# Properties

 Label 280.2.q.d Level $280$ Weight $2$ Character orbit 280.q Analytic conductor $2.236$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 280.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 + 1) q^{3} + \beta_{2} q^{5} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + (2 \beta_{3} + 2 \beta_1) q^{9}+O(q^{10})$$ q + (b2 + b1 + 1) * q^3 + b2 * q^5 + (2*b3 - b2 + b1 - 1) * q^7 + (2*b3 + 2*b1) * q^9 $$q + (\beta_{2} + \beta_1 + 1) q^{3} + \beta_{2} q^{5} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + (2 \beta_{3} + 2 \beta_1) q^{9} + (2 \beta_{2} + 2 \beta_1 + 2) q^{11} + 2 q^{13} + (\beta_{3} - 1) q^{15} + (2 \beta_{2} - 4 \beta_1 + 2) q^{17} + (4 \beta_{3} + 4 \beta_1) q^{19} + ( - 3 \beta_{2} - 2 \beta_1 - 4) q^{21} + ( - \beta_{3} - 7 \beta_{2} - \beta_1) q^{23} + ( - \beta_{2} - 1) q^{25} + ( - \beta_{3} - 1) q^{27} + ( - 2 \beta_{3} - 5) q^{29} + (2 \beta_{2} + 2 \beta_1 + 2) q^{31} + (4 \beta_{3} + 6 \beta_{2} + 4 \beta_1) q^{33} + ( - \beta_{3} - 2 \beta_1 + 1) q^{35} + ( - 4 \beta_{3} - 4 \beta_1) q^{37} + (2 \beta_{2} + 2 \beta_1 + 2) q^{39} + (2 \beta_{3} + 3) q^{41} + ( - 7 \beta_{3} + 3) q^{43} - 2 \beta_1 q^{45} + ( - 4 \beta_{3} + 6 \beta_{2} - 4 \beta_1) q^{47} + ( - 2 \beta_{3} - 5 \beta_{2} + 2 \beta_1) q^{49} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1) q^{51} - 4 \beta_1 q^{53} + (2 \beta_{3} - 2) q^{55} + (4 \beta_{3} - 8) q^{57} + (4 \beta_{2} + 4) q^{59} + ( - 4 \beta_{3} + \beta_{2} - 4 \beta_1) q^{61} + ( - 2 \beta_{3} - 8 \beta_{2} - 4) q^{63} + 2 \beta_{2} q^{65} + (3 \beta_{2} - 7 \beta_1 + 3) q^{67} + ( - 8 \beta_{3} + 9) q^{69} - 12 q^{71} + (2 \beta_{2} + 4 \beta_1 + 2) q^{73} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{75} + ( - 6 \beta_{2} - 4 \beta_1 - 8) q^{77} - 4 \beta_{2} q^{79} + (\beta_{2} + 6 \beta_1 + 1) q^{81} + (3 \beta_{3} - 9) q^{83} + ( - 4 \beta_{3} - 2) q^{85} + ( - \beta_{2} - 3 \beta_1 - 1) q^{87} + ( - 4 \beta_{3} - 11 \beta_{2} - 4 \beta_1) q^{89} + (4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{91} + (4 \beta_{3} + 6 \beta_{2} + 4 \beta_1) q^{93} - 4 \beta_1 q^{95} + 6 q^{97} + (4 \beta_{3} - 8) q^{99}+O(q^{100})$$ q + (b2 + b1 + 1) * q^3 + b2 * q^5 + (2*b3 - b2 + b1 - 1) * q^7 + (2*b3 + 2*b1) * q^9 + (2*b2 + 2*b1 + 2) * q^11 + 2 * q^13 + (b3 - 1) * q^15 + (2*b2 - 4*b1 + 2) * q^17 + (4*b3 + 4*b1) * q^19 + (-3*b2 - 2*b1 - 4) * q^21 + (-b3 - 7*b2 - b1) * q^23 + (-b2 - 1) * q^25 + (-b3 - 1) * q^27 + (-2*b3 - 5) * q^29 + (2*b2 + 2*b1 + 2) * q^31 + (4*b3 + 6*b2 + 4*b1) * q^33 + (-b3 - 2*b1 + 1) * q^35 + (-4*b3 - 4*b1) * q^37 + (2*b2 + 2*b1 + 2) * q^39 + (2*b3 + 3) * q^41 + (-7*b3 + 3) * q^43 - 2*b1 * q^45 + (-4*b3 + 6*b2 - 4*b1) * q^47 + (-2*b3 - 5*b2 + 2*b1) * q^49 + (-2*b3 - 6*b2 - 2*b1) * q^51 - 4*b1 * q^53 + (2*b3 - 2) * q^55 + (4*b3 - 8) * q^57 + (4*b2 + 4) * q^59 + (-4*b3 + b2 - 4*b1) * q^61 + (-2*b3 - 8*b2 - 4) * q^63 + 2*b2 * q^65 + (3*b2 - 7*b1 + 3) * q^67 + (-8*b3 + 9) * q^69 - 12 * q^71 + (2*b2 + 4*b1 + 2) * q^73 + (-b3 - b2 - b1) * q^75 + (-6*b2 - 4*b1 - 8) * q^77 - 4*b2 * q^79 + (b2 + 6*b1 + 1) * q^81 + (3*b3 - 9) * q^83 + (-4*b3 - 2) * q^85 + (-b2 - 3*b1 - 1) * q^87 + (-4*b3 - 11*b2 - 4*b1) * q^89 + (4*b3 - 2*b2 + 2*b1 - 2) * q^91 + (4*b3 + 6*b2 + 4*b1) * q^93 - 4*b1 * q^95 + 6 * q^97 + (4*b3 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} - 2 q^{5} - 2 q^{7}+O(q^{10})$$ 4 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 $$4 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 4 q^{11} + 8 q^{13} - 4 q^{15} + 4 q^{17} - 10 q^{21} + 14 q^{23} - 2 q^{25} - 4 q^{27} - 20 q^{29} + 4 q^{31} - 12 q^{33} + 4 q^{35} + 4 q^{39} + 12 q^{41} + 12 q^{43} - 12 q^{47} + 10 q^{49} + 12 q^{51} - 8 q^{55} - 32 q^{57} + 8 q^{59} - 2 q^{61} - 4 q^{65} + 6 q^{67} + 36 q^{69} - 48 q^{71} + 4 q^{73} + 2 q^{75} - 20 q^{77} + 8 q^{79} + 2 q^{81} - 36 q^{83} - 8 q^{85} - 2 q^{87} + 22 q^{89} - 4 q^{91} - 12 q^{93} + 24 q^{97} - 32 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 4 * q^11 + 8 * q^13 - 4 * q^15 + 4 * q^17 - 10 * q^21 + 14 * q^23 - 2 * q^25 - 4 * q^27 - 20 * q^29 + 4 * q^31 - 12 * q^33 + 4 * q^35 + 4 * q^39 + 12 * q^41 + 12 * q^43 - 12 * q^47 + 10 * q^49 + 12 * q^51 - 8 * q^55 - 32 * q^57 + 8 * q^59 - 2 * q^61 - 4 * q^65 + 6 * q^67 + 36 * q^69 - 48 * q^71 + 4 * q^73 + 2 * q^75 - 20 * q^77 + 8 * q^79 + 2 * q^81 - 36 * q^83 - 8 * q^85 - 2 * q^87 + 22 * q^89 - 4 * q^91 - 12 * q^93 + 24 * q^97 - 32 * q^99

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/280\mathbb{Z}\right)^\times$$.

