# Properties

 Label 390.2.y.b Level $390$ Weight $2$ Character orbit 390.y Analytic conductor $3.114$ Analytic rank $0$ Dimension $4$ 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.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.

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

## 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 1.23205 + 1.86603i −0.500000 0.866025i −0.633975 + 0.366025i 1.00000i 0.500000 + 0.866025i −0.133975 2.23205i
139.2 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −2.23205 0.133975i −0.500000 0.866025i −2.36603 + 1.36603i 1.00000i 0.500000 + 0.866025i −1.86603 1.23205i
289.1 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i 1.23205 1.86603i −0.500000 + 0.866025i −0.633975 0.366025i 1.00000i 0.500000 0.866025i −0.133975 + 2.23205i
289.2 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −2.23205 + 0.133975i −0.500000 + 0.866025i −2.36603 1.36603i 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
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.b 4
3.b odd 2 1 1170.2.bp.d 4
5.b even 2 1 390.2.y.c yes 4
5.c odd 4 1 1950.2.i.y 4
5.c odd 4 1 1950.2.i.bh 4
13.c even 3 1 390.2.y.c yes 4
15.d odd 2 1 1170.2.bp.e 4
39.i odd 6 1 1170.2.bp.e 4
65.n even 6 1 inner 390.2.y.b 4
65.q odd 12 1 1950.2.i.y 4
65.q odd 12 1 1950.2.i.bh 4
195.x odd 6 1 1170.2.bp.d 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 6 T_{7}^{3} + 14 T_{7}^{2} + 12 T_{7} + 4$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$25 + 10 T - T^{2} + 2 T^{3} + T^{4}$$
$7$ $$4 + 12 T + 14 T^{2} + 6 T^{3} + T^{4}$$
$11$ $$4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4}$$
$13$ $$169 + 52 T + 3 T^{2} + 4 T^{3} + T^{4}$$
$17$ $$121 + 132 T + 59 T^{2} + 12 T^{3} + T^{4}$$
$19$ $$36 - 36 T + 30 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$676 + 468 T + 134 T^{2} + 18 T^{3} + T^{4}$$
$29$ $$9 - 18 T + 39 T^{2} + 6 T^{3} + T^{4}$$
$31$ $$( -4 + T )^{4}$$
$37$ $$2209 - 1128 T + 239 T^{2} - 24 T^{3} + T^{4}$$
$41$ $$121 + 88 T + 75 T^{2} - 8 T^{3} + T^{4}$$
$43$ $$484 + 132 T - 10 T^{2} - 6 T^{3} + T^{4}$$
$47$ $$5476 + 152 T^{2} + T^{4}$$
$53$ $$1089 + 78 T^{2} + T^{4}$$
$59$ $$10816 - 416 T + 120 T^{2} + 4 T^{3} + T^{4}$$
$61$ $$20449 - 572 T + 159 T^{2} + 4 T^{3} + T^{4}$$
$67$ $$5476 - 2220 T + 374 T^{2} - 30 T^{3} + T^{4}$$
$71$ $$36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4}$$
$73$ $$3481 + 182 T^{2} + T^{4}$$
$79$ $$( 12 + T )^{4}$$
$83$ $$2116 + 104 T^{2} + T^{4}$$
$89$ $$1936 + 176 T + 60 T^{2} - 4 T^{3} + T^{4}$$
$97$ $$10000 - 100 T^{2} + T^{4}$$