LMFDB - Embedded newform 3311.1.gb.a.937.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3311,1,Mod(62,3311)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3311, base_ring=CyclotomicField(210))

chi = DirichletCharacter(H, H._module([105, 147, 95]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3311.62");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3311 = 7 \cdot 11 \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3311.gb (of order $210$, degree $48$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$1.65240425683$
Analytic rank:	$0$
Dimension:	$48$
Coefficient field:	$\Q(\zeta_{210})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{48} - x^{47} + x^{46} + x^{43} - x^{42} + 2 x^{41} - x^{40} + x^{39} + x^{36} - x^{35} + x^{34} + \cdots + 1 $ x^48 - x^47 + x^46 + x^43 - x^42 + 2x^41 - x^40 + x^39 + x^36 - x^35 + x^34 - x^33 + x^32 - x^31 - x^28 - x^26 - x^24 - x^22 - x^20 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 + x^9 - x^8 + 2x^7 - x^6 + x^5 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{210}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{210} - \cdots)$

Embedding invariants

Embedding label			937.1
Root			$0.337330 - 0.941386i$ of defining polynomial
Character	$\chi$	$=$	3311.937
Dual form			3311.1.gb.a.2470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.984327 - 0.369424i) q^{2} +(0.0793539 - 0.0693294i) q^{4} +(0.913545 - 0.406737i) q^{7} +(-0.445712 + 0.828271i) q^{8} +(-0.575617 + 0.817719i) q^{9} +O(q^{10})$	\(q+(0.984327 - 0.369424i) q^{2} +(0.0793539 - 0.0693294i) q^{4} +(0.913545 - 0.406737i) q^{7} +(-0.445712 + 0.828271i) q^{8} +(-0.575617 + 0.817719i) q^{9} +(-0.134233 + 0.990950i) q^{11} +(0.748969 - 0.737848i) q^{14} +(-0.146887 + 1.08437i) q^{16} +(-0.264510 + 1.01755i) q^{18} +(0.233951 + 1.02501i) q^{22} +(-0.494561 + 0.458885i) q^{23} +(0.842721 + 0.538351i) q^{25} +(0.0442946 - 0.0956117i) q^{28} +(-0.0265555 - 0.0137819i) q^{29} +(0.0467069 + 0.204636i) q^{32} +(0.0110146 + 0.104796i) q^{36} +(-0.355157 - 0.0373286i) q^{37} +(-0.420357 + 0.907359i) q^{43} +(0.0580501 + 0.0879421i) q^{44} +(-0.317286 + 0.634396i) q^{46} +(0.669131 - 0.743145i) q^{49} +(1.02839 + 0.218592i) q^{50} +(0.260813 + 0.0636481i) q^{53} +(-0.0702897 + 0.937951i) q^{56} +(-0.0312307 - 0.00375563i) q^{58} +(-0.193256 + 0.981148i) q^{63} +(-0.481258 - 0.729074i) q^{64} +(1.90772 - 0.588455i) q^{67} +(0.377301 + 0.174794i) q^{71} +(-0.420734 - 0.841234i) q^{72} +(-0.363381 + 0.0944602i) q^{74} +(0.280427 + 0.959875i) q^{77} +(1.34860 - 1.21428i) q^{79} +(-0.337330 - 0.941386i) q^{81} +(-0.0785688 + 1.04843i) q^{86} +(-0.760946 - 0.552860i) q^{88} +(-0.00743108 + 0.0707020i) q^{92} +(0.384108 - 0.978690i) q^{98} +(-0.733052 - 0.680173i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 2 q^{2} + 6 q^{7} - 21 q^{8} + q^{9}+O(q^{10}) $ 48 * q - 2 * q^2 + 6 * q^7 - 21 * q^8 + q^9	$ 48 q - 2 q^{2} + 6 q^{7} - 21 q^{8} + q^{9} - 2 q^{11} - 9 q^{14} + 3 q^{16} - 6 q^{18} + 2 q^{22} - q^{23} + q^{25} - 10 q^{28} - q^{29} - 4 q^{32} - 7 q^{36} - 5 q^{37} + q^{43} + 23 q^{44} + 4 q^{46} + 6 q^{49} - q^{50} + 10 q^{53} - 15 q^{56} + 4 q^{58} + q^{63} - 21 q^{64} + q^{67} - 7 q^{71} - 4 q^{72} - 14 q^{74} + q^{77} + 2 q^{79} - q^{81} + 13 q^{86} - 7 q^{88} + 20 q^{92} + q^{98} + 4 q^{99}+O(q^{100}) $ 48 * q - 2 * q^2 + 6 * q^7 - 21 * q^8 + q^9 - 2 * q^11 - 9 * q^14 + 3 * q^16 - 6 * q^18 + 2 * q^22 - q^23 + q^25 - 10 * q^28 - q^29 - 4 * q^32 - 7 * q^36 - 5 * q^37 + q^43 + 23 * q^44 + 4 * q^46 + 6 * q^49 - q^50 + 10 * q^53 - 15 * q^56 + 4 * q^58 + q^63 - 21 * q^64 + q^67 - 7 * q^71 - 4 * q^72 - 14 * q^74 + q^77 + 2 * q^79 - q^81 + 13 * q^86 - 7 * q^88 + 20 * q^92 + q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3311\mathbb{Z}\right)^\times$.

$n$	$904$	$1893$	$2927$
$\chi(n)$	$e\left(\frac{1}{10}\right)$	$-1$	$e\left(\frac{23}{42}\right)$

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 $

$2$	0.984327	−	0.369424i	0.984327	−	0.369424i	0.193256	−	0.981148i	$-0.438095\pi$
$2$	0.984327	−	0.369424i	0.984327	−	0.369424i	0.791071	+	0.611724i	$0.209524\pi$
$3$		0			0		−0.460642	−	0.887586i	$-0.652381\pi$
$3$		0			0		0.460642	+	0.887586i	$0.347619\pi$
$4$	0.0793539	−	0.0693294i	0.0793539	−	0.0693294i
$5$		0			0		−0.959875	−	0.280427i	$-0.909524\pi$
$5$		0			0		0.959875	+	0.280427i	$0.0904762\pi$
$6$		0			0
$7$	0.913545	−	0.406737i	0.913545	−	0.406737i
$7$	0.913545	−	0.406737i	0.913545	−	0.406737i
$8$	−0.445712	+	0.828271i	−0.445712	+	0.828271i
$9$	−0.575617	+	0.817719i	−0.575617	+	0.817719i
$10$		0			0
$11$	−0.134233	+	0.990950i	−0.134233	+	0.990950i
$11$	−0.134233	+	0.990950i	−0.134233	+	0.990950i
$12$		0			0
$13$		0			0		0.379225	−	0.925304i	$-0.376190\pi$
$13$		0			0		−0.379225	+	0.925304i	$0.623810\pi$
$14$	0.748969	−	0.737848i	0.748969	−	0.737848i
$15$		0			0
$16$	−0.146887	+	1.08437i	−0.146887	+	1.08437i
$17$		0			0		0.611724	−	0.791071i	$-0.290476\pi$
$17$		0			0		−0.611724	+	0.791071i	$0.709524\pi$
$18$	−0.264510	+	1.01755i	−0.264510	+	1.01755i
$19$		0			0		0.599822	−	0.800134i	$-0.295238\pi$
$19$		0			0		−0.599822	+	0.800134i	$0.704762\pi$
$20$		0			0
$21$		0			0
$22$	0.233951	+	1.02501i	0.233951	+	1.02501i
$23$	−0.494561	+	0.458885i	−0.494561	+	0.458885i	−0.887586	−	0.460642i	$-0.847619\pi$
$23$	−0.494561	+	0.458885i	−0.494561	+	0.458885i	0.393025	+	0.919528i	$0.371429\pi$
$24$		0			0
$25$	0.842721	+	0.538351i	0.842721	+	0.538351i
$26$		0			0
$27$		0			0
$28$	0.0442946	−	0.0956117i	0.0442946	−	0.0956117i
$29$	−0.0265555	−	0.0137819i	−0.0265555	−	0.0137819i	0.447313	−	0.894377i	$-0.352381\pi$
$29$	−0.0265555	−	0.0137819i	−0.0265555	−	0.0137819i	−0.473869	+	0.880596i	$0.657143\pi$
$30$		0			0
$31$		0			0		−0.772417	−	0.635116i	$-0.780952\pi$
$31$		0			0		0.772417	+	0.635116i	$0.219048\pi$
$32$	0.0467069	+	0.204636i	0.0467069	+	0.204636i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.0110146	+	0.104796i	0.0110146	+	0.104796i
