# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1260,2,Mod(881,1260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1260, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1260.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1260.d (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: $$10.0611506547$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 6x^{2} + 4$$ x^4 + 6*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + (\beta_{3} - \beta_1) q^{7}+O(q^{10})$$ q - q^5 + (b3 - b1) * q^7 $$q - q^{5} + (\beta_{3} - \beta_1) q^{7} - \beta_1 q^{11} + ( - \beta_{2} + \beta_1) q^{13} - 2 \beta_{3} q^{17} + ( - \beta_{2} + \beta_1) q^{19} - \beta_{2} q^{23} + q^{25} + (2 \beta_{2} - 3 \beta_1) q^{29} + ( - \beta_{2} - \beta_1) q^{31} + ( - \beta_{3} + \beta_1) q^{35} + ( - 2 \beta_{3} + 6) q^{37} + ( - 2 \beta_{3} + 4) q^{41} + ( - 2 \beta_{3} + 4) q^{43} + ( - 2 \beta_{3} + 2) q^{47} + ( - 2 \beta_{2} + 3) q^{49} + ( - \beta_{2} - 4 \beta_1) q^{53} + \beta_1 q^{55} + (2 \beta_{3} - 6) q^{59} + ( - 2 \beta_{2} + 2 \beta_1) q^{61} + (\beta_{2} - \beta_1) q^{65} + ( - 2 \beta_{2} - \beta_1) q^{71} + ( - 3 \beta_{2} - 5 \beta_1) q^{73} + ( - \beta_{2} - 2) q^{77} + 4 \beta_{3} q^{79} + ( - 2 \beta_{3} + 10) q^{83} + 2 \beta_{3} q^{85} + ( - 4 \beta_{3} - 2) q^{89} + ( - 2 \beta_{3} + \beta_{2} - 5 \beta_1 + 2) q^{91} + (\beta_{2} - \beta_1) q^{95} + ( - 3 \beta_{2} + 7 \beta_1) q^{97}+O(q^{100})$$ q - q^5 + (b3 - b1) * q^7 - b1 * q^11 + (-b2 + b1) * q^13 - 2*b3 * q^17 + (-b2 + b1) * q^19 - b2 * q^23 + q^25 + (2*b2 - 3*b1) * q^29 + (-b2 - b1) * q^31 + (-b3 + b1) * q^35 + (-2*b3 + 6) * q^37 + (-2*b3 + 4) * q^41 + (-2*b3 + 4) * q^43 + (-2*b3 + 2) * q^47 + (-2*b2 + 3) * q^49 + (-b2 - 4*b1) * q^53 + b1 * q^55 + (2*b3 - 6) * q^59 + (-2*b2 + 2*b1) * q^61 + (b2 - b1) * q^65 + (-2*b2 - b1) * q^71 + (-3*b2 - 5*b1) * q^73 + (-b2 - 2) * q^77 + 4*b3 * q^79 + (-2*b3 + 10) * q^83 + 2*b3 * q^85 + (-4*b3 - 2) * q^89 + (-2*b3 + b2 - 5*b1 + 2) * q^91 + (b2 - b1) * q^95 + (-3*b2 + 7*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{5}+O(q^{10})$$ 4 * q - 4 * q^5 $$4 q - 4 q^{5} + 4 q^{25} + 24 q^{37} + 16 q^{41} + 16 q^{43} + 8 q^{47} + 12 q^{49} - 24 q^{59} - 8 q^{77} + 40 q^{83} - 8 q^{89} + 8 q^{91}+O(q^{100})$$ 4 * q - 4 * q^5 + 4 * q^25 + 24 * q^37 + 16 * q^41 + 16 * q^43 + 8 * q^47 + 12 * q^49 - 24 * q^59 - 8 * q^77 + 40 * q^83 - 8 * q^89 + 8 * q^91

