Properties

Label 483.2.d.c
Level $483$
Weight $2$
Character orbit 483.d
Analytic conductor $3.857$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

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

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
−0.382683 + 0.923880i
0.923880 + 0.382683i
−0.923880 0.382683i
0.382683 0.923880i
−0.382683 0.923880i
0.923880 0.382683i
−0.923880 + 0.382683i
0.382683 + 0.923880i
1.00000i −0.923880 + 1.46508i 1.00000 3.37849 1.46508 + 0.923880i −2.41421 1.08239i 3.00000i −1.29289 2.70711i 3.37849i
461.2 1.00000i −0.382683 + 1.68925i 1.00000 −2.93015 1.68925 + 0.382683i 0.414214 2.61313i 3.00000i −2.70711 1.29289i 2.93015i
461.3 1.00000i 0.382683 1.68925i 1.00000 2.93015 −1.68925 0.382683i 0.414214 + 2.61313i 3.00000i −2.70711 1.29289i 2.93015i
461.4 1.00000i 0.923880 1.46508i 1.00000 −3.37849 −1.46508 0.923880i −2.41421 + 1.08239i 3.00000i −1.29289 2.70711i 3.37849i
461.5 1.00000i −0.923880 1.46508i 1.00000 3.37849 1.46508 0.923880i −2.41421 + 1.08239i 3.00000i −1.29289 + 2.70711i 3.37849i
461.6 1.00000i −0.382683 1.68925i 1.00000 −2.93015 1.68925 0.382683i 0.414214 + 2.61313i 3.00000i −2.70711 + 1.29289i 2.93015i
461.7 1.00000i 0.382683 + 1.68925i 1.00000 2.93015 −1.68925 + 0.382683i 0.414214 2.61313i 3.00000i −2.70711 + 1.29289i 2.93015i
461.8 1.00000i 0.923880 + 1.46508i 1.00000 −3.37849 −1.46508 + 0.923880i −2.41421 1.08239i 3.00000i −1.29289 + 2.70711i 3.37849i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 461.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
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.c 8
3.b odd 2 1 inner 483.2.d.c 8
7.b odd 2 1 inner 483.2.d.c 8
21.c even 2 1 inner 483.2.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.d.c 8 1.a even 1 1 trivial
483.2.d.c 8 3.b odd 2 1 inner
483.2.d.c 8 7.b odd 2 1 inner
483.2.d.c 8 21.c even 2 1 inner

Hecke kernels

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

\( T_{2}^{2} + 1 \)
\( T_{5}^{4} - 20 T_{5}^{2} + 98 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( 81 + 72 T^{2} + 32 T^{4} + 8 T^{6} + T^{8} \)
$5$ \( ( 98 - 20 T^{2} + T^{4} )^{2} \)
$7$ \( ( 49 + 28 T + 10 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$11$ \( ( 4 + 12 T^{2} + T^{4} )^{2} \)
$13$ \( ( 32 + 16 T^{2} + T^{4} )^{2} \)
$17$ \( ( 2 - 20 T^{2} + T^{4} )^{2} \)
$19$ \( ( 128 + 32 T^{2} + T^{4} )^{2} \)
$23$ \( ( 1 + T^{2} )^{4} \)
$29$ \( ( 18 + T^{2} )^{4} \)
$31$ \( ( 1058 + 68 T^{2} + T^{4} )^{2} \)
$37$ \( ( -68 - 4 T + T^{2} )^{4} \)
$41$ \( ( 6272 - 160 T^{2} + T^{4} )^{2} \)
$43$ \( ( -16 + 8 T + T^{2} )^{4} \)
$47$ \( ( 578 - 52 T^{2} + T^{4} )^{2} \)
$53$ \( ( 16 + 24 T^{2} + T^{4} )^{2} \)
$59$ \( ( 1922 - 100 T^{2} + T^{4} )^{2} \)
$61$ \( ( 10082 + 244 T^{2} + T^{4} )^{2} \)
$67$ \( ( -158 - 4 T + T^{2} )^{4} \)
$71$ \( ( 256 + 96 T^{2} + T^{4} )^{2} \)
$73$ \( ( 1568 + 80 T^{2} + T^{4} )^{2} \)
$79$ \( ( -158 - 4 T + T^{2} )^{4} \)
$83$ \( ( 32 - 80 T^{2} + T^{4} )^{2} \)
$89$ \( ( 7938 - 180 T^{2} + T^{4} )^{2} \)
$97$ \( ( 1058 + 148 T^{2} + T^{4} )^{2} \)
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