LMFDB - Newform orbit 475.2.l.a

Newspace parameters

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.l (of order $9$, degree $6$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$1$	$-\zeta_{18}^{5}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

101.1

0.939693	+	0.342020i
−0.173648	+	0.984808i
−0.766044	−	0.642788i
−0.173648	−	0.984808i
0.939693	−	0.342020i
−0.766044	+	0.642788i

1.93969

−

1.62760i

−0.613341

−

0.223238i

0.766044

−

4.34445i

−1.55303

0.565258i

0.766044

−

1.32683i

−3.05303

−

5.28801i

−1.97178

−

1.65452i

176.1

0.826352

−

0.300767i

−0.0923963

0.524005i

−0.939693

0.788496i

0.0812519

0.460802i

−0.939693

−

1.62760i

−1.41875

2.45734i

2.55303

0.929228i

226.1

0.233956

−

1.32683i

2.20574

1.85083i

0.173648

0.0632028i

2.97178

−

2.49362i

0.173648

0.300767i

1.47178

−

2.54920i

0.918748

5.21048i

251.1

0.826352

0.300767i

−0.0923963

−

0.524005i

−0.939693

−

0.788496i

0.0812519

−

0.460802i

−0.939693

1.62760i

−1.41875

−

2.45734i

2.55303

−

0.929228i

301.1

1.93969

1.62760i

−0.613341

0.223238i

0.766044

4.34445i

−1.55303

−

0.565258i

0.766044

1.32683i

−3.05303

5.28801i

−1.97178

1.65452i

351.1

0.233956

1.32683i

2.20574

−

1.85083i

0.173648

−

0.0632028i

2.97178

2.49362i

0.173648

−

0.300767i

1.47178

2.54920i

0.918748

−

5.21048i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	475.2.l.a		6
5.b	even	2	1		19.2.e.a	✓	6
5.c	odd	4	2		475.2.u.a		12
15.d	odd	2	1		171.2.u.c		6
19.e	even	9	1	inner	475.2.l.a		6
19.e	even	9	1		9025.2.a.bd		3
19.f	odd	18	1		9025.2.a.x		3
20.d	odd	2	1		304.2.u.b		6
35.c	odd	2	1		931.2.w.a		6
35.i	odd	6	1		931.2.v.a		6
35.i	odd	6	1		931.2.x.b		6
35.j	even	6	1		931.2.v.b		6
35.j	even	6	1		931.2.x.a		6
95.d	odd	2	1		361.2.e.h		6
95.h	odd	6	1		361.2.e.a		6
95.h	odd	6	1		361.2.e.b		6
95.i	even	6	1		361.2.e.f		6
95.i	even	6	1		361.2.e.g		6
95.o	odd	18	1		361.2.a.h		3
95.o	odd	18	2		361.2.c.h		6
95.o	odd	18	1		361.2.e.a		6
95.o	odd	18	1		361.2.e.b		6
95.o	odd	18	1		361.2.e.h		6
95.p	even	18	1		19.2.e.a	✓	6
95.p	even	18	1		361.2.a.g		3
95.p	even	18	2		361.2.c.i		6
95.p	even	18	1		361.2.e.f		6
95.p	even	18	1		361.2.e.g		6
95.q	odd	36	2		475.2.u.a		12
285.bd	odd	18	1		171.2.u.c		6
285.bd	odd	18	1		3249.2.a.z		3
285.bf	even	18	1		3249.2.a.s		3
380.ba	odd	18	1		304.2.u.b		6
380.ba	odd	18	1		5776.2.a.br		3
380.bb	even	18	1		5776.2.a.bi		3
665.cv	odd	18	1		931.2.w.a		6
665.cw	odd	18	1		931.2.x.b		6
665.db	even	18	1		931.2.x.a		6
665.dc	even	18	1		931.2.v.b		6
665.df	odd	18	1		931.2.v.a		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
19.2.e.a	✓	6	5.b	even	2	1
19.2.e.a	✓	6	95.p	even	18	1
