Properties

Label 700.2.a.a
Level $700$
Weight $2$
Character orbit 700.a
Self dual yes
Analytic conductor $5.590$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [700,2,Mod(1,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, 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("700.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.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: \(5.58952814149\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} - q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - q^{7} + 6 q^{9} + 3 q^{11} - q^{13} + 5 q^{17} - 8 q^{19} + 3 q^{21} - 2 q^{23} - 9 q^{27} - q^{29} - 2 q^{31} - 9 q^{33} - 10 q^{37} + 3 q^{39} - 6 q^{41} + 4 q^{43} - 11 q^{47} + q^{49} - 15 q^{51} - 6 q^{53} + 24 q^{57} - 10 q^{59} - 6 q^{63} + 10 q^{67} + 6 q^{69} + 10 q^{73} - 3 q^{77} - 7 q^{79} + 9 q^{81} - 12 q^{83} + 3 q^{87} + 8 q^{89} + q^{91} + 6 q^{93} - 3 q^{97} + 18 q^{99}+O(q^{100}) \) 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 −3.00000 0 0 0 −1.00000 0 6.00000 0
\(n\): 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 700.2.a.a 1
3.b odd 2 1 6300.2.a.c 1
4.b odd 2 1 2800.2.a.bf 1
5.b even 2 1 700.2.a.j 1
5.c odd 4 2 140.2.e.a 2
7.b odd 2 1 4900.2.a.w 1
15.d odd 2 1 6300.2.a.t 1
15.e even 4 2 1260.2.k.c 2
20.d odd 2 1 2800.2.a.a 1
20.e even 4 2 560.2.g.a 2
35.c odd 2 1 4900.2.a.b 1
35.f even 4 2 980.2.e.b 2
35.k even 12 4 980.2.q.c 4
35.l odd 12 4 980.2.q.f 4
40.i odd 4 2 2240.2.g.e 2
40.k even 4 2 2240.2.g.f 2
60.l odd 4 2 5040.2.t.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 5.c odd 4 2
560.2.g.a 2 20.e even 4 2
700.2.a.a 1 1.a even 1 1 trivial
700.2.a.j 1 5.b even 2 1
980.2.e.b 2 35.f even 4 2
980.2.q.c 4 35.k even 12 4
980.2.q.f 4 35.l odd 12 4
1260.2.k.c 2 15.e even 4 2
2240.2.g.e 2 40.i odd 4 2
2240.2.g.f 2 40.k even 4 2
2800.2.a.a 1 20.d odd 2 1
2800.2.a.bf 1 4.b odd 2 1
4900.2.a.b 1 35.c odd 2 1
4900.2.a.w 1 7.b odd 2 1
5040.2.t.s 2 60.l odd 4 2
6300.2.a.c 1 3.b odd 2 1
6300.2.a.t 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(700))\):

\( T_{3} + 3 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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