Properties

Label 462.2.j.a
Level $462$
Weight $2$
Character orbit 462.j
Analytic conductor $3.689$
Analytic rank $0$
Dimension $4$
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.j (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + \zeta_{10}^{2} q^{3} -\zeta_{10}^{3} q^{4} + ( 1 + \zeta_{10}^{2} ) q^{5} -\zeta_{10} q^{6} + \zeta_{10}^{3} q^{7} + \zeta_{10}^{2} q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + \zeta_{10}^{2} q^{3} -\zeta_{10}^{3} q^{4} + ( 1 + \zeta_{10}^{2} ) q^{5} -\zeta_{10} q^{6} + \zeta_{10}^{3} q^{7} + \zeta_{10}^{2} q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{10} + ( -1 - \zeta_{10} - \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{11} + q^{12} + ( -4 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{13} -\zeta_{10}^{2} q^{14} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{15} -\zeta_{10} q^{16} + ( 3 + 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{17} -\zeta_{10}^{3} q^{18} + ( \zeta_{10} - 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{19} + ( 1 - \zeta_{10}^{3} ) q^{20} - q^{21} + ( 2 - 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{22} + ( 2 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{23} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{24} + ( \zeta_{10} - 4 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{25} + ( 4 - 4 \zeta_{10} ) q^{26} -\zeta_{10} q^{27} + \zeta_{10} q^{28} + ( -2 + 2 \zeta_{10} + 8 \zeta_{10}^{3} ) q^{29} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{30} + ( 1 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{31} + q^{32} + ( -2 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{33} + ( -6 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{34} + ( -1 + \zeta_{10}^{3} ) q^{35} + \zeta_{10}^{2} q^{36} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{37} + ( -1 + 3 \zeta_{10} - \zeta_{10}^{2} ) q^{38} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{39} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{40} + ( 7 \zeta_{10} - 6 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{41} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{42} + ( -4 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{43} + ( -2 + 4 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{44} + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{45} + ( -2 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{46} + ( -6 \zeta_{10} + 2 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{47} -\zeta_{10}^{3} q^{48} -\zeta_{10} q^{49} + ( -1 + 4 \zeta_{10} - \zeta_{10}^{2} ) q^{50} + ( -3 + 3 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{51} + ( 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{52} + ( 6 \zeta_{10} - 6 \zeta_{10}^{2} ) q^{53} + q^{54} + ( -3 - 2 \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{55} - q^{56} + ( 2 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{57} + ( -2 \zeta_{10} - 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{58} + ( -4 + 4 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{59} + ( 1 + \zeta_{10}^{2} ) q^{60} + ( 4 - 10 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{61} + ( 5 - 5 \zeta_{10} + \zeta_{10}^{3} ) q^{62} -\zeta_{10}^{2} q^{63} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64} -4 q^{65} + ( 3 - 2 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{66} + ( -6 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{67} + ( 6 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{68} + ( -\zeta_{10} + 3 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{69} + ( 1 - \zeta_{10} - \zeta_{10}^{3} ) q^{70} + 12 \zeta_{10} q^{71} -\zeta_{10} q^{72} + ( -8 + 8 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{73} + ( 3 \zeta_{10} - 6 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{74} + ( 3 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{75} + ( -2 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{76} + ( 2 - 4 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{77} + ( 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{78} + ( 4 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{79} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{80} -\zeta_{10}^{3} q^{81} + ( -7 + 6 \zeta_{10} - 7 \zeta_{10}^{2} ) q^{82} + ( 12 - 8 \zeta_{10} + 12 \zeta_{10}^{2} ) q^{83} + \zeta_{10}^{3} q^{84} + ( 6 \zeta_{10} + 3 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{85} + ( 4 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{86} + ( -8 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{87} + ( -2 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{88} + ( 10 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{89} + ( 1 - \zeta_{10}^{3} ) q^{90} + ( -4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{91} + ( -1 + \zeta_{10} - 2 \zeta_{10}^{3} ) q^{92} + ( -5 + 6 \zeta_{10} - 5 \zeta_{10}^{2} ) q^{93} + ( 6 - 2 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{94} + ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{95} + \zeta_{10}^{2} q^{96} + ( 8 - 8 \zeta_{10}^{3} ) q^{97} + q^{98} + ( 2 - 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} - q^{3} - q^{4} + 3q^{5} - q^{6} + q^{7} - q^{8} - q^{9} + O(q^{10}) \) \( 4q - q^{2} - q^{3} - q^{4} + 3q^{5} - q^{6} + q^{7} - q^{8} - q^{9} - 2q^{10} - q^{11} + 4q^{12} - 8q^{13} + q^{14} - 2q^{15} - q^{16} + 12q^{17} - q^{18} + 5q^{19} + 3q^{20} - 4q^{21} + 9q^{22} + 10q^{23} - q^{24} + 6q^{25} + 12q^{26} - q^{27} + q^{28} + 2q^{29} - 2q^{30} - 9q^{31} + 4q^{32} - 11q^{33} - 18q^{34} - 3q^{35} - q^{36} + 12q^{37} - 8q^{39} - 2q^{40} + 20q^{41} + q^{42} - 24q^{43} - q^{44} - 2q^{45} - 14q^{47} - q^{48} - q^{49} + q^{50} - 3q^{51} + 12q^{52} + 12q^{53} + 4q^{54} - 12q^{55} - 4q^{56} + 2q^{58} - 18q^{59} + 3q^{60} + 2q^{61} + 16q^{62} + q^{63} - q^{64} - 16q^{65} + 4q^{66} - 32q^{67} + 12q^{68} - 5q^{69} + 2q^{70} + 12q^{71} - q^{72} - 26q^{73} + 12q^{74} + q^{75} - 10q^{76} + q^{77} - 8q^{78} + 8q^{79} - 2q^{80} - q^{81} - 15q^{82} + 28q^{83} + q^{84} + 9q^{85} - 4q^{86} - 28q^{87} - 11q^{88} + 34q^{89} + 3q^{90} - 12q^{91} - 5q^{92} - 9q^{93} + 16q^{94} + 5q^{95} - q^{96} + 24q^{97} + 4q^{98} + 9q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/462\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.809017 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i 1.30902 0.951057i −0.809017 + 0.587785i −0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i −1.61803
295.1 0.309017 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0.190983 + 0.587785i 0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i 0.618034
379.1 0.309017 + 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 0.190983 0.587785i 0.309017 0.951057i 0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i 0.618034
421.1 −0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i 1.30902 + 0.951057i −0.809017 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i −1.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.j.a 4
11.c even 5 1 inner 462.2.j.a 4
11.c even 5 1 5082.2.a.bt 2
11.d odd 10 1 5082.2.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.j.a 4 1.a even 1 1 trivial
462.2.j.a 4 11.c even 5 1 inner
5082.2.a.bj 2 11.d odd 10 1
5082.2.a.bt 2 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 3 T_{5}^{3} + 4 T_{5}^{2} - 2 T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$5$ \( 1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4} \)
$7$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$11$ \( 121 + 11 T - 9 T^{2} + T^{3} + T^{4} \)
$13$ \( 256 + 192 T + 64 T^{2} + 8 T^{3} + T^{4} \)
$17$ \( 81 + 27 T + 54 T^{2} - 12 T^{3} + T^{4} \)
$19$ \( 25 + 10 T^{2} - 5 T^{3} + T^{4} \)
$23$ \( ( 5 - 5 T + T^{2} )^{2} \)
$29$ \( 1936 - 528 T + 64 T^{2} - 2 T^{3} + T^{4} \)
$31$ \( 361 + 304 T + 106 T^{2} + 9 T^{3} + T^{4} \)
$37$ \( 81 + 27 T + 54 T^{2} - 12 T^{3} + T^{4} \)
$41$ \( 3025 - 825 T + 190 T^{2} - 20 T^{3} + T^{4} \)
$43$ \( ( 16 + 12 T + T^{2} )^{2} \)
$47$ \( 1936 + 704 T + 136 T^{2} + 14 T^{3} + T^{4} \)
$53$ \( 1296 - 648 T + 144 T^{2} - 12 T^{3} + T^{4} \)
$59$ \( 1936 - 88 T + 124 T^{2} + 18 T^{3} + T^{4} \)
$61$ \( 1936 + 792 T + 124 T^{2} - 2 T^{3} + T^{4} \)
$67$ \( ( 44 + 16 T + T^{2} )^{2} \)
$71$ \( 20736 - 1728 T + 144 T^{2} - 12 T^{3} + T^{4} \)
$73$ \( 1936 + 616 T + 276 T^{2} + 26 T^{3} + T^{4} \)
$79$ \( 16 + 8 T + 24 T^{2} - 8 T^{3} + T^{4} \)
$83$ \( 30976 - 5632 T + 544 T^{2} - 28 T^{3} + T^{4} \)
$89$ \( ( 61 - 17 T + T^{2} )^{2} \)
$97$ \( 4096 - 1024 T + 256 T^{2} - 24 T^{3} + T^{4} \)
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