# Properties

 Label 1638.2.bj.c 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 182) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{4} + \zeta_{12}^{3} q^{5} - \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ q + (z^3 - z) * q^2 + (-z^2 + 1) * q^4 + z^3 * q^5 - z * q^7 + z^3 * q^8 $$q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{2} + 1) q^{4} + \zeta_{12}^{3} q^{5} - \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{2} q^{10} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 2) q^{11} + (\zeta_{12}^{3} + 3 \zeta_{12}) q^{13} + q^{14} - \zeta_{12}^{2} q^{16} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + \zeta_{12} + 4) q^{17} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{19} + \zeta_{12} q^{20} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 1) q^{22} + (\zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12}) q^{23} + 4 q^{25} + ( - \zeta_{12}^{2} - 3) q^{26} + (\zeta_{12}^{3} - \zeta_{12}) q^{28} - 3 \zeta_{12}^{2} q^{29} + (7 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{31} + \zeta_{12} q^{32} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{34} + ( - \zeta_{12}^{2} + 1) q^{35} + ( - 3 \zeta_{12}^{2} + 6) q^{37} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 2) q^{38} - q^{40} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12} + 4) q^{41} + (6 \zeta_{12}^{3} + 7 \zeta_{12}^{2} - 3 \zeta_{12} - 7) q^{43} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{44} + ( - \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{46} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{47} + \zeta_{12}^{2} q^{49} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{50} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}) q^{52} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12} - 5) q^{53} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{55} + ( - \zeta_{12}^{2} + 1) q^{56} + 3 \zeta_{12} q^{58} + (\zeta_{12}^{2} + 9 \zeta_{12} + 1) q^{59} + ( - 2 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + \zeta_{12} + 10) q^{61} + (\zeta_{12}^{3} - 7 \zeta_{12}^{2} + \zeta_{12}) q^{62} - q^{64} + (3 \zeta_{12}^{2} - 4) q^{65} + ( - 3 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 3 \zeta_{12} + 10) q^{67} + ( - \zeta_{12}^{3} - 4 \zeta_{12}^{2} - \zeta_{12}) q^{68} + \zeta_{12}^{3} q^{70} + (8 \zeta_{12}^{2} + 8) q^{71} + (\zeta_{12}^{3} + 12 \zeta_{12}^{2} - 6) q^{73} + (6 \zeta_{12}^{3} - 3 \zeta_{12}) q^{74} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{76} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 1) q^{77} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12} - 9) q^{79} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{80} + (4 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{82} + (9 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{83} + (\zeta_{12}^{2} + 4 \zeta_{12} + 1) q^{85} + ( - 7 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{86} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{88} + (6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 