LMFDB - Embedded newform 2268.2.w.a.1349.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(269,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.269"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.w (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-3,0,-5,0,0,0,-9,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$18.1100711784$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1349.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2268.1349
Dual form			2268.2.w.a.269.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 2.59808i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(-4.50000 - 2.59808i) q^{11} +(-1.50000 - 2.59808i) q^{17} +(1.50000 + 0.866025i) q^{19} +(4.50000 - 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} -1.73205i q^{31} +(1.50000 + 7.79423i) q^{35} +(-3.50000 + 6.06218i) q^{37} +(-3.00000 + 5.19615i) q^{41} +(-2.00000 - 3.46410i) q^{43} +3.00000 q^{47} +(5.50000 + 4.33013i) q^{49} +(-4.50000 + 2.59808i) q^{53} +15.5885i q^{55} -3.00000 q^{59} +12.1244i q^{61} +5.00000 q^{67} -10.3923i q^{71} +(-10.5000 + 6.06218i) q^{73} +(9.00000 + 10.3923i) q^{77} -1.00000 q^{79} +(6.00000 + 10.3923i) q^{83} +(-4.50000 + 7.79423i) q^{85} +(4.50000 - 7.79423i) q^{89} -5.19615i q^{95} +(-6.00000 + 3.46410i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{5} - 5 q^{7} - 9 q^{11} - 3 q^{17} + 3 q^{19} + 9 q^{23} - 4 q^{25} + 3 q^{35} - 7 q^{37} - 6 q^{41} - 4 q^{43} + 6 q^{47} + 11 q^{49} - 9 q^{53} - 6 q^{59} + 10 q^{67} - 21 q^{73} + 18 q^{77}+ \cdots - 12 q^{97}+O(q^{100}) $ 2 * q - 3 * q^5 - 5 * q^7 - 9 * q^11 - 3 * q^17 + 3 * q^19 + 9 * q^23 - 4 * q^25 + 3 * q^35 - 7 * q^37 - 6 * q^41 - 4 * q^43 + 6 * q^47 + 11 * q^49 - 9 * q^53 - 6 * q^59 + 10 * q^67 - 21 * q^73 + 18 * q^77 - 2 * q^79 + 12 * q^83 - 9 * q^85 + 9 * q^89 - 12 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	−1.50000	−	2.59808i	−0.670820	−	1.16190i	−0.977672	−	0.210138i	$-0.932609\pi$
$5$	−1.50000	−	2.59808i	−0.670820	−	1.16190i	0.306851	−	0.951757i	$-0.400725\pi$
$6$		0			0
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.50000	−	2.59808i	−1.35680	−	0.783349i	−0.367610	−	0.929980i	$-0.619824\pi$
$11$	−4.50000	−	2.59808i	−1.35680	−	0.783349i	−0.989191	+	0.146631i	$0.953157\pi$
$12$		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
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	0.624780	−	0.780801i	$-0.285189\pi$
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	−0.988583	+	0.150675i	$0.951855\pi$
$18$		0			0
$19$	1.50000	+	0.866025i	0.344124	+	0.198680i	0.662094	−	0.749421i	$-0.269668\pi$
$19$	1.50000	+	0.866025i	0.344124	+	0.198680i	−0.317970	+	0.948101i	$0.603001\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.50000	−	2.59808i	0.938315	−	0.541736i	0.0488832	−	0.998805i	$-0.484434\pi$
$23$	4.50000	−	2.59808i	0.938315	−	0.541736i	0.889432	+	0.457068i	$0.151100\pi$
$24$		0			0
$25$	−2.00000	+	3.46410i	−0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$		−	1.73205i		−	0.311086i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$		−	1.73205i		−	0.311086i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.50000	+	7.79423i	0.253546	+	1.31747i
$36$		0			0
$37$	−3.50000	+	6.06218i	−0.575396	+	0.996616i	0.420602	+	0.907245i	$0.361819\pi$
$37$	−3.50000	+	6.06218i	−0.575396	+	0.996616i	−0.995998	+	0.0893706i	$0.971514\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	−0.999353	−	0.0359748i	$-0.988546\pi$
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	0.530831	+	0.847477i	$0.321880\pi$
$42$		0			0
$43$	−2.00000	−	3.46410i	−0.304997	−	0.528271i	0.672264	−	0.740312i	$-0.265322\pi$
$43$	−2.00000	−	3.46410i	−0.304997	−	0.528271i	−0.977261	+	0.212041i	$0.931989\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.00000			0.437595			0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000			0.437595			0.218797	+	0.975770i	$0.429787\pi$
$48$		0			0
$49$	5.50000	+	4.33013i	0.785714	+	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.50000	+	2.59808i	−0.618123	+	0.356873i	−0.776138	−	0.630563i	$-0.782824\pi$
$53$	−4.50000	+	2.59808i	−0.618123	+	0.356873i	0.158015	+	0.987437i	$0.449491\pi$
$54$		0			0
$55$			15.5885i			2.10195i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.00000			−0.390567			−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000			−0.390567			−0.195283	+	0.980747i	$0.562563\pi$
$60$		0			0
$61$			12.1244i			1.55236i	0.630509	+	0.776182i	$0.282846\pi$
$61$			12.1244i			1.55236i	−0.630509	+	0.776182i	$0.717154\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.00000			0.610847			0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000			0.610847			0.305424	+	0.952217i	$0.401202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.3923i		−	1.23334i	−0.787222	−	0.616670i	$-0.788481\pi$
$71$		−	10.3923i		−	1.23334i	0.787222	−	0.616670i	$-0.211519\pi$
$72$		0			0
$73$	−10.5000	+	6.06218i	−1.22893	+	0.709524i	−0.966807	−	0.255510i	$-0.917757\pi$
$73$	−10.5000	+	6.06218i	−1.22893	+	0.709524i	−0.262126	+	0.965034i	$0.584423\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.00000	+	10.3923i	1.02565	+	1.18431i
$78$		0			0
$79$	−1.00000			−0.112509			−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000			−0.112509			−0.0562544	+	0.998416i	$0.517916\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	0.980982	+	0.194099i	$0.0621783\pi$
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	−0.322396	+	0.946605i	$0.604488\pi$
$84$		0			0
$85$	−4.50000	+	7.79423i	−0.488094	+	0.845403i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.50000	−	7.79423i	0.476999	−	0.826187i	−0.522654	−	0.852545i	$-0.675058\pi$
