Properties

 Label 1040.2.f.f Level $1040$ Weight $2$ Character orbit 1040.f Analytic conductor $8.304$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

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: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} + 11x^{8} + 36x^{6} + 42x^{4} + 13x^{2} + 1$$ x^10 + 11*x^8 + 36*x^6 + 42*x^4 + 13*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 520) 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} - \beta_{5} q^{5} + \beta_{2} q^{7} + ( - \beta_1 - 1) q^{9}+O(q^{10})$$ q - b3 * q^3 - b5 * q^5 + b2 * q^7 + (-b1 - 1) * q^9 $$q - \beta_{3} q^{3} - \beta_{5} q^{5} + \beta_{2} q^{7} + ( - \beta_1 - 1) q^{9} + \beta_{4} q^{11} - \beta_{8} q^{13} + (\beta_{8} - \beta_{6} + \beta_{3} + \cdots - 2) q^{15}+ \cdots + ( - \beta_{9} - 2 \beta_{8} + \cdots + 4 \beta_{3}) q^{99}+O(q^{100})$$ q - b3 * q^3 - b5 * q^5 + b2 * q^7 + (-b1 - 1) * q^9 + b4 * q^11 - b8 * q^13 + (b8 - b6 + b3 - b1 - 2) * q^15 + (-b9 - b8 - b7 + b6 + b5 + b4) * q^17 + b9 * q^19 + (-b9 - b8 - b7 + b6 + b5 + b4) * q^21 + (b6 + b5 - b3) * q^23 + (-b7 + b5 + b4 - b3 - b2 - b1 - 1) * q^25 + (-b9 - b8 - b7 + b6 + b5 - b4) * q^27 + (-b8 + b7 + b6 - b5 + b1 + 2) * q^29 + (b9 + b8 + b7) * q^31 + (-b6 + b5 - b2 - 2*b1 - 2) * q^33 + (-b9 - b8 - b6 + b3 - b2) * q^35 + (-b2 + 2) * q^37 + (-2*b5 - b4 + 2*b3 + b1 + 2) * q^39 + (b9 + b4 - 2*b3) * q^41 + (-b9 + b4 - b3) * q^43 + (-b9 - b8 - 2*b7 + b6 + b5 + b3 - b1) * q^45 - b2 * q^47 + (-b8 + b7 - b6 + b5 + 1) * q^49 + (-b8 + b7 + b6 - b5 - 2*b2 + 2*b1 + 4) * q^51 + (-b9 + b8 + b7 - 2*b6 - 2*b5 - b4 + 2*b3) * q^53 + (b9 - 2*b6 - b4 - b3 - b1 - 2) * q^55 + (b6 - b5 + b2 + 2) * q^57 + (-b9 - 2*b8 - 2*b7 + 2*b4) * q^59 + (b8 - b7 + b6 - b5 - 2*b2 - b1 - 2) * q^61 + (-b8 + b7 + b6 - b5 + b2 + 2*b1 + 4) * q^63 + (-b9 - b8 - 2*b7 + b5 + b4 + b2 + b1 - 1) * q^65 + (b8 - b7 - b6 + b5 - b2 - 2*b1) * q^67 + (-b8 + b7 + b6 - b5 + b1) * q^69 + (-2*b9 - 2*b8 - 2*b7 + 3*b6 + 3*b5 + b4 - 2*b3) * q^71 + (-b8 + b7 + b6 - b5 + b2 + 2*b1 + 2) * q^73 + (-b8 + 2*b6 + b5 - b4 - b2 - b1 - 2) * q^75 + (b9 - 3*b4 - 2*b3) * q^77 + (2*b8 - 2*b7 - 2*b1 - 4) * q^79 + (-b8 + b7 + 3*b6 - 3*b5 + 3*b1 + 5) * q^81 + (-b8 + b7 + b6 - b5 - b2 - 2*b1) * q^83 + (b9 + b8 - 2*b6 - b5 - 2*b4 + b3 - b2 + b1 + 2) * q^85 + (b9 + 2*b8 + 2*b7 - 4*b6 - 4*b5 - b4 + 2*b3) * q^87 + (2*b8 + 2*b7 - 2*b6 - 2*b5 + 4*b3) * q^89 + (-b9 - b8 + b7 + 2*b6 - 2*b5 + b4 + 2*b2 + b1 + 2) * q^91 + (-b6 + b5 + b2 - 2*b1 - 2) * q^93 + (-2*b8 + 2*b4 + b3 + 2*b2 + b1 + 2) * q^95 + (b8 - b7 - 2*b1 - 8) * q^97 + (-b9 - 2*b8 - 2*b7 + 2*b6 + 2*b5 + 4*b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 3 q^{5} - 2 q^{7} - 8 q^{9}+O(q^{10})$$ 10 * q - 3 * q^5 - 2 * q^7 - 8 * q^9 $$10 q - 3 q^{5} - 2 q^{7} - 8 q^{9} - 2 q^{13} - 13 q^{15} - q^{25} + 8 q^{29} - 8 q^{33} + 3 q^{35} + 22 q^{37} + 12 q^{39} + 4 q^{45} + 2 q^{47} + 12 q^{49} + 30 q^{51} - 12 q^{55} + 12 q^{57} - 16 q^{61} + 24 q^{63} - 9 q^{65} + 16 q^{67} - 12 q^{69} + 4 q^{73} - 21 q^{75} - 28 q^{79} + 22 q^{81} - 4 q^{83} + 25 q^{85} - 2 q^{91} - 12 q^{93} + 10 q^{95} - 72 q^{97}+O(q^{100})$$ 10 * q - 3 * q^5 - 2 * q^7 - 8 * q^9 - 2 * q^13 - 13 * q^15 - q^25 + 8 * q^29 - 8 * q^33 + 3 * q^35 + 22 * q^37 + 12 * q^39 + 4 * q^45 + 2 * q^47 + 12 * q^49 + 30 * q^51 - 12 * q^55 + 12 * q^57 - 16 * q^61 + 24 * q^63 - 9 * q^65 + 16 * q^67 - 12 * q^69 + 4 * q^73 - 21 * q^75 - 28 * q^79 + 22 * q^81 - 4 * q^83 + 25 * q^85 - 2 * q^91 - 12 * q^93 + 10 * q^95 - 72 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 11x^{8} + 36x^{6} + 42x^{4} + 13x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{8} + 8\nu^{6} + 8\nu^{4} - 18\nu^{2} - 9 ) / 2$$ (v^8 + 8*v^6 + 8*v^4 - 18*v^2 - 9) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{8} + 12\nu^{6} + 44\nu^{4} + 50\nu^{2} + 3 ) / 2$$ (v^8 + 12*v^6 + 44*v^4 + 50*v^2 + 3) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{9} - 10\nu^{7} - 26\nu^{5} - 16\nu^{3} + 3\nu ) / 2$$ (-v^9 - 10*v^7 - 26*v^5 - 16*v^3 + 3*v) / 2 $$\beta_{4}$$ $$=$$ $$-\nu^{9} - 11\nu^{7} - 35\nu^{5} - 33\nu^{3} + 4\nu$$ -v^9 - 11*v^7 - 35*v^5 - 33*v^3 + 4*v $$\beta_{5}$$ $$=$$ $$( -5\nu^{9} - 5\nu^{8} - 52\nu^{7} - 52\nu^{6} - 152\nu^{5} - 152\nu^{4} - 146\nu^{3} - 146\nu^{2} - 27\nu - 23 ) / 4$$ (-5*v^9 - 5*v^8 - 52*v^7 - 52*v^6 - 152*v^5 - 152*v^4 - 146*v^3 - 146*v^2 - 27*v - 23) / 4 $$\beta_{6}$$ $$=$$ $$( -5\nu^{9} + 5\nu^{8} - 52\nu^{7} + 52\nu^{6} - 152\nu^{5} + 152\nu^{4} - 146\nu^{3} + 146\nu^{2} - 27\nu + 23 ) / 4$$ (-5*v^9 + 5*v^8 - 52*v^7 + 52*v^6 - 152*v^5 + 152*v^4 - 146*v^3 + 146*v^2 - 27*v + 23) / 4 $$\beta_{7}$$ $$=$$ $$-3\nu^{9} - \nu^{8} - 33\nu^{7} - 11\nu^{6} - 107\nu^{5} - 35\nu^{4} - 117\nu^{3} - 35\nu^{2} - 22\nu - 5$$ -3*v^9 - v^8 - 33*v^7 - 11*v^6 - 107*v^5 - 35*v^4 - 117*v^3 - 35*v^2 - 22*v - 5 $$\beta_{8}$$ $$=$$ $$-3\nu^{9} + \nu^{8} - 33\nu^{7} + 11\nu^{6} - 107\nu^{5} + 35\nu^{4} - 117\nu^{3} + 35\nu^{2} - 22\nu + 5$$ -3*v^9 + v^8 - 33*v^7 + 11*v^6 - 107*v^5 + 35*v^4 - 117*v^3 + 35*v^2 - 22*v + 5 $$\beta_{9}$$ $$=$$ $$6\nu^{9} + 65\nu^{7} + 205\nu^{5} + 217\nu^{3} + 41\nu$$ 6*v^9 + 65*v^7 + 205*v^5 + 217*v^3 + 41*v
 $$\nu$$ $$=$$ $$( -\beta_{9} - \beta_{8} - \beta_{7} + \beta_{4} - 2\beta_{3} ) / 4$$ (-b9 - b8 - b7 + b4 - 2*b3) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{8} - \beta_{7} - 3\beta_{2} - \beta _1 - 10 ) / 4$$ (b8 - b7 - 3*b2 - b1 - 10) / 4 $$\nu^{3}$$ $$=$$ $$( 6\beta_{9} + 5\beta_{8} + 5\beta_{7} + 2\beta_{6} + 2\beta_{5} - 2\beta_{4} + 6\beta_{3} ) / 4$$ (6*b9 + 5*b8 + 5*b7 + 2*b6 + 2*b5 - 2*b4 + 6*b3) / 4 $$\nu^{4}$$ $$=$$ $$( -4\beta_{8} + 4\beta_{7} + \beta_{6} - \beta_{5} + 9\beta_{2} + 2\beta _1 + 24 ) / 2$$ (-4*b8 + 4*b7 + b6 - b5 + 9*b2 + 2*b1 + 24) / 2 $$\nu^{5}$$ $$=$$ $$( -37\beta_{9} - 29\beta_{8} - 29\beta_{7} - 18\beta_{6} - 18\beta_{5} + 7\beta_{4} - 20\beta_{3} ) / 4$$ (-37*b9 - 29*b8 - 29*b7 - 18*b6 - 18*b5 + 7*b4 - 20*b3) / 4 $$\nu^{6}$$ $$=$$ $$( 55\beta_{8} - 55\beta_{7} - 18\beta_{6} + 18\beta_{5} - 109\beta_{2} - 21\beta _1 - 274 ) / 4$$ (55*b8 - 55*b7 - 18*b6 + 18*b5 - 109*b2 - 21*b1 - 274) / 4 $$\nu^{7}$$ $$=$$ $$( 230\beta_{9} + 175\beta_{8} + 175\beta_{7} + 128\beta_{6} + 128\beta_{5} - 32\beta_{4} + 84\beta_{3} ) / 4$$ (230*b9 + 175*b8 + 175*b7 + 128*b6 + 128*b5 - 32*b4 + 84*b3) / 4 $$\nu^{8}$$ $$=$$ $$( -179\beta_{8} + 179\beta_{7} + 64\beta_{6} - 64\beta_{5} + 337\beta_{2} + 63\beta _1 + 832 ) / 2$$ (-179*b8 + 179*b7 + 64*b6 - 64*b5 + 337*b2 + 63*b1 + 832) / 2 $$\nu^{9}$$ $$=$$ $$( -1437\beta_{9} - 1079\beta_{8} - 1079\beta_{7} - 844\beta_{6} - 844\beta_{5} + 173\beta_{4} - 430\beta_{3} ) / 4$$ (-1437*b9 - 1079*b8 - 1079*b7 - 844*b6 - 844*b5 + 173*b4 - 430*b3) / 4

