LMFDB - Embedded newform 140.2.i.a.81.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,2,Mod(81,140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("140.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 140 = 2^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	140.i (of order $3$, degree $2$, 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:	$1.11790562830$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	140.81
Dual form			140.2.i.a.121.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{9} +(-3.00000 + 5.19615i) q^{11} +2.00000 q^{13} -1.00000 q^{15} +(3.00000 - 5.19615i) q^{17} +(-4.00000 - 6.92820i) q^{19} +(-0.500000 + 2.59808i) q^{21} +(-1.50000 - 2.59808i) q^{23} +(-0.500000 + 0.866025i) q^{25} -5.00000 q^{27} +3.00000 q^{29} +(-1.00000 + 1.73205i) q^{31} +(-3.00000 - 5.19615i) q^{33} +(2.00000 + 1.73205i) q^{35} +(-4.00000 - 6.92820i) q^{37} +(-1.00000 + 1.73205i) q^{39} -3.00000 q^{41} +5.00000 q^{43} +(-1.00000 + 1.73205i) q^{45} +(5.50000 - 4.33013i) q^{49} +(3.00000 + 5.19615i) q^{51} +(-6.00000 + 10.3923i) q^{53} -6.00000 q^{55} +8.00000 q^{57} +(0.500000 + 0.866025i) q^{61} +(4.00000 + 3.46410i) q^{63} +(1.00000 + 1.73205i) q^{65} +(3.50000 - 6.06218i) q^{67} +3.00000 q^{69} +(5.00000 - 8.66025i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(-3.00000 + 15.5885i) q^{77} +(2.00000 + 3.46410i) q^{79} +(-0.500000 + 0.866025i) q^{81} +3.00000 q^{83} +6.00000 q^{85} +(-1.50000 + 2.59808i) q^{87} +(1.50000 + 2.59808i) q^{89} +(5.00000 - 1.73205i) q^{91} +(-1.00000 - 1.73205i) q^{93} +(4.00000 - 6.92820i) q^{95} -10.0000 q^{97} -12.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - q^3 + q^5 + 5 * q^7 + 2 * q^9	$ 2 q - q^{3} + q^{5} + 5 q^{7} + 2 q^{9} - 6 q^{11} + 4 q^{13} - 2 q^{15} + 6 q^{17} - 8 q^{19} - q^{21} - 3 q^{23} - q^{25} - 10 q^{27} + 6 q^{29} - 2 q^{31} - 6 q^{33} + 4 q^{35} - 8 q^{37} - 2 q^{39} - 6 q^{41} + 10 q^{43} - 2 q^{45} + 11 q^{49} + 6 q^{51} - 12 q^{53} - 12 q^{55} + 16 q^{57} + q^{61} + 8 q^{63} + 2 q^{65} + 7 q^{67} + 6 q^{69} + 10 q^{73} - q^{75} - 6 q^{77} + 4 q^{79} - q^{81} + 6 q^{83} + 12 q^{85} - 3 q^{87} + 3 q^{89} + 10 q^{91} - 2 q^{93} + 8 q^{95} - 20 q^{97} - 24 q^{99}+O(q^{100}) $ 2 * q - q^3 + q^5 + 5 * q^7 + 2 * q^9 - 6 * q^11 + 4 * q^13 - 2 * q^15 + 6 * q^17 - 8 * q^19 - q^21 - 3 * q^23 - q^25 - 10 * q^27 + 6 * q^29 - 2 * q^31 - 6 * q^33 + 4 * q^35 - 8 * q^37 - 2 * q^39 - 6 * q^41 + 10 * q^43 - 2 * q^45 + 11 * q^49 + 6 * q^51 - 12 * q^53 - 12 * q^55 + 16 * q^57 + q^61 + 8 * q^63 + 2 * q^65 + 7 * q^67 + 6 * q^69 + 10 * q^73 - q^75 - 6 * q^77 + 4 * q^79 - q^81 + 6 * q^83 + 12 * q^85 - 3 * q^87 + 3 * q^89 + 10 * q^91 - 2 * q^93 + 8 * q^95 - 20 * q^97 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/140\mathbb{Z}\right)^\times$.

$n$	$57$	$71$	$101$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i	−0.973494	−	0.228714i	$-0.926548\pi$
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i	0.684819	+	0.728714i	$0.259881\pi$
$4$		0			0
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$	2.50000	−	0.866025i	0.944911	−	0.327327i
$7$	2.50000	−	0.866025i	0.944911	−	0.327327i
$8$		0			0
$9$	1.00000	+	1.73205i	0.333333	+	0.577350i
$10$		0			0
$11$	−3.00000	+	5.19615i	−0.904534	+	1.56670i	−0.0829925	+	0.996550i	$0.526448\pi$
$11$	−3.00000	+	5.19615i	−0.904534	+	1.56670i	−0.821541	+	0.570149i	$0.806886\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	−0.230285	−	0.973123i	$-0.573966\pi$
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	0.957892	−	0.287129i	$-0.0927008\pi$
$18$		0			0
$19$	−4.00000	−	6.92820i	−0.917663	−	1.58944i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−4.00000	−	6.92820i	−0.917663	−	1.58944i	−0.114708	−	0.993399i	$-0.536593\pi$
$20$		0			0
$21$	−0.500000	+	2.59808i	−0.109109	+	0.566947i
$22$		0			0
$23$	−1.50000	−	2.59808i	−0.312772	−	0.541736i	0.666190	−	0.745782i	$-0.267924\pi$
$23$	−1.50000	−	2.59808i	−0.312772	−	0.541736i	−0.978961	+	0.204046i	$0.934591\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$	−5.00000			−0.962250
$28$		0			0
$29$	3.00000			0.557086			0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000			0.557086			0.278543	+	0.960424i	$0.410149\pi$
$30$		0			0
$31$	−1.00000	+	1.73205i	−0.179605	+	0.311086i	−0.941745	−	0.336327i	$-0.890815\pi$
$31$	−1.00000	+	1.73205i	−0.179605	+	0.311086i	0.762140	+	0.647412i	$0.224149\pi$
$32$		0			0
$33$	−3.00000	−	5.19615i	−0.522233	−	0.904534i
$34$		0			0
$35$	2.00000	+	1.73205i	0.338062	+	0.292770i
$36$		0			0
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	−0.981236	−	0.192809i	$-0.938240\pi$
$37$	−4.00000	−	6.92820i	−0.657596	−	1.13899i	0.323640	−	0.946180i	$-0.395093\pi$
$38$		0			0
$39$	−1.00000	+	1.73205i	−0.160128	+	0.277350i
$40$		0			0
$41$	−3.00000			−0.468521			−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000			−0.468521			−0.234261	+	0.972174i	$0.575267\pi$
$42$		0			0
