# Properties

 Label 387.2.h.a Level $387$ Weight $2$ Character orbit 387.h Analytic conductor $3.090$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$387 = 3^{2} \cdot 43$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 387.h (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.09021055822$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 43) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + ( -1 + \zeta_{6} ) q^{5} -3 \zeta_{6} q^{7} + 3 q^{8} +O(q^{10})$$ $$q - q^{2} - q^{4} + ( -1 + \zeta_{6} ) q^{5} -3 \zeta_{6} q^{7} + 3 q^{8} + ( 1 - \zeta_{6} ) q^{10} + 5 \zeta_{6} q^{13} + 3 \zeta_{6} q^{14} - q^{16} + 3 \zeta_{6} q^{17} + ( -1 + \zeta_{6} ) q^{19} + ( 1 - \zeta_{6} ) q^{20} + ( -7 + 7 \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} -5 \zeta_{6} q^{26} + 3 \zeta_{6} q^{28} + 3 \zeta_{6} q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} -5 q^{32} -3 \zeta_{6} q^{34} + 3 q^{35} + ( 9 - 9 \zeta_{6} ) q^{37} + ( 1 - \zeta_{6} ) q^{38} + ( -3 + 3 \zeta_{6} ) q^{40} + 10 q^{41} + ( -7 + 6 \zeta_{6} ) q^{43} + ( 7 - 7 \zeta_{6} ) q^{46} + 8 q^{47} + ( -2 + 2 \zeta_{6} ) q^{49} -4 \zeta_{6} q^{50} -5 \zeta_{6} q^{52} + ( -5 + 5 \zeta_{6} ) q^{53} -9 \zeta_{6} q^{56} -3 \zeta_{6} q^{58} -12 q^{59} + 13 \zeta_{6} q^{61} + ( 5 - 5 \zeta_{6} ) q^{62} + 7 q^{64} -5 q^{65} + ( 3 - 3 \zeta_{6} ) q^{67} -3 \zeta_{6} q^{68} -3 q^{70} -\zeta_{6} q^{71} -11 \zeta_{6} q^{73} + ( -9 + 9 \zeta_{6} ) q^{74} + ( 1 - \zeta_{6} ) q^{76} + 5 \zeta_{6} q^{79} + ( 1 - \zeta_{6} ) q^{80} -10 q^{82} + ( 9 - 9 \zeta_{6} ) q^{83} -3 q^{85} + ( 7 - 6 \zeta_{6} ) q^{86} + ( -1 + \zeta_{6} ) q^{89} + ( 15 - 15 \zeta_{6} ) q^{91} + ( 7 - 7 \zeta_{6} ) q^{92} -8 q^{94} -\zeta_{6} q^{95} -2 q^{97} + ( 2 - 2 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{4} - q^{5} - 3 q^{7} + 6 q^{8} + O(q^{10})$$ $$2 q - 2 q^{2} - 2 q^{4} - q^{5} - 3 q^{7} + 6 q^{8} + q^{10} + 5 q^{13} + 3 q^{14} - 2 q^{16} + 3 q^{17} - q^{19} + q^{20} - 7 q^{23} + 4 q^{25} - 5 q^{26} + 3 q^{28} + 3 q^{29} - 5 q^{31} - 10 q^{32} - 3 q^{34} + 6 q^{35} + 9 q^{37} + q^{38} - 3 q^{40} + 20 q^{41} - 8 q^{43} + 7 q^{46} + 16 q^{47} - 2 q^{49} - 4 q^{50} - 5 q^{52} - 5 q^{53} - 9 q^{56} - 3 q^{58} - 24 q^{59} + 13 q^{61} + 5 q^{62} + 14 q^{64} - 10 q^{65} + 3 q^{67} - 3 q^{68} - 6 q^{70} - q^{71} - 11 q^{73} - 9 q^{74} + q^{76} + 5 q^{79} + q^{80} - 20 q^{82} + 9 q^{83} - 6 q^{85} + 8 q^{86} - q^{89} + 15 q^{91} + 7 q^{92} - 16 q^{94} - q^{95} - 4 q^{97} + 2 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/387\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$173$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
208.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 0 −1.00000 −0.500000 0.866025i 0 −1.50000 + 2.59808i 3.00000 0 0.500000 + 0.866025i
307.1 −1.00000 0 −1.00000 −0.500000 + 0.866025i 0 −1.50000 2.59808i 3.00000 0 0.500000 0.866025i
 $$n$$: 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
43.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.2.h.a 2
3.b odd 2 1 43.2.c.a 2
12.b even 2 1 688.2.i.d 2
43.c even 3 1 inner 387.2.h.a 2
129.f odd 6 1 43.2.c.a 2
129.f odd 6 1 1849.2.a.c 1
129.h even 6 1 1849.2.a.a 1
516.p even 6 1 688.2.i.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.c.a 2 3.b odd 2 1
43.2.c.a 2 129.f odd 6 1
387.2.h.a 2 1.a even 1 1 trivial
387.2.h.a 2 43.c even 3 1 inner
688.2.i.d 2 12.b even 2 1
688.2.i.d 2 516.p even 6 1
1849.2.a.a 1 129.h even 6 1
1849.2.a.c 1 129.f odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(387, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$9 + 3 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$25 - 5 T + T^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$49 + 7 T + T^{2}$$
$29$ $$9 - 3 T + T^{2}$$
$31$ $$25 + 5 T + T^{2}$$
$37$ $$81 - 9 T + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$43 + 8 T + T^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$25 + 5 T + T^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$169 - 13 T + T^{2}$$
$67$ $$9 - 3 T + T^{2}$$
$71$ $$1 + T + T^{2}$$
$73$ $$121 + 11 T + T^{2}$$
$79$ $$25 - 5 T + T^{2}$$
$83$ $$81 - 9 T + T^{2}$$
$89$ $$1 + T + T^{2}$$
$97$ $$( 2 + T )^{2}$$