LMFDB - Embedded newform 650.2.b.g.599.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,2,Mod(599,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("650.599");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	650.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.19027613138$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.2.b.g.599.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -2.00000i q^{12} -1.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -6.00000 q^{19} -8.00000 q^{21} +2.00000i q^{22} +6.00000i q^{23} -2.00000 q^{24} -1.00000 q^{26} +4.00000i q^{27} -4.00000i q^{28} -2.00000 q^{29} -6.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +6.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} +8.00000i q^{42} -10.0000i q^{43} +2.00000 q^{44} +6.00000 q^{46} +12.0000i q^{47} +2.00000i q^{48} -9.00000 q^{49} +4.00000 q^{51} +1.00000i q^{52} +2.00000i q^{53} +4.00000 q^{54} -4.00000 q^{56} -12.0000i q^{57} +2.00000i q^{58} -10.0000 q^{59} +2.00000 q^{61} +6.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +12.0000i q^{67} +2.00000i q^{68} -12.0000 q^{69} +10.0000 q^{71} -1.00000i q^{72} +10.0000i q^{73} +2.00000 q^{74} +6.00000 q^{76} -8.00000i q^{77} -2.00000i q^{78} +4.00000 q^{79} -11.0000 q^{81} -10.0000i q^{82} +8.00000 q^{84} -10.0000 q^{86} -4.00000i q^{87} -2.00000i q^{88} +14.0000 q^{89} +4.00000 q^{91} -6.00000i q^{92} -12.0000i q^{93} +12.0000 q^{94} +2.00000 q^{96} -14.0000i q^{97} +9.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} - 4 q^{11} + 8 q^{14} + 2 q^{16} - 12 q^{19} - 16 q^{21} - 4 q^{24} - 2 q^{26} - 4 q^{29} - 12 q^{31} - 4 q^{34} + 2 q^{36} + 4 q^{39} + 20 q^{41} + 4 q^{44} + 12 q^{46} - 18 q^{49} + 8 q^{51} + 8 q^{54} - 8 q^{56} - 20 q^{59} + 4 q^{61} - 2 q^{64} - 8 q^{66} - 24 q^{69} + 20 q^{71} + 4 q^{74} + 12 q^{76} + 8 q^{79} - 22 q^{81} + 16 q^{84} - 20 q^{86} + 28 q^{89} + 8 q^{91} + 24 q^{94} + 4 q^{96} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 - 4 * q^11 + 8 * q^14 + 2 * q^16 - 12 * q^19 - 16 * q^21 - 4 * q^24 - 2 * q^26 - 4 * q^29 - 12 * q^31 - 4 * q^34 + 2 * q^36 + 4 * q^39 + 20 * q^41 + 4 * q^44 + 12 * q^46 - 18 * q^49 + 8 * q^51 + 8 * q^54 - 8 * q^56 - 20 * q^59 + 4 * q^61 - 2 * q^64 - 8 * q^66 - 24 * q^69 + 20 * q^71 + 4 * q^74 + 12 * q^76 + 8 * q^79 - 22 * q^81 + 16 * q^84 - 20 * q^86 + 28 * q^89 + 8 * q^91 + 24 * q^94 + 4 * q^96 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	− 2.00000i	− 0.577350i
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	1.00000i	0.235702i
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−8.00000	−1.74574
$22$	2.00000i	0.426401i
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	−2.00000	−0.408248
$25$	0	0
$26$	−1.00000	−0.196116
$27$	4.00000i	0.769800i
$28$	− 4.00000i	− 0.755929i
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.00000i	− 0.696311i
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	6.00000i	0.973329i
$39$	2.00000	0.320256
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	8.00000i	1.23443i
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	6.00000	0.884652
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	2.00000i	0.288675i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	4.00000	0.560112
$52$	1.00000i	0.138675i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	−4.00000	−0.534522
$57$	− 12.0000i	− 1.58944i
$58$	2.00000i	0.262613i
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	6.00000i	0.762001i
$63$	− 4.00000i	− 0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	2.00000i	0.242536i
$69$	−12.0000	−1.44463
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	− 1.00000i	− 0.117851i
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	6.00000	0.688247
$77$	− 8.00000i	− 0.911685i
$78$	− 2.00000i	− 0.226455i
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	− 10.0000i	− 1.10432i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	8.00000	0.872872
$85$	0	0
$86$	−10.0000	−1.07833
$87$	− 4.00000i	− 0.428845i
$88$	− 2.00000i	− 0.213201i
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	− 6.00000i	− 0.625543i
$93$	− 12.0000i	− 1.24434i
$94$	12.0000	1.23771
$95$	0	0
$96$	2.00000	0.204124
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	9.00000i	0.909137i
$99$	2.00000	0.201008
