LMFDB - Embedded newform 1800.2.k.i.901.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1800,2,Mod(901,1800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1800.901"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1800.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,2,0,0,0,0,8,-4,0,0,0,0,0,8,0,-8,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$14.3730723638$
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 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			901.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1800.901
Dual form			1800.2.k.i.901.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +4.00000 q^{7} +(-2.00000 - 2.00000i) q^{8} +2.00000i q^{11} +6.00000i q^{13} +(4.00000 - 4.00000i) q^{14} -4.00000 q^{16} +6.00000 q^{17} +4.00000i q^{19} +(2.00000 + 2.00000i) q^{22} +8.00000 q^{23} +(6.00000 + 6.00000i) q^{26} -8.00000i q^{28} -6.00000i q^{29} -2.00000 q^{31} +(-4.00000 + 4.00000i) q^{32} +(6.00000 - 6.00000i) q^{34} +2.00000i q^{37} +(4.00000 + 4.00000i) q^{38} -4.00000 q^{41} +4.00000i q^{43} +4.00000 q^{44} +(8.00000 - 8.00000i) q^{46} -4.00000 q^{47} +9.00000 q^{49} +12.0000 q^{52} -10.0000i q^{53} +(-8.00000 - 8.00000i) q^{56} +(-6.00000 - 6.00000i) q^{58} -2.00000i q^{59} +4.00000i q^{61} +(-2.00000 + 2.00000i) q^{62} +8.00000i q^{64} -12.0000i q^{67} -12.0000i q^{68} -8.00000 q^{71} -10.0000 q^{73} +(2.00000 + 2.00000i) q^{74} +8.00000 q^{76} +8.00000i q^{77} -2.00000 q^{79} +(-4.00000 + 4.00000i) q^{82} +(4.00000 + 4.00000i) q^{86} +(4.00000 - 4.00000i) q^{88} +4.00000 q^{89} +24.0000i q^{91} -16.0000i q^{92} +(-4.00000 + 4.00000i) q^{94} -2.00000 q^{97} +(9.00000 - 9.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 8 q^{7} - 4 q^{8} + 8 q^{14} - 8 q^{16} + 12 q^{17} + 4 q^{22} + 16 q^{23} + 12 q^{26} - 4 q^{31} - 8 q^{32} + 12 q^{34} + 8 q^{38} - 8 q^{41} + 8 q^{44} + 16 q^{46} - 8 q^{47} + 18 q^{49}+ \cdots + 18 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 8 * q^7 - 4 * q^8 + 8 * q^14 - 8 * q^16 + 12 * q^17 + 4 * q^22 + 16 * q^23 + 12 * q^26 - 4 * q^31 - 8 * q^32 + 12 * q^34 + 8 * q^38 - 8 * q^41 + 8 * q^44 + 16 * q^46 - 8 * q^47 + 18 * q^49 + 24 * q^52 - 16 * q^56 - 12 * q^58 - 4 * q^62 - 16 * q^71 - 20 * q^73 + 4 * q^74 + 16 * q^76 - 4 * q^79 - 8 * q^82 + 8 * q^86 + 8 * q^88 + 8 * q^89 - 8 * q^94 - 4 * q^97 + 18 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$1$	$-1$	$1$	$1$

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	−	1.00000i	0.707107	−	0.707107i
$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$3$		0			0
$3$		0			0
$4$		−	2.00000i		−	1.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	4.00000			1.51186			0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000			1.51186			0.755929	+	0.654654i	$0.227186\pi$
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$		0			0
$10$		0			0
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$		0			0
$13$			6.00000i			1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$			6.00000i			1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	4.00000	−	4.00000i	1.06904	−	1.06904i
$15$		0			0
$16$	−4.00000			−1.00000
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$		0			0
$22$	2.00000	+	2.00000i	0.426401	+	0.426401i
$23$	8.00000			1.66812			0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000			1.66812			0.834058	+	0.551677i	$0.186012\pi$
$24$		0			0
$25$		0			0
$26$	6.00000	+	6.00000i	1.17670	+	1.17670i
$27$		0			0
$28$		−	8.00000i		−	1.51186i
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$	−4.00000	+	4.00000i	−0.707107	+	0.707107i
$33$		0			0
$34$	6.00000	−	6.00000i	1.02899	−	1.02899i
$35$		0			0
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	4.00000	+	4.00000i	0.648886	+	0.648886i
$39$		0			0
$40$		0			0
$41$	−4.00000			−0.624695			−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000			−0.624695			−0.312348	+	0.949968i	$0.601115\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	4.00000			0.603023
$45$		0			0
$46$	8.00000	−	8.00000i	1.17954	−	1.17954i
$47$	−4.00000			−0.583460			−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000			−0.583460			−0.291730	+	0.956501i	$0.594231\pi$
$48$		0			0
$49$	9.00000			1.28571
$50$		0			0
$51$		0			0
$52$	12.0000			1.66410
$53$		−	10.0000i		−	1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$		−	10.0000i		−	1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$		0			0
$55$		0			0
$56$	−8.00000	−	8.00000i	−1.06904	−	1.06904i
$57$		0			0
$58$	−6.00000	−	6.00000i	−0.787839	−	0.787839i
$59$		−	2.00000i		−	0.260378i	−0.991489	−	0.130189i	$-0.958442\pi$
$59$		−	2.00000i		−	0.260378i	0.991489	−	0.130189i	$-0.0415584\pi$
$60$		0			0
$61$			4.00000i			0.512148i	0.966657	+	0.256074i	$0.0824290\pi$
$61$			4.00000i			0.512148i	−0.966657	+	0.256074i	$0.917571\pi$
$62$	−2.00000	+	2.00000i	−0.254000	+	0.254000i
$63$		0			0
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$		−	12.0000i		−	1.45521i
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$	−10.0000			−1.17041			−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000			−1.17041			−0.585206	+	0.810885i	$0.698986\pi$
$74$	2.00000	+	2.00000i	0.232495	+	0.232495i
$75$		0			0
$76$	8.00000			0.917663
$77$			8.00000i			0.911685i
$78$		0			0
$79$	−2.00000			−0.225018			−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000			−0.225018			−0.112509	+	0.993651i	$0.535889\pi$
$80$		0			0
$81$		0			0
$82$	−4.00000	+	4.00000i	−0.441726	+	0.441726i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$	4.00000	+	4.00000i	0.431331	+	0.431331i
$87$		0			0
$88$	4.00000	−	4.00000i	0.426401	−	0.426401i
$89$	4.00000			0.423999			0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000			0.423999			0.212000	+	0.977270i	$0.432002\pi$
