LMFDB - Embedded newform 418.2.b.a.417.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [418,2,Mod(417,418)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(418, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("418.417");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 418 = 2 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	418.b (of order $2$, degree $1$, 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:	$3.33774680449$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 10 $ x^2 + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.2
Root			$3.16228i$ of defining polynomial
Character	$\chi$	$=$	418.417
Dual form			418.2.b.a.417.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +3.16228i q^{7} -1.00000 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +3.16228i q^{7} -1.00000 q^{8} +3.00000 q^{9} +2.00000 q^{10} +(1.00000 - 3.16228i) q^{11} -6.00000 q^{13} -3.16228i q^{14} +1.00000 q^{16} +6.32456i q^{17} -3.00000 q^{18} +(-3.00000 + 3.16228i) q^{19} -2.00000 q^{20} +(-1.00000 + 3.16228i) q^{22} -4.00000 q^{23} -1.00000 q^{25} +6.00000 q^{26} +3.16228i q^{28} +6.00000 q^{29} +9.48683i q^{31} -1.00000 q^{32} -6.32456i q^{34} -6.32456i q^{35} +3.00000 q^{36} +9.48683i q^{37} +(3.00000 - 3.16228i) q^{38} +2.00000 q^{40} +(1.00000 - 3.16228i) q^{44} -6.00000 q^{45} +4.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} -9.48683i q^{53} +(-2.00000 + 6.32456i) q^{55} -3.16228i q^{56} -6.00000 q^{58} -3.16228i q^{61} -9.48683i q^{62} +9.48683i q^{63} +1.00000 q^{64} +12.0000 q^{65} +6.32456i q^{68} +6.32456i q^{70} +9.48683i q^{71} -3.00000 q^{72} -9.48683i q^{74} +(-3.00000 + 3.16228i) q^{76} +(10.0000 + 3.16228i) q^{77} -2.00000 q^{80} +9.00000 q^{81} -6.32456i q^{83} -12.6491i q^{85} +(-1.00000 + 3.16228i) q^{88} +6.00000 q^{90} -18.9737i q^{91} -4.00000 q^{92} +8.00000 q^{94} +(6.00000 - 6.32456i) q^{95} -18.9737i q^{97} +3.00000 q^{98} +(3.00000 - 9.48683i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 + 6 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 6 q^{9} + 4 q^{10} + 2 q^{11} - 12 q^{13} + 2 q^{16} - 6 q^{18} - 6 q^{19} - 4 q^{20} - 2 q^{22} - 8 q^{23} - 2 q^{25} + 12 q^{26} + 12 q^{29} - 2 q^{32} + 6 q^{36} + 6 q^{38} + 4 q^{40} + 2 q^{44} - 12 q^{45} + 8 q^{46} - 16 q^{47} - 6 q^{49} + 2 q^{50} - 12 q^{52} - 4 q^{55} - 12 q^{58} + 2 q^{64} + 24 q^{65} - 6 q^{72} - 6 q^{76} + 20 q^{77} - 4 q^{80} + 18 q^{81} - 2 q^{88} + 12 q^{90} - 8 q^{92} + 16 q^{94} + 12 q^{95} + 6 q^{98} + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 + 6 * q^9 + 4 * q^10 + 2 * q^11 - 12 * q^13 + 2 * q^16 - 6 * q^18 - 6 * q^19 - 4 * q^20 - 2 * q^22 - 8 * q^23 - 2 * q^25 + 12 * q^26 + 12 * q^29 - 2 * q^32 + 6 * q^36 + 6 * q^38 + 4 * q^40 + 2 * q^44 - 12 * q^45 + 8 * q^46 - 16 * q^47 - 6 * q^49 + 2 * q^50 - 12 * q^52 - 4 * q^55 - 12 * q^58 + 2 * q^64 + 24 * q^65 - 6 * q^72 - 6 * q^76 + 20 * q^77 - 4 * q^80 + 18 * q^81 - 2 * q^88 + 12 * q^90 - 8 * q^92 + 16 * q^94 + 12 * q^95 + 6 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/418\mathbb{Z}\right)^\times$.

$n$	$287$	$343$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	1.00000			0.500000
$5$	−2.00000			−0.894427			−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000			−0.894427			−0.447214	+	0.894427i	$0.647584\pi$
$6$		0			0
$7$			3.16228i			1.19523i	0.801784	+	0.597614i	$0.203885\pi$
$7$			3.16228i			1.19523i	−0.801784	+	0.597614i	$0.796115\pi$
$8$	−1.00000			−0.353553
$9$	3.00000			1.00000
$10$	2.00000			0.632456
$11$	1.00000	−	3.16228i	0.301511	−	0.953463i
$11$	1.00000	−	3.16228i	0.301511	−	0.953463i
$12$		0			0
$13$	−6.00000			−1.66410			−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000			−1.66410			−0.832050	+	0.554700i	$0.812833\pi$
$14$		−	3.16228i		−	0.845154i
$15$		0			0
$16$	1.00000			0.250000
$17$			6.32456i			1.53393i	0.641689	+	0.766965i	$0.278234\pi$
$17$			6.32456i			1.53393i	−0.641689	+	0.766965i	$0.721766\pi$
$18$	−3.00000			−0.707107
$19$	−3.00000	+	3.16228i	−0.688247	+	0.725476i
$19$	−3.00000	+	3.16228i	−0.688247	+	0.725476i
$20$	−2.00000			−0.447214
$21$		0			0
$22$	−1.00000	+	3.16228i	−0.213201	+	0.674200i
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	6.00000			1.17670
$27$		0			0
$28$			3.16228i			0.597614i
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$			9.48683i			1.70389i	0.523635	+	0.851943i	$0.324576\pi$
$31$			9.48683i			1.70389i	−0.523635	+	0.851943i	$0.675424\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$		−	6.32456i		−	1.08465i
$35$		−	6.32456i		−	1.06904i
$36$	3.00000			0.500000
$37$			9.48683i			1.55963i	0.626013	+	0.779813i	$0.284686\pi$
$37$			9.48683i			1.55963i	−0.626013	+	0.779813i	$0.715314\pi$
$38$	3.00000	−	3.16228i	0.486664	−	0.512989i
$39$		0			0
$40$	2.00000			0.316228
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	1.00000	−	3.16228i	0.150756	−	0.476731i
$45$	−6.00000			−0.894427
$46$	4.00000			0.589768
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$	1.00000			0.141421
$51$		0			0
$52$	−6.00000			−0.832050
$53$		−	9.48683i		−	1.30312i	−0.758599	−	0.651558i	$-0.774116\pi$
$53$		−	9.48683i		−	1.30312i	0.758599	−	0.651558i	$-0.225884\pi$
$54$		0			0
$55$	−2.00000	+	6.32456i	−0.269680	+	0.852803i
$56$		−	3.16228i		−	0.422577i
$57$		0			0
$58$	−6.00000			−0.787839
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$		−	3.16228i		−	0.404888i	−0.979294	−	0.202444i	$-0.935112\pi$
$61$		−	3.16228i		−	0.404888i	0.979294	−	0.202444i	$-0.0648884\pi$
$62$		−	9.48683i		−	1.20483i
$63$			9.48683i			1.19523i
$64$	1.00000			0.125000
$65$	12.0000			1.48842
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$			6.32456i			0.766965i
