# Properties

 Label 1260.2.f.a Level $1260$ Weight $2$ Character orbit 1260.f Analytic conductor $10.061$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1260,2,Mod(629,1260)]

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, 1, 1]))

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

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

chi := DirichletCharacter("1260.629");

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.f (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: $$8$$ Coefficient field: 8.0.10070523904.11 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 10x^{4} + 81$$ x^8 - 10*x^4 + 81 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{5} + \beta_{5} q^{7}+O(q^{10})$$ q - b3 * q^5 + b5 * q^7 $$q - \beta_{3} q^{5} + \beta_{5} q^{7} + (\beta_{6} + \beta_{5}) q^{11} + ( - \beta_{7} + \beta_1) q^{13} + ( - \beta_{7} - \beta_1) q^{17} + (\beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{2} + 3 \beta_1) q^{19} + (\beta_{4} - 5) q^{23} + ( - \beta_{4} - 3 \beta_{2}) q^{25} - \beta_{2} q^{29} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{31} + ( - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{35} + (\beta_{6} + \beta_{5} + \beta_{2}) q^{37} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} + \beta_1) q^{41} + (\beta_{6} + \beta_{5} - 5 \beta_{2}) q^{43} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_{2} - 4 \beta_1) q^{47} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 1) q^{49} + (\beta_{4} + 7) q^{53} + (\beta_{3} + 5 \beta_1) q^{55} + ( - 4 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} + 4 \beta_1) q^{59} + ( - \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{2} + 1) q^{65} + (2 \beta_{6} + 2 \beta_{5} - 4 \beta_{2}) q^{67} + (\beta_{6} + \beta_{5}) q^{71} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} - \beta_1) q^{73} + ( - 3 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 - 7) q^{77} + ( - 4 \beta_{4} + 4) q^{79} + (2 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} - 6 \beta_{3} - 3 \beta_{2} - 4 \beta_1) q^{83} + ( - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{2} + 1) q^{85} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} - \beta_1) q^{89} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 2) q^{91} + (2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - \beta_{2} + 3) q^{95} + ( - 5 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + 4 \beta_{2} + 5 \beta_1) q^{97}+O(q^{100})$$ q - b3 * q^5 + b5 * q^7 + (b6 + b5) * q^11 + (-b7 + b1) * q^13 + (-b7 - b1) * q^17 + (b7 + b6 - b5 + 2*b3 + b2 + 3*b1) * q^19 + (b4 - 5) * q^23 + (-b4 - 3*b2) * q^25 - b2 * q^29 + (-b7 + 2*b6 - 2*b5 + 4*b3 + 2*b2 + 