# Properties

 Label 2028.4.a.c Level $2028$ Weight $4$ Character orbit 2028.a Self dual yes Analytic conductor $119.656$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$2028 = 2^{2} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2028.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$119.655873492$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 12) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} + 18q^{5} - 8q^{7} + 9q^{9} + O(q^{10})$$ $$q + 3q^{3} + 18q^{5} - 8q^{7} + 9q^{9} - 36q^{11} + 54q^{15} + 18q^{17} + 100q^{19} - 24q^{21} + 72q^{23} + 199q^{25} + 27q^{27} - 234q^{29} + 16q^{31} - 108q^{33} - 144q^{35} + 226q^{37} - 90q^{41} + 452q^{43} + 162q^{45} - 432q^{47} - 279q^{49} + 54q^{51} + 414q^{53} - 648q^{55} + 300q^{57} + 684q^{59} + 422q^{61} - 72q^{63} - 332q^{67} + 216q^{69} + 360q^{71} - 26q^{73} + 597q^{75} + 288q^{77} + 512q^{79} + 81q^{81} + 1188q^{83} + 324q^{85} - 702q^{87} + 630q^{89} + 48q^{93} + 1800q^{95} + 1054q^{97} - 324q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 3.00000 0 18.0000 0 −8.00000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$13$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2028.4.a.c 1
13.b even 2 1 12.4.a.a 1
13.d odd 4 2 2028.4.b.c 2
39.d odd 2 1 36.4.a.a 1
52.b odd 2 1 48.4.a.a 1
65.d even 2 1 300.4.a.b 1
65.h odd 4 2 300.4.d.e 2
91.b odd 2 1 588.4.a.c 1
91.r even 6 2 588.4.i.d 2
91.s odd 6 2 588.4.i.e 2
104.e even 2 1 192.4.a.f 1
104.h odd 2 1 192.4.a.l 1
117.n odd 6 2 324.4.e.a 2
117.t even 6 2 324.4.e.h 2
143.d odd 2 1 1452.4.a.d 1
156.h even 2 1 144.4.a.g 1
195.e odd 2 1 900.4.a.g 1
195.s even 4 2 900.4.d.c 2
208.o odd 4 2 768.4.d.j 2
208.p even 4 2 768.4.d.g 2
260.g odd 2 1 1200.4.a.be 1
260.p even 4 2 1200.4.f.d 2
273.g even 2 1 1764.4.a.b 1
273.w odd 6 2 1764.4.k.b 2
273.ba even 6 2 1764.4.k.o 2
312.b odd 2 1 576.4.a.b 1
312.h even 2 1 576.4.a.a 1
364.h even 2 1 2352.4.a.bk 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 13.b even 2 1
36.4.a.a 1 39.d odd 2 1
48.4.a.a 1 52.b odd 2 1
144.4.a.g 1 156.h even 2 1
192.4.a.f 1 104.e even 2 1
192.4.a.l 1 104.h odd 2 1
300.4.a.b 1 65.d even 2 1
300.4.d.e 2 65.h odd 4 2
324.4.e.a 2 117.n odd 6 2
324.4.e.h 2 117.t even 6 2
576.4.a.a 1 312.h even 2 1
576.4.a.b 1 312.b odd 2 1
588.4.a.c 1 91.b odd 2 1
588.4.i.d 2 91.r even 6 2
588.4.i.e 2 91.s odd 6 2
768.4.d.g 2 208.p even 4 2
768.4.d.j 2 208.o odd 4 2
900.4.a.g 1 195.e odd 2 1
900.4.d.c 2 195.s even 4 2
1200.4.a.be 1 260.g odd 2 1
1200.4.f.d 2 260.p even 4 2
1452.4.a.d 1 143.d odd 2 1
1764.4.a.b 1 273.g even 2 1
1764.4.k.b 2 273.w odd 6 2
1764.4.k.o 2 273.ba even 6 2
2028.4.a.c 1 1.a even 1 1 trivial
2028.4.b.c 2 13.d odd 4 2
2352.4.a.bk 1 364.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 18$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2028))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$-18 + T$$
$7$ $$8 + T$$
$11$ $$36 + T$$
$13$ $$T$$
$17$ $$-18 + T$$
$19$ $$-100 + T$$
$23$ $$-72 + T$$
$29$ $$234 + T$$
$31$ $$-16 + T$$
$37$ $$-226 + T$$
$41$ $$90 + T$$
$43$ $$-452 + T$$
$47$ $$432 + T$$
$53$ $$-414 + T$$
$59$ $$-684 + T$$
$61$ $$-422 + T$$
$67$ $$332 + T$$
$71$ $$-360 + T$$
$73$ $$26 + T$$
$79$ $$-512 + T$$
$83$ $$-1188 + T$$
$89$ $$-630 + T$$
$97$ $$-1054 + T$$
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