− 0.109599i
$334$	−12.0000	−0.656611
$335$	4.00000	0.218543
$336$	− 2.00000i	− 0.109109i
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	0	0
$342$	2.00000	0.108148
$343$	− 20.0000i	− 1.07990i
$344$	4.00000i	0.215666i
$345$	6.00000i	0.323029i
$346$	18.0000i	0.967686i
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	− 20.0000i	− 1.07058i	−0.844670	−	0.535288i	$-0.820203\pi$
$349$	− 20.0000i	− 1.07058i	0.844670	−	0.535288i	$-0.179797\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	0	0
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	− 6.00000i	− 0.317999i
$357$	0	0
$358$	− 6.00000i	− 0.317110i
$359$	24.0000i	1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$	24.0000i	1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$	−1.00000	−0.0527046
$361$	15.0000	0.789474
$362$	2.00000i	0.105118i
$363$	11.0000	0.577350
$364$	0	0
$365$	−4.00000	−0.209370
$366$	− 14.0000i	− 0.731792i
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	6.00000	0.312772
$369$	6.00000i	0.312348i
$370$	− 2.00000i	− 0.103975i
$371$	12.0000i	0.623009i
$372$	− 8.00000i	− 0.414781i
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	− 1.00000i	− 0.0516398i
$376$	0	0
$377$	0	0
$378$	−2.00000	−0.102869
$379$	− 34.0000i	− 1.74646i	−0.487306	−	0.873231i	$-0.662020\pi$
$379$	− 34.0000i	− 1.74646i	0.487306	−	0.873231i	$-0.337980\pi$
$380$	2.00000	0.102598
$381$	16.0000	0.819705
$382$	0	0
$383$	12.0000i	0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$	12.0000i	0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	1.00000i	0.0510310i
$385$	0	0
$386$	−8.00000	−0.407189
$387$	4.00000	0.203331
$388$	4.00000i	0.203069i
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	0	0
$392$	3.00000i	0.151523i
$393$	6.00000	0.302660
$394$	18.0000	0.906827
$395$	− 16.0000i	− 0.805047i
$396$	0	0
$397$	− 26.0000i	− 1.30490i	−0.757831	−	0.652451i	$-0.773741\pi$
$397$	− 26.0000i	− 1.30490i	0.757831	−	0.652451i	$-0.226259\pi$
$398$	− 16.0000i	− 0.802008i
$399$	4.00000	0.200250
$400$	−1.00000	−0.0500000
$401$	18.0000i	0.898877i	0.893311	+	0.449439i	$0.148376\pi$
$401$	18.0000i	0.898877i	−0.893311	+	0.449439i	$0.851624\pi$
$402$	−4.00000	−0.199502
$403$	0	0
$404$	−12.0000	−0.597022
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	14.0000i	0.692255i	0.938187	+	0.346128i	$0.112504\pi$
$409$	14.0000i	0.692255i	−0.938187	+	0.346128i	$0.887496\pi$
$410$	6.00000i	0.296319i
$411$	− 6.00000i	− 0.295958i
$412$	−16.0000	−0.788263
$413$	0	0
$414$	− 6.00000i	− 0.294884i
$415$	12.0000	0.589057
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	−2.00000	−0.0975900
$421$	32.0000i	1.55958i	0.626038	+	0.779792i	$0.284675\pi$
$421$	32.0000i	1.55958i	−0.626038	+	0.779792i	$0.715325\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	− 6.00000i	− 0.291386i
$425$	0	0
$426$	0	0
$427$	− 28.0000i	− 1.35501i
$428$	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
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	− 16.0000i	− 0.768025i
$435$	0	0
$436$	4.00000i	0.191565i
$437$	12.0000i	0.574038i
$438$	4.00000	0.191127
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	2.00000i	0.0949158i
$445$	−6.00000	−0.284427
$446$	−10.0000	−0.473514
$447$	18.0000i	0.851371i
$448$	2.00000i	0.0944911i
$449$	30.0000i	1.41579i	0.706319	+	0.707894i	$0.250354\pi$
$449$	30.0000i	1.41579i	−0.706319	+	0.707894i	$0.749646\pi$
$450$	1.00000i	0.0471405i
$451$	0	0
$452$	−12.0000	−0.564433
$453$	− 8.00000i	− 0.375873i
$454$	12.0000	0.563188
