LMFDB - Embedded newform 1872.2.by.j.433.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1872,2,Mod(433,1872)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1872.433"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1872, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1872.by (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,3,0,0,0,12,0,6,0,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$14.9479952584$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-43})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} - 11x + 121 $ x^4 - x^3 - 10x^2 - 11x + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			433.1
Root			$3.08945 - 1.20635i$ of defining polynomial
Character	$\chi$	$=$	1872.433
Dual form			1872.2.by.j.1297.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41269i q^{5} +(3.58945 - 2.07237i) q^{7} +(3.00000 + 1.73205i) q^{11} +(1.50000 - 3.27872i) q^{13} +(-3.08945 - 5.35109i) q^{17} +(3.00000 - 1.73205i) q^{19} +(1.00000 - 1.73205i) q^{23} -0.821092 q^{25} +(-4.08945 + 7.08314i) q^{29} +7.60885i q^{31} +(-5.00000 - 8.66025i) q^{35} +(0.910546 + 0.525704i) q^{37} +(-5.08945 - 2.93840i) q^{41} +(-0.410546 - 0.711086i) q^{43} +10.3923i q^{47} +(5.08945 - 8.81519i) q^{49} +10.1789 q^{53} +(4.17891 - 7.23808i) q^{55} +(1.17891 - 0.680643i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(-7.91055 - 3.61904i) q^{65} +(-3.58945 - 2.07237i) q^{67} +(-3.00000 + 1.73205i) q^{71} +11.3828i q^{73} +14.3578 q^{77} -13.1789 q^{79} -11.7536i q^{83} +(-12.9105 + 7.45391i) q^{85} +(-6.00000 - 3.46410i) q^{89} +(-1.41055 - 14.8774i) q^{91} +(-4.17891 - 7.23808i) q^{95} +(-1.76836 + 1.02096i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{7} + 12 q^{11} + 6 q^{13} - q^{17} + 12 q^{19} + 4 q^{23} - 26 q^{25} - 5 q^{29} - 20 q^{35} + 15 q^{37} - 9 q^{41} - 13 q^{43} + 9 q^{49} + 18 q^{53} - 6 q^{55} - 18 q^{59} - 10 q^{61} - 43 q^{65}+ \cdots + 27 q^{97}+O(q^{100}) $ 4 * q + 3 * q^7 + 12 * q^11 + 6 * q^13 - q^17 + 12 * q^19 + 4 * q^23 - 26 * q^25 - 5 * q^29 - 20 * q^35 + 15 * q^37 - 9 * q^41 - 13 * q^43 + 9 * q^49 + 18 * q^53 - 6 * q^55 - 18 * q^59 - 10 * q^61 - 43 * q^65 - 3 * q^67 - 12 * q^71 + 12 * q^77 - 30 * q^79 - 63 * q^85 - 24 * q^89 - 17 * q^91 + 6 * q^95 + 27 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	2.41269i		−	1.07899i	−0.841989	−	0.539495i	$-0.818615\pi$
$5$		−	2.41269i		−	1.07899i	0.841989	−	0.539495i	$-0.181385\pi$
$6$		0			0
$7$	3.58945	−	2.07237i	1.35669	−	0.783283i	0.367511	−	0.930019i	$-0.380210\pi$
$7$	3.58945	−	2.07237i	1.35669	−	0.783283i	0.989176	+	0.146736i	$0.0468769\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.00000	+	1.73205i	0.904534	+	0.522233i	0.878668	−	0.477432i	$-0.158432\pi$
$11$	3.00000	+	1.73205i	0.904534	+	0.522233i	0.0258656	+	0.999665i	$0.491766\pi$
$12$		0			0
$13$	1.50000	−	3.27872i	0.416025	−	0.909353i
$13$	1.50000	−	3.27872i	0.416025	−	0.909353i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.08945	−	5.35109i	−0.749303	−	1.29783i	−0.948157	−	0.317801i	$-0.897055\pi$
$17$	−3.08945	−	5.35109i	−0.749303	−	1.29783i	0.198855	−	0.980029i	$-0.436278\pi$
$18$		0			0
$19$	3.00000	−	1.73205i	0.688247	−	0.397360i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	3.00000	−	1.73205i	0.688247	−	0.397360i	0.802955	+	0.596040i	$0.203260\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.00000	−	1.73205i	0.208514	−	0.361158i	−0.742732	−	0.669588i	$-0.766471\pi$
$23$	1.00000	−	1.73205i	0.208514	−	0.361158i	0.951247	+	0.308431i	$0.0998038\pi$
$24$		0			0
$25$	−0.821092			−0.164218
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.08945	+	7.08314i	−0.759393	+	1.31531i	0.183768	+	0.982970i	$0.441170\pi$
$29$	−4.08945	+	7.08314i	−0.759393	+	1.31531i	−0.943161	+	0.332337i	$0.892163\pi$
$30$		0			0
$31$			7.60885i			1.36659i	0.730143	+	0.683295i	$0.239454\pi$
$31$			7.60885i			1.36659i	−0.730143	+	0.683295i	$0.760546\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.00000	−	8.66025i	−0.845154	−	1.46385i
$36$		0			0
$37$	0.910546	+	0.525704i	0.149693	+	0.0864252i	0.572976	−	0.819572i	$-0.305789\pi$
$37$	0.910546	+	0.525704i	0.149693	+	0.0864252i	−0.423283	+	0.905998i	$0.639122\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.08945	−	2.93840i	−0.794839	−	0.458901i	0.0468242	−	0.998903i	$-0.485090\pi$
$41$	−5.08945	−	2.93840i	−0.794839	−	0.458901i	−0.841663	+	0.540003i	$0.818423\pi$
$42$		0			0
$43$	−0.410546	−	0.711086i	−0.0626077	−	0.108440i	0.833023	−	0.553239i	$-0.186608\pi$
$43$	−0.410546	−	0.711086i	−0.0626077	−	0.108440i	−0.895630	+	0.444799i	$0.853275\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.3923i			1.51587i	0.652328	+	0.757937i	$0.273792\pi$
$47$			10.3923i			1.51587i	−0.652328	+	0.757937i	$0.726208\pi$
$48$		0			0
$49$	5.08945	−	8.81519i	0.727065	−	1.25931i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	10.1789			1.39818			0.699090	−	0.715033i	$-0.253589\pi$
$53$	10.1789			1.39818			0.699090	+	0.715033i	$0.253589\pi$
$54$		0			0
$55$	4.17891	−	7.23808i	0.563484	−	0.975983i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.17891	−	0.680643i	0.153481	−	0.0886122i	−0.421293	−	0.906925i	$-0.638423\pi$
$59$	1.17891	−	0.680643i	0.153481	−	0.0886122i	0.574773	+	0.818313i	$0.305090\pi$
$60$		0			0
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	0.660415	−	0.750901i	$-0.270381\pi$
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	−0.980507	+	0.196485i	$0.937047\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.91055	−	3.61904i	−0.981182	−	0.448887i
$66$		0			0
$67$	−3.58945	−	2.07237i	−0.438522	−	0.253181i	0.264449	−	0.964400i	$-0.414810\pi$
$67$	−3.58945	−	2.07237i	−0.438522	−	0.253181i	−0.702970	+	0.711219i	$0.748143\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.00000	+	1.73205i	−0.356034	+	0.205557i	−0.667340	−	0.744753i	$-0.732567\pi$
$71$	−3.00000	+	1.73205i	−0.356034	+	0.205557i	0.311305	+	0.950310i	$0.399234\pi$
$72$		0			0
$73$			11.3828i			1.33226i	0.745836	+	0.666130i	$0.232050\pi$
