# Properties

 Label 336.3.bh.e Level $336$ Weight $3$ Character orbit 336.bh Analytic conductor $9.155$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 336.bh (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.15533688251$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{3} + ( 4 + 2 \beta_{1} + \beta_{3} ) q^{5} + ( -5 \beta_{1} + \beta_{3} ) q^{7} -3 \beta_{1} q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{3} + ( 4 + 2 \beta_{1} + \beta_{3} ) q^{5} + ( -5 \beta_{1} + \beta_{3} ) q^{7} -3 \beta_{1} q^{9} + ( 6 + 6 \beta_{1} ) q^{11} + ( 1 + 2 \beta_{1} - 4 \beta_{2} ) q^{13} + ( -6 + \beta_{2} - 2 \beta_{3} ) q^{15} + ( -8 + 8 \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + ( 14 + 7 \beta_{1} + \beta_{3} ) q^{19} + ( 5 + 10 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{21} + ( 12 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{23} + ( 11 + 11 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{25} + ( 3 + 6 \beta_{1} ) q^{27} + ( -4 \beta_{2} + 8 \beta_{3} ) q^{29} + ( -17 + 17 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{31} + ( -12 - 6 \beta_{1} ) q^{33} + ( 34 + 14 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} ) q^{35} + ( -11 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{37} + ( -3 - 3 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{39} + ( 26 + 52 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -7 + \beta_{2} - 2 \beta_{3} ) q^{43} + ( 6 - 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{45} + ( 44 + 22 \beta_{1} - \beta_{3} ) q^{47} + ( -1 - \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{49} + ( -24 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{51} + ( 60 + 60 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{53} + ( 12 + 24 \beta_{1} + 6 \beta_{2} ) q^{55} + ( -21 + \beta_{2} - 2 \beta_{3} ) q^{57} + ( 4 - 4 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} ) q^{59} + ( -24 - 12 \beta_{1} - 4 \beta_{3} ) q^{61} + ( -15 - 15 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{63} + ( -90 \beta_{1} - 14 \beta_{2} + 7 \beta_{3} ) q^{65} + ( -55 - 55 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} ) q^{67} + ( -12 - 24 \beta_{1} + 9 \beta_{2} ) q^{69} + ( -78 - 7 \beta_{2} + 14 \beta_{3} ) q^{71} + ( -11 + 11 \beta_{1} + 20 \beta_{2} - 20 \beta_{3} ) q^{73} + ( -22 - 11 \beta_{1} - 12 \beta_{3} ) q^{75} + ( 30 + 6 \beta_{2} ) q^{77} + ( -5 \beta_{1} + 22 \beta_{2} - 11 \beta_{3} ) q^{79} + ( -9 - 9 \beta_{1} ) q^{81} + ( 10 + 20 \beta_{1} - 19 \beta_{2} ) q^{83} + ( -72 + 10 \beta_{2} - 20 \beta_{3} ) q^{85} + ( 12 \beta_{2} - 12 \beta_{3} ) q^{87} + ( -24 - 12 \beta_{1} ) q^{89} + ( 10 - 91 \beta_{1} + 2 \beta_{2} - 21 \beta_{3} ) q^{91} + ( -51 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{93} + ( 66 + 66 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} ) q^{95} + ( 12 + 24 \beta_{1} + 2 \beta_{2} ) q^{97} + 18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{3} + 12 q^{5} + 10 q^{7} + 6 q^{9} + O(q^{10})$$ $$4 q - 6 q^{3} + 12 q^{5} + 10 q^{7} + 6 q^{9} + 12 q^{11} - 24 q^{15} - 48 q^{17} + 42 q^{19} - 24 q^{23} + 22 q^{25} - 102 q^{31} - 36 q^{33} + 108 q^{35} + 22 q^{37} - 6 q^{39} - 28 q^{43} + 36 q^{45} + 132 q^{47} - 2 q^{49} + 48 q^{51} + 120 q^{53} - 84 q^{57} + 24 q^{59} - 72 q^{61} - 30 q^{63} + 