# Properties

 Label 1638.2.p.i Level $1638$ Weight $2$ Character orbit 1638.p Analytic conductor $13.079$ Analytic rank $0$ Dimension $8$ 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.p (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.447703281.1 Defining polynomial: $$x^{8} - x^{7} - 2 x^{6} + 2 x^{5} + 3 x^{4} + 4 x^{3} - 8 x^{2} - 8 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 2 x^{6} + 2 x^{5} + 3 x^{4} + 4 x^{3} - 8 x^{2} - 8 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - 2 \nu^{5} + 2 \nu^{4} + 3 \nu^{3} + 4 \nu^{2} - 8 \nu - 8$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + \nu^{5} - 2 \nu^{3} - \nu^{2} + 6 \nu + 4$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} + 2 \nu^{4} + \nu^{3} - 6 \nu^{2} - 4 \nu$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + 6 \nu^{5} + 2 \nu^{4} - 3 \nu^{3} - 12 \nu^{2} + 4 \nu + 32$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{5} + \nu^{4} - \nu^{3} - 5 \nu^{2} - \nu + 8$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$2 \nu^{7} + \nu^{6} - 5 \nu^{5} - 4 \nu^{4} + 4 \nu^{3} + 17 \nu^{2} + 2 \nu - 24$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} - \beta_{5} + 2 \beta_{3} + 3 \beta_{2} + \beta_{1} + 1$$ $$\nu^{5}$$ $$=$$ $$\beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_{1} - 5$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{7} + 2 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} - 5 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$-\beta_{7} - 4 \beta_{6} + 4 \beta_{5} - \beta_{3} - 2 \beta_{1} - 5$$

