Properties

Label 740.1.g.b
Level $740$
Weight $1$
Character orbit 740.g
Self dual yes
Analytic conductor $0.369$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -4, -740, 185
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,1,Mod(739,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.739");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 740.g (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.369308109348\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{185})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.2960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + q^{5} + q^{8} - q^{9} + q^{10} - 2 q^{13} + q^{16} - 2 q^{17} - q^{18} + q^{20} + q^{25} - 2 q^{26} + q^{32} - 2 q^{34} - q^{36} + q^{37} + q^{40} + 2 q^{41} - q^{45} - q^{49} + q^{50} - 2 q^{52} + q^{64} - 2 q^{65} - 2 q^{68} - q^{72} + q^{74} + q^{80} + q^{81} + 2 q^{82} - 2 q^{85} - q^{90} + 2 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/740\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\) \(371\)
\(\chi(n)\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
739.1
0
1.00000 0 1.00000 1.00000 0 0 1.00000 −1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
185.d even 2 1 RM by \(\Q(\sqrt{185}) \)
740.g odd 2 1 CM by \(\Q(\sqrt{-185}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.1.g.b yes 1
4.b odd 2 1 CM 740.1.g.b yes 1
5.b even 2 1 740.1.g.a 1
5.c odd 4 2 3700.1.b.e 2
20.d odd 2 1 740.1.g.a 1
20.e even 4 2 3700.1.b.e 2
37.b even 2 1 740.1.g.a 1
148.b odd 2 1 740.1.g.a 1
185.d even 2 1 RM 740.1.g.b yes 1
185.h odd 4 2 3700.1.b.e 2
740.g odd 2 1 CM 740.1.g.b yes 1
740.m even 4 2 3700.1.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.g.a 1 5.b even 2 1
740.1.g.a 1 20.d odd 2 1
740.1.g.a 1 37.b even 2 1
740.1.g.a 1 148.b odd 2 1
740.1.g.b yes 1 1.a even 1 1 trivial
740.1.g.b yes 1 4.b odd 2 1 CM
740.1.g.b yes 1 185.d even 2 1 RM
740.1.g.b yes 1 740.g odd 2 1 CM
3700.1.b.e 2 5.c odd 4 2
3700.1.b.e 2 20.e even 4 2
3700.1.b.e 2 185.h odd 4 2
3700.1.b.e 2 740.m even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(740, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{97} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 1 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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