LMFDB - Embedded newform 2550.2.c.e.1801.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(1801,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2550.1801");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.c (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:	$20.3618525154$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1801.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2550.1801
Dual form			2550.2.c.e.1801.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +3.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +3.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} +5.00000i q^{11} -1.00000i q^{12} +6.00000 q^{13} -3.00000i q^{14} +1.00000 q^{16} +(-4.00000 - 1.00000i) q^{17} +1.00000 q^{18} +5.00000 q^{19} +3.00000 q^{21} -5.00000i q^{22} -4.00000i q^{23} +1.00000i q^{24} -6.00000 q^{26} +1.00000i q^{27} +3.00000i q^{28} +6.00000i q^{29} -5.00000i q^{31} -1.00000 q^{32} +5.00000 q^{33} +(4.00000 + 1.00000i) q^{34} -1.00000 q^{36} -7.00000i q^{37} -5.00000 q^{38} -6.00000i q^{39} +10.0000i q^{41} -3.00000 q^{42} -9.00000 q^{43} +5.00000i q^{44} +4.00000i q^{46} +7.00000 q^{47} -1.00000i q^{48} -2.00000 q^{49} +(-1.00000 + 4.00000i) q^{51} +6.00000 q^{52} -9.00000 q^{53} -1.00000i q^{54} -3.00000i q^{56} -5.00000i q^{57} -6.00000i q^{58} +10.0000i q^{61} +5.00000i q^{62} -3.00000i q^{63} +1.00000 q^{64} -5.00000 q^{66} -13.0000 q^{67} +(-4.00000 - 1.00000i) q^{68} -4.00000 q^{69} +10.0000i q^{71} +1.00000 q^{72} +16.0000i q^{73} +7.00000i q^{74} +5.00000 q^{76} -15.0000 q^{77} +6.00000i q^{78} +1.00000i q^{79} +1.00000 q^{81} -10.0000i q^{82} +6.00000 q^{83} +3.00000 q^{84} +9.00000 q^{86} +6.00000 q^{87} -5.00000i q^{88} -10.0000 q^{89} +18.0000i q^{91} -4.00000i q^{92} -5.00000 q^{93} -7.00000 q^{94} +1.00000i q^{96} +8.00000i q^{97} +2.00000 q^{98} -5.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} + 12 q^{13} + 2 q^{16} - 8 q^{17} + 2 q^{18} + 10 q^{19} + 6 q^{21} - 12 q^{26} - 2 q^{32} + 10 q^{33} + 8 q^{34} - 2 q^{36} - 10 q^{38} - 6 q^{42} - 18 q^{43} + 14 q^{47} - 4 q^{49} - 2 q^{51} + 12 q^{52} - 18 q^{53} + 2 q^{64} - 10 q^{66} - 26 q^{67} - 8 q^{68} - 8 q^{69} + 2 q^{72} + 10 q^{76} - 30 q^{77} + 2 q^{81} + 12 q^{83} + 6 q^{84} + 18 q^{86} + 12 q^{87} - 20 q^{89} - 10 q^{93} - 14 q^{94} + 4 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 + 12 * q^13 + 2 * q^16 - 8 * q^17 + 2 * q^18 + 10 * q^19 + 6 * q^21 - 12 * q^26 - 2 * q^32 + 10 * q^33 + 8 * q^34 - 2 * q^36 - 10 * q^38 - 6 * q^42 - 18 * q^43 + 14 * q^47 - 4 * q^49 - 2 * q^51 + 12 * q^52 - 18 * q^53 + 2 * q^64 - 10 * q^66 - 26 * q^67 - 8 * q^68 - 8 * q^69 + 2 * q^72 + 10 * q^76 - 30 * q^77 + 2 * q^81 + 12 * q^83 + 6 * q^84 + 18 * q^86 + 12 * q^87 - 20 * q^89 - 10 * q^93 - 14 * q^94 + 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$.

$n$	$751$	$851$	$1327$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$			1.00000i			0.408248i
$7$			3.00000i			1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$			3.00000i			1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	−1.00000			−0.353553
$9$	−1.00000			−0.333333
$10$		0			0
$11$			5.00000i			1.50756i	0.657129	+	0.753778i	$0.271771\pi$
$11$			5.00000i			1.50756i	−0.657129	+	0.753778i	$0.728229\pi$
$12$		−	1.00000i		−	0.288675i
$13$	6.00000			1.66410			0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000			1.66410			0.832050	+	0.554700i	$0.187167\pi$
$14$		−	3.00000i		−	0.801784i
$15$		0			0
$16$	1.00000			0.250000
$17$	−4.00000	−	1.00000i	−0.970143	−	0.242536i
$17$	−4.00000	−	1.00000i	−0.970143	−	0.242536i
$18$	1.00000			0.235702
$19$	5.00000			1.14708			0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000			1.14708			0.573539	+	0.819178i	$0.305570\pi$
$20$		0			0
$21$	3.00000			0.654654
$22$		−	5.00000i		−	1.06600i
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$			1.00000i			0.204124i
$25$		0			0
$26$	−6.00000			−1.17670
$27$			1.00000i			0.192450i
$28$			3.00000i			0.566947i
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$		−	5.00000i		−	0.898027i	−0.893525	−	0.449013i	$-0.851776\pi$
$31$		−	5.00000i		−	0.898027i	0.893525	−	0.449013i	$-0.148224\pi$
$32$	−1.00000			−0.176777
$33$	5.00000			0.870388
$34$	4.00000	+	1.00000i	0.685994	+	0.171499i
$35$		0			0
$36$	−1.00000			−0.166667
$37$		−	7.00000i		−	1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$		−	7.00000i		−	1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	−5.00000			−0.811107
$39$		−	6.00000i		−	0.960769i
$40$		0			0
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$	−3.00000			−0.462910
$43$	−9.00000			−1.37249			−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000			−1.37249			−0.686244	+	0.727372i	$0.740742\pi$
$44$			5.00000i			0.753778i
$45$		0			0
$46$			4.00000i			0.589768i
$47$	7.00000			1.02105			0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000			1.02105			0.510527	+	0.859861i	$0.329450\pi$
$48$		−	1.00000i		−	0.144338i
$49$	−2.00000			−0.285714
$50$		0			0
$51$	−1.00000	+	4.00000i	−0.140028	+	0.560112i
$52$	6.00000			0.832050
$53$	−9.00000			−1.23625			−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000			−1.23625			−0.618123	+	0.786082i	$0.712106\pi$
$54$		−	1.00000i		−	0.136083i
$55$		0			0
$56$		−	3.00000i		−	0.400892i
$57$		−	5.00000i		−	0.662266i
$58$		−	6.00000i		−	0.787839i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$			10.0000i			1.28037i	0.768221	+	0.640184i	$0.221142\pi$
$61$			10.0000i			1.28037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$			5.00000i			0.635001i
$63$		−	3.00000i		−	0.377964i
$64$	1.00000			0.125000
$65$		0			0
$66$	−5.00000			−0.615457
$67$	−13.0000			−1.58820			−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000			−1.58820			−0.794101	+	0.607785i	$0.792058\pi$
$68$	−4.00000	−	1.00000i	−0.485071	−	0.121268i
$69$	−4.00000			−0.481543
$70$		0			0
$71$			10.0000i			1.18678i	0.804914	+	0.593391i	$0.202211\pi$
