Properties

Label 2541.2.a.bf
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
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.

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

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
−0.618034
1.61803
0.381966 −1.00000 −1.85410 −0.618034 −0.381966 1.00000 −1.47214 1.00000 −0.236068
1.2 2.61803 −1.00000 4.85410 1.61803 −2.61803 1.00000 7.47214 1.00000 4.23607
\(n\): 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.bf 2
3.b odd 2 1 7623.2.a.u 2
11.b odd 2 1 2541.2.a.m 2
11.c even 5 2 231.2.j.a 4
33.d even 2 1 7623.2.a.by 2
33.h odd 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.c even 5 2
693.2.m.e 4 33.h odd 10 2
2541.2.a.m 2 11.b odd 2 1
2541.2.a.bf 2 1.a even 1 1 trivial
7623.2.a.u 2 3.b odd 2 1
7623.2.a.by 2 33.d even 2 1

Hecke kernels

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

\( T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} - 1 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 19 \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 5T - 25 \) Copy content Toggle raw display
$43$ \( (T + 9)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 9T - 11 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T - 29 \) Copy content Toggle raw display
$59$ \( T^{2} - 23T + 131 \) Copy content Toggle raw display
$61$ \( T^{2} - 45 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$71$ \( T^{2} - 20T + 95 \) Copy content Toggle raw display
$73$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$79$ \( T^{2} - 7T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T - 44 \) Copy content Toggle raw display
$89$ \( T^{2} - 21T + 49 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 171 \) Copy content Toggle raw display
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