LMFDB - Embedded newform 2352.2.q.a.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:	$0$
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 294)
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.a.1537.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{3} +(-2.00000 + 3.46410i) q^{5} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{3} +(-2.00000 + 3.46410i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(-2.00000 - 3.46410i) q^{11} +4.00000 q^{13} +4.00000 q^{15} +(2.00000 - 3.46410i) q^{19} +(-5.50000 - 9.52628i) q^{25} +1.00000 q^{27} +2.00000 q^{29} +(4.00000 + 6.92820i) q^{31} +(-2.00000 + 3.46410i) q^{33} +(3.00000 - 5.19615i) q^{37} +(-2.00000 - 3.46410i) q^{39} -4.00000 q^{43} +(-2.00000 - 3.46410i) q^{45} +(-4.00000 + 6.92820i) q^{47} +(5.00000 + 8.66025i) q^{53} +16.0000 q^{55} -4.00000 q^{57} +(2.00000 + 3.46410i) q^{59} +(2.00000 - 3.46410i) q^{61} +(-8.00000 + 13.8564i) q^{65} +(2.00000 + 3.46410i) q^{67} -8.00000 q^{71} +(8.00000 + 13.8564i) q^{73} +(-5.50000 + 9.52628i) q^{75} +(-4.00000 + 6.92820i) q^{79} +(-0.500000 - 0.866025i) q^{81} +12.0000 q^{83} +(-1.00000 - 1.73205i) q^{87} +(-4.00000 + 6.92820i) q^{89} +(4.00000 - 6.92820i) q^{93} +(8.00000 + 13.8564i) q^{95} +8.00000 q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 4 q^{5} - q^{9}+O(q^{10}) $ 2 * q - q^3 - 4 * q^5 - q^9	$ 2 q - q^{3} - 4 q^{5} - q^{9} - 4 q^{11} + 8 q^{13} + 8 q^{15} + 4 q^{19} - 11 q^{25} + 2 q^{27} + 4 q^{29} + 8 q^{31} - 4 q^{33} + 6 q^{37} - 4 q^{39} - 8 q^{43} - 4 q^{45} - 8 q^{47} + 10 q^{53} + 32 q^{55} - 8 q^{57} + 4 q^{59} + 4 q^{61} - 16 q^{65} + 4 q^{67} - 16 q^{71} + 16 q^{73} - 11 q^{75} - 8 q^{79} - q^{81} + 24 q^{83} - 2 q^{87} - 8 q^{89} + 8 q^{93} + 16 q^{95} + 16 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q - q^3 - 4 * q^5 - q^9 - 4 * q^11 + 8 * q^13 + 8 * q^15 + 4 * q^19 - 11 * q^25 + 2 * q^27 + 4 * q^29 + 8 * q^31 - 4 * q^33 + 6 * q^37 - 4 * q^39 - 8 * q^43 - 4 * q^45 - 8 * q^47 + 10 * q^53 + 32 * q^55 - 8 * q^57 + 4 * q^59 + 4 * q^61 - 16 * q^65 + 4 * q^67 - 16 * q^71 + 16 * q^73 - 11 * q^75 - 8 * q^79 - q^81 + 24 * q^83 - 2 * q^87 - 8 * q^89 + 8 * q^93 + 16 * q^95 + 16 * q^97 + 8 * 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
$5$	−2.00000	+	3.46410i	−0.894427	+	1.54919i	−0.0599153	+	0.998203i	$0.519083\pi$
$5$	−2.00000	+	3.46410i	−0.894427	+	1.54919i	−0.834512	+	0.550990i	$0.814250\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	−2.00000	−	3.46410i	−0.603023	−	1.04447i	−0.992361	−	0.123371i	$-0.960630\pi$
$11$	−2.00000	−	3.46410i	−0.603023	−	1.04447i	0.389338	−	0.921095i	$-0.372704\pi$
$12$		0			0
$13$	4.00000			1.10940			0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000			1.10940			0.554700	+	0.832050i	$0.312833\pi$
$14$		0			0
$15$	4.00000			1.03280
$16$		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
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	−5.50000	−	9.52628i	−1.10000	−	1.90526i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	4.00000	+	6.92820i	0.718421	+	1.24434i	0.961625	+	0.274367i	$0.0884683\pi$
$31$	4.00000	+	6.92820i	0.718421	+	1.24434i	−0.243204	+	0.969975i	$0.578198\pi$
$32$		0			0
$33$	−2.00000	+	3.46410i	−0.348155	+	0.603023i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.00000	−	5.19615i	0.493197	−	0.854242i	−0.506772	−	0.862080i	$-0.669162\pi$
$37$	3.00000	−	5.19615i	0.493197	−	0.854242i	0.999969	+	0.00783774i	$0.00249486\pi$
$38$		0			0
$39$	−2.00000	−	3.46410i	−0.320256	−	0.554700i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$	−2.00000	−	3.46410i	−0.298142	−	0.516398i
$46$		0			0
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	0.411606	+	0.911362i	$0.364968\pi$
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	−0.995066	+	0.0992202i	$0.968365\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.00000	+	8.66025i	0.686803	+	1.18958i	0.972867	+	0.231367i	$0.0743197\pi$
$53$	5.00000	+	8.66025i	0.686803	+	1.18958i	−0.286064	+	0.958211i	$0.592347\pi$
$54$		0			0
$55$	16.0000			2.15744
$56$		0			0
$57$	−4.00000			−0.529813
$58$		0			0
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	0.966342	−	0.257260i	$-0.0828195\pi$
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	−0.705965	+	0.708247i	$0.749486\pi$
$60$		0			0
$61$	2.00000	−	3.46410i	0.256074	−	0.443533i	−0.709113	−	0.705095i	$-0.750904\pi$
$61$	2.00000	−	3.46410i	0.256074	−	0.443533i	0.965187	+	0.261562i	$0.0842377\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.00000	+	13.8564i	−0.992278	+	1.71868i
$66$		0			0
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	0.961946	−	0.273241i	$-0.0880957\pi$
$67$	2.00000	+	3.46410i	0.244339	+	0.423207i	−0.717607	+	0.696449i	$0.754762\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$	8.00000	+	13.8564i	0.936329	+	1.62177i	0.772246	+	0.635323i	$0.219133\pi$
$73$	8.00000	+	13.8564i	0.936329	+	1.62177i	0.164083	+	0.986447i	$0.447534\pi$
$74$		0			0
$75$	−5.50000	+	9.52628i	−0.635085	+	1.10000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	0.548352	+	0.836247i	$0.315255\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.00000	−	1.73205i	−0.107211	−	0.185695i
$88$		0			0
$89$	−4.00000	+	6.92820i	−0.423999	+	0.734388i	−0.996326	−	0.0856373i	$-0.972707\pi$
