# Properties

 Label 1800.4.f.f Level $1800$ Weight $4$ Character orbit 1800.f Analytic conductor $106.203$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1800.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$106.203438010$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 360) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{7}+O(q^{10})$$ q + b * q^7 $$q + \beta q^{7} - 34 q^{11} + 34 \beta q^{13} - 19 \beta q^{17} - 4 q^{19} - 76 \beta q^{23} + 46 q^{29} - 260 q^{31} - 156 \beta q^{37} + 48 q^{41} + 100 \beta q^{43} + 52 \beta q^{47} + 339 q^{49} + 207 \beta q^{53} + 2 q^{59} - 38 q^{61} - 122 \beta q^{67} + 708 q^{71} + 189 \beta q^{73} - 34 \beta q^{77} + 852 q^{79} - 422 \beta q^{83} + 1380 q^{89} - 136 q^{91} + 257 \beta q^{97} +O(q^{100})$$ q + b * q^7 - 34 * q^11 + 34*b * q^13 - 19*b * q^17 - 4 * q^19 - 76*b * q^23 + 46 * q^29 - 260 * q^31 - 156*b * q^37 + 48 * q^41 + 100*b * q^43 + 52*b * q^47 + 339 * q^49 + 207*b * q^53 + 2 * q^59 - 38 * q^61 - 122*b * q^67 + 708 * q^71 + 189*b * q^73 - 34*b * q^77 + 852 * q^79 - 422*b * q^83 + 1380 * q^89 - 136 * q^91 + 257*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 68 q^{11} - 8 q^{19} + 92 q^{29} - 520 q^{31} + 96 q^{41} + 678 q^{49} + 4 q^{59} - 76 q^{61} + 1416 q^{71} + 1704 q^{79} + 2760 q^{89} - 272 q^{91}+O(q^{100})$$ 2 * q - 68 * q^11 - 8 * q^19 + 92 * q^29 - 520 * q^31 + 96 * q^41 + 678 * q^49 + 4 * q^59 - 76 * q^61 + 1416 * q^71 + 1704 * q^79 + 2760 * q^89 - 272 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1001$$ $$1351$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
0 0 0 0 0 2.00000i 0 0 0
649.2 0 0 0 0 0 2.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.4.f.f 2
3.b odd 2 1 1800.4.f.t 2
5.b even 2 1 inner 1800.4.f.f 2
5.c odd 4 1 360.4.a.k yes 1
5.c odd 4 1 1800.4.a.q 1
15.d odd 2 1 1800.4.f.t 2
15.e even 4 1 360.4.a.d 1
15.e even 4 1 1800.4.a.r 1
20.e even 4 1 720.4.a.x 1
60.l odd 4 1 720.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.a.d 1 15.e even 4 1
360.4.a.k yes 1 5.c odd 4 1
720.4.a.g 1 60.l odd 4 1
720.4.a.x 1 20.e even 4 1
1800.4.a.q 1 5.c odd 4 1
1800.4.a.r 1 15.e even 4 1
1800.4.f.f 2 1.a even 1 1 trivial
1800.4.f.f 2 5.b even 2 1 inner
1800.4.f.t 2 3.b odd 2 1
1800.4.f.t 2 15.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 $$S_{4}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11} + 34$$ T11 + 34

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 34)^{2}$$
$13$ $$T^{2} + 4624$$
$17$ $$T^{2} + 1444$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 23104$$
$29$ $$(T - 46)^{2}$$
$31$ $$(T + 260)^{2}$$
$37$ $$T^{2} + 97344$$
$41$ $$(T - 48)^{2}$$
$43$ $$T^{2} + 40000$$
$47$ $$T^{2} + 10816$$
$53$ $$T^{2} + 171396$$
$59$ $$(T - 2)^{2}$$
$61$ $$(T + 38)^{2}$$
$67$ $$T^{2} + 59536$$
$71$ $$(T - 708)^{2}$$
$73$ $$T^{2} + 142884$$
$79$ $$(T - 852)^{2}$$
$83$ $$T^{2} + 712336$$
$89$ $$(T - 1380)^{2}$$
$97$ $$T^{2} + 264196$$