Properties

Label 6006.2.a.cg
Level 6006
Weight 2
Character orbit 6006.a
Self dual Yes
Analytic conductor 47.958
Analytic rank 0
Dimension 7
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6006 = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6006.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.958151454\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{6} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( -\beta_{1} q^{5} \) \(+ q^{6}\) \(+ q^{7}\) \(+ q^{8}\) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( -\beta_{1} q^{5} \) \(+ q^{6}\) \(+ q^{7}\) \(+ q^{8}\) \(+ q^{9}\) \( -\beta_{1} q^{10} \) \(+ q^{11}\) \(+ q^{12}\) \(+ q^{13}\) \(+ q^{14}\) \( -\beta_{1} q^{15} \) \(+ q^{16}\) \( + ( 1 + \beta_{6} ) q^{17} \) \(+ q^{18}\) \( + ( 1 - \beta_{2} - \beta_{5} ) q^{19} \) \( -\beta_{1} q^{20} \) \(+ q^{21}\) \(+ q^{22}\) \( + \beta_{5} q^{23} \) \(+ q^{24}\) \( + ( 2 + \beta_{2} + \beta_{3} + \beta_{5} ) q^{25} \) \(+ q^{26}\) \(+ q^{27}\) \(+ q^{28}\) \( + ( 1 - \beta_{1} + \beta_{4} ) q^{29} \) \( -\beta_{1} q^{30} \) \( + ( 2 + \beta_{1} - \beta_{5} ) q^{31} \) \(+ q^{32}\) \(+ q^{33}\) \( + ( 1 + \beta_{6} ) q^{34} \) \( -\beta_{1} q^{35} \) \(+ q^{36}\) \( + ( 1 + \beta_{1} - \beta_{6} ) q^{37} \) \( + ( 1 - \beta_{2} - \beta_{5} ) q^{38} \) \(+ q^{39}\) \( -\beta_{1} q^{40} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{41} \) \(+ q^{42}\) \( + ( -\beta_{4} + \beta_{5} - \beta_{6} ) q^{43} \) \(+ q^{44}\) \( -\beta_{1} q^{45} \) \( + \beta_{5} q^{46} \) \( + ( -\beta_{3} - \beta_{5} ) q^{47} \) \(+ q^{48}\) \(+ q^{49}\) \( + ( 2 + \beta_{2} + \beta_{3} + \beta_{5} ) q^{50} \) \( + ( 1 + \beta_{6} ) q^{51} \) \(+ q^{52}\) \( + ( -1 - \beta_{2} - \beta_{3} ) q^{53} \) \(+ q^{54}\) \( -\beta_{1} q^{55} \) \(+ q^{56}\) \( + ( 1 - \beta_{2} - \beta_{5} ) q^{57} \) \( + ( 1 - \beta_{1} + \beta_{4} ) q^{58} \) \( + ( 1 - \beta_{1} - \beta_{5} + \beta_{6} ) q^{59} \) \( -\beta_{1} q^{60} \) \( + 2 q^{61} \) \( + ( 2 + \beta_{1} - \beta_{5} ) q^{62} \) \(+ q^{63}\) \(+ q^{64}\) \( -\beta_{1} q^{65} \) \(+ q^{66}\) \( + ( 4 + 2 \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{67} \) \( + ( 1 + \beta_{6} ) q^{68} \) \( + \beta_{5} q^{69} \) \( -\beta_{1} q^{70} \) \( + ( -1 - \beta_{1} + \beta_{5} - \beta_{6} ) q^{71} \) \(+ q^{72}\) \( + ( \beta_{1} - \beta_{4} - \beta_{6} ) q^{73} \) \( + ( 1 + \beta_{1} - \beta_{6} ) q^{74} \) \( + ( 2 + \beta_{2} + \beta_{3} + \beta_{5} ) q^{75} \) \( + ( 1 - \beta_{2} - \beta_{5} ) q^{76} \) \(+ q^{77}\) \(+ q^{78}\) \( + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{79} \) \( -\beta_{1} q^{80} \) \(+ q^{81}\) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{82} \) \( + ( \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{83} \) \(+ q^{84}\) \( + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 4 \beta_{5} + \beta_{6} ) q^{85} \) \( + ( -\beta_{4} + \beta_{5} - \beta_{6} ) q^{86} \) \( + ( 1 - \beta_{1} + \beta_{4} ) q^{87} \) \(+ q^{88}\) \( + ( -1 + \beta_{1} - \beta_{4} ) q^{89} \) \( -\beta_{1} q^{90} \) \(+ q^{91}\) \( + \beta_{5} q^{92} \) \( + ( 2 + \beta_{1} - \beta_{5} ) q^{93} \) \( + ( -\beta_{3} - \beta_{5} ) q^{94} \) \( + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{95} \) \(+ q^{96}\) \( + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{97} \) \(+ q^{98}\) \(+ q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(7q \) \(\mathstrut +\mathstrut 7q^{2} \) \(\mathstrut +\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 7q^{4} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 7q^{6} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 7q^{8} \) \(\mathstrut +\mathstrut 7q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut +\mathstrut 7q^{2} \) \(\mathstrut +\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 7q^{4} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 7q^{6} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 7q^{8} \) \(\mathstrut +\mathstrut 7q^{9} \) \(\mathstrut +\mathstrut 2q^{10} \) \(\mathstrut +\mathstrut 7q^{11} \) \(\mathstrut +\mathstrut 7q^{12} \) \(\mathstrut +\mathstrut 7q^{13} \) \(\mathstrut +\mathstrut 7q^{14} \) \(\mathstrut +\mathstrut 2q^{15} \) \(\mathstrut +\mathstrut 7q^{16} \) \(\mathstrut +\mathstrut 6q^{17} \) \(\mathstrut +\mathstrut 7q^{18} \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut +\mathstrut 2q^{20} \) \(\mathstrut +\mathstrut 7q^{21} \) \(\mathstrut +\mathstrut 7q^{22} \) \(\mathstrut +\mathstrut 2q^{23} \) \(\mathstrut +\mathstrut 7q^{24} \) \(\mathstrut +\mathstrut 17q^{25} \) \(\mathstrut +\mathstrut 7q^{26} \) \(\mathstrut +\mathstrut 7q^{27} \) \(\mathstrut +\mathstrut 7q^{28} \) \(\mathstrut +\mathstrut 6q^{29} \) \(\mathstrut +\mathstrut 2q^{30} \) \(\mathstrut +\mathstrut 10q^{31} \) \(\mathstrut +\mathstrut 7q^{32} \) \(\mathstrut +\mathstrut 7q^{33} \) \(\mathstrut +\mathstrut 6q^{34} \) \(\mathstrut +\mathstrut 2q^{35} \) \(\mathstrut +\mathstrut 7q^{36} \) \(\mathstrut +\mathstrut 6q^{37} \) \(\mathstrut +\mathstrut 4q^{38} \) \(\mathstrut +\mathstrut 7q^{39} \) \(\mathstrut +\mathstrut 2q^{40} \) \(\mathstrut -\mathstrut 4q^{41} \) \(\mathstrut +\mathstrut 7q^{42} \) \(\mathstrut +\mathstrut 6q^{43} \) \(\mathstrut +\mathstrut 7q^{44} \) \(\mathstrut +\mathstrut 2q^{45} \) \(\mathstrut +\mathstrut 2q^{46} \) \(\mathstrut -\mathstrut 2q^{47} \) \(\mathstrut +\mathstrut 7q^{48} \) \(\mathstrut +\mathstrut 7q^{49} \) \(\mathstrut +\mathstrut 17q^{50} \) \(\mathstrut +\mathstrut 6q^{51} \) \(\mathstrut +\mathstrut 7q^{52} \) \(\mathstrut -\mathstrut 8q^{53} \) \(\mathstrut +\mathstrut 7q^{54} \) \(\mathstrut +\mathstrut 2q^{55} \) \(\mathstrut +\mathstrut 7q^{56} \) \(\mathstrut +\mathstrut 