Properties

Label 3626.2.a.bb
Level $3626$
Weight $2$
Character orbit 3626.a
Self dual yes
Analytic conductor $28.954$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: 4.4.9792.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 2x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 518)
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.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_{2} + 1) q^{3} + q^{4} - \beta_{3} q^{5} + ( - \beta_{2} + 1) q^{6} + q^{8} - 2 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - \beta_{2} + 1) q^{3} + q^{4} - \beta_{3} q^{5} + ( - \beta_{2} + 1) q^{6} + q^{8} - 2 \beta_{2} q^{9} - \beta_{3} q^{10} + ( - \beta_1 + 1) q^{11} + ( - \beta_{2} + 1) q^{12} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{13} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{15} + q^{16} + (3 \beta_{2} - 2 \beta_1 + 1) q^{17} - 2 \beta_{2} q^{18} + (\beta_{3} + \beta_{2} + 2) q^{19} - \beta_{3} q^{20} + ( - \beta_1 + 1) q^{22} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{23} + ( - \beta_{2} + 1) q^{24} + ( - \beta_{3} - 3 \beta_{2}) q^{25} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{26} + (\beta_{2} + 1) q^{27} + ( - 2 \beta_{3} + 2 \beta_{2} + 2) q^{29} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{30} + (\beta_{2} - \beta_1 + 3) q^{31} + q^{32} + (\beta_{3} - \beta_{2} + 2) q^{33} + (3 \beta_{2} - 2 \beta_1 + 1) q^{34} - 2 \beta_{2} q^{36} - q^{37} + (\beta_{3} + \beta_{2} + 2) q^{38} + (\beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{39} - \beta_{3} q^{40} + ( - 2 \beta_{2} + 7) q^{41} + ( - \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{43} + ( - \beta_1 + 1) q^{44} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{45}+ \cdots + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{8} + 2 q^{10} + 2 q^{11} + 4 q^{12} + 8 q^{13} + 2 q^{15} + 4 q^{16} + 6 q^{19} + 2 q^{20} + 2 q^{22} - 8 q^{23} + 4 q^{24} + 2 q^{25} + 8 q^{26} + 4 q^{27} + 12 q^{29} + 2 q^{30} + 10 q^{31} + 4 q^{32} + 6 q^{33} - 4 q^{37} + 6 q^{38} + 20 q^{39} + 2 q^{40} + 28 q^{41} + 6 q^{43} + 2 q^{44} - 8 q^{46} + 10 q^{47} + 4 q^{48} + 2 q^{50} - 16 q^{51} + 8 q^{52} + 2 q^{53} + 4 q^{54} - 2 q^{55} - 2 q^{57} + 12 q^{58} + 8 q^{59} + 2 q^{60} + 38 q^{61} + 10 q^{62} + 4 q^{64} - 14 q^{65} + 6 q^{66} - 20 q^{69} + 12 q^{71} + 6 q^{73} - 4 q^{74} + 26 q^{75} + 6 q^{76} + 20 q^{78} - 6 q^{79} + 2 q^{80} - 4 q^{81} + 28 q^{82} + 2 q^{83} - 6 q^{85} + 6 q^{86} - 4 q^{87} + 2 q^{88} + 18 q^{89} - 8 q^{92} + 6 q^{93} + 10 q^{94} - 18 q^{95} + 4 q^{96} - 2 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 7x^{2} + 2x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 3\nu^{2} - 3\nu + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 4\nu^{2} + \nu - 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 2\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + 4\beta_{2} + 9\beta _1 + 8 \) 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
3.63019
−1.21597
−1.48330
1.06909
1.00000 −0.414214 1.00000 −0.503673 −0.414214 0 1.00000 −2.82843 −0.503673
1.2 1.00000 −0.414214 1.00000 1.50367 −0.414214 0 1.00000 −2.82843 1.50367
1.3 1.00000 2.41421 1.00000 −2.58101 2.41421 0 1.00000 2.82843 −2.58101
1.4 1.00000 2.41421 1.00000 3.58101 2.41421 0 1.00000 2.82843 3.58101
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3626.2.a.bb 4
7.b odd 2 1 3626.2.a.y 4
7.c even 3 2 518.2.e.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
518.2.e.f 8 7.c even 3 2
3626.2.a.y 4 7.b odd 2 1
3626.2.a.bb 4 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{2} - 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{4} - 2T_{5}^{3} - 9T_{5}^{2} + 10T_{5} + 7 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} - 7T_{11}^{2} + 14T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} + \cdots + 7 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} + \cdots - 161 \) Copy content Toggle raw display
$17$ \( T^{4} - 46 T^{2} + \cdots - 23 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + \cdots - 71 \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} + \cdots - 161 \) Copy content Toggle raw display
$29$ \( T^{4} - 12 T^{3} + \cdots - 1136 \) Copy content Toggle raw display
$31$ \( T^{4} - 10 T^{3} + \cdots + 7 \) Copy content Toggle raw display
$37$ \( (T + 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 14 T + 41)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 6 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$47$ \( T^{4} - 10 T^{3} + \cdots - 401 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} + \cdots + 391 \) Copy content Toggle raw display
$59$ \( T^{4} - 8 T^{3} + \cdots - 3728 \) Copy content Toggle raw display
$61$ \( T^{4} - 38 T^{3} + \cdots + 7327 \) Copy content Toggle raw display
$67$ \( T^{4} - 46 T^{2} + \cdots - 23 \) Copy content Toggle raw display
$71$ \( T^{4} - 12 T^{3} + \cdots + 3184 \) Copy content Toggle raw display
$73$ \( T^{4} - 6 T^{3} + \cdots - 2111 \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} + \cdots + 4743 \) Copy content Toggle raw display
$83$ \( T^{4} - 2 T^{3} + \cdots + 1897 \) Copy content Toggle raw display
$89$ \( T^{4} - 18 T^{3} + \cdots - 3647 \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + \cdots + 49 \) Copy content Toggle raw display
show more
show less