Properties

Label 8005.2.a.a
Level 8005
Weight 2
Character orbit 8005.a
Self dual yes
Analytic conductor 63.920
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 8005 = 5 \cdot 1601 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8005.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9202468180\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0
−1.00000 −2.00000 −1.00000 1.00000 2.00000 2.00000 3.00000 1.00000 −1.00000
\(n\): 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 8005.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8005.2.a.a 1 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(1601\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8005))\):

\( T_{2} + 1 \)
\( T_{3} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 2 T^{2} \)
$3$ \( 1 + 2 T + 3 T^{2} \)
$5$ \( 1 - T \)
$7$ \( 1 - 2 T + 7 T^{2} \)
$11$ \( 1 + 6 T + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 - 6 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 - 2 T + 23 T^{2} \)
$29$ \( 1 - 2 T + 29 T^{2} \)
$31$ \( 1 + 8 T + 31 T^{2} \)
$37$ \( 1 - 2 T + 37 T^{2} \)
$41$ \( 1 + 6 T + 41 T^{2} \)
$43$ \( 1 - 8 T + 43 T^{2} \)
$47$ \( 1 - 12 T + 47 T^{2} \)
$53$ \( 1 + 6 T + 53 T^{2} \)
$59$ \( 1 - 10 T + 59 T^{2} \)
$61$ \( 1 + 2 T + 61 T^{2} \)
$67$ \( 1 + 8 T + 67 T^{2} \)
$71$ \( 1 - 10 T + 71 T^{2} \)
$73$ \( 1 + 2 T + 73 T^{2} \)
$79$ \( 1 + 8 T + 79 T^{2} \)
$83$ \( 1 - 6 T + 83 T^{2} \)
$89$ \( 1 + 2 T + 89 T^{2} \)
$97$ \( 1 - 10 T + 97 T^{2} \)
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