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Properties

Label 1620.3.t.f
Level $1620$
Weight $3$
Character orbit 1620.t
Analytic conductor $44.142$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{25} + 288 q^{49} - 72 q^{55} - 120 q^{61} - 480 q^{79} + 24 q^{85} + 96 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 0 0 −4.98895 − 0.332280i 0 −0.607949 − 0.351000i 0 0 0
269.2 0 0 0 −4.98514 − 0.385228i 0 −6.51924 − 3.76389i 0 0 0
269.3 0 0 0 −4.91882 − 0.897339i 0 10.9644 + 6.33031i 0 0 0
269.4 0 0 0 −4.58175 + 2.00188i 0 9.28893 + 5.36297i 0 0 0
269.5 0 0 0 −4.45514 − 2.26973i 0 −2.97789 − 1.71928i 0 0 0
269.6 0 0 0 −4.02456 + 2.96697i 0 −9.28893 − 5.36297i 0 0 0
269.7 0 0 0 −2.76464 − 4.16614i 0 −4.03229 − 2.32805i 0 0 0
269.8 0 0 0 −2.22566 − 4.47732i 0 4.03229 + 2.32805i 0 0 0
269.9 0 0 0 −2.20671 + 4.48669i 0 0.607949 + 0.351000i 0 0 0
269.10 0 0 0 −2.15895 + 4.50987i 0 6.51924 + 3.76389i 0 0 0
269.11 0 0 0 −1.68229 + 4.70849i 0 −10.9644 − 6.33031i 0 0 0
269.12 0 0 0 −0.261925 + 4.99313i 0 2.97789 + 1.71928i 0 0 0
269.13 0 0 0 0.261925 − 4.99313i 0 2.97789 + 1.71928i 0 0 0
269.14 0 0 0 1.68229 − 4.70849i 0 −10.9644 − 6.33031i 0 0 0
269.15 0 0 0 2.15895 − 4.50987i 0 6.51924 + 3.76389i 0 0 0
269.16 0 0 0 2.20671 − 4.48669i 0 0.607949 + 0.351000i 0 0 0
269.17 0 0 0 2.22566 + 4.47732i 0 4.03229 + 2.32805i 0 0 0
269.18 0 0 0 2.76464 + 4.16614i 0 −4.03229 − 2.32805i 0 0 0
269.19 0 0 0 4.02456 − 2.96697i 0 −9.28893 − 5.36297i 0 0 0
269.20 0 0 0 4.45514 + 2.26973i 0 −2.97789 − 1.71928i 0 0 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1349.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.d 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 1620.3.t.f 48
3.b odd 2 1 inner 1620.3.t.f 48
5.b even 2 1 inner 1620.3.t.f 48
9.c even 3 1 1620.3.b.a ✓ 24
9.c even 3 1 inner 1620.3.t.f 48
9.d odd 6 1 1620.3.b.a ✓ 24
9.d odd 6 1 inner 1620.3.t.f 48
15.d odd 2 1 inner 1620.3.t.f 48
45.h odd 6 1 1620.3.b.a ✓ 24
45.h odd 6 1 inner 1620.3.t.f 48
45.j even 6 1 1620.3.b.a ✓ 24
45.j even 6 1 inner 1620.3.t.f 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.3.b.a ✓ 24 9.c even 3 1
1620.3.b.a ✓ 24 9.d odd 6 1
1620.3.b.a ✓ 24 45.h odd 6 1
1620.3.b.a ✓ 24 45.j even 6 1
1620.3.t.f 48 1.a even 1 1 trivial
1620.3.t.f 48 3.b odd 2 1 inner
1620.3.t.f 48 5.b even 2 1 inner
1620.3.t.f 48 9.c even 3 1 inner
1620.3.t.f 48 9.d odd 6 1 inner
1620.3.t.f 48 15.d odd 2 1 inner
1620.3.t.f 48 45.h odd 6 1 inner
1620.3.t.f 48 45.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{24} - 366 T_{7}^{22} + 88353 T_{7}^{20} - 12104630 T_{7}^{18} + 1195527555 T_{7}^{16} - 72024338430 T_{7}^{14} + 3106792494870 T_{7}^{12} - 76507326662148 T_{7}^{10} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
\( T_{17}^{12} - 1662 T_{17}^{10} + 827391 T_{17}^{8} - 157133482 T_{17}^{6} + 10180375374 T_{17}^{4} - 42366144912 T_{17}^{2} + 41987187424 \) Copy content Toggle raw display