# Properties

 Label 3648.2.a.i 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 114) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} - 4 q^{7} + q^{9}+O(q^{10})$$ q - q^3 - 4 * q^7 + q^9 $$q - q^{3} - 4 q^{7} + q^{9} + 4 q^{13} + 6 q^{17} - q^{19} + 4 q^{21} - 6 q^{23} - 5 q^{25} - q^{27} - 6 q^{29} + 2 q^{31} + 4 q^{37} - 4 q^{39} + 6 q^{41} + 4 q^{43} + 6 q^{47} + 9 q^{49} - 6 q^{51} - 6 q^{53} + q^{57} + 12 q^{59} - 14 q^{61} - 4 q^{63} - 8 q^{67} + 6 q^{69} + 14 q^{73} + 5 q^{75} - 10 q^{79} + q^{81} + 12 q^{83} + 6 q^{87} - 6 q^{89} - 16 q^{91} - 2 q^{93} - 10 q^{97}+O(q^{100})$$ q - q^3 - 4 * q^7 + q^9 + 4 * q^13 + 6 * q^17 - q^19 + 4 * q^21 - 6 * q^23 - 5 * q^25 - q^27 - 6 * q^29 + 2 * q^31 + 4 * q^37 - 4 * q^39 + 6 * q^41 + 4 * q^43 + 6 * q^47 + 9 * q^49 - 6 * q^51 - 6 * q^53 + q^57 + 12 * q^59 - 14 * q^61 - 4 * q^63 - 8 * q^67 + 6 * q^69 + 14 * q^73 + 5 * q^75 - 10 * q^79 + q^81 + 12 * q^83 + 6 * q^87 - 6 * q^89 - 16 * q^91 - 2 * q^93 - 10 * 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
 0
0 −1.00000 0 0 0 −4.00000 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.i 1
4.b odd 2 1 3648.2.a.bc 1
8.b even 2 1 114.2.a.c 1
8.d odd 2 1 912.2.a.c 1
24.f even 2 1 2736.2.a.o 1
24.h odd 2 1 342.2.a.c 1
40.f even 2 1 2850.2.a.g 1
40.i odd 4 2 2850.2.d.p 2
56.h odd 2 1 5586.2.a.u 1
120.i odd 2 1 8550.2.a.bj 1
152.g odd 2 1 2166.2.a.a 1
456.p even 2 1 6498.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.a.c 1 8.b even 2 1
342.2.a.c 1 24.h odd 2 1
912.2.a.c 1 8.d odd 2 1
2166.2.a.a 1 152.g odd 2 1
2736.2.a.o 1 24.f even 2 1
2850.2.a.g 1 40.f even 2 1
2850.2.d.p 2 40.i odd 4 2
3648.2.a.i 1 1.a even 1 1 trivial
3648.2.a.bc 1 4.b odd 2 1
5586.2.a.u 1 56.h odd 2 1
6498.2.a.t 1 456.p even 2 1
8550.2.a.bj 1 120.i odd 2 1

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{7} + 4$$ T7 + 4 $$T_{11}$$ T11 $$T_{23} + 6$$ T23 + 6 $$T_{31} - 2$$ T31 - 2

## Hecke characteristic polynomials

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