# Properties

 Label 336.3.bh.f Level $336$ Weight $3$ Character orbit 336.bh Analytic conductor $9.155$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{65})$$ Defining polynomial: $$x^{4} - x^{3} + 17x^{2} + 16x + 256$$ x^4 - x^3 + 17*x^2 + 16*x + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 2) q^{3} + ( - \beta_{3} - 2 \beta_1 - 1) q^{5} - \beta_{2} q^{7} + ( - 3 \beta_1 + 3) q^{9}+O(q^{10})$$ q + (-b1 + 2) * q^3 + (-b3 - 2*b1 - 1) * q^5 - b2 * q^7 + (-3*b1 + 3) * q^9 $$q + ( - \beta_1 + 2) q^{3} + ( - \beta_{3} - 2 \beta_1 - 1) q^{5} - \beta_{2} q^{7} + ( - 3 \beta_1 + 3) q^{9} + (\beta_{3} - \beta_{2} - 7 \beta_1) q^{11} + ( - \beta_{2} + \beta_1) q^{13} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{15} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{17} + ( - \beta_{3} + 13 \beta_1 + 14) q^{19} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{21} + (24 \beta_1 - 24) q^{23} + (3 \beta_{3} - 3 \beta_{2} + 32 \beta_1) q^{25} + ( - 6 \beta_1 + 3) q^{27} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{29} + ( - 6 \beta_{3} - 6 \beta_{2} + 5 \beta_1 - 10) q^{31} + (3 \beta_{3} - 6 \beta_1 - 9) q^{33} + (2 \beta_{3} + 3 \beta_{2} - 47 \beta_1 + 47) q^{35} + ( - \beta_{3} - 2 \beta_{2} - 37 \beta_1 + 38) q^{37} + (\beta_{3} - \beta_{2} + 2 \beta_1) q^{39} + (4 \beta_{2} - 24 \beta_1 + 10) q^{41} + (6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 22) q^{43} + ( - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 6) q^{45} + ( - 2 \beta_{3} + 14 \beta_1 + 16) q^{47} + (\beta_{2} - 49) q^{49} + ( - 2 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 6) q^{51} + ( - \beta_{3} + \beta_{2} + 49 \beta_1) q^{53} + ( - 3 \beta_{2} - 75 \beta_1 + 39) q^{55} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 42) q^{57} + (\beta_{3} + \beta_{2} + 41 \beta_1 - 82) q^{59} + ( - 8 \beta_{3} - 36 \beta_1 - 28) q^{61} + (3 \beta_{3} + 3 \beta_1 - 3) q^{63} + (2 \beta_{3} + 4 \beta_{2} - 50 \beta_1 + 48) q^{65} + (5 \beta_{3} - 5 \beta_{2} + 6 \beta_1) q^{67} + (48 \beta_1 - 24) q^{69} + ( - 12 \beta_{3} - 6 \beta_{2} - 6 \beta_1 + 60) q^{71} + ( - 7 \beta_{3} - 7 \beta_{2} + 4 \beta_1 - 8) q^{73} + (9 \beta_{3} + 35 \beta_1 + 26) q^{75} + (7 \beta_{3} + 8 \beta_{2} + 56 \beta_1 - 105) q^{77} + (2 \beta_{3} + 4 \beta_{2} + 29 \beta_1 - 31) q^{79} - 9 \beta_1 q^{81} + (17 \beta_{2} + 21 \beta_1 - 19) q^{83} + (12 \beta_{3} + 6 \beta_{2} + 6 \beta_1 + 102) q^{85} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 6) q^{87} + (6 \beta_{3} + 36 \beta_1 + 30) q^{89} + ( - \beta_{3} - \beta_1 - 48) q^{91} + ( - 6 \beta_{3} - 12 \beta_{2} + 21 \beta_1 - 15) q^{93} + ( - 12 \beta_{3} + 12 \beta_{2} - 18 \beta_1) q^{95} + ( - 7 \beta_{2} + 17 \beta_1 - 5) q^{97} + (6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 27) q^{99}+O(q^{100})$$ q + (-b1 + 2) * q^3 + (-b3 - 2*b1 - 1) * q^5 - b2 * q^7 + (-3*b1 + 3) * q^9 + (b3 - b2 - 7*b1) * q^11 + (-b2 + b1) * q^13 + (-2*b3 - b2 - b1 - 3) * q^15 + (-2*b3 - 2*b2 + 2*b1 - 4) * q^17 + (-b3 + 13*b1 + 14) * q^19 + (b3 - b2 + b1 - 1) * q^21 + (24*b1 - 