# Properties

 Label 4950.2.a.ba Level $4950$ Weight $2$ Character orbit 4950.a Self dual yes Analytic conductor $39.526$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4950,2,Mod(1,4950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4950, 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("4950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4950.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: $$39.5259490005$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - 3 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - 3 * q^7 + q^8 $$q + q^{2} + q^{4} - 3 q^{7} + q^{8} - q^{11} - 4 q^{13} - 3 q^{14} + q^{16} + 3 q^{17} - 5 q^{19} - q^{22} + 4 q^{23} - 4 q^{26} - 3 q^{28} - 5 q^{29} + 7 q^{31} + q^{32} + 3 q^{34} + 7 q^{37} - 5 q^{38} + 8 q^{41} + 6 q^{43} - q^{44} + 4 q^{46} + 8 q^{47} + 2 q^{49} - 4 q^{52} + 9 q^{53} - 3 q^{56} - 5 q^{58} - 13 q^{61} + 7 q^{62} + q^{64} + 12 q^{67} + 3 q^{68} + 3 q^{71} + 6 q^{73} + 7 q^{74} - 5 q^{76} + 3 q^{77} + 8 q^{82} + 4 q^{83} + 6 q^{86} - q^{88} + 15 q^{89} + 12 q^{91} + 4 q^{92} + 8 q^{94} + 12 q^{97} + 2 q^{98}+O(q^{100})$$ q + q^2 + q^4 - 3 * q^7 + q^8 - q^11 - 4 * q^13 - 3 * q^14 + q^16 + 3 * q^17 - 5 * q^19 - q^22 + 4 * q^23 - 4 * q^26 - 3 * q^28 - 5 * q^29 + 7 * q^31 + q^32 + 3 * q^34 + 7 * q^37 - 5 * q^38 + 8 * q^41 + 6 * q^43 - q^44 + 4 * q^46 + 8 * q^47 + 2 * q^49 - 4 * q^52 + 9 * q^53 - 3 * q^56 - 5 * q^58 - 13 * q^61 + 7 * q^62 + q^64 + 12 * q^67 + 3 * q^68 + 3 * q^71 + 6 * q^73 + 7 * q^74 - 5 * q^76 + 3 * q^77 + 8 * q^82 + 4 * q^83 + 6 * q^86 - q^88 + 15 * q^89 + 12 * q^91 + 4 * q^92 + 8 * q^94 + 12 * q^97 + 2 * q^98

## 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
 0
1.00000 0 1.00000 0 0 −3.00000 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4950.2.a.ba 1
3.b odd 2 1 550.2.a.e 1
5.b even 2 1 4950.2.a.q 1
5.c odd 4 2 990.2.c.d 2
12.b even 2 1 4400.2.a.k 1
15.d odd 2 1 550.2.a.j 1
15.e even 4 2 110.2.b.a 2
33.d even 2 1 6050.2.a.bk 1
60.h even 2 1 4400.2.a.s 1
60.l odd 4 2 880.2.b.a 2
165.d even 2 1 6050.2.a.f 1
165.l odd 4 2 1210.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.b.a 2 15.e even 4 2
550.2.a.e 1 3.b odd 2 1
550.2.a.j 1 15.d odd 2 1
880.2.b.a 2 60.l odd 4 2
990.2.c.d 2 5.c odd 4 2
1210.2.b.a 2 165.l odd 4 2
4400.2.a.k 1 12.b even 2 1
4400.2.a.s 1 60.h even 2 1
4950.2.a.q 1 5.b even 2 1
4950.2.a.ba 1 1.a even 1 1 trivial
6050.2.a.f 1 165.d even 2 1
6050.2.a.bk 1 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4950))$$:

 $$T_{7} + 3$$ T7 + 3 $$T_{13} + 4$$ T13 + 4 $$T_{17} - 3$$ T17 - 3 $$T_{19} + 5$$ T19 + 5 $$T_{23} - 4$$ T23 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 3$$
$11$ $$T + 1$$
$13$ $$T + 4$$
$17$ $$T - 3$$
$19$ $$T + 5$$
$23$ $$T - 4$$
$29$ $$T + 5$$
$31$ $$T - 7$$
$37$ $$T - 7$$
$41$ $$T - 8$$
$43$ $$T - 6$$
$47$ $$T - 8$$
$53$ $$T - 9$$
$59$ $$T$$
$61$ $$T + 13$$
$67$ $$T - 12$$
$71$ $$T - 3$$
$73$ $$T - 6$$
$79$ $$T$$
$83$ $$T - 4$$
$89$ $$T - 15$$
$97$ $$T - 12$$