LMFDB - Embedded newform 3330.2.d.k.1999.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1999,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3330.1999");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.d (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:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1999.4
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3330.1999
Dual form			3330.2.d.k.1999.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(0.707107 + 2.12132i) q^{5} +0.414214i q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +(0.707107 + 2.12132i) q^{5} +0.414214i q^{7} -1.00000i q^{8} +(-2.12132 + 0.707107i) q^{10} +3.58579 q^{11} +3.24264i q^{13} -0.414214 q^{14} +1.00000 q^{16} -6.41421i q^{17} +7.82843 q^{19} +(-0.707107 - 2.12132i) q^{20} +3.58579i q^{22} +3.00000i q^{23} +(-4.00000 + 3.00000i) q^{25} -3.24264 q^{26} -0.414214i q^{28} -7.07107 q^{29} +7.65685 q^{31} +1.00000i q^{32} +6.41421 q^{34} +(-0.878680 + 0.292893i) q^{35} -1.00000i q^{37} +7.82843i q^{38} +(2.12132 - 0.707107i) q^{40} +9.41421 q^{41} +6.48528i q^{43} -3.58579 q^{44} -3.00000 q^{46} -7.89949i q^{47} +6.82843 q^{49} +(-3.00000 - 4.00000i) q^{50} -3.24264i q^{52} +1.82843i q^{53} +(2.53553 + 7.60660i) q^{55} +0.414214 q^{56} -7.07107i q^{58} +1.07107 q^{59} +10.4853 q^{61} +7.65685i q^{62} -1.00000 q^{64} +(-6.87868 + 2.29289i) q^{65} +4.82843i q^{67} +6.41421i q^{68} +(-0.292893 - 0.878680i) q^{70} +6.58579 q^{71} -7.82843i q^{73} +1.00000 q^{74} -7.82843 q^{76} +1.48528i q^{77} -3.75736 q^{79} +(0.707107 + 2.12132i) q^{80} +9.41421i q^{82} -12.8995i q^{83} +(13.6066 - 4.53553i) q^{85} -6.48528 q^{86} -3.58579i q^{88} -16.8995 q^{89} -1.34315 q^{91} -3.00000i q^{92} +7.89949 q^{94} +(5.53553 + 16.6066i) q^{95} +18.7279i q^{97} +6.82843i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 20 q^{11} + 4 q^{14} + 4 q^{16} + 20 q^{19} - 16 q^{25} + 4 q^{26} + 8 q^{31} + 20 q^{34} - 12 q^{35} + 32 q^{41} - 20 q^{44} - 12 q^{46} + 16 q^{49} - 12 q^{50} - 4 q^{55} - 4 q^{56} - 24 q^{59} + 8 q^{61} - 4 q^{64} - 36 q^{65} - 4 q^{70} + 32 q^{71} + 4 q^{74} - 20 q^{76} - 32 q^{79} + 12 q^{85} + 8 q^{86} - 28 q^{89} - 28 q^{91} - 8 q^{94} + 8 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 + 20 * q^11 + 4 * q^14 + 4 * q^16 + 20 * q^19 - 16 * q^25 + 4 * q^26 + 8 * q^31 + 20 * q^34 - 12 * q^35 + 32 * q^41 - 20 * q^44 - 12 * q^46 + 16 * q^49 - 12 * q^50 - 4 * q^55 - 4 * q^56 - 24 * q^59 + 8 * q^61 - 4 * q^64 - 36 * q^65 - 4 * q^70 + 32 * q^71 + 4 * q^74 - 20 * q^76 - 32 * q^79 + 12 * q^85 + 8 * q^86 - 28 * q^89 - 28 * q^91 - 8 * q^94 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times$.

$n$	$371$	$631$	$667$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	0.707107	+	2.12132i	0.316228	+	0.948683i
$5$	0.707107	+	2.12132i	0.316228	+	0.948683i
$6$		0			0
$7$			0.414214i			0.156558i	0.996931	+	0.0782790i	$0.0249425\pi$
$7$			0.414214i			0.156558i	−0.996931	+	0.0782790i	$0.975058\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−2.12132	+	0.707107i	−0.670820	+	0.223607i
$11$	3.58579			1.08116			0.540578	−	0.841294i	$-0.318206\pi$
$11$	3.58579			1.08116			0.540578	+	0.841294i	$0.318206\pi$
$12$		0			0
$13$			3.24264i			0.899347i	0.893193	+	0.449673i	$0.148460\pi$
$13$			3.24264i			0.899347i	−0.893193	+	0.449673i	$0.851540\pi$
$14$	−0.414214			−0.110703
$15$		0			0
$16$	1.00000			0.250000
$17$		−	6.41421i		−	1.55568i	−0.628465	−	0.777838i	$-0.716317\pi$
$17$		−	6.41421i		−	1.55568i	0.628465	−	0.777838i	$-0.283683\pi$
$18$		0			0
$19$	7.82843			1.79596			0.897982	−	0.440032i	$-0.145033\pi$
$19$	7.82843			1.79596			0.897982	+	0.440032i	$0.145033\pi$
$20$	−0.707107	−	2.12132i	−0.158114	−	0.474342i
$21$		0			0
$22$			3.58579i			0.764492i
$23$			3.00000i			0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$			3.00000i			0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$		0			0
$25$	−4.00000	+	3.00000i	−0.800000	+	0.600000i
$26$	−3.24264			−0.635934
$27$		0			0
$28$		−	0.414214i		−	0.0782790i
$29$	−7.07107			−1.31306			−0.656532	−	0.754298i	$-0.727977\pi$
$29$	−7.07107			−1.31306			−0.656532	+	0.754298i	$0.727977\pi$
$30$		0			0
$31$	7.65685			1.37521			0.687606	−	0.726084i	$-0.258662\pi$
$31$	7.65685			1.37521			0.687606	+	0.726084i	$0.258662\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	6.41421			1.10003
$35$	−0.878680	+	0.292893i	−0.148524	+	0.0495080i
$36$		0			0
$37$		−	1.00000i		−	0.164399i
$37$		−	1.00000i		−	0.164399i
$38$			7.82843i			1.26994i
$39$		0			0
$40$	2.12132	−	0.707107i	0.335410	−	0.111803i
$41$	9.41421			1.47025			0.735127	−	0.677930i	$-0.237123\pi$
$41$	9.41421			1.47025			0.735127	+	0.677930i	$0.237123\pi$
$42$		0			0
$43$			6.48528i			0.988996i	0.869179	+	0.494498i	$0.164648\pi$
$43$			6.48528i			0.988996i	−0.869179	+	0.494498i	$0.835352\pi$
$44$	−3.58579			−0.540578
$45$		0			0
$46$	−3.00000			−0.442326
$47$		−	7.89949i		−	1.15226i	−0.817358	−	0.576130i	$-0.804562\pi$
$47$		−	7.89949i		−	1.15226i	0.817358	−	0.576130i	$-0.195438\pi$
$48$		0			0
$49$	6.82843			0.975490
$50$	−3.00000	−	4.00000i	−0.424264	−	0.565685i
$51$		0			0
$52$		−	3.24264i		−	0.449673i
$53$			1.82843i			0.251154i	0.992084	+	0.125577i	$0.0400782\pi$
$53$			1.82843i			0.251154i	−0.992084	+	0.125577i	$0.959922\pi$
$54$		0			0
$55$	2.53553	+	7.60660i	0.341891	+	1.02567i
$56$	0.414214			0.0553516
$57$		0			0
$58$		−	7.07107i		−	0.928477i
$59$	1.07107			0.139441			0.0697206	−	0.997567i	$-0.477789\pi$
$59$	1.07107			0.139441			0.0697206	+	0.997567i	$0.477789\pi$
$60$		0			0
$61$	10.4853			1.34250			0.671251	−	0.741230i	$-0.265757\pi$
$61$	10.4853			1.34250			0.671251	+	0.741230i	$0.265757\pi$
$62$			7.65685i			0.972421i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−6.87868	+	2.29289i	−0.853195	+	0.284398i
$66$		0			0
$67$			4.82843i			0.589886i	0.955515	+	0.294943i	$0.0953007\pi$
$67$			4.82843i			0.589886i	−0.955515	+	0.294943i	$0.904699\pi$
