Properties

Label 6030.2.a.bs
Level 6030
Weight 2
Character orbit 6030.a
Self dual Yes
Analytic conductor 48.150
Analytic rank 1
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.11324.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + \nu^{2} - 5 \nu - 3 \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/2\)

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
2.25619
1.18994
−0.356500
−2.08963
1.00000 0 1.00000 −1.00000 0 −4.51238 1.00000 0 −1.00000
1.2 1.00000 0 1.00000 −1.00000 0 −2.37988 1.00000 0 −1.00000
1.3 1.00000 0 1.00000 −1.00000 0 0.713000 1.00000 0 −1.00000
1.4 1.00000 0 1.00000 −1.00000 0 4.17926 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6030))\):

\(T_{7}^{4} \) \(\mathstrut +\mathstrut 2 T_{7}^{3} \) \(\mathstrut -\mathstrut 20 T_{7}^{2} \) \(\mathstrut -\mathstrut 32 T_{7} \) \(\mathstrut +\mathstrut 32 \)
\(T_{11}^{4} \) \(\mathstrut +\mathstrut 10 T_{11}^{3} \) \(\mathstrut +\mathstrut 14 T_{11}^{2} \) \(\mathstrut -\mathstrut 88 T_{11} \) \(\mathstrut -\mathstrut 160 \)
\(T_{13}^{4} \) \(\mathstrut +\mathstrut 2 T_{13}^{3} \) \(\mathstrut -\mathstrut 32 T_{13}^{2} \) \(\mathstrut -\mathstrut 24 T_{13} \) \(\mathstrut +\mathstrut 208 \)
\(T_{17}^{4} \) \(\mathstrut -\mathstrut 6 T_{17}^{3} \) \(\mathstrut -\mathstrut 42 T_{17}^{2} \) \(\mathstrut +\mathstrut 152 T_{17} \) \(\mathstrut +\mathstrut 608 \)
\(T_{23}^{4} \) \(\mathstrut +\mathstrut 6 T_{23}^{3} \) \(\mathstrut -\mathstrut 36 T_{23}^{2} \) \(\mathstrut -\mathstrut 256 T_{23} \) \(\mathstrut -\mathstrut 256 \)
\(T_{29}^{4} \) \(\mathstrut +\mathstrut 14 T_{29}^{3} \) \(\mathstrut +\mathstrut 4 T_{29}^{2} \) \(\mathstrut -\mathstrut 448 T_{29} \) \(\mathstrut -\mathstrut 800 \)