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: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) 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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + \beta_1 q^{5} + \beta_{3} q^{6} + (\beta_{2} + 2) q^{7} + (\beta_{3} - 4) q^{8} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + \beta_1 q^{5} + \beta_{3} q^{6} + (\beta_{2} + 2) q^{7} + (\beta_{3} - 4) q^{8} - \beta_{2} q^{9} + (\beta_{3} + 4 \beta_{2} + \beta_1 - 4) q^{10} + ( - 2 \beta_{2} + 2) q^{11} + ( - 2 \beta_{2} - \beta_1) q^{12} + ( - 2 \beta_{3} - 3) q^{13} + (\beta_{3} + 3 \beta_1) q^{14} + \beta_{3} q^{15} - 3 \beta_1 q^{16} + (\beta_{3} - 6 \beta_{2} + \beta_1 + 6) q^{17} + ( - \beta_{3} - \beta_1) q^{18} + ( - 3 \beta_{2} - \beta_1) q^{19} + (3 \beta_{3} - 4) q^{20} + (2 \beta_{2} - 3) q^{21} - 2 \beta_{3} q^{22} - \beta_{2} q^{23} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 4) q^{24} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{25} + (8 \beta_{2} - \beta_1) q^{26} + q^{27} + (3 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 6) q^{28} + ( - 2 \beta_{3} - 6) q^{29} + ( - 4 \beta_{2} - \beta_1) q^{30} + ( - 3 \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{31} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 4) q^{32} + 2 \beta_{2} q^{33} + ( - 5 \beta_{3} - 4) q^{34} + (\beta_{3} + 3 \beta_1) q^{35} + ( - \beta_{3} + 2) q^{36} + ( - 5 \beta_{2} - \beta_1) q^{37} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 4) q^{38} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{39} + ( - 4 \beta_{2} - 5 \beta_1) q^{40} + ( - 4 \beta_{3} - 4) q^{41} + (2 \beta_{3} - \beta_1) q^{42} + (3 \beta_{3} - 5) q^{43} + (4 \beta_{2} + 2 \beta_1) q^{44} + ( - \beta_{3} - \beta_1) q^{45} + ( - \beta_{3} - \beta_1) q^{46} + ( - 2 \beta_{2} + 3 \beta_1) q^{47} - 3 \beta_{3} q^{48} + (5 \beta_{2} + 3) q^{49} - 4 q^{50} + (6 \beta_{2} - \beta_1) q^{51} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{52} + (5 \beta_{3} + 5 \beta_1) q^{53} + \beta_1 q^{54} - 2 \beta_{3} q^{55} + (2 \beta_{3} - 4 \beta_{2} - \beta_1 - 8) q^{56} + ( - \beta_{3} + 3) q^{57} + (8 \beta_{2} - 4 \beta_1) q^{58} + ( - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1 - 6) q^{59} + ( - 3 \beta_{3} - 4 \beta_{2} - 3 \beta_1 + 4) q^{60} - 6 \beta_{2} q^{61} + ( - 2 \beta_{3} + 12) q^{62} + ( - 3 \beta_{2} + 1) q^{63} + (\beta_{3} + 4) q^{64} + (8 \beta_{2} - \beta_1) q^{65} + (2 \beta_{3} + 2 \beta_1) q^{66} + (2 \beta_{3} - 11 \beta_{2} + 2 \beta_1 + 11) q^{67} + (8 \beta_{2} + 3 \beta_1) q^{68} + q^{69} + (3 \beta_{3} + 8 \beta_{2} + 2 \beta_1 - 12) q^{70} + (5 \beta_{3} + 6) q^{71} + (4 \beta_{2} + \beta_1) q^{72} + (2 \beta_{3} + 9 \beta_{2} + 2 \beta_1 - 9) q^{73} + ( - 