LMFDB - Embedded newform 462.2.g.c.419.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [462,2,Mod(419,462)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

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

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

chi := DirichletCharacter("462.419");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 462 = 2 \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	462.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.68908857338$
Analytic rank:	$0$
Dimension:	$4$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			419.3
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	462.419
Dual form			462.2.g.c.419.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.73205 q^{3} -1.00000 q^{4} -1.73205i q^{6} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.73205 q^{3} -1.00000 q^{4} -1.73205i q^{6} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +3.00000 q^{9} +1.00000i q^{11} +1.73205 q^{12} +(-1.73205 + 2.00000i) q^{14} +1.00000 q^{16} -3.46410 q^{17} +3.00000i q^{18} +3.46410i q^{19} +(-3.46410 - 3.00000i) q^{21} -1.00000 q^{22} +6.00000i q^{23} +1.73205i q^{24} -5.00000 q^{25} -5.19615 q^{27} +(-2.00000 - 1.73205i) q^{28} +6.00000i q^{29} +3.46410i q^{31} +1.00000i q^{32} -1.73205i q^{33} -3.46410i q^{34} -3.00000 q^{36} -2.00000 q^{37} -3.46410 q^{38} -3.46410 q^{41} +(3.00000 - 3.46410i) q^{42} -2.00000 q^{43} -1.00000i q^{44} -6.00000 q^{46} +10.3923 q^{47} -1.73205 q^{48} +(1.00000 + 6.92820i) q^{49} -5.00000i q^{50} +6.00000 q^{51} -6.00000i q^{53} -5.19615i q^{54} +(1.73205 - 2.00000i) q^{56} -6.00000i q^{57} -6.00000 q^{58} +10.3923 q^{59} -6.92820i q^{61} -3.46410 q^{62} +(6.00000 + 5.19615i) q^{63} -1.00000 q^{64} +1.73205 q^{66} -4.00000 q^{67} +3.46410 q^{68} -10.3923i q^{69} +6.00000i q^{71} -3.00000i q^{72} +6.92820i q^{73} -2.00000i q^{74} +8.66025 q^{75} -3.46410i q^{76} +(-1.73205 + 2.00000i) q^{77} +8.00000 q^{79} +9.00000 q^{81} -3.46410i q^{82} -10.3923 q^{83} +(3.46410 + 3.00000i) q^{84} -2.00000i q^{86} -10.3923i q^{87} +1.00000 q^{88} +13.8564 q^{89} -6.00000i q^{92} -6.00000i q^{93} +10.3923i q^{94} -1.73205i q^{96} -6.92820i q^{97} +(-6.92820 + 1.00000i) q^{98} +3.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 8 q^{7} + 12 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 8 * q^7 + 12 * q^9	$ 4 q - 4 q^{4} + 8 q^{7} + 12 q^{9} + 4 q^{16} - 4 q^{22} - 20 q^{25} - 8 q^{28} - 12 q^{36} - 8 q^{37} + 12 q^{42} - 8 q^{43} - 24 q^{46} + 4 q^{49} + 24 q^{51} - 24 q^{58} + 24 q^{63} - 4 q^{64} - 16 q^{67} + 32 q^{79} + 36 q^{81} + 4 q^{88}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^7 + 12 * q^9 + 4 * q^16 - 4 * q^22 - 20 * q^25 - 8 * q^28 - 12 * q^36 - 8 * q^37 + 12 * q^42 - 8 * q^43 - 24 * q^46 + 4 * q^49 + 24 * q^51 - 24 * q^58 + 24 * q^63 - 4 * q^64 - 16 * q^67 + 32 * q^79 + 36 * q^81 + 4 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/462\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$211$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	−1.73205			−1.00000
$3$	−1.73205			−1.00000
$4$	−1.00000			−0.500000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		−	1.73205i		−	0.707107i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		−	1.00000i		−	0.353553i
$9$	3.00000			1.00000
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$	1.73205			0.500000
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−1.73205	+	2.00000i	−0.462910	+	0.534522i
$15$		0			0
$16$	1.00000			0.250000
$17$	−3.46410			−0.840168			−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410			−0.840168			−0.420084	+	0.907485i	$0.637999\pi$
$18$			3.00000i			0.707107i
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$	−3.46410	−	3.00000i	−0.755929	−	0.654654i
$22$	−1.00000			−0.213201
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$			1.73205i			0.353553i
$25$	−5.00000			−1.00000
$26$		0			0
$27$	−5.19615			−1.00000
$28$	−2.00000	−	1.73205i	−0.377964	−	0.327327i
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$			3.46410i			0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$			3.46410i			0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$			1.00000i			0.176777i
$33$		−	1.73205i		−	0.301511i
$34$		−	3.46410i		−	0.594089i
$35$		0			0
$36$	−3.00000			−0.500000
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$	−3.46410			−0.561951
$39$		0			0
$40$		0			0
$41$	−3.46410			−0.541002			−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410			−0.541002			−0.270501	+	0.962720i	$0.587189\pi$
$42$	3.00000	−	3.46410i	0.462910	−	0.534522i
$43$	−2.00000			−0.304997			−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000			−0.304997			−0.152499	+	0.988304i	$0.548732\pi$
$44$		−	1.00000i		−	0.150756i
$45$		0			0
$46$	−6.00000			−0.884652
$47$	10.3923			1.51587			0.757937	−	0.652328i	$-0.226208\pi$
$47$	10.3923			1.51587			0.757937	+	0.652328i	$0.226208\pi$
$48$	−1.73205			−0.250000
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		−	5.00000i		−	0.707107i
$51$	6.00000			0.840168
$52$		0			0
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		−	5.19615i		−	0.707107i
$55$		0			0
$56$	1.73205	−	2.00000i	0.231455	−	0.267261i
$57$		−	6.00000i		−	0.794719i
$58$	−6.00000			−0.787839
$59$	10.3923			1.35296			0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923			1.35296			0.676481	+	0.736460i	$0.263504\pi$
$60$		0			0
$61$		−	6.92820i		−	0.887066i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$		−	6.92820i		−	0.887066i	0.896258	−	0.443533i	$-0.146275\pi$
$62$	−3.46410			−0.439941
$63$	6.00000	+	5.19615i	0.755929	+	0.654654i
$64$	−1.00000			−0.125000
$65$		0			0
$66$	1.73205			0.213201
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$	3.46410			0.420084
$69$		−	10.3923i		−	1.25109i
$70$		0			0
$71$			6.00000i			0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$			6.00000i			0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$		−	3.00000i		−	0.353553i
