# Properties

 Label 336.2.bl.d 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: $$\Q(\sqrt{-3})$$ 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.d 2
3.b odd 2 1 1008.2.cs.b 2
4.b odd 2 1 336.2.bl.h yes 2
7.b odd 2 1 2352.2.bl.g 2
7.c even 3 1 2352.2.b.e 2
7.c even 3 1 2352.2.bl.a 2
7.d odd 6 1 336.2.bl.h yes 2
7.d odd 6 1 2352.2.b.d 2
8.b even 2 1 1344.2.bl.e 2
8.d odd 2 1 1344.2.bl.a 2
12.b even 2 1 1008.2.cs.a 2
21.g even 6 1 1008.2.cs.a 2
21.g even 6 1 7056.2.b.k 2
21.h odd 6 1 7056.2.b.d 2
28.d even 2 1 2352.2.bl.a 2
28.f even 6 1 inner 336.2.bl.d 2
28.f even 6 1 2352.2.b.e 2
28.g odd 6 1 2352.2.b.d 2
28.g odd 6 1 2352.2.bl.g 2
56.j odd 6 1 1344.2.bl.a 2
56.m even 6 1 1344.2.bl.e 2
84.j odd 6 1 1008.2.cs.b 2
84.j odd 6 1 7056.2.b.d 2
84.n even 6 1 7056.2.b.k 2

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + T + T^{2}$$
$5$ $$1 - 6 T + 17 T^{2} - 30 T^{3} + 25 T^{4}$$
$7$ $$1 - T + 7 T^{2}$$
$11$ $$1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$17$ $$1 + 17 T^{2} + 289 T^{4}$$
$19$ $$1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4}$$
$23$ $$1 - 12 T + 71 T^{2} - 276 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4}$$
$37$ $$( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} )$$
$41$ $$1 - 70 T^{2} + 1681 T^{4}$$
$43$ $$1 - 11 T^{2} + 1849 T^{4}$$
$47$ $$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$
$53$ $$1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4}$$
$59$ $$1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4}$$
$61$ $$1 + 24 T + 253 T^{2} + 1464 T^{3} + 3721 T^{4}$$
$67$ $$1 - 15 T + 142 T^{2} - 1005 T^{3} + 4489 T^{4}$$
$71$ $$1 - 130 T^{2} + 5041 T^{4}$$
$73$ $$1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4}$$
$79$ $$( 1 - 17 T + 79 T^{2} )( 1 - 4 T + 79 T^{2} )$$
$83$ $$( 1 + 18 T + 83 T^{2} )^{2}$$
$89$ $$1 - 12 T + 137 T^{2} - 1068 T^{3} + 7921 T^{4}$$
$97$ $$1 - 146 T^{2} + 9409 T^{4}$$