Properties

Label 2601.2.a.bb
Level $2601$
Weight $2$
Character orbit 2601.a
Self dual yes
Analytic conductor $20.769$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
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.

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

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
1.84776
0.765367
−0.765367
−2.41421 0 3.82843 −0.765367 0 2.61313 −4.41421 0 1.84776
1.2 −2.41421 0 3.82843 0.765367 0 −2.61313 −4.41421 0 −1.84776
1.3 0.414214 0 −1.82843 −1.84776 0 1.08239 −1.58579 0 −0.765367
1.4 0.414214 0 −1.82843 1.84776 0 −1.08239 −1.58579 0 0.765367
\(n\): 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.bb 4
3.b odd 2 1 289.2.a.f 4
12.b even 2 1 4624.2.a.bp 4
15.d odd 2 1 7225.2.a.u 4
17.b even 2 1 inner 2601.2.a.bb 4
17.e odd 16 2 153.2.l.c 4
51.c odd 2 1 289.2.a.f 4
51.f odd 4 2 289.2.b.b 4
51.g odd 8 4 289.2.c.c 8
51.i even 16 2 17.2.d.a 4
51.i even 16 2 289.2.d.a 4
51.i even 16 2 289.2.d.b 4
51.i even 16 2 289.2.d.c 4
204.h even 2 1 4624.2.a.bp 4
204.t odd 16 2 272.2.v.d 4
255.h odd 2 1 7225.2.a.u 4
255.bc odd 16 2 425.2.n.b 4
255.be even 16 2 425.2.m.a 4
255.bj odd 16 2 425.2.n.a 4
357.be odd 16 2 833.2.l.a 4
357.bm even 48 4 833.2.v.b 8
357.bn odd 48 4 833.2.v.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.d.a 4 51.i even 16 2
153.2.l.c 4 17.e odd 16 2
272.2.v.d 4 204.t odd 16 2
289.2.a.f 4 3.b odd 2 1
289.2.a.f 4 51.c odd 2 1
289.2.b.b 4 51.f odd 4 2
289.2.c.c 8 51.g odd 8 4
289.2.d.a 4 51.i even 16 2
289.2.d.b 4 51.i even 16 2
289.2.d.c 4 51.i even 16 2
425.2.m.a 4 255.be even 16 2
425.2.n.a 4 255.bj odd 16 2
425.2.n.b 4 255.bc odd 16 2
833.2.l.a 4 357.be odd 16 2
833.2.v.a 8 357.bn odd 48 4
833.2.v.b 8 357.bm even 48 4
2601.2.a.bb 4 1.a even 1 1 trivial
2601.2.a.bb 4 17.b even 2 1 inner
4624.2.a.bp 4 12.b even 2 1
4624.2.a.bp 4 204.h even 2 1
7225.2.a.u 4 15.d odd 2 1
7225.2.a.u 4 255.h odd 2 1

Hecke kernels

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

\( T_{2}^{2} + 2 T_{2} - 1 \)
\( T_{5}^{4} - 4 T_{5}^{2} + 2 \)
\( T_{7}^{4} - 8 T_{7}^{2} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + 2 T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 2 - 4 T^{2} + T^{4} \)
$7$ \( 8 - 8 T^{2} + T^{4} \)
$11$ \( 8 - 8 T^{2} + T^{4} \)
$13$ \( ( -2 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( -4 - 4 T + T^{2} )^{2} \)
$23$ \( 392 - 40 T^{2} + T^{4} \)
$29$ \( 2 - 20 T^{2} + T^{4} \)
$31$ \( 648 - 72 T^{2} + T^{4} \)
$37$ \( 1250 - 100 T^{2} + T^{4} \)
$41$ \( 98 - 68 T^{2} + T^{4} \)
$43$ \( ( -4 + 4 T + T^{2} )^{2} \)
$47$ \( ( 56 + 16 T + T^{2} )^{2} \)
$53$ \( ( -2 + T^{2} )^{2} \)
$59$ \( ( 6 + T )^{4} \)
$61$ \( 1250 - 100 T^{2} + T^{4} \)
$67$ \( ( 8 + 8 T + T^{2} )^{2} \)
$71$ \( 5000 - 200 T^{2} + T^{4} \)
$73$ \( 4802 - 196 T^{2} + T^{4} \)
$79$ \( 392 - 40 T^{2} + T^{4} \)
$83$ \( ( 4 + 12 T + T^{2} )^{2} \)
$89$ \( ( 62 + 16 T + T^{2} )^{2} \)
$97$ \( 4418 - 148 T^{2} + T^{4} \)
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