# Properties

 Label 4900.2.e.q Level $4900$ Weight $2$ Character orbit 4900.e Analytic conductor $39.127$ Analytic rank $0$ 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(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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 980) 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 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1) q^{3} - 2 \beta_{3} q^{9}+O(q^{10})$$ q + (b2 + b1) * q^3 - 2*b3 * q^9 $$q + (\beta_{2} + \beta_1) q^{3} - 2 \beta_{3} q^{9} + (2 \beta_{3} - 1) q^{11} + (\beta_{2} + 5 \beta_1) q^{13} + (\beta_{2} - 5 \beta_1) q^{17} + ( - 4 \beta_{3} - 2) q^{19} + ( - \beta_{2} - 2 \beta_1) q^{23} + (\beta_{2} - \beta_1) q^{27} + (4 \beta_{3} - 1) q^{29} + ( - \beta_{3} - 6) q^{31} + (\beta_{2} + 3 \beta_1) q^{33} + (\beta_{2} - 2 \beta_1) q^{37} + ( - 6 \beta_{3} - 7) q^{39} + ( - \beta_{3} - 2) q^{41} + (4 \beta_{2} - 6 \beta_1) q^{43} + ( - 7 \beta_{2} - \beta_1) q^{47} + (4 \beta_{3} + 3) q^{51} + (3 \beta_{2} + 8 \beta_1) q^{53} + ( - 6 \beta_{2} - 10 \beta_1) q^{57} + ( - \beta_{3} + 2) q^{59} + ( - 2 \beta_{3} - 8) q^{61} + (5 \beta_{2} - 4 \beta_1) q^{67} + (3 \beta_{3} + 4) q^{69} + ( - 6 \beta_{3} - 2) q^{71} + (2 \beta_{2} + 8 \beta_1) q^{73} + (10 \beta_{3} + 1) q^{79} + ( - 6 \beta_{3} - 1) q^{81} - 8 \beta_1 q^{83} + (3 \beta_{2} + 7 \beta_1) q^{87} - 12 \beta_{3} q^{89} + ( - 7 \beta_{2} - 8 \beta_1) q^{93} + (9 \beta_{2} - 3 \beta_1) q^{97} + (2 \beta_{3} - 8) q^{99}+O(q^{100})$$ q + (b2 + b1) * q^3 - 2*b3 * q^9 + (2*b3 - 1) * q^11 + (b2 + 5*b1) * q^13 + (b2 - 5*b1) * q^17 + (-4*b3 - 2) * q^19 + (-b2 - 2*b1) * q^23 + (b2 - b1) * q^27 + (4*b3 - 1) * q^29 + (-b3 - 6) * q^31 + (b2 + 3*b1) * q^33 + (b2 - 2*b1) * q^37 + (-6*b3 - 7) * q^39 + (-b3 - 2) * q^41 + (4*b2 - 6*b1) * q^43 + (-7*b2 - b1) * q^47 + (4*b3 + 3) * q^51 + (3*b2 + 8*b1) * q^53 + (-6*b2 - 10*b1) * q^57 + (-b3 + 2) * q^59 + (-2*b3 - 8) * q^61 + (5*b2 - 4*b1) * q^67 + (3*b3 + 4) * q^69 + (-6*b3 - 2) * q^71 + (2*b2 + 8*b1) * q^73 + (10*b3 + 1) * q^79 + (-6*b3 - 1) * q^81 - 8*b1 * q^83 + (3*b2 + 7*b1) * q^87 - 12*b3 * q^89 + (-7*b2 - 8*b1) * q^93 + (9*b2 - 3*b1) * q^97 + (2*b3 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 4 q^{11} - 8 q^{19} - 4 q^{29} - 24 q^{31} - 28 q^{39} - 8 q^{41} + 12 q^{51} + 8 q^{59} - 32 q^{61} + 16 q^{69} - 8 q^{71} + 4 q^{79} - 4 q^{81} - 32 q^{99}+O(q^{100})$$ 4 * q - 4 * q^11 - 8 * q^19 - 4 * q^29 - 24 * q^31 - 28 * q^39 - 8 * q^41 + 12 * q^51 + 8 * q^59 - 32 * q^61 + 16 * q^69 - 8 * q^71 + 4 * q^79 - 4 * q^81 - 32 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## 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
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
0 2.41421i 0 0 0 0 0 −2.82843 0
2549.2 0 0.414214i 0 0 0 0 0 2.82843 0
2549.3 0 0.414214i 0 0 0 0 0 2.82843 0
2549.4 0 2.41421i 0 0 0 0 0 −2.82843 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.q 4
5.b even 2 1 inner 4900.2.e.q 4
5.c odd 4 1 980.2.a.j 2
5.c odd 4 1 4900.2.a.z 2
7.b odd 2 1 4900.2.e.r 4
15.e even 4 1 8820.2.a.bg 2
20.e even 4 1 3920.2.a.bx 2
35.c odd 2 1 4900.2.e.r 4
35.f even 4 1 980.2.a.k yes 2
35.f even 4 1 4900.2.a.x 2
35.k even 12 2 980.2.i.k 4
35.l odd 12 2 980.2.i.l 4
105.k odd 4 1 8820.2.a.bl 2
140.j odd 4 1 3920.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.j 2 5.c odd 4 1
980.2.a.k yes 2 35.f even 4 1
980.2.i.k 4 35.k even 12 2
980.2.i.l 4 35.l odd 12 2
3920.2.a.bo 2 140.j odd 4 1
3920.2.a.bx 2 20.e even 4 1
4900.2.a.x 2 35.f even 4 1
4900.2.a.z 2 5.c odd 4 1
4900.2.e.q 4 1.a even 1 1 trivial
4900.2.e.q 4 5.b even 2 1 inner
4900.2.e.r 4 7.b odd 2 1
4900.2.e.r 4 35.c odd 2 1
8820.2.a.bg 2 15.e even 4 1
8820.2.a.bl 2 105.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 $$S_{2}^{\mathrm{new}}(4900, [\chi])$$:

 $$T_{3}^{4} + 6T_{3}^{2} + 1$$ T3^4 + 6*T3^2 + 1 $$T_{11}^{2} + 2T_{11} - 7$$ T11^2 + 2*T11 - 7 $$T_{19}^{2} + 4T_{19} - 28$$ T19^2 + 4*T19 - 28 $$T_{31}^{2} + 12T_{31} + 34$$ T31^2 + 12*T31 + 34

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 6T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 2 T - 7)^{2}$$
$13$ $$T^{4} + 54T^{2} + 529$$
$17$ $$T^{4} + 54T^{2} + 529$$
$19$ $$(T^{2} + 4 T - 28)^{2}$$
$23$ $$T^{4} + 12T^{2} + 4$$
$29$ $$(T^{2} + 2 T - 31)^{2}$$
$31$ $$(T^{2} + 12 T + 34)^{2}$$
$37$ $$T^{4} + 12T^{2} + 4$$
$41$ $$(T^{2} + 4 T + 2)^{2}$$
$43$ $$T^{4} + 136T^{2} + 16$$
$47$ $$T^{4} + 198T^{2} + 9409$$
$53$ $$T^{4} + 164T^{2} + 2116$$
$59$ $$(T^{2} - 4 T + 2)^{2}$$
$61$ $$(T^{2} + 16 T + 56)^{2}$$
$67$ $$T^{4} + 132T^{2} + 1156$$
$71$ $$(T^{2} + 4 T - 68)^{2}$$
$73$ $$T^{4} + 144T^{2} + 3136$$
$79$ $$(T^{2} - 2 T - 199)^{2}$$
$83$ $$(T^{2} + 64)^{2}$$
$89$ $$(T^{2} - 288)^{2}$$
$97$ $$T^{4} + 342 T^{2} + 23409$$