LMFDB - Embedded newform 1875.1.h.c.1124.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,1,Mod(374,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 3]))

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

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

chi := DirichletCharacter("1875.374");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1875.h (of order $10$, degree $4$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.935746898687$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 375)
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.140625.1

Embedding invariants

Embedding label			1124.1
Root			$0.809017 - 0.587785i$ of defining polynomial
Character	$\chi$	$=$	1875.1124
Dual form			1875.1.h.c.749.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 1.53884i) q^{2} +(0.809017 - 0.587785i) q^{3} +(-1.30902 + 0.951057i) q^{4} +(1.30902 + 0.951057i) q^{6} +(-0.809017 - 0.587785i) q^{8} +(0.309017 - 0.951057i) q^{9} +O(q^{10})$	\(q+(0.500000 + 1.53884i) q^{2} +(0.809017 - 0.587785i) q^{3} +(-1.30902 + 0.951057i) q^{4} +(1.30902 + 0.951057i) q^{6} +(-0.809017 - 0.587785i) q^{8} +(0.309017 - 0.951057i) q^{9} +(-0.500000 + 1.53884i) q^{12} +(0.500000 + 0.363271i) q^{17} +1.61803 q^{18} +(1.30902 + 0.951057i) q^{19} +(-0.190983 - 0.587785i) q^{23} -1.00000 q^{24} +(-0.309017 - 0.951057i) q^{27} +(-0.500000 - 0.363271i) q^{31} -1.00000 q^{32} +(-0.309017 + 0.951057i) q^{34} +(0.500000 + 1.53884i) q^{36} +(-0.809017 + 2.48990i) q^{38} +(0.809017 - 0.587785i) q^{46} +(-1.30902 + 0.951057i) q^{47} +1.00000 q^{49} +0.618034 q^{51} +(-1.30902 + 0.951057i) q^{53} +(1.30902 - 0.951057i) q^{54} +1.61803 q^{57} +(-0.500000 - 1.53884i) q^{61} +(0.309017 - 0.951057i) q^{62} +(-0.500000 - 1.53884i) q^{64} -1.00000 q^{68} +(-0.500000 - 0.363271i) q^{69} +(-0.809017 + 0.587785i) q^{72} -2.61803 q^{76} +(1.30902 - 0.951057i) q^{79} +(-0.809017 - 0.587785i) q^{81} +(-1.30902 - 0.951057i) q^{83} +(0.809017 + 0.587785i) q^{92} -0.618034 q^{93} +(-2.11803 - 1.53884i) q^{94} +(-0.809017 + 0.587785i) q^{96} +(0.500000 + 1.53884i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + q^{3} - 3 q^{4} + 3 q^{6} - q^{8} - q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 + q^3 - 3 * q^4 + 3 * q^6 - q^8 - q^9	$ 4 q + 2 q^{2} + q^{3} - 3 q^{4} + 3 q^{6} - q^{8} - q^{9} - 2 q^{12} + 2 q^{17} + 2 q^{18} + 3 q^{19} - 3 q^{23} - 4 q^{24} + q^{27} - 2 q^{31} - 4 q^{32} + q^{34} + 2 q^{36} - q^{38} + q^{46} - 3 q^{47} + 4 q^{49} - 2 q^{51} - 3 q^{53} + 3 q^{54} + 2 q^{57} - 2 q^{61} - q^{62} - 2 q^{64} - 4 q^{68} - 2 q^{69} - q^{72} - 6 q^{76} + 3 q^{79} - q^{81} - 3 q^{83} + q^{92} + 2 q^{93} - 4 q^{94} - q^{96} + 2 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + q^3 - 3 * q^4 + 3 * q^6 - q^8 - q^9 - 2 * q^12 + 2 * q^17 + 2 * q^18 + 3 * q^19 - 3 * q^23 - 4 * q^24 + q^27 - 2 * q^31 - 4 * q^32 + q^34 + 2 * q^36 - q^38 + q^46 - 3 * q^47 + 4 * q^49 - 2 * q^51 - 3 * q^53 + 3 * q^54 + 2 * q^57 - 2 * q^61 - q^62 - 2 * q^64 - 4 * q^68 - 2 * q^69 - q^72 - 6 * q^76 + 3 * q^79 - q^81 - 3 * q^83 + q^92 + 2 * q^93 - 4 * q^94 - q^96 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$.

