Properties

Label 2541.2.a.bj
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: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) 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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} + \beta_1) q^{5} + \beta_1 q^{6} + q^{7} + (\beta_{2} + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} + \beta_1) q^{5} + \beta_1 q^{6} + q^{7} + (\beta_{2} + 1) q^{8} + q^{9} + ( - \beta_1 + 2) q^{10} + (\beta_{2} + 1) q^{12} + (\beta_1 + 2) q^{13} + \beta_1 q^{14} + ( - \beta_{2} + \beta_1) q^{15} + ( - \beta_{2} + 2 \beta_1 - 1) q^{16} + ( - \beta_{2} + \beta_1) q^{17} + \beta_1 q^{18} + ( - \beta_{2} - \beta_1 + 3) q^{19} + (\beta_{2} - 3) q^{20} + q^{21} + (3 \beta_{2} + \beta_1 - 1) q^{23} + (\beta_{2} + 1) q^{24} - 2 \beta_{2} q^{25} + (\beta_{2} + 2 \beta_1 + 3) q^{26} + q^{27} + (\beta_{2} + 1) q^{28} + (\beta_1 + 2) q^{29} + ( - \beta_1 + 2) q^{30} + (2 \beta_{2} - 2 \beta_1) q^{31} + ( - \beta_{2} - 2 \beta_1 + 3) q^{32} + ( - \beta_1 + 2) q^{34} + ( - \beta_{2} + \beta_1) q^{35} + (\beta_{2} + 1) q^{36} + ( - \beta_{2} - 1) q^{37} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{38} + (\beta_1 + 2) q^{39} + (\beta_{2} - 3) q^{40} + (\beta_{2} - 2 \beta_1 + 5) q^{41} + \beta_1 q^{42} + ( - \beta_{2} - 4 \beta_1 + 4) q^{43} + ( - \beta_{2} + \beta_1) q^{45} + (4 \beta_{2} + 2 \beta_1 + 6) q^{46} + ( - 2 \beta_{2} + 3 \beta_1 - 3) q^{47} + ( - \beta_{2} + 2 \beta_1 - 1) q^{48} + q^{49} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{50} + ( - \beta_{2} + \beta_1) q^{51} + (3 \beta_{2} + 2 \beta_1 + 3) q^{52} + (\beta_{2} - 7) q^{53} + \beta_1 q^{54} + (\beta_{2} + 1) q^{56} + ( - \beta_{2} - \beta_1 + 3) q^{57} + (\beta_{2} + 2 \beta_1 + 3) q^{58} + (3 \beta_1 + 1) q^{59} + (\beta_{2} - 3) q^{60} + (3 \beta_{2} - 3 \beta_1 + 5) q^{61} + (2 \beta_1 - 4) q^{62} + q^{63} + ( - \beta_{2} - 2 \beta_1 - 5) q^{64} + ( - 2 \beta_{2} + \beta_1 + 2) q^{65} + ( - 5 \beta_{2} + 2 \beta_1) q^{67} + (\beta_{2} - 3) q^{68} + (3 \beta_{2} + \beta_1 - 1) q^{69} + ( - \beta_1 + 2) q^{70} + (\beta_{2} - 3 \beta_1 + 1) q^{71} + (\beta_{2} + 1) q^{72} + (3 \beta_{2} + 5 \beta_1 - 3) q^{73} + ( - \beta_{2} - 2 \beta_1 - 1) q^{74} - 2 \beta_{2} q^{75} + (2 \beta_{2} - 4 \beta_1 - 2) q^{76} + (\beta_{2} + 2 \beta_1 + 3) q^{78} + ( - 2 \beta_{2} + 6 \beta_1 + 4) q^{79} + ( - \beta_{2} - 2 \beta_1 + 7) q^{80} + q^{81} + ( - \beta_{2} + 6 \beta_1 - 5) q^{82} + ( - 6 \beta_{2} + 3 \beta_1 - 5) q^{83} + (\beta_{2} + 1) q^{84} + ( - 2 \beta_{2} + 5) q^{85} + ( - 5 \beta_{2} + 3 \beta_1 - 13) q^{86} + (\beta_1 + 2) q^{87} + ( - 3 \beta_{2} - \beta_1) q^{89} + ( - \beta_1 + 2) q^{90} + (\beta_1 + 2) q^{91} + (8 \beta_1 + 12) q^{92} + (2 \beta_{2} - 2 \beta_1) q^{93} + (\beta_{2} - 5 \beta_1 + 7) q^{94} + ( - 5 \beta_{2} + 5 \beta_1 + 1) q^{95} + ( - \beta_{2} - 2 \beta_1 + 3) q^{96} + (2 \beta_{2} - 7 \beta_1 - 6) q^{97} + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 3 q^{3} + 3 q^{4} + q^{5} + q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 3 q^{3} + 3 q^{4} + q^{5} + q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} + 5 q^{10} + 3 q^{12} + 7 q^{13} + q^{14} + q^{15} - q^{16} + q^{17} + q^{18} + 8 q^{19} - 9 q^{20} + 