# Properties

 Label 280.2.s.b Level $280$ Weight $2$ Character orbit 280.s Analytic conductor $2.236$ Analytic rank $0$ Dimension $8$ CM discriminant -56 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 280.s (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.40282095616.8 Defining polynomial: $$x^{8} - 4 x^{6} + 8 x^{4} - 36 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$5$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{4} ) q^{2} + \beta_{1} q^{3} + 2 \beta_{4} q^{4} -\beta_{5} q^{5} + ( \beta_{1} + \beta_{2} ) q^{6} -\beta_{7} q^{7} + ( -2 + 2 \beta_{4} ) q^{8} + ( -\beta_{3} - 5 \beta_{4} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{4} ) q^{2} + \beta_{1} q^{3} + 2 \beta_{4} q^{4} -\beta_{5} q^{5} + ( \beta_{1} + \beta_{2} ) q^{6} -\beta_{7} q^{7} + ( -2 + 2 \beta_{4} ) q^{8} + ( -\beta_{3} - 5 \beta_{4} + \beta_{7} ) q^{9} + ( -\beta_{5} + \beta_{6} ) q^{10} + 2 \beta_{2} q^{12} -\beta_{1} q^{13} + ( \beta_{3} - \beta_{7} ) q^{14} + ( -2 + 2 \beta_{3} - \beta_{4} + \beta_{7} ) q^{15} -4 q^{16} + ( 5 - 2 \beta_{3} - 5 \beta_{4} ) q^{18} + ( \beta_{1} + \beta_{5} - 3 \beta_{6} ) q^{19} + 2 \beta_{6} q^{20} + ( \beta_{1} + 3 \beta_{5} + \beta_{6} ) q^{21} + ( -3 - 2 \beta_{3} + 3 \beta_{4} ) q^{23} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{24} + ( 3 + \beta_{3} + 3 \beta_{4} ) q^{25} + ( -\beta_{1} - \beta_{2} ) q^{26} + ( -\beta_{1} - 3 \beta_{2} - 4 \beta_{5} + 2 \beta_{6} ) q^{27} + 2 \beta_{3} q^{28} + ( -1 + \beta_{3} - 3 \beta_{4} + 3 \beta_{7} ) q^{30} + ( -4 - 4 \beta_{4} ) q^{32} + ( \beta_{1} - 2 \beta_{2} + \beta_{5} ) q^{35} + ( 10 - 2 \beta_{3} - 2 \beta_{7} ) q^{36} + ( \beta_{1} + \beta_{2} - 2 \beta_{5} - 4 \beta_{6} ) q^{38} + ( \beta_{3} + 8 \beta_{4} - \beta_{7} ) q^{39} + ( 2 \beta_{5} + 2 \beta_{6} ) q^{40} + ( \beta_{1} + \beta_{2} + 4 \beta_{5} - 2 \beta_{6} ) q^{42} + ( -3 \beta_{1} + \beta_{2} - \beta_{5} - 4 \beta_{6} ) q^{45} + ( -6 - 2 \beta_{3} - 2 \beta_{7} ) q^{46} -4 \beta_{1} q^{48} + 7 \beta_{4} q^{49} + ( \beta_{3} + 6 \beta_{4} + \beta_{7} ) q^{50} -2 \beta_{2} q^{52} + ( 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{5} + 6 \beta_{6} ) q^{54} + ( 2 \beta_{3} + 2 \beta_{7} ) q^{56} + ( -1 - \beta_{4} - 6 \beta_{7} ) q^{57} + ( 2 \beta_{1} - \beta_{2} + \beta_{5} - 3 \beta_{6} ) q^{59} + ( 2 - 2 \beta_{3} - 4 \beta_{4} + 4 \beta_{7} ) q^{60} + ( -2 \beta_{1} - 3 \beta_{2} + 3 \beta_{5} + \beta_{6} ) q^{61} + ( 7 - 5 \beta_{3} - 7 \beta_{4} ) q^{63} -8 \beta_{4} q^{64} + ( 2 - 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{65} + ( -3 \beta_{1} + \beta_{2} - 2 \beta_{5} + 6 \beta_{6} ) q^{69} + ( 3 \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} ) q^{70} + ( 4 + 3 \beta_{3} + 3 \beta_{7} ) q^{71} + ( 10 + 10 \beta_{4} - 4 \beta_{7} ) q^{72} + ( 3 \beta_{1} + 4 \beta_{2} + \beta_{5} - 