171.2.u.c		6	15.d	odd	2	1
171.2.u.c		6	285.bd	odd	18	1
304.2.u.b		6	20.d	odd	2	1
304.2.u.b		6	380.ba	odd	18	1
361.2.a.g		3	95.p	even	18	1
361.2.a.h		3	95.o	odd	18	1
361.2.c.h		6	95.o	odd	18	2
361.2.c.i		6	95.p	even	18	2
361.2.e.a		6	95.h	odd	6	1
361.2.e.a		6	95.o	odd	18	1
361.2.e.b		6	95.h	odd	6	1
361.2.e.b		6	95.o	odd	18	1
361.2.e.f		6	95.i	even	6	1
361.2.e.f		6	95.p	even	18	1
361.2.e.g		6	95.i	even	6	1
361.2.e.g		6	95.p	even	18	1
361.2.e.h		6	95.d	odd	2	1
361.2.e.h		6	95.o	odd	18	1
475.2.l.a		6	1.a	even	1	1	trivial
475.2.l.a		6	19.e	even	9	1	inner
475.2.u.a		12	5.c	odd	4	2
475.2.u.a		12	95.q	odd	36	2
931.2.v.a		6	35.i	odd	6	1
931.2.v.a		6	665.df	odd	18	1
931.2.v.b		6	35.j	even	6	1
931.2.v.b		6	665.dc	even	18	1
931.2.w.a		6	35.c	odd	2	1
931.2.w.a		6	665.cv	odd	18	1
931.2.x.a		6	35.j	even	6	1
931.2.x.a		6	665.db	even	18	1
931.2.x.b		6	35.i	odd	6	1
931.2.x.b		6	665.cw	odd	18	1
3249.2.a.s		3	285.bf	even	18	1
3249.2.a.z		3	285.bd	odd	18	1
5776.2.a.bi		3	380.bb	even	18	1
5776.2.a.br		3	380.ba	odd	18	1
9025.2.a.x		3	19.f	odd	18	1
9025.2.a.bd		3	19.e	even	9	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} - 6T_{2}^{5} + 18T_{2}^{4} - 30T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9 $ T2^6 - 6*T2^5 + 18*T2^4 - 30*T2^3 + 36*T2^2 - 27*T2 + 9 acting on $S_{2}^{\mathrm{new}}(475, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} - 6 T^{5} + 18 T^{4} - 30 T^{3} + \cdots + 9 $ T^6 - 6T^5 + 18T^4 - 30T^3 + 36T^2 - 27*T + 9
$3$	$ T^{6} - 3 T^{5} + 3 T^{4} + 8 T^{3} + \cdots + 1 $ T^6 - 3T^5 + 3T^4 + 8T^3 + 6T^2 + 3*T + 1
$5$	$ T^{6} $ T^6
$7$	$ T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1 $ T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$11$	$ T^{6} + 9 T^{4} - 18 T^{3} + 81 T^{2} + \cdots + 81 $ T^6 + 9T^4 - 18T^3 + 81T^2 - 81T + 81
$13$	$ T^{6} - 3 T^{5} + 24 T^{4} + \cdots + 1369 $ T^6 - 3T^5 + 24T^4 - 26T^3 - 114T^2 - 222*T + 1369
$17$	$ T^{6} + 3 T^{5} - 30 T^{3} + 36 T^{2} + \cdots + 9 $ T^6 + 3T^5 - 30T^3 + 36*T^2 + 9
$19$	$ T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 6859 $ T^6 + 12T^5 + 78T^4 + 385T^3 + 1482T^2 + 4332*T + 6859
$23$	$ T^{6} + 6 T^{5} + 36 T^{4} + 192 T^{3} + \cdots + 576 $ T^6 + 6T^5 + 36T^4 + 192T^3 + 576T^2 + 864*T + 576
$29$	$ T^{6} + 3 T^{5} + 36 T^{4} + \cdots + 12321 $ T^6 + 3T^5 + 36T^4 - 57T^3 - 477T^2 - 1998*T + 12321
$31$	$ T^{6} - 9 T^{5} + 75 T^{4} + \cdots + 2809 $ T^6 - 9T^5 + 75T^4 - 160T^3 + 513T^2 + 318*T + 2809
$37$	$ (T^{3} - 21 T + 17)^{2} $ (T^3 - 21*T + 17)^2
$41$	$ T^{6} - 21 T^{5} + 162 T^{4} + \cdots + 12321 $ T^6 - 21T^5 + 162T^4 - 672T^3 + 3411T^2 - 8991*T + 12321
$43$	$ T^{6} - 3 T^{5} + 60 T^{4} + \cdots + 26569 $ T^6 - 3T^5 + 60T^4 - 8T^3 - 663T^2 - 5379*T + 26569
