LMFDB - Embedded newform 2028.2.q.d.1837.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2028,2,Mod(361,2028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("2028.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2028.q (of order $6$, 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:	$16.1936615299$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1837.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2028.1837
Dual form			2028.2.q.d.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(-1.73205 + 1.00000i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	$q+(-0.500000 + 0.866025i) q^{3} +(-1.73205 + 1.00000i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-3.00000 - 5.19615i) q^{17} +(1.73205 - 1.00000i) q^{19} -2.00000i q^{21} +5.00000 q^{25} +1.00000 q^{27} +(3.00000 - 5.19615i) q^{29} +2.00000i q^{31} +(1.73205 + 1.00000i) q^{37} +(10.3923 + 6.00000i) q^{41} +(-2.00000 - 3.46410i) q^{43} +(-1.50000 + 2.59808i) q^{49} +6.00000 q^{51} +6.00000 q^{53} +2.00000i q^{57} +(-10.3923 + 6.00000i) q^{59} +(-1.00000 - 1.73205i) q^{61} +(1.73205 + 1.00000i) q^{63} +(8.66025 + 5.00000i) q^{67} +(10.3923 - 6.00000i) q^{71} -14.0000i q^{73} +(-2.50000 + 4.33013i) q^{75} +8.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} +12.0000i q^{83} +(3.00000 + 5.19615i) q^{87} +(-1.73205 - 1.00000i) q^{93} +(-8.66025 + 5.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 2 * q^9	$ 4 q - 2 q^{3} - 2 q^{9} - 12 q^{17} + 20 q^{25} + 4 q^{27} + 12 q^{29} - 8 q^{43} - 6 q^{49} + 24 q^{51} + 24 q^{53} - 4 q^{61} - 10 q^{75} + 32 q^{79} - 2 q^{81} + 12 q^{87}+O(q^{100}) $ 4 * q - 2 * q^3 - 2 * q^9 - 12 * q^17 + 20 * q^25 + 4 * q^27 + 12 * q^29 - 8 * q^43 - 6 * q^49 + 24 * q^51 + 24 * q^53 - 4 * q^61 - 10 * q^75 + 32 * q^79 - 2 * q^81 + 12 * q^87

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1861$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$4$		0			0
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$	−1.73205	+	1.00000i	−0.654654	+	0.377964i	−0.790237	−	0.612801i	$-0.790043\pi$
$7$	−1.73205	+	1.00000i	−0.654654	+	0.377964i	0.135583	+	0.990766i	$0.456709\pi$
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$		0			0
$13$		0			0
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.00000	−	5.19615i	−0.727607	−	1.26025i	−0.957892	−	0.287129i	$-0.907299\pi$
$17$	−3.00000	−	5.19615i	−0.727607	−	1.26025i	0.230285	−	0.973123i	$-0.426034\pi$
$18$		0			0
$19$	1.73205	−	1.00000i	0.397360	−	0.229416i	−0.287984	−	0.957635i	$-0.592985\pi$
$19$	1.73205	−	1.00000i	0.397360	−	0.229416i	0.685344	+	0.728219i	$0.259652\pi$
$20$		0			0
$21$		−	2.00000i		−	0.436436i
$22$		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
$25$	5.00000			1.00000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	3.00000	−	5.19615i	0.557086	−	0.964901i	−0.440652	−	0.897678i	$-0.645253\pi$
$29$	3.00000	−	5.19615i	0.557086	−	0.964901i	0.997738	−	0.0672232i	$-0.0214140\pi$
$30$		0			0
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.73205	+	1.00000i	0.284747	+	0.164399i	0.635571	−	0.772043i	$-0.280765\pi$
$37$	1.73205	+	1.00000i	0.284747	+	0.164399i	−0.350823	+	0.936442i	$0.614098\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.3923	+	6.00000i	1.62301	+	0.937043i	0.986110	+	0.166092i	$0.0531149\pi$
$41$	10.3923	+	6.00000i	1.62301	+	0.937043i	0.636895	+	0.770950i	$0.280218\pi$
$42$		0			0
$43$	−2.00000	−	3.46410i	−0.304997	−	0.528271i	0.672264	−	0.740312i	$-0.265322\pi$
$43$	−2.00000	−	3.46410i	−0.304997	−	0.528271i	−0.977261	+	0.212041i	$0.931989\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.50000	+	2.59808i	−0.214286	+	0.371154i
$50$		0			0
$51$	6.00000			0.840168
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			2.00000i			0.264906i
$58$		0			0
$59$	−10.3923	+	6.00000i	−1.35296	+	0.781133i	−0.988663	−	0.150148i	$-0.952025\pi$
$59$	−10.3923	+	6.00000i	−1.35296	+	0.781133i	−0.364299	+	0.931282i	$0.618692\pi$
$60$		0			0
$61$	−1.00000	−	1.73205i	−0.128037	−	0.221766i	0.794879	−	0.606768i	$-0.207534\pi$
$61$	−1.00000	−	1.73205i	−0.128037	−	0.221766i	−0.922916	+	0.385002i	$0.874201\pi$
$62$		0			0
$63$	1.73205	+	1.00000i	0.218218	+	0.125988i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	8.66025	+	5.00000i	1.05802	+	0.610847i	0.924883	−	0.380251i	$-0.124162\pi$
$67$	8.66025	+	5.00000i	1.05802	+	0.610847i	0.133135	+	0.991098i	$0.457496\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.3923	−	6.00000i	1.23334	−	0.712069i	0.265615	−	0.964079i	$-0.414425\pi$
$71$	10.3923	−	6.00000i	1.23334	−	0.712069i	0.967725	+	0.252010i	$0.0810916\pi$
$72$		0			0
$73$		−	14.0000i		−	1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$		−	14.0000i		−	1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$		0			0
$75$	−2.50000	+	4.33013i	−0.288675	+	0.500000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$			12.0000i			1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$			12.0000i			1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	3.00000	+	5.19615i	0.321634	+	0.557086i
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−1.73205	−	1.00000i	−0.179605	−	0.103695i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.66025	+	5.00000i	−0.879316	+	0.507673i	−0.870433	−	0.492287i	$-0.836161\pi$
