# Properties

 Label 336.2.b.b 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: $$\Q(\sqrt{-6})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta q^{5} + ( 1 - \beta ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + \beta q^{5} + ( 1 - \beta ) q^{7} + q^{9} + \beta q^{11} + 2 \beta q^{13} -\beta q^{15} + \beta q^{17} + 2 q^{19} + ( -1 + \beta ) q^{21} + 3 \beta q^{23} - q^{25} - q^{27} + 6 q^{29} -8 q^{31} -\beta q^{33} + ( 6 + \beta ) q^{35} + 4 q^{37} -2 \beta q^{39} -3 \beta q^{41} -2 \beta q^{43} + \beta q^{45} + 12 q^{47} + ( -5 - 2 \beta ) q^{49} -\beta q^{51} -6 q^{53} -6 q^{55} -2 q^{57} -12 q^{59} + ( 1 - \beta ) q^{63} -12 q^{65} -3 \beta q^{69} + 5 \beta q^{71} -6 \beta q^{73} + q^{75} + ( 6 + \beta ) q^{77} -4 \beta q^{79} + q^{81} -6 q^{85} -6 q^{87} -5 \beta q^{89} + ( 12 + 2 \beta ) q^{91} + 8 q^{93} + 2 \beta q^{95} + 2 \beta q^{97} + \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{7} + 2q^{9} + 4q^{19} - 2q^{21} - 2q^{25} - 2q^{27} + 12q^{29} - 16q^{31} + 12q^{35} + 8q^{37} + 24q^{47} - 10q^{49} - 12q^{53} - 12q^{55} - 4q^{57} - 24q^{59} + 2q^{63} - 24q^{65} + 2q^{75} + 12q^{77} + 2q^{81} - 12q^{85} - 12q^{87} + 24q^{91} + 16q^{93} + O(q^{100})$$

## 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
 − 2.44949i 2.44949i
0 −1.00000 0 2.44949i 0 1.00000 + 2.44949i 0 1.00000 0
223.2 0 −1.00000 0 2.44949i 0 1.00000 2.44949i 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.b 2
3.b odd 2 1 1008.2.b.e 2
4.b odd 2 1 336.2.b.c yes 2
7.b odd 2 1 336.2.b.c yes 2
7.c even 3 2 2352.2.bl.n 4
7.d odd 6 2 2352.2.bl.m 4
8.b even 2 1 1344.2.b.d 2
8.d odd 2 1 1344.2.b.a 2
12.b even 2 1 1008.2.b.d 2
21.c even 2 1 1008.2.b.d 2
24.f even 2 1 4032.2.b.d 2
24.h odd 2 1 4032.2.b.f 2
28.d even 2 1 inner 336.2.b.b 2
28.f even 6 2 2352.2.bl.n 4
28.g odd 6 2 2352.2.bl.m 4
56.e even 2 1 1344.2.b.d 2
56.h odd 2 1 1344.2.b.a 2
84.h odd 2 1 1008.2.b.e 2
168.e odd 2 1 4032.2.b.f 2
168.i even 2 1 4032.2.b.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.b.b 2 1.a even 1 1 trivial
336.2.b.b 2 28.d even 2 1 inner
336.2.b.c yes 2 4.b odd 2 1
336.2.b.c yes 2 7.b odd 2 1
1008.2.b.d 2 12.b even 2 1
1008.2.b.d 2 21.c even 2 1
1008.2.b.e 2 3.b odd 2 1
1008.2.b.e 2 84.h odd 2 1
1344.2.b.a 2 8.d odd 2 1
1344.2.b.a 2 56.h odd 2 1
1344.2.b.d 2 8.b even 2 1
1344.2.b.d 2 56.e even 2 1
2352.2.bl.m 4 7.d odd 6 2
2352.2.bl.m 4 28.g odd 6 2
2352.2.bl.n 4 7.c even 3 2
2352.2.bl.n 4 28.f even 6 2
4032.2.b.d 2 24.f even 2 1
4032.2.b.d 2 168.i even 2 1
4032.2.b.f 2 24.h odd 2 1
4032.2.b.f 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 $$S_{2}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}^{2} + 6$$ $$T_{19} - 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$( 1 + T )^{2}$$
$5$ $$1 - 4 T^{2} + 25 T^{4}$$
$7$ $$1 - 2 T + 7 T^{2}$$
$11$ $$1 - 16 T^{2} + 121 T^{4}$$
$13$ $$1 - 2 T^{2} + 169 T^{4}$$
$17$ $$1 - 28 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 2 T + 19 T^{2} )^{2}$$
$23$ $$1 + 8 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 8 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 4 T + 37 T^{2} )^{2}$$
$41$ $$1 - 28 T^{2} + 1681 T^{4}$$
$43$ $$1 - 62 T^{2} + 1849 T^{4}$$
$47$ $$( 1 - 12 T + 47 T^{2} )^{2}$$
$53$ $$( 1 + 6 T + 53 T^{2} )^{2}$$
$59$ $$( 1 + 12 T + 59 T^{2} )^{2}$$
$61$ $$( 1 - 61 T^{2} )^{2}$$
$67$ $$( 1 - 67 T^{2} )^{2}$$
$71$ $$1 + 8 T^{2} + 5041 T^{4}$$
$73$ $$1 + 70 T^{2} + 5329 T^{4}$$
$79$ $$1 - 62 T^{2} + 6241 T^{4}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$1 - 28 T^{2} + 7921 T^{4}$$
$97$ $$1 - 170 T^{2} + 9409 T^{4}$$