# Properties

 Label 260.2.bg.b Level $260$ Weight $2$ Character orbit 260.bg Analytic conductor $2.076$ Analytic rank $0$ Dimension $4$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$260 = 2^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 260.bg (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07611045255$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{4} + (2 \zeta_{12}^{2} - \zeta_{12} - 2) q^{5} + (2 \zeta_{12}^{3} + 2) q^{8} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{9}+O(q^{10})$$ q + (-z^2 - z + 1) * q^2 + (2*z^3 - 2*z) * q^4 + (2*z^2 - z - 2) * q^5 + (2*z^3 + 2) * q^8 + (3*z^3 - 3*z) * q^9 $$q + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{4} + (2 \zeta_{12}^{2} - \zeta_{12} - 2) q^{5} + (2 \zeta_{12}^{3} + 2) q^{8} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{9} + ( - \zeta_{12}^{3} + 3 \zeta_{12}^{2} + \zeta_{12}) q^{10} + (3 \zeta_{12}^{2} + 2 \zeta_{12} - 3) q^{13} + ( - 4 \zeta_{12}^{2} + 4) q^{16} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - \zeta_{12} - 3) q^{17} + (3 \zeta_{12}^{3} + 3) q^{18} + ( - 4 \zeta_{12}^{3} + 2) q^{20} + ( - 4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 4 \zeta_{12}) q^{25} + ( - 5 \zeta_{12}^{3} + \zeta_{12}^{2} + 5 \zeta_{12}) q^{26} + ( - 5 \zeta_{12}^{2} + 2 \zeta_{12} - 5) q^{29} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{32} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12} + 5) q^{34} + ( - 6 \zeta_{12}^{2} + 6) q^{36} + ( - 5 \zeta_{12}^{3} + \zeta_{12}^{2} + 6 \zeta_{12} - 6) q^{37} + (2 \zeta_{12}^{2} - 6 \zeta_{12} - 2) q^{40} + (4 \zeta_{12}^{2} - 5 \zeta_{12} + 4) q^{41} + ( - 6 \zeta_{12}^{3} + 3) q^{45} + 7 \zeta_{12} q^{49} + ( - \zeta_{12}^{3} - 7) q^{50} + ( - 6 \zeta_{12}^{3} - 4) q^{52} + (7 \zeta_{12}^{3} + 9 \zeta_{12}^{2} - 9 \zeta_{12} - 7) q^{53} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 7 \zeta_{12} - 10) q^{58} + ( - 5 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 5 \zeta_{12}) q^{61} + 8 \zeta_{12}^{3} q^{64} + (\zeta_{12}^{3} - 8 \zeta_{12}^{2} - \zeta_{12}) q^{65} + ( - 6 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{68} + (6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12}) q^{72} + (3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} + 3) q^{73} + ( - 7 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 7 \zeta_{12} - 10) q^{74} + (4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4 \zeta_{12}) q^{80} + ( - 9 \zeta_{12}^{2} + 9) q^{81} + (\zeta_{12}^{3} + \zeta_{12}^{2} - 9 \zeta_{12} + 8) q^{82} + ( - 6 \zeta_{12}^{3} - 9 \zeta_{12}^{2} - 3 \zeta_{12} + 2) q^{85} + ( - 10 \zeta_{12}^{2} + 10) q^{89} + (3 \zeta_{12}^{2} - 9 \zeta_{12} - 3) q^{90} + ( - 13 \zeta_{12}^{3} + 13 \zeta_{12}^{2} + 13 \zeta_{12}) q^{97} + ( - 7 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 7 \zeta_{12}) q^{98}+O(q^{100})$$ q + (-z^2 - z + 1) * q^2 + (2*z^3 - 2*z) * q^4 + (2*z^2 - z - 2) * q^5 + (2*z^3 + 2) * q^8 + (3*z^3 - 3*z) * q^9 + (-z^3 + 3*z^2 + z) * q^10 + (3*z^2 + 2*z - 3) * q^13 + (-4*z^2 + 4) * q^16 + (4*z^3 + 4*z^2 - z - 3) * q^17 + (3*z^3 + 3) * q^18 + (-4*z^3 + 2) * q^20 + (-4*z^3 - 3*z^2 + 4*z) * q^25 + (-5*z^3 + z^2 + 5*z) * q^26 + (-5*z^2 + 2*z - 5) * q^29 + (4*z^3 - 4*z^2 - 4*z) * q^32 + (-3*z^3 + 6*z + 5) * q^34 + (-6*z^2 + 6) * q^36 + (-5*z^3 + z^2 + 6*z - 6) * q^37 + (2*z^2 - 6*z - 2) * q^40 + (4*z^2 - 5*z + 4) * q^41 + (-6*z^3 + 3) * q^45 + 7*z * q^49 + (-z^3 - 7) * q^50 + (-6*z^3 - 4) * q^52 + (7*z^3 + 9*z^2 - 9*z - 7) * q^53 + (3*z^3 + 3*z^2 + 7*z - 10) * q^58 + (-5*z^3 - 6*z^2 - 5*z) * q^61 + 8*z^3 * q^64 + (z^3 - 8*z^2 - z) * q^65 + (-6*z^3 - 8*z^2 - 2*z + 2) * q^68 + (6*z^3 - 6*z^2 - 6*z) * q^72 + (3*z^3 + 5*z^2 + 5*z + 3) * q^73 + (-7*z^3 + 5*z^2 + 7*z - 10) * q^74 + (4*z^3 + 8*z^2 - 4*z) * q^80 + (-9*z^2 + 9) * q^81 + (z^3 + z^2 - 9*z + 8) * q^82 + (-6*z^3 - 9*z^2 - 3*z + 2) * q^85 + (-10*z^2 + 10) * q^89 + (3*z^2 - 9*z - 3) * q^90 + (-13*z^3 + 13*z^2 + 13*z) * q^97 + (-7*z^3 - 7*z^2 + 7*z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 4 q^{5} + 8 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 4 * q^5 + 8 * q^8 $$4 q + 2 q^{2} - 4 q^{5} + 8 q^{8} + 6 q^{10} - 6 q^{13} + 8 q^{16} - 4 q^{17} + 12 q^{18} + 8 q^{20} - 6 q^{25} + 2 q^{26} - 30 q^{29} - 8 q^{32} + 20 q^{34} + 12 q^{36} - 22 q^{37} - 4 q^{40} + 24 q^{41} + 12 q^{45} - 28 q^{50} - 16 q^{52} - 10 q^{53} - 34 q^{58} - 12 q^{61} - 16 q^{65} - 8 q^{68} - 12 q^{72} + 22 q^{73} - 30 q^{74} + 16 q^{80} + 18 q^{81} + 34 q^{82} - 10 q^{85} + 20 q^{89} - 6 q^{90} + 26 q^{97} - 14 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 - 4 * q^5 + 8 * q^8 + 6 * q^10 - 6 * q^13 + 8 * q^16 - 4 * q^17 + 12 * q^18 + 8 * q^20 - 6 * q^25 + 2 * q^26 - 30 * q^29 - 8 * q^32 + 20 * q^34 + 12 * q^36 - 22 * q^37 - 4 * q^40 + 24 * q^41 + 12 * q^45 - 28 * q^50 - 16 * q^52 - 10 * q^53 - 34 * q^58 - 12 * q^61 - 16 * q^65 - 8 * q^68 - 12 * q^72 + 22 * q^73 - 30 * q^74 + 16 * q^80 + 18 * q^81 + 34 * q^82 - 10 * q^85 + 20 * q^89 - 6 * q^90 + 26 * q^97 - 14 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/260\mathbb{Z}\right)^\times$$.

