LMFDB - Embedded newform 1380.2.i.a.1241.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(1241,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("1380.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1380.i (of order $2$, degree $1$, 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:	$11.0193554789$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{14} - 4 x^{13} - 2 x^{12} + 12 x^{11} + 17 x^{10} + 16 x^{9} - 110 x^{8} + 48 x^{7} + \cdots + 6561 $ x^16 - x^14 - 4x^13 - 2x^12 + 12x^11 + 17x^10 + 16x^9 - 110x^8 + 48x^7 + 153x^6 + 324x^5 - 162x^4 - 972x^3 - 729x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.3
Root			$1.58090 - 0.707640i$ of defining polynomial
Character	$\chi$	$=$	1380.1241
Dual form			1380.2.i.a.1241.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.58090 - 0.707640i) q^{3} -1.00000 q^{5} -1.33803i q^{7} +(1.99849 + 2.23742i) q^{9} +O(q^{10})$	\(q+(-1.58090 - 0.707640i) q^{3} -1.00000 q^{5} -1.33803i q^{7} +(1.99849 + 2.23742i) q^{9} +2.74985 q^{11} -4.49229 q^{13} +(1.58090 + 0.707640i) q^{15} -3.35352 q^{17} +3.68097i q^{19} +(-0.946840 + 2.11528i) q^{21} +(-0.364431 - 4.78197i) q^{23} +1.00000 q^{25} +(-1.57613 - 4.95134i) q^{27} +3.60211i q^{29} -0.848670 q^{31} +(-4.34724 - 1.94590i) q^{33} +1.33803i q^{35} +2.13797i q^{37} +(7.10186 + 3.17892i) q^{39} -5.16250i q^{41} +12.2341i q^{43} +(-1.99849 - 2.23742i) q^{45} +0.193970i q^{47} +5.20969 q^{49} +(5.30158 + 2.37308i) q^{51} +5.60806 q^{53} -2.74985 q^{55} +(2.60480 - 5.81924i) q^{57} +6.74892i q^{59} +1.26463i q^{61} +(2.99372 - 2.67403i) q^{63} +4.49229 q^{65} +13.1252i q^{67} +(-2.80778 + 7.81770i) q^{69} +7.83428i q^{71} -6.09587 q^{73} +(-1.58090 - 0.707640i) q^{75} -3.67937i q^{77} +16.9228i q^{79} +(-1.01207 + 8.94291i) q^{81} +14.1183 q^{83} +3.35352 q^{85} +(2.54900 - 5.69458i) q^{87} -3.98744 q^{89} +6.01080i q^{91} +(1.34166 + 0.600553i) q^{93} -3.68097i q^{95} +13.7241i q^{97} +(5.49555 + 6.15256i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{5} + 2 q^{9}+O(q^{10}) $ 16 * q - 16 * q^5 + 2 * q^9	$ 16 q - 16 q^{5} + 2 q^{9} - 4 q^{21} + 10 q^{23} + 16 q^{25} - 12 q^{27} + 4 q^{31} + 2 q^{33} - 14 q^{39} - 2 q^{45} + 8 q^{49} - 2 q^{51} + 4 q^{53} - 2 q^{57} + 30 q^{63} - 4 q^{73} + 10 q^{81} - 4 q^{83} - 10 q^{87} - 28 q^{89} - 10 q^{93} - 26 q^{99}+O(q^{100}) $ 16 * q - 16 * q^5 + 2 * q^9 - 4 * q^21 + 10 * q^23 + 16 * q^25 - 12 * q^27 + 4 * q^31 + 2 * q^33 - 14 * q^39 - 2 * q^45 + 8 * q^49 - 2 * q^51 + 4 * q^53 - 2 * q^57 + 30 * q^63 - 4 * q^73 + 10 * q^81 - 4 * q^83 - 10 * q^87 - 28 * q^89 - 10 * q^93 - 26 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$691$	$1201$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.58090	−	0.707640i	−0.912733	−	0.408556i
$3$	−1.58090	−	0.707640i	−0.912733	−	0.408556i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	1.33803i		−	0.505726i	−0.967502	−	0.252863i	$-0.918628\pi$
$7$		−	1.33803i		−	0.505726i	0.967502	−	0.252863i	$-0.0813722\pi$
$8$		0			0
$9$	1.99849	+	2.23742i	0.666164	+	0.745806i
$10$		0			0
$11$	2.74985			0.829111			0.414556	−	0.910024i	$-0.363937\pi$
$11$	2.74985			0.829111			0.414556	+	0.910024i	$0.363937\pi$
$12$		0			0
$13$	−4.49229			−1.24594			−0.622968	−	0.782247i	$-0.714073\pi$
$13$	−4.49229			−1.24594			−0.622968	+	0.782247i	$0.714073\pi$
$14$		0			0
$15$	1.58090	+	0.707640i	0.408187	+	0.182712i
$16$		0			0
$17$	−3.35352			−0.813348			−0.406674	−	0.913573i	$-0.633311\pi$
$17$	−3.35352			−0.813348			−0.406674	+	0.913573i	$0.633311\pi$
$18$		0			0
$19$			3.68097i			0.844472i	0.906486	+	0.422236i	$0.138755\pi$
$19$			3.68097i			0.844472i	−0.906486	+	0.422236i	$0.861245\pi$
$20$		0			0
$21$	−0.946840	+	2.11528i	−0.206617	+	0.461593i
$22$		0			0
$23$	−0.364431	−	4.78197i	−0.0759891	−	0.997109i
$23$	−0.364431	−	4.78197i	−0.0759891	−	0.997109i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.57613	−	4.95134i	−0.303326	−	0.952887i
$28$		0			0
$29$			3.60211i			0.668896i	0.942414	+	0.334448i	$0.108550\pi$
$29$			3.60211i			0.668896i	−0.942414	+	0.334448i	$0.891450\pi$
$30$		0			0
$31$	−0.848670			−0.152426			−0.0762128	−	0.997092i	$-0.524283\pi$
$31$	−0.848670			−0.152426			−0.0762128	+	0.997092i	$0.524283\pi$
$32$		0			0
$33$	−4.34724	−	1.94590i	−0.756757	−	0.338738i
$34$		0			0
$35$			1.33803i			0.226168i
$36$		0			0
$37$			2.13797i			0.351479i	0.984437	+	0.175740i	$0.0562317\pi$
$37$			2.13797i			0.351479i	−0.984437	+	0.175740i	$0.943768\pi$
$38$		0			0
$39$	7.10186	+	3.17892i	1.13721	+	0.509035i
$40$		0			0
$41$		−	5.16250i		−	0.806246i	−0.915146	−	0.403123i	$-0.867925\pi$
$41$		−	5.16250i		−	0.806246i	0.915146	−	0.403123i	$-0.132075\pi$
$42$		0			0
$43$			12.2341i			1.86569i	0.360284	+	0.932843i	$0.382680\pi$
$43$			12.2341i			1.86569i	−0.360284	+	0.932843i	$0.617320\pi$
$44$		0			0
$45$	−1.99849	−	2.23742i	−0.297917	−	0.333534i
$46$		0			0
$47$			0.193970i			0.0282934i	0.999900	+	0.0141467i	$0.00450318\pi$
$47$			0.193970i			0.0282934i	−0.999900	+	0.0141467i	$0.995497\pi$
$48$		0			0
$49$	5.20969			0.744241
$50$		0			0
$51$	5.30158	+	2.37308i	0.742370	+	0.332298i
$52$		0			0
$53$	5.60806			0.770327			0.385163	−	0.922848i	$-0.374145\pi$
$53$	5.60806			0.770327			0.385163	+	0.922848i	$0.374145\pi$
$54$		0			0
$55$	−2.74985			−0.370790
$56$		0			0
$57$	2.60480	−	5.81924i	0.345014	−	0.770778i
$58$		0			0
$59$			6.74892i			0.878635i	0.898332	+	0.439318i	$0.144780\pi$
$59$			6.74892i			0.878635i	−0.898332	+	0.439318i	$0.855220\pi$
$60$		0			0
$61$			1.26463i			0.161919i	0.996717	+	0.0809594i	$0.0257984\pi$
$61$			1.26463i			0.161919i	−0.996717	+	0.0809594i	$0.974202\pi$
$62$		0			0
$63$	2.99372	−	2.67403i	0.377173	−	0.336896i
$64$		0			0
$65$	4.49229			0.557200
$66$		0			0
$67$			13.1252i			1.60349i	0.597664	+	0.801747i	$0.296096\pi$
