LMFDB - Embedded newform 1275.2.d.a.424.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1275,2,Mod(424,1275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1275.424"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1275 = 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1275.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$10.1809262577$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			424.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1275.424
Dual form			1275.2.d.a.424.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} -1.00000 q^{3} -2.00000 q^{4} +2.00000i q^{6} +2.00000 q^{7} +1.00000 q^{9} +5.00000i q^{11} +2.00000 q^{12} +1.00000i q^{13} -4.00000i q^{14} -4.00000 q^{16} +(1.00000 + 4.00000i) q^{17} -2.00000i q^{18} -5.00000 q^{19} -2.00000 q^{21} +10.0000 q^{22} +1.00000 q^{23} +2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} +6.00000i q^{29} +10.0000i q^{31} +8.00000i q^{32} -5.00000i q^{33} +(8.00000 - 2.00000i) q^{34} -2.00000 q^{36} +2.00000 q^{37} +10.0000i q^{38} -1.00000i q^{39} +5.00000i q^{41} +4.00000i q^{42} +1.00000i q^{43} -10.0000i q^{44} -2.00000i q^{46} -2.00000i q^{47} +4.00000 q^{48} -3.00000 q^{49} +(-1.00000 - 4.00000i) q^{51} -2.00000i q^{52} +6.00000i q^{53} +2.00000i q^{54} +5.00000 q^{57} +12.0000 q^{58} -10.0000i q^{61} +20.0000 q^{62} +2.00000 q^{63} +8.00000 q^{64} -10.0000 q^{66} -12.0000i q^{67} +(-2.00000 - 8.00000i) q^{68} -1.00000 q^{69} +6.00000 q^{73} -4.00000i q^{74} +10.0000 q^{76} +10.0000i q^{77} -2.00000 q^{78} -4.00000i q^{79} +1.00000 q^{81} +10.0000 q^{82} +6.00000i q^{83} +4.00000 q^{84} +2.00000 q^{86} -6.00000i q^{87} +10.0000 q^{89} +2.00000i q^{91} -2.00000 q^{92} -10.0000i q^{93} -4.00000 q^{94} -8.00000i q^{96} -8.00000 q^{97} +6.00000i q^{98} +5.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{4} + 4 q^{7} + 2 q^{9} + 4 q^{12} - 8 q^{16} + 2 q^{17} - 10 q^{19} - 4 q^{21} + 20 q^{22} + 2 q^{23} + 4 q^{26} - 2 q^{27} - 8 q^{28} + 16 q^{34} - 4 q^{36} + 4 q^{37} + 8 q^{48} - 6 q^{49}+ \cdots - 16 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^4 + 4 * q^7 + 2 * q^9 + 4 * q^12 - 8 * q^16 + 2 * q^17 - 10 * q^19 - 4 * q^21 + 20 * q^22 + 2 * q^23 + 4 * q^26 - 2 * q^27 - 8 * q^28 + 16 * q^34 - 4 * q^36 + 4 * q^37 + 8 * q^48 - 6 * q^49 - 2 * q^51 + 10 * q^57 + 24 * q^58 + 40 * q^62 + 4 * q^63 + 16 * q^64 - 20 * q^66 - 4 * q^68 - 2 * q^69 + 12 * q^73 + 20 * q^76 - 4 * q^78 + 2 * q^81 + 20 * q^82 + 8 * q^84 + 4 * q^86 + 20 * q^89 - 4 * q^92 - 8 * q^94 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1275\mathbb{Z}\right)^\times$.

$n$	$52$	$751$	$851$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$		−	2.00000i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$		−	2.00000i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$	−2.00000			−1.00000
$5$		0			0
$5$		0			0
$6$			2.00000i			0.816497i
$7$	2.00000			0.755929			0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000			0.755929			0.377964	+	0.925820i	$0.376624\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			5.00000i			1.50756i	0.657129	+	0.753778i	$0.271771\pi$
$11$			5.00000i			1.50756i	−0.657129	+	0.753778i	$0.728229\pi$
$12$	2.00000			0.577350
$13$			1.00000i			0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$			1.00000i			0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$		−	4.00000i		−	1.06904i
$15$		0			0
$16$	−4.00000			−1.00000
$17$	1.00000	+	4.00000i	0.242536	+	0.970143i
$17$	1.00000	+	4.00000i	0.242536	+	0.970143i
$18$		−	2.00000i		−	0.471405i
$19$	−5.00000			−1.14708			−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000			−1.14708			−0.573539	+	0.819178i	$0.694430\pi$
$20$		0			0
$21$	−2.00000			−0.436436
$22$	10.0000			2.13201
$23$	1.00000			0.208514			0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000			0.208514			0.104257	+	0.994550i	$0.466753\pi$
$24$		0			0
$25$		0			0
$26$	2.00000			0.392232
$27$	−1.00000			−0.192450
$28$	−4.00000			−0.755929
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$			10.0000i			1.79605i	0.439941	+	0.898027i	$0.354999\pi$
$31$			10.0000i			1.79605i	−0.439941	+	0.898027i	$0.645001\pi$
$32$			8.00000i			1.41421i
$33$		−	5.00000i		−	0.870388i
$34$	8.00000	−	2.00000i	1.37199	−	0.342997i
$35$		0			0
$36$	−2.00000			−0.333333
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$			10.0000i			1.62221i
$39$		−	1.00000i		−	0.160128i
$40$		0			0
$41$			5.00000i			0.780869i	0.920631	+	0.390434i	$0.127675\pi$
$41$			5.00000i			0.780869i	−0.920631	+	0.390434i	$0.872325\pi$
$42$			4.00000i			0.617213i
$43$			1.00000i			0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$			1.00000i			0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$		−	10.0000i		−	1.50756i
$45$		0			0
$46$		−	2.00000i		−	0.294884i
$47$		−	2.00000i		−	0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$		−	2.00000i		−	0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	4.00000			0.577350
$49$	−3.00000			−0.428571
$50$		0			0
$51$	−1.00000	−	4.00000i	−0.140028	−	0.560112i
$52$		−	2.00000i		−	0.277350i
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$			2.00000i			0.272166i
$55$		0			0
$56$		0			0
$57$	5.00000			0.662266
$58$	12.0000			1.57568
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		−	10.0000i		−	1.28037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$		−	10.0000i		−	1.28037i	0.768221	−	0.640184i	$-0.221142\pi$
$62$	20.0000			2.54000
$63$	2.00000			0.251976
$64$	8.00000			1.00000
$65$		0			0
$66$	−10.0000			−1.23091
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	−2.00000	−	8.00000i	−0.242536	−	0.970143i
$69$	−1.00000			−0.120386
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	6.00000			0.702247			0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000			0.702247			0.351123	+	0.936329i	$0.385800\pi$
$74$		−	4.00000i		−	0.464991i
$75$		0			0
$76$	10.0000			1.14708
$77$			10.0000i			1.13961i
