# Properties

 Label 4800.2.a.i Level $4800$ Weight $2$ Character orbit 4800.a Self dual yes Analytic conductor $38.328$ Analytic rank $1$ 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: $$1$$ 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 $$q - q^{3} - 3 q^{7} + q^{9} + 5 q^{13} - 5 q^{19} + 3 q^{21} + 4 q^{23} - q^{27} - 4 q^{29} - 5 q^{31} + 10 q^{37} - 5 q^{39} - 10 q^{41} - q^{43} - 2 q^{47} + 2 q^{49} + 10 q^{53} + 5 q^{57} + 10 q^{59} + 5 q^{61} - 3 q^{63} + 3 q^{67} - 4 q^{69} - 10 q^{71} - 10 q^{73} + q^{81} + 14 q^{83} + 4 q^{87} + 16 q^{89} - 15 q^{91} + 5 q^{93} - 5 q^{97}+O(q^{100})$$ q - q^3 - 3 * q^7 + q^9 + 5 * q^13 - 5 * q^19 + 3 * q^21 + 4 * q^23 - q^27 - 4 * q^29 - 5 * q^31 + 10 * q^37 - 5 * q^39 - 10 * q^41 - q^43 - 2 * q^47 + 2 * q^49 + 10 * q^53 + 5 * q^57 + 10 * q^59 + 5 * q^61 - 3 * q^63 + 3 * q^67 - 4 * q^69 - 10 * q^71 - 10 * q^73 + q^81 + 14 * q^83 + 4 * q^87 + 16 * q^89 - 15 * q^91 + 5 * q^93 - 5 * q^97

## 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.i 1
4.b odd 2 1 4800.2.a.cm 1
5.b even 2 1 4800.2.a.cl 1
5.c odd 4 2 4800.2.f.x 2
8.b even 2 1 2400.2.a.v yes 1
8.d odd 2 1 2400.2.a.l 1
20.d odd 2 1 4800.2.a.h 1
20.e even 4 2 4800.2.f.m 2
24.f even 2 1 7200.2.a.bt 1
24.h odd 2 1 7200.2.a.g 1
40.e odd 2 1 2400.2.a.w yes 1
40.f even 2 1 2400.2.a.m yes 1
40.i odd 4 2 2400.2.f.f 2
40.k even 4 2 2400.2.f.m 2
120.i odd 2 1 7200.2.a.bu 1
120.m even 2 1 7200.2.a.h 1
120.q odd 4 2 7200.2.f.t 2
120.w even 4 2 7200.2.f.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2400.2.a.l 1 8.d odd 2 1
2400.2.a.m yes 1 40.f even 2 1
2400.2.a.v yes 1 8.b even 2 1
2400.2.a.w yes 1 40.e odd 2 1
2400.2.f.f 2 40.i odd 4 2
2400.2.f.m 2 40.k even 4 2
4800.2.a.h 1 20.d odd 2 1
4800.2.a.i 1 1.a even 1 1 trivial
4800.2.a.cl 1 5.b even 2 1
4800.2.a.cm 1 4.b odd 2 1
4800.2.f.m 2 20.e even 4 2
4800.2.f.x 2 5.c odd 4 2
7200.2.a.g 1 24.h odd 2 1
7200.2.a.h 1 120.m even 2 1
7200.2.a.bt 1 24.f even 2 1
7200.2.a.bu 1 120.i odd 2 1
7200.2.f.j 2 120.w even 4 2
7200.2.f.t 2 120.q odd 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 $$T_{11}$$ T11 $$T_{13} - 5$$ T13 - 5 $$T_{19} + 5$$ T19 + 5 $$T_{23} - 4$$ T23 - 4

## Hecke characteristic polynomials

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