$73$			6.92820i			0.810885i	0.914121	+	0.405442i	$0.132883\pi$
$73$			6.92820i			0.810885i	−0.914121	+	0.405442i	$0.867117\pi$
$74$		−	2.00000i		−	0.232495i
$75$	8.66025			1.00000
$76$		−	3.46410i		−	0.397360i
$77$	−1.73205	+	2.00000i	−0.197386	+	0.227921i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		−	3.46410i		−	0.382546i
$83$	−10.3923			−1.14070			−0.570352	−	0.821401i	$-0.693193\pi$
$83$	−10.3923			−1.14070			−0.570352	+	0.821401i	$0.693193\pi$
$84$	3.46410	+	3.00000i	0.377964	+	0.327327i
$85$		0			0
$86$		−	2.00000i		−	0.215666i
$87$		−	10.3923i		−	1.11417i
$88$	1.00000			0.106600
$89$	13.8564			1.46878			0.734388	−	0.678730i	$-0.237469\pi$
$89$	13.8564			1.46878			0.734388	+	0.678730i	$0.237469\pi$
$90$		0			0
$91$		0			0
$92$		−	6.00000i		−	0.625543i
$93$		−	6.00000i		−	0.622171i
$94$			10.3923i			1.07188i
$95$		0			0
$96$		−	1.73205i		−	0.176777i
$97$		−	6.92820i		−	0.703452i	−0.936103	−	0.351726i	$-0.885595\pi$
$97$		−	6.92820i		−	0.703452i	0.936103	−	0.351726i	$-0.114405\pi$
$98$	−6.92820	+	1.00000i	−0.699854	+	0.101015i
$99$			3.00000i			0.301511i
$100$	5.00000			0.500000
$101$	6.92820			0.689382			0.344691	−	0.938716i	$-0.387984\pi$
$101$	6.92820			0.689382			0.344691	+	0.938716i	$0.387984\pi$
$102$			6.00000i			0.594089i
$103$			17.3205i			1.70664i	0.521387	+	0.853320i	$0.325415\pi$
$103$			17.3205i			1.70664i	−0.521387	+	0.853320i	$0.674585\pi$
$104$		0			0
$105$		0			0
$106$	6.00000			0.582772
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	5.19615			0.500000
$109$	−20.0000			−1.91565			−0.957826	−	0.287348i	$-0.907226\pi$
$109$	−20.0000			−1.91565			−0.957826	+	0.287348i	$0.907226\pi$
$110$		0			0
$111$	3.46410			0.328798
$112$	2.00000	+	1.73205i	0.188982	+	0.163663i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	6.00000			0.561951
$115$		0			0
$116$		−	6.00000i		−	0.557086i
$117$		0			0
$118$			10.3923i			0.956689i
$119$	−6.92820	−	6.00000i	−0.635107	−	0.550019i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	6.92820			0.627250
$123$	6.00000			0.541002
$124$		−	3.46410i		−	0.311086i
$125$		0			0
$126$	−5.19615	+	6.00000i	−0.462910	+	0.534522i
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	3.46410			0.304997
$130$		0			0
$131$	3.46410			0.302660			0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410			0.302660			0.151330	+	0.988483i	$0.451644\pi$
$132$			1.73205i			0.150756i
$133$	−6.00000	+	6.92820i	−0.520266	+	0.600751i
$134$		−	4.00000i		−	0.345547i
$135$		0			0
$136$			3.46410i			0.297044i
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	10.3923			0.884652
$139$			10.3923i			0.881464i	0.897639	+	0.440732i	$0.145281\pi$
$139$			10.3923i			0.881464i	−0.897639	+	0.440732i	$0.854719\pi$
$140$		0			0
$141$	−18.0000			−1.51587
$142$	−6.00000			−0.503509
$143$		0			0
$144$	3.00000			0.250000
$145$		0			0
$146$	−6.92820			−0.573382
$147$	−1.73205	−	12.0000i	−0.142857	−	0.989743i
$148$	2.00000			0.164399
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$			8.66025i			0.707107i
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$	3.46410			0.280976
$153$	−10.3923			−0.840168
$154$	−2.00000	−	1.73205i	−0.161165	−	0.139573i
$155$		0			0
$156$		0			0
$157$		−	24.2487i		−	1.93526i	−0.252377	−	0.967629i	$-0.581212\pi$
$157$		−	24.2487i		−	1.93526i	0.252377	−	0.967629i	$-0.418788\pi$
$158$			8.00000i			0.636446i
$159$			10.3923i			0.824163i
$160$		0			0
$161$	−10.3923	+	12.0000i	−0.819028	+	0.945732i
$162$			9.00000i			0.707107i
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$	3.46410			0.270501
$165$		0			0
$166$		−	10.3923i		−	0.806599i
$167$	6.92820			0.536120			0.268060	−	0.963402i	$-0.413617\pi$
$167$	6.92820			0.536120			0.268060	+	0.963402i	$0.413617\pi$
$168$	−3.00000	+	3.46410i	−0.231455	+	0.267261i
$169$	13.0000			1.00000
$170$		0			0
$171$			10.3923i			0.794719i
$172$	2.00000			0.152499
$173$	6.92820			0.526742			0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820			0.526742			0.263371	+	0.964695i	$0.415166\pi$
$174$	10.3923			0.787839
$175$	−10.0000	−	8.66025i	−0.755929	−	0.654654i
$176$			1.00000i			0.0753778i
$177$	−18.0000			−1.35296
$178$			13.8564i			1.03858i
$179$		−	12.0000i		−	0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$		−	12.0000i		−	0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$		0			0
$181$		−	17.3205i		−	1.28742i	−0.765268	−	0.643712i	$-0.777394\pi$
$181$		−	17.3205i		−	1.28742i	0.765268	−	0.643712i	$-0.222606\pi$
$182$		0			0
$183$			12.0000i			0.887066i
$184$	6.00000			0.442326
$185$		0			0
$186$	6.00000			0.439941
$187$		−	3.46410i		−	0.253320i
$188$	−10.3923			−0.757937
$189$	−10.3923	−	9.00000i	−0.755929	−	0.654654i
$190$		0			0
$191$			6.00000i			0.434145i	0.976156	+	0.217072i	$0.0696508\pi$
$191$			6.00000i			0.434145i	−0.976156	+	0.217072i	$0.930349\pi$
$192$	1.73205			0.125000
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$	6.92820			0.497416
$195$		0			0
$196$	−1.00000	−	6.92820i	−0.0714286	−	0.494872i
$197$			6.00000i			0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$			6.00000i			0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	−3.00000			−0.213201
$199$		−	3.46410i		−	0.245564i	−0.992434	−	0.122782i	$-0.960818\pi$
$199$		−	3.46410i		−	0.245564i	0.992434	−	0.122782i	$-0.0391815\pi$
$200$			5.00000i			0.353553i
$201$	6.92820			0.488678
$202$			6.92820i			0.487467i
$203$	−10.3923	+	12.0000i	−0.729397	+	0.842235i
$204$	−6.00000			−0.420084
$205$		0			0
$206$	−17.3205			−1.20678
$207$			18.0000i			1.25109i
$208$		0			0
$209$	−3.46410			−0.239617
$210$		0			0
$211$	22.0000			1.51454			0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000			1.51454			0.757271	+	0.653101i	$0.226532\pi$
$212$			6.00000i			0.412082i
$213$		−	10.3923i		−	0.712069i
$214$	12.0000			0.820303
$215$		0			0
$216$			5.19615i			0.353553i
$217$	−6.00000	+	6.92820i	−0.407307	+	0.470317i
