Properties

Label 2541.2.a.bo
Level $2541$
Weight $2$
Character orbit 2541.a
Self dual yes
Analytic conductor $20.290$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.7488.1
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - q^{3} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{3} + 1) q^{5} + \beta_{3} q^{6} + q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - q^{3} + ( - \beta_{2} + 1) q^{4} + ( - \beta_{3} + 1) q^{5} + \beta_{3} q^{6} + q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{8} + q^{9} + ( - \beta_{3} - \beta_{2} + 3) q^{10} + (\beta_{2} - 1) q^{12} + (2 \beta_{2} + \beta_1 - 1) q^{13} - \beta_{3} q^{14} + (\beta_{3} - 1) q^{15} + ( - 2 \beta_{3} - 2 \beta_1) q^{16} + ( - \beta_{2} - \beta_1 + 1) q^{17} - \beta_{3} q^{18} + ( - \beta_{3} - \beta_{2} + 1) q^{19} + ( - 3 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{20} - q^{21} + (\beta_{2} + \beta_1 + 2) q^{23} + (\beta_{3} + \beta_{2} + \beta_1) q^{24} + ( - 2 \beta_{3} - \beta_{2} - 1) q^{25} + (5 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 1) q^{26} - q^{27} + ( - \beta_{2} + 1) q^{28} + (2 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{29} + (\beta_{3} + \beta_{2} - 3) q^{30} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{31} + (2 \beta_{3} + 4) q^{32} + ( - 3 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{34} + ( - \beta_{3} + 1) q^{35} + ( - \beta_{2} + 1) q^{36} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{37} + ( - 3 \beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{38} + ( - 2 \beta_{2} - \beta_1 + 1) q^{39} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 2) q^{40} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{41} + \beta_{3} q^{42} + ( - \beta_{3} + \beta_1 + 4) q^{43} + ( - \beta_{3} + 1) q^{45} + (\beta_{2} + 2 \beta_1 + 1) q^{46} + (2 \beta_{2} - 3 \beta_1 + 2) q^{47} + (2 \beta_{3} + 2 \beta_1) q^{48} + q^{49} + ( - \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{50} + (\beta_{2} + \beta_1 - 1) q^{51} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 10) q^{52} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{53} + \beta_{3} q^{54} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{56} + (\beta_{3} + \beta_{2} - 1) q^{57} + ( - 6 \beta_{3} + \beta_{2} - 5) q^{58} + (7 \beta_{3} + 6 \beta_1 + 1) q^{59} + (3 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{60} + ( - 3 \beta_{3} - \beta_{2} - 1) q^{61} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{62} + q^{63} + (2 \beta_{2} + 4 \beta_1 - 6) q^{64} + (5 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{65} + ( - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 5) q^{67} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 5) q^{68} + ( - \beta_{2} - \beta_1 - 2) q^{69} + ( - \beta_{3} - \beta_{2} + 3) q^{70} + ( - \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 6) q^{71} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{72} + (2 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 1) q^{73} + (5 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 4) q^{74} + (2 \beta_{3} + \beta_{2} + 1) q^{75} + ( - 5 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 6) q^{76} + ( - 5 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 1) q^{78} + (3 \beta_{2} + 2 \beta_1 + 3) q^{79} + ( - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 4) q^{80} + q^{81} + ( - 7 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{82} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 6) q^{83} + (\beta_{2} - 1) q^{84} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{85} + ( - 4 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{86} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 - 4) q^{87} + ( - 3 \beta_{3} + 4 \beta_{2} - 5) q^{89} + ( - \beta_{3} - \beta_{2} + 3) q^{90} + (2 \beta_{2} + \beta_1 - 1) q^{91} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{92} + (\beta_{3} + \beta_{2} + \beta_1) q^{93} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 3) q^{94} + ( - 4 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{95} + ( - 2 \beta_{3} - 4) q^{96} + (2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 9) q^{97} - \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} + 6 q^{5} - 2 q^{6} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} + 6 q^{5} - 2 q^{6} + 4 q^{7} + 4 q^{9} + 14 q^{10} - 4 q^{12} - 2 q^{13} + 2 q^{14} - 6 q^{15} + 2 q^{17} + 2 q^{18} + 6 q^{19} + 8 q^{20} - 4 q^{21} + 10 q^{23} - 4 q^{27} + 4 q^{28} + 14 q^{29} - 14 q^{30} + 12 q^{32} - 2 q^{34} + 6 q^{35} + 4 q^{36} - 12 q^{37} + 16 q^{38} + 2 q^{39} + 8 q^{40} + 16 q^{41} - 2 q^{42} + 20 q^{43} + 6 q^{45} + 8 q^{46} + 2 q^{47} + 4 q^{49} + 24 q^{50} - 2 q^{51} - 40 q^{52} - 16 q^{53} - 2 q^{54} - 6 q^{57} - 8 q^{58} + 2 q^{59} - 8 q^{60} + 2 q^{61} + 8 q^{62} + 4 q^{63} - 16 q^{64} - 2 q^{65} - 20 q^{67} + 20 q^{68} - 10 q^{69} + 14 q^{70} - 10 q^{71} - 10 q^{73} - 20 q^{74} + 28 q^{76} + 16 q^{79} + 12 q^{80} + 4 q^{81} + 28 q^{82} + 18 q^{83} - 4 q^{84} + 26 q^{86} - 14 q^{87} - 14 q^{89} + 14 q^{90} - 2 q^{91} - 8 q^{92} - 18 q^{94} + 22 q^{95} - 12 q^{96} + 38 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 4x^{2} + 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 8\beta _1 + 3 \) Copy content Toggle raw display

