Properties

Label 38.2.a.b
Level $38$
Weight $2$
Character orbit 38.a
Self dual yes
Analytic conductor $0.303$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.303431527681\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - 4q^{5} - q^{6} + 3q^{7} + q^{8} - 2q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - 4q^{5} - q^{6} + 3q^{7} + q^{8} - 2q^{9} - 4q^{10} + 2q^{11} - q^{12} - q^{13} + 3q^{14} + 4q^{15} + q^{16} + 3q^{17} - 2q^{18} - q^{19} - 4q^{20} - 3q^{21} + 2q^{22} - q^{23} - q^{24} + 11q^{25} - q^{26} + 5q^{27} + 3q^{28} - 5q^{29} + 4q^{30} - 8q^{31} + q^{32} - 2q^{33} + 3q^{34} - 12q^{35} - 2q^{36} - 2q^{37} - q^{38} + q^{39} - 4q^{40} - 8q^{41} - 3q^{42} + 4q^{43} + 2q^{44} + 8q^{45} - q^{46} + 8q^{47} - q^{48} + 2q^{49} + 11q^{50} - 3q^{51} - q^{52} - q^{53} + 5q^{54} - 8q^{55} + 3q^{56} + q^{57} - 5q^{58} + 15q^{59} + 4q^{60} + 2q^{61} - 8q^{62} - 6q^{63} + q^{64} + 4q^{65} - 2q^{66} + 3q^{67} + 3q^{68} + q^{69} - 12q^{70} + 2q^{71} - 2q^{72} + 9q^{73} - 2q^{74} - 11q^{75} - q^{76} + 6q^{77} + q^{78} - 10q^{79} - 4q^{80} + q^{81} - 8q^{82} - 6q^{83} - 3q^{84} - 12q^{85} + 4q^{86} + 5q^{87} + 2q^{88} + 8q^{90} - 3q^{91} - q^{92} + 8q^{93} + 8q^{94} + 4q^{95} - q^{96} - 2q^{97} + 2q^{98} - 4q^{99} + O(q^{100}) \)

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
1.00000 −1.00000 1.00000 −4.00000 −1.00000 3.00000 1.00000 −2.00000 −4.00000
\(n\): 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 38.2.a.b 1
3.b odd 2 1 342.2.a.d 1
4.b odd 2 1 304.2.a.d 1
5.b even 2 1 950.2.a.b 1
5.c odd 4 2 950.2.b.c 2
7.b odd 2 1 1862.2.a.f 1
8.b even 2 1 1216.2.a.n 1
8.d odd 2 1 1216.2.a.g 1
11.b odd 2 1 4598.2.a.a 1
12.b even 2 1 2736.2.a.w 1
13.b even 2 1 6422.2.a.b 1
15.d odd 2 1 8550.2.a.u 1
19.b odd 2 1 722.2.a.b 1
19.c even 3 2 722.2.c.d 2
19.d odd 6 2 722.2.c.f 2
19.e even 9 6 722.2.e.c 6
19.f odd 18 6 722.2.e.d 6
20.d odd 2 1 7600.2.a.h 1
57.d even 2 1 6498.2.a.y 1
76.d even 2 1 5776.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 1.a even 1 1 trivial
304.2.a.d 1 4.b odd 2 1
342.2.a.d 1 3.b odd 2 1
722.2.a.b 1 19.b odd 2 1
722.2.c.d 2 19.c even 3 2
722.2.c.f 2 19.d odd 6 2
722.2.e.c 6 19.e even 9 6
722.2.e.d 6 19.f odd 18 6
950.2.a.b 1 5.b even 2 1
950.2.b.c 2 5.c odd 4 2
1216.2.a.g 1 8.d odd 2 1
1216.2.a.n 1 8.b even 2 1
1862.2.a.f 1 7.b odd 2 1
2736.2.a.w 1 12.b even 2 1
4598.2.a.a 1 11.b odd 2 1
5776.2.a.d 1 76.d even 2 1
6422.2.a.b 1 13.b even 2 1
6498.2.a.y 1 57.d even 2 1
7600.2.a.h 1 20.d odd 2 1
8550.2.a.u 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(38))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 + T + 3 T^{2} \)
$5$ \( 1 + 4 T + 5 T^{2} \)
$7$ \( 1 - 3 T + 7 T^{2} \)
$11$ \( 1 - 2 T + 11 T^{2} \)
$13$ \( 1 + T + 13 T^{2} \)
$17$ \( 1 - 3 T + 17 T^{2} \)
$19$ \( 1 + T \)
$23$ \( 1 + T + 23 T^{2} \)
$29$ \( 1 + 5 T + 29 T^{2} \)
$31$ \( 1 + 8 T + 31 T^{2} \)
$37$ \( 1 + 2 T + 37 T^{2} \)
$41$ \( 1 + 8 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 - 8 T + 47 T^{2} \)
$53$ \( 1 + T + 53 T^{2} \)
$59$ \( 1 - 15 T + 59 T^{2} \)
$61$ \( 1 - 2 T + 61 T^{2} \)
$67$ \( 1 - 3 T + 67 T^{2} \)
$71$ \( 1 - 2 T + 71 T^{2} \)
$73$ \( 1 - 9 T + 73 T^{2} \)
$79$ \( 1 + 10 T + 79 T^{2} \)
$83$ \( 1 + 6 T + 83 T^{2} \)
$89$ \( 1 + 89 T^{2} \)
$97$ \( 1 + 2 T + 97 T^{2} \)
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