Properties

Label 39.2.e.a
Level $39$
Weight $2$
Character orbit 39.e
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.e (of order \(3\), 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: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{4} - q^{5} + \zeta_{6} q^{6} - 2 \zeta_{6} q^{7} - 3 q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{4} - q^{5} + \zeta_{6} q^{6} - 2 \zeta_{6} q^{7} - 3 q^{8} - \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{10} + ( - 2 \zeta_{6} + 2) q^{11} + q^{12} + ( - \zeta_{6} - 3) q^{13} + 2 q^{14} + (\zeta_{6} - 1) q^{15} + ( - \zeta_{6} + 1) q^{16} + 7 \zeta_{6} q^{17} + q^{18} + 6 \zeta_{6} q^{19} - \zeta_{6} q^{20} - 2 q^{21} + 2 \zeta_{6} q^{22} + ( - 6 \zeta_{6} + 6) q^{23} + (3 \zeta_{6} - 3) q^{24} - 4 q^{25} + ( - 3 \zeta_{6} + 4) q^{26} - q^{27} + ( - 2 \zeta_{6} + 2) q^{28} + ( - \zeta_{6} + 1) q^{29} - \zeta_{6} q^{30} + 4 q^{31} - 5 \zeta_{6} q^{32} - 2 \zeta_{6} q^{33} - 7 q^{34} + 2 \zeta_{6} q^{35} + ( - \zeta_{6} + 1) q^{36} + (\zeta_{6} - 1) q^{37} - 6 q^{38} + (3 \zeta_{6} - 4) q^{39} + 3 q^{40} + (9 \zeta_{6} - 9) q^{41} + ( - 2 \zeta_{6} + 2) q^{42} - 6 \zeta_{6} q^{43} + 2 q^{44} + \zeta_{6} q^{45} + 6 \zeta_{6} q^{46} + 6 q^{47} - \zeta_{6} q^{48} + ( - 3 \zeta_{6} + 3) q^{49} + ( - 4 \zeta_{6} + 4) q^{50} + 7 q^{51} + ( - 4 \zeta_{6} + 1) q^{52} - 9 q^{53} + ( - \zeta_{6} + 1) q^{54} + (2 \zeta_{6} - 2) q^{55} + 6 \zeta_{6} q^{56} + 6 q^{57} + \zeta_{6} q^{58} - q^{60} - \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} + (2 \zeta_{6} - 2) q^{63} + 7 q^{64} + (\zeta_{6} + 3) q^{65} + 2 q^{66} + ( - 2 \zeta_{6} + 2) q^{67} + (7 \zeta_{6} - 7) q^{68} - 6 \zeta_{6} q^{69} - 2 q^{70} - 6 \zeta_{6} q^{71} + 3 \zeta_{6} q^{72} + 11 q^{73} - \zeta_{6} q^{74} + (4 \zeta_{6} - 4) q^{75} + (6 \zeta_{6} - 6) q^{76} - 4 q^{77} + ( - 4 \zeta_{6} + 1) q^{78} - 4 q^{79} + (\zeta_{6} - 1) q^{80} + (\zeta_{6} - 1) q^{81} - 9 \zeta_{6} q^{82} - 14 q^{83} - 2 \zeta_{6} q^{84} - 7 \zeta_{6} q^{85} + 6 q^{86} - \zeta_{6} q^{87} + (6 \zeta_{6} - 6) q^{88} + ( - 14 \zeta_{6} + 14) q^{89} - q^{90} + (8 \zeta_{6} - 2) q^{91} + 6 q^{92} + ( - 4 \zeta_{6} + 4) q^{93} + (6 \zeta_{6} - 6) q^{94} - 6 \zeta_{6} q^{95} - 5 q^{96} + 2 \zeta_{6} q^{97} + 3 \zeta_{6} q^{98} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} + q^{4} - 2 q^{5} + q^{6} - 2 q^{7} - 6 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{3} + q^{4} - 2 q^{5} + q^{6} - 2 q^{7} - 6 q^{8} - q^{9} + q^{10} + 2 q^{11} + 2 q^{12} - 7 q^{13} + 4 q^{14} - q^{15} + q^{16} + 7 q^{17} + 2 q^{18} + 6 q^{19} - q^{20} - 4 q^{21} + 2 q^{22} + 6 q^{23} - 3 q^{24} - 8 q^{25} + 5 q^{26} - 2 q^{27} + 