Properties

Label 2541.2.a.v
Level 2541
Weight 2
Character orbit 2541.a
Self dual yes
Analytic conductor 20.290
Analytic rank 0
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \beta ) q^{2} - q^{3} + 3 q^{4} + ( 1 - \beta ) q^{5} + ( -1 + 2 \beta ) q^{6} - q^{7} + ( 1 - 2 \beta ) q^{8} + q^{9} +O(q^{10})\) \( q + ( 1 - 2 \beta ) q^{2} - q^{3} + 3 q^{4} + ( 1 - \beta ) q^{5} + ( -1 + 2 \beta ) q^{6} - q^{7} + ( 1 - 2 \beta ) q^{8} + q^{9} + ( 3 - \beta ) q^{10} -3 q^{12} -2 \beta q^{13} + ( -1 + 2 \beta ) q^{14} + ( -1 + \beta ) q^{15} - q^{16} -3 \beta q^{17} + ( 1 - 2 \beta ) q^{18} + ( 2 - 3 \beta ) q^{19} + ( 3 - 3 \beta ) q^{20} + q^{21} + ( 6 - \beta ) q^{23} + ( -1 + 2 \beta ) q^{24} + ( -3 - \beta ) q^{25} + ( 4 + 2 \beta ) q^{26} - q^{27} -3 q^{28} + 6 q^{29} + ( -3 + \beta ) q^{30} + ( 5 - 5 \beta ) q^{31} + ( -3 + 6 \beta ) q^{32} + ( 6 + 3 \beta ) q^{34} + ( -1 + \beta ) q^{35} + 3 q^{36} + ( 3 + \beta ) q^{37} + ( 8 - \beta ) q^{38} + 2 \beta q^{39} + ( 3 - \beta ) q^{40} + ( -9 + \beta ) q^{41} + ( 1 - 2 \beta ) q^{42} -6 \beta q^{43} + ( 1 - \beta ) q^{45} + ( 8 - 11 \beta ) q^{46} + ( -2 + 4 \beta ) q^{47} + q^{48} + q^{49} + ( -1 + 7 \beta ) q^{50} + 3 \beta q^{51} -6 \beta q^{52} + ( 10 - 2 \beta ) q^{53} + ( -1 + 2 \beta ) q^{54} + ( -1 + 2 \beta ) q^{56} + ( -2 + 3 \beta ) q^{57} + ( 6 - 12 \beta ) q^{58} + 2 \beta q^{59} + ( -3 + 3 \beta ) q^{60} + ( 15 - 5 \beta ) q^{62} - q^{63} -13 q^{64} + 2 q^{65} -9 \beta q^{68} + ( -6 + \beta ) q^{69} + ( -3 + \beta ) q^{70} + ( 8 - 8 \beta ) q^{71} + ( 1 - 2 \beta ) q^{72} + ( -4 - 6 \beta ) q^{73} + ( 1 - 7 \beta ) q^{74} + ( 3 + \beta ) q^{75} + ( 6 - 9 \beta ) q^{76} + ( -4 - 2 \beta ) q^{78} + 10 q^{79} + ( -1 + \beta ) q^{80} + q^{81} + ( -11 + 17 \beta ) q^{82} + ( -10 - 4 \beta ) q^{83} + 3 q^{84} + 3 q^{85} + ( 12 + 6 \beta ) q^{86} -6 q^{87} + ( 4 + \beta ) q^{89} + ( 3 - \beta ) q^{90} + 2 \beta q^{91} + ( 18 - 3 \beta ) q^{92} + ( -5 + 5 \beta ) q^{93} -10 q^{94} + ( 5 - 2 \beta ) q^{95} + ( 3 - 6 \beta ) q^{96} -6 q^{97} + ( 1 - 2 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 6q^{4} + q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 6q^{4} + q^{5} - 2q^{7} + 2q^{9} + 5q^{10} - 6q^{12} - 2q^{13} - q^{15} - 2q^{16} - 3q^{17} + q^{19} + 3q^{20} + 2q^{21} + 11q^{23} - 7q^{25} + 10q^{26} - 2q^{27} - 6q^{28} + 12q^{29} - 5q^{30} + 5q^{31} + 15q^{34} - q^{35} + 6q^{36} + 7q^{37} + 15q^{38} + 2q^{39} + 5q^{40} - 17q^{41} - 6q^{43} + q^{45} + 5q^{46} + 2q^{48} + 2q^{49} + 5q^{50} + 3q^{51} - 6q^{52} + 18q^{53} - q^{57} + 2q^{59} - 3q^{60} + 25q^{62} - 2q^{63} - 26q^{64} + 4q^{65} - 9q^{68} - 11q^{69} - 5q^{70} + 8q^{71} - 14q^{73} - 5q^{74} + 7q^{75} + 3q^{76} - 10q^{78} + 20q^{79} - q^{80} + 2q^{81} - 5q^{82} - 24q^{83} + 6q^{84} + 6q^{85} + 30q^{86} - 12q^{87} + 9q^{89} + 5q^{90} + 2q^{91} + 33q^{92} - 5q^{93} - 20q^{94} + 8q^{95} - 12q^{97} + O(q^{100}) \)

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
−2.23607 −1.00000 3.00000 −0.618034 2.23607 −1.00000 −2.23607 1.00000 1.38197
1.2 2.23607 −1.00000 3.00000 1.61803 −2.23607 −1.00000 2.23607 1.00000 3.61803
\(n\): 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.v 2
3.b odd 2 1 7623.2.a.bj 2
11.b odd 2 1 2541.2.a.w 2
11.d odd 10 2 231.2.j.e 4
33.d even 2 1 7623.2.a.bk 2
33.f even 10 2 693.2.m.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.e 4 11.d odd 10 2
693.2.m.a 4 33.f even 10 2
2541.2.a.v 2 1.a even 1 1 trivial
2541.2.a.w 2 11.b odd 2 1
7623.2.a.bj 2 3.b odd 2 1
7623.2.a.bk 2 33.d even 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2541))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + 4 T^{4} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 - T + 9 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$ \( ( 1 + T )^{2} \)
$11$ 1
$13$ \( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( 1 + 3 T + 25 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( 1 - T + 27 T^{2} - 19 T^{3} + 361 T^{4} \)
$23$ \( 1 - 11 T + 75 T^{2} - 253 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 5 T + 37 T^{2} - 155 T^{3} + 961 T^{4} \)
$37$ \( 1 - 7 T + 85 T^{2} - 259 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 17 T + 153 T^{2} + 697 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 6 T + 50 T^{2} + 258 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 74 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 18 T + 182 T^{2} - 954 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 2 T + 114 T^{2} - 118 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 + 61 T^{2} )^{2} \)
$67$ \( ( 1 + 67 T^{2} )^{2} \)
$71$ \( 1 - 8 T + 78 T^{2} - 568 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 14 T + 150 T^{2} + 1022 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 10 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 24 T + 290 T^{2} + 1992 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 9 T + 197 T^{2} - 801 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 6 T + 97 T^{2} )^{2} \)
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