Properties

Label 462.2.n.c
Level $462$
Weight $2$
Character orbit 462.n
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.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} - 4 \nu - 5 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu + 5 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-4 \beta_{3} + 4 \beta_{1} + 5\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−1.63746 1.52274i
2.13746 + 0.656712i
−1.63746 + 1.52274i
2.13746 0.656712i
0.500000 0.866025i −1.50000 + 0.866025i −0.500000 0.866025i −1.13746 0.656712i 1.73205i 1.13746 + 2.38876i −1.00000 1.50000 2.59808i −1.13746 + 0.656712i
65.2 0.500000 0.866025i −1.50000 + 0.866025i −0.500000 0.866025i 2.63746 + 1.52274i 1.73205i −2.63746 + 0.209313i −1.00000 1.50000 2.59808i 2.63746 1.52274i
263.1 0.500000 + 0.866025i −1.50000 0.866025i −0.500000 + 0.866025i −1.13746 + 0.656712i 1.73205i 1.13746 2.38876i −1.00000 1.50000 + 2.59808i −1.13746 0.656712i
263.2 0.500000 + 0.866025i −1.50000 0.866025i −0.500000 + 0.866025i 2.63746 1.52274i 1.73205i −2.63746 0.209313i −1.00000 1.50000 + 2.59808i 2.63746 + 1.52274i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
231.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.n.c yes 4
3.b odd 2 1 462.2.n.b yes 4
7.c even 3 1 462.2.n.d yes 4
11.b odd 2 1 462.2.n.a 4
21.h odd 6 1 462.2.n.a 4
33.d even 2 1 462.2.n.d yes 4
77.h odd 6 1 462.2.n.b yes 4
231.l even 6 1 inner 462.2.n.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.n.a 4 11.b odd 2 1
462.2.n.a 4 21.h odd 6 1
462.2.n.b yes 4 3.b odd 2 1
462.2.n.b yes 4 77.h odd 6 1
462.2.n.c yes 4 1.a even 1 1 trivial
462.2.n.c yes 4 231.l even 6 1 inner
462.2.n.d yes 4 7.c even 3 1
462.2.n.d yes 4 33.d even 2 1

Hecke kernels

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

\( T_{5}^{4} - 3 T_{5}^{3} - T_{5}^{2} + 12 T_{5} + 16 \)
\( T_{17}^{4} - T_{17}^{3} + 15 T_{17}^{2} + 14 T_{17} + 196 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( ( 3 + 3 T + T^{2} )^{2} \)
$5$ \( 16 + 12 T - T^{2} - 3 T^{3} + T^{4} \)
$7$ \( 49 + 21 T + 2 T^{2} + 3 T^{3} + T^{4} \)
$11$ \( 121 - 55 T + 14 T^{2} - 5 T^{3} + T^{4} \)
$13$ \( 256 + 44 T^{2} + T^{4} \)
$17$ \( 196 + 14 T + 15 T^{2} - T^{3} + T^{4} \)
$19$ \( 196 + 210 T + 89 T^{2} + 15 T^{3} + T^{4} \)
$23$ \( 1764 + 126 T - 39 T^{2} - 3 T^{3} + T^{4} \)
$29$ \( ( -1 + T )^{4} \)
$31$ \( 784 - 364 T + 141 T^{2} - 13 T^{3} + T^{4} \)
$37$ \( 4 + 14 T + 51 T^{2} - 7 T^{3} + T^{4} \)
$41$ \( ( -56 - 2 T + T^{2} )^{2} \)
$43$ \( 3136 + 131 T^{2} + T^{4} \)
$47$ \( 576 - 360 T + 51 T^{2} + 15 T^{3} + T^{4} \)
$53$ \( 12544 + 1008 T - 85 T^{2} - 9 T^{3} + T^{4} \)
$59$ \( ( 3 - 3 T + T^{2} )^{2} \)
$61$ \( ( 48 + 12 T + T^{2} )^{2} \)
$67$ \( 3136 + 112 T + 60 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( 3136 + 131 T^{2} + T^{4} \)
$73$ \( 4096 + 768 T - 16 T^{2} - 12 T^{3} + T^{4} \)
$79$ \( 196 + 210 T + 89 T^{2} + 15 T^{3} + T^{4} \)
$83$ \( ( 28 - 13 T + T^{2} )^{2} \)
$89$ \( 256 - 96 T - 4 T^{2} + 6 T^{3} + T^{4} \)
$97$ \( ( -21 + 12 T + T^{2} )^{2} \)
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