Properties

Label 4027.2.a.a
Level 4027
Weight 2
Character orbit 4027.a
Self dual yes
Analytic conductor 32.156
Analytic rank 2
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4027 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4027.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1557568940\)
Analytic rank: \(2\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

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
−1.61803 1.23607 0.618034 −1.00000 −2.00000 −4.61803 2.23607 −1.47214 1.61803
1.2 0.618034 −3.23607 −1.61803 −1.00000 −2.00000 −2.38197 −2.23607 7.47214 −0.618034
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4027.2.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4027.2.a.a 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(4027\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4027))\).