Properties

Label 2898.2.a.bf
Level $2898$
Weight $2$
Character orbit 2898.a
Self dual yes
Analytic conductor $23.141$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.1406465058\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{2} q^{5} - q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta_{2} q^{5} - q^{7} - q^{8} -\beta_{2} q^{10} -2 \beta_{1} q^{11} + ( 2 - \beta_{1} ) q^{13} + q^{14} + q^{16} + ( -\beta_{1} + \beta_{2} ) q^{17} + ( 2 + \beta_{1} + \beta_{2} ) q^{19} + \beta_{2} q^{20} + 2 \beta_{1} q^{22} + q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{25} + ( -2 + \beta_{1} ) q^{26} - q^{28} + ( -4 + 2 \beta_{1} + \beta_{2} ) q^{29} + ( 2 - \beta_{1} + \beta_{2} ) q^{31} - q^{32} + ( \beta_{1} - \beta_{2} ) q^{34} -\beta_{2} q^{35} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{37} + ( -2 - \beta_{1} - \beta_{2} ) q^{38} -\beta_{2} q^{40} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{41} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{43} -2 \beta_{1} q^{44} - q^{46} + ( -4 - \beta_{1} - 2 \beta_{2} ) q^{47} + q^{49} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{50} + ( 2 - \beta_{1} ) q^{52} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 4 + 2 \beta_{2} ) q^{55} + q^{56} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{58} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( -2 + \beta_{1} - \beta_{2} ) q^{62} + q^{64} + ( 2 + 3 \beta_{2} ) q^{65} + 4 \beta_{1} q^{67} + ( -\beta_{1} + \beta_{2} ) q^{68} + \beta_{2} q^{70} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{71} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{74} + ( 2 + \beta_{1} + \beta_{2} ) q^{76} + 2 \beta_{1} q^{77} + ( 2 - 2 \beta_{1} ) q^{79} + \beta_{2} q^{80} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{82} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{83} + ( 8 - 2 \beta_{1} ) q^{85} + ( -2 - 2 \beta_{1} - 3 \beta_{2} ) q^{86} + 2 \beta_{1} q^{88} + ( -5 \beta_{1} - 5 \beta_{2} ) q^{89} + ( -2 + \beta_{1} ) q^{91} + q^{92} + ( 4 + \beta_{1} + 2 \beta_{2} ) q^{94} + ( 4 - 2 \beta_{1} ) q^{95} + ( 2 + \beta_{1} ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} - q^{5} - 3q^{7} - 3q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} - q^{5} - 3q^{7} - 3q^{8} + q^{10} - 2q^{11} + 5q^{13} + 3q^{14} + 3q^{16} - 2q^{17} + 6q^{19} - q^{20} + 2q^{22} + 3q^{23} + 2q^{25} - 5q^{26} - 3q^{28} - 11q^{29} + 4q^{31} - 3q^{32} + 2q^{34} + q^{35} + 5q^{37} - 6q^{38} + q^{40} - 9q^{41} + 5q^{43} - 2q^{44} - 3q^{46} - 11q^{47} + 3q^{49} - 2q^{50} + 5q^{52} + 2q^{53} + 10q^{55} + 3q^{56} + 11q^{58} + 10q^{61} - 4q^{62} + 3q^{64} + 3q^{65} + 4q^{67} - 2q^{68} - q^{70} + 10q^{71} + 18q^{73} - 5q^{74} + 6q^{76} + 2q^{77} + 4q^{79} - q^{80} + 9q^{82} - 4q^{83} + 22q^{85} - 5q^{86} + 2q^{88} - 5q^{91} + 3q^{92} + 11q^{94} + 10q^{95} + 7q^{97} - 3q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 6 x - 2\):

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

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
−0.363328
3.12489
−1.76156
−1.00000 0 1.00000 −3.14134 0 −1.00000 −1.00000 0 3.14134
1.2 −1.00000 0 1.00000 −0.484862 0 −1.00000 −1.00000 0 0.484862
1.3 −1.00000 0 1.00000 2.62620 0 −1.00000 −1.00000 0 −2.62620
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 2898.2.a.bf 3
3.b odd 2 1 2898.2.a.bg yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2898.2.a.bf 3 1.a even 1 1 trivial
2898.2.a.bg yes 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2898))\):

\( T_{5}^{3} + T_{5}^{2} - 8 T_{5} - 4 \)
\( T_{11}^{3} + 2 T_{11}^{2} - 24 T_{11} + 16 \)
\( T_{13}^{3} - 5 T_{13}^{2} + 2 T_{13} + 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( T^{3} \)
$5$ \( -4 - 8 T + T^{2} + T^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( 16 - 24 T + 2 T^{2} + T^{3} \)
$13$ \( 10 + 2 T - 5 T^{2} + T^{3} \)
$17$ \( -44 - 18 T + 2 T^{2} + T^{3} \)
$19$ \( 20 + 2 T - 6 T^{2} + T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( -68 + 16 T + 11 T^{2} + T^{3} \)
$31$ \( -8 - 14 T - 4 T^{2} + T^{3} \)
$37$ \( 64 - 16 T - 5 T^{2} + T^{3} \)
$41$ \( -44 - 16 T + 9 T^{2} + T^{3} \)
$43$ \( 352 - 64 T - 5 T^{2} + T^{3} \)
$47$ \( -122 + 10 T + 11 T^{2} + T^{3} \)
$53$ \( -160 - 96 T - 2 T^{2} + T^{3} \)
$59$ \( T^{3} \)
$61$ \( 472 - 44 T - 10 T^{2} + T^{3} \)
$67$ \( -128 - 96 T - 4 T^{2} + T^{3} \)
$71$ \( 848 - 88 T - 10 T^{2} + T^{3} \)
$73$ \( 88 + 68 T - 18 T^{2} + T^{3} \)
$79$ \( 64 - 20 T - 4 T^{2} + T^{3} \)
$83$ \( -232 - 62 T + 4 T^{2} + T^{3} \)
$89$ \( -1000 - 250 T + T^{3} \)
$97$ \( -2 + 10 T - 7 T^{2} + T^{3} \)
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