# Properties

 Label 390.2.y.e 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.

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

## 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 2.00000 1.00000i −0.500000 0.866025i 4.33013 2.50000i 1.00000i 0.500000 + 0.866025i −2.23205 0.133975i
139.2 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 2.00000 + 1.00000i −0.500000 0.866025i −4.33013 + 2.50000i 1.00000i 0.500000 + 0.866025i 1.23205 + 1.86603i
289.1 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i 2.00000 + 1.00000i −0.500000 + 0.866025i 4.33013 + 2.50000i 1.00000i 0.500000 0.866025i −2.23205 + 0.133975i
289.2 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 2.00000 1.00000i −0.500000 + 0.866025i −4.33013 2.50000i 1.00000i 0.500000 0.866025i 1.23205 1.86603i
 $$n$$: 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.e 4
3.b odd 2 1 1170.2.bp.b 4
5.b even 2 1 inner 390.2.y.e 4
5.c odd 4 1 1950.2.i.a 2
5.c odd 4 1 1950.2.i.x 2
13.c even 3 1 inner 390.2.y.e 4
15.d odd 2 1 1170.2.bp.b 4
39.i odd 6 1 1170.2.bp.b 4
65.n even 6 1 inner 390.2.y.e 4
65.q odd 12 1 1950.2.i.a 2
65.q odd 12 1 1950.2.i.x 2
195.x odd 6 1 1170.2.bp.b 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 25 T_{7}^{2} + 625$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$( 5 - 4 T + T^{2} )^{2}$$
$7$ $$625 - 25 T^{2} + T^{4}$$
$11$ $$( 9 + 3 T + T^{2} )^{2}$$
$13$ $$169 - T^{2} + T^{4}$$
$17$ $$256 - 16 T^{2} + T^{4}$$
$19$ $$( 1 + T + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$( 4 - 2 T + T^{2} )^{2}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$6561 - 81 T^{2} + T^{4}$$
$41$ $$( 100 + 10 T + T^{2} )^{2}$$
$43$ $$20736 - 144 T^{2} + T^{4}$$
$47$ $$( 49 + T^{2} )^{2}$$
$53$ $$( 9 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$1296 - 36 T^{2} + T^{4}$$
$71$ $$( 144 + 12 T + T^{2} )^{2}$$
$73$ $$( 256 + T^{2} )^{2}$$
$79$ $$( -14 + T )^{4}$$
$83$ $$( 100 + T^{2} )^{2}$$
$89$ $$( 1 + T + T^{2} )^{2}$$
$97$ $$10000 - 100 T^{2} + T^{4}$$