$71$			10.0000i			1.18678i	−0.804914	+	0.593391i	$0.797789\pi$
$72$	1.00000			0.117851
$73$			16.0000i			1.87266i	0.351123	+	0.936329i	$0.385800\pi$
$73$			16.0000i			1.87266i	−0.351123	+	0.936329i	$0.614200\pi$
$74$			7.00000i			0.813733i
$75$		0			0
$76$	5.00000			0.573539
$77$	−15.0000			−1.70941
$78$			6.00000i			0.679366i
$79$			1.00000i			0.112509i	0.998416	+	0.0562544i	$0.0179158\pi$
$79$			1.00000i			0.112509i	−0.998416	+	0.0562544i	$0.982084\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		−	10.0000i		−	1.10432i
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$	3.00000			0.327327
$85$		0			0
$86$	9.00000			0.970495
$87$	6.00000			0.643268
$88$		−	5.00000i		−	0.533002i
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$			18.0000i			1.88691i
$92$		−	4.00000i		−	0.417029i
$93$	−5.00000			−0.518476
$94$	−7.00000			−0.721995
$95$		0			0
$96$			1.00000i			0.102062i
$97$			8.00000i			0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$			8.00000i			0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	2.00000			0.202031
$99$		−	5.00000i		−	0.502519i
$100$		0			0
$101$	17.0000			1.69156			0.845782	−	0.533529i	$-0.179135\pi$
$101$	17.0000			1.69156			0.845782	+	0.533529i	$0.179135\pi$
$102$	1.00000	−	4.00000i	0.0990148	−	0.396059i
$103$	6.00000			0.591198			0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000			0.591198			0.295599	+	0.955312i	$0.404481\pi$
$104$	−6.00000			−0.588348
$105$		0			0
$106$	9.00000			0.874157
$107$			3.00000i			0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$			3.00000i			0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$			1.00000i			0.0962250i
$109$		−	9.00000i		−	0.862044i	−0.902342	−	0.431022i	$-0.858153\pi$
$109$		−	9.00000i		−	0.862044i	0.902342	−	0.431022i	$-0.141847\pi$
$110$		0			0
$111$	−7.00000			−0.664411
$112$			3.00000i			0.283473i
$113$			1.00000i			0.0940721i	0.998893	+	0.0470360i	$0.0149776\pi$
$113$			1.00000i			0.0940721i	−0.998893	+	0.0470360i	$0.985022\pi$
$114$			5.00000i			0.468293i
$115$		0			0
$116$			6.00000i			0.557086i
$117$	−6.00000			−0.554700
$118$		0			0
$119$	3.00000	−	12.0000i	0.275010	−	1.10004i
$120$		0			0
$121$	−14.0000			−1.27273
$122$		−	10.0000i		−	0.905357i
$123$	10.0000			0.901670
$124$		−	5.00000i		−	0.449013i
$125$		0			0
$126$			3.00000i			0.267261i
$127$	22.0000			1.95218			0.976092	−	0.217357i	$-0.0697436\pi$
$127$	22.0000			1.95218			0.976092	+	0.217357i	$0.0697436\pi$
$128$	−1.00000			−0.0883883
$129$			9.00000i			0.792406i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	5.00000			0.435194
$133$			15.0000i			1.30066i
$134$	13.0000			1.12303
$135$		0			0
$136$	4.00000	+	1.00000i	0.342997	+	0.0857493i
$137$	−18.0000			−1.53784			−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000			−1.53784			−0.768922	+	0.639343i	$0.779207\pi$
$138$	4.00000			0.340503
$139$		−	4.00000i		−	0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$		−	4.00000i		−	0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$		0			0
$141$		−	7.00000i		−	0.589506i
$142$		−	10.0000i		−	0.839181i
$143$			30.0000i			2.50873i
$144$	−1.00000			−0.0833333
$145$		0			0
$146$		−	16.0000i		−	1.32417i
$147$			2.00000i			0.164957i
$148$		−	7.00000i		−	0.575396i
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$	2.00000			0.162758			0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000			0.162758			0.0813788	+	0.996683i	$0.474068\pi$
$152$	−5.00000			−0.405554
$153$	4.00000	+	1.00000i	0.323381	+	0.0808452i
$154$	15.0000			1.20873
$155$		0			0
$156$		−	6.00000i		−	0.480384i
$157$	22.0000			1.75579			0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000			1.75579			0.877896	+	0.478852i	$0.158947\pi$
$158$		−	1.00000i		−	0.0795557i
$159$			9.00000i			0.713746i
$160$		0			0
$161$	12.0000			0.945732
$162$	−1.00000			−0.0785674
$163$		−	14.0000i		−	1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$		−	14.0000i		−	1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$			10.0000i			0.780869i
$165$		0			0
$166$	−6.00000			−0.465690
$167$		−	22.0000i		−	1.70241i	−0.524832	−	0.851206i	$-0.675872\pi$
$167$		−	22.0000i		−	1.70241i	0.524832	−	0.851206i	$-0.324128\pi$
$168$	−3.00000			−0.231455
$169$	23.0000			1.76923
$170$		0			0
$171$	−5.00000			−0.382360
$172$	−9.00000			−0.686244
$173$			6.00000i			0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$			6.00000i			0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	−6.00000			−0.454859
$175$		0			0
$176$			5.00000i			0.376889i
$177$		0			0
$178$	10.0000			0.749532
$179$	−10.0000			−0.747435			−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000			−0.747435			−0.373718	+	0.927543i	$0.621917\pi$
$180$		0			0
$181$			5.00000i			0.371647i	0.982583	+	0.185824i	$0.0594953\pi$
$181$			5.00000i			0.371647i	−0.982583	+	0.185824i	$0.940505\pi$
$182$		−	18.0000i		−	1.33425i
$183$	10.0000			0.739221
$184$			4.00000i			0.294884i
$185$		0			0
$186$	5.00000			0.366618
$187$	5.00000	−	20.0000i	0.365636	−	1.46254i
$188$	7.00000			0.510527
$189$	−3.00000			−0.218218
$190$		0			0
$191$	7.00000			0.506502			0.253251	−	0.967401i	$-0.418500\pi$
$191$	7.00000			0.506502			0.253251	+	0.967401i	$0.418500\pi$
$192$		−	1.00000i		−	0.0721688i
$193$		−	14.0000i		−	1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$		−	14.0000i		−	1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$		−	8.00000i		−	0.574367i
$195$		0			0
$196$	−2.00000			−0.142857
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$			5.00000i			0.355335i
$199$			11.0000i			0.779769i	0.920864	+	0.389885i	$0.127485\pi$
$199$			11.0000i			0.779769i	−0.920864	+	0.389885i	$0.872515\pi$
$200$		0			0
$201$			13.0000i			0.916949i
$202$	−17.0000			−1.19612
$203$	−18.0000			−1.26335
$204$	−1.00000	+	4.00000i	−0.0700140	+	0.280056i
$205$		0			0
$206$	−6.00000			−0.418040
$207$			4.00000i			0.278019i
$208$	6.00000			0.416025
