# Properties

 Label 4600.2.a.r Level $4600$ Weight $2$ Character orbit 4600.a Self dual yes Analytic conductor $36.731$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, 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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.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: $$36.7311849298$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} - 2 \beta q^{7} + (\beta + 1) q^{9} +O(q^{10})$$ q - b * q^3 - 2*b * q^7 + (b + 1) * q^9 $$q - \beta q^{3} - 2 \beta q^{7} + (\beta + 1) q^{9} - 4 q^{11} + (\beta - 2) q^{13} + (2 \beta - 2) q^{17} + 4 q^{19} + (2 \beta + 8) q^{21} - q^{23} + (\beta - 4) q^{27} + ( - \beta - 6) q^{29} + ( - \beta + 4) q^{31} + 4 \beta q^{33} + (2 \beta + 2) q^{37} + (\beta - 4) q^{39} + (\beta - 2) q^{41} + (2 \beta + 4) q^{43} + (3 \beta - 4) q^{47} + (4 \beta + 9) q^{49} - 8 q^{51} + (4 \beta - 6) q^{53} - 4 \beta q^{57} + ( - 4 \beta + 4) q^{59} + (2 \beta + 6) q^{61} + ( - 4 \beta - 8) q^{63} + ( - 4 \beta + 4) q^{67} + \beta q^{69} + (3 \beta - 4) q^{71} + ( - \beta - 14) q^{73} + 8 \beta q^{77} - 4 \beta q^{79} - 7 q^{81} - 12 q^{83} + (7 \beta + 4) q^{87} + 10 q^{89} + (2 \beta - 8) q^{91} + ( - 3 \beta + 4) q^{93} + (4 \beta + 6) q^{97} + ( - 4 \beta - 4) q^{99} +O(q^{100})$$ q - b * q^3 - 2*b * q^7 + (b + 1) * q^9 - 4 * q^11 + (b - 2) * q^13 + (2*b - 2) * q^17 + 4 * q^19 + (2*b + 8) * q^21 - q^23 + (b - 4) * q^27 + (-b - 6) * q^29 + (-b + 4) * q^31 + 4*b * q^33 + (2*b + 2) * q^37 + (b - 4) * q^39 + (b - 2) * q^41 + (2*b + 4) * q^43 + (3*b - 4) * q^47 + (4*b + 9) * q^49 - 8 * q^51 + (4*b - 6) * q^53 - 4*b * q^57 + (-4*b + 4) * q^59 + (2*b + 6) * q^61 + (-4*b - 8) * q^63 + (-4*b + 4) * q^67 + b * q^69 + (3*b - 4) * q^71 + (-b - 14) * q^73 + 8*b * q^77 - 4*b * q^79 - 7 * q^81 - 12 * q^83 + (7*b + 4) * q^87 + 10 * q^89 + (2*b - 8) * q^91 + (-3*b + 4) * q^93 + (4*b + 6) * q^97 + (-4*b - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 - 2 * q^7 + 3 * q^9 $$2 q - q^{3} - 2 q^{7} + 3 q^{9} - 8 q^{11} - 3 q^{13} - 2 q^{17} + 8 q^{19} + 18 q^{21} - 2 q^{23} - 7 q^{27} - 13 q^{29} + 7 q^{31} + 4 q^{33} + 6 q^{37} - 7 q^{39} - 3 q^{41} + 10 q^{43} - 5 q^{47} + 22 q^{49} - 16 q^{51} - 8 q^{53} - 4 q^{57} + 4 q^{59} + 14 q^{61} - 20 q^{63} + 4 q^{67} + q^{69} - 5 q^{71} - 29 q^{73} + 8 q^{77} - 4 q^{79} - 14 q^{81} - 24 q^{83} + 15 q^{87} + 20 q^{89} - 14 q^{91} + 5 q^{93} + 16 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q - q^3 - 2 * q^7 + 3 * q^9 - 8 * q^11 - 3 * q^13 - 2 * q^17 + 8 * q^19 + 18 * q^21 - 2 * q^23 - 7 * q^27 - 13 * q^29 + 7 * q^31 + 4 * q^33 + 6 * q^37 - 7 * q^39 - 3 * q^41 + 10 * q^43 - 5 * q^47 + 22 * q^49 - 16 * q^51 - 8 * q^53 - 4 * q^57 + 4 * q^59 + 14 * q^61 - 20 * q^63 + 4 * q^67 + q^69 - 5 * q^71 - 29 * q^73 + 8 * q^77 - 4 * q^79 - 14 * q^81 - 24 * q^83 + 15 * q^87 + 20 * q^89 - 14 * q^91 + 5 * q^93 + 16 * q^97 - 12 * 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
 2.56155 −1.56155
0 −2.56155 0 0 0 −5.12311 0 3.56155 0
1.2 0 1.56155 0 0 0 3.12311 0 −0.561553 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
$$2$$ $$1$$
$$5$$ $$1$$
$$23$$ $$1$$

## 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 4600.2.a.r 2
4.b odd 2 1 9200.2.a.bx 2
5.b even 2 1 920.2.a.f 2
5.c odd 4 2 4600.2.e.m 4
15.d odd 2 1 8280.2.a.bb 2
20.d odd 2 1 1840.2.a.k 2
40.e odd 2 1 7360.2.a.bm 2
40.f even 2 1 7360.2.a.bj 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.f 2 5.b even 2 1
1840.2.a.k 2 20.d odd 2 1
4600.2.a.r 2 1.a even 1 1 trivial
4600.2.e.m 4 5.c odd 4 2
7360.2.a.bj 2 40.f even 2 1
7360.2.a.bm 2 40.e odd 2 1
8280.2.a.bb 2 15.d odd 2 1
9200.2.a.bx 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4600))$$:

 $$T_{3}^{2} + T_{3} - 4$$ T3^2 + T3 - 4 $$T_{7}^{2} + 2T_{7} - 16$$ T7^2 + 2*T7 - 16 $$T_{11} + 4$$ T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2T - 16$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 3T - 2$$
$17$ $$T^{2} + 2T - 16$$
$19$ $$(T - 4)^{2}$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} + 13T + 38$$
$31$ $$T^{2} - 7T + 8$$
$37$ $$T^{2} - 6T - 8$$
$41$ $$T^{2} + 3T - 2$$
$43$ $$T^{2} - 10T + 8$$
$47$ $$T^{2} + 5T - 32$$
$53$ $$T^{2} + 8T - 52$$
$59$ $$T^{2} - 4T - 64$$
$61$ $$T^{2} - 14T + 32$$
$67$ $$T^{2} - 4T - 64$$
$71$ $$T^{2} + 5T - 32$$
$73$ $$T^{2} + 29T + 206$$
$79$ $$T^{2} + 4T - 64$$
$83$ $$(T + 12)^{2}$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} - 16T - 4$$