# Properties

 Label 4800.2.f.x 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 2400) 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.

 $$f(q)$$ $$=$$ $$q - i q^{3} + 3 i q^{7} - q^{9} +O(q^{10})$$ q - i * q^3 + 3*i * q^7 - q^9 $$q - i q^{3} + 3 i q^{7} - q^{9} + 5 i q^{13} + 5 q^{19} + 3 q^{21} + 4 i q^{23} + i q^{27} + 4 q^{29} - 5 q^{31} - 10 i q^{37} + 5 q^{39} - 10 q^{41} - i q^{43} + 2 i q^{47} - 2 q^{49} + 10 i q^{53} - 5 i q^{57} - 10 q^{59} + 5 q^{61} - 3 i q^{63} - 3 i q^{67} + 4 q^{69} - 10 q^{71} - 10 i q^{73} + q^{81} + 14 i q^{83} - 4 i q^{87} - 16 q^{89} - 15 q^{91} + 5 i q^{93} + 5 i q^{97} +O(q^{100})$$ q - i * q^3 + 3*i * q^7 - q^9 + 5*i * q^13 + 5 * q^19 + 3 * q^21 + 4*i * q^23 + i * q^27 + 4 * q^29 - 5 * q^31 - 10*i * q^37 + 5 * q^39 - 10 * q^41 - i * q^43 + 2*i * q^47 - 2 * q^49 + 10*i * q^53 - 5*i * q^57 - 10 * q^59 + 5 * q^61 - 3*i * q^63 - 3*i * q^67 + 4 * q^69 - 10 * q^71 - 10*i * q^73 + q^81 + 14*i * q^83 - 4*i * q^87 - 16 * q^89 - 15 * q^91 + 5*i * q^93 + 5*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 10 q^{19} + 6 q^{21} + 8 q^{29} - 10 q^{31} + 10 q^{39} - 20 q^{41} - 4 q^{49} - 20 q^{59} + 10 q^{61} + 8 q^{69} - 20 q^{71} + 2 q^{81} - 32 q^{89} - 30 q^{91}+O(q^{100})$$ 2 * q - 2 * q^9 + 10 * q^19 + 6 * q^21 + 8 * q^29 - 10 * q^31 + 10 * q^39 - 20 * q^41 - 4 * q^49 - 20 * q^59 + 10 * q^61 + 8 * q^69 - 20 * q^71 + 2 * q^81 - 32 * q^89 - 30 * q^91

## 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 3.00000i 0 −1.00000 0
3649.2 0 1.00000i 0 0 0 3.00000i 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.x 2
4.b odd 2 1 4800.2.f.m 2
5.b even 2 1 inner 4800.2.f.x 2
5.c odd 4 1 4800.2.a.i 1
5.c odd 4 1 4800.2.a.cl 1
8.b even 2 1 2400.2.f.f 2
8.d odd 2 1 2400.2.f.m 2
20.d odd 2 1 4800.2.f.m 2
20.e even 4 1 4800.2.a.h 1
20.e even 4 1 4800.2.a.cm 1
24.f even 2 1 7200.2.f.t 2
24.h odd 2 1 7200.2.f.j 2
40.e odd 2 1 2400.2.f.m 2
40.f even 2 1 2400.2.f.f 2
40.i odd 4 1 2400.2.a.m yes 1
40.i odd 4 1 2400.2.a.v yes 1
40.k even 4 1 2400.2.a.l 1
40.k even 4 1 2400.2.a.w yes 1
120.i odd 2 1 7200.2.f.j 2
120.m even 2 1 7200.2.f.t 2
120.q odd 4 1 7200.2.a.h 1
120.q odd 4 1 7200.2.a.bt 1
120.w even 4 1 7200.2.a.g 1
120.w even 4 1 7200.2.a.bu 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2400.2.a.l 1 40.k even 4 1
2400.2.a.m yes 1 40.i odd 4 1
2400.2.a.v yes 1 40.i odd 4 1
2400.2.a.w yes 1 40.k even 4 1
2400.2.f.f 2 8.b even 2 1
2400.2.f.f 2 40.f even 2 1
2400.2.f.m 2 8.d odd 2 1
2400.2.f.m 2 40.e odd 2 1
4800.2.a.h 1 20.e even 4 1
4800.2.a.i 1 5.c odd 4 1
4800.2.a.cl 1 5.c odd 4 1
4800.2.a.cm 1 20.e even 4 1
4800.2.f.m 2 4.b odd 2 1
4800.2.f.m 2 20.d odd 2 1
4800.2.f.x 2 1.a even 1 1 trivial
4800.2.f.x 2 5.b even 2 1 inner
7200.2.a.g 1 120.w even 4 1
7200.2.a.h 1 120.q odd 4 1
7200.2.a.bt 1 120.q odd 4 1
7200.2.a.bu 1 120.w even 4 1
7200.2.f.j 2 24.h odd 2 1
7200.2.f.j 2 120.i odd 2 1
7200.2.f.t 2 24.f even 2 1
7200.2.f.t 2 120.m even 2 1

## Hecke kernels

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

 $$T_{7}^{2} + 9$$ T7^2 + 9 $$T_{11}$$ T11 $$T_{13}^{2} + 25$$ T13^2 + 25 $$T_{19} - 5$$ T19 - 5 $$T_{23}^{2} + 16$$ T23^2 + 16 $$T_{31} + 5$$ T31 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 25$$
$17$ $$T^{2}$$
$19$ $$(T - 5)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 4)^{2}$$
$31$ $$(T + 5)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2} + 1$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 100$$
$59$ $$(T + 10)^{2}$$
$61$ $$(T - 5)^{2}$$
$67$ $$T^{2} + 9$$
$71$ $$(T + 10)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 196$$
$89$ $$(T + 16)^{2}$$
$97$ $$T^{2} + 25$$