$73$			11.3828i			1.33226i	−0.745836	+	0.666130i	$0.767950\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	14.3578			1.63623
$78$		0			0
$79$	−13.1789			−1.48274			−0.741372	−	0.671095i	$-0.765824\pi$
$79$	−13.1789			−1.48274			−0.741372	+	0.671095i	$0.765824\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	11.7536i		−	1.29012i	−0.764130	−	0.645062i	$-0.776831\pi$
$83$		−	11.7536i		−	1.29012i	0.764130	−	0.645062i	$-0.223169\pi$
$84$		0			0
$85$	−12.9105	+	7.45391i	−1.40035	+	0.808490i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.00000	−	3.46410i	−0.635999	−	0.367194i	0.147073	−	0.989126i	$-0.453015\pi$
$89$	−6.00000	−	3.46410i	−0.635999	−	0.367194i	−0.783072	+	0.621932i	$0.786348\pi$
$90$		0			0
$91$	−1.41055	−	14.8774i	−0.147865	−	1.55957i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.17891	−	7.23808i	−0.428747	−	0.742612i
$96$		0			0
$97$	−1.76836	+	1.02096i	−0.179550	+	0.103663i	−0.587081	−	0.809528i	$-0.699723\pi$
$97$	−1.76836	+	1.02096i	−0.179550	+	0.103663i	0.407531	+	0.913191i	$0.366390\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.08945	−	10.5472i	0.605923	−	1.04949i	−0.385981	−	0.922507i	$-0.626137\pi$
$101$	6.08945	−	10.5472i	0.605923	−	1.04949i	0.991905	−	0.126983i	$-0.0405295\pi$
$102$		0			0
$103$	−3.17891			−0.313227			−0.156614	−	0.987660i	$-0.550058\pi$
$103$	−3.17891			−0.313227			−0.156614	+	0.987660i	$0.550058\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.00000	−	8.66025i	0.483368	−	0.837218i	−0.516449	−	0.856318i	$-0.672747\pi$
$107$	5.00000	−	8.66025i	0.483368	−	0.837218i	0.999818	+	0.0190994i	$0.00607989\pi$
$108$		0			0
$109$		−	7.60885i		−	0.728795i	−0.931243	−	0.364398i	$-0.881275\pi$
$109$		−	7.60885i		−	0.728795i	0.931243	−	0.364398i	$-0.118725\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.08945	+	10.5472i	0.572848	+	0.992201i	0.996272	+	0.0862697i	$0.0274947\pi$
$113$	6.08945	+	10.5472i	0.572848	+	0.992201i	−0.423424	+	0.905932i	$0.639172\pi$
$114$		0			0
$115$	−4.17891	−	2.41269i	−0.389685	−	0.224985i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−22.1789	−	12.8050i	−2.03314	−	1.17383i
$120$		0			0
$121$	0.500000	+	0.866025i	0.0454545	+	0.0787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	10.0824i		−	0.901800i
$126$		0			0
$127$	−1.41055	+	2.44314i	−0.125166	+	0.216793i	−0.921798	−	0.387671i	$-0.873280\pi$
$127$	−1.41055	+	2.44314i	−0.125166	+	0.216793i	0.796632	+	0.604465i	$0.206613\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.35782			0.555485			0.277743	−	0.960656i	$-0.410414\pi$
$131$	6.35782			0.555485			0.277743	+	0.960656i	$0.410414\pi$
$132$		0			0
$133$	7.17891	−	12.4342i	0.622490	−	1.07818i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.0895	−	6.40250i	0.947436	−	0.547002i	0.0551525	−	0.998478i	$-0.482436\pi$
$137$	11.0895	−	6.40250i	0.947436	−	0.547002i	0.892284	+	0.451476i	$0.149102\pi$
$138$		0			0
$139$	8.58945	+	14.8774i	0.728548	+	1.26188i	0.957497	+	0.288444i	$0.0931378\pi$
$139$	8.58945	+	14.8774i	0.728548	+	1.26188i	−0.228949	+	0.973438i	$0.573529\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	10.1789	−	7.23808i	0.851203	−	0.605279i
$144$		0			0
$145$	17.0895	+	9.86660i	1.41920	+	0.819377i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.08945	+	2.93840i	−0.416944	+	0.240723i	−0.693769	−	0.720197i	$-0.744051\pi$
$149$	−5.08945	+	2.93840i	−0.416944	+	0.240723i	0.276825	+	0.960920i	$0.410718\pi$
$150$		0			0
$151$			20.0431i			1.63108i	0.578699	+	0.815541i	$0.303561\pi$
$151$			20.0431i			1.63108i	−0.578699	+	0.815541i	$0.696439\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	18.3578			1.47454
$156$		0			0
$157$	7.00000			0.558661			0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000			0.558661			0.279330	+	0.960195i	$0.409888\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	8.28949i		−	0.653303i
$162$		0			0
$163$	6.58945	−	3.80442i	0.516126	−	0.297985i	−0.219222	−	0.975675i	$-0.570352\pi$
$163$	6.58945	−	3.80442i	0.516126	−	0.297985i	0.735348	+	0.677690i	$0.237019\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000	+	3.46410i	0.464294	+	0.268060i	0.713848	−	0.700301i	$-0.246951\pi$
$167$	6.00000	+	3.46410i	0.464294	+	0.268060i	−0.249554	+	0.968361i	$0.580284\pi$
$168$		0			0
$169$	−8.50000	−	9.83616i	−0.653846	−	0.756628i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.00000	+	15.5885i	0.684257	+	1.18517i	0.973670	+	0.227964i	$0.0732068\pi$
$173$	9.00000	+	15.5885i	0.684257	+	1.18517i	−0.289412	+	0.957205i	$0.593460\pi$
$174$		0			0
$175$	−2.94727	+	1.70161i	−0.222793	+	0.128629i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.17891	−	7.23808i	0.312346	−	0.541000i	−0.666524	−	0.745484i	$-0.732218\pi$
$179$	4.17891	−	7.23808i	0.312346	−	0.541000i	0.978870	+	0.204484i	$0.0655518\pi$
$180$		0			0
$181$	−20.5367			−1.52648			−0.763241	−	0.646113i	$-0.776393\pi$
$181$	−20.5367			−1.52648			−0.763241	+	0.646113i	$0.776393\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.26836	−	2.19687i	0.0932519	−	0.161517i
$186$		0			0
$187$		−	21.4044i		−	1.56524i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.17891	+	2.04193i	0.0853028	+	0.147749i	0.905520	−	0.424303i	$-0.139481\pi$
$191$	1.17891	+	2.04193i	0.0853028	+	0.147749i	−0.820217	+	0.572052i	$0.806148\pi$
$192$		0			0
$193$	−11.6789	−	6.74282i	−0.840666	−	0.485359i	0.0168244	−	0.999858i	$-0.494644\pi$
$193$	−11.6789	−	6.74282i	−0.840666	−	0.485359i	−0.857491	+	0.514500i	$0.827978\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.35782	+	4.82539i	0.595470	+	0.343795i	0.767257	−	0.641339i	$-0.221621\pi$
$197$	8.35782	+	4.82539i	0.595470	+	0.343795i	−0.171788	+	0.985134i	$0.554954\pi$
$198$		0			0
$199$	−10.7684	−	18.6514i	−0.763349	−	1.32216i	−0.941115	−	0.338086i	$-0.890220\pi$
$199$	−10.7684	−	18.6514i	−0.763349	−	1.32216i	0.177766	−	0.984073i	$-0.443113\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			33.8995i			2.37928i
$204$		0			0
$205$	−7.08945	+	12.2793i	−0.495149	+	0.857623i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$	−7.76836	+	13.4552i	−0.534796	+	0.926294i	0.464377	+	0.885638i	$0.346278\pi$
