# Properties

 Label 128.4.b.b Level 128 Weight 4 Character orbit 128.b Analytic conductor 7.552 Analytic rank 0 Dimension 2 CM discriminant -8 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 128.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.55224448073$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 19 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 19 q^{9} + 25 \beta q^{11} -90 q^{17} + 45 \beta q^{19} + 125 q^{25} + 46 \beta q^{27} -200 q^{33} + 522 q^{41} -171 \beta q^{43} -343 q^{49} -90 \beta q^{51} -360 q^{57} -115 \beta q^{59} -387 \beta q^{67} + 430 q^{73} + 125 \beta q^{75} + 145 q^{81} + 241 \beta q^{83} -1026 q^{89} + 1910 q^{97} + 475 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 38q^{9} + O(q^{10})$$ $$2q + 38q^{9} - 180q^{17} + 250q^{25} - 400q^{33} + 1044q^{41} - 686q^{49} - 720q^{57} + 860q^{73} + 290q^{81} - 2052q^{89} + 3820q^{97} + O(q^{100})$$

## 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 19.0000 0
65.2 0 2.82843i 0 0 0 0 0 19.0000 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 128.4.b.b 2
3.b odd 2 1 1152.4.d.e 2
4.b odd 2 1 inner 128.4.b.b 2
8.b even 2 1 inner 128.4.b.b 2
8.d odd 2 1 CM 128.4.b.b 2
12.b even 2 1 1152.4.d.e 2
16.e even 4 2 256.4.a.k 2
16.f odd 4 2 256.4.a.k 2
24.f even 2 1 1152.4.d.e 2
24.h odd 2 1 1152.4.d.e 2
48.i odd 4 2 2304.4.a.bf 2
48.k even 4 2 2304.4.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.b.b 2 1.a even 1 1 trivial
128.4.b.b 2 4.b odd 2 1 inner
128.4.b.b 2 8.b even 2 1 inner
128.4.b.b 2 8.d odd 2 1 CM
256.4.a.k 2 16.e even 4 2
256.4.a.k 2 16.f odd 4 2
1152.4.d.e 2 3.b odd 2 1
1152.4.d.e 2 12.b even 2 1
1152.4.d.e 2 24.f even 2 1
1152.4.d.e 2 24.h odd 2 1
2304.4.a.bf 2 48.i odd 4 2
2304.4.a.bf 2 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(128, [\chi])$$:

 $$T_{3}^{2} + 8$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 10 T + 27 T^{2} )( 1 + 10 T + 27 T^{2} )$$
$5$ $$( 1 - 125 T^{2} )^{2}$$
$7$ $$( 1 + 343 T^{2} )^{2}$$
$11$ $$( 1 - 18 T + 1331 T^{2} )( 1 + 18 T + 1331 T^{2} )$$
$13$ $$( 1 - 2197 T^{2} )^{2}$$
$17$ $$( 1 + 90 T + 4913 T^{2} )^{2}$$
$19$ $$( 1 - 106 T + 6859 T^{2} )( 1 + 106 T + 6859 T^{2} )$$
$23$ $$( 1 + 12167 T^{2} )^{2}$$
$29$ $$( 1 - 24389 T^{2} )^{2}$$
$31$ $$( 1 + 29791 T^{2} )^{2}$$
$37$ $$( 1 - 50653 T^{2} )^{2}$$
$41$ $$( 1 - 522 T + 68921 T^{2} )^{2}$$
$43$ $$( 1 - 290 T + 79507 T^{2} )( 1 + 290 T + 79507 T^{2} )$$
$47$ $$( 1 + 103823 T^{2} )^{2}$$
$53$ $$( 1 - 148877 T^{2} )^{2}$$
$59$ $$( 1 - 846 T + 205379 T^{2} )( 1 + 846 T + 205379 T^{2} )$$
$61$ $$( 1 - 226981 T^{2} )^{2}$$
$67$ $$( 1 - 70 T + 300763 T^{2} )( 1 + 70 T + 300763 T^{2} )$$
$71$ $$( 1 + 357911 T^{2} )^{2}$$
$73$ $$( 1 - 430 T + 389017 T^{2} )^{2}$$
$79$ $$( 1 + 493039 T^{2} )^{2}$$
$83$ $$( 1 - 1350 T + 571787 T^{2} )( 1 + 1350 T + 571787 T^{2} )$$
$89$ $$( 1 + 1026 T + 704969 T^{2} )^{2}$$
$97$ $$( 1 - 1910 T + 912673 T^{2} )^{2}$$