Properties

 Label 1900.1.g.a Level $1900$ Weight $1$ Character orbit 1900.g Analytic conductor $0.948$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -19 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,1,Mod(949,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1900.949");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1900.g (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: $$0.948223524003$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.76.1 Artin image: $C_4\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} + \cdots)$$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - i q^{7} - q^{9} +O(q^{10})$$ q - z * q^7 - q^9 $$q - i q^{7} - q^{9} - q^{11} - i q^{17} - q^{19} - 2 i q^{23} + i q^{43} - i q^{47} - q^{61} + i q^{63} + i q^{73} + i q^{77} + q^{81} - 2 i q^{83} + q^{99} +O(q^{100})$$ q - z * q^7 - q^9 - q^11 - z * q^17 - q^19 - 2*z * q^23 + z * q^43 - z * q^47 - q^61 + z * q^63 + z * q^73 + z * q^77 + q^81 - 2*z * q^83 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 2 q^{11} - 2 q^{19} - 2 q^{61} + 2 q^{81} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 - 2 * q^11 - 2 * q^19 - 2 * q^61 + 2 * q^81 + 2 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
949.1
 1.00000i − 1.00000i
0 0 0 0 0 1.00000i 0 −1.00000 0
949.2 0 0 0 0 0 1.00000i 0 −1.00000 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
5.b even 2 1 inner
95.d odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.1.g.a 2
5.b even 2 1 inner 1900.1.g.a 2
5.c odd 4 1 76.1.c.a 1
5.c odd 4 1 1900.1.e.a 1
15.e even 4 1 684.1.h.a 1
19.b odd 2 1 CM 1900.1.g.a 2
20.e even 4 1 304.1.e.a 1
35.f even 4 1 3724.1.e.c 1
35.k even 12 2 3724.1.bc.b 2
35.l odd 12 2 3724.1.bc.c 2
40.i odd 4 1 1216.1.e.a 1
40.k even 4 1 1216.1.e.b 1
60.l odd 4 1 2736.1.o.b 1
95.d odd 2 1 inner 1900.1.g.a 2
95.g even 4 1 76.1.c.a 1
95.g even 4 1 1900.1.e.a 1
95.l even 12 2 1444.1.h.a 2
95.m odd 12 2 1444.1.h.a 2
95.q odd 36 6 1444.1.j.a 6
95.r even 36 6 1444.1.j.a 6
285.j odd 4 1 684.1.h.a 1
380.j odd 4 1 304.1.e.a 1
665.n odd 4 1 3724.1.e.c 1
665.ca odd 12 2 3724.1.bc.b 2
665.ck even 12 2 3724.1.bc.c 2
760.t even 4 1 1216.1.e.a 1
760.y odd 4 1 1216.1.e.b 1
1140.w even 4 1 2736.1.o.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.1.c.a 1 5.c odd 4 1
76.1.c.a 1 95.g even 4 1
304.1.e.a 1 20.e even 4 1
304.1.e.a 1 380.j odd 4 1
684.1.h.a 1 15.e even 4 1
684.1.h.a 1 285.j odd 4 1
1216.1.e.a 1 40.i odd 4 1
1216.1.e.a 1 760.t even 4 1
1216.1.e.b 1 40.k even 4 1
1216.1.e.b 1 760.y odd 4 1
1444.1.h.a 2 95.l even 12 2
1444.1.h.a 2 95.m odd 12 2
1444.1.j.a 6 95.q odd 36 6
1444.1.j.a 6 95.r even 36 6
1900.1.e.a 1 5.c odd 4 1
1900.1.e.a 1 95.g even 4 1
1900.1.g.a 2 1.a even 1 1 trivial
1900.1.g.a 2 5.b even 2 1 inner
1900.1.g.a 2 19.b odd 2 1 CM
1900.1.g.a 2 95.d odd 2 1 inner
2736.1.o.b 1 60.l odd 4 1
2736.1.o.b 1 1140.w even 4 1
3724.1.e.c 1 35.f even 4 1
3724.1.e.c 1 665.n odd 4 1
3724.1.bc.b 2 35.k even 12 2
3724.1.bc.b 2 665.ca odd 12 2
3724.1.bc.c 2 35.l odd 12 2
3724.1.bc.c 2 665.ck even 12 2

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1900, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 1$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 1$$
$47$ $$T^{2} + 1$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 1)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 1$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 4$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$