# Properties

 Label 1800.4.f.a 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 120) 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{7}+O(q^{10})$$ q + 2*b * q^7 $$q + 2 \beta q^{7} - 72 q^{11} + 3 \beta q^{13} - 19 \beta q^{17} - 52 q^{19} + 76 \beta q^{23} - 78 q^{29} + 120 q^{31} - 75 \beta q^{37} - 362 q^{41} + 242 \beta q^{43} - 140 \beta q^{47} + 327 q^{49} - 335 \beta q^{53} + 696 q^{59} + 222 q^{61} - 2 \beta q^{67} - 96 q^{71} - 89 \beta q^{73} - 144 \beta q^{77} + 632 q^{79} - 306 \beta q^{83} + 994 q^{89} - 24 q^{91} + 817 \beta q^{97} +O(q^{100})$$ q + 2*b * q^7 - 72 * q^11 + 3*b * q^13 - 19*b * q^17 - 52 * q^19 + 76*b * q^23 - 78 * q^29 + 120 * q^31 - 75*b * q^37 - 362 * q^41 + 242*b * q^43 - 140*b * q^47 + 327 * q^49 - 335*b * q^53 + 696 * q^59 + 222 * q^61 - 2*b * q^67 - 96 * q^71 - 89*b * q^73 - 144*b * q^77 + 632 * q^79 - 306*b * q^83 + 994 * q^89 - 24 * q^91 + 817*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 144 q^{11} - 104 q^{19} - 156 q^{29} + 240 q^{31} - 724 q^{41} + 654 q^{49} + 1392 q^{59} + 444 q^{61} - 192 q^{71} + 1264 q^{79} + 1988 q^{89} - 48 q^{91}+O(q^{100})$$ 2 * q - 144 * q^11 - 104 * q^19 - 156 * q^29 + 240 * q^31 - 724 * q^41 + 654 * q^49 + 1392 * q^59 + 444 * q^61 - 192 * q^71 + 1264 * q^79 + 1988 * q^89 - 48 * 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 4.00000i 0 0 0
649.2 0 0 0 0 0 4.00000i 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.a 2
3.b odd 2 1 600.4.f.i 2
5.b even 2 1 inner 1800.4.f.a 2
5.c odd 4 1 360.4.a.l 1
5.c odd 4 1 1800.4.a.n 1
12.b even 2 1 1200.4.f.a 2
15.d odd 2 1 600.4.f.i 2
15.e even 4 1 120.4.a.a 1
15.e even 4 1 600.4.a.l 1
20.e even 4 1 720.4.a.v 1
60.h even 2 1 1200.4.f.a 2
60.l odd 4 1 240.4.a.h 1
60.l odd 4 1 1200.4.a.k 1
120.q odd 4 1 960.4.a.o 1
120.w even 4 1 960.4.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.a 1 15.e even 4 1
240.4.a.h 1 60.l odd 4 1
360.4.a.l 1 5.c odd 4 1
600.4.a.l 1 15.e even 4 1
600.4.f.i 2 3.b odd 2 1
600.4.f.i 2 15.d odd 2 1
720.4.a.v 1 20.e even 4 1
960.4.a.o 1 120.q odd 4 1
960.4.a.bf 1 120.w even 4 1
1200.4.a.k 1 60.l odd 4 1
1200.4.f.a 2 12.b even 2 1
1200.4.f.a 2 60.h even 2 1
1800.4.a.n 1 5.c odd 4 1
1800.4.f.a 2 1.a even 1 1 trivial
1800.4.f.a 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 $$S_{4}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} + 72$$ T11 + 72

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 72)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 1444$$
$19$ $$(T + 52)^{2}$$
$23$ $$T^{2} + 23104$$
$29$ $$(T + 78)^{2}$$
$31$ $$(T - 120)^{2}$$
$37$ $$T^{2} + 22500$$
$41$ $$(T + 362)^{2}$$
$43$ $$T^{2} + 234256$$
$47$ $$T^{2} + 78400$$
$53$ $$T^{2} + 448900$$
$59$ $$(T - 696)^{2}$$
$61$ $$(T - 222)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T + 96)^{2}$$
$73$ $$T^{2} + 31684$$
$79$ $$(T - 632)^{2}$$
$83$ $$T^{2} + 374544$$
$89$ $$(T - 994)^{2}$$
$97$ $$T^{2} + 2669956$$