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

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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:

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

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