LMFDB - Embedded newform 270.2.c.b.109.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [270,2,Mod(109,270)] 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("270.109"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(270, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 270 = 2 \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	270.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$2.15596085457$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			109.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	270.109
Dual form			270.2.c.b.109.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +4.00000i q^{7} -1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} +5.00000 q^{11} +3.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +6.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} +5.00000i q^{22} -1.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -3.00000 q^{26} -4.00000i q^{28} -9.00000 q^{29} -5.00000 q^{31} +1.00000i q^{32} -1.00000 q^{34} +(8.00000 + 4.00000i) q^{35} -2.00000i q^{37} +6.00000i q^{38} +(-2.00000 - 1.00000i) q^{40} +2.00000 q^{41} -1.00000i q^{43} -5.00000 q^{44} +1.00000 q^{46} -13.0000i q^{47} -9.00000 q^{49} +(4.00000 - 3.00000i) q^{50} -3.00000i q^{52} +(5.00000 - 10.0000i) q^{55} +4.00000 q^{56} -9.00000i q^{58} -4.00000 q^{59} +8.00000 q^{61} -5.00000i q^{62} -1.00000 q^{64} +(6.00000 + 3.00000i) q^{65} -4.00000i q^{67} -1.00000i q^{68} +(-4.00000 + 8.00000i) q^{70} +6.00000 q^{71} +2.00000i q^{73} +2.00000 q^{74} -6.00000 q^{76} +20.0000i q^{77} -9.00000 q^{79} +(1.00000 - 2.00000i) q^{80} +2.00000i q^{82} -4.00000i q^{83} +(2.00000 + 1.00000i) q^{85} +1.00000 q^{86} -5.00000i q^{88} -14.0000 q^{89} -12.0000 q^{91} +1.00000i q^{92} +13.0000 q^{94} +(6.00000 - 12.0000i) q^{95} -10.0000i q^{97} -9.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{5} + 4 q^{10} + 10 q^{11} - 8 q^{14} + 2 q^{16} + 12 q^{19} - 2 q^{20} - 6 q^{25} - 6 q^{26} - 18 q^{29} - 10 q^{31} - 2 q^{34} + 16 q^{35} - 4 q^{40} + 4 q^{41} - 10 q^{44} + 2 q^{46}+ \cdots + 12 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^5 + 4 * q^10 + 10 * q^11 - 8 * q^14 + 2 * q^16 + 12 * q^19 - 2 * q^20 - 6 * q^25 - 6 * q^26 - 18 * q^29 - 10 * q^31 - 2 * q^34 + 16 * q^35 - 4 * q^40 + 4 * q^41 - 10 * q^44 + 2 * q^46 - 18 * q^49 + 8 * q^50 + 10 * q^55 + 8 * q^56 - 8 * q^59 + 16 * q^61 - 2 * q^64 + 12 * q^65 - 8 * q^70 + 12 * q^71 + 4 * q^74 - 12 * q^76 - 18 * q^79 + 2 * q^80 + 4 * q^85 + 2 * q^86 - 28 * q^89 - 24 * q^91 + 26 * q^94 + 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/270\mathbb{Z}\right)^\times$.

$n$	$191$	$217$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$		0			0
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	2.00000	+	1.00000i	0.632456	+	0.316228i
$11$	5.00000			1.50756			0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000			1.50756			0.753778	+	0.657129i	$0.228229\pi$
$12$		0			0
$13$			3.00000i			0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$			3.00000i			0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	−4.00000			−1.06904
$15$		0			0
$16$	1.00000			0.250000
$17$			1.00000i			0.242536i	0.992620	+	0.121268i	$0.0386960\pi$
$17$			1.00000i			0.242536i	−0.992620	+	0.121268i	$0.961304\pi$
$18$		0			0
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$	−1.00000	+	2.00000i	−0.223607	+	0.447214i
$21$		0			0
$22$			5.00000i			1.06600i
$23$		−	1.00000i		−	0.208514i	−0.994550	−	0.104257i	$-0.966753\pi$
$23$		−	1.00000i		−	0.208514i	0.994550	−	0.104257i	$-0.0332465\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$	−3.00000			−0.588348
$27$		0			0
$28$		−	4.00000i		−	0.755929i
$29$	−9.00000			−1.67126			−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000			−1.67126			−0.835629	+	0.549294i	$0.814897\pi$
$30$		0			0
$31$	−5.00000			−0.898027			−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000			−0.898027			−0.449013	+	0.893525i	$0.648224\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	−1.00000			−0.171499
$35$	8.00000	+	4.00000i	1.35225	+	0.676123i
$36$		0			0
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$			6.00000i			0.973329i
$39$		0			0
$40$	−2.00000	−	1.00000i	−0.316228	−	0.158114i
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$		−	1.00000i		−	0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$		−	1.00000i		−	0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	−5.00000			−0.753778
$45$		0			0
$46$	1.00000			0.147442
$47$		−	13.0000i		−	1.89624i	−0.317905	−	0.948122i	$-0.602979\pi$
$47$		−	13.0000i		−	1.89624i	0.317905	−	0.948122i	$-0.397021\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$	4.00000	−	3.00000i	0.565685	−	0.424264i
$51$		0			0
$52$		−	3.00000i		−	0.416025i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$	5.00000	−	10.0000i	0.674200	−	1.34840i
$56$	4.00000			0.534522
$57$		0			0
$58$		−	9.00000i		−	1.18176i
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$		−	5.00000i		−	0.635001i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	6.00000	+	3.00000i	0.744208	+	0.372104i
$66$		0			0
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$		−	1.00000i		−	0.121268i
$69$		0			0
$70$	−4.00000	+	8.00000i	−0.478091	+	0.956183i
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$			2.00000i			0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$			2.00000i			0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	2.00000			0.232495
$75$		0			0
$76$	−6.00000			−0.688247
$77$			20.0000i			2.27921i
$78$		0			0
$79$	−9.00000			−1.01258			−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000			−1.01258			−0.506290	+	0.862364i	$0.668983\pi$
$80$	1.00000	−	2.00000i	0.111803	−	0.223607i
$81$		0			0
$82$			2.00000i			0.220863i
