LMFDB - Embedded newform 2160.2.by.b.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(289,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4, 3]))

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

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

chi := DirichletCharacter("2160.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.by (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			289.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2160.289
Dual form			2160.2.by.b.1009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.23205 + 0.133975i) q^{5} +(0.866025 + 0.500000i) q^{7} +O(q^{10})$	$q+(-2.23205 + 0.133975i) q^{5} +(0.866025 + 0.500000i) q^{7} +(1.00000 - 1.73205i) q^{11} +(-5.19615 + 3.00000i) q^{13} -2.00000i q^{17} +6.00000 q^{19} +(0.866025 - 0.500000i) q^{23} +(4.96410 - 0.598076i) q^{25} +(-4.50000 + 7.79423i) q^{29} +(-1.00000 - 1.73205i) q^{31} +(-2.00000 - 1.00000i) q^{35} -2.00000i q^{37} +(-5.50000 - 9.52628i) q^{41} +(-3.46410 - 2.00000i) q^{43} +(-6.06218 - 3.50000i) q^{47} +(-3.00000 - 5.19615i) q^{49} +(-2.00000 + 4.00000i) q^{55} +(-2.00000 - 3.46410i) q^{59} +(3.50000 - 6.06218i) q^{61} +(11.1962 - 7.39230i) q^{65} +(-9.52628 + 5.50000i) q^{67} -6.00000 q^{71} -4.00000i q^{73} +(1.73205 - 1.00000i) q^{77} +(6.00000 - 10.3923i) q^{79} +(9.52628 + 5.50000i) q^{83} +(0.267949 + 4.46410i) q^{85} +1.00000 q^{89} -6.00000 q^{91} +(-13.3923 + 0.803848i) q^{95} +(-6.92820 - 4.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5}+O(q^{10}) $ 4 * q - 2 * q^5	$ 4 q - 2 q^{5} + 4 q^{11} + 24 q^{19} + 6 q^{25} - 18 q^{29} - 4 q^{31} - 8 q^{35} - 22 q^{41} - 12 q^{49} - 8 q^{55} - 8 q^{59} + 14 q^{61} + 24 q^{65} - 24 q^{71} + 24 q^{79} + 8 q^{85} + 4 q^{89} - 24 q^{91} - 12 q^{95}+O(q^{100}) $ 4 * q - 2 * q^5 + 4 * q^11 + 24 * q^19 + 6 * q^25 - 18 * q^29 - 4 * q^31 - 8 * q^35 - 22 * q^41 - 12 * q^49 - 8 * q^55 - 8 * q^59 + 14 * q^61 + 24 * q^65 - 24 * q^71 + 24 * q^79 + 8 * q^85 + 4 * q^89 - 24 * q^91 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.23205	+	0.133975i	−0.998203	+	0.0599153i
$5$	−2.23205	+	0.133975i	−0.998203	+	0.0599153i
$6$		0			0
$7$	0.866025	+	0.500000i	0.327327	+	0.188982i	0.654654	−	0.755929i	$-0.272814\pi$
$7$	0.866025	+	0.500000i	0.327327	+	0.188982i	−0.327327	+	0.944911i	$0.606148\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	−0.674967	−	0.737848i	$-0.735842\pi$
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	0.976478	+	0.215615i	$0.0691756\pi$
$12$		0			0
$13$	−5.19615	+	3.00000i	−1.44115	+	0.832050i	−0.997927	−	0.0643593i	$-0.979500\pi$
$13$	−5.19615	+	3.00000i	−1.44115	+	0.832050i	−0.443227	+	0.896410i	$0.646166\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.866025	−	0.500000i	0.180579	−	0.104257i	−0.406986	−	0.913434i	$-0.633420\pi$
$23$	0.866025	−	0.500000i	0.180579	−	0.104257i	0.587565	+	0.809177i	$0.300087\pi$
$24$		0			0
$25$	4.96410	−	0.598076i	0.992820	−	0.119615i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.50000	+	7.79423i	−0.835629	+	1.44735i	0.0578882	+	0.998323i	$0.481563\pi$
$29$	−4.50000	+	7.79423i	−0.835629	+	1.44735i	−0.893517	+	0.449029i	$0.851770\pi$
$30$		0			0
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	0.762140	−	0.647412i	$-0.224149\pi$
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	−0.941745	+	0.336327i	$0.890815\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.00000	−	1.00000i	−0.338062	−	0.169031i
$36$		0			0
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.50000	−	9.52628i	−0.858956	−	1.48775i	−0.872926	−	0.487852i	$-0.837780\pi$
$41$	−5.50000	−	9.52628i	−0.858956	−	1.48775i	0.0139704	−	0.999902i	$-0.495553\pi$
$42$		0			0
$43$	−3.46410	−	2.00000i	−0.528271	−	0.304997i	0.212041	−	0.977261i	$-0.431989\pi$
$43$	−3.46410	−	2.00000i	−0.528271	−	0.304997i	−0.740312	+	0.672264i	$0.765322\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.06218	−	3.50000i	−0.884260	−	0.510527i	−0.0121990	−	0.999926i	$-0.503883\pi$
$47$	−6.06218	−	3.50000i	−0.884260	−	0.510527i	−0.872060	+	0.489398i	$0.837217\pi$
$48$		0			0
$49$	−3.00000	−	5.19615i	−0.428571	−	0.742307i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$	−2.00000	+	4.00000i	−0.269680	+	0.539360i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	0.705965	−	0.708247i	$-0.250514\pi$
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	−0.966342	+	0.257260i	$0.917180\pi$
$60$		0			0
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	$-0.685424\pi$
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	0.998264	+	0.0588933i	$0.0187572\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	11.1962	−	7.39230i	1.38871	−	0.916903i
$66$		0			0
$67$	−9.52628	+	5.50000i	−1.16382	+	0.671932i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	−9.52628	+	5.50000i	−1.16382	+	0.671932i	−0.211604	+	0.977356i	$0.567869\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.73205	−	1.00000i	0.197386	−	0.113961i
$78$		0			0
$79$	6.00000	−	10.3923i	0.675053	−	1.16923i	−0.301401	−	0.953498i	$-0.597454\pi$
$79$	6.00000	−	10.3923i	0.675053	−	1.16923i	0.976453	−	0.215728i	$-0.0692125\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	9.52628	+	5.50000i	1.04565	+	0.603703i	0.921427	−	0.388552i	$-0.127024\pi$
$83$	9.52628	+	5.50000i	1.04565	+	0.603703i	0.124218	+	0.992255i	$0.460358\pi$
$84$		0			0
$85$	0.267949	+	4.46410i	0.0290632	+	0.484200i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.00000			0.106000			0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000			0.106000			0.0529999	+	0.998595i	$0.483122\pi$
$90$		0			0
$91$	−6.00000			−0.628971
