# Properties

 Label 1200.4.a.be Level $1200$ Weight $4$ Character orbit 1200.a Self dual yes Analytic conductor $70.802$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$70.8022920069$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 12) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} + 8q^{7} + 9q^{9} + O(q^{10})$$ $$q + 3q^{3} + 8q^{7} + 9q^{9} - 36q^{11} + 10q^{13} - 18q^{17} + 100q^{19} + 24q^{21} + 72q^{23} + 27q^{27} - 234q^{29} + 16q^{31} - 108q^{33} + 226q^{37} + 30q^{39} + 90q^{41} + 452q^{43} + 432q^{47} - 279q^{49} - 54q^{51} - 414q^{53} + 300q^{57} + 684q^{59} + 422q^{61} + 72q^{63} + 332q^{67} + 216q^{69} + 360q^{71} - 26q^{73} - 288q^{77} - 512q^{79} + 81q^{81} - 1188q^{83} - 702q^{87} - 630q^{89} + 80q^{91} + 48q^{93} + 1054q^{97} - 324q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 3.00000 0 0 0 8.00000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.4.a.be 1
4.b odd 2 1 300.4.a.b 1
5.b even 2 1 48.4.a.a 1
5.c odd 4 2 1200.4.f.d 2
12.b even 2 1 900.4.a.g 1
15.d odd 2 1 144.4.a.g 1
20.d odd 2 1 12.4.a.a 1
20.e even 4 2 300.4.d.e 2
35.c odd 2 1 2352.4.a.bk 1
40.e odd 2 1 192.4.a.f 1
40.f even 2 1 192.4.a.l 1
60.h even 2 1 36.4.a.a 1
60.l odd 4 2 900.4.d.c 2
80.k odd 4 2 768.4.d.g 2
80.q even 4 2 768.4.d.j 2
120.i odd 2 1 576.4.a.a 1
120.m even 2 1 576.4.a.b 1
140.c even 2 1 588.4.a.c 1
140.p odd 6 2 588.4.i.d 2
140.s even 6 2 588.4.i.e 2
180.n even 6 2 324.4.e.a 2
180.p odd 6 2 324.4.e.h 2
220.g even 2 1 1452.4.a.d 1
260.g odd 2 1 2028.4.a.c 1
260.u even 4 2 2028.4.b.c 2
420.o odd 2 1 1764.4.a.b 1
420.ba even 6 2 1764.4.k.b 2
420.be odd 6 2 1764.4.k.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 20.d odd 2 1
36.4.a.a 1 60.h even 2 1
48.4.a.a 1 5.b even 2 1
144.4.a.g 1 15.d odd 2 1
192.4.a.f 1 40.e odd 2 1
192.4.a.l 1 40.f even 2 1
300.4.a.b 1 4.b odd 2 1
300.4.d.e 2 20.e even 4 2
324.4.e.a 2 180.n even 6 2
324.4.e.h 2 180.p odd 6 2
576.4.a.a 1 120.i odd 2 1
576.4.a.b 1 120.m even 2 1
588.4.a.c 1 140.c even 2 1
588.4.i.d 2 140.p odd 6 2
588.4.i.e 2 140.s even 6 2
768.4.d.g 2 80.k odd 4 2
768.4.d.j 2 80.q even 4 2
900.4.a.g 1 12.b even 2 1
900.4.d.c 2 60.l odd 4 2
1200.4.a.be 1 1.a even 1 1 trivial
1200.4.f.d 2 5.c odd 4 2
1452.4.a.d 1 220.g even 2 1
1764.4.a.b 1 420.o odd 2 1
1764.4.k.b 2 420.ba even 6 2
1764.4.k.o 2 420.be odd 6 2
2028.4.a.c 1 260.g odd 2 1
2028.4.b.c 2 260.u even 4 2
2352.4.a.bk 1 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1200))$$:

 $$T_{7} - 8$$ $$T_{11} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$T$$
$7$ $$-8 + T$$
$11$ $$36 + T$$
$13$ $$-10 + T$$
$17$ $$18 + T$$
$19$ $$-100 + T$$
$23$ $$-72 + T$$
$29$ $$234 + T$$
$31$ $$-16 + T$$
$37$ $$-226 + T$$
$41$ $$-90 + T$$
$43$ $$-452 + T$$
$47$ $$-432 + T$$
$53$ $$414 + T$$
$59$ $$-684 + T$$
$61$ $$-422 + T$$
$67$ $$-332 + T$$
$71$ $$-360 + T$$
$73$ $$26 + T$$
$79$ $$512 + T$$
$83$ $$1188 + T$$
$89$ $$630 + T$$
$97$ $$-1054 + T$$