6 \zeta_{12} + 4) q^{89} + ( - 4 \zeta_{12}^{2} + 1) q^{91} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 3) q^{92} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{94} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{95} + ( - 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{97} - \zeta_{12} q^{98} +O(q^{100})$$ q + (z^3 - z) * q^2 + (-z^2 + 1) * q^4 + z^3 * q^5 - z * q^7 + z^3 * q^8 - z^2 * q^10 + (-z^3 + z^2 + z - 2) * q^11 + (z^3 + 3*z) * q^13 + q^14 - z^2 * q^16 + (-2*z^3 - 4*z^2 + z + 4) * q^17 + (-2*z^2 + 2*z - 2) * q^19 + z * q^20 + (-2*z^3 + z^2 + z - 1) * q^22 + (z^3 - 3*z^2 + z) * q^23 + 4 * q^25 + (-z^2 - 3) * q^26 + (z^3 - z) * q^28 - 3*z^2 * q^29 + (7*z^3 - 2*z^2 + 1) * q^31 + z * q^32 + (4*z^3 + 2*z^2 - 1) * q^34 + (-z^2 + 1) * q^35 + (-3*z^2 + 6) * q^37 + (-2*z^3 + 4*z - 2) * q^38 - q^40 + (z^3 - 2*z^2 - z + 4) * q^41 + (6*z^3 + 7*z^2 - 3*z - 7) * q^43 + (-z^3 + 2*z^2 - 1) * q^44 + (-z^2 + 3*z - 1) * q^46 + (-4*z^3 + 8*z^2 - 4) * q^47 + z^2 * q^49 + (4*z^3 - 4*z) * q^50 + (-3*z^3 + 4*z) * q^52 + (-2*z^3 + 4*z - 5) * q^53 + (-z^3 + z^2 - z) * q^55 + (-z^2 + 1) * q^56 + 3*z * q^58 + (z^2 + 9*z + 1) * q^59 + (-2*z^3 - 10*z^2 + z + 10) * q^61 + (z^3 - 7*z^2 + z) * q^62 - q^64 + (3*z^2 - 4) * q^65 + (-3*z^3 - 5*z^2 + 3*z + 10) * q^67 + (-z^3 - 4*z^2 - z) * q^68 + z^3 * q^70 + (8*z^2 + 8) * q^71 + (z^3 + 12*z^2 - 6) * q^73 + (6*z^3 - 3*z) * q^74 + (-2*z^3 + 2*z^2 + 2*z - 4) * q^76 + (-z^3 + 2*z - 1) * q^77 + (-3*z^3 + 6*z - 9) * q^79 + (-z^3 + z) * q^80 + (4*z^3 - z^2 - 2*z + 1) * q^82 + (9*z^3 - 6*z^2 + 3) * q^83 + (z^2 + 4*z + 1) * q^85 + (-7*z^3 - 6*z^2 + 3) * q^86 + (-z^3 + z^2 - z) * q^88 + (6*z^3 - 2*z^2 - 6*z + 4) * q^89 + (-4*z^2 + 1) * q^91 + (-z^3 + 2*z - 3) * q^92 + (-4*z^3 + 4*z^2 - 4*z) * q^94 + (-4*z^3 + 2*z^2 + 2*z - 2) * q^95 + (-2*z^2 - 2*z - 2) * q^97 - z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4}+O(q^{10})$$ 4 * q + 2 * q^4 $$4 q + 2 q^{4} - 2 q^{10} - 6 q^{11} + 4 q^{14} - 2 q^{16} + 8 q^{17} - 12 q^{19} - 2 q^{22} - 6 q^{23} + 16 q^{25} - 14 q^{26} - 6 q^{29} + 2 q^{35} + 18 q^{37} - 8 q^{38} - 4 q^{40} + 12 q^{41} - 14 q^{43} - 6 q^{46} + 2 q^{49} - 20 q^{53} + 2 q^{55} + 2 q^{56} + 6 q^{59} + 20 q^{61} - 14 q^{62} - 4 q^{64} - 10 q^{65} + 30 q^{67} - 8 q^{68} + 48 q^{71} - 12 q^{76} - 4 q^{77} - 36 q^{79} + 2 q^{82} + 6 q^{85} + 2 q^{88} + 12 q^{89} - 4 q^{91} - 12 q^{92} + 8 q^{94} - 4 q^{95} - 12 q^{97}+O(q^{100})$$ 4 * q + 2 * q^4 - 2 * q^10 - 6 * q^11 + 4 * q^14 - 2 * q^16 + 8 * q^17 - 12 * q^19 - 2 * q^22 - 6 * q^23 + 16 * q^25 - 14 * q^26 - 6 * q^29 + 2 * q^35 + 18 * q^37 - 8 * q^38 - 4 * q^40 + 12 * q^41 - 14 * q^43 - 6 * q^46 + 2 * q^49 - 20 * q^53 + 2 * q^55 + 2 * q^56 + 6 * q^59 + 20 * q^61 - 14 * q^62 - 4 * q^64 - 10 * q^65 + 30 * q^67 - 8 * q^68 + 48 * q^71 - 12 * q^76 - 4 * q^77 - 36 * q^79 + 2 * q^82 + 6 * q^85 + 2 * q^88 + 12 * q^89 - 4 * q^91 - 12 * q^92 + 8 * q^94 - 4 * q^95 - 12 * q^97