$89$	−4.00000	+	6.92820i	−0.423999	+	0.734388i	0.572327	+	0.820025i	$0.306041\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	4.00000	−	6.92820i	0.414781	−	0.718421i
$94$		0			0
$95$	8.00000	+	13.8564i	0.820783	+	1.42164i
$96$		0			0
$97$	8.00000			0.812277			0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000			0.812277			0.406138	+	0.913812i	$0.366875\pi$
$98$		0			0
$99$	4.00000			0.402015
$100$		0			0
$101$	−2.00000	−	3.46410i	−0.199007	−	0.344691i	0.749199	−	0.662344i	$-0.230438\pi$
$101$	−2.00000	−	3.46410i	−0.199007	−	0.344691i	−0.948207	+	0.317653i	$0.897105\pi$
$102$		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
$107$	2.00000	−	3.46410i	0.193347	−	0.334887i	−0.753010	−	0.658009i	$-0.771399\pi$
$107$	2.00000	−	3.46410i	0.193347	−	0.334887i	0.946357	+	0.323122i	$0.104732\pi$
$108$		0			0
$109$	7.00000	+	12.1244i	0.670478	+	1.16130i	0.977769	+	0.209687i	$0.0672444\pi$
$109$	7.00000	+	12.1244i	0.670478	+	1.16130i	−0.307290	+	0.951616i	$0.599422\pi$
$110$		0			0
$111$	−6.00000			−0.569495
$112$		0			0
$113$	−14.0000			−1.31701			−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000			−1.31701			−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.00000	+	3.46410i	−0.184900	+	0.320256i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.50000	+	4.33013i	−0.227273	+	0.393648i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	24.0000			2.14663
$126$		0			0
$127$	16.0000			1.41977			0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000			1.41977			0.709885	+	0.704317i	$0.248747\pi$
$128$		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
$135$	−2.00000	+	3.46410i	−0.172133	+	0.298142i
$136$		0			0
$137$	5.00000	+	8.66025i	0.427179	+	0.739895i	0.996621	−	0.0821359i	$-0.0261741\pi$
$137$	5.00000	+	8.66025i	0.427179	+	0.739895i	−0.569442	+	0.822031i	$0.692841\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$	8.00000			0.673722
$142$		0			0
$143$	−8.00000	−	13.8564i	−0.668994	−	1.15873i
$144$		0			0
$145$	−4.00000	+	6.92820i	−0.332182	+	0.575356i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.00000	−	8.66025i	0.409616	−	0.709476i	−0.585231	−	0.810867i	$-0.698996\pi$
$149$	5.00000	−	8.66025i	0.409616	−	0.709476i	0.994847	+	0.101391i	$0.0323294\pi$
$150$		0			0
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	0.656101	−	0.754673i	$-0.272204\pi$
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	−0.981617	+	0.190864i	$0.938871\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−32.0000			−2.57030
$156$		0			0
$157$	−2.00000	−	3.46410i	−0.159617	−	0.276465i	0.775113	−	0.631822i	$-0.217693\pi$
$157$	−2.00000	−	3.46410i	−0.159617	−	0.276465i	−0.934731	+	0.355357i	$0.884359\pi$
$158$		0			0
$159$	5.00000	−	8.66025i	0.396526	−	0.686803i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	6.00000	−	10.3923i	0.469956	−	0.813988i	−0.529454	−	0.848339i	$-0.677603\pi$
$163$	6.00000	−	10.3923i	0.469956	−	0.813988i	0.999410	+	0.0343508i	$0.0109363\pi$
$164$		0			0
$165$	−8.00000	−	13.8564i	−0.622799	−	1.07872i
$166$		0			0
$167$	−8.00000			−0.619059			−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000			−0.619059			−0.309529	+	0.950890i	$0.600171\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$	2.00000	+	3.46410i	0.152944	+	0.264906i
$172$		0			0
$173$	2.00000	−	3.46410i	0.152057	−	0.263371i	−0.779926	−	0.625871i	$-0.784744\pi$
$173$	2.00000	−	3.46410i	0.152057	−	0.263371i	0.931984	+	0.362500i	$0.118077\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	2.00000	−	3.46410i	0.150329	−	0.260378i
$178$		0			0
$179$	6.00000	+	10.3923i	0.448461	+	0.776757i	0.998286	−	0.0585225i	$-0.0186389\pi$
$179$	6.00000	+	10.3923i	0.448461	+	0.776757i	−0.549825	+	0.835280i	$0.685306\pi$
$180$		0			0
$181$	20.0000			1.48659			0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000			1.48659			0.743294	+	0.668965i	$0.233262\pi$
$182$		0			0
$183$	−4.00000			−0.295689
$184$		0			0
$185$	12.0000	+	20.7846i	0.882258	+	1.52811i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	−0.899770	+	0.436365i	$0.856266\pi$
$194$		0			0
$195$	16.0000			1.14578
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	4.00000	+	6.92820i	0.283552	+	0.491127i	0.972257	−	0.233915i	$-0.0751537\pi$
$199$	4.00000	+	6.92820i	0.283552	+	0.491127i	−0.688705	+	0.725042i	$0.741820\pi$
$200$		0			0
$201$	2.00000	−	3.46410i	0.141069	−	0.244339i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−16.0000			−1.10674
$210$		0			0
$211$	28.0000			1.92760			0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000			1.92760			0.963800	+	0.266627i	$0.0859092\pi$
$212$		0			0
$213$	4.00000	+	6.92820i	0.274075	+	0.474713i
$214$		0			0
$215$	8.00000	−	13.8564i	0.545595	−	0.944999i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	8.00000	−	13.8564i	0.540590	−	0.936329i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$		0			0
$225$	11.0000			0.733333
$226$		0			0
$227$	10.0000	+	17.3205i	0.663723	+	1.14960i	0.979630	+	0.200812i	$0.0643581\pi$
$227$	10.0000	+	17.3205i	0.663723	+	1.14960i	−0.315906	+	0.948790i	$0.602309\pi$
$228$		0			0
$229$	−2.00000	+	3.46410i	−0.132164	+	0.228914i	−0.924510	−	0.381157i	$-0.875526\pi$
$229$	−2.00000	+	3.46410i	−0.132164	+	0.228914i	0.792347	+	0.610071i	$0.208859\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.00000	−	8.66025i	0.327561	−	0.567352i	−0.654466	−	0.756091i	$-0.727107\pi$
