# Properties

 Label 390.2.e.c Level $390$ Weight $2$ Character orbit 390.e 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.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 2.00000 1.00000i 1.00000 2.00000i 1.00000i −1.00000 −1.00000 2.00000i
79.2 1.00000i 1.00000i −1.00000 2.00000 + 1.00000i 1.00000 2.00000i 1.00000i −1.00000 −1.00000 + 2.00000i
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.e.c 2
3.b odd 2 1 1170.2.e.a 2
4.b odd 2 1 3120.2.l.g 2
5.b even 2 1 inner 390.2.e.c 2
5.c odd 4 1 1950.2.a.c 1
5.c odd 4 1 1950.2.a.z 1
15.d odd 2 1 1170.2.e.a 2
15.e even 4 1 5850.2.a.t 1
15.e even 4 1 5850.2.a.bj 1
20.d odd 2 1 3120.2.l.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.e.c 2 1.a even 1 1 trivial
390.2.e.c 2 5.b even 2 1 inner
1170.2.e.a 2 3.b odd 2 1
1170.2.e.a 2 15.d odd 2 1
1950.2.a.c 1 5.c odd 4 1
1950.2.a.z 1 5.c odd 4 1
3120.2.l.g 2 4.b odd 2 1
3120.2.l.g 2 20.d odd 2 1
5850.2.a.t 1 15.e even 4 1
5850.2.a.bj 1 15.e even 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$5 - 4 T + T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( 4 + T )^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$4 + T^{2}$$
$59$ $$( 10 + T )^{2}$$
$61$ $$( 14 + T )^{2}$$
$67$ $$256 + T^{2}$$
$71$ $$( 4 + T )^{2}$$
$73$ $$64 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$144 + T^{2}$$