Properties

Label 2541.2.a.bm
Level $2541$
Weight $2$
Character orbit 2541.a
Self dual yes
Analytic conductor $20.290$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 - 1) q^{2} - q^{3} + (\beta_{3} + \beta_1) q^{4} - 2 \beta_{2} q^{5} + ( - \beta_{2} - \beta_1 + 1) q^{6} + q^{7} + (\beta_1 + 2) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 - 1) q^{2} - q^{3} + (\beta_{3} + \beta_1) q^{4} - 2 \beta_{2} q^{5} + ( - \beta_{2} - \beta_1 + 1) q^{6} + q^{7} + (\beta_1 + 2) q^{8} + q^{9} + ( - 2 \beta_1 - 2) q^{10} + ( - \beta_{3} - \beta_1) q^{12} + ( - 2 \beta_{2} - 2 \beta_1) q^{13} + (\beta_{2} + \beta_1 - 1) q^{14} + 2 \beta_{2} q^{15} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{16} + ( - 2 \beta_{3} + 4 \beta_1 - 2) q^{17} + (\beta_{2} + \beta_1 - 1) q^{18} + (2 \beta_{3} - 2 \beta_1 + 2) q^{19} + ( - 2 \beta_{3} - 4 \beta_1 + 2) q^{20} - q^{21} + (3 \beta_{3} - 4 \beta_1 - 3) q^{23} + ( - \beta_1 - 2) q^{24} + ( - 4 \beta_{3} + 4 \beta_{2} + 3) q^{25} + ( - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{26} - q^{27} + (\beta_{3} + \beta_1) q^{28} + (\beta_{3} + 4 \beta_{2} - 2 \beta_1 - 2) q^{29} + (2 \beta_1 + 2) q^{30} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{31} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{32} + (6 \beta_{3} - 2 \beta_1 + 2) q^{34} - 2 \beta_{2} q^{35} + (\beta_{3} + \beta_1) q^{36} + ( - \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 3) q^{37} + ( - 4 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{38} + (2 \beta_{2} + 2 \beta_1) q^{39} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 2) q^{40} + ( - 6 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{41} + ( - \beta_{2} - \beta_1 + 1) q^{42} + (5 \beta_{3} - 6 \beta_1) q^{43} - 2 \beta_{2} q^{45} + ( - 7 \beta_{3} - 4 \beta_{2} - \beta_1 + 3) q^{46} + ( - 6 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{47} + (\beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{48} + q^{49} + (4 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{50} + (2 \beta_{3} - 4 \beta_1 + 2) q^{51} + ( - 2 \beta_{3} - 4 \beta_{2} - 8 \beta_1) q^{52} + (5 \beta_{3} - 2 \beta_1 - 2) q^{53} + ( - \beta_{2} - \beta_1 + 1) q^{54} + (\beta_1 + 2) q^{56} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{57} + ( - 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 6) q^{58} + (2 \beta_{3} + 2 \beta_{2} - 6) q^{59} + (2 \beta_{3} + 4 \beta_1 - 2) q^{60} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 2) q^{61} + ( - 8 \beta_{2} - 8 \beta_1 + 6) q^{62} + q^{63} + ( - 2 \beta_{3} - 5 \beta_{2} - \beta_1 + 1) q^{64} + (4 \beta_{2} + 4 \beta_1 + 4) q^{65} + (3 \beta_{3} - 2 \beta_1 - 8) q^{67} + ( - 4 \beta_{3} + 6 \beta_{2} + 4 \beta_1 + 2) q^{68} + ( - 3 \beta_{3} + 4 \beta_1 + 3) q^{69} + ( - 2 \beta_1 - 2) q^{70} + ( - 3 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 6) q^{71} + (\beta_1 + 2) q^{72} + ( - 2 \beta_{2} + 2) q^{73} + (5 \beta_{3} + 3 \beta_1 + 7) q^{74} + (4 \beta_{3} - 4 \beta_{2} - 3) q^{75} + (4 \beta_{3} - 2 \beta_{2}) q^{76} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{78} + (\beta_{3} + 2 \beta_1 + 4) q^{79} + (4 \beta_{3} - 2 \beta_{2} - 10) q^{80} + q^{81} + (8 \beta_{3} - 2 \beta_{2} - 12 \beta_1 - 6) q^{82} + (2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 - 2) q^{83} + ( - \beta_{3} - \beta_1) q^{84} + ( - 8 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 8) q^{85} + ( - 11 \beta_{3} - \beta_{2} + 4 \beta_1) q^{86} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{87} + (4 \beta_{3} - 6 \beta_{2} - 4 \beta_1 - 2) q^{89} + ( - 2 \beta_1 - 2) q^{90} + ( - 2 \beta_{2} - 2 \beta_1) q^{91} + ( - 5 \beta_{2} - 8 \beta_1 - 1) q^{92} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{93} + (10 \beta_{3} - 4 \beta_1) q^{94} + (4 \beta_{3} - 4 \beta_{2} - 4) q^{95} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{96} + ( - 6 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 8) q^{97} + (\beta_{2} + \beta_1 - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{5} + q^{6} + 4 q^{7} + 9 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{5} + q^{6} + 4 q^{7} + 9 q^{8} + 4 q^{9} - 10 q^{10} - 3 q^{12} - 6 q^{13} - q^{14} + 4 q^{15} - 3 q^{16} - 8 q^{17} - q^{18} + 10 q^{19} - 4 q^{21} - 10 q^{23} - 9 q^{24} + 12 q^{25} - 20 q^{26} - 4 q^{27} + 3 q^{28} + 10 q^{30} - 18 q^{31} + 2 q^{32} + 18 q^{34} - 4 q^{35} + 3 q^{36} - 2 q^{37} - 8 q^{38} + 6 q^{39} - 6 q^{40} - 10 q^{41} + q^{42} + 4 q^{43} - 4 q^{45} - 11 q^{46} + 4 q^{47} + 3 q^{48} + 4 q^{49} + 9 q^{50} + 8 q^{51} - 20 q^{52} + q^{54} + 9 q^{56} - 10 q^{57} + 14 q^{58} - 16 q^{59} - 14 q^{61} + 4 q^{63} - 11 q^{64} + 28 q^{65} - 28 q^{67} + 16 q^{68} + 10 q^{69} - 10 q^{70} - 18 q^{71} + 9 q^{72} + 4 q^{73} + 41 q^{74} - 12 q^{75} + 4 q^{76} + 20 q^{78} + 20 q^{79} - 36 q^{80} + 4 q^{81} - 24 q^{82} - 6 q^{83} - 3 q^{84} + 20 q^{85} - 20 q^{86} - 16 q^{89} - 10 q^{90} - 6 q^{91} - 22 q^{92} + 18 q^{93} + 16 q^{94} - 16 q^{95} - 2 q^{96} + 32 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 2\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 3\beta_1 \) 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
−0.477260
0.737640
−1.35567
2.09529
−1.77222 −1.00000 1.14077 0.589926 1.77222 1.00000 1.52274 1.00000 −1.04548
1.2 −1.45589 −1.00000 0.119606 2.38705 1.45589 1.00000 2.73764 1.00000 −3.47528
1.3 −0.162147 −1.00000 −1.97371 −4.38705 0.162147 1.00000 0.644326 1.00000 0.711349
1.4 2.39026 −1.00000 3.71333 −2.58993 −2.39026 1.00000 4.09529 1.00000 −6.19059
\(n\): 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.bm 4
3.b odd 2 1 7623.2.a.cl 4
11.b odd 2 1 2541.2.a.bn 4
11.d odd 10 2 231.2.j.f 8
33.d even 2 1 7623.2.a.ci 4
33.f even 10 2 693.2.m.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.f 8 11.d odd 10 2
693.2.m.f 8 33.f even 10 2
2541.2.a.bm 4 1.a even 1 1 trivial
2541.2.a.bn 4 11.b odd 2 1
7623.2.a.ci 4 33.d even 2 1
7623.2.a.cl 4 3.b odd 2 1

