LMFDB - Embedded newform 1620.2.r.c.109.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(109,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

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

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

chi := DirichletCharacter("1620.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1620.r (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:	$12.9357651274$
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:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			109.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1620.109
Dual form			1620.2.r.c.1189.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.23205 + 1.86603i) q^{5} +(3.46410 - 2.00000i) q^{7} +O(q^{10})$	$q+(1.23205 + 1.86603i) q^{5} +(3.46410 - 2.00000i) q^{7} +(2.00000 + 3.46410i) q^{11} +4.00000i q^{17} +(3.46410 + 2.00000i) q^{23} +(-1.96410 + 4.59808i) q^{25} +(-3.00000 - 5.19615i) q^{29} +(-2.00000 + 3.46410i) q^{31} +(8.00000 + 4.00000i) q^{35} -8.00000i q^{37} +(5.00000 - 8.66025i) q^{41} +(-3.46410 + 2.00000i) q^{43} +(-3.46410 + 2.00000i) q^{47} +(4.50000 - 7.79423i) q^{49} +12.0000i q^{53} +(-4.00000 + 8.00000i) q^{55} +(2.00000 - 3.46410i) q^{59} +(-1.00000 - 1.73205i) q^{61} +(3.46410 + 2.00000i) q^{67} +8.00000i q^{73} +(13.8564 + 8.00000i) q^{77} +(-6.00000 - 10.3923i) q^{79} +(-3.46410 + 2.00000i) q^{83} +(-7.46410 + 4.92820i) q^{85} +10.0000 q^{89} +(6.92820 - 4.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5}+O(q^{10}) $ 4 * q - 2 * q^5	$ 4 q - 2 q^{5} + 8 q^{11} + 6 q^{25} - 12 q^{29} - 8 q^{31} + 32 q^{35} + 20 q^{41} + 18 q^{49} - 16 q^{55} + 8 q^{59} - 4 q^{61} - 24 q^{79} - 16 q^{85} + 40 q^{89}+O(q^{100}) $ 4 * q - 2 * q^5 + 8 * q^11 + 6 * q^25 - 12 * q^29 - 8 * q^31 + 32 * q^35 + 20 * q^41 + 18 * q^49 - 16 * q^55 + 8 * q^59 - 4 * q^61 - 24 * q^79 - 16 * q^85 + 40 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{3}\right)$

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
$5$	1.23205	+	1.86603i	0.550990	+	0.834512i
$5$	1.23205	+	1.86603i	0.550990	+	0.834512i
$6$		0			0
$7$	3.46410	−	2.00000i	1.30931	−	0.755929i	0.327327	−	0.944911i	$-0.393852\pi$
$7$	3.46410	−	2.00000i	1.30931	−	0.755929i	0.981981	+	0.188982i	$0.0605189\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	0.992361	+	0.123371i	$0.0393705\pi$
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	−0.389338	+	0.921095i	$0.627296\pi$
$12$		0			0
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.00000i			0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$			4.00000i			0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.46410	+	2.00000i	0.722315	+	0.417029i	0.815604	−	0.578610i	$-0.196405\pi$
$23$	3.46410	+	2.00000i	0.722315	+	0.417029i	−0.0932891	+	0.995639i	$0.529738\pi$
$24$		0			0
$25$	−1.96410	+	4.59808i	−0.392820	+	0.919615i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00000	−	5.19615i	−0.557086	−	0.964901i	−0.997738	−	0.0672232i	$-0.978586\pi$
$29$	−3.00000	−	5.19615i	−0.557086	−	0.964901i	0.440652	−	0.897678i	$-0.354747\pi$
$30$		0			0
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	8.00000	+	4.00000i	1.35225	+	0.676123i
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.00000	−	8.66025i	0.780869	−	1.35250i	−0.150567	−	0.988600i	$-0.548110\pi$
$41$	5.00000	−	8.66025i	0.780869	−	1.35250i	0.931436	−	0.363905i	$-0.118557\pi$
$42$		0			0
$43$	−3.46410	+	2.00000i	−0.528271	+	0.304997i	−0.740312	−	0.672264i	$-0.765322\pi$
$43$	−3.46410	+	2.00000i	−0.528271	+	0.304997i	0.212041	+	0.977261i	$0.431989\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.46410	+	2.00000i	−0.505291	+	0.291730i	−0.730896	−	0.682489i	$-0.760898\pi$
$47$	−3.46410	+	2.00000i	−0.505291	+	0.291730i	0.225605	+	0.974219i	$0.427564\pi$
$48$		0			0
$49$	4.50000	−	7.79423i	0.642857	−	1.11346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			12.0000i			1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$			12.0000i			1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$		0			0
$55$	−4.00000	+	8.00000i	−0.539360	+	1.07872i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.00000	−	3.46410i	0.260378	−	0.450988i	−0.705965	−	0.708247i	$-0.749486\pi$
$59$	2.00000	−	3.46410i	0.260378	−	0.450988i	0.966342	+	0.257260i	$0.0828195\pi$
$60$		0			0
$61$	−1.00000	−	1.73205i	−0.128037	−	0.221766i	0.794879	−	0.606768i	$-0.207534\pi$
$61$	−1.00000	−	1.73205i	−0.128037	−	0.221766i	−0.922916	+	0.385002i	$0.874201\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	3.46410	+	2.00000i	0.423207	+	0.244339i	0.696449	−	0.717607i	$-0.254762\pi$
$67$	3.46410	+	2.00000i	0.423207	+	0.244339i	−0.273241	+	0.961946i	$0.588096\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$			8.00000i			0.936329i	0.883641	+	0.468165i	$0.155085\pi$
$73$			8.00000i			0.936329i	−0.883641	+	0.468165i	$0.844915\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	13.8564	+	8.00000i	1.57908	+	0.911685i
$78$		0			0
$79$	−6.00000	−	10.3923i	−0.675053	−	1.16923i	−0.976453	−	0.215728i	$-0.930788\pi$
$79$	−6.00000	−	10.3923i	−0.675053	−	1.16923i	0.301401	−	0.953498i	$-0.402546\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.46410	+	2.00000i	−0.380235	+	0.219529i	−0.677920	−	0.735135i	$-0.737119\pi$
$83$	−3.46410	+	2.00000i	−0.380235	+	0.219529i	0.297686	+	0.954664i	$0.403785\pi$
$84$		0			0
$85$	−7.46410	+	4.92820i	−0.809595	+	0.534539i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	6.92820	−	4.00000i	0.703452	−	0.406138i	−0.105180	−	0.994453i	$-0.533542\pi$