$68$			6.41421i			0.777838i
$69$		0			0
$70$	−0.292893	−	0.878680i	−0.0350074	−	0.105022i
$71$	6.58579			0.781589			0.390795	−	0.920478i	$-0.372200\pi$
$71$	6.58579			0.781589			0.390795	+	0.920478i	$0.372200\pi$
$72$		0			0
$73$		−	7.82843i		−	0.916248i	−0.888888	−	0.458124i	$-0.848522\pi$
$73$		−	7.82843i		−	0.916248i	0.888888	−	0.458124i	$-0.151478\pi$
$74$	1.00000			0.116248
$75$		0			0
$76$	−7.82843			−0.897982
$77$			1.48528i			0.169264i
$78$		0			0
$79$	−3.75736			−0.422736			−0.211368	−	0.977407i	$-0.567792\pi$
$79$	−3.75736			−0.422736			−0.211368	+	0.977407i	$0.567792\pi$
$80$	0.707107	+	2.12132i	0.0790569	+	0.237171i
$81$		0			0
$82$			9.41421i			1.03963i
$83$		−	12.8995i		−	1.41590i	−0.706261	−	0.707952i	$-0.749619\pi$
$83$		−	12.8995i		−	1.41590i	0.706261	−	0.707952i	$-0.250381\pi$
$84$		0			0
$85$	13.6066	−	4.53553i	1.47584	−	0.491948i
$86$	−6.48528			−0.699326
$87$		0			0
$88$		−	3.58579i		−	0.382246i
$89$	−16.8995			−1.79134			−0.895671	−	0.444716i	$-0.853304\pi$
$89$	−16.8995			−1.79134			−0.895671	+	0.444716i	$0.853304\pi$
$90$		0			0
$91$	−1.34315			−0.140800
$92$		−	3.00000i		−	0.312772i
$93$		0			0
$94$	7.89949			0.814771
$95$	5.53553	+	16.6066i	0.567934	+	1.70380i
$96$		0			0
$97$			18.7279i			1.90153i	0.309909	+	0.950766i	$0.399701\pi$
$97$			18.7279i			1.90153i	−0.309909	+	0.950766i	$0.600299\pi$
$98$			6.82843i			0.689775i
$99$		0			0
$100$	4.00000	−	3.00000i	0.400000	−	0.300000i
$101$	−17.3137			−1.72278			−0.861389	−	0.507946i	$-0.830405\pi$
$101$	−17.3137			−1.72278			−0.861389	+	0.507946i	$0.830405\pi$
$102$		0			0
$103$			0.928932i			0.0915304i	0.998952	+	0.0457652i	$0.0145726\pi$
$103$			0.928932i			0.0915304i	−0.998952	+	0.0457652i	$0.985427\pi$
$104$	3.24264			0.317967
$105$		0			0
$106$	−1.82843			−0.177593
$107$		−	1.92893i		−	0.186477i	−0.995644	−	0.0932385i	$-0.970278\pi$
$107$		−	1.92893i		−	0.186477i	0.995644	−	0.0932385i	$-0.0297219\pi$
$108$		0			0
$109$	−13.2426			−1.26841			−0.634207	−	0.773163i	$-0.718673\pi$
$109$	−13.2426			−1.26841			−0.634207	+	0.773163i	$0.718673\pi$
$110$	−7.60660	+	2.53553i	−0.725261	+	0.241754i
$111$		0			0
$112$			0.414214i			0.0391395i
$113$			2.82843i			0.266076i	0.991111	+	0.133038i	$0.0424732\pi$
$113$			2.82843i			0.266076i	−0.991111	+	0.133038i	$0.957527\pi$
$114$		0			0
$115$	−6.36396	+	2.12132i	−0.593442	+	0.197814i
$116$	7.07107			0.656532
$117$		0			0
$118$			1.07107i			0.0985998i
$119$	2.65685			0.243553
$120$		0			0
$121$	1.85786			0.168897
$122$			10.4853i			0.949293i
$123$		0			0
$124$	−7.65685			−0.687606
$125$	−9.19239	−	6.36396i	−0.822192	−	0.569210i
$126$		0			0
$127$			12.4142i			1.10158i	0.834643	+	0.550792i	$0.185674\pi$
$127$			12.4142i			1.10158i	−0.834643	+	0.550792i	$0.814326\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−2.29289	−	6.87868i	−0.201100	−	0.603300i
$131$	17.8995			1.56389			0.781943	−	0.623350i	$-0.214229\pi$
$131$	17.8995			1.56389			0.781943	+	0.623350i	$0.214229\pi$
$132$		0			0
$133$			3.24264i			0.281173i
$134$	−4.82843			−0.417113
$135$		0			0
$136$	−6.41421			−0.550014
$137$			13.5563i			1.15820i	0.815258	+	0.579099i	$0.196595\pi$
$137$			13.5563i			1.15820i	−0.815258	+	0.579099i	$0.803405\pi$
$138$		0			0
$139$	−14.4853			−1.22863			−0.614313	−	0.789063i	$-0.710567\pi$
$139$	−14.4853			−1.22863			−0.614313	+	0.789063i	$0.710567\pi$
$140$	0.878680	−	0.292893i	0.0742620	−	0.0247540i
$141$		0			0
$142$			6.58579i			0.552667i
$143$			11.6274i			0.972333i
$144$		0			0
$145$	−5.00000	−	15.0000i	−0.415227	−	1.24568i
$146$	7.82843			0.647885
$147$		0			0
$148$			1.00000i			0.0821995i
$149$	20.4853			1.67822			0.839110	−	0.543962i	$-0.183076\pi$
$149$	20.4853			1.67822			0.839110	+	0.543962i	$0.183076\pi$
$150$		0			0
$151$	−0.414214			−0.0337082			−0.0168541	−	0.999858i	$-0.505365\pi$
$151$	−0.414214			−0.0337082			−0.0168541	+	0.999858i	$0.505365\pi$
$152$		−	7.82843i		−	0.634969i
$153$		0			0
$154$	−1.48528			−0.119687
$155$	5.41421	+	16.2426i	0.434880	+	1.30464i
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		−	3.75736i		−	0.298919i
$159$		0			0
$160$	−2.12132	+	0.707107i	−0.167705	+	0.0559017i
$161$	−1.24264			−0.0979338
$162$		0			0
$163$			13.1421i			1.02937i	0.857379	+	0.514686i	$0.172091\pi$
$163$			13.1421i			1.02937i	−0.857379	+	0.514686i	$0.827909\pi$
$164$	−9.41421			−0.735127
$165$		0			0
$166$	12.8995			1.00119
$167$			7.34315i			0.568230i	0.958790	+	0.284115i	$0.0916997\pi$
$167$			7.34315i			0.568230i	−0.958790	+	0.284115i	$0.908300\pi$
$168$		0			0
$169$	2.48528			0.191175
$170$	4.53553	+	13.6066i	0.347860	+	1.04358i
$171$		0			0
$172$		−	6.48528i		−	0.494498i
$173$		−	10.6569i		−	0.810226i	−0.914267	−	0.405113i	$-0.867232\pi$
$173$		−	10.6569i		−	0.810226i	0.914267	−	0.405113i	$-0.132768\pi$
$174$		0			0
$175$	−1.24264	−	1.65685i	−0.0939348	−	0.125246i
$176$	3.58579			0.270289
$177$		0			0
$178$		−	16.8995i		−	1.26667i
$179$	−13.1716			−0.984490			−0.492245	−	0.870457i	$-0.663824\pi$
$179$	−13.1716			−0.984490			−0.492245	+	0.870457i	$0.663824\pi$
$180$		0			0
$181$	23.4142			1.74036			0.870182	−	0.492730i	$-0.164001\pi$
$181$	23.4142			1.74036			0.870182	+	0.492730i	$0.164001\pi$
$182$		−	1.34315i		−	0.0995606i
$183$		0			0
$184$	3.00000			0.221163
$185$	2.12132	−	0.707107i	0.155963	−	0.0519875i
$186$		0			0
$187$		−	23.0000i		−	1.68193i
$188$			7.89949i			0.576130i
$189$		0			0
$190$	−16.6066	+	5.53553i	−1.20477	+	0.401590i
$191$	2.65685			0.192243			0.0961216	−	0.995370i	$-0.469356\pi$
$191$	2.65685			0.192243			0.0961216	+	0.995370i	$0.469356\pi$
$192$		0			0
$193$		−	13.3137i		−	0.958342i	−0.877722	−	0.479171i	$-0.840937\pi$
$193$		−	13.3137i		−	0.958342i	0.877722	−	0.479171i	$-0.159063\pi$
$194$	−18.7279			−1.34459
$195$		0			0
$196$	−6.82843			−0.487745
$197$			13.1421i			0.936338i	0.883639	+	0.468169i	$0.155086\pi$
$197$			13.1421i			0.936338i	−0.883639	+	0.468169i	$0.844914\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$	3.00000	+	4.00000i	0.212132	+	0.282843i