24) * q^23 + (3*b3 - 3*b2 + 32*b1) * q^25 + (-6*b1 + 3) * q^27 + (-2*b3 - b2 - b1 + 3) * q^29 + (-6*b3 - 6*b2 + 5*b1 - 10) * q^31 + (3*b3 - 6*b1 - 9) * q^33 + (2*b3 + 3*b2 - 47*b1 + 47) * q^35 + (-b3 - 2*b2 - 37*b1 + 38) * q^37 + (b3 - b2 + 2*b1) * q^39 + (4*b2 - 24*b1 + 10) * q^41 + (6*b3 + 3*b2 + 3*b1 - 22) * q^43 + (-3*b3 - 3*b2 + 3*b1 - 6) * q^45 + (-2*b3 + 14*b1 + 16) * q^47 + (b2 - 49) * q^49 + (-2*b3 - 4*b2 + 8*b1 - 6) * q^51 + (-b3 + b2 + 49*b1) * q^53 + (-3*b2 - 75*b1 + 39) * q^55 + (-2*b3 - b2 - b1 + 42) * q^57 + (b3 + b2 + 41*b1 - 82) * q^59 + (-8*b3 - 36*b1 - 28) * q^61 + (3*b3 + 3*b1 - 3) * q^63 + (2*b3 + 4*b2 - 50*b1 + 48) * q^65 + (5*b3 - 5*b2 + 6*b1) * q^67 + (48*b1 - 24) * q^69 + (-12*b3 - 6*b2 - 6*b1 + 60) * q^71 + (-7*b3 - 7*b2 + 4*b1 - 8) * q^73 + (9*b3 + 35*b1 + 26) * q^75 + (7*b3 + 8*b2 + 56*b1 - 105) * q^77 + (2*b3 + 4*b2 + 29*b1 - 31) * q^79 - 9*b1 * q^81 + (17*b2 + 21*b1 - 19) * q^83 + (12*b3 + 6*b2 + 6*b1 + 102) * q^85 + (-3*b3 - 3*b2 - 3*b1 + 6) * q^87 + (6*b3 + 36*b1 + 30) * q^89 + (-b3 - b1 - 48) * q^91 + (-6*b3 - 12*b2 + 21*b1 - 15) * q^93 + (-12*b3 + 12*b2 - 18*b1) * q^95 + (-7*b2 + 17*b1 - 5) * q^97 + (6*b3 + 3*b2 + 3*b1 - 27) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{3} - 9 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ 4 * q + 6 * q^3 - 9 * q^5 - 2 * q^7 + 6 * q^9 $$4 q + 6 q^{3} - 9 q^{5} - 2 q^{7} + 6 q^{9} - 15 q^{11} - 18 q^{15} - 18 q^{17} + 81 q^{19} - 3 q^{21} - 48 q^{23} + 61 q^{25} + 6 q^{29} - 48 q^{31} - 45 q^{33} + 102 q^{35} + 73 q^{37} + 3 q^{39} - 70 q^{43} - 27 q^{45} + 90 q^{47} - 194 q^{49} - 18 q^{51} + 99 q^{53} + 162 q^{57} - 243 q^{59} - 192 q^{61} - 3 q^{63} + 102 q^{65} + 7 q^{67} + 204 q^{71} - 45 q^{73} + 183 q^{75} - 285 q^{77} - 56 q^{79} - 18 q^{81} + 444 q^{85} + 9 q^{87} + 198 q^{89} - 195 q^{91} - 48 q^{93} - 24 q^{95} - 90 q^{99}+O(q^{100})$$ 4 * q + 6 * q^3 - 9 * q^5 - 2 * q^7 + 6 * q^9 - 15 * q^11 - 18 * q^15 - 18 * q^17 + 81 * q^19 - 3 * q^21 - 48 * q^23 + 61 * q^25 + 6 * q^29 - 48 * q^31 - 45 * q^33 + 102 * q^35 + 73 * q^37 + 3 * q^39 - 70 * q^43 - 27 * q^45 + 90 * q^47 - 194 * q^49 - 18 * q^51 + 99 * q^53 + 162 * q^57 - 243 * q^59 - 192 * q^61 - 3 * q^63 + 102 * q^65 + 7 * q^67 + 204 * q^71 - 45 * q^73 + 183 * q^75 - 285 * q^77 - 56 * q^79 - 18 * q^81 + 444 * q^85 + 9 * q^87 + 198 * q^89 - 195 * q^91 - 48 * q^93 - 24 * q^95 - 90 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 17x^{2} + 16x + 256$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 17\nu^{2} - 17\nu + 256 ) / 272$$ (-v^3 + 17*v^2 - 17*v + 256) / 272 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - \nu^{2} + 33\nu + 16 ) / 16$$ (v^3 - v^2 + 33*v + 16) / 16 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - 17\nu + 33 ) / 17$$ (v^3 - 17*v + 33) / 17
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} + \beta_1 ) / 3$$ (-b3 + b2 + b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 2\beta_{2} + 50\beta _1 - 51 ) / 3$$ (b3 + 2*b2 + 50*b1 - 51) / 3 $$\nu^{3}$$ $$=$$ $$( 34\beta_{3} + 17\beta_{2} + 17\beta _1 - 99 ) / 3$$ (34*b3 + 17*b2 + 17*b1 - 99) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$1 - \beta_{1}$$