$n$	$626$	$1252$
$\chi(n)$	$-1$	$e\left(\frac{9}{10}\right)$

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 $

$2$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$2$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$3$	0.809017	−	0.587785i	0.809017	−	0.587785i
$3$	0.809017	−	0.587785i	0.809017	−	0.587785i
$4$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$5$		0			0
$5$		0			0
$6$	1.30902	+	0.951057i	1.30902	+	0.951057i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$9$	0.309017	−	0.951057i	0.309017	−	0.951057i
$10$		0			0
$11$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$11$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$12$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$13$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$13$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$17$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$18$	1.61803			1.61803
$19$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$19$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$23$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$24$	−1.00000			−1.00000
$25$		0			0
$26$		0			0
$27$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$28$		0			0
$29$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$29$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$30$		0			0
$31$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$31$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$32$	−1.00000			−1.00000
$33$		0			0
$34$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$35$		0			0
$36$	0.500000	+	1.53884i	0.500000	+	1.53884i
$37$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$37$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$38$	−0.809017	+	2.48990i	−0.809017	+	2.48990i
$39$		0			0
$40$		0			0
$41$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$41$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$	0.809017	−	0.587785i	0.809017	−	0.587785i
$47$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$47$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$48$		0			0
$49$	1.00000			1.00000
$50$		0			0
$51$	0.618034			0.618034
$52$		0			0
$53$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$53$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$54$	1.30902	−	0.951057i	1.30902	−	0.951057i
$55$		0			0
$56$		0			0
$57$	1.61803			1.61803
$58$		0			0
$59$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$59$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$60$		0			0
$61$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$61$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$62$	0.309017	−	0.951057i	0.309017	−	0.951057i
$63$		0			0
$64$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$65$		0			0
$66$		0			0
$67$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$67$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$68$	−1.00000			−1.00000
$69$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$70$		0			0
$71$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$71$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$72$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$73$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$73$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$74$		0			0
$75$		0			0
$76$	−2.61803			−2.61803
$77$		0			0
$78$		0			0
$79$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$79$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$80$		0			0
$81$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$82$		0			0
$83$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$83$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$89$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$90$		0			0
$91$		0			0
$92$	0.809017	+	0.587785i	0.809017	+	0.587785i
$93$	−0.618034			−0.618034
$94$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$95$		0			0
$96$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$97$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$97$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$98$	0.500000	+	1.53884i	0.500000	+	1.53884i
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	0.309017	+	0.951057i	0.309017	+	0.951057i
$103$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$103$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$104$		0			0
$105$		0			0
$106$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$107$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$107$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$108$	1.30902	+	0.951057i	1.30902	+	0.951057i
$109$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$109$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$113$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$114$	0.809017	+	2.48990i	0.809017	+	2.48990i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$122$	2.11803	−	1.53884i	2.11803	−	1.53884i
$123$		0			0
$124$	1.00000			1.00000
$125$		0			0
$126$		0			0
$127$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$127$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$128$	1.30902	−	0.951057i	1.30902	−	0.951057i
