# Properties

 Label 4056.2.c.f Level $4056$ Weight $2$ Character orbit 4056.c Analytic conductor $32.387$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(337,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4056.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4056 = 2^{3} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4056.c (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: $$32.3873230598$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 312) 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + 2 \beta q^{7} + q^{9}+O(q^{10})$$ q - q^3 + 2*b * q^7 + q^9 $$q - q^{3} + 2 \beta q^{7} + q^{9} + \beta q^{11} + 6 q^{17} - 2 \beta q^{19} - 2 \beta q^{21} - 4 q^{23} + 5 q^{25} - q^{27} + 10 q^{29} - 4 \beta q^{31} - \beta q^{33} + \beta q^{37} + 4 q^{43} - \beta q^{47} - 9 q^{49} - 6 q^{51} - 2 q^{53} + 2 \beta q^{57} - 5 \beta q^{59} + 10 q^{61} + 2 \beta q^{63} + 4 \beta q^{67} + 4 q^{69} + \beta q^{71} + 5 \beta q^{73} - 5 q^{75} - 8 q^{77} + 8 q^{79} + q^{81} + 3 \beta q^{83} - 10 q^{87} + 6 \beta q^{89} + 4 \beta q^{93} - \beta q^{97} + \beta q^{99} +O(q^{100})$$ q - q^3 + 2*b * q^7 + q^9 + b * q^11 + 6 * q^17 - 2*b * q^19 - 2*b * q^21 - 4 * q^23 + 5 * q^25 - q^27 + 10 * q^29 - 4*b * q^31 - b * q^33 + b * q^37 + 4 * q^43 - b * q^47 - 9 * q^49 - 6 * q^51 - 2 * q^53 + 2*b * q^57 - 5*b * q^59 + 10 * q^61 + 2*b * q^63 + 4*b * q^67 + 4 * q^69 + b * q^71 + 5*b * q^73 - 5 * q^75 - 8 * q^77 + 8 * q^79 + q^81 + 3*b * q^83 - 10 * q^87 + 6*b * q^89 + 4*b * q^93 - b * q^97 + b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{9} + 12 q^{17} - 8 q^{23} + 10 q^{25} - 2 q^{27} + 20 q^{29} + 8 q^{43} - 18 q^{49} - 12 q^{51} - 4 q^{53} + 20 q^{61} + 8 q^{69} - 10 q^{75} - 16 q^{77} + 16 q^{79} + 2 q^{81} - 20 q^{87}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 + 12 * q^17 - 8 * q^23 + 10 * q^25 - 2 * q^27 + 20 * q^29 + 8 * q^43 - 18 * q^49 - 12 * q^51 - 4 * q^53 + 20 * q^61 + 8 * q^69 - 10 * q^75 - 16 * q^77 + 16 * q^79 + 2 * q^81 - 20 * q^87

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$$.

 $$n$$ $$1015$$ $$2029$$ $$2705$$ $$3889$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
0 −1.00000 0 0 0 4.00000i 0 1.00000 0
337.2 0 −1.00000 0 0 0 4.00000i 0 1.00000 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4056.2.c.f 2
13.b even 2 1 inner 4056.2.c.f 2
13.d odd 4 1 312.2.a.b 1
13.d odd 4 1 4056.2.a.f 1
39.f even 4 1 936.2.a.d 1
52.f even 4 1 624.2.a.g 1
52.f even 4 1 8112.2.a.y 1
65.g odd 4 1 7800.2.a.w 1
104.j odd 4 1 2496.2.a.u 1
104.m even 4 1 2496.2.a.j 1
156.l odd 4 1 1872.2.a.k 1
312.w odd 4 1 7488.2.a.bh 1
312.y even 4 1 7488.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.a.b 1 13.d odd 4 1
624.2.a.g 1 52.f even 4 1
936.2.a.d 1 39.f even 4 1
1872.2.a.k 1 156.l odd 4 1
2496.2.a.j 1 104.m even 4 1
2496.2.a.u 1 104.j odd 4 1
4056.2.a.f 1 13.d odd 4 1
4056.2.c.f 2 1.a even 1 1 trivial
4056.2.c.f 2 13.b even 2 1 inner
7488.2.a.y 1 312.y even 4 1
7488.2.a.bh 1 312.w odd 4 1
7800.2.a.w 1 65.g odd 4 1
8112.2.a.y 1 52.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4056, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11}^{2} + 4$$ T11^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2}$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T - 10)^{2}$$
$31$ $$T^{2} + 64$$
$37$ $$T^{2} + 4$$
$41$ $$T^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 4$$
$53$ $$(T + 2)^{2}$$
$59$ $$T^{2} + 100$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$T^{2} + 4$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2} + 144$$
$97$ $$T^{2} + 4$$