Properties

Label 2898.2.a.be
Level $2898$
Weight $2$
Character orbit 2898.a
Self dual yes
Analytic conductor $23.141$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 322)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 3 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 6\)\()/2\)

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
0.470683
2.34292
−1.81361
−1.00000 0 1.00000 −4.24914 0 −1.00000 −1.00000 0 4.24914
1.2 −1.00000 0 1.00000 −0.853635 0 −1.00000 −1.00000 0 0.853635
1.3 −1.00000 0 1.00000 1.10278 0 −1.00000 −1.00000 0 −1.10278
\(n\): 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.be 3
3.b odd 2 1 322.2.a.g 3
12.b even 2 1 2576.2.a.w 3
15.d odd 2 1 8050.2.a.bh 3
21.c even 2 1 2254.2.a.p 3
69.c even 2 1 7406.2.a.x 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.a.g 3 3.b odd 2 1
2254.2.a.p 3 21.c even 2 1
2576.2.a.w 3 12.b even 2 1
2898.2.a.be 3 1.a even 1 1 trivial
7406.2.a.x 3 69.c even 2 1
8050.2.a.bh 3 15.d odd 2 1

Hecke kernels

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

\( T_{5}^{3} + 4 T_{5}^{2} - 2 T_{5} - 4 \)
\( T_{11}^{3} - 4 T_{11}^{2} - 12 T_{11} + 16 \)
\( T_{13}^{3} - 2 T_{13}^{2} - 16 T_{13} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( T^{3} \)
$5$ \( -4 - 2 T + 4 T^{2} + T^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( 16 - 12 T - 4 T^{2} + T^{3} \)
$13$ \( 16 - 16 T - 2 T^{2} + T^{3} \)
$17$ \( -44 + 6 T + 8 T^{2} + T^{3} \)
$19$ \( -16 - 28 T + T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( 352 - 32 T - 10 T^{2} + T^{3} \)
$31$ \( -32 - 14 T + 2 T^{2} + T^{3} \)
$37$ \( -344 - 28 T + 10 T^{2} + T^{3} \)
$41$ \( -344 - 100 T + 6 T^{2} + T^{3} \)
$43$ \( -128 - 64 T + 4 T^{2} + T^{3} \)
$47$ \( -32 - 14 T + 2 T^{2} + T^{3} \)
$53$ \( -136 - 60 T - 2 T^{2} + T^{3} \)
$59$ \( -216 - 54 T + 6 T^{2} + T^{3} \)
$61$ \( -524 - 74 T + 8 T^{2} + T^{3} \)
$67$ \( 16 - 12 T - 4 T^{2} + T^{3} \)
$71$ \( -64 + 200 T - 28 T^{2} + T^{3} \)
$73$ \( -1448 - 220 T + 6 T^{2} + T^{3} \)
$79$ \( -2432 - 92 T + 20 T^{2} + T^{3} \)
$83$ \( 656 + 244 T + 28 T^{2} + T^{3} \)
$89$ \( -76 - 34 T + T^{3} \)
$97$ \( 172 - 26 T - 16 T^{2} + T^{3} \)
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