LMFDB - Newform orbit 475.2.u.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(24,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 16]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("475.24");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.u (of order $18$, degree $6$, 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:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	$\Q(\zeta_{36})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{6} + 1 $ x^12 - x^6 + 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:	$\mathrm{SU}(2)[C_{18}]$

Character values

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

$n$	$77$	$401$
$\chi(n)$	$-1$	$\zeta_{36}^{2} - \zeta_{36}^{8}$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

24.1

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

−0.300767

−

0.826352i

−0.524005

−

0.0923963i

0.939693

−

0.788496i

0.0812519

0.460802i

−1.62760

0.939693i

−2.45734

−

1.41875i

−2.55303

−

0.929228i

24.2

0.300767

0.826352i

0.524005

0.0923963i

0.939693

−

0.788496i

0.0812519

0.460802i

1.62760

−

0.939693i

2.45734

1.41875i

−2.55303

−

0.929228i

74.1

−1.32683

−

0.233956i

−1.85083

2.20574i

−0.173648

−

0.0632028i

2.97178

−

2.49362i

0.300767

−

0.173648i

2.54920

1.47178i

−0.918748

−

5.21048i

74.2

1.32683

0.233956i

1.85083

−

2.20574i

−0.173648

−

0.0632028i

2.97178

−

2.49362i

−0.300767

0.173648i

−2.54920

−

1.47178i

−0.918748

−

5.21048i

99.1

−0.300767

0.826352i

−0.524005

0.0923963i

0.939693

0.788496i

0.0812519

−

0.460802i

−1.62760

−

0.939693i

−2.45734

1.41875i

−2.55303

0.929228i

99.2

0.300767

−

0.826352i

0.524005

−

0.0923963i

0.939693

0.788496i

0.0812519

−

0.460802i

1.62760

0.939693i

2.45734

−

1.41875i

−2.55303

0.929228i

149.1

−1.62760

1.93969i

0.223238

0.613341i

−0.766044

−

4.34445i

−1.55303

−

0.565258i

−1.32683

0.766044i

5.28801

3.05303i

1.97178

−

1.65452i

149.2

1.62760

−

1.93969i

−0.223238

−

0.613341i

−0.766044

−

4.34445i

−1.55303

−

0.565258i

1.32683

−

0.766044i

−5.28801

−

3.05303i

1.97178