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 2$$ (v^3 + 4*v) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 8\nu ) / 2$$ (v^3 + 8*v) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 3$$ v^2 + 3
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 3$$ b3 - 3 $$\nu^{3}$$ $$=$$ $$-2\beta_{2} + 4\beta_1$$ -2*b2 + 4*b1

## Character values

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

 $$n$$ $$281$$ $$631$$ $$757$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 − 2.28825i 2.28825i 0.874032i − 0.874032i
0 0 0 −1.00000 0 −2.23607 1.41421i 0 0 0
881.2 0 0 0 −1.00000 0 −2.23607 + 1.41421i 0 0 0
881.3 0 0 0 −1.00000 0 2.23607 1.41421i 0 0 0
881.4 0 0 0 −1.00000 0 2.23607 + 1.41421i 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
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.d.a 4
3.b odd 2 1 1260.2.d.b yes 4
4.b odd 2 1 5040.2.f.b 4
5.b even 2 1 6300.2.d.b 4
5.c odd 4 2 6300.2.f.c 8
7.b odd 2 1 1260.2.d.b yes 4
12.b even 2 1 5040.2.f.d 4
15.d odd 2 1 6300.2.d.a 4
15.e even 4 2 6300.2.f.a 8
21.c even 2 1 inner 1260.2.d.a 4
28.d even 2 1 5040.2.f.d 4
35.c odd 2 1 6300.2.d.a 4
35.f even 4 2 6300.2.f.a 8
84.h odd 2 1 5040.2.f.b 4
105.g even 2 1 6300.2.d.b 4
105.k odd 4 2 6300.2.f.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.d.a 4 1.a even 1 1 trivial
1260.2.d.a 4 21.c even 2 1 inner
1260.2.d.b yes 4 3.b odd 2 1
1260.2.d.b yes 4 7.b odd 2 1
5040.2.f.b 4 4.b odd 2 1
5040.2.f.b 4 84.h odd 2 1
5040.2.f.d 4 12.b even 2 1
5040.2.f.d 4 28.d even 2 1
6300.2.d.a 4 15.d odd 2 1
6300.2.d.a 4 35.c odd 2 1
6300.2.d.b 4 5.b even 2 1
6300.2.d.b 4 105.g even 2 1
6300.2.f.a 8 15.e even 4 2
6300.2.f.a 8 35.f even 4 2
6300.2.f.c 8 5.c odd 4 2
6300.2.f.c 8 105.k odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{41}^{2} - 8T_{41} - 4$$ acting on $$S_{2}^{\mathrm{new}}(1260, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T + 1)^{4}$$
$7$ $$T^{4} - 6T^{2} + 49$$
$11$ $$(T^{2} + 2)^{2}$$
$13$ $$T^{4} + 24T^{2} + 64$$
$17$ $$(T^{2} - 20)^{2}$$
$19$ $$T^{4} + 24T^{2} + 64$$
$23$ $$(T^{2} + 10)^{2}$$
$29$ $$T^{4} + 116T^{2} + 484$$
$31$ $$T^{4} + 24T^{2} + 64$$
$37$ $$(T^{2} - 12 T + 16)^{2}$$
$41$ $$(T^{2} - 8 T - 4)^{2}$$
$43$ $$(T^{2} - 8 T - 4)^{2}$$
$47$ $$(T^{2} - 4 T - 16)^{2}$$
$53$ $$T^{4} + 84T^{2} + 484$$
$59$ $$(T^{2} + 12 T + 16)^{2}$$
$61$ $$T^{4} + 96T^{2} + 1024$$
$67$ $$T^{4}$$
$71$ $$T^{4} + 84T^{2} + 1444$$
$73$ $$T^{4} + 280T^{2} + 1600$$
$79$ $$(T^{2} - 80)^{2}$$
$83$ $$(T^{2} - 20 T + 80)^{2}$$
$89$ $$(T^{2} + 4 T - 76)^{2}$$
$97$ $$T^{4} + 376T^{2} + 64$$