Properties

Label 2898.2.a.x
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: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( 1 + \beta ) q^{5} - q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( 1 + \beta ) q^{5} - q^{7} - q^{8} + ( -1 - \beta ) q^{10} -4 q^{11} + ( -1 - \beta ) q^{13} + q^{14} + q^{16} -2 \beta q^{17} + 4 q^{19} + ( 1 + \beta ) q^{20} + 4 q^{22} + q^{23} + ( 4 + 3 \beta ) q^{25} + ( 1 + \beta ) q^{26} - q^{28} + ( -7 + \beta ) q^{29} + ( 2 - 2 \beta ) q^{31} - q^{32} + 2 \beta q^{34} + ( -1 - \beta ) q^{35} + ( 5 + \beta ) q^{37} -4 q^{38} + ( -1 - \beta ) q^{40} + ( 1 - 3 \beta ) q^{41} + ( -1 - 3 \beta ) q^{43} -4 q^{44} - q^{46} + ( -7 - \beta ) q^{47} + q^{49} + ( -4 - 3 \beta ) q^{50} + ( -1 - \beta ) q^{52} + 2 \beta q^{53} + ( -4 - 4 \beta ) q^{55} + q^{56} + ( 7 - \beta ) q^{58} + ( -2 - 2 \beta ) q^{59} -2 q^{61} + ( -2 + 2 \beta ) q^{62} + q^{64} + ( -9 - 3 \beta ) q^{65} -4 q^{67} -2 \beta q^{68} + ( 1 + \beta ) q^{70} + ( -10 + 2 \beta ) q^{71} + ( -8 + 2 \beta ) q^{73} + ( -5 - \beta ) q^{74} + 4 q^{76} + 4 q^{77} + ( -4 + 4 \beta ) q^{79} + ( 1 + \beta ) q^{80} + ( -1 + 3 \beta ) q^{82} -4 q^{83} + ( -16 - 4 \beta ) q^{85} + ( 1 + 3 \beta ) q^{86} + 4 q^{88} + ( 2 + 4 \beta ) q^{89} + ( 1 + \beta ) q^{91} + q^{92} + ( 7 + \beta ) q^{94} + ( 4 + 4 \beta ) q^{95} + ( -7 + \beta ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 3q^{5} - 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 3q^{5} - 2q^{7} - 2q^{8} - 3q^{10} - 8q^{11} - 3q^{13} + 2q^{14} + 2q^{16} - 2q^{17} + 8q^{19} + 3q^{20} + 8q^{22} + 2q^{23} + 11q^{25} + 3q^{26} - 2q^{28} - 13q^{29} + 2q^{31} - 2q^{32} + 2q^{34} - 3q^{35} + 11q^{37} - 8q^{38} - 3q^{40} - q^{41} - 5q^{43} - 8q^{44} - 2q^{46} - 15q^{47} + 2q^{49} - 11q^{50} - 3q^{52} + 2q^{53} - 12q^{55} + 2q^{56} + 13q^{58} - 6q^{59} - 4q^{61} - 2q^{62} + 2q^{64} - 21q^{65} - 8q^{67} - 2q^{68} + 3q^{70} - 18q^{71} - 14q^{73} - 11q^{74} + 8q^{76} + 8q^{77} - 4q^{79} + 3q^{80} + q^{82} - 8q^{83} - 36q^{85} + 5q^{86} + 8q^{88} + 8q^{89} + 3q^{91} + 2q^{92} + 15q^{94} + 12q^{95} - 13q^{97} - 2q^{98} + O(q^{100}) \)

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
−2.37228
3.37228
−1.00000 0 1.00000 −1.37228 0 −1.00000 −1.00000 0 1.37228
1.2 −1.00000 0 1.00000 4.37228 0 −1.00000 −1.00000 0 −4.37228
\(n\): 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.x 2
3.b odd 2 1 966.2.a.o 2
12.b even 2 1 7728.2.a.bh 2
21.c even 2 1 6762.2.a.cd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.o 2 3.b odd 2 1
2898.2.a.x 2 1.a even 1 1 trivial
6762.2.a.cd 2 21.c even 2 1
7728.2.a.bh 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2898))\):

\( T_{5}^{2} - 3 T_{5} - 6 \)
\( T_{11} + 4 \)
\( T_{13}^{2} + 3 T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -6 - 3 T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( -6 + 3 T + T^{2} \)
$17$ \( -32 + 2 T + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( 34 + 13 T + T^{2} \)
$31$ \( -32 - 2 T + T^{2} \)
$37$ \( 22 - 11 T + T^{2} \)
$41$ \( -74 + T + T^{2} \)
$43$ \( -68 + 5 T + T^{2} \)
$47$ \( 48 + 15 T + T^{2} \)
$53$ \( -32 - 2 T + T^{2} \)
$59$ \( -24 + 6 T + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( 48 + 18 T + T^{2} \)
$73$ \( 16 + 14 T + T^{2} \)
$79$ \( -128 + 4 T + T^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( -116 - 8 T + T^{2} \)
$97$ \( 34 + 13 T + T^{2} \)
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