LMFDB - Embedded newform 567.2.c.c.566.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,2,Mod(566,567)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(567, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("567.566");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 567 = 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	567.c (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:	$4.52751779461$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 $ x^12 - 7x^10 + 37x^8 - 78x^6 + 123x^4 - 36*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			566.8
Root			$-0.474636 - 0.274031i$ of defining polynomial
Character	$\chi$	$=$	567.566
Dual form			567.2.c.c.566.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.641589i q^{2} +1.58836 q^{4} +2.21105 q^{5} +(-1.69963 - 2.02763i) q^{7} +2.30225i q^{8} +O(q^{10})$	\(q+0.641589i q^{2} +1.58836 q^{4} +2.21105 q^{5} +(-1.69963 - 2.02763i) q^{7} +2.30225i q^{8} +1.41858i q^{10} +3.39272i q^{11} +1.80202i q^{13} +(1.30090 - 1.09046i) q^{14} +1.69963 q^{16} +5.96901 q^{17} -1.64419i q^{19} +3.51195 q^{20} -2.17673 q^{22} +2.37364i q^{23} -0.111264 q^{25} -1.15616 q^{26} +(-2.69963 - 3.22061i) q^{28} -2.82251i q^{29} -10.7221i q^{31} +5.69497i q^{32} +3.82965i q^{34} +(-3.75796 - 4.48318i) q^{35} +1.69963 q^{37} +1.05489 q^{38} +5.09039i q^{40} -0.910147 q^{41} +3.92216 q^{43} +5.38887i q^{44} -1.52290 q^{46} +0.246010 q^{47} +(-1.22253 + 6.89242i) q^{49} -0.0713858i q^{50} +2.86227i q^{52} +7.87589i q^{53} +7.50146i q^{55} +(4.66811 - 3.91298i) q^{56} +1.81089 q^{58} -10.7819 q^{59} +1.41858i q^{61} +6.87916 q^{62} -0.254572 q^{64} +3.98436i q^{65} -7.98762 q^{67} +9.48096 q^{68} +(2.87636 - 2.41106i) q^{70} -12.1743i q^{71} +0.426103i q^{73} +1.09046i q^{74} -2.61157i q^{76} +(6.87916 - 5.76636i) q^{77} -4.98762 q^{79} +3.75796 q^{80} -0.583940i q^{82} -8.57081 q^{83} +13.1978 q^{85} +2.51641i q^{86} -7.81089 q^{88} -10.5358 q^{89} +(3.65383 - 3.06277i) q^{91} +3.77020i q^{92} +0.157838i q^{94} -3.63537i q^{95} -7.27586i q^{97} +(-4.42210 - 0.784360i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{4} + 4 q^{7}+O(q^{10}) $ 12 * q - 4 * q^4 + 4 * q^7	$ 12 q - 4 q^{4} + 4 q^{7} - 4 q^{16} + 20 q^{22} - 8 q^{28} - 4 q^{37} + 20 q^{43} - 40 q^{46} - 12 q^{49} - 4 q^{58} + 16 q^{64} - 24 q^{67} - 36 q^{70} + 12 q^{79} + 12 q^{85} - 68 q^{88} - 24 q^{91}+O(q^{100}) $ 12 * q - 4 * q^4 + 4 * q^7 - 4 * q^16 + 20 * q^22 - 8 * q^28 - 4 * q^37 + 20 * q^43 - 40 * q^46 - 12 * q^49 - 4 * q^58 + 16 * q^64 - 24 * q^67 - 36 * q^70 + 12 * q^79 + 12 * q^85 - 68 * q^88 - 24 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/567\mathbb{Z}\right)^\times$.

$n$	$325$	$407$
$\chi(n)$	$-1$	$-1$

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 $

$2$			0.641589i			0.453672i	0.973933	+	0.226836i	$0.0728381\pi$
$2$			0.641589i			0.453672i	−0.973933	+	0.226836i	$0.927162\pi$
$3$		0			0
$3$		0			0
$4$	1.58836			0.794182
$5$	2.21105			0.988811			0.494405	−	0.869231i	$-0.335386\pi$
$5$	2.21105			0.988811			0.494405	+	0.869231i	$0.335386\pi$
$6$		0			0
$7$	−1.69963	−	2.02763i	−0.642399	−	0.766370i
$7$	−1.69963	−	2.02763i	−0.642399	−	0.766370i
$8$			2.30225i			0.813970i
$9$		0			0
$10$			1.41858i			0.448596i
$11$			3.39272i			1.02294i	0.859300	+	0.511471i	$0.170899\pi$
$11$			3.39272i			1.02294i	−0.859300	+	0.511471i	$0.829101\pi$
$12$		0			0
$13$			1.80202i			0.499791i	0.968273	+	0.249896i	$0.0803963\pi$
$13$			1.80202i			0.499791i	−0.968273	+	0.249896i	$0.919604\pi$
$14$	1.30090	−	1.09046i	0.347681	−	0.291438i
$15$		0			0
$16$	1.69963			0.424907
$17$	5.96901			1.44770			0.723849	−	0.689959i	$-0.242371\pi$
$17$	5.96901			1.44770			0.723849	+	0.689959i	$0.242371\pi$
$18$		0			0
$19$		−	1.64419i		−	0.377202i	−0.982054	−	0.188601i	$-0.939605\pi$
$19$		−	1.64419i		−	0.377202i	0.982054	−	0.188601i	$-0.0603953\pi$
$20$	3.51195			0.785296
$21$		0			0
$22$	−2.17673			−0.464080
$23$			2.37364i			0.494938i	0.968896	+	0.247469i	$0.0795988\pi$
$23$			2.37364i			0.494938i	−0.968896	+	0.247469i	$0.920401\pi$
$24$		0			0
$25$	−0.111264			−0.0222528
$26$	−1.15616			−0.226741
$27$		0			0
$28$	−2.69963	−	3.22061i	−0.510182	−	0.608638i
$29$		−	2.82251i		−	0.524127i	−0.965051	−	0.262064i	$-0.915597\pi$
$29$		−	2.82251i		−	0.524127i	0.965051	−	0.262064i	$-0.0844031\pi$
$30$		0			0
$31$		−	10.7221i		−	1.92574i	−0.269966	−	0.962870i	$-0.587012\pi$
$31$		−	10.7221i		−	1.92574i	0.269966	−	0.962870i	$-0.412988\pi$
$32$			5.69497i			1.00674i
$33$		0			0
$34$			3.82965i			0.656779i
$35$	−3.75796	−	4.48318i	−0.635211	−	0.757795i
$36$		0			0
$37$	1.69963			0.279417			0.139709	−	0.990193i	$-0.455383\pi$
$37$	1.69963			0.279417			0.139709	+	0.990193i	$0.455383\pi$
$38$	1.05489			0.171126
$39$		0			0
$40$			5.09039i			0.804862i
$41$	−0.910147			−0.142141			−0.0710706	−	0.997471i	$-0.522642\pi$
$41$	−0.910147			−0.142141			−0.0710706	+	0.997471i	$0.522642\pi$
$42$		0			0
$43$	3.92216			0.598123			0.299062	−	0.954234i	$-0.403326\pi$
$43$	3.92216			0.598123			0.299062	+	0.954234i	$0.403326\pi$
$44$			5.38887i			0.812402i
$45$		0			0
$46$	−1.52290			−0.224539
$47$	0.246010			0.0358843			0.0179422	−	0.999839i	$-0.494289\pi$
$47$	0.246010			0.0358843			0.0179422	+	0.999839i	$0.494289\pi$
$48$		0			0
$49$	−1.22253	+	6.89242i	−0.174647	+	0.984631i
$50$		−	0.0713858i		−	0.0100955i
$51$		0			0
$52$			2.86227i			0.396925i
$53$			7.87589i			1.08184i	0.841075	+	0.540919i	$0.181923\pi$
$53$			7.87589i			1.08184i	−0.841075	+	0.540919i	$0.818077\pi$
$54$		0			0
$55$			7.50146i			1.01150i
$56$	4.66811	−	3.91298i	0.623802	−	0.522893i
$57$		0			0
$58$	1.81089			0.237782
$59$	−10.7819			−1.40368			−0.701839	−	0.712335i	$-0.747638\pi$
$59$	−10.7819			−1.40368			−0.701839	+	0.712335i	$0.747638\pi$
$60$		0			0
$61$			1.41858i			0.181631i	0.995868	+	0.0908155i	$0.0289474\pi$
$61$			1.41858i			0.181631i	−0.995868	+	0.0908155i	$0.971053\pi$
$62$	6.87916			0.873654
$63$		0			0
$64$	−0.254572			−0.0318214
$65$			3.98436i			0.494199i
$66$		0			0
$67$	−7.98762			−0.975843			−0.487922	−	0.872887i	$-0.662245\pi$
$67$	−7.98762			−0.975843			−0.487922	+	0.872887i	$0.662245\pi$
$68$	9.48096			1.14974
$69$		0			0
