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

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

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

$p$ $F_p(T)$
$2$ \( 1 + 8 T \)
$3$ \( 1 - 46 T + 729 T^{2} \)
$5$ \( ( 1 - 125 T )( 1 + 125 T ) \)
$7$ \( ( 1 - 343 T )( 1 + 343 T ) \)
$11$ \( 1 + 2338 T + 1771561 T^{2} \)
$13$ \( ( 1 - 2197 T )( 1 + 2197 T ) \)
$17$ \( 1 + 1726 T + 24137569 T^{2} \)
$19$ \( 1 + 2482 T + 47045881 T^{2} \)
$23$ \( ( 1 - 12167 T )( 1 + 12167 T ) \)
$29$ \( ( 1 - 24389 T )( 1 + 24389 T ) \)
$31$ \( ( 1 - 29791 T )( 1 + 29791 T ) \)
$37$ \( ( 1 - 50653 T )( 1 + 50653 T ) \)
$41$ \( 1 - 134642 T + 4750104241 T^{2} \)
$43$ \( 1 + 74914 T + 6321363049 T^{2} \)
$47$ \( ( 1 - 103823 T )( 1 + 103823 T ) \)
$53$ \( ( 1 - 148877 T )( 1 + 148877 T ) \)
$59$ \( 1 - 304958 T + 42180533641 T^{2} \)
$61$ \( ( 1 - 226981 T )( 1 + 226981 T ) \)
$67$ \( 1 + 596626 T + 90458382169 T^{2} \)
$71$ \( ( 1 - 357911 T )( 1 + 357911 T ) \)
$73$ \( 1 + 593134 T + 151334226289 T^{2} \)
$79$ \( ( 1 - 493039 T )( 1 + 493039 T ) \)
$83$ \( 1 - 678926 T + 326940373369 T^{2} \)
$89$ \( 1 + 357262 T + 496981290961 T^{2} \)
$97$ \( 1 - 1822754 T + 832972004929 T^{2} \)
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