Properties

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

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{9} + O(q^{10})$$ $$q - q^{3} + q^{9} + 4q^{11} + 2q^{13} + 2q^{17} + 8q^{19} + 4q^{23} - q^{27} + 6q^{29} - 4q^{33} + 2q^{37} - 2q^{39} - 6q^{41} - 4q^{43} - 12q^{47} - 7q^{49} - 2q^{51} - 6q^{53} - 8q^{57} + 12q^{59} - 14q^{61} + 12q^{67} - 4q^{69} - 2q^{73} + 8q^{79} + q^{81} + 4q^{83} - 6q^{87} + 2q^{89} + 14q^{97} + 4q^{99} + O(q^{100})$$

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 0 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.u 1
4.b odd 2 1 4800.2.a.ca 1
5.b even 2 1 960.2.a.o 1
5.c odd 4 2 4800.2.f.ba 2
8.b even 2 1 2400.2.a.y 1
8.d odd 2 1 2400.2.a.j 1
15.d odd 2 1 2880.2.a.i 1
20.d odd 2 1 960.2.a.f 1
20.e even 4 2 4800.2.f.j 2
24.f even 2 1 7200.2.a.u 1
24.h odd 2 1 7200.2.a.bg 1
40.e odd 2 1 480.2.a.e yes 1
40.f even 2 1 480.2.a.b 1
40.i odd 4 2 2400.2.f.e 2
40.k even 4 2 2400.2.f.n 2
60.h even 2 1 2880.2.a.j 1
80.k odd 4 2 3840.2.k.k 2
80.q even 4 2 3840.2.k.p 2
120.i odd 2 1 1440.2.a.k 1
120.m even 2 1 1440.2.a.j 1
120.q odd 4 2 7200.2.f.b 2
120.w even 4 2 7200.2.f.bb 2

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

 $$T_{7}$$ $$T_{11} - 4$$ $$T_{13} - 2$$ $$T_{19} - 8$$ $$T_{23} - 4$$

Hecke characteristic polynomials

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