Properties

Label 6003.2.a.f
Level 6003
Weight 2
Character orbit 6003.a
Self dual yes
Analytic conductor 47.934
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(1\)
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: no (minimal twist has level 2001)
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.

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

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
1.61803
−0.618034
1.00000 0 −1.00000 2.00000 0 0 −3.00000 0 2.00000
1.2 1.00000 0 −1.00000 2.00000 0 0 −3.00000 0 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 6003.2.a.f 2
3.b odd 2 1 2001.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.d 2 3.b odd 2 1
6003.2.a.f 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + 2 T^{2} )^{2} \)
$3$ 1
$5$ \( ( 1 - 2 T + 5 T^{2} )^{2} \)
$7$ \( ( 1 + 7 T^{2} )^{2} \)
$11$ \( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 14 T^{2} + 289 T^{4} \)
$19$ \( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( ( 1 - T )^{2} \)
$29$ \( ( 1 + T )^{2} \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 54 T^{2} + 1369 T^{4} \)
$41$ \( 1 - 4 T + 6 T^{2} - 164 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 4 T + 70 T^{2} - 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 8 T + 30 T^{2} + 376 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 2 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 - 4 T + 59 T^{2} )^{2} \)
$61$ \( 1 + 16 T + 166 T^{2} + 976 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 + 4 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 8 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 4 T + 70 T^{2} + 292 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 12 T + 174 T^{2} - 948 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 4 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 2 T^{2} + 7921 T^{4} \)
$97$ \( 1 - 8 T + 30 T^{2} - 776 T^{3} + 9409 T^{4} \)
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