# Properties

 Label 280.2.g.b Level $280$ Weight $2$ Character orbit 280.g Analytic conductor $2.236$ Analytic rank $0$ Dimension $6$ 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.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.5161984.1 Defining polynomial: $$x^{6} - 4x^{3} + 25x^{2} - 20x + 8$$ x^6 - 4*x^3 + 25*x^2 - 20*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{3} + ( - \beta_{5} + \beta_1) q^{5} + \beta_{4} q^{7} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{9}+O(q^{10})$$ q - b5 * q^3 + (-b5 + b1) * q^5 + b4 * q^7 + (b3 - b2 - b1 - 1) * q^9 $$q - \beta_{5} q^{3} + ( - \beta_{5} + \beta_1) q^{5} + \beta_{4} q^{7} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{9} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{11} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 - 3) q^{15} + (\beta_{5} - \beta_{2} + \beta_1) q^{17} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{19} + \beta_{3} q^{21} + (2 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{23} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_1 - 2) q^{25} + (\beta_{5} - 6 \beta_{4} - \beta_{2} + \beta_1) q^{27} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{29} + (2 \beta_{3} + 4) q^{31} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1) q^{33} + (\beta_{3} + \beta_{2}) q^{35} + 6 \beta_{4} q^{37} + ( - 5 \beta_{3} - \beta_{2} - \beta_1) q^{39} + (2 \beta_{2} + 2 \beta_1 + 6) q^{41} + ( - 2 \beta_{2} + 2 \beta_1) q^{43} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 4) q^{45} + (3 \beta_{5} - 2 \beta_{4} + 3 \beta_{2} - 3 \beta_1) q^{47} - q^{49} + ( - 3 \beta_{3} + \beta_{2} + \beta_1 + 6) q^{51} + (4 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{53} + ( - \beta_{5} + 4 \beta_{4} + \beta_{2} + 3 \beta_1 - 2) q^{55} + (2 \beta_{5} + 6 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{57} + (3 \beta_{2} + 3 \beta_1 - 2) q^{59} + ( - \beta_{2} - \beta_1 + 8) q^{61} + (\beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{63} + (\beta_{5} - 5 \beta_{4} - 4 \beta_{3} + \beta_{2} - \beta_1 - 5) q^{65} + ( - 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{67} + (6 \beta_{3} - 4) q^{69} + (2 \beta_{3} + 2) q^{71} + (2 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{73} + (3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_1 - 6) q^{75} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} + \beta_1) q^{77} + (5 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{79} + ( - 6 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 3) q^{81} + (6 \beta_{5} + 6 \beta_{4} + \beta_{2} - \beta_1) q^{83} + (2 \beta_{5} - 4 \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{85} + ( - 3 \beta_{5} - 6 \beta_{4} - 3 \beta_{2} + 3 \beta_1) q^{87} + ( - 6 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{89} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{91} + ( - 2 \beta_{5} - 8 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{93} + (3 \beta_{5} + 7 \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{95} + ( - 3 \beta_{5} - 4 \beta_{4} - \beta_{2} + \beta_1) q^{97} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{99}+O(q^{100})$$ q - b5 * q^3 + (-b5 + b1) * q^5 + b4 * q^7 + (b3 - b2 - b1 - 1) * q^9 + (-b3 - b2 - b1 + 2) * q^11 + (-b5 - 2*b4 - 2*b2 + 2*b1) * q^13 + (-b5 + b4 - b2 - b1 - 3) * q^15 + (b5 - b2 + b1) * q^17 + (-2*b3 - b2 - b1 - 2) * q^19 + b3 * q^21 + (2*b4 + 2*b2 - 2*b1) * q^23 + (-b5 - b4 - b3 - 2*b1 - 2) * q^25 + (b5 - 6*b4 - b2 + b1) * q^27 + (3*b3 + 3*b2 + 3*b1) * q^29 + (2*b3 + 4) * q^31 + (-b5 + 2*b4 + b2 - b1) * q^33 + (b3 + b2) * q^35 + 6*b4 * q^37 + (-5*b3 - b2 - b1) * q^39 + (2*b2 + 2*b1 + 6) * q^41 + (-2*b2 + 2*b1) * q^43 + (2*b5 - 2*b4 + 2*b3 - b2 + 2*b1 - 4) * q^45 + (3*b5 - 2*b4 + 3*b2 - 3*b1) * q^47 - q^49 + (-3*b3 + b2 + b1 + 6) * q^51 + (4*b5 - 2*b4 + 2*b2 - 2*b1) * q^53 + (-b5 + 4*b4 + b2 + 3*b1 - 2) * q^55 + (2*b5 + 6*b4 + 2*b2 - 2*b1) * q^57 + (3*b2 + 3*b1 - 2) * q^59 + (-b2 - b1 + 8) * q^61 + (b5 - b4 - b2 + b1) * q^63 + (b5 - 5*b4 - 4*b3 + b2 - b1 - 5) * q^65 + (-2*b5 - 4*b4 + 2*b2 - 2*b1) * q^67 + (6*b3 - 4) * q^69 + (2*b3 + 2) * q^71 + (2*b4 + 2*b2 - 2*b1) * q^73 + (3*b5 + 2*b4 + 2*b3 - 2*b1 - 6) * q^75 + (-b5 + 2*b4 - b2 + b1) * q^77 + (5*b3 + b2 + b1 - 4) * q^79 + (-6*b3 - 2*b2 - 2*b1 + 3) * q^81 + (6*b5 + 6*b4 + b2 - b1) * q^83 + (2*b5 - 4*b4 - b3 + 3*b2 + b1 + 2) * q^85 + (-3*b5 - 6*b4 - 3*b2 + 3*b1) * q^87 + (-6*b3 - 2*b2 - 2*b1 - 2) * q^89 + (b3 + 2*b2 + 2*b1 + 2) * q^91 + (-2*b5 - 8*b4 - 2*b2 + 2*b1) * q^93 + (3*b5 + 7*b4 - b3 + 2*b2 - 2*b1 - 1) * q^95 + (-3*b5 - 4*b4 - b2 + b1) * q^97 + (2*b3 - 4*b2 - 4*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 8 q^{9}+O(q^{10})$$ 6 * q - 8 * q^9 $$6 q - 8 q^{9} + 14 q^{11} - 18 q^{15} - 8 q^{19} - 2 q^{21} - 10 q^{25} - 6 q^{29} + 20 q^{31} - 2 q^{35} + 10 q^{39} + 36 q^{41} - 28 q^{45} - 6 q^{49} + 42 q^{51} - 12 q^{55} - 12 q^{59} + 48 q^{61} - 22 q^{65} - 36 q^{69} + 8 q^{71} - 40 q^{75} - 34 q^{79} + 30 q^{81} + 14 q^{85} + 10 q^{91} - 4 q^{95} - 4 q^{99}+O(q^{100})$$ 6 * q - 8 * q^9 + 14 * q^11 - 18 * q^15 - 8 * q^19 - 2 * q^21 - 10 * q^25 - 6 * q^29 + 20 * q^31 - 2 * q^35 + 10 * q^39 + 36 * q^41 - 28 * q^45 - 6 * q^49 + 42 * q^51 - 12 * q^55 - 12 * q^59 + 48 * q^61 - 22 * q^65 - 36 * q^69 + 8 * q^71 - 40 * q^75 - 34 * q^79 + 30 * q^81 + 14 * q^85 + 10 * q^91 - 4 * q^95 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 4x^{3} + 25x^{2} - 20x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -5\nu^{5} - 2\nu^{4} - 25\nu^{3} + 10\nu^{2} - 121\nu + 100 ) / 121$$ (-5*v^5 - 2*v^4 - 25*v^3 + 10*v^2 - 121*v + 100) / 121 $$\beta_{3}$$ $$=$$ $$( 7\nu^{5} + 27\nu^{4} + 35\nu^{3} - 14\nu^{2} + 223 ) / 121$$ (7*v^5 + 27*v^4 + 35*v^3 - 14*v^2 + 223) / 121 $$\beta_{4}$$ $$=$$ $$( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242$$ (-25*v^5 - 10*v^4 - 4*v^3 + 50*v^2 - 605*v + 258) / 242 $$\beta_{5}$$ $$=$$ $$( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242$$ (-65*v^5 - 26*v^4 + 38*v^3 + 372*v^2 - 1331*v + 574) / 242
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - 3\beta_{4} + \beta_{2} - \beta_1$$ b5 - 3*b4 + b2 - b1 $$\nu^{3}$$ $$=$$ $$2\beta_{4} - 5\beta_{2} + 2$$ 2*b4 - 5*b2 + 2 $$\nu^{4}$$ $$=$$ $$5\beta_{3} + 7\beta_{2} + 7\beta _1 - 15$$ 5*b3 + 7*b2 + 7*b1 - 15 $$\nu^{5}$$ $$=$$ $$2\beta_{5} - 16\beta_{4} - 2\beta_{3} - 29\beta _1 + 16$$ 2*b5 - 16*b4 - 2*b3 - 29*b1 + 16