$209$			25.0000i			1.72929i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−9.00000			−0.618123
$213$	10.0000			0.685189
$214$		−	3.00000i		−	0.205076i
$215$		0			0
$216$		−	1.00000i		−	0.0680414i
$217$	15.0000			1.01827
$218$			9.00000i			0.609557i
$219$	16.0000			1.08118
$220$		0			0
$221$	−24.0000	−	6.00000i	−1.61441	−	0.403604i
$222$	7.00000			0.469809
$223$	−14.0000			−0.937509			−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000			−0.937509			−0.468755	+	0.883328i	$0.655297\pi$
$224$		−	3.00000i		−	0.200446i
$225$		0			0
$226$		−	1.00000i		−	0.0665190i
$227$			23.0000i			1.52656i	0.646066	+	0.763282i	$0.276413\pi$
$227$			23.0000i			1.52656i	−0.646066	+	0.763282i	$0.723587\pi$
$228$		−	5.00000i		−	0.331133i
$229$	−20.0000			−1.32164			−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000			−1.32164			−0.660819	+	0.750546i	$0.729791\pi$
$230$		0			0
$231$			15.0000i			0.986928i
$232$		−	6.00000i		−	0.393919i
$233$			6.00000i			0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$			6.00000i			0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	6.00000			0.392232
$235$		0			0
$236$		0			0
$237$	1.00000			0.0649570
$238$	−3.00000	+	12.0000i	−0.194461	+	0.777844i
$239$	−15.0000			−0.970269			−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000			−0.970269			−0.485135	+	0.874439i	$0.661229\pi$
$240$		0			0
$241$			20.0000i			1.28831i	0.764894	+	0.644157i	$0.222792\pi$
$241$			20.0000i			1.28831i	−0.764894	+	0.644157i	$0.777208\pi$
$242$	14.0000			0.899954
$243$		−	1.00000i		−	0.0641500i
$244$			10.0000i			0.640184i
$245$		0			0
$246$	−10.0000			−0.637577
$247$	30.0000			1.90885
$248$			5.00000i			0.317500i
$249$		−	6.00000i		−	0.380235i
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		−	3.00000i		−	0.188982i
$253$	20.0000			1.25739
$254$	−22.0000			−1.38040
$255$		0			0
$256$	1.00000			0.0625000
$257$	22.0000			1.37232			0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000			1.37232			0.686161	+	0.727450i	$0.259294\pi$
$258$		−	9.00000i		−	0.560316i
$259$	21.0000			1.30488
$260$		0			0
$261$		−	6.00000i		−	0.371391i
$262$		0			0
$263$	1.00000			0.0616626			0.0308313	−	0.999525i	$-0.490185\pi$
$263$	1.00000			0.0616626			0.0308313	+	0.999525i	$0.490185\pi$
$264$	−5.00000			−0.307729
$265$		0			0
$266$		−	15.0000i		−	0.919709i
$267$			10.0000i			0.611990i
$268$	−13.0000			−0.794101
$269$		−	24.0000i		−	1.46331i	−0.681677	−	0.731653i	$-0.738749\pi$
$269$		−	24.0000i		−	1.46331i	0.681677	−	0.731653i	$-0.261251\pi$
$270$		0			0
$271$	2.00000			0.121491			0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000			0.121491			0.0607457	+	0.998153i	$0.480652\pi$
$272$	−4.00000	−	1.00000i	−0.242536	−	0.0606339i
$273$	18.0000			1.08941
$274$	18.0000			1.08742
$275$		0			0
$276$	−4.00000			−0.240772
$277$			13.0000i			0.781094i	0.920583	+	0.390547i	$0.127714\pi$
$277$			13.0000i			0.781094i	−0.920583	+	0.390547i	$0.872286\pi$
$278$			4.00000i			0.239904i
$279$			5.00000i			0.299342i
$280$		0			0
$281$	12.0000			0.715860			0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000			0.715860			0.357930	+	0.933748i	$0.383483\pi$
$282$			7.00000i			0.416844i
$283$			16.0000i			0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$			16.0000i			0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$			10.0000i			0.593391i
$285$		0			0
$286$		−	30.0000i		−	1.77394i
$287$	−30.0000			−1.77084
$288$	1.00000			0.0589256
$289$	15.0000	+	8.00000i	0.882353	+	0.470588i
$290$		0			0
$291$	8.00000			0.468968
$292$			16.0000i			0.936329i
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		−	2.00000i		−	0.116642i
$295$		0			0
$296$			7.00000i			0.406867i
$297$	−5.00000			−0.290129
$298$	10.0000			0.579284
$299$		−	24.0000i		−	1.38796i
$300$		0			0
$301$		−	27.0000i		−	1.55625i
$302$	−2.00000			−0.115087
$303$		−	17.0000i		−	0.976624i
$304$	5.00000			0.286770
$305$		0			0
$306$	−4.00000	−	1.00000i	−0.228665	−	0.0571662i
$307$	−28.0000			−1.59804			−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000			−1.59804			−0.799022	+	0.601302i	$0.794649\pi$
$308$	−15.0000			−0.854704
$309$		−	6.00000i		−	0.341328i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$			6.00000i			0.339683i
$313$			26.0000i			1.46961i	0.678280	+	0.734803i	$0.262726\pi$
$313$			26.0000i			1.46961i	−0.678280	+	0.734803i	$0.737274\pi$
$314$	−22.0000			−1.24153
$315$		0			0
$316$			1.00000i			0.0562544i
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		−	9.00000i		−	0.504695i
$319$	−30.0000			−1.67968
$320$		0			0
$321$	3.00000			0.167444
$322$	−12.0000			−0.668734
$323$	−20.0000	−	5.00000i	−1.11283	−	0.278207i
$324$	1.00000			0.0555556
$325$		0			0
$326$			14.0000i			0.775388i
$327$	−9.00000			−0.497701
$328$		−	10.0000i		−	0.552158i
$329$			21.0000i			1.15777i
$330$		0			0
$331$	7.00000			0.384755			0.192377	−	0.981321i	$-0.438380\pi$
$331$	7.00000			0.384755			0.192377	+	0.981321i	$0.438380\pi$
$332$	6.00000			0.329293
$333$			7.00000i			0.383598i
$334$			22.0000i			1.20379i
$335$		0			0
$336$	3.00000			0.163663
$337$			28.0000i			1.52526i	0.646837	+	0.762629i	$0.276092\pi$
$337$			28.0000i			1.52526i	−0.646837	+	0.762629i	$0.723908\pi$
$338$	−23.0000			−1.25104
$339$	1.00000			0.0543125
$340$		0			0
$341$	25.0000			1.35383
$342$	5.00000			0.270369
$343$			15.0000i			0.809924i
$344$	9.00000			0.485247
$345$		0			0
$346$		−	6.00000i		−	0.322562i
$347$		−	17.0000i		−	0.912608i	−0.889824	−	0.456304i	$-0.849173\pi$
$347$		−	17.0000i		−	0.912608i	0.889824	−	0.456304i	$-0.150827\pi$
$348$	6.00000			0.321634
$349$	20.0000			1.07058			0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000			1.07058			0.535288	+	0.844670i	$0.320203\pi$
$350$		0			0
$351$			6.00000i			0.320256i
$352$		−	5.00000i		−	0.266501i
$353$	−14.0000			−0.745145			−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000			−0.745145			−0.372572	+	0.928003i	$0.621524\pi$
$354$		0			0