$37$	−0.355157	−	0.0373286i	−0.355157	−	0.0373286i	−0.0747301	−	0.997204i	$-0.523810\pi$
$37$	−0.355157	−	0.0373286i	−0.355157	−	0.0373286i	−0.280427	+	0.959875i	$0.590476\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.266037	−	0.963963i	$-0.414286\pi$
$41$		0			0		−0.266037	+	0.963963i	$0.585714\pi$
$42$		0			0
$43$	−0.420357	+	0.907359i	−0.420357	+	0.907359i
$43$	−0.420357	+	0.907359i	−0.420357	+	0.907359i
$44$	0.0580501	+	0.0879421i	0.0580501	+	0.0879421i
$45$		0			0
$46$	−0.317286	+	0.634396i	−0.317286	+	0.634396i
$47$		0			0		−0.393025	−	0.919528i	$-0.628571\pi$
$47$		0			0		0.393025	+	0.919528i	$0.371429\pi$
$48$		0			0
$49$	0.669131	−	0.743145i	0.669131	−	0.743145i
$50$	1.02839	+	0.218592i	1.02839	+	0.218592i
$51$		0			0
$52$		0			0
$53$	0.260813	+	0.0636481i	0.260813	+	0.0636481i	0.365341	−	0.930874i	$-0.380952\pi$
$53$	0.260813	+	0.0636481i	0.260813	+	0.0636481i	−0.104528	+	0.994522i	$0.533333\pi$
$54$		0			0
$55$		0			0
$56$	−0.0702897	+	0.937951i	−0.0702897	+	0.937951i
$57$		0			0
$58$	−0.0312307	−	0.00375563i	−0.0312307	−	0.00375563i
$59$		0			0		−0.473869	−	0.880596i	$-0.657143\pi$
$59$		0			0		0.473869	+	0.880596i	$0.342857\pi$
$60$		0			0
$61$		0			0		0.772417	−	0.635116i	$-0.219048\pi$
$61$		0			0		−0.772417	+	0.635116i	$0.780952\pi$
$62$		0			0
$63$	−0.193256	+	0.981148i	−0.193256	+	0.981148i
$64$	−0.481258	−	0.729074i	−0.481258	−	0.729074i
$65$		0			0
$66$		0			0
$67$	1.90772	−	0.588455i	1.90772	−	0.588455i	0.936235	−	0.351375i	$-0.114286\pi$
$67$	1.90772	−	0.588455i	1.90772	−	0.588455i	0.971490	−	0.237080i	$-0.0761905\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.377301	+	0.174794i	0.377301	+	0.174794i	0.599822	−	0.800134i	$-0.295238\pi$
$71$	0.377301	+	0.174794i	0.377301	+	0.174794i	−0.222521	+	0.974928i	$0.571429\pi$
$72$	−0.420734	−	0.841234i	−0.420734	−	0.841234i
$73$		0			0		0.646600	−	0.762830i	$-0.276190\pi$
$73$		0			0		−0.646600	+	0.762830i	$0.723810\pi$
$74$	−0.363381	+	0.0944602i	−0.363381	+	0.0944602i
$75$		0			0
$76$		0			0
$77$	0.280427	+	0.959875i	0.280427	+	0.959875i
$78$		0			0
$79$	1.34860	−	1.21428i	1.34860	−	1.21428i	0.393025	−	0.919528i	$-0.371429\pi$
$79$	1.34860	−	1.21428i	1.34860	−	1.21428i	0.955573	−	0.294755i	$-0.0952381\pi$
$80$		0			0
$81$	−0.337330	−	0.941386i	−0.337330	−	0.941386i
$82$		0			0
$83$		0			0		0.986491	−	0.163818i	$-0.0523810\pi$
$83$		0			0		−0.986491	+	0.163818i	$0.947619\pi$
$84$		0			0
$85$		0			0
$86$	−0.0785688	+	1.04843i	−0.0785688	+	1.04843i
$87$		0			0
$88$	−0.760946	−	0.552860i	−0.760946	−	0.552860i
$89$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$89$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$90$		0			0
$91$		0			0
$92$	−0.00743108	+	0.0707020i	−0.00743108	+	0.0707020i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.995974	−	0.0896393i	$-0.971429\pi$
$97$		0			0		0.995974	+	0.0896393i	$0.0285714\pi$
$98$	0.384108	−	0.978690i	0.384108	−	0.978690i
$99$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$100$	0.104197	−	0.0157051i	0.104197	−	0.0157051i
$101$		0			0		−0.486989	−	0.873408i	$-0.661905\pi$
$101$		0			0		0.486989	+	0.873408i	$0.338095\pi$
$102$		0			0
$103$		0			0		−0.999552	−	0.0299155i	$-0.990476\pi$
$103$		0			0		0.999552	+	0.0299155i	$0.00952381\pi$
$104$		0			0
$105$		0			0
$106$	0.280238	−	0.0336999i	0.280238	−	0.0336999i
$107$	−1.83720	−	0.0825089i	−1.83720	−	0.0825089i	−0.900969	−	0.433884i	$-0.857143\pi$
$107$	−1.83720	−	0.0825089i	−1.83720	−	0.0825089i	−0.936235	+	0.351375i	$0.885714\pi$
$108$		0			0
$109$	1.35289	+	1.45807i	1.35289	+	1.45807i	0.753071	+	0.657939i	$0.228571\pi$
$109$	1.35289	+	1.45807i	1.35289	+	1.45807i	0.599822	+	0.800134i	$0.295238\pi$
$110$		0			0
$111$		0			0
$112$	0.306863	+	1.05036i	0.306863	+	1.05036i
$113$	−0.226809	−	1.24982i	−0.226809	−	1.24982i	−0.873408	−	0.486989i	$-0.838095\pi$
$113$	−0.226809	−	1.24982i	−0.226809	−	1.24982i	0.646600	−	0.762830i	$-0.276190\pi$
$114$		0			0
$115$		0			0
$116$	−0.00306277		0.000747433i	−0.00306277		0.000747433i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.963963	−	0.266037i	−0.963963	−	0.266037i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	0.172233	+	1.03716i	0.172233	+	1.03716i
$127$	−0.678903	−	0.448140i	−0.678903	−	0.448140i	0.163818	−	0.986491i	$-0.447619\pi$
$127$	−0.678903	−	0.448140i	−0.678903	−	0.448140i	−0.842721	+	0.538351i	$0.819048\pi$
$128$	−0.912864	−	0.663235i	−0.912864	−	0.663235i
$129$		0			0
$130$		0			0
$131$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$131$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$132$		0			0
$133$		0			0
$134$	1.66044	−	1.28399i	1.66044	−	1.28399i
$135$		0			0
$136$		0			0
$137$	1.01451	−	0.969973i	1.01451	−	0.969973i	0.0149594	−	0.999888i	$-0.495238\pi$
$137$	1.01451	−	0.969973i	1.01451	−	0.969973i	0.999552	+	0.0299155i	$0.00952381\pi$
$138$		0			0
$139$		0			0		−0.850680	−	0.525684i	$-0.823810\pi$
$139$		0			0		0.850680	+	0.525684i	$0.176190\pi$
$140$		0			0
$141$		0			0
$142$	0.435961	+	0.0326707i	0.435961	+	0.0326707i
$143$		0			0
$144$	−0.802157	−	0.744293i	−0.802157	−	0.744293i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.0307711	+	0.0216607i	−0.0307711	+	0.0216607i
$149$	−1.28051	−	0.713976i	−1.28051	−	0.713976i	−0.309017	−	0.951057i	$-0.600000\pi$
$149$	−1.28051	−	0.713976i	−1.28051	−	0.713976i	−0.971490	+	0.237080i	$0.923810\pi$
$150$		0			0
$151$	−0.0639213	+	1.42332i	−0.0639213	+	1.42332i	0.669131	+	0.743145i	$0.266667\pi$
$151$	−0.0639213	+	1.42332i	−0.0639213	+	1.42332i	−0.733052	+	0.680173i	$0.761905\pi$
$152$		0			0
$153$		0			0
$154$	0.630633	+	0.841234i	0.630633	+	0.841234i
$155$		0			0
$156$		0			0
$157$		0			0		0.119394	−	0.992847i	$-0.461905\pi$
$157$		0			0		−0.119394	+	0.992847i	$0.538095\pi$
$158$	0.878876	−	1.69346i	0.878876	−	1.69346i
$159$		0			0
$160$		0			0
$161$	−0.265158	+	0.620369i	−0.265158	+	0.620369i
$162$	−0.679814	−	0.802014i	−0.679814	−	0.802014i
$163$	0.880439	+	0.619768i	0.880439	+	0.619768i	0.925304	−	0.379225i	$-0.123810\pi$