$233$	5.00000	−	8.66025i	0.327561	−	0.567352i	0.982027	+	0.188739i	$0.0604400\pi$
$234$		0			0
$235$	−16.0000	−	27.7128i	−1.04372	−	1.80778i
$236$		0			0
$237$	8.00000			0.519656
$238$		0			0
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	−4.00000	−	6.92820i	−0.257663	−	0.446285i	0.707953	−	0.706260i	$-0.249619\pi$
$241$	−4.00000	−	6.92820i	−0.257663	−	0.446285i	−0.965615	+	0.259975i	$0.916286\pi$
$242$		0			0
$243$	−0.500000	+	0.866025i	−0.0320750	+	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	8.00000	−	13.8564i	0.509028	−	0.881662i
$248$		0			0
$249$	−6.00000	−	10.3923i	−0.380235	−	0.658586i
$250$		0			0
$251$	20.0000			1.26239			0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000			1.26239			0.631194	+	0.775625i	$0.282565\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.00000	+	6.92820i	−0.249513	+	0.432169i	−0.963391	−	0.268101i	$-0.913604\pi$
$257$	−4.00000	+	6.92820i	−0.249513	+	0.432169i	0.713878	+	0.700270i	$0.246937\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−1.00000	+	1.73205i	−0.0618984	+	0.107211i
$262$		0			0
$263$	4.00000	+	6.92820i	0.246651	+	0.427211i	0.962594	−	0.270947i	$-0.0873367\pi$
$263$	4.00000	+	6.92820i	0.246651	+	0.427211i	−0.715944	+	0.698158i	$0.754003\pi$
$264$		0			0
$265$	−40.0000			−2.45718
$266$		0			0
$267$	8.00000			0.489592
$268$		0			0
$269$	−14.0000	−	24.2487i	−0.853595	−	1.47847i	−0.877942	−	0.478766i	$-0.841084\pi$
$269$	−14.0000	−	24.2487i	−0.853595	−	1.47847i	0.0243472	−	0.999704i	$-0.492249\pi$
$270$		0			0
$271$	−16.0000	+	27.7128i	−0.971931	+	1.68343i	−0.282218	+	0.959350i	$0.591070\pi$
$271$	−16.0000	+	27.7128i	−0.971931	+	1.68343i	−0.689713	+	0.724083i	$0.742263\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−22.0000	+	38.1051i	−1.32665	+	2.29783i
$276$		0			0
$277$	−11.0000	−	19.0526i	−0.660926	−	1.14476i	−0.980373	−	0.197153i	$-0.936830\pi$
$277$	−11.0000	−	19.0526i	−0.660926	−	1.14476i	0.319447	−	0.947604i	$-0.396503\pi$
$278$		0			0
$279$	−8.00000			−0.478947
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	−14.0000	−	24.2487i	−0.832214	−	1.44144i	−0.896279	−	0.443491i	$-0.853740\pi$
$283$	−14.0000	−	24.2487i	−0.832214	−	1.44144i	0.0640654	−	0.997946i	$-0.479593\pi$
$284$		0			0
$285$	8.00000	−	13.8564i	0.473879	−	0.820783i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$	−4.00000	−	6.92820i	−0.234484	−	0.406138i
$292$		0			0
$293$	−12.0000			−0.701047			−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000			−0.701047			−0.350524	+	0.936554i	$0.613996\pi$
$294$		0			0
$295$	−16.0000			−0.931556
$296$		0			0
$297$	−2.00000	−	3.46410i	−0.116052	−	0.201008i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−2.00000	+	3.46410i	−0.114897	+	0.199007i
$304$		0			0
$305$	8.00000	+	13.8564i	0.458079	+	0.793416i
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		0			0
$311$	16.0000	+	27.7128i	0.907277	+	1.57145i	0.817832	+	0.575458i	$0.195176\pi$
$311$	16.0000	+	27.7128i	0.907277	+	1.57145i	0.0894452	+	0.995992i	$0.471491\pi$
$312$		0			0
$313$	−12.0000	+	20.7846i	−0.678280	+	1.17482i	0.297218	+	0.954810i	$0.403941\pi$
$313$	−12.0000	+	20.7846i	−0.678280	+	1.17482i	−0.975499	+	0.220006i	$0.929392\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.0000	+	25.9808i	−0.842484	+	1.45922i	0.0453045	+	0.998973i	$0.485574\pi$
$317$	−15.0000	+	25.9808i	−0.842484	+	1.45922i	−0.887788	+	0.460252i	$0.847759\pi$
$318$		0			0
$319$	−4.00000	−	6.92820i	−0.223957	−	0.387905i
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−22.0000	−	38.1051i	−1.22034	−	2.11369i
$326$		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
$333$	3.00000	+	5.19615i	0.164399	+	0.284747i
$334$		0			0
$335$	−16.0000			−0.874173
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$	7.00000	+	12.1244i	0.380188	+	0.658505i
$340$		0			0
$341$	16.0000	−	27.7128i	0.866449	−	1.50073i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	0.980921	−	0.194409i	$-0.0622790\pi$
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	−0.658824	+	0.752297i	$0.728946\pi$
$348$		0			0
$349$	−12.0000			−0.642345			−0.321173	−	0.947021i	$-0.604077\pi$
$349$	−12.0000			−0.642345			−0.321173	+	0.947021i	$0.604077\pi$
$350$		0			0
$351$	4.00000			0.213504
$352$		0			0
$353$	12.0000	+	20.7846i	0.638696	+	1.10625i	0.985719	+	0.168397i	$0.0538590\pi$
$353$	12.0000	+	20.7846i	0.638696	+	1.10625i	−0.347024	+	0.937856i	$0.612808\pi$
$354$		0			0
$355$	16.0000	−	27.7128i	0.849192	−	1.47084i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	8.00000	−	13.8564i	0.422224	−	0.731313i	−0.573933	−	0.818902i	$-0.694583\pi$
$359$	8.00000	−	13.8564i	0.422224	−	0.731313i	0.996157	+	0.0875892i	$0.0279163\pi$
$360$		0			0
$361$	1.50000	+	2.59808i	0.0789474	+	0.136741i
$362$		0			0
$363$	5.00000			0.262432
$364$		0			0
$365$	−64.0000			−3.34991
$366$		0			0
$367$	−8.00000	−	13.8564i	−0.417597	−	0.723299i	0.578101	−	0.815966i	$-0.303794\pi$
$367$	−8.00000	−	13.8564i	−0.417597	−	0.723299i	−0.995697	+	0.0926670i	$0.970461\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−11.0000	+	19.0526i	−0.569558	+	0.986504i	0.427051	+	0.904227i	$0.359552\pi$
$373$	−11.0000	+	19.0526i	−0.569558	+	0.986504i	−0.996610	+	0.0822766i	$0.973781\pi$
$374$		0			0
$375$	−12.0000	−	20.7846i	−0.619677	−	1.07331i