$47$	$ T^{6} - 3 T^{5} + 54 T^{4} - 24 T^{3} + \cdots + 9 $ T^6 - 3T^5 + 54T^4 - 24T^3 - 18T^2 + 9
$53$	$ T^{6} - 3 T^{5} + 84 T^{3} + \cdots + 2601 $ T^6 - 3T^5 + 84T^3 + 387T^2 + 1377T + 2601
$59$	$ T^{6} - 12 T^{5} + 18 T^{4} + \cdots + 71289 $ T^6 - 12T^5 + 18T^4 - 159T^3 + 3006T^2 + 19224*T + 71289
$61$	$ T^{6} + 12 T^{5} + 24 T^{4} + \cdots + 32761 $ T^6 + 12T^5 + 24T^4 - 37T^3 + 984T^2 - 2172*T + 32761
$67$	$ T^{6} - 30 T^{5} + 348 T^{4} + \cdots + 179776 $ T^6 - 30T^5 + 348T^4 - 2528T^3 + 20928T^2 - 86496*T + 179776
$71$	$ T^{6} + 6 T^{5} - 36 T^{4} + \cdots + 788544 $ T^6 + 6T^5 - 36T^4 - 1536T^3 + 8352T^2 - 31968*T + 788544
$73$	$ T^{6} - 12 T^{5} + 96 T^{4} + \cdots + 4096 $ T^6 - 12T^5 + 96T^4 - 512T^3 + 768T^2 + 3072*T + 4096
$79$	$ T^{6} + 39 T^{5} + 708 T^{4} + \cdots + 654481 $ T^6 + 39T^5 + 708T^4 + 7487T^3 + 51663T^2 + 242700*T + 654481
$83$	$ T^{6} + 189 T^{4} - 918 T^{3} + \cdots + 210681 $ T^6 + 189T^4 - 918T^3 + 35721T^2 - 86751T + 210681
$89$	$ T^{6} + 12 T^{5} + 54 T^{4} + \cdots + 3249 $ T^6 + 12T^5 + 54T^4 - 300T^3 + 522T^2 - 1539*T + 3249
$97$	$ T^{6} + 18 T^{5} + 234 T^{4} + \cdots + 16129 $ T^6 + 18T^5 + 234T^4 + 1855T^3 + 9522T^2 + 20574*T + 16129
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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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.l (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} + 1) q^{2} + (\zeta_{18}^{5} - \zeta_{18}^{3} - \zeta_{18} + 1) q^{3} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18} + 1) q^{4} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18}^{2}) q^{6} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2}) q^{7} + (\zeta_{18}^{5} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + \zeta_{18}) q^{8} + (2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 3 \zeta_{18} + 2) q^{9}+O(q^{10}) \) q + (-z^5 - z^4 + z + 1) * q^2 + (z^5 - z^3 - z + 1) * q^3 + (-z^5 - 2z^4 - 2z^3 + z + 1) * q^4 + (2z^5 - z^4 + z^3 - 2z^2) * q^6 + (-z^5 - z^4 + z^2) * q^7 + (z^5 - 3z^4 - 2z^3 - 3z^2 + z) q^8 + (2z^4 - 3z^3 - 3z + 2) q^9	\( q + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} + 1) q^{2} + (\zeta_{18}^{5} - \zeta_{18}^{3} - \zeta_{18} + 1) q^{3} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18} + 1) q^{4} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18}^{2}) q^{6} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2}) q^{7} + (\zeta_{18}^{5} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + \zeta_{18}) q^{8} + (2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 3 \zeta_{18} + 