$163$	0.880439	+	0.619768i	0.880439	+	0.619768i	−0.0448648	+	0.998993i	$0.514286\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.611724	−	0.791071i	$-0.709524\pi$
$167$		0			0		0.611724	+	0.791071i	$0.290476\pi$
$168$		0			0
$169$	−0.712376	−	0.701798i	−0.712376	−	0.701798i
$170$		0			0
$171$		0			0
$172$	0.0295497	+	0.101146i	0.0295497	+	0.101146i
$173$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$173$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$174$		0			0
$175$	0.988831	+	0.149042i	0.988831	+	0.149042i
$176$	−1.05484	−	0.291116i	−1.05484	−	0.291116i
$177$		0			0
$178$		0			0
$179$	−0.0595032	+	0.00625404i	−0.0595032	+	0.00625404i	−0.134233	−	0.990950i	$-0.542857\pi$
$179$	−0.0595032	+	0.00625404i	−0.0595032	+	0.00625404i	0.0747301	+	0.997204i	$0.476190\pi$
$180$		0			0
$181$		0			0		−0.0149594	−	0.999888i	$-0.504762\pi$
$181$		0			0		0.0149594	+	0.999888i	$0.495238\pi$
$182$		0			0
$183$		0			0
$184$	−0.159650	−	0.614161i	−0.159650	−	0.614161i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.227135	−	0.665031i	0.227135	−	0.665031i	−0.772417	−	0.635116i	$-0.780952\pi$
$191$	0.227135	−	0.665031i	0.227135	−	0.665031i	0.999552	−	0.0299155i	$-0.00952381\pi$
$192$		0			0
$193$	0.342231	−	0.911872i	0.342231	−	0.911872i	−0.646600	−	0.762830i	$-0.723810\pi$
$193$	0.342231	−	0.911872i	0.342231	−	0.911872i	0.988831	−	0.149042i	$-0.0476190\pi$
$194$		0			0
$195$		0			0
$196$	0.00157633	−	0.105362i	0.00157633	−	0.105362i
$197$	1.80186	−	0.707179i	1.80186	−	0.707179i	0.809017	−	0.587785i	$-0.200000\pi$
$197$	1.80186	−	0.707179i	1.80186	−	0.707179i	0.992847	−	0.119394i	$-0.0380952\pi$
$198$	−0.972835	−	0.398705i	−0.972835	−	0.398705i
$199$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$199$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$200$	−0.821511	+	0.458052i	−0.821511	+	0.458052i
$201$		0			0
$202$		0			0
$203$	−0.0298653	−	0.00178927i	−0.0298653	−	0.00178927i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.0905618	−	0.668554i	−0.0905618	−	0.668554i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.331801	−	0.198242i	−0.331801	−	0.198242i	0.337330	−	0.941386i	$-0.390476\pi$
$211$	−0.331801	−	0.198242i	−0.331801	−	0.198242i	−0.669131	+	0.743145i	$0.733333\pi$
$212$	0.0251092	−	0.0130313i	0.0251092	−	0.0130313i
$213$		0			0
$214$	−1.83889	+	0.597492i	−1.83889	+	0.597492i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	1.87034	+	0.935428i	1.87034	+	0.935428i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.880596	−	0.473869i	$-0.842857\pi$
$223$		0			0		0.880596	+	0.473869i	$0.157143\pi$
$224$	0.125902	+	0.167947i	0.125902	+	0.167947i
$225$	−0.925304	+	0.379225i	−0.925304	+	0.379225i
$226$	−0.684967	−	1.14644i	−0.684967	−	1.14644i
$227$		0			0		0.842721	−	0.538351i	$-0.180952\pi$
$227$		0			0		−0.842721	+	0.538351i	$0.819048\pi$
$228$		0			0
$229$		0			0		0.163818	−	0.986491i	$-0.447619\pi$
$229$		0			0		−0.163818	+	0.986491i	$0.552381\pi$
$230$		0			0
$231$		0			0
$232$	0.0232512	−	0.0158524i	0.0232512	−	0.0158524i
$233$	−0.998210	+	0.0598042i	−0.998210	+	0.0598042i	−0.550897	−	0.834573i	$-0.685714\pi$
$233$	−0.998210	+	0.0598042i	−0.998210	+	0.0598042i	−0.447313	+	0.894377i	$0.647619\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.606862	−	0.454935i	−0.606862	−	0.454935i	0.251587	−	0.967835i	$-0.419048\pi$
$239$	−0.606862	−	0.454935i	−0.606862	−	0.454935i	−0.858449	+	0.512899i	$0.828571\pi$
$240$		0			0
$241$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$241$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$242$	−1.04714	+	0.0942439i	−1.04714	+	0.0942439i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.669131	−	0.743145i	$-0.733333\pi$
$251$		0			0		0.669131	+	0.743145i	$0.266667\pi$
$252$	0.0526869	+	0.0912563i	0.0526869	+	0.0912563i
$253$	−0.388346	−	0.551683i	−0.388346	−	0.551683i
$254$	−0.833816	−	0.190313i	−0.833816	−	0.190313i
$255$		0			0
$256$	−0.301464	−	0.0831989i	−0.301464	−	0.0831989i
$257$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$257$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$258$		0			0
$259$	−0.339635	+	0.110354i	−0.339635	+	0.110354i
$260$		0			0
$261$	0.0265555	−	0.0137819i	0.0265555	−	0.0137819i
$262$		0			0
$263$	1.44329	+	0.984017i	1.44329	+	0.984017i	0.995974	+	0.0896393i	$0.0285714\pi$
$263$	1.44329	+	0.984017i	1.44329	+	0.984017i	0.447313	+	0.894377i	$0.352381\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.110588	−	0.178958i	0.110588	−	0.178958i
$269$		0			0		0.858449	−	0.512899i	$-0.171429\pi$
$269$		0			0		−0.858449	+	0.512899i	$0.828571\pi$
$270$		0			0
$271$		0			0		−0.762830	−	0.646600i	$-0.776190\pi$
$271$		0			0		0.762830	+	0.646600i	$0.223810\pi$
$272$		0			0
$273$		0			0
$274$	0.640280	−	1.32956i	0.640280	−	1.32956i
$275$	−0.646600	+	0.762830i	−0.646600	+	0.762830i
$276$		0			0
$277$	−0.00312737	+	0.209034i	−0.00312737	+	0.209034i	0.992847	+	0.119394i	$0.0380952\pi$
$277$	−0.00312737	+	0.209034i	−0.00312737	+	0.209034i	−0.995974	+	0.0896393i	$0.971429\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.796440	+	1.42841i	−0.796440	+	1.42841i	0.104528	+	0.994522i	$0.466667\pi$
$281$	−0.796440	+	1.42841i	−0.796440	+	1.42841i	−0.900969	+	0.433884i	$0.857143\pi$
$282$		0			0
$283$		0			0		0.538351	−	0.842721i	$-0.319048\pi$
$283$		0			0		−0.538351	+	0.842721i	$0.680952\pi$
$284$	0.0420587	−	0.0122874i	0.0420587	−	0.0122874i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.194220	−	0.0795990i	−0.194220	−	0.0795990i
$289$	−0.251587	−	0.967835i	−0.251587	−	0.967835i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.178557	−	0.983930i	$-0.442857\pi$
$293$		0			0		−0.178557	+	0.983930i	$0.557143\pi$
$294$		0			0
$295$		0			0
$296$	0.189216	−	0.277529i	0.189216	−	0.277529i
$297$		0			0
$298$	−1.52420	−	0.229736i	−1.52420	−	0.229736i
$299$		0			0
$300$		0			0
$301$	−0.0149594	+	0.999888i	−0.0149594	+	0.999888i
