# 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: $$x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} - 2 \beta_{3} q^{4} + \beta_{6} q^{5} + ( - \beta_{6} + \beta_{2} + \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - 3 \beta_{3} + 3) q^{9}+O(q^{10})$$ q + b4 * q^2 - 2*b3 * q^4 + b6 * q^5 + (-b6 + b2 + b1) * q^7 + 2*b2 * q^8 + (-3*b3 + 3) * q^9 $$q + \beta_{4} q^{2} - 2 \beta_{3} q^{4} + \beta_{6} q^{5} + ( - \beta_{6} + \beta_{2} + \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - 3 \beta_{3} + 3) q^{9} - \beta_{5} q^{10} + (2 \beta_{7} + \beta_{5} - \beta_{3}) q^{11} + ( - \beta_{6} + 4 \beta_{4} + 2 \beta_{2} + \beta_1) q^{13} + ( - \beta_{7} + 2 \beta_{3} - 2) q^{14} + (4 \beta_{3} - 4) q^{16} + (3 \beta_{4} + 3 \beta_{2}) q^{18} + ( - \beta_{7} + 3 \beta_{3} + 3) q^{19} + ( - 2 \beta_{6} + 2 \beta_1) q^{20} + (2 \beta_{6} + \beta_{2} + 2 \beta_1) q^{22} + (\beta_{6} - 3 \beta_{4} - 2 \beta_1) q^{23} + 5 \beta_{3} q^{25} + ( - \beta_{7} - 4 \beta_{3} - 4) q^{26} + ( - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{28} + ( - 4 \beta_{4} - 4 \beta_{2}) q^{32} + ( - \beta_{7} - 5 \beta_{3} + 5) q^{35} - 6 q^{36} + (\beta_{6} + 4 \beta_{4} - 2 \beta_1) q^{37} + (3 \beta_{4} - 3 \beta_{2} - 2 \beta_1) q^{38} - 2 \beta_{7} q^{40} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} - 1) q^{41} + ( - 2 \beta_{7} - 4 \beta_{5} + 2 \beta_{3} - 2) q^{44} + 3 \beta_1 q^{45} + (2 \beta_{7} + \beta_{5} + 6 \beta_{3}) q^{46} + ( - 3 \beta_{6} - \beta_{4} - 2 \beta_{2}) q^{47} + (2 \beta_{7} + 2 \beta_{5} - 3) q^{49} - 5 \beta_{2} q^{50} + ( - 4 \beta_{4} + 4 \beta_{2} - 2 \beta_1) q^{52} + (6 \beta_{6} - 2 \beta_{4} - 2 \beta_{2} - 3 \beta_1) q^{53} + ( - \beta_{6} - 10 \beta_{4} - 5 \beta_{2} + \beta_1) q^{55} + (2 \beta_{7} + 2 \beta_{5} + 4) q^{56} - 2 \beta_{5} q^{59} + ( - 3 \beta_{6} - 3 \beta_{4}) q^{63} + 8 q^{64} + ( - 2 \beta_{7} - 4 \beta_{5} - 5 \beta_{3} + 5) q^{65} + (5 \beta_{4} + 5 \beta_{2} - 2 \beta_1) q^{70} - 6 \beta_{4} q^{72} + (2 \beta_{7} + \beta_{5} - 8 \beta_{3}) q^{74} + (2 \beta_{7} + 2 \beta_{5} - 12 \beta_{3} + 6) q^{76} + ( - 4 \beta_{6} + 4 \beta_{4} - 6 \beta_{2} + \beta_1) q^{77} - 4 \beta_1 q^{80} - 9 \beta_{3} q^{81} + (4 \beta_{6} - \beta_{4} - 2 \beta_{2}) q^{82} + ( - 8 \beta_{6} - 2 \beta_{4} - 2 \beta_{2} + 4 \beta_1) q^{88} - 4 \beta_{7} q^{89} + ( - 3 \beta_{7} - 3 \beta_{5}) q^{90} + ( - \beta_{7} + 3 \beta_{5} + 8 \beta_{3} - 9) q^{91} + (2 \beta_{6} - 6 \beta_{2} + 2 \beta_1) q^{92} + (3 \beta_{5} - 2 \beta_{3} + 4) q^{94} + (6 \beta_{6} + 5 \beta_{4} + 5 \beta_{2} - 3 \beta_1) q^{95} + (4 \beta_{6} - 3 \beta_{4}) q^{98} + (3 \beta_{7} - 3 \beta_{5} - 3) q^{99}+O(q^{100})$$ q + b4 * q^2 - 2*b3 * q^4 + b6 * q^5 + (-b6 + b2 + b1) * q^7 + 2*b2 * q^8 + (-3*b3 + 3) * q^9 - b5 * q^10 + (2*b7 + b5 - b3) * q^11 + (-b6 + 4*b4 + 2*b2 + b1) * q^13 + (-b7 + 2*b3 - 2) * q^14 + (4*b3 - 4) * q^16 + (3*b4 + 3*b2) * q^18 + (-b7 + 3*b3 + 3) * q^19 + (-2*b6 + 2*b1) * q^20 + (2*b6 + b2 + 2*b1) * q^22 + (b6 - 3*b4 - 2*b1) * q^23 + 5*b3 * q^25 + (-b7 - 4*b3 - 4) * q^26 + (-2*b4 - 2*b2 - 2*b1) * q^28 + (-4*b4 - 4*b2) * q^32 + (-b7 - 5*b3 + 5) * q^35 - 6 * q^36 + (b6 + 4*b4 - 2*b1) * q^37 + (3*b4 - 3*b2 - 2*b1) * q^38 - 2*b7 * q^40 + (2*b7 + 2*b5 + 2*b3 - 1) * q^41 + (-2*b7 - 4*b5 + 2*b3 - 2) * q^44 + 3*b1 * q^45 + (2*b7 + b5 + 6*b3) * q^46 + (-3*b6 - b4 - 2*b2) * q^47 + (2*b7 + 2*b5 - 3) * q^49 - 5*b2 * q^50 + (-4*b4 + 4*b2 - 2*b1) * q^52 + (6*b6 - 2*b4 - 2*b2 - 3*b1) * q^53 + (-b6 - 10*b4 - 5*b2 + b1) * q^55 + (2*b7 + 2*b5 + 4) * q^56 - 2*b5 * q^59 + (-3*b6 - 3*b4) * q^63 + 8 * q^64 + (-2*b7 - 4*b5 - 5*b3 + 5) * q^65 + (5*b4 + 5*b2 - 2*b1) * q^70 - 6*b4 * q^72 + (2*b7 + b5 - 8*b3) * q^74 + (2*b7 + 2*b5 - 12*b3 + 6) * q^76 + (-4*b6 + 4*b4 - 6*b2 + b1) * q^77 - 4*b1 * q^80 - 9*b3 * q^81 + (4*b6 - b4 - 2*b2) * q^82 + (-8*b6 - 2*b4 - 2*b2 + 4*b1) * q^88 - 4*b7 * q^89 + (-3*b7 - 3*b5) * q^90 + (-b7 + 3*b5 + 8*b3 - 9) * q^91 + (2*b6 - 6*b2 + 2*b1) * q^92 + (3*b5 - 2*b3 + 4) * q^94 + (6*b6 + 5*b4 + 5*b2 - 3*b1) * q^95 + (4*b6 - 3*b4) * q^98 + (3*b7 - 3*b5 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4} + 12 q^{9}+O(q^{10})$$ 8 * q - 8 * q^4 + 12 * q^9 $$8 q - 8 q^{4} + 12 q^{9} - 4 q^{11} - 8 q^{14} - 16 q^{16} + 36 q^{19} + 20 q^{25} - 48 q^{26} + 20 q^{35} - 48 q^{36} - 8 q^{44} + 24 q^{46} - 24 q^{49} + 32 q^{56} + 64 q^{64} + 20 q^{65} - 32 q^{74} - 36 q^{81} - 40 q^{91} + 24 q^{94} - 24 q^{99}+O(q^{100})$$ 8 * q - 8 * q^4 + 12 * q^9 - 4 * q^11 - 8 * q^14 - 16 * q^16 + 36 * q^19 + 20 * q^25 - 48 * q^26 + 20 * q^35 - 48 * q^36 - 8 * q^44 + 24 * q^46 - 24 * q^49 + 32 * q^56 + 64 * q^64 + 20 * q^65 - 32 * q^74 - 36 * q^81 - 40 * q^91 + 24 * q^94 - 24 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{6} + 14\nu^{4} + 7\nu^{2} - 36 ) / 63$$ (v^6 + 14*v^4 + 7*v^2 - 36) / 63 $$\beta_{2}$$ $$=$$ $$( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189$$ (-4*v^7 + 7*v^5 + 35*v^3 + 81*v) / 189 $$\beta_{3}$$ $$=$$ $$( -4\nu^{6} + 7\nu^{4} - 28\nu^{2} + 144 ) / 63$$ (-4*v^6 + 7*v^4 - 28*v^2 + 144) / 63 $$\beta_{4}$$ $$=$$ $$( -5\nu^{7} - 7\nu^{5} - 35\nu^{3} + 180\nu ) / 189$$ (-5*v^7 - 7*v^5 - 35*v^3 + 180*v) / 189 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + 13\nu ) / 21$$ (v^7 + 13*v) / 21 $$\beta_{6}$$ $$=$$ $$( -8\nu^{6} + 14\nu^{4} + 7\nu^{2} + 162 ) / 63$$ (-8*v^6 + 14*v^4 + 7*v^2 + 162) / 63 $$\beta_{7}$$ $$=$$ $$( 19\nu^{7} - 49\nu^{5} + 133\nu^{3} - 684\nu ) / 189$$ (19*v^7 - 49*v^5 + 133*v^3 - 684*v) / 189
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} + \beta_{2} ) / 2$$ (b5 + b4 + b2) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{6} - 2\beta_{3} + 2$$ b6 - 2*b3 + 2 $$\nu^{3}$$ $$=$$ $$( \beta_{7} + \beta_{5} + 7\beta_{2} ) / 2$$ (b7 + b5 + 7*b2) / 2 $$\nu^{4}$$ $$=$$ $$\beta_{3} + 4\beta_1$$ b3 + 4*b1 $$\nu^{5}$$ $$=$$ $$( -5\beta_{7} - 19\beta_{4} ) / 2$$ (-5*b7 - 19*b4) / 2 $$\nu^{6}$$ $$=$$ $$-7\beta_{6} + 7\beta _1 + 22$$ -7*b6 + 7*b1 + 22 $$\nu^{7}$$ $$=$$ $$( 29\beta_{5} - 13\beta_{4} - 13\beta_{2} ) / 2$$ (29*b5 - 13*b4 - 13*b2) / 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

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