$129$		0			0
$130$		0			0
$131$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$131$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$137$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$137$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$138$	0.309017	−	0.951057i	0.309017	−	0.951057i
$139$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$139$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$140$		0			0
$141$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	0.809017	−	0.587785i	0.809017	−	0.587785i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$151$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$152$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$153$	0.500000	−	0.363271i	0.500000	−	0.363271i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$	2.11803	+	1.53884i	2.11803	+	1.53884i
$159$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$160$		0			0
$161$		0			0
$162$	0.500000	−	1.53884i	0.500000	−	1.53884i
$163$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$163$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$164$		0			0
$165$		0			0
$166$	0.809017	−	2.48990i	0.809017	−	2.48990i
$167$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$167$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$168$		0			0
$169$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$170$		0			0
$171$	1.30902	−	0.951057i	1.30902	−	0.951057i
$172$		0			0
$173$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$173$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$179$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$180$		0			0
$181$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$181$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$182$		0			0
$183$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$184$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$185$		0			0
$186$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$187$		0			0
$188$	0.809017	−	2.48990i	0.809017	−	2.48990i
$189$		0			0
$190$		0			0
$191$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$191$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$192$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$197$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$197$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$198$		0			0
$199$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$199$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$205$		0			0
$206$		0			0
$207$	−0.618034			−0.618034
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$211$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$212$	0.809017	−	2.48990i	0.809017	−	2.48990i
$213$		0			0
$214$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$215$		0			0
$216$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$217$		0			0
$218$	1.00000			1.00000
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$223$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$224$		0			0
$225$		0			0
$226$	2.61803			2.61803
$227$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$227$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$228$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$229$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$229$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$233$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.500000	−	1.53884i	0.500000	−	1.53884i
$238$		0			0
$239$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$239$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$240$		0			0
$241$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$241$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$242$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$243$	−1.00000			−1.00000
$244$	2.11803	+	1.53884i	2.11803	+	1.53884i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	0.190983	+	0.587785i	0.190983	+	0.587785i
$249$	−1.61803			−1.61803
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.809017	+	0.587785i	0.809017	+	0.587785i
$257$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$257$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$263$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$269$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$270$		0			0
$271$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$271$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$272$		0			0
$273$		0			0
$274$	−3.23607			−3.23607
$275$		0			0
$276$	1.00000			1.00000
$277$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$277$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$278$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$279$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$280$		0			0
$281$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$281$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$282$	−2.61803			−2.61803
