Properties

Label 1120.3.c.g
Level $1120$
Weight $3$
Character orbit 1120.c
Analytic conductor $30.518$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1120.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.5177896084\)
Analytic rank: \(0\)
Dimension: \(80\)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80q - 224q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 80q - 224q^{9} + 72q^{15} - 104q^{25} + 112q^{39} + 192q^{49} + 472q^{65} - 800q^{71} - 480q^{79} - 896q^{81} - 1176q^{95} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1 0 3.68915i 0 −0.693644 4.95165i 0 4.51720 5.34742i 0 −4.60986 0
209.2 0 3.68915i 0 −0.693644 + 4.95165i 0 4.51720 + 5.34742i 0 −4.60986 0
209.3 0 1.25300i 0 3.15168 3.88161i 0 −5.63086 + 4.15854i 0 7.43000 0
209.4 0 1.25300i 0 3.15168 + 3.88161i 0 −5.63086 4.15854i 0 7.43000 0
209.5 0 3.68915i 0 0.693644 4.95165i 0 −4.51720 + 5.34742i 0 −4.60986 0
209.6 0 3.68915i 0 0.693644 + 4.95165i 0 −4.51720 5.34742i 0 −4.60986 0
209.7 0 3.23212i 0 −3.82711 3.21764i 0 6.97525 + 0.588096i 0 −1.44659 0
209.8 0 3.23212i 0 −3.82711 + 3.21764i 0 6.97525 0.588096i 0 −1.44659 0
209.9 0 3.23212i 0 3.82711 3.21764i 0 6.97525 0.588096i 0 −1.44659 0
209.10 0 3.23212i 0 3.82711 + 3.21764i 0 6.97525 + 0.588096i 0 −1.44659 0
209.11 0 2.74983i 0 −4.72637 + 1.63138i 0 1.41172 6.85617i 0 1.43843 0
209.12 0 2.74983i 0 −4.72637 1.63138i 0 1.41172 + 6.85617i 0 1.43843 0
209.13 0 3.68915i 0 0.693644 4.95165i 0 4.51720 + 5.34742i 0 −4.60986 0
209.14 0 3.68915i 0 0.693644 + 4.95165i 0 4.51720 5.34742i 0 −4.60986 0
209.15 0 2.74983i 0 4.72637 + 1.63138i 0 −1.41172 + 6.85617i 0 1.43843 0
209.16 0 2.74983i 0 4.72637 1.63138i 0 −1.41172 6.85617i 0 1.43843 0
209.17 0 1.25300i 0 3.15168 3.88161i 0 5.63086 + 4.15854i 0 7.43000 0
209.18 0 1.25300i 0 3.15168 + 3.88161i 0 5.63086 4.15854i 0 7.43000 0
209.19 0 5.26463i 0 4.99643 0.188883i 0 6.12587 + 3.38729i 0 −18.7163 0
209.20 0 5.26463i 0 4.99643 + 0.188883i 0 6.12587 3.38729i 0 −18.7163 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
35.c odd 2 1 inner
40.f even 2 1 inner
56.h odd 2 1 inner
280.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.3.c.g 80
4.b odd 2 1 280.3.c.g 80
5.b even 2 1 inner 1120.3.c.g 80
7.b odd 2 1 inner 1120.3.c.g 80
8.b even 2 1 inner 1120.3.c.g 80
8.d odd 2 1 280.3.c.g 80
20.d odd 2 1 280.3.c.g 80
28.d even 2 1 280.3.c.g 80
35.c odd 2 1 inner 1120.3.c.g 80
40.e odd 2 1 280.3.c.g 80
40.f even 2 1 inner 1120.3.c.g 80
56.e even 2 1 280.3.c.g 80
56.h odd 2 1 inner 1120.3.c.g 80
140.c even 2 1 280.3.c.g 80
280.c odd 2 1 inner 1120.3.c.g 80
280.n even 2 1 280.3.c.g 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.3.c.g 80 4.b odd 2 1
280.3.c.g 80 8.d odd 2 1
280.3.c.g 80 20.d odd 2 1
280.3.c.g 80 28.d even 2 1
280.3.c.g 80 40.e odd 2 1
280.3.c.g 80 56.e even 2 1
280.3.c.g 80 140.c even 2 1
280.3.c.g 80 280.n even 2 1
1120.3.c.g 80 1.a even 1 1 trivial
1120.3.c.g 80 5.b even 2 1 inner
1120.3.c.g 80 7.b odd 2 1 inner
1120.3.c.g 80 8.b even 2 1 inner
1120.3.c.g 80 35.c odd 2 1 inner
1120.3.c.g 80 40.f even 2 1 inner
1120.3.c.g 80 56.h odd 2 1 inner
1120.3.c.g 80 280.c odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1120, [\chi])\):

\(T_{3}^{20} + \cdots\)
\(19\!\cdots\!29\)\( T_{17}^{12} - \)\(64\!\cdots\!40\)\( T_{17}^{10} + \)\(13\!\cdots\!28\)\( T_{17}^{8} - \)\(17\!\cdots\!20\)\( T_{17}^{6} + \)\(12\!\cdots\!84\)\( T_{17}^{4} - \)\(36\!\cdots\!40\)\( T_{17}^{2} + \)\(11\!\cdots\!00\)\( \)">\(T_{17}^{20} - \cdots\)
\(11\!\cdots\!52\)\( T_{19}^{12} - \)\(25\!\cdots\!04\)\( T_{19}^{10} + \)\(32\!\cdots\!20\)\( T_{19}^{8} - \)\(22\!\cdots\!24\)\( T_{19}^{6} + \)\(85\!\cdots\!08\)\( T_{19}^{4} - \)\(16\!\cdots\!12\)\( T_{19}^{2} + \)\(10\!\cdots\!80\)\( \)">\(T_{19}^{20} - \cdots\)