Properties

Label 4008.2.a.f
Level $4008$
Weight $2$
Character orbit 4008.a
Self dual yes
Analytic conductor $32.004$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.284897.1
Defining polynomial: \(x^{5} - 2 x^{4} - 6 x^{3} + 5 x^{2} + 10 x + 3\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( -\beta_{3} + \beta_{4} ) q^{5} + ( -1 - \beta_{2} + \beta_{3} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -\beta_{3} + \beta_{4} ) q^{5} + ( -1 - \beta_{2} + \beta_{3} ) q^{7} + q^{9} + ( -\beta_{3} - \beta_{4} ) q^{11} + ( -2 - \beta_{1} + \beta_{2} - \beta_{4} ) q^{13} + ( -\beta_{3} + \beta_{4} ) q^{15} + ( -3 + 2 \beta_{3} ) q^{17} + ( -\beta_{1} + \beta_{2} ) q^{19} + ( -1 - \beta_{2} + \beta_{3} ) q^{21} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{23} + ( -1 + 3 \beta_{1} + \beta_{3} ) q^{25} + q^{27} + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{29} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{31} + ( -\beta_{3} - \beta_{4} ) q^{33} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{35} + ( -1 - 3 \beta_{2} ) q^{37} + ( -2 - \beta_{1} + \beta_{2} - \beta_{4} ) q^{39} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{41} + ( -4 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{43} + ( -\beta_{3} + \beta_{4} ) q^{45} + ( -1 + \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{47} + ( -2 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{49} + ( -3 + 2 \beta_{3} ) q^{51} + ( -2 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{53} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{55} + ( -\beta_{1} + \beta_{2} ) q^{57} + ( 2 - 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{59} + ( -4 - 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{61} + ( -1 - \beta_{2} + \beta_{3} ) q^{63} + ( -3 - \beta_{1} + 3 \beta_{3} - 4 \beta_{4} ) q^{65} + ( -4 + \beta_{1} - \beta_{2} - 4 \beta_{3} + 3 \beta_{4} ) q^{67} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{69} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{4} ) q^{71} + ( -5 + 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{73} + ( -1 + 3 \beta_{1} + \beta_{3} ) q^{75} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{77} + ( -2 - \beta_{1} - 4 \beta_{3} - 2 \beta_{4} ) q^{79} + q^{81} + ( -2 - 2 \beta_{1} - 5 \beta_{2} + \beta_{3} - \beta_{4} ) q^{83} + ( -4 - 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{85} + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{87} + ( -8 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{89} + ( 1 + 3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{91} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{93} + ( -1 - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{95} + ( -6 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{97} + ( -\beta_{3} - \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} + q^{5} - 4q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} + q^{5} - 4q^{7} + 5q^{9} - 3q^{11} - 14q^{13} + q^{15} - 13q^{17} - 2q^{19} - 4q^{21} - 5q^{23} + 2q^{25} + 5q^{27} + 13q^{29} + 2q^{31} - 3q^{33} - 12q^{35} - 5q^{37} - 14q^{39} - 20q^{41} - 