$201$		0			0
$202$		−	17.3137i		−	1.21819i
$203$		−	2.92893i		−	0.205571i
$204$		0			0
$205$	6.65685	+	19.9706i	0.464935	+	1.39480i
$206$	−0.928932			−0.0647218
$207$		0			0
$208$			3.24264i			0.224837i
$209$	28.0711			1.94172
$210$		0			0
$211$	−26.3848			−1.81640			−0.908201	−	0.418533i	$-0.862544\pi$
$211$	−26.3848			−1.81640			−0.908201	+	0.418533i	$0.862544\pi$
$212$		−	1.82843i		−	0.125577i
$213$		0			0
$214$	1.92893			0.131859
$215$	−13.7574	+	4.58579i	−0.938244	+	0.312748i
$216$		0			0
$217$			3.17157i			0.215300i
$218$		−	13.2426i		−	0.896905i
$219$		0			0
$220$	−2.53553	−	7.60660i	−0.170946	−	0.512837i
$221$	20.7990			1.39909
$222$		0			0
$223$			1.65685i			0.110951i	0.998460	+	0.0554756i	$0.0176675\pi$
$223$			1.65685i			0.110951i	−0.998460	+	0.0554756i	$0.982333\pi$
$224$	−0.414214			−0.0276758
$225$		0			0
$226$	−2.82843			−0.188144
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$	6.72792			0.444594			0.222297	−	0.974979i	$-0.428645\pi$
$229$	6.72792			0.444594			0.222297	+	0.974979i	$0.428645\pi$
$230$	−2.12132	−	6.36396i	−0.139876	−	0.419627i
$231$		0			0
$232$			7.07107i			0.464238i
$233$			14.4853i			0.948962i	0.880266	+	0.474481i	$0.157364\pi$
$233$			14.4853i			0.948962i	−0.880266	+	0.474481i	$0.842636\pi$
$234$		0			0
$235$	16.7574	−	5.58579i	1.09313	−	0.364377i
$236$	−1.07107			−0.0697206
$237$		0			0
$238$			2.65685i			0.172218i
$239$	−24.6274			−1.59302			−0.796508	−	0.604629i	$-0.793322\pi$
$239$	−24.6274			−1.59302			−0.796508	+	0.604629i	$0.793322\pi$
$240$		0			0
$241$	5.41421			0.348760			0.174380	−	0.984678i	$-0.444208\pi$
$241$	5.41421			0.348760			0.174380	+	0.984678i	$0.444208\pi$
$242$			1.85786i			0.119428i
$243$		0			0
$244$	−10.4853			−0.671251
$245$	4.82843	+	14.4853i	0.308477	+	0.925431i
$246$		0			0
$247$			25.3848i			1.61519i
$248$		−	7.65685i		−	0.486211i
$249$		0			0
$250$	6.36396	−	9.19239i	0.402492	−	0.581378i
$251$	0.828427			0.0522899			0.0261449	−	0.999658i	$-0.491677\pi$
$251$	0.828427			0.0522899			0.0261449	+	0.999658i	$0.491677\pi$
$252$		0			0
$253$			10.7574i			0.676309i
$254$	−12.4142			−0.778937
$255$		0			0
$256$	1.00000			0.0625000
$257$			12.4142i			0.774377i	0.922001	+	0.387189i	$0.126554\pi$
$257$			12.4142i			0.774377i	−0.922001	+	0.387189i	$0.873446\pi$
$258$		0			0
$259$	0.414214			0.0257380
$260$	6.87868	−	2.29289i	0.426598	−	0.142199i
$261$		0			0
$262$			17.8995i			1.10583i
$263$			28.5858i			1.76268i	0.472487	+	0.881338i	$0.343356\pi$
$263$			28.5858i			1.76268i	−0.472487	+	0.881338i	$0.656644\pi$
$264$		0			0
$265$	−3.87868	+	1.29289i	−0.238265	+	0.0794218i
$266$	−3.24264			−0.198819
$267$		0			0
$268$		−	4.82843i		−	0.294943i
$269$	19.6274			1.19670			0.598352	−	0.801233i	$-0.295822\pi$
$269$	19.6274			1.19670			0.598352	+	0.801233i	$0.295822\pi$
$270$		0			0
$271$	−1.17157			−0.0711680			−0.0355840	−	0.999367i	$-0.511329\pi$
$271$	−1.17157			−0.0711680			−0.0355840	+	0.999367i	$0.511329\pi$
$272$		−	6.41421i		−	0.388919i
$273$		0			0
$274$	−13.5563			−0.818969
$275$	−14.3431	+	10.7574i	−0.864924	+	0.648693i
$276$		0			0
$277$			16.8995i			1.01539i	0.861536	+	0.507696i	$0.169503\pi$
$277$			16.8995i			1.01539i	−0.861536	+	0.507696i	$0.830497\pi$
$278$		−	14.4853i		−	0.868769i
$279$		0			0
$280$	0.292893	+	0.878680i	0.0175037	+	0.0525112i
$281$	−23.3848			−1.39502			−0.697509	−	0.716576i	$-0.745708\pi$
$281$	−23.3848			−1.39502			−0.697509	+	0.716576i	$0.745708\pi$
$282$		0			0
$283$		−	20.7990i		−	1.23637i	−0.786032	−	0.618186i	$-0.787868\pi$
$283$		−	20.7990i		−	1.23637i	0.786032	−	0.618186i	$-0.212132\pi$
$284$	−6.58579			−0.390795
$285$		0			0
$286$	−11.6274			−0.687544
$287$			3.89949i			0.230180i
$288$		0			0
$289$	−24.1421			−1.42013
$290$	15.0000	−	5.00000i	0.880830	−	0.293610i
$291$		0			0
$292$			7.82843i			0.458124i
$293$			2.65685i			0.155215i	0.996984	+	0.0776075i	$0.0247281\pi$
$293$			2.65685i			0.155215i	−0.996984	+	0.0776075i	$0.975272\pi$
$294$		0			0
$295$	0.757359	+	2.27208i	0.0440952	+	0.132285i
$296$	−1.00000			−0.0581238
$297$		0			0
$298$			20.4853i			1.18668i
$299$	−9.72792			−0.562580
$300$		0			0
$301$	−2.68629			−0.154835
$302$		−	0.414214i		−	0.0238353i
$303$		0			0
$304$	7.82843			0.448991
$305$	7.41421	+	22.2426i	0.424537	+	1.27361i
$306$		0			0
$307$			5.89949i			0.336702i	0.985727	+	0.168351i	$0.0538442\pi$
$307$			5.89949i			0.336702i	−0.985727	+	0.168351i	$0.946156\pi$
$308$		−	1.48528i		−	0.0846318i
$309$		0			0
$310$	−16.2426	+	5.41421i	−0.922520	+	0.307507i
$311$	7.17157			0.406663			0.203331	−	0.979110i	$-0.434823\pi$
$311$	7.17157			0.406663			0.203331	+	0.979110i	$0.434823\pi$
$312$		0			0
$313$		−	22.7279i		−	1.28466i	−0.766429	−	0.642329i	$-0.777968\pi$
$313$		−	22.7279i		−	1.28466i	0.766429	−	0.642329i	$-0.222032\pi$
$314$		0			0
$315$		0			0
$316$	3.75736			0.211368
$317$		−	26.8284i		−	1.50683i	−0.657543	−	0.753417i	$-0.728404\pi$
$317$		−	26.8284i		−	1.50683i	0.657543	−	0.753417i	$-0.271596\pi$
$318$		0			0
$319$	−25.3553			−1.41963
$320$	−0.707107	−	2.12132i	−0.0395285	−	0.118585i
$321$		0			0
$322$		−	1.24264i		−	0.0692497i
$323$		−	50.2132i		−	2.79394i
$324$		0			0
$325$	−9.72792	−	12.9706i	−0.539608	−	0.719477i
$326$	−13.1421			−0.727876
$327$		0			0
$328$		−	9.41421i		−	0.519813i
$329$	3.27208			0.180395
$330$		0			0
$331$	−27.4558			−1.50911			−0.754555	−	0.656237i	$-0.772147\pi$
$331$	−27.4558			−1.50911			−0.754555	+	0.656237i	$0.772147\pi$
$332$			12.8995i			0.707952i
$333$		0			0
$334$	−7.34315			−0.401799
$335$	−10.2426	+	3.41421i	−0.559615	+	0.186538i
$336$		0			0
$337$		−	5.97056i		−	0.325237i	−0.986689	−	0.162619i	$-0.948006\pi$
$337$		−	5.97056i		−	0.325237i	0.986689	−	0.162619i	$-0.0519940\pi$
$338$			2.48528i			0.135181i
$339$		0			0
$340$	−13.6066	+	4.53553i	−0.737922	+	0.245974i
$341$	27.4558			1.48682
$342$		0			0
$343$			5.72792i			0.309279i
$344$	6.48528			0.349663
$345$		0			0
$346$	10.6569			0.572916
