Properties

Label 4030.2.a.b
Level 4030
Weight 2
Character orbit 4030.a
Self dual yes
Analytic conductor 32.180
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4030.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1797120146\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

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
−0.618034
1.61803
−1.00000 −1.23607 1.00000 −1.00000 1.23607 2.85410 −1.00000 −1.47214 1.00000
1.2 −1.00000 3.23607 1.00000 −1.00000 −3.23607 −3.85410 −1.00000 7.47214 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 4030.2.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4030.2.a.b 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(13\) \(-1\)
\(31\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2 T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).