LMFDB - Newform orbit 3375.1.o.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3375,1,Mod(26,3375)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3375.26"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3375, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$N$	$=$	$3375 = 3^{3} \cdot 5^{3}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3375.o (of order $10$ , degree $4$ , not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.68434441764$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{6} + x^{4} - x^{2} + 1$ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 675)
Projective image:	$A_{5}$
Projective field:	Galois closure of 5.1.31640625.2

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

The $q$ -expansion and trace form are shown below.

$f(q)$	$=$	$q + (\zeta_{20}^{9} + \zeta_{20}^{5}) q^{2} + ( - \zeta_{20}^{8} - \zeta_{20}^{4} - 1) q^{4} + ( - \zeta_{20}^{8} + \zeta_{20}^{2}) q^{7} + ( - \zeta_{20}^{9} + \cdots + \zeta_{20}^{3}) q^{8} - \zeta_{20}^{7} q^{11} + \cdots + (2 \zeta_{20}^{9} + 2 \zeta_{20}^{5} + \cdots + \zeta_{20}) q^{98} +O(q^{100})$ q + (z^9 + z^5) * q^2 + (-z^8 - z^4 - 1) * q^4 + (-z^8 + z^2) * q^7 + (-z^9 + z^7 - z^5 + z^3) * q^8 - z^7 * q^11 + (2z^7 + z^3 - z) q^14 + (z^8 - z^6 + z^4 - z^2 + 1) * q^16 - z * q^17 + z^6 * q^19 + (z^6 + z^2) * q^22 + z^7 * q^23 + (z^8 - 2z^6 - 2z^2 + 1) * q^28 + (-z^5 + z^3) * q^29 + (z^9 - z^7 + 2z^5 - z^3 + z) q^32 + (-z^6 + 1) * q^34 + (z^4 - z^2) * q^37 + (-z^5 - z) * q^38 + z^3 * q^41 + (z^7 - z^5 - z) * q^44 + (-z^6 - z^2) * q^46 + (-z^5 + z^3) * q^47 + (-z^6 + z^4 + 1) * q^49 + (-z^7 + z) * q^53 + (z^9 - 2z^7 + 2z^5 - z^3 + 2z) q^56 + (z^8 + z^4 - z^2 + 1) * q^58 + z^2 * q^61 + (-z^8 + z^6 - 2z^4 + z^2 - 2) q^64 + (-z^8 - z^4) * q^67 + (z^9 + z^5 + z) * q^68 + (-z^5 + z^3) * q^71 + (z^9 - z^7 - z^3 + z) * q^74 + (-z^6 + z^4 + 1) * q^76 + (-z^9 - z^5) * q^77 + (-z^8 - 1) * q^79 + (z^8 - z^2) * q^82 + (-z^9 + z^3) * q^83 + (-z^6 + z^4 - z^2 + 1) * q^88 - z^7 * q^89 + (-z^7 + z^5 + z) * q^92 + (z^8 + z^4 - z^2 + 1) * q^94 - z^4 * q^97 + (2z^9 + 2z^5 - z^3 + z) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 4 q^{4} + 4 q^{7} + 2 q^{19} + 4 q^{22} - 2 q^{28} + 6 q^{34} - 4 q^{37} - 4 q^{46} + 4 q^{49} + 2 q^{58} + 2 q^{61} - 6 q^{64} + 4 q^{67} + 4 q^{76} - 6 q^{79} - 4 q^{82} + 2 q^{88} + 2 q^{94} + 2 q^{97}+O(q^{100})$ 8 * q - 4 * q^4 + 4 * q^7 + 2 * q^19 + 4 * q^22 - 2 * q^28 + 6 * q^34 - 4 * q^37 - 4 * q^46 + 4 * q^49 + 2 * q^58 + 2 * q^61 - 6 * q^64 + 4 * q^67 + 4 * q^76 - 6 * q^79 - 4 * q^82 + 2 * q^88 + 2 * q^94 + 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3375\mathbb{Z}\right)^\times$ .

$n$	$1001$	$2377$
$\chi(n)$	$-1$	$\zeta_{20}^{4}$

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}

26.1

0.587785	−	0.809017i
−0.587785	+	0.809017i
0.951057	+	0.309017i
−0.951057	−	0.309017i
0.951057	−	0.309017i
−0.951057	+	0.309017i
0.587785	+	0.809017i
−0.587785	−	0.809017i

