LMFDB - Embedded newform 2448.2.be.u.1585.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2448,2,Mod(1441,2448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2448, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("2448.1441");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2448 = 2^{4} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2448.be (of order $4$, 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:	$19.5473784148$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 68)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1585.1
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	2448.1585
Dual form			2448.2.be.u.1441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{5} +(-2.30278 - 2.30278i) q^{7} +O(q^{10})$	$q+(1.00000 - 1.00000i) q^{5} +(-2.30278 - 2.30278i) q^{7} +(-0.302776 - 0.302776i) q^{11} +2.60555 q^{13} +(-3.60555 - 2.00000i) q^{17} -0.605551i q^{19} +(-4.30278 - 4.30278i) q^{23} +3.00000i q^{25} +(1.60555 - 1.60555i) q^{29} +(-4.30278 + 4.30278i) q^{31} -4.60555 q^{35} +(3.00000 - 3.00000i) q^{37} +(-1.00000 - 1.00000i) q^{41} +3.39445i q^{43} -4.00000 q^{47} +3.60555i q^{49} +5.21110i q^{53} -0.605551 q^{55} -8.60555i q^{59} +(-6.21110 - 6.21110i) q^{61} +(2.60555 - 2.60555i) q^{65} -9.21110 q^{67} +(2.90833 - 2.90833i) q^{71} +(-7.00000 + 7.00000i) q^{73} +1.39445i q^{77} +(0.302776 + 0.302776i) q^{79} +17.8167i q^{83} +(-5.60555 + 1.60555i) q^{85} -7.81665 q^{89} +(-6.00000 - 6.00000i) q^{91} +(-0.605551 - 0.605551i) q^{95} +(-7.60555 + 7.60555i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 2 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 - 2 * q^7	$ 4 q + 4 q^{5} - 2 q^{7} + 6 q^{11} - 4 q^{13} - 10 q^{23} - 8 q^{29} - 10 q^{31} - 4 q^{35} + 12 q^{37} - 4 q^{41} - 16 q^{47} + 12 q^{55} + 4 q^{61} - 4 q^{65} - 8 q^{67} - 10 q^{71} - 28 q^{73} - 6 q^{79} - 8 q^{85} + 12 q^{89} - 24 q^{91} + 12 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 2 * q^7 + 6 * q^11 - 4 * q^13 - 10 * q^23 - 8 * q^29 - 10 * q^31 - 4 * q^35 + 12 * q^37 - 4 * q^41 - 16 * q^47 + 12 * q^55 + 4 * q^61 - 4 * q^65 - 8 * q^67 - 10 * q^71 - 28 * q^73 - 6 * q^79 - 8 * q^85 + 12 * q^89 - 24 * q^91 + 12 * q^95 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2448\mathbb{Z}\right)^\times$.

$n$	$613$	$1361$	$1873$	$2143$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.00000	−	1.00000i	0.447214	−	0.447214i	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	1.00000	−	1.00000i	0.447214	−	0.447214i	0.894427	+	0.447214i	$0.147584\pi$
$6$		0			0
$7$	−2.30278	−	2.30278i	−0.870367	−	0.870367i	0.122145	−	0.992512i	$-0.461023\pi$
$7$	−2.30278	−	2.30278i	−0.870367	−	0.870367i	−0.992512	+	0.122145i	$0.961023\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.302776	−	0.302776i	−0.0912903	−	0.0912903i	0.659987	−	0.751277i	$-0.270562\pi$
$11$	−0.302776	−	0.302776i	−0.0912903	−	0.0912903i	−0.751277	+	0.659987i	$0.770562\pi$
$12$		0			0
$13$	2.60555			0.722650			0.361325	−	0.932440i	$-0.382325\pi$
$13$	2.60555			0.722650			0.361325	+	0.932440i	$0.382325\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.60555	−	2.00000i	−0.874475	−	0.485071i
$17$	−3.60555	−	2.00000i	−0.874475	−	0.485071i
$18$		0			0
$19$		−	0.605551i		−	0.138923i	−0.997585	−	0.0694615i	$-0.977872\pi$
$19$		−	0.605551i		−	0.138923i	0.997585	−	0.0694615i	$-0.0221281\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.30278	−	4.30278i	−0.897191	−	0.897191i	0.0979961	−	0.995187i	$-0.468757\pi$
$23$	−4.30278	−	4.30278i	−0.897191	−	0.897191i	−0.995187	+	0.0979961i	$0.968757\pi$
$24$		0			0
$25$			3.00000i			0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.60555	−	1.60555i	0.298143	−	0.298143i	−0.542143	−	0.840286i	$-0.682387\pi$
$29$	1.60555	−	1.60555i	0.298143	−	0.298143i	0.840286	+	0.542143i	$0.182387\pi$
$30$		0			0
$31$	−4.30278	+	4.30278i	−0.772801	+	0.772801i	−0.978595	−	0.205794i	$-0.934022\pi$
$31$	−4.30278	+	4.30278i	−0.772801	+	0.772801i	0.205794	+	0.978595i	$0.434022\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.60555			−0.778480
$36$		0			0
$37$	3.00000	−	3.00000i	0.493197	−	0.493197i	−0.416115	−	0.909312i	$-0.636609\pi$
$37$	3.00000	−	3.00000i	0.493197	−	0.493197i	0.909312	+	0.416115i	$0.136609\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.00000	−	1.00000i	−0.156174	−	0.156174i	0.624695	−	0.780869i	$-0.285223\pi$
$41$	−1.00000	−	1.00000i	−0.156174	−	0.156174i	−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$			3.39445i			0.517649i	0.965924	+	0.258824i	$0.0833351\pi$
$43$			3.39445i			0.517649i	−0.965924	+	0.258824i	$0.916665\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.00000			−0.583460			−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000			−0.583460			−0.291730	+	0.956501i	$0.594231\pi$
$48$		0			0
$49$			3.60555i			0.515079i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			5.21110i			0.715800i	0.933760	+	0.357900i	$0.116507\pi$
$53$			5.21110i			0.715800i	−0.933760	+	0.357900i	$0.883493\pi$
$54$		0			0
$55$	−0.605551			−0.0816525
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	8.60555i		−	1.12035i	−0.828375	−	0.560174i	$-0.810734\pi$
$59$		−	8.60555i		−	1.12035i	0.828375	−	0.560174i	$-0.189266\pi$
$60$		0			0
$61$	−6.21110	−	6.21110i	−0.795250	−	0.795250i	0.187092	−	0.982342i	$-0.440094\pi$
$61$	−6.21110	−	6.21110i	−0.795250	−	0.795250i	−0.982342	+	0.187092i	$0.940094\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.60555	−	2.60555i	0.323179	−	0.323179i
$66$		0			0
$67$	−9.21110			−1.12532			−0.562658	−	0.826690i	$-0.690221\pi$
$67$	−9.21110			−1.12532			−0.562658	+	0.826690i	$0.690221\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.90833	−	2.90833i	0.345155	−	0.345155i	−0.513146	−	0.858301i	$-0.671520\pi$
$71$	2.90833	−	2.90833i	0.345155	−	0.345155i	0.858301	+	0.513146i	$0.171520\pi$
$72$		0			0
$73$	−7.00000	+	7.00000i	−0.819288	+	0.819288i	−0.986005	−	0.166717i	$-0.946683\pi$
