# Properties

 Label 128.5.d.b Level 128 Weight 5 Character orbit 128.d Self dual yes Analytic conductor 13.231 Analytic rank 0 Dimension 2 CM discriminant -8 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 128.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.2313552747$$ 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: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 8\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 8 \beta q^{3} + 47 q^{9} +O(q^{10})$$ $$q + 8 \beta q^{3} + 47 q^{9} + 168 \beta q^{11} + 574 q^{17} -408 \beta q^{19} + 625 q^{25} -272 \beta q^{27} + 2688 q^{33} -1246 q^{41} + 840 \beta q^{43} + 2401 q^{49} + 4592 \beta q^{51} -6528 q^{57} -4920 \beta q^{59} -5208 \beta q^{67} -9506 q^{73} + 5000 \beta q^{75} -8159 q^{81} -5688 \beta q^{83} -5474 q^{89} + 9982 q^{97} + 7896 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 94q^{9} + O(q^{10})$$ $$2q + 94q^{9} + 1148q^{17} + 1250q^{25} + 5376q^{33} - 2492q^{41} + 4802q^{49} - 13056q^{57} - 19012q^{73} - 16318q^{81} - 10948q^{89} + 19964q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
63.1
 −1.41421 1.41421
0 −11.3137 0 0 0 0 0 47.0000 0
63.2 0 11.3137 0 0 0 0 0 47.0000 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 128.5.d.b 2
3.b odd 2 1 1152.5.b.a 2
4.b odd 2 1 inner 128.5.d.b 2
8.b even 2 1 inner 128.5.d.b 2
8.d odd 2 1 CM 128.5.d.b 2
12.b even 2 1 1152.5.b.a 2
16.e even 4 2 256.5.c.e 2
16.f odd 4 2 256.5.c.e 2
24.f even 2 1 1152.5.b.a 2
24.h odd 2 1 1152.5.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.5.d.b 2 1.a even 1 1 trivial
128.5.d.b 2 4.b odd 2 1 inner
128.5.d.b 2 8.b even 2 1 inner
128.5.d.b 2 8.d odd 2 1 CM
256.5.c.e 2 16.e even 4 2
256.5.c.e 2 16.f odd 4 2
1152.5.b.a 2 3.b odd 2 1
1152.5.b.a 2 12.b even 2 1
1152.5.b.a 2 24.f even 2 1
1152.5.b.a 2 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 128$$ acting on $$S_{5}^{\mathrm{new}}(128, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 34 T^{2} + 6561 T^{4}$$
$5$ $$( 1 - 25 T )^{2}( 1 + 25 T )^{2}$$
$7$ $$( 1 - 49 T )^{2}( 1 + 49 T )^{2}$$
$11$ $$1 - 27166 T^{2} + 214358881 T^{4}$$
$13$ $$( 1 - 169 T )^{2}( 1 + 169 T )^{2}$$
$17$ $$( 1 - 574 T + 83521 T^{2} )^{2}$$
$19$ $$1 - 72286 T^{2} + 16983563041 T^{4}$$
$23$ $$( 1 - 529 T )^{2}( 1 + 529 T )^{2}$$
$29$ $$( 1 - 841 T )^{2}( 1 + 841 T )^{2}$$
$31$ $$( 1 - 961 T )^{2}( 1 + 961 T )^{2}$$
$37$ $$( 1 - 1369 T )^{2}( 1 + 1369 T )^{2}$$
$41$ $$( 1 + 1246 T + 2825761 T^{2} )^{2}$$
$43$ $$1 + 5426402 T^{2} + 11688200277601 T^{4}$$
$47$ $$( 1 - 2209 T )^{2}( 1 + 2209 T )^{2}$$
$53$ $$( 1 - 2809 T )^{2}( 1 + 2809 T )^{2}$$
$59$ $$1 - 24178078 T^{2} + 146830437604321 T^{4}$$
$61$ $$( 1 - 3721 T )^{2}( 1 + 3721 T )^{2}$$
$67$ $$1 - 13944286 T^{2} + 406067677556641 T^{4}$$
$71$ $$( 1 - 5041 T )^{2}( 1 + 5041 T )^{2}$$
$73$ $$( 1 + 9506 T + 28398241 T^{2} )^{2}$$
$79$ $$( 1 - 6241 T )^{2}( 1 + 6241 T )^{2}$$
$83$ $$1 + 30209954 T^{2} + 2252292232139041 T^{4}$$
$89$ $$( 1 + 5474 T + 62742241 T^{2} )^{2}$$
$97$ $$( 1 - 9982 T + 88529281 T^{2} )^{2}$$