3*b1) * q^31 + (-b7 - b5 + b4 + b3 + b2 + 3*b1 - 2) * q^35 + (b6 + b5 + b2) * q^37 + (-b7 - 2*b6 + 2*b5 - 2*b2 + b1) * q^41 + (b6 + b5 - 5*b2) * q^43 + (-2*b7 - b6 + b5 - 2*b3 - b2 - 4*b1) * q^47 + (-2*b7 + b6 - b5 - 2*b4 + 2*b3 + b2 - 1) * q^49 + (b4 + 7) * q^53 + (b3 + 5*b1) * q^55 + (-4*b7 + b6 - b5 + b2 + 4*b1) * q^59 + (-b6 - b5 + b4 - 2*b2 + 1) * q^65 + (2*b6 + 2*b5 - 4*b2) * q^67 + (b6 + b5) * q^71 + (b7 - 2*b6 + 2*b5 - 2*b2 - b1) * q^73 + (-3*b7 + b6 - b5 - b4 + 2*b3 + b2 - b1 - 7) * q^77 + (-4*b4 + 4) * q^79 + (2*b7 - 3*b6 + 3*b5 - 6*b3 - 3*b2 - 4*b1) * q^83 + (-b6 - b5 - b4 + 2*b2 + 1) * q^85 + (b7 - b6 + b5 - b2 - b1) * q^89 + (-b7 + 2*b6 - 2*b5 + 4*b3 + 2*b2 + 3*b1 + 2) * q^91 + (2*b6 + 2*b5 + 3*b4 - b2 + 3) * q^95 + (-5*b7 + 4*b6 - 4*b5 + 4*b2 + 5*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 40 q^{23} - 16 q^{35} - 8 q^{49} + 56 q^{53} + 8 q^{65} - 56 q^{77} + 32 q^{79} + 8 q^{85} + 16 q^{91} + 24 q^{95}+O(q^{100})$$ 8 * q - 40 * q^23 - 16 * q^35 - 8 * q^49 + 56 * q^53 + 8 * q^65 - 56 * q^77 + 32 * q^79 + 8 * q^85 + 16 * q^91 + 24 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 10x^{4} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{6} - \nu^{2} ) / 18$$ (v^6 - v^2) / 18 $$\beta_{3}$$ $$=$$ $$( \nu^{5} - 7\nu ) / 6$$ (v^5 - 7*v) / 6 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} + 19\nu^{2} ) / 18$$ (-v^6 + 19*v^2) / 18 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} + \nu^{6} + 3\nu^{5} + 9\nu^{4} + \nu^{3} - \nu^{2} - 3\nu - 45 ) / 36$$ (-v^7 + v^6 + 3*v^5 + 9*v^4 + v^3 - v^2 - 3*v - 45) / 36 $$\beta_{6}$$ $$=$$ $$( \nu^{7} - \nu^{6} - 3\nu^{5} + 9\nu^{4} - \nu^{3} + \nu^{2} + 3\nu - 45 ) / 36$$ (v^7 - v^6 - 3*v^5 + 9*v^4 - v^3 + v^2 + 3*v - 45) / 36 $$\beta_{7}$$ $$=$$ $$( \nu^{7} - 10\nu^{3} ) / 27$$ (v^7 - 10*v^3) / 27
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{2}$$ b4 + b2 $$\nu^{3}$$ $$=$$ $$-3\beta_{7} + 2\beta_{6} - 2\beta_{5} + 2\beta_{3} + 2\beta_{2} + 2\beta_1$$ -3*b7 + 2*b6 - 2*b5 + 2*b3 + 2*b2 + 2*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{6} + 2\beta_{5} + 5$$ 2*b6 + 2*b5 + 5 $$\nu^{5}$$ $$=$$ $$6\beta_{3} + 7\beta_1$$ 6*b3 + 7*b1 $$\nu^{6}$$ $$=$$ $$\beta_{4} + 19\beta_{2}$$ b4 + 19*b2 $$\nu^{7}$$ $$=$$ $$-3\beta_{7} + 20\beta_{6} - 20\beta_{5} + 20\beta_{3} + 20\beta_{2} + 20\beta_1$$ -3*b7 + 20*b6 - 20*b5 + 20*b3 + 20*b2 + 20*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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
629.1
 0.420861 − 1.68014i 0.420861 + 1.68014i −1.68014 + 0.420861i −1.68014 − 0.420861i 1.68014 + 0.420861i 1.68014 − 0.420861i −0.420861 − 1.68014i −0.420861 + 1.68014i
0 0 0 −1.95522 1.08495i 0 2.37608 + 1.16372i 0 0 0
