# Properties

 Label 4640.2.a.i Level $4640$ Weight $2$ Character orbit 4640.a Self dual yes Analytic conductor $37.051$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4640,2,Mod(1,4640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4640, 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("4640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4640 = 2^{5} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4640.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: $$37.0505865379$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - q^{5} + ( - 3 \beta + 1) q^{7} + (\beta - 2) q^{9}+O(q^{10})$$ q + b * q^3 - q^5 + (-3*b + 1) * q^7 + (b - 2) * q^9 $$q + \beta q^{3} - q^{5} + ( - 3 \beta + 1) q^{7} + (\beta - 2) q^{9} + ( - 2 \beta + 2) q^{11} + (5 \beta - 2) q^{13} - \beta q^{15} + (\beta - 3) q^{17} + ( - 2 \beta - 4) q^{19} + ( - 2 \beta - 3) q^{21} + (3 \beta - 4) q^{23} + q^{25} + ( - 4 \beta + 1) q^{27} + q^{29} + ( - \beta + 1) q^{31} - 2 q^{33} + (3 \beta - 1) q^{35} + ( - 2 \beta + 8) q^{37} + (3 \beta + 5) q^{39} + 6 \beta q^{41} + ( - \beta + 7) q^{43} + ( - \beta + 2) q^{45} + 8 q^{47} + (3 \beta + 3) q^{49} + ( - 2 \beta + 1) q^{51} + ( - 7 \beta + 5) q^{53} + (2 \beta - 2) q^{55} + ( - 6 \beta - 2) q^{57} + (\beta + 10) q^{59} + (3 \beta + 4) q^{61} + (4 \beta - 5) q^{63} + ( - 5 \beta + 2) q^{65} - 4 \beta q^{67} + ( - \beta + 3) q^{69} + (8 \beta - 8) q^{71} + (5 \beta + 5) q^{73} + \beta q^{75} + ( - 2 \beta + 8) q^{77} + (5 \beta + 6) q^{79} + ( - 6 \beta + 2) q^{81} + (6 \beta - 12) q^{83} + ( - \beta + 3) q^{85} + \beta q^{87} + (6 \beta + 2) q^{89} + ( - 4 \beta - 17) q^{91} - q^{93} + (2 \beta + 4) q^{95} + ( - 11 \beta + 2) q^{97} + (4 \beta - 6) q^{99} +O(q^{100})$$ q + b * q^3 - q^5 + (-3*b + 1) * q^7 + (b - 2) * q^9 + (-2*b + 2) * q^11 + (5*b - 2) * q^13 - b * q^15 + (b - 3) * q^17 + (-2*b - 4) * q^19 + (-2*b - 3) * q^21 + (3*b - 4) * q^23 + q^25 + (-4*b + 1) * q^27 + q^29 + (-b + 1) * q^31 - 2 * q^33 + (3*b - 1) * q^35 + (-2*b + 8) * q^37 + (3*b + 5) * q^39 + 6*b * q^41 + (-b + 7) * q^43 + (-b + 2) * q^45 + 8 * q^47 + (3*b + 3) * q^49 + (-2*b + 1) * q^51 + (-7*b + 5) * q^53 + (2*b - 2) * q^55 + (-6*b - 2) * q^57 + (b + 10) * q^59 + (3*b + 4) * q^61 + (4*b - 5) * q^63 + (-5*b + 2) * q^65 - 4*b * q^67 + (-b + 3) * q^69 + (8*b - 8) * q^71 + (5*b + 5) * q^73 + b * q^75 + (-2*b + 8) * q^77 + (5*b + 6) * q^79 + (-6*b + 2) * q^81 + (6*b - 12) * q^83 + (-b + 3) * q^85 + b * q^87 + (6*b + 2) * q^89 + (-4*b - 17) * q^91 - q^93 + (2*b + 4) * q^95 + (-11*b + 2) * q^97 + (4*b - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 2 q^{5} - q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q + q^3 - 2 * q^5 - q^7 - 3 * q^9 $$2 q + q^{3} - 2 q^{5} - q^{7} - 3 q^{9} + 2 q^{11} + q^{13} - q^{15} - 5 q^{17} - 10 q^{19} - 8 q^{21} - 5 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{29} + q^{31} - 4 q^{33} + q^{35} + 14 q^{37} + 13 q^{39} + 6 q^{41} + 13 q^{43} + 3 q^{45} + 16 q^{47} + 9 q^{49} + 3 q^{53} - 2 q^{55} - 10 q^{57} + 21 q^{59} + 11 q^{61} - 6 q^{63} - q^{65} - 4 q^{67} + 5 q^{69} - 8 q^{71} + 15 q^{73} + q^{75} + 14 q^{77} + 17 q^{79} - 2 q^{81} - 18 q^{83} + 5 q^{85} + q^{87} + 10 q^{89} - 38 q^{91} - 2 q^{93} + 10 q^{95} - 7 q^{97} - 8 q^{99}+O(q^{100})$$ 2 * q + q^3 - 2 * q^5 - q^7 - 3 * q^9 + 2 * q^11 + q^13 - q^15 - 5 * q^17 - 10 * q^19 - 8 * q^21 - 5 * q^23 + 2 * q^25 - 2 * q^27 + 2 * q^29 + q^31 - 4 * q^33 + q^35 + 14 * q^37 + 13 * q^39 + 6 * q^41 + 13 * q^43 + 3 * q^45 + 16 * q^47 + 9 * q^49 + 3 * q^53 - 2 * q^55 - 10 * q^57 + 21 * q^59 + 11 * q^61 - 6 * q^63 - q^65 - 4 * q^67 + 5 * q^69 - 8 * q^71 + 15 * q^73 + q^75 + 14 * q^77 + 17 * q^79 - 2 * q^81 - 18 * q^83 + 5 * q^85 + q^87 + 10 * q^89 - 38 * q^91 - 2 * q^93 + 10 * q^95 - 7 * q^97 - 8 * 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
 −0.618034 1.61803
0 −0.618034 0 −1.00000 0 2.85410 0 −2.61803 0
1.2 0 1.61803 0 −1.00000 0 −3.85410 0 −0.381966 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
$$2$$ $$+1$$
$$5$$ $$+1$$
$$29$$ $$-1$$