$90$		0			0
$91$			24.0000i			2.51588i
$92$		−	16.0000i		−	1.66812i
$93$		0			0
$94$	−4.00000	+	4.00000i	−0.412568	+	0.412568i
$95$		0			0
$96$		0			0
$97$	−2.00000			−0.203069			−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000			−0.203069			−0.101535	+	0.994832i	$0.532375\pi$
$98$	9.00000	−	9.00000i	0.909137	−	0.909137i
$99$		0			0
$100$		0			0
$101$			10.0000i			0.995037i	0.867453	+	0.497519i	$0.165755\pi$
$101$			10.0000i			0.995037i	−0.867453	+	0.497519i	$0.834245\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$	12.0000	−	12.0000i	1.17670	−	1.17670i
$105$		0			0
$106$	−10.0000	−	10.0000i	−0.971286	−	0.971286i
$107$		−	20.0000i		−	1.93347i	−0.255774	−	0.966736i	$-0.582330\pi$
$107$		−	20.0000i		−	1.93347i	0.255774	−	0.966736i	$-0.417670\pi$
$108$		0			0
$109$		−	8.00000i		−	0.766261i	−0.923694	−	0.383131i	$-0.874846\pi$
$109$		−	8.00000i		−	0.766261i	0.923694	−	0.383131i	$-0.125154\pi$
$110$		0			0
$111$		0			0
$112$	−16.0000			−1.51186
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$		0			0
$116$	−12.0000			−1.11417
$117$		0			0
$118$	−2.00000	−	2.00000i	−0.184115	−	0.184115i
$119$	24.0000			2.20008
$120$		0			0
$121$	7.00000			0.636364
$122$	4.00000	+	4.00000i	0.362143	+	0.362143i
$123$		0			0
$124$			4.00000i			0.359211i
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	8.00000	+	8.00000i	0.707107	+	0.707107i
$129$		0			0
$130$		0			0
$131$		−	18.0000i		−	1.57267i	−0.617802	−	0.786334i	$-0.711977\pi$
$131$		−	18.0000i		−	1.57267i	0.617802	−	0.786334i	$-0.288023\pi$
$132$		0			0
$133$			16.0000i			1.38738i
$134$	−12.0000	−	12.0000i	−1.03664	−	1.03664i
$135$		0			0
$136$	−12.0000	−	12.0000i	−1.02899	−	1.02899i
$137$	−2.00000			−0.170872			−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000			−0.170872			−0.0854358	+	0.996344i	$0.527228\pi$
$138$		0			0
$139$		−	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
$141$		0			0
$142$	−8.00000	+	8.00000i	−0.671345	+	0.671345i
$143$	−12.0000			−1.00349
$144$		0			0
$145$		0			0
$146$	−10.0000	+	10.0000i	−0.827606	+	0.827606i
$147$		0			0
$148$	4.00000			0.328798
$149$		−	6.00000i		−	0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$		−	6.00000i		−	0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$		0			0
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$	8.00000	−	8.00000i	0.648886	−	0.648886i
$153$		0			0
$154$	8.00000	+	8.00000i	0.644658	+	0.644658i
$155$		0			0
$156$		0			0
$157$		−	14.0000i		−	1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$		−	14.0000i		−	1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	−2.00000	+	2.00000i	−0.159111	+	0.159111i
$159$		0			0
$160$		0			0
$161$	32.0000			2.52195
$162$		0			0
$163$			16.0000i			1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$			16.0000i			1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$			8.00000i			0.624695i
$165$		0			0
$166$		0			0
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$		0			0
$171$		0			0
$172$	8.00000			0.609994
$173$			14.0000i			1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$			14.0000i			1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$		0			0
$175$		0			0
$176$		−	8.00000i		−	0.603023i
$177$		0			0
$178$	4.00000	−	4.00000i	0.299813	−	0.299813i
$179$		−	6.00000i		−	0.448461i	−0.974536	−	0.224231i	$-0.928013\pi$
$179$		−	6.00000i		−	0.448461i	0.974536	−	0.224231i	$-0.0719869\pi$
$180$		0			0
$181$			16.0000i			1.18927i	0.803996	+	0.594635i	$0.202704\pi$
$181$			16.0000i			1.18927i	−0.803996	+	0.594635i	$0.797296\pi$
$182$	24.0000	+	24.0000i	1.77900	+	1.77900i
$183$		0			0
$184$	−16.0000	−	16.0000i	−1.17954	−	1.17954i
$185$		0			0
$186$		0			0
$187$			12.0000i			0.877527i
$188$			8.00000i			0.583460i
$189$		0			0
$190$		0			0
$191$	−16.0000			−1.15772			−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000			−1.15772			−0.578860	+	0.815427i	$0.696502\pi$
$192$		0			0
$193$	26.0000			1.87152			0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000			1.87152			0.935760	+	0.352636i	$0.114715\pi$
$194$	−2.00000	+	2.00000i	−0.143592	+	0.143592i
$195$		0			0
$196$		−	18.0000i		−	1.28571i
$197$			14.0000i			0.997459i	0.866758	+	0.498729i	$0.166200\pi$
$197$			14.0000i			0.997459i	−0.866758	+	0.498729i	$0.833800\pi$
$198$		0			0
$199$	−6.00000			−0.425329			−0.212664	−	0.977125i	$-0.568214\pi$
$199$	−6.00000			−0.425329			−0.212664	+	0.977125i	$0.568214\pi$
$200$		0			0
$201$		0			0
$202$	10.0000	+	10.0000i	0.703598	+	0.703598i
$203$		−	24.0000i		−	1.68447i
$204$		0			0
$205$		0			0
$206$	8.00000	−	8.00000i	0.557386	−	0.557386i
$207$		0			0
$208$		−	24.0000i		−	1.66410i
$209$	−8.00000			−0.553372
$210$		0			0
$211$		−	12.0000i		−	0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$		−	12.0000i		−	0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	−20.0000			−1.37361
$213$		0			0
$214$	−20.0000	−	20.0000i	−1.36717	−	1.36717i
$215$		0			0
$216$		0			0
$217$	−8.00000			−0.543075
$218$	−8.00000	−	8.00000i	−0.541828	−	0.541828i
$219$		0			0
$220$		0			0
$221$			36.0000i			2.42162i
$222$		0			0
$223$	−16.0000			−1.07144			−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000			−1.07144			−0.535720	+	0.844396i	$0.679960\pi$
$224$	−16.0000	+	16.0000i	−1.06904	+	1.06904i
$225$		0			0
$226$	6.00000	−	6.00000i	0.399114	−	0.399114i
$227$			12.0000i			0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$			12.0000i			0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$		0			0
$229$			20.0000i			1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$			20.0000i			1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$		0			0
$231$		0			0