$78$	−2.00000			−0.226455
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$	10.0000			1.10432
$83$			6.00000i			0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$			6.00000i			0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	4.00000			0.436436
$85$		0			0
$86$	2.00000			0.215666
$87$		−	6.00000i		−	0.643268i
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$			2.00000i			0.209657i
$92$	−2.00000			−0.208514
$93$		−	10.0000i		−	1.03695i
$94$	−4.00000			−0.412568
$95$		0			0
$96$		−	8.00000i		−	0.816497i
$97$	−8.00000			−0.812277			−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000			−0.812277			−0.406138	+	0.913812i	$0.633125\pi$
$98$			6.00000i			0.606092i
$99$			5.00000i			0.502519i
$100$		0			0
$101$	12.0000			1.19404			0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000			1.19404			0.597022	+	0.802225i	$0.296350\pi$
$102$	−8.00000	+	2.00000i	−0.792118	+	0.198030i
$103$		−	9.00000i		−	0.886796i	−0.896325	−	0.443398i	$-0.853773\pi$
$103$		−	9.00000i		−	0.886796i	0.896325	−	0.443398i	$-0.146227\pi$
$104$		0			0
$105$		0			0
$106$	12.0000			1.16554
$107$	−3.00000			−0.290021			−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000			−0.290021			−0.145010	+	0.989430i	$0.546322\pi$
$108$	2.00000			0.192450
$109$		−	4.00000i		−	0.383131i	−0.981480	−	0.191565i	$-0.938644\pi$
$109$		−	4.00000i		−	0.383131i	0.981480	−	0.191565i	$-0.0613564\pi$
$110$		0			0
$111$	−2.00000			−0.189832
$112$	−8.00000			−0.755929
$113$	11.0000			1.03479			0.517396	−	0.855746i	$-0.326901\pi$
$113$	11.0000			1.03479			0.517396	+	0.855746i	$0.326901\pi$
$114$		−	10.0000i		−	0.936586i
$115$		0			0
$116$		−	12.0000i		−	1.11417i
$117$			1.00000i			0.0924500i
$118$		0			0
$119$	2.00000	+	8.00000i	0.183340	+	0.733359i
$120$		0			0
$121$	−14.0000			−1.27273
$122$	−20.0000			−1.81071
$123$		−	5.00000i		−	0.450835i
$124$		−	20.0000i		−	1.79605i
$125$		0			0
$126$		−	4.00000i		−	0.356348i
$127$		−	7.00000i		−	0.621150i	−0.950549	−	0.310575i	$-0.899478\pi$
$127$		−	7.00000i		−	0.621150i	0.950549	−	0.310575i	$-0.100522\pi$
$128$		0			0
$129$		−	1.00000i		−	0.0880451i
$130$		0			0
$131$			5.00000i			0.436852i	0.975854	+	0.218426i	$0.0700922\pi$
$131$			5.00000i			0.436852i	−0.975854	+	0.218426i	$0.929908\pi$
$132$			10.0000i			0.870388i
$133$	−10.0000			−0.867110
$134$	−24.0000			−2.07328
$135$		0			0
$136$		0			0
$137$			8.00000i			0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$			8.00000i			0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$			2.00000i			0.170251i
$139$			16.0000i			1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$139$			16.0000i			1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$140$		0			0
$141$			2.00000i			0.168430i
$142$		0			0
$143$	−5.00000			−0.418121
$144$	−4.00000			−0.333333
$145$		0			0
$146$		−	12.0000i		−	0.993127i
$147$	3.00000			0.247436
$148$	−4.00000			−0.328798
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	12.0000			0.976546			0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000			0.976546			0.488273	+	0.872691i	$0.337627\pi$
$152$		0			0
$153$	1.00000	+	4.00000i	0.0808452	+	0.323381i
$154$	20.0000			1.61165
$155$		0			0
$156$			2.00000i			0.160128i
$157$		−	17.0000i		−	1.35675i	−0.734717	−	0.678374i	$-0.762685\pi$
$157$		−	17.0000i		−	1.35675i	0.734717	−	0.678374i	$-0.237315\pi$
$158$	−8.00000			−0.636446
$159$		−	6.00000i		−	0.475831i
$160$		0			0
$161$	2.00000			0.157622
$162$		−	2.00000i		−	0.157135i
$163$	−14.0000			−1.09656			−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000			−1.09656			−0.548282	+	0.836293i	$0.684718\pi$
$164$		−	10.0000i		−	0.780869i
$165$		0			0
$166$	12.0000			0.931381
$167$	−3.00000			−0.232147			−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000			−0.232147			−0.116073	+	0.993241i	$0.537031\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$	−5.00000			−0.382360
$172$		−	2.00000i		−	0.152499i
$173$	1.00000			0.0760286			0.0380143	−	0.999277i	$-0.487897\pi$
$173$	1.00000			0.0760286			0.0380143	+	0.999277i	$0.487897\pi$
$174$	−12.0000			−0.909718
$175$		0			0
$176$		−	20.0000i		−	1.50756i
$177$		0			0
$178$		−	20.0000i		−	1.49906i
$179$	−10.0000			−0.747435			−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000			−0.747435			−0.373718	+	0.927543i	$0.621917\pi$
$180$		0			0
$181$			20.0000i			1.48659i	0.668965	+	0.743294i	$0.266738\pi$
$181$			20.0000i			1.48659i	−0.668965	+	0.743294i	$0.733262\pi$
$182$	4.00000			0.296500
$183$			10.0000i			0.739221i
$184$		0			0
$185$		0			0
$186$	−20.0000			−1.46647
$187$	−20.0000	+	5.00000i	−1.46254	+	0.365636i
$188$			4.00000i			0.291730i
$189$	−2.00000			−0.145479
$190$		0			0
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$	−8.00000			−0.577350
$193$	6.00000			0.431889			0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000			0.431889			0.215945	+	0.976406i	$0.430717\pi$
$194$			16.0000i			1.14873i
$195$		0			0
$196$	6.00000			0.428571
$197$	−23.0000			−1.63868			−0.819341	−	0.573306i	$-0.805660\pi$
$197$	−23.0000			−1.63868			−0.819341	+	0.573306i	$0.805660\pi$
$198$	10.0000			0.710669
$199$			16.0000i			1.13421i	0.823646	+	0.567105i	$0.191937\pi$
$199$			16.0000i			1.13421i	−0.823646	+	0.567105i	$0.808063\pi$
$200$		0			0
$201$			12.0000i			0.846415i
$202$		−	24.0000i		−	1.68863i
$203$			12.0000i			0.842235i
$204$	2.00000	+	8.00000i	0.140028	+	0.560112i
$205$		0			0
$206$	−18.0000			−1.25412
$207$	1.00000			0.0695048
$208$		−	4.00000i		−	0.277350i
$209$		−	25.0000i		−	1.72929i
$210$		0			0
$211$		−	10.0000i		−	0.688428i	−0.938891	−	0.344214i	$-0.888145\pi$
$211$		−	10.0000i		−	0.688428i	0.938891	−	0.344214i	$-0.111855\pi$
$212$		−	12.0000i		−	0.824163i
$213$		0			0
$214$			6.00000i			0.410152i
$215$		0			0
$216$		0			0
$217$			20.0000i			1.35769i
$218$	−8.00000			−0.541828
$219$	−6.00000			−0.405442
$220$		0			0
$221$	−4.00000	+	1.00000i	−0.269069	+	0.0672673i
$222$			4.00000i			0.268462i
