# Properties

 Label 1040.2.bg.n Level $1040$ Weight $2$ Character orbit 1040.bg Analytic conductor $8.304$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(577,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 3]))

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

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

chi := DirichletCharacter("1040.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1040 = 2^{4} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1040.bg (of order $$4$$, degree $$2$$, 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.619810816.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1$$ x^8 - 2*x^5 + 14*x^4 - 8*x^3 + 2*x^2 + 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} - \beta_{4} - \beta_{2} + 1) q^{3} + (\beta_{7} + \beta_{6}) q^{5} + ( - 2 \beta_{6} + 2 \beta_{5} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{5} - \beta_{4} + \beta_{3} + \cdots + \beta_1) q^{9}+O(q^{10})$$ q + (b5 - b4 - b2 + 1) * q^3 + (b7 + b6) * q^5 + (-2*b6 + 2*b5 - 2*b4 + 2*b3 - b1) * q^7 + (b5 - b4 + b3 - b2 + b1) * q^9 $$q + (\beta_{5} - \beta_{4} - \beta_{2} + 1) q^{3} + (\beta_{7} + \beta_{6}) q^{5} + ( - 2 \beta_{6} + 2 \beta_{5} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{7} + \beta_1) q^{99}+O(q^{100})$$ q + (b5 - b4 - b2 + 1) * q^3 + (b7 + b6) * q^5 + (-2*b6 + 2*b5 - 2*b4 + 2*b3 - b1) * q^7 + (b5 - b4 + b3 - b2 + b1) * q^9 + (b6 - b5 - b3 - b2 - 1) * q^11 + (b5 - 2*b4 - b3 - 2*b2 - b1) * q^13 + (b7 + b5 - 3*b4 - b3) * q^15 + (-b7 - b6 - b4 + 2*b2 - b1 - 2) * q^17 + (b3 - 2*b2 - 2) * q^19 + (-b7 - b6 + b4 - b2 + b1 - 1) * q^21 + (b7 + b6 - b4 - b3 - 2*b2 - b1 - 2) * q^23 + (b7 - 2*b6 + b3 + 2*b2 - b1 + 1) * q^25 + (-2*b7 - b6 - b5 + 2*b4 - b2 + 2*b1 - 1) * q^27 + (2*b6 - b5 + b4 - b3 - 2*b2 + b1) * q^29 + (-b6 - 2*b5 + b4 + b2 - 1) * q^31 + (b7 - b5 - b4 - 2) * q^33 + (-b7 + b6 + b5 - 2*b4 + b3 + b2 + 3) * q^35 + (b5 - b4 + b3 - 6*b2 - b1) * q^37 + (2*b7 + b5 - 3*b4 - 4*b2 + b1) * q^39 + (-2*b7 - b6 - b5 - 2*b2 - 2*b1 + 2) * q^41 + (b7 - 3*b6 + 4*b5 - b4 + 3*b3 - b1) * q^43 + (2*b7 - 3*b6 + 6*b5 - 6*b4 + 3*b3 - 3*b2 + 1) * q^45 + (-2*b2 - 3*b1) * q^47 + (4*b5 - 2*b4 - 2*b3 - 1) * q^49 + (-3*b5 + 3*b4 - 3*b3 + 4*b2) * q^51 + (-b7 - 2*b6 + b5 - 3*b4 + 3*b2 - b1 - 3) * q^53 + (-2*b6 - b3 + 3*b2 + b1 - 1) * q^55 + (-b7 - 3*b5 + 3*b4 + 2*b3 - 6) * q^57 + (4*b6 - b5 + 5*b4 + 2*b2 - 2) * q^59 + (3*b7 - 2*b5 - 2*b4 + b3) * q^61 + (-5*b7 - 5*b5 + 3*b4 - 2*b3 - 2) * q^63 + (-b7 - 2*b5 - 2*b4 - 5*b3 + 5*b2 + b1) * q^65 + (b7 - 2*b5 + b3) * q^67 + (3*b7 - 4*b4 - b3 - 2) * q^69 + (-b7 - 2*b6 + 4*b5 - 6*b4 - 2*b2 - b1 + 2) * q^71 + (-3*b7 - 2*b5 + 2*b4 - b3) * q^73 + (b7 - 2*b6 + 4*b5 - 4*b4 - 2*b3 + 5*b2 - b1 + 5) * q^75 + (b6 - b5 + 2*b4 - 3*b2 + 3) * q^77 + (-6*b6 + 5*b5 - 5*b4 + 5*b3 + 2*b2 - 2*b1) * q^79 + (-b7 + 2*b5 + 4*b4 + 3*b3 - 1) * q^81 + (4*b6 + 4*b2 - 3*b1) * q^83 + (-3*b7 + 2*b5 + b3 + 3*b2 - b1 - 1) * q^85 + (b6 - b5 + 2*b3 - 3*b2 - 3) * q^87 + (-b7 - 2*b6 - 5*b5 + 3*b4 + 5*b2 - b1 - 5) * q^89 + (2*b7 + b6 + 2*b5 + 3*b4 - b3 - 5*b2 + 3*b1 - 1) * q^91 + (-2*b6 + 6*b2 - b1) * q^93 + (-b7 - 3*b6 - b4 + b3 + 2*b1) * q^95 + (2*b7 + 3*b5 + 3*b4 + 5*b3 + 2) * q^97 + (b7 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 