# Properties

 Label 462.2.i.c Level $462$ Weight $2$ Character orbit 462.i Analytic conductor $3.689$ Analytic rank $0$ Dimension $2$ 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.50000 + 2.59808i −1.00000 2.50000 + 0.866025i −1.00000 −0.500000 0.866025i −1.50000 + 2.59808i
331.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.50000 2.59808i −1.00000 2.50000 0.866025i −1.00000 −0.500000 + 0.866025i −1.50000 2.59808i
 $$n$$: 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
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.c 2
3.b odd 2 1 1386.2.k.c 2
7.c even 3 1 inner 462.2.i.c 2
7.c even 3 1 3234.2.a.h 1
7.d odd 6 1 3234.2.a.g 1
21.g even 6 1 9702.2.a.bd 1
21.h odd 6 1 1386.2.k.c 2
21.h odd 6 1 9702.2.a.cf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.c 2 1.a even 1 1 trivial
462.2.i.c 2 7.c even 3 1 inner
1386.2.k.c 2 3.b odd 2 1
1386.2.k.c 2 21.h odd 6 1
3234.2.a.g 1 7.d odd 6 1
3234.2.a.h 1 7.c even 3 1
9702.2.a.bd 1 21.g even 6 1
9702.2.a.cf 1 21.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 $$S_{2}^{\mathrm{new}}(462, [\chi])$$:

 $$T_{5}^{2} - 3 T_{5} + 9$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$4 + 2 T + T^{2}$$
$41$ $$( 3 + T )^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$81 - 9 T + T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$144 - 12 T + T^{2}$$
$61$ $$25 + 5 T + T^{2}$$
$67$ $$25 + 5 T + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$256 - 16 T + T^{2}$$
$79$ $$289 + 17 T + T^{2}$$
$83$ $$( -9 + T )^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$( -17 + T )^{2}$$