# Properties

 Label 1260.2.bm.a Level $1260$ Weight $2$ Character orbit 1260.bm Analytic conductor $10.061$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1260.bm (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.0611506547$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ x^4 - x^3 - 4*x^2 - 5*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_{2}) q^{5} + (2 \beta_{2} + \beta_1 - 2) q^{7}+O(q^{10})$$ q + (-b3 - b2) * q^5 + (2*b2 + b1 - 2) * q^7 $$q + ( - \beta_{3} - \beta_{2}) q^{5} + (2 \beta_{2} + \beta_1 - 2) q^{7} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{11} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{13} + (\beta_{3} + 2 \beta_{2} - 3) q^{17} + (2 \beta_{3} + \beta_{2} - \beta_1) q^{19} + (3 \beta_{2} + 2 \beta_1 + 1) q^{23} + ( - 4 \beta_{2} + \beta_1 + 4) q^{25} + (\beta_{3} + \beta_{2} + \beta_1) q^{29} + (\beta_{3} - 2 \beta_1 + 1) q^{31} + (2 \beta_{3} - 2 \beta_1 - 3) q^{35} + (5 \beta_{2} + \beta_1 + 4) q^{37} + ( - \beta_{3} - \beta_{2} - \beta_1 + 8) q^{41} + (\beta_{3} + 5 \beta_{2} - \beta_1 - 2) q^{43} + (3 \beta_{2} + \beta_1 + 2) q^{47} + ( - 3 \beta_{3} + \beta_{2}) q^{49} + ( - 3 \beta_{3} - 2 \beta_{2} + 1) q^{53} + ( - \beta_{3} + 6 \beta_{2} + 2 \beta_1 + 3) q^{55} + ( - \beta_{3} + 2 \beta_1 - 1) q^{59} + (4 \beta_{3} - 5 \beta_{2} - 2 \beta_1 + 7) q^{61} + (8 \beta_{2} - 2 \beta_1 + 2) q^{65} + (4 \beta_{3} + 5 \beta_{2} - 6) q^{67} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{71} + ( - \beta_{3} + 2 \beta_{2} - 5) q^{73} + (3 \beta_{3} - 8 \beta_{2} + 7) q^{77} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{79} + (3 \beta_{3} - \beta_{2} - 3 \beta_1 + 2) q^{83} + (3 \beta_{3} + 6 \beta_{2} - 2 \beta_1 - 3) q^{85} + ( - 7 \beta_{2} + 7) q^{89} + ( - 2 \beta_{3} - 10 \beta_{2} + 4 \beta_1 + 6) q^{91} + (9 \beta_{2} - \beta_1 - 4) q^{95} + ( - 8 \beta_{2} + 4) q^{97}+O(q^{100})$$ q + (-b3 - b2) * q^5 + (2*b2 + b1 - 2) * q^7 + (b3 - 2*b2 - 2*b1 + 1) * q^11 + (2*b3 + 2*b2 - 2*b1) * q^13 + (b3 + 2*b2 - 3) * q^17 + (2*b3 + b2 - b1) * q^19 + (3*b2 + 2*b1 + 1) * q^23 + (-4*b2 + b1 + 4) * q^25 + (b3 + b2 + b1) * q^29 + (b3 - 2*b1 + 1) * q^31 + (2*b3 - 2*b1 - 3) * q^35 + (5*b2 + b1 + 4) * q^37 + (-b3 - b2 - b1 + 8) * q^41 + (b3 + 5*b2 - b1 - 2) * q^43 + (3*b2 + b1 + 2) * q^47 + (-3*b3 + b2) * q^49 + (-3*b3 - 2*b2 + 