$283$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$283$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$289$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$293$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$294$	1.30902	+	0.951057i	1.30902	+	0.951057i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−0.809017	−	2.48990i	−0.809017	−	2.48990i
$303$		0			0
$304$		0			0
$305$		0			0
$306$	0.809017	+	0.587785i	0.809017	+	0.587785i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$311$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$312$		0			0
$313$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$313$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$314$		0			0
$315$		0			0
$316$	−0.809017	+	2.48990i	−0.809017	+	2.48990i
$317$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$317$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$318$	−2.61803			−2.61803
$319$		0			0
$320$		0			0
$321$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$322$		0			0
$323$	0.309017	+	0.951057i	0.309017	+	0.951057i
$324$	1.61803			1.61803
$325$		0			0
$326$		0			0
$327$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$331$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$332$	2.61803			2.61803
$333$		0			0
$334$	0.809017	−	2.48990i	0.809017	−	2.48990i
$335$		0			0
$336$		0			0
$337$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$337$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$338$	0.500000	−	1.53884i	0.500000	−	1.53884i
$339$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$340$		0			0
$341$		0			0
$342$	2.11803	+	1.53884i	2.11803	+	1.53884i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	2.61803	−	1.90211i	2.61803	−	1.90211i
$347$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$347$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$348$		0			0
$349$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$349$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$353$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$		0			0
$361$	0.500000	+	1.53884i	0.500000	+	1.53884i
$362$	−0.809017	+	2.48990i	−0.809017	+	2.48990i
$363$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$364$		0			0
$365$		0			0
$366$	0.809017	−	2.48990i	0.809017	−	2.48990i
$367$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$367$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.809017	−	0.587785i	0.809017	−	0.587785i
$373$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$373$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$374$		0			0
$375$		0			0
$376$	1.61803			1.61803
$377$		0			0
$378$		0			0
$379$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$379$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$383$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$384$	0.500000	−	1.53884i	0.500000	−	1.53884i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$389$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$390$		0			0
$391$	0.118034	−	0.363271i	0.118034	−	0.363271i
$392$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$393$		0			0
$394$	0.809017	+	0.587785i	0.809017	+	0.587785i
$395$		0			0
$396$		0			0
$397$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$397$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$398$	0.309017	+	0.951057i	0.309017	+	0.951057i
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$409$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$409$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$410$		0			0
$411$	0.618034	+	1.90211i	0.618034	+	1.90211i
$412$		0			0
$413$		0			0
$414$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$415$		0			0
$416$		0			0
$417$	0.500000	+	0.363271i	0.500000	+	0.363271i
$418$		0			0
$419$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$419$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$420$		0			0
$421$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$421$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$422$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$423$	0.500000	+	1.53884i	0.500000	+	1.53884i
$424$	1.61803			1.61803
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.809017	−	0.587785i	0.809017	−	0.587785i
$429$		0			0
$430$		0			0
$431$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$431$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$432$		0			0
$433$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$433$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$434$		0			0
$435$		0			0
$436$	0.309017	+	0.951057i	0.309017	+	0.951057i
$437$	0.309017	−	0.951057i	0.309017	−	0.951057i
$438$		0			0
$439$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$439$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$440$		0			0
$441$	0.309017	−	0.951057i	0.309017	−	0.951057i
$442$		0			0
$443$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$443$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$	0.809017	+	2.48990i	0.809017	+	2.48990i
$453$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$454$	0.809017	−	0.587785i	0.809017	−	0.587785i
$455$		0			0
$456$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$459$	0.190983	−	0.587785i	0.190983	−	0.587785i
$460$		0			0