## 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 4640.2.a.i yes 2
4.b odd 2 1 4640.2.a.g 2
8.b even 2 1 9280.2.a.y 2
8.d odd 2 1 9280.2.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4640.2.a.g 2 4.b odd 2 1
4640.2.a.i yes 2 1.a even 1 1 trivial
9280.2.a.y 2 8.b even 2 1
9280.2.a.bd 2 8.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4640))$$:

 $$T_{3}^{2} - T_{3} - 1$$ T3^2 - T3 - 1 $$T_{7}^{2} + T_{7} - 11$$ T7^2 + T7 - 11 $$T_{11}^{2} - 2T_{11} - 4$$ T11^2 - 2*T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 1$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + T - 11$$
$11$ $$T^{2} - 2T - 4$$
$13$ $$T^{2} - T - 31$$
$17$ $$T^{2} + 5T + 5$$
$19$ $$T^{2} + 10T + 20$$
$23$ $$T^{2} + 5T - 5$$
$29$ $$(T - 1)^{2}$$
$31$ $$T^{2} - T - 1$$
$37$ $$T^{2} - 14T + 44$$
$41$ $$T^{2} - 6T - 36$$
$43$ $$T^{2} - 13T + 41$$
$47$ $$(T - 8)^{2}$$
$53$ $$T^{2} - 3T - 59$$
$59$ $$T^{2} - 21T + 109$$
$61$ $$T^{2} - 11T + 19$$
$67$ $$T^{2} + 4T - 16$$
$71$ $$T^{2} + 8T - 64$$
$73$ $$T^{2} - 15T + 25$$
$79$ $$T^{2} - 17T + 41$$
$83$ $$T^{2} + 18T + 36$$
$89$ $$T^{2} - 10T - 20$$
$97$ $$T^{2} + 7T - 139$$
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