## Character values

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

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

## 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 1.00000i 0 −0.866025 0.500000i 1.00000i 0 −0.500000 0.866025i
127.2 0.866025 0.500000i 0 0.500000 0.866025i 1.00000i 0 0.866025 + 0.500000i 1.00000i 0 −0.500000 0.866025i
1135.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 −0.866025 + 0.500000i 1.00000i 0 −0.500000 + 0.866025i
1135.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 0.866025 0.500000i 1.00000i 0 −0.500000 + 0.866025i
 $$n$$: 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.c 4
3.b odd 2 1 182.2.m.a 4
12.b even 2 1 1456.2.cc.b 4
13.e even 6 1 inner 1638.2.bj.c 4
21.c even 2 1 1274.2.m.a 4
21.g even 6 1 1274.2.o.a 4
21.g even 6 1 1274.2.v.b 4
21.h odd 6 1 1274.2.o.b 4
21.h odd 6 1 1274.2.v.a 4
39.h odd 6 1 182.2.m.a 4
39.h odd 6 1 2366.2.d.k 4
39.i odd 6 1 2366.2.d.k 4
39.k even 12 1 2366.2.a.q 2
39.k even 12 1 2366.2.a.s 2
156.r even 6 1 1456.2.cc.b 4
273.u even 6 1 1274.2.m.a 4
273.x odd 6 1 1274.2.o.b 4
273.y even 6 1 1274.2.o.a 4
273.bp odd 6 1 1274.2.v.a 4
273.br even 6 1 1274.2.v.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.m.a 4 3.b odd 2 1
182.2.m.a 4 39.h odd 6 1
1274.2.m.a 4 21.c even 2 1
1274.2.m.a 4 273.u even 6 1
1274.2.o.a 4 21.g even 6 1
1274.2.o.a 4 273.y even 6 1
1274.2.o.b 4 21.h odd 6 1
1274.2.o.b 4 273.x odd 6 1
1274.2.v.a 4 21.h odd 6 1
1274.2.v.a 4 273.bp odd 6 1
1274.2.v.b 4 21.g even 6 1
1274.2.v.b 4 273.br even 6 1
1456.2.cc.b 4 12.b even 2 1
1456.2.cc.b 4 156.r even 6 1
1638.2.bj.c 4 1.a even 1 1 trivial
1638.2.bj.c 4 13.e even 6 1 inner
2366.2.a.q 2 39.k even 12 1
2366.2.a.s 2 39.k even 12 1
2366.2.d.k 4 39.h odd 6 1
2366.2.d.k 4 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 $$S_{2}^{\mathrm{new}}(1638, [\chi])$$:

 $$T_{5}^{2} + 1$$ T5^2 + 1 $$T_{11}^{4} + 6T_{11}^{3} + 14T_{11}^{2} + 12T_{11} + 4$$ T11^4 + 6*T11^3 + 14*T11^2 + 12*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} - T^{2} + 1$$
$11$ $$T^{4} + 6 T^{3} + 14 T^{2} + 12 T + 4$$
$13$ $$T^{4} - T^{2} + 169$$
$17$ $$T^{4} - 8 T^{3} + 51 T^{2} - 104 T + 169$$
$19$ $$T^{4} + 12 T^{3} + 56 T^{2} + 96 T + 64$$
$23$ $$T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36$$
$29$ $$(T^{2} + 3 T + 9)^{2}$$
$31$ $$T^{4} + 104T^{2} + 2116$$
$37$ $$(T^{2} - 9 T + 27)^{2}$$
$41$ $$T^{4} - 12 T^{3} + 59 T^{2} + \cdots + 121$$
$43$ $$T^{4} + 14 T^{3} + 174 T^{2} + \cdots + 484$$
$47$ $$T^{4} + 128T^{2} + 1024$$
$53$ $$(T^{2} + 10 T + 13)^{2}$$
$59$ $$T^{4} - 6 T^{3} - 66 T^{2} + \cdots + 6084$$
$61$ $$T^{4} - 20 T^{3} + 303 T^{2} + \cdots + 9409$$
$67$ $$T^{4} - 30 T^{3} + 366 T^{2} + \cdots + 4356$$
$71$ $$(T^{2} - 24 T + 192)^{2}$$
$73$ $$T^{4} + 218 T^{2} + 11449$$
$79$ $$(T^{2} + 18 T + 54)^{2}$$
$83$ $$T^{4} + 216T^{2} + 2916$$
$89$ $$T^{4} - 12 T^{3} + 24 T^{2} + \cdots + 576$$
$97$ $$T^{4} + 12 T^{3} + 56 T^{2} + 96 T + 64$$