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	− 28.0000i	− 1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$	− 28.0000i	− 1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$	4.00000	0.186908
$459$	0	0
$460$	− 6.00000i	− 0.279751i
$461$	− 42.0000i	− 1.95614i	−0.208288	−	0.978068i	$-0.566789\pi$
$461$	− 42.0000i	− 1.95614i	0.208288	−	0.978068i	$-0.433211\pi$
$462$	0	0
$463$	10.0000i	0.464739i	0.972628	+	0.232370i	$0.0746479\pi$
$463$	10.0000i	0.464739i	−0.972628	+	0.232370i	$0.925352\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	0	0
$474$	16.0000i	0.734904i
$475$	− 2.00000i	− 0.0917663i
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−26.0000	−1.18427
$483$	− 12.0000i	− 0.546019i
$484$	−11.0000	−0.500000
$485$	4.00000	0.181631
$486$	− 1.00000i	− 0.0453609i
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	14.0000i	0.633750i
$489$	16.0000i	0.723545i
$490$	3.00000	0.135526
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	− 6.00000i	− 0.270501i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	8.00000i	0.359211i
$497$	0	0
$498$	−12.0000	−0.537733
$499$	14.0000i	0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$	14.0000i	0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$	1.00000i	0.0447214i
$501$	− 12.0000i	− 0.536120i
$502$	− 30.0000i	− 1.33897i
$503$	−18.0000	−0.802580	−0.401290	−	0.915951i	$-0.631438\pi$
$503$	−18.0000	−0.802580	−0.401290	+	0.915951i	$0.631438\pi$
$504$	2.00000	0.0890871
$505$	12.0000i	0.533993i
$506$	0	0
$507$	0	0
$508$	−16.0000	−0.709885
$509$	6.00000i	0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$	6.00000i	0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	− 1.00000i	− 0.0441942i
$513$	2.00000i	0.0883022i
$514$	− 12.0000i	− 0.529297i
$515$	16.0000i	0.705044i
$516$	−4.00000	−0.176090
$517$	0	0
$518$	4.00000i	0.175750i
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−6.00000	−0.262111
$525$	2.00000i	0.0872872i
$526$	− 30.0000i	− 1.30806i
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	−6.00000	−0.260623
$531$	0	0
$532$	−4.00000	−0.173422
$533$	0	0
$534$	6.00000	0.259645
$535$	− 12.0000i	− 0.518805i
$536$	4.00000	0.172774
$537$	6.00000	0.258919
$538$	− 12.0000i	− 0.517357i
$539$	0	0
$540$	− 1.00000i	− 0.0430331i
$541$	− 8.00000i	− 0.343947i	−0.985102	−	0.171973i	$-0.944986\pi$
$541$	− 8.00000i	− 0.343947i	0.985102	−	0.171973i	$-0.0550143\pi$
$542$	−8.00000	−0.343629
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	6.00000i	0.256307i
$549$	14.0000	0.597505
$550$	0	0
$551$	0	0
$552$	6.00000i	0.255377i
$553$	32.0000i	1.36078i
$554$	26.0000i	1.10463i
$555$	2.00000	0.0848953
$556$	−8.00000	−0.339276
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	8.00000	0.338667
$559$	0	0
$560$	2.00000	0.0845154
$561$	0	0
$562$	30.0000	1.26547
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	12.0000i	0.504844i
$566$	− 28.0000i	− 1.17693i
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	2.00000i	0.0837708i
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	0	0
$574$	− 12.0000i	− 0.500870i
$575$	−6.00000	−0.250217
$576$	−1.00000	−0.0416667
$577$	− 16.0000i	− 0.666089i	−0.942911	−	0.333044i	$-0.891924\pi$
$577$	− 16.0000i	− 0.666089i	0.942911	−	0.333044i	$-0.108076\pi$
$578$	17.0000i	0.707107i
$579$	− 8.00000i	− 0.332469i
$580$	0	0
$581$	−24.0000	−0.995688
$582$	−4.00000	−0.165805
$583$	0	0
$584$	−4.00000	−0.165521
$585$	0	0
$586$	6.00000	0.247858
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	−3.00000	−0.123718
$589$	−16.0000	−0.659269
$590$	0	0
$591$	18.0000i	0.740421i
