Properties

Label 864.2.y.a
Level $864$
Weight $2$
Character orbit 864.y
Analytic conductor $6.899$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.y (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{9} - 12 q^{17} - 48 q^{21} + 24 q^{25} + 6 q^{29} - 6 q^{33} + 30 q^{37} - 12 q^{41} + 30 q^{45} - 6 q^{49} - 36 q^{53} - 6 q^{57} - 12 q^{61} - 60 q^{65} - 78 q^{69} + 48 q^{73} - 12 q^{77} - 36 q^{81} + 102 q^{85} - 66 q^{89} + 36 q^{93} + 6 q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 0 −1.72103 0.195105i 0 0.517524 + 0.188363i 0 0.780902 4.42872i 0 2.92387 + 0.671562i 0
97.2 0 −1.11085 + 1.32892i 0 −0.229433 0.0835067i 0 −0.342276 + 1.94115i 0 −0.532044 2.95244i 0
97.3 0 −0.602286 1.62396i 0 −2.89262 1.05283i 0 −0.0578408 + 0.328031i 0 −2.27450 + 1.95618i 0
97.4 0 −0.418682 + 1.68069i 0 3.54422 + 1.28999i 0 0.131496 0.745751i 0 −2.64941 1.40735i 0
97.5 0 0.418682 1.68069i 0 3.54422 + 1.28999i 0 −0.131496 + 0.745751i 0 −2.64941 1.40735i 0
97.6 0 0.602286 + 1.62396i 0 −2.89262 1.05283i 0 0.0578408 0.328031i 0 −2.27450 + 1.95618i 0
97.7 0 1.11085 1.32892i 0 −0.229433 0.0835067i 0 0.342276 1.94115i 0 −0.532044 2.95244i 0
97.8 0 1.72103 + 0.195105i 0 0.517524 + 0.188363i 0 −0.780902 + 4.42872i 0 2.92387 + 0.671562i 0
193.1 0 −1.57833 0.713349i 0 0.394613 2.23796i 0 3.88380 3.25889i 0 1.98227 + 2.25180i 0
193.2 0 −1.46485 0.924244i 0 −0.217354 + 1.23268i 0 −2.00770 + 1.68466i 0 1.29155 + 2.70775i 0
193.3 0 −1.34012 + 1.09730i 0 0.181867 1.03142i 0 −0.246536 + 0.206868i 0 0.591863 2.94104i 0
193.4 0 −0.0827900 + 1.73007i 0 −0.532774 + 3.02151i 0 −2.03971 + 1.71152i 0 −2.98629 0.286465i 0
193.5 0 0.0827900 1.73007i 0 −0.532774 + 3.02151i 0 2.03971 1.71152i 0 −2.98629 0.286465i 0
193.6 0 1.34012 1.09730i 0 0.181867 1.03142i 0 0.246536 0.206868i 0 0.591863 2.94104i 0
193.7 0 1.46485 + 0.924244i 0 −0.217354 + 1.23268i 0 2.00770 1.68466i 0 1.29155 + 2.70775i 0
193.8 0 1.57833 + 0.713349i 0 0.394613 2.23796i 0 −3.88380 + 3.25889i 0 1.98227 + 2.25180i 0
385.1 0 −1.57833 + 0.713349i 0 0.394613 + 2.23796i 0 3.88380 + 3.25889i 0 1.98227 2.25180i 0
385.2 0 −1.46485 + 0.924244i 0 −0.217354 1.23268i 0 −2.00770 1.68466i 0 1.29155 2.70775i 0
385.3 0 −1.34012 1.09730i 0 0.181867 + 1.03142i 0 −0.246536 0.206868i 0 0.591863 + 2.94104i 0
385.4 0 −0.0827900 1.73007i 0 −0.532774 3.02151i 0 −2.03971 1.71152i 0 −2.98629 + 0.286465i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
27.e even 9 1 inner
108.j odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.y.a 48
4.b odd 2 1 inner 864.2.y.a 48
27.e even 9 1 inner 864.2.y.a 48
108.j odd 18 1 inner 864.2.y.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.y.a 48 1.a even 1 1 trivial
864.2.y.a 48 4.b odd 2 1 inner
864.2.y.a 48 27.e even 9 1 inner
864.2.y.a 48 108.j odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\):

\(T_{5}^{24} - \cdots\)
\(19\!\cdots\!67\)\( T_{7}^{24} + \)\(72\!\cdots\!76\)\( T_{7}^{22} + \)\(16\!\cdots\!88\)\( T_{7}^{20} - \)\(14\!\cdots\!38\)\( T_{7}^{18} + \)\(37\!\cdots\!46\)\( T_{7}^{16} + \)\(14\!\cdots\!59\)\( T_{7}^{14} - \)\(13\!\cdots\!07\)\( T_{7}^{12} + \)\(93\!\cdots\!36\)\( T_{7}^{10} + \)\(61\!\cdots\!35\)\( T_{7}^{8} + \)\(10\!\cdots\!34\)\( T_{7}^{6} + 945447002172 T_{7}^{4} + 108970914168 T_{7}^{2} + 7695324729 \)">\(T_{7}^{48} + \cdots\)