$43$	5.00000			0.762493			0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000			0.762493			0.381246	+	0.924473i	$0.375495\pi$
$44$		0			0
$45$	−1.00000	+	1.73205i	−0.149071	+	0.258199i
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$	3.00000	+	5.19615i	0.420084	+	0.727607i
$52$		0			0
$53$	−6.00000	+	10.3923i	−0.824163	+	1.42749i	0.0783936	+	0.996922i	$0.475021\pi$
$53$	−6.00000	+	10.3923i	−0.824163	+	1.42749i	−0.902557	+	0.430570i	$0.858312\pi$
$54$		0			0
$55$	−6.00000			−0.809040
$56$		0			0
$57$	8.00000			1.05963
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$	4.00000	+	3.46410i	0.503953	+	0.436436i
$64$		0			0
$65$	1.00000	+	1.73205i	0.124035	+	0.214834i
$66$		0			0
$67$	3.50000	−	6.06218i	0.427593	−	0.740613i	−0.569066	−	0.822292i	$-0.692695\pi$
$67$	3.50000	−	6.06218i	0.427593	−	0.740613i	0.996659	+	0.0816792i	$0.0260283\pi$
$68$		0			0
$69$	3.00000			0.361158
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	5.00000	−	8.66025i	0.585206	−	1.01361i	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	5.00000	−	8.66025i	0.585206	−	1.01361i	0.994850	−	0.101361i	$-0.0323196\pi$
$74$		0			0
$75$	−0.500000	−	0.866025i	−0.0577350	−	0.100000i
$76$		0			0
$77$	−3.00000	+	15.5885i	−0.341882	+	1.77647i
$78$		0			0
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	0.956325	−	0.292306i	$-0.0944227\pi$
$79$	2.00000	+	3.46410i	0.225018	+	0.389742i	−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	3.00000			0.329293			0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000			0.329293			0.164646	+	0.986353i	$0.447352\pi$
$84$		0			0
$85$	6.00000			0.650791
$86$		0			0
$87$	−1.50000	+	2.59808i	−0.160817	+	0.278543i
$88$		0			0
$89$	1.50000	+	2.59808i	0.159000	+	0.275396i	0.934508	−	0.355942i	$-0.115840\pi$
$89$	1.50000	+	2.59808i	0.159000	+	0.275396i	−0.775509	+	0.631337i	$0.782506\pi$
$90$		0			0
$91$	5.00000	−	1.73205i	0.524142	−	0.181568i
$92$		0			0
$93$	−1.00000	−	1.73205i	−0.103695	−	0.179605i
$94$		0			0
$95$	4.00000	−	6.92820i	0.410391	−	0.710819i
$96$		0			0
$97$	−10.0000			−1.01535			−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000			−1.01535			−0.507673	+	0.861550i	$0.669494\pi$
$98$		0			0
$99$	−12.0000			−1.20605
$100$		0			0
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	−0.781697	−	0.623658i	$-0.785646\pi$
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	0.930953	+	0.365140i	$0.118979\pi$
$102$		0			0
$103$	3.50000	+	6.06218i	0.344865	+	0.597324i	0.985329	−	0.170664i	$-0.0545913\pi$
$103$	3.50000	+	6.06218i	0.344865	+	0.597324i	−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$	−2.50000	+	0.866025i	−0.243975	+	0.0845154i
$106$		0			0
$107$	1.50000	+	2.59808i	0.145010	+	0.251166i	0.929377	−	0.369132i	$-0.120345\pi$
$107$	1.50000	+	2.59808i	0.145010	+	0.251166i	−0.784366	+	0.620298i	$0.787012\pi$
$108$		0			0
$109$	−8.50000	+	14.7224i	−0.814152	+	1.41015i	0.0957826	+	0.995402i	$0.469465\pi$
$109$	−8.50000	+	14.7224i	−0.814152	+	1.41015i	−0.909935	+	0.414751i	$0.863869\pi$
$110$		0			0
$111$	8.00000			0.759326
$112$		0			0
$113$	−12.0000			−1.12887			−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000			−1.12887			−0.564433	+	0.825479i	$0.690905\pi$
$114$		0			0
$115$	1.50000	−	2.59808i	0.139876	−	0.242272i
$116$		0			0
$117$	2.00000	+	3.46410i	0.184900	+	0.320256i
$118$		0			0
$119$	3.00000	−	15.5885i	0.275010	−	1.42899i
$120$		0			0
$121$	−12.5000	−	21.6506i	−1.13636	−	1.96824i
$122$		0			0
$123$	1.50000	−	2.59808i	0.135250	−	0.234261i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$	−2.50000	+	4.33013i	−0.220113	+	0.381246i
$130$		0			0
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	−0.999602	−	0.0281993i	$-0.991023\pi$
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	0.475380	−	0.879781i	$-0.342311\pi$
$132$		0			0
$133$	−16.0000	−	13.8564i	−1.38738	−	1.20150i
$134$		0			0
$135$	−2.50000	−	4.33013i	−0.215166	−	0.372678i
$136$		0			0
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	−0.487278	−	0.873247i	$-0.662010\pi$
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	0.999893	−	0.0146279i	$-0.00465636\pi$
$138$		0			0
$139$	2.00000			0.169638			0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000			0.169638			0.0848189	+	0.996396i	$0.472969\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.00000	+	10.3923i	−0.501745	+	0.869048i
$144$		0			0
$145$	1.50000	+	2.59808i	0.124568	+	0.215758i
$146$		0			0
$147$	1.00000	+	6.92820i	0.0824786	+	0.571429i
$148$		0			0
$149$	7.50000	+	12.9904i	0.614424	+	1.06421i	0.990485	+	0.137619i	$0.0439449\pi$
$149$	7.50000	+	12.9904i	0.614424	+	1.06421i	−0.376061	+	0.926595i	$0.622722\pi$
$150$		0			0
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	−0.587646	−	0.809118i	$-0.699945\pi$
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	0.994540	+	0.104357i	$0.0332784\pi$
$152$		0			0
$153$	12.0000			0.970143
$154$		0			0
$155$	−2.00000			−0.160644
$156$		0			0
$157$	−7.00000	+	12.1244i	−0.558661	+	0.967629i	0.438948	+	0.898513i	$0.355351\pi$
