Properties

Label 567.2.f.l
Level $567$
Weight $2$
Character orbit 567.f
Analytic conductor $4.528$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1156923.1
Defining polynomial: \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{4} + (\beta_{5} - \beta_{3} - \beta_1) q^{5} + (\beta_{3} - 1) q^{7} + (2 \beta_{2} - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{4} + (\beta_{5} - \beta_{3} - \beta_1) q^{5} + (\beta_{3} - 1) q^{7} + (2 \beta_{2} - 3) q^{8} + (2 \beta_{2} + \beta_1 + 1) q^{10} + (\beta_{5} + 2 \beta_{3} - 2) q^{11} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{13} + ( - \beta_{4} - \beta_{2}) q^{14} + ( - 3 \beta_{4} + 4 \beta_{3} - 4) q^{16} + ( - 2 \beta_{2} + \beta_1 + 1) q^{17} + (2 \beta_{2} - 2 \beta_1) q^{19} + ( - \beta_{5} + 7 \beta_{3} - 7) q^{20} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{22} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{23} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2) q^{25} + (2 \beta_{2} - 3 \beta_1 - 9) q^{26} + ( - \beta_{2} + \beta_1 + 2) q^{28} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} - 1) q^{29} - 2 \beta_{3} q^{31} + ( - 3 \beta_{5} - 3 \beta_{4} + 6 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{32} + (\beta_{5} + 4 \beta_{4} - 7 \beta_{3} + 7) q^{34} + (\beta_1 + 1) q^{35} + 5 q^{37} + ( - 4 \beta_{4} + 6 \beta_{3} - 6) q^{38} + ( - \beta_{5} - 4 \beta_{4} + \beta_{3} - 4 \beta_{2} + \beta_1) q^{40} + (2 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{41} + (\beta_{5} + 4 \beta_{4} + 4 \beta_{3} - 4) q^{43} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{44} + ( - 4 \beta_{2} - 2 \beta_1 - 2) q^{46} - 2 \beta_{4} q^{47} - \beta_{3} q^{49} + (3 \beta_{5} - \beta_{4} - 9 \beta_{3} - \beta_{2} - 3 \beta_1) q^{50} + ( - \beta_{5} - 10 \beta_{4} + 3 \beta_{3} - 3) q^{52} + ( - 2 \beta_{2} + \beta_1 + 4) q^{53} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{55} + (2 \beta_{4} - 3 \beta_{3} + 3) q^{56} + (3 \beta_{5} - 9 \beta_{3} - 3 \beta_1) q^{58} + (4 \beta_{4} + 6 \beta_{3} + 4 \beta_{2}) q^{59} + (3 \beta_{5} - \beta_{3} + 1) q^{61} + 2 \beta_{2} q^{62} + ( - 6 \beta_{2} + 1) q^{64} + (\beta_{5} - 6 \beta_{4} - 7 \beta_{3} + 7) q^{65} + (3 \beta_{5} - 2 \beta_{3} - 3 \beta_1) q^{67} + (\beta_{5} + 8 \beta_{4} - 13 \beta_{3} + 8 \beta_{2} - \beta_1) q^{68} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} - 1) q^{70} - 3 \beta_1 q^{71} + (2 \beta_{2} + \beta_1 + 3) q^{73} + 5 \beta_{4} q^{74} + ( - 6 \beta_{4} + 16 \beta_{3} - 6 \beta_{2}) q^{76} + ( - \beta_{5} - 2 \beta_{3} + \beta_1) q^{77} + ( - 3 \beta_{5} + 2 \beta_{3} - 2) q^{79} + ( - 6 \beta_{2} + \beta_1 + 1) q^{80} - 6 q^{82} + (2 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 4) q^{83} + ( - \beta_{5} + 2 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} + \beta_1) q^{85} + (3 \beta_{5} + \beta_{4} - 15 \beta_{3} + \beta_{2} - 3 \beta_1) q^{86} + ( - \beta_{5} + 2 \beta_{4} - 8 \beta_{3} + 8) q^{88} + (2 \beta_{2} - \beta_1 + 5) q^{89} + ( - 2 \beta_{2} - \beta_1 + 1) q^{91} + (2 \beta_{5} - 14 \beta_{3} + 14) q^{92} + ( - 2 \beta_{5} - 2 \beta_{4} + 8 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{94} + (2 \beta_{5} + 10 \beta_{3} - 2 \beta_1) q^{95} + ( - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 4) q^{97} + \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 3 q^{5} - 3 q^{7} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} - 3 q^{5} - 3 q^{7} - 18 q^{8} + 6 q^{10} - 6 q^{11} - 3 q^{13} - 12 q^{16} + 6 q^{17} - 21 q^{20} + 3 q^{22} + 6 q^{23} - 6 q^{25} - 54 q^{26} + 12 q^{28} - 3 q^{29} - 6 q^{31} + 18 q^{32} + 21 q^{34} + 6 q^{35} + 30 q^{37} - 18 q^{38} + 3 q^{40} + 12 q^{41} - 12 q^{43} - 6 q^{44} - 12 q^{46} - 3 q^{49} - 27 q^{50} - 9 q^{52} + 24 q^{53} - 24 q^{55} + 9 q^{56} - 27 q^{58} + 18 q^{59} + 3 q^{61} + 6 q^{64} + 21 q^{65} - 6 q^{67} - 39 q^{68} - 3 q^{70} + 18 q^{73} + 48 q^{76} - 6 q^{77} - 6 q^{79} + 6 q^{80} - 36 q^{82} - 12 q^{83} - 15 q^{85} - 45 q^{86} + 24 q^{88} + 30 q^{89} + 6 q^{91} + 42 q^{92} + 24 q^{94} + 30 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 2\nu^{3} + 8\nu^{2} - 7\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 22\nu^{3} - 28\nu^{2} + 43\nu - 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{5} - 13\nu^{4} + 54\nu^{3} - 71\nu^{2} + 99\nu - 40 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 15\nu^{4} - 64\nu^{3} + 82\nu^{2} - 121\nu + 50 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} + 4\beta _1 - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{5} + 2\beta_{4} + 4\beta_{3} + \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 20\beta_{5} + 14\beta_{4} + 25\beta_{3} + 13\beta_{2} - 25\beta _1 + 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -34\beta_{5} - 16\beta_{4} - 59\beta_{3} + 7\beta_{2} - 28\beta _1 + 121 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(-\beta_{3}\)

