# Properties

 Label 342.2.s.b Level $342$ Weight $2$ Character orbit 342.s Analytic conductor $2.731$ Analytic rank $0$ Dimension $4$ 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.s (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.73088374913$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{7} - q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -1 + 2 \beta_{1} - \beta_{3} ) q^{7} - q^{8} + \beta_{1} q^{10} + ( -2 + 4 \beta_{2} - 2 \beta_{3} ) q^{11} + ( 1 + 3 \beta_{1} + \beta_{2} ) q^{13} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{14} -\beta_{2} q^{16} + ( -4 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{17} + ( 5 - 2 \beta_{2} ) q^{19} + \beta_{3} q^{20} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{22} -4 \beta_{1} q^{23} + ( -3 + 3 \beta_{2} ) q^{25} + ( -1 + 2 \beta_{2} + 3 \beta_{3} ) q^{26} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{28} + ( -\beta_{1} + 2 \beta_{3} ) q^{29} + ( 3 - 6 \beta_{2} + 3 \beta_{3} ) q^{31} + ( 1 - \beta_{2} ) q^{32} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{34} + ( 4 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{35} + ( -1 + 2 \beta_{2} - 3 \beta_{3} ) q^{37} + ( 2 + 3 \beta_{2} ) q^{38} + ( -\beta_{1} + \beta_{3} ) q^{40} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{43} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{44} -4 \beta_{3} q^{46} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{47} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{49} -3 q^{50} + ( -2 - 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{52} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{53} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{55} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{56} + ( -2 \beta_{1} + \beta_{3} ) q^{58} + ( -\beta_{1} - 12 \beta_{2} - \beta_{3} ) q^{59} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{61} + ( 6 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{62} + q^{64} + ( 6 + 2 \beta_{1} - \beta_{3} ) q^{65} + ( -1 - 6 \beta_{1} - \beta_{2} ) q^{67} + ( 2 - 4 \beta_{2} + \beta_{3} ) q^{68} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{70} -6 \beta_{2} q^{71} + ( -2 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} ) q^{73} + ( -2 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{74} + ( -3 + 5 \beta_{2} ) q^{76} + ( 6 - 12 \beta_{2} + 8 \beta_{3} ) q^{77} + ( 6 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{79} -\beta_{1} q^{80} + ( -6 + 12 \beta_{2} + \beta_{3} ) q^{83} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{85} + ( -1 + 2 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{86} + ( 2 - 4 \beta_{2} + 2 \beta_{3} ) q^{88} + ( 5 \beta_{1} - 10 \beta_{3} ) q^{89} + ( 5 + 5 \beta_{2} ) q^{91} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{92} + ( -2 + 4 \beta_{2} - \beta_{3} ) q^{94} + ( 3 \beta_{1} - 5 \beta_{3} ) q^{95} + ( -12 - 6 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{97} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 2q^{4} - 4q^{7} - 4q^{8} + O(q^{10})$$ $$4q + 2q^{2} - 2q^{4} - 4q^{7} - 4q^{8} + 6q^{13} - 2q^{14} - 2q^{16} - 12q^{17} + 16q^{19} - 12q^{22} - 6q^{25} + 2q^{28} + 2q^{32} - 12q^{34} + 12q^{35} + 14q^{38} + 2q^{43} - 12q^{44} + 12q^{47} - 12q^{50} - 6q^{52} - 8q^{55} + 4q^{56} - 24q^{59} + 2q^{61} + 18q^{62} + 4q^{64} + 24q^{65} - 6q^{67} + 12q^{70} - 12q^{71} + 14q^{73} - 6q^{74} - 2q^{76} + 18q^{79} + 4q^{85} - 2q^{86} + 30q^{91} - 36q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$191$$ $$325$$ $$\chi(n)$$ $$-1$$ $$1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.22474 + 0.707107i 0 −3.44949 −1.00000 0 −1.22474 0.707107i
107.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.22474 0.707107i 0 1.44949 −1.00000 0 1.22474 + 0.707107i
179.1 0.500000 0.866025i 0 −0.500000 0.866025i −1.22474 0.707107i 0 −3.44949 −1.00000 0 −1.22474 + 0.707107i
179.2 0.500000 0.866025i 0 −0.500000 0.866025i 1.22474 + 0.707107i 0 1.44949 −1.00000 0 1.22474 0.707107i
 $$n$$: 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
57.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.s.b yes 4
3.b odd 2 1 342.2.s.a 4
4.b odd 2 1 2736.2.dc.a 4
12.b even 2 1 2736.2.dc.b 4
19.d odd 6 1 342.2.s.a 4
57.f even 6 1 inner 342.2.s.b yes 4
76.f even 6 1 2736.2.dc.b 4
228.n odd 6 1 2736.2.dc.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.s.a 4 3.b odd 2 1
342.2.s.a 4 19.d odd 6 1
342.2.s.b yes 4 1.a even 1 1 trivial
342.2.s.b yes 4 57.f even 6 1 inner
2736.2.dc.a 4 4.b odd 2 1
2736.2.dc.a 4 228.n odd 6 1
2736.2.dc.b 4 12.b even 2 1
2736.2.dc.b 4 76.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{4} + 12 T_{17}^{3} + 58 T_{17}^{2} + 120 T_{17} + 100$$ acting on $$S_{2}^{\mathrm{new}}(342, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$4 - 2 T^{2} + T^{4}$$
$7$ $$( -5 + 2 T + T^{2} )^{2}$$
$11$ $$16 + 40 T^{2} + T^{4}$$
$13$ $$225 + 90 T - 3 T^{2} - 6 T^{3} + T^{4}$$
$17$ $$100 + 120 T + 58 T^{2} + 12 T^{3} + T^{4}$$
$19$ $$( 19 - 8 T + T^{2} )^{2}$$
$23$ $$1024 - 32 T^{2} + T^{4}$$
$29$ $$36 + 6 T^{2} + T^{4}$$
$31$ $$81 + 90 T^{2} + T^{4}$$
$37$ $$225 + 42 T^{2} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$529 + 46 T + 27 T^{2} - 2 T^{3} + T^{4}$$
$47$ $$100 - 120 T + 58 T^{2} - 12 T^{3} + T^{4}$$
$53$ $$576 + 24 T^{2} + T^{4}$$
$59$ $$19044 + 3312 T + 438 T^{2} + 24 T^{3} + T^{4}$$
$61$ $$25 + 10 T + 9 T^{2} - 2 T^{3} + T^{4}$$
$67$ $$4761 - 414 T - 57 T^{2} + 6 T^{3} + T^{4}$$
$71$ $$( 36 + 6 T + T^{2} )^{2}$$
$73$ $$625 - 350 T + 171 T^{2} - 14 T^{3} + T^{4}$$
$79$ $$81 - 162 T + 117 T^{2} - 18 T^{3} + T^{4}$$
$83$ $$11236 + 220 T^{2} + T^{4}$$
$89$ $$22500 + 150 T^{2} + T^{4}$$
$97$ $$1296 + 1296 T + 468 T^{2} + 36 T^{3} + T^{4}$$