# Properties

 Label 336.4.q.a 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) 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.

 $$f(q)$$ $$=$$ $$q + ( -3 + 3 \zeta_{6} ) q^{3} -7 \zeta_{6} q^{5} + ( 21 - 7 \zeta_{6} ) q^{7} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -3 + 3 \zeta_{6} ) q^{3} -7 \zeta_{6} q^{5} + ( 21 - 7 \zeta_{6} ) q^{7} -9 \zeta_{6} q^{9} + ( 7 - 7 \zeta_{6} ) q^{11} -52 q^{13} + 21 q^{15} + ( -72 + 72 \zeta_{6} ) q^{17} + 20 \zeta_{6} q^{19} + ( -42 + 63 \zeta_{6} ) q^{21} -48 \zeta_{6} q^{23} + ( 76 - 76 \zeta_{6} ) q^{25} + 27 q^{27} -243 q^{29} + ( 95 - 95 \zeta_{6} ) q^{31} + 21 \zeta_{6} q^{33} + ( -49 - 98 \zeta_{6} ) q^{35} -352 \zeta_{6} q^{37} + ( 156 - 156 \zeta_{6} ) q^{39} -296 q^{41} -158 q^{43} + ( -63 + 63 \zeta_{6} ) q^{45} -142 \zeta_{6} q^{47} + ( 392 - 245 \zeta_{6} ) q^{49} -216 \zeta_{6} q^{51} + ( 375 - 375 \zeta_{6} ) q^{53} -49 q^{55} -60 q^{57} + ( 279 - 279 \zeta_{6} ) q^{59} -246 \zeta_{6} q^{61} + ( -63 - 126 \zeta_{6} ) q^{63} + 364 \zeta_{6} q^{65} + ( -730 + 730 \zeta_{6} ) q^{67} + 144 q^{69} -338 q^{71} + ( 542 - 542 \zeta_{6} ) q^{73} + 228 \zeta_{6} q^{75} + ( 98 - 147 \zeta_{6} ) q^{77} -305 \zeta_{6} q^{79} + ( -81 + 81 \zeta_{6} ) q^{81} -1123 q^{83} + 504 q^{85} + ( 729 - 729 \zeta_{6} ) q^{87} + 426 \zeta_{6} q^{89} + ( -1092 + 364 \zeta_{6} ) q^{91} + 285 \zeta_{6} q^{93} + ( 140 - 140 \zeta_{6} ) q^{95} -369 q^{97} -63 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} - 7q^{5} + 35q^{7} - 9q^{9} + O(q^{10})$$ $$2q - 3q^{3} - 7q^{5} + 35q^{7} - 9q^{9} + 7q^{11} - 104q^{13} + 42q^{15} - 72q^{17} + 20q^{19} - 21q^{21} - 48q^{23} + 76q^{25} + 54q^{27} - 486q^{29} + 95q^{31} + 21q^{33} - 196q^{35} - 352q^{37} + 156q^{39} - 592q^{41} - 316q^{43} - 63q^{45} - 142q^{47} + 539q^{49} - 216q^{51} + 375q^{53} - 98q^{55} - 120q^{57} + 279q^{59} - 246q^{61} - 252q^{63} + 364q^{65} - 730q^{67} + 288q^{69} - 676q^{71} + 542q^{73} + 228q^{75} + 49q^{77} - 305q^{79} - 81q^{81} - 2246q^{83} + 1008q^{85} + 729q^{87} + 426q^{89} - 1820q^{91} + 285q^{93} + 140q^{95} - 738q^{97} - 126q^{99} + O(q^{100})$$

## 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 −3.50000 6.06218i 0 17.5000 6.06218i 0 −4.50000 7.79423i 0
289.1 0 −1.50000 2.59808i 0 −3.50000 + 6.06218i 0 17.5000 + 6.06218i 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.a 2
4.b odd 2 1 168.4.q.a 2
7.c even 3 1 inner 336.4.q.a 2
7.c even 3 1 2352.4.a.bg 1
7.d odd 6 1 2352.4.a.c 1
12.b even 2 1 504.4.s.e 2
28.f even 6 1 1176.4.a.i 1
28.g odd 6 1 168.4.q.a 2
28.g odd 6 1 1176.4.a.f 1
84.n even 6 1 504.4.s.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.a 2 4.b odd 2 1
168.4.q.a 2 28.g odd 6 1
336.4.q.a 2 1.a even 1 1 trivial
336.4.q.a 2 7.c even 3 1 inner
504.4.s.e 2 12.b even 2 1
504.4.s.e 2 84.n even 6 1
1176.4.a.f 1 28.g odd 6 1
1176.4.a.i 1 28.f even 6 1
2352.4.a.c 1 7.d odd 6 1
2352.4.a.bg 1 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 7 T_{5} + 49$$ acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$49 + 7 T + T^{2}$$
$7$ $$343 - 35 T + T^{2}$$
$11$ $$49 - 7 T + T^{2}$$
$13$ $$( 52 + T )^{2}$$
$17$ $$5184 + 72 T + T^{2}$$
$19$ $$400 - 20 T + T^{2}$$
$23$ $$2304 + 48 T + T^{2}$$
$29$ $$( 243 + T )^{2}$$
$31$ $$9025 - 95 T + T^{2}$$
$37$ $$123904 + 352 T + T^{2}$$
$41$ $$( 296 + T )^{2}$$
$43$ $$( 158 + T )^{2}$$
$47$ $$20164 + 142 T + T^{2}$$
$53$ $$140625 - 375 T + T^{2}$$
$59$ $$77841 - 279 T + T^{2}$$
$61$ $$60516 + 246 T + T^{2}$$
$67$ $$532900 + 730 T + T^{2}$$
$71$ $$( 338 + T )^{2}$$
$73$ $$293764 - 542 T + T^{2}$$
$79$ $$93025 + 305 T + T^{2}$$
$83$ $$( 1123 + T )^{2}$$
$89$ $$181476 - 426 T + T^{2}$$
$97$ $$( 369 + T )^{2}$$