# Properties

 Label 4928.2.a.bm Level 4928 Weight 2 Character orbit 4928.a Self dual yes Analytic conductor 39.350 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4928 = 2^{6} \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4928.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.3502781161$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \beta q^{3} + 2 q^{5} + q^{7} + ( 1 + 4 \beta ) q^{9} +O(q^{10})$$ $$q -2 \beta q^{3} + 2 q^{5} + q^{7} + ( 1 + 4 \beta ) q^{9} + q^{11} + ( -2 + 2 \beta ) q^{13} -4 \beta q^{15} + ( -2 + 2 \beta ) q^{17} + ( -4 + 4 \beta ) q^{19} -2 \beta q^{21} -4 \beta q^{23} - q^{25} + ( -8 - 4 \beta ) q^{27} + ( -6 + 4 \beta ) q^{29} + ( -4 - 2 \beta ) q^{31} -2 \beta q^{33} + 2 q^{35} + ( 6 - 4 \beta ) q^{37} -4 q^{39} + ( -10 + 2 \beta ) q^{41} -8 q^{43} + ( 2 + 8 \beta ) q^{45} + ( 4 + 2 \beta ) q^{47} + q^{49} -4 q^{51} + ( -2 - 4 \beta ) q^{53} + 2 q^{55} -8 q^{57} -2 \beta q^{59} + ( 6 - 2 \beta ) q^{61} + ( 1 + 4 \beta ) q^{63} + ( -4 + 4 \beta ) q^{65} + ( -12 + 4 \beta ) q^{67} + ( 8 + 8 \beta ) q^{69} + ( -8 + 4 \beta ) q^{71} + ( -2 - 2 \beta ) q^{73} + 2 \beta q^{75} + q^{77} + ( -4 + 8 \beta ) q^{79} + ( 5 + 12 \beta ) q^{81} + ( 4 - 12 \beta ) q^{83} + ( -4 + 4 \beta ) q^{85} + ( -8 + 4 \beta ) q^{87} + 2 q^{89} + ( -2 + 2 \beta ) q^{91} + ( 4 + 12 \beta ) q^{93} + ( -8 + 8 \beta ) q^{95} + ( 10 - 12 \beta ) q^{97} + ( 1 + 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 4q^{5} + 2q^{7} + 6q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 4q^{5} + 2q^{7} + 6q^{9} + 2q^{11} - 2q^{13} - 4q^{15} - 2q^{17} - 4q^{19} - 2q^{21} - 4q^{23} - 2q^{25} - 20q^{27} - 8q^{29} - 10q^{31} - 2q^{33} + 4q^{35} + 8q^{37} - 8q^{39} - 18q^{41} - 16q^{43} + 12q^{45} + 10q^{47} + 2q^{49} - 8q^{51} - 8q^{53} + 4q^{55} - 16q^{57} - 2q^{59} + 10q^{61} + 6q^{63} - 4q^{65} - 20q^{67} + 24q^{69} - 12q^{71} - 6q^{73} + 2q^{75} + 2q^{77} + 22q^{81} - 4q^{83} - 4q^{85} - 12q^{87} + 4q^{89} - 2q^{91} + 20q^{93} - 8q^{95} + 8q^{97} + 6q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
0 −3.23607 0 2.00000 0 1.00000 0 7.47214 0
1.2 0 1.23607 0 2.00000 0 1.00000 0 −1.47214 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 4928.2.a.bm 2
4.b odd 2 1 4928.2.a.bv 2
8.b even 2 1 77.2.a.d 2
8.d odd 2 1 1232.2.a.m 2
24.h odd 2 1 693.2.a.h 2
40.f even 2 1 1925.2.a.r 2
40.i odd 4 2 1925.2.b.h 4
56.e even 2 1 8624.2.a.ce 2
56.h odd 2 1 539.2.a.f 2
56.j odd 6 2 539.2.e.j 4
56.p even 6 2 539.2.e.i 4
88.b odd 2 1 847.2.a.f 2
88.o even 10 2 847.2.f.a 4
88.o even 10 2 847.2.f.n 4
88.p odd 10 2 847.2.f.b 4
88.p odd 10 2 847.2.f.m 4
168.i even 2 1 4851.2.a.y 2
264.m even 2 1 7623.2.a.bl 2
616.o even 2 1 5929.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 8.b even 2 1
539.2.a.f 2 56.h odd 2 1
539.2.e.i 4 56.p even 6 2
539.2.e.j 4 56.j odd 6 2
693.2.a.h 2 24.h odd 2 1
847.2.a.f 2 88.b odd 2 1
847.2.f.a 4 88.o even 10 2
847.2.f.b 4 88.p odd 10 2
847.2.f.m 4 88.p odd 10 2
847.2.f.n 4 88.o even 10 2
1232.2.a.m 2 8.d odd 2 1
1925.2.a.r 2 40.f even 2 1
1925.2.b.h 4 40.i odd 4 2
4851.2.a.y 2 168.i even 2 1
4928.2.a.bm 2 1.a even 1 1 trivial
4928.2.a.bv 2 4.b odd 2 1
5929.2.a.m 2 616.o even 2 1
7623.2.a.bl 2 264.m even 2 1
8624.2.a.ce 2 56.e even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4928))$$:

 $$T_{3}^{2} + 2 T_{3} - 4$$ $$T_{5} - 2$$ $$T_{13}^{2} + 2 T_{13} - 4$$ $$T_{17}^{2} + 2 T_{17} - 4$$ $$T_{19}^{2} + 4 T_{19} - 16$$ $$T_{23}^{2} + 4 T_{23} - 16$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$
$5$ $$( 1 - 2 T + 5 T^{2} )^{2}$$
$7$ $$( 1 - T )^{2}$$
$11$ $$( 1 - T )^{2}$$
$13$ $$1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4}$$
$17$ $$1 + 2 T + 30 T^{2} + 34 T^{3} + 289 T^{4}$$
$19$ $$1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4}$$
$23$ $$1 + 4 T + 30 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$1 + 8 T + 54 T^{2} + 232 T^{3} + 841 T^{4}$$
$31$ $$1 + 10 T + 82 T^{2} + 310 T^{3} + 961 T^{4}$$
$37$ $$1 - 8 T + 70 T^{2} - 296 T^{3} + 1369 T^{4}$$
$41$ $$1 + 18 T + 158 T^{2} + 738 T^{3} + 1681 T^{4}$$
$43$ $$( 1 + 8 T + 43 T^{2} )^{2}$$
$47$ $$1 - 10 T + 114 T^{2} - 470 T^{3} + 2209 T^{4}$$
$53$ $$1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4}$$
$59$ $$1 + 2 T + 114 T^{2} + 118 T^{3} + 3481 T^{4}$$
$61$ $$1 - 10 T + 142 T^{2} - 610 T^{3} + 3721 T^{4}$$
$67$ $$1 + 20 T + 214 T^{2} + 1340 T^{3} + 4489 T^{4}$$
$71$ $$1 + 12 T + 158 T^{2} + 852 T^{3} + 5041 T^{4}$$
$73$ $$1 + 6 T + 150 T^{2} + 438 T^{3} + 5329 T^{4}$$
$79$ $$1 + 78 T^{2} + 6241 T^{4}$$
$83$ $$1 + 4 T - 10 T^{2} + 332 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 2 T + 89 T^{2} )^{2}$$
$97$ $$1 - 8 T + 30 T^{2} - 776 T^{3} + 9409 T^{4}$$