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−13.3923	+	0.803848i	−1.37402	+	0.0824730i
$96$		0			0
$97$	−6.92820	−	4.00000i	−0.703452	−	0.406138i	0.105180	−	0.994453i	$-0.466458\pi$
$97$	−6.92820	−	4.00000i	−0.703452	−	0.406138i	−0.808632	+	0.588315i	$0.799792\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	−0.811976	−	0.583691i	$-0.801608\pi$
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	0.911479	+	0.411346i	$0.134941\pi$
$102$		0			0
$103$	6.92820	−	4.00000i	0.682656	−	0.394132i	−0.118199	−	0.992990i	$-0.537712\pi$
$103$	6.92820	−	4.00000i	0.682656	−	0.394132i	0.800855	+	0.598858i	$0.204379\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	3.00000i		−	0.290021i	−0.989430	−	0.145010i	$-0.953678\pi$
$107$		−	3.00000i		−	0.290021i	0.989430	−	0.145010i	$-0.0463216\pi$
$108$		0			0
$109$	−7.00000			−0.670478			−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000			−0.670478			−0.335239	+	0.942133i	$0.608817\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.3923	+	6.00000i	−0.977626	+	0.564433i	−0.901553	−	0.432670i	$-0.857572\pi$
$113$	−10.3923	+	6.00000i	−0.977626	+	0.564433i	−0.0760733	+	0.997102i	$0.524238\pi$
$114$		0			0
$115$	−1.86603	+	1.23205i	−0.174008	+	0.114889i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.00000	−	1.73205i	0.0916698	−	0.158777i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$		−	19.0000i		−	1.68598i	−0.537931	−	0.842989i	$-0.680794\pi$
$127$		−	19.0000i		−	1.68598i	0.537931	−	0.842989i	$-0.319206\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	−0.999602	−	0.0281993i	$-0.991023\pi$
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	0.475380	−	0.879781i	$-0.342311\pi$
$132$		0			0
$133$	5.19615	+	3.00000i	0.450564	+	0.260133i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.3923	−	6.00000i	−0.887875	−	0.512615i	−0.0146279	−	0.999893i	$-0.504656\pi$
$137$	−10.3923	−	6.00000i	−0.887875	−	0.512615i	−0.873247	+	0.487278i	$0.837990\pi$
$138$		0			0
$139$	−8.00000	−	13.8564i	−0.678551	−	1.17529i	−0.975417	−	0.220366i	$-0.929275\pi$
$139$	−8.00000	−	13.8564i	−0.678551	−	1.17529i	0.296866	−	0.954919i	$-0.404058\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			12.0000i			1.00349i
$144$		0			0
$145$	9.00000	−	18.0000i	0.747409	−	1.49482i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.500000	+	0.866025i	0.0409616	+	0.0709476i	0.885779	−	0.464107i	$-0.153625\pi$
$149$	0.500000	+	0.866025i	0.0409616	+	0.0709476i	−0.844818	+	0.535054i	$0.820291\pi$
$150$		0			0
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	−0.587646	−	0.809118i	$-0.699945\pi$
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	0.994540	+	0.104357i	$0.0332784\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.46410	+	3.73205i	0.197921	+	0.299766i
$156$		0			0
$157$	−3.46410	+	2.00000i	−0.276465	+	0.159617i	−0.631822	−	0.775113i	$-0.717693\pi$
$157$	−3.46410	+	2.00000i	−0.276465	+	0.159617i	0.355357	+	0.934731i	$0.384359\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.00000			0.0788110
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.59808	−	1.50000i	0.201045	−	0.116073i	−0.396098	−	0.918208i	$-0.629636\pi$
$167$	2.59808	−	1.50000i	0.201045	−	0.116073i	0.597143	+	0.802135i	$0.296303\pi$
$168$		0			0
$169$	11.5000	−	19.9186i	0.884615	−	1.53220i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.46410	−	2.00000i	−0.263371	−	0.152057i	0.362500	−	0.931984i	$-0.381923\pi$
$173$	−3.46410	−	2.00000i	−0.263371	−	0.152057i	−0.625871	+	0.779926i	$0.715256\pi$
$174$		0			0
$175$	4.59808	+	1.96410i	0.347582	+	0.148472i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2.00000			−0.149487			−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000			−0.149487			−0.0747435	+	0.997203i	$0.523814\pi$
$180$		0			0
$181$	−13.0000			−0.966282			−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000			−0.966282			−0.483141	+	0.875542i	$0.660504\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.267949	+	4.46410i	0.0197000	+	0.328207i
$186$		0			0
$187$	−3.46410	−	2.00000i	−0.253320	−	0.146254i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.00000	+	5.19615i	−0.217072	+	0.375980i	−0.953912	−	0.300088i	$-0.902984\pi$
$191$	−3.00000	+	5.19615i	−0.217072	+	0.375980i	0.736839	+	0.676068i	$0.236317\pi$
$192$		0			0
$193$	−8.66025	+	5.00000i	−0.623379	+	0.359908i	−0.778183	−	0.628037i	$-0.783859\pi$
$193$	−8.66025	+	5.00000i	−0.623379	+	0.359908i	0.154805	+	0.987945i	$0.450525\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.00000i			0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$			8.00000i			0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$		0			0
$199$	−18.0000			−1.27599			−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000			−1.27599			−0.637993	+	0.770042i	$0.720235\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.79423	+	4.50000i	−0.547048	+	0.315838i
$204$		0			0
$205$	13.5526	+	20.5263i	0.946552	+	1.43362i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.00000	−	10.3923i	0.415029	−	0.718851i
$210$		0			0
$211$	9.00000	+	15.5885i	0.619586	+	1.07315i	0.989561	+	0.144112i	$0.0460326\pi$
$211$	9.00000	+	15.5885i	0.619586	+	1.07315i	−0.369976	+	0.929041i	$0.620634\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.00000	+	4.00000i	0.545595	+	0.272798i
$216$		0			0
$217$		−	2.00000i		−	0.135769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.00000	+	10.3923i	0.403604	+	0.699062i
$222$		0			0
$223$	19.9186	+	11.5000i	1.33385	+	0.770097i	0.985887	−	0.167412i	$-0.0535411\pi$
$223$	19.9186	+	11.5000i	1.33385	+	0.770097i	0.347960	+	0.937509i	$0.386874\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.92820	−	4.00000i	−0.459841	−	0.265489i	0.252136	−	0.967692i	$-0.418867\pi$
