Properties

Label 483.2.a.e
Level $483$
Weight $2$
Character orbit 483.a
Self dual yes
Analytic conductor $3.857$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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.

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

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 1.00000 0.618034 1.38197 −1.61803 1.00000 2.23607 1.00000 −2.23607
1.2 0.618034 1.00000 −1.61803 3.61803 0.618034 1.00000 −2.23607 1.00000 2.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(23\) \(-1\)

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 483.2.a.e 2
3.b odd 2 1 1449.2.a.g 2
4.b odd 2 1 7728.2.a.be 2
7.b odd 2 1 3381.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.e 2 1.a even 1 1 trivial
1449.2.a.g 2 3.b odd 2 1
3381.2.a.o 2 7.b odd 2 1
7728.2.a.be 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(483))\):

\( T_{2}^{2} + T_{2} - 1 \)
\( T_{5}^{2} - 5 T_{5} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 5 - 5 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -1 + T + T^{2} \)
$17$ \( -19 + 2 T + T^{2} \)
$19$ \( -1 - 4 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 11 - 8 T + T^{2} \)
$31$ \( -41 + 4 T + T^{2} \)
$37$ \( -1 + 4 T + T^{2} \)
$41$ \( -19 - 2 T + T^{2} \)
$43$ \( -11 + T + T^{2} \)
$47$ \( -20 - 10 T + T^{2} \)
$53$ \( -1 - 11 T + T^{2} \)
$59$ \( 5 + 5 T + T^{2} \)
$61$ \( 9 + 9 T + T^{2} \)
$67$ \( -25 + 5 T + T^{2} \)
$71$ \( -55 + 5 T + T^{2} \)
$73$ \( 11 - 8 T + T^{2} \)
$79$ \( 139 + 24 T + T^{2} \)
$83$ \( 1 + 18 T + T^{2} \)
$89$ \( -89 - 7 T + T^{2} \)
$97$ \( -45 + T^{2} \)
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