$232$	−12.0000	+	12.0000i	−0.787839	+	0.787839i
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$		0			0
$236$	−4.00000			−0.260378
$237$		0			0
$238$	24.0000	−	24.0000i	1.55569	−	1.55569i
$239$	−16.0000			−1.03495			−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000			−1.03495			−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$	7.00000	−	7.00000i	0.449977	−	0.449977i
$243$		0			0
$244$	8.00000			0.512148
$245$		0			0
$246$		0			0
$247$	−24.0000			−1.52708
$248$	4.00000	+	4.00000i	0.254000	+	0.254000i
$249$		0			0
$250$		0			0
$251$		−	18.0000i		−	1.13615i	−0.822977	−	0.568075i	$-0.807688\pi$
$251$		−	18.0000i		−	1.13615i	0.822977	−	0.568075i	$-0.192312\pi$
$252$		0			0
$253$			16.0000i			1.00591i
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	22.0000			1.37232			0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000			1.37232			0.686161	+	0.727450i	$0.259294\pi$
$258$		0			0
$259$			8.00000i			0.497096i
$260$		0			0
$261$		0			0
$262$	−18.0000	−	18.0000i	−1.11204	−	1.11204i
$263$	−12.0000			−0.739952			−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000			−0.739952			−0.369976	+	0.929041i	$0.620634\pi$
$264$		0			0
$265$		0			0
$266$	16.0000	+	16.0000i	0.981023	+	0.981023i
$267$		0			0
$268$	−24.0000			−1.46603
$269$			2.00000i			0.121942i	0.998140	+	0.0609711i	$0.0194197\pi$
$269$			2.00000i			0.121942i	−0.998140	+	0.0609711i	$0.980580\pi$
$270$		0			0
$271$	−2.00000			−0.121491			−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000			−0.121491			−0.0607457	+	0.998153i	$0.519348\pi$
$272$	−24.0000			−1.45521
$273$		0			0
$274$	−2.00000	+	2.00000i	−0.120824	+	0.120824i
$275$		0			0
$276$		0			0
$277$		−	2.00000i		−	0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$		−	2.00000i		−	0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	−4.00000	−	4.00000i	−0.239904	−	0.239904i
$279$		0			0
$280$		0			0
$281$	28.0000			1.67034			0.835170	−	0.549992i	$-0.185369\pi$
$281$	28.0000			1.67034			0.835170	+	0.549992i	$0.185369\pi$
$282$		0			0
$283$			28.0000i			1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$			28.0000i			1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$			16.0000i			0.949425i
$285$		0			0
$286$	−12.0000	+	12.0000i	−0.709575	+	0.709575i
$287$	−16.0000			−0.944450
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$			20.0000i			1.17041i
$293$			22.0000i			1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$			22.0000i			1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$		0			0
$295$		0			0
$296$	4.00000	−	4.00000i	0.232495	−	0.232495i
$297$		0			0
$298$	−6.00000	−	6.00000i	−0.347571	−	0.347571i
$299$			48.0000i			2.77591i
$300$		0			0
$301$			16.0000i			0.922225i
$302$	−2.00000	+	2.00000i	−0.115087	+	0.115087i
$303$		0			0
$304$		−	16.0000i		−	0.917663i
$305$		0			0
$306$		0			0
$307$		−	16.0000i		−	0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$		−	16.0000i		−	0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	16.0000			0.911685
$309$		0			0
$310$		0			0
$311$	−32.0000			−1.81455			−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000			−1.81455			−0.907277	+	0.420534i	$0.861843\pi$
$312$		0			0
$313$	−18.0000			−1.01742			−0.508710	−	0.860938i	$-0.669877\pi$
$313$	−18.0000			−1.01742			−0.508710	+	0.860938i	$0.669877\pi$
$314$	−14.0000	−	14.0000i	−0.790066	−	0.790066i
$315$		0			0
$316$			4.00000i			0.225018i
$317$		−	6.00000i		−	0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$		−	6.00000i		−	0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$		0			0
$319$	12.0000			0.671871
$320$		0			0
$321$		0			0
$322$	32.0000	−	32.0000i	1.78329	−	1.78329i
$323$			24.0000i			1.33540i
$324$		0			0
$325$		0			0
$326$	16.0000	+	16.0000i	0.886158	+	0.886158i
$327$		0			0
$328$	8.00000	+	8.00000i	0.441726	+	0.441726i
$329$	−16.0000			−0.882109
$330$		0			0
$331$			4.00000i			0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$			4.00000i			0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$		0			0
$333$		0			0
$334$	−12.0000	+	12.0000i	−0.656611	+	0.656611i
$335$		0			0
$336$		0			0
$337$	18.0000			0.980522			0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000			0.980522			0.490261	+	0.871576i	$0.336901\pi$
$338$	−23.0000	+	23.0000i	−1.25104	+	1.25104i
$339$		0			0
$340$		0			0
$341$		−	4.00000i		−	0.216612i
$342$		0			0
$343$	8.00000			0.431959
$344$	8.00000	−	8.00000i	0.431331	−	0.431331i
$345$		0			0
$346$	14.0000	+	14.0000i	0.752645	+	0.752645i
$347$		−	20.0000i		−	1.07366i	−0.843692	−	0.536828i	$-0.819622\pi$
$347$		−	20.0000i		−	1.07366i	0.843692	−	0.536828i	$-0.180378\pi$
$348$		0			0
$349$			20.0000i			1.07058i	0.844670	+	0.535288i	$0.179797\pi$
$349$			20.0000i			1.07058i	−0.844670	+	0.535288i	$0.820203\pi$
$350$		0			0
$351$		0			0
$352$	−8.00000	−	8.00000i	−0.426401	−	0.426401i
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$		0			0
$355$		0			0
$356$		−	8.00000i		−	0.423999i
$357$		0			0
$358$	−6.00000	−	6.00000i	−0.317110	−	0.317110i
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	16.0000	+	16.0000i	0.840941	+	0.840941i
$363$		0			0
$364$	48.0000			2.51588
$365$		0			0
$366$		0			0
$367$	12.0000			0.626395			0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000			0.626395			0.313197	+	0.949688i	$0.398600\pi$
$368$	−32.0000			−1.66812
$369$		0			0
$370$		0			0
$371$		−	40.0000i		−	2.07670i
$372$		0			0
$373$		−	26.0000i		−	1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$		−	26.0000i		−	1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$	12.0000	+	12.0000i	0.620505	+	0.620505i
$375$		0			0
$376$	8.00000	+	8.00000i	0.412568	+	0.412568i
$377$	36.0000			1.85409
$378$		0			0