$355$		0			0
$356$	−10.0000			−0.529999
$357$	−12.0000	−	3.00000i	−0.635107	−	0.158777i
$358$	10.0000			0.528516
$359$	−15.0000			−0.791670			−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000			−0.791670			−0.395835	+	0.918322i	$0.629545\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		−	5.00000i		−	0.262794i
$363$			14.0000i			0.734809i
$364$			18.0000i			0.943456i
$365$		0			0
$366$	−10.0000			−0.522708
$367$			13.0000i			0.678594i	0.940679	+	0.339297i	$0.110189\pi$
$367$			13.0000i			0.678594i	−0.940679	+	0.339297i	$0.889811\pi$
$368$		−	4.00000i		−	0.208514i
$369$		−	10.0000i		−	0.520579i
$370$		0			0
$371$		−	27.0000i		−	1.40177i
$372$	−5.00000			−0.259238
$373$	−4.00000			−0.207112			−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000			−0.207112			−0.103556	+	0.994624i	$0.533022\pi$
$374$	−5.00000	+	20.0000i	−0.258544	+	1.03418i
$375$		0			0
$376$	−7.00000			−0.360997
$377$			36.0000i			1.85409i
$378$	3.00000			0.154303
$379$		−	14.0000i		−	0.719132i	−0.933120	−	0.359566i	$-0.882925\pi$
$379$		−	14.0000i		−	0.719132i	0.933120	−	0.359566i	$-0.117075\pi$
$380$		0			0
$381$		−	22.0000i		−	1.12709i
$382$	−7.00000			−0.358151
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$			1.00000i			0.0510310i
$385$		0			0
$386$			14.0000i			0.712581i
$387$	9.00000			0.457496
$388$			8.00000i			0.406138i
$389$	−5.00000			−0.253510			−0.126755	−	0.991934i	$-0.540456\pi$
$389$	−5.00000			−0.253510			−0.126755	+	0.991934i	$0.540456\pi$
$390$		0			0
$391$	−4.00000	+	16.0000i	−0.202289	+	0.809155i
$392$	2.00000			0.101015
$393$		0			0
$394$		−	18.0000i		−	0.906827i
$395$		0			0
$396$		−	5.00000i		−	0.251259i
$397$		−	17.0000i		−	0.853206i	−0.904439	−	0.426603i	$-0.859710\pi$
$397$		−	17.0000i		−	0.853206i	0.904439	−	0.426603i	$-0.140290\pi$
$398$		−	11.0000i		−	0.551380i
$399$	15.0000			0.750939
$400$		0			0
$401$		−	30.0000i		−	1.49813i	−0.662497	−	0.749064i	$-0.730503\pi$
$401$		−	30.0000i		−	1.49813i	0.662497	−	0.749064i	$-0.269497\pi$
$402$		−	13.0000i		−	0.648381i
$403$		−	30.0000i		−	1.49441i
$404$	17.0000			0.845782
$405$		0			0
$406$	18.0000			0.893325
$407$	35.0000			1.73489
$408$	1.00000	−	4.00000i	0.0495074	−	0.198030i
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$			18.0000i			0.887875i
$412$	6.00000			0.295599
$413$		0			0
$414$		−	4.00000i		−	0.196589i
$415$		0			0
$416$	−6.00000			−0.294174
$417$	−4.00000			−0.195881
$418$		−	25.0000i		−	1.22279i
$419$		−	4.00000i		−	0.195413i	−0.995215	−	0.0977064i	$-0.968849\pi$
$419$		−	4.00000i		−	0.195413i	0.995215	−	0.0977064i	$-0.0311506\pi$
$420$		0			0
$421$	−28.0000			−1.36464			−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000			−1.36464			−0.682318	+	0.731055i	$0.739028\pi$
$422$		0			0
$423$	−7.00000			−0.340352
$424$	9.00000			0.437079
$425$		0			0
$426$	−10.0000			−0.484502
$427$	−30.0000			−1.45180
$428$			3.00000i			0.145010i
$429$	30.0000			1.44841
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$			1.00000i			0.0481125i
$433$	11.0000			0.528626			0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000			0.528626			0.264313	+	0.964437i	$0.414855\pi$
$434$	−15.0000			−0.720023
$435$		0			0
$436$		−	9.00000i		−	0.431022i
$437$		−	20.0000i		−	0.956730i
$438$	−16.0000			−0.764510
$439$			16.0000i			0.763638i	0.924237	+	0.381819i	$0.124702\pi$
$439$			16.0000i			0.763638i	−0.924237	+	0.381819i	$0.875298\pi$
$440$		0			0
$441$	2.00000			0.0952381
$442$	24.0000	+	6.00000i	1.14156	+	0.285391i
$443$	−4.00000			−0.190046			−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000			−0.190046			−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−7.00000			−0.332205
$445$		0			0
$446$	14.0000			0.662919
$447$			10.0000i			0.472984i
$448$			3.00000i			0.141737i
$449$			21.0000i			0.991051i	0.868593	+	0.495526i	$0.165025\pi$
$449$			21.0000i			0.991051i	−0.868593	+	0.495526i	$0.834975\pi$
$450$		0			0
$451$	−50.0000			−2.35441
$452$			1.00000i			0.0470360i
$453$		−	2.00000i		−	0.0939682i
$454$		−	23.0000i		−	1.07944i
$455$		0			0
$456$			5.00000i			0.234146i
$457$	17.0000			0.795226			0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000			0.795226			0.397613	+	0.917553i	$0.369839\pi$
$458$	20.0000			0.934539
$459$	1.00000	−	4.00000i	0.0466760	−	0.186704i
$460$		0			0
$461$	−3.00000			−0.139724			−0.0698620	−	0.997557i	$-0.522256\pi$
$461$	−3.00000			−0.139724			−0.0698620	+	0.997557i	$0.522256\pi$
$462$		−	15.0000i		−	0.697863i
$463$	−14.0000			−0.650635			−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000			−0.650635			−0.325318	+	0.945605i	$0.605471\pi$
$464$			6.00000i			0.278543i
$465$		0			0
$466$		−	6.00000i		−	0.277945i
$467$	−8.00000			−0.370196			−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000			−0.370196			−0.185098	+	0.982720i	$0.559260\pi$
$468$	−6.00000			−0.277350
$469$		−	39.0000i		−	1.80085i
$470$		0			0
$471$		−	22.0000i		−	1.01371i
$472$		0			0
$473$		−	45.0000i		−	2.06910i
$474$	−1.00000			−0.0459315
$475$		0			0
$476$	3.00000	−	12.0000i	0.137505	−	0.550019i
$477$	9.00000			0.412082
$478$	15.0000			0.686084
$479$		−	24.0000i		−	1.09659i	−0.836286	−	0.548294i	$-0.815277\pi$
$479$		−	24.0000i		−	1.09659i	0.836286	−	0.548294i	$-0.184723\pi$
$480$		0			0
$481$		−	42.0000i		−	1.91504i
$482$		−	20.0000i		−	0.910975i
$483$		−	12.0000i		−	0.546019i
$484$	−14.0000			−0.636364
$485$		0			0
$486$			1.00000i			0.0453609i
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		−	10.0000i		−	0.452679i
$489$	−14.0000			−0.633102
$490$		0			0
$491$	22.0000			0.992846			0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000			0.992846			0.496423	+	0.868081i	$0.334646\pi$
$492$	10.0000			0.450835
$493$	6.00000	−	24.0000i	0.270226	−	1.08091i
$494$	−30.0000			−1.34976
$495$		0			0
$496$		−	5.00000i		−	0.224507i
$497$	−30.0000			−1.34568
$498$			6.00000i			0.268866i