$73$	−7.00000	+	7.00000i	−0.819288	+	0.819288i	0.166717	+	0.986005i	$0.446683\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			1.39445i			0.158912i
$78$		0			0
$79$	0.302776	+	0.302776i	0.0340649	+	0.0340649i	0.723934	−	0.689869i	$-0.242332\pi$
$79$	0.302776	+	0.302776i	0.0340649	+	0.0340649i	−0.689869	+	0.723934i	$0.742332\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			17.8167i			1.95563i	0.209466	+	0.977816i	$0.432827\pi$
$83$			17.8167i			1.95563i	−0.209466	+	0.977816i	$0.567173\pi$
$84$		0			0
$85$	−5.60555	+	1.60555i	−0.608007	+	0.174146i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.81665			−0.828564			−0.414282	−	0.910149i	$-0.635967\pi$
$89$	−7.81665			−0.828564			−0.414282	+	0.910149i	$0.635967\pi$
$90$		0			0
$91$	−6.00000	−	6.00000i	−0.628971	−	0.628971i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.605551	−	0.605551i	−0.0621282	−	0.0621282i
$96$		0			0
$97$	−7.60555	+	7.60555i	−0.772227	+	0.772227i	−0.978495	−	0.206269i	$-0.933868\pi$
$97$	−7.60555	+	7.60555i	−0.772227	+	0.772227i	0.206269	+	0.978495i	$0.433868\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.6056			−1.05529			−0.527646	−	0.849464i	$-0.676925\pi$
$101$	−10.6056			−1.05529			−0.527646	+	0.849464i	$0.676925\pi$
$102$		0			0
$103$	13.2111			1.30173			0.650864	−	0.759194i	$-0.274407\pi$
$103$	13.2111			1.30173			0.650864	+	0.759194i	$0.274407\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.69722	−	5.69722i	0.550771	−	0.550771i	−0.375892	−	0.926663i	$-0.622664\pi$
$107$	5.69722	−	5.69722i	0.550771	−	0.550771i	0.926663	+	0.375892i	$0.122664\pi$
$108$		0			0
$109$	−6.21110	−	6.21110i	−0.594916	−	0.594916i	0.344039	−	0.938955i	$-0.388205\pi$
$109$	−6.21110	−	6.21110i	−0.594916	−	0.594916i	−0.938955	+	0.344039i	$0.888205\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.60555	−	9.60555i	−0.903614	−	0.903614i	0.0921325	−	0.995747i	$-0.470632\pi$
$113$	−9.60555	−	9.60555i	−0.903614	−	0.903614i	−0.995747	+	0.0921325i	$0.970632\pi$
$114$		0			0
$115$	−8.60555			−0.802472
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.69722	+	12.9083i	0.338924	+	1.18330i
$120$		0			0
$121$		−	10.8167i		−	0.983332i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.00000	+	8.00000i	0.715542	+	0.715542i
$126$		0			0
$127$			0.605551i			0.0537340i	0.999639	+	0.0268670i	$0.00855306\pi$
$127$			0.605551i			0.0537340i	−0.999639	+	0.0268670i	$0.991447\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.30278	+	6.30278i	−0.550676	+	0.550676i	−0.926636	−	0.375960i	$-0.877313\pi$
$131$	−6.30278	+	6.30278i	−0.550676	+	0.550676i	0.375960	+	0.926636i	$0.377313\pi$
$132$		0			0
$133$	−1.39445	+	1.39445i	−0.120914	+	0.120914i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.60555			0.222607			0.111304	−	0.993786i	$-0.464497\pi$
$137$	2.60555			0.222607			0.111304	+	0.993786i	$0.464497\pi$
$138$		0			0
$139$	−0.302776	+	0.302776i	−0.0256811	+	0.0256811i	−0.719831	−	0.694150i	$-0.755781\pi$
$139$	−0.302776	+	0.302776i	−0.0256811	+	0.0256811i	0.694150	+	0.719831i	$0.255781\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.788897	−	0.788897i	−0.0659709	−	0.0659709i
$144$		0			0
$145$		−	3.21110i		−	0.266668i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.42221			−0.689974			−0.344987	−	0.938607i	$-0.612117\pi$
$149$	−8.42221			−0.689974			−0.344987	+	0.938607i	$0.612117\pi$
$150$		0			0
$151$		−	4.60555i		−	0.374794i	−0.982284	−	0.187397i	$-0.939995\pi$
$151$		−	4.60555i		−	0.374794i	0.982284	−	0.187397i	$-0.0600052\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			8.60555i			0.691215i
$156$		0			0
$157$	−8.42221			−0.672165			−0.336083	−	0.941833i	$-0.609102\pi$
$157$	−8.42221			−0.672165			−0.336083	+	0.941833i	$0.609102\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			19.8167i			1.56177i
$162$		0			0
$163$	4.30278	+	4.30278i	0.337019	+	0.337019i	0.855244	−	0.518225i	$-0.173407\pi$
$163$	4.30278	+	4.30278i	0.337019	+	0.337019i	−0.518225	+	0.855244i	$0.673407\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	14.9083	−	14.9083i	1.15364	−	1.15364i	0.167824	−	0.985817i	$-0.446326\pi$
$167$	14.9083	−	14.9083i	1.15364	−	1.15364i	0.985817	−	0.167824i	$-0.0536740\pi$
$168$		0			0
$169$	−6.21110			−0.477777
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14.8167	−	14.8167i	1.12649	−	1.12649i	0.135746	−	0.990744i	$-0.456657\pi$
$173$	14.8167	−	14.8167i	1.12649	−	1.12649i	0.990744	−	0.135746i	$-0.0433430\pi$
$174$		0			0
$175$	6.90833	−	6.90833i	0.522220	−	0.522220i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	15.0278i		−	1.12323i	−0.827400	−	0.561614i	$-0.810181\pi$
$179$		−	15.0278i		−	1.12323i	0.827400	−	0.561614i	$-0.189819\pi$
$180$		0			0
$181$	−13.0000	−	13.0000i	−0.966282	−	0.966282i	0.0331674	−	0.999450i	$-0.489441\pi$
$181$	−13.0000	−	13.0000i	−0.966282	−	0.966282i	−0.999450	+	0.0331674i	$0.989441\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	6.00000i		−	0.441129i
$186$		0			0
$187$	0.486122	+	1.69722i	0.0355487	+	0.124113i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.78890			0.491227			0.245614	−	0.969368i	$-0.421011\pi$
$191$	6.78890			0.491227			0.245614	+	0.969368i	$0.421011\pi$
$192$		0			0
$193$	−8.21110	−	8.21110i	−0.591048	−	0.591048i	0.346866	−	0.937915i	$-0.387246\pi$
$193$	−8.21110	−	8.21110i	−0.591048	−	0.591048i	−0.937915	+	0.346866i	$0.887246\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	17.6056	+	17.6056i	1.25434	+	1.25434i	0.953753	+	0.300590i	$0.0971836\pi$
$197$	17.6056	+	17.6056i	1.25434	+	1.25434i	0.300590	+	0.953753i	$0.402816\pi$
$198$		0			0
$199$	10.1194	−	10.1194i	0.717347	−	0.717347i	−0.250714	−	0.968061i	$-0.580665\pi$
$199$	10.1194	−	10.1194i	0.717347	−	0.717347i	0.968061	+	0.250714i	$0.0806653\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.39445			−0.518989
$204$		0			0
$205$	−2.00000			−0.139686
