Properties

Label 38.4.a.a
Level $38$
Weight $4$
Character orbit 38.a
Self dual yes
Analytic conductor $2.242$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.24207258022\)
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

\(f(q)\) \(=\) \( q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 9 q^{5} + 4 q^{6} - 31 q^{7} - 8 q^{8} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 9 q^{5} + 4 q^{6} - 31 q^{7} - 8 q^{8} - 23 q^{9} + 18 q^{10} + 57 q^{11} - 8 q^{12} - 52 q^{13} + 62 q^{14} + 18 q^{15} + 16 q^{16} + 69 q^{17} + 46 q^{18} + 19 q^{19} - 36 q^{20} + 62 q^{21} - 114 q^{22} - 72 q^{23} + 16 q^{24} - 44 q^{25} + 104 q^{26} + 100 q^{27} - 124 q^{28} - 150 q^{29} - 36 q^{30} + 32 q^{31} - 32 q^{32} - 114 q^{33} - 138 q^{34} + 279 q^{35} - 92 q^{36} - 226 q^{37} - 38 q^{38} + 104 q^{39} + 72 q^{40} - 258 q^{41} - 124 q^{42} - 67 q^{43} + 228 q^{44} + 207 q^{45} + 144 q^{46} + 579 q^{47} - 32 q^{48} + 618 q^{49} + 88 q^{50} - 138 q^{51} - 208 q^{52} - 432 q^{53} - 200 q^{54} - 513 q^{55} + 248 q^{56} - 38 q^{57} + 300 q^{58} - 330 q^{59} + 72 q^{60} - 13 q^{61} - 64 q^{62} + 713 q^{63} + 64 q^{64} + 468 q^{65} + 228 q^{66} - 856 q^{67} + 276 q^{68} + 144 q^{69} - 558 q^{70} + 642 q^{71} + 184 q^{72} - 487 q^{73} + 452 q^{74} + 88 q^{75} + 76 q^{76} - 1767 q^{77} - 208 q^{78} - 700 q^{79} - 144 q^{80} + 421 q^{81} + 516 q^{82} - 12 q^{83} + 248 q^{84} - 621 q^{85} + 134 q^{86} + 300 q^{87} - 456 q^{88} - 600 q^{89} - 414 q^{90} + 1612 q^{91} - 288 q^{92} - 64 q^{93} - 1158 q^{94} - 171 q^{95} + 64 q^{96} + 1424 q^{97} - 1236 q^{98} - 1311 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−2.00000 −2.00000 4.00000 −9.00000 4.00000 −31.0000 −8.00000 −23.0000 18.0000
\(n\): 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 38.4.a.a 1
3.b odd 2 1 342.4.a.d 1
4.b odd 2 1 304.4.a.a 1
5.b even 2 1 950.4.a.d 1
5.c odd 4 2 950.4.b.d 2
7.b odd 2 1 1862.4.a.a 1
8.b even 2 1 1216.4.a.e 1
8.d odd 2 1 1216.4.a.b 1
19.b odd 2 1 722.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.a 1 1.a even 1 1 trivial
304.4.a.a 1 4.b odd 2 1
342.4.a.d 1 3.b odd 2 1
722.4.a.d 1 19.b odd 2 1
950.4.a.d 1 5.b even 2 1
950.4.b.d 2 5.c odd 4 2
1216.4.a.b 1 8.d odd 2 1
1216.4.a.e 1 8.b even 2 1
1862.4.a.a 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(38))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T + 9 \) Copy content Toggle raw display
$7$ \( T + 31 \) Copy content Toggle raw display
$11$ \( T - 57 \) Copy content Toggle raw display
$13$ \( T + 52 \) Copy content Toggle raw display
$17$ \( T - 69 \) Copy content Toggle raw display
$19$ \( T - 19 \) Copy content Toggle raw display
$23$ \( T + 72 \) Copy content Toggle raw display
$29$ \( T + 150 \) Copy content Toggle raw display
$31$ \( T - 32 \) Copy content Toggle raw display
$37$ \( T + 226 \) Copy content Toggle raw display
$41$ \( T + 258 \) Copy content Toggle raw display
$43$ \( T + 67 \) Copy content Toggle raw display
$47$ \( T - 579 \) Copy content Toggle raw display
$53$ \( T + 432 \) Copy content Toggle raw display
$59$ \( T + 330 \) Copy content Toggle raw display
$61$ \( T + 13 \) Copy content Toggle raw display
$67$ \( T + 856 \) Copy content Toggle raw display
$71$ \( T - 642 \) Copy content Toggle raw display
$73$ \( T + 487 \) Copy content Toggle raw display
$79$ \( T + 700 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T + 600 \) Copy content Toggle raw display
$97$ \( T - 1424 \) Copy content Toggle raw display
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