$461$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$461$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$462$		0			0
$463$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$463$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$464$		0			0
$465$		0			0
$466$	−1.00000	+	3.07768i	−1.00000	+	3.07768i
$467$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$467$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$	2.61803			2.61803
$475$		0			0
$476$		0			0
$477$	0.500000	+	1.53884i	0.500000	+	1.53884i
$478$		0			0
$479$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$479$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$480$		0			0
$481$		0			0
$482$	1.00000			1.00000
$483$		0			0
$484$	0.500000	−	1.53884i	0.500000	−	1.53884i
$485$		0			0
$486$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$487$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$487$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$488$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$489$		0			0
$490$		0			0
$491$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$491$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−0.809017	−	2.48990i	−0.809017	−	2.48990i
$499$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$499$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$500$		0			0
$501$	−1.61803			−1.61803
$502$		0			0
$503$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$503$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.00000			−1.00000
$508$		0			0
$509$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$509$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	0.500000	−	1.53884i	0.500000	−	1.53884i
$514$	0.809017	+	2.48990i	0.809017	+	2.48990i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$520$		0			0
$521$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$521$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$522$		0			0
$523$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$523$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$524$		0			0
$525$		0			0
$526$	−1.00000			−1.00000
$527$	−0.118034	−	0.363271i	−0.118034	−	0.363271i
$528$		0			0
$529$	0.500000	−	0.363271i	0.500000	−	0.363271i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$541$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$542$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$543$	1.61803			1.61803
$544$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$545$		0			0
$546$		0			0
$547$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$547$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$548$	−1.00000	−	3.07768i	−1.00000	−	3.07768i
$549$	−1.61803			−1.61803
$550$		0			0
$551$		0			0
$552$	0.190983	+	0.587785i	0.190983	+	0.587785i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$557$	−2.00000			−2.00000			−1.00000			$\pi$
$557$	−2.00000			−2.00000			−1.00000			$\pi$
$558$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$563$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$564$	−0.809017	−	2.48990i	−0.809017	−	2.48990i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$569$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$570$		0			0
$571$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$571$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−1.61803			−1.61803
$577$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$577$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$578$	0.809017	−	0.587785i	0.809017	−	0.587785i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$587$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$587$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$588$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$589$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$590$		0			0
$591$	0.190983	−	0.587785i	0.190983	−	0.587785i
$592$		0			0
$593$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$593$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.500000	−	0.363271i	0.500000	−	0.363271i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$601$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$602$		0			0
$603$		0			0
$604$	2.11803	−	1.53884i	2.11803	−	1.53884i
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$613$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$613$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$617$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$618$		0			0
$619$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$619$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$620$		0			0
$621$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$631$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$632$	−1.61803			−1.61803
$633$	0.500000	+	0.363271i	0.500000	+	0.363271i
$634$	−1.00000	+	3.07768i	−1.00000	+	3.07768i
$635$		0			0
$636$	−0.809017	−	2.48990i	−0.809017	−	2.48990i