6 q^{3} + 2 q^{5}+O(q^{10})$$ 8 * q + 6 * q^3 + 2 * q^5 $$8 q + 6 q^{3} + 2 q^{5} - 6 q^{11} - 2 q^{13} + 2 q^{15} - 16 q^{17} - 14 q^{19} - 12 q^{21} - 14 q^{23} + 12 q^{25} - 12 q^{27} - 2 q^{31} - 8 q^{33} + 24 q^{35} + 6 q^{39} + 16 q^{41} - 6 q^{43} + 6 q^{45} - 24 q^{49} - 24 q^{53} - 10 q^{55} - 40 q^{57} - 22 q^{59} + 20 q^{61} - 16 q^{63} + 12 q^{67} - 4 q^{69} + 10 q^{71} - 4 q^{73} + 30 q^{75} + 24 q^{77} - 20 q^{81} - 20 q^{85} - 16 q^{87} - 28 q^{89} - 20 q^{91} + 2 q^{95} + 12 q^{97} + 2 q^{99}+O(q^{100})$$ 8 * q + 6 * q^3 + 2 * q^5 - 6 * q^11 - 2 * q^13 + 2 * q^15 - 16 * q^17 - 14 * q^19 - 12 * q^21 - 14 * q^23 + 12 * q^25 - 12 * q^27 - 2 * q^31 - 8 * q^33 + 24 * q^35 + 6 * q^39 + 16 * q^41 - 6 * q^43 + 6 * q^45 - 24 * q^49 - 24 * q^53 - 10 * q^55 - 40 * q^57 - 22 * q^59 + 20 * q^61 - 16 * q^63 + 12 * q^67 - 4 * q^69 + 10 * q^71 - 4 * q^73 + 30 * q^75 + 24 * q^77 - 20 * q^81 - 20 * q^85 - 16 * q^87 - 28 * q^89 - 20 * q^91 + 2 * q^95 + 12 * q^97 + 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 378\nu + 63 ) / 319$$ (64*v^7 + 16*v^6 + 4*v^5 - 127*v^4 + 944*v^3 - 276*v^2 + 378*v + 63) / 319 $$\beta_{2}$$ $$=$$ $$( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319$$ (-63*v^7 + 64*v^6 + 16*v^5 + 130*v^4 - 1009*v^3 + 1448*v^2 - 402*v - 67) / 319 $$\beta_{3}$$ $$=$$ $$( -67\nu^{7} + 63\nu^{6} - 64\nu^{5} + 118\nu^{4} - 1068\nu^{3} + 1545\nu^{2} - 1263\nu + 268 ) / 319$$ (-67*v^7 + 63*v^6 - 64*v^5 + 118*v^4 - 1068*v^3 + 1545*v^2 - 1263*v + 268) / 319 $$\beta_{4}$$ $$=$$ $$( 83\nu^{7} - 59\nu^{6} + 65\nu^{5} - 70\nu^{4} + 1304\nu^{3} - 1614\nu^{2} + 1198\nu + 306 ) / 319$$ (83*v^7 - 59*v^6 + 65*v^5 - 70*v^4 + 1304*v^3 - 1614*v^2 + 1198*v + 306) / 319 $$\beta_{5}$$ $$=$$ $$( -172\nu^{7} - 43\nu^{6} + 69\nu^{5} + 441\nu^{4} - 2218\nu^{3} + 662\nu^{2} + 619\nu - 269 ) / 319$$ (-172*v^7 - 43*v^6 + 69*v^5 + 441*v^4 - 2218*v^3 + 662*v^2 + 619*v - 269) / 319 $$\beta_{6}$$ $$=$$ $$( -196\nu^{7} - 49\nu^{6} - 92\nu^{5} + 369\nu^{4} - 2572\nu^{3} + 1244\nu^{2} - 1038\nu - 173 ) / 319$$ (-196*v^7 - 49*v^6 - 92*v^5 + 369*v^4 - 2572*v^3 + 1244*v^2 - 1038*v - 173) / 319 $$\beta_{7}$$ $$=$$ $$\nu^{7} - 2\nu^{4} + 14\nu^{3} - 8\nu^{2} + \nu + 2$$ v^7 - 2*v^4 + 14*v^3 - 8*v^2 + v + 2
 $$\nu$$ $$=$$ $$( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 ) / 2$$ (-b7 - b5 + b4 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2}$$ b6 - b5 + b4 - b3 + 2*b2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{7} + 5\beta_{5} - 5\beta_{4} - 2\beta_{2} + 3\beta _1 + 2 ) / 2$$ (3*b7 + 5*b5 - 5*b4 - 2*b2 + 3*b1 + 2) / 2 $$\nu^{4}$$ $$=$$ $$-\beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 7$$ -b7 - b5 + 5*b4 + 4*b3 - 7 $$\nu^{5}$$ $$=$$ $$( 11\beta_{7} + 2\beta_{6} + 9\beta_{5} - 11\beta_{4} - 12\beta_{3} + 12\beta_{2} - 11\beta _1 + 12 ) / 2$$ (11*b7 + 2*b6 + 9*b5 - 11*b4 - 12*b3 + 12*b2 - 11*b1 + 12) / 2 $$\nu^{6}$$ $$=$$ $$-15\beta_{6} + 16\beta_{5} - 16\beta_{4} + 16\beta_{3} - 28\beta_{2} + 7\beta_1$$ -15*b6 + 16*b5 - 16*b4 + 16*b3 - 28*b2 + 7*b1 $$\nu^{7}$$ $$=$$ $$( -43\beta_{7} + 16\beta_{6} - 89\beta_{5} + 105\beta_{4} + 60\beta_{2} - 43\beta _1 - 60 ) / 2$$ (-43*b7 + 16*b6 - 89*b5 + 105*b4 + 60*b2 - 43*b1 - 60) / 2