$223$			11.0000i			0.736614i	0.929704	+	0.368307i	$0.120063\pi$
$223$			11.0000i			0.736614i	−0.929704	+	0.368307i	$0.879937\pi$
$224$			16.0000i			1.06904i
$225$		0			0
$226$		−	22.0000i		−	1.46342i
$227$	−3.00000			−0.199117			−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000			−0.199117			−0.0995585	+	0.995032i	$0.531743\pi$
$228$	−10.0000			−0.662266
$229$	−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
$231$		−	10.0000i		−	0.657952i
$232$		0			0
$233$	−29.0000			−1.89985			−0.949927	−	0.312473i	$-0.898843\pi$
$233$	−29.0000			−1.89985			−0.949927	+	0.312473i	$0.898843\pi$
$234$	2.00000			0.130744
$235$		0			0
$236$		0			0
$237$			4.00000i			0.259828i
$238$	16.0000	−	4.00000i	1.03713	−	0.259281i
$239$	20.0000			1.29369			0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000			1.29369			0.646846	+	0.762620i	$0.276088\pi$
$240$		0			0
$241$			10.0000i			0.644157i	0.946713	+	0.322078i	$0.104381\pi$
$241$			10.0000i			0.644157i	−0.946713	+	0.322078i	$0.895619\pi$
$242$			28.0000i			1.79991i
$243$	−1.00000			−0.0641500
$244$			20.0000i			1.28037i
$245$		0			0
$246$	−10.0000			−0.637577
$247$		−	5.00000i		−	0.318142i
$248$		0			0
$249$		−	6.00000i		−	0.380235i
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$	−4.00000			−0.251976
$253$			5.00000i			0.314347i
$254$	−14.0000			−0.878438
$255$		0			0
$256$	16.0000			1.00000
$257$		−	2.00000i		−	0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$		−	2.00000i		−	0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	−2.00000			−0.124515
$259$	4.00000			0.248548
$260$		0			0
$261$			6.00000i			0.371391i
$262$	10.0000			0.617802
$263$			6.00000i			0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$			6.00000i			0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$		0			0
$265$		0			0
$266$			20.0000i			1.22628i
$267$	−10.0000			−0.611990
$268$			24.0000i			1.46603i
$269$			21.0000i			1.28039i	0.768211	+	0.640196i	$0.221147\pi$
$269$			21.0000i			1.28039i	−0.768211	+	0.640196i	$0.778853\pi$
$270$		0			0
$271$	27.0000			1.64013			0.820067	−	0.572268i	$-0.193936\pi$
$271$	27.0000			1.64013			0.820067	+	0.572268i	$0.193936\pi$
$272$	−4.00000	−	16.0000i	−0.242536	−	0.970143i
$273$		−	2.00000i		−	0.121046i
$274$	16.0000			0.966595
$275$		0			0
$276$	2.00000			0.120386
$277$	22.0000			1.32185			0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000			1.32185			0.660926	+	0.750451i	$0.270164\pi$
$278$	32.0000			1.91923
$279$			10.0000i			0.598684i
$280$		0			0
$281$	22.0000			1.31241			0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000			1.31241			0.656205	+	0.754583i	$0.272161\pi$
$282$	4.00000			0.238197
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$		0			0
$285$		0			0
$286$			10.0000i			0.591312i
$287$			10.0000i			0.590281i
$288$			8.00000i			0.471405i
$289$	−15.0000	+	8.00000i	−0.882353	+	0.470588i
$290$		0			0
$291$	8.00000			0.468968
$292$	−12.0000			−0.702247
$293$		−	14.0000i		−	0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$		−	14.0000i		−	0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$		−	6.00000i		−	0.349927i
$295$		0			0
$296$		0			0
$297$		−	5.00000i		−	0.290129i
$298$		0			0
$299$			1.00000i			0.0578315i
$300$		0			0
$301$			2.00000i			0.115278i
$302$		−	24.0000i		−	1.38104i
$303$	−12.0000			−0.689382
$304$	20.0000			1.14708
$305$		0			0
$306$	8.00000	−	2.00000i	0.457330	−	0.114332i
$307$		−	12.0000i		−	0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$		−	12.0000i		−	0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$		−	20.0000i		−	1.13961i
$309$			9.00000i			0.511992i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	16.0000			0.904373			0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000			0.904373			0.452187	+	0.891923i	$0.350644\pi$
$314$	−34.0000			−1.91873
$315$		0			0
$316$			8.00000i			0.450035i
$317$	22.0000			1.23564			0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000			1.23564			0.617822	+	0.786318i	$0.288015\pi$
$318$	−12.0000			−0.672927
$319$	−30.0000			−1.67968
$320$		0			0
$321$	3.00000			0.167444
$322$		−	4.00000i		−	0.222911i
$323$	−5.00000	−	20.0000i	−0.278207	−	1.11283i
$324$	−2.00000			−0.111111
$325$		0			0
$326$			28.0000i			1.55078i
$327$			4.00000i			0.221201i
$328$		0			0
$329$		−	4.00000i		−	0.220527i
$330$		0			0
$331$	−23.0000			−1.26419			−0.632097	−	0.774889i	$-0.717806\pi$
$331$	−23.0000			−1.26419			−0.632097	+	0.774889i	$0.717806\pi$
$332$		−	12.0000i		−	0.658586i
$333$	2.00000			0.109599
$334$			6.00000i			0.328305i
$335$		0			0
$336$	8.00000			0.436436
$337$	32.0000			1.74315			0.871576	−	0.490261i	$-0.163099\pi$
$337$	32.0000			1.74315			0.871576	+	0.490261i	$0.163099\pi$
$338$		−	24.0000i		−	1.30543i
$339$	−11.0000			−0.597438
$340$		0			0
$341$	−50.0000			−2.70765
$342$			10.0000i			0.540738i
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$		−	2.00000i		−	0.107521i
$347$	−8.00000			−0.429463			−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000			−0.429463			−0.214731	+	0.976673i	$0.568888\pi$
$348$			12.0000i			0.643268i
$349$	5.00000			0.267644			0.133822	−	0.991005i	$-0.457275\pi$
$349$	5.00000			0.267644			0.133822	+	0.991005i	$0.457275\pi$
$350$		0			0
$351$		−	1.00000i		−	0.0533761i
$352$	−40.0000			−2.13201
$353$		−	24.0000i		−	1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$		−	24.0000i		−	1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$		0			0
$355$		0			0
$356$	−20.0000			−1.06000
$357$	−2.00000	−	8.00000i	−0.105851	−	0.423405i
$358$			20.0000i			1.05703i
$359$	10.0000			0.527780			0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000			0.527780			0.263890	+	0.964553i	$0.414994\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$	40.0000			2.10235
$363$	14.0000			0.734809
$364$		−	4.00000i		−	0.209657i
$365$		0			0
$366$	20.0000			1.04542
$367$	−18.0000			−0.939592			−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000			−0.939592			−0.469796	+	0.882775i	$0.655673\pi$
