# Properties

 Label 1800.2.f.c Level $1800$ Weight $2$ Character orbit 1800.f Analytic conductor $14.373$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1800.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3730723638$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 24) 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.

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ q $$q - 4 q^{11} - \beta q^{13} + \beta q^{17} + 4 q^{19} + 4 \beta q^{23} + 6 q^{29} + 8 q^{31} - 3 \beta q^{37} + 6 q^{41} + 2 \beta q^{43} + 7 q^{49} + \beta q^{53} + 4 q^{59} - 2 q^{61} + 2 \beta q^{67} - 8 q^{71} + 5 \beta q^{73} + 8 q^{79} + 2 \beta q^{83} - 6 q^{89} - \beta q^{97} +O(q^{100})$$ q - 4 * q^11 - b * q^13 + b * q^17 + 4 * q^19 + 4*b * q^23 + 6 * q^29 + 8 * q^31 - 3*b * q^37 + 6 * q^41 + 2*b * q^43 + 7 * q^49 + b * q^53 + 4 * q^59 - 2 * q^61 + 2*b * q^67 - 8 * q^71 + 5*b * q^73 + 8 * q^79 + 2*b * q^83 - 6 * q^89 - b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 8 q^{11} + 8 q^{19} + 12 q^{29} + 16 q^{31} + 12 q^{41} + 14 q^{49} + 8 q^{59} - 4 q^{61} - 16 q^{71} + 16 q^{79} - 12 q^{89}+O(q^{100})$$ 2 * q - 8 * q^11 + 8 * q^19 + 12 * q^29 + 16 * q^31 + 12 * q^41 + 14 * q^49 + 8 * q^59 - 4 * q^61 - 16 * q^71 + 16 * q^79 - 12 * q^89

