LMFDB - Embedded newform 2268.2.j.a.1513.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(757,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2268.757");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.j (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:	$18.1100711784$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1513.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2268.1513
Dual form			2268.2.j.a.757.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 3.46410i) q^{5} +(0.500000 - 0.866025i) q^{7} +O(q^{10})$	$q+(-2.00000 - 3.46410i) q^{5} +(0.500000 - 0.866025i) q^{7} +(-1.00000 + 1.73205i) q^{11} +(3.00000 + 5.19615i) q^{13} -4.00000 q^{17} -4.00000 q^{19} +(-1.00000 - 1.73205i) q^{23} +(-5.50000 + 9.52628i) q^{25} +(1.00000 - 1.73205i) q^{29} -4.00000 q^{35} +2.00000 q^{37} +(2.00000 - 3.46410i) q^{43} +(-6.00000 + 10.3923i) q^{47} +(-0.500000 - 0.866025i) q^{49} -6.00000 q^{53} +8.00000 q^{55} +(4.00000 + 6.92820i) q^{59} +(-3.00000 + 5.19615i) q^{61} +(12.0000 - 20.7846i) q^{65} +(4.00000 + 6.92820i) q^{67} +14.0000 q^{71} -2.00000 q^{73} +(1.00000 + 1.73205i) q^{77} +(-6.00000 + 10.3923i) q^{79} +(2.00000 - 3.46410i) q^{83} +(8.00000 + 13.8564i) q^{85} +6.00000 q^{91} +(8.00000 + 13.8564i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 + q^7	$ 2 q - 4 q^{5} + q^{7} - 2 q^{11} + 6 q^{13} - 8 q^{17} - 8 q^{19} - 2 q^{23} - 11 q^{25} + 2 q^{29} - 8 q^{35} + 4 q^{37} + 4 q^{43} - 12 q^{47} - q^{49} - 12 q^{53} + 16 q^{55} + 8 q^{59} - 6 q^{61} + 24 q^{65} + 8 q^{67} + 28 q^{71} - 4 q^{73} + 2 q^{77} - 12 q^{79} + 4 q^{83} + 16 q^{85} + 12 q^{91} + 16 q^{95} + 2 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 + q^7 - 2 * q^11 + 6 * q^13 - 8 * q^17 - 8 * q^19 - 2 * q^23 - 11 * q^25 + 2 * q^29 - 8 * q^35 + 4 * q^37 + 4 * q^43 - 12 * q^47 - q^49 - 12 * q^53 + 16 * q^55 + 8 * q^59 - 6 * q^61 + 24 * q^65 + 8 * q^67 + 28 * q^71 - 4 * q^73 + 2 * q^77 - 12 * q^79 + 4 * q^83 + 16 * q^85 + 12 * q^91 + 16 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$1135$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.0599153	−	0.998203i	$-0.519083\pi$
$6$		0			0
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	3.00000	+	5.19615i	0.832050	+	1.44115i	0.896410	+	0.443227i	$0.146166\pi$
$13$	3.00000	+	5.19615i	0.832050	+	1.44115i	−0.0643593	+	0.997927i	$0.520500\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.00000			−0.970143			−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000			−0.970143			−0.485071	+	0.874475i	$0.661206\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.00000	−	1.73205i	−0.208514	−	0.361158i	0.742732	−	0.669588i	$-0.233529\pi$
$23$	−1.00000	−	1.73205i	−0.208514	−	0.361158i	−0.951247	+	0.308431i	$0.900196\pi$
$24$		0			0
$25$	−5.50000	+	9.52628i	−1.10000	+	1.90526i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.00000	−	1.73205i	0.185695	−	0.321634i	−0.758115	−	0.652121i	$-0.773880\pi$
$29$	1.00000	−	1.73205i	0.185695	−	0.321634i	0.943811	+	0.330487i	$0.107213\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.00000			−0.676123
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	2.00000	−	3.46410i	0.304997	−	0.528271i	−0.672264	−	0.740312i	$-0.734678\pi$
$43$	2.00000	−	3.46410i	0.304997	−	0.528271i	0.977261	+	0.212041i	$0.0680112\pi$
$44$		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
$49$	−0.500000	−	0.866025i	−0.0714286	−	0.123718i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	8.00000			1.07872
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.00000	+	6.92820i	0.520756	+	0.901975i	0.999709	+	0.0241347i	$0.00768307\pi$
$59$	4.00000	+	6.92820i	0.520756	+	0.901975i	−0.478953	+	0.877841i	$0.658984\pi$
$60$		0			0
$61$	−3.00000	+	5.19615i	−0.384111	+	0.665299i	−0.991645	−	0.128994i	$-0.958825\pi$
$61$	−3.00000	+	5.19615i	−0.384111	+	0.665299i	0.607535	+	0.794293i	$0.292159\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	12.0000	−	20.7846i	1.48842	−	2.57801i
$66$		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
$69$		0			0
$70$		0			0
$71$	14.0000			1.66149			0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000			1.66149			0.830747	+	0.556650i	$0.187914\pi$
$72$		0			0
$73$	−2.00000			−0.234082			−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000			−0.234082			−0.117041	+	0.993127i	$0.537341\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.00000	+	1.73205i	0.113961	+	0.197386i
$78$		0			0
$79$	−6.00000	+	10.3923i	−0.675053	+	1.16923i	0.301401	+	0.953498i	$0.402546\pi$
$79$	−6.00000	+	10.3923i	−0.675053	+	1.16923i	−0.976453	+	0.215728i	$0.930788\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.00000	−	3.46410i	0.219529	−	0.380235i	−0.735135	−	0.677920i	$-0.762881\pi$
$83$	2.00000	−	3.46410i	0.219529	−	0.380235i	0.954664	+	0.297686i	$0.0962148\pi$
$84$		0			0
$85$	8.00000	+	13.8564i	0.867722	+	1.50294i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.00000	+	13.8564i	0.820783	+	1.42164i
$96$		0			0
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	−0.810782	−	0.585348i	$-0.800958\pi$
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	0.912317	+	0.409484i	$0.134291\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.00000	+	13.8564i	−0.796030	+	1.37876i	0.126153	+	0.992011i	$0.459737\pi$
