Properties

Label 6027.2.a.bd
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 0
Dimension 10
CM No
Inner twists 1

Related objects

Downloads

Learn more about

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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + ( -1 - \beta_{4} ) q^{5} -\beta_{1} q^{6} + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} + \beta_{8} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + ( -1 - \beta_{4} ) q^{5} -\beta_{1} q^{6} + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} + \beta_{8} ) q^{8} + q^{9} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{10} -\beta_{6} q^{11} + ( -2 - \beta_{2} ) q^{12} + ( -\beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{13} + ( 1 + \beta_{4} ) q^{15} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{16} + ( -1 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{17} + \beta_{1} q^{18} + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{9} ) q^{19} + ( -2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{20} + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{6} - \beta_{7} - \beta_{9} ) q^{22} + ( -\beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{23} + ( -1 - \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} ) q^{24} + ( 1 + \beta_{1} + \beta_{4} + \beta_{6} + \beta_{8} ) q^{25} + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{8} + \beta_{9} ) q^{26} - q^{27} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{9} ) q^{29} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{30} + ( -1 + \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{31} + ( 3 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{32} + \beta_{6} q^{33} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( 2 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{37} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{38} + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{39} + ( -3 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{40} - q^{41} + ( 1 - \beta_{1} + 2 \beta_{5} - 2 \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{43} + ( -1 - \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{44} + ( -1 - \beta_{4} ) q^{45} + ( -1 - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{46} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{8} - \beta_{9} ) q^{47} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{48} + ( 4 + \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} ) q^{50} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{51} + ( 4 - 2 \beta_{1} - \beta_{3} - \beta_{7} - \beta_{8} ) q^{52} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} ) q^{53} -\beta_{1} q^{54} + ( 1 + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{55} + ( 1 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{9} ) q^{57} + ( -2 + \beta_{1} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{58} + ( -\beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{9} ) q^{59} + ( 2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{60} + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} ) q^{61} + ( 1 + \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{62} + ( 8 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{64} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{9} ) q^{65} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{6} + \beta_{7} + \beta_{9} ) q^{66} + ( 1 + 2 \beta_{1} + 4 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{67} + ( -1 + 3 \beta_{1} + \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{68} + ( \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{69} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{71} + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} + \beta_{8} ) q^{72} + ( 4 + 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{8} + 3 \beta_{9} ) q^{73} + ( 4 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} ) q^{74} + ( -1 - \beta_{1} - \beta_{4} - \beta_{6} - \beta_{8} ) q^{75} + ( 3 + 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} + \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + \beta_{9} ) q^{76} + ( 2 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{8} - \beta_{9} ) q^{78} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{79} + ( -5 - 5 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 8 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} - 4 \beta_{9} ) q^{80} + q^{81} -\beta_{1} q^{82} + ( -5 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} ) q^{83} + ( 4 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{85} + ( -5 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + \beta_{8} - 3 \beta_{9} ) q^{86} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{9} ) q^{87} + ( 2 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{9} ) q^{88} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{9} ) q^{89} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{90} + ( 3 - 4 \beta_{1} - \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 5 \beta_{6} - 4 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{92} + ( 1 - \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{93} + ( 6 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{94} + ( -2 - \beta_{1} - 3 \beta_{2} - \beta_{3} - 3 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{95} + ( -3 - 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{96} + ( -2 + 2 \beta_{2} + 3 \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{97} -\beta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 4q^{2} - 10q^{3} + 18q^{4} - 6q^{5} - 4q^{6} + 12q^{8} + 10q^{9} + O(q^{10}) \) \( 10q + 4q^{2} - 10q^{3} + 18q^{4} - 6q^{5} - 4q^{6} + 12q^{8} + 10q^{9} - 2q^{10} - 2q^{11} - 18q^{12} + 6q^{15} + 14q^{16} - 8q^{17} + 4q^{18} - 6q^{19} - 20q^{20} + 2q^{22} - 12q^{24} + 10q^{25} - 16q^{26} - 10q^{27} + 16q^{29} + 2q^{30} - 2q^{31} + 38q^{32} + 2q^{33} + 4q^{34} + 18q^{36} + 24q^{37} + 26q^{38} - 40q^{40} - 10q^{41} + 8q^{43} - 8q^{44} - 6q^{45} + 4q^{46} + 8q^{47} - 14q^{48} + 44q^{50} + 8q^{51} + 30q^{52} + 24q^{53} - 4q^{54} + 6q^{57} - 14q^{58} - 6q^{59} + 20q^{60} + 14q^{61} + 2q^{62} + 86q^{64} + 28q^{65} - 2q^{66} + 26q^{67} + 6q^{68} + 14q^{71} + 12q^{72} + 36q^{73} + 18q^{74} - 10q^{75} + 32q^{76} + 16q^{78} + 20q^{79} - 70q^{80} + 10q^{81} - 4q^{82} - 40q^{83} + 24q^{85} - 36q^{86} - 16q^{87} + 14q^{88} - 2q^{89} - 2q^{90} + 8q^{92} + 2q^{93} + 54q^{94} - 24q^{95} - 38q^{96} - 16q^{97} - 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 4 x^{9} - 11 x^{8} + 56 x^{7} + 26 x^{6} - 266 x^{5} + 52 x^{4} + 526 x^{3} - 255 x^{2} - 372 x + 239\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{8} - 4 \nu^{7} - 8 \nu^{6} + 46 \nu^{5} - 2 \nu^{4} - 152 \nu^{3} + 90 \nu^{2} + 154 \nu - 121 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{9} - 4 \nu^{8} - 8 \nu^{7} + 46 \nu^{6} - 4 \nu^{5} - 148 \nu^{4} + 110 \nu^{3} + 120 \nu^{2} - 163 \nu + 52 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{9} + 5 \nu^{8} + 4 \nu^{7} - 54 \nu^{6} + 50 \nu^{5} + 148 \nu^{4} - 262 \nu^{3} - 46 \nu^{2} + 315 \nu - 153 \)\()/2\)
\(\beta_{6}\)\(=\)\( -\nu^{8} + 4 \nu^{7} + 9 \nu^{6} - 47 \nu^{5} - 9 \nu^{4} + 160 \nu^{3} - 59 \nu^{2} - 165 \nu + 102 \)
\(\beta_{7}\)\(=\)\( \nu^{8} - 4 \nu^{7} - 9 \nu^{6} + 48 \nu^{5} + 8 \nu^{4} - 169 \nu^{3} + 67 \nu^{2} + 180 \nu - 114 \)
\(\beta_{8}\)\(=\)\( -\nu^{8} + 4 \nu^{7} + 9 \nu^{6} - 48 \nu^{5} - 8 \nu^{4} + 170 \nu^{3} - 68 \nu^{2} - 185 \nu + 117 \)
\(\beta_{9}\)\(=\)\((\)\( -2 \nu^{9} + 9 \nu^{8} + 14 \nu^{7} - 104 \nu^{6} + 28 \nu^{5} + 342 \nu^{4} - 268 \nu^{3} - 318 \nu^{2} + 348 \nu - 57 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{8} + \beta_{7} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{5} + \beta_{4} - \beta_{3} + 8 \beta_{2} + \beta_{1} + 22\)
\(\nu^{5}\)\(=\)\(9 \beta_{8} + 10 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 9 \beta_{2} + 31 \beta_{1} + 11\)
\(\nu^{6}\)\(=\)\(\beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 12 \beta_{5} + 12 \beta_{4} - 10 \beta_{3} + 58 \beta_{2} + 13 \beta_{1} + 140\)
\(\nu^{7}\)\(=\)\(\beta_{9} + 67 \beta_{8} + 82 \beta_{7} + 17 \beta_{6} + 13 \beta_{5} + 15 \beta_{4} - 10 \beta_{3} + 73 \beta_{2} + 211 \beta_{1} + 95\)
\(\nu^{8}\)\(=\)\(4 \beta_{9} + 14 \beta_{8} + 36 \beta_{7} + 38 \beta_{6} + 104 \beta_{5} + 112 \beta_{4} - 74 \beta_{3} + 420 \beta_{2} + 130 \beta_{1} + 951\)
\(\nu^{9}\)\(=\)\(24 \beta_{9} + 472 \beta_{8} + 638 \beta_{7} + 200 \beta_{6} + 120 \beta_{5} + 170 \beta_{4} - 68 \beta_{3} + 586 \beta_{2} + 1495 \beta_{1} + 782\)

