Properties

Label 2898.2.a.w
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−1.00000 0 1.00000 −2.73205 0 1.00000 −1.00000 0 2.73205
1.2 −1.00000 0 1.00000 0.732051 0 1.00000 −1.00000 0 −0.732051
\(n\): 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.w 2
3.b odd 2 1 322.2.a.f 2
12.b even 2 1 2576.2.a.u 2
15.d odd 2 1 8050.2.a.x 2
21.c even 2 1 2254.2.a.o 2
69.c even 2 1 7406.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.a.f 2 3.b odd 2 1
2254.2.a.o 2 21.c even 2 1
2576.2.a.u 2 12.b even 2 1
2898.2.a.w 2 1.a even 1 1 trivial
7406.2.a.s 2 69.c even 2 1
8050.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2898))\):

\( T_{5}^{2} + 2 T_{5} - 2 \)
\( T_{11}^{2} - 12 \)
\( T_{13}^{2} - 4 T_{13} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( -2 + 2 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -12 + T^{2} \)
$13$ \( -8 - 4 T + T^{2} \)
$17$ \( -2 + 2 T + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( 8 + T )^{2} \)
$31$ \( -2 + 10 T + T^{2} \)
$37$ \( 4 - 8 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( -18 - 6 T + T^{2} \)
$53$ \( -12 + T^{2} \)
$59$ \( -66 - 6 T + T^{2} \)
$61$ \( -18 - 6 T + T^{2} \)
$67$ \( 52 + 16 T + T^{2} \)
$71$ \( -104 - 4 T + T^{2} \)
$73$ \( 4 + 8 T + T^{2} \)
$79$ \( -12 + 12 T + T^{2} \)
$83$ \( -92 + 8 T + T^{2} \)
$89$ \( 54 + 18 T + T^{2} \)
$97$ \( 46 + 14 T + T^{2} \)
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