$227$	−6.92820	−	4.00000i	−0.459841	−	0.265489i	−0.711977	+	0.702202i	$0.752200\pi$
$228$		0			0
$229$	−3.50000	−	6.06218i	−0.231287	−	0.400600i	0.726900	−	0.686743i	$-0.240960\pi$
$229$	−3.50000	−	6.06218i	−0.231287	−	0.400600i	−0.958187	+	0.286143i	$0.907627\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			10.0000i			0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$			10.0000i			0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$		0			0
$235$	14.0000	+	7.00000i	0.913259	+	0.456630i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	14.0000	+	24.2487i	0.905585	+	1.56852i	0.820130	+	0.572177i	$0.193901\pi$
$239$	14.0000	+	24.2487i	0.905585	+	1.56852i	0.0854543	+	0.996342i	$0.472766\pi$
$240$		0			0
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	−0.881680	−	0.471848i	$-0.843587\pi$
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	0.849472	+	0.527633i	$0.176921\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.39230	+	11.1962i	0.472277	+	0.715296i
$246$		0			0
$247$	−31.1769	+	18.0000i	−1.98374	+	1.14531i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	18.0000			1.13615			0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000			1.13615			0.568075	+	0.822977i	$0.307688\pi$
$252$		0			0
$253$		−	2.00000i		−	0.125739i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.3923	−	6.00000i	0.648254	−	0.374270i	−0.139533	−	0.990217i	$-0.544560\pi$
$257$	10.3923	−	6.00000i	0.648254	−	0.374270i	0.787787	+	0.615948i	$0.211227\pi$
$258$		0			0
$259$	1.00000	−	1.73205i	0.0621370	−	0.107624i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−13.8564	−	8.00000i	−0.854423	−	0.493301i	0.00771799	−	0.999970i	$-0.497543\pi$
$263$	−13.8564	−	8.00000i	−0.854423	−	0.493301i	−0.862141	+	0.506669i	$0.830877\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.00000			0.182913			0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000			0.182913			0.0914566	+	0.995809i	$0.470848\pi$
$270$		0			0
$271$	14.0000			0.850439			0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000			0.850439			0.425220	+	0.905090i	$0.360197\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.92820	−	9.19615i	0.236880	−	0.554549i
$276$		0			0
$277$	−19.0526	−	11.0000i	−1.14476	−	0.660926i	−0.197153	−	0.980373i	$-0.563170\pi$
$277$	−19.0526	−	11.0000i	−1.14476	−	0.660926i	−0.947604	+	0.319447i	$0.896503\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.50000	+	2.59808i	−0.0894825	+	0.154988i	−0.907293	−	0.420500i	$-0.861855\pi$
$281$	−1.50000	+	2.59808i	−0.0894825	+	0.154988i	0.817810	+	0.575488i	$0.195188\pi$
$282$		0			0
$283$	−0.866025	+	0.500000i	−0.0514799	+	0.0297219i	−0.525519	−	0.850782i	$-0.676129\pi$
$283$	−0.866025	+	0.500000i	−0.0514799	+	0.0297219i	0.474039	+	0.880504i	$0.342796\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	11.0000i		−	0.649309i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	15.5885	−	9.00000i	0.910687	−	0.525786i	0.0300351	−	0.999549i	$-0.490438\pi$
$293$	15.5885	−	9.00000i	0.910687	−	0.525786i	0.880652	+	0.473763i	$0.157105\pi$
$294$		0			0
$295$	4.92820	+	7.46410i	0.286931	+	0.434577i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.00000	+	5.19615i	−0.173494	+	0.300501i
$300$		0			0
$301$	−2.00000	−	3.46410i	−0.115278	−	0.199667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.00000	+	14.0000i	−0.400819	+	0.801638i
$306$		0			0
$307$		−	9.00000i		−	0.513657i	−0.966457	−	0.256829i	$-0.917322\pi$
$307$		−	9.00000i		−	0.513657i	0.966457	−	0.256829i	$-0.0826776\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.00000	−	5.19615i	−0.170114	−	0.294647i	0.768345	−	0.640036i	$-0.221080\pi$
$311$	−3.00000	−	5.19615i	−0.170114	−	0.294647i	−0.938460	+	0.345389i	$0.887747\pi$
$312$		0			0
$313$	19.0526	+	11.0000i	1.07691	+	0.621757i	0.930062	−	0.367402i	$-0.119753\pi$
$313$	19.0526	+	11.0000i	1.07691	+	0.621757i	0.146852	+	0.989158i	$0.453086\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.73205	+	1.00000i	0.0972817	+	0.0561656i	0.547852	−	0.836576i	$-0.315446\pi$
$317$	1.73205	+	1.00000i	0.0972817	+	0.0561656i	−0.450570	+	0.892741i	$0.648779\pi$
$318$		0			0
$319$	9.00000	+	15.5885i	0.503903	+	0.872786i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$	−24.0000	+	18.0000i	−1.33128	+	0.998460i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.50000	−	6.06218i	−0.192961	−	0.334219i
$330$		0			0
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	−0.954765	−	0.297361i	$-0.903893\pi$
$331$	−4.00000	+	6.92820i	−0.219860	+	0.380808i	0.734905	+	0.678170i	$0.237227\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	20.5263	−	13.5526i	1.12147	−	0.740455i
$336$		0			0
$337$	−6.92820	+	4.00000i	−0.377403	+	0.217894i	−0.676688	−	0.736270i	$-0.736585\pi$
$337$	−6.92820	+	4.00000i	−0.377403	+	0.217894i	0.299285	+	0.954164i	$0.403252\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.00000			−0.216612
$342$		0			0
$343$		−	13.0000i		−	0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.3923	−	6.00000i	0.557888	−	0.322097i	−0.194409	−	0.980921i	$-0.562279\pi$
$347$	10.3923	−	6.00000i	0.557888	−	0.322097i	0.752297	+	0.658824i	$0.228946\pi$
$348$		0			0
$349$	−5.50000	+	9.52628i	−0.294408	+	0.509930i	−0.974847	−	0.222875i	$-0.928456\pi$
$349$	−5.50000	+	9.52628i	−0.294408	+	0.509930i	0.680439	+	0.732805i	$0.261789\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−13.8564	−	8.00000i	−0.737502	−	0.425797i	0.0836583	−	0.996495i	$-0.473340\pi$
$353$	−13.8564	−	8.00000i	−0.737502	−	0.425797i	−0.821160	+	0.570697i	$0.806673\pi$
$354$		0			0
