# Properties

 Label 260.2.p.a Level $260$ Weight $2$ Character orbit 260.p Analytic conductor $2.076$ Analytic rank $0$ 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.p (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07611045255$$ Analytic rank: $$0$$ 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.

 $$f(q)$$ $$=$$ $$q + (i - 1) q^{2} - 2 i q^{4} + (i + 2) q^{5} + (2 i + 2) q^{8} - 3 i q^{9} +O(q^{10})$$ q + (i - 1) * q^2 - 2*i * q^4 + (i + 2) * q^5 + (2*i + 2) * q^8 - 3*i * q^9 $$q + (i - 1) q^{2} - 2 i q^{4} + (i + 2) q^{5} + (2 i + 2) q^{8} - 3 i q^{9} + (i - 3) q^{10} + ( - 2 i + 3) q^{13} - 4 q^{16} + (5 i + 5) q^{17} + (3 i + 3) q^{18} + ( - 4 i + 2) q^{20} + (4 i + 3) q^{25} + (5 i - 1) q^{26} + 4 i q^{29} + ( - 4 i + 4) q^{32} - 10 q^{34} - 6 q^{36} + (7 i - 7) q^{37} + (6 i + 2) q^{40} - 10 i q^{41} + ( - 6 i + 3) q^{45} - 7 i q^{49} + ( - i - 7) q^{50} + ( - 6 i - 4) q^{52} + ( - 5 i + 5) q^{53} + ( - 4 i - 4) q^{58} - 12 q^{61} + 8 i q^{64} + ( - i + 8) q^{65} + ( - 10 i + 10) q^{68} + ( - 6 i + 6) q^{72} + ( - 11 i - 11) q^{73} - 14 i q^{74} + ( - 4 i - 8) q^{80} - 9 q^{81} + (10 i + 10) q^{82} + (15 i + 5) q^{85} - 10 q^{89} + (9 i + 3) q^{90} + (13 i - 13) q^{97} + (7 i + 7) q^{98} +O(q^{100})$$ q + (i - 1) * q^2 - 2*i * q^4 + (i + 2) * q^5 + (2*i + 2) * q^8 - 3*i * q^9 + (i - 3) * q^10 + (-2*i + 3) * q^13 - 4 * q^16 + (5*i + 5) * q^17 + (3*i + 3) * q^18 + (-4*i + 2) * q^20 + (4*i + 3) * q^25 + (5*i - 1) * q^26 + 4*i * q^29 + (-4*i + 4) * q^32 - 10 * q^34 - 6 * q^36 + (7*i - 7) * q^37 + (6*i + 2) * q^40 - 10*i * q^41 + (-6*i + 3) * q^45 - 7*i * q^49 + (-i - 7) * q^50 + (-6*i - 4) * q^52 + (-5*i + 5) * q^53 + (-4*i - 4) * q^58 - 12 * q^61 + 8*i * q^64 + (-i + 8) * q^65 + (-10*i + 10) * q^68 + (-6*i + 6) * q^72 + (-11*i - 11) * q^73 - 14*i * q^74 + (-4*i - 8) * q^80 - 9 * q^81 + (10*i + 10) * q^82 + (15*i + 5) * q^85 - 10 * q^89 + (9*i + 3) * q^90 + (13*i - 13) * q^97 + (7*i + 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 4 q^{5} + 4 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 4 * q^5 + 4 * q^8 $$2 q - 2 q^{2} + 4 q^{5} + 4 q^{8} - 6 q^{10} + 6 q^{13} - 8 q^{16} + 10 q^{17} + 6 q^{18} + 4 q^{20} + 6 q^{25} - 2 q^{26} + 8 q^{32} - 20 q^{34} - 12 q^{36} - 14 q^{37} + 4 q^{40} + 6 q^{45} - 14 q^{50} - 8 q^{52} + 10 q^{53} - 8 q^{58} - 24 q^{61} + 16 q^{65} + 20 q^{68} + 12 q^{72} - 22 q^{73} - 16 q^{80} - 18 q^{81} + 20 q^{82} + 10 q^{85} - 20 q^{89} + 6 q^{90} - 26 q^{97} + 14 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 4 * q^5 + 4 * q^8 - 6 * q^10 + 6 * q^13 - 8 * q^16 + 10 * q^17 + 6 * q^18 + 4 * q^20 + 6 * q^25 - 2 * q^26 + 8 * q^32 - 20 * q^34 - 12 * q^36 - 14 * q^37 + 4 * q^40 + 6 * q^45 - 14 * q^50 - 8 * q^52 + 10 * q^53 - 8 * q^58 - 24 * q^61 + 16 * q^65 + 20 * q^68 + 12 * q^72 - 22 * q^73 - 16 * q^80 - 18 * q^81 + 20 * 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 $$\left(\mathbb{Z}/260\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
103.1
 − 1.00000i 1.00000i
−1.00000 1.00000i 0 2.00000i 2.00000 1.00000i 0 0 2.00000 2.00000i 3.00000i −3.00000 1.00000i
207.1 −1.00000 + 1.00000i 0 2.00000i 2.00000 + 1.00000i 0 0 2.00000 + 2.00000i 3.00000i −3.00000 + 1.00000i
 $$n$$: 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.h odd 4 1 inner
260.p 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.p.a 2
4.b odd 2 1 CM 260.2.p.a 2
5.c odd 4 1 260.2.p.b yes 2
13.b even 2 1 260.2.p.b yes 2
20.e even 4 1 260.2.p.b yes 2
52.b odd 2 1 260.2.p.b yes 2
65.h odd 4 1 inner 260.2.p.a 2
260.p even 4 1 inner 260.2.p.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.p.a 2 1.a even 1 1 trivial
260.2.p.a 2 4.b odd 2 1 CM
260.2.p.a 2 65.h odd 4 1 inner
260.2.p.a 2 260.p even 4 1 inner
260.2.p.b yes 2 5.c odd 4 1
260.2.p.b yes 2 13.b even 2 1
260.2.p.b yes 2 20.e even 4 1
260.2.p.b yes 2 52.b odd 2 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 $$T_{7}$$ T7 $$T_{37}^{2} + 14T_{37} + 98$$ T37^2 + 14*T37 + 98

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 6T + 13$$
$17$ $$T^{2} - 10T + 50$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 16$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 14T + 98$$
$41$ $$T^{2} + 100$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 10T + 50$$
$59$ $$T^{2}$$
$61$ $$(T + 12)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 22T + 242$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 26T + 338$$