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: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

Character values

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

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

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 \(S_{2}^{\mathrm{new}}(462, [\chi])\):

\( T_{5}^{4} + 7T_{5}^{2} + 49 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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