# 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 $$6$$, degree $$2$$, minimal)

## 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: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ 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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.c even 3 1 inner
12.b even 2 1 inner
21.h odd 6 1 inner
28.g odd 6 1 inner
84.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.bj.f 8
3.b odd 2 1 inner 336.2.bj.f 8
4.b odd 2 1 inner 336.2.bj.f 8
7.c even 3 1 inner 336.2.bj.f 8
7.c even 3 1 2352.2.h.i 4
7.d odd 6 1 2352.2.h.j 4
12.b even 2 1 inner 336.2.bj.f 8
21.g even 6 1 2352.2.h.j 4
21.h odd 6 1 inner 336.2.bj.f 8
21.h odd 6 1 2352.2.h.i 4
28.f even 6 1 2352.2.h.j 4
28.g odd 6 1 inner 336.2.bj.f 8
28.g odd 6 1 2352.2.h.i 4
84.j odd 6 1 2352.2.h.j 4
84.n even 6 1 inner 336.2.bj.f 8
84.n even 6 1 2352.2.h.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bj.f 8 1.a even 1 1 trivial
336.2.bj.f 8 3.b odd 2 1 inner
336.2.bj.f 8 4.b odd 2 1 inner
336.2.bj.f 8 7.c even 3 1 inner
336.2.bj.f 8 12.b even 2 1 inner
336.2.bj.f 8 21.h odd 6 1 inner
336.2.bj.f 8 28.g odd 6 1 inner
336.2.bj.f 8 84.n even 6 1 inner
2352.2.h.i 4 7.c even 3 1
2352.2.h.i 4 21.h odd 6 1
2352.2.h.i 4 28.g odd 6 1
2352.2.h.i 4 84.n even 6 1
2352.2.h.j 4 7.d odd 6 1
2352.2.h.j 4 21.g even 6 1
2352.2.h.j 4 28.f even 6 1
2352.2.h.j 4 84.j odd 6 1

## Hecke kernels

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 5 T^{2} + 16 T^{4} + 45 T^{6} + 81 T^{8}$$
$5$ $$( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} )^{2}( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} )^{2}$$
$7$ $$( 1 + 2 T^{2} + 49 T^{4} )^{2}$$
$11$ $$( 1 - 11 T^{2} )^{4}( 1 + 11 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 4 T + 13 T^{2} )^{8}$$
$17$ $$( 1 + 23 T^{2} + 240 T^{4} + 6647 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 26 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 35 T^{2} + 696 T^{4} - 18515 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 14 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 + 53 T^{2} + 1848 T^{4} + 50933 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 11 T + 37 T^{2} )^{4}( 1 + 10 T + 37 T^{2} )^{4}$$
$41$ $$( 1 - 38 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 - 82 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 5 T^{2} - 2184 T^{4} + 11045 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 95 T^{2} + 6216 T^{4} + 266855 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 107 T^{2} + 7968 T^{4} - 372467 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4} )^{4}$$
$67$ $$( 1 + 53 T^{2} - 1680 T^{4} + 237917 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 34 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 10 T + 73 T^{2} )^{4}( 1 + 17 T + 73 T^{2} )^{4}$$
$79$ $$( 1 + 77 T^{2} - 312 T^{4} + 480557 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 - 10 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 + 167 T^{2} + 19968 T^{4} + 1322807 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 8 T + 97 T^{2} )^{8}$$