$355$	13.3923	−	0.803848i	0.710790	−	0.0426638i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−30.0000			−1.58334			−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000			−1.58334			−0.791670	+	0.610949i	$0.790788\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.535898	+	8.92820i	0.0280502	+	0.467324i
$366$		0			0
$367$	13.8564	+	8.00000i	0.723299	+	0.417597i	0.815966	−	0.578101i	$-0.196206\pi$
$367$	13.8564	+	8.00000i	0.723299	+	0.417597i	−0.0926670	+	0.995697i	$0.529539\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	10.3923	−	6.00000i	0.538093	−	0.310668i	−0.206213	−	0.978507i	$-0.566114\pi$
$373$	10.3923	−	6.00000i	0.538093	−	0.310668i	0.744306	+	0.667839i	$0.232781\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	54.0000i		−	2.78114i
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	27.7128	−	16.0000i	1.41606	−	0.817562i	0.420109	−	0.907474i	$-0.361992\pi$
$383$	27.7128	−	16.0000i	1.41606	−	0.817562i	0.995950	+	0.0899119i	$0.0286586\pi$
$384$		0			0
$385$	−3.73205	+	2.46410i	−0.190203	+	0.125582i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.50000	+	16.4545i	−0.481669	+	0.834275i	−0.999779	−	0.0210389i	$-0.993303\pi$
$389$	−9.50000	+	16.4545i	−0.481669	+	0.834275i	0.518110	+	0.855314i	$0.326636\pi$
$390$		0			0
$391$	−1.00000	−	1.73205i	−0.0505722	−	0.0875936i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−12.0000	+	24.0000i	−0.603786	+	1.20757i
$396$		0			0
$397$			4.00000i			0.200754i	0.994949	+	0.100377i	$0.0320049\pi$
$397$			4.00000i			0.200754i	−0.994949	+	0.100377i	$0.967995\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.00000	−	8.66025i	−0.249688	−	0.432472i	0.713751	−	0.700399i	$-0.246995\pi$
$401$	−5.00000	−	8.66025i	−0.249688	−	0.432472i	−0.963439	+	0.267927i	$0.913661\pi$
$402$		0			0
$403$	10.3923	+	6.00000i	0.517678	+	0.298881i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.46410	−	2.00000i	−0.171709	−	0.0991363i
$408$		0			0
$409$	19.0000	+	32.9090i	0.939490	+	1.62724i	0.766426	+	0.642333i	$0.222033\pi$
$409$	19.0000	+	32.9090i	0.939490	+	1.62724i	0.173064	+	0.984911i	$0.444633\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	4.00000i		−	0.196827i
$414$		0			0
$415$	−22.0000	−	11.0000i	−1.07994	−	0.539969i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	17.0000	+	29.4449i	0.830504	+	1.43848i	0.897639	+	0.440732i	$0.145281\pi$
$419$	17.0000	+	29.4449i	0.830504	+	1.43848i	−0.0671345	+	0.997744i	$0.521386\pi$
$420$		0			0
$421$	−11.0000	+	19.0526i	−0.536107	+	0.928565i	0.463002	+	0.886357i	$0.346772\pi$
$421$	−11.0000	+	19.0526i	−0.536107	+	0.928565i	−0.999109	+	0.0422075i	$0.986561\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.19615	−	9.92820i	−0.0580219	−	0.481589i
$426$		0			0
$427$	6.06218	−	3.50000i	0.293369	−	0.169377i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$		0			0
$433$			2.00000i			0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$			2.00000i			0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.19615	−	3.00000i	0.248566	−	0.143509i
$438$		0			0
$439$	−12.0000	+	20.7846i	−0.572729	+	0.991995i	0.423556	+	0.905870i	$0.360782\pi$
$439$	−12.0000	+	20.7846i	−0.572729	+	0.991995i	−0.996284	+	0.0861252i	$0.972552\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7.79423	+	4.50000i	0.370315	+	0.213801i	0.673596	−	0.739100i	$-0.264749\pi$
$443$	7.79423	+	4.50000i	0.370315	+	0.213801i	−0.303281	+	0.952901i	$0.598082\pi$
$444$		0			0
$445$	−2.23205	+	0.133975i	−0.105809	+	0.00635100i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		0			0
$451$	−22.0000			−1.03594
$452$		0			0
$453$		0			0
$454$		0			0
$455$	13.3923	−	0.803848i	0.627841	−	0.0376850i
$456$		0			0
$457$	−8.66025	−	5.00000i	−0.405110	−	0.233890i	0.283577	−	0.958950i	$-0.408479\pi$
$457$	−8.66025	−	5.00000i	−0.405110	−	0.233890i	−0.688686	+	0.725059i	$0.741812\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−10.5000	+	18.1865i	−0.489034	+	0.847031i	−0.999920	−	0.0126168i	$-0.995984\pi$
$461$	−10.5000	+	18.1865i	−0.489034	+	0.847031i	0.510887	+	0.859648i	$0.329317\pi$
$462$		0			0
$463$	−31.1769	+	18.0000i	−1.44891	+	0.836531i	−0.998417	−	0.0562469i	$-0.982087\pi$
$463$	−31.1769	+	18.0000i	−1.44891	+	0.836531i	−0.450497	+	0.892778i	$0.648753\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	36.0000i		−	1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$		−	36.0000i		−	1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$		0			0
$469$	−11.0000			−0.507933
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6.92820	+	4.00000i	−0.318559	+	0.183920i
$474$		0			0
$475$	29.7846	−	3.58846i	1.36661	−	0.164650i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	14.0000	−	24.2487i	0.639676	−	1.10795i	−0.345827	−	0.938298i	$-0.612402\pi$
$479$	14.0000	−	24.2487i	0.639676	−	1.10795i	0.985504	−	0.169654i	$-0.0542649\pi$
$480$		0			0
$481$	6.00000	+	10.3923i	0.273576	+	0.473848i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	16.0000	+	8.00000i	0.726523	+	0.363261i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$	15.5885	+	9.00000i	0.702069	+	0.405340i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5.19615	−	3.00000i	−0.233079	−	0.134568i
$498$		0			0
$499$	12.0000	+	20.7846i	0.537194	+	0.930447i	0.999054	+	0.0434940i	$0.0138489\pi$
$499$	12.0000	+	20.7846i	0.537194	+	0.930447i	−0.461860	+	0.886953i	$0.652818\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			27.0000i			1.20387i	0.798545	+	0.601935i	$0.205603\pi$
$503$			27.0000i			1.20387i	−0.798545	+	0.601935i	$0.794397\pi$
$504$		0			0
$505$	−2.00000	+	4.00000i	−0.0889988	+	0.177998i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.50000	+	12.9904i	0.332432	+	0.575789i	0.982988	−	0.183669i	$-0.0587976\pi$