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.183346	+	0.183346i	−0.0126823	+	0.0126823i
$210$		0			0
$211$	−15.5139	−	15.5139i	−1.06802	−	1.06802i	−0.997511	−	0.0705082i	$-0.977538\pi$
$211$	−15.5139	−	15.5139i	−1.06802	−	1.06802i	−0.0705082	−	0.997511i	$-0.522462\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.39445	+	3.39445i	0.231499	+	0.231499i
$216$		0			0
$217$	19.8167			1.34524
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−9.39445	−	5.21110i	−0.631939	−	0.350537i
$222$		0			0
$223$			17.8167i			1.19309i	0.802579	+	0.596546i	$0.203461\pi$
$223$			17.8167i			1.19309i	−0.802579	+	0.596546i	$0.796539\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.302776	−	0.302776i	−0.0200959	−	0.0200959i	0.696987	−	0.717083i	$-0.254523\pi$
$227$	−0.302776	−	0.302776i	−0.0200959	−	0.0200959i	−0.717083	+	0.696987i	$0.754523\pi$
$228$		0			0
$229$			4.60555i			0.304343i	0.988354	+	0.152172i	$0.0486267\pi$
$229$			4.60555i			0.304343i	−0.988354	+	0.152172i	$0.951373\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.60555	+	5.60555i	−0.367232	+	0.367232i	−0.866467	−	0.499235i	$-0.833614\pi$
$233$	−5.60555	+	5.60555i	−0.367232	+	0.367232i	0.499235	+	0.866467i	$0.333614\pi$
$234$		0			0
$235$	−4.00000	+	4.00000i	−0.260931	+	0.260931i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−14.7889			−0.956614			−0.478307	−	0.878193i	$-0.658749\pi$
$239$	−14.7889			−0.956614			−0.478307	+	0.878193i	$0.658749\pi$
$240$		0			0
$241$	9.00000	−	9.00000i	0.579741	−	0.579741i	−0.355091	−	0.934832i	$-0.615550\pi$
$241$	9.00000	−	9.00000i	0.579741	−	0.579741i	0.934832	+	0.355091i	$0.115550\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.60555	+	3.60555i	0.230350	+	0.230350i
$246$		0			0
$247$		−	1.57779i		−	0.100393i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$			2.60555i			0.163810i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	19.3944i		−	1.20979i	−0.796304	−	0.604896i	$-0.793215\pi$
$257$		−	19.3944i		−	1.20979i	0.796304	−	0.604896i	$-0.206785\pi$
$258$		0			0
$259$	−13.8167			−0.858525
$260$		0			0
$261$		0			0
$262$		0			0
$263$			23.0278i			1.41995i	0.704226	+	0.709976i	$0.251294\pi$
$263$			23.0278i			1.41995i	−0.704226	+	0.709976i	$0.748706\pi$
$264$		0			0
$265$	5.21110	+	5.21110i	0.320115	+	0.320115i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−17.4222	+	17.4222i	−1.06225	+	1.06225i	−0.0643214	+	0.997929i	$0.520488\pi$
$269$	−17.4222	+	17.4222i	−1.06225	+	1.06225i	−0.997929	+	0.0643214i	$0.979512\pi$
$270$		0			0
$271$	−1.21110			−0.0735692			−0.0367846	−	0.999323i	$-0.511712\pi$
$271$	−1.21110			−0.0735692			−0.0367846	+	0.999323i	$0.511712\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.908327	−	0.908327i	0.0547742	−	0.0547742i
$276$		0			0
$277$	3.60555	−	3.60555i	0.216637	−	0.216637i	−0.590443	−	0.807079i	$-0.701047\pi$
$277$	3.60555	−	3.60555i	0.216637	−	0.216637i	0.807079	+	0.590443i	$0.201047\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	18.4222i		−	1.09898i	−0.835501	−	0.549488i	$-0.814823\pi$
$281$		−	18.4222i		−	1.09898i	0.835501	−	0.549488i	$-0.185177\pi$
$282$		0			0
$283$	−2.11943	−	2.11943i	−0.125987	−	0.125987i	0.641302	−	0.767289i	$-0.278395\pi$
$283$	−2.11943	−	2.11943i	−0.125987	−	0.125987i	−0.767289	+	0.641302i	$0.778395\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			4.60555i			0.271857i
$288$		0			0
$289$	9.00000	+	14.4222i	0.529412	+	0.848365i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	21.6333			1.26383			0.631916	−	0.775037i	$-0.282269\pi$
$293$	21.6333			1.26383			0.631916	+	0.775037i	$0.282269\pi$
$294$		0			0
$295$	−8.60555	−	8.60555i	−0.501035	−	0.501035i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−11.2111	−	11.2111i	−0.648355	−	0.648355i
$300$		0			0
$301$	7.81665	−	7.81665i	0.450544	−	0.450544i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.4222			−0.711293
$306$		0			0
$307$	−21.2111			−1.21058			−0.605291	−	0.796004i	$-0.706943\pi$
$307$	−21.2111			−1.21058			−0.605291	+	0.796004i	$0.706943\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	18.9083	−	18.9083i	1.07219	−	1.07219i	0.0750101	−	0.997183i	$-0.476101\pi$
$311$	18.9083	−	18.9083i	1.07219	−	1.07219i	0.997183	−	0.0750101i	$-0.0238989\pi$
$312$		0			0
$313$	6.21110	+	6.21110i	0.351072	+	0.351072i	0.860508	−	0.509436i	$-0.170146\pi$
$313$	6.21110	+	6.21110i	0.351072	+	0.351072i	−0.509436	+	0.860508i	$0.670146\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−19.6056	−	19.6056i	−1.10116	−	1.10116i	−0.994271	−	0.106886i	$-0.965912\pi$
$317$	−19.6056	−	19.6056i	−1.10116	−	1.10116i	−0.106886	−	0.994271i	$-0.534088\pi$
$318$		0			0
$319$	−0.972244			−0.0544352
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.21110	+	2.18335i	−0.0673875	+	0.121485i
$324$		0			0
$325$			7.81665i			0.433590i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.21110	+	9.21110i	0.507825	+	0.507825i
$330$		0			0
$331$			3.39445i			0.186576i	0.995639	+	0.0932879i	$0.0297377\pi$
$331$			3.39445i			0.186576i	−0.995639	+	0.0932879i	$0.970262\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−9.21110	+	9.21110i	−0.503256	+	0.503256i
$336$		0			0
$337$	−16.2111	+	16.2111i	−0.883075	+	0.883075i	−0.993846	−	0.110771i	$-0.964668\pi$
$337$	−16.2111	+	16.2111i	−0.883075	+	0.883075i	0.110771	+	0.993846i	$0.464668\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.60555			0.141099
$342$		0			0
$343$	−7.81665	+	7.81665i	−0.422060	+	0.422060i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.51388	−	5.51388i	−0.296000	−	0.296000i	0.543445	−	0.839445i	$-0.317120\pi$
$347$	−5.51388	−	5.51388i	−0.296000	−	0.296000i	−0.839445	+	0.543445i	$0.817120\pi$
$348$		0			0
$349$			23.6333i			1.26506i	0.774535	+	0.632531i	$0.217984\pi$
$349$			23.6333i			1.26506i	−0.774535	+	0.632531i	$0.782016\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	31.2111			1.66120			0.830600	−	0.556870i	$-0.187998\pi$
