Properties

Label 567.2.f.b
Level $567$
Weight $2$
Character orbit 567.f
Analytic conductor $4.528$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.52751779461\)
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 21)
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.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
190.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0 0.500000 0.866025i −1.00000 + 1.73205i 0 0.500000 + 0.866025i −3.00000 0 2.00000
379.1 −0.500000 + 0.866025i 0 0.500000 + 0.866025i −1.00000 1.73205i 0 0.500000 0.866025i −3.00000 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.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.f.b 2
3.b odd 2 1 567.2.f.g 2
9.c even 3 1 63.2.a.a 1
9.c even 3 1 inner 567.2.f.b 2
9.d odd 6 1 21.2.a.a 1
9.d odd 6 1 567.2.f.g 2
36.f odd 6 1 1008.2.a.l 1
36.h even 6 1 336.2.a.a 1
45.h odd 6 1 525.2.a.d 1
45.j even 6 1 1575.2.a.c 1
45.k odd 12 2 1575.2.d.a 2
45.l even 12 2 525.2.d.a 2
63.g even 3 1 441.2.e.a 2
63.h even 3 1 441.2.e.a 2
63.i even 6 1 147.2.e.c 2
63.j odd 6 1 147.2.e.b 2
63.k odd 6 1 441.2.e.b 2
63.l odd 6 1 441.2.a.f 1
63.n odd 6 1 147.2.e.b 2
63.o even 6 1 147.2.a.a 1
63.s even 6 1 147.2.e.c 2
63.t odd 6 1 441.2.e.b 2
72.j odd 6 1 1344.2.a.g 1
72.l even 6 1 1344.2.a.s 1
72.n even 6 1 4032.2.a.h 1
72.p odd 6 1 4032.2.a.k 1
99.g even 6 1 2541.2.a.j 1
99.h odd 6 1 7623.2.a.g 1
117.n odd 6 1 3549.2.a.c 1
144.u even 12 2 5376.2.c.l 2
144.w odd 12 2 5376.2.c.r 2
153.i odd 6 1 6069.2.a.b 1
171.l even 6 1 7581.2.a.d 1
180.n even 6 1 8400.2.a.bn 1
252.o even 6 1 2352.2.q.x 2
252.r odd 6 1 2352.2.q.e 2
252.s odd 6 1 2352.2.a.v 1
252.bb even 6 1 2352.2.q.x 2
252.bi even 6 1 7056.2.a.p 1
252.bn odd 6 1 2352.2.q.e 2
315.z even 6 1 3675.2.a.n 1
504.cc even 6 1 9408.2.a.bv 1
504.co odd 6 1 9408.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 9.d odd 6 1
63.2.a.a 1 9.c even 3 1
147.2.a.a 1 63.o even 6 1
147.2.e.b 2 63.j odd 6 1
147.2.e.b 2 63.n odd 6 1
147.2.e.c 2 63.i even 6 1
147.2.e.c 2 63.s even 6 1
336.2.a.a 1 36.h even 6 1
441.2.a.f 1 63.l odd 6 1
441.2.e.a 2 63.g even 3 1
441.2.e.a 2 63.h even 3 1
441.2.e.b 2 63.k odd 6 1
441.2.e.b 2 63.t odd 6 1
525.2.a.d 1 45.h odd 6 1
525.2.d.a 2 45.l even 12 2
567.2.f.b 2 1.a even 1 1 trivial
567.2.f.b 2 9.c even 3 1 inner
567.2.f.g 2 3.b odd 2 1
567.2.f.g 2 9.d odd 6 1
1008.2.a.l 1 36.f odd 6 1
1344.2.a.g 1 72.j odd 6 1
1344.2.a.s 1 72.l even 6 1
1575.2.a.c 1 45.j even 6 1
1575.2.d.a 2 45.k odd 12 2
2352.2.a.v 1 252.s odd 6 1
2352.2.q.e 2 252.r odd 6 1
2352.2.q.e 2 252.bn odd 6 1
2352.2.q.x 2 252.o even 6 1
2352.2.q.x 2 252.bb even 6 1
2541.2.a.j 1 99.g even 6 1
3549.2.a.c 1 117.n odd 6 1
3675.2.a.n 1 315.z even 6 1
4032.2.a.h 1 72.n even 6 1
4032.2.a.k 1 72.p odd 6 1
5376.2.c.l 2 144.u even 12 2
5376.2.c.r 2 144.w odd 12 2
6069.2.a.b 1 153.i odd 6 1
7056.2.a.p 1 252.bi even 6 1
7581.2.a.d 1 171.l even 6 1
7623.2.a.g 1 99.h odd 6 1
8400.2.a.bn 1 180.n even 6 1
9408.2.a.m 1 504.co odd 6 1
9408.2.a.bv 1 504.cc even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\):

\( T_{2}^{2} + T_{2} + 1 \)
\( T_{5}^{2} + 2 T_{5} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 + 2 T + T^{2} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( 16 - 4 T + T^{2} \)
$13$ \( 4 - 2 T + T^{2} \)
$17$ \( ( -6 + T )^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( 4 + 2 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( 4 - 2 T + T^{2} \)
$43$ \( 16 - 4 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 144 - 12 T + T^{2} \)
$61$ \( 4 - 2 T + T^{2} \)
$67$ \( 16 + 4 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( 256 - 16 T + T^{2} \)
$83$ \( 144 + 12 T + T^{2} \)
$89$ \( ( -14 + T )^{2} \)
$97$ \( 324 + 18 T + T^{2} \)
show more
show less