Properties

Label 462.2.i.e
Level $462$
Weight $2$
Character orbit 462.i
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

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
1.32288 2.29129i
−1.32288 + 2.29129i
1.32288 + 2.29129i
−1.32288 2.29129i
0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.32288 2.29129i 1.00000 −1.32288 2.29129i −1.00000 −0.500000 0.866025i 1.32288 2.29129i
67.2 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.32288 + 2.29129i 1.00000 1.32288 + 2.29129i −1.00000 −0.500000 0.866025i −1.32288 + 2.29129i
331.1 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.32288 + 2.29129i 1.00000 −1.32288 + 2.29129i −1.00000 −0.500000 + 0.866025i 1.32288 + 2.29129i
331.2 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.32288 2.29129i 1.00000 1.32288 2.29129i −1.00000 −0.500000 + 0.866025i −1.32288 2.29129i
\(n\): e.g. 2-40 or 990-1000
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.e 4
3.b odd 2 1 1386.2.k.r 4
7.c even 3 1 inner 462.2.i.e 4
7.c even 3 1 3234.2.a.w 2
7.d odd 6 1 3234.2.a.ba 2
21.g even 6 1 9702.2.a.dm 2
21.h odd 6 1 1386.2.k.r 4
21.h odd 6 1 9702.2.a.db 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.e 4 1.a even 1 1 trivial
462.2.i.e 4 7.c even 3 1 inner
1386.2.k.r 4 3.b odd 2 1
1386.2.k.r 4 21.h odd 6 1
3234.2.a.w 2 7.c even 3 1
3234.2.a.ba 2 7.d odd 6 1
9702.2.a.db 2 21.h odd 6 1
9702.2.a.dm 2 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 \(S_{2}^{\mathrm{new}}(462, [\chi])\):

\( T_{5}^{4} + 7 T_{5}^{2} + 49 \)
\( T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( 49 + 7 T^{2} + T^{4} \)
$7$ \( 49 + 7 T^{2} + T^{4} \)
$11$ \( ( 1 + T + T^{2} )^{2} \)
$13$ \( ( 4 + T )^{4} \)
$17$ \( ( 9 - 3 T + T^{2} )^{2} \)
$19$ \( 784 + 28 T^{2} + T^{4} \)
$23$ \( 49 + 7 T^{2} + T^{4} \)
$29$ \( ( -2 + T )^{4} \)
$31$ \( ( 16 - 4 T + T^{2} )^{2} \)
$37$ \( 144 + 96 T + 76 T^{2} - 8 T^{3} + T^{4} \)
$41$ \( ( -9 + T )^{4} \)
$43$ \( ( -12 - 8 T + T^{2} )^{2} \)
$47$ \( 2209 - 376 T + 111 T^{2} + 8 T^{3} + T^{4} \)
$53$ \( ( 16 + 4 T + T^{2} )^{2} \)
$59$ \( 9216 + 768 T + 160 T^{2} - 8 T^{3} + T^{4} \)
$61$ \( 2209 - 376 T + 111 T^{2} + 8 T^{3} + T^{4} \)
$67$ \( 10609 + 618 T + 139 T^{2} - 6 T^{3} + T^{4} \)
$71$ \( ( 36 + 16 T + T^{2} )^{2} \)
$73$ \( 5184 - 1440 T + 328 T^{2} - 20 T^{3} + T^{4} \)
$79$ \( 3249 + 912 T + 199 T^{2} + 16 T^{3} + T^{4} \)
$83$ \( ( -87 - 10 T + T^{2} )^{2} \)
$89$ \( 1296 + 576 T + 220 T^{2} + 16 T^{3} + T^{4} \)
$97$ \( ( -111 + 2 T + T^{2} )^{2} \)
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