6 \beta_{3} - 4 \beta_{2} - 6 \beta_1 + 4) q^{74} + (\beta_{2} - \beta_1) q^{75} + ( - 6 \beta_{3} + 10) q^{76} + ( - 4 \beta_{2} + 6) q^{77} + ( - \beta_{3} - 8) q^{78} + (7 \beta_{2} + \beta_1) q^{79} + ( - 3 \beta_{3} - 12 \beta_{2} - 3 \beta_1 + 12) q^{80} + (\beta_{2} - 1) q^{81} + 16 \beta_{2} q^{82} + ( - 2 \beta_{3} + 8) q^{83} + ( - \beta_{3} - 6 \beta_{2} - 3 \beta_1 + 2) q^{84} + ( - 5 \beta_{3} - 4) q^{85} + ( - 12 \beta_{2} - 8 \beta_1) q^{86} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 6) q^{87} + (2 \beta_{3} + 8 \beta_{2} + 2 \beta_1 - 8) q^{88} + 10 \beta_{2} q^{89} + ( - \beta_{3} + 4) q^{90} + ( - 4 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 6) q^{91} + ( - \beta_{3} + 2) q^{92} + ( - \beta_{2} + 3 \beta_1) q^{93} + (\beta_{3} + 12 \beta_{2} + \beta_1 - 12) q^{94} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 4) q^{95} + (4 \beta_{2} + \beta_1) q^{96} + (6 \beta_{3} + 6) q^{97} + (5 \beta_{3} + 8 \beta_1) q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 2 q^{3} - 5 q^{4} + q^{5} - 2 q^{6} + 10 q^{7} - 18 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - 2 q^{3} - 5 q^{4} + q^{5} - 2 q^{6} + 10 q^{7} - 18 q^{8} - 2 q^{9} - 9 q^{10} + 4 q^{11} - 5 q^{12} - 8 q^{13} + q^{14} - 2 q^{15} - 3 q^{16} + 11 q^{17} + q^{18} - 7 q^{19} - 22 q^{20} - 8 q^{21} + 4 q^{22} - 2 q^{23} + 9 q^{24} + q^{25} + 15 q^{26} + 4 q^{27} - 20 q^{28} - 20 q^{29} - 9 q^{30} + q^{31} + 9 q^{32} + 4 q^{33} - 6 q^{34} + q^{35} + 10 q^{36} - 11 q^{37} + 12 q^{38} + 4 q^{39} - 13 q^{40} - 8 q^{41} - 5 q^{42} - 26 q^{43} + 10 q^{44} + q^{45} + q^{46} - q^{47} + 6 q^{48} + 22 q^{49} - 16 q^{50} + 11 q^{51} - 7 q^{52} - 5 q^{53} + q^{54} + 4 q^{55} - 45 q^{56} + 14 q^{57} + 12 q^{58} - 10 q^{59} + 11 q^{60} - 12 q^{61} + 52 q^{62} - 2 q^{63} + 14 q^{64} + 15 q^{65} - 2 q^{66} + 20 q^{67} + 19 q^{68} + 4 q^{69} - 36 q^{70} + 14 q^{71} + 9 q^{72} - 20 q^{73} + 14 q^{74} + q^{75} + 52 q^{76} + 16 q^{77} - 30 q^{78} + 15 q^{79} + 27 q^{80} - 2 q^{81} + 32 q^{82} + 36 q^{83} - 5 q^{84} - 6 q^{85} - 32 q^{86} + 10 q^{87} - 18 q^{88} + 20 q^{89} + 18 q^{90} - 20 q^{91} + 10 q^{92} + q^{93} - 25 q^{94} + 12 q^{95} + 9 q^{96} + 12 q^{97} - 2 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) Copy content Toggle raw display

Character values

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

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

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 \(S_{2}^{\mathrm{new}}(483, [\chi])\):

\( T_{2}^{4} - T_{2}^{3} + 5T_{2}^{2} + 4T_{2} + 16 \) Copy content Toggle raw display
\( T_{5}^{4} - T_{5}^{3} + 5T_{5}^{2} + 4T_{5} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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