Properties

Label 462.2.y.d
Level $462$
Weight $2$
Character orbit 462.y
Analytic conductor $3.689$
Analytic rank $0$
Dimension $40$
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: \(40\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40q + 5q^{2} - 5q^{3} + 5q^{4} - 3q^{5} + 10q^{6} - 7q^{7} - 10q^{8} + 5q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q + 5q^{2} - 5q^{3} + 5q^{4} - 3q^{5} + 10q^{6} - 7q^{7} - 10q^{8} + 5q^{9} + 12q^{10} + 2q^{11} + 20q^{12} + 10q^{13} + 5q^{14} - 6q^{15} + 5q^{16} + 5q^{18} - 13q^{19} + 6q^{20} - 2q^{21} - 4q^{22} - 20q^{23} - 5q^{24} + 12q^{25} + 10q^{27} - 3q^{28} + 12q^{29} + 3q^{30} - 10q^{31} - 20q^{32} + 8q^{33} - 13q^{35} - 10q^{36} + 3q^{37} + 2q^{38} + 5q^{39} - 3q^{40} - 44q^{41} - 3q^{42} + 36q^{43} - 3q^{44} + 12q^{45} + 11q^{47} + 10q^{48} + 33q^{49} - 4q^{50} + 26q^{53} + 20q^{54} - 20q^{55} + 8q^{56} + 4q^{57} - 6q^{58} + 4q^{59} + 3q^{60} - 23q^{61} + 5q^{63} - 10q^{64} - 50q^{65} - 7q^{66} - 108q^{67} + 20q^{69} - 86q^{71} + 5q^{72} - 35q^{73} + 3q^{74} - 2q^{75} - 44q^{76} - 37q^{77} + 20q^{78} + 3q^{79} - 3q^{80} + 5q^{81} - 28q^{82} - 88q^{83} - 5q^{84} + 96q^{85} - 13q^{86} + 6q^{87} - 8q^{88} + 6q^{89} + 6q^{90} + 40q^{91} - 20q^{92} + 10q^{93} - 24q^{94} + 36q^{95} - 5q^{96} + 60q^{97} + 2q^{98} - 4q^{99} + O(q^{100}) \)

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 −0.305851 + 2.90998i 0.809017 + 0.587785i −2.53596 0.754243i 0.309017 + 0.951057i 0.913545 + 0.406737i −1.46300 + 2.53399i
25.2 0.913545 + 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i −0.0415637 + 0.395452i 0.809017 + 0.587785i −0.168521 + 2.64038i 0.309017 + 0.951057i 0.913545 + 0.406737i −0.198815 + 0.344358i
25.3 0.913545 + 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i 0.0926977 0.881959i 0.809017 + 0.587785i 1.77880 1.95854i 0.309017 + 0.951057i 0.913545 + 0.406737i 0.443409 0.768006i
25.4 0.913545 + 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i 0.111151 1.05753i 0.809017 + 0.587785i 2.10806 + 1.59878i 0.309017 + 0.951057i 0.913545 + 0.406737i 0.531680 0.920896i
25.5 0.913545 + 0.406737i 0.978148 + 0.207912i 0.669131 + 0.743145i 0.457151 4.34950i 0.809017 + 0.587785i −2.04249 1.68174i 0.309017 + 0.951057i 0.913545 + 0.406737i 2.18673 3.78753i
37.1 0.913545 0.406737i 0.978148 0.207912i 0.669131 0.743145i −0.305851 2.90998i 0.809017 0.587785i −2.53596 + 0.754243i 0.309017 0.951057i 0.913545 0.406737i −1.46300 2.53399i
37.2 0.913545 0.406737i 0.978148 0.207912i 0.669131 0.743145i −0.0415637 0.395452i 0.809017 0.587785i −0.168521 2.64038i 0.309017 0.951057i 0.913545 0.406737i −0.198815 0.344358i
37.3 0.913545 0.406737i 0.978148 0.207912i 0.669131 0.743145i 0.0926977 + 0.881959i 0.809017 0.587785i 1.77880 + 1.95854i 0.309017 0.951057i 0.913545 0.406737i 0.443409 + 0.768006i
