# Properties

 Label 1040.2.k.c Level $1040$ Weight $2$ Character orbit 1040.k Analytic conductor $8.304$ 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] = [1040,2,Mod(961,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1040.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1040 = 2^{4} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1040.k (of order $$2$$, degree $$1$$, not 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: $$8.30444181021$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.9144576.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 12x^{4} + 36x^{2} + 4$$ x^6 + 12*x^4 + 36*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 260) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} - \beta_{2} q^{5} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{4} + 1) q^{9}+O(q^{10})$$ q - b3 * q^3 - b2 * q^5 + (b5 + b1) * q^7 + (-b4 + 1) * q^9 $$q - \beta_{3} q^{3} - \beta_{2} q^{5} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{4} + 1) q^{9} + (\beta_{5} - 2 \beta_{2}) q^{11} + (\beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{13} - \beta_1 q^{15} + (\beta_{5} - 2 \beta_1) q^{19} + (2 \beta_{2} - 2 \beta_1) q^{21} + ( - 2 \beta_{4} - \beta_{3} + 2) q^{23} - q^{25} - 2 q^{27} + ( - \beta_{4} - 2 \beta_{3} - 2) q^{29} + (\beta_{5} + 2 \beta_{2} + 2 \beta_1) q^{31} + (\beta_{5} - 2 \beta_{2} - 3 \beta_1) q^{33} + \beta_{4} q^{35} + (\beta_{5} + 4 \beta_{2} + 3 \beta_1) q^{37} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + \cdots - 2) q^{39}+ \cdots + (\beta_{5} - 8 \beta_{2}) q^{99}+O(q^{100})$$ q - b3 * q^3 - b2 * q^5 + (b5 + b1) * q^7 + (-b4 + 1) * q^9 + (b5 - 2*b2) * q^11 + (b4 + b3 - b2 + b1) * q^13 - b1 * q^15 + (b5 - 2*b1) * q^19 + (2*b2 - 2*b1) * q^21 + (-2*b4 - b3 + 2) * q^23 - q^25 - 2 * q^27 + (-b4 - 2*b3 - 2) * q^29 + (b5 + 2*b2 + 2*b1) * q^31 + (b5 - 2*b2 - 3*b1) * q^33 + b4 * q^35 + (b5 + 4*b2 + 3*b1) * q^37 + (-b5 + b4 + 2*b3 + 4*b2 - 2*b1 - 2) * q^39 + (2*b5 + 2*b2) * q^41 + (-3*b3 - 2) * q^43 + (b5 - b2 + b1) * q^45 + (-b5 - 4*b2 + 3*b1) * q^47 + (-2*b4 - 2*b3 - 1) * q^49 + (2*b4 - 2*b3 - 2) * q^53 + (b4 + b3 - 2) * q^55 + (3*b5 - 10*b2 + b1) * q^57 + (-b5 + 8*b2) * q^59 + (3*b4 - 2) * q^61 + (-b5 - 8*b2 + b1) * q^63 + (-b5 - b3 - 1) * q^65 + (3*b5 - 4*b2 + b1) * q^67 + (-b4 - 6*b3) * q^69 + (-3*b5 - 6*b1) * q^71 + (-b5 + 8*b2 - 3*b1) * q^73 + b3 * q^75 - 6 * q^77 + (2*b4 - 4*b3 + 4) * q^79 + (3*b4 + 2*b3 - 3) * q^81 + (b5 + 4*b2 + 3*b1) * q^83 + (-2*b4 + 6) * q^87 + (2*b5 + 8*b2 + 6*b1) * q^89 + (2*b5 + b4 - 2*b3 + 6*b2 + 2*b1 - 2) * q^91 + (-b5 + 6*b2 - b1) * q^93 + (b4 + 3*b3) * q^95 + (-2*b5 - 6*b2 - 2*b1) * q^97 + (b5 - 8*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{9}+O(q^{10})$$ 6 * q + 6 * q^9 $$6 q + 6 q^{9} + 12 q^{23} - 6 q^{25} - 12 q^{27} - 12 q^{29} - 12 q^{39} - 12 q^{43} - 6 q^{49} - 12 q^{53} - 12 q^{55} - 12 q^{61} - 6 q^{65} - 36 q^{77} + 24 q^{79} - 18 q^{81} + 36 q^{87} - 12 q^{91}+O(q^{100})$$ 6 * q + 6 * q^9 + 12 * q^23 - 6 * q^25 - 12 * q^27 - 12 * q^29 - 12 * q^39 - 12 * q^43 - 6 * q^49 - 12 * q^53 - 12 * q^55 - 12 * q^61 - 6 * q^65 - 36 * q^77 + 24 * q^79 - 18 * q^81 + 36 * q^87 - 12 