Properties

Label 3626.2.a.x
Level $3626$
Weight $2$
Character orbit 3626.a
Self dual yes
Analytic conductor $28.954$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{3} q^{3} + q^{4} + (2 \beta_{3} + \beta_1) q^{5} - \beta_{3} q^{6} - q^{8} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_{3} q^{3} + q^{4} + (2 \beta_{3} + \beta_1) q^{5} - \beta_{3} q^{6} - q^{8} + ( - \beta_{2} - 1) q^{9} + ( - 2 \beta_{3} - \beta_1) q^{10} + ( - \beta_{2} - 2) q^{11} + \beta_{3} q^{12} - 3 \beta_{3} q^{13} + ( - 2 \beta_{2} + 3) q^{15} + q^{16} + (2 \beta_{3} + 4 \beta_1) q^{17} + (\beta_{2} + 1) q^{18} + 2 \beta_{3} q^{19} + (2 \beta_{3} + \beta_1) q^{20} + (\beta_{2} + 2) q^{22} + ( - \beta_{2} - 1) q^{23} - \beta_{3} q^{24} + ( - 3 \beta_{2} + 2) q^{25} + 3 \beta_{3} q^{26} + ( - \beta_{3} + \beta_1) q^{27} + (\beta_{2} + 1) q^{29} + (2 \beta_{2} - 3) q^{30} - \beta_1 q^{31} - q^{32} + (\beta_{3} + \beta_1) q^{33} + ( - 2 \beta_{3} - 4 \beta_1) q^{34} + ( - \beta_{2} - 1) q^{36} - q^{37} - 2 \beta_{3} q^{38} + (3 \beta_{2} - 6) q^{39} + ( - 2 \beta_{3} - \beta_1) q^{40} + (5 \beta_{3} - 4 \beta_1) q^{41} + (2 \beta_{2} + 6) q^{43} + ( - \beta_{2} - 2) q^{44} + (3 \beta_{3} - \beta_1) q^{45} + (\beta_{2} + 1) q^{46} + (2 \beta_{3} + 4 \beta_1) q^{47} + \beta_{3} q^{48} + (3 \beta_{2} - 2) q^{50} - 2 \beta_{2} q^{51} - 3 \beta_{3} q^{52} + (\beta_{3} - \beta_1) q^{54} + (\beta_{3} - 2 \beta_1) q^{55} + ( - 2 \beta_{2} + 4) q^{57} + ( - \beta_{2} - 1) q^{58} + (4 \beta_{3} + 4 \beta_1) q^{59} + ( - 2 \beta_{2} + 3) q^{60} + ( - 2 \beta_{3} - 7 \beta_1) q^{61} + \beta_1 q^{62} + q^{64} + (6 \beta_{2} - 9) q^{65} + ( - \beta_{3} - \beta_1) q^{66} + (3 \beta_{2} + 2) q^{67} + (2 \beta_{3} + 4 \beta_1) q^{68} + (2 \beta_{3} + \beta_1) q^{69} + 12 q^{71} + (\beta_{2} + 1) q^{72} + (2 \beta_{3} + \beta_1) q^{73} + q^{74} + (11 \beta_{3} + 3 \beta_1) q^{75} + 2 \beta_{3} q^{76} + ( - 3 \beta_{2} + 6) q^{78} + (\beta_{2} + 5) q^{79} + (2 \beta_{3} + \beta_1) q^{80} + 4 \beta_{2} q^{81} + ( - 5 \beta_{3} + 4 \beta_1) q^{82} + (6 \beta_{3} + 6 \beta_1) q^{83} + 10 q^{85} + ( - 2 \beta_{2} - 6) q^{86} + ( - 2 \beta_{3} - \beta_1) q^{87} + (\beta_{2} + 2) q^{88} + (4 \beta_{3} - 6 \beta_1) q^{89} + ( - 3 \beta_{3} + \beta_1) q^{90} + ( - \beta_{2} - 1) q^{92} + q^{93} + ( - 2 \beta_{3} - 4 \beta_1) q^{94} + ( - 4 \beta_{2} + 6) q^{95} - \beta_{3} q^{96} + (10 \beta_{3} + 10 \beta_1) q^{97} + (2 \beta_{2} + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 2 q^{9} - 6 q^{11} + 16 q^{15} + 4 q^{16} + 2 q^{18} + 6 q^{22} - 2 q^{23} + 14 q^{25} + 2 q^{29} - 16 q^{30} - 4 q^{32} - 2 q^{36} - 4 q^{37} - 30 q^{39} + 20 q^{43} - 6 q^{44} + 2 q^{46} - 14 q^{50} + 4 q^{51} + 20 q^{57} - 2 q^{58} + 16 q^{60} + 4 q^{64} - 48 q^{65} + 2 q^{67} + 48 q^{71} + 2 q^{72} + 4 q^{74} + 30 q^{78} + 18 q^{79} - 8 q^{81} + 40 q^{85} - 20 q^{86} + 6 q^{88} - 2 q^{92} + 4 q^{93} + 32 q^{95} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) 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
0.456850
2.18890
−2.18890
−0.456850
−1.00000 −2.18890 1.00000 −3.92095 2.18890 0 −1.00000 1.79129 3.92095
1.2 −1.00000 −0.456850 1.00000 1.27520 0.456850 0 −1.00000 −2.79129 −1.27520
1.3 −1.00000 0.456850 1.00000 −1.27520 −0.456850 0 −1.00000 −2.79129 1.27520
1.4 −1.00000 2.18890 1.00000 3.92095 −2.18890 0 −1.00000 1.79129 −3.92095
\(n\): 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.x 4
7.b odd 2 1 inner 3626.2.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3626.2.a.x 4 1.a even 1 1 trivial
3626.2.a.x 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3626))\):

\( T_{3}^{4} - 5T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{4} - 17T_{5}^{2} + 25 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 5T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 17T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 45T^{2} + 81 \) Copy content Toggle raw display
$17$ \( T^{4} - 68T^{2} + 400 \) Copy content Toggle raw display
$19$ \( T^{4} - 20T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} + T - 5)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - T - 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 5T^{2} + 1 \) Copy content Toggle raw display
$37$ \( (T + 1)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 285 T^{2} + 19881 \) Copy content Toggle raw display
$43$ \( (T^{2} - 10 T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 68T^{2} + 400 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 209T^{2} + 289 \) Copy content Toggle raw display
$67$ \( (T^{2} - T - 47)^{2} \) Copy content Toggle raw display
$71$ \( (T - 12)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 17T^{2} + 25 \) Copy content Toggle raw display
$79$ \( (T^{2} - 9 T + 15)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 356 T^{2} + 29584 \) Copy content Toggle raw display
$97$ \( (T^{2} - 300)^{2} \) Copy content Toggle raw display
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