# Properties

 Label 1200.2.a.r 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 120) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + 4q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} + 4q^{7} + q^{9} + 6q^{13} + 2q^{17} - 4q^{19} + 4q^{21} - 8q^{23} + q^{27} - 6q^{29} + 6q^{37} + 6q^{39} + 10q^{41} - 4q^{43} + 8q^{47} + 9q^{49} + 2q^{51} - 10q^{53} - 4q^{57} + 6q^{61} + 4q^{63} - 4q^{67} - 8q^{69} + 14q^{73} - 16q^{79} + q^{81} + 12q^{83} - 6q^{87} + 2q^{89} + 24q^{91} - 2q^{97} + O(q^{100})$$

## 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.r 1
3.b odd 2 1 3600.2.a.bo 1
4.b odd 2 1 600.2.a.a 1
5.b even 2 1 240.2.a.a 1
5.c odd 4 2 1200.2.f.f 2
8.b even 2 1 4800.2.a.bh 1
8.d odd 2 1 4800.2.a.bl 1
12.b even 2 1 1800.2.a.c 1
15.d odd 2 1 720.2.a.f 1
15.e even 4 2 3600.2.f.l 2
20.d odd 2 1 120.2.a.a 1
20.e even 4 2 600.2.f.c 2
40.e odd 2 1 960.2.a.g 1
40.f even 2 1 960.2.a.n 1
40.i odd 4 2 4800.2.f.n 2
40.k even 4 2 4800.2.f.u 2
60.h even 2 1 360.2.a.e 1
60.l odd 4 2 1800.2.f.g 2
80.k odd 4 2 3840.2.k.a 2
80.q even 4 2 3840.2.k.z 2
120.i odd 2 1 2880.2.a.b 1
120.m even 2 1 2880.2.a.r 1
140.c even 2 1 5880.2.a.p 1
180.n even 6 2 3240.2.q.a 2
180.p odd 6 2 3240.2.q.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.a.a 1 20.d odd 2 1
240.2.a.a 1 5.b even 2 1
360.2.a.e 1 60.h even 2 1
600.2.a.a 1 4.b odd 2 1
600.2.f.c 2 20.e even 4 2
720.2.a.f 1 15.d odd 2 1
960.2.a.g 1 40.e odd 2 1
960.2.a.n 1 40.f even 2 1
1200.2.a.r 1 1.a even 1 1 trivial
1200.2.f.f 2 5.c odd 4 2
1800.2.a.c 1 12.b even 2 1
1800.2.f.g 2 60.l odd 4 2
2880.2.a.b 1 120.i odd 2 1
2880.2.a.r 1 120.m even 2 1
3240.2.q.a 2 180.n even 6 2
3240.2.q.m 2 180.p odd 6 2
3600.2.a.bo 1 3.b odd 2 1
3600.2.f.l 2 15.e even 4 2
3840.2.k.a 2 80.k odd 4 2
3840.2.k.z 2 80.q even 4 2
4800.2.a.bh 1 8.b even 2 1
4800.2.a.bl 1 8.d odd 2 1
4800.2.f.n 2 40.i odd 4 2
4800.2.f.u 2 40.k even 4 2
5880.2.a.p 1 140.c even 2 1

## Hecke kernels

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

 $$T_{7} - 4$$ $$T_{11}$$ $$T_{13} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T$$
$5$ 1
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 + 11 T^{2}$$
$13$ $$1 - 6 T + 13 T^{2}$$
$17$ $$1 - 2 T + 17 T^{2}$$
$19$ $$1 + 4 T + 19 T^{2}$$
$23$ $$1 + 8 T + 23 T^{2}$$
$29$ $$1 + 6 T + 29 T^{2}$$
$31$ $$1 + 31 T^{2}$$
$37$ $$1 - 6 T + 37 T^{2}$$
$41$ $$1 - 10 T + 41 T^{2}$$
$43$ $$1 + 4 T + 43 T^{2}$$
$47$ $$1 - 8 T + 47 T^{2}$$
$53$ $$1 + 10 T + 53 T^{2}$$
$59$ $$1 + 59 T^{2}$$
$61$ $$1 - 6 T + 61 T^{2}$$
$67$ $$1 + 4 T + 67 T^{2}$$
$71$ $$1 + 71 T^{2}$$
$73$ $$1 - 14 T + 73 T^{2}$$
$79$ $$1 + 16 T + 79 T^{2}$$
$83$ $$1 - 12 T + 83 T^{2}$$
$89$ $$1 - 2 T + 89 T^{2}$$
$97$ $$1 + 2 T + 97 T^{2}$$