# Properties

 Label 1560.2.l.c Level $1560$ Weight $2$ Character orbit 1560.l Analytic conductor $12.457$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,2,Mod(1249,1560)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1560.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1560.l (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$12.4566627153$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.5161984.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 4x^{3} + 25x^{2} - 20x + 8$$ x^6 - 4*x^3 + 25*x^2 - 20*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} - \beta_{5} q^{5} - q^{9}+O(q^{10})$$ q + b3 * q^3 - b5 * q^5 - q^9 $$q + \beta_{3} q^{3} - \beta_{5} q^{5} - q^{9} + (\beta_{5} + \beta_{4} - 2) q^{11} + \beta_{3} q^{13} + \beta_{2} q^{15} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{17}+ \cdots + ( - \beta_{5} - \beta_{4} + 2) q^{99}+O(q^{100})$$ q + b3 * q^3 - b5 * q^5 - q^9 + (b5 + b4 - 2) * q^11 + b3 * q^13 + b2 * q^15 + (-b5 + b4 + 2*b3 - b2 - b1) * q^17 + (-b5 - b4 + b2 - b1) * q^19 + (b5 - b4 - 4*b3 + b2 + b1) * q^23 + (b4 + b3 - b2 - 2*b1 - 2) * q^25 - b3 * q^27 + (2*b2 - 2*b1 + 2) * q^29 + (-2*b5 - 2*b4 + 4) * q^31 + (-2*b3 - b2 - b1) * q^33 + (-2*b3 - 2*b2 - 2*b1) * q^37 - q^39 + (2*b5 + 2*b4 + b2 - b1 - 4) * q^41 + (-b5 + b4 + 2*b3 - b2 - b1) * q^43 + b5 * q^45 + (-b5 + b4 + 6*b3 - 2*b2 - 2*b1) * q^47 + 7 * q^49 + (-b5 - b4 + b2 - b1 - 2) * q^51 + (-2*b5 + 2*b4 - 2*b3) * q^53 + (2*b5 - b4 - b3 + b2 + 2*b1 - 3) * q^55 + (b5 - b4 + b2 + b1) * q^57 + (3*b5 + 3*b4 + 2) * q^59 + (b5 + b4 + b2 - b1) * q^61 + b2 * q^65 + (2*b5 - 2*b4 - 2*b2 - 2*b1) * q^67 + (b5 + b4 - b2 + b1 + 4) * q^69 + (b2 - b1 + 2) * q^71 + (-2*b3 - 4*b2 - 4*b1) * q^73 + (-b5 - 2*b4 - 2*b3 - b1 - 1) * q^75 + (b5 + b4 - b2 + b1 - 6) * q^79 + q^81 + (b5 - b4 - 2*b3 - 2*b2 - 2*b1) * q^83 + (b5 + 3*b4 - 2*b3 + b2 - b1 - 6) * q^85 + (2*b5 - 2*b4 + 2*b3) * q^87 + (-2*b5 - 2*b4 - b2 + b1 + 8) * q^89 + (4*b3 + 2*b2 + 2*b1) * q^93 + (-b5 - b4 - 6*b3 - b2 - 3*b1 + 2) * q^95 + 6*b3 * q^97 + (-b5 - b4 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{9}+O(q^{10})$$ 6 * q - 6 * q^9 $$6 q - 6 q^{9} - 12 q^{11} + 2 q^{15} + 4 q^{19} - 10 q^{25} + 20 q^{29} + 24 q^{31} - 6 q^{39} - 20 q^{41} + 42 q^{49} - 8 q^{51} - 20 q^{55} + 12 q^{59} + 4 q^{61} + 2 q^{65} + 20 q^{69} + 16 q^{71} - 4 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} + 44 q^{89} + 16 q^{95} + 12 q^{99}+O(q^{100})$$ 6 * q - 6 * q^9 - 12 * q^11 + 2 * q^15 + 4 * q^19 - 10 * q^25 + 20 * q^29 + 24 * q^31 - 6 * q^39 - 20 * q^41 + 42 * q^49 - 8 * q^51 - 20 * q^55 + 12 * q^59 + 4 * q^61 + 2 * q^65 + 20 * q^69 + 16 * q^71 - 4 * q^75 - 40 * q^79 + 6 * q^81 - 32 * q^85 + 44 * q^89 + 16 * q^95 + 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 4x^{3} + 25x^{2} - 20x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{5} + 25\nu^{4} + 10\nu^{3} - 4\nu^{2} - 121\nu + 323 ) / 121$$ (2*v^5 + 25*v^4 + 10*v^3 - 4*v^2 - 121*v + 323) / 121 $$\beta_{2}$$ $$=$$ $$( -7\nu^{5} - 27\nu^{4} - 35\nu^{3} + 14\nu^{2} - 121\nu - 223 ) / 121$$ (-7*v^5 - 27*v^4 - 35*v^3 + 14*v^2 - 121*v - 223) / 121 $$\beta_{3}$$ $$=$$ $$( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242$$ (-25*v^5 - 10*v^4 - 4*v^3 + 50*v^2 - 605*v + 258) / 242 $$\beta_{4}$$ $$=$$ $$( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1573\nu + 574 ) / 242$$ (-65*v^5 - 26*v^4 + 38*v^3 + 372*v^2 - 1573*v + 574) / 242 $$\beta_{5}$$ $$=$$ $$( 75\nu^{5} + 30\nu^{4} + 12\nu^{3} - 392\nu^{2} + 1573\nu - 774 ) / 242$$ (75*v^5 + 30*v^4 + 12*v^3 - 392*v^2 + 1573*v - 774) / 242
