Properties

Label 539.2.e.b
Level 539
Weight 2
Character orbit 539.e
Analytic conductor 4.304
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( 2 - 2 \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} -2 \zeta_{6} q^{5} -2 q^{6} -3 q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( 2 - 2 \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} -2 \zeta_{6} q^{5} -2 q^{6} -3 q^{8} -\zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{10} + ( -1 + \zeta_{6} ) q^{11} -2 \zeta_{6} q^{12} -4 q^{13} -4 q^{15} + \zeta_{6} q^{16} + ( 4 - 4 \zeta_{6} ) q^{17} + ( -1 + \zeta_{6} ) q^{18} -2 q^{20} + q^{22} + 4 \zeta_{6} q^{23} + ( -6 + 6 \zeta_{6} ) q^{24} + ( 1 - \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{26} + 4 q^{27} -6 q^{29} + 4 \zeta_{6} q^{30} + ( 10 - 10 \zeta_{6} ) q^{31} + ( -5 + 5 \zeta_{6} ) q^{32} + 2 \zeta_{6} q^{33} -4 q^{34} - q^{36} + 6 \zeta_{6} q^{37} + ( -8 + 8 \zeta_{6} ) q^{39} + 6 \zeta_{6} q^{40} -4 q^{41} + 12 q^{43} + \zeta_{6} q^{44} + ( -2 + 2 \zeta_{6} ) q^{45} + ( 4 - 4 \zeta_{6} ) q^{46} -10 \zeta_{6} q^{47} + 2 q^{48} - q^{50} -8 \zeta_{6} q^{51} + ( -4 + 4 \zeta_{6} ) q^{52} + ( 6 - 6 \zeta_{6} ) q^{53} -4 \zeta_{6} q^{54} + 2 q^{55} + 6 \zeta_{6} q^{58} + ( 2 - 2 \zeta_{6} ) q^{59} + ( -4 + 4 \zeta_{6} ) q^{60} -10 q^{62} + 7 q^{64} + 8 \zeta_{6} q^{65} + ( 2 - 2 \zeta_{6} ) q^{66} + ( -8 + 8 \zeta_{6} ) q^{67} -4 \zeta_{6} q^{68} + 8 q^{69} -12 q^{71} + 3 \zeta_{6} q^{72} + ( -8 + 8 \zeta_{6} ) q^{73} + ( 6 - 6 \zeta_{6} ) q^{74} -2 \zeta_{6} q^{75} + 8 q^{78} -8 \zeta_{6} q^{79} + ( 2 - 2 \zeta_{6} ) q^{80} + ( 11 - 11 \zeta_{6} ) q^{81} + 4 \zeta_{6} q^{82} -8 q^{85} -12 \zeta_{6} q^{86} + ( -12 + 12 \zeta_{6} ) q^{87} + ( 3 - 3 \zeta_{6} ) q^{88} -6 \zeta_{6} q^{89} + 2 q^{90} + 4 q^{92} -20 \zeta_{6} q^{93} + ( -10 + 10 \zeta_{6} ) q^{94} + 10 \zeta_{6} q^{96} + 10 q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + 2q^{3} + q^{4} - 2q^{5} - 4q^{6} - 6q^{8} - q^{9} + O(q^{10}) \) \( 2q - q^{2} + 2q^{3} + q^{4} - 2q^{5} - 4q^{6} - 6q^{8} - q^{9} - 2q^{10} - q^{11} - 2q^{12} - 8q^{13} - 8q^{15} + q^{16} + 4q^{17} - q^{18} - 4q^{20} + 2q^{22} + 4q^{23} - 6q^{24} + q^{25} + 4q^{26} + 8q^{27} - 12q^{29} + 4q^{30} + 10q^{31} - 5q^{32} + 2q^{33} - 8q^{34} - 2q^{36} + 6q^{37} - 8q^{39} + 6q^{40} - 8q^{41} + 24q^{43} + q^{44} - 2q^{45} + 4q^{46} - 10q^{47} + 4q^{48} - 2q^{50} - 8q^{51} - 4q^{52} + 6q^{53} - 4q^{54} + 4q^{55} + 6q^{58} + 2q^{59} - 4q^{60} - 20q^{62} + 14q^{64} + 8q^{65} + 2q^{66} - 8q^{67} - 4q^{68} + 16q^{69} - 24q^{71} + 3q^{72} - 8q^{73} + 6q^{74} - 2q^{75} + 16q^{78} - 8q^{79} + 2q^{80} + 11q^{81} + 4q^{82} - 16q^{85} - 12q^{86} - 12q^{87} + 3q^{88} - 6q^{89} + 4q^{90} + 8q^{92} - 20q^{93} - 10q^{94} + 10q^{96} + 20q^{97} + 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(442\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 1.00000 1.73205i 0.500000 0.866025i −1.00000 1.73205i −2.00000 0 −3.00000 −0.500000 0.866025i −1.00000 + 1.73205i
177.1 −0.500000 + 0.866025i 1.00000 + 1.73205i 0.500000 + 0.866025i −1.00000 + 1.73205i −2.00000 0 −3.00000 −0.500000 + 0.866025i −1.00000 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.e.b 2
7.b odd 2 1 539.2.e.a 2
7.c even 3 1 539.2.a.d 1
7.c even 3 1 inner 539.2.e.b 2
7.d odd 6 1 77.2.a.c 1
7.d odd 6 1 539.2.e.a 2
21.g even 6 1 693.2.a.a 1
21.h odd 6 1 4851.2.a.a 1
28.f even 6 1 1232.2.a.a 1
28.g odd 6 1 8624.2.a.bc 1
35.i odd 6 1 1925.2.a.c 1
35.k even 12 2 1925.2.b.d 2
56.j odd 6 1 4928.2.a.g 1
56.m even 6 1 4928.2.a.bi 1
77.h odd 6 1 5929.2.a.b 1
77.i even 6 1 847.2.a.a 1
77.n even 30 4 847.2.f.k 4
77.p odd 30 4 847.2.f.e 4
231.k odd 6 1 7623.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.c 1 7.d odd 6 1
539.2.a.d 1 7.c even 3 1
539.2.e.a 2 7.b odd 2 1
539.2.e.a 2 7.d odd 6 1
539.2.e.b 2 1.a even 1 1 trivial
539.2.e.b 2 7.c even 3 1 inner
693.2.a.a 1 21.g even 6 1
847.2.a.a 1 77.i even 6 1
847.2.f.e 4 77.p odd 30 4
847.2.f.k 4 77.n even 30 4
1232.2.a.a 1 28.f even 6 1
1925.2.a.c 1 35.i odd 6 1
1925.2.b.d 2 35.k even 12 2
4851.2.a.a 1 21.h odd 6 1
4928.2.a.g 1 56.j odd 6 1
4928.2.a.bi 1 56.m even 6 1
5929.2.a.b 1 77.h odd 6 1
7623.2.a.n 1 231.k odd 6 1
8624.2.a.bc 1 28.g odd 6 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \)
\( T_{3}^{2} - 2 T_{3} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \)
$3$ \( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} \)
$5$ \( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 + T + T^{2} \)
$13$ \( ( 1 + 4 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \)
$19$ \( 1 - 19 T^{2} + 361 T^{4} \)
$23$ \( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} \)
$37$ \( 1 - 6 T - T^{2} - 222 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 4 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 12 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 10 T + 53 T^{2} + 470 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 2 T - 55 T^{2} - 118 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 61 T^{2} + 3721 T^{4} \)
$67$ \( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 8 T - 9 T^{2} + 584 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 10 T + 97 T^{2} )^{2} \)
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