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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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 \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

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

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

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