 $$\nu$$ $$=$$ $$( -\beta_{5} - \beta_{4} - \beta_{2} - \beta_1 ) / 2$$ (-b5 - b4 - b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{5} + \beta_{4} - 6\beta_{3} + \beta_{2} + \beta_1 ) / 2$$ (-b5 + b4 - 6*b3 + b2 + b1) / 2 $$\nu^{3}$$ $$=$$ $$( 5\beta_{5} + 5\beta_{4} + 4\beta_{3} - 5\beta_{2} - 5\beta _1 + 4 ) / 2$$ (5*b5 + 5*b4 + 4*b3 - 5*b2 - 5*b1 + 4) / 2 $$\nu^{4}$$ $$=$$ $$( -9\beta_{5} - 9\beta_{4} - 5\beta_{2} + 5\beta _1 - 30 ) / 2$$ (-9*b5 - 9*b4 - 5*b2 + 5*b1 - 30) / 2 $$\nu^{5}$$ $$=$$ $$( 25\beta_{5} + 29\beta_{4} - 32\beta_{3} + 29\beta_{2} + 25\beta _1 + 32 ) / 2$$ (25*b5 + 29*b4 - 32*b3 + 29*b2 + 25*b1 + 32) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times$$.

 $$n$$ $$391$$ $$521$$ $$781$$ $$937$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 −1.75233 − 1.75233i 0.432320 + 0.432320i 1.32001 + 1.32001i −1.75233 + 1.75233i 0.432320 − 0.432320i 1.32001 − 1.32001i
0 1.00000i 0 −1.75233 + 1.38900i 0 0 0 −1.00000 0
1249.2 0 1.00000i 0 0.432320 2.19388i 0 0 0 −1.00000 0
1249.3 0 1.00000i 0 1.32001 + 1.80487i 0 0 0 −1.00000 0
1249.4 0 1.00000i 0 −1.75233 1.38900i 0 0 0 −1.00000 0
1249.5 0 1.00000i 0 0.432320 + 2.19388i 0 0 0 −1.00000 0
1249.6 0 1.00000i 0 1.32001 1.80487i 0 0 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1249.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.2.l.c 6
3.b odd 2 1 4680.2.l.e 6
4.b odd 2 1 3120.2.l.m 6
5.b even 2 1 inner 1560.2.l.c 6
5.c odd 4 1 7800.2.a.bj 3
5.c odd 4 1 7800.2.a.bp 3
15.d odd 2 1 4680.2.l.e 6
20.d odd 2 1 3120.2.l.m 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.c 6 1.a even 1 1 trivial
1560.2.l.c 6 5.b even 2 1 inner
3120.2.l.m 6 4.b odd 2 1
3120.2.l.m 6 20.d odd 2 1
4680.2.l.e 6 3.b odd 2 1
4680.2.l.e 6 15.d odd 2 1
7800.2.a.bj 3 5.c odd 4 1
7800.2.a.bp 3 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}$$ acting on $$S_{2}^{\mathrm{new}}(1560, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{2} + 1)^{3}$$
$5$ $$T^{6} + 5 T^{4} + \cdots + 125$$
$7$ $$T^{6}$$
$11$ $$(T^{3} + 6 T^{2} + 2 T - 20)^{2}$$
$13$ $$(T^{2} + 1)^{3}$$
$17$ $$T^{6} + 56 T^{4} + \cdots + 4096$$
$19$ $$(T^{3} - 2 T^{2} - 24 T - 16)^{2}$$
$23$ $$T^{6} + 84 T^{4} + \cdots + 6400$$
$29$ $$(T^{3} - 10 T^{2} + \cdots + 472)^{2}$$
$31$ $$(T^{3} - 12 T^{2} + \cdots + 160)^{2}$$
$37$ $$T^{6} + 92 T^{4} + \cdots + 18496$$
$41$ $$(T^{3} + 10 T^{2} + \cdots - 332)^{2}$$
$43$ $$T^{6} + 56 T^{4} + \cdots + 4096$$
$47$ $$T^{6} + 188 T^{4} + \cdots + 65536$$
$53$ $$T^{6} + 188 T^{4} + \cdots + 222784$$
$59$ $$(T^{3} - 6 T^{2} - 78 T - 44)^{2}$$
$61$ $$(T^{3} - 2 T^{2} - 32 T + 32)^{2}$$
$67$ $$T^{6} + 272 T^{4} + \cdots + 65536$$
$71$ $$(T^{3} - 8 T^{2} + 2 T + 64)^{2}$$
$73$ $$T^{6} + 332 T^{4} + \cdots + 678976$$
$79$ $$(T^{3} + 20 T^{2} + \cdots + 160)^{2}$$
$83$ $$T^{6} + 140 T^{4} + \cdots + 53824$$
$89$ $$(T^{3} - 22 T^{2} + \cdots + 244)^{2}$$
$97$ $$(T^{2} + 36)^{3}$$