# Properties

 Label 128.2.e.a Level $128$ Weight $2$ Character orbit 128.e Analytic conductor $1.022$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,2,Mod(33,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("128.33");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 128.e (of order $$4$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$1.02208514587$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (i - 1) q^{3} + (i + 1) q^{5} + 2 i q^{7} + i q^{9}+O(q^{10})$$ q + (i - 1) * q^3 + (i + 1) * q^5 + 2*i * q^7 + i * q^9 $$q + (i - 1) q^{3} + (i + 1) q^{5} + 2 i q^{7} + i q^{9} + (i + 1) q^{11} + ( - i + 1) q^{13} - 2 q^{15} - 2 q^{17} + ( - 3 i + 3) q^{19} + ( - 2 i - 2) q^{21} - 6 i q^{23} - 3 i q^{25} + ( - 4 i - 4) q^{27} + (3 i - 3) q^{29} + 8 q^{31} - 2 q^{33} + (2 i - 2) q^{35} + ( - 3 i - 3) q^{37} + 2 i q^{39} + (5 i + 5) q^{43} + (i - 1) q^{45} - 8 q^{47} + 3 q^{49} + ( - 2 i + 2) q^{51} + (5 i + 5) q^{53} + 2 i q^{55} + 6 i q^{57} + ( - 3 i - 3) q^{59} + ( - 9 i + 9) q^{61} - 2 q^{63} + 2 q^{65} + (5 i - 5) q^{67} + (6 i + 6) q^{69} + 10 i q^{71} - 4 i q^{73} + (3 i + 3) q^{75} + (2 i - 2) q^{77} + 5 q^{81} + (i - 1) q^{83} + ( - 2 i - 2) q^{85} - 6 i q^{87} + 4 i q^{89} + (2 i + 2) q^{91} + (8 i - 8) q^{93} + 6 q^{95} - 2 q^{97} + (i - 1) q^{99} +O(q^{100})$$ q + (i - 1) * q^3 + (i + 1) * q^5 + 2*i * q^7 + i * q^9 + (i + 1) * q^11 + (-i + 1) * q^13 - 2 * q^15 - 2 * q^17 + (-3*i + 3) * q^19 + (-2*i - 2) * q^21 - 6*i * q^23 - 3*i * q^25 + (-4*i - 4) * q^27 + (3*i - 3) * q^29 + 8 * q^31 - 2 * q^33 + (2*i - 2) * q^35 + (-3*i - 3) * q^37 + 2*i * q^39 + (5*i + 5) * q^43 + (i - 1) * q^45 - 8 * q^47 + 3 * q^49 + (-2*i + 2) * q^51 + (5*i + 5) * q^53 + 2*i * q^55 + 6*i * q^57 + (-3*i - 3) * q^59 + (-9*i + 9) * q^61 - 2 * q^63 + 2 * q^65 + (5*i - 5) * q^67 + (6*i + 6) * q^69 + 10*i * q^71 - 4*i * q^73 + (3*i + 3) * q^75 + (2*i - 2) * q^77 + 5 * q^81 + (i - 1) * q^83 + (-2*i - 2) * q^85 - 6*i * q^87 + 4*i * q^89 + (2*i + 2) * q^91 + (8*i - 8) * q^93 + 6 * q^95 - 2 * q^97 + (i - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{11} + 2 q^{13} - 4 q^{15} - 4 q^{17} + 6 q^{19} - 4 q^{21} - 8 q^{27} - 6 q^{29} + 16 q^{31} - 4 q^{33} - 4 q^{35} - 6 q^{37} + 10 q^{43} - 2 q^{45} - 16 q^{47} + 6 q^{49} + 4 q^{51} + 10 q^{53} - 6 q^{59} + 18 q^{61} - 4 q^{63} + 4 q^{65} - 10 q^{67} + 12 q^{69} + 6 q^{75} - 4 q^{77} + 10 q^{81} - 2 q^{83} - 4 q^{85} + 4 q^{91} - 16 q^{93} + 12 q^{95} - 4 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^11 + 2 * q^13 - 4 * q^15 - 4 * q^17 + 6 * q^19 - 4 * q^21 - 8 * q^27 - 6 * q^29 + 16 * q^31 - 4 * q^33 - 4 * q^35 - 6 * q^37 + 10 * q^43 - 2 * q^45 - 16 * q^47 + 6 * q^49 + 4 * q^51 + 10 * q^53 - 6 * q^59 + 18 * q^61 - 4 * q^63 + 4 * q^65 - 10 * q^67 + 12 * q^69 + 6 * q^75 - 4 * q^77 + 10 * q^81 - 2 * q^83 - 4 * q^85 + 4 * q^91 - 16 * q^93 + 12 * q^95 - 4 * q^97 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$127$$ $$\chi(n)$$ $$i$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 − 1.00000i 1.00000i
