Properties

Label 8.14.a.a
Level 8
Weight 14
Character orbit 8.a
Self dual yes
Analytic conductor 8.578
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 14 \)
Character orbit: \([\chi]\) = 8.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.57847431615\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 12q^{3} - 4330q^{5} - 139992q^{7} - 1594179q^{9} + O(q^{10}) \) \( q - 12q^{3} - 4330q^{5} - 139992q^{7} - 1594179q^{9} - 6484324q^{11} - 22588034q^{13} + 51960q^{15} - 23732270q^{17} + 325344836q^{19} + 1679904q^{21} + 921600632q^{23} - 1201954225q^{25} + 38262024q^{27} - 3865879218q^{29} - 2253401440q^{31} + 77811888q^{33} + 606165360q^{35} + 18250384566q^{37} + 271056408q^{39} + 34422845322q^{41} - 17192501444q^{43} + 6902795070q^{45} - 67371749904q^{47} - 77291250343q^{49} + 284787240q^{51} - 87281218426q^{53} + 28077122920q^{55} - 3904138032q^{57} + 540214518668q^{59} - 51276568850q^{61} + 223172306568q^{63} + 97806187220q^{65} + 25519930676q^{67} - 11059207584q^{69} - 1387500699032q^{71} - 819049441238q^{73} + 14423450700q^{75} + 907753485408q^{77} - 4030935615344q^{79} + 2541177101529q^{81} + 4180823831428q^{83} + 102760729100q^{85} + 46390550616q^{87} + 2677027798266q^{89} + 3162144055728q^{91} + 27040817280q^{93} - 1408743139880q^{95} - 14039464316446q^{97} + 10337173149996q^{99} + O(q^{100}) \)

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
0 −12.0000 0 −4330.00 0 −139992. 0 −1.59418e6 0
\(n\): 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 8.14.a.a 1
3.b odd 2 1 72.14.a.a 1
4.b odd 2 1 16.14.a.c 1
5.b even 2 1 200.14.a.a 1
5.c odd 4 2 200.14.c.a 2
8.b even 2 1 64.14.a.f 1
8.d odd 2 1 64.14.a.d 1
12.b even 2 1 144.14.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.14.a.a 1 1.a even 1 1 trivial
16.14.a.c 1 4.b odd 2 1
64.14.a.d 1 8.d odd 2 1
64.14.a.f 1 8.b even 2 1
72.14.a.a 1 3.b odd 2 1
144.14.a.f 1 12.b even 2 1
200.14.a.a 1 5.b even 2 1
200.14.c.a 2 5.c odd 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 12 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(8))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 12 T + 1594323 T^{2} \)
$5$ \( 1 + 4330 T + 1220703125 T^{2} \)
$7$ \( 1 + 139992 T + 96889010407 T^{2} \)
$11$ \( 1 + 6484324 T + 34522712143931 T^{2} \)
$13$ \( 1 + 22588034 T + 302875106592253 T^{2} \)
$17$ \( 1 + 23732270 T + 9904578032905937 T^{2} \)
$19$ \( 1 - 325344836 T + 42052983462257059 T^{2} \)
$23$ \( 1 - 921600632 T + 504036361936467383 T^{2} \)
$29$ \( 1 + 3865879218 T + 10260628712958602189 T^{2} \)
$31$ \( 1 + 2253401440 T + 24417546297445042591 T^{2} \)
$37$ \( 1 - 18250384566 T + \)\(24\!\cdots\!97\)\( T^{2} \)
$41$ \( 1 - 34422845322 T + \)\(92\!\cdots\!21\)\( T^{2} \)
$43$ \( 1 + 17192501444 T + \)\(17\!\cdots\!43\)\( T^{2} \)
$47$ \( 1 + 67371749904 T + \)\(54\!\cdots\!27\)\( T^{2} \)
$53$ \( 1 + 87281218426 T + \)\(26\!\cdots\!73\)\( T^{2} \)
$59$ \( 1 - 540214518668 T + \)\(10\!\cdots\!79\)\( T^{2} \)
$61$ \( 1 + 51276568850 T + \)\(16\!\cdots\!81\)\( T^{2} \)
$67$ \( 1 - 25519930676 T + \)\(54\!\cdots\!87\)\( T^{2} \)
$71$ \( 1 + 1387500699032 T + \)\(11\!\cdots\!11\)\( T^{2} \)
$73$ \( 1 + 819049441238 T + \)\(16\!\cdots\!33\)\( T^{2} \)
$79$ \( 1 + 4030935615344 T + \)\(46\!\cdots\!39\)\( T^{2} \)
$83$ \( 1 - 4180823831428 T + \)\(88\!\cdots\!63\)\( T^{2} \)
$89$ \( 1 - 2677027798266 T + \)\(21\!\cdots\!69\)\( T^{2} \)
$97$ \( 1 + 14039464316446 T + \)\(67\!\cdots\!77\)\( T^{2} \)
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