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	− 4.00000i	− 0.396059i
$103$	− 18.0000i	− 1.77359i	−0.462160	−	0.886796i	$-0.652926\pi$
$103$	− 18.0000i	− 1.77359i	0.462160	−	0.886796i	$-0.347074\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	2.00000	0.194257
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	− 4.00000i	− 0.384900i
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	4.00000i	0.377964i
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	−12.0000	−1.12390
$115$	0	0
$116$	2.00000	0.185695
$117$	1.00000i	0.0924500i
$118$	10.0000i	0.920575i
$119$	8.00000	0.733359
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 2.00000i	− 0.181071i
$123$	20.0000i	1.80334i
$124$	6.00000	0.538816
$125$	0	0
$126$	−4.00000	−0.356348
$127$	14.0000i	1.24230i	0.783692	+	0.621150i	$0.213334\pi$
$127$	14.0000i	1.24230i	−0.783692	+	0.621150i	$0.786666\pi$
$128$	1.00000i	0.0883883i
$129$	20.0000	1.76090
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	4.00000i	0.348155i
$133$	− 24.0000i	− 2.08106i
$134$	12.0000	1.03664
$135$	0	0
$136$	2.00000	0.171499
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	12.0000i	1.02151i
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	−24.0000	−2.02116
$142$	− 10.0000i	− 0.839181i
$143$	2.00000i	0.167248i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	− 18.0000i	− 1.48461i
$148$	− 2.00000i	− 0.164399i
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	− 6.00000i	− 0.486664i
$153$	2.00000i	0.161690i
$154$	−8.00000	−0.644658
$155$	0	0
$156$	−2.00000	−0.160128
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	− 4.00000i	− 0.318223i
$159$	−4.00000	−0.317221
$160$	0	0
$161$	−24.0000	−1.89146
$162$	11.0000i	0.864242i
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	0	0
$167$	− 20.0000i	− 1.54765i	−0.633402	−	0.773823i	$-0.718342\pi$
$167$	− 20.0000i	− 1.54765i	0.633402	−	0.773823i	$-0.281658\pi$
$168$	− 8.00000i	− 0.617213i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	6.00000	0.458831
$172$	10.0000i	0.762493i
$173$	10.0000i	0.760286i	0.924928	+	0.380143i	$0.124125\pi$
$173$	10.0000i	0.760286i	−0.924928	+	0.380143i	$0.875875\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	−2.00000	−0.150756
$177$	− 20.0000i	− 1.50329i
$178$	− 14.0000i	− 1.04934i
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	− 4.00000i	− 0.296500i
$183$	4.00000i	0.295689i
$184$	−6.00000	−0.442326
$185$	0	0
$186$	−12.0000	−0.879883
$187$	4.00000i	0.292509i
$188$	− 12.0000i	− 0.875190i
$189$	−16.0000	−1.16383
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	− 2.00000i	− 0.144338i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	9.00000	0.642857
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	− 2.00000i	− 0.142134i
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−24.0000	−1.69283
$202$	− 14.0000i	− 0.985037i
$203$	− 8.00000i	− 0.561490i
$204$	−4.00000	−0.280056
$205$	0	0
$206$	−18.0000	−1.25412
$207$	− 6.00000i	− 0.417029i
$208$	− 1.00000i	− 0.0693375i
$209$	12.0000	0.830057
$210$	0	0
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	− 2.00000i	− 0.137361i
$213$	20.0000i	1.37038i
$214$	−6.00000	−0.410152
$215$	0	0
$216$	−4.00000	−0.272166
$217$	− 24.0000i	− 1.62923i
$218$	− 6.00000i	− 0.406371i
$219$	−20.0000	−1.35147
$220$	0	0
$221$	−2.00000	−0.134535
$222$	4.00000i	0.268462i
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	2.00000	0.133038
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	12.0000i	0.794719i
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	− 2.00000i	− 0.131306i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	10.0000	0.650945
$237$	8.00000i	0.519656i
$238$	− 8.00000i	− 0.518563i
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	7.00000i	0.449977i
$243$	− 10.0000i	− 0.641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	20.0000	1.27515
$247$	6.00000i	0.381771i
$248$	− 6.00000i	− 0.381000i
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	4.00000i	0.251976i
$253$	− 12.0000i	− 0.754434i
$254$	14.0000	0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000i	1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$	30.0000i	1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$	− 20.0000i	− 1.24515i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	2.00000	0.123797
$262$	− 4.00000i	− 0.247121i
