Properties

Label 3626.2.a.n
Level $3626$
Weight $2$
Character orbit 3626.a
Self dual yes
Analytic conductor $28.954$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3626,2,Mod(1,3626)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3626, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3626.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3626 = 2 \cdot 7^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3626.a (trivial)

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: \(28.9537557729\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + q^{8} - 3 q^{9} + q^{16} + 2 \beta q^{17} - 3 q^{18} - 3 \beta q^{19} - 6 q^{23} - 5 q^{25} - 6 q^{29} + 3 \beta q^{31} + q^{32} + 2 \beta q^{34} - 3 q^{36} + q^{37} - 3 \beta q^{38} - 5 \beta q^{41} - 4 q^{43} - 6 q^{46} + 2 \beta q^{47} - 5 q^{50} - 6 q^{58} + \beta q^{59} + 2 \beta q^{61} + 3 \beta q^{62} + q^{64} - 8 q^{67} + 2 \beta q^{68} - 3 q^{72} + \beta q^{73} + q^{74} - 3 \beta q^{76} - 10 q^{79} + 9 q^{81} - 5 \beta q^{82} - 4 \beta q^{83} - 4 q^{86} - 2 \beta q^{89} - 6 q^{92} + 2 \beta q^{94} - 12 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 6 q^{9} + 2 q^{16} - 6 q^{18} - 12 q^{23} - 10 q^{25} - 12 q^{29} + 2 q^{32} - 6 q^{36} + 2 q^{37} - 8 q^{43} - 12 q^{46} - 10 q^{50} - 12 q^{58} + 2 q^{64} - 16 q^{67} - 6 q^{72} + 2 q^{74} - 20 q^{79} + 18 q^{81} - 8 q^{86} - 12 q^{92}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
1.00000 0 1.00000 0 0 0 1.00000 −3.00000 0
1.2 1.00000 0 1.00000 0 0 0 1.00000 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)
\(37\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3626.2.a.n 2
7.b odd 2 1 inner 3626.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3626.2.a.n 2 1.a even 1 1 trivial
3626.2.a.n 2 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3626))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 8 \) Copy content Toggle raw display
$19$ \( T^{2} - 18 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 18 \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 50 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 2 \) Copy content Toggle raw display
$61$ \( T^{2} - 8 \) Copy content Toggle raw display
$67$ \( (T + 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 2 \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 32 \) Copy content Toggle raw display
$89$ \( T^{2} - 8 \) Copy content Toggle raw display
$97$ \( T^{2} - 288 \) Copy content Toggle raw display
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