$302$	0.462888	+	1.42462i	0.462888	+	1.42462i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$307$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$308$	0.0888006	+	0.0567280i	0.0888006	+	0.0567280i
$309$		0			0
$310$		0			0
$311$		0			0		−0.646600	−	0.762830i	$-0.723810\pi$
$311$		0			0		0.646600	+	0.762830i	$0.276190\pi$
$312$		0			0
$313$		0			0		0.0598042	−	0.998210i	$-0.480952\pi$
$313$		0			0		−0.0598042	+	0.998210i	$0.519048\pi$
$314$		0			0
$315$		0			0
$316$	0.0228310	−	0.189856i	0.0228310	−	0.189856i
$317$	−0.580986	+	0.607664i	−0.580986	+	0.607664i	−0.946327	−	0.323210i	$-0.895238\pi$
$317$	−0.580986	+	0.607664i	−0.580986	+	0.607664i	0.365341	+	0.930874i	$0.380952\pi$
$318$		0			0
$319$	0.0172218	−	0.0244652i	0.0172218	−	0.0244652i
$320$		0			0
$321$		0			0
$322$	−0.0318233	+	0.708601i	−0.0318233	+	0.708601i
$323$		0			0
$324$	−0.0920343	−	0.0513158i	−0.0920343	−	0.0513158i
$325$		0			0
$326$	1.09560	+	0.284798i	1.09560	+	0.284798i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.530586	−	0.0397619i	−0.530586	−	0.0397619i	−0.193256	−	0.981148i	$-0.561905\pi$
$331$	−0.530586	−	0.0397619i	−0.530586	−	0.0397619i	−0.337330	+	0.941386i	$0.609524\pi$
$332$		0			0
$333$	0.234959	−	0.268932i	0.234959	−	0.268932i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.692006	−	1.55427i	−0.692006	−	1.55427i	−0.826239	−	0.563320i	$-0.809524\pi$
$337$	−0.692006	−	1.55427i	−0.692006	−	1.55427i	0.134233	−	0.990950i	$-0.457143\pi$
$338$	−0.960472	−	0.427630i	−0.960472	−	0.427630i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.309017	−	0.951057i	0.309017	−	0.951057i
$344$	−0.564181	−	0.752590i	−0.564181	−	0.752590i
$345$		0			0
$346$		0			0
$347$	−0.0536726	−	0.323210i	−0.0536726	−	0.323210i	0.946327	−	0.323210i	$-0.104762\pi$
$347$	−0.0536726	−	0.323210i	−0.0536726	−	0.323210i	−1.00000			$\pi$
$348$		0			0
$349$		0			0		0.999552	−	0.0299155i	$-0.00952381\pi$
$349$		0			0		−0.999552	+	0.0299155i	$0.990476\pi$
$350$	1.02839	−	0.218592i	1.02839	−	0.218592i
$351$		0			0
$352$	−0.209054	+	0.0188152i	−0.209054	+	0.0188152i
$353$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$353$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.0562602	+	0.0281379i	−0.0562602	+	0.0281379i
$359$	−1.72799	+	0.340361i	−1.72799	+	0.340361i	−0.955573	−	0.294755i	$-0.904762\pi$
$359$	−1.72799	+	0.340361i	−1.72799	+	0.340361i	−0.772417	+	0.635116i	$0.780952\pi$
$360$		0			0
$361$	−0.280427	−	0.959875i	−0.280427	−	0.959875i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.992847	−	0.119394i	$-0.0380952\pi$
$367$		0			0		−0.992847	+	0.119394i	$0.961905\pi$
$368$	−0.424955	−	0.603690i	−0.424955	−	0.603690i
$369$		0			0
$370$		0			0
$371$	0.264152	−	0.0479366i	0.264152	−	0.0479366i
$372$		0			0
$373$	1.63402	−	0.246289i	1.63402	−	0.246289i	0.733052	−	0.680173i	$-0.238095\pi$
$373$	1.63402	−	0.246289i	1.63402	−	0.246289i	0.900969	+	0.433884i	$0.142857\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.893682	−	0.934718i	−0.893682	−	0.934718i	0.104528	−	0.994522i	$-0.466667\pi$
$379$	−0.893682	−	0.934718i	−0.893682	−	0.934718i	−0.998210	+	0.0598042i	$0.980952\pi$
$380$		0			0
$381$		0			0
$382$	−0.0221030	−	0.738517i	−0.0221030	−	0.738517i
$383$		0			0		0.512899	−	0.858449i	$-0.328571\pi$
$383$		0			0		−0.512899	+	0.858449i	$0.671429\pi$
$384$		0			0
$385$		0			0
$386$		−	1.02401i		−	1.02401i
$387$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$388$		0			0
$389$	0.920357	−	0.0413333i	0.920357	−	0.0413333i	0.420357	−	0.907359i	$-0.361905\pi$
$389$	0.920357	−	0.0413333i	0.920357	−	0.0413333i	0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	0.317286	+	0.885450i	0.317286	+	0.885450i
$393$		0			0
$394$	1.51237	−	1.36175i	1.51237	−	1.36175i
$395$		0			0
$396$	−0.105327	−	0.00315230i	−0.105327	−	0.00315230i
$397$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$397$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$398$		0			0
$399$		0			0
$400$	−0.707555	+	0.834742i	−0.707555	+	0.834742i
$401$	−0.225076	−	0.450027i	−0.225076	−	0.450027i	0.753071	−	0.657939i	$-0.228571\pi$
$401$	−0.225076	−	0.450027i	−0.225076	−	0.450027i	−0.978148	+	0.207912i	$0.933333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−0.0300582	+	0.00927172i	−0.0300582	+	0.00927172i
$407$	0.0846647	−	0.346932i	0.0846647	−	0.346932i
$408$		0			0
$409$		0			0		−0.550897	−	0.834573i	$-0.685714\pi$
$409$		0			0		0.550897	+	0.834573i	$0.314286\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−0.336122	−	0.624620i	−0.336122	−	0.624620i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$419$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$420$		0			0
$421$	0.425304	+	1.24525i	0.425304	+	1.24525i	0.925304	+	0.379225i	$0.123810\pi$
$421$	0.425304	+	1.24525i	0.425304	+	1.24525i	−0.500000	+	0.866025i	$0.666667\pi$
$422$	−0.399836	−	0.0725595i	−0.399836	−	0.0725595i
$423$		0			0
$424$	−0.168965	+	0.187655i	−0.168965	+	0.187655i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.151510	+	0.120825i	−0.151510	+	0.120825i
$429$		0			0
$430$		0			0
$431$	−1.52194	−	0.494510i	−1.52194	−	0.494510i	−0.575617	−	0.817719i	$-0.695238\pi$
$431$	−1.52194	−	0.494510i	−1.52194	−	0.494510i	−0.946327	+	0.323210i	$0.895238\pi$
$432$		0			0
$433$		0			0		0.701798	−	0.712376i	$-0.252381\pi$
$433$		0			0		−0.701798	+	0.712376i	$0.747619\pi$
$434$		0			0
$435$		0			0
$436$	0.208445	+	0.0219084i	0.208445	+	0.0219084i
$437$		0			0
$438$		0			0
$439$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$439$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$440$		0			0
$441$	0.222521	+	0.974928i	0.222521	+	0.974928i
$442$		0			0
$443$	−0.703502	−	1.13843i	−0.703502	−	1.13843i	−0.983930	−	0.178557i	$-0.942857\pi$
$443$	−0.703502	−	1.13843i	−0.703502	−	1.13843i	0.280427	−	0.959875i	$-0.409524\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.736192	−	0.470297i	−0.736192	−	0.470297i
$449$	1.96208	+	0.0293548i	1.96208	+	0.0293548i	0.983930	−	0.178557i	$-0.0571429\pi$
$449$	1.96208	+	0.0293548i	1.96208	+	0.0293548i	0.978148	+	0.207912i	$0.0666667\pi$
