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

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

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:

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

Atkin-Lehner signs

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 8 T \)
$3$ \( 1 - 12 T + 2187 T^{2} \)
$5$ \( 1 + 210 T + 78125 T^{2} \)
$7$ \( 1 - 1016 T + 823543 T^{2} \)
$11$ \( 1 - 1092 T + 19487171 T^{2} \)
$13$ \( 1 - 1382 T + 62748517 T^{2} \)
$17$ \( 1 - 14706 T + 410338673 T^{2} \)
$19$ \( 1 + 39940 T + 893871739 T^{2} \)
$23$ \( 1 - 68712 T + 3404825447 T^{2} \)
$29$ \( 1 + 102570 T + 17249876309 T^{2} \)
$31$ \( 1 - 227552 T + 27512614111 T^{2} \)
$37$ \( 1 - 160526 T + 94931877133 T^{2} \)
$41$ \( 1 - 10842 T + 194754273881 T^{2} \)
$43$ \( 1 + 630748 T + 271818611107 T^{2} \)
$47$ \( 1 - 472656 T + 506623120463 T^{2} \)
$53$ \( 1 + 1494018 T + 1174711139837 T^{2} \)
$59$ \( 1 - 2640660 T + 2488651484819 T^{2} \)
$61$ \( 1 - 827702 T + 3142742836021 T^{2} \)
$67$ \( 1 + 126004 T + 6060711605323 T^{2} \)
$71$ \( 1 + 1414728 T + 9095120158391 T^{2} \)
$73$ \( 1 - 980282 T + 11047398519097 T^{2} \)
$79$ \( 1 + 3566800 T + 19203908986159 T^{2} \)
$83$ \( 1 - 5672892 T + 27136050989627 T^{2} \)
$89$ \( 1 + 11951190 T + 44231334895529 T^{2} \)
$97$ \( 1 - 8682146 T + 80798284478113 T^{2} \)
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Additional information

eta-product

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