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 discriminant -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 \(2\), degree \(1\), minimal)

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

$q$-expansion

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

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
−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 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 8.7.d.a 1
3.b odd 2 1 72.7.b.a 1
4.b odd 2 1 32.7.d.a 1
8.b even 2 1 32.7.d.a 1
8.d odd 2 1 CM 8.7.d.a 1
12.b even 2 1 288.7.b.a 1
16.e even 4 2 256.7.c.d 2
16.f odd 4 2 256.7.c.d 2
24.f even 2 1 72.7.b.a 1
24.h odd 2 1 288.7.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.7.d.a 1 1.a even 1 1 trivial
8.7.d.a 1 8.d odd 2 1 CM
32.7.d.a 1 4.b odd 2 1
32.7.d.a 1 8.b even 2 1
72.7.b.a 1 3.b odd 2 1
72.7.b.a 1 24.f even 2 1
256.7.c.d 2 16.e even 4 2
256.7.c.d 2 16.f odd 4 2
288.7.b.a 1 12.b even 2 1
288.7.b.a 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 46 \) acting on \(S_{7}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 + T \)
$3$ \( -46 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 2338 + T \)
$13$ \( T \)
$17$ \( 1726 + T \)
$19$ \( 2482 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( T \)
$37$ \( T \)
$41$ \( -134642 + T \)
$43$ \( 74914 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( -304958 + T \)
$61$ \( T \)
$67$ \( 596626 + T \)
$71$ \( T \)
$73$ \( 593134 + T \)
$79$ \( T \)
$83$ \( -678926 + T \)
$89$ \( 357262 + T \)
$97$ \( -1822754 + T \)
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