Properties

Label 4830.2.a.bz
Level $4830$
Weight $2$
Character orbit 4830.a
Self dual yes
Analytic conductor $38.568$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4830.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.5677441763\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.48119
2.17009
0.311108
1.00000 −1.00000 1.00000 −1.00000 −1.00000 1.00000 1.00000 1.00000 −1.00000
1.2 1.00000 −1.00000 1.00000 −1.00000 −1.00000 1.00000 1.00000 1.00000 −1.00000
1.3 1.00000 −1.00000 1.00000 −1.00000 −1.00000 1.00000 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4830.2.a.bz 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4830.2.a.bz 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4830))\):

\( T_{11}^{3} - 2 T_{11}^{2} - 36 T_{11} + 104 \)
\( T_{13}^{3} + 2 T_{13}^{2} - 12 T_{13} - 8 \)
\( T_{17}^{3} + 8 T_{17}^{2} - 32 T_{17} - 272 \)
\( T_{19}^{3} + 6 T_{19}^{2} - 4 T_{19} - 40 \)
\( T_{29}^{3} - 16 T_{29}^{2} + 64 T_{29} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( 104 - 36 T - 2 T^{2} + T^{3} \)
$13$ \( -8 - 12 T + 2 T^{2} + T^{3} \)
$17$ \( -272 - 32 T + 8 T^{2} + T^{3} \)
$19$ \( -40 - 4 T + 6 T^{2} + T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( -32 + 64 T - 16 T^{2} + T^{3} \)
$31$ \( 40 - 4 T - 6 T^{2} + T^{3} \)
$37$ \( ( -2 + T )^{3} \)
$41$ \( 104 - 84 T - 2 T^{2} + T^{3} \)
$43$ \( 8 - 4 T - 10 T^{2} + T^{3} \)
$47$ \( 40 - 4 T - 6 T^{2} + T^{3} \)
$53$ \( -104 - 84 T + 2 T^{2} + T^{3} \)
$59$ \( 320 - 16 T - 12 T^{2} + T^{3} \)
$61$ \( 296 + 44 T - 18 T^{2} + T^{3} \)
$67$ \( 8 - 4 T - 10 T^{2} + T^{3} \)
$71$ \( -8 - 20 T + 2 T^{2} + T^{3} \)
$73$ \( -1528 - 244 T + 6 T^{2} + T^{3} \)
$79$ \( -128 - 64 T + T^{3} \)
$83$ \( -536 - 52 T + 10 T^{2} + T^{3} \)
$89$ \( 1816 - 220 T - 6 T^{2} + T^{3} \)
$97$ \( -8 - 12 T + 2 T^{2} + T^{3} \)
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