−0.587785

0.190983i

−0.500000

0.363271i

−0.618034

0.587785

−

0.809017i

26.2

0.587785

−

0.190983i

−0.500000

0.363271i

−0.618034

−0.587785

0.809017i

701.1

−0.951057

1.30902i

−0.500000

−

1.53884i

1.61803

0.951057

0.309017i

701.2

0.951057

−

1.30902i

−0.500000

−

1.53884i

1.61803

−0.951057

−

0.309017i

2051.1

−0.951057

−

1.30902i

−0.500000

1.53884i

1.61803

0.951057

−

0.309017i

2051.2

0.951057

1.30902i

−0.500000

1.53884i

1.61803

−0.951057

0.309017i

2726.1

−0.587785

−

0.190983i

−0.500000

−

0.363271i

−0.618034

0.587785

0.809017i

2726.2

0.587785

0.190983i

−0.500000

−

0.363271i

−0.618034

−0.587785

−

0.809017i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
25.d	even	5	1	inner
75.j	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3375.1.o.a		8
3.b	odd	2	1	inner	3375.1.o.a		8
5.b	even	2	1		675.1.o.a	✓	8
5.c	odd	4	1		3375.1.m.a		8
5.c	odd	4	1		3375.1.m.b		8
15.d	odd	2	1		675.1.o.a	✓	8
15.e	even	4	1		3375.1.m.a		8
15.e	even	4	1		3375.1.m.b		8
25.d	even	5	1	inner	3375.1.o.a		8
25.e	even	10	1		675.1.o.a	✓	8
25.f	odd	20	1		3375.1.m.a		8
25.f	odd	20	1		3375.1.m.b		8
45.h	odd	6	2		2025.1.y.a		16
45.j	even	6	2		2025.1.y.a		16
75.h	odd	10	1		675.1.o.a	✓	8
75.j	odd	10	1	inner	3375.1.o.a		8
75.l	even	20	1		3375.1.m.a		8
75.l	even	20	1		3375.1.m.b		8
225.u	even	30	2		2025.1.y.a		16
225.v	odd	30	2		2025.1.y.a		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
675.1.o.a	✓	8	5.b	even	2	1
675.1.o.a	✓	8	15.d	odd	2	1
675.1.o.a	✓	8	25.e	even	10	1
675.1.o.a	✓	8	75.h	odd	10	1
2025.1.y.a		16	45.h	odd	6	2
2025.1.y.a		16	45.j	even	6	2
2025.1.y.a		16	225.u	even	30	2
2025.1.y.a		16	225.v	odd	30	2
3375.1.m.a		8	5.c	odd	4	1
3375.1.m.a		8	15.e	even	4	1
3375.1.m.a		8	25.f	odd	20	1
3375.1.m.a		8	75.l	even	20	1
3375.1.m.b		8	5.c	odd	4	1
3375.1.m.b		8	15.e	even	4	1
3375.1.m.b		8	25.f	odd	20	1
3375.1.m.b		8	75.l	even	20	1
3375.1.o.a		8	1.a	even	1	1	trivial
3375.1.o.a		8	3.b	odd	2	1	inner
3375.1.o.a		8	25.d	even	5	1	inner
3375.1.o.a		8	75.j	odd	10	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3375, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8} + T^{6} + 6 T^{4} + \cdots + 1$ T^8 + T^6 + 6T^4 - 4T^2 + 1
$3$	$T^{8}$ T^8
$5$	$T^{8}$ T^8
$7$	$(T^{2} - T - 1)^{4}$ (T^2 - T - 1)^4
$11$	$T^{8} - T^{6} + T^{4} + \cdots + 1$ T^8 - T^6 + T^4 - T^2 + 1
$13$	$T^{8}$ T^8
$17$	$T^{8} - T^{6} + T^{4} + \cdots + 1$ T^8 - T^6 + T^4 - T^2 + 1
$19$	$(T^{4} - T^{3} + T^{2} + \cdots + 1)^{2}$ (T^4 - T^3 + T^2 - T + 1)^2
$23$	$T^{8} - T^{6} + T^{4} + \cdots + 1$ T^8 - T^6 + T^4 - T^2 + 1
$29$	$T^{8} + T^{6} + 6 T^{4} + \cdots + 1$ T^8 + T^6 + 6T^4 - 4T^2 + 1
$31$	$T^{8}$ T^8
$37$	$(T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2}$ (T^4 + 2T^3 + 4T^2 + 3*T + 1)^2
$41$	$T^{8} - T^{6} + T^{4} + \cdots + 1$ T^8 - T^6 + T^4 - T^2 + 1
$43$	$T^{8}$ T^8
$47$	$T^{8} + T^{6} + 6 T^{4} + \cdots + 1$ T^8 + T^6 + 6T^4 - 4T^2 + 1
$53$	$T^{8} - 4 T^{6} + \cdots + 1$ T^8 - 4T^6 + 6T^4 + T^2 + 1
$59$	$T^{8}$ T^8
$61$	$(T^{4} - T^{3} + T^{2} + \cdots + 1)^{2}$ (T^4 - T^3 + T^2 - T + 1)^2
$67$	$(T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2}$ (T^4 - 2T^3 + 4T^2 - 3*T + 1)^2
$71$	$T^{8} + T^{6} + 6 T^{4} + \cdots + 1$ T^8 + T^6 + 6T^4 - 4T^2 + 1
$73$	$T^{8}$ T^8
$79$	$(T^{4} + 3 T^{3} + 4 T^{2} + \cdots + 1)^{2}$ (T^4 + 3T^3 + 4T^2 + 2*T + 1)^2
$83$	$T^{8} - 4 T^{6} + \cdots + 1$ T^8 - 4T^6 + 6T^4 + T^2 + 1
$89$	$T^{8} - T^{6} + T^{4} + \cdots + 1$ T^8 - T^6 + T^4 - T^2 + 1
$97$	$(T^{4} - T^{3} + T^{2} + \cdots + 1)^{2}$ (T^4 - T^3 + T^2 - T + 1)^2
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
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$n$ :		e.g. 2-40 or 990-1000
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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

Inner twists

Twists

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	3375.1.o.a

Level	$3375$
Weight	$1$
Character orbit	3375.o
Analytic conductor	$1.684$
Analytic rank	$0$
Dimension	$8$
Projective image	$A_{5}$
CM/RM	no
Inner twists	$4$