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Properties

Label 360.4.k.e
Level $360$
Weight $4$
Character orbit 360.k
Analytic conductor $21.241$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,4,Mod(181,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("360.181");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 360.k (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: no
Analytic conductor: \(21.2406876021\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 14 q^{4} + 56 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 14 q^{4} + 56 q^{7} - 30 q^{10} + 50 q^{16} + 500 q^{22} - 600 q^{25} - 260 q^{28} + 264 q^{31} - 936 q^{34} - 30 q^{40} + 2108 q^{46} + 456 q^{49} - 976 q^{52} + 880 q^{55} - 2628 q^{58} + 3518 q^{64} + 700 q^{70} - 432 q^{73} - 3516 q^{76} - 1656 q^{79} - 3640 q^{82} + 5028 q^{88} + 2636 q^{94} + 1584 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1 −2.82506 − 0.137902i 0 7.96197 + 0.779165i − 5.00000i 0 −16.6486 −22.3856 − 3.29916i 0 −0.689511 + 14.1253i
181.2 −2.82506 + 0.137902i 0 7.96197 − 0.779165i 5.00000i 0 −16.6486 −22.3856 + 3.29916i 0 −0.689511 − 14.1253i
181.3 −2.73897 − 0.705728i 0 7.00390 + 3.86593i − 5.00000i 0 31.9487 −16.4552 − 15.5315i 0 −3.52864 + 13.6948i
181.4 −2.73897 + 0.705728i 0 7.00390 − 3.86593i 5.00000i 0 31.9487 −16.4552 + 15.5315i 0 −3.52864 − 13.6948i
181.5 −2.30129 − 1.64440i 0 2.59188 + 7.56850i 5.00000i 0 −10.8044 6.48100 − 21.6794i 0 8.22202 − 11.5065i
181.6 −2.30129 + 1.64440i 0 2.59188 − 7.56850i − 5.00000i 0 −10.8044 6.48100 + 21.6794i 0 8.22202 + 11.5065i
181.7 −1.76484 − 2.21028i 0 −1.77065 + 7.80159i − 5.00000i 0 −17.4426 20.3686 − 9.85498i 0 −11.0514 + 8.82422i
181.8 −1.76484 + 2.21028i 0 −1.77065 − 7.80159i 5.00000i 0 −17.4426 20.3686 + 9.85498i 0 −11.0514 − 8.82422i
181.9 −1.08113 − 2.61365i 0 −5.66231 + 5.65140i 5.00000i 0 20.1838 20.8925 + 8.68936i 0 13.0682 − 5.40566i
181.10 −1.08113 + 2.61365i 0 −5.66231 − 5.65140i − 5.00000i 0 20.1838 20.8925 − 8.68936i 0 13.0682 + 5.40566i
181.11 −0.829220 − 2.70414i 0 −6.62479 + 4.48466i − 5.00000i 0 6.76306 17.6206 + 14.1956i 0 −13.5207 + 4.14610i
181.12 −0.829220 + 2.70414i 0 −6.62479 − 4.48466i 5.00000i 0 6.76306 17.6206 − 14.1956i 0 −13.5207 − 4.14610i
181.13 0.829220 − 2.70414i 0 −6.62479 − 4.48466i − 5.00000i 0 6.76306 −17.6206 + 14.1956i 0 −13.5207 − 4.14610i
181.14 0.829220 + 2.70414i 0 −6.62479 + 4.48466i 5.00000i 0 6.76306 −17.6206 − 14.1956i 0 −13.5207 + 4.14610i
181.15 1.08113 − 2.61365i 0 −5.66231 − 5.65140i 5.00000i 0 20.1838 −20.8925 + 8.68936i 0 13.0682 + 5.40566i
181.16 1.08113 + 2.61365i 0 −5.66231 + 5.65140i − 5.00000i 0 20.1838 −20.8925 − 8.68936i 0 13.0682 − 5.40566i
181.17 1.76484 − 2.21028i 0 −1.77065 − 7.80159i − 5.00000i 0 −17.4426 −20.3686 − 9.85498i 0 −11.0514 − 8.82422i
181.18 1.76484 + 2.21028i 0 −1.77065 + 7.80159i 5.00000i 0 −17.4426 −20.3686 + 9.85498i 0 −11.0514 + 8.82422i
181.19 2.30129 − 1.64440i 0 2.59188 − 7.56850i 5.00000i 0 −10.8044 −6.48100 − 21.6794i 0 8.22202 + 11.5065i
181.20 2.30129 + 1.64440i 0 2.59188 + 7.56850i − 5.00000i 0 −10.8044 −6.48100 + 21.6794i 0 8.22202 − 11.5065i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.4.k.e ✓ 24
3.b odd 2 1 inner 360.4.k.e ✓ 24
4.b odd 2 1 1440.4.k.e 24
8.b even 2 1 inner 360.4.k.e ✓ 24
8.d odd 2 1 1440.4.k.e 24
12.b even 2 1 1440.4.k.e 24
24.f even 2 1 1440.4.k.e 24
24.h odd 2 1 inner 360.4.k.e ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.k.e ✓ 24 1.a even 1 1 trivial
360.4.k.e ✓ 24 3.b odd 2 1 inner
360.4.k.e ✓ 24 8.b even 2 1 inner
360.4.k.e ✓ 24 24.h odd 2 1 inner
1440.4.k.e 24 4.b odd 2 1
1440.4.k.e 24 8.d odd 2 1
1440.4.k.e 24 12.b even 2 1
1440.4.k.e 24 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 14T_{7}^{5} - 988T_{7}^{4} + 4760T_{7}^{3} + 276448T_{7}^{2} + 256640T_{7} - 13683200 \) acting on \(S_{4}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display