LMFDB - Newform orbit 1035.1.bd.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1035,1,Mod(19,1035)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1035, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 11, 15]))

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

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

chi := DirichletCharacter("1035.19");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1035 = 3^{2} \cdot 5 \cdot 23$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1035.bd (of order $22$ , degree $10$ , 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:	$0.516532288075$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{22}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{22} - \cdots)$

$q$ -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.

$f(q)$	$=$	$q + ( - \zeta_{22}^{10} + \zeta_{22}^{2}) q^{2} + ( - \zeta_{22}^{9} + \zeta_{22}^{4} + \zeta_{22}) q^{4} - \zeta_{22}^{3} q^{5} + ( - \zeta_{22}^{8} + \zeta_{22}^{6} + \cdots + 1) q^{8} + ( - \zeta_{22}^{5} - \zeta_{22}^{2}) q^{10} + \cdots + ( - \zeta_{22}^{6} - \zeta_{22}^{3}) q^{98} +O(q^{100})$ q + (-z^10 + z^2) * q^2 + (-z^9 + z^4 + z) * q^4 - z^3 * q^5 + (-z^8 + z^6 + z^3 + 1) * q^8 + (-z^5 - z^2) * q^10 + (-z^10 + z^8 - z^7 + z^5 + z^2) * q^16 + (z^9 - 1) * q^17 + (-z^10 - z^3) * q^19 + (-z^7 - z^4 - z) * q^20 + z^5 * q^23 + z^6 * q^25 + (z^9 + z^5) * q^31 + (z^10 - z^9 + z^7 - z^6 + z^4 + z) * q^32 + (z^10 + z^8 - z^2 - 1) * q^34 + (-z^9 - z^5 - z^2 + z) * q^38 + (-z^9 - z^6 - z^3 - 1) * q^40 + (z^7 + z^4) * q^46 + (-z^6 - z^5) * q^47 - z^4 * q^49 + (z^8 + z^5) * q^50 + (z^8 - z^7) * q^53 + (z^2 + z) * q^61 + (z^8 + z^7 + z^4 - 1) * q^62 + (z^9 - z^8 + z^6 - z^5 + z^3 - z + 1) * q^64 + (z^10 + z^9 + z^7 - z^4 - z^2 - z) * q^68 + (-z^8 - z^7 - z^4 + z^3 - z + 1) * q^76 + (-z^6 - z) * q^79 + (z^10 - z^8 - z^5 - z^2 + 1) * q^80 + (z^4 - z) * q^83 + (z^3 + z) * q^85 + (z^9 + z^6 + z^3) * q^92 + (-z^8 - z^7 - z^5 - z^4) * q^94 + (z^6 - z^2) * q^95 + (-z^6 - z^3) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$10 q - q^{4} - q^{5} + 11 q^{8} - q^{16} - 9 q^{17} - q^{20} + q^{23} - q^{25} + 2 q^{31} - 11 q^{34} - 11 q^{40} + q^{49} - 2 q^{53} - 11 q^{62} + 10 q^{64} + 2 q^{68} + 11 q^{76} + 10 q^{80} - 2 q^{83}+ \cdots + q^{92}+O(q^{100})$ 10 * q - q^4 - q^5 + 11 * q^8 - q^16 - 9 * q^17 - q^20 + q^23 - q^25 + 2 * q^31 - 11 * q^34 - 11 * q^40 + q^49 - 2 * q^53 - 11 * q^62 + 10 * q^64 + 2 * q^68 + 11 * q^76 + 10 * q^80 - 2 * q^83 + 2 * q^85 + q^92

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1035\mathbb{Z}\right)^\times$ .

$n$	$461$	$622$	$856$
$\chi(n)$	$1$	$-1$	$-\zeta_{22}^{6}$

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}

19.1

0.654861	−	0.755750i
0.654861	+	0.755750i
0.959493	+	0.281733i
−0.841254	+	0.540641i
−0.415415	+	0.909632i
0.142315	−	0.989821i
0.142315	+	0.989821i
−0.841254	−	0.540641i
−0.415415	−	0.909632i
0.959493	−	0.281733i