−

1.65452i

199.1

−1.32683

0.233956i

−1.85083

−

2.20574i

−0.173648

0.0632028i

2.97178

2.49362i

0.300767

0.173648i

2.54920

−

1.47178i

−0.918748

5.21048i

199.2

1.32683

−

0.233956i

1.85083

2.20574i

−0.173648

0.0632028i

2.97178

2.49362i

−0.300767

−

0.173648i

−2.54920

1.47178i

−0.918748

5.21048i

424.1

−1.62760

−

1.93969i

0.223238

−

0.613341i

−0.766044

4.34445i

−1.55303

0.565258i

−1.32683

−

0.766044i

5.28801

−

3.05303i

1.97178

1.65452i

424.2

1.62760

1.93969i

−0.223238

0.613341i

−0.766044

4.34445i

−1.55303

0.565258i

1.32683

0.766044i

−5.28801

3.05303i

1.97178

1.65452i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
19.e	even	9	1	inner
95.p	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	475.2.u.a		12
5.b	even	2	1	inner	475.2.u.a		12
5.c	odd	4	1		19.2.e.a	✓	6
5.c	odd	4	1		475.2.l.a		6
15.e	even	4	1		171.2.u.c		6
19.e	even	9	1	inner	475.2.u.a		12
20.e	even	4	1		304.2.u.b		6
35.f	even	4	1		931.2.w.a		6
35.k	even	12	1		931.2.v.a		6
35.k	even	12	1		931.2.x.b		6
35.l	odd	12	1		931.2.v.b		6
35.l	odd	12	1		931.2.x.a		6
95.g	even	4	1		361.2.e.h		6
95.l	even	12	1		361.2.e.a		6
95.l	even	12	1		361.2.e.b		6
95.m	odd	12	1		361.2.e.f		6
95.m	odd	12	1		361.2.e.g		6
95.p	even	18	1	inner	475.2.u.a		12
95.q	odd	36	1		19.2.e.a	✓	6
95.q	odd	36	1		361.2.a.g		3
95.q	odd	36	2		361.2.c.i		6
95.q	odd	36	1		361.2.e.f		6
95.q	odd	36	1		361.2.e.g		6
95.q	odd	36	1		475.2.l.a		6
95.q	odd	36	1		9025.2.a.bd		3
95.r	even	36	1		361.2.a.h		3
95.r	even	36	2		361.2.c.h		6
95.r	even	36	1		361.2.e.a		6
95.r	even	36	1		361.2.e.b		6
95.r	even	36	1		361.2.e.h		6
95.r	even	36	1		9025.2.a.x		3
285.bi	even	36	1		171.2.u.c		6
285.bi	even	36	1		3249.2.a.z		3
285.bj	odd	36	1		3249.2.a.s		3
380.bi	odd	36	1		5776.2.a.bi		3
380.bj	even	36	1		304.2.u.b		6
380.bj	even	36	1		5776.2.a.br		3
665.dj	odd	36	1		931.2.x.a		6
665.dk	even	36	1		931.2.x.b		6
665.dm	even	36	1		931.2.w.a		6
665.dp	even	36	1		931.2.v.a		6
665.dq	odd	36	1		931.2.v.b		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
19.2.e.a	✓	6	5.c	odd	4	1
19.2.e.a	✓	6	95.q	odd	36	1
171.2.u.c		6	15.e	even	4	1