$347$		−	7.55635i		−	0.405646i	−0.979215	−	0.202823i	$-0.934988\pi$
$347$		−	7.55635i		−	0.405646i	0.979215	−	0.202823i	$-0.0650116\pi$
$348$		0			0
$349$	20.0416			1.07280			0.536402	−	0.843963i	$-0.319783\pi$
$349$	20.0416			1.07280			0.536402	+	0.843963i	$0.319783\pi$
$350$	1.65685	−	1.24264i	0.0885626	−	0.0664219i
$351$		0			0
$352$			3.58579i			0.191123i
$353$		−	12.1421i		−	0.646261i	−0.946354	−	0.323130i	$-0.895265\pi$
$353$		−	12.1421i		−	0.646261i	0.946354	−	0.323130i	$-0.104735\pi$
$354$		0			0
$355$	4.65685	+	13.9706i	0.247160	+	0.741480i
$356$	16.8995			0.895671
$357$		0			0
$358$		−	13.1716i		−	0.696139i
$359$	14.9289			0.787919			0.393959	−	0.919128i	$-0.371105\pi$
$359$	14.9289			0.787919			0.393959	+	0.919128i	$0.371105\pi$
$360$		0			0
$361$	42.2843			2.22549
$362$			23.4142i			1.23062i
$363$		0			0
$364$	1.34315			0.0704000
$365$	16.6066	−	5.53553i	0.869229	−	0.289743i
$366$		0			0
$367$			6.75736i			0.352731i	0.984325	+	0.176366i	$0.0564341\pi$
$367$			6.75736i			0.352731i	−0.984325	+	0.176366i	$0.943566\pi$
$368$			3.00000i			0.156386i
$369$		0			0
$370$	0.707107	+	2.12132i	0.0367607	+	0.110282i
$371$	−0.757359			−0.0393201
$372$		0			0
$373$		−	10.2426i		−	0.530344i	−0.964201	−	0.265172i	$-0.914571\pi$
$373$		−	10.2426i		−	0.530344i	0.964201	−	0.265172i	$-0.0854287\pi$
$374$	23.0000			1.18930
$375$		0			0
$376$	−7.89949			−0.407385
$377$		−	22.9289i		−	1.18090i
$378$		0			0
$379$	3.27208			0.168075			0.0840377	−	0.996463i	$-0.473218\pi$
$379$	3.27208			0.168075			0.0840377	+	0.996463i	$0.473218\pi$
$380$	−5.53553	−	16.6066i	−0.283967	−	0.851901i
$381$		0			0
$382$			2.65685i			0.135936i
$383$		−	3.34315i		−	0.170827i	−0.996346	−	0.0854134i	$-0.972779\pi$
$383$		−	3.34315i		−	0.170827i	0.996346	−	0.0854134i	$-0.0272211\pi$
$384$		0			0
$385$	−3.15076	+	1.05025i	−0.160577	+	0.0535258i
$386$	13.3137			0.677650
$387$		0			0
$388$		−	18.7279i		−	0.950766i
$389$	10.9706			0.556230			0.278115	−	0.960548i	$-0.410290\pi$
$389$	10.9706			0.556230			0.278115	+	0.960548i	$0.410290\pi$
$390$		0			0
$391$	19.2426			0.973142
$392$		−	6.82843i		−	0.344888i
$393$		0			0
$394$	−13.1421			−0.662091
$395$	−2.65685	−	7.97056i	−0.133681	−	0.401043i
$396$		0			0
$397$			8.48528i			0.425864i	0.977067	+	0.212932i	$0.0683013\pi$
$397$			8.48528i			0.425864i	−0.977067	+	0.212932i	$0.931699\pi$
$398$		−	20.0000i		−	1.00251i
$399$		0			0
$400$	−4.00000	+	3.00000i	−0.200000	+	0.150000i
$401$	−33.0416			−1.65002			−0.825010	−	0.565118i	$-0.808831\pi$
$401$	−33.0416			−1.65002			−0.825010	+	0.565118i	$0.808831\pi$
$402$		0			0
$403$			24.8284i			1.23679i
$404$	17.3137			0.861389
$405$		0			0
$406$	2.92893			0.145360
$407$		−	3.58579i		−	0.177741i
$408$		0			0
$409$	4.38478			0.216813			0.108407	−	0.994107i	$-0.465425\pi$
$409$	4.38478			0.216813			0.108407	+	0.994107i	$0.465425\pi$
$410$	−19.9706	+	6.65685i	−0.986276	+	0.328759i
$411$		0			0
$412$		−	0.928932i		−	0.0457652i
$413$			0.443651i			0.0218306i
$414$		0			0
$415$	27.3640	−	9.12132i	1.34324	−	0.447748i
$416$	−3.24264			−0.158984
$417$		0			0
$418$			28.0711i			1.37300i
$419$	−2.27208			−0.110998			−0.0554991	−	0.998459i	$-0.517675\pi$
$419$	−2.27208			−0.110998			−0.0554991	+	0.998459i	$0.517675\pi$
$420$		0			0
$421$	37.4558			1.82549			0.912743	−	0.408534i	$-0.133960\pi$
$421$	37.4558			1.82549			0.912743	+	0.408534i	$0.133960\pi$
$422$		−	26.3848i		−	1.28439i
$423$		0			0
$424$	1.82843			0.0887963
$425$	19.2426	+	25.6569i	0.933405	+	1.24454i
$426$		0			0
$427$			4.34315i			0.210180i
$428$			1.92893i			0.0932385i
$429$		0			0
$430$	−4.58579	−	13.7574i	−0.221146	−	0.663439i
$431$	−24.1716			−1.16430			−0.582152	−	0.813080i	$-0.697789\pi$
$431$	−24.1716			−1.16430			−0.582152	+	0.813080i	$0.697789\pi$
$432$		0			0
$433$		−	6.51472i		−	0.313077i	−0.987672	−	0.156539i	$-0.949966\pi$
$433$		−	6.51472i		−	0.313077i	0.987672	−	0.156539i	$-0.0500336\pi$
$434$	−3.17157			−0.152240
$435$		0			0
$436$	13.2426			0.634207
$437$			23.4853i			1.12345i
$438$		0			0
$439$	0.727922			0.0347418			0.0173709	−	0.999849i	$-0.494470\pi$
$439$	0.727922			0.0347418			0.0173709	+	0.999849i	$0.494470\pi$
$440$	7.60660	−	2.53553i	0.362631	−	0.120877i
$441$		0			0
$442$			20.7990i			0.989307i
$443$			12.3431i			0.586441i	0.956045	+	0.293220i	$0.0947269\pi$
$443$			12.3431i			0.586441i	−0.956045	+	0.293220i	$0.905273\pi$
$444$		0			0
$445$	−11.9497	−	35.8492i	−0.566472	−	1.69942i
$446$	−1.65685			−0.0784543
$447$		0			0
$448$		−	0.414214i		−	0.0195698i
$449$	25.7990			1.21753			0.608765	−	0.793351i	$-0.291665\pi$
$449$	25.7990			1.21753			0.608765	+	0.793351i	$0.291665\pi$
$450$		0			0
$451$	33.7574			1.58957
$452$		−	2.82843i		−	0.133038i
$453$		0			0
$454$	12.0000			0.563188
$455$	−0.949747	−	2.84924i	−0.0445248	−	0.133575i
$456$		0			0
$457$		−	32.9706i		−	1.54230i	−0.636655	−	0.771149i	$-0.719682\pi$
$457$		−	32.9706i		−	1.54230i	0.636655	−	0.771149i	$-0.280318\pi$
$458$			6.72792i			0.314375i
$459$		0			0
$460$	6.36396	−	2.12132i	0.296721	−	0.0989071i
$461$	−15.1716			−0.706611			−0.353305	−	0.935508i	$-0.614942\pi$
$461$	−15.1716			−0.706611			−0.353305	+	0.935508i	$0.614942\pi$
$462$		0			0
$463$		−	10.1005i		−	0.469410i	−0.972067	−	0.234705i	$-0.924588\pi$
$463$		−	10.1005i		−	0.469410i	0.972067	−	0.234705i	$-0.0754125\pi$
$464$	−7.07107			−0.328266
$465$		0			0
$466$	−14.4853			−0.671018
$467$		−	31.7574i		−	1.46956i	−0.678308	−	0.734778i	$-0.737286\pi$
$467$		−	31.7574i		−	1.46956i	0.678308	−	0.734778i	$-0.262714\pi$
$468$		0			0
$469$	−2.00000			−0.0923514
$470$	5.58579	+	16.7574i	0.257653	+	0.772959i
$471$		0			0
$472$		−	1.07107i		−	0.0492999i
$473$			23.2548i			1.06926i
$474$		0			0
$475$	−31.3137	+	23.4853i	−1.43677	+	1.07758i
$476$	−2.65685			−0.121777
$477$		0			0
$478$		−	24.6274i		−	1.12643i
$479$	11.8284			0.540455			0.270227	−	0.962797i	$-0.412901\pi$
$479$	11.8284			0.540455			0.270227	+	0.962797i	$0.412901\pi$
$480$		0			0
$481$	3.24264			0.147852