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$		0			0
$85$	2.00000	+	1.00000i	0.216930	+	0.108465i
$86$	1.00000			0.107833
$87$		0			0
$88$		−	5.00000i		−	0.533002i
$89$	−14.0000			−1.48400			−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000			−1.48400			−0.741999	+	0.670402i	$0.766122\pi$
$90$		0			0
$91$	−12.0000			−1.25794
$92$			1.00000i			0.104257i
$93$		0			0
$94$	13.0000			1.34085
$95$	6.00000	−	12.0000i	0.615587	−	1.23117i
$96$		0			0
$97$		−	10.0000i		−	1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$		−	10.0000i		−	1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$		−	9.00000i		−	0.909137i
$99$		0			0
$100$	3.00000	+	4.00000i	0.300000	+	0.400000i
$101$	−17.0000			−1.69156			−0.845782	−	0.533529i	$-0.820865\pi$
$101$	−17.0000			−1.69156			−0.845782	+	0.533529i	$0.820865\pi$
$102$		0			0
$103$			10.0000i			0.985329i	0.870219	+	0.492665i	$0.163977\pi$
$103$			10.0000i			0.985329i	−0.870219	+	0.492665i	$0.836023\pi$
$104$	3.00000			0.294174
$105$		0			0
$106$		0			0
$107$			6.00000i			0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$			6.00000i			0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$		0			0
$109$	8.00000			0.766261			0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000			0.766261			0.383131	+	0.923694i	$0.374846\pi$
$110$	10.0000	+	5.00000i	0.953463	+	0.476731i
$111$		0			0
$112$			4.00000i			0.377964i
$113$			3.00000i			0.282216i	0.989994	+	0.141108i	$0.0450665\pi$
$113$			3.00000i			0.282216i	−0.989994	+	0.141108i	$0.954933\pi$
$114$		0			0
$115$	−2.00000	−	1.00000i	−0.186501	−	0.0932505i
$116$	9.00000			0.835629
$117$		0			0
$118$		−	4.00000i		−	0.368230i
$119$	−4.00000			−0.366679
$120$		0			0
$121$	14.0000			1.27273
$122$			8.00000i			0.724286i
$123$		0			0
$124$	5.00000			0.449013
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$			4.00000i			0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$			4.00000i			0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−3.00000	+	6.00000i	−0.263117	+	0.526235i
$131$	3.00000			0.262111			0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000			0.262111			0.131056	+	0.991375i	$0.458163\pi$
$132$		0			0
$133$			24.0000i			2.08106i
$134$	4.00000			0.345547
$135$		0			0
$136$	1.00000			0.0857493
$137$		−	18.0000i		−	1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$		0			0
$139$	8.00000			0.678551			0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000			0.678551			0.339276	+	0.940687i	$0.389818\pi$
$140$	−8.00000	−	4.00000i	−0.676123	−	0.338062i
$141$		0			0
$142$			6.00000i			0.503509i
$143$			15.0000i			1.25436i
$144$		0			0
$145$	−9.00000	+	18.0000i	−0.747409	+	1.49482i
$146$	−2.00000			−0.165521
$147$		0			0
$148$			2.00000i			0.164399i
$149$	−13.0000			−1.06500			−0.532501	−	0.846430i	$-0.678748\pi$
$149$	−13.0000			−1.06500			−0.532501	+	0.846430i	$0.678748\pi$
$150$		0			0
$151$	13.0000			1.05792			0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000			1.05792			0.528962	+	0.848645i	$0.322581\pi$
$152$		−	6.00000i		−	0.486664i
$153$		0			0
$154$	−20.0000			−1.61165
$155$	−5.00000	+	10.0000i	−0.401610	+	0.803219i
$156$		0			0
$157$		−	13.0000i		−	1.03751i	−0.854922	−	0.518756i	$-0.826395\pi$
$157$		−	13.0000i		−	1.03751i	0.854922	−	0.518756i	$-0.173605\pi$
$158$		−	9.00000i		−	0.716002i
$159$		0			0
$160$	2.00000	+	1.00000i	0.158114	+	0.0790569i
$161$	4.00000			0.315244
$162$		0			0
$163$			19.0000i			1.48819i	0.668071	+	0.744097i	$0.267120\pi$
$163$			19.0000i			1.48819i	−0.668071	+	0.744097i	$0.732880\pi$
$164$	−2.00000			−0.156174
$165$		0			0
$166$	4.00000			0.310460
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$	4.00000			0.307692
$170$	−1.00000	+	2.00000i	−0.0766965	+	0.153393i
$171$		0			0
$172$			1.00000i			0.0762493i
$173$			16.0000i			1.21646i	0.793762	+	0.608229i	$0.208120\pi$
$173$			16.0000i			1.21646i	−0.793762	+	0.608229i	$0.791880\pi$
$174$		0			0
$175$	16.0000	−	12.0000i	1.20949	−	0.907115i
$176$	5.00000			0.376889
$177$		0			0
$178$		−	14.0000i		−	1.04934i
$179$	20.0000			1.49487			0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000			1.49487			0.747435	+	0.664335i	$0.231285\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$		−	12.0000i		−	0.889499i
$183$		0			0
$184$	−1.00000			−0.0737210
$185$	−4.00000	−	2.00000i	−0.294086	−	0.147043i
$186$		0			0
$187$			5.00000i			0.365636i
$188$			13.0000i			0.948122i
$189$		0			0
$190$	12.0000	+	6.00000i	0.870572	+	0.435286i
$191$	6.00000			0.434145			0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000			0.434145			0.217072	+	0.976156i	$0.430349\pi$
$192$		0			0
$193$		−	4.00000i		−	0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$		−	4.00000i		−	0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	10.0000			0.717958
$195$		0			0
$196$	9.00000			0.642857
$197$		−	22.0000i		−	1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$		−	22.0000i		−	1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$		0			0
$199$	−3.00000			−0.212664			−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000			−0.212664			−0.106332	+	0.994331i	$0.533911\pi$
$200$	−4.00000	+	3.00000i	−0.282843	+	0.212132i
$201$		0			0
$202$		−	17.0000i		−	1.19612i
$203$		−	36.0000i		−	2.52670i
$204$		0			0
$205$	2.00000	−	4.00000i	0.139686	−	0.279372i
$206$	−10.0000			−0.696733
$207$		0			0
$208$			3.00000i			0.208013i
$209$	30.0000			2.07514
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$	−6.00000			−0.410152
$215$	−2.00000	−	1.00000i	−0.136399	−	0.0681994i
$216$		0			0
$217$		−	20.0000i		−	1.35769i
$218$			8.00000i			0.541828i
$219$		0			0
$220$	−5.00000	+	10.0000i	−0.337100	+	0.674200i
$221$	−3.00000			−0.201802
$222$		0			0
$223$		−	22.0000i		−	1.47323i	−0.676313	−	0.736614i	$-0.736423\pi$
