# Properties

 Label 4800.2.a.j 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

 $$f(q)$$ $$=$$ $$q - q^{3} - 3 q^{7} + q^{9}+O(q^{10})$$ q - q^3 - 3 * q^7 + q^9 $$q - q^{3} - 3 q^{7} + q^{9} + 4 q^{11} - 7 q^{13} - 4 q^{17} - q^{19} + 3 q^{21} - 8 q^{23} - q^{27} + 3 q^{31} - 4 q^{33} + 2 q^{37} + 7 q^{39} - 6 q^{41} + 11 q^{43} + 6 q^{47} + 2 q^{49} + 4 q^{51} + 6 q^{53} + q^{57} - 6 q^{59} + q^{61} - 3 q^{63} + 15 q^{67} + 8 q^{69} - 6 q^{71} - 2 q^{73} - 12 q^{77} + 8 q^{79} + q^{81} - 2 q^{83} - 16 q^{89} + 21 q^{91} - 3 q^{93} - 13 q^{97} + 4 q^{99}+O(q^{100})$$ q - q^3 - 3 * q^7 + q^9 + 4 * q^11 - 7 * q^13 - 4 * q^17 - q^19 + 3 * q^21 - 8 * q^23 - q^27 + 3 * q^31 - 4 * q^33 + 2 * q^37 + 7 * q^39 - 6 * q^41 + 11 * q^43 + 6 * q^47 + 2 * q^49 + 4 * q^51 + 6 * q^53 + q^57 - 6 * q^59 + q^61 - 3 * q^63 + 15 * q^67 + 8 * q^69 - 6 * q^71 - 2 * q^73 - 12 * q^77 + 8 * q^79 + q^81 - 2 * q^83 - 16 * q^89 + 21 * q^91 - 3 * q^93 - 13 * 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 −3.00000 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.j 1
4.b odd 2 1 4800.2.a.ck 1
5.b even 2 1 4800.2.a.cn 1
5.c odd 4 2 4800.2.f.be 2
8.b even 2 1 2400.2.a.u yes 1
8.d odd 2 1 2400.2.a.n yes 1
20.d odd 2 1 4800.2.a.g 1
20.e even 4 2 4800.2.f.f 2
24.f even 2 1 7200.2.a.bs 1
24.h odd 2 1 7200.2.a.i 1
40.e odd 2 1 2400.2.a.x yes 1
40.f even 2 1 2400.2.a.k 1
40.i odd 4 2 2400.2.f.b 2
40.k even 4 2 2400.2.f.q 2
120.i odd 2 1 7200.2.a.bv 1
120.m even 2 1 7200.2.a.f 1
120.q odd 4 2 7200.2.f.e 2
120.w even 4 2 7200.2.f.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2400.2.a.k 1 40.f even 2 1
2400.2.a.n yes 1 8.d odd 2 1
2400.2.a.u yes 1 8.b even 2 1
2400.2.a.x yes 1 40.e odd 2 1
2400.2.f.b 2 40.i odd 4 2
2400.2.f.q 2 40.k even 4 2
4800.2.a.g 1 20.d odd 2 1
4800.2.a.j 1 1.a even 1 1 trivial
4800.2.a.ck 1 4.b odd 2 1
4800.2.a.cn 1 5.b even 2 1
4800.2.f.f 2 20.e even 4 2
4800.2.f.be 2 5.c odd 4 2
7200.2.a.f 1 120.m even 2 1
7200.2.a.i 1 24.h odd 2 1
7200.2.a.bs 1 24.f even 2 1
7200.2.a.bv 1 120.i odd 2 1
7200.2.f.e 2 120.q odd 4 2
7200.2.f.y 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4800))$$:

 $$T_{7} + 3$$ T7 + 3 $$T_{11} - 4$$ T11 - 4 $$T_{13} + 7$$ T13 + 7 $$T_{19} + 1$$ T19 + 1 $$T_{23} + 8$$ T23 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 3$$
$11$ $$T - 4$$
$13$ $$T + 7$$
$17$ $$T + 4$$
$19$ $$T + 1$$
$23$ $$T + 8$$
$29$ $$T$$
$31$ $$T - 3$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T - 11$$
$47$ $$T - 6$$
$53$ $$T - 6$$
$59$ $$T + 6$$
$61$ $$T - 1$$
$67$ $$T - 15$$
$71$ $$T + 6$$
$73$ $$T + 2$$
$79$ $$T - 8$$
$83$ $$T + 2$$
$89$ $$T + 16$$
$97$ $$T + 13$$