# Properties

 Label 108.4.a.a Level $108$ Weight $4$ Character orbit 108.a Self dual yes Analytic conductor $6.372$ 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] = [108,4,Mod(1,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("108.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 108.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: $$6.37220628062$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q - 9 q^{5} - q^{7}+O(q^{10})$$ q - 9 * q^5 - q^7 $$q - 9 q^{5} - q^{7} - 63 q^{11} - 28 q^{13} - 72 q^{17} + 98 q^{19} - 126 q^{23} - 44 q^{25} + 126 q^{29} - 259 q^{31} + 9 q^{35} + 386 q^{37} + 450 q^{41} - 34 q^{43} + 54 q^{47} - 342 q^{49} + 693 q^{53} + 567 q^{55} - 180 q^{59} - 280 q^{61} + 252 q^{65} - 586 q^{67} - 504 q^{71} + 161 q^{73} + 63 q^{77} + 440 q^{79} - 999 q^{83} + 648 q^{85} - 882 q^{89} + 28 q^{91} - 882 q^{95} - 721 q^{97}+O(q^{100})$$ q - 9 * q^5 - q^7 - 63 * q^11 - 28 * q^13 - 72 * q^17 + 98 * q^19 - 126 * q^23 - 44 * q^25 + 126 * q^29 - 259 * q^31 + 9 * q^35 + 386 * q^37 + 450 * q^41 - 34 * q^43 + 54 * q^47 - 342 * q^49 + 693 * q^53 + 567 * q^55 - 180 * q^59 - 280 * q^61 + 252 * q^65 - 586 * q^67 - 504 * q^71 + 161 * q^73 + 63 * q^77 + 440 * q^79 - 999 * q^83 + 648 * q^85 - 882 * q^89 + 28 * q^91 - 882 * q^95 - 721 * 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 0 0 −9.00000 0 −1.00000 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 108.4.a.a 1
3.b odd 2 1 108.4.a.d yes 1
4.b odd 2 1 432.4.a.c 1
8.b even 2 1 1728.4.a.y 1
8.d odd 2 1 1728.4.a.z 1
9.c even 3 2 324.4.e.g 2
9.d odd 6 2 324.4.e.b 2
12.b even 2 1 432.4.a.l 1
24.f even 2 1 1728.4.a.h 1
24.h odd 2 1 1728.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.a.a 1 1.a even 1 1 trivial
108.4.a.d yes 1 3.b odd 2 1
324.4.e.b 2 9.d odd 6 2
324.4.e.g 2 9.c even 3 2
432.4.a.c 1 4.b odd 2 1
432.4.a.l 1 12.b even 2 1
1728.4.a.g 1 24.h odd 2 1
1728.4.a.h 1 24.f even 2 1
1728.4.a.y 1 8.b even 2 1
1728.4.a.z 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(108))$$:

 $$T_{5} + 9$$ T5 + 9 $$T_{7} + 1$$ T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 9$$
$7$ $$T + 1$$
$11$ $$T + 63$$
$13$ $$T + 28$$
$17$ $$T + 72$$
$19$ $$T - 98$$
$23$ $$T + 126$$
$29$ $$T - 126$$
$31$ $$T + 259$$
$37$ $$T - 386$$
$41$ $$T - 450$$
$43$ $$T + 34$$
$47$ $$T - 54$$
$53$ $$T - 693$$
$59$ $$T + 180$$
$61$ $$T + 280$$
$67$ $$T + 586$$
$71$ $$T + 504$$
$73$ $$T - 161$$
$79$ $$T - 440$$
$83$ $$T + 999$$
$89$ $$T + 882$$
$97$ $$T + 721$$