# Properties

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

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.02208514587$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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

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

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

## 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.

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.b 2
3.b odd 2 1 1152.2.k.b 2
4.b odd 2 1 128.2.e.a 2
8.b even 2 1 16.2.e.a 2
8.d odd 2 1 64.2.e.a 2
12.b even 2 1 1152.2.k.a 2
16.e even 4 1 16.2.e.a 2
16.e even 4 1 inner 128.2.e.b 2
16.f odd 4 1 64.2.e.a 2
16.f odd 4 1 128.2.e.a 2
24.f even 2 1 576.2.k.a 2
24.h odd 2 1 144.2.k.a 2
32.g even 8 2 1024.2.a.b 2
32.g even 8 2 1024.2.b.e 2
32.h odd 8 2 1024.2.a.e 2
32.h odd 8 2 1024.2.b.b 2
40.e odd 2 1 1600.2.l.a 2
40.f even 2 1 400.2.l.c 2
40.i odd 4 1 400.2.q.a 2
40.i odd 4 1 400.2.q.b 2
40.k even 4 1 1600.2.q.a 2
40.k even 4 1 1600.2.q.b 2
48.i odd 4 1 144.2.k.a 2
48.i odd 4 1 1152.2.k.b 2
48.k even 4 1 576.2.k.a 2
48.k even 4 1 1152.2.k.a 2
56.h odd 2 1 784.2.m.b 2
56.j odd 6 2 784.2.x.c 4
56.p even 6 2 784.2.x.f 4
80.i odd 4 1 400.2.q.a 2
80.j even 4 1 1600.2.q.a 2
80.k odd 4 1 1600.2.l.a 2
80.q even 4 1 400.2.l.c 2
80.s even 4 1 1600.2.q.b 2
80.t odd 4 1 400.2.q.b 2
96.o even 8 2 9216.2.a.s 2
96.p odd 8 2 9216.2.a.d 2
112.l odd 4 1 784.2.m.b 2
112.w even 12 2 784.2.x.f 4
112.x odd 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.b even 2 1
16.2.e.a 2 16.e even 4 1
64.2.e.a 2 8.d odd 2 1
64.2.e.a 2 16.f odd 4 1
128.2.e.a 2 4.b odd 2 1
128.2.e.a 2 16.f odd 4 1
128.2.e.b 2 1.a even 1 1 trivial
128.2.e.b 2 16.e even 4 1 inner
144.2.k.a 2 24.h odd 2 1
144.2.k.a 2 48.i odd 4 1
400.2.l.c 2 40.f even 2 1
400.2.l.c 2 80.q even 4 1
400.2.q.a 2 40.i odd 4 1
400.2.q.a 2 80.i odd 4 1
400.2.q.b 2 40.i odd 4 1
400.2.q.b 2 80.t odd 4 1
576.2.k.a 2 24.f even 2 1
576.2.k.a 2 48.k even 4 1
784.2.m.b 2 56.h odd 2 1
784.2.m.b 2 112.l odd 4 1
784.2.x.c 4 56.j odd 6 2
784.2.x.c 4 112.x odd 12 2
784.2.x.f 4 56.p even 6 2
784.2.x.f 4 112.w even 12 2
1024.2.a.b 2 32.g even 8 2
1024.2.a.e 2 32.h odd 8 2
1024.2.b.b 2 32.h odd 8 2
1024.2.b.e 2 32.g even 8 2
1152.2.k.a 2 12.b even 2 1
1152.2.k.a 2 48.k even 4 1
1152.2.k.b 2 3.b odd 2 1
1152.2.k.b 2 48.i odd 4 1
1600.2.l.a 2 40.e odd 2 1
1600.2.l.a 2 80.k odd 4 1
1600.2.q.a 2 40.k even 4 1
1600.2.q.a 2 80.j even 4 1
1600.2.q.b 2 40.k even 4 1
1600.2.q.b 2 80.s even 4 1
9216.2.a.d 2 96.p odd 8 2
9216.2.a.s 2 96.o even 8 2

## Hecke kernels

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

## Hecke characteristic polynomials

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