$368$	−4.00000			−0.208514
$369$			5.00000i			0.260290i
$370$		0			0
$371$			12.0000i			0.623009i
$372$			20.0000i			1.03695i
$373$		−	14.0000i		−	0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$		−	14.0000i		−	0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	10.0000	+	40.0000i	0.517088	+	2.06835i
$375$		0			0
$376$		0			0
$377$	−6.00000			−0.309016
$378$			4.00000i			0.205738i
$379$		−	34.0000i		−	1.74646i	−0.487306	−	0.873231i	$-0.662020\pi$
$379$		−	34.0000i		−	1.74646i	0.487306	−	0.873231i	$-0.337980\pi$
$380$		0			0
$381$			7.00000i			0.358621i
$382$			16.0000i			0.818631i
$383$			16.0000i			0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$			16.0000i			0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$		0			0
$385$		0			0
$386$		−	12.0000i		−	0.610784i
$387$			1.00000i			0.0508329i
$388$	16.0000			0.812277
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$	1.00000	+	4.00000i	0.0505722	+	0.202289i
$392$		0			0
$393$		−	5.00000i		−	0.252217i
$394$			46.0000i			2.31745i
$395$		0			0
$396$		−	10.0000i		−	0.502519i
$397$	−8.00000			−0.401508			−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000			−0.401508			−0.200754	+	0.979642i	$0.564339\pi$
$398$	32.0000			1.60402
$399$	10.0000			0.500626
$400$		0			0
$401$		−	15.0000i		−	0.749064i	−0.927214	−	0.374532i	$-0.877803\pi$
$401$		−	15.0000i		−	0.749064i	0.927214	−	0.374532i	$-0.122197\pi$
$402$	24.0000			1.19701
$403$	−10.0000			−0.498135
$404$	−24.0000			−1.19404
$405$		0			0
$406$	24.0000			1.19110
$407$			10.0000i			0.495682i
$408$		0			0
$409$	−25.0000			−1.23617			−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000			−1.23617			−0.618085	+	0.786111i	$0.712091\pi$
$410$		0			0
$411$		−	8.00000i		−	0.394611i
$412$			18.0000i			0.886796i
$413$		0			0
$414$		−	2.00000i		−	0.0982946i
$415$		0			0
$416$	−8.00000			−0.392232
$417$		−	16.0000i		−	0.783523i
$418$	−50.0000			−2.44558
$419$			16.0000i			0.781651i	0.920465	+	0.390826i	$0.127810\pi$
$419$			16.0000i			0.781651i	−0.920465	+	0.390826i	$0.872190\pi$
$420$		0			0
$421$	−13.0000			−0.633581			−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000			−0.633581			−0.316791	+	0.948495i	$0.602605\pi$
$422$	−20.0000			−0.973585
$423$		−	2.00000i		−	0.0972433i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	20.0000i		−	0.967868i
$428$	6.00000			0.290021
$429$	5.00000			0.241402
$430$		0			0
$431$			20.0000i			0.963366i	0.876346	+	0.481683i	$0.159974\pi$
$431$			20.0000i			0.963366i	−0.876346	+	0.481683i	$0.840026\pi$
$432$	4.00000			0.192450
$433$			11.0000i			0.528626i	0.964437	+	0.264313i	$0.0851452\pi$
$433$			11.0000i			0.528626i	−0.964437	+	0.264313i	$0.914855\pi$
$434$	40.0000			1.92006
$435$		0			0
$436$			8.00000i			0.383131i
$437$	−5.00000			−0.239182
$438$			12.0000i			0.573382i
$439$		−	24.0000i		−	1.14546i	−0.819745	−	0.572729i	$-0.805885\pi$
$439$		−	24.0000i		−	1.14546i	0.819745	−	0.572729i	$-0.194115\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$	2.00000	+	8.00000i	0.0951303	+	0.380521i
$443$		−	34.0000i		−	1.61539i	−0.589601	−	0.807694i	$-0.700715\pi$
$443$		−	34.0000i		−	1.61539i	0.589601	−	0.807694i	$-0.299285\pi$
$444$	4.00000			0.189832
$445$		0			0
$446$	22.0000			1.04173
$447$		0			0
$448$	16.0000			0.755929
$449$		−	34.0000i		−	1.60456i	−0.596948	−	0.802280i	$-0.703620\pi$
$449$		−	34.0000i		−	1.60456i	0.596948	−	0.802280i	$-0.296380\pi$
$450$		0			0
$451$	−25.0000			−1.17720
$452$	−22.0000			−1.03479
$453$	−12.0000			−0.563809
$454$			6.00000i			0.281594i
$455$		0			0
$456$		0			0
$457$			23.0000i			1.07589i	0.842978	+	0.537947i	$0.180800\pi$
$457$			23.0000i			1.07589i	−0.842978	+	0.537947i	$0.819200\pi$
$458$			20.0000i			0.934539i
$459$	−1.00000	−	4.00000i	−0.0466760	−	0.186704i
$460$		0			0
$461$	32.0000			1.49039			0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000			1.49039			0.745194	+	0.666847i	$0.232357\pi$
$462$	−20.0000			−0.930484
$463$			16.0000i			0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$			16.0000i			0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$		−	24.0000i		−	1.11417i
$465$		0			0
$466$			58.0000i			2.68680i
$467$			28.0000i			1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$			28.0000i			1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$		−	2.00000i		−	0.0924500i
$469$		−	24.0000i		−	1.10822i
$470$		0			0
$471$			17.0000i			0.783319i
$472$		0			0
$473$	−5.00000			−0.229900
$474$	8.00000			0.367452
$475$		0			0
$476$	−4.00000	−	16.0000i	−0.183340	−	0.733359i
$477$			6.00000i			0.274721i
$478$		−	40.0000i		−	1.82956i
$479$			21.0000i			0.959514i	0.877401	+	0.479757i	$0.159275\pi$
$479$			21.0000i			0.959514i	−0.877401	+	0.479757i	$0.840725\pi$
$480$		0			0
$481$			2.00000i			0.0911922i
$482$	20.0000			0.910975
$483$	−2.00000			−0.0910032
$484$	28.0000			1.27273
$485$		0			0
$486$			2.00000i			0.0907218i
$487$	32.0000			1.45006			0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000			1.45006			0.725029	+	0.688718i	$0.241826\pi$
$488$		0			0
$489$	14.0000			0.633102
$490$		0			0
$491$	−38.0000			−1.71492			−0.857458	−	0.514554i	$-0.827958\pi$
$491$	−38.0000			−1.71492			−0.857458	+	0.514554i	$0.827958\pi$
$492$			10.0000i			0.450835i
$493$	−24.0000	+	6.00000i	−1.08091	+	0.270226i
$494$	−10.0000			−0.449921
$495$		0			0
$496$		−	40.0000i		−	1.79605i
$497$		0			0
$498$	−12.0000			−0.537733
$499$			6.00000i			0.268597i	0.990941	+	0.134298i	$0.0428781\pi$
$499$			6.00000i			0.268597i	−0.990941	+	0.134298i	$0.957122\pi$
$500$		0			0
$501$	3.00000			0.134030
$502$			36.0000i			1.60676i
$503$	−9.00000			−0.401290			−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000			−0.401290			−0.200645	+	0.979664i	$0.564304\pi$
$504$		0			0
$505$		0			0
$506$	10.0000			0.444554
$507$	−12.0000			−0.532939
$508$			14.0000i			0.621150i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	12.0000			0.530849
