Properties

Label 462.2.j.h
Level $462$
Weight $2$
Character orbit 462.j
Analytic conductor $3.689$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.20164000000.8
Defining polynomial: \( x^{8} + 6x^{6} + 76x^{4} + 781x^{2} + 5041 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{2} q^{3} + (\beta_{6} - \beta_{3} - \beta_{2} - 1) q^{4} + ( - \beta_{6} + \beta_1) q^{5} - \beta_{6} q^{6} + ( - \beta_{6} + \beta_{3} + \beta_{2} + 1) q^{7} - \beta_{2} q^{8} + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{2} q^{3} + (\beta_{6} - \beta_{3} - \beta_{2} - 1) q^{4} + ( - \beta_{6} + \beta_1) q^{5} - \beta_{6} q^{6} + ( - \beta_{6} + \beta_{3} + \beta_{2} + 1) q^{7} - \beta_{2} q^{8} + \beta_{3} q^{9} + ( - \beta_{4} - 1) q^{10} + ( - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 2) q^{11} - q^{12} + ( - \beta_{7} + \beta_{6} - 3 \beta_{3} - \beta_{2} + \beta_1) q^{13} + \beta_{2} q^{14} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{15} - \beta_{6} q^{16} + (3 \beta_{6} - 3 \beta_{2} - 3) q^{17} + ( - \beta_{6} + \beta_{3} + \beta_{2} + 1) q^{18} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{19} + (\beta_{5} + \beta_{3}) q^{20} + q^{21} + ( - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{22} + ( - 4 \beta_{6} - \beta_{4} + 4 \beta_{3} + 2) q^{23} + \beta_{3} q^{24} + ( - 2 \beta_{7} + \beta_{3} + 5 \beta_{2} + 1) q^{25} + (2 \beta_{6} - \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{26} + \beta_{6} q^{27} + \beta_{6} q^{28} + ( - \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{29} + ( - \beta_{7} + \beta_{2}) q^{30} + (\beta_{7} - 2 \beta_{5} + 2 \beta_{3} - \beta_1) q^{31} - q^{32} + ( - \beta_{2} - \beta_1 + 1) q^{33} + ( - 3 \beta_{6} + 3 \beta_{3} + 3) q^{34} + ( - \beta_{5} - \beta_{3}) q^{35} + \beta_{2} q^{36} + (\beta_{7} - 3 \beta_{6} - \beta_{5} - \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - \beta_1 + 3) q^{37} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_1) q^{38} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} - \beta_{2} + \beta_1 - 1) q^{39} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{40} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 3) q^{41} - \beta_{3} q^{42} + ( - 5 \beta_{6} + \beta_{5} + \beta_{4} + 5 \beta_{3} + \beta_1) q^{43} + ( - \beta_{7} + 2 \beta_{6} - \beta_{3} - \beta_{2} - 1) q^{44} + (\beta_{4} + 1) q^{45} + ( - 4 \beta_{6} + \beta_{5} + 2 \beta_{3} + 4 \beta_{2}) q^{46} + (\beta_{7} + \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{47} + ( - \beta_{6} + \beta_{3} + \beta_{2} + 1) q^{48} - \beta_{6} q^{49} + (4 \beta_{6} + \beta_{2} - 2 \beta_1 + 1) q^{50} + (3 \beta_{6} - 3) q^{51} + (\beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} - 1) q^{52} + (2 \beta_{7} + 3 \beta_{6} + \beta_{5} - 6 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{53} + q^{54} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_1 + 8) q^{55} + q^{56} + ( - \beta_{7} + \beta_{3} + \beta_1) q^{57} + ( - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{2}) q^{58} + ( - \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 1) q^{59} + (\beta_{6} - \beta_1) q^{60} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{4} - 3 \beta_{2} - \beta_1 - 3) q^{61} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{62} - \beta_{2} q^{63} + \beta_{3} q^{64} + (7 \beta_{6} - 3 \beta_{4} - 7 \beta_{3} - 9) q^{65} + ( - \beta_{6} + \beta_{4} - \beta_{3}) q^{66} + ( - \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 - 1) q^{67} + ( - 3 \beta_{6} + 3 \beta_{2}) q^{68} + ( - \beta_{7} + 4 \beta_{3} + 2 \beta_{2} + 4) q^{69} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{70} + (8 \beta_{6} - 2 \beta_{2} - 2 \beta_1 - 2) q^{71} + \beta_{6} q^{72} + (\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + 7 \beta_{3} + 7 \beta_{2} - 2 \beta_1 - 1) q^{73} + (\beta_{7} - 6 \beta_{3} - 3 \beta_{2} - 6) q^{74} + (\beta_{6} - 2 \beta_{5} - 5 \beta_{3} - \beta_{2}) q^{75} + (\beta_{5} + \beta_{4} + \beta_1 - 1) q^{76} + (\beta_{7} - 2 \beta_{6} + \beta_{3} + \beta_{2} + 1) q^{77} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 - 2) q^{78} + (3 \beta_{7} - 6 \beta_{6} - 3 \beta_{5} - \beta_{3} + 6 \beta_{2} - 3 \beta_1) q^{79} + ( - \beta_{7} + \beta_{2}) q^{80} + (\beta_{6} - \beta_{3} - \beta_{2} - 1) q^{81} + (2 \beta_{7} + 6 \beta_{6} - 2 \beta_{4} - 3 \beta_{2} - 3) q^{82} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{4} + \beta_{2} - \beta_1 + 1) q^{83} + (\beta_{6} - \beta_{3} - \beta_{2} - 1) q^{84} + (3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 3) q^{85} + ( - \beta_{7} - 5 \beta_{6} + 5 \beta_{3} + 5 \beta_{2} + \beta_1) q^{86} + (\beta_{5} + 2 \beta_{4} + \beta_1 - 1) q^{87} + ( - \beta_{2} - \beta_1 + 1) q^{88} + ( - 7 \beta_{6} - 2 \beta_{4} + 7 \beta_{3} + 9) q^{89} + ( - \beta_{5} - \beta_{3}) q^{90} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_{2} + 1) q^{91} + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{92} + (\beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_1) q^{93} + ( - \beta_{7} + 3 \beta_{6} + \beta_{4} + 2 \beta_1) q^{94} + ( - \beta_{7} + 6 \beta_{6} + \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 6) q^{95} + \beta_{2} q^{96} + ( - \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{2}) q^{97} - q^{98} + (\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} - 2 q^{9} - 8 q^{10} + 10 q^{11} - 8 q^{12} + 10 q^{13} - 2 q^{14} + 2 q^{15} - 2 q^{16} - 12 q^{17} + 2 q^{18} + 2 q^{19} - 2 q^{20} + 8 q^{21} - 2 q^{24} - 4 q^{25} + 2 q^{27} + 2 q^{28} - 2 q^{29} - 2 q^{30} - 4 q^{31} - 8 q^{32} + 10 q^{33} + 12 q^{34} + 2 q^{35} - 2 q^{36} + 30 q^{37} - 2 q^{38} - 10 q^{39} + 2 q^{40} - 24 q^{41} + 2 q^{42} - 20 q^{43} + 8 q^{45} - 20 q^{46} - 6 q^{47} + 2 q^{48} - 2 q^{49} + 14 q^{50} - 18 q^{51} + 24 q^{53} + 8 q^{54} + 68 q^{55} + 8 q^{56} - 2 q^{57} + 2 q^{58} + 14 q^{59} + 2 q^{60} - 12 q^{61} + 4 q^{62} + 2 q^{63} - 2 q^{64} - 44 q^{65} - 12 q^{67} - 12 q^{68} + 20 q^{69} - 2 q^{70} + 4 q^{71} + 2 q^{72} - 34 q^{73} - 30 q^{74} + 14 q^{75} - 8 q^{76} - 20 q^{78} - 22 q^{79} - 2 q^{80} - 2 q^{81} - 6 q^{82} + 12 q^{83} - 2 q^{84} + 18 q^{85} - 30 q^{86} - 8 q^{87} + 10 q^{88} + 44 q^{89} + 2 q^{90} - 20 q^{92} + 4 q^{93} + 6 q^{94} - 44 q^{95} - 2 q^{96} - 8 q^{97} - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 6x^{6} + 76x^{4} + 781x^{2} + 5041 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17\nu^{6} - 537\nu^{4} + 44034\nu^{2} - 40328 ) / 383471 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -153\nu^{6} + 4833\nu^{4} - 12835\nu^{2} - 20519 ) / 383471 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -153\nu^{7} + 4833\nu^{5} - 12835\nu^{3} - 20519\nu ) / 383471 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -585\nu^{7} - 4078\nu^{5} - 49075\nu^{3} - 461926\nu ) / 383471 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -721\nu^{6} + 218\nu^{4} - 17876\nu^{2} - 139302 ) / 383471 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -721\nu^{7} + 218\nu^{5} - 17876\nu^{3} - 139302\nu ) / 383471 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 9\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 9\beta_{7} - 9\beta_{5} - 8\beta_{4} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -17\beta_{6} + 82\beta_{3} + 17\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -17\beta_{5} + 65\beta_{4} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -537\beta_{6} - 218\beta_{2} - 218 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -755\beta_{7} + 218\beta_{5} + 218\beta_{4} \) Copy content Toggle raw display

