# Properties

 Label 1800.2.f.g Level $1800$ Weight $2$ Character orbit 1800.f Analytic conductor $14.373$ 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] = [1800,2,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, 2, names="a")

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

chi := DirichletCharacter("1800.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

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: $$14.3730723638$$ 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 $$q + 2 \beta q^{7} + 3 \beta q^{13} + \beta q^{17} - 4 q^{19} - 4 \beta q^{23} - 6 q^{29} - 3 \beta q^{37} - 10 q^{41} + 2 \beta q^{43} - 4 \beta q^{47} - 9 q^{49} + 5 \beta q^{53} + 6 q^{61} - 2 \beta q^{67} + 7 \beta q^{73} - 16 q^{79} + 6 \beta q^{83} + 2 q^{89} - 24 q^{91} + \beta q^{97} +O(q^{100})$$ q + 2*b * q^7 + 3*b * q^13 + b * q^17 - 4 * q^19 - 4*b * q^23 - 6 * q^29 - 3*b * q^37 - 10 * q^41 + 2*b * q^43 - 4*b * q^47 - 9 * q^49 + 5*b * q^53 + 6 * q^61 - 2*b * q^67 + 7*b * q^73 - 16 * q^79 + 6*b * q^83 + 2 * q^89 - 24 * q^91 + b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 8 q^{19} - 12 q^{29} - 20 q^{41} - 18 q^{49} + 12 q^{61} - 32 q^{79} + 4 q^{89} - 48 q^{91}+O(q^{100})$$ 2 * q - 8 * q^19 - 12 * q^29 - 20 * q^41 - 18 * q^49 + 12 * q^61 - 32 * q^79 + 4 * 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 1800.2.f.g 2
3.b odd 2 1 600.2.f.c 2
4.b odd 2 1 3600.2.f.l 2
5.b even 2 1 inner 1800.2.f.g 2
5.c odd 4 1 360.2.a.e 1
5.c odd 4 1 1800.2.a.c 1
12.b even 2 1 1200.2.f.f 2
15.d odd 2 1 600.2.f.c 2
15.e even 4 1 120.2.a.a 1
15.e even 4 1 600.2.a.a 1
20.d odd 2 1 3600.2.f.l 2
20.e even 4 1 720.2.a.f 1
20.e even 4 1 3600.2.a.bo 1
24.f even 2 1 4800.2.f.n 2
24.h odd 2 1 4800.2.f.u 2
40.i odd 4 1 2880.2.a.r 1
40.k even 4 1 2880.2.a.b 1
45.k odd 12 2 3240.2.q.a 2
45.l even 12 2 3240.2.q.m 2
60.h even 2 1 1200.2.f.f 2
60.l odd 4 1 240.2.a.a 1
60.l odd 4 1 1200.2.a.r 1
105.k odd 4 1 5880.2.a.p 1
120.i odd 2 1 4800.2.f.u 2
120.m even 2 1 4800.2.f.n 2
120.q odd 4 1 960.2.a.n 1
120.q odd 4 1 4800.2.a.bh 1
120.w even 4 1 960.2.a.g 1
120.w even 4 1 4800.2.a.bl 1
240.z odd 4 1 3840.2.k.z 2
240.bb even 4 1 3840.2.k.a 2
240.bd odd 4 1 3840.2.k.z 2
240.bf even 4 1 3840.2.k.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.a.a 1 15.e even 4 1
240.2.a.a 1 60.l odd 4 1
360.2.a.e 1 5.c odd 4 1
600.2.a.a 1 15.e even 4 1
600.2.f.c 2 3.b odd 2 1
600.2.f.c 2 15.d odd 2 1
720.2.a.f 1 20.e even 4 1
960.2.a.g 1 120.w even 4 1
960.2.a.n 1 120.q odd 4 1
1200.2.a.r 1 60.l odd 4 1
1200.2.f.f 2 12.b even 2 1
1200.2.f.f 2 60.h even 2 1
1800.2.a.c 1 5.c odd 4 1
1800.2.f.g 2 1.a even 1 1 trivial
1800.2.f.g 2 5.b even 2 1 inner
2880.2.a.b 1 40.k even 4 1
2880.2.a.r 1 40.i odd 4 1
3240.2.q.a 2 45.k odd 12 2
3240.2.q.m 2 45.l even 12 2
3600.2.a.bo 1 20.e even 4 1
3600.2.f.l 2 4.b odd 2 1
3600.2.f.l 2 20.d odd 2 1
3840.2.k.a 2 240.bb even 4 1
3840.2.k.a 2 240.bf even 4 1
3840.2.k.z 2 240.z odd 4 1
3840.2.k.z 2 240.bd odd 4 1
4800.2.a.bh 1 120.q odd 4 1
4800.2.a.bl 1 120.w even 4 1
4800.2.f.n 2 24.f even 2 1
4800.2.f.n 2 120.m even 2 1
4800.2.f.u 2 24.h odd 2 1
4800.2.f.u 2 120.i odd 2 1
5880.2.a.p 1 105.k odd 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 4$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 64$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 100$$
$59$ $$T^{2}$$
$61$ $$(T - 6)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T + 16)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2} + 4$$