# Properties

 Label 1638.2.p.h 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} ) q^{5} + ( -\beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{7} - q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} ) q^{5} + ( -\beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{7} - q^{8} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{7} ) q^{10} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{7} ) q^{11} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{13} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{14} -\beta_{2} q^{16} + ( 2 - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{6} - \beta_{7} ) q^{19} + ( \beta_{1} + \beta_{4} ) q^{20} + ( -2 \beta_{1} - \beta_{4} ) q^{22} + ( \beta_{1} - \beta_{2} + 2 \beta_{5} - 2 \beta_{6} ) q^{23} + ( 3 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{25} + ( 2 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{26} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{28} + ( 1 - \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{29} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{31} + ( 1 - \beta_{2} ) q^{32} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{34} + ( -1 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{35} + ( \beta_{1} - \beta_{2} ) q^{37} + ( -\beta_{1} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{38} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{40} + ( -5 + 5 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{41} + ( \beta_{1} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{43} + ( -2 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{7} ) q^{44} + ( 2 - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{46} + ( -1 + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{47} + ( -1 + 3 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{49} + ( 1 - \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{7} ) q^{50} + ( 1 - \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{52} + ( -\beta_{2} + 2 \beta_{4} + 5 \beta_{5} - 5 \beta_{6} ) q^{53} + ( -9 + 9 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{55} + ( \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{56} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{58} + ( -6 + 6 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} ) q^{59} + ( 4 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{6} - 2 \beta_{7} ) q^{61} + ( -4 + 4 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{62} + q^{64} + ( -5 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{65} + ( -3 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - \beta_{6} - 3 \beta_{7} ) q^{67} + ( -2 \beta_{1} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{68} + ( -5 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{70} + ( \beta_{1} - 4 \beta_{2} - 5 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{71} + ( -4 \beta_{1} - 4 \beta_{2} + \beta_{5} - \beta_{6} ) q^{73} + ( 2 - 2 \beta_{2} - \beta_{3} ) q^{74} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{76} + ( 7 + \beta_{1} - 6 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{77} + ( 5 - 5 \beta_{2} + \beta_{3} - 5 \beta_{4} - 2 \beta_{5} + 5 \beta_{7} ) q^{79} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{7} ) q^{80} + ( -5 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} ) q^{82} + ( 7 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{7} ) q^{83} + ( -4 \beta_{1} + \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{85} + ( 1 - \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{86} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{7} ) q^{88} + ( -\beta_{1} + \beta_{2} + 2 \beta_{4} + 5 \beta_{5} - 5 \beta_{6} ) q^{89} + ( -3 + 3 \beta_{1} + 4 \beta_{2} - \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{91} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{6} ) q^{92} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{94} + ( -10 + 10 \beta_{2} + 5 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{95} + ( 5 \beta_{1} + 5 \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{97} + ( 2 - 3 \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{2} - 4q^{4} - 2q^{5} - 3q^{7} - 8q^{8} + O(q^{10})$$ $$8q + 4q^{2} - 4q^{4} - 2q^{5} - 3q^{7} - 8q^{8} - 4q^{10} + 8q^{11} + 3q^{13} - 3q^{14} - 4q^{16} + 2q^{17} + 8q^{19} - 2q^{20} + 4q^{22} - 4q^{23} + 2q^{25} + 12q^{26} - 2q^{29} + 14q^{31} + 4q^{32} + 4q^{34} + 4q^{35} - 6q^{37} + 4q^{38} + 2q^{40} - 12q^{41} - 4q^{44} + 4q^{46} - 7q^{47} - 7q^{49} - 2q^{50} + 9q^{52} + q^{53} - 25q^{55} + 3q^{56} - 4q^{58} - 16q^{59} + 8q^{61} - 14q^{62} + 8q^{64} - q^{65} - 38q^{67} + 2q^{68} - 22q^{70} - 20q^{71} - 7q^{73} + 6q^{74} - 4q^{76} + 24q^{77} + 24q^{79} + 4q^{80} - 24q^{82} + 64q^{83} + 15q^{85} - 8q^{88} + 11q^{89} - 20q^{91} + 8q^{92} - 14q^{94} - 28q^{95} + 11q^{97} - 2q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} + 4 \nu^{5} + 2 \nu^{4} + \nu^{3} - 10 \nu^{2} + 16$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + \nu^{5} - 2 \nu^{3} - \nu^{2} + 6 \nu + 8$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} - \nu^{6} + 2 \nu^{5} + 2 \nu^{4} - \nu^{3} - 16 \nu^{2} - 8 \nu + 8$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{7} + \nu^{6} - 6 \nu^{5} - 6 \nu^{4} + \nu^{3} + 16 \nu^{2} + 4 \nu - 24$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - 4 \nu^{5} + 3 \nu^{3} + 8 \nu^{2} - 2 \nu - 20$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{7} + \nu^{6} - 6 \nu^{5} - 6 \nu^{4} + \nu^{3} + 24 \nu^{2} + 12 \nu - 32$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$4 \nu^{7} + \nu^{6} - 9 \nu^{5} - 6 \nu^{4} + 6 \nu^{3} + 27 \nu^{2} + 4 \nu - 40$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{2} + 2 \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{6} - \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 3$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{7} - \beta_{6} - \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 4 \beta_{1}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$-\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1} - 1$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{6} - 4 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - 5 \beta_{2} + \beta_{1} - 9$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$6 \beta_{7} - 8 \beta_{6} - 5 \beta_{5} - 6 \beta_{4} + \beta_{3} + 5 \beta_{2} - 7 \beta_{1} - 12$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-3 \beta_{7} - 11 \beta_{6} + 4 \beta_{5} + 15 \beta_{4} - 11 \beta_{3} + 2 \beta_{2} + 5 \beta_{1} - 12$$$$)/3$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
919.1
 −1.38232 − 0.298668i 1.26359 + 0.635098i −0.571299 + 1.29368i 1.19003 − 0.764088i −1.38232 + 0.298668i 1.26359 − 0.635098i −0.571299 − 1.29368i 1.19003 + 0.764088i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.75410 + 3.03819i 0 1.12588 2.39424i −1.00000 0 −3.50820
919.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.611519 + 1.05918i 0 1.48662 + 2.18860i −1.00000 0 −1.22304
