# Properties

 Label 28.2.f.a Level 28 Weight 2 Character orbit 28.f Analytic conductor 0.224 Analytic rank 0 Dimension 4 CM No Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$28 = 2^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 28.f (of order $$6$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.22358112566$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2}$$ $$+ ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{3}$$ $$+ 2 \zeta_{12} q^{4}$$ $$+ ( -2 + \zeta_{12}^{2} ) q^{5}$$ $$+ ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{6}$$ $$+ ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7}$$ $$+ ( -2 - 2 \zeta_{12}^{3} ) q^{8}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2}$$ $$+ ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{3}$$ $$+ 2 \zeta_{12} q^{4}$$ $$+ ( -2 + \zeta_{12}^{2} ) q^{5}$$ $$+ ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{6}$$ $$+ ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7}$$ $$+ ( -2 - 2 \zeta_{12}^{3} ) q^{8}$$ $$+ ( 1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{10}$$ $$+ \zeta_{12} q^{11}$$ $$+ ( 4 - 2 \zeta_{12}^{2} ) q^{12}$$ $$+ ( 2 - 4 \zeta_{12}^{2} ) q^{13}$$ $$+ ( 2 + 3 \zeta_{12} - 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{14}$$ $$+ 3 \zeta_{12}^{3} q^{15}$$ $$+ 4 \zeta_{12}^{2} q^{16}$$ $$+ ( -1 - \zeta_{12}^{2} ) q^{17}$$ $$+ ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{19}$$ $$+ ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{20}$$ $$+ ( -1 + 5 \zeta_{12}^{2} ) q^{21}$$ $$+ ( -1 - \zeta_{12}^{3} ) q^{22}$$ $$+ ( \zeta_{12} - \zeta_{12}^{3} ) q^{23}$$ $$+ ( -2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{24}$$ $$+ ( -2 + 2 \zeta_{12}^{2} ) q^{25}$$ $$+ ( -4 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{26}$$ $$+ ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27}$$ $$+ ( -6 + 2 \zeta_{12}^{2} ) q^{28}$$ $$+ 4 q^{29}$$ $$+ ( 3 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{30}$$ $$+ ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{31}$$ $$+ ( 4 - 4 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{32}$$ $$+ ( 2 - \zeta_{12}^{2} ) q^{33}$$ $$+ ( -1 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{34}$$ $$+ ( \zeta_{12} - 5 \zeta_{12}^{3} ) q^{35}$$ $$-3 \zeta_{12}^{2} q^{37}$$ $$+ ( 3 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{38}$$ $$-6 \zeta_{12} q^{39}$$ $$+ ( 4 + 2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{40}$$ $$+ ( -2 + 4 \zeta_{12}^{2} ) q^{41}$$ $$+ ( 5 - 4 \zeta_{12} - 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{42}$$ $$+ 2 \zeta_{12}^{3} q^{43}$$ $$+ 2 \zeta_{12}^{2} q^{44}$$ $$+ ( -1 - \zeta_{12} + \zeta_{12}^{2} ) q^{46}$$ $$+ ( 5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{47}$$ $$+ ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{48}$$ $$+ ( 3 - 8 \zeta_{12}^{2} ) q^{49}$$ $$+ ( 2 - 2 \zeta_{12}^{3} ) q^{50}$$ $$+ ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{51}$$ $$+ ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{52}$$ $$+ ( 1 - \zeta_{12}^{2} ) q^{53}$$ $$+ ( -6 - 3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{54}$$ $$+ ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{55}$$ $$+ ( 2 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{56}$$ $$-9 q^{57}$$ $$+ ( -4 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{58}$$ $$+ ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{59}$$ $$+ ( -6 + 6 \zeta_{12}^{2} ) q^{60}$$ $$+ ( -6 + 3 \zeta_{12}^{2} ) q^{61}$$ $$+ ( 1 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{62}$$ $$+ 8 \zeta_{12}^{3} q^{64}$$ $$+ 6 \zeta_{12}^{2} q^{65}$$ $$+ ( -1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{66}$$ $$+ 3 \zeta_{12} q^{67}$$ $$+ ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{68}$$ $$+ ( 1 - 2 \zeta_{12}^{2} ) q^{69}$$ $$+ ( -1 - 5 \zeta_{12} + 5 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{70}$$ $$-14 \zeta_{12}^{3} q^{71}$$ $$+ ( 5 + 5 \zeta_{12}^{2} ) q^{73}$$ $$+ ( -3 + 3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{74}$$ $$+ ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{75}$$ $$+ ( 6 - 12 \zeta_{12}^{2} ) q^{76}$$ $$+ ( -3 + \zeta_{12}^{2} ) q^{77}$$ $$+ ( 6 + 6 \zeta_{12}^{3} ) q^{78}$$ $$+ ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{79}$$ $$+ ( -4 - 4 \zeta_{12}^{2} ) q^{80}$$ $$+ ( 9 - 9 \zeta_{12}^{2} ) q^{81}$$ $$+ ( 4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{82}$$ $$+ ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{83}$$ $$+ ( -2 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{84}$$ $$+ 3 q^{85}$$ $$+ ( 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{86}$$ $$+ ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{87}$$ $$+ ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{88}$$ $$+ ( 18 - 9 \zeta_{12}^{2} ) q^{89}$$ $$+ ( 8 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{91}$$ $$+ 2 q^{92}$$ $$+ 3 \zeta_{12}^{2} q^{93}$$ $$+ ( -5 + 5 \zeta_{12} - 5 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{94}$$ $$+ 9 \zeta_{12} q^{95}$$ $$+ ( -8 - 4 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{96}$$ $$+ ( -10 + 20 \zeta_{12}^{2} ) q^{97}$$ $$+ ( -8 + 5 \zeta_{12} + 5 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 6q^{5}$$ $$\mathstrut -\mathstrut 8q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 2q^{2}$$ $$\mathstrut -\mathstrut 6q^{5}$$ $$\mathstrut -\mathstrut 8q^{8}$$ $$\mathstrut +\mathstrut 6q^{10}$$ $$\mathstrut +\mathstrut 12q^{12}$$ $$\mathstrut +\mathstrut 2q^{14}$$ $$\mathstrut +\mathstrut 8q^{16}$$ $$\mathstrut -\mathstrut 6q^{17}$$ $$\mathstrut +\mathstrut 6q^{21}$$ $$\mathstrut -\mathstrut 4q^{22}$$ $$\mathstrut -\mathstrut 12q^{24}$$ $$\mathstrut -\mathstrut 4q^{25}$$ $$\mathstrut -\mathstrut 12q^{26}$$ $$\mathstrut -\mathstrut 20q^{28}$$ $$\mathstrut +\mathstrut 16q^{29}$$ $$\mathstrut -\mathstrut 6q^{30}$$ $$\mathstrut +\mathstrut 8q^{32}$$ $$\mathstrut +\mathstrut 6q^{33}$$ $$\mathstrut -\mathstrut 6q^{37}$$ $$\mathstrut +\mathstrut 18q^{38}$$ $$\mathstrut +\mathstrut 12q^{40}$$ $$\mathstrut +\mathstrut 12q^{42}$$ $$\mathstrut +\mathstrut 4q^{44}$$ $$\mathstrut -\mathstrut 2q^{46}$$ $$\mathstrut -\mathstrut 4q^{49}$$ $$\mathstrut +\mathstrut 8q^{50}$$ $$\mathstrut +\mathstrut 2q^{53}$$ $$\mathstrut -\mathstrut 18q^{54}$$ $$\mathstrut +\mathstrut 16q^{56}$$ $$\mathstrut -\mathstrut 36q^{57}$$ $$\mathstrut -\mathstrut 8q^{58}$$ $$\mathstrut -\mathstrut 12q^{60}$$ $$\mathstrut -\mathstrut 18q^{61}$$ $$\mathstrut +\mathstrut 12q^{65}$$ $$\mathstrut -\mathstrut 6q^{66}$$ $$\mathstrut +\mathstrut 6q^{70}$$ $$\mathstrut +\mathstrut 30q^{73}$$ $$\mathstrut -\mathstrut 6q^{74}$$ $$\mathstrut -\mathstrut 10q^{77}$$ $$\mathstrut +\mathstrut 24q^{78}$$ $$\mathstrut -\mathstrut 24q^{80}$$ $$\mathstrut +\mathstrut 18q^{81}$$ $$\mathstrut +\mathstrut 12q^{82}$$ $$\mathstrut +\mathstrut 12q^{85}$$ $$\mathstrut -\mathstrut 4q^{86}$$ $$\mathstrut +\mathstrut 4q^{88}$$ $$\mathstrut +\mathstrut 54q^{89}$$ $$\mathstrut +\mathstrut 8q^{92}$$ $$\mathstrut +\mathstrut 6q^{93}$$ $$\mathstrut -\mathstrut 30q^{94}$$ $$\mathstrut -\mathstrut 24q^{96}$$ $$\mathstrut -\mathstrut 22q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

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

 $$n$$ $$15$$ $$17$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−1.36603 0.366025i 0.866025 1.50000i 1.73205 + 1.00000i −1.50000 + 0.866025i −1.73205 + 1.73205i −1.73205 + 2.00000i −2.00000 2.00000i 0 2.36603 0.633975i
3.2 0.366025 1.36603i −0.866025 + 1.50000i −1.73205 1.00000i −1.50000 + 0.866025i 1.73205 + 1.73205i 1.73205 2.00000i −2.00000 + 2.00000i 0 0.633975 + 2.36603i
19.1 −1.36603 + 0.366025i 0.866025 + 1.50000i 1.73205 1.00000i −1.50000 0.866025i −1.73205 1.73205i −1.73205 2.00000i −2.00000 + 2.00000i 0 2.36603 + 0.633975i
19.2 0.366025 + 1.36603i −0.866025 1.50000i −1.73205 + 1.00000i −1.50000 0.866025i 1.73205 1.73205i 1.73205 + 2.00000i −2.00000 2.00000i 0 0.633975 2.36603i
 $$n$$: 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
4.b Odd 1 yes
7.d Odd 1 yes
28.f Even 1 yes

## Hecke kernels

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