Properties

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

\(f(q)\) \(=\) \( q + \beta q^{2} -\beta q^{5} + q^{7} -2 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} -\beta q^{5} + q^{7} -2 \beta q^{8} -2 q^{10} + 2 \beta q^{11} -2 q^{13} + \beta q^{14} -4 q^{16} + ( 3 - 2 \beta ) q^{17} + ( -4 + 3 \beta ) q^{19} + 4 q^{22} + 2 \beta q^{23} -3 q^{25} -2 \beta q^{26} + ( 3 + \beta ) q^{29} + 4 q^{31} + ( -4 + 3 \beta ) q^{34} -\beta q^{35} + ( 3 - 6 \beta ) q^{37} + ( 6 - 4 \beta ) q^{38} + 4 q^{40} + ( 3 - 4 \beta ) q^{41} -10 q^{43} + 4 q^{46} + ( 6 + 2 \beta ) q^{47} + q^{49} -3 \beta q^{50} + ( 9 + \beta ) q^{53} -4 q^{55} -2 \beta q^{56} + ( 2 + 3 \beta ) q^{58} -5 \beta q^{59} + ( -8 - 3 \beta ) q^{61} + 4 \beta q^{62} + 8 q^{64} + 2 \beta q^{65} -3 \beta q^{67} -2 q^{70} + ( -6 + \beta ) q^{71} + ( 2 + 9 \beta ) q^{73} + ( -12 + 3 \beta ) q^{74} + 2 \beta q^{77} -11 q^{79} + 4 \beta q^{80} + ( -8 + 3 \beta ) q^{82} + ( -6 + 4 \beta ) q^{83} + ( 4 - 3 \beta ) q^{85} -10 \beta q^{86} -8 q^{88} + ( -12 - 2 \beta ) q^{89} -2 q^{91} + ( 4 + 6 \beta ) q^{94} + ( -6 + 4 \beta ) q^{95} + ( 7 - 3 \beta ) q^{97} + \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} + O(q^{10}) \) \( 2q + 2q^{7} - 4q^{10} - 4q^{13} - 8q^{16} + 6q^{17} - 8q^{19} + 8q^{22} - 6q^{25} + 6q^{29} + 8q^{31} - 8q^{34} + 6q^{37} + 12q^{38} + 8q^{40} + 6q^{41} - 20q^{43} + 8q^{46} + 12q^{47} + 2q^{49} + 18q^{53} - 8q^{55} + 4q^{58} - 16q^{61} + 16q^{64} - 4q^{70} - 12q^{71} + 4q^{73} - 24q^{74} - 22q^{79} - 16q^{82} - 12q^{83} + 8q^{85} - 16q^{88} - 24q^{89} - 4q^{91} + 8q^{94} - 12q^{95} + 14q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
−1.41421 0 0 1.41421 0 1.00000 2.82843 0 −2.00000
1.2 1.41421 0 0 −1.41421 0 1.00000 −2.82843 0 −2.00000
\(n\): 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.j 2
3.b odd 2 1 2667.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.g 2 3.b odd 2 1
8001.2.a.j 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 4 T^{4} \)
$3$ 1
$5$ \( 1 + 8 T^{2} + 25 T^{4} \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( 1 + 14 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 6 T + 35 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 + 8 T + 36 T^{2} + 152 T^{3} + 361 T^{4} \)
$23$ \( 1 + 38 T^{2} + 529 T^{4} \)
$29$ \( 1 - 6 T + 65 T^{2} - 174 T^{3} + 841 T^{4} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 6 T + 11 T^{2} - 222 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 6 T + 59 T^{2} - 246 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 + 10 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 12 T + 122 T^{2} - 564 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 18 T + 185 T^{2} - 954 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 68 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 16 T + 168 T^{2} + 976 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 116 T^{2} + 4489 T^{4} \)
$71$ \( 1 + 12 T + 176 T^{2} + 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 4 T - 12 T^{2} - 292 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 + 11 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 12 T + 170 T^{2} + 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 24 T + 314 T^{2} + 2136 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 14 T + 225 T^{2} - 1358 T^{3} + 9409 T^{4} \)
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