Properties

Label 38.10.a.b
Level 38
Weight 10
Character orbit 38.a
Self dual yes
Analytic conductor 19.571
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5713617742\)
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

\(f(q)\) \(=\) \( q + 16q^{2} + 102q^{3} + 256q^{4} - 1581q^{5} + 1632q^{6} - 4865q^{7} + 4096q^{8} - 9279q^{9} + O(q^{10}) \) \( q + 16q^{2} + 102q^{3} + 256q^{4} - 1581q^{5} + 1632q^{6} - 4865q^{7} + 4096q^{8} - 9279q^{9} - 25296q^{10} - 64189q^{11} + 26112q^{12} - 48516q^{13} - 77840q^{14} - 161262q^{15} + 65536q^{16} + 314477q^{17} - 148464q^{18} + 130321q^{19} - 404736q^{20} - 496230q^{21} - 1027024q^{22} - 51088q^{23} + 417792q^{24} + 546436q^{25} - 776256q^{26} - 2954124q^{27} - 1245440q^{28} - 1543218q^{29} - 2580192q^{30} + 153108q^{31} + 1048576q^{32} - 6547278q^{33} + 5031632q^{34} + 7691565q^{35} - 2375424q^{36} + 71578q^{37} + 2085136q^{38} - 4948632q^{39} - 6475776q^{40} - 24190606q^{41} - 7939680q^{42} - 2906529q^{43} - 16432384q^{44} + 14670099q^{45} - 817408q^{46} + 14687405q^{47} + 6684672q^{48} - 16685382q^{49} + 8742976q^{50} + 32076654q^{51} - 12420096q^{52} + 107478052q^{53} - 47265984q^{54} + 101482809q^{55} - 19927040q^{56} + 13292742q^{57} - 24691488q^{58} + 138112586q^{59} - 41283072q^{60} - 122366017q^{61} + 2449728q^{62} + 45142335q^{63} + 16777216q^{64} + 76703796q^{65} - 104756448q^{66} + 67296612q^{67} + 80506112q^{68} - 5210976q^{69} + 123065040q^{70} + 253992790q^{71} - 38006784q^{72} + 25518121q^{73} + 1145248q^{74} + 55736472q^{75} + 33362176q^{76} + 312279485q^{77} - 79178112q^{78} - 264202112q^{79} - 103612416q^{80} - 118682091q^{81} - 387049696q^{82} - 724058420q^{83} - 127034880q^{84} - 497188137q^{85} - 46504464q^{86} - 157408236q^{87} - 262918144q^{88} - 1075037068q^{89} + 234721584q^{90} + 236030340q^{91} - 13078528q^{92} + 15617016q^{93} + 234998480q^{94} - 206037501q^{95} + 106954752q^{96} + 1173230648q^{97} - 266966112q^{98} + 595609731q^{99} + O(q^{100}) \)

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 102.000 256.000 −1581.00 1632.00 −4865.00 4096.00 −9279.00 −25296.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

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 38.10.a.b 1
3.b odd 2 1 342.10.a.b 1
4.b odd 2 1 304.10.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.a.b 1 1.a even 1 1 trivial
304.10.a.a 1 4.b odd 2 1
342.10.a.b 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 102 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(38))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 16 T \)
$3$ \( 1 - 102 T + 19683 T^{2} \)
$5$ \( 1 + 1581 T + 1953125 T^{2} \)
$7$ \( 1 + 4865 T + 40353607 T^{2} \)
$11$ \( 1 + 64189 T + 2357947691 T^{2} \)
$13$ \( 1 + 48516 T + 10604499373 T^{2} \)
$17$ \( 1 - 314477 T + 118587876497 T^{2} \)
$19$ \( 1 - 130321 T \)
$23$ \( 1 + 51088 T + 1801152661463 T^{2} \)
$29$ \( 1 + 1543218 T + 14507145975869 T^{2} \)
$31$ \( 1 - 153108 T + 26439622160671 T^{2} \)
$37$ \( 1 - 71578 T + 129961739795077 T^{2} \)
$41$ \( 1 + 24190606 T + 327381934393961 T^{2} \)
$43$ \( 1 + 2906529 T + 502592611936843 T^{2} \)
$47$ \( 1 - 14687405 T + 1119130473102767 T^{2} \)
$53$ \( 1 - 107478052 T + 3299763591802133 T^{2} \)
$59$ \( 1 - 138112586 T + 8662995818654939 T^{2} \)
$61$ \( 1 + 122366017 T + 11694146092834141 T^{2} \)
$67$ \( 1 - 67296612 T + 27206534396294947 T^{2} \)
$71$ \( 1 - 253992790 T + 45848500718449031 T^{2} \)
$73$ \( 1 - 25518121 T + 58871586708267913 T^{2} \)
$79$ \( 1 + 264202112 T + 119851595982618319 T^{2} \)
$83$ \( 1 + 724058420 T + 186940255267540403 T^{2} \)
$89$ \( 1 + 1075037068 T + 350356403707485209 T^{2} \)
$97$ \( 1 - 1173230648 T + 760231058654565217 T^{2} \)
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