Properties

Label 539.2.e.g
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + \zeta_{6} q^{5} - 2 q^{6} + 2 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + \zeta_{6} q^{5} - 2 q^{6} + 2 \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{10} + (\zeta_{6} - 1) q^{11} - 2 \zeta_{6} q^{12} - 4 q^{13} - q^{15} + 4 \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{17} + (4 \zeta_{6} - 4) q^{18} - 2 q^{20} - 2 q^{22} + \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} - 8 \zeta_{6} q^{26} - 5 q^{27} - 2 \zeta_{6} q^{30} + ( - 7 \zeta_{6} + 7) q^{31} + (8 \zeta_{6} - 8) q^{32} - \zeta_{6} q^{33} - 4 q^{34} - 4 q^{36} - 3 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{39} + 8 q^{41} - 6 q^{43} - 2 \zeta_{6} q^{44} + (2 \zeta_{6} - 2) q^{45} + (2 \zeta_{6} - 2) q^{46} + 8 \zeta_{6} q^{47} - 4 q^{48} + 8 q^{50} - 2 \zeta_{6} q^{51} + ( - 8 \zeta_{6} + 8) q^{52} + ( - 6 \zeta_{6} + 6) q^{53} - 10 \zeta_{6} q^{54} - q^{55} + ( - 5 \zeta_{6} + 5) q^{59} + ( - 2 \zeta_{6} + 2) q^{60} + 12 \zeta_{6} q^{61} + 14 q^{62} - 8 q^{64} - 4 \zeta_{6} q^{65} + ( - 2 \zeta_{6} + 2) q^{66} + ( - 7 \zeta_{6} + 7) q^{67} - 4 \zeta_{6} q^{68} - q^{69} - 3 q^{71} + ( - 4 \zeta_{6} + 4) q^{73} + ( - 6 \zeta_{6} + 6) q^{74} + 4 \zeta_{6} q^{75} + 8 q^{78} + 10 \zeta_{6} q^{79} + (4 \zeta_{6} - 4) q^{80} + (\zeta_{6} - 1) q^{81} + 16 \zeta_{6} q^{82} + 6 q^{83} - 2 q^{85} - 12 \zeta_{6} q^{86} + 15 \zeta_{6} q^{89} - 4 q^{90} - 2 q^{92} + 7 \zeta_{6} q^{93} + (16 \zeta_{6} - 16) q^{94} - 8 \zeta_{6} q^{96} + 7 q^{97} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} - 2 q^{4} + q^{5} - 4 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} - 2 q^{4} + q^{5} - 4 q^{6} + 2 q^{9} - 2 q^{10} - q^{11} - 2 q^{12} - 8 q^{13} - 2 q^{15} + 4 q^{16} - 2 q^{17} - 4 q^{18} - 4 q^{20} - 4 q^{22} + q^{23} + 4 q^{25} - 8 q^{26} - 10 q^{27} - 2 q^{30} + 7 q^{31} - 8 q^{32} - q^{33} - 8 q^{34} - 8 q^{36} - 3 q^{37} + 4 q^{39} + 16 q^{41} - 12 q^{43} - 2 q^{44} - 2 q^{45} - 2 q^{46} + 8 q^{47} - 8 q^{48} + 16 q^{50} - 2 q^{51} + 8 q^{52} + 6 q^{53} - 10 q^{54} - 2 q^{55} + 5 q^{59} + 2 q^{60} + 12 q^{61} + 28 q^{62} - 16 q^{64} - 4 q^{65} + 2 q^{66} + 7 q^{67} - 4 q^{68} - 2 q^{69} - 6 q^{71} + 4 q^{73} + 6 q^{74} + 4 q^{75} + 16 q^{78} + 10 q^{79} - 4 q^{80} - q^{81} + 16 q^{82} + 12 q^{83} - 4 q^{85} - 12 q^{86} + 15 q^{89} - 8 q^{90} - 4 q^{92} + 7 q^{93} - 16 q^{94} - 8 q^{96} + 14 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
1.00000 + 1.73205i −0.500000 + 0.866025i −1.00000 + 1.73205i 0.500000 + 0.866025i −2.00000 0 0 1.00000 + 1.73205i −1.00000 + 1.73205i
177.1 1.00000 1.73205i −0.500000 0.866025i −1.00000 1.73205i 0.500000 0.866025i −2.00000 0 0 1.00000 1.73205i −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.g 2
7.b odd 2 1 539.2.e.h 2
7.c even 3 1 539.2.a.a 1
