Properties

Label 567.2.a.c
Level $567$
Weight $2$
Character orbit 567.a
Self dual yes
Analytic conductor $4.528$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.52751779461\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + ( -1 - 2 \beta + \beta^{2} ) q^{4} + ( -3 - \beta + \beta^{2} ) q^{5} + q^{7} + ( 4 + 2 \beta - 3 \beta^{2} ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + ( -1 - 2 \beta + \beta^{2} ) q^{4} + ( -3 - \beta + \beta^{2} ) q^{5} + q^{7} + ( 4 + 2 \beta - 3 \beta^{2} ) q^{8} + ( 4 + \beta - 2 \beta^{2} ) q^{10} + ( \beta - \beta^{2} ) q^{11} + ( 7 + 2 \beta - 4 \beta^{2} ) q^{13} + ( -1 + \beta ) q^{14} + ( -5 - 3 \beta + 3 \beta^{2} ) q^{16} + ( -4 + \beta^{2} ) q^{17} + ( -3 - 2 \beta + \beta^{2} ) q^{19} + ( -\beta + \beta^{2} ) q^{20} + ( -1 - 4 \beta + 2 \beta^{2} ) q^{22} + ( -\beta - 2 \beta^{2} ) q^{23} + ( 2 + \beta - 2 \beta^{2} ) q^{25} + ( -11 - 7 \beta + 6 \beta^{2} ) q^{26} + ( -1 - 2 \beta + \beta^{2} ) q^{28} + ( -11 - 5 \beta + 4 \beta^{2} ) q^{29} + ( -7 + 3 \beta + 3 \beta^{2} ) q^{31} + 3 \beta q^{32} + ( 5 - \beta - \beta^{2} ) q^{34} + ( -3 - \beta + \beta^{2} ) q^{35} + ( -7 + 3 \beta + 3 \beta^{2} ) q^{37} + ( 4 + 2 \beta - 3 \beta^{2} ) q^{38} + ( -7 + 2 \beta + 2 \beta^{2} ) q^{40} + ( 2 - \beta - \beta^{2} ) q^{41} + ( 1 - \beta - \beta^{2} ) q^{43} + ( 3 + 7 \beta - 4 \beta^{2} ) q^{44} + ( -2 - 5 \beta + \beta^{2} ) q^{46} + ( 5 - 2 \beta - 3 \beta^{2} ) q^{47} + q^{49} + ( -4 - 5 \beta + 3 \beta^{2} ) q^{50} + ( 3 + 10 \beta - 5 \beta^{2} ) q^{52} + ( 2 + 3 \beta - 2 \beta^{2} ) q^{53} + ( 2 + 2 \beta - \beta^{2} ) q^{55} + ( 4 + 2 \beta - 3 \beta^{2} ) q^{56} + ( 15 + 6 \beta - 9 \beta^{2} ) q^{58} + ( 1 + 5 \beta ) q^{59} + ( 2 + 3 \beta ) q^{61} + ( 10 - \beta ) q^{62} + ( 10 + 3 \beta - 3 \beta^{2} ) q^{64} + ( -15 + \beta + 5 \beta^{2} ) q^{65} + ( -10 + 3 \beta^{2} ) q^{67} + ( 2 + 3 \beta - 2 \beta^{2} ) q^{68} + ( 4 + \beta - 2 \beta^{2} ) q^{70} + ( -15 - 3 \beta + 6 \beta^{2} ) q^{71} + ( -9 + 4 \beta + \beta^{2} ) q^{73} + ( 10 - \beta ) q^{74} + ( -1 - 3 \beta + 3 \beta^{2} ) q^{76} + ( \beta - \beta^{2} ) q^{77} + ( -13 + 3 \beta^{2} ) q^{79} + ( 9 - \beta - 2 \beta^{2} ) q^{80} -3 q^{82} + ( 4 + 4 \beta + \beta^{2} ) q^{83} + ( 11 + 2 \beta - 4 \beta^{2} ) q^{85} + ( -2 - \beta ) q^{86} + ( -5 - 8 \beta + 7 \beta^{2} ) q^{88} + ( 10 + 3 \beta - 7 \beta^{2} ) q^{89} + ( 7 + 2 \beta - 4 \beta^{2} ) q^{91} + ( 3 + 8 \beta - 2 \beta^{2} ) q^{92} + ( -8 - 2 \beta + \beta^{2} ) q^{94} + ( 6 + \beta - \beta^{2} ) q^{95} + ( -17 - 7 \beta + 8 \beta^{2} ) q^{97} + ( -1 + \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} - 6q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} - 6q^{8} - 6q^{11} - 3q^{13} - 3q^{14} + 3q^{16} - 6q^{17} - 3q^{19} + 6q^{20} + 9q^{22} - 12q^{23} - 6q^{25} + 3q^{26} + 3q^{28} - 9q^{29} - 3q^{31} + 9q^{34} - 3q^{35} - 3q^{37} - 6q^{38} - 9q^{40} - 3q^{43} - 15q^{44} - 3q^{47} + 3q^{49} + 6q^{50} - 21q^{52} - 6q^{53} - 6q^{56} - 9q^{58} + 3q^{59} + 6q^{61} + 30q^{62} + 12q^{64} - 15q^{65} - 12q^{67} - 6q^{68} - 9q^{71} - 21q^{73} + 30q^{74} + 15q^{76} - 6q^{77} - 21q^{79} + 15q^{80} - 9q^{82} + 18q^{83} + 9q^{85} - 6q^{86} + 27q^{88} - 12q^{89} - 3q^{91} - 3q^{92} - 18q^{94} + 12q^{95} - 3q^{97} - 3q^{98} + O(q^{100}) \)