$70$	2.87636	−	2.41106i	0.343790	−	0.288177i
$71$		−	12.1743i		−	1.44482i	−0.691463	−	0.722412i	$-0.743034\pi$
$71$		−	12.1743i		−	1.44482i	0.691463	−	0.722412i	$-0.256966\pi$
$72$		0			0
$73$			0.426103i			0.0498715i	0.999689	+	0.0249358i	$0.00793812\pi$
$73$			0.426103i			0.0498715i	−0.999689	+	0.0249358i	$0.992062\pi$
$74$			1.09046i			0.126764i
$75$		0			0
$76$		−	2.61157i		−	0.299567i
$77$	6.87916	−	5.76636i	0.783953	−	0.657137i
$78$		0			0
$79$	−4.98762			−0.561151			−0.280576	−	0.959832i	$-0.590525\pi$
$79$	−4.98762			−0.561151			−0.280576	+	0.959832i	$0.590525\pi$
$80$	3.75796			0.420153
$81$		0			0
$82$		−	0.583940i		−	0.0644854i
$83$	−8.57081			−0.940769			−0.470384	−	0.882462i	$-0.655885\pi$
$83$	−8.57081			−0.940769			−0.470384	+	0.882462i	$0.655885\pi$
$84$		0			0
$85$	13.1978			1.43150
$86$			2.51641i			0.271352i
$87$		0			0
$88$	−7.81089			−0.832644
$89$	−10.5358			−1.11680			−0.558399	−	0.829573i	$-0.688584\pi$
$89$	−10.5358			−1.11680			−0.558399	+	0.829573i	$0.688584\pi$
$90$		0			0
$91$	3.65383	−	3.06277i	0.383025	−	0.321065i
$92$			3.77020i			0.393071i
$93$		0			0
$94$			0.157838i			0.0162797i
$95$		−	3.63537i		−	0.372982i
$96$		0			0
$97$		−	7.27586i		−	0.738751i	−0.929280	−	0.369376i	$-0.879572\pi$
$97$		−	7.27586i		−	0.738751i	0.929280	−	0.369376i	$-0.120428\pi$
$98$	−4.42210	−	0.784360i	−0.446699	−	0.0792324i
$99$		0			0
$100$	−0.176728			−0.0176728
$101$	−4.66811			−0.464494			−0.232247	−	0.972657i	$-0.574608\pi$
$101$	−4.66811			−0.464494			−0.232247	+	0.972657i	$0.574608\pi$
$102$		0			0
$103$			6.24071i			0.614916i	0.951562	+	0.307458i	$0.0994783\pi$
$103$			6.24071i			0.614916i	−0.951562	+	0.307458i	$0.900522\pi$
$104$	−4.14871			−0.406815
$105$		0			0
$106$	−5.05308			−0.490799
$107$			1.48939i			0.143985i	0.997405	+	0.0719925i	$0.0229358\pi$
$107$			1.48939i			0.143985i	−0.997405	+	0.0719925i	$0.977064\pi$
$108$		0			0
$109$	−4.38688			−0.420187			−0.210093	−	0.977681i	$-0.567377\pi$
$109$	−4.38688			−0.420187			−0.210093	+	0.977681i	$0.567377\pi$
$110$	−4.81285			−0.458887
$111$		0			0
$112$	−2.88874	−	3.44621i	−0.272960	−	0.325636i
$113$		−	17.1777i		−	1.61595i	−0.589220	−	0.807973i	$-0.700565\pi$
$113$		−	17.1777i		−	1.61595i	0.589220	−	0.807973i	$-0.299435\pi$
$114$		0			0
$115$			5.24823i			0.489400i
$116$		−	4.48318i		−	0.416253i
$117$		0			0
$118$		−	6.91752i		−	0.636809i
$119$	−10.1451	−	12.1029i	−0.929999	−	1.10947i
$120$		0			0
$121$	−0.510520			−0.0464110
$122$	−0.910147			−0.0824009
$123$		0			0
$124$		−	17.0305i		−	1.52939i
$125$	−11.3013			−1.01081
$126$		0			0
$127$	6.32141			0.560935			0.280467	−	0.959864i	$-0.409511\pi$
$127$	6.32141			0.560935			0.280467	+	0.959864i	$0.409511\pi$
$128$			11.2266i			0.992301i
$129$		0			0
$130$	−2.55632			−0.224204
$131$	−17.0243			−1.48742			−0.743708	−	0.668505i	$-0.766935\pi$
$131$	−17.0243			−1.48742			−0.743708	+	0.668505i	$0.766935\pi$
$132$		0			0
$133$	−3.33379	+	2.79450i	−0.289076	+	0.242314i
$134$		−	5.12477i		−	0.442712i
$135$		0			0
$136$			13.7422i			1.17838i
$137$		−	6.26517i		−	0.535270i	−0.963520	−	0.267635i	$-0.913758\pi$
$137$		−	6.26517i		−	0.535270i	0.963520	−	0.267635i	$-0.0862421\pi$
$138$		0			0
$139$		−	7.68440i		−	0.651782i	−0.945407	−	0.325891i	$-0.894336\pi$
$139$		−	7.68440i		−	0.651782i	0.945407	−	0.325891i	$-0.105664\pi$
$140$	−5.96901	−	7.12092i	−0.504473	−	0.601827i
$141$		0			0
$142$	7.81089			0.655476
$143$	−6.11375			−0.511258
$144$		0			0
$145$		−	6.24071i		−	0.518263i
$146$	−0.273383			−0.0226253
$147$		0			0
$148$	2.69963			0.221908
$149$			15.4377i			1.26471i	0.774679	+	0.632355i	$0.217911\pi$
$149$			15.4377i			1.26471i	−0.774679	+	0.632355i	$0.782089\pi$
$150$		0			0
$151$	11.6872			0.951095			0.475547	−	0.879690i	$-0.342250\pi$
$151$	11.6872			0.951095			0.475547	+	0.879690i	$0.342250\pi$
$152$	3.78533			0.307031
$153$		0			0
$154$	3.69963	+	4.41359i	0.298125	+	0.355657i
$155$		−	23.7070i		−	1.90419i
$156$		0			0
$157$			5.69944i			0.454865i	0.973794	+	0.227432i	$0.0730330\pi$
$157$			5.69944i			0.454865i	−0.973794	+	0.227432i	$0.926967\pi$
$158$		−	3.20000i		−	0.254578i
$159$		0			0
$160$			12.5919i			0.995473i
$161$	4.81285	−	4.03430i	0.379306	−	0.317948i
$162$		0			0
$163$	−10.2101			−0.799721			−0.399860	−	0.916576i	$-0.630941\pi$
$163$	−10.2101			−0.799721			−0.399860	+	0.916576i	$0.630941\pi$
$164$	−1.44565			−0.112886
$165$		0			0
$166$		−	5.49894i		−	0.426800i
$167$	3.61322			0.279599			0.139800	−	0.990180i	$-0.455354\pi$
$167$	3.61322			0.279599			0.139800	+	0.990180i	$0.455354\pi$
$168$		0			0
$169$	9.75271			0.750209
$170$			8.46754i			0.649431i
$171$		0			0
$172$	6.22981			0.475019
$173$	18.0791			1.37453			0.687266	−	0.726406i	$-0.258811\pi$
$173$	18.0791			1.37453			0.687266	+	0.726406i	$0.258811\pi$
$174$		0			0
$175$	0.189108	+	0.225602i	0.0142952	+	0.0170539i
$176$			5.76636i			0.434655i
$177$		0			0
$178$		−	6.75968i		−	0.506660i
$179$		−	5.03194i		−	0.376105i	−0.982159	−	0.188052i	$-0.939783\pi$
$179$		−	5.03194i		−	0.376105i	0.982159	−	0.188052i	$-0.0602175\pi$
$180$		0			0
$181$		−	13.5592i		−	1.00785i	−0.863747	−	0.503925i	$-0.831889\pi$
$181$		−	13.5592i		−	1.00785i	0.863747	−	0.503925i	$-0.168111\pi$
$182$	1.96504	+	2.34425i	0.145658	+	0.173768i
$183$		0			0
$184$	−5.46472			−0.402865
$185$	3.75796			0.276291
$186$		0			0
$187$			20.2512i			1.48091i
$188$	0.390754			0.0284987
$189$		0			0
$190$	2.33242			0.169211
$191$			10.2416i			0.741055i	0.928821	+	0.370528i	$0.120823\pi$
$191$			10.2416i			0.741055i	−0.928821	+	0.370528i	$0.879177\pi$
$192$		0			0
$193$	16.1323			1.16123			0.580614	−	0.814179i	$-0.302812\pi$
$193$	16.1323			1.16123			0.580614	+	0.814179i	$0.302812\pi$
$194$	4.66811			0.335151
$195$		0			0
$196$	−1.94182	+	10.9477i	−0.138701	+	0.781976i
$197$		−	3.86303i		−	0.275230i	−0.990486	−	0.137615i	$-0.956056\pi$
$197$		−	3.86303i		−	0.275230i	0.990486	−	0.137615i	$-0.0439436\pi$
$198$		0			0
$199$		−	15.2034i		−	1.07774i	−0.842388	−	0.538871i	$-0.818851\pi$
$199$		−	15.2034i		−	1.07774i	0.842388	−	0.538871i	$-0.181149\pi$
$200$		−	0.256158i		−	0.0181131i
$201$		0			0
$202$		−	2.99500i		−	0.210728i
$203$	−5.72300	+	4.79722i	−0.401676	+	0.336699i
$204$		0			0
$205$	−2.01238			−0.140551
