# Properties

 Label 1638.2.bj.f Level $1638$ Weight $2$ Character orbit 1638.bj 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.bj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.195105024.2 Defining polynomial: $$x^{8} - 4 x^{7} + 5 x^{6} + 4 x^{5} - 20 x^{4} + 12 x^{3} + 45 x^{2} - 108 x + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} -\beta_{4} q^{4} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{5} -\beta_{6} q^{7} + ( -\beta_{2} - \beta_{6} ) q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} -\beta_{4} q^{4} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{5} -\beta_{6} q^{7} + ( -\beta_{2} - \beta_{6} ) q^{8} + ( 2 - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{10} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{11} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{13} - q^{14} + ( -1 - \beta_{4} ) q^{16} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{17} + ( -2 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{19} + ( 1 - \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} ) q^{20} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{22} + ( -1 - \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{23} + ( -1 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{25} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{26} + \beta_{2} q^{28} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - 4 \beta_{6} ) q^{29} + ( -2 - 2 \beta_{2} - 4 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{31} -\beta_{6} q^{32} + ( 1 - \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{34} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{35} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{37} + ( 2 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{38} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} ) q^{40} + ( -1 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{41} + ( -5 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{43} + ( \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{44} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{46} + ( 3 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} + \beta_{6} + 5 \beta_{7} ) q^{47} + ( 1 + \beta_{4} ) q^{49} + ( 2 - 2 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{50} + ( 2 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{52} + ( 3 + \beta_{1} + 4 \beta_{2} - \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} ) q^{53} + ( -6 + 2 \beta_{2} + \beta_{3} - 6 \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{55} + \beta_{4} q^{56} + ( -4 + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{58} + ( -3 - 3 \beta_{1} - 4 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{59} + ( 8 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 5 \beta_{6} ) q^{61} + ( -2 - \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - \beta_{5} - 5 \beta_{6} - \beta_{7} ) q^{62} - q^{64} + ( -9 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{65} + ( 2 + \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{67} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{68} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{70} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{71} + ( 2 - 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 8 \beta_{6} ) q^{73} + ( -1 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{74} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{76} + ( 1 + \beta_{1} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{77} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{79} + ( -2 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{80} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 4 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{82} + ( 4 + \beta_{1} + 6 \beta_{2} + \beta_{3} + 10 \beta_{4} + 2 \beta_{5} + 5 \beta_{6} + \beta_{7} ) q^{83} + ( -9 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{85} + ( -2 + 2 \beta_{2} - 4 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{86} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{88} + ( 3 - 2 \beta_{1} + 10 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{89} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{91} + ( -3 + \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{7} ) q^{92} + ( 1 - 5 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{94} + ( -7 + 7 \beta_{1} - 5 \beta_{2} + 6 \beta_{3} - 10 \beta_{4} + 6 \beta_{5} - 9 \beta_{6} - 7 \beta_{7} ) q^{95} + ( -9 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 5 \beta_{6} + \beta_{7} ) q^{97} + \beta_{6} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + O(q^{10})$$ $$8q + 4q^{4} + 6q^{10} - 6q^{11} + 12q^{13} - 8q^{14} - 4q^{16} - 2q^{17} - 12q^{19} + 6q^{20} - 4q^{22} - 8q^{23} - 24q^{25} - 6q^{26} - 2q^{29} + 6q^{35} + 18q^{37} + 4q^{38} + 12q^{40} - 12q^{41} - 8q^{43} + 18q^{46} + 4q^{49} - 12q^{50} + 12q^{52} + 12q^{53} - 22q^{55} - 4q^{56} - 24q^{58} - 18q^{59} - 8q^{61} - 8q^{62} - 8q^{64} - 46q^{65} + 18q^{67} + 2q^{68} - 6q^{71} + 6q^{74} - 12q^{76} + 8q^{77} - 4q^{79} + 6q^{80} + 10q^{82} - 54q^{85} + 4q^{88} + 18q^{89} + 6q^{91} - 16q^{92} - 2q^{94} + 50q^{95} - 54q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 5 x^{6} + 4 x^{5} - 20 x^{4} + 12 x^{3} + 45 x^{2} - 108 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + 2 \nu^{5} + 10 \nu^{4} + 10 \nu^{3} + 42 \nu^{2} - 45 \nu - 27$$$$)/216$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} - 2 \nu^{5} - 10 \nu^{4} - 10 \nu^{3} + 30 \nu^{2} - 27 \nu + 27$$$$)/72$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{7} + 8 \nu^{6} - 4 \nu^{5} - 26 \nu^{4} + 52 \nu^{3} + 18 \nu^{2} - 153 \nu + 162$$$$)/108$$ $$\beta_{5}$$ $$=$$ $$($$$$-7 \nu^{7} + 13 \nu^{6} - 2 \nu^{5} - 22 \nu^{4} + 26 \nu^{3} + 54 \nu^{2} - 225 \nu + 243$$$$)/108$$ $$\beta_{6}$$ $$=$$ $$($$$$19 \nu^{7} - 49 \nu^{6} + 14 \nu^{5} + 130 \nu^{4} - 218 \nu^{3} - 150 \nu^{2} + 801 \nu - 891$$$$)/216$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 14 \nu^{6} - 10 \nu^{5} - 32 \nu^{4} + 58 \nu^{3} + 24 \nu^{2} - 207 \nu + 270$$$$)/36$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 3 \beta_{6} + 3 \beta_{5} - 3 \beta_{3} + 6 \beta_{2} + \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-5 \beta_{7} - 9 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} - 5 \beta_{3} - 6 \beta_{2} + 3$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{7} - 6 \beta_{6} + 6 \beta_{5} - 24 \beta_{4} - 8 \beta_{3} - 6 \beta_{2} + 8 \beta_{1} - 15$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{7} - 18 \beta_{6} - 18 \beta_{5} - 36 \beta_{4} + 6 \beta_{2} - 9 \beta_{1} + 6$$