$263$	2.00000i	0.123325i	0.998097	+	0.0616626i	$0.0196403\pi$
$263$	2.00000i	0.123325i	−0.998097	+	0.0616626i	$0.980360\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	−24.0000	−1.47153
$267$	28.0000i	1.71357i
$268$	− 12.0000i	− 0.733017i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	− 2.00000i	− 0.121268i
$273$	8.00000i	0.484182i
$274$	18.0000	1.08742
$275$	0	0
$276$	12.0000	0.722315
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	− 8.00000i	− 0.479808i
$279$	6.00000	0.359211
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	24.0000i	1.42918i
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	−10.0000	−0.593391
$285$	0	0
$286$	2.00000	0.118262
$287$	40.0000i	2.36113i
$288$	1.00000i	0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	28.0000	1.64139
$292$	− 10.0000i	− 0.585206i
$293$	22.0000i	1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$	22.0000i	1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$	−18.0000	−1.04978
$295$	0	0
$296$	−2.00000	−0.116248
$297$	− 8.00000i	− 0.464207i
$298$	2.00000i	0.115857i
$299$	6.00000	0.346989
$300$	0	0
$301$	40.0000	2.30556
$302$	− 6.00000i	− 0.345261i
$303$	28.0000i	1.60856i
$304$	−6.00000	−0.344124
$305$	0	0
$306$	2.00000	0.114332
$307$	− 24.0000i	− 1.36975i	−0.728659	−	0.684876i	$-0.759856\pi$
$307$	− 24.0000i	− 1.36975i	0.728659	−	0.684876i	$-0.240144\pi$
$308$	8.00000i	0.455842i
$309$	36.0000	2.04797
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	2.00000i	0.113228i
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−4.00000	−0.225018
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	4.00000i	0.224309i
$319$	4.00000	0.223957
$320$	0	0
$321$	12.0000	0.669775
$322$	24.0000i	1.33747i
$323$	12.0000i	0.667698i
$324$	11.0000	0.611111
$325$	0	0
$326$	−4.00000	−0.221540
$327$	12.0000i	0.663602i
$328$	10.0000i	0.552158i
$329$	−48.0000	−2.64633
$330$	0	0
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	−20.0000	−1.09435
$335$	0	0
$336$	−8.00000	−0.436436
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	1.00000i	0.0543928i
$339$	−4.00000	−0.217250
$340$	0	0
$341$	12.0000	0.649836
$342$	− 6.00000i	− 0.324443i
$343$	− 8.00000i	− 0.431959i
$344$	10.0000	0.539164
$345$	0	0
$346$	10.0000	0.537603
$347$	− 6.00000i	− 0.322097i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$	− 6.00000i	− 0.322097i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$	4.00000i	0.214423i
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	2.00000i	0.106600i
$353$	34.0000i	1.80964i	0.425797	+	0.904819i	$0.359994\pi$
$353$	34.0000i	1.80964i	−0.425797	+	0.904819i	$0.640006\pi$
$354$	−20.0000	−1.06299
$355$	0	0
$356$	−14.0000	−0.741999
$357$	16.0000i	0.846810i
$358$	− 4.00000i	− 0.211407i
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	− 10.0000i	− 0.525588i
$363$	− 14.0000i	− 0.734809i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	4.00000	0.209083
$367$	− 30.0000i	− 1.56599i	−0.622030	−	0.782994i	$-0.713692\pi$
$367$	− 30.0000i	− 1.56599i	0.622030	−	0.782994i	$-0.286308\pi$
$368$	6.00000i	0.312772i
$369$	−10.0000	−0.520579
$370$	0	0
$371$	−8.00000	−0.415339
$372$	12.0000i	0.622171i
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	−12.0000	−0.618853
$377$	2.00000i	0.103005i
$378$	16.0000i	0.822951i
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	0	0
$381$	−28.0000	−1.43448
$382$	0	0
$383$	12.0000i	0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$	12.0000i	0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	14.0000	0.712581
$387$	10.0000i	0.508329i
$388$	14.0000i	0.710742i
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	− 9.00000i	− 0.454569i
$393$	8.00000i	0.403547i
$394$	6.00000	0.302276
$395$	0	0
$396$	−2.00000	−0.100504
$397$	− 38.0000i	− 1.90717i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	− 38.0000i	− 1.90717i	0.301131	−	0.953583i	$-0.402636\pi$
$398$	0	0
$399$	48.0000	2.40301
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	24.0000i	1.19701i
$403$	6.00000i	0.298881i
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−8.00000	−0.397033
$407$	− 4.00000i	− 0.198273i
$408$	4.00000i	0.198030i
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	−36.0000	−1.77575
$412$	18.0000i	0.886796i
$413$	− 40.0000i	− 1.96827i
$414$	−6.00000	−0.294884
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	16.0000i	0.783523i