## Character values

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

 $$n$$ $$261$$ $$417$$ $$561$$ $$911$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$-\beta_{2}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 −0.252709 − 0.252709i 1.18254 + 1.18254i 0.561103 + 0.561103i −1.49094 − 1.49094i −0.252709 + 0.252709i 1.18254 − 1.18254i 0.561103 − 0.561103i −1.49094 + 1.49094i
0 −0.725850 + 0.725850i 0 2.23127 0.146426i 0 4.24997i 0 1.94628i 0
577.2 0 0.240275 0.240275i 0 −1.60536 + 1.55654i 0 3.95872i 0 2.88454i 0
577.3 0 1.33000 1.33000i 0 −1.45220 1.70032i 0 1.61845i 0 0.537789i 0
577.4 0 2.15558 2.15558i 0 1.82630 + 1.29021i 0 1.90970i 0 6.29303i 0
593.1 0 −0.725850 0.725850i 0 2.23127 + 0.146426i 0 4.24997i 0 1.94628i 0
593.2 0 0.240275 + 0.240275i 0 −1.60536 1.55654i 0 3.95872i 0 2.88454i 0
593.3 0 1.33000 + 1.33000i 0 −1.45220 + 1.70032i 0 1.61845i 0 0.537789i 0
593.4 0 2.15558 + 2.15558i 0 1.82630 1.29021i 0 1.90970i 0 6.29303i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 577.4 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.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.bg.n 8
4.b odd 2 1 65.2.k.b yes 8
5.c odd 4 1 1040.2.cd.n 8
12.b even 2 1 585.2.w.e 8
13.d odd 4 1 1040.2.cd.n 8
20.d odd 2 1 325.2.k.b 8
20.e even 4 1 65.2.f.b 8
20.e even 4 1 325.2.f.b 8
52.b odd 2 1 845.2.k.b 8
52.f even 4 1 65.2.f.b 8
52.f even 4 1 845.2.f.b 8
52.i odd 6 2 845.2.o.c 16
52.j odd 6 2 845.2.o.d 16
52.l even 12 2 845.2.t.c 16
52.l even 12 2 845.2.t.d 16
60.l odd 4 1 585.2.n.e 8
65.k even 4 1 inner 1040.2.bg.n 8
156.l odd 4 1 585.2.n.e 8
260.l odd 4 1 325.2.k.b 8
260.l odd 4 1 845.2.k.b 8
260.p even 4 1 845.2.f.b 8
260.s odd 4 1 65.2.k.b yes 8
260.u even 4 1 325.2.f.b 8
260.be odd 12 2 845.2.o.d 16
260.bg even 12 2 845.2.t.d 16
260.bj even 12 2 845.2.t.c 16
260.bl odd 12 2 845.2.o.c 16
780.bn even 4 1 585.2.w.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.f.b 8 20.e even 4 1
65.2.f.b 8 52.f even 4 1
65.2.k.b yes 8 4.b odd 2 1
65.2.k.b yes 8 260.s odd 4 1
325.2.f.b 8 20.e even 4 1
325.2.f.b 8 260.u even 4 1
325.2.k.b 8 20.d odd 2 1
325.2.k.b 8 260.l odd 4 1
585.2.n.e 8 60.l odd 4 1
585.2.n.e 8 156.l odd 4 1
585.2.w.e 8 12.b even 2 1
585.2.w.e 8 780.bn even 4 1
845.2.f.b 8 52.f even 4 1
845.2.f.b 8 260.p even 4 1
845.2.k.b 8 52.b odd 2 1
845.2.k.b 8 260.l odd 4 1
845.2.o.c 16 52.i odd 6 2
845.2.o.c 16 260.bl odd 12 2
845.2.o.d 16 52.j odd 6 2
845.2.o.d 16 260.be odd 12 2
845.2.t.c 16 52.l even 12 2
845.2.t.c 16 260.bj even 12 2
845.2.t.d 16 52.l even 12 2
845.2.t.d 16 260.bg even 12 2
1040.2.bg.n 8 1.a even 1 1 trivial
1040.2.bg.n 8 65.k even 4 1 inner
1040.2.cd.n 8 5.c odd 4 1
1040.2.cd.n 8 13.d odd 4 1