$223$		−	22.0000i		−	1.47323i	0.676313	−	0.736614i	$-0.263577\pi$
$224$	−4.00000			−0.267261
$225$		0			0
$226$	−3.00000			−0.199557
$227$		−	2.00000i		−	0.132745i	−0.997795	−	0.0663723i	$-0.978857\pi$
$227$		−	2.00000i		−	0.132745i	0.997795	−	0.0663723i	$-0.0211425\pi$
$228$		0			0
$229$	−2.00000			−0.132164			−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000			−0.132164			−0.0660819	+	0.997814i	$0.521050\pi$
$230$	1.00000	−	2.00000i	0.0659380	−	0.131876i
$231$		0			0
$232$			9.00000i			0.590879i
$233$		−	14.0000i		−	0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$		−	14.0000i		−	0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$		0			0
$235$	−26.0000	−	13.0000i	−1.69605	−	0.848026i
$236$	4.00000			0.260378
$237$		0			0
$238$		−	4.00000i		−	0.259281i
$239$	16.0000			1.03495			0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000			1.03495			0.517477	+	0.855697i	$0.326871\pi$
$240$		0			0
$241$	−23.0000			−1.48156			−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000			−1.48156			−0.740780	+	0.671748i	$0.765544\pi$
$242$			14.0000i			0.899954i
$243$		0			0
$244$	−8.00000			−0.512148
$245$	−9.00000	+	18.0000i	−0.574989	+	1.14998i
$246$		0			0
$247$			18.0000i			1.14531i
$248$			5.00000i			0.317500i
$249$		0			0
$250$	−2.00000	−	11.0000i	−0.126491	−	0.695701i
$251$	−15.0000			−0.946792			−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000			−0.946792			−0.473396	+	0.880850i	$0.656972\pi$
$252$		0			0
$253$		−	5.00000i		−	0.314347i
$254$	−4.00000			−0.250982
$255$		0			0
$256$	1.00000			0.0625000
$257$			21.0000i			1.30994i	0.755653	+	0.654972i	$0.227320\pi$
$257$			21.0000i			1.30994i	−0.755653	+	0.654972i	$0.772680\pi$
$258$		0			0
$259$	8.00000			0.497096
$260$	−6.00000	−	3.00000i	−0.372104	−	0.186052i
$261$		0			0
$262$			3.00000i			0.185341i
$263$			8.00000i			0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$			8.00000i			0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$		0			0
$265$		0			0
$266$	−24.0000			−1.47153
$267$		0			0
$268$			4.00000i			0.244339i
$269$	9.00000			0.548740			0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000			0.548740			0.274370	+	0.961624i	$0.411531\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$			1.00000i			0.0606339i
$273$		0			0
$274$	18.0000			1.08742
$275$	−15.0000	−	20.0000i	−0.904534	−	1.20605i
$276$		0			0
$277$		−	26.0000i		−	1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$		−	26.0000i		−	1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$			8.00000i			0.479808i
$279$		0			0
$280$	4.00000	−	8.00000i	0.239046	−	0.478091i
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$		0			0
$283$		−	20.0000i		−	1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$		−	20.0000i		−	1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	−6.00000			−0.356034
$285$		0			0
$286$	−15.0000			−0.886969
$287$			8.00000i			0.472225i
$288$		0			0
$289$	16.0000			0.941176
$290$	−18.0000	−	9.00000i	−1.05700	−	0.528498i
$291$		0			0
$292$		−	2.00000i		−	0.117041i
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$		0			0
$295$	−4.00000	+	8.00000i	−0.232889	+	0.465778i
$296$	−2.00000			−0.116248
$297$		0			0
$298$		−	13.0000i		−	0.753070i
$299$	3.00000			0.173494
$300$		0			0
$301$	4.00000			0.230556
$302$			13.0000i			0.748066i
$303$		0			0
$304$	6.00000			0.344124
$305$	8.00000	−	16.0000i	0.458079	−	0.916157i
$306$		0			0
$307$			21.0000i			1.19853i	0.800549	+	0.599267i	$0.204541\pi$
$307$			21.0000i			1.19853i	−0.800549	+	0.599267i	$0.795459\pi$
$308$		−	20.0000i		−	1.13961i
$309$		0			0
$310$	−10.0000	−	5.00000i	−0.567962	−	0.283981i
$311$	−18.0000			−1.02069			−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000			−1.02069			−0.510343	+	0.859971i	$0.670482\pi$
$312$		0			0
$313$			14.0000i			0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$			14.0000i			0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	13.0000			0.733632
$315$		0			0
$316$	9.00000			0.506290
$317$			4.00000i			0.224662i	0.993671	+	0.112331i	$0.0358318\pi$
$317$			4.00000i			0.224662i	−0.993671	+	0.112331i	$0.964168\pi$
$318$		0			0
$319$	−45.0000			−2.51952
$320$	−1.00000	+	2.00000i	−0.0559017	+	0.111803i
$321$		0			0
$322$			4.00000i			0.222911i
$323$			6.00000i			0.333849i
$324$		0			0
$325$	12.0000	−	9.00000i	0.665640	−	0.499230i
$326$	−19.0000			−1.05231
$327$		0			0
$328$		−	2.00000i		−	0.110432i
$329$	52.0000			2.86685
$330$		0			0
$331$	10.0000			0.549650			0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000			0.549650			0.274825	+	0.961494i	$0.411380\pi$
$332$			4.00000i			0.219529i
$333$		0			0
$334$	−12.0000			−0.656611
$335$	−8.00000	−	4.00000i	−0.437087	−	0.218543i
$336$		0			0
$337$			16.0000i			0.871576i	0.900049	+	0.435788i	$0.143530\pi$
$337$			16.0000i			0.871576i	−0.900049	+	0.435788i	$0.856470\pi$
$338$			4.00000i			0.217571i
$339$		0			0
$340$	−2.00000	−	1.00000i	−0.108465	−	0.0542326i
$341$	−25.0000			−1.35383
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$	−1.00000			−0.0539164
$345$		0			0
$346$	−16.0000			−0.860165
$347$		−	6.00000i		−	0.322097i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$		−	6.00000i		−	0.322097i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$		0			0
$349$	−16.0000			−0.856460			−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000			−0.856460			−0.428230	+	0.903670i	$0.640863\pi$
$350$	12.0000	+	16.0000i	0.641427	+	0.855236i
$351$		0			0
$352$			5.00000i			0.266501i
$353$		−	11.0000i		−	0.585471i	−0.956193	−	0.292735i	$-0.905434\pi$
$353$		−	11.0000i		−	0.585471i	0.956193	−	0.292735i	$-0.0945655\pi$
$354$		0			0
$355$	6.00000	−	12.0000i	0.318447	−	0.636894i
$356$	14.0000			0.741999
$357$		0			0
$358$			20.0000i			1.05703i