$418$	− 12.0000i	− 0.586939i
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	− 28.0000i	− 1.36302i
$423$	− 12.0000i	− 0.583460i
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	20.0000	0.969003
$427$	8.00000i	0.387147i
$428$	6.00000i	0.290021i
$429$	−4.00000	−0.193122
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	4.00000i	0.192450i
$433$	− 38.0000i	− 1.82616i	−0.407777	−	0.913082i	$-0.633696\pi$
$433$	− 38.0000i	− 1.82616i	0.407777	−	0.913082i	$-0.366304\pi$
$434$	−24.0000	−1.15204
$435$	0	0
$436$	−6.00000	−0.287348
$437$	− 36.0000i	− 1.72211i
$438$	20.0000i	0.955637i
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	2.00000i	0.0951303i
$443$	− 14.0000i	− 0.665160i	−0.943075	−	0.332580i	$-0.892081\pi$
$443$	− 14.0000i	− 0.665160i	0.943075	−	0.332580i	$-0.107919\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	4.00000	0.189405
$447$	− 4.00000i	− 0.189194i
$448$	− 4.00000i	− 0.188982i
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−20.0000	−0.941763
$452$	− 2.00000i	− 0.0940721i
$453$	12.0000i	0.563809i
$454$	4.00000	0.187729
$455$	0	0
$456$	12.0000	0.561951
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	10.0000i	0.467269i
$459$	8.00000	0.373408
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	− 16.0000i	− 0.744387i
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 10.0000i	− 0.462745i	−0.972865	−	0.231372i	$-0.925678\pi$
$467$	− 10.0000i	− 0.462745i	0.972865	−	0.231372i	$-0.0743216\pi$
$468$	− 1.00000i	− 0.0462250i
$469$	−48.0000	−2.21643
$470$	0	0
$471$	20.0000	0.921551
$472$	− 10.0000i	− 0.460287i
$473$	20.0000i	0.919601i
$474$	8.00000	0.367452
$475$	0	0
$476$	−8.00000	−0.366679
$477$	− 2.00000i	− 0.0915737i
$478$	− 26.0000i	− 1.18921i
$479$	−2.00000	−0.0913823	−0.0456912	−	0.998956i	$-0.514549\pi$
$479$	−2.00000	−0.0913823	−0.0456912	+	0.998956i	$0.514549\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	22.0000i	1.00207i
$483$	− 48.0000i	− 2.18408i
$484$	7.00000	0.318182
$485$	0	0
$486$	−10.0000	−0.453609
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	2.00000i	0.0905357i
$489$	8.00000	0.361773
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	− 20.0000i	− 0.901670i
$493$	4.00000i	0.180151i
$494$	6.00000	0.269953
$495$	0	0
$496$	−6.00000	−0.269408
$497$	40.0000i	1.79425i
$498$	0	0
$499$	−38.0000	−1.70111	−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000	−1.70111	−0.850557	+	0.525883i	$0.823735\pi$
$500$	0	0
$501$	40.0000	1.78707
$502$	0	0
$503$	− 14.0000i	− 0.624229i	−0.950044	−	0.312115i	$-0.898963\pi$
$503$	− 14.0000i	− 0.624229i	0.950044	−	0.312115i	$-0.101037\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	−12.0000	−0.533465
$507$	− 2.00000i	− 0.0888231i
$508$	− 14.0000i	− 0.621150i
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−40.0000	−1.76950
$512$	− 1.00000i	− 0.0441942i
$513$	− 24.0000i	− 1.05963i
$514$	30.0000	1.32324
$515$	0	0
$516$	−20.0000	−0.880451
$517$	− 24.0000i	− 1.05552i
$518$	8.00000i	0.351500i
$519$	−20.0000	−0.877903
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	− 2.00000i	− 0.0875376i
$523$	− 6.00000i	− 0.262362i	−0.991358	−	0.131181i	$-0.958123\pi$
$523$	− 6.00000i	− 0.262362i	0.991358	−	0.131181i	$-0.0418769\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	2.00000	0.0872041
$527$	12.0000i	0.522728i
$528$	− 4.00000i	− 0.174078i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	10.0000	0.433963
$532$	24.0000i	1.04053i
$533$	− 10.0000i	− 0.433148i
$534$	28.0000	1.21168
$535$	0	0
$536$	−12.0000	−0.518321
$537$	8.00000i	0.345225i
$538$	6.00000i	0.258678i
$539$	18.0000	0.775315
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	2.00000i	0.0859074i
$543$	20.0000i	0.858282i
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	8.00000	0.342368
$547$	− 22.0000i	− 0.940652i	−0.882493	−	0.470326i	$-0.844136\pi$
$547$	− 22.0000i	− 0.940652i	0.882493	−	0.470326i	$-0.155864\pi$
$548$	− 18.0000i	− 0.768922i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	12.0000	0.511217
$552$	− 12.0000i	− 0.510754i
$553$	16.0000i	0.680389i
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	−8.00000	−0.339276
$557$	− 38.0000i	− 1.61011i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	− 38.0000i	− 1.61011i	0.593199	−	0.805056i	$-0.297865\pi$
$558$	− 6.00000i	− 0.254000i