## Hecke kernels

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

 $$T_{3}^{8} - 6T_{3}^{7} + 18T_{3}^{6} - 20T_{3}^{5} + 8T_{3}^{4} + 4T_{3}^{3} + 32T_{3}^{2} - 16T_{3} + 4$$ T3^8 - 6*T3^7 + 18*T3^6 - 20*T3^5 + 8*T3^4 + 4*T3^3 + 32*T3^2 - 16*T3 + 4 $$T_{7}^{8} + 40T_{7}^{6} + 504T_{7}^{4} + 2096T_{7}^{2} + 2704$$ T7^8 + 40*T7^6 + 504*T7^4 + 2096*T7^2 + 2704 $$T_{11}^{8} + 6T_{11}^{7} + 18T_{11}^{6} + 8T_{11}^{5} - 4T_{11}^{4} - 12T_{11}^{3} + 32T_{11}^{2} - 16T_{11} + 4$$ T11^8 + 6*T11^7 + 18*T11^6 + 8*T11^5 - 4*T11^4 - 12*T11^3 + 32*T11^2 - 16*T11 + 4 $$T_{19}^{8} + 14T_{19}^{7} + 98T_{19}^{6} + 396T_{19}^{5} + 1004T_{19}^{4} + 1524T_{19}^{3} + 1352T_{19}^{2} + 520T_{19} + 100$$ T19^8 + 14*T19^7 + 98*T19^6 + 396*T19^5 + 1004*T19^4 + 1524*T19^3 + 1352*T19^2 + 520*T19 + 100

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - 6 T^{7} + \cdots + 4$$
$5$ $$T^{8} - 2 T^{7} + \cdots + 625$$
$7$ $$T^{8} + 40 T^{6} + \cdots + 2704$$
$11$ $$T^{8} + 6 T^{7} + \cdots + 4$$
$13$ $$T^{8} + 2 T^{7} + \cdots + 28561$$
$17$ $$T^{8} + 16 T^{7} + \cdots + 13456$$
$19$ $$T^{8} + 14 T^{7} + \cdots + 100$$
$23$ $$T^{8} + 14 T^{7} + \cdots + 40804$$
$29$ $$T^{8} + 44 T^{6} + \cdots + 10000$$
$31$ $$T^{8} + 2 T^{7} + \cdots + 16900$$
$37$ $$T^{8} + 140 T^{6} + \cdots + 336400$$
$41$ $$T^{8} - 16 T^{7} + \cdots + 13456$$
$43$ $$T^{8} + 6 T^{7} + \cdots + 8836$$
$47$ $$T^{8} + 160 T^{6} + \cdots + 26896$$
$53$ $$T^{8} + 24 T^{7} + \cdots + 19600$$
$59$ $$T^{8} + 22 T^{7} + \cdots + 119716$$
$61$ $$(T^{4} - 10 T^{3} + \cdots - 3628)^{2}$$
$67$ $$(T^{4} - 6 T^{3} + \cdots - 148)^{2}$$
$71$ $$T^{8} - 10 T^{7} + \cdots + 1223236$$
$73$ $$(T^{4} + 2 T^{3} + \cdots + 740)^{2}$$
$79$ $$T^{8} + 292 T^{6} + \cdots + 13719616$$
$83$ $$T^{8} + 416 T^{6} + \cdots + 54346384$$
$89$ $$T^{8} + 28 T^{7} + \cdots + 1795600$$
$97$ $$(T^{4} - 6 T^{3} + \cdots + 3704)^{2}$$