# Properties

 Label 8.9.d.a Level 8 Weight 9 Character orbit 8.d Self dual yes Analytic conductor 3.259 Analytic rank 0 Dimension 1 CM discriminant -8 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$8 = 2^{3}$$ Weight: $$k$$ = $$9$$ Character orbit: $$[\chi]$$ = 8.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.25902888049$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 16q^{2} + 34q^{3} + 256q^{4} + 544q^{6} + 4096q^{8} - 5405q^{9} + O(q^{10})$$ $$q + 16q^{2} + 34q^{3} + 256q^{4} + 544q^{6} + 4096q^{8} - 5405q^{9} - 27166q^{11} + 8704q^{12} + 65536q^{16} + 162434q^{17} - 86480q^{18} - 72286q^{19} - 434656q^{22} + 139264q^{24} + 390625q^{25} - 406844q^{27} + 1048576q^{32} - 923644q^{33} + 2598944q^{34} - 1383680q^{36} - 1156576q^{38} - 4099006q^{41} + 5426402q^{43} - 6954496q^{44} + 2228224q^{48} + 5764801q^{49} + 6250000q^{50} + 5522756q^{51} - 6509504q^{54} - 2457724q^{57} - 24178078q^{59} + 16777216q^{64} - 14778304q^{66} - 13944286q^{67} + 41583104q^{68} - 22138880q^{72} + 33567554q^{73} + 13281250q^{75} - 18505216q^{76} + 21629509q^{81} - 65584096q^{82} + 30209954q^{83} + 86822432q^{86} - 111271936q^{88} - 95519806q^{89} + 35651584q^{96} - 77418238q^{97} + 92236816q^{98} + 146832230q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/8\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$7$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0
16.0000 34.0000 256.000 0 544.000 0 4096.00 −5405.00 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 $$\Q(\sqrt{-2})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.9.d.a 1
3.b odd 2 1 72.9.b.a 1
4.b odd 2 1 32.9.d.a 1
8.b even 2 1 32.9.d.a 1
8.d odd 2 1 CM 8.9.d.a 1
12.b even 2 1 288.9.b.a 1
16.e even 4 2 256.9.c.f 2
16.f odd 4 2 256.9.c.f 2
24.f even 2 1 72.9.b.a 1
24.h odd 2 1 288.9.b.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.9.d.a 1 1.a even 1 1 trivial
8.9.d.a 1 8.d odd 2 1 CM
32.9.d.a 1 4.b odd 2 1
32.9.d.a 1 8.b even 2 1
72.9.b.a 1 3.b odd 2 1
72.9.b.a 1 24.f even 2 1
256.9.c.f 2 16.e even 4 2
256.9.c.f 2 16.f odd 4 2
288.9.b.a 1 12.b even 2 1
288.9.b.a 1 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 34$$ acting on $$S_{9}^{\mathrm{new}}(8, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 16 T$$
$3$ $$1 - 34 T + 6561 T^{2}$$
$5$ $$( 1 - 625 T )( 1 + 625 T )$$
$7$ $$( 1 - 2401 T )( 1 + 2401 T )$$
$11$ $$1 + 27166 T + 214358881 T^{2}$$
$13$ $$( 1 - 28561 T )( 1 + 28561 T )$$
$17$ $$1 - 162434 T + 6975757441 T^{2}$$
$19$ $$1 + 72286 T + 16983563041 T^{2}$$
$23$ $$( 1 - 279841 T )( 1 + 279841 T )$$
$29$ $$( 1 - 707281 T )( 1 + 707281 T )$$
$31$ $$( 1 - 923521 T )( 1 + 923521 T )$$
$37$ $$( 1 - 1874161 T )( 1 + 1874161 T )$$
$41$ $$1 + 4099006 T + 7984925229121 T^{2}$$
$43$ $$1 - 5426402 T + 11688200277601 T^{2}$$
$47$ $$( 1 - 4879681 T )( 1 + 4879681 T )$$
$53$ $$( 1 - 7890481 T )( 1 + 7890481 T )$$
$59$ $$1 + 24178078 T + 146830437604321 T^{2}$$
$61$ $$( 1 - 13845841 T )( 1 + 13845841 T )$$
$67$ $$1 + 13944286 T + 406067677556641 T^{2}$$
$71$ $$( 1 - 25411681 T )( 1 + 25411681 T )$$
$73$ $$1 - 33567554 T + 806460091894081 T^{2}$$
$79$ $$( 1 - 38950081 T )( 1 + 38950081 T )$$
$83$ $$1 - 30209954 T + 2252292232139041 T^{2}$$
$89$ $$1 + 95519806 T + 3936588805702081 T^{2}$$
$97$ $$1 + 77418238 T + 7837433594376961 T^{2}$$