# Properties

 Label 3648.2.a.r Level $3648$ Weight $2$ Character orbit 3648.a Self dual yes Analytic conductor $29.129$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} + 3 q^{5} + 5 q^{7} + q^{9}+O(q^{10})$$ q - q^3 + 3 * q^5 + 5 * q^7 + q^9 $$q - q^{3} + 3 q^{5} + 5 q^{7} + q^{9} + q^{11} - 2 q^{13} - 3 q^{15} - q^{17} - q^{19} - 5 q^{21} + 4 q^{23} + 4 q^{25} - q^{27} + 2 q^{29} + 6 q^{31} - q^{33} + 15 q^{35} + 2 q^{39} - q^{43} + 3 q^{45} + 9 q^{47} + 18 q^{49} + q^{51} - 10 q^{53} + 3 q^{55} + q^{57} - 8 q^{59} + q^{61} + 5 q^{63} - 6 q^{65} + 8 q^{67} - 4 q^{69} + 12 q^{71} - 11 q^{73} - 4 q^{75} + 5 q^{77} - 16 q^{79} + q^{81} + 12 q^{83} - 3 q^{85} - 2 q^{87} - 6 q^{89} - 10 q^{91} - 6 q^{93} - 3 q^{95} - 10 q^{97} + q^{99}+O(q^{100})$$ q - q^3 + 3 * q^5 + 5 * q^7 + q^9 + q^11 - 2 * q^13 - 3 * q^15 - q^17 - q^19 - 5 * q^21 + 4 * q^23 + 4 * q^25 - q^27 + 2 * q^29 + 6 * q^31 - q^33 + 15 * q^35 + 2 * q^39 - q^43 + 3 * q^45 + 9 * q^47 + 18 * q^49 + q^51 - 10 * q^53 + 3 * q^55 + q^57 - 8 * q^59 + q^61 + 5 * q^63 - 6 * q^65 + 8 * q^67 - 4 * q^69 + 12 * q^71 - 11 * q^73 - 4 * q^75 + 5 * q^77 - 16 * q^79 + q^81 + 12 * q^83 - 3 * q^85 - 2 * q^87 - 6 * q^89 - 10 * q^91 - 6 * q^93 - 3 * q^95 - 10 * q^97 + 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 3.00000 0 5.00000 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.r 1
4.b odd 2 1 3648.2.a.bh 1
8.b even 2 1 912.2.a.g 1
8.d odd 2 1 57.2.a.a 1
24.f even 2 1 171.2.a.d 1
24.h odd 2 1 2736.2.a.v 1
40.e odd 2 1 1425.2.a.j 1
40.k even 4 2 1425.2.c.b 2
56.e even 2 1 2793.2.a.b 1
88.g even 2 1 6897.2.a.f 1
104.h odd 2 1 9633.2.a.o 1
120.m even 2 1 4275.2.a.b 1
152.b even 2 1 1083.2.a.e 1
168.e odd 2 1 8379.2.a.p 1
456.l odd 2 1 3249.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.a.a 1 8.d odd 2 1
171.2.a.d 1 24.f even 2 1
912.2.a.g 1 8.b even 2 1
1083.2.a.e 1 152.b even 2 1
1425.2.a.j 1 40.e odd 2 1
1425.2.c.b 2 40.k even 4 2
2736.2.a.v 1 24.h odd 2 1
2793.2.a.b 1 56.e even 2 1
3249.2.a.b 1 456.l odd 2 1
3648.2.a.r 1 1.a even 1 1 trivial
3648.2.a.bh 1 4.b odd 2 1
4275.2.a.b 1 120.m even 2 1
6897.2.a.f 1 88.g even 2 1
8379.2.a.p 1 168.e odd 2 1
9633.2.a.o 1 104.h odd 2 1

## Hecke kernels

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

 $$T_{5} - 3$$ T5 - 3 $$T_{7} - 5$$ T7 - 5 $$T_{11} - 1$$ T11 - 1 $$T_{23} - 4$$ T23 - 4 $$T_{31} - 6$$ T31 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T - 3$$
$7$ $$T - 5$$
$11$ $$T - 1$$
$13$ $$T + 2$$
$17$ $$T + 1$$
$19$ $$T + 1$$
$23$ $$T - 4$$
$29$ $$T - 2$$
$31$ $$T - 6$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T + 1$$
$47$ $$T - 9$$
$53$ $$T + 10$$
$59$ $$T + 8$$
$61$ $$T - 1$$
$67$ $$T - 8$$
$71$ $$T - 12$$
$73$ $$T + 11$$
$79$ $$T + 16$$
$83$ $$T - 12$$
$89$ $$T + 6$$
$97$ $$T + 10$$