$89$	4.50000	−	7.79423i	0.476999	−	0.826187i	0.999653	+	0.0263586i	$0.00839118\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	5.19615i		−	0.533114i
$96$		0			0
$97$	−6.00000	+	3.46410i	−0.609208	+	0.351726i	−0.772655	−	0.634826i	$-0.781072\pi$
$97$	−6.00000	+	3.46410i	−0.609208	+	0.351726i	0.163448	+	0.986552i	$0.447739\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.50000	−	7.79423i	0.447767	−	0.775555i	−0.550474	−	0.834853i	$-0.685553\pi$
$101$	4.50000	−	7.79423i	0.447767	−	0.775555i	0.998240	+	0.0592978i	$0.0188862\pi$
$102$		0			0
$103$	4.50000	−	2.59808i	0.443398	−	0.255996i	−0.261640	−	0.965166i	$-0.584263\pi$
$103$	4.50000	−	2.59808i	0.443398	−	0.255996i	0.705038	+	0.709170i	$0.250930\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.5000	+	7.79423i	1.30509	+	0.753497i	0.981273	−	0.192622i	$-0.0616990\pi$
$107$	13.5000	+	7.79423i	1.30509	+	0.753497i	0.323821	+	0.946118i	$0.395032\pi$
$108$		0			0
$109$	8.50000	+	14.7224i	0.814152	+	1.41015i	0.909935	+	0.414751i	$0.136131\pi$
$109$	8.50000	+	14.7224i	0.814152	+	1.41015i	−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−18.0000	−	10.3923i	−1.69330	−	0.977626i	−0.951825	−	0.306643i	$-0.900794\pi$
$113$	−18.0000	−	10.3923i	−1.69330	−	0.977626i	−0.741473	−	0.670983i	$-0.765872\pi$
$114$		0			0
$115$	−13.5000	−	7.79423i	−1.25888	−	0.726816i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.50000	+	7.79423i	0.137505	+	0.714496i
$120$		0			0
$121$	8.00000	+	13.8564i	0.727273	+	1.25967i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.00000			−0.268328
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.50000	+	7.79423i	0.393167	+	0.680985i	0.992865	−	0.119241i	$-0.0380462\pi$
$131$	4.50000	+	7.79423i	0.393167	+	0.680985i	−0.599699	+	0.800226i	$0.704713\pi$
$132$		0			0
$133$	−3.00000	−	3.46410i	−0.260133	−	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.50000	+	2.59808i	0.384461	+	0.221969i	0.679757	−	0.733437i	$-0.262085\pi$
$137$	4.50000	+	2.59808i	0.384461	+	0.221969i	−0.295296	+	0.955406i	$0.595418\pi$
$138$		0			0
$139$	−9.00000	−	5.19615i	−0.763370	−	0.440732i	0.0671344	−	0.997744i	$-0.478614\pi$
$139$	−9.00000	−	5.19615i	−0.763370	−	0.440732i	−0.830504	+	0.557012i	$0.811948\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.50000	−	2.59808i	0.368654	−	0.212843i	−0.304216	−	0.952603i	$-0.598394\pi$
$149$	4.50000	−	2.59808i	0.368654	−	0.212843i	0.672870	+	0.739760i	$0.265061\pi$
$150$		0			0
$151$	−6.50000	+	11.2583i	−0.528962	+	0.916190i	0.470467	+	0.882418i	$0.344085\pi$
$151$	−6.50000	+	11.2583i	−0.528962	+	0.916190i	−0.999430	+	0.0337724i	$0.989248\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.50000	+	2.59808i	−0.361449	+	0.208683i
$156$		0			0
$157$		−	5.19615i		−	0.414698i	−0.978267	−	0.207349i	$-0.933516\pi$
$157$		−	5.19615i		−	0.414698i	0.978267	−	0.207349i	$-0.0664836\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−13.5000	+	2.59808i	−1.06395	+	0.204757i
$162$		0			0
$163$	0.500000	−	0.866025i	0.0391630	−	0.0678323i	−0.845780	−	0.533533i	$-0.820864\pi$
$163$	0.500000	−	0.866025i	0.0391630	−	0.0678323i	0.884943	+	0.465700i	$0.154198\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000	+	10.3923i	−0.464294	+	0.804181i	−0.999169	−	0.0407502i	$-0.987025\pi$
$167$	−6.00000	+	10.3923i	−0.464294	+	0.804181i	0.534875	+	0.844931i	$0.320359\pi$
$168$		0			0
$169$	−6.50000	−	11.2583i	−0.500000	−	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.00000			−0.684257			−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000			−0.684257			−0.342129	+	0.939653i	$0.611148\pi$
$174$		0			0
$175$	8.00000	−	6.92820i	0.604743	−	0.523723i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.50000	−	2.59808i	0.336346	−	0.194189i	−0.322309	−	0.946634i	$-0.604459\pi$
$179$	4.50000	−	2.59808i	0.336346	−	0.194189i	0.658655	+	0.752445i	$0.271126\pi$
$180$		0			0
$181$			6.92820i			0.514969i	0.966282	+	0.257485i	$0.0828937\pi$
$181$			6.92820i			0.514969i	−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	21.0000			1.54395
$186$		0			0
$187$			15.5885i			1.13994i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	15.5885i		−	1.12794i	−0.825795	−	0.563971i	$-0.809273\pi$
$191$		−	15.5885i		−	1.12794i	0.825795	−	0.563971i	$-0.190727\pi$
$192$		0			0
$193$	−5.00000			−0.359908			−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000			−0.359908			−0.179954	+	0.983675i	$0.557595\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−1.50000	+	0.866025i	−0.106332	+	0.0613909i	−0.552223	−	0.833696i	$-0.686220\pi$
$199$	−1.50000	+	0.866025i	−0.106332	+	0.0613909i	0.445891	+	0.895087i	$0.352887\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	18.0000			1.25717
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.50000	−	7.79423i	−0.311272	−	0.539138i
$210$		0			0
$211$	10.0000	−	17.3205i	0.688428	−	1.19239i	−0.283918	−	0.958849i	$-0.591634\pi$
$211$	10.0000	−	17.3205i	0.688428	−	1.19239i	0.972346	−	0.233544i	$-0.0750324\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.00000	+	10.3923i	−0.409197	+	0.708749i
$216$		0			0
$217$	−1.50000	+	4.33013i	−0.101827	+	0.293948i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−9.00000	+	5.19615i	−0.602685	+	0.347960i	−0.770097	−	0.637927i	$-0.779792\pi$
$223$	−9.00000	+	5.19615i	−0.602685	+	0.347960i	0.167412	+	0.985887i	$0.446459\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.50000	+	7.79423i	−0.298675	+	0.517321i	−0.975833	−	0.218517i	$-0.929878\pi$
$227$	−4.50000	+	7.79423i	−0.298675	+	0.517321i	0.677158	+	0.735838i	$0.263211\pi$
$228$		0			0
