# Properties

 Label 4800.2.f.d Level $4800$ Weight $2$ Character orbit 4800.f Analytic conductor $38.328$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4800 = 2^{6} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4800.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$38.3281929702$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} - q^{9}+O(q^{10})$$ q + i * q^3 - q^9 $$q + i q^{3} - q^{9} - 4 q^{11} + 2 i q^{13} - 2 i q^{17} - 4 q^{19} - 8 i q^{23} - i q^{27} + 6 q^{29} + 8 q^{31} - 4 i q^{33} + 6 i q^{37} - 2 q^{39} - 6 q^{41} - 4 i q^{43} + 7 q^{49} + 2 q^{51} + 2 i q^{53} - 4 i q^{57} + 4 q^{59} + 2 q^{61} - 4 i q^{67} + 8 q^{69} + 8 q^{71} + 10 i q^{73} + 8 q^{79} + q^{81} + 4 i q^{83} + 6 i q^{87} + 6 q^{89} + 8 i q^{93} - 2 i q^{97} + 4 q^{99} +O(q^{100})$$ q + i * q^3 - q^9 - 4 * q^11 + 2*i * q^13 - 2*i * q^17 - 4 * q^19 - 8*i * q^23 - i * q^27 + 6 * q^29 + 8 * q^31 - 4*i * q^33 + 6*i * q^37 - 2 * q^39 - 6 * q^41 - 4*i * q^43 + 7 * q^49 + 2 * q^51 + 2*i * q^53 - 4*i * q^57 + 4 * q^59 + 2 * q^61 - 4*i * q^67 + 8 * q^69 + 8 * q^71 + 10*i * q^73 + 8 * q^79 + q^81 + 4*i * q^83 + 6*i * q^87 + 6 * q^89 + 8*i * q^93 - 2*i * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 8 q^{11} - 8 q^{19} + 12 q^{29} + 16 q^{31} - 4 q^{39} - 12 q^{41} + 14 q^{49} + 4 q^{51} + 8 q^{59} + 4 q^{61} + 16 q^{69} + 16 q^{71} + 16 q^{79} + 2 q^{81} + 12 q^{89} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 - 8 * q^11 - 8 * q^19 + 12 * q^29 + 16 * q^31 - 4 * q^39 - 12 * q^41 + 14 * q^49 + 4 * q^51 + 8 * q^59 + 4 * q^61 + 16 * q^69 + 16 * q^71 + 16 * q^79 + 2 * q^81 + 12 * q^89 + 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1601$$ $$4351$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3649.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 0 0 −1.00000 0
3649.2 0 1.00000i 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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4800.2.f.d 2
4.b odd 2 1 4800.2.f.bg 2
5.b even 2 1 inner 4800.2.f.d 2
5.c odd 4 1 192.2.a.d 1
5.c odd 4 1 4800.2.a.q 1
8.b even 2 1 600.2.f.e 2
8.d odd 2 1 1200.2.f.b 2
15.e even 4 1 576.2.a.d 1
20.d odd 2 1 4800.2.f.bg 2
20.e even 4 1 192.2.a.b 1
20.e even 4 1 4800.2.a.cc 1
24.f even 2 1 3600.2.f.r 2
24.h odd 2 1 1800.2.f.c 2
35.f even 4 1 9408.2.a.h 1
40.e odd 2 1 1200.2.f.b 2
40.f even 2 1 600.2.f.e 2
40.i odd 4 1 24.2.a.a 1
40.i odd 4 1 600.2.a.h 1
40.k even 4 1 48.2.a.a 1
40.k even 4 1 1200.2.a.d 1
60.l odd 4 1 576.2.a.b 1
80.i odd 4 1 768.2.d.e 2
80.j even 4 1 768.2.d.d 2
80.s even 4 1 768.2.d.d 2
80.t odd 4 1 768.2.d.e 2
120.i odd 2 1 1800.2.f.c 2
120.m even 2 1 3600.2.f.r 2
120.q odd 4 1 144.2.a.b 1
120.q odd 4 1 3600.2.a.v 1
120.w even 4 1 72.2.a.a 1
120.w even 4 1 1800.2.a.m 1
140.j odd 4 1 9408.2.a.cc 1
240.z odd 4 1 2304.2.d.k 2
240.bb even 4 1 2304.2.d.i 2