3 \beta_{6} ) q^{75} + ( 2 \beta_{2} - 6 \beta_{5} - 2 \beta_{6} ) q^{76} + ( -8 + 2 \beta_{3} + 8 \beta_{4} ) q^{78} + ( -\beta_{3} - 12 \beta_{4} + \beta_{7} ) q^{79} + 4 \beta_{5} q^{80} + ( -15 + 7 \beta_{3} + 7 \beta_{7} ) q^{81} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{5} + 4 \beta_{6} ) q^{83} + ( 2 \beta_{2} + 2 \beta_{5} - 6 \beta_{6} ) q^{84} + ( -4 \beta_{1} - 2 \beta_{2} - 5 \beta_{5} - 3 \beta_{6} ) q^{90} + ( -\beta_{1} - 3 \beta_{5} - \beta_{6} ) q^{91} + ( -6 - 6 \beta_{4} - 4 \beta_{7} ) q^{92} + ( 4 + \beta_{3} - 13 \beta_{4} - 2 \beta_{7} ) q^{95} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{96} + ( -7 + 7 \beta_{4} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{2} - 16q^{8} + O(q^{10})$$ $$8q + 8q^{2} - 16q^{8} - 16q^{15} - 32q^{16} + 40q^{18} - 24q^{23} + 24q^{25} - 8q^{30} - 32q^{32} + 80q^{36} - 48q^{46} - 8q^{57} + 16q^{60} + 56q^{63} + 16q^{65} + 32q^{71} + 80q^{72} - 64q^{78} - 120q^{81} - 48q^{92} + 32q^{95} - 56q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 8 x^{4} - 36 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 23 \nu^{5} - 46 \nu^{3} + 72 \nu$$$$)/135$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{7} - 2 \nu^{5} - 41 \nu^{3} + 297 \nu$$$$)/135$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{6} + \nu^{4} + 43 \nu^{2} - 81$$$$)/45$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{6} - \nu^{4} + 2 \nu^{2} + 36$$$$)/45$$ $$\beta_{5}$$ $$=$$ $$($$$$7 \nu^{7} - 19 \nu^{5} - 7 \nu^{3} - 126 \nu$$$$)/135$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{5} - 9 \nu^{3} + 23 \nu$$$$)/15$$ $$\beta_{7}$$ $$=$$ $$($$$$-7 \nu^{6} + 19 \nu^{4} - 38 \nu^{2} + 171$$$$)/45$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{6} - \beta_{5} + 3 \beta_{2} + \beta_{1}$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + 1$$ $$\nu^{3}$$ $$=$$ $$($$$$-8 \beta_{6} - 9 \beta_{5} + 2 \beta_{2} - \beta_{1}$$$$)/5$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} - 5 \beta_{4} + 2 \beta_{3}$$ $$\nu^{5}$$ $$=$$ $$($$$$-7 \beta_{6} - 16 \beta_{5} - 7 \beta_{2} + 21 \beta_{1}$$$$)/5$$ $$\nu^{6}$$ $$=$$ $$-\beta_{7} - 19 \beta_{4} + 19$$ $$\nu^{7}$$ $$=$$ $$($$$$-63 \beta_{6} + 26 \beta_{5} + 37 \beta_{2} + 74 \beta_{1}$$$$)/5$$

## Character values

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

 $$n$$ $$57$$ $$71$$ $$141$$ $$241$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 −1.03179 + 1.39119i −1.71331 − 0.254137i 1.71331 + 0.254137i 1.03179 − 1.39119i −1.03179 − 1.39119i −1.71331 + 0.254137i 1.71331 − 0.254137i 1.03179 + 1.39119i
1.00000 1.00000i −2.42298 2.42298i 2.00000i 1.75060 1.39119i −4.84596 1.87083 1.87083i −2.00000 2.00000i 8.74166i 0.359404 3.14179i
13.2 1.00000 1.00000i −1.45917 1.45917i 2.00000i −2.22158 + 0.254137i −2.91834 −1.87083 + 1.87083i −2.00000 2.00000i 1.25834i −1.96744 + 2.47572i