 $$n$$ $$57$$ $$71$$ $$141$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
0 −0.207107 + 0.358719i 0 −0.500000 0.866025i 0 1.62132 + 2.09077i 0 1.41421 + 2.44949i 0
81.2 0 1.20711 2.09077i 0 −0.500000 0.866025i 0 −2.62132 0.358719i 0 −1.41421 2.44949i 0
121.1 0 −0.207107 0.358719i 0 −0.500000 + 0.866025i 0 1.62132 2.09077i 0 1.41421 2.44949i 0
121.2 0 1.20711 + 2.09077i 0 −0.500000 + 0.866025i 0 −2.62132 + 0.358719i 0 −1.41421 + 2.44949i 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 280.2.q.d 4
3.b odd 2 1 2520.2.bi.k 4
4.b odd 2 1 560.2.q.j 4
5.b even 2 1 1400.2.q.h 4
5.c odd 4 2 1400.2.bh.g 8
7.b odd 2 1 1960.2.q.q 4
7.c even 3 1 inner 280.2.q.d 4
7.c even 3 1 1960.2.a.p 2
7.d odd 6 1 1960.2.a.t 2
7.d odd 6 1 1960.2.q.q 4
21.h odd 6 1 2520.2.bi.k 4
28.f even 6 1 3920.2.a.bp 2
28.g odd 6 1 560.2.q.j 4
28.g odd 6 1 3920.2.a.bz 2
35.i odd 6 1 9800.2.a.br 2
35.j even 6 1 1400.2.q.h 4
35.j even 6 1 9800.2.a.bz 2
35.l odd 12 2 1400.2.bh.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.d 4 1.a even 1 1 trivial
280.2.q.d 4 7.c even 3 1 inner
560.2.q.j 4 4.b odd 2 1
560.2.q.j 4 28.g odd 6 1
1400.2.q.h 4 5.b even 2 1
1400.2.q.h 4 35.j even 6 1
1400.2.bh.g 8 5.c odd 4 2
1400.2.bh.g 8 35.l odd 12 2
1960.2.a.p 2 7.c even 3 1
1960.2.a.t 2 7.d odd 6 1
1960.2.q.q 4 7.b odd 2 1
1960.2.q.q 4 7.d odd 6 1
2520.2.bi.k 4 3.b odd 2 1
2520.2.bi.k 4 21.h odd 6 1
3920.2.a.bp 2 28.f even 6 1
3920.2.a.bz 2 28.g odd 6 1
9800.2.a.br 2 35.i odd 6 1
9800.2.a.bz 2 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 2T_{3}^{3} + 5T_{3}^{2} + 2T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} + 2 T^{3} - 3 T^{2} + 14 T + 49$$
$11$ $$T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16$$
$13$ $$(T - 2)^{4}$$
$17$ $$T^{4} - 4 T^{3} + 44 T^{2} + 112 T + 784$$
$19$ $$T^{4} + 32T^{2} + 1024$$
$23$ $$T^{4} - 14 T^{3} + 149 T^{2} + \cdots + 2209$$
$29$ $$(T^{2} + 10 T + 17)^{2}$$
$31$ $$T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16$$
$37$ $$T^{4} + 32T^{2} + 1024$$
$41$ $$(T^{2} - 6 T + 1)^{2}$$
$43$ $$(T^{2} - 6 T - 89)^{2}$$
$47$ $$T^{4} + 12 T^{3} + 140 T^{2} + \cdots + 16$$
$53$ $$T^{4} + 32T^{2} + 1024$$
$59$ $$(T^{2} - 4 T + 16)^{2}$$
$61$ $$T^{4} + 2 T^{3} + 35 T^{2} - 62 T + 961$$
$67$ $$T^{4} - 6 T^{3} + 125 T^{2} + \cdots + 7921$$
$71$ $$(T + 12)^{4}$$
$73$ $$T^{4} - 4 T^{3} + 44 T^{2} + 112 T + 784$$
$79$ $$(T^{2} - 4 T + 16)^{2}$$
$83$ $$(T^{2} + 18 T + 63)^{2}$$
$89$ $$T^{4} - 22 T^{3} + 395 T^{2} + \cdots + 7921$$
$97$ $$(T - 6)^{4}$$