Properties

Label 39.2.a.b
Level $39$
Weight $2$
Character orbit 39.a
Self dual yes
Analytic conductor $0.311$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 39.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.311416567883\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + q^{3} + ( 1 - 2 \beta ) q^{4} -2 \beta q^{5} + ( -1 + \beta ) q^{6} + 2 \beta q^{7} + ( -3 + \beta ) q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + q^{3} + ( 1 - 2 \beta ) q^{4} -2 \beta q^{5} + ( -1 + \beta ) q^{6} + 2 \beta q^{7} + ( -3 + \beta ) q^{8} + q^{9} + ( -4 + 2 \beta ) q^{10} -2 q^{11} + ( 1 - 2 \beta ) q^{12} - q^{13} + ( 4 - 2 \beta ) q^{14} -2 \beta q^{15} + 3 q^{16} + ( 2 + 4 \beta ) q^{17} + ( -1 + \beta ) q^{18} -2 \beta q^{19} + ( 8 - 2 \beta ) q^{20} + 2 \beta q^{21} + ( 2 - 2 \beta ) q^{22} -4 q^{23} + ( -3 + \beta ) q^{24} + 3 q^{25} + ( 1 - \beta ) q^{26} + q^{27} + ( -8 + 2 \beta ) q^{28} + 2 q^{29} + ( -4 + 2 \beta ) q^{30} + ( -4 + 2 \beta ) q^{31} + ( 3 + \beta ) q^{32} -2 q^{33} + ( 6 - 2 \beta ) q^{34} -8 q^{35} + ( 1 - 2 \beta ) q^{36} + ( -2 - 4 \beta ) q^{37} + ( -4 + 2 \beta ) q^{38} - q^{39} + ( -4 + 6 \beta ) q^{40} + ( 8 - 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{42} + ( 4 - 4 \beta ) q^{43} + ( -2 + 4 \beta ) q^{44} -2 \beta q^{45} + ( 4 - 4 \beta ) q^{46} + ( -6 - 4 \beta ) q^{47} + 3 q^{48} + q^{49} + ( -3 + 3 \beta ) q^{50} + ( 2 + 4 \beta ) q^{51} + ( -1 + 2 \beta ) q^{52} -2 q^{53} + ( -1 + \beta ) q^{54} + 4 \beta q^{55} + ( 4 - 6 \beta ) q^{56} -2 \beta q^{57} + ( -2 + 2 \beta ) q^{58} + ( 2 + 4 \beta ) q^{59} + ( 8 - 2 \beta ) q^{60} + ( 2 + 8 \beta ) q^{61} + ( 8 - 6 \beta ) q^{62} + 2 \beta q^{63} + ( -7 + 2 \beta ) q^{64} + 2 \beta q^{65} + ( 2 - 2 \beta ) q^{66} + ( 4 + 2 \beta ) q^{67} -14 q^{68} -4 q^{69} + ( 8 - 8 \beta ) q^{70} + 2 q^{71} + ( -3 + \beta ) q^{72} + ( 6 - 4 \beta ) q^{73} + ( -6 + 2 \beta ) q^{74} + 3 q^{75} + ( 8 - 2 \beta ) q^{76} -4 \beta q^{77} + ( 1 - \beta ) q^{78} -8 \beta q^{79} -6 \beta q^{80} + q^{81} + ( -12 + 10 \beta ) q^{82} + ( -2 + 4 \beta ) q^{83} + ( -8 + 2 \beta ) q^{84} + ( -16 - 4 \beta ) q^{85} + ( -12 + 8 \beta ) q^{86} + 2 q^{87} + ( 6 - 2 \beta ) q^{88} + ( 12 + 2 \beta ) q^{89} + ( -4 + 2 \beta ) q^{90} -2 \beta q^{91} + ( -4 + 8 \beta ) q^{92} + ( -4 + 2 \beta ) q^{93} + ( -2 - 2 \beta ) q^{94} + 8 q^{95} + ( 3 + \beta ) q^{96} + ( -2 + 4 \beta ) q^{97} + ( -1 + \beta ) q^{98} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 6q^{8} + 2q^{9} - 8q^{10} - 4q^{11} + 2q^{12} - 2q^{13} + 8q^{14} + 6q^{16} + 4q^{17} - 2q^{18} + 16q^{20} + 4q^{22} - 8q^{23} - 6q^{24} + 6q^{25} + 2q^{26} + 2q^{27} - 16q^{28} + 4q^{29} - 8q^{30} - 8q^{31} + 6q^{32} - 4q^{33} + 12q^{34} - 16q^{35} + 2q^{36} - 4q^{37} - 8q^{38} - 2q^{39} - 8q^{40} + 16q^{41} + 8q^{42} + 8q^{43} - 4q^{44} + 