Properties

Label 55.9.d.a
Level $55$
Weight $9$
Character orbit 55.d
Self dual yes
Analytic conductor $22.406$
Analytic rank $0$
Dimension $1$
CM discriminant -55
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [55,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.54");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 9 \)
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: \(22.4058235534\)
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)
 
\(f(q)\) \(=\) \( q - 23 q^{2} + 273 q^{4} + 625 q^{5} + 1282 q^{7} - 391 q^{8} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 23 q^{2} + 273 q^{4} + 625 q^{5} + 1282 q^{7} - 391 q^{8} + 6561 q^{9} - 14375 q^{10} + 14641 q^{11} - 30878 q^{13} - 29486 q^{14} - 60895 q^{16} - 5438 q^{17} - 150903 q^{18} + 170625 q^{20} - 336743 q^{22} + 390625 q^{25} + 710194 q^{26} + 349986 q^{28} + 706562 q^{31} + 1500681 q^{32} + 125074 q^{34} + 801250 q^{35} + 1791153 q^{36} - 244375 q^{40} + 5566882 q^{43} + 3996993 q^{44} + 4100625 q^{45} - 4121277 q^{49} - 8984375 q^{50} - 8429694 q^{52} + 9150625 q^{55} - 501262 q^{56} - 12387358 q^{59} - 16250926 q^{62} + 8411202 q^{63} - 18926543 q^{64} - 19298750 q^{65} - 1484574 q^{68} - 18428750 q^{70} - 34839358 q^{71} - 2565351 q^{72} + 56370562 q^{73} + 18769762 q^{77} - 38059375 q^{80} + 43046721 q^{81} + 90097762 q^{83} - 3398750 q^{85} - 128038286 q^{86} - 5724631 q^{88} + 125357762 q^{89} - 94314375 q^{90} - 39585596 q^{91} + 94789371 q^{98} + 96059601 q^{99}+O(q^{100}) \) 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
−23.0000 0 273.000 625.000 0 1282.00 −391.000 6561.00 −14375.0
\(n\): 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 55.9.d.a 1
5.b even 2 1 55.9.d.b yes 1
11.b odd 2 1 55.9.d.b yes 1
55.d odd 2 1 CM 55.9.d.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.9.d.a 1 1.a even 1 1 trivial
55.9.d.a 1 55.d odd 2 1 CM
55.9.d.b yes 1 5.b even 2 1
55.9.d.b yes 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 23 \) acting on \(S_{9}^{\mathrm{new}}(55, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 23 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 625 \) Copy content Toggle raw display
$7$ \( T - 1282 \) Copy content Toggle raw display
$11$ \( T - 14641 \) Copy content Toggle raw display
$13$ \( T + 30878 \) Copy content Toggle raw display
$17$ \( T + 5438 \) 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
$31$ \( T - 706562 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 5566882 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T + 12387358 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T + 34839358 \) Copy content Toggle raw display
$73$ \( T - 56370562 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 90097762 \) Copy content Toggle raw display
$89$ \( T - 125357762 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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