$450$	−0.770707	+	0.715112i	−0.770707	+	0.715112i
$451$		0			0
$452$	−0.104647	−	0.0834535i	−0.104647	−	0.0834535i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.105514	−	0.778936i	0.105514	−	0.778936i	−0.858449	−	0.512899i	$-0.828571\pi$
$457$	0.105514	−	0.778936i	0.105514	−	0.778936i	0.963963	−	0.266037i	$-0.0857143\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$461$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$462$		0			0
$463$	1.99418	−	0.149443i	1.99418	−	0.149443i	0.995974	−	0.0896393i	$-0.0285714\pi$
$463$	1.99418	−	0.149443i	1.99418	−	0.149443i	0.998210	−	0.0598042i	$-0.0190476\pi$
$464$	0.0188453	−	0.0267715i	0.0188453	−	0.0267715i
$465$		0			0
$466$	−0.960472	+	0.427630i	−0.960472	+	0.427630i
$467$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$467$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$468$		0			0
$469$	1.50345	−	1.31352i	1.50345	−	1.31352i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.842721	−	0.538351i	−0.842721	−	0.538351i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.202174	+	0.176635i	−0.202174	+	0.176635i
$478$	−0.765415	−	0.223616i	−0.765415	−	0.223616i
$479$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$479$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.0949384	+	0.0457199i	−0.0949384	+	0.0457199i
$485$		0			0
$486$		0			0
$487$	0.820112	−	0.807934i	0.820112	−	0.807934i	−0.163818	−	0.986491i	$-0.552381\pi$
$487$	0.820112	−	0.807934i	0.820112	−	0.807934i	0.983930	+	0.178557i	$0.0571429\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.480819	+	1.84967i	−0.480819	+	1.84967i	0.0448648	+	0.998993i	$0.485714\pi$
$491$	−0.480819	+	1.84967i	−0.480819	+	1.84967i	−0.525684	+	0.850680i	$0.676190\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.415777	+	0.00622047i	0.415777	+	0.00622047i
$498$		0			0
$499$	0.00536320	−	0.179198i	0.00536320	−	0.179198i	−0.992847	−	0.119394i	$-0.961905\pi$
$499$	0.00536320	−	0.179198i	0.00536320	−	0.179198i	0.998210	−	0.0598042i	$-0.0190476\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.525684	−	0.850680i	$-0.676190\pi$
$503$		0			0		0.525684	+	0.850680i	$0.323810\pi$
$504$	−0.726521	−	0.597378i	−0.726521	−	0.597378i
$505$		0			0
$506$	−0.586064	−	0.399572i	−0.586064	−	0.399572i
$507$		0			0
$508$	−0.0849429	+	0.0115063i	−0.0849429	+	0.0115063i
$509$		0			0		−0.104528	−	0.994522i	$-0.533333\pi$
$509$		0			0		0.104528	+	0.994522i	$0.466667\pi$
$510$		0			0
$511$		0			0
$512$	0.796345	−	0.0716723i	0.796345	−	0.0716723i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−0.293545	+	0.234094i	−0.293545	+	0.234094i
$519$		0			0
$520$		0			0
$521$		0			0		0.941386	−	0.337330i	$-0.109524\pi$
$521$		0			0		−0.941386	+	0.337330i	$0.890476\pi$
$522$	0.0210480	−	0.0233761i	0.0210480	−	0.0233761i
$523$		0			0		−0.978148	−	0.207912i	$-0.933333\pi$
$523$		0			0		0.978148	+	0.207912i	$0.0666667\pi$
$524$		0			0
$525$		0			0
$526$	1.78419	+	0.435409i	1.78419	+	0.435409i
$527$		0			0
$528$		0			0
$529$	−0.0407155	+	0.543310i	−0.0407155	+	0.543310i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.362895	+	1.84240i	−0.362895	+	1.84240i
$537$		0			0
$538$		0			0
$539$	0.646600	+	0.762830i	0.646600	+	0.762830i
$540$		0			0
$541$	−0.0786949	−	1.31352i	−0.0786949	−	1.31352i	−0.791071	−	0.611724i	$-0.790476\pi$
$541$	−0.0786949	−	1.31352i	−0.0786949	−	1.31352i	0.712376	−	0.701798i	$-0.247619\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.93026	+	0.501767i	−1.93026	+	0.501767i	−0.946327	+	0.323210i	$0.895238\pi$
$547$	−1.93026	+	0.501767i	−1.93026	+	0.501767i	−0.983930	+	0.178557i	$0.942857\pi$
$548$	0.0132578	−	0.147307i	0.0132578	−	0.147307i
$549$		0			0
$550$	−0.354658	+	0.989743i	−0.354658	+	0.989743i
$551$		0			0
$552$		0			0
$553$	0.738112	−	1.65783i	0.738112	−	1.65783i
$554$	0.0741437	+	0.206913i	0.0741437	+	0.206913i
$555$		0			0
$556$		0			0
$557$	1.02477	−	0.0460223i	1.02477	−	0.0460223i	0.473869	−	0.880596i	$-0.342857\pi$
$557$	1.02477	−	0.0460223i	1.02477	−	0.0460223i	0.550897	+	0.834573i	$0.314286\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.256270	+	1.70024i	−0.256270	+	1.70024i
$563$		0			0		0.512899	−	0.858449i	$-0.328571\pi$
$563$		0			0		−0.512899	+	0.858449i	$0.671429\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.691063	−	0.722795i	−0.691063	−	0.722795i
$568$	−0.312945	+	0.234600i	−0.312945	+	0.234600i
$569$	0.859434	−	0.728485i	0.859434	−	0.728485i	−0.104528	−	0.994522i	$-0.533333\pi$
$569$	0.859434	−	0.728485i	0.859434	−	0.728485i	0.963963	+	0.266037i	$0.0857143\pi$
$570$		0			0
$571$	−0.550256	+	1.40203i	−0.550256	+	1.40203i	0.337330	+	0.941386i	$0.390476\pi$
$571$	−0.550256	+	1.40203i	−0.550256	+	1.40203i	−0.887586	+	0.460642i	$0.847619\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.663818	+	0.120465i	−0.663818	+	0.120465i
$576$	0.873198	+	0.0261338i	0.873198	+	0.0261338i
$577$		0			0		−0.635116	−	0.772417i	$-0.719048\pi$
$577$		0			0		0.635116	+	0.772417i	$0.280952\pi$
$578$	−0.605185	−	0.859724i	−0.605185	−	0.859724i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.0980818	+	0.249908i	−0.0980818	+	0.249908i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.981148	−	0.193256i	$-0.0619048\pi$
$587$		0			0		−0.981148	+	0.193256i	$0.938095\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.0926460	−	0.379638i	0.0926460	−	0.379638i
$593$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$593$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$594$		0			0
$595$		0			0
$596$	−0.151113	+	0.0321200i	−0.151113	+	0.0321200i
$597$		0			0
$598$		0			0
$599$	0.246733	+	1.48580i	0.246733	+	1.48580i	0.772417	+	0.635116i	$0.219048\pi$
$599$	0.246733	+	1.48580i	0.246733	+	1.48580i	−0.525684	+	0.850680i	$0.676190\pi$
$600$		0			0
$601$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$601$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$602$	0.354658	+	0.989743i	0.354658	+	0.989743i
$603$	−0.616928	+	1.89871i	−0.616928	+	1.89871i
$604$	0.0936054	+	0.117377i	0.0936054	+	0.117377i