$69$		0			0
$70$			6.32456i			0.755929i
$71$			9.48683i			1.12588i	0.826498	+	0.562940i	$0.190330\pi$
$71$			9.48683i			1.12588i	−0.826498	+	0.562940i	$0.809670\pi$
$72$	−3.00000			−0.353553
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		−	9.48683i		−	1.10282i
$75$		0			0
$76$	−3.00000	+	3.16228i	−0.344124	+	0.362738i
$77$	10.0000	+	3.16228i	1.13961	+	0.360375i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−2.00000			−0.223607
$81$	9.00000			1.00000
$82$		0			0
$83$		−	6.32456i		−	0.694210i	−0.937826	−	0.347105i	$-0.887165\pi$
$83$		−	6.32456i		−	0.694210i	0.937826	−	0.347105i	$-0.112835\pi$
$84$		0			0
$85$		−	12.6491i		−	1.37199i
$86$		0			0
$87$		0			0
$88$	−1.00000	+	3.16228i	−0.106600	+	0.337100i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	6.00000			0.632456
$91$		−	18.9737i		−	1.98898i
$92$	−4.00000			−0.417029
$93$		0			0
$94$	8.00000			0.825137
$95$	6.00000	−	6.32456i	0.615587	−	0.648886i
$96$		0			0
$97$		−	18.9737i		−	1.92648i	−0.268635	−	0.963242i	$-0.586573\pi$
$97$		−	18.9737i		−	1.92648i	0.268635	−	0.963242i	$-0.413427\pi$
$98$	3.00000			0.303046
$99$	3.00000	−	9.48683i	0.301511	−	0.953463i
$100$	−1.00000			−0.100000
$101$		−	15.8114i		−	1.57329i	−0.617404	−	0.786646i	$-0.711816\pi$
$101$		−	15.8114i		−	1.57329i	0.617404	−	0.786646i	$-0.288184\pi$
$102$		0			0
$103$			9.48683i			0.934765i	0.884055	+	0.467383i	$0.154803\pi$
$103$			9.48683i			0.934765i	−0.884055	+	0.467383i	$0.845197\pi$
$104$	6.00000			0.588348
$105$		0			0
$106$			9.48683i			0.921443i
$107$	18.0000			1.74013			0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000			1.74013			0.870063	+	0.492941i	$0.164078\pi$
$108$		0			0
$109$	−6.00000			−0.574696			−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000			−0.574696			−0.287348	+	0.957826i	$0.592774\pi$
$110$	2.00000	−	6.32456i	0.190693	−	0.603023i
$111$		0			0
$112$			3.16228i			0.298807i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	8.00000			0.746004
$116$	6.00000			0.557086
$117$	−18.0000			−1.66410
$118$		0			0
$119$	−20.0000			−1.83340
$120$		0			0
$121$	−9.00000	−	6.32456i	−0.818182	−	0.574960i
$122$			3.16228i			0.286299i
$123$		0			0
$124$			9.48683i			0.851943i
$125$	12.0000			1.07331
$126$		−	9.48683i		−	0.845154i
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$	−12.0000			−1.05247
$131$			6.32456i			0.552579i	0.961074	+	0.276289i	$0.0891049\pi$
$131$			6.32456i			0.552579i	−0.961074	+	0.276289i	$0.910895\pi$
$132$		0			0
$133$	−10.0000	−	9.48683i	−0.867110	−	0.822613i
$134$		0			0
$135$		0			0
$136$		−	6.32456i		−	0.542326i
$137$	8.00000			0.683486			0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000			0.683486			0.341743	+	0.939793i	$0.388983\pi$
$138$		0			0
$139$			12.6491i			1.07288i	0.843937	+	0.536442i	$0.180232\pi$
$139$			12.6491i			1.07288i	−0.843937	+	0.536442i	$0.819768\pi$
$140$		−	6.32456i		−	0.534522i
$141$		0			0
$142$		−	9.48683i		−	0.796117i
$143$	−6.00000	+	18.9737i	−0.501745	+	1.58666i
$144$	3.00000			0.250000
$145$	−12.0000			−0.996546
$146$		0			0
$147$		0			0
$148$			9.48683i			0.779813i
$149$			3.16228i			0.259064i	0.991575	+	0.129532i	$0.0413475\pi$
$149$			3.16228i			0.259064i	−0.991575	+	0.129532i	$0.958653\pi$
$150$		0			0
$151$	12.0000			0.976546			0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000			0.976546			0.488273	+	0.872691i	$0.337627\pi$
$152$	3.00000	−	3.16228i	0.243332	−	0.256495i
$153$			18.9737i			1.53393i
$154$	−10.0000	−	3.16228i	−0.805823	−	0.254824i
$155$		−	18.9737i		−	1.52400i
$156$		0			0
$157$	18.0000			1.43656			0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000			1.43656			0.718278	+	0.695756i	$0.244931\pi$
$158$		0			0
$159$		0			0
$160$	2.00000			0.158114
$161$		−	12.6491i		−	0.996890i
$162$	−9.00000			−0.707107
$163$	−6.00000			−0.469956			−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000			−0.469956			−0.234978	+	0.972001i	$0.575502\pi$
$164$		0			0
$165$		0			0
$166$			6.32456i			0.490881i
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$			12.6491i			0.970143i
$171$	−9.00000	+	9.48683i	−0.688247	+	0.725476i
$172$		0			0
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$		0			0
$175$		−	3.16228i		−	0.239046i
$176$	1.00000	−	3.16228i	0.0753778	−	0.238366i
$177$		0			0
$178$		0			0
$179$			18.9737i			1.41816i	0.705129	+	0.709079i	$0.250889\pi$
$179$			18.9737i			1.41816i	−0.705129	+	0.709079i	$0.749111\pi$
$180$	−6.00000			−0.447214
$181$			9.48683i			0.705151i	0.935783	+	0.352575i	$0.114694\pi$
$181$			9.48683i			0.705151i	−0.935783	+	0.352575i	$0.885306\pi$
$182$			18.9737i			1.40642i
$183$		0			0
$184$	4.00000			0.294884
$185$		−	18.9737i		−	1.39497i
$186$		0			0
$187$	20.0000	+	6.32456i	1.46254	+	0.462497i
$188$	−8.00000			−0.583460
$189$		0			0
$190$	−6.00000	+	6.32456i	−0.435286	+	0.458831i
$191$	−20.0000			−1.44715			−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000			−1.44715			−0.723575	+	0.690246i	$0.757502\pi$
$192$		0			0
$193$	24.0000			1.72756			0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000			1.72756			0.863779	+	0.503871i	$0.168091\pi$
$194$			18.9737i			1.36223i
$195$		0			0
$196$	−3.00000			−0.214286
$197$			15.8114i			1.12651i	0.826281	+	0.563257i	$0.190452\pi$
$197$			15.8114i			1.12651i	−0.826281	+	0.563257i	$0.809548\pi$
$198$	−3.00000	+	9.48683i	−0.213201	+	0.674200i
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$	1.00000			0.0707107
$201$		0			0
$202$			15.8114i			1.11249i
$203$			18.9737i			1.33169i
$204$		0			0
$205$		0			0
$206$		−	9.48683i		−	0.660979i
$207$	−12.0000			−0.834058
$208$	−6.00000			−0.416025