3 q^{21} - 2 q^{23} + 3 q^{24} + 11 q^{26} + 3 q^{27} + 3 q^{28} + 7 q^{29} + 5 q^{30} - 2 q^{31} + 7 q^{32} + 5 q^{34} + q^{35} + 3 q^{36} - 3 q^{37} - 10 q^{38} + 7 q^{39} - 9 q^{40} + 13 q^{41} + q^{42} + 8 q^{43} + q^{45} + 20 q^{46} - 6 q^{47} - q^{48} + 3 q^{49} - 8 q^{50} + q^{51} + 11 q^{52} - 21 q^{53} + q^{54} + 3 q^{56} + 8 q^{57} + 11 q^{58} + 6 q^{59} - 9 q^{60} + 12 q^{61} - 10 q^{62} + 3 q^{63} - 17 q^{64} + 7 q^{65} + 2 q^{67} - 9 q^{68} - 2 q^{69} + 5 q^{70} + 3 q^{72} - 4 q^{73} - 5 q^{74} - 10 q^{76} + 11 q^{78} + 18 q^{79} + 19 q^{80} + 3 q^{81} - 9 q^{82} - 12 q^{83} + 3 q^{84} + 15 q^{85} - 36 q^{86} + 7 q^{87} - q^{89} + 5 q^{90} + 7 q^{91} + 44 q^{92} - 2 q^{93} + 16 q^{94} + 8 q^{95} + 7 q^{96} - 25 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
−1.81361
0.470683
2.34292
−1.81361 1.00000 1.28917 −2.10278 −1.81361 1.00000 1.28917 1.00000 3.81361
1.2 0.470683 1.00000 −1.77846 3.24914 0.470683 1.00000 −1.77846 1.00000 1.52932
1.3 2.34292 1.00000 3.48929 −0.146365 2.34292 1.00000 3.48929 1.00000 −0.342923
\(n\): 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.bj yes 3
3.b odd 2 1 7623.2.a.ca 3
11.b odd 2 1 2541.2.a.bh 3
33.d even 2 1 7623.2.a.cc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.bh 3 11.b odd 2 1
2541.2.a.bj yes 3 1.a even 1 1 trivial
7623.2.a.ca 3 3.b odd 2 1
7623.2.a.cc 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2541))\):

\( T_{2}^{3} - T_{2}^{2} - 4T_{2} + 2 \) Copy content Toggle raw display
\( T_{5}^{3} - T_{5}^{2} - 7T_{5} - 1 \) Copy content Toggle raw display
\( T_{13}^{3} - 7T_{13}^{2} + 12T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 4T + 2 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - T^{2} - 7T - 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 7 T^{2} + 12 T - 2 \) Copy content Toggle raw display
$17$ \( T^{3} - T^{2} - 7T - 1 \) Copy content Toggle raw display
$19$ \( T^{3} - 8 T^{2} + 6 T + 44 \) Copy content Toggle raw display
$23$ \( T^{3} + 2 T^{2} - 78 T - 152 \) Copy content Toggle raw display
$29$ \( T^{3} - 7 T^{2} + 12 T - 2 \) Copy content Toggle raw display
$31$ \( T^{3} + 2 T^{2} - 28 T + 8 \) Copy content Toggle raw display
$37$ \( T^{3} + 3 T^{2} - 4 T - 8 \) Copy content Toggle raw display
$41$ \( T^{3} - 13 T^{2} + 40 T - 32 \) Copy content Toggle raw display
$43$ \( T^{3} - 8 T^{2} - 71 T + 422 \) Copy content Toggle raw display
$47$ \( T^{3} + 6 T^{2} - 31 T - 34 \) Copy content Toggle raw display
$53$ \( T^{3} + 21 T^{2} + 140 T + 296 \) Copy content Toggle raw display
$59$ \( T^{3} - 6 T^{2} - 27 T + 86 \) Copy content Toggle raw display
$61$ \( T^{3} - 12 T^{2} - 18 T + 292 \) Copy content Toggle raw display
$67$ \( T^{3} - 2 T^{2} - 151 T - 584 \) Copy content Toggle raw display
$71$ \( T^{3} - 34T - 76 \) Copy content Toggle raw display
$73$ \( T^{3} + 4 T^{2} - 226 T - 1628 \) Copy content Toggle raw display
$79$ \( T^{3} - 18 T^{2} - 28 T + 1208 \) Copy content Toggle raw display
$83$ \( T^{3} + 12 T^{2} - 171 T - 2056 \) Copy content Toggle raw display
$89$ \( T^{3} + T^{2} - 79 T + 73 \) Copy content Toggle raw display
$97$ \( T^{3} + 25 T^{2} + 24 T - 1882 \) Copy content Toggle raw display
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