$592$	− 2.00000i	− 0.0821995i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	− 18.0000i	− 0.737309i
$597$	16.0000	0.654836
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	−8.00000	−0.326056
$603$	− 4.00000i	− 0.162893i
$604$	8.00000i	0.325515i
$605$	11.0000i	0.447214i
$606$	− 12.0000i	− 0.487467i
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	14.0000	0.566843
$611$	0	0
$612$	0	0
$613$	− 22.0000i	− 0.888572i	−0.895885	−	0.444286i	$-0.853457\pi$
$613$	− 22.0000i	− 0.888572i	0.895885	−	0.444286i	$-0.146543\pi$
$614$	4.00000	0.161427
$615$	−6.00000	−0.241943
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	− 16.0000i	− 0.643614i
$619$	− 38.0000i	− 1.52735i	−0.645601	−	0.763674i	$-0.723393\pi$
$619$	− 38.0000i	− 1.52735i	0.645601	−	0.763674i	$-0.276607\pi$
$620$	8.00000	0.321288
$621$	6.00000	0.240772
$622$	24.0000i	0.962312i
$623$	12.0000	0.480770
$624$	0	0
$625$	1.00000	0.0400000
$626$	10.0000i	0.399680i
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	0	0
$630$	− 2.00000i	− 0.0796819i
$631$	− 20.0000i	− 0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$	− 20.0000i	− 0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$	− 16.0000i	− 0.636446i
$633$	−4.00000	−0.158986
$634$	−18.0000	−0.714871
$635$	16.0000i	0.634941i
$636$	6.00000	0.237915
$637$	0	0
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	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	$-0.0251319\pi$
$644$	12.0000i	0.472866i
$645$	4.00000i	0.157500i
$646$	0	0
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	16.0000	0.627089
$652$	− 16.0000i	− 0.626608i
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	−4.00000	−0.156412
$655$	6.00000i	0.234439i
$656$	6.00000i	0.234261i
$657$	4.00000i	0.156055i
$658$	0	0
$659$	−18.0000	−0.701180	−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000	−0.701180	−0.350590	+	0.936529i	$0.614019\pi$
$660$	0	0
$661$	16.0000i	0.622328i	0.950356	+	0.311164i	$0.100719\pi$
$661$	16.0000i	0.622328i	−0.950356	+	0.311164i	$0.899281\pi$
$662$	2.00000	0.0777322
$663$	0	0
$664$	12.0000	0.465690
$665$	4.00000i	0.155113i
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	12.0000i	0.464294i
$669$	− 10.0000i	− 0.386622i
$670$	− 4.00000i	− 0.154533i
$671$	0	0
$672$	−2.00000	−0.0771517
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	14.0000i	0.539260i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	− 12.0000i	− 0.460857i
$679$	−8.00000	−0.307012
$680$	0	0
$681$	12.0000i	0.459841i
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	− 2.00000i	− 0.0764719i
$685$	6.00000	0.229248
$686$	−20.0000	−0.763604
$687$	4.00000i	0.152610i
$688$	4.00000	0.152499
$689$	0	0
$690$	6.00000	0.228416
$691$	26.0000i	0.989087i	0.869153	+	0.494543i	$0.164665\pi$
$691$	26.0000i	0.989087i	−0.869153	+	0.494543i	$0.835335\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	− 24.0000i	− 0.911028i
$695$	8.00000i	0.303457i
$696$	0	0
$697$	0	0
$698$	−20.0000	−0.757011
$699$	0	0
$700$	− 2.00000i	− 0.0755929i
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	0	0
$705$	0	0
$706$	6.00000	0.225813
$707$	− 24.0000i	− 0.902613i
$708$	0	0
$709$	− 20.0000i	− 0.751116i	−0.926799	−	0.375558i	$-0.877451\pi$
$709$	− 20.0000i	− 0.751116i	0.926799	−	0.375558i	$-0.122549\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	−6.00000	−0.224860
$713$	48.0000i	1.79761i
$714$	0	0
$715$	0	0
$716$	−6.00000	−0.224231
$717$	0	0
$718$	24.0000	0.895672
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	1.00000i	0.0372678i
$721$	− 32.0000i	− 1.19174i
$722$	− 15.0000i	− 0.558242i