$206$	−4.00397			−0.278970
$207$		0			0
$208$			3.06277i			0.212365i
$209$	5.57825			0.385856
$210$		0			0
$211$	−23.9047			−1.64567			−0.822833	−	0.568283i	$-0.807608\pi$
$211$	−23.9047			−1.64567			−0.822833	+	0.568283i	$0.807608\pi$
$212$			12.5098i			0.859176i
$213$		0			0
$214$	−0.955577			−0.0653219
$215$	8.67208			0.591431
$216$		0			0
$217$	−21.7403	+	18.2235i	−1.47583	+	1.23709i
$218$		−	2.81457i		−	0.190627i
$219$		0			0
$220$			11.9150i			0.803312i
$221$			10.7563i			0.723547i
$222$		0			0
$223$		−	19.1909i		−	1.28512i	−0.766236	−	0.642559i	$-0.777873\pi$
$223$		−	19.1909i		−	1.28512i	0.766236	−	0.642559i	$-0.222127\pi$
$224$	11.5473	−	9.67933i	0.771534	−	0.646727i
$225$		0			0
$226$	11.0210			0.733109
$227$	8.67208			0.575586			0.287793	−	0.957693i	$-0.407078\pi$
$227$	8.67208			0.575586			0.287793	+	0.957693i	$0.407078\pi$
$228$		0			0
$229$		−	14.3688i		−	0.949515i	−0.880117	−	0.474758i	$-0.842536\pi$
$229$		−	14.3688i		−	0.949515i	0.880117	−	0.474758i	$-0.157464\pi$
$230$	−3.36721			−0.222027
$231$		0			0
$232$	6.49814			0.426624
$233$			29.7160i			1.94676i	0.229194	+	0.973381i	$0.426391\pi$
$233$			29.7160i			1.94676i	−0.229194	+	0.973381i	$0.573609\pi$
$234$		0			0
$235$	0.543941			0.0354828
$236$	−17.1255			−1.11478
$237$		0			0
$238$	7.76509	−	6.50898i	0.503336	−	0.421914i
$239$		−	15.8311i		−	1.02403i	−0.858977	−	0.512015i	$-0.828899\pi$
$239$		−	15.8311i		−	1.02403i	0.858977	−	0.512015i	$-0.171101\pi$
$240$		0			0
$241$		−	5.02263i		−	0.323536i	−0.986829	−	0.161768i	$-0.948280\pi$
$241$		−	5.02263i		−	0.323536i	0.986829	−	0.161768i	$-0.0517196\pi$
$242$		−	0.327544i		−	0.0210553i
$243$		0			0
$244$			2.25323i			0.144248i
$245$	−2.70307	+	15.2395i	−0.172693	+	0.973614i
$246$		0			0
$247$	2.96286			0.188522
$248$	24.6849			1.56749
$249$		0			0
$250$		−	7.25076i		−	0.458578i
$251$	7.29728			0.460600			0.230300	−	0.973120i	$-0.426029\pi$
$251$	7.29728			0.460600			0.230300	+	0.973120i	$0.426029\pi$
$252$		0			0
$253$	−8.05308			−0.506293
$254$			4.05575i			0.254480i
$255$		0			0
$256$	−7.71201			−0.482000
$257$	−8.00794			−0.499522			−0.249761	−	0.968308i	$-0.580352\pi$
$257$	−8.00794			−0.499522			−0.249761	+	0.968308i	$0.580352\pi$
$258$		0			0
$259$	−2.88874	−	3.44621i	−0.179497	−	0.214137i
$260$			6.32862i			0.392484i
$261$		0			0
$262$		−	10.9226i		−	0.674798i
$263$			15.7098i			0.968707i	0.874872	+	0.484353i	$0.160945\pi$
$263$			15.7098i			0.968707i	−0.874872	+	0.484353i	$0.839055\pi$
$264$		0			0
$265$			17.4140i			1.06973i
$266$	−1.79292	−	2.13892i	−0.109931	−	0.131146i
$267$		0			0
$268$	−12.6872			−0.774997
$269$	−10.4924			−0.639731			−0.319866	−	0.947463i	$-0.603638\pi$
$269$	−10.4924			−0.639731			−0.319866	+	0.947463i	$0.603638\pi$
$270$		0			0
$271$			22.2537i			1.35181i	0.736987	+	0.675907i	$0.236248\pi$
$271$			22.2537i			1.35181i	−0.736987	+	0.675907i	$0.763752\pi$
$272$	10.1451			0.615137
$273$		0			0
$274$	4.01966			0.242837
$275$		−	0.377488i		−	0.0227634i
$276$		0			0
$277$	−22.8502			−1.37294			−0.686468	−	0.727160i	$-0.740840\pi$
$277$	−22.8502			−1.37294			−0.686468	+	0.727160i	$0.740840\pi$
$278$	4.93022			0.295695
$279$		0			0
$280$	10.3214	−	8.65178i	0.616822	−	0.517043i
$281$		−	0.919189i		−	0.0548342i	−0.999624	−	0.0274171i	$-0.991272\pi$
$281$		−	0.919189i		−	0.0548342i	0.999624	−	0.0274171i	$-0.00872823\pi$
$282$		0			0
$283$			22.1209i			1.31495i	0.753475	+	0.657477i	$0.228376\pi$
$283$			22.1209i			1.31495i	−0.753475	+	0.657477i	$0.771624\pi$
$284$		−	19.3372i		−	1.14745i
$285$		0			0
$286$		−	3.92251i		−	0.231943i
$287$	1.54691	+	1.84544i	0.0913113	+	0.108933i
$288$		0			0
$289$	18.6291			1.09583
$290$	4.00397			0.235121
$291$		0			0
$292$			0.676806i			0.0396071i
$293$	29.2518			1.70891			0.854453	−	0.519529i	$-0.173892\pi$
$293$	29.2518			1.70891			0.854453	+	0.519529i	$0.173892\pi$
$294$		0			0
$295$	−23.8392			−1.38797
$296$			3.91298i			0.227437i
$297$		0			0
$298$	−9.90468			−0.573763
$299$	−4.27735			−0.247366
$300$		0			0
$301$	−6.66621	−	7.95266i	−0.384234	−	0.458384i
$302$			7.49841i			0.431485i
$303$		0			0
$304$		−	2.79450i		−	0.160276i
$305$			3.13656i			0.179599i
$306$		0			0
$307$			14.8451i			0.847254i	0.905837	+	0.423627i	$0.139243\pi$
$307$			14.8451i			0.847254i	−0.905837	+	0.423627i	$0.860757\pi$
$308$	10.9266	−	9.15907i	0.622601	−	0.521887i
$309$		0			0
$310$	15.2101			0.863878
$311$	19.3800			1.09894			0.549471	−	0.835513i	$-0.314829\pi$
$311$	19.3800			1.09894			0.549471	+	0.835513i	$0.314829\pi$
$312$		0			0
$313$			14.6195i			0.826342i	0.910653	+	0.413171i	$0.135579\pi$
$313$			14.6195i			0.826342i	−0.910653	+	0.413171i	$0.864421\pi$
$314$	−3.65669			−0.206359
$315$		0			0
$316$	−7.92216			−0.445656
$317$			16.9795i			0.953662i	0.878995	+	0.476831i	$0.158215\pi$
$317$			16.9795i			0.953662i	−0.878995	+	0.476831i	$0.841785\pi$
$318$		0			0
$319$	9.57598			0.536152
$320$	−0.562870			−0.0314654
$321$		0			0
$322$	2.58836	+	3.08787i	0.144244	+	0.172080i
$323$		−	9.81416i		−	0.546074i
$324$		0			0
$325$		−	0.200501i		−	0.0111218i
$326$		−	6.55072i		−	0.362811i
$327$		0			0
$328$		−	2.09539i		−	0.115699i
$329$	−0.418126	−	0.498817i	−0.0230520	−	0.0275007i
$330$		0			0
$331$	19.8960			1.09358			0.546792	−	0.837268i	$-0.315849\pi$
$331$	19.8960			1.09358			0.546792	+	0.837268i	$0.315849\pi$
$332$	−13.6136			−0.747142
$333$		0			0
$334$			2.31820i			0.126846i
$335$	−17.6610			−0.964924
$336$		0			0
$337$	−0.980337			−0.0534023			−0.0267012	−	0.999643i	$-0.508500\pi$
$337$	−0.980337			−0.0534023			−0.0267012	+	0.999643i	$0.508500\pi$
$338$			6.25723i			0.340348i
$339$		0			0
$340$	20.9629			1.13687
$341$	36.3769			1.96992
$342$		0			0
$343$	16.0531	−	9.23572i	0.866785	−	0.498682i
$344$			9.02980i			0.486854i
$345$		0			0
$346$			11.5994i			0.623586i
$347$			21.2120i			1.13872i	0.822088	+	0.569361i	$0.192809\pi$
$347$			21.2120i			1.13872i	−0.822088	+	0.569361i	$0.807191\pi$
$348$		0			0
$349$			10.0453i			0.537710i	0.963181	+	0.268855i	$0.0866453\pi$
$349$			10.0453i			0.537710i	−0.963181	+	0.268855i	$0.913355\pi$
$350$	−0.144744	+	0.121329i	−0.00773688	+	0.00648533i
$351$		0			0
$352$	−19.3214			−1.02983
$353$	2.74655			0.146184			0.0730920	−	0.997325i	$-0.476713\pi$