$499$		−	4.00000i		−	0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$		−	4.00000i		−	0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$		0			0
$501$	−22.0000			−0.982888
$502$	−12.0000			−0.535586
$503$		−	24.0000i		−	1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$		−	24.0000i		−	1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$			3.00000i			0.133631i
$505$		0			0
$506$	−20.0000			−0.889108
$507$		−	23.0000i		−	1.02147i
$508$	22.0000			0.976092
$509$	35.0000			1.55135			0.775674	−	0.631134i	$-0.217410\pi$
$509$	35.0000			1.55135			0.775674	+	0.631134i	$0.217410\pi$
$510$		0			0
$511$	−48.0000			−2.12339
$512$	−1.00000			−0.0441942
$513$			5.00000i			0.220755i
$514$	−22.0000			−0.970378
$515$		0			0
$516$			9.00000i			0.396203i
$517$			35.0000i			1.53930i
$518$	−21.0000			−0.922687
$519$	6.00000			0.263371
$520$		0			0
$521$		−	35.0000i		−	1.53338i	−0.642019	−	0.766689i	$-0.721903\pi$
$521$		−	35.0000i		−	1.53338i	0.642019	−	0.766689i	$-0.278097\pi$
$522$			6.00000i			0.262613i
$523$	−4.00000			−0.174908			−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000			−0.174908			−0.0874539	+	0.996169i	$0.527873\pi$
$524$		0			0
$525$		0			0
$526$	−1.00000			−0.0436021
$527$	−5.00000	+	20.0000i	−0.217803	+	0.871214i
$528$	5.00000			0.217597
$529$	7.00000			0.304348
$530$		0			0
$531$		0			0
$532$			15.0000i			0.650332i
$533$			60.0000i			2.59889i
$534$		−	10.0000i		−	0.432742i
$535$		0			0
$536$	13.0000			0.561514
$537$			10.0000i			0.431532i
$538$			24.0000i			1.03471i
$539$		−	10.0000i		−	0.430730i
$540$		0			0
$541$			15.0000i			0.644900i	0.946586	+	0.322450i	$0.104506\pi$
$541$			15.0000i			0.644900i	−0.946586	+	0.322450i	$0.895494\pi$
$542$	−2.00000			−0.0859074
$543$	5.00000			0.214571
$544$	4.00000	+	1.00000i	0.171499	+	0.0428746i
$545$		0			0
$546$	−18.0000			−0.770329
$547$			18.0000i			0.769624i	0.922995	+	0.384812i	$0.125734\pi$
$547$			18.0000i			0.769624i	−0.922995	+	0.384812i	$0.874266\pi$
$548$	−18.0000			−0.768922
$549$		−	10.0000i		−	0.426790i
$550$		0			0
$551$			30.0000i			1.27804i
$552$	4.00000			0.170251
$553$	−3.00000			−0.127573
$554$		−	13.0000i		−	0.552317i
$555$		0			0
$556$		−	4.00000i		−	0.169638i
$557$	7.00000			0.296600			0.148300	−	0.988942i	$-0.452620\pi$
$557$	7.00000			0.296600			0.148300	+	0.988942i	$0.452620\pi$
$558$		−	5.00000i		−	0.211667i
$559$	−54.0000			−2.28396
$560$		0			0
$561$	−20.0000	−	5.00000i	−0.844401	−	0.211100i
$562$	−12.0000			−0.506189
$563$	26.0000			1.09577			0.547885	−	0.836554i	$-0.315433\pi$
$563$	26.0000			1.09577			0.547885	+	0.836554i	$0.315433\pi$
$564$		−	7.00000i		−	0.294753i
$565$		0			0
$566$		−	16.0000i		−	0.672530i
$567$			3.00000i			0.125988i
$568$		−	10.0000i		−	0.419591i
$569$	20.0000			0.838444			0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000			0.838444			0.419222	+	0.907884i	$0.362303\pi$
$570$		0			0
$571$			30.0000i			1.25546i	0.778431	+	0.627730i	$0.216016\pi$
$571$			30.0000i			1.25546i	−0.778431	+	0.627730i	$0.783984\pi$
$572$			30.0000i			1.25436i
$573$		−	7.00000i		−	0.292429i
$574$	30.0000			1.25218
$575$		0			0
$576$	−1.00000			−0.0416667
$577$	17.0000			0.707719			0.353860	−	0.935299i	$-0.384869\pi$
$577$	17.0000			0.707719			0.353860	+	0.935299i	$0.384869\pi$
$578$	−15.0000	−	8.00000i	−0.623918	−	0.332756i
$579$	−14.0000			−0.581820
$580$		0			0
$581$			18.0000i			0.746766i
$582$	−8.00000			−0.331611
$583$		−	45.0000i		−	1.86371i
$584$		−	16.0000i		−	0.662085i
$585$		0			0
$586$	−6.00000			−0.247858
$587$	2.00000			0.0825488			0.0412744	−	0.999148i	$-0.486858\pi$
$587$	2.00000			0.0825488			0.0412744	+	0.999148i	$0.486858\pi$
$588$			2.00000i			0.0824786i
$589$		−	25.0000i		−	1.03011i
$590$		0			0
$591$	18.0000			0.740421
$592$		−	7.00000i		−	0.287698i
$593$	6.00000			0.246390			0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000			0.246390			0.123195	+	0.992382i	$0.460686\pi$
$594$	5.00000			0.205152
$595$		0			0
$596$	−10.0000			−0.409616
$597$	11.0000			0.450200
$598$			24.0000i			0.981433i
$599$	−5.00000			−0.204294			−0.102147	−	0.994769i	$-0.532571\pi$
$599$	−5.00000			−0.204294			−0.102147	+	0.994769i	$0.532571\pi$
$600$		0			0
$601$		−	10.0000i		−	0.407909i	−0.978980	−	0.203954i	$-0.934621\pi$
$601$		−	10.0000i		−	0.407909i	0.978980	−	0.203954i	$-0.0653794\pi$
$602$			27.0000i			1.10044i
$603$	13.0000			0.529401
$604$	2.00000			0.0813788
$605$		0			0
$606$			17.0000i			0.690578i
$607$		−	32.0000i		−	1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$		−	32.0000i		−	1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	−5.00000			−0.202777
$609$			18.0000i			0.729397i
$610$		0			0
$611$	42.0000			1.69914
$612$	4.00000	+	1.00000i	0.161690	+	0.0404226i
$613$	−24.0000			−0.969351			−0.484675	−	0.874694i	$-0.661062\pi$
$613$	−24.0000			−0.969351			−0.484675	+	0.874694i	$0.661062\pi$
$614$	28.0000			1.12999
$615$		0			0
$616$	15.0000			0.604367
$617$			13.0000i			0.523360i	0.965155	+	0.261680i	$0.0842766\pi$
$617$			13.0000i			0.523360i	−0.965155	+	0.261680i	$0.915723\pi$
$618$			6.00000i			0.241355i
$619$		−	14.0000i		−	0.562708i	−0.959604	−	0.281354i	$-0.909217\pi$
$619$		−	14.0000i		−	0.562708i	0.959604	−	0.281354i	$-0.0907834\pi$
$620$		0			0
$621$	4.00000			0.160514
$622$		0			0
$623$		−	30.0000i		−	1.20192i
$624$		−	6.00000i		−	0.240192i
$625$		0			0
$626$		−	26.0000i		−	1.03917i
$627$	25.0000			0.998404
$628$	22.0000			0.877896
$629$	−7.00000	+	28.0000i	−0.279108	+	1.11643i
$630$		0			0
$631$	22.0000			0.875806			0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000			0.875806			0.437903	+	0.899022i	$0.355721\pi$
$632$		−	1.00000i		−	0.0397779i
$633$		0			0
$634$		−	18.0000i		−	0.714871i
$635$		0			0
$636$			9.00000i			0.356873i
$637$	−12.0000			−0.475457
$638$	30.0000			1.18771
$639$		−	10.0000i		−	0.395594i
$640$		0			0
$641$		−	30.0000i		−	1.18493i	−0.805597	−	0.592464i	$-0.798155\pi$
$641$		−	30.0000i		−	1.18493i	0.805597	−	0.592464i	$-0.201845\pi$
