LMFDB - Embedded newform 2352.2.q.g.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(961,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2352.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$18.7808145554$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2352.961
Dual form			2352.2.q.g.1537.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{9} +(-3.00000 - 5.19615i) q^{11} -2.00000 q^{13} +(2.00000 - 3.46410i) q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} +1.00000 q^{27} +6.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(-3.00000 + 5.19615i) q^{33} +(-1.00000 + 1.73205i) q^{37} +(1.00000 + 1.73205i) q^{39} -12.0000 q^{41} +4.00000 q^{43} +(-6.00000 + 10.3923i) q^{47} +(3.00000 + 5.19615i) q^{53} -4.00000 q^{57} +(-5.00000 + 8.66025i) q^{61} +(4.00000 + 6.92820i) q^{67} +6.00000 q^{69} -6.00000 q^{71} +(-5.00000 - 8.66025i) q^{73} +(2.50000 - 4.33013i) q^{75} +(-2.00000 + 3.46410i) q^{79} +(-0.500000 - 0.866025i) q^{81} -12.0000 q^{83} +(-3.00000 - 5.19615i) q^{87} +(6.00000 - 10.3923i) q^{89} +(-4.00000 + 6.92820i) q^{93} +10.0000 q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{9}+O(q^{10}) $ 2 * q - q^3 - q^9	$ 2 q - q^{3} - q^{9} - 6 q^{11} - 4 q^{13} + 4 q^{19} - 6 q^{23} + 5 q^{25} + 2 q^{27} + 12 q^{29} - 8 q^{31} - 6 q^{33} - 2 q^{37} + 2 q^{39} - 24 q^{41} + 8 q^{43} - 12 q^{47} + 6 q^{53} - 8 q^{57} - 10 q^{61} + 8 q^{67} + 12 q^{69} - 12 q^{71} - 10 q^{73} + 5 q^{75} - 4 q^{79} - q^{81} - 24 q^{83} - 6 q^{87} + 12 q^{89} - 8 q^{93} + 20 q^{97} + 12 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^9 - 6 * q^11 - 4 * q^13 + 4 * q^19 - 6 * q^23 + 5 * q^25 + 2 * q^27 + 12 * q^29 - 8 * q^31 - 6 * q^33 - 2 * q^37 + 2 * q^39 - 24 * q^41 + 8 * q^43 - 12 * q^47 + 6 * q^53 - 8 * q^57 - 10 * q^61 + 8 * q^67 + 12 * q^69 - 12 * q^71 - 10 * q^73 + 5 * q^75 - 4 * q^79 - q^81 - 24 * q^83 - 6 * q^87 + 12 * q^89 - 8 * q^93 + 20 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i
$4$		0			0
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	−3.00000	−	5.19615i	−0.904534	−	1.56670i	−0.821541	−	0.570149i	$-0.806886\pi$
$11$	−3.00000	−	5.19615i	−0.904534	−	1.56670i	−0.0829925	−	0.996550i	$-0.526448\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	2.00000	−	3.46410i	0.458831	−	0.794719i	−0.540068	−	0.841621i	$-0.681602\pi$
$19$	2.00000	−	3.46410i	0.458831	−	0.794719i	0.998899	+	0.0469020i	$0.0149348\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	−0.961625	−	0.274367i	$-0.911532\pi$
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	0.243204	−	0.969975i	$-0.421802\pi$
$32$		0			0
$33$	−3.00000	+	5.19615i	−0.522233	+	0.904534i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.00000	+	1.73205i	−0.164399	+	0.284747i	−0.936442	−	0.350823i	$-0.885902\pi$
$37$	−1.00000	+	1.73205i	−0.164399	+	0.284747i	0.772043	+	0.635571i	$0.219235\pi$
$38$		0			0
$39$	1.00000	+	1.73205i	0.160128	+	0.277350i
$40$		0			0
$41$	−12.0000			−1.87409			−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000			−1.87409			−0.937043	+	0.349215i	$0.886448\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.00000	+	10.3923i	−0.875190	+	1.51587i	−0.0186297	+	0.999826i	$0.505930\pi$
$47$	−6.00000	+	10.3923i	−0.875190	+	1.51587i	−0.856560	+	0.516047i	$0.827403\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.00000	+	5.19615i	0.412082	+	0.713746i	0.995117	−	0.0987002i	$-0.0314685\pi$
$53$	3.00000	+	5.19615i	0.412082	+	0.713746i	−0.583036	+	0.812447i	$0.698135\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.00000			−0.529813
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−5.00000	+	8.66025i	−0.640184	+	1.10883i	0.345207	+	0.938527i	$0.387809\pi$
$61$	−5.00000	+	8.66025i	−0.640184	+	1.10883i	−0.985391	+	0.170305i	$0.945525\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.00000	+	6.92820i	0.488678	+	0.846415i	0.999915	−	0.0130248i	$-0.00414604\pi$
$67$	4.00000	+	6.92820i	0.488678	+	0.846415i	−0.511237	+	0.859440i	$0.670813\pi$
$68$		0			0
$69$	6.00000			0.722315
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	−5.00000	−	8.66025i	−0.585206	−	1.01361i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−5.00000	−	8.66025i	−0.585206	−	1.01361i	0.409644	−	0.912245i	$-0.365653\pi$
$74$		0			0
$75$	2.50000	−	4.33013i	0.288675	−	0.500000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.00000	+	3.46410i	−0.225018	+	0.389742i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−2.00000	+	3.46410i	−0.225018	+	0.389742i	0.731307	+	0.682048i	$0.238911\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−3.00000	−	5.19615i	−0.321634	−	0.557086i
$88$		0			0
$89$	6.00000	−	10.3923i	0.635999	−	1.10158i	−0.350304	−	0.936636i	$-0.613922\pi$
$89$	6.00000	−	10.3923i	0.635999	−	1.10158i	0.986303	−	0.164946i	$-0.0527450\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−4.00000	+	6.92820i	−0.414781	+	0.718421i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$		0			0
$101$	−6.00000	−	10.3923i	−0.597022	−	1.03407i	−0.993258	−	0.115924i	$-0.963017\pi$
$101$	−6.00000	−	10.3923i	−0.597022	−	1.03407i	0.396236	−	0.918149i	$-0.370316\pi$
$102$		0			0
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	−0.992990	−	0.118199i	$-0.962288\pi$
