# Properties

 Label 3960.2.d.d Level $3960$ Weight $2$ Character orbit 3960.d Analytic conductor $31.621$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.350464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 1320) 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} - \beta_{2}) q^{5} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{7}+O(q^{10})$$ q + (-b5 - b2) * q^5 + (b5 - b3 + b2) * q^7 $$q + ( - \beta_{5} - \beta_{2}) q^{5} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{7} - q^{11} + ( - \beta_{4} - \beta_{2}) q^{13} + \beta_{2} q^{17} + 2 q^{19} + ( - 2 \beta_{5} + 2 \beta_{3} - 4 \beta_{2}) q^{23} + ( - 2 \beta_{3} - \beta_1 - 1) q^{25} + (\beta_{5} + \beta_{3} - 6) q^{29} + ( - \beta_{5} - \beta_{3} - 2 \beta_1 - 2) q^{31} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + 4) q^{35} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2}) q^{37} + ( - \beta_{5} - \beta_{3} - 2 \beta_1 + 6) q^{41} + (3 \beta_{5} - 4 \beta_{4} + \cdots + 3 \beta_{2}) q^{43}+ \cdots + ( - 2 \beta_{4} - 4 \beta_{2}) q^{97}+O(q^{100})$$ q + (-b5 - b2) * q^5 + (b5 - b3 + b2) * q^7 - q^11 + (-b4 - b2) * q^13 + b2 * q^17 + 2 * q^19 + (-2*b5 + 2*b3 - 4*b2) * q^23 + (-2*b3 - b1 - 1) * q^25 + (b5 + b3 - 6) * q^29 + (-b5 - b3 - 2*b1 - 2) * q^31 + (-b4 + 2*b3 - b2 + 4) * q^35 + (-2*b5 + 2*b3 + 2*b2) * q^37 + (-b5 - b3 - 2*b1 + 6) * q^41 + (3*b5 - 4*b4 - 3*b3 + 3*b2) * q^43 + (2*b5 - 2*b3 - 2*b2) * q^47 + (-2*b5 - 2*b3 + b1 - 1) * q^49 + (b5 - 2*b4 - b3 - 2*b2) * q^53 + (b5 + b2) * q^55 + (2*b5 + 2*b3 + 3*b1) * q^59 + (-2*b5 - 2*b3 - 2*b1 - 6) * q^61 + (b5 - 2*b4 - b3 - b2 - 2) * q^65 + (2*b5 - 2*b3 - 4*b2) * q^67 + (4*b5 + 4*b3 - b1) * q^71 + (2*b5 + b4 - 2*b3 + 5*b2) * q^73 + (-b5 + b3 - b2) * q^77 + (2*b5 + 2*b3 + 10) * q^79 + (5*b4 + b2) * q^83 + (b4 + b2 + b1 + 2) * q^85 + (-3*b5 - 3*b3 - 2*b1 - 8) * q^89 + (b5 + b3 + 2) * q^91 + (-2*b5 - 2*b2) * q^95 + (-2*b4 - 4*b2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{5}+O(q^{10})$$ 6 * q + 2 * q^5 $$6 q + 2 q^{5} - 6 q^{11} + 12 q^{19} - 2 q^{25} - 40 q^{29} - 8 q^{31} + 20 q^{35} + 40 q^{41} + 2 q^{49} - 2 q^{55} - 8 q^{59} - 28 q^{61} - 12 q^{65} - 16 q^{71} + 52 q^{79} + 12 q^{85} - 36 q^{89} + 8 q^{91} + 4 q^{95}+O(q^{100})$$ 6 * q + 2 * q^5 - 6 * q^11 + 12 * q^19 - 2 * q^25 - 40 * q^29 - 8 * q^31 + 20 * q^35 + 40 * q^41 + 2 * q^49 - 2 * q^55 - 8 * q^59 - 28 * q^61 - 12 * q^65 - 16 * q^71 + 52 * q^79 + 12 * q^85 - 36 * q^89 + 8 * q^91 + 4 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{5} - 16\nu^{4} + 8\nu^{3} + 2\nu^{2} - 4\nu - 76 ) / 23$$ (2*v^5 - 16*v^4 + 8*v^3 + 2*v^2 - 4*v - 76) / 23 $$\beta_{2}$$ $$=$$ $$( 4\nu^{5} - 9\nu^{4} + 16\nu^{3} + 4\nu^{2} + 38\nu - 14 ) / 23$$ (4*v^5 - 9*v^4 + 16*v^3 + 4*v^2 + 38*v - 14) / 23 $$\beta_{3}$$ $$=$$ $$( 13\nu^{5} - 12\nu^{4} + 6\nu^{3} + 36\nu^{2} + 112\nu - 11 ) / 23$$ (13*v^5 - 12*v^4 + 6*v^3 + 36*v^2 + 112*v - 11) / 23 $$\beta_{4}$$ $$=$$ $$( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 60\nu^{2} - 64\nu + 26 ) / 23$$ (-14*v^5 + 20*v^4 - 10*v^3 - 60*v^2 - 64*v + 26) / 23 $$\beta_{5}$$ $$=$$ $$-\nu^{5} + 2\nu^{4} - 2\nu^{3} - 2\nu^{2} - 4\nu + 3$$ -v^5 + 2*v^4 - 2*v^3 - 2*v^2 - 4*v + 3
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{3} + 2\beta_{2} + \beta _1 + 2 ) / 4$$ (b5 + b3 + 2*b2 + b1 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - 2\beta_{4} - \beta_{3} + 2\beta_{2} ) / 2$$ (b5 - 2*b4 - b3 + 2*b2) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{4} - 2\beta_{3} + 4\beta_{2} - 2\beta _1 - 4 ) / 2$$ (-b4 - 2*b3 + 4*b2 - 2*b1 - 4) / 2 $$\nu^{4}$$ $$=$$ $$( -\beta_{5} - \beta_{3} - 5\beta _1 - 14 ) / 2$$ (-b5 - b3 - 5*b1 - 14) / 2 $$\nu^{5}$$ $$=$$ $$-4\beta_{5} + 3\beta_{4} + \beta_{3} - 8\beta_{2} - 4\beta _1 - 9$$ -4*b5 + 3*b4 + b3 - 8*b2 - 4*b1 - 9

