Properties

Label 2541.2.a.o
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: \(\Q(\sqrt{3}) \)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−2.73205 −1.00000 5.46410 −3.73205 2.73205 1.00000 −9.46410 1.00000 10.1962
1.2 0.732051 −1.00000 −1.46410 −0.267949 −0.732051 1.00000 −2.53590 1.00000 −0.196152
\(n\): 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.o 2
3.b odd 2 1 7623.2.a.bv 2
11.b odd 2 1 2541.2.a.be yes 2
33.d even 2 1 7623.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.o 2 1.a even 1 1 trivial
2541.2.a.be yes 2 11.b odd 2 1
7623.2.a.x 2 33.d even 2 1
7623.2.a.bv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2541))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( \)
$13$ \( 1 + 2 T + 24 T^{2} + 26 T^{3} + 169 T^{4} \)
$17$ \( 1 - 4 T + 35 T^{2} - 68 T^{3} + 289 T^{4} \)
$19$ \( 1 + 6 T + 20 T^{2} + 114 T^{3} + 361 T^{4} \)
$23$ \( 1 - 10 T + 68 T^{2} - 230 T^{3} + 529 T^{4} \)
$29$ \( 1 + 6 T + 64 T^{2} + 174 T^{3} + 841 T^{4} \)
$31$ \( 1 - 8 T + 66 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( 1 + 12 T + 98 T^{2} + 444 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 4 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 6 T + 83 T^{2} - 258 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 8 T + 83 T^{2} + 376 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 12 T + 130 T^{2} - 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 4 T - 25 T^{2} - 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 6 T + 104 T^{2} + 366 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 + 7 T + 67 T^{2} )^{2} \)
$71$ \( 1 + 6 T + 4 T^{2} + 426 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 10 T + 168 T^{2} - 730 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 8 T + 162 T^{2} - 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 12 T + 199 T^{2} + 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 12 T + 139 T^{2} + 1068 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 10 T + 216 T^{2} - 970 T^{3} + 9409 T^{4} \)
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