2 q^{28} + q^{29} - q^{30} + 8 q^{31} - 5 q^{32} - 2 q^{33} - 14 q^{34} + 2 q^{35} + q^{36} - q^{37} - 12 q^{38} - 5 q^{39} + 6 q^{40} - 9 q^{41} + 2 q^{42} - 6 q^{43} + 4 q^{44} + q^{45} + 6 q^{46} + 12 q^{47} - q^{48} + 3 q^{49} + 4 q^{50} + 14 q^{51} - 2 q^{52} - 18 q^{53} + q^{54} - 2 q^{55} + 6 q^{56} + 12 q^{57} + q^{58} - 2 q^{60} - q^{61} - 4 q^{62} - 2 q^{63} + 14 q^{64} + 7 q^{65} + 4 q^{66} + 2 q^{67} - 7 q^{68} - 6 q^{69} - 4 q^{70} - 6 q^{71} + 3 q^{72} + 22 q^{73} - q^{74} - 4 q^{75} - 6 q^{76} - 8 q^{77} - 2 q^{78} - 8 q^{79} - q^{80} - q^{81} - 9 q^{82} - 28 q^{83} - 2 q^{84} - 7 q^{85} + 12 q^{86} - q^{87} - 6 q^{88} + 14 q^{89} - 2 q^{90} + 4 q^{91} + 12 q^{92} + 4 q^{93} - 6 q^{94} - 6 q^{95} - 10 q^{96} + 2 q^{97} + 3 q^{98} - 4 q^{99}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i −1.00000 0.500000 0.866025i −1.00000 + 1.73205i −3.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
22.1 −0.500000 + 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i −1.00000 0.500000 + 0.866025i −1.00000 1.73205i −3.00000 −0.500000 0.866025i 0.500000 0.866025i
\(n\): 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.a 2
3.b odd 2 1 117.2.g.b 2
4.b odd 2 1 624.2.q.c 2
5.b even 2 1 975.2.i.f 2
5.c odd 4 2 975.2.bb.d 4
12.b even 2 1 1872.2.t.j 2
13.b even 2 1 507.2.e.c 2
13.c even 3 1 inner 39.2.e.a 2
13.c even 3 1 507.2.a.c 1
13.d odd 4 2 507.2.j.d 4
13.e even 6 1 507.2.a.b 1
13.e even 6 1 507.2.e.c 2
13.f odd 12 2 507.2.b.b 2
13.f odd 12 2 507.2.j.d 4
39.h odd 6 1 1521.2.a.d 1
39.i odd 6 1 117.2.g.b 2
39.i odd 6 1 1521.2.a.a 1
39.k even 12 2 1521.2.b.c 2
52.i odd 6 1 8112.2.a.bc 1
52.j odd 6 1 624.2.q.c 2
52.j odd 6 1 8112.2.a.w 1
65.n even 6 1 975.2.i.f 2
65.q odd 12 2 975.2.bb.d 4
156.p even 6 1 1872.2.t.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 1.a even 1 1 trivial
39.2.e.a 2 13.c even 3 1 inner
117.2.g.b 2 3.b odd 2 1
117.2.g.b 2 39.i odd 6 1
507.2.a.b 1 13.e even 6 1
507.2.a.c 1 13.c even 3 1
507.2.b.b 2 13.f odd 12 2
507.2.e.c 2 13.b even 2 1
507.2.e.c 2 13.e even 6 1
507.2.j.d 4 13.d odd 4 2
507.2.j.d 4 13.f odd 12 2
624.2.q.c 2 4.b odd 2 1
624.2.q.c 2 52.j odd 6 1
975.2.i.f 2 5.b even 2 1
975.2.i.f 2 65.n even 6 1
975.2.bb.d 4 5.c odd 4 2
975.2.bb.d 4 65.q odd 12 2
1521.2.a.a 1 39.i odd 6 1
1521.2.a.d 1 39.h odd 6 1
1521.2.b.c 2 39.k even 12 2
1872.2.t.j 2 12.b even 2 1
1872.2.t.j 2 156.p even 6 1
8112.2.a.w 1 52.j odd 6 1
8112.2.a.bc 1 52.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( (T - 11)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T + 14)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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