20q^{43} + q^{45} + q^{47} - 9q^{49} - 13q^{51} - 3q^{53} - 3q^{55} - 2q^{57} - q^{59} - 34q^{61} - 4q^{63} - 22q^{65} - 16q^{67} - 5q^{69} - 5q^{71} - 12q^{73} + 2q^{75} - 8q^{77} - 20q^{79} + 5q^{81} - 15q^{83} - 27q^{85} + 13q^{87} - 48q^{89} + 7q^{91} + 2q^{93} - 5q^{95} - 21q^{97} - 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 6 x^{3} + 5 x^{2} + 10 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - 3 \nu^{3} - 3 \nu^{2} + 8 \nu + 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 3 \nu^{3} - 4 \nu^{2} + 9 \nu + 6 \)
\(\beta_{4}\)\(=\)\( 2 \nu^{4} - 5 \nu^{3} - 9 \nu^{2} + 14 \nu + 11 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} - 3 \beta_{3} + \beta_{2} + 5 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(3 \beta_{4} - 12 \beta_{3} + 7 \beta_{2} + 10 \beta_{1} + 18\)

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
1.80442
−0.759623
−1.65065
−0.435916
3.04177
0 1.00000 0 −3.40664 0 0.548479 0 1.00000 0
1.2 0 1.00000 0 −0.473647 0 0.663349 0 1.00000 0
1.3 0 1.00000 0 −0.457884 0 −2.37531 0 1.00000 0
1.4 0 1.00000 0 2.07208 0 1.37406 0 1.00000 0
1.5 0 1.00000 0 3.26608 0 −4.21058 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.f 5
4.b odd 2 1 8016.2.a.s 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.f 5 1.a even 1 1 trivial
8016.2.a.s 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\( T_{5}^{5} - T_{5}^{4} - 13 T_{5}^{3} + 12 T_{5}^{2} + 19 T_{5} + 5 \)
\( T_{7}^{5} + 4 T_{7}^{4} - 5 T_{7}^{3} - 13 T_{7}^{2} + 17 T_{7} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( ( -1 + T )^{5} \)
$5$ \( 5 + 19 T + 12 T^{2} - 13 T^{3} - T^{4} + T^{5} \)
$7$ \( -5 + 17 T - 13 T^{2} - 5 T^{3} + 4 T^{4} + T^{5} \)
$11$ \( 55 + 47 T - 30 T^{2} - 19 T^{3} + 3 T^{4} + T^{5} \)
$13$ \( -215 - 132 T + 62 T^{2} + 61 T^{3} + 14 T^{4} + T^{5} \)
$17$ \( 9 - 35 T - 54 T^{2} + 34 T^{3} + 13 T^{4} + T^{5} \)
$19$ \( -9 - 40 T - 44 T^{2} - 13 T^{3} + 2 T^{4} + T^{5} \)
$23$ \( 283 + 167 T - 74 T^{2} - 37 T^{3} + 5 T^{4} + T^{5} \)
$29$ \( 2355 - 1853 T + 408 T^{2} + 13 T^{3} - 13 T^{4} + T^{5} \)
$31$ \( 697 + 620 T + 33 T^{2} - 50 T^{3} - 2 T^{4} + T^{5} \)
$37$ \( 919 - 103 T - 368 T^{2} - 71 T^{3} + 5 T^{4} + T^{5} \)
$41$ \( 21643 - 2306 T - 1325 T^{2} + 14 T^{3} + 20 T^{4} + T^{5} \)
$43$ \( 207 - 70 T - 237 T^{2} + 78 T^{3} + 20 T^{4} + T^{5} \)
$47$ \( -14915 + 4818 T + 293 T^{2} - 155 T^{3} - T^{4} + T^{5} \)
$53$ \( -387 + 1550 T - 213 T^{2} - 91 T^{3} + 3 T^{4} + T^{5} \)
$59$ \( 457 - 62 T - 625 T^{2} - 201 T^{3} + T^{4} + T^{5} \)
$61$ \( -20373 - 14696 T - 650 T^{2} + 301 T^{3} + 34 T^{4} + T^{5} \)
$67$ \( 2501 - 826 T - 1810 T^{2} - 103 T^{3} + 16 T^{4} + T^{5} \)
$71$ \( -81 - 863 T - 825 T^{2} - 140 T^{3} + 5 T^{4} + T^{5} \)
$73$ \( -2249 - 6472 T - 1848 T^{2} - 103 T^{3} + 12 T^{4} + T^{5} \)
$79$ \( 27429 - 3020 T - 1903 T^{2} - 32 T^{3} + 20 T^{4} + T^{5} \)
$83$ \( 9379 - 11516 T - 3815 T^{2} - 201 T^{3} + 15 T^{4} + T^{5} \)
$89$ \( 34985 + 26658 T + 7128 T^{2} + 861 T^{3} + 48 T^{4} + T^{5} \)
$97$ \( -361 - 521 T + 11 T^{2} + 108 T^{3} + 21 T^{4} + T^{5} \)
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