# Properties

 Label 63.4.e.d Level $63$ Weight $4$ Character orbit 63.e Analytic conductor $3.717$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 63.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.71712033036$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 19 x^{6} + 319 x^{4} + 798 x^{2} + 1764$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{3} ) q^{2} + ( -2 - 2 \beta_{2} - \beta_{6} ) q^{4} + ( \beta_{1} + \beta_{3} + \beta_{7} ) q^{5} + ( -2 + \beta_{2} - \beta_{4} - 2 \beta_{6} ) q^{7} + ( \beta_{3} + \beta_{5} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{3} ) q^{2} + ( -2 - 2 \beta_{2} - \beta_{6} ) q^{4} + ( \beta_{1} + \beta_{3} + \beta_{7} ) q^{5} + ( -2 + \beta_{2} - \beta_{4} - 2 \beta_{6} ) q^{7} + ( \beta_{3} + \beta_{5} - \beta_{7} ) q^{8} + ( -8 - 8 \beta_{2} - 5 \beta_{6} ) q^{10} + ( \beta_{1} - 3 \beta_{5} ) q^{11} + ( 25 - \beta_{4} ) q^{13} + ( -8 \beta_{1} - 17 \beta_{3} + 2 \beta_{5} - 3 \beta_{7} ) q^{14} + ( -28 \beta_{2} + 5 \beta_{4} - 5 \beta_{6} ) q^{16} + ( -26 \beta_{1} + 2 \beta_{5} ) q^{17} + ( 61 \beta_{2} - 11 \beta_{4} + 11 \beta_{6} ) q^{19} + ( -25 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} ) q^{20} + ( -16 + 11 \beta_{4} ) q^{22} + ( 46 \beta_{1} + 46 \beta_{3} + 2 \beta_{7} ) q^{23} + ( -83 - 83 \beta_{2} + 17 \beta_{6} ) q^{25} + ( 20 \beta_{1} + 20 \beta_{3} - \beta_{7} ) q^{26} + ( 54 + 148 \beta_{2} - \beta_{4} + 5 \beta_{6} ) q^{28} + ( 45 \beta_{3} - \beta_{5} + \beta_{7} ) q^{29} + ( -45 - 45 \beta_{2} + 20 \beta_{6} ) q^{31} + ( 45 \beta_{1} + 13 \beta_{5} ) q^{32} + ( 264 + 18 \beta_{4} ) q^{34} + ( -26 \beta_{1} - 71 \beta_{3} - 11 \beta_{5} + 13 \beta_{7} ) q^{35} + ( -81 \beta_{2} - 25 \beta_{4} + 25 \beta_{6} ) q^{37} + ( -116 \beta_{1} - 11 \beta_{5} ) q^{38} + ( 192 \beta_{2} + 27 \beta_{4} - 27 \beta_{6} ) q^{40} + ( -72 \beta_{3} + 16 \beta_{5} - 16 \beta_{7} ) q^{41} + ( -223 - 27 \beta_{4} ) q^{43} + ( 47 \beta_{1} + 47 \beta_{3} - 13 \beta_{7} ) q^{44} + ( -456 - 456 \beta_{2} - 54 \beta_{6} ) q^{46} + ( -44 \beta_{1} - 44 \beta_{3} - 16 \beta_{7} ) q^{47} + ( 243 + 379 \beta_{2} + 13 \beta_{4} - 2 \beta_{6} ) q^{49} + ( 2 \beta_{3} - 17 \beta_{5} + 17 \beta_{7} ) q^{50} + ( -2 - 2 \beta_{2} - 24 \beta_{6} ) q^{52} + ( \beta_{1} - 7 \beta_{5} ) q^{53} + ( 592 - 71 \beta_{4} ) q^{55} + ( -27 \beta_{1} + 10 \beta_{3} + 19 \beta_{5} - 4 \beta_{7} ) q^{56} + ( -448 \beta_{2} + 49 \beta_{4} - 49 \beta_{6} ) q^{58} + ( 73 \beta_{1} + 17 \beta_{5} ) q^{59} + ( 310 \beta_{2} + 46 \beta_{4} - 46 \beta_{6} ) q^{61} + ( 55 \beta_{3} - 20 \beta_{5} + 20 \beta_{7} ) q^{62} + ( -200 - 57 \beta_{4} ) q^{64} + ( 2 \beta_{1} + 2 \beta_{3} + 30 \beta_{7} ) q^{65} + ( -463 - 463 \beta_{2} + 19 \beta_{6} ) q^{67} + ( 146 \beta_{1} + 146 \beta_{3} + 34 \beta_{7} ) q^{68} + ( 264 + 736 \beta_{2} - \beta_{4} + 19 \beta_{6} ) q^{70} + ( 36 \beta_{5} - 36 \beta_{7} ) q^{71} + ( -441 - 441 \beta_{2} - 7 \beta_{6} ) q^{73} + ( -44 \beta_{1} - 25 \beta_{5} ) q^{74} + ( 650 + 72 \beta_{4} ) q^{76} + ( 145 \beta_{1} + 99 \beta_{3} - 10 \beta_{5} - 41 \beta_{7} ) q^{77} + ( 23 \beta_{2} - 42 \beta_{4} + 42 \beta_{6} ) q^{79} + ( 143 \beta_{1} + 3 \beta_{5} ) q^{80} + ( 688 \beta_{2} - 136 \beta_{4} + 136 \beta_{6} ) q^{82} + ( 189 \beta_{3} - 21 \beta_{5} + 21 \beta_{7} ) q^{83} + ( -192 + 174 \beta_{4} ) q^{85} + ( -358 \beta_{1} - 358 \beta_{3} - 27 \beta_{7} ) q^{86} + ( -624 - 624 \beta_{2} + 93 \beta_{6} ) q^{88} + ( -314 \beta_{1} - 314 \beta_{3} + 22 \beta_{7} ) q^{89} + ( 94 + 121 \beta_{2} - 23 \beta_{4} - 53 \beta_{6} ) q^{91} + ( -358 \beta_{3} + 38 \beta_{5} - 38 \beta_{7} ) q^{92} + ( 408 + 408 \beta_{2} + 108 \beta_{6} ) q^{94} + ( -314 \beta_{1} - 6 \beta_{5} ) q^{95} + ( 798 + 91 \beta_{4} ) q^{97} + ( -71 \beta_{1} + 298 \beta_{3} + 2 \beta_{5} + 11 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 6q^{4} - 12q^{7} + O(q^{10})$$ $$8q - 6q^{4} - 12q^{7} - 22q^{10} + 204q^{13} + 102q^{16} - 222q^{19} - 172q^{22} - 366q^{25} - 166q^{28} - 220q^{31} + 2040q^{34} + 374q^{37} - 822q^{40} - 1676q^{43} - 1716q^{46} + 380q^{49} + 40q^{52} + 5020q^{55} + 1694q^{58} - 1332q^{61} - 1372q^{64} - 1890q^{67} - 866q^{70} - 1750q^{73} + 4912q^{76} - 8q^{79} - 2480q^{82} - 2232q^{85} - 2682q^{88} + 466q^{91} + 1416q^{94} + 6020q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 19 x^{6} + 319 x^{4} + 798 x^{2} + 1764$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$19 \nu^{6} + 319 \nu^{4} + 6061 \nu^{2} + 1764$$$$)/13398$$ $$\beta_{3}$$ $$=$$ $$($$$$19 \nu^{7} + 319 \nu^{5} + 6061 \nu^{3} + 1764 \nu$$$$)/13398$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} - 2392$$$$)/319$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 3987 \nu$$$$)/319$$ $$\beta_{6}$$ $$=$$ $$($$$$-74 \nu^{6} - 1595 \nu^{4} - 23606 \nu^{2} - 59052$$$$)/6699$$ $$\beta_{7}$$ $$=$$ $$($$$$-81 \nu^{7} - 1595 \nu^{5} - 25839 \nu^{3} - 64638 \nu$$$$)/4466$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{4} + 10 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{5} + 15 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$-19 \beta_{6} - 148 \beta_{2} - 148$$ $$\nu^{5}$$ $$=$$ $$-19 \beta_{7} - 243 \beta_{3} - 243 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$319 \beta_{4} + 2392$$ $$\nu^{7}$$ $$=$$ $$319 \beta_{5} + 3987 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$$.