$353$	31.2111			1.66120			0.830600	+	0.556870i	$0.187998\pi$
$354$		0			0
$355$		−	5.81665i		−	0.308716i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			16.6056i			0.876407i	0.898876	+	0.438204i	$0.144385\pi$
$359$			16.6056i			0.876407i	−0.898876	+	0.438204i	$0.855615\pi$
$360$		0			0
$361$	18.6333			0.980700
$362$		0			0
$363$		0			0
$364$		0			0
$365$			14.0000i			0.732793i
$366$		0			0
$367$	6.90833	+	6.90833i	0.360612	+	0.360612i	0.864038	−	0.503426i	$-0.167927\pi$
$367$	6.90833	+	6.90833i	0.360612	+	0.360612i	−0.503426	+	0.864038i	$0.667927\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12.0000	−	12.0000i	0.623009	−	0.623009i
$372$		0			0
$373$	21.0278			1.08878			0.544388	−	0.838834i	$-0.316762\pi$
$373$	21.0278			1.08878			0.544388	+	0.838834i	$0.316762\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.18335	−	4.18335i	0.215453	−	0.215453i
$378$		0			0
$379$	20.7250	−	20.7250i	1.06457	−	1.06457i	0.0668047	−	0.997766i	$-0.478720\pi$
$379$	20.7250	−	20.7250i	1.06457	−	1.06457i	0.997766	−	0.0668047i	$-0.0212804\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	19.3944i		−	0.991010i	−0.868605	−	0.495505i	$-0.834983\pi$
$383$		−	19.3944i		−	0.991010i	0.868605	−	0.495505i	$-0.165017\pi$
$384$		0			0
$385$	1.39445	+	1.39445i	0.0710677	+	0.0710677i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	10.1833i		−	0.516316i	−0.966103	−	0.258158i	$-0.916884\pi$
$389$		−	10.1833i		−	0.516316i	0.966103	−	0.258158i	$-0.0831155\pi$
$390$		0			0
$391$	6.90833	+	24.1194i	0.349369	+	1.21977i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.605551			0.0304686
$396$		0			0
$397$	−9.00000	−	9.00000i	−0.451697	−	0.451697i	0.444220	−	0.895918i	$-0.353481\pi$
$397$	−9.00000	−	9.00000i	−0.451697	−	0.451697i	−0.895918	+	0.444220i	$0.853481\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.00000	+	7.00000i	0.349563	+	0.349563i	0.859947	−	0.510384i	$-0.170497\pi$
$401$	7.00000	+	7.00000i	0.349563	+	0.349563i	−0.510384	+	0.859947i	$0.670497\pi$
$402$		0			0
$403$	−11.2111	+	11.2111i	−0.558465	+	0.558465i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.81665			−0.0900482
$408$		0			0
$409$	23.2111			1.14772			0.573858	−	0.818955i	$-0.305446\pi$
$409$	23.2111			1.14772			0.573858	+	0.818955i	$0.305446\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−19.8167	+	19.8167i	−0.975114	+	0.975114i
$414$		0			0
$415$	17.8167	+	17.8167i	0.874585	+	0.874585i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.30278	+	2.30278i	0.112498	+	0.112498i	0.761115	−	0.648617i	$-0.224652\pi$
$419$	2.30278	+	2.30278i	0.112498	+	0.112498i	−0.648617	+	0.761115i	$0.724652\pi$
$420$		0			0
$421$	18.6056			0.906779			0.453390	−	0.891312i	$-0.350215\pi$
$421$	18.6056			0.906779			0.453390	+	0.891312i	$0.350215\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.00000	−	10.8167i	0.291043	−	0.524685i
$426$		0			0
$427$			28.6056i			1.38432i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.88057	−	1.88057i	−0.0905839	−	0.0905839i	0.660363	−	0.750947i	$-0.270403\pi$
$431$	−1.88057	−	1.88057i	−0.0905839	−	0.0905839i	−0.750947	+	0.660363i	$0.770403\pi$
$432$		0			0
$433$		−	15.0278i		−	0.722188i	−0.932529	−	0.361094i	$-0.882403\pi$
$433$		−	15.0278i		−	0.722188i	0.932529	−	0.361094i	$-0.117597\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.60555	+	2.60555i	−0.124640	+	0.124640i
$438$		0			0
$439$	26.3028	−	26.3028i	1.25536	−	1.25536i	0.302081	−	0.953282i	$-0.402319\pi$
$439$	26.3028	−	26.3028i	1.25536	−	1.25536i	0.953282	−	0.302081i	$-0.0976812\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.2111			−0.817724			−0.408862	−	0.912596i	$-0.634074\pi$
$443$	−17.2111			−0.817724			−0.408862	+	0.912596i	$0.634074\pi$
$444$		0			0
$445$	−7.81665	+	7.81665i	−0.370545	+	0.370545i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	16.8167	+	16.8167i	0.793627	+	0.793627i	0.982082	−	0.188455i	$-0.0603479\pi$
$449$	16.8167	+	16.8167i	0.793627	+	0.793627i	−0.188455	+	0.982082i	$0.560348\pi$
$450$		0			0
$451$			0.605551i			0.0285143i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−12.0000			−0.562569
$456$		0			0
$457$		−	36.6056i		−	1.71234i	−0.516698	−	0.856168i	$-0.672839\pi$
$457$		−	36.6056i		−	1.71234i	0.516698	−	0.856168i	$-0.327161\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	29.2111i		−	1.36050i	−0.732982	−	0.680248i	$-0.761872\pi$
$461$		−	29.2111i		−	1.36050i	0.732982	−	0.680248i	$-0.238128\pi$
$462$		0			0
$463$	39.6333			1.84192			0.920958	−	0.389662i	$-0.127408\pi$
$463$	39.6333			1.84192			0.920958	+	0.389662i	$0.127408\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			16.2389i			0.751445i	0.926732	+	0.375722i	$0.122605\pi$
$467$			16.2389i			0.751445i	−0.926732	+	0.375722i	$0.877395\pi$
$468$		0			0
$469$	21.2111	+	21.2111i	0.979438	+	0.979438i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.02776	−	1.02776i	0.0472563	−	0.0472563i
$474$		0			0
$475$	1.81665			0.0833538
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.3028	−	12.3028i	0.562128	−	0.562128i	−0.367783	−	0.929912i	$-0.619883\pi$
$479$	12.3028	−	12.3028i	0.562128	−	0.562128i	0.929912	+	0.367783i	$0.119883\pi$
$480$		0			0
$481$	7.81665	−	7.81665i	0.356409	−	0.356409i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			15.2111i			0.690701i
$486$		0			0
$487$	17.3305	+	17.3305i	0.785321	+	0.785321i	0.980723	−	0.195402i	$-0.0626011\pi$
$487$	17.3305	+	17.3305i	0.785321	+	0.785321i	−0.195402	+	0.980723i	$0.562601\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	9.81665i		−	0.443019i	−0.975158	−	0.221510i	$-0.928902\pi$
$491$		−	9.81665i		−	0.443019i	0.975158	−	0.221510i	$-0.0710984\pi$
$492$		0			0
$493$	−9.00000	+	2.57779i	−0.405340	+	0.116098i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−13.3944			−0.600823
$498$		0			0
$499$	−10.3028	−	10.3028i	−0.461216	−	0.461216i	0.437838	−	0.899054i	$-0.355744\pi$