## 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
 0.403032 − 0.403032i 0.403032 + 0.403032i 1.45161 + 1.45161i 1.45161 − 1.45161i −0.854638 + 0.854638i −0.854638 − 0.854638i
0 0 0 −1.48119 1.67513i 0 4.15633i 0 0 0
3169.2 0 0 0 −1.48119 + 1.67513i 0 4.15633i 0 0 0
3169.3 0 0 0 0.311108 2.21432i 0 1.52543i 0 0 0
3169.4 0 0 0 0.311108 + 2.21432i 0 1.52543i 0 0 0
3169.5 0 0 0 2.17009 0.539189i 0 0.630898i 0 0 0
3169.6 0 0 0 2.17009 + 0.539189i 0 0.630898i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3169.6 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.d 6
3.b odd 2 1 1320.2.d.b 6
5.b even 2 1 inner 3960.2.d.d 6
12.b even 2 1 2640.2.d.f 6
15.d odd 2 1 1320.2.d.b 6
15.e even 4 1 6600.2.a.bq 3
15.e even 4 1 6600.2.a.bu 3
60.h even 2 1 2640.2.d.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.d.b 6 3.b odd 2 1
1320.2.d.b 6 15.d odd 2 1
2640.2.d.f 6 12.b even 2 1
2640.2.d.f 6 60.h even 2 1
3960.2.d.d 6 1.a even 1 1 trivial
3960.2.d.d 6 5.b even 2 1 inner
6600.2.a.bq 3 15.e even 4 1
6600.2.a.bu 3 15.e even 4 1

## Hecke kernels

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

 $$T_{7}^{6} + 20T_{7}^{4} + 48T_{7}^{2} + 16$$ T7^6 + 20*T7^4 + 48*T7^2 + 16 $$T_{29}^{3} + 20T_{29}^{2} + 120T_{29} + 208$$ T29^3 + 20*T29^2 + 120*T29 + 208

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6} - 2 T^{5} + \cdots + 125$$
$7$ $$T^{6} + 20 T^{4} + \cdots + 16$$
$11$ $$(T + 1)^{6}$$
$13$ $$T^{6} + 16 T^{4} + \cdots + 16$$
$17$ $$T^{6} + 12 T^{4} + \cdots + 16$$
$19$ $$(T - 2)^{6}$$
$23$ $$T^{6} + 128 T^{4} + \cdots + 16384$$
$29$ $$(T^{3} + 20 T^{2} + \cdots + 208)^{2}$$
$31$ $$(T^{3} + 4 T^{2} - 56 T + 80)^{2}$$
$37$ $$T^{6} + 272 T^{4} + \cdots + 640000$$
$41$ $$(T^{3} - 20 T^{2} + \cdots + 272)^{2}$$
$43$ $$T^{6} + 276 T^{4} + \cdots + 309136$$
$47$ $$T^{6} + 272 T^{4} + \cdots + 640000$$
$53$ $$T^{6} + 112 T^{4} + \cdots + 6400$$
$59$ $$(T^{3} + 4 T^{2} + \cdots - 592)^{2}$$
$61$ $$(T^{3} + 14 T^{2} + \cdots - 344)^{2}$$
$67$ $$T^{6} + 512 T^{4} + \cdots + 4734976$$
$71$ $$(T^{3} + 8 T^{2} + \cdots - 1712)^{2}$$
$73$ $$T^{6} + 192 T^{4} + \cdots + 204304$$
$79$ $$(T^{3} - 26 T^{2} + \cdots - 184)^{2}$$
$83$ $$T^{6} + 272 T^{4} + \cdots + 583696$$
$89$ $$(T^{3} + 18 T^{2} + \cdots - 1096)^{2}$$
$97$ $$T^{6} + 176 T^{4} + \cdots + 4096$$