# 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 Defining polynomial: $$x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81$$ x^8 + 5*x^6 + 16*x^4 + 45*x^2 + 81 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (2 \beta_{6} + \beta_{4} - 2 \beta_{2} + 1) q^{5} + (3 \beta_{5} + \beta_{3}) q^{7} + (\beta_{6} - 2 \beta_{4} - \beta_{2} - 2) q^{9}+O(q^{10})$$ q + b1 * q^3 + (2*b6 + b4 - 2*b2 + 1) * q^5 + (3*b5 + b3) * q^7 + (b6 - 2*b4 - b2 - 2) * q^9 $$q + \beta_1 q^{3} + (2 \beta_{6} + \beta_{4} - 2 \beta_{2} + 1) q^{5} + (3 \beta_{5} + \beta_{3}) q^{7} + (\beta_{6} - 2 \beta_{4} - \beta_{2} - 2) q^{9} + (\beta_{5} - 2 \beta_1) q^{11} - 4 q^{13} + ( - \beta_{7} - 6 \beta_{5} - 6 \beta_{3} + \beta_1) q^{15} + (\beta_{4} - 2 \beta_{2}) q^{17} + 7 \beta_{3} q^{19} + (2 \beta_{6} + 3 \beta_{4} - 3 \beta_{2} + 2) q^{21} + (2 \beta_{7} + \beta_{3}) q^{23} - 6 \beta_{4} q^{25} + (2 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 2 \beta_1) q^{27} + (4 \beta_{6} + 2) q^{29} + 3 \beta_{5} q^{31} + ( - \beta_{6} + 5 \beta_{4} + \beta_{2} + 5) q^{33} + ( - 6 \beta_{7} - 2 \beta_{5} - 3 \beta_{3} + 4 \beta_1) q^{35} + (\beta_{4} + 1) q^{37} - 4 \beta_1 q^{39} + ( - 4 \beta_{6} - 2) q^{41} + (2 \beta_{5} + 2 \beta_{3}) q^{43} + ( - 8 \beta_{4} + 5 \beta_{2}) q^{45} + (6 \beta_{7} + 3 \beta_{3}) q^{47} + (8 \beta_{4} + 3) q^{49} + ( - \beta_{7} - 6 \beta_{3}) q^{51} + ( - \beta_{4} + 2 \beta_{2}) q^{53} + (11 \beta_{5} + 11 \beta_{3}) q^{55} + ( - 7 \beta_{6} - 7) q^{57} + ( - \beta_{5} + 2 \beta_1) q^{59} + ( - 3 \beta_{4} - 3) q^{61} + ( - 3 \beta_{7} - 6 \beta_{5} - 9 \beta_{3} + 2 \beta_1) q^{63} + ( - 8 \beta_{6} - 4 \beta_{4} + 8 \beta_{2} - 4) q^{65} + 9 \beta_{5} q^{67} + (\beta_{6} - 5) q^{69} + ( - 8 \beta_{7} - 4 \beta_{5} - 4 \beta_{3} + 8 \beta_1) q^{71} + 7 \beta_{4} q^{73} + 6 \beta_{7} q^{75} + ( - 4 \beta_{6} - 3 \beta_{4} + 6 \beta_{2} - 2) q^{77} + 9 \beta_{3} q^{79} + (\beta_{4} + 5 \beta_{2}) q^{81} + (8 \beta_{7} + 4 \beta_{5} + 4 \beta_{3} - 8 \beta_1) q^{83} + 11 q^{85} + ( - 12 \beta_{5} + 2 \beta_1) q^{87} + ( - 2 \beta_{6} - \beta_{4} + 2 \beta_{2} - 1) q^{89} + ( - 12 \beta_{5} - 4 \beta_{3}) q^{91} + (3 \beta_{6} + 3 \beta_{4} - 3 \beta_{2} + 3) q^{93} + (7 \beta_{5} - 14 \beta_1) q^{95} + 8 q^{97} + ( - 5 \beta_{7} + 3 \beta_{5} + 3 \beta_{3} + 5 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^3 + (2*b6 + b4 - 2*b2 + 1) * q^5 + (3*b5 + b3) * q^7 + (b6 - 2*b4 - b2 - 2) * q^9 + (b5 - 2*b1) * q^11 - 4 * q^13 + (-b7 - 6*b5 - 6*b3 + b1) * q^15 + (b4 - 2*b2) * q^17 + 7*b3 * q^19 + (2*b6 + 3*b4 - 3*b2 + 2) * q^21 + (2*b7 + b3) * q^23 - 6*b4 * q^25 + (2*b7 - 3*b5 - 3*b3 - 2*b1) * q^27 + (4*b6 + 2) * q^29 + 3*b5 * q^31 + (-b6 + 5*b4 + b2 + 5) * q^33 + (-6*b7 - 2*b5 - 3*b3 + 4*b1) * q^35 + (b4 + 1) * q^37 - 4*b1 * q^39 + (-4*b6 - 2) * q^41 + (2*b5 + 2*b3) * q^43 + (-8*b4 + 5*b2) * q^45 + (6*b7 + 3*b3) * q^47 + (8*b4 + 3) * q^49 + (-b7 - 6*b3) * q^51 + (-b4 + 2*b2) * q^53 + (11*b5 + 11*b3) * q^55 + (-7*b6 - 7) * q^57 + (-b5 + 2*b1) * q^59 + (-3*b4 - 3) * q^61 + (-3*b7 - 6*b5 - 9*b3 + 2*b1) * q^63 + (-8*b6 - 4*b4 + 8*b2 - 