$605$		0			0
$606$		0			0
$607$		0			0		0.791071	−	0.611724i	$-0.209524\pi$
$607$		0			0		−0.791071	+	0.611724i	$0.790476\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.26308	−	1.44571i	1.26308	−	1.44571i	0.420357	−	0.907359i	$-0.361905\pi$
$613$	1.26308	−	1.44571i	1.26308	−	1.44571i	0.842721	−	0.538351i	$-0.180952\pi$
$614$		0			0
$615$		0			0
$616$	−0.920027	−	0.195558i	−0.920027	−	0.195558i
$617$	1.35659	+	1.25873i	1.35659	+	1.25873i	0.936235	+	0.351375i	$0.114286\pi$
$617$	1.35659	+	1.25873i	1.35659	+	1.25873i	0.420357	+	0.907359i	$0.361905\pi$
$618$		0			0
$619$		0			0		0.337330	−	0.941386i	$-0.390476\pi$
$619$		0			0		−0.337330	+	0.941386i	$0.609524\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.420357	+	0.907359i	0.420357	+	0.907359i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.811279	+	1.56321i	−0.811279	+	1.56321i	0.0149594	+	0.999888i	$0.495238\pi$
$631$	−0.811279	+	1.56321i	−0.811279	+	1.56321i	−0.826239	+	0.563320i	$0.809524\pi$
$632$	0.404670	+	1.65823i	0.404670	+	1.65823i
$633$		0			0
$634$	−0.347395	+	0.812770i	−0.347395	+	0.812770i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	0.00791383	−	0.0304439i	0.00791383	−	0.0304439i
$639$	−0.360114	+	0.207912i	−0.360114	+	0.207912i
$640$		0			0
$641$	0.943250	−	0.403165i	0.943250	−	0.403165i	0.134233	−	0.990950i	$-0.457143\pi$
$641$	0.943250	−	0.403165i	0.943250	−	0.403165i	0.809017	+	0.587785i	$0.200000\pi$
$642$		0			0
$643$		0			0		0.550897	−	0.834573i	$-0.314286\pi$
$643$		0			0		−0.550897	+	0.834573i	$0.685714\pi$
$644$	0.0219685	+	0.0676120i	0.0219685	+	0.0676120i
$645$		0			0
$646$		0			0
$647$		0			0		−0.351375	−	0.936235i	$-0.614286\pi$
$647$		0			0		0.351375	+	0.936235i	$0.385714\pi$
$648$	0.930076	+	0.140186i	0.930076	+	0.140186i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.112834	−	0.0118594i	0.112834	−	0.0118594i
$653$	−0.336182	+	1.85252i	−0.336182	+	1.85252i	0.163818	+	0.986491i	$0.447619\pi$
$653$	−0.336182	+	1.85252i	−0.336182	+	1.85252i	−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.202749	+	1.34515i	0.202749	+	1.34515i	0.826239	+	0.563320i	$0.190476\pi$
$659$	0.202749	+	1.34515i	0.202749	+	1.34515i	−0.623490	+	0.781831i	$0.714286\pi$
$660$		0			0
$661$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$661$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$662$	−0.536959	+	0.156872i	−0.536959	+	0.156872i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.131926	−	0.351517i	0.131926	−	0.351517i
$667$	0.0194576	−	0.00536996i	0.0194576	−	0.00536996i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.962316	−	0.536561i	0.962316	−	0.536561i	0.0747301	−	0.997204i	$-0.476190\pi$
$673$	0.962316	−	0.536561i	0.962316	−	0.536561i	0.887586	+	0.460642i	$0.152381\pi$
$674$	−1.25534	−	1.27427i	−1.25534	−	1.27427i
$675$		0			0
$676$	−0.105185	−	0.00630179i	−0.105185	−	0.00630179i
$677$		0			0		0.858449	−	0.512899i	$-0.171429\pi$
$677$		0			0		−0.858449	+	0.512899i	$0.828571\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.50961	+	1.02924i	1.50961	+	1.02924i	0.983930	+	0.178557i	$0.0571429\pi$
$683$	1.50961	+	1.02924i	1.50961	+	1.02924i	0.525684	+	0.850680i	$0.323810\pi$
$684$		0			0
$685$		0			0
$686$	−0.0471694	−	1.05031i	−0.0471694	−	1.05031i
$687$		0			0
$688$	−0.922164	−	0.589101i	−0.922164	−	0.589101i
$689$		0			0
$690$		0			0
$691$		0			0		−0.894377	−	0.447313i	$-0.852381\pi$
$691$		0			0		0.894377	+	0.447313i	$0.147619\pi$
$692$		0			0
$693$	−0.946327	−	0.323210i	−0.946327	−	0.323210i
$694$	−0.172233	−	0.298316i	−0.172233	−	0.298316i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.0888006	−	0.0567280i	0.0888006	−	0.0567280i
$701$	0.203133	−	0.125527i	0.203133	−	0.125527i	−0.420357	−	0.907359i	$-0.638095\pi$
$701$	0.203133	−	0.125527i	0.203133	−	0.125527i	0.623490	+	0.781831i	$0.285714\pi$
$702$		0			0
$703$		0			0
$704$	0.787077	−	0.379036i	0.787077	−	0.379036i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.48126	−	0.555925i	−1.48126	−	0.555925i	−0.525684	−	0.850680i	$-0.676190\pi$
$709$	−1.48126	−	0.555925i	−1.48126	−	0.555925i	−0.955573	+	0.294755i	$0.904762\pi$
$710$		0			0
$711$	0.216667	+	1.80174i	0.216667	+	1.80174i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.00428822	+	0.00462160i	−0.00428822	+	0.00462160i
$717$		0			0
$718$	−1.57517	+	0.973387i	−1.57517	+	0.973387i
$719$		0			0		0.842721	−	0.538351i	$-0.180952\pi$
$719$		0			0		−0.842721	+	0.538351i	$0.819048\pi$
$720$		0			0
$721$		0			0
$722$	−0.630633	−	0.841234i	−0.630633	−	0.841234i
$723$		0			0
$724$		0			0
$725$	−0.0149594	−	0.0259105i	−0.0149594	−	0.0259105i
$726$		0			0
$727$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$727$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$728$		0			0
$729$	0.963963	+	0.266037i	0.963963	+	0.266037i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.0448648	−	0.998993i	$-0.514286\pi$
$733$		0			0		0.0448648	+	0.998993i	$0.485714\pi$
$734$		0			0
$735$		0			0
$736$	−0.117004	−	0.0797720i	−0.117004	−	0.0797720i
$737$	0.327049	+	1.96945i	0.327049	+	1.96945i
$738$		0			0
$739$	−0.185527	−	1.36962i	−0.185527	−	1.36962i	−0.809017	−	0.587785i	$-0.800000\pi$
$739$	−0.185527	−	1.36962i	−0.185527	−	1.36962i	0.623490	−	0.781831i	$-0.285714\pi$
$740$		0			0
$741$		0			0
$742$	0.242303	−	0.144769i	0.242303	−	0.144769i
$743$	−0.559851	−	0.0335414i	−0.559851	−	0.0335414i	−0.222521	−	0.974928i	$-0.571429\pi$
$743$	−0.559851	−	0.0335414i	−0.559851	−	0.0335414i	−0.337330	+	0.941386i	$0.609524\pi$
$744$		0			0
$745$		0			0
$746$	1.51743	−	0.846076i	1.51743	−	0.846076i
$747$		0			0
$748$		0			0
$749$	−1.71193	+	0.671882i	−1.71193	+	0.671882i
$750$		0			0
$751$	0.634233	+	0.124924i	0.634233	+	0.124924i	0.500000	−	0.866025i	$-0.333333\pi$
$751$	0.634233	+	0.124924i	0.634233	+	0.124924i	0.134233	+	0.990950i	$0.457143\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.42665	−	0.416796i	1.42665	−	0.416796i	0.525684	−	0.850680i	$-0.323810\pi$
$757$	1.42665	−	0.416796i	1.42665	−	0.416796i	0.900969	+	0.433884i	$0.142857\pi$
