Properties

Label 2624.2.a.r
Level $2624$
Weight $2$
Character orbit 2624.a
Self dual yes
Analytic conductor $20.953$
Analytic rank $0$
Dimension $3$
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] = [2624,2,Mod(1,2624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2624.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2624 = 2^{6} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2624.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: \(20.9527454904\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 41)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{2} + 2) q^{7} + ( - \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - \beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{2} + 2) q^{7} + ( - \beta_{2} - \beta_1) q^{9} + (\beta_{2} - 2 \beta_1) q^{11} + 2 \beta_1 q^{13} + ( - 2 \beta_{2} + 2) q^{15} - 2 q^{17} + ( - \beta_{2} + 2 \beta_1 - 2) q^{19} + ( - 3 \beta_{2} - \beta_1 + 3) q^{21} + (2 \beta_{2} - 2 \beta_1 + 2) q^{23} + ( - 2 \beta_{2} - 1) q^{25} + (2 \beta_{2} + 2) q^{27} + (2 \beta_{2} + 2) q^{29} + ( - 2 \beta_{2} - 2 \beta_1 + 6) q^{31} + (\beta_{2} + 3 \beta_1 - 5) q^{33} + ( - 4 \beta_{2} - 2 \beta_1 + 4) q^{35} + ( - 3 \beta_{2} - 3 \beta_1 + 3) q^{37} + ( - 2 \beta_1 + 2) q^{39} + q^{41} + ( - 2 \beta_1 + 2) q^{43} + ( - \beta_{2} + \beta_1 + 3) q^{45} + (3 \beta_{2} + 6 \beta_1 - 2) q^{47} + ( - 5 \beta_{2} - \beta_1) q^{49} + 2 \beta_{2} q^{51} + (2 \beta_{2} - 2) q^{53} + (4 \beta_{2} + 2 \beta_1) q^{55} + (\beta_{2} - 3 \beta_1 + 5) q^{57} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{59} + (2 \beta_{2} + 4 \beta_1 - 2) q^{61} + ( - 3 \beta_{2} - 2 \beta_1 + 2) q^{63} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{65} + ( - \beta_{2} + 2 \beta_1) q^{67} + (4 \beta_1 - 8) q^{69} + ( - \beta_{2} - 4 \beta_1 + 8) q^{71} + ( - \beta_{2} + 7 \beta_1 - 3) q^{73} + ( - \beta_{2} - 2 \beta_1 + 6) q^{75} + (3 \beta_{2} - \beta_1 - 5) q^{77} + (\beta_{2} + 2 \beta_1 + 10) q^{79} + (3 \beta_{2} + 5 \beta_1 - 6) q^{81} + 4 \beta_{2} q^{83} + (2 \beta_{2} + 2 \beta_1 - 2) q^{85} + (2 \beta_1 - 6) q^{87} + (2 \beta_{2} - 6 \beta_1) q^{89} + (2 \beta_1 + 2) q^{91} + ( - 8 \beta_{2} + 4) q^{93} + ( - 2 \beta_{2} - 2) q^{95} + (4 \beta_{2} + 2) q^{97} + (3 \beta_{2} + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{5} + 6 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{5} + 6 q^{7} - q^{9} - 2 q^{11} + 2 q^{13} + 6 q^{15} - 6 q^{17} - 4 q^{19} + 8 q^{21} + 4 q^{23} - 3 q^{25} + 6 q^{27} + 6 q^{29} + 16 q^{31} - 12 q^{33} + 10 q^{35} + 6 q^{37} + 4 q^{39} + 3 q^{41} + 4 q^{43} + 10 q^{45} - q^{49} - 6 q^{53} + 2 q^{55} + 12 q^{57} + 8 q^{59} - 2 q^{61} + 4 q^{63} - 8 q^{65} + 2 q^{67} - 20 q^{69} + 20 q^{71} - 2 q^{73} + 16 q^{75} - 16 q^{77} + 32 q^{79} - 13 q^{81} - 4 q^{85} - 16 q^{87} - 6 q^{89} + 8 q^{91} + 12 q^{93} - 6 q^{95} + 6 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
−1.48119
2.17009
0.311108
0 −1.67513 0 0.806063 0 0.324869 0 −0.193937 0
1.2 0 −0.539189 0 −1.70928 0 1.46081 0 −2.70928 0
1.3 0 2.21432 0 2.90321 0 4.21432 0 1.90321 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(41\) \(-1\)