$376$		0			0
$377$	8.00000			0.412021
$378$		0			0
$379$	−36.0000			−1.84920			−0.924598	−	0.380945i	$-0.875599\pi$
$379$	−36.0000			−1.84920			−0.924598	+	0.380945i	$0.875599\pi$
$380$		0			0
$381$	−8.00000	−	13.8564i	−0.409852	−	0.709885i
$382$		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
$389$	3.00000	+	5.19615i	0.152106	+	0.263455i	0.932002	−	0.362454i	$-0.118061\pi$
$389$	3.00000	+	5.19615i	0.152106	+	0.263455i	−0.779895	+	0.625910i	$0.784728\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	12.0000			0.605320
$394$		0			0
$395$	−16.0000	−	27.7128i	−0.805047	−	1.39438i
$396$		0			0
$397$	−2.00000	+	3.46410i	−0.100377	+	0.173858i	−0.911840	−	0.410546i	$-0.865338\pi$
$397$	−2.00000	+	3.46410i	−0.100377	+	0.173858i	0.811463	+	0.584404i	$0.198672\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.00000	−	15.5885i	0.449439	−	0.778450i	−0.548911	−	0.835881i	$-0.684957\pi$
$401$	9.00000	−	15.5885i	0.449439	−	0.778450i	0.998350	+	0.0574304i	$0.0182907\pi$
$402$		0			0
$403$	16.0000	+	27.7128i	0.797017	+	1.38047i
$404$		0			0
$405$	4.00000			0.198762
$406$		0			0
$407$	−24.0000			−1.18964
$408$		0			0
$409$	12.0000	+	20.7846i	0.593362	+	1.02773i	0.993776	+	0.111398i	$0.0355330\pi$
$409$	12.0000	+	20.7846i	0.593362	+	1.02773i	−0.400414	+	0.916334i	$0.631134\pi$
$410$		0			0
$411$	5.00000	−	8.66025i	0.246632	−	0.427179i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−24.0000	+	41.5692i	−1.17811	+	2.04055i
$416$		0			0
$417$	6.00000	+	10.3923i	0.293821	+	0.508913i
$418$		0			0
$419$	−28.0000			−1.36789			−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000			−1.36789			−0.683945	+	0.729534i	$0.739737\pi$
$420$		0			0
$421$	6.00000			0.292422			0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000			0.292422			0.146211	+	0.989253i	$0.453292\pi$
$422$		0			0
$423$	−4.00000	−	6.92820i	−0.194487	−	0.336861i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−8.00000	+	13.8564i	−0.386244	+	0.668994i
$430$		0			0
$431$	20.0000	+	34.6410i	0.963366	+	1.66860i	0.713942	+	0.700205i	$0.246908\pi$
$431$	20.0000	+	34.6410i	0.963366	+	1.66860i	0.249424	+	0.968394i	$0.419759\pi$
$432$		0			0
$433$	−8.00000			−0.384455			−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000			−0.384455			−0.192228	+	0.981350i	$0.561571\pi$
$434$		0			0
$435$	8.00000			0.383571
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.00000	+	10.3923i	−0.285069	+	0.493753i	−0.972626	−	0.232377i	$-0.925350\pi$
$443$	−6.00000	+	10.3923i	−0.285069	+	0.493753i	0.687557	+	0.726130i	$0.258683\pi$
$444$		0			0
$445$	−16.0000	−	27.7128i	−0.758473	−	1.31371i
$446$		0			0
$447$	−10.0000			−0.472984
$448$		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
$451$		0			0
$452$		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	$-0.409849\pi$
$461$	12.0000			0.558896			0.279448	+	0.960161i	$0.409849\pi$
$462$		0			0
$463$	−8.00000			−0.371792			−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000			−0.371792			−0.185896	+	0.982569i	$0.559519\pi$
$464$		0			0
$465$	16.0000	+	27.7128i	0.741982	+	1.28515i
$466$		0			0
$467$	10.0000	−	17.3205i	0.462745	−	0.801498i	−0.536352	−	0.843995i	$-0.680198\pi$
$467$	10.0000	−	17.3205i	0.462745	−	0.801498i	0.999097	+	0.0424970i	$0.0135313\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−2.00000	+	3.46410i	−0.0921551	+	0.159617i
$472$		0			0
$473$	8.00000	+	13.8564i	0.367840	+	0.637118i
$474$		0			0
$475$	−44.0000			−2.01886
$476$		0			0
$477$	−10.0000			−0.457869
$478$		0			0
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	−0.998392	−	0.0566937i	$-0.981944\pi$
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	0.450098	−	0.892979i	$-0.351389\pi$
$480$		0			0
$481$	12.0000	−	20.7846i	0.547153	−	0.947697i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−16.0000	+	27.7128i	−0.726523	+	1.25837i
$486$		0			0
$487$	−4.00000	−	6.92820i	−0.181257	−	0.313947i	0.761052	−	0.648691i	$-0.224683\pi$
$487$	−4.00000	−	6.92820i	−0.181257	−	0.313947i	−0.942309	+	0.334744i	$0.891350\pi$
$488$		0			0
$489$	−12.0000			−0.542659
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−8.00000	+	13.8564i	−0.359573	+	0.622799i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−6.00000	+	10.3923i	−0.268597	+	0.465223i	−0.968500	−	0.249015i	$-0.919893\pi$
$499$	−6.00000	+	10.3923i	−0.268597	+	0.465223i	0.699903	+	0.714238i	$0.253227\pi$
$500$		0			0
$501$	4.00000	+	6.92820i	0.178707	+	0.309529i
$502$		0			0
$503$	16.0000			0.713405			0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000			0.713405			0.356702	+	0.934218i	$0.383901\pi$
$504$		0			0
$505$	16.0000			0.711991
$506$		0			0
$507$	−1.50000	−	2.59808i	−0.0666173	−	0.115385i
$508$		0			0
$509$	2.00000	−	3.46410i	0.0886484	−	0.153544i	−0.818292	−	0.574803i	$-0.805079\pi$
$509$	2.00000	−	3.46410i	0.0886484	−	0.153544i	0.906940	+	0.421260i	$0.138412\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	2.00000	−	3.46410i	0.0883022	−	0.152944i
$514$		0			0
$515$	−16.0000	−	27.7128i	−0.705044	−	1.22117i
$516$		0			0
$517$	32.0000			1.40736
$518$		0			0
$519$	−4.00000			−0.175581
$520$		0			0
$521$	−16.0000	−	27.7128i	−0.700973	−	1.21412i	−0.968125	−	0.250466i	$-0.919416\pi$
$521$	−16.0000	−	27.7128i	−0.700973	−	1.21412i	0.267153	−	0.963654i	$-0.413917\pi$