$379$		−	12.0000i		−	0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$		−	12.0000i		−	0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$		0			0
$381$		0			0
$382$	−16.0000	+	16.0000i	−0.818631	+	0.818631i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	26.0000	−	26.0000i	1.32337	−	1.32337i
$387$		0			0
$388$			4.00000i			0.203069i
$389$		−	10.0000i		−	0.507020i	−0.967333	−	0.253510i	$-0.918415\pi$
$389$		−	10.0000i		−	0.507020i	0.967333	−	0.253510i	$-0.0815851\pi$
$390$		0			0
$391$	48.0000			2.42746
$392$	−18.0000	−	18.0000i	−0.909137	−	0.909137i
$393$		0			0
$394$	14.0000	+	14.0000i	0.705310	+	0.705310i
$395$		0			0
$396$		0			0
$397$		−	2.00000i		−	0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$		−	2.00000i		−	0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	−6.00000	+	6.00000i	−0.300753	+	0.300753i
$399$		0			0
$400$		0			0
$401$	−24.0000			−1.19850			−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000			−1.19850			−0.599251	+	0.800561i	$0.704535\pi$
$402$		0			0
$403$		−	12.0000i		−	0.597763i
$404$	20.0000			0.995037
$405$		0			0
$406$	−24.0000	−	24.0000i	−1.19110	−	1.19110i
$407$	−4.00000			−0.198273
$408$		0			0
$409$	−30.0000			−1.48340			−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000			−1.48340			−0.741702	+	0.670729i	$0.765981\pi$
$410$		0			0
$411$		0			0
$412$		−	16.0000i		−	0.788263i
$413$		−	8.00000i		−	0.393654i
$414$		0			0
$415$		0			0
$416$	−24.0000	−	24.0000i	−1.17670	−	1.17670i
$417$		0			0
$418$	−8.00000	+	8.00000i	−0.391293	+	0.391293i
$419$		−	14.0000i		−	0.683945i	−0.939710	−	0.341972i	$-0.888905\pi$
$419$		−	14.0000i		−	0.683945i	0.939710	−	0.341972i	$-0.111095\pi$
$420$		0			0
$421$		−	4.00000i		−	0.194948i	−0.995238	−	0.0974740i	$-0.968924\pi$
$421$		−	4.00000i		−	0.194948i	0.995238	−	0.0974740i	$-0.0310763\pi$
$422$	−12.0000	−	12.0000i	−0.584151	−	0.584151i
$423$		0			0
$424$	−20.0000	+	20.0000i	−0.971286	+	0.971286i
$425$		0			0
$426$		0			0
$427$			16.0000i			0.774294i
$428$	−40.0000			−1.93347
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	10.0000			0.480569			0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000			0.480569			0.240285	+	0.970702i	$0.422759\pi$
$434$	−8.00000	+	8.00000i	−0.384012	+	0.384012i
$435$		0			0
$436$	−16.0000			−0.766261
$437$			32.0000i			1.53077i
$438$		0			0
$439$	26.0000			1.24091			0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000			1.24091			0.620456	+	0.784241i	$0.286947\pi$
$440$		0			0
$441$		0			0
$442$	36.0000	+	36.0000i	1.71235	+	1.71235i
$443$			4.00000i			0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$			4.00000i			0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$		0			0
$445$		0			0
$446$	−16.0000	+	16.0000i	−0.757622	+	0.757622i
$447$		0			0
$448$			32.0000i			1.51186i
$449$	−24.0000			−1.13263			−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000			−1.13263			−0.566315	+	0.824189i	$0.691631\pi$
$450$		0			0
$451$		−	8.00000i		−	0.376705i
$452$		−	12.0000i		−	0.564433i
$453$		0			0
$454$	12.0000	+	12.0000i	0.563188	+	0.563188i
$455$		0			0
$456$		0			0
$457$	18.0000			0.842004			0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000			0.842004			0.421002	+	0.907060i	$0.361678\pi$
$458$	20.0000	+	20.0000i	0.934539	+	0.934539i
$459$		0			0
$460$		0			0
$461$		−	2.00000i		−	0.0931493i	−0.998915	−	0.0465746i	$-0.985169\pi$
$461$		−	2.00000i		−	0.0931493i	0.998915	−	0.0465746i	$-0.0148305\pi$
$462$		0			0
$463$	−20.0000			−0.929479			−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000			−0.929479			−0.464739	+	0.885448i	$0.653852\pi$
$464$			24.0000i			1.11417i
$465$		0			0
$466$	6.00000	−	6.00000i	0.277945	−	0.277945i
$467$		−	32.0000i		−	1.48078i	−0.672176	−	0.740392i	$-0.734640\pi$
$467$		−	32.0000i		−	1.48078i	0.672176	−	0.740392i	$-0.265360\pi$
$468$		0			0
$469$		−	48.0000i		−	2.21643i
$470$		0			0
$471$		0			0
$472$	−4.00000	+	4.00000i	−0.184115	+	0.184115i
$473$	−8.00000			−0.367840
$474$		0			0
$475$		0			0
$476$		−	48.0000i		−	2.20008i
$477$		0			0
$478$	−16.0000	+	16.0000i	−0.731823	+	0.731823i
$479$	32.0000			1.46212			0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000			1.46212			0.731059	+	0.682315i	$0.239027\pi$
$480$		0			0
$481$	−12.0000			−0.547153
$482$	−18.0000	+	18.0000i	−0.819878	+	0.819878i
$483$		0			0
$484$		−	14.0000i		−	0.636364i
$485$		0			0
$486$		0			0
$487$	24.0000			1.08754			0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000			1.08754			0.543772	+	0.839233i	$0.316996\pi$
$488$	8.00000	−	8.00000i	0.362143	−	0.362143i
$489$		0			0
$490$		0			0
$491$			34.0000i			1.53440i	0.641409	+	0.767199i	$0.278350\pi$
$491$			34.0000i			1.53440i	−0.641409	+	0.767199i	$0.721650\pi$
$492$		0			0
$493$		−	36.0000i		−	1.62136i
$494$	−24.0000	+	24.0000i	−1.07981	+	1.07981i
$495$		0			0
$496$	8.00000			0.359211
$497$	−32.0000			−1.43540
$498$		0			0
$499$		−	20.0000i		−	0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$		−	20.0000i		−	0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$		0			0
$501$		0			0
$502$	−18.0000	−	18.0000i	−0.803379	−	0.803379i
$503$	12.0000			0.535054			0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000			0.535054			0.267527	+	0.963550i	$0.413794\pi$
$504$		0			0
$505$		0			0
$506$	16.0000	+	16.0000i	0.711287	+	0.711287i
$507$		0			0
$508$		0			0
$509$			30.0000i			1.32973i	0.746965	+	0.664863i	$0.231510\pi$
$509$			30.0000i			1.32973i	−0.746965	+	0.664863i	$0.768490\pi$
$510$		0			0
$511$	−40.0000			−1.76950
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$		0			0
$514$	22.0000	−	22.0000i	0.970378	−	0.970378i
$515$		0			0
$516$		0			0
$517$		−	8.00000i		−	0.351840i
$518$	8.00000	+	8.00000i	0.351500	+	0.351500i
$519$		0			0
