Properties

 Label 1040.2.d.f Level $1040$ Weight $2$ Character orbit 1040.d Analytic conductor $8.304$ Analytic rank $0$ Dimension $10$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(209,1040)]

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, 0]))

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

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

chi := DirichletCharacter("1040.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1040 = 2^{4} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1040.d (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: $$x^{10} - 4x^{9} + 2x^{8} + 16x^{7} - 15x^{6} - 40x^{5} - 75x^{4} + 400x^{3} + 250x^{2} - 2500x + 3125$$ x^10 - 4*x^9 + 2*x^8 + 16*x^7 - 15*x^6 - 40*x^5 - 75*x^4 + 400*x^3 + 250*x^2 - 2500*x + 3125 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{9} q^{3} + \beta_{2} q^{5} - \beta_{4} q^{7} + ( - \beta_{8} - \beta_{6} - \beta_{5} - 4) q^{9}+O(q^{10})$$ q - b9 * q^3 + b2 * q^5 - b4 * q^7 + (-b8 - b6 - b5 - 4) * q^9 $$q - \beta_{9} q^{3} + \beta_{2} q^{5} - \beta_{4} q^{7} + ( - \beta_{8} - \beta_{6} - \beta_{5} - 4) q^{9} + ( - \beta_{8} - \beta_{7} - \beta_{2} + \cdots - 3) q^{11}+ \cdots + (3 \beta_{8} + 5 \beta_{7} - 4 \beta_{6} + \cdots + 7) q^{99}+O(q^{100})$$ q - b9 * q^3 + b2 * q^5 - b4 * q^7 + (-b8 - b6 - b5 - 4) * q^9 + (-b8 - b7 - b2 + b1 - 3) * q^11 - b3 * q^13 + (b9 + b6 + b5 + b4 - b3 + b2 + 2*b1) * q^15 + (b6 - b5 - 2*b3) * q^17 + (-b8 + b7 - b2 + b1 + 3) * q^19 + (-2*b7 + 2*b6 + 2*b5 + b2 - b1 + 2) * q^21 + (-b9 + b6 - b5 + 2*b3) * q^23 + (-b9 + b7 + b5 + b4 + 2) * q^25 + (2*b9 + b6 - b5 - 2*b3 + 2*b2 + 2*b1) * q^27 + (-b8 + 2*b6 + 2*b5 + 1) * q^29 + (b8 - b7 + 2*b6 + 2*b5 + b2 - b1 + 3) * q^31 + (2*b9 + 2*b6 - 2*b5 - b4 + 6*b3 - 3*b2 - 3*b1) * q^33 + (b9 + b8 + b7 + b5 + b4 + 4*b3 - 2*b1) * q^35 + (-2*b6 + 2*b5 + b4 + 2*b3) * q^37 - b7 * q^39 + (2*b8 - 2) * q^41 + (-b9 - 2*b3 + 2*b2 + 2*b1) * q^43 + (-b9 + b8 + b7 + 3*b6 + b5 - b4 + b3 - 3*b2 - b1 + 5) * q^45 + (b4 - 4*b3) * q^47 + (2*b8 + b6 + b5 + 2*b2 - 2*b1 - 1) * q^49 + (2*b8 - b6 - b5) * q^51 + (2*b9 + 2*b2 + 2*b1) * q^53 + (b9 + b8 + 2*b7 + b6 - b5 - 2*b4 - 2*b3 - 2*b2 - 3) * q^55 + (-4*b9 + 2*b6 - 2*b5 - 3*b4 - 6*b3 - 3*b2 - 3*b1) * q^57 + (-b8 - b7 + b2 - b1 + 3) * q^59 + (-3*b8 - 2*b7 - 2*b6 - 2*b5 - 1) * q^61 + (2*b6 - 2*b5 + b4 + 10*b3 - b2 - b1) * q^63 - b5 * q^65 + (-2*b9 - 2*b6 + 2*b5 + b4 + 2*b3 - b2 - b1) * q^67 + (b8 + 4*b7 - 2*b6 - 2*b5 - 7) * q^69 + (b8 - 3*b7 + 2*b6 + 2*b5 - 3) * q^71 + (2*b9 + b4 + 3*b2 + 3*b1) * q^73 + (-2*b9 - 2*b8 + b7 - 2*b6 - 3*b5 - b4 - 6*b3 - 2*b2 - 9) * q^75 + (2*b9 + 2*b6 - 2*b5 - 2*b3 + 4*b2 + 4*b1) * q^77 + (2*b7 + 2*b6 + 2*b5 + 4) * q^79 + (b8 + 2*b6 + 2*b5 - 2*b2 + 2*b1 + 2) * q^81 + (-3*b4 - 2*b3 - 3*b2 - 3*b1) * q^83 + (-b9 + b8 - b7 + b6 - b5 - 7*b3 + b2 + 1) * q^85 + (4*b6 - 4*b5 - 2*b3 - 4*b2 - 4*b1) * q^87 + (-2*b6 - 2*b5 - 2*b2 + 2*b1 - 2) * q^89 + (-b8 - b6 - b5 - 1) * q^91 + (-2*b9 + 3*b4 + 6*b3 - b2 - b1) * q^93 + (b9 - b8 + 3*b6 - b5 - 2*b4 - 2*b3 + 2*b2 - 2*b1 - 3) * q^95 + (2*b9 + 2*b4 + 2*b3) * q^97 + (3*b8 + 5*b7 - 4*b6 - 4*b5 + b2 - b1 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 4 q^{5} - 30 q^{9}+O(q^{10})$$ 10 * q - 4 * q^5 - 30 * q^9 $$10 q - 4 q^{5} - 30 q^{9} - 16 q^{11} - 4 q^{15} + 36 q^{19} + 4 q^{21} + 12 q^{25} - 4 q^{29} + 8 q^{31} - 18 q^{35} + 4 q^{39} - 24 q^{41} + 36 q^{45} - 38 q^{49} + 4 q^{51} - 32 q^{55} + 28 q^{59} + 20 q^{61} + 4 q^{65} - 72 q^{69} - 36 q^{71} - 62 q^{75} + 16 q^{79} + 18 q^{81} + 8 q^{85} + 12 q^{89} - 52 q^{95} + 68 q^{99}+O(q^{100})$$ 10 * q - 4 * q^5 - 30 * q^9 - 16 * q^11 - 4 * q^15 + 36 * q^19 + 4 * q^21 + 12 * q^25 - 4 * q^29 + 8 * q^31 - 18 * q^35 + 4 * q^39 - 24 * q^41 + 36 * q^45 - 38 * q^49 + 4 * q^51 - 32 * q^55 + 28 * q^59 + 20 * q^61 + 4 * q^65 - 72 * q^69 - 36 * q^71 - 62 * q^75 + 16 * q^79 + 18 * q^81 + 8 * q^85 + 12 * q^89 - 52 * q^95 + 68 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 4x^{9} + 2x^{8} + 16x^{7} - 15x^{6} - 40x^{5} - 75x^{4} + 400x^{3} + 250x^{2} - 2500x + 3125$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{9} - 4\nu^{8} + 2\nu^{7} + 16\nu^{6} - 15\nu^{5} - 40\nu^{4} - 75\nu^{3} + 400\nu^{2} + 250\nu - 2500 ) / 625$$ (v^9 - 4*v^8 + 2*v^7 + 16*v^6 - 15*v^5 - 40*v^4 - 75*v^3 + 400*v^2 + 250*v - 2500) / 625 $$\beta_{3}$$ $$=$$ $$( - 4 \nu^{9} + 11 \nu^{8} + 22 \nu^{7} - 39 \nu^{6} - 100 \nu^{5} + 120 \nu^{4} + 700 \nu^{3} + \cdots + 3125 ) / 2500$$ (-4*v^9 + 11*v^8 + 22*v^7 - 39*v^6 - 100*v^5 + 120*v^4 + 700*v^3 - 425*v^2 - 3250*v + 3125) / 2500 $$\beta_{4}$$ $$=$$ $$( \nu^{9} - 3\nu^{7} + 36\nu^{5} + 36\nu^{4} - 115\nu^{3} - 320\nu^{2} + 325\nu + 500 ) / 500$$ (v^9 - 3*v^7 + 36*v^5 + 36*v^4 - 115*v^3 - 320*v^2 + 325*v + 500) / 500 $$\beta_{5}$$ $$=$$ $$( \nu^{9} - 9\nu^{7} - 6\nu^{6} + 24\nu^{5} + 60\nu^{4} - 195\nu^{3} - 300\nu^{2} + 675\nu + 750 ) / 500$$ (v^9 - 9*v^7 - 6*v^6 + 24*v^5 + 60*v^4 - 195*v^3 - 300*v^2 + 675*v + 750) / 500 $$\beta_{6}$$ $$=$$ $$( -\nu^{9} + 6\nu^{8} + 5\nu^{7} - 32\nu^{6} - 8\nu^{5} + 80\nu^{4} + 235\nu^{3} - 450\nu^{2} - 1375\nu + 2500 ) / 500$$ (-v^9 + 6*v^8 + 5*v^7 - 32*v^6 - 8*v^5 + 80*v^4 + 235*v^3 - 450*v^2 - 1375*v + 2500) / 500 $$\beta_{7}$$ $$=$$ $$( - 8 \nu^{9} + 7 \nu^{8} + 34 \nu^{7} - 53 \nu^{6} - 80 \nu^{5} + 120 \nu^{4} + 900 \nu^{3} + \cdots + 4375 ) / 2500$$ (-8*v^9 + 7*v^8 + 34*v^7 - 53*v^6 - 80*v^5 + 120*v^4 + 900*v^3 + 675*v^2 - 4250*v + 4375) / 2500 $$\beta_{8}$$ $$=$$ $$( - 8 \nu^{9} + 27 \nu^{8} + 4 \nu^{7} - 113 \nu^{6} - 60 \nu^{5} + 320 \nu^{4} + 1700 \nu^{3} + \cdots + 14375 ) / 2500$$ (-8*v^9 + 27*v^8 + 4*v^7 - 113*v^6 - 60*v^5 + 320*v^4 + 1700*v^3 - 2825*v^2 - 4500*v + 14375) / 2500 $$\beta_{9}$$ $$=$$ $$( 18 \nu^{9} - 37 \nu^{8} - 74 \nu^{7} + 213 \nu^{6} + 300 \nu^{5} - 340 \nu^{4} - 2650 \nu^{3} + \cdots - 19375 ) / 2500$$ (18*v^9 - 37*v^8 - 74*v^7 + 213*v^6 + 300*v^5 - 340*v^4 - 2650*v^3 + 2475*v^2 + 12750*v - 19375) / 2500
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} + \beta_{7} + \beta_{6} - \beta_{4} + 2$$ b9 + b7 + b6 - b4 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{9} + 3\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - 3\beta_{3} + 3\beta_{2} + \beta _1 + 1$$ b9 + 3*b8 + b7 + b6 - b5 - 3*b3 + 3*b2 + b1 + 1 $$\nu^{4}$$ $$=$$ $$3\beta_{9} + 2\beta_{8} + 7\beta_{7} - 2\beta_{6} + 5\beta_{5} + 3\beta_{4} + 12\beta_{3} + 4\beta_{2}$$ 3*b9 + 2*b8 + 7*b7 - 2*b6 + 5*b5 + 3*b4 + 12*b3 + 4*b2 $$\nu^{5}$$ $$=$$ $$4 \beta_{9} + 2 \beta_{8} + 10 \beta_{7} + 4 \beta_{6} - 8 \beta_{5} + 8 \beta_{4} - 16 \beta_{3} + \cdots - 14$$ 4*b9 + 2*b8 + 10*b7 + 4*b6 - 8*b5 + 8*b4 - 16*b3 - 5*b2 - 2*b1 - 14 $$\nu^{6}$$ $$=$$ $$14\beta_{9} + 6\beta_{8} - 26\beta_{6} - 6\beta_{5} + 16\beta_{4} + 80\beta_{3} - 16\beta_{2} - 24\beta _1 + 33$$ 14*b9 + 6*b8 - 26*b6 - 6*b5 + 16*b4 + 80*b3 - 16*b2 - 24*b1 + 33 $$\nu^{7}$$ $$=$$ $$-20\beta_{9} - 42\beta_{8} - 2\beta_{7} - 28\beta_{5} + 60\beta_{4} + 40\beta_{3} + 2\beta_{2} + 73\beta _1 + 30$$ -20*b9 - 42*b8 - 2*b7 - 28*b5 + 60*b4 + 40*b3 + 2*b2 + 73*b1 + 30 $$\nu^{8}$$ $$=$$ $$113 \beta_{9} - 62 \beta_{8} - 73 \beta_{7} + 73 \beta_{6} - 62 \beta_{5} - 75 \beta_{4} + 316 \beta_{3} + \cdots - 32$$ 113*b9 - 62*b8 - 73*b7 + 73*b6 - 62*b5 - 75*b4 + 316*b3 - 200*b2 + 12*b1 - 32 $$\nu^{9}$$ $$=$$ $$123 \beta_{9} + 75 \beta_{8} - 183 \beta_{7} + 363 \beta_{6} - 91 \beta_{5} - 36 \beta_{4} - 81 \beta_{3} + \cdots + 849$$ 123*b9 + 75*b8 - 183*b7 + 363*b6 - 91*b5 - 36*b4 - 81*b3 + 387*b2 + 81*b1 + 849

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 1.64680 + 1.51263i 2.19360 + 0.433733i −1.11501 − 1.93823i −2.16128 − 0.573465i 1.43589 − 1.71412i 1.43589 + 1.71412i −2.16128 + 0.573465i −1.11501 + 1.93823i 2.19360 − 0.433733i 1.64680 − 1.51263i
0 3.08870i 0 −1.64680 + 1.51263i 0 3.54010i 0 −6.54010 0