## Character values

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

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$\beta_{3}$$ $$-1 - \beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
919.1
 1.19003 + 0.764088i −1.38232 + 0.298668i −0.571299 − 1.29368i 1.26359 − 0.635098i 1.19003 − 0.764088i −1.38232 − 0.298668i −0.571299 + 1.29368i 1.26359 + 0.635098i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.05781 + 3.56422i 0 1.65876 2.06119i −1.00000 0 −4.11561
919.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.14553 + 1.98411i 0 −1.12588 + 2.39424i −1.00000 0 −2.29105
919.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.228205 0.395262i 0 2.45374 + 0.989520i −1.00000 0 0.456409
919.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.97513 3.42102i 0 −1.48662 2.18860i −1.00000 0 3.95025
991.1 0.500000 0.866025i 0 −0.500000 0.866025i −2.05781 3.56422i 0 1.65876 + 2.06119i −1.00000 0 −4.11561
991.2 0.500000 0.866025i 0 −0.500000 0.866025i −1.14553 1.98411i 0 −1.12588 2.39424i −1.00000 0 −2.29105
991.3 0.500000 0.866025i 0 −0.500000 0.866025i 0.228205 + 0.395262i 0 2.45374 0.989520i −1.00000 0 0.456409
991.4 0.500000 0.866025i 0 −0.500000 0.866025i 1.97513 + 3.42102i 0 −1.48662 + 2.18860i −1.00000 0 3.95025
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 991.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g 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.p.i 8
3.b odd 2 1 546.2.k.b yes 8
7.c even 3 1 1638.2.m.g 8
13.c even 3 1 1638.2.m.g 8
21.h odd 6 1 546.2.j.d 8
39.i odd 6 1 546.2.j.d 8
91.g even 3 1 inner 1638.2.p.i 8
273.bm odd 6 1 546.2.k.b yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.d 8 21.h odd 6 1
546.2.j.d 8 39.i odd 6 1
546.2.k.b yes 8 3.b odd 2 1
546.2.k.b yes 8 273.bm odd 6 1
1638.2.m.g 8 7.c even 3 1
1638.2.m.g 8 13.c even 3 1
1638.2.p.i 8 1.a even 1 1 trivial
1638.2.p.i 8 91.g even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1638, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{4}$$
$3$ $$T^{8}$$
$5$ $$289 - 510 T + 1189 T^{2} + 442 T^{3} + 332 T^{4} + 26 T^{5} + 21 T^{6} + 2 T^{7} + T^{8}$$
$7$ $$2401 - 1029 T + 392 T^{2} - 231 T^{3} + 123 T^{4} - 33 T^{5} + 8 T^{6} - 3 T^{7} + T^{8}$$
$11$ $$( -67 - 89 T - 10 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$13$ $$28561 + 24167 T + 10478 T^{2} + 3471 T^{3} + 1031 T^{4} + 267 T^{5} + 62 T^{6} + 11 T^{7} + T^{8}$$
$17$ $$1 - 9 T + 67 T^{2} - 118 T^{3} + 161 T^{4} - 74 T^{5} + 30 T^{6} + 4 T^{7} + T^{8}$$
$19$ $$( -13 - 16 T + 5 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$23$ $$27889 - 25885 T + 22355 T^{2} - 4890 T^{3} + 1817 T^{4} - 210 T^{5} + 110 T^{6} - 10 T^{7} + T^{8}$$
$29$ $$4489 - 7437 T + 8971 T^{2} - 5282 T^{3} + 2345 T^{4} - 322 T^{5} + 54 T^{6} + 2 T^{7} + T^{8}$$
$31$ $$210681 + 37179 T + 28593 T^{2} + 1620 T^{3} + 2331 T^{4} + 126 T^{5} + 84 T^{6} - 6 T^{7} + T^{8}$$
$37$ $$44521 + 173653 T + 622469 T^{2} + 202164 T^{3} + 44345 T^{4} + 5634 T^{5} + 524 T^{6} + 28 T^{7} + T^{8}$$
$41$ $$1912689 + 136917 T + 137037 T^{2} - 9108 T^{3} + 7081 T^{4} - 198 T^{5} + 92 T^{6} + T^{8}$$
$43$ $$4489 + 1005 T + 2771 T^{2} - 1374 T^{3} + 1287 T^{4} - 258 T^{5} + 74 T^{6} + 6 T^{7} + T^{8}$$
$47$ $$2601 - 5814 T + 15852 T^{2} + 6282 T^{3} + 3199 T^{4} + 172 T^{5} + 57 T^{6} + T^{7} + T^{8}$$
$53$ $$6561 + 1458 T + 3888 T^{2} - 1926 T^{3} + 1729 T^{4} - 344 T^{5} + 93 T^{6} + 7 T^{7} + T^{8}$$
$59$ $$96721 - 205260 T + 480073 T^{2} + 93136 T^{3} + 21458 T^{4} + 1034 T^{5} + 147 T^{6} + 2 T^{7} + T^{8}$$
$61$ $$( -4176 - 456 T + 136 T^{2} + 24 T^{3} + T^{4} )^{2}$$
$67$ $$( 167 + 199 T + 29 T^{2} - 15 T^{3} + T^{4} )^{2}$$
$71$ $$674041 + 502452 T + 296549 T^{2} + 67992 T^{3} + 13518 T^{4} + 654 T^{5} + 131 T^{6} + 6 T^{7} + T^{8}$$
$73$ $$1 + 5 T + 32 T^{2} - 33 T^{3} + 53 T^{4} - 3 T^{5} + 8 T^{6} - T^{7} + T^{8}$$
$79$ $$6985449 - 1538226 T + 542235 T^{2} - 18618 T^{3} + 10270 T^{4} + 240 T^{5} + 221 T^{6} + 12 T^{7} + T^{8}$$
$83$ $$( 1919 + 1329 T - 86 T^{2} - 16 T^{3} + T^{4} )^{2}$$
$89$ $$729 + 4860 T + 27810 T^{2} + 29250 T^{3} + 24373 T^{4} + 3890 T^{5} + 455 T^{6} + 25 T^{7} + T^{8}$$
$97$ $$45873529 + 853398 T + 1289200 T^{2} - 37234 T^{3} + 28445 T^{4} - 440 T^{5} + 189 T^{6} + T^{7} + T^{8}$$