Properties

Label 539.2.a.a
Level $539$
Weight $2$
Character orbit 539.a
Self dual yes
Analytic conductor $4.304$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.30393666895\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{2} + q^{3} + 2 q^{4} - q^{5} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + q^{3} + 2 q^{4} - q^{5} - 2 q^{6} - 2 q^{9} + 2 q^{10} + q^{11} + 2 q^{12} - 4 q^{13} - q^{15} - 4 q^{16} + 2 q^{17} + 4 q^{18} - 2 q^{20} - 2 q^{22} - q^{23} - 4 q^{25} + 8 q^{26} - 5 q^{27} + 2 q^{30} - 7 q^{31} + 8 q^{32} + q^{33} - 4 q^{34} - 4 q^{36} + 3 q^{37} - 4 q^{39} + 8 q^{41} - 6 q^{43} + 2 q^{44} + 2 q^{45} + 2 q^{46} - 8 q^{47} - 4 q^{48} + 8 q^{50} + 2 q^{51} - 8 q^{52} - 6 q^{53} + 10 q^{54} - q^{55} - 5 q^{59} - 2 q^{60} - 12 q^{61} + 14 q^{62} - 8 q^{64} + 4 q^{65} - 2 q^{66} - 7 q^{67} + 4 q^{68} - q^{69} - 3 q^{71} - 4 q^{73} - 6 q^{74} - 4 q^{75} + 8 q^{78} - 10 q^{79} + 4 q^{80} + q^{81} - 16 q^{82} + 6 q^{83} - 2 q^{85} + 12 q^{86} - 15 q^{89} - 4 q^{90} - 2 q^{92} - 7 q^{93} + 16 q^{94} + 8 q^{96} + 7 q^{97} - 2 q^{99}+O(q^{100}) \) 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
−2.00000 1.00000 2.00000 −1.00000 −2.00000 0 0 −2.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 539.2.a.a 1
3.b odd 2 1 4851.2.a.t 1
4.b odd 2 1 8624.2.a.j 1
7.b odd 2 1 11.2.a.a 1
7.c even 3 2 539.2.e.g 2
7.d odd 6 2 539.2.e.h 2
11.b odd 2 1 5929.2.a.h 1
21.c even 2 1 99.2.a.d 1
28.d even 2 1 176.2.a.b 1
35.c odd 2 1 275.2.a.b 1
35.f even 4 2 275.2.b.a 2
56.e even 2 1 704.2.a.c 1
56.h odd 2 1 704.2.a.h 1
63.l odd 6 2 891.2.e.k 2
63.o even 6 2 891.2.e.b 2
77.b even 2 1 121.2.a.d 1
77.j odd 10 4 121.2.c.e 4
77.l even 10 4 121.2.c.a 4
84.h odd 2 1 1584.2.a.g 1
91.b odd 2 1 1859.2.a.b 1
105.g even 2 1 2475.2.a.a 1
105.k odd 4 2 2475.2.c.a 2
112.j even 4 2 2816.2.c.f 2
112.l odd 4 2 2816.2.c.j 2
119.d odd 2 1 3179.2.a.a 1
133.c even 2 1 3971.2.a.b 1
140.c even 2 1 4400.2.a.i 1
140.j odd 4 2 4400.2.b.h 2
161.c even 2 1 5819.2.a.a 1
168.e odd 2 1 6336.2.a.bu 1
168.i even 2 1 6336.2.a.br 1
203.c odd 2 1 9251.2.a.d 1
231.h odd 2 1 1089.2.a.b 1
308.g odd 2 1 1936.2.a.i 1
385.h even 2 1 3025.2.a.a 1
616.g odd 2 1 7744.2.a.k 1
616.o even 2 1 7744.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 7.b odd 2 1
99.2.a.d 1 21.c even 2 1
121.2.a.d 1 77.b even 2 1
121.2.c.a 4 77.l even 10 4
121.2.c.e 4 77.j odd 10 4
176.2.a.b 1 28.d even 2 1
275.2.a.b 1 35.c odd 2 1
275.2.b.a 2 35.f even 4 2
539.2.a.a 1 1.a even 1 1 trivial
539.2.e.g 2 7.c even 3 2
539.2.e.h 2 7.d odd 6 2
704.2.a.c 1 56.e even 2 1
704.2.a.h 1 56.h odd 2 1
891.2.e.b 2 63.o even 6 2
891.2.e.k 2 63.l odd 6 2
1089.2.a.b 1 231.h odd 2 1
1584.2.a.g 1 84.h odd 2 1
1859.2.a.b 1 91.b odd 2 1
1936.2.a.i 1 308.g odd 2 1
2475.2.a.a 1 105.g even 2 1
2475.2.c.a 2 105.k odd 4 2
2816.2.c.f 2 112.j even 4 2
2816.2.c.j 2 112.l odd 4 2
3025.2.a.a 1 385.h even 2 1
3179.2.a.a 1 119.d odd 2 1
3971.2.a.b 1 133.c even 2 1
4400.2.a.i 1 140.c even 2 1
4400.2.b.h 2 140.j odd 4 2
4851.2.a.t 1 3.b odd 2 1
5819.2.a.a 1 161.c even 2 1
5929.2.a.h 1 11.b odd 2 1
6336.2.a.br 1 168.i even 2 1
6336.2.a.bu 1 168.e odd 2 1
7744.2.a.k 1 616.g odd 2 1
7744.2.a.x 1 616.o even 2 1
8624.2.a.j 1 4.b odd 2 1
9251.2.a.d 1 203.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(539))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 7 \) Copy content Toggle raw display
$37$ \( T - 3 \) Copy content Toggle raw display
$41$ \( T - 8 \) Copy content Toggle raw display
$43$ \( T + 6 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 5 \) Copy content Toggle raw display
$61$ \( T + 12 \) Copy content Toggle raw display
$67$ \( T + 7 \) Copy content Toggle raw display
$71$ \( T + 3 \) Copy content Toggle raw display
$73$ \( T + 4 \) Copy content Toggle raw display
$79$ \( T + 10 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T + 15 \) Copy content Toggle raw display
$97$ \( T - 7 \) Copy content Toggle raw display
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