Properties

Label 2601.2.a.bf
Level $2601$
Weight $2$
Character orbit 2601.a
Self dual yes
Analytic conductor $20.769$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: 4.4.7232.1
Defining polynomial: \(x^{4} - 2 x^{3} - 5 x^{2} + 4 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 5 x^{2} + 4 x + 4\):

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

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
−0.652223
−1.63640
3.06644
1.22219
−2.06644 0 2.27016 0.347777 0 −4.33660 −0.558268 0 −0.718659
1.2 −0.222191 0 −1.95063 −0.636405 0 1.72844 0.877796 0 0.141404
1.3 1.65222 0 0.729840 4.06644 0 0.922382 −2.09859 0 6.71866
1.4 2.63640 0 4.95063 2.22219 0 −2.31423 7.77906 0 5.85860
\(n\): 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.bf 4
3.b odd 2 1 867.2.a.k 4
17.b even 2 1 2601.2.a.be 4
17.d even 8 2 153.2.f.b 8
51.c odd 2 1 867.2.a.l 4
51.f odd 4 2 867.2.d.f 8
51.g odd 8 2 51.2.e.a 8
51.g odd 8 2 867.2.e.g 8
51.i even 16 4 867.2.h.i 16
51.i even 16 4 867.2.h.k 16
68.g odd 8 2 2448.2.be.x 8
204.p even 8 2 816.2.bd.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.e.a 8 51.g odd 8 2
153.2.f.b 8 17.d even 8 2
816.2.bd.e 8 204.p even 8 2
867.2.a.k 4 3.b odd 2 1
867.2.a.l 4 51.c odd 2 1
867.2.d.f 8 51.f odd 4 2
867.2.e.g 8 51.g odd 8 2
867.2.h.i 16 51.i even 16 4
867.2.h.k 16 51.i even 16 4
2448.2.be.x 8 68.g odd 8 2
2601.2.a.be 4 17.b even 2 1
2601.2.a.bf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2601))\):

\( T_{2}^{4} - 2 T_{2}^{3} - 5 T_{2}^{2} + 8 T_{2} + 2 \)
\( T_{5}^{4} - 6 T_{5}^{3} + 7 T_{5}^{2} + 4 T_{5} - 2 \)
\( T_{7}^{4} + 4 T_{7}^{3} - 6 T_{7}^{2} - 16 T_{7} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + 8 T - 5 T^{2} - 2 T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( -2 + 4 T + 7 T^{2} - 6 T^{3} + T^{4} \)
$7$ \( 16 - 16 T - 6 T^{2} + 4 T^{3} + T^{4} \)
$11$ \( -16 + 24 T + T^{2} - 6 T^{3} + T^{4} \)
$13$ \( -64 + 80 T - 23 T^{2} - 2 T^{3} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( -32 + 25 T^{2} - 10 T^{3} + T^{4} \)
$23$ \( -184 - 136 T - 15 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( -16 - 80 T + 70 T^{2} - 16 T^{3} + T^{4} \)
$31$ \( 32 - 64 T - 20 T^{2} + 4 T^{3} + T^{4} \)
$37$ \( -772 - 344 T + 12 T^{3} + T^{4} \)
$41$ \( -1666 - 644 T - 17 T^{2} + 14 T^{3} + T^{4} \)
$43$ \( -1168 + 392 T + 17 T^{2} - 14 T^{3} + T^{4} \)
$47$ \( -64 - 160 T - 52 T^{2} + 4 T^{3} + T^{4} \)
$53$ \( 128 - 256 T + 124 T^{2} - 20 T^{3} + T^{4} \)
$59$ \( -4672 + 608 T + 126 T^{2} - 24 T^{3} + T^{4} \)
$61$ \( -356 - 152 T + 16 T^{2} + 12 T^{3} + T^{4} \)
$67$ \( 2048 + 1024 T - 196 T^{2} - 4 T^{3} + T^{4} \)
$71$ \( 32 + 64 T - 52 T^{2} - 4 T^{3} + T^{4} \)
$73$ \( -328 + 120 T + 114 T^{2} + 20 T^{3} + T^{4} \)
$79$ \( -16 + 48 T - 42 T^{2} + 8 T^{3} + T^{4} \)
$83$ \( 5248 + 128 T - 164 T^{2} - 4 T^{3} + T^{4} \)
$89$ \( 1088 - 352 T - 110 T^{2} + 4 T^{3} + T^{4} \)
$97$ \( 368 + 432 T + 166 T^{2} + 24 T^{3} + T^{4} \)
show more
show less