# Properties

 Label 1638.2.bj.b Level $1638$ Weight $2$ Character orbit 1638.bj 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.bj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( -1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{10} + ( -1 - \zeta_{12}^{2} ) q^{11} + ( 3 \zeta_{12} - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{13} + q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( -\zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{17} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{19} + ( 2 - \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{20} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{22} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{23} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{25} + ( 3 - 2 \zeta_{12}^{3} ) q^{26} + \zeta_{12} q^{28} + ( 3 + 2 \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{29} + ( -3 + 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + ( 1 - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{34} + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{35} + ( 1 + 5 \zeta_{12} + \zeta_{12}^{2} ) q^{37} + q^{38} + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{40} -3 \zeta_{12} q^{41} + ( -\zeta_{12} + 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{43} + ( 1 - 2 \zeta_{12}^{2} ) q^{44} + ( 4 - 2 \zeta_{12}^{2} ) q^{46} + ( 2 - 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{47} + ( 1 - \zeta_{12}^{2} ) q^{49} + ( 2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{50} + ( 2 + 3 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{52} -7 q^{53} + ( -3 + \zeta_{12} + 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{55} + \zeta_{12}^{2} q^{56} + ( 4 + 3 \zeta_{12} - 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{58} + ( -8 + 6 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{59} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{61} + ( 3 - 3 \zeta_{12} - 3 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{62} - q^{64} + ( -4 - \zeta_{12} + 5 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{65} + ( 4 + 2 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{67} + ( 2 + \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{68} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{70} + ( 6 - 3 \zeta_{12} - 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{71} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{73} + ( \zeta_{12} + 5 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{74} + \zeta_{12} q^{76} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{77} + ( 9 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{79} + ( 1 - \zeta_{12} + \zeta_{12}^{2} ) q^{80} -3 \zeta_{12}^{2} q^{82} + ( 3 - 6 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{83} + ( -2 - \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{85} + ( 1 - 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{86} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{88} + ( 2 + 7 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{89} + ( 3 - 2 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{91} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{92} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{94} + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{95} + ( 10 + 7 \zeta_{12} - 5 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{97} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + O(q^{10})$$ $$4q + 2q^{4} - 2q^{10} - 6q^{11} - 4q^{13} + 4q^{14} - 2q^{16} - 4q^{17} + 6q^{20} + 4q^{25} + 12q^{26} + 6q^{29} + 2q^{35} + 6q^{37} + 4q^{38} - 4q^{40} + 10q^{43} + 12q^{46} + 2q^{49} + 12q^{50} + 4q^{52} - 28q^{53} - 6q^{55} + 2q^{56} + 12q^{58} - 24q^{59} + 6q^{62} - 4q^{64} - 6q^{65} + 24q^{67} + 4q^{68} + 18q^{71} + 10q^{74} + 36q^{79} + 6q^{80} - 6q^{82} - 6q^{85} + 12q^{89} + 6q^{91} - 2q^{94} + 2q^{95} + 30q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i 2.73205i 0 −0.866025 0.500000i 1.00000i 0 −1.36603 2.36603i
127.2 0.866025 0.500000i 0 0.500000 0.866025i 0.732051i 0 0.866025 + 0.500000i 1.00000i 0 0.366025 + 0.633975i
1135.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 2.73205i 0 −0.866025 + 0.500000i 1.00000i 0 −1.36603 + 2.36603i
1135.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.732051i 0 0.866025 0.500000i 1.00000i 0 0.366025 0.633975i
 $$n$$: 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.bj.b 4
3.b odd 2 1 546.2.s.a 4
13.e even 6 1 inner 1638.2.bj.b 4
39.h odd 6 1 546.2.s.a 4
39.k even 12 1 7098.2.a.bp 2
39.k even 12 1 7098.2.a.bx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.a 4 3.b odd 2 1
546.2.s.a 4 39.h odd 6 1
1638.2.bj.b 4 1.a even 1 1 trivial
1638.2.bj.b 4 13.e even 6 1 inner
7098.2.a.bp 2 39.k even 12 1
7098.2.a.bx 2 39.k even 12 1

## Hecke kernels

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

 $$T_{5}^{4} + 8 T_{5}^{2} + 4$$ $$T_{11}^{2} + 3 T_{11} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$4 + 8 T^{2} + T^{4}$$
$7$ $$1 - T^{2} + T^{4}$$
$11$ $$( 3 + 3 T + T^{2} )^{2}$$
$13$ $$169 + 52 T + 3 T^{2} + 4 T^{3} + T^{4}$$
$17$ $$1 + 4 T + 15 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$1 - T^{2} + T^{4}$$
$23$ $$144 + 12 T^{2} + T^{4}$$
$29$ $$9 + 18 T + 39 T^{2} - 6 T^{3} + T^{4}$$
$31$ $$324 + 72 T^{2} + T^{4}$$
$37$ $$484 + 132 T - 10 T^{2} - 6 T^{3} + T^{4}$$
$41$ $$81 - 9 T^{2} + T^{4}$$
$43$ $$484 - 220 T + 78 T^{2} - 10 T^{3} + T^{4}$$
$47$ $$121 + 26 T^{2} + T^{4}$$
$53$ $$( 7 + T )^{4}$$
$59$ $$144 + 288 T + 204 T^{2} + 24 T^{3} + T^{4}$$
$61$ $$729 + 27 T^{2} + T^{4}$$
$67$ $$1936 - 1056 T + 236 T^{2} - 24 T^{3} + T^{4}$$
$71$ $$324 - 324 T + 126 T^{2} - 18 T^{3} + T^{4}$$
$73$ $$64 + 32 T^{2} + T^{4}$$
$79$ $$( 33 - 18 T + T^{2} )^{2}$$
$83$ $$4 + 104 T^{2} + T^{4}$$
$89$ $$1369 + 444 T + 11 T^{2} - 12 T^{3} + T^{4}$$
$97$ $$676 - 780 T + 326 T^{2} - 30 T^{3} + T^{4}$$