$359$	6.00000			0.316668			0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000			0.316668			0.158334	+	0.987386i	$0.449388\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		−	22.0000i		−	1.15629i
$363$		0			0
$364$	12.0000			0.628971
$365$	4.00000	+	2.00000i	0.209370	+	0.104685i
$366$		0			0
$367$			22.0000i			1.14839i	0.818718	+	0.574195i	$0.194685\pi$
$367$			22.0000i			1.14839i	−0.818718	+	0.574195i	$0.805315\pi$
$368$		−	1.00000i		−	0.0521286i
$369$		0			0
$370$	2.00000	−	4.00000i	0.103975	−	0.207950i
$371$		0			0
$372$		0			0
$373$			21.0000i			1.08734i	0.839299	+	0.543669i	$0.182965\pi$
$373$			21.0000i			1.08734i	−0.839299	+	0.543669i	$0.817035\pi$
$374$	−5.00000			−0.258544
$375$		0			0
$376$	−13.0000			−0.670424
$377$		−	27.0000i		−	1.39057i
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$	−6.00000	+	12.0000i	−0.307794	+	0.615587i
$381$		0			0
$382$			6.00000i			0.306987i
$383$			19.0000i			0.970855i	0.874277	+	0.485427i	$0.161336\pi$
$383$			19.0000i			0.970855i	−0.874277	+	0.485427i	$0.838664\pi$
$384$		0			0
$385$	40.0000	+	20.0000i	2.03859	+	1.01929i
$386$	4.00000			0.203595
$387$		0			0
$388$			10.0000i			0.507673i
$389$	−5.00000			−0.253510			−0.126755	−	0.991934i	$-0.540456\pi$
$389$	−5.00000			−0.253510			−0.126755	+	0.991934i	$0.540456\pi$
$390$		0			0
$391$	1.00000			0.0505722
$392$			9.00000i			0.454569i
$393$		0			0
$394$	22.0000			1.10834
$395$	−9.00000	+	18.0000i	−0.452839	+	0.905678i
$396$		0			0
$397$			13.0000i			0.652451i	0.945292	+	0.326226i	$0.105777\pi$
$397$			13.0000i			0.652451i	−0.945292	+	0.326226i	$0.894223\pi$
$398$		−	3.00000i		−	0.150376i
$399$		0			0
$400$	−3.00000	−	4.00000i	−0.150000	−	0.200000i
$401$	16.0000			0.799002			0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000			0.799002			0.399501	+	0.916733i	$0.369183\pi$
$402$		0			0
$403$		−	15.0000i		−	0.747203i
$404$	17.0000			0.845782
$405$		0			0
$406$	36.0000			1.78665
$407$		−	10.0000i		−	0.495682i
$408$		0			0
$409$	−11.0000			−0.543915			−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000			−0.543915			−0.271957	+	0.962309i	$0.587671\pi$
$410$	4.00000	+	2.00000i	0.197546	+	0.0987730i
$411$		0			0
$412$		−	10.0000i		−	0.492665i
$413$		−	16.0000i		−	0.787309i
$414$		0			0
$415$	−8.00000	−	4.00000i	−0.392705	−	0.196352i
$416$	−3.00000			−0.147087
$417$		0			0
$418$			30.0000i			1.46735i
$419$	−23.0000			−1.12362			−0.561812	−	0.827265i	$-0.689895\pi$
$419$	−23.0000			−1.12362			−0.561812	+	0.827265i	$0.689895\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.00000	−	3.00000i	0.194029	−	0.145521i
$426$		0			0
$427$			32.0000i			1.54859i
$428$		−	6.00000i		−	0.290021i
$429$		0			0
$430$	1.00000	−	2.00000i	0.0482243	−	0.0964486i
$431$	8.00000			0.385346			0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000			0.385346			0.192673	+	0.981263i	$0.438284\pi$
$432$		0			0
$433$		−	22.0000i		−	1.05725i	−0.848855	−	0.528626i	$-0.822707\pi$
$433$		−	22.0000i		−	1.05725i	0.848855	−	0.528626i	$-0.177293\pi$
$434$	20.0000			0.960031
$435$		0			0
$436$	−8.00000			−0.383131
$437$		−	6.00000i		−	0.287019i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	−10.0000	−	5.00000i	−0.476731	−	0.238366i
$441$		0			0
$442$		−	3.00000i		−	0.142695i
$443$			30.0000i			1.42534i	0.701498	+	0.712672i	$0.252515\pi$
$443$			30.0000i			1.42534i	−0.701498	+	0.712672i	$0.747485\pi$
$444$		0			0
$445$	−14.0000	+	28.0000i	−0.663664	+	1.32733i
$446$	22.0000			1.04173
$447$		0			0
$448$		−	4.00000i		−	0.188982i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$	10.0000			0.470882
$452$		−	3.00000i		−	0.141108i
$453$		0			0
$454$	2.00000			0.0938647
$455$	−12.0000	+	24.0000i	−0.562569	+	1.12514i
$456$		0			0
$457$			16.0000i			0.748448i	0.927338	+	0.374224i	$0.122091\pi$
$457$			16.0000i			0.748448i	−0.927338	+	0.374224i	$0.877909\pi$
$458$		−	2.00000i		−	0.0934539i
$459$		0			0
$460$	2.00000	+	1.00000i	0.0932505	+	0.0466252i
$461$	6.00000			0.279448			0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000			0.279448			0.139724	+	0.990190i	$0.455378\pi$
$462$		0			0
$463$			6.00000i			0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$			6.00000i			0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$	−9.00000			−0.417815
$465$		0			0
$466$	14.0000			0.648537
$467$			6.00000i			0.277647i	0.990317	+	0.138823i	$0.0443321\pi$
$467$			6.00000i			0.277647i	−0.990317	+	0.138823i	$0.955668\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$	13.0000	−	26.0000i	0.599645	−	1.19929i
$471$		0			0
$472$			4.00000i			0.184115i
$473$		−	5.00000i		−	0.229900i
$474$		0			0
$475$	−18.0000	−	24.0000i	−0.825897	−	1.10120i
$476$	4.00000			0.183340
$477$		0			0
$478$			16.0000i			0.731823i
$479$	−2.00000			−0.0913823			−0.0456912	−	0.998956i	$-0.514549\pi$
$479$	−2.00000			−0.0913823			−0.0456912	+	0.998956i	$0.514549\pi$
$480$		0			0
$481$	6.00000			0.273576
$482$		−	23.0000i		−	1.04762i
$483$		0			0
$484$	−14.0000			−0.636364
$485$	−20.0000	−	10.0000i	−0.908153	−	0.454077i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		−	8.00000i		−	0.362143i
$489$		0			0
$490$	−18.0000	−	9.00000i	−0.813157	−	0.406579i
$491$	−24.0000			−1.08310			−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000			−1.08310			−0.541552	+	0.840667i	$0.682163\pi$
$492$		0			0
$493$		−	9.00000i		−	0.405340i
$494$	−18.0000			−0.809858
$495$		0			0
$496$	−5.00000			−0.224507
$497$			24.0000i			1.07655i
$498$		0			0
$499$	12.0000			0.537194			0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000			0.537194			0.268597	+	0.963253i	$0.413440\pi$
$500$	11.0000	−	2.00000i	0.491935	−	0.0894427i
$501$		0			0
$502$		−	15.0000i		−	0.669483i
$503$		−	9.00000i		−	0.401290i	−0.979664	−	0.200645i	$-0.935696\pi$
$503$		−	9.00000i		−	0.401290i	0.979664	−	0.200645i	$-0.0643038\pi$
