# Properties

 Label 1638.2.j.n Level $1638$ Weight $2$ Character orbit 1638.j 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.j (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{2} + \beta_{2} q^{4} + \beta_1 q^{5} + (2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} - q^{8}+O(q^{10})$$ q + (b2 + 1) * q^2 + b2 * q^4 + b1 * q^5 + (2*b3 + b2 + b1 + 1) * q^7 - q^8 $$q + (\beta_{2} + 1) q^{2} + \beta_{2} q^{4} + \beta_1 q^{5} + (2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} - q^{8} + (\beta_{3} + \beta_1) q^{10} + (\beta_{3} + \beta_{2} + \beta_1) q^{11} + q^{13} + (\beta_{3} + \beta_{2} - \beta_1) q^{14} + ( - \beta_{2} - 1) q^{16} + (3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{17} + ( - 3 \beta_{2} - 3) q^{19} + \beta_{3} q^{20} + (\beta_{3} - 1) q^{22} + ( - 2 \beta_{2} + 5 \beta_1 - 2) q^{23} - 3 \beta_{2} q^{25} + (\beta_{2} + 1) q^{26} + ( - \beta_{3} - 2 \beta_1 - 1) q^{28} - 5 q^{29} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{31} - \beta_{2} q^{32} + (3 \beta_{3} + 1) q^{34} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 4) q^{35} + ( - 2 \beta_{2} - 5 \beta_1 - 2) q^{37} - 3 \beta_{2} q^{38} - \beta_1 q^{40} + (\beta_{3} - 6) q^{41} + (2 \beta_{3} + 2) q^{43} + ( - \beta_{2} - \beta_1 - 1) q^{44} + (5 \beta_{3} - 2 \beta_{2} + 5 \beta_1) q^{46} + (9 \beta_{2} + 9) q^{47} + (2 \beta_{3} - 5 \beta_{2} - 2 \beta_1) q^{49} + 3 q^{50} + \beta_{2} q^{52} + (8 \beta_{3} - \beta_{2} + 8 \beta_1) q^{53} + (\beta_{3} - 2) q^{55} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{56} + ( - 5 \beta_{2} - 5) q^{58} + ( - \beta_{3} - 9 \beta_{2} - \beta_1) q^{59} + (\beta_{2} - 3 \beta_1 + 1) q^{61} + ( - 2 \beta_{3} - 4) q^{62} + q^{64} + \beta_1 q^{65} + (4 \beta_{3} - 9 \beta_{2} + 4 \beta_1) q^{67} + (\beta_{2} - 3 \beta_1 + 1) q^{68} + (\beta_{3} - 4 \beta_{2} - 2) q^{70} + q^{71} + (5 \beta_{3} + 6 \beta_{2} + 5 \beta_1) q^{73} + ( - 5 \beta_{3} - 2 \beta_{2} - 5 \beta_1) q^{74} + 3 q^{76} + ( - 4 \beta_{2} - 2 \beta_1 - 3) q^{77} + ( - 2 \beta_{2} + 8 \beta_1 - 2) q^{79} + ( - \beta_{3} - \beta_1) q^{80} + ( - 6 \beta_{2} - \beta_1 - 6) q^{82} + (4 \beta_{3} + 2) q^{83} + ( - \beta_{3} - 6) q^{85} + (2 \beta_{2} - 2 \beta_1 + 2) q^{86} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{88} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{89} + (2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{91} + (5 \beta_{3} + 2) q^{92} + 9 \beta_{2} q^{94} + ( - 3 \beta_{3} - 3 \beta_1) q^{95} + (7 \beta_{3} - 2) q^{97} + ( - 2 \beta_{3} - 4 \beta_1 + 5) q^{98}+O(q^{100})$$ q + (b2 + 1) * q^2 + b2 * q^4 + b1 * q^5 + (2*b3 + b2 + b1 + 1) * q^7 - q^8 + (b3 + b1) * q^10 + (b3 + b2 + b1) * q^11 + q^13 + (b3 + b2 - b1) * q^14 + (-b2 - 1) * q^16 + (3*b3 - b2 + 3*b1) * q^17 + (-3*b2 - 3) * q^19 + b3 * q^20 + (b3 - 1) * q^22 + (-2*b2 + 5*b1 - 2) * q^23 - 3*b2 * q^25 + (b2 + 1) * q^26 + (-b3 - 2*b1 - 1) * q^28 - 5 * q^29 + (-2*b3 + 4*b2 - 2*b1) * q^31 - b2 * q^32 + (3*b3 + 1) * q^34 + (b3 - 2*b2 + b1 - 4) * q^35 + (-2*b2 - 5*b1 - 2) * q^37 - 3*b2 * q^38 - b1 * q^40 + (b3 - 6) * q^41 + (2*b3 + 2) * q^43 + (-b2 - b1 - 1) * q^44 + (5*b3 - 2*b2 + 5*b1) * q^46 + (9*b2 + 9) * q^47 + (2*b3 - 5*b2 - 2*b1) * q^49 + 3 * q^50 + b2 * q^52 + (8*b3 - b2 + 8*b1) * q^53 + (b3 - 2) * q^55 + (-2*b3 - b2 - b1 - 1) * q^56 + (-5*b2 - 5) * q^58 + (-b3 - 9*b2 - b1) * q^59 + (b2 - 3*b1 + 1) * q^61 + (-2*b3 - 4) * q^62 + q^64 + b1 * q^65 + (4*b3 - 9*b2 + 4*b1) * q^67 + (b2 - 3*b1 + 1) * q^68 + (b3 - 4*b2 - 2) * q^70 + q^71 + (5*b3 + 6*b2 + 5*b1) * q^73 + (-5*b3 - 2*b2 - 5*b1) * q^74 + 3 * q^76 + (-4*b2 - 2*b1 - 3) * q^77 + (-2*b2 + 8*b1 - 2) * q^79 + (-b3 - b1) * q^80 + (-6*b2 - b1 - 6) * q^82 + (4*b3 + 2) * q^83 + (-b3 - 6) * q^85 + (2*b2 - 2*b1 + 2) * q^86 + (-b3 - b2 - b1) * q^88 + (-2*b2 + 3*b1 - 2) * q^89 + (2*b3 + b2 + b1 + 1) * q^91 + (5*b3 + 2) * q^92 + 9*b2 * q^94 + (-3*b3 - 3*b1) * q^95 + (7*b3 - 2) * q^97 + (-2*b3 - 4*b1 + 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{4} + 2 q^{7} - 4 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^4 + 2 * q^7 - 4 * q^8 $$4 q + 2 q^{2} - 2 q^{4} + 2 q^{7} - 4 q^{8} - 2 q^{11} + 4 q^{13} - 2 q^{14} - 2 q^{16} + 2 q^{17} - 6 q^{19} - 4 q^{22} - 4 q^{23} + 6 q^{25} + 2 q^{26} - 4 q^{28} - 20 q^{29} - 8 q^{31} + 2 q^{32} + 4 q^{34} - 12 q^{35} - 4 q^{37} + 6 q^{38} - 24 q^{41} + 8 q^{43} - 2 q^{44} + 4 q^{46} + 18 q^{47} + 10 q^{49} + 12 q^{50} - 2 q^{52} + 2 q^{53} - 8 q^{55} - 2 q^{56} - 10 q^{58} + 18 q^{59} + 2 q^{61} - 16 q^{62} + 4 q^{64} + 18 q^{67} + 2 q^{68} + 4 q^{71} - 12 q^{73} + 4 q^{74} + 12 q^{76} - 4 q^{77} - 4 q^{79} - 12 q^{82} + 8 q^{83} - 24 q^{85} + 4 q^{86} + 2 q^{88} - 4 q^{89} + 2 q^{91} + 8 q^{92} - 18 q^{94} - 8 q^{97} + 20 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^4 + 2 * q^7 - 4 * q^8 - 2 * q^11 + 4 * q^13 - 2 * q^14 - 2 * q^16 + 2 * q^17 - 6 * q^19 - 4 * q^22 - 4 * q^23 + 6 * q^25 + 2 * q^26 - 4 * q^28 - 20 * q^29 - 8 * q^31 + 2 * q^32 + 4 * q^34 - 12 * q^35 - 4 * q^37 + 6 * q^38 - 24 * q^41 + 8 * q^43 - 2 * q^44 + 4 * q^46 + 18 * q^47 + 10 * q^49 + 12 * q^50 - 2 * q^52 + 2 * q^53 - 8 * q^55 - 2 * q^56 - 10 * q^58 + 18 * q^59 + 2 * q^61 - 16 * q^62 + 4 * q^64 + 18 * q^67 + 2 * q^68 + 4 * q^71 - 12 * q^73 + 4 * q^74 + 12 * q^76 - 4 * q^77 - 4 * q^79 - 12 * q^82 + 8 * q^83 - 24 * q^85 + 4 * q^86 + 2 * q^88 - 4 * q^89 + 2 * q^91 + 8 * q^92 - 18 * q^94 - 8 * q^97 + 20 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
