Properties

Label 1078.2.a.v
Level $1078$
Weight $2$
Character orbit 1078.a
Self dual yes
Analytic conductor $8.608$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + 2 \beta q^{3} + q^{4} + 2 \beta q^{6} + q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + 2 \beta q^{3} + q^{4} + 2 \beta q^{6} + q^{8} + 5 q^{9} - q^{11} + 2 \beta q^{12} + 3 \beta q^{13} + q^{16} - 2 \beta q^{17} + 5 q^{18} - 3 \beta q^{19} - q^{22} + 6 q^{23} + 2 \beta q^{24} - 5 q^{25} + 3 \beta q^{26} + 4 \beta q^{27} - 4 q^{29} - 5 \beta q^{31} + q^{32} - 2 \beta q^{33} - 2 \beta q^{34} + 5 q^{36} + 2 q^{37} - 3 \beta q^{38} + 12 q^{39} + 2 \beta q^{41} + 10 q^{43} - q^{44} + 6 q^{46} - 9 \beta q^{47} + 2 \beta q^{48} - 5 q^{50} - 8 q^{51} + 3 \beta q^{52} + 2 q^{53} + 4 \beta q^{54} - 12 q^{57} - 4 q^{58} + 8 \beta q^{59} - 7 \beta q^{61} - 5 \beta q^{62} + q^{64} - 2 \beta q^{66} + 8 q^{67} - 2 \beta q^{68} + 12 \beta q^{69} + 16 q^{71} + 5 q^{72} - 6 \beta q^{73} + 2 q^{74} - 10 \beta q^{75} - 3 \beta q^{76} + 12 q^{78} - 8 q^{79} + q^{81} + 2 \beta q^{82} - 9 \beta q^{83} + 10 q^{86} - 8 \beta q^{87} - q^{88} + 5 \beta q^{89} + 6 q^{92} - 20 q^{93} - 9 \beta q^{94} + 2 \beta q^{96} - 5 \beta q^{97} - 5 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 10 q^{9} - 2 q^{11} + 2 q^{16} + 10 q^{18} - 2 q^{22} + 12 q^{23} - 10 q^{25} - 8 q^{29} + 2 q^{32} + 10 q^{36} + 4 q^{37} + 24 q^{39} + 20 q^{43} - 2 q^{44} + 12 q^{46} - 10 q^{50} - 16 q^{51} + 4 q^{53} - 24 q^{57} - 8 q^{58} + 2 q^{64} + 16 q^{67} + 32 q^{71} + 10 q^{72} + 4 q^{74} + 24 q^{78} - 16 q^{79} + 2 q^{81} + 20 q^{86} - 2 q^{88} + 12 q^{92} - 40 q^{93} - 10 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
1.00000 −2.82843 1.00000 0 −2.82843 0 1.00000 5.00000 0
1.2 1.00000 2.82843 1.00000 0 2.82843 0 1.00000 5.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.a.v 2
3.b odd 2 1 9702.2.a.co 2
4.b odd 2 1 8624.2.a.bz 2
7.b odd 2 1 inner 1078.2.a.v 2
7.c even 3 2 1078.2.e.o 4
7.d odd 6 2 1078.2.e.o 4
21.c even 2 1 9702.2.a.co 2
28.d even 2 1 8624.2.a.bz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.v 2 1.a even 1 1 trivial
1078.2.a.v 2 7.b odd 2 1 inner
1078.2.e.o 4 7.c even 3 2
1078.2.e.o 4 7.d odd 6 2
8624.2.a.bz 2 4.b odd 2 1
8624.2.a.bz 2 28.d even 2 1
9702.2.a.co 2 3.b odd 2 1
9702.2.a.co 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1078))\):

\( T_{3}^{2} - 8 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{13}^{2} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 8 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 18 \) Copy content Toggle raw display
$17$ \( T^{2} - 8 \) Copy content Toggle raw display
$19$ \( T^{2} - 18 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 50 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 8 \) Copy content Toggle raw display
$43$ \( (T - 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 162 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 128 \) Copy content Toggle raw display
$61$ \( T^{2} - 98 \) Copy content Toggle raw display
$67$ \( (T - 8)^{2} \) Copy content Toggle raw display
$71$ \( (T - 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 72 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 162 \) Copy content Toggle raw display
$89$ \( T^{2} - 50 \) Copy content Toggle raw display
$97$ \( T^{2} - 50 \) Copy content Toggle raw display
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