# Properties

 Label 390.2.i.c Level $390$ Weight $2$ Character orbit 390.i Analytic conductor $3.114$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} - q^{5} + \zeta_{6} q^{6} -2 \zeta_{6} q^{7} - q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} - q^{5} + \zeta_{6} q^{6} -2 \zeta_{6} q^{7} - q^{8} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{10} + ( 3 - 3 \zeta_{6} ) q^{11} + q^{12} + ( 3 - 4 \zeta_{6} ) q^{13} -2 q^{14} + ( 1 - \zeta_{6} ) q^{15} + ( -1 + \zeta_{6} ) q^{16} -6 \zeta_{6} q^{17} - q^{18} -2 \zeta_{6} q^{19} + \zeta_{6} q^{20} + 2 q^{21} -3 \zeta_{6} q^{22} + ( -3 + 3 \zeta_{6} ) q^{23} + ( 1 - \zeta_{6} ) q^{24} + q^{25} + ( -1 - 3 \zeta_{6} ) q^{26} + q^{27} + ( -2 + 2 \zeta_{6} ) q^{28} + ( -3 + 3 \zeta_{6} ) q^{29} -\zeta_{6} q^{30} + 5 q^{31} + \zeta_{6} q^{32} + 3 \zeta_{6} q^{33} -6 q^{34} + 2 \zeta_{6} q^{35} + ( -1 + \zeta_{6} ) q^{36} + ( 7 - 7 \zeta_{6} ) q^{37} -2 q^{38} + ( 1 + 3 \zeta_{6} ) q^{39} + q^{40} + ( -6 + 6 \zeta_{6} ) q^{41} + ( 2 - 2 \zeta_{6} ) q^{42} + \zeta_{6} q^{43} -3 q^{44} + \zeta_{6} q^{45} + 3 \zeta_{6} q^{46} -3 q^{47} -\zeta_{6} q^{48} + ( 3 - 3 \zeta_{6} ) q^{49} + ( 1 - \zeta_{6} ) q^{50} + 6 q^{51} + ( -4 + \zeta_{6} ) q^{52} -6 q^{53} + ( 1 - \zeta_{6} ) q^{54} + ( -3 + 3 \zeta_{6} ) q^{55} + 2 \zeta_{6} q^{56} + 2 q^{57} + 3 \zeta_{6} q^{58} + 9 \zeta_{6} q^{59} - q^{60} -2 \zeta_{6} q^{61} + ( 5 - 5 \zeta_{6} ) q^{62} + ( -2 + 2 \zeta_{6} ) q^{63} + q^{64} + ( -3 + 4 \zeta_{6} ) q^{65} + 3 q^{66} + ( -8 + 8 \zeta_{6} ) q^{67} + ( -6 + 6 \zeta_{6} ) q^{68} -3 \zeta_{6} q^{69} + 2 q^{70} + 12 \zeta_{6} q^{71} + \zeta_{6} q^{72} + 14 q^{73} -7 \zeta_{6} q^{74} + ( -1 + \zeta_{6} ) q^{75} + ( -2 + 2 \zeta_{6} ) q^{76} -6 q^{77} + ( 4 - \zeta_{6} ) q^{78} + 5 q^{79} + ( 1 - \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + 6 \zeta_{6} q^{82} -6 q^{83} -2 \zeta_{6} q^{84} + 6 \zeta_{6} q^{85} + q^{86} -3 \zeta_{6} q^{87} + ( -3 + 3 \zeta_{6} ) q^{88} + ( 18 - 18 \zeta_{6} ) q^{89} + q^{90} + ( -8 + 2 \zeta_{6} ) q^{91} + 3 q^{92} + ( -5 + 5 \zeta_{6} ) q^{93} + ( -3 + 3 \zeta_{6} ) q^{94} + 2 \zeta_{6} q^{95} - q^{96} -14 \zeta_{6} q^{97} -3 \zeta_{6} q^{98} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} - 2q^{7} - 2q^{8} - q^{9} + O(q^{10})$$ $$2q + q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} - 2q^{7} - 2q^{8} - q^{9} - q^{10} + 3q^{11} + 2q^{12} + 2q^{13} - 4q^{14} + q^{15} - q^{16} - 6q^{17} - 2q^{18} - 2q^{19} + q^{20} + 4q^{21} - 3q^{22} - 3q^{23} + q^{24} + 2q^{25} - 5q^{26} + 2q^{27} - 2q^{28} - 3q^{29} - q^{30} + 10q^{31} + q^{32} + 3q^{33} - 12q^{34} + 2q^{35} - q^{36} + 7q^{37} - 4q^{38} + 5q^{39} + 2q^{40} - 6q^{41} + 2q^{42} + q^{43} - 6q^{44} + q^{45} + 3q^{46} - 6q^{47} - q^{48} + 3q^{49} + q^{50} + 12q^{51} - 7q^{52} - 12q^{53} + q^{54} - 3q^{55} + 2q^{56} + 4q^{57} + 3q^{58} + 9q^{59} - 2q^{60} - 2q^{61} + 5q^{62} - 2q^{63} + 2q^{64} - 2q^{65} + 6q^{66} - 8q^{67} - 6q^{68} - 3q^{69} + 4q^{70} + 12q^{71} + q^{72} + 28q^{73} - 7q^{74} - q^{75} - 2q^{76} - 12q^{77} + 7q^{78} + 10q^{79} + q^{80} - q^{81} + 6q^{82} - 12q^{83} - 2q^{84} + 6q^{85} + 2q^{86} - 3q^{87} - 3q^{88} + 18q^{89} + 2q^{90} - 14q^{91} + 6q^{92} - 5q^{93} - 3q^{94} + 2q^{95} - 2q^{96} - 14q^{97} - 3q^{98} - 6q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$$.

