# Properties

 Label 27.3.d.a Level 27 Weight 3 Character orbit 27.d Analytic conductor 0.736 Analytic rank 0 Dimension 2 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$27 = 3^{3}$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 27.d (of order $$6$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.735696713773$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 1 + \zeta_{6} ) q^{2}$$ $$-\zeta_{6} q^{4}$$ $$+ ( -4 + 2 \zeta_{6} ) q^{5}$$ $$+ ( -2 + 2 \zeta_{6} ) q^{7}$$ $$+ ( 5 - 10 \zeta_{6} ) q^{8}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 1 + \zeta_{6} ) q^{2}$$ $$-\zeta_{6} q^{4}$$ $$+ ( -4 + 2 \zeta_{6} ) q^{5}$$ $$+ ( -2 + 2 \zeta_{6} ) q^{7}$$ $$+ ( 5 - 10 \zeta_{6} ) q^{8}$$ $$-6 q^{10}$$ $$+ ( 1 + \zeta_{6} ) q^{11}$$ $$+ 4 \zeta_{6} q^{13}$$ $$+ ( -4 + 2 \zeta_{6} ) q^{14}$$ $$+ ( 11 - 11 \zeta_{6} ) q^{16}$$ $$+ ( -9 + 18 \zeta_{6} ) q^{17}$$ $$+ 11 q^{19}$$ $$+ ( 2 + 2 \zeta_{6} ) q^{20}$$ $$+ 3 \zeta_{6} q^{22}$$ $$+ ( 32 - 16 \zeta_{6} ) q^{23}$$ $$+ ( -13 + 13 \zeta_{6} ) q^{25}$$ $$+ ( -4 + 8 \zeta_{6} ) q^{26}$$ $$+ 2 q^{28}$$ $$+ ( -26 - 26 \zeta_{6} ) q^{29}$$ $$-32 \zeta_{6} q^{31}$$ $$+ ( -18 + 9 \zeta_{6} ) q^{32}$$ $$+ ( -27 + 27 \zeta_{6} ) q^{34}$$ $$+ ( 4 - 8 \zeta_{6} ) q^{35}$$ $$-34 q^{37}$$ $$+ ( 11 + 11 \zeta_{6} ) q^{38}$$ $$+ 30 \zeta_{6} q^{40}$$ $$+ ( 14 - 7 \zeta_{6} ) q^{41}$$ $$+ ( 61 - 61 \zeta_{6} ) q^{43}$$ $$+ ( 1 - 2 \zeta_{6} ) q^{44}$$ $$+ 48 q^{46}$$ $$+ ( 28 + 28 \zeta_{6} ) q^{47}$$ $$+ 45 \zeta_{6} q^{49}$$ $$+ ( -26 + 13 \zeta_{6} ) q^{50}$$ $$+ ( 4 - 4 \zeta_{6} ) q^{52}$$ $$-6 q^{55}$$ $$+ ( 10 + 10 \zeta_{6} ) q^{56}$$ $$-78 \zeta_{6} q^{58}$$ $$+ ( -58 + 29 \zeta_{6} ) q^{59}$$ $$+ ( -56 + 56 \zeta_{6} ) q^{61}$$ $$+ ( 32 - 64 \zeta_{6} ) q^{62}$$ $$-71 q^{64}$$ $$+ ( -8 - 8 \zeta_{6} ) q^{65}$$ $$+ 31 \zeta_{6} q^{67}$$ $$+ ( 18 - 9 \zeta_{6} ) q^{68}$$ $$+ ( 12 - 12 \zeta_{6} ) q^{70}$$ $$+ ( 18 - 36 \zeta_{6} ) q^{71}$$ $$+ 65 q^{73}$$ $$+ ( -34 - 34 \zeta_{6} ) q^{74}$$ $$-11 \zeta_{6} q^{76}$$ $$+ ( -4 + 2 \zeta_{6} ) q^{77}$$ $$+ ( -38 + 38 \zeta_{6} ) q^{79}$$ $$+ ( -22 + 44 \zeta_{6} ) q^{80}$$ $$+ 21 q^{82}$$ $$+ ( 28 + 28 \zeta_{6} ) q^{83}$$ $$-54 \zeta_{6} q^{85}$$ $$+ ( 122 - 61 \zeta_{6} ) q^{86}$$ $$+ ( 15 - 15 \zeta_{6} ) q^{88}$$ $$+ ( -72 + 144 \zeta_{6} ) q^{89}$$ $$-8 q^{91}$$ $$+ ( -16 - 16 \zeta_{6} ) q^{92}$$ $$+ 84 \zeta_{6} q^{94}$$ $$+ ( -44 + 22 \zeta_{6} ) q^{95}$$ $$+ ( 115 - 115 \zeta_{6} ) q^{97}$$ $$+ ( -45 + 90 \zeta_{6} ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut +\mathstrut 3q^{2}$$ $$\mathstrut -\mathstrut q^{4}$$ $$\mathstrut -\mathstrut 6q^{5}$$ $$\mathstrut -\mathstrut 2q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut +\mathstrut 3q^{2}$$ $$\mathstrut -\mathstrut q^{4}$$ $$\mathstrut -\mathstrut 6q^{5}$$ $$\mathstrut -\mathstrut 2q^{7}$$ $$\mathstrut -\mathstrut 12q^{10}$$ $$\mathstrut +\mathstrut 3q^{11}$$ $$\mathstrut +\mathstrut 4q^{13}$$ $$\mathstrut -\mathstrut 6q^{14}$$ $$\mathstrut +\mathstrut 11q^{16}$$ $$\mathstrut +\mathstrut 22q^{19}$$ $$\mathstrut +\mathstrut 6q^{20}$$ $$\mathstrut +\mathstrut 3q^{22}$$ $$\mathstrut +\mathstrut 48q^{23}$$ $$\mathstrut -\mathstrut 13q^{25}$$ $$\mathstrut +\mathstrut 4q^{28}$$ $$\mathstrut -\mathstrut 78q^{29}$$ $$\mathstrut -\mathstrut 32q^{31}$$ $$\mathstrut -\mathstrut 27q^{32}$$ $$\mathstrut -\mathstrut 27q^{34}$$ $$\mathstrut -\mathstrut 68q^{37}$$ $$\mathstrut +\mathstrut 33q^{38}$$ $$\mathstrut +\mathstrut 30q^{40}$$ $$\mathstrut +\mathstrut 21q^{41}$$ $$\mathstrut +\mathstrut 61q^{43}$$ $$\mathstrut +\mathstrut 96q^{46}$$ $$\mathstrut +\mathstrut 84q^{47}$$ $$\mathstrut +\mathstrut 45q^{49}$$ $$\mathstrut -\mathstrut 39q^{50}$$ $$\mathstrut +\mathstrut 4q^{52}$$ $$\mathstrut -\mathstrut 12q^{55}$$ $$\mathstrut +\mathstrut 30q^{56}$$ $$\mathstrut -\mathstrut 78q^{58}$$ $$\mathstrut -\mathstrut 87q^{59}$$ $$\mathstrut -\mathstrut 56q^{61}$$ $$\mathstrut -\mathstrut 142q^{64}$$ $$\mathstrut -\mathstrut 24q^{65}$$ $$\mathstrut +\mathstrut 31q^{67}$$ $$\mathstrut +\mathstrut 27q^{68}$$ $$\mathstrut +\mathstrut 12q^{70}$$ $$\mathstrut +\mathstrut 130q^{73}$$ $$\mathstrut -\mathstrut 102q^{74}$$ $$\mathstrut -\mathstrut 11q^{76}$$ $$\mathstrut -\mathstrut 6q^{77}$$ $$\mathstrut -\mathstrut 38q^{79}$$ $$\mathstrut +\mathstrut 42q^{82}$$ $$\mathstrut +\mathstrut 84q^{83}$$ $$\mathstrut -\mathstrut 54q^{85}$$ $$\mathstrut +\mathstrut 183q^{86}$$ $$\mathstrut +\mathstrut 15q^{88}$$ $$\mathstrut -\mathstrut 16q^{91}$$ $$\mathstrut -\mathstrut 48q^{92}$$ $$\mathstrut +\mathstrut 84q^{94}$$ $$\mathstrut -\mathstrut 66q^{95}$$ $$\mathstrut +\mathstrut 115q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.50000 + 0.866025i 0 −0.500000 0.866025i −3.00000 + 1.73205i 0 −1.00000 + 1.73205i 8.66025i 0 −6.00000
17.1 1.50000 0.866025i 0 −0.500000 + 0.866025i −3.00000 1.73205i 0 −1.00000 1.73205i 8.66025i 0 −6.00000
 $$n$$: 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
9.d Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(27, [\chi])$$.