Properties

Label 2541.2.a.y
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 \( q + \beta q^{2} - q^{3} + (\beta - 1) q^{4} + ( - 2 \beta + 1) q^{5} - \beta q^{6} + q^{7} + ( - 2 \beta + 1) q^{8} + q^{9} + ( - \beta - 2) q^{10} + ( - \beta + 1) q^{12} + (2 \beta - 3) q^{13} + \beta q^{14} + (2 \beta - 1) q^{15} - 3 \beta q^{16} + 2 q^{17} + \beta q^{18} + ( - 4 \beta + 5) q^{19} + (\beta - 3) q^{20} - q^{21} - 2 q^{23} + (2 \beta - 1) q^{24} + ( - \beta + 2) q^{26} - q^{27} + (\beta - 1) q^{28} + 5 q^{29} + (\beta + 2) q^{30} + (4 \beta + 4) q^{31} + (\beta - 5) q^{32} + 2 \beta q^{34} + ( - 2 \beta + 1) q^{35} + (\beta - 1) q^{36} + ( - 4 \beta + 5) q^{37} + (\beta - 4) q^{38} + ( - 2 \beta + 3) q^{39} + 5 q^{40} + (4 \beta - 4) q^{41} - \beta q^{42} + ( - 4 \beta - 2) q^{43} + ( - 2 \beta + 1) q^{45} - 2 \beta q^{46} + (4 \beta + 3) q^{47} + 3 \beta q^{48} + q^{49} - 2 q^{51} + ( - 3 \beta + 5) q^{52} + 4 \beta q^{53} - \beta q^{54} + ( - 2 \beta + 1) q^{56} + (4 \beta - 5) q^{57} + 5 \beta q^{58} + (8 \beta - 5) q^{59} + ( - \beta + 3) q^{60} + (4 \beta + 6) q^{61} + (8 \beta + 4) q^{62} + q^{63} + (2 \beta + 1) q^{64} + (4 \beta - 7) q^{65} + (10 \beta - 7) q^{67} + (2 \beta - 2) q^{68} + 2 q^{69} + ( - \beta - 2) q^{70} + ( - 4 \beta + 4) q^{71} + ( - 2 \beta + 1) q^{72} + ( - 6 \beta - 1) q^{73} + (\beta - 4) q^{74} + (5 \beta - 9) q^{76} + (\beta - 2) q^{78} + ( - 4 \beta + 6) q^{79} + (3 \beta + 6) q^{80} + q^{81} + 4 q^{82} + ( - 4 \beta + 14) q^{83} + ( - \beta + 1) q^{84} + ( - 4 \beta + 2) q^{85} + ( - 6 \beta - 4) q^{86} - 5 q^{87} + ( - 4 \beta + 6) q^{89} + ( - \beta - 2) q^{90} + (2 \beta - 3) q^{91} + ( - 2 \beta + 2) q^{92} + ( - 4 \beta - 4) q^{93} + (7 \beta + 4) q^{94} + ( - 6 \beta + 13) q^{95} + ( - \beta + 5) q^{96} + (12 \beta - 10) q^{97} + \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 2 q^{7} + 2 q^{9} - 5 q^{10} + q^{12} - 4 q^{13} + q^{14} - 3 q^{16} + 4 q^{17} + q^{18} + 6 q^{19} - 5 q^{20} - 2 q^{21} - 4 q^{23} + 3 q^{26} - 2 q^{27} - q^{28} + 10 q^{29} + 5 q^{30} + 12 q^{31} - 9 q^{32} + 2 q^{34} - q^{36} + 6 q^{37} - 7 q^{38} + 4 q^{39} + 10 q^{40} - 4 q^{41} - q^{42} - 8 q^{43} - 2 q^{46} + 10 q^{47} + 3 q^{48} + 2 q^{49} - 4 q^{51} + 7 q^{52} + 4 q^{53} - q^{54} - 6 q^{57} + 5 q^{58} - 2 q^{59} + 5 q^{60} + 16 q^{61} + 16 q^{62} + 2 q^{63} + 4 q^{64} - 10 q^{65} - 4 q^{67} - 2 q^{68} + 4 q^{69} - 5 q^{70} + 4 q^{71} - 8 q^{73} - 7 q^{74} - 13 q^{76} - 3 q^{78} + 8 q^{79} + 15 q^{80} + 2 q^{81} + 8 q^{82} + 24 q^{83} + q^{84} - 14 q^{86} - 10 q^{87} + 8 q^{89} - 5 q^{90} - 4 q^{91} + 2 q^{92} - 12 q^{93} + 15 q^{94} + 20 q^{95} + 9 q^{96} - 8 q^{97} + q^{98}+O(q^{100}) \) 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 2.23607 0.618034 1.00000 2.23607 1.00000 −1.38197
1.2 1.61803 −1.00000 0.618034 −2.23607 −1.61803 1.00000 −2.23607 1.00000 −3.61803
\(n\): 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.y yes 2
3.b odd 2 1 7623.2.a.y 2
11.b odd 2 1 2541.2.a.q 2
33.d even 2 1 7623.2.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.q 2 11.b odd 2 1
2541.2.a.y yes 2 1.a even 1 1 trivial
7623.2.a.y 2 3.b odd 2 1
7623.2.a.bn 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
\( T_{5}^{2} - 5 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 1 \) 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
$5$ \( T^{2} - 5 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$23$ \( (T + 2)^{2} \) Copy content Toggle raw display
$29$ \( (T - 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 10T + 5 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$59$ \( T^{2} + 2T - 79 \) Copy content Toggle raw display
$61$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 121 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 29 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$83$ \( T^{2} - 24T + 124 \) Copy content Toggle raw display
$89$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 164 \) Copy content Toggle raw display
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