Properties

Label 483.2.j.a
Level $483$
Weight $2$
Character orbit 483.j
Analytic conductor $3.857$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64q + 4q^{2} - 36q^{4} - 24q^{8} + 32q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 64q + 4q^{2} - 36q^{4} - 24q^{8} + 32q^{9} - 44q^{16} - 4q^{18} - 16q^{23} - 12q^{25} + 84q^{26} + 24q^{29} - 12q^{31} + 8q^{32} + 32q^{35} - 72q^{36} - 8q^{46} - 12q^{47} + 8q^{49} - 96q^{50} - 108q^{52} - 24q^{58} + 36q^{59} + 168q^{64} - 40q^{70} - 16q^{71} - 12q^{72} + 48q^{73} - 48q^{75} - 32q^{78} - 32q^{81} - 24q^{82} + 88q^{85} + 36q^{87} + 152q^{92} + 24q^{93} - 108q^{94} - 44q^{95} + 60q^{96} + 112q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1 −1.34482 + 2.32929i −0.866025 + 0.500000i −2.61707 4.53290i −0.428287 + 0.741815i 2.68964i −2.40961 + 1.09260i 8.69868 0.500000 0.866025i −1.15194 1.99521i
229.2 −1.34482 + 2.32929i −0.866025 + 0.500000i −2.61707 4.53290i 0.428287 0.741815i 2.68964i 2.40961 1.09260i 8.69868 0.500000 0.866025i 1.15194 + 1.99521i
229.3 −1.20973 + 2.09532i 0.866025 0.500000i −1.92691 3.33751i −1.20224 + 2.08234i 2.41947i 0.190002 2.63892i 4.48526 0.500000 0.866025i −2.90878 5.03816i
229.4 −1.20973 + 2.09532i 0.866025 0.500000i −1.92691 3.33751i 1.20224 2.08234i 2.41947i −0.190002 + 2.63892i 4.48526 0.500000 0.866025i 2.90878 + 5.03816i
229.5 −0.987485 + 1.71037i 0.866025 0.500000i −0.950252 1.64589i −0.274706 + 0.475805i 1.97497i 2.33704 1.24025i −0.196501 0.500000 0.866025i −0.542536 0.939701i
229.6 −0.987485 + 1.71037i 0.866025 0.500000i −0.950252 1.64589i 0.274706 0.475805i 1.97497i −2.33704 + 1.24025i −0.196501 0.500000 0.866025i 0.542536 + 0.939701i
229.7 −0.708878 + 1.22781i −0.866025 + 0.500000i −0.00501609 0.00868813i −1.44242 + 2.49834i 1.41776i −0.0745936 2.64470i −2.82129 0.500000 0.866025i −2.04500 3.54204i
229.8 −0.708878 + 1.22781i −0.866025 + 0.500000i −0.00501609 0.00868813i 1.44242 2.49834i 1.41776i 0.0745936 + 2.64470i −2.82129 0.500000 0.866025i 2.04500 + 3.54204i
229.9 −0.676219 + 1.17124i −0.866025 + 0.500000i 0.0854570 + 0.148016i −1.08513 + 1.87949i 1.35244i −2.39455 + 1.12522i −2.93602 0.500000 0.866025i −1.46757 2.54190i
229.10 −0.676219 + 1.17124i −0.866025 + 0.500000i 0.0854570 + 0.148016i 1.08513 1.87949i 1.35244i 2.39455 1.12522i −2.93602 0.500000 0.866025i 1.46757 + 2.54190i
229.11 −0.585633 + 1.01435i 0.866025 0.500000i 0.314069 + 0.543983i −1.31255 + 2.27340i 1.17127i 1.98677 + 1.74721i −3.07825 0.500000 0.866025i −1.53734 2.66275i
229.12 −0.585633 + 1.01435i 0.866025 0.500000i 0.314069 + 0.543983i 1.31255 2.27340i 1.17127i −1.98677 1.74721i −3.07825 0.500000 0.866025i 1.53734 + 2.66275i
229.13 −0.123216 + 0.213416i 0.866025 0.500000i 0.969636 + 1.67946i −1.99266 + 3.45140i 0.246432i −1.63265 2.08193i −0.970763 0.500000 0.866025i −0.491056 0.850535i
229.14 −0.123216 + 0.213416i 0.866025 0.500000i 0.969636 + 1.67946i 1.99266 3.45140i 0.246432i 1.63265 + 2.08193i −0.970763 0.500000 0.866025i 0.491056 + 0.850535i
229.15 −0.0934478 + 0.161856i −0.866025 + 0.500000i 0.982535 + 1.70180i −0.724626 + 1.25509i 0.186896i 2.60010 0.489382i −0.741054 0.500000 0.866025i −0.135429 0.234571i
229.16 −0.0934478 + 0.161856i −0.866025 + 0.500000i 0.982535 + 1.70180i 0.724626 1.25509i 0.186896i −2.60010 + 0.489382i −0.741054 0.500000 0.866025i 0.135429 + 0.234571i
229.17 0.208136 0.360502i −0.866025 + 0.500000i 0.913359 + 1.58198i −0.648045 + 1.12245i 0.416272i 0.918903 + 2.48105i 1.59295 0.500000 0.866025i 0.269763 + 0.467243i
229.18 0.208136 0.360502i −0.866025 + 0.500000i 0.913359 + 1.58198i 0.648045 1.12245i 0.416272i −0.918903 2.48105i 1.59295 0.500000 0.866025i −0.269763 0.467243i
229.19 0.287128 0.497320i 0.866025 0.500000i 0.835115 + 1.44646i −0.504635 + 0.874053i 0.574256i −1.18416 + 2.36596i 2.10765 0.500000 0.866025i 0.289790 + 0.501930i
229.20 0.287128 0.497320i 0.866025 0.500000i 0.835115 + 1.44646i 0.504635 0.874053i 0.574256i 1.18416 2.36596i 2.10765 0.500000 0.866025i −0.289790 0.501930i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 367.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.b odd 2 1 inner
161.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.j.a 64
7.d odd 6 1 inner 483.2.j.a 64
23.b odd 2 1 inner 483.2.j.a 64
161.g even 6 1 inner 483.2.j.a 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.j.a 64 1.a even 1 1 trivial
483.2.j.a 64 7.d odd 6 1 inner
483.2.j.a 64 23.b odd 2 1 inner
483.2.j.a 64 161.g even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(483, [\chi])\).