# Properties

 Label 280.3.c.f Level $280$ Weight $3$ Character orbit 280.c Analytic conductor $7.629$ Analytic rank $0$ Dimension $4$ CM discriminant -56 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 280.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.62944740209$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{14})$$ Defining polynomial: $$x^{4} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 \beta_{2} q^{2} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{3} -4 q^{4} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{5} + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{6} -7 \beta_{2} q^{7} + 8 \beta_{2} q^{8} + ( -9 - 4 \beta_{1} + 4 \beta_{3} ) q^{9} +O(q^{10})$$ $$q -2 \beta_{2} q^{2} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{3} -4 q^{4} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{5} + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{6} -7 \beta_{2} q^{7} + 8 \beta_{2} q^{8} + ( -9 - 4 \beta_{1} + 4 \beta_{3} ) q^{9} + ( -6 - 6 \beta_{2} + 2 \beta_{3} ) q^{10} + ( -4 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{12} + ( \beta_{1} - 18 \beta_{2} + \beta_{3} ) q^{13} -14 q^{14} + ( 13 + 6 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{15} + 16 q^{16} + ( 8 \beta_{1} + 18 \beta_{2} + 8 \beta_{3} ) q^{18} + ( -6 + 7 \beta_{1} - 7 \beta_{3} ) q^{19} + ( -12 + 4 \beta_{1} + 12 \beta_{2} ) q^{20} + ( 14 + 7 \beta_{1} - 7 \beta_{3} ) q^{21} + ( -12 \beta_{1} - 12 \beta_{3} ) q^{23} + ( -16 - 8 \beta_{1} + 8 \beta_{3} ) q^{24} + ( -6 \beta_{1} - 11 \beta_{2} + 6 \beta_{3} ) q^{25} + ( -36 + 2 \beta_{1} - 2 \beta_{3} ) q^{26} + ( -8 \beta_{1} - 56 \beta_{2} - 8 \beta_{3} ) q^{27} + 28 \beta_{2} q^{28} + ( -2 - 4 \beta_{1} - 26 \beta_{2} - 12 \beta_{3} ) q^{30} -32 \beta_{2} q^{32} + ( -21 - 21 \beta_{2} + 7 \beta_{3} ) q^{35} + ( 36 + 16 \beta_{1} - 16 \beta_{3} ) q^{36} + ( -14 \beta_{1} + 12 \beta_{2} - 14 \beta_{3} ) q^{38} + ( 22 + 16 \beta_{1} - 16 \beta_{3} ) q^{39} + ( 24 + 24 \beta_{2} - 8 \beta_{3} ) q^{40} + ( -14 \beta_{1} - 28 \beta_{2} - 14 \beta_{3} ) q^{42} + ( 1 + 9 \beta_{1} + 55 \beta_{2} + 24 \beta_{3} ) q^{45} + ( -24 \beta_{1} + 24 \beta_{3} ) q^{46} + ( 16 \beta_{1} + 32 \beta_{2} + 16 \beta_{3} ) q^{48} -49 q^{49} + ( -22 + 12 \beta_{1} + 12 \beta_{3} ) q^{50} + ( -4 \beta_{1} + 72 \beta_{2} - 4 \beta_{3} ) q^{52} + ( -112 - 16 \beta_{1} + 16 \beta_{3} ) q^{54} + 56 q^{56} + ( 8 \beta_{1} + 86 \beta_{2} + 8 \beta_{3} ) q^{57} + ( 54 + 17 \beta_{1} - 17 \beta_{3} ) q^{59} + ( -52 - 24 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} ) q^{60} + ( -6 - 23 \beta_{1} + 23 \beta_{3} ) q^{61} + ( 28 \beta_{1} + 63 \beta_{2} + 28 \beta_{3} ) q^{63} -64 q^{64} + ( -47 + 6 \beta_{1} - 61 \beta_{2} + 18 \beta_{3} ) q^{65} + ( 168 + 24 \beta_{1} - 24 \beta_{3} ) q^{69} + ( -42 + 14 \beta_{1} + 42 \beta_{2} ) q^{70} + ( -24 \beta_{1} + 24 \beta_{3} ) q^{71} + ( -32 \beta_{1} - 72 \beta_{2} - 32 \beta_{3} ) q^{72} + ( 22 - \beta_{1} - 84 \beta_{2} - 23 \beta_{3} ) q^{75} + ( 24 - 28 \beta_{1} + 28 \beta_{3} ) q^{76} + ( -32 \beta_{1} - 44 \beta_{2} - 32 \beta_{3} ) q^{78} + ( -24 \beta_{1} + 24 \beta_{3} ) q^{79} + ( 48 - 16 \beta_{1} - 48 \beta_{2} ) q^{80} + ( 143 + 36 \beta_{1} - 36 \beta_{3} ) q^{81} + ( 31 \beta_{1} - 18 \beta_{2} + 31 \beta_{3} ) q^{83} + ( -56 - 28 \beta_{1} + 28 \beta_{3} ) q^{84} + ( 110 + 48 \beta_{1} - 2 \beta_{2} - 18 \beta_{3} ) q^{90} + ( -126 + 7 \beta_{1} - 7 \beta_{3} ) q^{91} + ( 48 \beta_{1} + 48 \beta_{3} ) q^{92} + ( -67 + 6 \beta_{1} - 31 \beta_{2} - 42 \beta_{3} ) q^{95} + ( 64 + 32 \beta_{1} - 32 \beta_{3} ) q^{96} + 98 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 16q^{4} + 12q^{5} + 16q^{6} - 36q^{9} + O(q^{10})$$ $$4q - 16q^{4} + 12q^{5} + 16q^{6} - 36q^{9} - 24q^{10} - 56q^{14} + 52q^{15} + 64q^{16} - 24q^{19} - 48q^{20} + 56q^{21} - 64q^{24} - 144q^{26} - 8q^{30} - 84q^{35} + 144q^{36} + 88q^{39} + 96q^{40} + 4q^{45} - 196q^{49} - 88q^{50} - 448q^{54} + 224q^{56} + 216q^{59} - 208q^{60} - 24q^{61} - 256q^{64} - 188q^{65} + 672q^{69} - 168q^{70} + 88q^{75} + 96q^{76} + 192q^{80} + 572q^{81} - 224q^{84} + 440q^{90} - 504q^{91} - 268q^{95} + 256q^{96} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
69.1
 −1.87083 − 1.87083i 1.87083 + 1.87083i 1.87083 − 1.87083i −1.87083 + 1.87083i
2.00000i 1.74166i −4.00000 4.87083 1.12917i −3.48331 7.00000i 8.00000i 5.96663 −2.25834 9.74166i
69.2 2.00000i 5.74166i −4.00000 1.12917 4.87083i 11.4833 7.00000i 8.00000i −23.9666 −9.74166 2.25834i
69.3 2.00000i 5.74166i −4.00000 1.12917 + 4.87083i 11.4833 7.00000i 8.00000i −23.9666 −9.74166 + 2.25834i
69.4 2.00000i 1.74166i −4.00000 4.87083 + 1.12917i −3.48331 7.00000i 8.00000i 5.96663 −2.25834 + 9.74166i
 $$n$$: 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
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
5.b even 2 1 inner
280.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.3.c.f yes 4
4.b odd 2 1 1120.3.c.f 4
5.b even 2 1 inner 280.3.c.f yes 4
7.b odd 2 1 280.3.c.e 4
8.b even 2 1 280.3.c.e 4
8.d odd 2 1 1120.3.c.e 4
20.d odd 2 1 1120.3.c.f 4
28.d even 2 1 1120.3.c.e 4
35.c odd 2 1 280.3.c.e 4
40.e odd 2 1 1120.3.c.e 4
40.f even 2 1 280.3.c.e 4
56.e even 2 1 1120.3.c.f 4
56.h odd 2 1 CM 280.3.c.f yes 4
140.c even 2 1 1120.3.c.e 4
280.c odd 2 1 inner 280.3.c.f yes 4
280.n even 2 1 1120.3.c.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.3.c.e 4 7.b odd 2 1
280.3.c.e 4 8.b even 2 1
280.3.c.e 4 35.c odd 2 1
280.3.c.e 4 40.f even 2 1
280.3.c.f yes 4 1.a even 1 1 trivial
280.3.c.f yes 4 5.b even 2 1 inner
280.3.c.f yes 4 56.h odd 2 1 CM
280.3.c.f yes 4 280.c odd 2 1 inner
1120.3.c.e 4 8.d odd 2 1
1120.3.c.e 4 28.d even 2 1
1120.3.c.e 4 40.e odd 2 1
1120.3.c.e 4 140.c even 2 1
1120.3.c.f 4 4.b odd 2 1
1120.3.c.f 4 20.d odd 2 1
1120.3.c.f 4 56.e even 2 1
1120.3.c.f 4 280.n even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(280, [\chi])$$:

 $$T_{3}^{4} + 36 T_{3}^{2} + 100$$ $$T_{17}$$ $$T_{19}^{2} + 12 T_{19} - 650$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + T^{2} )^{2}$$
$3$ $$100 + 36 T^{2} + T^{4}$$
$5$ $$625 - 300 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$7$ $$( 49 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$96100 + 676 T^{2} + T^{4}$$
$17$ $$T^{4}$$
$19$ $$( -650 + 12 T + T^{2} )^{2}$$
$23$ $$( 2016 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$( -1130 - 108 T + T^{2} )^{2}$$
$61$ $$( -7370 + 12 T + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$( -8064 + T^{2} )^{2}$$
$73$ $$T^{4}$$
$79$ $$( -8064 + T^{2} )^{2}$$
$83$ $$172396900 + 27556 T^{2} + T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$