Properties

Label 6014.2.a.d
Level 6014
Weight 2
Character orbit 6014.a
Self dual Yes
Analytic conductor 48.022
Analytic rank 0
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6014 = 2 \cdot 31 \cdot 97 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6014.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.0220317756\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.380224.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( -\nu^{4} + 8 \nu^{2} + \nu - 4 \)
\(\beta_{3}\)\(=\)\( 2 \nu^{4} - \nu^{3} - 15 \nu^{2} + 4 \nu + 6 \)
\(\beta_{4}\)\(=\)\( 3 \nu^{4} - 2 \nu^{3} - 23 \nu^{2} + 9 \nu + 11 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(16\) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(20\)

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
−2.56491
−0.658084
−0.479072
0.874805
2.82726
1.00000 −2.56491 1.00000 −1.19216 −2.56491 0 1.00000 3.57875 −1.19216
1.2 1.00000 −0.658084 1.00000 2.75080 −0.658084 0 1.00000 −2.56693 2.75080
1.3 1.00000 −0.479072 1.00000 −2.78756 −0.479072 0 1.00000 −2.77049 −2.78756
1.4 1.00000 0.874805 1.00000 −2.68974 0.874805 0 1.00000 −2.23472 −2.68974
1.5 1.00000 2.82726 1.00000 −0.0813392 2.82726 0 1.00000 4.99338 −0.0813392
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(31\) \(1\)
\(97\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{5} \) \(\mathstrut -\mathstrut 8 T_{3}^{3} \) \(\mathstrut -\mathstrut 2 T_{3}^{2} \) \(\mathstrut +\mathstrut 5 T_{3} \) \(\mathstrut +\mathstrut 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6014))\).