$482$			5.41421i			0.246611i
$483$		0			0
$484$	−1.85786			−0.0844484
$485$	−39.7279	+	13.2426i	−1.80395	+	0.601317i
$486$		0			0
$487$			30.0000i			1.35943i	0.733476	+	0.679715i	$0.237896\pi$
$487$			30.0000i			1.35943i	−0.733476	+	0.679715i	$0.762104\pi$
$488$		−	10.4853i		−	0.474646i
$489$		0			0
$490$	−14.4853	+	4.82843i	−0.654378	+	0.218126i
$491$	13.5858			0.613118			0.306559	−	0.951852i	$-0.400822\pi$
$491$	13.5858			0.613118			0.306559	+	0.951852i	$0.400822\pi$
$492$		0			0
$493$			45.3553i			2.04270i
$494$	−25.3848			−1.14212
$495$		0			0
$496$	7.65685			0.343803
$497$			2.72792i			0.122364i
$498$		0			0
$499$	9.48528			0.424620			0.212310	−	0.977202i	$-0.431901\pi$
$499$	9.48528			0.424620			0.212310	+	0.977202i	$0.431901\pi$
$500$	9.19239	+	6.36396i	0.411096	+	0.284605i
$501$		0			0
$502$			0.828427i			0.0369745i
$503$			12.3431i			0.550354i	0.961394	+	0.275177i	$0.0887364\pi$
$503$			12.3431i			0.550354i	−0.961394	+	0.275177i	$0.911264\pi$
$504$		0			0
$505$	−12.2426	−	36.7279i	−0.544790	−	1.63437i
$506$	−10.7574			−0.478223
$507$		0			0
$508$		−	12.4142i		−	0.550792i
$509$	−7.68629			−0.340689			−0.170344	−	0.985385i	$-0.554488\pi$
$509$	−7.68629			−0.340689			−0.170344	+	0.985385i	$0.554488\pi$
$510$		0			0
$511$	3.24264			0.143446
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−12.4142			−0.547567
$515$	−1.97056	+	0.656854i	−0.0868334	+	0.0289445i
$516$		0			0
$517$		−	28.3259i		−	1.24577i
$518$			0.414214i			0.0181995i
$519$		0			0
$520$	2.29289	+	6.87868i	0.100550	+	0.301650i
$521$	−32.0416			−1.40377			−0.701885	−	0.712291i	$-0.747658\pi$
$521$	−32.0416			−1.40377			−0.701885	+	0.712291i	$0.747658\pi$
$522$		0			0
$523$		−	34.0000i		−	1.48672i	−0.668894	−	0.743358i	$-0.733232\pi$
$523$		−	34.0000i		−	1.48672i	0.668894	−	0.743358i	$-0.266768\pi$
$524$	−17.8995			−0.781943
$525$		0			0
$526$	−28.5858			−1.24640
$527$		−	49.1127i		−	2.13938i
$528$		0			0
$529$	14.0000			0.608696
$530$	−1.29289	−	3.87868i	−0.0561597	−	0.168479i
$531$		0			0
$532$		−	3.24264i		−	0.140586i
$533$			30.5269i			1.32227i
$534$		0			0
$535$	4.09188	−	1.36396i	0.176908	−	0.0589692i
$536$	4.82843			0.208556
$537$		0			0
$538$			19.6274i			0.846198i
$539$	24.4853			1.05466
$540$		0			0
$541$	−6.21320			−0.267126			−0.133563	−	0.991040i	$-0.542642\pi$
$541$	−6.21320			−0.267126			−0.133563	+	0.991040i	$0.542642\pi$
$542$		−	1.17157i		−	0.0503234i
$543$		0			0
$544$	6.41421			0.275007
$545$	−9.36396	−	28.0919i	−0.401108	−	1.20332i
$546$		0			0
$547$		−	45.6274i		−	1.95089i	−0.220249	−	0.975444i	$-0.570687\pi$
$547$		−	45.6274i		−	1.95089i	0.220249	−	0.975444i	$-0.429313\pi$
$548$		−	13.5563i		−	0.579099i
$549$		0			0
$550$	−10.7574	−	14.3431i	−0.458695	−	0.611594i
$551$	−55.3553			−2.35822
$552$		0			0
$553$		−	1.55635i		−	0.0661827i
$554$	−16.8995			−0.717991
$555$		0			0
$556$	14.4853			0.614313
$557$			6.44365i			0.273026i	0.990638	+	0.136513i	$0.0435896\pi$
$557$			6.44365i			0.273026i	−0.990638	+	0.136513i	$0.956410\pi$
$558$		0			0
$559$	−21.0294			−0.889450
$560$	−0.878680	+	0.292893i	−0.0371310	+	0.0123770i
$561$		0			0
$562$		−	23.3848i		−	0.986427i
$563$			19.4142i			0.818212i	0.912487	+	0.409106i	$0.134159\pi$
$563$			19.4142i			0.818212i	−0.912487	+	0.409106i	$0.865841\pi$
$564$		0			0
$565$	−6.00000	+	2.00000i	−0.252422	+	0.0841406i
$566$	20.7990			0.874247
$567$		0			0
$568$		−	6.58579i		−	0.276333i
$569$	−11.0416			−0.462889			−0.231445	−	0.972848i	$-0.574345\pi$
$569$	−11.0416			−0.462889			−0.231445	+	0.972848i	$0.574345\pi$
$570$		0			0
$571$	20.0416			0.838716			0.419358	−	0.907821i	$-0.362255\pi$
$571$	20.0416			0.838716			0.419358	+	0.907821i	$0.362255\pi$
$572$		−	11.6274i		−	0.486167i
$573$		0			0
$574$	−3.89949			−0.162762
$575$	−9.00000	−	12.0000i	−0.375326	−	0.500435i
$576$		0			0
$577$			26.4853i			1.10260i	0.834308	+	0.551298i	$0.185867\pi$
$577$			26.4853i			1.10260i	−0.834308	+	0.551298i	$0.814133\pi$
$578$		−	24.1421i		−	1.00418i
$579$		0			0
$580$	5.00000	+	15.0000i	0.207614	+	0.622841i
$581$	5.34315			0.221671
$582$		0			0
$583$			6.55635i			0.271536i
$584$	−7.82843			−0.323943
$585$		0			0
$586$	−2.65685			−0.109754
$587$		−	2.82843i		−	0.116742i	−0.998295	−	0.0583708i	$-0.981409\pi$
$587$		−	2.82843i		−	0.116742i	0.998295	−	0.0583708i	$-0.0185906\pi$
$588$		0			0
$589$	59.9411			2.46983
$590$	−2.27208	+	0.757359i	−0.0935400	+	0.0311800i
$591$		0			0
$592$		−	1.00000i		−	0.0410997i
$593$		−	28.9706i		−	1.18968i	−0.803845	−	0.594839i	$-0.797216\pi$
$593$		−	28.9706i		−	1.18968i	0.803845	−	0.594839i	$-0.202784\pi$
$594$		0			0
$595$	1.87868	+	5.63604i	0.0770184	+	0.231055i
$596$	−20.4853			−0.839110
$597$		0			0
$598$		−	9.72792i		−	0.397804i
$599$	−20.8701			−0.852727			−0.426364	−	0.904552i	$-0.640206\pi$
$599$	−20.8701			−0.852727			−0.426364	+	0.904552i	$0.640206\pi$
$600$		0			0
$601$	−23.9706			−0.977780			−0.488890	−	0.872346i	$-0.662598\pi$
$601$	−23.9706			−0.977780			−0.488890	+	0.872346i	$0.662598\pi$
$602$		−	2.68629i		−	0.109485i
$603$		0			0
$604$	0.414214			0.0168541
$605$	1.31371	+	3.94113i	0.0534098	+	0.160230i
$606$		0			0
$607$		−	40.1421i		−	1.62932i	−0.579940	−	0.814660i	$-0.696924\pi$
$607$		−	40.1421i		−	1.62932i	0.579940	−	0.814660i	$-0.303076\pi$
$608$			7.82843i			0.317485i
$609$		0			0
$610$	−22.2426	+	7.41421i	−0.900578	+	0.300193i
$611$	25.6152			1.03628
$612$		0			0
$613$			21.3137i			0.860853i	0.902626	+	0.430426i	$0.141637\pi$
$613$			21.3137i			0.860853i	−0.902626	+	0.430426i	$0.858363\pi$
$614$	−5.89949			−0.238084
$615$		0			0
$616$	1.48528			0.0598437
$617$		−	23.6985i		−	0.954065i	−0.878886	−	0.477033i	$-0.841712\pi$
$617$		−	23.6985i		−	0.954065i	0.878886	−	0.477033i	$-0.158288\pi$
$618$		0			0
$619$	−35.2132			−1.41534			−0.707669	−	0.706544i	$-0.750253\pi$
$619$	−35.2132			−1.41534			−0.707669	+	0.706544i	$0.750253\pi$
$620$	−5.41421	−	16.2426i	−0.217440	−	0.652320i
$621$		0			0
$622$			7.17157i			0.287554i