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
−1.53209
−0.347296
1.87939
−2.53209 0 4.41147 0.879385 0 1.00000 −6.10607 0 −2.22668
1.2 −1.34730 0 −0.184793 −2.53209 0 1.00000 2.94356 0 3.41147
1.3 0.879385 0 −1.22668 −1.34730 0 1.00000 −2.83750 0 −1.18479
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)

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 567.2.a.c 3
3.b odd 2 1 567.2.a.h 3
4.b odd 2 1 9072.2.a.bs 3
7.b odd 2 1 3969.2.a.l 3
9.c even 3 2 189.2.f.b 6
9.d odd 6 2 63.2.f.a 6
12.b even 2 1 9072.2.a.ca 3
21.c even 2 1 3969.2.a.q 3
36.f odd 6 2 3024.2.r.k 6
36.h even 6 2 1008.2.r.h 6
63.g even 3 2 1323.2.g.d 6
63.h even 3 2 1323.2.h.c 6
63.i even 6 2 441.2.h.e 6
63.j odd 6 2 441.2.h.d 6
63.k odd 6 2 1323.2.g.e 6
63.l odd 6 2 1323.2.f.d 6
63.n odd 6 2 441.2.g.c 6
63.o even 6 2 441.2.f.c 6
63.s even 6 2 441.2.g.b 6
63.t odd 6 2 1323.2.h.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a 6 9.d odd 6 2
189.2.f.b 6 9.c even 3 2
441.2.f.c 6 63.o even 6 2
441.2.g.b 6 63.s even 6 2
441.2.g.c 6 63.n odd 6 2
441.2.h.d 6 63.j odd 6 2
441.2.h.e 6 63.i even 6 2
567.2.a.c 3 1.a even 1 1 trivial
567.2.a.h 3 3.b odd 2 1
1008.2.r.h 6 36.h even 6 2
1323.2.f.d 6 63.l odd 6 2
1323.2.g.d 6 63.g even 3 2
1323.2.g.e 6 63.k odd 6 2
1323.2.h.b 6 63.t odd 6 2
1323.2.h.c 6 63.h even 3 2
3024.2.r.k 6 36.f odd 6 2
3969.2.a.l 3 7.b odd 2 1
3969.2.a.q 3 21.c even 2 1
9072.2.a.bs 3 4.b odd 2 1
9072.2.a.ca 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 3 T_{2}^{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(567))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + 3 T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( -3 + 3 T^{2} + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( 3 + 9 T + 6 T^{2} + T^{3} \)
$13$ \( -107 - 33 T + 3 T^{2} + T^{3} \)
$17$ \( 3 + 9 T + 6 T^{2} + T^{3} \)
$19$ \( -17 - 6 T + 3 T^{2} + T^{3} \)
$23$ \( -3 + 27 T + 12 T^{2} + T^{3} \)
$29$ \( -333 - 36 T + 9 T^{2} + T^{3} \)
$31$ \( -323 - 78 T + 3 T^{2} + T^{3} \)
$37$ \( -323 - 78 T + 3 T^{2} + T^{3} \)
$41$ \( 9 - 9 T + T^{3} \)
$43$ \( 1 - 6 T + 3 T^{2} + T^{3} \)
$47$ \( 51 - 54 T + 3 T^{2} + T^{3} \)
$53$ \( 3 - 9 T + 6 T^{2} + T^{3} \)
$59$ \( -51 - 72 T - 3 T^{2} + T^{3} \)
$61$ \( 19 - 15 T - 6 T^{2} + T^{3} \)
$67$ \( -17 + 21 T + 12 T^{2} + T^{3} \)
$71$ \( 27 - 54 T + 9 T^{2} + T^{3} \)
$73$ \( -269 + 84 T + 21 T^{2} + T^{3} \)
$79$ \( 181 + 120 T + 21 T^{2} + T^{3} \)
$83$ \( -9 + 45 T - 18 T^{2} + T^{3} \)
$89$ \( -813 - 63 T + 12 T^{2} + T^{3} \)
$97$ \( -323 - 168 T + 3 T^{2} + T^{3} \)
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