Properties

 Label 336.4.k.a Level $336$ Weight $4$ Character orbit 336.k Analytic conductor $19.825$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 336.k (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$19.8246417619$$ 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: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 21) 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.

 $$f(q)$$ $$=$$ $$q + ( -3 + 6 \zeta_{6} ) q^{3} + ( 19 - 18 \zeta_{6} ) q^{7} -27 q^{9} +O(q^{10})$$ $$q + ( -3 + 6 \zeta_{6} ) q^{3} + ( 19 - 18 \zeta_{6} ) q^{7} -27 q^{9} + ( -36 + 72 \zeta_{6} ) q^{13} + ( -90 + 180 \zeta_{6} ) q^{19} + ( 51 + 60 \zeta_{6} ) q^{21} -125 q^{25} + ( 81 - 162 \zeta_{6} ) q^{27} + ( -90 + 180 \zeta_{6} ) q^{31} -110 q^{37} -324 q^{39} -520 q^{43} + ( 37 - 360 \zeta_{6} ) q^{49} -810 q^{57} + ( -540 + 1080 \zeta_{6} ) q^{61} + ( -513 + 486 \zeta_{6} ) q^{63} + 880 q^{67} + ( 216 - 432 \zeta_{6} ) q^{73} + ( 375 - 750 \zeta_{6} ) q^{75} -884 q^{79} + 729 q^{81} + ( 612 + 720 \zeta_{6} ) q^{91} -810 q^{93} + ( -792 + 1584 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 20q^{7} - 54q^{9} + O(q^{10})$$ $$2q + 20q^{7} - 54q^{9} + 162q^{21} - 250q^{25} - 220q^{37} - 648q^{39} - 1040q^{43} - 286q^{49} - 1620q^{57} - 540q^{63} + 1760q^{67} - 1768q^{79} + 1458q^{81} + 1944q^{91} - 1620q^{93} + O(q^{100})$$

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 5.19615i 0 0 0 10.0000 + 15.5885i 0 −27.0000 0
209.2 0 5.19615i 0 0 0 10.0000 15.5885i 0 −27.0000 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 336.4.k.a 2
3.b odd 2 1 CM 336.4.k.a 2
4.b odd 2 1 21.4.c.a 2
7.b odd 2 1 inner 336.4.k.a 2
12.b even 2 1 21.4.c.a 2
21.c even 2 1 inner 336.4.k.a 2
28.d even 2 1 21.4.c.a 2
28.f even 6 1 147.4.g.a 2
28.f even 6 1 147.4.g.b 2
28.g odd 6 1 147.4.g.a 2
28.g odd 6 1 147.4.g.b 2
84.h odd 2 1 21.4.c.a 2
84.j odd 6 1 147.4.g.a 2
84.j odd 6 1 147.4.g.b 2
84.n even 6 1 147.4.g.a 2
84.n even 6 1 147.4.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.c.a 2 4.b odd 2 1
21.4.c.a 2 12.b even 2 1
21.4.c.a 2 28.d even 2 1
21.4.c.a 2 84.h odd 2 1
147.4.g.a 2 28.f even 6 1
147.4.g.a 2 28.g odd 6 1
147.4.g.a 2 84.j odd 6 1
147.4.g.a 2 84.n even 6 1
147.4.g.b 2 28.f even 6 1
147.4.g.b 2 28.g odd 6 1
147.4.g.b 2 84.j odd 6 1
147.4.g.b 2 84.n even 6 1
336.4.k.a 2 1.a even 1 1 trivial
336.4.k.a 2 3.b odd 2 1 CM
336.4.k.a 2 7.b odd 2 1 inner
336.4.k.a 2 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$27 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$343 - 20 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$3888 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$24300 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$24300 + T^{2}$$
$37$ $$( 110 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$( 520 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$874800 + T^{2}$$
$67$ $$( -880 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$139968 + T^{2}$$
$79$ $$( 884 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$1881792 + T^{2}$$