Properties

Label 2600.2.a.bc
Level $2600$
Weight $2$
Character orbit 2600.a
Self dual yes
Analytic conductor $20.761$
Analytic rank $0$
Dimension $5$
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] = [2600,2,Mod(1,2600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2600.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: \(20.7611045255\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.10592240.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 13x^{3} + 24x^{2} + 34x - 50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 520)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} + \beta_{2}) q^{7} + (\beta_{3} + \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} + \beta_{2}) q^{7} + (\beta_{3} + \beta_{2} + 3) q^{9} + (\beta_{4} - \beta_{2} - \beta_1 + 2) q^{11} + q^{13} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{17} + ( - \beta_{4} + \beta_{2} - \beta_1 + 4) q^{19} + (\beta_{4} - \beta_{3} + \beta_{2} + \cdots - 1) q^{21}+ \cdots + (\beta_{4} + 2 \beta_{3} - 5 \beta_{2} + \cdots + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{3} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{3} + 15 q^{9} + 8 q^{11} + 5 q^{13} - 6 q^{17} + 18 q^{19} + 2 q^{21} + 16 q^{23} + 2 q^{27} + 2 q^{29} - 4 q^{31} - 22 q^{33} + 2 q^{37} + 2 q^{39} - 12 q^{41} + 20 q^{47} + 19 q^{49} - 2 q^{51} - 4 q^{53} - 26 q^{57} + 14 q^{59} + 10 q^{61} + 54 q^{63} - 2 q^{67} + 36 q^{69} + 18 q^{71} - 8 q^{73} + 10 q^{77} + 8 q^{79} + 9 q^{81} - 22 q^{83} + 10 q^{87} - 6 q^{89} - 30 q^{93} + 6 q^{97} + 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 13x^{3} + 24x^{2} + 34x - 50 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 2\nu^{3} - 10\nu^{2} - 16\nu + 15 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} - 2\nu^{3} + 15\nu^{2} + 16\nu - 45 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{4} + \nu^{3} + 25\nu^{2} - 8\nu - 55 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + 8\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{4} + 12\beta_{3} + 13\beta_{2} + 47 \) 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
−3.05748
−1.76881
1.16232
2.57527
3.08870
0 −3.05748 0 0 0 3.34821 0 6.34821 0
1.2 0 −1.76881 0 0 0 −2.87131 0 0.128689 0
1.3 0 1.16232 0 0 0 −4.64901 0 −1.64901 0
1.4 0 2.57527 0 0 0 0.632021 0 3.63202 0
1.5 0 3.08870 0 0 0 3.54010 0 6.54010 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

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

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 2600.2.a.bc 5
4.b odd 2 1 5200.2.a.cm 5
5.b even 2 1 2600.2.a.bb 5
5.c odd 4 2 520.2.d.c 10
15.e even 4 2 4680.2.l.i 10
20.d odd 2 1 5200.2.a.cn 5
20.e even 4 2 1040.2.d.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.d.c 10 5.c odd 4 2
1040.2.d.f 10 20.e even 4 2
2600.2.a.bb 5 5.b even 2 1
2600.2.a.bc 5 1.a even 1 1 trivial
4680.2.l.i 10 15.e even 4 2
5200.2.a.cm 5 4.b odd 2 1
5200.2.a.cn 5 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2600))\):

\( T_{3}^{5} - 2T_{3}^{4} - 13T_{3}^{3} + 24T_{3}^{2} + 34T_{3} - 50 \) Copy content Toggle raw display
\( T_{7}^{5} - 27T_{7}^{3} + 14T_{7}^{2} + 160T_{7} - 100 \) Copy content Toggle raw display
\( T_{11}^{5} - 8T_{11}^{4} - 20T_{11}^{3} + 222T_{11}^{2} + 72T_{11} - 1544 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 2 T^{4} + \cdots - 50 \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 27 T^{3} + \cdots - 100 \) Copy content Toggle raw display
$11$ \( T^{5} - 8 T^{4} + \cdots - 1544 \) Copy content Toggle raw display
$13$ \( (T - 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{5} + 6 T^{4} + \cdots - 72 \) Copy content Toggle raw display
$19$ \( T^{5} - 18 T^{4} + \cdots + 5368 \) Copy content Toggle raw display
$23$ \( T^{5} - 16 T^{4} + \cdots + 1616 \) Copy content Toggle raw display
$29$ \( T^{5} - 2 T^{4} + \cdots - 9232 \) Copy content Toggle raw display
$31$ \( T^{5} + 4 T^{4} + \cdots + 1944 \) Copy content Toggle raw display
$37$ \( T^{5} - 2 T^{4} + \cdots + 1660 \) Copy content Toggle raw display
$41$ \( T^{5} + 12 T^{4} + \cdots + 5120 \) Copy content Toggle raw display
$43$ \( T^{5} - 89 T^{3} + \cdots + 5010 \) Copy content Toggle raw display
$47$ \( T^{5} - 20 T^{4} + \cdots - 60 \) Copy content Toggle raw display
$53$ \( T^{5} + 4 T^{4} + \cdots - 1408 \) Copy content Toggle raw display
$59$ \( T^{5} - 14 T^{4} + \cdots + 4008 \) Copy content Toggle raw display
$61$ \( T^{5} - 10 T^{4} + \cdots + 1472 \) Copy content Toggle raw display
$67$ \( T^{5} + 2 T^{4} + \cdots + 36304 \) Copy content Toggle raw display
$71$ \( T^{5} - 18 T^{4} + \cdots - 5770 \) Copy content Toggle raw display
$73$ \( T^{5} + 8 T^{4} + \cdots + 9232 \) Copy content Toggle raw display
$79$ \( T^{5} - 8 T^{4} + \cdots - 14720 \) Copy content Toggle raw display
$83$ \( T^{5} + 22 T^{4} + \cdots + 13472 \) Copy content Toggle raw display
$89$ \( T^{5} + 6 T^{4} + \cdots + 2208 \) Copy content Toggle raw display
$97$ \( T^{5} - 6 T^{4} + \cdots + 160 \) Copy content Toggle raw display
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