Properties

Label 567.2.e.a
Level 567
Weight 2
Character orbit 567.e
Analytic conductor 4.528
Analytic rank 1
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(1\)
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 63)
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} -3 q^{8} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{4} + \zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} -3 q^{8} + ( 1 - \zeta_{6} ) q^{10} + ( -5 + 5 \zeta_{6} ) q^{11} -5 q^{13} + ( -1 + 3 \zeta_{6} ) q^{14} + \zeta_{6} q^{16} + ( -3 + 3 \zeta_{6} ) q^{17} -\zeta_{6} q^{19} + q^{20} + 5 q^{22} -3 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + 5 \zeta_{6} q^{26} + ( -3 + 2 \zeta_{6} ) q^{28} - q^{29} + ( -5 + 5 \zeta_{6} ) q^{32} + 3 q^{34} + ( 1 - 3 \zeta_{6} ) q^{35} -3 \zeta_{6} q^{37} + ( -1 + \zeta_{6} ) q^{38} -3 \zeta_{6} q^{40} -5 q^{41} - q^{43} + 5 \zeta_{6} q^{44} + ( -3 + 3 \zeta_{6} ) q^{46} + ( 3 + 5 \zeta_{6} ) q^{49} -4 q^{50} + ( -5 + 5 \zeta_{6} ) q^{52} + ( 9 - 9 \zeta_{6} ) q^{53} -5 q^{55} + ( 6 + 3 \zeta_{6} ) q^{56} + \zeta_{6} q^{58} + 14 \zeta_{6} q^{61} + 7 q^{64} -5 \zeta_{6} q^{65} + ( -4 + 4 \zeta_{6} ) q^{67} + 3 \zeta_{6} q^{68} + ( -3 + 2 \zeta_{6} ) q^{70} -12 q^{71} + ( -3 + 3 \zeta_{6} ) q^{73} + ( -3 + 3 \zeta_{6} ) q^{74} - q^{76} + ( 15 - 10 \zeta_{6} ) q^{77} -8 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{80} + 5 \zeta_{6} q^{82} -9 q^{83} -3 q^{85} + \zeta_{6} q^{86} + ( 15 - 15 \zeta_{6} ) q^{88} + 13 \zeta_{6} q^{89} + ( 10 + 5 \zeta_{6} ) q^{91} -3 q^{92} + ( 1 - \zeta_{6} ) q^{95} -9 q^{97} + ( 5 - 8 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{4} + q^{5} - 5q^{7} - 6q^{8} + O(q^{10}) \) \( 2q - q^{2} + q^{4} + q^{5} - 5q^{7} - 6q^{8} + q^{10} - 5q^{11} - 10q^{13} + q^{14} + q^{16} - 3q^{17} - q^{19} + 2q^{20} + 10q^{22} - 3q^{23} + 4q^{25} + 5q^{26} - 4q^{28} - 2q^{29} - 5q^{32} + 6q^{34} - q^{35} - 3q^{37} - q^{38} - 3q^{40} - 10q^{41} - 2q^{43} + 5q^{44} - 3q^{46} + 11q^{49} - 8q^{50} - 5q^{52} + 9q^{53} - 10q^{55} + 15q^{56} + q^{58} + 14q^{61} + 14q^{64} - 5q^{65} - 4q^{67} + 3q^{68} - 4q^{70} - 24q^{71} - 3q^{73} - 3q^{74} - 2q^{76} + 20q^{77} - 8q^{79} - q^{80} + 5q^{82} - 18q^{83} - 6q^{85} + q^{86} + 15q^{88} + 13q^{89} + 25q^{91} - 6q^{92} + q^{95} - 18q^{97} + 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 0 0.500000 + 0.866025i 0.500000 0.866025i 0 −2.50000 + 0.866025i −3.00000 0 0.500000 + 0.866025i
487.1 −0.500000 0.866025i 0 0.500000 0.866025i 0.500000 + 0.866025i 0 −2.50000 0.866025i −3.00000 0 0.500000 0.866025i
\(n\): 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 567.2.e.a 2
3.b odd 2 1 567.2.e.b 2
7.c even 3 1 inner 567.2.e.a 2
7.c even 3 1 3969.2.a.d 1
7.d odd 6 1 3969.2.a.f 1
9.c even 3 1 63.2.g.a 2
