Properties

Label 539.2.e.j
Level $539$
Weight $2$
Character orbit 539.e
Analytic conductor $4.304$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{3} + \beta_{2} - \beta_1) q^{3} + 3 \beta_1 q^{4} + ( - 2 \beta_1 - 2) q^{5} + (\beta_{3} + 5) q^{6} - \beta_{3} q^{8} + (2 \beta_{2} - 3 \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{3} + \beta_{2} - \beta_1) q^{3} + 3 \beta_1 q^{4} + ( - 2 \beta_1 - 2) q^{5} + (\beta_{3} + 5) q^{6} - \beta_{3} q^{8} + (2 \beta_{2} - 3 \beta_1 - 3) q^{9} + (2 \beta_{3} + 2 \beta_{2}) q^{10} - \beta_1 q^{11} + ( - 3 \beta_{2} + 3 \beta_1 + 3) q^{12} + (\beta_{3} - 1) q^{13} + ( - 2 \beta_{3} - 2) q^{15} + (\beta_1 + 1) q^{16} + (\beta_{3} + \beta_{2} + \beta_1) q^{17} + (3 \beta_{3} + 3 \beta_{2} - 10 \beta_1) q^{18} + (2 \beta_{2} + 2 \beta_1 + 2) q^{19} + 6 q^{20} + \beta_{3} q^{22} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{23} + ( - \beta_{3} - \beta_{2} + 5 \beta_1) q^{24} - \beta_1 q^{25} + (\beta_{2} + 5 \beta_1 + 5) q^{26} + ( - 2 \beta_{3} - 10) q^{27} + ( - 2 \beta_{3} + 4) q^{29} + (2 \beta_{2} - 10 \beta_1 - 10) q^{30} + ( - \beta_{3} - \beta_{2} + 5 \beta_1) q^{31} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{32} + (\beta_{2} - \beta_1 - 1) q^{33} + ( - \beta_{3} + 5) q^{34} + (6 \beta_{3} + 9) q^{36} + (2 \beta_{2} + 4 \beta_1 + 4) q^{37} + ( - 2 \beta_{3} - 2 \beta_{2} - 10 \beta_1) q^{38} - 4 \beta_1 q^{39} - 2 \beta_{2} q^{40} + ( - \beta_{3} + 9) q^{41} + 8 q^{43} + (3 \beta_1 + 3) q^{44} + ( - 4 \beta_{3} - 4 \beta_{2} + 6 \beta_1) q^{45} + ( - 2 \beta_{3} - 2 \beta_{2} + 10 \beta_1) q^{46} + ( - \beta_{2} + 5 \beta_1 + 5) q^{47} + (\beta_{3} + 1) q^{48} + \beta_{3} q^{50} + ( - 4 \beta_1 - 4) q^{51} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{52} + ( - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{53} + (10 \beta_{2} - 10 \beta_1 - 10) q^{54} - 2 q^{55} - 8 q^{57} + ( - 4 \beta_{2} - 10 \beta_1 - 10) q^{58} + (\beta_{3} + \beta_{2} - \beta_1) q^{59} + (6 \beta_{3} + 6 \beta_{2} - 6 \beta_1) q^{60} + ( - \beta_{2} - 5 \beta_1 - 5) q^{61} + ( - 5 \beta_{3} - 5) q^{62} - 13 q^{64} + (2 \beta_{2} + 2 \beta_1 + 2) q^{65} + (\beta_{3} + \beta_{2} - 5 \beta_1) q^{66} + (2 \beta_{3} + 2 \beta_{2} + 10 \beta_1) q^{67} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{68} + (4 \beta_{3} + 12) q^{69} + (2 \beta_{3} - 6) q^{71} + ( - 3 \beta_{2} + 10 \beta_1 + 10) q^{72} + ( - \beta_{3} - \beta_{2} + 3 \beta_1) q^{73} + ( - 4 \beta_{3} - 4 \beta_{2} - 10 \beta_1) q^{74} + (\beta_{2} - \beta_1 - 1) q^{75} + (6 \beta_{3} - 6) q^{76} + 4 \beta_{3} q^{78} + 4 \beta_{2} q^{79} - 2 \beta_1 q^{80} + ( - 6 \beta_{3} - 6 \beta_{2} + 11 \beta_1) q^{81} + ( - 9 \beta_{2} - 5 \beta_1 - 5) q^{82} + ( - 6 \beta_{3} - 2) q^{83} + ( - 2 \beta_{3} + 2) q^{85} - 8 \beta_{2} q^{86} + (2 \beta_{3} + 2 \beta_{2} + 6 \beta_1) q^{87} + ( - \beta_{3} - \beta_{2}) q^{88} + (2 \beta_1 + 2) q^{89} + ( - 6 \beta_{3} - 20) q^{90} + ( - 6 \beta_{3} - 6) q^{92} + ( - 6 \beta_{2} + 10 \beta_1 + 10) q^{93} + ( - 5 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{94} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{95} + ( - 3 \beta_{2} + 15 \beta_1 + 15) q^{96} + (6 \beta_{3} - 4) q^{97} + ( - 2 \beta_{3} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 6 q^{4} - 4 q^{5} + 20 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 6 q^{4} - 4 q^{5} + 20 q^{6} - 6 q^{9} + 2 q^{11} + 6 q^{12} - 4 q^{13} - 8 q^{15} + 2 q^{16} - 2 q^{17} + 20 q^{18} + 4 q^{19} + 24 q^{20} + 4 q^{23} - 10 q^{24} + 2 q^{25} + 10 q^{26} - 40 q^{27} + 16 q^{29} - 20 q^{30} - 10 q^{31} - 2 q^{33} + 20 q^{34} + 36 q^{36} + 8 q^{37} + 20 q^{38} + 8 q^{39} + 36 q^{41} + 32 q^{43} + 6 q^{44} - 12 q^{45} - 20 q^{46} + 10 q^{47} + 4 q^{48} - 8 q^{51} + 6 q^{52} - 8 q^{53} - 20 q^{54} - 8 q^{55} - 32 q^{57} - 20 q^{58} + 2 q^{59} + 12 q^{60} - 10 q^{61} - 20 q^{62} - 52 q^{64} + 4 q^{65} + 10 q^{66} - 20 q^{67} - 6 q^{68} + 48 q^{69} - 24 q^{71} + 20 q^{72} - 6 q^{73} + 20 q^{74} - 2 q^{75} - 24 q^{76} + 4 q^{80} - 22 q^{81} - 10 q^{82} - 8 q^{83} + 8 q^{85} - 12 q^{87} + 4 q^{89} - 80 q^{90} - 24 q^{92} + 20 q^{93} - 10 q^{94} + 8 q^{95} + 30 q^{96} - 16 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} + 6\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(442\)
\(\chi(n)\) \(-1 - \beta_{1}\) \(1\)