$509$	7.50000	+	12.9904i	0.332432	+	0.575789i	−0.650556	+	0.759458i	$0.725464\pi$
$510$		0			0
$511$	2.00000	−	3.46410i	0.0884748	−	0.153243i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−14.9282	+	9.85641i	−0.657815	+	0.434325i
$516$		0			0
$517$	−12.1244	+	7.00000i	−0.533229	+	0.307860i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	37.0000			1.62100			0.810500	−	0.585739i	$-0.199196\pi$
$521$	37.0000			1.62100			0.810500	+	0.585739i	$0.199196\pi$
$522$		0			0
$523$			29.0000i			1.26808i	0.773300	+	0.634041i	$0.218605\pi$
$523$			29.0000i			1.26808i	−0.773300	+	0.634041i	$0.781395\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.46410	+	2.00000i	−0.150899	+	0.0871214i
$528$		0			0
$529$	−11.0000	+	19.0526i	−0.478261	+	0.828372i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	57.1577	+	33.0000i	2.47577	+	1.42939i
$534$		0			0
$535$	0.401924	+	6.69615i	0.0173767	+	0.289500i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.0000			−0.516877
$540$		0			0
$541$	−17.0000			−0.730887			−0.365444	−	0.930834i	$-0.619083\pi$
$541$	−17.0000			−0.730887			−0.365444	+	0.930834i	$0.619083\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	15.6244	−	0.937822i	0.669274	−	0.0401719i
$546$		0			0
$547$	−30.3109	−	17.5000i	−1.29600	−	0.748246i	−0.316289	−	0.948663i	$-0.602437\pi$
$547$	−30.3109	−	17.5000i	−1.29600	−	0.748246i	−0.979711	+	0.200417i	$0.935770\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−27.0000	+	46.7654i	−1.15024	+	1.99227i
$552$		0			0
$553$	10.3923	−	6.00000i	0.441926	−	0.255146i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	24.0000i		−	1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$557$		−	24.0000i		−	1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−32.0429	+	18.5000i	−1.35045	+	0.779682i	−0.988312	−	0.152443i	$-0.951286\pi$
$563$	−32.0429	+	18.5000i	−1.35045	+	0.779682i	−0.362137	+	0.932125i	$0.617953\pi$
$564$		0			0
$565$	22.3923	−	14.7846i	0.942051	−	0.621993i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.00000	−	1.73205i	0.0419222	−	0.0726113i	−0.844303	−	0.535866i	$-0.819985\pi$
$569$	1.00000	−	1.73205i	0.0419222	−	0.0726113i	0.886225	+	0.463255i	$0.153319\pi$
$570$		0			0
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	0.577301	−	0.816532i	$-0.304106\pi$
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	−0.995788	+	0.0916910i	$0.970773\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.00000	−	3.00000i	0.166812	−	0.125109i
$576$		0			0
$577$		−	32.0000i		−	1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$		−	32.0000i		−	1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.50000	+	9.52628i	0.228178	+	0.395217i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.59808	+	1.50000i	0.107234	+	0.0619116i	0.552658	−	0.833408i	$-0.313614\pi$
$587$	2.59808	+	1.50000i	0.107234	+	0.0619116i	−0.445424	+	0.895320i	$0.646947\pi$
$588$		0			0
$589$	−6.00000	−	10.3923i	−0.247226	−	0.428207i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			30.0000i			1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$			30.0000i			1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$		0			0
$595$	−2.00000	+	4.00000i	−0.0819920	+	0.163984i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.00000	−	10.3923i	−0.245153	−	0.424618i	0.717021	−	0.697051i	$-0.245505\pi$
$599$	−6.00000	−	10.3923i	−0.245153	−	0.424618i	−0.962175	+	0.272433i	$0.912172\pi$
$600$		0			0
$601$	11.0000	−	19.0526i	0.448699	−	0.777170i	−0.549602	−	0.835426i	$-0.685221\pi$
$601$	11.0000	−	19.0526i	0.448699	−	0.777170i	0.998302	+	0.0582563i	$0.0185541\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−8.62436	−	13.0622i	−0.350630	−	0.531053i
$606$		0			0
$607$	0.866025	−	0.500000i	0.0351509	−	0.0202944i	−0.482322	−	0.875994i	$-0.660206\pi$
$607$	0.866025	−	0.500000i	0.0351509	−	0.0202944i	0.517472	+	0.855700i	$0.326873\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	42.0000			1.69914
$612$		0			0
$613$		−	34.0000i		−	1.37325i	−0.727013	−	0.686624i	$-0.759092\pi$
$613$		−	34.0000i		−	1.37325i	0.727013	−	0.686624i	$-0.240908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−27.7128	+	16.0000i	−1.11568	+	0.644136i	−0.940294	−	0.340365i	$-0.889449\pi$
$617$	−27.7128	+	16.0000i	−1.11568	+	0.644136i	−0.175382	+	0.984500i	$0.556116\pi$
$618$		0			0
$619$	5.00000	−	8.66025i	0.200967	−	0.348085i	−0.747873	−	0.663842i	$-0.768925\pi$
$619$	5.00000	−	8.66025i	0.200967	−	0.348085i	0.948840	+	0.315757i	$0.102258\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.866025	+	0.500000i	0.0346966	+	0.0200321i
$624$		0			0
$625$	24.2846	−	5.93782i	0.971384	−	0.237513i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.00000			−0.159490
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.54552	+	42.4090i	0.101016	+	1.68295i
$636$		0			0
$637$	31.1769	+	18.0000i	1.23527	+	0.713186i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.50000	−	11.2583i	0.256735	−	0.444677i	−0.708631	−	0.705580i	$-0.750687\pi$
$641$	6.50000	−	11.2583i	0.256735	−	0.444677i	0.965365	+	0.260902i	$0.0840201\pi$
$642$		0			0
$643$	28.5788	−	16.5000i	1.12704	−	0.650696i	0.183851	−	0.982954i	$-0.441144\pi$
$643$	28.5788	−	16.5000i	1.12704	−	0.650696i	0.943189	+	0.332258i	$0.107810\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	33.0000i		−	1.29736i	−0.761060	−	0.648682i	$-0.775321\pi$
$647$		−	33.0000i		−	1.29736i	0.761060	−	0.648682i	$-0.224679\pi$
$648$		0			0
$649$	−8.00000			−0.314027
$650$		0			0
$651$		0			0
$652$		0			0
$653$	22.5167	−	13.0000i	0.881145	−	0.508729i	0.0101092	−	0.999949i	$-0.496782\pi$