$723$	− 26.0000i	− 0.966950i
$724$	2.00000	0.0743294
$725$	0	0
$726$	− 11.0000i	− 0.408248i
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	4.00000i	0.148047i
$731$	0	0
$732$	−14.0000	−0.517455
$733$	50.0000i	1.84679i	0.383849	+	0.923396i	$0.374598\pi$
$733$	50.0000i	1.84679i	−0.383849	+	0.923396i	$0.625402\pi$
$734$	− 8.00000i	− 0.295285i
$735$	3.00000i	0.110657i
$736$	− 6.00000i	− 0.221163i
$737$	0	0
$738$	6.00000	0.220863
$739$	− 38.0000i	− 1.39785i	−0.715194	−	0.698926i	$-0.753662\pi$
$739$	− 38.0000i	− 1.39785i	0.715194	−	0.698926i	$-0.246338\pi$
$740$	−2.00000	−0.0735215
$741$	0	0
$742$	12.0000	0.440534
$743$	36.0000i	1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$	36.0000i	1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$	−8.00000	−0.293294
$745$	−18.0000	−0.659469
$746$	− 26.0000i	− 0.951928i
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	24.0000i	0.876941i
$750$	−1.00000	−0.0365148
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	30.0000	1.09326
$754$	0	0
$755$	8.00000	0.291150
$756$	2.00000i	0.0727393i
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	−34.0000	−1.23494
$759$	0	0
$760$	− 2.00000i	− 0.0725476i
$761$	− 6.00000i	− 0.217500i	−0.994069	−	0.108750i	$-0.965315\pi$
$761$	− 6.00000i	− 0.217500i	0.994069	−	0.108750i	$-0.0346848\pi$
$762$	− 16.0000i	− 0.579619i
$763$	−8.00000	−0.289619
$764$	0	0
$765$	0	0
$766$	12.0000	0.433578
$767$	0	0
$768$	1.00000	0.0360844
$769$	− 22.0000i	− 0.793340i	−0.917961	−	0.396670i	$-0.870166\pi$
$769$	− 22.0000i	− 0.793340i	0.917961	−	0.396670i	$-0.129834\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	8.00000i	0.287926i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	− 4.00000i	− 0.143777i
$775$	− 8.00000i	− 0.287368i
$776$	4.00000	0.143592
$777$	−4.00000	−0.143499
$778$	36.0000i	1.29066i
$779$	−12.0000	−0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	3.00000	0.107143
$785$	2.00000i	0.0713831i
$786$	− 6.00000i	− 0.214013i
$787$	16.0000i	0.570338i	0.958477	+	0.285169i	$0.0920498\pi$
$787$	16.0000i	0.570338i	−0.958477	+	0.285169i	$0.907950\pi$
$788$	− 18.0000i	− 0.641223i
$789$	30.0000	1.06803
$790$	−16.0000	−0.569254
$791$	− 24.0000i	− 0.853342i
$792$	0	0
$793$	0	0
$794$	−26.0000	−0.922705
$795$	− 6.00000i	− 0.212798i
$796$	−16.0000	−0.567105
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	− 4.00000i	− 0.141598i
$799$	0	0
$800$	1.00000i	0.0353553i
$801$	6.00000i	0.212000i
$802$	18.0000	0.635602
$803$	0	0
$804$	4.00000i	0.141069i
$805$	12.0000	0.422944
$806$	0	0
$807$	12.0000	0.422420
$808$	12.0000i	0.422159i
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	1.00000	0.0351364
$811$	26.0000i	0.912983i	0.889728	+	0.456492i	$0.150894\pi$
$811$	26.0000i	0.912983i	−0.889728	+	0.456492i	$0.849106\pi$
$812$	0	0
$813$	− 8.00000i	− 0.280572i
$814$	0	0
$815$	−16.0000	−0.560456
$816$	0	0
$817$	8.00000i	0.279885i
$818$	14.0000	0.489499
$819$	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	$-0.0333885\pi$
$822$	−6.00000	−0.209274
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	16.0000i	0.557386i
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	−6.00000	−0.208514
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	− 12.0000i	− 0.416526i
$831$	−26.0000	−0.901930
$832$	0	0
$833$	0	0
$834$	− 8.00000i	− 0.277017i
$835$	12.0000	0.415277
$836$	0	0
$837$	8.00000i	0.276520i
$838$	− 6.00000i	− 0.207267i
$839$	48.0000i	1.65714i	0.559883	+	0.828572i	$0.310846\pi$
$839$	48.0000i	1.65714i	−0.559883	+	0.828572i	$0.689154\pi$
$840$	2.00000i	0.0690066i
