# Properties

 Label 336.2.bc.d Level 336 Weight 2 Character orbit 336.bc Analytic conductor 2.683 Analytic rank 0 Dimension 2 CM no Inner twists 2

# 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: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 - \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( 2 - \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} + ( 3 + 3 \zeta_{6} ) q^{11} + ( 3 + 3 \zeta_{6} ) q^{15} + ( 3 - 3 \zeta_{6} ) q^{17} + ( -2 + \zeta_{6} ) q^{19} + ( -4 - \zeta_{6} ) q^{21} + ( -6 + 3 \zeta_{6} ) q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 1 + \zeta_{6} ) q^{31} + 9 q^{33} + ( 6 - 9 \zeta_{6} ) q^{35} -7 \zeta_{6} q^{37} -6 q^{41} -4 q^{43} + 9 q^{45} -3 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 3 - 6 \zeta_{6} ) q^{51} + ( 3 + 3 \zeta_{6} ) q^{53} + ( -9 + 18 \zeta_{6} ) q^{55} + ( -3 + 3 \zeta_{6} ) q^{57} + ( 3 - 3 \zeta_{6} ) q^{59} + ( -14 + 7 \zeta_{6} ) q^{61} + ( -9 + 3 \zeta_{6} ) q^{63} + ( 5 - 5 \zeta_{6} ) q^{67} + ( -9 + 9 \zeta_{6} ) q^{69} + ( -6 + 12 \zeta_{6} ) q^{71} + ( -7 - 7 \zeta_{6} ) q^{73} + ( -4 + 8 \zeta_{6} ) q^{75} + ( 3 - 15 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} -9 \zeta_{6} q^{81} -12 q^{83} + 9 q^{85} -9 \zeta_{6} q^{89} + 3 q^{93} + ( -3 - 3 \zeta_{6} ) q^{95} + ( -4 + 8 \zeta_{6} ) q^{97} + ( 18 - 9 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} + 3q^{5} - 4q^{7} + 3q^{9} + O(q^{10})$$ $$2q + 3q^{3} + 3q^{5} - 4q^{7} + 3q^{9} + 9q^{11} + 9q^{15} + 3q^{17} - 3q^{19} - 9q^{21} - 9q^{23} - 4q^{25} + 3q^{31} + 18q^{33} + 3q^{35} - 7q^{37} - 12q^{41} - 8q^{43} + 18q^{45} - 3q^{47} + 2q^{49} + 9q^{53} - 3q^{57} + 3q^{59} - 21q^{61} - 15q^{63} + 5q^{67} - 9q^{69} - 21q^{73} - 9q^{77} - q^{79} - 9q^{81} - 24q^{83} + 18q^{85} - 9q^{89} + 6q^{93} - 9q^{95} + 27q^{99} + O(q^{100})$$

