# Properties

 Label 648.4.a.j Level $648$ Weight $4$ Character orbit 648.a Self dual yes Analytic conductor $38.233$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.29952.1 Defining polynomial: $$x^{4} - 8 x^{2} + 13$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 2 - \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} +O(q^{10})$$ $$q + ( 2 - \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{11} + ( 1 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{13} + ( 4 + 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{17} + ( 20 - 3 \beta_{1} - 5 \beta_{2} ) q^{19} + ( 50 - 5 \beta_{1} - 6 \beta_{2} - 7 \beta_{3} ) q^{23} + ( 2 - 6 \beta_{1} - 2 \beta_{2} - 16 \beta_{3} ) q^{25} + ( 54 + 9 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{29} + ( 20 + 2 \beta_{1} - 2 \beta_{2} + 16 \beta_{3} ) q^{31} + ( 102 - 3 \beta_{1} - 2 \beta_{2} + 7 \beta_{3} ) q^{35} + ( -69 + \beta_{1} + 17 \beta_{2} + 22 \beta_{3} ) q^{37} + ( 96 + 12 \beta_{1} + 4 \beta_{2} + 20 \beta_{3} ) q^{41} + ( 40 + 19 \beta_{1} - 3 \beta_{2} - 56 \beta_{3} ) q^{43} + ( 192 - 16 \beta_{1} + 12 \beta_{3} ) q^{47} + ( -67 + 4 \beta_{1} - 8 \beta_{2} + 44 \beta_{3} ) q^{49} + ( 236 + 10 \beta_{1} - 24 \beta_{2} - 34 \beta_{3} ) q^{53} + ( 76 - \beta_{1} + \beta_{2} + 32 \beta_{3} ) q^{55} + ( 248 + 12 \beta_{1} + 24 \beta_{2} - 28 \beta_{3} ) q^{59} + ( -137 + 3 \beta_{1} - \beta_{2} - 114 \beta_{3} ) q^{61} + ( 332 - 6 \beta_{1} - 21 \beta_{3} ) q^{65} + ( 116 + 19 \beta_{1} + 21 \beta_{2} + 96 \beta_{3} ) q^{67} + ( 430 - 27 \beta_{1} + 6 \beta_{2} + 43 \beta_{3} ) q^{71} + ( -191 + 44 \beta_{1} - 40 \beta_{2} + 36 \beta_{3} ) q^{73} + ( 432 + 4 \beta_{1} - 4 \beta_{2} + 84 \beta_{3} ) q^{77} + ( 172 - 23 \beta_{1} + 23 \beta_{2} - 160 \beta_{3} ) q^{79} + ( 532 + 38 \beta_{1} - 20 \beta_{2} + 6 \beta_{3} ) q^{83} + ( -331 + 11 \beta_{1} + 7 \beta_{2} + 126 \beta_{3} ) q^{85} + ( 528 - 40 \beta_{1} + 56 \beta_{2} - 55 \beta_{3} ) q^{89} + ( 444 - 5 \beta_{1} - 3 \beta_{2} + 64 \beta_{3} ) q^{91} + ( 514 - 37 \beta_{1} - 6 \beta_{2} - 163 \beta_{3} ) q^{95} + ( -454 + 40 \beta_{1} - 4 \beta_{2} - 204 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{5} + O(q^{10})$$ $$4q + 8q^{5} + 8q^{11} + 4q^{13} + 16q^{17} + 80q^{19} + 200q^{23} + 8q^{25} + 216q^{29} + 80q^{31} + 408q^{35} - 276q^{37} + 384q^{41} + 160q^{43} + 768q^{47} - 268q^{49} + 944q^{53} + 304q^{55} + 992q^{59} - 548q^{61} + 1328q^{65} + 464q^{67} + 1720q^{71} - 764q^{73} + 1728q^{77} + 688q^{79} + 2128q^{83} - 1324q^{85} + 2112q^{89} + 1776q^{91} + 2056q^{95} - 1816q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 8 x^{2} + 13$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3} + \nu^{2} - 14 \nu - 4$$ $$\beta_{2}$$ $$=$$ $$4 \nu^{3} - \nu^{2} - 16 \nu + 4$$ $$\beta_{3}$$ $$=$$ $$3 \nu^{2} - 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 12$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{3} + 7 \beta_{2} - 8 \beta_{1}$$$$)/12$$