$623$		−	7.00000i		−	0.280449i
$624$		0			0
$625$	7.00000	−	24.0000i	0.280000	−	0.960000i
$626$	22.7279			0.908390
$627$		0			0
$628$		0			0
$629$	−6.41421			−0.255751
$630$		0			0
$631$	24.4853			0.974744			0.487372	−	0.873195i	$-0.337956\pi$
$631$	24.4853			0.974744			0.487372	+	0.873195i	$0.337956\pi$
$632$			3.75736i			0.149460i
$633$		0			0
$634$	26.8284			1.06549
$635$	−26.3345	+	8.77817i	−1.04505	+	0.348351i
$636$		0			0
$637$			22.1421i			0.877303i
$638$		−	25.3553i		−	1.00383i
$639$		0			0
$640$	2.12132	−	0.707107i	0.0838525	−	0.0279508i
$641$	21.1716			0.836227			0.418113	−	0.908395i	$-0.362691\pi$
$641$	21.1716			0.836227			0.418113	+	0.908395i	$0.362691\pi$
$642$		0			0
$643$		−	32.4558i		−	1.27993i	−0.768403	−	0.639967i	$-0.778948\pi$
$643$		−	32.4558i		−	1.27993i	0.768403	−	0.639967i	$-0.221052\pi$
$644$	1.24264			0.0489669
$645$		0			0
$646$	50.2132			1.97561
$647$			21.0000i			0.825595i	0.910823	+	0.412798i	$0.135448\pi$
$647$			21.0000i			0.825595i	−0.910823	+	0.412798i	$0.864552\pi$
$648$		0			0
$649$	3.84062			0.150758
$650$	12.9706	−	9.72792i	0.508747	−	0.381560i
$651$		0			0
$652$		−	13.1421i		−	0.514686i
$653$			3.51472i			0.137542i	0.997632	+	0.0687708i	$0.0219077\pi$
$653$			3.51472i			0.137542i	−0.997632	+	0.0687708i	$0.978092\pi$
$654$		0			0
$655$	12.6569	+	37.9706i	0.494544	+	1.48363i
$656$	9.41421			0.367563
$657$		0			0
$658$			3.27208i			0.127559i
$659$	−11.1127			−0.432889			−0.216445	−	0.976295i	$-0.569446\pi$
$659$	−11.1127			−0.432889			−0.216445	+	0.976295i	$0.569446\pi$
$660$		0			0
$661$	16.7574			0.651786			0.325893	−	0.945407i	$-0.394335\pi$
$661$	16.7574			0.651786			0.325893	+	0.945407i	$0.394335\pi$
$662$		−	27.4558i		−	1.06710i
$663$		0			0
$664$	−12.8995			−0.500597
$665$	−6.87868	+	2.29289i	−0.266744	+	0.0889146i
$666$		0			0
$667$		−	21.2132i		−	0.821379i
$668$		−	7.34315i		−	0.284115i
$669$		0			0
$670$	−3.41421	−	10.2426i	−0.131903	−	0.395708i
$671$	37.5980			1.45145
$672$		0			0
$673$		−	33.9706i		−	1.30947i	−0.755859	−	0.654734i	$-0.772780\pi$
$673$		−	33.9706i		−	1.30947i	0.755859	−	0.654734i	$-0.227220\pi$
$674$	5.97056			0.229977
$675$		0			0
$676$	−2.48528			−0.0955877
$677$		−	40.7990i		−	1.56803i	−0.620740	−	0.784016i	$-0.713168\pi$
$677$		−	40.7990i		−	1.56803i	0.620740	−	0.784016i	$-0.286832\pi$
$678$		0			0
$679$	−7.75736			−0.297700
$680$	−4.53553	−	13.6066i	−0.173930	−	0.521789i
$681$		0			0
$682$			27.4558i			1.05134i
$683$		−	31.0711i		−	1.18890i	−0.804132	−	0.594451i	$-0.797370\pi$
$683$		−	31.0711i		−	1.18890i	0.804132	−	0.594451i	$-0.202630\pi$
$684$		0			0
$685$	−28.7574	+	9.58579i	−1.09876	+	0.366254i
$686$	−5.72792			−0.218693
$687$		0			0
$688$			6.48528i			0.247249i
$689$	−5.92893			−0.225874
$690$		0			0
$691$	33.4142			1.27114			0.635568	−	0.772045i	$-0.280766\pi$
$691$	33.4142			1.27114			0.635568	+	0.772045i	$0.280766\pi$
$692$			10.6569i			0.405113i
$693$		0			0
$694$	7.55635			0.286835
$695$	−10.2426	−	30.7279i	−0.388526	−	1.16558i
$696$		0			0
$697$		−	60.3848i		−	2.28724i
$698$			20.0416i			0.758587i
$699$		0			0
$700$	1.24264	+	1.65685i	0.0469674	+	0.0626232i
$701$	30.6274			1.15678			0.578391	−	0.815760i	$-0.303681\pi$
$701$	30.6274			1.15678			0.578391	+	0.815760i	$0.303681\pi$
$702$		0			0
$703$		−	7.82843i		−	0.295255i
$704$	−3.58579			−0.135144
$705$		0			0
$706$	12.1421			0.456975
$707$		−	7.17157i		−	0.269715i
$708$		0			0
$709$	−29.9289			−1.12400			−0.562002	−	0.827136i	$-0.689969\pi$
$709$	−29.9289			−1.12400			−0.562002	+	0.827136i	$0.689969\pi$
$710$	−13.9706	+	4.65685i	−0.524306	+	0.174769i
$711$		0			0
$712$			16.8995i			0.633335i
$713$			22.9706i			0.860254i
$714$		0			0
$715$	−24.6655	+	8.22183i	−0.922437	+	0.307479i
$716$	13.1716			0.492245
$717$		0			0
$718$			14.9289i			0.557143i
$719$	−1.41421			−0.0527413			−0.0263706	−	0.999652i	$-0.508395\pi$
$719$	−1.41421			−0.0527413			−0.0263706	+	0.999652i	$0.508395\pi$
$720$		0			0
$721$	−0.384776			−0.0143298
$722$			42.2843i			1.57366i
$723$		0			0
$724$	−23.4142			−0.870182
$725$	28.2843	−	21.2132i	1.05045	−	0.787839i
$726$		0			0
$727$		−	16.3848i		−	0.607678i	−0.952723	−	0.303839i	$-0.901732\pi$
$727$		−	16.3848i		−	0.607678i	0.952723	−	0.303839i	$-0.0982684\pi$
$728$			1.34315i			0.0497803i
$729$		0			0
$730$	5.53553	+	16.6066i	0.204879	+	0.614638i
$731$	41.5980			1.53856
$732$		0			0
$733$		−	22.9706i		−	0.848437i	−0.905560	−	0.424219i	$-0.860549\pi$
$733$		−	22.9706i		−	0.848437i	0.905560	−	0.424219i	$-0.139451\pi$
$734$	−6.75736			−0.249419
$735$		0			0
$736$	−3.00000			−0.110581
$737$			17.3137i			0.637759i
$738$		0			0
$739$	10.0000			0.367856			0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000			0.367856			0.183928	+	0.982940i	$0.441119\pi$
$740$	−2.12132	+	0.707107i	−0.0779813	+	0.0259938i
$741$		0			0
$742$		−	0.757359i		−	0.0278035i
$743$		−	19.3553i		−	0.710079i	−0.934851	−	0.355039i	$-0.884467\pi$
$743$		−	19.3553i		−	0.710079i	0.934851	−	0.355039i	$-0.115533\pi$
$744$		0			0
$745$	14.4853	+	43.4558i	0.530700	+	1.59210i
$746$	10.2426			0.375010
$747$		0			0
$748$			23.0000i			0.840963i
$749$	0.798990			0.0291945
$750$		0			0
$751$	27.1127			0.989356			0.494678	−	0.869076i	$-0.335286\pi$
$751$	27.1127			0.989356			0.494678	+	0.869076i	$0.335286\pi$
$752$		−	7.89949i		−	0.288065i
$753$		0			0
$754$	22.9289			0.835022
$755$	−0.292893	−	0.878680i	−0.0106595	−	0.0319784i
$756$		0			0
$757$			26.2132i			0.952735i	0.879246	+	0.476368i	$0.158047\pi$
$757$			26.2132i			0.952735i	−0.879246	+	0.476368i	$0.841953\pi$
$758$			3.27208i			0.118847i
$759$		0			0
$760$	16.6066	−	5.53553i	0.602385	−	0.200795i
$761$	−2.68629			−0.0973780			−0.0486890	−	0.998814i	$-0.515504\pi$
$761$	−2.68629			−0.0973780			−0.0486890	+	0.998814i	$0.515504\pi$
$762$		0			0
$763$		−	5.48528i		−	0.198581i
$764$	−2.65685			−0.0961216
$765$		0			0
$766$	3.34315			0.120793
$767$			3.47309i			0.125406i
$768$		0			0
$769$	−11.5563			−0.416733			−0.208366	−	0.978051i	$-0.566815\pi$
