Properties

Label 2601.2.a.bj
Level $2601$
Weight $2$
Character orbit 2601.a
Self dual yes
Analytic conductor $20.769$
Analytic rank $1$
Dimension $6$
CM no
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: \(6\)
Coefficient field: 6.6.45769536.1
Defining polynomial: \(x^{6} - 15 x^{4} + 72 x^{2} - 109\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 3 + \beta_{2} ) q^{4} -\beta_{3} q^{5} + ( -3 + \beta_{2} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 3 + \beta_{2} ) q^{4} -\beta_{3} q^{5} + ( -3 + \beta_{2} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + ( -2 - 5 \beta_{2} - \beta_{4} ) q^{10} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{11} + ( -1 - 2 \beta_{2} - 3 \beta_{4} ) q^{13} + ( -3 \beta_{1} + \beta_{3} ) q^{14} + ( 1 + 4 \beta_{2} + \beta_{4} ) q^{16} + ( -3 - \beta_{2} + 2 \beta_{4} ) q^{19} + ( -2 \beta_{1} - 3 \beta_{3} - \beta_{5} ) q^{20} + ( -4 + 5 \beta_{2} + 6 \beta_{4} ) q^{22} + ( -\beta_{1} - 2 \beta_{5} ) q^{23} + ( 4 + 3 \beta_{2} + 5 \beta_{4} ) q^{25} + ( -\beta_{1} - 2 \beta_{3} - 3 \beta_{5} ) q^{26} + ( -7 + \beta_{4} ) q^{28} + ( -\beta_{3} + \beta_{5} ) q^{29} + ( -6 - \beta_{2} - \beta_{4} ) q^{31} + ( -\beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{32} + ( -2 \beta_{1} + 3 \beta_{3} - \beta_{5} ) q^{35} + ( 1 + 3 \beta_{2} + 2 \beta_{4} ) q^{37} + ( -3 \beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{38} + ( -11 - 8 \beta_{2} - 6 \beta_{4} ) q^{40} + ( -\beta_{1} + \beta_{3} - 2 \beta_{5} ) q^{41} + ( 2 - 2 \beta_{2} ) q^{43} + ( -2 \beta_{1} + 3 \beta_{3} + 4 \beta_{5} ) q^{44} + ( -3 - 3 \beta_{2} - 10 \beta_{4} ) q^{46} + ( -2 \beta_{1} - \beta_{3} - \beta_{5} ) q^{47} + ( 4 - 6 \beta_{2} + \beta_{4} ) q^{49} + ( 4 \beta_{1} + 3 \beta_{3} + 5 \beta_{5} ) q^{50} + ( -4 - 10 \beta_{2} - 11 \beta_{4} ) q^{52} + ( -3 \beta_{1} + 2 \beta_{3} + \beta_{5} ) q^{53} + ( -4 - 2 \beta_{2} - 5 \beta_{4} ) q^{55} + ( -\beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{56} + ( -3 - 4 \beta_{2} + 4 \beta_{4} ) q^{58} + ( 3 \beta_{1} + \beta_{3} - \beta_{5} ) q^{59} + ( -5 + 2 \beta_{2} + \beta_{4} ) q^{61} + ( -6 \beta_{1} - \beta_{3} - \beta_{5} ) q^{62} + ( -4 + 2 \beta_{2} + 5 \beta_{4} ) q^{64} + ( \beta_{1} + 4 \beta_{3} + 2 \beta_{5} ) q^{65} + ( -6 \beta_{2} - 4 \beta_{4} ) q^{67} + ( -3 + 12 \beta_{2} - 2 \beta_{4} ) q^{70} + ( 2 \beta_{3} - \beta_{5} ) q^{71} + ( 1 - \beta_{2} - \beta_{4} ) q^{73} + ( \beta_{1} + 3 \beta_{3} + 2 \beta_{5} ) q^{74} + ( -13 - 4 \beta_{2} + 5 \beta_{4} ) q^{76} + ( 4 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} ) q^{77} + ( -7 + 2 \beta_{2} ) q^{79} + ( -7 \beta_{1} - 2 \beta_{3} - 4 \beta_{5} ) q^{80} + ( -1 + 2 \beta_{2} - 9 \beta_{4} ) q^{82} + ( -2 \beta_{1} + 3 \beta_{3} + 3 \beta_{5} ) q^{83} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{86} + ( 7 \beta_{2} + 11 \beta_{4} ) q^{88} + ( \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{89} + ( 2 + 2 \beta_{2} + 7 \beta_{4} ) q^{91} + ( -\beta_{1} - 3 \beta_{3} - 6 \beta_{5} ) q^{92} + ( -11 - 8 \beta_{2} - 6 \beta_{4} ) q^{94} + ( 4 \beta_{1} + \beta_{3} + \beta_{5} ) q^{95} + ( 1 + 6 \beta_{2} + 6 \beta_{4} ) q^{97} + ( 4 \beta_{1} - 6 \beta_{3} + \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 18q^{4} - 18q^{7} + O(q^{10}) \) \( 6q + 18q^{4} - 18q^{7} - 12q^{10} - 6q^{13} + 6q^{16} - 18q^{19} - 24q^{22} + 24q^{25} - 42q^{28} - 36q^{31} + 6q^{37} - 66q^{40} + 12q^{43} - 18q^{46} + 24q^{49} - 24q^{52} - 24q^{55} - 18q^{58} - 30q^{61} - 24q^{64} - 18q^{70} + 6q^{73} - 78q^{76} - 42q^{79} - 6q^{82} + 12q^{91} - 66q^{94} + 6q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 15 x^{4} + 72 x^{2} - 109\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 10 \nu^{2} + 23 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 10 \nu^{3} + 23 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 10 \beta_{2} + 27\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 10 \beta_{3} + 27 \beta_{1}\)

