Properties

Label 2550.2.a.c
Level $2550$
Weight $2$
Character orbit 2550.a
Self dual yes
Analytic conductor $20.362$
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] = [2550,2,Mod(1,2550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2550, 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("2550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2550.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: \(20.3618525154\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 102)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} - 4 q^{11} - q^{12} + 2 q^{13} + q^{16} - q^{17} - q^{18} + 4 q^{19} + 4 q^{22} + q^{24} - 2 q^{26} - q^{27} - 10 q^{29} + 8 q^{31} - q^{32} + 4 q^{33} + q^{34} + q^{36} + 2 q^{37} - 4 q^{38} - 2 q^{39} + 10 q^{41} - 12 q^{43} - 4 q^{44} - q^{48} - 7 q^{49} + q^{51} + 2 q^{52} - 6 q^{53} + q^{54} - 4 q^{57} + 10 q^{58} + 12 q^{59} - 10 q^{61} - 8 q^{62} + q^{64} - 4 q^{66} + 12 q^{67} - q^{68} - q^{72} - 10 q^{73} - 2 q^{74} + 4 q^{76} + 2 q^{78} - 8 q^{79} + q^{81} - 10 q^{82} - 4 q^{83} + 12 q^{86} + 10 q^{87} + 4 q^{88} - 6 q^{89} - 8 q^{93} + q^{96} + 14 q^{97} + 7 q^{98} - 4 q^{99}+O(q^{100}) \) 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
−1.00000 −1.00000 1.00000 0 1.00000 0 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(5\) \( +1 \)
\(17\) \( +1 \)

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 2550.2.a.c 1
3.b odd 2 1 7650.2.a.ca 1
5.b even 2 1 102.2.a.c 1
5.c odd 4 2 2550.2.d.m 2
15.d odd 2 1 306.2.a.b 1
20.d odd 2 1 816.2.a.b 1
35.c odd 2 1 4998.2.a.be 1
40.e odd 2 1 3264.2.a.bc 1
40.f even 2 1 3264.2.a.m 1
60.h even 2 1 2448.2.a.p 1
85.c even 2 1 1734.2.a.j 1
85.j even 4 2 1734.2.b.b 2
85.m even 8 4 1734.2.f.e 4
120.i odd 2 1 9792.2.a.k 1
120.m even 2 1 9792.2.a.l 1
255.h odd 2 1 5202.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.c 1 5.b even 2 1
306.2.a.b 1 15.d odd 2 1
816.2.a.b 1 20.d odd 2 1
1734.2.a.j 1 85.c even 2 1
1734.2.b.b 2 85.j even 4 2
1734.2.f.e 4 85.m even 8 4
2448.2.a.p 1 60.h even 2 1
2550.2.a.c 1 1.a even 1 1 trivial
2550.2.d.m 2 5.c odd 4 2
3264.2.a.m 1 40.f even 2 1
3264.2.a.bc 1 40.e odd 2 1
4998.2.a.be 1 35.c odd 2 1
5202.2.a.c 1 255.h odd 2 1
7650.2.a.ca 1 3.b odd 2 1
9792.2.a.k 1 120.i odd 2 1
9792.2.a.l 1 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2550))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 1 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 10 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T + 12 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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