# Properties

 Label 336.2.bl.c Level $336$ Weight $2$ Character orbit 336.bl 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.bl (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{3} + ( 2 - \zeta_{6} ) q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{3} + ( 2 - \zeta_{6} ) q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 1 + \zeta_{6} ) q^{11} + ( -1 + 2 \zeta_{6} ) q^{15} + ( 2 + 2 \zeta_{6} ) q^{17} + 2 \zeta_{6} q^{19} + ( -2 - \zeta_{6} ) q^{21} + ( -2 + 2 \zeta_{6} ) q^{25} + q^{27} + 9 q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} + ( -2 + \zeta_{6} ) q^{33} + ( 1 + 4 \zeta_{6} ) q^{35} -10 \zeta_{6} q^{37} + ( -6 + 12 \zeta_{6} ) q^{41} + ( 2 - 4 \zeta_{6} ) q^{43} + ( -1 - \zeta_{6} ) q^{45} -12 \zeta_{6} q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + ( -4 + 2 \zeta_{6} ) q^{51} + ( 9 - 9 \zeta_{6} ) q^{53} + 3 q^{55} -2 q^{57} + ( 9 - 9 \zeta_{6} ) q^{59} + ( 3 - 2 \zeta_{6} ) q^{63} + ( -8 - 8 \zeta_{6} ) q^{67} + ( 8 - 16 \zeta_{6} ) q^{71} + ( -4 - 4 \zeta_{6} ) q^{73} -2 \zeta_{6} q^{75} + ( -4 + 5 \zeta_{6} ) q^{77} + ( 6 - 3 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + 3 q^{83} + 6 q^{85} + ( -9 + 9 \zeta_{6} ) q^{87} + ( -4 + 2 \zeta_{6} ) q^{89} -5 \zeta_{6} q^{93} + ( 2 + 2 \zeta_{6} ) q^{95} + ( -11 + 22 \zeta_{6} ) q^{97} + ( 1 - 2 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} + 3q^{5} + q^{7} - q^{9} + O(q^{10})$$ $$2q - q^{3} + 3q^{5} + q^{7} - q^{9} + 3q^{11} + 6q^{17} + 2q^{19} - 5q^{21} - 2q^{25} + 2q^{27} + 18q^{29} - 5q^{31} - 3q^{33} + 6q^{35} - 10q^{37} - 3q^{45} - 12q^{47} - 13q^{49} - 6q^{51} + 9q^{53} + 6q^{55} - 4q^{57} + 9q^{59} + 4q^{63} - 24q^{67} - 12q^{73} - 2q^{75} - 3q^{77} + 9q^{79} - q^{81} + 6q^{83} + 12q^{85} - 9q^{87} - 6q^{89} - 5q^{93} + 6q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 + 0.866025i 0 1.50000 0.866025i 0 0.500000 + 2.59808i 0 −0.500000 0.866025i 0
271.1 0 −0.500000 0.866025i 0 1.50000 + 0.866025i 0 0.500000 2.59808i 0 −0.500000 + 0.866025i 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
28.f 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.bl.c 2
3.b odd 2 1 1008.2.cs.e 2
4.b odd 2 1 336.2.bl.g yes 2
7.b odd 2 1 2352.2.bl.h 2
7.c even 3 1 2352.2.b.g 2
7.c even 3 1 2352.2.bl.b 2
7.d odd 6 1 336.2.bl.g yes 2
7.d odd 6 1 2352.2.b.c 2
8.b even 2 1 1344.2.bl.f 2
8.d odd 2 1 1344.2.bl.b 2
12.b even 2 1 1008.2.cs.d 2
21.g even 6 1 1008.2.cs.d 2
21.g even 6 1 7056.2.b.i 2
21.h odd 6 1 7056.2.b.e 2
28.d even 2 1 2352.2.bl.b 2
28.f even 6 1 inner 336.2.bl.c 2
28.f even 6 1 2352.2.b.g 2
28.g odd 6 1 2352.2.b.c 2
28.g odd 6 1 2352.2.bl.h 2
56.j odd 6 1 1344.2.bl.b 2
56.m even 6 1 1344.2.bl.f 2
84.j odd 6 1 1008.2.cs.e 2
84.j odd 6 1 7056.2.b.e 2
84.n even 6 1 7056.2.b.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bl.c 2 1.a even 1 1 trivial
336.2.bl.c 2 28.f even 6 1 inner
336.2.bl.g yes 2 4.b odd 2 1
336.2.bl.g yes 2 7.d odd 6 1
1008.2.cs.d 2 12.b even 2 1
1008.2.cs.d 2 21.g even 6 1
1008.2.cs.e 2 3.b odd 2 1
1008.2.cs.e 2 84.j odd 6 1
1344.2.bl.b 2 8.d odd 2 1
1344.2.bl.b 2 56.j odd 6 1
1344.2.bl.f 2 8.b even 2 1
1344.2.bl.f 2 56.m even 6 1
2352.2.b.c 2 7.d odd 6 1
2352.2.b.c 2 28.g odd 6 1
2352.2.b.g 2 7.c even 3 1
2352.2.b.g 2 28.f even 6 1
2352.2.bl.b 2 7.c even 3 1
2352.2.bl.b 2 28.d even 2 1
2352.2.bl.h 2 7.b odd 2 1
2352.2.bl.h 2 28.g odd 6 1
7056.2.b.e 2 21.h odd 6 1
7056.2.b.e 2 84.j odd 6 1
7056.2.b.i 2 21.g even 6 1
7056.2.b.i 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 $$S_{2}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}^{2} - 3 T_{5} + 3$$ $$T_{11}^{2} - 3 T_{11} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$3 - 3 T + T^{2}$$
$7$ $$7 - T + T^{2}$$
$11$ $$3 - 3 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$12 - 6 T + T^{2}$$
$19$ $$4 - 2 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -9 + T )^{2}$$
$31$ $$25 + 5 T + T^{2}$$
$37$ $$100 + 10 T + T^{2}$$
$41$ $$108 + T^{2}$$
$43$ $$12 + T^{2}$$
$47$ $$144 + 12 T + T^{2}$$
$53$ $$81 - 9 T + T^{2}$$
$59$ $$81 - 9 T + T^{2}$$
$61$ $$T^{2}$$
$67$ $$192 + 24 T + T^{2}$$
$71$ $$192 + T^{2}$$
$73$ $$48 + 12 T + T^{2}$$
$79$ $$27 - 9 T + T^{2}$$
$83$ $$( -3 + T )^{2}$$
$89$ $$12 + 6 T + T^{2}$$
$97$ $$363 + T^{2}$$