# Properties

 Label 27.8.a.c Level 27 Weight 8 Character orbit 27.a Self dual Yes Analytic conductor 8.434 Analytic rank 1 Dimension 2 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$27 = 3^{3}$$ Weight: $$k$$ = $$8$$ Character orbit: $$[\chi]$$ = 27.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$8.43439568807$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 3$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 6 \beta q^{2} -20 q^{4} -204 \beta q^{5} -559 q^{7} -888 \beta q^{8} +O(q^{10})$$ $$q + 6 \beta q^{2} -20 q^{4} -204 \beta q^{5} -559 q^{7} -888 \beta q^{8} -3672 q^{10} + 2724 \beta q^{11} -8671 q^{13} -3354 \beta q^{14} -13424 q^{16} + 14508 \beta q^{17} -32461 q^{19} + 4080 \beta q^{20} + 49032 q^{22} -47580 \beta q^{23} + 46723 q^{25} -52026 \beta q^{26} + 11180 q^{28} + 91104 \beta q^{29} + 229892 q^{31} + 33120 \beta q^{32} + 261144 q^{34} + 114036 \beta q^{35} -541177 q^{37} -194766 \beta q^{38} + 543456 q^{40} + 204096 \beta q^{41} -465112 q^{43} -54480 \beta q^{44} -856440 q^{46} -479532 \beta q^{47} -511062 q^{49} + 280338 \beta q^{50} + 173420 q^{52} -592488 \beta q^{53} -1667088 q^{55} + 496392 \beta q^{56} + 1639872 q^{58} -453372 \beta q^{59} -137773 q^{61} + 1379352 \beta q^{62} + 2314432 q^{64} + 1768884 \beta q^{65} -314041 q^{67} -290160 \beta q^{68} + 2052648 q^{70} -1622232 \beta q^{71} + 2669537 q^{73} -3247062 \beta q^{74} + 649220 q^{76} -1522716 \beta q^{77} + 1101815 q^{79} + 2738496 \beta q^{80} + 3673728 q^{82} + 3509736 \beta q^{83} -8878896 q^{85} -2790672 \beta q^{86} -7256736 q^{88} -1897380 \beta q^{89} + 4847089 q^{91} + 951600 \beta q^{92} -8631576 q^{94} + 6622044 \beta q^{95} -2979379 q^{97} -3066372 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 40q^{4} - 1118q^{7} + O(q^{10})$$ $$2q - 40q^{4} - 1118q^{7} - 7344q^{10} - 17342q^{13} - 26848q^{16} - 64922q^{19} + 98064q^{22} + 93446q^{25} + 22360q^{28} + 459784q^{31} + 522288q^{34} - 1082354q^{37} + 1086912q^{40} - 930224q^{43} - 1712880q^{46} - 1022124q^{49} + 346840q^{52} - 3334176q^{55} + 3279744q^{58} - 275546q^{61} + 4628864q^{64} - 628082q^{67} + 4105296q^{70} + 5339074q^{73} + 1298440q^{76} + 2203630q^{79} + 7347456q^{82} - 17757792q^{85} - 14513472q^{88} + 9694178q^{91} - 17263152q^{94} - 5958758q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
−10.3923 0 −20.0000 353.338 0 −559.000 1538.06 0 −3672.00
1.2 10.3923 0 −20.0000 −353.338 0 −559.000 −1538.06 0 −3672.00
 $$n$$: 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
3.b Odd 1 yes

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{2} - 108$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(27))$$.