2) q^{9} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} + 2 \zeta_{18}) q^{11} + (\zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18} - 1) q^{12} + (\zeta_{18}^{4} - 3 \zeta_{18}^{3} + 2 \zeta_{18} + 2) q^{13} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3}) q^{14} + ( - 3 \zeta_{18}^{2} - \zeta_{18} - 3) q^{16} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} - 2 \zeta_{18} - 1) q^{17} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{2} - 4 \zeta_{18} + 1) q^{18} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} + \zeta_{18}^{2} + 2 \zeta_{18} - 2) q^{19} + (\zeta_{18}^{4} - \zeta_{18}) q^{21} + 3 \zeta_{18} q^{22} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}) q^{23} + (4 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{24} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - 5 \zeta_{18}^{3} + \zeta_{18} + 5) q^{26} + (3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2}) q^{27} + ( - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 2 \zeta_{18}) q^{28} + ( - \zeta_{18}^{4} + \zeta_{18}^{3} - 5 \zeta_{18}^{2} + \zeta_{18} - 1) q^{29} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 2 \zeta_{18} + 3) q^{31} + (3 \zeta_{18}^{3} - 3 \zeta_{18} - 3) q^{32} + (2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{33} + (3 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 2 \zeta_{18} - 2) q^{34} + (5 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 3 \zeta_{18} - 5) q^{36} + ( - 3 \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 2 \zeta_{18}) q^{37} + (\zeta_{18}^{5} + 3 \zeta_{18}^{4} - \zeta_{18}^{3} + 5 \zeta_{18}^{2} + 2 \zeta_{18} + 3) q^{38} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 4) q^{39} + (\zeta_{18}^{5} - \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 4 \zeta_{18} + 4) q^{41} + (\zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}) q^{42} + (3 \zeta_{18}^{5} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2) q^{43} + ( - 3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{3} + 3 \zeta_{18}^{2} + \zeta_{18} + 1) q^{44} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 6 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 2 \zeta_{18}) q^{46} + (2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 3 \zeta_{18} + 2) q^{47} + ( - 2 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 5 \zeta_{18} - 2) q^{48} + (\zeta_{18}^{5} - \zeta_{18}^{4} + 5 \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{49} + ( - \zeta_{18}^{5} - \zeta_{18}^{3} + \zeta_{18}^{2} + \zeta_{18} + 1) q^{51} + ( - 4 \zeta_{18}^{5} - 6 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{52} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 