# Properties

 Label 336.2.k.a Level 336 Weight 2 Character orbit 336.k Analytic conductor 2.683 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.k (of order $$2$$, degree $$1$$, not 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: $$2$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} + ( 4 - 8 \zeta_{6} ) q^{13} + ( -2 + 4 \zeta_{6} ) q^{19} + ( -5 + 4 \zeta_{6} ) q^{21} -5 q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 6 - 12 \zeta_{6} ) q^{31} + 10 q^{37} -12 q^{39} + 8 q^{43} + ( -3 + 8 \zeta_{6} ) q^{49} + 6 q^{57} + ( -4 + 8 \zeta_{6} ) q^{61} + ( 3 + 6 \zeta_{6} ) q^{63} + 16 q^{67} + ( 8 - 16 \zeta_{6} ) q^{73} + ( -5 + 10 \zeta_{6} ) q^{75} + 4 q^{79} + 9 q^{81} + ( -20 + 16 \zeta_{6} ) q^{91} -18 q^{93} + ( -8 + 16 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{7} - 6q^{9} + O(q^{10})$$ $$2q - 4q^{7} - 6q^{9} - 6q^{21} - 10q^{25} + 20q^{37} - 24q^{39} + 16q^{43} + 2q^{49} + 12q^{57} + 12q^{63} + 32q^{67} + 8q^{79} + 18q^{81} - 24q^{91} - 36q^{93} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 0 0 0 −2.00000 1.73205i 0 −3.00000 0
209.2 0 1.73205i 0 0 0 −2.00000 + 1.73205i 0 −3.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.b odd 2 1 inner
21.c 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.k.a 2
3.b odd 2 1 CM 336.2.k.a 2
4.b odd 2 1 84.2.f.a 2
7.b odd 2 1 inner 336.2.k.a 2
8.b even 2 1 1344.2.k.a 2
8.d odd 2 1 1344.2.k.b 2
12.b even 2 1 84.2.f.a 2
20.d odd 2 1 2100.2.d.b 2
20.e even 4 2 2100.2.f.e 4
21.c even 2 1 inner 336.2.k.a 2
24.f even 2 1 1344.2.k.b 2
24.h odd 2 1 1344.2.k.a 2
28.d even 2 1 84.2.f.a 2
28.f even 6 1 588.2.k.a 2
28.f even 6 1 588.2.k.e 2
28.g odd 6 1 588.2.k.a 2
28.g odd 6 1 588.2.k.e 2
36.f odd 6 1 2268.2.x.c 2
36.f odd 6 1 2268.2.x.e 2
36.h even 6 1 2268.2.x.c 2
36.h even 6 1 2268.2.x.e 2
56.e even 2 1 1344.2.k.b 2
56.h odd 2 1 1344.2.k.a 2
60.h even 2 1 2100.2.d.b 2
60.l odd 4 2 2100.2.f.e 4
84.h odd 2 1 84.2.f.a 2
84.j odd 6 1 588.2.k.a 2
84.j odd 6 1 588.2.k.e 2
84.n even 6 1 588.2.k.a 2
84.n even 6 1 588.2.k.e 2
140.c even 2 1 2100.2.d.b 2
140.j odd 4 2 2100.2.f.e 4
168.e odd 2 1 1344.2.k.b 2
168.i even 2 1 1344.2.k.a 2
252.s odd 6 1 2268.2.x.c 2
252.s odd 6 1 2268.2.x.e 2
252.bi even 6 1 2268.2.x.c 2
252.bi even 6 1 2268.2.x.e 2
420.o odd 2 1 2100.2.d.b 2
420.w even 4 2 2100.2.f.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.f.a 2 4.b odd 2 1
84.2.f.a 2 12.b even 2 1
84.2.f.a 2 28.d even 2 1
84.2.f.a 2 84.h odd 2 1
336.2.k.a 2 1.a even 1 1 trivial
336.2.k.a 2 3.b odd 2 1 CM
336.2.k.a 2 7.b odd 2 1 inner
336.2.k.a 2 21.c even 2 1 inner
588.2.k.a 2 28.f even 6 1
588.2.k.a 2 28.g odd 6 1
588.2.k.a 2 84.j odd 6 1
588.2.k.a 2 84.n even 6 1
588.2.k.e 2 28.f even 6 1
588.2.k.e 2 28.g odd 6 1
588.2.k.e 2 84.j odd 6 1
588.2.k.e 2 84.n even 6 1
1344.2.k.a 2 8.b even 2 1
1344.2.k.a 2 24.h odd 2 1
1344.2.k.a 2 56.h odd 2 1
1344.2.k.a 2 168.i even 2 1
1344.2.k.b 2 8.d odd 2 1
1344.2.k.b 2 24.f even 2 1
1344.2.k.b 2 56.e even 2 1
1344.2.k.b 2 168.e odd 2 1
2100.2.d.b 2 20.d odd 2 1
2100.2.d.b 2 60.h even 2 1
2100.2.d.b 2 140.c even 2 1
2100.2.d.b 2 420.o odd 2 1
2100.2.f.e 4 20.e even 4 2
2100.2.f.e 4 60.l odd 4 2
2100.2.f.e 4 140.j odd 4 2
2100.2.f.e 4 420.w even 4 2
2268.2.x.c 2 36.f odd 6 1
2268.2.x.c 2 36.h even 6 1
2268.2.x.c 2 252.s odd 6 1
2268.2.x.c 2 252.bi even 6 1
2268.2.x.e 2 36.f odd 6 1
2268.2.x.e 2 36.h even 6 1
2268.2.x.e 2 252.s odd 6 1
2268.2.x.e 2 252.bi even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 3 T^{2}$$
$5$ $$( 1 + 5 T^{2} )^{2}$$
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ $$( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$( 1 - 29 T^{2} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$
$37$ $$( 1 - 10 T + 37 T^{2} )^{2}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$( 1 - 53 T^{2} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 - 14 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} )$$
$67$ $$( 1 - 16 T + 67 T^{2} )^{2}$$
$71$ $$( 1 - 71 T^{2} )^{2}$$
$73$ $$( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$
$79$ $$( 1 - 4 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$