$218$		−	20.0000i		−	1.35457i
$219$		−	12.0000i		−	0.810885i
$220$		0			0
$221$		0			0
$222$			3.46410i			0.232495i
$223$		−	24.2487i		−	1.62381i	−0.583787	−	0.811907i	$-0.698430\pi$
$223$		−	24.2487i		−	1.62381i	0.583787	−	0.811907i	$-0.301570\pi$
$224$	−1.73205	+	2.00000i	−0.115728	+	0.133631i
$225$	−15.0000			−1.00000
$226$		0			0
$227$	−17.3205			−1.14960			−0.574801	−	0.818293i	$-0.694921\pi$
$227$	−17.3205			−1.14960			−0.574801	+	0.818293i	$0.694921\pi$
$228$			6.00000i			0.397360i
$229$		−	10.3923i		−	0.686743i	−0.939200	−	0.343371i	$-0.888431\pi$
$229$		−	10.3923i		−	0.686743i	0.939200	−	0.343371i	$-0.111569\pi$
$230$		0			0
$231$	3.00000	−	3.46410i	0.197386	−	0.227921i
$232$	6.00000			0.393919
$233$			18.0000i			1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$			18.0000i			1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$		0			0
$235$		0			0
$236$	−10.3923			−0.676481
$237$	−13.8564			−0.900070
$238$	6.00000	−	6.92820i	0.388922	−	0.449089i
$239$		−	12.0000i		−	0.776215i	−0.921614	−	0.388108i	$-0.873129\pi$
$239$		−	12.0000i		−	0.776215i	0.921614	−	0.388108i	$-0.126871\pi$
$240$		0			0
$241$		−	20.7846i		−	1.33885i	−0.742878	−	0.669427i	$-0.766540\pi$
$241$		−	20.7846i		−	1.33885i	0.742878	−	0.669427i	$-0.233460\pi$
$242$		−	1.00000i		−	0.0642824i
$243$	−15.5885			−1.00000
$244$			6.92820i			0.443533i
$245$		0			0
$246$			6.00000i			0.382546i
$247$		0			0
$248$	3.46410			0.219971
$249$	18.0000			1.14070
$250$		0			0
$251$	−17.3205			−1.09326			−0.546630	−	0.837374i	$-0.684090\pi$
$251$	−17.3205			−1.09326			−0.546630	+	0.837374i	$0.684090\pi$
$252$	−6.00000	−	5.19615i	−0.377964	−	0.327327i
$253$	−6.00000			−0.377217
$254$			8.00000i			0.501965i
$255$		0			0
$256$	1.00000			0.0625000
$257$	6.92820			0.432169			0.216085	−	0.976375i	$-0.430671\pi$
$257$	6.92820			0.432169			0.216085	+	0.976375i	$0.430671\pi$
$258$			3.46410i			0.215666i
$259$	−4.00000	−	3.46410i	−0.248548	−	0.215249i
$260$		0			0
$261$			18.0000i			1.11417i
$262$			3.46410i			0.214013i
$263$			12.0000i			0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$			12.0000i			0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	−1.73205			−0.106600
$265$		0			0
$266$	−6.92820	−	6.00000i	−0.424795	−	0.367884i
$267$	−24.0000			−1.46878
$268$	4.00000			0.244339
$269$	27.7128			1.68968			0.844840	−	0.535019i	$-0.179696\pi$
$269$	27.7128			1.68968			0.844840	+	0.535019i	$0.179696\pi$
$270$		0			0
$271$			31.1769i			1.89386i	0.321436	+	0.946931i	$0.395835\pi$
$271$			31.1769i			1.89386i	−0.321436	+	0.946931i	$0.604165\pi$
$272$	−3.46410			−0.210042
$273$		0			0
$274$	12.0000			0.724947
$275$		−	5.00000i		−	0.301511i
$276$			10.3923i			0.625543i
$277$	8.00000			0.480673			0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000			0.480673			0.240337	+	0.970690i	$0.422742\pi$
$278$	−10.3923			−0.623289
$279$			10.3923i			0.622171i
$280$		0			0
$281$			18.0000i			1.07379i	0.843649	+	0.536895i	$0.180403\pi$
$281$			18.0000i			1.07379i	−0.843649	+	0.536895i	$0.819597\pi$
$282$		−	18.0000i		−	1.07188i
$283$		−	17.3205i		−	1.02960i	−0.857311	−	0.514799i	$-0.827867\pi$
$283$		−	17.3205i		−	1.02960i	0.857311	−	0.514799i	$-0.172133\pi$
$284$		−	6.00000i		−	0.356034i
$285$		0			0
$286$		0			0
$287$	−6.92820	−	6.00000i	−0.408959	−	0.354169i
$288$			3.00000i			0.176777i
$289$	−5.00000			−0.294118
$290$		0			0
$291$			12.0000i			0.703452i
$292$		−	6.92820i		−	0.405442i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	12.0000	−	1.73205i	0.699854	−	0.101015i
$295$		0			0
$296$			2.00000i			0.116248i
$297$		−	5.19615i		−	0.301511i
$298$	−6.00000			−0.347571
$299$		0			0
$300$	−8.66025			−0.500000
$301$	−4.00000	−	3.46410i	−0.230556	−	0.199667i
$302$			16.0000i			0.920697i
$303$	−12.0000			−0.689382
$304$			3.46410i			0.198680i
$305$		0			0
$306$		−	10.3923i		−	0.594089i
$307$			24.2487i			1.38395i	0.721923	+	0.691974i	$0.243259\pi$
$307$			24.2487i			1.38395i	−0.721923	+	0.691974i	$0.756741\pi$
$308$	1.73205	−	2.00000i	0.0986928	−	0.113961i
$309$		−	30.0000i		−	1.70664i
$310$		0			0
$311$	−3.46410			−0.196431			−0.0982156	−	0.995165i	$-0.531313\pi$
$311$	−3.46410			−0.196431			−0.0982156	+	0.995165i	$0.531313\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	24.2487			1.36843
$315$		0			0
$316$	−8.00000			−0.450035
$317$			30.0000i			1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$			30.0000i			1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$	−10.3923			−0.582772
$319$	−6.00000			−0.335936
$320$		0			0
$321$			20.7846i			1.16008i
$322$	−12.0000	−	10.3923i	−0.668734	−	0.579141i
$323$		−	12.0000i		−	0.667698i
$324$	−9.00000			−0.500000
$325$		0			0
$326$		−	16.0000i		−	0.886158i
$327$	34.6410			1.91565
$328$			3.46410i			0.191273i
$329$	20.7846	+	18.0000i	1.14589	+	0.992372i
$330$		0			0
$331$	28.0000			1.53902			0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000			1.53902			0.769510	+	0.638635i	$0.220501\pi$
$332$	10.3923			0.570352
$333$	−6.00000			−0.328798
$334$			6.92820i			0.379094i
$335$		0			0
$336$	−3.46410	−	3.00000i	−0.188982	−	0.163663i
$337$	22.0000			1.19842			0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000			1.19842			0.599208	+	0.800593i	$0.295482\pi$
$338$			13.0000i			0.707107i
$339$		0			0
$340$		0			0
$341$	−3.46410			−0.187592
$342$	−10.3923			−0.561951
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$			2.00000i			0.107833i
$345$		0			0
$346$			6.92820i			0.372463i
$347$			12.0000i			0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$			12.0000i			0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$			10.3923i			0.557086i
$349$			34.6410i			1.85429i	0.374701	+	0.927146i	$0.377745\pi$
$349$			34.6410i			1.85429i	−0.374701	+	0.927146i	$0.622255\pi$
$350$	8.66025	−	10.0000i	0.462910	−	0.534522i
$351$		0			0
$352$	−1.00000			−0.0533002
$353$	−6.92820			−0.368751			−0.184376	−	0.982856i	$-0.559026\pi$