$101$	−8.00000	+	13.8564i	−0.796030	+	1.37876i	−0.922183	+	0.386753i	$0.873597\pi$
$102$		0			0
$103$	8.00000	+	13.8564i	0.788263	+	1.36531i	0.927030	+	0.374987i	$0.122353\pi$
$103$	8.00000	+	13.8564i	0.788263	+	1.36531i	−0.138767	+	0.990325i	$0.544314\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	18.0000			1.74013			0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000			1.74013			0.870063	+	0.492941i	$0.164078\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.00000	−	8.66025i	−0.470360	−	0.814688i	0.529065	−	0.848581i	$-0.322543\pi$
$113$	−5.00000	−	8.66025i	−0.470360	−	0.814688i	−0.999425	+	0.0338931i	$0.989209\pi$
$114$		0			0
$115$	−4.00000	+	6.92820i	−0.373002	+	0.646058i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.00000	+	3.46410i	−0.183340	+	0.317554i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	24.0000			2.14663
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	0.765331	−	0.643637i	$-0.222575\pi$
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	−0.940072	+	0.340977i	$0.889242\pi$
$132$		0			0
$133$	−2.00000	+	3.46410i	−0.173422	+	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.00000	−	1.73205i	0.0854358	−	0.147979i	−0.820141	−	0.572161i	$-0.806105\pi$
$137$	1.00000	−	1.73205i	0.0854358	−	0.147979i	0.905577	+	0.424182i	$0.139438\pi$
$138$		0			0
$139$	−2.00000	−	3.46410i	−0.169638	−	0.293821i	0.768655	−	0.639664i	$-0.220926\pi$
$139$	−2.00000	−	3.46410i	−0.169638	−	0.293821i	−0.938293	+	0.345843i	$0.887593\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−12.0000			−1.00349
$144$		0			0
$145$	−8.00000			−0.664364
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	0.962348	−	0.271821i	$-0.0876260\pi$
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	−0.716578	+	0.697507i	$0.754293\pi$
$150$		0			0
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	−0.656101	−	0.754673i	$-0.727796\pi$
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	0.981617	+	0.190864i	$0.0611289\pi$
$152$		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	$-0.977982\pi$
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	0.438948	−	0.898513i	$-0.355351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.00000			−0.157622
$162$		0			0
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.00000	−	3.46410i	−0.154765	−	0.268060i	0.778209	−	0.628006i	$-0.216129\pi$
$167$	−2.00000	−	3.46410i	−0.154765	−	0.268060i	−0.932973	+	0.359946i	$0.882795\pi$
$168$		0			0
$169$	−11.5000	+	19.9186i	−0.884615	+	1.53220i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.00000	+	13.8564i	−0.608229	+	1.05348i	0.383304	+	0.923622i	$0.374786\pi$
$173$	−8.00000	+	13.8564i	−0.608229	+	1.05348i	−0.991532	+	0.129861i	$0.958547\pi$
$174$		0			0
$175$	5.50000	+	9.52628i	0.415761	+	0.720119i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−18.0000			−1.34538			−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000			−1.34538			−0.672692	+	0.739923i	$0.734862\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.00000	−	6.92820i	−0.294086	−	0.509372i
$186$		0			0
$187$	4.00000	−	6.92820i	0.292509	−	0.506640i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.00000	−	15.5885i	0.651217	−	1.12794i	−0.331611	−	0.943416i	$-0.607592\pi$
$191$	9.00000	−	15.5885i	0.651217	−	1.12794i	0.982828	−	0.184525i	$-0.0590746\pi$
$192$		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
$197$	−22.0000			−1.56744			−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000			−1.56744			−0.783718	+	0.621117i	$0.786679\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.00000	−	1.73205i	−0.0701862	−	0.121566i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.00000	−	6.92820i	0.276686	−	0.479234i
$210$		0			0
$211$	2.00000	+	3.46410i	0.137686	+	0.238479i	0.926620	−	0.375999i	$-0.122700\pi$
$211$	2.00000	+	3.46410i	0.137686	+	0.238479i	−0.788935	+	0.614477i	$0.789367\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−16.0000			−1.09119
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.0000	−	20.7846i	−0.807207	−	1.39812i
$222$		0			0
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	−0.968309	−	0.249756i	$-0.919650\pi$
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	0.700449	+	0.713702i	$0.252983\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	−5.00000	−	8.66025i	−0.330409	−	0.572286i	0.652183	−	0.758062i	$-0.273853\pi$
$229$	−5.00000	−	8.66025i	−0.330409	−	0.572286i	−0.982592	+	0.185776i	$0.940520\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	14.0000			0.917170			0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000			0.917170			0.458585	+	0.888650i	$0.348356\pi$
$234$		0			0
$235$	48.0000			3.13117
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	0.946590	−	0.322440i	$-0.104503\pi$
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	−0.752536	+	0.658551i	$0.771170\pi$
$240$		0			0
$241$	−7.00000	+	12.1244i	−0.450910	+	0.780998i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−7.00000	+	12.1244i	−0.450910	+	0.780998i	0.547533	+	0.836784i	$0.315567\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.00000	+	3.46410i	−0.127775	+	0.221313i
$246$		0			0
$247$	−12.0000	−	20.7846i	−0.763542	−	1.32249i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−24.0000			−1.51487			−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$		0			0
$253$	4.00000			0.251478
