Properties

 Label 336.2.bj.f Level 336 Weight 2 Character orbit 336.bj Analytic conductor 2.683 Analytic rank 0 Dimension 8 CM No Inner twists 8

Related objects

Newspace parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.bj (of order $$6$$ and degree $$2$$)

Newform invariants

 Self dual: No Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.303595776.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 1 - 2 \beta_{2} + \beta_{4} + 2 \beta_{6} ) q^{5} + ( \beta_{3} + 3 \beta_{5} ) q^{7} + ( -2 - \beta_{2} - 2 \beta_{4} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 1 - 2 \beta_{2} + \beta_{4} + 2 \beta_{6} ) q^{5} + ( \beta_{3} + 3 \beta_{5} ) q^{7} + ( -2 - \beta_{2} - 2 \beta_{4} + \beta_{6} ) q^{9} + ( -2 \beta_{1} + \beta_{5} ) q^{11} -4 q^{13} + ( \beta_{1} - 6 \beta_{3} - 6 \beta_{5} - \beta_{7} ) q^{15} + ( -2 \beta_{2} + \beta_{4} ) q^{17} + 7 \beta_{3} q^{19} + ( 2 - 3 \beta_{2} + 3 \beta_{4} + 2 \beta_{6} ) q^{21} + ( \beta_{3} + 2 \beta_{7} ) q^{23} -6 \beta_{4} q^{25} + ( -2 \beta_{1} - 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{7} ) q^{27} + ( 2 + 4 \beta_{6} ) q^{29} + 3 \beta_{5} q^{31} + ( 5 + \beta_{2} + 5 \beta_{4} - \beta_{6} ) q^{33} + ( 4 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} - 6 \beta_{7} ) q^{35} + ( 1 + \beta_{4} ) q^{37} -4 \beta_{1} q^{39} + ( -2 - 4 \beta_{6} ) q^{41} + ( 2 \beta_{3} + 2 \beta_{5} ) q^{43} + ( 5 \beta_{2} - 8 \beta_{4} ) q^{45} + ( 3 \beta_{3} + 6 \beta_{7} ) q^{47} + ( 3 + 8 \beta_{4} ) q^{49} + ( -6 \beta_{3} - \beta_{7} ) q^{51} + ( 2 \beta_{2} - \beta_{4} ) q^{53} + ( 11 \beta_{3} + 11 \beta_{5} ) q^{55} + ( -7 - 7 \beta_{6} ) q^{57} + ( 2 \beta_{1} - \beta_{5} ) q^{59} + ( -3 - 3 \beta_{4} ) q^{61} + ( 2 \beta_{1} - 9 \beta_{3} - 6 \beta_{5} - 3 \beta_{7} ) q^{63} + ( -4 + 8 \beta_{2} - 4 \beta_{4} - 8 \beta_{6} ) q^{65} + 9 \beta_{5} q^{67} + ( -5 + \beta_{6} ) q^{69} + ( 8 \beta_{1} - 4 \beta_{3} - 4 \beta_{5} - 8 \beta_{7} ) q^{71} + 7 \beta_{4} q^{73} + 6 \beta_{7} q^{75} + ( -2 + 6 \beta_{2} - 3 \beta_{4} - 4 \beta_{6} ) q^{77} + 9 \beta_{3} q^{79} + ( 5 \beta_{2} + \beta_{4} ) q^{81} + ( -8 \beta_{1} + 4 \beta_{3} + 4 \beta_{5} + 8 \beta_{7} ) q^{83} + 11 q^{85} + ( 2 \beta_{1} - 12 \beta_{5} ) q^{87} + ( -1 + 2 \beta_{2} - \beta_{4} - 2 \beta_{6} ) q^{89} + ( -4 \beta_{3} - 12 \beta_{5} ) q^{91} + ( 3 - 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{6} ) q^{93} + ( -14 \beta_{1} + 7 \beta_{5} ) q^{95} + 8 q^{97} + ( 5 \beta_{1} + 3 \beta_{3} + 3 \beta_{5} - 5 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 10q^{9} + O(q^{10})$$ $$8q - 10q^{9} - 32q^{13} + 2q^{21} + 24q^{25} + 22q^{33} + 4q^{37} + 22q^{45} - 8q^{49} - 28q^{57} - 12q^{61} - 44q^{69} - 28q^{73} - 14q^{81} + 88q^{85} + 6q^{93} + 64q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 32 \nu^{4} + 16 \nu^{2} + 45$$$$)/144$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 32 \nu^{5} + 16 \nu^{3} + 45 \nu$$$$)/432$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{6} - 16 \nu^{4} - 80 \nu^{2} - 225$$$$)/144$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu$$$$)/48$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - 13$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} + 16 \nu^{5} + 80 \nu^{3} + 225 \nu$$$$)/144$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - 2 \beta_{4} - \beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$-\beta_{7} + 15 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$-16 \beta_{6} - 13$$ $$\nu^{7}$$ $$=$$ $$48 \beta_{5} - 13 \beta_{1}$$

Character Values

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

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-1 - \beta_{4}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
95.1
 −1.26217 + 1.18614i −0.396143 + 1.68614i 0.396143 − 1.68614i 1.26217 − 1.18614i −1.26217 − 1.18614i −0.396143 − 1.68614i 0.396143 + 1.68614i 1.26217 + 1.18614i
0 −1.26217 + 1.18614i 0 2.87228 1.65831i 0 −1.73205 2.00000i 0 0.186141 2.99422i 0
95.2 0 −0.396143 + 1.68614i 0 −2.87228 + 1.65831i 0 1.73205 + 2.00000i 0 −2.68614 1.33591i 0
95.3 0 0.396143 1.68614i 0 −2.87228 + 1.65831i 0 −1.73205 2.00000i 0 −2.68614 1.33591i 0
95.4 0 1.26217 1.18614i 0 2.87228 1.65831i 0 1.73205 + 2.00000i 0 0.186141 2.99422i 0
191.1 0 −1.26217 1.18614i 0 2.87228 + 1.65831i 0 −1.73205 + 2.00000i 0 0.186141 + 2.99422i 0
191.2 0 −0.396143 1.68614i 0 −2.87228 1.65831i 0 1.73205 2.00000i 0 −2.68614 + 1.33591i 0
191.3 0 0.396143 + 1.68614i 0 −2.87228 1.65831i 0 −1.73205 + 2.00000i 0 −2.68614 + 1.33591i 0
191.4 0 1.26217 + 1.18614i 0 2.87228 + 1.65831i 0 1.73205 2.00000i 0 0.186141 + 2.99422i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 191.4 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
4.b Odd 1 yes
7.c Even 1 yes
12.b Even 1 yes
21.h Odd 1 yes
28.g Odd 1 yes
84.n Even 1 yes

Hecke kernels

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

 $$T_{5}^{4} - 11 T_{5}^{2} + 121$$ $$T_{13} + 4$$ $$T_{19}^{4} - 49 T_{19}^{2} + 2401$$