Properties

Label 8001.2.a.h
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( 2 + \beta ) q^{4} + ( -1 - \beta ) q^{5} - q^{7} + ( -4 - \beta ) q^{8} +O(q^{10})\) \( q -\beta q^{2} + ( 2 + \beta ) q^{4} + ( -1 - \beta ) q^{5} - q^{7} + ( -4 - \beta ) q^{8} + ( 4 + 2 \beta ) q^{10} + ( 2 - 2 \beta ) q^{11} + ( -1 - \beta ) q^{13} + \beta q^{14} + 3 \beta q^{16} -4 q^{17} -4 q^{19} + ( -6 - 4 \beta ) q^{20} + 8 q^{22} + ( 3 - \beta ) q^{23} + 3 \beta q^{25} + ( 4 + 2 \beta ) q^{26} + ( -2 - \beta ) q^{28} + ( -5 + \beta ) q^{29} + ( 5 - \beta ) q^{31} + ( -4 - \beta ) q^{32} + 4 \beta q^{34} + ( 1 + \beta ) q^{35} + ( -3 + \beta ) q^{37} + 4 \beta q^{38} + ( 8 + 6 \beta ) q^{40} + ( -2 - 2 \beta ) q^{41} + ( 2 + 2 \beta ) q^{43} + ( -4 - 4 \beta ) q^{44} + ( 4 - 2 \beta ) q^{46} + 10 q^{47} + q^{49} + ( -12 - 3 \beta ) q^{50} + ( -6 - 4 \beta ) q^{52} + ( 1 + 3 \beta ) q^{53} + ( 6 + 2 \beta ) q^{55} + ( 4 + \beta ) q^{56} + ( -4 + 4 \beta ) q^{58} + ( 1 + 3 \beta ) q^{59} + ( -7 + \beta ) q^{61} + ( 4 - 4 \beta ) q^{62} + ( 4 - \beta ) q^{64} + ( 5 + 3 \beta ) q^{65} + ( 2 + 2 \beta ) q^{67} + ( -8 - 4 \beta ) q^{68} + ( -4 - 2 \beta ) q^{70} + 12 q^{71} + ( -7 + \beta ) q^{73} + ( -4 + 2 \beta ) q^{74} + ( -8 - 4 \beta ) q^{76} + ( -2 + 2 \beta ) q^{77} + ( -4 + 4 \beta ) q^{79} + ( -12 - 6 \beta ) q^{80} + ( 8 + 4 \beta ) q^{82} + ( -3 + 3 \beta ) q^{83} + ( 4 + 4 \beta ) q^{85} + ( -8 - 4 \beta ) q^{86} + 8 \beta q^{88} + ( -13 + 3 \beta ) q^{89} + ( 1 + \beta ) q^{91} + 2 q^{92} -10 \beta q^{94} + ( 4 + 4 \beta ) q^{95} + ( -12 - 2 \beta ) q^{97} -\beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + 5q^{4} - 3q^{5} - 2q^{7} - 9q^{8} + O(q^{10}) \) \( 2q - q^{2} + 5q^{4} - 3q^{5} - 2q^{7} - 9q^{8} + 10q^{10} + 2q^{11} - 3q^{13} + q^{14} + 3q^{16} - 8q^{17} - 8q^{19} - 16q^{20} + 16q^{22} + 5q^{23} + 3q^{25} + 10q^{26} - 5q^{28} - 9q^{29} + 9q^{31} - 9q^{32} + 4q^{34} + 3q^{35} - 5q^{37} + 4q^{38} + 22q^{40} - 6q^{41} + 6q^{43} - 12q^{44} + 6q^{46} + 20q^{47} + 2q^{49} - 27q^{50} - 16q^{52} + 5q^{53} + 14q^{55} + 9q^{56} - 4q^{58} + 5q^{59} - 13q^{61} + 4q^{62} + 7q^{64} + 13q^{65} + 6q^{67} - 20q^{68} - 10q^{70} + 24q^{71} - 13q^{73} - 6q^{74} - 20q^{76} - 2q^{77} - 4q^{79} - 30q^{80} + 20q^{82} - 3q^{83} + 12q^{85} - 20q^{86} + 8q^{88} - 23q^{89} + 3q^{91} + 4q^{92} - 10q^{94} + 12q^{95} - 26q^{97} - q^{98} + O(q^{100}) \)

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
2.56155
−1.56155
−2.56155 0 4.56155 −3.56155 0 −1.00000 −6.56155 0 9.12311
1.2 1.56155 0 0.438447 0.561553 0 −1.00000 −2.43845 0 0.876894
\(n\): 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.h 2
3.b odd 2 1 2667.2.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.i 2 3.b odd 2 1
8001.2.a.h 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{2} + T_{2} - 4 \)
\( T_{5}^{2} + 3 T_{5} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 2 T^{3} + 4 T^{4} \)
$3$ 1
$5$ \( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 - 2 T + 6 T^{2} - 22 T^{3} + 121 T^{4} \)
$13$ \( 1 + 3 T + 24 T^{2} + 39 T^{3} + 169 T^{4} \)
$17$ \( ( 1 + 4 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 5 T + 48 T^{2} - 115 T^{3} + 529 T^{4} \)
$29$ \( 1 + 9 T + 74 T^{2} + 261 T^{3} + 841 T^{4} \)
$31$ \( 1 - 9 T + 78 T^{2} - 279 T^{3} + 961 T^{4} \)
$37$ \( 1 + 5 T + 76 T^{2} + 185 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 6 T + 74 T^{2} + 246 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 6 T + 78 T^{2} - 258 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 - 10 T + 47 T^{2} )^{2} \)
$53$ \( 1 - 5 T + 74 T^{2} - 265 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 5 T + 86 T^{2} - 295 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 13 T + 160 T^{2} + 793 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 6 T + 126 T^{2} - 402 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 13 T + 184 T^{2} + 949 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 4 T + 94 T^{2} + 316 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 3 T + 130 T^{2} + 249 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 23 T + 272 T^{2} + 2047 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 26 T + 346 T^{2} + 2522 T^{3} + 9409 T^{4} \)
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