# 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 $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.22358112566$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

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

## 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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.c Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{2}^{\mathrm{new}}(28, [\chi])$$.