Properties

Label 567.2.o.c
Level 567
Weight 2
Character orbit 567.o
Analytic conductor 4.528
Analytic rank 0
Dimension 4
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 3 \beta_{2} q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( 1 - 3 \beta_{2} ) q^{7} + \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 3 \beta_{2} q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( 1 - 3 \beta_{2} ) q^{7} + \beta_{3} q^{8} + ( -5 + 10 \beta_{2} ) q^{10} + \beta_{1} q^{11} + ( -4 + 2 \beta_{2} ) q^{13} + ( \beta_{1} - 3 \beta_{3} ) q^{14} + ( 1 - \beta_{2} ) q^{16} + ( 3 - 6 \beta_{2} ) q^{19} + ( -3 \beta_{1} + 6 \beta_{3} ) q^{20} + 5 \beta_{2} q^{22} + ( -\beta_{1} + \beta_{3} ) q^{23} + ( -10 + 10 \beta_{2} ) q^{25} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{26} + ( 9 - 6 \beta_{2} ) q^{28} -2 \beta_{1} q^{29} + ( 2 - \beta_{2} ) q^{31} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{32} + ( 4 \beta_{1} - 5 \beta_{3} ) q^{35} - q^{37} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{38} + ( -10 + 5 \beta_{2} ) q^{40} + ( \beta_{1} + \beta_{3} ) q^{41} + ( -2 + 2 \beta_{2} ) q^{43} + 3 \beta_{3} q^{44} -5 q^{46} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{47} + ( -8 + 3 \beta_{2} ) q^{49} + ( -10 \beta_{1} + 10 \beta_{3} ) q^{50} + ( -6 - 6 \beta_{2} ) q^{52} -4 \beta_{3} q^{53} + ( -5 + 10 \beta_{2} ) q^{55} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{56} -10 \beta_{2} q^{58} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{59} + ( 4 + 4 \beta_{2} ) q^{61} + ( 2 \beta_{1} - \beta_{3} ) q^{62} + 13 q^{64} -6 \beta_{1} q^{65} + 10 \beta_{2} q^{67} + ( 25 - 5 \beta_{2} ) q^{70} + 5 \beta_{3} q^{71} + ( 6 - 12 \beta_{2} ) q^{73} -\beta_{1} q^{74} + ( 18 - 9 \beta_{2} ) q^{76} + ( \beta_{1} - 3 \beta_{3} ) q^{77} + ( -2 + 2 \beta_{2} ) q^{79} + ( 2 \beta_{1} - \beta_{3} ) q^{80} + ( -5 + 10 \beta_{2} ) q^{82} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{83} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{86} + ( -5 + 5 \beta_{2} ) q^{88} + ( 6 \beta_{1} - 3 \beta_{3} ) q^{89} + ( 2 + 8 \beta_{2} ) q^{91} -3 \beta_{1} q^{92} + ( 20 - 10 \beta_{2} ) q^{94} + ( 9 \beta_{1} - 9 \beta_{3} ) q^{95} + ( -8 - 8 \beta_{2} ) q^{97} + ( -8 \beta_{1} + 3 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{4} - 2q^{7} + O(q^{10}) \) \( 4q + 6q^{4} - 2q^{7} - 12q^{13} + 2q^{16} + 10q^{22} - 20q^{25} + 24q^{28} + 6q^{31} - 4q^{37} - 30q^{40} - 4q^{43} - 20q^{46} - 26q^{49} - 36q^{52} - 20q^{58} + 24q^{61} + 52q^{64} + 20q^{67} + 90q^{70} + 54q^{76} - 4q^{79} - 10q^{88} + 24q^{91} + 60q^{94} - 48q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(5 \beta_{3}\)

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1
−1.93649 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i
1.93649 1.11803i
−1.93649 1.11803i 0 1.50000 + 2.59808i −1.93649 3.35410i 0 −0.500000 2.59808i 2.23607i 0 8.66025i
188.2 1.93649 + 1.11803i 0 1.50000 + 2.59808i 1.93649 + 3.35410i 0 −0.500000 2.59808i 2.23607i 0 8.66025i
377.1 −1.93649 + 1.11803i 0 1.50000 2.59808i −1.93649 + 3.35410i 0 −0.500000 + 2.59808i 2.23607i 0 8.66025i
377.2 1.93649 1.11803i 0 1.50000 2.59808i 1.93649 3.35410i 0 −0.500000 + 2.59808i 2.23607i 0 8.66025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.l odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.o.c 4
3.b odd 2 1 inner 567.2.o.c 4
7.b odd 2 1 567.2.o.d 4
9.c even 3 1 189.2.c.b 4
9.c even 3 1 567.2.o.d 4
9.d odd 6 1 189.2.c.b 4
9.d odd 6 1 567.2.o.d 4
21.c even 2 1 567.2.o.d 4
36.f odd 6 1 3024.2.k.i 4
36.h even 6 1 3024.2.k.i 4
63.l odd 6 1 189.2.c.b 4
63.l odd 6 1 inner 567.2.o.c 4
63.o even 6 1 189.2.c.b 4
63.o even 6 1 inner 567.2.o.c 4
252.s odd 6 1 3024.2.k.i 4
252.bi even 6 1 3024.2.k.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.c.b 4 9.c even 3 1
189.2.c.b 4 9.d odd 6 1
189.2.c.b 4 63.l odd 6 1
189.2.c.b 4 63.o even 6 1
567.2.o.c 4 1.a even 1 1 trivial
567.2.o.c 4 3.b odd 2 1 inner
567.2.o.c 4 63.l odd 6 1 inner
567.2.o.c 4 63.o even 6 1 inner
567.2.o.d 4 7.b odd 2 1
567.2.o.d 4 9.c even 3 1
567.2.o.d 4 9.d odd 6 1
567.2.o.d 4 21.c even 2 1
3024.2.k.i 4 36.f odd 6 1
3024.2.k.i 4 36.h even 6 1
3024.2.k.i 4 252.s odd 6 1
3024.2.k.i 4 252.bi even 6 1

Hecke kernels

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

\( T_{2}^{4} - 5 T_{2}^{2} + 25 \)
\( T_{13}^{2} + 6 T_{13} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8} \)
$3$ 1
$5$ \( ( 1 + 5 T^{2} )^{2}( 1 - 5 T^{2} + 25 T^{4} ) \)
$7$ \( ( 1 + T + 7 T^{2} )^{2} \)
$11$ \( 1 + 17 T^{2} + 168 T^{4} + 2057 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 + 6 T + 25 T^{2} + 78 T^{3} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 17 T^{2} )^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 41 T^{2} + 1152 T^{4} + 21689 T^{6} + 279841 T^{8} \)
$29$ \( 1 + 38 T^{2} + 603 T^{4} + 31958 T^{6} + 707281 T^{8} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + T + 37 T^{2} )^{4} \)
$41$ \( 1 - 67 T^{2} + 2808 T^{4} - 112627 T^{6} + 2825761 T^{8} \)
$43$ \( ( 1 + 2 T - 39 T^{2} + 86 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 34 T^{2} - 1053 T^{4} - 75106 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 - 26 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 58 T^{2} - 117 T^{4} - 201898 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )^{2}( 1 + T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 17 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 38 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( 1 - 106 T^{2} + 4347 T^{4} - 730234 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 + 43 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 5 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2} \)
show more
show less