## Character values

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

 $$n$$ $$379$$ $$703$$ $$911$$ $$\chi(n)$$ $$-\beta_{4}$$ $$1$$ $$1$$

## 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
 1.72124 − 0.193255i −1.58726 + 0.693255i 1.30512 + 1.13871i 0.560908 − 1.63871i −1.58726 − 0.693255i 1.72124 + 0.193255i 0.560908 + 1.63871i 1.30512 − 1.13871i
−0.866025 + 0.500000i 0 0.500000 0.866025i 3.05596i 0 0.866025 + 0.500000i 1.00000i 0 1.52798 + 2.64654i
127.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.78801i 0 0.866025 + 0.500000i 1.00000i 0 −0.894007 1.54846i
127.3 0.866025 0.500000i 0 0.500000 0.866025i 0.332808i 0 −0.866025 0.500000i 1.00000i 0 0.166404 + 0.288220i
127.4 0.866025 0.500000i 0 0.500000 0.866025i 4.39924i 0 −0.866025 0.500000i 1.00000i 0 2.19962 + 3.80986i
1135.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.78801i 0 0.866025 0.500000i 1.00000i 0 −0.894007 + 1.54846i
1135.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 3.05596i 0 0.866025 0.500000i 1.00000i 0 1.52798 2.64654i
1135.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i 4.39924i 0 −0.866025 + 0.500000i 1.00000i 0 2.19962 3.80986i
1135.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.332808i 0 −0.866025 + 0.500000i 1.00000i 0 0.166404 0.288220i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1135.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.f 8
3.b odd 2 1 546.2.s.e 8
13.e even 6 1 inner 1638.2.bj.f 8
39.h odd 6 1 546.2.s.e 8
39.k even 12 1 7098.2.a.cn 4
39.k even 12 1 7098.2.a.co 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.e 8 3.b odd 2 1
546.2.s.e 8 39.h odd 6 1
1638.2.bj.f 8 1.a even 1 1 trivial
1638.2.bj.f 8 13.e even 6 1 inner
7098.2.a.cn 4 39.k even 12 1
7098.2.a.co 4 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 $$S_{2}^{\mathrm{new}}(1638, [\chi])$$:

 $$T_{5}^{8} + 32 T_{5}^{6} + 276 T_{5}^{4} + 608 T_{5}^{2} + 64$$ $$T_{11}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$64 + 608 T^{2} + 276 T^{4} + 32 T^{6} + T^{8}$$
$7$ $$( 1 - T^{2} + T^{4} )^{2}$$
$11$ $$1024 + 2304 T + 2016 T^{2} + 648 T^{3} - 31 T^{4} - 54 T^{5} + 3 T^{6} + 6 T^{7} + T^{8}$$
$13$ $$28561 - 26364 T + 11154 T^{2} - 3120 T^{3} + 815 T^{4} - 240 T^{5} + 66 T^{6} - 12 T^{7} + T^{8}$$
$17$ $$37249 - 4246 T + 6274 T^{2} - 112 T^{3} + 751 T^{4} - 16 T^{5} + 34 T^{6} + 2 T^{7} + T^{8}$$
$19$ $$234256 - 191664 T + 30492 T^{2} + 17820 T^{3} - 43 T^{4} - 540 T^{5} + 3 T^{6} + 12 T^{7} + T^{8}$$
$23$ $$183184 + 99296 T + 47404 T^{2} + 10328 T^{3} + 2509 T^{4} + 344 T^{5} + 79 T^{6} + 8 T^{7} + T^{8}$$
$29$ $$4096 + 1024 T + 2240 T^{2} - 752 T^{3} + 865 T^{4} - 94 T^{5} + 35 T^{6} + 2 T^{7} + T^{8}$$
$31$ $$47524 + 53616 T^{2} + 4253 T^{4} + 114 T^{6} + T^{8}$$
$37$ $$5184 - 15552 T + 15120 T^{2} + 1296 T^{3} - 1188 T^{4} - 108 T^{5} + 114 T^{6} - 18 T^{7} + T^{8}$$
$41$ $$256 + 384 T - 272 T^{2} - 696 T^{3} + 921 T^{4} - 348 T^{5} + 19 T^{6} + 12 T^{7} + T^{8}$$
$43$ $$20164 - 10508 T + 9026 T^{2} - 422 T^{3} + 1075 T^{4} - 52 T^{5} + 89 T^{6} + 8 T^{7} + T^{8}$$
$47$ $$59783824 + 2914872 T^{2} + 51041 T^{4} + 378 T^{6} + T^{8}$$
$53$ $$( 2461 + 426 T - 106 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$59$ $$4477456 - 2336064 T + 234876 T^{2} + 89424 T^{3} - 2179 T^{4} - 1458 T^{5} + 27 T^{6} + 18 T^{7} + T^{8}$$
$61$ $$2283121 + 1680232 T + 1058246 T^{2} + 155392 T^{3} + 24331 T^{4} + 1280 T^{5} + 182 T^{6} + 8 T^{7} + T^{8}$$
$67$ $$11916304 - 8201952 T + 2258060 T^{2} - 258984 T^{3} + 1077 T^{4} + 1962 T^{5} - T^{6} - 18 T^{7} + T^{8}$$
$71$ $$324 - 1620 T + 3402 T^{2} - 3510 T^{3} + 1719 T^{4} - 234 T^{5} - 27 T^{6} + 6 T^{7} + T^{8}$$
$73$ $$65536 + 368640 T^{2} + 22592 T^{4} + 288 T^{6} + T^{8}$$
$79$ $$( -104 - 148 T - 39 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$83$ $$22146436 + 1924816 T^{2} + 46077 T^{4} + 382 T^{6} + T^{8}$$
$89$ $$20584369 + 7268274 T + 84178 T^{2} - 272340 T^{3} + 14751 T^{4} + 3060 T^{5} - 62 T^{6} - 18 T^{7} + T^{8}$$
$97$ $$1290496 - 1063296 T - 19232 T^{2} + 256464 T^{3} + 93060 T^{4} + 14796 T^{5} + 1246 T^{6} + 54 T^{7} + T^{8}$$