# Properties

 Label 1200.2.a.p 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 75) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + 3 q^{7} + q^{9}+O(q^{10})$$ q + q^3 + 3 * q^7 + q^9 $$q + q^{3} + 3 q^{7} + q^{9} - 2 q^{11} + q^{13} + 2 q^{17} + 5 q^{19} + 3 q^{21} - 6 q^{23} + q^{27} + 10 q^{29} + 3 q^{31} - 2 q^{33} + 2 q^{37} + q^{39} - 8 q^{41} - q^{43} - 2 q^{47} + 2 q^{49} + 2 q^{51} - 4 q^{53} + 5 q^{57} + 10 q^{59} + 7 q^{61} + 3 q^{63} + 3 q^{67} - 6 q^{69} + 8 q^{71} - 14 q^{73} - 6 q^{77} + q^{81} - 6 q^{83} + 10 q^{87} + 3 q^{91} + 3 q^{93} + 17 q^{97} - 2 q^{99}+O(q^{100})$$ q + q^3 + 3 * q^7 + q^9 - 2 * q^11 + q^13 + 2 * q^17 + 5 * q^19 + 3 * q^21 - 6 * q^23 + q^27 + 10 * q^29 + 3 * q^31 - 2 * q^33 + 2 * q^37 + q^39 - 8 * q^41 - q^43 - 2 * q^47 + 2 * q^49 + 2 * q^51 - 4 * q^53 + 5 * q^57 + 10 * q^59 + 7 * q^61 + 3 * q^63 + 3 * q^67 - 6 * q^69 + 8 * q^71 - 14 * q^73 - 6 * q^77 + q^81 - 6 * q^83 + 10 * q^87 + 3 * q^91 + 3 * q^93 + 17 * q^97 - 2 * q^99

## 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.p 1
3.b odd 2 1 3600.2.a.bk 1
4.b odd 2 1 75.2.a.c yes 1
5.b even 2 1 1200.2.a.c 1
5.c odd 4 2 1200.2.f.d 2
8.b even 2 1 4800.2.a.be 1
8.d odd 2 1 4800.2.a.bq 1
12.b even 2 1 225.2.a.a 1
15.d odd 2 1 3600.2.a.j 1
15.e even 4 2 3600.2.f.p 2
20.d odd 2 1 75.2.a.a 1
20.e even 4 2 75.2.b.a 2
28.d even 2 1 3675.2.a.q 1
40.e odd 2 1 4800.2.a.bb 1
40.f even 2 1 4800.2.a.br 1
40.i odd 4 2 4800.2.f.y 2
40.k even 4 2 4800.2.f.l 2
44.c even 2 1 9075.2.a.a 1
60.h even 2 1 225.2.a.e 1
60.l odd 4 2 225.2.b.a 2
140.c even 2 1 3675.2.a.b 1
220.g even 2 1 9075.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.a.a 1 20.d odd 2 1
75.2.a.c yes 1 4.b odd 2 1
75.2.b.a 2 20.e even 4 2
225.2.a.a 1 12.b even 2 1
225.2.a.e 1 60.h even 2 1
225.2.b.a 2 60.l odd 4 2
1200.2.a.c 1 5.b even 2 1
1200.2.a.p 1 1.a even 1 1 trivial
1200.2.f.d 2 5.c odd 4 2
3600.2.a.j 1 15.d odd 2 1
3600.2.a.bk 1 3.b odd 2 1
3600.2.f.p 2 15.e even 4 2
3675.2.a.b 1 140.c even 2 1
3675.2.a.q 1 28.d even 2 1
4800.2.a.bb 1 40.e odd 2 1
4800.2.a.be 1 8.b even 2 1
4800.2.a.bq 1 8.d odd 2 1
4800.2.a.br 1 40.f even 2 1
4800.2.f.l 2 40.k even 4 2
4800.2.f.y 2 40.i odd 4 2
9075.2.a.a 1 44.c even 2 1
9075.2.a.s 1 220.g even 2 1

## Hecke kernels

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

 $$T_{7} - 3$$ T7 - 3 $$T_{11} + 2$$ T11 + 2 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

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