Properties

Label 4830.2.a.cd
Level $4830$
Weight $2$
Character orbit 4830.a
Self dual yes
Analytic conductor $38.568$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.10273.1
Defining polynomial: \(x^{4} - 2 x^{3} - 5 x^{2} + x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2,\beta_3\) 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_{3} ) q^{11} - q^{12} + ( 1 + \beta_{2} - \beta_{3} ) q^{13} - q^{14} + q^{15} + q^{16} + ( -2 \beta_{1} + \beta_{2} ) q^{17} - q^{18} + ( 4 + \beta_{2} ) q^{19} - q^{20} - q^{21} + ( -1 - \beta_{3} ) q^{22} - q^{23} + q^{24} + q^{25} + ( -1 - \beta_{2} + \beta_{3} ) q^{26} - q^{27} + q^{28} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{29} - q^{30} + ( 5 - \beta_{1} ) q^{31} - q^{32} + ( -1 - \beta_{3} ) q^{33} + ( 2 \beta_{1} - \beta_{2} ) q^{34} - q^{35} + q^{36} + ( 1 + \beta_{1} + \beta_{2} ) q^{37} + ( -4 - \beta_{2} ) q^{38} + ( -1 - \beta_{2} + \beta_{3} ) q^{39} + q^{40} + 2 \beta_{1} q^{41} + q^{42} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + ( 1 + \beta_{3} ) q^{44} - q^{45} + q^{46} + ( -2 + \beta_{2} - 2 \beta_{3} ) q^{47} - q^{48} + q^{49} - q^{50} + ( 2 \beta_{1} - \beta_{2} ) q^{51} + ( 1 + \beta_{2} - \beta_{3} ) q^{52} + ( 1 + \beta_{1} - \beta_{2} ) q^{53} + q^{54} + ( -1 - \beta_{3} ) q^{55} - q^{56} + ( -4 - \beta_{2} ) q^{57} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{58} + ( 1 - \beta_{1} - \beta_{2} ) q^{59} + q^{60} -2 \beta_{3} q^{61} + ( -5 + \beta_{1} ) q^{62} + q^{63} + q^{64} + ( -1 - \beta_{2} + \beta_{3} ) q^{65} + ( 1 + \beta_{3} ) q^{66} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{67} + ( -2 \beta_{1} + \beta_{2} ) q^{68} + q^{69} + q^{70} + ( -4 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{71} - q^{72} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{73} + ( -1 - \beta_{1} - \beta_{2} ) q^{74} - q^{75} + ( 4 + \beta_{2} ) q^{76} + ( 1 + \beta_{3} ) q^{77} + ( 1 + \beta_{2} - \beta_{3} ) q^{78} + ( 5 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{79} - q^{80} + q^{81} -2 \beta_{1} q^{82} + ( 2 + \beta_{2} + 2 \beta_{3} ) q^{83} - q^{84} + ( 2 \beta_{1} - \beta_{2} ) q^{85} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{86} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{87} + ( -1 - \beta_{3} ) q^{88} + ( 2 - \beta_{1} - \beta_{3} ) q^{89} + q^{90} + ( 1 + \beta_{2} - \beta_{3} ) q^{91} - q^{92} + ( -5 + \beta_{1} ) q^{93} + ( 2 - \beta_{2} + 2 \beta_{3} ) q^{94} + ( -4 - \beta_{2} ) q^{95} + q^{96} + ( -6 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{97} - q^{98} + ( 1 + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + 4q^{7} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + 4q^{7} - 4q^{8} + 4q^{9} + 4q^{10} + 2q^{11} - 4q^{12} + 4q^{13} - 4q^{14} + 4q^{15} + 4q^{16} - 2q^{17} - 4q^{18} + 14q^{19} - 4q^{20} - 4q^{21} - 2q^{22} - 4q^{23} + 4q^{24} + 4q^{25} - 4q^{26} - 4q^{27} + 4q^{28} - 4q^{29} - 4q^{30} + 20q^{31} - 4q^{32} - 2q^{33} + 2q^{34} - 4q^{35} + 4q^{36} + 2q^{37} - 14q^{38} - 4q^{39} + 4q^{40} + 4q^{42} + 8q^{43} + 2q^{44} - 4q^{45} + 4q^{46} - 6q^{47} - 4q^{48} + 4q^{49} - 4q^{50} + 2q^{51} + 4q^{52} + 6q^{53} + 4q^{54} - 2q^{55} - 4q^{56} - 14q^{57} + 4q^{58} + 6q^{59} + 4q^{60} + 4q^{61} - 20q^{62} + 4q^{63} + 4q^{64} - 4q^{65} + 2q^{66} - 4q^{67} - 2q^{68} + 4q^{69} + 4q^{70} - 16q^{71} - 4q^{72} - 14q^{73} - 2q^{74} - 4q^{75} + 14q^{76} + 2q^{77} + 4q^{78} + 18q^{79} - 4q^{80} + 4q^{81} + 2q^{83} - 4q^{84} + 2q^{85} - 8q^{86} + 4q^{87} - 2q^{88} + 10q^{89} + 4q^{90} + 4q^{91} - 4q^{92} - 20q^{93} + 6q^{94} - 14q^{95} + 4q^{96} - 18q^{97} - 4q^{98} + 2q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} - 4 \nu^{2} - 8 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{3} - 6 \nu^{2} - 4 \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + 2 \beta_{1} + 7\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3} + 3 \beta_{2} + 8 \beta_{1} + 18\)\()/2\)

