Properties

Label 8007.2.a.b
Level 8007
Weight 2
Character orbit 8007.a
Self dual yes
Analytic conductor 63.936
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8007 = 3 \cdot 17 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8007.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9362168984\)
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: yes
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + q^{3} + ( 1 - 2 \beta ) q^{4} + ( -2 - \beta ) q^{5} + ( -1 + \beta ) q^{6} + ( -2 - \beta ) q^{7} + ( -3 + \beta ) q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + q^{3} + ( 1 - 2 \beta ) q^{4} + ( -2 - \beta ) q^{5} + ( -1 + \beta ) q^{6} + ( -2 - \beta ) q^{7} + ( -3 + \beta ) q^{8} + q^{9} -\beta q^{10} + 2 \beta q^{11} + ( 1 - 2 \beta ) q^{12} -4 \beta q^{13} -\beta q^{14} + ( -2 - \beta ) q^{15} + 3 q^{16} - q^{17} + ( -1 + \beta ) q^{18} + ( -2 + 2 \beta ) q^{19} + ( 2 + 3 \beta ) q^{20} + ( -2 - \beta ) q^{21} + ( 4 - 2 \beta ) q^{22} + ( 4 + \beta ) q^{23} + ( -3 + \beta ) q^{24} + ( 1 + 4 \beta ) q^{25} + ( -8 + 4 \beta ) q^{26} + q^{27} + ( 2 + 3 \beta ) q^{28} + ( 2 + 5 \beta ) q^{29} -\beta q^{30} + ( 4 + 2 \beta ) q^{31} + ( 3 + \beta ) q^{32} + 2 \beta q^{33} + ( 1 - \beta ) q^{34} + ( 6 + 4 \beta ) q^{35} + ( 1 - 2 \beta ) q^{36} + ( 2 - 2 \beta ) q^{37} + ( 6 - 4 \beta ) q^{38} -4 \beta q^{39} + ( 4 + \beta ) q^{40} + ( 2 - \beta ) q^{41} -\beta q^{42} -8 \beta q^{43} + ( -8 + 2 \beta ) q^{44} + ( -2 - \beta ) q^{45} + ( -2 + 3 \beta ) q^{46} + ( 6 + 4 \beta ) q^{47} + 3 q^{48} + ( -1 + 4 \beta ) q^{49} + ( 7 - 3 \beta ) q^{50} - q^{51} + ( 16 - 4 \beta ) q^{52} -4 q^{53} + ( -1 + \beta ) q^{54} + ( -4 - 4 \beta ) q^{55} + ( 4 + \beta ) q^{56} + ( -2 + 2 \beta ) q^{57} + ( 8 - 3 \beta ) q^{58} -6 q^{59} + ( 2 + 3 \beta ) q^{60} + ( 4 - \beta ) q^{61} + 2 \beta q^{62} + ( -2 - \beta ) q^{63} + ( -7 + 2 \beta ) q^{64} + ( 8 + 8 \beta ) q^{65} + ( 4 - 2 \beta ) q^{66} + ( 4 - 8 \beta ) q^{67} + ( -1 + 2 \beta ) q^{68} + ( 4 + \beta ) q^{69} + ( 2 + 2 \beta ) q^{70} -8 q^{71} + ( -3 + \beta ) q^{72} + ( -8 + \beta ) q^{73} + ( -6 + 4 \beta ) q^{74} + ( 1 + 4 \beta ) q^{75} + ( -10 + 6 \beta ) q^{76} + ( -4 - 4 \beta ) q^{77} + ( -8 + 4 \beta ) q^{78} + ( 6 + 7 \beta ) q^{79} + ( -6 - 3 \beta ) q^{80} + q^{81} + ( -4 + 3 \beta ) q^{82} + ( -2 - 2 \beta ) q^{83} + ( 2 + 3 \beta ) q^{84} + ( 2 + \beta ) q^{85} + ( -16 + 8 \beta ) q^{86} + ( 2 + 5 \beta ) q^{87} + ( 4 - 6 \beta ) q^{88} + ( 6 + 8 \beta ) q^{89} -\beta q^{90} + ( 8 + 8 \beta ) q^{91} -7 \beta q^{92} + ( 4 + 2 \beta ) q^{93} + ( 2 + 2 \beta ) q^{94} -2 \beta q^{95} + ( 3 + \beta ) q^{96} -3 \beta q^{97} + ( 9 - 5 \beta ) q^{98} + 2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 4q^{5} - 2q^{6} - 4q^{7} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 