Properties

Label 8.15.d.a
Level 8
Weight 15
Character orbit 8.d
Self dual yes
Analytic conductor 9.946
Analytic rank 0
Dimension 1
CM discriminant -8
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 8.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(9.94631745215\)
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

\(f(q)\) \(=\) \( q - 128q^{2} + 3022q^{3} + 16384q^{4} - 386816q^{6} - 2097152q^{8} + 4349515q^{9} + O(q^{10}) \) \( q - 128q^{2} + 3022q^{3} + 16384q^{4} - 386816q^{6} - 2097152q^{8} + 4349515q^{9} + 38712254q^{11} + 49512448q^{12} + 268435456q^{16} - 328636222q^{17} - 556737920q^{18} + 1778973806q^{19} - 4955168512q^{22} - 6337593344q^{24} + 6103515625q^{25} - 1309897988q^{27} - 34359738368q^{32} + 116988431588q^{33} + 42065436416q^{34} + 71262453760q^{36} - 227708647168q^{38} - 333393570766q^{41} - 495012562114q^{43} + 634261569536q^{44} + 811211948032q^{48} + 678223072849q^{49} - 781250000000q^{50} - 993138662884q^{51} + 167666942464q^{54} + 5376058841732q^{57} - 3914494552162q^{59} + 4398046511104q^{64} - 14974519243264q^{66} + 2711103884558q^{67} - 5384375861248q^{68} - 9121594081280q^{72} + 1579402558802q^{73} + 18444824218750q^{75} + 29146706837504q^{76} - 24762107129771q^{81} + 42674377058048q^{82} - 31146255762898q^{83} + 63361607950592q^{86} - 81185480900608q^{88} - 38433671549134q^{89} - 103835129348096q^{96} - 62815034524126q^{97} - 86812553324672q^{98} + 168379529456810q^{99} + O(q^{100}) \)

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
−128.000 3022.00 16384.0 0 −386816. 0 −2.09715e6 4.34952e6 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 8.15.d.a 1
3.b odd 2 1 72.15.b.a 1
4.b odd 2 1 32.15.d.a 1
8.b even 2 1 32.15.d.a 1
8.d odd 2 1 CM 8.15.d.a 1
24.f even 2 1 72.15.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.15.d.a 1 1.a even 1 1 trivial
8.15.d.a 1 8.d odd 2 1 CM
32.15.d.a 1 4.b odd 2 1
32.15.d.a 1 8.b even 2 1
72.15.b.a 1 3.b odd 2 1
72.15.b.a 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 3022 \) acting on \(S_{15}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 128 T \)
$3$ \( 1 - 3022 T + 4782969 T^{2} \)
$5$ \( ( 1 - 78125 T )( 1 + 78125 T ) \)
$7$ \( ( 1 - 823543 T )( 1 + 823543 T ) \)
$11$ \( 1 - 38712254 T + 379749833583241 T^{2} \)
$13$ \( ( 1 - 62748517 T )( 1 + 62748517 T ) \)
$17$ \( 1 + 328636222 T + 168377826559400929 T^{2} \)
$19$ \( 1 - 1778973806 T + 799006685782884121 T^{2} \)
$23$ \( ( 1 - 3404825447 T )( 1 + 3404825447 T ) \)
$29$ \( ( 1 - 17249876309 T )( 1 + 17249876309 T ) \)
$31$ \( ( 1 - 27512614111 T )( 1 + 27512614111 T ) \)
$37$ \( ( 1 - 94931877133 T )( 1 + 94931877133 T ) \)
$41$ \( 1 + 333393570766 T + \)\(37\!\cdots\!61\)\( T^{2} \)
$43$ \( 1 + 495012562114 T + \)\(73\!\cdots\!49\)\( T^{2} \)
$47$ \( ( 1 - 506623120463 T )( 1 + 506623120463 T ) \)
$53$ \( ( 1 - 1174711139837 T )( 1 + 1174711139837 T ) \)
$59$ \( 1 + 3914494552162 T + \)\(61\!\cdots\!61\)\( T^{2} \)
$61$ \( ( 1 - 3142742836021 T )( 1 + 3142742836021 T ) \)
$67$ \( 1 - 2711103884558 T + \)\(36\!\cdots\!29\)\( T^{2} \)
$71$ \( ( 1 - 9095120158391 T )( 1 + 9095120158391 T ) \)
$73$ \( 1 - 1579402558802 T + \)\(12\!\cdots\!09\)\( T^{2} \)
$79$ \( ( 1 - 19203908986159 T )( 1 + 19203908986159 T ) \)
$83$ \( 1 + 31146255762898 T + \)\(73\!\cdots\!29\)\( T^{2} \)
$89$ \( 1 + 38433671549134 T + \)\(19\!\cdots\!41\)\( T^{2} \)
$97$ \( 1 + 62815034524126 T + \)\(65\!\cdots\!69\)\( T^{2} \)
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