Properties

Label 2898.2.a.ba
Level $2898$
Weight $2$
Character orbit 2898.a
Self dual yes
Analytic conductor $23.141$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} -\beta q^{5} - q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} -\beta q^{5} - q^{7} + q^{8} -\beta q^{10} + ( -2 + 2 \beta ) q^{11} + ( -2 + \beta ) q^{13} - q^{14} + q^{16} + ( -2 - 2 \beta ) q^{17} + ( -2 + 2 \beta ) q^{19} -\beta q^{20} + ( -2 + 2 \beta ) q^{22} - q^{23} + ( -1 + \beta ) q^{25} + ( -2 + \beta ) q^{26} - q^{28} + ( -6 + \beta ) q^{29} -2 \beta q^{31} + q^{32} + ( -2 - 2 \beta ) q^{34} + \beta q^{35} + \beta q^{37} + ( -2 + 2 \beta ) q^{38} -\beta q^{40} + ( -2 + \beta ) q^{41} + ( 2 - 3 \beta ) q^{43} + ( -2 + 2 \beta ) q^{44} - q^{46} + ( -4 - \beta ) q^{47} + q^{49} + ( -1 + \beta ) q^{50} + ( -2 + \beta ) q^{52} + ( -8 + 4 \beta ) q^{53} -8 q^{55} - q^{56} + ( -6 + \beta ) q^{58} + ( -8 - 2 \beta ) q^{59} + ( 4 + 2 \beta ) q^{61} -2 \beta q^{62} + q^{64} + ( -4 + \beta ) q^{65} + ( 2 + 2 \beta ) q^{67} + ( -2 - 2 \beta ) q^{68} + \beta q^{70} -6 \beta q^{71} + ( 6 - 2 \beta ) q^{73} + \beta q^{74} + ( -2 + 2 \beta ) q^{76} + ( 2 - 2 \beta ) q^{77} + ( 8 - 4 \beta ) q^{79} -\beta q^{80} + ( -2 + \beta ) q^{82} + ( 6 - 2 \beta ) q^{83} + ( 8 + 4 \beta ) q^{85} + ( 2 - 3 \beta ) q^{86} + ( -2 + 2 \beta ) q^{88} -14 q^{89} + ( 2 - \beta ) q^{91} - q^{92} + ( -4 - \beta ) q^{94} -8 q^{95} + ( -14 - \beta ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - q^{5} - 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - q^{5} - 2q^{7} + 2q^{8} - q^{10} - 2q^{11} - 3q^{13} - 2q^{14} + 2q^{16} - 6q^{17} - 2q^{19} - q^{20} - 2q^{22} - 2q^{23} - q^{25} - 3q^{26} - 2q^{28} - 11q^{29} - 2q^{31} + 2q^{32} - 6q^{34} + q^{35} + q^{37} - 2q^{38} - q^{40} - 3q^{41} + q^{43} - 2q^{44} - 2q^{46} - 9q^{47} + 2q^{49} - q^{50} - 3q^{52} - 12q^{53} - 16q^{55} - 2q^{56} - 11q^{58} - 18q^{59} + 10q^{61} - 2q^{62} + 2q^{64} - 7q^{65} + 6q^{67} - 6q^{68} + q^{70} - 6q^{71} + 10q^{73} + q^{74} - 2q^{76} + 2q^{77} + 12q^{79} - q^{80} - 3q^{82} + 10q^{83} + 20q^{85} + q^{86} - 2q^{88} - 28q^{89} + 3q^{91} - 2q^{92} - 9q^{94} - 16q^{95} - 29q^{97} + 2q^{98} + O(q^{100}) \)

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
2.56155
−1.56155
1.00000 0 1.00000 −2.56155 0 −1.00000 1.00000 0 −2.56155
1.2 1.00000 0 1.00000 1.56155 0 −1.00000 1.00000 0 1.56155
\(n\): 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.ba 2
3.b odd 2 1 966.2.a.l 2
12.b even 2 1 7728.2.a.bm 2
21.c even 2 1 6762.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.l 2 3.b odd 2 1
2898.2.a.ba 2 1.a even 1 1 trivial
6762.2.a.bw 2 21.c even 2 1
7728.2.a.bm 2 12.b even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -4 + T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -16 + 2 T + T^{2} \)
$13$ \( -2 + 3 T + T^{2} \)
$17$ \( -8 + 6 T + T^{2} \)
$19$ \( -16 + 2 T + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( 26 + 11 T + T^{2} \)
$31$ \( -16 + 2 T + T^{2} \)
$37$ \( -4 - T + T^{2} \)
$41$ \( -2 + 3 T + T^{2} \)
$43$ \( -38 - T + T^{2} \)
$47$ \( 16 + 9 T + T^{2} \)
$53$ \( -32 + 12 T + T^{2} \)
$59$ \( 64 + 18 T + T^{2} \)
$61$ \( 8 - 10 T + T^{2} \)
$67$ \( -8 - 6 T + T^{2} \)
$71$ \( -144 + 6 T + T^{2} \)
$73$ \( 8 - 10 T + T^{2} \)
$79$ \( -32 - 12 T + T^{2} \)
$83$ \( 8 - 10 T + T^{2} \)
$89$ \( ( 14 + T )^{2} \)
$97$ \( 206 + 29 T + T^{2} \)
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