# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.k (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{3} ) q^{7} + \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{3} ) q^{7} + \beta_{2} q^{9} + ( -\beta_{1} - \beta_{3} ) q^{13} + ( 3 + \beta_{2} ) q^{15} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{17} + ( -\beta_{1} - \beta_{3} ) q^{19} + ( 3 + \beta_{1} - \beta_{2} ) q^{21} + 2 \beta_{2} q^{23} + q^{25} + 3 \beta_{3} q^{27} + 2 \beta_{2} q^{29} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{35} -2 q^{37} + ( 3 - \beta_{2} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{41} -4 q^{43} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{45} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{47} + ( -5 - 2 \beta_{1} - 2 \beta_{3} ) q^{49} + ( -6 - 2 \beta_{2} ) q^{51} -2 \beta_{2} q^{53} + ( 3 - \beta_{2} ) q^{57} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{59} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{61} + ( 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{63} -2 \beta_{2} q^{65} -8 q^{67} + 6 \beta_{3} q^{69} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{73} + \beta_{1} q^{75} + 10 q^{79} -9 q^{81} + ( \beta_{1} - \beta_{3} ) q^{83} -12 q^{85} + 6 \beta_{3} q^{87} + ( -6 - \beta_{1} - \beta_{3} ) q^{91} -2 \beta_{2} q^{95} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} + O(q^{10})$$ $$4q + 4q^{7} + 12q^{15} + 12q^{21} + 4q^{25} - 8q^{37} + 12q^{39} - 16q^{43} - 20q^{49} - 24q^{51} + 12q^{57} - 32q^{67} + 40q^{79} - 36q^{81} - 48q^{85} - 24q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

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

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 −1.22474 − 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i 1.22474 + 1.22474i
0 −1.22474 1.22474i 0 −2.44949 0 1.00000 + 2.44949i 0 3.00000i 0
209.2 0 −1.22474 + 1.22474i 0 −2.44949 0 1.00000 2.44949i 0 3.00000i 0
209.3 0 1.22474 1.22474i 0 2.44949 0 1.00000 + 2.44949i 0 3.00000i 0
209.4 0 1.22474 + 1.22474i 0 2.44949 0 1.00000 2.44949i 0 3.00000i 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
3.b Odd 1 yes
7.b Odd 1 yes
21.c Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{5}^{2} - 6$$ acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$.