# Properties

 Label 2028.2.b.a Level $2028$ Weight $2$ Character orbit 2028.b Analytic conductor $16.194$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2028 = 2^{2} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2028.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.1936615299$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + 4 i q^{5} -2 i q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + 4 i q^{5} -2 i q^{7} + q^{9} -4 i q^{11} -4 i q^{15} -2 q^{17} + 2 i q^{19} + 2 i q^{21} -11 q^{25} - q^{27} -6 q^{29} + 10 i q^{31} + 4 i q^{33} + 8 q^{35} + 10 i q^{37} -8 i q^{41} -4 q^{43} + 4 i q^{45} -4 i q^{47} + 3 q^{49} + 2 q^{51} -10 q^{53} + 16 q^{55} -2 i q^{57} -8 i q^{59} -14 q^{61} -2 i q^{63} -2 i q^{67} -16 i q^{71} -10 i q^{73} + 11 q^{75} -8 q^{77} -16 q^{79} + q^{81} -8 i q^{85} + 6 q^{87} -4 i q^{89} -10 i q^{93} -8 q^{95} + 2 i q^{97} -4 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{9} - 4q^{17} - 22q^{25} - 2q^{27} - 12q^{29} + 16q^{35} - 8q^{43} + 6q^{49} + 4q^{51} - 20q^{53} + 32q^{55} - 28q^{61} + 22q^{75} - 16q^{77} - 32q^{79} + 2q^{81} + 12q^{87} - 16q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
0 −1.00000 0 4.00000i 0 2.00000i 0 1.00000 0
337.2 0 −1.00000 0 4.00000i 0 2.00000i 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 2028.2.b.a 2
3.b odd 2 1 6084.2.b.j 2
13.b even 2 1 inner 2028.2.b.a 2
13.c even 3 2 2028.2.q.h 4
13.d odd 4 1 156.2.a.a 1
13.d odd 4 1 2028.2.a.c 1
13.e even 6 2 2028.2.q.h 4
13.f odd 12 2 2028.2.i.e 2
13.f odd 12 2 2028.2.i.g 2
39.d odd 2 1 6084.2.b.j 2
39.f even 4 1 468.2.a.d 1
39.f even 4 1 6084.2.a.b 1
52.f even 4 1 624.2.a.e 1
52.f even 4 1 8112.2.a.bi 1
65.f even 4 1 3900.2.h.b 2
65.g odd 4 1 3900.2.a.m 1
65.k even 4 1 3900.2.h.b 2
91.i even 4 1 7644.2.a.k 1
104.j odd 4 1 2496.2.a.bc 1
104.m even 4 1 2496.2.a.o 1
117.y odd 12 2 4212.2.i.l 2
117.z even 12 2 4212.2.i.b 2
156.l odd 4 1 1872.2.a.s 1
312.w odd 4 1 7488.2.a.d 1
312.y even 4 1 7488.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.a 1 13.d odd 4 1
468.2.a.d 1 39.f even 4 1
624.2.a.e 1 52.f even 4 1
1872.2.a.s 1 156.l odd 4 1
2028.2.a.c 1 13.d odd 4 1
2028.2.b.a 2 1.a even 1 1 trivial
2028.2.b.a 2 13.b even 2 1 inner
2028.2.i.e 2 13.f odd 12 2
2028.2.i.g 2 13.f odd 12 2
2028.2.q.h 4 13.c even 3 2
2028.2.q.h 4 13.e even 6 2
2496.2.a.o 1 104.m even 4 1
2496.2.a.bc 1 104.j odd 4 1
3900.2.a.m 1 65.g odd 4 1
3900.2.h.b 2 65.f even 4 1
3900.2.h.b 2 65.k even 4 1
4212.2.i.b 2 117.z even 12 2
4212.2.i.l 2 117.y odd 12 2
6084.2.a.b 1 39.f even 4 1
6084.2.b.j 2 3.b odd 2 1
6084.2.b.j 2 39.d odd 2 1
7488.2.a.c 1 312.y even 4 1
7488.2.a.d 1 312.w odd 4 1
7644.2.a.k 1 91.i even 4 1
8112.2.a.bi 1 52.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(2028, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$16 + T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$4 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$100 + T^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$64 + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$16 + T^{2}$$
$53$ $$( 10 + T )^{2}$$
$59$ $$64 + T^{2}$$
$61$ $$( 14 + T )^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$256 + T^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$( 16 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$16 + T^{2}$$
$97$ $$4 + T^{2}$$