# Properties

 Label 1040.2.f.e Level $1040$ Weight $2$ Character orbit 1040.f Analytic conductor $8.304$ 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] = [1040,2,Mod(129,1040)]

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, 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("1040.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1040 = 2^{4} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1040.f (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: $$8$$ Coefficient field: 8.0.12745506816.3 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 16x^{6} + 79x^{4} + 120x^{2} + 9$$ x^8 + 16*x^6 + 79*x^4 + 120*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{3} + \beta_{4} q^{5} + (\beta_{7} - \beta_{4}) q^{7} + ( - \beta_1 - 2) q^{9}+O(q^{10})$$ q + b6 * q^3 + b4 * q^5 + (b7 - b4) * q^7 + (-b1 - 2) * q^9 $$q + \beta_{6} q^{3} + \beta_{4} q^{5} + (\beta_{7} - \beta_{4}) q^{7} + ( - \beta_1 - 2) q^{9} + \beta_{5} q^{11} + (\beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{13}+ \cdots + ( - \beta_{5} - 5 \beta_{4} - 5 \beta_{3}) q^{99}+O(q^{100})$$ q + b6 * q^3 + b4 * q^5 + (b7 - b4) * q^7 + (-b1 - 2) * q^9 + b5 * q^11 + (b7 + b6 - b4 - b3 - b2) * q^13 + (-b7 - b5 - b3) * q^15 + (-b4 - b3 - 2*b2) * q^17 + (b5 + b4 + b3) * q^19 + (2*b5 + 3*b4 + 3*b3) * q^21 + (b6 - b4 - b3 - 2*b2) * q^23 + (-2*b6 + b4 + b3 + 2*b2 + 1) * q^25 + (-4*b6 + b4 + b3 + 2*b2) * q^27 + (-b1 - 3) * q^29 + (-b5 + 2*b4 + 2*b3) * q^31 + (-b7 + 2*b4 - b3) * q^33 + (2*b6 - b1 - 3) * q^35 + (-b7 - b4 + 2*b3) * q^37 + (b5 + 2*b4 + 2*b3 - b1 - 3) * q^39 + (-2*b5 - b4 - b3) * q^41 + (-b6 - b4 - b3 - 2*b2) * q^43 + (3*b7 - 2*b5 - 4*b4 - 2*b3) * q^45 + (3*b7 - b4 - 2*b3) * q^47 + (2*b1 + 3) * q^49 + 4 * q^51 - 2*b6 * q^53 + (-b6 - b4 - b3 - 2*b2 - b1 + 1) * q^55 + (-3*b7 + 4*b4 - b3) * q^57 + (-3*b5 - b4 - b3) * q^59 + (b1 - 1) * q^61 + (-5*b7 + 7*b4 - 2*b3) * q^63 + (b6 - 2*b3 + b2 - b1) * q^65 + (b7 + b4 - 2*b3) * q^67 + (-b1 - 1) * q^69 + (-b5 + b4 + b3) * q^71 + (-b7 - b4 + 2*b3) * q^73 + (b6 + 2*b1 + 6) * q^75 + (4*b6 - b4 - b3 - 2*b2) * q^77 + (-2*b1 - 2) * q^79 + (b1 + 10) * q^81 + (-3*b7 - b4 + 4*b3) * q^83 + (2*b7 + 2*b5 - b4 - 3*b3) * q^85 + (-8*b6 + b4 + b3 + 2*b2) * q^87 + (4*b4 + 4*b3) * q^89 + (2*b5 + 2*b4 + 2*b3 + b1 + 7) * q^91 + (-3*b7 + 2*b4 + b3) * q^93 + (-3*b6 - b1 - 3) * q^95 + (2*b7 - 3*b4 + b3) * q^97 + (-b5 - 5*b4 - 5*b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 16 q^{9}+O(q^{10})$$ 8 * q - 16 * q^9 $$8 q - 16 q^{9} + 8 q^{25} - 24 q^{29} - 24 q^{35} - 24 q^{39} + 24 q^{49} + 32 q^{51} + 8 q^{55} - 8 q^{61} - 8 q^{69} + 48 q^{75} - 16 q^{79} + 80 q^{81} + 56 q^{91} - 24 q^{95}+O(q^{100})$$ 8 * q - 16 * q^9 + 8 * q^25 - 24 * q^29 - 24 * q^35 - 24 * q^39 + 24 * q^49 + 32 * q^51 + 8 * q^55 - 8 * q^61 - 8 * q^69 + 48 * q^75 - 16 * q^79 + 80 * q^81 + 56 * q^91 - 24 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 16x^{6} + 79x^{4} + 120x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{4} + 16\nu^{2} + 15 ) / 3$$ (2*v^4 + 16*v^2 + 15) / 3 $$\beta_{2}$$ $$=$$ $$( -\nu^{7} - 12\nu^{5} - 26\nu^{3} + 39\nu ) / 15$$ (-v^7 - 12*v^5 - 26*v^3 + 39*v) / 15 $$\beta_{3}$$ $$=$$ $$( -2\nu^{7} - 3\nu^{6} - 29\nu^{5} - 36\nu^{4} - 137\nu^{3} - 78\nu^{2} - 237\nu + 72 ) / 45$$ (-2*v^7 - 3*v^6 - 29*v^5 - 36*v^4 - 137*v^3 - 78*v^2 - 237*v + 72) / 45 $$\beta_{4}$$ $$=$$ $$( -2\nu^{7} + 3\nu^{6} - 29\nu^{5} + 36\nu^{4} - 137\nu^{3} + 78\nu^{2} - 237\nu - 72 ) / 45$$ (-2*v^7 + 3*v^6 - 29*v^5 + 36*v^4 - 137*v^3 + 78*v^2 - 237*v - 72) / 45 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} - 16\nu^{5} - 76\nu^{3} - 87\nu ) / 9$$ (-v^7 - 16*v^5 - 76*v^3 - 87*v) / 9 $$\beta_{6}$$ $$=$$ $$( -\nu^{7} - 16\nu^{5} - 76\nu^{3} - 105\nu ) / 9$$ (-v^7 - 16*v^5 - 76*v^3 - 105*v) / 9 $$\beta_{7}$$ $$=$$ $$( -2\nu^{7} + 9\nu^{6} - 29\nu^{5} + 108\nu^{4} - 137\nu^{3} + 324\nu^{2} - 237\nu + 144 ) / 45$$ (-2*v^7 + 9*v^6 - 29*v^5 + 108*v^4 - 137*v^3 + 324*v^2 - 237*v + 144) / 45
 $$\nu$$ $$=$$ $$( -\beta_{6} + \beta_{5} ) / 2$$ (-b6 + b5) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{7} - 2\beta_{4} + \beta_{3} - 8 ) / 2$$ (b7 - 2*b4 + b3 - 8) / 2 $$\nu^{3}$$ $$=$$ $$( 7\beta_{6} - 6\beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_{2} ) / 2$$ (7*b6 - 6*b5 - 2*b4 - 2*b3 + b2) / 2 $$\nu^{4}$$ $$=$$ $$( -8\beta_{7} + 16\beta_{4} - 8\beta_{3} + 3\beta _1 + 49 ) / 2$$ (-8*b7 + 16*b4 - 8*b3 + 3*b1 + 49) / 2 $$\nu^{5}$$ $$=$$ $$( -56\beta_{6} + 39\beta_{5} + 25\beta_{4} + 25\beta_{3} - 5\beta_{2} ) / 2$$ (-56*b6 + 39*b5 + 25*b4 + 25*b3 - 5*b2) / 2 $$\nu^{6}$$ $$=$$ $$( 70\beta_{7} - 125\beta_{4} + 55\beta_{3} - 36\beta _1 - 332 ) / 2$$ (70*b7 - 125*b4 + 55*b3 - 36*b1 - 332) / 2 $$\nu^{7}$$ $$=$$ $$( 451\beta_{6} - 273\beta_{5} - 248\beta_{4} - 248\beta_{3} + 4\beta_{2} ) / 2$$ (451*b6 - 273*b5 - 248*b4 - 248*b3 + 4*b2) / 2

