# Properties

 Label 3648.2.a.bd Level $3648$ Weight $2$ Character orbit 3648.a Self dual yes Analytic conductor $29.129$ 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] = [3648,2,Mod(1,3648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3648, 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("3648.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3648 = 2^{6} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3648.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: $$29.1294266574$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1824) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} - q^{7} + q^{9}+O(q^{10})$$ q + q^3 + q^5 - q^7 + q^9 $$q + q^{3} + q^{5} - q^{7} + q^{9} - 3 q^{11} + q^{15} - 7 q^{17} - q^{19} - q^{21} + 8 q^{23} - 4 q^{25} + q^{27} - 2 q^{31} - 3 q^{33} - q^{35} - 4 q^{37} - 4 q^{41} + q^{43} + q^{45} - 3 q^{47} - 6 q^{49} - 7 q^{51} - 6 q^{53} - 3 q^{55} - q^{57} + 6 q^{59} + 5 q^{61} - q^{63} + 2 q^{67} + 8 q^{69} - 2 q^{71} - 11 q^{73} - 4 q^{75} + 3 q^{77} - 10 q^{79} + q^{81} - 16 q^{83} - 7 q^{85} - 14 q^{89} - 2 q^{93} - q^{95} - 8 q^{97} - 3 q^{99}+O(q^{100})$$ q + q^3 + q^5 - q^7 + q^9 - 3 * q^11 + q^15 - 7 * q^17 - q^19 - q^21 + 8 * q^23 - 4 * q^25 + q^27 - 2 * q^31 - 3 * q^33 - q^35 - 4 * q^37 - 4 * q^41 + q^43 + q^45 - 3 * q^47 - 6 * q^49 - 7 * q^51 - 6 * q^53 - 3 * q^55 - q^57 + 6 * q^59 + 5 * q^61 - q^63 + 2 * q^67 + 8 * q^69 - 2 * q^71 - 11 * q^73 - 4 * q^75 + 3 * q^77 - 10 * q^79 + q^81 - 16 * q^83 - 7 * q^85 - 14 * q^89 - 2 * q^93 - q^95 - 8 * q^97 - 3 * 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
0 1.00000 0 1.00000 0 −1.00000 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$1$$

## 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 3648.2.a.bd 1
4.b odd 2 1 3648.2.a.m 1
8.b even 2 1 1824.2.a.b 1
8.d odd 2 1 1824.2.a.i yes 1
24.f even 2 1 5472.2.a.r 1
24.h odd 2 1 5472.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1824.2.a.b 1 8.b even 2 1
1824.2.a.i yes 1 8.d odd 2 1
3648.2.a.m 1 4.b odd 2 1
3648.2.a.bd 1 1.a even 1 1 trivial
5472.2.a.n 1 24.h odd 2 1
5472.2.a.r 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3648))$$:

 $$T_{5} - 1$$ T5 - 1 $$T_{7} + 1$$ T7 + 1 $$T_{11} + 3$$ T11 + 3 $$T_{23} - 8$$ T23 - 8 $$T_{31} + 2$$ T31 + 2

## Hecke characteristic polynomials

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