$512$		−	32.0000i		−	1.41421i
$513$	5.00000			0.220755
$514$	−4.00000			−0.176432
$515$		0			0
$516$			2.00000i			0.0880451i
$517$	10.0000			0.439799
$518$		−	8.00000i		−	0.351500i
$519$	−1.00000			−0.0438951
$520$		0			0
$521$		−	15.0000i		−	0.657162i	−0.944476	−	0.328581i	$-0.893430\pi$
$521$		−	15.0000i		−	0.657162i	0.944476	−	0.328581i	$-0.106570\pi$
$522$	12.0000			0.525226
$523$			36.0000i			1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$			36.0000i			1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$		−	10.0000i		−	0.436852i
$525$		0			0
$526$	12.0000			0.523225
$527$	−40.0000	+	10.0000i	−1.74243	+	0.435607i
$528$			20.0000i			0.870388i
$529$	−22.0000			−0.956522
$530$		0			0
$531$		0			0
$532$	20.0000			0.867110
$533$	−5.00000			−0.216574
$534$			20.0000i			0.865485i
$535$		0			0
$536$		0			0
$537$	10.0000			0.431532
$538$	42.0000			1.81075
$539$		−	15.0000i		−	0.646096i
$540$		0			0
$541$		−	10.0000i		−	0.429934i	−0.976621	−	0.214967i	$-0.931036\pi$
$541$		−	10.0000i		−	0.429934i	0.976621	−	0.214967i	$-0.0689643\pi$
$542$		−	54.0000i		−	2.31950i
$543$		−	20.0000i		−	0.858282i
$544$	−32.0000	+	8.00000i	−1.37199	+	0.342997i
$545$		0			0
$546$	−4.00000			−0.171184
$547$	−18.0000			−0.769624			−0.384812	−	0.922995i	$-0.625734\pi$
$547$	−18.0000			−0.769624			−0.384812	+	0.922995i	$0.625734\pi$
$548$		−	16.0000i		−	0.683486i
$549$		−	10.0000i		−	0.426790i
$550$		0			0
$551$		−	30.0000i		−	1.27804i
$552$		0			0
$553$		−	8.00000i		−	0.340195i
$554$		−	44.0000i		−	1.86938i
$555$		0			0
$556$		−	32.0000i		−	1.35710i
$557$		−	12.0000i		−	0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$		−	12.0000i		−	0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$	20.0000			0.846668
$559$	−1.00000			−0.0422955
$560$		0			0
$561$	20.0000	−	5.00000i	0.844401	−	0.211100i
$562$		−	44.0000i		−	1.85603i
$563$		−	14.0000i		−	0.590030i	−0.955493	−	0.295015i	$-0.904675\pi$
$563$		−	14.0000i		−	0.590030i	0.955493	−	0.295015i	$-0.0953246\pi$
$564$		−	4.00000i		−	0.168430i
$565$		0			0
$566$			8.00000i			0.336265i
$567$	2.00000			0.0839921
$568$		0			0
$569$	40.0000			1.67689			0.838444	−	0.544988i	$-0.183466\pi$
$569$	40.0000			1.67689			0.838444	+	0.544988i	$0.183466\pi$
$570$		0			0
$571$		−	20.0000i		−	0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$		−	20.0000i		−	0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$	10.0000			0.418121
$573$	8.00000			0.334205
$574$	20.0000			0.834784
$575$		0			0
$576$	8.00000			0.333333
$577$		−	27.0000i		−	1.12402i	−0.827129	−	0.562012i	$-0.810027\pi$
$577$		−	27.0000i		−	1.12402i	0.827129	−	0.562012i	$-0.189973\pi$
$578$	16.0000	+	30.0000i	0.665512	+	1.24784i
$579$	−6.00000			−0.249351
$580$		0			0
$581$			12.0000i			0.497844i
$582$		−	16.0000i		−	0.663221i
$583$	−30.0000			−1.24247
$584$		0			0
$585$		0			0
$586$	−28.0000			−1.15667
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	−6.00000			−0.247436
$589$		−	50.0000i		−	2.06021i
$590$		0			0
$591$	23.0000			0.946094
$592$	−8.00000			−0.328798
$593$		−	14.0000i		−	0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$		−	14.0000i		−	0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$	−10.0000			−0.410305
$595$		0			0
$596$		0			0
$597$		−	16.0000i		−	0.654836i
$598$	2.00000			0.0817861
$599$	20.0000			0.817178			0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000			0.817178			0.408589	+	0.912719i	$0.366021\pi$
$600$		0			0
$601$		−	10.0000i		−	0.407909i	−0.978980	−	0.203954i	$-0.934621\pi$
$601$		−	10.0000i		−	0.407909i	0.978980	−	0.203954i	$-0.0653794\pi$
$602$	4.00000			0.163028
$603$		−	12.0000i		−	0.488678i
$604$	−24.0000			−0.976546
$605$		0			0
$606$			24.0000i			0.974933i
$607$	−28.0000			−1.13648			−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000			−1.13648			−0.568242	+	0.822861i	$0.692376\pi$
$608$		−	40.0000i		−	1.62221i
$609$		−	12.0000i		−	0.486265i
$610$		0			0
$611$	2.00000			0.0809113
$612$	−2.00000	−	8.00000i	−0.0808452	−	0.323381i
$613$			21.0000i			0.848182i	0.905620	+	0.424091i	$0.139406\pi$
$613$			21.0000i			0.848182i	−0.905620	+	0.424091i	$0.860594\pi$
$614$	−24.0000			−0.968561
$615$		0			0
$616$		0			0
$617$	42.0000			1.69086			0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000			1.69086			0.845428	+	0.534089i	$0.179345\pi$
$618$	18.0000			0.724066
$619$			36.0000i			1.44696i	0.690344	+	0.723481i	$0.257459\pi$
$619$			36.0000i			1.44696i	−0.690344	+	0.723481i	$0.742541\pi$
$620$		0			0
$621$	−1.00000			−0.0401286
$622$		0			0
$623$	20.0000			0.801283
$624$			4.00000i			0.160128i
$625$		0			0
$626$		−	32.0000i		−	1.27898i
$627$			25.0000i			0.998404i
$628$			34.0000i			1.35675i
$629$	2.00000	+	8.00000i	0.0797452	+	0.318981i
$630$		0			0
$631$	−23.0000			−0.915616			−0.457808	−	0.889051i	$-0.651365\pi$
$631$	−23.0000			−0.915616			−0.457808	+	0.889051i	$0.651365\pi$
$632$		0			0
$633$			10.0000i			0.397464i
$634$		−	44.0000i		−	1.74746i
$635$		0			0
$636$			12.0000i			0.475831i
$637$		−	3.00000i		−	0.118864i
$638$			60.0000i			2.37542i
$639$		0			0
$640$		0			0
$641$		−	35.0000i		−	1.38242i	−0.722655	−	0.691208i	$-0.757079\pi$
$641$		−	35.0000i		−	1.38242i	0.722655	−	0.691208i	$-0.242921\pi$
$642$		−	6.00000i		−	0.236801i
$643$	−4.00000			−0.157745			−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000			−0.157745			−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−4.00000			−0.157622
$645$		0			0
$646$	−40.0000	+	10.0000i	−1.57378	+	0.393445i
$647$			8.00000i			0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$			8.00000i			0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		−	20.0000i		−	0.783862i
$652$	28.0000			1.09656
$653$	11.0000			0.430463			0.215232	−	0.976563i	$-0.430949\pi$
$653$	11.0000			0.430463			0.215232	+	0.976563i	$0.430949\pi$
$654$	8.00000			0.312825
$655$		0			0
$656$		−	20.0000i		−	0.780869i
$657$	6.00000			0.234082
$658$	−8.00000			−0.311872
