# Properties

 Label 1638.2.r.z 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: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( 1 - \beta_{3} ) q^{5} + ( 1 - \beta_{2} ) q^{7} - q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( 1 - \beta_{3} ) q^{5} + ( 1 - \beta_{2} ) q^{7} - q^{8} + ( \beta_{1} + \beta_{2} ) q^{10} + \beta_{1} q^{11} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + q^{14} -\beta_{2} q^{16} + ( -5 + 5 \beta_{2} ) q^{17} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{20} + ( \beta_{1} + \beta_{3} ) q^{22} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{23} -3 \beta_{3} q^{25} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{26} + \beta_{2} q^{28} -\beta_{2} q^{29} -2 \beta_{3} q^{31} + ( 1 - \beta_{2} ) q^{32} -5 q^{34} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{35} + ( 5 \beta_{1} - 3 \beta_{2} ) q^{37} + 3 \beta_{3} q^{38} + ( -1 + \beta_{3} ) q^{40} + ( 2 \beta_{1} - \beta_{2} ) q^{41} + ( 8 + 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{43} + \beta_{3} q^{44} + ( 4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{46} -3 \beta_{3} q^{47} -\beta_{2} q^{49} + 3 \beta_{1} q^{50} + ( 1 - \beta_{1} + \beta_{3} ) q^{52} + ( 9 + 2 \beta_{3} ) q^{53} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{55} + ( -1 + \beta_{2} ) q^{56} + ( 1 - \beta_{2} ) q^{58} + ( 8 - 2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 1 - \beta_{2} ) q^{61} + 2 \beta_{1} q^{62} + q^{64} + ( -4 + 3 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{65} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{67} -5 \beta_{2} q^{68} + ( 1 - \beta_{3} ) q^{70} + ( -4 + 4 \beta_{2} ) q^{71} + ( -5 - \beta_{3} ) q^{73} + ( 3 + 5 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} ) q^{74} -3 \beta_{1} q^{76} -\beta_{3} q^{77} + ( 4 + \beta_{3} ) q^{79} + ( -\beta_{1} - \beta_{2} ) q^{80} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{82} -4 \beta_{3} q^{83} + ( -5 + 5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{85} + ( 8 + 2 \beta_{3} ) q^{86} -\beta_{1} q^{88} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -1 + \beta_{1} - \beta_{3} ) q^{91} + ( 4 - 2 \beta_{3} ) q^{92} + 3 \beta_{1} q^{94} + ( -12 + 6 \beta_{1} + 12 \beta_{2} + 6 \beta_{3} ) q^{95} + ( -2 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{97} + ( 1 - \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 2q^{4} + 6q^{5} + 2q^{7} - 4q^{8} + O(q^{10})$$ $$4q + 2q^{2} - 2q^{4} + 6q^{5} + 2q^{7} - 4q^{8} + 3q^{10} + q^{11} - 2q^{13} + 4q^{14} - 2q^{16} - 10q^{17} - 3q^{19} - 3q^{20} - q^{22} - 10q^{23} + 6q^{25} - q^{26} + 2q^{28} - 2q^{29} + 4q^{31} + 2q^{32} - 20q^{34} + 3q^{35} - q^{37} - 6q^{38} - 6q^{40} + 14q^{43} - 2q^{44} + 10q^{46} + 6q^{47} - 2q^{49} + 3q^{50} + q^{52} + 32q^{53} + 10q^{55} - 2q^{56} + 2q^{58} + 18q^{59} + 2q^{61} + 2q^{62} + 4q^{64} - 3q^{65} + 6q^{67} - 10q^{68} + 6q^{70} - 8q^{71} - 18q^{73} + q^{74} - 3q^{76} + 2q^{77} + 14q^{79} - 3q^{80} + 8q^{83} - 15q^{85} + 28q^{86} - q^{88} - 7q^{89} - q^{91} + 20q^{92} + 3q^{94} - 30q^{95} - 8q^{97} + 2q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 5 \nu^{2} - 5 \nu + 16$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4 \beta_{2} + \beta_{1} - 4$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3} - 4$$

## 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
 −0.780776 − 1.35234i 1.28078 + 2.21837i −0.780776 + 1.35234i 1.28078 − 2.21837i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.561553 0 0.500000 0.866025i −1.00000 0 −0.280776 0.486319i
757.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 3.56155 0 0.500000 0.866025i −1.00000 0 1.78078 + 3.08440i
1387.1 0.500000 0.866025i 0 −0.500000 0.866025i −0.561553 0 0.500000 + 0.866025i −1.00000 0 −0.280776 + 0.486319i
1387.2 0.500000 0.866025i 0 −0.500000 0.866025i 3.56155 0 0.500000 + 0.866025i −1.00000 0 1.78078 3.08440i
 $$n$$: 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.z 4
3.b odd 2 1 546.2.l.i 4
13.c even 3 1 inner 1638.2.r.z 4
39.h odd 6 1 7098.2.a.bq 2
39.i odd 6 1 546.2.l.i 4
39.i odd 6 1 7098.2.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.i 4 3.b odd 2 1
546.2.l.i 4 39.i odd 6 1
1638.2.r.z 4 1.a even 1 1 trivial
1638.2.r.z 4 13.c even 3 1 inner
7098.2.a.bq 2 39.h odd 6 1
7098.2.a.bw 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 $$S_{2}^{\mathrm{new}}(1638, [\chi])$$:

 $$T_{5}^{2} - 3 T_{5} - 2$$ $$T_{11}^{4} - T_{11}^{3} + 5 T_{11}^{2} + 4 T_{11} + 16$$ $$T_{17}^{2} + 5 T_{17} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( -2 - 3 T + T^{2} )^{2}$$
$7$ $$( 1 - T + T^{2} )^{2}$$
$11$ $$16 + 4 T + 5 T^{2} - T^{3} + T^{4}$$
$13$ $$( 13 + T + T^{2} )^{2}$$
$17$ $$( 25 + 5 T + T^{2} )^{2}$$
$19$ $$1296 - 108 T + 45 T^{2} + 3 T^{3} + T^{4}$$
$23$ $$64 + 80 T + 92 T^{2} + 10 T^{3} + T^{4}$$
$29$ $$( 1 + T + T^{2} )^{2}$$
$31$ $$( -16 - 2 T + T^{2} )^{2}$$
$37$ $$11236 - 106 T + 107 T^{2} + T^{3} + T^{4}$$
$41$ $$289 + 17 T^{2} + T^{4}$$
$43$ $$1024 - 448 T + 164 T^{2} - 14 T^{3} + T^{4}$$
$47$ $$( -36 - 3 T + T^{2} )^{2}$$
$53$ $$( 47 - 16 T + T^{2} )^{2}$$
$59$ $$4096 - 1152 T + 260 T^{2} - 18 T^{3} + T^{4}$$
$61$ $$( 1 - T + T^{2} )^{2}$$
$67$ $$64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4}$$
$71$ $$( 16 + 4 T + T^{2} )^{2}$$
$73$ $$( 16 + 9 T + T^{2} )^{2}$$
$79$ $$( 8 - 7 T + T^{2} )^{2}$$
$83$ $$( -64 - 4 T + T^{2} )^{2}$$
$89$ $$676 - 182 T + 75 T^{2} + 7 T^{3} + T^{4}$$
$97$ $$2704 - 416 T + 116 T^{2} + 8 T^{3} + T^{4}$$