1) * q^53 + (-b3 + 6*b2 + 2*b1 + 3) * q^55 + (-b3 + 2*b1 - 1) * q^59 + (4*b3 - 5*b2 - 2*b1 + 7) * q^61 + (8*b2 - 2*b1 + 2) * q^65 + (4*b3 + 5*b2 - 6) * q^67 + (2*b3 + 2*b2 + 2*b1 - 4) * q^71 + (-b3 + 2*b2 - 5) * q^73 + (3*b3 - 8*b2 + 7) * q^77 + (2*b3 - 3*b2 - b1 + 4) * q^79 + (3*b3 - b2 - 3*b1 + 2) * q^83 + (3*b3 + 6*b2 - 2*b1 - 3) * q^85 + (-7*b2 + 7) * q^89 + (-2*b3 - 10*b2 + 4*b1 + 6) * q^91 + (9*b2 - b1 - 4) * q^95 + (-8*b2 + 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{5} - 3 q^{7}+O(q^{10})$$ 4 * q - q^5 - 3 * q^7 $$4 q - q^{5} - 3 q^{7} - 3 q^{11} - 9 q^{17} - q^{19} + 12 q^{23} + 9 q^{25} + 2 q^{29} + q^{31} - 16 q^{35} + 27 q^{37} + 30 q^{41} + 15 q^{47} + 5 q^{49} + 3 q^{53} + 27 q^{55} - q^{59} + 12 q^{61} + 22 q^{65} - 18 q^{67} - 12 q^{71} - 15 q^{73} + 9 q^{77} + 7 q^{79} - 5 q^{85} + 14 q^{89} + 10 q^{91} + q^{95}+O(q^{100})$$ 4 * q - q^5 - 3 * q^7 - 3 * q^11 - 9 * q^17 - q^19 + 12 * q^23 + 9 * q^25 + 2 * q^29 + q^31 - 16 * q^35 + 27 * q^37 + 30 * q^41 + 15 * q^47 + 5 * q^49 + 3 * q^53 + 27 * q^55 - q^59 + 12 * q^61 + 22 * q^65 - 18 * q^67 - 12 * q^71 - 15 * q^73 + 9 * q^77 + 7 * q^79 - 5 * q^85 + 14 * q^89 + 10 * q^91 + q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 4\nu^{2} - 4\nu - 5 ) / 20$$ (v^3 + 4*v^2 - 4*v - 5) / 20 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 4\nu + 5 ) / 4$$ (-v^3 + 4*v + 5) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 5\beta_{2}$$ b3 + 5*b2 $$\nu^{3}$$ $$=$$ $$-4\beta_{3} + 4\beta _1 + 5$$ -4*b3 + 4*b1 + 5

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$757$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 2.13746 − 0.656712i −1.63746 + 1.52274i 2.13746 + 0.656712i −1.63746 − 1.52274i
0 0 0 −2.13746 0.656712i 0 1.13746 2.38876i 0 0 0
109.2 0 0 0 1.63746 + 1.52274i 0 −2.63746 0.209313i 0 0 0
289.1 0 0 0 −2.13746 + 0.656712i 0 1.13746 + 2.38876i 0 0 0
289.2 0 0 0 1.63746 1.52274i 0 −2.63746 + 0.209313i 0 0 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
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.bm.a 4
3.b odd 2 1 140.2.q.a 4
5.b even 2 1 1260.2.bm.b 4
7.c even 3 1 1260.2.bm.b 4
12.b even 2 1 560.2.bw.e 4
15.d odd 2 1 140.2.q.b yes 4
15.e even 4 2 700.2.i.f 8
21.c even 2 1 980.2.q.g 4
21.g even 6 1 980.2.e.c 4
21.g even 6 1 980.2.q.b 4
21.h odd 6 1 140.2.q.b yes 4
21.h odd 6 1 980.2.e.f 4
