Properties

Label 2601.2.a.o
Level $2601$
Weight $2$
Character orbit 2601.a
Self dual yes
Analytic conductor $20.769$
Analytic rank $1$
Dimension $2$
CM discriminant -51
Inner twists $4$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 153)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

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

\(f(q)\) \(=\) \( q -2 q^{4} -\beta q^{5} +O(q^{10})\) \( q -2 q^{4} -\beta q^{5} + \beta q^{11} - q^{13} + 4 q^{16} -5 q^{19} + 2 \beta q^{20} + \beta q^{23} + 12 q^{25} + 2 \beta q^{29} + \beta q^{41} -11 q^{43} -2 \beta q^{44} -7 q^{49} + 2 q^{52} -17 q^{55} -8 q^{64} + \beta q^{65} + 8 q^{67} -4 \beta q^{71} + 10 q^{76} -4 \beta q^{80} -2 \beta q^{92} + 5 \beta q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + O(q^{10}) \) \( 2q - 4q^{4} - 2q^{13} + 8q^{16} - 10q^{19} + 24q^{25} - 22q^{43} - 14q^{49} + 4q^{52} - 34q^{55} - 16q^{64} + 16q^{67} + 20q^{76} + O(q^{100}) \)

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
0 0 −2.00000 −4.12311 0 0 0 0 0
1.2 0 0 −2.00000 4.12311 0 0 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
51.c odd 2 1 CM by \(\Q(\sqrt{-51}) \)
3.b odd 2 1 inner
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2601.2.a.o 2
3.b odd 2 1 inner 2601.2.a.o 2
17.b even 2 1 inner 2601.2.a.o 2
17.c even 4 2 153.2.d.b 2
51.c odd 2 1 CM 2601.2.a.o 2
51.f odd 4 2 153.2.d.b 2
68.f odd 4 2 2448.2.c.g 2
204.l even 4 2 2448.2.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.2.d.b 2 17.c even 4 2
153.2.d.b 2 51.f odd 4 2
2448.2.c.g 2 68.f odd 4 2
2448.2.c.g 2 204.l even 4 2
2601.2.a.o 2 1.a even 1 1 trivial
2601.2.a.o 2 3.b odd 2 1 inner
2601.2.a.o 2 17.b even 2 1 inner
2601.2.a.o 2 51.c odd 2 1 CM

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -17 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -17 + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 5 + T )^{2} \)
$23$ \( -17 + T^{2} \)
$29$ \( -68 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( -17 + T^{2} \)
$43$ \( ( 11 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( ( -8 + T )^{2} \)
$71$ \( -272 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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