Properties

Label 1502.2.a.c
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

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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: \(\Z[a_1, \ldots, a_{5}]\)
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.

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

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 −2.23607 1.00000 −0.381966 2.23607 −2.61803 −1.00000 2.00000 0.381966
1.2 −1.00000 2.23607 1.00000 −2.61803 −2.23607 −0.381966 −1.00000 2.00000 2.61803
\(n\): 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.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1502.2.a.c 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 \( T_{3}^{2} - 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1502))\).