Properties

Label 2541.2.a.bg
Level 2541
Weight 2
Character orbit 2541.a
Self dual yes
Analytic conductor 20.290
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2541.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.2899871536\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{2} + q^{3} + ( 2 + \beta_{1} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} ) q^{5} + ( -1 - \beta_{2} ) q^{6} + q^{7} + ( -1 - 2 \beta_{1} ) q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{2} + q^{3} + ( 2 + \beta_{1} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} ) q^{5} + ( -1 - \beta_{2} ) q^{6} + q^{7} + ( -1 - 2 \beta_{1} ) q^{8} + q^{9} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{10} + ( 2 + \beta_{1} ) q^{12} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{13} + ( -1 - \beta_{2} ) q^{14} + ( 1 - \beta_{1} - \beta_{2} ) q^{15} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{16} + ( -2 - 4 \beta_{1} + 2 \beta_{2} ) q^{17} + ( -1 - \beta_{2} ) q^{18} + ( 3 - \beta_{1} + \beta_{2} ) q^{19} + ( -2 - 2 \beta_{1} - 3 \beta_{2} ) q^{20} + q^{21} + ( 4 + 2 \beta_{2} ) q^{23} + ( -1 - 2 \beta_{1} ) q^{24} + ( 4 + \beta_{1} - 3 \beta_{2} ) q^{25} + ( -1 - 5 \beta_{1} - 2 \beta_{2} ) q^{26} + q^{27} + ( 2 + \beta_{1} ) q^{28} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{29} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{30} + ( -2 + 2 \beta_{1} - 4 \beta_{2} ) q^{31} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{32} + ( 6 \beta_{1} + 4 \beta_{2} ) q^{34} + ( 1 - \beta_{1} - \beta_{2} ) q^{35} + ( 2 + \beta_{1} ) q^{36} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{37} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{38} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{39} + ( 7 + \beta_{1} + 3 \beta_{2} ) q^{40} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -1 - \beta_{2} ) q^{42} + ( 4 - 4 \beta_{1} - 2 \beta_{2} ) q^{43} + ( 1 - \beta_{1} - \beta_{2} ) q^{45} + ( -10 - 2 \beta_{1} - 2 \beta_{2} ) q^{46} + ( 1 + \beta_{1} + 3 \beta_{2} ) q^{47} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{48} + q^{49} + ( 4 + \beta_{1} - 7 \beta_{2} ) q^{50} + ( -2 - 4 \beta_{1} + 2 \beta_{2} ) q^{51} + ( 10 + 6 \beta_{1} + \beta_{2} ) q^{52} -2 \beta_{1} q^{53} + ( -1 - \beta_{2} ) q^{54} + ( -1 - 2 \beta_{1} ) q^{56} + ( 3 - \beta_{1} + \beta_{2} ) q^{57} + ( -1 - 5 \beta_{1} - 2 \beta_{2} ) q^{58} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{1} - 3 \beta_{2} ) q^{60} + 2 q^{61} + ( 12 - 2 \beta_{2} ) q^{62} + q^{63} + ( -1 - 4 \beta_{1} + 2 \beta_{2} ) q^{64} + ( -7 + \beta_{1} - 7 \beta_{2} ) q^{65} + ( -1 - 5 \beta_{1} + \beta_{2} ) q^{67} + ( -14 - 8 \beta_{1} ) q^{68} + ( 4 + 2 \beta_{2} ) q^{69} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{70} + ( -4 + 4 \beta_{1} ) q^{71} + ( -1 - 2 \beta_{1} ) q^{72} + ( 7 - 3 \beta_{1} + \beta_{2} ) q^{73} + ( -9 - \beta_{1} + 2 \beta_{2} ) q^{74} + ( 4 + \beta_{1} - 3 \beta_{2} ) q^{75} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{76} + ( -1 - 5 \beta_{1} - 2 \beta_{2} ) q^{78} + ( -4 + 4 \beta_{1} ) q^{79} + ( -13 - \beta_{1} + 2 \beta_{2} ) q^{80} + q^{81} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{82} + ( -2 + 6 \beta_{1} ) q^{83} + ( 2 + \beta_{1} ) q^{84} + ( 6 - 2 \beta_{1} + 12 \beta_{2} ) q^{85} + ( 6 + 10 \beta_{1} - 6 \beta_{2} ) q^{86} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{87} + ( 10 + 4 \beta_{2} ) q^{89} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{90} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{91} + ( 10 + 6 \beta_{1} + 4 \beta_{2} ) q^{92} + ( -2 + 2 \beta_{1} - 4 \beta_{2} ) q^{93} + ( -11 - 5 \beta_{1} + 2 \beta_{2} ) q^{94} + ( 3 - 5 \beta_{1} + \beta_{2} ) q^{95} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{96} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{97} + ( -1 - \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{2} + 3q^{3} + 6q^{4} + 4q^{5} - 2q^{6} + 3q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 2q^{2} + 3q^{3} + 6q^{4} + 4q^{5} - 2q^{6} + 3q^{7} - 3q^{8} + 3q^{9} + 11q^{10} + 6q^{12} + 4q^{13} - 2q^{14} + 4q^{15} - 4q^{16} - 8q^{17} - 2q^{18} + 8q^{19} - 3q^{20} + 3q^{21} + 10q^{23} - 3q^{24} + 15q^{25} - q^{26} + 3q^{27} + 6q^{28} + 4q^{29} + 11q^{30} - 2q^{31} - 8q^{32} - 4q^{34} + 4q^{35} + 6q^{36} - 13q^{38} + 4q^{39} + 18q^{40} - 14q^{41} - 2q^{42} + 14q^{43} + 4q^{45} - 28q^{46} - 4q^{48} + 3q^{49} + 19q^{50} - 8q^{51} + 29q^{52} - 2q^{54} - 3q^{56} + 8q^{57} - q^{58} - 3q^{60} + 6q^{61} + 38q^{62} + 3q^{63} - 5q^{64} - 14q^{65} - 4q^{67} - 42q^{68} + 10q^{69} + 11q^{70} - 12q^{71} - 3q^{72} + 20q^{73} - 29q^{74} + 15q^{75} + 11q^{76} - q^{78} - 12q^{79} - 41q^{80} + 3q^{81} - 6q^{82} - 6q^{83} + 6q^{84} + 6q^{85} + 24q^{86} + 4q^{87} + 26q^{89} + 11q^{90} + 4q^{91} + 26q^{92} - 2q^{93} - 35q^{94} + 8q^{95} - 8q^{96} - 4q^{97} - 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 4 x - 1\):

