# Properties

 Label 4056.2.c.e 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 24) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta q^{5} + q^{9}+O(q^{10})$$ q - q^3 + b * q^5 + q^9 $$q - q^{3} + \beta q^{5} + q^{9} + 2 \beta q^{11} - \beta q^{15} - 2 q^{17} + 2 \beta q^{19} + 8 q^{23} + q^{25} - q^{27} + 6 q^{29} - 4 \beta q^{31} - 2 \beta q^{33} + 3 \beta q^{37} + 3 \beta q^{41} - 4 q^{43} + \beta q^{45} + 7 q^{49} + 2 q^{51} - 2 q^{53} - 8 q^{55} - 2 \beta q^{57} + 2 \beta q^{59} - 2 q^{61} + 2 \beta q^{67} - 8 q^{69} - 4 \beta q^{71} + 5 \beta q^{73} - q^{75} - 8 q^{79} + q^{81} + 2 \beta q^{83} - 2 \beta q^{85} - 6 q^{87} - 3 \beta q^{89} + 4 \beta q^{93} - 8 q^{95} - \beta q^{97} + 2 \beta q^{99} +O(q^{100})$$ q - q^3 + b * q^5 + q^9 + 2*b * q^11 - b * q^15 - 2 * q^17 + 2*b * q^19 + 8 * q^23 + q^25 - q^27 + 6 * q^29 - 4*b * q^31 - 2*b * q^33 + 3*b * q^37 + 3*b * q^41 - 4 * q^43 + b * q^45 + 7 * q^49 + 2 * q^51 - 2 * q^53 - 8 * q^55 - 2*b * q^57 + 2*b * q^59 - 2 * q^61 + 2*b * q^67 - 8 * q^69 - 4*b * q^71 + 5*b * q^73 - q^75 - 8 * q^79 + q^81 + 2*b * q^83 - 2*b * q^85 - 6 * q^87 - 3*b * q^89 + 4*b * q^93 - 8 * q^95 - b * q^97 + 2*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 $$2 q - 2 q^{3} + 2 q^{9} - 4 q^{17} + 16 q^{23} + 2 q^{25} - 2 q^{27} + 12 q^{29} - 8 q^{43} + 14 q^{49} + 4 q^{51} - 4 q^{53} - 16 q^{55} - 4 q^{61} - 16 q^{69} - 2 q^{75} - 16 q^{79} + 2 q^{81} - 12 q^{87} - 16 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 - 4 * q^17 + 16 * q^23 + 2 * q^25 - 2 * q^27 + 12 * q^29 - 8 * q^43 + 14 * q^49 + 4 * q^51 - 4 * q^53 - 16 * q^55 - 4 * q^61 - 16 * q^69 - 2 * q^75 - 16 * q^79 + 2 * q^81 - 12 * q^87 - 16 * q^95

## 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 2.00000i 0 0 0 1.00000 0
337.2 0 −1.00000 0 2.00000i 0 0 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.e 2
13.b even 2 1 inner 4056.2.c.e 2
13.d odd 4 1 24.2.a.a 1
13.d odd 4 1 4056.2.a.i 1
39.f even 4 1 72.2.a.a 1
52.f even 4 1 48.2.a.a 1
52.f even 4 1 8112.2.a.be 1
65.f even 4 1 600.2.f.e 2
65.g odd 4 1 600.2.a.h 1
65.k even 4 1 600.2.f.e 2
91.i even 4 1 1176.2.a.i 1
91.z odd 12 2 1176.2.q.i 2
91.bb even 12 2 1176.2.q.a 2
104.j odd 4 1 192.2.a.d 1
104.m even 4 1 192.2.a.b 1
117.y odd 12 2 648.2.i.g 2
117.z even 12 2 648.2.i.b 2
143.g even 4 1 2904.2.a.c 1
156.l odd 4 1 144.2.a.b 1
195.j odd 4 1 1800.2.f.c 2
195.n even 4 1 1800.2.a.m 1
195.u odd 4 1 1800.2.f.c 2
208.l even 4 1 768.2.d.d 2
208.m odd 4 1 768.2.d.e 2
208.r odd 4 1 768.2.d.e 2
208.s even 4 1 768.2.d.d 2
221.g odd 4 1 6936.2.a.p 1
247.i even 4 1 8664.2.a.j 1
260.l odd 4 1 1200.2.f.b 2