629.2 0 0 0 −1.95522 + 1.08495i 0 2.37608 1.16372i 0 0 0
629.3 0 0 0 −1.08495 1.95522i 0 −0.595188 2.57794i 0 0 0
629.4 0 0 0 −1.08495 + 1.95522i 0 −0.595188 + 2.57794i 0 0 0
629.5 0 0 0 1.08495 1.95522i 0 0.595188 + 2.57794i 0 0 0
629.6 0 0 0 1.08495 + 1.95522i 0 0.595188 2.57794i 0 0 0
629.7 0 0 0 1.95522 1.08495i 0 −2.37608 1.16372i 0 0 0
629.8 0 0 0 1.95522 + 1.08495i 0 −2.37608 + 1.16372i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 629.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g 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.f.a 8
3.b odd 2 1 1260.2.f.b yes 8
4.b odd 2 1 5040.2.k.f 8
5.b even 2 1 1260.2.f.b yes 8
5.c odd 4 2 6300.2.d.f 16
7.b odd 2 1 inner 1260.2.f.a 8
12.b even 2 1 5040.2.k.e 8
15.d odd 2 1 inner 1260.2.f.a 8
15.e even 4 2 6300.2.d.f 16
20.d odd 2 1 5040.2.k.e 8
21.c even 2 1 1260.2.f.b yes 8
28.d even 2 1 5040.2.k.f 8
35.c odd 2 1 1260.2.f.b yes 8
35.f even 4 2 6300.2.d.f 16
60.h even 2 1 5040.2.k.f 8
84.h odd 2 1 5040.2.k.e 8
105.g even 2 1 inner 1260.2.f.a 8
105.k odd 4 2 6300.2.d.f 16
140.c even 2 1 5040.2.k.e 8
420.o odd 2 1 5040.2.k.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.f.a 8 1.a even 1 1 trivial
1260.2.f.a 8 7.b odd 2 1 inner
1260.2.f.a 8 15.d odd 2 1 inner
1260.2.f.a 8 105.g even 2 1 inner
1260.2.f.b yes 8 3.b odd 2 1
1260.2.f.b yes 8 5.b even 2 1
1260.2.f.b yes 8 21.c even 2 1
1260.2.f.b yes 8 35.c odd 2 1
5040.2.k.e 8 12.b even 2 1
5040.2.k.e 8 20.d odd 2 1
5040.2.k.e 8 84.h odd 2 1
5040.2.k.e 8 140.c even 2 1
5040.2.k.f 8 4.b odd 2 1
5040.2.k.f 8 28.d even 2 1
5040.2.k.f 8 60.h even 2 1
5040.2.k.f 8 420.o odd 2 1
6300.2.d.f 16 5.c odd 4 2
6300.2.d.f 16 15.e even 4 2
6300.2.d.f 16 35.f even 4 2
6300.2.d.f 16 105.k odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{23}^{2} + 10T_{23} + 18$$ acting on $$S_{2}^{\mathrm{new}}(1260, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 22T^{4} + 625$$
$7$ $$T^{8} + 4 T^{6} - 10 T^{4} + \cdots + 2401$$
$11$ $$(T^{2} + 14)^{4}$$
$13$ $$(T^{4} - 12 T^{2} + 8)^{2}$$
$17$ $$(T^{4} + 12 T^{2} + 8)^{2}$$
$19$ $$(T^{4} + 52 T^{2} + 648)^{2}$$
$23$ $$(T^{2} + 10 T + 18)^{4}$$
$29$ $$(T^{2} + 2)^{4}$$
$31$ $$(T^{4} + 76 T^{2} + 72)^{2}$$
$37$ $$(T^{4} + 32 T^{2} + 144)^{2}$$
$41$ $$(T^{4} - 140 T^{2} + 3528)^{2}$$
$43$ $$(T^{4} + 128 T^{2} + 1296)^{2}$$
$47$ $$(T^{4} + 104 T^{2} + 2592)^{2}$$
$53$ $$(T^{2} - 14 T + 42)^{4}$$
$59$ $$(T^{4} - 152 T^{2} + 288)^{2}$$
$61$ $$T^{8}$$
$67$ $$(T^{4} + 176 T^{2} + 576)^{2}$$
$71$ $$(T^{2} + 14)^{4}$$
$73$ $$(T^{4} - 76 T^{2} + 72)^{2}$$
$79$ $$(T^{2} - 8 T - 96)^{4}$$
$83$ $$(T^{4} + 168 T^{2} + 1568)^{2}$$
$89$ $$(T^{4} - 20 T^{2} + 72)^{2}$$
$97$ $$(T^{4} - 364 T^{2} + 31752)^{2}$$