# Properties

 Label 900.2.d.b Level $900$ Weight $2$ Character orbit 900.d Analytic conductor $7.187$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,2,Mod(649,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("900.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 900.d (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: $$7.18653618192$$ 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 36) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 $$q + 2 \beta q^{7} + \beta q^{13} - 8 q^{19} - 4 q^{31} + 5 \beta q^{37} + 4 \beta q^{43} - 9 q^{49} + 14 q^{61} + 8 \beta q^{67} - 5 \beta q^{73} + 4 q^{79} - 8 q^{91} - 7 \beta q^{97} +O(q^{100})$$ q + 2*b * q^7 + b * q^13 - 8 * q^19 - 4 * q^31 + 5*b * q^37 + 4*b * q^43 - 9 * q^49 + 14 * q^61 + 8*b * q^67 - 5*b * q^73 + 4 * q^79 - 8 * q^91 - 7*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 16 q^{19} - 8 q^{31} - 18 q^{49} + 28 q^{61} + 8 q^{79} - 16 q^{91}+O(q^{100})$$ 2 * q - 16 * q^19 - 8 * q^31 - 18 * q^49 + 28 * q^61 + 8 * q^79 - 16 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.d.b 2
3.b odd 2 1 CM 900.2.d.b 2
4.b odd 2 1 3600.2.f.m 2
5.b even 2 1 inner 900.2.d.b 2
5.c odd 4 1 36.2.a.a 1
5.c odd 4 1 900.2.a.g 1
12.b even 2 1 3600.2.f.m 2
15.d odd 2 1 inner 900.2.d.b 2
15.e even 4 1 36.2.a.a 1
15.e even 4 1 900.2.a.g 1
20.d odd 2 1 3600.2.f.m 2
20.e even 4 1 144.2.a.a 1
20.e even 4 1 3600.2.a.e 1
35.f even 4 1 1764.2.a.e 1
35.k even 12 2 1764.2.k.g 2
35.l odd 12 2 1764.2.k.h 2
40.i odd 4 1 576.2.a.e 1
40.k even 4 1 576.2.a.f 1
45.k odd 12 2 324.2.e.c 2
45.l even 12 2 324.2.e.c 2
55.e even 4 1 4356.2.a.g 1
60.h even 2 1 3600.2.f.m 2
60.l odd 4 1 144.2.a.a 1
60.l odd 4 1 3600.2.a.e 1
65.f even 4 1 6084.2.b.f 2
65.h odd 4 1 6084.2.a.i 1
65.k even 4 1 6084.2.b.f 2
80.i odd 4 1 2304.2.d.q 2
80.j even 4 1 2304.2.d.a 2
80.s even 4 1 2304.2.d.a 2
80.t odd 4 1 2304.2.d.q 2
105.k odd 4 1 1764.2.a.e 1
105.w odd 12 2 1764.2.k.g 2
105.x even 12 2 1764.2.k.h 2
120.q odd 4 1 576.2.a.f 1
120.w even 4 1 576.2.a.e 1
140.j odd 4 1 7056.2.a.bb 1
165.l odd 4 1 4356.2.a.g 1
180.v odd 12 2 1296.2.i.h 2
180.x even 12 2 1296.2.i.h 2
195.j odd 4 1 6084.2.b.f 2
195.s even 4 1 6084.2.a.i 1
195.u odd 4 1 6084.2.b.f 2
240.z odd 4 1 2304.2.d.a 2
240.bb even 4 1 2304.2.d.q 2
240.bd odd 4 1 2304.2.d.a 2
240.bf even 4 1 2304.2.d.q 2
420.w even 4 1 7056.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.a.a 1 5.c odd 4 1
36.2.a.a 1 15.e even 4 1
144.2.a.a 1 20.e even 4 1
144.2.a.a 1 60.l odd 4 1
324.2.e.c 2 45.k odd 12 2
324.2.e.c 2 45.l even 12 2
576.2.a.e 1 40.i odd 4 1
576.2.a.e 1 120.w even 4 1
576.2.a.f 1 40.k even 4 1
576.2.a.f 1 120.q odd 4 1
900.2.a.g 1 5.c odd 4 1
900.2.a.g 1 15.e even 4 1
900.2.d.b 2 1.a even 1 1 trivial
900.2.d.b 2 3.b odd 2 1 CM
900.2.d.b 2 5.b even 2 1 inner
900.2.d.b 2 15.d odd 2 1 inner
1296.2.i.h 2 180.v odd 12 2
1296.2.i.h 2 180.x even 12 2
1764.2.a.e 1 35.f even 4 1
1764.2.a.e 1 105.k odd 4 1
1764.2.k.g 2 35.k even 12 2
1764.2.k.g 2 105.w odd 12 2
1764.2.k.h 2 35.l odd 12 2
1764.2.k.h 2 105.x even 12 2
2304.2.d.a 2 80.j even 4 1
2304.2.d.a 2 80.s even 4 1
2304.2.d.a 2 240.z odd 4 1
2304.2.d.a 2 240.bd odd 4 1
2304.2.d.q 2 80.i odd 4 1
2304.2.d.q 2 80.t odd 4 1
2304.2.d.q 2 240.bb even 4 1
2304.2.d.q 2 240.bf even 4 1
3600.2.a.e 1 20.e even 4 1
3600.2.a.e 1 60.l odd 4 1
3600.2.f.m 2 4.b odd 2 1
3600.2.f.m 2 12.b even 2 1
3600.2.f.m 2 20.d odd 2 1
3600.2.f.m 2 60.h even 2 1
4356.2.a.g 1 55.e even 4 1
4356.2.a.g 1 165.l odd 4 1
6084.2.a.i 1 65.h odd 4 1
6084.2.a.i 1 195.s even 4 1
6084.2.b.f 2 65.f even 4 1
6084.2.b.f 2 65.k even 4 1
6084.2.b.f 2 195.j odd 4 1
6084.2.b.f 2 195.u odd 4 1
7056.2.a.bb 1 140.j odd 4 1
7056.2.a.bb 1 420.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(900, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2}$$
$19$ $$(T + 8)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 14)^{2}$$
$67$ $$T^{2} + 256$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 196$$