$209$	7.00000	+	12.6491i	0.484200	+	0.874957i
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		−	9.48683i		−	0.651558i
$213$		0			0
$214$	−18.0000			−1.23045
$215$		0			0
$216$		0			0
$217$	−30.0000			−2.03653
$218$	6.00000			0.406371
$219$		0			0
$220$	−2.00000	+	6.32456i	−0.134840	+	0.426401i
$221$		−	37.9473i		−	2.55261i
$222$		0			0
$223$			9.48683i			0.635285i	0.948210	+	0.317643i	$0.102891\pi$
$223$			9.48683i			0.635285i	−0.948210	+	0.317643i	$0.897109\pi$
$224$		−	3.16228i		−	0.211289i
$225$	−3.00000			−0.200000
$226$		0			0
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$	6.00000			0.396491			0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000			0.396491			0.198246	+	0.980152i	$0.436476\pi$
$230$	−8.00000			−0.527504
$231$		0			0
$232$	−6.00000			−0.393919
$233$		−	12.6491i		−	0.828671i	−0.910124	−	0.414335i	$-0.864014\pi$
$233$		−	12.6491i		−	0.828671i	0.910124	−	0.414335i	$-0.135986\pi$
$234$	18.0000			1.17670
$235$	16.0000			1.04372
$236$		0			0
$237$		0			0
$238$	20.0000			1.29641
$239$			3.16228i			0.204551i	0.994756	+	0.102275i	$0.0326123\pi$
$239$			3.16228i			0.204551i	−0.994756	+	0.102275i	$0.967388\pi$
$240$		0			0
$241$	18.0000			1.15948			0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000			1.15948			0.579741	+	0.814801i	$0.303154\pi$
$242$	9.00000	+	6.32456i	0.578542	+	0.406558i
$243$		0			0
$244$		−	3.16228i		−	0.202444i
$245$	6.00000			0.383326
$246$		0			0
$247$	18.0000	−	18.9737i	1.14531	−	1.20727i
$248$		−	9.48683i		−	0.602414i
$249$		0			0
$250$	−12.0000			−0.758947
$251$	−2.00000			−0.126239			−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000			−0.126239			−0.0631194	+	0.998006i	$0.520105\pi$
$252$			9.48683i			0.597614i
$253$	−4.00000	+	12.6491i	−0.251478	+	0.795243i
$254$	−12.0000			−0.752947
$255$		0			0
$256$	1.00000			0.0625000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	−30.0000			−1.86411
$260$	12.0000			0.744208
$261$	18.0000			1.11417
$262$		−	6.32456i		−	0.390732i
$263$		−	15.8114i		−	0.974972i	−0.873131	−	0.487486i	$-0.837914\pi$
$263$		−	15.8114i		−	0.974972i	0.873131	−	0.487486i	$-0.162086\pi$
$264$		0			0
$265$			18.9737i			1.16554i
$266$	10.0000	+	9.48683i	0.613139	+	0.581675i
$267$		0			0
$268$		0			0
$269$		−	28.4605i		−	1.73527i	−0.497204	−	0.867634i	$-0.665640\pi$
$269$		−	28.4605i		−	1.73527i	0.497204	−	0.867634i	$-0.334360\pi$
$270$		0			0
$271$		−	9.48683i		−	0.576284i	−0.957588	−	0.288142i	$-0.906962\pi$
$271$		−	9.48683i		−	0.576284i	0.957588	−	0.288142i	$-0.0930375\pi$
$272$			6.32456i			0.383482i
$273$		0			0
$274$	−8.00000			−0.483298
$275$	−1.00000	+	3.16228i	−0.0603023	+	0.190693i
$276$		0			0
$277$		−	3.16228i		−	0.190003i	−0.995477	−	0.0950014i	$-0.969714\pi$
$277$		−	3.16228i		−	0.190003i	0.995477	−	0.0950014i	$-0.0302856\pi$
$278$		−	12.6491i		−	0.758643i
$279$			28.4605i			1.70389i
$280$			6.32456i			0.377964i
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$			18.9737i			1.12787i	0.825820	+	0.563934i	$0.190713\pi$
$283$			18.9737i			1.12787i	−0.825820	+	0.563934i	$0.809287\pi$
$284$			9.48683i			0.562940i
$285$		0			0
$286$	6.00000	−	18.9737i	0.354787	−	1.12194i
$287$		0			0
$288$	−3.00000			−0.176777
$289$	−23.0000			−1.35294
$290$	12.0000			0.704664
$291$		0			0
$292$		0			0
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$		0			0
$296$		−	9.48683i		−	0.551411i
$297$		0			0
$298$		−	3.16228i		−	0.183186i
$299$	24.0000			1.38796
$300$		0			0
$301$		0			0
$302$	−12.0000			−0.690522
$303$		0			0
$304$	−3.00000	+	3.16228i	−0.172062	+	0.181369i
$305$			6.32456i			0.362143i
$306$		−	18.9737i		−	1.08465i
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$	10.0000	+	3.16228i	0.569803	+	0.180187i
$309$		0			0
$310$			18.9737i			1.07763i
$311$	20.0000			1.13410			0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000			1.13410			0.567048	+	0.823685i	$0.308085\pi$
$312$		0			0
$313$	24.0000			1.35656			0.678280	−	0.734803i	$-0.262726\pi$
$313$	24.0000			1.35656			0.678280	+	0.734803i	$0.262726\pi$
$314$	−18.0000			−1.01580
$315$		−	18.9737i		−	1.06904i
$316$		0			0
$317$			28.4605i			1.59850i	0.600998	+	0.799250i	$0.294770\pi$
$317$			28.4605i			1.59850i	−0.600998	+	0.799250i	$0.705230\pi$
$318$		0			0
$319$	6.00000	−	18.9737i	0.335936	−	1.06232i
$320$	−2.00000			−0.111803
$321$		0			0
$322$			12.6491i			0.704907i
$323$	−20.0000	−	18.9737i	−1.11283	−	1.05572i
$324$	9.00000			0.500000
$325$	6.00000			0.332820
$326$	6.00000			0.332309
$327$		0			0
$328$		0			0
$329$		−	25.2982i		−	1.39474i
$330$		0			0
$331$		−	18.9737i		−	1.04289i	−0.853286	−	0.521443i	$-0.825394\pi$
$331$		−	18.9737i		−	1.04289i	0.853286	−	0.521443i	$-0.174606\pi$
$332$		−	6.32456i		−	0.347105i
$333$			28.4605i			1.55963i
$334$	12.0000			0.656611
$335$		0			0
$336$		0			0
$337$	−18.0000			−0.980522			−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000			−0.980522			−0.490261	+	0.871576i	$0.663099\pi$
$338$	−23.0000			−1.25104
$339$		0			0
$340$		−	12.6491i		−	0.685994i
$341$	30.0000	+	9.48683i	1.62459	+	0.513741i
$342$	9.00000	−	9.48683i	0.486664	−	0.512989i
$343$			12.6491i			0.682988i
$344$		0			0
$345$		0			0
$346$	6.00000			0.322562
$347$		−	12.6491i		−	0.679040i	−0.940599	−	0.339520i	$-0.889735\pi$
$347$		−	12.6491i		−	0.679040i	0.940599	−	0.339520i	$-0.110265\pi$
$348$		0			0
$349$			28.4605i			1.52346i	0.647897	+	0.761728i	$0.275649\pi$
$349$			28.4605i			1.52346i	−0.647897	+	0.761728i	$0.724351\pi$
$350$			3.16228i			0.169031i
$351$		0			0
$352$	−1.00000	+	3.16228i	−0.0533002	+	0.168550i
$353$	−4.00000			−0.212899			−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000			−0.212899			−0.106449	+	0.994318i	$0.533948\pi$
