Properties

Label 38.10.a.a
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

\(f(q)\) \(=\) \( q + 16q^{2} - 119q^{3} + 256q^{4} - 684q^{5} - 1904q^{6} + 9149q^{7} + 4096q^{8} - 5522q^{9} + O(q^{10}) \) \( q + 16q^{2} - 119q^{3} + 256q^{4} - 684q^{5} - 1904q^{6} + 9149q^{7} + 4096q^{8} - 5522q^{9} - 10944q^{10} + 5790q^{11} - 30464q^{12} - 179881q^{13} + 146384q^{14} + 81396q^{15} + 65536q^{16} - 594093q^{17} - 88352q^{18} + 130321q^{19} - 175104q^{20} - 1088731q^{21} + 92640q^{22} - 1744767q^{23} - 487424q^{24} - 1485269q^{25} - 2878096q^{26} + 2999395q^{27} + 2342144q^{28} + 4314387q^{29} + 1302336q^{30} + 160232q^{31} + 1048576q^{32} - 689010q^{33} - 9505488q^{34} - 6257916q^{35} - 1413632q^{36} - 21943090q^{37} + 2085136q^{38} + 21405839q^{39} - 2801664q^{40} + 294816q^{41} - 17419696q^{42} - 39393148q^{43} + 1482240q^{44} + 3777048q^{45} - 27916272q^{46} + 46596360q^{47} - 7798784q^{48} + 43350594q^{49} - 23764304q^{50} + 70697067q^{51} - 46049536q^{52} + 22121703q^{53} + 47990320q^{54} - 3960360q^{55} + 37474304q^{56} - 15508199q^{57} + 69030192q^{58} + 33070233q^{59} + 20837376q^{60} + 188535938q^{61} + 2563712q^{62} - 50520778q^{63} + 16777216q^{64} + 123038604q^{65} - 11024160q^{66} - 20769067q^{67} - 152087808q^{68} + 207627273q^{69} - 100126656q^{70} - 232299978q^{71} - 22618112q^{72} - 3022183q^{73} - 351089440q^{74} + 176747011q^{75} + 33362176q^{76} + 52972710q^{77} + 342493424q^{78} - 446379406q^{79} - 44826624q^{80} - 248238479q^{81} + 4717056q^{82} + 794022846q^{83} - 278715136q^{84} + 406359612q^{85} - 630290368q^{86} - 513412053q^{87} + 23715840q^{88} + 90999336q^{89} + 60432768q^{90} - 1645731269q^{91} - 446660352q^{92} - 19067608q^{93} + 745541760q^{94} - 89139564q^{95} - 124780544q^{96} - 123974170q^{97} + 693609504q^{98} - 31972380q^{99} + O(q^{100}) \)

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 −119.000 256.000 −684.000 −1904.00 9149.00 4096.00 −5522.00 −10944.0
\(n\): 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.a 1
3.b odd 2 1 342.10.a.a 1
4.b odd 2 1 304.10.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.a.a 1 1.a even 1 1 trivial
304.10.a.b 1 4.b odd 2 1
342.10.a.a 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 119 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(38))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 16 T \)
$3$ \( 1 + 119 T + 19683 T^{2} \)
$5$ \( 1 + 684 T + 1953125 T^{2} \)
$7$ \( 1 - 9149 T + 40353607 T^{2} \)
$11$ \( 1 - 5790 T + 2357947691 T^{2} \)
$13$ \( 1 + 179881 T + 10604499373 T^{2} \)
$17$ \( 1 + 594093 T + 118587876497 T^{2} \)
$19$ \( 1 - 130321 T \)
$23$ \( 1 + 1744767 T + 1801152661463 T^{2} \)
$29$ \( 1 - 4314387 T + 14507145975869 T^{2} \)
$31$ \( 1 - 160232 T + 26439622160671 T^{2} \)
$37$ \( 1 + 21943090 T + 129961739795077 T^{2} \)
$41$ \( 1 - 294816 T + 327381934393961 T^{2} \)
$43$ \( 1 + 39393148 T + 502592611936843 T^{2} \)
$47$ \( 1 - 46596360 T + 1119130473102767 T^{2} \)
$53$ \( 1 - 22121703 T + 3299763591802133 T^{2} \)
$59$ \( 1 - 33070233 T + 8662995818654939 T^{2} \)
$61$ \( 1 - 188535938 T + 11694146092834141 T^{2} \)
$67$ \( 1 + 20769067 T + 27206534396294947 T^{2} \)
$71$ \( 1 + 232299978 T + 45848500718449031 T^{2} \)
$73$ \( 1 + 3022183 T + 58871586708267913 T^{2} \)
$79$ \( 1 + 446379406 T + 119851595982618319 T^{2} \)
$83$ \( 1 - 794022846 T + 186940255267540403 T^{2} \)
$89$ \( 1 - 90999336 T + 350356403707485209 T^{2} \)
$97$ \( 1 + 123974170 T + 760231058654565217 T^{2} \)
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