# Properties

 Label 336.6.q.a 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + (9 \zeta_{6} - 9) q^{3} - 86 \zeta_{6} q^{5} + (147 \zeta_{6} - 98) q^{7} - 81 \zeta_{6} q^{9} +O(q^{10})$$ q + (9*z - 9) * q^3 - 86*z * q^5 + (147*z - 98) * q^7 - 81*z * q^9 $$q + (9 \zeta_{6} - 9) q^{3} - 86 \zeta_{6} q^{5} + (147 \zeta_{6} - 98) q^{7} - 81 \zeta_{6} q^{9} + ( - 34 \zeta_{6} + 34) q^{11} - 3 q^{13} + 774 q^{15} + ( - 1904 \zeta_{6} + 1904) q^{17} - 1489 \zeta_{6} q^{19} + ( - 882 \zeta_{6} - 441) q^{21} - 224 \zeta_{6} q^{23} + (4271 \zeta_{6} - 4271) q^{25} + 729 q^{27} - 6508 q^{29} + ( - 1731 \zeta_{6} + 1731) q^{31} + 306 \zeta_{6} q^{33} + ( - 4214 \zeta_{6} + 12642) q^{35} + 7633 \zeta_{6} q^{37} + ( - 27 \zeta_{6} + 27) q^{39} + 15414 q^{41} - 18491 q^{43} + (6966 \zeta_{6} - 6966) q^{45} + 18462 \zeta_{6} q^{47} + ( - 7203 \zeta_{6} - 12005) q^{49} + 17136 \zeta_{6} q^{51} + ( - 19956 \zeta_{6} + 19956) q^{53} - 2924 q^{55} + 13401 q^{57} + (31828 \zeta_{6} - 31828) q^{59} + 57654 \zeta_{6} q^{61} + ( - 3969 \zeta_{6} + 11907) q^{63} + 258 \zeta_{6} q^{65} + (60563 \zeta_{6} - 60563) q^{67} + 2016 q^{69} + 44834 q^{71} + (20821 \zeta_{6} - 20821) q^{73} - 38439 \zeta_{6} q^{75} + (3332 \zeta_{6} + 1666) q^{77} - 30531 \zeta_{6} q^{79} + (6561 \zeta_{6} - 6561) q^{81} - 110602 q^{83} - 163744 q^{85} + ( - 58572 \zeta_{6} + 58572) q^{87} + 58992 \zeta_{6} q^{89} + ( - 441 \zeta_{6} + 294) q^{91} + 15579 \zeta_{6} q^{93} + (128054 \zeta_{6} - 128054) q^{95} - 119846 q^{97} - 2754 q^{99} +O(q^{100})$$ q + (9*z - 9) * q^3 - 86*z * q^5 + (147*z - 98) * q^7 - 81*z * q^9 + (-34*z + 34) * q^11 - 3 * q^13 + 774 * q^15 + (-1904*z + 1904) * q^17 - 1489*z * q^19 + (-882*z - 441) * q^21 - 224*z * q^23 + (4271*z - 4271) * q^25 + 729 * q^27 - 6508 * q^29 + (-1731*z + 1731) * q^31 + 306*z * q^33 + (-4214*z + 12642) * q^35 + 7633*z * q^37 + (-27*z + 27) * q^39 + 15414 * q^41 - 18491 * q^43 + (6966*z - 6966) * q^45 + 18462*z * q^47 + (-7203*z - 12005) * q^49 + 17136*z * q^51 + (-19956*z + 19956) * q^53 - 2924 * q^55 + 13401 * q^57 + (31828*z - 31828) * q^59 + 57654*z * q^61 + (-3969*z + 11907) * q^63 + 258*z * q^65 + (60563*z - 60563) * q^67 + 2016 * q^69 + 44834 * q^71 + (20821*z - 20821) * q^73 - 38439*z * q^75 + (3332*z + 1666) * q^77 - 30531*z * q^79 + (6561*z - 6561) * q^81 - 110602 * q^83 - 163744 * q^85 + (-58572*z + 58572) * q^87 + 58992*z * q^89 + (-441*z + 294) * q^91 + 15579*z * q^93 + (128054*z - 128054) * q^95 - 119846 * q^97 - 2754 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 9 q^{3} - 86 q^{5} - 49 q^{7} - 81 q^{9}+O(q^{10})$$ 2 * q - 9 * q^3 - 86 * q^5 - 49 * q^7 - 81 * q^9 $$2 q - 9 q^{3} - 86 q^{5} - 49 q^{7} - 81 q^{9} + 34 q^{11} - 6 q^{13} + 1548 