4q^{57} \) \(\mathstrut +\mathstrut 6q^{58} \) \(\mathstrut +\mathstrut 6q^{59} \) \(\mathstrut +\mathstrut 2q^{60} \) \(\mathstrut +\mathstrut 14q^{61} \) \(\mathstrut +\mathstrut 10q^{62} \) \(\mathstrut +\mathstrut 7q^{63} \) \(\mathstrut +\mathstrut 7q^{64} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut 7q^{66} \) \(\mathstrut +\mathstrut 18q^{67} \) \(\mathstrut +\mathstrut 6q^{68} \) \(\mathstrut +\mathstrut 2q^{69} \) \(\mathstrut +\mathstrut 2q^{70} \) \(\mathstrut -\mathstrut 2q^{71} \) \(\mathstrut +\mathstrut 7q^{72} \) \(\mathstrut +\mathstrut 2q^{73} \) \(\mathstrut +\mathstrut 6q^{74} \) \(\mathstrut +\mathstrut 17q^{75} \) \(\mathstrut +\mathstrut 4q^{76} \) \(\mathstrut +\mathstrut 7q^{77} \) \(\mathstrut +\mathstrut 7q^{78} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 2q^{80} \) \(\mathstrut +\mathstrut 7q^{81} \) \(\mathstrut -\mathstrut 4q^{82} \) \(\mathstrut -\mathstrut 2q^{83} \) \(\mathstrut +\mathstrut 7q^{84} \) \(\mathstrut -\mathstrut 12q^{85} \) \(\mathstrut +\mathstrut 6q^{86} \) \(\mathstrut +\mathstrut 6q^{87} \) \(\mathstrut +\mathstrut 7q^{88} \) \(\mathstrut -\mathstrut 6q^{89} \) \(\mathstrut +\mathstrut 2q^{90} \) \(\mathstrut +\mathstrut 7q^{91} \) \(\mathstrut +\mathstrut 2q^{92} \) \(\mathstrut +\mathstrut 10q^{93} \) \(\mathstrut -\mathstrut 2q^{94} \) \(\mathstrut +\mathstrut 20q^{95} \) \(\mathstrut +\mathstrut 7q^{96} \) \(\mathstrut -\mathstrut 12q^{97} \) \(\mathstrut +\mathstrut 7q^{98} \) \(\mathstrut +\mathstrut 7q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(3\) \(x^{6}\mathstrut -\mathstrut \) \(8\) \(x^{5}\mathstrut +\mathstrut \) \(23\) \(x^{4}\mathstrut +\mathstrut \) \(5\) \(x^{3}\mathstrut -\mathstrut \) \(31\) \(x^{2}\mathstrut +\mathstrut \) \(18\) \(x\mathstrut -\mathstrut \) \(3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 10 \nu^{4} + 13 \nu^{3} + 18 \nu^{2} - 13 \nu + 3 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{5} - 4 \nu^{4} - 20 \nu^{3} + 26 \nu^{2} + 38 \nu - 25 \)
\(\beta_{3}\)\(=\)\( -8 \nu^{6} + 20 \nu^{5} + 74 \nu^{4} - 148 \nu^{3} - 112 \nu^{2} + 200 \nu - 54 \)
\(\beta_{4}\)\(=\)\( -9 \nu^{6} + 22 \nu^{5} + 84 \nu^{4} - 159 \nu^{3} - 134 \nu^{2} + 199 \nu - 46 \)
\(\beta_{5}\)\(=\)\( 16 \nu^{6} - 40 \nu^{5} - 148 \nu^{4} + 294 \nu^{3} + 226 \nu^{2} - 382 \nu + 102 \)
\(\beta_{6}\)\(=\)\( 18 \nu^{6} - 44 \nu^{5} - 168 \nu^{4} + 320 \nu^{3} + 264 \nu^{2} - 408 \nu + 101 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(11\) \(\beta_{6}\mathstrut -\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(20\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(16\) \(\beta_{6}\mathstrut -\mathstrut \) \(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(20\) \(\beta_{1}\mathstrut +\mathstrut \) \(60\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(129\) \(\beta_{6}\mathstrut -\mathstrut \) \(70\) \(\beta_{5}\mathstrut +\mathstrut \) \(87\) \(\beta_{4}\mathstrut +\mathstrut \) \(35\) \(\beta_{3}\mathstrut -\mathstrut \) \(6\) \(\beta_{2}\mathstrut -\mathstrut \) \(139\) \(\beta_{1}\mathstrut +\mathstrut \) \(270\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(206\) \(\beta_{6}\mathstrut -\mathstrut \) \(166\) \(\beta_{5}\mathstrut +\mathstrut \) \(75\) \(\beta_{4}\mathstrut +\mathstrut \) \(19\) \(\beta_{3}\mathstrut -\mathstrut \) \(26\) \(\beta_{2}\mathstrut -\mathstrut \) \(223\) \(\beta_{1}\mathstrut +\mathstrut \) \(621\)\()/2\)

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
0.677533
0.532847
0.340108
−1.43468
−2.52204
3.44766
1.95857
1.00000 1.00000 1.00000 −4.20218 1.00000 1.00000 1.00000 1.00000 −4.20218
1.2 1.00000 1.00000 1.00000 −2.28125 1.00000 1.00000 1.00000 1.00000 −2.28125
1.3 1.00000 1.00000 1.00000 −1.03080 1.00000 1.00000 1.00000 1.00000 −1.03080
1.4 1.00000 1.00000 1.00000 1.17854 1.00000 1.00000 1.00000 1.00000 1.17854
1.5 1.00000 1.00000 1.00000 1.43215 1.00000 1.00000 1.00000 1.00000 1.43215
1.6 1.00000 1.00000 1.00000 2.81711 1.00000 1.00000 1.00000 1.00000 2.81711
1.7 1.00000 1.00000 1.00000 4.08644 1.00000 1.00000 1.00000 1.00000 4.08644
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)
\(13\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6006))\):

\(T_{5}^{7} \) \(\mathstrut -\mathstrut 2 T_{5}^{6} \) \(\mathstrut -\mathstrut 24 T_{5}^{5} \) \(\mathstrut +\mathstrut 48 T_{5}^{4} \) \(\mathstrut +\mathstrut 120 T_{5}^{3} \) \(\mathstrut -\mathstrut 224 T_{5}^{2} \) \(\mathstrut -\mathstrut 96 T_{5} \) \(\mathstrut +\mathstrut 192 \)
\(T_{17}^{7} \) \(\mathstrut -\mathstrut 6 T_{17}^{6} \) \(\mathstrut -\mathstrut 72 T_{17}^{5} \) \(\mathstrut +\mathstrut 424 T_{17}^{4} \) \(\mathstrut +\mathstrut 1152 T_{17}^{3} \) \(\mathstrut -\mathstrut 6528 T_{17}^{2} \) \(\mathstrut -\mathstrut 2304 T_{17} \) \(\mathstrut +\mathstrut 13824 \)
\(T_{19}^{7} \) \(\mathstrut -\mathstrut 4 T_{19}^{6} \) \(\mathstrut -\mathstrut 84 T_{19}^{5} \) \(\mathstrut +\mathstrut 392 T_{19}^{4} \) \(\mathstrut +\mathstrut 1120 T_{19}^{3} \) \(\mathstrut -\mathstrut 8576 T_{19}^{2} \) \(\mathstrut +\mathstrut 14592 T_{19} \) \(\mathstrut -\mathstrut 7680 \)
\(T_{23}^{7} \) \(\mathstrut -\mathstrut 2 T_{23}^{6} \) \(\mathstrut -\mathstrut 72 T_{23}^{5} \) \(\mathstrut +\mathstrut 104 T_{23}^{4} \) \(\mathstrut +\mathstrut 1248 T_{23}^{3} \) \(\mathstrut -\mathstrut 2304 T_{23}^{2} \) \(\mathstrut -\mathstrut 4608 T_{23} \) \(\mathstrut +\mathstrut 9216 \)
\(T_{31}^{7} \) \(\mathstrut -\mathstrut 10 T_{31}^{6} \) \(\mathstrut -\mathstrut 44 T_{31}^{5} \) \(\mathstrut +\mathstrut 408 T_{31}^{4} \) \(\mathstrut +\mathstrut 472 T_{31}^{3} \) \(\mathstrut -\mathstrut 4896 T_{31}^{2} \) \(\mathstrut -\mathstrut 96 T_{31} \) \(\mathstrut +\mathstrut 14272 \)