Properties

Label 6027.2.a.bg
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 0
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6027.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1258372982\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{4} ) q^{5} -\beta_{1} q^{6} + ( \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{11} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{4} ) q^{5} -\beta_{1} q^{6} + ( \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{11} ) q^{8} + q^{9} + ( 1 - \beta_{1} + \beta_{2} + \beta_{7} - \beta_{8} ) q^{10} + ( -\beta_{4} + \beta_{6} + \beta_{10} ) q^{11} + ( 1 + \beta_{2} ) q^{12} + ( 2 + \beta_{3} + \beta_{8} ) q^{13} + ( 1 + \beta_{4} ) q^{15} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} ) q^{16} + ( \beta_{1} + \beta_{4} - \beta_{5} - \beta_{9} ) q^{17} -\beta_{1} q^{18} + ( -\beta_{3} + \beta_{4} + \beta_{11} ) q^{19} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{20} + ( -\beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{11} ) q^{22} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{8} - \beta_{10} ) q^{23} + ( \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{11} ) q^{24} + ( 2 - 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{25} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{26} + q^{27} + ( 2 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{29} + ( 1 - \beta_{1} + \beta_{2} + \beta_{7} - \beta_{8} ) q^{30} + ( 2 \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{31} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} ) q^{32} + ( -\beta_{4} + \beta_{6} + \beta_{10} ) q^{33} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{10} ) q^{34} + ( 1 + \beta_{2} ) q^{36} + ( -2 + 3 \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{11} ) q^{37} + ( -2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{38} + ( 2 + \beta_{3} + \beta_{8} ) q^{39} + ( 3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} - 3 \beta_{7} + \beta_{10} + 2 \beta_{11} ) q^{40} - q^{41} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{43} + ( 2 - \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{11} ) q^{44} + ( 1 + \beta_{4} ) q^{45} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - 3 \beta_{8} - \beta_{10} ) q^{46} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{47} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} ) q^{48} + ( 5 - 4 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{50} + ( \beta_{1} + \beta_{4} - \beta_{5} - \beta_{9} ) q^{51} + ( 3 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{52} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{53} -\beta_{1} q^{54} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{10} ) q^{55} + ( -\beta_{3} + \beta_{4} + \beta_{11} ) q^{57} + ( -1 - \beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{58} + ( 4 - \beta_{1} + \beta_{2} - 3 \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{59} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{60} + ( 3 + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{61} + ( -3 - 2 \beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{62} + ( 2 - \beta_{1} - \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{8} - 2 \beta_{11} ) q^{64} + ( 1 + 3 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{65} + ( -\beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{11} ) q^{66} + ( -3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} + \beta_{7} + 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{67} + ( 4 