# Properties

 Label 260.2.r.b Level $260$ Weight $2$ Character orbit 260.r Analytic conductor $2.076$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.07611045255$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + (\beta_{3} - \beta_1 - 1) q^{5} + (\beta_{3} + \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q + b3 * q^3 + (b3 - b1 - 1) * q^5 + (b3 + b2 + b1) * q^9 $$q + \beta_{3} q^{3} + (\beta_{3} - \beta_1 - 1) q^{5} + (\beta_{3} + \beta_{2} + \beta_1) q^{9} + (\beta_{3} + 2 \beta_{2} - 2) q^{11} + (2 \beta_{2} - 3) q^{13} + (\beta_{3} - 2 \beta_{2} + 2) q^{15} + ( - 2 \beta_{3} - \beta_{2} + 1) q^{17} + (3 \beta_{3} + 4 \beta_{2} - 4) q^{19} + ( - 4 \beta_{2} + \beta_1 - 4) q^{23} + 5 q^{25} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{27} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{29} + (4 \beta_{2} + 3 \beta_1 + 4) q^{31} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{33} + ( - 3 \beta_{3} + 3 \beta_1 + 6) q^{37} + ( - 3 \beta_{3} - 2 \beta_1) q^{39} + (5 \beta_{2} - 2 \beta_1 + 5) q^{41} + (2 \beta_{2} + 3 \beta_1 + 2) q^{43} - 5 \beta_{2} q^{45} + (4 \beta_{3} - 4 \beta_1 - 4) q^{47} - 7 q^{49} + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{51} + ( - 4 \beta_{3} + \beta_{2} - 1) q^{53} + ( - 3 \beta_{3} - 4 \beta_{2} + 4) q^{55} + ( - \beta_{3} - 6 \beta_{2} - \beta_1) q^{57} + (4 \beta_{2} - \beta_1 + 4) q^{59} + ( - 3 \beta_{3} + 3 \beta_1 + 2) q^{61} + ( - 5 \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{65} + (3 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{67} + ( - 5 \beta_{3} + 5 \beta_1 - 2) q^{69} + (2 \beta_{2} + \beta_1 + 2) q^{71} + ( - 3 \beta_{3} - 8 \beta_{2} - 3 \beta_1) q^{73} + 5 \beta_{3} q^{75} + ( - 3 \beta_{3} - 3 \beta_1) q^{79} + (3 \beta_{3} - 3 \beta_1 + 1) q^{81} + (2 \beta_{3} - 2 \beta_1 - 8) q^{83} + (5 \beta_{2} - 5) q^{85} + (2 \beta_{2} + 2 \beta_1 + 2) q^{87} + (\beta_{2} + 2 \beta_1 + 1) q^{89} + (\beta_{3} - \beta_1 - 6) q^{93} + ( - 5 \beta_{3} - 10 \beta_{2} + 10) q^{95} + (6 \beta_{3} + 8 \beta_{2} + 6 \beta_1) q^{97} + ( - 4 \beta_{2} - 3 \beta_1 - 4) q^{99}+O(q^{100})$$ q + b3 * q^3 + (b3 - b1 - 1) * q^5 + (b3 + b2 + b1) * q^9 + (b3 + 2*b2 - 2) * q^11 + (2*b2 - 3) * q^13 + (b3 - 2*b2 + 2) * q^15 + (-2*b3 - b2 + 1) * q^17 + (3*b3 + 4*b2 - 4) * q^19 + (-4*b2 + b1 - 4) * q^23 + 5 * q^25 + (-2*b2 - 2*b1 - 2) * q^27 + (-b3 - 4*b2 - b1) * q^29 + (4*b2 + 3*b1 + 4) * q^31 + (-b3 - 2*b2 - b1) * q^33 + (-3*b3 + 3*b1 + 6) * q^37 + (-3*b3 - 2*b1) * q^39 + (5*b2 - 2*b1 + 5) * q^41 + (2*b2 + 3*b1 + 2) * q^43 - 5*b2 * q^45 + (4*b3 - 4*b1 - 4) * q^47 - 7 * q^49 + (-b3 + 4*b2 - b1) * q^51 + (-4*b3 + b2 - 1) * q^53 + (-3*b3 - 4*b2 + 4) * q^55 + (-b3 - 6*b2 - b1) * q^57 + (4*b2 - b1 + 4) * q^59 + (-3*b3 + 3*b1 + 2) * q^61 + (-5*b3 - 2*b2 + b1 + 3) * q^65 + (3*b3 - 4*b2 + 3*b1) * q^67 + (-5*b3 + 5*b1 - 2) * q^69 + (2*b2 + b1 + 2) * q^71 + (-3*b3 - 8*b2 - 3*b1) * q^73 + 5*b3 * q^75 + (-3*b3 - 3*b1) * q^79 + (3*b3 - 3*b1 + 1) * q^81 + (2*b3 - 2*b1 - 8) * q^83 + (5*b2 - 5) * q^85 + (2*b2 + 2*b1 + 2) * q^87 + (b2 + 2*b1 + 1) * q^89 + (b3 - b1 - 6) * q^93 + (-5*b3 - 10*b2 + 10) * q^95 + (6*b3 + 8*b2 + 6*b1) * q^97 + (-4*b2 - 3*b1 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3}+O(q^{10})$$ 4 * q + 2 * q^3 $$4 q + 2 q^{3} - 6 q^{11} - 12 q^{13} + 10 q^{15} - 10 q^{19} - 18 q^{23} + 20 q^{25} - 4 q^{27} + 10 q^{31} + 12 q^{37} - 2 q^{39} + 24 q^{41} + 2 q^{43} - 28 q^{49} - 12 q^{53} + 10 q^{55} + 18 q^{59} - 4 q^{61} - 28 q^{69} + 6 q^{71} + 10 q^{75} + 16 q^{81} - 24 q^{83} - 20 q^{85} + 4 q^{87} - 20 q^{93} + 30 q^{95} - 10 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 - 6 * q^11 - 12 * q^13 + 10 * q^15 - 10 * q^19 - 18 * q^23 + 20 * q^25 - 4 * q^27 + 10 * q^31 + 12 * q^37 - 2 * q^39 + 24 * q^41 + 2 * q^43 - 28 * q^49 - 12 * q^53 + 10 * q^55 + 18 * q^59 - 4 * q^61 - 28 * q^69 + 6 * q^71 + 10 * q^75 + 16 * q^81 - 24 * q^83 - 20 * q^85 + 4 * q^87 - 20 * q^93 + 30 * q^95 - 10 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu + 1$$ v^2 + v + 1 $$\beta_{2}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v $$\beta_{3}$$ $$=$$ $$-\nu^{2} + \nu - 1$$ -v^2 + v - 1
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + \beta _1 - 2 ) / 2$$ (-b3 + b1 - 2) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{3} + \beta_{2} - \beta_1$$ -b3 + b2 - b1