$157$	−7.00000	+	12.1244i	−0.558661	+	0.967629i	−0.997609	+	0.0691164i	$0.977982\pi$
$158$		0			0
$159$	−6.00000	−	10.3923i	−0.475831	−	0.824163i
$160$		0			0
$161$	−6.00000	−	5.19615i	−0.472866	−	0.409514i
$162$		0			0
$163$	8.00000	+	13.8564i	0.626608	+	1.08532i	0.988227	+	0.152992i	$0.0488907\pi$
$163$	8.00000	+	13.8564i	0.626608	+	1.08532i	−0.361619	+	0.932326i	$0.617776\pi$
$164$		0			0
$165$	3.00000	−	5.19615i	0.233550	−	0.404520i
$166$		0			0
$167$	−21.0000			−1.62503			−0.812514	−	0.582941i	$-0.801902\pi$
$167$	−21.0000			−1.62503			−0.812514	+	0.582941i	$0.801902\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	8.00000	−	13.8564i	0.611775	−	1.05963i
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−0.500000	+	2.59808i	−0.0377964	+	0.196396i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.00000	−	5.19615i	0.224231	−	0.388379i	−0.731858	−	0.681457i	$-0.761346\pi$
$179$	3.00000	−	5.19615i	0.224231	−	0.388379i	0.956088	+	0.293079i	$0.0946798\pi$
$180$		0			0
$181$	17.0000			1.26360			0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000			1.26360			0.631800	+	0.775131i	$0.282316\pi$
$182$		0			0
$183$	−1.00000			−0.0739221
$184$		0			0
$185$	4.00000	−	6.92820i	0.294086	−	0.509372i
$186$		0			0
$187$	18.0000	+	31.1769i	1.31629	+	2.27988i
$188$		0			0
$189$	−12.5000	+	4.33013i	−0.909241	+	0.314970i
$190$		0			0
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	0.982828	+	0.184525i	$0.0590746\pi$
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	−0.331611	+	0.943416i	$0.607592\pi$
$192$		0			0
$193$	−1.00000	+	1.73205i	−0.0719816	+	0.124676i	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−1.00000	+	1.73205i	−0.0719816	+	0.124676i	0.827788	+	0.561041i	$0.189599\pi$
$194$		0			0
$195$	−2.00000			−0.143223
$196$		0			0
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		0			0
$199$	−10.0000	+	17.3205i	−0.708881	+	1.22782i	0.256391	+	0.966573i	$0.417466\pi$
$199$	−10.0000	+	17.3205i	−0.708881	+	1.22782i	−0.965272	+	0.261245i	$0.915867\pi$
$200$		0			0
$201$	3.50000	+	6.06218i	0.246871	+	0.427593i
$202$		0			0
$203$	7.50000	−	2.59808i	0.526397	−	0.182349i
$204$		0			0
$205$	−1.50000	−	2.59808i	−0.104765	−	0.181458i
$206$		0			0
$207$	3.00000	−	5.19615i	0.208514	−	0.361158i
$208$		0			0
$209$	48.0000			3.32023
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.50000	+	4.33013i	0.170499	+	0.295312i
$216$		0			0
$217$	−1.00000	+	5.19615i	−0.0678844	+	0.352738i
$218$		0			0
$219$	5.00000	+	8.66025i	0.337869	+	0.585206i
$220$		0			0
$221$	6.00000	−	10.3923i	0.403604	−	0.699062i
$222$		0			0
$223$	8.00000			0.535720			0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000			0.535720			0.267860	+	0.963458i	$0.413684\pi$
$224$		0			0
$225$	−2.00000			−0.133333
$226$		0			0
$227$	6.00000	−	10.3923i	0.398234	−	0.689761i	−0.595274	−	0.803523i	$-0.702957\pi$
$227$	6.00000	−	10.3923i	0.398234	−	0.689761i	0.993508	+	0.113761i	$0.0362899\pi$
$228$		0			0
$229$	−1.00000	−	1.73205i	−0.0660819	−	0.114457i	0.831092	−	0.556136i	$-0.187717\pi$
$229$	−1.00000	−	1.73205i	−0.0660819	−	0.114457i	−0.897173	+	0.441679i	$0.854383\pi$
$230$		0			0
$231$	−12.0000	−	10.3923i	−0.789542	−	0.683763i
$232$		0			0
$233$	−3.00000	−	5.19615i	−0.196537	−	0.340411i	0.750867	−	0.660454i	$-0.229636\pi$
$233$	−3.00000	−	5.19615i	−0.196537	−	0.340411i	−0.947403	+	0.320043i	$0.896303\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−4.00000			−0.259828
$238$		0			0
$239$	6.00000			0.388108			0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000			0.388108			0.194054	+	0.980991i	$0.437836\pi$
$240$		0			0
$241$	−13.0000	+	22.5167i	−0.837404	+	1.45043i	0.0546547	+	0.998505i	$0.482594\pi$
$241$	−13.0000	+	22.5167i	−0.837404	+	1.45043i	−0.892058	+	0.451920i	$0.850739\pi$
$242$		0			0
$243$	−8.00000	−	13.8564i	−0.513200	−	0.888889i
$244$		0			0
$245$	6.50000	+	2.59808i	0.415270	+	0.165985i
$246$		0			0
$247$	−8.00000	−	13.8564i	−0.509028	−	0.881662i
$248$		0			0
$249$	−1.50000	+	2.59808i	−0.0950586	+	0.164646i
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$	18.0000			1.13165
$254$		0			0
$255$	−3.00000	+	5.19615i	−0.187867	+	0.325396i
$256$		0			0
$257$	12.0000	+	20.7846i	0.748539	+	1.29651i	0.948523	+	0.316709i	$0.102578\pi$
$257$	12.0000	+	20.7846i	0.748539	+	1.29651i	−0.199983	+	0.979799i	$0.564089\pi$
$258$		0			0
$259$	−16.0000	−	13.8564i	−0.994192	−	0.860995i
$260$		0			0
$261$	3.00000	+	5.19615i	0.185695	+	0.321634i
$262$		0			0
$263$	1.50000	−	2.59808i	0.0924940	−	0.160204i	−0.816066	−	0.577959i	$-0.803849\pi$
$263$	1.50000	−	2.59808i	0.0924940	−	0.160204i	0.908560	+	0.417755i	$0.137183\pi$
$264$		0			0
$265$	−12.0000			−0.737154
$266$		0			0
$267$	−3.00000			−0.183597
$268$		0			0
$269$	7.50000	−	12.9904i	0.457283	−	0.792038i	−0.541533	−	0.840679i	$-0.682156\pi$
$269$	7.50000	−	12.9904i	0.457283	−	0.792038i	0.998816	+	0.0486418i	$0.0154893\pi$
$270$		0			0
$271$	8.00000	+	13.8564i	0.485965	+	0.841717i	0.999870	−	0.0161307i	$-0.00513477\pi$