$254$		0			0
$255$		0			0
$256$		0			0
$257$	12.0000	+	20.7846i	0.748539	+	1.29651i	0.948523	+	0.316709i	$0.102578\pi$
$257$	12.0000	+	20.7846i	0.748539	+	1.29651i	−0.199983	+	0.979799i	$0.564089\pi$
$258$		0			0
$259$	1.00000	−	1.73205i	0.0621370	−	0.107624i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.00000	+	15.5885i	−0.554964	+	0.961225i	0.442943	+	0.896550i	$0.353935\pi$
$263$	−9.00000	+	15.5885i	−0.554964	+	0.961225i	−0.997906	+	0.0646755i	$0.979399\pi$
$264$		0			0
$265$	12.0000	+	20.7846i	0.737154	+	1.27679i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−20.0000			−1.21942			−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000			−1.21942			−0.609711	+	0.792624i	$0.708714\pi$
$270$		0			0
$271$	24.0000			1.45790			0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000			1.45790			0.728948	+	0.684569i	$0.240010\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−11.0000	−	19.0526i	−0.663325	−	1.14891i
$276$		0			0
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	−0.319447	−	0.947604i	$-0.603497\pi$
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	0.980373	−	0.197153i	$-0.0631696\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.00000	−	8.66025i	0.298275	−	0.516627i	−0.677466	−	0.735554i	$-0.736922\pi$
$281$	5.00000	−	8.66025i	0.298275	−	0.516627i	0.975741	+	0.218926i	$0.0702554\pi$
$282$		0			0
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	0.800439	−	0.599414i	$-0.204600\pi$
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	−0.919327	+	0.393494i	$0.871266\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$		0			0
$295$	16.0000	−	27.7128i	0.931556	−	1.61350i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.00000	−	10.3923i	0.346989	−	0.601003i
$300$		0			0
$301$	−2.00000	−	3.46410i	−0.115278	−	0.199667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	24.0000			1.37424
$306$		0			0
$307$	4.00000			0.228292			0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000			0.228292			0.114146	+	0.993464i	$0.463587\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.00000	−	3.46410i	−0.113410	−	0.196431i	0.803733	−	0.594990i	$-0.202844\pi$
$311$	−2.00000	−	3.46410i	−0.113410	−	0.196431i	−0.917143	+	0.398559i	$0.869511\pi$
$312$		0			0
$313$	−5.00000	+	8.66025i	−0.282617	+	0.489506i	−0.972028	−	0.234863i	$-0.924536\pi$
$313$	−5.00000	+	8.66025i	−0.282617	+	0.489506i	0.689412	+	0.724370i	$0.257869\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	0.494489	+	0.869184i	$0.335355\pi$
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	−0.999980	+	0.00635137i	$0.997978\pi$
$318$		0			0
$319$	2.00000	+	3.46410i	0.111979	+	0.193952i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.0000			0.890264
$324$		0			0
$325$	−66.0000			−3.66102
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.00000	+	10.3923i	0.330791	+	0.572946i
$330$		0			0
$331$	14.0000	−	24.2487i	0.769510	−	1.33283i	−0.168320	−	0.985732i	$-0.553834\pi$
$331$	14.0000	−	24.2487i	0.769510	−	1.33283i	0.937829	−	0.347097i	$-0.112833\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	16.0000	−	27.7128i	0.874173	−	1.51411i
$336$		0			0
$337$	3.00000	+	5.19615i	0.163420	+	0.283052i	0.936093	−	0.351752i	$-0.114414\pi$
$337$	3.00000	+	5.19615i	0.163420	+	0.283052i	−0.772673	+	0.634804i	$0.781081\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.00000	−	5.19615i	−0.161048	−	0.278944i	0.774197	−	0.632945i	$-0.218154\pi$
$347$	−3.00000	−	5.19615i	−0.161048	−	0.278944i	−0.935245	+	0.354001i	$0.884821\pi$
$348$		0			0
$349$	−11.0000	+	19.0526i	−0.588817	+	1.01986i	0.405571	+	0.914063i	$0.367073\pi$
$349$	−11.0000	+	19.0526i	−0.588817	+	1.01986i	−0.994388	+	0.105797i	$0.966261\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.00000	+	10.3923i	−0.319348	+	0.553127i	−0.980352	−	0.197256i	$-0.936797\pi$
$353$	−6.00000	+	10.3923i	−0.319348	+	0.553127i	0.661004	+	0.750382i	$0.270130\pi$
$354$		0			0
$355$	−28.0000	−	48.4974i	−1.48609	−	2.57398i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.00000			−0.105556			−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000			−0.105556			−0.0527780	+	0.998606i	$0.516808\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.00000	+	6.92820i	0.209370	+	0.362639i
$366$		0			0
$367$	8.00000	−	13.8564i	0.417597	−	0.723299i	−0.578101	−	0.815966i	$-0.696206\pi$
$367$	8.00000	−	13.8564i	0.417597	−	0.723299i	0.995697	+	0.0926670i	$0.0295392\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.00000	+	5.19615i	−0.155752	+	0.269771i
$372$		0			0
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	−0.996610	−	0.0822766i	$-0.973781\pi$
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	0.427051	−	0.904227i	$-0.359552\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.0000			0.618031
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.00000	−	6.92820i	−0.204390	−	0.354015i	0.745548	−	0.666452i	$-0.232188\pi$
$383$	−4.00000	−	6.92820i	−0.204390	−	0.354015i	−0.949938	+	0.312437i	$0.898855\pi$
$384$		0			0
$385$	4.00000	−	6.92820i	0.203859	−	0.353094i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−11.0000	+	19.0526i	−0.557722	+	0.966003i	0.439964	+	0.898015i	$0.354991\pi$
$389$	−11.0000	+	19.0526i	−0.557722	+	0.966003i	−0.997686	+	0.0679877i	$0.978342\pi$
$390$		0			0
