Properties

Label 2541.2.a.ba
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: yes
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{3} + (\beta - 1) q^{4} + ( - 2 \beta - 1) q^{5} + \beta q^{6} - q^{7} + ( - 2 \beta + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + q^{3} + (\beta - 1) q^{4} + ( - 2 \beta - 1) q^{5} + \beta q^{6} - q^{7} + ( - 2 \beta + 1) q^{8} + q^{9} + ( - 3 \beta - 2) q^{10} + (\beta - 1) q^{12} + (2 \beta + 3) q^{13} - \beta q^{14} + ( - 2 \beta - 1) q^{15} - 3 \beta q^{16} + ( - 4 \beta + 2) q^{17} + \beta q^{18} + 3 q^{19} + ( - \beta - 1) q^{20} - q^{21} + (4 \beta + 2) q^{23} + ( - 2 \beta + 1) q^{24} + 8 \beta q^{25} + (5 \beta + 2) q^{26} + q^{27} + ( - \beta + 1) q^{28} - 3 q^{29} + ( - 3 \beta - 2) q^{30} + (\beta - 5) q^{32} + ( - 2 \beta - 4) q^{34} + (2 \beta + 1) q^{35} + (\beta - 1) q^{36} + (4 \beta - 3) q^{37} + 3 \beta q^{38} + (2 \beta + 3) q^{39} + (4 \beta + 3) q^{40} + ( - 4 \beta + 8) q^{41} - \beta q^{42} + (8 \beta - 2) q^{43} + ( - 2 \beta - 1) q^{45} + (6 \beta + 4) q^{46} - 3 q^{47} - 3 \beta q^{48} + q^{49} + (8 \beta + 8) q^{50} + ( - 4 \beta + 2) q^{51} + (3 \beta - 1) q^{52} + (8 \beta - 4) q^{53} + \beta q^{54} + (2 \beta - 1) q^{56} + 3 q^{57} - 3 \beta q^{58} + ( - 4 \beta + 5) q^{59} + ( - \beta - 1) q^{60} + ( - 4 \beta + 10) q^{61} - q^{63} + (2 \beta + 1) q^{64} + ( - 12 \beta - 7) q^{65} + ( - 6 \beta + 1) q^{67} + (2 \beta - 6) q^{68} + (4 \beta + 2) q^{69} + (3 \beta + 2) q^{70} + ( - 4 \beta + 8) q^{71} + ( - 2 \beta + 1) q^{72} + (2 \beta + 9) q^{73} + (\beta + 4) q^{74} + 8 \beta q^{75} + (3 \beta - 3) q^{76} + (5 \beta + 2) q^{78} + ( - 12 \beta + 6) q^{79} + (9 \beta + 6) q^{80} + q^{81} + (4 \beta - 4) q^{82} + 6 q^{83} + ( - \beta + 1) q^{84} + (8 \beta + 6) q^{85} + (6 \beta + 8) q^{86} - 3 q^{87} + ( - 4 \beta + 2) q^{89} + ( - 3 \beta - 2) q^{90} + ( - 2 \beta - 3) q^{91} + (2 \beta + 2) q^{92} - 3 \beta q^{94} + ( - 6 \beta - 3) q^{95} + (\beta - 5) q^{96} + 2 q^{97} + \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} - q^{4} - 4 q^{5} + q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 2 q^{3} - q^{4} - 4 q^{5} + q^{6} - 2 q^{7} + 2 q^{9} - 7 q^{10} - q^{12} + 8 q^{13} - q^{14} - 4 q^{15} - 3 q^{16} + q^{18} + 6 q^{19} - 3 q^{20} - 2 q^{21} + 8 q^{23} + 8 q^{25} + 9 q^{26} + 2 q^{27} + q^{28} - 6 q^{29} - 7 q^{30} - 9 q^{32} - 10 q^{34} + 4 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} + 8 q^{39} + 10 q^{40} + 12 q^{41} - q^{42} + 4 q^{43} - 4 q^{45} + 14 q^{46} - 6 q^{47} - 3 q^{48} + 2 q^{49} + 24 q^{50} + q^{52} + q^{54} + 6 q^{57} - 3 q^{58} + 6 q^{59} - 3 q^{60} + 16 q^{61} - 2 q^{63} + 4 q^{64} - 26 q^{65} - 4 q^{67} - 10 q^{68} + 8 q^{69} + 7 q^{70} + 12 q^{71} + 20 q^{73} + 9 q^{74} + 8 q^{75} - 3 q^{76} + 9 q^{78} + 21 q^{80} + 2 q^{81} - 4 q^{82} + 12 q^{83} + q^{84} + 20 q^{85} + 22 q^{86} - 6 q^{87} - 7 q^{90} - 8 q^{91} + 6 q^{92} - 3 q^{94} - 12 q^{95} - 9 q^{96} + 4 q^{97} + q^{98}+O(q^{100}) \) 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
−0.618034 1.00000 −1.61803 0.236068 −0.618034 −1.00000 2.23607 1.00000 −0.145898
1.2 1.61803 1.00000 0.618034 −4.23607 1.61803 −1.00000 −2.23607 1.00000 −6.85410
\(n\): 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.ba yes 2
3.b odd 2 1 7623.2.a.bb 2
11.b odd 2 1 2541.2.a.r 2
33.d even 2 1 7623.2.a.bq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.r 2 11.b odd 2 1
2541.2.a.ba yes 2 1.a even 1 1 trivial
7623.2.a.bb 2 3.b odd 2 1
7623.2.a.bq 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2541))\):

\( T_{2}^{2} - T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 4T_{5} - 1 \) Copy content Toggle raw display
\( T_{13}^{2} - 8T_{13} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 8T + 11 \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( (T - 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$29$ \( (T + 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$47$ \( (T + 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 80 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$61$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 41 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$73$ \( T^{2} - 20T + 95 \) Copy content Toggle raw display
$79$ \( T^{2} - 180 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 20 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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