# Properties

 Label 4800.2.f.z 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 600) 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 $$q + i q^{3} + 3 i q^{7} - q^{9} + 2 q^{11} + 3 i q^{13} - 6 i q^{17} + 7 q^{19} - 3 q^{21} - 6 i q^{23} - i q^{27} - 2 q^{29} + 5 q^{31} + 2 i q^{33} + 10 i q^{37} - 3 q^{39} + 12 q^{41} + 3 i q^{43} - 10 i q^{47} - 2 q^{49} + 6 q^{51} + 7 i q^{57} + 6 q^{59} + 13 q^{61} - 3 i q^{63} - 7 i q^{67} + 6 q^{69} + 4 q^{71} - 6 i q^{73} + 6 i q^{77} - 8 q^{79} + q^{81} - 6 i q^{83} - 2 i q^{87} - 16 q^{89} - 9 q^{91} + 5 i q^{93} + 7 i q^{97} - 2 q^{99} +O(q^{100})$$ q + i * q^3 + 3*i * q^7 - q^9 + 2 * q^11 + 3*i * q^13 - 6*i * q^17 + 7 * q^19 - 3 * q^21 - 6*i * q^23 - i * q^27 - 2 * q^29 + 5 * q^31 + 2*i * q^33 + 10*i * q^37 - 3 * q^39 + 12 * q^41 + 3*i * q^43 - 10*i * q^47 - 2 * q^49 + 6 * q^51 + 7*i * q^57 + 6 * q^59 + 13 * q^61 - 3*i * q^63 - 7*i * q^67 + 6 * q^69 + 4 * q^71 - 6*i * q^73 + 6*i * q^77 - 8 * q^79 + q^81 - 6*i * q^83 - 2*i * q^87 - 16 * q^89 - 9 * q^91 + 5*i * q^93 + 7*i * q^97 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 4 q^{11} + 14 q^{19} - 6 q^{21} - 4 q^{29} + 10 q^{31} - 6 q^{39} + 24 q^{41} - 4 q^{49} + 12 q^{51} + 12 q^{59} + 26 q^{61} + 12 q^{69} + 8 q^{71} - 16 q^{79} + 2 q^{81} - 32 q^{89} - 18 q^{91} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 + 4 * q^11 + 14 * q^19 - 6 * q^21 - 4 * q^29 + 10 * q^31 - 6 * q^39 + 24 * q^41 - 4 * q^49 + 12 * q^51 + 12 * q^59 + 26 * q^61 + 12 * q^69 + 8 * q^71 - 16 * q^79 + 2 * q^81 - 32 * q^89 - 18 * 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.z 2
4.b odd 2 1 4800.2.f.k 2
5.b even 2 1 inner 4800.2.f.z 2
5.c odd 4 1 4800.2.a.bd 1
5.c odd 4 1 4800.2.a.bs 1
8.b even 2 1 1200.2.f.c 2
8.d odd 2 1 600.2.f.d 2
20.d odd 2 1 4800.2.f.k 2
20.e even 4 1 4800.2.a.bc 1
20.e even 4 1 4800.2.a.bp 1
24.f even 2 1 1800.2.f.e 2
24.h odd 2 1 3600.2.f.o 2
40.e odd 2 1 600.2.f.d 2
40.f even 2 1 1200.2.f.c 2
40.i odd 4 1 1200.2.a.b 1
40.i odd 4 1 1200.2.a.q 1
40.k even 4 1 600.2.a.b 1
40.k even 4 1 600.2.a.i yes 1
120.i odd 2 1 3600.2.f.o 2
120.m even 2 1 1800.2.f.e 2
120.q odd 4 1 1800.2.a.e 1
120.q odd 4 1 1800.2.a.t 1
120.w even 4 1 3600.2.a.i 1
120.w even 4 1 3600.2.a.bl 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.a.b 1 40.k even 4 1
600.2.a.i yes 1 40.k even 4 1
600.2.f.d 2 8.d odd 2 1
600.2.f.d 2 40.e odd 2 1
1200.2.a.b 1 40.i odd 4 1
1200.2.a.q 1 40.i odd 4 1
1200.2.f.c 2 8.b even 2 1
1200.2.f.c 2 40.f even 2 1
1800.2.a.e 1 120.q odd 4 1
1800.2.a.t 1 120.q odd 4 1
1800.2.f.e 2 24.f even 2 1
1800.2.f.e 2 120.m even 2 1
3600.2.a.i 1 120.w even 4 1
3600.2.a.bl 1 120.w even 4 1
3600.2.f.o 2 24.h odd 2 1
3600.2.f.o 2 120.i odd 2 1
4800.2.a.bc 1 20.e even 4 1
4800.2.a.bd 1 5.c odd 4 1
4800.2.a.bp 1 20.e even 4 1
4800.2.a.bs 1 5.c odd 4 1
4800.2.f.k 2 4.b odd 2 1
4800.2.f.k 2 20.d odd 2 1
4800.2.f.z 2 1.a even 1 1 trivial
4800.2.f.z 2 5.b even 2 1 inner

## 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 $$T_{13}^{2} + 9$$ T13^2 + 9 $$T_{19} - 7$$ T19 - 7 $$T_{23}^{2} + 36$$ T23^2 + 36 $$T_{31} - 5$$ T31 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 9$$
$17$ $$T^{2} + 36$$
$19$ $$(T - 7)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T + 2)^{2}$$
$31$ $$(T - 5)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T - 12)^{2}$$
$43$ $$T^{2} + 9$$
$47$ $$T^{2} + 100$$
$53$ $$T^{2}$$
$59$ $$(T - 6)^{2}$$
$61$ $$(T - 13)^{2}$$
$67$ $$T^{2} + 49$$
$71$ $$(T - 4)^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T + 16)^{2}$$
$97$ $$T^{2} + 49$$