$520$		0			0
$521$	24.0000			1.05146			0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000			1.05146			0.525730	+	0.850652i	$0.323792\pi$
$522$		0			0
$523$		−	40.0000i		−	1.74908i	−0.484955	−	0.874539i	$-0.661164\pi$
$523$		−	40.0000i		−	1.74908i	0.484955	−	0.874539i	$-0.338836\pi$
$524$	−36.0000			−1.57267
$525$		0			0
$526$	−12.0000	+	12.0000i	−0.523225	+	0.523225i
$527$	−12.0000			−0.522728
$528$		0			0
$529$	41.0000			1.78261
$530$		0			0
$531$		0			0
$532$	32.0000			1.38738
$533$		−	24.0000i		−	1.03956i
$534$		0			0
$535$		0			0
$536$	−24.0000	+	24.0000i	−1.03664	+	1.03664i
$537$		0			0
$538$	2.00000	+	2.00000i	0.0862261	+	0.0862261i
$539$			18.0000i			0.775315i
$540$		0			0
$541$		−	32.0000i		−	1.37579i	−0.725811	−	0.687894i	$-0.758536\pi$
$541$		−	32.0000i		−	1.37579i	0.725811	−	0.687894i	$-0.241464\pi$
$542$	−2.00000	+	2.00000i	−0.0859074	+	0.0859074i
$543$		0			0
$544$	−24.0000	+	24.0000i	−1.02899	+	1.02899i
$545$		0			0
$546$		0			0
$547$			28.0000i			1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$			28.0000i			1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$			4.00000i			0.170872i
$549$		0			0
$550$		0			0
$551$	24.0000			1.02243
$552$		0			0
$553$	−8.00000			−0.340195
$554$	−2.00000	−	2.00000i	−0.0849719	−	0.0849719i
$555$		0			0
$556$	−8.00000			−0.339276
$557$			30.0000i			1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$			30.0000i			1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$		0			0
$562$	28.0000	−	28.0000i	1.18111	−	1.18111i
$563$			36.0000i			1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$			36.0000i			1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$		0			0
$565$		0			0
$566$	28.0000	+	28.0000i	1.17693	+	1.17693i
$567$		0			0
$568$	16.0000	+	16.0000i	0.671345	+	0.671345i
$569$	−32.0000			−1.34151			−0.670755	−	0.741679i	$-0.734030\pi$
$569$	−32.0000			−1.34151			−0.670755	+	0.741679i	$0.734030\pi$
$570$		0			0
$571$		−	20.0000i		−	0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$		−	20.0000i		−	0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$			24.0000i			1.00349i
$573$		0			0
$574$	−16.0000	+	16.0000i	−0.667827	+	0.667827i
$575$		0			0
$576$		0			0
$577$	−10.0000			−0.416305			−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000			−0.416305			−0.208153	+	0.978096i	$0.566745\pi$
$578$	19.0000	−	19.0000i	0.790296	−	0.790296i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	20.0000			0.828315
$584$	20.0000	+	20.0000i	0.827606	+	0.827606i
$585$		0			0
$586$	22.0000	+	22.0000i	0.908812	+	0.908812i
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$		0			0
$589$		−	8.00000i		−	0.329634i
$590$		0			0
$591$		0			0
$592$		−	8.00000i		−	0.328798i
$593$	−26.0000			−1.06769			−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000			−1.06769			−0.533846	+	0.845582i	$0.679254\pi$
$594$		0			0
$595$		0			0
$596$	−12.0000			−0.491539
$597$		0			0
$598$	48.0000	+	48.0000i	1.96287	+	1.96287i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$	16.0000	+	16.0000i	0.652111	+	0.652111i
$603$		0			0
$604$			4.00000i			0.162758i
$605$		0			0
$606$		0			0
$607$	4.00000			0.162355			0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000			0.162355			0.0811775	+	0.996700i	$0.474132\pi$
$608$	−16.0000	−	16.0000i	−0.648886	−	0.648886i
$609$		0			0
$610$		0			0
$611$		−	24.0000i		−	0.970936i
$612$		0			0
$613$			26.0000i			1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$			26.0000i			1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	−16.0000	−	16.0000i	−0.645707	−	0.645707i
$615$		0			0
$616$	16.0000	−	16.0000i	0.644658	−	0.644658i
$617$	6.00000			0.241551			0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000			0.241551			0.120775	+	0.992680i	$0.461462\pi$
$618$		0			0
$619$			20.0000i			0.803868i	0.915669	+	0.401934i	$0.131662\pi$
$619$			20.0000i			0.803868i	−0.915669	+	0.401934i	$0.868338\pi$
$620$		0			0
$621$		0			0
$622$	−32.0000	+	32.0000i	−1.28308	+	1.28308i
$623$	16.0000			0.641026
$624$		0			0
$625$		0			0
$626$	−18.0000	+	18.0000i	−0.719425	+	0.719425i
$627$		0			0
$628$	−28.0000			−1.11732
$629$			12.0000i			0.478471i
$630$		0			0
$631$	−30.0000			−1.19428			−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000			−1.19428			−0.597141	+	0.802137i	$0.703697\pi$
$632$	4.00000	+	4.00000i	0.159111	+	0.159111i
$633$		0			0
$634$	−6.00000	−	6.00000i	−0.238290	−	0.238290i
$635$		0			0
$636$		0			0
$637$			54.0000i			2.13956i
$638$	12.0000	−	12.0000i	0.475085	−	0.475085i
$639$		0			0
$640$		0			0
$641$	4.00000			0.157991			0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000			0.157991			0.0789953	+	0.996875i	$0.474829\pi$
$642$		0			0
$643$		−	24.0000i		−	0.946468i	−0.880937	−	0.473234i	$-0.843087\pi$
$643$		−	24.0000i		−	0.946468i	0.880937	−	0.473234i	$-0.156913\pi$
$644$		−	64.0000i		−	2.52195i
$645$		0			0
$646$	24.0000	+	24.0000i	0.944267	+	0.944267i
$647$	−12.0000			−0.471769			−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000			−0.471769			−0.235884	+	0.971781i	$0.575799\pi$
$648$		0			0
$649$	4.00000			0.157014
$650$		0			0
$651$		0			0
$652$	32.0000			1.25322
$653$			6.00000i			0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$			6.00000i			0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$		0			0
$655$		0			0
$656$	16.0000			0.624695
$657$		0			0
$658$	−16.0000	+	16.0000i	−0.623745	+	0.623745i
$659$		−	18.0000i		−	0.701180i	−0.936529	−	0.350590i	$-0.885981\pi$
$659$		−	18.0000i		−	0.701180i	0.936529	−	0.350590i	$-0.114019\pi$
$660$		0			0
$661$		−	28.0000i		−	1.08907i	−0.838737	−	0.544537i	$-0.816705\pi$
$661$		−	28.0000i		−	1.08907i	0.838737	−	0.544537i	$-0.183295\pi$