180 q^{65} - 110 q^{67} - 312 q^{71} - 66 q^{73} - 66 q^{75} + 120 q^{77} + 10 q^{79} - 18 q^{81} - 288 q^{85} - 72 q^{89} + 222 q^{91} + 102 q^{93} + 132 q^{95} + 72 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4 \nu$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 2 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{3} + \beta_{2}$$$$)/3$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 −1.50000 + 0.866025i 0 −1.24264 0.717439i 0 −1.74264 6.77962i 0 1.50000 2.59808i 0
145.2 0 −1.50000 + 0.866025i 0 7.24264 + 4.18154i 0 6.74264 1.88064i 0 1.50000 2.59808i 0
241.1 0 −1.50000 0.866025i 0 −1.24264 + 0.717439i 0 −1.74264 + 6.77962i 0 1.50000 + 2.59808i 0
241.2 0 −1.50000 0.866025i 0 7.24264 4.18154i 0 6.74264 + 1.88064i 0 1.50000 + 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.3.bh.e 4
3.b odd 2 1 1008.3.cg.h 4
4.b odd 2 1 42.3.g.a 4
7.c even 3 1 2352.3.f.e 4
7.d odd 6 1 inner 336.3.bh.e 4
7.d odd 6 1 2352.3.f.e 4
12.b even 2 1 126.3.n.a 4
20.d odd 2 1 1050.3.p.a 4
20.e even 4 2 1050.3.q.a 8
21.g even 6 1 1008.3.cg.h 4
28.d even 2 1 294.3.g.a 4
28.f even 6 1 42.3.g.a 4
28.f even 6 1 294.3.c.a 4
28.g odd 6 1 294.3.c.a 4
28.g odd 6 1 294.3.g.a 4
84.h odd 2 1 882.3.n.e 4
84.j odd 6 1 126.3.n.a 4
84.j odd 6 1 882.3.c.b 4
84.n even 6 1 882.3.c.b 4
84.n even 6 1 882.3.n.e 4
140.s even 6 1 1050.3.p.a 4
140.x odd 12 2 1050.3.q.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.g.a 4 4.b odd 2 1
42.3.g.a 4 28.f even 6 1
126.3.n.a 4 12.b even 2 1
126.3.n.a 4 84.j odd 6 1
294.3.c.a 4 28.f even 6 1
294.3.c.a 4 28.g odd 6 1
294.3.g.a 4 28.d even 2 1
294.3.g.a 4 28.g odd 6 1
336.3.bh.e 4 1.a even 1 1 trivial
336.3.bh.e 4 7.d odd 6 1 inner
882.3.c.b 4 84.j odd 6 1
882.3.c.b 4 84.n even 6 1
882.3.n.e 4 84.h odd 2 1
882.3.n.e 4 84.n even 6 1
1008.3.cg.h 4 3.b odd 2 1
1008.3.cg.h 4 21.g even 6 1
1050.3.p.a 4 20.d odd 2 1
1050.3.p.a 4 140.s even 6 1
1050.3.q.a 8 20.e even 4 2
1050.3.q.a 8 140.x odd 12 2
2352.3.f.e 4 7.c even 3 1
2352.3.f.e 4 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 12 T_{5}^{3} + 36 T_{5}^{2} + 144 T_{5} + 144$$ acting on $$S_{3}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + 3 T + T^{2} )^{2}$$
$5$ $$144 + 144 T + 36 T^{2} - 12 T^{3} + T^{4}$$
$7$ $$2401 - 490 T + 51 T^{2} - 10 T^{3} + T^{4}$$
$11$ $$( 36 - 6 T + T^{2} )^{2}$$
$13$ $$145161 + 774 T^{2} + T^{4}$$
$17$ $$28224 + 8064 T + 936 T^{2} + 48 T^{3} + T^{4}$$
$19$ $$15129 - 5166 T + 711 T^{2} - 42 T^{3} + T^{4}$$
$23$ $$254016 - 12096 T + 1080 T^{2} + 24 T^{3} + T^{4}$$
$29$ $$( -1152 + T^{2} )^{2}$$
$31$ $$423801 + 66402 T + 4119 T^{2} + 102 T^{3} + T^{4}$$
$37$ $$27889 + 3674 T + 651 T^{2} - 22 T^{3} + T^{4}$$
$41$ $$3732624 + 4248 T^{2} + T^{4}$$
$43$ $$( -23 + 14 T + T^{2} )^{2}$$
$47$ $$2039184 - 188496 T + 7236 T^{2} - 132 T^{3} + T^{4}$$
$53$ $$8714304 - 354240 T + 11448 T^{2} - 120 T^{3} + T^{4}$$
$59$ $$1272384 + 27072 T - 936 T^{2} - 24 T^{3} + T^{4}$$
$61$ $$2304 + 3456 T + 1776 T^{2} + 72 T^{3} + T^{4}$$
$67$ $$253009 - 55330 T + 12603 T^{2} + 110 T^{3} + T^{4}$$
$71$ $$( 2556 + 156 T + T^{2} )^{2}$$
$73$ $$85322169 - 609642 T - 7785 T^{2} + 66 T^{3} + T^{4}$$
$79$ $$75463969 + 86870 T + 8787 T^{2} - 10 T^{3} + T^{4}$$
$83$ $$69956496 + 17928 T^{2} + T^{4}$$
$89$ $$( 432 + 36 T + T^{2} )^{2}$$
$97$ $$112896 + 1056 T^{2} + T^{4}$$