$211$	−7.76836	+	13.4552i	−0.534796	+	0.926294i	−0.999173	+	0.0406565i	$0.987055\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.71563	+	0.990521i	−0.117005	+	0.0675530i
$216$		0			0
$217$	15.7684	+	27.3116i	1.07043	+	1.85403i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−22.1789	+	2.10282i	−1.49191	+	0.141451i
$222$		0			0
$223$	13.1789	+	7.60885i	0.882525	+	0.509526i	0.871490	−	0.490413i	$-0.163154\pi$
$223$	13.1789	+	7.60885i	0.882525	+	0.509526i	0.0110349	+	0.999939i	$0.496487\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.82109	−	4.51551i	0.519104	−	0.299705i	−0.217464	−	0.976068i	$-0.569778\pi$
$227$	7.82109	−	4.51551i	0.519104	−	0.299705i	0.736568	+	0.676363i	$0.236445\pi$
$228$		0			0
$229$		−	9.65078i		−	0.637741i	−0.947798	−	0.318871i	$-0.896696\pi$
$229$		−	9.65078i		−	0.637741i	0.947798	−	0.318871i	$-0.103304\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−22.7156			−1.48815			−0.744075	−	0.668096i	$-0.767110\pi$
$233$	−22.7156			−1.48815			−0.744075	+	0.668096i	$0.767110\pi$
$234$		0			0
$235$	25.0735			1.63561
$236$		0			0
$237$		0			0
$238$		0			0
$239$			2.10282i			0.136020i	0.997685	+	0.0680099i	$0.0216650\pi$
$239$			2.10282i			0.136020i	−0.997685	+	0.0680099i	$0.978335\pi$
$240$		0			0
$241$	5.08945	−	2.93840i	0.327841	−	0.189279i	−0.327041	−	0.945010i	$-0.606052\pi$
$241$	5.08945	−	2.93840i	0.327841	−	0.189279i	0.654882	+	0.755731i	$0.272718\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−21.2684	−	12.2793i	−1.35879	−	0.784495i
$246$		0			0
$247$	−1.17891	−	12.4342i	−0.0750121	−	0.791171i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.3578	+	17.9403i	0.653780	+	1.13238i	0.982198	+	0.187847i	$0.0601510\pi$
$251$	10.3578	+	17.9403i	0.653780	+	1.13238i	−0.328419	+	0.944532i	$0.606516\pi$
$252$		0			0
$253$	6.00000	−	3.46410i	0.377217	−	0.217786i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	11.0895	−	19.2075i	0.691741	−	1.19813i	−0.279526	−	0.960138i	$-0.590177\pi$
$257$	11.0895	−	19.2075i	0.691741	−	1.19813i	0.971267	−	0.237993i	$-0.0764894\pi$
$258$		0			0
$259$	4.35782			0.270782
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−16.1789	+	28.0227i	−0.997634	+	1.72795i	−0.439282	+	0.898349i	$0.644767\pi$
$263$	−16.1789	+	28.0227i	−0.997634	+	1.72795i	−0.558352	+	0.829604i	$0.688566\pi$
$264$		0			0
$265$		−	24.5586i		−	1.50862i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.17891	−	10.7022i	−0.376735	−	0.652524i	0.613850	−	0.789422i	$-0.289620\pi$
$269$	−6.17891	−	10.7022i	−0.376735	−	0.652524i	−0.990585	+	0.136899i	$0.956286\pi$
$270$		0			0
$271$	12.5895	+	7.26852i	0.764755	+	0.441531i	0.831000	−	0.556272i	$-0.187769\pi$
$271$	12.5895	+	7.26852i	0.764755	+	0.441531i	−0.0662455	+	0.997803i	$0.521102\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.46327	−	1.42217i	−0.148541	−	0.0857602i
$276$		0			0
$277$	0.0894542	+	0.154939i	0.00537478	+	0.00930939i	0.868700	−	0.495338i	$-0.164956\pi$
$277$	0.0894542	+	0.154939i	0.00537478	+	0.00930939i	−0.863326	+	0.504647i	$0.831622\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			11.4437i			0.682675i	0.939941	+	0.341337i	$0.110880\pi$
$281$			11.4437i			0.682675i	−0.939941	+	0.341337i	$0.889120\pi$
$282$		0			0
$283$	−13.5895	+	23.5376i	−0.807809	+	1.39917i	0.106569	+	0.994305i	$0.466014\pi$
$283$	−13.5895	+	23.5376i	−0.807809	+	1.39917i	−0.914378	+	0.404861i	$0.867320\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−24.3578			−1.43780
$288$		0			0
$289$	−10.5895	+	18.3415i	−0.622909	+	1.07891i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−28.4473	+	16.4240i	−1.66191	+	0.959503i	−0.690104	+	0.723710i	$0.742435\pi$
$293$	−28.4473	+	16.4240i	−1.66191	+	0.959503i	−0.971803	+	0.235792i	$0.924232\pi$
$294$		0			0
$295$	−1.64218	−	2.84434i	−0.0956116	−	0.165604i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.17891	−	5.87680i	−0.241673	−	0.339864i
$300$		0			0
$301$	−2.94727	−	1.70161i	−0.169878	−	0.0980790i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−10.4473	+	6.03173i	−0.598209	+	0.345376i
$306$		0			0
$307$			11.0729i			0.631967i	0.948765	+	0.315983i	$0.102334\pi$
$307$			11.0729i			0.631967i	−0.948765	+	0.315983i	$0.897666\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	11.6422			0.660168			0.330084	−	0.943952i	$-0.392923\pi$
$311$	11.6422			0.660168			0.330084	+	0.943952i	$0.392923\pi$
$312$		0			0
$313$	27.8945			1.57669			0.788346	−	0.615232i	$-0.210938\pi$
$313$	27.8945			1.57669			0.788346	+	0.615232i	$0.210938\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			3.77398i			0.211968i	0.994368	+	0.105984i	$0.0337992\pi$
$317$			3.77398i			0.211968i	−0.994368	+	0.105984i	$0.966201\pi$
$318$		0			0
$319$	−24.5367	+	14.1663i	−1.37379	+	0.793160i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−18.5367	−	10.7022i	−1.03141	−	0.595485i
$324$		0			0
$325$	−1.23164	+	2.69213i	−0.0683190	+	0.149332i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	21.5367	+	37.3027i	1.18736	+	2.05656i
$330$		0			0
$331$	17.4105	−	10.0520i	0.956970	−	0.552507i	0.0617309	−	0.998093i	$-0.480338\pi$
$331$	17.4105	−	10.0520i	0.956970	−	0.552507i	0.895239	+	0.445586i	$0.147005\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.00000	+	8.66025i	−0.273179	+	0.473160i
$336$		0			0
$337$	−25.3578			−1.38133			−0.690664	−	0.723176i	$-0.742682\pi$
$337$	−25.3578			−1.38133			−0.690664	+	0.723176i	$0.742682\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−13.1789	+	22.8265i	−0.713678	+	1.23613i
$342$		0			0
$343$		−	13.1758i		−	0.711424i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.1789	−	17.6304i	−0.546432	−	0.946449i	−0.998515	−	0.0544728i	$-0.982652\pi$
$347$	−10.1789	−	17.6304i	−0.546432	−	0.946449i	0.452083	−	0.891976i	$-0.350681\pi$
$348$		0			0
$349$	8.94727	+	5.16571i	0.478936	+	0.276514i	0.719973	−	0.694002i	$-0.244154\pi$
$349$	8.94727	+	5.16571i	0.478936	+	0.276514i	−0.241037	+	0.970516i	$0.577487\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−16.4473	−	9.49584i	−0.875400	−	0.505412i	−0.00626098	−	0.999980i	$-0.501993\pi$
