Properties

Label 8.5.d.a
Level 8
Weight 5
Character orbit 8.d
Self dual yes
Analytic conductor 0.827
Analytic rank 0
Dimension 1
CM discriminant -8
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 8.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.826959704671\)
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

\(f(q)\) \(=\) \( q + 4q^{2} - 14q^{3} + 16q^{4} - 56q^{6} + 64q^{8} + 115q^{9} + O(q^{10}) \) \( q + 4q^{2} - 14q^{3} + 16q^{4} - 56q^{6} + 64q^{8} + 115q^{9} - 46q^{11} - 224q^{12} + 256q^{16} - 574q^{17} + 460q^{18} + 434q^{19} - 184q^{22} - 896q^{24} + 625q^{25} - 476q^{27} + 1024q^{32} + 644q^{33} - 2296q^{34} + 1840q^{36} + 1736q^{38} - 1246q^{41} - 3502q^{43} - 736q^{44} - 3584q^{48} + 2401q^{49} + 2500q^{50} + 8036q^{51} - 1904q^{54} - 6076q^{57} - 238q^{59} + 4096q^{64} + 2576q^{66} - 5134q^{67} - 9184q^{68} + 7360q^{72} + 9506q^{73} - 8750q^{75} + 6944q^{76} - 2651q^{81} - 4984q^{82} + 11186q^{83} - 14008q^{86} - 2944q^{88} + 5474q^{89} - 14336q^{96} - 9982q^{97} + 9604q^{98} - 5290q^{99} + O(q^{100}) \)

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
4.00000 −14.0000 16.0000 0 −56.0000 0 64.0000 115.000 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 8.5.d.a 1
3.b odd 2 1 72.5.b.a 1
4.b odd 2 1 32.5.d.a 1
5.b even 2 1 200.5.g.a 1
5.c odd 4 2 200.5.e.a 2
8.b even 2 1 32.5.d.a 1
8.d odd 2 1 CM 8.5.d.a 1
12.b even 2 1 288.5.b.a 1
16.e even 4 2 256.5.c.d 2
16.f odd 4 2 256.5.c.d 2
20.d odd 2 1 800.5.g.a 1
20.e even 4 2 800.5.e.a 2
24.f even 2 1 72.5.b.a 1
24.h odd 2 1 288.5.b.a 1
40.e odd 2 1 200.5.g.a 1
40.f even 2 1 800.5.g.a 1
40.i odd 4 2 800.5.e.a 2
40.k even 4 2 200.5.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.5.d.a 1 1.a even 1 1 trivial
8.5.d.a 1 8.d odd 2 1 CM
32.5.d.a 1 4.b odd 2 1
32.5.d.a 1 8.b even 2 1
72.5.b.a 1 3.b odd 2 1
72.5.b.a 1 24.f even 2 1
200.5.e.a 2 5.c odd 4 2
200.5.e.a 2 40.k even 4 2
200.5.g.a 1 5.b even 2 1
200.5.g.a 1 40.e odd 2 1
256.5.c.d 2 16.e even 4 2
256.5.c.d 2 16.f odd 4 2
288.5.b.a 1 12.b even 2 1
288.5.b.a 1 24.h odd 2 1
800.5.e.a 2 20.e even 4 2
800.5.e.a 2 40.i odd 4 2
800.5.g.a 1 20.d odd 2 1
800.5.g.a 1 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 14 \) acting on \(S_{5}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T \)
$3$ \( 1 + 14 T + 81 T^{2} \)
$5$ \( ( 1 - 25 T )( 1 + 25 T ) \)
$7$ \( ( 1 - 49 T )( 1 + 49 T ) \)
$11$ \( 1 + 46 T + 14641 T^{2} \)
$13$ \( ( 1 - 169 T )( 1 + 169 T ) \)
$17$ \( 1 + 574 T + 83521 T^{2} \)
$19$ \( 1 - 434 T + 130321 T^{2} \)
$23$ \( ( 1 - 529 T )( 1 + 529 T ) \)
$29$ \( ( 1 - 841 T )( 1 + 841 T ) \)
$31$ \( ( 1 - 961 T )( 1 + 961 T ) \)
$37$ \( ( 1 - 1369 T )( 1 + 1369 T ) \)
$41$ \( 1 + 1246 T + 2825761 T^{2} \)
$43$ \( 1 + 3502 T + 3418801 T^{2} \)
$47$ \( ( 1 - 2209 T )( 1 + 2209 T ) \)
$53$ \( ( 1 - 2809 T )( 1 + 2809 T ) \)
$59$ \( 1 + 238 T + 12117361 T^{2} \)
$61$ \( ( 1 - 3721 T )( 1 + 3721 T ) \)
$67$ \( 1 + 5134 T + 20151121 T^{2} \)
$71$ \( ( 1 - 5041 T )( 1 + 5041 T ) \)
$73$ \( 1 - 9506 T + 28398241 T^{2} \)
$79$ \( ( 1 - 6241 T )( 1 + 6241 T ) \)
$83$ \( 1 - 11186 T + 47458321 T^{2} \)
$89$ \( 1 - 5474 T + 62742241 T^{2} \)
$97$ \( 1 + 9982 T + 88529281 T^{2} \)
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