Properties

Label 6003.2.a.h
Level 6003
Weight 2
Character orbit 6003.a
Self dual Yes
Analytic conductor 47.934
Analytic rank 0
Dimension 5
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.312617.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( 2 - \beta_{1} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} ) q^{5} + ( -1 - \beta_{2} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{8} +O(q^{10})\) \( q + \beta_{3} q^{2} + ( 2 - \beta_{1} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} ) q^{5} + ( -1 - \beta_{2} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{8} + ( 3 - 2 \beta_{1} - \beta_{4} ) q^{10} + ( 2 - \beta_{4} ) q^{11} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{13} + ( 2 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{14} + ( -2 - 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{16} -\beta_{1} q^{17} + ( -1 - 2 \beta_{4} ) q^{19} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{20} + ( 2 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{22} - q^{23} + \beta_{3} q^{25} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{26} + ( -4 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{28} - q^{29} + ( -2 - \beta_{2} + 2 \beta_{3} ) q^{31} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{32} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{34} + ( 3 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{35} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{37} + ( 4 - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{38} + ( 4 - 2 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} ) q^{40} + ( 1 - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{41} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{43} + ( 4 - \beta_{1} - \beta_{4} ) q^{44} -\beta_{3} q^{46} + ( 5 + \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} ) q^{47} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{49} + ( 4 - \beta_{1} ) q^{50} + ( 2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{52} + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{53} + ( 3 - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{55} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{56} -\beta_{3} q^{58} + ( -2 - 4 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} ) q^{59} + ( 3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{61} + ( 10 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{62} + ( -3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{64} + ( 4 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{65} + ( -\beta_{1} - 4 \beta_{4} ) q^{67} + ( 2 - 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{68} + ( -6 + 5 \beta_{1} + 3 \beta_{3} - \beta_{4} ) q^{70} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{71} + ( 7 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{73} + ( 2 - 5 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 5 \beta_{4} ) q^{74} + ( -2 + 3 \beta_{1} - 2 \beta_{4} ) q^{76} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{77} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{79} + ( -2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{80} + ( 4 - \beta_{1} + 4 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{82} + ( -5 + 7 \beta_{1} + 2 \beta_{3} ) q^{83} + ( \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{85} + ( 12 - 5 \beta_{1} - \beta_{2} - 4 \beta_{3} - 3 \beta_{4} ) q^{86} + ( -1 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{88} + ( -1 + 4 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{89} + ( 2 + \beta_{1} - 2 \beta_{3} ) q^{91} + ( -2 + \beta_{1} ) q^{92} + ( -7 + 6 \beta_{1} - 8 \beta_{4} ) q^{94} + ( 1 + 5 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} ) q^{95} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{97} + ( -10 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 2q^{2} + 8q^{4} + 3q^{5} - 5q^{7} + 3q^{8} + O(q^{10}) \) \( 5q + 2q^{2} + 8q^{4} + 3q^{5} - 5q^{7} + 3q^{8} + 9q^{10} + 8q^{11} + 5q^{13} + 2q^{14} - 10q^{16} - 2q^{17} - 9q^{19} + 12q^{20} + 10q^{22} - 5q^{23} + 2q^{25} + 3q^{26} - 14q^{28} - 5q^{29} - 6q^{31} + 8q^{32} + 3q^{34} + 15q^{35} + 10q^{37} + 10q^{38} + 26q^{40} + 11q^{41} - 9q^{43} + 16q^{44} - 2q^{46} + 13q^{47} - 6q^{49} + 18q^{50} + 12q^{52} - q^{53} + 13q^{55} + 4q^{56} - 2q^{58} - 6q^{59} + 23q^{61} + 36q^{62} - q^{64} + 20q^{65} - 10q^{67} + 10q^{68} - 16q^{70} + 11q^{71} + 31q^{73} + 18q^{74} - 8q^{76} - 3q^{77} + 8q^{79} + 8q^{80} + 16q^{82} - 7q^{83} + 6q^{85} + 36q^{86} - 3q^{88} - 3q^{89} + 8q^{91} - 8q^{92} - 39q^{94} + 11q^{95} + 3q^{97} - 38q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 5 x^{3} + 11 x^{2} - x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 5 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 5 \nu + 1 \)
\(\beta_{4}\)\(=\)\( -\nu^{4} + \nu^{3} + 6 \nu^{2} - 5 \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 5 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(5 \beta_{4} + 6 \beta_{3} + \beta_{2} + 8\)

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
−0.447481
1.70431
2.26093
0.762877
−2.28064
−2.10891 0 2.44748 −1.70032 0 −4.14780 −0.943696 0 3.58582
1.2 −1.51515 0 0.295689 1.86677 0 1.57109 2.58229 0 −2.82845
1.3 1.31874 0 −0.260930 −2.51371 0 −2.25278 −2.98157 0 −3.31493
1.4 1.79920 0 1.23712 2.60753 0 1.37040 −1.37257 0 4.69147
1.5 2.50612 0 4.28064 2.73973 0 −1.54091 5.71555 0 6.86609
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

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

Hecke kernels

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

\( T_{2}^{5} - 2 T_{2}^{4} - 7 T_{2}^{3} + 13 T_{2}^{2} + 11 T_{2} - 19 \)
\( T_{5}^{5} - 3 T_{5}^{4} - 9 T_{5}^{3} + 28 T_{5}^{2} + 17 T_{5} - 57 \)