8q^{46} - 12q^{47} + 6q^{48} + 2q^{49} - 6q^{50} + 4q^{51} - 2q^{52} - 4q^{53} - 2q^{54} + 8q^{56} - 4q^{58} + 4q^{59} + 16q^{60} + 4q^{61} + 16q^{62} - 14q^{64} + 4q^{66} + 8q^{67} - 28q^{68} - 8q^{69} + 16q^{70} + 4q^{71} - 6q^{72} + 12q^{73} - 12q^{74} + 6q^{75} + 16q^{76} + 2q^{78} + 2q^{81} - 24q^{82} - 4q^{83} - 16q^{84} - 32q^{85} - 24q^{86} + 4q^{87} + 12q^{88} + 24q^{89} - 8q^{90} - 8q^{92} - 8q^{93} - 4q^{94} + 16q^{95} + 6q^{96} - 4q^{97} - 2q^{98} - 4q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
−2.41421 1.00000 3.82843 2.82843 −2.41421 −2.82843 −4.41421 1.00000 −6.82843
1.2 0.414214 1.00000 −1.82843 −2.82843 0.414214 2.82843 −1.58579 1.00000 −1.17157
\(n\): 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 39.2.a.b 2
3.b odd 2 1 117.2.a.c 2
4.b odd 2 1 624.2.a.k 2
5.b even 2 1 975.2.a.l 2
5.c odd 4 2 975.2.c.h 4
7.b odd 2 1 1911.2.a.h 2
8.b even 2 1 2496.2.a.bf 2
8.d odd 2 1 2496.2.a.bi 2
9.c even 3 2 1053.2.e.m 4
9.d odd 6 2 1053.2.e.e 4
11.b odd 2 1 4719.2.a.p 2
12.b even 2 1 1872.2.a.w 2
13.b even 2 1 507.2.a.h 2
13.c even 3 2 507.2.e.h 4
13.d odd 4 2 507.2.b.e 4
13.e even 6 2 507.2.e.d 4
13.f odd 12 4 507.2.j.f 8
15.d odd 2 1 2925.2.a.v 2
15.e even 4 2 2925.2.c.u 4
21.c even 2 1 5733.2.a.u 2
24.f even 2 1 7488.2.a.co 2
24.h odd 2 1 7488.2.a.cl 2
39.d odd 2 1 1521.2.a.f 2
39.f even 4 2 1521.2.b.j 4
52.b odd 2 1 8112.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 1.a even 1 1 trivial
117.2.a.c 2 3.b odd 2 1
507.2.a.h 2 13.b even 2 1
507.2.b.e 4 13.d odd 4 2
507.2.e.d 4 13.e even 6 2
507.2.e.h 4 13.c even 3 2
507.2.j.f 8 13.f odd 12 4
624.2.a.k 2 4.b odd 2 1
975.2.a.l 2 5.b even 2 1
975.2.c.h 4 5.c odd 4 2
1053.2.e.e 4 9.d odd 6 2
1053.2.e.m 4 9.c even 3 2
1521.2.a.f 2 39.d odd 2 1
1521.2.b.j 4 39.f even 4 2
1872.2.a.w 2 12.b even 2 1
1911.2.a.h 2 7.b odd 2 1
2496.2.a.bf 2 8.b even 2 1
2496.2.a.bi 2 8.d odd 2 1
2925.2.a.v 2 15.d odd 2 1
2925.2.c.u 4 15.e even 4 2
4719.2.a.p 2 11.b odd 2 1
5733.2.a.u 2 21.c even 2 1
7488.2.a.cl 2 24.h odd 2 1
7488.2.a.co 2 24.f even 2 1
8112.2.a.bm 2 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(39))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 2 T + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -8 + T^{2} \)
$7$ \( -8 + T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -28 - 4 T + T^{2} \)
$19$ \( -8 + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( 8 + 8 T + T^{2} \)
$37$ \( -28 + 4 T + T^{2} \)
$41$ \( 56 - 16 T + T^{2} \)
$43$ \( -16 - 8 T + T^{2} \)
$47$ \( 4 + 12 T + T^{2} \)
$53$ \( ( 2 + T )^{2} \)
$59$ \( -28 - 4 T + T^{2} \)
$61$ \( -124 - 4 T + T^{2} \)
$67$ \( 8 - 8 T + T^{2} \)
$71$ \( ( -2 + T )^{2} \)
$73$ \( 4 - 12 T + T^{2} \)
$79$ \( -128 + T^{2} \)
$83$ \( -28 + 4 T + T^{2} \)
$89$ \( 136 - 24 T + T^{2} \)
$97$ \( -28 + 4 T + T^{2} \)
show more
show less