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
−2.55580
−2.31242
−1.76653
1.76653
2.31242
2.55580
−2.55580 0 4.53209 3.91571 0 −1.46791 −6.47150 0 −10.0077
1.2 −2.31242 0 3.34730 0.803096 0 −2.65270 −3.11552 0 −1.85710
1.3 −1.76653 0 1.12061 −3.31998 0 −4.87939 1.55346 0 5.86484
1.4 1.76653 0 1.12061 3.31998 0 −4.87939 −1.55346 0 5.86484
1.5 2.31242 0 3.34730 −0.803096 0 −2.65270 3.11552 0 −1.85710
1.6 2.55580 0 4.53209 −3.91571 0 −1.46791 6.47150 0 −10.0077
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2601.2.a.bj 6
3.b odd 2 1 inner 2601.2.a.bj 6
17.b even 2 1 2601.2.a.bk yes 6
51.c odd 2 1 2601.2.a.bk yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2601.2.a.bj 6 1.a even 1 1 trivial
2601.2.a.bj 6 3.b odd 2 1 inner
2601.2.a.bk yes 6 17.b even 2 1
2601.2.a.bk yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2601))\):

\( T_{2}^{6} - 15 T_{2}^{4} + 72 T_{2}^{2} - 109 \)
\( T_{5}^{6} - 27 T_{5}^{4} + 186 T_{5}^{2} - 109 \)
\( T_{7}^{3} + 9 T_{7}^{2} + 24 T_{7} + 19 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -109 + 72 T^{2} - 15 T^{4} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( -109 + 186 T^{2} - 27 T^{4} + T^{6} \)
$7$ \( ( 19 + 24 T + 9 T^{2} + T^{3} )^{2} \)
$11$ \( -981 + 396 T^{2} - 45 T^{4} + T^{6} \)
$13$ \( ( -37 - 18 T + 3 T^{2} + T^{3} )^{2} \)
$17$ \( T^{6} \)
$19$ \( ( -53 + 6 T + 9 T^{2} + T^{3} )^{2} \)
$23$ \( -39349 + 3828 T^{2} - 111 T^{4} + T^{6} \)
$29$ \( -8829 + 1539 T^{2} - 72 T^{4} + T^{6} \)
$31$ \( ( 199 + 105 T + 18 T^{2} + T^{3} )^{2} \)
$37$ \( ( -17 - 18 T - 3 T^{2} + T^{3} )^{2} \)
$41$ \( -981 + 5715 T^{2} - 162 T^{4} + T^{6} \)
$43$ \( ( 8 - 6 T^{2} + T^{3} )^{2} \)
$47$ \( -981 + 963 T^{2} - 108 T^{4} + T^{6} \)
$53$ \( -8829 + 8181 T^{2} - 180 T^{4} + T^{6} \)
$59$ \( -8829 + 16848 T^{2} - 261 T^{4} + T^{6} \)
$61$ \( ( 71 + 66 T + 15 T^{2} + T^{3} )^{2} \)
$67$ \( ( 296 - 84 T + T^{3} )^{2} \)
$71$ \( -149221 + 9015 T^{2} - 171 T^{4} + T^{6} \)
$73$ \( ( 3 - 3 T^{2} + T^{3} )^{2} \)
$79$ \( ( 267 + 135 T + 21 T^{2} + T^{3} )^{2} \)
$83$ \( -580861 + 29331 T^{2} - 348 T^{4} + T^{6} \)
$89$ \( -31501 + 4269 T^{2} - 156 T^{4} + T^{6} \)
$97$ \( ( -109 - 105 T - 3 T^{2} + T^{3} )^{2} \)
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