Properties

Label 9600.2.a.df
Level $9600$
Weight $2$
Character orbit 9600.a
Self dual yes
Analytic conductor $76.656$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9600,2,Mod(1,9600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9600, 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("9600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9600 = 2^{7} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9600.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: \(76.6563859404\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) 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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + (\beta + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + (\beta + 1) q^{7} + q^{9} + (\beta - 2) q^{11} - 3 q^{13} - 3 \beta q^{17} + (\beta - 3) q^{19} + (\beta + 1) q^{21} + ( - 3 \beta + 2) q^{23} + q^{27} + (\beta + 4) q^{29} + ( - 3 \beta - 1) q^{31} + (\beta - 2) q^{33} + (2 \beta - 4) q^{37} - 3 q^{39} + (3 \beta - 2) q^{41} + ( - \beta + 1) q^{43} + ( - 2 \beta + 6) q^{47} + 2 \beta q^{49} - 3 \beta q^{51} + (\beta - 6) q^{53} + (\beta - 3) q^{57} + ( - 2 \beta - 2) q^{59} + ( - 2 \beta - 7) q^{61} + (\beta + 1) q^{63} + ( - \beta - 13) q^{67} + ( - 3 \beta + 2) q^{69} + ( - \beta + 4) q^{71} + (2 \beta + 8) q^{73} + ( - \beta + 4) q^{77} + q^{81} + ( - 2 \beta - 6) q^{83} + (\beta + 4) q^{87} - 4 q^{89} + ( - 3 \beta - 3) q^{91} + ( - 3 \beta - 1) q^{93} + (4 \beta - 9) q^{97} + (\beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} - 4 q^{11} - 6 q^{13} - 6 q^{19} + 2 q^{21} + 4 q^{23} + 2 q^{27} + 8 q^{29} - 2 q^{31} - 4 q^{33} - 8 q^{37} - 6 q^{39} - 4 q^{41} + 2 q^{43} + 12 q^{47} - 12 q^{53} - 6 q^{57} - 4 q^{59} - 14 q^{61} + 2 q^{63} - 26 q^{67} + 4 q^{69} + 8 q^{71} + 16 q^{73} + 8 q^{77} + 2 q^{81} - 12 q^{83} + 8 q^{87} - 8 q^{89} - 6 q^{91} - 2 q^{93} - 18 q^{97} - 4 q^{99}+O(q^{100}) \) 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
−2.44949
2.44949
0 1.00000 0 0 0 −1.44949 0 1.00000 0
1.2 0 1.00000 0 0 0 3.44949 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 9600.2.a.df yes 2
4.b odd 2 1 9600.2.a.ck yes 2
5.b even 2 1 9600.2.a.cf 2
8.b even 2 1 9600.2.a.cv yes 2
8.d odd 2 1 9600.2.a.cy yes 2
20.d odd 2 1 9600.2.a.dn yes 2
40.e odd 2 1 9600.2.a.cn yes 2
40.f even 2 1 9600.2.a.dd yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9600.2.a.cf 2 5.b even 2 1
9600.2.a.ck yes 2 4.b odd 2 1
9600.2.a.cn yes 2 40.e odd 2 1
9600.2.a.cv yes 2 8.b even 2 1
9600.2.a.cy yes 2 8.d odd 2 1
9600.2.a.dd yes 2 40.f even 2 1
9600.2.a.df yes 2 1.a even 1 1 trivial
9600.2.a.dn yes 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9600))\):

\( T_{7}^{2} - 2T_{7} - 5 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 2 \) Copy content Toggle raw display
\( T_{13} + 3 \) Copy content Toggle raw display
\( T_{17}^{2} - 54 \) Copy content Toggle raw display
\( T_{29}^{2} - 8T_{29} + 10 \) Copy content Toggle raw display
\( T_{31}^{2} + 2T_{31} - 53 \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 5 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 2 \) Copy content Toggle raw display
$13$ \( (T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 54 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 3 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T - 50 \) Copy content Toggle raw display
$29$ \( T^{2} - 8T + 10 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 53 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T - 8 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 50 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T - 5 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 12 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 30 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 20 \) Copy content Toggle raw display
$61$ \( T^{2} + 14T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} + 26T + 163 \) Copy content Toggle raw display
$71$ \( T^{2} - 8T + 10 \) Copy content Toggle raw display
$73$ \( T^{2} - 16T + 40 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 12 \) Copy content Toggle raw display
$89$ \( (T + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 18T - 15 \) Copy content Toggle raw display
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