Properties

Label 567.2.f.c
Level $567$
Weight $2$
Character orbit 567.f
Analytic conductor $4.528$
Analytic rank $0$
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.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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} + \zeta_{6} q^{4} + \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} + \zeta_{6} q^{4} + \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - 3 q^{8} - q^{10} + (2 \zeta_{6} - 2) q^{11} + 5 \zeta_{6} q^{13} + \zeta_{6} q^{14} + ( - \zeta_{6} + 1) q^{16} - 3 q^{17} - 2 q^{19} + (\zeta_{6} - 1) q^{20} - 2 \zeta_{6} q^{22} + 6 \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} - 5 q^{26} + q^{28} + (5 \zeta_{6} - 5) q^{29} + 6 \zeta_{6} q^{31} - 5 \zeta_{6} q^{32} + ( - 3 \zeta_{6} + 3) q^{34} + q^{35} - 3 q^{37} + ( - 2 \zeta_{6} + 2) q^{38} - 3 \zeta_{6} q^{40} - 10 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} - 2 q^{44} - 6 q^{46} + (6 \zeta_{6} - 6) q^{47} - \zeta_{6} q^{49} + 4 \zeta_{6} q^{50} + (5 \zeta_{6} - 5) q^{52} + 6 q^{53} - 2 q^{55} + (3 \zeta_{6} - 3) q^{56} - 5 \zeta_{6} q^{58} + 6 \zeta_{6} q^{59} + (7 \zeta_{6} - 7) q^{61} - 6 q^{62} + 7 q^{64} + (5 \zeta_{6} - 5) q^{65} + 2 \zeta_{6} q^{67} - 3 \zeta_{6} q^{68} + (\zeta_{6} - 1) q^{70} + 12 q^{71} - 15 q^{73} + ( - 3 \zeta_{6} + 3) q^{74} - 2 \zeta_{6} q^{76} + 2 \zeta_{6} q^{77} + (14 \zeta_{6} - 14) q^{79} + q^{80} + 10 q^{82} + ( - 18 \zeta_{6} + 18) q^{83} - 3 \zeta_{6} q^{85} + 4 \zeta_{6} q^{86} + ( - 6 \zeta_{6} + 6) q^{88} + 5 q^{89} + 5 q^{91} + (6 \zeta_{6} - 6) q^{92} - 6 \zeta_{6} q^{94} - 2 \zeta_{6} q^{95} + ( - 18 \zeta_{6} + 18) q^{97} + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{4} + q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{4} + q^{5} + q^{7} - 6 q^{8} - 2 q^{10} - 2 q^{11} + 5 q^{13} + q^{14} + q^{16} - 6 q^{17} - 4 q^{19} - q^{20} - 2 q^{22} + 6 q^{23} + 4 q^{25} - 10 q^{26} + 2 q^{28} - 5 q^{29} + 6 q^{31} - 5 q^{32} + 3 q^{34} + 2 q^{35} - 6 q^{37} + 2 q^{38} - 3 q^{40} - 10 q^{41} + 4 q^{43} - 4 q^{44} - 12 q^{46} - 6 q^{47} - q^{49} + 4 q^{50} - 5 q^{52} + 12 q^{53} - 4 q^{55} - 3 q^{56} - 5 q^{58} + 6 q^{59} - 7 q^{61} - 12 q^{62} + 14 q^{64} - 5 q^{65} + 2 q^{67} - 3 q^{68} - q^{70} + 24 q^{71} - 30 q^{73} + 3 q^{74} - 2 q^{76} + 2 q^{77} - 14 q^{79} + 2 q^{80} + 20 q^{82} + 18 q^{83} - 3 q^{85} + 4 q^{86} + 6 q^{88} + 10 q^{89} + 10 q^{91} - 6 q^{92} - 6 q^{94} - 2 q^{95} + 18 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 0.500000 0.866025i 0 0.500000 + 0.866025i −3.00000 0 −1.00000
379.1 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.500000 0.866025i −3.00000 0 −1.00000
\(n\): 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.c 2
3.b odd 2 1 567.2.f.f 2
9.c even 3 1 567.2.a.b yes 1
9.c even 3 1 inner 567.2.f.c 2
9.d odd 6 1 567.2.a.a 1
9.d odd 6 1 567.2.f.f 2
36.f odd 6 1 9072.2.a.j 1
36.h even 6 1 9072.2.a.q 1
63.l odd 6 1 3969.2.a.e 1
63.o even 6 1 3969.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.a.a 1 9.d odd 6 1
567.2.a.b yes 1 9.c even 3 1
567.2.f.c 2 1.a even 1 1 trivial
567.2.f.c 2 9.c even 3 1 inner
567.2.f.f 2 3.b odd 2 1
567.2.f.f 2 9.d odd 6 1
3969.2.a.b 1 63.o even 6 1
3969.2.a.e 1 63.l odd 6 1
9072.2.a.j 1 36.f odd 6 1
9072.2.a.q 1 36.h even 6 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$31$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$37$ \( (T + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 15)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$83$ \( T^{2} - 18T + 324 \) Copy content Toggle raw display
$89$ \( (T - 5)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 18T + 324 \) Copy content Toggle raw display
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