Properties

Label 89.2.a.c
Level 89
Weight 2
Character orbit 89.a
Self dual Yes
Analytic conductor 0.711
Analytic rank 0
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 89.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.710668577989\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.535120.1
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 2 \nu + 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{4} + 2 \nu^{3} + 6 \nu^{2} - 3 \nu - 4 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 8 \nu^{2} - \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 5\)
\(\nu^{4}\)\(=\)\(2 \beta_{4} + 11 \beta_{3} + 10 \beta_{2} + 15 \beta_{1} + 24\)

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
−1.75935
3.34526
0.450090
0.851093
−0.887094
−2.47725 −2.75935 4.13678 −2.13678 6.83562 3.26017 −5.29335 4.61403 5.29335
1.2 −2.09813 2.34526 2.40214 −0.402140 −4.92067 0.678986 −0.843741 2.50027 0.843741
1.3 −0.745954 −0.549910 −1.44355 3.44355 0.410207 5.17498 2.56873 −2.69760 −2.56873
1.4 1.62791 −0.148907 0.650080 1.34992 −0.242407 −2.21676 −2.19754 −2.97783 2.19754
1.5 2.69343 −1.88709 5.25455 −3.25455 −5.08275 1.10263 8.76590 0.561124 −8.76590
\(n\): 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
\(89\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{5} + T_{2}^{4} - 10 T_{2}^{3} - 10 T_{2}^{2} + 21 T_{2} + 17 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(89))\).