Properties

Label 2640.2.a.bc
Level $2640$
Weight $2$
Character orbit 2640.a
Self dual yes
Analytic conductor $21.081$
Analytic rank $0$
Dimension $2$
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] = [2640,2,Mod(1,2640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2640, 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("2640.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2640.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: \(21.0805061336\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 660)
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 \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + ( - \beta - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{5} + ( - \beta - 1) q^{7} + q^{9} - q^{11} + ( - \beta - 1) q^{13} + q^{15} + (\beta + 3) q^{17} + 2 \beta q^{19} + ( - \beta - 1) q^{21} + q^{25} + q^{27} + 8 q^{29} + ( - 2 \beta - 2) q^{31} - q^{33} + ( - \beta - 1) q^{35} + ( - 2 \beta + 4) q^{37} + ( - \beta - 1) q^{39} + 8 q^{41} + ( - \beta + 7) q^{43} + q^{45} + (2 \beta - 2) q^{47} + (2 \beta + 7) q^{49} + (\beta + 3) q^{51} + 2 q^{53} - q^{55} + 2 \beta q^{57} - 8 q^{59} + 2 \beta q^{61} + ( - \beta - 1) q^{63} + ( - \beta - 1) q^{65} + 4 q^{67} + 4 \beta q^{71} + (\beta - 3) q^{73} + q^{75} + (\beta + 1) q^{77} + (2 \beta + 4) q^{79} + q^{81} + (\beta + 7) q^{83} + (\beta + 3) q^{85} + 8 q^{87} + 6 q^{89} + (2 \beta + 14) q^{91} + ( - 2 \beta - 2) q^{93} + 2 \beta q^{95} + ( - 4 \beta + 2) q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} - 2 q^{21} + 2 q^{25} + 2 q^{27} + 16 q^{29} - 4 q^{31} - 2 q^{33} - 2 q^{35} + 8 q^{37} - 2 q^{39} + 16 q^{41} + 14 q^{43} + 2 q^{45} - 4 q^{47} + 14 q^{49} + 6 q^{51} + 4 q^{53} - 2 q^{55} - 16 q^{59} - 2 q^{63} - 2 q^{65} + 8 q^{67} - 6 q^{73} + 2 q^{75} + 2 q^{77} + 8 q^{79} + 2 q^{81} + 14 q^{83} + 6 q^{85} + 16 q^{87} + 12 q^{89} + 28 q^{91} - 4 q^{93} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
0 1.00000 0 1.00000 0 −4.60555 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 2.60555 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)

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 2640.2.a.bc 2
3.b odd 2 1 7920.2.a.bo 2
4.b odd 2 1 660.2.a.e 2
12.b even 2 1 1980.2.a.h 2
20.d odd 2 1 3300.2.a.w 2
20.e even 4 2 3300.2.c.l 4
44.c even 2 1 7260.2.a.w 2
60.h even 2 1 9900.2.a.bl 2
60.l odd 4 2 9900.2.c.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.a.e 2 4.b odd 2 1
1980.2.a.h 2 12.b even 2 1
2640.2.a.bc 2 1.a even 1 1 trivial
3300.2.a.w 2 20.d odd 2 1
3300.2.c.l 4 20.e even 4 2
7260.2.a.w 2 44.c even 2 1
7920.2.a.bo 2 3.b odd 2 1
9900.2.a.bl 2 60.h even 2 1
9900.2.c.q 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2640))\):

\( T_{7}^{2} + 2T_{7} - 12 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 12 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} - 4 \) Copy content Toggle raw display
\( T_{19}^{2} - 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 52 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
$41$ \( (T - 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 14T + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( (T + 8)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 52 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 208 \) Copy content Toggle raw display
$73$ \( T^{2} + 6T - 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
$83$ \( T^{2} - 14T + 36 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 204 \) Copy content Toggle raw display
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