$353$	2.74655			0.146184			0.0730920	+	0.997325i	$0.476713\pi$
$354$		0			0
$355$		−	26.9180i		−	1.42866i
$356$	−16.7348			−0.886941
$357$		0			0
$358$	3.22843			0.170628
$359$		−	10.0013i		−	0.527849i	−0.964543	−	0.263925i	$-0.914983\pi$
$359$		−	10.0013i		−	0.527849i	0.964543	−	0.263925i	$-0.0850170\pi$
$360$		0			0
$361$	16.2967			0.857719
$362$	8.69945			0.457233
$363$		0			0
$364$	5.80361	−	4.86479i	0.304192	−	0.254984i
$365$			0.942134i			0.0493135i
$366$		0			0
$367$		−	5.81461i		−	0.303520i	−0.988417	−	0.151760i	$-0.951506\pi$
$367$		−	5.81461i		−	0.303520i	0.988417	−	0.151760i	$-0.0484941\pi$
$368$			4.03430i			0.210303i
$369$		0			0
$370$			2.41106i			0.125345i
$371$	15.9694	−	13.3861i	0.829088	−	0.694971i
$372$		0			0
$373$	−15.5192			−0.803553			−0.401776	−	0.915738i	$-0.631607\pi$
$373$	−15.5192			−0.803553			−0.401776	+	0.915738i	$0.631607\pi$
$374$	−12.9929			−0.671847
$375$		0			0
$376$			0.566378i			0.0292087i
$377$	5.08623			0.261954
$378$		0			0
$379$	2.79714			0.143679			0.0718396	−	0.997416i	$-0.477113\pi$
$379$	2.79714			0.143679			0.0718396	+	0.997416i	$0.477113\pi$
$380$		−	5.77430i		−	0.296215i
$381$		0			0
$382$	−6.57089			−0.336196
$383$	−3.48458			−0.178054			−0.0890268	−	0.996029i	$-0.528376\pi$
$383$	−3.48458			−0.178054			−0.0890268	+	0.996029i	$0.528376\pi$
$384$		0			0
$385$	15.2101	−	12.7497i	0.775181	−	0.649784i
$386$			10.3503i			0.526817i
$387$		0			0
$388$		−	11.5567i		−	0.586703i
$389$			7.35563i			0.372945i	0.982460	+	0.186473i	$0.0597056\pi$
$389$			7.35563i			0.372945i	−0.982460	+	0.186473i	$0.940294\pi$
$390$		0			0
$391$			14.1683i			0.716520i
$392$	−15.8681	−	2.81457i	−0.801460	−	0.142157i
$393$		0			0
$394$	2.47848			0.124864
$395$	−11.0279			−0.554872
$396$		0			0
$397$		−	19.2838i		−	0.967825i	−0.875116	−	0.483912i	$-0.839215\pi$
$397$		−	19.2838i		−	0.967825i	0.875116	−	0.483912i	$-0.160785\pi$
$398$	9.75434			0.488941
$399$		0			0
$400$	−0.189108			−0.00945538
$401$			11.0918i			0.553897i	0.960885	+	0.276949i	$0.0893232\pi$
$401$			11.0918i			0.553897i	−0.960885	+	0.276949i	$0.910677\pi$
$402$		0			0
$403$	19.3214			0.962468
$404$	−7.41465			−0.368893
$405$		0			0
$406$	−3.07784	−	3.67181i	−0.152751	−	0.182229i
$407$			5.76636i			0.285828i
$408$		0			0
$409$		−	20.2763i		−	1.00260i	−0.865275	−	0.501298i	$-0.832856\pi$
$409$		−	20.2763i		−	1.00260i	0.865275	−	0.501298i	$-0.167144\pi$
$410$		−	1.29112i		−	0.0637639i
$411$		0			0
$412$			9.91252i			0.488355i
$413$	18.3252	+	21.8616i	0.901722	+	1.07574i
$414$		0			0
$415$	−18.9505			−0.930243
$416$	−10.2625			−0.503159
$417$		0			0
$418$			3.57895i			0.175052i
$419$	−11.0987			−0.542208			−0.271104	−	0.962550i	$-0.587389\pi$
$419$	−11.0987			−0.542208			−0.271104	+	0.962550i	$0.587389\pi$
$420$		0			0
$421$	−9.18539			−0.447668			−0.223834	−	0.974627i	$-0.571857\pi$
$421$	−9.18539			−0.447668			−0.223834	+	0.974627i	$0.571857\pi$
$422$		−	15.3370i		−	0.746592i
$423$		0			0
$424$	−18.1323			−0.880583
$425$	−0.664137			−0.0322154
$426$		0			0
$427$	2.87636	−	2.41106i	0.139197	−	0.116680i
$428$			2.36570i			0.114350i
$429$		0			0
$430$			5.56391i			0.268315i
$431$			15.1102i			0.727833i	0.931432	+	0.363916i	$0.118561\pi$
$431$			15.1102i			0.727833i	−0.931432	+	0.363916i	$0.881439\pi$
$432$		0			0
$433$			3.33578i			0.160307i	0.996783	+	0.0801537i	$0.0255411\pi$
$433$			3.33578i			0.160307i	−0.996783	+	0.0801537i	$0.974459\pi$
$434$	−11.6920	−	13.9484i	−0.561234	−	0.669542i
$435$		0			0
$436$	−6.96796			−0.333705
$437$	3.90270			0.186692
$438$		0			0
$439$			6.82465i			0.325723i	0.986649	+	0.162861i	$0.0520724\pi$
$439$			6.82465i			0.325723i	−0.986649	+	0.162861i	$0.947928\pi$
$440$	−17.2703			−0.823327
$441$		0			0
$442$	−6.90112			−0.328253
$443$		−	11.2901i		−	0.536407i	−0.963362	−	0.268203i	$-0.913570\pi$
$443$		−	11.2901i		−	0.536407i	0.963362	−	0.268203i	$-0.0864299\pi$
$444$		0			0
$445$	−23.2953			−1.10430
$446$	12.3127			0.583022
$447$		0			0
$448$	0.432677	+	0.516176i	0.0204421	+	0.0243870i
$449$		−	24.8554i		−	1.17300i	−0.809950	−	0.586498i	$-0.800506\pi$
$449$		−	24.8554i		−	1.17300i	0.809950	−	0.586498i	$-0.199494\pi$
$450$		0			0
$451$		−	3.08787i		−	0.145402i
$452$		−	27.2845i		−	1.28335i
$453$		0			0
$454$			5.56391i			0.261127i
$455$	8.07879	−	6.77193i	0.378740	−	0.317473i
$456$		0			0
$457$	−12.6094			−0.589843			−0.294922	−	0.955521i	$-0.595294\pi$
$457$	−12.6094			−0.589843			−0.294922	+	0.955521i	$0.595294\pi$
$458$	9.21884			0.430768
$459$		0			0
$460$			8.33610i			0.388673i
$461$	28.8063			1.34164			0.670821	−	0.741620i	$-0.265942\pi$
$461$	28.8063			1.34164			0.670821	+	0.741620i	$0.265942\pi$
$462$		0			0
$463$	25.1716			1.16982			0.584912	−	0.811096i	$-0.301129\pi$
$463$	25.1716			1.16982			0.584912	+	0.811096i	$0.301129\pi$
$464$		−	4.79722i		−	0.222705i
$465$		0			0
$466$	−19.0655			−0.883191
$467$	−25.5951			−1.18440			−0.592199	−	0.805792i	$-0.701740\pi$
$467$	−25.5951			−1.18440			−0.592199	+	0.805792i	$0.701740\pi$
$468$		0			0
$469$	13.5760	+	16.1959i	0.626881	+	0.747857i
$470$			0.348986i			0.0160975i
$471$		0			0
$472$		−	24.8226i		−	1.14255i
$473$			13.3068i			0.611846i
$474$		0			0
$475$			0.182939i			0.00839381i
$476$	−16.1141	−	19.2238i	−0.738589	−	0.881123i
$477$		0			0
$478$	10.1571			0.464573
$479$	−0.535498			−0.0244675			−0.0122338	−	0.999925i	$-0.503894\pi$
$479$	−0.535498			−0.0244675			−0.0122338	+	0.999925i	$0.503894\pi$
$480$		0			0
$481$			3.06277i			0.139650i
$482$	3.22246			0.146779
$483$		0			0
$484$	−0.810892			−0.0368587
$485$		−	16.0873i		−	0.730485i
$486$		0			0
$487$	34.1323			1.54668			0.773341	−	0.633990i	$-0.218584\pi$
$487$	34.1323			1.54668			0.773341	+	0.633990i	$0.218584\pi$
$488$	−3.26594			−0.147842
$489$		0			0
$490$	−9.77747	−	1.73426i	−0.441701	−	0.0783458i
$491$			6.77749i			0.305864i	0.988237	+	0.152932i	$0.0488715\pi$
$491$			6.77749i			0.305864i	−0.988237	+	0.152932i	$0.951128\pi$
$492$		0			0
$493$		−	16.8476i		−	0.758778i
$494$			1.90094i			0.0855272i
$495$		0			0
$496$		−	18.2235i		−	0.818260i
$497$	−24.6849	+	20.6918i	−1.10727	+	0.928153i
$498$		0			0
$499$	8.60074			0.385022			0.192511	−	0.981295i	$-0.438337\pi$