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 2624.2.a.r 3
4.b odd 2 1 2624.2.a.q 3
8.b even 2 1 41.2.a.a 3
8.d odd 2 1 656.2.a.f 3
24.f even 2 1 5904.2.a.bk 3
24.h odd 2 1 369.2.a.f 3
40.f even 2 1 1025.2.a.j 3
40.i odd 4 2 1025.2.b.h 6
56.h odd 2 1 2009.2.a.g 3
88.b odd 2 1 4961.2.a.d 3
104.e even 2 1 6929.2.a.b 3
120.i odd 2 1 9225.2.a.bv 3
328.g even 2 1 1681.2.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.2.a.a 3 8.b even 2 1
369.2.a.f 3 24.h odd 2 1
656.2.a.f 3 8.d odd 2 1
1025.2.a.j 3 40.f even 2 1
1025.2.b.h 6 40.i odd 4 2
1681.2.a.d 3 328.g even 2 1
2009.2.a.g 3 56.h odd 2 1
2624.2.a.q 3 4.b odd 2 1
2624.2.a.r 3 1.a even 1 1 trivial
4961.2.a.d 3 88.b odd 2 1
5904.2.a.bk 3 24.f even 2 1
6929.2.a.b 3 104.e even 2 1
9225.2.a.bv 3 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2624))\):

\( T_{3}^{3} - 4T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{3} - 2T_{5}^{2} - 4T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{3} - 6T_{7}^{2} + 8T_{7} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 4T - 2 \) Copy content Toggle raw display
$5$ \( T^{3} - 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$7$ \( T^{3} - 6 T^{2} + 8 T - 2 \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} - 20 T - 50 \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} - 12 T + 8 \) Copy content Toggle raw display
$17$ \( (T + 2)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} + 4 T^{2} - 16 T + 10 \) Copy content Toggle raw display
$23$ \( T^{3} - 4 T^{2} - 32 T - 32 \) Copy content Toggle raw display
$29$ \( T^{3} - 6 T^{2} - 4 T + 40 \) Copy content Toggle raw display
$31$ \( T^{3} - 16 T^{2} + 64 T - 32 \) Copy content Toggle raw display
$37$ \( T^{3} - 6 T^{2} - 36 T + 108 \) Copy content Toggle raw display
$41$ \( (T - 1)^{3} \) Copy content Toggle raw display
$43$ \( T^{3} - 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$47$ \( T^{3} - 120T - 502 \) Copy content Toggle raw display
$53$ \( T^{3} + 6 T^{2} - 4 T - 8 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 16 T + 160 \) Copy content Toggle raw display
$61$ \( T^{3} + 2 T^{2} - 52 T - 184 \) Copy content Toggle raw display
$67$ \( T^{3} - 2 T^{2} - 20 T + 50 \) Copy content Toggle raw display
$71$ \( T^{3} - 20 T^{2} + 84 T + 134 \) Copy content Toggle raw display
$73$ \( T^{3} + 2 T^{2} - 180 T + 244 \) Copy content Toggle raw display
$79$ \( T^{3} - 32 T^{2} + 328 T - 1090 \) Copy content Toggle raw display
$83$ \( T^{3} - 64T + 128 \) Copy content Toggle raw display
$89$ \( T^{3} + 6 T^{2} - 148 T - 920 \) Copy content Toggle raw display
$97$ \( T^{3} - 6 T^{2} - 52 T + 248 \) Copy content Toggle raw display
show more
show less