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	0.598858	+	0.800855i	$0.295621\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.00000	+	5.19615i	−0.290021	+	0.502331i	−0.973814	−	0.227345i	$-0.926996\pi$
$107$	−3.00000	+	5.19615i	−0.290021	+	0.502331i	0.683793	+	0.729676i	$0.260329\pi$
$108$		0			0
$109$	−7.00000	−	12.1244i	−0.670478	−	1.16130i	−0.977769	−	0.209687i	$-0.932756\pi$
$109$	−7.00000	−	12.1244i	−0.670478	−	1.16130i	0.307290	−	0.951616i	$-0.400578\pi$
$110$		0			0
$111$	2.00000			0.189832
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.00000	−	1.73205i	0.0924500	−	0.160128i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−12.5000	+	21.6506i	−1.13636	+	1.96824i
$122$		0			0
$123$	6.00000	+	10.3923i	0.541002	+	0.937043i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	4.00000			0.354943			0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000			0.354943			0.177471	+	0.984126i	$0.443208\pi$
$128$		0			0
$129$	−2.00000	−	3.46410i	−0.176090	−	0.304997i
$130$		0			0
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	0.475380	+	0.879781i	$0.342311\pi$
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	−0.999602	+	0.0281993i	$0.991023\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.00000	−	5.19615i	−0.256307	−	0.443937i	0.708942	−	0.705266i	$-0.249173\pi$
$137$	−3.00000	−	5.19615i	−0.256307	−	0.443937i	−0.965250	+	0.261329i	$0.915839\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$	12.0000			1.01058
$142$		0			0
$143$	6.00000	+	10.3923i	0.501745	+	0.869048i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	−0.716578	−	0.697507i	$-0.754293\pi$
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	0.962348	+	0.271821i	$0.0876260\pi$
$150$		0			0
$151$	4.00000	+	6.92820i	0.325515	+	0.563809i	0.981617	−	0.190864i	$-0.0611289\pi$
$151$	4.00000	+	6.92820i	0.325515	+	0.563809i	−0.656101	+	0.754673i	$0.727796\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	7.00000	+	12.1244i	0.558661	+	0.967629i	0.997609	+	0.0691164i	$0.0220180\pi$
$157$	7.00000	+	12.1244i	0.558661	+	0.967629i	−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$	3.00000	−	5.19615i	0.237915	−	0.412082i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−8.00000	+	13.8564i	−0.626608	+	1.08532i	0.361619	+	0.932326i	$0.382224\pi$
$163$	−8.00000	+	13.8564i	−0.626608	+	1.08532i	−0.988227	+	0.152992i	$0.951109\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	2.00000	+	3.46410i	0.152944	+	0.264906i
$172$		0			0
$173$	−6.00000	+	10.3923i	−0.456172	+	0.790112i	−0.998755	−	0.0498898i	$-0.984113\pi$
$173$	−6.00000	+	10.3923i	−0.456172	+	0.790112i	0.542583	+	0.840002i	$0.317446\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.00000	+	5.19615i	0.224231	+	0.388379i	0.956088	−	0.293079i	$-0.0946798\pi$
$179$	3.00000	+	5.19615i	0.224231	+	0.388379i	−0.731858	+	0.681457i	$0.761346\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$	10.0000			0.739221
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.00000	−	5.19615i	0.217072	−	0.375980i	−0.736839	−	0.676068i	$-0.763683\pi$
$191$	3.00000	−	5.19615i	0.217072	−	0.375980i	0.953912	+	0.300088i	$0.0970159\pi$
$192$		0			0
$193$	5.00000	+	8.66025i	0.359908	+	0.623379i	0.987945	−	0.154805i	$-0.0494748\pi$
$193$	5.00000	+	8.66025i	0.359908	+	0.623379i	−0.628037	+	0.778183i	$0.716141\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	8.00000	+	13.8564i	0.567105	+	0.982255i	0.996850	+	0.0793045i	$0.0252700\pi$
$199$	8.00000	+	13.8564i	0.567105	+	0.982255i	−0.429745	+	0.902950i	$0.641397\pi$
$200$		0			0
$201$	4.00000	−	6.92820i	0.282138	−	0.488678i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.00000	−	5.19615i	−0.208514	−	0.361158i
$208$		0			0
$209$	−24.0000			−1.66011
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$	3.00000	+	5.19615i	0.205557	+	0.356034i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−5.00000	+	8.66025i	−0.337869	+	0.585206i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	8.00000			0.535720			0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000			0.535720			0.267860	+	0.963458i	$0.413684\pi$
$224$		0			0
$225$	−5.00000			−0.333333
$226$		0			0
$227$	−12.0000	−	20.7846i	−0.796468	−	1.37952i	−0.921903	−	0.387421i	$-0.873366\pi$
$227$	−12.0000	−	20.7846i	−0.796468	−	1.37952i	0.125435	−	0.992102i	$-0.459967\pi$
$228$		0			0
$229$	1.00000	−	1.73205i	0.0660819	−	0.114457i	−0.831092	−	0.556136i	$-0.812283\pi$
$229$	1.00000	−	1.73205i	0.0660819	−	0.114457i	0.897173	+	0.441679i	$0.145617\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.00000	+	5.19615i	−0.196537	+	0.340411i	−0.947403	−	0.320043i	$-0.896303\pi$
$233$	−3.00000	+	5.19615i	−0.196537	+	0.340411i	0.750867	+	0.660454i	$0.229636\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	4.00000			0.259828
$238$		0			0
$239$	−18.0000			−1.16432			−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000			−1.16432			−0.582162	+	0.813073i	$0.697793\pi$
$240$		0			0
$241$	−5.00000	−	8.66025i	−0.322078	−	0.557856i	0.658838	−	0.752285i	$-0.271048\pi$
$241$	−5.00000	−	8.66025i	−0.322078	−	0.557856i	−0.980917	+	0.194429i	$0.937715\pi$
$242$		0			0
$243$	−0.500000	+	0.866025i	−0.0320750	+	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.00000	+	6.92820i	−0.254514	+	0.440831i