$504$		0			0
$505$	−17.0000	+	34.0000i	−0.756490	+	1.51298i
$506$	5.00000			0.222277
$507$		0			0
$508$		−	4.00000i		−	0.177471i
$509$	−21.0000			−0.930809			−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000			−0.930809			−0.465404	+	0.885098i	$0.654091\pi$
$510$		0			0
$511$	−8.00000			−0.353899
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−21.0000			−0.926270
$515$	20.0000	+	10.0000i	0.881305	+	0.440653i
$516$		0			0
$517$		−	65.0000i		−	2.85870i
$518$			8.00000i			0.351500i
$519$		0			0
$520$	3.00000	−	6.00000i	0.131559	−	0.263117i
$521$	22.0000			0.963837			0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000			0.963837			0.481919	+	0.876216i	$0.339940\pi$
$522$		0			0
$523$		−	23.0000i		−	1.00572i	−0.864368	−	0.502860i	$-0.832281\pi$
$523$		−	23.0000i		−	1.00572i	0.864368	−	0.502860i	$-0.167719\pi$
$524$	−3.00000			−0.131056
$525$		0			0
$526$	−8.00000			−0.348817
$527$		−	5.00000i		−	0.217803i
$528$		0			0
$529$	22.0000			0.956522
$530$		0			0
$531$		0			0
$532$		−	24.0000i		−	1.04053i
$533$			6.00000i			0.259889i
$534$		0			0
$535$	12.0000	+	6.00000i	0.518805	+	0.259403i
$536$	−4.00000			−0.172774
$537$		0			0
$538$			9.00000i			0.388018i
$539$	−45.0000			−1.93829
$540$		0			0
$541$	10.0000			0.429934			0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000			0.429934			0.214967	+	0.976621i	$0.431036\pi$
$542$			16.0000i			0.687259i
$543$		0			0
$544$	−1.00000			−0.0428746
$545$	8.00000	−	16.0000i	0.342682	−	0.685365i
$546$		0			0
$547$			25.0000i			1.06892i	0.845193	+	0.534461i	$0.179486\pi$
$547$			25.0000i			1.06892i	−0.845193	+	0.534461i	$0.820514\pi$
$548$			18.0000i			0.768922i
$549$		0			0
$550$	20.0000	−	15.0000i	0.852803	−	0.639602i
$551$	−54.0000			−2.30048
$552$		0			0
$553$		−	36.0000i		−	1.53088i
$554$	26.0000			1.10463
$555$		0			0
$556$	−8.00000			−0.339276
$557$			42.0000i			1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$			42.0000i			1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$		0			0
$559$	3.00000			0.126886
$560$	8.00000	+	4.00000i	0.338062	+	0.169031i
$561$		0			0
$562$			18.0000i			0.759284i
$563$			4.00000i			0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$			4.00000i			0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$		0			0
$565$	6.00000	+	3.00000i	0.252422	+	0.126211i
$566$	20.0000			0.840663
$567$		0			0
$568$		−	6.00000i		−	0.251754i
$569$	−20.0000			−0.838444			−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000			−0.838444			−0.419222	+	0.907884i	$0.637697\pi$
$570$		0			0
$571$	−32.0000			−1.33916			−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000			−1.33916			−0.669579	+	0.742741i	$0.733526\pi$
$572$		−	15.0000i		−	0.627182i
$573$		0			0
$574$	−8.00000			−0.333914
$575$	−4.00000	+	3.00000i	−0.166812	+	0.125109i
$576$		0			0
$577$		−	32.0000i		−	1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$		−	32.0000i		−	1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$			16.0000i			0.665512i
$579$		0			0
$580$	9.00000	−	18.0000i	0.373705	−	0.747409i
$581$	16.0000			0.663792
$582$		0			0
$583$		0			0
$584$	2.00000			0.0827606
$585$		0			0
$586$	−6.00000			−0.247858
$587$			36.0000i			1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$			36.0000i			1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$		0			0
$589$	−30.0000			−1.23613
$590$	−8.00000	−	4.00000i	−0.329355	−	0.164677i
$591$		0			0
$592$		−	2.00000i		−	0.0821995i
$593$		−	21.0000i		−	0.862367i	−0.902264	−	0.431183i	$-0.858096\pi$
$593$		−	21.0000i		−	0.862367i	0.902264	−	0.431183i	$-0.141904\pi$
$594$		0			0
$595$	−4.00000	+	8.00000i	−0.163984	+	0.327968i
$596$	13.0000			0.532501
$597$		0			0
$598$			3.00000i			0.122679i
$599$	−42.0000			−1.71607			−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000			−1.71607			−0.858037	+	0.513588i	$0.828316\pi$
$600$		0			0
$601$	−19.0000			−0.775026			−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000			−0.775026			−0.387513	+	0.921864i	$0.626666\pi$
$602$			4.00000i			0.163028i
$603$		0			0
$604$	−13.0000			−0.528962
$605$	14.0000	−	28.0000i	0.569181	−	1.13836i
$606$		0			0
$607$		−	10.0000i		−	0.405887i	−0.979190	−	0.202944i	$-0.934949\pi$
$607$		−	10.0000i		−	0.405887i	0.979190	−	0.202944i	$-0.0650509\pi$
$608$			6.00000i			0.243332i
$609$		0			0
$610$	16.0000	+	8.00000i	0.647821	+	0.323911i
$611$	39.0000			1.57777
$612$		0			0
$613$			23.0000i			0.928961i	0.885583	+	0.464481i	$0.153759\pi$
$613$			23.0000i			0.928961i	−0.885583	+	0.464481i	$0.846241\pi$
$614$	−21.0000			−0.847491
$615$		0			0
$616$	20.0000			0.805823
$617$		−	29.0000i		−	1.16750i	−0.811935	−	0.583748i	$-0.801586\pi$
$617$		−	29.0000i		−	1.16750i	0.811935	−	0.583748i	$-0.198414\pi$
$618$		0			0
$619$	−26.0000			−1.04503			−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000			−1.04503			−0.522514	+	0.852631i	$0.675006\pi$
$620$	5.00000	−	10.0000i	0.200805	−	0.401610i
$621$		0			0
$622$		−	18.0000i		−	0.721734i
$623$		−	56.0000i		−	2.24359i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−14.0000			−0.559553
$627$		0			0
$628$			13.0000i			0.518756i
$629$	2.00000			0.0797452
$630$		0			0
$631$	−4.00000			−0.159237			−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000			−0.159237			−0.0796187	+	0.996825i	$0.525370\pi$
$632$			9.00000i			0.358001i
$633$		0			0
$634$	−4.00000			−0.158860
$635$	8.00000	+	4.00000i	0.317470	+	0.158735i
$636$		0			0
$637$		−	27.0000i		−	1.06978i
$638$		−	45.0000i		−	1.78157i
$639$		0			0
$640$	−2.00000	−	1.00000i	−0.0790569	−	0.0395285i
$641$	20.0000			0.789953			0.394976	−	0.918691i	$-0.370753\pi$
$641$	20.0000			0.789953			0.394976	+	0.918691i	$0.370753\pi$
$642$		0			0
$643$		−	21.0000i		−	0.828159i	−0.910241	−	0.414080i	$-0.864104\pi$