$662$	4.00000	+	4.00000i	0.155464	+	0.155464i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	48.0000i		−	1.85857i
$668$			24.0000i			0.928588i
$669$		0			0
$670$		0			0
$671$	−8.00000			−0.308837
$672$		0			0
$673$	2.00000			0.0770943			0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000			0.0770943			0.0385472	+	0.999257i	$0.487727\pi$
$674$	18.0000	−	18.0000i	0.693334	−	0.693334i
$675$		0			0
$676$			46.0000i			1.76923i
$677$		−	14.0000i		−	0.538064i	−0.963131	−	0.269032i	$-0.913296\pi$
$677$		−	14.0000i		−	0.538064i	0.963131	−	0.269032i	$-0.0867037\pi$
$678$		0			0
$679$	−8.00000			−0.307012
$680$		0			0
$681$		0			0
$682$	−4.00000	−	4.00000i	−0.153168	−	0.153168i
$683$		−	24.0000i		−	0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$		−	24.0000i		−	0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$		0			0
$685$		0			0
$686$	8.00000	−	8.00000i	0.305441	−	0.305441i
$687$		0			0
$688$		−	16.0000i		−	0.609994i
$689$	60.0000			2.28582
$690$		0			0
$691$			12.0000i			0.456502i	0.973602	+	0.228251i	$0.0733006\pi$
$691$			12.0000i			0.456502i	−0.973602	+	0.228251i	$0.926699\pi$
$692$	28.0000			1.06440
$693$		0			0
$694$	−20.0000	−	20.0000i	−0.759190	−	0.759190i
$695$		0			0
$696$		0			0
$697$	−24.0000			−0.909065
$698$	20.0000	+	20.0000i	0.757011	+	0.757011i
$699$		0			0
$700$		0			0
$701$		−	2.00000i		−	0.0755390i	−0.999286	−	0.0377695i	$-0.987975\pi$
$701$		−	2.00000i		−	0.0755390i	0.999286	−	0.0377695i	$-0.0120253\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$	−16.0000			−0.603023
$705$		0			0
$706$	−6.00000	+	6.00000i	−0.225813	+	0.225813i
$707$			40.0000i			1.50435i
$708$		0			0
$709$			44.0000i			1.65245i	0.563337	+	0.826227i	$0.309517\pi$
$709$			44.0000i			1.65245i	−0.563337	+	0.826227i	$0.690483\pi$
$710$		0			0
$711$		0			0
$712$	−8.00000	−	8.00000i	−0.299813	−	0.299813i
$713$	−16.0000			−0.599205
$714$		0			0
$715$		0			0
$716$	−12.0000			−0.448461
$717$		0			0
$718$	−24.0000	+	24.0000i	−0.895672	+	0.895672i
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$		0			0
$721$	32.0000			1.19174
$722$	3.00000	−	3.00000i	0.111648	−	0.111648i
$723$		0			0
$724$	32.0000			1.18927
$725$		0			0
$726$		0			0
$727$	−40.0000			−1.48352			−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000			−1.48352			−0.741759	+	0.670667i	$0.766008\pi$
$728$	48.0000	−	48.0000i	1.77900	−	1.77900i
$729$		0			0
$730$		0			0
$731$			24.0000i			0.887672i
$732$		0			0
$733$		−	6.00000i		−	0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$		−	6.00000i		−	0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	12.0000	−	12.0000i	0.442928	−	0.442928i
$735$		0			0
$736$	−32.0000	+	32.0000i	−1.17954	+	1.17954i
$737$	24.0000			0.884051
$738$		0			0
$739$			12.0000i			0.441427i	0.975339	+	0.220714i	$0.0708386\pi$
$739$			12.0000i			0.441427i	−0.975339	+	0.220714i	$0.929161\pi$
$740$		0			0
$741$		0			0
$742$	−40.0000	−	40.0000i	−1.46845	−	1.46845i
$743$	−48.0000			−1.76095			−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000			−1.76095			−0.880475	+	0.474093i	$0.842776\pi$
$744$		0			0
$745$		0			0
$746$	−26.0000	−	26.0000i	−0.951928	−	0.951928i
$747$		0			0
$748$	24.0000			0.877527
$749$		−	80.0000i		−	2.92314i
$750$		0			0
$751$	−10.0000			−0.364905			−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000			−0.364905			−0.182453	+	0.983215i	$0.558404\pi$
$752$	16.0000			0.583460
$753$		0			0
$754$	36.0000	−	36.0000i	1.31104	−	1.31104i
$755$		0			0
$756$		0			0
$757$		−	10.0000i		−	0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$		−	10.0000i		−	0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	−12.0000	−	12.0000i	−0.435860	−	0.435860i
$759$		0			0
$760$		0			0
$761$	8.00000			0.290000			0.145000	−	0.989432i	$-0.453682\pi$
$761$	8.00000			0.290000			0.145000	+	0.989432i	$0.453682\pi$
$762$		0			0
$763$		−	32.0000i		−	1.15848i
$764$			32.0000i			1.15772i
$765$		0			0
$766$		0			0
$767$	12.0000			0.433295
$768$		0			0
$769$	−18.0000			−0.649097			−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000			−0.649097			−0.324548	+	0.945869i	$0.605212\pi$
$770$		0			0
$771$		0			0
$772$		−	52.0000i		−	1.87152i
$773$			6.00000i			0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$			6.00000i			0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$		0			0
$775$		0			0
$776$	4.00000	+	4.00000i	0.143592	+	0.143592i
$777$		0			0
$778$	−10.0000	−	10.0000i	−0.358517	−	0.358517i
$779$		−	16.0000i		−	0.573259i
$780$		0			0
$781$		−	16.0000i		−	0.572525i
$782$	48.0000	−	48.0000i	1.71648	−	1.71648i
$783$		0			0
$784$	−36.0000			−1.28571
$785$		0			0
$786$		0			0
$787$			32.0000i			1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$			32.0000i			1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	28.0000			0.997459
$789$		0			0
$790$		0			0
$791$	24.0000			0.853342
$792$		0			0
$793$	−24.0000			−0.852265
$794$	−2.00000	−	2.00000i	−0.0709773	−	0.0709773i
$795$		0			0
$796$			12.0000i			0.425329i
$797$		−	18.0000i		−	0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$		−	18.0000i		−	0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0
$801$		0			0
$802$	−24.0000	+	24.0000i	−0.847469	+	0.847469i
$803$		−	20.0000i		−	0.705785i
$804$		0			0
$805$		0			0
$806$	−12.0000	−	12.0000i	−0.422682	−	0.422682i
$807$		0			0
$808$	20.0000	−	20.0000i	0.703598	−	0.703598i
$809$	44.0000			1.54696			0.773479	−	0.633822i	$-0.218515\pi$
$809$	44.0000			1.54696			0.773479	+	0.633822i	$0.218515\pi$
$810$		0			0
$811$		−	52.0000i		−	1.82597i	−0.407997	−	0.912983i	$-0.633772\pi$
$811$		−	52.0000i		−	1.82597i	0.407997	−	0.912983i	$-0.366228\pi$
$812$	−48.0000			−1.68447
$813$		0			0
$814$	−4.00000	+	4.00000i	−0.140200	+	0.140200i
$815$		0			0
$816$		0			0
