# Properties

 Label 462.2.k.a Level $462$ Weight $2$ Character orbit 462.k Analytic conductor $3.689$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.k (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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 0.866025 + 1.50000i 0.500000 + 0.866025i −1.73205 + 3.00000i 1.73205i 2.00000 + 1.73205i 1.00000i −1.50000 + 2.59808i 3.00000 1.73205i
89.2 0.866025 + 0.500000i −0.866025 1.50000i 0.500000 + 0.866025i 1.73205 3.00000i 1.73205i 2.00000 + 1.73205i 1.00000i −1.50000 + 2.59808i 3.00000 1.73205i
353.1 −0.866025 + 0.500000i 0.866025 1.50000i 0.500000 0.866025i −1.73205 3.00000i 1.73205i 2.00000 1.73205i 1.00000i −1.50000 2.59808i 3.00000 + 1.73205i
353.2 0.866025 0.500000i −0.866025 + 1.50000i 0.500000 0.866025i 1.73205 + 3.00000i 1.73205i 2.00000 1.73205i 1.00000i −1.50000 2.59808i 3.00000 + 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g 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.k.a 4
3.b odd 2 1 inner 462.2.k.a 4
7.d odd 6 1 inner 462.2.k.a 4
21.g even 6 1 inner 462.2.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.k.a 4 1.a even 1 1 trivial
462.2.k.a 4 3.b odd 2 1 inner
462.2.k.a 4 7.d odd 6 1 inner
462.2.k.a 4 21.g even 6 1 inner

## Hecke kernels

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

 $$T_{5}^{4} + 12 T_{5}^{2} + 144$$ $$T_{17}^{4} + 3 T_{17}^{2} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$144 + 12 T^{2} + T^{4}$$
$7$ $$( 7 - 4 T + T^{2} )^{2}$$
$11$ $$1 - T^{2} + T^{4}$$
$13$ $$( 12 + T^{2} )^{2}$$
$17$ $$9 + 3 T^{2} + T^{4}$$
$19$ $$( 3 + 3 T + T^{2} )^{2}$$
$23$ $$81 - 9 T^{2} + T^{4}$$
$29$ $$( 9 + T^{2} )^{2}$$
$31$ $$( 12 - 6 T + T^{2} )^{2}$$
$37$ $$( 49 + 7 T + T^{2} )^{2}$$
$41$ $$( -48 + T^{2} )^{2}$$
$43$ $$( -1 + T )^{4}$$
$47$ $$9 + 3 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$21609 + 147 T^{2} + T^{4}$$
$61$ $$( 108 - 18 T + T^{2} )^{2}$$
$67$ $$( 256 - 16 T + T^{2} )^{2}$$
$71$ $$( 225 + T^{2} )^{2}$$
$73$ $$( 192 + 24 T + T^{2} )^{2}$$
$79$ $$( 64 + 8 T + T^{2} )^{2}$$
$83$ $$( -300 + T^{2} )^{2}$$
$89$ $$11664 + 108 T^{2} + T^{4}$$
$97$ $$( 243 + T^{2} )^{2}$$