Properties

Label 368.4.a.e
Level $368$
Weight $4$
Character orbit 368.a
Self dual yes
Analytic conductor $21.713$
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] = [368,4,Mod(1,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 368.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: \(21.7127028821\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 46)
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)
 
\(f(q)\) \(=\) \( q + 9 q^{3} - 20 q^{5} - 2 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} - 20 q^{5} - 2 q^{7} + 54 q^{9} + 52 q^{11} + 43 q^{13} - 180 q^{15} - 50 q^{17} + 74 q^{19} - 18 q^{21} + 23 q^{23} + 275 q^{25} + 243 q^{27} - 7 q^{29} + 273 q^{31} + 468 q^{33} + 40 q^{35} - 4 q^{37} + 387 q^{39} + 123 q^{41} + 152 q^{43} - 1080 q^{45} - 75 q^{47} - 339 q^{49} - 450 q^{51} + 86 q^{53} - 1040 q^{55} + 666 q^{57} + 444 q^{59} + 262 q^{61} - 108 q^{63} - 860 q^{65} - 764 q^{67} + 207 q^{69} + 21 q^{71} + 681 q^{73} + 2475 q^{75} - 104 q^{77} - 426 q^{79} + 729 q^{81} - 902 q^{83} + 1000 q^{85} - 63 q^{87} - 1272 q^{89} - 86 q^{91} + 2457 q^{93} - 1480 q^{95} - 342 q^{97} + 2808 q^{99}+O(q^{100}) \) 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
0 9.00000 0 −20.0000 0 −2.00000 0 54.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(23\) \( -1 \)

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 368.4.a.e 1
4.b odd 2 1 46.4.a.b 1
8.b even 2 1 1472.4.a.a 1
8.d odd 2 1 1472.4.a.j 1
12.b even 2 1 414.4.a.b 1
20.d odd 2 1 1150.4.a.d 1
20.e even 4 2 1150.4.b.a 2
28.d even 2 1 2254.4.a.b 1
92.b even 2 1 1058.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.4.a.b 1 4.b odd 2 1
368.4.a.e 1 1.a even 1 1 trivial
414.4.a.b 1 12.b even 2 1
1058.4.a.b 1 92.b even 2 1
1150.4.a.d 1 20.d odd 2 1
1150.4.b.a 2 20.e even 4 2
1472.4.a.a 1 8.b even 2 1
1472.4.a.j 1 8.d odd 2 1
2254.4.a.b 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 9 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(368))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T + 20 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 52 \) Copy content Toggle raw display
$13$ \( T - 43 \) Copy content Toggle raw display
$17$ \( T + 50 \) Copy content Toggle raw display
$19$ \( T - 74 \) Copy content Toggle raw display
$23$ \( T - 23 \) Copy content Toggle raw display
$29$ \( T + 7 \) Copy content Toggle raw display
$31$ \( T - 273 \) Copy content Toggle raw display
$37$ \( T + 4 \) Copy content Toggle raw display
$41$ \( T - 123 \) Copy content Toggle raw display
$43$ \( T - 152 \) Copy content Toggle raw display
$47$ \( T + 75 \) Copy content Toggle raw display
$53$ \( T - 86 \) Copy content Toggle raw display
$59$ \( T - 444 \) Copy content Toggle raw display
$61$ \( T - 262 \) Copy content Toggle raw display
$67$ \( T + 764 \) Copy content Toggle raw display
$71$ \( T - 21 \) Copy content Toggle raw display
$73$ \( T - 681 \) Copy content Toggle raw display
$79$ \( T + 426 \) Copy content Toggle raw display
$83$ \( T + 902 \) Copy content Toggle raw display
$89$ \( T + 1272 \) Copy content Toggle raw display
$97$ \( T + 342 \) Copy content Toggle raw display
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