Properties

Label 8.7.d.a
Level 8
Weight 7
Character orbit 8.d
Self dual Yes
Analytic conductor 1.840
Analytic rank 0
Dimension 1
CM disc. -8
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(1.84043266896\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 46q^{3} \) \(\mathstrut +\mathstrut 64q^{4} \) \(\mathstrut -\mathstrut 368q^{6} \) \(\mathstrut -\mathstrut 512q^{8} \) \(\mathstrut +\mathstrut 1387q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 46q^{3} \) \(\mathstrut +\mathstrut 64q^{4} \) \(\mathstrut -\mathstrut 368q^{6} \) \(\mathstrut -\mathstrut 512q^{8} \) \(\mathstrut +\mathstrut 1387q^{9} \) \(\mathstrut -\mathstrut 2338q^{11} \) \(\mathstrut +\mathstrut 2944q^{12} \) \(\mathstrut +\mathstrut 4096q^{16} \) \(\mathstrut -\mathstrut 1726q^{17} \) \(\mathstrut -\mathstrut 11096q^{18} \) \(\mathstrut -\mathstrut 2482q^{19} \) \(\mathstrut +\mathstrut 18704q^{22} \) \(\mathstrut -\mathstrut 23552q^{24} \) \(\mathstrut +\mathstrut 15625q^{25} \) \(\mathstrut +\mathstrut 30268q^{27} \) \(\mathstrut -\mathstrut 32768q^{32} \) \(\mathstrut -\mathstrut 107548q^{33} \) \(\mathstrut +\mathstrut 13808q^{34} \) \(\mathstrut +\mathstrut 88768q^{36} \) \(\mathstrut +\mathstrut 19856q^{38} \) \(\mathstrut +\mathstrut 134642q^{41} \) \(\mathstrut -\mathstrut 74914q^{43} \) \(\mathstrut -\mathstrut 149632q^{44} \) \(\mathstrut +\mathstrut 188416q^{48} \) \(\mathstrut +\mathstrut 117649q^{49} \) \(\mathstrut -\mathstrut 125000q^{50} \) \(\mathstrut -\mathstrut 79396q^{51} \) \(\mathstrut -\mathstrut 242144q^{54} \) \(\mathstrut -\mathstrut 114172q^{57} \) \(\mathstrut +\mathstrut 304958q^{59} \) \(\mathstrut +\mathstrut 262144q^{64} \) \(\mathstrut +\mathstrut 860384q^{66} \) \(\mathstrut -\mathstrut 596626q^{67} \) \(\mathstrut -\mathstrut 110464q^{68} \) \(\mathstrut -\mathstrut 710144q^{72} \) \(\mathstrut -\mathstrut 593134q^{73} \) \(\mathstrut +\mathstrut 718750q^{75} \) \(\mathstrut -\mathstrut 158848q^{76} \) \(\mathstrut +\mathstrut 381205q^{81} \) \(\mathstrut -\mathstrut 1077136q^{82} \) \(\mathstrut +\mathstrut 678926q^{83} \) \(\mathstrut +\mathstrut 599312q^{86} \) \(\mathstrut +\mathstrut 1197056q^{88} \) \(\mathstrut -\mathstrut 357262q^{89} \) \(\mathstrut -\mathstrut 1507328q^{96} \) \(\mathstrut +\mathstrut 1822754q^{97} \) \(\mathstrut -\mathstrut 941192q^{98} \) \(\mathstrut -\mathstrut 3242806q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−8.00000 46.0000 64.0000 0 −368.000 0 −512.000 1387.00 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 \(T_{3} \) \(\mathstrut -\mathstrut 46 \) acting on \(S_{7}^{\mathrm{new}}(8, [\chi])\).