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{68} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{8} - \beta_{10} ) q^{69} + ( 3 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{71} + ( \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{11} ) q^{72} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{73} + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} ) q^{74} + ( 2 - 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{75} + ( -3 - 2 \beta_{1} - \beta_{2} - 4 \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} + 2 \beta_{10} ) q^{76} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{78} + ( \beta_{1} + \beta_{3} + \beta_{7} - 2 \beta_{8} - 3 \beta_{10} ) q^{79} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{80} + q^{81} + \beta_{1} q^{82} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{11} ) q^{83} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} + 3 \beta_{10} + 2 \beta_{11} ) q^{85} + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{11} ) q^{86} + ( 2 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{87} + ( -4 - \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 5 \beta_{5} - \beta_{7} - 4 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{88} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + 3 \beta_{9} + 2 \beta_{10} ) q^{89} + ( 1 - \beta_{1} + \beta_{2} + \beta_{7} - \beta_{8} ) q^{90} + ( 5 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} + 4 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{92} + ( 2 \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{93} + ( 3 + 3 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - \beta_{7} - 2 \beta_{10} + \beta_{11} ) q^{94} + ( 2 - 3 \beta_{1} - 4 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{95} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} ) q^{96} + ( 3 - 4 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{97} + ( -\beta_{4} + \beta_{6} + \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 2q^{2} + 12q^{3} + 10q^{4} + 12q^{5} - 2q^{6} + 12q^{9} + O(q^{10}) \) \( 12q - 2q^{2} + 12q^{3} + 10q^{4} + 12q^{5} - 2q^{6} + 12q^{9} + 11q^{10} + 10q^{11} + 10q^{12} + 15q^{13} + 12q^{15} + 14q^{16} + 8q^{17} - 2q^{18} + 2q^{19} + 16q^{20} - 7q^{22} + 5q^{23} + 20q^{25} + 12q^{27} + 20q^{29} + 11q^{30} + 10q^{31} + 3q^{32} + 10q^{33} - 23q^{34} + 10q^{36} - 17q^{37} + 6q^{38} + 15q^{39} + 39q^{40} - 12q^{41} + 12q^{43} + 20q^{44} + 12q^{45} - 36q^{46} + 34q^{47} + 14q^{48} + 59q^{50} + 8q^{51} + 26q^{52} + 6q^{53} - 2q^{54} - q^{55} + 2q^{57} - 11q^{58} + 27q^{59} + 16q^{60} + 22q^{61} - 45q^{62} + 26q^{64} - 7q^{66} - 26q^{67} + 33q^{68} + 5q^{69} + 50q^{71} + 21q^{73} - 35q^{74} + 20q^{75} - 24q^{76} - 10q^{79} + 22q^{80} + 12q^{81} + 2q^{82} + 8q^{83} + 8q^{85} - 17q^{86} + 20q^{87} - 46q^{88} + 11q^{89} + 11q^{90} + 63q^{92} + 10q^{93} + 10q^{94} + 35q^{95} + 3q^{96} + 32q^{97} + 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} - 15 x^{10} + 30 x^{9} + 74 x^{8} - 149 x^{7} - 140 x^{6} + 278 x^{5} + 126 x^{4} - 211 x^{3} - 64 x^{2} + 53 x + 18\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 8 \nu^{4} + 14 \nu^{3} + 12 \nu^{2} - 13 \nu - 6 \)
\(\beta_{4}\)\(=\)\( \nu^{9} - 2 \nu^{8} - 13 \nu^{7} + 27 \nu^{6} + 47 \nu^{5} - 109 \nu^{4} - 30 \nu^{3} + 115 \nu^{2} - 9 \nu - 25 \)
\(\beta_{5}\)\(=\)\( \nu^{11} - 2 \nu^{10} - 14 \nu^{9} + 28 \nu^{8} + 60 \nu^{7} - 121 \nu^{6} - 80 \nu^{5} + 158 \nu^{4} + 45 \nu^{3} - 61 \nu^{2} - 12 \nu + 3 \)
\(\beta_{6}\)\(=\)\( -\nu^{10} + 2 \nu^{9} + 14 \nu^{8} - 28 \nu^{7} - 60 \nu^{6} + 121 \nu^{5} + 81 \nu^{4} - 158 \nu^{3} - 53 \nu^{2} + 60 \nu + 23 \)
