# Properties

 Label 28.2.e.a Level $28$ Weight $2$ Character orbit 28.e Analytic conductor $0.224$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$28 = 2^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 28.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.223581125660$$ Analytic rank: $$0$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{3} + ( -3 + 3 \zeta_{6} ) q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + ( 2 - 2 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{3} + ( -3 + 3 \zeta_{6} ) q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + ( 2 - 2 \zeta_{6} ) q^{9} + 3 \zeta_{6} q^{11} + 2 q^{13} + 3 q^{15} -3 \zeta_{6} q^{17} + ( 1 - \zeta_{6} ) q^{19} + ( -2 + 3 \zeta_{6} ) q^{21} + ( -3 + 3 \zeta_{6} ) q^{23} -4 \zeta_{6} q^{25} -5 q^{27} -6 q^{29} + 7 \zeta_{6} q^{31} + ( 3 - 3 \zeta_{6} ) q^{33} + ( 9 - 3 \zeta_{6} ) q^{35} + ( 1 - \zeta_{6} ) q^{37} -2 \zeta_{6} q^{39} + 6 q^{41} -4 q^{43} + 6 \zeta_{6} q^{45} + ( 9 - 9 \zeta_{6} ) q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( -3 + 3 \zeta_{6} ) q^{51} -3 \zeta_{6} q^{53} -9 q^{55} - q^{57} -9 \zeta_{6} q^{59} + ( 1 - \zeta_{6} ) q^{61} + ( -6 + 2 \zeta_{6} ) q^{63} + ( -6 + 6 \zeta_{6} ) q^{65} + 7 \zeta_{6} q^{67} + 3 q^{69} + \zeta_{6} q^{73} + ( -4 + 4 \zeta_{6} ) q^{75} + ( 6 - 9 \zeta_{6} ) q^{77} + ( 13 - 13 \zeta_{6} ) q^{79} -\zeta_{6} q^{81} + 12 q^{83} + 9 q^{85} + 6 \zeta_{6} q^{87} + ( -15 + 15 \zeta_{6} ) q^{89} + ( -2 - 4 \zeta_{6} ) q^{91} + ( 7 - 7 \zeta_{6} ) q^{93} + 3 \zeta_{6} q^{95} -10 q^{97} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} - 3q^{5} - 4q^{7} + 2q^{9} + O(q^{10})$$ $$2q - q^{3} - 3q^{5} - 4q^{7} + 2q^{9} + 3q^{11} + 4q^{13} + 6q^{15} - 3q^{17} + q^{19} - q^{21} - 3q^{23} - 4q^{25} - 10q^{27} - 12q^{29} + 7q^{31} + 3q^{33} + 15q^{35} + q^{37} - 2q^{39} + 12q^{41} - 8q^{43} + 6q^{45} + 9q^{47} + 2q^{49} - 3q^{51} - 3q^{53} - 18q^{55} - 2q^{57} - 9q^{59} + q^{61} - 10q^{63} - 6q^{65} + 7q^{67} + 6q^{69} + q^{73} - 4q^{75} + 3q^{77} + 13q^{79} - q^{81} + 24q^{83} + 18q^{85} + 6q^{87} - 15q^{89} - 8q^{91} + 7q^{93} + 3q^{95} - 20q^{97} + 12q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/28\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$17$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 0.866025i 0 −1.50000 + 2.59808i 0 −2.00000 1.73205i 0 1.00000 1.73205i 0
25.1 0 −0.500000 + 0.866025i 0 −1.50000 2.59808i 0 −2.00000 + 1.73205i 0 1.00000 + 1.73205i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.2.e.a 2
3.b odd 2 1 252.2.k.c 2
4.b odd 2 1 112.2.i.b 2
5.b even 2 1 700.2.i.c 2
5.c odd 4 2 700.2.r.b 4
7.b odd 2 1 196.2.e.a 2
7.c even 3 1 inner 28.2.e.a 2
7.c even 3 1 196.2.a.b 1
7.d odd 6 1 196.2.a.a 1
7.d odd 6 1 196.2.e.a 2