$353$	−16.4473	−	9.49584i	−0.875400	−	0.505412i	−0.869139	+	0.494568i	$0.835326\pi$
$354$		0			0
$355$	4.17891	+	7.23808i	0.221793	+	0.384157i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			18.6818i			0.985987i	0.870033	+	0.492994i	$0.164097\pi$
$359$			18.6818i			0.985987i	−0.870033	+	0.492994i	$0.835903\pi$
$360$		0			0
$361$	−3.50000	+	6.06218i	−0.184211	+	0.319062i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	27.4633			1.43749
$366$		0			0
$367$	6.41055	−	11.1034i	0.334628	−	0.579592i	−0.648785	−	0.760971i	$-0.724723\pi$
$367$	6.41055	−	11.1034i	0.334628	−	0.579592i	0.983413	+	0.181379i	$0.0580561\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	36.5367	−	21.0945i	1.89689	−	1.09517i
$372$		0			0
$373$	5.67891	+	9.83616i	0.294043	+	0.509297i	0.974762	−	0.223248i	$-0.0716659\pi$
$373$	5.67891	+	9.83616i	0.294043	+	0.509297i	−0.680719	+	0.732545i	$0.738333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	17.0895	+	24.0329i	0.880152	+	1.23776i
$378$		0			0
$379$	5.41055	+	3.12378i	0.277921	+	0.160458i	0.632482	−	0.774575i	$-0.282036\pi$
$379$	5.41055	+	3.12378i	0.277921	+	0.160458i	−0.354561	+	0.935033i	$0.615370\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−16.7156	+	9.65078i	−0.854129	+	0.493132i	−0.862042	−	0.506837i	$-0.830815\pi$
$383$	−16.7156	+	9.65078i	−0.854129	+	0.493132i	0.00791288	+	0.999969i	$0.497481\pi$
$384$		0			0
$385$		−	34.6410i		−	1.76547i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.53673			0.331425			0.165713	−	0.986174i	$-0.447008\pi$
$389$	6.53673			0.331425			0.165713	+	0.986174i	$0.447008\pi$
$390$		0			0
$391$	−12.3578			−0.624962
$392$		0			0
$393$		0			0
$394$		0			0
$395$			31.7967i			1.59986i
$396$		0			0
$397$	18.5895	−	10.7326i	0.932978	−	0.538655i	0.0452258	−	0.998977i	$-0.485599\pi$
$397$	18.5895	−	10.7326i	0.932978	−	0.538655i	0.887752	+	0.460322i	$0.152266\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.26836	−	5.35109i	−0.462840	−	0.267221i	0.250398	−	0.968143i	$-0.419439\pi$
$401$	−9.26836	−	5.35109i	−0.462840	−	0.267221i	−0.713238	+	0.700922i	$0.752772\pi$
$402$		0			0
$403$	24.9473	+	11.4133i	1.24271	+	0.568535i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.82109	+	3.15422i	0.0902682	+	0.156349i
$408$		0			0
$409$	−0.857817	+	0.495261i	−0.0424163	+	0.0244891i	−0.521058	−	0.853521i	$-0.674463\pi$
$409$	−0.857817	+	0.495261i	−0.0424163	+	0.0244891i	0.478642	+	0.878010i	$0.341129\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.82109	−	4.88627i	0.138817	−	0.240438i
$414$		0			0
$415$	−28.3578			−1.39203
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.35782	−	11.0121i	0.310600	−	0.537974i	−0.667893	−	0.744258i	$-0.732803\pi$
$419$	6.35782	−	11.0121i	0.310600	−	0.537974i	0.978492	+	0.206283i	$0.0661368\pi$
$420$		0			0
$421$			32.1674i			1.56774i	0.620922	+	0.783872i	$0.286758\pi$
$421$			32.1674i			1.56774i	−0.620922	+	0.783872i	$0.713242\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.53673	+	4.39374i	0.123049	+	0.213128i
$426$		0			0
$427$	−17.9473	−	10.3619i	−0.868529	−	0.501446i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−16.1789	−	9.34090i	−0.779311	−	0.449935i	0.0568753	−	0.998381i	$-0.481886\pi$
$431$	−16.1789	−	9.34090i	−0.779311	−	0.449935i	−0.836186	+	0.548446i	$0.815220\pi$
$432$		0			0
$433$	1.32109	+	2.28820i	0.0634876	+	0.109964i	0.896022	−	0.444009i	$-0.146444\pi$
$433$	1.32109	+	2.28820i	0.0634876	+	0.109964i	−0.832535	+	0.553973i	$0.813111\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	6.92820i		−	0.331421i
$438$		0			0
$439$	11.9473	−	20.6933i	0.570212	−	0.987636i	−0.426332	−	0.904567i	$-0.640194\pi$
$439$	11.9473	−	20.6933i	0.570212	−	0.987636i	0.996544	−	0.0830694i	$-0.0264723\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	8.00000			0.380091			0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000			0.380091			0.190046	+	0.981775i	$0.439136\pi$
$444$		0			0
$445$	−8.35782	+	14.4762i	−0.396199	+	0.686236i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9.53673	−	5.50603i	0.450066	−	0.259846i	−0.257792	−	0.966200i	$-0.582995\pi$
$449$	9.53673	−	5.50603i	0.450066	−	0.259846i	0.707858	+	0.706355i	$0.249662\pi$
$450$		0			0
$451$	−10.1789	−	17.6304i	−0.479306	−	0.830182i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−35.8945	+	3.40322i	−1.68276	+	0.159545i
$456$		0			0
$457$	6.32109	+	3.64948i	0.295688	+	0.170716i	0.640504	−	0.767955i	$-0.278725\pi$
$457$	6.32109	+	3.64948i	0.295688	+	0.170716i	−0.344816	+	0.938670i	$0.612059\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	21.2684	−	12.2793i	0.990566	−	0.571904i	0.0851228	−	0.996370i	$-0.472872\pi$
$461$	21.2684	−	12.2793i	0.990566	−	0.571904i	0.905444	+	0.424467i	$0.139538\pi$
$462$		0			0
$463$			1.42217i			0.0660940i	0.999454	+	0.0330470i	$0.0105211\pi$
$463$			1.42217i			0.0660940i	−0.999454	+	0.0330470i	$0.989479\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.00000			−0.277647			−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000			−0.277647			−0.138823	+	0.990317i	$0.544332\pi$
$468$		0			0
$469$	−17.1789			−0.793248
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	2.84434i		−	0.130783i
$474$		0			0
$475$	−2.46327	+	1.42217i	−0.113023	+	0.0652537i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.00000	−	3.46410i	−0.274147	−	0.158279i	0.356624	−	0.934248i	$-0.383928\pi$
$479$	−6.00000	−	3.46410i	−0.274147	−	0.158279i	−0.630771	+	0.775969i	$0.717261\pi$
$480$		0			0
$481$	3.08945	−	2.19687i	0.140867	−	0.100169i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.46327	+	4.26652i	0.111852	+	0.193733i
$486$		0			0
$487$	−21.0000	+	12.1244i	−0.951601	+	0.549407i	−0.893578	−	0.448908i	$-0.851813\pi$
$487$	−21.0000	+	12.1244i	−0.951601	+	0.549407i	−0.0580230	+	0.998315i	$0.518480\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.35782	+	16.2082i	−0.422312	+	0.731466i	−0.996165	−	0.0874920i	$-0.972115\pi$
$491$	−9.35782	+	16.2082i	−0.422312	+	0.731466i	0.573853	+	0.818958i	$0.305448\pi$