$353$	−6.92820			−0.368751			−0.184376	+	0.982856i	$0.559026\pi$
$354$		−	18.0000i		−	0.956689i
$355$		0			0
$356$	−13.8564			−0.734388
$357$	12.0000	+	10.3923i	0.635107	+	0.550019i
$358$	12.0000			0.634220
$359$		−	36.0000i		−	1.90001i	−0.312239	−	0.950004i	$-0.601079\pi$
$359$		−	36.0000i		−	1.90001i	0.312239	−	0.950004i	$-0.398921\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$	17.3205			0.910346
$363$	1.73205			0.0909091
$364$		0			0
$365$		0			0
$366$	−12.0000			−0.627250
$367$		−	10.3923i		−	0.542474i	−0.962513	−	0.271237i	$-0.912567\pi$
$367$		−	10.3923i		−	0.542474i	0.962513	−	0.271237i	$-0.0874327\pi$
$368$			6.00000i			0.312772i
$369$	−10.3923			−0.541002
$370$		0			0
$371$	10.3923	−	12.0000i	0.539542	−	0.623009i
$372$			6.00000i			0.311086i
$373$	−4.00000			−0.207112			−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000			−0.207112			−0.103556	+	0.994624i	$0.533022\pi$
$374$	3.46410			0.179124
$375$		0			0
$376$		−	10.3923i		−	0.535942i
$377$		0			0
$378$	9.00000	−	10.3923i	0.462910	−	0.534522i
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$	−13.8564			−0.709885
$382$	−6.00000			−0.306987
$383$	−17.3205			−0.885037			−0.442518	−	0.896759i	$-0.645915\pi$
$383$	−17.3205			−0.885037			−0.442518	+	0.896759i	$0.645915\pi$
$384$			1.73205i			0.0883883i
$385$		0			0
$386$		−	2.00000i		−	0.101797i
$387$	−6.00000			−0.304997
$388$			6.92820i			0.351726i
$389$		−	18.0000i		−	0.912636i	−0.889817	−	0.456318i	$-0.849168\pi$
$389$		−	18.0000i		−	0.912636i	0.889817	−	0.456318i	$-0.150832\pi$
$390$		0			0
$391$		−	20.7846i		−	1.05112i
$392$	6.92820	−	1.00000i	0.349927	−	0.0505076i
$393$	−6.00000			−0.302660
$394$	−6.00000			−0.302276
$395$		0			0
$396$		−	3.00000i		−	0.150756i
$397$		−	17.3205i		−	0.869291i	−0.900602	−	0.434646i	$-0.856874\pi$
$397$		−	17.3205i		−	0.869291i	0.900602	−	0.434646i	$-0.143126\pi$
$398$	3.46410			0.173640
$399$	10.3923	−	12.0000i	0.520266	−	0.600751i
$400$	−5.00000			−0.250000
$401$		−	36.0000i		−	1.79775i	−0.438201	−	0.898877i	$-0.644384\pi$
$401$		−	36.0000i		−	1.79775i	0.438201	−	0.898877i	$-0.355616\pi$
$402$			6.92820i			0.345547i
$403$		0			0
$404$	−6.92820			−0.344691
$405$		0			0
$406$	−12.0000	−	10.3923i	−0.595550	−	0.515761i
$407$		−	2.00000i		−	0.0991363i
$408$		−	6.00000i		−	0.297044i
$409$		−	6.92820i		−	0.342578i	−0.985221	−	0.171289i	$-0.945207\pi$
$409$		−	6.92820i		−	0.342578i	0.985221	−	0.171289i	$-0.0547931\pi$
$410$		0			0
$411$			20.7846i			1.02523i
$412$		−	17.3205i		−	0.853320i
$413$	20.7846	+	18.0000i	1.02274	+	0.885722i
$414$	−18.0000			−0.884652
$415$		0			0
$416$		0			0
$417$		−	18.0000i		−	0.881464i
$418$		−	3.46410i		−	0.169435i
$419$	10.3923			0.507697			0.253849	−	0.967244i	$-0.418303\pi$
$419$	10.3923			0.507697			0.253849	+	0.967244i	$0.418303\pi$
$420$		0			0
$421$	2.00000			0.0974740			0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000			0.0974740			0.0487370	+	0.998812i	$0.484480\pi$
$422$			22.0000i			1.07094i
$423$	31.1769			1.51587
$424$	−6.00000			−0.291386
$425$	17.3205			0.840168
$426$	10.3923			0.503509
$427$	12.0000	−	13.8564i	0.580721	−	0.670559i
$428$			12.0000i			0.580042i
$429$		0			0
$430$		0			0
$431$			12.0000i			0.578020i	0.957326	+	0.289010i	$0.0933260\pi$
$431$			12.0000i			0.578020i	−0.957326	+	0.289010i	$0.906674\pi$
$432$	−5.19615			−0.250000
$433$			27.7128i			1.33179i	0.746044	+	0.665896i	$0.231951\pi$
$433$			27.7128i			1.33179i	−0.746044	+	0.665896i	$0.768049\pi$
$434$	−6.92820	−	6.00000i	−0.332564	−	0.288009i
$435$		0			0
$436$	20.0000			0.957826
$437$	−20.7846			−0.994263
$438$	12.0000			0.573382
$439$			3.46410i			0.165333i	0.996577	+	0.0826663i	$0.0263436\pi$
$439$			3.46410i			0.165333i	−0.996577	+	0.0826663i	$0.973656\pi$
$440$		0			0
$441$	3.00000	+	20.7846i	0.142857	+	0.989743i
$442$		0			0
$443$		−	24.0000i		−	1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$		−	24.0000i		−	1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	−3.46410			−0.164399
$445$		0			0
$446$	24.2487			1.14821
$447$		−	10.3923i		−	0.491539i
$448$	−2.00000	−	1.73205i	−0.0944911	−	0.0818317i
$449$		−	24.0000i		−	1.13263i	−0.824189	−	0.566315i	$-0.808369\pi$
$449$		−	24.0000i		−	1.13263i	0.824189	−	0.566315i	$-0.191631\pi$
$450$		−	15.0000i		−	0.707107i
$451$		−	3.46410i		−	0.163118i
$452$		0			0
$453$	−27.7128			−1.30206
$454$		−	17.3205i		−	0.812892i
$455$		0			0
$456$	−6.00000			−0.280976
$457$	34.0000			1.59045			0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000			1.59045			0.795226	+	0.606313i	$0.207352\pi$
$458$	10.3923			0.485601
$459$	18.0000			0.840168
$460$		0			0
$461$	34.6410			1.61339			0.806696	−	0.590966i	$-0.201253\pi$
$461$	34.6410			1.61339			0.806696	+	0.590966i	$0.201253\pi$
$462$	3.46410	+	3.00000i	0.161165	+	0.139573i
$463$	8.00000			0.371792			0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000			0.371792			0.185896	+	0.982569i	$0.440481\pi$
$464$			6.00000i			0.278543i
$465$		0			0
$466$	−18.0000			−0.833834
$467$	−10.3923			−0.480899			−0.240449	−	0.970662i	$-0.577295\pi$
$467$	−10.3923			−0.480899			−0.240449	+	0.970662i	$0.577295\pi$
$468$		0			0
$469$	−8.00000	−	6.92820i	−0.369406	−	0.319915i
$470$		0			0
$471$			42.0000i			1.93526i
$472$		−	10.3923i		−	0.478345i
$473$		−	2.00000i		−	0.0919601i
$474$		−	13.8564i		−	0.636446i
$475$		−	17.3205i		−	0.794719i
$476$	6.92820	+	6.00000i	0.317554	+	0.275010i
$477$		−	18.0000i		−	0.824163i
$478$	12.0000			0.548867
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	20.7846			0.946713
$483$	18.0000	−	20.7846i	0.819028	−	0.945732i
$484$	1.00000			0.0454545
$485$		0			0
$486$		−	15.5885i		−	0.707107i
$487$	−32.0000			−1.45006			−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000			−1.45006			−0.725029	+	0.688718i	$0.758174\pi$
$488$	−6.92820			−0.313625
$489$	27.7128			1.25322
$490$		0			0
$491$			12.0000i			0.541552i	0.962642	+	0.270776i	$0.0872803\pi$