$653$	22.5167	−	13.0000i	0.881145	−	0.508729i	0.871036	+	0.491220i	$0.163449\pi$
$654$		0			0
$655$	14.7846	+	22.3923i	0.577683	+	0.874940i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	10.0000	−	17.3205i	0.389545	−	0.674711i	−0.602844	−	0.797859i	$-0.705966\pi$
$659$	10.0000	−	17.3205i	0.389545	−	0.674711i	0.992388	+	0.123148i	$0.0392990\pi$
$660$		0			0
$661$	5.00000	+	8.66025i	0.194477	+	0.336845i	0.946729	−	0.322031i	$-0.104366\pi$
$661$	5.00000	+	8.66025i	0.194477	+	0.336845i	−0.752252	+	0.658876i	$0.771032\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−12.0000	−	6.00000i	−0.465340	−	0.232670i
$666$		0			0
$667$			9.00000i			0.348481i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−7.00000	−	12.1244i	−0.270232	−	0.468056i
$672$		0			0
$673$	−5.19615	−	3.00000i	−0.200297	−	0.115642i	0.396497	−	0.918036i	$-0.370226\pi$
$673$	−5.19615	−	3.00000i	−0.200297	−	0.115642i	−0.596794	+	0.802395i	$0.703559\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−19.0526	−	11.0000i	−0.732249	−	0.422764i	0.0869952	−	0.996209i	$-0.472274\pi$
$677$	−19.0526	−	11.0000i	−0.732249	−	0.422764i	−0.819244	+	0.573444i	$0.805607\pi$
$678$		0			0
$679$	−4.00000	−	6.92820i	−0.153506	−	0.265880i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			4.00000i			0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$			4.00000i			0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$		0			0
$685$	24.0000	+	12.0000i	0.916993	+	0.458496i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−22.0000	+	38.1051i	−0.836919	+	1.44959i	0.0555386	+	0.998457i	$0.482312\pi$
$691$	−22.0000	+	38.1051i	−0.836919	+	1.44959i	−0.892458	+	0.451130i	$0.851021\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	19.7128	+	29.8564i	0.747750	+	1.13252i
$696$		0			0
$697$	−19.0526	+	11.0000i	−0.721667	+	0.416655i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−13.0000			−0.491003			−0.245502	−	0.969396i	$-0.578953\pi$
$701$	−13.0000			−0.491003			−0.245502	+	0.969396i	$0.578953\pi$
$702$		0			0
$703$		−	12.0000i		−	0.452589i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.73205	−	1.00000i	0.0651405	−	0.0376089i
$708$		0			0
$709$	−13.5000	+	23.3827i	−0.507003	+	0.878155i	0.492964	+	0.870050i	$0.335913\pi$
$709$	−13.5000	+	23.3827i	−0.507003	+	0.878155i	−0.999967	+	0.00810550i	$0.997420\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.73205	−	1.00000i	−0.0648658	−	0.0374503i
$714$		0			0
$715$	−1.60770	−	26.7846i	−0.0601244	−	1.00169i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	44.0000			1.64092			0.820462	−	0.571702i	$-0.193717\pi$
$719$	44.0000			1.64092			0.820462	+	0.571702i	$0.193717\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−17.6769	+	41.3827i	−0.656504	+	1.53691i
$726$		0			0
$727$	−18.1865	−	10.5000i	−0.674501	−	0.389423i	0.123279	−	0.992372i	$-0.460659\pi$
$727$	−18.1865	−	10.5000i	−0.674501	−	0.389423i	−0.797780	+	0.602949i	$0.793992\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4.00000	+	6.92820i	−0.147945	+	0.256249i
$732$		0			0
$733$	3.46410	−	2.00000i	0.127950	−	0.0738717i	−0.434659	−	0.900595i	$-0.643131\pi$
$733$	3.46410	−	2.00000i	0.127950	−	0.0738717i	0.562609	+	0.826723i	$0.309798\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			22.0000i			0.810380i
$738$		0			0
$739$	40.0000			1.47142			0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000			1.47142			0.735712	+	0.677295i	$0.236848\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12.9904	+	7.50000i	−0.476571	+	0.275148i	−0.718986	−	0.695024i	$-0.755394\pi$
$743$	−12.9904	+	7.50000i	−0.476571	+	0.275148i	0.242415	+	0.970173i	$0.422060\pi$
$744$		0			0
$745$	−1.23205	−	1.86603i	−0.0451388	−	0.0683659i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.50000	−	2.59808i	0.0548088	−	0.0949316i
$750$		0			0
$751$	13.0000	+	22.5167i	0.474377	+	0.821645i	0.999570	−	0.0293387i	$-0.00934013\pi$
$751$	13.0000	+	22.5167i	0.474377	+	0.821645i	−0.525193	+	0.850983i	$0.676007\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10.0000	+	20.0000i	−0.363937	+	0.727875i
$756$		0			0
$757$			10.0000i			0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$			10.0000i			0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.50000	−	7.79423i	−0.163125	−	0.282541i	0.772863	−	0.634573i	$-0.218824\pi$
$761$	−4.50000	−	7.79423i	−0.163125	−	0.282541i	−0.935988	+	0.352032i	$0.885491\pi$
$762$		0			0
$763$	−6.06218	−	3.50000i	−0.219466	−	0.126709i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	20.7846	+	12.0000i	0.750489	+	0.433295i
$768$		0			0
$769$	−7.50000	−	12.9904i	−0.270457	−	0.468445i	0.698522	−	0.715589i	$-0.253841\pi$
$769$	−7.50000	−	12.9904i	−0.270457	−	0.468445i	−0.968979	+	0.247143i	$0.920508\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	12.0000i		−	0.431610i	−0.976436	−	0.215805i	$-0.930762\pi$
$773$		−	12.0000i		−	0.431610i	0.976436	−	0.215805i	$-0.0692376\pi$
$774$		0			0
$775$	−6.00000	−	8.00000i	−0.215526	−	0.287368i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−33.0000	−	57.1577i	−1.18235	−	2.04789i
$780$		0			0
$781$	−6.00000	+	10.3923i	−0.214697	+	0.371866i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.46410	−	4.92820i	0.266405	−	0.175895i
$786$		0			0
$787$	38.1051	−	22.0000i	1.35830	−	0.784215i	0.368906	−	0.929467i	$-0.379732\pi$
$787$	38.1051	−	22.0000i	1.35830	−	0.784215i	0.989395	+	0.145251i	$0.0463991\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$			42.0000i			1.49146i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.73205	+	1.00000i	−0.0613524	+	0.0354218i	−0.530362	−	0.847771i	$-0.677944\pi$