$559$	−10.0000	−0.422955
$560$	0	0
$561$	−8.00000	−0.337760
$562$	6.00000i	0.253095i
$563$	6.00000i	0.252870i	0.991975	+	0.126435i	$0.0403535\pi$
$563$	6.00000i	0.252870i	−0.991975	+	0.126435i	$0.959647\pi$
$564$	24.0000	1.01058
$565$	0	0
$566$	14.0000	0.588464
$567$	− 44.0000i	− 1.84783i
$568$	10.0000i	0.419591i
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	− 2.00000i	− 0.0836242i
$573$	0	0
$574$	40.0000	1.66957
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 18.0000i	− 0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$	− 18.0000i	− 0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$	− 13.0000i	− 0.540729i
$579$	−28.0000	−1.16364
$580$	0	0
$581$	0	0
$582$	− 28.0000i	− 1.16064i
$583$	− 4.00000i	− 0.165663i
$584$	−10.0000	−0.413803
$585$	0	0
$586$	22.0000	0.908812
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	18.0000i	0.742307i
$589$	36.0000	1.48335
$590$	0	0
$591$	−12.0000	−0.493614
$592$	2.00000i	0.0821995i
$593$	− 2.00000i	− 0.0821302i	−0.999156	−	0.0410651i	$-0.986925\pi$
$593$	− 2.00000i	− 0.0821302i	0.999156	−	0.0410651i	$-0.0130751\pi$
$594$	−8.00000	−0.328244
$595$	0	0
$596$	2.00000	0.0819232
$597$	0	0
$598$	− 6.00000i	− 0.245358i
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	− 40.0000i	− 1.63028i
$603$	− 12.0000i	− 0.488678i
$604$	−6.00000	−0.244137
$605$	0	0
$606$	28.0000	1.13742
$607$	34.0000i	1.38002i	0.723801	+	0.690009i	$0.242393\pi$
$607$	34.0000i	1.38002i	−0.723801	+	0.690009i	$0.757607\pi$
$608$	6.00000i	0.243332i
$609$	16.0000	0.648353
$610$	0	0
$611$	12.0000	0.485468
$612$	− 2.00000i	− 0.0808452i
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	−24.0000	−0.968561
$615$	0	0
$616$	8.00000	0.322329
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	− 36.0000i	− 1.44813i
$619$	46.0000	1.84890	0.924448	−	0.381308i	$-0.124526\pi$
$619$	46.0000	1.84890	0.924448	+	0.381308i	$0.124526\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	12.0000i	0.481156i
$623$	56.0000i	2.24359i
$624$	2.00000	0.0800641
$625$	0	0
$626$	−6.00000	−0.239808
$627$	24.0000i	0.958468i
$628$	10.0000i	0.399043i
$629$	4.00000	0.159490
$630$	0	0
$631$	30.0000	1.19428	0.597141	−	0.802137i	$-0.296303\pi$
$631$	30.0000	1.19428	0.597141	+	0.802137i	$0.296303\pi$
$632$	4.00000i	0.159111i
$633$	56.0000i	2.22580i
$634$	18.0000	0.714871
$635$	0	0
$636$	4.00000	0.158610
$637$	9.00000i	0.356593i
$638$	− 4.00000i	− 0.158362i
$639$	−10.0000	−0.395594
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	− 12.0000i	− 0.473602i
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	12.0000	0.472134
$647$	− 42.0000i	− 1.65119i	−0.564263	−	0.825595i	$-0.690840\pi$
$647$	− 42.0000i	− 1.65119i	0.564263	−	0.825595i	$-0.309160\pi$
$648$	− 11.0000i	− 0.432121i
$649$	20.0000	0.785069
$650$	0	0
$651$	48.0000	1.88127
$652$	4.00000i	0.156652i
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	12.0000	0.469237
$655$	0	0
$656$	10.0000	0.390434
$657$	− 10.0000i	− 0.390137i
$658$	48.0000i	1.87123i
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	14.0000i	0.544125i
$663$	− 4.00000i	− 0.155347i
$664$	0	0
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	− 12.0000i	− 0.464642i
$668$	20.0000i	0.773823i
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−4.00000	−0.154418
$672$	8.00000i	0.308607i
$673$	2.00000i	0.0770943i	0.999257	+	0.0385472i	$0.0122730\pi$
$673$	2.00000i	0.0770943i	−0.999257	+	0.0385472i	$0.987727\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	1.00000	0.0384615
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	4.00000i	0.153619i
$679$	56.0000	2.14908
$680$	0	0
$681$	−8.00000	−0.306561
$682$	− 12.0000i	− 0.459504i
$683$	− 44.0000i	− 1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$	− 44.0000i	− 1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	−8.00000	−0.305441
$687$	− 20.0000i	− 0.763048i
$688$	− 10.0000i	− 0.381246i
$689$	2.00000	0.0761939
$690$	0	0
$691$	−38.0000	−1.44559	−0.722794	−	0.691063i	$-0.757142\pi$
$691$	−38.0000	−1.44559	−0.722794	+	0.691063i	$0.757142\pi$
$692$	− 10.0000i	− 0.380143i
$693$	8.00000i	0.303895i
$694$	−6.00000	−0.227757
$695$	0	0
$696$	4.00000	0.151620
$697$	− 20.0000i	− 0.757554i
$698$	2.00000i	0.0757011i
$699$	12.0000	0.453882
$700$	0	0
$701$	38.0000	1.43524	0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000	1.43524	0.717620	+	0.696435i	$0.245231\pi$
