Properties

Label 1386.2.r.d
Level $1386$
Weight $2$
Character orbit 1386.r
Analytic conductor $11.067$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24q + 12q^{4} + 8q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 12q^{4} + 8q^{7} - 12q^{16} + 12q^{19} - 24q^{22} + 4q^{25} + 4q^{28} + 20q^{37} + 24q^{43} + 12q^{46} + 40q^{49} - 28q^{58} - 120q^{61} - 24q^{64} - 32q^{67} + 48q^{70} - 48q^{73} - 64q^{79} - 48q^{82} + 32q^{85} - 12q^{88} + 56q^{91} + 60q^{94} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.36138 2.35797i 0 −2.13605 + 1.56119i 1.00000i 0 −2.35797 + 1.36138i
89.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.651304 1.12809i 0 −0.212626 2.63719i 1.00000i 0 −1.12809 + 0.651304i
89.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.645853 1.11865i 0 2.63647 + 0.221466i 1.00000i 0 −1.11865 + 0.645853i
89.4 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.171460 + 0.296977i 0 −2.33970 1.23523i 1.00000i 0 0.296977 0.171460i
89.5 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.474393 + 0.821673i 0 1.41034 + 2.23851i 1.00000i 0 0.821673 0.474393i
89.6 −0.866025 0.500000i 0 0.500000 + 0.866025i −2.01268 + 3.48607i 0 2.64157 0.148737i 1.00000i 0 3.48607 2.01268i
89.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.01268 3.48607i 0 2.64157 0.148737i 1.00000i 0 3.48607 2.01268i
89.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.474393 0.821673i 0 1.41034 + 2.23851i 1.00000i 0 0.821673 0.474393i
89.9 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.171460 0.296977i 0 −2.33970 1.23523i 1.00000i 0 0.296977 0.171460i
89.10 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.645853 + 1.11865i 0 2.63647 + 0.221466i 1.00000i 0 −1.11865 + 0.645853i
89.11 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.651304 + 1.12809i 0 −0.212626 2.63719i 1.00000i 0 −1.12809 + 0.651304i
89.12 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.36138 + 2.35797i 0 −2.13605 + 1.56119i 1.00000i 0 −2.35797 + 1.36138i
1277.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.36138 + 2.35797i 0 −2.13605 1.56119i 1.00000i 0 −2.35797 1.36138i
1277.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.651304 + 1.12809i 0 −0.212626 + 2.63719i 1.00000i 0 −1.12809 0.651304i
1277.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.645853 + 1.11865i 0 2.63647 0.221466i 1.00000i 0 −1.11865 0.645853i
1277.4 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.171460 0.296977i 0 −2.33970 + 1.23523i 1.00000i 0 0.296977 + 0.171460i
1277.5 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.474393 0.821673i 0 1.41034 2.23851i 1.00000i 0 0.821673 + 0.474393i
1277.6 −0.866025 + 0.500000i 0 0.500000 0.866025i −2.01268 3.48607i 0 2.64157 + 0.148737i 1.00000i 0 3.48607 + 2.01268i
1277.7 0.866025 0.500000i 0 0.500000 0.866025i 2.01268 + 3.48607i 0 2.64157 + 0.148737i 1.00000i 0 3.48607 + 2.01268i
1277.8 0.866025 0.500000i 0 0.500000 0.866025i 0.474393 + 0.821673i 0 1.41034 2.23851i 1.00000i 0 0.821673 + 0.474393i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1277.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.r.d 24
3.b odd 2 1 inner 1386.2.r.d 24
7.d odd 6 1 inner 1386.2.r.d 24
21.g even 6 1 inner 1386.2.r.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.r.d 24 1.a even 1 1 trivial
1386.2.r.d 24 3.b odd 2 1 inner
1386.2.r.d 24 7.d odd 6 1 inner
1386.2.r.d 24 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{24} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1386, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database