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: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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 \( T_{5}^{4} - 3T_{5}^{3} + 4T_{5}^{2} - 2T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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