Properties

Label 70.2.a.a
Level $70$
Weight $2$
Character orbit 70.a
Self dual yes
Analytic conductor $0.559$
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] = [70,2,Mod(1,70)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(70, 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("70.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 70.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: \(0.558952814149\)
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)
 
\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - 3 q^{9} - q^{10} + 4 q^{11} - 6 q^{13} - q^{14} + q^{16} + 2 q^{17} - 3 q^{18} - q^{20} + 4 q^{22} + q^{25} - 6 q^{26} - q^{28} + 6 q^{29} + 8 q^{31}+ \cdots - 12 q^{99}+O(q^{100}) \) 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 0 1.00000 −1.00000 0 −1.00000 1.00000 −3.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(7\) \( +1 \)

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 70.2.a.a 1
3.b odd 2 1 630.2.a.d 1
4.b odd 2 1 560.2.a.d 1
5.b even 2 1 350.2.a.b 1
5.c odd 4 2 350.2.c.b 2
7.b odd 2 1 490.2.a.h 1
7.c even 3 2 490.2.e.d 2
7.d odd 6 2 490.2.e.c 2
8.b even 2 1 2240.2.a.n 1
8.d odd 2 1 2240.2.a.q 1
11.b odd 2 1 8470.2.a.j 1
12.b even 2 1 5040.2.a.bm 1
15.d odd 2 1 3150.2.a.bj 1
15.e even 4 2 3150.2.g.c 2
20.d odd 2 1 2800.2.a.m 1
20.e even 4 2 2800.2.g.n 2
21.c even 2 1 4410.2.a.b 1
28.d even 2 1 3920.2.a.t 1
35.c odd 2 1 2450.2.a.l 1
35.f even 4 2 2450.2.c.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 1.a even 1 1 trivial
350.2.a.b 1 5.b even 2 1
350.2.c.b 2 5.c odd 4 2
490.2.a.h 1 7.b odd 2 1
490.2.e.c 2 7.d odd 6 2
490.2.e.d 2 7.c even 3 2
560.2.a.d 1 4.b odd 2 1
630.2.a.d 1 3.b odd 2 1
2240.2.a.n 1 8.b even 2 1
2240.2.a.q 1 8.d odd 2 1
2450.2.a.l 1 35.c odd 2 1
2450.2.c.k 2 35.f even 4 2
2800.2.a.m 1 20.d odd 2 1
2800.2.g.n 2 20.e even 4 2
3150.2.a.bj 1 15.d odd 2 1
3150.2.g.c 2 15.e even 4 2
3920.2.a.t 1 28.d even 2 1
4410.2.a.b 1 21.c even 2 1
5040.2.a.bm 1 12.b even 2 1
8470.2.a.j 1 11.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(70))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T + 6 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T + 8 \) Copy content Toggle raw display
$61$ \( T + 14 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T + 16 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T - 8 \) Copy content Toggle raw display
$89$ \( T - 10 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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