Properties

Label 2541.2.a.u
Level 2541
Weight 2
Character orbit 2541.a
Self dual yes
Analytic conductor 20.290
Analytic rank 1
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.56155
−1.56155
−2.56155 1.00000 4.56155 1.00000 −2.56155 −1.00000 −6.56155 1.00000 −2.56155
1.2 1.56155 1.00000 0.438447 1.00000 1.56155 −1.00000 −2.43845 1.00000 1.56155
\(n\): 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.u 2
3.b odd 2 1 7623.2.a.bt 2
11.b odd 2 1 2541.2.a.bc yes 2
33.d even 2 1 7623.2.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.u 2 1.a even 1 1 trivial
2541.2.a.bc yes 2 11.b odd 2 1
7623.2.a.be 2 33.d even 2 1
7623.2.a.bt 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2541))\):

\( T_{2}^{2} + T_{2} - 4 \)
\( T_{5} - 1 \)
\( T_{13}^{2} + 7 T_{13} + 8 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 2 T^{3} + 4 T^{4} \)
$3$ \( ( 1 - T )^{2} \)
$5$ \( ( 1 - T + 5 T^{2} )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( \)
$13$ \( 1 + 7 T + 34 T^{2} + 91 T^{3} + 169 T^{4} \)
$17$ \( 1 + 17 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 6 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 + 4 T + 23 T^{2} )^{2} \)
$29$ \( 1 + T + 20 T^{2} + 29 T^{3} + 841 T^{4} \)
$31$ \( 1 - 6 T^{2} + 961 T^{4} \)
$37$ \( 1 - 7 T + 48 T^{2} - 259 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 11 T + 108 T^{2} - 451 T^{3} + 1681 T^{4} \)
$43$ \( 1 + T + 48 T^{2} + 43 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 11 T + 120 T^{2} - 517 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 5 T + 6 T^{2} + 265 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 21 T + 224 T^{2} + 1239 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 2 T - 30 T^{2} - 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 3 T + 30 T^{2} + 201 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 2 T + 126 T^{2} + 142 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 6 T + 138 T^{2} + 438 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 2 T + 142 T^{2} - 158 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 3 T + 62 T^{2} + 249 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 6 T + 119 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 15 T + 212 T^{2} - 1455 T^{3} + 9409 T^{4} \)
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