Properties

Label 6003.2.a.m
Level 6003
Weight 2
Character orbit 6003.a
Self dual Yes
Analytic conductor 47.934
Analytic rank 1
Dimension 11
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + \beta_{4} q^{5} -\beta_{6} q^{7} + ( -1 - \beta_{1} - \beta_{3} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + \beta_{4} q^{5} -\beta_{6} q^{7} + ( -1 - \beta_{1} - \beta_{3} ) q^{8} + ( 1 - \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{10} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} ) q^{11} + \beta_{7} q^{13} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{10} ) q^{14} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} ) q^{16} + ( -2 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{10} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{9} - \beta_{10} ) q^{19} + ( -2 + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{10} ) q^{20} + ( -3 - \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} ) q^{22} - q^{23} + ( 1 + \beta_{2} - \beta_{3} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{25} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} ) q^{26} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{10} ) q^{28} + q^{29} + ( 4 - \beta_{3} + \beta_{5} + \beta_{6} ) q^{31} + ( -2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{32} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{34} + ( -2 - 2 \beta_{1} + \beta_{3} - \beta_{6} - \beta_{8} ) q^{35} + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{37} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 3 \beta_{8} - \beta_{9} + \beta_{10} ) q^{38} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{40} + ( \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{7} - 2 \beta_{9} ) q^{41} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{43} + ( -4 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + 4 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{44} + \beta_{1} q^{46} + ( -1 + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{47} + ( 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{49} + ( -\beta_{1} + 2 \beta_{2} + 3 \beta_{4} - \beta_{6} + \beta_{8} + \beta_{10} ) q^{50} + ( -3 - \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{52} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{6} + \beta_{9} + \beta_{10} ) q^{53} + ( -1 - \beta_{1} - 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{55} + ( -5 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{56} -\beta_{1} q^{58} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{59} + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{61} + ( 3 - 4 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + \beta_{10} ) q^{62} + ( 3 + 4 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{64} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} ) q^{65} + ( -4 - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{9} ) q^{67} + ( -6 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{10} ) q^{68} + ( 3 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{9} - \beta_{10} ) q^{70} + ( -5 - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{71} + ( \beta_{1} - \beta_{2} + 4 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{73} + ( -3 + 4 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - \beta_{10} ) q^{74} + ( -2 + 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} + 8 \beta_{6} + \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} ) q^{76} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} ) q^{77} + ( 4 + 5 \beta_{1} - \beta_{2} + \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{79} + ( -7 - 2 \beta_{1} - \beta_{2} + \beta_{4} - 3 \beta_{5} - 5 \beta_{6} - 3 \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{80} + ( -2 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{6} + 4 \beta_{8} + \beta_{9} + \beta_{10} ) q^{82} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{83} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} ) q^{85} + ( -5 - 2 \beta_{2} - 2 \beta_{3} + \beta_{7} ) q^{86} + ( -5 - 2 \beta_{1} - \beta_{3} + 5 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} - \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} ) q^{88} + ( -4 + 3 \beta_{1} + \beta_{3} + \beta_{6} + 2 \beta_{9} ) q^{89} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{91} + ( -2 - \beta_{2} ) q^{92} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} + 6 \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} ) q^{94} + ( -3 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} - 3 \beta_{8} - \beta_{9} - \beta_{10} ) q^{95} + ( -4 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} ) q^{97} + ( -9 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} + \beta_{6} - 3 \beta_{7} - \beta_{9} - \beta_{10} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 2q^{2} + 18q^{4} - 2q^{5} + 3q^{7} - 18q^{8} + O(q^{10}) \) \( 11q - 2q^{2} + 18q^{4} - 2q^{5} + 3q^{7} - 18q^{8} + 14q^{10} - 11q^{11} - 5q^{13} - 17q^{14} + 20q^{16} - 15q^{17} - 6q^{19} - 21q^{20} - 10q^{22} - 11q^{23} + 3q^{25} + 5q^{26} + 7q^{28} + 11q^{29} + 35q^{31} - 28q^{32} + 28q^{34} - 15q^{35} - 28q^{37} + 2q^{38} - q^{40} - 10q^{41} - 6q^{43} - 18q^{44} + 2q^{46} - 15q^{47} + 22q^{49} - 15q^{50} - 36q^{52} + 7q^{53} - 12q^{55} - 56q^{56} - 2q^{58} + 20q^{59} - 20q^{61} + 11q^{62} + 36q^{64} - 11q^{65} - 39q^{67} - 35q^{68} + 38q^{70} - 49q^{71} - 3q^{73} - 37q^{74} - 18q^{76} - 25q^{77} + 41q^{79} - 51q^{80} - 19q^{82} - 13q^{83} - 62q^{86} - 40q^{88} - 34q^{89} + 2q^{91} - 18q^{92} - 14q^{94} - 25q^{95} - 11q^{97} - 53q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 2 x^{10} - 18 x^{9} + 30 x^{8} + 124 x^{7} - 152 x^{6} - 408 x^{5} + 285 x^{4} + 634 x^{3} - 93 x^{2} - 369 x - 108\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu - 1 \)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{10} + 5 \nu^{9} + 9 \nu^{8} - 63 \nu^{7} - 13 \nu^{6} + 251 \nu^{5} - 27 \nu^{4} - 360 \nu^{3} - 4 \nu^{2} + 165 \nu + 48 \)\()/6\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{10} + 7 \nu^{9} + 21 \nu^{8} - 87 \nu^{7} - 59 \nu^{6} + 343 \nu^{5} + 63 \nu^{4} - 498 \nu^{3} - 170 \nu^{2} + 261 \nu + 153 \)\()/9\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{10} - \nu^{9} + 24 \nu^{8} + 6 \nu^{7} - 187 \nu^{6} + 23 \nu^{5} + 567 \nu^{4} - 150 \nu^{3} - 598 \nu^{2} + 117 \nu + 162 \)\()/9\)
\(\beta_{7}\)\(=\)\( \nu^{8} - 2 \nu^{7} - 12 \nu^{6} + 22 \nu^{5} + 44 \nu^{4} - 68 \nu^{3} - 57 \nu^{2} + 52 \nu + 31 \)
\(\beta_{8}\)\(=\)\((\)\( -4 \nu^{10} + 14 \nu^{9} + 42 \nu^{8} - 174 \nu^{7} - 109 \nu^{6} + 686 \nu^{5} + 18 \nu^{4} - 996 \nu^{3} - 7 \nu^{2} + 504 \nu + 135 \)\()/9\)
\(\beta_{9}\)\(=\)\((\)\( 2 \nu^{10} - 7 \nu^{9} - 30 \nu^{8} + 105 \nu^{7} + 167 \nu^{6} - 541 \nu^{5} - 450 \nu^{4} + 1101 \nu^{3} + 620 \nu^{2} - 693 \nu - 369 \)\()/9\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{10} + 5 \nu^{9} + 15 \nu^{8} - 75 \nu^{7} - 91 \nu^{6} + 389 \nu^{5} + 303 \nu^{4} - 822 \nu^{3} - 538 \nu^{2} + 591 \nu + 354 \)\()/6\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{9} + \beta_{7} + \beta_{5} + \beta_{3} + 7 \beta_{2} + \beta_{1} + 22\)
\(\nu^{5}\)\(=\)\(\beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} + 10 \beta_{3} + 2 \beta_{2} + 29 \beta_{1} + 10\)
\(\nu^{6}\)\(=\)\(12 \beta_{9} + \beta_{8} + 12 \beta_{7} + 10 \beta_{5} + 12 \beta_{3} + 47 \beta_{2} + 14 \beta_{1} + 135\)
\(\nu^{7}\)\(=\)\(12 \beta_{10} + 16 \beta_{9} + 12 \beta_{8} + 3 \beta_{7} + \beta_{6} - 10 \beta_{5} - 10 \beta_{4} + 82 \beta_{3} + 25 \beta_{2} + 184 \beta_{1} + 89\)
\(\nu^{8}\)\(=\)\(2 \beta_{10} + 110 \beta_{9} + 14 \beta_{8} + 107 \beta_{7} + 2 \beta_{6} + 78 \beta_{5} + 2 \beta_{4} + 112 \beta_{3} + 319 \beta_{2} + 142 \beta_{1} + 875\)
\(\nu^{9}\)\(=\)\(105 \beta_{10} + 172 \beta_{9} + 106 \beta_{8} + 53 \beta_{7} + 15 \beta_{6} - 73 \beta_{5} - 71 \beta_{4} + 629 \beta_{3} + 240 \beta_{2} + 1229 \beta_{1} + 752\)
\(\nu^{10}\)\(=\)\(38 \beta_{10} + 910 \beta_{9} + 138 \beta_{8} + 856 \beta_{7} + 30 \beta_{6} + 559 \beta_{5} + 36 \beta_{4} + 954 \beta_{3} + 2194 \beta_{2} + 1266 \beta_{1} + 5861\)

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
2.72479
2.70316
2.24285
1.76484
1.17662
−0.467085
−0.661934
−1.05971
−1.94502
−1.96728
−2.51124
−2.72479 0 5.42450 1.02492 0 4.32726 −9.33107 0 −2.79269
1.2 −2.70316 0 5.30708 −2.79988 0 0.880738 −8.93955 0 7.56852
1.3 −2.24285 0 3.03039 −3.33222 0 0.336371 −2.31101 0 7.47368
1.4 −1.76484 0 1.11465 2.54815 0 −4.97592 1.56250 0 −4.49707
1.5 −1.17662 0 −0.615555 −2.85352 0 3.62966 3.07753 0 3.35752
1.6 0.467085 0 −1.78183 0.0105419 0 −1.85912 −1.76644 0 0.00492395
1.7 0.661934 0 −1.56184 1.16115 0 4.80000 −2.35770 0 0.768607
1.8 1.05971 0 −0.877023 1.30384 0 0.720797 −3.04880 0 1.38169
1.9 1.94502 0 1.78310 −0.890641 0 −3.69089 −0.421884 0 −1.73231
1.10 1.96728 0 1.87020 3.90206 0 −0.839519 −0.255353 0 7.67646
1.11 2.51124 0 4.30634 −2.07440 0 −0.329384 5.79178 0 −5.20933
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\(T_{2}^{11} + \cdots\)
\(T_{5}^{11} + \cdots\)