$229$	−19.5000	+	11.2583i	−1.28860	+	0.743971i	−0.978404	−	0.206702i	$-0.933727\pi$
$229$	−19.5000	+	11.2583i	−1.28860	+	0.743971i	−0.310192	+	0.950674i	$0.600393\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.50000	−	2.59808i	−0.294805	−	0.170206i	0.345302	−	0.938492i	$-0.387777\pi$
$233$	−4.50000	−	2.59808i	−0.294805	−	0.170206i	−0.640107	+	0.768286i	$0.721110\pi$
$234$		0			0
$235$	−4.50000	−	7.79423i	−0.293548	−	0.508439i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.00000	−	5.19615i	−0.582162	−	0.336111i	0.179830	−	0.983698i	$-0.442445\pi$
$239$	−9.00000	−	5.19615i	−0.582162	−	0.336111i	−0.761992	+	0.647586i	$0.775778\pi$
$240$		0			0
$241$	−1.50000	−	0.866025i	−0.0966235	−	0.0557856i	0.450910	−	0.892570i	$-0.351100\pi$
$241$	−1.50000	−	0.866025i	−0.0966235	−	0.0557856i	−0.547533	+	0.836784i	$0.684433\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.00000	−	20.7846i	0.191663	−	1.32788i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−24.0000			−1.51487			−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$		0			0
$253$	−27.0000			−1.69748
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.5000	−	23.3827i	−0.842107	−	1.45857i	−0.888110	−	0.459631i	$-0.847982\pi$
$257$	−13.5000	−	23.3827i	−0.842107	−	1.45857i	0.0460033	−	0.998941i	$-0.485352\pi$
$258$		0			0
$259$	14.0000	−	12.1244i	0.869918	−	0.753371i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.50000	−	2.59808i	−0.277482	−	0.160204i	0.354801	−	0.934942i	$-0.384549\pi$
$263$	−4.50000	−	2.59808i	−0.277482	−	0.160204i	−0.632283	+	0.774738i	$0.717882\pi$
$264$		0			0
$265$	13.5000	+	7.79423i	0.829298	+	0.478796i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.5000	+	18.1865i	0.640196	+	1.10885i	0.985389	+	0.170321i	$0.0544803\pi$
$269$	10.5000	+	18.1865i	0.640196	+	1.10885i	−0.345192	+	0.938532i	$0.612186\pi$
$270$		0			0
$271$	19.5000	+	11.2583i	1.18454	+	0.683895i	0.957061	−	0.289888i	$-0.0936180\pi$
$271$	19.5000	+	11.2583i	1.18454	+	0.683895i	0.227480	+	0.973783i	$0.426951\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	18.0000	−	10.3923i	1.08544	−	0.626680i
$276$		0			0
$277$	6.50000	−	11.2583i	0.390547	−	0.676448i	−0.601975	−	0.798515i	$-0.705619\pi$
$277$	6.50000	−	11.2583i	0.390547	−	0.676448i	0.992522	+	0.122068i	$0.0389525\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18.0000	+	10.3923i	−1.07379	+	0.619953i	−0.929214	−	0.369541i	$-0.879515\pi$
$281$	−18.0000	+	10.3923i	−1.07379	+	0.619953i	−0.144575	+	0.989494i	$0.546182\pi$
$282$		0			0
$283$		−	8.66025i		−	0.514799i	−0.966305	−	0.257399i	$-0.917134\pi$
$283$		−	8.66025i		−	0.514799i	0.966305	−	0.257399i	$-0.0828656\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.0000	−	10.3923i	0.708338	−	0.613438i
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.00000	+	5.19615i	−0.175262	+	0.303562i	−0.940252	−	0.340480i	$-0.889411\pi$
$293$	−3.00000	+	5.19615i	−0.175262	+	0.303562i	0.764990	+	0.644042i	$0.222744\pi$
$294$		0			0
$295$	4.50000	+	7.79423i	0.262000	+	0.453798i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	2.00000	+	10.3923i	0.115278	+	0.599002i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	31.5000	−	18.1865i	1.80368	−	1.04136i
$306$		0			0
$307$		−	17.3205i		−	0.988534i	−0.869310	−	0.494267i	$-0.835437\pi$
$307$		−	17.3205i		−	0.988534i	0.869310	−	0.494267i	$-0.164563\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−15.0000			−0.850572			−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000			−0.850572			−0.425286	+	0.905059i	$0.639826\pi$
$312$		0			0
$313$			12.1244i			0.685309i	0.939461	+	0.342655i	$0.111326\pi$
$313$			12.1244i			0.685309i	−0.939461	+	0.342655i	$0.888674\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			25.9808i			1.45922i	0.683861	+	0.729612i	$0.260300\pi$
$317$			25.9808i			1.45922i	−0.683861	+	0.729612i	$0.739700\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	5.19615i		−	0.289122i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7.50000	−	2.59808i	−0.413488	−	0.143237i
$330$		0			0
$331$	31.0000			1.70391			0.851957	−	0.523612i	$-0.175416\pi$
$331$	31.0000			1.70391			0.851957	+	0.523612i	$0.175416\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.50000	−	12.9904i	−0.409769	−	0.709740i
$336$		0			0
$337$	1.00000	−	1.73205i	0.0544735	−	0.0943508i	−0.837503	−	0.546433i	$-0.815985\pi$
$337$	1.00000	−	1.73205i	0.0544735	−	0.0943508i	0.891976	+	0.452082i	$0.149319\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.50000	+	7.79423i	−0.243689	+	0.422081i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	15.5885i		−	0.836832i	−0.908255	−	0.418416i	$-0.862585\pi$
$347$		−	15.5885i		−	0.836832i	0.908255	−	0.418416i	$-0.137415\pi$
$348$		0			0
$349$	−12.0000	+	6.92820i	−0.642345	+	0.370858i	−0.785517	−	0.618840i	$-0.787603\pi$
$349$	−12.0000	+	6.92820i	−0.642345	+	0.370858i	0.143172	+	0.989698i	$0.454270\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.50000	+	2.59808i	−0.0798369	+	0.138282i	−0.903179	−	0.429263i	$-0.858773\pi$
$353$	−1.50000	+	2.59808i	−0.0798369	+	0.138282i	0.823343	+	0.567545i	$0.192107\pi$
$354$		0			0
$355$	−27.0000	+	15.5885i	−1.43301	+	0.827349i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−22.5000	−	12.9904i	−1.18750	−	0.685606i	−0.229766	−	0.973246i	$-0.573796\pi$
$359$	−22.5000	−	12.9904i	−1.18750	−	0.685606i	−0.957739	+	0.287640i	$0.907129\pi$
$360$		0			0
$361$	−8.00000	−	13.8564i	−0.421053	−	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	31.5000	+	18.1865i	1.64879	+	0.951927i
$366$		0			0
$367$	−28.5000	−	16.4545i	−1.48769	−	0.858917i	−0.487787	−	0.872963i	$-0.662196\pi$