$97$	6.92820	−	4.00000i	0.703452	−	0.406138i	0.808632	+	0.588315i	$0.200208\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.00000	+	1.73205i	0.0995037	+	0.172345i	0.911479	−	0.411346i	$-0.134941\pi$
$101$	1.00000	+	1.73205i	0.0995037	+	0.172345i	−0.811976	+	0.583691i	$0.801608\pi$
$102$		0			0
$103$	−3.46410	−	2.00000i	−0.341328	−	0.197066i	0.319531	−	0.947576i	$-0.396475\pi$
$103$	−3.46410	−	2.00000i	−0.341328	−	0.197066i	−0.660859	+	0.750510i	$0.729808\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.3923	+	6.00000i	0.977626	+	0.564433i	0.901553	−	0.432670i	$-0.142428\pi$
$113$	10.3923	+	6.00000i	0.977626	+	0.564433i	0.0760733	+	0.997102i	$0.475762\pi$
$114$		0			0
$115$	0.535898	+	8.92820i	0.0499728	+	0.832559i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8.00000	+	13.8564i	0.733359	+	1.27021i
$120$		0			0
$121$	−2.50000	+	4.33013i	−0.227273	+	0.393648i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$			4.00000i			0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$			4.00000i			0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	0.475380	+	0.879781i	$0.342311\pi$
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	−0.999602	+	0.0281993i	$0.991023\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.3923	−	6.00000i	0.887875	−	0.512615i	0.0146279	−	0.999893i	$-0.495344\pi$
$137$	10.3923	−	6.00000i	0.887875	−	0.512615i	0.873247	+	0.487278i	$0.162010\pi$
$138$		0			0
$139$	8.00000	−	13.8564i	0.678551	−	1.17529i	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	8.00000	−	13.8564i	0.678551	−	1.17529i	0.975417	−	0.220366i	$-0.0707252\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	6.00000	−	12.0000i	0.498273	−	0.996546i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.00000	+	1.73205i	−0.0819232	+	0.141895i	−0.904076	−	0.427372i	$-0.859440\pi$
$149$	−1.00000	+	1.73205i	−0.0819232	+	0.141895i	0.822153	+	0.569267i	$0.192773\pi$
$150$		0			0
$151$	10.0000	+	17.3205i	0.813788	+	1.40952i	0.910195	+	0.414181i	$0.135932\pi$
$151$	10.0000	+	17.3205i	0.813788	+	1.40952i	−0.0964061	+	0.995342i	$0.530735\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.92820	+	0.535898i	−0.717131	+	0.0430444i
$156$		0			0
$157$	−6.92820	−	4.00000i	−0.552931	−	0.319235i	0.197372	−	0.980329i	$-0.436759\pi$
$157$	−6.92820	−	4.00000i	−0.552931	−	0.319235i	−0.750303	+	0.661094i	$0.770093\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	16.0000			1.26098
$162$		0			0
$163$		−	20.0000i		−	1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$		−	20.0000i		−	1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.3923	−	6.00000i	−0.804181	−	0.464294i	0.0407502	−	0.999169i	$-0.487025\pi$
$167$	−10.3923	−	6.00000i	−0.804181	−	0.464294i	−0.844931	+	0.534875i	$0.820359\pi$
$168$		0			0
$169$	−6.50000	−	11.2583i	−0.500000	−	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.46410	−	2.00000i	0.263371	−	0.152057i	−0.362500	−	0.931984i	$-0.618077\pi$
$173$	3.46410	−	2.00000i	0.263371	−	0.152057i	0.625871	+	0.779926i	$0.284744\pi$
$174$		0			0
$175$	2.39230	+	19.8564i	0.180841	+	1.50100i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.00000			−0.298974			−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000			−0.298974			−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	14.9282	−	9.85641i	1.09754	−	0.724657i
$186$		0			0
$187$	−13.8564	+	8.00000i	−1.01328	+	0.585018i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.0000	−	20.7846i	−0.868290	−	1.50392i	−0.863743	−	0.503932i	$-0.831886\pi$
$191$	−12.0000	−	20.7846i	−0.868290	−	1.50392i	−0.00454614	−	0.999990i	$-0.501447\pi$
$192$		0			0
$193$	13.8564	+	8.00000i	0.997406	+	0.575853i	0.907480	−	0.420096i	$-0.138004\pi$
$193$	13.8564	+	8.00000i	0.997406	+	0.575853i	0.0899262	+	0.995948i	$0.471337\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	4.00000i		−	0.284988i	−0.989796	−	0.142494i	$-0.954488\pi$
$197$		−	4.00000i		−	0.284988i	0.989796	−	0.142494i	$-0.0455122\pi$
$198$		0			0
$199$	12.0000			0.850657			0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000			0.850657			0.425329	+	0.905039i	$0.360158\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−20.7846	−	12.0000i	−1.45879	−	0.842235i
$204$		0			0
$205$	22.3205	−	1.33975i	1.55893	−	0.0935719i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$211$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.00000	−	4.00000i	−0.545595	−	0.272798i
$216$		0			0
$217$			16.0000i			1.08615i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	17.3205	−	10.0000i	1.15987	−	0.669650i	0.208595	−	0.978002i	$-0.433111\pi$
$223$	17.3205	−	10.0000i	1.15987	−	0.669650i	0.951272	+	0.308353i	$0.0997777\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−17.3205	+	10.0000i	−1.14960	+	0.663723i	−0.948790	−	0.315906i	$-0.897691\pi$
$227$	−17.3205	+	10.0000i	−1.14960	+	0.663723i	−0.200812	+	0.979630i	$0.564358\pi$
$228$		0			0
$229$	13.0000	−	22.5167i	0.859064	−	1.48794i	−0.0137585	−	0.999905i	$-0.504380\pi$
$229$	13.0000	−	22.5167i	0.859064	−	1.48794i	0.872823	−	0.488037i	$-0.162287\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	20.0000i		−	1.31024i	−0.755523	−	0.655122i	$-0.772617\pi$
$233$		−	20.0000i		−	1.31024i	0.755523	−	0.655122i	$-0.227383\pi$
$234$		0			0
$235$	−8.00000	−	4.00000i	−0.521862	−	0.260931i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.00000	−	6.92820i	0.258738	−	0.448148i	−0.707166	−	0.707048i	$-0.750027\pi$