## Character values

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

 $$n$$ $$41$$ $$131$$ $$157$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
177.1
 0.618034i − 1.61803i − 0.618034i 1.61803i
0 −0.618034 + 0.618034i 0 −2.23607 0 0 0 2.23607i 0
177.2 0 1.61803 1.61803i 0 2.23607 0 0 0 2.23607i 0
213.1 0 −0.618034 0.618034i 0 −2.23607 0 0 0 2.23607i 0
213.2 0 1.61803 + 1.61803i 0 2.23607 0 0 0 2.23607i 0
 $$n$$: 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
65.f 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.r.b yes 4
3.b odd 2 1 2340.2.bp.e 4
4.b odd 2 1 1040.2.cd.j 4
5.b even 2 1 1300.2.r.b 4
5.c odd 4 1 260.2.m.b 4
5.c odd 4 1 1300.2.m.b 4
13.d odd 4 1 260.2.m.b 4
15.e even 4 1 2340.2.u.f 4
20.e even 4 1 1040.2.bg.j 4
39.f even 4 1 2340.2.u.f 4
52.f even 4 1 1040.2.bg.j 4
65.f even 4 1 inner 260.2.r.b yes 4
65.g odd 4 1 1300.2.m.b 4
65.k even 4 1 1300.2.r.b 4
195.u odd 4 1 2340.2.bp.e 4
260.l odd 4 1 1040.2.cd.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.m.b 4 5.c odd 4 1
260.2.m.b 4 13.d odd 4 1
260.2.r.b yes 4 1.a even 1 1 trivial
260.2.r.b yes 4 65.f even 4 1 inner
1040.2.bg.j 4 20.e even 4 1
1040.2.bg.j 4 52.f even 4 1
1040.2.cd.j 4 4.b odd 2 1
1040.2.cd.j 4 260.l odd 4 1
1300.2.m.b 4 5.c odd 4 1
1300.2.m.b 4 65.g odd 4 1
1300.2.r.b 4 5.b even 2 1
1300.2.r.b 4 65.k even 4 1
2340.2.u.f 4 15.e even 4 1
2340.2.u.f 4 39.f even 4 1
2340.2.bp.e 4 3.b odd 2 1
2340.2.bp.e 4 195.u odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} + 4T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(260, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 2 T^{3} + 2 T^{2} + 4 T + 4$$
$5$ $$(T^{2} - 5)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 6 T^{3} + 18 T^{2} + 12 T + 4$$
$13$ $$(T^{2} + 6 T + 13)^{2}$$
$17$ $$T^{4} + 100$$
$19$ $$T^{4} + 10 T^{3} + 50 T^{2} + \cdots + 100$$
$23$ $$T^{4} + 18 T^{3} + 162 T^{2} + \cdots + 1444$$
$29$ $$T^{4} + 28T^{2} + 16$$
$31$ $$T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 100$$
$37$ $$(T^{2} - 6 T - 36)^{2}$$
$41$ $$T^{4} - 24 T^{3} + 288 T^{2} + \cdots + 3844$$
$43$ $$T^{4} - 2 T^{3} + 2 T^{2} + 44 T + 484$$
$47$ $$(T^{2} - 80)^{2}$$
$53$ $$T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 484$$
$59$ $$T^{4} - 18 T^{3} + 162 T^{2} + \cdots + 1444$$
$61$ $$(T^{2} + 2 T - 44)^{2}$$
$67$ $$T^{4} + 188T^{2} + 16$$
$71$ $$T^{4} - 6 T^{3} + 18 T^{2} - 12 T + 4$$
$73$ $$T^{4} + 140T^{2} + 400$$
$79$ $$T^{4} + 108T^{2} + 1296$$
$83$ $$(T^{2} + 12 T + 16)^{2}$$
$89$ $$T^{4} + 100$$
$97$ $$T^{4} + 368 T^{2} + 30976$$