$354$		0			0
$355$		−	18.9737i		−	1.00702i
$356$		0			0
$357$		0			0
$358$		−	18.9737i		−	1.00279i
$359$			22.1359i			1.16829i	0.811649	+	0.584145i	$0.198570\pi$
$359$			22.1359i			1.16829i	−0.811649	+	0.584145i	$0.801430\pi$
$360$	6.00000			0.316228
$361$	−1.00000	−	18.9737i	−0.0526316	−	0.998614i
$362$		−	9.48683i		−	0.498617i
$363$		0			0
$364$		−	18.9737i		−	0.994490i
$365$		0			0
$366$		0			0
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$	−4.00000			−0.208514
$369$		0			0
$370$			18.9737i			0.986394i
$371$	30.0000			1.55752
$372$		0			0
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$	−20.0000	−	6.32456i	−1.03418	−	0.327035i
$375$		0			0
$376$	8.00000			0.412568
$377$	−36.0000			−1.85409
$378$		0			0
$379$			18.9737i			0.974612i	0.873231	+	0.487306i	$0.162020\pi$
$379$			18.9737i			0.974612i	−0.873231	+	0.487306i	$0.837980\pi$
$380$	6.00000	−	6.32456i	0.307794	−	0.324443i
$381$		0			0
$382$	20.0000			1.02329
$383$		−	9.48683i		−	0.484755i	−0.970182	−	0.242377i	$-0.922073\pi$
$383$		−	9.48683i		−	0.484755i	0.970182	−	0.242377i	$-0.0779272\pi$
$384$		0			0
$385$	−20.0000	−	6.32456i	−1.01929	−	0.322329i
$386$	−24.0000			−1.22157
$387$		0			0
$388$		−	18.9737i		−	0.963242i
$389$	−10.0000			−0.507020			−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000			−0.507020			−0.253510	+	0.967333i	$0.581585\pi$
$390$		0			0
$391$		−	25.2982i		−	1.27939i
$392$	3.00000			0.151523
$393$		0			0
$394$		−	15.8114i		−	0.796566i
$395$		0			0
$396$	3.00000	−	9.48683i	0.150756	−	0.476731i
$397$	−2.00000			−0.100377			−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000			−0.100377			−0.0501886	+	0.998740i	$0.515982\pi$
$398$	20.0000			1.00251
$399$		0			0
$400$	−1.00000			−0.0500000
$401$			18.9737i			0.947500i	0.880659	+	0.473750i	$0.157100\pi$
$401$			18.9737i			0.947500i	−0.880659	+	0.473750i	$0.842900\pi$
$402$		0			0
$403$		−	56.9210i		−	2.83544i
$404$		−	15.8114i		−	0.786646i
$405$	−18.0000			−0.894427
$406$		−	18.9737i		−	0.941647i
$407$	30.0000	+	9.48683i	1.48704	+	0.470245i
$408$		0			0
$409$	−6.00000			−0.296681			−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000			−0.296681			−0.148340	+	0.988936i	$0.547393\pi$
$410$		0			0
$411$		0			0
$412$			9.48683i			0.467383i
$413$		0			0
$414$	12.0000			0.589768
$415$			12.6491i			0.620920i
$416$	6.00000			0.294174
$417$		0			0
$418$	−7.00000	−	12.6491i	−0.342381	−	0.618688i
$419$	4.00000			0.195413			0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000			0.195413			0.0977064	+	0.995215i	$0.468849\pi$
$420$		0			0
$421$		−	9.48683i		−	0.462360i	−0.972911	−	0.231180i	$-0.925741\pi$
$421$		−	9.48683i		−	0.462360i	0.972911	−	0.231180i	$-0.0742586\pi$
$422$	12.0000			0.584151
$423$	−24.0000			−1.16692
$424$			9.48683i			0.460721i
$425$		−	6.32456i		−	0.306786i
$426$		0			0
$427$	10.0000			0.483934
$428$	18.0000			0.870063
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$			18.9737i			0.911816i	0.890027	+	0.455908i	$0.150685\pi$
$433$			18.9737i			0.911816i	−0.890027	+	0.455908i	$0.849315\pi$
$434$	30.0000			1.44005
$435$		0			0
$436$	−6.00000			−0.287348
$437$	12.0000	−	12.6491i	0.574038	−	0.605089i
$438$		0			0
$439$	36.0000			1.71819			0.859093	−	0.511819i	$-0.171028\pi$
$439$	36.0000			1.71819			0.859093	+	0.511819i	$0.171028\pi$
$440$	2.00000	−	6.32456i	0.0953463	−	0.301511i
$441$	−9.00000			−0.428571
$442$			37.9473i			1.80497i
$443$	4.00000			0.190046			0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000			0.190046			0.0950229	+	0.995475i	$0.469708\pi$
$444$		0			0
$445$		0			0
$446$		−	9.48683i		−	0.449215i
$447$		0			0
$448$			3.16228i			0.149404i
$449$			18.9737i			0.895423i	0.894178	+	0.447711i	$0.147761\pi$
$449$			18.9737i			0.895423i	−0.894178	+	0.447711i	$0.852239\pi$
$450$	3.00000			0.141421
$451$		0			0
$452$		0			0
$453$		0			0
$454$	12.0000			0.563188
$455$			37.9473i			1.77900i
$456$		0			0
$457$			18.9737i			0.887551i	0.896138	+	0.443775i	$0.146361\pi$
$457$			18.9737i			0.887551i	−0.896138	+	0.443775i	$0.853639\pi$
$458$	−6.00000			−0.280362
$459$		0			0
$460$	8.00000			0.373002
$461$		−	3.16228i		−	0.147282i	−0.997285	−	0.0736410i	$-0.976538\pi$
$461$		−	3.16228i		−	0.147282i	0.997285	−	0.0736410i	$-0.0234619\pi$
$462$		0			0
$463$	4.00000			0.185896			0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000			0.185896			0.0929479	+	0.995671i	$0.470371\pi$
$464$	6.00000			0.278543
$465$		0			0
$466$			12.6491i			0.585959i
$467$	−28.0000			−1.29569			−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000			−1.29569			−0.647843	+	0.761774i	$0.724329\pi$
$468$	−18.0000			−0.832050
$469$		0			0
$470$	−16.0000			−0.738025
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	3.00000	−	3.16228i	0.137649	−	0.145095i
$476$	−20.0000			−0.916698
$477$		−	28.4605i		−	1.30312i
$478$		−	3.16228i		−	0.144639i
$479$			3.16228i			0.144488i	0.997387	+	0.0722441i	$0.0230161\pi$
$479$			3.16228i			0.144488i	−0.997387	+	0.0722441i	$0.976984\pi$
$480$		0			0
$481$		−	56.9210i		−	2.59537i
$482$	−18.0000			−0.819878
$483$		0			0
$484$	−9.00000	−	6.32456i	−0.409091	−	0.287480i
$485$			37.9473i			1.72310i
$486$		0			0
$487$		−	28.4605i		−	1.28967i	−0.764323	−	0.644834i	$-0.776926\pi$
$487$		−	28.4605i		−	1.28967i	0.764323	−	0.644834i	$-0.223074\pi$
$488$			3.16228i			0.143150i
$489$		0			0
$490$	−6.00000			−0.271052
$491$			25.2982i			1.14169i	0.821057	+	0.570846i	$0.193385\pi$
$491$			25.2982i			1.14169i	−0.821057	+	0.570846i	$0.806615\pi$
$492$		0			0
$493$			37.9473i			1.70906i
$494$	−18.0000	+	18.9737i	−0.809858	+	0.853666i
$495$	−6.00000	+	18.9737i	−0.269680	+	0.852803i
