# Properties

 Label 336.4.bc.b Level $336$ Weight $4$ Character orbit 336.bc Analytic conductor $19.825$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$19.8246417619$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 3 + 3 \zeta_{6} ) q^{3} + ( 18 - 19 \zeta_{6} ) q^{7} + 27 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 3 + 3 \zeta_{6} ) q^{3} + ( 18 - 19 \zeta_{6} ) q^{7} + 27 \zeta_{6} q^{9} + ( -53 + 106 \zeta_{6} ) q^{13} + ( 146 - 73 \zeta_{6} ) q^{19} + ( 111 - 60 \zeta_{6} ) q^{21} + ( 125 - 125 \zeta_{6} ) q^{25} + ( -81 + 162 \zeta_{6} ) q^{27} + ( 109 + 109 \zeta_{6} ) q^{31} + 323 \zeta_{6} q^{37} + ( -477 + 477 \zeta_{6} ) q^{39} + 71 q^{43} + ( -37 - 323 \zeta_{6} ) q^{49} + 657 q^{57} + ( -1080 + 540 \zeta_{6} ) q^{61} + ( 513 - 27 \zeta_{6} ) q^{63} + ( -127 + 127 \zeta_{6} ) q^{67} + ( 703 + 703 \zeta_{6} ) q^{73} + ( 750 - 375 \zeta_{6} ) q^{75} -1387 \zeta_{6} q^{79} + ( -729 + 729 \zeta_{6} ) q^{81} + ( 1060 + 901 \zeta_{6} ) q^{91} + 981 \zeta_{6} q^{93} + ( 792 - 1584 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 9q^{3} + 17q^{7} + 27q^{9} + O(q^{10})$$ $$2q + 9q^{3} + 17q^{7} + 27q^{9} + 219q^{19} + 162q^{21} + 125q^{25} + 327q^{31} + 323q^{37} - 477q^{39} + 142q^{43} - 397q^{49} + 1314q^{57} - 1620q^{61} + 999q^{63} - 127q^{67} + 2109q^{73} + 1125q^{75} - 1387q^{79} - 729q^{81} + 3021q^{91} + 981q^{93} + O(q^{100})$$

## Character values

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

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 4.50000 + 2.59808i 0 0 0 8.50000 16.4545i 0 13.5000 + 23.3827i 0
257.1 0 4.50000 2.59808i 0 0 0 8.50000 + 16.4545i 0 13.5000 23.3827i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.bc.b 2
3.b odd 2 1 CM 336.4.bc.b 2
4.b odd 2 1 84.4.k.a 2
7.d odd 6 1 inner 336.4.bc.b 2
12.b even 2 1 84.4.k.a 2
21.g even 6 1 inner 336.4.bc.b 2
28.d even 2 1 588.4.k.b 2
28.f even 6 1 84.4.k.a 2
28.f even 6 1 588.4.f.a 2
28.g odd 6 1 588.4.f.a 2
28.g odd 6 1 588.4.k.b 2
84.h odd 2 1 588.4.k.b 2
84.j odd 6 1 84.4.k.a 2
84.j odd 6 1 588.4.f.a 2
84.n even 6 1 588.4.f.a 2
84.n even 6 1 588.4.k.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.k.a 2 4.b odd 2 1
84.4.k.a 2 12.b even 2 1
84.4.k.a 2 28.f even 6 1
84.4.k.a 2 84.j odd 6 1
336.4.bc.b 2 1.a even 1 1 trivial
336.4.bc.b 2 3.b odd 2 1 CM
336.4.bc.b 2 7.d odd 6 1 inner
336.4.bc.b 2 21.g even 6 1 inner
588.4.f.a 2 28.f even 6 1
588.4.f.a 2 28.g odd 6 1
588.4.f.a 2 84.j odd 6 1
588.4.f.a 2 84.n even 6 1
588.4.k.b 2 28.d even 2 1
588.4.k.b 2 28.g odd 6 1
588.4.k.b 2 84.h odd 2 1
588.4.k.b 2 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}$$ $$T_{13}^{2} + 8427$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$27 - 9 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$343 - 17 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$8427 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$15987 - 219 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$35643 - 327 T + T^{2}$$
$37$ $$104329 - 323 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -71 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$874800 + 1620 T + T^{2}$$
$67$ $$16129 + 127 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$1482627 - 2109 T + T^{2}$$
$79$ $$1923769 + 1387 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$1881792 + T^{2}$$