$367$	−28.5000	−	16.4545i	−1.48769	−	0.858917i	−0.999901	+	0.0140459i	$0.995529\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	13.5000	−	2.59808i	0.700885	−	0.134885i
$372$		0			0
$373$	12.5000	+	21.6506i	0.647225	+	1.12103i	0.983783	+	0.179364i	$0.0574041\pi$
$373$	12.5000	+	21.6506i	0.647225	+	1.12103i	−0.336557	+	0.941663i	$0.609263\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	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
$381$		0			0
$382$		0			0
$383$	10.5000	+	18.1865i	0.536525	+	0.929288i	0.999088	+	0.0427020i	$0.0135966\pi$
$383$	10.5000	+	18.1865i	0.536525	+	0.929288i	−0.462563	+	0.886586i	$0.653070\pi$
$384$		0			0
$385$	13.5000	−	38.9711i	0.688024	−	1.98615i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	22.5000	+	12.9904i	1.14080	+	0.658638i	0.946627	−	0.322330i	$-0.104466\pi$
$389$	22.5000	+	12.9904i	1.14080	+	0.658638i	0.194168	+	0.980968i	$0.437799\pi$
$390$		0			0
$391$	−13.5000	−	7.79423i	−0.682724	−	0.394171i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.50000	+	2.59808i	0.0754732	+	0.130723i
$396$		0			0
$397$	7.50000	+	4.33013i	0.376414	+	0.217323i	0.676257	−	0.736666i	$-0.263601\pi$
$397$	7.50000	+	4.33013i	0.376414	+	0.217323i	−0.299843	+	0.953989i	$0.596934\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−31.5000	+	18.1865i	−1.57303	+	0.908192i	−0.577241	+	0.816574i	$0.695871\pi$
$401$	−31.5000	+	18.1865i	−1.57303	+	0.908192i	−0.995794	+	0.0916181i	$0.970796\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	31.5000	−	18.1865i	1.56140	−	0.901473i
$408$		0			0
$409$		−	5.19615i		−	0.256933i	−0.991714	−	0.128467i	$-0.958994\pi$
$409$		−	5.19615i		−	0.256933i	0.991714	−	0.128467i	$-0.0410055\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.50000	+	2.59808i	0.369051	+	0.127843i
$414$		0			0
$415$	18.0000	−	31.1769i	0.883585	−	1.53041i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	−0.974546	−	0.224189i	$-0.928027\pi$
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	0.681426	+	0.731887i	$0.261360\pi$
$420$		0			0
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	0.999109	+	0.0422075i	$0.0134391\pi$
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	12.0000			0.582086
$426$		0			0
$427$	10.5000	−	30.3109i	0.508131	−	1.46685i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−13.5000	+	7.79423i	−0.650272	+	0.375435i	−0.788560	−	0.614957i	$-0.789173\pi$
$431$	−13.5000	+	7.79423i	−0.650272	+	0.375435i	0.138288	+	0.990392i	$0.455840\pi$
$432$		0			0
$433$		−	34.6410i		−	1.66474i	−0.554220	−	0.832370i	$-0.686983\pi$
$433$		−	34.6410i		−	1.66474i	0.554220	−	0.832370i	$-0.313017\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9.00000			0.430528
$438$		0			0
$439$		−	15.5885i		−	0.743996i	−0.928233	−	0.371998i	$-0.878673\pi$
$439$		−	15.5885i		−	0.743996i	0.928233	−	0.371998i	$-0.121327\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	15.5885i		−	0.740630i	−0.928906	−	0.370315i	$-0.879250\pi$
$443$		−	15.5885i		−	0.740630i	0.928906	−	0.370315i	$-0.120750\pi$
$444$		0			0
$445$	−27.0000			−1.27992
$446$		0			0
$447$		0			0
$448$		0			0
$449$			20.7846i			0.980886i	0.871473	+	0.490443i	$0.163165\pi$
$449$			20.7846i			0.980886i	−0.871473	+	0.490443i	$0.836835\pi$
$450$		0			0
$451$	27.0000	−	15.5885i	1.27138	−	0.734032i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	31.0000			1.45012			0.725059	−	0.688686i	$-0.241812\pi$
$457$	31.0000			1.45012			0.725059	+	0.688686i	$0.241812\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.0000	+	25.9808i	0.698620	+	1.21004i	0.968945	+	0.247276i	$0.0795353\pi$
$461$	15.0000	+	25.9808i	0.698620	+	1.21004i	−0.270326	+	0.962769i	$0.587131\pi$
$462$		0			0
$463$	8.00000	−	13.8564i	0.371792	−	0.643962i	−0.618050	−	0.786139i	$-0.712077\pi$
$463$	8.00000	−	13.8564i	0.371792	−	0.643962i	0.989841	+	0.142177i	$0.0454103\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.50000	−	7.79423i	0.208235	−	0.360674i	−0.742923	−	0.669376i	$-0.766561\pi$
$467$	4.50000	−	7.79423i	0.208235	−	0.360674i	0.951159	+	0.308702i	$0.0998947\pi$
$468$		0			0
$469$	−12.5000	−	4.33013i	−0.577196	−	0.199947i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			20.7846i			0.955677i
$474$		0			0
$475$	−6.00000	+	3.46410i	−0.275299	+	0.158944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−16.5000	+	28.5788i	−0.753904	+	1.30580i	0.192013	+	0.981392i	$0.438498\pi$
$479$	−16.5000	+	28.5788i	−0.753904	+	1.30580i	−0.945917	+	0.324408i	$0.894835\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	18.0000	+	10.3923i	0.817338	+	0.471890i
$486$		0			0
$487$	−0.500000	−	0.866025i	−0.0226572	−	0.0392434i	0.854475	−	0.519493i	$-0.173879\pi$
$487$	−0.500000	−	0.866025i	−0.0226572	−	0.0392434i	−0.877132	+	0.480250i	$0.840546\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.00000	+	5.19615i	0.406164	+	0.234499i	0.689140	−	0.724628i	$-0.257988\pi$
$491$	9.00000	+	5.19615i	0.406164	+	0.234499i	−0.282976	+	0.959127i	$0.591322\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.00000	+	25.9808i	−0.403705	+	1.16540i
$498$		0			0
$499$	−3.50000	−	6.06218i	−0.156682	−	0.271380i	0.776989	−	0.629515i	$-0.216746\pi$
$499$	−3.50000	−	6.06218i	−0.156682	−	0.271380i	−0.933670	+	0.358134i	$0.883413\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$	−27.0000			−1.20148
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10.5000	+	18.1865i	0.465404	+	0.806104i	0.999220	−	0.0394971i	$-0.0125756\pi$
$509$	10.5000	+	18.1865i	0.465404	+	0.806104i	−0.533815	+	0.845601i	$0.679242\pi$
$510$		0			0
$511$	31.5000	−	6.06218i	1.39348	−	0.268175i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−13.5000	−	7.79423i	−0.594881	−	0.343455i
