Properties

Label 55.5.d.b
Level $55$
Weight $5$
Character orbit 55.d
Self dual yes
Analytic conductor $5.685$
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] = [55,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.54");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(5.68534796961\)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{2} - 7 q^{4} + 25 q^{5} + 78 q^{7} - 69 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} - 7 q^{4} + 25 q^{5} + 78 q^{7} - 69 q^{8} + 81 q^{9} + 75 q^{10} + 121 q^{11} - 162 q^{13} + 234 q^{14} - 95 q^{16} - 402 q^{17} + 243 q^{18} - 175 q^{20} + 363 q^{22} + 625 q^{25} - 486 q^{26} - 546 q^{28} - 1598 q^{31} + 819 q^{32} - 1206 q^{34} + 1950 q^{35} - 567 q^{36} - 1725 q^{40} - 3522 q^{43} - 847 q^{44} + 2025 q^{45} + 3683 q^{49} + 1875 q^{50} + 1134 q^{52} + 3025 q^{55} - 5382 q^{56} + 3442 q^{59} - 4794 q^{62} + 6318 q^{63} + 3977 q^{64} - 4050 q^{65} + 2814 q^{68} + 5850 q^{70} - 3998 q^{71} - 5589 q^{72} + 10638 q^{73} + 9438 q^{77} - 2375 q^{80} + 6561 q^{81} - 13602 q^{83} - 10050 q^{85} - 10566 q^{86} - 8349 q^{88} - 15838 q^{89} + 6075 q^{90} - 12636 q^{91} + 11049 q^{98} + 9801 q^{99}+O(q^{100}) \) 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
3.00000 0 −7.00000 25.0000 0 78.0000 −69.0000 81.0000 75.0000
\(n\): 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 55.5.d.b yes 1
5.b even 2 1 55.5.d.a 1
5.c odd 4 2 275.5.c.c 2
11.b odd 2 1 55.5.d.a 1
55.d odd 2 1 CM 55.5.d.b yes 1
55.e even 4 2 275.5.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.5.d.a 1 5.b even 2 1
55.5.d.a 1 11.b odd 2 1
55.5.d.b yes 1 1.a even 1 1 trivial
55.5.d.b yes 1 55.d odd 2 1 CM
275.5.c.c 2 5.c odd 4 2
275.5.c.c 2 55.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 3 \) acting on \(S_{5}^{\mathrm{new}}(55, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 3 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T - 78 \) Copy content Toggle raw display
$11$ \( T - 121 \) Copy content Toggle raw display
$13$ \( T + 162 \) Copy content Toggle raw display
$17$ \( T + 402 \) 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
$31$ \( T + 1598 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 3522 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T - 3442 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T + 3998 \) Copy content Toggle raw display
$73$ \( T - 10638 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 13602 \) Copy content Toggle raw display
$89$ \( T + 15838 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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