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 0.0585812i
0.500000 + 2.43956i
0.500000 1.51496i
0.500000 + 0.0585812i
0.500000 2.43956i
0.500000 + 1.51496i
−1.07255 1.85771i 0 −1.30073 + 2.25294i −1.87328 + 3.24462i 0 −0.500000 0.866025i 1.29021 0 8.03677
190.2 −0.261988 0.453777i 0 0.862724 1.49428i 1.10074 1.90653i 0 −0.500000 0.866025i −1.95205 0 −1.15352
190.3 1.33454 + 2.31149i 0 −2.56199 + 4.43750i −0.727452 + 1.25998i 0 −0.500000 0.866025i −8.33816 0 −3.88325
379.1 −1.07255 + 1.85771i 0 −1.30073 2.25294i −1.87328 3.24462i 0 −0.500000 + 0.866025i 1.29021 0 8.03677
379.2 −0.261988 + 0.453777i 0 0.862724 + 1.49428i 1.10074 + 1.90653i 0 −0.500000 + 0.866025i −1.95205 0 −1.15352
379.3 1.33454 2.31149i 0 −2.56199 4.43750i −0.727452 1.25998i 0 −0.500000 + 0.866025i −8.33816 0 −3.88325
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.3
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.l 6
3.b odd 2 1 567.2.f.m 6
9.c even 3 1 567.2.a.f yes 3
9.c even 3 1 inner 567.2.f.l 6
9.d odd 6 1 567.2.a.e 3
9.d odd 6 1 567.2.f.m 6
36.f odd 6 1 9072.2.a.cb 3
36.h even 6 1 9072.2.a.bu 3
63.l odd 6 1 3969.2.a.n 3
63.o even 6 1 3969.2.a.o 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.a.e 3 9.d odd 6 1
567.2.a.f yes 3 9.c even 3 1
567.2.f.l 6 1.a even 1 1 trivial
567.2.f.l 6 9.c even 3 1 inner
567.2.f.m 6 3.b odd 2 1
567.2.f.m 6 9.d odd 6 1
3969.2.a.n 3 63.l odd 6 1
3969.2.a.o 3 63.o even 6 1
9072.2.a.bu 3 36.h even 6 1
9072.2.a.cb 3 36.f odd 6 1

Hecke kernels

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

\( T_{2}^{6} + 6T_{2}^{4} + 6T_{2}^{3} + 36T_{2}^{2} + 18T_{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{6} + 3T_{5}^{5} + 15T_{5}^{4} + 6T_{5}^{3} + 72T_{5}^{2} + 72T_{5} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 6 T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + 15 T^{4} + 6 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 6 T^{5} + 33 T^{4} + 30 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + 45 T^{4} + \cdots + 12544 \) Copy content Toggle raw display
$17$ \( (T^{3} - 3 T^{2} - 24 T - 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 48 T - 56)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 6 T^{5} + 60 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} + 45 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T + 4)^{3} \) Copy content Toggle raw display
$37$ \( (T - 5)^{6} \) Copy content Toggle raw display
$41$ \( T^{6} - 12 T^{5} + 144 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$43$ \( T^{6} + 12 T^{5} + 189 T^{4} + \cdots + 425104 \) Copy content Toggle raw display
$47$ \( T^{6} + 24 T^{4} - 48 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$53$ \( (T^{3} - 12 T^{2} + 21 T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 18 T^{5} + 312 T^{4} + \cdots + 304704 \) Copy content Toggle raw display
$61$ \( T^{6} - 3 T^{5} + 87 T^{4} + \cdots + 35344 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + 105 T^{4} + \cdots + 68644 \) Copy content Toggle raw display
$71$ \( (T^{3} - 81 T - 108)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 9 T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 6 T^{5} + 105 T^{4} + \cdots + 68644 \) Copy content Toggle raw display
$83$ \( T^{6} + 12 T^{5} + 144 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$89$ \( (T^{3} - 15 T^{2} + 48 T + 48)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 12 T^{5} + 144 T^{4} + \cdots + 33856 \) Copy content Toggle raw display
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