# Properties

 Label 4650.2.a.x Level $4650$ Weight $2$ Character orbit 4650.a Self dual yes Analytic conductor $37.130$ Analytic rank $1$ 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] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, 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("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.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: $$37.1304369399$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} - 4 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 - q^6 - 4 * q^7 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} - q^{6} - 4 q^{7} + q^{8} + q^{9} + 2 q^{11} - q^{12} - 2 q^{13} - 4 q^{14} + q^{16} + q^{18} + 4 q^{21} + 2 q^{22} + 6 q^{23} - q^{24} - 2 q^{26} - q^{27} - 4 q^{28} + q^{31} + q^{32} - 2 q^{33} + q^{36} + 2 q^{37} + 2 q^{39} - 10 q^{41} + 4 q^{42} + 4 q^{43} + 2 q^{44} + 6 q^{46} - 4 q^{47} - q^{48} + 9 q^{49} - 2 q^{52} - 6 q^{53} - q^{54} - 4 q^{56} - 4 q^{59} + q^{62} - 4 q^{63} + q^{64} - 2 q^{66} - 4 q^{67} - 6 q^{69} - 16 q^{71} + q^{72} - 4 q^{73} + 2 q^{74} - 8 q^{77} + 2 q^{78} + 4 q^{79} + q^{81} - 10 q^{82} - 8 q^{83} + 4 q^{84} + 4 q^{86} + 2 q^{88} + 6 q^{89} + 8 q^{91} + 6 q^{92} - q^{93} - 4 q^{94} - q^{96} - 14 q^{97} + 9 q^{98} + 2 q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 - q^6 - 4 * q^7 + q^8 + q^9 + 2 * q^11 - q^12 - 2 * q^13 - 4 * q^14 + q^16 + q^18 + 4 * q^21 + 2 * q^22 + 6 * q^23 - q^24 - 2 * q^26 - q^27 - 4 * q^28 + q^31 + q^32 - 2 * q^33 + q^36 + 2 * q^37 + 2 * q^39 - 10 * q^41 + 4 * q^42 + 4 * q^43 + 2 * q^44 + 6 * q^46 - 4 * q^47 - q^48 + 9 * q^49 - 2 * q^52 - 6 * q^53 - q^54 - 4 * q^56 - 4 * q^59 + q^62 - 4 * q^63 + q^64 - 2 * q^66 - 4 * q^67 - 6 * q^69 - 16 * q^71 + q^72 - 4 * q^73 + 2 * q^74 - 8 * q^77 + 2 * q^78 + 4 * q^79 + q^81 - 10 * q^82 - 8 * q^83 + 4 * q^84 + 4 * q^86 + 2 * q^88 + 6 * q^89 + 8 * q^91 + 6 * q^92 - q^93 - 4 * q^94 - q^96 - 14 * q^97 + 9 * q^98 + 2 * q^99

## 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 −1.00000 1.00000 0 −1.00000 −4.00000 1.00000 1.00000 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
$$2$$ $$-1$$
$$3$$ $$+1$$
$$5$$ $$+1$$
$$31$$ $$-1$$

## 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 4650.2.a.x 1
5.b even 2 1 930.2.a.j 1
5.c odd 4 2 4650.2.d.h 2
15.d odd 2 1 2790.2.a.v 1
20.d odd 2 1 7440.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.j 1 5.b even 2 1
2790.2.a.v 1 15.d odd 2 1
4650.2.a.x 1 1.a even 1 1 trivial
4650.2.d.h 2 5.c odd 4 2
7440.2.a.g 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4650))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11} - 2$$ T11 - 2 $$T_{13} + 2$$ T13 + 2 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T - 2$$
$13$ $$T + 2$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 6$$
$29$ $$T$$
$31$ $$T - 1$$
$37$ $$T - 2$$
$41$ $$T + 10$$
$43$ $$T - 4$$
$47$ $$T + 4$$
$53$ $$T + 6$$
$59$ $$T + 4$$
$61$ $$T$$
$67$ $$T + 4$$
$71$ $$T + 16$$
$73$ $$T + 4$$
$79$ $$T - 4$$
$83$ $$T + 8$$
$89$ $$T - 6$$
$97$ $$T + 14$$