Properties

Label 462.2.i.g
Level $462$
Weight $2$
Character orbit 462.i
Analytic conductor $3.689$
Analytic rank $0$
Dimension $6$
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.21870000.1
Defining polynomial: \(x^{6} - 3 x^{5} + 24 x^{4} - 43 x^{3} + 138 x^{2} - 117 x + 73\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 24 x^{4} - 43 x^{3} + 138 x^{2} - 117 x + 73\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 30 \nu^{3} + 40 \nu^{2} - 70 \nu + 13 \)\()/31\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + 18 \nu^{4} - 15 \nu^{3} + 144 \nu^{2} + 151 \nu - 164 \)\()/62\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} - 13 \nu^{4} + 47 \nu^{3} - 197 \nu^{2} + 461 \nu - 133 \)\()/62\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} - 13 \nu^{4} + 16 \nu^{3} - 166 \nu^{2} + 151 \nu - 102 \)\()/31\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_{1} - 8\)
\(\nu^{3}\)\(=\)\(-3 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + \beta_{2} - 8 \beta_{1} - 7\)
\(\nu^{4}\)\(=\)\(16 \beta_{5} - 16 \beta_{4} + 20 \beta_{3} - 9 \beta_{2} - 28 \beta_{1} + 75\)
\(\nu^{5}\)\(=\)\(45 \beta_{5} - 60 \beta_{4} + 40 \beta_{3} - 33 \beta_{2} + 55 \beta_{1} + 139\)

Character values

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

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 3.23735i
0.500000 3.05087i
0.500000 + 0.679547i
0.500000 3.23735i
0.500000 + 3.05087i
0.500000 0.679547i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −2.20942 3.82682i −1.00000 2.20942 1.45550i 1.00000 −0.500000 0.866025i −2.20942 + 3.82682i
67.2 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.806615 + 1.39710i −1.00000 −0.806615 2.51980i 1.00000 −0.500000 0.866025i 0.806615 1.39710i
67.3 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.40280 + 2.42972i −1.00000 −1.40280 + 2.24325i 1.00000 −0.500000 0.866025i 1.40280 2.42972i
331.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −2.20942 + 3.82682i −1.00000 2.20942 + 1.45550i 1.00000 −0.500000 + 0.866025i −2.20942 3.82682i
331.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.806615 1.39710i −1.00000 −0.806615 + 2.51980i 1.00000 −0.500000 + 0.866025i 0.806615 + 1.39710i
331.3 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.40280 2.42972i −1.00000 −1.40280 2.24325i 1.00000 −0.500000 + 0.866025i 1.40280 + 2.42972i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.i.g 6
3.b odd 2 1 1386.2.k.v 6
7.c even 3 1 inner 462.2.i.g 6
7.c even 3 1 3234.2.a.bf 3
7.d odd 6 1 3234.2.a.bh 3
21.g even 6 1 9702.2.a.dw 3
21.h odd 6 1 1386.2.k.v 6
21.h odd 6 1 9702.2.a.dv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.g 6 1.a even 1 1 trivial
462.2.i.g 6 7.c even 3 1 inner
1386.2.k.v 6 3.b odd 2 1
1386.2.k.v 6 21.h odd 6 1
3234.2.a.bf 3 7.c even 3 1
3234.2.a.bh 3 7.d odd 6 1
9702.2.a.dv 3 21.h odd 6 1
9702.2.a.dw 3 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\):

\( T_{5}^{6} + 15 T_{5}^{4} - 40 T_{5}^{3} + 225 T_{5}^{2} - 300 T_{5} + 400 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{3} \)
$3$ \( ( 1 - T + T^{2} )^{3} \)
$5$ \( 400 - 300 T + 225 T^{2} - 40 T^{3} + 15 T^{4} + T^{6} \)
$7$ \( 343 + 42 T^{2} - 20 T^{3} + 6 T^{4} + T^{6} \)
$11$ \( ( 1 - T + T^{2} )^{3} \)
$13$ \( T^{6} \)
$17$ \( 19321 + 7923 T + 3666 T^{2} + 107 T^{3} + 66 T^{4} + 3 T^{5} + T^{6} \)
$19$ \( 784 + 336 T + 396 T^{2} - 164 T^{3} + 69 T^{4} - 9 T^{5} + T^{6} \)
$23$ \( 7921 + 2403 T + 996 T^{2} + 97 T^{3} + 36 T^{4} + 3 T^{5} + T^{6} \)
$29$ \( ( 348 - 48 T - 9 T^{2} + T^{3} )^{2} \)
$31$ \( 36864 - 9216 T + 3456 T^{2} - 96 T^{3} + 84 T^{4} - 6 T^{5} + T^{6} \)
$37$ \( 51984 + 30096 T + 12636 T^{2} + 2316 T^{3} + 309 T^{4} + 21 T^{5} + T^{6} \)
$41$ \( ( 306 - 27 T - 12 T^{2} + T^{3} )^{2} \)
$43$ \( ( 404 - 132 T - 3 T^{2} + T^{3} )^{2} \)
$47$ \( 81 + 243 T + 756 T^{2} - 63 T^{3} + 36 T^{4} + 3 T^{5} + T^{6} \)
$53$ \( ( 64 + 8 T + T^{2} )^{3} \)
$59$ \( 2304 - 2304 T + 2736 T^{2} + 336 T^{3} + 129 T^{4} - 9 T^{5} + T^{6} \)
$61$ \( 9604 + 6174 T + 3381 T^{2} + 574 T^{3} + 99 T^{4} - 6 T^{5} + T^{6} \)
$67$ \( 44944 + 13356 T + 5241 T^{2} + 46 T^{3} + 99 T^{4} + 6 T^{5} + T^{6} \)
$71$ \( ( 108 - 108 T - 9 T^{2} + T^{3} )^{2} \)
$73$ \( 230400 - 57600 T + 14400 T^{2} - 960 T^{3} + 120 T^{4} + T^{6} \)
$79$ \( 256 - 528 T + 1281 T^{2} + 428 T^{3} + 111 T^{4} + 12 T^{5} + T^{6} \)
$83$ \( ( 164 + 33 T - 18 T^{2} + T^{3} )^{2} \)
$89$ \( 256 - 192 T + 336 T^{2} + 112 T^{3} + 156 T^{4} - 12 T^{5} + T^{6} \)
$97$ \( ( -7 + T )^{6} \)
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