# Properties

 Label 27.3.b.b Level 27 Weight 3 Character orbit 27.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.735696713773$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{2} -5 q^{4} -3 i q^{5} + 5 q^{7} -3 i q^{8} +O(q^{10})$$ $$q + 3 i q^{2} -5 q^{4} -3 i q^{5} + 5 q^{7} -3 i q^{8} + 9 q^{10} -15 i q^{11} -10 q^{13} + 15 i q^{14} -11 q^{16} + 18 i q^{17} -16 q^{19} + 15 i q^{20} + 45 q^{22} -12 i q^{23} + 16 q^{25} -30 i q^{26} -25 q^{28} + 30 i q^{29} - q^{31} -45 i q^{32} -54 q^{34} -15 i q^{35} + 20 q^{37} -48 i q^{38} -9 q^{40} + 60 i q^{41} + 50 q^{43} + 75 i q^{44} + 36 q^{46} -6 i q^{47} -24 q^{49} + 48 i q^{50} + 50 q^{52} -27 i q^{53} -45 q^{55} -15 i q^{56} -90 q^{58} -30 i q^{59} -76 q^{61} -3 i q^{62} + 91 q^{64} + 30 i q^{65} -10 q^{67} -90 i q^{68} + 45 q^{70} -90 i q^{71} + 65 q^{73} + 60 i q^{74} + 80 q^{76} -75 i q^{77} + 14 q^{79} + 33 i q^{80} -180 q^{82} + 3 i q^{83} + 54 q^{85} + 150 i q^{86} -45 q^{88} + 90 i q^{89} -50 q^{91} + 60 i q^{92} + 18 q^{94} + 48 i q^{95} -85 q^{97} -72 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 10q^{4} + 10q^{7} + O(q^{10})$$ $$2q - 10q^{4} + 10q^{7} + 18q^{10} - 20q^{13} - 22q^{16} - 32q^{19} + 90q^{22} + 32q^{25} - 50q^{28} - 2q^{31} - 108q^{34} + 40q^{37} - 18q^{40} + 100q^{43} + 72q^{46} - 48q^{49} + 100q^{52} - 90q^{55} - 180q^{58} - 152q^{61} + 182q^{64} - 20q^{67} + 90q^{70} + 130q^{73} + 160q^{76} + 28q^{79} - 360q^{82} + 108q^{85} - 90q^{88} - 100q^{91} + 36q^{94} - 170q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 − 1.00000i 1.00000i
3.00000i 0 −5.00000 3.00000i 0 5.00000 3.00000i 0 9.00000
26.2 3.00000i 0 −5.00000 3.00000i 0 5.00000 3.00000i 0 9.00000
 $$n$$: 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.3.b.b 2
3.b odd 2 1 inner 27.3.b.b 2
4.b odd 2 1 432.3.e.c 2
5.b even 2 1 675.3.c.h 2
5.c odd 4 1 675.3.d.a 2
5.c odd 4 1 675.3.d.d 2
8.b even 2 1 1728.3.e.m 2
8.d odd 2 1 1728.3.e.g 2
9.c even 3 2 81.3.d.b 4
9.d odd 6 2 81.3.d.b 4
12.b even 2 1 432.3.e.c 2
15.d odd 2 1 675.3.c.h 2
15.e even 4 1 675.3.d.a 2
15.e even 4 1 675.3.d.d 2
24.f even 2 1 1728.3.e.g 2
24.h odd 2 1 1728.3.e.m 2
36.f odd 6 2 1296.3.q.j 4
36.h even 6 2 1296.3.q.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.3.b.b 2 1.a even 1 1 trivial
27.3.b.b 2 3.b odd 2 1 inner
81.3.d.b 4 9.c even 3 2
81.3.d.b 4 9.d odd 6 2
432.3.e.c 2 4.b odd 2 1
432.3.e.c 2 12.b even 2 1
675.3.c.h 2 5.b even 2 1
675.3.c.h 2 15.d odd 2 1
675.3.d.a 2 5.c odd 4 1
675.3.d.a 2 15.e even 4 1
675.3.d.d 2 5.c odd 4 1
675.3.d.d 2 15.e even 4 1
1296.3.q.j 4 36.f odd 6 2
1296.3.q.j 4 36.h even 6 2
1728.3.e.g 2 8.d odd 2 1
1728.3.e.g 2 24.f even 2 1
1728.3.e.m 2 8.b even 2 1
1728.3.e.m 2 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 9$$ acting on $$S_{3}^{\mathrm{new}}(27, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2} + 16 T^{4}$$
$3$ 1
$5$ $$1 - 41 T^{2} + 625 T^{4}$$
$7$ $$( 1 - 5 T + 49 T^{2} )^{2}$$
$11$ $$1 - 17 T^{2} + 14641 T^{4}$$
$13$ $$( 1 + 10 T + 169 T^{2} )^{2}$$
$17$ $$1 - 254 T^{2} + 83521 T^{4}$$
$19$ $$( 1 + 16 T + 361 T^{2} )^{2}$$
$23$ $$1 - 914 T^{2} + 279841 T^{4}$$
$29$ $$1 - 782 T^{2} + 707281 T^{4}$$
$31$ $$( 1 + T + 961 T^{2} )^{2}$$
$37$ $$( 1 - 20 T + 1369 T^{2} )^{2}$$
$41$ $$1 + 238 T^{2} + 2825761 T^{4}$$
$43$ $$( 1 - 50 T + 1849 T^{2} )^{2}$$
$47$ $$1 - 4382 T^{2} + 4879681 T^{4}$$
$53$ $$1 - 4889 T^{2} + 7890481 T^{4}$$
$59$ $$1 - 6062 T^{2} + 12117361 T^{4}$$
$61$ $$( 1 + 76 T + 3721 T^{2} )^{2}$$
$67$ $$( 1 + 10 T + 4489 T^{2} )^{2}$$
$71$ $$1 - 1982 T^{2} + 25411681 T^{4}$$
$73$ $$( 1 - 65 T + 5329 T^{2} )^{2}$$
$79$ $$( 1 - 14 T + 6241 T^{2} )^{2}$$
$83$ $$1 - 13769 T^{2} + 47458321 T^{4}$$
$89$ $$1 - 7742 T^{2} + 62742241 T^{4}$$
$97$ $$( 1 + 85 T + 9409 T^{2} )^{2}$$