# Properties

 Label 648.4.a.f Level $648$ Weight $4$ Character orbit 648.a Self dual yes Analytic conductor $38.233$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{201})$$ Defining polynomial: $$x^{2} - x - 50$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{201}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 4 + \beta ) q^{5} + ( -15 + \beta ) q^{7} +O(q^{10})$$ $$q + ( 4 + \beta ) q^{5} + ( -15 + \beta ) q^{7} + ( -23 - 3 \beta ) q^{11} + ( 46 - \beta ) q^{13} + ( 11 + 8 \beta ) q^{17} + ( -43 + 5 \beta ) q^{19} + ( -29 - \beta ) q^{23} + ( 92 + 8 \beta ) q^{25} + ( 54 - \beta ) q^{29} + ( 68 + 16 \beta ) q^{31} + ( 141 - 11 \beta ) q^{35} + ( -90 - \beta ) q^{37} + ( 336 - 2 \beta ) q^{41} + ( -35 - 23 \beta ) q^{43} + ( 426 - 2 \beta ) q^{47} + ( 83 - 30 \beta ) q^{49} + ( 334 - 16 \beta ) q^{53} + ( -695 - 35 \beta ) q^{55} + ( 274 - 6 \beta ) q^{59} + ( 142 - 17 \beta ) q^{61} + ( -17 + 42 \beta ) q^{65} + ( 581 - 11 \beta ) q^{67} + ( -403 - 3 \beta ) q^{71} + ( -755 + 18 \beta ) q^{73} + ( -258 + 22 \beta ) q^{77} + ( -11 + 41 \beta ) q^{79} + ( 770 + 38 \beta ) q^{83} + ( 1652 + 43 \beta ) q^{85} + 1323 q^{89} + ( -891 + 61 \beta ) q^{91} + ( 833 - 23 \beta ) q^{95} + ( 236 + 50 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{5} - 30q^{7} + O(q^{10})$$ $$2q + 8q^{5} - 30q^{7} - 46q^{11} + 92q^{13} + 22q^{17} - 86q^{19} - 58q^{23} + 184q^{25} + 108q^{29} + 136q^{31} + 282q^{35} - 180q^{37} + 672q^{41} - 70q^{43} + 852q^{47} + 166q^{49} + 668q^{53} - 1390q^{55} + 548q^{59} + 284q^{61} - 34q^{65} + 1162q^{67} - 806q^{71} - 1510q^{73} - 516q^{77} - 22q^{79} + 1540q^{83} + 3304q^{85} + 2646q^{89} - 1782q^{91} + 1666q^{95} + 472q^{97} + O(q^{100})$$

## 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
 −6.58872 7.58872
0 0 0 −10.1774 0 −29.1774 0 0 0
1.2 0 0 0 18.1774 0 −0.822553 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.f yes 2
3.b odd 2 1 648.4.a.c 2
4.b odd 2 1 1296.4.a.q 2
9.c even 3 2 648.4.i.n 4
9.d odd 6 2 648.4.i.t 4
12.b even 2 1 1296.4.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.c 2 3.b odd 2 1
648.4.a.f yes 2 1.a even 1 1 trivial
648.4.i.n 4 9.c even 3 2
648.4.i.t 4 9.d odd 6 2
1296.4.a.m 2 12.b even 2 1
1296.4.a.q 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 8 T_{5} - 185$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(648))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-185 - 8 T + T^{2}$$
$7$ $$24 + 30 T + T^{2}$$
$11$ $$-1280 + 46 T + T^{2}$$
$13$ $$1915 - 92 T + T^{2}$$
$17$ $$-12743 - 22 T + T^{2}$$
$19$ $$-3176 + 86 T + T^{2}$$
$23$ $$640 + 58 T + T^{2}$$
$29$ $$2715 - 108 T + T^{2}$$
$31$ $$-46832 - 136 T + T^{2}$$
$37$ $$7899 + 180 T + T^{2}$$
$41$ $$112092 - 672 T + T^{2}$$
$43$ $$-105104 + 70 T + T^{2}$$
$47$ $$180672 - 852 T + T^{2}$$
$53$ $$60100 - 668 T + T^{2}$$
$59$ $$67840 - 548 T + T^{2}$$
$61$ $$-37925 - 284 T + T^{2}$$
$67$ $$313240 - 1162 T + T^{2}$$
$71$ $$160600 + 806 T + T^{2}$$
$73$ $$504901 + 1510 T + T^{2}$$
$79$ $$-337760 + 22 T + T^{2}$$
$83$ $$302656 - 1540 T + T^{2}$$
$89$ $$( -1323 + T )^{2}$$
$97$ $$-446804 - 472 T + T^{2}$$