# Properties

 Label 4800.2.a.bg 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 480) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + 4 q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} + 4 q^{7} + q^{9} - 4 q^{11} + 6 q^{13} - 2 q^{17} - 4 q^{19} - 4 q^{21} - q^{27} - 10 q^{29} - 4 q^{31} + 4 q^{33} - 10 q^{37} - 6 q^{39} + 2 q^{41} - 4 q^{43} - 8 q^{47} + 9 q^{49} + 2 q^{51} + 2 q^{53} + 4 q^{57} - 12 q^{59} + 10 q^{61} + 4 q^{63} + 12 q^{67} - 10 q^{73} - 16 q^{77} - 4 q^{79} + q^{81} + 4 q^{83} + 10 q^{87} - 6 q^{89} + 24 q^{91} + 4 q^{93} + 14 q^{97} - 4 q^{99} + O(q^{100})$$

## 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 −1.00000 0 0 0 4.00000 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.bg 1
4.b odd 2 1 4800.2.a.bo 1
5.b even 2 1 960.2.a.m 1
5.c odd 4 2 4800.2.f.h 2
8.b even 2 1 2400.2.a.bh 1
8.d odd 2 1 2400.2.a.a 1
15.d odd 2 1 2880.2.a.c 1
20.d odd 2 1 960.2.a.h 1
20.e even 4 2 4800.2.f.bb 2
24.f even 2 1 7200.2.a.d 1
24.h odd 2 1 7200.2.a.bw 1
40.e odd 2 1 480.2.a.f yes 1
40.f even 2 1 480.2.a.a 1
40.i odd 4 2 2400.2.f.o 2
40.k even 4 2 2400.2.f.d 2
60.h even 2 1 2880.2.a.p 1
80.k odd 4 2 3840.2.k.c 2
80.q even 4 2 3840.2.k.bb 2
120.i odd 2 1 1440.2.a.g 1
120.m even 2 1 1440.2.a.n 1
120.q odd 4 2 7200.2.f.ba 2
120.w even 4 2 7200.2.f.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.a.a 1 40.f even 2 1
480.2.a.f yes 1 40.e odd 2 1
960.2.a.h 1 20.d odd 2 1
960.2.a.m 1 5.b even 2 1
1440.2.a.g 1 120.i odd 2 1
1440.2.a.n 1 120.m even 2 1
2400.2.a.a 1 8.d odd 2 1
2400.2.a.bh 1 8.b even 2 1
2400.2.f.d 2 40.k even 4 2
2400.2.f.o 2 40.i odd 4 2
2880.2.a.c 1 15.d odd 2 1
2880.2.a.p 1 60.h even 2 1
3840.2.k.c 2 80.k odd 4 2
3840.2.k.bb 2 80.q even 4 2
4800.2.a.bg 1 1.a even 1 1 trivial
4800.2.a.bo 1 4.b odd 2 1
4800.2.f.h 2 5.c odd 4 2
4800.2.f.bb 2 20.e even 4 2
7200.2.a.d 1 24.f even 2 1
7200.2.a.bw 1 24.h odd 2 1
7200.2.f.c 2 120.w even 4 2
7200.2.f.ba 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4800))$$:

 $$T_{7} - 4$$ $$T_{11} + 4$$ $$T_{13} - 6$$ $$T_{19} + 4$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$-4 + T$$
$11$ $$4 + T$$
$13$ $$-6 + T$$
$17$ $$2 + T$$
$19$ $$4 + T$$
$23$ $$T$$
$29$ $$10 + T$$
$31$ $$4 + T$$
$37$ $$10 + T$$
$41$ $$-2 + T$$
$43$ $$4 + T$$
$47$ $$8 + T$$
$53$ $$-2 + T$$
$59$ $$12 + T$$
$61$ $$-10 + T$$
$67$ $$-12 + T$$
$71$ $$T$$
$73$ $$10 + T$$
$79$ $$4 + T$$
$83$ $$-4 + T$$
$89$ $$6 + T$$
$97$ $$-14 + T$$