# Properties

 Label 1380.2.f.a Level $1380$ Weight $2$ Character orbit 1380.f Analytic conductor $11.019$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1380.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0193554789$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{3} + ( -\beta_{1} + \beta_{4} ) q^{5} -\beta_{3} q^{7} - q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{3} + ( -\beta_{1} + \beta_{4} ) q^{5} -\beta_{3} q^{7} - q^{9} -\beta_{1} q^{11} + ( \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{13} + ( -\beta_{2} - \beta_{5} ) q^{15} + ( -\beta_{4} + 2 \beta_{5} ) q^{17} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{19} - q^{21} -\beta_{3} q^{23} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{25} + \beta_{3} q^{27} + ( 2 - 3 \beta_{2} ) q^{29} + ( -4 - \beta_{1} + \beta_{2} ) q^{31} -\beta_{5} q^{33} + ( -\beta_{2} - \beta_{5} ) q^{35} + ( -3 \beta_{4} + \beta_{5} ) q^{37} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{39} + ( -3 - \beta_{1} - 4 \beta_{2} ) q^{41} + ( \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{43} + ( \beta_{1} - \beta_{4} ) q^{45} + ( -4 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{47} + 6 q^{49} + ( -2 \beta_{1} + \beta_{2} ) q^{51} + ( -\beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{53} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{55} + ( -\beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{57} + ( -2 - \beta_{2} ) q^{59} + ( -7 + 4 \beta_{1} - \beta_{2} ) q^{61} + \beta_{3} q^{63} + ( \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{65} + ( 2 \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{67} - q^{69} + ( -1 - 3 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -7 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{73} + ( 2 + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{75} -\beta_{5} q^{77} + ( 2 - 6 \beta_{1} + 2 \beta_{2} ) q^{79} + q^{81} + ( 8 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{83} + ( \beta_{1} - 3 \beta_{2} + 5 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{85} + ( -2 \beta_{3} - 3 \beta_{4} ) q^{87} + ( -2 - 4 \beta_{2} ) q^{89} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{91} + ( 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{93} + ( 5 - 3 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{95} + ( -3 \beta_{3} + 7 \beta_{4} - 5 \beta_{5} ) q^{97} + \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{9} + O(q^{10})$$ $$6q - 6q^{9} + 2q^{15} + 4q^{19} - 6q^{21} + 2q^{25} + 18q^{29} - 26q^{31} + 2q^{35} + 4q^{39} - 10q^{41} + 36q^{49} - 2q^{51} + 16q^{55} - 10q^{59} - 40q^{61} + 4q^{65} - 6q^{69} - 14q^{71} + 8q^{75} + 8q^{79} + 6q^{81} + 6q^{85} - 4q^{89} + 4q^{91} + 28q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 8 \nu^{4} - 4 \nu^{3} - \nu^{2} + 2 \nu + 38$$$$)/23$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{5} + 17 \nu^{4} - 20 \nu^{3} - 5 \nu^{2} + 10 \nu + 29$$$$)/23$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13$$$$)/23$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{5} + 19 \nu^{4} - 21 \nu^{3} - 11 \nu^{2} - 70 \nu + 27$$$$)/23$$ $$\beta_{5}$$ $$=$$ $$($$$$-14 \nu^{5} + 20 \nu^{4} - 10 \nu^{3} - 37 \nu^{2} - 64 \nu + 26$$$$)/23$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{2} + 5 \beta_{1} - 7$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 3 \beta_{2} + 8 \beta_{1} - 9$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$461$$ $$691$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
829.1
 0.403032 + 0.403032i −0.854638 − 0.854638i 1.45161 + 1.45161i 0.403032 − 0.403032i −0.854638 + 0.854638i 1.45161 − 1.45161i
0 1.00000i 0 −1.67513 1.48119i 0 1.00000i 0 −1.00000 0
829.2 0 1.00000i 0 −0.539189 + 2.17009i 0 1.00000i 0 −1.00000 0
829.3 0 1.00000i 0 2.21432 + 0.311108i 0 1.00000i 0 −1.00000 0
829.4 0 1.00000i 0 −1.67513 + 1.48119i 0 1.00000i 0 −1.00000 0
829.5 0 1.00000i 0 −0.539189 2.17009i 0 1.00000i 0 −1.00000 0
829.6 0 1.00000i 0 2.21432 0.311108i 0 1.00000i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 829.6 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 1380.2.f.a 6
3.b odd 2 1 4140.2.f.a 6
5.b even 2 1 inner 1380.2.f.a 6
5.c odd 4 1 6900.2.a.y 3
5.c odd 4 1 6900.2.a.z 3
15.d odd 2 1 4140.2.f.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.f.a 6 1.a even 1 1 trivial
1380.2.f.a 6 5.b even 2 1 inner
4140.2.f.a 6 3.b odd 2 1
4140.2.f.a 6 15.d odd 2 1
6900.2.a.y 3 5.c odd 4 1
6900.2.a.z 3 5.c odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 1 + T^{2} )^{3}$$
$5$ $$125 - 5 T^{2} - 16 T^{3} - T^{4} + T^{6}$$
$7$ $$( 1 + T^{2} )^{3}$$
$11$ $$( -2 - 4 T + T^{3} )^{2}$$
$13$ $$100 + 156 T^{2} + 32 T^{4} + T^{6}$$
$17$ $$625 + 275 T^{2} + 31 T^{4} + T^{6}$$
$19$ $$( -10 - 14 T - 2 T^{2} + T^{3} )^{2}$$
$23$ $$( 1 + T^{2} )^{3}$$
$29$ $$( 61 - 3 T - 9 T^{2} + T^{3} )^{2}$$
$31$ $$( 59 + 51 T + 13 T^{2} + T^{3} )^{2}$$
$37$ $$625 + 475 T^{2} + 59 T^{4} + T^{6}$$
$41$ $$( 25 - 57 T + 5 T^{2} + T^{3} )^{2}$$
$43$ $$5776 + 1600 T^{2} + 80 T^{4} + T^{6}$$
$47$ $$3364 + 1080 T^{2} + 92 T^{4} + T^{6}$$
$53$ $$4489 + 839 T^{2} + 51 T^{4} + T^{6}$$
$59$ $$( -1 + 5 T + 5 T^{2} + T^{3} )^{2}$$
$61$ $$( 74 + 74 T + 20 T^{2} + T^{3} )^{2}$$
$67$ $$26569 + 9083 T^{2} + 195 T^{4} + T^{6}$$
$71$ $$( -5 - 49 T + 7 T^{2} + T^{3} )^{2}$$
$73$ $$42436 + 9980 T^{2} + 208 T^{4} + T^{6}$$
$79$ $$( -416 - 128 T - 4 T^{2} + T^{3} )^{2}$$
$83$ $$1849 + 10963 T^{2} + 215 T^{4} + T^{6}$$
$89$ $$( -40 - 52 T + 2 T^{2} + T^{3} )^{2}$$
$97$ $$781456 + 40400 T^{2} + 388 T^{4} + T^{6}$$