# Properties

 Label 1380.2.a.f Level $1380$ Weight $2$ Character orbit 1380.a Self dual yes Analytic conductor $11.019$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1380.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.0193554789$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} + q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} - q^{5} + q^{7} + q^{9} + ( -2 + \beta ) q^{11} -\beta q^{13} + q^{15} + ( -1 + \beta ) q^{17} -3 \beta q^{19} - q^{21} + q^{23} + q^{25} - q^{27} + ( -7 + \beta ) q^{29} + ( 3 + 2 \beta ) q^{31} + ( 2 - \beta ) q^{33} - q^{35} + ( -5 + 2 \beta ) q^{37} + \beta q^{39} + ( -3 - \beta ) q^{41} + ( 2 + 4 \beta ) q^{43} - q^{45} -3 \beta q^{47} -6 q^{49} + ( 1 - \beta ) q^{51} + ( -3 - 3 \beta ) q^{53} + ( 2 - \beta ) q^{55} + 3 \beta q^{57} + ( -9 - \beta ) q^{59} + ( -2 - \beta ) q^{61} + q^{63} + \beta q^{65} + ( -1 - 4 \beta ) q^{67} - q^{69} + ( -5 - 3 \beta ) q^{71} + 3 \beta q^{73} - q^{75} + ( -2 + \beta ) q^{77} + ( -4 + 4 \beta ) q^{79} + q^{81} + ( -13 + \beta ) q^{83} + ( 1 - \beta ) q^{85} + ( 7 - \beta ) q^{87} + ( -8 - 2 \beta ) q^{89} -\beta q^{91} + ( -3 - 2 \beta ) q^{93} + 3 \beta q^{95} -2 \beta q^{97} + ( -2 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} - 4q^{11} + 2q^{15} - 2q^{17} - 2q^{21} + 2q^{23} + 2q^{25} - 2q^{27} - 14q^{29} + 6q^{31} + 4q^{33} - 2q^{35} - 10q^{37} - 6q^{41} + 4q^{43} - 2q^{45} - 12q^{49} + 2q^{51} - 6q^{53} + 4q^{55} - 18q^{59} - 4q^{61} + 2q^{63} - 2q^{67} - 2q^{69} - 10q^{71} - 2q^{75} - 4q^{77} - 8q^{79} + 2q^{81} - 26q^{83} + 2q^{85} + 14q^{87} - 16q^{89} - 6q^{93} - 4q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.44949 2.44949
0 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$23$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.2.a.f 2
3.b odd 2 1 4140.2.a.r 2
4.b odd 2 1 5520.2.a.bl 2
5.b even 2 1 6900.2.a.r 2
5.c odd 4 2 6900.2.f.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.f 2 1.a even 1 1 trivial
4140.2.a.r 2 3.b odd 2 1
5520.2.a.bl 2 4.b odd 2 1
6900.2.a.r 2 5.b even 2 1
6900.2.f.h 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1380))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-2 + 4 T + T^{2}$$
$13$ $$-6 + T^{2}$$
$17$ $$-5 + 2 T + T^{2}$$
$19$ $$-54 + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$43 + 14 T + T^{2}$$
$31$ $$-15 - 6 T + T^{2}$$
$37$ $$1 + 10 T + T^{2}$$
$41$ $$3 + 6 T + T^{2}$$
$43$ $$-92 - 4 T + T^{2}$$
$47$ $$-54 + T^{2}$$
$53$ $$-45 + 6 T + T^{2}$$
$59$ $$75 + 18 T + T^{2}$$
$61$ $$-2 + 4 T + T^{2}$$
$67$ $$-95 + 2 T + T^{2}$$
$71$ $$-29 + 10 T + T^{2}$$
$73$ $$-54 + T^{2}$$
$79$ $$-80 + 8 T + T^{2}$$
$83$ $$163 + 26 T + T^{2}$$
$89$ $$40 + 16 T + T^{2}$$
$97$ $$-24 + T^{2}$$