$522$		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
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$	−4.00000			−0.173585
$532$		0			0
$533$		0			0
$534$		0			0
$535$	8.00000	+	13.8564i	0.345870	+	0.599065i
$536$		0			0
$537$	6.00000	−	10.3923i	0.258919	−	0.448461i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.00000	−	1.73205i	0.0429934	−	0.0744667i	−0.843728	−	0.536771i	$-0.819644\pi$
$541$	1.00000	−	1.73205i	0.0429934	−	0.0744667i	0.886721	+	0.462304i	$0.152977\pi$
$542$		0			0
$543$	−10.0000	−	17.3205i	−0.429141	−	0.743294i
$544$		0			0
$545$	−56.0000			−2.39878
$546$		0			0
$547$	20.0000			0.855138			0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000			0.855138			0.427569	+	0.903983i	$0.359370\pi$
$548$		0			0
$549$	2.00000	+	3.46410i	0.0853579	+	0.147844i
$550$		0			0
$551$	4.00000	−	6.92820i	0.170406	−	0.295151i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	12.0000	−	20.7846i	0.509372	−	0.882258i
$556$		0			0
$557$	9.00000	+	15.5885i	0.381342	+	0.660504i	0.991254	−	0.131965i	$-0.0421286\pi$
$557$	9.00000	+	15.5885i	0.381342	+	0.660504i	−0.609912	+	0.792469i	$0.708795\pi$
$558$		0			0
$559$	−16.0000			−0.676728
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.00000	−	3.46410i	−0.0842900	−	0.145994i	0.820798	−	0.571218i	$-0.193529\pi$
$563$	−2.00000	−	3.46410i	−0.0842900	−	0.145994i	−0.905088	+	0.425223i	$0.860196\pi$
$564$		0			0
$565$	28.0000	−	48.4974i	1.17797	−	2.04030i
$566$		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
$571$	18.0000	+	31.1769i	0.753277	+	1.30471i	0.946227	+	0.323505i	$0.104861\pi$
$571$	18.0000	+	31.1769i	0.753277	+	1.30471i	−0.192950	+	0.981209i	$0.561806\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	8.00000	+	13.8564i	0.333044	+	0.576850i	0.983107	−	0.183031i	$-0.0585908\pi$
$577$	8.00000	+	13.8564i	0.333044	+	0.576850i	−0.650063	+	0.759880i	$0.725257\pi$
$578$		0			0
$579$	−1.00000	+	1.73205i	−0.0415586	+	0.0719816i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	20.0000	−	34.6410i	0.828315	−	1.43468i
$584$		0			0
$585$	−8.00000	−	13.8564i	−0.330759	−	0.572892i
$586$		0			0
$587$	−28.0000			−1.15568			−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000			−1.15568			−0.577842	+	0.816149i	$0.696105\pi$
$588$		0			0
$589$	32.0000			1.31854
$590$		0			0
$591$	−3.00000	−	5.19615i	−0.123404	−	0.213741i
$592$		0			0
$593$	12.0000	−	20.7846i	0.492781	−	0.853522i	−0.507184	−	0.861838i	$-0.669314\pi$
$593$	12.0000	−	20.7846i	0.492781	−	0.853522i	0.999965	+	0.00831589i	$0.00264706\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	4.00000	−	6.92820i	0.163709	−	0.283552i
$598$		0			0
$599$	12.0000	+	20.7846i	0.490307	+	0.849236i	0.999938	−	0.0111569i	$-0.00355143\pi$
$599$	12.0000	+	20.7846i	0.490307	+	0.849236i	−0.509631	+	0.860393i	$0.670218\pi$
$600$		0			0
$601$	−32.0000			−1.30531			−0.652654	−	0.757656i	$-0.726344\pi$
$601$	−32.0000			−1.30531			−0.652654	+	0.757656i	$0.726344\pi$
$602$		0			0
$603$	−4.00000			−0.162893
$604$		0			0
$605$	−10.0000	−	17.3205i	−0.406558	−	0.704179i
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−16.0000	+	27.7128i	−0.647291	+	1.12114i
$612$		0			0
$613$	−13.0000	−	22.5167i	−0.525065	−	0.909439i	−0.999574	−	0.0291886i	$-0.990708\pi$
$613$	−13.0000	−	22.5167i	−0.525065	−	0.909439i	0.474509	−	0.880251i	$-0.342626\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	22.0000			0.885687			0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000			0.885687			0.442843	+	0.896599i	$0.353970\pi$
$618$		0			0
$619$	10.0000	+	17.3205i	0.401934	+	0.696170i	0.993959	−	0.109749i	$-0.0350048\pi$
$619$	10.0000	+	17.3205i	0.401934	+	0.696170i	−0.592025	+	0.805919i	$0.701671\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−20.5000	+	35.5070i	−0.820000	+	1.42028i
$626$		0			0
$627$	8.00000	+	13.8564i	0.319489	+	0.553372i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$	−14.0000	−	24.2487i	−0.556450	−	0.963800i
$634$		0			0
$635$	−32.0000	+	55.4256i	−1.26988	+	2.19950i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	4.00000	−	6.92820i	0.158238	−	0.274075i
$640$		0			0
$641$	1.00000	+	1.73205i	0.0394976	+	0.0684119i	0.885098	−	0.465404i	$-0.154091\pi$
$641$	1.00000	+	1.73205i	0.0394976	+	0.0684119i	−0.845601	+	0.533816i	$0.820758\pi$
$642$		0			0
$643$	36.0000			1.41970			0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000			1.41970			0.709851	+	0.704352i	$0.248762\pi$
$644$		0			0
$645$	−16.0000			−0.629999
$646$		0			0
$647$	−12.0000	−	20.7846i	−0.471769	−	0.817127i	0.527710	−	0.849425i	$-0.323051\pi$
$647$	−12.0000	−	20.7846i	−0.471769	−	0.817127i	−0.999478	+	0.0322975i	$0.989718\pi$
$648$		0			0
$649$	8.00000	−	13.8564i	0.314027	−	0.543912i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	23.0000	−	39.8372i	0.900060	−	1.55895i	0.0726446	−	0.997358i	$-0.476856\pi$
$653$	23.0000	−	39.8372i	0.900060	−	1.55895i	0.827415	−	0.561591i	$-0.189811\pi$
$654$		0			0
$655$	−24.0000	−	41.5692i	−0.937758	−	1.62424i
$656$		0			0
$657$	−16.0000			−0.624219
$658$		0			0
$659$	−4.00000			−0.155818			−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000			−0.155818			−0.0779089	+	0.996960i	$0.524824\pi$
$660$		0			0
$661$	−14.0000	−	24.2487i	−0.544537	−	0.943166i	−0.998636	−	0.0522143i	$-0.983372\pi$
