Properties

Label 6003.2.a.d
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

Related objects

Downloads

Learn more about

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: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + ( 1 + \beta ) q^{5} + 3 q^{8} +O(q^{10})\) \( q - q^{2} - q^{4} + ( 1 + \beta ) q^{5} + 3 q^{8} + ( -1 - \beta ) q^{10} + ( 1 + \beta ) q^{11} + ( -1 - \beta ) q^{13} - q^{16} -2 \beta q^{19} + ( -1 - \beta ) q^{20} + ( -1 - \beta ) q^{22} - q^{23} + 3 \beta q^{25} + ( 1 + \beta ) q^{26} + q^{29} + ( -5 + \beta ) q^{31} -5 q^{32} + ( 5 - \beta ) q^{37} + 2 \beta q^{38} + ( 3 + 3 \beta ) q^{40} + ( -5 - \beta ) q^{41} + ( 6 - 4 \beta ) q^{43} + ( -1 - \beta ) q^{44} + q^{46} + ( -2 - 2 \beta ) q^{47} -7 q^{49} -3 \beta q^{50} + ( 1 + \beta ) q^{52} + ( -6 + 4 \beta ) q^{53} + ( 5 + 3 \beta ) q^{55} - q^{58} + ( -7 - \beta ) q^{59} + ( 1 - 5 \beta ) q^{61} + ( 5 - \beta ) q^{62} + 7 q^{64} + ( -5 - 3 \beta ) q^{65} + ( 9 - \beta ) q^{67} + ( -13 + \beta ) q^{71} -2 \beta q^{73} + ( -5 + \beta ) q^{74} + 2 \beta q^{76} -2 q^{79} + ( -1 - \beta ) q^{80} + ( 5 + \beta ) q^{82} + ( 2 + 2 \beta ) q^{83} + ( -6 + 4 \beta ) q^{86} + ( 3 + 3 \beta ) q^{88} + ( -2 - 6 \beta ) q^{89} + q^{92} + ( 2 + 2 \beta ) q^{94} + ( -8 - 4 \beta ) q^{95} + ( 2 - 6 \beta ) q^{97} + 7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{4} + 3q^{5} + 6q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{4} + 3q^{5} + 6q^{8} - 3q^{10} + 3q^{11} - 3q^{13} - 2q^{16} - 2q^{19} - 3q^{20} - 3q^{22} - 2q^{23} + 3q^{25} + 3q^{26} + 2q^{29} - 9q^{31} - 10q^{32} + 9q^{37} + 2q^{38} + 9q^{40} - 11q^{41} + 8q^{43} - 3q^{44} + 2q^{46} - 6q^{47} - 14q^{49} - 3q^{50} + 3q^{52} - 8q^{53} + 13q^{55} - 2q^{58} - 15q^{59} - 3q^{61} + 9q^{62} + 14q^{64} - 13q^{65} + 17q^{67} - 25q^{71} - 2q^{73} - 9q^{74} + 2q^{76} - 4q^{79} - 3q^{80} + 11q^{82} + 6q^{83} - 8q^{86} + 9q^{88} - 10q^{89} + 2q^{92} + 6q^{94} - 20q^{95} - 2q^{97} + 14q^{98} + O(q^{100}) \)

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
−1.56155
2.56155
−1.00000 0 −1.00000 −0.561553 0 0 3.00000 0 0.561553
1.2 −1.00000 0 −1.00000 3.56155 0 0 3.00000 0 −3.56155
\(n\): 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.d 2
3.b odd 2 1 2001.2.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.f 2 3.b odd 2 1
6003.2.a.d 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\( T_{2} + 1 \)
\( T_{5}^{2} - 3 T_{5} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + 2 T^{2} )^{2} \)
$3$ 1
$5$ \( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \)
$7$ \( ( 1 + 7 T^{2} )^{2} \)
$11$ \( 1 - 3 T + 20 T^{2} - 33 T^{3} + 121 T^{4} \)
$13$ \( 1 + 3 T + 24 T^{2} + 39 T^{3} + 169 T^{4} \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( 1 + 2 T + 22 T^{2} + 38 T^{3} + 361 T^{4} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( 1 - T )^{2} \)
$31$ \( 1 + 9 T + 78 T^{2} + 279 T^{3} + 961 T^{4} \)
$37$ \( 1 - 9 T + 90 T^{2} - 333 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 11 T + 108 T^{2} + 451 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 8 T + 34 T^{2} - 344 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 6 T + 86 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 8 T + 54 T^{2} + 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 15 T + 170 T^{2} + 885 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 3 T + 18 T^{2} + 183 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 17 T + 202 T^{2} - 1139 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 25 T + 294 T^{2} + 1775 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 2 T + 130 T^{2} + 146 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 + 2 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 6 T + 158 T^{2} - 498 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 10 T + 50 T^{2} + 890 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 2 T + 42 T^{2} + 194 T^{3} + 9409 T^{4} \)
show more
show less