Properties

Label 273.2.g.a
Level $273$
Weight $2$
Character orbit 273.g
Analytic conductor $2.180$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32q + 16q^{4} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 16q^{4} - 16q^{16} - 16q^{25} + 16q^{30} - 32q^{36} - 48q^{42} + 48q^{43} - 32q^{49} - 16q^{51} - 80q^{64} + 32q^{78} + 80q^{79} - 48q^{81} - 96q^{88} + 32q^{91} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
272.1 −2.35085 −0.813690 1.52902i 3.52651 0.514868i 1.91287 + 3.59451i −1.49028 + 2.18611i −3.58861 −1.67582 + 2.48830i 1.21038i
272.2 −2.35085 −0.813690 + 1.52902i 3.52651 0.514868i 1.91287 3.59451i −1.49028 2.18611i −3.58861 −1.67582 2.48830i 1.21038i
272.3 −2.35085 0.813690 1.52902i 3.52651 0.514868i −1.91287 + 3.59451i 1.49028 2.18611i −3.58861 −1.67582 2.48830i 1.21038i
272.4 −2.35085 0.813690 + 1.52902i 3.52651 0.514868i −1.91287 3.59451i 1.49028 + 2.18611i −3.58861 −1.67582 + 2.48830i 1.21038i
272.5 −1.77583 −1.65032 0.525794i 1.15356 3.85624i 2.93067 + 0.933719i −1.56982 2.12971i 1.50313 2.44708 + 1.73545i 6.84801i
272.6 −1.77583 −1.65032 + 0.525794i 1.15356 3.85624i 2.93067 0.933719i −1.56982 + 2.12971i 1.50313 2.44708 1.73545i 6.84801i
272.7 −1.77583 1.65032 0.525794i 1.15356 3.85624i −2.93067 + 0.933719i 1.56982 + 2.12971i 1.50313 2.44708 1.73545i 6.84801i
272.8 −1.77583 1.65032 + 0.525794i 1.15356 3.85624i −2.93067 0.933719i 1.56982 2.12971i 1.50313 2.44708 + 1.73545i 6.84801i
272.9 −1.10741 −0.669305 1.59751i −0.773652 1.62666i 0.741192 + 1.76909i 2.53337 + 0.762910i 3.07156 −2.10406 + 2.13844i 1.80138i
272.10 −1.10741 −0.669305 + 1.59751i −0.773652 1.62666i 0.741192 1.76909i 2.53337 0.762910i 3.07156 −2.10406 2.13844i 1.80138i
272.11 −1.10741 0.669305 1.59751i −0.773652 1.62666i −0.741192 + 1.76909i −2.53337 0.762910i 3.07156 −2.10406 2.13844i 1.80138i
272.12 −1.10741 0.669305 + 1.59751i −0.773652 1.62666i −0.741192 1.76909i −2.53337 + 0.762910i 3.07156 −2.10406 + 2.13844i 1.80138i
272.13 −0.305902 −1.47187 0.913018i −1.90642 2.05385i 0.450247 + 0.279294i −0.946976 2.47047i 1.19498 1.33280 + 2.68769i 0.628276i
272.14 −0.305902 −1.47187 + 0.913018i −1.90642 2.05385i 0.450247 0.279294i −0.946976 + 2.47047i 1.19498 1.33280 2.68769i 0.628276i
272.15 −0.305902 1.47187 0.913018i −1.90642 2.05385i −0.450247 + 0.279294i 0.946976 + 2.47047i 1.19498 1.33280 2.68769i 0.628276i
272.16 −0.305902 1.47187 + 0.913018i −1.90642 2.05385i −0.450247 0.279294i 0.946976 2.47047i 1.19498 1.33280 + 2.68769i 0.628276i
272.17 0.305902 −1.47187 0.913018i −1.90642 2.05385i −0.450247 0.279294i 0.946976 + 2.47047i −1.19498 1.33280 + 2.68769i 0.628276i
272.18 0.305902 −1.47187 + 0.913018i −1.90642 2.05385i −0.450247 + 0.279294i 0.946976 2.47047i −1.19498 1.33280 2.68769i 0.628276i
272.19 0.305902 1.47187 0.913018i −1.90642 2.05385i 0.450247 0.279294i −0.946976 2.47047i −1.19498 1.33280 2.68769i 0.628276i
272.20 0.305902 1.47187 + 0.913018i −1.90642 2.05385i 0.450247 + 0.279294i −0.946976 + 2.47047i −1.19498 1.33280 + 2.68769i 0.628276i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 272.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
13.b even 2 1 inner
21.c even 2 1 inner
39.d odd 2 1 inner
91.b odd 2 1 inner
273.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.g.a 32
3.b odd 2 1 inner 273.2.g.a 32
7.b odd 2 1 inner 273.2.g.a 32
13.b even 2 1 inner 273.2.g.a 32
21.c even 2 1 inner 273.2.g.a 32
39.d odd 2 1 inner 273.2.g.a 32
91.b odd 2 1 inner 273.2.g.a 32
273.g even 2 1 inner 273.2.g.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.g.a 32 1.a even 1 1 trivial
273.2.g.a 32 3.b odd 2 1 inner
273.2.g.a 32 7.b odd 2 1 inner
273.2.g.a 32 13.b even 2 1 inner
273.2.g.a 32 21.c even 2 1 inner
273.2.g.a 32 39.d odd 2 1 inner
273.2.g.a 32 91.b odd 2 1 inner
273.2.g.a 32 273.g even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(273, [\chi])\).