Properties

Label 8008.2.a.g
Level 8008
Weight 2
Character orbit 8008.a
Self dual yes
Analytic conductor 63.944
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9442019386\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
0 −0.618034 0 1.61803 0 1.00000 0 −2.61803 0
1.2 0 1.61803 0 −0.618034 0 1.00000 0 −0.381966 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)
\(13\) \(-1\)

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 8008.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.g 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{2} - T_{3} - 1 \)
\( T_{5}^{2} - T_{5} - 1 \)
\( T_{17}^{2} + 9 T_{17} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T + 5 T^{2} - 3 T^{3} + 9 T^{4} \)
$5$ \( 1 - T + 9 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( ( 1 - T )^{2} \)
$13$ \( ( 1 - T )^{2} \)
$17$ \( 1 + 9 T + 43 T^{2} + 153 T^{3} + 289 T^{4} \)
$19$ \( 1 - 3 T + 39 T^{2} - 57 T^{3} + 361 T^{4} \)
$23$ \( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4} \)
$29$ \( 1 - 8 T + 54 T^{2} - 232 T^{3} + 841 T^{4} \)
$31$ \( 1 + 10 T + 82 T^{2} + 310 T^{3} + 961 T^{4} \)
$37$ \( 1 - 10 T + 54 T^{2} - 370 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 6 T + 86 T^{2} + 246 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 3 T + 87 T^{2} + 129 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 14 T + 138 T^{2} + 658 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 3 T + 107 T^{2} - 159 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 2 T + 114 T^{2} + 118 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 19 T + 201 T^{2} + 1159 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 15 T + 159 T^{2} - 1005 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 5 T + 47 T^{2} + 355 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 126 T^{2} + 5329 T^{4} \)
$79$ \( 1 + T - 53 T^{2} + 79 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 3 T + 167 T^{2} - 249 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 35 T + 483 T^{2} + 3115 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 2 T + 97 T^{2} )^{2} \)
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