$491$			12.0000i			0.541552i	−0.962642	+	0.270776i	$0.912720\pi$
$492$	−6.00000			−0.270501
$493$		−	20.7846i		−	0.936092i
$494$		0			0
$495$		0			0
$496$			3.46410i			0.155543i
$497$	−10.3923	+	12.0000i	−0.466159	+	0.538274i
$498$			18.0000i			0.806599i
$499$	16.0000			0.716258			0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000			0.716258			0.358129	+	0.933672i	$0.383415\pi$
$500$		0			0
$501$	−12.0000			−0.536120
$502$		−	17.3205i		−	0.773052i
$503$	−20.7846			−0.926740			−0.463370	−	0.886165i	$-0.653360\pi$
$503$	−20.7846			−0.926740			−0.463370	+	0.886165i	$0.653360\pi$
$504$	5.19615	−	6.00000i	0.231455	−	0.267261i
$505$		0			0
$506$		−	6.00000i		−	0.266733i
$507$	−22.5167			−1.00000
$508$	−8.00000			−0.354943
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−12.0000	+	13.8564i	−0.530849	+	0.612971i
$512$			1.00000i			0.0441942i
$513$		−	18.0000i		−	0.794719i
$514$			6.92820i			0.305590i
$515$		0			0
$516$	−3.46410			−0.152499
$517$			10.3923i			0.457053i
$518$	3.46410	−	4.00000i	0.152204	−	0.175750i
$519$	−12.0000			−0.526742
$520$		0			0
$521$	−6.92820			−0.303530			−0.151765	−	0.988417i	$-0.548496\pi$
$521$	−6.92820			−0.303530			−0.151765	+	0.988417i	$0.548496\pi$
$522$	−18.0000			−0.787839
$523$			17.3205i			0.757373i	0.925525	+	0.378686i	$0.123624\pi$
$523$			17.3205i			0.757373i	−0.925525	+	0.378686i	$0.876376\pi$
$524$	−3.46410			−0.151330
$525$	17.3205	+	15.0000i	0.755929	+	0.654654i
$526$	−12.0000			−0.523225
$527$		−	12.0000i		−	0.522728i
$528$		−	1.73205i		−	0.0753778i
$529$	−13.0000			−0.565217
$530$		0			0
$531$	31.1769			1.35296
$532$	6.00000	−	6.92820i	0.260133	−	0.300376i
$533$		0			0
$534$		−	24.0000i		−	1.03858i
$535$		0			0
$536$			4.00000i			0.172774i
$537$			20.7846i			0.896922i
$538$			27.7128i			1.19478i
$539$	−6.92820	+	1.00000i	−0.298419	+	0.0430730i
$540$		0			0
$541$	−16.0000			−0.687894			−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000			−0.687894			−0.343947	+	0.938989i	$0.611764\pi$
$542$	−31.1769			−1.33916
$543$			30.0000i			1.28742i
$544$		−	3.46410i		−	0.148522i
$545$		0			0
$546$		0			0
$547$	2.00000			0.0855138			0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000			0.0855138			0.0427569	+	0.999086i	$0.486386\pi$
$548$			12.0000i			0.512615i
$549$		−	20.7846i		−	0.887066i
$550$	5.00000			0.213201
$551$	−20.7846			−0.885454
$552$	−10.3923			−0.442326
$553$	16.0000	+	13.8564i	0.680389	+	0.589234i
$554$			8.00000i			0.339887i
$555$		0			0
$556$		−	10.3923i		−	0.440732i
$557$		−	18.0000i		−	0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$		−	18.0000i		−	0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	−10.3923			−0.439941
$559$		0			0
$560$		0			0
$561$			6.00000i			0.253320i
$562$	−18.0000			−0.759284
$563$	17.3205			0.729972			0.364986	−	0.931013i	$-0.381074\pi$
$563$	17.3205			0.729972			0.364986	+	0.931013i	$0.381074\pi$
$564$	18.0000			0.757937
$565$		0			0
$566$	17.3205			0.728035
$567$	18.0000	+	15.5885i	0.755929	+	0.654654i
$568$	6.00000			0.251754
$569$		−	18.0000i		−	0.754599i	−0.926091	−	0.377300i	$-0.876853\pi$
$569$		−	18.0000i		−	0.754599i	0.926091	−	0.377300i	$-0.123147\pi$
$570$		0			0
$571$	−22.0000			−0.920671			−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000			−0.920671			−0.460336	+	0.887745i	$0.652271\pi$
$572$		0			0
$573$		−	10.3923i		−	0.434145i
$574$	6.00000	−	6.92820i	0.250435	−	0.289178i
$575$		−	30.0000i		−	1.25109i
$576$	−3.00000			−0.125000
$577$			34.6410i			1.44212i	0.692870	+	0.721062i	$0.256346\pi$
$577$			34.6410i			1.44212i	−0.692870	+	0.721062i	$0.743654\pi$
$578$		−	5.00000i		−	0.207973i
$579$	3.46410			0.143963
$580$		0			0
$581$	−20.7846	−	18.0000i	−0.862291	−	0.746766i
$582$	−12.0000			−0.497416
$583$	6.00000			0.248495
$584$	6.92820			0.286691
$585$		0			0
$586$		0			0
$587$	17.3205			0.714894			0.357447	−	0.933933i	$-0.383647\pi$
$587$	17.3205			0.714894			0.357447	+	0.933933i	$0.383647\pi$
$588$	1.73205	+	12.0000i	0.0714286	+	0.494872i
$589$	−12.0000			−0.494451
$590$		0			0
$591$		−	10.3923i		−	0.427482i
$592$	−2.00000			−0.0821995
$593$	−31.1769			−1.28028			−0.640141	−	0.768257i	$-0.721124\pi$
$593$	−31.1769			−1.28028			−0.640141	+	0.768257i	$0.721124\pi$
$594$	5.19615			0.213201
$595$		0			0
$596$		−	6.00000i		−	0.245770i
$597$			6.00000i			0.245564i
$598$		0			0
$599$		−	30.0000i		−	1.22577i	−0.790173	−	0.612883i	$-0.790010\pi$
$599$		−	30.0000i		−	1.22577i	0.790173	−	0.612883i	$-0.209990\pi$
$600$		−	8.66025i		−	0.353553i
$601$			6.92820i			0.282607i	0.989966	+	0.141304i	$0.0451294\pi$
$601$			6.92820i			0.282607i	−0.989966	+	0.141304i	$0.954871\pi$
$602$	3.46410	−	4.00000i	0.141186	−	0.163028i
$603$	−12.0000			−0.488678
$604$	−16.0000			−0.651031
$605$		0			0
$606$		−	12.0000i		−	0.487467i
$607$			10.3923i			0.421811i	0.977506	+	0.210905i	$0.0676412\pi$
$607$			10.3923i			0.421811i	−0.977506	+	0.210905i	$0.932359\pi$
$608$	−3.46410			−0.140488
$609$	18.0000	−	20.7846i	0.729397	−	0.842235i
$610$		0			0
$611$		0			0
$612$	10.3923			0.420084
$613$	−16.0000			−0.646234			−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000			−0.646234			−0.323117	+	0.946359i	$0.604731\pi$
$614$	−24.2487			−0.978598
$615$		0			0
$616$	2.00000	+	1.73205i	0.0805823	+	0.0697863i
$617$			36.0000i			1.44931i	0.689114	+	0.724653i	$0.258000\pi$
$617$			36.0000i			1.44931i	−0.689114	+	0.724653i	$0.742000\pi$
$618$	30.0000			1.20678
$619$			41.5692i			1.67081i	0.549636	+	0.835404i	$0.314766\pi$
$619$			41.5692i			1.67081i	−0.549636	+	0.835404i	$0.685234\pi$
$620$		0			0
$621$		−	31.1769i		−	1.25109i
$622$		−	3.46410i		−	0.138898i
$623$	27.7128	+	24.0000i	1.11029	+	0.961540i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	6.00000			0.239617
$628$			24.2487i			0.967629i
$629$	6.92820			0.276246
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		−	8.00000i		−	0.318223i
$633$	−38.1051			−1.51454
$634$	−30.0000			−1.19145
$635$		0			0
$636$		−	10.3923i		−	0.412082i
