Properties

Label 2541.2.a.m
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

Related objects

Downloads

Learn more about

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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{2} - q^{3} + 3 \beta q^{4} + \beta q^{5} + ( 1 + \beta ) q^{6} - q^{7} + ( -1 - 4 \beta ) q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 - \beta ) q^{2} - q^{3} + 3 \beta q^{4} + \beta q^{5} + ( 1 + \beta ) q^{6} - q^{7} + ( -1 - 4 \beta ) q^{8} + q^{9} + ( -1 - 2 \beta ) q^{10} -3 \beta q^{12} + ( 1 - 4 \beta ) q^{13} + ( 1 + \beta ) q^{14} -\beta q^{15} + ( 5 + 3 \beta ) q^{16} + 3 q^{17} + ( -1 - \beta ) q^{18} -4 q^{19} + ( 3 + 3 \beta ) q^{20} + q^{21} + ( -4 - 2 \beta ) q^{23} + ( 1 + 4 \beta ) q^{24} + ( -4 + \beta ) q^{25} + ( 3 + 7 \beta ) q^{26} - q^{27} -3 \beta q^{28} -3 q^{29} + ( 1 + 2 \beta ) q^{30} + ( -3 + 2 \beta ) q^{31} + ( -6 - 3 \beta ) q^{32} + ( -3 - 3 \beta ) q^{34} -\beta q^{35} + 3 \beta q^{36} + ( -2 - 4 \beta ) q^{37} + ( 4 + 4 \beta ) q^{38} + ( -1 + 4 \beta ) q^{39} + ( -4 - 5 \beta ) q^{40} + ( -5 + 5 \beta ) q^{41} + ( -1 - \beta ) q^{42} + 9 q^{43} + \beta q^{45} + ( 6 + 8 \beta ) q^{46} + ( 2 + 5 \beta ) q^{47} + ( -5 - 3 \beta ) q^{48} + q^{49} + ( 3 + 2 \beta ) q^{50} -3 q^{51} + ( -12 - 9 \beta ) q^{52} + ( -4 + 5 \beta ) q^{53} + ( 1 + \beta ) q^{54} + ( 1 + 4 \beta ) q^{56} + 4 q^{57} + ( 3 + 3 \beta ) q^{58} + ( 11 + \beta ) q^{59} + ( -3 - 3 \beta ) q^{60} + ( -3 + 6 \beta ) q^{61} + ( 1 - \beta ) q^{62} - q^{63} + ( -1 + 6 \beta ) q^{64} + ( -4 - 3 \beta ) q^{65} + ( 6 - 6 \beta ) q^{67} + 9 \beta q^{68} + ( 4 + 2 \beta ) q^{69} + ( 1 + 2 \beta ) q^{70} + ( 9 + 2 \beta ) q^{71} + ( -1 - 4 \beta ) q^{72} + ( -1 + 3 \beta ) q^{73} + ( 6 + 10 \beta ) q^{74} + ( 4 - \beta ) q^{75} -12 \beta q^{76} + ( -3 - 7 \beta ) q^{78} + ( -2 - 3 \beta ) q^{79} + ( 3 + 8 \beta ) q^{80} + q^{81} -5 \beta q^{82} + ( 10 - 8 \beta ) q^{83} + 3 \beta q^{84} + 3 \beta q^{85} + ( -9 - 9 \beta ) q^{86} + 3 q^{87} + ( 14 - 7 \beta ) q^{89} + ( -1 - 2 \beta ) q^{90} + ( -1 + 4 \beta ) q^{91} + ( -6 - 18 \beta ) q^{92} + ( 3 - 2 \beta ) q^{93} + ( -7 - 12 \beta ) q^{94} -4 \beta q^{95} + ( 6 + 3 \beta ) q^{96} + ( 3 - 12 \beta ) q^{97} + ( -1 - \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} - 2q^{3} + 3q^{4} + q^{5} + 3q^{6} - 2q^{7} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 3q^{2} - 2q^{3} + 3q^{4} + q^{5} + 3q^{6} - 2q^{7} - 6q^{8} + 2q^{9} - 4q^{10} - 3q^{12} - 2q^{13} + 3q^{14} - q^{15} + 13q^{16} + 6q^{17} - 3q^{18} - 8q^{19} + 9q^{20} + 2q^{21} - 10q^{23} + 6q^{24} - 7q^{25} + 13q^{26} - 2q^{27} - 3q^{28} - 6q^{29} + 4q^{30} - 4q^{31} - 15q^{32} - 9q^{34} - q^{35} + 3q^{36} - 8q^{37} + 12q^{38} + 2q^{39} - 13q^{40} - 5q^{41} - 3q^{42} + 18q^{43} + q^{45} + 20q^{46} + 9q^{47} - 13q^{48} + 2q^{49} + 8q^{50} - 6q^{51} - 33q^{52} - 3q^{53} + 3q^{54} + 6q^{56} + 8q^{57} + 9q^{58} + 23q^{59} - 9q^{60} + q^{62} - 2q^{63} + 4q^{64} - 11q^{65} + 6q^{67} + 9q^{68} + 10q^{69} + 4q^{70} + 20q^{71} - 6q^{72} + q^{73} + 22q^{74} + 7q^{75} - 12q^{76} - 13q^{78} - 7q^{79} + 14q^{80} + 2q^{81} - 5q^{82} + 12q^{83} + 3q^{84} + 3q^{85} - 27q^{86} + 6q^{87} + 21q^{89} - 4q^{90} + 2q^{91} - 30q^{92} + 4q^{93} - 26q^{94} - 4q^{95} + 15q^{96} - 6q^{97} - 3q^{98} + O(q^{100}) \)

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
−2.61803 −1.00000 4.85410 1.61803 2.61803 −1.00000 −7.47214 1.00000 −4.23607
1.2 −0.381966 −1.00000 −1.85410 −0.618034 0.381966 −1.00000 1.47214 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.m 2
3.b odd 2 1 7623.2.a.by 2
11.b odd 2 1 2541.2.a.bf 2
11.d odd 10 2 231.2.j.a 4
33.d even 2 1 7623.2.a.u 2
33.f even 10 2 693.2.m.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.a 4 11.d odd 10 2
693.2.m.e 4 33.f even 10 2
2541.2.a.m 2 1.a even 1 1 trivial
2541.2.a.bf 2 11.b odd 2 1
7623.2.a.u 2 33.d even 2 1
7623.2.a.by 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2541))\):

\( T_{2}^{2} + 3 T_{2} + 1 \)
\( T_{5}^{2} - T_{5} - 1 \)
\( T_{13}^{2} + 2 T_{13} - 19 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 - T + 9 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( \)
$13$ \( 1 + 2 T + 7 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 3 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 10 T + 66 T^{2} + 230 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 3 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 4 T + 61 T^{2} + 124 T^{3} + 961 T^{4} \)
$37$ \( 1 + 8 T + 70 T^{2} + 296 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 5 T + 57 T^{2} + 205 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 - 9 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 9 T + 83 T^{2} - 423 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 3 T + 77 T^{2} + 159 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 23 T + 249 T^{2} - 1357 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 77 T^{2} + 3721 T^{4} \)
$67$ \( 1 - 6 T + 98 T^{2} - 402 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 20 T + 237 T^{2} - 1420 T^{3} + 5041 T^{4} \)
$73$ \( 1 - T + 135 T^{2} - 73 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 7 T + 159 T^{2} + 553 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 12 T + 122 T^{2} - 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 21 T + 227 T^{2} - 1869 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 6 T + 23 T^{2} + 582 T^{3} + 9409 T^{4} \)
show more
show less