$516$		0			0
$517$	−13.5000	−	7.79423i	−0.593729	−	0.342790i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7.50000	−	12.9904i	−0.328581	−	0.569119i	0.653650	−	0.756797i	$-0.273237\pi$
$521$	−7.50000	−	12.9904i	−0.328581	−	0.569119i	−0.982231	+	0.187678i	$0.939904\pi$
$522$		0			0
$523$	−22.5000	−	12.9904i	−0.983856	−	0.568030i	−0.0804241	−	0.996761i	$-0.525627\pi$
$523$	−22.5000	−	12.9904i	−0.983856	−	0.568030i	−0.903432	+	0.428731i	$0.858961\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.50000	+	2.59808i	−0.196023	+	0.113174i
$528$		0			0
$529$	2.00000	−	3.46410i	0.0869565	−	0.150613i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	46.7654i		−	2.02184i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−13.5000	−	33.7750i	−0.581486	−	1.45479i
$540$		0			0
$541$	18.5000	−	32.0429i	0.795377	−	1.37763i	−0.127222	−	0.991874i	$-0.540606\pi$
$541$	18.5000	−	32.0429i	0.795377	−	1.37763i	0.922599	−	0.385759i	$-0.126061\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	25.5000	−	44.1673i	1.09230	−	1.89192i
$546$		0			0
$547$	−16.0000	−	27.7128i	−0.684111	−	1.18491i	−0.973715	−	0.227768i	$-0.926857\pi$
$547$	−16.0000	−	27.7128i	−0.684111	−	1.18491i	0.289605	−	0.957146i	$-0.406476\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	2.50000	+	0.866025i	0.106311	+	0.0368271i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−22.5000	+	12.9904i	−0.953356	+	0.550420i	−0.894122	−	0.447824i	$-0.852199\pi$
$557$	−22.5000	+	12.9904i	−0.953356	+	0.550420i	−0.0592339	+	0.998244i	$0.518866\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−27.0000			−1.13791			−0.568957	−	0.822367i	$-0.692653\pi$
$563$	−27.0000			−1.13791			−0.568957	+	0.822367i	$0.692653\pi$
$564$		0			0
$565$			62.3538i			2.62325i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	15.5885i		−	0.653502i	−0.945110	−	0.326751i	$-0.894046\pi$
$569$		−	15.5885i		−	0.653502i	0.945110	−	0.326751i	$-0.105954\pi$
$570$		0			0
$571$	−7.00000			−0.292941			−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000			−0.292941			−0.146470	+	0.989215i	$0.546791\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			20.7846i			0.866778i
$576$		0			0
$577$	−22.5000	+	12.9904i	−0.936687	+	0.540797i	−0.888920	−	0.458062i	$-0.848544\pi$
$577$	−22.5000	+	12.9904i	−0.936687	+	0.540797i	−0.0477669	+	0.998859i	$0.515210\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.00000	−	31.1769i	−0.248922	−	1.29344i
$582$		0			0
$583$	27.0000			1.11823
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−18.0000	−	31.1769i	−0.742940	−	1.28681i	−0.951151	−	0.308725i	$-0.900098\pi$
$587$	−18.0000	−	31.1769i	−0.742940	−	1.28681i	0.208212	−	0.978084i	$-0.433236\pi$
$588$		0			0
$589$	1.50000	−	2.59808i	0.0618064	−	0.107052i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	10.5000	−	18.1865i	0.431183	−	0.746831i	−0.565792	−	0.824548i	$-0.691430\pi$
$593$	10.5000	−	18.1865i	0.431183	−	0.746831i	0.996976	+	0.0777165i	$0.0247629\pi$
$594$		0			0
$595$	18.0000	−	15.5885i	0.737928	−	0.639064i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			5.19615i			0.212309i	0.994350	+	0.106155i	$0.0338538\pi$
$599$			5.19615i			0.212309i	−0.994350	+	0.106155i	$0.966146\pi$
$600$		0			0
$601$	30.0000	−	17.3205i	1.22373	−	0.706518i	0.258015	−	0.966141i	$-0.416931\pi$
$601$	30.0000	−	17.3205i	1.22373	−	0.706518i	0.965710	+	0.259623i	$0.0835982\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	24.0000	−	41.5692i	0.975739	−	1.69003i
$606$		0			0
$607$	−25.5000	+	14.7224i	−1.03501	+	0.597565i	−0.918417	−	0.395614i	$-0.870532\pi$
$607$	−25.5000	+	14.7224i	−1.03501	+	0.597565i	−0.116596	+	0.993179i	$0.537198\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.500000	+	0.866025i	0.0201948	+	0.0349784i	0.875946	−	0.482409i	$-0.160238\pi$
$613$	0.500000	+	0.866025i	0.0201948	+	0.0349784i	−0.855751	+	0.517387i	$0.826905\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−18.0000	−	10.3923i	−0.724653	−	0.418378i	0.0918100	−	0.995777i	$-0.470735\pi$
$617$	−18.0000	−	10.3923i	−0.724653	−	0.418378i	−0.816463	+	0.577398i	$0.804068\pi$
$618$		0			0
$619$	−22.5000	−	12.9904i	−0.904351	−	0.522127i	−0.0257420	−	0.999669i	$-0.508195\pi$
$619$	−22.5000	−	12.9904i	−0.904351	−	0.522127i	−0.878609	+	0.477541i	$0.841528\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−18.0000	+	15.5885i	−0.721155	+	0.624538i
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	21.0000			0.837325
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.0000	−	20.7846i	−0.476205	−	0.824812i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−13.5000	−	7.79423i	−0.533218	−	0.307854i	0.209108	−	0.977893i	$-0.432944\pi$
$641$	−13.5000	−	7.79423i	−0.533218	−	0.307854i	−0.742326	+	0.670039i	$0.766277\pi$
$642$		0			0
$643$	3.00000	+	1.73205i	0.118308	+	0.0683054i	0.557986	−	0.829850i	$-0.311574\pi$
$643$	3.00000	+	1.73205i	0.118308	+	0.0683054i	−0.439678	+	0.898155i	$0.644907\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−4.50000	−	7.79423i	−0.176913	−	0.306423i	0.763908	−	0.645325i	$-0.223278\pi$
$647$	−4.50000	−	7.79423i	−0.176913	−	0.306423i	−0.940822	+	0.338902i	$0.889945\pi$
$648$		0			0
$649$	13.5000	+	7.79423i	0.529921	+	0.305950i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−13.5000	+	7.79423i	−0.528296	+	0.305012i	−0.740322	−	0.672252i	$-0.765327\pi$
$653$	−13.5000	+	7.79423i	−0.528296	+	0.305012i	0.212026	+	0.977264i	$0.431994\pi$
$654$		0			0
$655$	13.5000	−	23.3827i	0.527489	−	0.913637i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.00000	−	5.19615i	0.350590	−	0.202413i	−0.314355	−	0.949306i	$-0.601788\pi$
