# Properties

 Label 128.2.a.a Level $128$ Weight $2$ Character orbit 128.a Self dual yes Analytic conductor $1.022$ 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] = [128,2,Mod(1,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("128.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 128.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: $$1.02208514587$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{3} - 2 q^{5} - 4 q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^3 - 2 * q^5 - 4 * q^7 + q^9 $$q - 2 q^{3} - 2 q^{5} - 4 q^{7} + q^{9} + 2 q^{11} - 2 q^{13} + 4 q^{15} - 2 q^{17} - 2 q^{19} + 8 q^{21} + 4 q^{23} - q^{25} + 4 q^{27} + 6 q^{29} - 4 q^{33} + 8 q^{35} - 10 q^{37} + 4 q^{39} - 6 q^{41} - 6 q^{43} - 2 q^{45} - 8 q^{47} + 9 q^{49} + 4 q^{51} + 6 q^{53} - 4 q^{55} + 4 q^{57} - 14 q^{59} - 2 q^{61} - 4 q^{63} + 4 q^{65} - 10 q^{67} - 8 q^{69} + 12 q^{71} + 14 q^{73} + 2 q^{75} - 8 q^{77} - 8 q^{79} - 11 q^{81} + 6 q^{83} + 4 q^{85} - 12 q^{87} - 2 q^{89} + 8 q^{91} + 4 q^{95} - 2 q^{97} + 2 q^{99}+O(q^{100})$$ q - 2 * q^3 - 2 * q^5 - 4 * q^7 + q^9 + 2 * q^11 - 2 * q^13 + 4 * q^15 - 2 * q^17 - 2 * q^19 + 8 * q^21 + 4 * q^23 - q^25 + 4 * q^27 + 6 * q^29 - 4 * q^33 + 8 * q^35 - 10 * q^37 + 4 * q^39 - 6 * q^41 - 6 * q^43 - 2 * q^45 - 8 * q^47 + 9 * q^49 + 4 * q^51 + 6 * q^53 - 4 * q^55 + 4 * q^57 - 14 * q^59 - 2 * q^61 - 4 * q^63 + 4 * q^65 - 10 * q^67 - 8 * q^69 + 12 * q^71 + 14 * q^73 + 2 * q^75 - 8 * q^77 - 8 * q^79 - 11 * q^81 + 6 * q^83 + 4 * q^85 - 12 * q^87 - 2 * q^89 + 8 * q^91 + 4 * q^95 - 2 * q^97 + 2 * 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 −2.00000 0 −2.00000 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
$$2$$ $$+1$$

## 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 128.2.a.a 1
3.b odd 2 1 1152.2.a.m 1
4.b odd 2 1 128.2.a.c yes 1
5.b even 2 1 3200.2.a.x 1
5.c odd 4 2 3200.2.c.l 2
7.b odd 2 1 6272.2.a.h 1
8.b even 2 1 128.2.a.d yes 1
8.d odd 2 1 128.2.a.b yes 1
12.b even 2 1 1152.2.a.r 1
16.e even 4 2 256.2.b.c 2
16.f odd 4 2 256.2.b.a 2
20.d odd 2 1 3200.2.a.e 1
20.e even 4 2 3200.2.c.f 2
24.f even 2 1 1152.2.a.h 1
24.h odd 2 1 1152.2.a.c 1
28.d even 2 1 6272.2.a.b 1
32.g even 8 4 1024.2.e.i 4
32.h odd 8 4 1024.2.e.m 4
40.e odd 2 1 3200.2.a.u 1
40.f even 2 1 3200.2.a.h 1
40.i odd 4 2 3200.2.c.e 2
40.k even 4 2 3200.2.c.k 2
48.i odd 4 2 2304.2.d.r 2
48.k even 4 2 2304.2.d.b 2
56.e even 2 1 6272.2.a.g 1
56.h odd 2 1 6272.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 1.a even 1 1 trivial
128.2.a.b yes 1 8.d odd 2 1
128.2.a.c yes 1 4.b odd 2 1
128.2.a.d yes 1 8.b even 2 1
256.2.b.a 2 16.f odd 4 2
256.2.b.c 2 16.e even 4 2
1024.2.e.i 4 32.g even 8 4
1024.2.e.m 4 32.h odd 8 4
1152.2.a.c 1 24.h odd 2 1
1152.2.a.h 1 24.f even 2 1
1152.2.a.m 1 3.b odd 2 1
1152.2.a.r 1 12.b even 2 1
2304.2.d.b 2 48.k even 4 2
2304.2.d.r 2 48.i odd 4 2
3200.2.a.e 1 20.d odd 2 1
3200.2.a.h 1 40.f even 2 1
3200.2.a.u 1 40.e odd 2 1
3200.2.a.x 1 5.b even 2 1
3200.2.c.e 2 40.i odd 4 2
3200.2.c.f 2 20.e even 4 2
3200.2.c.k 2 40.k even 4 2
3200.2.c.l 2 5.c odd 4 2
6272.2.a.a 1 56.h odd 2 1
6272.2.a.b 1 28.d even 2 1
6272.2.a.g 1 56.e even 2 1
6272.2.a.h 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(128))$$:

 $$T_{3} + 2$$ T3 + 2 $$T_{5} + 2$$ T5 + 2

## Hecke characteristic polynomials

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