Properties

Label 8001.2.a.i
Level 8001
Weight 2
Character orbit 8001.a
Self dual yes
Analytic conductor 63.888
Analytic rank 1
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{4} + ( 1 + \beta ) q^{5} - q^{7} +O(q^{10})\) \( q -2 q^{4} + ( 1 + \beta ) q^{5} - q^{7} + ( 2 - 2 \beta ) q^{11} + ( -1 - \beta ) q^{13} + 4 q^{16} + ( -4 - \beta ) q^{17} + 4 q^{19} + ( -2 - 2 \beta ) q^{20} + ( -3 + 3 \beta ) q^{23} + 3 \beta q^{25} + 2 q^{28} + ( 1 - 2 \beta ) q^{29} + ( -3 + 3 \beta ) q^{31} + ( -1 - \beta ) q^{35} + ( -7 + 2 \beta ) q^{37} + ( -2 + 5 \beta ) q^{41} + ( 6 - 4 \beta ) q^{43} + ( -4 + 4 \beta ) q^{44} + ( 10 - 2 \beta ) q^{47} + q^{49} + ( 2 + 2 \beta ) q^{52} + 11 q^{53} + ( -6 - 2 \beta ) q^{55} + ( 7 - \beta ) q^{59} + ( -7 - \beta ) q^{61} -8 q^{64} + ( -5 - 3 \beta ) q^{65} -2 q^{67} + ( 8 + 2 \beta ) q^{68} + ( 8 + 2 \beta ) q^{71} + ( -7 + \beta ) q^{73} -8 q^{76} + ( -2 + 2 \beta ) q^{77} + ( -8 + 3 \beta ) q^{79} + ( 4 + 4 \beta ) q^{80} + ( -5 - 5 \beta ) q^{83} + ( -8 - 6 \beta ) q^{85} + ( 5 - 5 \beta ) q^{89} + ( 1 + \beta ) q^{91} + ( 6 - 6 \beta ) q^{92} + ( 4 + 4 \beta ) q^{95} + ( 4 - 5 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + 3q^{5} - 2q^{7} + O(q^{10}) \) \( 2q - 4q^{4} + 3q^{5} - 2q^{7} + 2q^{11} - 3q^{13} + 8q^{16} - 9q^{17} + 8q^{19} - 6q^{20} - 3q^{23} + 3q^{25} + 4q^{28} - 3q^{31} - 3q^{35} - 12q^{37} + q^{41} + 8q^{43} - 4q^{44} + 18q^{47} + 2q^{49} + 6q^{52} + 22q^{53} - 14q^{55} + 13q^{59} - 15q^{61} - 16q^{64} - 13q^{65} - 4q^{67} + 18q^{68} + 18q^{71} - 13q^{73} - 16q^{76} - 2q^{77} - 13q^{79} + 12q^{80} - 15q^{83} - 22q^{85} + 5q^{89} + 3q^{91} + 6q^{92} + 12q^{95} + 3q^{97} + O(q^{100}) \)

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
−1.56155
2.56155
0 0 −2.00000 −0.561553 0 −1.00000 0 0 0
1.2 0 0 −2.00000 3.56155 0 −1.00000 0 0 0
\(n\): 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.i 2
3.b odd 2 1 2667.2.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.f 2 3.b odd 2 1
8001.2.a.i 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2} \)
\( T_{5}^{2} - 3 T_{5} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{2} \)
$3$ 1
$5$ \( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 - 2 T + 6 T^{2} - 22 T^{3} + 121 T^{4} \)
$13$ \( 1 + 3 T + 24 T^{2} + 39 T^{3} + 169 T^{4} \)
$17$ \( 1 + 9 T + 50 T^{2} + 153 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 3 T + 10 T^{2} + 69 T^{3} + 529 T^{4} \)
$29$ \( 1 + 41 T^{2} + 841 T^{4} \)
$31$ \( 1 + 3 T + 26 T^{2} + 93 T^{3} + 961 T^{4} \)
$37$ \( 1 + 12 T + 93 T^{2} + 444 T^{3} + 1369 T^{4} \)
$41$ \( 1 - T - 24 T^{2} - 41 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 8 T + 34 T^{2} - 344 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 18 T + 158 T^{2} - 846 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 11 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 13 T + 156 T^{2} - 767 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 15 T + 174 T^{2} + 915 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 + 2 T + 67 T^{2} )^{2} \)
$71$ \( 1 - 18 T + 206 T^{2} - 1278 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 13 T + 184 T^{2} + 949 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 13 T + 162 T^{2} + 1027 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 15 T + 116 T^{2} + 1245 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 5 T + 78 T^{2} - 445 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 3 T + 90 T^{2} - 291 T^{3} + 9409 T^{4} \)
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