2 \zeta_{18} - 1) q^{53} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{3} + 4 \zeta_{18}^{2} + \zeta_{18} + 6) q^{54} + (2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 5 \zeta_{18}^{2} - 5 \zeta_{18} - 1) q^{56} + ( - 5 \zeta_{18}^{5} - \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 4 \zeta_{18} - 5) q^{57} + ( - \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 4 \zeta_{18}^{2} - 4 \zeta_{18} - 6) q^{58} + ( - 2 \zeta_{18}^{5} + 7 \zeta_{18}^{4} - 7 \zeta_{18} + 2) q^{59} + ( - 3 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 4) q^{61} + (\zeta_{18}^{5} - 7 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3) q^{62} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 3 \zeta_{18} - 3) q^{63} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 3 \zeta_{18} - 4) q^{64} + ( - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3 \zeta_{18} - 3) q^{66} + (4 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 6 \zeta_{18}^{2} + 2 \zeta_{18} + 4) q^{67} + (3 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 5 \zeta_{18}^{3} + 4 \zeta_{18}^{2} + 3 \zeta_{18}) q^{68} + (4 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2 \zeta_{18} - 4) q^{69} + ( - 10 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2) q^{71} + (13 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 3 \zeta_{18} + 1) q^{72} + ( - 4 \zeta_{18}^{5} + 4 \zeta_{18}^{3} + 4 \zeta_{18}^{2}) q^{73} + ( - 5 \zeta_{18}^{5} - \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{2} + \zeta_{18} + 5) q^{74} + ( - 7 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 9 \zeta_{18}^{2} + 6 \zeta_{18} + 5) q^{76} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} + 3) q^{77} + (\zeta_{18}^{5} + 4 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 4 \zeta_{18} - 1) q^{78} + (\zeta_{18}^{5} - \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 3 \zeta_{18} - 6) q^{79} + (\zeta_{18}^{5} + 5 \zeta_{18}^{4} - \zeta_{18}^{2} + 1) q^{81} + ( - 6 \zeta_{18}^{5} - 7 \zeta_{18}^{4} - 4 \zeta_{18}^{3} + 6 \zeta_{18}^{2} + 11 \zeta_{18} + 11) q^{82} + ( - 6 \zeta_{18}^{5} - 6 \zeta_{18}^{4} - 3 \zeta_{18}^{2} + 9 \zeta_{18}) q^{83} + (\zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18}^{2} + \zeta_{18}) q^{84} + (8 \zeta_{18}^{3} + 7 \zeta_{18}^{2} + 8 \zeta_{18}) q^{86} + (6 \zeta_{18}^{5} - 7 \zeta_{18}^{4} + 7 \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 6 \zeta_{18}) q^{87} + ( - 6 \zeta_{18}^{5} - 6 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 3 \zeta_{18} - 3) q^{88} + ( - 5 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - \zeta_{18} - 1) q^{89} + ( - 5 