$637$		0			0
$638$		−	6.00000i		−	0.237542i
$639$			18.0000i			0.712069i
$640$		0			0
$641$			12.0000i			0.473972i	0.971513	+	0.236986i	$0.0761595\pi$
$641$			12.0000i			0.473972i	−0.971513	+	0.236986i	$0.923841\pi$
$642$	−20.7846			−0.820303
$643$		−	34.6410i		−	1.36611i	−0.730368	−	0.683054i	$-0.760651\pi$
$643$		−	34.6410i		−	1.36611i	0.730368	−	0.683054i	$-0.239349\pi$
$644$	10.3923	−	12.0000i	0.409514	−	0.472866i
$645$		0			0
$646$	12.0000			0.472134
$647$	−38.1051			−1.49807			−0.749033	−	0.662532i	$-0.769482\pi$
$647$	−38.1051			−1.49807			−0.749033	+	0.662532i	$0.769482\pi$
$648$		−	9.00000i		−	0.353553i
$649$			10.3923i			0.407934i
$650$		0			0
$651$	10.3923	−	12.0000i	0.407307	−	0.470317i
$652$	16.0000			0.626608
$653$			30.0000i			1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$			30.0000i			1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$			34.6410i			1.35457i
$655$		0			0
$656$	−3.46410			−0.135250
$657$			20.7846i			0.810885i
$658$	−18.0000	+	20.7846i	−0.701713	+	0.810268i
$659$			36.0000i			1.40236i	0.712984	+	0.701180i	$0.247343\pi$
$659$			36.0000i			1.40236i	−0.712984	+	0.701180i	$0.752657\pi$
$660$		0			0
$661$			24.2487i			0.943166i	0.881822	+	0.471583i	$0.156317\pi$
$661$			24.2487i			0.943166i	−0.881822	+	0.471583i	$0.843683\pi$
$662$			28.0000i			1.08825i
$663$		0			0
$664$			10.3923i			0.403300i
$665$		0			0
$666$		−	6.00000i		−	0.232495i
$667$	−36.0000			−1.39393
$668$	−6.92820			−0.268060
$669$			42.0000i			1.62381i
$670$		0			0
$671$	6.92820			0.267460
$672$	3.00000	−	3.46410i	0.115728	−	0.133631i
$673$	10.0000			0.385472			0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000			0.385472			0.192736	+	0.981251i	$0.438264\pi$
$674$			22.0000i			0.847408i
$675$	25.9808			1.00000
$676$	−13.0000			−0.500000
$677$	−13.8564			−0.532545			−0.266272	−	0.963898i	$-0.585792\pi$
$677$	−13.8564			−0.532545			−0.266272	+	0.963898i	$0.585792\pi$
$678$		0			0
$679$	12.0000	−	13.8564i	0.460518	−	0.531760i
$680$		0			0
$681$	30.0000			1.14960
$682$		−	3.46410i		−	0.132647i
$683$			36.0000i			1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$			36.0000i			1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$		−	10.3923i		−	0.397360i
$685$		0			0
$686$	−15.5885	−	10.0000i	−0.595170	−	0.381802i
$687$			18.0000i			0.686743i
$688$	−2.00000			−0.0762493
$689$		0			0
$690$		0			0
$691$			34.6410i			1.31781i	0.752228	+	0.658903i	$0.228979\pi$
$691$			34.6410i			1.31781i	−0.752228	+	0.658903i	$0.771021\pi$
$692$	−6.92820			−0.263371
$693$	−5.19615	+	6.00000i	−0.197386	+	0.227921i
$694$	−12.0000			−0.455514
$695$		0			0
$696$	−10.3923			−0.393919
$697$	12.0000			0.454532
$698$	−34.6410			−1.31118
$699$		−	31.1769i		−	1.17922i
$700$	10.0000	+	8.66025i	0.377964	+	0.327327i
$701$		−	6.00000i		−	0.226617i	−0.993560	−	0.113308i	$-0.963855\pi$
$701$		−	6.00000i		−	0.226617i	0.993560	−	0.113308i	$-0.0361448\pi$
$702$		0			0
$703$		−	6.92820i		−	0.261302i
$704$		−	1.00000i		−	0.0376889i
$705$		0			0
$706$		−	6.92820i		−	0.260746i
$707$	13.8564	+	12.0000i	0.521124	+	0.451306i
$708$	18.0000			0.676481
$709$	−2.00000			−0.0751116			−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000			−0.0751116			−0.0375558	+	0.999295i	$0.511957\pi$
$710$		0			0
$711$	24.0000			0.900070
$712$		−	13.8564i		−	0.519291i
$713$	−20.7846			−0.778390
$714$	−10.3923	+	12.0000i	−0.388922	+	0.449089i
$715$		0			0
$716$			12.0000i			0.448461i
$717$			20.7846i			0.776215i
$718$	36.0000			1.34351
$719$	−45.0333			−1.67946			−0.839730	−	0.543005i	$-0.817287\pi$
$719$	−45.0333			−1.67946			−0.839730	+	0.543005i	$0.817287\pi$
$720$		0			0
$721$	−30.0000	+	34.6410i	−1.11726	+	1.29010i
$722$			7.00000i			0.260513i
$723$			36.0000i			1.33885i
$724$			17.3205i			0.643712i
$725$		−	30.0000i		−	1.11417i
$726$			1.73205i			0.0642824i
$727$		−	51.9615i		−	1.92715i	−0.267445	−	0.963573i	$-0.586179\pi$
$727$		−	51.9615i		−	1.92715i	0.267445	−	0.963573i	$-0.413821\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	6.92820			0.256249
$732$		−	12.0000i		−	0.443533i
$733$		−	20.7846i		−	0.767697i	−0.923396	−	0.383849i	$-0.874598\pi$
$733$		−	20.7846i		−	0.767697i	0.923396	−	0.383849i	$-0.125402\pi$
$734$	10.3923			0.383587
$735$		0			0
$736$	−6.00000			−0.221163
$737$		−	4.00000i		−	0.147342i
$738$		−	10.3923i		−	0.382546i
$739$	−10.0000			−0.367856			−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000			−0.367856			−0.183928	+	0.982940i	$0.558881\pi$
$740$		0			0
$741$		0			0
$742$	12.0000	+	10.3923i	0.440534	+	0.381514i
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$	−6.00000			−0.219971
$745$		0			0
$746$		−	4.00000i		−	0.146450i
$747$	−31.1769			−1.14070
$748$			3.46410i			0.126660i
$749$	20.7846	−	24.0000i	0.759453	−	0.876941i
$750$		0			0
$751$	40.0000			1.45962			0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000			1.45962			0.729810	+	0.683650i	$0.239608\pi$
$752$	10.3923			0.378968
$753$	30.0000			1.09326
$754$		0			0
$755$		0			0
$756$	10.3923	+	9.00000i	0.377964	+	0.327327i
$757$	34.0000			1.23575			0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000			1.23575			0.617876	+	0.786276i	$0.287994\pi$
$758$			16.0000i			0.581146i
$759$	10.3923			0.377217
$760$		0			0
$761$	−51.9615			−1.88360			−0.941802	−	0.336168i	$-0.890869\pi$
$761$	−51.9615			−1.88360			−0.941802	+	0.336168i	$0.890869\pi$
$762$		−	13.8564i		−	0.501965i
$763$	−40.0000	−	34.6410i	−1.44810	−	1.25409i
$764$		−	6.00000i		−	0.217072i
$765$		0			0
$766$		−	17.3205i		−	0.625815i
$767$		0			0
$768$	−1.73205			−0.0625000
$769$			48.4974i			1.74886i	0.485150	+	0.874431i	$0.338765\pi$
$769$			48.4974i			1.74886i	−0.485150	+	0.874431i	$0.661235\pi$
$770$		0			0
$771$	−12.0000			−0.432169
$772$	2.00000			0.0719816
$773$	−13.8564			−0.498380			−0.249190	−	0.968455i	$-0.580164\pi$
$773$	−13.8564			−0.498380			−0.249190	+	0.968455i	$0.580164\pi$
$774$		−	6.00000i		−	0.215666i
