# Properties

 Label 4900.2.a.be Level $4900$ Weight $2$ Character orbit 4900.a Self dual yes Analytic conductor $39.127$ Analytic rank $1$ Dimension $4$ 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(1,4900)]

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, 0, 0]))

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

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

chi := DirichletCharacter("4900.1");

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.a (trivial)

## 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: yes Analytic conductor: $$39.1266969904$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 11x^{2} + 16$$ x^4 - 11*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3}+O(q^{10})$$ q + b2 * q^3 $$q + \beta_{2} q^{3} + ( - \beta_{3} - 2) q^{11} + ( - 2 \beta_{2} + 2 \beta_1) q^{13} + (2 \beta_{2} - \beta_1) q^{17} + \beta_{3} q^{19} + ( - \beta_{2} - 2 \beta_1) q^{23} - 3 \beta_{2} q^{27} + (\beta_{3} + 1) q^{29} + \beta_{3} q^{31} - 3 \beta_1 q^{33} + ( - 4 \beta_{2} - \beta_1) q^{37} + (2 \beta_{3} - 2) q^{39} + (\beta_{3} - 7) q^{41} + ( - 3 \beta_{2} + \beta_1) q^{43} + (2 \beta_{2} + \beta_1) q^{47} + ( - \beta_{3} + 4) q^{51} + (2 \beta_{2} - 3 \beta_1) q^{53} + ( - 2 \beta_{2} + 3 \beta_1) q^{57} - \beta_{3} q^{59} + ( - 2 \beta_{3} - 7) q^{61} + ( - 5 \beta_{2} + 4 \beta_1) q^{67} + ( - 2 \beta_{3} - 7) q^{69} + ( - 2 \beta_{3} + 2) q^{71} + (2 \beta_{2} + \beta_1) q^{73} + (\beta_{3} + 4) q^{79} - 9 q^{81} + (\beta_{2} - 3 \beta_1) q^{83} + ( - \beta_{2} + 3 \beta_1) q^{87} - 7 q^{89} + ( - 2 \beta_{2} + 3 \beta_1) q^{93} - 4 \beta_{2} q^{97}+O(q^{100})$$ q + b2 * q^3 + (-b3 - 2) * q^11 + (-2*b2 + 2*b1) * q^13 + (2*b2 - b1) * q^17 + b3 * q^19 + (-b2 - 2*b1) * q^23 - 3*b2 * q^27 + (b3 + 1) * q^29 + b3 * q^31 - 3*b1 * q^33 + (-4*b2 - b1) * q^37 + (2*b3 - 2) * q^39 + (b3 - 7) * q^41 + (-3*b2 + b1) * q^43 + (2*b2 + b1) * q^47 + (-b3 + 4) * q^51 + (2*b2 - 3*b1) * q^53 + (-2*b2 + 3*b1) * q^57 - b3 * q^59 + (-2*b3 - 7) * q^61 + (-5*b2 + 4*b1) * q^67 + (-2*b3 - 7) * q^69 + (-2*b3 + 2) * q^71 + (2*b2 + b1) * q^73 + (b3 + 4) * q^79 - 9 * q^81 + (b2 - 3*b1) * q^83 + (-b2 + 3*b1) * q^87 - 7 * q^89 + (-2*b2 + 3*b1) * q^93 - 4*b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 6 q^{11} - 2 q^{19} + 2 q^{29} - 2 q^{31} - 12 q^{39} - 30 q^{41} + 18 q^{51} + 2 q^{59} - 24 q^{61} - 24 q^{69} + 12 q^{71} + 14 q^{79} - 36 q^{81} - 28 q^{89}+O(q^{100})$$ 4 * q - 6 * q^11 - 2 * q^19 + 2 * q^29 - 2 * q^31 - 12 * q^39 - 30 * q^41 + 18 * q^51 + 2 * q^59 - 24 * q^61 - 24 * q^69 + 12 * q^71 + 14 * q^79 - 36 * q^81 - 28 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 11x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 7\nu ) / 4$$ (v^3 - 7*v) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 6$$ v^2 - 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 6$$ b3 + 6 $$\nu^{3}$$ $$=$$ $$4\beta_{2} + 7\beta_1$$ 4*b2 + 7*b1