$758$	−1.22498	−	0.589920i	−1.22498	−	0.589920i
$759$		0			0
$760$		0			0
$761$		0			0		−0.925304	−	0.379225i	$-0.876190\pi$
$761$		0			0		0.925304	+	0.379225i	$0.123810\pi$
$762$		0			0
$763$	1.82898	+	0.781744i	1.82898	+	0.781744i
$764$	−0.0280822	−	0.0685200i	−0.0280822	−	0.0685200i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$769$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$770$		0			0
$771$		0			0
$772$	−0.0360622	−	0.0960873i	−0.0360622	−	0.0960873i
$773$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$773$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$774$	−0.812094	−	0.667740i	−0.812094	−	0.667740i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	0.890663	−	0.380688i	0.890663	−	0.380688i
$779$		0			0
$780$		0			0
$781$	−0.223859	+	0.350423i	−0.223859	+	0.350423i
$782$		0			0
$783$		0			0
$784$	0.707555	+	0.834742i	0.707555	+	0.834742i
$785$		0			0
$786$		0			0
$787$		0			0		−0.237080	−	0.971490i	$-0.576190\pi$
$787$		0			0		0.237080	+	0.971490i	$0.423810\pi$
$788$	0.0939566	−	0.181040i	0.0939566	−	0.181040i
$789$		0			0
$790$		0			0
$791$	−0.715547	−	1.04951i	−0.715547	−	1.04951i
$792$	0.890098	−	0.304005i	0.890098	−	0.304005i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.873408	−	0.486989i	$-0.838095\pi$
$797$		0			0		0.873408	+	0.486989i	$0.161905\pi$
$798$		0			0
$799$		0			0
$800$	−0.0708052	+	0.197596i	−0.0708052	+	0.197596i
$801$		0			0
$802$	−0.387799	−	0.359825i	−0.387799	−	0.359825i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.665900	−	0.636666i	0.665900	−	0.636666i	−0.280427	−	0.959875i	$-0.590476\pi$
$809$	0.665900	−	0.636666i	0.665900	−	0.636666i	0.946327	+	0.323210i	$0.104762\pi$
$810$		0			0
$811$		0			0		−0.913545	−	0.406737i	$-0.866667\pi$
$811$		0			0		0.913545	+	0.406737i	$0.133333\pi$
$812$	−0.00249398	+	0.00192856i	−0.00249398	+	0.00192856i
$813$		0			0
$814$	−0.0448275	−	0.372772i	−0.0448275	−	0.372772i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.387862	−	0.443943i	−0.387862	−	0.443943i	0.525684	−	0.850680i	$-0.323810\pi$
$821$	−0.387862	−	0.443943i	−0.387862	−	0.443943i	−0.913545	+	0.406737i	$0.866667\pi$
$822$		0			0
$823$	−0.320476	+	0.0681193i	−0.320476	+	0.0681193i	−0.365341	−	0.930874i	$-0.619048\pi$
$823$	−0.320476	+	0.0681193i	−0.320476	+	0.0681193i	0.0448648	+	0.998993i	$0.485714\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.0706700	−	0.289586i	0.0706700	−	0.289586i	−0.925304	−	0.379225i	$-0.876190\pi$
$827$	0.0706700	−	0.289586i	0.0706700	−	0.289586i	0.995974	+	0.0896393i	$0.0285714\pi$
$828$	−0.0535369	−	0.0467738i	−0.0535369	−	0.0467738i
$829$		0			0		−0.538351	−	0.842721i	$-0.680952\pi$
$829$		0			0		0.538351	+	0.842721i	$0.319048\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.998993	−	0.0448648i	$-0.985714\pi$
$839$		0			0		0.998993	+	0.0448648i	$0.0142857\pi$
$840$		0			0
$841$	−0.575102	−	0.816987i	−0.575102	−	0.816987i
$842$	0.878664	+	1.06862i	0.878664	+	1.06862i
$843$		0			0
$844$	−0.0400737	+	0.00727230i	−0.0400737	+	0.00727230i
$845$		0			0
$846$		0			0
$847$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$848$	−0.107328	+	0.273467i	−0.107328	+	0.273467i
$849$		0			0
$850$		0			0
$851$	0.192776	−	0.144515i	0.192776	−	0.144515i
$852$		0			0
$853$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$853$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$854$		0			0
$855$		0			0
$856$	0.887203	−	1.48493i	0.887203	−	1.48493i
$857$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$857$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	−1.68077	+	0.0754837i	−1.68077	+	0.0754837i
$863$	1.34197	−	0.222849i	1.34197	−	0.222849i	0.550897	−	0.834573i	$-0.314286\pi$
$863$	1.34197	−	0.222849i	1.34197	−	0.222849i	0.791071	+	0.611724i	$0.209524\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.02227	+	1.49939i	1.02227	+	1.49939i
$870$		0			0
$871$		0			0
$872$	−1.81068	+	0.470683i	−1.81068	+	0.470683i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.68391	+	0.603401i	1.68391	+	0.603401i	0.992847	−	0.119394i	$-0.0380952\pi$
$877$	1.68391	+	0.603401i	1.68391	+	0.603401i	0.691063	+	0.722795i	$0.257143\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$881$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$882$	0.579195	+	0.877443i	0.579195	+	0.877443i
$883$	−0.348235	+	1.76797i	−0.348235	+	1.76797i	0.251587	+	0.967835i	$0.419048\pi$
$883$	−0.348235	+	1.76797i	−0.348235	+	1.76797i	−0.599822	+	0.800134i	$0.704762\pi$
$884$		0			0
$885$		0			0
$886$	−1.11304	−	0.860699i	−1.11304	−	0.860699i
$887$		0			0		−0.473869	−	0.880596i	$-0.657143\pi$
$887$		0			0		0.473869	+	0.880596i	$0.342857\pi$
$888$		0			0
$889$	−0.802484	−	0.133261i	−0.802484	−	0.133261i
$890$		0			0
$891$	0.978148	−	0.207912i	0.978148	−	0.207912i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−1.10370	−	0.234600i	−1.10370	−	0.234600i
$897$		0			0
$898$	1.94217	−	0.695944i	1.94217	−	0.695944i
$899$		0			0
$900$	−0.0471350	+	0.0942439i	−0.0471350	+	0.0942439i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.13628	+	0.369200i	1.13628	+	0.369200i
$905$		0			0
$906$		0			0
$907$	−1.73978	+	0.156583i	−1.73978	+	0.156583i	−0.913545	−	0.406737i	$-0.866667\pi$
$907$	−1.73978	+	0.156583i	−1.73978	+	0.156583i	−0.826239	+	0.563320i	$0.809524\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.84490	−	0.249908i	1.84490	−	0.249908i	0.873408	−	0.486989i	$-0.161905\pi$
$911$	1.84490	−	0.249908i	1.84490	−	0.249908i	0.971490	+	0.237080i	$0.0761905\pi$
$912$		0			0
$913$		0			0
$914$	−0.183897	−	0.805707i	−0.183897	−	0.805707i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.118526	+	0.0160554i	0.118526	+	0.0160554i	0.193256	−	0.981148i	$-0.438095\pi$
$919$	0.118526	+	0.0160554i	0.118526	+	0.0160554i	−0.0747301	+	0.997204i	$0.523810\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.279203	−	0.222657i	−0.279203	−	0.222657i
$926$	1.90772	−	0.883801i	1.90772	−	0.883801i
$927$		0			0