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$641$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$642$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$647$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$647$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$648$	0.309017	+	0.951057i	0.309017	+	0.951057i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$653$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$654$	0.809017	−	0.587785i	0.809017	−	0.587785i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$659$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$660$		0			0
$661$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$661$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$662$	−0.809017	+	2.48990i	−0.809017	+	2.48990i
$663$		0			0
$664$	0.500000	+	1.53884i	0.500000	+	1.53884i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	2.61803			2.61803
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$673$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$674$		0			0
$675$		0			0
$676$	1.61803			1.61803
$677$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$677$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$678$	2.11803	−	1.53884i	2.11803	−	1.53884i
$679$		0			0
$680$		0			0
$681$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$682$		0			0
$683$	0.500000	+	0.363271i	0.500000	+	0.363271i	0.809017	−	0.587785i	$-0.200000\pi$
$683$	0.500000	+	0.363271i	0.500000	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$684$	−0.809017	+	2.48990i	−0.809017	+	2.48990i
$685$		0			0
$686$		0			0
$687$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$691$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$692$	2.61803	+	1.90211i	2.61803	+	1.90211i
$693$		0			0
$694$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	0.309017	+	0.951057i	0.309017	+	0.951057i
$699$	2.00000			2.00000
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.809017	+	0.587785i	0.809017	+	0.587785i
$707$		0			0
$708$		0			0
$709$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$709$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$710$		0			0
$711$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$712$		0			0
$713$	−0.118034	+	0.363271i	−0.118034	+	0.363271i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$719$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$720$		0			0
$721$		0			0
$722$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$723$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$724$	−2.61803			−2.61803
$725$		0			0
$726$	−1.61803			−1.61803
$727$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$727$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$728$		0			0
$729$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$730$		0			0
$731$		0			0
$732$	2.61803			2.61803
$733$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$733$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$734$		0			0
$735$		0			0
$736$	0.190983	+	0.587785i	0.190983	+	0.587785i
$737$		0			0
$738$		0			0
$739$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$739$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.00000			−2.00000			−1.00000			$\pi$
$743$	−2.00000			−2.00000			−1.00000			$\pi$
$744$	0.500000	+	0.363271i	0.500000	+	0.363271i
$745$		0			0
$746$		0			0
$747$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$751$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−2.61803	−	1.90211i	−2.61803	−	1.90211i
$759$		0			0
$760$		0			0
$761$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$761$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$767$		0			0
$768$	1.00000			1.00000
$769$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$769$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$770$		0			0
$771$	1.30902	−	0.951057i	1.30902	−	0.951057i
$772$		0			0
$773$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$773$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	0.618034			0.618034
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$787$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$788$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$789$	0.190983	+	0.587785i	0.190983	+	0.587785i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$797$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$797$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$798$		0			0
$799$	−1.00000			−1.00000
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$809$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$810$		0			0
$811$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$811$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$812$		0			0
$813$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−2.61803			−2.61803
$819$		0			0
$820$		0			0
$821$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$821$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$822$	−2.61803	+	1.90211i	−2.61803	+	1.90211i
$823$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$823$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.500000	+	1.53884i	0.500000	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$827$	0.500000	+	1.53884i	0.500000	+	1.53884i	−0.309017	+	0.951057i	$0.600000\pi$
$828$	0.809017	−	0.587785i	0.809017	−	0.587785i