$702$	− 4.00000i	− 0.150970i
$703$	− 12.0000i	− 0.452589i
$704$	2.00000	0.0753778
$705$	0	0
$706$	34.0000	1.27961
$707$	56.0000i	2.10610i
$708$	20.0000i	0.751646i
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	14.0000i	0.524672i
$713$	− 36.0000i	− 1.34821i
$714$	16.0000	0.598785
$715$	0	0
$716$	−4.00000	−0.149487
$717$	52.0000i	1.94198i
$718$	− 6.00000i	− 0.223918i
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	72.0000	2.68142
$722$	− 17.0000i	− 0.632674i
$723$	− 44.0000i	− 1.63638i
$724$	−10.0000	−0.371647
$725$	0	0
$726$	−14.0000	−0.519589
$727$	14.0000i	0.519231i	0.965712	+	0.259616i	$0.0835959\pi$
$727$	14.0000i	0.519231i	−0.965712	+	0.259616i	$0.916404\pi$
$728$	4.00000i	0.148250i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−20.0000	−0.739727
$732$	− 4.00000i	− 0.147844i
$733$	2.00000i	0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$	2.00000i	0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$	−30.0000	−1.10732
$735$	0	0
$736$	6.00000	0.221163
$737$	− 24.0000i	− 0.884051i
$738$	10.0000i	0.368105i
$739$	−42.0000	−1.54499	−0.772497	−	0.635018i	$-0.780993\pi$
$739$	−42.0000	−1.54499	−0.772497	+	0.635018i	$0.780993\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	8.00000i	0.293689i
$743$	− 12.0000i	− 0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$	− 12.0000i	− 0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$	12.0000	0.439941
$745$	0	0
$746$	−14.0000	−0.512576
$747$	0	0
$748$	− 4.00000i	− 0.146254i
$749$	24.0000	0.876941
$750$	0	0
$751$	44.0000	1.60558	0.802791	−	0.596260i	$-0.203347\pi$
$751$	44.0000	1.60558	0.802791	+	0.596260i	$0.203347\pi$
$752$	12.0000i	0.437595i
$753$	0	0
$754$	2.00000	0.0728357
$755$	0	0
$756$	16.0000	0.581914
$757$	− 18.0000i	− 0.654221i	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	− 18.0000i	− 0.654221i	0.944986	−	0.327111i	$-0.106075\pi$
$758$	6.00000i	0.217930i
$759$	24.0000	0.871145
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	28.0000i	1.01433i
$763$	24.0000i	0.868858i
$764$	0	0
$765$	0	0
$766$	12.0000	0.433578
$767$	10.0000i	0.361079i
$768$	2.00000i	0.0721688i
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−60.0000	−2.16085
$772$	− 14.0000i	− 0.503871i
$773$	− 34.0000i	− 1.22290i	−0.791285	−	0.611448i	$-0.790588\pi$
$773$	− 34.0000i	− 1.22290i	0.791285	−	0.611448i	$-0.209412\pi$
$774$	10.0000	0.359443
$775$	0	0
$776$	14.0000	0.502571
$777$	− 16.0000i	− 0.573997i
$778$	− 10.0000i	− 0.358517i
$779$	−60.0000	−2.14972
$780$	0	0
$781$	−20.0000	−0.715656
$782$	− 12.0000i	− 0.429119i
$783$	− 8.00000i	− 0.285897i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	8.00000	0.285351
$787$	8.00000i	0.285169i	0.989783	+	0.142585i	$0.0455413\pi$
$787$	8.00000i	0.285169i	−0.989783	+	0.142585i	$0.954459\pi$
$788$	− 6.00000i	− 0.213741i
$789$	−4.00000	−0.142404
$790$	0	0
$791$	−8.00000	−0.284447
$792$	2.00000i	0.0710669i
$793$	− 2.00000i	− 0.0710221i
$794$	−38.0000	−1.34857
$795$	0	0
$796$	0	0
$797$	22.0000i	0.779280i	0.920967	+	0.389640i	$0.127401\pi$
$797$	22.0000i	0.779280i	−0.920967	+	0.389640i	$0.872599\pi$
$798$	− 48.0000i	− 1.69918i
$799$	24.0000	0.849059
$800$	0	0
$801$	−14.0000	−0.494666
$802$	6.00000i	0.211867i
$803$	− 20.0000i	− 0.705785i
$804$	24.0000	0.846415
$805$	0	0
$806$	6.00000	0.211341
$807$	− 12.0000i	− 0.422420i
$808$	14.0000i	0.492518i
$809$	34.0000	1.19538	0.597688	−	0.801729i	$-0.296086\pi$
$809$	34.0000	1.19538	0.597688	+	0.801729i	$0.296086\pi$
$810$	0	0
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	8.00000i	0.280745i
$813$	− 4.00000i	− 0.140286i
$814$	−4.00000	−0.140200
$815$	0	0
$816$	4.00000	0.140028
$817$	60.0000i	2.09913i
$818$	34.0000i	1.18878i
$819$	−4.00000	−0.139771
$820$	0	0
$821$	50.0000	1.74501	0.872506	−	0.488603i	$-0.162493\pi$
$821$	50.0000	1.74501	0.872506	+	0.488603i	$0.162493\pi$
$822$	36.0000i	1.25564i
$823$	14.0000i	0.488009i	0.969774	+	0.244005i	$0.0784612\pi$
$823$	14.0000i	0.488009i	−0.969774	+	0.244005i	$0.921539\pi$
$824$	18.0000	0.627060
$825$	0	0
$826$	−40.0000	−1.39178
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	6.00000i	0.208514i
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	4.00000	0.138758
$832$	1.00000i	0.0346688i
$833$	18.0000i	0.623663i
$834$	16.0000	0.554035
$835$	0	0
$836$	−12.0000	−0.415029