$67$			13.1252i			1.60349i	−0.597664	+	0.801747i	$0.703904\pi$
$68$		0			0
$69$	−2.80778	+	7.81770i	−0.338017	+	0.941140i
$70$		0			0
$71$			7.83428i			0.929758i	0.885374	+	0.464879i	$0.153902\pi$
$71$			7.83428i			0.929758i	−0.885374	+	0.464879i	$0.846098\pi$
$72$		0			0
$73$	−6.09587			−0.713468			−0.356734	−	0.934206i	$-0.616110\pi$
$73$	−6.09587			−0.713468			−0.356734	+	0.934206i	$0.616110\pi$
$74$		0			0
$75$	−1.58090	−	0.707640i	−0.182547	−	0.0817112i
$76$		0			0
$77$		−	3.67937i		−	0.419303i
$78$		0			0
$79$			16.9228i			1.90397i	0.306148	+	0.951984i	$0.400960\pi$
$79$			16.9228i			1.90397i	−0.306148	+	0.951984i	$0.599040\pi$
$80$		0			0
$81$	−1.01207	+	8.94291i	−0.112452	+	0.993657i
$82$		0			0
$83$	14.1183			1.54969			0.774844	−	0.632153i	$-0.217829\pi$
$83$	14.1183			1.54969			0.774844	+	0.632153i	$0.217829\pi$
$84$		0			0
$85$	3.35352			0.363740
$86$		0			0
$87$	2.54900	−	5.69458i	0.273281	−	0.610523i
$88$		0			0
$89$	−3.98744			−0.422668			−0.211334	−	0.977414i	$-0.567781\pi$
$89$	−3.98744			−0.422668			−0.211334	+	0.977414i	$0.567781\pi$
$90$		0			0
$91$			6.01080i			0.630103i
$92$		0			0
$93$	1.34166	+	0.600553i	0.139124	+	0.0622744i
$94$		0			0
$95$		−	3.68097i		−	0.377659i
$96$		0			0
$97$			13.7241i			1.39347i	0.717327	+	0.696736i	$0.245365\pi$
$97$			13.7241i			1.39347i	−0.717327	+	0.696736i	$0.754635\pi$
$98$		0			0
$99$	5.49555	+	6.15256i	0.552324	+	0.618356i
$100$		0			0
$101$		−	12.7233i		−	1.26602i	−0.774144	−	0.633009i	$-0.781819\pi$
$101$		−	12.7233i		−	1.26602i	0.774144	−	0.633009i	$-0.218181\pi$
$102$		0			0
$103$			5.13554i			0.506019i	0.967464	+	0.253010i	$0.0814205\pi$
$103$			5.13554i			0.506019i	−0.967464	+	0.253010i	$0.918580\pi$
$104$		0			0
$105$	0.946840	−	2.11528i	0.0924021	−	0.206431i
$106$		0			0
$107$	20.1748			1.95037			0.975187	−	0.221384i	$-0.0710575\pi$
$107$	20.1748			1.95037			0.975187	+	0.221384i	$0.0710575\pi$
$108$		0			0
$109$			0.279628i			0.0267835i	0.999910	+	0.0133917i	$0.00426285\pi$
$109$			0.279628i			0.0267835i	−0.999910	+	0.0133917i	$0.995737\pi$
$110$		0			0
$111$	1.51291	−	3.37991i	0.143599	−	0.320807i
$112$		0			0
$113$	−5.92514			−0.557390			−0.278695	−	0.960380i	$-0.589902\pi$
$113$	−5.92514			−0.557390			−0.278695	+	0.960380i	$0.589902\pi$
$114$		0			0
$115$	0.364431	+	4.78197i	0.0339834	+	0.445921i
$116$		0			0
$117$	−8.97780	−	10.0511i	−0.829998	−	0.929227i
$118$		0			0
$119$			4.48709i			0.411331i
$120$		0			0
$121$	−3.43832			−0.312575
$122$		0			0
$123$	−3.65319	+	8.16139i	−0.329397	+	0.735888i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−15.2571			−1.35385			−0.676924	−	0.736053i	$-0.736687\pi$
$127$	−15.2571			−1.35385			−0.676924	+	0.736053i	$0.736687\pi$
$128$		0			0
$129$	8.65735	−	19.3409i	0.762237	−	1.70287i
$130$		0			0
$131$		−	4.88905i		−	0.427158i	−0.976926	−	0.213579i	$-0.931488\pi$
$131$		−	4.88905i		−	0.427158i	0.976926	−	0.213579i	$-0.0685122\pi$
$132$		0			0
$133$	4.92523			0.427071
$134$		0			0
$135$	1.57613	+	4.95134i	0.135652	+	0.426144i
$136$		0			0
$137$	−9.25821			−0.790982			−0.395491	−	0.918470i	$-0.629426\pi$
$137$	−9.25821			−0.790982			−0.395491	+	0.918470i	$0.629426\pi$
$138$		0			0
$139$	21.2373			1.80133			0.900663	−	0.434519i	$-0.143082\pi$
$139$	21.2373			1.80133			0.900663	+	0.434519i	$0.143082\pi$
$140$		0			0
$141$	0.137261	−	0.306647i	0.0115594	−	0.0258243i
$142$		0			0
$143$	−12.3531			−1.03302
$144$		0			0
$145$		−	3.60211i		−	0.299139i
$146$		0			0
$147$	−8.23600	−	3.68658i	−0.679294	−	0.304064i
$148$		0			0
$149$	2.24729			0.184105			0.0920525	−	0.995754i	$-0.470657\pi$
$149$	2.24729			0.184105			0.0920525	+	0.995754i	$0.470657\pi$
$150$		0			0
$151$	−4.52212			−0.368005			−0.184002	−	0.982926i	$-0.558905\pi$
$151$	−4.52212			−0.368005			−0.184002	+	0.982926i	$0.558905\pi$
$152$		0			0
$153$	−6.70198	−	7.50322i	−0.541823	−	0.606599i
$154$		0			0
$155$	0.848670			0.0681668
$156$		0			0
$157$		−	8.66622i		−	0.691639i	−0.938301	−	0.345820i	$-0.887601\pi$
$157$		−	8.66622i		−	0.691639i	0.938301	−	0.345820i	$-0.112399\pi$
$158$		0			0
$159$	−8.86579	−	3.96849i	−0.703103	−	0.314722i
$160$		0			0
$161$	−6.39839	+	0.487618i	−0.504264	+	0.0384297i
$162$		0			0
$163$	2.24414			0.175775			0.0878873	−	0.996130i	$-0.471988\pi$
$163$	2.24414			0.175775			0.0878873	+	0.996130i	$0.471988\pi$
$164$		0			0
$165$	4.34724	+	1.94590i	0.338432	+	0.151488i
$166$		0			0
$167$		−	9.38176i		−	0.725982i	−0.931793	−	0.362991i	$-0.881756\pi$
$167$		−	9.38176i		−	0.725982i	0.931793	−	0.362991i	$-0.118244\pi$
$168$		0			0
$169$	7.18066			0.552358
$170$		0			0
$171$	−8.23586	+	7.35638i	−0.629812	+	0.562556i
$172$		0			0
$173$			18.1796i			1.38217i	0.722775	+	0.691084i	$0.242866\pi$
$173$			18.1796i			1.38217i	−0.722775	+	0.691084i	$0.757134\pi$
$174$		0			0
$175$		−	1.33803i		−	0.101145i
$176$		0			0
$177$	4.77581	−	10.6694i	0.358972	−	0.801959i
$178$		0			0
$179$			18.3058i			1.36824i	0.729369	+	0.684120i	$0.239814\pi$
$179$			18.3058i			1.36824i	−0.729369	+	0.684120i	$0.760186\pi$
$180$		0			0
$181$		−	2.77431i		−	0.206213i	−0.994670	−	0.103107i	$-0.967122\pi$
$181$		−	2.77431i		−	0.206213i	0.994670	−	0.103107i	$-0.0328783\pi$
$182$		0			0
$183$	0.894900	−	1.99925i	0.0661529	−	0.147789i
$184$		0			0
$185$		−	2.13797i		−	0.157186i
$186$		0			0
$187$	−9.22168			−0.674356
$188$		0			0
$189$	−6.62502	+	2.10890i	−0.481900	+	0.153400i
$190$		0			0
$191$	−3.45036			−0.249660			−0.124830	−	0.992178i	$-0.539838\pi$
$191$	−3.45036			−0.249660			−0.124830	+	0.992178i	$0.539838\pi$
$192$		0			0
$193$	12.7334			0.916573			0.458286	−	0.888805i	$-0.348463\pi$
$193$	12.7334			0.916573			0.458286	+	0.888805i	$0.348463\pi$
$194$		0			0
$195$	−7.10186	−	3.17892i	−0.508575	−	0.227647i
$196$		0			0
