# Properties

 Label 4800.2.f.f 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 $$q - i q^{3} + 3 i q^{7} - q^{9} - 4 q^{11} + 7 i q^{13} - 4 i q^{17} - q^{19} + 3 q^{21} - 8 i q^{23} + i q^{27} - 3 q^{31} + 4 i q^{33} + 2 i q^{37} + 7 q^{39} - 6 q^{41} + 11 i q^{43} - 6 i q^{47} - 2 q^{49} - 4 q^{51} - 6 i q^{53} + i q^{57} - 6 q^{59} + q^{61} - 3 i q^{63} - 15 i q^{67} - 8 q^{69} + 6 q^{71} + 2 i q^{73} - 12 i q^{77} + 8 q^{79} + q^{81} - 2 i q^{83} + 16 q^{89} - 21 q^{91} + 3 i q^{93} - 13 i q^{97} + 4 q^{99} +O(q^{100})$$ q - i * q^3 + 3*i * q^7 - q^9 - 4 * q^11 + 7*i * q^13 - 4*i * q^17 - q^19 + 3 * q^21 - 8*i * q^23 + i * q^27 - 3 * q^31 + 4*i * q^33 + 2*i * q^37 + 7 * q^39 - 6 * q^41 + 11*i * q^43 - 6*i * q^47 - 2 * q^49 - 4 * q^51 - 6*i * q^53 + i * q^57 - 6 * q^59 + q^61 - 3*i * q^63 - 15*i * q^67 - 8 * q^69 + 6 * q^71 + 2*i * q^73 - 12*i * q^77 + 8 * q^79 + q^81 - 2*i * q^83 + 16 * q^89 - 21 * q^91 + 3*i * q^93 - 13*i * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 8 q^{11} - 2 q^{19} + 6 q^{21} - 6 q^{31} + 14 q^{39} - 12 q^{41} - 4 q^{49} - 8 q^{51} - 12 q^{59} + 2 q^{61} - 16 q^{69} + 12 q^{71} + 16 q^{79} + 2 q^{81} + 32 q^{89} - 42 q^{91} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 - 8 * q^11 - 2 * q^19 + 6 * q^21 - 6 * q^31 + 14 * q^39 - 12 * q^41 - 4 * q^49 - 8 * q^51 - 12 * q^59 + 2 * q^61 - 16 * q^69 + 12 * q^71 + 16 * q^79 + 2 * q^81 + 32 * q^89 - 42 * q^91 + 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 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.f 2
4.b odd 2 1 4800.2.f.be 2
5.b even 2 1 inner 4800.2.f.f 2
5.c odd 4 1 4800.2.a.g 1
5.c odd 4 1 4800.2.a.ck 1
8.b even 2 1 2400.2.f.q 2
8.d odd 2 1 2400.2.f.b 2
20.d odd 2 1 4800.2.f.be 2
20.e even 4 1 4800.2.a.j 1
20.e even 4 1 4800.2.a.cn 1
24.f even 2 1 7200.2.f.y 2
24.h odd 2 1 7200.2.f.e 2
40.e odd 2 1 2400.2.f.b 2
40.f even 2 1 2400.2.f.q 2
40.i odd 4 1 2400.2.a.n yes 1
40.i odd 4 1 2400.2.a.x yes 1
40.k even 4 1 2400.2.a.k 1
40.k even 4 1 2400.2.a.u yes 1
120.i odd 2 1 7200.2.f.e 2
120.m even 2 1 7200.2.f.y 2
120.q odd 4 1 7200.2.a.i 1
120.q odd 4 1 7200.2.a.bv 1
120.w even 4 1 7200.2.a.f 1
120.w even 4 1 7200.2.a.bs 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2400.2.a.k 1 40.k even 4 1
2400.2.a.n yes 1 40.i odd 4 1
2400.2.a.u yes 1 40.k even 4 1
2400.2.a.x yes 1 40.i odd 4 1
2400.2.f.b 2 8.d odd 2 1
2400.2.f.b 2 40.e odd 2 1
2400.2.f.q 2 8.b even 2 1
2400.2.f.q 2 40.f even 2 1
4800.2.a.g 1 5.c odd 4 1
4800.2.a.j 1 20.e even 4 1
4800.2.a.ck 1 5.c odd 4 1
4800.2.a.cn 1 20.e even 4 1
4800.2.f.f 2 1.a even 1 1 trivial
4800.2.f.f 2 5.b even 2 1 inner
4800.2.f.be 2 4.b odd 2 1
4800.2.f.be 2 20.d odd 2 1
7200.2.a.f 1 120.w even 4 1
7200.2.a.i 1 120.q odd 4 1
7200.2.a.bs 1 120.w even 4 1
7200.2.a.bv 1 120.q odd 4 1
7200.2.f.e 2 24.h odd 2 1
7200.2.f.e 2 120.i odd 2 1
7200.2.f.y 2 24.f even 2 1
7200.2.f.y 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 $$T_{11} + 4$$ T11 + 4 $$T_{13}^{2} + 49$$ T13^2 + 49 $$T_{19} + 1$$ T19 + 1 $$T_{23}^{2} + 64$$ T23^2 + 64 $$T_{31} + 3$$ T31 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 49$$
$17$ $$T^{2} + 16$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 64$$
$29$ $$T^{2}$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 121$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 6)^{2}$$
$61$ $$(T - 1)^{2}$$
$67$ $$T^{2} + 225$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 4$$
$89$ $$(T - 16)^{2}$$
$97$ $$T^{2} + 169$$