260.s odd 4 1 1200.2.f.b 2
260.u even 4 1 1200.2.a.d 1
273.o odd 4 1 3528.2.a.d 1
273.cb odd 12 2 3528.2.s.y 2
273.cd even 12 2 3528.2.s.j 2
312.w odd 4 1 576.2.a.b 1
312.y even 4 1 576.2.a.d 1
364.p odd 4 1 2352.2.a.i 1
364.bw odd 12 2 2352.2.q.r 2
364.ce even 12 2 2352.2.q.l 2
429.l odd 4 1 8712.2.a.u 1
468.bs even 12 2 1296.2.i.m 2
468.ch odd 12 2 1296.2.i.e 2
520.t even 4 1 4800.2.a.cc 1
520.x odd 4 1 4800.2.f.bg 2
520.y even 4 1 4800.2.f.d 2
520.bj even 4 1 4800.2.f.d 2
520.bk odd 4 1 4800.2.f.bg 2
520.bo odd 4 1 4800.2.a.q 1
572.k odd 4 1 5808.2.a.s 1
624.s odd 4 1 2304.2.d.k 2
624.u even 4 1 2304.2.d.i 2
624.bm even 4 1 2304.2.d.i 2
624.bo odd 4 1 2304.2.d.k 2
728.x odd 4 1 9408.2.a.cc 1
728.ba even 4 1 9408.2.a.h 1
780.u even 4 1 3600.2.f.r 2
780.bb odd 4 1 3600.2.a.v 1
780.bn even 4 1 3600.2.f.r 2
1092.u even 4 1 7056.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 13.d odd 4 1
48.2.a.a 1 52.f even 4 1
72.2.a.a 1 39.f even 4 1
144.2.a.b 1 156.l odd 4 1
192.2.a.b 1 104.m even 4 1
192.2.a.d 1 104.j odd 4 1
576.2.a.b 1 312.w odd 4 1
576.2.a.d 1 312.y even 4 1
600.2.a.h 1 65.g odd 4 1
600.2.f.e 2 65.f even 4 1
600.2.f.e 2 65.k even 4 1
648.2.i.b 2 117.z even 12 2
648.2.i.g 2 117.y odd 12 2
768.2.d.d 2 208.l even 4 1
768.2.d.d 2 208.s even 4 1
768.2.d.e 2 208.m odd 4 1
768.2.d.e 2 208.r odd 4 1
1176.2.a.i 1 91.i even 4 1
1176.2.q.a 2 91.bb even 12 2
1176.2.q.i 2 91.z odd 12 2
1200.2.a.d 1 260.u even 4 1
1200.2.f.b 2 260.l odd 4 1
1200.2.f.b 2 260.s odd 4 1
1296.2.i.e 2 468.ch odd 12 2
1296.2.i.m 2 468.bs even 12 2
1800.2.a.m 1 195.n even 4 1
1800.2.f.c 2 195.j odd 4 1
1800.2.f.c 2 195.u odd 4 1
2304.2.d.i 2 624.u even 4 1
2304.2.d.i 2 624.bm even 4 1
2304.2.d.k 2 624.s odd 4 1
2304.2.d.k 2 624.bo odd 4 1
2352.2.a.i 1 364.p odd 4 1
2352.2.q.l 2 364.ce even 12 2
2352.2.q.r 2 364.bw odd 12 2
2904.2.a.c 1 143.g even 4 1
3528.2.a.d 1 273.o odd 4 1
3528.2.s.j 2 273.cd even 12 2
3528.2.s.y 2 273.cb odd 12 2
3600.2.a.v 1 780.bb odd 4 1
3600.2.f.r 2 780.u even 4 1
3600.2.f.r 2 780.bn even 4 1
4056.2.a.i 1 13.d odd 4 1
4056.2.c.e 2 1.a even 1 1 trivial
4056.2.c.e 2 13.b even 2 1 inner
4800.2.a.q 1 520.bo odd 4 1
4800.2.a.cc 1 520.t even 4 1
4800.2.f.d 2 520.y even 4 1
4800.2.f.d 2 520.bj even 4 1
4800.2.f.bg 2 520.x odd 4 1
4800.2.f.bg 2 520.bk odd 4 1
5808.2.a.s 1 572.k odd 4 1
6936.2.a.p 1 221.g odd 4 1
7056.2.a.q 1 1092.u even 4 1
8112.2.a.be 1 52.f even 4 1
8664.2.a.j 1 247.i even 4 1
8712.2.a.u 1 429.l odd 4 1
9408.2.a.h 1 728.ba even 4 1
9408.2.a.cc 1 728.x odd 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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