$197$		−	2.53785i		−	0.180814i	−0.995905	−	0.0904072i	$-0.971183\pi$
$197$		−	2.53785i		−	0.180814i	0.995905	−	0.0904072i	$-0.0288169\pi$
$198$		0			0
$199$		−	19.0376i		−	1.34954i	−0.738029	−	0.674769i	$-0.764243\pi$
$199$		−	19.0376i		−	1.34954i	0.738029	−	0.674769i	$-0.235757\pi$
$200$		0			0
$201$	9.28789	−	20.7496i	0.655117	−	1.46356i
$202$		0			0
$203$	4.81972			0.338278
$204$		0			0
$205$			5.16250i			0.360564i
$206$		0			0
$207$	9.97094	−	10.3721i	0.693028	−	0.720911i
$208$		0			0
$209$			10.1221i			0.700161i
$210$		0			0
$211$	−15.8055			−1.08810			−0.544049	−	0.839054i	$-0.683109\pi$
$211$	−15.8055			−1.08810			−0.544049	+	0.839054i	$0.683109\pi$
$212$		0			0
$213$	5.54385	−	12.3852i	0.379858	−	0.848621i
$214$		0			0
$215$		−	12.2341i		−	0.834360i
$216$		0			0
$217$			1.13554i			0.0770856i
$218$		0			0
$219$	9.63696	+	4.31368i	0.651205	+	0.291492i
$220$		0			0
$221$	15.0650			1.01338
$222$		0			0
$223$	−5.05448			−0.338473			−0.169237	−	0.985575i	$-0.554130\pi$
$223$	−5.05448			−0.338473			−0.169237	+	0.985575i	$0.554130\pi$
$224$		0			0
$225$	1.99849	+	2.23742i	0.133233	+	0.149161i
$226$		0			0
$227$	−15.0937			−1.00180			−0.500902	−	0.865504i	$-0.666998\pi$
$227$	−15.0937			−1.00180			−0.500902	+	0.865504i	$0.666998\pi$
$228$		0			0
$229$			8.15559i			0.538937i	0.963009	+	0.269468i	$0.0868479\pi$
$229$			8.15559i			0.538937i	−0.963009	+	0.269468i	$0.913152\pi$
$230$		0			0
$231$	−2.60367	+	5.81671i	−0.171309	+	0.382712i
$232$		0			0
$233$			3.14664i			0.206143i	0.994674	+	0.103072i	$0.0328671\pi$
$233$			3.14664i			0.206143i	−0.994674	+	0.103072i	$0.967133\pi$
$234$		0			0
$235$		−	0.193970i		−	0.0126532i
$236$		0			0
$237$	11.9753	−	26.7533i	0.777878	−	1.73781i
$238$		0			0
$239$			13.1632i			0.851457i	0.904851	+	0.425729i	$0.139982\pi$
$239$			13.1632i			0.851457i	−0.904851	+	0.425729i	$0.860018\pi$
$240$		0			0
$241$			14.7351i			0.949169i	0.880210	+	0.474584i	$0.157402\pi$
$241$			14.7351i			0.949169i	−0.880210	+	0.474584i	$0.842598\pi$
$242$		0			0
$243$	7.92834	−	13.4217i	0.508603	−	0.861001i
$244$		0			0
$245$	−5.20969			−0.332835
$246$		0			0
$247$		−	16.5360i		−	1.05216i
$248$		0			0
$249$	−22.3197	−	9.99069i	−1.41445	−	0.633134i
$250$		0			0
$251$	−11.4540			−0.722972			−0.361486	−	0.932377i	$-0.617731\pi$
$251$	−11.4540			−0.722972			−0.361486	+	0.932377i	$0.617731\pi$
$252$		0			0
$253$	−1.00213	−	13.1497i	−0.0630034	−	0.826714i
$254$		0			0
$255$	−5.30158	−	2.37308i	−0.331998	−	0.148608i
$256$		0			0
$257$		−	17.2711i		−	1.07734i	−0.842516	−	0.538671i	$-0.818927\pi$
$257$		−	17.2711i		−	1.07734i	0.842516	−	0.538671i	$-0.181073\pi$
$258$		0			0
$259$	2.86065			0.177752
$260$		0			0
$261$	−8.05943	+	7.19879i	−0.498866	+	0.445594i
$262$		0			0
$263$	1.15448			0.0711881			0.0355940	−	0.999366i	$-0.488668\pi$
$263$	1.15448			0.0711881			0.0355940	+	0.999366i	$0.488668\pi$
$264$		0			0
$265$	−5.60806			−0.344501
$266$		0			0
$267$	6.30374	+	2.82167i	0.385783	+	0.172684i
$268$		0			0
$269$			17.6983i			1.07909i	0.841958	+	0.539543i	$0.181403\pi$
$269$			17.6983i			1.07909i	−0.841958	+	0.539543i	$0.818597\pi$
$270$		0			0
$271$	16.8107			1.02117			0.510587	−	0.859826i	$-0.329428\pi$
$271$	16.8107			1.02117			0.510587	+	0.859826i	$0.329428\pi$
$272$		0			0
$273$	4.25348	−	9.50247i	0.257432	−	0.575115i
$274$		0			0
$275$	2.74985			0.165822
$276$		0			0
$277$	−22.9798			−1.38072			−0.690361	−	0.723465i	$-0.742548\pi$
$277$	−22.9798			−1.38072			−0.690361	+	0.723465i	$0.742548\pi$
$278$		0			0
$279$	−1.69606	−	1.89883i	−0.101540	−	0.113680i
$280$		0			0
$281$	−29.4848			−1.75891			−0.879457	−	0.475978i	$-0.842094\pi$
$281$	−29.4848			−1.75891			−0.879457	+	0.475978i	$0.842094\pi$
$282$		0			0
$283$			6.98137i			0.414999i	0.978235	+	0.207500i	$0.0665326\pi$
$283$			6.98137i			0.414999i	−0.978235	+	0.207500i	$0.933467\pi$
$284$		0			0
$285$	−2.60480	+	5.81924i	−0.154295	+	0.344702i
$286$		0			0
$287$	−6.90755			−0.407740
$288$		0			0
$289$	−5.75391			−0.338465
$290$		0			0
$291$	9.71173	−	21.6965i	0.569312	−	1.27187i
$292$		0			0
$293$	−25.6671			−1.49949			−0.749745	−	0.661727i	$-0.769824\pi$
$293$	−25.6671			−1.49949			−0.749745	+	0.661727i	$0.769824\pi$
$294$		0			0
$295$		−	6.74892i		−	0.392938i
$296$		0			0
$297$	−4.33412	−	13.6155i	−0.251491	−	0.790049i
$298$		0			0
$299$	1.63713	+	21.4820i	0.0946776	+	1.24233i
$300$		0			0
$301$	16.3696			0.943525
$302$		0			0
$303$	−9.00354	+	20.1143i	−0.517240	+	1.15554i
$304$		0			0
$305$		−	1.26463i		−	0.0724123i
$306$		0			0
$307$	12.4508			0.710607			0.355303	−	0.934751i	$-0.384378\pi$
$307$	12.4508			0.710607			0.355303	+	0.934751i	$0.384378\pi$
$308$		0			0
$309$	3.63411	−	8.11877i	0.206737	−	0.461861i
$310$		0			0
$311$		−	22.2670i		−	1.26264i	−0.775521	−	0.631322i	$-0.782513\pi$
$311$		−	22.2670i		−	1.26264i	0.775521	−	0.631322i	$-0.217487\pi$
$312$		0			0
$313$		−	23.6062i		−	1.33430i	−0.744923	−	0.667150i	$-0.767514\pi$
$313$		−	23.6062i		−	1.33430i	0.744923	−	0.667150i	$-0.232486\pi$
$314$		0			0
$315$	−2.99372	+	2.67403i	−0.168677	+	0.150665i
$316$		0			0
$317$		−	29.2244i		−	1.64141i	−0.571356	−	0.820703i	$-0.693582\pi$
$317$		−	29.2244i		−	1.64141i	0.571356	−	0.820703i	$-0.306418\pi$
$318$		0			0
$319$			9.90527i			0.554589i
$320$		0			0
$321$	−31.8944	−	14.2765i	−1.78017	−	0.796837i
$322$		0			0
$323$		−	12.3442i		−	0.686849i
$324$		0			0
$325$	−4.49229			−0.249187
$326$		0			0
$327$	0.197876	−	0.442064i	0.0109426	−	0.0244462i
$328$		0			0
$329$	0.259536			0.0143087
$330$		0			0
$331$	−1.62651			−0.0894010			−0.0447005	−	0.999000i	$-0.514233\pi$
$331$	−1.62651			−0.0894010			−0.0447005	+	0.999000i	$0.514233\pi$
$332$		0			0
$333$	−4.78352	+	4.27271i	−0.262135	+	0.234143i
$334$		0			0
$335$		−	13.1252i		−	0.717104i
$336$		0			0
