# Properties

 Label 1800.4.a.bc Level $1800$ Weight $4$ Character orbit 1800.a Self dual yes Analytic conductor $106.203$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$106.203438010$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + 18 q^{7}+O(q^{10})$$ q + 18 * q^7 $$q + 18 q^{7} - 34 q^{11} - 12 q^{13} - 102 q^{17} + 164 q^{19} + 48 q^{23} - 146 q^{29} + 100 q^{31} - 328 q^{37} + 288 q^{41} - 120 q^{43} + 16 q^{47} - 19 q^{49} - 126 q^{53} - 642 q^{59} + 602 q^{61} - 436 q^{67} - 652 q^{71} - 1062 q^{73} - 612 q^{77} + 388 q^{79} - 444 q^{83} + 820 q^{89} - 216 q^{91} + 766 q^{97}+O(q^{100})$$ q + 18 * q^7 - 34 * q^11 - 12 * q^13 - 102 * q^17 + 164 * q^19 + 48 * q^23 - 146 * q^29 + 100 * q^31 - 328 * q^37 + 288 * q^41 - 120 * q^43 + 16 * q^47 - 19 * q^49 - 126 * q^53 - 642 * q^59 + 602 * q^61 - 436 * q^67 - 652 * q^71 - 1062 * q^73 - 612 * q^77 + 388 * q^79 - 444 * q^83 + 820 * q^89 - 216 * q^91 + 766 * 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 18.0000 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 1800.4.a.bc 1
3.b odd 2 1 1800.4.a.be 1
5.b even 2 1 360.4.a.a 1
5.c odd 4 2 1800.4.f.e 2
15.d odd 2 1 360.4.a.j yes 1
15.e even 4 2 1800.4.f.s 2
20.d odd 2 1 720.4.a.m 1
60.h even 2 1 720.4.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.a.a 1 5.b even 2 1
360.4.a.j yes 1 15.d odd 2 1
720.4.a.m 1 20.d odd 2 1
720.4.a.z 1 60.h even 2 1
1800.4.a.bc 1 1.a even 1 1 trivial
1800.4.a.be 1 3.b odd 2 1
1800.4.f.e 2 5.c odd 4 2
1800.4.f.s 2 15.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1800))$$:

 $$T_{7} - 18$$ T7 - 18 $$T_{11} + 34$$ T11 + 34 $$T_{17} + 102$$ T17 + 102

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 18$$
$11$ $$T + 34$$
$13$ $$T + 12$$
$17$ $$T + 102$$
$19$ $$T - 164$$
$23$ $$T - 48$$
$29$ $$T + 146$$
$31$ $$T - 100$$
$37$ $$T + 328$$
$41$ $$T - 288$$
$43$ $$T + 120$$
$47$ $$T - 16$$
$53$ $$T + 126$$
$59$ $$T + 642$$
$61$ $$T - 602$$
$67$ $$T + 436$$
$71$ $$T + 652$$
$73$ $$T + 1062$$
$79$ $$T - 388$$
$83$ $$T + 444$$
$89$ $$T - 820$$
$97$ $$T - 766$$
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