Properties

Label 39.4.a.a
Level $39$
Weight $4$
Character orbit 39.a
Self dual yes
Analytic conductor $2.301$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 39.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.30107449022\)
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

\(f(q)\) \(=\) \( q - 3 q^{3} - 8 q^{4} - 12 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 8 q^{4} - 12 q^{5} + 2 q^{7} + 9 q^{9} - 36 q^{11} + 24 q^{12} + 13 q^{13} + 36 q^{15} + 64 q^{16} - 78 q^{17} + 74 q^{19} + 96 q^{20} - 6 q^{21} - 96 q^{23} + 19 q^{25} - 27 q^{27} - 16 q^{28} + 18 q^{29} - 214 q^{31} + 108 q^{33} - 24 q^{35} - 72 q^{36} - 286 q^{37} - 39 q^{39} - 384 q^{41} + 524 q^{43} + 288 q^{44} - 108 q^{45} + 300 q^{47} - 192 q^{48} - 339 q^{49} + 234 q^{51} - 104 q^{52} + 558 q^{53} + 432 q^{55} - 222 q^{57} + 576 q^{59} - 288 q^{60} + 74 q^{61} + 18 q^{63} - 512 q^{64} - 156 q^{65} + 38 q^{67} + 624 q^{68} + 288 q^{69} - 456 q^{71} - 682 q^{73} - 57 q^{75} - 592 q^{76} - 72 q^{77} + 704 q^{79} - 768 q^{80} + 81 q^{81} - 888 q^{83} + 48 q^{84} + 936 q^{85} - 54 q^{87} - 1020 q^{89} + 26 q^{91} + 768 q^{92} + 642 q^{93} - 888 q^{95} + 110 q^{97} - 324 q^{99}+O(q^{100}) \) 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
0 −3.00000 −8.00000 −12.0000 0 2.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(13\) \(-1\)

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 39.4.a.a 1
3.b odd 2 1 117.4.a.a 1
4.b odd 2 1 624.4.a.g 1
5.b even 2 1 975.4.a.e 1
7.b odd 2 1 1911.4.a.f 1
8.b even 2 1 2496.4.a.o 1
8.d odd 2 1 2496.4.a.f 1
12.b even 2 1 1872.4.a.m 1
13.b even 2 1 507.4.a.c 1
13.d odd 4 2 507.4.b.b 2
39.d odd 2 1 1521.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.a 1 1.a even 1 1 trivial
117.4.a.a 1 3.b odd 2 1
507.4.a.c 1 13.b even 2 1
507.4.b.b 2 13.d odd 4 2
624.4.a.g 1 4.b odd 2 1
975.4.a.e 1 5.b even 2 1
1521.4.a.f 1 39.d odd 2 1
1872.4.a.m 1 12.b even 2 1
1911.4.a.f 1 7.b odd 2 1
2496.4.a.f 1 8.d odd 2 1
2496.4.a.o 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(39))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 12 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T + 36 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T + 78 \) Copy content Toggle raw display
$19$ \( T - 74 \) Copy content Toggle raw display
$23$ \( T + 96 \) Copy content Toggle raw display
$29$ \( T - 18 \) Copy content Toggle raw display
$31$ \( T + 214 \) Copy content Toggle raw display
$37$ \( T + 286 \) Copy content Toggle raw display
$41$ \( T + 384 \) Copy content Toggle raw display
$43$ \( T - 524 \) Copy content Toggle raw display
$47$ \( T - 300 \) Copy content Toggle raw display
$53$ \( T - 558 \) Copy content Toggle raw display
$59$ \( T - 576 \) Copy content Toggle raw display
$61$ \( T - 74 \) Copy content Toggle raw display
$67$ \( T - 38 \) Copy content Toggle raw display
$71$ \( T + 456 \) Copy content Toggle raw display
$73$ \( T + 682 \) Copy content Toggle raw display
$79$ \( T - 704 \) Copy content Toggle raw display
$83$ \( T + 888 \) Copy content Toggle raw display
$89$ \( T + 1020 \) Copy content Toggle raw display
$97$ \( T - 110 \) Copy content Toggle raw display
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