$499$	8.60074			0.385022			0.192511	+	0.981295i	$0.438337\pi$
$500$	−17.9505			−0.802771
$501$		0			0
$502$			4.68185i			0.208961i
$503$	2.96518			0.132211			0.0661055	−	0.997813i	$-0.478943\pi$
$503$	2.96518			0.132211			0.0661055	+	0.997813i	$0.478943\pi$
$504$		0			0
$505$	−10.3214			−0.459297
$506$		−	5.16677i		−	0.229691i
$507$		0			0
$508$	10.0407			0.445484
$509$	−6.09765			−0.270273			−0.135137	−	0.990827i	$-0.543147\pi$
$509$	−6.09765			−0.270273			−0.135137	+	0.990827i	$0.543147\pi$
$510$		0			0
$511$	0.863976	−	0.724216i	0.0382201	−	0.0320374i
$512$			17.5053i			0.773631i
$513$		0			0
$514$		−	5.13780i		−	0.226619i
$515$			13.7985i			0.608035i
$516$		0			0
$517$			0.834644i			0.0367076i
$518$	2.21105	−	1.85338i	0.0971479	−	0.0814328i
$519$		0			0
$520$	−9.17301			−0.402263
$521$	32.6929			1.43230			0.716150	−	0.697946i	$-0.245903\pi$
$521$	32.6929			1.43230			0.716150	+	0.697946i	$0.245903\pi$
$522$		0			0
$523$		−	2.00252i		−	0.0875643i	−0.999041	−	0.0437821i	$-0.986059\pi$
$523$		−	2.00252i		−	0.0875643i	0.999041	−	0.0437821i	$-0.0139407\pi$
$524$	−27.0407			−1.18128
$525$		0			0
$526$	−10.0792			−0.439475
$527$		−	64.0001i		−	2.78789i
$528$		0			0
$529$	17.3658			0.755036
$530$	−11.1726			−0.485307
$531$		0			0
$532$	−5.29528	+	4.43869i	−0.229579	+	0.192442i
$533$		−	1.64011i		−	0.0710409i
$534$		0			0
$535$			3.29312i			0.142374i
$536$		−	18.3895i		−	0.794307i
$537$		0			0
$538$		−	6.73179i		−	0.290228i
$539$	−23.3840	−	4.14769i	−1.00722	−	0.178654i
$540$		0			0
$541$	−11.4451			−0.492061			−0.246031	−	0.969262i	$-0.579126\pi$
$541$	−11.4451			−0.492061			−0.246031	+	0.969262i	$0.579126\pi$
$542$	−14.2777			−0.613280
$543$		0			0
$544$			33.9933i			1.45745i
$545$	−9.69960			−0.415485
$546$		0			0
$547$	7.83922			0.335181			0.167590	−	0.985857i	$-0.446401\pi$
$547$	7.83922			0.335181			0.167590	+	0.985857i	$0.446401\pi$
$548$		−	9.95137i		−	0.425102i
$549$		0			0
$550$	0.242192			0.0103271
$551$	−4.64074			−0.197702
$552$		0			0
$553$	8.47710	+	10.1130i	0.360483	+	0.430050i
$554$		−	14.6604i		−	0.622862i
$555$		0			0
$556$		−	12.2056i		−	0.517634i
$557$		−	0.0134996i		−	0.000571997i	−1.00000		0.000285998i	$-0.999909\pi$
$557$		−	0.0134996i		−	0.000571997i	1.00000		0.000285998i	$-9.10361e-5\pi$
$558$		0			0
$559$			7.06782i			0.298937i
$560$	−6.38714	−	7.61974i	−0.269906	−	0.321993i
$561$		0			0
$562$	0.589741			0.0248767
$563$	−19.0906			−0.804571			−0.402286	−	0.915514i	$-0.631784\pi$
$563$	−19.0906			−0.804571			−0.402286	+	0.915514i	$0.631784\pi$
$564$		0			0
$565$		−	37.9808i		−	1.59786i
$566$	−14.1925			−0.596557
$567$		0			0
$568$	28.0283			1.17604
$569$		−	37.3437i		−	1.56553i	−0.622318	−	0.782765i	$-0.713809\pi$
$569$		−	37.3437i		−	1.56553i	0.622318	−	0.782765i	$-0.286191\pi$
$570$		0			0
$571$	45.2843			1.89509			0.947544	−	0.319626i	$-0.103557\pi$
$571$	45.2843			1.89509			0.947544	+	0.319626i	$0.103557\pi$
$572$	−9.71086			−0.406032
$573$		0			0
$574$	−1.18401	+	0.992481i	−0.0494197	+	0.0414254i
$575$		−	0.264101i		−	0.0110138i
$576$		0			0
$577$			37.0988i			1.54444i	0.635354	+	0.772221i	$0.280854\pi$
$577$			37.0988i			1.54444i	−0.635354	+	0.772221i	$0.719146\pi$
$578$			11.9522i			0.497146i
$579$		0			0
$580$		−	9.91252i		−	0.411595i
$581$	14.5672	+	17.3784i	0.604349	+	0.720977i
$582$		0			0
$583$	−26.7207			−1.10666
$584$	−0.980996			−0.0405939
$585$		0			0
$586$			18.7676i			0.775282i
$587$	−34.1224			−1.40838			−0.704191	−	0.710011i	$-0.748690\pi$
$587$	−34.1224			−1.40838			−0.704191	+	0.710011i	$0.748690\pi$
$588$		0			0
$589$	−17.6291			−0.726393
$590$		−	15.2950i		−	0.629684i
$591$		0			0
$592$	2.88874			0.118726
$593$	−19.6999			−0.808980			−0.404490	−	0.914542i	$-0.632551\pi$
$593$	−19.6999			−0.808980			−0.404490	+	0.914542i	$0.632551\pi$
$594$		0			0
$595$	−22.4313	−	26.7601i	−0.919594	−	1.09706i
$596$			24.5208i			1.00441i
$597$		0			0
$598$		−	2.74430i		−	0.112223i
$599$		−	11.2472i		−	0.459547i	−0.973244	−	0.229773i	$-0.926201\pi$
$599$		−	11.2472i		−	0.459547i	0.973244	−	0.229773i	$-0.0737985\pi$
$600$		0			0
$601$			34.3692i			1.40195i	0.713186	+	0.700975i	$0.247252\pi$
$601$			34.3692i			1.40195i	−0.713186	+	0.700975i	$0.752748\pi$
$602$	5.10234	−	4.27696i	0.207956	−	0.174316i
$603$		0			0
$604$	18.5636			0.755342
$605$	−1.12879			−0.0458917
$606$		0			0
$607$			39.0160i			1.58361i	0.610775	+	0.791804i	$0.290858\pi$
$607$			39.0160i			1.58361i	−0.610775	+	0.791804i	$0.709142\pi$
$608$	9.36359			0.379744
$609$		0			0
$610$	−2.01238			−0.0814789
$611$			0.443317i			0.0179347i
$612$		0			0
$613$	−16.1099			−0.650672			−0.325336	−	0.945598i	$-0.605477\pi$
$613$	−16.1099			−0.650672			−0.325336	+	0.945598i	$0.605477\pi$
$614$	−9.52444			−0.384375
$615$		0			0
$616$	13.2756	+	15.8376i	0.534890	+	0.638114i
$617$			8.12411i			0.327064i	0.986538	+	0.163532i	$0.0522888\pi$
$617$			8.12411i			0.327064i	−0.986538	+	0.163532i	$0.947711\pi$
$618$		0			0
$619$			37.4144i			1.50381i	0.659270	+	0.751906i	$0.270866\pi$
$619$			37.4144i			1.50381i	−0.659270	+	0.751906i	$0.729134\pi$
$620$		−	37.6554i		−	1.51228i
$621$		0			0
$622$			12.4340i			0.498559i
$623$	17.9070	+	21.3628i	0.717430	+	0.855881i
$624$		0			0
$625$	−24.4313			−0.977252
$626$	−9.37969			−0.374888
$627$		0			0
$628$			9.05278i			0.361245i
$629$	10.1451			0.404511
$630$		0			0
$631$	−19.8268			−0.789294			−0.394647	−	0.918833i	$-0.629133\pi$
$631$	−19.8268			−0.789294			−0.394647	+	0.918833i	$0.629133\pi$
$632$		−	11.4828i		−	0.456760i
$633$		0			0
$634$	−10.8938			−0.432649
$635$	13.9770			0.554658
$636$		0			0
$637$	−12.4203	−	2.20302i	−0.492110	−	0.0872870i
$638$			6.14384i			0.243237i
$639$		0			0
$640$			24.8226i			0.981199i
$641$			9.25896i			0.365707i	0.983140	+	0.182853i	$0.0585334\pi$
$641$			9.25896i			0.365707i	−0.983140	+	0.182853i	$0.941467\pi$
$642$		0			0
$643$		−	41.9683i		−	1.65507i	−0.561416	−	0.827534i	$-0.689743\pi$
$643$		−	41.9683i		−	1.65507i	0.561416	−	0.827534i	$-0.310257\pi$
$644$	7.64456	−	6.40794i	0.301238	−	0.252508i
$645$		0			0
$646$	6.29665			0.247739
$647$	−6.28587			−0.247123			−0.123561	−	0.992337i	$-0.539432\pi$
$647$	−6.28587			−0.247123			−0.123561	+	0.992337i	$0.539432\pi$