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
0.809017 + 1.40126i
−0.309017 0.535233i
0.809017 1.40126i
−0.309017 + 0.535233i
−1.11803 1.93649i −0.618034 + 1.07047i −1.50000 + 2.59808i −1.00000 1.73205i 2.76393 0 2.23607 0.736068 + 1.27491i −2.23607 + 3.87298i
67.2 1.11803 + 1.93649i 1.61803 2.80252i −1.50000 + 2.59808i −1.00000 1.73205i 7.23607 0 −2.23607 −3.73607 6.47106i 2.23607 3.87298i
177.1 −1.11803 + 1.93649i −0.618034 1.07047i −1.50000 2.59808i −1.00000 + 1.73205i 2.76393 0 2.23607 0.736068 1.27491i −2.23607 3.87298i
177.2 1.11803 1.93649i 1.61803 + 2.80252i −1.50000 2.59808i −1.00000 + 1.73205i 7.23607 0 −2.23607 −3.73607 + 6.47106i 2.23607 + 3.87298i
\(n\): 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.j 4
7.b odd 2 1 539.2.e.i 4
7.c even 3 1 539.2.a.f 2
7.c even 3 1 inner 539.2.e.j 4
7.d odd 6 1 77.2.a.d 2
7.d odd 6 1 539.2.e.i 4
21.g even 6 1 693.2.a.h 2
21.h odd 6 1 4851.2.a.y 2
28.f even 6 1 1232.2.a.m 2
28.g odd 6 1 8624.2.a.ce 2
35.i odd 6 1 1925.2.a.r 2
35.k even 12 2 1925.2.b.h 4
56.j odd 6 1 4928.2.a.bm 2
56.m even 6 1 4928.2.a.bv 2
77.h odd 6 1 5929.2.a.m 2
77.i even 6 1 847.2.a.f 2
77.n even 30 2 847.2.f.b 4
77.n even 30 2 847.2.f.m 4
77.p odd 30 2 847.2.f.a 4
77.p odd 30 2 847.2.f.n 4
231.k odd 6 1 7623.2.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 7.d odd 6 1
539.2.a.f 2 7.c even 3 1
539.2.e.i 4 7.b odd 2 1
539.2.e.i 4 7.d odd 6 1
539.2.e.j 4 1.a even 1 1 trivial
539.2.e.j 4 7.c even 3 1 inner
693.2.a.h 2 21.g even 6 1
847.2.a.f 2 77.i even 6 1
847.2.f.a 4 77.p odd 30 2
847.2.f.b 4 77.n even 30 2
847.2.f.m 4 77.n even 30 2
847.2.f.n 4 77.p odd 30 2
1232.2.a.m 2 28.f even 6 1
1925.2.a.r 2 35.i odd 6 1
1925.2.b.h 4 35.k even 12 2
4851.2.a.y 2 21.h odd 6 1
4928.2.a.bm 2 56.j odd 6 1
4928.2.a.bv 2 56.m even 6 1
5929.2.a.m 2 77.h odd 6 1
7623.2.a.bl 2 231.k odd 6 1
8624.2.a.ce 2 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 \(S_{2}^{\mathrm{new}}(539, [\chi])\):

\( T_{2}^{4} + 5T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{3}^{4} - 2T_{3}^{3} + 8T_{3}^{2} + 8T_{3} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 5T^{2} + 25 \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 8 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + 8 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} + 32 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} + 32 T^{2} + 64 T + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 10 T^{3} + 80 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} + 68 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$41$ \( (T^{2} - 18 T + 76)^{2} \) Copy content Toggle raw display
$43$ \( (T - 8)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 10 T^{3} + 80 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$53$ \( T^{4} + 8 T^{3} + 68 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$59$ \( T^{4} - 2 T^{3} + 8 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$61$ \( T^{4} + 10 T^{3} + 80 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$67$ \( T^{4} + 20 T^{3} + 320 T^{2} + \cdots + 6400 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 6 T^{3} + 32 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$79$ \( T^{4} + 80T^{2} + 6400 \) Copy content Toggle raw display
$83$ \( (T^{2} + 4 T - 176)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 164)^{2} \) Copy content Toggle raw display
show more
show less