## 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
 −1.76556 − 3.05805i 2.26556 + 3.92407i −1.76556 + 3.05805i 2.26556 − 3.92407i
0 1.50000 0.866025i 0 −8.29669 4.79010i 0 −0.500000 + 6.98212i 0 1.50000 2.59808i 0
145.2 0 1.50000 0.866025i 0 3.79669 + 2.19202i 0 −0.500000 6.98212i 0 1.50000 2.59808i 0
241.1 0 1.50000 + 0.866025i 0 −8.29669 + 4.79010i 0 −0.500000 6.98212i 0 1.50000 + 2.59808i 0
241.2 0 1.50000 + 0.866025i 0 3.79669 2.19202i 0 −0.500000 + 6.98212i 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.f 4
3.b odd 2 1 1008.3.cg.m 4
4.b odd 2 1 84.3.m.b 4
7.c even 3 1 2352.3.f.f 4
7.d odd 6 1 inner 336.3.bh.f 4
7.d odd 6 1 2352.3.f.f 4
12.b even 2 1 252.3.z.e 4
20.d odd 2 1 2100.3.bd.f 4
20.e even 4 2 2100.3.be.d 8
21.g even 6 1 1008.3.cg.m 4
28.d even 2 1 588.3.m.d 4
28.f even 6 1 84.3.m.b 4
28.f even 6 1 588.3.d.b 4
28.g odd 6 1 588.3.d.b 4
28.g odd 6 1 588.3.m.d 4
84.h odd 2 1 1764.3.z.h 4
84.j odd 6 1 252.3.z.e 4
84.j odd 6 1 1764.3.d.f 4
84.n even 6 1 1764.3.d.f 4
84.n even 6 1 1764.3.z.h 4
140.s even 6 1 2100.3.bd.f 4
140.x odd 12 2 2100.3.be.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.m.b 4 4.b odd 2 1
84.3.m.b 4 28.f even 6 1
252.3.z.e 4 12.b even 2 1
252.3.z.e 4 84.j odd 6 1
336.3.bh.f 4 1.a even 1 1 trivial
336.3.bh.f 4 7.d odd 6 1 inner
588.3.d.b 4 28.f even 6 1
588.3.d.b 4 28.g odd 6 1
588.3.m.d 4 28.d even 2 1
588.3.m.d 4 28.g odd 6 1
1008.3.cg.m 4 3.b odd 2 1
1008.3.cg.m 4 21.g even 6 1
1764.3.d.f 4 84.j odd 6 1
1764.3.d.f 4 84.n even 6 1
1764.3.z.h 4 84.h odd 2 1
1764.3.z.h 4 84.n even 6 1
2100.3.bd.f 4 20.d odd 2 1
2100.3.bd.f 4 140.s even 6 1
2100.3.be.d 8 20.e even 4 2
2100.3.be.d 8 140.x odd 12 2
2352.3.f.f 4 7.c even 3 1
2352.3.f.f 4 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 9T_{5}^{3} - 15T_{5}^{2} - 378T_{5} + 1764$$ acting on $$S_{3}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 3 T + 3)^{2}$$
$5$ $$T^{4} + 9 T^{3} - 15 T^{2} + \cdots + 1764$$
$7$ $$(T^{2} + T + 49)^{2}$$
$11$ $$T^{4} + 15 T^{3} + 315 T^{2} + \cdots + 8100$$
$13$ $$T^{4} + 99T^{2} + 2304$$
$17$ $$T^{4} + 18 T^{3} - 60 T^{2} + \cdots + 28224$$
$19$ $$T^{4} - 81 T^{3} + 2685 T^{2} + \cdots + 248004$$
$23$ $$(T^{2} + 24 T + 576)^{2}$$
$29$ $$(T^{2} - 3 T - 144)^{2}$$
$31$ $$T^{4} + 48 T^{3} - 795 T^{2} + \cdots + 2442969$$
$37$ $$T^{4} - 73 T^{3} + 4143 T^{2} + \cdots + 1406596$$
$41$ $$T^{4} + 2424 T^{2} + 121104$$
$43$ $$(T^{2} + 35 T - 1010)^{2}$$
$47$ $$T^{4} - 90 T^{3} + 3180 T^{2} + \cdots + 230400$$
$53$ $$T^{4} - 99 T^{3} + 7497 T^{2} + \cdots + 5308416$$
$59$ $$T^{4} + 243 T^{3} + \cdots + 23736384$$
$61$ $$T^{4} + 192 T^{3} + 12240 T^{2} + \cdots + 2304$$
$67$ $$T^{4} - 7 T^{3} + 3693 T^{2} + \cdots + 13278736$$
$71$ $$(T^{2} - 102 T - 2664)^{2}$$
$73$ $$T^{4} + 45 T^{3} - 1545 T^{2} + \cdots + 4928400$$
$79$ $$T^{4} + 56 T^{3} + 2937 T^{2} + \cdots + 39601$$
$83$ $$T^{4} + 28839 T^{2} + \cdots + 189282564$$
$89$ $$T^{4} - 198 T^{3} + 14580 T^{2} + \cdots + 2286144$$
$97$ $$T^{4} + 5211 T^{2} + \cdots + 4717584$$