171.2.u.c		6	285.bi	even	36	1
304.2.u.b		6	20.e	even	4	1
304.2.u.b		6	380.bj	even	36	1
361.2.a.g		3	95.q	odd	36	1
361.2.a.h		3	95.r	even	36	1
361.2.c.h		6	95.r	even	36	2
361.2.c.i		6	95.q	odd	36	2
361.2.e.a		6	95.l	even	12	1
361.2.e.a		6	95.r	even	36	1
361.2.e.b		6	95.l	even	12	1
361.2.e.b		6	95.r	even	36	1
361.2.e.f		6	95.m	odd	12	1
361.2.e.f		6	95.q	odd	36	1
361.2.e.g		6	95.m	odd	12	1
361.2.e.g		6	95.q	odd	36	1
361.2.e.h		6	95.g	even	4	1
361.2.e.h		6	95.r	even	36	1
475.2.l.a		6	5.c	odd	4	1
475.2.l.a		6	95.q	odd	36	1
475.2.u.a		12	1.a	even	1	1	trivial
475.2.u.a		12	5.b	even	2	1	inner
475.2.u.a		12	19.e	even	9	1	inner
475.2.u.a		12	95.p	even	18	1	inner
931.2.v.a		6	35.k	even	12	1
931.2.v.a		6	665.dp	even	36	1
931.2.v.b		6	35.l	odd	12	1
931.2.v.b		6	665.dq	odd	36	1
931.2.w.a		6	35.f	even	4	1
931.2.w.a		6	665.dm	even	36	1
931.2.x.a		6	35.l	odd	12	1
931.2.x.a		6	665.dj	odd	36	1
931.2.x.b		6	35.k	even	12	1
931.2.x.b		6	665.dk	even	36	1
3249.2.a.s		3	285.bj	odd	36	1
3249.2.a.z		3	285.bi	even	36	1
5776.2.a.bi		3	380.bi	odd	36	1
5776.2.a.br		3	380.bj	even	36	1
9025.2.a.x		3	95.r	even	36	1
9025.2.a.bd		3	95.q	odd	36	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{12} + 36T_{2}^{8} - 90T_{2}^{6} + 81T_{2}^{2} + 81 $ T2^12 + 36*T2^8 - 90*T2^6 + 81*T2^2 + 81 acting on $S_{2}^{\mathrm{new}}(475, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} + 36 T^{8} + \cdots + 81 $ T^12 + 36T^8 - 90T^6 + 81*T^2 + 81
$3$	$ T^{12} + 3 T^{10} + \cdots + 1 $ T^12 + 3T^10 + 69T^8 + 8T^6 - 6T^4 - 3*T^2 + 1
$5$	$ T^{12} $ T^12
$7$	$ T^{12} - 6 T^{10} + \cdots + 1 $ T^12 - 6T^10 + 27T^8 - 52T^6 + 75T^4 - 9*T^2 + 1
$11$	$ (T^{6} + 9 T^{4} - 18 T^{3} + \cdots + 81)^{2} $ (T^6 + 9T^4 - 18T^3 + 81T^2 - 81T + 81)^2
$13$	$ T^{12} - 39 T^{10} + \cdots + 1874161 $ T^12 - 39T^10 + 192T^8 + 4742T^6 + 67164T^4 + 361416*T^2 + 1874161
$17$	$ T^{12} + 9 T^{10} + \cdots + 81 $ T^12 + 9T^10 + 252T^8 + 882T^6 + 1296T^4 - 648*T^2 + 81
$19$	$ (T^{6} - 12 T^{5} + \cdots + 6859)^{2} $ (T^6 - 12T^5 + 78T^4 - 385T^3 + 1482T^2 - 4332*T + 6859)^2
$23$	$ T^{12} - 36 T^{10} + \cdots + 331776 $ T^12 - 36T^10 + 144T^8 + 4608T^6 + 41472T^4 + 82944*T^2 + 331776