$499$	−10.3028	−	10.3028i	−0.461216	−	0.461216i	−0.899054	+	0.437838i	$0.855744\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	27.5139	+	27.5139i	1.22678	+	1.22678i	0.965174	+	0.261609i	$0.0842532\pi$
$503$	27.5139	+	27.5139i	1.22678	+	1.22678i	0.261609	+	0.965174i	$0.415747\pi$
$504$		0			0
$505$	−10.6056	+	10.6056i	−0.471941	+	0.471941i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.21110			0.142330			0.0711648	−	0.997465i	$-0.477328\pi$
$509$	3.21110			0.142330			0.0711648	+	0.997465i	$0.477328\pi$
$510$		0			0
$511$	32.2389			1.42616
$512$		0			0
$513$		0			0
$514$		0			0
$515$	13.2111	−	13.2111i	0.582151	−	0.582151i
$516$		0			0
$517$	1.21110	+	1.21110i	0.0532642	+	0.0532642i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13.0000	−	13.0000i	−0.569540	−	0.569540i	0.362459	−	0.932000i	$-0.381937\pi$
$521$	−13.0000	−	13.0000i	−0.569540	−	0.569540i	−0.932000	+	0.362459i	$0.881937\pi$
$522$		0			0
$523$	12.0000			0.524723			0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000			0.524723			0.262362	+	0.964970i	$0.415499\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	24.1194	−	6.90833i	1.05066	−	0.300931i
$528$		0			0
$529$			14.0278i			0.609902i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.60555	−	2.60555i	−0.112859	−	0.112859i
$534$		0			0
$535$		−	11.3944i		−	0.492625i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.09167	−	1.09167i	0.0470217	−	0.0470217i
$540$		0			0
$541$	3.00000	−	3.00000i	0.128980	−	0.128980i	−0.639670	−	0.768650i	$-0.720929\pi$
$541$	3.00000	−	3.00000i	0.128980	−	0.128980i	0.768650	+	0.639670i	$0.220929\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.4222			−0.532109
$546$		0			0
$547$	11.5139	−	11.5139i	0.492298	−	0.492298i	−0.416732	−	0.909030i	$-0.636825\pi$
$547$	11.5139	−	11.5139i	0.492298	−	0.492298i	0.909030	+	0.416732i	$0.136825\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.972244	−	0.972244i	−0.0414190	−	0.0414190i
$552$		0			0
$553$		−	1.39445i		−	0.0592980i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	21.3944			0.906512			0.453256	−	0.891380i	$-0.350262\pi$
$557$	21.3944			0.906512			0.453256	+	0.891380i	$0.350262\pi$
$558$		0			0
$559$			8.84441i			0.374079i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			20.6056i			0.868420i	0.900812	+	0.434210i	$0.142972\pi$
$563$			20.6056i			0.868420i	−0.900812	+	0.434210i	$0.857028\pi$
$564$		0			0
$565$	−19.2111			−0.808217
$566$		0			0
$567$		0			0
$568$		0			0
$569$			30.4222i			1.27537i	0.770299	+	0.637683i	$0.220107\pi$
$569$			30.4222i			1.27537i	−0.770299	+	0.637683i	$0.779893\pi$
$570$		0			0
$571$	−2.30278	−	2.30278i	−0.0963682	−	0.0963682i	0.657279	−	0.753647i	$-0.271707\pi$
$571$	−2.30278	−	2.30278i	−0.0963682	−	0.0963682i	−0.753647	+	0.657279i	$0.771707\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	12.9083	−	12.9083i	0.538314	−	0.538314i
$576$		0			0
$577$	−31.4500			−1.30928			−0.654640	−	0.755941i	$-0.727180\pi$
$577$	−31.4500			−1.30928			−0.654640	+	0.755941i	$0.727180\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	41.0278	−	41.0278i	1.70212	−	1.70212i
$582$		0			0
$583$	1.57779	−	1.57779i	0.0653456	−	0.0653456i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	4.60555i		−	0.190091i	−0.995473	−	0.0950457i	$-0.969700\pi$
$587$		−	4.60555i		−	0.190091i	0.995473	−	0.0950457i	$-0.0302997\pi$
$588$		0			0
$589$	2.60555	+	2.60555i	0.107360	+	0.107360i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	1.57779i		−	0.0647923i	−0.999475	−	0.0323961i	$-0.989686\pi$
$593$		−	1.57779i		−	0.0647923i	0.999475	−	0.0323961i	$-0.0103138\pi$
$594$		0			0
$595$	16.6056	+	9.21110i	0.680761	+	0.377618i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	26.0555			1.06460			0.532300	−	0.846556i	$-0.321328\pi$
$599$	26.0555			1.06460			0.532300	+	0.846556i	$0.321328\pi$
$600$		0			0
$601$	−4.21110	−	4.21110i	−0.171774	−	0.171774i	0.615984	−	0.787759i	$-0.288759\pi$
$601$	−4.21110	−	4.21110i	−0.171774	−	0.171774i	−0.787759	+	0.615984i	$0.788759\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.8167	−	10.8167i	−0.439760	−	0.439760i
$606$		0			0
$607$	−9.51388	+	9.51388i	−0.386156	+	0.386156i	−0.873314	−	0.487158i	$-0.838034\pi$
$607$	−9.51388	+	9.51388i	−0.386156	+	0.386156i	0.487158	+	0.873314i	$0.338034\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10.4222			−0.421637
$612$		0			0
$613$	40.4222			1.63264			0.816319	−	0.577602i	$-0.196011\pi$
$613$	40.4222			1.63264			0.816319	+	0.577602i	$0.196011\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.60555	+	1.60555i	−0.0646371	+	0.0646371i	−0.738686	−	0.674049i	$-0.764554\pi$
$617$	−1.60555	+	1.60555i	−0.0646371	+	0.0646371i	0.674049	+	0.738686i	$0.264554\pi$
$618$		0			0
$619$	−14.4861	−	14.4861i	−0.582246	−	0.582246i	0.353274	−	0.935520i	$-0.385068\pi$
$619$	−14.4861	−	14.4861i	−0.582246	−	0.582246i	−0.935520	+	0.353274i	$0.885068\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18.0000	+	18.0000i	0.721155	+	0.721155i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−16.8167	+	4.81665i	−0.670524	+	0.192053i
$630$		0			0
$631$		−	24.2389i		−	0.964934i	−0.875914	−	0.482467i	$-0.839741\pi$
$631$		−	24.2389i		−	0.964934i	0.875914	−	0.482467i	$-0.160259\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.605551	+	0.605551i	0.0240306	+	0.0240306i
$636$		0			0
$637$			9.39445i			0.372222i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−30.8167	+	30.8167i	−1.21718	+	1.21718i	−0.248571	+	0.968614i	$0.579961\pi$
$641$	−30.8167	+	30.8167i	−1.21718	+	1.21718i	−0.968614	+	0.248571i	$0.920039\pi$
$642$		0			0
$643$	−4.11943	+	4.11943i	−0.162454	+	0.162454i	−0.783653	−	0.621199i	$-0.786646\pi$
$643$	−4.11943	+	4.11943i	−0.162454	+	0.162454i	0.621199	+	0.783653i	$0.286646\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−17.2111			−0.676638			−0.338319	−	0.941031i	$-0.609858\pi$
$647$	−17.2111			−0.676638			−0.338319	+	0.941031i	$0.609858\pi$
$648$		0			0