$239$	4.00000	−	6.92820i	0.258738	−	0.448148i	0.965904	+	0.258900i	$0.0833599\pi$
$240$		0			0
$241$	1.00000	+	1.73205i	0.0644157	+	0.111571i	0.896435	−	0.443176i	$-0.146148\pi$
$241$	1.00000	+	1.73205i	0.0644157	+	0.111571i	−0.832019	+	0.554747i	$0.812815\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	20.0885	−	1.20577i	1.28340	−	0.0770339i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$			16.0000i			1.00591i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.3923	−	6.00000i	−0.648254	−	0.374270i	0.139533	−	0.990217i	$-0.455440\pi$
$257$	−10.3923	−	6.00000i	−0.648254	−	0.374270i	−0.787787	+	0.615948i	$0.788773\pi$
$258$		0			0
$259$	−16.0000	−	27.7128i	−0.994192	−	1.72199i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.46410	+	2.00000i	−0.213606	+	0.123325i	−0.602986	−	0.797752i	$-0.706023\pi$
$263$	−3.46410	+	2.00000i	−0.213606	+	0.123325i	0.389380	+	0.921077i	$0.372689\pi$
$264$		0			0
$265$	−22.3923	+	14.7846i	−1.37555	+	0.908211i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.00000			−0.365826			−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000			−0.365826			−0.182913	+	0.983129i	$0.558553\pi$
$270$		0			0
$271$	4.00000			0.242983			0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000			0.242983			0.121491	+	0.992592i	$0.461232\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−19.8564	+	2.39230i	−1.19739	+	0.144261i
$276$		0			0
$277$	−27.7128	+	16.0000i	−1.66510	+	0.961347i	−0.694881	+	0.719125i	$0.744543\pi$
$277$	−27.7128	+	16.0000i	−1.66510	+	0.961347i	−0.970221	+	0.242222i	$0.922124\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.00000	−	5.19615i	−0.178965	−	0.309976i	0.762561	−	0.646916i	$-0.223942\pi$
$281$	−3.00000	−	5.19615i	−0.178965	−	0.309976i	−0.941526	+	0.336939i	$0.890608\pi$
$282$		0			0
$283$	−24.2487	−	14.0000i	−1.44144	−	0.832214i	−0.443491	−	0.896279i	$-0.646260\pi$
$283$	−24.2487	−	14.0000i	−1.44144	−	0.832214i	−0.997946	+	0.0640654i	$0.979593\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	40.0000i		−	2.36113i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.3923	+	6.00000i	0.607125	+	0.350524i	0.771839	−	0.635818i	$-0.219337\pi$
$293$	10.3923	+	6.00000i	0.607125	+	0.350524i	−0.164714	+	0.986341i	$0.552670\pi$
$294$		0			0
$295$	8.92820	−	0.535898i	0.519820	−	0.0312012i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−8.00000	+	13.8564i	−0.461112	+	0.798670i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.00000	−	4.00000i	0.114520	−	0.229039i
$306$		0			0
$307$			12.0000i			0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$			12.0000i			0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.0000	+	20.7846i	−0.680458	+	1.17859i	0.294384	+	0.955687i	$0.404886\pi$
$311$	−12.0000	+	20.7846i	−0.680458	+	1.17859i	−0.974841	+	0.222900i	$0.928448\pi$
$312$		0			0
$313$	−13.8564	+	8.00000i	−0.783210	+	0.452187i	−0.837567	−	0.546335i	$-0.816023\pi$
$313$	−13.8564	+	8.00000i	−0.783210	+	0.452187i	0.0543564	+	0.998522i	$0.482689\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.46410	−	2.00000i	0.194563	−	0.112331i	−0.399554	−	0.916710i	$-0.630835\pi$
$317$	3.46410	−	2.00000i	0.194563	−	0.112331i	0.594117	+	0.804379i	$0.297502\pi$
$318$		0			0
$319$	12.0000	−	20.7846i	0.671871	−	1.16371i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8.00000	+	13.8564i	−0.441054	+	0.763928i
$330$		0			0
$331$	4.00000	+	6.92820i	0.219860	+	0.380808i	0.954765	−	0.297361i	$-0.0961066\pi$
$331$	4.00000	+	6.92820i	0.219860	+	0.380808i	−0.734905	+	0.678170i	$0.762773\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.535898	+	8.92820i	0.0292793	+	0.487800i
$336$		0			0
$337$	6.92820	+	4.00000i	0.377403	+	0.217894i	0.676688	−	0.736270i	$-0.263415\pi$
$337$	6.92820	+	4.00000i	0.377403	+	0.217894i	−0.299285	+	0.954164i	$0.596748\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−16.0000			−0.866449
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.3923	−	6.00000i	−0.557888	−	0.322097i	0.194409	−	0.980921i	$-0.437721\pi$
$347$	−10.3923	−	6.00000i	−0.557888	−	0.322097i	−0.752297	+	0.658824i	$0.771054\pi$
$348$		0			0
$349$	−7.00000	−	12.1244i	−0.374701	−	0.649002i	0.615581	−	0.788074i	$-0.288921\pi$
$349$	−7.00000	−	12.1244i	−0.374701	−	0.649002i	−0.990282	+	0.139072i	$0.955588\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	24.2487	−	14.0000i	1.29063	−	0.745145i	0.311863	−	0.950127i	$-0.399047\pi$
$353$	24.2487	−	14.0000i	1.29063	−	0.745145i	0.978766	+	0.204982i	$0.0657137\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−14.9282	+	9.85641i	−0.781378	+	0.515908i
$366$		0			0
$367$	3.46410	−	2.00000i	0.180825	−	0.104399i	−0.406855	−	0.913493i	$-0.633375\pi$
$367$	3.46410	−	2.00000i	0.180825	−	0.104399i	0.587680	+	0.809093i	$0.300041\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	24.0000	+	41.5692i	1.24602	+	2.15817i
$372$		0			0
$373$	20.7846	+	12.0000i	1.07619	+	0.621336i	0.929865	−	0.367901i	$-0.119923\pi$
$373$	20.7846	+	12.0000i	1.07619	+	0.621336i	0.146321	+	0.989237i	$0.453257\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−8.00000			−0.410932			−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000			−0.410932			−0.205466	+	0.978664i	$0.565871\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−24.2487	−	14.0000i	−1.23905	−	0.715367i	−0.270151	−	0.962818i	$-0.587074\pi$