$29$	$ (T^{6} - 3 T^{5} + \cdots + 12321)^{2} $ (T^6 - 3T^5 + 36T^4 + 57T^3 - 477T^2 + 1998*T + 12321)^2
$31$	$ (T^{6} - 9 T^{5} + \cdots + 2809)^{2} $ (T^6 - 9T^5 + 75T^4 - 160T^3 + 513T^2 + 318*T + 2809)^2
$37$	$ (T^{6} + 42 T^{4} + \cdots + 289)^{2} $ (T^6 + 42T^4 + 441T^2 + 289)^2
$41$	$ (T^{6} - 21 T^{5} + \cdots + 12321)^{2} $ (T^6 - 21T^5 + 162T^4 - 672T^3 + 3411T^2 - 8991*T + 12321)^2
$43$	$ T^{12} + \cdots + 705911761 $ T^12 - 111T^10 + 2226T^8 + 58760T^6 + 3541785T^4 + 64164135*T^2 + 705911761
$47$	$ T^{12} - 99 T^{10} + \cdots + 81 $ T^12 - 99T^10 + 2736T^8 + 2502T^6 + 1296T^4 + 324*T^2 + 81
$53$	$ T^{12} + 9 T^{10} + \cdots + 6765201 $ T^12 + 9T^10 + 1278T^8 - 6408T^6 - 81567T^4 - 117045*T^2 + 6765201
$59$	$ (T^{6} + 12 T^{5} + \cdots + 71289)^{2} $ (T^6 + 12T^5 + 18T^4 + 159T^3 + 3006T^2 - 19224*T + 71289)^2
$61$	$ (T^{6} + 12 T^{5} + \cdots + 32761)^{2} $ (T^6 + 12T^5 + 24T^4 - 37T^3 + 984T^2 - 2172*T + 32761)^2
$67$	$ T^{12} + \cdots + 32319410176 $ T^12 + 204T^10 + 11280T^8 - 3344896T^6 + 125781504T^4 - 43146240*T^2 + 32319410176
$71$	$ (T^{6} + 6 T^{5} + \cdots + 788544)^{2} $ (T^6 + 6T^5 - 36T^4 - 1536T^3 + 8352T^2 - 31968*T + 788544)^2
$73$	$ T^{12} - 48 T^{10} + \cdots + 16777216 $ T^12 - 48T^10 - 1536T^8 + 32768T^6 + 4521984T^4 + 3145728*T^2 + 16777216
$79$	$ (T^{6} - 39 T^{5} + \cdots + 654481)^{2} $ (T^6 - 39T^5 + 708T^4 - 7487T^3 + 51663T^2 - 242700*T + 654481)^2
$83$	$ T^{12} + \cdots + 44386483761 $ T^12 - 378T^10 + 107163T^8 - 13081176T^6 + 1196352423T^4 - 7525736001*T^2 + 44386483761
$89$	$ (T^{6} - 12 T^{5} + \cdots + 3249)^{2} $ (T^6 - 12T^5 + 54T^4 + 300T^3 + 522T^2 + 1539*T + 3249)^2
$97$	$ T^{12} + \cdots + 260144641 $ T^12 - 144T^10 + 7020T^8 - 306865T^6 + 21887316T^4 + 116128800*T^2 + 260144641
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.u (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{36}^{9} + \cdots - \zeta_{36}^{5}) q^{2}+ \cdots + (3 \zeta_{36}^{10} - 3 \zeta_{36}^{6} + \cdots + 1) q^{9}+O(q^{10}) \) q + (-z^9 + z^7 - z^5) * q^2 + (z^9 - z^5 - z^3 + z) * q^3 + (-z^10 + z^8 - 2z^6 + 2z^4 - z^2 + 1) * q^4 + (2z^8 - z^6 - z^4 + 1) q^6 + (z^11 + z^7 - z^5 - z) * q^7 + (3z^11 - 3z^7 - z^5 - 2z^3 + 2z) * q^8 + (3z^10 - 3z^6 - 2z^4 + 1) q^9	\( q + ( - \zeta_{36}^{9} + \cdots - \zeta_{36}^{5}) q^{2}+ \cdots + ( - \zeta_{36}^{10} - 6 \zeta_{36}^{8} + \cdots + 1) q^{99}+O(q^{100}) \) q + (-z^9 + z^7 - z^5) * q^2 + (z^9 - z^5 - z^3 + z) * q^3 + (-z^10 + z^8 - 2z^6 + 2z^4 - z^2 + 1) * q^4 + (2z^8 - z^6 - z^4 + 1) q^6 + (z^11 + z^7 - z^5 - z) * q^7 + (3z^11 - 3z^7 - z^5 - 2z^3 + 2z) * q^8 + (3z^10 - 3z^6 - 2z^4 + 1) q^9 + (2z^10 + 2z^8 - z^4 - z^2) * q^11 + (z^9 - z^7 + z^5 - z^3) * q^12 + (2z^9 + z^7 - 3z^3 - 3z) q^13 + (z^8 - z^6 + 2z^4 - z^2 + 1) q^14 + (-z^10 + 3z^2 - 3) q^16 + (-z^11 + z^9 - 2z^7 + z^5 - z^3) q^17 + (4z^11 + z^9 - 4z^5 + 4z) q^18 + (-2z^10 - z^8 - 2z^4 + 2z^2 + 2) q^19 + (-z^10 + z^4) * q^21 + 3z q^22 + (-2z^7 + 2z^5 - 2z^3) q^23 + (-4z^8 + 3z^6 + 2z^4 + 3z^2 - 4) * q^24 + (z^10 - z^8 + 5z^6 - z^4 + z^2) q^26 + (3z^11 - 3z^7 + 2z^3 + 3z) * q^27 + (3z^11 - 2z^3 + 2z) q^28 + (-z^10 + z^6 + z^4 - 5z^2) q^29 + (-2z^10 - z^8 + 3z^6 - z^4 - 2z^2) q^31 + (3z^9 - 3z^3 - 3z) q^32 + (z^11 + z^9 - 3z^7 - 2z^5 + z^3 + 3z) q^33 + (2z^10 - 3z^8 + 4z^6 - 4z^4 + 3z^2 - 2) q^34 + (-3z^10 + 5z^8 - 2z^6 + 3z^4 - 3z^2 - 3) q^36 + (2z^11 - z^7 - 3z^5 + 3z) q^37 + (-5z^11 + 3z^9 + 3z^7 - z^5 - z^3 - 5z) * q^38 + (-z^10 + 2z^8 - z^4 - z^2 + 4) q^39 + (4z^10 + z^8 + z^6 - 4z^2 + 3) * q^41 + (-z^7 + z^5 - z^3) * q^42 + (-2z^11 - 2z^9 - 3z^5 + 5z^3) * q^43 + (-z^10 + 3z^8 + z^6 + 2z^4 - 2) * q^44 + (-2z^10 - 2z^8 + 6z^6 - 2z^4 + 4z^2 - 6) q^46 + (2z^11 - 2z^9 - 2z^7 + 3z^3 - z) * q^47 + (2z^11 - 2z^9 - 2z^7 + 5z^3 - 3z) q^48 + (-z^10 - z^8 + 5z^6 + z^4 - 5) q^49 + (z^10 - z^8 + z^6) * q^51 + (z^11 + z^9 + 6z^7 - 4z^5 + 3z^3 - 6z) * q^52 + (-z^9 + 3z^7 + 2z^5 + 3z^3 - z) q^53 + (-z^10 - 2z^8 - 2z^6 + 6z^2 - 4) q^54 + (-5z^10 + 2z^8 + 3z^4 + 3z^2 - 1) * q^56 + (5z^9 + z^7 - 5z^5 - 2z^3 + 3z) * q^57 + (4z^11 - 6z^9 + 5z^7 + z^5 - z) q^58 + (7z^10 + 2z^8 - 7z^4 - 2z^2 - 2) * q^59 + (-3z^8 - 4z^6 + 4z^4 + 3z^2) * q^61 + (-3z^11 - 3z^9 + 7z^7 + z^5 + 2z^3 - 7z) q^62 + (-z^11 - 3z^9 + z^7 + z^5 + 2z^3 + 2z) q^63 + (-3z^10 + 3z^8 + 4z^6 + 3z^4 - 3z^2) q^64 + (3z^10 - 3z^6 - 3z^4 + 3z^2) * q^66 + (-6z^11 - 4z^9 - 4z^7 - 2z^3 + 6z) q^67 + (-4z^11 + 4z^7 - 3z^5 + 5z^3 - 7z) q^68 + (-2z^10 + 4z^6 - 2z^2) q^69 + (-2z^6 - 10z^4 - 2z^2) q^71 + (-z^9 - 2z^7 + 13z^5 - 2z^3 - z) q^72 + (-4z^11 + 4z^5 + 4z^3) q^73 + (-z^10 + 5z^8 - 5z^6 + z^4) * q^74 + (6z^10 - 7z^8 - 2z^6 - 3z^4 - 2z^2 + 7) q^76 + (-z^11 - 3z^9 - 2z^7 - z^5 + z) * q^77 + (-3z^11 - z^9 + 4z^7 - z^5 - 3z^3) q^78 + (-3z^10 - z^8 - z^6 - 6z^2 + 7) * q^79 + (z^8 + 5z^4 + 1) q^81 + (6z^11 - 11z^9 + 7z^7 - 6z^5 + 4z^3 + 4z) * q^82 + (3z^11 - 6z^7 + 6z^5 - 3z) * q^83 + (-z^10 - z^8 + z^6 - z^4 + 2z^2 - 1) q^84 + (8z^10 - 8z^6 - 7z^2 + 8) q^86 + (-7z^11 + 7z^7 + 6z^5 - 7z^3 - z) * q^87 + (-3z^11 - 3z^9 - 6z^7 + 6z^5 + 3z^3 + 3z) * q^88 + (z^10 + 5z^8 - 2z^6 - 3z^4 + 3) q^89 + (-3z^10 - 5z^8 - z^6 + z^4 + 5z^2 + 3) q^91 + (-8z^11 + 6z^9 + 2z^5 + 2z^3 - 8z) q^92 + (-7z^11 + 2z^9 + 8z^7 + 2z^5 - 7z^3) q^93 + (2z^10 - 4z^8 + 2z^4 + 2z^2 - 3) * q^94 + 3 * q^96 + (2z^11 + 2z^9 - 5z^7 + 2z^5 + 2z^3) q^97 + (-4z^11 - z^9 + 5z^5 + 5z^3 - 6z) * q^98 + (-z^10 - 6z^8 - 5z^6 + 5z^4 + 6z^2 + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{6} - 6 q^{9}+O(q^{10}) \) 12 * q + 6 * q^6 - 6 * q^9	\( 12 q + 6 q^{6} - 6 q^{9} + 6 q^{14} - 36 q^{16} + 24 q^{19} - 30 q^{24} + 30 q^{26} + 6 q^{29} + 18 q^{31} - 48 q^{36} + 48 q^{39} + 42 q^{41} - 18 q^{44} - 36 q^{46} - 30 q^{49} + 6 q^{51} - 60 q^{54} - 12 q^{56} - 24 q^{59} - 24 q^{61} + 24 q^{64} - 18 q^{66} + 24 q^{69} - 12 q^{71} - 30 q^{74} + 72 q^{76} + 78 q^{79} + 12 q^{81} - 6 q^{84} + 48 q^{86} + 24 q^{89} + 30 q^{91} - 36 q^{94} + 36 q^{96} - 18 q^{99}+O(q^{100}) \) 12 * q + 6 * q^6 - 6 * q^9 + 6 * q^14 - 36 * q^16 + 24 * q^19 - 30 * q^24 + 30 * q^26 + 6 * q^29 + 18 * q^31 - 48 * q^36 + 48 * q^39 + 42 * q^41 - 18 * q^44 - 36 * q^46 - 30 * q^49 + 6 * q^51 - 60 * q^54 - 12 * q^56 - 24 * q^59 - 24 * q^61 + 24 * q^64 - 18 * q^66 + 24 * q^69 - 12 * q^71 - 30 * q^74 + 72 * q^76 + 78 * q^79 + 12 * q^81 - 6 * q^84 + 48 * q^86 + 24 * q^89 + 30 * q^91 - 36 * q^94 + 36 * q^96 - 18 * q^99