$829$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$829$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.500000	+	0.363271i	0.500000	+	0.363271i
$834$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$835$		0			0
$836$		0			0
$837$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$838$		0			0
$839$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$839$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$840$		0			0
$841$	0.309017	−	0.951057i	0.309017	−	0.951057i
$842$	2.11803	+	1.53884i	2.11803	+	1.53884i
$843$		0			0
$844$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$845$		0			0
$846$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$853$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$854$		0			0
$855$		0			0
$856$	0.500000	+	0.363271i	0.500000	+	0.363271i
$857$	1.61803			1.61803			0.809017	−	0.587785i	$-0.200000\pi$
$857$	1.61803			1.61803			0.809017	+	0.587785i	$0.200000\pi$
$858$		0			0
$859$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$859$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$863$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$864$	0.309017	+	0.951057i	0.309017	+	0.951057i
$865$		0			0
$866$		0			0
$867$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$873$		0			0
$874$	1.61803			1.61803
$875$		0			0
$876$		0			0
$877$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$877$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$878$	−2.61803	+	1.90211i	−2.61803	+	1.90211i
$879$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$880$		0			0
$881$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$881$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$882$	1.61803			1.61803
$883$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$883$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$884$		0			0
$885$		0			0
$886$	0.809017	+	2.48990i	0.809017	+	2.48990i
$887$	0.500000	−	1.53884i	0.500000	−	1.53884i	−0.309017	−	0.951057i	$-0.600000\pi$
$887$	0.500000	−	1.53884i	0.500000	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.61803			−2.61803
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−1.00000			−1.00000
$902$		0			0
$903$		0			0
$904$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$905$		0			0
$906$	−2.11803	−	1.53884i	−2.11803	−	1.53884i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$	0.809017	+	0.587785i	0.809017	+	0.587785i
$909$		0			0
$910$		0			0
$911$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$911$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.309017	−	0.951057i	0.309017	−	0.951057i
$917$		0			0
$918$	1.00000			1.00000
$919$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$919$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$929$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$930$		0			0
$931$	1.30902	+	0.951057i	1.30902	+	0.951057i
$932$	−3.23607			−3.23607
$933$		0			0
$934$	0.809017	−	2.48990i	0.809017	−	2.48990i
$935$		0			0
$936$		0			0
$937$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$937$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$941$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$947$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$948$	0.809017	+	2.48990i	0.809017	+	2.48990i
$949$		0			0
$950$		0			0
$951$	2.00000			2.00000
$952$		0			0
$953$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$953$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$954$	−2.11803	+	1.53884i	−2.11803	+	1.53884i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$962$		0			0
$963$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$964$	0.309017	+	0.951057i	0.309017	+	0.951057i
$965$		0			0
$966$		0			0
$967$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$967$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$968$	1.00000			1.00000
$969$	0.809017	+	0.587785i	0.809017	+	0.587785i
$970$		0			0
$971$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$971$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$972$	1.30902	−	0.951057i	1.30902	−	0.951057i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	0.809017	−	0.587785i	$-0.200000\pi$
$977$	−0.190983	−	0.587785i	−0.190983	−	0.587785i	−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$982$		0			0
$983$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−0.309017	−	0.951057i	$-0.600000\pi$
$983$	−1.30902	−	0.951057i	−1.30902	−	0.951057i	−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$991$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$992$	0.500000	+	0.363271i	0.500000	+	0.363271i
$993$	1.61803			1.61803
$994$		0			0
$995$		0			0
$996$	2.11803	−	1.53884i	2.11803	−	1.53884i
$997$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$997$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$998$	−0.809017	−	2.48990i	−0.809017	−	2.48990i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.1.h.c.1124.1		4
3.2	odd	2		1875.1.h.b.1124.1		4
5.2	odd	4		1875.1.j.f.1376.1		8
5.3	odd	4		1875.1.j.f.1376.2		8
5.4	even	2		1875.1.h.b.1124.1		4
15.2	even	4		1875.1.j.f.1376.2		8
15.8	even	4		1875.1.j.f.1376.1		8
15.14	odd	2	CM	1875.1.h.c.1124.1		4