$391$	4.00000	+	6.92820i	0.202289	+	0.350374i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	48.0000			2.41514
$396$		0			0
$397$	−18.0000			−0.903394			−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000			−0.903394			−0.451697	+	0.892171i	$0.649181\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	0.998350	−	0.0574304i	$-0.0182907\pi$
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	−0.548911	+	0.835881i	$0.684957\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.00000	+	3.46410i	−0.0991363	+	0.171709i
$408$		0			0
$409$	−19.0000	−	32.9090i	−0.939490	−	1.62724i	−0.766426	−	0.642333i	$-0.777967\pi$
$409$	−19.0000	−	32.9090i	−0.939490	−	1.62724i	−0.173064	−	0.984911i	$-0.555367\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.00000			0.393654
$414$		0			0
$415$	−16.0000			−0.785409
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$	3.00000	−	5.19615i	0.146211	−	0.253245i	−0.783613	−	0.621249i	$-0.786625\pi$
$421$	3.00000	−	5.19615i	0.146211	−	0.253245i	0.929824	+	0.368004i	$0.119959\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	22.0000	−	38.1051i	1.06716	−	1.84837i
$426$		0			0
$427$	3.00000	+	5.19615i	0.145180	+	0.251459i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	30.0000			1.44505			0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000			1.44505			0.722525	+	0.691345i	$0.242982\pi$
$432$		0			0
$433$	26.0000			1.24948			0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000			1.24948			0.624740	+	0.780833i	$0.285205\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.00000	+	6.92820i	0.191346	+	0.331421i
$438$		0			0
$439$	12.0000	−	20.7846i	0.572729	−	0.991995i	−0.423556	−	0.905870i	$-0.639218\pi$
$439$	12.0000	−	20.7846i	0.572729	−	0.991995i	0.996284	−	0.0861252i	$-0.0274485\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.0000	+	32.9090i	−0.902717	+	1.56355i	−0.0787648	+	0.996893i	$0.525098\pi$
$443$	−19.0000	+	32.9090i	−0.902717	+	1.56355i	−0.823952	+	0.566659i	$0.808236\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−14.0000			−0.660701			−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000			−0.660701			−0.330350	+	0.943858i	$0.607167\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−12.0000	−	20.7846i	−0.562569	−	0.974398i
$456$		0			0
$457$	−5.00000	+	8.66025i	−0.233890	+	0.405110i	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−5.00000	+	8.66025i	−0.233890	+	0.405110i	0.725059	+	0.688686i	$0.241812\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−20.0000	+	34.6410i	−0.931493	+	1.61339i	−0.150721	+	0.988576i	$0.548160\pi$
$461$	−20.0000	+	34.6410i	−0.931493	+	1.61339i	−0.780771	+	0.624817i	$0.785174\pi$
$462$		0			0
$463$	−2.00000	−	3.46410i	−0.0929479	−	0.160990i	0.815802	−	0.578331i	$-0.196296\pi$
$463$	−2.00000	−	3.46410i	−0.0929479	−	0.160990i	−0.908750	+	0.417340i	$0.862962\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.00000			−0.370196			−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000			−0.370196			−0.185098	+	0.982720i	$0.559260\pi$
$468$		0			0
$469$	8.00000			0.369406
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.00000	+	6.92820i	0.183920	+	0.318559i
$474$		0			0
$475$	22.0000	−	38.1051i	1.00943	−	1.74838i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	14.0000	−	24.2487i	0.639676	−	1.10795i	−0.345827	−	0.938298i	$-0.612402\pi$
$479$	14.0000	−	24.2487i	0.639676	−	1.10795i	0.985504	−	0.169654i	$-0.0542649\pi$
$480$		0			0
$481$	6.00000	+	10.3923i	0.273576	+	0.473848i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.00000			−0.363261
$486$		0			0
$487$	−16.0000			−0.725029			−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000			−0.725029			−0.362515	+	0.931978i	$0.618082\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	21.0000	+	36.3731i	0.947717	+	1.64149i	0.750218	+	0.661190i	$0.229948\pi$
$491$	21.0000	+	36.3731i	0.947717	+	1.64149i	0.197499	+	0.980303i	$0.436718\pi$
$492$		0			0
$493$	−4.00000	+	6.92820i	−0.180151	+	0.312031i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.00000	−	12.1244i	0.313993	−	0.543852i
$498$		0			0
$499$	10.0000	+	17.3205i	0.447661	+	0.775372i	0.998233	−	0.0594153i	$-0.0189236\pi$
$499$	10.0000	+	17.3205i	0.447661	+	0.775372i	−0.550572	+	0.834788i	$0.685590\pi$
$500$		0			0
$501$		0			0
$502$		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
$505$	64.0000			2.84796
$506$		0			0
$507$		0			0
$508$		0			0
$509$	18.0000	+	31.1769i	0.797836	+	1.38189i	0.921023	+	0.389509i	$0.127355\pi$
$509$	18.0000	+	31.1769i	0.797836	+	1.38189i	−0.123187	+	0.992384i	$0.539311\pi$
$510$		0			0
$511$	−1.00000	+	1.73205i	−0.0442374	+	0.0766214i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	32.0000	−	55.4256i	1.41009	−	2.44234i
$516$		0			0
$517$	−12.0000	−	20.7846i	−0.527759	−	0.914106i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.0000			0.525730			0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000			0.525730			0.262865	+	0.964833i	$0.415333\pi$
$522$		0			0
$523$	−28.0000			−1.22435			−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000			−1.22435			−0.612177	+	0.790721i	$0.709706\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	9.50000	−	16.4545i	0.413043	−	0.715412i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−36.0000	−	62.3538i	−1.55642	−	2.69579i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.00000			0.0861461