$271$	8.00000	+	13.8564i	0.485965	+	0.841717i	−0.513905	+	0.857847i	$0.671801\pi$
$272$		0			0
$273$	−1.00000	+	5.19615i	−0.0605228	+	0.314485i
$274$		0			0
$275$	−3.00000	−	5.19615i	−0.180907	−	0.313340i
$276$		0			0
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	−0.319447	−	0.947604i	$-0.603497\pi$
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	0.980373	−	0.197153i	$-0.0631696\pi$
$278$		0			0
$279$	−4.00000			−0.239474
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$	−10.0000	+	17.3205i	−0.594438	+	1.02960i	0.399188	+	0.916869i	$0.369292\pi$
$283$	−10.0000	+	17.3205i	−0.594438	+	1.02960i	−0.993626	+	0.112728i	$0.964041\pi$
$284$		0			0
$285$	4.00000	+	6.92820i	0.236940	+	0.410391i
$286$		0			0
$287$	−7.50000	+	2.59808i	−0.442711	+	0.153360i
$288$		0			0
$289$	−9.50000	−	16.4545i	−0.558824	−	0.967911i
$290$		0			0
$291$	5.00000	−	8.66025i	0.293105	−	0.507673i
$292$		0			0
$293$	−12.0000			−0.701047			−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000			−0.701047			−0.350524	+	0.936554i	$0.613996\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	15.0000	−	25.9808i	0.870388	−	1.50756i
$298$		0			0
$299$	−3.00000	−	5.19615i	−0.173494	−	0.300501i
$300$		0			0
$301$	12.5000	−	4.33013i	0.720488	−	0.249584i
$302$		0			0
$303$	1.50000	+	2.59808i	0.0861727	+	0.149256i
$304$		0			0
$305$	−0.500000	+	0.866025i	−0.0286299	+	0.0495885i
$306$		0			0
$307$	−19.0000			−1.08439			−0.542194	−	0.840254i	$-0.682406\pi$
$307$	−19.0000			−1.08439			−0.542194	+	0.840254i	$0.682406\pi$
$308$		0			0
$309$	−7.00000			−0.398216
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	2.00000	+	3.46410i	0.113047	+	0.195803i	0.916997	−	0.398894i	$-0.130606\pi$
$313$	2.00000	+	3.46410i	0.113047	+	0.195803i	−0.803951	+	0.594696i	$0.797272\pi$
$314$		0			0
$315$	−1.00000	+	5.19615i	−0.0563436	+	0.292770i
$316$		0			0
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	−0.999980	−	0.00635137i	$-0.997978\pi$
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	0.494489	−	0.869184i	$-0.335355\pi$
$318$		0			0
$319$	−9.00000	+	15.5885i	−0.503903	+	0.872786i
$320$		0			0
$321$	−3.00000			−0.167444
$322$		0			0
$323$	−48.0000			−2.67079
$324$		0			0
$325$	−1.00000	+	1.73205i	−0.0554700	+	0.0960769i
$326$		0			0
$327$	−8.50000	−	14.7224i	−0.470051	−	0.814152i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	11.0000	+	19.0526i	0.604615	+	1.04722i	0.992112	+	0.125353i	$0.0400062\pi$
$331$	11.0000	+	19.0526i	0.604615	+	1.04722i	−0.387498	+	0.921871i	$0.626660\pi$
$332$		0			0
$333$	8.00000	−	13.8564i	0.438397	−	0.759326i
$334$		0			0
$335$	7.00000			0.382451
$336$		0			0
$337$	20.0000			1.08947			0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000			1.08947			0.544735	+	0.838608i	$0.316630\pi$
$338$		0			0
$339$	6.00000	−	10.3923i	0.325875	−	0.564433i
$340$		0			0
$341$	−6.00000	−	10.3923i	−0.324918	−	0.562775i
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$		0			0
$345$	1.50000	+	2.59808i	0.0807573	+	0.139876i
$346$		0			0
$347$	13.5000	−	23.3827i	0.724718	−	1.25525i	−0.234372	−	0.972147i	$-0.575303\pi$
$347$	13.5000	−	23.3827i	0.724718	−	1.25525i	0.959090	−	0.283101i	$-0.0913633\pi$
$348$		0			0
$349$	−1.00000			−0.0535288			−0.0267644	−	0.999642i	$-0.508520\pi$
$349$	−1.00000			−0.0535288			−0.0267644	+	0.999642i	$0.508520\pi$
$350$		0			0
$351$	−10.0000			−0.533761
$352$		0			0
$353$	6.00000	−	10.3923i	0.319348	−	0.553127i	−0.661004	−	0.750382i	$-0.729870\pi$
$353$	6.00000	−	10.3923i	0.319348	−	0.553127i	0.980352	+	0.197256i	$0.0632029\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	12.0000	+	10.3923i	0.635107	+	0.550019i
$358$		0			0
$359$	−3.00000	−	5.19615i	−0.158334	−	0.274242i	0.775934	−	0.630814i	$-0.217279\pi$
$359$	−3.00000	−	5.19615i	−0.158334	−	0.274242i	−0.934268	+	0.356572i	$0.883946\pi$
$360$		0			0
$361$	−22.5000	+	38.9711i	−1.18421	+	2.05111i
$362$		0			0
$363$	25.0000			1.31216
$364$		0			0
$365$	10.0000			0.523424
$366$		0			0
$367$	−2.50000	+	4.33013i	−0.130499	+	0.226031i	−0.923869	−	0.382709i	$-0.874991\pi$
$367$	−2.50000	+	4.33013i	−0.130499	+	0.226031i	0.793370	+	0.608740i	$0.208325\pi$
$368$		0			0
$369$	−3.00000	−	5.19615i	−0.156174	−	0.270501i
$370$		0			0
$371$	−6.00000	+	31.1769i	−0.311504	+	1.61862i
$372$		0			0
$373$	8.00000	+	13.8564i	0.414224	+	0.717458i	0.995347	−	0.0963587i	$-0.0307196\pi$
$373$	8.00000	+	13.8564i	0.414224	+	0.717458i	−0.581122	+	0.813816i	$0.697386\pi$
$374$		0			0
$375$	0.500000	−	0.866025i	0.0258199	−	0.0447214i
$376$		0			0
$377$	6.00000			0.309016
$378$		0			0
$379$	8.00000			0.410932			0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000			0.410932			0.205466	+	0.978664i	$0.434129\pi$
$380$		0			0
$381$	−4.00000	+	6.92820i	−0.204926	+	0.354943i
$382$		0			0
$383$	−1.50000	−	2.59808i	−0.0766464	−	0.132755i	0.825155	−	0.564907i	$-0.191088\pi$
$383$	−1.50000	−	2.59808i	−0.0766464	−	0.132755i	−0.901801	+	0.432151i	$0.857755\pi$
$384$		0			0