$841$	−29.0000	−1.00000
$842$	32.0000	1.10279
$843$	30.0000i	1.03325i
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	− 22.0000i	− 0.755929i
$848$	−6.00000	−0.206041
$849$	28.0000	0.960958
$850$	0	0
$851$	− 12.0000i	− 0.411355i
$852$	0	0
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	−28.0000	−0.958140
$855$	−2.00000	−0.0683986
$856$	− 12.0000i	− 0.410152i
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	− 4.00000i	− 0.136399i
$861$	12.0000	0.408959
$862$	24.0000	0.817443
$863$	− 12.0000i	− 0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$	− 12.0000i	− 0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	− 18.0000i	− 0.612018i
$866$	14.0000i	0.475739i
$867$	−17.0000	−0.577350
$868$	−16.0000	−0.543075
$869$	0	0
$870$	0	0
$871$	0	0
$872$	4.00000	0.135457
$873$	− 4.00000i	− 0.135379i
$874$	12.0000	0.405906
$875$	−2.00000	−0.0676123
$876$	− 4.00000i	− 0.135147i
$877$	− 58.0000i	− 1.95852i	−0.202606	−	0.979260i	$-0.564941\pi$
$877$	− 58.0000i	− 1.95852i	0.202606	−	0.979260i	$-0.435059\pi$
$878$	32.0000i	1.07995i
$879$	6.00000i	0.202375i
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	− 3.00000i	− 0.101015i
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.0000	−1.00730	−0.503651	−	0.863907i	$-0.668010\pi$
$887$	−30.0000	−1.00730	−0.503651	+	0.863907i	$0.668010\pi$
$888$	2.00000	0.0671156
$889$	− 32.0000i	− 1.07325i
$890$	6.00000i	0.201120i
$891$	0	0
$892$	10.0000i	0.334825i
$893$	0	0
$894$	18.0000	0.602010
$895$	6.00000i	0.200558i
$896$	2.00000	0.0668153
$897$	0	0
$898$	30.0000	1.00111
$899$	0	0
$900$	1.00000	0.0333333
$901$	0	0
$902$	0	0
$903$	− 8.00000i	− 0.266223i
$904$	12.0000i	0.399114i
$905$	− 2.00000i	− 0.0664822i
$906$	−8.00000	−0.265782
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	− 12.0000i	− 0.398234i
$909$	12.0000	0.398015
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	2.00000i	0.0662266i
$913$	0	0
$914$	−28.0000	−0.926158
$915$	14.0000i	0.462826i
$916$	− 4.00000i	− 0.132164i
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−6.00000	−0.197814
$921$	4.00000i	0.131804i
$922$	−42.0000	−1.38320
$923$	0	0
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	10.0000	0.328620
$927$	16.0000	0.525509
$928$	0	0
$929$	− 54.0000i	− 1.77168i	−0.463988	−	0.885841i	$-0.653582\pi$
$929$	− 54.0000i	− 1.77168i	0.463988	−	0.885841i	$-0.346418\pi$
$930$	8.00000i	0.262330i
$931$	6.00000i	0.196642i
$932$	0	0
$933$	−24.0000	−0.785725
$934$	36.0000i	1.17796i
$935$	0	0
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	8.00000i	0.261209i
$939$	−10.0000	−0.326338
$940$	0	0
$941$	42.0000i	1.36916i	0.728937	+	0.684580i	$0.240015\pi$
$941$	42.0000i	1.36916i	−0.728937	+	0.684580i	$0.759985\pi$
$942$	− 2.00000i	− 0.0651635i
$943$	36.0000i	1.17232i
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	− 36.0000i	− 1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$	− 36.0000i	− 1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$	16.0000	0.519656
$949$	0	0
$950$	−2.00000	−0.0648886
$951$	− 18.0000i	− 0.583690i
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	6.00000i	0.194257i
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.0000	−0.387500
$960$	− 1.00000i	− 0.0322749i
$961$	−33.0000	−1.06452
$962$	0	0
$963$	−12.0000	−0.386695
$964$	26.0000i	0.837404i
$965$	8.00000	0.257529
$966$	−12.0000	−0.386094
$967$	50.0000i	1.60789i	0.594703	+	0.803946i	$0.297270\pi$
$967$	50.0000i	1.60789i	−0.594703	+	0.803946i	$0.702730\pi$
$968$	11.0000i	0.353553i
$969$	0	0
$970$	− 4.00000i	− 0.128432i
