# Properties

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

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## 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} ) q^{2} -\beta_{2} q^{4} + ( 1 + \beta_{3} ) q^{5} -\beta_{2} q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{2} ) q^{2} -\beta_{2} q^{4} + ( 1 + \beta_{3} ) q^{5} -\beta_{2} q^{7} - q^{8} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{10} + ( \beta_{1} + \beta_{3} ) q^{11} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{13} - q^{14} + ( -1 + \beta_{2} ) q^{16} + ( 2 \beta_{1} - 5 \beta_{2} ) q^{17} + \beta_{1} q^{19} + ( \beta_{1} - \beta_{2} ) q^{20} + \beta_{1} q^{22} + ( 4 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{23} + \beta_{3} q^{25} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{26} + ( -1 + \beta_{2} ) q^{28} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{29} + \beta_{2} q^{32} + ( -5 - 2 \beta_{3} ) q^{34} + ( \beta_{1} - \beta_{2} ) q^{35} + ( 5 + \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{37} -\beta_{3} q^{38} + ( -1 - \beta_{3} ) q^{40} + ( -1 + 4 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{41} + 8 \beta_{2} q^{43} -\beta_{3} q^{44} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{46} + ( 8 + 3 \beta_{3} ) q^{47} + ( -1 + \beta_{2} ) q^{49} + ( \beta_{1} + \beta_{3} ) q^{50} + ( 1 + \beta_{1} + 2 \beta_{3} ) q^{52} -7 q^{53} + ( 4 - 4 \beta_{2} ) q^{55} + \beta_{2} q^{56} + ( 2 \beta_{1} + \beta_{2} ) q^{58} + 2 \beta_{1} q^{59} + ( 4 \beta_{1} + \beta_{2} ) q^{61} + q^{64} + ( -5 - \beta_{1} + 9 \beta_{2} - \beta_{3} ) q^{65} + ( -4 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{67} + ( -5 - 2 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{68} + ( -1 - \beta_{3} ) q^{70} + ( -6 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -5 - \beta_{3} ) q^{73} + ( \beta_{1} - 5 \beta_{2} ) q^{74} + ( -\beta_{1} - \beta_{3} ) q^{76} -\beta_{3} q^{77} + ( -4 + \beta_{3} ) q^{79} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{80} + ( 4 \beta_{1} + \beta_{2} ) q^{82} -2 \beta_{3} q^{83} + ( 5 \beta_{1} - 13 \beta_{2} ) q^{85} + 8 q^{86} + ( -\beta_{1} - \beta_{3} ) q^{88} + ( 6 + \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{89} + ( 1 + \beta_{1} + 2 \beta_{3} ) q^{91} + ( -4 - 2 \beta_{3} ) q^{92} + ( 8 + 3 \beta_{1} - 8 \beta_{2} + 3 \beta_{3} ) q^{94} -4 \beta_{2} q^{95} + ( -2 \beta_{1} + 6 \beta_{2} ) q^{97} + \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 2q^{4} + 2q^{5} - 2q^{7} - 4q^{8} + O(q^{10})$$ $$4q + 2q^{2} - 2q^{4} + 2q^{5} - 2q^{7} - 4q^{8} + q^{10} - q^{11} - 2q^{13} - 4q^{14} - 2q^{16} - 8q^{17} + q^{19} - q^{20} + q^{22} + 6q^{23} - 2q^{25} - q^{26} - 2q^{28} - 4q^{29} + 2q^{32} - 16q^{34} - q^{35} + 9q^{37} + 2q^{38} - 2q^{40} - 6q^{41} + 16q^{43} + 2q^{44} - 6q^{46} + 26q^{47} - 2q^{49} - q^{50} + q^{52} - 28q^{53} + 8q^{55} + 2q^{56} + 4q^{58} + 2q^{59} + 6q^{61} + 4q^{64} - q^{65} - 10q^{67} - 8q^{68} - 2q^{70} + 2q^{71} - 18q^{73} - 9q^{74} + q^{76} + 2q^{77} - 18q^{79} - q^{80} + 6q^{82} + 4q^{83} - 21q^{85} + 32q^{86} + q^{88} + 11q^{89} + q^{91} - 12q^{92} + 13q^{94} - 8q^{95} + 10q^{97} + 2q^{98} + O(q^{100})$$

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 $$\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$$.

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

## 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
 1.28078 − 2.21837i −0.780776 + 1.35234i 1.28078 + 2.21837i −0.780776 − 1.35234i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.56155 0 −0.500000 + 0.866025i −1.00000 0 −0.780776 1.35234i
757.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.56155 0 −0.500000 + 0.866025i −1.00000 0 1.28078 + 2.21837i
1387.1 0.500000 0.866025i 0 −0.500000 0.866025i −1.56155 0 −0.500000 0.866025i −1.00000 0 −0.780776 + 1.35234i
1387.2 0.500000 0.866025i 0 −0.500000 0.866025i 2.56155 0 −0.500000 0.866025i −1.00000 0 1.28078 2.21837i
 $$n$$: 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.y 4
3.b odd 2 1 546.2.l.l 4
13.c even 3 1 inner 1638.2.r.y 4
39.h odd 6 1 7098.2.a.bi 2
39.i odd 6 1 546.2.l.l 4
39.i odd 6 1 7098.2.a.bt 2

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

 $$T_{5}^{2} - T_{5} - 4$$ $$T_{11}^{4} + T_{11}^{3} + 5 T_{11}^{2} - 4 T_{11} + 16$$ $$T_{17}^{4} + 8 T_{17}^{3} + 65 T_{17}^{2} - 8 T_{17} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( -4 - T + T^{2} )^{2}$$
$7$ $$( 1 + T + T^{2} )^{2}$$
$11$ $$16 - 4 T + 5 T^{2} + T^{3} + T^{4}$$
$13$ $$( 13 + T + T^{2} )^{2}$$
$17$ $$1 - 8 T + 65 T^{2} + 8 T^{3} + T^{4}$$
$19$ $$16 + 4 T + 5 T^{2} - T^{3} + T^{4}$$
$23$ $$64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4}$$
$29$ $$169 - 52 T + 29 T^{2} + 4 T^{3} + T^{4}$$
$31$ $$T^{4}$$
$37$ $$256 - 144 T + 65 T^{2} - 9 T^{3} + T^{4}$$
$41$ $$3481 - 354 T + 95 T^{2} + 6 T^{3} + T^{4}$$
$43$ $$( 64 - 8 T + T^{2} )^{2}$$
$47$ $$( 4 - 13 T + T^{2} )^{2}$$
$53$ $$( 7 + T )^{4}$$
$59$ $$256 + 32 T + 20 T^{2} - 2 T^{3} + T^{4}$$
$61$ $$3481 + 354 T + 95 T^{2} - 6 T^{3} + T^{4}$$
$67$ $$64 + 80 T + 92 T^{2} + 10 T^{3} + T^{4}$$
$71$ $$23104 + 304 T + 156 T^{2} - 2 T^{3} + T^{4}$$
$73$ $$( 16 + 9 T + T^{2} )^{2}$$
$79$ $$( 16 + 9 T + T^{2} )^{2}$$
$83$ $$( -16 - 2 T + T^{2} )^{2}$$
$89$ $$676 - 286 T + 95 T^{2} - 11 T^{3} + T^{4}$$
$97$ $$64 - 80 T + 92 T^{2} - 10 T^{3} + T^{4}$$
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