$661$	−14.0000	−	24.2487i	−0.544537	−	0.943166i	0.454099	−	0.890951i	$-0.349961\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−8.00000	−	13.8564i	−0.309298	−	0.535720i
$670$		0			0
$671$	−16.0000			−0.617673
$672$		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
$675$	−5.50000	−	9.52628i	−0.211695	−	0.366667i
$676$		0			0
$677$	−6.00000	+	10.3923i	−0.230599	+	0.399409i	−0.957984	−	0.286820i	$-0.907402\pi$
$677$	−6.00000	+	10.3923i	−0.230599	+	0.399409i	0.727386	+	0.686229i	$0.240735\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	10.0000	−	17.3205i	0.383201	−	0.663723i
$682$		0			0
$683$	−22.0000	−	38.1051i	−0.841807	−	1.45805i	−0.888366	−	0.459136i	$-0.848159\pi$
$683$	−22.0000	−	38.1051i	−0.841807	−	1.45805i	0.0465592	−	0.998916i	$-0.485174\pi$
$684$		0			0
$685$	−40.0000			−1.52832
$686$		0			0
$687$	4.00000			0.152610
$688$		0			0
$689$	20.0000	+	34.6410i	0.761939	+	1.31972i
$690$		0			0
$691$	10.0000	−	17.3205i	0.380418	−	0.658903i	−0.610704	−	0.791859i	$-0.709113\pi$
$691$	10.0000	−	17.3205i	0.380418	−	0.658903i	0.991122	+	0.132956i	$0.0424468\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	24.0000	−	41.5692i	0.910372	−	1.57681i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−10.0000			−0.378235
$700$		0			0
$701$	34.0000			1.28416			0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000			1.28416			0.642081	+	0.766637i	$0.278071\pi$
$702$		0			0
$703$	−12.0000	−	20.7846i	−0.452589	−	0.783906i
$704$		0			0
$705$	−16.0000	+	27.7128i	−0.602595	+	1.04372i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	19.0000	−	32.9090i	0.713560	−	1.23592i	−0.249952	−	0.968258i	$-0.580415\pi$
$709$	19.0000	−	32.9090i	0.713560	−	1.23592i	0.963512	−	0.267664i	$-0.0862517\pi$
$710$		0			0
$711$	−4.00000	−	6.92820i	−0.150012	−	0.259828i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	64.0000			2.39346
$716$		0			0
$717$	−12.0000	−	20.7846i	−0.448148	−	0.776215i
$718$		0			0
$719$	−12.0000	+	20.7846i	−0.447524	+	0.775135i	−0.998224	−	0.0595683i	$-0.981028\pi$
$719$	−12.0000	+	20.7846i	−0.447524	+	0.775135i	0.550700	+	0.834703i	$0.314361\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−4.00000	+	6.92820i	−0.148762	+	0.257663i
$724$		0			0
$725$	−11.0000	−	19.0526i	−0.408530	−	0.707594i
$726$		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
$733$	2.00000	−	3.46410i	0.0738717	−	0.127950i	−0.826723	−	0.562609i	$-0.809798\pi$
$733$	2.00000	−	3.46410i	0.0738717	−	0.127950i	0.900595	+	0.434659i	$0.143131\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	8.00000	−	13.8564i	0.294684	−	0.510407i
$738$		0			0
$739$	−10.0000	−	17.3205i	−0.367856	−	0.637145i	0.621374	−	0.783514i	$-0.286575\pi$
$739$	−10.0000	−	17.3205i	−0.367856	−	0.637145i	−0.989230	+	0.146369i	$0.953241\pi$
$740$		0			0
$741$	−16.0000			−0.587775
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	20.0000	+	34.6410i	0.732743	+	1.26915i
$746$		0			0
$747$	−6.00000	+	10.3923i	−0.219529	+	0.380235i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$751$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$752$		0			0
$753$	−10.0000	−	17.3205i	−0.364420	−	0.631194i
$754$		0			0
$755$	32.0000			1.16460
$756$		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
$761$	−8.00000	+	13.8564i	−0.290000	+	0.502294i	−0.973809	−	0.227366i	$-0.926989\pi$
$761$	−8.00000	+	13.8564i	−0.290000	+	0.502294i	0.683810	+	0.729661i	$0.260322\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.00000	+	13.8564i	0.288863	+	0.500326i
$768$		0			0
$769$	40.0000			1.44244			0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000			1.44244			0.721218	+	0.692708i	$0.243582\pi$
$770$		0			0
$771$	8.00000			0.288113
$772$		0			0
$773$	−18.0000	−	31.1769i	−0.647415	−	1.12136i	−0.983738	−	0.179609i	$-0.942517\pi$
$773$	−18.0000	−	31.1769i	−0.647415	−	1.12136i	0.336323	−	0.941747i	$-0.390817\pi$
$774$		0			0
$775$	44.0000	−	76.2102i	1.58053	−	2.73755i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	16.0000	+	27.7128i	0.572525	+	0.991642i
$782$		0			0
$783$	2.00000			0.0714742
$784$		0			0
$785$	16.0000			0.571064
$786$		0			0
$787$	10.0000	+	17.3205i	0.356462	+	0.617409i	0.987367	−	0.158450i	$-0.0506498\pi$
$787$	10.0000	+	17.3205i	0.356462	+	0.617409i	−0.630905	+	0.775860i	$0.717316\pi$
$788$		0			0
$789$	4.00000	−	6.92820i	0.142404	−	0.246651i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	8.00000	−	13.8564i	0.284088	−	0.492055i
$794$		0			0
$795$	20.0000	+	34.6410i	0.709327	+	1.22859i
$796$		0			0
$797$	12.0000			0.425062			0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000			0.425062			0.212531	+	0.977154i	$0.431829\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−4.00000	−	6.92820i	−0.141333	−	0.244796i
$802$		0			0
$803$	32.0000	−	55.4256i	1.12926	−	1.95593i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−14.0000	+	24.2487i	−0.492823	+	0.853595i
$808$		0			0
$809$	−21.0000	−	36.3731i	−0.738321	−	1.27881i	−0.953251	−	0.302180i	$-0.902286\pi$
$809$	−21.0000	−	36.3731i	−0.738321	−	1.27881i	0.214930	−	0.976629i	$-0.431048\pi$
$810$		0			0
$811$	44.0000			1.54505			0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000			1.54505			0.772524	+	0.634985i	$0.218994\pi$
$812$		0			0
$813$	32.0000			1.12229
$814$		0			0
$815$	24.0000	+	41.5692i	0.840683	+	1.45611i