$492$		0			0
$493$	50.5367			2.27606
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7.17891	+	12.4342i	−0.322018	+	0.557752i
$498$		0			0
$499$			1.36129i			0.0609395i	0.999536	+	0.0304698i	$0.00970033\pi$
$499$			1.36129i			0.0609395i	−0.999536	+	0.0304698i	$0.990300\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.00000	−	5.19615i	−0.133763	−	0.231685i	0.791361	−	0.611349i	$-0.209373\pi$
$503$	−3.00000	−	5.19615i	−0.133763	−	0.231685i	−0.925124	+	0.379664i	$0.876040\pi$
$504$		0			0
$505$	−25.4473	−	14.6920i	−1.13239	−	0.653785i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.2684	+	7.08314i	0.543786	+	0.313955i	0.746612	−	0.665260i	$-0.231679\pi$
$509$	12.2684	+	7.08314i	0.543786	+	0.313955i	−0.202826	+	0.979215i	$0.565013\pi$
$510$		0			0
$511$	23.5895	+	40.8581i	1.04354	+	1.80746i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			7.66973i			0.337969i
$516$		0			0
$517$	−18.0000	+	31.1769i	−0.791639	+	1.37116i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	38.8945			1.70400			0.852000	−	0.523541i	$-0.175389\pi$
$521$	38.8945			1.70400			0.852000	+	0.523541i	$0.175389\pi$
$522$		0			0
$523$	12.1789	−	21.0945i	0.532546	−	0.922398i	−0.466731	−	0.884399i	$-0.654568\pi$
$523$	12.1789	−	21.0945i	0.532546	−	0.922398i	0.999278	−	0.0379984i	$-0.0120982\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	40.7156	−	23.5072i	1.77360	−	1.02399i
$528$		0			0
$529$	9.50000	+	16.4545i	0.413043	+	0.715412i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−17.2684	+	12.2793i	−0.747976	+	0.531875i
$534$		0			0
$535$	−20.8945	−	12.0635i	−0.903350	−	0.521549i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	30.5367	−	17.6304i	1.31531	−	0.759395i
$540$		0			0
$541$			3.09334i			0.132993i	0.997787	+	0.0664965i	$0.0211821\pi$
$541$			3.09334i			0.132993i	−0.997787	+	0.0664965i	$0.978818\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−18.3578			−0.786362
$546$		0			0
$547$	−13.5367			−0.578789			−0.289394	−	0.957210i	$-0.593454\pi$
$547$	−13.5367			−0.578789			−0.289394	+	0.957210i	$0.593454\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			28.3326i			1.20701i
$552$		0			0
$553$	−47.3051	+	27.3116i	−2.01162	+	1.16141i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	27.8051	+	16.0533i	1.17814	+	0.680199i	0.955583	−	0.294721i	$-0.0952267\pi$
$557$	27.8051	+	16.0533i	1.17814	+	0.680199i	0.222556	+	0.974920i	$0.428560\pi$
$558$		0			0
$559$	−2.94727	+	0.279435i	−0.124656	+	0.0118188i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9.17891	+	15.8983i	0.386845	+	0.670035i	0.992023	−	0.126055i	$-0.0402316\pi$
$563$	9.17891	+	15.8983i	0.386845	+	0.670035i	−0.605178	+	0.796090i	$0.706898\pi$
$564$		0			0
$565$	25.4473	−	14.6920i	1.07057	−	0.618097i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15.3578	+	26.6005i	−0.643833	+	1.11515i	0.340737	+	0.940159i	$0.389324\pi$
$569$	−15.3578	+	26.6005i	−0.643833	+	1.11515i	−0.984570	+	0.174993i	$0.944010\pi$
$570$		0			0
$571$	7.64218			0.319815			0.159908	−	0.987132i	$-0.448880\pi$
$571$	7.64218			0.319815			0.159908	+	0.987132i	$0.448880\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.821092	+	1.42217i	−0.0342419	+	0.0593087i
$576$		0			0
$577$		−	43.2404i		−	1.80012i	−0.435765	−	0.900060i	$-0.643522\pi$
$577$		−	43.2404i		−	1.80012i	0.435765	−	0.900060i	$-0.356478\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−24.3578	−	42.1890i	−1.01053	−	1.75029i
$582$		0			0
$583$	30.5367	+	17.6304i	1.26470	+	0.730176i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−19.1789	−	11.0729i	−0.791598	−	0.457029i	0.0489266	−	0.998802i	$-0.484420\pi$
$587$	−19.1789	−	11.0729i	−0.791598	−	0.457029i	−0.840525	+	0.541773i	$0.817753\pi$
$588$		0			0
$589$	13.1789	+	22.8265i	0.543027	+	0.940551i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			32.8481i			1.34891i	0.738316	+	0.674454i	$0.235621\pi$
$593$			32.8481i			1.34891i	−0.738316	+	0.674454i	$0.764379\pi$
$594$		0			0
$595$	−30.8945	+	53.5109i	−1.26655	+	2.19373i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9.64218			0.393969			0.196984	−	0.980407i	$-0.436885\pi$
$599$	9.64218			0.393969			0.196984	+	0.980407i	$0.436885\pi$
$600$		0			0
$601$	0.0894542	−	0.154939i	0.00364891	−	0.00632010i	−0.864195	−	0.503157i	$-0.832172\pi$
$601$	0.0894542	−	0.154939i	0.00364891	−	0.00632010i	0.867844	+	0.496837i	$0.165505\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.08945	−	1.20635i	0.0849484	−	0.0490450i
$606$		0			0
$607$	8.00000	+	13.8564i	0.324710	+	0.562414i	0.981454	−	0.191700i	$-0.0614000\pi$
$607$	8.00000	+	13.8564i	0.324710	+	0.562414i	−0.656744	+	0.754114i	$0.728067\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	34.0735	+	15.5885i	1.37846	+	0.630641i
$612$		0			0
$613$	−2.14218	−	1.23679i	−0.0865220	−	0.0499535i	0.456115	−	0.889921i	$-0.349241\pi$
$613$	−2.14218	−	1.23679i	−0.0865220	−	0.0499535i	−0.542637	+	0.839967i	$0.682574\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.08945	+	1.20635i	−0.0841183	+	0.0485657i	−0.541469	−	0.840721i	$-0.682132\pi$
$617$	−2.08945	+	1.20635i	−0.0841183	+	0.0485657i	0.457351	+	0.889286i	$0.348798\pi$
$618$		0			0
$619$		−	10.3314i		−	0.415255i	−0.978208	−	0.207627i	$-0.933426\pi$
$619$		−	10.3314i		−	0.415255i	0.978208	−	0.207627i	$-0.0665742\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−28.7156			−1.15047
$624$		0			0
$625$	−28.4313			−1.13725
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	6.49655i		−	0.259034i
$630$		0			0
$631$	−5.41055	+	3.12378i	−0.215390	+	0.124356i	−0.603814	−	0.797125i	$-0.706353\pi$
$631$	−5.41055	+	3.12378i	−0.215390	+	0.124356i	0.388424	+	0.921481i	$0.373020\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.89454	+	3.40322i	0.233918	+	0.135052i
$636$		0			0
$637$	−21.2684	−	29.9097i	−0.842683	−	1.18506i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.268363	−	0.464818i	−0.0105997	−	0.0183592i	0.860677	−	0.509152i	$-0.170041\pi$
$641$	−0.268363	−	0.464818i	−0.0105997	−	0.0183592i	−0.871277	+	0.490792i	$0.836707\pi$
$642$		0			0