$642$	−3.00000			−0.118401
$643$		−	24.0000i		−	0.946468i	−0.880937	−	0.473234i	$-0.843087\pi$
$643$		−	24.0000i		−	0.946468i	0.880937	−	0.473234i	$-0.156913\pi$
$644$	12.0000			0.472866
$645$		0			0
$646$	20.0000	+	5.00000i	0.786889	+	0.196722i
$647$	−8.00000			−0.314512			−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000			−0.314512			−0.157256	+	0.987558i	$0.550265\pi$
$648$	−1.00000			−0.0392837
$649$		0			0
$650$		0			0
$651$		−	15.0000i		−	0.587896i
$652$		−	14.0000i		−	0.548282i
$653$			36.0000i			1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$			36.0000i			1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$	9.00000			0.351928
$655$		0			0
$656$			10.0000i			0.390434i
$657$		−	16.0000i		−	0.624219i
$658$		−	21.0000i		−	0.818665i
$659$	10.0000			0.389545			0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000			0.389545			0.194772	+	0.980848i	$0.437603\pi$
$660$		0			0
$661$	12.0000			0.466746			0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000			0.466746			0.233373	+	0.972387i	$0.425024\pi$
$662$	−7.00000			−0.272063
$663$	−6.00000	+	24.0000i	−0.233021	+	0.932083i
$664$	−6.00000			−0.232845
$665$		0			0
$666$		−	7.00000i		−	0.271244i
$667$	24.0000			0.929284
$668$		−	22.0000i		−	0.851206i
$669$			14.0000i			0.541271i
$670$		0			0
$671$	−50.0000			−1.93023
$672$	−3.00000			−0.115728
$673$		−	14.0000i		−	0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$		−	14.0000i		−	0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$		−	28.0000i		−	1.07852i
$675$		0			0
$676$	23.0000			0.884615
$677$			8.00000i			0.307465i	0.988113	+	0.153732i	$0.0491294\pi$
$677$			8.00000i			0.307465i	−0.988113	+	0.153732i	$0.950871\pi$
$678$	−1.00000			−0.0384048
$679$	−24.0000			−0.921035
$680$		0			0
$681$	23.0000			0.881362
$682$	−25.0000			−0.957299
$683$		−	44.0000i		−	1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$		−	44.0000i		−	1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$	−5.00000			−0.191180
$685$		0			0
$686$		−	15.0000i		−	0.572703i
$687$			20.0000i			0.763048i
$688$	−9.00000			−0.343122
$689$	−54.0000			−2.05724
$690$		0			0
$691$		−	20.0000i		−	0.760836i	−0.924815	−	0.380418i	$-0.875780\pi$
$691$		−	20.0000i		−	0.760836i	0.924815	−	0.380418i	$-0.124220\pi$
$692$			6.00000i			0.228086i
$693$	15.0000			0.569803
$694$			17.0000i			0.645311i
$695$		0			0
$696$	−6.00000			−0.227429
$697$	10.0000	−	40.0000i	0.378777	−	1.51511i
$698$	−20.0000			−0.757011
$699$	6.00000			0.226941
$700$		0			0
$701$	22.0000			0.830929			0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000			0.830929			0.415464	+	0.909610i	$0.363619\pi$
$702$		−	6.00000i		−	0.226455i
$703$		−	35.0000i		−	1.32005i
$704$			5.00000i			0.188445i
$705$		0			0
$706$	14.0000			0.526897
$707$			51.0000i			1.91805i
$708$		0			0
$709$		−	19.0000i		−	0.713560i	−0.934188	−	0.356780i	$-0.883875\pi$
$709$		−	19.0000i		−	0.713560i	0.934188	−	0.356780i	$-0.116125\pi$
$710$		0			0
$711$		−	1.00000i		−	0.0375029i
$712$	10.0000			0.374766
$713$	−20.0000			−0.749006
$714$	12.0000	+	3.00000i	0.449089	+	0.112272i
$715$		0			0
$716$	−10.0000			−0.373718
$717$			15.0000i			0.560185i
$718$	15.0000			0.559795
$719$		−	4.00000i		−	0.149175i	−0.997214	−	0.0745874i	$-0.976236\pi$
$719$		−	4.00000i		−	0.149175i	0.997214	−	0.0745874i	$-0.0237640\pi$
$720$		0			0
$721$			18.0000i			0.670355i
$722$	−6.00000			−0.223297
$723$	20.0000			0.743808
$724$			5.00000i			0.185824i
$725$		0			0
$726$		−	14.0000i		−	0.519589i
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$		−	18.0000i		−	0.667124i
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	36.0000	+	9.00000i	1.33151	+	0.332877i
$732$	10.0000			0.369611
$733$	−34.0000			−1.25582			−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000			−1.25582			−0.627909	+	0.778287i	$0.716089\pi$
$734$		−	13.0000i		−	0.479839i
$735$		0			0
$736$			4.00000i			0.147442i
$737$		−	65.0000i		−	2.39431i
$738$			10.0000i			0.368105i
$739$	25.0000			0.919640			0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000			0.919640			0.459820	+	0.888012i	$0.347914\pi$
$740$		0			0
$741$		−	30.0000i		−	1.10208i
$742$			27.0000i			0.991201i
$743$		−	24.0000i		−	0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$		−	24.0000i		−	0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	5.00000			0.183309
$745$		0			0
$746$	4.00000			0.146450
$747$	−6.00000			−0.219529
$748$	5.00000	−	20.0000i	0.182818	−	0.731272i
$749$	−9.00000			−0.328853
$750$		0			0
$751$		−	20.0000i		−	0.729810i	−0.931045	−	0.364905i	$-0.881101\pi$
$751$		−	20.0000i		−	0.729810i	0.931045	−	0.364905i	$-0.118899\pi$
$752$	7.00000			0.255264
$753$		−	12.0000i		−	0.437304i
$754$		−	36.0000i		−	1.31104i
$755$		0			0
$756$	−3.00000			−0.109109
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$			14.0000i			0.508503i
$759$		−	20.0000i		−	0.725954i
$760$		0			0
$761$	12.0000			0.435000			0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000			0.435000			0.217500	+	0.976060i	$0.430210\pi$
$762$			22.0000i			0.796976i
$763$	27.0000			0.977466
$764$	7.00000			0.253251
$765$		0			0
$766$	24.0000			0.867155
$767$		0			0
$768$		−	1.00000i		−	0.0360844i
$769$	15.0000			0.540914			0.270457	−	0.962732i	$-0.412825\pi$
$769$	15.0000			0.540914			0.270457	+	0.962732i	$0.412825\pi$
$770$		0			0
$771$		−	22.0000i		−	0.792311i
$772$		−	14.0000i		−	0.503871i
$773$	6.00000			0.215805			0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000			0.215805			0.107903	+	0.994161i	$0.465587\pi$
$774$	−9.00000			−0.323498
$775$		0			0
$776$		−	8.00000i		−	0.287183i
$777$		−	21.0000i		−	0.753371i
$778$	5.00000			0.179259
$779$			50.0000i			1.79144i
$780$		0			0
$781$	−50.0000			−1.78914
$782$	4.00000	−	16.0000i	0.143040	−	0.572159i
$783$	−6.00000			−0.214423
$784$	−2.00000			−0.0714286
$785$		0			0
$786$		0			0
$787$		−	2.00000i		−	0.0712923i	−0.999364	−	0.0356462i	$-0.988651\pi$