$383$	−24.2487	−	14.0000i	−1.23905	−	0.715367i	−0.968900	+	0.247451i	$0.920407\pi$
$384$		0			0
$385$	2.14359	+	35.7128i	0.109248	+	1.82009i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−17.0000	−	29.4449i	−0.861934	−	1.49291i	−0.870059	−	0.492947i	$-0.835920\pi$
$389$	−17.0000	−	29.4449i	−0.861934	−	1.49291i	0.00812520	−	0.999967i	$-0.497414\pi$
$390$		0			0
$391$	−8.00000	+	13.8564i	−0.404577	+	0.700749i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.0000	−	24.0000i	0.603786	−	1.20757i
$396$		0			0
$397$		−	8.00000i		−	0.401508i	−0.979642	−	0.200754i	$-0.935661\pi$
$397$		−	8.00000i		−	0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.00000	−	12.1244i	0.349563	−	0.605461i	−0.636609	−	0.771187i	$-0.719663\pi$
$401$	7.00000	−	12.1244i	0.349563	−	0.605461i	0.986172	+	0.165726i	$0.0529966\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	27.7128	−	16.0000i	1.37367	−	0.793091i
$408$		0			0
$409$	13.0000	−	22.5167i	0.642809	−	1.11338i	−0.341994	−	0.939702i	$-0.611102\pi$
$409$	13.0000	−	22.5167i	0.642809	−	1.11338i	0.984803	−	0.173675i	$-0.0555643\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	16.0000i		−	0.787309i
$414$		0			0
$415$	−8.00000	−	4.00000i	−0.392705	−	0.196352i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−14.0000	+	24.2487i	−0.683945	+	1.18463i	0.289822	+	0.957080i	$0.406404\pi$
$419$	−14.0000	+	24.2487i	−0.683945	+	1.18463i	−0.973767	+	0.227547i	$0.926930\pi$
$420$		0			0
$421$	−5.00000	−	8.66025i	−0.243685	−	0.422075i	0.718076	−	0.695965i	$-0.245023\pi$
$421$	−5.00000	−	8.66025i	−0.243685	−	0.422075i	−0.961761	+	0.273890i	$0.911690\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−18.3923	−	7.85641i	−0.892158	−	0.381092i
$426$		0			0
$427$	−6.92820	−	4.00000i	−0.335279	−	0.193574i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.00000			0.385346			0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000			0.385346			0.192673	+	0.981263i	$0.438284\pi$
$432$		0			0
$433$		−	16.0000i		−	0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$		−	16.0000i		−	0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−6.00000	−	10.3923i	−0.286364	−	0.495998i	0.686575	−	0.727059i	$-0.259113\pi$
$439$	−6.00000	−	10.3923i	−0.286364	−	0.495998i	−0.972939	+	0.231062i	$0.925780\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−31.1769	+	18.0000i	−1.48126	+	0.855206i	−0.999774	−	0.0212481i	$-0.993236\pi$
$443$	−31.1769	+	18.0000i	−1.48126	+	0.855206i	−0.481486	+	0.876454i	$0.659903\pi$
$444$		0			0
$445$	12.3205	+	18.6603i	0.584048	+	0.884581i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	40.0000			1.88353
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	34.6410	−	20.0000i	1.62044	−	0.935561i	0.633636	−	0.773631i	$-0.281562\pi$
$457$	34.6410	−	20.0000i	1.62044	−	0.935561i	0.986802	−	0.161929i	$-0.0517716\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.00000	−	5.19615i	−0.139724	−	0.242009i	0.787668	−	0.616100i	$-0.211288\pi$
$461$	−3.00000	−	5.19615i	−0.139724	−	0.242009i	−0.927392	+	0.374091i	$0.877955\pi$
$462$		0			0
$463$	10.3923	+	6.00000i	0.482971	+	0.278844i	0.721654	−	0.692254i	$-0.243382\pi$
$463$	10.3923	+	6.00000i	0.482971	+	0.278844i	−0.238683	+	0.971098i	$0.576716\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−13.8564	−	8.00000i	−0.637118	−	0.367840i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.00000	−	13.8564i	−0.365529	−	0.633115i	0.623332	−	0.781958i	$-0.285779\pi$
$479$	−8.00000	−	13.8564i	−0.365529	−	0.633115i	−0.988861	+	0.148842i	$0.952445\pi$
$480$		0			0
$481$		0			0
$482$		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
$491$	18.0000	−	31.1769i	0.812329	−	1.40699i	−0.0989017	−	0.995097i	$-0.531533\pi$
$491$	18.0000	−	31.1769i	0.812329	−	1.40699i	0.911230	−	0.411897i	$-0.135134\pi$
$492$		0			0
$493$	20.7846	−	12.0000i	0.936092	−	0.540453i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	12.0000	−	20.7846i	0.537194	−	0.930447i	−0.461860	−	0.886953i	$-0.652818\pi$
$499$	12.0000	−	20.7846i	0.537194	−	0.930447i	0.999054	−	0.0434940i	$-0.0138489\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	36.0000i		−	1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$		−	36.0000i		−	1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$		0			0
$505$	−2.00000	+	4.00000i	−0.0889988	+	0.177998i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	21.0000	−	36.3731i	0.930809	−	1.61221i	0.148866	−	0.988857i	$-0.452438\pi$
$509$	21.0000	−	36.3731i	0.930809	−	1.61221i	0.781943	−	0.623350i	$-0.214229\pi$
$510$		0			0
$511$	16.0000	+	27.7128i	0.707798	+	1.22594i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.535898	−	8.92820i	−0.0236145	−	0.393424i
$516$		0			0
$517$	−13.8564	−	8.00000i	−0.609404	−	0.351840i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−38.0000			−1.66481			−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000			−1.66481			−0.832405	+	0.554168i	$0.813037\pi$
$522$		0			0
$523$			28.0000i			1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$			28.0000i			1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−13.8564	−	8.00000i	−0.603595	−	0.348485i
$528$		0			0
$529$	−3.50000	−	6.06218i	−0.152174	−	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−22.3923	+	14.7846i	−0.968104	+	0.639194i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	36.0000			1.55063
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.46410	+	3.73205i	0.105551	+	0.159863i
$546$		0			0
