# Properties

 Label 280.2.ba.a Level $280$ Weight $2$ Character orbit 280.ba Analytic conductor $2.236$ Analytic rank $0$ Dimension $8$ CM discriminant -40 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 280.ba (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.3317760000.3 Defining polynomial: $$x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} -2 \beta_{3} q^{4} + \beta_{6} q^{5} + ( \beta_{1} + \beta_{2} - \beta_{6} ) q^{7} + 2 \beta_{2} q^{8} + ( 3 - 3 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{2} -2 \beta_{3} q^{4} + \beta_{6} q^{5} + ( \beta_{1} + \beta_{2} - \beta_{6} ) q^{7} + 2 \beta_{2} q^{8} + ( 3 - 3 \beta_{3} ) q^{9} -\beta_{5} q^{10} + ( -\beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{11} + ( \beta_{1} + 2 \beta_{2} + 4 \beta_{4} - \beta_{6} ) q^{13} + ( -2 + 2 \beta_{3} - \beta_{7} ) q^{14} + ( -4 + 4 \beta_{3} ) q^{16} + ( 3 \beta_{2} + 3 \beta_{4} ) q^{18} + ( 3 + 3 \beta_{3} - \beta_{7} ) q^{19} + ( 2 \beta_{1} - 2 \beta_{6} ) q^{20} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{6} ) q^{22} + ( -2 \beta_{1} - 3 \beta_{4} + \beta_{6} ) q^{23} + 5 \beta_{3} q^{25} + ( -4 - 4 \beta_{3} - \beta_{7} ) q^{26} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{28} + ( -4 \beta_{2} - 4 \beta_{4} ) q^{32} + ( 5 - 5 \beta_{3} - \beta_{7} ) q^{35} -6 q^{36} + ( -2 \beta_{1} + 4 \beta_{4} + \beta_{6} ) q^{37} + ( -2 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} ) q^{38} -2 \beta_{7} q^{40} + ( -1 + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{41} + ( -2 + 2 \beta_{3} - 4 \beta_{5} - 2 \beta_{7} ) q^{44} + 3 \beta_{1} q^{45} + ( 6 \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{46} + ( -2 \beta_{2} - \beta_{4} - 3 \beta_{6} ) q^{47} + ( -3 + 2 \beta_{5} + 2 \beta_{7} ) q^{49} -5 \beta_{2} q^{50} + ( -2 \beta_{1} + 4 \beta_{2} - 4 \beta_{4} ) q^{52} + ( -3 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 6 \beta_{6} ) q^{53} + ( \beta_{1} - 5 \beta_{2} - 10 \beta_{4} - \beta_{6} ) q^{55} + ( 4 + 2 \beta_{5} + 2 \beta_{7} ) q^{56} -2 \beta_{5} q^{59} + ( -3 \beta_{4} - 3 \beta_{6} ) q^{63} + 8 q^{64} + ( 5 - 5 \beta_{3} - 4 \beta_{5} - 2 \beta_{7} ) q^{65} + ( -2 \beta_{1} + 5 \beta_{2} + 5 \beta_{4} ) q^{70} -6 \beta_{4} q^{72} + ( -8 \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{74} + ( 6 - 12 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{76} + ( \beta_{1} - 6 \beta_{2} + 4 \beta_{4} - 4 \beta_{6} ) q^{77} -4 \beta_{1} q^{80} -9 \beta_{3} q^{81} + ( -2 \beta_{2} - \beta_{4} + 4 \beta_{6} ) q^{82} + ( 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 8 \beta_{6} ) q^{88} -4 \beta_{7} q^{89} + ( -3 \beta_{5} - 3 \beta_{7} ) q^{90} + ( -9 + 8 \beta_{3} + 3 \beta_{5} - \beta_{7} ) q^{91} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{6} ) q^{92} + ( 4 - 2 \beta_{3} + 3 \beta_{5} ) q^{94} + ( -3 \beta_{1} + 5 \beta_{2} + 5 \beta_{4} + 6 \beta_{6} ) q^{95} + ( -3 \beta_{4} + 4 \beta_{6} ) q^{98} + ( -3 - 3 \beta_{5} + 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} + 12q^{9} + O(q^{10})$$ $$8q - 8q^{4} + 12q^{9} - 4q^{11} - 8q^{14} - 16q^{16} + 36q^{19} + 20q^{25} - 48q^{26} + 20q^{35} - 48q^{36} - 8q^{44} + 24q^{46} - 24q^{49} + 32q^{56} + 64q^{64} + 20q^{65} - 32q^{74} - 36q^{81} - 40q^{91} + 24q^{94} - 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 7 x^{4} - 36 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 14 \nu^{4} + 7 \nu^{2} - 36$$$$)/63$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{7} + 7 \nu^{5} + 35 \nu^{3} + 81 \nu$$$$)/189$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 144$$$$)/63$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{7} - 7 \nu^{5} - 35 \nu^{3} + 180 \nu$$$$)/189$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 13 \nu$$$$)/21$$ $$\beta_{6}$$ $$=$$ $$($$$$-8 \nu^{6} + 14 \nu^{4} + 7 \nu^{2} + 162$$$$)/63$$ $$\beta_{7}$$ $$=$$ $$($$$$19 \nu^{7} - 49 \nu^{5} + 133 \nu^{3} - 684 \nu$$$$)/189$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - 2 \beta_{3} + 2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + \beta_{5} + 7 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} + 4 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{7} - 19 \beta_{4}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-7 \beta_{6} + 7 \beta_{1} + 22$$ $$\nu^{7}$$ $$=$$ $$($$$$29 \beta_{5} - 13 \beta_{4} - 13 \beta_{2}$$$$)/2$$