## 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 1.50000 + 2.59808i 0 −2.00000 1.73205i 0 1.50000 2.59808i 0
257.1 0 1.50000 + 0.866025i 0 1.50000 2.59808i 0 −2.00000 + 1.73205i 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
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.d 2
3.b odd 2 1 336.2.bc.b 2
4.b odd 2 1 84.2.k.a 2
7.c even 3 1 2352.2.k.a 2
7.d odd 6 1 336.2.bc.b 2
7.d odd 6 1 2352.2.k.d 2
12.b even 2 1 84.2.k.b yes 2
20.d odd 2 1 2100.2.bi.f 2
20.e even 4 2 2100.2.bo.a 4
21.g even 6 1 inner 336.2.bc.d 2
21.g even 6 1 2352.2.k.a 2
21.h odd 6 1 2352.2.k.d 2
28.d even 2 1 588.2.k.d 2
28.f even 6 1 84.2.k.b yes 2
28.f even 6 1 588.2.f.a 2
28.g odd 6 1 588.2.f.c 2
28.g odd 6 1 588.2.k.c 2
36.f odd 6 1 2268.2.w.f 2
36.f odd 6 1 2268.2.bm.a 2
36.h even 6 1 2268.2.w.a 2
36.h even 6 1 2268.2.bm.f 2
60.h even 2 1 2100.2.bi.e 2
60.l odd 4 2 2100.2.bo.f 4
84.h odd 2 1 588.2.k.c 2
84.j odd 6 1 84.2.k.a 2
84.j odd 6 1 588.2.f.c 2
84.n even 6 1 588.2.f.a 2
84.n even 6 1 588.2.k.d 2
140.s even 6 1 2100.2.bi.e 2
140.x odd 12 2 2100.2.bo.f 4
252.n even 6 1 2268.2.w.a 2
252.r odd 6 1 2268.2.bm.a 2
252.bj even 6 1 2268.2.bm.f 2
252.bn odd 6 1 2268.2.w.f 2
420.be odd 6 1 2100.2.bi.f 2
420.br even 12 2 2100.2.bo.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.k.a 2 4.b odd 2 1
84.2.k.a 2 84.j odd 6 1
84.2.k.b yes 2 12.b even 2 1
84.2.k.b yes 2 28.f even 6 1
336.2.bc.b 2 3.b odd 2 1
336.2.bc.b 2 7.d odd 6 1
336.2.bc.d 2 1.a even 1 1 trivial
336.2.bc.d 2 21.g even 6 1 inner
588.2.f.a 2 28.f even 6 1
588.2.f.a 2 84.n even 6 1
588.2.f.c 2 28.g odd 6 1
588.2.f.c 2 84.j odd 6 1
588.2.k.c 2 28.g odd 6 1
588.2.k.c 2 84.h odd 2 1
588.2.k.d 2 28.d even 2 1
588.2.k.d 2 84.n even 6 1
2100.2.bi.e 2 60.h even 2 1
2100.2.bi.e 2 140.s even 6 1
2100.2.bi.f 2 20.d odd 2 1
2100.2.bi.f 2 420.be odd 6 1
2100.2.bo.a 4 20.e even 4 2
2100.2.bo.a 4 420.br even 12 2
2100.2.bo.f 4 60.l odd 4 2
2100.2.bo.f 4 140.x odd 12 2
2268.2.w.a 2 36.h even 6 1
2268.2.w.a 2 252.n even 6 1
2268.2.w.f 2 36.f odd 6 1
2268.2.w.f 2 252.bn odd 6 1
2268.2.bm.a 2 36.f odd 6 1
2268.2.bm.a 2 252.r odd 6 1
2268.2.bm.f 2 36.h even 6 1
2268.2.bm.f 2 252.bj even 6 1
2352.2.k.a 2 7.c even 3 1
2352.2.k.a 2 21.g even 6 1
2352.2.k.d 2 7.d odd 6 1
2352.2.k.d 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 $$S_{2}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}^{2} - 3 T_{5} + 9$$ $$T_{13}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 3 T + 3 T^{2}$$
$5$ $$1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4}$$
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 13 T^{2} )^{2}$$
$17$ $$1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4}$$
$19$ $$1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4}$$
$23$ $$1 + 9 T + 50 T^{2} + 207 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 29 T^{2} )^{2}$$
$31$ $$( 1 - 7 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$
$37$ $$1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{2}$$
$47$ $$1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4}$$
$53$ $$1 - 9 T + 80 T^{2} - 477 T^{3} + 2809 T^{4}$$
$59$ $$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$
$61$ $$1 + 21 T + 208 T^{2} + 1281 T^{3} + 3721 T^{4}$$
$67$ $$( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} )$$
$71$ $$1 - 34 T^{2} + 5041 T^{4}$$
$73$ $$1 + 21 T + 220 T^{2} + 1533 T^{3} + 5329 T^{4}$$
$79$ $$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 12 T + 83 T^{2} )^{2}$$
$89$ $$1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4}$$
$97$ $$1 - 146 T^{2} + 9409 T^{4}$$