\(\beta_{7}\)\(=\)\( \nu^{11} - 2 \nu^{10} - 14 \nu^{9} + 28 \nu^{8} + 60 \nu^{7} - 122 \nu^{6} - 79 \nu^{5} + 167 \nu^{4} + 39 \nu^{3} - 79 \nu^{2} - 10 \nu + 11 \)
\(\beta_{8}\)\(=\)\( \nu^{11} - \nu^{10} - 16 \nu^{9} + 15 \nu^{8} + 87 \nu^{7} - 75 \nu^{6} - 188 \nu^{5} + 137 \nu^{4} + 154 \nu^{3} - 87 \nu^{2} - 35 \nu + 9 \)
\(\beta_{9}\)\(=\)\( \nu^{11} - 4 \nu^{10} - 11 \nu^{9} + 58 \nu^{8} + 18 \nu^{7} - 269 \nu^{6} + 104 \nu^{5} + 435 \nu^{4} - 207 \nu^{3} - 281 \nu^{2} + 87 \nu + 62 \)
\(\beta_{10}\)\(=\)\( -2 \nu^{11} + 4 \nu^{10} + 29 \nu^{9} - 59 \nu^{8} - 132 \nu^{7} + 283 \nu^{6} + 194 \nu^{5} - 484 \nu^{4} - 71 \nu^{3} + 309 \nu^{2} - 21 \nu - 54 \)
\(\beta_{11}\)\(=\)\( 3 \nu^{11} - 5 \nu^{10} - 45 \nu^{9} + 73 \nu^{8} + 220 \nu^{7} - 345 \nu^{6} - 394 \nu^{5} + 571 \nu^{4} + 267 \nu^{3} - 342 \nu^{2} - 44 \nu + 48 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{10} + \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} + 7 \beta_{2} - \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(-8 \beta_{11} + \beta_{10} + 9 \beta_{8} + 7 \beta_{7} + \beta_{6} + 10 \beta_{5} - 9 \beta_{4} - \beta_{3} + \beta_{2} + 20 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(-2 \beta_{11} + 10 \beta_{10} + 12 \beta_{8} + 10 \beta_{6} + 14 \beta_{5} - 12 \beta_{4} - \beta_{3} + 46 \beta_{2} - 11 \beta_{1} + 88\)
\(\nu^{7}\)\(=\)\(-56 \beta_{11} + 15 \beta_{10} + \beta_{9} + 71 \beta_{8} + 43 \beta_{7} + 13 \beta_{6} + 83 \beta_{5} - 70 \beta_{4} - 12 \beta_{3} + 14 \beta_{2} + 109 \beta_{1} - 4\)
\(\nu^{8}\)\(=\)\(-29 \beta_{11} + 82 \beta_{10} + \beta_{9} + 112 \beta_{8} + \beta_{7} + 81 \beta_{6} + 137 \beta_{5} - 110 \beta_{4} - 13 \beta_{3} + 304 \beta_{2} - 86 \beta_{1} + 549\)
\(\nu^{9}\)\(=\)\(-386 \beta_{11} + 151 \beta_{10} + 15 \beta_{9} + 539 \beta_{8} + 262 \beta_{7} + 123 \beta_{6} + 644 \beta_{5} - 521 \beta_{4} - 108 \beta_{3} + 149 \beta_{2} + 622 \beta_{1} + 32\)
\(\nu^{10}\)\(=\)\(-300 \beta_{11} + 632 \beta_{10} + 16 \beta_{9} + 950 \beta_{8} + 23 \beta_{7} + 617 \beta_{6} + 1175 \beta_{5} - 914 \beta_{4} - 123 \beta_{3} + 2037 \beta_{2} - 585 \beta_{1} + 3540\)
\(\nu^{11}\)\(=\)\(-2669 \beta_{11} + 1314 \beta_{10} + 154 \beta_{9} + 4019 \beta_{8} + 1621 \beta_{7} + 1040 \beta_{6} + 4842 \beta_{5} - 3811 \beta_{4} - 875 \beta_{3} + 1409 \beta_{2} + 3665 \beta_{1} + 774\)

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
2.71780
2.11101
1.94261
1.22157
1.14271
0.837060
−0.420502
−0.467496
−1.05133
−1.09852
−2.44902
−2.48589
−2.71780 1.00000 5.38643 0.137890 −2.71780 0 −9.20364 1.00000 −0.374758
1.2 −2.11101 1.00000 2.45637 −2.44925 −2.11101 0 −0.963413 1.00000 5.17041
1.3 −1.94261 1.00000 1.77372 2.73840 −1.94261 0 0.439573 1.00000 −5.31964
1.4 −1.22157 1.00000 −0.507765 0.145411 −1.22157 0 3.06341 1.00000 −0.177630
1.5 −1.14271 1.00000 −0.694209 1.38626 −1.14271 0 3.07871 1.00000 −1.58409
1.6 −0.837060 1.00000 −1.29933 2.51468 −0.837060 0 2.76174 1.00000 −2.10494
1.7 0.420502 1.00000 −1.82318 −1.49913 0.420502 0 −1.60765 1.00000 −0.630387
1.8 0.467496 1.00000 −1.78145 2.48991 0.467496 0 −1.76781 1.00000 1.16402
1.9 1.05133 1.00000 −0.894704 4.26226 1.05133 0 −3.04329 1.00000 4.48105
1.10 1.09852 1.00000 −0.793257 −3.51189 1.09852 0 −3.06844 1.00000 −3.85788
1.11 2.44902 1.00000 3.99771 4.01828 2.44902 0 4.89242 1.00000 9.84085
1.12 2.48589 1.00000 4.17966 1.76717 2.48589 0 5.41840 1.00000 4.39300
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(41\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2}^{12} + \cdots\)
\(T_{5}^{12} - \cdots\)
\(T_{13}^{12} - \cdots\)