$643$	13.7684	−	7.94917i	0.542971	−	0.313485i	−0.203311	−	0.979114i	$-0.565170\pi$
$643$	13.7684	−	7.94917i	0.542971	−	0.313485i	0.746282	+	0.665630i	$0.231837\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.82109	+	11.8145i	−0.268165	+	0.464475i	−0.968388	−	0.249449i	$-0.919750\pi$
$647$	−6.82109	+	11.8145i	−0.268165	+	0.464475i	0.700223	+	0.713924i	$0.253084\pi$
$648$		0			0
$649$	4.71563			0.185105
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.178908	−	0.309878i	0.00700122	−	0.0121265i	−0.862504	−	0.506051i	$-0.831105\pi$
$653$	0.178908	−	0.309878i	0.00700122	−	0.0121265i	0.869505	+	0.493925i	$0.164438\pi$
$654$		0			0
$655$		−	15.3395i		−	0.599363i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.35782	+	7.54796i	0.169756	+	0.294027i	0.938334	−	0.345730i	$-0.112369\pi$
$659$	4.35782	+	7.54796i	0.169756	+	0.294027i	−0.768578	+	0.639756i	$0.779035\pi$
$660$		0			0
$661$	11.1422	+	6.43294i	0.433381	+	0.250212i	0.700786	−	0.713372i	$-0.252833\pi$
$661$	11.1422	+	6.43294i	0.433381	+	0.250212i	−0.267405	+	0.963584i	$0.586166\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−30.0000	−	17.3205i	−1.16335	−	0.671660i
$666$		0			0
$667$	8.17891	+	14.1663i	0.316689	+	0.548521i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	17.3205i		−	0.668651i
$672$		0			0
$673$	2.67891	−	4.64001i	0.103264	−	0.178859i	−0.809763	−	0.586757i	$-0.800405\pi$
$673$	2.67891	−	4.64001i	0.103264	−	0.178859i	0.913028	+	0.407897i	$0.133738\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.64218			0.139980			0.0699902	−	0.997548i	$-0.477703\pi$
$677$	3.64218			0.139980			0.0699902	+	0.997548i	$0.477703\pi$
$678$		0			0
$679$	−4.23164	+	7.32941i	−0.162395	+	0.281277i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−15.5367	+	8.97013i	−0.594496	+	0.343233i	−0.766873	−	0.641798i	$-0.778189\pi$
$683$	−15.5367	+	8.97013i	−0.594496	+	0.343233i	0.172377	+	0.985031i	$0.444855\pi$
$684$		0			0
$685$	−15.4473	−	26.7555i	−0.590210	−	1.02227i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	15.2684	−	33.3738i	0.581678	−	1.27144i
$690$		0			0
$691$	7.12618	+	4.11430i	0.271093	+	0.156515i	0.629384	−	0.777094i	$-0.283307\pi$
$691$	7.12618	+	4.11430i	0.271093	+	0.156515i	−0.358291	+	0.933610i	$0.616641\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	35.8945	−	20.7237i	1.36156	−	0.786096i
$696$		0			0
$697$			36.3122i			1.37542i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−48.3578			−1.82645			−0.913225	−	0.407456i	$-0.866416\pi$
$701$	−48.3578			−1.82645			−0.913225	+	0.407456i	$0.866416\pi$
$702$		0			0
$703$	3.64218			0.137368
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	50.4785i		−	1.89844i
$708$		0			0
$709$	25.5000	−	14.7224i	0.957673	−	0.552913i	0.0622167	−	0.998063i	$-0.480183\pi$
$709$	25.5000	−	14.7224i	0.957673	−	0.552913i	0.895456	+	0.445150i	$0.146850\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13.1789	+	7.60885i	0.493554	+	0.284954i
$714$		0			0
$715$	−17.4633	−	24.5586i	−0.653089	−	0.918439i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	15.1789	+	26.2906i	0.566078	+	0.980475i	0.996949	+	0.0780618i	$0.0248731\pi$
$719$	15.1789	+	26.2906i	0.566078	+	0.980475i	−0.430871	+	0.902414i	$0.641794\pi$
$720$		0			0
$721$	−11.4105	+	6.58788i	−0.424951	+	0.245346i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.35782	−	5.81591i	0.124706	−	0.215997i
$726$		0			0
$727$	−17.5367			−0.650401			−0.325201	−	0.945645i	$-0.605432\pi$
$727$	−17.5367			−0.650401			−0.325201	+	0.945645i	$0.605432\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.53673	+	4.39374i	−0.0938242	+	0.162508i
$732$		0			0
$733$			10.6413i			0.393045i	0.980499	+	0.196523i	$0.0629649\pi$
$733$			10.6413i			0.393045i	−0.980499	+	0.196523i	$0.937035\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7.17891	−	12.4342i	−0.264438	−	0.458021i
$738$		0			0
$739$	13.1789	+	7.60885i	0.484794	+	0.279896i	0.722412	−	0.691463i	$-0.243033\pi$
$739$	13.1789	+	7.60885i	0.484794	+	0.279896i	−0.237618	+	0.971359i	$0.576367\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.17891	−	0.680643i	−0.0432500	−	0.0249704i	0.478219	−	0.878241i	$-0.341282\pi$
$743$	−1.17891	−	0.680643i	−0.0432500	−	0.0249704i	−0.521469	+	0.853270i	$0.674616\pi$
$744$		0			0
$745$	7.08945	+	12.2793i	0.259738	+	0.449879i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	41.4474i		−	1.51446i
$750$		0			0
$751$	−9.82109	+	17.0106i	−0.358377	+	0.620727i	−0.987690	−	0.156425i	$-0.950003\pi$
$751$	−9.82109	+	17.0106i	−0.358377	+	0.620727i	0.629313	+	0.777152i	$0.283336\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	48.3578			1.75992
$756$		0			0
$757$	19.3578	−	33.5287i	0.703572	−	1.21862i	−0.263633	−	0.964623i	$-0.584921\pi$
$757$	19.3578	−	33.5287i	0.703572	−	1.21862i	0.967204	−	0.253999i	$-0.0817460\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	42.0000	−	24.2487i	1.52250	−	0.879015i	0.522852	−	0.852423i	$-0.324868\pi$
$761$	42.0000	−	24.2487i	1.52250	−	0.879015i	0.999646	−	0.0265919i	$-0.00846546\pi$
$762$		0			0
$763$	−15.7684	−	27.3116i	−0.570853	−	0.988746i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.463275	−	4.88627i	−0.0167279	−	0.176433i
$768$		0			0
$769$	−6.00000	−	3.46410i	−0.216366	−	0.124919i	0.387901	−	0.921701i	$-0.373200\pi$
$769$	−6.00000	−	3.46410i	−0.216366	−	0.124919i	−0.604266	+	0.796782i	$0.706534\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	27.5367	−	15.8983i	0.990427	−	0.571823i	0.0850252	−	0.996379i	$-0.472903\pi$
$773$	27.5367	−	15.8983i	0.990427	−	0.571823i	0.905402	+	0.424555i	$0.139570\pi$
$774$		0			0
$775$		−	6.24756i		−	0.224419i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−20.3578			−0.729394
$780$		0			0
$781$	−12.0000			−0.429394
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	16.8889i		−	0.602789i
$786$		0			0
$787$	29.4105	−	16.9802i	1.04837	−	0.605278i	0.126179	−	0.992007i	$-0.459729\pi$
$787$	29.4105	−	16.9802i	1.04837	−	0.605278i	0.922193	+	0.386729i	$0.126395\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	43.7156	+	25.2392i	1.55435	+	0.897404i