$816$		0			0
$817$	−8.00000	+	13.8564i	−0.279885	+	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.00000	−	8.66025i	0.174501	−	0.302245i	−0.765487	−	0.643451i	$-0.777502\pi$
$821$	5.00000	−	8.66025i	0.174501	−	0.302245i	0.939989	+	0.341206i	$0.110835\pi$
$822$		0			0
$823$	8.00000	+	13.8564i	0.278862	+	0.483004i	0.971102	−	0.238664i	$-0.0767093\pi$
$823$	8.00000	+	13.8564i	0.278862	+	0.483004i	−0.692240	+	0.721668i	$0.743376\pi$
$824$		0			0
$825$	44.0000			1.53188
$826$		0			0
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$	26.0000	+	45.0333i	0.903017	+	1.56407i	0.823557	+	0.567234i	$0.191986\pi$
$829$	26.0000	+	45.0333i	0.903017	+	1.56407i	0.0794606	+	0.996838i	$0.474680\pi$
$830$		0			0
$831$	−11.0000	+	19.0526i	−0.381586	+	0.660926i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	16.0000	−	27.7128i	0.553703	−	0.959041i
$836$		0			0
$837$	4.00000	+	6.92820i	0.138260	+	0.239474i
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$	−3.00000	−	5.19615i	−0.103325	−	0.178965i
$844$		0			0
$845$	−6.00000	+	10.3923i	−0.206406	+	0.357506i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−14.0000	+	24.2487i	−0.480479	+	0.832214i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	52.0000			1.78045			0.890223	−	0.455525i	$-0.150548\pi$
$853$	52.0000			1.78045			0.890223	+	0.455525i	$0.150548\pi$
$854$		0			0
$855$	−16.0000			−0.547188
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−18.0000	+	31.1769i	−0.614152	+	1.06374i	0.376381	+	0.926465i	$0.377169\pi$
$859$	−18.0000	+	31.1769i	−0.614152	+	1.06374i	−0.990533	+	0.137277i	$0.956165\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24.0000	−	41.5692i	0.816970	−	1.41503i	−0.0909355	−	0.995857i	$-0.528986\pi$
$863$	24.0000	−	41.5692i	0.816970	−	1.41503i	0.907905	−	0.419176i	$-0.137681\pi$
$864$		0			0
$865$	8.00000	+	13.8564i	0.272008	+	0.471132i
$866$		0			0
$867$	−17.0000			−0.577350
$868$		0			0
$869$	32.0000			1.08553
$870$		0			0
$871$	8.00000	+	13.8564i	0.271070	+	0.469506i
$872$		0			0
$873$	−4.00000	+	6.92820i	−0.135379	+	0.234484i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−9.00000	+	15.5885i	−0.303908	+	0.526385i	−0.977018	−	0.213158i	$-0.931625\pi$
$877$	−9.00000	+	15.5885i	−0.303908	+	0.526385i	0.673109	+	0.739543i	$0.264958\pi$
$878$		0			0
$879$	6.00000	+	10.3923i	0.202375	+	0.350524i
$880$		0			0
$881$	−8.00000			−0.269527			−0.134763	−	0.990878i	$-0.543027\pi$
$881$	−8.00000			−0.269527			−0.134763	+	0.990878i	$0.543027\pi$
$882$		0			0
$883$	−4.00000			−0.134611			−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000			−0.134611			−0.0673054	+	0.997732i	$0.521440\pi$
$884$		0			0
$885$	8.00000	+	13.8564i	0.268917	+	0.465778i
$886$		0			0
$887$	12.0000	−	20.7846i	0.402921	−	0.697879i	−0.591156	−	0.806557i	$-0.701328\pi$
$887$	12.0000	−	20.7846i	0.402921	−	0.697879i	0.994077	+	0.108678i	$0.0346618\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−2.00000	+	3.46410i	−0.0670025	+	0.116052i
$892$		0			0
$893$	16.0000	+	27.7128i	0.535420	+	0.927374i
$894$		0			0
$895$	−48.0000			−1.60446
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.00000	+	13.8564i	0.266815	+	0.462137i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−40.0000	+	69.2820i	−1.32964	+	2.30301i
$906$		0			0
$907$	−10.0000	−	17.3205i	−0.332045	−	0.575118i	0.650868	−	0.759191i	$-0.274405\pi$
$907$	−10.0000	−	17.3205i	−0.332045	−	0.575118i	−0.982913	+	0.184073i	$0.941072\pi$
$908$		0			0
$909$	4.00000			0.132672
$910$		0			0
$911$	8.00000			0.265052			0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000			0.265052			0.132526	+	0.991180i	$0.457691\pi$
$912$		0			0
$913$	−24.0000	−	41.5692i	−0.794284	−	1.37574i
$914$		0			0
$915$	8.00000	−	13.8564i	0.264472	−	0.458079i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−24.0000	+	41.5692i	−0.791687	+	1.37124i	0.133235	+	0.991084i	$0.457464\pi$
$919$	−24.0000	+	41.5692i	−0.791687	+	1.37124i	−0.924922	+	0.380158i	$0.875870\pi$
$920$		0			0
$921$	10.0000	+	17.3205i	0.329511	+	0.570730i
$922$		0			0
$923$	−32.0000			−1.05329
$924$		0			0
$925$	−66.0000			−2.17007
$926$		0			0
$927$	−4.00000	−	6.92820i	−0.131377	−	0.227552i
$928$		0			0
$929$	−24.0000	+	41.5692i	−0.787414	+	1.36384i	0.140132	+	0.990133i	$0.455247\pi$
$929$	−24.0000	+	41.5692i	−0.787414	+	1.36384i	−0.927546	+	0.373709i	$0.878086\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	16.0000	−	27.7128i	0.523816	−	0.907277i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$	24.0000			0.783210
$940$		0			0
$941$	6.00000	+	10.3923i	0.195594	+	0.338779i	0.947095	−	0.320953i	$-0.104003\pi$
$941$	6.00000	+	10.3923i	0.195594	+	0.338779i	−0.751501	+	0.659732i	$0.770670\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−14.0000	+	24.2487i	−0.454939	+	0.787977i	−0.998685	−	0.0512727i	$-0.983672\pi$
$947$	−14.0000	+	24.2487i	−0.454939	+	0.787977i	0.543746	+	0.839250i	$0.317006\pi$
$948$		0			0
$949$	32.0000	+	55.4256i	1.03876	+	1.79919i
$950$		0			0
$951$	30.0000			0.972817
$952$		0			0
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−4.00000	+	6.92820i	−0.129302	+	0.223957i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−16.5000	+	28.5788i	−0.532258	+	0.921898i
$962$		0			0
$963$	2.00000	+	3.46410i	0.0644491	+	0.111629i
$964$		0			0
$965$	8.00000			0.257529