25.2	odd	20		375.1.c.a.251.4	yes	4
25.3	odd	20		1875.1.j.f.1751.1		8
25.4	even	10		1875.1.h.b.749.1		4
25.6	even	5		1875.1.h.a.1499.1		4
25.8	odd	20		1875.1.j.e.1001.1		8
25.9	even	10		1875.1.h.d.374.1		4
25.11	even	5		375.1.d.b.374.2		2
25.12	odd	20		1875.1.j.e.251.1		8
25.13	odd	20		1875.1.j.e.251.2		8
25.14	even	10		375.1.d.a.374.1		2
25.16	even	5		1875.1.h.a.374.1		4
25.17	odd	20		1875.1.j.e.1001.2		8
25.19	even	10		1875.1.h.d.1499.1		4
25.21	even	5	inner	1875.1.h.c.749.1		4
25.22	odd	20		1875.1.j.f.1751.2		8
25.23	odd	20		375.1.c.a.251.1	✓	4
75.2	even	20		375.1.c.a.251.1	✓	4
75.8	even	20		1875.1.j.e.1001.2		8
75.11	odd	10		375.1.d.a.374.1		2
75.14	odd	10		375.1.d.b.374.2		2
75.17	even	20		1875.1.j.e.1001.1		8
75.23	even	20		375.1.c.a.251.4	yes	4
75.29	odd	10	inner	1875.1.h.c.749.1		4
75.38	even	20		1875.1.j.e.251.1		8
75.41	odd	10		1875.1.h.d.374.1		4
75.44	odd	10		1875.1.h.a.1499.1		4
75.47	even	20		1875.1.j.f.1751.1		8
75.53	even	20		1875.1.j.f.1751.2		8
75.56	odd	10		1875.1.h.d.1499.1		4
75.59	odd	10		1875.1.h.a.374.1		4
75.62	even	20		1875.1.j.e.251.2		8
75.71	odd	10		1875.1.h.b.749.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
375.1.c.a.251.1	✓	4	25.23	odd	20
375.1.c.a.251.1	✓	4	75.2	even	20
375.1.c.a.251.4	yes	4	25.2	odd	20
375.1.c.a.251.4	yes	4	75.23	even	20
375.1.d.a.374.1		2	25.14	even	10
375.1.d.a.374.1		2	75.11	odd	10
375.1.d.b.374.2		2	25.11	even	5
375.1.d.b.374.2		2	75.14	odd	10
1875.1.h.a.374.1		4	25.16	even	5
1875.1.h.a.374.1		4	75.59	odd	10
1875.1.h.a.1499.1		4	25.6	even	5
1875.1.h.a.1499.1		4	75.44	odd	10
1875.1.h.b.749.1		4	25.4	even	10
1875.1.h.b.749.1		4	75.71	odd	10
1875.1.h.b.1124.1		4	3.2	odd	2
1875.1.h.b.1124.1		4	5.4	even	2
1875.1.h.c.749.1		4	25.21	even	5	inner
1875.1.h.c.749.1		4	75.29	odd	10	inner
1875.1.h.c.1124.1		4	1.1	even	1	trivial
1875.1.h.c.1124.1		4	15.14	odd	2	CM
1875.1.h.d.374.1		4	25.9	even	10
1875.1.h.d.374.1		4	75.41	odd	10
1875.1.h.d.1499.1		4	25.19	even	10
1875.1.h.d.1499.1		4	75.56	odd	10
1875.1.j.e.251.1		8	25.12	odd	20
1875.1.j.e.251.1		8	75.38	even	20
1875.1.j.e.251.2		8	25.13	odd	20
1875.1.j.e.251.2		8	75.62	even	20
1875.1.j.e.1001.1		8	25.8	odd	20
1875.1.j.e.1001.1		8	75.17	even	20
1875.1.j.e.1001.2		8	25.17	odd	20
1875.1.j.e.1001.2		8	75.8	even	20
1875.1.j.f.1376.1		8	5.2	odd	4
1875.1.j.f.1376.1		8	15.8	even	4
1875.1.j.f.1376.2		8	5.3	odd	4
1875.1.j.f.1376.2		8	15.2	even	4
1875.1.j.f.1751.1		8	25.3	odd	20
1875.1.j.f.1751.1		8	75.47	even	20
1875.1.j.f.1751.2		8	25.22	odd	20
1875.1.j.f.1751.2		8	75.53	even	20

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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1875.h (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 1.53884i) q^{2} +(0.809017 - 0.587785i) q^{3} +(-1.30902 + 0.951057i) q^{4} +(1.30902 + 0.951057i) q^{6} +(-0.809017 - 0.587785i) q^{8} +(0.309017 - 0.951057i) q^{9} +O(q^{10})\)	\(q+(0.500000 + 1.53884i) q^{2} +(0.809017 - 0.587785i) q^{3} +(-1.30902 + 0.951057i) q^{4} +(1.30902 + 0.951057i) q^{6} +(-0.809017 - 0.587785i) q^{8} +(0.309017 - 0.951057i) q^{9} +(-0.500000 + 1.53884i) q^{12} +(0.500000 + 0.363271i) q^{17} +1.61803 q^{18} +(1.30902 + 0.951057i) q^{19} +(-0.190983 - 0.587785i) q^{23} -1.00000 q^{24} +(-0.309017 - 0.951057i) q^{27} +(-0.500000 - 0.363271i) q^{31} -1.00000 q^{32} +(-0.309017 + 0.951057i) q^{34} +(0.500000 + 1.53884i) q^{36} +(-0.809017 + 2.48990i) q^{38} +(0.809017 - 0.587785i) q^{46} +(-1.30902 + 0.951057i) q^{47} +1.00000 q^{49} +0.618034 q^{51} +(-1.30902 + 0.951057i) q^{53} +(1.30902 - 0.951057i) q^{54} +1.61803 q^{57} +(-0.500000 - 1.53884i) q^{61} +(0.309017 - 0.951057i) q^{62} +(-0.500000 - 1.53884i) q^{64} -1.00000 q^{68} +(-0.500000 - 0.363271i) q^{69} +(-0.809017 + 0.587785i) q^{72} -2.61803 q^{76} +(1.30902 - 0.951057i) q^{79} +(-0.809017 - 0.587785i) q^{81} +(-1.30902 - 0.951057i) q^{83} +(0.809017 + 0.587785i) q^{92} -0.618034 q^{93} +(-2.11803 - 1.53884i) q^{94} +(-0.809017 + 0.587785i) q^{96} +(0.500000 + 1.53884i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + q^{3} - 3 q^{4} + 3 q^{6} - q^{8} - q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 + q^3 - 3 * q^4 + 3 * q^6 - q^8 - q^9	\( 4 q + 2 q^{2} + q^{3} - 3 q^{4} + 3 q^{6} - q^{8} - q^{9} - 2 q^{12} + 2 q^{17} + 2 q^{18} + 3 q^{19} - 3 q^{23} - 4 q^{24} + q^{27} - 2 q^{31} - 4 q^{32} + q^{34} + 2 q^{36} - q^{38} + q^{46} - 3 q^{47} + 4 q^{49} - 2 q^{51} - 3 q^{53} + 3 q^{54} + 2 q^{57} - 2 q^{61} - q^{62} - 2 q^{64} - 4 q^{68} - 2 q^{69} - q^{72} - 6 q^{76} + 3 q^{79} - q^{81} - 3 q^{83} + q^{92} + 2 q^{93} - 4 q^{94} - q^{96} + 2 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + q^3 - 3 * q^4 + 3 * q^6 - q^8 - q^9 - 2 * q^12 + 2 * q^17 + 2 * q^18 + 3 * q^19 - 3 * q^23 - 4 * q^24 + q^27 - 2 * q^31 - 4 * q^32 + q^34 + 2 * q^36 - q^38 + q^46 - 3 * q^47 + 4 * q^49 - 2 * q^51 - 3 * q^53 + 3 * q^54 + 2 * q^57 - 2 * q^61 - q^62 - 2 * q^64 - 4 * q^68 - 2 * q^69 - q^72 - 6 * q^76 + 3 * q^79 - q^81 - 3 * q^83 + q^92 + 2 * q^93 - 4 * q^94 - q^96 + 2 * q^98

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