$248$		0			0
$249$	6.00000	+	10.3923i	0.380235	+	0.658586i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	36.0000			2.26330
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.00000	+	10.3923i	−0.374270	+	0.648254i	−0.990217	−	0.139533i	$-0.955440\pi$
$257$	−6.00000	+	10.3923i	−0.374270	+	0.648254i	0.615948	+	0.787787i	$0.288773\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−3.00000	+	5.19615i	−0.185695	+	0.321634i
$262$		0			0
$263$	−3.00000	−	5.19615i	−0.184988	−	0.320408i	0.758585	−	0.651575i	$-0.225891\pi$
$263$	−3.00000	−	5.19615i	−0.184988	−	0.320408i	−0.943572	+	0.331166i	$0.892558\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−12.0000			−0.734388
$268$		0			0
$269$	12.0000	+	20.7846i	0.731653	+	1.26726i	0.956176	+	0.292791i	$0.0945841\pi$
$269$	12.0000	+	20.7846i	0.731653	+	1.26726i	−0.224523	+	0.974469i	$0.572083\pi$
$270$		0			0
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	−0.513905	−	0.857847i	$-0.671801\pi$
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	0.999870	+	0.0161307i	$0.00513477\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.0000	−	25.9808i	0.904534	−	1.56670i
$276$		0			0
$277$	11.0000	+	19.0526i	0.660926	+	1.14476i	0.980373	+	0.197153i	$0.0631696\pi$
$277$	11.0000	+	19.0526i	0.660926	+	1.14476i	−0.319447	+	0.947604i	$0.603497\pi$
$278$		0			0
$279$	8.00000			0.478947
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	−10.0000	−	17.3205i	−0.594438	−	1.02960i	−0.993626	−	0.112728i	$-0.964041\pi$
$283$	−10.0000	−	17.3205i	−0.594438	−	1.02960i	0.399188	−	0.916869i	$-0.369292\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$	−5.00000	−	8.66025i	−0.293105	−	0.507673i
$292$		0			0
$293$	12.0000			0.701047			0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000			0.701047			0.350524	+	0.936554i	$0.386004\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−3.00000	−	5.19615i	−0.174078	−	0.301511i
$298$		0			0
$299$	6.00000	−	10.3923i	0.346989	−	0.601003i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−6.00000	+	10.3923i	−0.344691	+	0.597022i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	20.0000			1.14146			0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000			1.14146			0.570730	+	0.821138i	$0.306660\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		0			0
$311$	6.00000	+	10.3923i	0.340229	+	0.589294i	0.984475	−	0.175525i	$-0.0561621\pi$
$311$	6.00000	+	10.3923i	0.340229	+	0.589294i	−0.644246	+	0.764818i	$0.722829\pi$
$312$		0			0
$313$	13.0000	−	22.5167i	0.734803	−	1.27272i	−0.220006	−	0.975499i	$-0.570608\pi$
$313$	13.0000	−	22.5167i	0.734803	−	1.27272i	0.954810	−	0.297218i	$-0.0960589\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.0000	−	25.9808i	0.842484	−	1.45922i	−0.0453045	−	0.998973i	$-0.514426\pi$
$317$	15.0000	−	25.9808i	0.842484	−	1.45922i	0.887788	−	0.460252i	$-0.152241\pi$
$318$		0			0
$319$	−18.0000	−	31.1769i	−1.00781	−	1.74557i
$320$		0			0
$321$	6.00000			0.334887
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−5.00000	−	8.66025i	−0.277350	−	0.480384i
$326$		0			0
$327$	−7.00000	+	12.1244i	−0.387101	+	0.670478i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	10.0000	−	17.3205i	0.549650	−	0.952021i	−0.448649	−	0.893708i	$-0.648095\pi$
$331$	10.0000	−	17.3205i	0.549650	−	0.952021i	0.998298	−	0.0583130i	$-0.0185721\pi$
$332$		0			0
$333$	−1.00000	−	1.73205i	−0.0547997	−	0.0949158i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$		0			0
$339$	3.00000	+	5.19615i	0.162938	+	0.282216i
$340$		0			0
$341$	−24.0000	+	41.5692i	−1.29967	+	2.25110i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.00000	−	15.5885i	−0.483145	−	0.836832i	0.516667	−	0.856186i	$-0.327172\pi$
$347$	−9.00000	−	15.5885i	−0.483145	−	0.836832i	−0.999813	+	0.0193540i	$0.993839\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$	−2.00000			−0.106752
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3.00000	−	5.19615i	0.158334	−	0.274242i	−0.775934	−	0.630814i	$-0.782721\pi$
$359$	3.00000	−	5.19615i	0.158334	−	0.274242i	0.934268	+	0.356572i	$0.116054\pi$
$360$		0			0
$361$	1.50000	+	2.59808i	0.0789474	+	0.136741i
$362$		0			0
$363$	25.0000			1.31216
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−16.0000	−	27.7128i	−0.835193	−	1.44660i	−0.893873	−	0.448320i	$-0.852022\pi$
$367$	−16.0000	−	27.7128i	−0.835193	−	1.44660i	0.0586798	−	0.998277i	$-0.481311\pi$
$368$		0			0
$369$	6.00000	−	10.3923i	0.312348	−	0.541002i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	−0.707055	−	0.707159i	$-0.749977\pi$
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	0.965945	+	0.258748i	$0.0833099\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.0000			−0.618031
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$	−2.00000	−	3.46410i	−0.102463	−	0.177471i
$382$		0			0
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	−0.377531	−	0.925997i	$-0.623227\pi$
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	0.990702	−	0.136047i	$-0.0434398\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2.00000	+	3.46410i	−0.101666	+	0.176090i
$388$		0			0
$389$	−15.0000	−	25.9808i	−0.760530	−	1.31728i	−0.942578	−	0.333987i	$-0.891606\pi$
$389$	−15.0000	−	25.9808i	−0.760530	−	1.31728i	0.182047	−	0.983290i	$-0.441728\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	12.0000			0.605320
