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: \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 5 \nu^{2} - 5 \nu + 16 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4 \beta_{2} + \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(5 \beta_{3} - 4\)

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 \( T_{2}^{4} + T_{2}^{3} + 5 T_{2}^{2} - 4 T_{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(39, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 4 T + 5 T^{2} + T^{3} + T^{4} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( -2 + 3 T + T^{2} )^{2} \)
$7$ \( 4 + 6 T + 11 T^{2} - 3 T^{3} + T^{4} \)
$11$ \( ( 4 - 2 T + T^{2} )^{2} \)
$13$ \( ( 13 - T + T^{2} )^{2} \)
$17$ \( 16 - 4 T + 5 T^{2} + T^{3} + T^{4} \)
$19$ \( 64 - 48 T + 44 T^{2} + 6 T^{3} + T^{4} \)
$23$ \( ( 4 + 2 T + T^{2} )^{2} \)
$29$ \( 1444 - 38 T + 39 T^{2} + T^{3} + T^{4} \)
$31$ \( ( -4 - T + T^{2} )^{2} \)
$37$ \( 676 + 286 T + 95 T^{2} + 11 T^{3} + T^{4} \)
$41$ \( 16 - 4 T + 5 T^{2} + T^{3} + T^{4} \)
$43$ \( 4 + 10 T + 23 T^{2} + 5 T^{3} + T^{4} \)
$47$ \( ( -68 + T^{2} )^{2} \)
$53$ \( ( -8 - 11 T + T^{2} )^{2} \)
$59$ \( 1024 - 448 T + 164 T^{2} - 14 T^{3} + T^{4} \)
$61$ \( 2209 + 752 T + 209 T^{2} + 16 T^{3} + T^{4} \)
$67$ \( 4 + 10 T + 23 T^{2} + 5 T^{3} + T^{4} \)
$71$ \( ( 196 + 14 T + T^{2} )^{2} \)
$73$ \( ( 19 + 12 T + T^{2} )^{2} \)
$79$ \( ( 52 - 15 T + T^{2} )^{2} \)
$83$ \( ( 8 + 10 T + T^{2} )^{2} \)
$89$ \( 4096 + 1152 T + 260 T^{2} + 18 T^{3} + T^{4} \)
$97$ \( 1444 - 494 T + 131 T^{2} - 13 T^{3} + T^{4} \)
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