$496$			9.48683i			0.425971i
$497$	−30.0000			−1.34568
$498$		0			0
$499$	36.0000			1.61158			0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000			1.61158			0.805791	+	0.592200i	$0.201741\pi$
$500$	12.0000			0.536656
$501$		0			0
$502$	2.00000			0.0892644
$503$			34.7851i			1.55099i	0.631354	+	0.775494i	$0.282499\pi$
$503$			34.7851i			1.55099i	−0.631354	+	0.775494i	$0.717501\pi$
$504$		−	9.48683i		−	0.422577i
$505$			31.6228i			1.40720i
$506$	4.00000	−	12.6491i	0.177822	−	0.562322i
$507$		0			0
$508$	12.0000			0.532414
$509$		−	28.4605i		−	1.26149i	−0.775990	−	0.630745i	$-0.782750\pi$
$509$		−	28.4605i		−	1.26149i	0.775990	−	0.630745i	$-0.217250\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−0.0441942
$513$		0			0
$514$		0			0
$515$		−	18.9737i		−	0.836080i
$516$		0			0
$517$	−8.00000	+	25.2982i	−0.351840	+	1.11261i
$518$	30.0000			1.31812
$519$		0			0
$520$	−12.0000			−0.526235
$521$		−	18.9737i		−	0.831251i	−0.909536	−	0.415626i	$-0.863563\pi$
$521$		−	18.9737i		−	0.831251i	0.909536	−	0.415626i	$-0.136437\pi$
$522$	−18.0000			−0.787839
$523$	6.00000			0.262362			0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000			0.262362			0.131181	+	0.991358i	$0.458123\pi$
$524$			6.32456i			0.276289i
$525$		0			0
$526$			15.8114i			0.689409i
$527$	−60.0000			−2.61364
$528$		0			0
$529$	−7.00000			−0.304348
$530$		−	18.9737i		−	0.824163i
$531$		0			0
$532$	−10.0000	−	9.48683i	−0.433555	−	0.411306i
$533$		0			0
$534$		0			0
$535$	−36.0000			−1.55642
$536$		0			0
$537$		0			0
$538$			28.4605i			1.22702i
$539$	−3.00000	+	9.48683i	−0.129219	+	0.408627i
$540$		0			0
$541$		−	9.48683i		−	0.407871i	−0.978984	−	0.203935i	$-0.934627\pi$
$541$		−	9.48683i		−	0.407871i	0.978984	−	0.203935i	$-0.0653733\pi$
$542$			9.48683i			0.407494i
$543$		0			0
$544$		−	6.32456i		−	0.271163i
$545$	12.0000			0.514024
$546$		0			0
$547$	−12.0000			−0.513083			−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000			−0.513083			−0.256541	+	0.966533i	$0.582583\pi$
$548$	8.00000			0.341743
$549$		−	9.48683i		−	0.404888i
$550$	1.00000	−	3.16228i	0.0426401	−	0.134840i
$551$	−18.0000	+	18.9737i	−0.766826	+	0.808305i
$552$		0			0
$553$		0			0
$554$			3.16228i			0.134352i
$555$		0			0
$556$			12.6491i			0.536442i
$557$		−	15.8114i		−	0.669950i	−0.942227	−	0.334975i	$-0.891272\pi$
$557$		−	15.8114i		−	0.669950i	0.942227	−	0.334975i	$-0.108728\pi$
$558$		−	28.4605i		−	1.20483i
$559$		0			0
$560$		−	6.32456i		−	0.267261i
$561$		0			0
$562$	18.0000			0.759284
$563$	6.00000			0.252870			0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000			0.252870			0.126435	+	0.991975i	$0.459647\pi$
$564$		0			0
$565$		0			0
$566$		−	18.9737i		−	0.797523i
$567$			28.4605i			1.19523i
$568$		−	9.48683i		−	0.398059i
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		−	18.9737i		−	0.794023i	−0.917814	−	0.397012i	$-0.870047\pi$
$571$		−	18.9737i		−	0.794023i	0.917814	−	0.397012i	$-0.129953\pi$
$572$	−6.00000	+	18.9737i	−0.250873	+	0.793329i
$573$		0			0
$574$		0			0
$575$	4.00000			0.166812
$576$	3.00000			0.125000
$577$	12.0000			0.499567			0.249783	−	0.968302i	$-0.419641\pi$
$577$	12.0000			0.499567			0.249783	+	0.968302i	$0.419641\pi$
$578$	23.0000			0.956674
$579$		0			0
$580$	−12.0000			−0.498273
$581$	20.0000			0.829740
$582$		0			0
$583$	−30.0000	−	9.48683i	−1.24247	−	0.392904i
$584$		0			0
$585$	36.0000			1.48842
$586$	−6.00000			−0.247858
$587$	28.0000			1.15568			0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000			1.15568			0.577842	+	0.816149i	$0.303895\pi$
$588$		0			0
$589$	−30.0000	−	28.4605i	−1.23613	−	1.17269i
$590$		0			0
$591$		0			0
$592$			9.48683i			0.389906i
$593$			25.2982i			1.03887i	0.854509	+	0.519437i	$0.173858\pi$
$593$			25.2982i			1.03887i	−0.854509	+	0.519437i	$0.826142\pi$
$594$		0			0
$595$	40.0000			1.63984
$596$			3.16228i			0.129532i
$597$		0			0
$598$	−24.0000			−0.981433
$599$		−	9.48683i		−	0.387621i	−0.981039	−	0.193811i	$-0.937915\pi$
$599$		−	9.48683i		−	0.387621i	0.981039	−	0.193811i	$-0.0620848\pi$
$600$		0			0
$601$	30.0000			1.22373			0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000			1.22373			0.611863	+	0.790964i	$0.290420\pi$
$602$		0			0
$603$		0			0
$604$	12.0000			0.488273
$605$	18.0000	+	12.6491i	0.731804	+	0.514259i
$606$		0			0
$607$	12.0000			0.487065			0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000			0.487065			0.243532	+	0.969893i	$0.421694\pi$
$608$	3.00000	−	3.16228i	0.121666	−	0.128247i
$609$		0			0
$610$		−	6.32456i		−	0.256074i
$611$	48.0000			1.94187
$612$			18.9737i			0.766965i
$613$			22.1359i			0.894062i	0.894518	+	0.447031i	$0.147519\pi$
$613$			22.1359i			0.894062i	−0.894518	+	0.447031i	$0.852481\pi$
$614$	12.0000			0.484281
$615$		0			0
$616$	−10.0000	−	3.16228i	−0.402911	−	0.127412i
$617$	28.0000			1.12724			0.563619	−	0.826035i	$-0.309409\pi$
$617$	28.0000			1.12724			0.563619	+	0.826035i	$0.309409\pi$
$618$		0			0
$619$	−30.0000			−1.20580			−0.602901	−	0.797816i	$-0.705989\pi$
$619$	−30.0000			−1.20580			−0.602901	+	0.797816i	$0.705989\pi$
$620$		−	18.9737i		−	0.762001i
$621$		0			0
$622$	−20.0000			−0.801927
$623$		0			0
$624$		0			0
$625$	−19.0000			−0.760000
$626$	−24.0000			−0.959233
$627$		0			0
$628$	18.0000			0.718278
$629$	−60.0000			−2.39236
$630$			18.9737i			0.755929i
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		−	28.4605i		−	1.13031i
$635$	−24.0000			−0.952411
$636$		0			0
$637$	18.0000			0.713186
$638$	−6.00000	+	18.9737i	−0.237542	+	0.751175i
$639$			28.4605i			1.12588i
$640$	2.00000			0.0790569
$641$			18.9737i			0.749415i	0.927143	+	0.374707i	$0.122257\pi$
$641$			18.9737i			0.749415i	−0.927143	+	0.374707i	$0.877743\pi$
