# Properties

 Label 1638.2.r.ba Level $1638$ Weight $2$ Character orbit 1638.r Analytic conductor $13.079$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.r (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-43})$$ Defining polynomial: $$x^{4} - x^{3} - 10 x^{2} - 11 x + 121$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -1 + \beta_{2} ) q^{4} + 2 q^{5} + ( 1 - \beta_{2} ) q^{7} - q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -1 + \beta_{2} ) q^{4} + 2 q^{5} + ( 1 - \beta_{2} ) q^{7} - q^{8} + 2 \beta_{2} q^{10} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{11} + ( -2 + \beta_{1} - \beta_{3} ) q^{13} + q^{14} -\beta_{2} q^{16} + ( 5 - 5 \beta_{2} ) q^{17} + ( 3 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{19} + ( -2 + 2 \beta_{2} ) q^{20} + ( -1 + \beta_{1} - 2 \beta_{3} ) q^{22} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{23} - q^{25} + ( -2 \beta_{2} - \beta_{3} ) q^{26} + \beta_{2} q^{28} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{29} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{31} + ( 1 - \beta_{2} ) q^{32} + 5 q^{34} + ( 2 - 2 \beta_{2} ) q^{35} -2 \beta_{2} q^{37} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{38} -2 q^{40} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{41} + ( -6 - \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{43} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{44} + ( 2 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{46} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{47} -\beta_{2} q^{49} -\beta_{2} q^{50} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{52} + 3 q^{53} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{55} + ( -1 + \beta_{2} ) q^{56} + ( -3 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{58} + ( -6 - \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{59} + ( 3 - 3 \beta_{2} ) q^{61} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{62} + q^{64} + ( -4 + 2 \beta_{1} - 2 \beta_{3} ) q^{65} + ( -1 + 2 \beta_{1} + 10 \beta_{2} - \beta_{3} ) q^{67} + 5 \beta_{2} q^{68} + 2 q^{70} + ( 4 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{71} + 4 q^{73} + ( 2 - 2 \beta_{2} ) q^{74} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{76} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{77} + ( 8 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{79} -2 \beta_{2} q^{80} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{82} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{83} + ( 10 - 10 \beta_{2} ) q^{85} + ( -7 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{86} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{88} + 9 \beta_{2} q^{89} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{91} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{92} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{94} + ( 6 - 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{95} + ( -2 + 2 \beta_{1} - 4 \beta_{3} ) q^{97} + ( 1 - \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 2q^{4} + 8q^{5} + 2q^{7} - 4q^{8} + O(q^{10})$$ $$4q + 2q^{2} - 2q^{4} + 8q^{5} + 2q^{7} - 4q^{8} + 4q^{10} + q^{11} - 6q^{13} + 4q^{14} - 2q^{16} + 10q^{17} + 5q^{19} - 4q^{20} - q^{22} - 5q^{23} - 4q^{25} - 3q^{26} + 2q^{28} + 5q^{29} - 6q^{31} + 2q^{32} + 20q^{34} + 4q^{35} - 4q^{37} + 10q^{38} - 8q^{40} - 3q^{41} - 13q^{43} - 2q^{44} + 5q^{46} + 6q^{47} - 2q^{49} - 2q^{50} + 3q^{52} + 12q^{53} + 2q^{55} - 2q^{56} - 5q^{58} - 13q^{59} + 6q^{61} - 3q^{62} + 4q^{64} - 12q^{65} + 19q^{67} + 10q^{68} + 8q^{70} + 7q^{71} + 16q^{73} + 4q^{74} + 5q^{76} + 2q^{77} + 30q^{79} - 4q^{80} + 3q^{82} - 2q^{83} + 20q^{85} - 26q^{86} - q^{88} + 18q^{89} - 3q^{91} + 10q^{92} + 3q^{94} + 10q^{95} - 2q^{97} + 2q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 10 x^{2} - 11 x + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 10 \nu^{2} - 10 \nu - 11$$$$)/110$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 10 \nu + 11$$$$)/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 11 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-10 \beta_{3} + 10 \beta_{1} + 11$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$$.

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$-1 + \beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
757.1
 −2.58945 − 2.07237i 3.08945 + 1.20635i −2.58945 + 2.07237i 3.08945 − 1.20635i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.00000 0 0.500000 0.866025i −1.00000 0 1.00000 + 1.73205i
757.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.00000 0 0.500000 0.866025i −1.00000 0 1.00000 + 1.73205i
1387.1 0.500000 0.866025i 0 −0.500000 0.866025i 2.00000 0 0.500000 + 0.866025i −1.00000 0 1.00000 1.73205i
1387.2 0.500000 0.866025i 0 −0.500000 0.866025i 2.00000 0 0.500000 + 0.866025i −1.00000 0 1.00000 1.73205i
 $$n$$: 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.r.ba 4
3.b odd 2 1 546.2.l.k 4
13.c even 3 1 inner 1638.2.r.ba 4
39.h odd 6 1 7098.2.a.bk 2
39.i odd 6 1 546.2.l.k 4
39.i odd 6 1 7098.2.a.br 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.k 4 3.b odd 2 1
546.2.l.k 4 39.i odd 6 1
1638.2.r.ba 4 1.a even 1 1 trivial
1638.2.r.ba 4 13.c even 3 1 inner
7098.2.a.bk 2 39.h odd 6 1
7098.2.a.br 2 39.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1638, [\chi])$$:

 $$T_{5} - 2$$ $$T_{11}^{4} - T_{11}^{3} + 33 T_{11}^{2} + 32 T_{11} + 1024$$ $$T_{17}^{2} - 5 T_{17} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( -2 + T )^{4}$$
$7$ $$( 1 - T + T^{2} )^{2}$$
$11$ $$1024 + 32 T + 33 T^{2} - T^{3} + T^{4}$$
$13$ $$( 13 + 3 T + T^{2} )^{2}$$
$17$ $$( 25 - 5 T + T^{2} )^{2}$$
$19$ $$676 + 130 T + 51 T^{2} - 5 T^{3} + T^{4}$$
$23$ $$676 - 130 T + 51 T^{2} + 5 T^{3} + T^{4}$$
$29$ $$676 + 130 T + 51 T^{2} - 5 T^{3} + T^{4}$$
$31$ $$( -30 + 3 T + T^{2} )^{2}$$
$37$ $$( 4 + 2 T + T^{2} )^{2}$$
$41$ $$900 - 90 T + 39 T^{2} + 3 T^{3} + T^{4}$$
$43$ $$100 + 130 T + 159 T^{2} + 13 T^{3} + T^{4}$$
$47$ $$( -30 - 3 T + T^{2} )^{2}$$
$53$ $$( -3 + T )^{4}$$
$59$ $$100 + 130 T + 159 T^{2} + 13 T^{3} + T^{4}$$
$61$ $$( 9 - 3 T + T^{2} )^{2}$$
$67$ $$3364 - 1102 T + 303 T^{2} - 19 T^{3} + T^{4}$$
$71$ $$400 + 140 T + 69 T^{2} - 7 T^{3} + T^{4}$$
$73$ $$( -4 + T )^{4}$$
$79$ $$( 24 - 15 T + T^{2} )^{2}$$
$83$ $$( -32 + T + T^{2} )^{2}$$
$89$ $$( 81 - 9 T + T^{2} )^{2}$$
$97$ $$16384 - 256 T + 132 T^{2} + 2 T^{3} + T^{4}$$