## 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
 −1.50597 −2.39417 2.39417 1.50597
0 0 0 −10.5206 0 −0.354888 0 0 0
1.2 0 0 0 −5.80342 0 −26.1228 0 0 0
1.3 0 0 0 6.33932 0 19.1946 0 0 0
1.4 0 0 0 17.9847 0 7.28309 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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.j yes 4
3.b odd 2 1 648.4.a.g 4
4.b odd 2 1 1296.4.a.bb 4
9.c even 3 2 648.4.i.u 8
9.d odd 6 2 648.4.i.v 8
12.b even 2 1 1296.4.a.x 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.g 4 3.b odd 2 1
648.4.a.j yes 4 1.a even 1 1 trivial
648.4.i.u 8 9.c even 3 2
648.4.i.v 8 9.d odd 6 2
1296.4.a.x 4 12.b even 2 1
1296.4.a.bb 4 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 8 T_{5}^{3} - 222 T_{5}^{2} + 376 T_{5} + 6961$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(648))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$6961 + 376 T - 222 T^{2} - 8 T^{3} + T^{4}$$
$7$ $$1296 + 3456 T - 552 T^{2} + T^{4}$$
$11$ $$61072 + 6304 T - 1560 T^{2} - 8 T^{3} + T^{4}$$
$13$ $$1237417 + 3404 T - 2274 T^{2} - 4 T^{3} + T^{4}$$
$17$ $$14319721 + 69680 T - 7782 T^{2} - 16 T^{3} + T^{4}$$
$19$ $$31790992 + 530368 T - 10824 T^{2} - 80 T^{3} + T^{4}$$
$23$ $$48919312 + 1100320 T - 10104 T^{2} - 200 T^{3} + T^{4}$$
$29$ $$54863217 + 1735848 T - 4638 T^{2} - 216 T^{3} + T^{4}$$
$31$ $$48805888 + 342016 T - 15936 T^{2} - 80 T^{3} + T^{4}$$
$37$ $$948334041 - 23154588 T - 99282 T^{2} + 276 T^{3} + T^{4}$$
$41$ $$-2045323008 + 21141504 T - 17760 T^{2} - 384 T^{3} + T^{4}$$
$43$ $$513055504 + 1524224 T - 202920 T^{2} - 160 T^{3} + T^{4}$$
$47$ $$-1142228736 - 847872 T + 157344 T^{2} - 768 T^{3} + T^{4}$$
$53$ $$-22310316032 + 112215040 T + 84288 T^{2} - 944 T^{3} + T^{4}$$
$59$ $$593268736 + 83089408 T + 30336 T^{2} - 992 T^{3} + T^{4}$$
$61$ $$104164416841 - 177955180 T - 575106 T^{2} + 548 T^{3} + T^{4}$$
$67$ $$-30700073072 + 359901376 T - 704328 T^{2} - 464 T^{3} + T^{4}$$
$71$ $$-9807661424 - 110047648 T + 880872 T^{2} - 1720 T^{3} + T^{4}$$
$73$ $$-194677777919 - 951038212 T - 912378 T^{2} + 764 T^{3} + T^{4}$$
$79$ $$320004558736 + 1127611712 T - 1761864 T^{2} - 688 T^{3} + T^{4}$$
$83$ $$6134511616 - 164967424 T + 1236288 T^{2} - 2128 T^{3} + T^{4}$$
$89$ $$-1312975087623 + 2534079168 T - 109206 T^{2} - 2112 T^{3} + T^{4}$$
$97$ $$2989754128 - 2084846240 T - 1073832 T^{2} + 1816 T^{3} + T^{4}$$