# Properties

 Label 1200.2.a.k 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 30) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - 4q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} - 4q^{7} + q^{9} - 2q^{13} - 6q^{17} + 4q^{19} - 4q^{21} + q^{27} - 6q^{29} - 8q^{31} - 2q^{37} - 2q^{39} - 6q^{41} - 4q^{43} + 9q^{49} - 6q^{51} + 6q^{53} + 4q^{57} - 10q^{61} - 4q^{63} - 4q^{67} - 2q^{73} - 8q^{79} + q^{81} + 12q^{83} - 6q^{87} + 18q^{89} + 8q^{91} - 8q^{93} - 2q^{97} + O(q^{100})$$

## 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.k 1
3.b odd 2 1 3600.2.a.f 1
4.b odd 2 1 150.2.a.b 1
5.b even 2 1 240.2.a.b 1
5.c odd 4 2 1200.2.f.e 2
8.b even 2 1 4800.2.a.d 1
8.d odd 2 1 4800.2.a.cq 1
12.b even 2 1 450.2.a.d 1
15.d odd 2 1 720.2.a.j 1
15.e even 4 2 3600.2.f.i 2
20.d odd 2 1 30.2.a.a 1
20.e even 4 2 150.2.c.a 2
28.d even 2 1 7350.2.a.ct 1
40.e odd 2 1 960.2.a.e 1
40.f even 2 1 960.2.a.p 1
40.i odd 4 2 4800.2.f.w 2
40.k even 4 2 4800.2.f.p 2
60.h even 2 1 90.2.a.c 1
60.l odd 4 2 450.2.c.b 2
80.k odd 4 2 3840.2.k.y 2
80.q even 4 2 3840.2.k.f 2
120.i odd 2 1 2880.2.a.q 1
120.m even 2 1 2880.2.a.a 1
140.c even 2 1 1470.2.a.d 1
140.p odd 6 2 1470.2.i.o 2
140.s even 6 2 1470.2.i.q 2
180.n even 6 2 810.2.e.b 2
180.p odd 6 2 810.2.e.l 2
220.g even 2 1 3630.2.a.w 1
260.g odd 2 1 5070.2.a.w 1
260.u even 4 2 5070.2.b.k 2
340.d odd 2 1 8670.2.a.g 1
420.o odd 2 1 4410.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 20.d odd 2 1
90.2.a.c 1 60.h even 2 1
150.2.a.b 1 4.b odd 2 1
150.2.c.a 2 20.e even 4 2
240.2.a.b 1 5.b even 2 1
450.2.a.d 1 12.b even 2 1
450.2.c.b 2 60.l odd 4 2
720.2.a.j 1 15.d odd 2 1
810.2.e.b 2 180.n even 6 2
810.2.e.l 2 180.p odd 6 2
960.2.a.e 1 40.e odd 2 1
960.2.a.p 1 40.f even 2 1
1200.2.a.k 1 1.a even 1 1 trivial
1200.2.f.e 2 5.c odd 4 2
1470.2.a.d 1 140.c even 2 1
1470.2.i.o 2 140.p odd 6 2
1470.2.i.q 2 140.s even 6 2
2880.2.a.a 1 120.m even 2 1
2880.2.a.q 1 120.i odd 2 1
3600.2.a.f 1 3.b odd 2 1
3600.2.f.i 2 15.e even 4 2
3630.2.a.w 1 220.g even 2 1
3840.2.k.f 2 80.q even 4 2
3840.2.k.y 2 80.k odd 4 2
4410.2.a.z 1 420.o odd 2 1
4800.2.a.d 1 8.b even 2 1
4800.2.a.cq 1 8.d odd 2 1
4800.2.f.p 2 40.k even 4 2
4800.2.f.w 2 40.i odd 4 2
5070.2.a.w 1 260.g odd 2 1
5070.2.b.k 2 260.u even 4 2
7350.2.a.ct 1 28.d even 2 1
8670.2.a.g 1 340.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1200))$$:

 $$T_{7} + 4$$ $$T_{11}$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$4 + T$$
$11$ $$T$$
$13$ $$2 + T$$
$17$ $$6 + T$$
$19$ $$-4 + T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$8 + T$$
$37$ $$2 + T$$
$41$ $$6 + T$$
$43$ $$4 + T$$
$47$ $$T$$
$53$ $$-6 + T$$
$59$ $$T$$
$61$ $$10 + T$$
$67$ $$4 + T$$
$71$ $$T$$
$73$ $$2 + T$$
$79$ $$8 + T$$
$83$ $$-12 + T$$
$89$ $$-18 + T$$
$97$ $$2 + T$$