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7.00000	−	12.1244i	0.351320	−	0.608504i	−0.635161	−	0.772380i	$-0.719066\pi$
$397$	7.00000	−	12.1244i	0.351320	−	0.608504i	0.986481	+	0.163876i	$0.0523996\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.00000	+	5.19615i	−0.149813	+	0.259483i	−0.931158	−	0.364615i	$-0.881200\pi$
$401$	−3.00000	+	5.19615i	−0.149813	+	0.259483i	0.781345	+	0.624099i	$0.214534\pi$
$402$		0			0
$403$	8.00000	+	13.8564i	0.398508	+	0.690237i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.0000			0.594818
$408$		0			0
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	0.985558	−	0.169338i	$-0.0541630\pi$
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	−0.639430	+	0.768849i	$0.720830\pi$
$410$		0			0
$411$	−3.00000	+	5.19615i	−0.147979	+	0.256307i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	2.00000	+	3.46410i	0.0979404	+	0.169638i
$418$		0			0
$419$	−24.0000			−1.17248			−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000			−1.17248			−0.586238	+	0.810139i	$0.699392\pi$
$420$		0			0
$421$	26.0000			1.26716			0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000			1.26716			0.633581	+	0.773676i	$0.281584\pi$
$422$		0			0
$423$	−6.00000	−	10.3923i	−0.291730	−	0.505291i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	6.00000	−	10.3923i	0.289683	−	0.501745i
$430$		0			0
$431$	3.00000	+	5.19615i	0.144505	+	0.250290i	0.929188	−	0.369607i	$-0.120508\pi$
$431$	3.00000	+	5.19615i	0.144505	+	0.250290i	−0.784683	+	0.619897i	$0.787174\pi$
$432$		0			0
$433$	−26.0000			−1.24948			−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000			−1.24948			−0.624740	+	0.780833i	$0.714795\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.0000	+	20.7846i	0.574038	+	0.994263i
$438$		0			0
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	−0.945552	−	0.325471i	$-0.894477\pi$
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	0.754642	+	0.656136i	$0.227810\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.0000	−	25.9808i	0.712672	−	1.23438i	−0.251179	−	0.967941i	$-0.580818\pi$
$443$	15.0000	−	25.9808i	0.712672	−	1.23438i	0.963851	−	0.266443i	$-0.0858483\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−6.00000			−0.283790
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$	36.0000	+	62.3538i	1.69517	+	2.93613i
$452$		0			0
$453$	4.00000	−	6.92820i	0.187936	−	0.325515i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.0000	−	19.0526i	0.514558	−	0.891241i	−0.485299	−	0.874348i	$-0.661289\pi$
$457$	11.0000	−	19.0526i	0.514558	−	0.891241i	0.999857	−	0.0168929i	$-0.00537742\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$	28.0000			1.30127			0.650635	−	0.759390i	$-0.274503\pi$
$463$	28.0000			1.30127			0.650635	+	0.759390i	$0.274503\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	7.00000	−	12.1244i	0.322543	−	0.558661i
$472$		0			0
$473$	−12.0000	−	20.7846i	−0.551761	−	0.955677i
$474$		0			0
$475$	20.0000			0.917663
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$	6.00000	+	10.3923i	0.274147	+	0.474837i	0.969920	−	0.243426i	$-0.0782712\pi$
$479$	6.00000	+	10.3923i	0.274147	+	0.474837i	−0.695773	+	0.718262i	$0.744938\pi$
$480$		0			0
$481$	2.00000	−	3.46410i	0.0911922	−	0.157949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	16.0000	+	27.7128i	0.725029	+	1.25579i	0.958962	+	0.283535i	$0.0915071\pi$
$487$	16.0000	+	27.7128i	0.725029	+	1.25579i	−0.233933	+	0.972253i	$0.575160\pi$
$488$		0			0
$489$	16.0000			0.723545
$490$		0			0
$491$	18.0000			0.812329			0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000			0.812329			0.406164	+	0.913800i	$0.366866\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2.00000	+	3.46410i	−0.0895323	+	0.155074i	−0.907314	−	0.420455i	$-0.861871\pi$
$499$	−2.00000	+	3.46410i	−0.0895323	+	0.155074i	0.817781	+	0.575529i	$0.195204\pi$
$500$		0			0
$501$	6.00000	+	10.3923i	0.268060	+	0.464294i
$502$		0			0
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	4.50000	+	7.79423i	0.199852	+	0.346154i
$508$		0			0
$509$	12.0000	−	20.7846i	0.531891	−	0.921262i	−0.467416	−	0.884037i	$-0.654815\pi$
$509$	12.0000	−	20.7846i	0.531891	−	0.921262i	0.999307	−	0.0372243i	$-0.0118516\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	2.00000	−	3.46410i	0.0883022	−	0.152944i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	72.0000			3.16656
$518$		0			0
$519$	12.0000			0.526742
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	2.00000	−	3.46410i	0.0874539	−	0.151475i	−0.818980	−	0.573822i	$-0.805460\pi$
$523$	2.00000	−	3.46410i	0.0874539	−	0.151475i	0.906434	+	0.422347i	$0.138794\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	24.0000			1.03956
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.00000	−	5.19615i	0.129460	−	0.224231i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.00000	+	1.73205i	−0.0429934	+	0.0744667i	−0.886721	−	0.462304i	$-0.847023\pi$
$541$	−1.00000	+	1.73205i	−0.0429934	+	0.0744667i	0.843728	+	0.536771i	$0.180356\pi$
$542$		0			0
$543$	1.00000	+	1.73205i	0.0429141	+	0.0743294i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−8.00000			−0.342055			−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000			−0.342055			−0.171028	+	0.985266i	$0.554709\pi$