$966$		0			0
$967$	8.00000			0.257263			0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000			0.257263			0.128631	+	0.991692i	$0.458942\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.00000	−	3.46410i	0.0641831	−	0.111168i	−0.832148	−	0.554553i	$-0.812889\pi$
$971$	2.00000	−	3.46410i	0.0641831	−	0.111168i	0.896331	+	0.443385i	$0.146223\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−22.0000	+	38.1051i	−0.704564	+	1.22034i
$976$		0			0
$977$	−7.00000	−	12.1244i	−0.223950	−	0.387893i	0.732054	−	0.681247i	$-0.238562\pi$
$977$	−7.00000	−	12.1244i	−0.223950	−	0.387893i	−0.956004	+	0.293354i	$0.905229\pi$
$978$		0			0
$979$	32.0000			1.02272
$980$		0			0
$981$	−14.0000			−0.446986
$982$		0			0
$983$	−4.00000	−	6.92820i	−0.127580	−	0.220975i	0.795158	−	0.606402i	$-0.207388\pi$
$983$	−4.00000	−	6.92820i	−0.127580	−	0.220975i	−0.922739	+	0.385426i	$0.874054\pi$
$984$		0			0
$985$	−12.0000	+	20.7846i	−0.382352	+	0.662253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	4.00000	+	6.92820i	0.127064	+	0.220082i	0.922538	−	0.385906i	$-0.126111\pi$
$991$	4.00000	+	6.92820i	0.127064	+	0.220082i	−0.795474	+	0.605988i	$0.792778\pi$
$992$		0			0
$993$	−20.0000			−0.634681
$994$		0			0
$995$	−32.0000			−1.01447
$996$		0			0
$997$	−14.0000	−	24.2487i	−0.443384	−	0.767964i	0.554554	−	0.832148i	$-0.312889\pi$
$997$	−14.0000	−	24.2487i	−0.443384	−	0.767964i	−0.997938	+	0.0641836i	$0.979556\pi$
$998$		0			0
$999$	3.00000	−	5.19615i	0.0949158	−	0.164399i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.q.a.961.1		2
4.3	odd	2		294.2.e.e.79.1		2
7.2	even	3		2352.2.a.y.1.1		1
7.3	odd	6		2352.2.q.y.1537.1		2
7.4	even	3	inner	2352.2.q.a.1537.1		2
7.5	odd	6		2352.2.a.b.1.1		1
7.6	odd	2		2352.2.q.y.961.1		2
12.11	even	2		882.2.g.f.667.1		2
21.2	odd	6		7056.2.a.a.1.1		1
21.5	even	6		7056.2.a.ca.1.1		1
28.3	even	6		294.2.e.d.67.1		2
28.11	odd	6		294.2.e.e.67.1		2
28.19	even	6		294.2.a.c.1.1	yes	1
28.23	odd	6		294.2.a.b.1.1	✓	1
28.27	even	2		294.2.e.d.79.1		2
56.5	odd	6		9408.2.a.de.1.1		1
56.19	even	6		9408.2.a.bo.1.1		1
56.37	even	6		9408.2.a.b.1.1		1
56.51	odd	6		9408.2.a.br.1.1		1
84.11	even	6		882.2.g.f.361.1		2
84.23	even	6		882.2.a.f.1.1		1
84.47	odd	6		882.2.a.l.1.1		1
84.59	odd	6		882.2.g.a.361.1		2
84.83	odd	2		882.2.g.a.667.1		2
140.19	even	6		7350.2.a.br.1.1		1
140.79	odd	6		7350.2.a.cj.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.2.a.b.1.1	✓	1	28.23	odd	6
294.2.a.c.1.1	yes	1	28.19	even	6
294.2.e.d.67.1		2	28.3	even	6
294.2.e.d.79.1		2	28.27	even	2
294.2.e.e.67.1		2	28.11	odd	6
294.2.e.e.79.1		2	4.3	odd	2
882.2.a.f.1.1		1	84.23	even	6
882.2.a.l.1.1		1	84.47	odd	6
882.2.g.a.361.1		2	84.59	odd	6
882.2.g.a.667.1		2	84.83	odd	2
882.2.g.f.361.1		2	84.11	even	6
882.2.g.f.667.1		2	12.11	even	2
2352.2.a.b.1.1		1	7.5	odd	6
2352.2.a.y.1.1		1	7.2	even	3
2352.2.q.a.961.1		2	1.1	even	1	trivial
2352.2.q.a.1537.1		2	7.4	even	3	inner
2352.2.q.y.961.1		2	7.6	odd	2
2352.2.q.y.1537.1		2	7.3	odd	6
7056.2.a.a.1.1		1	21.2	odd	6
7056.2.a.ca.1.1		1	21.5	even	6
7350.2.a.br.1.1		1	140.19	even	6
7350.2.a.cj.1.1		1	140.79	odd	6
9408.2.a.b.1.1		1	56.37	even	6
9408.2.a.bo.1.1		1	56.19	even	6
9408.2.a.br.1.1		1	56.51	odd	6
9408.2.a.de.1.1		1	56.5	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)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} +(-2.00000 + 3.46410i) q^{5} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{3} +(-2.00000 + 3.46410i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(-2.00000 - 3.46410i) q^{11} +4.00000 q^{13} +4.00000 q^{15} +(2.00000 - 3.46410i) q^{19} +(-5.50000 - 9.52628i) q^{25} +1.00000 q^{27} +2.00000 q^{29} +(4.00000 + 6.92820i) q^{31} +(-2.00000 + 3.46410i) q^{33} +(3.00000 - 5.19615i) q^{37} +(-2.00000 - 3.46410i) q^{39} -4.00000 q^{43} +(-2.00000 - 3.46410i) q^{45} +(-4.00000 + 6.92820i) q^{47} +(5.00000 + 8.66025i) q^{53} +16.0000 q^{55} -4.00000 q^{57} +(2.00000 + 3.46410i) q^{59} +(2.00000 - 3.46410i) q^{61} +(-8.00000 + 13.8564i) q^{65} +(2.00000 + 3.46410i) q^{67} -8.00000 q^{71} +(8.00000 + 13.8564i) q^{73} +(-5.50000 + 9.52628i) q^{75} +(-4.00000 + 6.92820i) q^{79} +(-0.500000 - 0.866025i) q^{81} +12.0000 q^{83} +(-1.00000 - 1.73205i) q^{87} +(-4.00000 + 6.92820i) q^{89} +(4.00000 - 6.92820i) q^{93} +(8.00000 + 13.8564i) q^{95} +8.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 4 q^{5} - q^{9}+O(q^{10}) \) 2 * q - q^3 - 4 * q^5 - q^9	\( 2 q - q^{3} - 4 q^{5} - q^{9} - 4 q^{11} + 8 q^{13} + 8 q^{15} + 4 q^{19} - 11 q^{25} + 2 q^{27} + 4 q^{29} + 8 q^{31} - 4 q^{33} + 6 q^{37} - 4 q^{39} - 8 q^{43} - 4 q^{45} - 8 q^{47} + 10 q^{53} + 32 q^{55} - 8 q^{57} + 4 q^{59} + 4 q^{61} - 16 q^{65} + 4 q^{67} - 16 q^{71} + 16 q^{73} - 11 q^{75} - 8 q^{79} - q^{81} + 24 q^{83} - 2 q^{87} - 8 q^{89} + 8 q^{93} + 16 q^{95} + 16 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q - q^3 - 4 * q^5 - q^9 - 4 * q^11 + 8 * q^13 + 8 * q^15 + 4 * q^19 - 11 * q^25 + 2 * q^27 + 4 * q^29 + 8 * q^31 - 4 * q^33 + 6 * q^37 - 4 * q^39 - 8 * q^43 - 4 * q^45 - 8 * q^47 + 10 * q^53 + 32 * q^55 - 8 * q^57 + 4 * q^59 + 4 * q^61 - 16 * q^65 + 4 * q^67 - 16 * q^71 + 16 * q^73 - 11 * q^75 - 8 * q^79 - q^81 + 24 * q^83 - 2 * q^87 - 8 * q^89 + 8 * q^93 + 16 * q^95 + 16 * q^97 + 8 * q^99

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