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
−1.38266
3.35017
0.673533
−0.641043
−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
1.4 −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.cd 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4830.2.a.cd 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4830))\):

\( T_{11}^{4} - 2 T_{11}^{3} - 28 T_{11}^{2} + 48 T_{11} - 16 \)
\( T_{13}^{4} - 4 T_{13}^{3} - 36 T_{13}^{2} + 136 T_{13} + 96 \)
\( T_{17}^{4} + 2 T_{17}^{3} - 100 T_{17}^{2} - 112 T_{17} + 2448 \)
\( T_{19}^{4} - 14 T_{19}^{3} + 40 T_{19}^{2} + 104 T_{19} - 288 \)
\( T_{29}^{4} + 4 T_{29}^{3} - 76 T_{29}^{2} - 504 T_{29} - 784 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( -16 + 48 T - 28 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( 96 + 136 T - 36 T^{2} - 4 T^{3} + T^{4} \)
$17$ \( 2448 - 112 T - 100 T^{2} + 2 T^{3} + T^{4} \)
$19$ \( -288 + 104 T + 40 T^{2} - 14 T^{3} + T^{4} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( -784 - 504 T - 76 T^{2} + 4 T^{3} + T^{4} \)
$31$ \( -208 - 200 T + 124 T^{2} - 20 T^{3} + T^{4} \)
$37$ \( 416 - 120 T - 76 T^{2} - 2 T^{3} + T^{4} \)
$41$ \( 272 - 320 T - 104 T^{2} + T^{4} \)
$43$ \( -544 + 472 T - 52 T^{2} - 8 T^{3} + T^{4} \)
$47$ \( 1632 - 440 T - 96 T^{2} + 6 T^{3} + T^{4} \)
$53$ \( 96 + 88 T - 28 T^{2} - 6 T^{3} + T^{4} \)
$59$ \( -128 + 416 T - 64 T^{2} - 6 T^{3} + T^{4} \)
$61$ \( 48 + 80 T - 112 T^{2} - 4 T^{3} + T^{4} \)
$67$ \( 3264 - 184 T - 132 T^{2} + 4 T^{3} + T^{4} \)
$71$ \( -416 - 168 T + 36 T^{2} + 16 T^{3} + T^{4} \)
$73$ \( 2592 - 728 T - 60 T^{2} + 14 T^{3} + T^{4} \)
$79$ \( -6272 + 1568 T - 16 T^{2} - 18 T^{3} + T^{4} \)
$83$ \( 3232 + 72 T - 192 T^{2} - 2 T^{3} + T^{4} \)
$89$ \( 16 + 48 T - 28 T^{2} - 10 T^{3} + T^{4} \)
$97$ \( 13264 - 2048 T - 156 T^{2} + 18 T^{3} + T^{4} \)
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