$540$		0			0
$541$	−30.0000			−1.28980			−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000			−1.28980			−0.644900	+	0.764267i	$0.723101\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.00000	+	6.92820i	0.171341	+	0.296772i
$546$		0			0
$547$	−12.0000	+	20.7846i	−0.513083	+	0.888686i	0.486802	+	0.873512i	$0.338163\pi$
$547$	−12.0000	+	20.7846i	−0.513083	+	0.888686i	−0.999885	+	0.0151734i	$0.995170\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.00000	+	6.92820i	−0.170406	+	0.295151i
$552$		0			0
$553$	6.00000	+	10.3923i	0.255146	+	0.441926i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−38.0000			−1.61011			−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000			−1.61011			−0.805056	+	0.593199i	$0.797865\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.00000	−	6.92820i	−0.168580	−	0.291989i	0.769341	−	0.638838i	$-0.220585\pi$
$563$	−4.00000	−	6.92820i	−0.168580	−	0.291989i	−0.937921	+	0.346850i	$0.887251\pi$
$564$		0			0
$565$	−20.0000	+	34.6410i	−0.841406	+	1.45736i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15.0000	+	25.9808i	−0.628833	+	1.08917i	0.358954	+	0.933355i	$0.383134\pi$
$569$	−15.0000	+	25.9808i	−0.628833	+	1.08917i	−0.987786	+	0.155815i	$0.950200\pi$
$570$		0			0
$571$	12.0000	+	20.7846i	0.502184	+	0.869809i	0.999997	+	0.00252413i	$0.000803457\pi$
$571$	12.0000	+	20.7846i	0.502184	+	0.869809i	−0.497812	+	0.867285i	$0.665863\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	22.0000			0.917463
$576$		0			0
$577$	−38.0000			−1.58196			−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000			−1.58196			−0.790980	+	0.611842i	$0.790429\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.00000	−	3.46410i	−0.0829740	−	0.143715i
$582$		0			0
$583$	6.00000	−	10.3923i	0.248495	−	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16.0000	+	27.7128i	−0.660391	+	1.14383i	0.320122	+	0.947376i	$0.396276\pi$
$587$	−16.0000	+	27.7128i	−0.660391	+	1.14383i	−0.980513	+	0.196454i	$0.937057\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	12.0000			0.492781			0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000			0.492781			0.246390	+	0.969171i	$0.420755\pi$
$594$		0			0
$595$	16.0000			0.655936
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.00000	−	5.19615i	−0.122577	−	0.212309i	0.798206	−	0.602384i	$-0.205782\pi$
$599$	−3.00000	−	5.19615i	−0.122577	−	0.212309i	−0.920783	+	0.390075i	$0.872449\pi$
$600$		0			0
$601$	23.0000	−	39.8372i	0.938190	−	1.62499i	0.169344	−	0.985557i	$-0.445835\pi$
$601$	23.0000	−	39.8372i	0.938190	−	1.62499i	0.768845	−	0.639435i	$-0.220832\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14.0000	−	24.2487i	0.569181	−	0.985850i
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−72.0000			−2.91281
$612$		0			0
$613$	6.00000			0.242338			0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000			0.242338			0.121169	+	0.992632i	$0.461336\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−19.0000	−	32.9090i	−0.764911	−	1.32487i	−0.940294	−	0.340365i	$-0.889449\pi$
$617$	−19.0000	−	32.9090i	−0.764911	−	1.32487i	0.175382	−	0.984500i	$-0.443884\pi$
$618$		0			0
$619$	10.0000	−	17.3205i	0.401934	−	0.696170i	−0.592025	−	0.805919i	$-0.701671\pi$
$619$	10.0000	−	17.3205i	0.401934	−	0.696170i	0.993959	+	0.109749i	$0.0350048\pi$
$620$		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
$627$		0			0
$628$		0			0
$629$	−8.00000			−0.318981
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.0000	−	41.5692i	−0.952411	−	1.64962i
$636$		0			0
$637$	3.00000	−	5.19615i	0.118864	−	0.205879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5.00000	−	8.66025i	0.197488	−	0.342059i	−0.750225	−	0.661182i	$-0.770055\pi$
$641$	5.00000	−	8.66025i	0.197488	−	0.342059i	0.947713	+	0.319123i	$0.103388\pi$
$642$		0			0
$643$	−22.0000	−	38.1051i	−0.867595	−	1.50272i	−0.864447	−	0.502724i	$-0.832331\pi$
$643$	−22.0000	−	38.1051i	−0.867595	−	1.50272i	−0.00314839	−	0.999995i	$-0.501002\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	12.0000			0.471769			0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000			0.471769			0.235884	+	0.971781i	$0.424201\pi$
$648$		0			0
$649$	−16.0000			−0.628055
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.00000	−	5.19615i	−0.117399	−	0.203341i	0.801337	−	0.598213i	$-0.204122\pi$
$653$	−3.00000	−	5.19615i	−0.117399	−	0.203341i	−0.918736	+	0.394872i	$0.870789\pi$
$654$		0			0
$655$	−8.00000	+	13.8564i	−0.312586	+	0.541415i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.00000	+	1.73205i	−0.0389545	+	0.0674711i	−0.884845	−	0.465885i	$-0.845736\pi$
$659$	−1.00000	+	1.73205i	−0.0389545	+	0.0674711i	0.845891	+	0.533356i	$0.179069\pi$
$660$		0			0
$661$	−11.0000	−	19.0526i	−0.427850	−	0.741059i	0.568831	−	0.822454i	$-0.307396\pi$
$661$	−11.0000	−	19.0526i	−0.427850	−	0.741059i	−0.996682	+	0.0813955i	$0.974062\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	16.0000			0.620453
$666$		0			0
$667$	−4.00000			−0.154881
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6.00000	−	10.3923i	−0.231627	−	0.401190i
$672$		0			0
$673$	17.0000	−	29.4449i	0.655302	−	1.13502i	−0.326516	−	0.945192i	$-0.605875\pi$
$673$	17.0000	−	29.4449i	0.655302	−	1.13502i	0.981818	−	0.189824i	$-0.0607919\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.0000	−	20.7846i	0.461197	−	0.798817i	−0.537823	−	0.843057i	$-0.680753\pi$
