# Properties

 Label 336.6.q.c Level $336$ Weight $6$ Character orbit 336.q Analytic conductor $53.889$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 336.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$53.8889634572$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 9 \zeta_{6} + 9) q^{3} + 6 \zeta_{6} q^{5} + (133 \zeta_{6} - 126) q^{7} - 81 \zeta_{6} q^{9} +O(q^{10})$$ q + (-9*z + 9) * q^3 + 6*z * q^5 + (133*z - 126) * q^7 - 81*z * q^9 $$q + ( - 9 \zeta_{6} + 9) q^{3} + 6 \zeta_{6} q^{5} + (133 \zeta_{6} - 126) q^{7} - 81 \zeta_{6} q^{9} + (666 \zeta_{6} - 666) q^{11} - 559 q^{13} + 54 q^{15} + ( - 1740 \zeta_{6} + 1740) q^{17} + 1157 \zeta_{6} q^{19} + (1134 \zeta_{6} + 63) q^{21} - 3468 \zeta_{6} q^{23} + ( - 3089 \zeta_{6} + 3089) q^{25} - 729 q^{27} + 3372 q^{29} + ( - 6293 \zeta_{6} + 6293) q^{31} + 5994 \zeta_{6} q^{33} + (42 \zeta_{6} - 798) q^{35} - 3131 \zeta_{6} q^{37} + (5031 \zeta_{6} - 5031) q^{39} - 4866 q^{41} + 11407 q^{43} + ( - 486 \zeta_{6} + 486) q^{45} + 2310 \zeta_{6} q^{47} + ( - 15827 \zeta_{6} - 1813) q^{49} - 15660 \zeta_{6} q^{51} + ( - 28296 \zeta_{6} + 28296) q^{53} - 3996 q^{55} + 10413 q^{57} + ( - 20544 \zeta_{6} + 20544) q^{59} + 4630 \zeta_{6} q^{61} + ( - 567 \zeta_{6} + 10773) q^{63} - 3354 \zeta_{6} q^{65} + (18745 \zeta_{6} - 18745) q^{67} - 31212 q^{69} + 38226 q^{71} + (70589 \zeta_{6} - 70589) q^{73} - 27801 \zeta_{6} q^{75} + ( - 83916 \zeta_{6} - 4662) q^{77} - 62293 \zeta_{6} q^{79} + (6561 \zeta_{6} - 6561) q^{81} - 79818 q^{83} + 10440 q^{85} + ( - 30348 \zeta_{6} + 30348) q^{87} + 18120 \zeta_{6} q^{89} + ( - 74347 \zeta_{6} + 70434) q^{91} - 56637 \zeta_{6} q^{93} + (6942 \zeta_{6} - 6942) q^{95} + 124754 q^{97} + 53946 q^{99} +O(q^{100})$$ q + (-9*z + 9) * q^3 + 6*z * q^5 + (133*z - 126) * q^7 - 81*z * q^9 + (666*z - 666) * q^11 - 559 * q^13 + 54 * q^15 + (-1740*z + 1740) * q^17 + 1157*z * q^19 + (1134*z + 63) * q^21 - 3468*z * q^23 + (-3089*z + 3089) * q^25 - 729 * q^27 + 3372 * q^29 + (-6293*z + 6293) * q^31 + 5994*z * q^33 + (42*z - 798) * q^35 - 3131*z * q^37 + (5031*z - 5031) * q^39 - 4866 * q^41 + 11407 * q^43 + (-486*z + 486) * q^45 + 2310*z * q^47 + (-15827*z - 1813) * q^49 - 15660*z * q^51 + (-28296*z + 28296) * q^53 - 3996 * q^55 + 10413 * q^57 + (-20544*z + 20544) * q^59 + 4630*z * q^61 + (-567*z + 10773) * q^63 - 3354*z * q^65 + (18745*z - 18745) * q^67 - 31212 * q^69 + 38226 * q^71 + (70589*z - 70589) * q^73 - 27801*z * q^75 + (-83916*z - 4662) * q^77 - 62293*z * q^79 + (6561*z - 6561) * q^81 - 79818 * q^83 + 10440 * q^85 + (-30348*z + 30348) * q^87 + 18120*z * q^89 + (-74347*z + 70434) * q^91 - 56637*z * q^93 + (6942*z - 6942) * q^95 + 124754 * q^97 + 53946 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 9 q^{3} + 6 q^{5} - 119 q^{7} - 81 q^{9}+O(q^{10})$$ 2 * q + 9 * q^3 + 6 * q^5 - 119 * q^7 - 81 * q^9 $$2 q + 9 q^{3} + 6 q^{5} - 119 q^{7} - 81 q^{9} - 666 q^{11} - 1118 q^{13} + 108 q^{15} + 1740 q^{17} + 1157 q^{19} + 1260 q^{21} - 3468 q^{23} + 