Properties

Label 360.2.w.e
Level $360$
Weight $2$
Character orbit 360.w
Analytic conductor $2.875$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 120)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24q + 12q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 12q^{8} + 8q^{10} - 20q^{16} - 8q^{17} + 20q^{20} - 28q^{22} - 8q^{25} + 16q^{26} + 4q^{28} - 20q^{32} + 48q^{35} - 40q^{38} + 8q^{40} - 32q^{43} + 48q^{46} - 56q^{50} - 48q^{52} + 32q^{56} - 12q^{58} + 16q^{62} + 8q^{65} + 48q^{67} - 72q^{68} + 4q^{70} - 40q^{73} + 48q^{76} - 76q^{80} + 24q^{82} - 80q^{83} + 32q^{86} + 12q^{88} + 64q^{91} - 16q^{92} - 24q^{97} + 88q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1 −1.31581 + 0.518298i 0 1.46273 1.36397i 2.22965 + 0.169312i 0 0.645414 + 0.645414i −1.21775 + 2.55286i 0 −3.02156 + 0.932839i
163.2 −1.25738 + 0.647304i 0 1.16200 1.62781i −1.28903 1.82713i 0 −1.45533 1.45533i −0.407381 + 2.79894i 0 2.80350 + 1.46300i
163.3 −1.08304 0.909406i 0 0.345961 + 1.96985i −0.780766 + 2.09533i 0 −2.10796 2.10796i 1.41670 2.44805i 0 2.75111 1.55930i
163.4 −0.647304 + 1.25738i 0 −1.16200 1.62781i 1.28903 + 1.82713i 0 1.45533 + 1.45533i 2.79894 0.407381i 0 −3.13178 + 0.438090i
163.5 −0.624608 1.26880i 0 −1.21973 + 1.58501i −2.11218 0.733965i 0 1.93078 + 1.93078i 2.77292 + 0.557590i 0 0.388025 + 3.13838i
163.6 −0.518298 + 1.31581i 0 −1.46273 1.36397i −2.22965 0.169312i 0 −0.645414 0.645414i 2.55286 1.21775i 0 1.37841 2.84605i
163.7 −0.109339 1.40998i 0 −1.97609 + 0.308331i 0.0696909 2.23498i 0 −1.21782 1.21782i 0.650804 + 2.75254i 0 −3.15890 + 0.146107i
163.8 0.804501 1.16309i 0 −0.705556 1.87141i −1.51371 + 1.64581i 0 −3.43671 3.43671i −2.74424 0.684930i 0 0.696440 + 3.08463i
163.9 0.909406 + 1.08304i 0 −0.345961 + 1.96985i 0.780766 2.09533i 0 2.10796 + 2.10796i −2.44805 + 1.41670i 0 2.97936 1.05990i
163.10 1.16309 0.804501i 0 0.705556 1.87141i 1.51371 1.64581i 0 3.43671 + 3.43671i −0.684930 2.74424i 0 0.436527 3.13200i
163.11 1.26880 + 0.624608i 0 1.21973 + 1.58501i 2.11218 + 0.733965i 0 −1.93078 1.93078i 0.557590 + 2.77292i 0 2.22150 + 2.25054i
163.12 1.40998 + 0.109339i 0 1.97609 + 0.308331i −0.0696909 + 2.23498i 0 1.21782 + 1.21782i 2.75254 + 0.650804i 0 −0.342633 + 3.14366i
307.1 −1.31581 0.518298i 0 1.46273 + 1.36397i 2.22965 0.169312i 0 0.645414 0.645414i −1.21775 2.55286i 0 −3.02156 0.932839i
307.2 −1.25738 0.647304i 0 1.16200 + 1.62781i −1.28903 + 1.82713i 0 −1.45533 + 1.45533i −0.407381 2.79894i 0 2.80350 1.46300i