$659$	−20.0000			−0.779089			−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000			−0.779089			−0.389545	+	0.921008i	$0.627368\pi$
$660$		0			0
$661$	27.0000			1.05018			0.525089	−	0.851047i	$-0.324032\pi$
$661$	27.0000			1.05018			0.525089	+	0.851047i	$0.324032\pi$
$662$			46.0000i			1.78784i
$663$	4.00000	−	1.00000i	0.155347	−	0.0388368i
$664$		0			0
$665$		0			0
$666$		−	4.00000i		−	0.154997i
$667$			6.00000i			0.232321i
$668$	6.00000			0.232147
$669$		−	11.0000i		−	0.425285i
$670$		0			0
$671$	50.0000			1.93023
$672$		−	16.0000i		−	0.617213i
$673$	46.0000			1.77317			0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000			1.77317			0.886585	+	0.462566i	$0.153071\pi$
$674$		−	64.0000i		−	2.46519i
$675$		0			0
$676$	−24.0000			−0.923077
$677$	−3.00000			−0.115299			−0.0576497	−	0.998337i	$-0.518361\pi$
$677$	−3.00000			−0.115299			−0.0576497	+	0.998337i	$0.518361\pi$
$678$			22.0000i			0.844905i
$679$	−16.0000			−0.614024
$680$		0			0
$681$	3.00000			0.114960
$682$			100.000i			3.82920i
$683$	21.0000			0.803543			0.401771	−	0.915740i	$-0.368395\pi$
$683$	21.0000			0.803543			0.401771	+	0.915740i	$0.368395\pi$
$684$	10.0000			0.382360
$685$		0			0
$686$			40.0000i			1.52721i
$687$	10.0000			0.381524
$688$		−	4.00000i		−	0.152499i
$689$	−6.00000			−0.228582
$690$		0			0
$691$		−	20.0000i		−	0.760836i	−0.924815	−	0.380418i	$-0.875780\pi$
$691$		−	20.0000i		−	0.760836i	0.924815	−	0.380418i	$-0.124220\pi$
$692$	−2.00000			−0.0760286
$693$			10.0000i			0.379869i
$694$			16.0000i			0.607352i
$695$		0			0
$696$		0			0
$697$	−20.0000	+	5.00000i	−0.757554	+	0.189389i
$698$		−	10.0000i		−	0.378506i
$699$	29.0000			1.09688
$700$		0			0
$701$	−8.00000			−0.302156			−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000			−0.302156			−0.151078	+	0.988522i	$0.548274\pi$
$702$	−2.00000			−0.0754851
$703$	−10.0000			−0.377157
$704$			40.0000i			1.50756i
$705$		0			0
$706$	−48.0000			−1.80650
$707$	24.0000			0.902613
$708$		0			0
$709$		−	34.0000i		−	1.27690i	−0.769665	−	0.638448i	$-0.779577\pi$
$709$		−	34.0000i		−	1.27690i	0.769665	−	0.638448i	$-0.220423\pi$
$710$		0			0
$711$		−	4.00000i		−	0.150012i
$712$		0			0
$713$			10.0000i			0.374503i
$714$	−16.0000	+	4.00000i	−0.598785	+	0.149696i
$715$		0			0
$716$	20.0000			0.747435
$717$	−20.0000			−0.746914
$718$		−	20.0000i		−	0.746393i
$719$		−	9.00000i		−	0.335643i	−0.985817	−	0.167822i	$-0.946327\pi$
$719$		−	9.00000i		−	0.335643i	0.985817	−	0.167822i	$-0.0536733\pi$
$720$		0			0
$721$		−	18.0000i		−	0.670355i
$722$		−	12.0000i		−	0.446594i
$723$		−	10.0000i		−	0.371904i
$724$		−	40.0000i		−	1.48659i
$725$		0			0
$726$		−	28.0000i		−	1.03918i
$727$			8.00000i			0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$			8.00000i			0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−4.00000	+	1.00000i	−0.147945	+	0.0369863i
$732$		−	20.0000i		−	0.739221i
$733$		−	14.0000i		−	0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$		−	14.0000i		−	0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$			36.0000i			1.32878i
$735$		0			0
$736$			8.00000i			0.294884i
$737$	60.0000			2.21013
$738$	10.0000			0.368105
$739$	−5.00000			−0.183928			−0.0919640	−	0.995762i	$-0.529314\pi$
$739$	−5.00000			−0.183928			−0.0919640	+	0.995762i	$0.529314\pi$
$740$		0			0
$741$			5.00000i			0.183680i
$742$	24.0000			0.881068
$743$	−24.0000			−0.880475			−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000			−0.880475			−0.440237	+	0.897881i	$0.645106\pi$
$744$		0			0
$745$		0			0
$746$	−28.0000			−1.02515
$747$			6.00000i			0.219529i
$748$	40.0000	−	10.0000i	1.46254	−	0.365636i
$749$	−6.00000			−0.219235
$750$		0			0
$751$		−	10.0000i		−	0.364905i	−0.983215	−	0.182453i	$-0.941596\pi$
$751$		−	10.0000i		−	0.364905i	0.983215	−	0.182453i	$-0.0584036\pi$
$752$			8.00000i			0.291730i
$753$	18.0000			0.655956
$754$			12.0000i			0.437014i
$755$		0			0
$756$	4.00000			0.145479
$757$			43.0000i			1.56286i	0.623992	+	0.781431i	$0.285510\pi$
$757$			43.0000i			1.56286i	−0.623992	+	0.781431i	$0.714490\pi$
$758$	−68.0000			−2.46987
$759$		−	5.00000i		−	0.181489i
$760$		0			0
$761$	42.0000			1.52250			0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000			1.52250			0.761249	+	0.648459i	$0.224586\pi$
$762$	14.0000			0.507166
$763$		−	8.00000i		−	0.289619i
$764$	16.0000			0.578860
$765$		0			0
$766$	32.0000			1.15621
$767$		0			0
$768$	−16.0000			−0.577350
$769$	−5.00000			−0.180305			−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000			−0.180305			−0.0901523	+	0.995928i	$0.528735\pi$
$770$		0			0
$771$			2.00000i			0.0720282i
$772$	−12.0000			−0.431889
$773$			36.0000i			1.29483i	0.762138	+	0.647415i	$0.224150\pi$
$773$			36.0000i			1.29483i	−0.762138	+	0.647415i	$0.775850\pi$
$774$	2.00000			0.0718885
$775$		0			0
$776$		0			0
$777$	−4.00000			−0.143499
$778$		−	60.0000i		−	2.15110i
$779$		−	25.0000i		−	0.895718i
$780$		0			0
$781$		0			0
$782$	8.00000	−	2.00000i	0.286079	−	0.0715199i
$783$		−	6.00000i		−	0.214423i
$784$	12.0000			0.428571
$785$		0			0
$786$	−10.0000			−0.356688
$787$	12.0000			0.427754			0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000			0.427754			0.213877	+	0.976861i	$0.431391\pi$
$788$	46.0000			1.63868
$789$		−	6.00000i		−	0.213606i
$790$		0			0
$791$	22.0000			0.782230
$792$		0			0
$793$	10.0000			0.355110
$794$			16.0000i			0.567819i
$795$		0			0
$796$		−	32.0000i		−	1.13421i
$797$		−	12.0000i		−	0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$		−	12.0000i		−	0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$		−	20.0000i		−	0.707992i
$799$	8.00000	−	2.00000i	0.283020	−	0.0707549i
$800$		0			0
$801$	10.0000			0.353333
$802$	−30.0000			−1.05934
$803$			30.0000i			1.05868i
$804$		−	24.0000i		−	0.846415i
$805$		0			0
$806$			20.0000i			0.704470i
$807$		−	21.0000i		−	0.739235i
$808$		0			0
$809$			11.0000i			0.386739i	0.981126	+	0.193370i	$0.0619417\pi$
$809$			11.0000i			0.386739i	−0.981126	+	0.193370i	$0.938058\pi$
