# Properties

 Label 3960.2.d.a Level $3960$ Weight $2$ Character orbit 3960.d Analytic conductor $31.621$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Learn more

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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - i - 2) q^{5} + i q^{7}+O(q^{10})$$ q + (-i - 2) * q^5 + i * q^7 $$q + ( - i - 2) q^{5} + i q^{7} + q^{11} - i q^{17} - q^{19} + (4 i + 3) q^{25} - q^{29} - q^{31} + ( - 2 i + 1) q^{35} + i q^{37} - 6 i q^{43} - 8 i q^{47} + 6 q^{49} + 9 i q^{53} + ( - i - 2) q^{55} + 4 q^{59} - 7 q^{61} - 4 i q^{67} - 5 q^{71} - 14 i q^{73} + i q^{77} - 4 q^{79} + 16 i q^{83} + (2 i - 1) q^{85} - 7 q^{89} + (i + 2) q^{95} - 16 i q^{97} +O(q^{100})$$ q + (-i - 2) * q^5 + i * q^7 + q^11 - i * q^17 - q^19 + (4*i + 3) * q^25 - q^29 - q^31 + (-2*i + 1) * q^35 + i * q^37 - 6*i * q^43 - 8*i * q^47 + 6 * q^49 + 9*i * q^53 + (-i - 2) * q^55 + 4 * q^59 - 7 * q^61 - 4*i * q^67 - 5 * q^71 - 14*i * q^73 + i * q^77 - 4 * q^79 + 16*i * q^83 + (2*i - 1) * q^85 - 7 * q^89 + (i + 2) * q^95 - 16*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5}+O(q^{10})$$ 2 * q - 4 * q^5 $$2 q - 4 q^{5} + 2 q^{11} - 2 q^{19} + 6 q^{25} - 2 q^{29} - 2 q^{31} + 2 q^{35} + 12 q^{49} - 4 q^{55} + 8 q^{59} - 14 q^{61} - 10 q^{71} - 8 q^{79} - 2 q^{85} - 14 q^{89} + 4 q^{95}+O(q^{100})$$ 2 * q - 4 * q^5 + 2 * q^11 - 2 * q^19 + 6 * q^25 - 2 * q^29 - 2 * q^31 + 2 * q^35 + 12 * q^49 - 4 * q^55 + 8 * q^59 - 14 * q^61 - 10 * q^71 - 8 * q^79 - 2 * q^85 - 14 * q^89 + 4 * q^95

## 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
 1.00000i − 1.00000i
0 0 0 −2.00000 1.00000i 0 1.00000i 0 0 0
3169.2 0 0 0 −2.00000 + 1.00000i 0 1.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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.a 2
3.b odd 2 1 440.2.b.c 2
5.b even 2 1 inner 3960.2.d.a 2
12.b even 2 1 880.2.b.g 2
15.d odd 2 1 440.2.b.c 2
15.e even 4 1 2200.2.a.d 1
15.e even 4 1 2200.2.a.h 1
60.h even 2 1 880.2.b.g 2
60.l odd 4 1 4400.2.a.j 1
60.l odd 4 1 4400.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.b.c 2 3.b odd 2 1
440.2.b.c 2 15.d odd 2 1
880.2.b.g 2 12.b even 2 1
880.2.b.g 2 60.h even 2 1
2200.2.a.d 1 15.e even 4 1
2200.2.a.h 1 15.e even 4 1
3960.2.d.a 2 1.a even 1 1 trivial
3960.2.d.a 2 5.b even 2 1 inner
4400.2.a.j 1 60.l odd 4 1
4400.2.a.u 1 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 $$S_{2}^{\mathrm{new}}(3960, [\chi])$$:

 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{29} + 1$$ T29 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 1$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T + 1)^{2}$$
$31$ $$(T + 1)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 81$$
$59$ $$(T - 4)^{2}$$
$61$ $$(T + 7)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T + 5)^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 256$$
$89$ $$(T + 7)^{2}$$
$97$ $$T^{2} + 256$$
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