$797$	−1.73205	+	1.00000i	−0.0613524	+	0.0354218i	0.469010	+	0.883193i	$0.344611\pi$
$798$		0			0
$799$	−7.00000	+	12.1244i	−0.247642	+	0.428929i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.92820	−	4.00000i	−0.244491	−	0.141157i
$804$		0			0
$805$	−2.23205	+	0.133975i	−0.0786695	+	0.00472198i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.535898	+	8.92820i	0.0187717	+	0.312741i
$816$		0			0
$817$	−20.7846	−	12.0000i	−0.727161	−	0.419827i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.5000	+	26.8468i	−0.540954	+	0.936959i	0.457896	+	0.889006i	$0.348603\pi$
$821$	−15.5000	+	26.8468i	−0.540954	+	0.936959i	−0.998850	+	0.0479535i	$0.984730\pi$
$822$		0			0
$823$	−12.9904	+	7.50000i	−0.452816	+	0.261434i	−0.709019	−	0.705190i	$-0.750862\pi$
$823$	−12.9904	+	7.50000i	−0.452816	+	0.261434i	0.256203	+	0.966623i	$0.417529\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			35.0000i			1.21707i	0.793527	+	0.608535i	$0.208242\pi$
$827$			35.0000i			1.21707i	−0.793527	+	0.608535i	$0.791758\pi$
$828$		0			0
$829$	−15.0000			−0.520972			−0.260486	−	0.965478i	$-0.583883\pi$
$829$	−15.0000			−0.520972			−0.260486	+	0.965478i	$0.583883\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−10.3923	+	6.00000i	−0.360072	+	0.207888i
$834$		0			0
$835$	−5.59808	+	3.69615i	−0.193729	+	0.127911i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	19.0000	−	32.9090i	0.655953	−	1.13614i	−0.325701	−	0.945473i	$-0.605600\pi$
$839$	19.0000	−	32.9090i	0.655953	−	1.13614i	0.981654	−	0.190671i	$-0.0610663\pi$
$840$		0			0
$841$	−26.0000	−	45.0333i	−0.896552	−	1.55287i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−23.0000	+	46.0000i	−0.791224	+	1.58245i
$846$		0			0
$847$			7.00000i			0.240523i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.00000	−	1.73205i	−0.0342796	−	0.0593739i
$852$		0			0
$853$	−39.8372	−	23.0000i	−1.36400	−	0.787505i	−0.373845	−	0.927491i	$-0.621961\pi$
$853$	−39.8372	−	23.0000i	−1.36400	−	0.787505i	−0.990153	+	0.139986i	$0.955294\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15.5885	+	9.00000i	0.532492	+	0.307434i	0.742030	−	0.670366i	$-0.233863\pi$
$857$	15.5885	+	9.00000i	0.532492	+	0.307434i	−0.209539	+	0.977800i	$0.567196\pi$
$858$		0			0
$859$	−7.00000	−	12.1244i	−0.238837	−	0.413678i	0.721544	−	0.692369i	$-0.243433\pi$
$859$	−7.00000	−	12.1244i	−0.238837	−	0.413678i	−0.960381	+	0.278691i	$0.910099\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	45.0000i		−	1.53182i	−0.642949	−	0.765909i	$-0.722289\pi$
$863$		−	45.0000i		−	1.53182i	0.642949	−	0.765909i	$-0.277711\pi$
$864$		0			0
$865$	8.00000	+	4.00000i	0.272008	+	0.136004i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−12.0000	−	20.7846i	−0.407072	−	0.705070i
$870$		0			0
$871$	33.0000	−	57.1577i	1.11816	−	1.93671i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−10.5263	−	3.76795i	−0.355853	−	0.127380i
$876$		0			0
$877$	−39.8372	+	23.0000i	−1.34521	+	0.776655i	−0.987566	−	0.157205i	$-0.949752\pi$
$877$	−39.8372	+	23.0000i	−1.34521	+	0.776655i	−0.357640	+	0.933860i	$0.616418\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	15.0000			0.505363			0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000			0.505363			0.252681	+	0.967550i	$0.418688\pi$
$882$		0			0
$883$		−	3.00000i		−	0.100958i	−0.998725	−	0.0504790i	$-0.983925\pi$
$883$		−	3.00000i		−	0.100958i	0.998725	−	0.0504790i	$-0.0160748\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	9.50000	−	16.4545i	0.318620	−	0.551866i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−36.3731	−	21.0000i	−1.21718	−	0.702738i
$894$		0			0
$895$	4.46410	−	0.267949i	0.149218	−	0.00895655i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	18.0000			0.600334
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	29.0167	−	1.74167i	0.964546	−	0.0578951i
$906$		0			0
$907$	−28.5788	−	16.5000i	−0.948945	−	0.547874i	−0.0561918	−	0.998420i	$-0.517896\pi$
$907$	−28.5788	−	16.5000i	−0.948945	−	0.547874i	−0.892753	+	0.450546i	$0.851229\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6.00000	+	10.3923i	−0.198789	+	0.344312i	−0.948136	−	0.317865i	$-0.897034\pi$
$911$	−6.00000	+	10.3923i	−0.198789	+	0.344312i	0.749347	+	0.662177i	$0.230367\pi$
$912$		0			0
$913$	19.0526	−	11.0000i	0.630548	−	0.364047i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	12.0000i		−	0.396275i
$918$		0			0
$919$	−36.0000			−1.18753			−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000			−1.18753			−0.593765	+	0.804638i	$0.702359\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	31.1769	−	18.0000i	1.02620	−	0.592477i
$924$		0			0
$925$	−1.19615	−	9.92820i	−0.0393292	−	0.326437i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9.00000	−	15.5885i	0.295280	−	0.511441i	−0.679770	−	0.733426i	$-0.737920\pi$
$929$	9.00000	−	15.5885i	0.295280	−	0.511441i	0.975050	+	0.221985i	$0.0712536\pi$
$930$		0			0
$931$	−18.0000	−	31.1769i	−0.589926	−	1.02178i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	8.00000	+	4.00000i	0.261628	+	0.130814i
$936$		0			0
$937$		−	52.0000i		−	1.69877i	−0.527777	−	0.849383i	$-0.676974\pi$
$937$		−	52.0000i		−	1.69877i	0.527777	−	0.849383i	$-0.323026\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−20.5000	−	35.5070i	−0.668281	−	1.15750i	−0.978385	−	0.206794i	$-0.933697\pi$
$941$	−20.5000	−	35.5070i	−0.668281	−	1.15750i	0.310104	−	0.950703i	$-0.399636\pi$
$942$		0			0
$943$	−9.52628	−	5.50000i	−0.310218	−	0.179105i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−44.1673	−	25.5000i	−1.43524	−	0.828639i	−0.437730	−	0.899106i	$-0.644217\pi$
$947$	−44.1673	−	25.5000i	−1.43524	−	0.828639i	−0.997514	+	0.0704677i	$0.977551\pi$
$948$		0			0
