Properties

Label 3864.2.a.t
Level $3864$
Weight $2$
Character orbit 3864.a
Self dual yes
Analytic conductor $30.854$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.8541953410\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.39605.1
Defining polynomial: \(x^{4} - 2 x^{3} - 8 x^{2} + 9 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 + \beta_{1} ) q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 + \beta_{1} ) q^{5} - q^{7} + q^{9} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{11} + ( \beta_{1} - \beta_{3} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( -3 - 2 \beta_{2} - \beta_{3} ) q^{17} + ( 1 - \beta_{3} ) q^{19} - q^{21} - q^{23} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{25} + q^{27} + ( 3 + \beta_{2} + 2 \beta_{3} ) q^{29} + ( 3 - 2 \beta_{2} - \beta_{3} ) q^{31} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{33} + ( 1 - \beta_{1} ) q^{35} + ( 3 - \beta_{2} ) q^{37} + ( \beta_{1} - \beta_{3} ) q^{39} + ( 1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{41} + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{43} + ( -1 + \beta_{1} ) q^{45} + ( 4 + \beta_{2} - \beta_{3} ) q^{47} + q^{49} + ( -3 - 2 \beta_{2} - \beta_{3} ) q^{51} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{53} + ( 9 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{55} + ( 1 - \beta_{3} ) q^{57} + ( 1 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{59} + ( 2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{61} - q^{63} + ( 7 + 3 \beta_{2} + 2 \beta_{3} ) q^{65} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{67} - q^{69} + ( \beta_{1} - 3 \beta_{2} ) q^{71} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{73} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{75} + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{77} + ( 7 + 2 \beta_{2} + 5 \beta_{3} ) q^{79} + q^{81} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{83} + ( 5 - 7 \beta_{1} + \beta_{2} + \beta_{3} ) q^{85} + ( 3 + \beta_{2} + 2 \beta_{3} ) q^{87} + ( 3 \beta_{1} - \beta_{2} ) q^{89} + ( -\beta_{1} + \beta_{3} ) q^{91} + ( 3 - 2 \beta_{2} - \beta_{3} ) q^{93} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{95} + ( -7 - \beta_{2} - 4 \beta_{3} ) q^{97} + ( -1 + 2 \beta_{1} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 2q^{5} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 2q^{5} - 4q^{7} + 4q^{9} + 2q^{13} - 2q^{15} - 10q^{17} + 4q^{19} - 4q^{21} - 4q^{23} + 4q^{27} + 11q^{29} + 14q^{31} + 2q^{35} + 13q^{37} + 2q^{39} + 7q^{41} + 12q^{43} - 2q^{45} + 15q^{47} + 4q^{49} - 10q^{51} + q^{53} + 31q^{55} + 4q^{57} + 5q^{59} + 3q^{61} - 4q^{63} + 25q^{65} + 5q^{67} - 4q^{69} + 5q^{71} - 6q^{73} + 26q^{79} + 4q^{81} + 2q^{83} + 5q^{85} + 11q^{87} + 7q^{89} - 2q^{91} + 14q^{93} + 5q^{95} - 27q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 8 x^{2} + 9 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 9 \nu + 2 \)
\(\beta_{3}\)\(=\)\( -2 \nu^{3} + 3 \nu^{2} + 17 \nu - 9 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 2 \beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 3 \beta_{2} + 10 \beta_{1} + 3\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.57685
0.318315
0.681685
3.57685
0 1.00000 0 −3.57685 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −0.681685 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 −0.318315 0 −1.00000 0 1.00000 0
1.4 0 1.00000 0 2.57685 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3864.2.a.t 4
4.b odd 2 1 7728.2.a.bz 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.t 4 1.a even 1 1 trivial
7728.2.a.bz 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3864))\):

\( T_{5}^{4} + 2 T_{5}^{3} - 8 T_{5}^{2} - 9 T_{5} - 2 \)
\( T_{11}^{4} - 37 T_{11}^{2} + 320 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( -2 - 9 T - 8 T^{2} + 2 T^{3} + T^{4} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( 320 - 37 T^{2} + T^{4} \)
$13$ \( 188 + 29 T - 28 T^{2} - 2 T^{3} + T^{4} \)
$17$ \( ( -16 + 5 T + T^{2} )^{2} \)
$19$ \( 8 + 22 T - 7 T^{2} - 4 T^{3} + T^{4} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( -554 + 239 T + 5 T^{2} - 11 T^{3} + T^{4} \)
$31$ \( ( -10 - 7 T + T^{2} )^{2} \)
$37$ \( 10 - 55 T + 49 T^{2} - 13 T^{3} + T^{4} \)
$41$ \( 10 + 235 T - 39 T^{2} - 7 T^{3} + T^{4} \)
$43$ \( -2636 + 889 T - 38 T^{2} - 12 T^{3} + T^{4} \)
$47$ \( -776 + 192 T + 44 T^{2} - 15 T^{3} + T^{4} \)
$53$ \( -16 - 42 T - 27 T^{2} - T^{3} + T^{4} \)
$59$ \( 1336 + 450 T - 107 T^{2} - 5 T^{3} + T^{4} \)
$61$ \( 8020 + 800 T - 227 T^{2} - 3 T^{3} + T^{4} \)
$67$ \( -16 + 32 T - 11 T^{2} - 5 T^{3} + T^{4} \)
$71$ \( 1384 - 54 T - 119 T^{2} - 5 T^{3} + T^{4} \)
$73$ \( -1048 - 662 T - 93 T^{2} + 6 T^{3} + T^{4} \)
$79$ \( -17152 + 3608 T + T^{2} - 26 T^{3} + T^{4} \)
$83$ \( 776 + 518 T - 161 T^{2} - 2 T^{3} + T^{4} \)
$89$ \( -872 + 614 T - 71 T^{2} - 7 T^{3} + T^{4} \)
$97$ \( -7220 - 1615 T + 103 T^{2} + 27 T^{3} + T^{4} \)
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