4) * q^65 + 9*b5 * q^67 + (b6 - 5) * q^69 + (-8*b7 - 4*b5 - 4*b3 + 8*b1) * q^71 + 7*b4 * q^73 + 6*b7 * q^75 + (-4*b6 - 3*b4 + 6*b2 - 2) * q^77 + 9*b3 * q^79 + (b4 + 5*b2) * q^81 + (8*b7 + 4*b5 + 4*b3 - 8*b1) * q^83 + 11 * q^85 + (-12*b5 + 2*b1) * q^87 + (-2*b6 - b4 + 2*b2 - 1) * q^89 + (-12*b5 - 4*b3) * q^91 + (3*b6 + 3*b4 - 3*b2 + 3) * q^93 + (7*b5 - 14*b1) * q^95 + 8 * q^97 + (-5*b7 + 3*b5 + 3*b3 + 5*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 10 q^{9}+O(q^{10})$$ 8 * q - 10 * q^9 $$8 q - 10 q^{9} - 32 q^{13} + 2 q^{21} + 24 q^{25} + 22 q^{33} + 4 q^{37} + 22 q^{45} - 8 q^{49} - 28 q^{57} - 12 q^{61} - 44 q^{69} - 28 q^{73} - 14 q^{81} + 88 q^{85} + 6 q^{93} + 64 q^{97}+O(q^{100})$$ 8 * q - 10 * q^9 - 32 * q^13 + 2 * q^21 + 24 * q^25 + 22 * q^33 + 4 * q^37 + 22 * q^45 - 8 * q^49 - 28 * q^57 - 12 * q^61 - 44 * q^69 - 28 * q^73 - 14 * q^81 + 88 * q^85 + 6 * q^93 + 64 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{6} + 32\nu^{4} + 16\nu^{2} + 45 ) / 144$$ (v^6 + 32*v^4 + 16*v^2 + 45) / 144 $$\beta_{3}$$ $$=$$ $$( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432$$ (v^7 + 32*v^5 + 16*v^3 + 45*v) / 432 $$\beta_{4}$$ $$=$$ $$( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144$$ (-5*v^6 - 16*v^4 - 80*v^2 - 225) / 144 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + 13\nu ) / 48$$ (v^7 + 13*v) / 48 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} - 13 ) / 16$$ (-v^6 - 13) / 16 $$\beta_{7}$$ $$=$$ $$( 5\nu^{7} + 16\nu^{5} + 80\nu^{3} + 225\nu ) / 144$$ (5*v^7 + 16*v^5 + 80*v^3 + 225*v) / 144
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} - 2\beta_{4} - \beta_{2} - 2$$ b6 - 2*b4 - b2 - 2 $$\nu^{3}$$ $$=$$ $$2\beta_{7} - 3\beta_{5} - 3\beta_{3} - 2\beta_1$$ 2*b7 - 3*b5 - 3*b3 - 2*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5\beta_{2}$$ b4 + 5*b2 $$\nu^{5}$$ $$=$$ $$-\beta_{7} + 15\beta_{3}$$ -b7 + 15*b3 $$\nu^{6}$$ $$=$$ $$-16\beta_{6} - 13$$ -16*b6 - 13 $$\nu^{7}$$ $$=$$ $$48\beta_{5} - 13\beta_1$$ 48*b5 - 13*b1

## 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 $$S_{2}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}^{4} - 11T_{5}^{2} + 121$$ T5^4 - 11*T5^2 + 121 $$T_{13} + 4$$ T13 + 4 $$T_{19}^{4} - 49T_{19}^{2} + 2401$$ T19^4 - 49*T19^2 + 2401

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + 5 T^{6} + 16 T^{4} + 45 T^{2} + \cdots + 81$$
$5$ $$(T^{4} - 11 T^{2} + 121)^{2}$$
$7$ $$(T^{4} + 2 T^{2} + 49)^{2}$$
$11$ $$(T^{4} + 11 T^{2} + 121)^{2}$$
$13$ $$(T + 4)^{8}$$
$17$ $$(T^{4} - 11 T^{2} + 121)^{2}$$
$19$ $$(T^{4} - 49 T^{2} + 2401)^{2}$$
$23$ $$(T^{4} + 11 T^{2} + 121)^{2}$$
$29$ $$(T^{2} + 44)^{4}$$
$31$ $$(T^{4} - 9 T^{2} + 81)^{2}$$
$37$ $$(T^{2} - T + 1)^{4}$$
$41$ $$(T^{2} + 44)^{4}$$
$43$ $$(T^{2} + 4)^{4}$$
$47$ $$(T^{4} + 99 T^{2} + 9801)^{2}$$
$53$ $$(T^{4} - 11 T^{2} + 121)^{2}$$
$59$ $$(T^{4} + 11 T^{2} + 121)^{2}$$
$61$ $$(T^{2} + 3 T + 9)^{4}$$
$67$ $$(T^{4} - 81 T^{2} + 6561)^{2}$$
$71$ $$(T^{2} - 176)^{4}$$
$73$ $$(T^{2} + 7 T + 49)^{4}$$
$79$ $$(T^{4} - 81 T^{2} + 6561)^{2}$$
$83$ $$(T^{2} - 176)^{4}$$
$89$ $$(T^{4} - 11 T^{2} + 121)^{2}$$
$97$ $$(T - 8)^{8}$$