Properties

Label 67.2.a.b
Level 67
Weight 2
Character orbit 67.a
Self dual Yes
Analytic conductor 0.535
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 67.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.534997693543\)
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.

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

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 −0.381966 4.85410 −3.00000 1.00000 −3.85410 −7.47214 −2.85410 7.85410
1.2 −0.381966 −2.61803 −1.85410 −3.00000 1.00000 2.85410 1.47214 3.85410 1.14590
\(n\): 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
\(67\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(67))\).