Properties

Label 2541.2.a.bl
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: 4.4.7488.1
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} + (\beta_1 - 3) q^{13} + \beta_{3} q^{14} + ( - \beta_{3} - 1) q^{15} + ( - 2 \beta_{3} - 2 \beta_1) q^{16} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{17} + \beta_{3} q^{18} + ( - \beta_{3} + \beta_{2} - 5) q^{19} + ( - 3 \beta_{3} - \beta_1 - 1) q^{20} + q^{21} + ( - \beta_{2} + 3 \beta_1 - 2) q^{23} + (\beta_{3} + \beta_{2} + \beta_1) q^{24} + (2 \beta_{3} - \beta_{2} - 1) q^{25} + ( - 3 \beta_{3} - \beta_1 - 1) q^{26} + q^{27} + ( - \beta_{2} + 1) q^{28} + (2 \beta_{3} - 3 \beta_{2} - \beta_1) q^{29} + ( - \beta_{3} + \beta_{2} - 3) q^{30} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{31} + ( - 2 \beta_{3} - 4) q^{32} + ( - 3 \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{34} + ( - \beta_{3} - 1) q^{35} + ( - \beta_{2} + 1) q^{36} + (2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{37} + ( - 7 \beta_{3} - \beta_1 - 3) q^{38} + (\beta_1 - 3) q^{39} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{40} + (3 \beta_{3} + \beta_1 + 1) q^{41} + \beta_{3} q^{42} + (3 \beta_{3} + \beta_1 - 4) q^{43} + ( - \beta_{3} - 1) q^{45} + (\beta_{2} - 2 \beta_1 - 3) q^{46} + (4 \beta_{2} + \beta_1 - 2) q^{47} + ( - 2 \beta_{3} - 2 \beta_1) q^{48} + q^{49} + (\beta_{3} - \beta_{2} + \beta_1 + 6) q^{50} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{51} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{52} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{53} + \beta_{3} q^{54} + (\beta_{3} + \beta_{2} + \beta_1) q^{56} + ( - \beta_{3} + \beta_{2} - 5) q^{57} + (6 \beta_{3} + \beta_{2} + 4 \beta_1 + 7) q^{58} + (3 \beta_{3} + 2 \beta_1 - 1) q^{59} + ( - 3 \beta_{3} - \beta_1 - 1) q^{60} + (\beta_{3} - 3 \beta_{2} - 4 \beta_1 + 5) q^{61} + ( - 6 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{62} + q^{63} + (2 \beta_{2} + 4 \beta_1 - 6) q^{64} + (3 \beta_{3} + 4) q^{65} + ( - 2 \beta_{2} + 4 \beta_1 - 1) q^{67} + ( - \beta_{3} + 3 \beta_1 - 9) q^{68} + ( - \beta_{2} + 3 \beta_1 - 2) q^{69} + ( - \beta_{3} + \beta_{2} - 3) q^{70} + ( - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{71} + (\beta_{3} + \beta_{2} + \beta_1) q^{72} + ( - 2 \beta_{3} - \beta_1 - 9) q^{73} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 + 8) q^{74} + (2 \beta_{3} - \beta_{2} - 1) q^{75} + ( - \beta_{3} + 5 \beta_{2} + \beta_1 - 10) q^{76} + ( - 3 \beta_{3} - \beta_1 - 1) q^{78} + (3 \beta_{2} + 6 \beta_1 - 9) q^{79} + (2 \beta_{3} - 2 \beta_{2} + 4) q^{80} + q^{81} + (\beta_{3} - 3 \beta_{2} - \beta_1 + 8) q^{82} + (3 \beta_{3} + \beta_{2} + 4 \beta_1 - 2) q^{83} + ( - \beta_{2} + 1) q^{84} + (5 \beta_{3} - 2 \beta_{2} + \beta_1 + 4) q^{85} + ( - 4 \beta_{3} - 3 \beta_{2} - \beta_1 + 8) q^{86} + (2 \beta_{3} - 3 \beta_{2} - \beta_1) q^{87} + (5 \beta_{3} + 4 \beta_1 + 5) q^{89} + ( - \beta_{3} + \beta_{2} - 3) q^{90} + (\beta_1 - 3) q^{91} + ( - 5 \beta_{3} + \beta_{2} - 5 \beta_1 + 6) q^{92} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{93} + ( - 10 \beta_{3} - 4 \beta_{2} - 5 \beta_1 - 1) q^{94} + (8 \beta_{3} - \beta_{2} + \beta_1 + 8) q^{95} + ( - 2 \beta_{3} - 4) q^{96} + (2 \beta_{3} - \beta_1 - 1) q^{97} + \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 4 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 4 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{7} + 4 q^{9} - 10 q^{10} + 4 q^{12} - 10 q^{13} - 2 q^{14} - 2 q^{15} - 6 q^{17} - 2 q^{18} - 