 $$n$$ $$41$$ $$131$$ $$157$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$-1$$ $$\zeta_{12}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 0.866025 − 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
−0.366025 + 1.36603i 0 −1.73205 1.00000i −1.86603 1.23205i 0 0 2.00000 2.00000i −2.59808 1.50000i 2.36603 2.09808i
43.1 1.36603 0.366025i 0 1.73205 1.00000i −0.133975 + 2.23205i 0 0 2.00000 2.00000i 2.59808 1.50000i 0.633975 + 3.09808i
127.1 1.36603 + 0.366025i 0 1.73205 + 1.00000i −0.133975 2.23205i 0 0 2.00000 + 2.00000i 2.59808 + 1.50000i 0.633975 3.09808i
147.1 −0.366025 1.36603i 0 −1.73205 + 1.00000i −1.86603 + 1.23205i 0 0 2.00000 + 2.00000i −2.59808 + 1.50000i 2.36603 + 2.09808i
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
65.r odd 12 1 inner
260.bg even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.bg.b yes 4
4.b odd 2 1 CM 260.2.bg.b yes 4
5.c odd 4 1 260.2.bg.a 4
13.e even 6 1 260.2.bg.a 4
20.e even 4 1 260.2.bg.a 4
52.i odd 6 1 260.2.bg.a 4
65.r odd 12 1 inner 260.2.bg.b yes 4
260.bg even 12 1 inner 260.2.bg.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.bg.a 4 5.c odd 4 1
260.2.bg.a 4 13.e even 6 1
260.2.bg.a 4 20.e even 4 1
260.2.bg.a 4 52.i odd 6 1
260.2.bg.b yes 4 1.a even 1 1 trivial
260.2.bg.b yes 4 4.b odd 2 1 CM
260.2.bg.b yes 4 65.r odd 12 1 inner
260.2.bg.b yes 4 260.bg even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(260, [\chi])$$:

 $$T_{3}$$ T3 $$T_{17}^{4} + 4T_{17}^{3} + 53T_{17}^{2} + 14T_{17} + 1$$ T17^4 + 4*T17^3 + 53*T17^2 + 14*T17 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 4 T^{3} + 11 T^{2} + 20 T + 25$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 6 T^{3} + 23 T^{2} + 78 T + 169$$
$17$ $$T^{4} + 4 T^{3} + 53 T^{2} + 14 T + 1$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4} + 30 T^{3} + 371 T^{2} + \cdots + 5041$$
$31$ $$T^{4}$$
$37$ $$T^{4} + 22 T^{3} + 137 T^{2} + \cdots + 169$$
$41$ $$T^{4} - 24 T^{3} + 215 T^{2} + \cdots + 529$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 10 T^{3} + 50 T^{2} + \cdots + 11881$$
$59$ $$T^{4}$$
$61$ $$T^{4} + 12 T^{3} + 183 T^{2} + \cdots + 1521$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} - 22 T^{3} + 242 T^{2} + \cdots + 529$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$(T^{2} - 10 T + 100)^{2}$$
$97$ $$T^{4} - 26 T^{3} + 338 T^{2} + \cdots + 114244$$