Properties

Label 483.2.d.a
Level $483$
Weight $2$
Character orbit 483.d
Analytic conductor $3.857$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
461.1
1.61803i
0.618034i
0.618034i
1.61803i
1.61803i 0.618034 1.61803i −0.618034 −2.61803 −2.61803 1.00000i 2.61803 0.381966i 2.23607i −2.23607 2.00000i 4.23607i
461.2 0.618034i −1.61803 0.618034i 1.61803 −0.381966 −0.381966 + 1.00000i 0.381966 + 2.61803i 2.23607i 2.23607 + 2.00000i 0.236068i
461.3 0.618034i −1.61803 + 0.618034i 1.61803 −0.381966 −0.381966 1.00000i 0.381966 2.61803i 2.23607i 2.23607 2.00000i 0.236068i
461.4 1.61803i 0.618034 + 1.61803i −0.618034 −2.61803 −2.61803 + 1.00000i 2.61803 + 0.381966i 2.23607i −2.23607 + 2.00000i 4.23607i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.d.a 4
3.b odd 2 1 483.2.d.b yes 4
7.b odd 2 1 483.2.d.b yes 4
21.c even 2 1 inner 483.2.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.d.a 4 1.a even 1 1 trivial
483.2.d.a 4 21.c even 2 1 inner
483.2.d.b yes 4 3.b odd 2 1
483.2.d.b yes 4 7.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 \(S_{2}^{\mathrm{new}}(483, [\chi])\):

\( T_{2}^{4} + 3 T_{2}^{2} + 1 \)
\( T_{5}^{2} + 3 T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} + T^{4} \)
$3$ \( 9 + 6 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( ( 1 + 3 T + T^{2} )^{2} \)
$7$ \( 49 - 42 T + 18 T^{2} - 6 T^{3} + T^{4} \)
$11$ \( ( 5 + T^{2} )^{2} \)
$13$ \( 1 + 47 T^{2} + T^{4} \)
$17$ \( ( -11 - 6 T + T^{2} )^{2} \)
$19$ \( 361 + 42 T^{2} + T^{4} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( 121 + 42 T^{2} + T^{4} \)
$31$ \( 25 + 90 T^{2} + T^{4} \)
$37$ \( ( 59 - 16 T + T^{2} )^{2} \)
$41$ \( ( -1 - 4 T + T^{2} )^{2} \)
$43$ \( ( 41 - 13 T + T^{2} )^{2} \)
$47$ \( ( -8 + T )^{4} \)
$53$ \( 7921 + 227 T^{2} + T^{4} \)
$59$ \( ( -95 + 5 T + T^{2} )^{2} \)
$61$ \( 25 + 35 T^{2} + T^{4} \)
$67$ \( ( 89 + 19 T + T^{2} )^{2} \)
$71$ \( 3025 + 135 T^{2} + T^{4} \)
$73$ \( 961 + 82 T^{2} + T^{4} \)
$79$ \( ( 95 + 20 T + T^{2} )^{2} \)
$83$ \( ( -29 - 8 T + T^{2} )^{2} \)
$89$ \( ( -25 + 5 T + T^{2} )^{2} \)
$97$ \( 961 + 258 T^{2} + T^{4} \)
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