# Properties

 Label 4641.2.a.b Level $4641$ Weight $2$ Character orbit 4641.a Self dual yes Analytic conductor $37.059$ 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] = [4641,2,Mod(1,4641)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4641, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4641.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4641 = 3 \cdot 7 \cdot 13 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4641.a (trivial)

## Newform invariants

comment: select newform

sage: f = N # Warning: the index may be different

gp: f = lf \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$37.0585715781$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} - q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 - q^4 - 2 * q^5 + q^6 - q^7 + 3 * q^8 + q^9 $$q - q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} - q^{7} + 3 q^{8} + q^{9} + 2 q^{10} - 4 q^{11} + q^{12} + q^{13} + q^{14} + 2 q^{15} - q^{16} + q^{17} - q^{18} - 4 q^{19} + 2 q^{20} + q^{21} + 4 q^{22} - 3 q^{24} - q^{25} - q^{26} - q^{27} + q^{28} + 6 q^{29} - 2 q^{30} - 5 q^{32} + 4 q^{33} - q^{34} + 2 q^{35} - q^{36} - 2 q^{37} + 4 q^{38} - q^{39} - 6 q^{40} + 2 q^{41} - q^{42} + 12 q^{43} + 4 q^{44} - 2 q^{45} + q^{48} + q^{49} + q^{50} - q^{51} - q^{52} - 10 q^{53} + q^{54} + 8 q^{55} - 3 q^{56} + 4 q^{57} - 6 q^{58} + 12 q^{59} - 2 q^{60} - 10 q^{61} - q^{63} + 7 q^{64} - 2 q^{65} - 4 q^{66} + 4 q^{67} - q^{68} - 2 q^{70} + 8 q^{71} + 3 q^{72} + 10 q^{73} + 2 q^{74} + q^{75} + 4 q^{76} + 4 q^{77} + q^{78} - 8 q^{79} + 2 q^{80} + q^{81} - 2 q^{82} + 4 q^{83} - q^{84} - 2 q^{85} - 12 q^{86} - 6 q^{87} - 12 q^{88} + 10 q^{89} + 2 q^{90} - q^{91} + 8 q^{95} + 5 q^{96} + 2 q^{97} - q^{98} - 4 q^{99}+O(q^{100})$$ q - q^2 - q^3 - q^4 - 2 * q^5 + q^6 - q^7 + 3 * q^8 + q^9 + 2 * q^10 - 4 * q^11 + q^12 + q^13 + q^14 + 2 * q^15 - q^16 + q^17 - q^18 - 4 * q^19 + 2 * q^20 + q^21 + 4 * q^22 - 3 * q^24 - q^25 - q^26 - q^27 + q^28 + 6 * q^29 - 2 * q^30 - 5 * q^32 + 4 * q^33 - q^34 + 2 * q^35 - q^36 - 2 * q^37 + 4 * q^38 - q^39 - 6 * q^40 + 2 * q^41 - q^42 + 12 * q^43 + 4 * q^44 - 2 * q^45 + q^48 + q^49 + q^50 - q^51 - q^52 - 10 * q^53 + q^54 + 8 * q^55 - 3 * q^56 + 4 * q^57 - 6 * q^58 + 12 * q^59 - 2 * q^60 - 10 * q^61 - q^63 + 7 * q^64 - 2 * q^65 - 4 * q^66 + 4 * q^67 - q^68 - 2 * q^70 + 8 * q^71 + 3 * q^72 + 10 * q^73 + 2 * q^74 + q^75 + 4 * q^76 + 4 * q^77 + q^78 - 8 * q^79 + 2 * q^80 + q^81 - 2 * q^82 + 4 * q^83 - q^84 - 2 * q^85 - 12 * q^86 - 6 * q^87 - 12 * q^88 + 10 * q^89 + 2 * q^90 - q^91 + 8 * q^95 + 5 * q^96 + 2 * q^97 - 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 −2.00000 1.00000 −1.00000 3.00000 1.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$1$$
$$13$$ $$-1$$
$$17$$ $$-1$$

## 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 4641.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4641.2.a.b 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4641))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{5} + 2$$ T5 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T + 2$$
$7$ $$T + 1$$
$11$ $$T + 4$$
$13$ $$T - 1$$
$17$ $$T - 1$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T$$
$37$ $$T + 2$$
$41$ $$T - 2$$
$43$ $$T - 12$$
$47$ $$T$$
$53$ $$T + 10$$
$59$ $$T - 12$$
$61$ $$T + 10$$
$67$ $$T - 4$$
$71$ $$T - 8$$
$73$ $$T - 10$$
$79$ $$T + 8$$
$83$ $$T - 4$$
$89$ $$T - 10$$
$97$ $$T - 2$$