# Properties

 Label 336.4.q.e Level $336$ Weight $4$ Character orbit 336.q Analytic conductor $19.825$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 336.q (of order $$3$$, 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{5} + ( -7 + 21 \zeta_{6} ) q^{7} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{5} + ( -7 + 21 \zeta_{6} ) q^{7} -9 \zeta_{6} q^{9} + ( -15 + 15 \zeta_{6} ) q^{11} -64 q^{13} + 9 q^{15} + ( -84 + 84 \zeta_{6} ) q^{17} -16 \zeta_{6} q^{19} + ( 42 + 21 \zeta_{6} ) q^{21} -84 \zeta_{6} q^{23} + ( 116 - 116 \zeta_{6} ) q^{25} -27 q^{27} -297 q^{29} + ( -253 + 253 \zeta_{6} ) q^{31} + 45 \zeta_{6} q^{33} + ( -63 + 42 \zeta_{6} ) q^{35} + 316 \zeta_{6} q^{37} + ( -192 + 192 \zeta_{6} ) q^{39} + 360 q^{41} -26 q^{43} + ( 27 - 27 \zeta_{6} ) q^{45} -30 \zeta_{6} q^{47} + ( -392 + 147 \zeta_{6} ) q^{49} + 252 \zeta_{6} q^{51} + ( -363 + 363 \zeta_{6} ) q^{53} -45 q^{55} -48 q^{57} + ( -15 + 15 \zeta_{6} ) q^{59} + 118 \zeta_{6} q^{61} + ( 189 - 126 \zeta_{6} ) q^{63} -192 \zeta_{6} q^{65} + ( -370 + 370 \zeta_{6} ) q^{67} -252 q^{69} + 342 q^{71} + ( -362 + 362 \zeta_{6} ) q^{73} -348 \zeta_{6} q^{75} + ( -210 - 105 \zeta_{6} ) q^{77} + 467 \zeta_{6} q^{79} + ( -81 + 81 \zeta_{6} ) q^{81} -477 q^{83} -252 q^{85} + ( -891 + 891 \zeta_{6} ) q^{87} -906 \zeta_{6} q^{89} + ( 448 - 1344 \zeta_{6} ) q^{91} + 759 \zeta_{6} q^{93} + ( 48 - 48 \zeta_{6} ) q^{95} + 503 q^{97} + 135 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} + 3q^{5} + 7q^{7} - 9q^{9} + O(q^{10})$$ $$2q + 3q^{3} + 3q^{5} + 7q^{7} - 9q^{9} - 15q^{11} - 128q^{13} + 18q^{15} - 84q^{17} - 16q^{19} + 105q^{21} - 84q^{23} + 116q^{25} - 54q^{27} - 594q^{29} - 253q^{31} + 45q^{33} - 84q^{35} + 316q^{37} - 192q^{39} + 720q^{41} - 52q^{43} + 27q^{45} - 30q^{47} - 637q^{49} + 252q^{51} - 363q^{53} - 90q^{55} - 96q^{57} - 15q^{59} + 118q^{61} + 252q^{63} - 192q^{65} - 370q^{67} - 504q^{69} + 684q^{71} - 362q^{73} - 348q^{75} - 525q^{77} + 467q^{79} - 81q^{81} - 954q^{83} - 504q^{85} - 891q^{87} - 906q^{89} - 448q^{91} + 759q^{93} + 48q^{95} + 1006q^{97} + 270q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 2.59808i 0 1.50000 + 2.59808i 0 3.50000 + 18.1865i 0 −4.50000 7.79423i 0
289.1 0 1.50000 + 2.59808i 0 1.50000 2.59808i 0 3.50000 18.1865i 0 −4.50000 + 7.79423i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.q.e 2
4.b odd 2 1 21.4.e.a 2
7.c even 3 1 inner 336.4.q.e 2
7.c even 3 1 2352.4.a.i 1
7.d odd 6 1 2352.4.a.bd 1
12.b even 2 1 63.4.e.a 2
28.d even 2 1 147.4.e.h 2
28.f even 6 1 147.4.a.a 1
28.f even 6 1 147.4.e.h 2
28.g odd 6 1 21.4.e.a 2
28.g odd 6 1 147.4.a.b 1
84.h odd 2 1 441.4.e.c 2
84.j odd 6 1 441.4.a.k 1
84.j odd 6 1 441.4.e.c 2
84.n even 6 1 63.4.e.a 2
84.n even 6 1 441.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.a 2 4.b odd 2 1
21.4.e.a 2 28.g odd 6 1
63.4.e.a 2 12.b even 2 1
63.4.e.a 2 84.n even 6 1
147.4.a.a 1 28.f even 6 1
147.4.a.b 1 28.g odd 6 1
147.4.e.h 2 28.d even 2 1
147.4.e.h 2 28.f even 6 1
336.4.q.e 2 1.a even 1 1 trivial
336.4.q.e 2 7.c even 3 1 inner
441.4.a.k 1 84.j odd 6 1
441.4.a.l 1 84.n even 6 1
441.4.e.c 2 84.h odd 2 1
441.4.e.c 2 84.j odd 6 1
2352.4.a.i 1 7.c even 3 1
2352.4.a.bd 1 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 3 T_{5} + 9$$ acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$343 - 7 T + T^{2}$$
$11$ $$225 + 15 T + T^{2}$$
$13$ $$( 64 + T )^{2}$$
$17$ $$7056 + 84 T + T^{2}$$
$19$ $$256 + 16 T + T^{2}$$
$23$ $$7056 + 84 T + T^{2}$$
$29$ $$( 297 + T )^{2}$$
$31$ $$64009 + 253 T + T^{2}$$
$37$ $$99856 - 316 T + T^{2}$$
$41$ $$( -360 + T )^{2}$$
$43$ $$( 26 + T )^{2}$$
$47$ $$900 + 30 T + T^{2}$$
$53$ $$131769 + 363 T + T^{2}$$
$59$ $$225 + 15 T + T^{2}$$
$61$ $$13924 - 118 T + T^{2}$$
$67$ $$136900 + 370 T + T^{2}$$
$71$ $$( -342 + T )^{2}$$
$73$ $$131044 + 362 T + T^{2}$$
$79$ $$218089 - 467 T + T^{2}$$
$83$ $$( 477 + T )^{2}$$
$89$ $$820836 + 906 T + T^{2}$$
$97$ $$( -503 + T )^{2}$$