307.3 −1.08304 + 0.909406i 0 0.345961 1.96985i −0.780766 2.09533i 0 −2.10796 + 2.10796i 1.41670 + 2.44805i 0 2.75111 + 1.55930i
307.4 −0.647304 1.25738i 0 −1.16200 + 1.62781i 1.28903 1.82713i 0 1.45533 1.45533i 2.79894 + 0.407381i 0 −3.13178 0.438090i
307.5 −0.624608 + 1.26880i 0 −1.21973 1.58501i −2.11218 + 0.733965i 0 1.93078 1.93078i 2.77292 0.557590i 0 0.388025 3.13838i
307.6 −0.518298 1.31581i 0 −1.46273 + 1.36397i −2.22965 + 0.169312i 0 −0.645414 + 0.645414i 2.55286 + 1.21775i 0 1.37841 + 2.84605i
307.7 −0.109339 + 1.40998i 0 −1.97609 0.308331i 0.0696909 + 2.23498i 0 −1.21782 + 1.21782i 0.650804 2.75254i 0 −3.15890 0.146107i
307.8 0.804501 + 1.16309i 0 −0.705556 + 1.87141i −1.51371 1.64581i 0 −3.43671 + 3.43671i −2.74424 + 0.684930i 0 0.696440 3.08463i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k 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.w.e 24
3.b odd 2 1 120.2.v.a 24
4.b odd 2 1 1440.2.bi.e 24
5.c odd 4 1 inner 360.2.w.e 24
8.b even 2 1 1440.2.bi.e 24
8.d odd 2 1 inner 360.2.w.e 24
12.b even 2 1 480.2.bh.a 24
15.d odd 2 1 600.2.v.b 24
15.e even 4 1 120.2.v.a 24
15.e even 4 1 600.2.v.b 24
20.e even 4 1 1440.2.bi.e 24
24.f even 2 1 120.2.v.a 24
24.h odd 2 1 480.2.bh.a 24
40.i odd 4 1 1440.2.bi.e 24
40.k even 4 1 inner 360.2.w.e 24
60.h even 2 1 2400.2.bh.b 24
60.l odd 4 1 480.2.bh.a 24
60.l odd 4 1 2400.2.bh.b 24
120.i odd 2 1 2400.2.bh.b 24
120.m even 2 1 600.2.v.b 24
120.q odd 4 1 120.2.v.a 24
120.q odd 4 1 600.2.v.b 24
120.w even 4 1 480.2.bh.a 24
120.w even 4 1 2400.2.bh.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.v.a 24 3.b odd 2 1
120.2.v.a 24 15.e even 4 1
120.2.v.a 24 24.f even 2 1
120.2.v.a 24 120.q odd 4 1
360.2.w.e 24 1.a even 1 1 trivial
360.2.w.e 24 5.c odd 4 1 inner
360.2.w.e 24 8.d odd 2 1 inner
360.2.w.e 24 40.k even 4 1 inner
480.2.bh.a 24 12.b even 2 1
480.2.bh.a 24 24.h odd 2 1
480.2.bh.a 24 60.l odd 4 1
480.2.bh.a 24 120.w even 4 1
600.2.v.b 24 15.d odd 2 1
600.2.v.b 24 15.e even 4 1
600.2.v.b 24 120.m even 2 1
600.2.v.b 24 120.q odd 4 1
1440.2.bi.e 24 4.b odd 2 1
1440.2.bi.e 24 8.b even 2 1
1440.2.bi.e 24 20.e even 4 1
1440.2.bi.e 24 40.i odd 4 1
2400.2.bh.b 24 60.h even 2 1
2400.2.bh.b 24 60.l odd 4 1
2400.2.bh.b 24 120.i odd 2 1
2400.2.bh.b 24 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 720 T_{7}^{20} + 98656 T_{7}^{16} + 4752640 T_{7}^{12} + 81309952 T_{7}^{8} + 440926208 T_{7}^{4} + 268435456 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).