# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.n (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: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 5$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu + 5$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-4 \beta_{3} + 4 \beta_{1} + 5$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 2.13746 − 0.656712i −1.63746 + 1.52274i 2.13746 + 0.656712i −1.63746 − 1.52274i
−0.500000 + 0.866025i −1.50000 + 0.866025i −0.500000 0.866025i −1.13746 0.656712i 1.73205i −1.13746 2.38876i 1.00000 1.50000 2.59808i 1.13746 0.656712i
65.2 −0.500000 + 0.866025i −1.50000 + 0.866025i −0.500000 0.866025i 2.63746 + 1.52274i 1.73205i 2.63746 0.209313i 1.00000 1.50000 2.59808i −2.63746 + 1.52274i
263.1 −0.500000 0.866025i −1.50000 0.866025i −0.500000 + 0.866025i −1.13746 + 0.656712i 1.73205i −1.13746 + 2.38876i 1.00000 1.50000 + 2.59808i 1.13746 + 0.656712i
263.2 −0.500000 0.866025i −1.50000 0.866025i −0.500000 + 0.866025i 2.63746 1.52274i 1.73205i 2.63746 + 0.209313i 1.00000 1.50000 + 2.59808i −2.63746 1.52274i
 $$n$$: 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
231.l 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.n.a 4
3.b odd 2 1 462.2.n.d yes 4
7.c even 3 1 462.2.n.b yes 4
11.b odd 2 1 462.2.n.c yes 4
21.h odd 6 1 462.2.n.c yes 4
33.d even 2 1 462.2.n.b yes 4
77.h odd 6 1 462.2.n.d yes 4
231.l even 6 1 inner 462.2.n.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.n.a 4 1.a even 1 1 trivial
462.2.n.a 4 231.l even 6 1 inner
462.2.n.b yes 4 7.c even 3 1
462.2.n.b yes 4 33.d even 2 1
462.2.n.c yes 4 11.b odd 2 1
462.2.n.c yes 4 21.h odd 6 1
462.2.n.d yes 4 3.b odd 2 1
462.2.n.d yes 4 77.h odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} - 3 T_{5}^{3} - T_{5}^{2} + 12 T_{5} + 16$$ $$T_{17}^{4} + T_{17}^{3} + 15 T_{17}^{2} - 14 T_{17} + 196$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$( 3 + 3 T + T^{2} )^{2}$$
$5$ $$16 + 12 T - T^{2} - 3 T^{3} + T^{4}$$
$7$ $$49 - 21 T + 2 T^{2} - 3 T^{3} + T^{4}$$
$11$ $$( 11 - 5 T + T^{2} )^{2}$$
$13$ $$256 + 44 T^{2} + T^{4}$$
$17$ $$196 - 14 T + 15 T^{2} + T^{3} + T^{4}$$
$19$ $$196 - 210 T + 89 T^{2} - 15 T^{3} + T^{4}$$
$23$ $$1764 + 126 T - 39 T^{2} - 3 T^{3} + T^{4}$$
$29$ $$( 1 + T )^{4}$$
$31$ $$784 - 364 T + 141 T^{2} - 13 T^{3} + T^{4}$$
$37$ $$4 + 14 T + 51 T^{2} - 7 T^{3} + T^{4}$$
$41$ $$( -56 + 2 T + T^{2} )^{2}$$
$43$ $$3136 + 131 T^{2} + T^{4}$$
$47$ $$576 - 360 T + 51 T^{2} + 15 T^{3} + T^{4}$$
$53$ $$12544 + 1008 T - 85 T^{2} - 9 T^{3} + T^{4}$$
$59$ $$( 3 - 3 T + T^{2} )^{2}$$
$61$ $$( 48 - 12 T + T^{2} )^{2}$$
$67$ $$3136 + 112 T + 60 T^{2} - 2 T^{3} + T^{4}$$
$71$ $$3136 + 131 T^{2} + T^{4}$$
$73$ $$4096 - 768 T - 16 T^{2} + 12 T^{3} + T^{4}$$
$79$ $$196 - 210 T + 89 T^{2} - 15 T^{3} + T^{4}$$
$83$ $$( 28 + 13 T + T^{2} )^{2}$$
$89$ $$256 - 96 T - 4 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$( -21 + 12 T + T^{2} )^{2}$$