$97$	−8.66025	+	5.00000i	−0.879316	+	0.507673i	−0.00888289	+	0.999961i	$0.502828\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.0623905	−	0.998052i	$-0.480128\pi$
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.833143	−	0.553058i	$-0.186539\pi$
$102$		0			0
$103$	16.0000			1.57653			0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000			1.57653			0.788263	+	0.615338i	$0.210980\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.00000	−	10.3923i	0.580042	−	1.00466i	−0.415432	−	0.909624i	$-0.636370\pi$
$107$	6.00000	−	10.3923i	0.580042	−	1.00466i	0.995474	−	0.0950377i	$-0.0302972\pi$
$108$		0			0
$109$			14.0000i			1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$			14.0000i			1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$		0			0
$111$	−1.73205	+	1.00000i	−0.164399	+	0.0949158i
$112$		0			0
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	0.689714	−	0.724082i	$-0.257736\pi$
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	−0.971930	+	0.235269i	$0.924403\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	10.3923	+	6.00000i	0.952661	+	0.550019i
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$		0			0
$123$	−10.3923	+	6.00000i	−0.937043	+	0.541002i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−2.00000	+	3.46410i	−0.177471	+	0.307389i	−0.941014	−	0.338368i	$-0.890125\pi$
$127$	−2.00000	+	3.46410i	−0.177471	+	0.307389i	0.763542	+	0.645758i	$0.223458\pi$
$128$		0			0
$129$	4.00000			0.352180
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	−2.00000	+	3.46410i	−0.173422	+	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.3923	−	6.00000i	0.887875	−	0.512615i	0.0146279	−	0.999893i	$-0.495344\pi$
$137$	10.3923	−	6.00000i	0.887875	−	0.512615i	0.873247	+	0.487278i	$0.162010\pi$
$138$		0			0
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	0.975417	+	0.220366i	$0.0707252\pi$
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	−0.296866	+	0.954919i	$0.595942\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−1.50000	−	2.59808i	−0.123718	−	0.214286i
$148$		0			0
$149$	10.3923	−	6.00000i	0.851371	−	0.491539i	−0.00974235	−	0.999953i	$-0.503101\pi$
$149$	10.3923	−	6.00000i	0.851371	−	0.491539i	0.861113	+	0.508413i	$0.169768\pi$
$150$		0			0
$151$			10.0000i			0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$			10.0000i			0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$		0			0
$153$	−3.00000	+	5.19615i	−0.242536	+	0.420084i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$	−3.00000	+	5.19615i	−0.237915	+	0.412082i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−12.1244	+	7.00000i	−0.949653	+	0.548282i	−0.892973	−	0.450110i	$-0.851385\pi$
$163$	−12.1244	+	7.00000i	−0.949653	+	0.548282i	−0.0566798	+	0.998392i	$0.518051\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−20.7846	−	12.0000i	−1.60836	−	0.928588i	−0.989737	−	0.142901i	$-0.954357\pi$
$167$	−20.7846	−	12.0000i	−1.60836	−	0.928588i	−0.618624	−	0.785687i	$-0.712310\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	−1.73205	−	1.00000i	−0.132453	−	0.0764719i
$172$		0			0
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	0.729155	−	0.684349i	$-0.239913\pi$
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	−0.957241	+	0.289292i	$0.906580\pi$
$174$		0			0
$175$	−8.66025	+	5.00000i	−0.654654	+	0.377964i
$176$		0			0
$177$		−	12.0000i		−	0.901975i
$178$		0			0
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	−0.549825	−	0.835280i	$-0.685306\pi$
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	0.998286	+	0.0585225i	$0.0186389\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$	2.00000			0.147844
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.73205	+	1.00000i	−0.125988	+	0.0727393i
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	1.73205	+	1.00000i	0.124676	+	0.0719816i	0.561041	−	0.827788i	$-0.310401\pi$
$193$	1.73205	+	1.00000i	0.124676	+	0.0719816i	−0.436365	+	0.899770i	$0.643734\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	20.7846	+	12.0000i	1.48084	+	0.854965i	0.999764	−	0.0217133i	$-0.00691209\pi$
$197$	20.7846	+	12.0000i	1.48084	+	0.854965i	0.481078	+	0.876678i	$0.340245\pi$
$198$		0			0
$199$	10.0000	+	17.3205i	0.708881	+	1.22782i	0.965272	+	0.261245i	$0.0841331\pi$
$199$	10.0000	+	17.3205i	0.708881	+	1.22782i	−0.256391	+	0.966573i	$0.582534\pi$
$200$		0			0
$201$	−8.66025	+	5.00000i	−0.610847	+	0.352673i
$202$		0			0
$203$			12.0000i			0.842235i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	2.00000	−	3.46410i	0.137686	−	0.238479i	−0.788935	−	0.614477i	$-0.789367\pi$
$211$	2.00000	−	3.46410i	0.137686	−	0.238479i	0.926620	+	0.375999i	$0.122700\pi$
$212$		0			0
$213$			12.0000i			0.822226i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.00000	−	3.46410i	−0.135769	−	0.235159i
$218$		0			0
$219$	12.1244	+	7.00000i	0.819288	+	0.473016i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−22.5167	−	13.0000i	−1.50783	−	0.870544i	−0.999959	−	0.00910984i	$-0.997100\pi$
$223$	−22.5167	−	13.0000i	−1.50783	−	0.870544i	−0.507869	−	0.861435i	$-0.669566\pi$
$224$		0			0
$225$	−2.50000	−	4.33013i	−0.166667	−	0.288675i
$226$		0			0
$227$	20.7846	−	12.0000i	1.37952	−	0.796468i	0.387421	−	0.921903i	$-0.373366\pi$
$227$	20.7846	−	12.0000i	1.37952	−	0.796468i	0.992102	+	0.125435i	$0.0400326\pi$
$228$		0			0
