Properties

Label 2695.1.g.b
Level $2695$
Weight $1$
Character orbit 2695.g
Self dual yes
Analytic conductor $1.345$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -55
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2695,1,Mod(1814,2695)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2695, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2695.1814");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2695 = 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2695.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: \(1.34498020905\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.2695.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.2695.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{5} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{5} + q^{8} + q^{9} - q^{10} + q^{11} - q^{13} - q^{16} + 2 q^{17} - q^{18} - q^{22} + q^{25} + q^{26} - q^{31} - 2 q^{34} + q^{40} - q^{43} + q^{45} - q^{50} + q^{55} - q^{59} + q^{62} + q^{64} - q^{65} - q^{71} + q^{72} - q^{73} - q^{80} + q^{81} - q^{83} + 2 q^{85} + q^{86} + q^{88} - q^{89} - q^{90} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2695\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1816\) \(2157\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1814.1
0
−1.00000 0 0 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
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2695.1.g.b 1
5.b even 2 1 2695.1.g.e 1
7.b odd 2 1 2695.1.g.a 1
7.c even 3 2 385.1.q.b yes 2
7.d odd 6 2 2695.1.q.d 2
11.b odd 2 1 2695.1.g.e 1
21.h odd 6 2 3465.1.cd.a 2
35.c odd 2 1 2695.1.g.d 1
35.i odd 6 2 2695.1.q.a 2
35.j even 6 2 385.1.q.a 2
35.l odd 12 4 1925.1.w.c 4
55.d odd 2 1 CM 2695.1.g.b 1
77.b even 2 1 2695.1.g.d 1
77.h odd 6 2 385.1.q.a 2
77.i even 6 2 2695.1.q.a 2
105.o odd 6 2 3465.1.cd.b 2
231.l even 6 2 3465.1.cd.b 2
385.h even 2 1 2695.1.g.a 1
385.o even 6 2 2695.1.q.d 2
385.q odd 6 2 385.1.q.b yes 2
385.bc even 12 4 1925.1.w.c 4
1155.bo even 6 2 3465.1.cd.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.1.q.a 2 35.j even 6 2
385.1.q.a 2 77.h odd 6 2
385.1.q.b yes 2 7.c even 3 2
385.1.q.b yes 2 385.q odd 6 2
1925.1.w.c 4 35.l odd 12 4
1925.1.w.c 4 385.bc even 12 4
2695.1.g.a 1 7.b odd 2 1
2695.1.g.a 1 385.h even 2 1
2695.1.g.b 1 1.a even 1 1 trivial
2695.1.g.b 1 55.d odd 2 1 CM
2695.1.g.d 1 35.c odd 2 1
2695.1.g.d 1 77.b even 2 1
2695.1.g.e 1 5.b even 2 1
2695.1.g.e 1 11.b odd 2 1
2695.1.q.a 2 35.i odd 6 2
2695.1.q.a 2 77.i even 6 2
2695.1.q.d 2 7.d odd 6 2
2695.1.q.d 2 385.o even 6 2
3465.1.cd.a 2 21.h odd 6 2
3465.1.cd.a 2 1155.bo even 6 2
3465.1.cd.b 2 105.o odd 6 2
3465.1.cd.b 2 231.l even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2695, [\chi])\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display
\( T_{31} + 1 \) 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
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T + 1 \) 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
$31$ \( T + 1 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 1 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T + 1 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T + 1 \) Copy content Toggle raw display
$73$ \( T + 1 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 1 \) Copy content Toggle raw display
$89$ \( T + 1 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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