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
−2.58399
−1.69562
−1.49626
−1.44855
0.775610
1.32624
1.55584
2.13183
2.62834
2.80657
−2.58399 −1.00000 4.67700 −0.798812 2.58399 0 −6.91733 1.00000 2.06412
1.2 −1.69562 −1.00000 0.875138 0.871357 1.69562 0 1.90734 1.00000 −1.47749
1.3 −1.49626 −1.00000 0.238800 0.660548 1.49626 0 2.63522 1.00000 −0.988352
1.4 −1.44855 −1.00000 0.0982847 −3.93238 1.44855 0 2.75472 1.00000 5.69623
1.5 0.775610 −1.00000 −1.39843 −2.32372 −0.775610 0 −2.63586 1.00000 −1.80230
1.6 1.32624 −1.00000 −0.241096 0.903630 −1.32624 0 −2.97222 1.00000 1.19843
1.7 1.55584 −1.00000 0.420641 3.26436 −1.55584 0 −2.45723 1.00000 5.07882
1.8 2.13183 −1.00000 2.54468 −2.36072 −2.13183 0 1.16116 1.00000 −5.03264
1.9 2.62834 −1.00000 4.90817 1.82844 −2.62834 0 7.64366 1.00000 4.80575
1.10 2.80657 −1.00000 5.87681 −4.11270 −2.80657 0 10.8805 1.00000 −11.5426
\(n\): 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
\(3\) \(1\)
\(7\) \(-1\)
\(41\) \(1\)

Hecke kernels

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

\(T_{2}^{10} - \cdots\)
\(T_{5}^{10} + \cdots\)
\(T_{13}^{10} - \cdots\)