Properties

Label 8470.2.a.j
Level $8470$
Weight $2$
Character orbit 8470.a
Self dual yes
Analytic conductor $67.633$
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] = [8470,2,Mod(1,8470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8470, 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("8470.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.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: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} - 3 q^{9} + q^{10} + 6 q^{13} - q^{14} + q^{16} - 2 q^{17} + 3 q^{18} - q^{20} + q^{25} - 6 q^{26} + q^{28} - 6 q^{29} + 8 q^{31} - q^{32} + 2 q^{34} - q^{35} - 3 q^{36} - 10 q^{37} + q^{40} - 2 q^{41} - 4 q^{43} + 3 q^{45} + 8 q^{47} + q^{49} - q^{50} + 6 q^{52} - 2 q^{53} - q^{56} + 6 q^{58} - 8 q^{59} + 14 q^{61} - 8 q^{62} - 3 q^{63} + q^{64} - 6 q^{65} - 12 q^{67} - 2 q^{68} + q^{70} - 16 q^{71} + 3 q^{72} - 2 q^{73} + 10 q^{74} + 8 q^{79} - q^{80} + 9 q^{81} + 2 q^{82} - 8 q^{83} + 2 q^{85} + 4 q^{86} + 10 q^{89} - 3 q^{90} + 6 q^{91} - 8 q^{94} + 2 q^{97} - q^{98}+O(q^{100}) \) 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
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 8470.2.a.j 1
11.b odd 2 1 70.2.a.a 1
33.d even 2 1 630.2.a.d 1
44.c even 2 1 560.2.a.d 1
55.d odd 2 1 350.2.a.b 1
55.e even 4 2 350.2.c.b 2
77.b even 2 1 490.2.a.h 1
77.h odd 6 2 490.2.e.d 2
77.i even 6 2 490.2.e.c 2
88.b odd 2 1 2240.2.a.n 1
88.g even 2 1 2240.2.a.q 1
132.d odd 2 1 5040.2.a.bm 1
165.d even 2 1 3150.2.a.bj 1
165.l odd 4 2 3150.2.g.c 2
220.g even 2 1 2800.2.a.m 1
220.i odd 4 2 2800.2.g.n 2
231.h odd 2 1 4410.2.a.b 1
308.g odd 2 1 3920.2.a.t 1
385.h even 2 1 2450.2.a.l 1
385.l odd 4 2 2450.2.c.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 11.b odd 2 1
350.2.a.b 1 55.d odd 2 1
350.2.c.b 2 55.e even 4 2
490.2.a.h 1 77.b even 2 1
490.2.e.c 2 77.i even 6 2
490.2.e.d 2 77.h odd 6 2
560.2.a.d 1 44.c even 2 1
630.2.a.d 1 33.d even 2 1
2240.2.a.n 1 88.b odd 2 1
2240.2.a.q 1 88.g even 2 1
2450.2.a.l 1 385.h even 2 1
2450.2.c.k 2 385.l odd 4 2
2800.2.a.m 1 220.g even 2 1
2800.2.g.n 2 220.i odd 4 2
3150.2.a.bj 1 165.d even 2 1
3150.2.g.c 2 165.l odd 4 2
3920.2.a.t 1 308.g odd 2 1
4410.2.a.b 1 231.h odd 2 1
5040.2.a.bm 1 132.d odd 2 1
8470.2.a.j 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{13} - 6 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

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
$11$ \( T \) 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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