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
 − 1.35685i − 1.56822i 0.549054i − 2.50630i 0.341517i − 0.341517i 2.50630i − 0.549054i 1.56822i 1.35685i
0 3.22659i 0 0.589653 2.15692i 0 −1.65488 0 −7.41088 0
129.2 0 2.14926i 0 −1.82442 + 1.29287i 0 2.12171 0 −1.61930 0
129.3 0 1.57201i 0 −2.14575 0.629081i 0 −4.19743 0 0.528799 0
129.4 0 0.949078i 0 −0.108880 2.23342i 0 3.85660 0 2.09925 0
129.5 0 0.773218i 0 1.98940 + 1.02093i 0 −1.12600 0 2.40213 0
129.6 0 0.773218i 0 1.98940 1.02093i 0 −1.12600 0 2.40213 0
129.7 0 0.949078i 0 −0.108880 + 2.23342i 0 3.85660 0 2.09925 0
129.8 0 1.57201i 0 −2.14575 + 0.629081i 0 −4.19743 0 0.528799 0
129.9 0 2.14926i 0 −1.82442 1.29287i 0 2.12171 0 −1.61930 0
129.10 0 3.22659i 0 0.589653 + 2.15692i 0 −1.65488 0 −7.41088 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 129.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.f 10
4.b odd 2 1 520.2.f.a 10
5.b even 2 1 1040.2.f.g 10
13.b even 2 1 1040.2.f.g 10
20.d odd 2 1 520.2.f.b yes 10
20.e even 4 2 2600.2.k.f 20
52.b odd 2 1 520.2.f.b yes 10
65.d even 2 1 inner 1040.2.f.f 10
260.g odd 2 1 520.2.f.a 10
260.p even 4 2 2600.2.k.f 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.f.a 10 4.b odd 2 1
520.2.f.a 10 260.g odd 2 1
520.2.f.b yes 10 20.d odd 2 1
520.2.f.b yes 10 52.b odd 2 1
1040.2.f.f 10 1.a even 1 1 trivial
1040.2.f.f 10 65.d even 2 1 inner
1040.2.f.g 10 5.b even 2 1
1040.2.f.g 10 13.b even 2 1
2600.2.k.f 20 20.e even 4 2
2600.2.k.f 20 260.p even 4 2

