# Properties

 Label 4800.2.a.cc 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 24) 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 $$q + q^{3} + q^{9} + 4 q^{11} - 2 q^{13} - 2 q^{17} - 4 q^{19} - 8 q^{23} + q^{27} - 6 q^{29} - 8 q^{31} + 4 q^{33} + 6 q^{37} - 2 q^{39} - 6 q^{41} - 4 q^{43} - 7 q^{49} - 2 q^{51} - 2 q^{53} - 4 q^{57} + 4 q^{59} + 2 q^{61} + 4 q^{67} - 8 q^{69} - 8 q^{71} - 10 q^{73} + 8 q^{79} + q^{81} + 4 q^{83} - 6 q^{87} - 6 q^{89} - 8 q^{93} - 2 q^{97} + 4 q^{99}+O(q^{100})$$ q + q^3 + q^9 + 4 * q^11 - 2 * q^13 - 2 * q^17 - 4 * q^19 - 8 * q^23 + q^27 - 6 * q^29 - 8 * q^31 + 4 * q^33 + 6 * q^37 - 2 * q^39 - 6 * q^41 - 4 * q^43 - 7 * q^49 - 2 * q^51 - 2 * q^53 - 4 * q^57 + 4 * q^59 + 2 * q^61 + 4 * q^67 - 8 * q^69 - 8 * q^71 - 10 * q^73 + 8 * q^79 + q^81 + 4 * q^83 - 6 * q^87 - 6 * q^89 - 8 * q^93 - 2 * 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 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.cc 1
4.b odd 2 1 4800.2.a.q 1
5.b even 2 1 192.2.a.b 1
5.c odd 4 2 4800.2.f.bg 2
8.b even 2 1 1200.2.a.d 1
8.d odd 2 1 600.2.a.h 1
15.d odd 2 1 576.2.a.b 1
20.d odd 2 1 192.2.a.d 1
20.e even 4 2 4800.2.f.d 2
24.f even 2 1 1800.2.a.m 1
24.h odd 2 1 3600.2.a.v 1
35.c odd 2 1 9408.2.a.cc 1
40.e odd 2 1 24.2.a.a 1
40.f even 2 1 48.2.a.a 1
40.i odd 4 2 1200.2.f.b 2
40.k even 4 2 600.2.f.e 2
60.h even 2 1 576.2.a.d 1
80.k odd 4 2 768.2.d.e 2
80.q even 4 2 768.2.d.d 2
120.i odd 2 1 144.2.a.b 1
120.m even 2 1 72.2.a.a 1
120.q odd 4 2 1800.2.f.c 2
120.w even 4 2 3600.2.f.r 2
140.c even 2 1 9408.2.a.h 1
240.t even 4 2 2304.2.d.i 2
240.bm odd 4 2 2304.2.d.k 2
280.c odd 2 1 2352.2.a.i 1
280.n even 2 1 1176.2.a.i 1
280.ba even 6 2 1176.2.q.a 2
280.bf even 6 2 2352.2.q.l 2
280.bi odd 6 2 1176.2.q.i 2
280.bk odd 6 2 2352.2.q.r 2
360.z odd 6 2 648.2.i.g 2
360.bd even 6 2 648.2.i.b 2
360.bh odd 6 2 1296.2.i.e 2
360.bk even 6 2 1296.2.i.m 2
440.c even 2 1 2904.2.a.c 1
440.o odd 2 1 5808.2.a.s 1
520.b odd 2 1 4056.2.a.i 1
520.p even 2 1 8112.2.a.be 1
520.t even 4 2 4056.2.c.e 2
680.k odd 2 1 6936.2.a.p 1
760.p even 2 1 8664.2.a.j 1
840.b odd 2 1 3528.2.a.d 1
840.u even 2 1 7056.2.a.q 1
840.ct odd 6 2 3528.2.s.y 2
840.cv even 6 2 3528.2.s.j 2
1320.b odd 2 1 8712.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 40.e odd 2 1
48.2.a.a 1 40.f even 2 1
72.2.a.a 1 120.m even 2 1
144.2.a.b 1 120.i odd 2 1
192.2.a.b 1 5.b even 2 1
192.2.a.d 1 20.d odd 2 1
576.2.a.b 1 15.d odd 2 1
576.2.a.d 1 60.h even 2 1
600.2.a.h 1 8.d odd 2 1
600.2.f.e 2 40.k even 4 2
648.2.i.b 2 360.bd even 6 2
648.2.i.g 2 360.z odd 6 2
768.2.d.d 2 80.q even 4 2
768.2.d.e 2 80.k odd 4 2
1176.2.a.i 1 280.n even 2 1
1176.2.q.a 2 280.ba even 6 2
1176.2.q.i 2 280.bi odd 6 2
1200.2.a.d 1 8.b even 2 1
1200.2.f.b 2 40.i odd 4 2
1296.2.i.e 2 360.bh odd 6 2
1296.2.i.m 2 360.bk even 6 2
1800.2.a.m 1 24.f even 2 1
1800.2.f.c 2 120.q odd 4 2
2304.2.d.i 2 240.t even 4 2
2304.2.d.k 2 240.bm odd 4 2
2352.2.a.i 1 280.c odd 2 1
2352.2.q.l 2 280.bf even 6 2
2352.2.q.r 2 280.bk odd 6 2
2904.2.a.c 1 440.c even 2 1
3528.2.a.d 1 840.b odd 2 1
3528.2.s.j 2 840.cv even 6 2
3528.2.s.y 2 840.ct odd 6 2
3600.2.a.v 1 24.h odd 2 1
3600.2.f.r 2 120.w even 4 2
4056.2.a.i 1 520.b odd 2 1
4056.2.c.e 2 520.t even 4 2
4800.2.a.q 1 4.b odd 2 1
4800.2.a.cc 1 1.a even 1 1 trivial
4800.2.f.d 2 20.e even 4 2
4800.2.f.bg 2 5.c odd 4 2
5808.2.a.s 1 440.o odd 2 1
6936.2.a.p 1 680.k odd 2 1
7056.2.a.q 1 840.u even 2 1
8112.2.a.be 1 520.p even 2 1
8664.2.a.j 1 760.p even 2 1
8712.2.a.u 1 1320.b odd 2 1
9408.2.a.h 1 140.c even 2 1
9408.2.a.cc 1 35.c odd 2 1

## Hecke kernels

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

 $$T_{7}$$ T7 $$T_{11} - 4$$ T11 - 4 $$T_{13} + 2$$ T13 + 2 $$T_{19} + 4$$ T19 + 4 $$T_{23} + 8$$ T23 + 8

## Hecke characteristic polynomials

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