Properties

 Label 4928.2.a.bk 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 154) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{3} + ( -1 - \beta ) q^{5} - q^{7} + ( 3 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{3} + ( -1 - \beta ) q^{5} - q^{7} + ( 3 + 2 \beta ) q^{9} + q^{11} + ( 1 - \beta ) q^{13} + ( 6 + 2 \beta ) q^{15} + ( -2 - 2 \beta ) q^{17} + ( -5 + \beta ) q^{19} + ( 1 + \beta ) q^{21} -4 q^{23} + ( 1 + 2 \beta ) q^{25} + ( -10 - 2 \beta ) q^{27} + 2 \beta q^{29} -2 q^{31} + ( -1 - \beta ) q^{33} + ( 1 + \beta ) q^{35} + ( 2 + 4 \beta ) q^{37} + 4 q^{39} + ( 2 + 2 \beta ) q^{41} + ( -6 + 2 \beta ) q^{43} + ( -13 - 5 \beta ) q^{45} + 2 q^{47} + q^{49} + ( 12 + 4 \beta ) q^{51} + ( -4 + 2 \beta ) q^{53} + ( -1 - \beta ) q^{55} + 4 \beta q^{57} + ( 5 + \beta ) q^{59} + ( 3 + \beta ) q^{61} + ( -3 - 2 \beta ) q^{63} + 4 q^{65} + ( -2 - 6 \beta ) q^{67} + ( 4 + 4 \beta ) q^{69} + ( -2 + 2 \beta ) q^{71} + ( 4 - 4 \beta ) q^{73} + ( -11 - 3 \beta ) q^{75} - q^{77} + ( 11 + 6 \beta ) q^{81} + ( -1 + 5 \beta ) q^{83} + ( 12 + 4 \beta ) q^{85} + ( -10 - 2 \beta ) q^{87} + 10 q^{89} + ( -1 + \beta ) q^{91} + ( 2 + 2 \beta ) q^{93} + 4 \beta q^{95} + ( 8 - 2 \beta ) q^{97} + ( 3 + 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 6 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 6 q^{9} + 2 q^{11} + 2 q^{13} + 12 q^{15} - 4 q^{17} - 10 q^{19} + 2 q^{21} - 8 q^{23} + 2 q^{25} - 20 q^{27} - 4 q^{31} - 2 q^{33} + 2 q^{35} + 4 q^{37} + 8 q^{39} + 4 q^{41} - 12 q^{43} - 26 q^{45} + 4 q^{47} + 2 q^{49} + 24 q^{51} - 8 q^{53} - 2 q^{55} + 10 q^{59} + 6 q^{61} - 6 q^{63} + 8 q^{65} - 4 q^{67} + 8 q^{69} - 4 q^{71} + 8 q^{73} - 22 q^{75} - 2 q^{77} + 22 q^{81} - 2 q^{83} + 24 q^{85} - 20 q^{87} + 20 q^{89} - 2 q^{91} + 4 q^{93} + 16 q^{97} + 6 q^{99} + O(q^{100})$$

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 −3.23607 0 −1.00000 0 7.47214 0
1.2 0 1.23607 0 1.23607 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

Atkin-Lehner signs

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

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 4928.2.a.bk 2
4.b odd 2 1 4928.2.a.bt 2
8.b even 2 1 1232.2.a.p 2
8.d odd 2 1 154.2.a.d 2
24.f even 2 1 1386.2.a.m 2
40.e odd 2 1 3850.2.a.bj 2
40.k even 4 2 3850.2.c.q 4
56.e even 2 1 1078.2.a.w 2
56.h odd 2 1 8624.2.a.bf 2
56.k odd 6 2 1078.2.e.q 4
56.m even 6 2 1078.2.e.n 4
88.g even 2 1 1694.2.a.l 2
168.e odd 2 1 9702.2.a.cu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 8.d odd 2 1
1078.2.a.w 2 56.e even 2 1
1078.2.e.n 4 56.m even 6 2
1078.2.e.q 4 56.k odd 6 2
1232.2.a.p 2 8.b even 2 1
1386.2.a.m 2 24.f even 2 1
1694.2.a.l 2 88.g even 2 1
3850.2.a.bj 2 40.e odd 2 1
3850.2.c.q 4 40.k even 4 2
4928.2.a.bk 2 1.a even 1 1 trivial
4928.2.a.bt 2 4.b odd 2 1
8624.2.a.bf 2 56.h odd 2 1
9702.2.a.cu 2 168.e odd 2 1

Hecke kernels

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

 $$T_{3}^{2} + 2 T_{3} - 4$$ $$T_{5}^{2} + 2 T_{5} - 4$$ $$T_{13}^{2} - 2 T_{13} - 4$$ $$T_{17}^{2} + 4 T_{17} - 16$$ $$T_{19}^{2} + 10 T_{19} + 20$$ $$T_{23} + 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-4 + 2 T + T^{2}$$
$5$ $$-4 + 2 T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-4 - 2 T + T^{2}$$
$17$ $$-16 + 4 T + T^{2}$$
$19$ $$20 + 10 T + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$-20 + T^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$-76 - 4 T + T^{2}$$
$41$ $$-16 - 4 T + T^{2}$$
$43$ $$16 + 12 T + T^{2}$$
$47$ $$( -2 + T )^{2}$$
$53$ $$-4 + 8 T + T^{2}$$
$59$ $$20 - 10 T + T^{2}$$
$61$ $$4 - 6 T + T^{2}$$
$67$ $$-176 + 4 T + T^{2}$$
$71$ $$-16 + 4 T + T^{2}$$
$73$ $$-64 - 8 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$-124 + 2 T + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$44 - 16 T + T^{2}$$