$337$		−	14.8106i		−	0.806785i	−0.915027	−	0.403393i	$-0.867831\pi$
$337$		−	14.8106i		−	0.806785i	0.915027	−	0.403393i	$-0.132169\pi$
$338$		0			0
$339$	9.36705	+	4.19287i	0.508749	+	0.227725i
$340$		0			0
$341$	−2.33372			−0.126378
$342$		0			0
$343$		−	16.3369i		−	0.882108i
$344$		0			0
$345$	2.80778	−	7.81770i	0.151166	−	0.420891i
$346$		0			0
$347$			13.3560i			0.716987i	0.933532	+	0.358493i	$0.116709\pi$
$347$			13.3560i			0.716987i	−0.933532	+	0.358493i	$0.883291\pi$
$348$		0			0
$349$	−3.68952			−0.197495			−0.0987476	−	0.995113i	$-0.531484\pi$
$349$	−3.68952			−0.197495			−0.0987476	+	0.995113i	$0.531484\pi$
$350$		0			0
$351$	7.08043	+	22.2429i	0.377925	+	1.18724i
$352$		0			0
$353$			27.0193i			1.43809i	0.694962	+	0.719047i	$0.255421\pi$
$353$			27.0193i			1.43809i	−0.694962	+	0.719047i	$0.744579\pi$
$354$		0			0
$355$		−	7.83428i		−	0.415800i
$356$		0			0
$357$	3.17525	−	7.09365i	0.168052	−	0.375436i
$358$		0			0
$359$	1.66364			0.0878034			0.0439017	−	0.999036i	$-0.486021\pi$
$359$	1.66364			0.0878034			0.0439017	+	0.999036i	$0.486021\pi$
$360$		0			0
$361$	5.45048			0.286867
$362$		0			0
$363$	5.43565	+	2.43310i	0.285297	+	0.127704i
$364$		0			0
$365$	6.09587			0.319072
$366$		0			0
$367$		−	0.263847i		−	0.0137727i	−0.999976	−	0.00688635i	$-0.997808\pi$
$367$		−	0.263847i		−	0.0137727i	0.999976	−	0.00688635i	$-0.00219201\pi$
$368$		0			0
$369$	11.5507	−	10.3172i	0.601303	−	0.537092i
$370$		0			0
$371$		−	7.50373i		−	0.389574i
$372$		0			0
$373$			33.2551i			1.72188i	0.508705	+	0.860941i	$0.330124\pi$
$373$			33.2551i			1.72188i	−0.508705	+	0.860941i	$0.669876\pi$
$374$		0			0
$375$	1.58090	+	0.707640i	0.0816373	+	0.0365424i
$376$		0			0
$377$		−	16.1817i		−	0.833402i
$378$		0			0
$379$		−	7.43841i		−	0.382086i	−0.981582	−	0.191043i	$-0.938813\pi$
$379$		−	7.43841i		−	0.382086i	0.981582	−	0.191043i	$-0.0611869\pi$
$380$		0			0
$381$	24.1199	+	10.7965i	1.23570	+	0.553123i
$382$		0			0
$383$	21.3283			1.08982			0.544912	−	0.838493i	$-0.316563\pi$
$383$	21.3283			1.08982			0.544912	+	0.838493i	$0.316563\pi$
$384$		0			0
$385$			3.67937i			0.187518i
$386$		0			0
$387$	−27.3728	+	24.4498i	−1.39144	+	1.24285i
$388$		0			0
$389$	16.3543			0.829196			0.414598	−	0.910005i	$-0.363922\pi$
$389$	16.3543			0.829196			0.414598	+	0.910005i	$0.363922\pi$
$390$		0			0
$391$	1.22213	+	16.0364i	0.0618056	+	0.810996i
$392$		0			0
$393$	−3.45969	+	7.72910i	−0.174518	+	0.389882i
$394$		0			0
$395$		−	16.9228i		−	0.851480i
$396$		0			0
$397$	−8.50156			−0.426681			−0.213341	−	0.976978i	$-0.568434\pi$
$397$	−8.50156			−0.426681			−0.213341	+	0.976978i	$0.568434\pi$
$398$		0			0
$399$	−7.78629	−	3.48529i	−0.389802	−	0.174483i
$400$		0			0
$401$	25.3023			1.26354			0.631768	−	0.775158i	$-0.282330\pi$
$401$	25.3023			1.26354			0.631768	+	0.775158i	$0.282330\pi$
$402$		0			0
$403$	3.81247			0.189913
$404$		0			0
$405$	1.01207	−	8.94291i	0.0502900	−	0.444377i
$406$		0			0
$407$			5.87909i			0.291415i
$408$		0			0
$409$	−19.1762			−0.948204			−0.474102	−	0.880470i	$-0.657227\pi$
$409$	−19.1762			−0.948204			−0.474102	+	0.880470i	$0.657227\pi$
$410$		0			0
$411$	14.6363	+	6.55148i	0.721955	+	0.323161i
$412$		0			0
$413$	9.03023			0.444349
$414$		0			0
$415$	−14.1183			−0.693041
$416$		0			0
$417$	−33.5741	−	15.0284i	−1.64413	−	0.735943i
$418$		0			0
$419$	−40.6862			−1.98765			−0.993826	−	0.110952i	$-0.964610\pi$
$419$	−40.6862			−1.98765			−0.993826	+	0.110952i	$0.964610\pi$
$420$		0			0
$421$		−	12.6373i		−	0.615902i	−0.951402	−	0.307951i	$-0.900357\pi$
$421$		−	12.6373i		−	0.615902i	0.951402	−	0.307951i	$-0.0996433\pi$
$422$		0			0
$423$	−0.433991	+	0.387647i	−0.0211014	+	0.0188480i
$424$		0			0
$425$	−3.35352			−0.162670
$426$		0			0
$427$	1.69210			0.0818865
$428$		0			0
$429$	19.5291	+	8.74156i	0.942872	+	0.422047i
$430$		0			0
$431$	23.9254			1.15245			0.576224	−	0.817292i	$-0.304526\pi$
$431$	23.9254			1.15245			0.576224	+	0.817292i	$0.304526\pi$
$432$		0			0
$433$			17.9946i			0.864765i	0.901690	+	0.432382i	$0.142327\pi$
$433$			17.9946i			0.864765i	−0.901690	+	0.432382i	$0.857673\pi$
$434$		0			0
$435$	−2.54900	+	5.69458i	−0.122215	+	0.273034i
$436$		0			0
$437$	17.6023	−	1.34146i	0.842030	−	0.0641707i
$438$		0			0
$439$	9.85116			0.470170			0.235085	−	0.971975i	$-0.424463\pi$
$439$	9.85116			0.470170			0.235085	+	0.971975i	$0.424463\pi$
$440$		0			0
$441$	10.4115	+	11.6562i	0.495786	+	0.555059i
$442$		0			0
$443$		−	3.74236i		−	0.177805i	−0.996040	−	0.0889024i	$-0.971664\pi$
$443$		−	3.74236i		−	0.177805i	0.996040	−	0.0889024i	$-0.0283359\pi$
$444$		0			0
$445$	3.98744			0.189023
$446$		0			0
$447$	−3.55274	−	1.59027i	−0.168039	−	0.0752172i
$448$		0			0
$449$		−	30.5353i		−	1.44105i	−0.693428	−	0.720526i	$-0.743900\pi$
$449$		−	30.5353i		−	1.44105i	0.693428	−	0.720526i	$-0.256100\pi$
$450$		0			0
$451$		−	14.1961i		−	0.668468i
$452$		0			0
$453$	7.14902	+	3.20003i	0.335890	+	0.150351i
$454$		0			0
$455$		−	6.01080i		−	0.281790i
$456$		0			0
$457$			14.2902i			0.668467i	0.942490	+	0.334234i	$0.108477\pi$
$457$			14.2902i			0.668467i	−0.942490	+	0.334234i	$0.891523\pi$
$458$		0			0
$459$	5.28558	+	16.6044i	0.246710	+	0.775028i
$460$		0			0
$461$			1.17858i			0.0548917i	0.999623	+	0.0274459i	$0.00873738\pi$
$461$			1.17858i			0.0548917i	−0.999623	+	0.0274459i	$0.991263\pi$
$462$		0			0
$463$	5.02783			0.233663			0.116831	−	0.993152i	$-0.462726\pi$
$463$	5.02783			0.233663			0.116831	+	0.993152i	$0.462726\pi$
$464$		0			0
$465$	−1.34166	−	0.600553i	−0.0622181	−	0.0278500i
$466$		0			0
$467$	17.3944			0.804917			0.402459	−	0.915438i	$-0.368156\pi$
$467$	17.3944			0.804917			0.402459	+	0.915438i	$0.368156\pi$
$468$		0			0
$469$	17.5618			0.810928
$470$		0			0
$471$	−6.13256	+	13.7004i	−0.282574	+	0.631282i
$472$		0			0
