Properties

 Label 3960.2.d.f Level $3960$ Weight $2$ Character orbit 3960.d Analytic conductor $31.621$ 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] = [3960,2,Mod(3169,3960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3960.3169");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3960.d (of order $$2$$, degree $$1$$, 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: $$31.6207592004$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.47985531136.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 19x^{6} + 91x^{4} + 45x^{2} + 4$$ x^8 + 19*x^6 + 91*x^4 + 45*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 440) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{5} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{7}+O(q^{10})$$ q - b3 * q^5 + (-b4 - b3 + b1) * q^7 $$q - \beta_{3} q^{5} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{7} - q^{11} + (\beta_{7} - \beta_{6} + \cdots + \beta_{3}) q^{13}+ \cdots + (3 \beta_{4} + 3 \beta_{3} - 2 \beta_1) q^{97}+O(q^{100})$$ q - b3 * q^5 + (-b4 - b3 + b1) * q^7 - q^11 + (b7 - b6 + b4 + b3) * q^13 + (b7 - b6 + b4 + b3 - b2 - b1) * q^17 + (b7 - b5 + b3 + 3) * q^19 + (-b2 + 2*b1) * q^23 + (-b7 + b5 - b1 - 1) * q^25 + (-b6 - b5 + b4 + 1) * q^29 + (b7 - b5 + b4 - 3) * q^31 + (-b6 + 2*b4 + b3 - 2*b1 - 4) * q^35 + (b4 + b3 - b2 + b1) * q^37 + (b7 + b6 - b4 + b3 - 2) * q^41 + (b7 - b6 - b2 + 2*b1) * q^43 + (-b7 + b6 - 2*b4 - 2*b3 + b2) * q^47 + (-b6 - b5 + 3*b4 - 2*b3 - 2) * q^49 + (2*b4 + 2*b3 - b2 - b1) * q^53 + b3 * q^55 + (b7 + b6 + 2) * q^59 + (b6 + b5 - b4 + 7) * q^61 + (-b7 - b6 - b5 - b4 - b3 + b2 + 2*b1 + 1) * q^65 + (2*b4 + 2*b3 - b2 - 2*b1) * q^67 + (b7 - b5 + 3*b4 - 2*b3 + 1) * q^71 + (b7 - b6 + 3*b4 + 3*b3 - 2*b1) * q^73 + (b4 + b3 - b1) * q^77 + (-2*b7 - 2*b6 + 4) * q^79 + (-b7 + b6 - b2 - 4*b1) * q^83 + (-3*b7 - 2*b4 - b3 - b2 + 2*b1 - 4) * q^85 + (b6 + b5 - b3 + 5) * q^89 + (-2*b7 - 2*b6 - 2*b4 + 2*b3) * q^91 + (b7 - b6 - b5 - 3*b4 - b3 - b1 - 3) * q^95 + (3*b4 + 3*b3 - 2*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 8 q^{11} + 18 q^{19} - 2 q^{25} + 6 q^{29} - 30 q^{31} - 30 q^{35} - 20 q^{41} - 18 q^{49} + 12 q^{59} + 58 q^{61} + 8 q^{65} + 2 q^{71} + 40 q^{79} - 26 q^{85} + 42 q^{89} + 8 q^{91} - 28 q^{95}+O(q^{100})$$ 8 * q - 8 * q^11 + 18 * q^19 - 2 * q^25 + 6 * q^29 - 30 * q^31 - 30 * q^35 - 20 * q^41 - 18 * q^49 + 12 * q^59 + 58 * q^61 + 8 * q^65 + 2 * q^71 + 40 * q^79 - 26 * q^85 + 42 * q^89 + 8 * q^91 - 28 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 19x^{6} + 91x^{4} + 45x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{7} - 18\nu^{5} - 73\nu^{3} + 32\nu ) / 4$$ (-v^7 - 18*v^5 - 73*v^3 + 32*v) / 4 $$\beta_{3}$$ $$=$$ $$( 2\nu^{7} + \nu^{6} + 38\nu^{5} + 18\nu^{4} + 180\nu^{3} + 77\nu^{2} + 72\nu + 12 ) / 8$$ (2*v^7 + v^6 + 38*v^5 + 18*v^4 + 180*v^3 + 77*v^2 + 72*v + 12) / 8 $$\beta_{4}$$ $$=$$ $$( 2\nu^{7} - \nu^{6} + 38\nu^{5} - 18\nu^{4} + 180\nu^{3} - 77\nu^{2} + 72\nu - 12 ) / 8$$ (2*v^7 - v^6 + 38*v^5 - 18*v^4 + 180*v^3 - 77*v^2 + 72*v - 12) / 8 $$\beta_{5}$$ $$=$$ $$( -3\nu^{7} - \nu^{6} - 56\nu^{5} - 20\nu^{4} - 257\nu^{3} - 95\nu^{2} - 76\nu - 4 ) / 8$$ (-3*v^7 - v^6 - 56*v^5 - 20*v^4 - 257*v^3 - 95*v^2 - 76*v - 4) / 8 $$\beta_{6}$$ $$=$$ $$( 5\nu^{7} + 2\nu^{6} + 94\nu^{5} + 38\nu^{4} + 437\nu^{3} + 180\nu^{2} + 148\nu + 56 ) / 8$$ (5*v^7 + 2*v^6 + 94*v^5 + 38*v^4 + 437*v^3 + 180*v^2 + 148*v + 56) / 8 $$\beta_{7}$$ $$=$$ $$( -5\nu^{7} + 2\nu^{6} - 94\nu^{5} + 38\nu^{4} - 437\nu^{3} + 180\nu^{2} - 148\nu + 56 ) / 8$$ (-5*v^7 + 2*v^6 - 94*v^5 + 38*v^4 - 437*v^3 + 180*v^2 - 148*v + 56) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} - \beta_{3} - 5$$ b6 + b5 - b3 - 5 $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{6} - 2\beta_{4} - 2\beta_{3} + \beta_{2} - 9\beta_1$$ -b7 + b6 - 2*b4 - 2*b3 + b2 - 9*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{7} - 11\beta_{6} - 13\beta_{5} + 4\beta_{4} + 9\beta_{3} + 49$$ 2*b7 - 11*b6 - 13*b5 + 4*b4 + 9*b3 + 49 $$\nu^{5}$$ $$=$$ $$17\beta_{7} - 17\beta_{6} + 36\beta_{4} + 36\beta_{3} - 13\beta_{2} + 85\beta_1$$ 17*b7 - 17*b6 + 36*b4 + 36*b3 - 13*b2 + 85*b1 $$\nu^{6}$$ $$=$$ $$-36\beta_{7} + 121\beta_{6} + 157\beta_{5} - 76\beta_{4} - 81\beta_{3} - 509$$ -36*b7 + 121*b6 + 157*b5 - 76*b4 - 81*b3 - 509 $$\nu^{7}$$ $$=$$ $$-233\beta_{7} + 233\beta_{6} - 502\beta_{4} - 502\beta_{3} + 157\beta_{2} - 841\beta_1$$ -233*b7 + 233*b6 - 502*b4 - 502*b3 + 157*b2 - 841*b1

