Properties

Label 2898.2.a.bh
Level $2898$
Weight $2$
Character orbit 2898.a
Self dual yes
Analytic conductor $23.141$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.271296.1
Defining polynomial: \(x^{4} - 18 x^{2} - 8 x + 24\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{1} q^{5} + q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta_{1} q^{5} + q^{7} - q^{8} -\beta_{1} q^{10} + ( -2 + \beta_{1} ) q^{11} + \beta_{2} q^{13} - q^{14} + q^{16} + \beta_{2} q^{17} + ( \beta_{1} + \beta_{2} ) q^{19} + \beta_{1} q^{20} + ( 2 - \beta_{1} ) q^{22} - q^{23} + ( 4 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} -\beta_{2} q^{26} + q^{28} -2 q^{29} + ( 1 - \beta_{1} - \beta_{3} ) q^{31} - q^{32} -\beta_{2} q^{34} + \beta_{1} q^{35} + ( 1 - \beta_{2} - \beta_{3} ) q^{37} + ( -\beta_{1} - \beta_{2} ) q^{38} -\beta_{1} q^{40} + 2 q^{41} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{43} + ( -2 + \beta_{1} ) q^{44} + q^{46} + ( 3 + \beta_{1} + \beta_{3} ) q^{47} + q^{49} + ( -4 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{50} + \beta_{2} q^{52} + ( -3 - \beta_{2} - \beta_{3} ) q^{53} + ( 9 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{55} - q^{56} + 2 q^{58} + ( 4 - 2 \beta_{1} ) q^{59} + 3 \beta_{1} q^{61} + ( -1 + \beta_{1} + \beta_{3} ) q^{62} + q^{64} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{65} + ( 6 + 3 \beta_{1} + 2 \beta_{2} ) q^{67} + \beta_{2} q^{68} -\beta_{1} q^{70} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{71} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{73} + ( -1 + \beta_{2} + \beta_{3} ) q^{74} + ( \beta_{1} + \beta_{2} ) q^{76} + ( -2 + \beta_{1} ) q^{77} + ( 7 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{79} + \beta_{1} q^{80} -2 q^{82} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{85} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{86} + ( 2 - \beta_{1} ) q^{88} + ( -2 - \beta_{2} - 2 \beta_{3} ) q^{89} + \beta_{2} q^{91} - q^{92} + ( -3 - \beta_{1} - \beta_{3} ) q^{94} + ( 9 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{95} + ( -2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{4} + 4q^{7} - 4q^{8} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{4} + 4q^{7} - 4q^{8} - 8q^{11} - 4q^{14} + 4q^{16} + 8q^{22} - 4q^{23} + 16q^{25} + 4q^{28} - 8q^{29} + 4q^{31} - 4q^{32} + 4q^{37} + 8q^{41} + 8q^{43} - 8q^{44} + 4q^{46} + 12q^{47} + 4q^{49} - 16q^{50} - 12q^{53} + 36q^{55} - 4q^{56} + 8q^{58} + 16q^{59} - 4q^{62} + 4q^{64} + 24q^{67} + 4q^{71} + 12q^{73} - 4q^{74} - 8q^{77} + 28q^{79} - 8q^{82} - 8q^{83} - 8q^{86} + 8q^{88} - 8q^{89} - 4q^{92} - 12q^{94} + 36q^{95} - 8q^{97} - 4q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 18 x^{2} - 8 x + 24\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 2 \nu^{2} - 14 \nu + 12 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 18 \nu - 24 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - \beta_{2} + \beta_{1} + 9\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} + 2 \beta_{2} + 16 \beta_{1} + 6\)

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
−3.76613
−1.51727
0.974808
4.30859
−1.00000 0 1.00000 −3.76613 0 1.00000 −1.00000 0 3.76613
1.2 −1.00000 0 1.00000 −1.51727 0 1.00000 −1.00000 0 1.51727
1.3 −1.00000 0 1.00000 0.974808 0 1.00000 −1.00000 0 −0.974808
1.4 −1.00000 0 1.00000 4.30859 0 1.00000 −1.00000 0 −4.30859
\(n\): 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.bh 4
3.b odd 2 1 2898.2.a.bi yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2898.2.a.bh 4 1.a even 1 1 trivial
2898.2.a.bi yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2898))\):

\( T_{5}^{4} - 18 T_{5}^{2} - 8 T_{5} + 24 \)
\( T_{11}^{4} + 8 T_{11}^{3} + 6 T_{11}^{2} - 48 T_{11} - 48 \)
\( T_{13}^{4} - 30 T_{13}^{2} - 56 T_{13} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( T^{4} \)
$5$ \( 24 - 8 T - 18 T^{2} + T^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( -48 - 48 T + 6 T^{2} + 8 T^{3} + T^{4} \)
$13$ \( -24 - 56 T - 30 T^{2} + T^{4} \)
$17$ \( -24 - 56 T - 30 T^{2} + T^{4} \)
$19$ \( -36 + 128 T - 48 T^{2} + T^{4} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( ( 2 + T )^{4} \)
$31$ \( -8 + 104 T - 66 T^{2} - 4 T^{3} + T^{4} \)
$37$ \( 64 - 64 T - 78 T^{2} - 4 T^{3} + T^{4} \)
$41$ \( ( -2 + T )^{4} \)
$43$ \( -3168 + 1296 T - 114 T^{2} - 8 T^{3} + T^{4} \)
$47$ \( -648 + 360 T - 18 T^{2} - 12 T^{3} + T^{4} \)
$53$ \( -1440 - 624 T - 30 T^{2} + 12 T^{3} + T^{4} \)
$59$ \( -768 + 384 T + 24 T^{2} - 16 T^{3} + T^{4} \)
$61$ \( 1944 - 216 T - 162 T^{2} + T^{4} \)
$67$ \( -23904 + 4592 T - 66 T^{2} - 24 T^{3} + T^{4} \)
$71$ \( 416 + 448 T - 108 T^{2} - 4 T^{3} + T^{4} \)
$73$ \( 192 + 544 T - 36 T^{2} - 12 T^{3} + T^{4} \)
$79$ \( -1536 + 204 T^{2} - 28 T^{3} + T^{4} \)
$83$ \( 10364 - 2000 T - 288 T^{2} + 8 T^{3} + T^{4} \)
$89$ \( -11304 - 3384 T - 246 T^{2} + 8 T^{3} + T^{4} \)
$97$ \( 872 - 1928 T - 270 T^{2} + 8 T^{3} + T^{4} \)
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