$547$	24.2487	−	14.0000i	1.03680	−	0.598597i	0.117875	−	0.993028i	$-0.462392\pi$
$547$	24.2487	−	14.0000i	1.03680	−	0.598597i	0.918925	+	0.394432i	$0.129059\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−41.5692	−	24.0000i	−1.76770	−	1.02058i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	12.0000i		−	0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$		−	12.0000i		−	0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	17.3205	+	10.0000i	0.729972	+	0.421450i	0.818412	−	0.574632i	$-0.194855\pi$
$563$	17.3205	+	10.0000i	0.729972	+	0.421450i	−0.0884397	+	0.996082i	$0.528188\pi$
$564$		0			0
$565$	1.60770	+	26.7846i	0.0676362	+	1.12684i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.0000	+	22.5167i	0.544988	+	0.943948i	0.998608	+	0.0527519i	$0.0167993\pi$
$569$	13.0000	+	22.5167i	0.544988	+	0.943948i	−0.453619	+	0.891196i	$0.649867\pi$
$570$		0			0
$571$	−20.0000	+	34.6410i	−0.836974	+	1.44968i	0.0554391	+	0.998462i	$0.482344\pi$
$571$	−20.0000	+	34.6410i	−0.836974	+	1.44968i	−0.892413	+	0.451219i	$0.850989\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−16.0000	+	12.0000i	−0.667246	+	0.500435i
$576$		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
$581$	−8.00000	+	13.8564i	−0.331896	+	0.574861i
$582$		0			0
$583$	−41.5692	+	24.0000i	−1.72162	+	0.993978i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−31.1769	+	18.0000i	−1.28681	+	0.742940i	−0.978084	−	0.208212i	$-0.933236\pi$
$587$	−31.1769	+	18.0000i	−1.28681	+	0.742940i	−0.308725	+	0.951151i	$0.599902\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			36.0000i			1.47834i	0.673517	+	0.739171i	$0.264783\pi$
$593$			36.0000i			1.47834i	−0.673517	+	0.739171i	$0.735217\pi$
$594$		0			0
$595$	−16.0000	+	32.0000i	−0.655936	+	1.31187i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	−0.999938	−	0.0111569i	$-0.996449\pi$
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	0.509631	+	0.860393i	$0.329782\pi$
$600$		0			0
$601$	−19.0000	−	32.9090i	−0.775026	−	1.34238i	−0.934780	−	0.355228i	$-0.884403\pi$
$601$	−19.0000	−	32.9090i	−0.775026	−	1.34238i	0.159754	−	0.987157i	$-0.448930\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.1603	+	0.669873i	−0.453729	+	0.0272342i
$606$		0			0
$607$	24.2487	+	14.0000i	0.984225	+	0.568242i	0.903543	−	0.428497i	$-0.140957\pi$
$607$	24.2487	+	14.0000i	0.984225	+	0.568242i	0.0806818	+	0.996740i	$0.474290\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			8.00000i			0.323117i	0.986863	+	0.161558i	$0.0516520\pi$
$613$			8.00000i			0.323117i	−0.986863	+	0.161558i	$0.948348\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	38.1051	+	22.0000i	1.53405	+	0.885687i	0.999169	+	0.0407620i	$0.0129786\pi$
$617$	38.1051	+	22.0000i	1.53405	+	0.885687i	0.534885	+	0.844925i	$0.320355\pi$
$618$		0			0
$619$	−8.00000	−	13.8564i	−0.321547	−	0.556936i	0.659260	−	0.751915i	$-0.270870\pi$
$619$	−8.00000	−	13.8564i	−0.321547	−	0.556936i	−0.980807	+	0.194979i	$0.937536\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	34.6410	−	20.0000i	1.38786	−	0.801283i
$624$		0			0
$625$	−17.2846	−	18.0622i	−0.691384	−	0.722487i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	32.0000			1.27592
$630$		0			0
$631$	−28.0000			−1.11466			−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000			−1.11466			−0.557331	+	0.830290i	$0.688175\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.46410	+	4.92820i	−0.296204	+	0.195570i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7.00000	−	12.1244i	−0.276483	−	0.478883i	0.694025	−	0.719951i	$-0.255836\pi$
$641$	−7.00000	−	12.1244i	−0.276483	−	0.478883i	−0.970508	+	0.241068i	$0.922502\pi$
$642$		0			0
$643$	31.1769	+	18.0000i	1.22950	+	0.709851i	0.966925	−	0.255062i	$-0.0820957\pi$
$643$	31.1769	+	18.0000i	1.22950	+	0.709851i	0.262573	+	0.964912i	$0.415429\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			12.0000i			0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$			12.0000i			0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$		0			0
$649$	16.0000			0.628055
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.46410	+	2.00000i	0.135561	+	0.0782660i	0.566247	−	0.824236i	$-0.308395\pi$
$653$	3.46410	+	2.00000i	0.135561	+	0.0782660i	−0.430686	+	0.902502i	$0.641728\pi$
$654$		0			0
$655$	−26.7846	+	1.60770i	−1.04656	+	0.0628178i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14.0000	+	24.2487i	0.545363	+	0.944596i	0.998584	+	0.0531977i	$0.0169414\pi$
$659$	14.0000	+	24.2487i	0.545363	+	0.944596i	−0.453221	+	0.891398i	$0.649725\pi$
$660$		0			0
$661$	11.0000	−	19.0526i	0.427850	−	0.741059i	−0.568831	−	0.822454i	$-0.692604\pi$
$661$	11.0000	−	19.0526i	0.427850	−	0.741059i	0.996682	+	0.0813955i	$0.0259377\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	24.0000i		−	0.929284i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.00000	−	6.92820i	0.154418	−	0.267460i
$672$		0			0
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.46410	−	2.00000i	0.133136	−	0.0768662i	−0.431953	−	0.901896i	$-0.642175\pi$
$677$	3.46410	−	2.00000i	0.133136	−	0.0768662i	0.565089	+	0.825030i	$0.308842\pi$
$678$		0			0
$679$	16.0000	−	27.7128i	0.614024	−	1.06352i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			44.0000i			1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$			44.0000i			1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$		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