$229$		−	14.0000i		−	0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$		−	14.0000i		−	0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−4.00000	+	6.92820i	−0.259828	+	0.450035i
$238$		0			0
$239$		−	12.0000i		−	0.776215i	−0.921614	−	0.388108i	$-0.873129\pi$
$239$		−	12.0000i		−	0.776215i	0.921614	−	0.388108i	$-0.126871\pi$
$240$		0			0
$241$	8.66025	−	5.00000i	0.557856	−	0.322078i	−0.194429	−	0.980917i	$-0.562285\pi$
$241$	8.66025	−	5.00000i	0.557856	−	0.322078i	0.752285	+	0.658838i	$0.228952\pi$
$242$		0			0
$243$	−0.500000	−	0.866025i	−0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−10.3923	−	6.00000i	−0.658586	−	0.380235i
$250$		0			0
$251$	6.00000	+	10.3923i	0.378717	+	0.655956i	0.990876	−	0.134778i	$-0.0430322\pi$
$251$	6.00000	+	10.3923i	0.378717	+	0.655956i	−0.612159	+	0.790735i	$0.709699\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.00000	−	5.19615i	0.187135	−	0.324127i	−0.757159	−	0.653231i	$-0.773413\pi$
$257$	3.00000	−	5.19615i	0.187135	−	0.324127i	0.944294	+	0.329104i	$0.106747\pi$
$258$		0			0
$259$	−4.00000			−0.248548
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$	12.0000	−	20.7846i	0.739952	−	1.28163i	−0.212565	−	0.977147i	$-0.568182\pi$
$263$	12.0000	−	20.7846i	0.739952	−	1.28163i	0.952517	−	0.304487i	$-0.0984850\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.00000	−	15.5885i	−0.548740	−	0.950445i	−0.998361	−	0.0572259i	$-0.981774\pi$
$269$	−9.00000	−	15.5885i	−0.548740	−	0.950445i	0.449622	−	0.893219i	$-0.351559\pi$
$270$		0			0
$271$	22.5167	+	13.0000i	1.36779	+	0.789694i	0.990645	−	0.136461i	$-0.0435728\pi$
$271$	22.5167	+	13.0000i	1.36779	+	0.789694i	0.377144	+	0.926155i	$0.376906\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−5.00000	−	8.66025i	−0.300421	−	0.520344i	0.675810	−	0.737075i	$-0.263794\pi$
$277$	−5.00000	−	8.66025i	−0.300421	−	0.520344i	−0.976231	+	0.216731i	$0.930460\pi$
$278$		0			0
$279$	1.73205	−	1.00000i	0.103695	−	0.0598684i
$280$		0			0
$281$			12.0000i			0.715860i	0.933748	+	0.357930i	$0.116517\pi$
$281$			12.0000i			0.715860i	−0.933748	+	0.357930i	$0.883483\pi$
$282$		0			0
$283$	−8.00000	+	13.8564i	−0.475551	+	0.823678i	−0.999608	−	0.0280052i	$-0.991084\pi$
$283$	−8.00000	+	13.8564i	−0.475551	+	0.823678i	0.524057	+	0.851683i	$0.324418\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−24.0000			−1.41668
$288$		0			0
$289$	−9.50000	+	16.4545i	−0.558824	+	0.967911i
$290$		0			0
$291$		−	10.0000i		−	0.586210i
$292$		0			0
$293$	10.3923	−	6.00000i	0.607125	−	0.350524i	−0.164714	−	0.986341i	$-0.552670\pi$
$293$	10.3923	−	6.00000i	0.607125	−	0.350524i	0.771839	+	0.635818i	$0.219337\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	6.92820	+	4.00000i	0.399335	+	0.230556i
$302$		0			0
$303$	9.00000	+	15.5885i	0.517036	+	0.895533i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	2.00000i		−	0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$		−	2.00000i		−	0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$		0			0
$309$	−8.00000	+	13.8564i	−0.455104	+	0.788263i
$310$		0			0
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$	−22.0000			−1.24351			−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000			−1.24351			−0.621757	+	0.783210i	$0.713581\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	24.0000i		−	1.34797i	−0.738743	−	0.673987i	$-0.764580\pi$
$317$		−	24.0000i		−	1.34797i	0.738743	−	0.673987i	$-0.235420\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	6.00000	+	10.3923i	0.334887	+	0.580042i
$322$		0			0
$323$	−10.3923	−	6.00000i	−0.578243	−	0.333849i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−12.1244	−	7.00000i	−0.670478	−	0.387101i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−8.66025	+	5.00000i	−0.476011	+	0.274825i	−0.718752	−	0.695266i	$-0.755287\pi$
$331$	−8.66025	+	5.00000i	−0.476011	+	0.274825i	0.242742	+	0.970091i	$0.421953\pi$
$332$		0			0
$333$		−	2.00000i		−	0.109599i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$		0			0
$339$	6.00000			0.325875
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		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
$349$	−19.0526	−	11.0000i	−1.01986	−	0.588817i	−0.105797	−	0.994388i	$-0.533739\pi$
$349$	−19.0526	−	11.0000i	−1.01986	−	0.588817i	−0.914063	+	0.405571i	$0.867073\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−10.3923	−	6.00000i	−0.553127	−	0.319348i	0.197256	−	0.980352i	$-0.436797\pi$
$353$	−10.3923	−	6.00000i	−0.553127	−	0.319348i	−0.750382	+	0.661004i	$0.770130\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−10.3923	+	6.00000i	−0.550019	+	0.317554i
$358$		0			0
$359$			24.0000i			1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$			24.0000i			1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$		0			0
$361$	−7.50000	+	12.9904i	−0.394737	+	0.683704i
$362$		0			0
$363$	11.0000			0.577350
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−10.0000	+	17.3205i	−0.521996	+	0.904123i	0.477677	+	0.878536i	$0.341479\pi$
$367$	−10.0000	+	17.3205i	−0.521996	+	0.904123i	−0.999673	+	0.0255875i	$0.991854\pi$
$368$		0			0
$369$		−	12.0000i		−	0.624695i
$370$		0			0
$371$	−10.3923	+	6.00000i	−0.539542	+	0.311504i
$372$		0			0
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	0.625917	−	0.779890i	$-0.284725\pi$
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	−0.988363	+	0.152115i	$0.951392\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−22.5167	−	13.0000i	−1.15660	−	0.667765i	−0.206116	−	0.978528i	$-0.566082\pi$
