Properties

Label 2601.2.a.u
Level $2601$
Weight $2$
Character orbit 2601.a
Self dual yes
Analytic conductor $20.769$
Analytic rank $1$
Dimension $3$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2601 = 3^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2601.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.7690895657\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-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.

\(f(q)\) \(=\) \( q -2 q^{4} + ( 3 \beta_{1} - \beta_{2} ) q^{7} +O(q^{10})\) \( q -2 q^{4} + ( 3 \beta_{1} - \beta_{2} ) q^{7} + ( -4 \beta_{1} + 3 \beta_{2} ) q^{13} + 4 q^{16} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{19} -5 q^{25} + ( -6 \beta_{1} + 2 \beta_{2} ) q^{28} + ( \beta_{1} - 6 \beta_{2} ) q^{31} + ( -4 \beta_{1} + 7 \beta_{2} ) q^{37} + ( 7 \beta_{1} - \beta_{2} ) q^{43} + ( 7 - 5 \beta_{1} + 8 \beta_{2} ) q^{49} + ( 8 \beta_{1} - 6 \beta_{2} ) q^{52} + ( 5 \beta_{1} + 4 \beta_{2} ) q^{61} -8 q^{64} + ( -2 \beta_{1} - 7 \beta_{2} ) q^{67} -17 q^{73} + ( 6 \beta_{1} + 4 \beta_{2} ) q^{76} -17 q^{79} + ( -17 + 10 \beta_{1} - 9 \beta_{2} ) q^{91} + ( -8 \beta_{1} - 3 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 6q^{4} + O(q^{10}) \) \( 3q - 6q^{4} + 12q^{16} - 15q^{25} + 21q^{49} - 24q^{64} - 51q^{73} - 51q^{79} - 51q^{91} + O(q^{100}) \)

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
−1.53209
−0.347296
1.87939
0 0 −2.00000 0 0 −4.94356 0 0 0
1.2 0 0 −2.00000 0 0 0.837496 0 0 0
1.3 0 0 −2.00000 0 0 4.10607 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(17\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2601.2.a.u 3
3.b odd 2 1 CM 2601.2.a.u 3
17.b even 2 1 2601.2.a.v yes 3
51.c odd 2 1 2601.2.a.v yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2601.2.a.u 3 1.a even 1 1 trivial
2601.2.a.u 3 3.b odd 2 1 CM
2601.2.a.v yes 3 17.b even 2 1
2601.2.a.v yes 3 51.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2601))\):

\( T_{2} \)
\( T_{5} \)
\( T_{7}^{3} - 21 T_{7} + 17 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( T^{3} \)
$7$ \( 17 - 21 T + T^{3} \)
$11$ \( T^{3} \)
$13$ \( -89 - 39 T + T^{3} \)
$17$ \( T^{3} \)
$19$ \( 163 - 57 T + T^{3} \)
$23$ \( T^{3} \)
$29$ \( T^{3} \)
$31$ \( -289 - 93 T + T^{3} \)
$37$ \( 323 - 111 T + T^{3} \)
$41$ \( T^{3} \)
$43$ \( -71 - 129 T + T^{3} \)
$47$ \( T^{3} \)
$53$ \( T^{3} \)
$59$ \( T^{3} \)
$61$ \( -901 - 183 T + T^{3} \)
$67$ \( 127 - 201 T + T^{3} \)
$71$ \( T^{3} \)
$73$ \( ( 17 + T )^{3} \)
$79$ \( ( 17 + T )^{3} \)
$83$ \( T^{3} \)
$89$ \( T^{3} \)
$97$ \( 1853 - 291 T + T^{3} \)
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