Properties

Label 810.3.j.g
Level $810$
Weight $3$
Character orbit 810.j
Analytic conductor $22.071$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 810.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.0709014132\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} - 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} - 24 q^{7} - 12 q^{10} - 48 q^{13} - 48 q^{16} + 120 q^{22} + 24 q^{25} + 60 q^{34} + 24 q^{43} - 36 q^{49} + 96 q^{52} + 216 q^{55} - 396 q^{58} - 60 q^{61} + 192 q^{64} + 1032 q^{67} - 480 q^{70} - 240 q^{79} - 396 q^{85} - 240 q^{88} + 48 q^{91} - 48 q^{94} + 1440 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1 −0.707107 + 1.22474i 0 −1.00000 1.73205i 4.43710 + 2.30480i 0 6.62225 + 3.82336i 2.82843 0 −5.96030 + 3.80458i
269.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i 2.51279 + 4.32272i 0 5.12645 + 2.95976i 2.82843 0 −7.07104 + 0.0209028i
269.3 −0.707107 + 1.22474i 0 −1.00000 1.73205i −1.24727 4.84193i 0 0.315907 + 0.182389i 2.82843 0 6.81209 + 1.89618i
269.4 −0.707107 + 1.22474i 0 −1.00000 1.73205i −4.52502 + 2.12701i 0 −1.70200 0.982651i 2.82843 0 0.594626 7.04602i
269.5 −0.707107 + 1.22474i 0 −1.00000 1.73205i 3.23698 3.81077i 0 −8.15650 4.70916i 2.82843 0 2.37833 + 6.65909i
269.6 −0.707107 + 1.22474i 0 −1.00000 1.73205i −4.41458 + 2.34766i 0 −8.20611 4.73780i 2.82843 0 0.246293 7.06678i
269.7 0.707107 1.22474i 0 −1.00000 1.73205i −4.43710 2.30480i 0 6.62225 + 3.82336i −2.82843 0 −5.96030 + 3.80458i
269.8 0.707107 1.22474i 0 −1.00000 1.73205i 4.52502 2.12701i 0 −1.70200 0.982651i −2.82843 0 0.594626 7.04602i
269.9 0.707107 1.22474i 0 −1.00000 1.73205i −2.51279 4.32272i 0 5.12645 + 2.95976i −2.82843 0 −7.07104 + 0.0209028i
269.10 0.707107 1.22474i 0 −1.00000 1.73205i −3.23698 + 3.81077i 0 −8.15650 4.70916i −2.82843 0 2.37833 + 6.65909i
269.11 0.707107 1.22474i 0 −1.00000 1.73205i 1.24727 + 4.84193i 0 0.315907 + 0.182389i −2.82843 0 6.81209 + 1.89618i
269.12 0.707107 1.22474i 0 −1.00000 1.73205i 4.41458 2.34766i 0 −8.20611 4.73780i −2.82843 0 0.246293 7.06678i
539.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i 4.43710 2.30480i 0 6.62225 3.82336i 2.82843 0 −5.96030 3.80458i
539.2 −0.707107 1.22474i 0 −1.00000 + 1.73205i 2.51279 4.32272i 0 5.12645 2.95976i 2.82843 0 −7.07104 0.0209028i
539.3 −0.707107 1.22474i 0 −1.00000 + 1.73205i −1.24727 + 4.84193i 0 0.315907 0.182389i 2.82843 0 6.81209 1.89618i
539.4 −0.707107 1.22474i 0 −1.00000 + 1.73205i −4.52502 2.12701i 0 −1.70200 + 0.982651i 2.82843 0 0.594626 + 7.04602i
539.5 −0.707107 1.22474i 0 −1.00000 + 1.73205i 3.23698 + 3.81077i 0 −8.15650 + 4.70916i 2.82843 0 2.37833 6.65909i
539.6 −0.707107 1.22474i 0 −1.00000 + 1.73205i −4.41458 2.34766i 0 −8.20611 + 4.73780i 2.82843 0 0.246293 + 7.06678i
539.7 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −4.43710 + 2.30480i 0 6.62225 3.82336i −2.82843 0 −5.96030 3.80458i
539.8 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −2.51279 + 4.32272i 0 5.12645 2.95976i −2.82843 0 −7.07104 0.0209028i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 539.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
45.h odd 6 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.3.j.g 24
3.b odd 2 1 inner 810.3.j.g 24
5.b even 2 1 810.3.j.h 24
9.c even 3 1 810.3.b.c 24
9.c even 3 1 810.3.j.h 24
9.d odd 6 1 810.3.b.c 24
9.d odd 6 1 810.3.j.h 24
15.d odd 2 1 810.3.j.h 24
45.h odd 6 1 810.3.b.c 24
45.h odd 6 1 inner 810.3.j.g 24
45.j even 6 1 810.3.b.c 24
45.j even 6 1 inner 810.3.j.g 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
810.3.b.c 24 9.c even 3 1
810.3.b.c 24 9.d odd 6 1
810.3.b.c 24 45.h odd 6 1
810.3.b.c 24 45.j even 6 1
810.3.j.g 24 1.a even 1 1 trivial
810.3.j.g 24 3.b odd 2 1 inner
810.3.j.g 24 45.h odd 6 1 inner
810.3.j.g 24 45.j even 6 1 inner
810.3.j.h 24 5.b even 2 1
810.3.j.h 24 9.c even 3 1
810.3.j.h 24 9.d odd 6 1
810.3.j.h 24 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{12} + 12 T_{7}^{11} - 66 T_{7}^{10} - 1368 T_{7}^{9} + 5598 T_{7}^{8} + 97800 T_{7}^{7} - 139180 T_{7}^{6} - 3380928 T_{7}^{5} + 8604420 T_{7}^{4} + 41415120 T_{7}^{3} + 34859232 T_{7}^{2} + \cdots + 8386816 \) Copy content Toggle raw display
\( T_{17}^{12} - 1464 T_{17}^{10} + 793137 T_{17}^{8} - 199784716 T_{17}^{6} + 23433607236 T_{17}^{4} - 1040615269800 T_{17}^{2} + 1460109722500 \) Copy content Toggle raw display