$473$			33.6420i			1.54686i
$474$		0			0
$475$			3.68097i			0.168894i
$476$		0			0
$477$	11.2077	+	12.5476i	0.513164	+	0.574514i
$478$		0			0
$479$	−4.28103			−0.195605			−0.0978027	−	0.995206i	$-0.531181\pi$
$479$	−4.28103			−0.195605			−0.0978027	+	0.995206i	$0.531181\pi$
$480$		0			0
$481$		−	9.60436i		−	0.437921i
$482$		0			0
$483$	10.4603	+	3.75688i	0.475959	+	0.170944i
$484$		0			0
$485$		−	13.7241i		−	0.623180i
$486$		0			0
$487$	34.0100			1.54114			0.770570	−	0.637355i	$-0.219971\pi$
$487$	34.0100			1.54114			0.770570	+	0.637355i	$0.219971\pi$
$488$		0			0
$489$	−3.54776	−	1.58804i	−0.160435	−	0.0718138i
$490$		0			0
$491$			5.96723i			0.269297i	0.990893	+	0.134649i	$0.0429906\pi$
$491$			5.96723i			0.269297i	−0.990893	+	0.134649i	$0.957009\pi$
$492$		0			0
$493$		−	12.0798i		−	0.544045i
$494$		0			0
$495$	−5.49555	−	6.15256i	−0.247007	−	0.276537i
$496$		0			0
$497$	10.4825			0.470203
$498$		0			0
$499$	−28.5573			−1.27840			−0.639200	−	0.769041i	$-0.720734\pi$
$499$	−28.5573			−1.27840			−0.639200	+	0.769041i	$0.720734\pi$
$500$		0			0
$501$	−6.63891	+	14.8316i	−0.296605	+	0.662628i
$502$		0			0
$503$	−19.0457			−0.849207			−0.424604	−	0.905379i	$-0.639587\pi$
$503$	−19.0457			−0.849207			−0.424604	+	0.905379i	$0.639587\pi$
$504$		0			0
$505$			12.7233i			0.566181i
$506$		0			0
$507$	−11.3519	−	5.08132i	−0.504156	−	0.225669i
$508$		0			0
$509$			21.3015i			0.944174i	0.881552	+	0.472087i	$0.156499\pi$
$509$			21.3015i			0.944174i	−0.881552	+	0.472087i	$0.843501\pi$
$510$		0			0
$511$			8.15643i			0.360819i
$512$		0			0
$513$	18.2257	−	5.80168i	0.804686	−	0.256150i
$514$		0			0
$515$		−	5.13554i		−	0.226299i
$516$		0			0
$517$			0.533388i			0.0234584i
$518$		0			0
$519$	12.8646	−	28.7401i	0.564693	−	1.26155i
$520$		0			0
$521$	10.9035			0.477689			0.238844	−	0.971058i	$-0.423231\pi$
$521$	10.9035			0.477689			0.238844	+	0.971058i	$0.423231\pi$
$522$		0			0
$523$		−	23.1696i		−	1.01313i	−0.862201	−	0.506567i	$-0.830914\pi$
$523$		−	23.1696i		−	1.01313i	0.862201	−	0.506567i	$-0.169086\pi$
$524$		0			0
$525$	−0.946840	+	2.11528i	−0.0413235	+	0.0923186i
$526$		0			0
$527$	2.84603			0.123975
$528$		0			0
$529$	−22.7344	+	3.48539i	−0.988451	+	0.151539i
$530$		0			0
$531$	−15.1002	+	13.4877i	−0.655291	+	0.585315i
$532$		0			0
$533$			23.1914i			1.00453i
$534$		0			0
$535$	−20.1748			−0.872233
$536$		0			0
$537$	12.9539	−	28.9396i	0.559003	−	1.24884i
$538$		0			0
$539$	14.3259			0.617059
$540$		0			0
$541$	1.98002			0.0851277			0.0425638	−	0.999094i	$-0.486447\pi$
$541$	1.98002			0.0851277			0.0425638	+	0.999094i	$0.486447\pi$
$542$		0			0
$543$	−1.96322	+	4.38591i	−0.0842497	+	0.188218i
$544$		0			0
$545$		−	0.279628i		−	0.0119779i
$546$		0			0
$547$	25.0329			1.07033			0.535164	−	0.844748i	$-0.320250\pi$
$547$	25.0329			1.07033			0.535164	+	0.844748i	$0.320250\pi$
$548$		0			0
$549$	−2.82950	+	2.52734i	−0.120760	+	0.107864i
$550$		0			0
$551$	−13.2593			−0.564864
$552$		0			0
$553$	22.6432			0.962886
$554$		0			0
$555$	−1.51291	+	3.37991i	−0.0642195	+	0.143469i
$556$		0			0
$557$	−32.8210			−1.39067			−0.695334	−	0.718687i	$-0.744744\pi$
$557$	−32.8210			−1.39067			−0.695334	+	0.718687i	$0.744744\pi$
$558$		0			0
$559$		−	54.9592i		−	2.32453i
$560$		0			0
$561$	14.5785	+	6.52563i	0.615507	+	0.275512i
$562$		0			0
$563$	−25.6554			−1.08125			−0.540623	−	0.841265i	$-0.681811\pi$
$563$	−25.6554			−1.08125			−0.540623	+	0.841265i	$0.681811\pi$
$564$		0			0
$565$	5.92514			0.249272
$566$		0			0
$567$	11.9658	+	1.35417i	0.502518	+	0.0568699i
$568$		0			0
$569$	−10.8031			−0.452890			−0.226445	−	0.974024i	$-0.572710\pi$
$569$	−10.8031			−0.452890			−0.226445	+	0.974024i	$0.572710\pi$
$570$		0			0
$571$			35.0494i			1.46677i	0.679812	+	0.733386i	$0.262061\pi$
$571$			35.0494i			1.46677i	−0.679812	+	0.733386i	$0.737939\pi$
$572$		0			0
$573$	5.45468	+	2.44162i	0.227873	+	0.102000i
$574$		0			0
$575$	−0.364431	−	4.78197i	−0.0151978	−	0.199422i
$576$		0			0
$577$	−24.4613			−1.01834			−0.509168	−	0.860667i	$-0.670047\pi$
$577$	−24.4613			−1.01834			−0.509168	+	0.860667i	$0.670047\pi$
$578$		0			0
$579$	−20.1303	−	9.01069i	−0.836586	−	0.374471i
$580$		0			0
$581$		−	18.8907i		−	0.783717i
$582$		0			0
$583$	15.4213			0.638686
$584$		0			0
$585$	8.97780	+	10.0511i	0.371186	+	0.415563i
$586$		0			0
$587$			31.2882i			1.29140i	0.763591	+	0.645700i	$0.223434\pi$
$587$			31.2882i			1.29140i	−0.763591	+	0.645700i	$0.776566\pi$
$588$		0			0
$589$		−	3.12393i		−	0.128719i
$590$		0			0
$591$	−1.79589	+	4.01209i	−0.0738729	+	0.165035i
$592$		0			0
$593$			10.7744i			0.442450i	0.975223	+	0.221225i	$0.0710055\pi$
$593$			10.7744i			0.442450i	−0.975223	+	0.221225i	$0.928995\pi$
$594$		0			0
$595$		−	4.48709i		−	0.183953i
$596$		0			0
$597$	−13.4718	+	30.0965i	−0.551362	+	1.23177i
$598$		0			0
$599$			3.30496i			0.135037i	0.997718	+	0.0675185i	$0.0215082\pi$
$599$			3.30496i			0.135037i	−0.997718	+	0.0675185i	$0.978492\pi$
$600$		0			0
$601$	3.82938			0.156204			0.0781018	−	0.996945i	$-0.475114\pi$
$601$	3.82938			0.156204			0.0781018	+	0.996945i	$0.475114\pi$
$602$		0			0
$603$	−29.3664	+	26.2305i	−1.19589	+	1.06819i
$604$		0			0
$605$	3.43832			0.139788
$606$		0			0
$607$	−22.8992			−0.929450			−0.464725	−	0.885455i	$-0.653847\pi$
$607$	−22.8992			−0.929450			−0.464725	+	0.885455i	$0.653847\pi$
$608$		0			0
$609$	−7.61949	−	3.41063i	−0.308757	−	0.138206i
$610$		0			0
$611$		−	0.871368i		−	0.0352518i
$612$		0			0
$613$			29.6253i			1.19656i	0.801289	+	0.598278i	$0.204148\pi$
$613$			29.6253i			1.19656i	−0.801289	+	0.598278i	$0.795852\pi$
$614$		0			0
$615$	3.65319	−	8.16139i	0.147311	−	0.329099i
$616$		0			0
$617$	−20.4826			−0.824597			−0.412298	−	0.911049i	$-0.635274\pi$
$617$	−20.4826			−0.824597			−0.412298	+	0.911049i	$0.635274\pi$
