# Properties

 Label 1200.2.a.q 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 600) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + 3q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} + 3q^{7} + q^{9} - 2q^{11} + 3q^{13} - 6q^{17} + 7q^{19} + 3q^{21} + 6q^{23} + q^{27} - 2q^{29} + 5q^{31} - 2q^{33} - 10q^{37} + 3q^{39} + 12q^{41} + 3q^{43} - 10q^{47} + 2q^{49} - 6q^{51} + 7q^{57} + 6q^{59} - 13q^{61} + 3q^{63} + 7q^{67} + 6q^{69} + 4q^{71} + 6q^{73} - 6q^{77} + 8q^{79} + q^{81} - 6q^{83} - 2q^{87} + 16q^{89} + 9q^{91} + 5q^{93} + 7q^{97} - 2q^{99} + O(q^{100})$$

## 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 3.00000 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.q 1
3.b odd 2 1 3600.2.a.bl 1
4.b odd 2 1 600.2.a.b 1
5.b even 2 1 1200.2.a.b 1
5.c odd 4 2 1200.2.f.c 2
8.b even 2 1 4800.2.a.bd 1
8.d odd 2 1 4800.2.a.bp 1
12.b even 2 1 1800.2.a.e 1
15.d odd 2 1 3600.2.a.i 1
15.e even 4 2 3600.2.f.o 2
20.d odd 2 1 600.2.a.i yes 1
20.e even 4 2 600.2.f.d 2
40.e odd 2 1 4800.2.a.bc 1
40.f even 2 1 4800.2.a.bs 1
40.i odd 4 2 4800.2.f.z 2
40.k even 4 2 4800.2.f.k 2
60.h even 2 1 1800.2.a.t 1
60.l odd 4 2 1800.2.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.a.b 1 4.b odd 2 1
600.2.a.i yes 1 20.d odd 2 1
600.2.f.d 2 20.e even 4 2
1200.2.a.b 1 5.b even 2 1
1200.2.a.q 1 1.a even 1 1 trivial
1200.2.f.c 2 5.c odd 4 2
1800.2.a.e 1 12.b even 2 1
1800.2.a.t 1 60.h even 2 1
1800.2.f.e 2 60.l odd 4 2
3600.2.a.i 1 15.d odd 2 1
3600.2.a.bl 1 3.b odd 2 1
3600.2.f.o 2 15.e even 4 2
4800.2.a.bc 1 40.e odd 2 1
4800.2.a.bd 1 8.b even 2 1
4800.2.a.bp 1 8.d odd 2 1
4800.2.a.bs 1 40.f even 2 1
4800.2.f.k 2 40.k even 4 2
4800.2.f.z 2 40.i odd 4 2

## Hecke kernels

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

 $$T_{7} - 3$$ $$T_{11} + 2$$ $$T_{13} - 3$$

## Hecke characteristic polynomials

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