# Properties

 Label 342.2.g.d Level $342$ Weight $2$ Character orbit 342.g Analytic conductor $2.731$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 342.g (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.73088374913$$ 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: no (minimal twist has level 114) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -4 + 4 \zeta_{6} ) q^{5} -3 q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -4 + 4 \zeta_{6} ) q^{5} -3 q^{7} - q^{8} + 4 \zeta_{6} q^{10} -2 q^{11} + 7 \zeta_{6} q^{13} + ( -3 + 3 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( -5 + 2 \zeta_{6} ) q^{19} + 4 q^{20} + ( -2 + 2 \zeta_{6} ) q^{22} -4 \zeta_{6} q^{23} -11 \zeta_{6} q^{25} + 7 q^{26} + 3 \zeta_{6} q^{28} + 4 \zeta_{6} q^{29} + q^{31} + \zeta_{6} q^{32} + ( 12 - 12 \zeta_{6} ) q^{35} + 7 q^{37} + ( -3 + 5 \zeta_{6} ) q^{38} + ( 4 - 4 \zeta_{6} ) q^{40} + ( 4 - 4 \zeta_{6} ) q^{41} + ( -7 + 7 \zeta_{6} ) q^{43} + 2 \zeta_{6} q^{44} -4 q^{46} + 2 \zeta_{6} q^{47} + 2 q^{49} -11 q^{50} + ( 7 - 7 \zeta_{6} ) q^{52} -4 \zeta_{6} q^{53} + ( 8 - 8 \zeta_{6} ) q^{55} + 3 q^{56} + 4 q^{58} + ( -6 + 6 \zeta_{6} ) q^{59} + \zeta_{6} q^{61} + ( 1 - \zeta_{6} ) q^{62} + q^{64} -28 q^{65} -3 \zeta_{6} q^{67} -12 \zeta_{6} q^{70} + ( 2 - 2 \zeta_{6} ) q^{71} + ( 3 - 3 \zeta_{6} ) q^{73} + ( 7 - 7 \zeta_{6} ) q^{74} + ( 2 + 3 \zeta_{6} ) q^{76} + 6 q^{77} + ( -5 + 5 \zeta_{6} ) q^{79} -4 \zeta_{6} q^{80} -4 \zeta_{6} q^{82} + 12 q^{83} + 7 \zeta_{6} q^{86} + 2 q^{88} + 18 \zeta_{6} q^{89} -21 \zeta_{6} q^{91} + ( -4 + 4 \zeta_{6} ) q^{92} + 2 q^{94} + ( 12 - 20 \zeta_{6} ) q^{95} + ( -10 + 10 \zeta_{6} ) q^{97} + ( 2 - 2 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} - 4q^{5} - 6q^{7} - 2q^{8} + O(q^{10})$$ $$2q + q^{2} - q^{4} - 4q^{5} - 6q^{7} - 2q^{8} + 4q^{10} - 4q^{11} + 7q^{13} - 3q^{14} - q^{16} - 8q^{19} + 8q^{20} - 2q^{22} - 4q^{23} - 11q^{25} + 14q^{26} + 3q^{28} + 4q^{29} + 2q^{31} + q^{32} + 12q^{35} + 14q^{37} - q^{38} + 4q^{40} + 4q^{41} - 7q^{43} + 2q^{44} - 8q^{46} + 2q^{47} + 4q^{49} - 22q^{50} + 7q^{52} - 4q^{53} + 8q^{55} + 6q^{56} + 8q^{58} - 6q^{59} + q^{61} + q^{62} + 2q^{64} - 56q^{65} - 3q^{67} - 12q^{70} + 2q^{71} + 3q^{73} + 7q^{74} + 7q^{76} + 12q^{77} - 5q^{79} - 4q^{80} - 4q^{82} + 24q^{83} + 7q^{86} + 4q^{88} + 18q^{89} - 21q^{91} - 4q^{92} + 4q^{94} + 4q^{95} - 10q^{97} + 2q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$325$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i −2.00000 + 3.46410i 0 −3.00000 −1.00000 0 2.00000 + 3.46410i
235.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.00000 3.46410i 0 −3.00000 −1.00000 0 2.00000 3.46410i
 $$n$$: 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.g.d 2
3.b odd 2 1 114.2.e.a 2
4.b odd 2 1 2736.2.s.c 2
12.b even 2 1 912.2.q.d 2
19.c even 3 1 inner 342.2.g.d 2
19.c even 3 1 6498.2.a.l 1
19.d odd 6 1 6498.2.a.x 1
57.f even 6 1 2166.2.a.c 1
57.h odd 6 1 114.2.e.a 2
57.h odd 6 1 2166.2.a.f 1
76.g odd 6 1 2736.2.s.c 2
228.m even 6 1 912.2.q.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.e.a 2 3.b odd 2 1
114.2.e.a 2 57.h odd 6 1
342.2.g.d 2 1.a even 1 1 trivial
342.2.g.d 2 19.c even 3 1 inner
912.2.q.d 2 12.b even 2 1
912.2.q.d 2 228.m even 6 1
2166.2.a.c 1 57.f even 6 1
2166.2.a.f 1 57.h odd 6 1
2736.2.s.c 2 4.b odd 2 1
2736.2.s.c 2 76.g odd 6 1
6498.2.a.l 1 19.c even 3 1
6498.2.a.x 1 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(342, [\chi])$$:

 $$T_{5}^{2} + 4 T_{5} + 16$$ $$T_{7} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$16 + 4 T + T^{2}$$
$7$ $$( 3 + T )^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$49 - 7 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$19 + 8 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$16 - 4 T + T^{2}$$
$31$ $$( -1 + T )^{2}$$
$37$ $$( -7 + T )^{2}$$
$41$ $$16 - 4 T + T^{2}$$
$43$ $$49 + 7 T + T^{2}$$
$47$ $$4 - 2 T + T^{2}$$
$53$ $$16 + 4 T + T^{2}$$
$59$ $$36 + 6 T + T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$9 + 3 T + T^{2}$$
$71$ $$4 - 2 T + T^{2}$$
$73$ $$9 - 3 T + T^{2}$$
$79$ $$25 + 5 T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$324 - 18 T + T^{2}$$
$97$ $$100 + 10 T + T^{2}$$