# Properties

 Label 2028.2.q.h Level $2028$ Weight $2$ Character orbit 2028.q Analytic conductor $16.194$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2028 = 2^{2} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2028.q (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.1936615299$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 156) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{2} q^{3} + 4 \zeta_{12}^{3} q^{5} + 2 \zeta_{12} q^{7} + ( -1 + \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{2} q^{3} + 4 \zeta_{12}^{3} q^{5} + 2 \zeta_{12} q^{7} + ( -1 + \zeta_{12}^{2} ) q^{9} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{11} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{15} + ( 2 - 2 \zeta_{12}^{2} ) q^{17} -2 \zeta_{12} q^{19} + 2 \zeta_{12}^{3} q^{21} -11 q^{25} - q^{27} + 6 \zeta_{12}^{2} q^{29} + 10 \zeta_{12}^{3} q^{31} -4 \zeta_{12} q^{33} + ( -8 + 8 \zeta_{12}^{2} ) q^{35} + ( 10 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{37} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{41} + ( 4 - 4 \zeta_{12}^{2} ) q^{43} -4 \zeta_{12} q^{45} -4 \zeta_{12}^{3} q^{47} -3 \zeta_{12}^{2} q^{49} + 2 q^{51} -10 q^{53} -16 \zeta_{12}^{2} q^{55} -2 \zeta_{12}^{3} q^{57} + 8 \zeta_{12} q^{59} + ( 14 - 14 \zeta_{12}^{2} ) q^{61} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{63} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{67} + 16 \zeta_{12} q^{71} -10 \zeta_{12}^{3} q^{73} -11 \zeta_{12}^{2} q^{75} -8 q^{77} -16 q^{79} -\zeta_{12}^{2} q^{81} + 8 \zeta_{12} q^{85} + ( -6 + 6 \zeta_{12}^{2} ) q^{87} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{89} + ( -10 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{93} + ( 8 - 8 \zeta_{12}^{2} ) q^{95} -2 \zeta_{12} q^{97} -4 \zeta_{12}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{3} - 2q^{9} + 4q^{17} - 44q^{25} - 4q^{27} + 12q^{29} - 16q^{35} + 8q^{43} - 6q^{49} + 8q^{51} - 40q^{53} - 32q^{55} + 28q^{61} - 22q^{75} - 32q^{77} - 64q^{79} - 2q^{81} - 12q^{87} + 16q^{95} + O(q^{100})$$

## Character values

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

 $$n$$ $$677$$ $$1015$$ $$1861$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 - \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 0.500000 + 0.866025i 0 4.00000i 0 −1.73205 1.00000i 0 −0.500000 + 0.866025i 0
361.2 0 0.500000 + 0.866025i 0 4.00000i 0 1.73205 + 1.00000i 0 −0.500000 + 0.866025i 0
1837.1 0 0.500000 0.866025i 0 4.00000i 0 1.73205 1.00000i 0 −0.500000 0.866025i 0
1837.2 0 0.500000 0.866025i 0 4.00000i 0 −1.73205 + 1.00000i 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.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{5}^{2} + 16$$ $$T_{7}^{4} - 4 T_{7}^{2} + 16$$ $$T_{11}^{4} - 16 T_{11}^{2} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 - T + T^{2} )^{2}$$
$5$ $$( 16 + T^{2} )^{2}$$
$7$ $$16 - 4 T^{2} + T^{4}$$
$11$ $$256 - 16 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( 4 - 2 T + T^{2} )^{2}$$
$19$ $$16 - 4 T^{2} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$( 36 - 6 T + T^{2} )^{2}$$
$31$ $$( 100 + T^{2} )^{2}$$
$37$ $$10000 - 100 T^{2} + T^{4}$$
$41$ $$4096 - 64 T^{2} + T^{4}$$
$43$ $$( 16 - 4 T + T^{2} )^{2}$$
$47$ $$( 16 + T^{2} )^{2}$$
$53$ $$( 10 + T )^{4}$$
$59$ $$4096 - 64 T^{2} + T^{4}$$
$61$ $$( 196 - 14 T + T^{2} )^{2}$$
$67$ $$16 - 4 T^{2} + T^{4}$$
$71$ $$65536 - 256 T^{2} + T^{4}$$
$73$ $$( 100 + T^{2} )^{2}$$
$79$ $$( 16 + T )^{4}$$
$83$ $$T^{4}$$
$89$ $$256 - 16 T^{2} + T^{4}$$
$97$ $$16 - 4 T^{2} + T^{4}$$