$787$		−	2.00000i		−	0.0712923i	0.999364	−	0.0356462i	$-0.0113489\pi$
$788$			18.0000i			0.641223i
$789$		−	1.00000i		−	0.0356009i
$790$		0			0
$791$	−3.00000			−0.106668
$792$			5.00000i			0.177667i
$793$			60.0000i			2.13066i
$794$			17.0000i			0.603307i
$795$		0			0
$796$			11.0000i			0.389885i
$797$	37.0000			1.31061			0.655304	−	0.755366i	$-0.272541\pi$
$797$	37.0000			1.31061			0.655304	+	0.755366i	$0.272541\pi$
$798$	−15.0000			−0.530994
$799$	−28.0000	−	7.00000i	−0.990569	−	0.247642i
$800$		0			0
$801$	10.0000			0.353333
$802$			30.0000i			1.05934i
$803$	−80.0000			−2.82314
$804$			13.0000i			0.458475i
$805$		0			0
$806$			30.0000i			1.05670i
$807$	−24.0000			−0.844840
$808$	−17.0000			−0.598058
$809$		−	19.0000i		−	0.668004i	−0.942572	−	0.334002i	$-0.891601\pi$
$809$		−	19.0000i		−	0.668004i	0.942572	−	0.334002i	$-0.108399\pi$
$810$		0			0
$811$		−	20.0000i		−	0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$		−	20.0000i		−	0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$	−18.0000			−0.631676
$813$		−	2.00000i		−	0.0701431i
$814$	−35.0000			−1.22675
$815$		0			0
$816$	−1.00000	+	4.00000i	−0.0350070	+	0.140028i
$817$	−45.0000			−1.57435
$818$	−10.0000			−0.349642
$819$		−	18.0000i		−	0.628971i
$820$		0			0
$821$			30.0000i			1.04701i	0.852023	+	0.523504i	$0.175375\pi$
$821$			30.0000i			1.04701i	−0.852023	+	0.523504i	$0.824625\pi$
$822$		−	18.0000i		−	0.627822i
$823$		−	24.0000i		−	0.836587i	−0.908312	−	0.418294i	$-0.862628\pi$
$823$		−	24.0000i		−	0.836587i	0.908312	−	0.418294i	$-0.137372\pi$
$824$	−6.00000			−0.209020
$825$		0			0
$826$		0			0
$827$		−	27.0000i		−	0.938882i	−0.882964	−	0.469441i	$-0.844455\pi$
$827$		−	27.0000i		−	0.938882i	0.882964	−	0.469441i	$-0.155545\pi$
$828$			4.00000i			0.139010i
$829$	−20.0000			−0.694629			−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000			−0.694629			−0.347314	+	0.937749i	$0.612906\pi$
$830$		0			0
$831$	13.0000			0.450965
$832$	6.00000			0.208013
$833$	8.00000	+	2.00000i	0.277184	+	0.0692959i
$834$	4.00000			0.138509
$835$		0			0
$836$			25.0000i			0.864643i
$837$	5.00000			0.172825
$838$			4.00000i			0.138178i
$839$			16.0000i			0.552381i	0.961103	+	0.276191i	$0.0890721\pi$
$839$			16.0000i			0.552381i	−0.961103	+	0.276191i	$0.910928\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$	28.0000			0.964944
$843$		−	12.0000i		−	0.413302i
$844$		0			0
$845$		0			0
$846$	7.00000			0.240665
$847$		−	42.0000i		−	1.44314i
$848$	−9.00000			−0.309061
$849$	16.0000			0.549119
$850$		0			0
$851$	−28.0000			−0.959828
$852$	10.0000			0.342594
$853$			11.0000i			0.376633i	0.982108	+	0.188316i	$0.0603030\pi$
$853$			11.0000i			0.376633i	−0.982108	+	0.188316i	$0.939697\pi$
$854$	30.0000			1.02658
$855$		0			0
$856$		−	3.00000i		−	0.102538i
$857$		−	17.0000i		−	0.580709i	−0.956919	−	0.290354i	$-0.906227\pi$
$857$		−	17.0000i		−	0.580709i	0.956919	−	0.290354i	$-0.0937732\pi$
$858$	−30.0000			−1.02418
$859$	−35.0000			−1.19418			−0.597092	−	0.802173i	$-0.703677\pi$
$859$	−35.0000			−1.19418			−0.597092	+	0.802173i	$0.703677\pi$
$860$		0			0
$861$			30.0000i			1.02240i
$862$		0			0
$863$	31.0000			1.05525			0.527626	−	0.849477i	$-0.323082\pi$
$863$	31.0000			1.05525			0.527626	+	0.849477i	$0.323082\pi$
$864$		−	1.00000i		−	0.0340207i
$865$		0			0
$866$	−11.0000			−0.373795
$867$	8.00000	−	15.0000i	0.271694	−	0.509427i
$868$	15.0000			0.509133
$869$	−5.00000			−0.169613
$870$		0			0
$871$	−78.0000			−2.64293
$872$			9.00000i			0.304778i
$873$		−	8.00000i		−	0.270759i
$874$			20.0000i			0.676510i
$875$		0			0
$876$	16.0000			0.540590
$877$			18.0000i			0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$			18.0000i			0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$		−	16.0000i		−	0.539974i
$879$		−	6.00000i		−	0.202375i
$880$		0			0
$881$		−	35.0000i		−	1.17918i	−0.807703	−	0.589590i	$-0.799289\pi$
$881$		−	35.0000i		−	1.17918i	0.807703	−	0.589590i	$-0.200711\pi$
$882$	−2.00000			−0.0673435
$883$	−4.00000			−0.134611			−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000			−0.134611			−0.0673054	+	0.997732i	$0.521440\pi$
$884$	−24.0000	−	6.00000i	−0.807207	−	0.201802i
$885$		0			0
$886$	4.00000			0.134383
$887$			8.00000i			0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$			8.00000i			0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$	7.00000			0.234905
$889$			66.0000i			2.21357i
$890$		0			0
$891$			5.00000i			0.167506i
$892$	−14.0000			−0.468755
$893$	35.0000			1.17123
$894$		−	10.0000i		−	0.334450i
$895$		0			0
$896$		−	3.00000i		−	0.100223i
$897$	−24.0000			−0.801337
$898$		−	21.0000i		−	0.700779i
$899$	30.0000			1.00056
$900$		0			0
$901$	36.0000	+	9.00000i	1.19933	+	0.299833i
$902$	50.0000			1.66482
$903$	−27.0000			−0.898504
$904$		−	1.00000i		−	0.0332595i
$905$		0			0
$906$			2.00000i			0.0664455i
$907$		−	2.00000i		−	0.0664089i	−0.999449	−	0.0332045i	$-0.989429\pi$
$907$		−	2.00000i		−	0.0664089i	0.999449	−	0.0332045i	$-0.0105712\pi$
$908$			23.0000i			0.763282i
$909$	−17.0000			−0.563854
$910$		0			0
$911$		−	30.0000i		−	0.993944i	−0.867766	−	0.496972i	$-0.834445\pi$
$911$		−	30.0000i		−	0.993944i	0.867766	−	0.496972i	$-0.165555\pi$
$912$		−	5.00000i		−	0.165567i
$913$			30.0000i			0.992855i
$914$	−17.0000			−0.562310
$915$		0			0
$916$	−20.0000			−0.660819
$917$		0			0
$918$	−1.00000	+	4.00000i	−0.0330049	+	0.132020i
$919$	−40.0000			−1.31948			−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000			−1.31948			−0.659739	+	0.751495i	$0.729333\pi$
$920$		0			0
$921$			28.0000i			0.922631i
$922$	3.00000			0.0987997
$923$			60.0000i			1.97492i
$924$			15.0000i			0.493464i
$925$		0			0
$926$	14.0000			0.460069
$927$	−6.00000			−0.197066
$928$		−	6.00000i		−	0.196960i
$929$		−	49.0000i		−	1.60764i	−0.594874	−	0.803819i	$-0.702798\pi$
$929$		−	49.0000i		−	1.60764i	0.594874	−	0.803819i	$-0.297202\pi$
$930$		0			0
$931$	−10.0000			−0.327737