## Character values

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

 $$n$$ $$57$$ $$71$$ $$141$$ $$241$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 1.32001 + 1.32001i 0.432320 − 0.432320i −1.75233 + 1.75233i −1.75233 − 1.75233i 0.432320 + 0.432320i 1.32001 − 1.32001i
0 3.12489i 0 1.32001 1.80487i 0 1.00000i 0 −6.76491 0
169.2 0 1.76156i 0 0.432320 2.19388i 0 1.00000i 0 −0.103084 0
169.3 0 0.363328i 0 −1.75233 + 1.38900i 0 1.00000i 0 2.86799 0
169.4 0 0.363328i 0 −1.75233 1.38900i 0 1.00000i 0 2.86799 0
169.5 0 1.76156i 0 0.432320 + 2.19388i 0 1.00000i 0 −0.103084 0
169.6 0 3.12489i 0 1.32001 + 1.80487i 0 1.00000i 0 −6.76491 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 169.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.g.b 6
3.b odd 2 1 2520.2.t.g 6
4.b odd 2 1 560.2.g.f 6
5.b even 2 1 inner 280.2.g.b 6
5.c odd 4 1 1400.2.a.s 3
5.c odd 4 1 1400.2.a.t 3
7.b odd 2 1 1960.2.g.c 6
8.b even 2 1 2240.2.g.l 6
8.d odd 2 1 2240.2.g.m 6
12.b even 2 1 5040.2.t.y 6
15.d odd 2 1 2520.2.t.g 6
20.d odd 2 1 560.2.g.f 6
20.e even 4 1 2800.2.a.bq 3
20.e even 4 1 2800.2.a.br 3
35.c odd 2 1 1960.2.g.c 6
35.f even 4 1 9800.2.a.cd 3
35.f even 4 1 9800.2.a.cg 3
40.e odd 2 1 2240.2.g.m 6
40.f even 2 1 2240.2.g.l 6
60.h even 2 1 5040.2.t.y 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.g.b 6 1.a even 1 1 trivial
280.2.g.b 6 5.b even 2 1 inner
560.2.g.f 6 4.b odd 2 1
560.2.g.f 6 20.d odd 2 1
1400.2.a.s 3 5.c odd 4 1
1400.2.a.t 3 5.c odd 4 1
1960.2.g.c 6 7.b odd 2 1
1960.2.g.c 6 35.c odd 2 1
2240.2.g.l 6 8.b even 2 1
2240.2.g.l 6 40.f even 2 1
2240.2.g.m 6 8.d odd 2 1
2240.2.g.m 6 40.e odd 2 1
2520.2.t.g 6 3.b odd 2 1
2520.2.t.g 6 15.d odd 2 1
2800.2.a.bq 3 20.e even 4 1
2800.2.a.br 3 20.e even 4 1
5040.2.t.y 6 12.b even 2 1
5040.2.t.y 6 60.h even 2 1
9800.2.a.cd 3 35.f even 4 1
9800.2.a.cg 3 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 13T_{3}^{4} + 32T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 13 T^{4} + 32 T^{2} + 4$$
$5$ $$T^{6} + 5 T^{4} + 8 T^{3} + 25 T^{2} + \cdots + 125$$
$7$ $$(T^{2} + 1)^{3}$$
$11$ $$(T^{3} - 7 T^{2} + 8 T + 8)^{2}$$
$13$ $$T^{6} + 69 T^{4} + 1544 T^{2} + \cdots + 11236$$
$17$ $$T^{6} + 49 T^{4} + 536 T^{2} + \cdots + 400$$
$19$ $$(T^{3} + 4 T^{2} - 14 T + 8)^{2}$$
$23$ $$T^{6} + 92 T^{4} + 2416 T^{2} + \cdots + 18496$$
$29$ $$(T^{3} + 3 T^{2} - 72 T - 108)^{2}$$
$31$ $$(T^{3} - 10 T^{2} + 8 T + 80)^{2}$$
$37$ $$(T^{2} + 36)^{3}$$
$41$ $$(T^{3} - 18 T^{2} + 68 T + 88)^{2}$$
$43$ $$T^{6} + 80 T^{4} + 1600 T^{2} + \cdots + 4096$$
$47$ $$T^{6} + 177 T^{4} + 6480 T^{2} + \cdots + 53824$$
$53$ $$T^{6} + 188 T^{4} + 11376 T^{2} + \cdots + 222784$$
$59$ $$(T^{3} + 6 T^{2} - 78 T + 44)^{2}$$
$61$ $$(T^{3} - 24 T^{2} + 182 T - 440)^{2}$$
$67$ $$T^{6} + 228 T^{4} + 14336 T^{2} + \cdots + 262144$$
$71$ $$(T^{3} - 4 T^{2} - 20 T + 64)^{2}$$
$73$ $$T^{6} + 92 T^{4} + 2416 T^{2} + \cdots + 18496$$
$79$ $$(T^{3} + 17 T^{2} - 32 T - 548)^{2}$$
$83$ $$T^{6} + 428 T^{4} + 39940 T^{2} + \cdots + 678976$$
$89$ $$(T^{3} - 172 T + 464)^{2}$$
$97$ $$T^{6} + 113 T^{4} + 1048 T^{2} + \cdots + 1936$$