Hecke kernels

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

 $$T_{3}^{10} + 19T_{3}^{8} + 112T_{3}^{6} + 256T_{3}^{4} + 224T_{3}^{2} + 64$$ T3^10 + 19*T3^8 + 112*T3^6 + 256*T3^4 + 224*T3^2 + 64 $$T_{7}^{5} + T_{7}^{4} - 20T_{7}^{3} - 16T_{7}^{2} + 64T_{7} + 64$$ T7^5 + T7^4 - 20*T7^3 - 16*T7^2 + 64*T7 + 64 $$T_{37}^{5} - 11T_{37}^{4} + 28T_{37}^{3} + 32T_{37}^{2} - 128T_{37} - 16$$ T37^5 - 11*T37^4 + 28*T37^3 + 32*T37^2 - 128*T37 - 16

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$T^{10} + 19 T^{8} + \cdots + 64$$
$5$ $$T^{10} + 3 T^{9} + \cdots + 3125$$
$7$ $$(T^{5} + T^{4} - 20 T^{3} + \cdots + 64)^{2}$$
$11$ $$T^{10} + 64 T^{8} + \cdots + 1024$$
$13$ $$T^{10} + 2 T^{9} + \cdots + 371293$$
$17$ $$T^{10} + 107 T^{8} + \cdots + 262144$$
$19$ $$T^{10} + 92 T^{8} + \cdots + 256$$
$23$ $$T^{10} + 44 T^{8} + \cdots + 256$$
$29$ $$(T^{5} - 4 T^{4} - 56 T^{3} + \cdots - 64)^{2}$$
$31$ $$T^{10} + 108 T^{8} + \cdots + 256$$
$37$ $$(T^{5} - 11 T^{4} + \cdots - 16)^{2}$$
$41$ $$T^{10} + 208 T^{8} + \cdots + 4194304$$
$43$ $$T^{10} + 211 T^{8} + \cdots + 40246336$$
$47$ $$(T^{5} - T^{4} - 20 T^{3} + \cdots - 64)^{2}$$
$53$ $$T^{10} + \cdots + 294191104$$
$59$ $$T^{10} + 316 T^{8} + \cdots + 59228416$$
$61$ $$(T^{5} + 8 T^{4} + \cdots + 128)^{2}$$
$67$ $$(T^{5} - 8 T^{4} + \cdots - 10496)^{2}$$
$71$ $$T^{10} + \cdots + 583319104$$
$73$ $$(T^{5} - 2 T^{4} + \cdots + 6752)^{2}$$
$79$ $$(T^{5} + 14 T^{4} + \cdots + 83968)^{2}$$
$83$ $$(T^{5} + 2 T^{4} + \cdots - 132736)^{2}$$
$89$ $$T^{10} + 524 T^{8} + \cdots + 16777216$$
$97$ $$(T^{5} + 36 T^{4} + \cdots - 4736)^{2}$$