$810$		0			0
$811$			50.0000i			1.75574i	0.478901	+	0.877869i	$0.341035\pi$
$811$			50.0000i			1.75574i	−0.478901	+	0.877869i	$0.658965\pi$
$812$		−	24.0000i		−	0.842235i
$813$	−27.0000			−0.946931
$814$	20.0000			0.701000
$815$		0			0
$816$	4.00000	+	16.0000i	0.140028	+	0.560112i
$817$		−	5.00000i		−	0.174928i
$818$			50.0000i			1.74821i
$819$			2.00000i			0.0698857i
$820$		0			0
$821$			25.0000i			0.872506i	0.899824	+	0.436253i	$0.143695\pi$
$821$			25.0000i			0.872506i	−0.899824	+	0.436253i	$0.856305\pi$
$822$	−16.0000			−0.558064
$823$	46.0000			1.60346			0.801730	−	0.597687i	$-0.203913\pi$
$823$	46.0000			1.60346			0.801730	+	0.597687i	$0.203913\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.00000			−0.104320			−0.0521601	−	0.998639i	$-0.516611\pi$
$827$	−3.00000			−0.104320			−0.0521601	+	0.998639i	$0.516611\pi$
$828$	−2.00000			−0.0695048
$829$	50.0000			1.73657			0.868286	−	0.496064i	$-0.165222\pi$
$829$	50.0000			1.73657			0.868286	+	0.496064i	$0.165222\pi$
$830$		0			0
$831$	−22.0000			−0.763172
$832$			8.00000i			0.277350i
$833$	−3.00000	−	12.0000i	−0.103944	−	0.415775i
$834$	−32.0000			−1.10807
$835$		0			0
$836$			50.0000i			1.72929i
$837$		−	10.0000i		−	0.345651i
$838$	32.0000			1.10542
$839$			41.0000i			1.41548i	0.706474	+	0.707739i	$0.250285\pi$
$839$			41.0000i			1.41548i	−0.706474	+	0.707739i	$0.749715\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$			26.0000i			0.896019i
$843$	−22.0000			−0.757720
$844$			20.0000i			0.688428i
$845$		0			0
$846$	−4.00000			−0.137523
$847$	−28.0000			−0.962091
$848$		−	24.0000i		−	0.824163i
$849$	4.00000			0.137280
$850$		0			0
$851$	2.00000			0.0685591
$852$		0			0
$853$	36.0000			1.23262			0.616308	−	0.787505i	$-0.288628\pi$
$853$	36.0000			1.23262			0.616308	+	0.787505i	$0.288628\pi$
$854$	−40.0000			−1.36877
$855$		0			0
$856$		0			0
$857$	2.00000			0.0683187			0.0341593	−	0.999416i	$-0.489125\pi$
$857$	2.00000			0.0683187			0.0341593	+	0.999416i	$0.489125\pi$
$858$		−	10.0000i		−	0.341394i
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$		−	10.0000i		−	0.340799i
$862$	40.0000			1.36241
$863$			6.00000i			0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$			6.00000i			0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$		−	8.00000i		−	0.272166i
$865$		0			0
$866$	22.0000			0.747590
$867$	15.0000	−	8.00000i	0.509427	−	0.271694i
$868$		−	40.0000i		−	1.35769i
$869$	20.0000			0.678454
$870$		0			0
$871$	12.0000			0.406604
$872$		0			0
$873$	−8.00000			−0.270759
$874$			10.0000i			0.338255i
$875$		0			0
$876$	12.0000			0.405442
$877$	32.0000			1.08056			0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000			1.08056			0.540282	+	0.841484i	$0.318318\pi$
$878$	−48.0000			−1.61992
$879$			14.0000i			0.472208i
$880$		0			0
$881$			30.0000i			1.01073i	0.862907	+	0.505363i	$0.168641\pi$
$881$			30.0000i			1.01073i	−0.862907	+	0.505363i	$0.831359\pi$
$882$			6.00000i			0.202031i
$883$		−	19.0000i		−	0.639401i	−0.947519	−	0.319700i	$-0.896418\pi$
$883$		−	19.0000i		−	0.639401i	0.947519	−	0.319700i	$-0.103582\pi$
$884$	8.00000	−	2.00000i	0.269069	−	0.0672673i
$885$		0			0
$886$	−68.0000			−2.28450
$887$	−13.0000			−0.436497			−0.218249	−	0.975893i	$-0.570034\pi$
$887$	−13.0000			−0.436497			−0.218249	+	0.975893i	$0.570034\pi$
$888$		0			0
$889$		−	14.0000i		−	0.469545i
$890$		0			0
$891$			5.00000i			0.167506i
$892$		−	22.0000i		−	0.736614i
$893$			10.0000i			0.334637i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	1.00000i		−	0.0333890i
$898$	−68.0000			−2.26919
$899$	−60.0000			−2.00111
$900$		0			0
$901$	−24.0000	+	6.00000i	−0.799556	+	0.199889i
$902$			50.0000i			1.66482i
$903$		−	2.00000i		−	0.0665558i
$904$		0			0
$905$		0			0
$906$			24.0000i			0.797347i
$907$	32.0000			1.06254			0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000			1.06254			0.531271	+	0.847202i	$0.321714\pi$
$908$	6.00000			0.199117
$909$	12.0000			0.398015
$910$		0			0
$911$			5.00000i			0.165657i	0.996564	+	0.0828287i	$0.0263954\pi$
$911$			5.00000i			0.165657i	−0.996564	+	0.0828287i	$0.973605\pi$
$912$	−20.0000			−0.662266
$913$	−30.0000			−0.992855
$914$	46.0000			1.52154
$915$		0			0
$916$	20.0000			0.660819
$917$			10.0000i			0.330229i
$918$	−8.00000	+	2.00000i	−0.264039	+	0.0660098i
$919$	−15.0000			−0.494804			−0.247402	−	0.968913i	$-0.579577\pi$
$919$	−15.0000			−0.494804			−0.247402	+	0.968913i	$0.579577\pi$
$920$		0			0
$921$			12.0000i			0.395413i
$922$		−	64.0000i		−	2.10773i
$923$		0			0
$924$			20.0000i			0.657952i
$925$		0			0
$926$	32.0000			1.05159
$927$		−	9.00000i		−	0.295599i
$928$	−48.0000			−1.57568
$929$			21.0000i			0.688988i	0.938789	+	0.344494i	$0.111949\pi$
$929$			21.0000i			0.688988i	−0.938789	+	0.344494i	$0.888051\pi$
$930$		0			0
$931$	15.0000			0.491605
$932$	58.0000			1.89985
$933$		0			0
$934$	56.0000			1.83238
$935$		0			0
$936$		0			0
$937$			38.0000i			1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$			38.0000i			1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$	−48.0000			−1.56726
$939$	−16.0000			−0.522140
$940$		0			0
$941$			30.0000i			0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$			30.0000i			0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$	34.0000			1.10778
$943$			5.00000i			0.162822i
$944$		0			0
$945$		0			0
$946$			10.0000i			0.325128i
$947$	−48.0000			−1.55979			−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000			−1.55979			−0.779895	+	0.625910i	$0.784728\pi$
$948$		−	8.00000i		−	0.259828i
$949$			6.00000i			0.194768i
$950$		0			0
$951$	−22.0000			−0.713399
$952$		0			0
$953$			36.0000i			1.16615i	0.812417	+	0.583077i	$0.198151\pi$
$953$			36.0000i			1.16615i	−0.812417	+	0.583077i	$0.801849\pi$
$954$	12.0000			0.388514
$955$		0			0
$956$	−40.0000			−1.29369
$957$	30.0000			0.969762
$958$	42.0000			1.35696
$959$			16.0000i			0.516667i
$960$		0			0
$961$	−69.0000			−2.22581
$962$	4.00000			0.128965
