Properties

Label 3626.2.a.be
Level $3626$
Weight $2$
Character orbit 3626.a
Self dual yes
Analytic conductor $28.954$
Analytic rank $1$
Dimension $6$
CM no
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: \(6\)
Coefficient field: 6.6.1229312.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 24x^{2} - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3} \)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_1 q^{6} - q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_1 q^{6} - q^{8} + \beta_{2} q^{9} + ( - \beta_{3} - 1) q^{11} + \beta_1 q^{12} + ( - \beta_{5} + \beta_{4} - \beta_1) q^{13} + q^{16} - \beta_1 q^{17} - \beta_{2} q^{18} + (\beta_{5} - 2 \beta_{4} + \beta_1) q^{19} + (\beta_{3} + 1) q^{22} + (\beta_{3} - 2 \beta_{2} + 1) q^{23} - \beta_1 q^{24} - 5 q^{25} + (\beta_{5} - \beta_{4} + \beta_1) q^{26} + (\beta_{5} + \beta_{4} - \beta_1) q^{27} + (3 \beta_{3} - \beta_{2} + 2) q^{29} + (\beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{31} - q^{32} + ( - \beta_{5} - 3 \beta_{4} - \beta_1) q^{33} + \beta_1 q^{34} + \beta_{2} q^{36} - q^{37} + ( - \beta_{5} + 2 \beta_{4} - \beta_1) q^{38} + ( - 2 \beta_{2} - 2) q^{39} + (\beta_{5} + 2 \beta_{4} - \beta_1) q^{41} + (2 \beta_{3} - 3 \beta_{2} + 1) q^{43} + ( - \beta_{3} - 1) q^{44} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{46} + (\beta_{5} - 3 \beta_{4} - \beta_1) q^{47} + \beta_1 q^{48} + 5 q^{50} + ( - \beta_{2} - 3) q^{51} + ( - \beta_{5} + \beta_{4} - \beta_1) q^{52} + ( - 3 \beta_{3} + 4 \beta_{2} - 3) q^{53} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{54} + ( - \beta_{3} + 2 \beta_{2} + 1) q^{57} + ( - 3 \beta_{3} + \beta_{2} - 2) q^{58} + ( - 2 \beta_{5} + \beta_{4}) q^{59} + ( - 2 \beta_{5} - 4 \beta_{4}) q^{61} + ( - \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{62} + q^{64} + (\beta_{5} + 3 \beta_{4} + \beta_1) q^{66} + ( - 2 \beta_{3} - 6) q^{67} - \beta_1 q^{68} + ( - \beta_{5} + \beta_{4} - 3 \beta_1) q^{69} + ( - \beta_{3} + 3 \beta_{2}) q^{71} - \beta_{2} q^{72} + (2 \beta_{5} + 3 \beta_{4} - 2 \beta_1) q^{73} + q^{74} - 5 \beta_1 q^{75} + (\beta_{5} - 2 \beta_{4} + \beta_1) q^{76} + (2 \beta_{2} + 2) q^{78} + ( - \beta_{3} - 7) q^{79} + (2 \beta_{3} - 3 \beta_{2} - 2) q^{81} + ( - \beta_{5} - 2 \beta_{4} + \beta_1) q^{82} + ( - \beta_{5} - \beta_{4} + 2 \beta_1) q^{83} + ( - 2 \beta_{3} + 3 \beta_{2} - 1) q^{86} + (2 \beta_{5} + 8 \beta_{4}) q^{87} + (\beta_{3} + 1) q^{88} + ( - 3 \beta_{5} + \beta_{4} - 2 \beta_1) q^{89} + (\beta_{3} - 2 \beta_{2} + 1) q^{92} + (3 \beta_{3} - \beta_{2} - 4) q^{93} + ( - \beta_{5} + 3 \beta_{4} + \beta_1) q^{94} - \beta_1 q^{96} + (\beta_{5} + 3 \beta_{4} - 4 \beta_1) q^{97} + ( - \beta_{3} - 2 \beta_{2} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} + 2 q^{9} - 4 q^{11} + 6 q^{16} - 2 q^{18} + 4 q^{22} - 30 q^{25} + 4 q^{29} - 6 q^{32} + 2 q^{36} - 6 q^{37} - 16 q^{39} - 4 q^{43} - 4 q^{44} + 30 q^{50} - 20 q^{51} - 4 q^{53} + 12 q^{57} - 4 q^{58} + 6 q^{64} - 32 q^{67} + 8 q^{71} - 2 q^{72} + 6 q^{74} + 16 q^{78} - 40 q^{79} - 22 q^{81} + 4 q^{86} + 4 q^{88} - 32 q^{93} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 10x^{4} + 24x^{2} - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 6\nu^{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 8\nu^{3} + 12\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 12\nu^{3} - 32\nu ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{3} + 6\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{5} + 12\beta_{4} + 28\beta_1 \) 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
−2.54832
−1.76350
−0.629384
0.629384
1.76350
2.54832
−1.00000 −2.54832 1.00000 0 2.54832 0 −1.00000 3.49396 0
1.2 −1.00000 −1.76350 1.00000 0 1.76350 0 −1.00000 0.109916 0
1.3 −1.00000 −0.629384 1.00000 0 0.629384 0 −1.00000 −2.60388 0
1.4 −1.00000 0.629384 1.00000 0 −0.629384 0 −1.00000 −2.60388 0
1.5 −1.00000 1.76350 1.00000 0 −1.76350 0 −1.00000 0.109916 0
1.6 −1.00000 2.54832 1.00000 0 −2.54832 0 −1.00000 3.49396 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.be 6
7.b odd 2 1 inner 3626.2.a.be 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3626.2.a.be 6 1.a even 1 1 trivial
3626.2.a.be 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3626))\):

\( T_{3}^{6} - 10T_{3}^{4} + 24T_{3}^{2} - 8 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{11}^{3} + 2T_{11}^{2} - 8T_{11} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 10 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} + 2 T^{2} - 8 T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 40 T^{4} + \cdots - 512 \) Copy content Toggle raw display
$17$ \( T^{6} - 10 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$19$ \( T^{6} - 54 T^{4} + \cdots - 1352 \) Copy content Toggle raw display
$23$ \( (T^{3} - 28 T - 56)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 2 T^{2} + \cdots + 232)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 76 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$37$ \( (T + 1)^{6} \) Copy content Toggle raw display
$41$ \( T^{6} - 54 T^{4} + \cdots - 1352 \) Copy content Toggle raw display
$43$ \( (T^{3} + 2 T^{2} + \cdots - 232)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 104 T^{4} + \cdots - 512 \) Copy content Toggle raw display
$53$ \( (T^{3} + 2 T^{2} + \cdots + 328)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 118 T^{4} + \cdots - 6728 \) Copy content Toggle raw display
$61$ \( T^{6} - 208 T^{4} + \cdots - 86528 \) Copy content Toggle raw display
$67$ \( (T^{3} + 16 T^{2} + \cdots - 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 4 T^{2} + \cdots + 232)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 182 T^{4} + \cdots - 66248 \) Copy content Toggle raw display
$79$ \( (T^{3} + 20 T^{2} + \cdots + 232)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 66 T^{4} + \cdots - 1352 \) Copy content Toggle raw display
$89$ \( T^{6} - 290 T^{4} + \cdots - 13448 \) Copy content Toggle raw display
$97$ \( T^{6} - 194 T^{4} + \cdots - 14792 \) Copy content Toggle raw display
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