$618$		0			0
$619$			26.8265i			1.07825i	0.842227	+	0.539124i	$0.181244\pi$
$619$			26.8265i			1.07825i	−0.842227	+	0.539124i	$0.818756\pi$
$620$		0			0
$621$	−23.1028	+	9.34142i	−0.927082	+	0.374858i
$622$		0			0
$623$			5.33529i			0.213754i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	7.16281	−	16.0020i	0.286055	−	0.639060i
$628$		0			0
$629$		−	7.16971i		−	0.285875i
$630$		0			0
$631$		−	37.6576i		−	1.49912i	−0.661934	−	0.749562i	$-0.730264\pi$
$631$		−	37.6576i		−	1.49912i	0.661934	−	0.749562i	$-0.269736\pi$
$632$		0			0
$633$	24.9870	+	11.1846i	0.993143	+	0.444549i
$634$		0			0
$635$	15.2571			0.605459
$636$		0			0
$637$	−23.4034			−0.927278
$638$		0			0
$639$	−17.5286	+	15.6567i	−0.693419	+	0.619371i
$640$		0			0
$641$	43.1803			1.70552			0.852759	−	0.522305i	$-0.174928\pi$
$641$	43.1803			1.70552			0.852759	+	0.522305i	$0.174928\pi$
$642$		0			0
$643$			13.8356i			0.545622i	0.962068	+	0.272811i	$0.0879534\pi$
$643$			13.8356i			0.545622i	−0.962068	+	0.272811i	$0.912047\pi$
$644$		0			0
$645$	−8.65735	+	19.3409i	−0.340883	+	0.761548i
$646$		0			0
$647$			30.6706i			1.20579i	0.797822	+	0.602893i	$0.205985\pi$
$647$			30.6706i			1.20579i	−0.797822	+	0.602893i	$0.794015\pi$
$648$		0			0
$649$			18.5585i			0.728486i
$650$		0			0
$651$	0.803555	−	1.79518i	0.0314938	−	0.0703586i
$652$		0			0
$653$			7.78046i			0.304473i	0.988344	+	0.152236i	$0.0486475\pi$
$653$			7.78046i			0.304473i	−0.988344	+	0.152236i	$0.951352\pi$
$654$		0			0
$655$			4.88905i			0.191031i
$656$		0			0
$657$	−12.1825	−	13.6390i	−0.475286	−	0.532108i
$658$		0			0
$659$	30.8646			1.20231			0.601156	−	0.799132i	$-0.294707\pi$
$659$	30.8646			1.20231			0.601156	+	0.799132i	$0.294707\pi$
$660$		0			0
$661$			11.7589i			0.457370i	0.973501	+	0.228685i	$0.0734425\pi$
$661$			11.7589i			0.457370i	−0.973501	+	0.228685i	$0.926557\pi$
$662$		0			0
$663$	−23.8162	−	10.6606i	−0.924945	−	0.414023i
$664$		0			0
$665$	−4.92523			−0.190992
$666$		0			0
$667$	17.2252	−	1.31272i	0.666962	−	0.0508288i
$668$		0			0
$669$	7.99063	+	3.57675i	0.308936	+	0.138285i
$670$		0			0
$671$			3.47753i			0.134249i
$672$		0			0
$673$	24.3156			0.937297			0.468649	−	0.883385i	$-0.344741\pi$
$673$	24.3156			0.937297			0.468649	+	0.883385i	$0.344741\pi$
$674$		0			0
$675$	−1.57613	−	4.95134i	−0.0606652	−	0.190577i
$676$		0			0
$677$	35.6764			1.37115			0.685577	−	0.728000i	$-0.259550\pi$
$677$	35.6764			1.37115			0.685577	+	0.728000i	$0.259550\pi$
$678$		0			0
$679$	18.3632			0.704715
$680$		0			0
$681$	23.8616	+	10.6809i	0.914380	+	0.409293i
$682$		0			0
$683$		−	32.0532i		−	1.22648i	−0.789896	−	0.613241i	$-0.789866\pi$
$683$		−	32.0532i		−	1.22648i	0.789896	−	0.613241i	$-0.210134\pi$
$684$		0			0
$685$	9.25821			0.353738
$686$		0			0
$687$	5.77122	−	12.8932i	0.220186	−	0.491905i
$688$		0			0
$689$	−25.1930			−0.959778
$690$		0			0
$691$	3.89302			0.148097			0.0740487	−	0.997255i	$-0.476408\pi$
$691$	3.89302			0.148097			0.0740487	+	0.997255i	$0.476408\pi$
$692$		0			0
$693$	8.23228	−	7.35319i	0.312718	−	0.279324i
$694$		0			0
$695$	−21.2373			−0.805577
$696$		0			0
$697$			17.3125i			0.655759i
$698$		0			0
$699$	2.22669	−	4.97452i	0.0842211	−	0.188154i
$700$		0			0
$701$	−27.9970			−1.05743			−0.528716	−	0.848799i	$-0.677326\pi$
$701$	−27.9970			−1.05743			−0.528716	+	0.848799i	$0.677326\pi$
$702$		0			0
$703$	−7.86978			−0.296814
$704$		0			0
$705$	−0.137261	+	0.306647i	−0.00516954	+	0.0115490i
$706$		0			0
$707$	−17.0241			−0.640259
$708$		0			0
$709$			5.10459i			0.191707i	0.995395	+	0.0958534i	$0.0305580\pi$
$709$			5.10459i			0.191707i	−0.995395	+	0.0958534i	$0.969442\pi$
$710$		0			0
$711$	−37.8634	+	33.8201i	−1.41999	+	1.26835i
$712$		0			0
$713$	0.309282	+	4.05831i	0.0115827	+	0.151985i
$714$		0			0
$715$	12.3531			0.461981
$716$		0			0
$717$	9.31481	−	20.8097i	0.347868	−	0.777153i
$718$		0			0
$719$			29.5426i			1.10175i	0.834587	+	0.550876i	$0.185706\pi$
$719$			29.5426i			1.10175i	−0.834587	+	0.550876i	$0.814294\pi$
$720$		0			0
$721$	6.87148			0.255907
$722$		0			0
$723$	10.4271	−	23.2947i	0.387789	−	0.866338i
$724$		0			0
$725$			3.60211i			0.133779i
$726$		0			0
$727$		−	17.9066i		−	0.664119i	−0.943259	−	0.332059i	$-0.892257\pi$
$727$		−	17.9066i		−	0.664119i	0.943259	−	0.332059i	$-0.107743\pi$
$728$		0			0
$729$	−22.0316	+	15.6079i	−0.815986	+	0.578071i
$730$		0			0
$731$		−	41.0273i		−	1.51745i
$732$		0			0
$733$		−	3.48200i		−	0.128611i	−0.997930	−	0.0643054i	$-0.979517\pi$
$733$		−	3.48200i		−	0.128611i	0.997930	−	0.0643054i	$-0.0204832\pi$
$734$		0			0
$735$	8.23600	+	3.68658i	0.303789	+	0.135982i
$736$		0			0
$737$			36.0922i			1.32947i
$738$		0			0
$739$	−1.95416			−0.0718850			−0.0359425	−	0.999354i	$-0.511443\pi$
$739$	−1.95416			−0.0718850			−0.0359425	+	0.999354i	$0.511443\pi$
$740$		0			0
$741$	−11.7015	+	26.1417i	−0.429866	+	0.960340i
$742$		0			0
$743$	36.9407			1.35522			0.677611	−	0.735421i	$-0.263015\pi$
$743$	36.9407			1.35522			0.677611	+	0.735421i	$0.263015\pi$
$744$		0			0
$745$	−2.24729			−0.0823342
$746$		0			0
$747$	28.2153	+	31.5886i	1.03235	+	1.15577i
$748$		0			0
$749$		−	26.9944i		−	0.986354i
$750$		0			0
$751$		−	22.7485i		−	0.830106i	−0.909797	−	0.415053i	$-0.863763\pi$
$751$		−	22.7485i		−	0.830106i	0.909797	−	0.415053i	$-0.136237\pi$
$752$		0			0
$753$	18.1077	+	8.10533i	0.659881	+	0.295375i
$754$		0			0
$755$	4.52212			0.164577
$756$		0			0
$757$		−	37.6633i		−	1.36889i	−0.729063	−	0.684447i	$-0.760044\pi$
$757$		−	37.6633i		−	1.36889i	0.729063	−	0.684447i	$-0.239956\pi$
$758$		0			0
$759$	−7.72098	+	21.4975i	−0.280254	+	0.780310i
$760$		0			0
$761$		−	43.0364i		−	1.56007i	−0.625737	−	0.780034i	$-0.715202\pi$
$761$		−	43.0364i		−	1.56007i	0.625737	−	0.780034i	$-0.284798\pi$
$762$		0			0
