Properties

Label 462.2.y.a
Level $462$
Weight $2$
Character orbit 462.y
Analytic conductor $3.689$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.y (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24q - 3q^{2} - 3q^{3} + 3q^{4} + 5q^{5} - 6q^{6} + 11q^{7} + 6q^{8} + 3q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q - 3q^{2} - 3q^{3} + 3q^{4} + 5q^{5} - 6q^{6} + 11q^{7} + 6q^{8} + 3q^{9} + 10q^{10} + 6q^{11} + 12q^{12} - 2q^{13} - 5q^{14} + 3q^{16} - 2q^{17} - 3q^{18} + 7q^{19} - 10q^{20} - 4q^{21} + 2q^{22} + 24q^{23} + 3q^{24} + 4q^{25} + 4q^{26} + 6q^{27} + 14q^{28} - 6q^{29} - 7q^{31} + 12q^{32} - q^{33} - 24q^{34} + 4q^{35} - 6q^{36} - q^{37} + 8q^{38} - q^{39} + 16q^{41} - 4q^{42} + 52q^{43} - 4q^{44} - 10q^{45} - 4q^{46} + 27q^{47} + 6q^{48} - 33q^{49} - 22q^{50} - 8q^{51} - 4q^{52} + 13q^{53} - 12q^{54} + 30q^{55} + 14q^{56} - 16q^{57} - 3q^{58} + 14q^{59} - 5q^{60} + 9q^{61} - 4q^{62} + 5q^{63} - 6q^{64} - 50q^{65} - 4q^{66} - 20q^{67} - 2q^{68} - 32q^{69} - 17q^{70} - 18q^{71} - 3q^{72} + 7q^{73} + q^{74} + 11q^{75} - 4q^{76} - 34q^{77} - 12q^{78} + q^{79} + 3q^{81} - 2q^{82} - 68q^{83} - 5q^{84} - 19q^{86} - 8q^{87} - q^{88} - 42q^{89} + 10q^{90} - 16q^{91} + 32q^{92} + 7q^{93} + 8q^{94} + 38q^{95} + 3q^{96} - 80q^{97} - 2q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.913545 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i −0.269783 + 2.56681i −0.809017 0.587785i −0.146873 + 2.64167i −0.309017 0.951057i 0.913545 + 0.406737i 1.29048 2.23517i
25.2 −0.913545 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i −0.126123 + 1.19998i −0.809017 0.587785i 0.836823 2.50993i −0.309017 0.951057i 0.913545 + 0.406737i 0.603294 1.04494i
25.3 −0.913545 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i 0.0177181 0.168577i −0.809017 0.587785i 2.46190 + 0.969054i −0.309017 0.951057i 0.913545 + 0.406737i −0.0847527 + 0.146796i
37.1 −0.913545 + 0.406737i 0.978148 0.207912i 0.669131 0.743145i −0.269783 2.56681i −0.809017 + 0.587785i −0.146873 2.64167i −0.309017 + 0.951057i 0.913545 0.406737i 1.29048 + 2.23517i
37.2 −0.913545 + 0.406737i 0.978148 0.207912i 0.669131 0.743145i −0.126123 1.19998i −0.809017 + 0.587785i 0.836823 + 2.50993i −0.309017 + 0.951057i 0.913545 0.406737i 0.603294 + 1.04494i
37.3 −0.913545 + 0.406737i 0.978148 0.207912i 0.669131 0.743145i 0.0177181 + 0.168577i −0.809017 + 0.587785i 2.46190 0.969054i −0.309017 + 0.951057i 0.913545 0.406737i −0.0847527 0.146796i
163.1 −0.669131 0.743145i −0.913545 0.406737i −0.104528 + 0.994522i −2.39103 0.508229i 0.309017 + 0.951057i −0.0798814 + 2.64455i 0.809017 0.587785i 0.669131 + 0.743145i 1.22222 + 2.11695i
163.2 −0.669131 0.743145i −0.913545 0.406737i −0.104528 + 0.994522i −1.47473 0.313464i 0.309017 + 0.951057i 2.56873 0.633736i 0.809017 0.587785i 0.669131 + 0.743145i 0.753838 + 1.30569i