$677$	12.0000	−	20.7846i	0.461197	−	0.798817i	0.999021	+	0.0442400i	$0.0140866\pi$
$678$		0			0
$679$	−1.00000	−	1.73205i	−0.0383765	−	0.0664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	30.0000			1.14792			0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000			1.14792			0.573959	+	0.818884i	$0.305407\pi$
$684$		0			0
$685$	−8.00000			−0.305664
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−18.0000	−	31.1769i	−0.685745	−	1.18775i
$690$		0			0
$691$	14.0000	−	24.2487i	0.532585	−	0.922464i	−0.466691	−	0.884420i	$-0.654554\pi$
$691$	14.0000	−	24.2487i	0.532585	−	0.922464i	0.999276	−	0.0380440i	$-0.0121127\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.00000	+	13.8564i	−0.303457	+	0.525603i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−50.0000			−1.88847			−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000			−1.88847			−0.944237	+	0.329267i	$0.893198\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8.00000	+	13.8564i	0.300871	+	0.521124i
$708$		0			0
$709$	−19.0000	+	32.9090i	−0.713560	+	1.23592i	0.249952	+	0.968258i	$0.419585\pi$
$709$	−19.0000	+	32.9090i	−0.713560	+	1.23592i	−0.963512	+	0.267664i	$0.913748\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	24.0000	+	41.5692i	0.897549	+	1.55460i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	11.0000	+	19.0526i	0.408530	+	0.707594i
$726$		0			0
$727$	−8.00000	+	13.8564i	−0.296704	+	0.513906i	−0.975380	−	0.220532i	$-0.929221\pi$
$727$	−8.00000	+	13.8564i	−0.296704	+	0.513906i	0.678676	+	0.734438i	$0.262554\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.00000	+	13.8564i	−0.295891	+	0.512498i
$732$		0			0
$733$	3.00000	+	5.19615i	0.110808	+	0.191924i	0.916096	−	0.400959i	$-0.131323\pi$
$733$	3.00000	+	5.19615i	0.110808	+	0.191924i	−0.805289	+	0.592883i	$0.797990\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−16.0000			−0.589368
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.0000	−	25.9808i	−0.550297	−	0.953142i	−0.998253	−	0.0590862i	$-0.981181\pi$
$743$	−15.0000	−	25.9808i	−0.550297	−	0.953142i	0.447956	−	0.894055i	$-0.352152\pi$
$744$		0			0
$745$	12.0000	−	20.7846i	0.439646	−	0.761489i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	9.00000	−	15.5885i	0.328853	−	0.569590i
$750$		0			0
$751$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$751$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−32.0000			−1.16460
$756$		0			0
$757$	−42.0000			−1.52652			−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000			−1.52652			−0.763258	+	0.646094i	$0.776401\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−16.0000	−	27.7128i	−0.580000	−	1.00459i	−0.995479	−	0.0949859i	$-0.969719\pi$
$761$	−16.0000	−	27.7128i	−0.580000	−	1.00459i	0.415479	−	0.909603i	$-0.363614\pi$
$762$		0			0
$763$	−1.00000	+	1.73205i	−0.0362024	+	0.0627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−24.0000	+	41.5692i	−0.866590	+	1.50098i
$768$		0			0
$769$	19.0000	+	32.9090i	0.685158	+	1.18673i	0.973387	+	0.229166i	$0.0736000\pi$
$769$	19.0000	+	32.9090i	0.685158	+	1.18673i	−0.288230	+	0.957561i	$0.593067\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8.00000			−0.287740			−0.143870	−	0.989597i	$-0.545955\pi$
$773$	−8.00000			−0.287740			−0.143870	+	0.989597i	$0.545955\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−14.0000	+	24.2487i	−0.500959	+	0.867687i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−28.0000	+	48.4974i	−0.999363	+	1.73095i
$786$		0			0
$787$	6.00000	+	10.3923i	0.213877	+	0.370446i	0.952925	−	0.303207i	$-0.0980575\pi$
$787$	6.00000	+	10.3923i	0.213877	+	0.370446i	−0.739048	+	0.673653i	$0.764724\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−10.0000			−0.355559
$792$		0			0
$793$	−36.0000			−1.27840
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$	24.0000	−	41.5692i	0.849059	−	1.47061i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.00000	−	3.46410i	0.0705785	−	0.122245i
$804$		0			0
$805$	4.00000	+	6.92820i	0.140981	+	0.244187i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	10.0000			0.351581			0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000			0.351581			0.175791	+	0.984428i	$0.443752\pi$
$810$		0			0
$811$	28.0000			0.983213			0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000			0.983213			0.491606	+	0.870817i	$0.336410\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	32.0000	+	55.4256i	1.12091	+	1.94147i
$816$		0			0
$817$	−8.00000	+	13.8564i	−0.279885	+	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.00000	+	1.73205i	−0.0349002	+	0.0604490i	−0.882948	−	0.469471i	$-0.844445\pi$
$821$	−1.00000	+	1.73205i	−0.0349002	+	0.0604490i	0.848048	+	0.529920i	$0.177778\pi$
$822$		0			0
$823$	10.0000	+	17.3205i	0.348578	+	0.603755i	0.985997	−	0.166762i	$-0.0533313\pi$
$823$	10.0000	+	17.3205i	0.348578	+	0.603755i	−0.637419	+	0.770517i	$0.719998\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	18.0000			0.625921			0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000			0.625921			0.312961	+	0.949766i	$0.398679\pi$
$828$		0			0
$829$	−14.0000			−0.486240			−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000			−0.486240			−0.243120	+	0.969996i	$0.578171\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.00000	+	3.46410i	0.0692959	+	0.120024i
