# Properties

 Label 1800.4.f.x Level $1800$ Weight $4$ Character orbit 1800.f Analytic conductor $106.203$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,4,Mod(649,1800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1800.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$106.203438010$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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 72) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 6 \beta q^{7}+O(q^{10})$$ q + 6*b * q^7 $$q + 6 \beta q^{7} + 64 q^{11} + 29 \beta q^{13} - 16 \beta q^{17} + 136 q^{19} - 64 \beta q^{23} + 144 q^{29} + 20 q^{31} + 9 \beta q^{37} - 288 q^{41} - 100 \beta q^{43} - 192 \beta q^{47} + 199 q^{49} + 248 \beta q^{53} + 128 q^{59} - 458 q^{61} + 248 \beta q^{67} + 512 q^{71} - 301 \beta q^{73} + 384 \beta q^{77} - 1108 q^{79} + 352 \beta q^{83} + 960 q^{89} - 696 q^{91} - 103 \beta q^{97} +O(q^{100})$$ q + 6*b * q^7 + 64 * q^11 + 29*b * q^13 - 16*b * q^17 + 136 * q^19 - 64*b * q^23 + 144 * q^29 + 20 * q^31 + 9*b * q^37 - 288 * q^41 - 100*b * q^43 - 192*b * q^47 + 199 * q^49 + 248*b * q^53 + 128 * q^59 - 458 * q^61 + 248*b * q^67 + 512 * q^71 - 301*b * q^73 + 384*b * q^77 - 1108 * q^79 + 352*b * q^83 + 960 * q^89 - 696 * q^91 - 103*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 128 q^{11} + 272 q^{19} + 288 q^{29} + 40 q^{31} - 576 q^{41} + 398 q^{49} + 256 q^{59} - 916 q^{61} + 1024 q^{71} - 2216 q^{79} + 1920 q^{89} - 1392 q^{91}+O(q^{100})$$ 2 * q + 128 * q^11 + 272 * q^19 + 288 * q^29 + 40 * q^31 - 576 * q^41 + 398 * q^49 + 256 * q^59 - 916 * q^61 + 1024 * q^71 - 2216 * q^79 + 1920 * q^89 - 1392 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 12.0000i 0 0 0
649.2 0 0 0 0 0 12.0000i 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.x 2
3.b odd 2 1 1800.4.f.b 2
5.b even 2 1 inner 1800.4.f.x 2
5.c odd 4 1 72.4.a.d yes 1
5.c odd 4 1 1800.4.a.ba 1
15.d odd 2 1 1800.4.f.b 2
15.e even 4 1 72.4.a.a 1
15.e even 4 1 1800.4.a.z 1
20.e even 4 1 144.4.a.f 1
40.i odd 4 1 576.4.a.c 1
40.k even 4 1 576.4.a.d 1
45.k odd 12 2 648.4.i.a 2
45.l even 12 2 648.4.i.l 2
60.l odd 4 1 144.4.a.a 1
120.q odd 4 1 576.4.a.x 1
120.w even 4 1 576.4.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.a.a 1 15.e even 4 1
72.4.a.d yes 1 5.c odd 4 1
144.4.a.a 1 60.l odd 4 1
144.4.a.f 1 20.e even 4 1
576.4.a.c 1 40.i odd 4 1
576.4.a.d 1 40.k even 4 1
576.4.a.w 1 120.w even 4 1
576.4.a.x 1 120.q odd 4 1
648.4.i.a 2 45.k odd 12 2
648.4.i.l 2 45.l even 12 2
1800.4.a.z 1 15.e even 4 1
1800.4.a.ba 1 5.c odd 4 1
1800.4.f.b 2 3.b odd 2 1
1800.4.f.b 2 15.d odd 2 1
1800.4.f.x 2 1.a even 1 1 trivial
1800.4.f.x 2 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7}^{2} + 144$$ T7^2 + 144 $$T_{11} - 64$$ T11 - 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 144$$
$11$ $$(T - 64)^{2}$$
$13$ $$T^{2} + 3364$$
$17$ $$T^{2} + 1024$$
$19$ $$(T - 136)^{2}$$
$23$ $$T^{2} + 16384$$
$29$ $$(T - 144)^{2}$$
$31$ $$(T - 20)^{2}$$
$37$ $$T^{2} + 324$$
$41$ $$(T + 288)^{2}$$
$43$ $$T^{2} + 40000$$
$47$ $$T^{2} + 147456$$
$53$ $$T^{2} + 246016$$
$59$ $$(T - 128)^{2}$$
$61$ $$(T + 458)^{2}$$
$67$ $$T^{2} + 246016$$
$71$ $$(T - 512)^{2}$$
$73$ $$T^{2} + 362404$$
$79$ $$(T + 1108)^{2}$$
$83$ $$T^{2} + 495616$$
$89$ $$(T - 960)^{2}$$
$97$ $$T^{2} + 42436$$