$949$	12.0000	+	20.7846i	0.389536	+	0.674697i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	2.00000i		−	0.0647864i	−0.999475	−	0.0323932i	$-0.989687\pi$
$953$		−	2.00000i		−	0.0647864i	0.999475	−	0.0323932i	$-0.0103129\pi$
$954$		0			0
$955$	6.00000	−	12.0000i	0.194155	−	0.388311i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.00000	−	10.3923i	−0.193750	−	0.335585i
$960$		0			0
$961$	13.5000	−	23.3827i	0.435484	−	0.754280i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	18.6603	−	12.3205i	0.600695	−	0.396611i
$966$		0			0
$967$	−14.7224	+	8.50000i	−0.473441	+	0.273342i	−0.717679	−	0.696374i	$-0.754796\pi$
$967$	−14.7224	+	8.50000i	−0.473441	+	0.273342i	0.244238	+	0.969715i	$0.421462\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−42.0000			−1.34784			−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000			−1.34784			−0.673922	+	0.738802i	$0.735392\pi$
$972$		0			0
$973$		−	16.0000i		−	0.512936i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	46.7654	−	27.0000i	1.49616	−	0.863807i	0.496167	−	0.868227i	$-0.334741\pi$
$977$	46.7654	−	27.0000i	1.49616	−	0.863807i	0.999990	+	0.00442082i	$0.00140720\pi$
$978$		0			0
$979$	1.00000	−	1.73205i	0.0319601	−	0.0553566i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.4545	+	9.50000i	0.524816	+	0.303003i	0.738903	−	0.673812i	$-0.235344\pi$
$983$	16.4545	+	9.50000i	0.524816	+	0.303003i	−0.214087	+	0.976815i	$0.568677\pi$
$984$		0			0
$985$	−1.07180	−	17.8564i	−0.0341503	−	0.568952i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.00000			−0.127193
$990$		0			0
$991$	−4.00000			−0.127064			−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000			−0.127064			−0.0635321	+	0.997980i	$0.520237\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	40.1769	−	2.41154i	1.27369	−	0.0764510i
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.by.b.289.1		4
3.2	odd	2		720.2.by.a.529.2		4
4.3	odd	2		270.2.i.a.19.1		4
5.4	even	2	inner	2160.2.by.b.289.2		4
9.4	even	3	inner	2160.2.by.b.1009.2		4
9.5	odd	6		720.2.by.a.49.1		4
12.11	even	2		90.2.i.a.79.2	yes	4
15.14	odd	2		720.2.by.a.529.1		4
20.3	even	4		1350.2.e.a.451.1		2
20.7	even	4		1350.2.e.i.451.1		2
20.19	odd	2		270.2.i.a.19.2		4
36.7	odd	6		810.2.c.c.649.2		2
36.11	even	6		810.2.c.b.649.1		2
36.23	even	6		90.2.i.a.49.1	✓	4
36.31	odd	6		270.2.i.a.199.2		4
45.4	even	6	inner	2160.2.by.b.1009.1		4
45.14	odd	6		720.2.by.a.49.2		4
60.23	odd	4		450.2.e.g.151.1		2
60.47	odd	4		450.2.e.b.151.1		2
60.59	even	2		90.2.i.a.79.1	yes	4
180.7	even	12		4050.2.a.g.1.1		1
180.23	odd	12		450.2.e.g.301.1		2
180.43	even	12		4050.2.a.be.1.1		1
180.47	odd	12		4050.2.a.x.1.1		1
180.59	even	6		90.2.i.a.49.2	yes	4
180.67	even	12		1350.2.e.i.901.1		2
180.79	odd	6		810.2.c.c.649.1		2
180.83	odd	12		4050.2.a.j.1.1		1
180.103	even	12		1350.2.e.a.901.1		2
180.119	even	6		810.2.c.b.649.2		2
180.139	odd	6		270.2.i.a.199.1		4
180.167	odd	12		450.2.e.b.301.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.i.a.49.1	✓	4	36.23	even	6
90.2.i.a.49.2	yes	4	180.59	even	6
90.2.i.a.79.1	yes	4	60.59	even	2
90.2.i.a.79.2	yes	4	12.11	even	2
270.2.i.a.19.1		4	4.3	odd	2
270.2.i.a.19.2		4	20.19	odd	2
270.2.i.a.199.1		4	180.139	odd	6
270.2.i.a.199.2		4	36.31	odd	6
450.2.e.b.151.1		2	60.47	odd	4
450.2.e.b.301.1		2	180.167	odd	12
450.2.e.g.151.1		2	60.23	odd	4
450.2.e.g.301.1		2	180.23	odd	12
720.2.by.a.49.1		4	9.5	odd	6
720.2.by.a.49.2		4	45.14	odd	6
720.2.by.a.529.1		4	15.14	odd	2
720.2.by.a.529.2		4	3.2	odd	2
810.2.c.b.649.1		2	36.11	even	6
810.2.c.b.649.2		2	180.119	even	6
810.2.c.c.649.1		2	180.79	odd	6
810.2.c.c.649.2		2	36.7	odd	6
1350.2.e.a.451.1		2	20.3	even	4
1350.2.e.a.901.1		2	180.103	even	12
1350.2.e.i.451.1		2	20.7	even	4
1350.2.e.i.901.1		2	180.67	even	12
2160.2.by.b.289.1		4	1.1	even	1	trivial
2160.2.by.b.289.2		4	5.4	even	2	inner
2160.2.by.b.1009.1		4	45.4	even	6	inner
2160.2.by.b.1009.2		4	9.4	even	3	inner
4050.2.a.g.1.1		1	180.7	even	12
4050.2.a.j.1.1		1	180.83	odd	12
4050.2.a.x.1.1		1	180.47	odd	12
4050.2.a.be.1.1		1	180.43	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.by (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.23205 + 0.133975i) q^{5} +(0.866025 + 0.500000i) q^{7} +O(q^{10})\)	\(q+(-2.23205 + 0.133975i) q^{5} +(0.866025 + 0.500000i) q^{7} +(1.00000 - 1.73205i) q^{11} +(-5.19615 + 3.00000i) q^{13} -2.00000i q^{17} +6.00000 q^{19} +(0.866025 - 0.500000i) q^{23} +(4.96410 - 0.598076i) q^{25} +(-4.50000 + 7.79423i) q^{29} +(-1.00000 - 1.73205i) q^{31} +(-2.00000 - 1.00000i) q^{35} -2.00000i q^{37} +(-5.50000 - 9.52628i) q^{41} +(-3.46410 - 2.00000i) q^{43} +(-6.06218 - 3.50000i) q^{47} +(-3.00000 - 5.19615i) q^{49} +(-2.00000 + 4.00000i) q^{55} +(-2.00000 - 3.46410i) q^{59} +(3.50000 - 6.06218i) q^{61} +(11.1962 - 7.39230i) q^{65} +(-9.52628 + 5.50000i) q^{67} -6.00000 q^{71} -4.00000i q^{73} +(1.73205 - 1.00000i) q^{77} +(6.00000 - 10.3923i) q^{79} +(9.52628 + 5.50000i) q^{83} +(0.267949 + 4.46410i) q^{85} +1.00000 q^{89} -6.00000 q^{91} +(-13.3923 + 0.803848i) q^{95} +(-6.92820 - 4.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5}+O(q^{10}) \) 4 * q - 2 * q^5	\( 4 q - 2 q^{5} + 4 q^{11} + 24 q^{19} + 6 q^{25} - 18 q^{29} - 4 q^{31} - 8 q^{35} - 22 q^{41} - 12 q^{49} - 8 q^{55} - 8 q^{59} + 14 q^{61} + 24 q^{65} - 24 q^{71} + 24 q^{79} + 8 q^{85} + 4 q^{89} - 24 q^{91} - 12 q^{95}+O(q^{100}) \) 4 * q - 2 * q^5 + 4 * q^11 + 24 * q^19 + 6 * q^25 - 18 * q^29 - 4 * q^31 - 8 * q^35 - 22 * q^41 - 12 * q^49 - 8 * q^55 - 8 * q^59 + 14 * q^61 + 24 * q^65 - 24 * q^71 + 24 * q^79 + 8 * q^85 + 4 * q^89 - 24 * q^91 - 12 * q^95

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