# Properties

 Label 1638.2.r.f Level $1638$ Weight $2$ Character orbit 1638.r Analytic conductor $13.079$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -\zeta_{6} q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -\zeta_{6} q^{7} + q^{8} + ( -3 + 3 \zeta_{6} ) q^{11} + ( 4 - 3 \zeta_{6} ) q^{13} + q^{14} + ( -1 + \zeta_{6} ) q^{16} -3 \zeta_{6} q^{17} -5 \zeta_{6} q^{19} -3 \zeta_{6} q^{22} + ( -6 + 6 \zeta_{6} ) q^{23} -5 q^{25} + ( -1 + 4 \zeta_{6} ) q^{26} + ( -1 + \zeta_{6} ) q^{28} + ( -9 + 9 \zeta_{6} ) q^{29} + 8 q^{31} -\zeta_{6} q^{32} + 3 q^{34} + ( -8 + 8 \zeta_{6} ) q^{37} + 5 q^{38} + ( 3 - 3 \zeta_{6} ) q^{41} -8 \zeta_{6} q^{43} + 3 q^{44} -6 \zeta_{6} q^{46} -3 q^{47} + ( -1 + \zeta_{6} ) q^{49} + ( 5 - 5 \zeta_{6} ) q^{50} + ( -3 - \zeta_{6} ) q^{52} + 3 q^{53} -\zeta_{6} q^{56} -9 \zeta_{6} q^{58} -6 \zeta_{6} q^{59} + 7 \zeta_{6} q^{61} + ( -8 + 8 \zeta_{6} ) q^{62} + q^{64} + ( -2 + 2 \zeta_{6} ) q^{67} + ( -3 + 3 \zeta_{6} ) q^{68} -6 \zeta_{6} q^{71} -16 q^{73} -8 \zeta_{6} q^{74} + ( -5 + 5 \zeta_{6} ) q^{76} + 3 q^{77} -13 q^{79} + 3 \zeta_{6} q^{82} -18 q^{83} + 8 q^{86} + ( -3 + 3 \zeta_{6} ) q^{88} + ( -15 + 15 \zeta_{6} ) q^{89} + ( -3 - \zeta_{6} ) q^{91} + 6 q^{92} + ( 3 - 3 \zeta_{6} ) q^{94} + 16 \zeta_{6} q^{97} -\zeta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} - q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} - q^{7} + 2q^{8} - 3q^{11} + 5q^{13} + 2q^{14} - q^{16} - 3q^{17} - 5q^{19} - 3q^{22} - 6q^{23} - 10q^{25} + 2q^{26} - q^{28} - 9q^{29} + 16q^{31} - q^{32} + 6q^{34} - 8q^{37} + 10q^{38} + 3q^{41} - 8q^{43} + 6q^{44} - 6q^{46} - 6q^{47} - q^{49} + 5q^{50} - 7q^{52} + 6q^{53} - q^{56} - 9q^{58} - 6q^{59} + 7q^{61} - 8q^{62} + 2q^{64} - 2q^{67} - 3q^{68} - 6q^{71} - 32q^{73} - 8q^{74} - 5q^{76} + 6q^{77} - 26q^{79} + 3q^{82} - 36q^{83} + 16q^{86} - 3q^{88} - 15q^{89} - 7q^{91} + 12q^{92} + 3q^{94} + 16q^{97} - q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.500000 + 0.866025i 1.00000 0 0
1387.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −0.500000 0.866025i 1.00000 0 0
 $$n$$: 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.f 2
3.b odd 2 1 546.2.l.e 2
13.c even 3 1 inner 1638.2.r.f 2
39.h odd 6 1 7098.2.a.ba 1
39.i odd 6 1 546.2.l.e 2
39.i odd 6 1 7098.2.a.m 1

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

## Hecke kernels

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

 $$T_{5}$$ $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{17}^{2} + 3 T_{17} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$13 - 5 T + T^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$25 + 5 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$81 + 9 T + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$9 - 3 T + T^{2}$$
$43$ $$64 + 8 T + T^{2}$$
$47$ $$( 3 + T )^{2}$$
$53$ $$( -3 + T )^{2}$$
$59$ $$36 + 6 T + T^{2}$$
$61$ $$49 - 7 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$36 + 6 T + T^{2}$$
$73$ $$( 16 + T )^{2}$$
$79$ $$( 13 + T )^{2}$$
$83$ $$( 18 + T )^{2}$$
$89$ $$225 + 15 T + T^{2}$$
$97$ $$256 - 16 T + T^{2}$$