$548$		0			0
$549$	−5.00000	−	8.66025i	−0.213395	−	0.369611i
$550$		0			0
$551$	12.0000	−	20.7846i	0.511217	−	0.885454i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−21.0000	−	36.3731i	−0.889799	−	1.54118i	−0.840113	−	0.542411i	$-0.817511\pi$
$557$	−21.0000	−	36.3731i	−0.889799	−	1.54118i	−0.0496855	−	0.998765i	$-0.515822\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	−0.922032	−	0.387113i	$-0.873472\pi$
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	0.796266	+	0.604947i	$0.206806\pi$
$570$		0			0
$571$	−20.0000	−	34.6410i	−0.836974	−	1.44968i	−0.892413	−	0.451219i	$-0.850989\pi$
$571$	−20.0000	−	34.6410i	−0.836974	−	1.44968i	0.0554391	−	0.998462i	$-0.482344\pi$
$572$		0			0
$573$	−6.00000			−0.250654
$574$		0			0
$575$	−30.0000			−1.25109
$576$		0			0
$577$	−11.0000	−	19.0526i	−0.457936	−	0.793168i	0.540916	−	0.841077i	$-0.318078\pi$
$577$	−11.0000	−	19.0526i	−0.457936	−	0.793168i	−0.998852	+	0.0479084i	$0.984744\pi$
$578$		0			0
$579$	5.00000	−	8.66025i	0.207793	−	0.359908i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	18.0000	−	31.1769i	0.745484	−	1.29122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−24.0000			−0.990586			−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000			−0.990586			−0.495293	+	0.868726i	$0.664939\pi$
$588$		0			0
$589$	−32.0000			−1.31854
$590$		0			0
$591$	3.00000	+	5.19615i	0.123404	+	0.213741i
$592$		0			0
$593$	24.0000	−	41.5692i	0.985562	−	1.70704i	0.346149	−	0.938179i	$-0.387489\pi$
$593$	24.0000	−	41.5692i	0.985562	−	1.70704i	0.639413	−	0.768864i	$-0.279178\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	8.00000	−	13.8564i	0.327418	−	0.567105i
$598$		0			0
$599$	15.0000	+	25.9808i	0.612883	+	1.06155i	0.990752	+	0.135686i	$0.0433238\pi$
$599$	15.0000	+	25.9808i	0.612883	+	1.06155i	−0.377869	+	0.925859i	$0.623343\pi$
$600$		0			0
$601$	−2.00000			−0.0815817			−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000			−0.0815817			−0.0407909	+	0.999168i	$0.512988\pi$
$602$		0			0
$603$	−8.00000			−0.325785
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	0.333842	+	0.942629i	$0.391655\pi$
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	−0.983262	+	0.182199i	$0.941678\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.0000	−	20.7846i	0.485468	−	0.840855i
$612$		0			0
$613$	−19.0000	−	32.9090i	−0.767403	−	1.32918i	−0.938967	−	0.344008i	$-0.888215\pi$
$613$	−19.0000	−	32.9090i	−0.767403	−	1.32918i	0.171564	−	0.985173i	$-0.445118\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−18.0000			−0.724653			−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000			−0.724653			−0.362326	+	0.932051i	$0.618017\pi$
$618$		0			0
$619$	−10.0000	−	17.3205i	−0.401934	−	0.696170i	0.592025	−	0.805919i	$-0.298329\pi$
$619$	−10.0000	−	17.3205i	−0.401934	−	0.696170i	−0.993959	+	0.109749i	$0.964995\pi$
$620$		0			0
$621$	−3.00000	+	5.19615i	−0.120386	+	0.208514i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$	12.0000	+	20.7846i	0.479234	+	0.830057i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−20.0000			−0.796187			−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000			−0.796187			−0.398094	+	0.917345i	$0.630328\pi$
$632$		0			0
$633$	−2.00000	−	3.46410i	−0.0794929	−	0.137686i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	3.00000	−	5.19615i	0.118678	−	0.205557i
$640$		0			0
$641$	9.00000	+	15.5885i	0.355479	+	0.615707i	0.987200	−	0.159489i	$-0.0509845\pi$
$641$	9.00000	+	15.5885i	0.355479	+	0.615707i	−0.631721	+	0.775196i	$0.717651\pi$
$642$		0			0
$643$	−4.00000			−0.157745			−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000			−0.157745			−0.0788723	+	0.996885i	$0.525132\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	18.0000	+	31.1769i	0.707653	+	1.22569i	0.965726	+	0.259565i	$0.0835793\pi$
$647$	18.0000	+	31.1769i	0.707653	+	1.22569i	−0.258073	+	0.966126i	$0.583087\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.00000	−	15.5885i	0.352197	−	0.610023i	−0.634437	−	0.772975i	$-0.718768\pi$
$653$	9.00000	−	15.5885i	0.352197	−	0.610023i	0.986634	+	0.162951i	$0.0521013\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	10.0000			0.390137
$658$		0			0
$659$	6.00000			0.233727			0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000			0.233727			0.116863	+	0.993148i	$0.462716\pi$
$660$		0			0
$661$	19.0000	+	32.9090i	0.739014	+	1.28001i	0.952940	+	0.303160i	$0.0980418\pi$
$661$	19.0000	+	32.9090i	0.739014	+	1.28001i	−0.213925	+	0.976850i	$0.568625\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−18.0000	+	31.1769i	−0.696963	+	1.20717i
$668$		0			0
$669$	−4.00000	−	6.92820i	−0.154649	−	0.267860i
$670$		0			0
$671$	60.0000			2.31627
$672$		0			0
$673$	−34.0000			−1.31060			−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000			−1.31060			−0.655302	+	0.755367i	$0.727459\pi$
$674$		0			0
$675$	2.50000	+	4.33013i	0.0962250	+	0.166667i
$676$		0			0
$677$	6.00000	−	10.3923i	0.230599	−	0.399409i	−0.727386	−	0.686229i	$-0.759265\pi$
$677$	6.00000	−	10.3923i	0.230599	−	0.399409i	0.957984	+	0.286820i	$0.0925982\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−12.0000	+	20.7846i	−0.459841	+	0.796468i
$682$		0			0