$379$	−22.5167	−	13.0000i	−1.15660	−	0.667765i	−0.950488	+	0.310763i	$0.899416\pi$
$380$		0			0
$381$	−2.00000	−	3.46410i	−0.102463	−	0.177471i
$382$		0			0
$383$	20.7846	−	12.0000i	1.06204	−	0.613171i	0.136047	−	0.990702i	$-0.456560\pi$
$383$	20.7846	−	12.0000i	1.06204	−	0.613171i	0.925997	+	0.377531i	$0.123227\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2.00000	+	3.46410i	−0.101666	+	0.176090i
$388$		0			0
$389$	−30.0000			−1.52106			−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000			−1.52106			−0.760530	+	0.649303i	$0.775061\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−6.00000	+	10.3923i	−0.302660	+	0.524222i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	19.0526	−	11.0000i	0.956221	−	0.552074i	0.0612128	−	0.998125i	$-0.480503\pi$
$397$	19.0526	−	11.0000i	0.956221	−	0.552074i	0.895008	+	0.446051i	$0.147170\pi$
$398$		0			0
$399$	−2.00000	−	3.46410i	−0.100125	−	0.173422i
$400$		0			0
$401$	20.7846	+	12.0000i	1.03793	+	0.599251i	0.919247	−	0.393680i	$-0.128798\pi$
$401$	20.7846	+	12.0000i	1.03793	+	0.599251i	0.118686	+	0.992932i	$0.462132\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−8.66025	+	5.00000i	−0.428222	+	0.247234i	−0.698589	−	0.715523i	$-0.746188\pi$
$409$	−8.66025	+	5.00000i	−0.428222	+	0.247234i	0.270367	+	0.962757i	$0.412855\pi$
$410$		0			0
$411$			12.0000i			0.591916i
$412$		0			0
$413$	12.0000	−	20.7846i	0.590481	−	1.02274i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−16.0000			−0.783523
$418$		0			0
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	−0.681426	−	0.731887i	$-0.738640\pi$
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	0.974546	+	0.224189i	$0.0719734\pi$
$420$		0			0
$421$		−	34.0000i		−	1.65706i	−0.559946	−	0.828529i	$-0.689178\pi$
$421$		−	34.0000i		−	1.65706i	0.559946	−	0.828529i	$-0.310822\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−15.0000	−	25.9808i	−0.727607	−	1.26025i
$426$		0			0
$427$	3.46410	+	2.00000i	0.167640	+	0.0967868i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−20.7846	−	12.0000i	−1.00116	−	0.578020i	−0.0925683	−	0.995706i	$-0.529508\pi$
$431$	−20.7846	−	12.0000i	−1.00116	−	0.578020i	−0.908591	+	0.417687i	$0.862841\pi$
$432$		0			0
$433$	7.00000	+	12.1244i	0.336399	+	0.582659i	0.983752	−	0.179530i	$-0.0574578\pi$
$433$	7.00000	+	12.1244i	0.336399	+	0.582659i	−0.647354	+	0.762190i	$0.724124\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	4.00000	−	6.92820i	0.190910	−	0.330665i	−0.754642	−	0.656136i	$-0.772190\pi$
$439$	4.00000	−	6.92820i	0.190910	−	0.330665i	0.945552	+	0.325471i	$0.105523\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$		0			0
$443$	−12.0000			−0.570137			−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000			−0.570137			−0.285069	+	0.958507i	$0.592016\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			12.0000i			0.567581i
$448$		0			0
$449$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$449$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−8.66025	−	5.00000i	−0.406894	−	0.234920i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−32.9090	−	19.0000i	−1.53942	−	0.888783i	−0.998873	−	0.0474665i	$-0.984885\pi$
$457$	−32.9090	−	19.0000i	−1.53942	−	0.888783i	−0.540544	−	0.841316i	$-0.681781\pi$
$458$		0			0
$459$	−3.00000	−	5.19615i	−0.140028	−	0.242536i
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$			22.0000i			1.02243i	0.859454	+	0.511213i	$0.170804\pi$
$463$			22.0000i			1.02243i	−0.859454	+	0.511213i	$0.829196\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−36.0000			−1.66588			−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000			−1.66588			−0.832941	+	0.553362i	$0.813345\pi$
$468$		0			0
$469$	−20.0000			−0.923514
$470$		0			0
$471$	−1.00000	+	1.73205i	−0.0460776	+	0.0798087i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	8.66025	−	5.00000i	0.397360	−	0.229416i
$476$		0			0
$477$	−3.00000	−	5.19615i	−0.137361	−	0.237915i
$478$		0			0
$479$	−10.3923	−	6.00000i	−0.474837	−	0.274147i	0.243426	−	0.969920i	$-0.421729\pi$
$479$	−10.3923	−	6.00000i	−0.474837	−	0.274147i	−0.718262	+	0.695773i	$0.755062\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	1.73205	−	1.00000i	0.0784867	−	0.0453143i	−0.460243	−	0.887793i	$-0.652238\pi$
$487$	1.73205	−	1.00000i	0.0784867	−	0.0453143i	0.538730	+	0.842479i	$0.318904\pi$
$488$		0			0
$489$		−	14.0000i		−	0.633102i
$490$		0			0
$491$	6.00000	−	10.3923i	0.270776	−	0.468998i	−0.698285	−	0.715820i	$-0.746053\pi$
$491$	6.00000	−	10.3923i	0.270776	−	0.468998i	0.969061	+	0.246822i	$0.0793863\pi$
$492$		0			0
$493$	−36.0000			−1.62136
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.0000	+	20.7846i	−0.538274	+	0.932317i
$498$		0			0
$499$			14.0000i			0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$			14.0000i			0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$		0			0
$501$	20.7846	−	12.0000i	0.928588	−	0.536120i
$502$		0			0
$503$	12.0000	+	20.7846i	0.535054	+	0.926740i	0.999161	+	0.0409609i	$0.0130419\pi$
$503$	12.0000	+	20.7846i	0.535054	+	0.926740i	−0.464107	+	0.885779i	$0.653625\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	14.0000	+	24.2487i	0.619324	+	1.07270i
$512$		0			0
$513$	1.73205	−	1.00000i	0.0764719	−	0.0441511i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	6.00000			0.263371