4q^{5} - 2q^{6} - 4q^{7} - 6q^{8} + 2q^{9} + 2q^{12} - 4q^{15} + 6q^{16} - 2q^{17} - 2q^{18} - 4q^{19} + 4q^{20} - 4q^{21} + 8q^{22} + 8q^{23} - 6q^{24} + 2q^{25} - 16q^{26} + 2q^{27} + 4q^{28} + 4q^{29} + 8q^{31} + 6q^{32} + 2q^{34} + 12q^{35} + 2q^{36} + 4q^{37} + 12q^{38} + 8q^{40} + 4q^{41} - 16q^{44} - 4q^{45} - 4q^{46} + 12q^{47} + 6q^{48} - 2q^{49} + 14q^{50} - 2q^{51} + 32q^{52} - 8q^{53} - 2q^{54} - 8q^{55} + 8q^{56} - 4q^{57} + 16q^{58} - 12q^{59} + 4q^{60} + 8q^{61} - 4q^{63} - 14q^{64} + 16q^{65} + 8q^{66} + 8q^{67} - 2q^{68} + 8q^{69} + 4q^{70} - 16q^{71} - 6q^{72} - 16q^{73} - 12q^{74} + 2q^{75} - 20q^{76} - 8q^{77} - 16q^{78} + 12q^{79} - 12q^{80} + 2q^{81} - 8q^{82} - 4q^{83} + 4q^{84} + 4q^{85} - 32q^{86} + 4q^{87} + 8q^{88} + 12q^{89} + 16q^{91} + 8q^{93} + 4q^{94} + 6q^{96} + 18q^{98} + O(q^{100}) \)

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
−1.41421
1.41421
−2.41421 1.00000 3.82843 −0.585786 −2.41421 −0.585786 −4.41421 1.00000 1.41421
1.2 0.414214 1.00000 −1.82843 −3.41421 0.414214 −3.41421 −1.58579 1.00000 −1.41421
\(n\): 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 8007.2.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8007.2.a.b 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(1\)
\(157\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4} \)
$3$ \( ( 1 - T )^{2} \)
$5$ \( 1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4} \)
$7$ \( 1 + 4 T + 16 T^{2} + 28 T^{3} + 49 T^{4} \)
$11$ \( 1 + 14 T^{2} + 121 T^{4} \)
$13$ \( 1 - 6 T^{2} + 169 T^{4} \)
$17$ \( ( 1 + T )^{2} \)
$19$ \( 1 + 4 T + 34 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 8 T + 60 T^{2} - 184 T^{3} + 529 T^{4} \)
$29$ \( 1 - 4 T + 12 T^{2} - 116 T^{3} + 841 T^{4} \)
$31$ \( 1 - 8 T + 70 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( 1 - 4 T + 70 T^{2} - 148 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 4 T + 84 T^{2} - 164 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 42 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 12 T + 98 T^{2} - 564 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 4 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 + 6 T + 59 T^{2} )^{2} \)
$61$ \( 1 - 8 T + 136 T^{2} - 488 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 8 T + 22 T^{2} - 536 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 8 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 16 T + 208 T^{2} + 1168 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 12 T + 96 T^{2} - 948 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 4 T + 162 T^{2} + 332 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 12 T + 86 T^{2} - 1068 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 176 T^{2} + 9409 T^{4} \)
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