# Properties

 Label 1200.2.a.s Level $1200$ Weight $2$ Character orbit 1200.a Self dual yes Analytic conductor $9.582$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.58204824255$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 60) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + 4q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} + 4q^{7} + q^{9} + 4q^{11} - 4q^{17} + 4q^{21} + 4q^{23} + q^{27} - 6q^{29} - 4q^{31} + 4q^{33} + 8q^{37} - 10q^{41} + 4q^{43} - 4q^{47} + 9q^{49} - 4q^{51} + 12q^{53} - 4q^{59} + 2q^{61} + 4q^{63} - 4q^{67} + 4q^{69} + 8q^{73} + 16q^{77} + 12q^{79} + q^{81} + 4q^{83} - 6q^{87} - 10q^{89} - 4q^{93} - 8q^{97} + 4q^{99} + O(q^{100})$$

## 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 1.00000 0 0 0 4.00000 0 1.00000 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 1200.2.a.s 1
3.b odd 2 1 3600.2.a.bm 1
4.b odd 2 1 300.2.a.a 1
5.b even 2 1 1200.2.a.a 1
5.c odd 4 2 240.2.f.b 2
8.b even 2 1 4800.2.a.bf 1
8.d odd 2 1 4800.2.a.bn 1
12.b even 2 1 900.2.a.a 1
15.d odd 2 1 3600.2.a.d 1
15.e even 4 2 720.2.f.c 2
20.d odd 2 1 300.2.a.d 1
20.e even 4 2 60.2.d.a 2
40.e odd 2 1 4800.2.a.bj 1
40.f even 2 1 4800.2.a.bk 1
40.i odd 4 2 960.2.f.c 2
40.k even 4 2 960.2.f.f 2
60.h even 2 1 900.2.a.h 1
60.l odd 4 2 180.2.d.a 2
80.i odd 4 2 3840.2.d.b 2
80.j even 4 2 3840.2.d.o 2
80.s even 4 2 3840.2.d.r 2
80.t odd 4 2 3840.2.d.be 2
120.q odd 4 2 2880.2.f.l 2
120.w even 4 2 2880.2.f.p 2
140.j odd 4 2 2940.2.k.c 2
140.w even 12 4 2940.2.bb.d 4
140.x odd 12 4 2940.2.bb.e 4
180.v odd 12 4 1620.2.r.d 4
180.x even 12 4 1620.2.r.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.d.a 2 20.e even 4 2
180.2.d.a 2 60.l odd 4 2
240.2.f.b 2 5.c odd 4 2
300.2.a.a 1 4.b odd 2 1
300.2.a.d 1 20.d odd 2 1
720.2.f.c 2 15.e even 4 2
900.2.a.a 1 12.b even 2 1
900.2.a.h 1 60.h even 2 1
960.2.f.c 2 40.i odd 4 2
960.2.f.f 2 40.k even 4 2
1200.2.a.a 1 5.b even 2 1
1200.2.a.s 1 1.a even 1 1 trivial
1620.2.r.c 4 180.x even 12 4
1620.2.r.d 4 180.v odd 12 4
2880.2.f.l 2 120.q odd 4 2
2880.2.f.p 2 120.w even 4 2
2940.2.k.c 2 140.j odd 4 2
2940.2.bb.d 4 140.w even 12 4
2940.2.bb.e 4 140.x odd 12 4
3600.2.a.d 1 15.d odd 2 1
3600.2.a.bm 1 3.b odd 2 1
3840.2.d.b 2 80.i odd 4 2
3840.2.d.o 2 80.j even 4 2
3840.2.d.r 2 80.s even 4 2
3840.2.d.be 2 80.t odd 4 2
4800.2.a.bf 1 8.b even 2 1
4800.2.a.bj 1 40.e odd 2 1
4800.2.a.bk 1 40.f even 2 1
4800.2.a.bn 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1200))$$:

 $$T_{7} - 4$$ $$T_{11} - 4$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T$$
$5$ 1
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 - 4 T + 11 T^{2}$$
$13$ $$1 + 13 T^{2}$$
$17$ $$1 + 4 T + 17 T^{2}$$
$19$ $$1 + 19 T^{2}$$
$23$ $$1 - 4 T + 23 T^{2}$$
$29$ $$1 + 6 T + 29 T^{2}$$
$31$ $$1 + 4 T + 31 T^{2}$$
$37$ $$1 - 8 T + 37 T^{2}$$
$41$ $$1 + 10 T + 41 T^{2}$$
$43$ $$1 - 4 T + 43 T^{2}$$
$47$ $$1 + 4 T + 47 T^{2}$$
$53$ $$1 - 12 T + 53 T^{2}$$
$59$ $$1 + 4 T + 59 T^{2}$$
$61$ $$1 - 2 T + 61 T^{2}$$
$67$ $$1 + 4 T + 67 T^{2}$$
$71$ $$1 + 71 T^{2}$$
$73$ $$1 - 8 T + 73 T^{2}$$
$79$ $$1 - 12 T + 79 T^{2}$$
$83$ $$1 - 4 T + 83 T^{2}$$
$89$ $$1 + 10 T + 89 T^{2}$$
$97$ $$1 + 8 T + 97 T^{2}$$