Properties

Label 483.2.i.e
Level $483$
Weight $2$
Character orbit 483.i
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\)
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 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 5 \nu^{2} - 5 \nu + 16 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4 \beta_{2} + \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(5 \beta_{3} - 4\)

Character values

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

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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
−0.780776 1.35234i
1.28078 + 2.21837i
−0.780776 + 1.35234i
1.28078 2.21837i
−0.780776 1.35234i −0.500000 + 0.866025i −0.219224 + 0.379706i −0.780776 1.35234i 1.56155 2.50000 + 0.866025i −2.43845 −0.500000 0.866025i −1.21922 + 2.11176i
277.2 1.28078 + 2.21837i −0.500000 + 0.866025i −2.28078 + 3.95042i 1.28078 + 2.21837i −2.56155 2.50000 + 0.866025i −6.56155 −0.500000 0.866025i −3.28078 + 5.68247i
415.1 −0.780776 + 1.35234i −0.500000 0.866025i −0.219224 0.379706i −0.780776 + 1.35234i 1.56155 2.50000 0.866025i −2.43845 −0.500000 + 0.866025i −1.21922 2.11176i
415.2 1.28078 2.21837i −0.500000 0.866025i −2.28078 3.95042i 1.28078 2.21837i −2.56155 2.50000 0.866025i −6.56155 −0.500000 + 0.866025i −3.28078 5.68247i
\(n\): 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.e 4
7.c even 3 1 inner 483.2.i.e 4
7.c even 3 1 3381.2.a.s 2
7.d odd 6 1 3381.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.e 4 1.a even 1 1 trivial
483.2.i.e 4 7.c even 3 1 inner
3381.2.a.q 2 7.d odd 6 1
3381.2.a.s 2 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 \(S_{2}^{\mathrm{new}}(483, [\chi])\):

\( T_{2}^{4} - T_{2}^{3} + 5 T_{2}^{2} + 4 T_{2} + 16 \)
\( T_{5}^{4} - T_{5}^{3} + 5 T_{5}^{2} + 4 T_{5} + 16 \)
\( T_{11}^{2} - 2 T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 4 T + 5 T^{2} - T^{3} + T^{4} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( 16 + 4 T + 5 T^{2} - T^{3} + T^{4} \)
$7$ \( ( 7 - 5 T + T^{2} )^{2} \)
$11$ \( ( 4 - 2 T + T^{2} )^{2} \)
$13$ \( ( -13 + 4 T + T^{2} )^{2} \)
$17$ \( 676 - 286 T + 95 T^{2} - 11 T^{3} + T^{4} \)
$19$ \( 64 + 56 T + 41 T^{2} + 7 T^{3} + T^{4} \)
$23$ \( ( 1 + T + T^{2} )^{2} \)
$29$ \( ( 8 + 10 T + T^{2} )^{2} \)
$31$ \( 1444 + 38 T + 39 T^{2} - T^{3} + T^{4} \)
$37$ \( 676 + 286 T + 95 T^{2} + 11 T^{3} + T^{4} \)
$41$ \( ( -64 + 4 T + T^{2} )^{2} \)
$43$ \( ( 4 + 13 T + T^{2} )^{2} \)
$47$ \( 1444 - 38 T + 39 T^{2} + T^{3} + T^{4} \)
$53$ \( 10000 - 500 T + 125 T^{2} + 5 T^{3} + T^{4} \)
$59$ \( 64 + 80 T + 92 T^{2} + 10 T^{3} + T^{4} \)
$61$ \( ( 36 + 6 T + T^{2} )^{2} \)
$67$ \( 6889 - 1660 T + 317 T^{2} - 20 T^{3} + T^{4} \)
$71$ \( ( -94 - 7 T + T^{2} )^{2} \)
$73$ \( 6889 + 1660 T + 317 T^{2} + 20 T^{3} + T^{4} \)
$79$ \( 2704 - 780 T + 173 T^{2} - 15 T^{3} + T^{4} \)
$83$ \( ( 64 - 18 T + T^{2} )^{2} \)
$89$ \( ( 100 - 10 T + T^{2} )^{2} \)
$97$ \( ( -144 - 6 T + T^{2} )^{2} \)
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