# 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: $$x^{4} - x^{2} + 1$$ 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.

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

## 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 $$S_{2}^{\mathrm{new}}(1638, [\chi])$$:

 $$T_{5}^{4} + 8T_{5}^{2} + 4$$ T5^4 + 8*T5^2 + 4 $$T_{11}^{2} + 3T_{11} + 3$$ T11^2 + 3*T11 + 3

## Hecke characteristic polynomials

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