$648$		0			0
$649$		−	36.5798i		−	1.43588i
$650$	0.128639			0.00504563
$651$		0			0
$652$	−16.2174			−0.635124
$653$			23.2866i			0.911277i	0.890165	+	0.455638i	$0.150589\pi$
$653$			23.2866i			0.911277i	−0.890165	+	0.455638i	$0.849411\pi$
$654$		0			0
$655$	−37.6414			−1.47077
$656$	−1.54691			−0.0603968
$657$		0			0
$658$	0.320035	−	0.268265i	0.0124763	−	0.0104581i
$659$		−	29.8929i		−	1.16446i	−0.813024	−	0.582230i	$-0.802180\pi$
$659$		−	29.8929i		−	1.16446i	0.813024	−	0.582230i	$-0.197820\pi$
$660$		0			0
$661$		−	20.3440i		−	0.791291i	−0.918403	−	0.395645i	$-0.870521\pi$
$661$		−	20.3440i		−	0.791291i	0.918403	−	0.395645i	$-0.129479\pi$
$662$			12.7651i			0.496128i
$663$		0			0
$664$		−	19.7322i		−	0.765757i
$665$	−7.37118	+	6.17878i	−0.285842	+	0.239603i
$666$		0			0
$667$	6.69963			0.259411
$668$	5.73910			0.222053
$669$		0			0
$670$		−	11.3311i		−	0.437759i
$671$	−4.81285			−0.185798
$672$		0			0
$673$	17.1199			0.659925			0.329962	−	0.943994i	$-0.392964\pi$
$673$	17.1199			0.659925			0.329962	+	0.943994i	$0.392964\pi$
$674$		−	0.628973i		−	0.0242271i
$675$		0			0
$676$	15.4909			0.595802
$677$	28.4155			1.09210			0.546048	−	0.837754i	$-0.316132\pi$
$677$	28.4155			1.09210			0.546048	+	0.837754i	$0.316132\pi$
$678$		0			0
$679$	−14.7527	+	12.3663i	−0.566157	+	0.474573i
$680$			30.3846i			1.16520i
$681$		0			0
$682$			23.3390i			0.893697i
$683$			20.9274i			0.800764i	0.916348	+	0.400382i	$0.131123\pi$
$683$			20.9274i			0.800764i	−0.916348	+	0.400382i	$0.868877\pi$
$684$		0			0
$685$		−	13.8526i		−	0.529281i
$686$	5.92553	+	10.2995i	0.226238	+	0.393236i
$687$		0			0
$688$	6.66621			0.254147
$689$	−14.1925			−0.540693
$690$		0			0
$691$		−	24.0083i		−	0.913318i	−0.889642	−	0.456659i	$-0.849046\pi$
$691$		−	24.0083i		−	0.913318i	0.889642	−	0.456659i	$-0.150954\pi$
$692$	28.7163			1.09163
$693$		0			0
$694$	−13.6094			−0.516606
$695$		−	16.9906i		−	0.644489i
$696$		0			0
$697$	−5.43268			−0.205777
$698$	−6.44493			−0.243944
$699$		0			0
$700$	0.300372	+	0.358338i	0.0113530	+	0.0135439i
$701$			42.0117i			1.58676i	0.608728	+	0.793379i	$0.291680\pi$
$701$			42.0117i			1.58676i	−0.608728	+	0.793379i	$0.708320\pi$
$702$		0			0
$703$		−	2.79450i		−	0.105397i
$704$		−	0.863689i		−	0.0325515i
$705$		0			0
$706$			1.76215i			0.0663195i
$707$	7.93405	+	9.46517i	0.298391	+	0.355974i
$708$		0			0
$709$	37.2188			1.39778			0.698891	−	0.715228i	$-0.253677\pi$
$709$	37.2188			1.39778			0.698891	+	0.715228i	$0.253677\pi$
$710$	17.2703			0.648141
$711$		0			0
$712$		−	24.2562i		−	0.909039i
$713$	25.4503			0.953122
$714$		0			0
$715$	−13.5178			−0.505537
$716$		−	7.99255i		−	0.298696i
$717$		0			0
$718$	6.41673			0.239470
$719$	−18.2978			−0.682392			−0.341196	−	0.939992i	$-0.610832\pi$
$719$	−18.2978			−0.682392			−0.341196	+	0.939992i	$0.610832\pi$
$720$		0			0
$721$	12.6538	−	10.6069i	0.471253	−	0.395021i
$722$			10.4557i			0.389123i
$723$		0			0
$724$		−	21.5370i		−	0.800416i
$725$			0.314045i			0.0116633i
$726$		0			0
$727$		−	32.7027i		−	1.21288i	−0.795130	−	0.606439i	$-0.792598\pi$
$727$		−	32.7027i		−	1.21288i	0.795130	−	0.606439i	$-0.207402\pi$
$728$	7.05127	+	8.41204i	0.261338	+	0.311771i
$729$		0			0
$730$	−0.604462			−0.0223722
$731$	23.4114			0.865901
$732$		0			0
$733$		−	0.498614i		−	0.0184167i	−0.999958	−	0.00920836i	$-0.997069\pi$
$733$		−	0.498614i		−	0.0184167i	0.999958	−	0.00920836i	$-0.00293115\pi$
$734$	3.73059			0.137699
$735$		0			0
$736$	−13.5178			−0.498273
$737$		−	27.0997i		−	0.998231i
$738$		0			0
$739$	−47.7046			−1.75484			−0.877421	−	0.479722i	$-0.840737\pi$
$739$	−47.7046			−1.75484			−0.877421	+	0.479722i	$0.840737\pi$
$740$	5.96901			0.219425
$741$		0			0
$742$	8.58836	+	10.2458i	0.315289	+	0.376134i
$743$			10.6294i			0.389955i	0.980808	+	0.194978i	$0.0624634\pi$
$743$			10.6294i			0.389955i	−0.980808	+	0.194978i	$0.937537\pi$
$744$		0			0
$745$			34.1336i			1.25056i
$746$		−	9.95693i		−	0.364549i
$747$		0			0
$748$			32.1662i			1.17611i
$749$	3.01993	−	2.53141i	0.110346	−	0.0924958i
$750$		0			0
$751$	19.1185			0.697646			0.348823	−	0.937189i	$-0.386581\pi$
$751$	19.1185			0.697646			0.348823	+	0.937189i	$0.386581\pi$
$752$	0.418126			0.0152475
$753$		0			0
$754$			3.26327i			0.118841i
$755$	25.8411			0.940453
$756$		0			0
$757$	28.5388			1.03726			0.518631	−	0.854998i	$-0.326442\pi$
$757$	28.5388			1.03726			0.518631	+	0.854998i	$0.326442\pi$
$758$			1.79461i			0.0651832i
$759$		0			0
$760$	8.36955			0.303596
$761$	43.3300			1.57071			0.785355	−	0.619045i	$-0.212480\pi$
$761$	43.3300			1.57071			0.785355	+	0.619045i	$0.212480\pi$
$762$		0			0
$763$	7.45606	+	8.89494i	0.269927	+	0.322019i
$764$			16.2674i			0.588533i
$765$		0			0
$766$		−	2.23567i		−	0.0807779i
$767$		−	19.4292i		−	0.701546i
$768$		0			0
$769$		−	6.64171i		−	0.239506i	−0.992804	−	0.119753i	$-0.961790\pi$
$769$		−	6.64171i		−	0.239506i	0.992804	−	0.119753i	$-0.0382103\pi$
$770$	8.18006	+	9.75866i	0.294789	+	0.351678i
$771$		0			0
$772$	25.6240			0.922227
$773$	44.4831			1.59995			0.799973	−	0.600036i	$-0.204847\pi$
$773$	44.4831			1.59995			0.799973	+	0.600036i	$0.204847\pi$
$774$		0			0
$775$			1.19298i			0.0428532i
$776$	16.7509			0.601321
$777$		0			0
$778$	−4.71929			−0.169195
$779$			1.49645i			0.0536159i
$780$		0			0
$781$	41.3039			1.47797
$782$	−9.09020			−0.325065
$783$		0			0
$784$	−2.07784	+	11.7145i	−0.0742087	+	0.418377i
$785$			12.6017i			0.449775i
$786$		0			0
$787$			21.9854i			0.783695i	0.920030	+	0.391848i	$0.128164\pi$
$787$			21.9854i			0.783695i	−0.920030	+	0.391848i	$0.871836\pi$
$788$		−	6.13590i		−	0.218582i
$789$		0			0
$790$		−	7.07536i		−	0.251730i
$791$	−34.8300	+	29.1958i	−1.23841	+	1.03808i
$792$		0			0
$793$	−2.55632			−0.0907776
$794$	12.3722			0.439075
$795$		0			0
$796$		−	24.1486i		−	0.855923i
$797$	−19.4378			−0.688523			−0.344262	−	0.938874i	$-0.611871\pi$
$797$	−19.4378			−0.688523			−0.344262	+	0.938874i	$0.611871\pi$
$798$		0			0
$799$	1.46844			0.0519496
$800$		−	0.633646i		−	0.0224028i
$801$		0			0
$802$	−7.11636			−0.251287
$803$	−1.44565			−0.0510157
$804$		0			0
$805$	10.6414	−	8.92004i	0.375062	−	0.314390i
$806$			12.3964i			0.436645i
$807$		0			0
$808$		−	10.7472i		−	0.378084i
