Properties

Label 8018.2.a.b
Level 8018
Weight 2
Character orbit 8018.a
Self dual Yes
Analytic conductor 64.024
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8018 = 2 \cdot 19 \cdot 211 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8018.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0240523407\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
−1.00000 −0.618034 1.00000 −2.23607 0.618034 2.00000 −1.00000 −2.61803 2.23607
1.2 −1.00000 1.61803 1.00000 2.23607 −1.61803 2.00000 −1.00000 −0.381966 −2.23607
\(n\): 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
\(2\) \(1\)
\(19\) \(1\)
\(211\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut -\mathstrut T_{3} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8018))\).