Properties

Label 8.3.d.a
Level 8
Weight 3
Character orbit 8.d
Self dual yes
Analytic conductor 0.218
Analytic rank 0
Dimension 1
CM discriminant -8
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.217984211488\)
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

\(f(q)\) \(=\) \( q - 2q^{2} - 2q^{3} + 4q^{4} + 4q^{6} - 8q^{8} - 5q^{9} + O(q^{10}) \) \( q - 2q^{2} - 2q^{3} + 4q^{4} + 4q^{6} - 8q^{8} - 5q^{9} + 14q^{11} - 8q^{12} + 16q^{16} + 2q^{17} + 10q^{18} - 34q^{19} - 28q^{22} + 16q^{24} + 25q^{25} + 28q^{27} - 32q^{32} - 28q^{33} - 4q^{34} - 20q^{36} + 68q^{38} - 46q^{41} + 14q^{43} + 56q^{44} - 32q^{48} + 49q^{49} - 50q^{50} - 4q^{51} - 56q^{54} + 68q^{57} - 82q^{59} + 64q^{64} + 56q^{66} + 62q^{67} + 8q^{68} + 40q^{72} - 142q^{73} - 50q^{75} - 136q^{76} - 11q^{81} + 92q^{82} + 158q^{83} - 28q^{86} - 112q^{88} + 146q^{89} + 64q^{96} - 94q^{97} - 98q^{98} - 70q^{99} + O(q^{100}) \)

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
−2.00000 −2.00000 4.00000 0 4.00000 0 −8.00000 −5.00000 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 8.3.d.a 1
3.b odd 2 1 72.3.b.a 1
4.b odd 2 1 32.3.d.a 1
5.b even 2 1 200.3.g.a 1
5.c odd 4 2 200.3.e.a 2
7.b odd 2 1 392.3.g.a 1
7.c even 3 2 392.3.k.d 2
7.d odd 6 2 392.3.k.b 2
8.b even 2 1 32.3.d.a 1
8.d odd 2 1 CM 8.3.d.a 1
12.b even 2 1 288.3.b.a 1
16.e even 4 2 256.3.c.e 2
16.f odd 4 2 256.3.c.e 2
20.d odd 2 1 800.3.g.a 1
20.e even 4 2 800.3.e.a 2
24.f even 2 1 72.3.b.a 1
24.h odd 2 1 288.3.b.a 1
28.d even 2 1 1568.3.g.a 1
40.e odd 2 1 200.3.g.a 1
40.f even 2 1 800.3.g.a 1
40.i odd 4 2 800.3.e.a 2
40.k even 4 2 200.3.e.a 2
48.i odd 4 2 2304.3.g.j 2
48.k even 4 2 2304.3.g.j 2
56.e even 2 1 392.3.g.a 1
56.h odd 2 1 1568.3.g.a 1
56.k odd 6 2 392.3.k.d 2
56.m even 6 2 392.3.k.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.3.d.a 1 1.a even 1 1 trivial
8.3.d.a 1 8.d odd 2 1 CM
32.3.d.a 1 4.b odd 2 1
32.3.d.a 1 8.b even 2 1
72.3.b.a 1 3.b odd 2 1
72.3.b.a 1 24.f even 2 1
200.3.e.a 2 5.c odd 4 2
200.3.e.a 2 40.k even 4 2
200.3.g.a 1 5.b even 2 1
200.3.g.a 1 40.e odd 2 1
256.3.c.e 2 16.e even 4 2
256.3.c.e 2 16.f odd 4 2
288.3.b.a 1 12.b even 2 1
288.3.b.a 1 24.h odd 2 1
392.3.g.a 1 7.b odd 2 1
392.3.g.a 1 56.e even 2 1
392.3.k.b 2 7.d odd 6 2
392.3.k.b 2 56.m even 6 2
392.3.k.d 2 7.c even 3 2
392.3.k.d 2 56.k odd 6 2
800.3.e.a 2 20.e even 4 2
800.3.e.a 2 40.i odd 4 2
800.3.g.a 1 20.d odd 2 1
800.3.g.a 1 40.f even 2 1
1568.3.g.a 1 28.d even 2 1
1568.3.g.a 1 56.h odd 2 1
2304.3.g.j 2 48.i odd 4 2
2304.3.g.j 2 48.k even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T \)
$3$ \( 1 + 2 T + 9 T^{2} \)
$5$ \( ( 1 - 5 T )( 1 + 5 T ) \)
$7$ \( ( 1 - 7 T )( 1 + 7 T ) \)
$11$ \( 1 - 14 T + 121 T^{2} \)
$13$ \( ( 1 - 13 T )( 1 + 13 T ) \)
$17$ \( 1 - 2 T + 289 T^{2} \)
$19$ \( 1 + 34 T + 361 T^{2} \)
$23$ \( ( 1 - 23 T )( 1 + 23 T ) \)
$29$ \( ( 1 - 29 T )( 1 + 29 T ) \)
$31$ \( ( 1 - 31 T )( 1 + 31 T ) \)
$37$ \( ( 1 - 37 T )( 1 + 37 T ) \)
$41$ \( 1 + 46 T + 1681 T^{2} \)
$43$ \( 1 - 14 T + 1849 T^{2} \)
$47$ \( ( 1 - 47 T )( 1 + 47 T ) \)
$53$ \( ( 1 - 53 T )( 1 + 53 T ) \)
$59$ \( 1 + 82 T + 3481 T^{2} \)
$61$ \( ( 1 - 61 T )( 1 + 61 T ) \)
$67$ \( 1 - 62 T + 4489 T^{2} \)
$71$ \( ( 1 - 71 T )( 1 + 71 T ) \)
$73$ \( 1 + 142 T + 5329 T^{2} \)
$79$ \( ( 1 - 79 T )( 1 + 79 T ) \)
$83$ \( 1 - 158 T + 6889 T^{2} \)
$89$ \( 1 - 146 T + 7921 T^{2} \)
$97$ \( 1 + 94 T + 9409 T^{2} \)
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Additional information

This cusp form has an eta product $\eta(z)^2\eta(2z)\eta(4z)\eta(8z)^2=q\prod_{n=1}^\infty (1-q^n)^2(1-q^{2n})(1-q^{4n})(1-q^{8n})^2$, where $q=\exp(2\pi i z)$.