$659$	9.00000	−	5.19615i	0.350590	−	0.202413i	0.664945	+	0.746892i	$0.268455\pi$
$660$		0			0
$661$			22.5167i			0.875797i	0.899025	+	0.437898i	$0.144277\pi$
$661$			22.5167i			0.875797i	−0.899025	+	0.437898i	$0.855723\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.50000	+	12.9904i	−0.174503	+	0.503745i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	31.5000	−	54.5596i	1.21604	−	2.10625i
$672$		0			0
$673$	−11.0000	−	19.0526i	−0.424019	−	0.734422i	0.572309	−	0.820038i	$-0.306048\pi$
$673$	−11.0000	−	19.0526i	−0.424019	−	0.734422i	−0.996328	+	0.0856156i	$0.972714\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−45.0000			−1.72949			−0.864745	−	0.502211i	$-0.832520\pi$
$677$	−45.0000			−1.72949			−0.864745	+	0.502211i	$0.832520\pi$
$678$		0			0
$679$	18.0000	−	3.46410i	0.690777	−	0.132940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−31.5000	+	18.1865i	−1.20531	+	0.695888i	−0.961732	−	0.273992i	$-0.911656\pi$
$683$	−31.5000	+	18.1865i	−1.20531	+	0.695888i	−0.243582	+	0.969880i	$0.578323\pi$
$684$		0			0
$685$		−	15.5885i		−	0.595604i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$			32.9090i			1.25192i	0.779857	+	0.625958i	$0.215292\pi$
$691$			32.9090i			1.25192i	−0.779857	+	0.625958i	$0.784708\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			31.1769i			1.18261i
$696$		0			0
$697$	18.0000			0.681799
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	20.7846i		−	0.785024i	−0.919747	−	0.392512i	$-0.871606\pi$
$701$		−	20.7846i		−	0.785024i	0.919747	−	0.392512i	$-0.128394\pi$
$702$		0			0
$703$	−10.5000	+	6.06218i	−0.396015	+	0.228639i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−18.0000	+	15.5885i	−0.676960	+	0.586264i
$708$		0			0
$709$	31.0000			1.16423			0.582115	−	0.813107i	$-0.302225\pi$
$709$	31.0000			1.16423			0.582115	+	0.813107i	$0.302225\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.50000	−	7.79423i	−0.168526	−	0.291896i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7.50000	+	12.9904i	−0.279703	+	0.484459i	−0.971311	−	0.237814i	$-0.923569\pi$
$719$	−7.50000	+	12.9904i	−0.279703	+	0.484459i	0.691608	+	0.722273i	$0.256903\pi$
$720$		0			0
$721$	−13.5000	+	2.59808i	−0.502766	+	0.0967574i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−21.0000	+	12.1244i	−0.778847	+	0.449667i	−0.836021	−	0.548697i	$-0.815124\pi$
$727$	−21.0000	+	12.1244i	−0.778847	+	0.449667i	0.0571746	+	0.998364i	$0.481791\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−6.00000	+	10.3923i	−0.221918	+	0.384373i
$732$		0			0
$733$	−25.5000	+	14.7224i	−0.941864	+	0.543785i	−0.890544	−	0.454897i	$-0.849676\pi$
$733$	−25.5000	+	14.7224i	−0.941864	+	0.543785i	−0.0513199	+	0.998682i	$0.516343\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−22.5000	−	12.9904i	−0.828798	−	0.478507i
$738$		0			0
$739$	−8.50000	−	14.7224i	−0.312678	−	0.541573i	0.666264	−	0.745716i	$-0.267893\pi$
$739$	−8.50000	−	14.7224i	−0.312678	−	0.541573i	−0.978941	+	0.204143i	$0.934559\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	45.0000	+	25.9808i	1.65089	+	0.953142i	0.976707	+	0.214577i	$0.0688374\pi$
$743$	45.0000	+	25.9808i	1.65089	+	0.953142i	0.674183	+	0.738564i	$0.264496\pi$
$744$		0			0
$745$	−13.5000	−	7.79423i	−0.494602	−	0.285558i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−27.0000	−	31.1769i	−0.986559	−	1.13918i
$750$		0			0
$751$	14.5000	+	25.1147i	0.529113	+	0.916450i	0.999424	+	0.0339490i	$0.0108084\pi$
$751$	14.5000	+	25.1147i	0.529113	+	0.916450i	−0.470311	+	0.882501i	$0.655858\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	39.0000			1.41936
$756$		0			0
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	10.5000	+	18.1865i	0.380625	+	0.659261i	0.991152	−	0.132734i	$-0.0423756\pi$
$761$	10.5000	+	18.1865i	0.380625	+	0.659261i	−0.610527	+	0.791995i	$0.709042\pi$
$762$		0			0
$763$	−8.50000	−	44.1673i	−0.307721	−	1.59896i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−30.0000	−	17.3205i	−1.08183	−	0.624593i	−0.150439	−	0.988619i	$-0.548069\pi$
$769$	−30.0000	−	17.3205i	−1.08183	−	0.624593i	−0.931389	+	0.364026i	$0.881402\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.5000	−	33.7750i	−0.701366	−	1.21480i	−0.967987	−	0.251000i	$-0.919240\pi$
$773$	−19.5000	−	33.7750i	−0.701366	−	1.21480i	0.266621	−	0.963802i	$-0.414093\pi$
$774$		0			0
$775$	6.00000	+	3.46410i	0.215526	+	0.124434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.00000	+	5.19615i	−0.322458	+	0.186171i
$780$		0			0
$781$	−27.0000	+	46.7654i	−0.966136	+	1.67340i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−13.5000	+	7.79423i	−0.481836	+	0.278188i
$786$		0			0
$787$		−	1.73205i		−	0.0617409i	−0.999523	−	0.0308705i	$-0.990172\pi$
$787$		−	1.73205i		−	0.0617409i	0.999523	−	0.0308705i	$-0.00982794\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	36.0000	+	41.5692i	1.28001	+	1.47803i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.00000	+	15.5885i	−0.318796	+	0.552171i	−0.980237	−	0.197826i	$-0.936612\pi$
$797$	−9.00000	+	15.5885i	−0.318796	+	0.552171i	0.661441	+	0.749997i	$0.269945\pi$
$798$		0			0
$799$	−4.50000	−	7.79423i	−0.159199	−	0.275740i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	63.0000			2.22322
$804$		0			0
$805$	27.0000	+	31.1769i	0.951625	+	1.09884i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−22.5000	+	12.9904i	−0.791058	+	0.456717i	−0.840335	−	0.542068i	$-0.817642\pi$
$809$	−22.5000	+	12.9904i	−0.791058	+	0.456717i	0.0492770	+	0.998785i	$0.484308\pi$
$810$		0			0
$811$			51.9615i			1.82462i	0.409505	+	0.912308i	$0.365701\pi$
