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Properties

Label 360.4.d.g
Level $360$
Weight $4$
Character orbit 360.d
Analytic conductor $21.241$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,4,Mod(109,360)]
 
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, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.109");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 360.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: 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 16 q^{4} - 32 q^{10} - 416 q^{16} - 328 q^{25} - 1504 q^{31} - 2352 q^{34} - 2608 q^{40} - 1232 q^{46} - 5016 q^{49} - 4256 q^{55} - 1280 q^{64} + 400 q^{70} - 672 q^{76} - 224 q^{79} - 5648 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −2.65298 − 0.980667i 0 6.07658 + 5.20338i 8.16211 − 7.64068i 0 − 15.4432i −11.0183 − 19.7636i 0 −29.1469 + 12.2663i
109.2 −2.65298 − 0.980667i 0 6.07658 + 5.20338i 8.16211 + 7.64068i 0 15.4432i −11.0183 − 19.7636i 0 −14.1609 − 28.2749i
109.3 −2.65298 + 0.980667i 0 6.07658 − 5.20338i 8.16211 − 7.64068i 0 − 15.4432i −11.0183 + 19.7636i 0 −14.1609 + 28.2749i
109.4 −2.65298 + 0.980667i 0 6.07658 − 5.20338i 8.16211 + 7.64068i 0 15.4432i −11.0183 + 19.7636i 0 −29.1469 − 12.2663i
109.5 −2.17723 − 1.80546i 0 1.48065 + 7.86179i −3.30797 − 10.6798i 0 20.2999i 10.9704 − 19.7902i 0 −12.0796 + 29.2247i
109.6 −2.17723 − 1.80546i 0 1.48065 + 7.86179i −3.30797 + 10.6798i 0 − 20.2999i 10.9704 − 19.7902i 0 26.4841 − 17.2799i
109.7 −2.17723 + 1.80546i 0 1.48065 − 7.86179i −3.30797 − 10.6798i 0 20.2999i 10.9704 + 19.7902i 0 26.4841 + 17.2799i
109.8 −2.17723 + 1.80546i 0 1.48065 − 7.86179i −3.30797 + 10.6798i 0 − 20.2999i 10.9704 + 19.7902i 0 −12.0796 − 29.2247i
109.9 −1.10516 − 2.60358i 0 −5.55723 + 5.75475i −9.45713 − 5.96344i 0 − 31.7084i 21.1246 + 8.10875i 0 −5.07462 + 31.2129i
109.10 −1.10516 − 2.60358i 0 −5.55723 + 5.75475i −9.45713 + 5.96344i 0 31.7084i 21.1246 + 8.10875i 0 25.9780 + 18.0318i
109.11 −1.10516 + 2.60358i 0 −5.55723 − 5.75475i −9.45713 − 5.96344i 0 − 31.7084i 21.1246 − 8.10875i 0 25.9780 − 18.0318i
109.12 −1.10516 + 2.60358i 0 −5.55723 − 5.75475i −9.45713 + 5.96344i 0 31.7084i 21.1246 − 8.10875i 0 −5.07462 − 31.2129i
109.13 1.10516 − 2.60358i 0 −5.55723 − 5.75475i 9.45713 − 5.96344i 0 31.7084i −21.1246 + 8.10875i 0 −5.07462 − 31.2129i
109.14 1.10516 − 2.60358i 0 −5.55723 − 5.75475i 9.45713 + 5.96344i 0 − 31.7084i −21.1246 + 8.10875i 0 25.9780 − 18.0318i
109.15 1.10516 + 2.60358i 0 −5.55723 + 5.75475i 9.45713 − 5.96344i 0 31.7084i −21.1246 − 8.10875i 0 25.9780 + 18.0318i
109.16 1.10516 + 2.60358i 0 −5.55723 + 5.75475i 9.45713 + 5.96344i 0 − 31.7084i −21.1246 − 8.10875i 0 −5.07462 + 31.2129i
109.17 2.17723 − 1.80546i 0 1.48065 − 7.86179i 3.30797 − 10.6798i 0 − 20.2999i −10.9704 − 19.7902i 0 −12.0796 − 29.2247i
109.18 2.17723 − 1.80546i 0 1.48065 − 7.86179i 3.30797 + 10.6798i 0 20.2999i −10.9704 − 19.7902i 0 26.4841 + 17.2799i
109.19 2.17723 + 1.80546i 0 1.48065 + 7.86179i 3.30797 − 10.6798i 0 − 20.2999i −10.9704 + 19.7902i 0 26.4841 − 17.2799i
109.20 2.17723 + 1.80546i 0 1.48065 + 7.86179i 3.30797 + 10.6798i 0 20.2999i −10.9704 + 19.7902i 0 −12.0796 + 29.2247i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
15.d odd 2 1 inner
24.h odd 2 1 inner
40.f even 2 1 inner
120.i 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.d.g ✓ 24
3.b odd 2 1 inner 360.4.d.g ✓ 24
4.b odd 2 1 1440.4.d.g 24
5.b even 2 1 inner 360.4.d.g ✓ 24
8.b even 2 1 inner 360.4.d.g ✓ 24
8.d odd 2 1 1440.4.d.g 24
12.b even 2 1 1440.4.d.g 24
15.d odd 2 1 inner 360.4.d.g ✓ 24
20.d odd 2 1 1440.4.d.g 24
24.f even 2 1 1440.4.d.g 24
24.h odd 2 1 inner 360.4.d.g ✓ 24
40.e odd 2 1 1440.4.d.g 24
40.f even 2 1 inner 360.4.d.g ✓ 24
60.h even 2 1 1440.4.d.g 24
120.i odd 2 1 inner 360.4.d.g ✓ 24
120.m even 2 1 1440.4.d.g 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.d.g ✓ 24 1.a even 1 1 trivial
360.4.d.g ✓ 24 3.b odd 2 1 inner
360.4.d.g ✓ 24 5.b even 2 1 inner
360.4.d.g ✓ 24 8.b even 2 1 inner
360.4.d.g ✓ 24 15.d odd 2 1 inner
360.4.d.g ✓ 24 24.h odd 2 1 inner
360.4.d.g ✓ 24 40.f even 2 1 inner
360.4.d.g ✓ 24 120.i odd 2 1 inner
1440.4.d.g 24 4.b odd 2 1
1440.4.d.g 24 8.d odd 2 1
1440.4.d.g 24 12.b even 2 1
1440.4.d.g 24 20.d odd 2 1
1440.4.d.g 24 24.f even 2 1
1440.4.d.g 24 40.e odd 2 1
1440.4.d.g 24 60.h even 2 1
1440.4.d.g 24 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(360, [\chi])\):

\( T_{7}^{6} + 1656T_{7}^{4} + 752384T_{7}^{2} + 98811904 \) Copy content Toggle raw display
\( T_{11}^{6} + 2240T_{11}^{4} + 1038192T_{11}^{2} + 106427392 \) Copy content Toggle raw display
\( T_{13}^{6} - 10824T_{13}^{4} + 24930560T_{13}^{2} - 5152768000 \) Copy content Toggle raw display