Properties

Label 6006.2.a.cf
Level 6006
Weight 2
Character orbit 6006.a
Self dual Yes
Analytic conductor 47.958
Analytic rank 0
Dimension 6
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: \(6\)
Coefficient field: 6.6.72306708.1
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{4} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(11\) \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut +\mathstrut \) \(10\) \(x^{2}\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{5} + 11 \nu^{3} + \nu^{2} - 8 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - 11 \nu^{3} - \nu^{2} + 10 \nu \)
\(\beta_{3}\)\(=\)\( 4 \nu^{5} - 42 \nu^{3} - 4 \nu^{2} + 20 \nu - 1 \)
\(\beta_{4}\)\(=\)\( -4 \nu^{5} - 2 \nu^{4} + 42 \nu^{3} + 26 \nu^{2} - 18 \nu - 12 \)
\(\beta_{5}\)\(=\)\( 5 \nu^{5} + 4 \nu^{4} - 53 \nu^{3} - 47 \nu^{2} + 24 \nu + 18 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(21\) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(64\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(58\) \(\beta_{2}\mathstrut +\mathstrut \) \(101\) \(\beta_{1}\mathstrut +\mathstrut \) \(18\)\()/2\)

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
−0.508679
−0.916622
−3.10887
3.21938
0.717433
0.597364
1.00000 −1.00000 1.00000 −3.93175 −1.00000 1.00000 1.00000 1.00000 −3.93175
1.2 1.00000 −1.00000 1.00000 −2.18192 −1.00000 1.00000 1.00000 1.00000 −2.18192
1.3 1.00000 −1.00000 1.00000 −0.643320 −1.00000 1.00000 1.00000 1.00000 −0.643320
1.4 1.00000 −1.00000 1.00000 0.621238 −1.00000 1.00000 1.00000 1.00000 0.621238
1.5 1.00000 −1.00000 1.00000 2.78772 −1.00000 1.00000 1.00000 1.00000 2.78772
1.6 1.00000 −1.00000 1.00000 3.34804 −1.00000 1.00000 1.00000 1.00000 3.34804
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6006))\):

\(T_{5}^{6} \) \(\mathstrut -\mathstrut 20 T_{5}^{4} \) \(\mathstrut +\mathstrut 4 T_{5}^{3} \) \(\mathstrut +\mathstrut 88 T_{5}^{2} \) \(\mathstrut -\mathstrut 32 \)
\(T_{17}^{6} \) \(\mathstrut -\mathstrut 44 T_{17}^{4} \) \(\mathstrut -\mathstrut 8 T_{17}^{3} \) \(\mathstrut +\mathstrut 160 T_{17}^{2} \) \(\mathstrut -\mathstrut 128 \)
\(T_{19}^{6} \) \(\mathstrut -\mathstrut 2 T_{19}^{5} \) \(\mathstrut -\mathstrut 64 T_{19}^{4} \) \(\mathstrut +\mathstrut 24 T_{19}^{3} \) \(\mathstrut +\mathstrut 864 T_{19}^{2} \) \(\mathstrut -\mathstrut 384 T_{19} \) \(\mathstrut -\mathstrut 2432 \)
\(T_{23}^{6} \) \(\mathstrut -\mathstrut 10 T_{23}^{5} \) \(\mathstrut -\mathstrut 16 T_{23}^{4} \) \(\mathstrut +\mathstrut 376 T_{23}^{3} \) \(\mathstrut -\mathstrut 640 T_{23}^{2} \) \(\mathstrut -\mathstrut 1024 T_{23} \) \(\mathstrut -\mathstrut 256 \)
\(T_{31}^{6} \) \(\mathstrut -\mathstrut 2 T_{31}^{5} \) \(\mathstrut -\mathstrut 116 T_{31}^{4} \) \(\mathstrut +\mathstrut 204 T_{31}^{3} \) \(\mathstrut +\mathstrut 2728 T_{31}^{2} \) \(\mathstrut +\mathstrut 576 T_{31} \) \(\mathstrut -\mathstrut 5088 \)