$385$	−15.0000	+	5.19615i	−0.764471	+	0.264820i
$386$		0			0
$387$	5.00000	+	8.66025i	0.254164	+	0.440225i
$388$		0			0
$389$	−3.00000	+	5.19615i	−0.152106	+	0.263455i	−0.932002	−	0.362454i	$-0.881939\pi$
$389$	−3.00000	+	5.19615i	−0.152106	+	0.263455i	0.779895	+	0.625910i	$0.215272\pi$
$390$		0			0
$391$	−18.0000			−0.910299
$392$		0			0
$393$	12.0000			0.605320
$394$		0			0
$395$	−2.00000	+	3.46410i	−0.100631	+	0.174298i
$396$		0			0
$397$	−13.0000	−	22.5167i	−0.652451	−	1.13008i	−0.982526	−	0.186124i	$-0.940407\pi$
$397$	−13.0000	−	22.5167i	−0.652451	−	1.13008i	0.330075	−	0.943955i	$-0.392926\pi$
$398$		0			0
$399$	20.0000	−	6.92820i	1.00125	−	0.346844i
$400$		0			0
$401$	−1.50000	−	2.59808i	−0.0749064	−	0.129742i	0.826139	−	0.563466i	$-0.190532\pi$
$401$	−1.50000	−	2.59808i	−0.0749064	−	0.129742i	−0.901046	+	0.433724i	$0.857199\pi$
$402$		0			0
$403$	−2.00000	+	3.46410i	−0.0996271	+	0.172559i
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$	48.0000			2.37927
$408$		0			0
$409$	−5.50000	+	9.52628i	−0.271957	+	0.471044i	−0.969363	−	0.245633i	$-0.921004\pi$
$409$	−5.50000	+	9.52628i	−0.271957	+	0.471044i	0.697406	+	0.716677i	$0.254338\pi$
$410$		0			0
$411$	6.00000	+	10.3923i	0.295958	+	0.512615i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	1.50000	+	2.59808i	0.0736321	+	0.127535i
$416$		0			0
$417$	−1.00000	+	1.73205i	−0.0489702	+	0.0848189i
$418$		0			0
$419$	24.0000			1.17248			0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000			1.17248			0.586238	+	0.810139i	$0.300608\pi$
$420$		0			0
$421$	23.0000			1.12095			0.560476	−	0.828171i	$-0.310618\pi$
$421$	23.0000			1.12095			0.560476	+	0.828171i	$0.310618\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.00000	+	5.19615i	0.145521	+	0.252050i
$426$		0			0
$427$	2.00000	+	1.73205i	0.0967868	+	0.0838198i
$428$		0			0
$429$	−6.00000	−	10.3923i	−0.289683	−	0.501745i
$430$		0			0
$431$	−15.0000	+	25.9808i	−0.722525	+	1.25145i	0.237460	+	0.971397i	$0.423685\pi$
$431$	−15.0000	+	25.9808i	−0.722525	+	1.25145i	−0.959985	+	0.280052i	$0.909648\pi$
$432$		0			0
$433$	−28.0000			−1.34559			−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000			−1.34559			−0.672797	+	0.739827i	$0.734907\pi$
$434$		0			0
$435$	−3.00000			−0.143839
$436$		0			0
$437$	−12.0000	+	20.7846i	−0.574038	+	0.994263i
$438$		0			0
$439$	14.0000	+	24.2487i	0.668184	+	1.15733i	0.978412	+	0.206666i	$0.0662612\pi$
$439$	14.0000	+	24.2487i	0.668184	+	1.15733i	−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$	13.0000	+	5.19615i	0.619048	+	0.247436i
$442$		0			0
$443$	−4.50000	−	7.79423i	−0.213801	−	0.370315i	0.739100	−	0.673596i	$-0.235251\pi$
$443$	−4.50000	−	7.79423i	−0.213801	−	0.370315i	−0.952901	+	0.303281i	$0.901918\pi$
$444$		0			0
$445$	−1.50000	+	2.59808i	−0.0711068	+	0.123161i
$446$		0			0
$447$	−15.0000			−0.709476
$448$		0			0
$449$	−9.00000			−0.424736			−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000			−0.424736			−0.212368	+	0.977190i	$0.568118\pi$
$450$		0			0
$451$	9.00000	−	15.5885i	0.423793	−	0.734032i
$452$		0			0
$453$	5.00000	+	8.66025i	0.234920	+	0.406894i
$454$		0			0
$455$	4.00000	+	3.46410i	0.187523	+	0.162400i
$456$		0			0
$457$	2.00000	+	3.46410i	0.0935561	+	0.162044i	0.909005	−	0.416785i	$-0.136843\pi$
$457$	2.00000	+	3.46410i	0.0935561	+	0.162044i	−0.815449	+	0.578829i	$0.803510\pi$
$458$		0			0
$459$	−15.0000	+	25.9808i	−0.700140	+	1.21268i
$460$		0			0
$461$	30.0000			1.39724			0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000			1.39724			0.698620	+	0.715493i	$0.253798\pi$
$462$		0			0
$463$	29.0000			1.34774			0.673872	−	0.738848i	$-0.264630\pi$
$463$	29.0000			1.34774			0.673872	+	0.738848i	$0.264630\pi$
$464$		0			0
$465$	1.00000	−	1.73205i	0.0463739	−	0.0803219i
$466$		0			0
$467$	−16.5000	−	28.5788i	−0.763529	−	1.32247i	−0.941021	−	0.338349i	$-0.890132\pi$
$467$	−16.5000	−	28.5788i	−0.763529	−	1.32247i	0.177492	−	0.984122i	$-0.443202\pi$
$468$		0			0
$469$	3.50000	−	18.1865i	0.161615	−	0.839776i
$470$		0			0
$471$	−7.00000	−	12.1244i	−0.322543	−	0.558661i
$472$		0			0
$473$	−15.0000	+	25.9808i	−0.689701	+	1.19460i
$474$		0			0
$475$	8.00000			0.367065
$476$		0			0
$477$	−24.0000			−1.09888
$478$		0			0
$479$	15.0000	−	25.9808i	0.685367	−	1.18709i	−0.287954	−	0.957644i	$-0.592975\pi$
$479$	15.0000	−	25.9808i	0.685367	−	1.18709i	0.973321	−	0.229447i	$-0.0736918\pi$
$480$		0			0
$481$	−8.00000	−	13.8564i	−0.364769	−	0.631798i
$482$		0			0
$483$	7.50000	−	2.59808i	0.341262	−	0.118217i
$484$		0			0
$485$	−5.00000	−	8.66025i	−0.227038	−	0.393242i
$486$		0			0
$487$	−4.00000	+	6.92820i	−0.181257	+	0.313947i	−0.942309	−	0.334744i	$-0.891350\pi$
$487$	−4.00000	+	6.92820i	−0.181257	+	0.313947i	0.761052	+	0.648691i	$0.224683\pi$
$488$		0			0
$489$	−16.0000			−0.723545
$490$		0			0
$491$	6.00000			0.270776			0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000			0.270776			0.135388	+	0.990793i	$0.456772\pi$
$492$		0			0
$493$	9.00000	−	15.5885i	0.405340	−	0.702069i
