# Properties

 Label 4650.2.a.bp 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + q^6 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} - 4 q^{11} + q^{12} - 6 q^{13} + q^{16} - 2 q^{17} + q^{18} - 4 q^{19} - 4 q^{22} + 4 q^{23} + q^{24} - 6 q^{26} + q^{27} + 2 q^{29} - q^{31} + q^{32} - 4 q^{33} - 2 q^{34} + q^{36} + 2 q^{37} - 4 q^{38} - 6 q^{39} - 6 q^{41} + 4 q^{43} - 4 q^{44} + 4 q^{46} + q^{48} - 7 q^{49} - 2 q^{51} - 6 q^{52} - 2 q^{53} + q^{54} - 4 q^{57} + 2 q^{58} - 4 q^{59} - 6 q^{61} - q^{62} + q^{64} - 4 q^{66} - 16 q^{67} - 2 q^{68} + 4 q^{69} - 12 q^{71} + q^{72} + 6 q^{73} + 2 q^{74} - 4 q^{76} - 6 q^{78} - 16 q^{79} + q^{81} - 6 q^{82} + 12 q^{83} + 4 q^{86} + 2 q^{87} - 4 q^{88} - 18 q^{89} + 4 q^{92} - q^{93} + q^{96} + 14 q^{97} - 7 q^{98} - 4 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 + q^8 + q^9 - 4 * q^11 + q^12 - 6 * q^13 + q^16 - 2 * q^17 + q^18 - 4 * q^19 - 4 * q^22 + 4 * q^23 + q^24 - 6 * q^26 + q^27 + 2 * q^29 - q^31 + q^32 - 4 * q^33 - 2 * q^34 + q^36 + 2 * q^37 - 4 * q^38 - 6 * q^39 - 6 * q^41 + 4 * q^43 - 4 * q^44 + 4 * q^46 + q^48 - 7 * q^49 - 2 * q^51 - 6 * q^52 - 2 * q^53 + q^54 - 4 * q^57 + 2 * q^58 - 4 * q^59 - 6 * q^61 - q^62 + q^64 - 4 * q^66 - 16 * q^67 - 2 * q^68 + 4 * q^69 - 12 * q^71 + q^72 + 6 * q^73 + 2 * q^74 - 4 * q^76 - 6 * q^78 - 16 * q^79 + q^81 - 6 * q^82 + 12 * q^83 + 4 * q^86 + 2 * q^87 - 4 * q^88 - 18 * q^89 + 4 * q^92 - q^93 + q^96 + 14 * q^97 - 7 * q^98 - 4 * 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 0 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.bp 1
5.b even 2 1 930.2.a.b 1
5.c odd 4 2 4650.2.d.o 2
15.d odd 2 1 2790.2.a.ba 1
20.d odd 2 1 7440.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.b 1 5.b even 2 1
2790.2.a.ba 1 15.d odd 2 1
4650.2.a.bp 1 1.a even 1 1 trivial
4650.2.d.o 2 5.c odd 4 2
7440.2.a.q 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4650))$$:

 $$T_{7}$$ T7 $$T_{11} + 4$$ T11 + 4 $$T_{13} + 6$$ T13 + 6 $$T_{19} + 4$$ T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T + 6$$
$17$ $$T + 2$$
$19$ $$T + 4$$
$23$ $$T - 4$$
$29$ $$T - 2$$
$31$ $$T + 1$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T - 4$$
$47$ $$T$$
$53$ $$T + 2$$
$59$ $$T + 4$$
$61$ $$T + 6$$
$67$ $$T + 16$$
$71$ $$T + 12$$
$73$ $$T - 6$$
$79$ $$T + 16$$
$83$ $$T - 12$$
$89$ $$T + 18$$
$97$ $$T - 14$$