# Properties

 Label 336.2.bl.h 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( 4 - 2 \zeta_{6} ) q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( 4 - 2 \zeta_{6} ) q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 2 + 2 \zeta_{6} ) q^{11} + ( 1 - 2 \zeta_{6} ) q^{13} + ( 2 - 4 \zeta_{6} ) q^{15} -5 \zeta_{6} q^{19} + ( 1 + 2 \zeta_{6} ) q^{21} + ( -8 + 4 \zeta_{6} ) q^{23} + ( 7 - 7 \zeta_{6} ) q^{25} - q^{27} + ( 5 - 5 \zeta_{6} ) q^{31} + ( 4 - 2 \zeta_{6} ) q^{33} + ( -2 + 10 \zeta_{6} ) q^{35} + 11 \zeta_{6} q^{37} + ( -1 - \zeta_{6} ) q^{39} + ( 2 - 4 \zeta_{6} ) q^{41} + ( 5 - 10 \zeta_{6} ) q^{43} + ( -2 - 2 \zeta_{6} ) q^{45} + 6 \zeta_{6} q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} + ( -12 + 12 \zeta_{6} ) q^{53} + 12 q^{55} -5 q^{57} + ( -12 + 12 \zeta_{6} ) q^{59} + ( -16 + 8 \zeta_{6} ) q^{61} + ( 3 - \zeta_{6} ) q^{63} -6 \zeta_{6} q^{65} + ( -5 - 5 \zeta_{6} ) q^{67} + ( -4 + 8 \zeta_{6} ) q^{69} + ( 2 - 4 \zeta_{6} ) q^{71} + ( -3 - 3 \zeta_{6} ) q^{73} -7 \zeta_{6} q^{75} + ( -10 + 8 \zeta_{6} ) q^{77} + ( -14 + 7 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + 18 q^{83} + ( 8 - 4 \zeta_{6} ) q^{89} + ( 4 + \zeta_{6} ) q^{91} -5 \zeta_{6} q^{93} + ( -10 - 10 \zeta_{6} ) q^{95} + ( -4 + 8 \zeta_{6} ) q^{97} + ( 2 - 4 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + 6q^{5} - q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} + 6q^{5} - q^{7} - q^{9} + 6q^{11} - 5q^{19} + 4q^{21} - 12q^{23} + 7q^{25} - 2q^{27} + 5q^{31} + 6q^{33} + 6q^{35} + 11q^{37} - 3q^{39} - 6q^{45} + 6q^{47} - 13q^{49} - 12q^{53} + 24q^{55} - 10q^{57} - 12q^{59} - 24q^{61} + 5q^{63} - 6q^{65} - 15q^{67} - 9q^{73} - 7q^{75} - 12q^{77} - 21q^{79} - q^{81} + 36q^{83} + 12q^{89} + 9q^{91} - 5q^{93} - 30q^{95} + O(q^{100})$$

## 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 3.00000 1.73205i 0 −0.500000 + 2.59808i 0 −0.500000 0.866025i 0
271.1 0 0.500000 + 0.866025i 0 3.00000 + 1.73205i 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.h yes 2
3.b odd 2 1 1008.2.cs.a 2
4.b odd 2 1 336.2.bl.d 2
7.b odd 2 1 2352.2.bl.a 2
7.c even 3 1 2352.2.b.d 2
7.c even 3 1 2352.2.bl.g 2
7.d odd 6 1 336.2.bl.d 2
7.d odd 6 1 2352.2.b.e 2
8.b even 2 1 1344.2.bl.a 2
8.d odd 2 1 1344.2.bl.e 2
12.b even 2 1 1008.2.cs.b 2
21.g even 6 1 1008.2.cs.b 2
21.g even 6 1 7056.2.b.d 2
21.h odd 6 1 7056.2.b.k 2
28.d even 2 1 2352.2.bl.g 2
28.f even 6 1 inner 336.2.bl.h yes 2
28.f even 6 1 2352.2.b.d 2
28.g odd 6 1 2352.2.b.e 2
28.g odd 6 1 2352.2.bl.a 2
56.j odd 6 1 1344.2.bl.e 2
56.m even 6 1 1344.2.bl.a 2
84.j odd 6 1 1008.2.cs.a 2
84.j odd 6 1 7056.2.b.k 2
84.n even 6 1 7056.2.b.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bl.d 2 4.b odd 2 1
336.2.bl.d 2 7.d odd 6 1
336.2.bl.h yes 2 1.a even 1 1 trivial
336.2.bl.h yes 2 28.f even 6 1 inner
1008.2.cs.a 2 3.b odd 2 1
1008.2.cs.a 2 84.j odd 6 1
1008.2.cs.b 2 12.b even 2 1
1008.2.cs.b 2 21.g even 6 1
1344.2.bl.a 2 8.b even 2 1
1344.2.bl.a 2 56.m even 6 1
1344.2.bl.e 2 8.d odd 2 1
1344.2.bl.e 2 56.j odd 6 1
2352.2.b.d 2 7.c even 3 1
2352.2.b.d 2 28.f even 6 1
2352.2.b.e 2 7.d odd 6 1
2352.2.b.e 2 28.g odd 6 1
2352.2.bl.a 2 7.b odd 2 1
2352.2.bl.a 2 28.g odd 6 1
2352.2.bl.g 2 7.c even 3 1
2352.2.bl.g 2 28.d even 2 1
7056.2.b.d 2 21.g even 6 1
7056.2.b.d 2 84.n even 6 1
7056.2.b.k 2 21.h odd 6 1
7056.2.b.k 2 84.j odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 6 T_{5} + 12$$ $$T_{11}^{2} - 6 T_{11} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$12 - 6 T + T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$12 - 6 T + T^{2}$$
$13$ $$3 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$25 + 5 T + T^{2}$$
$23$ $$48 + 12 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$25 - 5 T + T^{2}$$
$37$ $$121 - 11 T + T^{2}$$
$41$ $$12 + T^{2}$$
$43$ $$75 + T^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$144 + 12 T + T^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$192 + 24 T + T^{2}$$
$67$ $$75 + 15 T + T^{2}$$
$71$ $$12 + T^{2}$$
$73$ $$27 + 9 T + T^{2}$$
$79$ $$147 + 21 T + T^{2}$$
$83$ $$( -18 + T )^{2}$$
$89$ $$48 - 12 T + T^{2}$$
$97$ $$48 + T^{2}$$