Properties

Label 39.4.f.a
Level $39$
Weight $4$
Character orbit 39.f
Analytic conductor $2.301$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 39.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.30107449022\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + 8 \beta_1 q^{4} + ( - 3 \beta_{3} + 3 \beta_{2} + 10 \beta_1 - 10) q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + 8 \beta_1 q^{4} + ( - 3 \beta_{3} + 3 \beta_{2} + 10 \beta_1 - 10) q^{7} + 27 q^{9} - 8 \beta_{2} q^{12} + (6 \beta_{3} - 35 \beta_1) q^{13} - 64 q^{16} + (15 \beta_{3} + 15 \beta_{2} - 28 \beta_1 - 28) q^{19} + (10 \beta_{3} - 10 \beta_{2} - 81 \beta_1 + 81) q^{21} + 125 \beta_1 q^{25} - 27 \beta_{3} q^{27} + ( - 24 \beta_{3} - 24 \beta_{2} - 80 \beta_1 - 80) q^{28} + (15 \beta_{3} + 15 \beta_{2} + 154 \beta_1 + 154) q^{31} + 216 \beta_1 q^{36} + ( - 42 \beta_{3} + 42 \beta_{2} - 55 \beta_1 + 55) q^{37} + (35 \beta_{2} - 162) q^{39} - 42 \beta_{2} q^{43} + 64 \beta_{3} q^{48} + ( - 120 \beta_{2} - 343 \beta_1) q^{49} + (48 \beta_{2} + 280) q^{52} + (28 \beta_{3} + 28 \beta_{2} - 405 \beta_1 - 405) q^{57} + 180 \beta_{3} q^{61} + ( - 81 \beta_{3} + 81 \beta_{2} + 270 \beta_1 - 270) q^{63} - 512 \beta_1 q^{64} + ( - 63 \beta_{3} - 63 \beta_{2} + 440 \beta_1 + 440) q^{67} + (36 \beta_{3} - 36 \beta_{2} + 595 \beta_1 - 595) q^{73} - 125 \beta_{2} q^{75} + ( - 120 \beta_{3} + 120 \beta_{2} - 224 \beta_1 + 224) q^{76} - 210 \beta_{3} q^{79} + 729 q^{81} + (80 \beta_{3} + 80 \beta_{2} + 648 \beta_1 + 648) q^{84} + (45 \beta_{3} + 165 \beta_{2} + 836 \beta_1 - 136) q^{91} + ( - 154 \beta_{3} - 154 \beta_{2} - 405 \beta_1 - 405) q^{93} + (132 \beta_{3} + 132 \beta_{2} - 665 \beta_1 - 665) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 40 q^{7} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 40 q^{7} + 108 q^{9} - 256 q^{16} - 112 q^{19} + 324 q^{21} - 320 q^{28} + 616 q^{31} + 220 q^{37} - 648 q^{39} + 1120 q^{52} - 1620 q^{57} - 1080 q^{63} + 1760 q^{67} - 2380 q^{73} + 896 q^{76} + 2916 q^{81} + 2592 q^{84} - 544 q^{91} - 1620 q^{93} - 2660 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 6\zeta_{12}^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\zeta_{12}^{3} + 6\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 3 ) / 6 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(14\) \(28\)
\(\chi(n)\) \(-1\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 −5.19615 8.00000i 0 0 −25.5885 25.5885i 0 27.0000 0
5.2 0 5.19615 8.00000i 0 0 5.58846 + 5.58846i 0 27.0000 0
8.1 0 −5.19615 8.00000i 0 0 −25.5885 + 25.5885i 0 27.0000 0
8.2 0 5.19615 8.00000i 0 0 5.58846 5.58846i 0 27.0000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.4.f.a 4
3.b odd 2 1 CM 39.4.f.a 4
13.d odd 4 1 inner 39.4.f.a 4
39.f even 4 1 inner 39.4.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.f.a 4 1.a even 1 1 trivial
39.4.f.a 4 3.b odd 2 1 CM
39.4.f.a 4 13.d odd 4 1 inner
39.4.f.a 4 39.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{4}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 40 T^{3} + 800 T^{2} + \cdots + 81796 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 506 T^{2} + 4826809 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 112 T^{3} + \cdots + 111978724 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 616 T^{3} + \cdots + 1244819524 \) Copy content Toggle raw display
$37$ \( T^{4} - 220 T^{3} + \cdots + 7957710436 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 47628)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 874800)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 1760 T^{3} + \cdots + 29885419876 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 2380 T^{3} + \cdots + 407128220356 \) Copy content Toggle raw display
$79$ \( (T^{2} - 1190700)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 2660 T^{3} + \cdots + 3186150916 \) Copy content Toggle raw display
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