$932$			6.00000i			0.196537i
$933$		0			0
$934$	8.00000			0.261768
$935$		0			0
$936$	6.00000			0.196116
$937$	42.0000			1.37208			0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000			1.37208			0.686040	+	0.727564i	$0.259347\pi$
$938$			39.0000i			1.27340i
$939$	26.0000			0.848478
$940$		0			0
$941$		−	10.0000i		−	0.325991i	−0.986627	−	0.162995i	$-0.947884\pi$
$941$		−	10.0000i		−	0.325991i	0.986627	−	0.162995i	$-0.0521156\pi$
$942$			22.0000i			0.716799i
$943$	40.0000			1.30258
$944$		0			0
$945$		0			0
$946$			45.0000i			1.46308i
$947$		−	47.0000i		−	1.52729i	−0.645634	−	0.763647i	$-0.723407\pi$
$947$		−	47.0000i		−	1.52729i	0.645634	−	0.763647i	$-0.276593\pi$
$948$	1.00000			0.0324785
$949$			96.0000i			3.11629i
$950$		0			0
$951$	18.0000			0.583690
$952$	−3.00000	+	12.0000i	−0.0972306	+	0.388922i
$953$	−34.0000			−1.10137			−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000			−1.10137			−0.550684	+	0.834714i	$0.685633\pi$
$954$	−9.00000			−0.291386
$955$		0			0
$956$	−15.0000			−0.485135
$957$			30.0000i			0.969762i
$958$			24.0000i			0.775405i
$959$		−	54.0000i		−	1.74375i
$960$		0			0
$961$	6.00000			0.193548
$962$			42.0000i			1.35413i
$963$		−	3.00000i		−	0.0966736i
$964$			20.0000i			0.644157i
$965$		0			0
$966$			12.0000i			0.386094i
$967$	22.0000			0.707472			0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000			0.707472			0.353736	+	0.935345i	$0.384911\pi$
$968$	14.0000			0.449977
$969$	−5.00000	+	20.0000i	−0.160623	+	0.642493i
$970$		0			0
$971$	22.0000			0.706014			0.353007	−	0.935621i	$-0.385159\pi$
$971$	22.0000			0.706014			0.353007	+	0.935621i	$0.385159\pi$
$972$		−	1.00000i		−	0.0320750i
$973$	12.0000			0.384702
$974$			12.0000i			0.384505i
$975$		0			0
$976$			10.0000i			0.320092i
$977$	12.0000			0.383914			0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000			0.383914			0.191957	+	0.981403i	$0.438517\pi$
$978$	14.0000			0.447671
$979$		−	50.0000i		−	1.59801i
$980$		0			0
$981$			9.00000i			0.287348i
$982$	−22.0000			−0.702048
$983$		−	24.0000i		−	0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$		−	24.0000i		−	0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	−10.0000			−0.318788
$985$		0			0
$986$	−6.00000	+	24.0000i	−0.191079	+	0.764316i
$987$	21.0000			0.668437
$988$	30.0000			0.954427
$989$			36.0000i			1.14473i
$990$		0			0
$991$		−	40.0000i		−	1.27064i	−0.772248	−	0.635321i	$-0.780868\pi$
$991$		−	40.0000i		−	1.27064i	0.772248	−	0.635321i	$-0.219132\pi$
$992$			5.00000i			0.158750i
$993$		−	7.00000i		−	0.222138i
$994$	30.0000			0.951542
$995$		0			0
$996$		−	6.00000i		−	0.190117i
$997$			63.0000i			1.99523i	0.0690239	+	0.997615i	$0.478012\pi$
$997$			63.0000i			1.99523i	−0.0690239	+	0.997615i	$0.521988\pi$
$998$			4.00000i			0.126618i
$999$	7.00000			0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.c.e.1801.1	✓	2
5.2	odd	4		2550.2.f.a.1699.1		2
5.3	odd	4		2550.2.f.m.1699.2		2
5.4	even	2		2550.2.c.g.1801.2	yes	2
17.16	even	2	inner	2550.2.c.e.1801.2	yes	2
85.33	odd	4		2550.2.f.a.1699.2		2
85.67	odd	4		2550.2.f.m.1699.1		2
85.84	even	2		2550.2.c.g.1801.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2550.2.c.e.1801.1	✓	2	1.1	even	1	trivial
2550.2.c.e.1801.2	yes	2	17.16	even	2	inner
2550.2.c.g.1801.1	yes	2	85.84	even	2
2550.2.c.g.1801.2	yes	2	5.4	even	2
2550.2.f.a.1699.1		2	5.2	odd	4
2550.2.f.a.1699.2		2	85.33	odd	4
2550.2.f.m.1699.1		2	85.67	odd	4
2550.2.f.m.1699.2		2	5.3	odd	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:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +3.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000i q^{6} +3.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} +5.00000i q^{11} -1.00000i q^{12} +6.00000 q^{13} -3.00000i q^{14} +1.00000 q^{16} +(-4.00000 - 1.00000i) q^{17} +1.00000 q^{18} +5.00000 q^{19} +3.00000 q^{21} -5.00000i q^{22} -4.00000i q^{23} +1.00000i q^{24} -6.00000 q^{26} +1.00000i q^{27} +3.00000i q^{28} +6.00000i q^{29} -5.00000i q^{31} -1.00000 q^{32} +5.00000 q^{33} +(4.00000 + 1.00000i) q^{34} -1.00000 q^{36} -7.00000i q^{37} -5.00000 q^{38} -6.00000i q^{39} +10.0000i q^{41} -3.00000 q^{42} -9.00000 q^{43} +5.00000i q^{44} +4.00000i q^{46} +7.00000 q^{47} -1.00000i q^{48} -2.00000 q^{49} +(-1.00000 + 4.00000i) q^{51} +6.00000 q^{52} -9.00000 q^{53} -1.00000i q^{54} -3.00000i q^{56} -5.00000i q^{57} -6.00000i q^{58} +10.0000i q^{61} +5.00000i q^{62} -3.00000i q^{63} +1.00000 q^{64} -5.00000 q^{66} -13.0000 q^{67} +(-4.00000 - 1.00000i) q^{68} -4.00000 q^{69} +10.0000i q^{71} +1.00000 q^{72} +16.0000i q^{73} +7.00000i q^{74} +5.00000 q^{76} -15.0000 q^{77} +6.00000i q^{78} +1.00000i q^{79} +1.00000 q^{81} -10.0000i q^{82} +6.00000 q^{83} +3.00000 q^{84} +9.00000 q^{86} +6.00000 q^{87} -5.00000i q^{88} -10.0000 q^{89} +18.0000i q^{91} -4.00000i q^{92} -5.00000 q^{93} -7.00000 q^{94} +1.00000i q^{96} +8.00000i q^{97} +2.00000 q^{98} -5.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} + 12 q^{13} + 2 q^{16} - 8 q^{17} + 2 q^{18} + 10 q^{19} + 6 q^{21} - 12 q^{26} - 2 q^{32} + 10 q^{33} + 8 q^{34} - 2 q^{36} - 10 q^{38} - 6 q^{42} - 18 q^{43} + 14 q^{47} - 4 q^{49} - 2 q^{51} + 12 q^{52} - 18 q^{53} + 2 q^{64} - 10 q^{66} - 26 q^{67} - 8 q^{68} - 8 q^{69} + 2 q^{72} + 10 q^{76} - 30 q^{77} + 2 q^{81} + 12 q^{83} + 6 q^{84} + 18 q^{86} + 12 q^{87} - 20 q^{89} - 10 q^{93} - 14 q^{94} + 4 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 + 12 * q^13 + 2 * q^16 - 8 * q^17 + 2 * q^18 + 10 * q^19 + 6 * q^21 - 12 * q^26 - 2 * q^32 + 10 * q^33 + 8 * q^34 - 2 * q^36 - 10 * q^38 - 6 * q^42 - 18 * q^43 + 14 * q^47 - 4 * q^49 - 2 * q^51 + 12 * q^52 - 18 * q^53 + 2 * q^64 - 10 * q^66 - 26 * q^67 - 8 * q^68 - 8 * q^69 + 2 * q^72 + 10 * q^76 - 30 * q^77 + 2 * q^81 + 12 * q^83 + 6 * q^84 + 18 * q^86 + 12 * q^87 - 20 * q^89 - 10 * q^93 - 14 * q^94 + 4 * q^98

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