# Properties

 Label 4800.2.a.by Level $4800$ Weight $2$ Character orbit 4800.a Self dual yes Analytic conductor $38.328$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4800 = 2^{6} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.3281929702$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2400) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{7} + q^{9}+O(q^{10})$$ q + q^3 - q^7 + q^9 $$q + q^{3} - q^{7} + q^{9} + 4 q^{11} - 3 q^{13} + 4 q^{17} + q^{19} - q^{21} + q^{27} + 8 q^{29} + q^{31} + 4 q^{33} + 2 q^{37} - 3 q^{39} + 2 q^{41} - 11 q^{43} + 2 q^{47} - 6 q^{49} + 4 q^{51} - 10 q^{53} + q^{57} + 6 q^{59} - 11 q^{61} - q^{63} + 9 q^{67} + 6 q^{71} + 14 q^{73} - 4 q^{77} + 16 q^{79} + q^{81} + 2 q^{83} + 8 q^{87} + 3 q^{91} + q^{93} + 11 q^{97} + 4 q^{99}+O(q^{100})$$ q + q^3 - q^7 + q^9 + 4 * q^11 - 3 * q^13 + 4 * q^17 + q^19 - q^21 + q^27 + 8 * q^29 + q^31 + 4 * q^33 + 2 * q^37 - 3 * q^39 + 2 * q^41 - 11 * q^43 + 2 * q^47 - 6 * q^49 + 4 * q^51 - 10 * q^53 + q^57 + 6 * q^59 - 11 * q^61 - q^63 + 9 * q^67 + 6 * q^71 + 14 * q^73 - 4 * q^77 + 16 * q^79 + q^81 + 2 * q^83 + 8 * q^87 + 3 * q^91 + q^93 + 11 * q^97 + 4 * q^99

## 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 1.00000 0 0 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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$

## 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 4800.2.a.by 1
4.b odd 2 1 4800.2.a.v 1
5.b even 2 1 4800.2.a.y 1
5.c odd 4 2 4800.2.f.bd 2
8.b even 2 1 2400.2.a.e 1
8.d odd 2 1 2400.2.a.bd yes 1
20.d odd 2 1 4800.2.a.bv 1
20.e even 4 2 4800.2.f.g 2
24.f even 2 1 7200.2.a.bh 1
24.h odd 2 1 7200.2.a.t 1
40.e odd 2 1 2400.2.a.h yes 1
40.f even 2 1 2400.2.a.ba yes 1
40.i odd 4 2 2400.2.f.c 2
40.k even 4 2 2400.2.f.p 2
120.i odd 2 1 7200.2.a.bk 1
120.m even 2 1 7200.2.a.q 1
120.q odd 4 2 7200.2.f.d 2
120.w even 4 2 7200.2.f.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2400.2.a.e 1 8.b even 2 1
2400.2.a.h yes 1 40.e odd 2 1
2400.2.a.ba yes 1 40.f even 2 1
2400.2.a.bd yes 1 8.d odd 2 1
2400.2.f.c 2 40.i odd 4 2
2400.2.f.p 2 40.k even 4 2
4800.2.a.v 1 4.b odd 2 1
4800.2.a.y 1 5.b even 2 1
4800.2.a.bv 1 20.d odd 2 1
4800.2.a.by 1 1.a even 1 1 trivial
4800.2.f.g 2 20.e even 4 2
4800.2.f.bd 2 5.c odd 4 2
7200.2.a.q 1 120.m even 2 1
7200.2.a.t 1 24.h odd 2 1
7200.2.a.bh 1 24.f even 2 1
7200.2.a.bk 1 120.i odd 2 1
7200.2.f.d 2 120.q odd 4 2
7200.2.f.z 2 120.w even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4800))$$:

 $$T_{7} + 1$$ T7 + 1 $$T_{11} - 4$$ T11 - 4 $$T_{13} + 3$$ T13 + 3 $$T_{19} - 1$$ T19 - 1 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T - 4$$
$13$ $$T + 3$$
$17$ $$T - 4$$
$19$ $$T - 1$$
$23$ $$T$$
$29$ $$T - 8$$
$31$ $$T - 1$$
$37$ $$T - 2$$
$41$ $$T - 2$$
$43$ $$T + 11$$
$47$ $$T - 2$$
$53$ $$T + 10$$
$59$ $$T - 6$$
$61$ $$T + 11$$
$67$ $$T - 9$$
$71$ $$T - 6$$
$73$ $$T - 14$$
$79$ $$T - 16$$
$83$ $$T - 2$$
$89$ $$T$$
$97$ $$T - 11$$