Properties

Label 1600.2.a.p
Level $1600$
Weight $2$
Character orbit 1600.a
Self dual yes
Analytic conductor $12.776$
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] = [1600,2,Mod(1,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, 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("1600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.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: \(12.7760643234\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
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)
 
\(f(q)\) \(=\) \( q + q^{3} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - 2 q^{7} - 2 q^{9} - 3 q^{11} + 4 q^{13} - 3 q^{17} + 5 q^{19} - 2 q^{21} - 6 q^{23} - 5 q^{27} - 2 q^{31} - 3 q^{33} - 2 q^{37} + 4 q^{39} - 3 q^{41} - 4 q^{43} - 12 q^{47} - 3 q^{49} - 3 q^{51} - 6 q^{53} + 5 q^{57} - 2 q^{61} + 4 q^{63} - 13 q^{67} - 6 q^{69} - 12 q^{71} + 11 q^{73} + 6 q^{77} + 10 q^{79} + q^{81} - 9 q^{83} + 15 q^{89} - 8 q^{91} - 2 q^{93} + 2 q^{97} + 6 q^{99}+O(q^{100}) \) 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 1.00000 0 0 0 −2.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1600.2.a.p 1
4.b odd 2 1 1600.2.a.j 1
5.b even 2 1 1600.2.a.i 1
5.c odd 4 2 1600.2.c.h 2
8.b even 2 1 400.2.a.d 1
8.d odd 2 1 50.2.a.a 1
20.d odd 2 1 1600.2.a.q 1
20.e even 4 2 1600.2.c.i 2
24.f even 2 1 450.2.a.g 1
24.h odd 2 1 3600.2.a.l 1
40.e odd 2 1 50.2.a.b yes 1
40.f even 2 1 400.2.a.f 1
40.i odd 4 2 400.2.c.c 2
40.k even 4 2 50.2.b.a 2
56.e even 2 1 2450.2.a.g 1
88.g even 2 1 6050.2.a.bi 1
104.h odd 2 1 8450.2.a.v 1
120.i odd 2 1 3600.2.a.bc 1
120.m even 2 1 450.2.a.c 1
120.q odd 4 2 450.2.c.c 2
120.w even 4 2 3600.2.f.f 2
280.n even 2 1 2450.2.a.bd 1
280.y odd 4 2 2450.2.c.m 2
440.c even 2 1 6050.2.a.h 1
520.b odd 2 1 8450.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.a.a 1 8.d odd 2 1
50.2.a.b yes 1 40.e odd 2 1
50.2.b.a 2 40.k even 4 2
400.2.a.d 1 8.b even 2 1
400.2.a.f 1 40.f even 2 1
400.2.c.c 2 40.i odd 4 2
450.2.a.c 1 120.m even 2 1
450.2.a.g 1 24.f even 2 1
450.2.c.c 2 120.q odd 4 2
1600.2.a.i 1 5.b even 2 1
1600.2.a.j 1 4.b odd 2 1
1600.2.a.p 1 1.a even 1 1 trivial
1600.2.a.q 1 20.d odd 2 1
1600.2.c.h 2 5.c odd 4 2
1600.2.c.i 2 20.e even 4 2
2450.2.a.g 1 56.e even 2 1
2450.2.a.bd 1 280.n even 2 1
2450.2.c.m 2 280.y odd 4 2
3600.2.a.l 1 24.h odd 2 1
3600.2.a.bc 1 120.i odd 2 1
3600.2.f.f 2 120.w even 4 2
6050.2.a.h 1 440.c even 2 1
6050.2.a.bi 1 88.g even 2 1
8450.2.a.d 1 520.b odd 2 1
8450.2.a.v 1 104.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1600))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} + 3 \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T + 3 \) Copy content Toggle raw display
$13$ \( T - 4 \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T - 5 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 2 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 3 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 12 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T + 13 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T - 11 \) Copy content Toggle raw display
$79$ \( T - 10 \) Copy content Toggle raw display
$83$ \( T + 9 \) Copy content Toggle raw display
$89$ \( T - 15 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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