$769$	−11.5563			−0.416733			−0.208366	+	0.978051i	$0.566815\pi$
$770$	−1.05025	−	3.15076i	−0.0378485	−	0.113545i
$771$		0			0
$772$			13.3137i			0.479171i
$773$			3.34315i			0.120245i	0.998191	+	0.0601223i	$0.0191491\pi$
$773$			3.34315i			0.120245i	−0.998191	+	0.0601223i	$0.980851\pi$
$774$		0			0
$775$	−30.6274	+	22.9706i	−1.10017	+	0.825127i
$776$	18.7279			0.672293
$777$		0			0
$778$			10.9706i			0.393314i
$779$	73.6985			2.64052
$780$		0			0
$781$	23.6152			0.845019
$782$			19.2426i			0.688115i
$783$		0			0
$784$	6.82843			0.243872
$785$		0			0
$786$		0			0
$787$		−	21.5147i		−	0.766917i	−0.923558	−	0.383458i	$-0.874733\pi$
$787$		−	21.5147i		−	0.766917i	0.923558	−	0.383458i	$-0.125267\pi$
$788$		−	13.1421i		−	0.468169i
$789$		0			0
$790$	7.97056	−	2.65685i	0.283580	−	0.0945266i
$791$	−1.17157			−0.0416563
$792$		0			0
$793$			34.0000i			1.20738i
$794$	−8.48528			−0.301131
$795$		0			0
$796$	20.0000			0.708881
$797$		−	2.87006i		−	0.101663i	−0.998707	−	0.0508313i	$-0.983813\pi$
$797$		−	2.87006i		−	0.101663i	0.998707	−	0.0508313i	$-0.0161871\pi$
$798$		0			0
$799$	−50.6690			−1.79254
$800$	−3.00000	−	4.00000i	−0.106066	−	0.141421i
$801$		0			0
$802$		−	33.0416i		−	1.16674i
$803$		−	28.0711i		−	0.990606i
$804$		0			0
$805$	−0.878680	−	2.63604i	−0.0309694	−	0.0929082i
$806$	−24.8284			−0.874544
$807$		0			0
$808$			17.3137i			0.609094i
$809$	54.2132			1.90603			0.953017	−	0.302916i	$-0.0979601\pi$
$809$	54.2132			1.90603			0.953017	+	0.302916i	$0.0979601\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$			2.92893i			0.102785i
$813$		0			0
$814$	3.58579			0.125682
$815$	−27.8787	+	9.29289i	−0.976548	+	0.325516i
$816$		0			0
$817$			50.7696i			1.77620i
$818$			4.38478i			0.153310i
$819$		0			0
$820$	−6.65685	−	19.9706i	−0.232467	−	0.697402i
$821$	−5.28427			−0.184422			−0.0922112	−	0.995739i	$-0.529393\pi$
$821$	−5.28427			−0.184422			−0.0922112	+	0.995739i	$0.529393\pi$
$822$		0			0
$823$			17.7279i			0.617957i	0.951069	+	0.308978i	$0.0999870\pi$
$823$			17.7279i			0.617957i	−0.951069	+	0.308978i	$0.900013\pi$
$824$	0.928932			0.0323609
$825$		0			0
$826$	−0.443651			−0.0154366
$827$		−	7.07107i		−	0.245885i	−0.992414	−	0.122943i	$-0.960767\pi$
$827$		−	7.07107i		−	0.245885i	0.992414	−	0.122943i	$-0.0392331\pi$
$828$		0			0
$829$	10.7574			0.373619			0.186809	−	0.982396i	$-0.440185\pi$
$829$	10.7574			0.373619			0.186809	+	0.982396i	$0.440185\pi$
$830$	9.12132	+	27.3640i	0.316606	+	0.949817i
$831$		0			0
$832$		−	3.24264i		−	0.112418i
$833$		−	43.7990i		−	1.51755i
$834$		0			0
$835$	−15.5772	+	5.19239i	−0.539070	+	0.179690i
$836$	−28.0711			−0.970858
$837$		0			0
$838$		−	2.27208i		−	0.0784876i
$839$	−21.2132			−0.732361			−0.366181	−	0.930544i	$-0.619335\pi$
$839$	−21.2132			−0.732361			−0.366181	+	0.930544i	$0.619335\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$			37.4558i			1.29081i
$843$		0			0
$844$	26.3848			0.908201
$845$	1.75736	+	5.27208i	0.0604550	+	0.181365i
$846$		0			0
$847$			0.769553i			0.0264421i
$848$			1.82843i			0.0627884i
$849$		0			0
$850$	−25.6569	+	19.2426i	−0.880023	+	0.660017i
$851$	3.00000			0.102839
$852$		0			0
$853$			44.8406i			1.53531i	0.640861	+	0.767657i	$0.278577\pi$
$853$			44.8406i			1.53531i	−0.640861	+	0.767657i	$0.721423\pi$
$854$	−4.34315			−0.148619
$855$		0			0
$856$	−1.92893			−0.0659295
$857$			42.4975i			1.45169i	0.687860	+	0.725843i	$0.258550\pi$
$857$			42.4975i			1.45169i	−0.687860	+	0.725843i	$0.741450\pi$
$858$		0			0
$859$	6.51472			0.222279			0.111140	−	0.993805i	$-0.464550\pi$
$859$	6.51472			0.222279			0.111140	+	0.993805i	$0.464550\pi$
$860$	13.7574	−	4.58579i	0.469122	−	0.156374i
$861$		0			0
$862$		−	24.1716i		−	0.823287i
$863$		−	5.55635i		−	0.189140i	−0.995518	−	0.0945702i	$-0.969852\pi$
$863$		−	5.55635i		−	0.189140i	0.995518	−	0.0945702i	$-0.0301477\pi$
$864$		0			0
$865$	22.6066	−	7.53553i	0.768648	−	0.256216i
$866$	6.51472			0.221379
$867$		0			0
$868$		−	3.17157i		−	0.107650i
$869$	−13.4731			−0.457043
$870$		0			0
$871$	−15.6569			−0.530512
$872$			13.2426i			0.448452i
$873$		0			0
$874$	−23.4853			−0.794401
$875$	2.63604	−	3.80761i	0.0891144	−	0.128721i
$876$		0			0
$877$		−	31.7574i		−	1.07237i	−0.844101	−	0.536185i	$-0.819865\pi$
$877$		−	31.7574i		−	1.07237i	0.844101	−	0.536185i	$-0.180135\pi$
$878$			0.727922i			0.0245662i
$879$		0			0
$880$	2.53553	+	7.60660i	0.0854728	+	0.256418i
$881$	26.6274			0.897100			0.448550	−	0.893758i	$-0.351941\pi$
$881$	26.6274			0.897100			0.448550	+	0.893758i	$0.351941\pi$
$882$		0			0
$883$		−	55.9706i		−	1.88356i	−0.336231	−	0.941780i	$-0.609152\pi$
$883$		−	55.9706i		−	1.88356i	0.336231	−	0.941780i	$-0.390848\pi$
$884$	−20.7990			−0.699546
$885$		0			0
$886$	−12.3431			−0.414676
$887$		−	8.14214i		−	0.273386i	−0.990613	−	0.136693i	$-0.956353\pi$
$887$		−	8.14214i		−	0.273386i	0.990613	−	0.136693i	$-0.0436474\pi$
$888$		0			0
$889$	−5.14214			−0.172462
$890$	35.8492	−	11.9497i	1.20167	−	0.400556i
$891$		0			0
$892$		−	1.65685i		−	0.0554756i
$893$		−	61.8406i		−	2.06942i
$894$		0			0
$895$	−9.31371	−	27.9411i	−0.311323	−	0.933969i
$896$	0.414214			0.0138379
$897$		0			0
$898$			25.7990i			0.860923i
$899$	−54.1421			−1.80574
$900$		0			0
$901$	11.7279			0.390714
$902$			33.7574i			1.12400i
$903$		0			0
$904$	2.82843			0.0940721
$905$	16.5563	+	49.6690i	0.550352	+	1.65106i
$906$		0			0
$907$			36.7990i			1.22189i	0.791673	+	0.610945i	$0.209210\pi$
$907$			36.7990i			1.22189i	−0.791673	+	0.610945i	$0.790790\pi$
$908$			12.0000i			0.398234i
$909$		0			0
$910$	2.84924	−	0.949747i	0.0944515	−	0.0314838i
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		−	46.2548i		−	1.53081i
$914$	32.9706			1.09057
$915$		0			0
$916$	−6.72792			−0.222297
$917$			7.41421i			0.244839i
$918$		0			0
$919$	−7.75736			−0.255892			−0.127946	−	0.991781i	$-0.540838\pi$
$919$	−7.75736			−0.255892			−0.127946	+	0.991781i	$0.540838\pi$
$920$	2.12132	+	6.36396i	0.0699379	+	0.209814i
