# Properties

 Label 336.2.bc.a Level 336 Weight 2 Character orbit 336.bc Analytic conductor 2.683 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: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.bc (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{6} ) q^{3} + ( 2 + \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{6} ) q^{3} + ( 2 + \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( 3 - 6 \zeta_{6} ) q^{13} + ( 10 - 5 \zeta_{6} ) q^{19} + ( -1 - 4 \zeta_{6} ) q^{21} + ( 5 - 5 \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 1 + \zeta_{6} ) q^{31} + 11 \zeta_{6} q^{37} + ( -9 + 9 \zeta_{6} ) q^{39} -13 q^{43} + ( 3 + 5 \zeta_{6} ) q^{49} -15 q^{57} + ( -8 + 4 \zeta_{6} ) q^{61} + ( -3 + 9 \zeta_{6} ) q^{63} + ( 5 - 5 \zeta_{6} ) q^{67} + ( -1 - \zeta_{6} ) q^{73} + ( -10 + 5 \zeta_{6} ) q^{75} + 17 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 12 - 15 \zeta_{6} ) q^{91} -3 \zeta_{6} q^{93} + ( 8 - 16 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} + 5q^{7} + 3q^{9} + O(q^{10})$$ $$2q - 3q^{3} + 5q^{7} + 3q^{9} + 15q^{19} - 6q^{21} + 5q^{25} + 3q^{31} + 11q^{37} - 9q^{39} - 26q^{43} + 11q^{49} - 30q^{57} - 12q^{61} + 3q^{63} + 5q^{67} - 3q^{73} - 15q^{75} + 17q^{79} - 9q^{81} + 9q^{91} - 3q^{93} + O(q^{100})$$

## 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 −1.50000 0.866025i 0 0 0 2.50000 + 0.866025i 0 1.50000 + 2.59808i 0
257.1 0 −1.50000 + 0.866025i 0 0 0 2.50000 0.866025i 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
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 336.2.bc.a 2
3.b odd 2 1 CM 336.2.bc.a 2
4.b odd 2 1 84.2.k.c 2
7.c even 3 1 2352.2.k.b 2
7.d odd 6 1 inner 336.2.bc.a 2
7.d odd 6 1 2352.2.k.b 2
12.b even 2 1 84.2.k.c 2
20.d odd 2 1 2100.2.bi.d 2
20.e even 4 2 2100.2.bo.e 4
21.g even 6 1 inner 336.2.bc.a 2
21.g even 6 1 2352.2.k.b 2
21.h odd 6 1 2352.2.k.b 2
28.d even 2 1 588.2.k.b 2
28.f even 6 1 84.2.k.c 2
28.f even 6 1 588.2.f.b 2
28.g odd 6 1 588.2.f.b 2
28.g odd 6 1 588.2.k.b 2
36.f odd 6 1 2268.2.w.e 2
36.f odd 6 1 2268.2.bm.d 2
36.h even 6 1 2268.2.w.e 2
36.h even 6 1 2268.2.bm.d 2
60.h even 2 1 2100.2.bi.d 2
60.l odd 4 2 2100.2.bo.e 4
84.h odd 2 1 588.2.k.b 2
84.j odd 6 1 84.2.k.c 2
84.j odd 6 1 588.2.f.b 2
84.n even 6 1 588.2.f.b 2
84.n even 6 1 588.2.k.b 2
140.s even 6 1 2100.2.bi.d 2
140.x odd 12 2 2100.2.bo.e 4
252.n even 6 1 2268.2.w.e 2
252.r odd 6 1 2268.2.bm.d 2
252.bj even 6 1 2268.2.bm.d 2
252.bn odd 6 1 2268.2.w.e 2
420.be odd 6 1 2100.2.bi.d 2
420.br even 12 2 2100.2.bo.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.k.c 2 4.b odd 2 1
84.2.k.c 2 12.b even 2 1
84.2.k.c 2 28.f even 6 1
84.2.k.c 2 84.j odd 6 1
336.2.bc.a 2 1.a even 1 1 trivial
336.2.bc.a 2 3.b odd 2 1 CM
336.2.bc.a 2 7.d odd 6 1 inner
336.2.bc.a 2 21.g even 6 1 inner
588.2.f.b 2 28.f even 6 1
588.2.f.b 2 28.g odd 6 1
588.2.f.b 2 84.j odd 6 1
588.2.f.b 2 84.n even 6 1
588.2.k.b 2 28.d even 2 1
588.2.k.b 2 28.g odd 6 1
588.2.k.b 2 84.h odd 2 1
588.2.k.b 2 84.n even 6 1
2100.2.bi.d 2 20.d odd 2 1
2100.2.bi.d 2 60.h even 2 1
2100.2.bi.d 2 140.s even 6 1
2100.2.bi.d 2 420.be odd 6 1
2100.2.bo.e 4 20.e even 4 2
2100.2.bo.e 4 60.l odd 4 2
2100.2.bo.e 4 140.x odd 12 2
2100.2.bo.e 4 420.br even 12 2
2268.2.w.e 2 36.f odd 6 1
2268.2.w.e 2 36.h even 6 1
2268.2.w.e 2 252.n even 6 1
2268.2.w.e 2 252.bn odd 6 1
2268.2.bm.d 2 36.f odd 6 1
2268.2.bm.d 2 36.h even 6 1
2268.2.bm.d 2 252.r odd 6 1
2268.2.bm.d 2 252.bj even 6 1
2352.2.k.b 2 7.c even 3 1
2352.2.k.b 2 7.d odd 6 1
2352.2.k.b 2 21.g even 6 1
2352.2.k.b 2 21.h odd 6 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{13}^{2} + 27$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 3 T + 3 T^{2}$$
$5$ $$1 - 5 T^{2} + 25 T^{4}$$
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$1 + 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 - 7 T + 19 T^{2} )$$
$23$ $$1 + 23 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 29 T^{2} )^{2}$$
$31$ $$( 1 - 7 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$
$37$ $$( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} )$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 + 13 T + 43 T^{2} )^{2}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 + 53 T^{2} + 2809 T^{4}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$
$67$ $$( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} )$$
$71$ $$( 1 - 71 T^{2} )^{2}$$
$73$ $$( 1 - 7 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$
$79$ $$( 1 - 13 T + 79 T^{2} )( 1 - 4 T + 79 T^{2} )$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$1 - 89 T^{2} + 7921 T^{4}$$
$97$ $$( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$