Properties

Label 966.2.a.d
Level $966$
Weight $2$
Character orbit 966.a
Self dual yes
Analytic conductor $7.714$
Analytic rank $0$
Dimension $1$
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] = [966,2,Mod(1,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, 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("966.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.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: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + 3 q^{5} + q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + 3 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - 3 q^{10} + 4 q^{11} - q^{12} + 3 q^{13} - q^{14} - 3 q^{15} + q^{16} - q^{18} + 3 q^{20} - q^{21} - 4 q^{22} - q^{23} + q^{24} + 4 q^{25} - 3 q^{26} - q^{27} + q^{28} + q^{29} + 3 q^{30} - 2 q^{31} - q^{32} - 4 q^{33} + 3 q^{35} + q^{36} - 5 q^{37} - 3 q^{39} - 3 q^{40} + 5 q^{41} + q^{42} - 7 q^{43} + 4 q^{44} + 3 q^{45} + q^{46} - 3 q^{47} - q^{48} + q^{49} - 4 q^{50} + 3 q^{52} + 12 q^{53} + q^{54} + 12 q^{55} - q^{56} - q^{58} - 2 q^{59} - 3 q^{60} - 6 q^{61} + 2 q^{62} + q^{63} + q^{64} + 9 q^{65} + 4 q^{66} - 12 q^{67} + q^{69} - 3 q^{70} + 10 q^{71} - q^{72} + 5 q^{74} - 4 q^{75} + 4 q^{77} + 3 q^{78} + 4 q^{79} + 3 q^{80} + q^{81} - 5 q^{82} + 4 q^{83} - q^{84} + 7 q^{86} - q^{87} - 4 q^{88} + 10 q^{89} - 3 q^{90} + 3 q^{91} - q^{92} + 2 q^{93} + 3 q^{94} + q^{96} + 19 q^{97} - q^{98} + 4 q^{99}+O(q^{100}) \) 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
0
−1.00000 −1.00000 1.00000 3.00000 1.00000 1.00000 −1.00000 1.00000 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(7\) \( -1 \)
\(23\) \( +1 \)

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 966.2.a.d 1
3.b odd 2 1 2898.2.a.k 1
4.b odd 2 1 7728.2.a.u 1
7.b odd 2 1 6762.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.d 1 1.a even 1 1 trivial
2898.2.a.k 1 3.b odd 2 1
6762.2.a.l 1 7.b odd 2 1
7728.2.a.u 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(966))\):

\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{13} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T - 3 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T - 3 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T - 1 \) Copy content Toggle raw display
$31$ \( T + 2 \) Copy content Toggle raw display
$37$ \( T + 5 \) Copy content Toggle raw display
$41$ \( T - 5 \) Copy content Toggle raw display
$43$ \( T + 7 \) Copy content Toggle raw display
$47$ \( T + 3 \) Copy content Toggle raw display
$53$ \( T - 12 \) Copy content Toggle raw display
$59$ \( T + 2 \) Copy content Toggle raw display
$61$ \( T + 6 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T - 10 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T - 4 \) Copy content Toggle raw display
$83$ \( T - 4 \) Copy content Toggle raw display
$89$ \( T - 10 \) Copy content Toggle raw display
$97$ \( T - 19 \) Copy content Toggle raw display
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