# Properties

 Label 1680.2.bg.t Level $1680$ Weight $2$ Character orbit 1680.bg Analytic conductor $13.415$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1680.bg (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4148675396$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 420) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{3} - \beta_1 q^{5} + (\beta_{3} - \beta_1) q^{7} + \beta_1 q^{9}+O(q^{10})$$ q + (b1 + 1) * q^3 - b1 * q^5 + (b3 - b1) * q^7 + b1 * q^9 $$q + (\beta_1 + 1) q^{3} - \beta_1 q^{5} + (\beta_{3} - \beta_1) q^{7} + \beta_1 q^{9} + (\beta_{3} + \beta_{2}) q^{11} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{13} + q^{15} + (\beta_{3} + \beta_{2}) q^{17} + 7 \beta_1 q^{19} + (\beta_{2} + 1) q^{21} + (\beta_{3} - 2 \beta_{2}) q^{23} + ( - \beta_1 - 1) q^{25} - q^{27} + ( - 2 \beta_{3} + \beta_{2} - 6) q^{29} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{31} + ( - \beta_{3} + 2 \beta_{2}) q^{33} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{35} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{37} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{39} + (2 \beta_{3} - \beta_{2}) q^{41} + (2 \beta_{3} - \beta_{2} + 1) q^{43} + (\beta_1 + 1) q^{45} + 6 \beta_1 q^{47} + (2 \beta_{3} - 2 \beta_{2} + 5 \beta_1 + 5) q^{49} + ( - \beta_{3} + 2 \beta_{2}) q^{51} + (2 \beta_{3} + 2 \beta_{2}) q^{53} + (2 \beta_{3} - \beta_{2}) q^{55} - 7 q^{57} + (\beta_{3} + \beta_{2} - 6 \beta_1 - 6) q^{59} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{61} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{63} + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{65} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{67} + (2 \beta_{3} - \beta_{2}) q^{69} + ( - 6 \beta_{3} + 3 \beta_{2}) q^{71} + (\beta_{3} + \beta_{2} - 5 \beta_1 - 5) q^{73} - \beta_1 q^{75} + (2 \beta_{3} - \beta_{2} + 12 \beta_1 + 6) q^{77} - 11 \beta_1 q^{79} + ( - \beta_1 - 1) q^{81} + ( - 2 \beta_{3} + \beta_{2} + 6) q^{83} + (2 \beta_{3} - \beta_{2}) q^{85} + ( - \beta_{3} - \beta_{2} - 6 \beta_1 - 6) q^{87} + ( - \beta_{3} + 2 \beta_{2} - 6 \beta_1) q^{89} + ( - 2 \beta_{3} + 2 \beta_{2} - 5 \beta_1 - 12) q^{91} + ( - 2 \beta_{3} + 4 \beta_{2} - \beta_1) q^{93} + (7 \beta_1 + 7) q^{95} + (4 \beta_{3} - 2 \beta_{2} + 8) q^{97} + ( - 2 \beta_{3} + \beta_{2}) q^{99}+O(q^{100})$$ q + (b1 + 1) * q^3 - b1 * q^5 + (b3 - b1) * q^7 + b1 * q^9 + (b3 + b2) * q^11 + (-2*b3 + b2 - 1) * q^13 + q^15 + (b3 + b2) * q^17 + 7*b1 * q^19 + (b2 + 1) * q^21 + (b3 - 2*b2) * q^23 + (-b1 - 1) * q^25 - q^27 + (-2*b3 + b2 - 6) * q^29 + (2*b3 + 2*b2 - b1 - 1) * q^31 + (-b3 + 2*b2) * q^33 + (b3 - b2 - b1 - 1) * q^35 + (b3 - 2*b2 - b1) * q^37 + (-b3 - b2 - b1 - 1) * q^39 + (2*b3 - b2) * q^41 + (2*b3 - b2 + 1) * q^43 + (b1 + 1) * q^45 + 6*b1 * q^47 + (2*b3 - 2*b2 + 5*b1 + 5) * q^49 + (-b3 + 2*b2) * q^51 + (2*b3 + 2*b2) * q^53 + (2*b3 - b2) * q^55 - 7 * q^57 + (b3 + b2 - 6*b1 - 6) * q^59 + (2*b3 - 4*b2 - 4*b1) * q^61 + (-b3 + b2 + b1 + 1) * q^63 + (-b3 + 2*b2 + b1) * q^65 + (b3 + b2 - b1 - 1) * q^67 + (2*b3 - b2) * q^69 + (-6*b3 + 3*b2) * q^71 + (b3 + b2 - 5*b1 - 5) * q^73 - b1 * q^75 + (2*b3 - b2 + 12*b1 + 6) * q^77 - 11*b1 * q^79 + (-b1 - 1) * q^81 + (-2*b3 + b2 + 6) * q^83 + (2*b3 - b2) * q^85 + (-b3 - b2 - 6*b1 - 6) * q^87 + (-b3 + 2*b2 - 6*b1) * q^89 + (-2*b3 + 2*b2 - 5*b1 - 12) * q^91 + (-2*b3 + 4*b2 - b1) * q^93 + (7*b1 + 7) * q^95 + (4*b3 - 2*b2 + 8) * q^97 + (-2*b3 + b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 2 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 - 2 * q^9 $$4 q + 2 q^{3} + 2 q^{5} + 2 q^{7} - 2 q^{9} - 4 q^{13} + 4 q^{15} - 14 q^{19} + 4 q^{21} - 2 q^{25} - 4 q^{27} - 24 q^{29} - 2 q^{31} - 2 q^{35} + 2 q^{37} - 2 q^{39} + 4 q^{43} + 2 q^{45} - 12 q^{47} + 10 q^{49} - 28 q^{57} - 12 q^{59} + 8 q^{61} + 2 q^{63} - 2 q^{65} - 2 q^{67} - 10 q^{73} + 2 q^{75} + 22 q^{79} - 2 q^{81} + 24 q^{83} - 12 q^{87} + 12 q^{89} - 38 q^{91} + 2 q^{93} + 14 q^{95} + 32 q^{97}+O(q^{100})$$ 4 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 - 2 * q^9 - 4 * q^13 + 4 * q^15 - 14 * q^19 + 4 * q^21 - 2 * q^25 - 4 * q^27 - 24 * q^29 - 2 * q^31 - 2 * q^35 + 2 * q^37 - 2 * q^39 + 4 * q^43 + 2 * q^45 - 12 * q^47 + 10 * q^49 - 28 * q^57 - 12 * q^59 + 8 * q^61 + 2 * q^63 - 2 * q^65 - 2 * q^67 - 10 * q^73 + 2 * q^75 + 22 * q^79 - 2 * q^81 + 24 * q^83 - 12 * q^87 + 12 * q^89 - 38 * q^91 + 2 * q^93 + 14 * q^95 + 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 2$$ (v^3 + 4*v) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu ) / 2$$ (-v^3 + 2*v) / 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$2\beta_1$$ 2*b1 $$\nu^{3}$$ $$=$$ $$( -4\beta_{3} + 2\beta_{2} ) / 3$$ (-4*b3 + 2*b2) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$421$$ $$1121$$ $$1471$$ $$\chi(n)$$ $$\beta_{1}$$ $$1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 −1.62132 2.09077i 0 −0.500000 + 0.866025i 0
961.2 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 2.62132 + 0.358719i 0 −0.500000 + 0.866025i 0
1201.1 0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 −1.62132 + 2.09077i 0 −0.500000 0.866025i 0
1201.2 0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 2.62132 0.358719i 0 −0.500000 0.866025i 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 1680.2.bg.t 4
4.b odd 2 1 420.2.q.d 4
7.c even 3 1 inner 1680.2.bg.t 4
12.b even 2 1 1260.2.s.e 4
20.d odd 2 1 2100.2.q.k 4
20.e even 4 2 2100.2.bc.f 8
28.d even 2 1 2940.2.q.q 4
28.f even 6 1 2940.2.a.p 2
28.f even 6 1 2940.2.q.q 4
28.g odd 6 1 420.2.q.d 4
28.g odd 6 1 2940.2.a.r 2
84.j odd 6 1 8820.2.a.bf 2
84.n even 6 1 1260.2.s.e 4
84.n even 6 1 8820.2.a.bk 2
140.p odd 6 1 2100.2.q.k 4
140.w even 12 2 2100.2.bc.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.d 4 4.b odd 2 1
420.2.q.d 4 28.g odd 6 1
1260.2.s.e 4 12.b even 2 1
1260.2.s.e 4 84.n even 6 1
1680.2.bg.t 4 1.a even 1 1 trivial
1680.2.bg.t 4 7.c even 3 1 inner
2100.2.q.k 4 20.d odd 2 1
2100.2.q.k 4 140.p odd 6 1
2100.2.bc.f 8 20.e even 4 2
2100.2.bc.f 8 140.w even 12 2
2940.2.a.p 2 28.f even 6 1
2940.2.a.r 2 28.g odd 6 1
2940.2.q.q 4 28.d even 2 1
2940.2.q.q 4 28.f even 6 1
8820.2.a.bf 2 84.j odd 6 1
8820.2.a.bk 2 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1680, [\chi])$$:

 $$T_{11}^{4} + 18T_{11}^{2} + 324$$ T11^4 + 18*T11^2 + 324 $$T_{13}^{2} + 2T_{13} - 17$$ T13^2 + 2*T13 - 17 $$T_{17}^{4} + 18T_{17}^{2} + 324$$ T17^4 + 18*T17^2 + 324

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$T^{4} - 2 T^{3} - 3 T^{2} - 14 T + 49$$
$11$ $$T^{4} + 18T^{2} + 324$$
$13$ $$(T^{2} + 2 T - 17)^{2}$$
$17$ $$T^{4} + 18T^{2} + 324$$
$19$ $$(T^{2} + 7 T + 49)^{2}$$
$23$ $$T^{4} + 18T^{2} + 324$$
$29$ $$(T^{2} + 12 T + 18)^{2}$$
$31$ $$T^{4} + 2 T^{3} + 75 T^{2} + \cdots + 5041$$
$37$ $$T^{4} - 2 T^{3} + 21 T^{2} + 34 T + 289$$
$41$ $$(T^{2} - 18)^{2}$$
$43$ $$(T^{2} - 2 T - 17)^{2}$$
$47$ $$(T^{2} + 6 T + 36)^{2}$$
$53$ $$T^{4} + 72T^{2} + 5184$$
$59$ $$T^{4} + 12 T^{3} + 126 T^{2} + \cdots + 324$$
$61$ $$T^{4} - 8 T^{3} + 120 T^{2} + \cdots + 3136$$
$67$ $$T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289$$
$71$ $$(T^{2} - 162)^{2}$$
$73$ $$T^{4} + 10 T^{3} + 93 T^{2} + 70 T + 49$$
$79$ $$(T^{2} - 11 T + 121)^{2}$$
$83$ $$(T^{2} - 12 T + 18)^{2}$$
$89$ $$T^{4} - 12 T^{3} + 126 T^{2} + \cdots + 324$$
$97$ $$(T^{2} - 16 T - 8)^{2}$$