$691$	16.0000	+	27.7128i	0.608669	+	1.05425i	0.991460	+	0.130410i	$0.0416295\pi$
$691$	16.0000	+	27.7128i	0.608669	+	1.05425i	−0.382791	+	0.923835i	$0.625037\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	35.7128	−	2.14359i	1.35466	−	0.0813111i
$696$		0			0
$697$	34.6410	+	20.0000i	1.31212	+	0.757554i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	26.0000			0.982006			0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000			0.982006			0.491003	+	0.871158i	$0.336630\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.92820	+	4.00000i	0.260562	+	0.150435i
$708$		0			0
$709$	−3.00000	−	5.19615i	−0.112667	−	0.195146i	0.804178	−	0.594389i	$-0.202606\pi$
$709$	−3.00000	−	5.19615i	−0.112667	−	0.195146i	−0.916845	+	0.399244i	$0.869273\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−13.8564	+	8.00000i	−0.518927	+	0.299602i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8.00000			−0.298350			−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000			−0.298350			−0.149175	+	0.988811i	$0.547662\pi$
$720$		0			0
$721$	−16.0000			−0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	29.7846	−	3.58846i	1.10617	−	0.133272i
$726$		0			0
$727$	−10.3923	+	6.00000i	−0.385429	+	0.222528i	−0.680178	−	0.733047i	$-0.738097\pi$
$727$	−10.3923	+	6.00000i	−0.385429	+	0.222528i	0.294749	+	0.955575i	$0.404764\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.00000	−	13.8564i	−0.295891	−	0.512498i
$732$		0			0
$733$	−13.8564	−	8.00000i	−0.511798	−	0.295487i	0.221774	−	0.975098i	$-0.428815\pi$
$733$	−13.8564	−	8.00000i	−0.511798	−	0.295487i	−0.733572	+	0.679611i	$0.762148\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			16.0000i			0.589368i
$738$		0			0
$739$	−40.0000			−1.47142			−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000			−1.47142			−0.735712	+	0.677295i	$0.763152\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	31.1769	+	18.0000i	1.14377	+	0.660356i	0.947361	−	0.320166i	$-0.103739\pi$
$743$	31.1769	+	18.0000i	1.14377	+	0.660356i	0.196409	+	0.980522i	$0.437072\pi$
$744$		0			0
$745$	−4.46410	+	0.267949i	−0.163552	+	0.00981690i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	24.0000	+	41.5692i	0.876941	+	1.51891i
$750$		0			0
$751$	14.0000	−	24.2487i	0.510867	−	0.884848i	−0.489053	−	0.872254i	$-0.662658\pi$
$751$	14.0000	−	24.2487i	0.510867	−	0.884848i	0.999921	−	0.0125942i	$-0.00400897\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−20.0000	+	40.0000i	−0.727875	+	1.45575i
$756$		0			0
$757$			16.0000i			0.581530i	0.956795	+	0.290765i	$0.0939098\pi$
$757$			16.0000i			0.581530i	−0.956795	+	0.290765i	$0.906090\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−21.0000	+	36.3731i	−0.761249	+	1.31852i	0.180957	+	0.983491i	$0.442080\pi$
$761$	−21.0000	+	36.3731i	−0.761249	+	1.31852i	−0.942207	+	0.335032i	$0.891253\pi$
$762$		0			0
$763$	6.92820	−	4.00000i	0.250818	−	0.144810i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−9.00000	+	15.5885i	−0.324548	+	0.562134i	−0.981421	−	0.191867i	$-0.938546\pi$
$769$	−9.00000	+	15.5885i	−0.324548	+	0.562134i	0.656873	+	0.754002i	$0.271879\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			12.0000i			0.431610i	0.976436	+	0.215805i	$0.0692376\pi$
$773$			12.0000i			0.431610i	−0.976436	+	0.215805i	$0.930762\pi$
$774$		0			0
$775$	−12.0000	−	16.0000i	−0.431053	−	0.574737i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.07180	−	17.8564i	−0.0382541	−	0.637322i
$786$		0			0
$787$	−24.2487	−	14.0000i	−0.864373	−	0.499046i	0.00110111	−	0.999999i	$-0.499650\pi$
$787$	−24.2487	−	14.0000i	−0.864373	−	0.499046i	−0.865474	+	0.500953i	$0.832983\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	48.0000			1.70668
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−24.2487	−	14.0000i	−0.858933	−	0.495905i	0.00472155	−	0.999989i	$-0.498497\pi$
$797$	−24.2487	−	14.0000i	−0.858933	−	0.495905i	−0.863655	+	0.504083i	$0.831830\pi$
$798$		0			0
$799$	−8.00000	−	13.8564i	−0.283020	−	0.490204i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−27.7128	+	16.0000i	−0.977964	+	0.564628i
$804$		0			0
$805$	19.7128	+	29.8564i	0.694785	+	1.05230i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$	40.0000			1.40459			0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000			1.40459			0.702295	+	0.711886i	$0.252159\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	37.3205	−	24.6410i	1.30728	−	0.863137i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7.00000	+	12.1244i	0.244302	+	0.423143i	0.961935	−	0.273278i	$-0.0881079\pi$
$821$	7.00000	+	12.1244i	0.244302	+	0.423143i	−0.717633	+	0.696421i	$0.754775\pi$
$822$		0			0
$823$	10.3923	+	6.00000i	0.362253	+	0.209147i	0.670069	−	0.742299i	$-0.266265\pi$
$823$	10.3923	+	6.00000i	0.362253	+	0.209147i	−0.307816	+	0.951446i	$0.599598\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			28.0000i			0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$			28.0000i			0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$		0			0
$829$	18.0000			0.625166			0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000			0.625166			0.312583	+	0.949890i	$0.398806\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	31.1769	+	18.0000i	1.08022	+	0.623663i
$834$		0			0
$835$	−1.60770	−	26.7846i	−0.0556366	−	0.926920i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−28.0000	−	48.4974i	−0.966667	−	1.67432i	−0.705067	−	0.709141i	$-0.749083\pi$
$839$	−28.0000	−	48.4974i	−0.966667	−	1.67432i	−0.261600	−	0.965176i	$-0.584250\pi$