235.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.707107 1.22474i 0 2.62132 0.358719i −1.00000 0 0.707107 1.22474i
235.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.707107 + 1.22474i 0 −1.62132 + 2.09077i −1.00000 0 −0.707107 + 1.22474i
1171.1 0.500000 0.866025i 0 −0.500000 0.866025i −0.707107 + 1.22474i 0 2.62132 + 0.358719i −1.00000 0 0.707107 + 1.22474i
1171.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.707107 1.22474i 0 −1.62132 2.09077i −1.00000 0 −0.707107 1.22474i
 $$n$$: 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
7.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.j.n 4
3.b odd 2 1 546.2.i.h 4
7.c even 3 1 inner 1638.2.j.n 4
21.g even 6 1 3822.2.a.bp 2
21.h odd 6 1 546.2.i.h 4
21.h odd 6 1 3822.2.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.h 4 3.b odd 2 1
546.2.i.h 4 21.h odd 6 1
1638.2.j.n 4 1.a even 1 1 trivial
1638.2.j.n 4 7.c even 3 1 inner
3822.2.a.bp 2 21.g even 6 1
3822.2.a.bs 2 21.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 $$S_{2}^{\mathrm{new}}(1638, [\chi])$$:

 $$T_{5}^{4} + 2T_{5}^{2} + 4$$ T5^4 + 2*T5^2 + 4 $$T_{11}^{4} + 2T_{11}^{3} + 5T_{11}^{2} - 2T_{11} + 1$$ T11^4 + 2*T11^3 + 5*T11^2 - 2*T11 + 1 $$T_{17}^{4} - 2T_{17}^{3} + 21T_{17}^{2} + 34T_{17} + 289$$ T17^4 - 2*T17^3 + 21*T17^2 + 34*T17 + 289

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 2T^{2} + 4$$
$7$ $$T^{4} - 2 T^{3} - 3 T^{2} - 14 T + 49$$
$11$ $$T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1$$
$13$ $$(T - 1)^{4}$$
$17$ $$T^{4} - 2 T^{3} + 21 T^{2} + 34 T + 289$$
$19$ $$(T^{2} + 3 T + 9)^{2}$$
$23$ $$T^{4} + 4 T^{3} + 62 T^{2} + \cdots + 2116$$
$29$ $$(T + 5)^{4}$$
$31$ $$T^{4} + 8 T^{3} + 56 T^{2} + 64 T + 64$$
$37$ $$T^{4} + 4 T^{3} + 62 T^{2} + \cdots + 2116$$
$41$ $$(T^{2} + 12 T + 34)^{2}$$
$43$ $$(T^{2} - 4 T - 4)^{2}$$
$47$ $$(T^{2} - 9 T + 81)^{2}$$
$53$ $$T^{4} - 2 T^{3} + 131 T^{2} + \cdots + 16129$$
$59$ $$T^{4} - 18 T^{3} + 245 T^{2} + \cdots + 6241$$
$61$ $$T^{4} - 2 T^{3} + 21 T^{2} + 34 T + 289$$
$67$ $$T^{4} - 18 T^{3} + 275 T^{2} + \cdots + 2401$$
$71$ $$(T - 1)^{4}$$
$73$ $$T^{4} + 12 T^{3} + 158 T^{2} + \cdots + 196$$
$79$ $$T^{4} + 4 T^{3} + 140 T^{2} + \cdots + 15376$$
$83$ $$(T^{2} - 4 T - 28)^{2}$$
$89$ $$T^{4} + 4 T^{3} + 30 T^{2} - 56 T + 196$$
$97$ $$(T^{2} + 4 T - 94)^{2}$$