$643$		−	21.0000i		−	0.828159i	0.910241	−	0.414080i	$-0.135896\pi$
$644$	−4.00000			−0.157622
$645$		0			0
$646$	−6.00000			−0.236067
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−20.0000			−0.785069
$650$	9.00000	+	12.0000i	0.353009	+	0.470679i
$651$		0			0
$652$		−	19.0000i		−	0.744097i
$653$			26.0000i			1.01746i	0.860927	+	0.508729i	$0.169885\pi$
$653$			26.0000i			1.01746i	−0.860927	+	0.508729i	$0.830115\pi$
$654$		0			0
$655$	3.00000	−	6.00000i	0.117220	−	0.234439i
$656$	2.00000			0.0780869
$657$		0			0
$658$			52.0000i			2.02717i
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$	44.0000			1.71140			0.855701	−	0.517471i	$-0.173126\pi$
$661$	44.0000			1.71140			0.855701	+	0.517471i	$0.173126\pi$
$662$			10.0000i			0.388661i
$663$		0			0
$664$	−4.00000			−0.155230
$665$	48.0000	+	24.0000i	1.86136	+	0.930680i
$666$		0			0
$667$			9.00000i			0.348481i
$668$		−	12.0000i		−	0.464294i
$669$		0			0
$670$	4.00000	−	8.00000i	0.154533	−	0.309067i
$671$	40.0000			1.54418
$672$		0			0
$673$		−	48.0000i		−	1.85026i	−0.379646	−	0.925132i	$-0.623954\pi$
$673$		−	48.0000i		−	1.85026i	0.379646	−	0.925132i	$-0.376046\pi$
$674$	−16.0000			−0.616297
$675$		0			0
$676$	−4.00000			−0.153846
$677$			4.00000i			0.153732i	0.997041	+	0.0768662i	$0.0244914\pi$
$677$			4.00000i			0.153732i	−0.997041	+	0.0768662i	$0.975509\pi$
$678$		0			0
$679$	40.0000			1.53506
$680$	1.00000	−	2.00000i	0.0383482	−	0.0766965i
$681$		0			0
$682$		−	25.0000i		−	0.957299i
$683$			44.0000i			1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$			44.0000i			1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$		0			0
$685$	−36.0000	−	18.0000i	−1.37549	−	0.687745i
$686$	8.00000			0.305441
$687$		0			0
$688$		−	1.00000i		−	0.0381246i
$689$		0			0
$690$		0			0
$691$	22.0000			0.836919			0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000			0.836919			0.418460	+	0.908235i	$0.362570\pi$
$692$		−	16.0000i		−	0.608229i
$693$		0			0
$694$	6.00000			0.227757
$695$	8.00000	−	16.0000i	0.303457	−	0.606915i
$696$		0			0
$697$			2.00000i			0.0757554i
$698$		−	16.0000i		−	0.605609i
$699$		0			0
$700$	−16.0000	+	12.0000i	−0.604743	+	0.453557i
$701$	−31.0000			−1.17085			−0.585427	−	0.810725i	$-0.699073\pi$
$701$	−31.0000			−1.17085			−0.585427	+	0.810725i	$0.699073\pi$
$702$		0			0
$703$		−	12.0000i		−	0.452589i
$704$	−5.00000			−0.188445
$705$		0			0
$706$	11.0000			0.413990
$707$		−	68.0000i		−	2.55740i
$708$		0			0
$709$	24.0000			0.901339			0.450669	−	0.892691i	$-0.351185\pi$
$709$	24.0000			0.901339			0.450669	+	0.892691i	$0.351185\pi$
$710$	12.0000	+	6.00000i	0.450352	+	0.225176i
$711$		0			0
$712$			14.0000i			0.524672i
$713$			5.00000i			0.187251i
$714$		0			0
$715$	30.0000	+	15.0000i	1.12194	+	0.560968i
$716$	−20.0000			−0.747435
$717$		0			0
$718$			6.00000i			0.223918i
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$	−40.0000			−1.48968
$722$			17.0000i			0.632674i
$723$		0			0
$724$	22.0000			0.817624
$725$	27.0000	+	36.0000i	1.00275	+	1.33701i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$			12.0000i			0.444750i
$729$		0			0
$730$	−2.00000	+	4.00000i	−0.0740233	+	0.148047i
$731$	1.00000			0.0369863
$732$		0			0
$733$			46.0000i			1.69905i	0.527549	+	0.849524i	$0.323111\pi$
$733$			46.0000i			1.69905i	−0.527549	+	0.849524i	$0.676889\pi$
$734$	−22.0000			−0.812035
$735$		0			0
$736$	1.00000			0.0368605
$737$		−	20.0000i		−	0.736709i
$738$		0			0
$739$	−10.0000			−0.367856			−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000			−0.367856			−0.183928	+	0.982940i	$0.558881\pi$
$740$	4.00000	+	2.00000i	0.147043	+	0.0735215i
$741$		0			0
$742$		0			0
$743$			9.00000i			0.330178i	0.986279	+	0.165089i	$0.0527911\pi$
$743$			9.00000i			0.330178i	−0.986279	+	0.165089i	$0.947209\pi$
$744$		0			0
$745$	−13.0000	+	26.0000i	−0.476283	+	0.952566i
$746$	−21.0000			−0.768865
$747$		0			0
$748$		−	5.00000i		−	0.182818i
$749$	−24.0000			−0.876941
$750$		0			0
$751$	−7.00000			−0.255434			−0.127717	−	0.991811i	$-0.540765\pi$
$751$	−7.00000			−0.255434			−0.127717	+	0.991811i	$0.540765\pi$
$752$		−	13.0000i		−	0.474061i
$753$		0			0
$754$	27.0000			0.983282
$755$	13.0000	−	26.0000i	0.473118	−	0.946237i
$756$		0			0
$757$			43.0000i			1.56286i	0.623992	+	0.781431i	$0.285510\pi$
$757$			43.0000i			1.56286i	−0.623992	+	0.781431i	$0.714490\pi$
$758$			28.0000i			1.01701i
$759$		0			0
$760$	−12.0000	−	6.00000i	−0.435286	−	0.217643i
$761$	−12.0000			−0.435000			−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000			−0.435000			−0.217500	+	0.976060i	$0.569790\pi$
$762$		0			0
$763$			32.0000i			1.15848i
$764$	−6.00000			−0.217072
$765$		0			0
$766$	−19.0000			−0.686498
$767$		−	12.0000i		−	0.433295i
$768$		0			0
$769$	−3.00000			−0.108183			−0.0540914	−	0.998536i	$-0.517226\pi$
$769$	−3.00000			−0.108183			−0.0540914	+	0.998536i	$0.517226\pi$
$770$	−20.0000	+	40.0000i	−0.720750	+	1.44150i
$771$		0			0
$772$			4.00000i			0.143963i
$773$			18.0000i			0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$			18.0000i			0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$		0			0
$775$	15.0000	+	20.0000i	0.538816	+	0.718421i
$776$	−10.0000			−0.358979
$777$		0			0
$778$		−	5.00000i		−	0.179259i
$779$	12.0000			0.429945
$780$		0			0
$781$	30.0000			1.07348
$782$			1.00000i			0.0357599i
$783$		0			0
$784$	−9.00000			−0.321429
$785$	−26.0000	−	13.0000i	−0.927980	−	0.463990i
$786$		0			0
$787$			13.0000i			0.463400i	0.972787	+	0.231700i	$0.0744288\pi$
$787$			13.0000i			0.463400i	−0.972787	+	0.231700i	$0.925571\pi$
$788$			22.0000i			0.783718i
$789$		0			0
$790$	−18.0000	−	9.00000i	−0.640411	−	0.320206i
$791$	−12.0000			−0.426671
$792$		0			0
$793$			24.0000i			0.852265i
$794$	−13.0000			−0.461353
$795$		0			0
$796$	3.00000			0.106332