$494$		0			0
$495$	−6.00000	−	10.3923i	−0.269680	−	0.467099i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1.00000	−	1.73205i	−0.0447661	−	0.0775372i	0.842774	−	0.538267i	$-0.180921\pi$
$499$	−1.00000	−	1.73205i	−0.0447661	−	0.0775372i	−0.887540	+	0.460730i	$0.847588\pi$
$500$		0			0
$501$	10.5000	−	18.1865i	0.469105	−	0.812514i
$502$		0			0
$503$	−9.00000			−0.401290			−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000			−0.401290			−0.200645	+	0.979664i	$0.564304\pi$
$504$		0			0
$505$	3.00000			0.133498
$506$		0			0
$507$	4.50000	−	7.79423i	0.199852	−	0.346154i
$508$		0			0
$509$	−19.5000	−	33.7750i	−0.864322	−	1.49705i	−0.867719	−	0.497056i	$-0.834414\pi$
$509$	−19.5000	−	33.7750i	−0.864322	−	1.49705i	0.00339621	−	0.999994i	$-0.498919\pi$
$510$		0			0
$511$	5.00000	−	25.9808i	0.221187	−	1.14932i
$512$		0			0
$513$	20.0000	+	34.6410i	0.883022	+	1.52944i
$514$		0			0
$515$	−3.50000	+	6.06218i	−0.154228	+	0.267131i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.0000	+	25.9808i	−0.657162	+	1.13824i	0.324185	+	0.945994i	$0.394910\pi$
$521$	−15.0000	+	25.9808i	−0.657162	+	1.13824i	−0.981347	+	0.192244i	$0.938423\pi$
$522$		0			0
$523$	−22.0000	−	38.1051i	−0.961993	−	1.66622i	−0.717486	−	0.696573i	$-0.754707\pi$
$523$	−22.0000	−	38.1051i	−0.961993	−	1.66622i	−0.244507	−	0.969648i	$-0.578626\pi$
$524$		0			0
$525$	−2.00000	−	1.73205i	−0.0872872	−	0.0755929i
$526$		0			0
$527$	6.00000	+	10.3923i	0.261364	+	0.452696i
$528$		0			0
$529$	7.00000	−	12.1244i	0.304348	−	0.527146i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.00000			−0.259889
$534$		0			0
$535$	−1.50000	+	2.59808i	−0.0648507	+	0.112325i
$536$		0			0
$537$	3.00000	+	5.19615i	0.129460	+	0.224231i
$538$		0			0
$539$	6.00000	+	41.5692i	0.258438	+	1.79051i
$540$		0			0
$541$	−8.50000	−	14.7224i	−0.365444	−	0.632967i	0.623404	−	0.781900i	$-0.285749\pi$
$541$	−8.50000	−	14.7224i	−0.365444	−	0.632967i	−0.988847	+	0.148933i	$0.952416\pi$
$542$		0			0
$543$	−8.50000	+	14.7224i	−0.364770	+	0.631800i
$544$		0			0
$545$	−17.0000			−0.728200
$546$		0			0
$547$	−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
$549$	−1.00000	+	1.73205i	−0.0426790	+	0.0739221i
$550$		0			0
$551$	−12.0000	−	20.7846i	−0.511217	−	0.885454i
$552$		0			0
$553$	8.00000	+	6.92820i	0.340195	+	0.294617i
$554$		0			0
$555$	4.00000	+	6.92820i	0.169791	+	0.294086i
$556$		0			0
$557$	−15.0000	+	25.9808i	−0.635570	+	1.10084i	0.350824	+	0.936442i	$0.385902\pi$
$557$	−15.0000	+	25.9808i	−0.635570	+	1.10084i	−0.986394	+	0.164399i	$0.947432\pi$
$558$		0			0
$559$	10.0000			0.422955
$560$		0			0
$561$	−36.0000			−1.51992
$562$		0			0
$563$	4.50000	−	7.79423i	0.189652	−	0.328488i	−0.755482	−	0.655169i	$-0.772597\pi$
$563$	4.50000	−	7.79423i	0.189652	−	0.328488i	0.945134	+	0.326682i	$0.105931\pi$
$564$		0			0
$565$	−6.00000	−	10.3923i	−0.252422	−	0.437208i
$566$		0			0
$567$	−0.500000	+	2.59808i	−0.0209980	+	0.109109i
$568$		0			0
$569$	9.00000	+	15.5885i	0.377300	+	0.653502i	0.990668	−	0.136295i	$-0.0435194\pi$
$569$	9.00000	+	15.5885i	0.377300	+	0.653502i	−0.613369	+	0.789797i	$0.710186\pi$
$570$		0			0
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	0.983444	+	0.181210i	$0.0580014\pi$
$572$		0			0
$573$	−18.0000			−0.751961
$574$		0			0
$575$	3.00000			0.125109
$576$		0			0
$577$	5.00000	−	8.66025i	0.208153	−	0.360531i	−0.742980	−	0.669314i	$-0.766588\pi$
$577$	5.00000	−	8.66025i	0.208153	−	0.360531i	0.951133	+	0.308783i	$0.0999216\pi$
$578$		0			0
$579$	−1.00000	−	1.73205i	−0.0415586	−	0.0719816i
$580$		0			0
$581$	7.50000	−	2.59808i	0.311152	−	0.107786i
$582$		0			0
$583$	−36.0000	−	62.3538i	−1.49097	−	2.58243i
$584$		0			0
$585$	−2.00000	+	3.46410i	−0.0826898	+	0.143223i
$586$		0			0
$587$	12.0000			0.495293			0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000			0.495293			0.247647	+	0.968850i	$0.420343\pi$
$588$		0			0
$589$	16.0000			0.659269
$590$		0			0
$591$	9.00000	−	15.5885i	0.370211	−	0.641223i
$592$		0			0
$593$	6.00000	+	10.3923i	0.246390	+	0.426761i	0.962522	−	0.271205i	$-0.0874221\pi$
$593$	6.00000	+	10.3923i	0.246390	+	0.426761i	−0.716131	+	0.697966i	$0.754089\pi$
$594$		0			0
$595$	15.0000	−	5.19615i	0.614940	−	0.213021i
$596$		0			0
$597$	−10.0000	−	17.3205i	−0.409273	−	0.708881i
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	−22.0000			−0.897399			−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000			−0.897399			−0.448699	+	0.893683i	$0.648113\pi$
$602$		0			0
$603$	14.0000			0.570124
$604$		0			0
$605$	12.5000	−	21.6506i	0.508197	−	0.880223i
$606$		0			0
$607$	21.5000	+	37.2391i	0.872658	+	1.51149i	0.859237	+	0.511578i	$0.170939\pi$
$607$	21.5000	+	37.2391i	0.872658	+	1.51149i	0.0134214	+	0.999910i	$0.495728\pi$
$608$		0			0
$609$	−1.50000	+	7.79423i	−0.0607831	+	0.315838i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.00000	+	1.73205i	−0.0403896	+	0.0699569i	−0.885514	−	0.464614i	$-0.846193\pi$
$613$	−1.00000	+	1.73205i	−0.0403896	+	0.0699569i	0.845124	+	0.534570i	$0.179527\pi$