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.04547 1.31342 3.04547 −1.31342
0 −1.73205 0 0 0 0 0 0 0
1.2 0 −1.73205 0 0 0 0 0 0 0
1.3 0 1.73205 0 0 0 0 0 0 0
1.4 0 1.73205 0 0 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## 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.a.be 4
5.b even 2 1 inner 4900.2.a.be 4
5.c odd 4 2 980.2.e.f 4
7.b odd 2 1 4900.2.a.bf 4
7.c even 3 2 700.2.i.f 8
35.c odd 2 1 4900.2.a.bf 4
35.f even 4 2 980.2.e.c 4
35.j even 6 2 700.2.i.f 8
35.k even 12 2 980.2.q.b 4
35.k even 12 2 980.2.q.g 4
35.l odd 12 2 140.2.q.a 4
35.l odd 12 2 140.2.q.b yes 4
105.x even 12 2 1260.2.bm.a 4
105.x even 12 2 1260.2.bm.b 4
140.w even 12 2 560.2.bw.a 4
140.w even 12 2 560.2.bw.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.q.a 4 35.l odd 12 2
140.2.q.b yes 4 35.l odd 12 2
560.2.bw.a 4 140.w even 12 2
560.2.bw.e 4 140.w even 12 2
700.2.i.f 8 7.c even 3 2
700.2.i.f 8 35.j even 6 2
980.2.e.c 4 35.f even 4 2
980.2.e.f 4 5.c odd 4 2
980.2.q.b 4 35.k even 12 2
980.2.q.g 4 35.k even 12 2
1260.2.bm.a 4 105.x even 12 2
1260.2.bm.b 4 105.x even 12 2
4900.2.a.be 4 1.a even 1 1 trivial
4900.2.a.be 4 5.b even 2 1 inner
4900.2.a.bf 4 7.b odd 2 1
4900.2.a.bf 4 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4900))$$:

 $$T_{3}^{2} - 3$$ T3^2 - 3 $$T_{11}^{2} + 3T_{11} - 12$$ T11^2 + 3*T11 - 12 $$T_{13}^{4} - 44T_{13}^{2} + 256$$ T13^4 - 44*T13^2 + 256 $$T_{19}^{2} + T_{19} - 14$$ T19^2 + T19 - 14 $$T_{23}^{4} - 62T_{23}^{2} + 49$$ T23^4 - 62*T23^2 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 3 T - 12)^{2}$$
$13$ $$T^{4} - 44T^{2} + 256$$
$17$ $$T^{4} - 23T^{2} + 4$$
$19$ $$(T^{2} + T - 14)^{2}$$
$23$ $$T^{4} - 62T^{2} + 49$$
$29$ $$(T^{2} - T - 14)^{2}$$
$31$ $$(T^{2} + T - 14)^{2}$$
$37$ $$T^{4} - 131T^{2} + 3136$$
$41$ $$(T^{2} + 15 T + 42)^{2}$$
$43$ $$T^{4} - 47T^{2} + 196$$
$47$ $$T^{4} - 47T^{2} + 196$$
$53$ $$T^{4} - 87T^{2} + 1764$$
$59$ $$(T^{2} - T - 14)^{2}$$
$61$ $$(T^{2} + 12 T - 21)^{2}$$
$67$ $$T^{4} - 206T^{2} + 2401$$
$71$ $$(T^{2} - 6 T - 48)^{2}$$
$73$ $$T^{4} - 47T^{2} + 196$$
$79$ $$(T^{2} - 7 T - 2)^{2}$$
$83$ $$T^{4} - 87T^{2} + 1764$$
$89$ $$(T + 7)^{4}$$
$97$ $$(T^{2} - 48)^{2}$$