# Properties

 Label 4800.2.f.l Level $4800$ Weight $2$ Character orbit 4800.f Analytic conductor $38.328$ Analytic rank $1$ 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: $$1$$ 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 75) 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 $$q + i q^{3} + 3 i q^{7} - q^{9} - 2 q^{11} - i q^{13} - 2 i q^{17} - 5 q^{19} - 3 q^{21} + 6 i q^{23} - i q^{27} + 10 q^{29} - 3 q^{31} - 2 i q^{33} + 2 i q^{37} + q^{39} - 8 q^{41} - i q^{43} - 2 i q^{47} - 2 q^{49} + 2 q^{51} + 4 i q^{53} - 5 i q^{57} - 10 q^{59} - 7 q^{61} - 3 i q^{63} - 3 i q^{67} - 6 q^{69} - 8 q^{71} - 14 i q^{73} - 6 i q^{77} + q^{81} - 6 i q^{83} + 10 i q^{87} + 3 q^{91} - 3 i q^{93} - 17 i q^{97} + 2 q^{99} +O(q^{100})$$ q + i * q^3 + 3*i * q^7 - q^9 - 2 * q^11 - i * q^13 - 2*i * q^17 - 5 * q^19 - 3 * q^21 + 6*i * q^23 - i * q^27 + 10 * q^29 - 3 * q^31 - 2*i * q^33 + 2*i * q^37 + q^39 - 8 * q^41 - i * q^43 - 2*i * q^47 - 2 * q^49 + 2 * q^51 + 4*i * q^53 - 5*i * q^57 - 10 * q^59 - 7 * q^61 - 3*i * q^63 - 3*i * q^67 - 6 * q^69 - 8 * q^71 - 14*i * q^73 - 6*i * q^77 + q^81 - 6*i * q^83 + 10*i * q^87 + 3 * q^91 - 3*i * q^93 - 17*i * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 4 q^{11} - 10 q^{19} - 6 q^{21} + 20 q^{29} - 6 q^{31} + 2 q^{39} - 16 q^{41} - 4 q^{49} + 4 q^{51} - 20 q^{59} - 14 q^{61} - 12 q^{69} - 16 q^{71} + 2 q^{81} + 6 q^{91} + 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 - 4 * q^11 - 10 * q^19 - 6 * q^21 + 20 * q^29 - 6 * q^31 + 2 * q^39 - 16 * q^41 - 4 * q^49 + 4 * q^51 - 20 * q^59 - 14 * q^61 - 12 * q^69 - 16 * q^71 + 2 * q^81 + 6 * q^91 + 4 * 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.l 2
4.b odd 2 1 4800.2.f.y 2
5.b even 2 1 inner 4800.2.f.l 2
5.c odd 4 1 4800.2.a.bb 1
5.c odd 4 1 4800.2.a.bq 1
8.b even 2 1 75.2.b.a 2
8.d odd 2 1 1200.2.f.d 2
20.d odd 2 1 4800.2.f.y 2
20.e even 4 1 4800.2.a.be 1
20.e even 4 1 4800.2.a.br 1
24.f even 2 1 3600.2.f.p 2
24.h odd 2 1 225.2.b.a 2
40.e odd 2 1 1200.2.f.d 2
40.f even 2 1 75.2.b.a 2
40.i odd 4 1 75.2.a.a 1
40.i odd 4 1 75.2.a.c yes 1
40.k even 4 1 1200.2.a.c 1
40.k even 4 1 1200.2.a.p 1
120.i odd 2 1 225.2.b.a 2
120.m even 2 1 3600.2.f.p 2
120.q odd 4 1 3600.2.a.j 1
120.q odd 4 1 3600.2.a.bk 1
120.w even 4 1 225.2.a.a 1
120.w even 4 1 225.2.a.e 1
280.s even 4 1 3675.2.a.b 1
280.s even 4 1 3675.2.a.q 1
440.t even 4 1 9075.2.a.a 1
440.t even 4 1 9075.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.a.a 1 40.i odd 4 1
75.2.a.c yes 1 40.i odd 4 1
75.2.b.a 2 8.b even 2 1
75.2.b.a 2 40.f even 2 1
225.2.a.a 1 120.w even 4 1
225.2.a.e 1 120.w even 4 1
225.2.b.a 2 24.h odd 2 1
225.2.b.a 2 120.i odd 2 1
1200.2.a.c 1 40.k even 4 1
1200.2.a.p 1 40.k even 4 1
1200.2.f.d 2 8.d odd 2 1
1200.2.f.d 2 40.e odd 2 1
3600.2.a.j 1 120.q odd 4 1
3600.2.a.bk 1 120.q odd 4 1
3600.2.f.p 2 24.f even 2 1
3600.2.f.p 2 120.m even 2 1
3675.2.a.b 1 280.s even 4 1
3675.2.a.q 1 280.s even 4 1
4800.2.a.bb 1 5.c odd 4 1
4800.2.a.be 1 20.e even 4 1
4800.2.a.bq 1 5.c odd 4 1
4800.2.a.br 1 20.e even 4 1
4800.2.f.l 2 1.a even 1 1 trivial
4800.2.f.l 2 5.b even 2 1 inner
4800.2.f.y 2 4.b odd 2 1
4800.2.f.y 2 20.d odd 2 1
9075.2.a.a 1 440.t even 4 1
9075.2.a.s 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 $$S_{2}^{\mathrm{new}}(4800, [\chi])$$:

 $$T_{7}^{2} + 9$$ T7^2 + 9 $$T_{11} + 2$$ T11 + 2 $$T_{13}^{2} + 1$$ T13^2 + 1 $$T_{19} + 5$$ T19 + 5 $$T_{23}^{2} + 36$$ T23^2 + 36 $$T_{31} + 3$$ T31 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 4$$
$19$ $$(T + 5)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 10)^{2}$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2} + 1$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 16$$
$59$ $$(T + 10)^{2}$$
$61$ $$(T + 7)^{2}$$
$67$ $$T^{2} + 9$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 289$$