Properties

Label 93.2.a.a
Level 93
Weight 2
Character orbit 93.a
Self dual Yes
Analytic conductor 0.743
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 93 = 3 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 93.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.742608738798\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
−2.61803 −1.00000 4.85410 0.236068 2.61803 −4.23607 −7.47214 1.00000 −0.618034
1.2 −0.381966 −1.00000 −1.85410 −4.23607 0.381966 0.236068 1.47214 1.00000 1.61803
\(n\): 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
\(3\) \(1\)
\(31\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(93))\).