Properties

Label 2800.2.a.bd
Level $2800$
Weight $2$
Character orbit 2800.a
Self dual yes
Analytic conductor $22.358$
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] = [2800,2,Mod(1,2800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.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: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{3} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} + q^{7} + q^{9} + 2 q^{17} + 2 q^{19} + 2 q^{21} + 8 q^{23} - 4 q^{27} + 2 q^{29} - 4 q^{31} + 6 q^{37} - 2 q^{41} + 8 q^{43} - 4 q^{47} + q^{49} + 4 q^{51} + 10 q^{53} + 4 q^{57} - 6 q^{59} + 4 q^{61} + q^{63} - 12 q^{67} + 16 q^{69} + 14 q^{73} + 8 q^{79} - 11 q^{81} + 6 q^{83} + 4 q^{87} + 10 q^{89} - 8 q^{93} + 2 q^{97}+O(q^{100}) \) 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 0 0 1.00000 0 1.00000 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 2800.2.a.bd 1
4.b odd 2 1 1400.2.a.a 1
5.b even 2 1 112.2.a.a 1
5.c odd 4 2 2800.2.g.g 2
15.d odd 2 1 1008.2.a.m 1
20.d odd 2 1 56.2.a.b 1
20.e even 4 2 1400.2.g.b 2
28.d even 2 1 9800.2.a.bj 1
35.c odd 2 1 784.2.a.i 1
35.i odd 6 2 784.2.i.b 2
35.j even 6 2 784.2.i.j 2
40.e odd 2 1 448.2.a.c 1
40.f even 2 1 448.2.a.h 1
60.h even 2 1 504.2.a.h 1
80.k odd 4 2 1792.2.b.a 2
80.q even 4 2 1792.2.b.h 2
105.g even 2 1 7056.2.a.c 1
120.i odd 2 1 4032.2.a.a 1
120.m even 2 1 4032.2.a.d 1
140.c even 2 1 392.2.a.b 1
140.p odd 6 2 392.2.i.a 2
140.s even 6 2 392.2.i.e 2
220.g even 2 1 6776.2.a.h 1
260.g odd 2 1 9464.2.a.h 1
280.c odd 2 1 3136.2.a.c 1
280.n even 2 1 3136.2.a.w 1
420.o odd 2 1 3528.2.a.b 1
420.ba even 6 2 3528.2.s.a 2
420.be odd 6 2 3528.2.s.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 20.d odd 2 1
112.2.a.a 1 5.b even 2 1
392.2.a.b 1 140.c even 2 1
392.2.i.a 2 140.p odd 6 2
392.2.i.e 2 140.s even 6 2
448.2.a.c 1 40.e odd 2 1
448.2.a.h 1 40.f even 2 1
504.2.a.h 1 60.h even 2 1
784.2.a.i 1 35.c odd 2 1
784.2.i.b 2 35.i odd 6 2
784.2.i.j 2 35.j even 6 2
1008.2.a.m 1 15.d odd 2 1
1400.2.a.a 1 4.b odd 2 1
1400.2.g.b 2 20.e even 4 2
1792.2.b.a 2 80.k odd 4 2
1792.2.b.h 2 80.q even 4 2
2800.2.a.bd 1 1.a even 1 1 trivial
2800.2.g.g 2 5.c odd 4 2
3136.2.a.c 1 280.c odd 2 1
3136.2.a.w 1 280.n even 2 1
3528.2.a.b 1 420.o odd 2 1
3528.2.s.a 2 420.ba even 6 2
3528.2.s.ba 2 420.be odd 6 2
4032.2.a.a 1 120.i odd 2 1
4032.2.a.d 1 120.m even 2 1
6776.2.a.h 1 220.g even 2 1
7056.2.a.c 1 105.g even 2 1
9464.2.a.h 1 260.g odd 2 1
9800.2.a.bj 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2800))\):

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