Properties

Label 2601.2.a.bd
Level $2601$
Weight $2$
Character orbit 2601.a
Self dual yes
Analytic conductor $20.769$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
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.

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

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
−1.84776 0 1.41421 3.61313 0 −3.84776 1.08239 0 −6.67619
1.2 −0.765367 0 −1.41421 −0.0823922 0 −2.76537 2.61313 0 0.0630603
1.3 0.765367 0 −1.41421 2.08239 0 −1.23463 −2.61313 0 1.59379
1.4 1.84776 0 1.41421 −1.61313 0 −0.152241 −1.08239 0 −2.98067
\(n\): 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.bd 4
3.b odd 2 1 867.2.a.n 4
17.b even 2 1 2601.2.a.bc 4
17.e odd 16 2 153.2.l.e 8
51.c odd 2 1 867.2.a.m 4
51.f odd 4 2 867.2.d.e 8
51.g odd 8 2 867.2.e.h 8
51.g odd 8 2 867.2.e.i 8
51.i even 16 2 51.2.h.a 8
51.i even 16 2 867.2.h.b 8
51.i even 16 2 867.2.h.f 8
51.i even 16 2 867.2.h.g 8
204.t odd 16 2 816.2.bq.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.h.a 8 51.i even 16 2
153.2.l.e 8 17.e odd 16 2
816.2.bq.a 8 204.t odd 16 2
867.2.a.m 4 51.c odd 2 1
867.2.a.n 4 3.b odd 2 1
867.2.d.e 8 51.f odd 4 2
867.2.e.h 8 51.g odd 8 2
867.2.e.i 8 51.g odd 8 2
867.2.h.b 8 51.i even 16 2
867.2.h.f 8 51.i even 16 2
867.2.h.g 8 51.i even 16 2
2601.2.a.bc 4 17.b even 2 1
2601.2.a.bd 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2601))\):

\( T_{2}^{4} - 4 T_{2}^{2} + 2 \)
\( T_{5}^{4} - 4 T_{5}^{3} - 2 T_{5}^{2} + 12 T_{5} + 1 \)
\( T_{7}^{4} + 8 T_{7}^{3} + 20 T_{7}^{2} + 16 T_{7} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 - 4 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 1 + 12 T - 2 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( 2 + 16 T + 20 T^{2} + 8 T^{3} + T^{4} \)
$11$ \( 1 + 4 T - 6 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( -47 - 68 T - 14 T^{2} + 4 T^{3} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( -367 - 172 T + 18 T^{2} + 12 T^{3} + T^{4} \)
$23$ \( -271 + 132 T + 14 T^{2} - 12 T^{3} + T^{4} \)
$29$ \( ( -14 - 4 T + T^{2} )^{2} \)
$31$ \( 632 - 192 T - 48 T^{2} + 8 T^{3} + T^{4} \)
$37$ \( 706 + 640 T + 196 T^{2} + 24 T^{3} + T^{4} \)
$41$ \( 1 + 4 T - 2 T^{2} - 12 T^{3} + T^{4} \)
$43$ \( 1297 + 196 T - 78 T^{2} - 4 T^{3} + T^{4} \)
$47$ \( -752 - 608 T - 72 T^{2} + 8 T^{3} + T^{4} \)
$53$ \( 496 - 160 T - 56 T^{2} + 8 T^{3} + T^{4} \)
$59$ \( 514 + 608 T + 196 T^{2} + 24 T^{3} + T^{4} \)
$61$ \( -1054 - 112 T + 132 T^{2} + 24 T^{3} + T^{4} \)
$67$ \( 4 + 16 T + 4 T^{2} - 8 T^{3} + T^{4} \)
$71$ \( 68 + 336 T + 164 T^{2} + 24 T^{3} + T^{4} \)
$73$ \( 752 - 1952 T - 248 T^{2} + 8 T^{3} + T^{4} \)
$79$ \( -62 - 144 T - 52 T^{2} + T^{4} \)
$83$ \( -272 - 416 T - 120 T^{2} + 8 T^{3} + T^{4} \)
$89$ \( -2558 - 896 T - 20 T^{2} + 16 T^{3} + T^{4} \)
$97$ \( -632 + 224 T + 40 T^{2} - 16 T^{3} + T^{4} \)
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