# Properties

 Label 1800.2.a.f Level $1800$ Weight $2$ Character orbit 1800.a Self dual yes Analytic conductor $14.373$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.3730723638$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 360) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{7}+O(q^{10})$$ q - 2 * q^7 $$q - 2 q^{7} - 2 q^{11} - 4 q^{13} - 2 q^{17} + 4 q^{19} + 8 q^{23} + 10 q^{29} + 4 q^{31} + 8 q^{43} + 8 q^{47} - 3 q^{49} + 6 q^{53} + 14 q^{59} - 14 q^{61} + 4 q^{67} - 12 q^{71} - 6 q^{73} + 4 q^{77} - 12 q^{79} + 4 q^{83} + 12 q^{89} + 8 q^{91} + 14 q^{97}+O(q^{100})$$ q - 2 * q^7 - 2 * q^11 - 4 * q^13 - 2 * q^17 + 4 * q^19 + 8 * q^23 + 10 * q^29 + 4 * q^31 + 8 * q^43 + 8 * q^47 - 3 * q^49 + 6 * q^53 + 14 * q^59 - 14 * q^61 + 4 * q^67 - 12 * q^71 - 6 * q^73 + 4 * q^77 - 12 * q^79 + 4 * q^83 + 12 * q^89 + 8 * q^91 + 14 * q^97

## 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 0 0 0 0 −2.00000 0 0 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 1800.2.a.f 1
3.b odd 2 1 1800.2.a.i 1
4.b odd 2 1 3600.2.a.bh 1
5.b even 2 1 360.2.a.d yes 1
5.c odd 4 2 1800.2.f.d 2
12.b even 2 1 3600.2.a.bd 1
15.d odd 2 1 360.2.a.c 1
15.e even 4 2 1800.2.f.h 2
20.d odd 2 1 720.2.a.i 1
20.e even 4 2 3600.2.f.q 2
40.e odd 2 1 2880.2.a.e 1
40.f even 2 1 2880.2.a.n 1
45.h odd 6 2 3240.2.q.n 2
45.j even 6 2 3240.2.q.d 2
60.h even 2 1 720.2.a.a 1
60.l odd 4 2 3600.2.f.g 2
120.i odd 2 1 2880.2.a.bd 1
120.m even 2 1 2880.2.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.a.c 1 15.d odd 2 1
360.2.a.d yes 1 5.b even 2 1
720.2.a.a 1 60.h even 2 1
720.2.a.i 1 20.d odd 2 1
1800.2.a.f 1 1.a even 1 1 trivial
1800.2.a.i 1 3.b odd 2 1
1800.2.f.d 2 5.c odd 4 2
1800.2.f.h 2 15.e even 4 2
2880.2.a.e 1 40.e odd 2 1
2880.2.a.n 1 40.f even 2 1
2880.2.a.w 1 120.m even 2 1
2880.2.a.bd 1 120.i odd 2 1
3240.2.q.d 2 45.j even 6 2
3240.2.q.n 2 45.h odd 6 2
3600.2.a.bd 1 12.b even 2 1
3600.2.a.bh 1 4.b odd 2 1
3600.2.f.g 2 60.l odd 4 2
3600.2.f.q 2 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1800))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{11} + 2$$ T11 + 2 $$T_{13} + 4$$ T13 + 4

## Hecke characteristic polynomials

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