919.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.441221 0.764218i 0 −2.45374 0.989520i −1.00000 0 0.882443
919.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.924396 1.60110i 0 −1.65876 + 2.06119i −1.00000 0 1.84879
991.1 0.500000 0.866025i 0 −0.500000 0.866025i −1.75410 3.03819i 0 1.12588 + 2.39424i −1.00000 0 −3.50820
991.2 0.500000 0.866025i 0 −0.500000 0.866025i −0.611519 1.05918i 0 1.48662 2.18860i −1.00000 0 −1.22304
991.3 0.500000 0.866025i 0 −0.500000 0.866025i 0.441221 + 0.764218i 0 −2.45374 + 0.989520i −1.00000 0 0.882443
991.4 0.500000 0.866025i 0 −0.500000 0.866025i 0.924396 + 1.60110i 0 −1.65876 2.06119i −1.00000 0 1.84879
 $$n$$: 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.h 8
3.b odd 2 1 546.2.k.c yes 8
7.c even 3 1 1638.2.m.h 8
13.c even 3 1 1638.2.m.h 8
21.h odd 6 1 546.2.j.c 8
39.i odd 6 1 546.2.j.c 8
91.g even 3 1 inner 1638.2.p.h 8
273.bm odd 6 1 546.2.k.c yes 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 2 T_{5}^{7} + 11 T_{5}^{6} - 6 T_{5}^{5} + 50 T_{5}^{4} + 65 T_{5}^{2} - 28 T_{5} + 49$$ acting on $$S_{2}^{\mathrm{new}}(1638, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{4}$$
$3$ $$T^{8}$$
$5$ $$49 - 28 T + 65 T^{2} + 50 T^{4} - 6 T^{5} + 11 T^{6} + 2 T^{7} + T^{8}$$
$7$ $$2401 + 1029 T + 392 T^{2} + 231 T^{3} + 123 T^{4} + 33 T^{5} + 8 T^{6} + 3 T^{7} + T^{8}$$
$11$ $$( -1 - 9 T - 14 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$13$ $$28561 - 6591 T + 3380 T^{2} - 195 T^{3} + 231 T^{4} - 15 T^{5} + 20 T^{6} - 3 T^{7} + T^{8}$$
$17$ $$729 - 4131 T + 22221 T^{2} - 6840 T^{3} + 2269 T^{4} - 218 T^{5} + 48 T^{6} - 2 T^{7} + T^{8}$$
$19$ $$( -153 + 210 T - 41 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$23$ $$9 - 243 T + 6663 T^{2} + 2730 T^{3} + 1477 T^{4} + 26 T^{5} + 50 T^{6} + 4 T^{7} + T^{8}$$
$29$ $$1121481 - 244629 T + 150789 T^{2} + 17016 T^{3} + 7867 T^{4} + 278 T^{5} + 96 T^{6} + 2 T^{7} + T^{8}$$
$31$ $$219961 - 97083 T + 55981 T^{2} - 7336 T^{3} + 4151 T^{4} - 806 T^{5} + 168 T^{6} - 14 T^{7} + T^{8}$$
$37$ $$1 + T + 9 T^{2} + 4 T^{3} + 71 T^{4} + 50 T^{5} + 28 T^{6} + 6 T^{7} + T^{8}$$
$41$ $$881721 + 397197 T + 165783 T^{2} + 28458 T^{3} + 6211 T^{4} + 678 T^{5} + 158 T^{6} + 12 T^{7} + T^{8}$$
$43$ $$169 + 949 T + 4887 T^{2} + 2482 T^{3} + 1169 T^{4} + 146 T^{5} + 34 T^{6} + T^{8}$$
$47$ $$3969 + 7182 T + 10980 T^{2} + 4530 T^{3} + 1885 T^{4} + 4 T^{5} + 81 T^{6} + 7 T^{7} + T^{8}$$
$53$ $$2283121 - 625554 T + 392002 T^{2} + 63466 T^{3} + 19391 T^{4} + 974 T^{5} + 147 T^{6} - T^{7} + T^{8}$$
$59$ $$1164241 + 513604 T + 244919 T^{2} + 26436 T^{3} + 8984 T^{4} + 1224 T^{5} + 239 T^{6} + 16 T^{7} + T^{8}$$
$61$ $$( -432 + 1224 T - 176 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$67$ $$( -3701 - 1013 T + 17 T^{2} + 19 T^{3} + T^{4} )^{2}$$
$71$ $$79762761 + 20041164 T + 4722951 T^{2} + 435780 T^{3} + 55036 T^{4} + 3788 T^{5} + 435 T^{6} + 20 T^{7} + T^{8}$$
$73$ $$388129 + 290941 T + 176348 T^{2} + 40011 T^{3} + 8381 T^{4} + 465 T^{5} + 116 T^{6} + 7 T^{7} + T^{8}$$
$79$ $$4313929 - 2297162 T + 1370703 T^{2} - 21170 T^{3} + 33662 T^{4} - 3916 T^{5} + 505 T^{6} - 24 T^{7} + T^{8}$$
$83$ $$( 2429 - 1683 T + 364 T^{2} - 32 T^{3} + T^{4} )^{2}$$
$89$ $$8696601 + 1486296 T + 560712 T^{2} + 12462 T^{3} + 13411 T^{4} + 136 T^{5} + 225 T^{6} - 11 T^{7} + T^{8}$$
$97$ $$4239481 - 2112534 T + 887956 T^{2} - 127378 T^{3} + 19745 T^{4} - 1172 T^{5} + 201 T^{6} - 11 T^{7} + T^{8}$$