Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 39.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.311416567883\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{3} - 2 \zeta_{6} q^{4} + (4 \zeta_{6} - 2) q^{5} + (\zeta_{6} - 2) q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{3} - 2 \zeta_{6} q^{4} + (4 \zeta_{6} - 2) q^{5} + (\zeta_{6} - 2) q^{7} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} - 2) q^{11} - 2 q^{12} + ( - \zeta_{6} + 4) q^{13} + (2 \zeta_{6} + 2) q^{15} + (4 \zeta_{6} - 4) q^{16} + ( - 2 \zeta_{6} + 4) q^{19} + ( - 4 \zeta_{6} + 8) q^{20} + (2 \zeta_{6} - 1) q^{21} + ( - 6 \zeta_{6} + 6) q^{23} - 7 q^{25} - q^{27} + (2 \zeta_{6} + 2) q^{28} + (6 \zeta_{6} - 6) q^{29} + (2 \zeta_{6} - 1) q^{31} + (2 \zeta_{6} - 4) q^{33} - 6 \zeta_{6} q^{35} + (2 \zeta_{6} - 2) q^{36} + ( - 4 \zeta_{6} + 3) q^{39} + ( - 4 \zeta_{6} - 4) q^{41} + \zeta_{6} q^{43} + (8 \zeta_{6} - 4) q^{44} + ( - 2 \zeta_{6} + 4) q^{45} + (4 \zeta_{6} - 2) q^{47} + 4 \zeta_{6} q^{48} + (4 \zeta_{6} - 4) q^{49} + ( - 6 \zeta_{6} - 2) q^{52} + 12 q^{53} + ( - 12 \zeta_{6} + 12) q^{55} + ( - 4 \zeta_{6} + 2) q^{57} + (2 \zeta_{6} - 4) q^{59} + ( - 8 \zeta_{6} + 4) q^{60} - \zeta_{6} q^{61} + (\zeta_{6} + 1) q^{63} + 8 q^{64} + (14 \zeta_{6} - 4) q^{65} + (5 \zeta_{6} + 5) q^{67} - 6 \zeta_{6} q^{69} + (6 \zeta_{6} - 12) q^{71} + (2 \zeta_{6} - 1) q^{73} + (7 \zeta_{6} - 7) q^{75} + ( - 4 \zeta_{6} - 4) q^{76} + 6 q^{77} - 11 q^{79} + ( - 8 \zeta_{6} - 8) q^{80} + (\zeta_{6} - 1) q^{81} + ( - 16 \zeta_{6} + 8) q^{83} + ( - 2 \zeta_{6} + 4) q^{84} + 6 \zeta_{6} q^{87} + (4 \zeta_{6} + 4) q^{89} + (5 \zeta_{6} - 7) q^{91} - 12 q^{92} + (\zeta_{6} + 1) q^{93} + 12 \zeta_{6} q^{95} + (3 \zeta_{6} - 6) q^{97} + (4 \zeta_{6} - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 2 q^{4} - 3 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 2 q^{4} - 3 q^{7} - q^{9} - 6 q^{11} - 4 q^{12} + 7 q^{13} + 6 q^{15} - 4 q^{16} + 6 q^{19} + 12 q^{20} + 6 q^{23} - 14 q^{25} - 2 q^{27} + 6 q^{28} - 6 q^{29} - 6 q^{33} - 6 q^{35} - 2 q^{36} + 2 q^{39} - 12 q^{41} + q^{43} + 6 q^{45} + 4 q^{48} - 4 q^{49} - 10 q^{52} + 24 q^{53} + 12 q^{55} - 6 q^{59} - q^{61} + 3 q^{63} + 16 q^{64} + 6 q^{65} + 15 q^{67} - 6 q^{69} - 18 q^{71} - 7 q^{75} - 12 q^{76} + 12 q^{77} - 22 q^{79} - 24 q^{80} - q^{81} + 6 q^{84} + 6 q^{87} + 12 q^{89} - 9 q^{91} - 24 q^{92} + 3 q^{93} + 12 q^{95} - 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(14\) \(28\)
\(\chi(n)\) \(1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i −1.00000 1.73205i 3.46410i 0 −1.50000 + 0.866025i 0 −0.500000 0.866025i 0
10.1 0 0.500000 + 0.866025i −1.00000 + 1.73205i 3.46410i 0 −1.50000 0.866025i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.2.j.a 2
3.b odd 2 1 117.2.q.a 2
4.b odd 2 1 624.2.bv.b 2
5.b even 2 1 975.2.bc.c 2
5.c odd 4 2 975.2.w.d 4
12.b even 2 1 1872.2.by.f 2
13.b even 2 1 507.2.j.b 2
13.c even 3 1 507.2.b.c 2
13.c even 3 1 507.2.j.b 2
13.d odd 4 2 507.2.e.f 4
13.e even 6 1 inner 39.2.j.a 2
13.e even 6 1 507.2.b.c 2
13.f odd 12 2 507.2.a.e 2
13.f odd 12 2 507.2.e.f 4
39.h odd 6 1 117.2.q.a 2
39.h odd 6 1 1521.2.b.f 2
39.i odd 6 1 1521.2.b.f 2
39.k even 12 2 1521.2.a.h 2
52.i odd 6 1 624.2.bv.b 2
52.l even 12 2 8112.2.a.bu 2
65.l even 6 1 975.2.bc.c 2
65.r odd 12 2 975.2.w.d 4
156.r even 6 1 1872.2.by.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a 2 1.a even 1 1 trivial
39.2.j.a 2 13.e even 6 1 inner
117.2.q.a 2 3.b odd 2 1
117.2.q.a 2 39.h odd 6 1
507.2.a.e 2 13.f odd 12 2
507.2.b.c 2 13.c even 3 1
507.2.b.c 2 13.e even 6 1
507.2.e.f 4 13.d odd 4 2
507.2.e.f 4 13.f odd 12 2
507.2.j.b 2 13.b even 2 1
507.2.j.b 2 13.c even 3 1
624.2.bv.b 2 4.b odd 2 1
624.2.bv.b 2 52.i odd 6 1
975.2.w.d 4 5.c odd 4 2
975.2.w.d 4 65.r odd 12 2
975.2.bc.c 2 5.b even 2 1
975.2.bc.c 2 65.l even 6 1
1521.2.a.h 2 39.k even 12 2
1521.2.b.f 2 39.h odd 6 1
1521.2.b.f 2 39.i odd 6 1
1872.2.by.f 2 12.b even 2 1
1872.2.by.f 2 156.r even 6 1
8112.2.a.bu 2 52.l even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(39, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 12 \) Copy content Toggle raw display
$7$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$13$ \( T^{2} - 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 3 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 12 \) Copy content Toggle raw display
$53$ \( (T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$61$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$71$ \( T^{2} + 18T + 108 \) Copy content Toggle raw display
$73$ \( T^{2} + 3 \) Copy content Toggle raw display
$79$ \( (T + 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 192 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
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