Properties

Label 2541.2.a.z
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: \(\Q(\sqrt{21}) \)
Defining polynomial: \(x^{2} - x - 5\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{3} + ( 3 + \beta ) q^{4} + 3 q^{5} -\beta q^{6} - q^{7} + ( 5 + 2 \beta ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta q^{2} - q^{3} + ( 3 + \beta ) q^{4} + 3 q^{5} -\beta q^{6} - q^{7} + ( 5 + 2 \beta ) q^{8} + q^{9} + 3 \beta q^{10} + ( -3 - \beta ) q^{12} - q^{13} -\beta q^{14} -3 q^{15} + ( 4 + 5 \beta ) q^{16} + ( -4 + 2 \beta ) q^{17} + \beta q^{18} + ( 3 - 2 \beta ) q^{19} + ( 9 + 3 \beta ) q^{20} + q^{21} + ( -2 + 2 \beta ) q^{23} + ( -5 - 2 \beta ) q^{24} + 4 q^{25} -\beta q^{26} - q^{27} + ( -3 - \beta ) q^{28} + ( 1 - 4 \beta ) q^{29} -3 \beta q^{30} -2 \beta q^{31} + ( 15 + 5 \beta ) q^{32} + ( 10 - 2 \beta ) q^{34} -3 q^{35} + ( 3 + \beta ) q^{36} + q^{37} + ( -10 + \beta ) q^{38} + q^{39} + ( 15 + 6 \beta ) q^{40} + ( 4 - 4 \beta ) q^{41} + \beta q^{42} + ( 2 + 2 \beta ) q^{43} + 3 q^{45} + 10 q^{46} + ( 5 + 2 \beta ) q^{47} + ( -4 - 5 \beta ) q^{48} + q^{49} + 4 \beta q^{50} + ( 4 - 2 \beta ) q^{51} + ( -3 - \beta ) q^{52} + ( -6 + 2 \beta ) q^{53} -\beta q^{54} + ( -5 - 2 \beta ) q^{56} + ( -3 + 2 \beta ) q^{57} + ( -20 - 3 \beta ) q^{58} + ( 1 - 2 \beta ) q^{59} + ( -9 - 3 \beta ) q^{60} -10 q^{61} + ( -10 - 2 \beta ) q^{62} - q^{63} + ( 17 + 10 \beta ) q^{64} -3 q^{65} + ( 5 - 2 \beta ) q^{67} + ( -2 + 4 \beta ) q^{68} + ( 2 - 2 \beta ) q^{69} -3 \beta q^{70} + ( 4 - 4 \beta ) q^{71} + ( 5 + 2 \beta ) q^{72} -7 q^{73} + \beta q^{74} -4 q^{75} + ( -1 - 5 \beta ) q^{76} + \beta q^{78} + 4 \beta q^{79} + ( 12 + 15 \beta ) q^{80} + q^{81} -20 q^{82} + ( 8 - 2 \beta ) q^{83} + ( 3 + \beta ) q^{84} + ( -12 + 6 \beta ) q^{85} + ( 10 + 4 \beta ) q^{86} + ( -1 + 4 \beta ) q^{87} + ( 2 - 4 \beta ) q^{89} + 3 \beta q^{90} + q^{91} + ( 4 + 6 \beta ) q^{92} + 2 \beta q^{93} + ( 10 + 7 \beta ) q^{94} + ( 9 - 6 \beta ) q^{95} + ( -15 - 5 \beta ) q^{96} + ( -6 - 2 \beta ) q^{97} + \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} + 7 q^{4} + 6 q^{5} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q + q^{2} - 2 q^{3} + 7 q^{4} + 6 q^{5} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} + 3 q^{10} - 7 q^{12} - 2 q^{13} - q^{14} - 6 q^{15} + 13 q^{16} - 6 q^{17} + q^{18} + 4 q^{19} + 21 q^{20} + 2 q^{21} - 2 q^{23} - 12 q^{24} + 8 q^{25} - q^{26} - 2 q^{27} - 7 q^{28} - 2 q^{29} - 3 q^{30} - 2 q^{31} + 35 q^{32} + 18 q^{34} - 6 q^{35} + 7 q^{36} + 2 q^{37} - 19 q^{38} + 2 q^{39} + 36 q^{40} + 4 q^{41} + q^{42} + 6 q^{43} + 6 q^{45} + 20 q^{46} + 12 q^{47} - 13 q^{48} + 2 q^{49} + 4 q^{50} + 6 q^{51} - 7 q^{52} - 10 q^{53} - q^{54} - 12 q^{56} - 4 q^{57} - 43 q^{58} - 21 q^{60} - 20 q^{61} - 22 q^{62} - 2 q^{63} + 44 q^{64} - 6 q^{65} + 8 q^{67} + 2 q^{69} - 3 q^{70} + 4 q^{71} + 12 q^{72} - 14 q^{73} + q^{74} - 8 q^{75} - 7 q^{76} + q^{78} + 4 q^{79} + 39 q^{80} + 2 q^{81} - 40 q^{82} + 14 q^{83} + 7 q^{84} - 18 q^{85} + 24 q^{86} + 2 q^{87} + 3 q^{90} + 2 q^{91} + 14 q^{92} + 2 q^{93} + 27 q^{94} + 12 q^{95} - 35 q^{96} - 14 q^{97} + q^{98} + O(q^{100}) \)

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
−1.79129
2.79129
−1.79129 −1.00000 1.20871 3.00000 1.79129 −1.00000 1.41742 1.00000 −5.37386
1.2 2.79129 −1.00000 5.79129 3.00000 −2.79129 −1.00000 10.5826 1.00000 8.37386
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)

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.z 2
3.b odd 2 1 7623.2.a.bf 2
11.b odd 2 1 231.2.a.b 2
33.d even 2 1 693.2.a.j 2
44.c even 2 1 3696.2.a.bl 2
55.d odd 2 1 5775.2.a.bn 2
77.b even 2 1 1617.2.a.o 2
231.h odd 2 1 4851.2.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.b 2 11.b odd 2 1
693.2.a.j 2 33.d even 2 1
1617.2.a.o 2 77.b even 2 1
2541.2.a.z 2 1.a even 1 1 trivial
3696.2.a.bl 2 44.c even 2 1
4851.2.a.ba 2 231.h odd 2 1
5775.2.a.bn 2 55.d odd 2 1
7623.2.a.bf 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2541))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -5 - T + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -3 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -12 + 6 T + T^{2} \)
$19$ \( -17 - 4 T + T^{2} \)
$23$ \( -20 + 2 T + T^{2} \)
$29$ \( -83 + 2 T + T^{2} \)
$31$ \( -20 + 2 T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( -80 - 4 T + T^{2} \)
$43$ \( -12 - 6 T + T^{2} \)
$47$ \( 15 - 12 T + T^{2} \)
$53$ \( 4 + 10 T + T^{2} \)
$59$ \( -21 + T^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( -5 - 8 T + T^{2} \)
$71$ \( -80 - 4 T + T^{2} \)
$73$ \( ( 7 + T )^{2} \)
$79$ \( -80 - 4 T + T^{2} \)
$83$ \( 28 - 14 T + T^{2} \)
$89$ \( -84 + T^{2} \)
$97$ \( 28 + 14 T + T^{2} \)
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