# Properties

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

# Learn more

## 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.

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

## 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 1.50000 + 2.59808i 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 1.50000 2.59808i 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.b 2
3.b odd 2 1 1170.2.i.j 2
5.b even 2 1 1950.2.i.o 2
5.c odd 4 2 1950.2.z.i 4
13.c even 3 1 inner 390.2.i.b 2
13.c even 3 1 5070.2.a.q 1
13.e even 6 1 5070.2.a.c 1
13.f odd 12 2 5070.2.b.a 2
39.i odd 6 1 1170.2.i.j 2
65.n even 6 1 1950.2.i.o 2
65.q odd 12 2 1950.2.z.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.b 2 1.a even 1 1 trivial
390.2.i.b 2 13.c even 3 1 inner
1170.2.i.j 2 3.b odd 2 1
1170.2.i.j 2 39.i odd 6 1
1950.2.i.o 2 5.b even 2 1
1950.2.i.o 2 65.n even 6 1
1950.2.z.i 4 5.c odd 4 2
1950.2.z.i 4 65.q odd 12 2
5070.2.a.c 1 13.e even 6 1
5070.2.a.q 1 13.c even 3 1
5070.2.b.a 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 $$S_{2}^{\mathrm{new}}(390, [\chi])$$:

 $$T_{7}^{2} - 3 T_{7} + 9$$ $$T_{11}^{2} + T_{11} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$9 - 3 T + T^{2}$$
$11$ $$1 + T + T^{2}$$
$13$ $$13 - 5 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$25 - 5 T + T^{2}$$
$23$ $$16 - 4 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -10 + T )^{2}$$
$37$ $$1 - T + T^{2}$$
$41$ $$36 + 6 T + T^{2}$$
$43$ $$4 - 2 T + T^{2}$$
$47$ $$( 9 + T )^{2}$$
$53$ $$( 13 + T )^{2}$$
$59$ $$16 + 4 T + T^{2}$$
$61$ $$4 - 2 T + T^{2}$$
$67$ $$144 - 12 T + T^{2}$$
$71$ $$4 - 2 T + T^{2}$$
$73$ $$( 16 + T )^{2}$$
$79$ $$( 10 + T )^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$1 + T + T^{2}$$
$97$ $$144 + 12 T + T^{2}$$
show more
show less