$840$		0			0
$841$	−3.50000	+	6.06218i	−0.120690	+	0.209041i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	13.0000	−	26.0000i	0.447214	−	0.894427i
$846$		0			0
$847$			20.0000i			0.687208i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	16.0000	−	27.7128i	0.548473	−	0.949983i
$852$		0			0
$853$	−6.92820	+	4.00000i	−0.237217	+	0.136957i	−0.613897	−	0.789386i	$-0.710399\pi$
$853$	−6.92820	+	4.00000i	−0.237217	+	0.136957i	0.376680	+	0.926343i	$0.377066\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−10.3923	+	6.00000i	−0.354994	+	0.204956i	−0.666883	−	0.745163i	$-0.732372\pi$
$857$	−10.3923	+	6.00000i	−0.354994	+	0.204956i	0.311888	+	0.950119i	$0.399038\pi$
$858$		0			0
$859$	−20.0000	+	34.6410i	−0.682391	+	1.18194i	0.291858	+	0.956462i	$0.405727\pi$
$859$	−20.0000	+	34.6410i	−0.682391	+	1.18194i	−0.974249	+	0.225475i	$0.927607\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			12.0000i			0.408485i	0.978920	+	0.204242i	$0.0654731\pi$
$863$			12.0000i			0.408485i	−0.978920	+	0.204242i	$0.934527\pi$
$864$		0			0
$865$	8.00000	+	4.00000i	0.272008	+	0.136004i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	24.0000	−	41.5692i	0.814144	−	1.41014i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−34.1051	+	28.9282i	−1.15296	+	0.977952i
$876$		0			0
$877$	−48.4974	−	28.0000i	−1.63764	−	0.945493i	−0.981642	−	0.190731i	$-0.938914\pi$
$877$	−48.4974	−	28.0000i	−1.63764	−	0.945493i	−0.655999	−	0.754761i	$-0.727753\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$			12.0000i			0.403832i	0.979403	+	0.201916i	$0.0647168\pi$
$883$			12.0000i			0.403832i	−0.979403	+	0.201916i	$0.935283\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	31.1769	+	18.0000i	1.04682	+	0.604381i	0.921757	−	0.387768i	$-0.126754\pi$
$887$	31.1769	+	18.0000i	1.04682	+	0.604381i	0.125061	+	0.992149i	$0.460087\pi$
$888$		0			0
$889$	8.00000	+	13.8564i	0.268311	+	0.464729i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−4.92820	−	7.46410i	−0.164732	−	0.249497i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	24.0000			0.800445
$900$		0			0
$901$	−48.0000			−1.59911
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12.3205	−	18.6603i	−0.409548	−	0.620288i
$906$		0			0
$907$	10.3923	−	6.00000i	0.345071	−	0.199227i	−0.317441	−	0.948278i	$-0.602824\pi$
$907$	10.3923	−	6.00000i	0.345071	−	0.199227i	0.662512	+	0.749051i	$0.269490\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−24.0000	−	41.5692i	−0.795155	−	1.37725i	−0.922740	−	0.385422i	$-0.874056\pi$
$911$	−24.0000	−	41.5692i	−0.795155	−	1.37725i	0.127585	−	0.991828i	$-0.459277\pi$
$912$		0			0
$913$	−13.8564	−	8.00000i	−0.458580	−	0.264761i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			48.0000i			1.58510i
$918$		0			0
$919$	12.0000			0.395843			0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000			0.395843			0.197922	+	0.980218i	$0.436581\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	36.7846	+	15.7128i	1.20947	+	0.516634i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9.00000	+	15.5885i	0.295280	+	0.511441i	0.975050	−	0.221985i	$-0.0712536\pi$
$929$	9.00000	+	15.5885i	0.295280	+	0.511441i	−0.679770	+	0.733426i	$0.737920\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−32.0000	−	16.0000i	−1.04651	−	0.523256i
$936$		0			0
$937$			32.0000i			1.04539i	0.852518	+	0.522697i	$0.175074\pi$
$937$			32.0000i			1.04539i	−0.852518	+	0.522697i	$0.824926\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5.00000	−	8.66025i	0.162995	−	0.282316i	−0.772946	−	0.634472i	$-0.781218\pi$
$941$	5.00000	−	8.66025i	0.162995	−	0.282316i	0.935942	+	0.352155i	$0.114551\pi$
$942$		0			0
$943$	34.6410	−	20.0000i	1.12807	−	0.651290i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	10.3923	−	6.00000i	0.337705	−	0.194974i	−0.321552	−	0.946892i	$-0.604204\pi$
$947$	10.3923	−	6.00000i	0.337705	−	0.194974i	0.659256	+	0.751918i	$0.270871\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			4.00000i			0.129573i	0.997899	+	0.0647864i	$0.0206366\pi$
$953$			4.00000i			0.129573i	−0.997899	+	0.0647864i	$0.979363\pi$
$954$		0			0
$955$	24.0000	−	48.0000i	0.776622	−	1.55324i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	24.0000	−	41.5692i	0.775000	−	1.34234i
$960$		0			0
$961$	7.50000	+	12.9904i	0.241935	+	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.14359	+	35.7128i	0.0690047	+	1.14964i
$966$		0			0
$967$	−17.3205	−	10.0000i	−0.556990	−	0.321578i	0.194946	−	0.980814i	$-0.437547\pi$
$967$	−17.3205	−	10.0000i	−0.556990	−	0.321578i	−0.751936	+	0.659236i	$0.770880\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$		−	64.0000i		−	2.05175i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	10.3923	+	6.00000i	0.332479	+	0.191957i	0.656941	−	0.753942i	$-0.271850\pi$
$977$	10.3923	+	6.00000i	0.332479	+	0.191957i	−0.324462	+	0.945899i	$0.605183\pi$
$978$		0			0
$979$	20.0000	+	34.6410i	0.639203	+	1.10713i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	24.2487	−	14.0000i	0.773414	−	0.446531i	−0.0606773	−	0.998157i	$-0.519326\pi$
$983$	24.2487	−	14.0000i	0.773414	−	0.446531i	0.834091	+	0.551627i	$0.185993\pi$
$984$		0			0
$985$	7.46410	−	4.92820i	0.237826	−	0.157026i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.0000			−0.508770
$990$		0			0
$991$	28.0000			0.889449			0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000			0.889449			0.444725	+	0.895667i	$0.353302\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	14.7846	+	22.3923i	0.468704	+	0.709884i
