Properties

Label 39.2.b.a
Level $39$
Weight $2$
Character orbit 39.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

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

$q$-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.

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

Character values

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

\(n\) \(14\) \(28\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 + 0.866025i
0.500000 0.866025i
1.73205i −1.00000 −1.00000 0 1.73205i 3.46410i 1.73205i 1.00000 0
25.2 1.73205i −1.00000 −1.00000 0 1.73205i 3.46410i 1.73205i 1.00000 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.2.b.a 2
3.b odd 2 1 117.2.b.a 2
4.b odd 2 1 624.2.c.e 2
5.b even 2 1 975.2.b.d 2
5.c odd 4 2 975.2.h.f 4
7.b odd 2 1 1911.2.c.d 2
8.b even 2 1 2496.2.c.k 2
8.d odd 2 1 2496.2.c.d 2
12.b even 2 1 1872.2.c.e 2
13.b even 2 1 inner 39.2.b.a 2
13.c even 3 1 507.2.j.a 2
13.c even 3 1 507.2.j.c 2
13.d odd 4 2 507.2.a.f 2
13.e even 6 1 507.2.j.a 2
13.e even 6 1 507.2.j.c 2
13.f odd 12 4 507.2.e.e 4
39.d odd 2 1 117.2.b.a 2
39.f even 4 2 1521.2.a.l 2
52.b odd 2 1 624.2.c.e 2
52.f even 4 2 8112.2.a.bv 2
65.d even 2 1 975.2.b.d 2
65.h odd 4 2 975.2.h.f 4
91.b odd 2 1 1911.2.c.d 2
104.e even 2 1 2496.2.c.k 2
104.h odd 2 1 2496.2.c.d 2
156.h even 2 1 1872.2.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 1.a even 1 1 trivial
39.2.b.a 2 13.b even 2 1 inner
117.2.b.a 2 3.b odd 2 1
117.2.b.a 2 39.d odd 2 1
507.2.a.f 2 13.d odd 4 2
507.2.e.e 4 13.f odd 12 4
507.2.j.a 2 13.c even 3 1
507.2.j.a 2 13.e even 6 1
507.2.j.c 2 13.c even 3 1
507.2.j.c 2 13.e even 6 1
624.2.c.e 2 4.b odd 2 1
624.2.c.e 2 52.b odd 2 1
975.2.b.d 2 5.b even 2 1
975.2.b.d 2 65.d even 2 1
975.2.h.f 4 5.c odd 4 2
975.2.h.f 4 65.h odd 4 2
1521.2.a.l 2 39.f even 4 2
1872.2.c.e 2 12.b even 2 1
1872.2.c.e 2 156.h even 2 1
1911.2.c.d 2 7.b odd 2 1
1911.2.c.d 2 91.b odd 2 1
2496.2.c.d 2 8.d odd 2 1
2496.2.c.d 2 104.h odd 2 1
2496.2.c.k 2 8.b even 2 1
2496.2.c.k 2 104.e even 2 1
8112.2.a.bv 2 52.f even 4 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( 12 + T^{2} \)
$11$ \( 12 + T^{2} \)
$13$ \( 13 + 2 T + T^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( 12 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( 12 + T^{2} \)
$37$ \( 48 + T^{2} \)
$41$ \( 48 + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( 12 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 108 + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 108 + T^{2} \)
$71$ \( 12 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 12 + T^{2} \)
$89$ \( 48 + T^{2} \)
$97$ \( 192 + T^{2} \)
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