Properties

Label 4830.2.a.cc
Level $4830$
Weight $2$
Character orbit 4830.a
Self dual yes
Analytic conductor $38.568$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
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} + ( -2 + \beta_{1} ) q^{11} + q^{12} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{13} - q^{14} - q^{15} + q^{16} + ( \beta_{1} + \beta_{2} ) q^{17} + q^{18} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{19} - q^{20} - q^{21} + ( -2 + \beta_{1} ) q^{22} - q^{23} + q^{24} + q^{25} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{26} + q^{27} - q^{28} + ( -\beta_{1} - 2 \beta_{2} ) q^{29} - q^{30} + ( -6 - \beta_{1} + \beta_{2} ) q^{31} + q^{32} + ( -2 + \beta_{1} ) q^{33} + ( \beta_{1} + \beta_{2} ) q^{34} + q^{35} + q^{36} + ( -2 - 2 \beta_{1} ) q^{37} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{38} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{39} - q^{40} + ( -2 + 2 \beta_{2} ) q^{41} - q^{42} + ( -6 + \beta_{1} - 2 \beta_{2} ) q^{43} + ( -2 + \beta_{1} ) q^{44} - q^{45} - q^{46} + ( -6 - \beta_{1} + \beta_{2} ) q^{47} + q^{48} + q^{49} + q^{50} + ( \beta_{1} + \beta_{2} ) q^{51} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{52} + ( 6 + 2 \beta_{1} ) q^{53} + q^{54} + ( 2 - \beta_{1} ) q^{55} - q^{56} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{57} + ( -\beta_{1} - 2 \beta_{2} ) q^{58} -8 q^{59} - q^{60} + ( 2 + 2 \beta_{2} ) q^{61} + ( -6 - \beta_{1} + \beta_{2} ) q^{62} - q^{63} + q^{64} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{65} + ( -2 + \beta_{1} ) q^{66} + ( -6 + \beta_{1} + 2 \beta_{2} ) q^{67} + ( \beta_{1} + \beta_{2} ) q^{68} - q^{69} + q^{70} + ( -2 + 3 \beta_{1} ) q^{71} + q^{72} + ( 2 + 4 \beta_{1} - 4 \beta_{2} ) q^{73} + ( -2 - 2 \beta_{1} ) q^{74} + q^{75} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{76} + ( 2 - \beta_{1} ) q^{77} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{78} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{79} - q^{80} + q^{81} + ( -2 + 2 \beta_{2} ) q^{82} + ( -6 + 3 \beta_{1} - 3 \beta_{2} ) q^{83} - q^{84} + ( -\beta_{1} - \beta_{2} ) q^{85} + ( -6 + \beta_{1} - 2 \beta_{2} ) q^{86} + ( -\beta_{1} - 2 \beta_{2} ) q^{87} + ( -2 + \beta_{1} ) q^{88} + ( -2 - 4 \beta_{1} + 5 \beta_{2} ) q^{89} - q^{90} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{91} - q^{92} + ( -6 - \beta_{1} + \beta_{2} ) q^{93} + ( -6 - \beta_{1} + \beta_{2} ) q^{94} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{95} + q^{96} + ( -6 - 4 \beta_{1} + \beta_{2} ) q^{97} + q^{98} + ( -2 + \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} + 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} + 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} - 3q^{10} - 6q^{11} + 3q^{12} - 6q^{13} - 3q^{14} - 3q^{15} + 3q^{16} + 3q^{18} - 6q^{19} - 3q^{20} - 3q^{21} - 6q^{22} - 3q^{23} + 3q^{24} + 3q^{25} - 6q^{26} + 3q^{27} - 3q^{28} - 3q^{30} - 18q^{31} + 3q^{32} - 6q^{33} + 3q^{35} + 3q^{36} - 6q^{37} - 6q^{38} - 6q^{39} - 3q^{40} - 6q^{41} - 3q^{42} - 18q^{43} - 6q^{44} - 3q^{45} - 3q^{46} - 18q^{47} + 3q^{48} + 3q^{49} + 3q^{50} - 6q^{52} + 18q^{53} + 3q^{54} + 6q^{55} - 3q^{56} - 6q^{57} - 24q^{59} - 3q^{60} + 6q^{61} - 18q^{62} - 3q^{63} + 3q^{64} + 6q^{65} - 6q^{66} - 18q^{67} - 3q^{69} + 3q^{70} - 6q^{71} + 3q^{72} + 6q^{73} - 6q^{74} + 3q^{75} - 6q^{76} + 6q^{77} - 6q^{78} - 3q^{80} + 3q^{81} - 6q^{82} - 18q^{83} - 3q^{84} - 18q^{86} - 6q^{88} - 6q^{89} - 3q^{90} + 6q^{91} - 3q^{92} - 18q^{93} - 18q^{94} + 6q^{95} + 3q^{96} - 18q^{97} + 3q^{98} - 6q^{99} + O(q^{100}) \)

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
−1.53209
−0.347296
1.87939
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.cc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4830.2.a.cc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4830))\):

\( T_{11}^{3} + 6 T_{11}^{2} - 24 \)
\( T_{13}^{3} + 6 T_{13}^{2} - 24 T_{13} - 136 \)
\( T_{17}^{3} - 36 T_{17} - 72 \)
\( T_{19}^{3} + 6 T_{19}^{2} - 72 T_{19} - 456 \)
\( T_{29}^{3} - 84 T_{29} + 136 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( -24 + 6 T^{2} + T^{3} \)
$13$ \( -136 - 24 T + 6 T^{2} + T^{3} \)
$17$ \( -72 - 36 T + T^{3} \)
$19$ \( -456 - 72 T + 6 T^{2} + T^{3} \)
$23$ \( ( 1 + T )^{3} \)
$29$ \( 136 - 84 T + T^{3} \)
$31$ \( 136 + 96 T + 18 T^{2} + T^{3} \)
$37$ \( -24 - 36 T + 6 T^{2} + T^{3} \)
$41$ \( -24 - 36 T + 6 T^{2} + T^{3} \)
$43$ \( -72 + 72 T + 18 T^{2} + T^{3} \)
$47$ \( 136 + 96 T + 18 T^{2} + T^{3} \)
$53$ \( 8 + 60 T - 18 T^{2} + T^{3} \)
$59$ \( ( 8 + T )^{3} \)
$61$ \( 152 - 36 T - 6 T^{2} + T^{3} \)
$67$ \( -424 + 24 T + 18 T^{2} + T^{3} \)
$71$ \( -424 - 96 T + 6 T^{2} + T^{3} \)
$73$ \( 888 - 180 T - 6 T^{2} + T^{3} \)
$79$ \( 576 - 144 T + T^{3} \)
$83$ \( -216 + 18 T^{2} + T^{3} \)
$89$ \( -424 - 240 T + 6 T^{2} + T^{3} \)
$97$ \( -872 - 48 T + 18 T^{2} + T^{3} \)
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