$649$	−2.60555	+	2.60555i	−0.102277	+	0.102277i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−24.8167	−	24.8167i	−0.971151	−	0.971151i	0.0284447	−	0.999595i	$-0.490945\pi$
$653$	−24.8167	−	24.8167i	−0.971151	−	0.971151i	−0.999595	+	0.0284447i	$0.990945\pi$
$654$		0			0
$655$			12.6056i			0.492540i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−32.0000			−1.24654			−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000			−1.24654			−0.623272	+	0.782006i	$0.714197\pi$
$660$		0			0
$661$		−	1.21110i		−	0.0471064i	−0.999723	−	0.0235532i	$-0.992502\pi$
$661$		−	1.21110i		−	0.0471064i	0.999723	−	0.0235532i	$-0.00749791\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			2.78890i			0.108149i
$666$		0			0
$667$	−13.8167			−0.534983
$668$		0			0
$669$		0			0
$670$		0			0
$671$			3.76114i			0.145197i
$672$		0			0
$673$	5.60555	+	5.60555i	0.216078	+	0.216078i	0.806843	−	0.590765i	$-0.201174\pi$
$673$	5.60555	+	5.60555i	0.216078	+	0.216078i	−0.590765	+	0.806843i	$0.701174\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	16.0278	−	16.0278i	0.615997	−	0.615997i	−0.328505	−	0.944502i	$-0.606545\pi$
$677$	16.0278	−	16.0278i	0.615997	−	0.615997i	0.944502	+	0.328505i	$0.106545\pi$
$678$		0			0
$679$	35.0278			1.34424
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.724981	+	0.724981i	−0.0277406	+	0.0277406i	−0.720841	−	0.693100i	$-0.756244\pi$
$683$	−0.724981	+	0.724981i	−0.0277406	+	0.0277406i	0.693100	+	0.720841i	$0.256244\pi$
$684$		0			0
$685$	2.60555	−	2.60555i	0.0995530	−	0.0995530i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			13.5778i			0.517273i
$690$		0			0
$691$	5.51388	+	5.51388i	0.209758	+	0.209758i	0.804165	−	0.594407i	$-0.202613\pi$
$691$	5.51388	+	5.51388i	0.209758	+	0.209758i	−0.594407	+	0.804165i	$0.702613\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			0.605551i			0.0229699i
$696$		0			0
$697$	1.60555	+	5.60555i	0.0608146	+	0.212325i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	7.81665			0.295231			0.147615	−	0.989045i	$-0.452840\pi$
$701$	7.81665			0.295231			0.147615	+	0.989045i	$0.452840\pi$
$702$		0			0
$703$	−1.81665	−	1.81665i	−0.0685164	−	0.0685164i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	24.4222	+	24.4222i	0.918492	+	0.918492i
$708$		0			0
$709$	14.3944	−	14.3944i	0.540595	−	0.540595i	−0.383108	−	0.923703i	$-0.625146\pi$
$709$	14.3944	−	14.3944i	0.540595	−	0.540595i	0.923703	+	0.383108i	$0.125146\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	37.0278			1.38670
$714$		0			0
$715$	−1.57779			−0.0590062
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.72498	+	4.72498i	−0.176212	+	0.176212i	−0.789702	−	0.613490i	$-0.789765\pi$
$719$	−4.72498	+	4.72498i	−0.176212	+	0.176212i	0.613490	+	0.789702i	$0.289765\pi$
$720$		0			0
$721$	−30.4222	−	30.4222i	−1.13298	−	1.13298i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.81665	+	4.81665i	0.178886	+	0.178886i
$726$		0			0
$727$	6.42221			0.238186			0.119093	−	0.992883i	$-0.462001\pi$
$727$	6.42221			0.238186			0.119093	+	0.992883i	$0.462001\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.78890	−	12.2389i	0.251096	−	0.452671i
$732$		0			0
$733$		−	46.0555i		−	1.70110i	−0.525895	−	0.850550i	$-0.676269\pi$
$733$		−	46.0555i		−	1.70110i	0.525895	−	0.850550i	$-0.323731\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.78890	+	2.78890i	0.102730	+	0.102730i
$738$		0			0
$739$		−	34.6611i		−	1.27503i	−0.770439	−	0.637514i	$-0.779963\pi$
$739$		−	34.6611i		−	1.27503i	0.770439	−	0.637514i	$-0.220037\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	14.7250	−	14.7250i	0.540207	−	0.540207i	−0.383383	−	0.923590i	$-0.625241\pi$
$743$	14.7250	−	14.7250i	0.540207	−	0.540207i	0.923590	+	0.383383i	$0.125241\pi$
$744$		0			0
$745$	−8.42221	+	8.42221i	−0.308566	+	0.308566i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−26.2389			−0.958747
$750$		0			0
$751$	−1.69722	+	1.69722i	−0.0619326	+	0.0619326i	−0.737395	−	0.675462i	$-0.763944\pi$
$751$	−1.69722	+	1.69722i	−0.0619326	+	0.0619326i	0.675462	+	0.737395i	$0.263944\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−4.60555	−	4.60555i	−0.167613	−	0.167613i
$756$		0			0
$757$			39.0278i			1.41849i	0.704963	+	0.709244i	$0.250964\pi$
$757$			39.0278i			1.41849i	−0.704963	+	0.709244i	$0.749036\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	31.4500			1.14006			0.570030	−	0.821624i	$-0.306932\pi$
$761$	31.4500			1.14006			0.570030	+	0.821624i	$0.306932\pi$
$762$		0			0
$763$			28.6056i			1.03559i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	22.4222i		−	0.809619i
$768$		0			0
$769$	−18.6056			−0.670933			−0.335467	−	0.942052i	$-0.608894\pi$
$769$	−18.6056			−0.670933			−0.335467	+	0.942052i	$0.608894\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			28.6056i			1.02887i	0.857529	+	0.514435i	$0.171998\pi$
$773$			28.6056i			1.02887i	−0.857529	+	0.514435i	$0.828002\pi$
$774$		0			0
$775$	−12.9083	−	12.9083i	−0.463681	−	0.463681i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.605551	+	0.605551i	−0.0216961	+	0.0216961i
$780$		0			0
$781$	−1.76114			−0.0630186
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−8.42221	+	8.42221i	−0.300601	+	0.300601i
$786$		0			0
$787$	−14.7250	+	14.7250i	−0.524889	+	0.524889i	−0.919044	−	0.394155i	$-0.871037\pi$
$787$	−14.7250	+	14.7250i	−0.524889	+	0.524889i	0.394155	+	0.919044i	$0.371037\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			44.2389i			1.57295i
$792$		0			0
$793$	−16.1833	−	16.1833i	−0.574687	−	0.574687i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	3.63331i		−	0.128698i	−0.997927	−	0.0643492i	$-0.979503\pi$
$797$		−	3.63331i		−	0.128698i	0.997927	−	0.0643492i	$-0.0204971\pi$
$798$		0			0
$799$	14.4222	+	8.00000i	0.510221	+	0.283020i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.23886			0.149586
$804$		0			0