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times$$.

 $$n$$ $$991$$ $$1981$$ $$2377$$ $$2521$$ $$3521$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3169.1
 − 2.67673i 2.67673i 0.339102i − 0.339102i 3.36007i − 3.36007i 0.655762i − 0.655762i
0 0 0 −2.06639 0.854430i 0 4.38559i 0 0 0
3169.2 0 0 0 −2.06639 + 0.854430i 0 4.38559i 0 0 0
3169.3 0 0 0 −0.422782 2.19574i 0 4.05237i 0 0 0
3169.4 0 0 0 −0.422782 + 2.19574i 0 4.05237i 0 0 0
3169.5 0 0 0 0.256321 2.22133i 0 1.08258i 0 0 0
3169.6 0 0 0 0.256321 + 2.22133i 0 1.08258i 0 0 0
3169.7 0 0 0 2.23285 0.119978i 0 0.415806i 0 0 0
3169.8 0 0 0 2.23285 + 0.119978i 0 0.415806i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3169.8 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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3960.2.d.f 8
3.b odd 2 1 440.2.b.d 8
5.b even 2 1 inner 3960.2.d.f 8
12.b even 2 1 880.2.b.j 8
15.d odd 2 1 440.2.b.d 8
15.e even 4 1 2200.2.a.x 4
15.e even 4 1 2200.2.a.y 4
60.h even 2 1 880.2.b.j 8
60.l odd 4 1 4400.2.a.cb 4
60.l odd 4 1 4400.2.a.ce 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.b.d 8 3.b odd 2 1
440.2.b.d 8 15.d odd 2 1
880.2.b.j 8 12.b even 2 1
880.2.b.j 8 60.h even 2 1
2200.2.a.x 4 15.e even 4 1
2200.2.a.y 4 15.e even 4 1
3960.2.d.f 8 1.a even 1 1 trivial
3960.2.d.f 8 5.b even 2 1 inner
4400.2.a.cb 4 60.l odd 4 1
4400.2.a.ce 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3960, [\chi])$$:

 $$T_{7}^{8} + 37T_{7}^{6} + 364T_{7}^{4} + 432T_{7}^{2} + 64$$ T7^8 + 37*T7^6 + 364*T7^4 + 432*T7^2 + 64 $$T_{29}^{4} - 3T_{29}^{3} - 38T_{29}^{2} + 32$$ T29^4 - 3*T29^3 - 38*T29^2 + 32

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + T^{6} + \cdots + 625$$
$7$ $$T^{8} + 37 T^{6} + \cdots + 64$$
$11$ $$(T + 1)^{8}$$
$13$ $$(T^{2} + 16)^{4}$$
$17$ $$T^{8} + 137 T^{6} + \cdots + 1024$$
$19$ $$(T^{4} - 9 T^{3} + \cdots - 128)^{2}$$
$23$ $$T^{8} + 154 T^{6} + \cdots + 732736$$
$29$ $$(T^{4} - 3 T^{3} - 38 T^{2} + 32)^{2}$$
$31$ $$(T^{4} + 15 T^{3} + \cdots + 16)^{2}$$
$37$ $$T^{8} + 135 T^{6} + \cdots + 21904$$
$41$ $$(T^{4} + 10 T^{3} + \cdots - 1024)^{2}$$
$43$ $$T^{8} + 192 T^{6} + \cdots + 1290496$$
$47$ $$T^{8} + 124 T^{6} + \cdots + 409600$$
$53$ $$T^{8} + 177 T^{6} + \cdots + 1638400$$
$59$ $$(T^{4} - 6 T^{3} - 47 T^{2} + \cdots - 32)^{2}$$
$61$ $$(T^{4} - 29 T^{3} + \cdots + 160)^{2}$$
$67$ $$T^{8} + 202 T^{6} + \cdots + 322624$$
$71$ $$(T^{4} - T^{3} - 137 T^{2} + \cdots - 1336)^{2}$$
$73$ $$T^{8} + 228 T^{6} + \cdots + 1048576$$
$79$ $$(T^{4} - 20 T^{3} + \cdots - 19456)^{2}$$
$83$ $$T^{8} + 524 T^{6} + \cdots + 38142976$$
$89$ $$(T^{4} - 21 T^{3} + \cdots - 346)^{2}$$
$97$ $$T^{8} + 310 T^{6} + \cdots + 262144$$