# Properties

 Label 648.4.a.k Level $648$ Weight $4$ Character orbit 648.a Self dual yes Analytic conductor $38.233$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 648.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.2332376837$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Defining polynomial: $$x^{5} - x^{4} - 19 x^{3} + 4 x^{2} + 81 x + 12$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 72) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} ) q^{5} + ( 1 + \beta_{1} ) q^{7} +O(q^{10})$$ $$q + ( -1 - \beta_{2} ) q^{5} + ( 1 + \beta_{1} ) q^{7} + ( 5 - 2 \beta_{2} + \beta_{4} ) q^{11} + ( 7 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{13} + ( -6 - 3 \beta_{2} + \beta_{3} - \beta_{4} ) q^{17} + ( 12 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{19} + ( -17 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{23} + ( 62 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{25} + ( -23 + 6 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{29} + ( 49 - \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{31} + ( 41 - 11 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} - 6 \beta_{4} ) q^{35} + ( 72 - 6 \beta_{1} + 12 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} ) q^{37} + ( -33 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} + \beta_{4} ) q^{41} + ( 161 - 8 \beta_{2} - \beta_{3} + 6 \beta_{4} ) q^{43} + ( 99 + 7 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} + 10 \beta_{4} ) q^{47} + ( 218 + 4 \beta_{1} + 11 \beta_{2} - 5 \beta_{3} + 9 \beta_{4} ) q^{49} + ( -68 + 10 \beta_{1} + 16 \beta_{2} - 7 \beta_{3} + 7 \beta_{4} ) q^{53} + ( 289 - 5 \beta_{1} - 18 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} ) q^{55} + ( 121 - 6 \beta_{1} + 12 \beta_{2} - 5 \beta_{3} ) q^{59} + ( 267 - 2 \beta_{1} + 23 \beta_{2} + 5 \beta_{3} - 13 \beta_{4} ) q^{61} + ( -121 - 32 \beta_{1} + 9 \beta_{2} + 7 \beta_{3} - 11 \beta_{4} ) q^{65} + ( 305 - 6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - 15 \beta_{4} ) q^{67} + ( 170 + 18 \beta_{1} + 18 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{71} + ( 390 + 8 \beta_{1} - 23 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} ) q^{73} + ( -187 + 4 \beta_{1} + 3 \beta_{2} + 20 \beta_{3} - 4 \beta_{4} ) q^{77} + ( 359 + 17 \beta_{1} - 18 \beta_{2} - 17 \beta_{3} + 3 \beta_{4} ) q^{79} + ( 215 + 23 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} ) q^{83} + ( 576 + 6 \beta_{1} - 12 \beta_{2} + 17 \beta_{3} - \beta_{4} ) q^{85} + ( -464 - 22 \beta_{1} - 14 \beta_{2} - 17 \beta_{3} + 9 \beta_{4} ) q^{89} + ( 613 + 37 \beta_{1} + 76 \beta_{2} - 3 \beta_{3} + 13 \beta_{4} ) q^{91} + ( 408 + 20 \beta_{1} - 16 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} ) q^{95} + ( 577 - 2 \beta_{1} - 48 \beta_{2} - 7 \beta_{3} + 11 \beta_{4} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 5q^{5} + 3q^{7} + O(q^{10})$$ $$5q - 5q^{5} + 3q^{7} + 25q^{11} + 29q^{13} - 28q^{17} + 64q^{19} - 89q^{23} + 322q^{25} - 129q^{29} + 241q^{31} + 243q^{35} + 366q^{37} - 171q^{41} + 803q^{43} + 477q^{47} + 1072q^{49} - 374q^{53} + 1469q^{55} + 607q^{59} + 1349q^{61} - 527q^{65} + 1549q^{67} + 812q^{71} + 1928q^{73} - 903q^{77} + 1727q^{79} + 1025q^{83} + 2902q^{85} - 2310q^{89} + 2985q^{91} + 2012q^{95} + 2875q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 19 x^{3} + 4 x^{2} + 81 x + 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{4} - 7 \nu^{3} - 14 \nu^{2} + 43 \nu - 19$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{4} - 7 \nu^{3} - 20 \nu^{2} + 49 \nu + 27$$ $$\beta_{3}$$ $$=$$ $$6 \nu^{4} - 18 \nu^{3} - 60 \nu^{2} + 126 \nu + 58$$ $$\beta_{4}$$ $$=$$ $$10 \nu^{4} - 38 \nu^{3} - 88 \nu^{2} + 290 \nu + 60$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - 6 \beta_{2} - 2 \beta_{1} + 6$$$$)/36$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - 12 \beta_{2} + 4 \beta_{1} + 282$$$$)/36$$ $$\nu^{3}$$ $$=$$ $$($$$$7 \beta_{4} + 19 \beta_{3} - 78 \beta_{2} - 14 \beta_{1} + 318$$$$)/36$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{4} + 26 \beta_{3} - 114 \beta_{2} + 20 \beta_{1} + 1650$$$$)/18$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.150045 −2.80998 −2.71954 2.50994 4.16963
0 0 0 −20.2222 0 −24.7425 0 0 0
1.2 0 0 0 −12.3969 0 30.6325 0 0 0
1.3 0 0 0 2.98196 0 11.7109 0 0 0
1.4 0 0 0 6.31871 0 −29.5796 0 0 0
1.5 0 0 0 18.3184 0 14.9787 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.4.a.k 5
3.b odd 2 1 648.4.a.l 5
4.b odd 2 1 1296.4.a.bc 5
9.c even 3 2 216.4.i.b 10
9.d odd 6 2 72.4.i.b 10
12.b even 2 1 1296.4.a.bd 5
36.f odd 6 2 432.4.i.f 10
36.h even 6 2 144.4.i.f 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.i.b 10 9.d odd 6 2
144.4.i.f 10 36.h even 6 2
216.4.i.b 10 9.c even 3 2
432.4.i.f 10 36.f odd 6 2
648.4.a.k 5 1.a even 1 1 trivial
648.4.a.l 5 3.b odd 2 1
1296.4.a.bc 5 4.b odd 2 1
1296.4.a.bd 5 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{5} + 5 T_{5}^{4} - 461 T_{5}^{3} - 1097 T_{5}^{2} + 36176 T_{5} - 86528$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(648))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5}$$
$5$ $$-86528 + 36176 T - 1097 T^{2} - 461 T^{3} + 5 T^{4} + T^{5}$$
$7$ $$-3932604 + 434844 T + 6615 T^{2} - 1389 T^{3} - 3 T^{4} + T^{5}$$
$11$ $$2450383 + 2671697 T + 77482 T^{2} - 3890 T^{3} - 25 T^{4} + T^{5}$$
$13$ $$11723252 + 2020232 T - 43435 T^{2} - 5405 T^{3} - 29 T^{4} + T^{5}$$
$17$ $$465026264 + 9589868 T - 387766 T^{2} - 8927 T^{3} + 28 T^{4} + T^{5}$$
$19$ $$-219796928 + 25870544 T + 365212 T^{2} - 10133 T^{3} - 64 T^{4} + T^{5}$$
$23$ $$54657912052 + 580906268 T - 4432661 T^{2} - 48365 T^{3} + 89 T^{4} + T^{5}$$
$29$ $$-2405376324 + 204856704 T - 2562561 T^{2} - 36093 T^{3} + 129 T^{4} + T^{5}$$
$31$ $$-2225150096 + 158063312 T + 3945385 T^{2} - 33665 T^{3} - 241 T^{4} + T^{5}$$
$37$ $$-525481299072 - 3133248768 T + 69985944 T^{2} - 163524 T^{3} - 366 T^{4} + T^{5}$$
$41$ $$-816147765 - 56379231 T - 1096182 T^{2} + 222 T^{3} + 171 T^{4} + T^{5}$$
$43$ $$236167016117 - 5502242383 T + 18306062 T^{2} + 143086 T^{3} - 803 T^{4} + T^{5}$$
$47$ $$134565454332 - 9738525204 T + 94739769 T^{2} - 184845 T^{3} - 477 T^{4} + T^{5}$$
$53$ $$758189779072 - 3898947712 T - 95073272 T^{2} - 242948 T^{3} + 374 T^{4} + T^{5}$$
$59$ $$804795801121 - 19481707399 T + 112363102 T^{2} - 93914 T^{3} - 607 T^{4} + T^{5}$$
$61$ $$-4452174344512 - 64161140320 T + 348549065 T^{2} + 110179 T^{3} - 1349 T^{4} + T^{5}$$
$67$ $$-6065728349525 - 39525998239 T + 147555010 T^{2} + 428158 T^{3} - 1549 T^{4} + T^{5}$$
$71$ $$-24420797440 + 6494698496 T + 243239552 T^{2} - 329744 T^{3} - 812 T^{4} + T^{5}$$
$73$ $$28702127905256 - 130341479668 T - 56743786 T^{2} + 1119481 T^{3} - 1928 T^{4} + T^{5}$$
$79$ $$85346901126224 - 861490732960 T + 1726168679 T^{2} - 226097 T^{3} - 1727 T^{4} + T^{5}$$
$83$ $$-70154840960560 + 156684449096 T + 655571969 T^{2} - 735857 T^{3} - 1025 T^{4} + T^{5}$$
$89$ $$-887800297007520 - 3596560713072 T - 4312041264 T^{2} - 277992 T^{3} + 2310 T^{4} + T^{5}$$
$97$ $$27675431349397 - 286750735375 T + 373099366 T^{2} + 1887694 T^{3} - 2875 T^{4} + T^{5}$$