Properties

Label 360.2.x.a
Level $360$
Weight $2$
Character orbit 360.x
Analytic conductor $2.875$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48q + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q + 16q^{10} - 8q^{16} + 16q^{22} - 32q^{28} + 32q^{31} - 56q^{40} - 16q^{46} - 56q^{52} - 80q^{58} - 64q^{70} + 48q^{76} - 48q^{82} + 64q^{88} - 32q^{97} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −1.40514 + 0.159967i 0 1.94882 0.449551i −0.215413 2.22567i 0 −2.32063 + 2.32063i −2.66645 + 0.943427i 0 0.658719 + 3.09291i
53.2 −1.37774 0.319115i 0 1.79633 + 0.879316i 1.42597 1.72238i 0 3.11972 3.11972i −2.19427 1.78470i 0 −2.51426 + 1.91795i
53.3 −1.33270 + 0.473189i 0 1.55218 1.26124i 1.45381 + 1.69895i 0 1.53029 1.53029i −1.47179 + 2.41533i 0 −2.74142 1.57626i
53.4 −1.20878 0.734068i 0 0.922287 + 1.77465i −2.03368 0.929591i 0 −2.49469 + 2.49469i 0.187875 2.82218i 0 1.77589 + 2.61653i
53.5 −1.19248 + 0.760261i 0 0.844006 1.81319i −2.23604 + 0.0113158i 0 0.471963 0.471963i 0.372039 + 2.80385i 0 2.65782 1.71347i
53.6 −1.16389 0.803348i 0 0.709264 + 1.87001i −1.29263 + 1.82458i 0 −0.306649 + 0.306649i 0.676767 2.74627i 0 2.97025 1.08518i
53.7 −0.803348 1.16389i 0 −0.709264 + 1.87001i 1.29263 1.82458i 0 −0.306649 + 0.306649i 2.74627 0.676767i 0 −3.16204 0.0386996i
53.8 −0.760261 + 1.19248i 0 −0.844006 1.81319i −2.23604 + 0.0113158i 0 0.471963 0.471963i 2.80385 + 0.372039i 0 1.68648 2.67503i
53.9 −0.734068 1.20878i 0 −0.922287 + 1.77465i 2.03368 + 0.929591i 0 −2.49469 + 2.49469i 2.82218 0.187875i 0 −0.369193 3.14065i
53.10 −0.473189 + 1.33270i 0 −1.55218 1.26124i 1.45381 + 1.69895i 0 1.53029 1.53029i 2.41533 1.47179i 0 −2.95212 + 1.13358i
53.11 −0.319115 1.37774i 0 −1.79633 + 0.879316i −1.42597 + 1.72238i 0 3.11972 3.11972i 1.78470 + 2.19427i 0 2.82804 + 1.41498i
53.12 −0.159967 + 1.40514i 0 −1.94882 0.449551i −0.215413 2.22567i 0 −2.32063 + 2.32063i 0.943427 2.66645i 0 3.16183 + 0.0533477i
53.13 0.159967 1.40514i 0 −1.94882 0.449551i 0.215413 + 2.22567i 0 −2.32063 + 2.32063i −0.943427 + 2.66645i 0 3.16183 + 0.0533477i
53.14 0.319115 + 1.37774i 0 −1.79633 + 0.879316i 1.42597 1.72238i 0 3.11972 3.11972i −1.78470 2.19427i 0 2.82804 + 1.41498i
53.15 0.473189 1.33270i 0 −1.55218 1.26124i −1.45381 1.69895i 0 1.53029 1.53029i −2.41533 + 1.47179i 0 −2.95212 + 1.13358i
53.16 0.734068 + 1.20878i 0 −0.922287 + 1.77465i −2.03368 0.929591i 0 −2.49469 + 2.49469i −2.82218 + 0.187875i 0 −0.369193 3.14065i
53.17 0.760261 1.19248i 0 −0.844006 1.81319i 2.23604 0.0113158i 0 0.471963 0.471963i −2.80385 0.372039i 0 1.68648 2.67503i
53.18 0.803348 + 1.16389i 0 −0.709264 + 1.87001i −1.29263 + 1.82458i 0 −0.306649 + 0.306649i −2.74627 + 0.676767i 0 −3.16204 0.0386996i
53.19 1.16389 + 0.803348i 0 0.709264 + 1.87001i 1.29263 1.82458i 0 −0.306649 + 0.306649i −0.676767 + 2.74627i 0 2.97025 1.08518i
53.20 1.19248 0.760261i 0 0.844006 1.81319i 2.23604 0.0113158i 0 0.471963 0.471963i −0.372039 2.80385i 0 2.65782 1.71347i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
8.b even 2 1 inner
15.e even 4 1 inner
24.h odd 2 1 inner
40.i odd 4 1 inner
120.w even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.x.a 48
3.b odd 2 1 inner 360.2.x.a 48
4.b odd 2 1 1440.2.bj.a 48
5.c odd 4 1 inner 360.2.x.a 48
8.b even 2 1 inner 360.2.x.a 48
8.d odd 2 1 1440.2.bj.a 48
12.b even 2 1 1440.2.bj.a 48
15.e even 4 1 inner 360.2.x.a 48
20.e even 4 1 1440.2.bj.a 48
24.f even 2 1 1440.2.bj.a 48
24.h odd 2 1 inner 360.2.x.a 48
40.i odd 4 1 inner 360.2.x.a 48
40.k even 4 1 1440.2.bj.a 48
60.l odd 4 1 1440.2.bj.a 48
120.q odd 4 1 1440.2.bj.a 48
120.w even 4 1 inner 360.2.x.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.x.a 48 1.a even 1 1 trivial
360.2.x.a 48 3.b odd 2 1 inner
360.2.x.a 48 5.c odd 4 1 inner
360.2.x.a 48 8.b even 2 1 inner
360.2.x.a 48 15.e even 4 1 inner
360.2.x.a 48 24.h odd 2 1 inner
360.2.x.a 48 40.i odd 4 1 inner
360.2.x.a 48 120.w even 4 1 inner
1440.2.bj.a 48 4.b odd 2 1
1440.2.bj.a 48 8.d odd 2 1
1440.2.bj.a 48 12.b even 2 1
1440.2.bj.a 48 20.e even 4 1
1440.2.bj.a 48 24.f even 2 1
1440.2.bj.a 48 40.k even 4 1
1440.2.bj.a 48 60.l odd 4 1
1440.2.bj.a 48 120.q odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(360, [\chi])\).