Properties

Label 2541.2.a.n
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: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} - q^{4} + ( - 3 \beta + 2) q^{5} + q^{6} + q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} - q^{4} + ( - 3 \beta + 2) q^{5} + q^{6} + q^{7} + 3 q^{8} + q^{9} + (3 \beta - 2) q^{10} + q^{12} + ( - 2 \beta + 2) q^{13} - q^{14} + (3 \beta - 2) q^{15} - q^{16} + (3 \beta + 3) q^{17} - q^{18} + (\beta + 1) q^{19} + (3 \beta - 2) q^{20} - q^{21} + ( - 5 \beta + 5) q^{23} - 3 q^{24} + ( - 3 \beta + 8) q^{25} + (2 \beta - 2) q^{26} - q^{27} - q^{28} - 2 q^{29} + ( - 3 \beta + 2) q^{30} + ( - 7 \beta + 4) q^{31} - 5 q^{32} + ( - 3 \beta - 3) q^{34} + ( - 3 \beta + 2) q^{35} - q^{36} + ( - 5 \beta - 4) q^{37} + ( - \beta - 1) q^{38} + (2 \beta - 2) q^{39} + ( - 9 \beta + 6) q^{40} + ( - 3 \beta - 6) q^{41} + q^{42} + ( - 2 \beta + 2) q^{43} + ( - 3 \beta + 2) q^{45} + (5 \beta - 5) q^{46} + 2 q^{47} + q^{48} + q^{49} + (3 \beta - 8) q^{50} + ( - 3 \beta - 3) q^{51} + (2 \beta - 2) q^{52} + (10 \beta - 4) q^{53} + q^{54} + 3 q^{56} + ( - \beta - 1) q^{57} + 2 q^{58} + ( - 2 \beta + 10) q^{59} + ( - 3 \beta + 2) q^{60} + (8 \beta - 4) q^{61} + (7 \beta - 4) q^{62} + q^{63} + 7 q^{64} + ( - 4 \beta + 10) q^{65} + 8 q^{67} + ( - 3 \beta - 3) q^{68} + (5 \beta - 5) q^{69} + (3 \beta - 2) q^{70} + (8 \beta - 4) q^{71} + 3 q^{72} + ( - 2 \beta - 2) q^{73} + (5 \beta + 4) q^{74} + (3 \beta - 8) q^{75} + ( - \beta - 1) q^{76} + ( - 2 \beta + 2) q^{78} - 14 q^{79} + (3 \beta - 2) q^{80} + q^{81} + (3 \beta + 6) q^{82} + (8 \beta - 10) q^{83} + q^{84} + ( - 12 \beta - 3) q^{85} + (2 \beta - 2) q^{86} + 2 q^{87} + (5 \beta - 7) q^{89} + (3 \beta - 2) q^{90} + ( - 2 \beta + 2) q^{91} + (5 \beta - 5) q^{92} + (7 \beta - 4) q^{93} - 2 q^{94} + ( - 4 \beta - 1) q^{95} + 5 q^{96} + (4 \beta - 10) q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{7} + 6 q^{8} + 2 q^{9} - q^{10} + 2 q^{12} + 2 q^{13} - 2 q^{14} - q^{15} - 2 q^{16} + 9 q^{17} - 2 q^{18} + 3 q^{19} - q^{20} - 2 q^{21} + 5 q^{23} - 6 q^{24} + 13 q^{25} - 2 q^{26} - 2 q^{27} - 2 q^{28} - 4 q^{29} + q^{30} + q^{31} - 10 q^{32} - 9 q^{34} + q^{35} - 2 q^{36} - 13 q^{37} - 3 q^{38} - 2 q^{39} + 3 q^{40} - 15 q^{41} + 2 q^{42} + 2 q^{43} + q^{45} - 5 q^{46} + 4 q^{47} + 2 q^{48} + 2 q^{49} - 13 q^{50} - 9 q^{51} - 2 q^{52} + 2 q^{53} + 2 q^{54} + 6 q^{56} - 3 q^{57} + 4 q^{58} + 18 q^{59} + q^{60} - q^{62} + 2 q^{63} + 14 q^{64} + 16 q^{65} + 16 q^{67} - 9 q^{68} - 5 q^{69} - q^{70} + 6 q^{72} - 6 q^{73} + 13 q^{74} - 13 q^{75} - 3 q^{76} + 2 q^{78} - 28 q^{79} - q^{80} + 2 q^{81} + 15 q^{82} - 12 q^{83} + 2 q^{84} - 18 q^{85} - 2 q^{86} + 4 q^{87} - 9 q^{89} - q^{90} + 2 q^{91} - 5 q^{92} - q^{93} - 4 q^{94} - 6 q^{95} + 10 q^{96} - 16 q^{97} - 2 q^{98}+O(q^{100}) \) 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
1.61803
−0.618034
−1.00000 −1.00000 −1.00000 −2.85410 1.00000 1.00000 3.00000 1.00000 2.85410
1.2 −1.00000 −1.00000 −1.00000 3.85410 1.00000 1.00000 3.00000 1.00000 −3.85410
\(n\): 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.n 2
3.b odd 2 1 7623.2.a.bu 2
11.b odd 2 1 2541.2.a.bd 2
11.c even 5 2 231.2.j.c 4
33.d even 2 1 7623.2.a.w 2
33.h odd 10 2 693.2.m.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.c 4 11.c even 5 2
693.2.m.c 4 33.h odd 10 2
2541.2.a.n 2 1.a even 1 1 trivial
2541.2.a.bd 2 11.b odd 2 1
7623.2.a.w 2 33.d even 2 1
7623.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2541))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 9T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} - 5T - 25 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - T - 61 \) Copy content Toggle raw display
$37$ \( T^{2} + 13T + 11 \) Copy content Toggle raw display
$41$ \( T^{2} + 15T + 45 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$47$ \( (T - 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 2T - 124 \) Copy content Toggle raw display
$59$ \( T^{2} - 18T + 76 \) Copy content Toggle raw display
$61$ \( T^{2} - 80 \) Copy content Toggle raw display
$67$ \( (T - 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 80 \) Copy content Toggle raw display
$73$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$79$ \( (T + 14)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 12T - 44 \) Copy content Toggle raw display
$89$ \( T^{2} + 9T - 11 \) Copy content Toggle raw display
$97$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
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