$834$		0			0
$835$	−8.00000	+	13.8564i	−0.276851	+	0.479521i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−22.0000	+	38.1051i	−0.759524	+	1.31553i	0.183569	+	0.983007i	$0.441235\pi$
$839$	−22.0000	+	38.1051i	−0.759524	+	1.31553i	−0.943093	+	0.332528i	$0.892098\pi$
$840$		0			0
$841$	12.5000	+	21.6506i	0.431034	+	0.746574i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	92.0000			3.16490
$846$		0			0
$847$	7.00000			0.240523
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.00000	−	3.46410i	−0.0685591	−	0.118748i
$852$		0			0
$853$	−7.00000	+	12.1244i	−0.239675	+	0.415130i	−0.960621	−	0.277862i	$-0.910374\pi$
$853$	−7.00000	+	12.1244i	−0.239675	+	0.415130i	0.720946	+	0.692992i	$0.243708\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	24.0000	−	41.5692i	0.819824	−	1.41998i	−0.0859870	−	0.996296i	$-0.527404\pi$
$857$	24.0000	−	41.5692i	0.819824	−	1.41998i	0.905811	−	0.423681i	$-0.139262\pi$
$858$		0			0
$859$	−22.0000	−	38.1051i	−0.750630	−	1.30013i	−0.947518	−	0.319704i	$-0.896417\pi$
$859$	−22.0000	−	38.1051i	−0.750630	−	1.30013i	0.196887	−	0.980426i	$-0.436917\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−14.0000			−0.476566			−0.238283	−	0.971196i	$-0.576585\pi$
$863$	−14.0000			−0.476566			−0.238283	+	0.971196i	$0.576585\pi$
$864$		0			0
$865$	64.0000			2.17607
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−12.0000	−	20.7846i	−0.407072	−	0.705070i
$870$		0			0
$871$	−24.0000	+	41.5692i	−0.813209	+	1.40852i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	12.0000	−	20.7846i	0.405674	−	0.702648i
$876$		0			0
$877$	−23.0000	−	39.8372i	−0.776655	−	1.34521i	−0.933860	−	0.357640i	$-0.883582\pi$
$877$	−23.0000	−	39.8372i	−0.776655	−	1.34521i	0.157205	−	0.987566i	$-0.449752\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	36.0000			1.21287			0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000			1.21287			0.606435	+	0.795133i	$0.292599\pi$
$882$		0			0
$883$	44.0000			1.48072			0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000			1.48072			0.740359	+	0.672212i	$0.234656\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6.00000	+	10.3923i	0.201460	+	0.348939i	0.948999	−	0.315279i	$-0.102098\pi$
$887$	6.00000	+	10.3923i	0.201460	+	0.348939i	−0.747539	+	0.664218i	$0.768765\pi$
$888$		0			0
$889$	6.00000	−	10.3923i	0.201234	−	0.348547i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	24.0000	−	41.5692i	0.803129	−	1.39106i
$894$		0			0
$895$	36.0000	+	62.3538i	1.20335	+	2.08426i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	24.0000			0.799556
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.0000	+	20.7846i	0.398893	+	0.690904i
$906$		0			0
$907$	−6.00000	+	10.3923i	−0.199227	+	0.345071i	−0.948278	−	0.317441i	$-0.897176\pi$
$907$	−6.00000	+	10.3923i	−0.199227	+	0.345071i	0.749051	+	0.662512i	$0.230510\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.00000	−	15.5885i	0.298183	−	0.516469i	−0.677537	−	0.735489i	$-0.736953\pi$
$911$	9.00000	−	15.5885i	0.298183	−	0.516469i	0.975720	+	0.219020i	$0.0702860\pi$
$912$		0			0
$913$	4.00000	+	6.92820i	0.132381	+	0.229290i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4.00000			−0.132092
$918$		0			0
$919$	16.0000			0.527791			0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000			0.527791			0.263896	+	0.964551i	$0.414993\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	42.0000	+	72.7461i	1.38245	+	2.39447i
$924$		0			0
$925$	−11.0000	+	19.0526i	−0.361678	+	0.626444i
$926$		0			0
$927$		0			0
$928$		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
$931$	2.00000	+	3.46410i	0.0655474	+	0.113531i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−32.0000			−1.04651
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14.0000	−	24.2487i	−0.456387	−	0.790485i	0.542380	−	0.840133i	$-0.317523\pi$
$941$	−14.0000	−	24.2487i	−0.456387	−	0.790485i	−0.998767	+	0.0496480i	$0.984190\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−5.00000	+	8.66025i	−0.162478	+	0.281420i	−0.935757	−	0.352646i	$-0.885282\pi$
$947$	−5.00000	+	8.66025i	−0.162478	+	0.281420i	0.773279	+	0.634066i	$0.218615\pi$
$948$		0			0
$949$	−6.00000	−	10.3923i	−0.194768	−	0.337348i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−22.0000			−0.712650			−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000			−0.712650			−0.356325	+	0.934362i	$0.615970\pi$
$954$		0			0
$955$	−72.0000			−2.32987
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.00000	−	1.73205i	−0.0322917	−	0.0559308i
$960$		0			0
$961$	15.5000	−	26.8468i	0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	20.0000	−	34.6410i	0.643823	−	1.11513i
$966$		0			0
$967$	−10.0000	−	17.3205i	−0.321578	−	0.556990i	0.659236	−	0.751936i	$-0.270880\pi$
$967$	−10.0000	−	17.3205i	−0.321578	−	0.556990i	−0.980814	+	0.194946i	$0.937547\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	36.0000			1.15529			0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000			1.15529			0.577647	+	0.816286i	$0.303971\pi$
$972$		0			0
$973$	−4.00000			−0.128234
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−15.0000	−	25.9808i	−0.479893	−	0.831198i	0.519841	−	0.854263i	$-0.325991\pi$
