Properties

Label 462.2.i.f
Level $462$
Weight $2$
Character orbit 462.i
Analytic conductor $3.689$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: 6.0.1156923.1
Defining polynomial: \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 22\nu^{3} - 28\nu^{2} + 43\nu - 18 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\nu^{5} + 7\nu^{4} - 31\nu^{3} + 37\nu^{2} - 56\nu + 22 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{5} + 15\nu^{4} - 64\nu^{3} + 82\nu^{2} - 121\nu + 50 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -3\nu^{5} + 8\nu^{4} - 33\nu^{3} + 44\nu^{2} - 62\nu + 24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 2\beta_{4} + \beta_{3} + 3\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{5} + 8\beta_{4} - 3\beta_{3} + 6\beta_{2} - \beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{5} + 18\beta_{4} - 9\beta_{3} + 12\beta_{2} - 17\beta _1 + 27 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 13\beta_{5} - 28\beta_{4} + 3\beta_{3} - 34\beta_{2} - 11\beta _1 + 49 ) / 2 \) Copy content Toggle raw display

Character values

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

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

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 + 0.0585812i
0.500000 + 1.51496i
0.500000 2.43956i
0.500000 0.0585812i
0.500000 1.51496i
0.500000 + 2.43956i
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.37328 2.37860i 1.00000 −1.37328 + 2.26144i 1.00000 −0.500000 0.866025i −1.37328 + 2.37860i
67.2 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.227452 0.393958i 1.00000 −0.227452 2.63596i 1.00000 −0.500000 0.866025i −0.227452 + 0.393958i
67.3 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.60074 + 2.77256i 1.00000 1.60074 + 2.10657i 1.00000 −0.500000 0.866025i 1.60074 2.77256i
331.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.37328 + 2.37860i 1.00000 −1.37328 2.26144i 1.00000 −0.500000 + 0.866025i −1.37328 2.37860i
331.2 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.227452 + 0.393958i 1.00000 −0.227452 + 2.63596i 1.00000 −0.500000 + 0.866025i −0.227452 0.393958i
331.3 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.60074 2.77256i 1.00000 1.60074 2.10657i 1.00000 −0.500000 + 0.866025i 1.60074 + 2.77256i
\(n\): 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.f 6
3.b odd 2 1 1386.2.k.w 6
7.c even 3 1 inner 462.2.i.f 6
7.c even 3 1 3234.2.a.bi 3
7.d odd 6 1 3234.2.a.bg 3
21.g even 6 1 9702.2.a.du 3
21.h odd 6 1 1386.2.k.w 6
21.h odd 6 1 9702.2.a.dt 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.f 6 1.a even 1 1 trivial
462.2.i.f 6 7.c even 3 1 inner
1386.2.k.w 6 3.b odd 2 1
1386.2.k.w 6 21.h odd 6 1
3234.2.a.bg 3 7.d odd 6 1
3234.2.a.bi 3 7.c even 3 1
9702.2.a.dt 3 21.h odd 6 1
9702.2.a.du 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 \(S_{2}^{\mathrm{new}}(462, [\chi])\):

\( T_{5}^{6} + 9T_{5}^{4} + 8T_{5}^{3} + 81T_{5}^{2} + 36T_{5} + 16 \) Copy content Toggle raw display
\( T_{13}^{3} - 36T_{13} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 9 T^{4} + 8 T^{3} + 81 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{6} + 12 T^{4} - 4 T^{3} + 84 T^{2} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$13$ \( (T^{3} - 36 T + 32)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$23$ \( T^{6} + 9 T^{5} + 78 T^{4} + 69 T^{3} + \cdots + 441 \) Copy content Toggle raw display
$29$ \( (T^{3} - 9 T^{2} + 92)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 6 T^{5} + 132 T^{4} + \cdots + 262144 \) Copy content Toggle raw display
$37$ \( T^{6} + 9 T^{5} + 93 T^{4} + \cdots + 26896 \) Copy content Toggle raw display
$41$ \( (T^{3} - 12 T^{2} + 21 T + 82)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 9 T^{2} - 12 T + 164)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 15 T^{5} + 174 T^{4} + \cdots + 361 \) Copy content Toggle raw display
$53$ \( T^{6} + 36 T^{4} + 64 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$59$ \( T^{6} + 9 T^{5} + 177 T^{4} + \cdots + 589824 \) Copy content Toggle raw display
$61$ \( T^{6} - 6 T^{5} + 165 T^{4} + \cdots + 51076 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + 51 T^{4} + \cdots + 7056 \) Copy content Toggle raw display
$71$ \( (T^{3} - 3 T^{2} - 24 T + 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 96 T^{4} - 384 T^{3} + \cdots + 36864 \) Copy content Toggle raw display
$79$ \( T^{6} + 12 T^{5} + 177 T^{4} + \cdots + 135424 \) Copy content Toggle raw display
$83$ \( (T^{3} + 6 T^{2} - 111 T + 188)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 36 T^{5} + 900 T^{4} + \cdots + 1763584 \) Copy content Toggle raw display
$97$ \( (T^{3} - 9 T^{2} - 81 T - 7)^{2} \) Copy content Toggle raw display
show more
show less