$642$		0			0
$643$	−4.00000			−0.157745			−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000			−0.157745			−0.0788723	+	0.996885i	$0.525132\pi$
$644$		−	12.6491i		−	0.498445i
$645$		0			0
$646$	20.0000	+	18.9737i	0.786889	+	0.746509i
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$	−9.00000			−0.353553
$649$		0			0
$650$	−6.00000			−0.235339
$651$		0			0
$652$	−6.00000			−0.234978
$653$	−46.0000			−1.80012			−0.900060	−	0.435767i	$-0.856477\pi$
$653$	−46.0000			−1.80012			−0.900060	+	0.435767i	$0.856477\pi$
$654$		0			0
$655$		−	12.6491i		−	0.494242i
$656$		0			0
$657$		0			0
$658$			25.2982i			0.986227i
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$		−	47.4342i		−	1.84498i	−0.386027	−	0.922488i	$-0.626153\pi$
$661$		−	47.4342i		−	1.84498i	0.386027	−	0.922488i	$-0.373847\pi$
$662$			18.9737i			0.737432i
$663$		0			0
$664$			6.32456i			0.245440i
$665$	20.0000	+	18.9737i	0.775567	+	0.735767i
$666$		−	28.4605i		−	1.10282i
$667$	−24.0000			−0.929284
$668$	−12.0000			−0.464294
$669$		0			0
$670$		0			0
$671$	−10.0000	−	3.16228i	−0.386046	−	0.122078i
$672$		0			0
$673$	24.0000			0.925132			0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000			0.925132			0.462566	+	0.886585i	$0.346929\pi$
$674$	18.0000			0.693334
$675$		0			0
$676$	23.0000			0.884615
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$	60.0000			2.30259
$680$			12.6491i			0.485071i
$681$		0			0
$682$	−30.0000	−	9.48683i	−1.14876	−	0.363270i
$683$		−	18.9737i		−	0.726007i	−0.931788	−	0.363004i	$-0.881751\pi$
$683$		−	18.9737i		−	0.726007i	0.931788	−	0.363004i	$-0.118249\pi$
$684$	−9.00000	+	9.48683i	−0.344124	+	0.362738i
$685$	−16.0000			−0.611329
$686$		−	12.6491i		−	0.482945i
$687$		0			0
$688$		0			0
$689$			56.9210i			2.16852i
$690$		0			0
$691$	10.0000			0.380418			0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000			0.380418			0.190209	+	0.981744i	$0.439083\pi$
$692$	−6.00000			−0.228086
$693$	30.0000	+	9.48683i	1.13961	+	0.360375i
$694$			12.6491i			0.480154i
$695$		−	25.2982i		−	0.959616i
$696$		0			0
$697$		0			0
$698$		−	28.4605i		−	1.07725i
$699$		0			0
$700$		−	3.16228i		−	0.119523i
$701$			15.8114i			0.597188i	0.954380	+	0.298594i	$0.0965176\pi$
$701$			15.8114i			0.597188i	−0.954380	+	0.298594i	$0.903482\pi$
$702$		0			0
$703$	−30.0000	−	28.4605i	−1.13147	−	1.07341i
$704$	1.00000	−	3.16228i	0.0376889	−	0.119183i
$705$		0			0
$706$	4.00000			0.150542
$707$	50.0000			1.88044
$708$		0			0
$709$	−46.0000			−1.72757			−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000			−1.72757			−0.863783	+	0.503864i	$0.831911\pi$
$710$			18.9737i			0.712069i
$711$		0			0
$712$		0			0
$713$		−	37.9473i		−	1.42114i
$714$		0			0
$715$	12.0000	−	37.9473i	0.448775	−	1.41915i
$716$			18.9737i			0.709079i
$717$		0			0
$718$		−	22.1359i		−	0.826106i
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$	−6.00000			−0.223607
$721$	−30.0000			−1.11726
$722$	1.00000	+	18.9737i	0.0372161	+	0.706127i
$723$		0			0
$724$			9.48683i			0.352575i
$725$	−6.00000			−0.222834
$726$		0			0
$727$	−28.0000			−1.03846			−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000			−1.03846			−0.519231	+	0.854634i	$0.673782\pi$
$728$			18.9737i			0.703211i
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$			9.48683i			0.350404i	0.984532	+	0.175202i	$0.0560579\pi$
$733$			9.48683i			0.350404i	−0.984532	+	0.175202i	$0.943942\pi$
$734$	−8.00000			−0.295285
$735$		0			0
$736$	4.00000			0.147442
$737$		0			0
$738$		0			0
$739$			6.32456i			0.232653i	0.993211	+	0.116326i	$0.0371118\pi$
$739$			6.32456i			0.232653i	−0.993211	+	0.116326i	$0.962888\pi$
$740$		−	18.9737i		−	0.697486i
$741$		0			0
$742$	−30.0000			−1.10133
$743$	−24.0000			−0.880475			−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000			−0.880475			−0.440237	+	0.897881i	$0.645106\pi$
$744$		0			0
$745$		−	6.32456i		−	0.231714i
$746$	−6.00000			−0.219676
$747$		−	18.9737i		−	0.694210i
$748$	20.0000	+	6.32456i	0.731272	+	0.231249i
$749$			56.9210i			2.07985i
$750$		0			0
$751$			28.4605i			1.03854i	0.854611	+	0.519269i	$0.173796\pi$
$751$			28.4605i			1.03854i	−0.854611	+	0.519269i	$0.826204\pi$
$752$	−8.00000			−0.291730
$753$		0			0
$754$	36.0000			1.31104
$755$	−24.0000			−0.873449
$756$		0			0
$757$	42.0000			1.52652			0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000			1.52652			0.763258	+	0.646094i	$0.223599\pi$
$758$		−	18.9737i		−	0.689155i
$759$		0			0
$760$	−6.00000	+	6.32456i	−0.217643	+	0.229416i
$761$			6.32456i			0.229265i	0.993408	+	0.114632i	$0.0365690\pi$
$761$			6.32456i			0.229265i	−0.993408	+	0.114632i	$0.963431\pi$
$762$		0			0
$763$		−	18.9737i		−	0.686893i
$764$	−20.0000			−0.723575
$765$		−	37.9473i		−	1.37199i
$766$			9.48683i			0.342773i
$767$		0			0
$768$		0			0
$769$			44.2719i			1.59649i	0.602336	+	0.798243i	$0.294237\pi$
$769$			44.2719i			1.59649i	−0.602336	+	0.798243i	$0.705763\pi$
$770$	20.0000	+	6.32456i	0.720750	+	0.227921i
$771$		0			0
$772$	24.0000			0.863779
$773$			9.48683i			0.341218i	0.985339	+	0.170609i	$0.0545734\pi$
$773$			9.48683i			0.341218i	−0.985339	+	0.170609i	$0.945427\pi$
$774$		0			0
$775$		−	9.48683i		−	0.340777i
$776$			18.9737i			0.681115i
$777$		0			0
$778$	10.0000			0.358517
$779$		0			0
$780$		0			0
$781$	30.0000	+	9.48683i	1.07348	+	0.339466i
$782$			25.2982i			0.904663i
$783$		0			0
$784$	−3.00000			−0.107143
$785$	−36.0000			−1.28490
$786$		0			0
$787$	−12.0000			−0.427754			−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000			−0.427754			−0.213877	+	0.976861i	$0.568609\pi$
$788$			15.8114i			0.563257i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−3.00000	+	9.48683i	−0.106600	+	0.337100i