$797$		−	14.0000i		−	0.495905i	−0.968772	−	0.247953i	$-0.920242\pi$
$797$		−	14.0000i		−	0.495905i	0.968772	−	0.247953i	$-0.0797578\pi$
$798$		0			0
$799$	13.0000			0.459907
$800$	4.00000	−	3.00000i	0.141421	−	0.106066i
$801$		0			0
$802$			16.0000i			0.564980i
$803$			10.0000i			0.352892i
$804$		0			0
$805$	4.00000	−	8.00000i	0.140981	−	0.281963i
$806$	15.0000			0.528352
$807$		0			0
$808$			17.0000i			0.598058i
$809$	12.0000			0.421898			0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000			0.421898			0.210949	+	0.977497i	$0.432345\pi$
$810$		0			0
$811$	−26.0000			−0.912983			−0.456492	−	0.889728i	$-0.650894\pi$
$811$	−26.0000			−0.912983			−0.456492	+	0.889728i	$0.650894\pi$
$812$			36.0000i			1.26335i
$813$		0			0
$814$	10.0000			0.350500
$815$	38.0000	+	19.0000i	1.33108	+	0.665541i
$816$		0			0
$817$		−	6.00000i		−	0.209913i
$818$		−	11.0000i		−	0.384606i
$819$		0			0
$820$	−2.00000	+	4.00000i	−0.0698430	+	0.139686i
$821$	46.0000			1.60541			0.802706	−	0.596376i	$-0.203393\pi$
$821$	46.0000			1.60541			0.802706	+	0.596376i	$0.203393\pi$
$822$		0			0
$823$		−	24.0000i		−	0.836587i	−0.908312	−	0.418294i	$-0.862628\pi$
$823$		−	24.0000i		−	0.836587i	0.908312	−	0.418294i	$-0.137372\pi$
$824$	10.0000			0.348367
$825$		0			0
$826$	16.0000			0.556711
$827$			22.0000i			0.765015i	0.923952	+	0.382507i	$0.124939\pi$
$827$			22.0000i			0.765015i	−0.923952	+	0.382507i	$0.875061\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$	4.00000	−	8.00000i	0.138842	−	0.277684i
$831$		0			0
$832$		−	3.00000i		−	0.104006i
$833$		−	9.00000i		−	0.311832i
$834$		0			0
$835$	24.0000	+	12.0000i	0.830554	+	0.415277i
$836$	−30.0000			−1.03757
$837$		0			0
$838$		−	23.0000i		−	0.794522i
$839$	−34.0000			−1.17381			−0.586905	−	0.809656i	$-0.699654\pi$
$839$	−34.0000			−1.17381			−0.586905	+	0.809656i	$0.699654\pi$
$840$		0			0
$841$	52.0000			1.79310
$842$		−	26.0000i		−	0.896019i
$843$		0			0
$844$		0			0
$845$	4.00000	−	8.00000i	0.137604	−	0.275208i
$846$		0			0
$847$			56.0000i			1.92418i
$848$		0			0
$849$		0			0
$850$	3.00000	+	4.00000i	0.102899	+	0.137199i
$851$	−2.00000			−0.0685591
$852$		0			0
$853$			19.0000i			0.650548i	0.945620	+	0.325274i	$0.105456\pi$
$853$			19.0000i			0.650548i	−0.945620	+	0.325274i	$0.894544\pi$
$854$	−32.0000			−1.09502
$855$		0			0
$856$	6.00000			0.205076
$857$		−	6.00000i		−	0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$		−	6.00000i		−	0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$		0			0
$859$	−2.00000			−0.0682391			−0.0341196	−	0.999418i	$-0.510863\pi$
$859$	−2.00000			−0.0682391			−0.0341196	+	0.999418i	$0.510863\pi$
$860$	2.00000	+	1.00000i	0.0681994	+	0.0340997i
$861$		0			0
$862$			8.00000i			0.272481i
$863$		−	39.0000i		−	1.32758i	−0.747921	−	0.663788i	$-0.768948\pi$
$863$		−	39.0000i		−	1.32758i	0.747921	−	0.663788i	$-0.231052\pi$
$864$		0			0
$865$	32.0000	+	16.0000i	1.08803	+	0.544016i
$866$	22.0000			0.747590
$867$		0			0
$868$			20.0000i			0.678844i
$869$	−45.0000			−1.52652
$870$		0			0
$871$	12.0000			0.406604
$872$		−	8.00000i		−	0.270914i
$873$		0			0
$874$	6.00000			0.202953
$875$	−8.00000	−	44.0000i	−0.270449	−	1.48747i
$876$		0			0
$877$		−	1.00000i		−	0.0337676i	−0.999857	−	0.0168838i	$-0.994625\pi$
$877$		−	1.00000i		−	0.0337676i	0.999857	−	0.0168838i	$-0.00537454\pi$
$878$		0			0
$879$		0			0
$880$	5.00000	−	10.0000i	0.168550	−	0.337100i
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$		−	36.0000i		−	1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$		−	36.0000i		−	1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$	3.00000			0.100901
$885$		0			0
$886$	−30.0000			−1.00787
$887$			51.0000i			1.71241i	0.516634	+	0.856206i	$0.327185\pi$
$887$			51.0000i			1.71241i	−0.516634	+	0.856206i	$0.672815\pi$
$888$		0			0
$889$	−16.0000			−0.536623
$890$	−28.0000	−	14.0000i	−0.938562	−	0.469281i
$891$		0			0
$892$			22.0000i			0.736614i
$893$		−	78.0000i		−	2.61017i
$894$		0			0
$895$	20.0000	−	40.0000i	0.668526	−	1.33705i
$896$	4.00000			0.133631
$897$		0			0
$898$		0			0
$899$	45.0000			1.50083
$900$		0			0
$901$		0			0
$902$			10.0000i			0.332964i
$903$		0			0
$904$	3.00000			0.0997785
$905$	−22.0000	+	44.0000i	−0.731305	+	1.46261i
$906$		0			0
$907$		−	3.00000i		−	0.0996134i	−0.998759	−	0.0498067i	$-0.984139\pi$
$907$		−	3.00000i		−	0.0996134i	0.998759	−	0.0498067i	$-0.0158605\pi$
$908$			2.00000i			0.0663723i
$909$		0			0
$910$	−24.0000	−	12.0000i	−0.795592	−	0.397796i
$911$	36.0000			1.19273			0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000			1.19273			0.596367	+	0.802712i	$0.296610\pi$
$912$		0			0
$913$		−	20.0000i		−	0.661903i
$914$	−16.0000			−0.529233
$915$		0			0
$916$	2.00000			0.0660819
$917$			12.0000i			0.396275i
$918$		0			0
$919$	−33.0000			−1.08857			−0.544285	−	0.838901i	$-0.683199\pi$
$919$	−33.0000			−1.08857			−0.544285	+	0.838901i	$0.683199\pi$
$920$	−1.00000	+	2.00000i	−0.0329690	+	0.0659380i
$921$		0			0
$922$			6.00000i			0.197599i
$923$			18.0000i			0.592477i
$924$		0			0
$925$	−8.00000	+	6.00000i	−0.263038	+	0.197279i
$926$	−6.00000			−0.197172
$927$		0			0
$928$		−	9.00000i		−	0.295439i
$929$	−6.00000			−0.196854			−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000			−0.196854			−0.0984268	+	0.995144i	$0.531381\pi$
$930$		0			0
$931$	−54.0000			−1.76978
$932$			14.0000i			0.458585i
$933$		0			0
$934$	−6.00000			−0.196326
$935$	10.0000	+	5.00000i	0.327035	+	0.163517i
$936$		0			0
$937$		−	40.0000i		−	1.30674i	−0.757037	−	0.653372i	$-0.773354\pi$
$937$		−	40.0000i		−	1.30674i	0.757037	−	0.653372i	$-0.226646\pi$
$938$			16.0000i			0.522419i
$939$		0			0
$940$	26.0000	+	13.0000i	0.848026	+	0.424013i
$941$	−31.0000			−1.01057			−0.505286	−	0.862952i	$-0.668613\pi$
