Properties

Label 462.2.u.b
Level $462$
Weight $2$
Character orbit 462.u
Analytic conductor $3.689$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.u (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32q + 8q^{4} + 10q^{5} - 8q^{6} - 10q^{7} + 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 8q^{4} + 10q^{5} - 8q^{6} - 10q^{7} + 8q^{9} + 4q^{10} + 8q^{11} - 2q^{14} + 6q^{15} - 8q^{16} - 12q^{17} - 16q^{19} + 10q^{20} - 8q^{21} - 4q^{22} + 8q^{23} + 8q^{24} + 6q^{25} + 10q^{28} + 20q^{29} - 50q^{31} - 16q^{33} - 12q^{35} - 8q^{36} - 16q^{37} + 6q^{40} + 40q^{41} + 12q^{44} + 52q^{49} + 40q^{51} - 32q^{54} - 40q^{55} - 8q^{56} + 10q^{58} + 60q^{59} + 4q^{60} - 4q^{61} + 20q^{62} - 10q^{63} + 8q^{64} + 8q^{66} - 16q^{67} + 12q^{68} + 30q^{69} - 28q^{70} - 48q^{71} - 74q^{73} - 40q^{74} - 24q^{76} + 6q^{77} - 60q^{79} - 8q^{81} + 20q^{82} + 4q^{83} - 2q^{84} - 10q^{85} - 36q^{86} + 20q^{87} - 16q^{88} - 4q^{90} - 20q^{91} - 8q^{92} - 10q^{93} - 20q^{95} - 8q^{96} + 60q^{97} - 40q^{98} - 8q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −0.951057 0.309017i −0.587785 0.809017i 0.809017 + 0.587785i −2.13481 + 0.693642i 0.309017 + 0.951057i 1.47539 2.19618i −0.587785 0.809017i −0.309017 + 0.951057i 2.24467
13.2 −0.951057 0.309017i −0.587785 0.809017i 0.809017 + 0.587785i −1.43844 + 0.467378i 0.309017 + 0.951057i −2.25182 1.38899i −0.587785 0.809017i −0.309017 + 0.951057i 1.51247
13.3 −0.951057 0.309017i −0.587785 0.809017i 0.809017 + 0.587785i 0.0789999 0.0256686i 0.309017 + 0.951057i −1.88075 + 1.86086i −0.587785 0.809017i −0.309017 + 0.951057i −0.0830654
13.4 −0.951057 0.309017i −0.587785 0.809017i 0.809017 + 0.587785i 3.76443 1.22314i 0.309017 + 0.951057i 2.55398 + 0.690781i −0.587785 0.809017i −0.309017 + 0.951057i −3.95815
13.5 0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 + 0.587785i −2.13310 + 0.693087i 0.309017 + 0.951057i −2.54873 + 0.709910i 0.587785 + 0.809017i −0.309017 + 0.951057i −2.24288
13.6 0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 + 0.587785i −0.169394 + 0.0550394i 0.309017 + 0.951057i 2.60497 0.462752i 0.587785 + 0.809017i −0.309017 + 0.951057i −0.178111
13.7 0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 + 0.587785i 2.46351 0.800443i 0.309017 + 0.951057i −0.0416492 + 2.64542i 0.587785 + 0.809017i −0.309017 + 0.951057i 2.59029
13.8 0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 + 0.587785i 3.18685 1.03547i 0.309017 + 0.951057i −1.29336 2.30808i 0.587785 + 0.809017i −0.309017 + 0.951057i 3.35085
139.1 −0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 0.951057i −1.12216 1.54452i −0.809017 + 0.587785i 1.92333 + 1.81681i 0.951057 + 0.309017i 0.809017 + 0.587785i 1.90913
139.2 −0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 0.951057i −0.946241 1.30239i −0.809017 + 0.587785i −2.49603 0.877400i 0.951057 + 0.309017i 0.809017 + 0.587785i 1.60984
139.3 −0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 0.951057i 0.884493 + 1.21740i −0.809017 + 0.587785i −2.53318 0.763555i 0.951057 + 0.309017i 0.809017 + 0.587785i −1.50479
139.4 −0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 0.951057i 2.23816 + 3.08056i −0.809017 + 0.587785i 0.345808 + 2.62305i 0.951057 + 0.309017i 0.809017 + 0.587785i −3.80779
139.5 0.587785 0.809017i −0.951057 0.309017i −0.309017 0.951057i −1.30266 1.79296i −0.809017 + 0.587785i 2.27066 1.35798i −0.951057 0.309017i 0.809017 + 0.587785i −2.21622
139.6 0.587785 0.809017i −0.951057 0.309017i −0.309017 0.951057i −1.08816 1.49773i −0.809017 + 0.587785i −2.64058 + 0.165270i −0.951057 0.309017i 0.809017 + 0.587785i −1.85129
139.7 0.587785 0.809017i −0.951057 0.309017i −0.309017 0.951057i 0.531600 + 0.731684i −0.809017 + 0.587785i 1.75879 + 1.97653i −0.951057 0.309017i 0.809017 + 0.587785i 0.904411
139.8 0.587785 0.809017i −0.951057 0.309017i −0.309017 0.951057i 2.18694 + 3.01006i −0.809017 + 0.587785i −2.24683 + 1.39706i −0.951057 0.309017i 0.809017 + 0.587785i 3.72064
349.1 −0.587785 0.809017i 0.951057 0.309017i −0.309017 + 0.951057i −1.12216 + 1.54452i −0.809017 0.587785i 1.92333 1.81681i 0.951057 0.309017i 0.809017 0.587785i 1.90913
349.2 −0.587785 0.809017i 0.951057 0.309017i −0.309017 + 0.951057i −0.946241 + 1.30239i −0.809017 0.587785i −2.49603 + 0.877400i 0.951057 0.309017i 0.809017 0.587785i 1.60984
349.3 −0.587785 0.809017i 0.951057 0.309017i −0.309017 + 0.951057i 0.884493 1.21740i −0.809017 0.587785i −2.53318 + 0.763555i 0.951057 0.309017i 0.809017 0.587785i −1.50479
349.4 −0.587785 0.809017i 0.951057 0.309017i −0.309017 + 0.951057i 2.23816 3.08056i −0.809017 0.587785i 0.345808 2.62305i 0.951057 0.309017i 0.809017 0.587785i −3.80779
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.l even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.u.b yes 32
7.b odd 2 1 462.2.u.a 32
11.d odd 10 1 462.2.u.a 32
77.l even 10 1 inner 462.2.u.b yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.u.a 32 7.b odd 2 1
462.2.u.a 32 11.d odd 10 1
462.2.u.b yes 32 1.a even 1 1 trivial
462.2.u.b yes 32 77.l even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{32} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).