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	−1.00000	−0.0320750
$973$	− 16.0000i	− 0.512936i
$974$	2.00000	0.0640841
$975$	0	0
$976$	14.0000	0.448129
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	16.0000	0.511624
$979$	0	0
$980$	− 3.00000i	− 0.0958315i
$981$	− 4.00000i	− 0.127710i
$982$	6.00000i	0.191468i
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	−6.00000	−0.191273
$985$	−18.0000	−0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24.0000	0.763156
$990$	0	0
$991$	−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
$993$	2.00000i	0.0634681i
$994$	0	0
$995$	16.0000i	0.507234i
$996$	12.0000i	0.380235i
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	14.0000	0.443162
$999$	− 2.00000i	− 0.0632772i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.m.1351.1		2
13.5	odd	4		390.2.a.d.1.1	✓	1
13.8	odd	4		5070.2.a.t.1.1		1
13.12	even	2	inner	5070.2.b.m.1351.2		2
39.5	even	4		1170.2.a.k.1.1		1
52.31	even	4		3120.2.a.j.1.1		1
65.18	even	4		1950.2.e.d.1249.2		2
65.44	odd	4		1950.2.a.o.1.1		1
65.57	even	4		1950.2.e.d.1249.1		2
156.83	odd	4		9360.2.a.g.1.1		1
195.44	even	4		5850.2.a.g.1.1		1
195.83	odd	4		5850.2.e.o.5149.1		2
195.122	odd	4		5850.2.e.o.5149.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.d.1.1	✓	1	13.5	odd	4
1170.2.a.k.1.1		1	39.5	even	4
1950.2.a.o.1.1		1	65.44	odd	4
1950.2.e.d.1249.1		2	65.57	even	4
1950.2.e.d.1249.2		2	65.18	even	4
3120.2.a.j.1.1		1	52.31	even	4
5070.2.a.t.1.1		1	13.8	odd	4
5070.2.b.m.1351.1		2	1.1	even	1	trivial
5070.2.b.m.1351.2		2	13.12	even	2	inner
5850.2.a.g.1.1		1	195.44	even	4
5850.2.e.o.5149.1		2	195.83	odd	4
5850.2.e.o.5149.2		2	195.122	odd	4
9360.2.a.g.1.1		1	156.83	odd	4

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} -2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -1.00000i q^{18} +2.00000i q^{19} -1.00000i q^{20} -2.00000i q^{21} +6.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +2.00000i q^{28} +1.00000 q^{30} +8.00000i q^{31} -1.00000i q^{32} +2.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} +2.00000 q^{38} -1.00000 q^{40} +6.00000i q^{41} -2.00000 q^{42} +4.00000 q^{43} +1.00000i q^{45} -6.00000i q^{46} +1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} -6.00000 q^{53} -1.00000i q^{54} +2.00000 q^{56} +2.00000i q^{57} -1.00000i q^{60} +14.0000 q^{61} +8.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} +6.00000 q^{69} -2.00000i q^{70} +1.00000i q^{72} +4.00000i q^{73} -2.00000 q^{74} -1.00000 q^{75} -2.00000i q^{76} -16.0000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} -12.0000i q^{83} +2.00000i q^{84} -4.00000i q^{86} +6.00000i q^{89} +1.00000 q^{90} -6.00000 q^{92} +8.00000i q^{93} -2.00000 q^{95} -1.00000i q^{96} -4.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} - 2 q^{12} - 4 q^{14} + 2 q^{16} + 12 q^{23} - 2 q^{25} + 2 q^{27} + 2 q^{30} + 4 q^{35} - 2 q^{36} + 4 q^{38} - 2 q^{40} - 4 q^{42} + 8 q^{43} + 2 q^{48} + 6 q^{49} - 12 q^{53} + 4 q^{56} + 28 q^{61} + 16 q^{62} - 2 q^{64} + 12 q^{69} - 4 q^{74} - 2 q^{75} - 32 q^{79} + 2 q^{81} + 12 q^{82} + 2 q^{90} - 12 q^{92} - 4 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 - 2 * q^12 - 4 * q^14 + 2 * q^16 + 12 * q^23 - 2 * q^25 + 2 * q^27 + 2 * q^30 + 4 * q^35 - 2 * q^36 + 4 * q^38 - 2 * q^40 - 4 * q^42 + 8 * q^43 + 2 * q^48 + 6 * q^49 - 12 * q^53 + 4 * q^56 + 28 * q^61 + 16 * q^62 - 2 * q^64 + 12 * q^69 - 4 * q^74 - 2 * q^75 - 32 * q^79 + 2 * q^81 + 12 * q^82 + 2 * q^90 - 12 * q^92 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more