13.3 1.00000 1.00000i 1.45917 + 1.45917i 2.00000i 2.22158 0.254137i 2.91834 −1.87083 + 1.87083i −2.00000 2.00000i 1.25834i 1.96744 2.47572i
13.4 1.00000 1.00000i 2.42298 + 2.42298i 2.00000i −1.75060 + 1.39119i 4.84596 1.87083 1.87083i −2.00000 2.00000i 8.74166i −0.359404 + 3.14179i
237.1 1.00000 + 1.00000i −2.42298 + 2.42298i 2.00000i 1.75060 + 1.39119i −4.84596 1.87083 + 1.87083i −2.00000 + 2.00000i 8.74166i 0.359404 + 3.14179i
237.2 1.00000 + 1.00000i −1.45917 + 1.45917i 2.00000i −2.22158 0.254137i −2.91834 −1.87083 1.87083i −2.00000 + 2.00000i 1.25834i −1.96744 2.47572i
237.3 1.00000 + 1.00000i 1.45917 1.45917i 2.00000i 2.22158 + 0.254137i 2.91834 −1.87083 1.87083i −2.00000 + 2.00000i 1.25834i 1.96744 + 2.47572i
237.4 1.00000 + 1.00000i 2.42298 2.42298i 2.00000i −1.75060 1.39119i 4.84596 1.87083 + 1.87083i −2.00000 + 2.00000i 8.74166i −0.359404 3.14179i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 237.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
5.c odd 4 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
35.f even 4 1 inner
40.i odd 4 1 inner
280.s even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.s.b 8
4.b odd 2 1 1120.2.w.b 8
5.c odd 4 1 inner 280.2.s.b 8
7.b odd 2 1 inner 280.2.s.b 8
8.b even 2 1 inner 280.2.s.b 8
8.d odd 2 1 1120.2.w.b 8
20.e even 4 1 1120.2.w.b 8
28.d even 2 1 1120.2.w.b 8
35.f even 4 1 inner 280.2.s.b 8
40.i odd 4 1 inner 280.2.s.b 8
40.k even 4 1 1120.2.w.b 8
56.e even 2 1 1120.2.w.b 8
56.h odd 2 1 CM 280.2.s.b 8
140.j odd 4 1 1120.2.w.b 8
280.s even 4 1 inner 280.2.s.b 8
280.y odd 4 1 1120.2.w.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.s.b 8 1.a even 1 1 trivial
280.2.s.b 8 5.c odd 4 1 inner
280.2.s.b 8 7.b odd 2 1 inner
280.2.s.b 8 8.b even 2 1 inner
280.2.s.b 8 35.f even 4 1 inner
280.2.s.b 8 40.i odd 4 1 inner
280.2.s.b 8 56.h odd 2 1 CM
280.2.s.b 8 280.s even 4 1 inner
1120.2.w.b 8 4.b odd 2 1
1120.2.w.b 8 8.d odd 2 1
1120.2.w.b 8 20.e even 4 1
1120.2.w.b 8 28.d even 2 1
1120.2.w.b 8 40.k even 4 1
1120.2.w.b 8 56.e even 2 1
1120.2.w.b 8 140.j odd 4 1
1120.2.w.b 8 280.y odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 156 T_{3}^{4} + 2500$$ acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 - 2 T + T^{2} )^{4}$$
$3$ $$2500 + 156 T^{4} + T^{8}$$
$5$ $$625 - 300 T^{2} + 72 T^{4} - 12 T^{6} + T^{8}$$
$7$ $$( 49 + T^{4} )^{2}$$
$11$ $$T^{8}$$
$13$ $$2500 + 156 T^{4} + T^{8}$$
$17$ $$T^{8}$$
$19$ $$( 1250 + 88 T^{2} + T^{4} )^{2}$$
$23$ $$( 100 - 120 T + 72 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$( 50 + 128 T^{2} + T^{4} )^{2}$$
$61$ $$( 6050 - 232 T^{2} + T^{4} )^{2}$$
$67$ $$T^{8}$$
$71$ $$( -110 - 8 T + T^{2} )^{4}$$
$73$ $$T^{8}$$
$79$ $$( 16900 + 316 T^{2} + T^{4} )^{2}$$
$83$ $$71402500 + 70716 T^{4} + T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$