# Properties

 Label 840.2.bg.j Level $840$ Weight $2$ Character orbit 840.bg Analytic conductor $6.707$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$840 = 2^{3} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 840.bg (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.70743376979$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.29428272.1 Defining polynomial: $$x^{6} - 6x^{4} - 4x^{3} - 42x^{2} + 343$$ x^6 - 6*x^4 - 4*x^3 - 42*x^2 + 343 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + (\beta_{3} + 1) q^{5} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{3} - 1) q^{9}+O(q^{10})$$ q + b3 * q^3 + (b3 + 1) * q^5 + (b5 + b1) * q^7 + (-b3 - 1) * q^9 $$q + \beta_{3} q^{3} + (\beta_{3} + 1) q^{5} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{3} - 1) q^{9} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{11} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{13} - q^{15} + ( - \beta_{5} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{17} + ( - \beta_{3} - 1) q^{19} + ( - \beta_1 + 1) q^{21} + (\beta_{5} + \beta_{4} + \beta_{3} + 2) q^{23} + \beta_{3} q^{25} + q^{27} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{29} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{31} + (\beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{33} + (\beta_{5} + 1) q^{35} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 3) q^{37} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1) q^{39} + (\beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{41} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{43} - \beta_{3} q^{45} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{47} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{49} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1) q^{51} + ( - \beta_{5} - 5 \beta_{3} + \beta_{2} - 1) q^{53} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{55} + q^{57} + (\beta_{5} - \beta_{4} + 11 \beta_{3} + \beta_1) q^{59} + ( - \beta_{5} - 1) q^{63} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{65} + (2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{67} + ( - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{69} + (\beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_1 - 4) q^{71} + (\beta_{4} + 5 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{73} + ( - \beta_{3} - 1) q^{75} + ( - 5 \beta_{4} + 8 \beta_{3} + 4 \beta_{2} + 5) q^{77} + (\beta_{5} + \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 1) q^{79} + \beta_{3} q^{81} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{83} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{85} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{87} + ( - 2 \beta_{5} - 2 \beta_{4} + 3 \beta_{2} - 3 \beta_1 - 2) q^{89} + ( - \beta_{5} + 2 \beta_{4} - 6 \beta_{3} - 3 \beta_{2} + 4) q^{91} + ( - \beta_{5} - \beta_{4} - 4 \beta_{3} - \beta_{2} + \beta_1 - 5) q^{93} - \beta_{3} q^{95} + 8 q^{97} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100})$$ q + b3 * q^3 + (b3 + 1) * q^5 + (b5 + b1) * q^7 + (-b3 - 1) * q^9 + (-2*b5 + b4 - 2*b3 + b2 - b1 - 1) * q^11 + (b5 - b4 + b3 + b2 - 1) * q^13 - q^15 + (-b5 - b4 + 3*b3 + 2*b2 + b1 - 2) * q^17 + (-b3 - 1) * q^19 + (-b1 + 1) * q^21 + (b5 + b4 + b3 + 2) * q^23 + b3 * q^25 + q^27 + (b5 - b4 + b3 + b2 + 2) * q^29 + (-b5 + 2*b4 + 2*b3 - b2 - 2*b1 + 1) * q^31 + (b5 + b4 + b3 - 2*b2 + 2*b1 + 2) * q^33 + (b5 + 1) * q^35 + (-2*b5 - 2*b4 - b3 + 3*b2 - 3*b1 - 3) * q^37 + (-b5 + b4 - 2*b3 - b1) * q^39 + (b5 + b3 + b2 + b1 - 3) * q^41 + (-b5 + 2*b4 - b3 - b2 + b1) * q^43 - b3 * q^45 + (b5 + b4 - b3 + b2 - b1) * q^47 + (-b5 + 3*b4 - 2*b3 - b2 + b1 + 2) * q^49 + (2*b5 + 2*b4 - 2*b3 - b2 + b1) * q^51 + (-b5 - 5*b3 + b2 - 1) * q^53 + (-b5 + 2*b4 - b3 - b2 + b1 + 1) * q^55 + q^57 + (b5 - b4 + 11*b3 + b1) * q^59 + (-b5 - 1) * q^63 + (-b3 + b2 - b1 - 1) * q^65 + (2*b5 - b4 + b3 - b2 + b1 + 1) * q^67 + (-b5 - b3 - b2 - b1 - 1) * q^69 + (b5 + 3*b4 + b3 + b2 + 4*b1 - 4) * q^71 + (b4 + 5*b3 - b2 - b1 + 1) * q^73 + (-b3 - 1) * q^75 + (-5*b4 + 8*b3 + 4*b2 + 5) * q^77 + (b5 + b4 - 2*b3 - 3*b2 + 3*b1 - 1) * q^79 + b3 * q^81 + (b5 - b4 + b3 + b2 + 2) * q^83 + (b5 + b4 + b3 + b2 + 2*b1 - 2) * q^85 + (-b5 + b4 + b3 - b1) * q^87 + (-2*b5 - 2*b4 + 3*b2 - 3*b1 - 2) * q^89 + (-b5 + 2*b4 - 6*b3 - 3*b2 + 4) * q^91 + (-b5 - b4 - 4*b3 - b2 + b1 - 5) * q^93 - b3 * q^95 + 8 * q^97 + (b5 - 2*b4 + b3 + b2 - b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{3} + 3 q^{5} - 3 q^{9}+O(q^{10})$$ 6 * q - 3 * q^3 + 3 * q^5 - 3 * q^9 $$6 q - 3 q^{3} + 3 q^{5} - 3 q^{9} + 3 q^{11} - 6 q^{13} - 6 q^{15} - 6 q^{17} - 3 q^{19} + 3 q^{21} + 3 q^{23} - 3 q^{25} + 6 q^{27} + 12 q^{29} - 12 q^{31} + 3 q^{33} + 3 q^{35} - 3 q^{37} + 3 q^{39} - 18 q^{41} + 3 q^{45} - 3 q^{47} + 12 q^{49} - 6 q^{51} + 15 q^{53} + 6 q^{55} + 6 q^{57} - 30 q^{59} - 3 q^{63} - 3 q^{65} - 6 q^{69} - 24 q^{71} - 18 q^{73} - 3 q^{75} + 33 q^{77} - 6 q^{79} - 3 q^{81} + 12 q^{83} - 12 q^{85} - 6 q^{87} + 30 q^{91} - 12 q^{93} + 3 q^{95} + 48 q^{97} - 6 q^{99}+O(q^{100})$$ 6 * q - 3 * q^3 + 3 * q^5 - 3 * q^9 + 3 * q^11 - 6 * q^13 - 6 * q^15 - 6 * q^17 - 3 * q^19 + 3 * q^21 + 3 * q^23 - 3 * q^25 + 6 * q^27 + 12 * q^29 - 12 * q^31 + 3 * q^33 + 3 * q^35 - 3 * q^37 + 3 * q^39 - 18 * q^41 + 3 * q^45 - 3 * q^47 + 12 * q^49 - 6 * q^51 + 15 * q^53 + 6 * q^55 + 6 * q^57 - 30 * q^59 - 3 * q^63 - 3 * q^65 - 6 * q^69 - 24 * q^71 - 18 * q^73 - 3 * q^75 + 33 * q^77 - 6 * q^79 - 3 * q^81 + 12 * q^83 - 12 * q^85 - 6 * q^87 + 30 * q^91 - 12 * q^93 + 3 * q^95 + 48 * q^97 - 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 6x^{4} - 4x^{3} - 42x^{2} + 343$$ :

 $$\beta_{1}$$ $$=$$ $$( 3\nu^{5} + 35\nu^{4} + 31\nu^{3} + 121\nu^{2} - 217\nu - 1519 ) / 490$$ (3*v^5 + 35*v^4 + 31*v^3 + 121*v^2 - 217*v - 1519) / 490 $$\beta_{2}$$ $$=$$ $$( 5\nu^{5} + 7\nu^{4} + 19\nu^{3} - 13\nu^{2} + 203\nu - 147 ) / 490$$ (5*v^5 + 7*v^4 + 19*v^3 - 13*v^2 + 203*v - 147) / 490 $$\beta_{3}$$ $$=$$ $$( -5\nu^{5} - 7\nu^{4} - 19\nu^{3} + 13\nu^{2} + 287\nu + 147 ) / 490$$ (-5*v^5 - 7*v^4 - 19*v^3 + 13*v^2 + 287*v + 147) / 490 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} - 7\nu^{4} - \nu^{3} + 11\nu^{2} + 91\nu + 245 ) / 70$$ (-v^5 - 7*v^4 - v^3 + 11*v^2 + 91*v + 245) / 70 $$\beta_{5}$$ $$=$$ $$( -13\nu^{5} - 35\nu^{4} + 29\nu^{3} - 81\nu^{2} + 637\nu + 1519 ) / 490$$ (-13*v^5 - 35*v^4 + 29*v^3 - 81*v^2 + 637*v + 1519) / 490