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 3 \zeta_{18} + 2) q^{91} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{3} - 8 \zeta_{18}^{2} - 8 \zeta_{18} - 6) q^{92} + ( - 2 \zeta_{18}^{5} + 8 \zeta_{18}^{4} - 7 \zeta_{18}^{3} + 7 \zeta_{18}^{2} - 8 \zeta_{18} + 2) q^{93} + (4 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 3) q^{94} + 3 q^{96} + (2 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 5 \zeta_{18} - 2) q^{97} + ( - 5 \zeta_{18}^{5} + 5 \zeta_{18}^{3} + 4 \zeta_{18}^{2} + 6 \zeta_{18} - 1) q^{98} + (6 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + \zeta_{18} + 4) q^{99}+O(q^{100}) \) q + (-z^5 - z^4 + z + 1) * q^2 + (z^5 - z^3 - z + 1) * q^3 + (-z^5 - 2z^4 - 2z^3 + z + 1) * q^4 + (2z^5 - z^4 + z^3 - 2z^2) * q^6 + (-z^5 - z^4 + z^2) * q^7 + (z^5 - 3z^4 - 2z^3 - 3z^2 + z) q^8 + (2z^4 - 3z^3 - 3z + 2) q^9 + (2z^5 - z^4 - z^2 + 2z) * q^11 + (z^5 + z^4 + z^3 - z - 1) * q^12 + (z^4 - 3z^3 + 2z + 2) * q^13 + (-z^5 - 2z^4 - z^3) q^14 + (-3z^2 - z - 3) q^16 + (z^5 + 2z^4 + z^3 - z^2 - 2z - 1) * q^17 + (4z^5 - 4z^2 - 4z + 1) q^18 + (z^5 + 2z^4 + z^2 + 2z - 2) * q^19 + (z^4 - z) * q^21 + 3z q^22 + (-2z^5 - 2z^4 - 2z^3 + 2z) * q^23 + (4z^5 - 2z^4 + 3z^3 - z^2 + 1) q^24 + (-z^5 - z^4 - 5z^3 + z + 5) q^26 + (3z^4 - 2z^3 + 3z^2) q^27 + (-2z^3 - 3z^2 - 2z) q^28 + (-z^4 + z^3 - 5z^2 + z - 1) q^29 + (-z^5 - z^4 - 3z^3 + 3z^2 - 2z + 3) q^31 + (3z^3 - 3z - 3) * q^32 + (2z^5 - 3z^4 + z^3 - z^2 + 1) * q^33 + (3z^5 + 4z^4 + 4z^3 - 2z - 2) * q^34 + (5z^5 + 3z^4 + 2z^3 - 2z^2 - 3z - 5) q^36 + (-3z^5 + z^4 + 2z^2 + 2z) q^37 + (z^5 + 3z^4 - z^3 + 5z^2 + 2z + 3) q^38 + (-2z^5 + z^4 + z^2 + z - 4) q^39 + (z^5 - z^3 + 3z^2 + 4z + 4) * q^41 + (z^5 + z^4 + z^3 - z) * q^42 + (3z^5 + 5z^3 + 2z^2 - 2) q^43 + (-3z^5 - 2z^4 + z^3 + 3z^2 + z + 1) q^44 + (-2z^5 - 2z^4 - 6z^3 - 2z^2 - 2z) q^46 + (2z^4 - 3z^3 + 2z^2 - 3z + 2) * q^47 + (-2z^4 + 5z^3 - 2z^2 + 5z - 2) * q^48 + (z^5 - z^4 + 5z^3 - z^2 + z) q^49 + (-z^5 - z^3 + z^2 + z + 1) * q^51 + (-4z^5 - 6z^4 - 3z^3 + z^2 - 1) q^52 + (-2z^5 + 3z^4 + 3z^3 - 2z - 1) * q^53 + (2z^5 - 2z^3 + 4z^2 + z + 6) q^54 + (2z^5 + 3z^4 - 5z^2 - 5z - 1) * q^56 + (-5z^5 - z^4 + 2z^3 + 4z - 5) q^57 + (-z^5 + 5z^4 - 4z^2 - 4z - 6) q^58 + (-2z^5 + 7z^4 - 7z + 2) q^59 + (-3z^5 + 4z^4 + 4z^3 - 4) q^61 + (z^5 - 7z^4 - 2z^3 - 3z^2 + 3) q^62 + (-z^5 + z^4 + 2z^3 + z^2 - 3z - 3) * q^63 + (-3z^5 - 3z^4 + 4z^3 + 3z - 4) * q^64 + (-3z^4 + 3z^3 - 3z^2 + 3z - 3) * q^66 + (4z^4 + 2z^3 - 6z^2 + 2z + 4) * q^67 + (3z^5 + 4z^4 + 5z^3 + 4z^2 + 3z) q^68 + (4z^3 - 2z^2 + 2z - 4) q^69 + (-10z^4 + 2z^3 + 2z^2 - 2) q^71 + (13z^5 + 2z^4 + 2z^3 - 3z + 1) * q^72 + (-4z^5 + 4z^3 + 4z^2) q^73 + (-5z^5 - z^4 - 5z^3 + 