$775$		−	17.3205i		−	0.622171i
$776$	−6.92820			−0.248708
$777$	6.92820	+	6.00000i	0.248548	+	0.215249i
$778$	18.0000			0.645331
$779$		−	12.0000i		−	0.429945i
$780$		0			0
$781$	−6.00000			−0.214697
$782$	20.7846			0.743256
$783$		−	31.1769i		−	1.11417i
$784$	1.00000	+	6.92820i	0.0357143	+	0.247436i
$785$		0			0
$786$		−	6.00000i		−	0.214013i
$787$			10.3923i			0.370446i	0.982697	+	0.185223i	$0.0593007\pi$
$787$			10.3923i			0.370446i	−0.982697	+	0.185223i	$0.940699\pi$
$788$		−	6.00000i		−	0.213741i
$789$		−	20.7846i		−	0.739952i
$790$		0			0
$791$		0			0
$792$	3.00000			0.106600
$793$		0			0
$794$	17.3205			0.614682
$795$		0			0
$796$			3.46410i			0.122782i
$797$	−13.8564			−0.490819			−0.245410	−	0.969419i	$-0.578922\pi$
$797$	−13.8564			−0.490819			−0.245410	+	0.969419i	$0.578922\pi$
$798$	12.0000	+	10.3923i	0.424795	+	0.367884i
$799$	−36.0000			−1.27359
$800$		−	5.00000i		−	0.176777i
$801$	41.5692			1.46878
$802$	36.0000			1.27120
$803$	−6.92820			−0.244491
$804$	−6.92820			−0.244339
$805$		0			0
$806$		0			0
$807$	−48.0000			−1.68968
$808$		−	6.92820i		−	0.243733i
$809$			30.0000i			1.05474i	0.849635	+	0.527372i	$0.176823\pi$
$809$			30.0000i			1.05474i	−0.849635	+	0.527372i	$0.823177\pi$
$810$		0			0
$811$			3.46410i			0.121641i	0.998149	+	0.0608205i	$0.0193717\pi$
$811$			3.46410i			0.121641i	−0.998149	+	0.0608205i	$0.980628\pi$
$812$	10.3923	−	12.0000i	0.364698	−	0.421117i
$813$		−	54.0000i		−	1.89386i
$814$	2.00000			0.0701000
$815$		0			0
$816$	6.00000			0.210042
$817$		−	6.92820i		−	0.242387i
$818$	6.92820			0.242239
$819$		0			0
$820$		0			0
$821$		−	6.00000i		−	0.209401i	−0.994504	−	0.104701i	$-0.966612\pi$
$821$		−	6.00000i		−	0.209401i	0.994504	−	0.104701i	$-0.0333885\pi$
$822$	−20.7846			−0.724947
$823$	−40.0000			−1.39431			−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000			−1.39431			−0.697156	+	0.716919i	$0.745552\pi$
$824$	17.3205			0.603388
$825$			8.66025i			0.301511i
$826$	−18.0000	+	20.7846i	−0.626300	+	0.723189i
$827$		−	36.0000i		−	1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$		−	36.0000i		−	1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$		−	18.0000i		−	0.625543i
$829$		−	24.2487i		−	0.842193i	−0.907016	−	0.421096i	$-0.861645\pi$
$829$		−	24.2487i		−	0.842193i	0.907016	−	0.421096i	$-0.138355\pi$
$830$		0			0
$831$	−13.8564			−0.480673
$832$		0			0
$833$	−3.46410	−	24.0000i	−0.120024	−	0.831551i
$834$	18.0000			0.623289
$835$		0			0
$836$	3.46410			0.119808
$837$		−	18.0000i		−	0.622171i
$838$			10.3923i			0.358996i
$839$	17.3205			0.597970			0.298985	−	0.954258i	$-0.403352\pi$
$839$	17.3205			0.597970			0.298985	+	0.954258i	$0.403352\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$			2.00000i			0.0689246i
$843$		−	31.1769i		−	1.07379i
$844$	−22.0000			−0.757271
$845$		0			0
$846$			31.1769i			1.07188i
$847$	−2.00000	−	1.73205i	−0.0687208	−	0.0595140i
$848$		−	6.00000i		−	0.206041i
$849$			30.0000i			1.02960i
$850$			17.3205i			0.594089i
$851$		−	12.0000i		−	0.411355i
$852$			10.3923i			0.356034i
$853$		−	27.7128i		−	0.948869i	−0.880291	−	0.474434i	$-0.842653\pi$
$853$		−	27.7128i		−	0.948869i	0.880291	−	0.474434i	$-0.157347\pi$
$854$	13.8564	+	12.0000i	0.474156	+	0.410632i
$855$		0			0
$856$	−12.0000			−0.410152
$857$	51.9615			1.77497			0.887486	−	0.460835i	$-0.152450\pi$
$857$	51.9615			1.77497			0.887486	+	0.460835i	$0.152450\pi$
$858$		0			0
$859$		−	34.6410i		−	1.18194i	−0.806695	−	0.590968i	$-0.798746\pi$
$859$		−	34.6410i		−	1.18194i	0.806695	−	0.590968i	$-0.201254\pi$
$860$		0			0
$861$	12.0000	+	10.3923i	0.408959	+	0.354169i
$862$	−12.0000			−0.408722
$863$		−	42.0000i		−	1.42970i	−0.699280	−	0.714848i	$-0.746496\pi$
$863$		−	42.0000i		−	1.42970i	0.699280	−	0.714848i	$-0.253504\pi$
$864$		−	5.19615i		−	0.176777i
$865$		0			0
$866$	−27.7128			−0.941720
$867$	8.66025			0.294118
$868$	6.00000	−	6.92820i	0.203653	−	0.235159i
$869$			8.00000i			0.271381i
$870$		0			0
$871$		0			0
$872$			20.0000i			0.677285i
$873$		−	20.7846i		−	0.703452i
$874$		−	20.7846i		−	0.703050i
$875$		0			0
$876$			12.0000i			0.405442i
$877$	−40.0000			−1.35070			−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000			−1.35070			−0.675352	+	0.737496i	$0.736008\pi$
$878$	−3.46410			−0.116908
$879$		0			0
$880$		0			0
$881$	41.5692			1.40050			0.700251	−	0.713896i	$-0.253071\pi$
$881$	41.5692			1.40050			0.700251	+	0.713896i	$0.253071\pi$
$882$	−20.7846	+	3.00000i	−0.699854	+	0.101015i
$883$	−8.00000			−0.269221			−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000			−0.269221			−0.134611	+	0.990899i	$0.542978\pi$
$884$		0			0
$885$		0			0
$886$	24.0000			0.806296
$887$	−13.8564			−0.465253			−0.232626	−	0.972566i	$-0.574732\pi$
$887$	−13.8564			−0.465253			−0.232626	+	0.972566i	$0.574732\pi$
$888$		−	3.46410i		−	0.116248i
$889$	16.0000	+	13.8564i	0.536623	+	0.464729i
$890$		0			0
$891$			9.00000i			0.301511i
$892$			24.2487i			0.811907i
$893$			36.0000i			1.20469i
$894$	10.3923			0.347571
$895$		0			0
$896$	1.73205	−	2.00000i	0.0578638	−	0.0668153i
$897$		0			0
$898$	24.0000			0.800890
$899$	−20.7846			−0.693206
$900$	15.0000			0.500000
$901$			20.7846i			0.692436i
$902$	3.46410			0.115342
$903$	6.92820	+	6.00000i	0.230556	+	0.199667i
$904$		0			0
$905$		0			0
$906$		−	27.7128i		−	0.920697i
$907$	−8.00000			−0.265636			−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000			−0.265636			−0.132818	+	0.991140i	$0.542403\pi$
$908$	17.3205			0.574801
$909$	20.7846			0.689382
$910$		0			0
$911$			6.00000i			0.198789i	0.995048	+	0.0993944i	$0.0316906\pi$
$911$			6.00000i			0.198789i	−0.995048	+	0.0993944i	$0.968309\pi$
$912$		−	6.00000i		−	0.198680i
$913$		−	10.3923i		−	0.343935i
$914$			34.0000i			1.12462i
$915$		0			0
$916$			10.3923i			0.343371i
$917$	6.92820	+	6.00000i	0.228789	+	0.198137i
$918$			18.0000i			0.594089i
$919$	−56.0000			−1.84727			−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000			−1.84727			−0.923635	+	0.383274i	$0.874797\pi$
