# Properties

 Label 128.2.b.a Level 128 Weight 2 Character orbit 128.b Analytic conductor 1.022 Analytic rank 0 Dimension 2 CM discriminant -8 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 128.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.02208514587$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} -5 q^{9} +O(q^{10})$$ $$q + \beta q^{3} -5 q^{9} + \beta q^{11} + 6 q^{17} -3 \beta q^{19} + 5 q^{25} -2 \beta q^{27} -8 q^{33} -6 q^{41} -3 \beta q^{43} -7 q^{49} + 6 \beta q^{51} + 24 q^{57} + 5 \beta q^{59} -3 \beta q^{67} -2 q^{73} + 5 \beta q^{75} + q^{81} + \beta q^{83} -18 q^{89} -10 q^{97} -5 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 10q^{9} + O(q^{10})$$ $$2q - 10q^{9} + 12q^{17} + 10q^{25} - 16q^{33} - 12q^{41} - 14q^{49} + 48q^{57} - 4q^{73} + 2q^{81} - 36q^{89} - 20q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 − 1.41421i 1.41421i
0 2.82843i 0 0 0 0 0 −5.00000 0
65.2 0 2.82843i 0 0 0 0 0 −5.00000 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.2.b.a 2
3.b odd 2 1 1152.2.d.c 2
4.b odd 2 1 inner 128.2.b.a 2
5.b even 2 1 3200.2.d.c 2
5.c odd 4 2 3200.2.f.o 4
8.b even 2 1 inner 128.2.b.a 2
8.d odd 2 1 CM 128.2.b.a 2
12.b even 2 1 1152.2.d.c 2
16.e even 4 2 256.2.a.e 2
16.f odd 4 2 256.2.a.e 2
20.d odd 2 1 3200.2.d.c 2
20.e even 4 2 3200.2.f.o 4
24.f even 2 1 1152.2.d.c 2
24.h odd 2 1 1152.2.d.c 2
32.g even 8 2 1024.2.e.a 2
32.g even 8 2 1024.2.e.f 2
32.h odd 8 2 1024.2.e.a 2
32.h odd 8 2 1024.2.e.f 2
40.e odd 2 1 3200.2.d.c 2
40.f even 2 1 3200.2.d.c 2
40.i odd 4 2 3200.2.f.o 4
40.k even 4 2 3200.2.f.o 4
48.i odd 4 2 2304.2.a.t 2
48.k even 4 2 2304.2.a.t 2
80.k odd 4 2 6400.2.a.by 2
80.q even 4 2 6400.2.a.by 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.a 2 1.a even 1 1 trivial
128.2.b.a 2 4.b odd 2 1 inner
128.2.b.a 2 8.b even 2 1 inner
128.2.b.a 2 8.d odd 2 1 CM
256.2.a.e 2 16.e even 4 2
256.2.a.e 2 16.f odd 4 2
1024.2.e.a 2 32.g even 8 2
1024.2.e.a 2 32.h odd 8 2
1024.2.e.f 2 32.g even 8 2
1024.2.e.f 2 32.h odd 8 2
1152.2.d.c 2 3.b odd 2 1
1152.2.d.c 2 12.b even 2 1
1152.2.d.c 2 24.f even 2 1
1152.2.d.c 2 24.h odd 2 1
2304.2.a.t 2 48.i odd 4 2
2304.2.a.t 2 48.k even 4 2
3200.2.d.c 2 5.b even 2 1
3200.2.d.c 2 20.d odd 2 1
3200.2.d.c 2 40.e odd 2 1
3200.2.d.c 2 40.f even 2 1
3200.2.f.o 4 5.c odd 4 2
3200.2.f.o 4 20.e even 4 2
3200.2.f.o 4 40.i odd 4 2
3200.2.f.o 4 40.k even 4 2
6400.2.a.by 2 80.k odd 4 2
6400.2.a.by 2 80.q even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 8$$ acting on $$S_{2}^{\mathrm{new}}(128, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$$
$5$ $$( 1 - 5 T^{2} )^{2}$$
$7$ $$( 1 + 7 T^{2} )^{2}$$
$11$ $$( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )$$
$13$ $$( 1 - 13 T^{2} )^{2}$$
$17$ $$( 1 - 6 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 2 T + 19 T^{2} )( 1 + 2 T + 19 T^{2} )$$
$23$ $$( 1 + 23 T^{2} )^{2}$$
$29$ $$( 1 - 29 T^{2} )^{2}$$
$31$ $$( 1 + 31 T^{2} )^{2}$$
$37$ $$( 1 - 37 T^{2} )^{2}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 10 T + 43 T^{2} )( 1 + 10 T + 43 T^{2} )$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$( 1 - 53 T^{2} )^{2}$$
$59$ $$( 1 - 6 T + 59 T^{2} )( 1 + 6 T + 59 T^{2} )$$
$61$ $$( 1 - 61 T^{2} )^{2}$$
$67$ $$( 1 - 14 T + 67 T^{2} )( 1 + 14 T + 67 T^{2} )$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 + 2 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 79 T^{2} )^{2}$$
$83$ $$( 1 - 18 T + 83 T^{2} )( 1 + 18 T + 83 T^{2} )$$
$89$ $$( 1 + 18 T + 89 T^{2} )^{2}$$
$97$ $$( 1 + 10 T + 97 T^{2} )^{2}$$