$837$	− 24.0000i	− 0.829561i
$838$	16.0000i	0.552711i
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	− 10.0000i	− 0.344623i
$843$	− 12.0000i	− 0.413302i
$844$	−28.0000	−0.963800
$845$	0	0
$846$	−12.0000	−0.412568
$847$	− 28.0000i	− 0.962091i
$848$	2.00000i	0.0686803i
$849$	−28.0000	−0.960958
$850$	0	0
$851$	−12.0000	−0.411355
$852$	− 20.0000i	− 0.685189i
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	6.00000	0.205076
$857$	14.0000i	0.478231i	0.970991	+	0.239115i	$0.0768574\pi$
$857$	14.0000i	0.478231i	−0.970991	+	0.239115i	$0.923143\pi$
$858$	4.00000i	0.136558i
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	−80.0000	−2.72639
$862$	− 18.0000i	− 0.613082i
$863$	− 36.0000i	− 1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$	− 36.0000i	− 1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	−38.0000	−1.29129
$867$	26.0000i	0.883006i
$868$	24.0000i	0.814613i
$869$	−8.00000	−0.271381
$870$	0	0
$871$	12.0000	0.406604
$872$	6.00000i	0.203186i
$873$	14.0000i	0.473828i
$874$	−36.0000	−1.21772
$875$	0	0
$876$	20.0000	0.675737
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	− 32.0000i	− 1.07995i
$879$	−44.0000	−1.48408
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	− 9.00000i	− 0.303046i
$883$	38.0000i	1.27880i	0.768874	+	0.639401i	$0.220818\pi$
$883$	38.0000i	1.27880i	−0.768874	+	0.639401i	$0.779182\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	−14.0000	−0.470339
$887$	− 22.0000i	− 0.738688i	−0.929293	−	0.369344i	$-0.879582\pi$
$887$	− 22.0000i	− 0.738688i	0.929293	−	0.369344i	$-0.120418\pi$
$888$	− 4.00000i	− 0.134231i
$889$	−56.0000	−1.87818
$890$	0	0
$891$	22.0000	0.737028
$892$	− 4.00000i	− 0.133930i
$893$	− 72.0000i	− 2.40939i
$894$	−4.00000	−0.133780
$895$	0	0
$896$	−4.00000	−0.133631
$897$	12.0000i	0.400668i
$898$	− 6.00000i	− 0.200223i
$899$	12.0000	0.400222
$900$	0	0
$901$	4.00000	0.133259
$902$	20.0000i	0.665927i
$903$	80.0000i	2.66223i
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	12.0000	0.398673
$907$	14.0000i	0.464862i	0.972613	+	0.232431i	$0.0746680\pi$
$907$	14.0000i	0.464862i	−0.972613	+	0.232431i	$0.925332\pi$
$908$	− 4.00000i	− 0.132745i
$909$	−14.0000	−0.464351
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	− 12.0000i	− 0.397360i
$913$	0	0
$914$	10.0000	0.330771
$915$	0	0
$916$	10.0000	0.330409
$917$	16.0000i	0.528367i
$918$	− 8.00000i	− 0.264039i
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	48.0000	1.58165
$922$	6.00000i	0.197599i
$923$	− 10.0000i	− 0.329154i
$924$	−16.0000	−0.526361
$925$	0	0
$926$	16.0000	0.525793
$927$	18.0000i	0.591198i
$928$	2.00000i	0.0656532i
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	54.0000	1.76978
$932$	6.00000i	0.196537i
$933$	− 24.0000i	− 0.785725i
$934$	−10.0000	−0.327210
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	− 18.0000i	− 0.588034i	−0.955800	−	0.294017i	$-0.905008\pi$
$937$	− 18.0000i	− 0.588034i	0.955800	−	0.294017i	$-0.0949923\pi$
$938$	48.0000i	1.56726i
$939$	12.0000	0.391605
$940$	0	0
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	− 20.0000i	− 0.651635i
$943$	60.0000i	1.95387i
$944$	−10.0000	−0.325472
$945$	0	0
$946$	20.0000	0.650256
$947$	8.00000i	0.259965i	0.991516	+	0.129983i	$0.0414921\pi$
$947$	8.00000i	0.259965i	−0.991516	+	0.129983i	$0.958508\pi$
$948$	− 8.00000i	− 0.259828i
$949$	10.0000	0.324614
$950$	0	0
$951$	−36.0000	−1.16738
$952$	8.00000i	0.259281i
$953$	18.0000i	0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$	18.0000i	0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	−26.0000	−0.840900
$957$	8.00000i	0.258603i
$958$	2.00000i	0.0646171i
$959$	−72.0000	−2.32500
$960$	0	0
$961$	5.00000	0.161290
$962$	− 2.00000i	− 0.0644826i
$963$	6.00000i	0.193347i
$964$	22.0000	0.708572
$965$	0	0
$966$	−48.0000	−1.54437
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	− 7.00000i	− 0.224989i
$969$	−24.0000	−0.770991
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	10.0000i	0.320750i
$973$	32.0000i	1.02587i
$974$	16.0000	0.512673
$975$	0	0
$976$	2.00000	0.0640184
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	− 8.00000i	− 0.255812i
$979$	−28.0000	−0.894884
$980$	0	0
$981$	−6.00000	−0.191565
$982$	0	0
$983$	− 16.0000i	− 0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$	− 16.0000i	− 0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$	−20.0000	−0.637577