$811$			51.9615i			1.82462i	−0.409505	+	0.912308i	$0.634299\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.00000			−0.105085
$816$		0			0
$817$		−	6.92820i		−	0.242387i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			5.19615i			0.181347i	0.995881	+	0.0906735i	$0.0289020\pi$
$821$			5.19615i			0.181347i	−0.995881	+	0.0906735i	$0.971098\pi$
$822$		0			0
$823$	−7.00000			−0.244005			−0.122002	−	0.992530i	$-0.538932\pi$
$823$	−7.00000			−0.244005			−0.122002	+	0.992530i	$0.538932\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			10.3923i			0.361376i	0.983540	+	0.180688i	$0.0578324\pi$
$827$			10.3923i			0.361376i	−0.983540	+	0.180688i	$0.942168\pi$
$828$		0			0
$829$	1.50000	−	0.866025i	0.0520972	−	0.0300783i	−0.473725	−	0.880673i	$-0.657091\pi$
$829$	1.50000	−	0.866025i	0.0520972	−	0.0300783i	0.525822	+	0.850594i	$0.323758\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.00000	−	20.7846i	0.103944	−	0.720144i
$834$		0			0
$835$	36.0000			1.24583
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12.0000	+	20.7846i	0.414286	+	0.717564i	0.995353	−	0.0962912i	$-0.0306980\pi$
$839$	12.0000	+	20.7846i	0.414286	+	0.717564i	−0.581067	+	0.813856i	$0.697365\pi$
$840$		0			0
$841$	−14.5000	+	25.1147i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−19.5000	+	33.7750i	−0.670820	+	1.16190i
$846$		0			0
$847$	−8.00000	−	41.5692i	−0.274883	−	1.42834i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			36.3731i			1.24685i
$852$		0			0
$853$	−12.0000	+	6.92820i	−0.410872	+	0.237217i	−0.691164	−	0.722698i	$-0.742902\pi$
$853$	−12.0000	+	6.92820i	−0.410872	+	0.237217i	0.280292	+	0.959915i	$0.409569\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	16.5000	−	28.5788i	0.563629	−	0.976235i	−0.433546	−	0.901131i	$-0.642738\pi$
$857$	16.5000	−	28.5788i	0.563629	−	0.976235i	0.997176	−	0.0751033i	$-0.0239287\pi$
$858$		0			0
$859$	34.5000	−	19.9186i	1.17712	−	0.679613i	0.221777	−	0.975097i	$-0.428814\pi$
$859$	34.5000	−	19.9186i	1.17712	−	0.679613i	0.955348	+	0.295484i	$0.0954809\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.50000	−	2.59808i	−0.153182	−	0.0884395i	0.421450	−	0.906852i	$-0.361521\pi$
$863$	−4.50000	−	2.59808i	−0.153182	−	0.0884395i	−0.574632	+	0.818412i	$0.694855\pi$
$864$		0			0
$865$	13.5000	+	23.3827i	0.459014	+	0.795035i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.50000	+	2.59808i	0.152652	+	0.0881337i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.50000	+	2.59808i	0.253546	+	0.0878310i
$876$		0			0
$877$	−5.50000	−	9.52628i	−0.185722	−	0.321680i	0.758098	−	0.652141i	$-0.226129\pi$
$877$	−5.50000	−	9.52628i	−0.185722	−	0.321680i	−0.943820	+	0.330461i	$0.892796\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	28.0000			0.942275			0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000			0.942275			0.471138	+	0.882060i	$0.343844\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.50000	−	2.59808i	−0.0503651	−	0.0872349i	0.839744	−	0.542983i	$-0.182705\pi$
$887$	−1.50000	−	2.59808i	−0.0503651	−	0.0872349i	−0.890109	+	0.455748i	$0.849372\pi$
$888$		0			0
$889$	−20.0000	−	6.92820i	−0.670778	−	0.232364i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.50000	+	2.59808i	0.150587	+	0.0869413i
$894$		0			0
$895$	−13.5000	−	7.79423i	−0.451255	−	0.260532i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	13.5000	+	7.79423i	0.449750	+	0.259663i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	18.0000	−	10.3923i	0.598340	−	0.345452i
$906$		0			0
$907$	−18.5000	+	32.0429i	−0.614282	+	1.06397i	0.376228	+	0.926527i	$0.377221\pi$
$907$	−18.5000	+	32.0429i	−0.614282	+	1.06397i	−0.990510	+	0.137441i	$0.956112\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−9.00000	+	5.19615i	−0.298183	+	0.172156i	−0.641626	−	0.767017i	$-0.721740\pi$
$911$	−9.00000	+	5.19615i	−0.298183	+	0.172156i	0.343443	+	0.939173i	$0.388407\pi$
$912$		0			0
$913$		−	62.3538i		−	2.06361i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4.50000	−	23.3827i	−0.148603	−	0.772164i
$918$		0			0
$919$	−9.50000	+	16.4545i	−0.313376	+	0.542783i	−0.979091	−	0.203423i	$-0.934793\pi$
$919$	−9.50000	+	16.4545i	−0.313376	+	0.542783i	0.665715	+	0.746206i	$0.268127\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−14.0000	−	24.2487i	−0.460317	−	0.797293i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	39.0000			1.27955			0.639774	−	0.768563i	$-0.279028\pi$
$929$	39.0000			1.27955			0.639774	+	0.768563i	$0.279028\pi$
$930$		0			0
$931$	4.50000	+	11.2583i	0.147482	+	0.368977i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	40.5000	−	23.3827i	1.32449	−	0.764696i
$936$		0			0
$937$			41.5692i			1.35801i	0.734135	+	0.679004i	$0.237588\pi$
$937$			41.5692i			1.35801i	−0.734135	+	0.679004i	$0.762412\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−21.0000			−0.684580			−0.342290	−	0.939594i	$-0.611203\pi$
$941$	−21.0000			−0.684580			−0.342290	+	0.939594i	$0.611203\pi$
$942$		0			0
$943$			31.1769i			1.01526i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			46.7654i			1.51967i	0.650116	+	0.759835i	$0.274720\pi$
$947$			46.7654i			1.51967i	−0.650116	+	0.759835i	$0.725280\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			20.7846i			0.673280i	0.941634	+	0.336640i	$0.109290\pi$
$953$			20.7846i			0.673280i	−0.941634	+	0.336640i	$0.890710\pi$
$954$		0			0
$955$	−40.5000	+	23.3827i	−1.31055	+	0.756646i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−9.00000	−	10.3923i	−0.290625	−	0.335585i
$960$		0			0
$961$	28.0000			0.903226
$962$		0			0
$963$		0			0
$964$		0			0
$965$	7.50000	+	12.9904i	0.241434	+	0.418175i
$966$		0			0
$967$	20.0000	−	34.6410i	0.643157	−	1.11398i	−0.341567	−	0.939857i	$-0.610958\pi$
