Properties

Label 8001.2.a.k
Level 8001
Weight 2
Character orbit 8001.a
Self dual yes
Analytic conductor 63.888
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 4 q^{4} + \beta q^{5} + q^{7} + 2 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 4 q^{4} + \beta q^{5} + q^{7} + 2 \beta q^{8} + 6 q^{10} + ( 2 + 2 \beta ) q^{13} + \beta q^{14} + 4 q^{16} + ( 3 - \beta ) q^{17} + ( 2 - \beta ) q^{19} + 4 \beta q^{20} -6 q^{23} + q^{25} + ( 12 + 2 \beta ) q^{26} + 4 q^{28} + ( 3 - 2 \beta ) q^{29} + 8 q^{31} + ( -6 + 3 \beta ) q^{34} + \beta q^{35} -7 q^{37} + ( -6 + 2 \beta ) q^{38} + 12 q^{40} + ( -3 + \beta ) q^{41} + 2 q^{43} -6 \beta q^{46} + 12 q^{47} + q^{49} + \beta q^{50} + ( 8 + 8 \beta ) q^{52} + ( 3 + 2 \beta ) q^{53} + 2 \beta q^{56} + ( -12 + 3 \beta ) q^{58} -\beta q^{59} + ( 2 - \beta ) q^{61} + 8 \beta q^{62} -8 q^{64} + ( 12 + 2 \beta ) q^{65} + ( 2 + 5 \beta ) q^{67} + ( 12 - 4 \beta ) q^{68} + 6 q^{70} + ( -6 - \beta ) q^{71} + ( -4 - 3 \beta ) q^{73} -7 \beta q^{74} + ( 8 - 4 \beta ) q^{76} + ( -1 - 2 \beta ) q^{79} + 4 \beta q^{80} + ( 6 - 3 \beta ) q^{82} -6 q^{83} + ( -6 + 3 \beta ) q^{85} + 2 \beta q^{86} -4 \beta q^{89} + ( 2 + 2 \beta ) q^{91} -24 q^{92} + 12 \beta q^{94} + ( -6 + 2 \beta ) q^{95} + ( 5 + 3 \beta ) q^{97} + \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{4} + 2q^{7} + O(q^{10}) \) \( 2q + 8q^{4} + 2q^{7} + 12q^{10} + 4q^{13} + 8q^{16} + 6q^{17} + 4q^{19} - 12q^{23} + 2q^{25} + 24q^{26} + 8q^{28} + 6q^{29} + 16q^{31} - 12q^{34} - 14q^{37} - 12q^{38} + 24q^{40} - 6q^{41} + 4q^{43} + 24q^{47} + 2q^{49} + 16q^{52} + 6q^{53} - 24q^{58} + 4q^{61} - 16q^{64} + 24q^{65} + 4q^{67} + 24q^{68} + 12q^{70} - 12q^{71} - 8q^{73} + 16q^{76} - 2q^{79} + 12q^{82} - 12q^{83} - 12q^{85} + 4q^{91} - 48q^{92} - 12q^{95} + 10q^{97} + O(q^{100}) \)

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
−2.44949
2.44949
−2.44949 0 4.00000 −2.44949 0 1.00000 −4.89898 0 6.00000
1.2 2.44949 0 4.00000 2.44949 0 1.00000 4.89898 0 6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 8001.2.a.k 2
3.b odd 2 1 2667.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.h 2 3.b odd 2 1
8001.2.a.k 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{2} - 6 \)
\( T_{5}^{2} - 6 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 4 T^{4} \)
$3$ 1
$5$ \( 1 + 4 T^{2} + 25 T^{4} \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( 1 - 4 T + 6 T^{2} - 52 T^{3} + 169 T^{4} \)
$17$ \( 1 - 6 T + 37 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 - 4 T + 36 T^{2} - 76 T^{3} + 361 T^{4} \)
$23$ \( ( 1 + 6 T + 23 T^{2} )^{2} \)
$29$ \( 1 - 6 T + 43 T^{2} - 174 T^{3} + 841 T^{4} \)
$31$ \( ( 1 - 8 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 7 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 6 T + 85 T^{2} + 246 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 - 2 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 - 12 T + 47 T^{2} )^{2} \)
$53$ \( 1 - 6 T + 91 T^{2} - 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 112 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 4 T + 120 T^{2} - 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 4 T - 12 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 12 T + 172 T^{2} + 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 8 T + 108 T^{2} + 584 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 2 T + 135 T^{2} + 158 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 82 T^{2} + 7921 T^{4} \)
$97$ \( 1 - 10 T + 165 T^{2} - 970 T^{3} + 9409 T^{4} \)
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