Properties

Label 6030.2.a.bu
Level 6030
Weight 2
Character orbit 6030.a
Self dual Yes
Analytic conductor 48.150
Analytic rank 0
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6030.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1497924188\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.70292.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(8\) \(x^{2}\mathstrut +\mathstrut \) \(10\) \(x\mathstrut +\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{2}\)\(=\)\( \nu^{3} + \nu^{2} - 6 \nu - 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 8 \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\mathstrut -\mathstrut \) \(3\)

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
−0.175890
1.59676
2.46640
−2.88727
1.00000 0 1.00000 1.00000 0 −3.96906 1.00000 0 1.00000
1.2 1.00000 0 1.00000 1.00000 0 −1.45037 1.00000 0 1.00000
1.3 1.00000 0 1.00000 1.00000 0 2.08313 1.00000 0 1.00000
1.4 1.00000 0 1.00000 1.00000 0 4.33630 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(67\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6030))\):

\(T_{7}^{4} \) \(\mathstrut -\mathstrut T_{7}^{3} \) \(\mathstrut -\mathstrut 20 T_{7}^{2} \) \(\mathstrut +\mathstrut 12 T_{7} \) \(\mathstrut +\mathstrut 52 \)
\(T_{11}^{4} \) \(\mathstrut -\mathstrut 3 T_{11}^{3} \) \(\mathstrut -\mathstrut 20 T_{11}^{2} \) \(\mathstrut +\mathstrut 32 T_{11} \) \(\mathstrut +\mathstrut 112 \)
\(T_{13}^{4} \) \(\mathstrut -\mathstrut 2 T_{13}^{3} \) \(\mathstrut -\mathstrut 20 T_{13}^{2} \) \(\mathstrut +\mathstrut 28 T_{13} \) \(\mathstrut -\mathstrut 8 \)
\(T_{17}^{4} \) \(\mathstrut -\mathstrut 2 T_{17}^{3} \) \(\mathstrut -\mathstrut 32 T_{17}^{2} \) \(\mathstrut +\mathstrut 80 T_{17} \) \(\mathstrut +\mathstrut 32 \)
\(T_{23}^{4} \) \(\mathstrut -\mathstrut 12 T_{23}^{3} \) \(\mathstrut +\mathstrut 8 T_{23}^{2} \) \(\mathstrut +\mathstrut 204 T_{23} \) \(\mathstrut -\mathstrut 448 \)
\(T_{29}^{4} \) \(\mathstrut +\mathstrut 2 T_{29}^{3} \) \(\mathstrut -\mathstrut 52 T_{29}^{2} \) \(\mathstrut +\mathstrut 148 T_{29} \) \(\mathstrut -\mathstrut 112 \)