# Properties

 Label 390.2.y.d Level $390$ Weight $2$ Character orbit 390.y Analytic conductor $3.114$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.y (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 1.00000 + 2.00000i 0.500000 + 0.866025i 0 1.00000i 0.500000 + 0.866025i 0.133975 2.23205i
139.2 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 1.00000 2.00000i 0.500000 + 0.866025i 0 1.00000i 0.500000 + 0.866025i 1.86603 1.23205i
289.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 1.00000 2.00000i 0.500000 0.866025i 0 1.00000i 0.500000 0.866025i 0.133975 + 2.23205i
289.2 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 1.00000 + 2.00000i 0.500000 0.866025i 0 1.00000i 0.500000 0.866025i 1.86603 + 1.23205i
 $$n$$: 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
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.y.d 4
3.b odd 2 1 1170.2.bp.c 4
5.b even 2 1 inner 390.2.y.d 4
5.c odd 4 1 1950.2.i.l 2
5.c odd 4 1 1950.2.i.p 2
13.c even 3 1 inner 390.2.y.d 4
15.d odd 2 1 1170.2.bp.c 4
39.i odd 6 1 1170.2.bp.c 4
65.n even 6 1 inner 390.2.y.d 4
65.q odd 12 1 1950.2.i.l 2
65.q odd 12 1 1950.2.i.p 2
195.x odd 6 1 1170.2.bp.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.y.d 4 1.a even 1 1 trivial
390.2.y.d 4 5.b even 2 1 inner
390.2.y.d 4 13.c even 3 1 inner
390.2.y.d 4 65.n even 6 1 inner
1170.2.bp.c 4 3.b odd 2 1
1170.2.bp.c 4 15.d odd 2 1
1170.2.bp.c 4 39.i odd 6 1
1170.2.bp.c 4 195.x odd 6 1
1950.2.i.l 2 5.c odd 4 1
1950.2.i.l 2 65.q odd 12 1
1950.2.i.p 2 5.c odd 4 1
1950.2.i.p 2 65.q odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$( 5 - 2 T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( 9 + 3 T + T^{2} )^{2}$$
$13$ $$169 - 22 T^{2} + T^{4}$$
$17$ $$1296 - 36 T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$6561 - 81 T^{2} + T^{4}$$
$29$ $$( 49 + 7 T + T^{2} )^{2}$$
$31$ $$( -1 + T )^{4}$$
$37$ $$1 - T^{2} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$28561 - 169 T^{2} + T^{4}$$
$47$ $$( 121 + T^{2} )^{2}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( 121 + 11 T + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$256 - 16 T^{2} + T^{4}$$
$71$ $$( 16 + 4 T + T^{2} )^{2}$$
$73$ $$T^{4}$$
$79$ $$( 1 + T )^{4}$$
$83$ $$( 16 + T^{2} )^{2}$$
$89$ $$( 16 + 4 T + T^{2} )^{2}$$
$97$ $$256 - 16 T^{2} + T^{4}$$