Properties

Label 39.2.e.b
Level $39$
Weight $2$
Character orbit 39.e
Analytic conductor $0.311$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 39.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.311416567883\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + ( - \beta_{3} - 2) q^{5} + (\beta_{3} + \beta_1) q^{6} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + ( - \beta_{3} + 4) q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + ( - \beta_{3} - 2) q^{5} + (\beta_{3} + \beta_1) q^{6} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + ( - \beta_{3} + 4) q^{8} + (\beta_{2} - 1) q^{9} + ( - 4 \beta_{2} + \beta_1) q^{10} + 2 \beta_{2} q^{11} + ( - \beta_{3} + 2) q^{12} + (\beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{13} + (2 \beta_{3} - 4) q^{14} + (2 \beta_{2} - \beta_1) q^{15} - 3 \beta_1 q^{16} + (\beta_{3} + \beta_1) q^{17} - \beta_{3} q^{18} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 4) q^{19} + (\beta_{3} + \beta_1) q^{20} + (\beta_{3} - 1) q^{21} + ( - 2 \beta_{3} - 2 \beta_1) q^{22} - 2 \beta_{2} q^{23} + ( - 4 \beta_{2} - \beta_1) q^{24} + (3 \beta_{3} + 3) q^{25} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 8) q^{26} + q^{27} + (6 \beta_{2} + 4 \beta_1) q^{28} + ( - 2 \beta_{2} + 3 \beta_1) q^{29} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 - 4) q^{30} + (\beta_{3} + 1) q^{31} + (\beta_{3} + 4 \beta_{2} + \beta_1 - 4) q^{32} + ( - 2 \beta_{2} + 2) q^{33} + ( - \beta_{3} + 4) q^{34} + ( - 2 \beta_{2} + 2) q^{35} + ( - 2 \beta_{2} - \beta_1) q^{36} + ( - 6 \beta_{2} + \beta_1) q^{37} + ( - 2 \beta_{3} - 8) q^{38} + ( - 2 \beta_{3} - \beta_1 - 1) q^{39} + ( - 3 \beta_{3} - 4) q^{40} - \beta_1 q^{41} + (4 \beta_{2} + 2 \beta_1) q^{42} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{43} + (2 \beta_{3} - 4) q^{44} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{45} + (2 \beta_{3} + 2 \beta_1) q^{46} + (4 \beta_{3} + 2) q^{47} + (3 \beta_{3} + 3 \beta_1) q^{48} + (2 \beta_{2} - 3 \beta_1) q^{49} + 12 \beta_{2} q^{50} - \beta_{3} q^{51} + (3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{52} + ( - 3 \beta_{3} + 4) q^{53} - \beta_1 q^{54} + ( - 4 \beta_{2} + 2 \beta_1) q^{55} + ( - 6 \beta_{3} - 8 \beta_{2} - 6 \beta_1 + 8) q^{56} + (2 \beta_{3} + 4) q^{57} + ( - \beta_{3} - 12 \beta_{2} - \beta_1 + 12) q^{58} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 6) q^{59} - \beta_{3} q^{60} + (2 \beta_{3} + 7 \beta_{2} + 2 \beta_1 - 7) q^{61} + 4 \beta_{2} q^{62} + (\beta_{2} + \beta_1) q^{63} + (\beta_{3} + 4) q^{64} + ( - 2 \beta_{3} + 10 \beta_{2} - 3 \beta_1 - 6) q^{65} + 2 \beta_{3} q^{66} + ( - 3 \beta_{2} + \beta_1) q^{67} + ( - 4 \beta_{2} - 3 \beta_1) q^{68} + (2 \beta_{2} - 2) q^{69} + 2 \beta_{3} q^{70} + (14 \beta_{2} - 14) q^{71} + (\beta_{3} + 4 \beta_{2} + \beta_1 - 4) q^{72} + ( - 2 \beta_{3} - 7) q^{73} + (5 \beta_{3} - 4 \beta_{2} + 5 \beta_1 + 4) q^{74} + ( - 3 \beta_{2} + 3 \beta_1) q^{75} + 2 \beta_1 q^{76} + ( - 2 \beta_{3} + 