# Properties

 Label 4928.2.a.bi Level 4928 Weight 2 Character orbit 4928.a Self dual yes Analytic conductor 39.350 Analytic rank 0 Dimension 1 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
0 2.00000 0 2.00000 0 1.00000 0 1.00000 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.bi 1
4.b odd 2 1 4928.2.a.g 1
8.b even 2 1 1232.2.a.a 1
8.d odd 2 1 77.2.a.c 1
24.f even 2 1 693.2.a.a 1
40.e odd 2 1 1925.2.a.c 1
40.k even 4 2 1925.2.b.d 2
56.e even 2 1 539.2.a.d 1
56.h odd 2 1 8624.2.a.bc 1
56.k odd 6 2 539.2.e.a 2
56.m even 6 2 539.2.e.b 2
88.g even 2 1 847.2.a.a 1
88.k even 10 4 847.2.f.k 4
88.l odd 10 4 847.2.f.e 4
168.e odd 2 1 4851.2.a.a 1
264.p odd 2 1 7623.2.a.n 1
616.g odd 2 1 5929.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.c 1 8.d odd 2 1
539.2.a.d 1 56.e even 2 1
539.2.e.a 2 56.k odd 6 2
539.2.e.b 2 56.m even 6 2
693.2.a.a 1 24.f even 2 1
847.2.a.a 1 88.g even 2 1
847.2.f.e 4 88.l odd 10 4
847.2.f.k 4 88.k even 10 4
1232.2.a.a 1 8.b even 2 1
1925.2.a.c 1 40.e odd 2 1
1925.2.b.d 2 40.k even 4 2
4851.2.a.a 1 168.e odd 2 1
4928.2.a.g 1 4.b odd 2 1
4928.2.a.bi 1 1.a even 1 1 trivial
5929.2.a.b 1 616.g odd 2 1
7623.2.a.n 1 264.p odd 2 1
8624.2.a.bc 1 56.h odd 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4928))$$:

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

## Hecke characteristic polynomials

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