9.c even 3 1 63.2.h.a yes 2
9.d odd 6 1 189.2.g.a 2
9.d odd 6 1 189.2.h.a 2
21.g even 6 1 3969.2.a.a 1
21.h odd 6 1 567.2.e.b 2
21.h odd 6 1 3969.2.a.c 1
36.f odd 6 1 1008.2.q.c 2
36.f odd 6 1 1008.2.t.d 2
36.h even 6 1 3024.2.q.b 2
36.h even 6 1 3024.2.t.d 2
63.g even 3 1 63.2.h.a yes 2
63.g even 3 1 441.2.f.b 2
63.h even 3 1 63.2.g.a 2
63.h even 3 1 441.2.f.b 2
63.i even 6 1 1323.2.f.b 2
63.i even 6 1 1323.2.g.a 2
63.j odd 6 1 189.2.g.a 2
63.j odd 6 1 1323.2.f.a 2
63.k odd 6 1 441.2.f.a 2
63.k odd 6 1 441.2.h.a 2
63.l odd 6 1 441.2.g.a 2
63.l odd 6 1 441.2.h.a 2
63.n odd 6 1 189.2.h.a 2
63.n odd 6 1 1323.2.f.a 2
63.o even 6 1 1323.2.g.a 2
63.o even 6 1 1323.2.h.a 2
63.s even 6 1 1323.2.f.b 2
63.s even 6 1 1323.2.h.a 2
63.t odd 6 1 441.2.f.a 2
63.t odd 6 1 441.2.g.a 2
252.o even 6 1 3024.2.q.b 2
252.u odd 6 1 1008.2.t.d 2
252.bb even 6 1 3024.2.t.d 2
252.bl odd 6 1 1008.2.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 9.c even 3 1
63.2.g.a 2 63.h even 3 1
63.2.h.a yes 2 9.c even 3 1
63.2.h.a yes 2 63.g even 3 1
189.2.g.a 2 9.d odd 6 1
189.2.g.a 2 63.j odd 6 1
189.2.h.a 2 9.d odd 6 1
189.2.h.a 2 63.n odd 6 1
441.2.f.a 2 63.k odd 6 1
441.2.f.a 2 63.t odd 6 1
441.2.f.b 2 63.g even 3 1
441.2.f.b 2 63.h even 3 1
441.2.g.a 2 63.l odd 6 1
441.2.g.a 2 63.t odd 6 1
441.2.h.a 2 63.k odd 6 1
441.2.h.a 2 63.l odd 6 1
567.2.e.a 2 1.a even 1 1 trivial
567.2.e.a 2 7.c even 3 1 inner
567.2.e.b 2 3.b odd 2 1
567.2.e.b 2 21.h odd 6 1
1008.2.q.c 2 36.f odd 6 1
1008.2.q.c 2 252.bl odd 6 1
1008.2.t.d 2 36.f odd 6 1
1008.2.t.d 2 252.u odd 6 1
1323.2.f.a 2 63.j odd 6 1
1323.2.f.a 2 63.n odd 6 1
1323.2.f.b 2 63.i even 6 1
1323.2.f.b 2 63.s even 6 1
1323.2.g.a 2 63.i even 6 1
1323.2.g.a 2 63.o even 6 1
1323.2.h.a 2 63.o even 6 1
1323.2.h.a 2 63.s even 6 1
3024.2.q.b 2 36.h even 6 1
3024.2.q.b 2 252.o even 6 1
3024.2.t.d 2 36.h even 6 1
3024.2.t.d 2 252.bb even 6 1
3969.2.a.a 1 21.g even 6 1
3969.2.a.c 1 21.h odd 6 1
3969.2.a.d 1 7.c even 3 1
3969.2.a.f 1 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \)
$3$ 1
$5$ \( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$ \( 1 + 5 T + 7 T^{2} \)
$11$ \( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 5 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \)
$23$ \( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + T + 29 T^{2} )^{2} \)
$31$ \( 1 - 31 T^{2} + 961 T^{4} \)
$37$ \( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 5 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \)
$67$ \( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 9 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 13 T + 80 T^{2} - 1157 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 9 T + 97 T^{2} )^{2} \)
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