2) q^{77} + (\beta_{3} - 4 \beta_{2} - 4) q^{78} + ( - \beta_{3} + 7) q^{79} + ( - 12 \beta_{2} + 3 \beta_1) q^{80} - \beta_{2} q^{81} + (\beta_{3} + 4 \beta_{2} + \beta_1 - 4) q^{82} + (2 \beta_{3} - 4) q^{83} + ( - 4 \beta_{3} - 6 \beta_{2} - 4 \beta_1 + 6) q^{84} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 - 4) q^{85} + ( - 2 \beta_{3} - 4) q^{86} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 2) q^{87} + (8 \beta_{2} + 2 \beta_1) q^{88} + ( - 8 \beta_{2} - 2 \beta_1) q^{89} + (\beta_{3} + 4) q^{90} + ( - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 4) q^{91} + ( - 2 \beta_{3} + 4) q^{92} + ( - \beta_{2} + \beta_1) q^{93} + (16 \beta_{2} + 2 \beta_1) q^{94} + (6 \beta_{3} - 16 \beta_{2} + 6 \beta_1 + 16) q^{95} + ( - \beta_{3} + 4) q^{96} + (\beta_{3} - 7 \beta_{2} + \beta_1 + 7) q^{97} + (\beta_{3} + 12 \beta_{2} + \beta_1 - 12) q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 2 q^{3} - 5 q^{4} - 6 q^{5} - q^{6} + 3 q^{7} + 18 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 2 q^{3} - 5 q^{4} - 6 q^{5} - q^{6} + 3 q^{7} + 18 q^{8} - 2 q^{9} - 7 q^{10} + 4 q^{11} + 10 q^{12} + 2 q^{13} - 20 q^{14} + 3 q^{15} - 3 q^{16} - q^{17} + 2 q^{18} - 6 q^{19} - q^{20} - 6 q^{21} + 2 q^{22} - 4 q^{23} - 9 q^{24} + 6 q^{25} + 25 q^{26} + 4 q^{27} + 16 q^{28} - q^{29} - 7 q^{30} + 2 q^{31} - 9 q^{32} + 4 q^{33} + 18 q^{34} + 4 q^{35} - 5 q^{36} - 11 q^{37} - 28 q^{38} - q^{39} - 10 q^{40} - q^{41} + 10 q^{42} - 5 q^{43} - 20 q^{44} + 3 q^{45} - 2 q^{46} - 3 q^{48} + q^{49} + 24 q^{50} + 2 q^{51} - 28 q^{52} + 22 q^{53} - q^{54} - 6 q^{55} + 22 q^{56} + 12 q^{57} + 25 q^{58} + 14 q^{59} + 2 q^{60} - 16 q^{61} + 8 q^{62} + 3 q^{63} + 14 q^{64} - 3 q^{65} - 4 q^{66} - 5 q^{67} - 11 q^{68} - 4 q^{69} - 4 q^{70} - 28 q^{71} - 9 q^{72} - 24 q^{73} + 3 q^{74} - 3 q^{75} + 2 q^{76} + 12 q^{77} - 26 q^{78} + 30 q^{79} - 21 q^{80} - 2 q^{81} - 9 q^{82} - 20 q^{83} + 16 q^{84} - 7 q^{85} - 12 q^{86} - q^{87} + 18 q^{88} - 18 q^{89} + 14 q^{90} + 27 q^{91} + 20 q^{92} - q^{93} + 34 q^{94} + 26 q^{95} + 18 q^{96} + 13 q^{97} - 25 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\) \(14\) \(28\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
1.28078 + 2.21837i
−0.780776 1.35234i
1.28078 2.21837i
−0.780776 + 1.35234i
−1.28078 2.21837i −0.500000 0.866025i −2.28078 + 3.95042i 0.561553 −1.28078 + 2.21837i 1.78078 3.08440i 6.56155 −0.500000 + 0.866025i −0.719224 1.24573i
16.2 0.780776 + 1.35234i −0.500000 0.866025i −0.219224 + 0.379706i −3.56155 0.780776 1.35234i −0.280776 + 0.486319i 2.43845 −0.500000 + 0.866025i −2.78078 4.81645i
22.1 −1.28078 + 2.21837i −0.500000 + 0.866025i −2.28078 3.95042i 0.561553 −1.28078 2.21837i 1.78078 + 3.08440i 6.56155 −0.500000 0.866025i −0.719224 + 1.24573i
22.2 0.780776 1.35234i −0.500000 + 0.866025i −0.219224 0.379706i −3.56155 0.780776 + 1.35234i −0.280776 0.486319i 2.43845 −0.500000 0.866025i −2.78078 + 4.81645i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.2.e.b 4