$920$		0			0
$921$		−	42.0000i		−	1.38395i
$922$			34.6410i			1.14084i
$923$		0			0
$924$	−3.00000	+	3.46410i	−0.0986928	+	0.113961i
$925$	10.0000			0.328798
$926$			8.00000i			0.262896i
$927$			51.9615i			1.70664i
$928$	−6.00000			−0.196960
$929$	−13.8564			−0.454614			−0.227307	−	0.973823i	$-0.572992\pi$
$929$	−13.8564			−0.454614			−0.227307	+	0.973823i	$0.572992\pi$
$930$		0			0
$931$	−24.0000	+	3.46410i	−0.786568	+	0.113531i
$932$		−	18.0000i		−	0.589610i
$933$	6.00000			0.196431
$934$		−	10.3923i		−	0.340047i
$935$		0			0
$936$		0			0
$937$			6.92820i			0.226335i	0.993576	+	0.113167i	$0.0360996\pi$
$937$			6.92820i			0.226335i	−0.993576	+	0.113167i	$0.963900\pi$
$938$	6.92820	−	8.00000i	0.226214	−	0.261209i
$939$		0			0
$940$		0			0
$941$	−55.4256			−1.80682			−0.903412	−	0.428774i	$-0.858946\pi$
$941$	−55.4256			−1.80682			−0.903412	+	0.428774i	$0.858946\pi$
$942$	−42.0000			−1.36843
$943$		−	20.7846i		−	0.676840i
$944$	10.3923			0.338241
$945$		0			0
$946$	2.00000			0.0650256
$947$			24.0000i			0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$			24.0000i			0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	13.8564			0.450035
$949$		0			0
$950$	17.3205			0.561951
$951$		−	51.9615i		−	1.68497i
$952$	−6.00000	+	6.92820i	−0.194461	+	0.224544i
$953$			30.0000i			0.971795i	0.874016	+	0.485898i	$0.161507\pi$
$953$			30.0000i			0.971795i	−0.874016	+	0.485898i	$0.838493\pi$
$954$	18.0000			0.582772
$955$		0			0
$956$			12.0000i			0.388108i
$957$	10.3923			0.335936
$958$		0			0
$959$	20.7846	−	24.0000i	0.671170	−	0.775000i
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$		−	36.0000i		−	1.16008i
$964$			20.7846i			0.669427i
$965$		0			0
$966$	20.7846	+	18.0000i	0.668734	+	0.579141i
$967$	32.0000			1.02905			0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000			1.02905			0.514525	+	0.857475i	$0.327968\pi$
$968$			1.00000i			0.0321412i
$969$			20.7846i			0.667698i
$970$		0			0
$971$	45.0333			1.44519			0.722594	−	0.691273i	$-0.242950\pi$
$971$	45.0333			1.44519			0.722594	+	0.691273i	$0.242950\pi$
$972$	15.5885			0.500000
$973$	−18.0000	+	20.7846i	−0.577054	+	0.666324i
$974$		−	32.0000i		−	1.02535i
$975$		0			0
$976$		−	6.92820i		−	0.221766i
$977$		−	48.0000i		−	1.53566i	−0.640656	−	0.767828i	$-0.721338\pi$
$977$		−	48.0000i		−	1.53566i	0.640656	−	0.767828i	$-0.278662\pi$
$978$			27.7128i			0.886158i
$979$			13.8564i			0.442853i
$980$		0			0
$981$	−60.0000			−1.91565
$982$	−12.0000			−0.382935
$983$	17.3205			0.552438			0.276219	−	0.961095i	$-0.410918\pi$
$983$	17.3205			0.552438			0.276219	+	0.961095i	$0.410918\pi$
$984$		−	6.00000i		−	0.191273i
$985$		0			0
$986$	20.7846			0.661917
$987$	−36.0000	−	31.1769i	−1.14589	−	0.992372i
$988$		0			0
$989$		−	12.0000i		−	0.381578i
$990$		0			0
$991$	40.0000			1.27064			0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000			1.27064			0.635321	+	0.772248i	$0.280868\pi$
$992$	−3.46410			−0.109985
$993$	−48.4974			−1.53902
$994$	−12.0000	−	10.3923i	−0.380617	−	0.329624i
$995$		0			0
$996$	−18.0000			−0.570352
$997$		−	27.7128i		−	0.877674i	−0.898567	−	0.438837i	$-0.855391\pi$
$997$		−	27.7128i		−	0.877674i	0.898567	−	0.438837i	$-0.144609\pi$
$998$			16.0000i			0.506471i
$999$	10.3923			0.328798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	462.2.g.c.419.3	yes	4
3.2	odd	2	inner	462.2.g.c.419.2	yes	4
7.6	odd	2	inner	462.2.g.c.419.4	yes	4
21.20	even	2	inner	462.2.g.c.419.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.g.c.419.1	✓	4	21.20	even	2	inner
462.2.g.c.419.2	yes	4	3.2	odd	2	inner
462.2.g.c.419.3	yes	4	1.1	even	1	trivial
462.2.g.c.419.4	yes	4	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	462.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.73205 q^{3} -1.00000 q^{4} -1.73205i q^{6} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.73205 q^{3} -1.00000 q^{4} -1.73205i q^{6} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +3.00000 q^{9} +1.00000i q^{11} +1.73205 q^{12} +(-1.73205 + 2.00000i) q^{14} +1.00000 q^{16} -3.46410 q^{17} +3.00000i q^{18} +3.46410i q^{19} +(-3.46410 - 3.00000i) q^{21} -1.00000 q^{22} +6.00000i q^{23} +1.73205i q^{24} -5.00000 q^{25} -5.19615 q^{27} +(-2.00000 - 1.73205i) q^{28} +6.00000i q^{29} +3.46410i q^{31} +1.00000i q^{32} -1.73205i q^{33} -3.46410i q^{34} -3.00000 q^{36} -2.00000 q^{37} -3.46410 q^{38} -3.46410 q^{41} +(3.00000 - 3.46410i) q^{42} -2.00000 q^{43} -1.00000i q^{44} -6.00000 q^{46} +10.3923 q^{47} -1.73205 q^{48} +(1.00000 + 6.92820i) q^{49} -5.00000i q^{50} +6.00000 q^{51} -6.00000i q^{53} -5.19615i q^{54} +(1.73205 - 2.00000i) q^{56} -6.00000i q^{57} -6.00000 q^{58} +10.3923 q^{59} -6.92820i q^{61} -3.46410 q^{62} +(6.00000 + 5.19615i) q^{63} -1.00000 q^{64} +1.73205 q^{66} -4.00000 q^{67} +3.46410 q^{68} -10.3923i q^{69} +6.00000i q^{71} -3.00000i q^{72} +6.92820i q^{73} -2.00000i q^{74} +8.66025 q^{75} -3.46410i q^{76} +(-1.73205 + 2.00000i) q^{77} +8.00000 q^{79} +9.00000 q^{81} -3.46410i q^{82} -10.3923 q^{83} +(3.46410 + 3.00000i) q^{84} -2.00000i q^{86} -10.3923i q^{87} +1.00000 q^{88} +13.8564 q^{89} -6.00000i q^{92} -6.00000i q^{93} +10.3923i q^{94} -1.73205i q^{96} -6.92820i q^{97} +(-6.92820 + 1.00000i) q^{98} +3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 8 * q^7 + 12 * q^9	\( 4 q - 4 q^{4} + 8 q^{7} + 12 q^{9} + 4 q^{16} - 4 q^{22} - 20 q^{25} - 8 q^{28} - 12 q^{36} - 8 q^{37} + 12 q^{42} - 8 q^{43} - 24 q^{46} + 4 q^{49} + 24 q^{51} - 24 q^{58} + 24 q^{63} - 4 q^{64} - 16 q^{67} + 32 q^{79} + 36 q^{81} + 4 q^{88}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^7 + 12 * q^9 + 4 * q^16 - 4 * q^22 - 20 * q^25 - 8 * q^28 - 12 * q^36 - 8 * q^37 + 12 * q^42 - 8 * q^43 - 24 * q^46 + 4 * q^49 + 24 * q^51 - 24 * q^58 + 24 * q^63 - 4 * q^64 - 16 * q^67 + 32 * q^79 + 36 * q^81 + 4 * q^88

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