$520$		0			0
$521$	30.0000			1.31432			0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000			1.31432			0.657162	+	0.753749i	$0.271757\pi$
$522$		0			0
$523$	8.00000	−	13.8564i	0.349816	−	0.605898i	−0.636401	−	0.771358i	$-0.719578\pi$
$523$	8.00000	−	13.8564i	0.349816	−	0.605898i	0.986216	+	0.165460i	$0.0529109\pi$
$524$		0			0
$525$		−	10.0000i		−	0.436436i
$526$		0			0
$527$	10.3923	−	6.00000i	0.452696	−	0.261364i
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$	10.3923	+	6.00000i	0.450988	+	0.260378i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	6.00000	+	10.3923i	0.258919	+	0.448461i
$538$		0			0
$539$		0			0
$540$		0			0
$541$			10.0000i			0.429934i	0.976621	+	0.214967i	$0.0689643\pi$
$541$			10.0000i			0.429934i	−0.976621	+	0.214967i	$0.931036\pi$
$542$		0			0
$543$	1.00000	−	1.73205i	0.0429141	−	0.0743294i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−16.0000			−0.684111			−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000			−0.684111			−0.342055	+	0.939680i	$0.611123\pi$
$548$		0			0
$549$	−1.00000	+	1.73205i	−0.0426790	+	0.0739221i
$550$		0			0
$551$		−	12.0000i		−	0.511217i
$552$		0			0
$553$	−13.8564	+	8.00000i	−0.589234	+	0.340195i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.3923	−	6.00000i	−0.440336	−	0.254228i	0.263404	−	0.964686i	$-0.415155\pi$
$557$	−10.3923	−	6.00000i	−0.440336	−	0.254228i	−0.703740	+	0.710457i	$0.748488\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.00000	+	10.3923i	0.252870	+	0.437983i	0.964315	−	0.264758i	$-0.0852922\pi$
$563$	6.00000	+	10.3923i	0.252870	+	0.437983i	−0.711445	+	0.702742i	$0.751959\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	2.00000i		−	0.0839921i
$568$		0			0
$569$	−9.00000	+	15.5885i	−0.377300	+	0.653502i	−0.990668	−	0.136295i	$-0.956481\pi$
$569$	−9.00000	+	15.5885i	−0.377300	+	0.653502i	0.613369	+	0.789797i	$0.289814\pi$
$570$		0			0
$571$	40.0000			1.67395			0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000			1.67395			0.836974	+	0.547243i	$0.184323\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$			2.00000i			0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$			2.00000i			0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$		0			0
$579$	−1.73205	+	1.00000i	−0.0719816	+	0.0415586i
$580$		0			0
$581$	−12.0000	−	20.7846i	−0.497844	−	0.862291i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.3923	+	6.00000i	0.428936	+	0.247647i	0.698893	−	0.715226i	$-0.253676\pi$
$587$	10.3923	+	6.00000i	0.428936	+	0.247647i	−0.269957	+	0.962872i	$0.587010\pi$
$588$		0			0
$589$	2.00000	+	3.46410i	0.0824086	+	0.142736i
$590$		0			0
$591$	−20.7846	+	12.0000i	−0.854965	+	0.493614i
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−20.0000			−0.818546
$598$		0			0
$599$	−48.0000			−1.96123			−0.980613	−	0.195952i	$-0.937220\pi$
$599$	−48.0000			−1.96123			−0.980613	+	0.195952i	$0.937220\pi$
$600$		0			0
$601$	−13.0000	+	22.5167i	−0.530281	+	0.918474i	0.469095	+	0.883148i	$0.344580\pi$
$601$	−13.0000	+	22.5167i	−0.530281	+	0.918474i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$		0			0
$603$		−	10.0000i		−	0.407231i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	2.00000	+	3.46410i	0.0811775	+	0.140604i	0.903756	−	0.428048i	$-0.140799\pi$
$607$	2.00000	+	3.46410i	0.0811775	+	0.140604i	−0.822578	+	0.568652i	$0.807465\pi$
$608$		0			0
$609$	−10.3923	−	6.00000i	−0.421117	−	0.243132i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	19.0526	+	11.0000i	0.769526	+	0.444286i	0.832705	−	0.553716i	$-0.186791\pi$
$613$	19.0526	+	11.0000i	0.769526	+	0.444286i	−0.0631797	+	0.998002i	$0.520124\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−31.1769	+	18.0000i	−1.25514	+	0.724653i	−0.972125	−	0.234464i	$-0.924666\pi$
$617$	−31.1769	+	18.0000i	−1.25514	+	0.724653i	−0.283011	+	0.959117i	$0.591333\pi$
$618$		0			0
$619$			22.0000i			0.884255i	0.896952	+	0.442127i	$0.145776\pi$
$619$			22.0000i			0.884255i	−0.896952	+	0.442127i	$0.854224\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	12.0000i		−	0.478471i
$630$		0			0
$631$	8.66025	−	5.00000i	0.344759	−	0.199047i	−0.317615	−	0.948220i	$-0.602882\pi$
$631$	8.66025	−	5.00000i	0.344759	−	0.199047i	0.662375	+	0.749173i	$0.269549\pi$
$632$		0			0
$633$	2.00000	+	3.46410i	0.0794929	+	0.137686i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−10.3923	−	6.00000i	−0.411113	−	0.237356i
$640$		0			0
$641$	9.00000	+	15.5885i	0.355479	+	0.615707i	0.987200	−	0.159489i	$-0.0509845\pi$
$641$	9.00000	+	15.5885i	0.355479	+	0.615707i	−0.631721	+	0.775196i	$0.717651\pi$
$642$		0			0
$643$	12.1244	−	7.00000i	0.478138	−	0.276053i	−0.241502	−	0.970400i	$-0.577640\pi$
$643$	12.1244	−	7.00000i	0.478138	−	0.276053i	0.719640	+	0.694347i	$0.244307\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−12.0000	+	20.7846i	−0.471769	+	0.817127i	−0.999478	−	0.0322975i	$-0.989718\pi$
$647$	−12.0000	+	20.7846i	−0.471769	+	0.817127i	0.527710	+	0.849425i	$0.323051\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	4.00000			0.156772
$652$		0			0
$653$	15.0000	−	25.9808i	0.586995	−	1.01671i	−0.407628	−	0.913148i	$-0.633644\pi$
$653$	15.0000	−	25.9808i	0.586995	−	1.01671i	0.994623	−	0.103558i	$-0.0330227\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−12.1244	+	7.00000i	−0.473016	+	0.273096i
