Properties

Label 3311.1.h.a
Level $3311$
Weight $1$
Character orbit 3311.h
Self dual yes
Analytic conductor $1.652$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -7, -3311, 473
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3311,1,Mod(3310,3311)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3311, 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("3311.3310");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3311 = 7 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3311.h (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: \(1.65240425683\)
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 \(\Q(\sqrt{-7}, \sqrt{473})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.23177.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 3 q^{4} - q^{7} - 4 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 3 q^{4} - q^{7} - 4 q^{8} - q^{9} - q^{11} + 2 q^{14} + 5 q^{16} + 2 q^{18} + 2 q^{22} - 2 q^{23} - q^{25} - 3 q^{28} + 2 q^{29} - 6 q^{32} - 3 q^{36} + q^{43} - 3 q^{44} + 4 q^{46} + q^{49} + 2 q^{50} + 2 q^{53} + 4 q^{56} - 4 q^{58} + q^{63} + 7 q^{64} + 2 q^{67} + 4 q^{72} + q^{77} + q^{81} - 2 q^{86} + 4 q^{88} - 6 q^{92} - 2 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3311\mathbb{Z}\right)^\times\).

\(n\) \(904\) \(1893\) \(2927\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3310.1
0
−2.00000 0 3.00000 0 0 −1.00000 −4.00000 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
473.d even 2 1 RM by \(\Q(\sqrt{473}) \)
3311.h odd 2 1 CM by \(\Q(\sqrt{-3311}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3311.1.h.a 1
7.b odd 2 1 CM 3311.1.h.a 1
11.b odd 2 1 3311.1.h.f yes 1
43.b odd 2 1 3311.1.h.f yes 1
77.b even 2 1 3311.1.h.f yes 1
301.c even 2 1 3311.1.h.f yes 1
473.d even 2 1 RM 3311.1.h.a 1
3311.h odd 2 1 CM 3311.1.h.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3311.1.h.a 1 1.a even 1 1 trivial
3311.1.h.a 1 7.b odd 2 1 CM
3311.1.h.a 1 473.d even 2 1 RM
3311.1.h.a 1 3311.h odd 2 1 CM
3311.1.h.f yes 1 11.b odd 2 1
3311.1.h.f yes 1 43.b odd 2 1
3311.1.h.f yes 1 77.b even 2 1
3311.1.h.f yes 1 301.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3311, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 2 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T \) 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
$53$ \( T - 2 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 2 \) 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
$97$ \( T \) Copy content Toggle raw display
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