Properties

Label 2541.2.a.s
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: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
−1.61803 1.00000 0.618034 −2.61803 −1.61803 −1.00000 2.23607 1.00000 4.23607
1.2 0.618034 1.00000 −1.61803 −0.381966 0.618034 −1.00000 −2.23607 1.00000 −0.236068
\(n\): 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.s 2
3.b odd 2 1 7623.2.a.bp 2
11.b odd 2 1 2541.2.a.bb 2
11.d odd 10 2 231.2.j.d 4
33.d even 2 1 7623.2.a.ba 2
33.f even 10 2 693.2.m.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.d 4 11.d odd 10 2
693.2.m.b 4 33.f even 10 2
2541.2.a.s 2 1.a even 1 1 trivial
2541.2.a.bb 2 11.b odd 2 1
7623.2.a.ba 2 33.d even 2 1
7623.2.a.bp 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2541))\):

\( T_{2}^{2} + T_{2} - 1 \)
\( T_{5}^{2} + 3 T_{5} + 1 \)
\( T_{13} - 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4} \)
$3$ \( ( 1 - T )^{2} \)
$5$ \( 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( \)
$13$ \( ( 1 - T + 13 T^{2} )^{2} \)
$17$ \( 1 - 4 T + 33 T^{2} - 68 T^{3} + 289 T^{4} \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( 1 + 2 T + 42 T^{2} + 46 T^{3} + 529 T^{4} \)
$29$ \( 1 + 13 T^{2} + 841 T^{4} \)
$31$ \( 1 + 16 T + 121 T^{2} + 496 T^{3} + 961 T^{4} \)
$37$ \( 1 + 4 T - 2 T^{2} + 148 T^{3} + 1369 T^{4} \)
$41$ \( 1 - T + 51 T^{2} - 41 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 - T + 43 T^{2} )^{2} \)
$47$ \( 1 - T + 33 T^{2} - 47 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 13 T + 147 T^{2} - 689 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 15 T + 163 T^{2} + 885 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 16 T + 181 T^{2} - 976 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 14 T + 178 T^{2} + 938 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 6 T + 131 T^{2} + 426 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 3 T + 47 T^{2} + 219 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 15 T + 203 T^{2} + 1185 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 15 T + 233 T^{2} + 1335 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 17 T + 97 T^{2} )^{2} \)
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