Properties

Label 7942.2.a.i
Level $7942$
Weight $2$
Character orbit 7942.a
Self dual yes
Analytic conductor $63.417$
Analytic rank $2$
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] = [7942,2,Mod(1,7942)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7942, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7942.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.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: \(63.4171892853\)
Analytic rank: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
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)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - q^{8} - 2 q^{9} + 2 q^{10} - q^{11} + q^{12} - q^{13} + 3 q^{14} - 2 q^{15} + q^{16} - 7 q^{17} + 2 q^{18} - 2 q^{20} - 3 q^{21} + q^{22} - 5 q^{23} - q^{24} - q^{25} + q^{26} - 5 q^{27} - 3 q^{28} - q^{29} + 2 q^{30} - 10 q^{31} - q^{32} - q^{33} + 7 q^{34} + 6 q^{35} - 2 q^{36} + 6 q^{37} - q^{39} + 2 q^{40} - 6 q^{41} + 3 q^{42} - 4 q^{43} - q^{44} + 4 q^{45} + 5 q^{46} + q^{48} + 2 q^{49} + q^{50} - 7 q^{51} - q^{52} + q^{53} + 5 q^{54} + 2 q^{55} + 3 q^{56} + q^{58} - 3 q^{59} - 2 q^{60} - 12 q^{61} + 10 q^{62} + 6 q^{63} + q^{64} + 2 q^{65} + q^{66} - 3 q^{67} - 7 q^{68} - 5 q^{69} - 6 q^{70} + 10 q^{71} + 2 q^{72} + 3 q^{73} - 6 q^{74} - q^{75} + 3 q^{77} + q^{78} - 8 q^{79} - 2 q^{80} + q^{81} + 6 q^{82} + 8 q^{83} - 3 q^{84} + 14 q^{85} + 4 q^{86} - q^{87} + q^{88} + 8 q^{89} - 4 q^{90} + 3 q^{91} - 5 q^{92} - 10 q^{93} - q^{96} - 8 q^{97} - 2 q^{98} + 2 q^{99}+O(q^{100}) \) 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 −2.00000 −1.00000 −3.00000 −1.00000 −2.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(1\)
\(19\) \(-1\)

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 7942.2.a.i 1
19.b odd 2 1 418.2.a.a 1
57.d even 2 1 3762.2.a.g 1
76.d even 2 1 3344.2.a.h 1
209.d even 2 1 4598.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.a 1 19.b odd 2 1
3344.2.a.h 1 76.d even 2 1
3762.2.a.g 1 57.d even 2 1
4598.2.a.b 1 209.d even 2 1
7942.2.a.i 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{13} + 1 \) 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
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T + 3 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T + 7 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 5 \) Copy content Toggle raw display
$29$ \( T + 1 \) Copy content Toggle raw display
$31$ \( T + 10 \) Copy content Toggle raw display
$37$ \( T - 6 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 1 \) Copy content Toggle raw display
$59$ \( T + 3 \) Copy content Toggle raw display
$61$ \( T + 12 \) Copy content Toggle raw display
$67$ \( T + 3 \) Copy content Toggle raw display
$71$ \( T - 10 \) Copy content Toggle raw display
$73$ \( T - 3 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T - 8 \) Copy content Toggle raw display
$89$ \( T - 8 \) Copy content Toggle raw display
$97$ \( T + 8 \) Copy content Toggle raw display
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