# Properties

 Label 336.4.q.f 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 42) 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.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} + 6 \zeta_{6} q^{5} + ( 14 - 21 \zeta_{6} ) q^{7} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 3 - 3 \zeta_{6} ) q^{3} + 6 \zeta_{6} q^{5} + ( 14 - 21 \zeta_{6} ) q^{7} -9 \zeta_{6} q^{9} + ( -30 + 30 \zeta_{6} ) q^{11} + 53 q^{13} + 18 q^{15} + ( 84 - 84 \zeta_{6} ) q^{17} -97 \zeta_{6} q^{19} + ( -21 - 42 \zeta_{6} ) q^{21} + 84 \zeta_{6} q^{23} + ( 89 - 89 \zeta_{6} ) q^{25} -27 q^{27} -180 q^{29} + ( 179 - 179 \zeta_{6} ) q^{31} + 90 \zeta_{6} q^{33} + ( 126 - 42 \zeta_{6} ) q^{35} + 145 \zeta_{6} q^{37} + ( 159 - 159 \zeta_{6} ) q^{39} + 126 q^{41} + 325 q^{43} + ( 54 - 54 \zeta_{6} ) q^{45} -366 \zeta_{6} q^{47} + ( -245 - 147 \zeta_{6} ) q^{49} -252 \zeta_{6} q^{51} + ( 768 - 768 \zeta_{6} ) q^{53} -180 q^{55} -291 q^{57} + ( -264 + 264 \zeta_{6} ) q^{59} -818 \zeta_{6} q^{61} + ( -189 + 63 \zeta_{6} ) q^{63} + 318 \zeta_{6} q^{65} + ( -523 + 523 \zeta_{6} ) q^{67} + 252 q^{69} + 342 q^{71} + ( 43 - 43 \zeta_{6} ) q^{73} -267 \zeta_{6} q^{75} + ( 210 + 420 \zeta_{6} ) q^{77} -1171 \zeta_{6} q^{79} + ( -81 + 81 \zeta_{6} ) q^{81} + 810 q^{83} + 504 q^{85} + ( -540 + 540 \zeta_{6} ) q^{87} + 600 \zeta_{6} q^{89} + ( 742 - 1113 \zeta_{6} ) q^{91} -537 \zeta_{6} q^{93} + ( 582 - 582 \zeta_{6} ) q^{95} + 386 q^{97} + 270 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} + 6q^{5} + 7q^{7} - 9q^{9} + O(q^{10})$$ $$2q + 3q^{3} + 6q^{5} + 7q^{7} - 9q^{9} - 30q^{11} + 106q^{13} + 36q^{15} + 84q^{17} - 97q^{19} - 84q^{21} + 84q^{23} + 89q^{25} - 54q^{27} - 360q^{29} + 179q^{31} + 90q^{33} + 210q^{35} + 145q^{37} + 159q^{39} + 252q^{41} + 650q^{43} + 54q^{45} - 366q^{47} - 637q^{49} - 252q^{51} + 768q^{53} - 360q^{55} - 582q^{57} - 264q^{59} - 818q^{61} - 315q^{63} + 318q^{65} - 523q^{67} + 504q^{69} + 684q^{71} + 43q^{73} - 267q^{75} + 840q^{77} - 1171q^{79} - 81q^{81} + 1620q^{83} + 1008q^{85} - 540q^{87} + 600q^{89} + 371q^{91} - 537q^{93} + 582q^{95} + 772q^{97} + 540q^{99} + O(q^{100})$$

## 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 3.00000 + 5.19615i 0 3.50000 18.1865i 0 −4.50000 7.79423i 0
289.1 0 1.50000 + 2.59808i 0 3.00000 5.19615i 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.f 2
4.b odd 2 1 42.4.e.a 2
7.c even 3 1 inner 336.4.q.f 2
7.c even 3 1 2352.4.a.f 1
7.d odd 6 1 2352.4.a.bf 1
12.b even 2 1 126.4.g.b 2
28.d even 2 1 294.4.e.i 2
28.f even 6 1 294.4.a.c 1
28.f even 6 1 294.4.e.i 2
28.g odd 6 1 42.4.e.a 2
28.g odd 6 1 294.4.a.d 1
84.h odd 2 1 882.4.g.g 2
84.j odd 6 1 882.4.a.l 1
84.j odd 6 1 882.4.g.g 2
84.n even 6 1 126.4.g.b 2
84.n even 6 1 882.4.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.a 2 4.b odd 2 1
42.4.e.a 2 28.g odd 6 1
126.4.g.b 2 12.b even 2 1
126.4.g.b 2 84.n even 6 1
294.4.a.c 1 28.f even 6 1
294.4.a.d 1 28.g odd 6 1
294.4.e.i 2 28.d even 2 1
294.4.e.i 2 28.f even 6 1
336.4.q.f 2 1.a even 1 1 trivial
336.4.q.f 2 7.c even 3 1 inner
882.4.a.l 1 84.j odd 6 1
882.4.a.o 1 84.n even 6 1
882.4.g.g 2 84.h odd 2 1
882.4.g.g 2 84.j odd 6 1
2352.4.a.f 1 7.c even 3 1
2352.4.a.bf 1 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 6 T_{5} + 36$$ acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$36 - 6 T + T^{2}$$
$7$ $$343 - 7 T + T^{2}$$
$11$ $$900 + 30 T + T^{2}$$
$13$ $$( -53 + T )^{2}$$
$17$ $$7056 - 84 T + T^{2}$$
$19$ $$9409 + 97 T + T^{2}$$
$23$ $$7056 - 84 T + T^{2}$$
$29$ $$( 180 + T )^{2}$$
$31$ $$32041 - 179 T + T^{2}$$
$37$ $$21025 - 145 T + T^{2}$$
$41$ $$( -126 + T )^{2}$$
$43$ $$( -325 + T )^{2}$$
$47$ $$133956 + 366 T + T^{2}$$
$53$ $$589824 - 768 T + T^{2}$$
$59$ $$69696 + 264 T + T^{2}$$
$61$ $$669124 + 818 T + T^{2}$$
$67$ $$273529 + 523 T + T^{2}$$
$71$ $$( -342 + T )^{2}$$
$73$ $$1849 - 43 T + T^{2}$$
$79$ $$1371241 + 1171 T + T^{2}$$
$83$ $$( -810 + T )^{2}$$
$89$ $$360000 - 600 T + T^{2}$$
$97$ $$( -386 + T )^{2}$$