0.512546

−

0.234072i

−0.446947

0.515804i

0.841254

0.540641i

−0.267092

0.909632i

0.557730

0.0801894i

109.1

0.512546

0.234072i

−0.446947

−

0.515804i

0.841254

−

0.540641i

−0.267092

−

0.909632i

0.557730

−

0.0801894i

199.1

1.80075

0.258908i

2.21616

0.650724i

−0.654861

−

0.755750i

2.16741

0.989821i

−0.983568

−

1.53046i

244.1

−0.425839

−

1.45027i

−1.08070

0.694523i

−0.142315

−

0.989821i

0.325137

0.281733i

−1.37491

0.627899i

379.1

−1.07028

−

1.66538i

−1.21259

2.65520i

−0.959493

0.281733i

3.76024

−

0.540641i

1.49611

1.29639i

424.1

−0.817178

0.708089i

0.0240754

−

0.167448i

0.415415

−

0.909632i

−0.485691

−

0.755750i

0.304632

1.03748i

559.1

−0.817178

−

0.708089i

0.0240754

0.167448i

0.415415

0.909632i

−0.485691

0.755750i

0.304632

−

1.03748i

649.1

−0.425839

1.45027i

−1.08070

−

0.694523i

−0.142315

0.989821i

0.325137

−

0.281733i

−1.37491

−

0.627899i

964.1

−1.07028

1.66538i

−1.21259

−

2.65520i

−0.959493

−

0.281733i

3.76024

0.540641i

1.49611

−

1.29639i

1009.1

1.80075

−

0.258908i

2.21616

−

0.650724i

−0.654861

0.755750i

2.16741

−

0.989821i

−0.983568

1.53046i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15})$
69.g	even	22	1	inner
115.i	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1035.1.bd.a	✓	10
3.b	odd	2	1		1035.1.bd.b	yes	10
5.b	even	2	1		1035.1.bd.b	yes	10
15.d	odd	2	1	CM	1035.1.bd.a	✓	10
23.d	odd	22	1		1035.1.bd.b	yes	10
69.g	even	22	1	inner	1035.1.bd.a	✓	10
115.i	odd	22	1	inner	1035.1.bd.a	✓	10
345.n	even	22	1		1035.1.bd.b	yes	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1035.1.bd.a	✓	10	1.a	even	1	1	trivial
1035.1.bd.a	✓	10	15.d	odd	2	1	CM
1035.1.bd.a	✓	10	69.g	even	22	1	inner
1035.1.bd.a	✓	10	115.i	odd	22	1	inner
1035.1.bd.b	yes	10	3.b	odd	2	1
1035.1.bd.b	yes	10	5.b	even	2	1
1035.1.bd.b	yes	10	23.d	odd	22	1
1035.1.bd.b	yes	10	345.n	even	22	1

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{10} - 11T_{2}^{7} + 33T_{2}^{4} + 11T_{2}^{3} - 22T_{2} + 11$ T2^10 - 11*T2^7 + 33*T2^4 + 11*T2^3 - 22*T2 + 11 acting on $S_{1}^{\mathrm{new}}(1035, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{10} - 11 T^{7} + \cdots + 11$ T^10 - 11T^7 + 33T^4 + 11T^3 - 22T + 11
$3$	$T^{10}$ T^10
$5$	$T^{10} + T^{9} + \cdots + 1$ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$7$	$T^{10}$ T^10
$11$	$T^{10}$ T^10
$13$	$T^{10}$ T^10
$17$	$T^{10} + 9 T^{9} + \cdots + 1$ T^10 + 9T^9 + 37T^8 + 91T^7 + 148T^6 + 166T^5 + 130T^4 + 70T^3 + 25T^2 + 5*T + 1
$19$	$T^{10} + 11 T^{6} + \cdots + 11$ T^10 + 11T^6 + 11T^5 + 22T^2 - 33T + 11
$23$	$T^{10} - T^{9} + \cdots + 1$ T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$29$	$T^{10}$ T^10
$31$	$T^{10} - 2 T^{9} + \cdots + 1$ T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 10T^5 + 20T^4 - 7T^3 + 3T^2 + 5*T + 1
$37$	$T^{10}$ T^10
$41$	$T^{10}$ T^10
$43$	$T^{10}$ T^10
$47$	$T^{10} + 11 T^{8} + \cdots + 11$ T^10 + 11T^8 + 44T^6 + 77T^4 + 55T^2 + 11
$53$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 - 3T^7 - 6T^6 - 12T^5 + 9T^4 + 7T^3 + 14T^2 + 6*T + 1
$59$	$T^{10}$ T^10
$61$	$T^{10} + 11 T^{4} + \cdots + 11$ T^10 + 11T^4 + 55T^3 + 77T^2 + 44T + 11
$67$	$T^{10}$ T^10
$71$	$T^{10}$ T^10
$73$	$T^{10}$ T^10
$79$	$T^{10} + 11 T^{4} + \cdots + 11$ T^10 + 11T^4 + 55T^3 + 77T^2 + 44T + 11
$83$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 10T^5 + 20T^4 + 7T^3 + 3T^2 - 5*T + 1
$89$	$T^{10}$ T^10
$97$	$T^{10}$ T^10
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Elliptic curves over $\Q$
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Introduction

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Properties

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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	1035.1.bd.a

Level	$1035$
Weight	$1$
Character orbit	1035.bd
Analytic conductor	$0.517$
Analytic rank	$0$
Dimension	$10$
Projective image	$D_{22}$
CM discriminant	-15
Inner twists	$4$