7.c even 3 1 inner 539.2.e.g 2
7.d odd 6 1 11.2.a.a 1
7.d odd 6 1 539.2.e.h 2
21.g even 6 1 99.2.a.d 1
21.h odd 6 1 4851.2.a.t 1
28.f even 6 1 176.2.a.b 1
28.g odd 6 1 8624.2.a.j 1
35.i odd 6 1 275.2.a.b 1
35.k even 12 2 275.2.b.a 2
56.j odd 6 1 704.2.a.h 1
56.m even 6 1 704.2.a.c 1
63.i even 6 1 891.2.e.b 2
63.k odd 6 1 891.2.e.k 2
63.s even 6 1 891.2.e.b 2
63.t odd 6 1 891.2.e.k 2
77.h odd 6 1 5929.2.a.h 1
77.i even 6 1 121.2.a.d 1
77.n even 30 4 121.2.c.a 4
77.p odd 30 4 121.2.c.e 4
84.j odd 6 1 1584.2.a.g 1
91.s odd 6 1 1859.2.a.b 1
105.p even 6 1 2475.2.a.a 1
105.w odd 12 2 2475.2.c.a 2
112.v even 12 2 2816.2.c.f 2
112.x odd 12 2 2816.2.c.j 2
119.h odd 6 1 3179.2.a.a 1
133.o even 6 1 3971.2.a.b 1
140.s even 6 1 4400.2.a.i 1
140.x odd 12 2 4400.2.b.h 2
161.g even 6 1 5819.2.a.a 1
168.ba even 6 1 6336.2.a.br 1
168.be odd 6 1 6336.2.a.bu 1
203.i odd 6 1 9251.2.a.d 1
231.k odd 6 1 1089.2.a.b 1
308.m odd 6 1 1936.2.a.i 1
385.o even 6 1 3025.2.a.a 1
616.s even 6 1 7744.2.a.x 1
616.z odd 6 1 7744.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 7.d odd 6 1
99.2.a.d 1 21.g even 6 1
121.2.a.d 1 77.i even 6 1
121.2.c.a 4 77.n even 30 4
121.2.c.e 4 77.p odd 30 4
176.2.a.b 1 28.f even 6 1
275.2.a.b 1 35.i odd 6 1
275.2.b.a 2 35.k even 12 2
539.2.a.a 1 7.c even 3 1
539.2.e.g 2 1.a even 1 1 trivial
539.2.e.g 2 7.c even 3 1 inner
539.2.e.h 2 7.b odd 2 1
539.2.e.h 2 7.d odd 6 1
704.2.a.c 1 56.m even 6 1
704.2.a.h 1 56.j odd 6 1
891.2.e.b 2 63.i even 6 1
891.2.e.b 2 63.s even 6 1
891.2.e.k 2 63.k odd 6 1
891.2.e.k 2 63.t odd 6 1
1089.2.a.b 1 231.k odd 6 1
1584.2.a.g 1 84.j odd 6 1
1859.2.a.b 1 91.s odd 6 1
1936.2.a.i 1 308.m odd 6 1
2475.2.a.a 1 105.p even 6 1
2475.2.c.a 2 105.w odd 12 2
2816.2.c.f 2 112.v even 12 2
2816.2.c.j 2 112.x odd 12 2
3025.2.a.a 1 385.o even 6 1
3179.2.a.a 1 119.h odd 6 1
3971.2.a.b 1 133.o even 6 1
4400.2.a.i 1 140.s even 6 1
4400.2.b.h 2 140.x odd 12 2
4851.2.a.t 1 21.h odd 6 1
5819.2.a.a 1 161.g even 6 1
5929.2.a.h 1 77.h odd 6 1
6336.2.a.br 1 168.ba even 6 1
6336.2.a.bu 1 168.be odd 6 1
7744.2.a.k 1 616.z odd 6 1
7744.2.a.x 1 616.s even 6 1
8624.2.a.j 1 28.g odd 6 1
9251.2.a.d 1 203.i odd 6 1

Hecke kernels

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

\( T_{2}^{2} - 2T_{2} + 4 \) Copy content Toggle raw display
\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$37$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$41$ \( (T - 8)^{2} \) Copy content Toggle raw display
$43$ \( (T + 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$61$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$67$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$71$ \( (T + 3)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$97$ \( (T - 7)^{2} \) Copy content Toggle raw display
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