$658$		0			0
$659$	18.0000	+	31.1769i	0.701180	+	1.21448i	0.968052	+	0.250748i	$0.0806766\pi$
$659$	18.0000	+	31.1769i	0.701180	+	1.21448i	−0.266872	+	0.963732i	$0.585990\pi$
$660$		0			0
$661$	1.73205	+	1.00000i	0.0673690	+	0.0388955i	0.533306	−	0.845922i	$-0.320949\pi$
$661$	1.73205	+	1.00000i	0.0673690	+	0.0388955i	−0.465937	+	0.884818i	$0.654283\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	22.5167	−	13.0000i	0.870544	−	0.502609i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−5.00000	+	8.66025i	−0.192736	+	0.333828i	−0.946156	−	0.323711i	$-0.895069\pi$
$673$	−5.00000	+	8.66025i	−0.192736	+	0.333828i	0.753420	+	0.657539i	$0.228403\pi$
$674$		0			0
$675$	5.00000			0.192450
$676$		0			0
$677$	6.00000			0.230599			0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000			0.230599			0.115299	+	0.993331i	$0.463217\pi$
$678$		0			0
$679$	10.0000	−	17.3205i	0.383765	−	0.664700i
$680$		0			0
$681$			24.0000i			0.919682i
$682$		0			0
$683$	20.7846	−	12.0000i	0.795301	−	0.459167i	−0.0465244	−	0.998917i	$-0.514815\pi$
$683$	20.7846	−	12.0000i	0.795301	−	0.459167i	0.841825	+	0.539750i	$0.181481\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	12.1244	+	7.00000i	0.462573	+	0.267067i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−12.1244	−	7.00000i	−0.461232	−	0.266293i	0.251330	−	0.967901i	$-0.419132\pi$
$691$	−12.1244	−	7.00000i	−0.461232	−	0.266293i	−0.712562	+	0.701609i	$0.752465\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	72.0000i		−	2.72719i
$698$		0			0
$699$	3.00000	−	5.19615i	0.113470	−	0.196537i
$700$		0			0
$701$	42.0000			1.58632			0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000			1.58632			0.793159	+	0.609015i	$0.208435\pi$
$702$		0			0
$703$	4.00000			0.150863
$704$		0			0
$705$		0			0
$706$		0			0
$707$			36.0000i			1.35392i
$708$		0			0
$709$	8.66025	−	5.00000i	0.325243	−	0.187779i	−0.328484	−	0.944509i	$-0.606538\pi$
$709$	8.66025	−	5.00000i	0.325243	−	0.187779i	0.653727	+	0.756730i	$0.273204\pi$
$710$		0			0
$711$	−4.00000	−	6.92820i	−0.150012	−	0.259828i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	10.3923	+	6.00000i	0.388108	+	0.224074i
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	−27.7128	+	16.0000i	−1.03208	+	0.595871i
$722$		0			0
$723$			10.0000i			0.371904i
$724$		0			0
$725$	15.0000	−	25.9808i	0.557086	−	0.964901i
$726$		0			0
$727$	−44.0000			−1.63187			−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000			−1.63187			−0.815935	+	0.578144i	$0.803777\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−12.0000	+	20.7846i	−0.443836	+	0.768747i
$732$		0			0
$733$			38.0000i			1.40356i	0.712393	+	0.701781i	$0.247612\pi$
$733$			38.0000i			1.40356i	−0.712393	+	0.701781i	$0.752388\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.73205	+	1.00000i	0.0637145	+	0.0367856i	0.531519	−	0.847046i	$-0.321621\pi$
$739$	1.73205	+	1.00000i	0.0637145	+	0.0367856i	−0.467804	+	0.883832i	$0.654955\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.3923	−	6.00000i	−0.381257	−	0.220119i	0.297108	−	0.954844i	$-0.403978\pi$
$743$	−10.3923	−	6.00000i	−0.381257	−	0.220119i	−0.678365	+	0.734725i	$0.737311\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	10.3923	−	6.00000i	0.380235	−	0.219529i
$748$		0			0
$749$			24.0000i			0.876941i
$750$		0			0
$751$	16.0000	−	27.7128i	0.583848	−	1.01125i	−0.411170	−	0.911559i	$-0.634880\pi$
$751$	16.0000	−	27.7128i	0.583848	−	1.01125i	0.995018	−	0.0996961i	$-0.0317870\pi$
$752$		0			0
$753$	−12.0000			−0.437304
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.00000	+	1.73205i	−0.0363456	+	0.0629525i	−0.883626	−	0.468193i	$-0.844905\pi$
$757$	−1.00000	+	1.73205i	−0.0363456	+	0.0629525i	0.847280	+	0.531146i	$0.178238\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	10.3923	−	6.00000i	0.376721	−	0.217500i	−0.299670	−	0.954043i	$-0.596877\pi$
$761$	10.3923	−	6.00000i	0.376721	−	0.217500i	0.676391	+	0.736543i	$0.263543\pi$
$762$		0			0
$763$	−14.0000	−	24.2487i	−0.506834	−	0.877862i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	19.0526	+	11.0000i	0.687053	+	0.396670i	0.802507	−	0.596643i	$-0.203499\pi$
$769$	19.0526	+	11.0000i	0.687053	+	0.396670i	−0.115454	+	0.993313i	$0.536832\pi$
$770$		0			0
$771$	3.00000	+	5.19615i	0.108042	+	0.187135i
$772$		0			0
$773$	10.3923	−	6.00000i	0.373785	−	0.215805i	−0.301326	−	0.953521i	$-0.597429\pi$
$773$	10.3923	−	6.00000i	0.373785	−	0.215805i	0.675111	+	0.737716i	$0.264096\pi$
$774$		0			0
$775$			10.0000i			0.359211i
$776$		0			0
$777$	2.00000	−	3.46410i	0.0717496	−	0.124274i
$778$		0			0
$779$	24.0000			0.859889
$780$		0			0
$781$		0			0
$782$		0			0
$783$	3.00000	−	5.19615i	0.107211	−	0.185695i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−22.5167	+	13.0000i	−0.802632	+	0.463400i	−0.844391	−	0.535728i	$-0.820037\pi$
$787$	−22.5167	+	13.0000i	−0.802632	+	0.463400i	0.0417585	+	0.999128i	$0.486704\pi$
$788$		0			0
$789$	12.0000	+	20.7846i	0.427211	+	0.739952i
$790$		0			0
$791$	10.3923	+	6.00000i	0.369508	+	0.213335i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.00000	−	15.5885i	−0.318796	−	0.552171i	0.661441	−	0.749997i	$-0.269945\pi$
$797$	−9.00000	−	15.5885i	−0.318796	−	0.552171i	−0.980237	+	0.197826i	$0.936612\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	18.0000			0.633630
$808$		0			0