35.j even 6 1 inner 1260.2.bm.a 4
60.h even 2 1 560.2.bw.a 4
84.n even 6 1 560.2.bw.a 4
105.g even 2 1 980.2.q.b 4
105.o odd 6 1 140.2.q.a 4
105.o odd 6 1 980.2.e.f 4
105.p even 6 1 980.2.e.c 4
105.p even 6 1 980.2.q.g 4
105.w odd 12 2 4900.2.a.bf 4
105.x even 12 2 700.2.i.f 8
105.x even 12 2 4900.2.a.be 4
420.ba even 6 1 560.2.bw.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.q.a 4 3.b odd 2 1
140.2.q.a 4 105.o odd 6 1
140.2.q.b yes 4 15.d odd 2 1
140.2.q.b yes 4 21.h odd 6 1
560.2.bw.a 4 60.h even 2 1
560.2.bw.a 4 84.n even 6 1
560.2.bw.e 4 12.b even 2 1
560.2.bw.e 4 420.ba even 6 1
700.2.i.f 8 15.e even 4 2
700.2.i.f 8 105.x even 12 2
980.2.e.c 4 21.g even 6 1
980.2.e.c 4 105.p even 6 1
980.2.e.f 4 21.h odd 6 1
980.2.e.f 4 105.o odd 6 1
980.2.q.b 4 21.g even 6 1
980.2.q.b 4 105.g even 2 1
980.2.q.g 4 21.c even 2 1
980.2.q.g 4 105.p even 6 1
1260.2.bm.a 4 1.a even 1 1 trivial
1260.2.bm.a 4 35.j even 6 1 inner
1260.2.bm.b 4 5.b even 2 1
1260.2.bm.b 4 7.c even 3 1
4900.2.a.be 4 105.x even 12 2
4900.2.a.bf 4 105.w odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1260, [\chi])$$:

 $$T_{11}^{4} + 3T_{11}^{3} + 21T_{11}^{2} - 36T_{11} + 144$$ T11^4 + 3*T11^3 + 21*T11^2 - 36*T11 + 144 $$T_{17}^{4} + 9T_{17}^{3} + 29T_{17}^{2} + 18T_{17} + 4$$ T17^4 + 9*T17^3 + 29*T17^2 + 18*T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + T^{3} - 4 T^{2} + 5 T + 25$$
$7$ $$T^{4} + 3 T^{3} + 2 T^{2} + 21 T + 49$$
$11$ $$T^{4} + 3 T^{3} + 21 T^{2} - 36 T + 144$$
$13$ $$T^{4} + 44T^{2} + 256$$
$17$ $$T^{4} + 9 T^{3} + 29 T^{2} + 18 T + 4$$
$19$ $$T^{4} + T^{3} + 15 T^{2} - 14 T + 196$$
$23$ $$T^{4} - 12 T^{3} + 41 T^{2} + 84 T + 49$$
$29$ $$(T^{2} - T - 14)^{2}$$
$31$ $$T^{4} - T^{3} + 15 T^{2} + 14 T + 196$$
$37$ $$T^{4} - 27 T^{3} + 299 T^{2} + \cdots + 3136$$
$41$ $$(T^{2} - 15 T + 42)^{2}$$
$43$ $$T^{4} + 47T^{2} + 196$$
$47$ $$T^{4} - 15 T^{3} + 89 T^{2} + \cdots + 196$$
$53$ $$T^{4} - 3 T^{3} - 39 T^{2} + \cdots + 1764$$
$59$ $$T^{4} + T^{3} + 15 T^{2} - 14 T + 196$$
$61$ $$T^{4} - 12 T^{3} + 165 T^{2} + \cdots + 441$$
$67$ $$T^{4} + 18 T^{3} + 59 T^{2} + \cdots + 2401$$
$71$ $$(T^{2} + 6 T - 48)^{2}$$
$73$ $$T^{4} + 15 T^{3} + 89 T^{2} + \cdots + 196$$
$79$ $$T^{4} - 7 T^{3} + 51 T^{2} + 14 T + 4$$
$83$ $$T^{4} + 87T^{2} + 1764$$
$89$ $$(T^{2} - 7 T + 49)^{2}$$
$97$ $$(T^{2} + 48)^{2}$$