$805$	19.8167	+	19.8167i	0.698445	+	0.698445i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	20.8167	+	20.8167i	0.731875	+	0.731875i	0.970991	−	0.239116i	$-0.0768577\pi$
$809$	20.8167	+	20.8167i	0.731875	+	0.731875i	−0.239116	+	0.970991i	$0.576858\pi$
$810$		0			0
$811$	−10.7250	+	10.7250i	−0.376605	+	0.376605i	−0.869876	−	0.493271i	$-0.835801\pi$
$811$	−10.7250	+	10.7250i	−0.376605	+	0.376605i	0.493271	+	0.869876i	$0.335801\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.60555			0.301439
$816$		0			0
$817$	2.05551			0.0719133
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12.8167	+	12.8167i	−0.447304	+	0.447304i	−0.894457	−	0.447153i	$-0.852438\pi$
$821$	−12.8167	+	12.8167i	−0.447304	+	0.447304i	0.447153	+	0.894457i	$0.352438\pi$
$822$		0			0
$823$	−12.9083	−	12.9083i	−0.449956	−	0.449956i	0.445384	−	0.895340i	$-0.353067\pi$
$823$	−12.9083	−	12.9083i	−0.449956	−	0.449956i	−0.895340	+	0.445384i	$0.853067\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.30278	+	6.30278i	0.219169	+	0.219169i	0.808148	−	0.588979i	$-0.200470\pi$
$827$	6.30278	+	6.30278i	0.219169	+	0.219169i	−0.588979	+	0.808148i	$0.700470\pi$
$828$		0			0
$829$	−18.8444			−0.654493			−0.327247	−	0.944939i	$-0.606121\pi$
$829$	−18.8444			−0.654493			−0.327247	+	0.944939i	$0.606121\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	7.21110	−	13.0000i	0.249850	−	0.450423i
$834$		0			0
$835$		−	29.8167i		−	1.03185i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−14.9083	−	14.9083i	−0.514693	−	0.514693i	0.401268	−	0.915961i	$-0.368570\pi$
$839$	−14.9083	−	14.9083i	−0.514693	−	0.514693i	−0.915961	+	0.401268i	$0.868570\pi$
$840$		0			0
$841$			23.8444i			0.822221i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−6.21110	+	6.21110i	−0.213668	+	0.213668i
$846$		0			0
$847$	−24.9083	+	24.9083i	−0.855860	+	0.855860i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−25.8167			−0.884983
$852$		0			0
$853$	29.4222	−	29.4222i	1.00740	−	1.00740i	0.00742468	−	0.999972i	$-0.497637\pi$
$853$	29.4222	−	29.4222i	1.00740	−	1.00740i	0.999972	−	0.00742468i	$-0.00236337\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.6333	+	18.6333i	0.636502	+	0.636502i	0.949691	−	0.313189i	$-0.101397\pi$
$857$	18.6333	+	18.6333i	0.636502	+	0.636502i	−0.313189	+	0.949691i	$0.601397\pi$
$858$		0			0
$859$		−	1.81665i		−	0.0619834i	−0.999520	−	0.0309917i	$-0.990133\pi$
$859$		−	1.81665i		−	0.0619834i	0.999520	−	0.0309917i	$-0.00986655\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−52.4777			−1.78636			−0.893181	−	0.449697i	$-0.851532\pi$
$863$	−52.4777			−1.78636			−0.893181	+	0.449697i	$0.851532\pi$
$864$		0			0
$865$		−	29.6333i		−	1.00756i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	0.183346i		−	0.00621959i
$870$		0			0
$871$	−24.0000			−0.813209
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	36.8444i		−	1.24557i
$876$		0			0
$877$	1.78890	+	1.78890i	0.0604068	+	0.0604068i	0.736665	−	0.676258i	$-0.236400\pi$
$877$	1.78890	+	1.78890i	0.0604068	+	0.0604068i	−0.676258	+	0.736665i	$0.736400\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	24.2111	−	24.2111i	0.815693	−	0.815693i	−0.169788	−	0.985481i	$-0.554308\pi$
$881$	24.2111	−	24.2111i	0.815693	−	0.815693i	0.985481	+	0.169788i	$0.0543083\pi$
$882$		0			0
$883$	−25.5778			−0.860761			−0.430381	−	0.902647i	$-0.641621\pi$
$883$	−25.5778			−0.860761			−0.430381	+	0.902647i	$0.641621\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−11.6972	+	11.6972i	−0.392754	+	0.392754i	−0.875668	−	0.482914i	$-0.839579\pi$
$887$	−11.6972	+	11.6972i	−0.392754	+	0.392754i	0.482914	+	0.875668i	$0.339579\pi$
$888$		0			0
$889$	1.39445	−	1.39445i	0.0467683	−	0.0467683i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			2.42221i			0.0810560i
$894$		0			0
$895$	−15.0278	−	15.0278i	−0.502322	−	0.502322i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			13.8167i			0.460811i
$900$		0			0
$901$	10.4222	−	18.7889i	0.347214	−	0.625949i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−26.0000			−0.864269
$906$		0			0
$907$	−24.5416	−	24.5416i	−0.814892	−	0.814892i	0.170471	−	0.985363i	$-0.445471\pi$
$907$	−24.5416	−	24.5416i	−0.814892	−	0.814892i	−0.985363	+	0.170471i	$0.945471\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4.72498	+	4.72498i	0.156546	+	0.156546i	0.781034	−	0.624488i	$-0.214693\pi$
$911$	4.72498	+	4.72498i	0.156546	+	0.156546i	−0.624488	+	0.781034i	$0.714693\pi$
$912$		0			0
$913$	5.39445	−	5.39445i	0.178530	−	0.178530i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	29.0278			0.958581
$918$		0			0
$919$	−40.8444			−1.34733			−0.673666	−	0.739036i	$-0.735282\pi$
$919$	−40.8444			−1.34733			−0.673666	+	0.739036i	$0.735282\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	7.57779	−	7.57779i	0.249426	−	0.249426i
$924$		0			0
$925$	9.00000	+	9.00000i	0.295918	+	0.295918i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.00000	−	1.00000i	−0.0328089	−	0.0328089i	0.690512	−	0.723321i	$-0.257385\pi$
$929$	−1.00000	−	1.00000i	−0.0328089	−	0.0328089i	−0.723321	+	0.690512i	$0.757385\pi$
$930$		0			0
$931$	2.18335			0.0715563
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.18335	+	1.21110i	0.0714031	+	0.0396073i
$936$		0			0
$937$			6.42221i			0.209804i	0.994483	+	0.104902i	$0.0334529\pi$
$937$			6.42221i			0.209804i	−0.994483	+	0.104902i	$0.966547\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	16.0278	+	16.0278i	0.522490	+	0.522490i	0.918323	−	0.395833i	$-0.129544\pi$
$941$	16.0278	+	16.0278i	0.522490	+	0.522490i	−0.395833	+	0.918323i	$0.629544\pi$
$942$		0			0
$943$			8.60555i			0.280235i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.27502	+	9.27502i	−0.301398	+	0.301398i	−0.841560	−	0.540163i	$-0.818363\pi$
$947$	−9.27502	+	9.27502i	−0.301398	+	0.301398i	0.540163	+	0.841560i	$0.318363\pi$
$948$		0			0
$949$	−18.2389	+	18.2389i	−0.592058	+	0.592058i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−50.2389			−1.62740			−0.813698	−	0.581288i	$-0.802549\pi$