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 0.281155i 2.81442i − 1.65070i 2.29678i 1.65070i − 2.29678i − 0.281155i − 2.81442i
0 3.09557i 0 −1.73205 1.41421i 0 4.37780 0 −6.58258 0
129.2 0 3.09557i 0 1.73205 + 1.41421i 0 −4.37780 0 −6.58258 0
129.3 0 0.646084i 0 −1.73205 + 1.41421i 0 −0.913701 0 2.58258 0
129.4 0 0.646084i 0 1.73205 1.41421i 0 0.913701 0 2.58258 0
129.5 0 0.646084i 0 −1.73205 1.41421i 0 −0.913701 0 2.58258 0
129.6 0 0.646084i 0 1.73205 + 1.41421i 0 0.913701 0 2.58258 0
129.7 0 3.09557i 0 −1.73205 + 1.41421i 0 4.37780 0 −6.58258 0
129.8 0 3.09557i 0 1.73205 1.41421i 0 −4.37780 0 −6.58258 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 129.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.b even 2 1 inner
65.d 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.f.e 8
4.b odd 2 1 260.2.d.a 8
5.b even 2 1 inner 1040.2.f.e 8
12.b even 2 1 2340.2.j.d 8
13.b even 2 1 inner 1040.2.f.e 8
20.d odd 2 1 260.2.d.a 8
20.e even 4 2 1300.2.f.f 8
52.b odd 2 1 260.2.d.a 8
52.f even 4 2 3380.2.c.d 8
60.h even 2 1 2340.2.j.d 8
65.d even 2 1 inner 1040.2.f.e 8
156.h even 2 1 2340.2.j.d 8
260.g odd 2 1 260.2.d.a 8
260.p even 4 2 1300.2.f.f 8
260.u even 4 2 3380.2.c.d 8
780.d even 2 1 2340.2.j.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.d.a 8 4.b odd 2 1
260.2.d.a 8 20.d odd 2 1
260.2.d.a 8 52.b odd 2 1
260.2.d.a 8 260.g odd 2 1
1040.2.f.e 8 1.a even 1 1 trivial
1040.2.f.e 8 5.b even 2 1 inner
1040.2.f.e 8 13.b even 2 1 inner
1040.2.f.e 8 65.d even 2 1 inner
1300.2.f.f 8 20.e even 4 2
1300.2.f.f 8 260.p even 4 2
2340.2.j.d 8 12.b even 2 1
2340.2.j.d 8 60.h even 2 1
2340.2.j.d 8 156.h even 2 1
2340.2.j.d 8 780.d even 2 1
3380.2.c.d 8 52.f even 4 2
3380.2.c.d 8 260.u even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1040, [\chi])$$:

 $$T_{3}^{4} + 10T_{3}^{2} + 4$$ T3^4 + 10*T3^2 + 4 $$T_{7}^{4} - 20T_{7}^{2} + 16$$ T7^4 - 20*T7^2 + 16 $$T_{37}^{4} - 68T_{37}^{2} + 400$$ T37^4 - 68*T37^2 + 400

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} + 10 T^{2} + 4)^{2}$$
$5$ $$(T^{4} - 2 T^{2} + 25)^{2}$$
$7$ $$(T^{4} - 20 T^{2} + 16)^{2}$$
$11$ $$(T^{4} + 22 T^{2} + 100)^{2}$$
$13$ $$(T^{4} - 2 T^{2} + 169)^{2}$$
$17$ $$(T^{4} + 40 T^{2} + 64)^{2}$$
$19$ $$(T^{4} + 30 T^{2} + 36)^{2}$$
$23$ $$(T^{4} + 34 T^{2} + 100)^{2}$$
$29$ $$(T^{2} + 6 T - 12)^{4}$$
$31$ $$(T^{4} + 102 T^{2} + 900)^{2}$$
$37$ $$(T^{4} - 68 T^{2} + 400)^{2}$$
$41$ $$(T^{4} + 88 T^{2} + 1600)^{2}$$
$43$ $$(T^{4} + 66 T^{2} + 900)^{2}$$
$47$ $$(T^{4} - 132 T^{2} + 3600)^{2}$$
$53$ $$(T^{4} + 40 T^{2} + 64)^{2}$$
$59$ $$(T^{4} + 190 T^{2} + 8836)^{2}$$
$61$ $$(T^{2} + 2 T - 20)^{4}$$
$67$ $$(T^{4} - 68 T^{2} + 400)^{2}$$
$71$ $$(T^{4} + 46 T^{2} + 4)^{2}$$
$73$ $$(T^{4} - 68 T^{2} + 400)^{2}$$
$79$ $$(T^{2} + 4 T - 80)^{4}$$
$83$ $$(T^{4} - 276 T^{2} + 144)^{2}$$
$89$ $$(T^{2} + 128)^{4}$$
$97$ $$(T^{4} - 152 T^{2} + 400)^{2}$$