5z^2 + z + 5) q^74 + (-7z^5 - 3z^4 + 2z^3 + 9z^2 + 6z + 5) q^76 + (-z^5 + 2z^4 - z^2 - z + 3) q^77 + (z^5 + 4z^4 - 3z^3 + 3z^2 - 4z - 1) * q^78 + (z^5 - z^3 - 7z^2 + 3z - 6) * q^79 + (z^5 + 5z^4 - z^2 + 1) q^81 + (-6z^5 - 7z^4 - 4z^3 + 6z^2 + 11z + 11) q^82 + (-6z^5 - 6z^4 - 3z^2 + 9z) * q^83 + (z^5 + z^4 + z^3 + z^2 + z) * q^84 + (8z^3 + 7z^2 + 8z) q^86 + (6z^5 - 7z^4 + 7z^3 - 7z^2 + 6z) q^87 + (-6z^5 - 6z^4 + 3z^3 + 3z^2 + 3z - 3) q^88 + (-5z^5 + 3z^4 - 2z^3 + 5z^2 - z - 1) * q^89 + (-5z^5 + z^4 + z^3 - 3z + 2) * q^91 + (2z^5 - 2z^3 - 8z^2 - 8z - 6) * q^92 + (-2z^5 + 8z^4 - 7z^3 + 7z^2 - 8z + 2) q^93 + (4z^5 - 2z^4 - 2z^2 - 2z + 3) * q^94 + 3 * q^96 + (2z^5 + 5z^4 - 2z^3 + 2z^2 - 5z - 2) q^97 + (-5z^5 + 5z^3 + 4z^2 + 6z - 1) * q^98 + (6z^5 - 5z^4 - 5z^3 + z + 4) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 3 q^{3} + 3 q^{6} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 + 3 * q^3 + 3 * q^6 - 6 * q^8 + 3 * q^9	\( 6 q + 6 q^{2} + 3 q^{3} + 3 q^{6} - 6 q^{8} + 3 q^{9} - 3 q^{12} + 3 q^{13} - 3 q^{14} - 18 q^{16} - 3 q^{17} + 6 q^{18} - 12 q^{19} - 6 q^{23} + 15 q^{24} + 15 q^{26} - 6 q^{27} - 6 q^{28} - 3 q^{29} + 9 q^{31} - 9 q^{32} + 9 q^{33} - 24 q^{36} + 15 q^{38} - 24 q^{39} + 21 q^{41} + 3 q^{42} + 3 q^{43} + 9 q^{44} - 18 q^{46} + 3 q^{47} + 3 q^{48} + 15 q^{49} + 3 q^{51} - 15 q^{52} + 3 q^{53} + 30 q^{54} - 6 q^{56} - 24 q^{57} - 36 q^{58} + 12 q^{59} - 12 q^{61} + 12 q^{62} - 12 q^{63} - 12 q^{64} - 9 q^{66} + 30 q^{67} + 15 q^{68} - 12 q^{69} - 6 q^{71} + 12 q^{72} + 12 q^{73} + 15 q^{74} + 36 q^{76} + 18 q^{77} - 15 q^{78} - 39 q^{79} + 6 q^{81} + 54 q^{82} + 3 q^{84} + 24 q^{86} + 21 q^{87} - 9 q^{88} - 12 q^{89} + 15 q^{91} - 42 q^{92} - 9 q^{93} + 18 q^{94} + 18 q^{96} - 18 q^{97} + 9 q^{98} + 9 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 + 3 * q^3 + 3 * q^6 - 6 * q^8 + 3 * q^9 - 3 * q^12 + 3 * q^13 - 3 * q^14 - 18 * q^16 - 3 * q^17 + 6 * q^18 - 12 * q^19 - 6 * q^23 + 15 * q^24 + 15 * q^26 - 6 * q^27 - 6 * q^28 - 3 * q^29 + 9 * q^31 - 9 * q^32 + 9 * q^33 - 24 * q^36 + 15 * q^38 - 24 * q^39 + 21 * q^41 + 3 * q^42 + 3 * q^43 + 9 * q^44 - 18 * q^46 + 3 * q^47 + 3 * q^48 + 15 * q^49 + 3 * q^51 - 15 * q^52 + 3 * q^53 + 30 * q^54 - 6 * q^56 - 24 * q^57 - 36 * q^58 + 12 * q^59 - 12 * q^61 + 12 * q^62 - 12 * q^63 - 12 * q^64 - 9 * q^66 + 30 * q^67 + 15 * q^68 - 12 * q^69 - 6 * q^71 + 12 * q^72 + 12 * q^73 + 15 * q^74 + 36 * q^76 + 18 * q^77 - 15 * q^78 - 39 * q^79 + 6 * q^81 + 54 * q^82 + 3 * q^84 + 24 * q^86 + 21 * q^87 - 9 * q^88 - 12 * q^89 + 15 * q^91 - 42 * q^92 - 9 * q^93 + 18 * q^94 + 18 * q^96 - 18 * q^97 + 9 * q^98 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

