Properties

Label 550.2.a.j
Level $550$
Weight $2$
Character orbit 550.a
Self dual yes
Analytic conductor $4.392$
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] = [550,2,Mod(1,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, 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("550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.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: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
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)
 
\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} + 3 q^{7} + q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - q^{6} + 3 q^{7} + q^{8} - 2 q^{9} + q^{11} - q^{12} + 4 q^{13} + 3 q^{14} + q^{16} + 3 q^{17} - 2 q^{18} - 5 q^{19} - 3 q^{21} + q^{22} + 4 q^{23} - q^{24} + 4 q^{26} + 5 q^{27} + 3 q^{28} + 5 q^{29} + 7 q^{31} + q^{32} - q^{33} + 3 q^{34} - 2 q^{36} - 7 q^{37} - 5 q^{38} - 4 q^{39} - 8 q^{41} - 3 q^{42} - 6 q^{43} + q^{44} + 4 q^{46} + 8 q^{47} - q^{48} + 2 q^{49} - 3 q^{51} + 4 q^{52} + 9 q^{53} + 5 q^{54} + 3 q^{56} + 5 q^{57} + 5 q^{58} - 13 q^{61} + 7 q^{62} - 6 q^{63} + q^{64} - q^{66} - 12 q^{67} + 3 q^{68} - 4 q^{69} - 3 q^{71} - 2 q^{72} - 6 q^{73} - 7 q^{74} - 5 q^{76} + 3 q^{77} - 4 q^{78} + q^{81} - 8 q^{82} + 4 q^{83} - 3 q^{84} - 6 q^{86} - 5 q^{87} + q^{88} - 15 q^{89} + 12 q^{91} + 4 q^{92} - 7 q^{93} + 8 q^{94} - q^{96} - 12 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) \) 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 3.00000 1.00000 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(11\) \( -1 \)

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 550.2.a.j 1
3.b odd 2 1 4950.2.a.q 1
4.b odd 2 1 4400.2.a.s 1
5.b even 2 1 550.2.a.e 1
5.c odd 4 2 110.2.b.a 2
11.b odd 2 1 6050.2.a.f 1
15.d odd 2 1 4950.2.a.ba 1
15.e even 4 2 990.2.c.d 2
20.d odd 2 1 4400.2.a.k 1
20.e even 4 2 880.2.b.a 2
55.d odd 2 1 6050.2.a.bk 1
55.e even 4 2 1210.2.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.b.a 2 5.c odd 4 2
550.2.a.e 1 5.b even 2 1
550.2.a.j 1 1.a even 1 1 trivial
880.2.b.a 2 20.e even 4 2
990.2.c.d 2 15.e even 4 2
1210.2.b.a 2 55.e even 4 2
4400.2.a.k 1 20.d odd 2 1
4400.2.a.s 1 4.b odd 2 1
4950.2.a.q 1 3.b odd 2 1
4950.2.a.ba 1 15.d odd 2 1
6050.2.a.f 1 11.b odd 2 1
6050.2.a.bk 1 55.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(550))\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{7} - 3 \) Copy content Toggle raw display
\( T_{13} - 4 \) 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
$7$ \( T - 3 \) Copy content Toggle raw display
$11$ \( T - 1 \) 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
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T - 5 \) Copy content Toggle raw display
$31$ \( T - 7 \) Copy content Toggle raw display
$37$ \( T + 7 \) Copy content Toggle raw display
$41$ \( T + 8 \) Copy content Toggle raw display
$43$ \( T + 6 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T - 9 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 13 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T + 3 \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 4 \) Copy content Toggle raw display
$89$ \( T + 15 \) Copy content Toggle raw display
$97$ \( T + 12 \) Copy content Toggle raw display
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