 $$\nu$$ $$=$$ $$\beta_{3} + \beta_{2}$$ b3 + b2 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + 2\beta_{4} - \beta_{2} + 2\beta _1 + 2$$ -b5 + 2*b4 - b2 + 2*b1 + 2 $$\nu^{3}$$ $$=$$ $$5\beta_{5} + \beta_{4} - 11\beta_{3} + \beta_{2} + 4\beta _1 - 3$$ 5*b5 + b4 - 11*b3 + b2 + 4*b1 - 3 $$\nu^{4}$$ $$=$$ $$\beta_{5} - 9\beta_{4} + 18\beta_{3} + 5\beta_{2} + 5\beta _1 + 40$$ b5 - 9*b4 + 18*b3 + 5*b2 + 5*b1 + 40 $$\nu^{5}$$ $$=$$ $$-23\beta_{5} + 14\beta_{4} - 24\beta_{3} + 44\beta_{2} - 17\beta _1 - 10$$ -23*b5 + 14*b4 - 24*b3 + 44*b2 - 17*b1 - 10

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$281$$ $$337$$ $$421$$ $$631$$ $$\chi(n)$$ $$-1 - \beta_{3}$$ $$1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 −2.56022 + 0.667305i −0.0741344 − 2.64471i 2.63435 + 0.245357i −2.56022 − 0.667305i −0.0741344 + 2.64471i 2.63435 − 0.245357i
0 −0.500000 0.866025i 0 0.500000 0.866025i 0 −2.56022 0.667305i 0 −0.500000 + 0.866025i 0
121.2 0 −0.500000 0.866025i 0 0.500000 0.866025i 0 −0.0741344 + 2.64471i 0 −0.500000 + 0.866025i 0
121.3 0 −0.500000 0.866025i 0 0.500000 0.866025i 0 2.63435 0.245357i 0 −0.500000 + 0.866025i 0
361.1 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 −2.56022 + 0.667305i 0 −0.500000 0.866025i 0
361.2 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 −0.0741344 2.64471i 0 −0.500000 0.866025i 0
361.3 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 2.63435 + 0.245357i 0 −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.3 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 840.2.bg.j 6
3.b odd 2 1 2520.2.bi.n 6
4.b odd 2 1 1680.2.bg.v 6
7.c even 3 1 inner 840.2.bg.j 6
7.c even 3 1 5880.2.a.bv 3
7.d odd 6 1 5880.2.a.bu 3
21.h odd 6 1 2520.2.bi.n 6
28.g odd 6 1 1680.2.bg.v 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bg.j 6 1.a even 1 1 trivial
840.2.bg.j 6 7.c even 3 1 inner
1680.2.bg.v 6 4.b odd 2 1
1680.2.bg.v 6 28.g odd 6 1
2520.2.bi.n 6 3.b odd 2 1
2520.2.bi.n 6 21.h odd 6 1
5880.2.a.bu 3 7.d odd 6 1
5880.2.a.bv 3 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{6} - 3T_{11}^{5} + 45T_{11}^{4} - 144T_{11}^{3} + 1674T_{11}^{2} - 4536T_{11} + 15876$$ acting on $$S_{2}^{\mathrm{new}}(840, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{2} + T + 1)^{3}$$
$5$ $$(T^{2} - T + 1)^{3}$$
$7$ $$T^{6} - 6 T^{4} - 4 T^{3} - 42 T^{2} + \cdots + 343$$
$11$ $$T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 15876$$
$13$ $$(T^{3} + 3 T^{2} - 15 T + 3)^{2}$$
$17$ $$T^{6} + 6 T^{5} + 90 T^{4} + \cdots + 102400$$
$19$ $$(T^{2} + T + 1)^{3}$$
$23$ $$T^{6} - 3 T^{5} + 33 T^{4} + 28 T^{3} + \cdots + 484$$
$29$ $$(T^{3} - 6 T^{2} - 6 T + 48)^{2}$$
$31$ $$T^{6} + 12 T^{5} + 171 T^{4} + \cdots + 202500$$
$37$ $$T^{6} + 3 T^{5} + 96 T^{4} + \cdots + 145161$$
$41$ $$(T^{3} + 9 T^{2} - 58)^{2}$$
$43$ $$(T^{3} - 39 T + 88)^{2}$$
$47$ $$T^{6} + 3 T^{5} + 81 T^{4} + \cdots + 19600$$
$53$ $$T^{6} - 15 T^{5} + 165 T^{4} + \cdots + 3136$$
$59$ $$T^{6} + 30 T^{5} + 618 T^{4} + \cdots + 705600$$
$61$ $$T^{6}$$
$67$ $$T^{6} + 39 T^{4} + 176 T^{3} + \cdots + 7744$$
$71$ $$(T^{3} + 12 T^{2} - 186 T - 2100)^{2}$$
$73$ $$T^{6} + 18 T^{5} + 243 T^{4} + \cdots + 3364$$
$79$ $$T^{6} + 6 T^{5} + 123 T^{4} + \cdots + 1024$$
$83$ $$(T^{3} - 6 T^{2} - 6 T + 48)^{2}$$
$89$ $$T^{6} + 90 T^{4} + 584 T^{3} + \cdots + 85264$$
$97$ $$(T - 8)^{6}$$