Properties

Label 4008.2.a.e
Level $4008$
Weight $2$
Character orbit 4008.a
Self dual yes
Analytic conductor $32.004$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0040411301\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( 2 - \beta_{2} ) q^{5} + ( -2 + 2 \beta_{1} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 2 - \beta_{2} ) q^{5} + ( -2 + 2 \beta_{1} ) q^{7} + q^{9} + ( -1 - \beta_{1} + \beta_{2} ) q^{11} + 2 q^{13} + ( -2 + \beta_{2} ) q^{15} + 3 \beta_{2} q^{17} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{19} + ( 2 - 2 \beta_{1} ) q^{21} -8 q^{23} + ( 2 - \beta_{1} - 5 \beta_{2} ) q^{25} - q^{27} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{31} + ( 1 + \beta_{1} - \beta_{2} ) q^{33} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{35} + ( -4 + 2 \beta_{1} ) q^{37} -2 q^{39} + ( 6 - 6 \beta_{1} - 3 \beta_{2} ) q^{41} + ( -2 - 2 \beta_{1} - 3 \beta_{2} ) q^{43} + ( 2 - \beta_{2} ) q^{45} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{47} + ( 5 - 4 \beta_{1} + 4 \beta_{2} ) q^{49} -3 \beta_{2} q^{51} + ( -4 + 4 \beta_{1} + 3 \beta_{2} ) q^{53} + ( -6 + 4 \beta_{2} ) q^{55} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{57} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( -1 + 5 \beta_{1} + 5 \beta_{2} ) q^{61} + ( -2 + 2 \beta_{1} ) q^{63} + ( 4 - 2 \beta_{2} ) q^{65} + ( -8 - 2 \beta_{1} - \beta_{2} ) q^{67} + 8 q^{69} + ( -4 - 4 \beta_{1} - 6 \beta_{2} ) q^{71} + ( 8 - 2 \beta_{1} - 4 \beta_{2} ) q^{73} + ( -2 + \beta_{1} + 5 \beta_{2} ) q^{75} + ( -4 - 4 \beta_{2} ) q^{77} + ( 4 - 2 \beta_{1} + 5 \beta_{2} ) q^{79} + q^{81} + ( -4 + 8 \beta_{1} + 6 \beta_{2} ) q^{83} + ( -9 + 3 \beta_{1} + 9 \beta_{2} ) q^{85} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{87} + ( -6 \beta_{1} + 2 \beta_{2} ) q^{89} + ( -4 + 4 \beta_{1} ) q^{91} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{93} + ( 8 - 4 \beta_{1} - 8 \beta_{2} ) q^{95} + ( 2 - 6 \beta_{2} ) q^{97} + ( -1 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 6q^{5} - 4q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 6q^{5} - 4q^{7} + 3q^{9} - 4q^{11} + 6q^{13} - 6q^{15} + 4q^{19} + 4q^{21} - 24q^{23} + 5q^{25} - 3q^{27} - 14q^{29} - 4q^{31} + 4q^{33} - 4q^{35} - 10q^{37} - 6q^{39} + 12q^{41} - 8q^{43} + 6q^{45} + 11q^{49} - 8q^{53} - 18q^{55} - 4q^{57} + 4q^{59} + 2q^{61} - 4q^{63} + 12q^{65} - 26q^{67} + 24q^{69} - 16q^{71} + 22q^{73} - 5q^{75} - 12q^{77} + 10q^{79} + 3q^{81} - 4q^{83} - 24q^{85} + 14q^{87} - 6q^{89} - 8q^{91} + 4q^{93} + 20q^{95} + 6q^{97} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)

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
−1.48119
2.17009
0.311108
0 −1.00000 0 0.324869 0 −4.96239 0 1.00000 0
1.2 0 −1.00000 0 1.46081 0 2.34017 0 1.00000 0
1.3 0 −1.00000 0 4.21432 0 −1.37778 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(167\) \(-1\)

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 4008.2.a.e 3
4.b odd 2 1 8016.2.a.n 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.e 3 1.a even 1 1 trivial
8016.2.a.n 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\( T_{5}^{3} - 6 T_{5}^{2} + 8 T_{5} - 2 \)
\( T_{7}^{3} + 4 T_{7}^{2} - 8 T_{7} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -2 + 8 T - 6 T^{2} + T^{3} \)
$7$ \( -16 - 8 T + 4 T^{2} + T^{3} \)
$11$ \( -20 - 4 T + 4 T^{2} + T^{3} \)
$13$ \( ( -2 + T )^{3} \)
$17$ \( 54 - 36 T + T^{3} \)
$19$ \( 32 - 16 T - 4 T^{2} + T^{3} \)
$23$ \( ( 8 + T )^{3} \)
$29$ \( -152 + 28 T + 14 T^{2} + T^{3} \)
$31$ \( -32 - 16 T + 4 T^{2} + T^{3} \)
$37$ \( -8 + 20 T + 10 T^{2} + T^{3} \)
$41$ \( 918 - 72 T - 12 T^{2} + T^{3} \)
$43$ \( -130 - 16 T + 8 T^{2} + T^{3} \)
$47$ \( -268 - 84 T + T^{3} \)
$53$ \( -290 - 44 T + 8 T^{2} + T^{3} \)
$59$ \( -32 - 32 T - 4 T^{2} + T^{3} \)
$61$ \( -4 - 132 T - 2 T^{2} + T^{3} \)
$67$ \( 554 + 212 T + 26 T^{2} + T^{3} \)
$71$ \( -1040 - 64 T + 16 T^{2} + T^{3} \)
$73$ \( -104 + 100 T - 22 T^{2} + T^{3} \)
$79$ \( 278 - 100 T - 10 T^{2} + T^{3} \)
$83$ \( -1424 - 256 T + 4 T^{2} + T^{3} \)
$89$ \( -920 - 148 T + 6 T^{2} + T^{3} \)
$97$ \( -152 - 132 T - 6 T^{2} + T^{3} \)
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