209.2 0 3.05748i 0 −2.19360 + 0.433733i 0 3.34821i 0 −6.34821 0
209.3 0 2.57527i 0 1.11501 1.93823i 0 0.632021i 0 −3.63202 0
209.4 0 1.76881i 0 2.16128 0.573465i 0 2.87131i 0 −0.128689 0
209.5 0 1.16232i 0 −1.43589 1.71412i 0 4.64901i 0 1.64901 0
209.6 0 1.16232i 0 −1.43589 + 1.71412i 0 4.64901i 0 1.64901 0
209.7 0 1.76881i 0 2.16128 + 0.573465i 0 2.87131i 0 −0.128689 0
209.8 0 2.57527i 0 1.11501 + 1.93823i 0 0.632021i 0 −3.63202 0
209.9 0 3.05748i 0 −2.19360 0.433733i 0 3.34821i 0 −6.34821 0
209.10 0 3.08870i 0 −1.64680 1.51263i 0 3.54010i 0 −6.54010 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 209.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.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.d.f 10
4.b odd 2 1 520.2.d.c 10
5.b even 2 1 inner 1040.2.d.f 10
5.c odd 4 1 5200.2.a.cm 5
5.c odd 4 1 5200.2.a.cn 5
12.b even 2 1 4680.2.l.i 10
20.d odd 2 1 520.2.d.c 10
20.e even 4 1 2600.2.a.bb 5
20.e even 4 1 2600.2.a.bc 5
60.h even 2 1 4680.2.l.i 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.d.c 10 4.b odd 2 1
520.2.d.c 10 20.d odd 2 1
1040.2.d.f 10 1.a even 1 1 trivial
1040.2.d.f 10 5.b even 2 1 inner
2600.2.a.bb 5 20.e even 4 1
2600.2.a.bc 5 20.e even 4 1
4680.2.l.i 10 12.b even 2 1
4680.2.l.i 10 60.h even 2 1
5200.2.a.cm 5 5.c odd 4 1
5200.2.a.cn 5 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{10} + 30T_{3}^{8} + 333T_{3}^{6} + 1660T_{3}^{4} + 3556T_{3}^{2} + 2500$$ acting on $$S_{2}^{\mathrm{new}}(1040, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$T^{10} + 30 T^{8} + \cdots + 2500$$
$5$ $$T^{10} + 4 T^{9} + \cdots + 3125$$
$7$ $$T^{10} + 54 T^{8} + \cdots + 10000$$
$11$ $$(T^{5} + 8 T^{4} + \cdots + 1544)^{2}$$
$13$ $$(T^{2} + 1)^{5}$$
$17$ $$T^{10} + 66 T^{8} + \cdots + 5184$$
$19$ $$(T^{5} - 18 T^{4} + \cdots + 5368)^{2}$$
$23$ $$T^{10} + 140 T^{8} + \cdots + 2611456$$
$29$ $$(T^{5} + 2 T^{4} + \cdots + 9232)^{2}$$
$31$ $$(T^{5} - 4 T^{4} + \cdots - 1944)^{2}$$
$37$ $$T^{10} + 194 T^{8} + \cdots + 2755600$$
$41$ $$(T^{5} + 12 T^{4} + \cdots + 5120)^{2}$$
$43$ $$T^{10} + 178 T^{8} + \cdots + 25100100$$
$47$ $$T^{10} + 134 T^{8} + \cdots + 3600$$
$53$ $$T^{10} + 240 T^{8} + \cdots + 1982464$$
$59$ $$(T^{5} - 14 T^{4} + \cdots + 4008)^{2}$$
$61$ $$(T^{5} - 10 T^{4} + \cdots + 1472)^{2}$$
$67$ $$T^{10} + \cdots + 1317980416$$
$71$ $$(T^{5} + 18 T^{4} + \cdots + 5770)^{2}$$
$73$ $$T^{10} + 352 T^{8} + \cdots + 85229824$$
$79$ $$(T^{5} - 8 T^{4} + \cdots - 14720)^{2}$$
$83$ $$T^{10} + \cdots + 181494784$$
$89$ $$(T^{5} - 6 T^{4} + \cdots - 2208)^{2}$$
$97$ $$T^{10} + 324 T^{8} + \cdots + 25600$$