0 −1.00000 1.00000i 0 1.00000 1.00000i 0 2.00000i 0 1.00000i 0
97.1 0 −1.00000 + 1.00000i 0 1.00000 + 1.00000i 0 2.00000i 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.2.e.a 2
3.b odd 2 1 1152.2.k.a 2
4.b odd 2 1 128.2.e.b 2
8.b even 2 1 64.2.e.a 2
8.d odd 2 1 16.2.e.a 2
12.b even 2 1 1152.2.k.b 2
16.e even 4 1 64.2.e.a 2
16.e even 4 1 inner 128.2.e.a 2
16.f odd 4 1 16.2.e.a 2
16.f odd 4 1 128.2.e.b 2
24.f even 2 1 144.2.k.a 2
24.h odd 2 1 576.2.k.a 2
32.g even 8 2 1024.2.a.e 2
32.g even 8 2 1024.2.b.b 2
32.h odd 8 2 1024.2.a.b 2
32.h odd 8 2 1024.2.b.e 2
40.e odd 2 1 400.2.l.c 2
40.f even 2 1 1600.2.l.a 2
40.i odd 4 1 1600.2.q.a 2
40.i odd 4 1 1600.2.q.b 2
40.k even 4 1 400.2.q.a 2
40.k even 4 1 400.2.q.b 2
48.i odd 4 1 576.2.k.a 2
48.i odd 4 1 1152.2.k.a 2
48.k even 4 1 144.2.k.a 2
48.k even 4 1 1152.2.k.b 2
56.e even 2 1 784.2.m.b 2
56.k odd 6 2 784.2.x.f 4
56.m even 6 2 784.2.x.c 4
80.i odd 4 1 1600.2.q.b 2
80.j even 4 1 400.2.q.b 2
80.k odd 4 1 400.2.l.c 2
80.q even 4 1 1600.2.l.a 2
80.s even 4 1 400.2.q.a 2
80.t odd 4 1 1600.2.q.a 2
96.o even 8 2 9216.2.a.d 2
96.p odd 8 2 9216.2.a.s 2
112.j even 4 1 784.2.m.b 2
112.u odd 12 2 784.2.x.f 4
112.v even 12 2 784.2.x.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 8.d odd 2 1
16.2.e.a 2 16.f odd 4 1
64.2.e.a 2 8.b even 2 1
64.2.e.a 2 16.e even 4 1
128.2.e.a 2 1.a even 1 1 trivial
128.2.e.a 2 16.e even 4 1 inner
128.2.e.b 2 4.b odd 2 1
128.2.e.b 2 16.f odd 4 1
144.2.k.a 2 24.f even 2 1
144.2.k.a 2 48.k even 4 1
400.2.l.c 2 40.e odd 2 1
400.2.l.c 2 80.k odd 4 1
400.2.q.a 2 40.k even 4 1
400.2.q.a 2 80.s even 4 1
400.2.q.b 2 40.k even 4 1
400.2.q.b 2 80.j even 4 1
576.2.k.a 2 24.h odd 2 1
576.2.k.a 2 48.i odd 4 1
784.2.m.b 2 56.e even 2 1
784.2.m.b 2 112.j even 4 1
784.2.x.c 4 56.m even 6 2
784.2.x.c 4 112.v even 12 2
784.2.x.f 4 56.k odd 6 2
784.2.x.f 4 112.u odd 12 2
1024.2.a.b 2 32.h odd 8 2
1024.2.a.e 2 32.g even 8 2
1024.2.b.b 2 32.g even 8 2
1024.2.b.e 2 32.h odd 8 2
1152.2.k.a 2 3.b odd 2 1
1152.2.k.a 2 48.i odd 4 1
1152.2.k.b 2 12.b even 2 1
1152.2.k.b 2 48.k even 4 1
1600.2.l.a 2 40.f even 2 1
1600.2.l.a 2 80.q even 4 1
1600.2.q.a 2 40.i odd 4 1
1600.2.q.a 2 80.t odd 4 1
1600.2.q.b 2 40.i odd 4 1
1600.2.q.b 2 80.i odd 4 1
9216.2.a.d 2 96.o even 8 2
9216.2.a.s 2 96.p odd 8 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 2T_{3} + 2$$ acting on $$S_{2}^{\mathrm{new}}(128, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 2$$
$5$ $$T^{2} - 2T + 2$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} - 2T + 2$$
$13$ $$T^{2} - 2T + 2$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} - 6T + 18$$
$23$ $$T^{2} + 36$$
$29$ $$T^{2} + 6T + 18$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 6T + 18$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 10T + 50$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} - 10T + 50$$
$59$ $$T^{2} + 6T + 18$$
$61$ $$T^{2} - 18T + 162$$
$67$ $$T^{2} + 10T + 50$$
$71$ $$T^{2} + 100$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 2T + 2$$
$89$ $$T^{2} + 16$$
$97$ $$(T + 2)^{2}$$