# Properties

 Label 4800.2.f.e 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 96) 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.

 $$f(q)$$ $$=$$ $$q - i q^{3} + 4 i q^{7} - q^{9} +O(q^{10})$$ q - i * q^3 + 4*i * q^7 - q^9 $$q - i q^{3} + 4 i q^{7} - q^{9} - 4 q^{11} + 2 i q^{13} + 6 i q^{17} - 4 q^{19} + 4 q^{21} + i q^{27} + 2 q^{29} + 4 q^{31} + 4 i q^{33} - 2 i q^{37} + 2 q^{39} + 2 q^{41} - 4 i q^{43} - 8 i q^{47} - 9 q^{49} + 6 q^{51} - 10 i q^{53} + 4 i q^{57} - 4 q^{59} - 6 q^{61} - 4 i q^{63} + 4 i q^{67} - 16 q^{71} - 6 i q^{73} - 16 i q^{77} - 4 q^{79} + q^{81} - 12 i q^{83} - 2 i q^{87} - 10 q^{89} - 8 q^{91} - 4 i q^{93} + 14 i q^{97} + 4 q^{99} +O(q^{100})$$ q - i * q^3 + 4*i * q^7 - q^9 - 4 * q^11 + 2*i * q^13 + 6*i * q^17 - 4 * q^19 + 4 * q^21 + i * q^27 + 2 * q^29 + 4 * q^31 + 4*i * q^33 - 2*i * q^37 + 2 * q^39 + 2 * q^41 - 4*i * q^43 - 8*i * q^47 - 9 * q^49 + 6 * q^51 - 10*i * q^53 + 4*i * q^57 - 4 * q^59 - 6 * q^61 - 4*i * q^63 + 4*i * q^67 - 16 * q^71 - 6*i * q^73 - 16*i * q^77 - 4 * q^79 + q^81 - 12*i * q^83 - 2*i * q^87 - 10 * q^89 - 8 * q^91 - 4*i * q^93 + 14*i * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 8 q^{11} - 8 q^{19} + 8 q^{21} + 4 q^{29} + 8 q^{31} + 4 q^{39} + 4 q^{41} - 18 q^{49} + 12 q^{51} - 8 q^{59} - 12 q^{61} - 32 q^{71} - 8 q^{79} + 2 q^{81} - 20 q^{89} - 16 q^{91} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 - 8 * q^11 - 8 * q^19 + 8 * q^21 + 4 * q^29 + 8 * q^31 + 4 * q^39 + 4 * q^41 - 18 * q^49 + 12 * q^51 - 8 * q^59 - 12 * q^61 - 32 * q^71 - 8 * q^79 + 2 * q^81 - 20 * q^89 - 16 * 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 4.00000i 0 −1.00000 0
3649.2 0 1.00000i 0 0 0 4.00000i 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.e 2
4.b odd 2 1 4800.2.f.bh 2
5.b even 2 1 inner 4800.2.f.e 2
5.c odd 4 1 192.2.a.a 1
5.c odd 4 1 4800.2.a.co 1
8.b even 2 1 2400.2.f.r 2
8.d odd 2 1 2400.2.f.a 2
15.e even 4 1 576.2.a.g 1
20.d odd 2 1 4800.2.f.bh 2
20.e even 4 1 192.2.a.c 1
20.e even 4 1 4800.2.a.f 1
24.f even 2 1 7200.2.f.x 2
24.h odd 2 1 7200.2.f.f 2
35.f even 4 1 9408.2.a.ct 1
40.e odd 2 1 2400.2.f.a 2
40.f even 2 1 2400.2.f.r 2
40.i odd 4 1 96.2.a.b yes 1
40.i odd 4 1 2400.2.a.q 1
40.k even 4 1 96.2.a.a 1
40.k even 4 1 2400.2.a.r 1
60.l odd 4 1 576.2.a.h 1
80.i odd 4 1 768.2.d.h 2
80.j even 4 1 768.2.d.a 2
80.s even 4 1 768.2.d.a 2
80.t odd 4 1 768.2.d.h 2
120.i odd 2 1 7200.2.f.f 2
120.m even 2 1 7200.2.f.x 2
120.q odd 4 1 288.2.a.c 1
120.q odd 4 1 7200.2.a.e 1
120.w even 4 1 288.2.a.b 1
120.w even 4 1 7200.2.a.bx 1
140.j odd 4 1 9408.2.a.bj 1
240.z odd 4 1 2304.2.d.c 2
240.bb even 4 1 2304.2.d.s 2
240.bd odd 4 1 2304.2.d.c 2
240.bf even 4 1 2304.2.d.s 2
280.s even 4 1 4704.2.a.e 1
280.y odd 4 1 4704.2.a.t 1
360.bo even 12 2 2592.2.i.b 2
360.br even 12 2 2592.2.i.w 2
360.bt odd 12 2 2592.2.i.q 2
360.bu odd 12 2 2592.2.i.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.a.a 1 40.k even 4 1
96.2.a.b yes 1 40.i odd 4 1
192.2.a.a 1 5.c odd 4 1
192.2.a.c 1 20.e even 4 1
288.2.a.b 1 120.w even 4 1
288.2.a.c 1 120.q odd 4 1
576.2.a.g 1 15.e even 4 1
576.2.a.h 1 60.l odd 4 1
768.2.d.a 2 80.j even 4 1
768.2.d.a 2 80.s even 4 1
768.2.d.h 2 80.i odd 4 1
768.2.d.h 2 80.t odd 4 1
2304.2.d.c 2 240.z odd 4 1
2304.2.d.c 2 240.bd odd 4 1
2304.2.d.s 2 240.bb even 4 1
2304.2.d.s 2 240.bf even 4 1
2400.2.a.q 1 40.i odd 4 1
2400.2.a.r 1 40.k even 4 1
2400.2.f.a 2 8.d odd 2 1
2400.2.f.a 2 40.e odd 2 1
2400.2.f.r 2 8.b even 2 1
2400.2.f.r 2 40.f even 2 1
2592.2.i.b 2 360.bo even 12 2
2592.2.i.h 2 360.bu odd 12 2
2592.2.i.q 2 360.bt odd 12 2
2592.2.i.w 2 360.br even 12 2
4704.2.a.e 1 280.s even 4 1
4704.2.a.t 1 280.y odd 4 1
4800.2.a.f 1 20.e even 4 1
4800.2.a.co 1 5.c odd 4 1
4800.2.f.e 2 1.a even 1 1 trivial
4800.2.f.e 2 5.b even 2 1 inner
4800.2.f.bh 2 4.b odd 2 1
4800.2.f.bh 2 20.d odd 2 1
7200.2.a.e 1 120.q odd 4 1
7200.2.a.bx 1 120.w even 4 1
7200.2.f.f 2 24.h odd 2 1
7200.2.f.f 2 120.i odd 2 1
7200.2.f.x 2 24.f even 2 1
7200.2.f.x 2 120.m even 2 1
9408.2.a.bj 1 140.j odd 4 1
9408.2.a.ct 1 35.f even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} + 4$$ T11 + 4 $$T_{13}^{2} + 4$$ T13^2 + 4 $$T_{19} + 4$$ T19 + 4 $$T_{23}$$ T23 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 36$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 100$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T + 6)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T + 16)^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 196$$