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
−0.326909
−1.43091
3.05896
0.698857
−2.05896 −1.00000 2.23931 −1.05896 2.05896 1.00000 −0.492737 1.00000 2.18035
1.2 0.301143 −1.00000 −1.90931 1.30114 −0.301143 1.00000 −1.17726 1.00000 0.391830
1.3 1.32691 −1.00000 −0.239314 2.32691 −1.32691 1.00000 −2.97136 1.00000 3.08759
1.4 2.43091 −1.00000 3.90931 3.43091 −2.43091 1.00000 4.64136 1.00000 8.34022
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 2541.2.a.bo yes 4
3.b odd 2 1 7623.2.a.cf 4
11.b odd 2 1 2541.2.a.bk 4
33.d even 2 1 7623.2.a.cm 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.bk 4 11.b odd 2 1
2541.2.a.bo yes 4 1.a even 1 1 trivial
7623.2.a.cf 4 3.b odd 2 1
7623.2.a.cm 4 33.d even 2 1

Hecke kernels

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

\( T_{2}^{4} - 2T_{2}^{3} - 4T_{2}^{2} + 8T_{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{4} - 6T_{5}^{3} + 8T_{5}^{2} + 6T_{5} - 11 \) Copy content Toggle raw display
\( T_{13}^{4} + 2T_{13}^{3} - 40T_{13}^{2} - 32T_{13} + 358 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 4 T^{2} + 8 T - 2 \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} + 8 T^{2} + 6 T - 11 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} - 40 T^{2} - 32 T + 358 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} - 12 T^{2} + 22 T + 13 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} - 4 T^{2} + 12 T - 2 \) Copy content Toggle raw display
$23$ \( T^{4} - 10 T^{3} + 24 T^{2} - 4 T - 2 \) Copy content Toggle raw display
$29$ \( T^{4} - 14 T^{3} + 48 T^{2} - 20 T - 74 \) Copy content Toggle raw display
$31$ \( T^{4} - 16 T^{2} - 24 T - 8 \) Copy content Toggle raw display
$37$ \( T^{4} + 12 T^{3} + 20 T^{2} - 48 T - 44 \) Copy content Toggle raw display
$41$ \( T^{4} - 16 T^{3} + 32 T^{2} + \cdots - 716 \) Copy content Toggle raw display
$43$ \( T^{4} - 20 T^{3} + 134 T^{2} + \cdots + 277 \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} - 100 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$53$ \( T^{4} + 16 T^{3} + 32 T^{2} + \cdots - 1664 \) Copy content Toggle raw display
$59$ \( T^{4} - 2 T^{3} - 256 T^{2} + \cdots + 15517 \) Copy content Toggle raw display
$61$ \( T^{4} - 2 T^{3} - 64 T^{2} - 52 T + 286 \) Copy content Toggle raw display
$67$ \( T^{4} + 20 T^{3} + 14 T^{2} + \cdots - 4103 \) Copy content Toggle raw display
$71$ \( T^{4} + 10 T^{3} - 208 T^{2} + \cdots - 8882 \) Copy content Toggle raw display
$73$ \( T^{4} + 10 T^{3} - 240 T^{2} + \cdots + 10894 \) Copy content Toggle raw display
$79$ \( T^{4} - 16 T^{3} - 4 T^{2} + 520 T + 676 \) Copy content Toggle raw display
$83$ \( T^{4} - 18 T^{3} + 68 T^{2} + 78 T + 13 \) Copy content Toggle raw display
$89$ \( T^{4} + 14 T^{3} - 112 T^{2} + \cdots + 4477 \) Copy content Toggle raw display
$97$ \( T^{4} - 38 T^{3} + 456 T^{2} + \cdots - 3938 \) Copy content Toggle raw display
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