Properties

Label 1502.2.a.d
Level 1502
Weight 2
Character orbit 1502.a
Self dual yes
Analytic conductor 11.994
Analytic rank 1
Dimension 2
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 1502 = 2 \cdot 751 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1502.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.9935303836\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

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
1.00000 −1.61803 1.00000 1.00000 −1.61803 −0.763932 1.00000 −0.381966 1.00000
1.2 1.00000 0.618034 1.00000 1.00000 0.618034 −5.23607 1.00000 −2.61803 1.00000
\(n\): 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 1502.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1502.2.a.d 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(751\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1502))\).