Properties

Label 483.2.i.b
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} + (\zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} + (\zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} + (6 \zeta_{6} - 6) q^{11} + 2 \zeta_{6} q^{12} + 5 q^{13} - 4 \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + \zeta_{6} q^{19} + (2 \zeta_{6} - 3) q^{21} + \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} + q^{27} + ( - 4 \zeta_{6} + 6) q^{28} + 6 q^{29} + (5 \zeta_{6} - 5) q^{31} - 6 \zeta_{6} q^{33} - 2 q^{36} + 7 \zeta_{6} q^{37} + (5 \zeta_{6} - 5) q^{39} - q^{43} + 12 \zeta_{6} q^{44} - 6 \zeta_{6} q^{47} + 4 q^{48} + (5 \zeta_{6} + 3) q^{49} + 6 \zeta_{6} q^{51} + ( - 10 \zeta_{6} + 10) q^{52} + (12 \zeta_{6} - 12) q^{53} - q^{57} + ( - 6 \zeta_{6} + 6) q^{59} - 14 \zeta_{6} q^{61} + ( - 3 \zeta_{6} + 1) q^{63} - 8 q^{64} + (5 \zeta_{6} - 5) q^{67} - 12 \zeta_{6} q^{68} - q^{69} - 6 q^{71} + ( - 7 \zeta_{6} + 7) q^{73} + 5 \zeta_{6} q^{75} + 2 q^{76} + (12 \zeta_{6} - 18) q^{77} - 5 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 12 q^{83} + (6 \zeta_{6} - 2) q^{84} + (6 \zeta_{6} - 6) q^{87} + 6 \zeta_{6} q^{89} + (5 \zeta_{6} + 10) q^{91} + 2 q^{92} - 5 \zeta_{6} q^{93} - 10 q^{97} + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{4} + 5 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 2 q^{4} + 5 q^{7} - q^{9} - 6 q^{11} + 2 q^{12} + 10 q^{13} - 4 q^{16} + 6 q^{17} + q^{19} - 4 q^{21} + q^{23} + 5 q^{25} + 2 q^{27} + 8 q^{28} + 12 q^{29} - 5 q^{31} - 6 q^{33} - 4 q^{36} + 7 q^{37} - 5 q^{39} - 2 q^{43} + 12 q^{44} - 6 q^{47} + 8 q^{48} + 11 q^{49} + 6 q^{51} + 10 q^{52} - 12 q^{53} - 2 q^{57} + 6 q^{59} - 14 q^{61} - q^{63} - 16 q^{64} - 5 q^{67} - 12 q^{68} - 2 q^{69} - 12 q^{71} + 7 q^{73} + 5 q^{75} + 4 q^{76} - 24 q^{77} - 5 q^{79} - q^{81} - 24 q^{83} + 2 q^{84} - 6 q^{87} + 6 q^{89} + 25 q^{91} + 4 q^{92} - 5 q^{93} - 20 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 2.50000 + 0.866025i 0 −0.500000 0.866025i 0
415.1 0 −0.500000 0.866025i 1.00000 + 1.73205i 0 0 2.50000 0.866025i 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.b 2
7.c even 3 1 inner 483.2.i.b 2
7.c even 3 1 3381.2.a.j 1
7.d odd 6 1 3381.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.b 2 1.a even 1 1 trivial
483.2.i.b 2 7.c even 3 1 inner
3381.2.a.g 1 7.d odd 6 1
3381.2.a.j 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 \(S_{2}^{\mathrm{new}}(483, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$13$ \( (T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$37$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$67$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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