Properties

Label 2601.2.a.z
Level $2601$
Weight $2$
Character orbit 2601.a
Self dual yes
Analytic conductor $20.769$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 867)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} - \beta_{2} ) q^{5} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{7} + ( 2 - 2 \beta_{1} + 3 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{1} - \beta_{2} ) q^{5} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{7} + ( 2 - 2 \beta_{1} + 3 \beta_{2} ) q^{8} + ( \beta_{1} - 2 \beta_{2} ) q^{10} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{11} + ( -5 - \beta_{1} + \beta_{2} ) q^{13} + ( -4 + 4 \beta_{1} - 3 \beta_{2} ) q^{14} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{16} + ( -1 - 3 \beta_{2} ) q^{19} + ( -2 + \beta_{1} - \beta_{2} ) q^{20} + ( -6 + 2 \beta_{1} - \beta_{2} ) q^{22} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{23} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{25} + ( -4 + 3 \beta_{1} + 2 \beta_{2} ) q^{26} + ( -7 + 7 \beta_{1} - 5 \beta_{2} ) q^{28} + ( 2 - 3 \beta_{1} + 5 \beta_{2} ) q^{29} + ( -4 - \beta_{1} + 2 \beta_{2} ) q^{31} -3 \beta_{1} q^{32} + ( 2 - \beta_{1} + \beta_{2} ) q^{35} + ( 1 + 5 \beta_{2} ) q^{37} + ( 2 + 4 \beta_{1} - 3 \beta_{2} ) q^{38} + ( -3 + 2 \beta_{1} + 2 \beta_{2} ) q^{40} + ( -6 - 2 \beta_{1} + \beta_{2} ) q^{41} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -7 + 5 \beta_{1} - 5 \beta_{2} ) q^{44} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{46} + ( \beta_{1} - 5 \beta_{2} ) q^{47} + ( -7 \beta_{1} + 5 \beta_{2} ) q^{49} + ( -2 + 5 \beta_{1} - 3 \beta_{2} ) q^{50} + ( -2 + 7 \beta_{1} - 3 \beta_{2} ) q^{52} + ( 8 + \beta_{1} ) q^{53} + ( -\beta_{1} + 5 \beta_{2} ) q^{55} + ( -8 + 11 \beta_{1} - 6 \beta_{2} ) q^{56} + ( 3 - 10 \beta_{1} + 8 \beta_{2} ) q^{58} + ( 3 + 6 \beta_{1} - 3 \beta_{2} ) q^{59} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{61} + ( -4 + \beta_{1} + 3 \beta_{2} ) q^{62} + ( 4 + 3 \beta_{1} - 3 \beta_{2} ) q^{64} + ( -7 - 5 \beta_{1} + 6 \beta_{2} ) q^{65} + ( -6 + 4 \beta_{2} ) q^{67} + ( 3 - 4 \beta_{1} + 2 \beta_{2} ) q^{70} + ( 5 + 4 \beta_{1} ) q^{71} + ( 5 + \beta_{1} - 2 \beta_{2} ) q^{73} + ( -4 - 6 \beta_{1} + 5 \beta_{2} ) q^{74} + ( -1 + 5 \beta_{1} - \beta_{2} ) q^{76} + ( 7 - 5 \beta_{1} + 5 \beta_{2} ) q^{77} + ( -1 - 2 \beta_{1} + 4 \beta_{2} ) q^{79} + ( -5 + \beta_{1} + 2 \beta_{2} ) q^{80} + ( -3 + 3 \beta_{1} + 3 \beta_{2} ) q^{82} + ( -9 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -10 + 12 \beta_{1} - 4 \beta_{2} ) q^{86} + ( 13 \beta_{1} - 8 \beta_{2} ) q^{88} + ( -7 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 2 - 7 \beta_{1} + 3 \beta_{2} ) q^{91} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{92} + ( 