Properties

Label 2.8.a.a
Level $2$
Weight $8$
Character orbit 2.a
Self dual yes
Analytic conductor $0.625$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.624770050968\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 8 q^{2} + 12 q^{3} + 64 q^{4} - 210 q^{5} - 96 q^{6} + 1016 q^{7} - 512 q^{8} - 2043 q^{9} + O(q^{10}) \) \( q - 8 q^{2} + 12 q^{3} + 64 q^{4} - 210 q^{5} - 96 q^{6} + 1016 q^{7} - 512 q^{8} - 2043 q^{9} + 1680 q^{10} + 1092 q^{11} + 768 q^{12} + 1382 q^{13} - 8128 q^{14} - 2520 q^{15} + 4096 q^{16} + 14706 q^{17} + 16344 q^{18} - 39940 q^{19} - 13440 q^{20} + 12192 q^{21} - 8736 q^{22} + 68712 q^{23} - 6144 q^{24} - 34025 q^{25} - 11056 q^{26} - 50760 q^{27} + 65024 q^{28} - 102570 q^{29} + 20160 q^{30} + 227552 q^{31} - 32768 q^{32} + 13104 q^{33} - 117648 q^{34} - 213360 q^{35} - 130752 q^{36} + 160526 q^{37} + 319520 q^{38} + 16584 q^{39} + 107520 q^{40} + 10842 q^{41} - 97536 q^{42} - 630748 q^{43} + 69888 q^{44} + 429030 q^{45} - 549696 q^{46} + 472656 q^{47} + 49152 q^{48} + 208713 q^{49} + 272200 q^{50} + 176472 q^{51} + 88448 q^{52} - 1494018 q^{53} + 406080 q^{54} - 229320 q^{55} - 520192 q^{56} - 479280 q^{57} + 820560 q^{58} + 2640660 q^{59} - 161280 q^{60} + 827702 q^{61} - 1820416 q^{62} - 2075688 q^{63} + 262144 q^{64} - 290220 q^{65} - 104832 q^{66} - 126004 q^{67} + 941184 q^{68} + 824544 q^{69} + 1706880 q^{70} - 1414728 q^{71} + 1046016 q^{72} + 980282 q^{73} - 1284208 q^{74} - 408300 q^{75} - 2556160 q^{76} + 1109472 q^{77} - 132672 q^{78} - 3566800 q^{79} - 860160 q^{80} + 3858921 q^{81} - 86736 q^{82} + 5672892 q^{83} + 780288 q^{84} - 3088260 q^{85} + 5045984 q^{86} - 1230840 q^{87} - 559104 q^{88} - 11951190 q^{89} - 3432240 q^{90} + 1404112 q^{91} + 4397568 q^{92} + 2730624 q^{93} - 3781248 q^{94} + 8387400 q^{95} - 393216 q^{96} + 8682146 q^{97} - 1669704 q^{98} - 2230956 q^{99} + O(q^{100}) \)

Expression as an eta quotient

\(f(z) = \eta(z)^{8}\eta(2z)^{8}=q\prod_{n=1}^\infty(1 - q^{n})^{8}(1 - q^{2n})^{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−8.00000 12.0000 64.0000 −210.000 −96.0000 1016.00 −512.000 −2043.00 1680.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2.8.a.a 1
3.b odd 2 1 18.8.a.b 1
4.b odd 2 1 16.8.a.b 1
5.b even 2 1 50.8.a.g 1
5.c odd 4 2 50.8.b.c 2
7.b odd 2 1 98.8.a.a 1
7.c even 3 2 98.8.c.d 2
7.d odd 6 2 98.8.c.e 2
8.b even 2 1 64.8.a.c 1
8.d odd 2 1 64.8.a.e 1
9.c even 3 2 162.8.c.l 2
9.d odd 6 2 162.8.c.a 2
11.b odd 2 1 242.8.a.e 1
12.b even 2 1 144.8.a.i 1
13.b even 2 1 338.8.a.d 1
13.d odd 4 2 338.8.b.d 2
15.d odd 2 1 450.8.a.c 1
15.e even 4 2 450.8.c.g 2
16.e even 4 2 256.8.b.b 2
16.f odd 4 2 256.8.b.f 2
17.b even 2 1 578.8.a.b 1
20.d odd 2 1 400.8.a.l 1
20.e even 4 2 400.8.c.j 2
24.f even 2 1 576.8.a.f 1
24.h odd 2 1 576.8.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.8.a.a 1 1.a even 1 1 trivial
16.8.a.b 1 4.b odd 2 1
18.8.a.b 1 3.b odd 2 1
50.8.a.g 1 5.b even 2 1
50.8.b.c 2 5.c odd 4 2
64.8.a.c 1 8.b even 2 1
64.8.a.e 1 8.d odd 2 1
98.8.a.a 1 7.b odd 2 1
98.8.c.d 2 7.c even 3 2
98.8.c.e 2 7.d odd 6 2
144.8.a.i 1 12.b even 2 1
162.8.c.a 2 9.d odd 6 2
162.8.c.l 2 9.c even 3 2
242.8.a.e 1 11.b odd 2 1
256.8.b.b 2 16.e even 4 2
256.8.b.f 2 16.f odd 4 2
338.8.a.d 1 13.b even 2 1
338.8.b.d 2 13.d odd 4 2
400.8.a.l 1 20.d odd 2 1
400.8.c.j 2 20.e even 4 2
450.8.a.c 1 15.d odd 2 1
450.8.c.g 2 15.e even 4 2
576.8.a.f 1 24.f even 2 1
576.8.a.g 1 24.h odd 2 1
578.8.a.b 1 17.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 + T \)
$3$ \( -12 + T \)
$5$ \( 210 + T \)
$7$ \( -1016 + T \)
$11$ \( -1092 + T \)
$13$ \( -1382 + T \)
$17$ \( -14706 + T \)
$19$ \( 39940 + T \)
$23$ \( -68712 + T \)
$29$ \( 102570 + T \)
$31$ \( -227552 + T \)
$37$ \( -160526 + T \)
$41$ \( -10842 + T \)
$43$ \( 630748 + T \)
$47$ \( -472656 + T \)
$53$ \( 1494018 + T \)
$59$ \( -2640660 + T \)
$61$ \( -827702 + T \)
$67$ \( 126004 + T \)
$71$ \( 1414728 + T \)
$73$ \( -980282 + T \)
$79$ \( 3566800 + T \)
$83$ \( -5672892 + T \)
$89$ \( 11951190 + T \)
$97$ \( -8682146 + T \)
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