Properties

Label 6003.2.a.l
Level 6003
Weight 2
Character orbit 6003.a
Self dual Yes
Analytic conductor 47.934
Analytic rank 0
Dimension 10
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{7} ) q^{5} + ( \beta_{2} - \beta_{3} + \beta_{7} - \beta_{9} ) q^{7} + ( 1 + 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{7} ) q^{5} + ( \beta_{2} - \beta_{3} + \beta_{7} - \beta_{9} ) q^{7} + ( 1 + 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} ) q^{8} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{9} ) q^{10} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{9} ) q^{11} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{9} ) q^{13} + ( 1 + 2 \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{14} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{16} + ( 2 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{17} + ( \beta_{1} + \beta_{4} - \beta_{8} + 2 \beta_{9} ) q^{19} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{20} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{22} - q^{23} + ( 1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{9} ) q^{25} + ( 3 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{26} + ( 2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{9} ) q^{28} - q^{29} + ( -2 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{31} + ( 3 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{32} + ( 3 \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} - 2 \beta_{6} - \beta_{7} + 3 \beta_{8} ) q^{34} + ( 1 + \beta_{2} - 2 \beta_{3} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{35} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{37} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{7} + \beta_{8} ) q^{38} + ( -2 \beta_{1} - \beta_{2} + 4 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{40} + ( 2 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{8} ) q^{41} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{43} + ( 3 - \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{44} -\beta_{1} q^{46} + ( 1 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{47} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{49} + ( -2 \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{50} + ( -2 + 2 \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{52} + ( 5 + \beta_{2} - 2 \beta_{3} + \beta_{5} + 3 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} ) q^{53} + ( -1 - 4 \beta_{1} + \beta_{2} + \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{55} + ( \beta_{1} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{56} -\beta_{1} q^{58} + ( 1 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{59} + ( -2 \beta_{1} - \beta_{4} + \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{61} + ( 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} + 4 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{62} + ( 3 + 3 \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - \beta_{7} + 3 \beta_{8} + 2 \beta_{9} ) q^{64} + ( -2 + \beta_{1} - 3 \beta_{2} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{9} ) q^{65} + ( 1 - 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{67} + ( 6 + 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} + \beta_{7} + \beta_{9} ) q^{68} + ( 3 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} + \beta_{8} - \beta_{9} ) q^{70} + ( 1 + 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} + 3 \beta_{8} ) q^{71} + ( -4 - \beta_{2} + \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{73} + ( -1 - 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{74} + ( 2 + \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{76} + ( 7 + \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{77} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + \beta_{9} ) q^{79} + ( -4 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{80} + ( 3 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{82} + ( 3 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} ) q^{83} + ( 2 + \beta_{1} - 3 \beta_{2} + 4 \beta_{4} + 2 \beta_{6} + \beta_{7} - 4 \beta_{8} + 2 \beta_{9} ) q^{85} + ( 1 + 4 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{86} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{88} + ( 1 - 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + \beta_{6} - \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{89} + ( \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{91} + ( -1 - \beta_{2} ) q^{92} + ( 5 + \beta_{1} - \beta_{2} - 3 \beta_{4} + 2 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{94} + ( 2 - 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} ) q^{95} + ( -3 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{97} + ( 7 - 3 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + 8 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - \beta_{9} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 3q^{2} + 9q^{4} + 10q^{5} + q^{7} + 9q^{8} + O(q^{10}) \) \( 10q + 3q^{2} + 9q^{4} + 10q^{5} + q^{7} + 9q^{8} - 6q^{10} - 13q^{13} + 12q^{14} - 5q^{16} + 22q^{17} - 2q^{19} - 3q^{20} + 3q^{22} - 10q^{23} + 10q^{25} + 25q^{26} + 19q^{28} - 10q^{29} - 22q^{31} + 31q^{32} + 13q^{34} + 15q^{35} - 9q^{37} + 10q^{38} - 6q^{40} + 25q^{41} + 3q^{43} + 27q^{44} - 3q^{46} + 17q^{47} + 17q^{49} - 2q^{50} - 18q^{52} + 43q^{53} - 11q^{55} + 7q^{56} - 3q^{58} + 7q^{59} - 6q^{61} - 3q^{62} + 33q^{64} - 11q^{65} + 11q^{67} + 51q^{68} + 34q^{70} + 17q^{71} - 44q^{73} - 9q^{74} + 24q^{76} + 71q^{77} + 5q^{79} - 38q^{80} + 33q^{82} + 32q^{83} + 16q^{85} + 9q^{86} + 18q^{88} + 10q^{89} - 3q^{91} - 9q^{92} + 47q^{94} + 8q^{95} + 6q^{97} + 73q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 3 x^{9} - 10 x^{8} + 32 x^{7} + 32 x^{6} - 118 x^{5} - 29 x^{4} + 182 x^{3} - 28 x^{2} - 101 x + 43\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{8} - 3 \nu^{7} - 8 \nu^{6} + 26 \nu^{5} + 16 \nu^{4} - 66 \nu^{3} + 3 \nu^{2} + 50 \nu - 22 \)
\(\beta_{4}\)\(=\)\( \nu^{8} - 4 \nu^{7} - 6 \nu^{6} + 36 \nu^{5} - \nu^{4} - 98 \nu^{3} + 45 \nu^{2} + 83 \nu - 53 \)
\(\beta_{5}\)\(=\)\( -\nu^{9} + 4 \nu^{8} + 5 \nu^{7} - 34 \nu^{6} + 10 \nu^{5} + 82 \nu^{4} - 69 \nu^{3} - 47 \nu^{2} + 72 \nu - 22 \)
\(\beta_{6}\)\(=\)\( 2 \nu^{8} - 7 \nu^{7} - 14 \nu^{6} + 63 \nu^{5} + 14 \nu^{4} - 172 \nu^{3} + 54 \nu^{2} + 146 \nu - 83 \)
\(\beta_{7}\)\(=\)\( 3 \nu^{9} - 10 \nu^{8} - 23 \nu^{7} + 91 \nu^{6} + 39 \nu^{5} - 253 \nu^{4} + 32 \nu^{3} + 226 \nu^{2} - 84 \nu - 17 \)
\(\beta_{8}\)\(=\)\( \nu^{9} - \nu^{8} - 16 \nu^{7} + 14 \nu^{6} + 89 \nu^{5} - 69 \nu^{4} - 200 \nu^{3} + 144 \nu^{2} + 153 \nu - 109 \)
\(\beta_{9}\)\(=\)\( 2 \nu^{9} - 9 \nu^{8} - 7 \nu^{7} + 77 \nu^{6} - 49 \nu^{5} - 186 \nu^{4} + 225 \nu^{3} + 94 \nu^{2} - 226 \nu + 78 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + \beta_{7} + \beta_{5} - 2 \beta_{4} - \beta_{3} + 8 \beta_{2} + 2 \beta_{1} + 13\)
\(\nu^{5}\)\(=\)\(-\beta_{9} + 8 \beta_{8} + \beta_{7} - 7 \beta_{6} + 9 \beta_{5} - 11 \beta_{4} - 2 \beta_{3} + 18 \beta_{2} + 21 \beta_{1} + 11\)
\(\nu^{6}\)\(=\)\(-8 \beta_{9} + 3 \beta_{8} + 9 \beta_{7} + 14 \beta_{5} - 24 \beta_{4} - 11 \beta_{3} + 57 \beta_{2} + 23 \beta_{1} + 69\)
\(\nu^{7}\)\(=\)\(-9 \beta_{9} + 54 \beta_{8} + 11 \beta_{7} - 38 \beta_{6} + 69 \beta_{5} - 93 \beta_{4} - 24 \beta_{3} + 136 \beta_{2} + 127 \beta_{1} + 90\)
\(\nu^{8}\)\(=\)\(-49 \beta_{9} + 44 \beta_{8} + 63 \beta_{7} + 2 \beta_{6} + 135 \beta_{5} - 219 \beta_{4} - 91 \beta_{3} + 397 \beta_{2} + 201 \beta_{1} + 407\)
\(\nu^{9}\)\(=\)\(-61 \beta_{9} + 355 \beta_{8} + 93 \beta_{7} - 183 \beta_{6} + 511 \beta_{5} - 730 \beta_{4} - 212 \beta_{3} + 981 \beta_{2} + 827 \beta_{1} + 676\)

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
−2.17460
−1.57663
−1.35371
−1.31926
0.685481
0.788514
1.21378
1.68137
2.37954
2.67549
−2.17460 0 2.72887 1.26068 0 1.97284 −1.58501 0 −2.74147
1.2 −1.57663 0 0.485749 1.88241 0 −0.868755 2.38741 0 −2.96785
1.3 −1.35371 0 −0.167476 −1.38626 0 −1.97610 2.93413 0 1.87659
1.4 −1.31926 0 −0.259562 4.11119 0 −2.74867 2.98094 0 −5.42371
1.5 0.685481 0 −1.53012 0.380940 0 −0.405330 −2.41983 0 0.261127
1.6 0.788514 0 −1.37825 4.42591 0 4.56405 −2.66379 0 3.48989
1.7 1.21378 0 −0.526738 −2.40259 0 −0.534912 −3.06690 0 −2.91621
1.8 1.68137 0 0.827018 1.60084 0 −4.81497 −1.97222 0 2.69161
1.9 2.37954 0 3.66223 2.05928 0 5.08987 3.95536 0 4.90014
1.10 2.67549 0 5.15827 −1.93239 0 0.721979 8.44992 0 −5.17011
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\(T_{2}^{10} - \cdots\)
\(T_{5}^{10} - \cdots\)