# Properties

 Label 4608.2.a.r Level $4608$ Weight $2$ Character orbit 4608.a Self dual yes Analytic conductor $36.795$ Analytic rank $0$ 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] = [4608,2,Mod(1,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, 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("4608.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4608 = 2^{9} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4608.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: $$36.7950652514$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1536) 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 2) q^{5} + ( - \beta + 2) q^{7}+O(q^{10})$$ q + (b + 2) * q^5 + (-b + 2) * q^7 $$q + (\beta + 2) q^{5} + ( - \beta + 2) q^{7} + 2 q^{11} + 2 \beta q^{13} + (4 \beta + 2) q^{17} - 4 \beta q^{19} + ( - 2 \beta - 4) q^{23} + (4 \beta + 1) q^{25} + (\beta + 2) q^{29} + (\beta + 6) q^{31} + 2 q^{35} + ( - 4 \beta + 4) q^{37} + ( - 4 \beta + 6) q^{41} + (4 \beta + 4) q^{43} + (6 \beta - 4) q^{47} + ( - 4 \beta - 1) q^{49} + ( - 7 \beta + 2) q^{53} + (2 \beta + 4) q^{55} - 4 q^{59} + ( - 4 \beta + 4) q^{61} + (4 \beta + 4) q^{65} + 8 q^{67} + ( - 2 \beta - 12) q^{71} + (4 \beta + 4) q^{73} + ( - 2 \beta + 4) q^{77} + (3 \beta + 10) q^{79} + ( - 8 \beta - 2) q^{83} + (10 \beta + 12) q^{85} - 2 q^{89} + (4 \beta - 4) q^{91} + ( - 8 \beta - 8) q^{95} + ( - 8 \beta + 2) q^{97} +O(q^{100})$$ q + (b + 2) * q^5 + (-b + 2) * q^7 + 2 * q^11 + 2*b * q^13 + (4*b + 2) * q^17 - 4*b * q^19 + (-2*b - 4) * q^23 + (4*b + 1) * q^25 + (b + 2) * q^29 + (b + 6) * q^31 + 2 * q^35 + (-4*b + 4) * q^37 + (-4*b + 6) * q^41 + (4*b + 4) * q^43 + (6*b - 4) * q^47 + (-4*b - 1) * q^49 + (-7*b + 2) * q^53 + (2*b + 4) * q^55 - 4 * q^59 + (-4*b + 4) * q^61 + (4*b + 4) * q^65 + 8 * q^67 + (-2*b - 12) * q^71 + (4*b + 4) * q^73 + (-2*b + 4) * q^77 + (3*b + 10) * q^79 + (-8*b - 2) * q^83 + (10*b + 12) * q^85 - 2 * q^89 + (4*b - 4) * q^91 + (-8*b - 8) * q^95 + (-8*b + 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} + 4 q^{7}+O(q^{10})$$ 2 * q + 4 * q^5 + 4 * q^7 $$2 q + 4 q^{5} + 4 q^{7} + 4 q^{11} + 4 q^{17} - 8 q^{23} + 2 q^{25} + 4 q^{29} + 12 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{41} + 8 q^{43} - 8 q^{47} - 2 q^{49} + 4 q^{53} + 8 q^{55} - 8 q^{59} + 8 q^{61} + 8 q^{65} + 16 q^{67} - 24 q^{71} + 8 q^{73} + 8 q^{77} + 20 q^{79} - 4 q^{83} + 24 q^{85} - 4 q^{89} - 8 q^{91} - 16 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + 4 * q^5 + 4 * q^7 + 4 * q^11 + 4 * q^17 - 8 * q^23 + 2 * q^25 + 4 * q^29 + 12 * q^31 + 4 * q^35 + 8 * q^37 + 12 * q^41 + 8 * q^43 - 8 * q^47 - 2 * q^49 + 4 * q^53 + 8 * q^55 - 8 * q^59 + 8 * q^61 + 8 * q^65 + 16 * q^67 - 24 * q^71 + 8 * q^73 + 8 * q^77 + 20 * q^79 - 4 * q^83 + 24 * q^85 - 4 * q^89 - 8 * q^91 - 16 * q^95 + 4 * q^97

## 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
 −1.41421 1.41421
0 0 0 0.585786 0 3.41421 0 0 0
1.2 0 0 0 3.41421 0 0.585786 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$

## 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 4608.2.a.r 2
3.b odd 2 1 1536.2.a.b 2
4.b odd 2 1 4608.2.a.n 2
8.b even 2 1 4608.2.a.e 2
8.d odd 2 1 4608.2.a.a 2
12.b even 2 1 1536.2.a.g yes 2
16.e even 4 2 4608.2.d.c 4
16.f odd 4 2 4608.2.d.o 4
24.f even 2 1 1536.2.a.e yes 2
24.h odd 2 1 1536.2.a.l yes 2
48.i odd 4 2 1536.2.d.a 4
48.k even 4 2 1536.2.d.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.b 2 3.b odd 2 1
1536.2.a.e yes 2 24.f even 2 1
1536.2.a.g yes 2 12.b even 2 1
1536.2.a.l yes 2 24.h odd 2 1
1536.2.d.a 4 48.i odd 4 2
1536.2.d.f 4 48.k even 4 2
4608.2.a.a 2 8.d odd 2 1
4608.2.a.e 2 8.b even 2 1
4608.2.a.n 2 4.b odd 2 1
4608.2.a.r 2 1.a even 1 1 trivial
4608.2.d.c 4 16.e even 4 2
4608.2.d.o 4 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4608))$$:

 $$T_{5}^{2} - 4T_{5} + 2$$ T5^2 - 4*T5 + 2 $$T_{7}^{2} - 4T_{7} + 2$$ T7^2 - 4*T7 + 2 $$T_{11} - 2$$ T11 - 2 $$T_{17}^{2} - 4T_{17} - 28$$ T17^2 - 4*T17 - 28 $$T_{23}^{2} + 8T_{23} + 8$$ T23^2 + 8*T23 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 2$$
$7$ $$T^{2} - 4T + 2$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} - 8$$
$17$ $$T^{2} - 4T - 28$$
$19$ $$T^{2} - 32$$
$23$ $$T^{2} + 8T + 8$$
$29$ $$T^{2} - 4T + 2$$
$31$ $$T^{2} - 12T + 34$$
$37$ $$T^{2} - 8T - 16$$
$41$ $$T^{2} - 12T + 4$$
$43$ $$T^{2} - 8T - 16$$
$47$ $$T^{2} + 8T - 56$$
$53$ $$T^{2} - 4T - 94$$
$59$ $$(T + 4)^{2}$$
$61$ $$T^{2} - 8T - 16$$
$67$ $$(T - 8)^{2}$$
$71$ $$T^{2} + 24T + 136$$
$73$ $$T^{2} - 8T - 16$$
$79$ $$T^{2} - 20T + 82$$
$83$ $$T^{2} + 4T - 124$$
$89$ $$(T + 2)^{2}$$
$97$ $$T^{2} - 4T - 124$$