Properties

Label 14.10.a.b
Level 14
Weight 10
Character orbit 14.a
Self dual yes
Analytic conductor 7.211
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 14 = 2 \cdot 7 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 14.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.21050170629\)
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

\(f(q)\) \(=\) \( q + 16q^{2} + 170q^{3} + 256q^{4} + 544q^{5} + 2720q^{6} - 2401q^{7} + 4096q^{8} + 9217q^{9} + O(q^{10}) \) \( q + 16q^{2} + 170q^{3} + 256q^{4} + 544q^{5} + 2720q^{6} - 2401q^{7} + 4096q^{8} + 9217q^{9} + 8704q^{10} + 48824q^{11} + 43520q^{12} - 15876q^{13} - 38416q^{14} + 92480q^{15} + 65536q^{16} - 21418q^{17} + 147472q^{18} - 716410q^{19} + 139264q^{20} - 408170q^{21} + 781184q^{22} - 2470000q^{23} + 696320q^{24} - 1657189q^{25} - 254016q^{26} - 1779220q^{27} - 614656q^{28} + 5556826q^{29} + 1479680q^{30} + 5799348q^{31} + 1048576q^{32} + 8300080q^{33} - 342688q^{34} - 1306144q^{35} + 2359552q^{36} - 3894430q^{37} - 11462560q^{38} - 2698920q^{39} + 2228224q^{40} - 6360858q^{41} - 6530720q^{42} - 18701296q^{43} + 12498944q^{44} + 5014048q^{45} - 39520000q^{46} + 56539068q^{47} + 11141120q^{48} + 5764801q^{49} - 26515024q^{50} - 3641060q^{51} - 4064256q^{52} - 59894682q^{53} - 28467520q^{54} + 26560256q^{55} - 9834496q^{56} - 121789700q^{57} + 88909216q^{58} + 165629662q^{59} + 23674880q^{60} + 51419016q^{61} + 92789568q^{62} - 22130017q^{63} + 16777216q^{64} - 8636544q^{65} + 132801280q^{66} + 93546508q^{67} - 5483008q^{68} - 419900000q^{69} - 20898304q^{70} - 95633536q^{71} + 37752832q^{72} + 306496402q^{73} - 62310880q^{74} - 281722130q^{75} - 183400960q^{76} - 117226424q^{77} - 43182720q^{78} + 496474152q^{79} + 35651584q^{80} - 483885611q^{81} - 101773728q^{82} - 371486962q^{83} - 104491520q^{84} - 11651392q^{85} - 299220736q^{86} + 944660420q^{87} + 199983104q^{88} - 165482550q^{89} + 80224768q^{90} + 38118276q^{91} - 632320000q^{92} + 985889160q^{93} + 904625088q^{94} - 389727040q^{95} + 178257920q^{96} + 758016742q^{97} + 92236816q^{98} + 450010808q^{99} + O(q^{100}) \)

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
16.0000 170.000 256.000 544.000 2720.00 −2401.00 4096.00 9217.00 8704.00
\(n\): 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 14.10.a.b 1
3.b odd 2 1 126.10.a.a 1
4.b odd 2 1 112.10.a.a 1
5.b even 2 1 350.10.a.a 1
5.c odd 4 2 350.10.c.d 2
7.b odd 2 1 98.10.a.b 1
7.c even 3 2 98.10.c.a 2
7.d odd 6 2 98.10.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.b 1 1.a even 1 1 trivial
98.10.a.b 1 7.b odd 2 1
98.10.c.a 2 7.c even 3 2
98.10.c.d 2 7.d odd 6 2
112.10.a.a 1 4.b odd 2 1
126.10.a.a 1 3.b odd 2 1
350.10.a.a 1 5.b even 2 1
350.10.c.d 2 5.c odd 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 170 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(14))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 16 T \)
$3$ \( 1 - 170 T + 19683 T^{2} \)
$5$ \( 1 - 544 T + 1953125 T^{2} \)
$7$ \( 1 + 2401 T \)
$11$ \( 1 - 48824 T + 2357947691 T^{2} \)
$13$ \( 1 + 15876 T + 10604499373 T^{2} \)
$17$ \( 1 + 21418 T + 118587876497 T^{2} \)
$19$ \( 1 + 716410 T + 322687697779 T^{2} \)
$23$ \( 1 + 2470000 T + 1801152661463 T^{2} \)
$29$ \( 1 - 5556826 T + 14507145975869 T^{2} \)
$31$ \( 1 - 5799348 T + 26439622160671 T^{2} \)
$37$ \( 1 + 3894430 T + 129961739795077 T^{2} \)
$41$ \( 1 + 6360858 T + 327381934393961 T^{2} \)
$43$ \( 1 + 18701296 T + 502592611936843 T^{2} \)
$47$ \( 1 - 56539068 T + 1119130473102767 T^{2} \)
$53$ \( 1 + 59894682 T + 3299763591802133 T^{2} \)
$59$ \( 1 - 165629662 T + 8662995818654939 T^{2} \)
$61$ \( 1 - 51419016 T + 11694146092834141 T^{2} \)
$67$ \( 1 - 93546508 T + 27206534396294947 T^{2} \)
$71$ \( 1 + 95633536 T + 45848500718449031 T^{2} \)
$73$ \( 1 - 306496402 T + 58871586708267913 T^{2} \)
$79$ \( 1 - 496474152 T + 119851595982618319 T^{2} \)
$83$ \( 1 + 371486962 T + 186940255267540403 T^{2} \)
$89$ \( 1 + 165482550 T + 350356403707485209 T^{2} \)
$97$ \( 1 - 758016742 T + 760231058654565217 T^{2} \)
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