$985$	0	0
$986$	4.00000	0.127386
$987$	− 96.0000i	− 3.05571i
$988$	− 6.00000i	− 0.190885i
$989$	60.0000	1.90789
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	6.00000i	0.190500i
$993$	− 28.0000i	− 0.888553i
$994$	40.0000	1.26872
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	38.0000i	1.20287i
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.2.b.g.599.1		2
3.2	odd	2		5850.2.e.u.5149.2		2
5.2	odd	4		130.2.a.c.1.1	✓	1
5.3	odd	4		650.2.a.c.1.1		1
5.4	even	2	inner	650.2.b.g.599.2		2
15.2	even	4		1170.2.a.d.1.1		1
15.8	even	4		5850.2.a.cb.1.1		1
15.14	odd	2		5850.2.e.u.5149.1		2
20.3	even	4		5200.2.a.bd.1.1		1
20.7	even	4		1040.2.a.b.1.1		1
35.27	even	4		6370.2.a.l.1.1		1
40.27	even	4		4160.2.a.t.1.1		1
40.37	odd	4		4160.2.a.c.1.1		1
60.47	odd	4		9360.2.a.by.1.1		1
65.2	even	12		1690.2.l.a.1161.2		4
65.7	even	12		1690.2.l.a.361.2		4
65.12	odd	4		1690.2.a.e.1.1		1
65.17	odd	12		1690.2.e.g.991.1		2
65.22	odd	12		1690.2.e.a.991.1		2
65.32	even	12		1690.2.l.a.361.1		4
65.37	even	12		1690.2.l.a.1161.1		4
65.38	odd	4		8450.2.a.n.1.1		1
65.42	odd	12		1690.2.e.a.191.1		2
65.47	even	4		1690.2.d.e.1351.2		2
65.57	even	4		1690.2.d.e.1351.1		2
65.62	odd	12		1690.2.e.g.191.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.a.c.1.1	✓	1	5.2	odd	4
650.2.a.c.1.1		1	5.3	odd	4
650.2.b.g.599.1		2	1.1	even	1	trivial
650.2.b.g.599.2		2	5.4	even	2	inner
1040.2.a.b.1.1		1	20.7	even	4
1170.2.a.d.1.1		1	15.2	even	4
1690.2.a.e.1.1		1	65.12	odd	4
1690.2.d.e.1351.1		2	65.57	even	4
1690.2.d.e.1351.2		2	65.47	even	4
1690.2.e.a.191.1		2	65.42	odd	12
1690.2.e.a.991.1		2	65.22	odd	12
1690.2.e.g.191.1		2	65.62	odd	12
1690.2.e.g.991.1		2	65.17	odd	12
1690.2.l.a.361.1		4	65.32	even	12
1690.2.l.a.361.2		4	65.7	even	12
1690.2.l.a.1161.1		4	65.37	even	12
1690.2.l.a.1161.2		4	65.2	even	12
4160.2.a.c.1.1		1	40.37	odd	4
4160.2.a.t.1.1		1	40.27	even	4
5200.2.a.bd.1.1		1	20.3	even	4
5850.2.a.cb.1.1		1	15.8	even	4
5850.2.e.u.5149.1		2	15.14	odd	2
5850.2.e.u.5149.2		2	3.2	odd	2
6370.2.a.l.1.1		1	35.27	even	4
8450.2.a.n.1.1		1	65.38	odd	4
9360.2.a.by.1.1		1	60.47	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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	650.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -2.00000i q^{12} -1.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -6.00000 q^{19} -8.00000 q^{21} +2.00000i q^{22} +6.00000i q^{23} -2.00000 q^{24} -1.00000 q^{26} +4.00000i q^{27} -4.00000i q^{28} -2.00000 q^{29} -6.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +6.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} +8.00000i q^{42} -10.0000i q^{43} +2.00000 q^{44} +6.00000 q^{46} +12.0000i q^{47} +2.00000i q^{48} -9.00000 q^{49} +4.00000 q^{51} +1.00000i q^{52} +2.00000i q^{53} +4.00000 q^{54} -4.00000 q^{56} -12.0000i q^{57} +2.00000i q^{58} -10.0000 q^{59} +2.00000 q^{61} +6.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} +12.0000i q^{67} +2.00000i q^{68} -12.0000 q^{69} +10.0000 q^{71} -1.00000i q^{72} +10.0000i q^{73} +2.00000 q^{74} +6.00000 q^{76} -8.00000i q^{77} -2.00000i q^{78} +4.00000 q^{79} -11.0000 q^{81} -10.0000i q^{82} +8.00000 q^{84} -10.0000 q^{86} -4.00000i q^{87} -2.00000i q^{88} +14.0000 q^{89} +4.00000 q^{91} -6.00000i q^{92} -12.0000i q^{93} +12.0000 q^{94} +2.00000 q^{96} -14.0000i q^{97} +9.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} - 4 q^{11} + 8 q^{14} + 2 q^{16} - 12 q^{19} - 16 q^{21} - 4 q^{24} - 2 q^{26} - 4 q^{29} - 12 q^{31} - 4 q^{34} + 2 q^{36} + 4 q^{39} + 20 q^{41} + 4 q^{44} + 12 q^{46} - 18 q^{49} + 8 q^{51} + 8 q^{54} - 8 q^{56} - 20 q^{59} + 4 q^{61} - 2 q^{64} - 8 q^{66} - 24 q^{69} + 20 q^{71} + 4 q^{74} + 12 q^{76} + 8 q^{79} - 22 q^{81} + 16 q^{84} - 20 q^{86} + 28 q^{89} + 8 q^{91} + 24 q^{94} + 4 q^{96} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 - 4 * q^11 + 8 * q^14 + 2 * q^16 - 12 * q^19 - 16 * q^21 - 4 * q^24 - 2 * q^26 - 4 * q^29 - 12 * q^31 - 4 * q^34 + 2 * q^36 + 4 * q^39 + 20 * q^41 + 4 * q^44 + 12 * q^46 - 18 * q^49 + 8 * q^51 + 8 * q^54 - 8 * q^56 - 20 * q^59 + 4 * q^61 - 2 * q^64 - 8 * q^66 - 24 * q^69 + 20 * q^71 + 4 * q^74 + 12 * q^76 + 8 * q^79 - 22 * q^81 + 16 * q^84 - 20 * q^86 + 28 * q^89 + 8 * q^91 + 24 * q^94 + 4 * q^96 + 4 * q^99

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