# Properties

 Label 1680.2.bg.q 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: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{3} + \beta_{2} q^{5} + (\beta_{3} + \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10})$$ q + (b2 + 1) * q^3 + b2 * q^5 + (b3 + b1) * q^7 + b2 * q^9 $$q + (\beta_{2} + 1) q^{3} + \beta_{2} q^{5} + (\beta_{3} + \beta_1) q^{7} + \beta_{2} q^{9} + (\beta_{2} + \beta_1 + 1) q^{11} - \beta_{3} q^{13} - q^{15} + ( - \beta_{2} - \beta_1 - 1) q^{17} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{19} + \beta_{3} q^{21} + (\beta_{3} + \beta_{2} + \beta_1) q^{23} + ( - \beta_{2} - 1) q^{25} - q^{27} + (\beta_{3} + 5) q^{29} + (\beta_{2} + 2 \beta_1 + 1) q^{31} + (\beta_{3} + \beta_{2} + \beta_1) q^{33} - \beta_1 q^{35} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{37} + \beta_1 q^{39} + (3 \beta_{3} - 3) q^{41} + (3 \beta_{3} - 2) q^{43} + ( - \beta_{2} - 1) q^{45} - 6 \beta_{2} q^{47} + ( - 7 \beta_{2} - 7) q^{49} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{51} + (2 \beta_{2} + 2 \beta_1 + 2) q^{53} + (\beta_{3} - 1) q^{55} + (2 \beta_{3} + 3) q^{57} + (3 \beta_{2} - 3 \beta_1 + 3) q^{59} + 8 \beta_{2} q^{61} - \beta_1 q^{63} + (\beta_{3} + \beta_1) q^{65} + ( - 2 \beta_{2} - \beta_1 - 2) q^{67} + (\beta_{3} - 1) q^{69} + ( - \beta_{3} - 11) q^{71} + 5 \beta_1 q^{73} - \beta_{2} q^{75} + (\beta_{3} - 7) q^{77} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{79} + ( - \beta_{2} - 1) q^{81} + (3 \beta_{3} + 3) q^{83} + ( - \beta_{3} + 1) q^{85} + (5 \beta_{2} - \beta_1 + 5) q^{87} + (5 \beta_{3} - \beta_{2} + 5 \beta_1) q^{89} + 7 \beta_{2} q^{91} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{93} + (3 \beta_{2} - 2 \beta_1 + 3) q^{95} + 8 q^{97} + (\beta_{3} - 1) q^{99}+O(q^{100})$$ q + (b2 + 1) * q^3 + b2 * q^5 + (b3 + b1) * q^7 + b2 * q^9 + (b2 + b1 + 1) * q^11 - b3 * q^13 - q^15 + (-b2 - b1 - 1) * q^17 + (2*b3 - 3*b2 + 2*b1) * q^19 + b3 * q^21 + (b3 + b2 + b1) * q^23 + (-b2 - 1) * q^25 - q^27 + (b3 + 5) * q^29 + (b2 + 2*b1 + 1) * q^31 + (b3 + b2 + b1) * q^33 - b1 * q^35 + (-b3 - 2*b2 - b1) * q^37 + b1 * q^39 + (3*b3 - 3) * q^41 + (3*b3 - 2) * q^43 + (-b2 - 1) * q^45 - 6*b2 * q^47 + (-7*b2 - 7) * q^49 + (-b3 - b2 - b1) * q^51 + (2*b2 + 2*b1 + 2) * q^53 + (b3 - 1) * q^55 + (2*b3 + 3) * q^57 + (3*b2 - 3*b1 + 3) * q^59 + 8*b2 * q^61 - b1 * q^63 + (b3 + b1) * q^65 + (-2*b2 - b1 - 2) * q^67 + (b3 - 1) * q^69 + (-b3 - 11) * q^71 + 5*b1 * q^73 - b2 * q^75 + (b3 - 7) * q^77 + (2*b3 + 3*b2 + 2*b1) * q^79 + (-b2 - 1) * q^81 + (3*b3 + 3) * q^83 + (-b3 + 1) * q^85 + (5*b2 - b1 + 5) * q^87 + (5*b3 - b2 + 5*b1) * q^89 + 7*b2 * q^91 + (2*b3 + b2 + 2*b1) * q^93 + (3*b2 - 2*b1 + 3) * q^95 + 8 * q^97 + (b3 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} - 2 q^{5} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 - 2 * q^5 - 2 * q^9 $$4 q + 2 q^{3} - 2 q^{5} - 2 q^{9} + 2 q^{11} - 4 q^{15} - 2 q^{17} + 6 q^{19} - 2 q^{23} - 2 q^{25} - 4 q^{27} + 20 q^{29} + 2 q^{31} - 2 q^{33} + 4 q^{37} - 12 q^{41} - 8 q^{43} - 2 q^{45} + 12 q^{47} - 14 q^{49} + 2 q^{51} + 4 q^{53} - 4 q^{55} + 12 q^{57} + 6 q^{59} - 16 q^{61} - 4 q^{67} - 4 q^{69} - 44 q^{71} + 2 q^{75} - 28 q^{77} - 6 q^{79} - 2 q^{81} + 12 q^{83} + 4 q^{85} + 10 q^{87} + 2 q^{89} - 14 q^{91} - 2 q^{93} + 6 q^{95} + 32 q^{97} - 4 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 - 2 * q^5 - 2 * q^9 + 2 * q^11 - 4 * q^15 - 2 * q^17 + 6 * q^19 - 2 * q^23 - 2 * q^25 - 4 * q^27 + 20 * q^29 + 2 * q^31 - 2 * q^33 + 4 * q^37 - 12 * q^41 - 8 * q^43 - 2 * q^45 + 12 * q^47 - 14 * q^49 + 2 * q^51 + 4 * q^53 - 4 * q^55 + 12 * q^57 + 6 * q^59 - 16 * q^61 - 4 * q^67 - 4 * q^69 - 44 * q^71 + 2 * q^75 - 28 * q^77 - 6 * q^79 - 2 * q^81 + 12 * q^83 + 4 * q^85 + 10 * q^87 + 2 * q^89 - 14 * q^91 - 2 * q^93 + 6 * q^95 + 32 * q^97 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## Character values

