Properties

Label 38.6.a.b
Level $38$
Weight $6$
Character orbit 38.a
Self dual yes
Analytic conductor $6.095$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.09458515289\)
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

\(f(q)\) \(=\) \( q + 4q^{2} - 14q^{3} + 16q^{4} - 45q^{5} - 56q^{6} - 121q^{7} + 64q^{8} - 47q^{9} + O(q^{10}) \) \( q + 4q^{2} - 14q^{3} + 16q^{4} - 45q^{5} - 56q^{6} - 121q^{7} + 64q^{8} - 47q^{9} - 180q^{10} - 381q^{11} - 224q^{12} - 100q^{13} - 484q^{14} + 630q^{15} + 256q^{16} + 933q^{17} - 188q^{18} + 361q^{19} - 720q^{20} + 1694q^{21} - 1524q^{22} - 552q^{23} - 896q^{24} - 1100q^{25} - 400q^{26} + 4060q^{27} - 1936q^{28} + 2394q^{29} + 2520q^{30} - 4024q^{31} + 1024q^{32} + 5334q^{33} + 3732q^{34} + 5445q^{35} - 752q^{36} + 9182q^{37} + 1444q^{38} + 1400q^{39} - 2880q^{40} - 2250q^{41} + 6776q^{42} - 23377q^{43} - 6096q^{44} + 2115q^{45} - 2208q^{46} - 26595q^{47} - 3584q^{48} - 2166q^{49} - 4400q^{50} - 13062q^{51} - 1600q^{52} - 16008q^{53} + 16240q^{54} + 17145q^{55} - 7744q^{56} - 5054q^{57} + 9576q^{58} - 126q^{59} + 10080q^{60} + 21335q^{61} - 16096q^{62} + 5687q^{63} + 4096q^{64} + 4500q^{65} + 21336q^{66} - 51760q^{67} + 14928q^{68} + 7728q^{69} + 21780q^{70} + 8574q^{71} - 3008q^{72} + 11153q^{73} + 36728q^{74} + 15400q^{75} + 5776q^{76} + 46101q^{77} + 5600q^{78} - 1660q^{79} - 11520q^{80} - 45419q^{81} - 9000q^{82} + 95964q^{83} + 27104q^{84} - 41985q^{85} - 93508q^{86} - 33516q^{87} - 24384q^{88} + 118848q^{89} + 8460q^{90} + 12100q^{91} - 8832q^{92} + 56336q^{93} - 106380q^{94} - 16245q^{95} - 14336q^{96} - 153760q^{97} - 8664q^{98} + 17907q^{99} + O(q^{100}) \)

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
4.00000 −14.0000 16.0000 −45.0000 −56.0000 −121.000 64.0000 −47.0000 −180.000
\(n\): 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 38.6.a.b 1
3.b odd 2 1 342.6.a.b 1
4.b odd 2 1 304.6.a.e 1
5.b even 2 1 950.6.a.a 1
19.b odd 2 1 722.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.a.b 1 1.a even 1 1 trivial
304.6.a.e 1 4.b odd 2 1
342.6.a.b 1 3.b odd 2 1
722.6.a.a 1 19.b odd 2 1
950.6.a.a 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 14 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(38))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T \)
$3$ \( 1 + 14 T + 243 T^{2} \)
$5$ \( 1 + 45 T + 3125 T^{2} \)
$7$ \( 1 + 121 T + 16807 T^{2} \)
$11$ \( 1 + 381 T + 161051 T^{2} \)
$13$ \( 1 + 100 T + 371293 T^{2} \)
$17$ \( 1 - 933 T + 1419857 T^{2} \)
$19$ \( 1 - 361 T \)
$23$ \( 1 + 552 T + 6436343 T^{2} \)
$29$ \( 1 - 2394 T + 20511149 T^{2} \)
$31$ \( 1 + 4024 T + 28629151 T^{2} \)
$37$ \( 1 - 9182 T + 69343957 T^{2} \)
$41$ \( 1 + 2250 T + 115856201 T^{2} \)
$43$ \( 1 + 23377 T + 147008443 T^{2} \)
$47$ \( 1 + 26595 T + 229345007 T^{2} \)
$53$ \( 1 + 16008 T + 418195493 T^{2} \)
$59$ \( 1 + 126 T + 714924299 T^{2} \)
$61$ \( 1 - 21335 T + 844596301 T^{2} \)
$67$ \( 1 + 51760 T + 1350125107 T^{2} \)
$71$ \( 1 - 8574 T + 1804229351 T^{2} \)
$73$ \( 1 - 11153 T + 2073071593 T^{2} \)
$79$ \( 1 + 1660 T + 3077056399 T^{2} \)
$83$ \( 1 - 95964 T + 3939040643 T^{2} \)
$89$ \( 1 - 118848 T + 5584059449 T^{2} \)
$97$ \( 1 + 153760 T + 8587340257 T^{2} \)
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