Character values

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

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{2} - \beta_{3} + \beta_{6}\)

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
−2.50900 + 1.82290i
2.50900 1.82290i
−0.839592 2.58400i
0.839592 + 2.58400i
−0.839592 + 2.58400i
0.839592 2.58400i
−2.50900 1.82290i
2.50900 + 1.82290i
0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i −3.31802 + 2.41068i −0.809017 + 0.587785i −0.309017 0.951057i −0.309017 + 0.951057i −0.809017 0.587785i −4.10130
169.2 0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i 1.69998 1.23511i −0.809017 + 0.587785i −0.309017 0.951057i −0.309017 + 0.951057i −0.809017 0.587785i 2.10130
295.1 −0.309017 + 0.951057i 0.809017 0.587785i −0.809017 0.587785i −0.530575 1.63294i 0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 0.587785i 0.309017 0.951057i 1.71698
295.2 −0.309017 + 0.951057i 0.809017 0.587785i −0.809017 0.587785i 1.14861 + 3.53506i 0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 0.587785i 0.309017 0.951057i −3.71698
379.1 −0.309017 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.530575 + 1.63294i 0.309017 0.951057i 0.809017 0.587785i 0.809017 + 0.587785i 0.309017 + 0.951057i 1.71698
379.2 −0.309017 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i 1.14861 3.53506i 0.309017 0.951057i 0.809017 0.587785i 0.809017 + 0.587785i 0.309017 + 0.951057i −3.71698
421.1 0.809017 0.587785i −0.309017 0.951057i 0.309017 0.951057i −3.31802 2.41068i −0.809017 0.587785i −0.309017 + 0.951057i −0.309017 0.951057i −0.809017 + 0.587785i −4.10130
421.2 0.809017 0.587785i −0.309017 0.951057i 0.309017 0.951057i 1.69998 + 1.23511i −0.809017 0.587785i −0.309017 + 0.951057i −0.309017 0.951057i −0.809017 + 0.587785i 2.10130
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 421.2
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.h 8
11.c even 5 1 inner 462.2.j.h 8
11.c even 5 1 5082.2.a.by 4
11.d odd 10 1 5082.2.a.cd 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.j.h 8 1.a even 1 1 trivial
462.2.j.h 8 11.c even 5 1 inner
5082.2.a.by 4 11.c even 5 1
5082.2.a.cd 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2T_{5}^{7} + 9T_{5}^{6} + 28T_{5}^{5} + 156T_{5}^{4} - 370T_{5}^{3} + 790T_{5}^{2} - 550T_{5} + 3025 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + 9 T^{6} + 28 T^{5} + \cdots + 3025 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 10 T^{7} + 51 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 10 T^{7} + 74 T^{6} + \cdots + 26896 \) Copy content Toggle raw display
$17$ \( (T^{4} + 6 T^{3} + 36 T^{2} + 81 T + 81)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 2 T^{7} - 3 T^{6} + 20 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( (T^{4} - 57 T^{2} + 20 T + 131)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 2 T^{7} - 26 T^{6} - 112 T^{5} + \cdots + 400 \) Copy content Toggle raw display
$31$ \( T^{8} + 4 T^{7} - 27 T^{6} + \cdots + 32761 \) Copy content Toggle raw display
$37$ \( T^{8} - 30 T^{7} + 506 T^{6} + \cdots + 1771561 \) Copy content Toggle raw display
$41$ \( T^{8} + 24 T^{7} + 408 T^{6} + \cdots + 25110121 \) Copy content Toggle raw display
$43$ \( (T^{4} + 10 T^{3} - 48 T^{2} - 540 T - 604)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 6 T^{7} - 2 T^{6} + \cdots + 99856 \) Copy content Toggle raw display
$53$ \( T^{8} - 24 T^{7} + 466 T^{6} + \cdots + 64320400 \) Copy content Toggle raw display
$59$ \( T^{8} - 14 T^{7} + 158 T^{6} + \cdots + 6948496 \) Copy content Toggle raw display
$61$ \( T^{8} + 12 T^{7} + 138 T^{6} + \cdots + 355216 \) Copy content Toggle raw display
$67$ \( (T^{4} + 6 T^{3} - 74 T^{2} - 224 T + 1516)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 4 T^{7} + 156 T^{6} + \cdots + 774400 \) Copy content Toggle raw display
$73$ \( T^{8} + 34 T^{7} + 758 T^{6} + \cdots + 4648336 \) Copy content Toggle raw display
$79$ \( T^{8} + 22 T^{7} + \cdots + 383376400 \) Copy content Toggle raw display
$83$ \( T^{8} - 12 T^{7} + 98 T^{6} + \cdots + 698896 \) Copy content Toggle raw display
$89$ \( (T^{4} - 22 T^{3} - 9 T^{2} + 1570 T - 4895)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 8 T^{7} + 34 T^{6} + 72 T^{5} + \cdots + 400 \) Copy content Toggle raw display
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