$921$		0			0
$922$		−	15.1716i		−	0.499649i
$923$			21.3553i			0.702920i
$924$		0			0
$925$	3.00000	+	4.00000i	0.0986394	+	0.131519i
$926$	10.1005			0.331923
$927$		0			0
$928$		−	7.07107i		−	0.232119i
$929$	−14.4853			−0.475247			−0.237623	−	0.971357i	$-0.576368\pi$
$929$	−14.4853			−0.475247			−0.237623	+	0.971357i	$0.576368\pi$
$930$		0			0
$931$	53.4558			1.75194
$932$		−	14.4853i		−	0.474481i
$933$		0			0
$934$	31.7574			1.03913
$935$	48.7904	−	16.2635i	1.59562	−	0.531872i
$936$		0			0
$937$			17.9411i			0.586111i	0.956095	+	0.293056i	$0.0946721\pi$
$937$			17.9411i			0.586111i	−0.956095	+	0.293056i	$0.905328\pi$
$938$		−	2.00000i		−	0.0653023i
$939$		0			0
$940$	−16.7574	+	5.58579i	−0.546565	+	0.182188i
$941$	26.3431			0.858762			0.429381	−	0.903123i	$-0.358732\pi$
$941$	26.3431			0.858762			0.429381	+	0.903123i	$0.358732\pi$
$942$		0			0
$943$			28.2426i			0.919707i
$944$	1.07107			0.0348603
$945$		0			0
$946$	−23.2548			−0.756080
$947$		−	20.0416i		−	0.651265i	−0.945496	−	0.325633i	$-0.894423\pi$
$947$		−	20.0416i		−	0.651265i	0.945496	−	0.325633i	$-0.105577\pi$
$948$		0			0
$949$	25.3848			0.824025
$950$	−23.4853	−	31.3137i	−0.761963	−	1.01595i
$951$		0			0
$952$		−	2.65685i		−	0.0861091i
$953$		−	3.41421i		−	0.110597i	−0.998470	−	0.0552986i	$-0.982389\pi$
$953$		−	3.41421i		−	0.110597i	0.998470	−	0.0552986i	$-0.0176111\pi$
$954$		0			0
$955$	1.87868	+	5.63604i	0.0607926	+	0.182378i
$956$	24.6274			0.796508
$957$		0			0
$958$			11.8284i			0.382159i
$959$	−5.61522			−0.181325
$960$		0			0
$961$	27.6274			0.891207
$962$			3.24264i			0.104547i
$963$		0			0
$964$	−5.41421			−0.174380
$965$	28.2426	−	9.41421i	0.909163	−	0.303054i
$966$		0			0
$967$			40.1005i			1.28955i	0.764374	+	0.644773i	$0.223048\pi$
$967$			40.1005i			1.28955i	−0.764374	+	0.644773i	$0.776952\pi$
$968$		−	1.85786i		−	0.0597140i
$969$		0			0
$970$	−13.2426	−	39.7279i	−0.425196	−	1.27559i
$971$	22.9706			0.737160			0.368580	−	0.929596i	$-0.379844\pi$
$971$	22.9706			0.737160			0.368580	+	0.929596i	$0.379844\pi$
$972$		0			0
$973$		−	6.00000i		−	0.192351i
$974$	−30.0000			−0.961262
$975$		0			0
$976$	10.4853			0.335626
$977$			20.6985i			0.662203i	0.943595	+	0.331102i	$0.107420\pi$
$977$			20.6985i			0.662203i	−0.943595	+	0.331102i	$0.892580\pi$
$978$		0			0
$979$	−60.5980			−1.93672
$980$	−4.82843	−	14.4853i	−0.154238	−	0.462715i
$981$		0			0
$982$			13.5858i			0.433540i
$983$		−	37.1127i		−	1.18371i	−0.806044	−	0.591856i	$-0.798396\pi$
$983$		−	37.1127i		−	1.18371i	0.806044	−	0.591856i	$-0.201604\pi$
$984$		0			0
$985$	−27.8787	+	9.29289i	−0.888289	+	0.296096i
$986$	−45.3553			−1.44441
$987$		0			0
$988$		−	25.3848i		−	0.807597i
$989$	−19.4558			−0.618660
$990$		0			0
$991$	−30.6274			−0.972912			−0.486456	−	0.873705i	$-0.661711\pi$
$991$	−30.6274			−0.972912			−0.486456	+	0.873705i	$0.661711\pi$
$992$			7.65685i			0.243105i
$993$		0			0
$994$	−2.72792			−0.0865244
$995$	−14.1421	−	42.4264i	−0.448336	−	1.34501i
$996$		0			0
$997$			9.78680i			0.309951i	0.987918	+	0.154975i	$0.0495299\pi$
$997$			9.78680i			0.309951i	−0.987918	+	0.154975i	$0.950470\pi$
$998$			9.48528i			0.300251i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.d.k.1999.4		4
3.2	odd	2		1110.2.d.g.889.1	✓	4
5.4	even	2	inner	3330.2.d.k.1999.2		4
15.2	even	4		5550.2.a.cb.1.1		2
15.8	even	4		5550.2.a.bs.1.2		2
15.14	odd	2		1110.2.d.g.889.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.d.g.889.1	✓	4	3.2	odd	2
1110.2.d.g.889.3	yes	4	15.14	odd	2
3330.2.d.k.1999.2		4	5.4	even	2	inner
3330.2.d.k.1999.4		4	1.1	even	1	trivial
5550.2.a.bs.1.2		2	15.8	even	4
5550.2.a.cb.1.1		2	15.2	even	4

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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(0.707107 + 2.12132i) q^{5} +0.414214i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(0.707107 + 2.12132i) q^{5} +0.414214i q^{7} -1.00000i q^{8} +(-2.12132 + 0.707107i) q^{10} +3.58579 q^{11} +3.24264i q^{13} -0.414214 q^{14} +1.00000 q^{16} -6.41421i q^{17} +7.82843 q^{19} +(-0.707107 - 2.12132i) q^{20} +3.58579i q^{22} +3.00000i q^{23} +(-4.00000 + 3.00000i) q^{25} -3.24264 q^{26} -0.414214i q^{28} -7.07107 q^{29} +7.65685 q^{31} +1.00000i q^{32} +6.41421 q^{34} +(-0.878680 + 0.292893i) q^{35} -1.00000i q^{37} +7.82843i q^{38} +(2.12132 - 0.707107i) q^{40} +9.41421 q^{41} +6.48528i q^{43} -3.58579 q^{44} -3.00000 q^{46} -7.89949i q^{47} +6.82843 q^{49} +(-3.00000 - 4.00000i) q^{50} -3.24264i q^{52} +1.82843i q^{53} +(2.53553 + 7.60660i) q^{55} +0.414214 q^{56} -7.07107i q^{58} +1.07107 q^{59} +10.4853 q^{61} +7.65685i q^{62} -1.00000 q^{64} +(-6.87868 + 2.29289i) q^{65} +4.82843i q^{67} +6.41421i q^{68} +(-0.292893 - 0.878680i) q^{70} +6.58579 q^{71} -7.82843i q^{73} +1.00000 q^{74} -7.82843 q^{76} +1.48528i q^{77} -3.75736 q^{79} +(0.707107 + 2.12132i) q^{80} +9.41421i q^{82} -12.8995i q^{83} +(13.6066 - 4.53553i) q^{85} -6.48528 q^{86} -3.58579i q^{88} -16.8995 q^{89} -1.34315 q^{91} -3.00000i q^{92} +7.89949 q^{94} +(5.53553 + 16.6066i) q^{95} +18.7279i q^{97} +6.82843i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 20 q^{11} + 4 q^{14} + 4 q^{16} + 20 q^{19} - 16 q^{25} + 4 q^{26} + 8 q^{31} + 20 q^{34} - 12 q^{35} + 32 q^{41} - 20 q^{44} - 12 q^{46} + 16 q^{49} - 12 q^{50} - 4 q^{55} - 4 q^{56} - 24 q^{59} + 8 q^{61} - 4 q^{64} - 36 q^{65} - 4 q^{70} + 32 q^{71} + 4 q^{74} - 20 q^{76} - 32 q^{79} + 12 q^{85} + 8 q^{86} - 28 q^{89} - 28 q^{91} - 8 q^{94} + 8 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 + 20 * q^11 + 4 * q^14 + 4 * q^16 + 20 * q^19 - 16 * q^25 + 4 * q^26 + 8 * q^31 + 20 * q^34 - 12 * q^35 + 32 * q^41 - 20 * q^44 - 12 * q^46 + 16 * q^49 - 12 * q^50 - 4 * q^55 - 4 * q^56 - 24 * q^59 + 8 * q^61 - 4 * q^64 - 36 * q^65 - 4 * q^70 + 32 * q^71 + 4 * q^74 - 20 * q^76 - 32 * q^79 + 12 * q^85 + 8 * q^86 - 28 * q^89 - 28 * q^91 - 8 * q^94 + 8 * q^95

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