$809$	−3.00000	+	5.19615i	−0.105474	+	0.182687i	−0.913932	−	0.405868i	$-0.866969\pi$
$809$	−3.00000	+	5.19615i	−0.105474	+	0.182687i	0.808458	+	0.588555i	$0.200303\pi$
$810$		0			0
$811$		−	22.0000i		−	0.772524i	−0.922389	−	0.386262i	$-0.873766\pi$
$811$		−	22.0000i		−	0.772524i	0.922389	−	0.386262i	$-0.126234\pi$
$812$		0			0
$813$	−22.5167	+	13.0000i	−0.789694	+	0.455930i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−6.92820	−	4.00000i	−0.242387	−	0.139942i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	10.3923	+	6.00000i	0.362694	+	0.209401i	0.670262	−	0.742125i	$-0.266182\pi$
$821$	10.3923	+	6.00000i	0.362694	+	0.209401i	−0.307568	+	0.951526i	$0.599515\pi$
$822$		0			0
$823$	−20.0000	−	34.6410i	−0.697156	−	1.20751i	−0.969448	−	0.245295i	$-0.921115\pi$
$823$	−20.0000	−	34.6410i	−0.697156	−	1.20751i	0.272292	−	0.962215i	$-0.412218\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	48.0000i		−	1.66912i	−0.550914	−	0.834562i	$-0.685721\pi$
$827$		−	48.0000i		−	1.66912i	0.550914	−	0.834562i	$-0.314279\pi$
$828$		0			0
$829$	1.00000	−	1.73205i	0.0347314	−	0.0601566i	−0.848137	−	0.529777i	$-0.822276\pi$
$829$	1.00000	−	1.73205i	0.0347314	−	0.0601566i	0.882869	+	0.469620i	$0.155609\pi$
$830$		0			0
$831$	10.0000			0.346896
$832$		0			0
$833$	18.0000			0.623663
$834$		0			0
$835$		0			0
$836$		0			0
$837$			2.00000i			0.0691301i
$838$		0			0
$839$	−31.1769	+	18.0000i	−1.07635	+	0.621429i	−0.929909	−	0.367791i	$-0.880114\pi$
$839$	−31.1769	+	18.0000i	−1.07635	+	0.621429i	−0.146438	+	0.989220i	$0.546781\pi$
$840$		0			0
$841$	−3.50000	−	6.06218i	−0.120690	−	0.209041i
$842$		0			0
$843$	−10.3923	−	6.00000i	−0.357930	−	0.206651i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	19.0526	+	11.0000i	0.654654	+	0.377964i
$848$		0			0
$849$	−8.00000	−	13.8564i	−0.274559	−	0.475551i
$850$		0			0
$851$		0			0
$852$		0			0
$853$			22.0000i			0.753266i	0.926363	+	0.376633i	$0.122918\pi$
$853$			22.0000i			0.753266i	−0.926363	+	0.376633i	$0.877082\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	42.0000			1.43469			0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000			1.43469			0.717346	+	0.696717i	$0.245357\pi$
$858$		0			0
$859$	44.0000			1.50126			0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000			1.50126			0.750630	+	0.660722i	$0.229750\pi$
$860$		0			0
$861$	12.0000	−	20.7846i	0.408959	−	0.708338i
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−9.50000	−	16.4545i	−0.322637	−	0.558824i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	8.66025	+	5.00000i	0.293105	+	0.169224i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	43.3013	−	25.0000i	1.46218	−	0.844190i	0.463068	−	0.886323i	$-0.346749\pi$
$877$	43.3013	−	25.0000i	1.46218	−	0.844190i	0.999112	+	0.0421327i	$0.0134152\pi$
$878$		0			0
$879$			12.0000i			0.404750i
$880$		0			0
$881$	−21.0000	+	36.3731i	−0.707508	+	1.22544i	0.258271	+	0.966073i	$0.416847\pi$
$881$	−21.0000	+	36.3731i	−0.707508	+	1.22544i	−0.965779	+	0.259367i	$0.916486\pi$
$882$		0			0
$883$	16.0000			0.538443			0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000			0.538443			0.269221	+	0.963078i	$0.413234\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12.0000	+	20.7846i	−0.402921	+	0.697879i	−0.994077	−	0.108678i	$-0.965338\pi$
$887$	−12.0000	+	20.7846i	−0.402921	+	0.697879i	0.591156	+	0.806557i	$0.298672\pi$
$888$		0			0
$889$		−	8.00000i		−	0.268311i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	10.3923	+	6.00000i	0.346603	+	0.200111i
$900$		0			0
$901$	−18.0000	−	31.1769i	−0.599667	−	1.03865i
$902$		0			0
$903$	−6.92820	+	4.00000i	−0.230556	+	0.133112i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	4.00000	−	6.92820i	0.132818	−	0.230047i	−0.791944	−	0.610594i	$-0.790931\pi$
$907$	4.00000	−	6.92820i	0.132818	−	0.230047i	0.924762	+	0.380547i	$0.124264\pi$
$908$		0			0
$909$	−18.0000			−0.597022
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−20.7846	+	12.0000i	−0.686368	+	0.396275i
$918$		0			0
$919$	8.00000	+	13.8564i	0.263896	+	0.457081i	0.967274	−	0.253735i	$-0.0816592\pi$
$919$	8.00000	+	13.8564i	0.263896	+	0.457081i	−0.703378	+	0.710816i	$0.748326\pi$
$920$		0			0
$921$	1.73205	+	1.00000i	0.0570730	+	0.0329511i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	8.66025	+	5.00000i	0.284747	+	0.164399i
$926$		0			0
$927$	−8.00000	−	13.8564i	−0.262754	−	0.455104i
$928$		0			0
$929$	−31.1769	+	18.0000i	−1.02288	+	0.590561i	−0.914937	−	0.403596i	$-0.867760\pi$
$929$	−31.1769	+	18.0000i	−1.02288	+	0.590561i	−0.107944	+	0.994157i	$0.534427\pi$
$930$		0			0
$931$			6.00000i			0.196642i
$932$		0			0
$933$	12.0000	−	20.7846i	0.392862	−	0.680458i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$	11.0000	−	19.0526i	0.358971	−	0.621757i
$940$		0			0
$941$		−	12.0000i		−	0.391189i	−0.980685	−	0.195594i	$-0.937336\pi$
$941$		−	12.0000i		−	0.391189i	0.980685	−	0.195594i	$-0.0626636\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−31.1769	−	18.0000i	−1.01311	−	0.584921i	−0.101012	−	0.994885i	$-0.532208\pi$
$947$	−31.1769	−	18.0000i	−1.01311	−	0.584921i	−0.912102	+	0.409964i	$0.865541\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	20.7846	+	12.0000i	0.673987	+	0.389127i
$952$		0			0
$953$	−9.00000	−	15.5885i	−0.291539	−	0.504960i	0.682635	−	0.730759i	$-0.260834\pi$