163.3 −0.669131 0.743145i −0.913545 0.406737i −0.104528 + 0.994522i 2.51399 + 0.534365i 0.309017 + 0.951057i 1.73654 1.99611i 0.809017 0.587785i 0.669131 + 0.743145i −1.28508 2.22582i
235.1 0.104528 0.994522i −0.669131 0.743145i −0.978148 0.207912i −0.154851 0.0689440i −0.809017 + 0.587785i −2.56131 + 0.663085i −0.309017 + 0.951057i −0.104528 + 0.994522i −0.0847527 + 0.146796i
235.2 0.104528 0.994522i −0.669131 0.743145i −0.978148 0.207912i 1.10227 + 0.490763i −0.809017 + 0.587785i 0.798293 + 2.52244i −0.309017 + 0.951057i −0.104528 + 0.994522i 0.603294 1.04494i
235.3 0.104528 0.994522i −0.669131 0.743145i −0.978148 0.207912i 2.35782 + 1.04977i −0.809017 + 0.587785i −1.43391 2.22349i −0.309017 + 0.951057i −0.104528 + 0.994522i 1.29048 2.23517i
247.1 0.978148 + 0.207912i 0.104528 0.994522i 0.913545 + 0.406737i −1.71977 1.91000i 0.309017 0.951057i −1.36179 2.26838i 0.809017 + 0.587785i −0.978148 0.207912i −1.28508 2.22582i
247.2 0.978148 + 0.207912i 0.104528 0.994522i 0.913545 + 0.406737i 1.00883 + 1.12042i 0.309017 0.951057i 0.191063 2.63884i 0.809017 + 0.587785i −0.978148 0.207912i 0.753838 + 1.30569i
247.3 0.978148 + 0.207912i 0.104528 0.994522i 0.913545 + 0.406737i 1.63565 + 1.81658i 0.309017 0.951057i 2.49043 + 0.893181i 0.809017 + 0.587785i −0.978148 0.207912i 1.22222 + 2.11695i
289.1 0.104528 + 0.994522i −0.669131 + 0.743145i −0.978148 + 0.207912i −0.154851 + 0.0689440i −0.809017 0.587785i −2.56131 0.663085i −0.309017 0.951057i −0.104528 0.994522i −0.0847527 0.146796i
289.2 0.104528 + 0.994522i −0.669131 + 0.743145i −0.978148 + 0.207912i 1.10227 0.490763i −0.809017 0.587785i 0.798293 2.52244i −0.309017 0.951057i −0.104528 0.994522i 0.603294 + 1.04494i
289.3 0.104528 + 0.994522i −0.669131 + 0.743145i −0.978148 + 0.207912i 2.35782 1.04977i −0.809017 0.587785i −1.43391 + 2.22349i −0.309017 0.951057i −0.104528 0.994522i 1.29048 + 2.23517i
361.1 0.978148 0.207912i 0.104528 + 0.994522i 0.913545 0.406737i −1.71977 + 1.91000i 0.309017 + 0.951057i −1.36179 + 2.26838i 0.809017 0.587785i −0.978148 + 0.207912i −1.28508 + 2.22582i
361.2 0.978148 0.207912i 0.104528 + 0.994522i 0.913545 0.406737i 1.00883 1.12042i 0.309017 + 0.951057i 0.191063 + 2.63884i 0.809017 0.587785i −0.978148 + 0.207912i 0.753838 1.30569i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 445.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.y.a 24
7.c even 3 1 inner 462.2.y.a 24
11.c even 5 1 inner 462.2.y.a 24
77.m even 15 1 inner 462.2.y.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.y.a 24 1.a even 1 1 trivial
462.2.y.a 24 7.c even 3 1 inner
462.2.y.a 24 11.c even 5 1 inner
462.2.y.a 24 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{24} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).