Properties

Label 2898.2.a.bd
Level $2898$
Weight $2$
Character orbit 2898.a
Self dual yes
Analytic conductor $23.141$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( 1 + \beta ) q^{5} - q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( 1 + \beta ) q^{5} - q^{7} + q^{8} + ( 1 + \beta ) q^{10} + ( -2 - 2 \beta ) q^{13} - q^{14} + q^{16} + ( 5 - \beta ) q^{17} + 2 \beta q^{19} + ( 1 + \beta ) q^{20} + q^{23} + ( 1 + 2 \beta ) q^{25} + ( -2 - 2 \beta ) q^{26} - q^{28} + 2 q^{29} + ( 1 + \beta ) q^{31} + q^{32} + ( 5 - \beta ) q^{34} + ( -1 - \beta ) q^{35} + 6 q^{37} + 2 \beta q^{38} + ( 1 + \beta ) q^{40} + 10 q^{41} + ( -2 + 2 \beta ) q^{43} + q^{46} + ( 9 + \beta ) q^{47} + q^{49} + ( 1 + 2 \beta ) q^{50} + ( -2 - 2 \beta ) q^{52} + 6 q^{53} - q^{56} + 2 q^{58} + ( -9 + \beta ) q^{59} + ( 5 - 3 \beta ) q^{61} + ( 1 + \beta ) q^{62} + q^{64} + ( -12 - 4 \beta ) q^{65} + 4 q^{67} + ( 5 - \beta ) q^{68} + ( -1 - \beta ) q^{70} + ( -2 - 2 \beta ) q^{71} + 6 \beta q^{73} + 6 q^{74} + 2 \beta q^{76} -4 \beta q^{79} + ( 1 + \beta ) q^{80} + 10 q^{82} + ( -2 - 4 \beta ) q^{83} + 4 \beta q^{85} + ( -2 + 2 \beta ) q^{86} + ( 5 - 5 \beta ) q^{89} + ( 2 + 2 \beta ) q^{91} + q^{92} + ( 9 + \beta ) q^{94} + ( 10 + 2 \beta ) q^{95} + ( -3 - 5 \beta ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + 2q^{8} + 2q^{10} - 4q^{13} - 2q^{14} + 2q^{16} + 10q^{17} + 2q^{20} + 2q^{23} + 2q^{25} - 4q^{26} - 2q^{28} + 4q^{29} + 2q^{31} + 2q^{32} + 10q^{34} - 2q^{35} + 12q^{37} + 2q^{40} + 20q^{41} - 4q^{43} + 2q^{46} + 18q^{47} + 2q^{49} + 2q^{50} - 4q^{52} + 12q^{53} - 2q^{56} + 4q^{58} - 18q^{59} + 10q^{61} + 2q^{62} + 2q^{64} - 24q^{65} + 8q^{67} + 10q^{68} - 2q^{70} - 4q^{71} + 12q^{74} + 2q^{80} + 20q^{82} - 4q^{83} - 4q^{86} + 10q^{89} + 4q^{91} + 2q^{92} + 18q^{94} + 20q^{95} - 6q^{97} + 2q^{98} + O(q^{100}) \)

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
−0.618034
1.61803
1.00000 0 1.00000 −1.23607 0 −1.00000 1.00000 0 −1.23607
1.2 1.00000 0 1.00000 3.23607 0 −1.00000 1.00000 0 3.23607
\(n\): 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.bd 2
3.b odd 2 1 322.2.a.e 2
12.b even 2 1 2576.2.a.t 2
15.d odd 2 1 8050.2.a.bf 2
21.c even 2 1 2254.2.a.k 2
69.c even 2 1 7406.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.a.e 2 3.b odd 2 1
2254.2.a.k 2 21.c even 2 1
2576.2.a.t 2 12.b even 2 1
2898.2.a.bd 2 1.a even 1 1 trivial
7406.2.a.j 2 69.c even 2 1
8050.2.a.bf 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2898))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -4 - 2 T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( -16 + 4 T + T^{2} \)
$17$ \( 20 - 10 T + T^{2} \)
$19$ \( -20 + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( -4 - 2 T + T^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( ( -10 + T )^{2} \)
$43$ \( -16 + 4 T + T^{2} \)
$47$ \( 76 - 18 T + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 76 + 18 T + T^{2} \)
$61$ \( -20 - 10 T + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( -16 + 4 T + T^{2} \)
$73$ \( -180 + T^{2} \)
$79$ \( -80 + T^{2} \)
$83$ \( -76 + 4 T + T^{2} \)
$89$ \( -100 - 10 T + T^{2} \)
$97$ \( -116 + 6 T + T^{2} \)
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