# Properties

 Label 2028.2.i.e Level $2028$ Weight $2$ Character orbit 2028.i 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.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.1936615299$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 156) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} -4 q^{5} + 2 \zeta_{6} q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} -4 q^{5} + 2 \zeta_{6} q^{7} -\zeta_{6} q^{9} + ( 4 - 4 \zeta_{6} ) q^{11} + ( -4 + 4 \zeta_{6} ) q^{15} -2 \zeta_{6} q^{17} + 2 \zeta_{6} q^{19} + 2 q^{21} + 11 q^{25} - q^{27} + ( 6 - 6 \zeta_{6} ) q^{29} -10 q^{31} -4 \zeta_{6} q^{33} -8 \zeta_{6} q^{35} + ( -10 + 10 \zeta_{6} ) q^{37} + ( -8 + 8 \zeta_{6} ) q^{41} -4 \zeta_{6} q^{43} + 4 \zeta_{6} q^{45} -4 q^{47} + ( 3 - 3 \zeta_{6} ) q^{49} -2 q^{51} -10 q^{53} + ( -16 + 16 \zeta_{6} ) q^{55} + 2 q^{57} + 8 \zeta_{6} q^{59} + 14 \zeta_{6} q^{61} + ( 2 - 2 \zeta_{6} ) q^{63} + ( -2 + 2 \zeta_{6} ) q^{67} -16 \zeta_{6} q^{71} -10 q^{73} + ( 11 - 11 \zeta_{6} ) q^{75} + 8 q^{77} -16 q^{79} + ( -1 + \zeta_{6} ) q^{81} + 8 \zeta_{6} q^{85} -6 \zeta_{6} q^{87} + ( 4 - 4 \zeta_{6} ) q^{89} + ( -10 + 10 \zeta_{6} ) q^{93} -8 \zeta_{6} q^{95} + 2 \zeta_{6} q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - 8q^{5} + 2q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} - 8q^{5} + 2q^{7} - q^{9} + 4q^{11} - 4q^{15} - 2q^{17} + 2q^{19} + 4q^{21} + 22q^{25} - 2q^{27} + 6q^{29} - 20q^{31} - 4q^{33} - 8q^{35} - 10q^{37} - 8q^{41} - 4q^{43} + 4q^{45} - 8q^{47} + 3q^{49} - 4q^{51} - 20q^{53} - 16q^{55} + 4q^{57} + 8q^{59} + 14q^{61} + 2q^{63} - 2q^{67} - 16q^{71} - 20q^{73} + 11q^{75} + 16q^{77} - 32q^{79} - q^{81} + 8q^{85} - 6q^{87} + 4q^{89} - 10q^{93} - 8q^{95} + 2q^{97} - 8q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$677$$ $$1015$$ $$1861$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
529.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 −4.00000 0 1.00000 + 1.73205i 0 −0.500000 0.866025i 0
2005.1 0 0.500000 + 0.866025i 0 −4.00000 0 1.00000 1.73205i 0 −0.500000 + 0.866025i 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.c even 3 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2028, [\chi])$$:

 $$T_{5} + 4$$ $$T_{7}^{2} - 2 T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$( 4 + T )^{2}$$
$7$ $$4 - 2 T + T^{2}$$
$11$ $$16 - 4 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$4 + 2 T + T^{2}$$
$19$ $$4 - 2 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$36 - 6 T + T^{2}$$
$31$ $$( 10 + T )^{2}$$
$37$ $$100 + 10 T + T^{2}$$
$41$ $$64 + 8 T + T^{2}$$
$43$ $$16 + 4 T + T^{2}$$
$47$ $$( 4 + T )^{2}$$
$53$ $$( 10 + T )^{2}$$
$59$ $$64 - 8 T + T^{2}$$
$61$ $$196 - 14 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$256 + 16 T + T^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$( 16 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$16 - 4 T + T^{2}$$
$97$ $$4 - 2 T + T^{2}$$