$763$	0.374149			0.0135451
$764$		0			0
$765$	6.70198	+	7.50322i	0.242311	+	0.271279i
$766$		0			0
$767$		−	30.3181i		−	1.09472i
$768$		0			0
$769$		−	2.73819i		−	0.0987416i	−0.998781	−	0.0493708i	$-0.984278\pi$
$769$		−	2.73819i		−	0.0987416i	0.998781	−	0.0493708i	$-0.0157216\pi$
$770$		0			0
$771$	−12.2217	+	27.3039i	−0.440154	+	0.983325i
$772$		0			0
$773$	46.2313			1.66283			0.831413	−	0.555655i	$-0.187532\pi$
$773$	46.2313			1.66283			0.831413	+	0.555655i	$0.187532\pi$
$774$		0			0
$775$	−0.848670			−0.0304851
$776$		0			0
$777$	−4.52240	−	2.02431i	−0.162240	−	0.0726218i
$778$		0			0
$779$	19.0030			0.680852
$780$		0			0
$781$			21.5431i			0.770873i
$782$		0			0
$783$	17.8353	−	5.67740i	0.637382	−	0.202894i
$784$		0			0
$785$			8.66622i			0.309311i
$786$		0			0
$787$			20.9501i			0.746792i	0.927672	+	0.373396i	$0.121807\pi$
$787$			20.9501i			0.746792i	−0.927672	+	0.373396i	$0.878193\pi$
$788$		0			0
$789$	−1.82511	−	0.816954i	−0.0649757	−	0.0290843i
$790$		0			0
$791$			7.92798i			0.281887i
$792$		0			0
$793$		−	5.68107i		−	0.201741i
$794$		0			0
$795$	8.86579	+	3.96849i	0.314437	+	0.140748i
$796$		0			0
$797$	−9.01320			−0.319264			−0.159632	−	0.987177i	$-0.551031\pi$
$797$	−9.01320			−0.319264			−0.159632	+	0.987177i	$0.551031\pi$
$798$		0			0
$799$		−	0.650481i		−	0.0230124i
$800$		0			0
$801$	−7.96886	−	8.92156i	−0.281566	−	0.315228i
$802$		0			0
$803$	−16.7627			−0.591544
$804$		0			0
$805$	6.39839	−	0.487618i	0.225514	−	0.0171863i
$806$		0			0
$807$	12.5241	−	27.9793i	0.440868	−	0.984918i
$808$		0			0
$809$		−	23.9666i		−	0.842622i	−0.906916	−	0.421311i	$-0.861570\pi$
$809$		−	23.9666i		−	0.842622i	0.906916	−	0.421311i	$-0.138430\pi$
$810$		0			0
$811$	−48.9023			−1.71719			−0.858596	−	0.512653i	$-0.828663\pi$
$811$	−48.9023			−1.71719			−0.858596	+	0.512653i	$0.828663\pi$
$812$		0			0
$813$	−26.5760	−	11.8959i	−0.932060	−	0.417207i
$814$		0			0
$815$	−2.24414			−0.0786088
$816$		0			0
$817$	−45.0334			−1.57552
$818$		0			0
$819$	−13.4487	+	12.0125i	−0.469934	+	0.419751i
$820$		0			0
$821$		−	34.2034i		−	1.19371i	−0.802350	−	0.596854i	$-0.796417\pi$
$821$		−	34.2034i		−	1.19371i	0.802350	−	0.596854i	$-0.203583\pi$
$822$		0			0
$823$	−37.2997			−1.30019			−0.650093	−	0.759855i	$-0.725270\pi$
$823$	−37.2997			−1.30019			−0.650093	+	0.759855i	$0.725270\pi$
$824$		0			0
$825$	−4.34724	−	1.94590i	−0.151351	−	0.0677477i
$826$		0			0
$827$	17.0591			0.593204			0.296602	−	0.955001i	$-0.404146\pi$
$827$	17.0591			0.593204			0.296602	+	0.955001i	$0.404146\pi$
$828$		0			0
$829$	21.0900			0.732486			0.366243	−	0.930519i	$-0.380644\pi$
$829$	21.0900			0.732486			0.366243	+	0.930519i	$0.380644\pi$
$830$		0			0
$831$	36.3288	+	16.2614i	1.26023	+	0.564103i
$832$		0			0
$833$	−17.4708			−0.605327
$834$		0			0
$835$			9.38176i			0.324669i
$836$		0			0
$837$	1.33761	+	4.20206i	0.0462347	+	0.145244i
$838$		0			0
$839$	−22.5935			−0.780014			−0.390007	−	0.920812i	$-0.627527\pi$
$839$	−22.5935			−0.780014			−0.390007	+	0.920812i	$0.627527\pi$
$840$		0			0
$841$	16.0248			0.552579
$842$		0			0
$843$	46.6125	+	20.8646i	1.60542	+	0.718615i
$844$		0			0
$845$	−7.18066			−0.247022
$846$		0			0
$847$			4.60056i			0.158077i
$848$		0			0
$849$	4.94030	−	11.0368i	0.169551	−	0.378784i
$850$		0			0
$851$	10.2237	−	0.779141i	0.350463	−	0.0267086i
$852$		0			0
$853$	−5.08828			−0.174219			−0.0871097	−	0.996199i	$-0.527763\pi$
$853$	−5.08828			−0.174219			−0.0871097	+	0.996199i	$0.527763\pi$
$854$		0			0
$855$	8.23586	−	7.35638i	0.281660	−	0.251583i
$856$		0			0
$857$			46.8919i			1.60180i	0.598800	+	0.800899i	$0.295645\pi$
$857$			46.8919i			1.60180i	−0.598800	+	0.800899i	$0.704355\pi$
$858$		0			0
$859$	−6.90124			−0.235467			−0.117734	−	0.993045i	$-0.537563\pi$
$859$	−6.90124			−0.235467			−0.117734	+	0.993045i	$0.537563\pi$
$860$		0			0
$861$	10.9201	+	4.88806i	0.372158	+	0.166585i
$862$		0			0
$863$		−	35.4546i		−	1.20689i	−0.797406	−	0.603444i	$-0.793795\pi$
$863$		−	35.4546i		−	1.20689i	0.797406	−	0.603444i	$-0.206205\pi$
$864$		0			0
$865$		−	18.1796i		−	0.618124i
$866$		0			0
$867$	9.09636	+	4.07170i	0.308929	+	0.138282i
$868$		0			0
$869$			46.5353i			1.57860i
$870$		0			0
$871$		−	58.9620i		−	1.99785i
$872$		0			0
$873$	−30.7066	+	27.4275i	−1.03926	+	0.928281i
$874$		0			0
$875$			1.33803i			0.0452335i
$876$		0			0
$877$	22.5282			0.760725			0.380362	−	0.924838i	$-0.375799\pi$
$877$	22.5282			0.760725			0.380362	+	0.924838i	$0.375799\pi$
$878$		0			0
$879$	40.5772	+	18.1631i	1.36863	+	0.612626i
$880$		0			0
$881$	−16.6378			−0.560542			−0.280271	−	0.959921i	$-0.590424\pi$
$881$	−16.6378			−0.560542			−0.280271	+	0.959921i	$0.590424\pi$
$882$		0			0
$883$	−18.9543			−0.637864			−0.318932	−	0.947778i	$-0.603324\pi$
$883$	−18.9543			−0.637864			−0.318932	+	0.947778i	$0.603324\pi$
$884$		0			0
$885$	−4.77581	+	10.6694i	−0.160537	+	0.358647i
$886$		0			0
$887$			39.9447i			1.34121i	0.741813	+	0.670606i	$0.233966\pi$
$887$			39.9447i			1.34121i	−0.741813	+	0.670606i	$0.766034\pi$
$888$		0			0
$889$			20.4144i			0.684676i
$890$		0			0
$891$	−2.78303	+	24.5917i	−0.0932351	+	0.823852i
$892$		0			0
$893$	−0.713996			−0.0238930
$894$		0			0
$895$		−	18.3058i		−	0.611896i
$896$		0			0
$897$	12.6134	−	35.1193i	0.421148	−	1.17260i
$898$		0			0
$899$		−	3.05701i		−	0.101957i
$900$		0			0
$901$	−18.8067			−0.626543
$902$		0			0
$903$	−25.8786	−	11.5838i	−0.861187	−	0.385483i
$904$		0			0
$905$			2.77431i			0.0922213i
$906$		0			0
$907$			12.6752i			0.420874i	0.977607	+	0.210437i	$0.0674887\pi$
$907$			12.6752i			0.420874i	−0.977607	+	0.210437i	$0.932511\pi$
$908$		0			0
$909$	28.4674	−	25.4275i	0.944204	−	0.843376i
$910$		0			0
$911$	38.2884			1.26855			0.634275	−	0.773107i	$-0.281299\pi$
$911$	38.2884			1.26855			0.634275	+	0.773107i	$0.281299\pi$
