# Properties

 Label 260.2.bj.a Level $260$ Weight $2$ Character orbit 260.bj 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.bj (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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{2} - 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + 3 \zeta_{12} q^{9} +O(q^{10})$$ q + (z^3 + z^2 - z) * q^2 - 2*z * q^4 + (-z^3 + 2*z^2 + z) * q^5 + (-2*z^3 + 2) * q^8 + 3*z * q^9 $$q + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{2} - 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + 3 \zeta_{12} q^{9} + (3 \zeta_{12}^{2} - \zeta_{12} - 3) q^{10} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 2 \zeta_{12}) q^{13} + 4 \zeta_{12}^{2} q^{16} + (5 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - \zeta_{12} - 1) q^{17} + (3 \zeta_{12}^{3} - 3) q^{18} + ( - 4 \zeta_{12}^{3} - 2) q^{20} + (3 \zeta_{12}^{2} + 4 \zeta_{12} - 3) q^{25} + (\zeta_{12}^{2} - 5 \zeta_{12} - 1) q^{26} + (2 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 2 \zeta_{12} + 10) q^{29} + (4 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{32} + (3 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 5) q^{34} - 6 \zeta_{12}^{2} q^{36} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 6 \zeta_{12} - 5) q^{37} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 6 \zeta_{12}) q^{40} + ( - 5 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 5 \zeta_{12}) q^{41} + (6 \zeta_{12}^{3} + 3) q^{45} + ( - 7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{49} + (\zeta_{12}^{3} - 7) q^{50} + ( - 6 \zeta_{12}^{3} + 4) q^{52} + (2 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} - 2) q^{53} + (10 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 7 \zeta_{12} + 7) q^{58} + ( - 10 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 5 \zeta_{12} + 6) q^{61} - 8 \zeta_{12}^{3} q^{64} + (8 \zeta_{12}^{2} - \zeta_{12} - 8) q^{65} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 2 \zeta_{12} + 10) q^{68} + ( - 6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{72} + ( - 8 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} + 8) q^{73} + ( - 5 \zeta_{12}^{2} + 7 \zeta_{12} - 5) q^{74} + (8 \zeta_{12}^{2} + 4 \zeta_{12} - 8) q^{80} + 9 \zeta_{12}^{2} q^{81} + ( - 10 \zeta_{12}^{3} + \zeta_{12}^{2} + 9 \zeta_{12} + 9) q^{82} + (9 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 15 \zeta_{12} + 7) q^{85} + (16 \zeta_{12}^{3} - 16 \zeta_{12}) q^{89} + (9 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 9 \zeta_{12}) q^{90} + (13 \zeta_{12}^{2} - 13 \zeta_{12} - 13) q^{97} + (7 \zeta_{12}^{2} + 7 \zeta_{12} - 7) q^{98} +O(q^{100})$$ q + (z^3 + z^2 - z) * q^2 - 2*z * q^4 + (-z^3 + 2*z^2 + z) * q^5 + (-2*z^3 + 2) * q^8 + 3*z * q^9 + (3*z^2 - z - 3) * q^10 + (2*z^3 + 3*z^2 - 2*z) * q^13 + 4*z^2 * q^16 + (5*z^3 - 4*z^2 - z - 1) * q^17 + (3*z^3 - 3) * q^18 + (-4*z^3 - 2) * q^20 + (3*z^2 + 4*z - 3) * q^25 + (z^2 - 5*z - 1) * q^26 + (2*z^3 - 5*z^2 - 2*z + 10) * q^29 + (4*z^2 - 4*z - 4) * q^32 + (3*z^3 - 10*z^2 + 5) * q^34 - 6*z^2 * q^36 + (-z^3 - z^2 + 6*z - 5) * q^37 + (-6*z^3 + 2*z^2 + 6*z) * q^40 + (-5*z^3 - 4*z^2 - 5*z) * q^41 + (6*z^3 + 3) * q^45 + (-7*z^3 + 7*z) * q^49 + (z^3 - 7) * q^50 + (-6*z^3 + 4) * q^52 + (2*z^3 - 5*z^2 + 5*z - 2) * q^53 + (10*z^3 + 3*z^2 - 7*z + 7) * q^58 + (-10*z^3 - 6*z^2 + 5*z + 6) * q^61 - 8*z^3 * q^64 + (8*z^2 - z - 8) * q^65 + (8*z^3 - 8*z^2 + 2*z + 10) * q^68 + (-6*z^2 + 6*z + 6) * q^72 + (-8*z^3 - 5*z^2 + 5*z + 8) * q^73 + (-5*z^2 + 7*z - 5) * q^74 + (8*z^2 + 4*z - 8) * q^80 + 9*z^2 * q^81 + (-10*z^3 + z^2 + 9*z + 9) * q^82 + (9*z^3 - 5*z^2 - 15*z + 7) * q^85 + (16*z^3 - 16*z) * q^89 + (9*z^3 - 3*z^2 - 9*z) * q^90 + (13*z^2 - 13*z - 13) * q^97 + (7*z^2 + 7*z - 7) * 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 $$4 q + 2 q^{2} + 4 q^{5} + 8 q^{8} - 6 q^{10} + 6 q^{13} + 8 q^{16} - 12 q^{17} - 12 q^{18} - 8 q^{20} - 6 q^{25} - 2 q^{26} + 30 q^{29} - 8 q^{32} - 12 q^{36} - 22 q^{37} + 4 q^{40} - 8 q^{41} + 12 q^{45} - 28 q^{50} + 16 q^{52} - 18 q^{53} + 34 q^{58} + 12 q^{61} - 16 q^{65} + 24 q^{68} + 12 q^{72} + 22 q^{73} - 30 q^{74} - 16 q^{80} + 18 q^{81} + 38 q^{82} + 18 q^{85} - 6 q^{90} - 26 q^{97} - 14 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 + 4 * q^5 + 8 * q^8 - 6 * q^10 + 6 * q^13 + 8 * q^16 - 12 * q^17 - 12 * q^18 - 8 * q^20 - 6 * q^25 - 2 * q^26 + 30 * q^29 - 8 * q^32 - 12 * q^36 - 22 * q^37 + 4 * q^40 - 8 * q^41 + 12 * q^45 - 28 * q^50 + 16 * q^52 - 18 * q^53 + 34 * q^58 + 12 * q^61 - 16 * q^65 + 24 * q^68 + 12 * q^72 + 22 * q^73 - 30 * q^74 - 16 * q^80 + 18 * q^81 + 38 * q^82 + 18 * q^85 - 6 * q^90 - 26 * q^97 - 14 * q^98

