Properties

Label 462.2.j.g
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: 8.0.324000000.3
Defining polynomial: \(x^{8} + 3 x^{6} + 9 x^{4} + 27 x^{2} + 81\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 3 x^{6} + 9 x^{4} + 27 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(\beta_{4}\)\(=\)\( \nu^{4} \)\(/9\)
\(\beta_{5}\)\(=\)\( \nu^{5} \)\(/9\)
\(\beta_{6}\)\(=\)\( \nu^{6} \)\(/27\)
\(\beta_{7}\)\(=\)\( \nu^{7} \)\(/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)
\(\nu^{4}\)\(=\)\(9 \beta_{4}\)
\(\nu^{5}\)\(=\)\(9 \beta_{5}\)
\(\nu^{6}\)\(=\)\(27 \beta_{6}\)
\(\nu^{7}\)\(=\)\(27 \beta_{7}\)

Character values

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

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{4}\)

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
0.535233 1.64728i
−0.535233 + 1.64728i
−1.40126 + 1.01807i
1.40126 1.01807i
−1.40126 1.01807i
1.40126 + 1.01807i
0.535233 + 1.64728i
−0.535233 1.64728i
0.809017 + 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i −2.26728 + 1.64728i 0.809017 0.587785i 0.309017 + 0.951057i −0.309017 + 0.951057i −0.809017 0.587785i −2.80252
169.2 0.809017 + 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i 2.26728 1.64728i 0.809017 0.587785i 0.309017 + 0.951057i −0.309017 + 0.951057i −0.809017 0.587785i 2.80252
295.1 −0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i −0.330792 1.01807i −0.309017 0.951057i −0.809017 0.587785i 0.809017 0.587785i 0.309017 0.951057i 1.07047
295.2 −0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0.330792 + 1.01807i −0.309017 0.951057i −0.809017 0.587785i 0.809017 0.587785i 0.309017 0.951057i −1.07047
379.1 −0.309017 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i −0.330792 + 1.01807i −0.309017 + 0.951057i −0.809017 + 0.587785i 0.809017 + 0.587785i 0.309017 + 0.951057i 1.07047
379.2 −0.309017 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 0.330792 1.01807i −0.309017 + 0.951057i −0.809017 + 0.587785i 0.809017 + 0.587785i 0.309017 + 0.951057i −1.07047
421.1 0.809017 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i −2.26728 1.64728i 0.809017 + 0.587785i 0.309017 0.951057i −0.309017 0.951057i −0.809017 + 0.587785i −2.80252
421.2 0.809017 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i 2.26728 + 1.64728i 0.809017 + 0.587785i 0.309017 0.951057i −0.309017 0.951057i −0.809017 + 0.587785i 2.80252
\(n\): 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.g 8
11.c even 5 1 inner 462.2.j.g 8
11.c even 5 1 5082.2.a.bz 4
11.d odd 10 1 5082.2.a.ce 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.j.g 8 1.a even 1 1 trivial
462.2.j.g 8 11.c even 5 1 inner
5082.2.a.bz 4 11.c even 5 1
5082.2.a.ce 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 3 T_{5}^{6} + 54 T_{5}^{4} + 108 T_{5}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$3$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$5$ \( 81 + 108 T^{2} + 54 T^{4} - 3 T^{6} + T^{8} \)
$7$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$11$ \( 14641 - 10648 T + 1573 T^{2} + 814 T^{3} - 435 T^{4} + 74 T^{5} + 13 T^{6} - 8 T^{7} + T^{8} \)
$13$ \( 1296 + 1296 T + 1512 T^{2} + 1728 T^{3} + 1980 T^{4} - 288 T^{5} + 42 T^{6} - 6 T^{7} + T^{8} \)
$17$ \( 121 + 550 T + 5373 T^{2} + 2740 T^{3} + 354 T^{4} - 170 T^{5} + 32 T^{6} + T^{8} \)
$19$ \( 421201 - 66198 T + 29096 T^{2} + 676 T^{3} + 354 T^{4} - 32 T^{5} + 39 T^{6} + 4 T^{7} + T^{8} \)
$23$ \( ( -71 - 90 T - 31 T^{2} + T^{4} )^{2} \)
$29$ \( 274576 + 163488 T + 91328 T^{2} + 20296 T^{3} + 2280 T^{4} - 104 T^{5} + 18 T^{6} - 2 T^{7} + T^{8} \)
$31$ \( 421201 - 66198 T + 29096 T^{2} + 676 T^{3} + 354 T^{4} - 32 T^{5} + 39 T^{6} + 4 T^{7} + T^{8} \)
$37$ \( 4157521 + 2560984 T + 778125 T^{2} + 138736 T^{3} + 27174 T^{4} + 3736 T^{5} + 380 T^{6} + 24 T^{7} + T^{8} \)
$41$ \( 121 + 550 T + 5373 T^{2} + 2740 T^{3} + 354 T^{4} - 170 T^{5} + 32 T^{6} + T^{8} \)
$43$ \( ( 244 + 88 T - 40 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$47$ \( 62031376 - 2142272 T + 1381120 T^{2} + 68888 T^{3} + 8424 T^{4} - 488 T^{5} + 70 T^{6} + 2 T^{7} + T^{8} \)
$53$ \( 4648336 - 120736 T + 200704 T^{2} - 39032 T^{3} + 7464 T^{4} + 1736 T^{5} + 286 T^{6} + 22 T^{7} + T^{8} \)
$59$ \( 59536 + 175680 T + 189152 T^{2} - 45400 T^{3} + 10584 T^{4} - 1240 T^{5} + 138 T^{6} - 10 T^{7} + T^{8} \)
$61$ \( 56010256 - 22571744 T + 5077152 T^{2} - 742072 T^{3} + 81080 T^{4} - 6568 T^{5} + 462 T^{6} - 26 T^{7} + T^{8} \)
$67$ \( ( 2596 - 172 T - 142 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$71$ \( 952576 - 562176 T + 631232 T^{2} - 57728 T^{3} - 3120 T^{4} + 928 T^{5} + 252 T^{6} + 16 T^{7} + T^{8} \)
$73$ \( 28344976 + 8901728 T + 2671680 T^{2} + 285208 T^{3} + 10184 T^{4} - 1448 T^{5} + 150 T^{6} + 2 T^{7} + T^{8} \)
$79$ \( 156816 + 128304 T + 291384 T^{2} - 9072 T^{3} - 216 T^{4} + 36 T^{5} + 96 T^{6} + 12 T^{7} + T^{8} \)
$83$ \( 2835856 + 2128576 T + 1512544 T^{2} + 466952 T^{3} + 84504 T^{4} + 9784 T^{5} + 766 T^{6} + 38 T^{7} + T^{8} \)
$89$ \( ( 229 + 18 T - 85 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$97$ \( 1008016 - 979904 T + 498912 T^{2} - 154472 T^{3} + 32280 T^{4} - 3848 T^{5} + 242 T^{6} - 6 T^{7} + T^{8} \)
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