# Properties

 Label 336.2.bc.e Level 336 Weight 2 Character orbit 336.bc Analytic conductor 2.683 Analytic rank 0 Dimension 4 CM no 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( 2 + \zeta_{12}^{2} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( 2 + \zeta_{12}^{2} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{11} + ( 2 - 4 \zeta_{12}^{2} ) q^{13} -3 q^{15} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + ( -2 - 2 \zeta_{12}^{2} ) q^{19} + ( -\zeta_{12} - 4 \zeta_{12}^{3} ) q^{21} -6 \zeta_{12} q^{23} + 2 \zeta_{12}^{2} q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + 3 \zeta_{12}^{3} q^{29} + ( 2 - \zeta_{12}^{2} ) q^{31} + ( -3 - 3 \zeta_{12}^{2} ) q^{33} + ( 4 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{35} + ( 2 - 2 \zeta_{12}^{2} ) q^{37} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{39} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{41} + 8 q^{43} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{45} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{47} + ( 3 + 5 \zeta_{12}^{2} ) q^{49} + ( -6 + 6 \zeta_{12}^{2} ) q^{51} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{53} + ( 3 - 6 \zeta_{12}^{2} ) q^{55} + 6 \zeta_{12}^{3} q^{57} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{59} + ( -9 + 6 \zeta_{12}^{2} ) q^{63} -6 \zeta_{12} q^{65} + 2 \zeta_{12}^{2} q^{67} + ( -6 + 12 \zeta_{12}^{2} ) q^{69} + 12 \zeta_{12}^{3} q^{71} + ( 8 - 4 \zeta_{12}^{2} ) q^{73} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{75} + ( 9 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{77} + ( -1 + \zeta_{12}^{2} ) q^{79} -9 \zeta_{12}^{2} q^{81} + ( 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{83} -6 q^{85} + ( 6 - 3 \zeta_{12}^{2} ) q^{87} + ( -6 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{89} + ( 8 - 10 \zeta_{12}^{2} ) q^{91} -3 \zeta_{12} q^{93} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{95} + ( 3 - 6 \zeta_{12}^{2} ) q^{97} + 9 \zeta_{12}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 10q^{7} - 6q^{9} + O(q^{10})$$ $$4q + 10q^{7} - 6q^{9} - 12q^{15} - 12q^{19} + 4q^{25} + 6q^{31} - 18q^{33} + 4q^{37} + 32q^{43} + 22q^{49} - 12q^{51} - 24q^{63} + 4q^{67} + 24q^{73} - 2q^{79} - 18q^{81} - 24q^{85} + 18q^{87} + 12q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1 - \zeta_{12}$$

## 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
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −0.866025 + 1.50000i 0 0.866025 + 1.50000i 0 2.50000 0.866025i 0 −1.50000 2.59808i 0
17.2 0 0.866025 1.50000i 0 −0.866025 1.50000i 0 2.50000 0.866025i 0 −1.50000 2.59808i 0
257.1 0 −0.866025 1.50000i 0 0.866025 1.50000i 0 2.50000 + 0.866025i 0 −1.50000 + 2.59808i 0
257.2 0 0.866025 + 1.50000i 0 −0.866025 + 1.50000i 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 inner
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.e 4
3.b odd 2 1 inner 336.2.bc.e 4
4.b odd 2 1 42.2.f.a 4
7.c even 3 1 2352.2.k.e 4
7.d odd 6 1 inner 336.2.bc.e 4
7.d odd 6 1 2352.2.k.e 4
12.b even 2 1 42.2.f.a 4
20.d odd 2 1 1050.2.s.b 4