$996$		0			0
$997$	−20.7846	+	12.0000i	−0.658255	+	0.380044i	−0.791612	−	0.611024i	$-0.790758\pi$
$997$	−20.7846	+	12.0000i	−0.658255	+	0.380044i	0.133357	+	0.991068i	$0.457424\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.2.r.c.109.2		4
3.2	odd	2		1620.2.r.d.109.1		4
5.4	even	2	inner	1620.2.r.c.109.1		4
9.2	odd	6		1620.2.r.d.1189.2		4
9.4	even	3		60.2.d.a.49.1	✓	2
9.5	odd	6		180.2.d.a.109.2		2
9.7	even	3	inner	1620.2.r.c.1189.1		4
15.14	odd	2		1620.2.r.d.109.2		4
36.23	even	6		720.2.f.c.289.2		2
36.31	odd	6		240.2.f.b.49.2		2
45.4	even	6		60.2.d.a.49.2	yes	2
45.13	odd	12		300.2.a.d.1.1		1
45.14	odd	6		180.2.d.a.109.1		2
45.22	odd	12		300.2.a.a.1.1		1
45.23	even	12		900.2.a.h.1.1		1
45.29	odd	6		1620.2.r.d.1189.1		4
45.32	even	12		900.2.a.a.1.1		1
45.34	even	6	inner	1620.2.r.c.1189.2		4
63.4	even	3		2940.2.bb.d.1549.1		4
63.13	odd	6		2940.2.k.c.589.2		2
63.31	odd	6		2940.2.bb.e.1549.2		4
63.40	odd	6		2940.2.bb.e.949.1		4
63.58	even	3		2940.2.bb.d.949.2		4
72.5	odd	6		2880.2.f.l.1729.1		2
72.13	even	6		960.2.f.f.769.2		2
72.59	even	6		2880.2.f.p.1729.1		2
72.67	odd	6		960.2.f.c.769.1		2
144.13	even	12		3840.2.d.r.2689.1		2
144.67	odd	12		3840.2.d.b.2689.1		2
144.85	even	12		3840.2.d.o.2689.2		2
144.139	odd	12		3840.2.d.be.2689.2		2
180.23	odd	12		3600.2.a.d.1.1		1
180.59	even	6		720.2.f.c.289.1		2
180.67	even	12		1200.2.a.s.1.1		1
180.103	even	12		1200.2.a.a.1.1		1
180.139	odd	6		240.2.f.b.49.1		2
180.167	odd	12		3600.2.a.bm.1.1		1
315.4	even	6		2940.2.bb.d.1549.2		4
315.94	odd	6		2940.2.bb.e.1549.1		4
315.139	odd	6		2940.2.k.c.589.1		2
315.184	even	6		2940.2.bb.d.949.1		4
315.229	odd	6		2940.2.bb.e.949.2		4
360.13	odd	12		4800.2.a.bj.1.1		1
360.59	even	6		2880.2.f.p.1729.2		2
360.67	even	12		4800.2.a.bf.1.1		1
360.139	odd	6		960.2.f.c.769.2		2
360.149	odd	6		2880.2.f.l.1729.2		2
360.157	odd	12		4800.2.a.bn.1.1		1
360.229	even	6		960.2.f.f.769.1		2
360.283	even	12		4800.2.a.bk.1.1		1
720.139	odd	12		3840.2.d.b.2689.2		2
720.229	even	12		3840.2.d.r.2689.2		2
720.499	odd	12		3840.2.d.be.2689.1		2
720.589	even	12		3840.2.d.o.2689.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.2.d.a.49.1	✓	2	9.4	even	3
60.2.d.a.49.2	yes	2	45.4	even	6
180.2.d.a.109.1		2	45.14	odd	6
180.2.d.a.109.2		2	9.5	odd	6
240.2.f.b.49.1		2	180.139	odd	6
240.2.f.b.49.2		2	36.31	odd	6
300.2.a.a.1.1		1	45.22	odd	12
300.2.a.d.1.1		1	45.13	odd	12
720.2.f.c.289.1		2	180.59	even	6
720.2.f.c.289.2		2	36.23	even	6
900.2.a.a.1.1		1	45.32	even	12
900.2.a.h.1.1		1	45.23	even	12
960.2.f.c.769.1		2	72.67	odd	6
960.2.f.c.769.2		2	360.139	odd	6
960.2.f.f.769.1		2	360.229	even	6
960.2.f.f.769.2		2	72.13	even	6
1200.2.a.a.1.1		1	180.103	even	12
1200.2.a.s.1.1		1	180.67	even	12
1620.2.r.c.109.1		4	5.4	even	2	inner
1620.2.r.c.109.2		4	1.1	even	1	trivial
1620.2.r.c.1189.1		4	9.7	even	3	inner
1620.2.r.c.1189.2		4	45.34	even	6	inner
1620.2.r.d.109.1		4	3.2	odd	2
1620.2.r.d.109.2		4	15.14	odd	2
1620.2.r.d.1189.1		4	45.29	odd	6
1620.2.r.d.1189.2		4	9.2	odd	6
2880.2.f.l.1729.1		2	72.5	odd	6
2880.2.f.l.1729.2		2	360.149	odd	6
2880.2.f.p.1729.1		2	72.59	even	6
2880.2.f.p.1729.2		2	360.59	even	6
2940.2.k.c.589.1		2	315.139	odd	6
2940.2.k.c.589.2		2	63.13	odd	6
2940.2.bb.d.949.1		4	315.184	even	6
2940.2.bb.d.949.2		4	63.58	even	3
2940.2.bb.d.1549.1		4	63.4	even	3
2940.2.bb.d.1549.2		4	315.4	even	6
2940.2.bb.e.949.1		4	63.40	odd	6
2940.2.bb.e.949.2		4	315.229	odd	6
2940.2.bb.e.1549.1		4	315.94	odd	6
2940.2.bb.e.1549.2		4	63.31	odd	6
3600.2.a.d.1.1		1	180.23	odd	12
3600.2.a.bm.1.1		1	180.167	odd	12
3840.2.d.b.2689.1		2	144.67	odd	12
3840.2.d.b.2689.2		2	720.139	odd	12
3840.2.d.o.2689.1		2	720.589	even	12
3840.2.d.o.2689.2		2	144.85	even	12
3840.2.d.r.2689.1		2	144.13	even	12
3840.2.d.r.2689.2		2	720.229	even	12
3840.2.d.be.2689.1		2	720.499	odd	12
3840.2.d.be.2689.2		2	144.139	odd	12
4800.2.a.bf.1.1		1	360.67	even	12
4800.2.a.bj.1.1		1	360.13	odd	12
4800.2.a.bk.1.1		1	360.283	even	12
4800.2.a.bn.1.1		1	360.157	odd	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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.23205 + 1.86603i) q^{5} +(3.46410 - 2.00000i) q^{7} +O(q^{10})\)	\(q+(1.23205 + 1.86603i) q^{5} +(3.46410 - 2.00000i) q^{7} +(2.00000 + 3.46410i) q^{11} +4.00000i q^{17} +(3.46410 + 2.00000i) q^{23} +(-1.96410 + 4.59808i) q^{25} +(-3.00000 - 5.19615i) q^{29} +(-2.00000 + 3.46410i) q^{31} +(8.00000 + 4.00000i) q^{35} -8.00000i q^{37} +(5.00000 - 8.66025i) q^{41} +(-3.46410 + 2.00000i) q^{43} +(-3.46410 + 2.00000i) q^{47} +(4.50000 - 7.79423i) q^{49} +12.0000i q^{53} +(-4.00000 + 8.00000i) q^{55} +(2.00000 - 3.46410i) q^{59} +(-1.00000 - 1.73205i) q^{61} +(3.46410 + 2.00000i) q^{67} +8.00000i q^{73} +(13.8564 + 8.00000i) q^{77} +(-6.00000 - 10.3923i) q^{79} +(-3.46410 + 2.00000i) q^{83} +(-7.46410 + 4.92820i) q^{85} +10.0000 q^{89} +(6.92820 - 4.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5}+O(q^{10}) \) 4 * q - 2 * q^5	\( 4 q - 2 q^{5} + 8 q^{11} + 6 q^{25} - 12 q^{29} - 8 q^{31} + 32 q^{35} + 20 q^{41} + 18 q^{49} - 16 q^{55} + 8 q^{59} - 4 q^{61} - 24 q^{79} - 16 q^{85} + 40 q^{89}+O(q^{100}) \) 4 * q - 2 * q^5 + 8 * q^11 + 6 * q^25 - 12 * q^29 - 8 * q^31 + 32 * q^35 + 20 * q^41 + 18 * q^49 - 16 * q^55 + 8 * q^59 - 4 * q^61 - 24 * q^79 - 16 * q^85 + 40 * q^89

Introduction

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