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 disc. -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 \(2\) and degree \(1\))

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: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes

Hecke kernels

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