$912$		0			0
$913$	38.8233			1.28486
$914$		0			0
$915$	−0.894900	+	1.99925i	−0.0295845	+	0.0660931i
$916$		0			0
$917$	−6.54167			−0.216025
$918$		0			0
$919$		−	15.8263i		−	0.522061i	−0.965331	−	0.261030i	$-0.915938\pi$
$919$		−	15.8263i		−	0.522061i	0.965331	−	0.261030i	$-0.0840622\pi$
$920$		0			0
$921$	−19.6835	−	8.81071i	−0.648594	−	0.290323i
$922$		0			0
$923$		−	35.1939i		−	1.15842i
$924$		0			0
$925$			2.13797i			0.0702959i
$926$		0			0
$927$	−11.4903	+	10.2633i	−0.377392	+	0.337092i
$928$		0			0
$929$		−	10.1512i		−	0.333049i	−0.986037	−	0.166525i	$-0.946745\pi$
$929$		−	10.1512i		−	0.333049i	0.986037	−	0.166525i	$-0.0532545\pi$
$930$		0			0
$931$			19.1767i			0.628491i
$932$		0			0
$933$	−15.7570	+	35.2018i	−0.515861	+	1.15246i
$934$		0			0
$935$	9.22168			0.301581
$936$		0			0
$937$			34.9391i			1.14141i	0.821155	+	0.570706i	$0.193330\pi$
$937$			34.9391i			1.14141i	−0.821155	+	0.570706i	$0.806670\pi$
$938$		0			0
$939$	−16.7047	+	37.3190i	−0.545137	+	1.21786i
$940$		0			0
$941$	−22.0562			−0.719010			−0.359505	−	0.933143i	$-0.617054\pi$
$941$	−22.0562			−0.719010			−0.359505	+	0.933143i	$0.617054\pi$
$942$		0			0
$943$	−24.6869	+	1.88137i	−0.803915	+	0.0612659i
$944$		0			0
$945$	6.62502	−	2.10890i	0.215512	−	0.0686025i
$946$		0			0
$947$			35.9711i			1.16890i	0.811428	+	0.584452i	$0.198690\pi$
$947$			35.9711i			1.16890i	−0.811428	+	0.584452i	$0.801310\pi$
$948$		0			0
$949$	27.3844			0.888935
$950$		0			0
$951$	−20.6804	+	46.2008i	−0.670606	+	1.49817i
$952$		0			0
$953$	−8.45502			−0.273885			−0.136943	−	0.990579i	$-0.543728\pi$
$953$	−8.45502			−0.273885			−0.136943	+	0.990579i	$0.543728\pi$
$954$		0			0
$955$	3.45036			0.111651
$956$		0			0
$957$	7.00937	−	15.6592i	0.226581	−	0.506192i
$958$		0			0
$959$			12.3877i			0.400020i
$960$		0			0
$961$	−30.2798			−0.976766
$962$		0			0
$963$	40.3192	+	45.1395i	1.29927	+	1.45460i
$964$		0			0
$965$	−12.7334			−0.409904
$966$		0			0
$967$	−5.70261			−0.183384			−0.0916918	−	0.995787i	$-0.529227\pi$
$967$	−5.70261			−0.183384			−0.0916918	+	0.995787i	$0.529227\pi$
$968$		0			0
$969$	−8.73525	+	19.5149i	−0.280617	+	0.626910i
$970$		0			0
$971$	−5.51098			−0.176856			−0.0884279	−	0.996083i	$-0.528184\pi$
$971$	−5.51098			−0.176856			−0.0884279	+	0.996083i	$0.528184\pi$
$972$		0			0
$973$		−	28.4161i		−	0.910977i
$974$		0			0
$975$	7.10186	+	3.17892i	0.227442	+	0.101807i
$976$		0			0
$977$	23.6088			0.755314			0.377657	−	0.925946i	$-0.376730\pi$
$977$	23.6088			0.755314			0.377657	+	0.925946i	$0.376730\pi$
$978$		0			0
$979$	−10.9649			−0.350439
$980$		0			0
$981$	−0.625644	+	0.558834i	−0.0199753	+	0.0178422i
$982$		0			0
$983$	−51.2747			−1.63541			−0.817705	−	0.575637i	$-0.804754\pi$
$983$	−51.2747			−1.63541			−0.817705	+	0.575637i	$0.804754\pi$
$984$		0			0
$985$			2.53785i			0.0808627i
$986$		0			0
$987$	−0.410301	−	0.183658i	−0.0130600	−	0.00584591i
$988$		0			0
$989$	58.5031	−	4.45849i	1.86029	−	0.141772i
$990$		0			0
$991$	39.0207			1.23953			0.619766	−	0.784786i	$-0.287227\pi$
$991$	39.0207			1.23953			0.619766	+	0.784786i	$0.287227\pi$
$992$		0			0
$993$	2.57135	+	1.15098i	0.0815993	+	0.0365253i
$994$		0			0
$995$			19.0376i			0.603532i
$996$		0			0
$997$	30.7630			0.974273			0.487137	−	0.873326i	$-0.338041\pi$
$997$	30.7630			0.974273			0.487137	+	0.873326i	$0.338041\pi$
$998$		0			0
$999$	10.5858	−	3.36971i	0.334920	−	0.106613i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.i.a.1241.3	✓	16
3.2	odd	2		1380.2.i.b.1241.4	yes	16
23.22	odd	2		1380.2.i.b.1241.3	yes	16
69.68	even	2	inner	1380.2.i.a.1241.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.i.a.1241.3	✓	16	1.1	even	1	trivial
1380.2.i.a.1241.4	yes	16	69.68	even	2	inner
1380.2.i.b.1241.3	yes	16	23.22	odd	2
1380.2.i.b.1241.4	yes	16	3.2	odd	2

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:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.58090 - 0.707640i) q^{3} -1.00000 q^{5} -1.33803i q^{7} +(1.99849 + 2.23742i) q^{9} +O(q^{10})\)	\(q+(-1.58090 - 0.707640i) q^{3} -1.00000 q^{5} -1.33803i q^{7} +(1.99849 + 2.23742i) q^{9} +2.74985 q^{11} -4.49229 q^{13} +(1.58090 + 0.707640i) q^{15} -3.35352 q^{17} +3.68097i q^{19} +(-0.946840 + 2.11528i) q^{21} +(-0.364431 - 4.78197i) q^{23} +1.00000 q^{25} +(-1.57613 - 4.95134i) q^{27} +3.60211i q^{29} -0.848670 q^{31} +(-4.34724 - 1.94590i) q^{33} +1.33803i q^{35} +2.13797i q^{37} +(7.10186 + 3.17892i) q^{39} -5.16250i q^{41} +12.2341i q^{43} +(-1.99849 - 2.23742i) q^{45} +0.193970i q^{47} +5.20969 q^{49} +(5.30158 + 2.37308i) q^{51} +5.60806 q^{53} -2.74985 q^{55} +(2.60480 - 5.81924i) q^{57} +6.74892i q^{59} +1.26463i q^{61} +(2.99372 - 2.67403i) q^{63} +4.49229 q^{65} +13.1252i q^{67} +(-2.80778 + 7.81770i) q^{69} +7.83428i q^{71} -6.09587 q^{73} +(-1.58090 - 0.707640i) q^{75} -3.67937i q^{77} +16.9228i q^{79} +(-1.01207 + 8.94291i) q^{81} +14.1183 q^{83} +3.35352 q^{85} +(2.54900 - 5.69458i) q^{87} -3.98744 q^{89} +6.01080i q^{91} +(1.34166 + 0.600553i) q^{93} -3.68097i q^{95} +13.7241i q^{97} +(5.49555 + 6.15256i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5} + 2 q^{9}+O(q^{10}) \) 16 * q - 16 * q^5 + 2 * q^9	\( 16 q - 16 q^{5} + 2 q^{9} - 4 q^{21} + 10 q^{23} + 16 q^{25} - 12 q^{27} + 4 q^{31} + 2 q^{33} - 14 q^{39} - 2 q^{45} + 8 q^{49} - 2 q^{51} + 4 q^{53} - 2 q^{57} + 30 q^{63} - 4 q^{73} + 10 q^{81} - 4 q^{83} - 10 q^{87} - 28 q^{89} - 10 q^{93} - 26 q^{99}+O(q^{100}) \) 16 * q - 16 * q^5 + 2 * q^9 - 4 * q^21 + 10 * q^23 + 16 * q^25 - 12 * q^27 + 4 * q^31 + 2 * q^33 - 14 * q^39 - 2 * q^45 + 8 * q^49 - 2 * q^51 + 4 * q^53 - 2 * q^57 + 30 * q^63 - 4 * q^73 + 10 * q^81 - 4 * q^83 - 10 * q^87 - 28 * q^89 - 10 * q^93 - 26 * q^99

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