# Properties

 Label 280.3.c.e 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{2} q^{2} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{3} -4 q^{4} + ( -3 - 3 \beta_{2} - \beta_{3} ) q^{5} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} -8 \beta_{2} q^{8} + ( -9 - 4 \beta_{1} + 4 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + 2 \beta_{2} q^{2} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{3} -4 q^{4} + ( -3 - 3 \beta_{2} - \beta_{3} ) 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 + 2 \beta_{1} - 6 \beta_{2} ) 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 + 2 \beta_{1} + \beta_{2} - 6 \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 + 12 \beta_{2} + 4 \beta_{3} ) 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 + 12 \beta_{1} + 26 \beta_{2} + 4 \beta_{3} ) q^{30} + 32 \beta_{2} q^{32} + ( 21 + 7 \beta_{1} - 21 \beta_{2} ) 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 - 8 \beta_{1} + 24 \beta_{2} ) q^{40} + ( -14 \beta_{1} - 28 \beta_{2} - 14 \beta_{3} ) q^{42} + ( -1 + 24 \beta_{1} + 55 \beta_{2} + 9 \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 - 8 \beta_{1} - 4 \beta_{2} + 24 \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 - 18 \beta_{1} + 61 \beta_{2} - 6 \beta_{3} ) q^{65} + ( -168 - 24 \beta_{1} + 24 \beta_{3} ) q^{69} + ( 42 + 42 \beta_{2} + 14 \beta_{3} ) q^{70} + ( -24 \beta_{1} + 24 \beta_{3} ) q^{71} + ( 32 \beta_{1} + 72 \beta_{2} + 32 \beta_{3} ) q^{72} + ( -22 - 23 \beta_{1} - 84 \beta_{2} - \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 - 48 \beta_{2} - 16 \beta_{3} ) 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 - 18 \beta_{1} - 2 \beta_{2} + 48 \beta_{3} ) q^{90} + ( 126 - 7 \beta_{1} + 7 \beta_{3} ) q^{91} + ( -48 \beta_{1} - 48 \beta_{3} ) q^{92} + ( -67 + 42 \beta_{1} + 31 \beta_{2} - 6 \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 5.74166i −4.00000 −1.12917 + 4.87083i −11.4833 7.00000i 8.00000i −23.9666 9.74166 + 2.25834i
69.2 2.00000i 1.74166i −4.00000 −4.87083 + 1.12917i 3.48331 7.00000i 8.00000i 5.96663 2.25834 + 9.74166i
69.3 2.00000i 1.74166i −4.00000 −4.87083 1.12917i 3.48331 7.00000i 8.00000i 5.96663 2.25834 9.74166i
69.4 2.00000i 5.74166i −4.00000 −1.12917 4.87083i −11.4833 7.00000i 8.00000i −23.9666 9.74166 2.25834i
 $$n$$: 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.e 4
4.b odd 2 1 1120.3.c.e 4
5.b even 2 1 inner 280.3.c.e 4
7.b odd 2 1 280.3.c.f yes 4
8.b even 2 1 280.3.c.f yes 4
8.d odd 2 1 1120.3.c.f 4
20.d odd 2 1 1120.3.c.e 4
28.d even 2 1 1120.3.c.f 4
35.c odd 2 1 280.3.c.f yes 4
40.e odd 2 1 1120.3.c.f 4
40.f even 2 1 280.3.c.f yes 4
56.e even 2 1 1120.3.c.e 4
56.h odd 2 1 CM 280.3.c.e 4
140.c even 2 1 1120.3.c.f 4
280.c odd 2 1 inner 280.3.c.e 4
280.n even 2 1 1120.3.c.e 4

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

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