3 + 6 \beta_{1} - 6 \beta_{2} ) q^{94} + ( 2 - \beta_{1} - 5 \beta_{2} ) q^{95} + ( -5 + 2 \beta_{2} ) q^{97} + ( 9 - 12 \beta_{1} + 12 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} + 3q^{5} - 3q^{7} + 6q^{8} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} + 3q^{5} - 3q^{7} + 6q^{8} - 3q^{11} - 15q^{13} - 12q^{14} + 3q^{16} - 3q^{19} - 6q^{20} - 18q^{22} - 3q^{23} - 6q^{25} - 12q^{26} - 21q^{28} + 6q^{29} - 12q^{31} + 6q^{35} + 3q^{37} + 6q^{38} - 9q^{40} - 18q^{41} - 24q^{43} - 21q^{44} + 9q^{46} - 6q^{50} - 6q^{52} + 24q^{53} - 24q^{56} + 9q^{58} + 9q^{59} - 9q^{61} - 12q^{62} + 12q^{64} - 21q^{65} - 18q^{67} + 9q^{70} + 15q^{71} + 15q^{73} - 12q^{74} - 3q^{76} + 21q^{77} - 3q^{79} - 15q^{80} - 9q^{82} - 30q^{86} + 6q^{91} + 6q^{92} + 9q^{94} + 6q^{95} - 15q^{97} + 27q^{98} + O(q^{100}) \)

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
1.87939
−0.347296
−1.53209
−0.879385 0 −1.22668 1.34730 0 1.22668 2.83750 0 −1.18479
1.2 1.34730 0 −0.184793 2.53209 0 0.184793 −2.94356 0 3.41147
1.3 2.53209 0 4.41147 −0.879385 0 −4.41147 6.10607 0 −2.22668
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 2601.2.a.z 3
3.b odd 2 1 867.2.a.j yes 3
17.b even 2 1 2601.2.a.y 3
51.c odd 2 1 867.2.a.i 3
51.f odd 4 2 867.2.d.d 6
51.g odd 8 4 867.2.e.j 12
51.i even 16 8 867.2.h.l 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
867.2.a.i 3 51.c odd 2 1
867.2.a.j yes 3 3.b odd 2 1
867.2.d.d 6 51.f odd 4 2
867.2.e.j 12 51.g odd 8 4
867.2.h.l 24 51.i even 16 8
2601.2.a.y 3 17.b even 2 1
2601.2.a.z 3 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{3} - 3 T_{2}^{2} + 3 \)
\( T_{5}^{3} - 3 T_{5}^{2} + 3 \)
\( T_{7}^{3} + 3 T_{7}^{2} - 6 T_{7} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 - 3 T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( 3 - 3 T^{2} + T^{3} \)
$7$ \( 1 - 6 T + 3 T^{2} + T^{3} \)
$11$ \( -57 - 18 T + 3 T^{2} + T^{3} \)
$13$ \( 109 + 72 T + 15 T^{2} + T^{3} \)
$17$ \( T^{3} \)
$19$ \( -53 - 24 T + 3 T^{2} + T^{3} \)
$23$ \( -3 - 18 T + 3 T^{2} + T^{3} \)
$29$ \( 213 - 45 T - 6 T^{2} + T^{3} \)
$31$ \( 37 + 39 T + 12 T^{2} + T^{3} \)
$37$ \( 199 - 72 T - 3 T^{2} + T^{3} \)
$41$ \( 153 + 99 T + 18 T^{2} + T^{3} \)
$43$ \( 424 + 180 T + 24 T^{2} + T^{3} \)
$47$ \( -171 - 63 T + T^{3} \)
$53$ \( -489 + 189 T - 24 T^{2} + T^{3} \)
$59$ \( 459 - 54 T - 9 T^{2} + T^{3} \)
$61$ \( -71 - 12 T + 9 T^{2} + T^{3} \)
$67$ \( -8 + 60 T + 18 T^{2} + T^{3} \)
$71$ \( 51 + 27 T - 15 T^{2} + T^{3} \)
$73$ \( -89 + 66 T - 15 T^{2} + T^{3} \)
$79$ \( 37 - 33 T + 3 T^{2} + T^{3} \)
$83$ \( -459 - 189 T + T^{3} \)
$89$ \( -153 - 117 T + T^{3} \)
$97$ \( 73 + 63 T + 15 T^{2} + T^{3} \)
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