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 2.02770 + 3.51207i 0.799027 + 1.38396i −0.799027 − 1.38396i −2.02770 − 3.51207i 2.02770 − 3.51207i 0.799027 − 1.38396i −0.799027 + 1.38396i −2.02770 + 3.51207i
−2.02770 + 3.51207i 0 −4.22311 7.31464i −4.96020 + 8.59131i 0 −15.3924 10.2992i 1.80961 0 −20.1156 34.8412i
37.2 −0.799027 + 1.38396i 0 2.72311 + 4.71657i 9.14584 15.8411i 0 12.3924 + 13.7633i −21.4878 0 14.6156 + 25.3149i
37.3 0.799027 1.38396i 0 2.72311 + 4.71657i −9.14584 + 15.8411i 0 12.3924 + 13.7633i 21.4878 0 14.6156 + 25.3149i
37.4 2.02770 3.51207i 0 −4.22311 7.31464i 4.96020 8.59131i 0 −15.3924 10.2992i −1.80961 0 −20.1156 34.8412i
46.1 −2.02770 3.51207i 0 −4.22311 + 7.31464i −4.96020 8.59131i 0 −15.3924 + 10.2992i 1.80961 0 −20.1156 + 34.8412i
46.2 −0.799027 1.38396i 0 2.72311 4.71657i 9.14584 + 15.8411i 0 12.3924 13.7633i −21.4878 0 14.6156 25.3149i
46.3 0.799027 + 1.38396i 0 2.72311 4.71657i −9.14584 15.8411i 0 12.3924 13.7633i 21.4878 0 14.6156 25.3149i
46.4 2.02770 + 3.51207i 0 −4.22311 + 7.31464i 4.96020 + 8.59131i 0 −15.3924 + 10.2992i −1.80961 0 −20.1156 + 34.8412i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 46.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.4.e.d 8
3.b odd 2 1 inner 63.4.e.d 8
7.b odd 2 1 441.4.e.x 8
7.c even 3 1 inner 63.4.e.d 8
7.c even 3 1 441.4.a.w 4
7.d odd 6 1 441.4.a.v 4
7.d odd 6 1 441.4.e.x 8
21.c even 2 1 441.4.e.x 8
21.g even 6 1 441.4.a.v 4
21.g even 6 1 441.4.e.x 8
21.h odd 6 1 inner 63.4.e.d 8
21.h odd 6 1 441.4.a.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.e.d 8 1.a even 1 1 trivial
63.4.e.d 8 3.b odd 2 1 inner
63.4.e.d 8 7.c even 3 1 inner
63.4.e.d 8 21.h odd 6 1 inner
441.4.a.v 4 7.d odd 6 1
441.4.a.v 4 21.g even 6 1
441.4.a.w 4 7.c even 3 1
441.4.a.w 4 21.h odd 6 1
441.4.e.x 8 7.b odd 2 1
441.4.e.x 8 7.d odd 6 1
441.4.e.x 8 21.c even 2 1
441.4.e.x 8 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 19 T_{2}^{6} + 319 T_{2}^{4} + 798 T_{2}^{2} + 1764$$ acting on $$S_{4}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1764 + 798 T^{2} + 319 T^{4} + 19 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$1084253184 + 14257824 T^{2} + 154561 T^{4} + 433 T^{6} + T^{8}$$
$7$ $$( 117649 + 2058 T - 77 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$11$ $$473520144384 + 2709159936 T^{2} + 14811841 T^{4} + 3937 T^{6} + T^{8}$$
$13$ $$( 602 - 51 T + T^{2} )^{4}$$
$17$ $$3373866295640064 + 894276536832 T^{2} + 178951824 T^{4} + 15396 T^{6} + T^{8}$$
$19$ $$( 7606564 - 306138 T + 15079 T^{2} + 111 T^{3} + T^{4} )^{2}$$
$23$ $$2000863076696064 + 1809458735616 T^{2} + 1591633296 T^{4} + 40452 T^{6} + T^{8}$$
$29$ $$( 96018048 - 38185 T^{2} + T^{4} )^{2}$$
$31$ $$( 264875625 - 1790250 T + 28375 T^{2} + 110 T^{3} + T^{4} )^{2}$$
$37$ $$( 458559396 + 4004418 T + 56383 T^{2} - 187 T^{3} + T^{4} )^{2}$$
$41$ $$( 6145155072 - 190144 T^{2} + T^{4} )^{2}$$
$43$ $$( 8716 + 419 T + T^{2} )^{4}$$
$47$ $$20763591005186555904 + 617889652531200 T^{2} + 13830651648 T^{4} + 135600 T^{6} + T^{8}$$
$53$ $$737093829878784 + 575595955872 T^{2} + 422332929 T^{4} + 21201 T^{6} + T^{8}$$
$59$ $$59009132377388196864 + 1579865096587680 T^{2} + 34616352033 T^{4} + 205665 T^{6} + T^{8}$$
$61$ $$( 77299264 + 5855472 T + 434764 T^{2} + 666 T^{3} + T^{4} )^{2}$$
$67$ $$( 42369282244 + 194516910 T + 687187 T^{2} + 945 T^{3} + T^{4} )^{2}$$
$71$ $$( 22856214528 - 557280 T^{2} + T^{4} )^{2}$$
$73$ $$( 35736877764 + 165411750 T + 576583 T^{2} + 875 T^{3} + T^{4} )^{2}$$
$79$ $$( 7243541881 - 340436 T + 85125 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$83$ $$( 10585989792 - 804825 T^{2} + T^{4} )^{2}$$
$89$ $$13\!\cdots\!84$$$$+ 2532972384732604416 T^{2} + 3649173392656 T^{4} + 2191972 T^{6} + T^{8}$$
$97$ $$( 166698 - 1505 T + T^{2} )^{4}$$