Properties

Label 2450.2.a.bh
Level $2450$
Weight $2$
Character orbit 2450.a
Self dual yes
Analytic conductor $19.563$
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] = [2450,2,Mod(1,2450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.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: \(19.5633484952\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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)
 
\(f(q)\) \(=\) \( q + q^{2} + 3 q^{3} + q^{4} + 3 q^{6} + q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + 3 q^{3} + q^{4} + 3 q^{6} + q^{8} + 6 q^{9} + 3 q^{12} + 2 q^{13} + q^{16} + 2 q^{17} + 6 q^{18} + 2 q^{19} - q^{23} + 3 q^{24} + 2 q^{26} + 9 q^{27} - q^{29} - 10 q^{31} + q^{32} + 2 q^{34} + 6 q^{36} - 8 q^{37} + 2 q^{38} + 6 q^{39} + 3 q^{41} + 5 q^{43} - q^{46} - 8 q^{47} + 3 q^{48} + 6 q^{51} + 2 q^{52} - 6 q^{53} + 9 q^{54} + 6 q^{57} - q^{58} - 2 q^{59} + 9 q^{61} - 10 q^{62} + q^{64} - 7 q^{67} + 2 q^{68} - 3 q^{69} + 6 q^{71} + 6 q^{72} + 10 q^{73} - 8 q^{74} + 2 q^{76} + 6 q^{78} - 10 q^{79} + 9 q^{81} + 3 q^{82} + 9 q^{83} + 5 q^{86} - 3 q^{87} + 7 q^{89} - q^{92} - 30 q^{93} - 8 q^{94} + 3 q^{96}+O(q^{100}) \) 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 3.00000 1.00000 0 3.00000 0 1.00000 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 2450.2.a.bh 1
5.b even 2 1 2450.2.a.c 1
5.c odd 4 2 490.2.c.b 2
7.b odd 2 1 2450.2.a.s 1
7.d odd 6 2 350.2.e.f 2
35.c odd 2 1 2450.2.a.r 1
35.f even 4 2 490.2.c.c 2
35.i odd 6 2 350.2.e.g 2
35.k even 12 4 70.2.i.a 4
35.l odd 12 4 490.2.i.b 4
105.w odd 12 4 630.2.u.b 4
140.x odd 12 4 560.2.bw.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.a 4 35.k even 12 4
350.2.e.f 2 7.d odd 6 2
350.2.e.g 2 35.i odd 6 2
490.2.c.b 2 5.c odd 4 2
490.2.c.c 2 35.f even 4 2
490.2.i.b 4 35.l odd 12 4
560.2.bw.c 4 140.x odd 12 4
630.2.u.b 4 105.w odd 12 4
2450.2.a.c 1 5.b even 2 1
2450.2.a.r 1 35.c odd 2 1
2450.2.a.s 1 7.b odd 2 1
2450.2.a.bh 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3} - 3 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display
\( T_{19} - 2 \) Copy content Toggle raw display
\( T_{23} + 1 \) Copy content Toggle raw display
\( T_{37} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T + 1 \) Copy content Toggle raw display
$31$ \( T + 10 \) Copy content Toggle raw display
$37$ \( T + 8 \) Copy content Toggle raw display
$41$ \( T - 3 \) Copy content Toggle raw display
$43$ \( T - 5 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 2 \) Copy content Toggle raw display
$61$ \( T - 9 \) Copy content Toggle raw display
$67$ \( T + 7 \) Copy content Toggle raw display
$71$ \( T - 6 \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T + 10 \) Copy content Toggle raw display
$83$ \( T - 9 \) Copy content Toggle raw display
$89$ \( T - 7 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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