$953$	−50.2389			−1.62740			−0.813698	+	0.581288i	$0.802549\pi$
$954$		0			0
$955$	6.78890	−	6.78890i	0.219684	−	0.219684i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.00000	−	6.00000i	−0.193750	−	0.193750i
$960$		0			0
$961$		−	6.02776i		−	0.194444i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16.4222			−0.528649
$966$		0			0
$967$		−	21.8167i		−	0.701576i	−0.936455	−	0.350788i	$-0.885914\pi$
$967$		−	21.8167i		−	0.701576i	0.936455	−	0.350788i	$-0.114086\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			55.0278i			1.76592i	0.469444	+	0.882962i	$0.344454\pi$
$971$			55.0278i			1.76592i	−0.469444	+	0.882962i	$0.655546\pi$
$972$		0			0
$973$	1.39445			0.0447040
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	16.8444i		−	0.538900i	−0.963014	−	0.269450i	$-0.913158\pi$
$977$		−	16.8444i		−	0.538900i	0.963014	−	0.269450i	$-0.0868420\pi$
$978$		0			0
$979$	2.36669	+	2.36669i	0.0756398	+	0.0756398i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	8.11943	−	8.11943i	0.258970	−	0.258970i	−0.565665	−	0.824635i	$-0.691381\pi$
$983$	8.11943	−	8.11943i	0.258970	−	0.258970i	0.824635	+	0.565665i	$0.191381\pi$
$984$		0			0
$985$	35.2111			1.12192
$986$		0			0
$987$		0			0
$988$		0			0
$989$	14.6056	−	14.6056i	0.464430	−	0.464430i
$990$		0			0
$991$	23.5139	−	23.5139i	0.746943	−	0.746943i	−0.226961	−	0.973904i	$-0.572879\pi$
$991$	23.5139	−	23.5139i	0.746943	−	0.746943i	0.973904	+	0.226961i	$0.0728790\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	20.2389i		−	0.641615i
$996$		0			0
$997$	39.2389	+	39.2389i	1.24271	+	1.24271i	0.958874	+	0.283834i	$0.0916063\pi$
$997$	39.2389	+	39.2389i	1.24271	+	1.24271i	0.283834	+	0.958874i	$0.408394\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2448.2.be.u.1585.1		4
3.2	odd	2		272.2.o.g.225.2		4
4.3	odd	2		612.2.k.e.361.2		4
12.11	even	2		68.2.e.a.21.1	yes	4
17.13	even	4	inner	2448.2.be.u.1441.1		4
24.5	odd	2		1088.2.o.s.769.1		4
24.11	even	2		1088.2.o.t.769.2		4
51.8	odd	8		4624.2.a.bq.1.1		4
51.26	odd	8		4624.2.a.bq.1.4		4
51.47	odd	4		272.2.o.g.81.2		4
60.23	odd	4		1700.2.m.a.1449.1		4
60.47	odd	4		1700.2.m.b.1449.2		4
60.59	even	2		1700.2.o.c.701.2		4
68.47	odd	4		612.2.k.e.217.2		4
204.11	odd	16		1156.2.h.e.977.4		16
204.23	odd	16		1156.2.h.e.977.1		16
204.47	even	4		68.2.e.a.13.1	✓	4
204.59	even	8		1156.2.a.h.1.4		4
204.71	odd	16		1156.2.h.e.733.1		16
204.83	even	8		1156.2.b.a.577.1		4
204.95	odd	16		1156.2.h.e.757.1		16
204.107	odd	16		1156.2.h.e.1001.1		16
204.131	odd	16		1156.2.h.e.1001.4		16
204.143	odd	16		1156.2.h.e.757.4		16
204.155	even	8		1156.2.b.a.577.4		4
204.167	odd	16		1156.2.h.e.733.4		16
204.179	even	8		1156.2.a.h.1.1		4
204.191	even	4		1156.2.e.c.829.2		4
204.203	even	2		1156.2.e.c.905.2		4
408.149	odd	4		1088.2.o.s.897.1		4
408.251	even	4		1088.2.o.t.897.2		4
1020.47	odd	4		1700.2.m.a.149.1		4
1020.659	even	4		1700.2.o.c.1101.2		4
1020.863	odd	4		1700.2.m.b.149.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.e.a.13.1	✓	4	204.47	even	4
68.2.e.a.21.1	yes	4	12.11	even	2
272.2.o.g.81.2		4	51.47	odd	4
272.2.o.g.225.2		4	3.2	odd	2
612.2.k.e.217.2		4	68.47	odd	4
612.2.k.e.361.2		4	4.3	odd	2
1088.2.o.s.769.1		4	24.5	odd	2
1088.2.o.s.897.1		4	408.149	odd	4
1088.2.o.t.769.2		4	24.11	even	2
1088.2.o.t.897.2		4	408.251	even	4
1156.2.a.h.1.1		4	204.179	even	8
1156.2.a.h.1.4		4	204.59	even	8
1156.2.b.a.577.1		4	204.83	even	8
1156.2.b.a.577.4		4	204.155	even	8
1156.2.e.c.829.2		4	204.191	even	4
1156.2.e.c.905.2		4	204.203	even	2
1156.2.h.e.733.1		16	204.71	odd	16
1156.2.h.e.733.4		16	204.167	odd	16
1156.2.h.e.757.1		16	204.95	odd	16
1156.2.h.e.757.4		16	204.143	odd	16
1156.2.h.e.977.1		16	204.23	odd	16
1156.2.h.e.977.4		16	204.11	odd	16
1156.2.h.e.1001.1		16	204.107	odd	16
1156.2.h.e.1001.4		16	204.131	odd	16
1700.2.m.a.149.1		4	1020.47	odd	4
1700.2.m.a.1449.1		4	60.23	odd	4
1700.2.m.b.149.2		4	1020.863	odd	4
1700.2.m.b.1449.2		4	60.47	odd	4
1700.2.o.c.701.2		4	60.59	even	2
1700.2.o.c.1101.2		4	1020.659	even	4
2448.2.be.u.1441.1		4	17.13	even	4	inner
2448.2.be.u.1585.1		4	1.1	even	1	trivial
4624.2.a.bq.1.1		4	51.8	odd	8
4624.2.a.bq.1.4		4	51.26	odd	8

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:	\( N \)	\(=\)	\( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2448.be (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{5} +(-2.30278 - 2.30278i) q^{7} +O(q^{10})\)	\(q+(1.00000 - 1.00000i) q^{5} +(-2.30278 - 2.30278i) q^{7} +(-0.302776 - 0.302776i) q^{11} +2.60555 q^{13} +(-3.60555 - 2.00000i) q^{17} -0.605551i q^{19} +(-4.30278 - 4.30278i) q^{23} +3.00000i q^{25} +(1.60555 - 1.60555i) q^{29} +(-4.30278 + 4.30278i) q^{31} -4.60555 q^{35} +(3.00000 - 3.00000i) q^{37} +(-1.00000 - 1.00000i) q^{41} +3.39445i q^{43} -4.00000 q^{47} +3.60555i q^{49} +5.21110i q^{53} -0.605551 q^{55} -8.60555i q^{59} +(-6.21110 - 6.21110i) q^{61} +(2.60555 - 2.60555i) q^{65} -9.21110 q^{67} +(2.90833 - 2.90833i) q^{71} +(-7.00000 + 7.00000i) q^{73} +1.39445i q^{77} +(0.302776 + 0.302776i) q^{79} +17.8167i q^{83} +(-5.60555 + 1.60555i) q^{85} -7.81665 q^{89} +(-6.00000 - 6.00000i) q^{91} +(-0.605551 - 0.605551i) q^{95} +(-7.60555 + 7.60555i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 2 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 - 2 * q^7	\( 4 q + 4 q^{5} - 2 q^{7} + 6 q^{11} - 4 q^{13} - 10 q^{23} - 8 q^{29} - 10 q^{31} - 4 q^{35} + 12 q^{37} - 4 q^{41} - 16 q^{47} + 12 q^{55} + 4 q^{61} - 4 q^{65} - 8 q^{67} - 10 q^{71} - 28 q^{73} - 6 q^{79} - 8 q^{85} + 12 q^{89} - 24 q^{91} + 12 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 2 * q^7 + 6 * q^11 - 4 * q^13 - 10 * q^23 - 8 * q^29 - 10 * q^31 - 4 * q^35 + 12 * q^37 - 4 * q^41 - 16 * q^47 + 12 * q^55 + 4 * q^61 - 4 * q^65 - 8 * q^67 - 10 * q^71 - 28 * q^73 - 6 * q^79 - 8 * q^85 + 12 * q^89 - 24 * q^91 + 12 * q^95 - 16 * q^97

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