Properties

Label 1620.3.o.g
Level $1620$
Weight $3$
Character orbit 1620.o
Analytic conductor $44.142$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 8 q^{7} + 40 q^{13} + 112 q^{19} + 80 q^{25} + 64 q^{31} - 176 q^{37} - 128 q^{43} - 216 q^{49} - 8 q^{61} + 40 q^{67} + 112 q^{73} + 136 q^{79} - 784 q^{91} - 128 q^{97}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1 0 0 0 −1.93649 1.11803i 0 −2.26486 3.92285i 0 0 0
701.2 0 0 0 −1.93649 1.11803i 0 5.78429 + 10.0187i 0 0 0
701.3 0 0 0 −1.93649 1.11803i 0 −0.973755 1.68659i 0 0 0
701.4 0 0 0 −1.93649 1.11803i 0 3.57283 + 6.18832i 0 0 0
701.5 0 0 0 −1.93649 1.11803i 0 −3.54568 6.14130i 0 0 0
701.6 0 0 0 −1.93649 1.11803i 0 −6.81144 11.7978i 0 0 0
701.7 0 0 0 −1.93649 1.11803i 0 3.48846 + 6.04219i 0 0 0
701.8 0 0 0 −1.93649 1.11803i 0 −1.24985 2.16480i 0 0 0
701.9 0 0 0 1.93649 + 1.11803i 0 3.48846 + 6.04219i 0 0 0
701.10 0 0 0 1.93649 + 1.11803i 0 −1.24985 2.16480i 0 0 0
701.11 0 0 0 1.93649 + 1.11803i 0 −6.81144 11.7978i 0 0 0
701.12 0 0 0 1.93649 + 1.11803i 0 3.57283 + 6.18832i 0 0 0
701.13 0 0 0 1.93649 + 1.11803i 0 −3.54568 6.14130i 0 0 0
701.14 0 0 0 1.93649 + 1.11803i 0 5.78429 + 10.0187i 0 0 0
701.15 0 0 0 1.93649 + 1.11803i 0 −0.973755 1.68659i 0 0 0
701.16 0 0 0 1.93649 + 1.11803i 0 −2.26486 3.92285i 0 0 0
1241.1 0 0 0 −1.93649 + 1.11803i 0 −2.26486 + 3.92285i 0 0 0
1241.2 0 0 0 −1.93649 + 1.11803i 0 5.78429 10.0187i 0 0 0
1241.3 0 0 0 −1.93649 + 1.11803i 0 −0.973755 + 1.68659i 0 0 0
1241.4 0 0 0 −1.93649 + 1.11803i 0 3.57283 6.18832i 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1241.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.3.o.g 32
3.b odd 2 1 inner 1620.3.o.g 32
9.c even 3 1 1620.3.g.c 16
9.c even 3 1 inner 1620.3.o.g 32
9.d odd 6 1 1620.3.g.c 16
9.d odd 6 1 inner 1620.3.o.g 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.3.g.c 16 9.c even 3 1
1620.3.g.c 16 9.d odd 6 1
1620.3.o.g 32 1.a even 1 1 trivial
1620.3.o.g 32 3.b odd 2 1 inner
1620.3.o.g 32 9.c even 3 1 inner
1620.3.o.g 32 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 4 T_{7}^{15} + 258 T_{7}^{14} + 512 T_{7}^{13} + 46349 T_{7}^{12} + 108708 T_{7}^{11} + 3508258 T_{7}^{10} + 11250064 T_{7}^{9} + 198230175 T_{7}^{8} + 684910540 T_{7}^{7} + \cdots + 1509541505956 \) acting on \(S_{3}^{\mathrm{new}}(1620, [\chi])\). Copy content Toggle raw display