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i
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
87.1 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
107.1 −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
243.1 −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.q odd 12 1 inner
260.bj 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.bj.a 4
4.b odd 2 1 CM 260.2.bj.a 4
5.c odd 4 1 260.2.bj.b yes 4
13.c even 3 1 260.2.bj.b yes 4
20.e even 4 1 260.2.bj.b yes 4
52.j odd 6 1 260.2.bj.b yes 4
65.q odd 12 1 inner 260.2.bj.a 4
260.bj even 12 1 inner 260.2.bj.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.bj.a 4 1.a even 1 1 trivial
260.2.bj.a 4 4.b odd 2 1 CM
260.2.bj.a 4 65.q odd 12 1 inner
260.2.bj.a 4 260.bj even 12 1 inner
260.2.bj.b yes 4 5.c odd 4 1
260.2.bj.b yes 4 13.c even 3 1
260.2.bj.b yes 4 20.e even 4 1
260.2.bj.b yes 4 52.j odd 6 1

## 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 $$T_{17}^{4} + 12T_{17}^{3} + 117T_{17}^{2} + 594T_{17} + 1089$$ T17^4 + 12*T17^3 + 117*T17^2 + 594*T17 + 1089

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 4 T^{3} + 11 T^{2} - 20 T + 25$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 6 T^{3} + 23 T^{2} - 78 T + 169$$
$17$ $$T^{4} + 12 T^{3} + 117 T^{2} + \cdots + 1089$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4} - 30 T^{3} + 371 T^{2} + \cdots + 5041$$
$31$ $$T^{4}$$
$37$ $$T^{4} + 22 T^{3} + 137 T^{2} + \cdots + 169$$
$41$ $$T^{4} + 8 T^{3} + 123 T^{2} + \cdots + 3481$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 18 T^{3} + 162 T^{2} + 54 T + 9$$
$59$ $$T^{4}$$
$61$ $$T^{4} - 12 T^{3} + 183 T^{2} + \cdots + 1521$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} - 22 T^{3} + 242 T^{2} + \cdots + 529$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4} - 256 T^{2} + 65536$$
$97$ $$T^{4} + 26 T^{3} + 338 T^{2} + \cdots + 114244$$