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Properties

Label 230.5.d.a
Level $230$
Weight $5$
Character orbit 230.d
Analytic conductor $23.775$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 230.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.7750915093\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32q + 256q^{4} + 64q^{6} + 832q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 256q^{4} + 64q^{6} + 832q^{9} - 408q^{13} + 2048q^{16} + 1024q^{18} + 1332q^{23} + 512q^{24} - 4000q^{25} + 2208q^{27} + 3732q^{29} - 412q^{31} + 300q^{35} + 6656q^{36} - 9208q^{39} - 6156q^{41} + 4480q^{46} + 5184q^{47} - 13820q^{49} - 3264q^{52} - 3328q^{54} - 6000q^{55} + 3200q^{58} - 30468q^{59} - 10752q^{62} + 16384q^{64} - 1168q^{69} + 4800q^{70} - 37644q^{71} + 8192q^{72} + 19984q^{73} + 27528q^{77} + 15744q^{78} + 69056q^{81} - 17408q^{82} + 3300q^{85} + 28936q^{87} + 10656q^{92} + 40648q^{93} - 23808q^{94} - 21600q^{95} + 4096q^{96} + 43008q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −2.82843 −5.93906 8.00000 − 11.1803i 16.7982 − 85.3294i −22.6274 −45.7275 31.6228i
91.2 −2.82843 −11.3966 8.00000 11.1803i 32.2344 − 85.1794i −22.6274 48.8818 − 31.6228i
91.3 −2.82843 7.33271 8.00000 11.1803i −20.7400 − 75.6542i −22.6274 −27.2314 − 31.6228i
91.4 −2.82843 15.6893 8.00000 11.1803i −44.3761 65.6030i −22.6274 165.154 − 31.6228i
91.5 −2.82843 −5.71864 8.00000 − 11.1803i 16.1747 − 36.1607i −22.6274 −48.2972 31.6228i
91.6 −2.82843 −14.3217 8.00000 11.1803i 40.5079 16.7293i −22.6274 124.111 − 31.6228i
91.7 −2.82843 0.828614 8.00000 − 11.1803i −2.34367 16.9191i −22.6274 −80.3134 31.6228i
91.8 −2.82843 7.86848 8.00000 − 11.1803i −22.2554 − 5.16947i −22.6274 −19.0870 31.6228i
91.9 −2.82843 7.86848 8.00000 11.1803i −22.2554 5.16947i −22.6274 −19.0870 − 31.6228i
91.10 −2.82843 0.828614 8.00000 11.1803i −2.34367 − 16.9191i −22.6274 −80.3134 − 31.6228i
91.11 −2.82843 −14.3217 8.00000 − 11.1803i 40.5079 − 16.7293i −22.6274 124.111 31.6228i
91.12 −2.82843 −5.71864 8.00000 11.1803i 16.1747 36.1607i −22.6274 −48.2972 − 31.6228i
91.13 −2.82843 15.6893 8.00000 − 11.1803i −44.3761 − 65.6030i −22.6274 165.154 31.6228i
91.14 −2.82843 7.33271 8.00000 − 11.1803i −20.7400 75.6542i −22.6274 −27.2314 31.6228i
91.15 −2.82843 −11.3966 8.00000 − 11.1803i 32.2344 85.1794i −22.6274 48.8818 31.6228i
91.16 −2.82843 −5.93906 8.00000 11.1803i 16.7982 85.3294i −22.6274 −45.7275 − 31.6228i
91.17 2.82843 17.4598 8.00000 11.1803i 49.3837 − 69.3646i 22.6274 223.844 31.6228i
91.18 2.82843 −0.0560696 8.00000 − 11.1803i −0.158589 58.4359i 22.6274 −80.9969 − 31.6228i
91.19 2.82843 11.3074 8.00000 − 11.1803i 31.9821 − 52.5241i 22.6274 46.8567 − 31.6228i
91.20 2.82843 −2.20532 8.00000 − 11.1803i −6.23758 − 53.7736i 22.6274 −76.1366 − 31.6228i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.5.d.a ✓ 32
23.b odd 2 1 inner 230.5.d.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.5.d.a ✓ 32 1.a even 1 1 trivial
230.5.d.a ✓ 32 23.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(230, [\chi])\).