Properties

Label 483.2.i.a
Level $483$
Weight $2$
Character orbit 483.i
Analytic conductor $3.857$
Analytic rank $0$
Dimension $2$
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} + ( 2 - 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} + ( 2 - 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{12} -3 q^{13} -4 \zeta_{6} q^{16} + ( -2 + 2 \zeta_{6} ) q^{17} -3 \zeta_{6} q^{19} + ( 1 + 2 \zeta_{6} ) q^{21} -\zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} + q^{27} + ( -2 - 4 \zeta_{6} ) q^{28} + 6 q^{29} + ( 3 - 3 \zeta_{6} ) q^{31} + 2 \zeta_{6} q^{33} -2 q^{36} + 3 \zeta_{6} q^{37} + ( 3 - 3 \zeta_{6} ) q^{39} + 3 q^{43} -4 \zeta_{6} q^{44} + 10 \zeta_{6} q^{47} + 4 q^{48} + ( -5 - 3 \zeta_{6} ) q^{49} -2 \zeta_{6} q^{51} + ( -6 + 6 \zeta_{6} ) q^{52} + ( 4 - 4 \zeta_{6} ) q^{53} + 3 q^{57} + ( 6 - 6 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} + ( -3 + \zeta_{6} ) q^{63} -8 q^{64} + ( -1 + \zeta_{6} ) q^{67} + 4 \zeta_{6} q^{68} + q^{69} -6 q^{71} + ( -1 + \zeta_{6} ) q^{73} + 5 \zeta_{6} q^{75} -6 q^{76} + ( -2 - 4 \zeta_{6} ) q^{77} + 7 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 4 q^{83} + ( 6 - 2 \zeta_{6} ) q^{84} + ( -6 + 6 \zeta_{6} ) q^{87} + 14 \zeta_{6} q^{89} + ( -6 + 9 \zeta_{6} ) q^{91} -2 q^{92} + 3 \zeta_{6} q^{93} -10 q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 2q^{4} + q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} + 2q^{4} + q^{7} - q^{9} + 2q^{11} + 2q^{12} - 6q^{13} - 4q^{16} - 2q^{17} - 3q^{19} + 4q^{21} - q^{23} + 5q^{25} + 2q^{27} - 8q^{28} + 12q^{29} + 3q^{31} + 2q^{33} - 4q^{36} + 3q^{37} + 3q^{39} + 6q^{43} - 4q^{44} + 10q^{47} + 8q^{48} - 13q^{49} - 2q^{51} - 6q^{52} + 4q^{53} + 6q^{57} + 6q^{59} + 2q^{61} - 5q^{63} - 16q^{64} - q^{67} + 4q^{68} + 2q^{69} - 12q^{71} - q^{73} + 5q^{75} - 12q^{76} - 8q^{77} + 7q^{79} - q^{81} + 8q^{83} + 10q^{84} - 6q^{87} + 14q^{89} - 3q^{91} - 4q^{92} + 3q^{93} - 20q^{97} - 4q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 1.00000 1.73205i 0 0 0.500000 2.59808i 0 −0.500000 0.866025i 0
415.1 0 −0.500000 0.866025i 1.00000 + 1.73205i 0 0 0.500000 + 2.59808i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.i.a 2
7.c even 3 1 inner 483.2.i.a 2
7.c even 3 1 3381.2.a.h 1
7.d odd 6 1 3381.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.a 2 1.a even 1 1 trivial
483.2.i.a 2 7.c even 3 1 inner
3381.2.a.e 1 7.d odd 6 1
3381.2.a.h 1 7.c even 3 1

Hecke kernels

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

\( T_{2} \)
\( T_{5} \)
\( T_{11}^{2} - 2 T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( 4 - 2 T + T^{2} \)
$13$ \( ( 3 + T )^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( 9 + 3 T + T^{2} \)
$23$ \( 1 + T + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( 9 - 3 T + T^{2} \)
$37$ \( 9 - 3 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -3 + T )^{2} \)
$47$ \( 100 - 10 T + T^{2} \)
$53$ \( 16 - 4 T + T^{2} \)
$59$ \( 36 - 6 T + T^{2} \)
$61$ \( 4 - 2 T + T^{2} \)
$67$ \( 1 + T + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( 1 + T + T^{2} \)
$79$ \( 49 - 7 T + T^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( 196 - 14 T + T^{2} \)
$97$ \( ( 10 + T )^{2} \)
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