$683$	−21.0000	−	36.3731i	−0.803543	−	1.39178i	−0.917270	−	0.398265i	$-0.869613\pi$
$683$	−21.0000	−	36.3731i	−0.803543	−	1.39178i	0.113728	−	0.993512i	$-0.463721\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−2.00000			−0.0763048
$688$		0			0
$689$	−6.00000	−	10.3923i	−0.228582	−	0.395915i
$690$		0			0
$691$	2.00000	−	3.46410i	0.0760836	−	0.131781i	−0.825473	−	0.564441i	$-0.809092\pi$
$691$	2.00000	−	3.46410i	0.0760836	−	0.131781i	0.901557	+	0.432660i	$0.142425\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	6.00000			0.226941
$700$		0			0
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$	4.00000	+	6.92820i	0.150863	+	0.261302i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	−0.756730	−	0.653727i	$-0.773204\pi$
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	0.944509	+	0.328484i	$0.106538\pi$
$710$		0			0
$711$	−2.00000	−	3.46410i	−0.0750059	−	0.129914i
$712$		0			0
$713$	48.0000			1.79761
$714$		0			0
$715$		0			0
$716$		0			0
$717$	9.00000	+	15.5885i	0.336111	+	0.582162i
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−5.00000	+	8.66025i	−0.185952	+	0.322078i
$724$		0			0
$725$	15.0000	+	25.9808i	0.557086	+	0.964901i
$726$		0			0
$727$	8.00000			0.296704			0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000			0.296704			0.148352	+	0.988935i	$0.452603\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−23.0000	+	39.8372i	−0.849524	+	1.47142i	0.0321090	+	0.999484i	$0.489778\pi$
$733$	−23.0000	+	39.8372i	−0.849524	+	1.47142i	−0.881633	+	0.471935i	$0.843556\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.0000	−	41.5692i	0.884051	−	1.53122i
$738$		0			0
$739$	16.0000	+	27.7128i	0.588570	+	1.01943i	0.994420	+	0.105493i	$0.0336420\pi$
$739$	16.0000	+	27.7128i	0.588570	+	1.01943i	−0.405851	+	0.913939i	$0.633025\pi$
$740$		0			0
$741$	8.00000			0.293887
$742$		0			0
$743$	−6.00000			−0.220119			−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000			−0.220119			−0.110059	+	0.993925i	$0.535104\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	6.00000	−	10.3923i	0.219529	−	0.380235i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	16.0000	−	27.7128i	0.583848	−	1.01125i	−0.411170	−	0.911559i	$-0.634880\pi$
$751$	16.0000	−	27.7128i	0.583848	−	1.01125i	0.995018	−	0.0996961i	$-0.0317870\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$	−18.0000	−	31.1769i	−0.653359	−	1.13165i
$760$		0			0
$761$	−18.0000	+	31.1769i	−0.652499	+	1.13016i	0.330015	+	0.943976i	$0.392946\pi$
$761$	−18.0000	+	31.1769i	−0.652499	+	1.13016i	−0.982514	+	0.186187i	$0.940387\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−26.0000			−0.937584			−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000			−0.937584			−0.468792	+	0.883309i	$0.655311\pi$
$770$		0			0
$771$	12.0000			0.432169
$772$		0			0
$773$	6.00000	+	10.3923i	0.215805	+	0.373785i	0.953521	−	0.301326i	$-0.0974291\pi$
$773$	6.00000	+	10.3923i	0.215805	+	0.373785i	−0.737716	+	0.675111i	$0.764096\pi$
$774$		0			0
$775$	20.0000	−	34.6410i	0.718421	−	1.24434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−24.0000	+	41.5692i	−0.859889	+	1.48937i
$780$		0			0
$781$	18.0000	+	31.1769i	0.644091	+	1.11560i
$782$		0			0
$783$	6.00000			0.214423
$784$		0			0
$785$		0			0
$786$		0			0
$787$	2.00000	+	3.46410i	0.0712923	+	0.123482i	0.899468	−	0.436987i	$-0.143954\pi$
$787$	2.00000	+	3.46410i	0.0712923	+	0.123482i	−0.828176	+	0.560469i	$0.810621\pi$
$788$		0			0
$789$	−3.00000	+	5.19615i	−0.106803	+	0.184988i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	10.0000	−	17.3205i	0.355110	−	0.615069i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−36.0000			−1.27519			−0.637593	−	0.770374i	$-0.720070\pi$
$797$	−36.0000			−1.27519			−0.637593	+	0.770374i	$0.720070\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	6.00000	+	10.3923i	0.212000	+	0.367194i
$802$		0			0
$803$	−30.0000	+	51.9615i	−1.05868	+	1.83368i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	12.0000	−	20.7846i	0.422420	−	0.731653i
$808$		0			0
$809$	27.0000	+	46.7654i	0.949269	+	1.64418i	0.746968	+	0.664860i	$0.231509\pi$
$809$	27.0000	+	46.7654i	0.949269	+	1.64418i	0.202301	+	0.979323i	$0.435158\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$		0			0
$813$	−16.0000			−0.561144
$814$		0			0
$815$		0			0
$816$		0			0
$817$	8.00000	−	13.8564i	0.279885	−	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.00000	+	15.5885i	−0.314102	+	0.544041i	−0.979246	−	0.202674i	$-0.935037\pi$
$821$	−9.00000	+	15.5885i	−0.314102	+	0.544041i	0.665144	+	0.746715i	$0.268370\pi$
$822$		0			0
$823$	−2.00000	−	3.46410i	−0.0697156	−	0.120751i	0.829060	−	0.559159i	$-0.188876\pi$
$823$	−2.00000	−	3.46410i	−0.0697156	−	0.120751i	−0.898776	+	0.438408i	$0.855543\pi$
$824$		0			0
$825$	−30.0000			−1.04447
$826$		0			0
$827$	6.00000			0.208640			0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000			0.208640			0.104320	+	0.994544i	$0.466733\pi$
$828$		0			0
$829$	−11.0000	−	19.0526i	−0.382046	−	0.661723i	0.609309	−	0.792933i	$-0.291447\pi$
$829$	−11.0000	−	19.0526i	−0.382046	−	0.661723i	−0.991355	+	0.131210i	$0.958114\pi$
$830$		0			0
$831$	11.0000	−	19.0526i	0.381586	−	0.660926i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−4.00000	−	6.92820i	−0.138260	−	0.239474i
