# Properties

 Label 336.3.bh.d Level $336$ Weight $3$ Character orbit 336.bh Analytic conductor $9.155$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.15533688251$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} + 1) q^{3} + ( - 3 \zeta_{6} + 6) q^{5} + ( - 3 \zeta_{6} - 5) q^{7} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (z + 1) * q^3 + (-3*z + 6) * q^5 + (-3*z - 5) * q^7 + 3*z * q^9 $$q + (\zeta_{6} + 1) q^{3} + ( - 3 \zeta_{6} + 6) q^{5} + ( - 3 \zeta_{6} - 5) q^{7} + 3 \zeta_{6} q^{9} + ( - 15 \zeta_{6} + 15) q^{11} + ( - 16 \zeta_{6} + 8) q^{13} + 9 q^{15} + (6 \zeta_{6} + 6) q^{17} + ( - 6 \zeta_{6} + 12) q^{19} + ( - 11 \zeta_{6} - 2) q^{21} + ( - 2 \zeta_{6} + 2) q^{25} + (6 \zeta_{6} - 3) q^{27} - 9 q^{29} + (7 \zeta_{6} + 7) q^{31} + ( - 15 \zeta_{6} + 30) q^{33} + (6 \zeta_{6} - 39) q^{35} - 10 \zeta_{6} q^{37} + ( - 24 \zeta_{6} + 24) q^{39} + ( - 12 \zeta_{6} + 6) q^{41} + 74 q^{43} + (9 \zeta_{6} + 9) q^{45} + (39 \zeta_{6} + 16) q^{49} + 18 \zeta_{6} q^{51} + (33 \zeta_{6} - 33) q^{53} + ( - 90 \zeta_{6} + 45) q^{55} + 18 q^{57} + ( - 9 \zeta_{6} - 9) q^{59} + ( - 52 \zeta_{6} + 104) q^{61} + ( - 24 \zeta_{6} + 9) q^{63} - 72 \zeta_{6} q^{65} + (76 \zeta_{6} - 76) q^{67} - 84 q^{71} + ( - 36 \zeta_{6} - 36) q^{73} + ( - 2 \zeta_{6} + 4) q^{75} + (75 \zeta_{6} - 120) q^{77} - 43 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} + (138 \zeta_{6} - 69) q^{83} + 54 q^{85} + ( - 9 \zeta_{6} - 9) q^{87} + (42 \zeta_{6} - 84) q^{89} + (104 \zeta_{6} - 88) q^{91} + 21 \zeta_{6} q^{93} + ( - 54 \zeta_{6} + 54) q^{95} + (214 \zeta_{6} - 107) q^{97} + 45 q^{99} +O(q^{100})$$ q + (z + 1) * q^3 + (-3*z + 6) * q^5 + (-3*z - 5) * q^7 + 3*z * q^9 + (-15*z + 15) * q^11 + (-16*z + 8) * q^13 + 9 * q^15 + (6*z + 6) * q^17 + (-6*z + 12) * q^19 + (-11*z - 2) * q^21 + (-2*z + 2) * q^25 + (6*z - 3) * q^27 - 9 * q^29 + (7*z + 7) * q^31 + (-15*z + 30) * q^33 + (6*z - 39) * q^35 - 10*z * q^37 + (-24*z + 24) * q^39 + (-12*z + 6) * q^41 + 74 * q^43 + (9*z + 9) * q^45 + (39*z + 16) * q^49 + 18*z * q^51 + (33*z - 33) * q^53 + (-90*z + 45) * q^55 + 18 * q^57 + (-9*z - 9) * q^59 + (-52*z + 104) * q^61 + (-24*z + 9) * q^63 - 72*z * q^65 + (76*z - 76) * q^67 - 84 * q^71 + (-36*z - 36) * q^73 + (-2*z + 4) * q^75 + (75*z - 120) * q^77 - 43*z * q^79 + (9*z - 9) * q^81 + (138*z - 69) * q^83 + 54 * q^85 + (-9*z - 9) * q^87 + (42*z - 84) * q^89 + (104*z - 88) * q^91 + 21*z * q^93 + (-54*z + 54) * q^95 + (214*z - 107) * q^97 + 45 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + 9 q^{5} - 13 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + 9 * q^5 - 13 * q^7 + 3 * q^9 $$2 q + 3 q^{3} + 9 q^{5} - 13 q^{7} + 3 q^{9} + 15 q^{11} + 18 q^{15} + 18 q^{17} + 18 q^{19} - 15 q^{21} + 2 q^{25} - 18 q^{29} + 21 q^{31} + 45 q^{33} - 72 q^{35} - 10 q^{37} + 24 q^{39} + 148 q^{43} + 27 q^{45} + 71 q^{49} + 18 q^{51} - 33 q^{53} + 36 