# Properties

 Label 336.2.bl.f 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( -2 + \zeta_{6} ) q^{5} + ( 3 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( -2 + \zeta_{6} ) q^{5} + ( 3 - \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 3 + 3 \zeta_{6} ) q^{11} + ( 4 - 8 \zeta_{6} ) q^{13} + ( -1 + 2 \zeta_{6} ) q^{15} + ( 2 + 2 \zeta_{6} ) q^{17} -2 \zeta_{6} q^{19} + ( 2 - 3 \zeta_{6} ) q^{21} + ( 8 - 4 \zeta_{6} ) q^{23} + ( -2 + 2 \zeta_{6} ) q^{25} - q^{27} -9 q^{29} + ( -1 + \zeta_{6} ) q^{31} + ( 6 - 3 \zeta_{6} ) q^{33} + ( -5 + 4 \zeta_{6} ) q^{35} + 2 \zeta_{6} q^{37} + ( -4 - 4 \zeta_{6} ) q^{39} + ( -2 + 4 \zeta_{6} ) q^{41} + ( -2 + 4 \zeta_{6} ) q^{43} + ( 1 + \zeta_{6} ) q^{45} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 4 - 2 \zeta_{6} ) q^{51} + ( -9 + 9 \zeta_{6} ) q^{53} -9 q^{55} -2 q^{57} + ( 3 - 3 \zeta_{6} ) q^{59} + ( -8 + 4 \zeta_{6} ) q^{61} + ( -1 - 2 \zeta_{6} ) q^{63} + 12 \zeta_{6} q^{65} + ( 4 - 8 \zeta_{6} ) q^{69} + ( -4 + 8 \zeta_{6} ) q^{71} + ( 4 + 4 \zeta_{6} ) q^{73} + 2 \zeta_{6} q^{75} + ( 12 + 3 \zeta_{6} ) q^{77} + ( -2 + \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} -15 q^{83} -6 q^{85} + ( -9 + 9 \zeta_{6} ) q^{87} + ( -12 + 6 \zeta_{6} ) q^{89} + ( 4 - 20 \zeta_{6} ) q^{91} + \zeta_{6} q^{93} + ( 2 + 2 \zeta_{6} ) q^{95} + ( 5 - 10 \zeta_{6} ) q^{97} + ( 3 - 6 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - 3q^{5} + 5q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} - 3q^{5} + 5q^{7} - q^{9} + 9q^{11} + 6q^{17} - 2q^{19} + q^{21} + 12q^{23} - 2q^{25} - 2q^{27} - 18q^{29} - q^{31} + 9q^{33} - 6q^{35} + 2q^{37} - 12q^{39} + 3q^{45} + 11q^{49} + 6q^{51} - 9q^{53} - 18q^{55} - 4q^{57} + 3q^{59} - 12q^{61} - 4q^{63} + 12q^{65} + 12q^{73} + 2q^{75} + 27q^{77} - 3q^{79} - q^{81} - 30q^{83} - 12q^{85} - 9q^{87} - 18q^{89} - 12q^{91} + q^{93} + 6q^{95} + O(q^{100})$$

## 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 2.50000 0.866025i 0 −0.500000 0.866025i 0
271.1 0 0.500000 + 0.866025i 0 −1.50000 0.866025i 0 2.50000 + 0.866025i 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.f yes 2
3.b odd 2 1 1008.2.cs.l 2
4.b odd 2 1 336.2.bl.b 2
7.b odd 2 1 2352.2.bl.e 2
7.c even 3 1 2352.2.b.b 2
7.c even 3 1 2352.2.bl.k 2
7.d odd 6 1 336.2.bl.b 2
7.d odd 6 1 2352.2.b.f 2
8.b even 2 1 1344.2.bl.c 2
8.d odd 2 1 1344.2.bl.g 2
12.b even 2 1 1008.2.cs.k 2
21.g even 6 1 1008.2.cs.k 2
21.g even 6 1 7056.2.b.f 2
21.h odd 6 1 7056.2.b.j 2
28.d even 2 1 2352.2.bl.k 2
28.f even 6 1 inner 336.2.bl.f yes 2
28.f even 6 1 2352.2.b.b 2
28.g odd 6 1 2352.2.b.f 2
28.g odd 6 1 2352.2.bl.e 2
56.j odd 6 1 1344.2.bl.g 2
56.m even 6 1 1344.2.bl.c 2
84.j odd 6 1 1008.2.cs.l 2
84.j odd 6 1 7056.2.b.j 2
84.n even 6 1 7056.2.b.f 2

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

 $$T_{5}^{2} + 3 T_{5} + 3$$ $$T_{11}^{2} - 9 T_{11} + 27$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T + T^{2}$$
$5$ $$1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4}$$
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$1 - 9 T + 38 T^{2} - 99 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$17$ $$1 - 6 T + 29 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4}$$
$23$ $$1 - 12 T + 71 T^{2} - 276 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 9 T + 29 T^{2} )^{2}$$
$31$ $$1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4}$$
$37$ $$1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4}$$
$41$ $$1 - 70 T^{2} + 1681 T^{4}$$
$43$ $$1 - 74 T^{2} + 1849 T^{4}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4}$$
$59$ $$1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4}$$
$61$ $$( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$
$67$ $$1 + 67 T^{2} + 4489 T^{4}$$
$71$ $$1 - 94 T^{2} + 5041 T^{4}$$
$73$ $$1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4}$$
$79$ $$1 + 3 T + 82 T^{2} + 237 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 15 T + 83 T^{2} )^{2}$$
$89$ $$1 + 18 T + 197 T^{2} + 1602 T^{3} + 7921 T^{4}$$
$97$ $$1 - 119 T^{2} + 9409 T^{4}$$