# Properties

 Label 2028.1.d.c Level $2028$ Weight $1$ Character orbit 2028.d Self dual yes Analytic conductor $1.012$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2028 = 2^{2} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2028.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.01210384562$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 156) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.160398576.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} -\beta q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} -\beta q^{7} + q^{9} + \beta q^{21} + q^{25} - q^{27} + \beta q^{31} + q^{43} + 2 q^{49} + q^{61} -\beta q^{63} + \beta q^{67} -\beta q^{73} - q^{75} + q^{79} + q^{81} -\beta q^{93} + \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{9} + 2q^{25} - 2q^{27} + 2q^{43} + 4q^{49} + 2q^{61} - 2q^{75} + 2q^{79} + 2q^{81} + O(q^{100})$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
677.1
 1.73205 −1.73205
0 −1.00000 0 0 0 −1.73205 0 1.00000 0
677.2 0 −1.00000 0 0 0 1.73205 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.b even 2 1 inner
39.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2028.1.d.c 2
3.b odd 2 1 CM 2028.1.d.c 2
13.b even 2 1 inner 2028.1.d.c 2
13.c even 3 2 2028.1.o.b 4
13.d odd 4 2 2028.1.g.a 2
13.e even 6 2 2028.1.o.b 4
13.f odd 12 2 156.1.s.a 2
13.f odd 12 2 2028.1.s.a 2
39.d odd 2 1 inner 2028.1.d.c 2
39.f even 4 2 2028.1.g.a 2
39.h odd 6 2 2028.1.o.b 4
39.i odd 6 2 2028.1.o.b 4
39.k even 12 2 156.1.s.a 2
39.k even 12 2 2028.1.s.a 2
52.l even 12 2 624.1.cb.a 2
65.o even 12 2 3900.1.br.b 4
65.s odd 12 2 3900.1.ca.b 2
65.t even 12 2 3900.1.br.b 4
104.u even 12 2 2496.1.cb.b 2
104.x odd 12 2 2496.1.cb.a 2
156.v odd 12 2 624.1.cb.a 2
195.bc odd 12 2 3900.1.br.b 4
195.bh even 12 2 3900.1.ca.b 2
195.bn odd 12 2 3900.1.br.b 4
312.bo even 12 2 2496.1.cb.a 2
312.bq odd 12 2 2496.1.cb.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.1.s.a 2 13.f odd 12 2
156.1.s.a 2 39.k even 12 2
624.1.cb.a 2 52.l even 12 2
624.1.cb.a 2 156.v odd 12 2
2028.1.d.c 2 1.a even 1 1 trivial
2028.1.d.c 2 3.b odd 2 1 CM
2028.1.d.c 2 13.b even 2 1 inner
2028.1.d.c 2 39.d odd 2 1 inner
2028.1.g.a 2 13.d odd 4 2
2028.1.g.a 2 39.f even 4 2
2028.1.o.b 4 13.c even 3 2
2028.1.o.b 4 13.e even 6 2
2028.1.o.b 4 39.h odd 6 2
2028.1.o.b 4 39.i odd 6 2
2028.1.s.a 2 13.f odd 12 2
2028.1.s.a 2 39.k even 12 2
2496.1.cb.a 2 104.x odd 12 2
2496.1.cb.a 2 312.bo even 12 2
2496.1.cb.b 2 104.u even 12 2
2496.1.cb.b 2 312.bq odd 12 2
3900.1.br.b 4 65.o even 12 2
3900.1.br.b 4 65.t even 12 2
3900.1.br.b 4 195.bc odd 12 2
3900.1.br.b 4 195.bn odd 12 2
3900.1.ca.b 2 65.s odd 12 2
3900.1.ca.b 2 195.bh even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 3$$ acting on $$S_{1}^{\mathrm{new}}(2028, [\chi])$$.

## Hecke characteristic polynomials

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