q^{57} - 27 q^{59} + 156 q^{61} - 6 q^{63} - 72 q^{65} - 76 q^{67} - 168 q^{71} - 108 q^{73} + 6 q^{75} - 165 q^{77} - 43 q^{79} - 9 q^{81} + 108 q^{85} - 27 q^{87} - 126 q^{89} - 72 q^{91} + 21 q^{93} + 54 q^{95} + 90 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 + 9 * q^5 - 13 * q^7 + 3 * q^9 + 15 * q^11 + 18 * q^15 + 18 * q^17 + 18 * q^19 - 15 * q^21 + 2 * q^25 - 18 * q^29 + 21 * q^31 + 45 * q^33 - 72 * q^35 - 10 * q^37 + 24 * q^39 + 148 * q^43 + 27 * q^45 + 71 * q^49 + 18 * q^51 - 33 * q^53 + 36 * q^57 - 27 * q^59 + 156 * q^61 - 6 * q^63 - 72 * q^65 - 76 * q^67 - 168 * q^71 - 108 * q^73 + 6 * q^75 - 165 * q^77 - 43 * q^79 - 9 * q^81 + 108 * q^85 - 27 * q^87 - 126 * q^89 - 72 * q^91 + 21 * q^93 + 54 * q^95 + 90 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.50000 0.866025i 0 4.50000 + 2.59808i 0 −6.50000 + 2.59808i 0 1.50000 2.59808i 0
241.1 0 1.50000 + 0.866025i 0 4.50000 2.59808i 0 −6.50000 2.59808i 0 1.50000 + 2.59808i 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.3.bh.d 2
3.b odd 2 1 1008.3.cg.a 2
4.b odd 2 1 21.3.f.a 2
7.c even 3 1 2352.3.f.a 2
7.d odd 6 1 inner 336.3.bh.d 2
7.d odd 6 1 2352.3.f.a 2
12.b even 2 1 63.3.m.d 2
20.d odd 2 1 525.3.o.h 2
20.e even 4 2 525.3.s.e 4
21.g even 6 1 1008.3.cg.a 2
28.d even 2 1 147.3.f.a 2
28.f even 6 1 21.3.f.a 2
28.f even 6 1 147.3.d.c 2
28.g odd 6 1 147.3.d.c 2
28.g odd 6 1 147.3.f.a 2
84.h odd 2 1 441.3.m.g 2
84.j odd 6 1 63.3.m.d 2
84.j odd 6 1 441.3.d.a 2
84.n even 6 1 441.3.d.a 2
84.n even 6 1 441.3.m.g 2
140.s even 6 1 525.3.o.h 2
140.x odd 12 2 525.3.s.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.a 2 4.b odd 2 1
21.3.f.a 2 28.f even 6 1
63.3.m.d 2 12.b even 2 1
63.3.m.d 2 84.j odd 6 1
147.3.d.c 2 28.f even 6 1
147.3.d.c 2 28.g odd 6 1
147.3.f.a 2 28.d even 2 1
147.3.f.a 2 28.g odd 6 1
336.3.bh.d 2 1.a even 1 1 trivial
336.3.bh.d 2 7.d odd 6 1 inner
441.3.d.a 2 84.j odd 6 1
441.3.d.a 2 84.n even 6 1
441.3.m.g 2 84.h odd 2 1
441.3.m.g 2 84.n even 6 1
525.3.o.h 2 20.d odd 2 1
525.3.o.h 2 140.s even 6 1
525.3.s.e 4 20.e even 4 2
525.3.s.e 4 140.x odd 12 2
1008.3.cg.a 2 3.b odd 2 1
1008.3.cg.a 2 21.g even 6 1
2352.3.f.a 2 7.c even 3 1
2352.3.f.a 2 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 9T_{5} + 27$$ acting on $$S_{3}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$T^{2} - 9T + 27$$
$7$ $$T^{2} + 13T + 49$$
$11$ $$T^{2} - 15T + 225$$
$13$ $$T^{2} + 192$$
$17$ $$T^{2} - 18T + 108$$
$19$ $$T^{2} - 18T + 108$$
$23$ $$T^{2}$$
$29$ $$(T + 9)^{2}$$
$31$ $$T^{2} - 21T + 147$$
$37$ $$T^{2} + 10T + 100$$
$41$ $$T^{2} + 108$$
$43$ $$(T - 74)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 33T + 1089$$
$59$ $$T^{2} + 27T + 243$$
$61$ $$T^{2} - 156T + 8112$$
$67$ $$T^{2} + 76T + 5776$$
$71$ $$(T + 84)^{2}$$
$73$ $$T^{2} + 108T + 3888$$
$79$ $$T^{2} + 43T + 1849$$
$83$ $$T^{2} + 14283$$
$89$ $$T^{2} + 126T + 5292$$
$97$ $$T^{2} + 34347$$