Properties

Label 483.2.u.a
Level $483$
Weight $2$
Character orbit 483.u
Analytic conductor $3.857$
Analytic rank $0$
Dimension $480$
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.u (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(48\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480q + 4q^{3} + 40q^{4} - 6q^{6} - 4q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 480q + 4q^{3} + 40q^{4} - 6q^{6} - 4q^{9} - 22q^{12} + 8q^{13} - 22q^{15} - 24q^{16} - 30q^{18} - 120q^{24} - 88q^{25} + 16q^{27} - 44q^{30} + 8q^{31} - 22q^{33} - 44q^{34} + 10q^{36} - 44q^{37} - 308q^{40} - 44q^{43} - 184q^{46} + 98q^{48} + 48q^{49} - 28q^{52} + 28q^{54} - 44q^{55} + 66q^{57} + 4q^{58} + 220q^{60} + 84q^{64} + 176q^{66} + 44q^{67} + 102q^{69} - 8q^{70} - 60q^{72} + 4q^{73} - 8q^{75} + 176q^{76} + 18q^{78} - 16q^{81} + 20q^{82} - 154q^{84} + 84q^{85} + 28q^{87} - 418q^{90} - 188q^{93} + 12q^{94} - 412q^{96} - 132q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 −1.50106 + 2.33570i 0.521192 1.65177i −2.37146 5.19277i −1.06257 0.311999i 3.07570 + 3.69676i 0.755750 + 0.654861i 10.1921 + 1.46540i −2.45672 1.72178i 2.32371 2.01351i
113.2 −1.48162 + 2.30545i −0.121691 + 1.72777i −2.28907 5.01236i −4.08876 1.20057i −3.80299 2.84046i −0.755750 0.654861i 9.52207 + 1.36907i −2.97038 0.420508i 8.82585 7.64764i
113.3 −1.39817 + 2.17560i 1.72494 + 0.156831i −1.94751 4.26445i 1.23964 + 0.363993i −2.75296 + 3.53349i −0.755750 0.654861i 6.88105 + 0.989346i 2.95081 + 0.541047i −2.52514 + 2.18804i
113.4 −1.39646 + 2.17293i −1.51591 + 0.837856i −1.94070 4.24954i 3.72219 + 1.09293i 0.296307 4.46400i −0.755750 0.654861i 6.83069 + 0.982105i 1.59599 2.54024i −7.57275 + 6.56182i
113.5 −1.33937 + 2.08410i −1.69763 0.343591i −1.71873 3.76350i −1.11280 0.326747i 2.98983 3.07784i 0.755750 + 0.654861i 5.24122 + 0.753574i 2.76389 + 1.16658i 2.17142 1.88155i
113.6 −1.30013 + 2.02305i −0.657657 1.60234i −1.57155 3.44121i 1.14892 + 0.337353i 4.09665 + 0.752782i −0.755750 0.654861i 4.24430 + 0.610238i −2.13497 + 2.10758i −2.17623 + 1.88572i
113.7 −1.20683 + 1.87786i 1.44817 + 0.950153i −1.23910 2.71324i −0.604163 0.177398i −3.53195 + 1.57280i 0.755750 + 0.654861i 2.17147 + 0.312210i 1.19442 + 2.75197i 1.06225 0.920445i
113.8 −1.18675 + 1.84662i −1.11165 + 1.32824i −1.17080 2.56370i −0.291860 0.0856978i −1.13351 3.62910i 0.755750 + 0.654861i 1.77815 + 0.255660i −0.528462 2.95309i 0.504617 0.437253i
113.9 −1.07812 + 1.67759i 1.58707 0.693695i −0.821124 1.79801i −1.95772 0.574839i −0.547316 + 3.41033i −0.755750 0.654861i −0.0461173 0.00663067i 2.03757 2.20188i 3.07500 2.66450i
113.10 −1.07128 + 1.66695i −1.25731 1.19129i −0.800245 1.75229i 3.65018 + 1.07179i 3.33276 0.819664i 0.755750 + 0.654861i −0.144404 0.0207621i 0.161658 + 2.99564i −5.69700 + 4.93648i
113.11 −1.06610 + 1.65888i −0.893651 1.48371i −0.784489 1.71779i −3.45226 1.01367i 3.41401 + 0.0993161i −0.755750 0.654861i −0.217736 0.0313058i −1.40277 + 2.65183i 5.36201 4.64621i
113.12 −0.908799 + 1.41412i 1.31791 1.12388i −0.342987 0.751037i −2.80823 0.824571i 0.391583 + 2.88507i 0.755750 + 0.654861i −1.95395 0.280936i 0.473786 2.96235i 3.71816 3.22180i
113.13 −0.853002 + 1.32730i −1.46051 + 0.931084i −0.203275 0.445110i −1.12254 0.329608i 0.00999087 2.73274i −0.755750 0.654861i −2.35922 0.339204i 1.26617 2.71971i 1.39502 1.20879i
113.14 −0.732726 + 1.14014i 1.28310 + 1.16347i 0.0677890 + 0.148437i 3.54035 + 1.03954i −2.26668 + 0.610418i −0.755750 0.654861i −2.90190 0.417231i 0.292696 + 2.98569i −3.77934 + 3.27481i
113.15 −0.645581 + 1.00454i −0.549837 + 1.64246i 0.238496 + 0.522234i 3.49538 + 1.02634i −1.29496 1.61268i 0.755750 + 0.654861i −3.04248 0.437442i −2.39536 1.80617i −3.28755 + 2.84868i
113.16 −0.621877 + 0.967660i −1.59011 0.686701i 0.281195 + 0.615732i 0.670051 + 0.196745i 1.65334 1.11164i −0.755750 0.654861i −3.04779 0.438206i 2.05688 + 2.18386i −0.607071 + 0.526030i
113.17 −0.552629 + 0.859908i 1.69926 + 0.335441i 0.396788 + 0.868844i 1.96993 + 0.578425i −1.22751 + 1.27583i 0.755750 + 0.654861i −2.98994 0.429889i 2.77496 + 1.14000i −1.58604 + 1.37431i
113.18 −0.542374 + 0.843951i 0.351081 + 1.69610i 0.412747 + 0.903790i −2.37324 0.696847i −1.62184 0.623624i 0.755750 + 0.654861i −2.97261 0.427396i −2.75348 + 1.19093i 1.87529 1.62495i
113.19 −0.497545 + 0.774196i −0.368438 1.69241i 0.479003 + 1.04887i −0.349963 0.102758i 1.49357 + 0.556808i 0.755750 + 0.654861i −2.87220 0.412960i −2.72851 + 1.24710i 0.253678 0.219813i
113.20 −0.437059 + 0.680077i 1.47311 0.911006i 0.559346 + 1.22480i 2.18862 + 0.642637i −0.0242835 + 1.39999i −0.755750 0.654861i −2.67778 0.385007i 1.34013 2.68403i −1.39360 + 1.20756i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 470.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.d odd 22 1 inner
69.g even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.u.a 480
3.b odd 2 1 inner 483.2.u.a 480
23.d odd 22 1 inner 483.2.u.a 480
69.g even 22 1 inner 483.2.u.a 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.u.a 480 1.a even 1 1 trivial
483.2.u.a 480 3.b odd 2 1 inner
483.2.u.a 480 23.d odd 22 1 inner
483.2.u.a 480 69.g even 22 1 inner

Hecke kernels

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