$838$		0			0
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	9.00000	+	15.5885i	0.309976	+	0.536895i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−10.0000	+	17.3205i	−0.343199	+	0.594438i
$850$		0			0
$851$	−6.00000	−	10.3923i	−0.205677	−	0.356244i
$852$		0			0
$853$	−14.0000			−0.479351			−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000			−0.479351			−0.239675	+	0.970853i	$0.577041\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.00000	+	10.3923i	0.204956	+	0.354994i	0.950119	−	0.311888i	$-0.100962\pi$
$857$	6.00000	+	10.3923i	0.204956	+	0.354994i	−0.745163	+	0.666883i	$0.767628\pi$
$858$		0			0
$859$	2.00000	−	3.46410i	0.0682391	−	0.118194i	−0.829887	−	0.557931i	$-0.811595\pi$
$859$	2.00000	−	3.46410i	0.0682391	−	0.118194i	0.898126	+	0.439738i	$0.144929\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.00000	+	5.19615i	−0.102121	+	0.176879i	−0.912558	−	0.408946i	$-0.865896\pi$
$863$	−3.00000	+	5.19615i	−0.102121	+	0.176879i	0.810437	+	0.585826i	$0.199230\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−17.0000			−0.577350
$868$		0			0
$869$	24.0000			0.814144
$870$		0			0
$871$	−8.00000	−	13.8564i	−0.271070	−	0.469506i
$872$		0			0
$873$	−5.00000	+	8.66025i	−0.169224	+	0.293105i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−7.00000	+	12.1244i	−0.236373	+	0.409410i	−0.959671	−	0.281126i	$-0.909292\pi$
$877$	−7.00000	+	12.1244i	−0.236373	+	0.409410i	0.723298	+	0.690536i	$0.242625\pi$
$878$		0			0
$879$	−6.00000	−	10.3923i	−0.202375	−	0.350524i
$880$		0			0
$881$	−24.0000			−0.808581			−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000			−0.808581			−0.404290	+	0.914631i	$0.632481\pi$
$882$		0			0
$883$	4.00000			0.134611			0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000			0.134611			0.0673054	+	0.997732i	$0.478560\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18.0000	+	31.1769i	−0.604381	+	1.04682i	0.387768	+	0.921757i	$0.373246\pi$
$887$	−18.0000	+	31.1769i	−0.604381	+	1.04682i	−0.992149	+	0.125061i	$0.960087\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−3.00000	+	5.19615i	−0.100504	+	0.174078i
$892$		0			0
$893$	24.0000	+	41.5692i	0.803129	+	1.39106i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−12.0000			−0.400668
$898$		0			0
$899$	−24.0000	−	41.5692i	−0.800445	−	1.38641i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−2.00000	−	3.46410i	−0.0664089	−	0.115024i	0.830909	−	0.556408i	$-0.187821\pi$
$907$	−2.00000	−	3.46410i	−0.0664089	−	0.115024i	−0.897318	+	0.441384i	$0.854488\pi$
$908$		0			0
$909$	12.0000			0.398015
$910$		0			0
$911$	−54.0000			−1.78910			−0.894550	−	0.446968i	$-0.852504\pi$
$911$	−54.0000			−1.78910			−0.894550	+	0.446968i	$0.852504\pi$
$912$		0			0
$913$	36.0000	+	62.3538i	1.19143	+	2.06361i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−8.00000	+	13.8564i	−0.263896	+	0.457081i	−0.967274	−	0.253735i	$-0.918341\pi$
$919$	−8.00000	+	13.8564i	−0.263896	+	0.457081i	0.703378	+	0.710816i	$0.251674\pi$
$920$		0			0
$921$	−10.0000	−	17.3205i	−0.329511	−	0.570730i
$922$		0			0
$923$	12.0000			0.394985
$924$		0			0
$925$	−10.0000			−0.328798
$926$		0			0
$927$	−4.00000	−	6.92820i	−0.131377	−	0.227552i
$928$		0			0
$929$	6.00000	−	10.3923i	0.196854	−	0.340960i	−0.750653	−	0.660697i	$-0.770261\pi$
$929$	6.00000	−	10.3923i	0.196854	−	0.340960i	0.947507	+	0.319736i	$0.103594\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	6.00000	−	10.3923i	0.196431	−	0.340229i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		0			0
$939$	−26.0000			−0.848478
$940$		0			0
$941$	−12.0000	−	20.7846i	−0.391189	−	0.677559i	0.601418	−	0.798935i	$-0.294603\pi$
$941$	−12.0000	−	20.7846i	−0.391189	−	0.677559i	−0.992607	+	0.121376i	$0.961269\pi$
$942$		0			0
$943$	36.0000	−	62.3538i	1.17232	−	2.03052i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−15.0000	+	25.9808i	−0.487435	+	0.844261i	−0.999896	−	0.0144491i	$-0.995401\pi$
$947$	−15.0000	+	25.9808i	−0.487435	+	0.844261i	0.512461	+	0.858710i	$0.328734\pi$
$948$		0			0
$949$	10.0000	+	17.3205i	0.324614	+	0.562247i
$950$		0			0
$951$	−30.0000			−0.972817
$952$		0			0
$953$	42.0000			1.36051			0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000			1.36051			0.680257	+	0.732974i	$0.261868\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−18.0000	+	31.1769i	−0.581857	+	1.00781i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−16.5000	+	28.5788i	−0.532258	+	0.921898i
$962$		0			0
$963$	−3.00000	−	5.19615i	−0.0966736	−	0.167444i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	28.0000			0.900419			0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000			0.900419			0.450210	+	0.892923i	$0.351349\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6.00000	+	10.3923i	−0.192549	+	0.333505i	−0.946094	−	0.323891i	$-0.895009\pi$
$971$	−6.00000	+	10.3923i	−0.192549	+	0.333505i	0.753545	+	0.657396i	$0.228342\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−5.00000	+	8.66025i	−0.160128	+	0.277350i
$976$		0			0
$977$	−3.00000	−	5.19615i	−0.0959785	−	0.166240i	0.814038	−	0.580812i	$-0.197265\pi$
$977$	−3.00000	−	5.19615i	−0.0959785	−	0.166240i	−0.910017	+	0.414572i	$0.863931\pi$
$978$		0			0
$979$	−72.0000			−2.30113
$980$		0			0
