Properties

Label 55.13.d.a
Level $55$
Weight $13$
Character orbit 55.d
Self dual yes
Analytic conductor $50.270$
Analytic rank $0$
Dimension $1$
CM discriminant -55
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [55,13,Mod(54,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.54");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 55.d (of order \(2\), degree \(1\), minimal)

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: \(50.2696599502\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 117 q^{2} + 9593 q^{4} + 15625 q^{5} - 87282 q^{7} - 643149 q^{8} + 531441 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 117 q^{2} + 9593 q^{4} + 15625 q^{5} - 87282 q^{7} - 643149 q^{8} + 531441 q^{9} - 1828125 q^{10} + 1771561 q^{11} + 9629118 q^{13} + 10211994 q^{14} + 35955505 q^{16} + 35761518 q^{17} - 62178597 q^{18} + 149890625 q^{20} - 207272637 q^{22} + 244140625 q^{25} - 1126606806 q^{26} - 837296226 q^{28} + 346700482 q^{31} - 1572455781 q^{32} - 4184097606 q^{34} - 1363781250 q^{35} + 5098113513 q^{36} - 10049203125 q^{40} - 7565541282 q^{43} + 16994584673 q^{44} + 8303765625 q^{45} - 6223139677 q^{49} - 28564453125 q^{50} + 92372128974 q^{52} + 27680640625 q^{55} + 56135331018 q^{56} - 84345242798 q^{59} - 40563956394 q^{62} - 46385233362 q^{63} + 36703577897 q^{64} + 150454968750 q^{65} + 343060242174 q^{68} + 159562406250 q^{70} + 240883653922 q^{71} - 341795747709 q^{72} + 297569550798 q^{73} - 154625387202 q^{77} + 561804765625 q^{80} + 282429536481 q^{81} - 579981676482 q^{83} + 558773718750 q^{85} + 885168329994 q^{86} - 1139377685589 q^{88} - 991704621598 q^{89} - 971540578125 q^{90} - 840448677276 q^{91} + 728107342209 q^{98} + 941480149401 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
54.1
0
−117.000 0 9593.00 15625.0 0 −87282.0 −643149. 531441. −1.82812e6
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.13.d.a 1
5.b even 2 1 55.13.d.b yes 1
11.b odd 2 1 55.13.d.b yes 1
55.d odd 2 1 CM 55.13.d.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.13.d.a 1 1.a even 1 1 trivial
55.13.d.a 1 55.d odd 2 1 CM
55.13.d.b yes 1 5.b even 2 1
55.13.d.b yes 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 117 \) acting on \(S_{13}^{\mathrm{new}}(55, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 117 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 15625 \) Copy content Toggle raw display
$7$ \( T + 87282 \) Copy content Toggle raw display
$11$ \( T - 1771561 \) Copy content Toggle raw display
$13$ \( T - 9629118 \) Copy content Toggle raw display
$17$ \( T - 35761518 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 346700482 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 7565541282 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T + 84345242798 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T - 240883653922 \) Copy content Toggle raw display
$73$ \( T - 297569550798 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 579981676482 \) Copy content Toggle raw display
$89$ \( T + 991704621598 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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