$977$	−15.0000	−	25.9808i	−0.479893	−	0.831198i	−0.999734	+	0.0230645i	$0.992658\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	24.0000	−	41.5692i	0.765481	−	1.32585i	−0.174511	−	0.984655i	$-0.555834\pi$
$983$	24.0000	−	41.5692i	0.765481	−	1.32585i	0.939992	−	0.341197i	$-0.110832\pi$
$984$		0			0
$985$	44.0000	+	76.2102i	1.40196	+	2.42826i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−8.00000			−0.254385
$990$		0			0
$991$	56.0000			1.77890			0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000			1.77890			0.889449	+	0.457034i	$0.151088\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	32.0000	+	55.4256i	1.01447	+	1.75711i
$996$		0			0
$997$	−19.0000	+	32.9090i	−0.601736	+	1.04224i	0.390822	+	0.920466i	$0.372191\pi$
$997$	−19.0000	+	32.9090i	−0.601736	+	1.04224i	−0.992558	+	0.121771i	$0.961143\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.j.a.1513.1		2
3.2	odd	2		2268.2.j.n.1513.1		2
9.2	odd	6		252.2.a.a.1.1		1
9.4	even	3	inner	2268.2.j.a.757.1		2
9.5	odd	6		2268.2.j.n.757.1		2
9.7	even	3		84.2.a.a.1.1	✓	1
36.7	odd	6		336.2.a.f.1.1		1
36.11	even	6		1008.2.a.a.1.1		1
45.2	even	12		6300.2.k.g.6049.1		2
45.7	odd	12		2100.2.k.i.1849.2		2
45.29	odd	6		6300.2.a.w.1.1		1
45.34	even	6		2100.2.a.r.1.1		1
45.38	even	12		6300.2.k.g.6049.2		2
45.43	odd	12		2100.2.k.i.1849.1		2
63.2	odd	6		1764.2.k.k.361.1		2
63.11	odd	6		1764.2.k.k.1549.1		2
63.16	even	3		588.2.i.e.361.1		2
63.20	even	6		1764.2.a.k.1.1		1
63.25	even	3		588.2.i.e.373.1		2
63.34	odd	6		588.2.a.d.1.1		1
63.38	even	6		1764.2.k.a.1549.1		2
63.47	even	6		1764.2.k.a.361.1		2
63.52	odd	6		588.2.i.d.373.1		2
63.61	odd	6		588.2.i.d.361.1		2
72.11	even	6		4032.2.a.bn.1.1		1
72.29	odd	6		4032.2.a.bm.1.1		1
72.43	odd	6		1344.2.a.a.1.1		1
72.61	even	6		1344.2.a.k.1.1		1
144.43	odd	12		5376.2.c.p.2689.1		2
144.61	even	12		5376.2.c.q.2689.1		2
144.115	odd	12		5376.2.c.p.2689.2		2
144.133	even	12		5376.2.c.q.2689.2		2
180.79	odd	6		8400.2.a.e.1.1		1
252.79	odd	6		2352.2.q.b.1537.1		2
252.83	odd	6		7056.2.a.cd.1.1		1
252.115	even	6		2352.2.q.z.961.1		2
252.151	odd	6		2352.2.q.b.961.1		2
252.187	even	6		2352.2.q.z.1537.1		2
252.223	even	6		2352.2.a.a.1.1		1
504.349	odd	6		9408.2.a.bn.1.1		1
504.475	even	6		9408.2.a.df.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.2.a.a.1.1	✓	1	9.7	even	3
252.2.a.a.1.1		1	9.2	odd	6
336.2.a.f.1.1		1	36.7	odd	6
588.2.a.d.1.1		1	63.34	odd	6
588.2.i.d.361.1		2	63.61	odd	6
588.2.i.d.373.1		2	63.52	odd	6
588.2.i.e.361.1		2	63.16	even	3
588.2.i.e.373.1		2	63.25	even	3
1008.2.a.a.1.1		1	36.11	even	6
1344.2.a.a.1.1		1	72.43	odd	6
1344.2.a.k.1.1		1	72.61	even	6
1764.2.a.k.1.1		1	63.20	even	6
1764.2.k.a.361.1		2	63.47	even	6
1764.2.k.a.1549.1		2	63.38	even	6
1764.2.k.k.361.1		2	63.2	odd	6
1764.2.k.k.1549.1		2	63.11	odd	6
2100.2.a.r.1.1		1	45.34	even	6
2100.2.k.i.1849.1		2	45.43	odd	12
2100.2.k.i.1849.2		2	45.7	odd	12
2268.2.j.a.757.1		2	9.4	even	3	inner
2268.2.j.a.1513.1		2	1.1	even	1	trivial
2268.2.j.n.757.1		2	9.5	odd	6
2268.2.j.n.1513.1		2	3.2	odd	2
2352.2.a.a.1.1		1	252.223	even	6
2352.2.q.b.961.1		2	252.151	odd	6
2352.2.q.b.1537.1		2	252.79	odd	6
2352.2.q.z.961.1		2	252.115	even	6
2352.2.q.z.1537.1		2	252.187	even	6
4032.2.a.bm.1.1		1	72.29	odd	6
4032.2.a.bn.1.1		1	72.11	even	6
5376.2.c.p.2689.1		2	144.43	odd	12
5376.2.c.p.2689.2		2	144.115	odd	12
5376.2.c.q.2689.1		2	144.61	even	12
5376.2.c.q.2689.2		2	144.133	even	12
6300.2.a.w.1.1		1	45.29	odd	6
6300.2.k.g.6049.1		2	45.2	even	12
6300.2.k.g.6049.2		2	45.38	even	12
7056.2.a.cd.1.1		1	252.83	odd	6
8400.2.a.e.1.1		1	180.79	odd	6
9408.2.a.bn.1.1		1	504.349	odd	6
9408.2.a.df.1.1		1	504.475	even	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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.j (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 3.46410i) q^{5} +(0.500000 - 0.866025i) q^{7} +O(q^{10})\)	\(q+(-2.00000 - 3.46410i) q^{5} +(0.500000 - 0.866025i) q^{7} +(-1.00000 + 1.73205i) q^{11} +(3.00000 + 5.19615i) q^{13} -4.00000 q^{17} -4.00000 q^{19} +(-1.00000 - 1.73205i) q^{23} +(-5.50000 + 9.52628i) q^{25} +(1.00000 - 1.73205i) q^{29} -4.00000 q^{35} +2.00000 q^{37} +(2.00000 - 3.46410i) q^{43} +(-6.00000 + 10.3923i) q^{47} +(-0.500000 - 0.866025i) q^{49} -6.00000 q^{53} +8.00000 q^{55} +(4.00000 + 6.92820i) q^{59} +(-3.00000 + 5.19615i) q^{61} +(12.0000 - 20.7846i) q^{65} +(4.00000 + 6.92820i) q^{67} +14.0000 q^{71} -2.00000 q^{73} +(1.00000 + 1.73205i) q^{77} +(-6.00000 + 10.3923i) q^{79} +(2.00000 - 3.46410i) q^{83} +(8.00000 + 13.8564i) q^{85} +6.00000 q^{91} +(8.00000 + 13.8564i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 + q^7	\( 2 q - 4 q^{5} + q^{7} - 2 q^{11} + 6 q^{13} - 8 q^{17} - 8 q^{19} - 2 q^{23} - 11 q^{25} + 2 q^{29} - 8 q^{35} + 4 q^{37} + 4 q^{43} - 12 q^{47} - q^{49} - 12 q^{53} + 16 q^{55} + 8 q^{59} - 6 q^{61} + 24 q^{65} + 8 q^{67} + 28 q^{71} - 4 q^{73} + 2 q^{77} - 12 q^{79} + 4 q^{83} + 16 q^{85} + 12 q^{91} + 16 q^{95} + 2 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 + q^7 - 2 * q^11 + 6 * q^13 - 8 * q^17 - 8 * q^19 - 2 * q^23 - 11 * q^25 + 2 * q^29 - 8 * q^35 + 4 * q^37 + 4 * q^43 - 12 * q^47 - q^49 - 12 * q^53 + 16 * q^55 + 8 * q^59 - 6 * q^61 + 24 * q^65 + 8 * q^67 + 28 * q^71 - 4 * q^73 + 2 * q^77 - 12 * q^79 + 4 * q^83 + 16 * q^85 + 12 * q^91 + 16 * q^95 + 2 * q^97

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