18 q^{19} + 4 q^{21} - 2 q^{23} - 8 q^{25} + 4 q^{27} + 4 q^{28} - 6 q^{29} - 10 q^{30} - 12 q^{32} - 2 q^{34} - 2 q^{35} + 4 q^{36} - 4 q^{37} - 10 q^{39} - 8 q^{40} - 2 q^{42} - 20 q^{43} - 2 q^{45} - 16 q^{46} - 6 q^{47} + 4 q^{49} + 24 q^{50} - 6 q^{51} - 8 q^{52} - 2 q^{54} - 18 q^{57} + 24 q^{58} - 6 q^{59} + 10 q^{61} + 4 q^{63} - 16 q^{64} + 10 q^{65} + 4 q^{67} - 28 q^{68} - 2 q^{69} - 10 q^{70} - 6 q^{71} - 34 q^{73} + 36 q^{74} - 8 q^{75} - 36 q^{76} - 24 q^{79} + 12 q^{80} + 4 q^{81} + 28 q^{82} - 6 q^{83} + 4 q^{84} + 8 q^{85} + 38 q^{86} - 6 q^{87} + 18 q^{89} - 10 q^{90} - 10 q^{91} + 24 q^{92} + 6 q^{94} + 18 q^{95} - 12 q^{96} - 10 q^{97} - 2 q^{98}+O(q^{100}) \) 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
0.698857
3.05896
−1.43091
−0.326909
−2.43091 1.00000 3.90931 1.43091 −2.43091 1.00000 −4.64136 1.00000 −3.47841
1.2 −1.32691 1.00000 −0.239314 0.326909 −1.32691 1.00000 2.97136 1.00000 −0.433778
1.3 −0.301143 1.00000 −1.90931 −0.698857 −0.301143 1.00000 1.17726 1.00000 0.210456
1.4 2.05896 1.00000 2.23931 −3.05896 2.05896 1.00000 0.492737 1.00000 −6.29827
\(n\): 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.bl 4
3.b odd 2 1 7623.2.a.cn 4
11.b odd 2 1 2541.2.a.bp yes 4
33.d even 2 1 7623.2.a.cg 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.bl 4 1.a even 1 1 trivial
2541.2.a.bp yes 4 11.b odd 2 1
7623.2.a.cg 4 33.d even 2 1
7623.2.a.cn 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2541))\):

\( T_{2}^{4} + 2T_{2}^{3} - 4T_{2}^{2} - 8T_{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{4} + 2T_{5}^{3} - 4T_{5}^{2} - 2T_{5} + 1 \) Copy content Toggle raw display
\( T_{13}^{4} + 10T_{13}^{3} + 32T_{13}^{2} + 32T_{13} - 2 \) 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
$5$ \( T^{4} + 2 T^{3} - 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 10 T^{3} + 32 T^{2} + 32 T - 2 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} - 40 T^{2} - 282 T - 263 \) Copy content Toggle raw display
$19$ \( T^{4} + 18 T^{3} + 108 T^{2} + \cdots + 198 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} - 64 T^{2} + 52 T + 286 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} - 96 T^{2} + \cdots + 1926 \) Copy content Toggle raw display
$31$ \( T^{4} - 96 T^{2} - 216 T + 792 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} - 76 T^{2} + \cdots + 1492 \) Copy content Toggle raw display
$41$ \( T^{4} - 40 T^{2} - 48 T + 4 \) Copy content Toggle raw display
$43$ \( T^{4} + 20 T^{3} + 110 T^{2} + \cdots - 611 \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} - 144 T^{2} + \cdots + 4653 \) Copy content Toggle raw display
$53$ \( T^{4} - 64 T^{2} - 192 T - 128 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} - 28 T^{2} - 126 T + 169 \) Copy content Toggle raw display
$61$ \( T^{4} - 10 T^{3} - 136 T^{2} + \cdots - 2138 \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} - 138 T^{2} + \cdots - 143 \) Copy content Toggle raw display
$71$ \( T^{4} + 6 T^{3} - 64 T^{2} - 264 T - 242 \) Copy content Toggle raw display
$73$ \( T^{4} + 34 T^{3} + 416 T^{2} + \cdots + 4054 \) Copy content Toggle raw display
$79$ \( T^{4} + 24 T^{3} - 36 T^{2} + \cdots - 23004 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} - 72 T^{2} - 306 T + 657 \) Copy content Toggle raw display
$89$ \( T^{4} - 18 T^{3} - 4 T^{2} + \cdots + 1441 \) Copy content Toggle raw display
$97$ \( T^{4} + 10 T^{3} - 152 T - 26 \) Copy content Toggle raw display
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