$817$	−16.0000			−0.559769
$818$	−30.0000	+	30.0000i	−1.04893	+	1.04893i
$819$		0			0
$820$		0			0
$821$			46.0000i			1.60541i	0.596376	+	0.802706i	$0.296607\pi$
$821$			46.0000i			1.60541i	−0.596376	+	0.802706i	$0.703393\pi$
$822$		0			0
$823$	48.0000			1.67317			0.836587	−	0.547833i	$-0.184547\pi$
$823$	48.0000			1.67317			0.836587	+	0.547833i	$0.184547\pi$
$824$	−16.0000	−	16.0000i	−0.557386	−	0.557386i
$825$		0			0
$826$	−8.00000	−	8.00000i	−0.278356	−	0.278356i
$827$		−	16.0000i		−	0.556375i	−0.960527	−	0.278187i	$-0.910266\pi$
$827$		−	16.0000i		−	0.556375i	0.960527	−	0.278187i	$-0.0897336\pi$
$828$		0			0
$829$		−	8.00000i		−	0.277851i	−0.990303	−	0.138926i	$-0.955635\pi$
$829$		−	8.00000i		−	0.277851i	0.990303	−	0.138926i	$-0.0443649\pi$
$830$		0			0
$831$		0			0
$832$	−48.0000			−1.66410
$833$	54.0000			1.87099
$834$		0			0
$835$		0			0
$836$			16.0000i			0.553372i
$837$		0			0
$838$	−14.0000	−	14.0000i	−0.483622	−	0.483622i
$839$	−16.0000			−0.552381			−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000			−0.552381			−0.276191	+	0.961103i	$0.589072\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$	−4.00000	−	4.00000i	−0.137849	−	0.137849i
$843$		0			0
$844$	−24.0000			−0.826114
$845$		0			0
$846$		0			0
$847$	28.0000			0.962091
$848$			40.0000i			1.37361i
$849$		0			0
$850$		0			0
$851$			16.0000i			0.548473i
$852$		0			0
$853$		−	46.0000i		−	1.57501i	−0.616308	−	0.787505i	$-0.711372\pi$
$853$		−	46.0000i		−	1.57501i	0.616308	−	0.787505i	$-0.288628\pi$
$854$	16.0000	+	16.0000i	0.547509	+	0.547509i
$855$		0			0
$856$	−40.0000	+	40.0000i	−1.36717	+	1.36717i
$857$	−6.00000			−0.204956			−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000			−0.204956			−0.102478	+	0.994735i	$0.532677\pi$
$858$		0			0
$859$			28.0000i			0.955348i	0.878537	+	0.477674i	$0.158520\pi$
$859$			28.0000i			0.955348i	−0.878537	+	0.477674i	$0.841480\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	36.0000			1.22545			0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000			1.22545			0.612727	+	0.790295i	$0.290072\pi$
$864$		0			0
$865$		0			0
$866$	10.0000	−	10.0000i	0.339814	−	0.339814i
$867$		0			0
$868$			16.0000i			0.543075i
$869$		−	4.00000i		−	0.135691i
$870$		0			0
$871$	72.0000			2.43963
$872$	−16.0000	+	16.0000i	−0.541828	+	0.541828i
$873$		0			0
$874$	32.0000	+	32.0000i	1.08242	+	1.08242i
$875$		0			0
$876$		0			0
$877$			10.0000i			0.337676i	0.985644	+	0.168838i	$0.0540015\pi$
$877$			10.0000i			0.337676i	−0.985644	+	0.168838i	$0.945999\pi$
$878$	26.0000	−	26.0000i	0.877457	−	0.877457i
$879$		0			0
$880$		0			0
$881$	−12.0000			−0.404290			−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000			−0.404290			−0.202145	+	0.979356i	$0.564791\pi$
$882$		0			0
$883$		−	8.00000i		−	0.269221i	−0.990899	−	0.134611i	$-0.957022\pi$
$883$		−	8.00000i		−	0.269221i	0.990899	−	0.134611i	$-0.0429784\pi$
$884$	72.0000			2.42162
$885$		0			0
$886$	4.00000	+	4.00000i	0.134383	+	0.134383i
$887$	−20.0000			−0.671534			−0.335767	−	0.941945i	$-0.608996\pi$
$887$	−20.0000			−0.671534			−0.335767	+	0.941945i	$0.608996\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$			32.0000i			1.07144i
$893$		−	16.0000i		−	0.535420i
$894$		0			0
$895$		0			0
$896$	32.0000	+	32.0000i	1.06904	+	1.06904i
$897$		0			0
$898$	−24.0000	+	24.0000i	−0.800890	+	0.800890i
$899$			12.0000i			0.400222i
$900$		0			0
$901$		−	60.0000i		−	1.99889i
$902$	−8.00000	−	8.00000i	−0.266371	−	0.266371i
$903$		0			0
$904$	−12.0000	−	12.0000i	−0.399114	−	0.399114i
$905$		0			0
$906$		0			0
$907$			4.00000i			0.132818i	0.997792	+	0.0664089i	$0.0211542\pi$
$907$			4.00000i			0.132818i	−0.997792	+	0.0664089i	$0.978846\pi$
$908$	24.0000			0.796468
$909$		0			0
$910$		0			0
$911$	−48.0000			−1.59031			−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000			−1.59031			−0.795155	+	0.606406i	$0.792611\pi$
$912$		0			0
$913$		0			0
$914$	18.0000	−	18.0000i	0.595387	−	0.595387i
$915$		0			0
$916$	40.0000			1.32164
$917$		−	72.0000i		−	2.37765i
$918$		0			0
$919$	−14.0000			−0.461817			−0.230909	−	0.972975i	$-0.574170\pi$
$919$	−14.0000			−0.461817			−0.230909	+	0.972975i	$0.574170\pi$
$920$		0			0
$921$		0			0
$922$	−2.00000	−	2.00000i	−0.0658665	−	0.0658665i
$923$		−	48.0000i		−	1.57994i
$924$		0			0
$925$		0			0
$926$	−20.0000	+	20.0000i	−0.657241	+	0.657241i
$927$		0			0
$928$	24.0000	+	24.0000i	0.787839	+	0.787839i
$929$	−16.0000			−0.524943			−0.262471	−	0.964940i	$-0.584538\pi$
$929$	−16.0000			−0.524943			−0.262471	+	0.964940i	$0.584538\pi$
$930$		0			0
$931$			36.0000i			1.17985i
$932$		−	12.0000i		−	0.393073i
$933$		0			0
$934$	−32.0000	−	32.0000i	−1.04707	−	1.04707i
$935$		0			0
$936$		0			0
$937$	−22.0000			−0.718709			−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000			−0.718709			−0.359354	+	0.933201i	$0.617003\pi$
$938$	−48.0000	−	48.0000i	−1.56726	−	1.56726i
$939$		0			0
$940$		0			0
$941$		−	46.0000i		−	1.49956i	−0.661689	−	0.749779i	$-0.730160\pi$
$941$		−	46.0000i		−	1.49956i	0.661689	−	0.749779i	$-0.269840\pi$
$942$		0			0
$943$	−32.0000			−1.04206
$944$			8.00000i			0.260378i
$945$		0			0
$946$	−8.00000	+	8.00000i	−0.260102	+	0.260102i
$947$			48.0000i			1.55979i	0.625910	+	0.779895i	$0.284728\pi$
$947$			48.0000i			1.55979i	−0.625910	+	0.779895i	$0.715272\pi$
$948$		0			0
$949$		−	60.0000i		−	1.94768i
$950$		0			0
$951$		0			0
$952$	−48.0000	−	48.0000i	−1.55569	−	1.55569i
$953$	46.0000			1.49009			0.745043	−	0.667016i	$-0.232429\pi$
$953$	46.0000			1.49009			0.745043	+	0.667016i	$0.232429\pi$
$954$		0			0
$955$		0			0
$956$			32.0000i			1.03495i
$957$		0			0
$958$	32.0000	−	32.0000i	1.03387	−	1.03387i
$959$	−8.00000			−0.258333
$960$		0			0
