# Properties

 Label 2028.4.b.c Level $2028$ Weight $4$ Character orbit 2028.b Analytic conductor $119.656$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2028 = 2^{2} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2028.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$119.655873492$$ Analytic rank: $$0$$ 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 12) 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.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 18 i q^{5} + 8 i q^{7} + 9 q^{9} +O(q^{10})$$ $$q + 3 q^{3} + 18 i q^{5} + 8 i q^{7} + 9 q^{9} + 36 i q^{11} + 54 i q^{15} -18 q^{17} + 100 i q^{19} + 24 i q^{21} -72 q^{23} -199 q^{25} + 27 q^{27} -234 q^{29} + 16 i q^{31} + 108 i q^{33} -144 q^{35} -226 i q^{37} -90 i q^{41} -452 q^{43} + 162 i q^{45} + 432 i q^{47} + 279 q^{49} -54 q^{51} + 414 q^{53} -648 q^{55} + 300 i q^{57} -684 i q^{59} + 422 q^{61} + 72 i q^{63} -332 i q^{67} -216 q^{69} + 360 i q^{71} + 26 i q^{73} -597 q^{75} -288 q^{77} + 512 q^{79} + 81 q^{81} + 1188 i q^{83} -324 i q^{85} -702 q^{87} -630 i q^{89} + 48 i q^{93} -1800 q^{95} + 1054 i q^{97} + 324 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} + 18q^{9} - 36q^{17} - 144q^{23} - 398q^{25} + 54q^{27} - 468q^{29} - 288q^{35} - 904q^{43} + 558q^{49} - 108q^{51} + 828q^{53} - 1296q^{55} + 844q^{61} - 432q^{69} - 1194q^{75} - 576q^{77} + 1024q^{79} + 162q^{81} - 1404q^{87} - 3600q^{95} + O(q^{100})$$

## 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 3.00000 0 18.0000i 0 8.00000i 0 9.00000 0
337.2 0 3.00000 0 18.0000i 0 8.00000i 0 9.00000 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 2028.4.b.c 2
13.b even 2 1 inner 2028.4.b.c 2
13.d odd 4 1 12.4.a.a 1
13.d odd 4 1 2028.4.a.c 1
39.f even 4 1 36.4.a.a 1
52.f even 4 1 48.4.a.a 1
65.f even 4 1 300.4.d.e 2
65.g odd 4 1 300.4.a.b 1
65.k even 4 1 300.4.d.e 2
91.i even 4 1 588.4.a.c 1
91.z odd 12 2 588.4.i.d 2
91.bb even 12 2 588.4.i.e 2
104.j odd 4 1 192.4.a.f 1
104.m even 4 1 192.4.a.l 1
117.y odd 12 2 324.4.e.h 2
117.z even 12 2 324.4.e.a 2
143.g even 4 1 1452.4.a.d 1
156.l odd 4 1 144.4.a.g 1
195.j odd 4 1 900.4.d.c 2
195.n even 4 1 900.4.a.g 1
195.u odd 4 1 900.4.d.c 2
208.l even 4 1 768.4.d.j 2
208.m odd 4 1 768.4.d.g 2
208.r odd 4 1 768.4.d.g 2
208.s even 4 1 768.4.d.j 2
260.l odd 4 1 1200.4.f.d 2
260.s odd 4 1 1200.4.f.d 2
260.u even 4 1 1200.4.a.be 1
273.o odd 4 1 1764.4.a.b 1
273.cb odd 12 2 1764.4.k.o 2
273.cd even 12 2 1764.4.k.b 2
312.w odd 4 1 576.4.a.a 1
312.y even 4 1 576.4.a.b 1
364.p odd 4 1 2352.4.a.bk 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 13.d odd 4 1
36.4.a.a 1 39.f even 4 1
48.4.a.a 1 52.f even 4 1
144.4.a.g 1 156.l odd 4 1
192.4.a.f 1 104.j odd 4 1
192.4.a.l 1 104.m even 4 1
300.4.a.b 1 65.g odd 4 1
300.4.d.e 2 65.f even 4 1
300.4.d.e 2 65.k even 4 1
324.4.e.a 2 117.z even 12 2
324.4.e.h 2 117.y odd 12 2
576.4.a.a 1 312.w odd 4 1
576.4.a.b 1 312.y even 4 1
588.4.a.c 1 91.i even 4 1
588.4.i.d 2 91.z odd 12 2
588.4.i.e 2 91.bb even 12 2
768.4.d.g 2 208.m odd 4 1
768.4.d.g 2 208.r odd 4 1
768.4.d.j 2 208.l even 4 1
768.4.d.j 2 208.s even 4 1
900.4.a.g 1 195.n even 4 1
900.4.d.c 2 195.j odd 4 1
900.4.d.c 2 195.u odd 4 1
1200.4.a.be 1 260.u even 4 1
1200.4.f.d 2 260.l odd 4 1
1200.4.f.d 2 260.s odd 4 1
1452.4.a.d 1 143.g even 4 1
1764.4.a.b 1 273.o odd 4 1
1764.4.k.b 2 273.cd even 12 2
1764.4.k.o 2 273.cb odd 12 2
2028.4.a.c 1 13.d odd 4 1
2028.4.b.c 2 1.a even 1 1 trivial
2028.4.b.c 2 13.b even 2 1 inner
2352.4.a.bk 1 364.p odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 324$$ acting on $$S_{4}^{\mathrm{new}}(2028, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$324 + T^{2}$$
$7$ $$64 + T^{2}$$
$11$ $$1296 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 18 + T )^{2}$$
$19$ $$10000 + T^{2}$$
$23$ $$( 72 + T )^{2}$$
$29$ $$( 234 + T )^{2}$$
$31$ $$256 + T^{2}$$
$37$ $$51076 + T^{2}$$
$41$ $$8100 + T^{2}$$
$43$ $$( 452 + T )^{2}$$
$47$ $$186624 + T^{2}$$
$53$ $$( -414 + T )^{2}$$
$59$ $$467856 + T^{2}$$
$61$ $$( -422 + T )^{2}$$
$67$ $$110224 + T^{2}$$
$71$ $$129600 + T^{2}$$
$73$ $$676 + T^{2}$$
$79$ $$( -512 + T )^{2}$$
$83$ $$1411344 + T^{2}$$
$89$ $$396900 + T^{2}$$
$97$ $$1110916 + T^{2}$$