# Properties

 Label 4800.2.f.bf 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: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} - q^{9} +O(q^{10})$$ $$q + i q^{3} - q^{9} + 4 q^{11} + 2 i q^{13} -2 i q^{17} + 4 q^{19} -i q^{27} -2 q^{29} + 4 i q^{33} -10 i q^{37} -2 q^{39} + 10 q^{41} -4 i q^{43} -8 i q^{47} + 7 q^{49} + 2 q^{51} + 10 i q^{53} + 4 i q^{57} -4 q^{59} + 2 q^{61} + 12 i q^{67} -8 q^{71} + 10 i q^{73} + q^{81} -12 i q^{83} -2 i q^{87} + 6 q^{89} -2 i q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} + 8q^{11} + 8q^{19} - 4q^{29} - 4q^{39} + 20q^{41} + 14q^{49} + 4q^{51} - 8q^{59} + 4q^{61} - 16q^{71} + 2q^{81} + 12q^{89} - 8q^{99} + O(q^{100})$$

## 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.bf 2
4.b odd 2 1 4800.2.f.c 2
5.b even 2 1 inner 4800.2.f.bf 2
5.c odd 4 1 960.2.a.l 1
5.c odd 4 1 4800.2.a.t 1
8.b even 2 1 75.2.b.b 2
8.d odd 2 1 1200.2.f.h 2
15.e even 4 1 2880.2.a.y 1
20.d odd 2 1 4800.2.f.c 2
20.e even 4 1 960.2.a.a 1
20.e even 4 1 4800.2.a.bz 1
24.f even 2 1 3600.2.f.e 2
24.h odd 2 1 225.2.b.b 2
40.e odd 2 1 1200.2.f.h 2
40.f even 2 1 75.2.b.b 2
40.i odd 4 1 15.2.a.a 1
40.i odd 4 1 75.2.a.b 1
40.k even 4 1 240.2.a.d 1
40.k even 4 1 1200.2.a.e 1
60.l odd 4 1 2880.2.a.bc 1
80.i odd 4 1 3840.2.k.m 2
80.j even 4 1 3840.2.k.r 2
80.s even 4 1 3840.2.k.r 2
80.t odd 4 1 3840.2.k.m 2
120.i odd 2 1 225.2.b.b 2
120.m even 2 1 3600.2.f.e 2
120.q odd 4 1 720.2.a.c 1
120.q odd 4 1 3600.2.a.u 1
120.w even 4 1 45.2.a.a 1
120.w even 4 1 225.2.a.b 1
280.s even 4 1 735.2.a.c 1
280.s even 4 1 3675.2.a.j 1
280.bt odd 12 2 735.2.i.e 2
280.bv even 12 2 735.2.i.d 2
360.br even 12 2 405.2.e.c 2
360.bu odd 12 2 405.2.e.f 2
440.t even 4 1 1815.2.a.d 1
440.t even 4 1 9075.2.a.g 1
520.bg odd 4 1 2535.2.a.j 1
680.bi odd 4 1 4335.2.a.c 1
760.t even 4 1 5415.2.a.j 1
840.bp odd 4 1 2205.2.a.i 1
920.x even 4 1 7935.2.a.d 1
1320.bn odd 4 1 5445.2.a.c 1
1560.bq even 4 1 7605.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 40.i odd 4 1
45.2.a.a 1 120.w even 4 1
75.2.a.b 1 40.i odd 4 1
75.2.b.b 2 8.b even 2 1
75.2.b.b 2 40.f even 2 1
225.2.a.b 1 120.w even 4 1
225.2.b.b 2 24.h odd 2 1
225.2.b.b 2 120.i odd 2 1
240.2.a.d 1 40.k even 4 1
405.2.e.c 2 360.br even 12 2
405.2.e.f 2 360.bu odd 12 2
720.2.a.c 1 120.q odd 4 1
735.2.a.c 1 280.s even 4 1
735.2.i.d 2 280.bv even 12 2
735.2.i.e 2 280.bt odd 12 2
960.2.a.a 1 20.e even 4 1
960.2.a.l 1 5.c odd 4 1
1200.2.a.e 1 40.k even 4 1
1200.2.f.h 2 8.d odd 2 1
1200.2.f.h 2 40.e odd 2 1
1815.2.a.d 1 440.t even 4 1
2205.2.a.i 1 840.bp odd 4 1
2535.2.a.j 1 520.bg odd 4 1
2880.2.a.y 1 15.e even 4 1
2880.2.a.bc 1 60.l odd 4 1
3600.2.a.u 1 120.q odd 4 1
3600.2.f.e 2 24.f even 2 1
3600.2.f.e 2 120.m even 2 1
3675.2.a.j 1 280.s even 4 1
3840.2.k.m 2 80.i odd 4 1
3840.2.k.m 2 80.t odd 4 1
3840.2.k.r 2 80.j even 4 1
3840.2.k.r 2 80.s even 4 1
4335.2.a.c 1 680.bi odd 4 1
4800.2.a.t 1 5.c odd 4 1
4800.2.a.bz 1 20.e even 4 1
4800.2.f.c 2 4.b odd 2 1
4800.2.f.c 2 20.d odd 2 1
4800.2.f.bf 2 1.a even 1 1 trivial
4800.2.f.bf 2 5.b even 2 1 inner
5415.2.a.j 1 760.t even 4 1
5445.2.a.c 1 1320.bn odd 4 1
7605.2.a.g 1 1560.bq even 4 1
7935.2.a.d 1 920.x even 4 1
9075.2.a.g 1 440.t even 4 1

## Hecke kernels

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

 $$T_{7}$$ $$T_{11} - 4$$ $$T_{13}^{2} + 4$$ $$T_{19} - 4$$ $$T_{23}$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$100 + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$4 + T^{2}$$