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Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	475.2.l.a

Level	$475$
Weight	$2$
Character orbit	475.l
Analytic conductor	$3.793$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{18})\)
Defining polynomial:	\( x^{6} - x^{3} + 1 \) x^6 - x^3 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 351.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} - 6 T^{5} + 18 T^{4} - 30 T^{3} + \cdots + 9 \) T^6 - 6T^5 + 18T^4 - 30T^3 + 36T^2 - 27*T + 9
$3$	\( T^{6} - 3 T^{5} + 3 T^{4} + 8 T^{3} + \cdots + 1 \) T^6 - 3T^5 + 3T^4 + 8T^3 + 6T^2 + 3*T + 1
$5$	\( T^{6} \) T^6
$7$	\( T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1 \) T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$11$	\( T^{6} + 9 T^{4} - 18 T^{3} + 81 T^{2} + \cdots + 81 \) T^6 + 9T^4 - 18T^3 + 81T^2 - 81T + 81
$13$	\( T^{6} - 3 T^{5} + 24 T^{4} + \cdots + 1369 \) T^6 - 3T^5 + 24T^4 - 26T^3 - 114T^2 - 222*T + 1369
$17$	\( T^{6} + 3 T^{5} - 30 T^{3} + 36 T^{2} + \cdots + 9 \) T^6 + 3T^5 - 30T^3 + 36*T^2 + 9
$19$	\( T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 6859 \) T^6 + 12T^5 + 78T^4 + 385T^3 + 1482T^2 + 4332*T + 6859
$23$	\( T^{6} + 6 T^{5} + 36 T^{4} + 192 T^{3} + \cdots + 576 \) T^6 + 6T^5 + 36T^4 + 192T^3 + 576T^2 + 864*T + 576
$29$	\( T^{6} + 3 T^{5} + 36 T^{4} + \cdots + 12321 \) T^6 + 3T^5 + 36T^4 - 57T^3 - 477T^2 - 1998*T + 12321
$31$	\( T^{6} - 9 T^{5} + 75 T^{4} + \cdots + 2809 \) T^6 - 9T^5 + 75T^4 - 160T^3 + 513T^2 + 318*T + 2809
$37$	\( (T^{3} - 21 T + 17)^{2} \) (T^3 - 21*T + 17)^2
$41$	\( T^{6} - 21 T^{5} + 162 T^{4} + \cdots + 12321 \) T^6 - 21T^5 + 162T^4 - 672T^3 + 3411T^2 - 8991*T + 12321
$43$	\( T^{6} - 3 T^{5} + 60 T^{4} + \cdots + 26569 \) T^6 - 3T^5 + 60T^4 - 8T^3 - 663T^2 - 5379*T + 26569
$47$	\( T^{6} - 3 T^{5} + 54 T^{4} - 24 T^{3} + \cdots + 9 \) T^6 - 3T^5 + 54T^4 - 24T^3 - 18T^2 + 9
$53$	\( T^{6} - 3 T^{5} + 84 T^{3} + \cdots + 2601 \) T^6 - 3T^5 + 84T^3 + 387T^2 + 1377T + 2601
$59$	\( T^{6} - 12 T^{5} + 18 T^{4} + \cdots + 71289 \) T^6 - 12T^5 + 18T^4 - 159T^3 + 3006T^2 + 19224*T + 71289
$61$	\( T^{6} + 12 T^{5} + 24 T^{4} + \cdots + 32761 \) T^6 + 12T^5 + 24T^4 - 37T^3 + 984T^2 - 2172*T + 32761
$67$	\( T^{6} - 30 T^{5} + 348 T^{4} + \cdots + 179776 \) T^6 - 30T^5 + 348T^4 - 2528T^3 + 20928T^2 - 86496*T + 179776
$71$	\( T^{6} + 6 T^{5} - 36 T^{4} + \cdots + 788544 \) T^6 + 6T^5 - 36T^4 - 1536T^3 + 8352T^2 - 31968*T + 788544
$73$	\( T^{6} - 12 T^{5} + 96 T^{4} + \cdots + 4096 \) T^6 - 12T^5 + 96T^4 - 512T^3 + 768T^2 + 3072*T + 4096
$79$	\( T^{6} + 39 T^{5} + 708 T^{4} + \cdots + 654481 \) T^6 + 39T^5 + 708T^4 + 7487T^3 + 51663T^2 + 242700*T + 654481
$83$	\( T^{6} + 189 T^{4} - 918 T^{3} + \cdots + 210681 \) T^6 + 189T^4 - 918T^3 + 35721T^2 - 86751T + 210681
$89$	\( T^{6} + 12 T^{5} + 54 T^{4} + \cdots + 3249 \) T^6 + 12T^5 + 54T^4 - 300T^3 + 522T^2 - 1539*T + 3249
$97$	\( T^{6} + 18 T^{5} + 234 T^{4} + \cdots + 16129 \) T^6 + 18T^5 + 234T^4 + 1855T^3 + 9522T^2 + 20574*T + 16129
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\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(1\)	\(-\zeta_{18}^{5}\)