Hecke kernels

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

\( T_{2}^{4} + T_{2}^{3} - 5T_{2}^{2} - 7T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{4} + 4T_{5}^{3} - 8T_{5}^{2} - 24T_{5} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} + 6T_{13}^{3} - 8T_{13}^{2} - 16T_{13} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} - 5 T^{2} - 7 T - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} - 8 T^{2} - 24 T + 16 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} - 8 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} - 20 T^{2} - 144 T + 304 \) Copy content Toggle raw display
$19$ \( T^{4} - 10 T^{3} + 24 T^{2} - 16 \) Copy content Toggle raw display
$23$ \( T^{4} + 10 T^{3} - 9 T^{2} - 190 T + 109 \) Copy content Toggle raw display
$29$ \( T^{4} - 79T^{2} + 29 \) Copy content Toggle raw display
$31$ \( T^{4} + 18 T^{3} + 68 T^{2} + \cdots - 1744 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} - 77 T^{2} - 218 T + 281 \) Copy content Toggle raw display
$41$ \( T^{4} + 10 T^{3} - 104 T^{2} + \cdots - 4496 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} - 103 T^{2} + \cdots + 1861 \) Copy content Toggle raw display
$47$ \( T^{4} - 4 T^{3} - 80 T^{2} + 568 T - 976 \) Copy content Toggle raw display
$53$ \( T^{4} - 51 T^{2} - 20 T + 209 \) Copy content Toggle raw display
$59$ \( T^{4} + 16 T^{3} + 72 T^{2} + 104 T + 16 \) Copy content Toggle raw display
$61$ \( T^{4} + 14 T^{3} + 16 T^{2} - 256 T + 16 \) Copy content Toggle raw display
$67$ \( T^{4} + 28 T^{3} + 273 T^{2} + \cdots + 1301 \) Copy content Toggle raw display
$71$ \( T^{4} + 18 T^{3} + 43 T^{2} + \cdots - 179 \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} - 8 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$79$ \( T^{4} - 20 T^{3} + 129 T^{2} + \cdots + 149 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} - 120 T^{2} + \cdots + 2864 \) Copy content Toggle raw display
$89$ \( T^{4} + 16 T^{3} - 48 T^{2} + \cdots - 304 \) Copy content Toggle raw display
$97$ \( T^{4} - 32 T^{3} + 268 T^{2} + \cdots - 8464 \) Copy content Toggle raw display
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