Introduction

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Newspace parameters

Newform invariants

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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.u.a

Level	$475$
Weight	$2$
Character orbit	475.u
Analytic conductor	$3.793$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{18})\)
Coefficient field:	\(\Q(\zeta_{36})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{6} + 1 \) x^12 - x^6 + 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:	$\mathrm{SU}(2)[C_{18}]$

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(-1\)	\(\zeta_{36}^{2} - \zeta_{36}^{8}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 24.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} + 36 T^{8} + \cdots + 81 \) T^12 + 36T^8 - 90T^6 + 81*T^2 + 81
$3$	\( T^{12} + 3 T^{10} + \cdots + 1 \) T^12 + 3T^10 + 69T^8 + 8T^6 - 6T^4 - 3*T^2 + 1
$5$	\( T^{12} \) T^12
$7$	\( T^{12} - 6 T^{10} + \cdots + 1 \) T^12 - 6T^10 + 27T^8 - 52T^6 + 75T^4 - 9*T^2 + 1
$11$	\( (T^{6} + 9 T^{4} - 18 T^{3} + \cdots + 81)^{2} \) (T^6 + 9T^4 - 18T^3 + 81T^2 - 81T + 81)^2
$13$	\( T^{12} - 39 T^{10} + \cdots + 1874161 \) T^12 - 39T^10 + 192T^8 + 4742T^6 + 67164T^4 + 361416*T^2 + 1874161
$17$	\( T^{12} + 9 T^{10} + \cdots + 81 \) T^12 + 9T^10 + 252T^8 + 882T^6 + 1296T^4 - 648*T^2 + 81
$19$	\( (T^{6} - 12 T^{5} + \cdots + 6859)^{2} \) (T^6 - 12T^5 + 78T^4 - 385T^3 + 1482T^2 - 4332*T + 6859)^2
$23$	\( T^{12} - 36 T^{10} + \cdots + 331776 \) T^12 - 36T^10 + 144T^8 + 4608T^6 + 41472T^4 + 82944*T^2 + 331776
$29$	\( (T^{6} - 3 T^{5} + \cdots + 12321)^{2} \) (T^6 - 3T^5 + 36T^4 + 57T^3 - 477T^2 + 1998*T + 12321)^2
$31$	\( (T^{6} - 9 T^{5} + \cdots + 2809)^{2} \) (T^6 - 9T^5 + 75T^4 - 160T^3 + 513T^2 + 318*T + 2809)^2
$37$	\( (T^{6} + 42 T^{4} + \cdots + 289)^{2} \) (T^6 + 42T^4 + 441T^2 + 289)^2
$41$	\( (T^{6} - 21 T^{5} + \cdots + 12321)^{2} \) (T^6 - 21T^5 + 162T^4 - 672T^3 + 3411T^2 - 8991*T + 12321)^2
$43$	\( T^{12} + \cdots + 705911761 \) T^12 - 111T^10 + 2226T^8 + 58760T^6 + 3541785T^4 + 64164135*T^2 + 705911761
$47$	\( T^{12} - 99 T^{10} + \cdots + 81 \) T^12 - 99T^10 + 2736T^8 + 2502T^6 + 1296T^4 + 324*T^2 + 81
$53$	\( T^{12} + 9 T^{10} + \cdots + 6765201 \) T^12 + 9T^10 + 1278T^8 - 6408T^6 - 81567T^4 - 117045*T^2 + 6765201
$59$	\( (T^{6} + 12 T^{5} + \cdots + 71289)^{2} \) (T^6 + 12T^5 + 18T^4 + 159T^3 + 3006T^2 - 19224*T + 71289)^2
$61$	\( (T^{6} + 12 T^{5} + \cdots + 32761)^{2} \) (T^6 + 12T^5 + 24T^4 - 37T^3 + 984T^2 - 2172*T + 32761)^2
$67$	\( T^{12} + \cdots + 32319410176 \) T^12 + 204T^10 + 11280T^8 - 3344896T^6 + 125781504T^4 - 43146240*T^2 + 32319410176
$71$	\( (T^{6} + 6 T^{5} + \cdots + 788544)^{2} \) (T^6 + 6T^5 - 36T^4 - 1536T^3 + 8352T^2 - 31968*T + 788544)^2
$73$	\( T^{12} - 48 T^{10} + \cdots + 16777216 \) T^12 - 48T^10 - 1536T^8 + 32768T^6 + 4521984T^4 + 3145728*T^2 + 16777216
$79$	\( (T^{6} - 39 T^{5} + \cdots + 654481)^{2} \) (T^6 - 39T^5 + 708T^4 - 7487T^3 + 51663T^2 - 242700*T + 654481)^2
$83$	\( T^{12} + \cdots + 44386483761 \) T^12 - 378T^10 + 107163T^8 - 13081176T^6 + 1196352423T^4 - 7525736001*T^2 + 44386483761
$89$	\( (T^{6} - 12 T^{5} + \cdots + 3249)^{2} \) (T^6 - 12T^5 + 54T^4 + 300T^3 + 522T^2 + 1539*T + 3249)^2
$97$	\( T^{12} + \cdots + 260144641 \) T^12 - 144T^10 + 7020T^8 - 306865T^6 + 21887316T^4 + 116128800*T^2 + 260144641
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