3089 q^{25} - 1458 q^{27} + 6744 q^{29} + 6293 q^{31} + 5994 q^{33} - 1554 q^{35} - 3131 q^{37} - 5031 q^{39} - 9732 q^{41} + 22814 q^{43} + 486 q^{45} + 2310 q^{47} - 19453 q^{49} - 15660 q^{51} + 28296 q^{53} - 7992 q^{55} + 20826 q^{57} + 20544 q^{59} + 4630 q^{61} + 20979 q^{63} - 3354 q^{65} - 18745 q^{67} - 62424 q^{69} + 76452 q^{71} - 70589 q^{73} - 27801 q^{75} - 93240 q^{77} - 62293 q^{79} - 6561 q^{81} - 159636 q^{83} + 20880 q^{85} + 30348 q^{87} + 18120 q^{89} + 66521 q^{91} - 56637 q^{93} - 6942 q^{95} + 249508 q^{97} + 107892 q^{99}+O(q^{100})$$ 2 * q + 9 * q^3 + 6 * q^5 - 119 * q^7 - 81 * q^9 - 666 * q^11 - 1118 * q^13 + 108 * q^15 + 1740 * q^17 + 1157 * q^19 + 1260 * q^21 - 3468 * q^23 + 3089 * q^25 - 1458 * q^27 + 6744 * q^29 + 6293 * q^31 + 5994 * q^33 - 1554 * q^35 - 3131 * q^37 - 5031 * q^39 - 9732 * q^41 + 22814 * q^43 + 486 * q^45 + 2310 * q^47 - 19453 * q^49 - 15660 * q^51 + 28296 * q^53 - 7992 * q^55 + 20826 * q^57 + 20544 * q^59 + 4630 * q^61 + 20979 * q^63 - 3354 * q^65 - 18745 * q^67 - 62424 * q^69 + 76452 * q^71 - 70589 * q^73 - 27801 * q^75 - 93240 * q^77 - 62293 * q^79 - 6561 * q^81 - 159636 * q^83 + 20880 * q^85 + 30348 * q^87 + 18120 * q^89 + 66521 * q^91 - 56637 * q^93 - 6942 * q^95 + 249508 * q^97 + 107892 * q^99

## 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 4.50000 7.79423i 0 3.00000 + 5.19615i 0 −59.5000 + 115.181i 0 −40.5000 70.1481i 0
289.1 0 4.50000 + 7.79423i 0 3.00000 5.19615i 0 −59.5000 115.181i 0 −40.5000 + 70.1481i 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 336.6.q.c 2
4.b odd 2 1 42.6.e.a 2
7.c even 3 1 inner 336.6.q.c 2
12.b even 2 1 126.6.g.c 2
28.d even 2 1 294.6.e.e 2
28.f even 6 1 294.6.a.j 1
28.f even 6 1 294.6.e.e 2
28.g odd 6 1 42.6.e.a 2
28.g odd 6 1 294.6.a.l 1
84.j odd 6 1 882.6.a.e 1
84.n even 6 1 126.6.g.c 2
84.n even 6 1 882.6.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.a 2 4.b odd 2 1
42.6.e.a 2 28.g odd 6 1
126.6.g.c 2 12.b even 2 1
126.6.g.c 2 84.n even 6 1
294.6.a.j 1 28.f even 6 1
294.6.a.l 1 28.g odd 6 1
294.6.e.e 2 28.d even 2 1
294.6.e.e 2 28.f even 6 1
336.6.q.c 2 1.a even 1 1 trivial
336.6.q.c 2 7.c even 3 1 inner
882.6.a.e 1 84.j odd 6 1
882.6.a.f 1 84.n even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 9T + 81$$
$5$ $$T^{2} - 6T + 36$$
$7$ $$T^{2} + 119T + 16807$$
$11$ $$T^{2} + 666T + 443556$$
$13$ $$(T + 559)^{2}$$
$17$ $$T^{2} - 1740 T + 3027600$$
$19$ $$T^{2} - 1157 T + 1338649$$
$23$ $$T^{2} + 3468 T + 12027024$$
$29$ $$(T - 3372)^{2}$$
$31$ $$T^{2} - 6293 T + 39601849$$
$37$ $$T^{2} + 3131 T + 9803161$$
$41$ $$(T + 4866)^{2}$$
$43$ $$(T - 11407)^{2}$$
$47$ $$T^{2} - 2310 T + 5336100$$
$53$ $$T^{2} - 28296 T + 800663616$$
$59$ $$T^{2} - 20544 T + 422055936$$
$61$ $$T^{2} - 4630 T + 21436900$$
$67$ $$T^{2} + 18745 T + 351375025$$
$71$ $$(T - 38226)^{2}$$
$73$ $$T^{2} + 70589 T + 4982806921$$
$79$ $$T^{2} + 62293 T + 3880417849$$
$83$ $$(T + 79818)^{2}$$
$89$ $$T^{2} - 18120 T + 328334400$$
$97$ $$(T - 124754)^{2}$$