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.

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

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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

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