$967$	20.0000	−	34.6410i	0.643157	−	1.11398i	0.984724	−	0.174123i	$-0.0557089\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	28.5000	−	49.3634i	0.914609	−	1.58415i	0.107135	−	0.994244i	$-0.465832\pi$
$971$	28.5000	−	49.3634i	0.914609	−	1.58415i	0.807473	−	0.589904i	$-0.200834\pi$
$972$		0			0
$973$	18.0000	+	20.7846i	0.577054	+	0.666324i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			36.3731i			1.16368i	0.813304	+	0.581839i	$0.197667\pi$
$977$			36.3731i			1.16368i	−0.813304	+	0.581839i	$0.802333\pi$
$978$		0			0
$979$	−40.5000	+	23.3827i	−1.29439	+	0.747314i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.50000	−	2.59808i	0.0478426	−	0.0828658i	−0.841112	−	0.540860i	$-0.818099\pi$
$983$	1.50000	−	2.59808i	0.0478426	−	0.0828658i	0.888955	+	0.457995i	$0.151432\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18.0000	−	10.3923i	−0.572367	−	0.330456i
$990$		0			0
$991$	−12.5000	−	21.6506i	−0.397076	−	0.687755i	0.596288	−	0.802771i	$-0.296642\pi$
$991$	−12.5000	−	21.6506i	−0.397076	−	0.687755i	−0.993364	+	0.115015i	$0.963308\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.50000	+	2.59808i	0.142660	+	0.0823646i
$996$		0			0
$997$	−19.5000	−	11.2583i	−0.617571	−	0.356555i	0.158352	−	0.987383i	$-0.449382\pi$
$997$	−19.5000	−	11.2583i	−0.617571	−	0.356555i	−0.775923	+	0.630828i	$0.782715\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.w.a.1349.1		2
3.2	odd	2		2268.2.w.f.1349.1		2
7.3	odd	6		2268.2.bm.a.1025.1		2
9.2	odd	6		2268.2.bm.a.593.1		2
9.4	even	3		84.2.k.b.5.1	yes	2
9.5	odd	6		84.2.k.a.5.1	✓	2
9.7	even	3		2268.2.bm.f.593.1		2
21.17	even	6		2268.2.bm.f.1025.1		2
36.23	even	6		336.2.bc.d.257.1		2
36.31	odd	6		336.2.bc.b.257.1		2
45.4	even	6		2100.2.bi.e.1601.1		2
45.13	odd	12		2100.2.bo.f.1349.1		4
45.14	odd	6		2100.2.bi.f.1601.1		2
45.22	odd	12		2100.2.bo.f.1349.2		4
45.23	even	12		2100.2.bo.a.1349.2		4
45.32	even	12		2100.2.bo.a.1349.1		4
63.4	even	3		588.2.k.d.521.1		2
63.5	even	6		588.2.f.a.293.2		2
63.13	odd	6		588.2.k.c.509.1		2
63.23	odd	6		588.2.f.c.293.1		2
63.31	odd	6		84.2.k.a.17.1	yes	2
63.32	odd	6		588.2.k.c.521.1		2
63.38	even	6	inner	2268.2.w.a.269.1		2
63.40	odd	6		588.2.f.c.293.2		2
63.41	even	6		588.2.k.d.509.1		2
63.52	odd	6		2268.2.w.f.269.1		2
63.58	even	3		588.2.f.a.293.1		2
63.59	even	6		84.2.k.b.17.1	yes	2
252.23	even	6		2352.2.k.a.881.2		2
252.31	even	6		336.2.bc.d.17.1		2
252.59	odd	6		336.2.bc.b.17.1		2
252.103	even	6		2352.2.k.a.881.1		2
252.131	odd	6		2352.2.k.d.881.1		2
252.247	odd	6		2352.2.k.d.881.2		2
315.59	even	6		2100.2.bi.e.101.1		2
315.94	odd	6		2100.2.bi.f.101.1		2
315.122	odd	12		2100.2.bo.f.1949.1		4
315.157	even	12		2100.2.bo.a.1949.2		4
315.248	odd	12		2100.2.bo.f.1949.2		4
315.283	even	12		2100.2.bo.a.1949.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.2.k.a.5.1	✓	2	9.5	odd	6
84.2.k.a.17.1	yes	2	63.31	odd	6
84.2.k.b.5.1	yes	2	9.4	even	3
84.2.k.b.17.1	yes	2	63.59	even	6
336.2.bc.b.17.1		2	252.59	odd	6
336.2.bc.b.257.1		2	36.31	odd	6
336.2.bc.d.17.1		2	252.31	even	6
336.2.bc.d.257.1		2	36.23	even	6
588.2.f.a.293.1		2	63.58	even	3
588.2.f.a.293.2		2	63.5	even	6
588.2.f.c.293.1		2	63.23	odd	6
588.2.f.c.293.2		2	63.40	odd	6
588.2.k.c.509.1		2	63.13	odd	6
588.2.k.c.521.1		2	63.32	odd	6
588.2.k.d.509.1		2	63.41	even	6
588.2.k.d.521.1		2	63.4	even	3
2100.2.bi.e.101.1		2	315.59	even	6
2100.2.bi.e.1601.1		2	45.4	even	6
2100.2.bi.f.101.1		2	315.94	odd	6
2100.2.bi.f.1601.1		2	45.14	odd	6
2100.2.bo.a.1349.1		4	45.32	even	12
2100.2.bo.a.1349.2		4	45.23	even	12
2100.2.bo.a.1949.1		4	315.283	even	12
2100.2.bo.a.1949.2		4	315.157	even	12
2100.2.bo.f.1349.1		4	45.13	odd	12
2100.2.bo.f.1349.2		4	45.22	odd	12
2100.2.bo.f.1949.1		4	315.122	odd	12
2100.2.bo.f.1949.2		4	315.248	odd	12
2268.2.w.a.269.1		2	63.38	even	6	inner
2268.2.w.a.1349.1		2	1.1	even	1	trivial
2268.2.w.f.269.1		2	63.52	odd	6
2268.2.w.f.1349.1		2	3.2	odd	2
2268.2.bm.a.593.1		2	9.2	odd	6
2268.2.bm.a.1025.1		2	7.3	odd	6
2268.2.bm.f.593.1		2	9.7	even	3
2268.2.bm.f.1025.1		2	21.17	even	6
2352.2.k.a.881.1		2	252.103	even	6
2352.2.k.a.881.2		2	252.23	even	6
2352.2.k.d.881.1		2	252.131	odd	6
2352.2.k.d.881.2		2	252.247	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.w (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 2.59808i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(-4.50000 - 2.59808i) q^{11} +(-1.50000 - 2.59808i) q^{17} +(1.50000 + 0.866025i) q^{19} +(4.50000 - 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} -1.73205i q^{31} +(1.50000 + 7.79423i) q^{35} +(-3.50000 + 6.06218i) q^{37} +(-3.00000 + 5.19615i) q^{41} +(-2.00000 - 3.46410i) q^{43} +3.00000 q^{47} +(5.50000 + 4.33013i) q^{49} +(-4.50000 + 2.59808i) q^{53} +15.5885i q^{55} -3.00000 q^{59} +12.1244i q^{61} +5.00000 q^{67} -10.3923i q^{71} +(-10.5000 + 6.06218i) q^{73} +(9.00000 + 10.3923i) q^{77} -1.00000 q^{79} +(6.00000 + 10.3923i) q^{83} +(-4.50000 + 7.79423i) q^{85} +(4.50000 - 7.79423i) q^{89} -5.19615i q^{95} +(-6.00000 + 3.46410i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} - 5 q^{7} - 9 q^{11} - 3 q^{17} + 3 q^{19} + 9 q^{23} - 4 q^{25} + 3 q^{35} - 7 q^{37} - 6 q^{41} - 4 q^{43} + 6 q^{47} + 11 q^{49} - 9 q^{53} - 6 q^{59} + 10 q^{67} - 21 q^{73} + 18 q^{77}+ \cdots - 12 q^{97}+O(q^{100}) \) 2 * q - 3 * q^5 - 5 * q^7 - 9 * q^11 - 3 * q^17 + 3 * q^19 + 9 * q^23 - 4 * q^25 + 3 * q^35 - 7 * q^37 - 6 * q^41 - 4 * q^43 + 6 * q^47 + 11 * q^49 - 9 * q^53 - 6 * q^59 + 10 * q^67 - 21 * q^73 + 18 * q^77 - 2 * q^79 + 12 * q^83 - 9 * q^85 + 9 * q^89 - 12 * q^97

Introduction

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