Properties

Label 184.5.e.a
Level $184$
Weight $5$
Character orbit 184.e
Self dual yes
Analytic conductor $19.020$
Analytic rank $0$
Dimension $1$
CM discriminant -184
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [184,5,Mod(45,184)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(184, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("184.45");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 184 = 2^{3} \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 184.e (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: \(19.0200732074\)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 16 q^{4} - 42 q^{5} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 16 q^{4} - 42 q^{5} + 64 q^{8} + 81 q^{9} - 168 q^{10} + 150 q^{11} + 256 q^{16} + 324 q^{18} + 630 q^{19} - 672 q^{20} + 600 q^{22} + 529 q^{23} + 1139 q^{25} - 1022 q^{31} + 1024 q^{32} + 1296 q^{36} - 1770 q^{37} + 2520 q^{38} - 2688 q^{40} - 3262 q^{41} - 810 q^{43} + 2400 q^{44} - 3402 q^{45} + 2116 q^{46} + 3682 q^{47} + 2401 q^{49} + 4556 q^{50} + 1110 q^{53} - 6300 q^{55} + 7350 q^{61} - 4088 q^{62} + 4096 q^{64} + 4470 q^{67} - 8318 q^{71} + 5184 q^{72} - 7742 q^{73} - 7080 q^{74} + 10080 q^{76} - 10752 q^{80} + 6561 q^{81} - 13048 q^{82} - 12810 q^{83} - 3240 q^{86} + 9600 q^{88} - 13608 q^{90} + 8464 q^{92} + 14728 q^{94} - 26460 q^{95} + 9604 q^{98} + 12150 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/184\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(93\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1
0
4.00000 0 16.0000 −42.0000 0 0 64.0000 81.0000 −168.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
184.e odd 2 1 CM by \(\Q(\sqrt{-46}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 184.5.e.a 1
4.b odd 2 1 736.5.e.a 1
8.b even 2 1 184.5.e.b yes 1
8.d odd 2 1 736.5.e.b 1
23.b odd 2 1 184.5.e.b yes 1
92.b even 2 1 736.5.e.b 1
184.e odd 2 1 CM 184.5.e.a 1
184.h even 2 1 736.5.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.5.e.a 1 1.a even 1 1 trivial
184.5.e.a 1 184.e odd 2 1 CM
184.5.e.b yes 1 8.b even 2 1
184.5.e.b yes 1 23.b odd 2 1
736.5.e.a 1 4.b odd 2 1
736.5.e.a 1 184.h even 2 1
736.5.e.b 1 8.d odd 2 1
736.5.e.b 1 92.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(184, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} + 42 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 42 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 150 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 630 \) Copy content Toggle raw display
$23$ \( T - 529 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 1022 \) Copy content Toggle raw display
$37$ \( T + 1770 \) Copy content Toggle raw display
$41$ \( T + 3262 \) Copy content Toggle raw display
$43$ \( T + 810 \) Copy content Toggle raw display
$47$ \( T - 3682 \) Copy content Toggle raw display
$53$ \( T - 1110 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 7350 \) Copy content Toggle raw display
$67$ \( T - 4470 \) Copy content Toggle raw display
$71$ \( T + 8318 \) Copy content Toggle raw display
$73$ \( T + 7742 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 12810 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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