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

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
2.11491
−1.86081
−0.254102
−2.47283 1.00000 4.11491 −2.58774 −2.47283 1.00000 −5.22982 1.00000 6.39905
1.2 −1.46260 1.00000 0.139194 2.39821 −1.46260 1.00000 2.72161 1.00000 −3.50761
1.3 1.93543 1.00000 1.74590 4.18953 1.93543 1.00000 −0.491797 1.00000 8.10856
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 2541.2.a.bg 3
3.b odd 2 1 7623.2.a.cd 3
11.b odd 2 1 231.2.a.e 3
33.d even 2 1 693.2.a.l 3
44.c even 2 1 3696.2.a.bo 3
55.d odd 2 1 5775.2.a.bp 3
77.b even 2 1 1617.2.a.t 3
231.h odd 2 1 4851.2.a.bi 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.e 3 11.b odd 2 1
693.2.a.l 3 33.d even 2 1
1617.2.a.t 3 77.b even 2 1
2541.2.a.bg 3 1.a even 1 1 trivial
3696.2.a.bo 3 44.c even 2 1
4851.2.a.bi 3 231.h odd 2 1
5775.2.a.bp 3 55.d odd 2 1
7623.2.a.cd 3 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(11\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2541))\):

\( T_{2}^{3} + 2 T_{2}^{2} - 4 T_{2} - 7 \)
\( T_{5}^{3} - 4 T_{5}^{2} - 7 T_{5} + 26 \)
\( T_{13}^{3} - 4 T_{13}^{2} - 27 T_{13} + 94 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} + T^{3} + 4 T^{4} + 8 T^{5} + 8 T^{6} \)
$3$ \( ( 1 - T )^{3} \)
$5$ \( 1 - 4 T + 8 T^{2} - 14 T^{3} + 40 T^{4} - 100 T^{5} + 125 T^{6} \)
$7$ \( ( 1 - T )^{3} \)
$11$ \( \)
$13$ \( 1 - 4 T + 12 T^{2} - 10 T^{3} + 156 T^{4} - 676 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 8 T + 11 T^{2} - 56 T^{3} + 187 T^{4} + 2312 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 8 T + 72 T^{2} - 308 T^{3} + 1368 T^{4} - 2888 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 10 T + 81 T^{2} - 396 T^{3} + 1863 T^{4} - 5290 T^{5} + 12167 T^{6} \)
$29$ \( 1 - 4 T + 60 T^{2} - 138 T^{3} + 1740 T^{4} - 3364 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 2 T + 17 T^{2} - 132 T^{3} + 527 T^{4} + 1922 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 68 T^{2} + 106 T^{3} + 2516 T^{4} + 50653 T^{6} \)
$41$ \( 1 + 14 T + 163 T^{2} + 1116 T^{3} + 6683 T^{4} + 23534 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 14 T + 85 T^{2} - 356 T^{3} + 3655 T^{4} - 25886 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 80 T^{2} + 32 T^{3} + 3760 T^{4} + 103823 T^{6} \)
$53$ \( 1 + 143 T^{2} + 8 T^{3} + 7579 T^{4} + 148877 T^{6} \)
$59$ \( 1 + 120 T^{2} - 52 T^{3} + 7080 T^{4} + 205379 T^{6} \)
$61$ \( ( 1 - 2 T + 61 T^{2} )^{3} \)
$67$ \( 1 + 4 T + 116 T^{2} + 300 T^{3} + 7772 T^{4} + 17956 T^{5} + 300763 T^{6} \)
$71$ \( 1 + 12 T + 197 T^{2} + 1448 T^{3} + 13987 T^{4} + 60492 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 20 T + 320 T^{2} - 3054 T^{3} + 23360 T^{4} - 106580 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 12 T + 221 T^{2} + 1640 T^{3} + 17459 T^{4} + 74892 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 6 T + 117 T^{2} + 500 T^{3} + 9711 T^{4} + 41334 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 26 T + 407 T^{2} - 4300 T^{3} + 36223 T^{4} - 205946 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 4 T + 171 T^{2} + 544 T^{3} + 16587 T^{4} + 37636 T^{5} + 912673 T^{6} \)
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