Properties

Label 3640.2.a.v
Level $3640$
Weight $2$
Character orbit 3640.a
Self dual yes
Analytic conductor $29.066$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3640,2,Mod(1,3640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3640, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 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("3640.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3640.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: \(29.0655463357\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} - q^{7} + (\beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - q^{5} - q^{7} + (\beta_{2} + \beta_1) q^{9} + (\beta_{3} - 1) q^{11} - q^{13} - \beta_1 q^{15} + ( - 3 \beta_{2} - \beta_1 - 3) q^{17} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{19} - \beta_1 q^{21} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{23} + q^{25} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{27} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{29} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{31} + (2 \beta_{2} + 2) q^{33} + q^{35} + ( - \beta_{3} + \beta_1 + 3) q^{37} - \beta_1 q^{39} + ( - 2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 1) q^{41} + ( - \beta_{3} + 6 \beta_1 - 3) q^{43} + ( - \beta_{2} - \beta_1) q^{45} + (2 \beta_{2} + 2 \beta_1 + 6) q^{47} + q^{49} + ( - 3 \beta_{3} - \beta_{2} - 4 \beta_1) q^{51} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{53} + ( - \beta_{3} + 1) q^{55} + (\beta_{3} - \beta_{2}) q^{57} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 + 4) q^{59} + (4 \beta_{3} + 2 \beta_{2} + 4) q^{61} + ( - \beta_{2} - \beta_1) q^{63} + q^{65} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 - 3) q^{67} + (2 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 6) q^{69} + ( - 3 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 3) q^{71} + (4 \beta_{3} - 2 \beta_1 + 6) q^{73} + \beta_1 q^{75} + ( - \beta_{3} + 1) q^{77} + (4 \beta_{3} - 5 \beta_{2} - 5 \beta_1 - 7) q^{79} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 - 5) q^{81} + (2 \beta_{3} + 2 \beta_1 + 2) q^{83} + (3 \beta_{2} + \beta_1 + 3) q^{85} + ( - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 7) q^{87} + ( - 3 \beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{89} + q^{91} + ( - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 11) q^{93} + (\beta_{3} - \beta_{2} - \beta_1) q^{95} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{97} + ( - \beta_{3} + 2 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} - 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} - 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} - 4 q^{13} - q^{15} - 7 q^{17} - q^{19} - q^{21} + 2 q^{23} + 4 q^{25} + 4 q^{27} + q^{29} - q^{31} + 4 q^{33} + 4 q^{35} + 13 q^{37} - q^{39} + q^{41} - 6 q^{43} + q^{45} + 22 q^{47} + 4 q^{49} - 2 q^{51} + 14 q^{53} + 4 q^{55} + 2 q^{57} + 21 q^{59} + 12 q^{61} + q^{63} + 4 q^{65} - 7 q^{67} + 20 q^{69} - 4 q^{71} + 22 q^{73} + q^{75} + 4 q^{77} - 23 q^{79} - 16 q^{81} + 10 q^{83} + 7 q^{85} + 23 q^{87} + 15 q^{89} + 4 q^{91} + 39 q^{93} + q^{95} - 8 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 2 \) 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
−1.75660
−0.820249
1.13856
2.43828
0 −1.75660 0 −1.00000 0 −1.00000 0 0.0856374 0
1.2 0 −0.820249 0 −1.00000 0 −1.00000 0 −2.32719 0
1.3 0 1.13856 0 −1.00000 0 −1.00000 0 −1.70367 0
1.4 0 2.43828 0 −1.00000 0 −1.00000 0 2.94523 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)
\(13\) \(1\)

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 3640.2.a.v 4
4.b odd 2 1 7280.2.a.bt 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3640.2.a.v 4 1.a even 1 1 trivial
7280.2.a.bt 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3640))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} - 5 T^{2} + 2 T + 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 7 T^{3} - 35 T^{2} - 176 T + 484 \) Copy content Toggle raw display
$19$ \( T^{4} + T^{3} - 13 T^{2} + 14 T - 4 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} - 52 T^{2} - 112 T - 64 \) Copy content Toggle raw display
$29$ \( T^{4} - T^{3} - 53 T^{2} + 236 T - 284 \) Copy content Toggle raw display
$31$ \( T^{4} + T^{3} - 95 T^{2} + 198 T + 484 \) Copy content Toggle raw display
$37$ \( T^{4} - 13 T^{3} + 53 T^{2} - 68 T - 4 \) Copy content Toggle raw display
$41$ \( T^{4} - T^{3} - 85 T^{2} + 352 T - 116 \) Copy content Toggle raw display
$43$ \( T^{4} + 6 T^{3} - 160 T^{2} + \cdots + 6544 \) Copy content Toggle raw display
$47$ \( T^{4} - 22 T^{3} + 148 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{4} - 14 T^{3} - 80 T^{2} + \cdots - 1136 \) Copy content Toggle raw display
$59$ \( T^{4} - 21 T^{3} + 77 T^{2} + \cdots - 3964 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} - 132 T^{2} + \cdots + 1856 \) Copy content Toggle raw display
$67$ \( T^{4} + 7 T^{3} - 65 T^{2} - 196 T - 16 \) Copy content Toggle raw display
$71$ \( T^{4} + 4 T^{3} - 164 T^{2} + \cdots - 944 \) Copy content Toggle raw display
$73$ \( T^{4} - 22 T^{3} + 40 T^{2} + \cdots - 496 \) Copy content Toggle raw display
$79$ \( T^{4} + 23 T^{3} - 71 T^{2} + \cdots - 21104 \) Copy content Toggle raw display
$83$ \( T^{4} - 10 T^{3} - 44 T^{2} + 320 T + 64 \) Copy content Toggle raw display
$89$ \( T^{4} - 15 T^{3} - 45 T^{2} + \cdots - 100 \) Copy content Toggle raw display
$97$ \( T^{4} + 8 T^{3} - 60 T^{2} + \cdots - 1216 \) Copy content Toggle raw display
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