# Properties

 Label 1200.2.a.f Level $1200$ Weight $2$ Character orbit 1200.a Self dual yes Analytic conductor $9.582$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 300) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} + q^{7} + q^{9} - 6q^{11} + 5q^{13} - 6q^{17} - 5q^{19} - q^{21} + 6q^{23} - q^{27} - 6q^{29} + q^{31} + 6q^{33} + 2q^{37} - 5q^{39} + q^{43} - 6q^{47} - 6q^{49} + 6q^{51} - 12q^{53} + 5q^{57} + 6q^{59} - 13q^{61} + q^{63} - 11q^{67} - 6q^{69} + 2q^{73} - 6q^{77} - 8q^{79} + q^{81} + 6q^{83} + 6q^{87} + 5q^{91} - q^{93} - 7q^{97} - 6q^{99} + O(q^{100})$$

## 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 1.00000 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.f 1
3.b odd 2 1 3600.2.a.z 1
4.b odd 2 1 300.2.a.c yes 1
5.b even 2 1 1200.2.a.n 1
5.c odd 4 2 1200.2.f.a 2
8.b even 2 1 4800.2.a.cf 1
8.d odd 2 1 4800.2.a.o 1
12.b even 2 1 900.2.a.c 1
15.d odd 2 1 3600.2.a.s 1
15.e even 4 2 3600.2.f.v 2
20.d odd 2 1 300.2.a.b 1
20.e even 4 2 300.2.d.a 2
40.e odd 2 1 4800.2.a.ce 1
40.f even 2 1 4800.2.a.p 1
40.i odd 4 2 4800.2.f.bi 2
40.k even 4 2 4800.2.f.b 2
60.h even 2 1 900.2.a.e 1
60.l odd 4 2 900.2.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.a.b 1 20.d odd 2 1
300.2.a.c yes 1 4.b odd 2 1
300.2.d.a 2 20.e even 4 2
900.2.a.c 1 12.b even 2 1
900.2.a.e 1 60.h even 2 1
900.2.d.a 2 60.l odd 4 2
1200.2.a.f 1 1.a even 1 1 trivial
1200.2.a.n 1 5.b even 2 1
1200.2.f.a 2 5.c odd 4 2
3600.2.a.s 1 15.d odd 2 1
3600.2.a.z 1 3.b odd 2 1
3600.2.f.v 2 15.e even 4 2
4800.2.a.o 1 8.d odd 2 1
4800.2.a.p 1 40.f even 2 1
4800.2.a.ce 1 40.e odd 2 1
4800.2.a.cf 1 8.b even 2 1
4800.2.f.b 2 40.k even 4 2
4800.2.f.bi 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1200))$$:

 $$T_{7} - 1$$ $$T_{11} + 6$$ $$T_{13} - 5$$

## Hecke characteristic polynomials

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