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 12x^{4} + 36x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 6\nu ) / 2$$ (v^3 + 6*v) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{4} + 6\nu^{2} ) / 2$$ (v^4 + 6*v^2) / 2 $$\beta_{4}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4 $$\beta_{5}$$ $$=$$ $$( \nu^{5} + 10\nu^{3} + 22\nu ) / 2$$ (v^5 + 10*v^3 + 22*v) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - 4$$ b4 - 4 $$\nu^{3}$$ $$=$$ $$2\beta_{2} - 6\beta_1$$ 2*b2 - 6*b1 $$\nu^{4}$$ $$=$$ $$-6\beta_{4} + 2\beta_{3} + 24$$ -6*b4 + 2*b3 + 24 $$\nu^{5}$$ $$=$$ $$2\beta_{5} - 20\beta_{2} + 38\beta_1$$ 2*b5 - 20*b2 + 38*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$417$$ $$561$$ $$911$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 − 2.60168i 2.60168i 0.339877i − 0.339877i 2.26180i − 2.26180i
0 −2.60168 0 1.00000i 0 2.76873i 0 3.76873 0
961.2 0 −2.60168 0 1.00000i 0 2.76873i 0 3.76873 0
961.3 0 0.339877 0 1.00000i 0 3.88448i 0 −2.88448 0
961.4 0 0.339877 0 1.00000i 0 3.88448i 0 −2.88448 0
961.5 0 2.26180 0 1.00000i 0 1.11575i 0 2.11575 0
961.6 0 2.26180 0 1.00000i 0 1.11575i 0 2.11575 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 961.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.k.c 6
4.b odd 2 1 260.2.f.a 6
12.b even 2 1 2340.2.c.d 6
13.b even 2 1 inner 1040.2.k.c 6
20.d odd 2 1 1300.2.f.e 6
20.e even 4 1 1300.2.d.c 6
20.e even 4 1 1300.2.d.d 6
52.b odd 2 1 260.2.f.a 6
52.f even 4 1 3380.2.a.m 3
52.f even 4 1 3380.2.a.n 3
156.h even 2 1 2340.2.c.d 6
260.g odd 2 1 1300.2.f.e 6
260.p even 4 1 1300.2.d.c 6
260.p even 4 1 1300.2.d.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.f.a 6 4.b odd 2 1
260.2.f.a 6 52.b odd 2 1
1040.2.k.c 6 1.a even 1 1 trivial
1040.2.k.c 6 13.b even 2 1 inner
1300.2.d.c 6 20.e even 4 1
1300.2.d.c 6 260.p even 4 1
1300.2.d.d 6 20.e even 4 1
1300.2.d.d 6 260.p even 4 1
1300.2.f.e 6 20.d odd 2 1
1300.2.f.e 6 260.g odd 2 1
2340.2.c.d 6 12.b even 2 1
2340.2.c.d 6 156.h even 2 1
3380.2.a.m 3 52.f even 4 1
3380.2.a.n 3 52.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} - 6T_{3} + 2$$ acting on $$S_{2}^{\mathrm{new}}(1040, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{3} - 6 T + 2)^{2}$$
$5$ $$(T^{2} + 1)^{3}$$
$7$ $$T^{6} + 24 T^{4} + \cdots + 144$$
$11$ $$T^{6} + 36 T^{4} + \cdots + 324$$
$13$ $$T^{6} - 9 T^{4} + \cdots + 2197$$
$17$ $$T^{6}$$
$19$ $$T^{6} + 96 T^{4} + \cdots + 12996$$
$23$ $$(T^{3} - 6 T^{2} + \cdots + 174)^{2}$$
$29$ $$(T^{3} + 6 T^{2} - 12 T - 84)^{2}$$
$31$ $$T^{6} + 60 T^{4} + \cdots + 4356$$
$37$ $$T^{6} + 144 T^{4} + \cdots + 63504$$
$41$ $$T^{6} + 108 T^{4} + \cdots + 5184$$
$43$ $$(T^{3} + 6 T^{2} - 42 T - 46)^{2}$$
$47$ $$T^{6} + 216 T^{4} + \cdots + 219024$$
$53$ $$(T^{3} + 6 T^{2} - 84 T + 24)^{2}$$
$59$ $$T^{6} + 216 T^{4} + \cdots + 171396$$
$61$ $$(T^{3} + 6 T^{2} + \cdots - 532)^{2}$$
$67$ $$T^{6} + 240 T^{4} + \cdots + 345744$$
$71$ $$T^{6} + 432 T^{4} + \cdots + 492804$$
$73$ $$T^{6} + 288 T^{4} + \cdots + 63504$$
$79$ $$(T^{3} - 12 T^{2} + \cdots + 1696)^{2}$$
$83$ $$T^{6} + 144 T^{4} + \cdots + 63504$$
$89$ $$T^{6} + 576 T^{4} + \cdots + 4064256$$
$97$ $$T^{6} + 204 T^{4} + \cdots + 576$$