## Character values

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

 $$n$$ $$57$$ $$71$$ $$141$$ $$241$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1 - \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −1.01575 + 1.40294i 1.72286 − 0.178197i 1.01575 − 1.40294i −1.72286 + 0.178197i −1.01575 − 1.40294i 1.72286 + 0.178197i 1.01575 + 1.40294i −1.72286 − 0.178197i
−0.707107 + 1.22474i 0 −1.00000 1.73205i −1.93649 1.11803i 0 1.41421 + 2.23607i 2.82843 1.50000 2.59808i 2.73861 1.58114i
19.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i 1.93649 + 1.11803i 0 1.41421 2.23607i 2.82843 1.50000 2.59808i −2.73861 + 1.58114i
19.3 0.707107 1.22474i 0 −1.00000 1.73205i −1.93649 1.11803i 0 −1.41421 + 2.23607i −2.82843 1.50000 2.59808i −2.73861 + 1.58114i
19.4 0.707107 1.22474i 0 −1.00000 1.73205i 1.93649 + 1.11803i 0 −1.41421 2.23607i −2.82843 1.50000 2.59808i 2.73861 1.58114i
59.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i −1.93649 + 1.11803i 0 1.41421 2.23607i 2.82843 1.50000 + 2.59808i 2.73861 + 1.58114i
59.2 −0.707107 1.22474i 0 −1.00000 + 1.73205i 1.93649 1.11803i 0 1.41421 + 2.23607i 2.82843 1.50000 + 2.59808i −2.73861 1.58114i
59.3 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −1.93649 + 1.11803i 0 −1.41421 2.23607i −2.82843 1.50000 + 2.59808i −2.73861 1.58114i
59.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 1.93649 1.11803i 0 −1.41421 + 2.23607i −2.82843 1.50000 + 2.59808i 2.73861 + 1.58114i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 59.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
5.b even 2 1 inner
7.d odd 6 1 inner
8.d odd 2 1 inner
35.i odd 6 1 inner
56.m even 6 1 inner
280.ba even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.ba.a 8
4.b odd 2 1 1120.2.bq.a 8
5.b even 2 1 inner 280.2.ba.a 8
7.d odd 6 1 inner 280.2.ba.a 8
8.b even 2 1 1120.2.bq.a 8
8.d odd 2 1 inner 280.2.ba.a 8
20.d odd 2 1 1120.2.bq.a 8
28.f even 6 1 1120.2.bq.a 8
35.i odd 6 1 inner 280.2.ba.a 8
40.e odd 2 1 CM 280.2.ba.a 8
40.f even 2 1 1120.2.bq.a 8
56.j odd 6 1 1120.2.bq.a 8
56.m even 6 1 inner 280.2.ba.a 8
140.s even 6 1 1120.2.bq.a 8
280.ba even 6 1 inner 280.2.ba.a 8
280.bk odd 6 1 1120.2.bq.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.ba.a 8 1.a even 1 1 trivial
280.2.ba.a 8 5.b even 2 1 inner
280.2.ba.a 8 7.d odd 6 1 inner
280.2.ba.a 8 8.d odd 2 1 inner
280.2.ba.a 8 35.i odd 6 1 inner
280.2.ba.a 8 40.e odd 2 1 CM
280.2.ba.a 8 56.m even 6 1 inner
280.2.ba.a 8 280.ba even 6 1 inner
1120.2.bq.a 8 4.b odd 2 1
1120.2.bq.a 8 8.b even 2 1
1120.2.bq.a 8 20.d odd 2 1
1120.2.bq.a 8 28.f even 6 1
1120.2.bq.a 8 40.f even 2 1
1120.2.bq.a 8 56.j odd 6 1
1120.2.bq.a 8 140.s even 6 1
1120.2.bq.a 8 280.bk odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 25 - 5 T^{2} + T^{4} )^{2}$$
$7$ $$( 49 + 6 T^{2} + T^{4} )^{2}$$
$11$ $$( 841 - 58 T + 33 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$13$ $$( 361 + 58 T^{2} + T^{4} )^{2}$$
$17$ $$T^{8}$$
$19$ $$( 289 - 306 T + 125 T^{2} - 18 T^{3} + T^{4} )^{2}$$
$23$ $$81 + 594 T^{2} + 4347 T^{4} + 66 T^{6} + T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$83521 + 27166 T^{2} + 8547 T^{4} + 94 T^{6} + T^{8}$$
$41$ $$( 1369 + 86 T^{2} + T^{4} )^{2}$$
$43$ $$T^{8}$$
$47$ $$2313441 - 155142 T^{2} + 8883 T^{4} - 102 T^{6} + T^{8}$$
$53$ $$260144641 + 4612894 T^{2} + 65667 T^{4} + 286 T^{6} + T^{8}$$
$59$ $$( 1600 - 40 T^{2} + T^{4} )^{2}$$
$61$ $$T^{8}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$( 25600 - 160 T^{2} + T^{4} )^{2}$$
$97$ $$T^{8}$$