q^{15} + 1904 q^{17} - 1489 q^{19} - 1764 q^{21} - 224 q^{23} - 4271 q^{25} + 1458 q^{27} - 13016 q^{29} + 1731 q^{31} + 306 q^{33} + 21070 q^{35} + 7633 q^{37} + 27 q^{39} + 30828 q^{41} - 36982 q^{43} - 6966 q^{45} + 18462 q^{47} - 31213 q^{49} + 17136 q^{51} + 19956 q^{53} - 5848 q^{55} + 26802 q^{57} - 31828 q^{59} + 57654 q^{61} + 19845 q^{63} + 258 q^{65} - 60563 q^{67} + 4032 q^{69} + 89668 q^{71} - 20821 q^{73} - 38439 q^{75} + 6664 q^{77} - 30531 q^{79} - 6561 q^{81} - 221204 q^{83} - 327488 q^{85} + 58572 q^{87} + 58992 q^{89} + 147 q^{91} + 15579 q^{93} - 128054 q^{95} - 239692 q^{97} - 5508 q^{99}+O(q^{100})$$ 2 * q - 9 * q^3 - 86 * q^5 - 49 * q^7 - 81 * q^9 + 34 * q^11 - 6 * q^13 + 1548 * q^15 + 1904 * q^17 - 1489 * q^19 - 1764 * q^21 - 224 * q^23 - 4271 * q^25 + 1458 * q^27 - 13016 * q^29 + 1731 * q^31 + 306 * q^33 + 21070 * q^35 + 7633 * q^37 + 27 * q^39 + 30828 * q^41 - 36982 * q^43 - 6966 * q^45 + 18462 * q^47 - 31213 * q^49 + 17136 * q^51 + 19956 * q^53 - 5848 * q^55 + 26802 * q^57 - 31828 * q^59 + 57654 * q^61 + 19845 * q^63 + 258 * q^65 - 60563 * q^67 + 4032 * q^69 + 89668 * q^71 - 20821 * q^73 - 38439 * q^75 + 6664 * q^77 - 30531 * q^79 - 6561 * q^81 - 221204 * q^83 - 327488 * q^85 + 58572 * q^87 + 58992 * q^89 + 147 * q^91 + 15579 * q^93 - 128054 * q^95 - 239692 * q^97 - 5508 * 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 −43.0000 74.4782i 0 −24.5000 + 127.306i 0 −40.5000 70.1481i 0
289.1 0 −4.50000 7.79423i 0 −43.0000 + 74.4782i 0 −24.5000 127.306i 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.a 2
4.b odd 2 1 42.6.e.b 2
7.c even 3 1 inner 336.6.q.a 2
12.b even 2 1 126.6.g.b 2
28.d even 2 1 294.6.e.m 2
28.f even 6 1 294.6.a.e 1
28.f even 6 1 294.6.e.m 2
28.g odd 6 1 42.6.e.b 2
28.g odd 6 1 294.6.a.d 1
84.j odd 6 1 882.6.a.y 1
84.n even 6 1 126.6.g.b 2
84.n even 6 1 882.6.a.m 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 86T_{5} + 7396$$ acting on $$S_{6}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9T + 81$$
$5$ $$T^{2} + 86T + 7396$$
$7$ $$T^{2} + 49T + 16807$$
$11$ $$T^{2} - 34T + 1156$$
$13$ $$(T + 3)^{2}$$
$17$ $$T^{2} - 1904 T + 3625216$$
$19$ $$T^{2} + 1489 T + 2217121$$
$23$ $$T^{2} + 224T + 50176$$
$29$ $$(T + 6508)^{2}$$
$31$ $$T^{2} - 1731 T + 2996361$$
$37$ $$T^{2} - 7633 T + 58262689$$
$41$ $$(T - 15414)^{2}$$
$43$ $$(T + 18491)^{2}$$
$47$ $$T^{2} - 18462 T + 340845444$$
$53$ $$T^{2} - 19956 T + 398241936$$
$59$ $$T^{2} + 31828 T + 1013021584$$
$61$ $$T^{2} - 57654 T + 3323983716$$
$67$ $$T^{2} + 60563 T + 3667876969$$
$71$ $$(T - 44834)^{2}$$
$73$ $$T^{2} + 20821 T + 433514041$$
$79$ $$T^{2} + 30531 T + 932141961$$
$83$ $$(T + 110602)^{2}$$
$89$ $$T^{2} - 58992 T + 3480056064$$
$97$ $$(T + 119846)^{2}$$