# Properties

 Label 336.2.k.b Level 336 Weight 2 Character orbit 336.k Analytic conductor 2.683 Analytic rank 0 Dimension 4 CM no 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{3} ) q^{7} + \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{3} ) q^{7} + \beta_{2} q^{9} + ( -\beta_{1} - \beta_{3} ) q^{13} + ( 3 + \beta_{2} ) q^{15} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{17} + ( -\beta_{1} - \beta_{3} ) q^{19} + ( 3 + \beta_{1} - \beta_{2} ) q^{21} + 2 \beta_{2} q^{23} + q^{25} + 3 \beta_{3} q^{27} + 2 \beta_{2} q^{29} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{35} -2 q^{37} + ( 3 - \beta_{2} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{41} -4 q^{43} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{45} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{47} + ( -5 - 2 \beta_{1} - 2 \beta_{3} ) q^{49} + ( -6 - 2 \beta_{2} ) q^{51} -2 \beta_{2} q^{53} + ( 3 - \beta_{2} ) q^{57} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{59} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{61} + ( 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{63} -2 \beta_{2} q^{65} -8 q^{67} + 6 \beta_{3} q^{69} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{73} + \beta_{1} q^{75} + 10 q^{79} -9 q^{81} + ( \beta_{1} - \beta_{3} ) q^{83} -12 q^{85} + 6 \beta_{3} q^{87} + ( -6 - \beta_{1} - \beta_{3} ) q^{91} -2 \beta_{2} q^{95} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} + O(q^{10})$$ $$4q + 4q^{7} + 12q^{15} + 12q^{21} + 4q^{25} - 8q^{37} + 12q^{39} - 16q^{43} - 20q^{49} - 24q^{51} + 12q^{57} - 32q^{67} + 40q^{79} - 36q^{81} - 48q^{85} - 24q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## 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
 −1.22474 − 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i 1.22474 + 1.22474i
0 −1.22474 1.22474i 0 −2.44949 0 1.00000 + 2.44949i 0 3.00000i 0
209.2 0 −1.22474 + 1.22474i 0 −2.44949 0 1.00000 2.44949i 0 3.00000i 0
209.3 0 1.22474 1.22474i 0 2.44949 0 1.00000 + 2.44949i 0 3.00000i 0
209.4 0 1.22474 + 1.22474i 0 2.44949 0 1.00000 2.44949i 0 3.00000i 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 inner
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.b 4
3.b odd 2 1 inner 336.2.k.b 4
4.b odd 2 1 42.2.d.a 4
7.b odd 2 1 inner 336.2.k.b 4
8.b even 2 1 1344.2.k.d 4
8.d odd 2 1 1344.2.k.c 4
12.b even 2 1 42.2.d.a 4
20.d odd 2 1 1050.2.b.b 4
20.e even 4 1 1050.2.d.b 4
20.e even 4 1 1050.2.d.e 4
21.c even 2 1 inner 336.2.k.b 4
24.f even 2 1 1344.2.k.c 4
24.h odd 2 1 1344.2.k.d 4
28.d even 2 1 42.2.d.a 4
28.f even 6 2 294.2.f.b 8
28.g odd 6 2 294.2.f.b 8
36.f odd 6 2 1134.2.m.g 8
36.h even 6 2 1134.2.m.g 8
56.e even 2 1 1344.2.k.c 4
56.h odd 2 1 1344.2.k.d 4
60.h even 2 1 1050.2.b.b 4
60.l odd 4 1 1050.2.d.b 4
60.l odd 4 1 1050.2.d.e 4
84.h odd 2 1 42.2.d.a 4
84.j odd 6 2 294.2.f.b 8
84.n even 6 2 294.2.f.b 8
140.c even 2 1 1050.2.b.b 4
140.j odd 4 1 1050.2.d.b 4
140.j odd 4 1 1050.2.d.e 4
168.e odd 2 1 1344.2.k.c 4
168.i even 2 1 1344.2.k.d 4
252.s odd 6 2 1134.2.m.g 8
252.bi even 6 2 1134.2.m.g 8
420.o odd 2 1 1050.2.b.b 4
420.w even 4 1 1050.2.d.b 4
420.w even 4 1 1050.2.d.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.d.a 4 4.b odd 2 1
42.2.d.a 4 12.b even 2 1
42.2.d.a 4 28.d even 2 1
42.2.d.a 4 84.h odd 2 1
294.2.f.b 8 28.f even 6 2
294.2.f.b 8 28.g odd 6 2
294.2.f.b 8 84.j odd 6 2
294.2.f.b 8 84.n even 6 2
336.2.k.b 4 1.a even 1 1 trivial
336.2.k.b 4 3.b odd 2 1 inner
336.2.k.b 4 7.b odd 2 1 inner
336.2.k.b 4 21.c even 2 1 inner
1050.2.b.b 4 20.d odd 2 1
1050.2.b.b 4 60.h even 2 1
1050.2.b.b 4 140.c even 2 1
1050.2.b.b 4 420.o odd 2 1
1050.2.d.b 4 20.e even 4 1
1050.2.d.b 4 60.l odd 4 1
1050.2.d.b 4 140.j odd 4 1
1050.2.d.b 4 420.w even 4 1
1050.2.d.e 4 20.e even 4 1
1050.2.d.e 4 60.l odd 4 1
1050.2.d.e 4 140.j odd 4 1
1050.2.d.e 4 420.w even 4 1
1134.2.m.g 8 36.f odd 6 2
1134.2.m.g 8 36.h even 6 2
1134.2.m.g 8 252.s odd 6 2
1134.2.m.g 8 252.bi even 6 2
1344.2.k.c 4 8.d odd 2 1
1344.2.k.c 4 24.f even 2 1
1344.2.k.c 4 56.e even 2 1
1344.2.k.c 4 168.e odd 2 1
1344.2.k.d 4 8.b even 2 1
1344.2.k.d 4 24.h odd 2 1
1344.2.k.d 4 56.h odd 2 1
1344.2.k.d 4 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 6$$ acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 9 T^{4}$$
$5$ $$( 1 + 4 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 - 2 T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 11 T^{2} )^{4}$$
$13$ $$( 1 - 20 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 10 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 32 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 10 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 22 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 31 T^{2} )^{4}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{4}$$
$41$ $$( 1 + 58 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{4}$$
$47$ $$( 1 + 70 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 70 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 32 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 28 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 8 T + 67 T^{2} )^{4}$$
$71$ $$( 1 - 71 T^{2} )^{4}$$
$73$ $$( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2}$$
$79$ $$( 1 - 10 T + 79 T^{2} )^{4}$$
$83$ $$( 1 + 160 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{4}$$
$97$ $$( 1 - 170 T^{2} + 9409 T^{4} )^{2}$$