$614$		0			0
$615$	3.00000			0.120972
$616$		0			0
$617$	−12.0000			−0.483102			−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000			−0.483102			−0.241551	+	0.970388i	$0.577656\pi$
$618$		0			0
$619$	−7.00000	+	12.1244i	−0.281354	+	0.487319i	−0.971718	−	0.236143i	$-0.924117\pi$
$619$	−7.00000	+	12.1244i	−0.281354	+	0.487319i	0.690365	+	0.723462i	$0.257450\pi$
$620$		0			0
$621$	7.50000	+	12.9904i	0.300965	+	0.521286i
$622$		0			0
$623$	6.00000	+	5.19615i	0.240385	+	0.208179i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$	−24.0000	+	41.5692i	−0.958468	+	1.66011i
$628$		0			0
$629$	−48.0000			−1.91389
$630$		0			0
$631$	38.0000			1.51276			0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000			1.51276			0.756378	+	0.654135i	$0.226967\pi$
$632$		0			0
$633$	2.00000	−	3.46410i	0.0794929	−	0.137686i
$634$		0			0
$635$	4.00000	+	6.92820i	0.158735	+	0.274937i
$636$		0			0
$637$	11.0000	−	8.66025i	0.435836	−	0.343132i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19.5000	−	33.7750i	0.770204	−	1.33403i	−0.167247	−	0.985915i	$-0.553488\pi$
$641$	19.5000	−	33.7750i	0.770204	−	1.33403i	0.937451	−	0.348117i	$-0.113179\pi$
$642$		0			0
$643$	20.0000			0.788723			0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000			0.788723			0.394362	+	0.918955i	$0.370966\pi$
$644$		0			0
$645$	−5.00000			−0.196875
$646$		0			0
$647$	−10.5000	+	18.1865i	−0.412798	+	0.714986i	−0.995194	−	0.0979182i	$-0.968782\pi$
$647$	−10.5000	+	18.1865i	−0.412798	+	0.714986i	0.582397	+	0.812905i	$0.302115\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−4.00000	−	3.46410i	−0.156772	−	0.135769i
$652$		0			0
$653$	21.0000	+	36.3731i	0.821794	+	1.42339i	0.904345	+	0.426801i	$0.140360\pi$
$653$	21.0000	+	36.3731i	0.821794	+	1.42339i	−0.0825519	+	0.996587i	$0.526307\pi$
$654$		0			0
$655$	6.00000	−	10.3923i	0.234439	−	0.406061i
$656$		0			0
$657$	20.0000			0.780274
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$	12.5000	−	21.6506i	0.486194	−	0.842112i	−0.513680	−	0.857982i	$-0.671718\pi$
$661$	12.5000	−	21.6506i	0.486194	−	0.842112i	0.999874	+	0.0158695i	$0.00505163\pi$
$662$		0			0
$663$	6.00000	+	10.3923i	0.233021	+	0.403604i
$664$		0			0
$665$	4.00000	−	20.7846i	0.155113	−	0.805993i
$666$		0			0
$667$	−4.50000	−	7.79423i	−0.174241	−	0.301794i
$668$		0			0
$669$	−4.00000	+	6.92820i	−0.154649	+	0.267860i
$670$		0			0
$671$	−6.00000			−0.231627
$672$		0			0
$673$	32.0000			1.23351			0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000			1.23351			0.616755	+	0.787155i	$0.288447\pi$
$674$		0			0
$675$	2.50000	−	4.33013i	0.0962250	−	0.166667i
$676$		0			0
$677$	9.00000	+	15.5885i	0.345898	+	0.599113i	0.985517	−	0.169580i	$-0.0542410\pi$
$677$	9.00000	+	15.5885i	0.345898	+	0.599113i	−0.639618	+	0.768693i	$0.720908\pi$
$678$		0			0
$679$	−25.0000	+	8.66025i	−0.959412	+	0.332350i
$680$		0			0
$681$	6.00000	+	10.3923i	0.229920	+	0.398234i
$682$		0			0
$683$	7.50000	−	12.9904i	0.286980	−	0.497063i	−0.686108	−	0.727500i	$-0.740682\pi$
$683$	7.50000	−	12.9904i	0.286980	−	0.497063i	0.973087	+	0.230437i	$0.0740155\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$		0			0
$687$	2.00000			0.0763048
$688$		0			0
$689$	−12.0000	+	20.7846i	−0.457164	+	0.791831i
$690$		0			0
$691$	−7.00000	−	12.1244i	−0.266293	−	0.461232i	0.701609	−	0.712562i	$-0.252465\pi$
$691$	−7.00000	−	12.1244i	−0.266293	−	0.461232i	−0.967901	+	0.251330i	$0.919132\pi$
$692$		0			0
$693$	−30.0000	+	10.3923i	−1.13961	+	0.394771i
$694$		0			0
$695$	1.00000	+	1.73205i	0.0379322	+	0.0657004i
$696$		0			0
$697$	−9.00000	+	15.5885i	−0.340899	+	0.590455i
$698$		0			0
$699$	6.00000			0.226941
$700$		0			0
$701$	27.0000			1.01978			0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000			1.01978			0.509888	+	0.860241i	$0.329687\pi$
$702$		0			0
$703$	−32.0000	+	55.4256i	−1.20690	+	2.09042i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.50000	−	7.79423i	0.0564133	−	0.293132i
$708$		0			0
$709$	0.500000	+	0.866025i	0.0187779	+	0.0325243i	0.875262	−	0.483650i	$-0.160689\pi$
$709$	0.500000	+	0.866025i	0.0187779	+	0.0325243i	−0.856484	+	0.516174i	$0.827356\pi$
$710$		0			0
$711$	−4.00000	+	6.92820i	−0.150012	+	0.259828i
$712$		0			0
$713$	6.00000			0.224702
$714$		0			0
$715$	−12.0000			−0.448775
$716$		0			0
$717$	−3.00000	+	5.19615i	−0.112037	+	0.194054i
$718$		0			0
$719$	15.0000	+	25.9808i	0.559406	+	0.968919i	0.997546	+	0.0700124i	$0.0223039\pi$
$719$	15.0000	+	25.9808i	0.559406	+	0.968919i	−0.438141	+	0.898906i	$0.644363\pi$
$720$		0			0
$721$	14.0000	+	12.1244i	0.521387	+	0.451535i
$722$		0			0
$723$	−13.0000	−	22.5167i	−0.483475	−	0.837404i
$724$		0			0
$725$	−1.50000	+	2.59808i	−0.0557086	+	0.0964901i
$726$		0			0
$727$	23.0000			0.853023			0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000			0.853023			0.426511	+	0.904482i	$0.359742\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	15.0000	−	25.9808i	0.554795	−	0.960933i
$732$		0			0