240.bd odd 4 1 2304.2.d.k 2
240.bf even 4 1 2304.2.d.i 2
280.s even 4 1 1176.2.a.i 1
280.y odd 4 1 2352.2.a.i 1
280.bp odd 12 2 2352.2.q.r 2
280.br even 12 2 2352.2.q.l 2
280.bt odd 12 2 1176.2.q.i 2
280.bv even 12 2 1176.2.q.a 2
360.bo even 12 2 1296.2.i.m 2
360.br even 12 2 648.2.i.b 2
360.bt odd 12 2 1296.2.i.e 2
360.bu odd 12 2 648.2.i.g 2
440.t even 4 1 2904.2.a.c 1
440.w odd 4 1 5808.2.a.s 1
520.y even 4 1 4056.2.c.e 2
520.bc even 4 1 8112.2.a.be 1
520.bg odd 4 1 4056.2.a.i 1
520.bj even 4 1 4056.2.c.e 2
680.bi odd 4 1 6936.2.a.p 1
760.t even 4 1 8664.2.a.j 1
840.bm even 4 1 7056.2.a.q 1
840.bp odd 4 1 3528.2.a.d 1
840.dc even 12 2 3528.2.s.j 2
840.dh odd 12 2 3528.2.s.y 2
1320.bn odd 4 1 8712.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 40.i odd 4 1
48.2.a.a 1 40.k even 4 1
72.2.a.a 1 120.w even 4 1
144.2.a.b 1 120.q odd 4 1
192.2.a.b 1 20.e even 4 1
192.2.a.d 1 5.c odd 4 1
576.2.a.b 1 60.l odd 4 1
576.2.a.d 1 15.e even 4 1
600.2.a.h 1 40.i odd 4 1
600.2.f.e 2 8.b even 2 1
600.2.f.e 2 40.f even 2 1
648.2.i.b 2 360.br even 12 2
648.2.i.g 2 360.bu odd 12 2
768.2.d.d 2 80.j even 4 1
768.2.d.d 2 80.s even 4 1
768.2.d.e 2 80.i odd 4 1
768.2.d.e 2 80.t odd 4 1
1176.2.a.i 1 280.s even 4 1
1176.2.q.a 2 280.bv even 12 2
1176.2.q.i 2 280.bt odd 12 2
1200.2.a.d 1 40.k even 4 1
1200.2.f.b 2 8.d odd 2 1
1200.2.f.b 2 40.e odd 2 1
1296.2.i.e 2 360.bt odd 12 2
1296.2.i.m 2 360.bo even 12 2
1800.2.a.m 1 120.w even 4 1
1800.2.f.c 2 24.h odd 2 1
1800.2.f.c 2 120.i odd 2 1
2304.2.d.i 2 240.bb even 4 1
2304.2.d.i 2 240.bf even 4 1
2304.2.d.k 2 240.z odd 4 1
2304.2.d.k 2 240.bd odd 4 1
2352.2.a.i 1 280.y odd 4 1
2352.2.q.l 2 280.br even 12 2
2352.2.q.r 2 280.bp odd 12 2
2904.2.a.c 1 440.t even 4 1
3528.2.a.d 1 840.bp odd 4 1
3528.2.s.j 2 840.dc even 12 2
3528.2.s.y 2 840.dh odd 12 2
3600.2.a.v 1 120.q odd 4 1
3600.2.f.r 2 24.f even 2 1
3600.2.f.r 2 120.m even 2 1
4056.2.a.i 1 520.bg odd 4 1
4056.2.c.e 2 520.y even 4 1
4056.2.c.e 2 520.bj even 4 1
4800.2.a.q 1 5.c odd 4 1
4800.2.a.cc 1 20.e even 4 1
4800.2.f.d 2 1.a even 1 1 trivial
4800.2.f.d 2 5.b even 2 1 inner
4800.2.f.bg 2 4.b odd 2 1
4800.2.f.bg 2 20.d odd 2 1
5808.2.a.s 1 440.w odd 4 1
6936.2.a.p 1 680.bi odd 4 1
7056.2.a.q 1 840.bm even 4 1
8112.2.a.be 1 520.bc even 4 1
8664.2.a.j 1 760.t even 4 1
8712.2.a.u 1 1320.bn odd 4 1
9408.2.a.h 1 35.f even 4 1
9408.2.a.cc 1 140.j odd 4 1

## Hecke kernels

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

 $$T_{7}$$ T7 $$T_{11} + 4$$ T11 + 4 $$T_{13}^{2} + 4$$ T13^2 + 4 $$T_{19} + 4$$ T19 + 4 $$T_{23}^{2} + 64$$ T23^2 + 64 $$T_{31} - 8$$ T31 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 4$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 64$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 4$$
$59$ $$(T - 4)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 4$$