$928$	0.00157995	−	0.00607793i	0.00157995	−	0.00607793i
$929$		0			0		0.611724	−	0.791071i	$-0.290476\pi$
$929$		0			0		−0.611724	+	0.791071i	$0.709524\pi$
$930$		0			0
$931$		0			0
$932$	−0.0750657	+	0.0739510i	−0.0750657	+	0.0739510i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.575617	−	0.817719i	$-0.304762\pi$
$937$		0			0		−0.575617	+	0.817719i	$0.695238\pi$
$938$	0.994637	−	1.84834i	0.994637	−	1.84834i
$939$		0			0
$940$		0			0
$941$		0			0		−0.959875	−	0.280427i	$-0.909524\pi$
$941$		0			0		0.959875	+	0.280427i	$0.0904762\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−1.02839	−	0.218592i	−1.02839	−	0.218592i
$947$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$947$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.76125	−	0.784158i	1.76125	−	0.784158i	0.772417	−	0.635116i	$-0.219048\pi$
$953$	1.76125	−	0.784158i	1.76125	−	0.784158i	0.988831	−	0.149042i	$-0.0476190\pi$
$954$	−0.133753	+	0.248554i	−0.133753	+	0.248554i
$955$		0			0
$956$	−0.0796973	+	0.00597249i	−0.0796973	+	0.00597249i
$957$		0			0
$958$		0			0
$959$	0.532279	−	1.29875i	0.532279	−	1.29875i
$960$		0			0
$961$	0.193256	+	0.981148i	0.193256	+	0.981148i
$962$		0			0
$963$	1.12499	−	1.45482i	1.12499	−	1.45482i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.19281	−	0.951233i	−1.19281	−	0.951233i	−0.193256	−	0.981148i	$-0.561905\pi$
$967$	−1.19281	−	0.951233i	−1.19281	−	0.951233i	−0.999552	+	0.0299155i	$0.990476\pi$
$968$	0.650000	−	0.679847i	0.650000	−	0.679847i
$969$		0			0
$970$		0			0
$971$		0			0		−0.842721	−	0.538351i	$-0.819048\pi$
$971$		0			0		0.842721	+	0.538351i	$0.180952\pi$
$972$		0			0
$973$		0			0
$974$	0.508788	−	1.09824i	0.508788	−	1.09824i
$975$		0			0
$976$		0			0
$977$	−0.732048	−	0.601923i	−0.732048	−	0.601923i	0.193256	−	0.981148i	$-0.438095\pi$
$977$	−0.732048	−	0.601923i	−0.732048	−	0.601923i	−0.925304	+	0.379225i	$0.876190\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.97104	+	0.266996i	−1.97104	+	0.266996i
$982$	0.210031	+	1.99831i	0.210031	+	1.99831i
$983$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$983$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.208481	−	0.641640i	−0.208481	−	0.641640i
$990$		0			0
$991$	1.52446	−	1.21572i	1.52446	−	1.21572i	0.623490	−	0.781831i	$-0.285714\pi$
$991$	1.52446	−	1.21572i	1.52446	−	1.21572i	0.900969	−	0.433884i	$-0.142857\pi$
$992$		0			0
$993$		0			0
$994$	0.411558	−	0.147475i	0.411558	−	0.147475i
$995$		0			0
$996$		0			0
$997$		0			0		−0.983930	−	0.178557i	$-0.942857\pi$
$997$		0			0		0.983930	+	0.178557i	$0.0571429\pi$
$998$	−0.0609211	−	0.178371i	−0.0609211	−	0.178371i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3311.1.gb.a.937.1	✓	48
7.6	odd	2	CM	3311.1.gb.a.937.1	✓	48
11.6	odd	10		3311.1.gb.b.1238.1	yes	48
43.19	odd	42		3311.1.gb.b.2169.1	yes	48
77.6	even	10		3311.1.gb.b.1238.1	yes	48
301.62	even	42		3311.1.gb.b.2169.1	yes	48
473.105	even	210	inner	3311.1.gb.a.2470.1	yes	48
3311.2470	odd	210	inner	3311.1.gb.a.2470.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3311.1.gb.a.937.1	✓	48	1.1	even	1	trivial
3311.1.gb.a.937.1	✓	48	7.6	odd	2	CM
3311.1.gb.a.2470.1	yes	48	473.105	even	210	inner
3311.1.gb.a.2470.1	yes	48	3311.2470	odd	210	inner
3311.1.gb.b.1238.1	yes	48	11.6	odd	10
3311.1.gb.b.1238.1	yes	48	77.6	even	10
3311.1.gb.b.2169.1	yes	48	43.19	odd	42
3311.1.gb.b.2169.1	yes	48	301.62	even	42

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3311 = 7 \cdot 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3311.gb (of order \(210\), degree \(48\), minimal)

\(f(q)\)	\(=\)	\(q+(0.984327 - 0.369424i) q^{2} +(0.0793539 - 0.0693294i) q^{4} +(0.913545 - 0.406737i) q^{7} +(-0.445712 + 0.828271i) q^{8} +(-0.575617 + 0.817719i) q^{9} +O(q^{10})\)	\(q+(0.984327 - 0.369424i) q^{2} +(0.0793539 - 0.0693294i) q^{4} +(0.913545 - 0.406737i) q^{7} +(-0.445712 + 0.828271i) q^{8} +(-0.575617 + 0.817719i) q^{9} +(-0.134233 + 0.990950i) q^{11} +(0.748969 - 0.737848i) q^{14} +(-0.146887 + 1.08437i) q^{16} +(-0.264510 + 1.01755i) q^{18} +(0.233951 + 1.02501i) q^{22} +(-0.494561 + 0.458885i) q^{23} +(0.842721 + 0.538351i) q^{25} +(0.0442946 - 0.0956117i) q^{28} +(-0.0265555 - 0.0137819i) q^{29} +(0.0467069 + 0.204636i) q^{32} +(0.0110146 + 0.104796i) q^{36} +(-0.355157 - 0.0373286i) q^{37} +(-0.420357 + 0.907359i) q^{43} +(0.0580501 + 0.0879421i) q^{44} +(-0.317286 + 0.634396i) q^{46} +(0.669131 - 0.743145i) q^{49} +(1.02839 + 0.218592i) q^{50} +(0.260813 + 0.0636481i) q^{53} +(-0.0702897 + 0.937951i) q^{56} +(-0.0312307 - 0.00375563i) q^{58} +(-0.193256 + 0.981148i) q^{63} +(-0.481258 - 0.729074i) q^{64} +(1.90772 - 0.588455i) q^{67} +(0.377301 + 0.174794i) q^{71} +(-0.420734 - 0.841234i) q^{72} +(-0.363381 + 0.0944602i) q^{74} +(0.280427 + 0.959875i) q^{77} +(1.34860 - 1.21428i) q^{79} +(-0.337330 - 0.941386i) q^{81} +(-0.0785688 + 1.04843i) q^{86} +(-0.760946 - 0.552860i) q^{88} +(-0.00743108 + 0.0707020i) q^{92} +(0.384108 - 0.978690i) q^{98} +(-0.733052 - 0.680173i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{2} + 6 q^{7} - 21 q^{8} + q^{9}+O(q^{10}) \) 48 * q - 2 * q^2 + 6 * q^7 - 21 * q^8 + q^9	\( 48 q - 2 q^{2} + 6 q^{7} - 21 q^{8} + q^{9} - 2 q^{11} - 9 q^{14} + 3 q^{16} - 6 q^{18} + 2 q^{22} - q^{23} + q^{25} - 10 q^{28} - q^{29} - 4 q^{32} - 7 q^{36} - 5 q^{37} + q^{43} + 23 q^{44} + 4 q^{46} + 6 q^{49} - q^{50} + 10 q^{53} - 15 q^{56} + 4 q^{58} + q^{63} - 21 q^{64} + q^{67} - 7 q^{71} - 4 q^{72} - 14 q^{74} + q^{77} + 2 q^{79} - q^{81} + 13 q^{86} - 7 q^{88} + 20 q^{92} + q^{98} + 4 q^{99}+O(q^{100}) \) 48 * q - 2 * q^2 + 6 * q^7 - 21 * q^8 + q^9 - 2 * q^11 - 9 * q^14 + 3 * q^16 - 6 * q^18 + 2 * q^22 - q^23 + q^25 - 10 * q^28 - q^29 - 4 * q^32 - 7 * q^36 - 5 * q^37 + q^43 + 23 * q^44 + 4 * q^46 + 6 * q^49 - q^50 + 10 * q^53 - 15 * q^56 + 4 * q^58 + q^63 - 21 * q^64 + q^67 - 7 * q^71 - 4 * q^72 - 14 * q^74 + q^77 + 2 * q^79 - q^81 + 13 * q^86 - 7 * q^88 + 20 * q^92 + q^98 + 4 * q^99

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