$793$			18.9737i			0.673775i
$794$	2.00000			0.0709773
$795$		0			0
$796$	−20.0000			−0.708881
$797$		−	9.48683i		−	0.336041i	−0.985784	−	0.168020i	$-0.946263\pi$
$797$		−	9.48683i		−	0.336041i	0.985784	−	0.168020i	$-0.0537375\pi$
$798$		0			0
$799$		−	50.5964i		−	1.78997i
$800$	1.00000			0.0353553
$801$		0			0
$802$		−	18.9737i		−	0.669983i
$803$		0			0
$804$		0			0
$805$			25.2982i			0.891645i
$806$			56.9210i			2.00496i
$807$		0			0
$808$			15.8114i			0.556243i
$809$		−	31.6228i		−	1.11180i	−0.831250	−	0.555899i	$-0.812374\pi$
$809$		−	31.6228i		−	1.11180i	0.831250	−	0.555899i	$-0.187626\pi$
$810$	18.0000			0.632456
$811$	18.0000			0.632065			0.316033	−	0.948748i	$-0.397649\pi$
$811$	18.0000			0.632065			0.316033	+	0.948748i	$0.397649\pi$
$812$			18.9737i			0.665845i
$813$		0			0
$814$	−30.0000	−	9.48683i	−1.05150	−	0.332513i
$815$	12.0000			0.420342
$816$		0			0
$817$		0			0
$818$	6.00000			0.209785
$819$		−	56.9210i		−	1.98898i
$820$		0			0
$821$			22.1359i			0.772550i	0.922384	+	0.386275i	$0.126238\pi$
$821$			22.1359i			0.772550i	−0.922384	+	0.386275i	$0.873762\pi$
$822$		0			0
$823$	24.0000			0.836587			0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000			0.836587			0.418294	+	0.908312i	$0.362628\pi$
$824$		−	9.48683i		−	0.330489i
$825$		0			0
$826$		0			0
$827$	−42.0000			−1.46048			−0.730242	−	0.683189i	$-0.760592\pi$
$827$	−42.0000			−1.46048			−0.730242	+	0.683189i	$0.760592\pi$
$828$	−12.0000			−0.417029
$829$			47.4342i			1.64746i	0.566985	+	0.823728i	$0.308110\pi$
$829$			47.4342i			1.64746i	−0.566985	+	0.823728i	$0.691890\pi$
$830$		−	12.6491i		−	0.439057i
$831$		0			0
$832$	−6.00000			−0.208013
$833$		−	18.9737i		−	0.657399i
$834$		0			0
$835$	24.0000			0.830554
$836$	7.00000	+	12.6491i	0.242100	+	0.437479i
$837$		0			0
$838$	−4.00000			−0.138178
$839$		−	9.48683i		−	0.327522i	−0.986500	−	0.163761i	$-0.947637\pi$
$839$		−	9.48683i		−	0.327522i	0.986500	−	0.163761i	$-0.0523626\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$			9.48683i			0.326938i
$843$		0			0
$844$	−12.0000			−0.413057
$845$	−46.0000			−1.58245
$846$	24.0000			0.825137
$847$	20.0000	−	28.4605i	0.687208	−	0.977914i
$848$		−	9.48683i		−	0.325779i
$849$		0			0
$850$			6.32456i			0.216930i
$851$		−	37.9473i		−	1.30082i
$852$		0			0
$853$		−	9.48683i		−	0.324823i	−0.986723	−	0.162411i	$-0.948073\pi$
$853$		−	9.48683i		−	0.324823i	0.986723	−	0.162411i	$-0.0519272\pi$
$854$	−10.0000			−0.342193
$855$	18.0000	−	18.9737i	0.615587	−	0.648886i
$856$	−18.0000			−0.615227
$857$	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
$859$	36.0000			1.22830			0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000			1.22830			0.614152	+	0.789188i	$0.289498\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			9.48683i			0.322936i	0.986878	+	0.161468i	$0.0516228\pi$
$863$			9.48683i			0.322936i	−0.986878	+	0.161468i	$0.948377\pi$
$864$		0			0
$865$	12.0000			0.408012
$866$		−	18.9737i		−	0.644751i
$867$		0			0
$868$	−30.0000			−1.01827
$869$		0			0
$870$		0			0
$871$		0			0
$872$	6.00000			0.203186
$873$		−	56.9210i		−	1.92648i
$874$	−12.0000	+	12.6491i	−0.405906	+	0.427863i
$875$			37.9473i			1.28285i
$876$		0			0
$877$	18.0000			0.607817			0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000			0.607817			0.303908	+	0.952701i	$0.401708\pi$
$878$	−36.0000			−1.21494
$879$		0			0
$880$	−2.00000	+	6.32456i	−0.0674200	+	0.213201i
$881$	20.0000			0.673817			0.336909	−	0.941537i	$-0.390619\pi$
$881$	20.0000			0.673817			0.336909	+	0.941537i	$0.390619\pi$
$882$	9.00000			0.303046
$883$	6.00000			0.201916			0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000			0.201916			0.100958	+	0.994891i	$0.467809\pi$
$884$		−	37.9473i		−	1.27631i
$885$		0			0
$886$	−4.00000			−0.134383
$887$	48.0000			1.61168			0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000			1.61168			0.805841	+	0.592132i	$0.201714\pi$
$888$		0			0
$889$			37.9473i			1.27271i
$890$		0			0
$891$	9.00000	−	28.4605i	0.301511	−	0.953463i
$892$			9.48683i			0.317643i
$893$	24.0000	−	25.2982i	0.803129	−	0.846573i
$894$		0			0
$895$		−	37.9473i		−	1.26844i
$896$		−	3.16228i		−	0.105644i
$897$		0			0
$898$		−	18.9737i		−	0.633159i
$899$			56.9210i			1.89842i
$900$	−3.00000			−0.100000
$901$	60.0000			1.99889
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	18.9737i		−	0.630706i
$906$		0			0
$907$		−	37.9473i		−	1.26002i	−0.776587	−	0.630010i	$-0.783051\pi$
$907$		−	37.9473i		−	1.26002i	0.776587	−	0.630010i	$-0.216949\pi$
$908$	−12.0000			−0.398234
$909$		−	47.4342i		−	1.57329i
$910$		−	37.9473i		−	1.25794i
$911$		−	9.48683i		−	0.314313i	−0.987574	−	0.157156i	$-0.949767\pi$
$911$		−	9.48683i		−	0.314313i	0.987574	−	0.157156i	$-0.0502327\pi$
$912$		0			0
$913$	−20.0000	−	6.32456i	−0.661903	−	0.209312i
$914$		−	18.9737i		−	0.627593i
$915$		0			0
$916$	6.00000			0.198246
$917$	−20.0000			−0.660458
$918$		0			0
$919$			22.1359i			0.730197i	0.930969	+	0.365099i	$0.118965\pi$
$919$			22.1359i			0.730197i	−0.930969	+	0.365099i	$0.881035\pi$
$920$	−8.00000			−0.263752
$921$		0			0
$922$			3.16228i			0.104144i
$923$		−	56.9210i		−	1.87358i
$924$		0			0
$925$		−	9.48683i		−	0.311925i
$926$	−4.00000			−0.131448
$927$			28.4605i			0.934765i
$928$	−6.00000			−0.196960
$929$	−40.0000			−1.31236			−0.656179	−	0.754606i	$-0.727828\pi$
$929$	−40.0000			−1.31236			−0.656179	+	0.754606i	$0.727828\pi$
$930$		0			0
$931$	9.00000	−	9.48683i	0.294963	−	0.310918i
$932$		−	12.6491i		−	0.414335i
$933$		0			0
$934$	28.0000			0.916188
$935$	−40.0000	−	12.6491i	−1.30814	−	0.413670i
$936$	18.0000			0.588348
