# Properties

 Label 336.2.b.d Level 336 Weight 2 Character orbit 336.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( 1 + 2 \zeta_{6} ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( 1 + 2 \zeta_{6} ) q^{7} + q^{9} + ( -2 + 4 \zeta_{6} ) q^{11} + ( 4 - 8 \zeta_{6} ) q^{17} + 4 q^{19} + ( 1 + 2 \zeta_{6} ) q^{21} + ( -2 + 4 \zeta_{6} ) q^{23} + 5 q^{25} + q^{27} -6 q^{29} -4 q^{31} + ( -2 + 4 \zeta_{6} ) q^{33} -2 q^{37} + ( 4 - 8 \zeta_{6} ) q^{41} + ( 2 - 4 \zeta_{6} ) q^{43} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 4 - 8 \zeta_{6} ) q^{51} -6 q^{53} + 4 q^{57} -12 q^{59} + ( -8 + 16 \zeta_{6} ) q^{61} + ( 1 + 2 \zeta_{6} ) q^{63} + ( 2 - 4 \zeta_{6} ) q^{67} + ( -2 + 4 \zeta_{6} ) q^{69} + ( 6 - 12 \zeta_{6} ) q^{71} + ( 8 - 16 \zeta_{6} ) q^{73} + 5 q^{75} + ( -10 + 8 \zeta_{6} ) q^{77} + ( 6 - 12 \zeta_{6} ) q^{79} + q^{81} -12 q^{83} -6 q^{87} + ( -4 + 8 \zeta_{6} ) q^{89} -4 q^{93} + ( -2 + 4 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 4q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 4q^{7} + 2q^{9} + 8q^{19} + 4q^{21} + 10q^{25} + 2q^{27} - 12q^{29} - 8q^{31} - 4q^{37} + 2q^{49} - 12q^{53} + 8q^{57} - 24q^{59} + 4q^{63} + 10q^{75} - 12q^{77} + 2q^{81} - 24q^{83} - 12q^{87} - 8q^{93} + O(q^{100})$$

## Character values

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

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
223.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.00000 0 0 0 2.00000 1.73205i 0 1.00000 0
223.2 0 1.00000 0 0 0 2.00000 + 1.73205i 0 1.00000 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.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.b.d yes 2
3.b odd 2 1 1008.2.b.h 2
4.b odd 2 1 336.2.b.a 2
7.b odd 2 1 336.2.b.a 2
7.c even 3 1 2352.2.bl.c 2
7.c even 3 1 2352.2.bl.d 2
7.d odd 6 1 2352.2.bl.i 2
7.d odd 6 1 2352.2.bl.j 2
8.b even 2 1 1344.2.b.b 2
8.d odd 2 1 1344.2.b.c 2
12.b even 2 1 1008.2.b.a 2
21.c even 2 1 1008.2.b.a 2
24.f even 2 1 4032.2.b.c 2
24.h odd 2 1 4032.2.b.g 2
28.d even 2 1 inner 336.2.b.d yes 2
28.f even 6 1 2352.2.bl.c 2
28.f even 6 1 2352.2.bl.d 2
28.g odd 6 1 2352.2.bl.i 2
28.g odd 6 1 2352.2.bl.j 2
56.e even 2 1 1344.2.b.b 2
56.h odd 2 1 1344.2.b.c 2
84.h odd 2 1 1008.2.b.h 2
168.e odd 2 1 4032.2.b.g 2
168.i even 2 1 4032.2.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.b.a 2 4.b odd 2 1
336.2.b.a 2 7.b odd 2 1
336.2.b.d yes 2 1.a even 1 1 trivial
336.2.b.d yes 2 28.d even 2 1 inner
1008.2.b.a 2 12.b even 2 1
1008.2.b.a 2 21.c even 2 1
1008.2.b.h 2 3.b odd 2 1
1008.2.b.h 2 84.h odd 2 1
1344.2.b.b 2 8.b even 2 1
1344.2.b.b 2 56.e even 2 1
1344.2.b.c 2 8.d odd 2 1
1344.2.b.c 2 56.h odd 2 1
2352.2.bl.c 2 7.c even 3 1
2352.2.bl.c 2 28.f even 6 1
2352.2.bl.d 2 7.c even 3 1
2352.2.bl.d 2 28.f even 6 1
2352.2.bl.i 2 7.d odd 6 1
2352.2.bl.i 2 28.g odd 6 1
2352.2.bl.j 2 7.d odd 6 1
2352.2.bl.j 2 28.g odd 6 1
4032.2.b.c 2 24.f even 2 1
4032.2.b.c 2 168.i even 2 1
4032.2.b.g 2 24.h odd 2 1
4032.2.b.g 2 168.e odd 2 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{19} - 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$( 1 - T )^{2}$$
$5$ $$( 1 - 5 T^{2} )^{2}$$
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 - 10 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 13 T^{2} )^{2}$$
$17$ $$1 + 14 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 4 T + 19 T^{2} )^{2}$$
$23$ $$1 - 34 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 4 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{2}$$
$41$ $$1 - 34 T^{2} + 1681 T^{4}$$
$43$ $$1 - 74 T^{2} + 1849 T^{4}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$( 1 + 6 T + 53 T^{2} )^{2}$$
$59$ $$( 1 + 12 T + 59 T^{2} )^{2}$$
$61$ $$1 + 70 T^{2} + 3721 T^{4}$$
$67$ $$( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$
$71$ $$1 - 34 T^{2} + 5041 T^{4}$$
$73$ $$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$
$79$ $$1 - 50 T^{2} + 6241 T^{4}$$
$83$ $$( 1 + 12 T + 83 T^{2} )^{2}$$
$89$ $$1 - 130 T^{2} + 7921 T^{4}$$
$97$ $$( 1 - 97 T^{2} )^{2}$$