$792$		0			0
$793$	−17.9473	+	1.70161i	−0.637326	+	0.0604259i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8.53673	+	14.7860i	0.302386	+	0.523748i	0.976676	−	0.214718i	$-0.0688834\pi$
$797$	8.53673	+	14.7860i	0.302386	+	0.523748i	−0.674290	+	0.738467i	$0.735550\pi$
$798$		0			0
$799$	55.6102	−	32.1065i	1.96735	−	1.13585i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−19.7156	+	34.1485i	−0.695750	+	1.20507i
$804$		0			0
$805$	−20.0000			−0.704907
$806$		0			0
$807$		0			0
$808$		0			0
$809$	23.2684	−	40.3020i	0.818072	−	1.41694i	−0.0890288	−	0.996029i	$-0.528376\pi$
$809$	23.2684	−	40.3020i	0.818072	−	1.41694i	0.907101	−	0.420913i	$-0.138290\pi$
$810$		0			0
$811$			0.680643i			0.0239006i	0.999929	+	0.0119503i	$0.00380399\pi$
$811$			0.680643i			0.0239006i	−0.999929	+	0.0119503i	$0.996196\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.17891	−	15.8983i	−0.321523	−	0.556894i
$816$		0			0
$817$	−2.46327	−	1.42217i	−0.0861791	−	0.0497555i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−27.5367	−	15.8983i	−0.961038	−	0.554856i	−0.0645459	−	0.997915i	$-0.520560\pi$
$821$	−27.5367	−	15.8983i	−0.961038	−	0.554856i	−0.896492	+	0.443059i	$0.853893\pi$
$822$		0			0
$823$	−12.0000	−	20.7846i	−0.418294	−	0.724506i	0.577474	−	0.816409i	$-0.304038\pi$
$823$	−12.0000	−	20.7846i	−0.418294	−	0.724506i	−0.995768	+	0.0919029i	$0.970705\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	35.8805i		−	1.24769i	−0.781549	−	0.623844i	$-0.785570\pi$
$827$		−	35.8805i		−	1.24769i	0.781549	−	0.623844i	$-0.214430\pi$
$828$		0			0
$829$	−11.6789	+	20.2285i	−0.405625	+	0.702564i	−0.994394	−	0.105738i	$-0.966280\pi$
$829$	−11.6789	+	20.2285i	−0.405625	+	0.702564i	0.588769	+	0.808302i	$0.299613\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−62.8945			−2.17917
$834$		0			0
$835$	8.35782	−	14.4762i	0.289234	−	0.500968i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−13.1789	+	7.60885i	−0.454986	+	0.262687i	−0.709934	−	0.704268i	$-0.751275\pi$
$839$	−13.1789	+	7.60885i	−0.454986	+	0.262687i	0.254947	+	0.966955i	$0.417942\pi$
$840$		0			0
$841$	−18.9473	−	32.8176i	−0.653354	−	1.13164i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−23.7316	+	20.5079i	−0.816393	+	0.705493i
$846$		0			0
$847$	3.58945	+	2.07237i	0.123335	+	0.0712076i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.82109	−	1.05141i	0.0624262	−	0.0360418i
$852$		0			0
$853$			24.6195i			0.842955i	0.906839	+	0.421477i	$0.138488\pi$
$853$			24.6195i			0.842955i	−0.906839	+	0.421477i	$0.861512\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	20.1789			0.689298			0.344649	−	0.938732i	$-0.387998\pi$
$857$	20.1789			0.689298			0.344649	+	0.938732i	$0.387998\pi$
$858$		0			0
$859$	−54.2524			−1.85107			−0.925533	−	0.378666i	$-0.876383\pi$
$859$	−54.2524			−1.85107			−0.925533	+	0.378666i	$0.876383\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	20.0431i		−	0.682274i	−0.940014	−	0.341137i	$-0.889188\pi$
$863$		−	20.0431i		−	0.682274i	0.940014	−	0.341137i	$-0.110812\pi$
$864$		0			0
$865$	37.6102	−	21.7142i	1.27878	−	0.738306i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−39.5367	−	22.8265i	−1.34119	−	0.774337i
$870$		0			0
$871$	−12.1789	+	8.66025i	−0.412667	+	0.293442i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−20.8945	−	36.1904i	−0.706364	−	1.22346i
$876$		0			0
$877$	11.0895	−	6.40250i	0.374464	−	0.216197i	−0.300943	−	0.953642i	$-0.597301\pi$
$877$	11.0895	−	6.40250i	0.374464	−	0.216197i	0.675407	+	0.737445i	$0.263968\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.9105	+	32.7540i	−0.637112	+	1.10351i	0.348951	+	0.937141i	$0.386538\pi$
$881$	−18.9105	+	32.7540i	−0.637112	+	1.10351i	−0.986063	+	0.166370i	$0.946795\pi$
$882$		0			0
$883$	17.8945			0.602199			0.301100	−	0.953593i	$-0.402646\pi$
$883$	17.8945			0.602199			0.301100	+	0.953593i	$0.402646\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.8211	+	18.7427i	−0.363337	+	0.629318i	−0.988508	−	0.151170i	$-0.951696\pi$
$887$	−10.8211	+	18.7427i	−0.363337	+	0.629318i	0.625171	+	0.780488i	$0.285029\pi$
$888$		0			0
$889$			11.6927i			0.392161i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	18.0000	+	31.1769i	0.602347	+	1.04330i
$894$		0			0
$895$	−17.4633	−	10.0824i	−0.583733	−	0.337018i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−53.8945	−	31.1160i	−1.79748	−	1.03778i
$900$		0			0
$901$	−31.4473	−	54.4683i	−1.04766	−	1.81460i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			49.5488i			1.64706i
$906$		0			0
$907$	14.0000	−	24.2487i	0.464862	−	0.805165i	−0.534333	−	0.845274i	$-0.679437\pi$
$907$	14.0000	−	24.2487i	0.464862	−	0.805165i	0.999195	+	0.0401089i	$0.0127705\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.71563			0.156236			0.0781180	−	0.996944i	$-0.475109\pi$
$911$	4.71563			0.156236			0.0781180	+	0.996944i	$0.475109\pi$
$912$		0			0
$913$	20.3578	−	35.2608i	0.673745	−	1.16696i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	22.8211	−	13.1758i	0.753619	−	0.435102i
$918$		0			0
$919$	12.0000	+	20.7846i	0.395843	+	0.685621i	0.993208	−	0.116348i	$-0.0371189\pi$
$919$	12.0000	+	20.7846i	0.395843	+	0.685621i	−0.597365	+	0.801970i	$0.703786\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.17891	+	12.4342i	0.0388042	+	0.409278i
$924$		0			0
$925$	−0.747642	−	0.431651i	−0.0245823	−	0.0141926i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	27.2684	−	15.7434i	0.894646	−	0.516524i	0.0191866	−	0.999816i	$-0.493892\pi$
$929$	27.2684	−	15.7434i	0.894646	−	0.516524i	0.875459	+	0.483292i	$0.160559\pi$
$930$		0			0
$931$		−	35.2608i		−	1.15563i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−51.6422			−1.68888
$936$		0			0
$937$	48.8945			1.59732			0.798658	−	0.601786i	$-0.205544\pi$
$937$	48.8945			1.59732			0.798658	+	0.601786i	$0.205544\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			37.2418i			1.21405i	0.794683	+	0.607024i	$0.207637\pi$
$941$			37.2418i			1.21405i	−0.794683	+	0.607024i	$0.792363\pi$
$942$		0			0
