Properties

Label 1078.2.a.g
Level $1078$
Weight $2$
Character orbit 1078.a
Self dual yes
Analytic conductor $8.608$
Analytic rank $1$
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] = [1078,2,Mod(1,1078)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1078, 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("1078.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.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: \(8.60787333789\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
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)
 
\(f(q)\) \(=\) \( q + q^{2} - 3 q^{3} + q^{4} - 2 q^{5} - 3 q^{6} + q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 3 q^{3} + q^{4} - 2 q^{5} - 3 q^{6} + q^{8} + 6 q^{9} - 2 q^{10} - q^{11} - 3 q^{12} + 7 q^{13} + 6 q^{15} + q^{16} - 2 q^{17} + 6 q^{18} - 2 q^{20} - q^{22} - 8 q^{23} - 3 q^{24} - q^{25} + 7 q^{26} - 9 q^{27} - 5 q^{29} + 6 q^{30} - 4 q^{31} + q^{32} + 3 q^{33} - 2 q^{34} + 6 q^{36} + 4 q^{37} - 21 q^{39} - 2 q^{40} - 4 q^{41} - 8 q^{43} - q^{44} - 12 q^{45} - 8 q^{46} - 2 q^{47} - 3 q^{48} - q^{50} + 6 q^{51} + 7 q^{52} - 6 q^{53} - 9 q^{54} + 2 q^{55} - 5 q^{58} - 3 q^{59} + 6 q^{60} - q^{61} - 4 q^{62} + q^{64} - 14 q^{65} + 3 q^{66} + 9 q^{67} - 2 q^{68} + 24 q^{69} - 2 q^{71} + 6 q^{72} - 4 q^{73} + 4 q^{74} + 3 q^{75} - 21 q^{78} + 9 q^{79} - 2 q^{80} + 9 q^{81} - 4 q^{82} - 6 q^{83} + 4 q^{85} - 8 q^{86} + 15 q^{87} - q^{88} - 6 q^{89} - 12 q^{90} - 8 q^{92} + 12 q^{93} - 2 q^{94} - 3 q^{96} - 7 q^{97} - 6 q^{99}+O(q^{100}) \) 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 −3.00000 1.00000 −2.00000 −3.00000 0 1.00000 6.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

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 1078.2.a.g 1
3.b odd 2 1 9702.2.a.y 1
4.b odd 2 1 8624.2.a.be 1
7.b odd 2 1 1078.2.a.m 1
7.c even 3 2 1078.2.e.f 2
7.d odd 6 2 154.2.e.a 2
21.c even 2 1 9702.2.a.i 1
21.g even 6 2 1386.2.k.o 2
28.d even 2 1 8624.2.a.b 1
28.f even 6 2 1232.2.q.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.a 2 7.d odd 6 2
1078.2.a.g 1 1.a even 1 1 trivial
1078.2.a.m 1 7.b odd 2 1
1078.2.e.f 2 7.c even 3 2
1232.2.q.e 2 28.f even 6 2
1386.2.k.o 2 21.g even 6 2
8624.2.a.b 1 28.d even 2 1
8624.2.a.be 1 4.b odd 2 1
9702.2.a.i 1 21.c even 2 1
9702.2.a.y 1 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1078))\):

\( T_{3} + 3 \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{13} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T - 7 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 8 \) Copy content Toggle raw display
$29$ \( T + 5 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 4 \) Copy content Toggle raw display
$41$ \( T + 4 \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T + 2 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 3 \) Copy content Toggle raw display
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T - 9 \) Copy content Toggle raw display
$71$ \( T + 2 \) Copy content Toggle raw display
$73$ \( T + 4 \) Copy content Toggle raw display
$79$ \( T - 9 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T + 7 \) Copy content Toggle raw display
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