# Properties

 Label 462.2.g.c Level $462$ Weight $2$ Character orbit 462.g 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.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

## Character values

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

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
419.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
1.00000i −1.73205 −1.00000 0 1.73205i 2.00000 1.73205i 1.00000i 3.00000 0
419.2 1.00000i 1.73205 −1.00000 0 1.73205i 2.00000 + 1.73205i 1.00000i 3.00000 0
419.3 1.00000i −1.73205 −1.00000 0 1.73205i 2.00000 + 1.73205i 1.00000i 3.00000 0
419.4 1.00000i 1.73205 −1.00000 0 1.73205i 2.00000 1.73205i 1.00000i 3.00000 0
 $$n$$: 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.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.g.c 4
3.b odd 2 1 inner 462.2.g.c 4
7.b odd 2 1 inner 462.2.g.c 4
21.c even 2 1 inner 462.2.g.c 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 7 - 4 T + T^{2} )^{2}$$
$11$ $$( 1 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( -12 + T^{2} )^{2}$$
$19$ $$( 12 + T^{2} )^{2}$$
$23$ $$( 36 + T^{2} )^{2}$$
$29$ $$( 36 + T^{2} )^{2}$$
$31$ $$( 12 + T^{2} )^{2}$$
$37$ $$( 2 + T )^{4}$$
$41$ $$( -12 + T^{2} )^{2}$$
$43$ $$( 2 + T )^{4}$$
$47$ $$( -108 + T^{2} )^{2}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( -108 + T^{2} )^{2}$$
$61$ $$( 48 + T^{2} )^{2}$$
$67$ $$( 4 + T )^{4}$$
$71$ $$( 36 + T^{2} )^{2}$$
$73$ $$( 48 + T^{2} )^{2}$$
$79$ $$( -8 + T )^{4}$$
$83$ $$( -108 + T^{2} )^{2}$$
$89$ $$( -192 + T^{2} )^{2}$$
$97$ $$( 48 + T^{2} )^{2}$$