$943$	−10.1789	+	5.87680i	−0.331471	+	0.191375i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−26.3578	−	15.2177i	−0.856514	−	0.494509i	0.00632940	−	0.999980i	$-0.497985\pi$
$947$	−26.3578	−	15.2177i	−0.856514	−	0.494509i	−0.862843	+	0.505471i	$0.831319\pi$
$948$		0			0
$949$	37.3211	+	17.0742i	1.21149	+	0.554253i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	3.00000	+	5.19615i	0.0971795	+	0.168320i	0.910516	−	0.413473i	$-0.135685\pi$
$953$	3.00000	+	5.19615i	0.0971795	+	0.168320i	−0.813337	+	0.581793i	$0.802351\pi$
$954$		0			0
$955$	4.92655	−	2.84434i	0.159419	−	0.0920408i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	26.5367	−	45.9630i	0.856916	−	1.48422i
$960$		0			0
$961$	−26.8945			−0.867566
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16.2684	+	28.1776i	−0.523697	+	0.907070i
$966$		0			0
$967$			35.3825i			1.13783i	0.822398	+	0.568913i	$0.192636\pi$
$967$			35.3825i			1.13783i	−0.822398	+	0.568913i	$0.807364\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2.35782	−	4.08386i	−0.0756659	−	0.131057i	0.825710	−	0.564095i	$-0.190775\pi$
$971$	−2.35782	−	4.08386i	−0.0756659	−	0.131057i	−0.901376	+	0.433038i	$0.857442\pi$
$972$		0			0
$973$	61.6629	+	35.6011i	1.97682	+	1.14132i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−14.7316	−	8.50531i	−0.471307	−	0.272109i	0.245480	−	0.969402i	$-0.421055\pi$
$977$	−14.7316	−	8.50531i	−0.471307	−	0.272109i	−0.716787	+	0.697293i	$0.754388\pi$
$978$		0			0
$979$	−12.0000	−	20.7846i	−0.383522	−	0.664279i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	24.8685i		−	0.793181i	−0.917996	−	0.396590i	$-0.870193\pi$
$983$		−	24.8685i		−	0.793181i	0.917996	−	0.396590i	$-0.129807\pi$
$984$		0			0
$985$	11.6422	−	20.1649i	0.370951	−	0.642506i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.64218			−0.0522184
$990$		0			0
$991$	−22.5367	+	39.0348i	−0.715903	+	1.23998i	0.246708	+	0.969090i	$0.420651\pi$
$991$	−22.5367	+	39.0348i	−0.715903	+	1.23998i	−0.962610	+	0.270890i	$0.912682\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−45.0000	+	25.9808i	−1.42660	+	0.823646i
$996$		0			0
$997$	−17.6789	−	30.6208i	−0.559897	−	0.969769i	−0.997504	−	0.0706035i	$-0.977507\pi$
$997$	−17.6789	−	30.6208i	−0.559897	−	0.969769i	0.437608	−	0.899166i	$-0.355826\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.2.by.j.433.1		4
3.2	odd	2		624.2.bv.f.433.2		4
4.3	odd	2		468.2.t.d.433.1		4
12.11	even	2		156.2.q.b.121.2	yes	4
13.10	even	6	inner	1872.2.by.j.1297.2		4
39.20	even	12		8112.2.a.cr.1.2		4
39.23	odd	6		624.2.bv.f.49.1		4
39.32	even	12		8112.2.a.cr.1.3		4
52.7	even	12		6084.2.a.bd.1.3		4
52.19	even	12		6084.2.a.bd.1.2		4
52.23	odd	6		468.2.t.d.361.2		4
52.35	odd	6		6084.2.b.o.4393.2		4
52.43	odd	6		6084.2.b.o.4393.3		4
60.23	odd	4		3900.2.bw.j.2149.2		8
60.47	odd	4		3900.2.bw.j.2149.3		8
60.59	even	2		3900.2.cd.i.901.2		4
156.11	odd	12		2028.2.i.n.2005.2		8
156.23	even	6		156.2.q.b.49.1	✓	4
156.35	even	6		2028.2.b.e.337.3		4
156.47	odd	4		2028.2.i.n.529.2		8
156.59	odd	12		2028.2.a.m.1.2		4
156.71	odd	12		2028.2.a.m.1.3		4
156.83	odd	4		2028.2.i.n.529.3		8
156.95	even	6		2028.2.b.e.337.2		4
156.107	even	6		2028.2.q.f.361.2		4
156.119	odd	12		2028.2.i.n.2005.3		8
156.155	even	2		2028.2.q.f.1837.1		4
780.23	odd	12		3900.2.bw.j.49.3		8
780.179	even	6		3900.2.cd.i.2701.2		4
780.647	odd	12		3900.2.bw.j.49.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.q.b.49.1	✓	4	156.23	even	6
156.2.q.b.121.2	yes	4	12.11	even	2
468.2.t.d.361.2		4	52.23	odd	6
468.2.t.d.433.1		4	4.3	odd	2
624.2.bv.f.49.1		4	39.23	odd	6
624.2.bv.f.433.2		4	3.2	odd	2
1872.2.by.j.433.1		4	1.1	even	1	trivial
1872.2.by.j.1297.2		4	13.10	even	6	inner
2028.2.a.m.1.2		4	156.59	odd	12
2028.2.a.m.1.3		4	156.71	odd	12
2028.2.b.e.337.2		4	156.95	even	6
2028.2.b.e.337.3		4	156.35	even	6
2028.2.i.n.529.2		8	156.47	odd	4
2028.2.i.n.529.3		8	156.83	odd	4
2028.2.i.n.2005.2		8	156.11	odd	12
2028.2.i.n.2005.3		8	156.119	odd	12
2028.2.q.f.361.2		4	156.107	even	6
2028.2.q.f.1837.1		4	156.155	even	2
3900.2.bw.j.49.2		8	780.647	odd	12
3900.2.bw.j.49.3		8	780.23	odd	12
3900.2.bw.j.2149.2		8	60.23	odd	4
3900.2.bw.j.2149.3		8	60.47	odd	4
3900.2.cd.i.901.2		4	60.59	even	2
3900.2.cd.i.2701.2		4	780.179	even	6
6084.2.a.bd.1.2		4	52.19	even	12
6084.2.a.bd.1.3		4	52.7	even	12
6084.2.b.o.4393.2		4	52.35	odd	6
6084.2.b.o.4393.3		4	52.43	odd	6
8112.2.a.cr.1.2		4	39.20	even	12
8112.2.a.cr.1.3		4	39.32	even	12

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1872.by (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-2.41269i q^{5} +(3.58945 - 2.07237i) q^{7} +(3.00000 + 1.73205i) q^{11} +(1.50000 - 3.27872i) q^{13} +(-3.08945 - 5.35109i) q^{17} +(3.00000 - 1.73205i) q^{19} +(1.00000 - 1.73205i) q^{23} -0.821092 q^{25} +(-4.08945 + 7.08314i) q^{29} +7.60885i q^{31} +(-5.00000 - 8.66025i) q^{35} +(0.910546 + 0.525704i) q^{37} +(-5.08945 - 2.93840i) q^{41} +(-0.410546 - 0.711086i) q^{43} +10.3923i q^{47} +(5.08945 - 8.81519i) q^{49} +10.1789 q^{53} +(4.17891 - 7.23808i) q^{55} +(1.17891 - 0.680643i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(-7.91055 - 3.61904i) q^{65} +(-3.58945 - 2.07237i) q^{67} +(-3.00000 + 1.73205i) q^{71} +11.3828i q^{73} +14.3578 q^{77} -13.1789 q^{79} -11.7536i q^{83} +(-12.9105 + 7.45391i) q^{85} +(-6.00000 - 3.46410i) q^{89} +(-1.41055 - 14.8774i) q^{91} +(-4.17891 - 7.23808i) q^{95} +(-1.76836 + 1.02096i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{7} + 12 q^{11} + 6 q^{13} - q^{17} + 12 q^{19} + 4 q^{23} - 26 q^{25} - 5 q^{29} - 20 q^{35} + 15 q^{37} - 9 q^{41} - 13 q^{43} + 9 q^{49} + 18 q^{53} - 6 q^{55} - 18 q^{59} - 10 q^{61} - 43 q^{65}+ \cdots + 27 q^{97}+O(q^{100}) \) 4 * q + 3 * q^7 + 12 * q^11 + 6 * q^13 - q^17 + 12 * q^19 + 4 * q^23 - 26 * q^25 - 5 * q^29 - 20 * q^35 + 15 * q^37 - 9 * q^41 - 13 * q^43 + 9 * q^49 + 18 * q^53 - 6 * q^55 - 18 * q^59 - 10 * q^61 - 43 * q^65 - 3 * q^67 - 12 * q^71 + 12 * q^77 - 30 * q^79 - 63 * q^85 - 24 * q^89 - 17 * q^91 + 6 * q^95 + 27 * q^97

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