# Properties

 Label 2028.4.b.a Level $2028$ Weight $4$ Character orbit 2028.b Analytic conductor $119.656$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2028,4,Mod(337,2028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2028.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2028 = 2^{2} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2028.b (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: $$119.655873492$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 156) 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.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 3 \beta q^{5} - 2 \beta q^{7} + 9 q^{9} +O(q^{10})$$ q - 3 * q^3 + 3*b * q^5 - 2*b * q^7 + 9 * q^9 $$q - 3 q^{3} + 3 \beta q^{5} - 2 \beta q^{7} + 9 q^{9} + 18 \beta q^{11} - 9 \beta q^{15} - 66 q^{17} - 28 \beta q^{19} + 6 \beta q^{21} - 96 q^{23} + 89 q^{25} - 27 q^{27} + 222 q^{29} - 130 \beta q^{31} - 54 \beta q^{33} + 24 q^{35} - 53 \beta q^{37} + 45 \beta q^{41} - 44 q^{43} + 27 \beta q^{45} + 84 \beta q^{47} + 327 q^{49} + 198 q^{51} + 30 q^{53} - 216 q^{55} + 84 \beta q^{57} + 174 \beta q^{59} - 346 q^{61} - 18 \beta q^{63} + 128 \beta q^{67} + 288 q^{69} + 84 \beta q^{71} - 407 \beta q^{73} - 267 q^{75} + 144 q^{77} + 200 q^{79} + 81 q^{81} - 618 \beta q^{83} - 198 \beta q^{85} - 666 q^{87} + 159 \beta q^{89} + 390 \beta q^{93} + 336 q^{95} + 251 \beta q^{97} + 162 \beta q^{99} +O(q^{100})$$ q - 3 * q^3 + 3*b * q^5 - 2*b * q^7 + 9 * q^9 + 18*b * q^11 - 9*b * q^15 - 66 * q^17 - 28*b * q^19 + 6*b * q^21 - 96 * q^23 + 89 * q^25 - 27 * q^27 + 222 * q^29 - 130*b * q^31 - 54*b * q^33 + 24 * q^35 - 53*b * q^37 + 45*b * q^41 - 44 * q^43 + 27*b * q^45 + 84*b * q^47 + 327 * q^49 + 198 * q^51 + 30 * q^53 - 216 * q^55 + 84*b * q^57 + 174*b * q^59 - 346 * q^61 - 18*b * q^63 + 128*b * q^67 + 288 * q^69 + 84*b * q^71 - 407*b * q^73 - 267 * q^75 + 144 * q^77 + 200 * q^79 + 81 * q^81 - 618*b * q^83 - 198*b * q^85 - 666 * q^87 + 159*b * q^89 + 390*b * q^93 + 336 * q^95 + 251*b * q^97 + 162*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 + 18 * q^9 $$2 q - 6 q^{3} + 18 q^{9} - 132 q^{17} - 192 q^{23} + 178 q^{25} - 54 q^{27} + 444 q^{29} + 48 q^{35} - 88 q^{43} + 654 q^{49} + 396 q^{51} + 60 q^{53} - 432 q^{55} - 692 q^{61} + 576 q^{69} - 534 q^{75} + 288 q^{77} + 400 q^{79} + 162 q^{81} - 1332 q^{87} + 672 q^{95}+O(q^{100})$$ 2 * q - 6 * q^3 + 18 * q^9 - 132 * q^17 - 192 * q^23 + 178 * q^25 - 54 * q^27 + 444 * q^29 + 48 * q^35 - 88 * q^43 + 654 * q^49 + 396 * q^51 + 60 * q^53 - 432 * q^55 - 692 * q^61 + 576 * q^69 - 534 * q^75 + 288 * q^77 + 400 * q^79 + 162 * q^81 - 1332 * q^87 + 672 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$1015$$ $$1861$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 −3.00000 0 6.00000i 0 4.00000i 0 9.00000 0
337.2 0 −3.00000 0 6.00000i 0 4.00000i 0 9.00000 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 2028.4.b.a 2
13.b even 2 1 inner 2028.4.b.a 2
13.d odd 4 1 156.4.a.a 1
13.d odd 4 1 2028.4.a.a 1
39.f even 4 1 468.4.a.b 1
52.f even 4 1 624.4.a.h 1
104.j odd 4 1 2496.4.a.n 1
104.m even 4 1 2496.4.a.e 1
156.l odd 4 1 1872.4.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.4.a.a 1 13.d odd 4 1
468.4.a.b 1 39.f even 4 1
624.4.a.h 1 52.f even 4 1
1872.4.a.j 1 156.l odd 4 1
2028.4.a.a 1 13.d odd 4 1
2028.4.b.a 2 1.a even 1 1 trivial
2028.4.b.a 2 13.b even 2 1 inner
2496.4.a.e 1 104.m even 4 1
2496.4.a.n 1 104.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 36$$ acting on $$S_{4}^{\mathrm{new}}(2028, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} + 36$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2} + 1296$$
$13$ $$T^{2}$$
$17$ $$(T + 66)^{2}$$
$19$ $$T^{2} + 3136$$
$23$ $$(T + 96)^{2}$$
$29$ $$(T - 222)^{2}$$
$31$ $$T^{2} + 67600$$
$37$ $$T^{2} + 11236$$
$41$ $$T^{2} + 8100$$
$43$ $$(T + 44)^{2}$$
$47$ $$T^{2} + 28224$$
$53$ $$(T - 30)^{2}$$
$59$ $$T^{2} + 121104$$
$61$ $$(T + 346)^{2}$$
$67$ $$T^{2} + 65536$$
$71$ $$T^{2} + 28224$$
$73$ $$T^{2} + 662596$$
$79$ $$(T - 200)^{2}$$
$83$ $$T^{2} + 1527696$$
$89$ $$T^{2} + 101124$$
$97$ $$T^{2} + 252004$$