# Properties

 Label 336.4.bc.a Level $336$ Weight $4$ Character orbit 336.bc Analytic conductor $19.825$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 336.bc (of order $$6$$, degree $$2$$, 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -3 - 3 \zeta_{6} ) q^{3} + ( -18 - \zeta_{6} ) q^{7} + 27 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -3 - 3 \zeta_{6} ) q^{3} + ( -18 - \zeta_{6} ) q^{7} + 27 \zeta_{6} q^{9} + ( -17 + 34 \zeta_{6} ) q^{13} + ( -34 + 17 \zeta_{6} ) q^{19} + ( 51 + 60 \zeta_{6} ) q^{21} + ( 125 - 125 \zeta_{6} ) q^{25} + ( 81 - 162 \zeta_{6} ) q^{27} + ( 199 + 199 \zeta_{6} ) q^{31} -433 \zeta_{6} q^{37} + ( 153 - 153 \zeta_{6} ) q^{39} + 449 q^{43} + ( 323 + 37 \zeta_{6} ) q^{49} + 153 q^{57} + ( 1080 - 540 \zeta_{6} ) q^{61} + ( 27 - 513 \zeta_{6} ) q^{63} + ( 1007 - 1007 \zeta_{6} ) q^{67} + ( 487 + 487 \zeta_{6} ) q^{73} + ( -750 + 375 \zeta_{6} ) q^{75} + 503 \zeta_{6} q^{79} + ( -729 + 729 \zeta_{6} ) q^{81} + ( 340 - 629 \zeta_{6} ) q^{91} -1791 \zeta_{6} q^{93} + ( -792 + 1584 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 9q^{3} - 37q^{7} + 27q^{9} + O(q^{10})$$ $$2q - 9q^{3} - 37q^{7} + 27q^{9} - 51q^{19} + 162q^{21} + 125q^{25} + 597q^{31} - 433q^{37} + 153q^{39} + 898q^{43} + 683q^{49} + 306q^{57} + 1620q^{61} - 459q^{63} + 1007q^{67} + 1461q^{73} - 1125q^{75} + 503q^{79} - 729q^{81} + 51q^{91} - 1791q^{93} + O(q^{100})$$

## Character values

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

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −4.50000 2.59808i 0 0 0 −18.5000 0.866025i 0 13.5000 + 23.3827i 0
257.1 0 −4.50000 + 2.59808i 0 0 0 −18.5000 + 0.866025i 0 13.5000 23.3827i 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.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.bc.a 2
3.b odd 2 1 CM 336.4.bc.a 2
4.b odd 2 1 84.4.k.b 2
7.d odd 6 1 inner 336.4.bc.a 2
12.b even 2 1 84.4.k.b 2
21.g even 6 1 inner 336.4.bc.a 2
28.d even 2 1 588.4.k.a 2
28.f even 6 1 84.4.k.b 2
28.f even 6 1 588.4.f.b 2
28.g odd 6 1 588.4.f.b 2
28.g odd 6 1 588.4.k.a 2
84.h odd 2 1 588.4.k.a 2
84.j odd 6 1 84.4.k.b 2
84.j odd 6 1 588.4.f.b 2
84.n even 6 1 588.4.f.b 2
84.n even 6 1 588.4.k.a 2

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}$$ $$T_{13}^{2} + 867$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$27 + 9 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$343 + 37 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$867 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$867 + 51 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$118803 - 597 T + T^{2}$$
$37$ $$187489 + 433 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -449 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$874800 - 1620 T + T^{2}$$
$67$ $$1014049 - 1007 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$711507 - 1461 T + T^{2}$$
$79$ $$253009 - 503 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$1881792 + T^{2}$$
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