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.00000 0.500000 + 0.866025i −1.00000 1.73205i −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
211.1 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −1.00000 0.500000 0.866025i −1.00000 + 1.73205i −1.00000 −0.500000 + 0.866025i −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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.i.c 2
3.b odd 2 1 1170.2.i.d 2
5.b even 2 1 1950.2.i.n 2
5.c odd 4 2 1950.2.z.k 4
13.c even 3 1 inner 390.2.i.c 2
13.c even 3 1 5070.2.a.j 1
13.e even 6 1 5070.2.a.v 1
13.f odd 12 2 5070.2.b.j 2
39.i odd 6 1 1170.2.i.d 2
65.n even 6 1 1950.2.i.n 2
65.q odd 12 2 1950.2.z.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.c 2 1.a even 1 1 trivial
390.2.i.c 2 13.c even 3 1 inner
1170.2.i.d 2 3.b odd 2 1
1170.2.i.d 2 39.i odd 6 1
1950.2.i.n 2 5.b even 2 1
1950.2.i.n 2 65.n even 6 1
1950.2.z.k 4 5.c odd 4 2
1950.2.z.k 4 65.q odd 12 2
5070.2.a.j 1 13.c even 3 1
5070.2.a.v 1 13.e even 6 1
5070.2.b.j 2 13.f odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$:

 $$T_{7}^{2} + 2 T_{7} + 4$$ $$T_{11}^{2} - 3 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$4 + 2 T + T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$13 - 2 T + T^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$9 + 3 T + T^{2}$$
$31$ $$( -5 + T )^{2}$$
$37$ $$49 - 7 T + T^{2}$$
$41$ $$36 + 6 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$( 3 + T )^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$81 - 9 T + T^{2}$$
$61$ $$4 + 2 T + T^{2}$$
$67$ $$64 + 8 T + T^{2}$$
$71$ $$144 - 12 T + T^{2}$$
$73$ $$( -14 + T )^{2}$$
$79$ $$( -5 + T )^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$324 - 18 T + T^{2}$$
$97$ $$196 + 14 T + T^{2}$$