8.b even 2 1 448.2.i.e 2
8.d odd 2 1 448.2.i.c 2
9.c even 3 1 2268.2.i.a 2
9.c even 3 1 2268.2.l.h 2
9.d odd 6 1 2268.2.i.h 2
9.d odd 6 1 2268.2.l.a 2
12.b even 2 1 1008.2.s.p 2
21.c even 2 1 1764.2.k.b 2
21.g even 6 1 1764.2.a.j 1
21.g even 6 1 1764.2.k.b 2
21.h odd 6 1 252.2.k.c 2
21.h odd 6 1 1764.2.a.a 1
28.d even 2 1 784.2.i.d 2
28.f even 6 1 784.2.a.g 1
28.f even 6 1 784.2.i.d 2
28.g odd 6 1 112.2.i.b 2
28.g odd 6 1 784.2.a.d 1
35.i odd 6 1 4900.2.a.n 1
35.j even 6 1 700.2.i.c 2
35.j even 6 1 4900.2.a.g 1
35.k even 12 2 4900.2.e.h 2
35.l odd 12 2 700.2.r.b 4
35.l odd 12 2 4900.2.e.i 2
56.j odd 6 1 3136.2.a.v 1
56.k odd 6 1 448.2.i.c 2
56.k odd 6 1 3136.2.a.s 1
56.m even 6 1 3136.2.a.k 1
56.p even 6 1 448.2.i.e 2
56.p even 6 1 3136.2.a.h 1
63.g even 3 1 2268.2.i.a 2
63.h even 3 1 2268.2.l.h 2
63.j odd 6 1 2268.2.l.a 2
63.n odd 6 1 2268.2.i.h 2
84.j odd 6 1 7056.2.a.bw 1
84.n even 6 1 1008.2.s.p 2
84.n even 6 1 7056.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.e.a 2 1.a even 1 1 trivial
28.2.e.a 2 7.c even 3 1 inner
112.2.i.b 2 4.b odd 2 1
112.2.i.b 2 28.g odd 6 1
196.2.a.a 1 7.d odd 6 1
196.2.a.b 1 7.c even 3 1
196.2.e.a 2 7.b odd 2 1
196.2.e.a 2 7.d odd 6 1
252.2.k.c 2 3.b odd 2 1
252.2.k.c 2 21.h odd 6 1
448.2.i.c 2 8.d odd 2 1
448.2.i.c 2 56.k odd 6 1
448.2.i.e 2 8.b even 2 1
448.2.i.e 2 56.p even 6 1
700.2.i.c 2 5.b even 2 1
700.2.i.c 2 35.j even 6 1
700.2.r.b 4 5.c odd 4 2
700.2.r.b 4 35.l odd 12 2
784.2.a.d 1 28.g odd 6 1
784.2.a.g 1 28.f even 6 1
784.2.i.d 2 28.d even 2 1
784.2.i.d 2 28.f even 6 1
1008.2.s.p 2 12.b even 2 1
1008.2.s.p 2 84.n even 6 1
1764.2.a.a 1 21.h odd 6 1
1764.2.a.j 1 21.g even 6 1
1764.2.k.b 2 21.c even 2 1
1764.2.k.b 2 21.g even 6 1
2268.2.i.a 2 9.c even 3 1
2268.2.i.a 2 63.g even 3 1
2268.2.i.h 2 9.d odd 6 1
2268.2.i.h 2 63.n odd 6 1
2268.2.l.a 2 9.d odd 6 1
2268.2.l.a 2 63.j odd 6 1
2268.2.l.h 2 9.c even 3 1
2268.2.l.h 2 63.h even 3 1
3136.2.a.h 1 56.p even 6 1
3136.2.a.k 1 56.m even 6 1
3136.2.a.s 1 56.k odd 6 1
3136.2.a.v 1 56.j odd 6 1
4900.2.a.g 1 35.j even 6 1
4900.2.a.n 1 35.i odd 6 1
4900.2.e.h 2 35.k even 12 2
4900.2.e.i 2 35.l odd 12 2
7056.2.a.f 1 84.n even 6 1
7056.2.a.bw 1 84.j odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(28, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$9 + 3 T + T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$49 - 7 T + T^{2}$$
$37$ $$1 - T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$81 - 9 T + T^{2}$$
$53$ $$9 + 3 T + T^{2}$$
$59$ $$81 + 9 T + T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$49 - 7 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$1 - T + T^{2}$$
$79$ $$169 - 13 T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$225 + 15 T + T^{2}$$
$97$ $$( 10 + T )^{2}$$