20.e even 4 1 1050.2.u.a 4
20.e even 4 1 1050.2.u.d 4
21.g even 6 1 inner 336.2.bc.e 4
21.g even 6 1 2352.2.k.e 4
21.h odd 6 1 2352.2.k.e 4
28.d even 2 1 294.2.f.a 4
28.f even 6 1 42.2.f.a 4
28.f even 6 1 294.2.d.a 4
28.g odd 6 1 294.2.d.a 4
28.g odd 6 1 294.2.f.a 4
36.f odd 6 1 1134.2.l.c 4
36.f odd 6 1 1134.2.t.d 4
36.h even 6 1 1134.2.l.c 4
36.h even 6 1 1134.2.t.d 4
60.h even 2 1 1050.2.s.b 4
60.l odd 4 1 1050.2.u.a 4
60.l odd 4 1 1050.2.u.d 4
84.h odd 2 1 294.2.f.a 4
84.j odd 6 1 42.2.f.a 4
84.j odd 6 1 294.2.d.a 4
84.n even 6 1 294.2.d.a 4
84.n even 6 1 294.2.f.a 4
140.s even 6 1 1050.2.s.b 4
140.x odd 12 1 1050.2.u.a 4
140.x odd 12 1 1050.2.u.d 4
252.n even 6 1 1134.2.l.c 4
252.r odd 6 1 1134.2.t.d 4
252.bj even 6 1 1134.2.t.d 4
252.bn odd 6 1 1134.2.l.c 4
420.be odd 6 1 1050.2.s.b 4
420.br even 12 1 1050.2.u.a 4
420.br even 12 1 1050.2.u.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.f.a 4 4.b odd 2 1
42.2.f.a 4 12.b even 2 1
42.2.f.a 4 28.f even 6 1
42.2.f.a 4 84.j odd 6 1
294.2.d.a 4 28.f even 6 1
294.2.d.a 4 28.g odd 6 1
294.2.d.a 4 84.j odd 6 1
294.2.d.a 4 84.n even 6 1
294.2.f.a 4 28.d even 2 1
294.2.f.a 4 28.g odd 6 1
294.2.f.a 4 84.h odd 2 1
294.2.f.a 4 84.n even 6 1
336.2.bc.e 4 1.a even 1 1 trivial
336.2.bc.e 4 3.b odd 2 1 inner
336.2.bc.e 4 7.d odd 6 1 inner
336.2.bc.e 4 21.g even 6 1 inner
1050.2.s.b 4 20.d odd 2 1
1050.2.s.b 4 60.h even 2 1
1050.2.s.b 4 140.s even 6 1
1050.2.s.b 4 420.be odd 6 1
1050.2.u.a 4 20.e even 4 1
1050.2.u.a 4 60.l odd 4 1
1050.2.u.a 4 140.x odd 12 1
1050.2.u.a 4 420.br even 12 1
1050.2.u.d 4 20.e even 4 1
1050.2.u.d 4 60.l odd 4 1
1050.2.u.d 4 140.x odd 12 1
1050.2.u.d 4 420.br even 12 1
1134.2.l.c 4 36.f odd 6 1
1134.2.l.c 4 36.h even 6 1
1134.2.l.c 4 252.n even 6 1
1134.2.l.c 4 252.bn odd 6 1
1134.2.t.d 4 36.f odd 6 1
1134.2.t.d 4 36.h even 6 1
1134.2.t.d 4 252.r odd 6 1
1134.2.t.d 4 252.bj even 6 1
2352.2.k.e 4 7.c even 3 1
2352.2.k.e 4 7.d odd 6 1
2352.2.k.e 4 21.g even 6 1
2352.2.k.e 4 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 $$S_{2}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}^{4} + 3 T_{5}^{2} + 9$$ $$T_{13}^{2} + 12$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 3 T^{2} + 9 T^{4}$$
$5$ $$1 - 7 T^{2} + 24 T^{4} - 175 T^{6} + 625 T^{8}$$
$7$ $$( 1 - 5 T + 7 T^{2} )^{2}$$
$11$ $$1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$
$17$ $$1 - 22 T^{2} + 195 T^{4} - 6358 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2}$$
$23$ $$1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 49 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 7 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 34 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{4}$$
$47$ $$1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8}$$
$53$ $$1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8}$$
$59$ $$1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 + 61 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 2 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 91 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2}$$