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 $1$

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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: \( x^{2} - 2 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + q^{3} + ( - 2 \beta + 1) q^{4} - 2 \beta q^{5} + (\beta - 1) q^{6} + 2 \beta q^{7} + (\beta - 3) q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + q^{3} + ( - 2 \beta + 1) q^{4} - 2 \beta q^{5} + (\beta - 1) q^{6} + 2 \beta q^{7} + (\beta - 3) q^{8} + q^{9} + (2 \beta - 4) q^{10} - 2 q^{11} + ( - 2 \beta + 1) q^{12} - q^{13} + ( - 2 \beta + 4) q^{14} - 2 \beta q^{15} + 3 q^{16} + (4 \beta + 2) q^{17} + (\beta - 1) q^{18} - 2 \beta q^{19} + ( - 2 \beta + 8) q^{20} + 2 \beta q^{21} + ( - 2 \beta + 2) q^{22} - 4 q^{23} + (\beta - 3) q^{24} + 3 q^{25} + ( - \beta + 1) q^{26} + q^{27} + (2 \beta - 8) q^{28} + 2 q^{29} + (2 \beta - 4) q^{30} + (2 \beta - 4) q^{31} + (\beta + 3) q^{32} - 2 q^{33} + ( - 2 \beta + 6) q^{34} - 8 q^{35} + ( - 2 \beta + 1) q^{36} + ( - 4 \beta - 2) q^{37} + (2 \beta - 4) q^{38} - q^{39} + (6 \beta - 4) q^{40} + ( - 2 \beta + 8) q^{41} + ( - 2 \beta + 4) q^{42} + ( - 4 \beta + 4) q^{43} + (4 \beta - 2) q^{44} - 2 \beta q^{45} + ( - 4 \beta + 4) q^{46} + ( - 4 \beta - 6) q^{47} + 3 q^{48} + q^{49} + (3 \beta - 3) q^{50} + (4 \beta + 2) q^{51} + (2 \beta - 1) q^{52} - 2 q^{53} + (\beta - 1) q^{54} + 4 \beta q^{55} + ( - 6 \beta + 4) q^{56} - 2 \beta q^{57} + (2 \beta - 2) q^{58} + (4 \beta + 2) q^{59} + ( - 2 \beta + 8) q^{60} + (8 \beta + 2) q^{61} + ( - 6 \beta + 8) q^{62} + 2 \beta q^{63} + (2 \beta - 7) q^{64} + 2 \beta q^{65} + ( - 2 \beta + 2) q^{66} + (2 \beta + 4) q^{67} - 14 q^{68} - 4 q^{69} + ( - 8 \beta + 8) q^{70} + 2 q^{71} + (\beta - 3) q^{72} + ( - 4 \beta + 6) q^{73} + (2 \beta - 6) q^{74} + 3 q^{75} + ( - 2 \beta + 8) q^{76} - 4 \beta q^{77} + ( - \beta + 1) q^{78} - 8 \beta q^{79} - 6 \beta q^{80} + q^{81} + (10 \beta - 12) q^{82} + (4 \beta - 2) q^{83} + (2 \beta - 8) q^{84} + ( - 4 \beta - 16) q^{85} + (8 \beta - 12) q^{86} + 2 q^{87} + ( - 2 \beta + 6) q^{88} + (2 \beta + 12) q^{89} + (2 \beta - 4) q^{90} - 2 \beta q^{91} + (8 \beta - 4) q^{92} + (2 \beta - 4) q^{93} + ( - 2 \beta - 2) q^{94} + 8 q^{95} + (\beta + 3) q^{96} + (4 \beta - 2) q^{97} + (\beta - 1) q^{98} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{8} + 2 q^{9} - 8 q^{10} - 4 q^{11} + 2 q^{12} - 2 q^{13} + 8 q^{14} + 6 q^{16} + 4 q^{17} - 2 q^{18} + 16 q^{20} + 4 q^{22} - 8 q^{23} - 6 q^{24} + 6 q^{25} + 2 q^{26} + 2 q^{27} - 16 q^{28} + 4 q^{29} - 8 q^{30} - 8 q^{31} + 6 q^{32} - 4 q^{33} + 12 q^{34} - 16 q^{35} + 2 q^{36} - 4 q^{37} - 8 q^{38} - 2 q^{39} - 8 q^{40} + 16 q^{41} + 8 q^{42} + 8 q^{43} - 4 q^{44} + 8 q^{46} - 12 q^{47} + 6 q^{48} + 2 q^{49} - 6 q^{50} + 4 q^{51} - 2 q^{52} - 4 q^{53} - 2 q^{54} + 8 q^{56} - 4 q^{58} + 4 q^{59} + 16 q^{60} + 4 q^{61} + 16 q^{62} - 14 q^{64} + 4 q^{66} + 8 q^{67} - 28 q^{68} - 8 q^{69} + 16 q^{70} + 4 q^{71} - 6 q^{72} + 12 q^{73} - 12 q^{74} + 6 q^{75} + 16 q^{76} + 2 q^{78} + 2 q^{81} - 24 q^{82} - 4 q^{83} - 16 q^{84} - 32 q^{85} - 24 q^{86} + 4 q^{87} + 12 q^{88} + 24 q^{89} - 8 q^{90} - 8 q^{92} - 8 q^{93} - 4 q^{94} + 16 q^{95} + 6 q^{96} - 4 q^{97} - 2 q^{98} - 4 q^{99}+O(q^{100}) \) 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
−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 \( T_{2}^{2} + 2T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(39))\). Copy content Toggle raw display

Hecke characteristic polynomials

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