37.4 0.913545 0.406737i 0.978148 0.207912i 0.669131 0.743145i 0.111151 + 1.05753i 0.809017 0.587785i 2.10806 1.59878i 0.309017 0.951057i 0.913545 0.406737i 0.531680 + 0.920896i
37.5 0.913545 0.406737i 0.978148 0.207912i 0.669131 0.743145i 0.457151 + 4.34950i 0.809017 0.587785i −2.04249 + 1.68174i 0.309017 0.951057i 0.913545 0.406737i 2.18673 + 3.78753i
163.1 0.669131 + 0.743145i −0.913545 0.406737i −0.104528 + 0.994522i −2.21083 0.469926i −0.309017 0.951057i 2.39247 + 1.12965i −0.809017 + 0.587785i 0.669131 + 0.743145i −1.13011 1.95741i
163.2 0.669131 + 0.743145i −0.913545 0.406737i −0.104528 + 0.994522i −1.55422 0.330360i −0.309017 0.951057i −1.63560 2.07962i −0.809017 + 0.587785i 0.669131 + 0.743145i −0.794473 1.37607i
163.3 0.669131 + 0.743145i −0.913545 0.406737i −0.104528 + 0.994522i −0.408781 0.0868890i −0.309017 0.951057i −2.24058 1.40705i −0.809017 + 0.587785i 0.669131 + 0.743145i −0.208957 0.361923i
163.4 0.669131 + 0.743145i −0.913545 0.406737i −0.104528 + 0.994522i 3.44957 + 0.733230i −0.309017 0.951057i −2.03992 + 1.68486i −0.809017 + 0.587785i 0.669131 + 0.743145i 1.76332 + 3.05416i
163.5 0.669131 + 0.743145i −0.913545 0.406737i −0.104528 + 0.994522i 3.65870 + 0.777681i −0.309017 0.951057i 2.31914 1.27341i −0.809017 + 0.587785i 0.669131 + 0.743145i 1.87022 + 3.23932i
235.1 −0.104528 + 0.994522i −0.669131 0.743145i −0.978148 0.207912i −3.99535 1.77885i 0.809017 0.587785i 2.64091 + 0.160013i 0.309017 0.951057i −0.104528 + 0.994522i 2.18673 3.78753i
235.2 −0.104528 + 0.994522i −0.669131 0.743145i −0.978148 0.207912i −0.971427 0.432507i 0.809017 0.587785i −2.64519 0.0543560i 0.309017 0.951057i −0.104528 + 0.994522i 0.531680 0.920896i
235.3 −0.104528 + 0.994522i −0.669131 0.743145i −0.978148 0.207912i −0.810148 0.360701i 0.809017 0.587785i −0.287883 + 2.63004i 0.309017 0.951057i −0.104528 + 0.994522i 0.443409 0.768006i
235.4 −0.104528 + 0.994522i −0.669131 0.743145i −0.978148 0.207912i 0.363254 + 0.161731i 0.809017 0.587785i −1.41564 2.23517i 0.309017 0.951057i −0.104528 + 0.994522i −0.198815 + 0.344358i
235.5 −0.104528 + 0.994522i −0.669131 0.743145i −0.978148 0.207912i 2.67304 + 1.19011i 0.809017 0.587785i 2.49497 0.880407i 0.309017 0.951057i −0.104528 + 0.994522i −1.46300 + 2.53399i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 445.5
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.d 40
7.c even 3 1 inner 462.2.y.d 40
11.c even 5 1 inner 462.2.y.d 40
77.m even 15 1 inner 462.2.y.d 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.y.d 40 1.a even 1 1 trivial
462.2.y.d 40 7.c even 3 1 inner
462.2.y.d 40 11.c even 5 1 inner
462.2.y.d 40 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{40} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).