$733$	−1.00000	−	1.73205i	−0.0369358	−	0.0639748i	0.846967	−	0.531646i	$-0.178426\pi$
$733$	−1.00000	−	1.73205i	−0.0369358	−	0.0639748i	−0.883902	+	0.467671i	$0.845093\pi$
$734$		0			0
$735$	−5.50000	+	4.33013i	−0.202871	+	0.159719i
$736$		0			0
$737$	21.0000	+	36.3731i	0.773545	+	1.33982i
$738$		0			0
$739$	23.0000	−	39.8372i	0.846069	−	1.46543i	−0.0386212	−	0.999254i	$-0.512297\pi$
$739$	23.0000	−	39.8372i	0.846069	−	1.46543i	0.884690	−	0.466180i	$-0.154370\pi$
$740$		0			0
$741$	16.0000			0.587775
$742$		0			0
$743$	−39.0000			−1.43077			−0.715386	−	0.698730i	$-0.753749\pi$
$743$	−39.0000			−1.43077			−0.715386	+	0.698730i	$0.753749\pi$
$744$		0			0
$745$	−7.50000	+	12.9904i	−0.274779	+	0.475931i
$746$		0			0
$747$	3.00000	+	5.19615i	0.109764	+	0.190117i
$748$		0			0
$749$	6.00000	+	5.19615i	0.219235	+	0.189863i
$750$		0			0
$751$	−10.0000	−	17.3205i	−0.364905	−	0.632034i	0.623856	−	0.781540i	$-0.285565\pi$
$751$	−10.0000	−	17.3205i	−0.364905	−	0.632034i	−0.988761	+	0.149505i	$0.952232\pi$
$752$		0			0
$753$	9.00000	−	15.5885i	0.327978	−	0.568075i
$754$		0			0
$755$	10.0000			0.363937
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$	−9.00000	+	15.5885i	−0.326679	+	0.565825i
$760$		0			0
$761$	−15.0000	−	25.9808i	−0.543750	−	0.941802i	−0.998684	−	0.0512772i	$-0.983671\pi$
$761$	−15.0000	−	25.9808i	−0.543750	−	0.941802i	0.454935	−	0.890525i	$-0.349663\pi$
$762$		0			0
$763$	−8.50000	+	44.1673i	−0.307721	+	1.59896i
$764$		0			0
$765$	6.00000	+	10.3923i	0.216930	+	0.375735i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−34.0000			−1.22607			−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000			−1.22607			−0.613036	+	0.790055i	$0.710052\pi$
$770$		0			0
$771$	−24.0000			−0.864339
$772$		0			0
$773$	21.0000	−	36.3731i	0.755318	−	1.30825i	−0.189899	−	0.981804i	$-0.560816\pi$
$773$	21.0000	−	36.3731i	0.755318	−	1.30825i	0.945216	−	0.326445i	$-0.105851\pi$
$774$		0			0
$775$	−1.00000	−	1.73205i	−0.0359211	−	0.0622171i
$776$		0			0
$777$	20.0000	−	6.92820i	0.717496	−	0.248548i
$778$		0			0
$779$	12.0000	+	20.7846i	0.429945	+	0.744686i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−15.0000			−0.536056
$784$		0			0
$785$	−14.0000			−0.499681
$786$		0			0
$787$	15.5000	−	26.8468i	0.552515	−	0.956985i	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	15.5000	−	26.8468i	0.552515	−	0.956985i	0.998092	−	0.0617409i	$-0.0196653\pi$
$788$		0			0
$789$	1.50000	+	2.59808i	0.0534014	+	0.0924940i
$790$		0			0
$791$	−30.0000	+	10.3923i	−1.06668	+	0.369508i
$792$

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:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	140.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{9} +(-3.00000 + 5.19615i) q^{11} +2.00000 q^{13} -1.00000 q^{15} +(3.00000 - 5.19615i) q^{17} +(-4.00000 - 6.92820i) q^{19} +(-0.500000 + 2.59808i) q^{21} +(-1.50000 - 2.59808i) q^{23} +(-0.500000 + 0.866025i) q^{25} -5.00000 q^{27} +3.00000 q^{29} +(-1.00000 + 1.73205i) q^{31} +(-3.00000 - 5.19615i) q^{33} +(2.00000 + 1.73205i) q^{35} +(-4.00000 - 6.92820i) q^{37} +(-1.00000 + 1.73205i) q^{39} -3.00000 q^{41} +5.00000 q^{43} +(-1.00000 + 1.73205i) q^{45} +(5.50000 - 4.33013i) q^{49} +(3.00000 + 5.19615i) q^{51} +(-6.00000 + 10.3923i) q^{53} -6.00000 q^{55} +8.00000 q^{57} +(0.500000 + 0.866025i) q^{61} +(4.00000 + 3.46410i) q^{63} +(1.00000 + 1.73205i) q^{65} +(3.50000 - 6.06218i) q^{67} +3.00000 q^{69} +(5.00000 - 8.66025i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(-3.00000 + 15.5885i) q^{77} +(2.00000 + 3.46410i) q^{79} +(-0.500000 + 0.866025i) q^{81} +3.00000 q^{83} +6.00000 q^{85} +(-1.50000 + 2.59808i) q^{87} +(1.50000 + 2.59808i) q^{89} +(5.00000 - 1.73205i) q^{91} +(-1.00000 - 1.73205i) q^{93} +(4.00000 - 6.92820i) q^{95} -10.0000 q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - q^3 + q^5 + 5 * q^7 + 2 * q^9	\( 2 q - q^{3} + q^{5} + 5 q^{7} + 2 q^{9} - 6 q^{11} + 4 q^{13} - 2 q^{15} + 6 q^{17} - 8 q^{19} - q^{21} - 3 q^{23} - q^{25} - 10 q^{27} + 6 q^{29} - 2 q^{31} - 6 q^{33} + 4 q^{35} - 8 q^{37} - 2 q^{39} - 6 q^{41} + 10 q^{43} - 2 q^{45} + 11 q^{49} + 6 q^{51} - 12 q^{53} - 12 q^{55} + 16 q^{57} + q^{61} + 8 q^{63} + 2 q^{65} + 7 q^{67} + 6 q^{69} + 10 q^{73} - q^{75} - 6 q^{77} + 4 q^{79} - q^{81} + 6 q^{83} + 12 q^{85} - 3 q^{87} + 3 q^{89} + 10 q^{91} - 2 q^{93} + 8 q^{95} - 20 q^{97} - 24 q^{99}+O(q^{100}) \) 2 * q - q^3 + q^5 + 5 * q^7 + 2 * q^9 - 6 * q^11 + 4 * q^13 - 2 * q^15 + 6 * q^17 - 8 * q^19 - q^21 - 3 * q^23 - q^25 - 10 * q^27 + 6 * q^29 - 2 * q^31 - 6 * q^33 + 4 * q^35 - 8 * q^37 - 2 * q^39 - 6 * q^41 + 10 * q^43 - 2 * q^45 + 11 * q^49 + 6 * q^51 - 12 * q^53 - 12 * q^55 + 16 * q^57 + q^61 + 8 * q^63 + 2 * q^65 + 7 * q^67 + 6 * q^69 + 10 * q^73 - q^75 - 6 * q^77 + 4 * q^79 - q^81 + 6 * q^83 + 12 * q^85 - 3 * q^87 + 3 * q^89 + 10 * q^91 - 2 * q^93 + 8 * q^95 - 20 * q^97 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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