$809$		−	21.0058i		−	0.738526i	−0.929325	−	0.369263i	$-0.879610\pi$
$809$		−	21.0058i		−	0.738526i	0.929325	−	0.369263i	$-0.120390\pi$
$810$		0			0
$811$		−	37.3291i		−	1.31080i	−0.755281	−	0.655401i	$-0.772500\pi$
$811$		−	37.3291i		−	1.31080i	0.755281	−	0.655401i	$-0.227500\pi$
$812$	−9.09020	+	7.61974i	−0.319004	+	0.267400i
$813$		0			0
$814$	−3.69963			−0.129672
$815$	−22.5751			−0.790772
$816$		0			0
$817$		−	6.44875i		−	0.225613i
$818$	13.0090			0.454849
$819$		0			0
$820$	−3.19639			−0.111623
$821$			12.5882i			0.439332i	0.975575	+	0.219666i	$0.0704968\pi$
$821$			12.5882i			0.439332i	−0.975575	+	0.219666i	$0.929503\pi$
$822$		0			0
$823$	−44.8378			−1.56295			−0.781474	−	0.623937i	$-0.785532\pi$
$823$	−44.8378			−1.56295			−0.781474	+	0.623937i	$0.785532\pi$
$824$	−14.3677			−0.500523
$825$		0			0
$826$	−14.0261	+	11.7572i	−0.488032	+	0.409086i
$827$			25.7293i			0.894695i	0.894360	+	0.447347i	$0.147631\pi$
$827$			25.7293i			0.894695i	−0.894360	+	0.447347i	$0.852369\pi$
$828$		0			0
$829$			16.9628i			0.589142i	0.955630	+	0.294571i	$0.0951767\pi$
$829$			16.9628i			0.589142i	−0.955630	+	0.294571i	$0.904823\pi$
$830$		−	12.1584i		−	0.422025i
$831$		0			0
$832$		−	0.458744i		−	0.0159041i
$833$	−7.29728	+	41.1409i	−0.252836	+	1.42545i
$834$		0			0
$835$	7.98900			0.276471
$836$	8.86030			0.306440
$837$		0			0
$838$		−	7.12081i		−	0.245984i
$839$	−26.7077			−0.922054			−0.461027	−	0.887386i	$-0.652519\pi$
$839$	−26.7077			−0.922054			−0.461027	+	0.887386i	$0.652519\pi$
$840$		0			0
$841$	21.0334			0.725290
$842$		−	5.89324i		−	0.203095i
$843$		0			0
$844$	−37.9693			−1.30696
$845$	21.5637			0.741815
$846$		0			0
$847$	0.867695	+	1.03514i	0.0298144	+	0.0355680i
$848$			13.3861i			0.459680i
$849$		0			0
$850$		−	0.426103i		−	0.0146152i
$851$			4.03430i			0.138294i
$852$		0			0
$853$			43.4499i			1.48770i	0.668349	+	0.743848i	$0.267001\pi$
$853$			43.4499i			1.48770i	−0.668349	+	0.743848i	$0.732999\pi$
$854$	1.54691	+	1.84544i	0.0529342	+	0.0631496i
$855$		0			0
$856$	−3.42896			−0.117199
$857$	−15.6686			−0.535229			−0.267615	−	0.963526i	$-0.586235\pi$
$857$	−15.6686			−0.535229			−0.267615	+	0.963526i	$0.586235\pi$
$858$		0			0
$859$			20.0431i			0.683862i	0.939725	+	0.341931i	$0.111081\pi$
$859$			20.0431i			0.683862i	−0.939725	+	0.341931i	$0.888919\pi$
$860$	13.7744			0.469704
$861$		0			0
$862$	−9.69453			−0.330197
$863$		−	40.0219i		−	1.36236i	−0.732115	−	0.681181i	$-0.761467\pi$
$863$		−	40.0219i		−	1.36236i	0.732115	−	0.681181i	$-0.238533\pi$
$864$		0			0
$865$	39.9739			1.35915
$866$	−2.14020			−0.0727269
$867$		0			0
$868$	−34.5316	+	28.9456i	−1.17208	+	0.982477i
$869$		−	16.9216i		−	0.574025i
$870$		0			0
$871$		−	14.3939i		−	0.487718i
$872$		−	10.0997i		−	0.342019i
$873$		0			0
$874$			2.50393i			0.0846967i
$875$	19.2079	+	22.9147i	0.649346	+	0.774658i
$876$		0			0
$877$	45.2705			1.52868			0.764338	−	0.644815i	$-0.223066\pi$
$877$	45.2705			1.52868			0.764338	+	0.644815i	$0.223066\pi$
$878$	−4.37862			−0.147771
$879$		0			0
$880$			12.7497i			0.429792i
$881$	−45.3385			−1.52749			−0.763746	−	0.645517i	$-0.776642\pi$
$881$	−45.3385			−1.52749			−0.763746	+	0.645517i	$0.776642\pi$
$882$		0			0
$883$	12.5650			0.422845			0.211423	−	0.977395i	$-0.432190\pi$
$883$	12.5650			0.422845			0.211423	+	0.977395i	$0.432190\pi$
$884$			17.0849i			0.574628i
$885$		0			0
$886$	7.24357			0.243352
$887$	35.7241			1.19950			0.599748	−	0.800189i	$-0.295267\pi$
$887$	35.7241			1.19950			0.599748	+	0.800189i	$0.295267\pi$
$888$		0			0
$889$	−10.7441	−	12.8175i	−0.360344	−	0.429884i
$890$		−	14.9460i		−	0.500991i
$891$		0			0
$892$		−	30.4821i		−	1.02062i
$893$		−	0.404487i		−	0.0135356i
$894$		0			0
$895$		−	11.1259i		−	0.371897i
$896$	22.7634	−	19.0811i	0.760470	−	0.637453i
$897$		0			0
$898$	15.9469			0.532155
$899$	−30.2632			−1.00933
$900$		0			0
$901$			47.0113i			1.56617i
$902$	1.98114			0.0659648
$903$		0			0
$904$	39.5475			1.31533
$905$		−	29.9801i		−	0.996573i
$906$		0			0
$907$	−9.04208			−0.300237			−0.150119	−	0.988668i	$-0.547966\pi$
$907$	−9.04208			−0.300237			−0.150119	+	0.988668i	$0.547966\pi$
$908$	13.7744			0.457120
$909$		0			0
$910$	4.34479	+	5.18326i	0.144029	+	0.171823i
$911$			41.0116i			1.35878i	0.733779	+	0.679388i	$0.237755\pi$
$911$			41.0116i			1.35878i	−0.733779	+	0.679388i	$0.762245\pi$
$912$		0			0
$913$		−	29.0783i		−	0.962352i
$914$		−	8.09005i		−	0.267595i
$915$		0			0
$916$		−	22.8228i		−	0.754088i
$917$	28.9349	+	34.5188i	0.955515	+	1.13991i
$918$		0			0
$919$	10.2326			0.337541			0.168771	−	0.985655i	$-0.446020\pi$
$919$	10.2326			0.337541			0.168771	+	0.985655i	$0.446020\pi$
$920$	−12.0828			−0.398357
$921$		0			0
$922$			18.4818i			0.608665i
$923$	21.9384			0.722110
$924$		0			0
$925$	−0.189108			−0.00621782
$926$			16.1498i			0.530716i
$927$		0			0
$928$	16.0741			0.527659
$929$	−25.6659			−0.842071			−0.421036	−	0.907044i	$-0.638333\pi$
$929$	−25.6659			−0.842071			−0.421036	+	0.907044i	$0.638333\pi$
$930$		0			0
$931$	11.3324	+	2.01006i	0.371405	+	0.0658772i
$932$			47.1999i			1.54608i
$933$		0			0
$934$		−	16.4215i		−	0.537328i
$935$			44.7763i			1.46434i
$936$		0			0
$937$		−	15.9276i		−	0.520333i	−0.965564	−	0.260167i	$-0.916223\pi$
$937$		−	15.9276i		−	0.520333i	0.965564	−	0.260167i	$-0.0837775\pi$
$938$	−10.3911	+	8.71020i	−0.339282	+	0.284398i
$939$		0			0
$940$	0.863976			0.0281798
$941$	39.3534			1.28288			0.641442	−	0.767172i	$-0.278337\pi$
$941$	39.3534			1.28288			0.641442	+	0.767172i	$0.278337\pi$
$942$		0			0
$943$		−	2.16036i		−	0.0703510i
$944$	−18.3252			−0.596433
$945$		0			0
$946$	−8.53747			−0.277577
$947$		−	33.3808i		−	1.08473i	−0.840143	−	0.542365i	$-0.817529\pi$
$947$		−	33.3808i		−	1.08473i	0.840143	−	0.542365i	$-0.182471\pi$
$948$		0			0
$949$	−0.767847			−0.0249254
$950$	−0.117372			−0.00380804
$951$		0			0
$952$	27.8640	−	23.3566i	0.903077	−	0.756991i
$953$		−	44.4622i		−	1.44027i	−0.693832	−	0.720137i	$-0.744079\pi$
$953$		−	44.4622i		−	1.44027i	0.693832	−	0.720137i	$-0.255921\pi$
$954$		0			0
$955$			22.6447i			0.732764i
$956$		−	25.1456i		−	0.813266i
$957$		0			0
$958$		−	0.343569i		−	0.0111002i
$959$	−12.7034	+	10.6485i	−0.410215	+	0.343857i
$960$		0			0
