Properties

Label 38.6.a.a
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

\(f(q)\) \(=\) \( q - 4q^{2} - 6q^{3} + 16q^{4} + 31q^{5} + 24q^{6} - 27q^{7} - 64q^{8} - 207q^{9} + O(q^{10}) \) \( q - 4q^{2} - 6q^{3} + 16q^{4} + 31q^{5} + 24q^{6} - 27q^{7} - 64q^{8} - 207q^{9} - 124q^{10} - 323q^{11} - 96q^{12} - 676q^{13} + 108q^{14} - 186q^{15} + 256q^{16} - 1107q^{17} + 828q^{18} - 361q^{19} + 496q^{20} + 162q^{21} + 1292q^{22} + 1384q^{23} + 384q^{24} - 2164q^{25} + 2704q^{26} + 2700q^{27} - 432q^{28} + 2870q^{29} + 744q^{30} + 1372q^{31} - 1024q^{32} + 1938q^{33} + 4428q^{34} - 837q^{35} - 3312q^{36} - 7982q^{37} + 1444q^{38} + 4056q^{39} - 1984q^{40} + 1202q^{41} - 648q^{42} - 9911q^{43} - 5168q^{44} - 6417q^{45} - 5536q^{46} + 3463q^{47} - 1536q^{48} - 16078q^{49} + 8656q^{50} + 6642q^{51} - 10816q^{52} + 17764q^{53} - 10800q^{54} - 10013q^{55} + 1728q^{56} + 2166q^{57} - 11480q^{58} + 27270q^{59} - 2976q^{60} + 20867q^{61} - 5488q^{62} + 5589q^{63} + 4096q^{64} - 20956q^{65} - 7752q^{66} + 15228q^{67} - 17712q^{68} - 8304q^{69} + 3348q^{70} + 40642q^{71} + 13248q^{72} - 66711q^{73} + 31928q^{74} + 12984q^{75} - 5776q^{76} + 8721q^{77} - 16224q^{78} + 68960q^{79} + 7936q^{80} + 34101q^{81} - 4808q^{82} - 12396q^{83} + 2592q^{84} - 34317q^{85} + 39644q^{86} - 17220q^{87} + 20672q^{88} + 41220q^{89} + 25668q^{90} + 18252q^{91} + 22144q^{92} - 8232q^{93} - 13852q^{94} - 11191q^{95} + 6144q^{96} - 113432q^{97} + 64312q^{98} + 66861q^{99} + O(q^{100}) \)

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 −6.00000 16.0000 31.0000 24.0000 −27.0000 −64.0000 −207.000 −124.000
\(n\): 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.a 1
3.b odd 2 1 342.6.a.e 1
4.b odd 2 1 304.6.a.c 1
5.b even 2 1 950.6.a.b 1
19.b odd 2 1 722.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.a.a 1 1.a even 1 1 trivial
304.6.a.c 1 4.b odd 2 1
342.6.a.e 1 3.b odd 2 1
722.6.a.b 1 19.b odd 2 1
950.6.a.b 1 5.b even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T \)
$3$ \( 1 + 6 T + 243 T^{2} \)
$5$ \( 1 - 31 T + 3125 T^{2} \)
$7$ \( 1 + 27 T + 16807 T^{2} \)
$11$ \( 1 + 323 T + 161051 T^{2} \)
$13$ \( 1 + 676 T + 371293 T^{2} \)
$17$ \( 1 + 1107 T + 1419857 T^{2} \)
$19$ \( 1 + 361 T \)
$23$ \( 1 - 1384 T + 6436343 T^{2} \)
$29$ \( 1 - 2870 T + 20511149 T^{2} \)
$31$ \( 1 - 1372 T + 28629151 T^{2} \)
$37$ \( 1 + 7982 T + 69343957 T^{2} \)
$41$ \( 1 - 1202 T + 115856201 T^{2} \)
$43$ \( 1 + 9911 T + 147008443 T^{2} \)
$47$ \( 1 - 3463 T + 229345007 T^{2} \)
$53$ \( 1 - 17764 T + 418195493 T^{2} \)
$59$ \( 1 - 27270 T + 714924299 T^{2} \)
$61$ \( 1 - 20867 T + 844596301 T^{2} \)
$67$ \( 1 - 15228 T + 1350125107 T^{2} \)
$71$ \( 1 - 40642 T + 1804229351 T^{2} \)
$73$ \( 1 + 66711 T + 2073071593 T^{2} \)
$79$ \( 1 - 68960 T + 3077056399 T^{2} \)
$83$ \( 1 + 12396 T + 3939040643 T^{2} \)
$89$ \( 1 - 41220 T + 5584059449 T^{2} \)
$97$ \( 1 + 113432 T + 8587340257 T^{2} \)
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