## 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 0 0 0 0
649.2 0 0 0 0 0 0 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 1800.2.f.c 2
3.b odd 2 1 600.2.f.e 2
4.b odd 2 1 3600.2.f.r 2
5.b even 2 1 inner 1800.2.f.c 2
5.c odd 4 1 72.2.a.a 1
5.c odd 4 1 1800.2.a.m 1
12.b even 2 1 1200.2.f.b 2
15.d odd 2 1 600.2.f.e 2
15.e even 4 1 24.2.a.a 1
15.e even 4 1 600.2.a.h 1
20.d odd 2 1 3600.2.f.r 2
20.e even 4 1 144.2.a.b 1
20.e even 4 1 3600.2.a.v 1
24.f even 2 1 4800.2.f.bg 2
24.h odd 2 1 4800.2.f.d 2
35.f even 4 1 3528.2.a.d 1
35.k even 12 2 3528.2.s.y 2
35.l odd 12 2 3528.2.s.j 2
40.i odd 4 1 576.2.a.d 1
40.k even 4 1 576.2.a.b 1
45.k odd 12 2 648.2.i.b 2
45.l even 12 2 648.2.i.g 2
55.e even 4 1 8712.2.a.u 1
60.h even 2 1 1200.2.f.b 2
60.l odd 4 1 48.2.a.a 1
60.l odd 4 1 1200.2.a.d 1
80.i odd 4 1 2304.2.d.i 2
80.j even 4 1 2304.2.d.k 2
80.s even 4 1 2304.2.d.k 2
80.t odd 4 1 2304.2.d.i 2
105.k odd 4 1 1176.2.a.i 1
105.w odd 12 2 1176.2.q.a 2
105.x even 12 2 1176.2.q.i 2
120.i odd 2 1 4800.2.f.d 2
120.m even 2 1 4800.2.f.bg 2
120.q odd 4 1 192.2.a.b 1
120.q odd 4 1 4800.2.a.cc 1
120.w even 4 1 192.2.a.d 1
120.w even 4 1 4800.2.a.q 1
140.j odd 4 1 7056.2.a.q 1
165.l odd 4 1 2904.2.a.c 1
180.v odd 12 2 1296.2.i.m 2
180.x even 12 2 1296.2.i.e 2
195.j odd 4 1 4056.2.c.e 2
195.s even 4 1 4056.2.a.i 1
195.u odd 4 1 4056.2.c.e 2
240.z odd 4 1 768.2.d.d 2
240.bb even 4 1 768.2.d.e 2
240.bd odd 4 1 768.2.d.d 2
240.bf even 4 1 768.2.d.e 2
255.o even 4 1 6936.2.a.p 1
285.j odd 4 1 8664.2.a.j 1
420.w even 4 1 2352.2.a.i 1
420.bp odd 12 2 2352.2.q.l 2
420.br even 12 2 2352.2.q.r 2
660.q even 4 1 5808.2.a.s 1
780.w odd 4 1 8112.2.a.be 1
840.bm even 4 1 9408.2.a.cc 1
840.bp odd 4 1 9408.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 15.e even 4 1
48.2.a.a 1 60.l odd 4 1
72.2.a.a 1 5.c odd 4 1
144.2.a.b 1 20.e even 4 1
192.2.a.b 1 120.q odd 4 1
192.2.a.d 1 120.w even 4 1
576.2.a.b 1 40.k even 4 1
576.2.a.d 1 40.i odd 4 1
600.2.a.h 1 15.e even 4 1
600.2.f.e 2 3.b odd 2 1
600.2.f.e 2 15.d odd 2 1
648.2.i.b 2 45.k odd 12 2
648.2.i.g 2 45.l even 12 2
768.2.d.d 2 240.z odd 4 1
768.2.d.d 2 240.bd odd 4 1
768.2.d.e 2 240.bb even 4 1
768.2.d.e 2 240.bf even 4 1
1176.2.a.i 1 105.k odd 4 1
1176.2.q.a 2 105.w odd 12 2
1176.2.q.i 2 105.x even 12 2
1200.2.a.d 1 60.l odd 4 1
1200.2.f.b 2 12.b even 2 1
1200.2.f.b 2 60.h even 2 1
1296.2.i.e 2 180.x even 12 2
1296.2.i.m 2 180.v odd 12 2
1800.2.a.m 1 5.c odd 4 1
1800.2.f.c 2 1.a even 1 1 trivial
1800.2.f.c 2 5.b even 2 1 inner
2304.2.d.i 2 80.i odd 4 1
2304.2.d.i 2 80.t odd 4 1
2304.2.d.k 2 80.j even 4 1
2304.2.d.k 2 80.s even 4 1
2352.2.a.i 1 420.w even 4 1
2352.2.q.l 2 420.bp odd 12 2
2352.2.q.r 2 420.br even 12 2
2904.2.a.c 1 165.l odd 4 1
3528.2.a.d 1 35.f even 4 1
3528.2.s.j 2 35.l odd 12 2
3528.2.s.y 2 35.k even 12 2
3600.2.a.v 1 20.e even 4 1
3600.2.f.r 2 4.b odd 2 1
3600.2.f.r 2 20.d odd 2 1
4056.2.a.i 1 195.s even 4 1
4056.2.c.e 2 195.j odd 4 1
4056.2.c.e 2 195.u odd 4 1
4800.2.a.q 1 120.w even 4 1
4800.2.a.cc 1 120.q odd 4 1
4800.2.f.d 2 24.h odd 2 1
4800.2.f.d 2 120.i odd 2 1
4800.2.f.bg 2 24.f even 2 1
4800.2.f.bg 2 120.m even 2 1
5808.2.a.s 1 660.q even 4 1
6936.2.a.p 1 255.o even 4 1
7056.2.a.q 1 140.j odd 4 1
8112.2.a.be 1 780.w odd 4 1
8664.2.a.j 1 285.j odd 4 1
8712.2.a.u 1 55.e even 4 1
9408.2.a.h 1 840.bp odd 4 1
9408.2.a.cc 1 840.bm even 4 1

## Hecke kernels

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

 $$T_{7}$$ T7 $$T_{11} + 4$$ T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 4$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 64$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 4$$
$59$ $$(T - 4)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 4$$