Properties

Label 6045.2.a.s
Level 6045
Weight 2
Character orbit 6045.a
Self dual Yes
Analytic conductor 48.270
Analytic rank 1
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6045.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2695680219\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.230224.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 3 \nu^{2} + 4 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)

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
0.424945
1.49592
−0.757366
2.49889
−1.66240
−2.24437 1.00000 3.03718 1.00000 −2.24437 3.68511 −2.32782 1.00000 −2.24437
1.2 −1.25814 1.00000 −0.417093 1.00000 −1.25814 −1.35765 3.04103 1.00000 −1.25814
1.3 −0.669031 1.00000 −1.55240 1.00000 −0.669031 4.35950 2.37666 1.00000 −0.669031
1.4 1.74558 1.00000 1.04704 1.00000 1.74558 −3.12693 −1.66347 1.00000 1.74558
1.5 2.42596 1.00000 3.88527 1.00000 2.42596 −4.56003 4.57359 1.00000 2.42596
\(n\): 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
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(1\)
\(31\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\(T_{2}^{5} \) \(\mathstrut -\mathstrut 8 T_{2}^{3} \) \(\mathstrut -\mathstrut 2 T_{2}^{2} \) \(\mathstrut +\mathstrut 14 T_{2} \) \(\mathstrut +\mathstrut 8 \)
\(T_{7}^{5} \) \(\mathstrut +\mathstrut T_{7}^{4} \) \(\mathstrut -\mathstrut 32 T_{7}^{3} \) \(\mathstrut -\mathstrut 34 T_{7}^{2} \) \(\mathstrut +\mathstrut 241 T_{7} \) \(\mathstrut +\mathstrut 311 \)
\(T_{11}^{5} \) \(\mathstrut +\mathstrut 9 T_{11}^{4} \) \(\mathstrut +\mathstrut 12 T_{11}^{3} \) \(\mathstrut -\mathstrut 48 T_{11}^{2} \) \(\mathstrut -\mathstrut 64 T_{11} \) \(\mathstrut -\mathstrut 16 \)