# Properties

 Label 1638.2.c.h Level $1638$ Weight $2$ Character orbit 1638.c 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} - q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} -\beta_{2} q^{7} + \beta_{2} q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} - q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} -\beta_{2} q^{7} + \beta_{2} q^{8} + ( -2 + \beta_{3} ) q^{10} + ( \beta_{1} - 3 \beta_{2} ) q^{11} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{13} - q^{14} + q^{16} + ( 2 - 3 \beta_{3} ) q^{17} + ( \beta_{1} + \beta_{2} ) q^{19} + ( -\beta_{1} + \beta_{2} ) q^{20} + ( -4 + \beta_{3} ) q^{22} + ( 2 - 3 \beta_{3} ) q^{23} + ( -3 + 3 \beta_{3} ) q^{25} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{26} + \beta_{2} q^{28} + \beta_{3} q^{29} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{31} -\beta_{2} q^{32} + ( 3 \beta_{1} + \beta_{2} ) q^{34} + ( -2 + \beta_{3} ) q^{35} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{37} + \beta_{3} q^{38} + ( 2 - \beta_{3} ) q^{40} -4 \beta_{2} q^{41} + ( -8 - \beta_{3} ) q^{43} + ( -\beta_{1} + 3 \beta_{2} ) q^{44} + ( 3 \beta_{1} + \beta_{2} ) q^{46} -4 \beta_{1} q^{47} - q^{49} -3 \beta_{1} q^{50} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{52} + ( -8 - 2 \beta_{3} ) q^{53} + ( -12 + 5 \beta_{3} ) q^{55} + q^{56} + ( -\beta_{1} - \beta_{2} ) q^{58} + ( 4 \beta_{1} + 6 \beta_{2} ) q^{59} + \beta_{3} q^{61} + 4 \beta_{3} q^{62} - q^{64} + ( -2 + 3 \beta_{1} - 5 \beta_{2} ) q^{65} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{67} + ( -2 + 3 \beta_{3} ) q^{68} + ( -\beta_{1} + \beta_{2} ) q^{70} + ( 6 \beta_{1} + 6 \beta_{2} ) q^{71} + ( -\beta_{1} + 9 \beta_{2} ) q^{73} + ( 6 - 3 \beta_{3} ) q^{74} + ( -\beta_{1} - \beta_{2} ) q^{76} + ( -4 + \beta_{3} ) q^{77} + 16 q^{79} + ( \beta_{1} - \beta_{2} ) q^{80} -4 q^{82} + 2 \beta_{2} q^{83} + ( 5 \beta_{1} - 11 \beta_{2} ) q^{85} + ( \beta_{1} + 9 \beta_{2} ) q^{86} + ( 4 - \beta_{3} ) q^{88} + 8 \beta_{2} q^{89} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{91} + ( -2 + 3 \beta_{3} ) q^{92} + ( 4 - 4 \beta_{3} ) q^{94} + ( -4 + \beta_{3} ) q^{95} + 10 \beta_{2} q^{97} + \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + O(q^{10})$$ $$4q - 4q^{4} - 6q^{10} + 6q^{13} - 4q^{14} + 4q^{16} + 2q^{17} - 14q^{22} + 2q^{23} - 6q^{25} + 6q^{26} + 2q^{29} - 6q^{35} + 2q^{38} + 6q^{40} - 34q^{43} - 4q^{49} - 6q^{52} - 36q^{53} - 38q^{55} + 4q^{56} + 2q^{61} + 8q^{62} - 4q^{64} - 8q^{65} - 2q^{68} + 18q^{74} - 14q^{77} + 64q^{79} - 16q^{82} + 14q^{88} + 6q^{91} - 2q^{92} + 8q^{94} - 14q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

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

## Character values

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

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
883.1
 − 2.56155i 1.56155i − 1.56155i 2.56155i
1.00000i 0 −1.00000 3.56155i 0 1.00000i 1.00000i 0 −3.56155
883.2 1.00000i 0 −1.00000 0.561553i 0 1.00000i 1.00000i 0 0.561553
883.3 1.00000i 0 −1.00000 0.561553i 0 1.00000i 1.00000i 0 0.561553
883.4 1.00000i 0 −1.00000 3.56155i 0 1.00000i 1.00000i 0 −3.56155
 $$n$$: 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.c.h 4
3.b odd 2 1 546.2.c.e 4
12.b even 2 1 4368.2.h.n 4
13.b even 2 1 inner 1638.2.c.h 4
21.c even 2 1 3822.2.c.h 4
39.d odd 2 1 546.2.c.e 4
39.f even 4 1 7098.2.a.bg 2
39.f even 4 1 7098.2.a.bv 2
156.h even 2 1 4368.2.h.n 4
273.g even 2 1 3822.2.c.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.e 4 3.b odd 2 1
546.2.c.e 4 39.d odd 2 1
1638.2.c.h 4 1.a even 1 1 trivial
1638.2.c.h 4 13.b even 2 1 inner
3822.2.c.h 4 21.c even 2 1
3822.2.c.h 4 273.g even 2 1
4368.2.h.n 4 12.b even 2 1
4368.2.h.n 4 156.h even 2 1
7098.2.a.bg 2 39.f even 4 1
7098.2.a.bv 2 39.f even 4 1

## Hecke kernels

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

 $$T_{5}^{4} + 13 T_{5}^{2} + 4$$ $$T_{11}^{4} + 33 T_{11}^{2} + 64$$ $$T_{17}^{2} - T_{17} - 38$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$4 + 13 T^{2} + T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$64 + 33 T^{2} + T^{4}$$
$13$ $$169 - 78 T + 18 T^{2} - 6 T^{3} + T^{4}$$
$17$ $$( -38 - T + T^{2} )^{2}$$
$19$ $$16 + 9 T^{2} + T^{4}$$
$23$ $$( -38 - T + T^{2} )^{2}$$
$29$ $$( -4 - T + T^{2} )^{2}$$
$31$ $$4096 + 144 T^{2} + T^{4}$$
$37$ $$324 + 117 T^{2} + T^{4}$$
$41$ $$( 16 + T^{2} )^{2}$$
$43$ $$( 68 + 17 T + T^{2} )^{2}$$
$47$ $$4096 + 144 T^{2} + T^{4}$$
$53$ $$( 64 + 18 T + T^{2} )^{2}$$
$59$ $$2704 + 168 T^{2} + T^{4}$$
$61$ $$( -4 - T + T^{2} )^{2}$$
$67$ $$64 + 52 T^{2} + T^{4}$$
$71$ $$20736 + 324 T^{2} + T^{4}$$
$73$ $$7396 + 189 T^{2} + T^{4}$$
$79$ $$( -16 + T )^{4}$$
$83$ $$( 4 + T^{2} )^{2}$$
$89$ $$( 64 + T^{2} )^{2}$$
$97$ $$( 100 + T^{2} )^{2}$$