# Properties

 Label 336.2.q.g Level 336 Weight 2 Character orbit 336.q Analytic conductor 2.683 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.q (of order $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} + ( 2 - \beta_{1} + \beta_{3} ) q^{7} -\beta_{2} q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} + ( 2 - \beta_{1} + \beta_{3} ) q^{7} -\beta_{2} q^{9} + ( -1 + \beta_{1} - 2 \beta_{3} ) q^{11} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{15} + ( 4 - 4 \beta_{2} ) q^{17} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{19} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{21} + 4 \beta_{2} q^{23} + ( -10 + \beta_{1} + 9 \beta_{2} - 2 \beta_{3} ) q^{25} + q^{27} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{29} + ( -1 + \beta_{2} ) q^{31} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{33} + ( 8 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{35} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{37} + ( -2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{39} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{43} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{45} -6 \beta_{2} q^{47} + ( -1 - 3 \beta_{1} + 3 \beta_{3} ) q^{49} + 4 \beta_{2} q^{51} + ( -5 - \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -14 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{55} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{57} + ( 3 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{59} -10 \beta_{2} q^{61} + ( -2 \beta_{2} - \beta_{3} ) q^{63} + ( -2 + 4 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} ) q^{65} + ( 4 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{67} -4 q^{69} -2 q^{71} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{73} + ( 1 - 2 \beta_{1} - 10 \beta_{2} + \beta_{3} ) q^{75} + ( 3 + 2 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{77} + ( 2 - 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{79} + ( -1 + \beta_{2} ) q^{81} + ( -4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{83} + ( 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{85} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{87} + ( 2 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 1 - 4 \beta_{1} + 8 \beta_{2} + \beta_{3} ) q^{91} -\beta_{2} q^{93} + ( -12 - 2 \beta_{1} + 14 \beta_{2} + 4 \beta_{3} ) q^{95} + ( 12 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{97} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + q^{5} + 6q^{7} - 2q^{9} + O(q^{10})$$ $$4q - 2q^{3} + q^{5} + 6q^{7} - 2q^{9} - q^{11} + 10q^{13} - 2q^{15} + 8q^{17} - 5q^{19} - 3q^{21} + 8q^{23} - 19q^{25} + 4q^{27} + 6q^{29} - 2q^{31} - q^{33} + 30q^{35} - 3q^{37} - 5q^{39} + 12q^{41} + 14q^{43} + q^{45} - 12q^{47} - 10q^{49} + 8q^{51} - 11q^{53} - 58q^{55} + 10q^{57} + 5q^{59} - 20q^{61} - 3q^{63} - 26q^{65} + 7q^{67} - 16q^{69} - 8q^{71} + q^{73} - 19q^{75} + 27q^{77} + 8q^{79} - 2q^{81} - 14q^{83} + 8q^{85} - 3q^{87} + 6q^{89} + 15q^{91} - 2q^{93} - 26q^{95} + 50q^{97} + 2q^{99} + O(q^{100})$$

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 $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 −1.63746 − 1.52274i 2.13746 + 0.656712i −1.63746 + 1.52274i 2.13746 − 0.656712i
0 −0.500000 + 0.866025i 0 −1.63746 2.83616i 0 1.50000 + 2.17945i 0 −0.500000 0.866025i 0
193.2 0 −0.500000 + 0.866025i 0 2.13746 + 3.70219i 0 1.50000 2.17945i 0 −0.500000 0.866025i 0
289.1 0 −0.500000 0.866025i 0 −1.63746 + 2.83616i 0 1.50000 2.17945i 0 −0.500000 + 0.866025i 0
289.2 0 −0.500000 0.866025i 0 2.13746 3.70219i 0 1.50000 + 2.17945i 0 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.c Even 1 yes

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}^{4} - T_{5}^{3} + 15 T_{5}^{2} + 14 T_{5} + 196$$ $$T_{11}^{4} + T_{11}^{3} + 15 T_{11}^{2} - 14 T_{11} + 196$$