# Properties

 Label 260.2.u.a Level $260$ Weight $2$ Character orbit 260.u Analytic conductor $2.076$ Analytic rank $1$ Dimension $2$ 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.u (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07611045255$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (i - 1) q^{2} - 2 i q^{4} + (i - 2) q^{5} + (2 i + 2) q^{8} - 3 q^{9} +O(q^{10})$$ q + (i - 1) * q^2 - 2*i * q^4 + (i - 2) * q^5 + (2*i + 2) * q^8 - 3 * q^9 $$q + (i - 1) q^{2} - 2 i q^{4} + (i - 2) q^{5} + (2 i + 2) q^{8} - 3 q^{9} + ( - 3 i + 1) q^{10} + ( - 3 i - 2) q^{13} - 4 q^{16} - 8 q^{17} + ( - 3 i + 3) q^{18} + (4 i + 2) q^{20} + ( - 4 i + 3) q^{25} + (i + 5) q^{26} - 4 q^{29} + ( - 4 i + 4) q^{32} + ( - 8 i + 8) q^{34} + 6 i q^{36} + ( - 5 i - 5) q^{37} + ( - 2 i - 6) q^{40} + (9 i + 9) q^{41} + ( - 3 i + 6) q^{45} + 7 i q^{49} + (7 i + 1) q^{50} + (4 i - 6) q^{52} + 4 i q^{53} + ( - 4 i + 4) q^{58} - 10 q^{61} + 8 i q^{64} + (4 i + 7) q^{65} + 16 i q^{68} + ( - 6 i - 6) q^{72} + (11 i + 11) q^{73} + 10 q^{74} + ( - 4 i + 8) q^{80} + 9 q^{81} - 18 q^{82} + ( - 8 i + 16) q^{85} + (13 i - 13) q^{89} + (9 i - 3) q^{90} + ( - 5 i + 5) q^{97} + ( - 7 i - 7) q^{98} +O(q^{100})$$ q + (i - 1) * q^2 - 2*i * q^4 + (i - 2) * q^5 + (2*i + 2) * q^8 - 3 * q^9 + (-3*i + 1) * q^10 + (-3*i - 2) * q^13 - 4 * q^16 - 8 * q^17 + (-3*i + 3) * q^18 + (4*i + 2) * q^20 + (-4*i + 3) * q^25 + (i + 5) * q^26 - 4 * q^29 + (-4*i + 4) * q^32 + (-8*i + 8) * q^34 + 6*i * q^36 + (-5*i - 5) * q^37 + (-2*i - 6) * q^40 + (9*i + 9) * q^41 + (-3*i + 6) * q^45 + 7*i * q^49 + (7*i + 1) * q^50 + (4*i - 6) * q^52 + 4*i * q^53 + (-4*i + 4) * q^58 - 10 * q^61 + 8*i * q^64 + (4*i + 7) * q^65 + 16*i * q^68 + (-6*i - 6) * q^72 + (11*i + 11) * q^73 + 10 * q^74 + (-4*i + 8) * q^80 + 9 * q^81 - 18 * q^82 + (-8*i + 16) * q^85 + (13*i - 13) * q^89 + (9*i - 3) * q^90 + (-5*i + 5) * q^97 + (-7*i - 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{5} + 4 q^{8} - 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^5 + 4 * q^8 - 6 * q^9 $$2 q - 2 q^{2} - 4 q^{5} + 4 q^{8} - 6 q^{9} + 2 q^{10} - 4 q^{13} - 8 q^{16} - 16 q^{17} + 6 q^{18} + 4 q^{20} + 6 q^{25} + 10 q^{26} - 8 q^{29} + 8 q^{32} + 16 q^{34} - 10 q^{37} - 12 q^{40} + 18 q^{41} + 12 q^{45} + 2 q^{50} - 12 q^{52} + 8 q^{58} - 20 q^{61} + 14 q^{65} - 12 q^{72} + 22 q^{73} + 20 q^{74} + 16 q^{80} + 18 q^{81} - 36 q^{82} + 32 q^{85} - 26 q^{89} - 6 q^{90} + 10 q^{97} - 14 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^5 + 4 * q^8 - 6 * q^9 + 2 * q^10 - 4 * q^13 - 8 * q^16 - 16 * q^17 + 6 * q^18 + 4 * q^20 + 6 * q^25 + 10 * q^26 - 8 * q^29 + 8 * q^32 + 16 * q^34 - 10 * q^37 - 12 * q^40 + 18 * q^41 + 12 * q^45 + 2 * q^50 - 12 * q^52 + 8 * q^58 - 20 * q^61 + 14 * q^65 - 12 * q^72 + 22 * q^73 + 20 * q^74 + 16 * q^80 + 18 * q^81 - 36 * q^82 + 32 * q^85 - 26 * q^89 - 6 * q^90 + 10 * q^97 - 14 * q^98

## Character values

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

 $$n$$ $$41$$ $$131$$ $$157$$ $$\chi(n)$$ $$i$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 1.00000i − 1.00000i
−1.00000 + 1.00000i 0 2.00000i −2.00000 + 1.00000i 0 0 2.00000 + 2.00000i −3.00000 1.00000 3.00000i
239.1 −1.00000 1.00000i 0 2.00000i −2.00000 1.00000i 0 0 2.00000 2.00000i −3.00000 1.00000 + 3.00000i
 $$n$$: 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.g odd 4 1 inner
260.u even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.u.a 2
4.b odd 2 1 CM 260.2.u.a 2
5.b even 2 1 260.2.u.b yes 2
13.d odd 4 1 260.2.u.b yes 2
20.d odd 2 1 260.2.u.b yes 2
52.f even 4 1 260.2.u.b yes 2
65.g odd 4 1 inner 260.2.u.a 2
260.u even 4 1 inner 260.2.u.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.u.a 2 1.a even 1 1 trivial
260.2.u.a 2 4.b odd 2 1 CM
260.2.u.a 2 65.g odd 4 1 inner
260.2.u.a 2 260.u even 4 1 inner
260.2.u.b yes 2 5.b even 2 1
260.2.u.b yes 2 13.d odd 4 1
260.2.u.b yes 2 20.d odd 2 1
260.2.u.b yes 2 52.f even 4 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 $$T_{17} + 8$$ T17 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4T + 13$$
$17$ $$(T + 8)^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T + 4)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 10T + 50$$
$41$ $$T^{2} - 18T + 162$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 16$$
$59$ $$T^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 22T + 242$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 26T + 338$$
$97$ $$T^{2} - 10T + 50$$