Properties

Label 2001.2.a.d
Level $2001$
Weight $2$
Character orbit 2001.a
Self dual yes
Analytic conductor $15.978$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.9780654445\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
−1.00000 −1.00000 −1.00000 −2.00000 1.00000 0 3.00000 1.00000 2.00000
1.2 −1.00000 −1.00000 −1.00000 −2.00000 1.00000 0 3.00000 1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(23\) \(1\)
\(29\) \(-1\)

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 2001.2.a.d 2
3.b odd 2 1 6003.2.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.d 2 1.a even 1 1 trivial
6003.2.a.f 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2001))\):

\( T_{2} + 1 \)
\( T_{5} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -16 + 4 T + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( -20 + T^{2} \)
$19$ \( -16 + 4 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( -1 + T )^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( -20 + T^{2} \)
$41$ \( -76 + 4 T + T^{2} \)
$43$ \( -16 - 4 T + T^{2} \)
$47$ \( -64 - 8 T + T^{2} \)
$53$ \( ( 2 + T )^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( 44 + 16 T + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( -76 + 4 T + T^{2} \)
$79$ \( 16 - 12 T + T^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( -180 + T^{2} \)
$97$ \( -164 - 8 T + T^{2} \)
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