$963$	−3.00000			−0.0966736
$964$		−	20.0000i		−	0.644157i
$965$		0			0
$966$			4.00000i			0.128698i
$967$		−	57.0000i		−	1.83300i	−0.400039	−	0.916498i	$-0.631003\pi$
$967$		−	57.0000i		−	1.83300i	0.400039	−	0.916498i	$-0.368997\pi$
$968$		0			0
$969$	5.00000	+	20.0000i	0.160623	+	0.642493i
$970$		0			0
$971$	42.0000			1.34784			0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000			1.34784			0.673922	+	0.738802i	$0.264608\pi$
$972$	2.00000			0.0641500
$973$			32.0000i			1.02587i
$974$		−	64.0000i		−	2.05069i
$975$		0			0
$976$			40.0000i			1.28037i
$977$			8.00000i			0.255943i	0.991778	+	0.127971i	$0.0408466\pi$
$977$			8.00000i			0.255943i	−0.991778	+	0.127971i	$0.959153\pi$
$978$		−	28.0000i		−	0.895341i
$979$			50.0000i			1.59801i
$980$		0			0
$981$		−	4.00000i		−	0.127710i
$982$			76.0000i			2.42526i
$983$	−19.0000			−0.606006			−0.303003	−	0.952990i	$-0.597989\pi$
$983$	−19.0000			−0.606006			−0.303003	+	0.952990i	$0.597989\pi$
$984$		0			0
$985$		0			0
$986$	12.0000	+	48.0000i	0.382158	+	1.52863i
$987$			4.00000i			0.127321i
$988$			10.0000i			0.318142i
$989$			1.00000i			0.0317982i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−80.0000			−2.54000
$993$	23.0000			0.729883
$994$		0			0
$995$		0			0
$996$			12.0000i			0.380235i
$997$	−28.0000			−0.886769			−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000			−0.886769			−0.443384	+	0.896332i	$0.646222\pi$
$998$	12.0000			0.379853
$999$	−2.00000			−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1275.2.d.a.424.1		2
5.2	odd	4		1275.2.g.b.526.2		2
5.3	odd	4		51.2.d.a.16.1	✓	2
5.4	even	2		1275.2.d.c.424.2		2
15.8	even	4		153.2.d.c.118.2		2
17.16	even	2		1275.2.d.c.424.1		2
20.3	even	4		816.2.c.b.577.2		2
40.3	even	4		3264.2.c.h.577.1		2
40.13	odd	4		3264.2.c.g.577.2		2
60.23	odd	4		2448.2.c.f.577.2		2
85.3	even	16		867.2.h.e.688.2		8
85.8	odd	8		867.2.e.a.616.1		4
85.13	odd	4		867.2.a.e.1.1		1
85.23	even	16		867.2.h.e.712.1		8
85.28	even	16		867.2.h.e.712.2		8
85.33	odd	4		51.2.d.a.16.2	yes	2
85.38	odd	4		867.2.a.d.1.1		1
85.43	odd	8		867.2.e.a.616.2		4
85.48	even	16		867.2.h.e.688.1		8
85.53	odd	8		867.2.e.a.829.1		4
85.58	even	16		867.2.h.e.733.1		8
85.63	even	16		867.2.h.e.757.1		8
85.67	odd	4		1275.2.g.b.526.1		2
85.73	even	16		867.2.h.e.757.2		8
85.78	even	16		867.2.h.e.733.2		8
85.83	odd	8		867.2.e.a.829.2		4
85.84	even	2	inner	1275.2.d.a.424.2		2
255.38	even	4		2601.2.a.c.1.1		1
255.98	even	4		2601.2.a.a.1.1		1
255.203	even	4		153.2.d.c.118.1		2
340.203	even	4		816.2.c.b.577.1		2
680.203	even	4		3264.2.c.h.577.2		2
680.373	odd	4		3264.2.c.g.577.1		2
1020.203	odd	4		2448.2.c.f.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.d.a.16.1	✓	2	5.3	odd	4
51.2.d.a.16.2	yes	2	85.33	odd	4
153.2.d.c.118.1		2	255.203	even	4
153.2.d.c.118.2		2	15.8	even	4
816.2.c.b.577.1		2	340.203	even	4
816.2.c.b.577.2		2	20.3	even	4
867.2.a.d.1.1		1	85.38	odd	4
867.2.a.e.1.1		1	85.13	odd	4
867.2.e.a.616.1		4	85.8	odd	8
867.2.e.a.616.2		4	85.43	odd	8
867.2.e.a.829.1		4	85.53	odd	8
867.2.e.a.829.2		4	85.83	odd	8
867.2.h.e.688.1		8	85.48	even	16
867.2.h.e.688.2		8	85.3	even	16
867.2.h.e.712.1		8	85.23	even	16
867.2.h.e.712.2		8	85.28	even	16
867.2.h.e.733.1		8	85.58	even	16
867.2.h.e.733.2		8	85.78	even	16
867.2.h.e.757.1		8	85.63	even	16
867.2.h.e.757.2		8	85.73	even	16
1275.2.d.a.424.1		2	1.1	even	1	trivial
1275.2.d.a.424.2		2	85.84	even	2	inner
1275.2.d.c.424.1		2	17.16	even	2
1275.2.d.c.424.2		2	5.4	even	2
1275.2.g.b.526.1		2	85.67	odd	4
1275.2.g.b.526.2		2	5.2	odd	4
2448.2.c.f.577.1		2	1020.203	odd	4
2448.2.c.f.577.2		2	60.23	odd	4
2601.2.a.a.1.1		1	255.98	even	4
2601.2.a.c.1.1		1	255.38	even	4
3264.2.c.g.577.1		2	680.373	odd	4
3264.2.c.g.577.2		2	40.13	odd	4
3264.2.c.h.577.1		2	40.3	even	4
3264.2.c.h.577.2		2	680.203	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1275.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -1.00000 q^{3} -2.00000 q^{4} +2.00000i q^{6} +2.00000 q^{7} +1.00000 q^{9} +5.00000i q^{11} +2.00000 q^{12} +1.00000i q^{13} -4.00000i q^{14} -4.00000 q^{16} +(1.00000 + 4.00000i) q^{17} -2.00000i q^{18} -5.00000 q^{19} -2.00000 q^{21} +10.0000 q^{22} +1.00000 q^{23} +2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} +6.00000i q^{29} +10.0000i q^{31} +8.00000i q^{32} -5.00000i q^{33} +(8.00000 - 2.00000i) q^{34} -2.00000 q^{36} +2.00000 q^{37} +10.0000i q^{38} -1.00000i q^{39} +5.00000i q^{41} +4.00000i q^{42} +1.00000i q^{43} -10.0000i q^{44} -2.00000i q^{46} -2.00000i q^{47} +4.00000 q^{48} -3.00000 q^{49} +(-1.00000 - 4.00000i) q^{51} -2.00000i q^{52} +6.00000i q^{53} +2.00000i q^{54} +5.00000 q^{57} +12.0000 q^{58} -10.0000i q^{61} +20.0000 q^{62} +2.00000 q^{63} +8.00000 q^{64} -10.0000 q^{66} -12.0000i q^{67} +(-2.00000 - 8.00000i) q^{68} -1.00000 q^{69} +6.00000 q^{73} -4.00000i q^{74} +10.0000 q^{76} +10.0000i q^{77} -2.00000 q^{78} -4.00000i q^{79} +1.00000 q^{81} +10.0000 q^{82} +6.00000i q^{83} +4.00000 q^{84} +2.00000 q^{86} -6.00000i q^{87} +10.0000 q^{89} +2.00000i q^{91} -2.00000 q^{92} -10.0000i q^{93} -4.00000 q^{94} -8.00000i q^{96} -8.00000 q^{97} +6.00000i q^{98} +5.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{4} + 4 q^{7} + 2 q^{9} + 4 q^{12} - 8 q^{16} + 2 q^{17} - 10 q^{19} - 4 q^{21} + 20 q^{22} + 2 q^{23} + 4 q^{26} - 2 q^{27} - 8 q^{28} + 16 q^{34} - 4 q^{36} + 4 q^{37} + 8 q^{48} - 6 q^{49}+ \cdots - 16 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^4 + 4 * q^7 + 2 * q^9 + 4 * q^12 - 8 * q^16 + 2 * q^17 - 10 * q^19 - 4 * q^21 + 20 * q^22 + 2 * q^23 + 4 * q^26 - 2 * q^27 - 8 * q^28 + 16 * q^34 - 4 * q^36 + 4 * q^37 + 8 * q^48 - 6 * q^49 - 2 * q^51 + 10 * q^57 + 24 * q^58 + 40 * q^62 + 4 * q^63 + 16 * q^64 - 20 * q^66 - 4 * q^68 - 2 * q^69 + 12 * q^73 + 20 * q^76 - 4 * q^78 + 2 * q^81 + 20 * q^82 + 8 * q^84 + 4 * q^86 + 20 * q^89 - 4 * q^92 - 8 * q^94 - 16 * q^97

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