# Properties

 Label 4900.2.e.m Level $4900$ Weight $2$ Character orbit 4900.e Analytic conductor $39.127$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4900,2,Mod(2549,4900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4900, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4900.2549");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4900 = 2^{2} \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4900.e (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: $$39.1266969904$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 2 q^{9}+O(q^{10})$$ q + i * q^3 + 2 * q^9 $$q + i q^{3} + 2 q^{9} + 6 q^{11} + 2 i q^{13} + 6 i q^{17} - 8 q^{19} + 3 i q^{23} + 5 i q^{27} - 3 q^{29} + 2 q^{31} + 6 i q^{33} - 8 i q^{37} - 2 q^{39} - 3 q^{41} + 5 i q^{43} - 6 q^{51} + 12 i q^{53} - 8 i q^{57} - q^{61} + 7 i q^{67} - 3 q^{69} - 10 i q^{73} + 4 q^{79} + q^{81} + 3 i q^{83} - 3 i q^{87} + 3 q^{89} + 2 i q^{93} + 10 i q^{97} + 12 q^{99} +O(q^{100})$$ q + i * q^3 + 2 * q^9 + 6 * q^11 + 2*i * q^13 + 6*i * q^17 - 8 * q^19 + 3*i * q^23 + 5*i * q^27 - 3 * q^29 + 2 * q^31 + 6*i * q^33 - 8*i * q^37 - 2 * q^39 - 3 * q^41 + 5*i * q^43 - 6 * q^51 + 12*i * q^53 - 8*i * q^57 - q^61 + 7*i * q^67 - 3 * q^69 - 10*i * q^73 + 4 * q^79 + q^81 + 3*i * q^83 - 3*i * q^87 + 3 * q^89 + 2*i * q^93 + 10*i * q^97 + 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} + 12 q^{11} - 16 q^{19} - 6 q^{29} + 4 q^{31} - 4 q^{39} - 6 q^{41} - 12 q^{51} - 2 q^{61} - 6 q^{69} + 8 q^{79} + 2 q^{81} + 6 q^{89} + 24 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 + 12 * q^11 - 16 * q^19 - 6 * q^29 + 4 * q^31 - 4 * q^39 - 6 * q^41 - 12 * q^51 - 2 * q^61 - 6 * q^69 + 8 * q^79 + 2 * q^81 + 6 * q^89 + 24 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$2451$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2549.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 0 0 2.00000 0
2549.2 0 1.00000i 0 0 0 0 0 2.00000 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
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 4900.2.e.m 2
5.b even 2 1 inner 4900.2.e.m 2
5.c odd 4 1 980.2.a.g 1
5.c odd 4 1 4900.2.a.i 1
7.b odd 2 1 4900.2.e.n 2
7.c even 3 2 700.2.r.a 4
15.e even 4 1 8820.2.a.p 1
20.e even 4 1 3920.2.a.k 1
35.c odd 2 1 4900.2.e.n 2
35.f even 4 1 980.2.a.e 1
35.f even 4 1 4900.2.a.q 1
35.j even 6 2 700.2.r.a 4
35.k even 12 2 980.2.i.f 2
35.l odd 12 2 140.2.i.a 2
35.l odd 12 2 700.2.i.b 2
105.k odd 4 1 8820.2.a.a 1
105.x even 12 2 1260.2.s.c 2
140.j odd 4 1 3920.2.a.w 1
140.w even 12 2 560.2.q.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.a 2 35.l odd 12 2
560.2.q.f 2 140.w even 12 2
700.2.i.b 2 35.l odd 12 2
700.2.r.a 4 7.c even 3 2
700.2.r.a 4 35.j even 6 2
980.2.a.e 1 35.f even 4 1
980.2.a.g 1 5.c odd 4 1
980.2.i.f 2 35.k even 12 2
1260.2.s.c 2 105.x even 12 2
3920.2.a.k 1 20.e even 4 1
3920.2.a.w 1 140.j odd 4 1
4900.2.a.i 1 5.c odd 4 1
4900.2.a.q 1 35.f even 4 1
4900.2.e.m 2 1.a even 1 1 trivial
4900.2.e.m 2 5.b even 2 1 inner
4900.2.e.n 2 7.b odd 2 1
4900.2.e.n 2 35.c odd 2 1
8820.2.a.a 1 105.k odd 4 1
8820.2.a.p 1 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4900, [\chi])$$:

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{11} - 6$$ T11 - 6 $$T_{19} + 8$$ T19 + 8 $$T_{31} - 2$$ T31 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 6)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 36$$
$19$ $$(T + 8)^{2}$$
$23$ $$T^{2} + 9$$
$29$ $$(T + 3)^{2}$$
$31$ $$(T - 2)^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T + 3)^{2}$$
$43$ $$T^{2} + 25$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 144$$
$59$ $$T^{2}$$
$61$ $$(T + 1)^{2}$$
$67$ $$T^{2} + 49$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 9$$
$89$ $$(T - 3)^{2}$$
$97$ $$T^{2} + 100$$