3.b odd 2 1 117.2.g.c 4
4.b odd 2 1 624.2.q.h 4
5.b even 2 1 975.2.i.k 4
5.c odd 4 2 975.2.bb.i 8
12.b even 2 1 1872.2.t.r 4
13.b even 2 1 507.2.e.g 4
13.c even 3 1 inner 39.2.e.b 4
13.c even 3 1 507.2.a.g 2
13.d odd 4 2 507.2.j.g 8
13.e even 6 1 507.2.a.d 2
13.e even 6 1 507.2.e.g 4
13.f odd 12 2 507.2.b.d 4
13.f odd 12 2 507.2.j.g 8
39.h odd 6 1 1521.2.a.m 2
39.i odd 6 1 117.2.g.c 4
39.i odd 6 1 1521.2.a.g 2
39.k even 12 2 1521.2.b.h 4
52.i odd 6 1 8112.2.a.bo 2
52.j odd 6 1 624.2.q.h 4
52.j odd 6 1 8112.2.a.bk 2
65.n even 6 1 975.2.i.k 4
65.q odd 12 2 975.2.bb.i 8
156.p even 6 1 1872.2.t.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 1.a even 1 1 trivial
39.2.e.b 4 13.c even 3 1 inner
117.2.g.c 4 3.b odd 2 1
117.2.g.c 4 39.i odd 6 1
507.2.a.d 2 13.e even 6 1
507.2.a.g 2 13.c even 3 1
507.2.b.d 4 13.f odd 12 2
507.2.e.g 4 13.b even 2 1
507.2.e.g 4 13.e even 6 1
507.2.j.g 8 13.d odd 4 2
507.2.j.g 8 13.f odd 12 2
624.2.q.h 4 4.b odd 2 1
624.2.q.h 4 52.j odd 6 1
975.2.i.k 4 5.b even 2 1
975.2.i.k 4 65.n even 6 1
975.2.bb.i 8 5.c odd 4 2
975.2.bb.i 8 65.q odd 12 2
1521.2.a.g 2 39.i odd 6 1
1521.2.a.m 2 39.h odd 6 1
1521.2.b.h 4 39.k even 12 2
1872.2.t.r 4 12.b even 2 1
1872.2.t.r 4 156.p even 6 1
8112.2.a.bk 2 52.j odd 6 1
8112.2.a.bo 2 52.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + T_{2}^{3} + 5T_{2}^{2} - 4T_{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + 5 T^{2} - 4 T + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 3 T^{3} + 11 T^{2} + 6 T + 4 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + T^{3} + 5 T^{2} - 4 T + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + 44 T^{2} - 48 T + 64 \) Copy content Toggle raw display
$23$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + T^{3} + 39 T^{2} - 38 T + 1444 \) Copy content Toggle raw display
$31$ \( (T^{2} - T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 11 T^{3} + 95 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$41$ \( T^{4} + T^{3} + 5 T^{2} - 4 T + 16 \) Copy content Toggle raw display
$43$ \( T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4 \) Copy content Toggle raw display
$47$ \( (T^{2} - 68)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 11 T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 14 T^{3} + 164 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$61$ \( T^{4} + 16 T^{3} + 209 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$67$ \( T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4 \) Copy content Toggle raw display
$71$ \( (T^{2} + 14 T + 196)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 19)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 15 T + 52)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 10 T + 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 18 T^{3} + 260 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$97$ \( T^{4} - 13 T^{3} + 131 T^{2} + \cdots + 1444 \) Copy content Toggle raw display
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