$981$	14.0000			0.446986
$982$		0			0
$983$	24.0000	+	41.5692i	0.765481	+	1.32585i	0.939992	+	0.341197i	$0.110832\pi$
$983$	24.0000	+	41.5692i	0.765481	+	1.32585i	−0.174511	+	0.984655i	$0.555834\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.0000	+	20.7846i	−0.381578	+	0.660912i
$990$		0			0
$991$	−20.0000	−	34.6410i	−0.635321	−	1.10041i	−0.986447	−	0.164080i	$-0.947534\pi$
$991$	−20.0000	−	34.6410i	−0.635321	−	1.10041i	0.351126	−	0.936328i	$-0.385799\pi$
$992$		0			0
$993$	−20.0000			−0.634681
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−5.00000	−	8.66025i	−0.158352	−	0.274273i	0.775923	−	0.630828i	$-0.217285\pi$
$997$	−5.00000	−	8.66025i	−0.158352	−	0.274273i	−0.934274	+	0.356555i	$0.883951\pi$
$998$		0			0
$999$	−1.00000	+	1.73205i	−0.0316386	+	0.0547997i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.q.g.961.1		2
4.3	odd	2		588.2.i.f.373.1		2
7.2	even	3		2352.2.a.s.1.1		1
7.3	odd	6		2352.2.q.s.1537.1		2
7.4	even	3	inner	2352.2.q.g.1537.1		2
7.5	odd	6		336.2.a.b.1.1		1
7.6	odd	2		2352.2.q.s.961.1		2
12.11	even	2		1764.2.k.d.1549.1		2
21.2	odd	6		7056.2.a.x.1.1		1
21.5	even	6		1008.2.a.g.1.1		1
28.3	even	6		588.2.i.c.361.1		2
28.11	odd	6		588.2.i.f.361.1		2
28.19	even	6		84.2.a.b.1.1	✓	1
28.23	odd	6		588.2.a.c.1.1		1
28.27	even	2		588.2.i.c.373.1		2
35.19	odd	6		8400.2.a.ct.1.1		1
56.5	odd	6		1344.2.a.o.1.1		1
56.19	even	6		1344.2.a.f.1.1		1
56.37	even	6		9408.2.a.r.1.1		1
56.51	odd	6		9408.2.a.co.1.1		1
84.11	even	6		1764.2.k.d.361.1		2
84.23	even	6		1764.2.a.g.1.1		1
84.47	odd	6		252.2.a.b.1.1		1
84.59	odd	6		1764.2.k.e.361.1		2
84.83	odd	2		1764.2.k.e.1549.1		2
112.5	odd	12		5376.2.c.x.2689.2		2
112.19	even	12		5376.2.c.i.2689.2		2
112.61	odd	12		5376.2.c.x.2689.1		2
112.75	even	12		5376.2.c.i.2689.1		2
140.19	even	6		2100.2.a.a.1.1		1
140.47	odd	12		2100.2.k.a.1849.1		2
140.103	odd	12		2100.2.k.a.1849.2		2
168.5	even	6		4032.2.a.t.1.1		1
168.131	odd	6		4032.2.a.u.1.1		1
252.47	odd	6		2268.2.j.f.757.1		2
252.103	even	6		2268.2.j.i.1513.1		2
252.131	odd	6		2268.2.j.f.1513.1		2
252.187	even	6		2268.2.j.i.757.1		2
420.47	even	12		6300.2.k.r.6049.2		2
420.299	odd	6		6300.2.a.p.1.1		1
420.383	even	12		6300.2.k.r.6049.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.2.a.b.1.1	✓	1	28.19	even	6
252.2.a.b.1.1		1	84.47	odd	6
336.2.a.b.1.1		1	7.5	odd	6
588.2.a.c.1.1		1	28.23	odd	6
588.2.i.c.361.1		2	28.3	even	6
588.2.i.c.373.1		2	28.27	even	2
588.2.i.f.361.1		2	28.11	odd	6
588.2.i.f.373.1		2	4.3	odd	2
1008.2.a.g.1.1		1	21.5	even	6
1344.2.a.f.1.1		1	56.19	even	6
1344.2.a.o.1.1		1	56.5	odd	6
1764.2.a.g.1.1		1	84.23	even	6
1764.2.k.d.361.1		2	84.11	even	6
1764.2.k.d.1549.1		2	12.11	even	2
1764.2.k.e.361.1		2	84.59	odd	6
1764.2.k.e.1549.1		2	84.83	odd	2
2100.2.a.a.1.1		1	140.19	even	6
2100.2.k.a.1849.1		2	140.47	odd	12
2100.2.k.a.1849.2		2	140.103	odd	12
2268.2.j.f.757.1		2	252.47	odd	6
2268.2.j.f.1513.1		2	252.131	odd	6
2268.2.j.i.757.1		2	252.187	even	6
2268.2.j.i.1513.1		2	252.103	even	6
2352.2.a.s.1.1		1	7.2	even	3
2352.2.q.g.961.1		2	1.1	even	1	trivial
2352.2.q.g.1537.1		2	7.4	even	3	inner
2352.2.q.s.961.1		2	7.6	odd	2
2352.2.q.s.1537.1		2	7.3	odd	6
4032.2.a.t.1.1		1	168.5	even	6
4032.2.a.u.1.1		1	168.131	odd	6
5376.2.c.i.2689.1		2	112.75	even	12
5376.2.c.i.2689.2		2	112.19	even	12
5376.2.c.x.2689.1		2	112.61	odd	12
5376.2.c.x.2689.2		2	112.5	odd	12
6300.2.a.p.1.1		1	420.299	odd	6
6300.2.k.r.6049.1		2	420.383	even	12
6300.2.k.r.6049.2		2	420.47	even	12
7056.2.a.x.1.1		1	21.2	odd	6
8400.2.a.ct.1.1		1	35.19	odd	6
9408.2.a.r.1.1		1	56.37	even	6
9408.2.a.co.1.1		1	56.51	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{9} +(-3.00000 - 5.19615i) q^{11} -2.00000 q^{13} +(2.00000 - 3.46410i) q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} +1.00000 q^{27} +6.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(-3.00000 + 5.19615i) q^{33} +(-1.00000 + 1.73205i) q^{37} +(1.00000 + 1.73205i) q^{39} -12.0000 q^{41} +4.00000 q^{43} +(-6.00000 + 10.3923i) q^{47} +(3.00000 + 5.19615i) q^{53} -4.00000 q^{57} +(-5.00000 + 8.66025i) q^{61} +(4.00000 + 6.92820i) q^{67} +6.00000 q^{69} -6.00000 q^{71} +(-5.00000 - 8.66025i) q^{73} +(2.50000 - 4.33013i) q^{75} +(-2.00000 + 3.46410i) q^{79} +(-0.500000 - 0.866025i) q^{81} -12.0000 q^{83} +(-3.00000 - 5.19615i) q^{87} +(6.00000 - 10.3923i) q^{89} +(-4.00000 + 6.92820i) q^{93} +10.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{9}+O(q^{10}) \) 2 * q - q^3 - q^9	\( 2 q - q^{3} - q^{9} - 6 q^{11} - 4 q^{13} + 4 q^{19} - 6 q^{23} + 5 q^{25} + 2 q^{27} + 12 q^{29} - 8 q^{31} - 6 q^{33} - 2 q^{37} + 2 q^{39} - 24 q^{41} + 8 q^{43} - 12 q^{47} + 6 q^{53} - 8 q^{57} - 10 q^{61} + 8 q^{67} + 12 q^{69} - 12 q^{71} - 10 q^{73} + 5 q^{75} - 4 q^{79} - q^{81} - 24 q^{83} - 6 q^{87} + 12 q^{89} - 8 q^{93} + 20 q^{97} + 12 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^9 - 6 * q^11 - 4 * q^13 + 4 * q^19 - 6 * q^23 + 5 * q^25 + 2 * q^27 + 12 * q^29 - 8 * q^31 - 6 * q^33 - 2 * q^37 + 2 * q^39 - 24 * q^41 + 8 * q^43 - 12 * q^47 + 6 * q^53 - 8 * q^57 - 10 * q^61 + 8 * q^67 + 12 * q^69 - 12 * q^71 - 10 * q^73 + 5 * q^75 - 4 * q^79 - q^81 - 24 * q^83 - 6 * q^87 + 12 * q^89 - 8 * q^93 + 20 * q^97 + 12 * q^99

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