Properties

Label 2600.2.a.r
Level $2600$
Weight $2$
Character orbit 2600.a
Self dual yes
Analytic conductor $20.761$
Analytic rank $1$
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] = [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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{7} - 3 q^{9} + (\beta + 3) q^{11} + q^{13} + (2 \beta - 1) q^{17} - 2 q^{19} + ( - 2 \beta - 2) q^{23} + q^{29} + (\beta + 3) q^{31} - 6 q^{37} + (2 \beta - 6) q^{43} + (\beta - 5) q^{47} + (2 \beta + 4) q^{49} + ( - 2 \beta - 1) q^{53} + ( - \beta - 9) q^{59} + ( - 2 \beta - 3) q^{61} + (3 \beta + 3) q^{63} + ( - \beta - 1) q^{67} + ( - 4 \beta - 2) q^{71} + (4 \beta + 2) q^{73} + ( - 4 \beta - 13) q^{77} + (2 \beta - 2) q^{79} + 9 q^{81} + ( - \beta - 13) q^{83} + ( - 4 \beta + 4) q^{89} + ( - \beta - 1) q^{91} - 10 q^{97} + ( - 3 \beta - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{7} - 6 q^{9} + 6 q^{11} + 2 q^{13} - 2 q^{17} - 4 q^{19} - 4 q^{23} + 2 q^{29} + 6 q^{31} - 12 q^{37} - 12 q^{43} - 10 q^{47} + 8 q^{49} - 2 q^{53} - 18 q^{59} - 6 q^{61} + 6 q^{63} - 2 q^{67} - 4 q^{71} + 4 q^{73} - 26 q^{77} - 4 q^{79} + 18 q^{81} - 26 q^{83} + 8 q^{89} - 2 q^{91} - 20 q^{97} - 18 q^{99}+O(q^{100}) \) 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
3.16228
−3.16228
0 0 0 0 0 −4.16228 0 −3.00000 0
1.2 0 0 0 0 0 2.16228 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
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.r 2
4.b odd 2 1 5200.2.a.bs 2
5.b even 2 1 2600.2.a.s yes 2
5.c odd 4 2 2600.2.d.o 4
20.d odd 2 1 5200.2.a.bq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2600.2.a.r 2 1.a even 1 1 trivial
2600.2.a.s yes 2 5.b even 2 1
2600.2.d.o 4 5.c odd 4 2
5200.2.a.bq 2 20.d odd 2 1
5200.2.a.bs 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2600))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 9 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 9 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T - 1 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T - 39 \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 36 \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 6T - 1 \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 12T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 10T + 15 \) Copy content Toggle raw display
$53$ \( T^{2} + 2T - 39 \) Copy content Toggle raw display
$59$ \( T^{2} + 18T + 71 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T - 31 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 9 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 156 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 156 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 36 \) Copy content Toggle raw display
$83$ \( T^{2} + 26T + 159 \) Copy content Toggle raw display
$89$ \( T^{2} - 8T - 144 \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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