$961$	−27.0000			−0.870968
$962$	−12.0000	+	12.0000i	−0.386896	+	0.386896i
$963$		0			0
$964$			36.0000i			1.15948i
$965$		0			0
$966$		0			0
$967$	8.00000			0.257263			0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000			0.257263			0.128631	+	0.991692i	$0.458942\pi$
$968$	−14.0000	−	14.0000i	−0.449977	−	0.449977i
$969$		0			0
$970$		0			0
$971$		−	42.0000i		−	1.34784i	−0.738802	−	0.673922i	$-0.764608\pi$
$971$		−	42.0000i		−	1.34784i	0.738802	−	0.673922i	$-0.235392\pi$
$972$		0			0
$973$		−	16.0000i		−	0.512936i
$974$	24.0000	−	24.0000i	0.769010	−	0.769010i
$975$		0			0
$976$		−	16.0000i		−	0.512148i
$977$	−34.0000			−1.08776			−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000			−1.08776			−0.543878	+	0.839164i	$0.683045\pi$
$978$		0			0
$979$			8.00000i			0.255681i
$980$		0			0
$981$		0			0
$982$	34.0000	+	34.0000i	1.08498	+	1.08498i
$983$	16.0000			0.510321			0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000			0.510321			0.255160	+	0.966899i	$0.417872\pi$
$984$		0			0
$985$		0			0
$986$	−36.0000	−	36.0000i	−1.14647	−	1.14647i
$987$		0			0
$988$			48.0000i			1.52708i
$989$			32.0000i			1.01754i
$990$		0			0
$991$	−46.0000			−1.46124			−0.730619	−	0.682785i	$-0.760768\pi$
$991$	−46.0000			−1.46124			−0.730619	+	0.682785i	$0.760768\pi$
$992$	8.00000	−	8.00000i	0.254000	−	0.254000i
$993$		0			0
$994$	−32.0000	+	32.0000i	−1.01498	+	1.01498i
$995$		0			0
$996$		0			0
$997$			46.0000i			1.45683i	0.685134	+	0.728417i	$0.259744\pi$
$997$			46.0000i			1.45683i	−0.685134	+	0.728417i	$0.740256\pi$
$998$	−20.0000	−	20.0000i	−0.633089	−	0.633089i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.2.k.i.901.1		2
3.2	odd	2		1800.2.k.c.901.2		2
4.3	odd	2		7200.2.k.c.3601.1		2
5.2	odd	4		1800.2.d.j.1549.2		2
5.3	odd	4		1800.2.d.a.1549.1		2
5.4	even	2		360.2.k.a.181.2	yes	2
8.3	odd	2		7200.2.k.c.3601.2		2
8.5	even	2	inner	1800.2.k.i.901.2		2
12.11	even	2		7200.2.k.a.3601.2		2
15.2	even	4		1800.2.d.e.1549.1		2
15.8	even	4		1800.2.d.g.1549.2		2
15.14	odd	2		360.2.k.c.181.1	yes	2
20.3	even	4		7200.2.d.a.2449.2		2
20.7	even	4		7200.2.d.h.2449.1		2
20.19	odd	2		1440.2.k.b.721.2		2
24.5	odd	2		1800.2.k.c.901.1		2
24.11	even	2		7200.2.k.a.3601.1		2
40.3	even	4		7200.2.d.h.2449.2		2
40.13	odd	4		1800.2.d.j.1549.1		2
40.19	odd	2		1440.2.k.b.721.1		2
40.27	even	4		7200.2.d.a.2449.1		2
40.29	even	2		360.2.k.a.181.1	✓	2
40.37	odd	4		1800.2.d.a.1549.2		2
60.23	odd	4		7200.2.d.b.2449.2		2
60.47	odd	4		7200.2.d.i.2449.1		2
60.59	even	2		1440.2.k.c.721.1		2
120.29	odd	2		360.2.k.c.181.2	yes	2
120.53	even	4		1800.2.d.e.1549.2		2
120.59	even	2		1440.2.k.c.721.2		2
120.77	even	4		1800.2.d.g.1549.1		2
120.83	odd	4		7200.2.d.i.2449.2		2
120.107	odd	4		7200.2.d.b.2449.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.k.a.181.1	✓	2	40.29	even	2
360.2.k.a.181.2	yes	2	5.4	even	2
360.2.k.c.181.1	yes	2	15.14	odd	2
360.2.k.c.181.2	yes	2	120.29	odd	2
1440.2.k.b.721.1		2	40.19	odd	2
1440.2.k.b.721.2		2	20.19	odd	2
1440.2.k.c.721.1		2	60.59	even	2
1440.2.k.c.721.2		2	120.59	even	2
1800.2.d.a.1549.1		2	5.3	odd	4
1800.2.d.a.1549.2		2	40.37	odd	4
1800.2.d.e.1549.1		2	15.2	even	4
1800.2.d.e.1549.2		2	120.53	even	4
1800.2.d.g.1549.1		2	120.77	even	4
1800.2.d.g.1549.2		2	15.8	even	4
1800.2.d.j.1549.1		2	40.13	odd	4
1800.2.d.j.1549.2		2	5.2	odd	4
1800.2.k.c.901.1		2	24.5	odd	2
1800.2.k.c.901.2		2	3.2	odd	2
1800.2.k.i.901.1		2	1.1	even	1	trivial
1800.2.k.i.901.2		2	8.5	even	2	inner
7200.2.d.a.2449.1		2	40.27	even	4
7200.2.d.a.2449.2		2	20.3	even	4
7200.2.d.b.2449.1		2	120.107	odd	4
7200.2.d.b.2449.2		2	60.23	odd	4
7200.2.d.h.2449.1		2	20.7	even	4
7200.2.d.h.2449.2		2	40.3	even	4
7200.2.d.i.2449.1		2	60.47	odd	4
7200.2.d.i.2449.2		2	120.83	odd	4
7200.2.k.a.3601.1		2	24.11	even	2
7200.2.k.a.3601.2		2	12.11	even	2
7200.2.k.c.3601.1		2	4.3	odd	2
7200.2.k.c.3601.2		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +4.00000 q^{7} +(-2.00000 - 2.00000i) q^{8} +2.00000i q^{11} +6.00000i q^{13} +(4.00000 - 4.00000i) q^{14} -4.00000 q^{16} +6.00000 q^{17} +4.00000i q^{19} +(2.00000 + 2.00000i) q^{22} +8.00000 q^{23} +(6.00000 + 6.00000i) q^{26} -8.00000i q^{28} -6.00000i q^{29} -2.00000 q^{31} +(-4.00000 + 4.00000i) q^{32} +(6.00000 - 6.00000i) q^{34} +2.00000i q^{37} +(4.00000 + 4.00000i) q^{38} -4.00000 q^{41} +4.00000i q^{43} +4.00000 q^{44} +(8.00000 - 8.00000i) q^{46} -4.00000 q^{47} +9.00000 q^{49} +12.0000 q^{52} -10.0000i q^{53} +(-8.00000 - 8.00000i) q^{56} +(-6.00000 - 6.00000i) q^{58} -2.00000i q^{59} +4.00000i q^{61} +(-2.00000 + 2.00000i) q^{62} +8.00000i q^{64} -12.0000i q^{67} -12.0000i q^{68} -8.00000 q^{71} -10.0000 q^{73} +(2.00000 + 2.00000i) q^{74} +8.00000 q^{76} +8.00000i q^{77} -2.00000 q^{79} +(-4.00000 + 4.00000i) q^{82} +(4.00000 + 4.00000i) q^{86} +(4.00000 - 4.00000i) q^{88} +4.00000 q^{89} +24.0000i q^{91} -16.0000i q^{92} +(-4.00000 + 4.00000i) q^{94} -2.00000 q^{97} +(9.00000 - 9.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 8 q^{7} - 4 q^{8} + 8 q^{14} - 8 q^{16} + 12 q^{17} + 4 q^{22} + 16 q^{23} + 12 q^{26} - 4 q^{31} - 8 q^{32} + 12 q^{34} + 8 q^{38} - 8 q^{41} + 8 q^{44} + 16 q^{46} - 8 q^{47} + 18 q^{49}+ \cdots + 18 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 8 * q^7 - 4 * q^8 + 8 * q^14 - 8 * q^16 + 12 * q^17 + 4 * q^22 + 16 * q^23 + 12 * q^26 - 4 * q^31 - 8 * q^32 + 12 * q^34 + 8 * q^38 - 8 * q^41 + 8 * q^44 + 16 * q^46 - 8 * q^47 + 18 * q^49 + 24 * q^52 - 16 * q^56 - 12 * q^58 - 4 * q^62 - 16 * q^71 - 20 * q^73 + 4 * q^74 + 16 * q^76 - 4 * q^79 - 8 * q^82 + 8 * q^86 + 8 * q^88 + 8 * q^89 - 8 * q^94 - 4 * q^97 + 18 * q^98

Introduction

L-functions

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