$961$	−83.9627			−2.70847
$962$	−1.96504			−0.0633554
$963$		0			0
$964$		−	7.97776i		−	0.256947i
$965$	35.6693			1.14824
$966$		0			0
$967$	40.1111			1.28989			0.644943	−	0.764231i	$-0.276881\pi$
$967$	40.1111			1.28989			0.644943	+	0.764231i	$0.276881\pi$
$968$		−	1.17535i		−	0.0377771i
$969$		0			0
$970$	10.3214			0.331401
$971$	46.0026			1.47629			0.738147	−	0.674640i	$-0.235701\pi$
$971$	46.0026			1.47629			0.738147	+	0.674640i	$0.235701\pi$
$972$		0			0
$973$	−15.5811	+	13.0606i	−0.499506	+	0.418704i
$974$			21.8989i			0.701686i
$975$		0			0
$976$			2.41106i			0.0771763i
$977$		−	54.0772i		−	1.73008i	−0.501699	−	0.865042i	$-0.667292\pi$
$977$		−	54.0772i		−	1.73008i	0.501699	−	0.865042i	$-0.332708\pi$
$978$		0			0
$979$		−	35.7451i		−	1.14242i
$980$	−4.29346	+	24.2058i	−0.137149	+	0.773227i
$981$		0			0
$982$	−4.34836			−0.138762
$983$	−13.9578			−0.445185			−0.222592	−	0.974912i	$-0.571452\pi$
$983$	−13.9578			−0.445185			−0.222592	+	0.974912i	$0.571452\pi$
$984$		0			0
$985$		−	8.54135i		−	0.272150i
$986$	10.8092			0.344236
$987$		0			0
$988$	4.70610			0.149721
$989$			9.30979i			0.296034i
$990$		0			0
$991$	−37.0297			−1.17629			−0.588144	−	0.808756i	$-0.700141\pi$
$991$	−37.0297			−1.17629			−0.588144	+	0.808756i	$0.700141\pi$
$992$	61.0618			1.93872
$993$		0			0
$994$	−13.2756	−	15.8376i	−0.421077	−	0.502337i
$995$		−	33.6155i		−	1.06568i
$996$		0			0
$997$		−	50.1466i		−	1.58816i	−0.607815	−	0.794079i	$-0.707954\pi$
$997$		−	50.1466i		−	1.58816i	0.607815	−	0.794079i	$-0.292046\pi$
$998$			5.51814i			0.174674i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.2.c.c.566.8		12
3.2	odd	2	inner	567.2.c.c.566.5		12
7.6	odd	2	inner	567.2.c.c.566.7		12
9.2	odd	6		63.2.o.a.41.4	yes	12
9.4	even	3		63.2.o.a.20.3	✓	12
9.5	odd	6		189.2.o.a.62.4		12
9.7	even	3		189.2.o.a.125.3		12
21.20	even	2	inner	567.2.c.c.566.6		12
36.7	odd	6		3024.2.cc.a.881.2		12
36.11	even	6		1008.2.cc.a.545.1		12
36.23	even	6		3024.2.cc.a.2897.5		12
36.31	odd	6		1008.2.cc.a.209.6		12
63.2	odd	6		441.2.s.c.374.3		12
63.4	even	3		441.2.s.c.362.4		12
63.5	even	6		1323.2.i.c.521.3		12
63.11	odd	6		441.2.i.c.68.3		12
63.13	odd	6		63.2.o.a.20.4	yes	12
63.16	even	3		1323.2.s.c.962.4		12
63.20	even	6		63.2.o.a.41.3	yes	12
63.23	odd	6		1323.2.i.c.521.4		12
63.25	even	3		1323.2.i.c.1097.3		12
63.31	odd	6		441.2.s.c.362.3		12
63.32	odd	6		1323.2.s.c.656.3		12
63.34	odd	6		189.2.o.a.125.4		12
63.38	even	6		441.2.i.c.68.4		12
63.40	odd	6		441.2.i.c.227.3		12
63.41	even	6		189.2.o.a.62.3		12
63.47	even	6		441.2.s.c.374.4		12
63.52	odd	6		1323.2.i.c.1097.4		12
63.58	even	3		441.2.i.c.227.4		12
63.59	even	6		1323.2.s.c.656.4		12
63.61	odd	6		1323.2.s.c.962.3		12
252.83	odd	6		1008.2.cc.a.545.6		12
252.139	even	6		1008.2.cc.a.209.1		12
252.167	odd	6		3024.2.cc.a.2897.2		12
252.223	even	6		3024.2.cc.a.881.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.o.a.20.3	✓	12	9.4	even	3
63.2.o.a.20.4	yes	12	63.13	odd	6
63.2.o.a.41.3	yes	12	63.20	even	6
63.2.o.a.41.4	yes	12	9.2	odd	6
189.2.o.a.62.3		12	63.41	even	6
189.2.o.a.62.4		12	9.5	odd	6
189.2.o.a.125.3		12	9.7	even	3
189.2.o.a.125.4		12	63.34	odd	6
441.2.i.c.68.3		12	63.11	odd	6
441.2.i.c.68.4		12	63.38	even	6
441.2.i.c.227.3		12	63.40	odd	6
441.2.i.c.227.4		12	63.58	even	3
441.2.s.c.362.3		12	63.31	odd	6
441.2.s.c.362.4		12	63.4	even	3
441.2.s.c.374.3		12	63.2	odd	6
441.2.s.c.374.4		12	63.47	even	6
567.2.c.c.566.5		12	3.2	odd	2	inner
567.2.c.c.566.6		12	21.20	even	2	inner
567.2.c.c.566.7		12	7.6	odd	2	inner
567.2.c.c.566.8		12	1.1	even	1	trivial
1008.2.cc.a.209.1		12	252.139	even	6
1008.2.cc.a.209.6		12	36.31	odd	6
1008.2.cc.a.545.1		12	36.11	even	6
1008.2.cc.a.545.6		12	252.83	odd	6
1323.2.i.c.521.3		12	63.5	even	6
1323.2.i.c.521.4		12	63.23	odd	6
1323.2.i.c.1097.3		12	63.25	even	3
1323.2.i.c.1097.4		12	63.52	odd	6
1323.2.s.c.656.3		12	63.32	odd	6
1323.2.s.c.656.4		12	63.59	even	6
1323.2.s.c.962.3		12	63.61	odd	6
1323.2.s.c.962.4		12	63.16	even	3
3024.2.cc.a.881.2		12	36.7	odd	6
3024.2.cc.a.881.5		12	252.223	even	6
3024.2.cc.a.2897.2		12	252.167	odd	6
3024.2.cc.a.2897.5		12	36.23	even	6

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:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	567.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.641589i q^{2} +1.58836 q^{4} +2.21105 q^{5} +(-1.69963 - 2.02763i) q^{7} +2.30225i q^{8} +O(q^{10})\)	\(q+0.641589i q^{2} +1.58836 q^{4} +2.21105 q^{5} +(-1.69963 - 2.02763i) q^{7} +2.30225i q^{8} +1.41858i q^{10} +3.39272i q^{11} +1.80202i q^{13} +(1.30090 - 1.09046i) q^{14} +1.69963 q^{16} +5.96901 q^{17} -1.64419i q^{19} +3.51195 q^{20} -2.17673 q^{22} +2.37364i q^{23} -0.111264 q^{25} -1.15616 q^{26} +(-2.69963 - 3.22061i) q^{28} -2.82251i q^{29} -10.7221i q^{31} +5.69497i q^{32} +3.82965i q^{34} +(-3.75796 - 4.48318i) q^{35} +1.69963 q^{37} +1.05489 q^{38} +5.09039i q^{40} -0.910147 q^{41} +3.92216 q^{43} +5.38887i q^{44} -1.52290 q^{46} +0.246010 q^{47} +(-1.22253 + 6.89242i) q^{49} -0.0713858i q^{50} +2.86227i q^{52} +7.87589i q^{53} +7.50146i q^{55} +(4.66811 - 3.91298i) q^{56} +1.81089 q^{58} -10.7819 q^{59} +1.41858i q^{61} +6.87916 q^{62} -0.254572 q^{64} +3.98436i q^{65} -7.98762 q^{67} +9.48096 q^{68} +(2.87636 - 2.41106i) q^{70} -12.1743i q^{71} +0.426103i q^{73} +1.09046i q^{74} -2.61157i q^{76} +(6.87916 - 5.76636i) q^{77} -4.98762 q^{79} +3.75796 q^{80} -0.583940i q^{82} -8.57081 q^{83} +13.1978 q^{85} +2.51641i q^{86} -7.81089 q^{88} -10.5358 q^{89} +(3.65383 - 3.06277i) q^{91} +3.77020i q^{92} +0.157838i q^{94} -3.63537i q^{95} -7.27586i q^{97} +(-4.42210 - 0.784360i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{4} + 4 q^{7}+O(q^{10}) \) 12 * q - 4 * q^4 + 4 * q^7	\( 12 q - 4 q^{4} + 4 q^{7} - 4 q^{16} + 20 q^{22} - 8 q^{28} - 4 q^{37} + 20 q^{43} - 40 q^{46} - 12 q^{49} - 4 q^{58} + 16 q^{64} - 24 q^{67} - 36 q^{70} + 12 q^{79} + 12 q^{85} - 68 q^{88} - 24 q^{91}+O(q^{100}) \) 12 * q - 4 * q^4 + 4 * q^7 - 4 * q^16 + 20 * q^22 - 8 * q^28 - 4 * q^37 + 20 * q^43 - 40 * q^46 - 12 * q^49 - 4 * q^58 + 16 * q^64 - 24 * q^67 - 36 * q^70 + 12 * q^79 + 12 * q^85 - 68 * q^88 - 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more