Properties

Label 688.2.a.b
Level $688$
Weight $2$
Character orbit 688.a
Self dual yes
Analytic conductor $5.494$
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] = [688,2,Mod(1,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, 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("688.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.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: \(5.49370765906\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 43)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{3} - 4 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} - 4 q^{5} + q^{9} - 3 q^{11} - 5 q^{13} - 8 q^{15} - 3 q^{17} + 2 q^{19} + q^{23} + 11 q^{25} - 4 q^{27} - 6 q^{29} + q^{31} - 6 q^{33} - 10 q^{39} + 5 q^{41} + q^{43} - 4 q^{45} - 4 q^{47} - 7 q^{49} - 6 q^{51} - 5 q^{53} + 12 q^{55} + 4 q^{57} + 12 q^{59} + 2 q^{61} + 20 q^{65} + 3 q^{67} + 2 q^{69} - 2 q^{71} + 2 q^{73} + 22 q^{75} + 8 q^{79} - 11 q^{81} - 15 q^{83} + 12 q^{85} - 12 q^{87} - 4 q^{89} + 2 q^{93} - 8 q^{95} + 7 q^{97} - 3 q^{99}+O(q^{100}) \) 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
0 2.00000 0 −4.00000 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(43\) \(-1\)

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 688.2.a.b 1
3.b odd 2 1 6192.2.a.ba 1
4.b odd 2 1 43.2.a.a 1
8.b even 2 1 2752.2.a.b 1
8.d odd 2 1 2752.2.a.f 1
12.b even 2 1 387.2.a.e 1
20.d odd 2 1 1075.2.a.h 1
20.e even 4 2 1075.2.b.b 2
28.d even 2 1 2107.2.a.a 1
44.c even 2 1 5203.2.a.a 1
52.b odd 2 1 7267.2.a.a 1
60.h even 2 1 9675.2.a.b 1
172.d even 2 1 1849.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.a.a 1 4.b odd 2 1
387.2.a.e 1 12.b even 2 1
688.2.a.b 1 1.a even 1 1 trivial
1075.2.a.h 1 20.d odd 2 1
1075.2.b.b 2 20.e even 4 2
1849.2.a.d 1 172.d even 2 1
2107.2.a.a 1 28.d even 2 1
2752.2.a.b 1 8.b even 2 1
2752.2.a.f 1 8.d odd 2 1
5203.2.a.a 1 44.c even 2 1
6192.2.a.ba 1 3.b odd 2 1
7267.2.a.a 1 52.b odd 2 1
9675.2.a.b 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(688))\):

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{5} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 2 \) Copy content Toggle raw display
$5$ \( T + 4 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 3 \) Copy content Toggle raw display
$13$ \( T + 5 \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T - 1 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 1 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 5 \) Copy content Toggle raw display
$43$ \( T - 1 \) Copy content Toggle raw display
$47$ \( T + 4 \) Copy content Toggle raw display
$53$ \( T + 5 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T - 3 \) Copy content Toggle raw display
$71$ \( T + 2 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T + 15 \) Copy content Toggle raw display
$89$ \( T + 4 \) Copy content Toggle raw display
$97$ \( T - 7 \) Copy content Toggle raw display
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