Properties

Label 2601.2.a.ba
Level $2601$
Weight $2$
Character orbit 2601.a
Self dual yes
Analytic conductor $20.769$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 153)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + ( -\beta_{1} - 2 \beta_{3} ) q^{5} + 2 \beta_{1} q^{7} + 3 q^{8} +O(q^{10})\) \( q - q^{2} - q^{4} + ( -\beta_{1} - 2 \beta_{3} ) q^{5} + 2 \beta_{1} q^{7} + 3 q^{8} + ( \beta_{1} + 2 \beta_{3} ) q^{10} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{11} -3 \beta_{2} q^{13} -2 \beta_{1} q^{14} - q^{16} + 2 \beta_{2} q^{19} + ( \beta_{1} + 2 \beta_{3} ) q^{20} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{22} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{23} + ( 5 + \beta_{2} ) q^{25} + 3 \beta_{2} q^{26} -2 \beta_{1} q^{28} + ( -2 \beta_{1} - \beta_{3} ) q^{29} -2 \beta_{3} q^{31} -5 q^{32} + ( -4 - 6 \beta_{2} ) q^{35} + ( 3 \beta_{1} - 4 \beta_{3} ) q^{37} -2 \beta_{2} q^{38} + ( -3 \beta_{1} - 6 \beta_{3} ) q^{40} + ( -2 \beta_{1} - \beta_{3} ) q^{41} -4 \beta_{2} q^{43} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{44} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{46} + 2 \beta_{2} q^{47} + ( 1 + 4 \beta_{2} ) q^{49} + ( -5 - \beta_{2} ) q^{50} + 3 \beta_{2} q^{52} + ( -4 + 3 \beta_{2} ) q^{53} + ( 12 + 4 \beta_{2} ) q^{55} + 6 \beta_{1} q^{56} + ( 2 \beta_{1} + \beta_{3} ) q^{58} + ( -4 + 6 \beta_{2} ) q^{59} + ( -8 \beta_{1} + 3 \beta_{3} ) q^{61} + 2 \beta_{3} q^{62} + 7 q^{64} + ( 9 \beta_{1} - 3 \beta_{3} ) q^{65} + ( 4 - 6 \beta_{2} ) q^{67} + ( 4 + 6 \beta_{2} ) q^{70} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{71} + 5 \beta_{1} q^{73} + ( -3 \beta_{1} + 4 \beta_{3} ) q^{74} -2 \beta_{2} q^{76} + ( -8 - 8 \beta_{2} ) q^{77} + ( -6 \beta_{1} + 4 \beta_{3} ) q^{79} + ( \beta_{1} + 2 \beta_{3} ) q^{80} + ( 2 \beta_{1} + \beta_{3} ) q^{82} + ( 4 + 6 \beta_{2} ) q^{83} + 4 \beta_{2} q^{86} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{88} + ( -12 + \beta_{2} ) q^{89} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{91} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{92} -2 \beta_{2} q^{94} + ( -6 \beta_{1} + 2 \beta_{3} ) q^{95} + ( 4 \beta_{1} + 3 \beta_{3} ) q^{97} + ( -1 - 4 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{4} + 12q^{8} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{4} + 12q^{8} - 4q^{16} + 20q^{25} - 20q^{32} - 16q^{35} + 4q^{49} - 20q^{50} - 16q^{53} + 48q^{55} - 16q^{59} + 28q^{64} + 16q^{67} + 16q^{70} - 32q^{77} + 16q^{83} - 48q^{89} - 4q^{98} + O(q^{100}) \)

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
1.84776
−0.765367
0.765367
−1.84776
−1.00000 0 −1.00000 −3.37849 0 3.69552 3.00000 0 3.37849
1.2 −1.00000 0 −1.00000 −2.93015 0 −1.53073 3.00000 0 2.93015
1.3 −1.00000 0 −1.00000 2.93015 0 1.53073 3.00000 0 −2.93015
1.4 −1.00000 0 −1.00000 3.37849 0 −3.69552 3.00000 0 −3.37849
\(n\): 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
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.ba 4
3.b odd 2 1 2601.2.a.bg 4
17.b even 2 1 inner 2601.2.a.ba 4
17.e odd 16 2 153.2.l.b yes 4
51.c odd 2 1 2601.2.a.bg 4
51.i even 16 2 153.2.l.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.2.l.a 4 51.i even 16 2
153.2.l.b yes 4 17.e odd 16 2
2601.2.a.ba 4 1.a even 1 1 trivial
2601.2.a.ba 4 17.b even 2 1 inner
2601.2.a.bg 4 3.b odd 2 1
2601.2.a.bg 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2601))\):

\( T_{2} + 1 \)
\( T_{5}^{4} - 20 T_{5}^{2} + 98 \)
\( T_{7}^{4} - 16 T_{7}^{2} + 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( T^{4} \)
$5$ \( 98 - 20 T^{2} + T^{4} \)
$7$ \( 32 - 16 T^{2} + T^{4} \)
$11$ \( 128 - 32 T^{2} + T^{4} \)
$13$ \( ( -18 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( -8 + T^{2} )^{2} \)
$23$ \( 32 - 80 T^{2} + T^{4} \)
$29$ \( 2 - 20 T^{2} + T^{4} \)
$31$ \( 32 - 16 T^{2} + T^{4} \)
$37$ \( 578 - 100 T^{2} + T^{4} \)
$41$ \( 2 - 20 T^{2} + T^{4} \)
$43$ \( ( -32 + T^{2} )^{2} \)
$47$ \( ( -8 + T^{2} )^{2} \)
$53$ \( ( -2 + 8 T + T^{2} )^{2} \)
$59$ \( ( -56 + 8 T + T^{2} )^{2} \)
$61$ \( 21218 - 292 T^{2} + T^{4} \)
$67$ \( ( -56 - 8 T + T^{2} )^{2} \)
$71$ \( 32 - 80 T^{2} + T^{4} \)
$73$ \( 1250 - 100 T^{2} + T^{4} \)
$79$ \( 9248 - 208 T^{2} + T^{4} \)
$83$ \( ( -56 - 8 T + T^{2} )^{2} \)
$89$ \( ( 142 + 24 T + T^{2} )^{2} \)
$97$ \( 578 - 100 T^{2} + T^{4} \)
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