$953$	−9.00000	−	15.5885i	−0.291539	−	0.504960i	−0.974174	+	0.225800i	$0.927501\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−12.0000	+	20.7846i	−0.387500	+	0.671170i
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$	−12.0000			−0.386695
$964$		0			0
$965$		0			0
$966$		0			0
$967$			50.0000i			1.60789i	0.594703	+	0.803946i	$0.297270\pi$
$967$			50.0000i			1.60789i	−0.594703	+	0.803946i	$0.702730\pi$
$968$		0			0
$969$	10.3923	−	6.00000i	0.333849	−	0.192748i
$970$		0			0
$971$	−6.00000	−	10.3923i	−0.192549	−	0.333505i	0.753545	−	0.657396i	$-0.228342\pi$
$971$	−6.00000	−	10.3923i	−0.192549	−	0.333505i	−0.946094	+	0.323891i	$0.895009\pi$
$972$		0			0
$973$	−27.7128	−	16.0000i	−0.888432	−	0.512936i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−20.7846	−	12.0000i	−0.664959	−	0.383914i	0.129205	−	0.991618i	$-0.458757\pi$
$977$	−20.7846	−	12.0000i	−0.664959	−	0.383914i	−0.794164	+	0.607704i	$0.792091\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	12.1244	−	7.00000i	0.387101	−	0.223493i
$982$		0			0
$983$		−	60.0000i		−	1.91370i	−0.290578	−	0.956851i	$-0.593847\pi$
$983$		−	60.0000i		−	1.91370i	0.290578	−	0.956851i	$-0.406153\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	26.0000	−	45.0333i	0.825917	−	1.43053i	−0.0752991	−	0.997161i	$-0.523991\pi$
$991$	26.0000	−	45.0333i	0.825917	−	1.43053i	0.901216	−	0.433370i	$-0.142676\pi$
$992$		0			0
$993$		−	10.0000i		−	0.317340i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.00000	+	8.66025i	0.158352	+	0.274273i	0.934274	−	0.356555i	$-0.116049\pi$
$997$	5.00000	+	8.66025i	0.158352	+	0.274273i	−0.775923	+	0.630828i	$0.782715\pi$
$998$		0			0
$999$	1.73205	+	1.00000i	0.0547997	+	0.0316386i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.2.q.d.1837.1		4
13.2	odd	12		2028.2.i.b.2005.1		2
13.3	even	3	inner	2028.2.q.d.361.2		4
13.4	even	6		2028.2.b.d.337.2		2
13.5	odd	4		2028.2.i.b.529.1		2
13.6	odd	12		156.2.a.b.1.1	✓	1
13.7	odd	12		2028.2.a.e.1.1		1
13.8	odd	4		2028.2.i.c.529.1		2
13.9	even	3		2028.2.b.d.337.1		2
13.10	even	6	inner	2028.2.q.d.361.1		4
13.11	odd	12		2028.2.i.c.2005.1		2
13.12	even	2	inner	2028.2.q.d.1837.2		4
39.17	odd	6		6084.2.b.a.4393.2		2
39.20	even	12		6084.2.a.h.1.1		1
39.32	even	12		468.2.a.c.1.1		1
39.35	odd	6		6084.2.b.a.4393.1		2
52.7	even	12		8112.2.a.i.1.1		1
52.19	even	12		624.2.a.b.1.1		1
65.19	odd	12		3900.2.a.a.1.1		1
65.32	even	12		3900.2.h.e.1249.1		2
65.58	even	12		3900.2.h.e.1249.2		2
91.6	even	12		7644.2.a.a.1.1		1
104.19	even	12		2496.2.a.v.1.1		1
104.45	odd	12		2496.2.a.h.1.1		1
117.32	even	12		4212.2.i.g.2809.1		2
117.58	odd	12		4212.2.i.f.2809.1		2
117.97	odd	12		4212.2.i.f.1405.1		2
117.110	even	12		4212.2.i.g.1405.1		2
156.71	odd	12		1872.2.a.i.1.1		1
312.149	even	12		7488.2.a.bf.1.1		1
312.227	odd	12		7488.2.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.a.b.1.1	✓	1	13.6	odd	12
468.2.a.c.1.1		1	39.32	even	12
624.2.a.b.1.1		1	52.19	even	12
1872.2.a.i.1.1		1	156.71	odd	12
2028.2.a.e.1.1		1	13.7	odd	12
2028.2.b.d.337.1		2	13.9	even	3
2028.2.b.d.337.2		2	13.4	even	6
2028.2.i.b.529.1		2	13.5	odd	4
2028.2.i.b.2005.1		2	13.2	odd	12
2028.2.i.c.529.1		2	13.8	odd	4
2028.2.i.c.2005.1		2	13.11	odd	12
2028.2.q.d.361.1		4	13.10	even	6	inner
2028.2.q.d.361.2		4	13.3	even	3	inner
2028.2.q.d.1837.1		4	1.1	even	1	trivial
2028.2.q.d.1837.2		4	13.12	even	2	inner
2496.2.a.h.1.1		1	104.45	odd	12
2496.2.a.v.1.1		1	104.19	even	12
3900.2.a.a.1.1		1	65.19	odd	12
3900.2.h.e.1249.1		2	65.32	even	12
3900.2.h.e.1249.2		2	65.58	even	12
4212.2.i.f.1405.1		2	117.97	odd	12
4212.2.i.f.2809.1		2	117.58	odd	12
4212.2.i.g.1405.1		2	117.110	even	12
4212.2.i.g.2809.1		2	117.32	even	12
6084.2.a.h.1.1		1	39.20	even	12
6084.2.b.a.4393.1		2	39.35	odd	6
6084.2.b.a.4393.2		2	39.17	odd	6
7488.2.a.bb.1.1		1	312.227	odd	12
7488.2.a.bf.1.1		1	312.149	even	12
7644.2.a.a.1.1		1	91.6	even	12
8112.2.a.i.1.1		1	52.7	even	12

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:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2028.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-1.73205 + 1.00000i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-1.73205 + 1.00000i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-3.00000 - 5.19615i) q^{17} +(1.73205 - 1.00000i) q^{19} -2.00000i q^{21} +5.00000 q^{25} +1.00000 q^{27} +(3.00000 - 5.19615i) q^{29} +2.00000i q^{31} +(1.73205 + 1.00000i) q^{37} +(10.3923 + 6.00000i) q^{41} +(-2.00000 - 3.46410i) q^{43} +(-1.50000 + 2.59808i) q^{49} +6.00000 q^{51} +6.00000 q^{53} +2.00000i q^{57} +(-10.3923 + 6.00000i) q^{59} +(-1.00000 - 1.73205i) q^{61} +(1.73205 + 1.00000i) q^{63} +(8.66025 + 5.00000i) q^{67} +(10.3923 - 6.00000i) q^{71} -14.0000i q^{73} +(-2.50000 + 4.33013i) q^{75} +8.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} +12.0000i q^{83} +(3.00000 + 5.19615i) q^{87} +(-1.73205 - 1.00000i) q^{93} +(-8.66025 + 5.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 2 * q^9	\( 4 q - 2 q^{3} - 2 q^{9} - 12 q^{17} + 20 q^{25} + 4 q^{27} + 12 q^{29} - 8 q^{43} - 6 q^{49} + 24 q^{51} + 24 q^{53} - 4 q^{61} - 10 q^{75} + 32 q^{79} - 2 q^{81} + 12 q^{87}+O(q^{100}) \) 4 * q - 2 * q^3 - 2 * q^9 - 12 * q^17 + 20 * q^25 + 4 * q^27 + 12 * q^29 - 8 * q^43 - 6 * q^49 + 24 * q^51 + 24 * q^53 - 4 * q^61 - 10 * q^75 + 32 * q^79 - 2 * q^81 + 12 * q^87

Introduction

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