Properties

Label 252.9.bk.a
Level $252$
Weight $9$
Character orbit 252.bk
Analytic conductor $102.659$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(102.659409735\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 1230 q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 1230 q^{7} - 101380 q^{13} + 62770 q^{19} + 2247666 q^{25} + 1389254 q^{31} - 2136026 q^{37} + 11510140 q^{43} - 3824398 q^{49} - 42646528 q^{55} + 27346232 q^{61} + 14239194 q^{67} - 64344138 q^{73} + 7061786 q^{79} - 54198208 q^{85} - 45697066 q^{91} + 476543496 q^{97} + O(q^{100}) \)

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 0 0 0 −1025.51 + 592.080i 0 2369.75 + 386.132i 0 0 0
53.2 0 0 0 −1009.83 + 583.025i 0 −1699.40 1696.12i 0 0 0
53.3 0 0 0 −724.791 + 418.458i 0 613.419 + 2321.32i 0 0 0
53.4 0 0 0 −700.103 + 404.204i 0 −641.390 + 2313.75i 0 0 0
53.5 0 0 0 −674.478 + 389.410i 0 −1634.98 1758.31i 0 0 0
53.6 0 0 0 −517.869 + 298.992i 0 2338.25 545.349i 0 0 0
53.7 0 0 0 −349.907 + 202.019i 0 −1821.32 + 1564.48i 0 0 0
53.8 0 0 0 −318.126 + 183.670i 0 1593.54 + 1795.95i 0 0 0
53.9 0 0 0 −173.159 + 99.9732i 0 1613.03 1778.47i 0 0 0
53.10 0 0 0 −36.7886 + 21.2399i 0 −115.865 2398.20i 0 0 0
53.11 0 0 0 −23.4272 + 13.5257i 0 −2307.52 + 663.449i 0 0 0
53.12 0 0 0 23.4272 13.5257i 0 −2307.52 + 663.449i 0 0 0
53.13 0 0 0 36.7886 21.2399i 0 −115.865 2398.20i 0 0 0
53.14 0 0 0 173.159 99.9732i 0 1613.03 1778.47i 0 0 0
53.15 0 0 0 318.126 183.670i 0 1593.54 + 1795.95i 0 0 0
53.16 0 0 0 349.907 202.019i 0 −1821.32 + 1564.48i 0 0 0
53.17 0 0 0 517.869 298.992i 0 2338.25 545.349i 0 0 0
53.18 0 0 0 674.478 389.410i 0 −1634.98 1758.31i 0 0 0
53.19 0 0 0 700.103 404.204i 0 −641.390 + 2313.75i 0 0 0
53.20 0 0 0 724.791 418.458i 0 613.419 + 2321.32i 0 0 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 233.22
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.9.bk.a 44
3.b odd 2 1 inner 252.9.bk.a 44
7.c even 3 1 inner 252.9.bk.a 44
21.h odd 6 1 inner 252.9.bk.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.9.bk.a 44 1.a even 1 1 trivial
252.9.bk.a 44 3.b odd 2 1 inner
252.9.bk.a 44 7.c even 3 1 inner
252.9.bk.a 44 21.h odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(252, [\chi])\).