$937$		−	37.9473i		−	1.23969i	−0.784726	−	0.619843i	$-0.787196\pi$
$937$		−	37.9473i		−	1.23969i	0.784726	−	0.619843i	$-0.212804\pi$
$938$		0			0
$939$		0			0
$940$	16.0000			0.521862
$941$	−18.0000			−0.586783			−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000			−0.586783			−0.293392	+	0.955992i	$0.594784\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	28.0000			0.909878			0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000			0.909878			0.454939	+	0.890523i	$0.349661\pi$
$948$		0			0
$949$		0			0
$950$	−3.00000	+	3.16228i	−0.0973329	+	0.102598i
$951$		0			0
$952$	20.0000			0.648204
$953$	24.0000			0.777436			0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000			0.777436			0.388718	+	0.921357i	$0.372918\pi$
$954$			28.4605i			0.921443i
$955$	40.0000			1.29437
$956$			3.16228i			0.102275i
$957$		0			0
$958$		−	3.16228i		−	0.102169i
$959$			25.2982i			0.816922i
$960$		0			0
$961$	−59.0000			−1.90323
$962$			56.9210i			1.83521i
$963$	54.0000			1.74013
$964$	18.0000			0.579741
$965$	−48.0000			−1.54517
$966$		0			0
$967$			41.1096i			1.32200i	0.750388	+	0.660998i	$0.229867\pi$
$967$			41.1096i			1.32200i	−0.750388	+	0.660998i	$0.770133\pi$
$968$	9.00000	+	6.32456i	0.289271	+	0.203279i
$969$		0			0
$970$		−	37.9473i		−	1.21842i
$971$			56.9210i			1.82668i	0.407196	+	0.913341i	$0.366507\pi$
$971$			56.9210i			1.82668i	−0.407196	+	0.913341i	$0.633493\pi$
$972$		0			0
$973$	−40.0000			−1.28234
$974$			28.4605i			0.911933i
$975$		0			0
$976$		−	3.16228i		−	0.101222i
$977$			18.9737i			0.607021i	0.952828	+	0.303511i	$0.0981588\pi$
$977$			18.9737i			0.607021i	−0.952828	+	0.303511i	$0.901841\pi$
$978$		0			0
$979$		0			0
$980$	6.00000			0.191663
$981$	−18.0000			−0.574696
$982$		−	25.2982i		−	0.807299i
$983$		−	47.4342i		−	1.51291i	−0.654043	−	0.756457i	$-0.726928\pi$
$983$		−	47.4342i		−	1.51291i	0.654043	−	0.756457i	$-0.273072\pi$
$984$		0			0
$985$		−	31.6228i		−	1.00759i
$986$		−	37.9473i		−	1.20849i
$987$		0			0
$988$	18.0000	−	18.9737i	0.572656	−	0.603633i
$989$		0			0
$990$	6.00000	−	18.9737i	0.190693	−	0.603023i
$991$			47.4342i			1.50680i	0.657565	+	0.753398i	$0.271587\pi$
$991$			47.4342i			1.50680i	−0.657565	+	0.753398i	$0.728413\pi$
$992$		−	9.48683i		−	0.301207i
$993$		0			0
$994$	30.0000			0.951542
$995$	40.0000			1.26809
$996$		0			0
$997$			3.16228i			0.100150i	0.998745	+	0.0500752i	$0.0159461\pi$
$997$			3.16228i			0.100150i	−0.998745	+	0.0500752i	$0.984054\pi$
$998$	−36.0000			−1.13956
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	418.2.b.a.417.2	yes	2
3.2	odd	2		3762.2.g.f.2089.2		2
11.10	odd	2		418.2.b.b.417.1	yes	2
19.18	odd	2		418.2.b.b.417.2	yes	2
33.32	even	2		3762.2.g.c.2089.1		2
57.56	even	2		3762.2.g.c.2089.2		2
209.208	even	2	inner	418.2.b.a.417.1	✓	2
627.626	odd	2		3762.2.g.f.2089.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.b.a.417.1	✓	2	209.208	even	2	inner
418.2.b.a.417.2	yes	2	1.1	even	1	trivial
418.2.b.b.417.1	yes	2	11.10	odd	2
418.2.b.b.417.2	yes	2	19.18	odd	2
3762.2.g.c.2089.1		2	33.32	even	2
3762.2.g.c.2089.2		2	57.56	even	2
3762.2.g.f.2089.1		2	627.626	odd	2
3762.2.g.f.2089.2		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 418 = 2 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	418.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +3.16228i q^{7} -1.00000 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +3.16228i q^{7} -1.00000 q^{8} +3.00000 q^{9} +2.00000 q^{10} +(1.00000 - 3.16228i) q^{11} -6.00000 q^{13} -3.16228i q^{14} +1.00000 q^{16} +6.32456i q^{17} -3.00000 q^{18} +(-3.00000 + 3.16228i) q^{19} -2.00000 q^{20} +(-1.00000 + 3.16228i) q^{22} -4.00000 q^{23} -1.00000 q^{25} +6.00000 q^{26} +3.16228i q^{28} +6.00000 q^{29} +9.48683i q^{31} -1.00000 q^{32} -6.32456i q^{34} -6.32456i q^{35} +3.00000 q^{36} +9.48683i q^{37} +(3.00000 - 3.16228i) q^{38} +2.00000 q^{40} +(1.00000 - 3.16228i) q^{44} -6.00000 q^{45} +4.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} -9.48683i q^{53} +(-2.00000 + 6.32456i) q^{55} -3.16228i q^{56} -6.00000 q^{58} -3.16228i q^{61} -9.48683i q^{62} +9.48683i q^{63} +1.00000 q^{64} +12.0000 q^{65} +6.32456i q^{68} +6.32456i q^{70} +9.48683i q^{71} -3.00000 q^{72} -9.48683i q^{74} +(-3.00000 + 3.16228i) q^{76} +(10.0000 + 3.16228i) q^{77} -2.00000 q^{80} +9.00000 q^{81} -6.32456i q^{83} -12.6491i q^{85} +(-1.00000 + 3.16228i) q^{88} +6.00000 q^{90} -18.9737i q^{91} -4.00000 q^{92} +8.00000 q^{94} +(6.00000 - 6.32456i) q^{95} -18.9737i q^{97} +3.00000 q^{98} +(3.00000 - 9.48683i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 + 6 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 6 q^{9} + 4 q^{10} + 2 q^{11} - 12 q^{13} + 2 q^{16} - 6 q^{18} - 6 q^{19} - 4 q^{20} - 2 q^{22} - 8 q^{23} - 2 q^{25} + 12 q^{26} + 12 q^{29} - 2 q^{32} + 6 q^{36} + 6 q^{38} + 4 q^{40} + 2 q^{44} - 12 q^{45} + 8 q^{46} - 16 q^{47} - 6 q^{49} + 2 q^{50} - 12 q^{52} - 4 q^{55} - 12 q^{58} + 2 q^{64} + 24 q^{65} - 6 q^{72} - 6 q^{76} + 20 q^{77} - 4 q^{80} + 18 q^{81} - 2 q^{88} + 12 q^{90} - 8 q^{92} + 16 q^{94} + 12 q^{95} + 6 q^{98} + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 + 6 * q^9 + 4 * q^10 + 2 * q^11 - 12 * q^13 + 2 * q^16 - 6 * q^18 - 6 * q^19 - 4 * q^20 - 2 * q^22 - 8 * q^23 - 2 * q^25 + 12 * q^26 + 12 * q^29 - 2 * q^32 + 6 * q^36 + 6 * q^38 + 4 * q^40 + 2 * q^44 - 12 * q^45 + 8 * q^46 - 16 * q^47 - 6 * q^49 + 2 * q^50 - 12 * q^52 - 4 * q^55 - 12 * q^58 + 2 * q^64 + 24 * q^65 - 6 * q^72 - 6 * q^76 + 20 * q^77 - 4 * q^80 + 18 * q^81 - 2 * q^88 + 12 * q^90 - 8 * q^92 + 16 * q^94 + 12 * q^95 + 6 * q^98 + 6 * q^99

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