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

 $$n$$ $$241$$ $$337$$ $$421$$ $$1121$$ $$1471$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$1$$ $$1$$ $$1$$

## 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
 1.32288 + 2.29129i −1.32288 − 2.29129i 1.32288 − 2.29129i −1.32288 + 2.29129i
0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 −1.32288 + 2.29129i 0 −0.500000 + 0.866025i 0
961.2 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 1.32288 2.29129i 0 −0.500000 + 0.866025i 0
1201.1 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 −1.32288 2.29129i 0 −0.500000 0.866025i 0
1201.2 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 1.32288 + 2.29129i 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.q 4
4.b odd 2 1 420.2.q.c 4
7.c even 3 1 inner 1680.2.bg.q 4
12.b even 2 1 1260.2.s.f 4
20.d odd 2 1 2100.2.q.h 4
20.e even 4 2 2100.2.bc.e 8
28.d even 2 1 2940.2.q.t 4
28.f even 6 1 2940.2.a.m 2
28.f even 6 1 2940.2.q.t 4
28.g odd 6 1 420.2.q.c 4
28.g odd 6 1 2940.2.a.s 2
84.j odd 6 1 8820.2.a.bj 2
84.n even 6 1 1260.2.s.f 4
84.n even 6 1 8820.2.a.be 2
140.p odd 6 1 2100.2.q.h 4
140.w even 12 2 2100.2.bc.e 8

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

## Hecke kernels

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

 $$T_{11}^{4} - 2T_{11}^{3} + 10T_{11}^{2} + 12T_{11} + 36$$ T11^4 - 2*T11^3 + 10*T11^2 + 12*T11 + 36 $$T_{13}^{2} - 7$$ T13^2 - 7 $$T_{17}^{4} + 2T_{17}^{3} + 10T_{17}^{2} - 12T_{17} + 36$$ T17^4 + 2*T17^3 + 10*T17^2 - 12*T17 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} + 7T^{2} + 49$$
$11$ $$T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36$$
$13$ $$(T^{2} - 7)^{2}$$
$17$ $$T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36$$
$19$ $$T^{4} - 6 T^{3} + 55 T^{2} + 114 T + 361$$
$23$ $$T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36$$
$29$ $$(T^{2} - 10 T + 18)^{2}$$
$31$ $$T^{4} - 2 T^{3} + 31 T^{2} + 54 T + 729$$
$37$ $$T^{4} - 4 T^{3} + 19 T^{2} + 12 T + 9$$
$41$ $$(T^{2} + 6 T - 54)^{2}$$
$43$ $$(T^{2} + 4 T - 59)^{2}$$
$47$ $$(T^{2} - 6 T + 36)^{2}$$
$53$ $$T^{4} - 4 T^{3} + 40 T^{2} + 96 T + 576$$
$59$ $$T^{4} - 6 T^{3} + 90 T^{2} + \cdots + 2916$$
$61$ $$(T^{2} + 8 T + 64)^{2}$$
$67$ $$T^{4} + 4 T^{3} + 19 T^{2} - 12 T + 9$$
$71$ $$(T^{2} + 22 T + 114)^{2}$$
$73$ $$T^{4} + 175 T^{2} + 30625$$
$79$ $$T^{4} + 6 T^{3} + 55 T^{2} - 114 T + 361$$
$83$ $$(T^{2} - 6 T - 54)^{2}$$
$89$ $$T^{4} - 2 T^{3} + 178 T^{2} + \cdots + 30276$$
$97$ $$(T - 8)^{4}$$