$941$	−31.0000			−1.01057			−0.505286	+	0.862952i	$0.668613\pi$
$942$		0			0
$943$		−	2.00000i		−	0.0651290i
$944$	−4.00000			−0.130189
$945$		0			0
$946$	5.00000			0.162564
$947$			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
$949$	−6.00000			−0.194768
$950$	24.0000	−	18.0000i	0.778663	−	0.583997i
$951$		0			0
$952$			4.00000i			0.129641i
$953$			61.0000i			1.97598i	0.154506	+	0.987992i	$0.450622\pi$
$953$			61.0000i			1.97598i	−0.154506	+	0.987992i	$0.549378\pi$
$954$		0			0
$955$	6.00000	−	12.0000i	0.194155	−	0.388311i
$956$	−16.0000			−0.517477
$957$		0			0
$958$		−	2.00000i		−	0.0646171i
$959$	72.0000			2.32500
$960$		0			0
$961$	−6.00000			−0.193548
$962$			6.00000i			0.193448i
$963$		0			0
$964$	23.0000			0.740780
$965$	−8.00000	−	4.00000i	−0.257529	−	0.128765i
$966$		0			0
$967$			2.00000i			0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$			2.00000i			0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$		−	14.0000i		−	0.449977i
$969$		0			0
$970$	10.0000	−	20.0000i	0.321081	−	0.642161i
$971$	9.00000			0.288824			0.144412	−	0.989518i	$-0.453871\pi$
$971$	9.00000			0.288824			0.144412	+	0.989518i	$0.453871\pi$
$972$		0			0
$973$			32.0000i			1.02587i
$974$	12.0000			0.384505
$975$		0			0
$976$	8.00000			0.256074
$977$		−	39.0000i		−	1.24772i	−0.781536	−	0.623860i	$-0.785563\pi$
$977$		−	39.0000i		−	1.24772i	0.781536	−	0.623860i	$-0.214437\pi$
$978$		0			0
$979$	−70.0000			−2.23721
$980$	9.00000	−	18.0000i	0.287494	−	0.574989i
$981$		0			0
$982$		−	24.0000i		−	0.765871i
$983$		−	29.0000i		−	0.924956i	−0.886631	−	0.462478i	$-0.846960\pi$
$983$		−	29.0000i		−	0.924956i	0.886631	−	0.462478i	$-0.153040\pi$
$984$		0			0
$985$	−44.0000	−	22.0000i	−1.40196	−	0.700978i
$986$	9.00000			0.286618
$987$		0			0
$988$		−	18.0000i		−	0.572656i
$989$	−1.00000			−0.0317982
$990$		0			0
$991$	−17.0000			−0.540023			−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000			−0.540023			−0.270011	+	0.962857i	$0.587027\pi$
$992$		−	5.00000i		−	0.158750i
$993$		0			0
$994$	−24.0000			−0.761234
$995$	−3.00000	+	6.00000i	−0.0951064	+	0.190213i
$996$		0			0
$997$			15.0000i			0.475055i	0.971381	+	0.237527i	$0.0763369\pi$
$997$			15.0000i			0.475055i	−0.971381	+	0.237527i	$0.923663\pi$
$998$			12.0000i			0.379853i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.c.b.109.2	yes	2
3.2	odd	2		270.2.c.a.109.1	✓	2
4.3	odd	2		2160.2.f.e.1729.1		2
5.2	odd	4		1350.2.a.b.1.1		1
5.3	odd	4		1350.2.a.v.1.1		1
5.4	even	2	inner	270.2.c.b.109.1	yes	2
9.2	odd	6		810.2.i.d.109.1		4
9.4	even	3		810.2.i.c.379.1		4
9.5	odd	6		810.2.i.d.379.2		4
9.7	even	3		810.2.i.c.109.2		4
12.11	even	2		2160.2.f.d.1729.2		2
15.2	even	4		1350.2.a.l.1.1		1
15.8	even	4		1350.2.a.j.1.1		1
15.14	odd	2		270.2.c.a.109.2	yes	2
20.19	odd	2		2160.2.f.e.1729.2		2
45.4	even	6		810.2.i.c.379.2		4
45.14	odd	6		810.2.i.d.379.1		4
45.29	odd	6		810.2.i.d.109.2		4
45.34	even	6		810.2.i.c.109.1		4
60.59	even	2		2160.2.f.d.1729.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.c.a.109.1	✓	2	3.2	odd	2
270.2.c.a.109.2	yes	2	15.14	odd	2
270.2.c.b.109.1	yes	2	5.4	even	2	inner
270.2.c.b.109.2	yes	2	1.1	even	1	trivial
810.2.i.c.109.1		4	45.34	even	6
810.2.i.c.109.2		4	9.7	even	3
810.2.i.c.379.1		4	9.4	even	3
810.2.i.c.379.2		4	45.4	even	6
810.2.i.d.109.1		4	9.2	odd	6
810.2.i.d.109.2		4	45.29	odd	6
810.2.i.d.379.1		4	45.14	odd	6
810.2.i.d.379.2		4	9.5	odd	6
1350.2.a.b.1.1		1	5.2	odd	4
1350.2.a.j.1.1		1	15.8	even	4
1350.2.a.l.1.1		1	15.2	even	4
1350.2.a.v.1.1		1	5.3	odd	4
2160.2.f.d.1729.1		2	60.59	even	2
2160.2.f.d.1729.2		2	12.11	even	2
2160.2.f.e.1729.1		2	4.3	odd	2
2160.2.f.e.1729.2		2	20.19	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:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	270.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +4.00000i q^{7} -1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} +5.00000 q^{11} +3.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +6.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} +5.00000i q^{22} -1.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -3.00000 q^{26} -4.00000i q^{28} -9.00000 q^{29} -5.00000 q^{31} +1.00000i q^{32} -1.00000 q^{34} +(8.00000 + 4.00000i) q^{35} -2.00000i q^{37} +6.00000i q^{38} +(-2.00000 - 1.00000i) q^{40} +2.00000 q^{41} -1.00000i q^{43} -5.00000 q^{44} +1.00000 q^{46} -13.0000i q^{47} -9.00000 q^{49} +(4.00000 - 3.00000i) q^{50} -3.00000i q^{52} +(5.00000 - 10.0000i) q^{55} +4.00000 q^{56} -9.00000i q^{58} -4.00000 q^{59} +8.00000 q^{61} -5.00000i q^{62} -1.00000 q^{64} +(6.00000 + 3.00000i) q^{65} -4.00000i q^{67} -1.00000i q^{68} +(-4.00000 + 8.00000i) q^{70} +6.00000 q^{71} +2.00000i q^{73} +2.00000 q^{74} -6.00000 q^{76} +20.0000i q^{77} -9.00000 q^{79} +(1.00000 - 2.00000i) q^{80} +2.00000i q^{82} -4.00000i q^{83} +(2.00000 + 1.00000i) q^{85} +1.00000 q^{86} -5.00000i q^{88} -14.0000 q^{89} -12.0000 q^{91} +1.00000i q^{92} +13.0000 q^{94} +(6.00000 - 12.0000i) q^{95} -10.0000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{5} + 4 q^{10} + 10 q^{11} - 8 q^{14} + 2 q^{16} + 12 q^{19} - 2 q^{20} - 6 q^{25} - 6 q^{26} - 18 q^{29} - 10 q^{31} - 2 q^{34} + 16 q^{35} - 4 q^{40} + 4 q^{41} - 10 q^{44} + 2 q^{46}+ \cdots + 12 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^5 + 4 * q^10 + 10 * q^11 - 8 * q^14 + 2 * q^16 + 12 * q^19 - 2 * q^20 - 6 * q^25 - 6 * q^26 - 18 * q^29 - 10 * q^31 - 2 * q^34 + 16 * q^35 - 4 * q^40 + 4 * q^41 - 10 * q^44 + 2 * q^46 - 18 * q^49 + 8 * q^50 + 10 * q^55 + 8 * q^56 - 8 * q^59 + 16 * q^61 - 2 * q^64 + 12 * q^65 - 8 * q^70 + 12 * q^71 + 4 * q^74 - 12 * q^76 - 18 * q^79 + 2 * q^80 + 4 * q^85 + 2 * q^86 - 28 * q^89 - 24 * q^91 + 26 * q^94 + 12 * q^95

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