Properties

Label 567.2.i.c
Level $567$
Weight $2$
Character orbit 567.i
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.i (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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} -3 q^{4} + ( 2 + \beta_{2} ) q^{7} -\beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{3} q^{2} -3 q^{4} + ( 2 + \beta_{2} ) q^{7} -\beta_{3} q^{8} + 2 \beta_{1} q^{11} + ( -1 - \beta_{2} ) q^{13} + ( -\beta_{1} + 3 \beta_{3} ) q^{14} - q^{16} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{17} + ( -2 - 2 \beta_{2} ) q^{19} + ( -10 + 10 \beta_{2} ) q^{22} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{23} + ( 5 - 5 \beta_{2} ) q^{25} + ( \beta_{1} - 2 \beta_{3} ) q^{26} + ( -6 - 3 \beta_{2} ) q^{28} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{29} + ( 1 - 2 \beta_{2} ) q^{31} -3 \beta_{3} q^{32} + ( -20 + 10 \beta_{2} ) q^{34} + ( -5 + 5 \beta_{2} ) q^{37} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{38} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{41} + 7 \beta_{2} q^{43} -6 \beta_{1} q^{44} -10 \beta_{2} q^{46} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{47} + ( 3 + 5 \beta_{2} ) q^{49} + 5 \beta_{1} q^{50} + ( 3 + 3 \beta_{2} ) q^{52} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{53} + ( \beta_{1} - 3 \beta_{3} ) q^{56} -10 \beta_{2} q^{58} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{59} + ( 5 - 10 \beta_{2} ) q^{61} + ( 2 \beta_{1} - \beta_{3} ) q^{62} + 13 q^{64} - q^{67} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{68} + 4 \beta_{3} q^{71} + ( 8 - 4 \beta_{2} ) q^{73} -5 \beta_{1} q^{74} + ( 6 + 6 \beta_{2} ) q^{76} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{77} + 11 q^{79} + ( 10 + 10 \beta_{2} ) q^{82} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{83} + ( -7 \beta_{1} + 7 \beta_{3} ) q^{86} + ( 10 - 10 \beta_{2} ) q^{88} + ( 4 \beta_{1} - 8 \beta_{3} ) q^{89} + ( -1 - 4 \beta_{2} ) q^{91} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{92} + ( 10 - 20 \beta_{2} ) q^{94} + ( -2 + \beta_{2} ) q^{97} + ( -5 \beta_{1} + 8 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{4} + 10q^{7} + O(q^{10}) \) \( 4q - 12q^{4} + 10q^{7} - 6q^{13} - 4q^{16} - 12q^{19} - 20q^{22} + 10q^{25} - 30q^{28} - 60q^{34} - 10q^{37} + 14q^{43} - 20q^{46} + 22q^{49} + 18q^{52} - 20q^{58} + 52q^{64} - 4q^{67} + 24q^{73} + 36q^{76} + 44q^{79} + 60q^{82} + 20q^{88} - 12q^{91} - 6q^{97} + O(q^{100}) \)

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 \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
−1.93649 1.11803i
1.93649 + 1.11803i
1.93649 1.11803i
−1.93649 + 1.11803i
2.23607i 0 −3.00000 0 0 2.50000 + 0.866025i 2.23607i 0 0
215.2 2.23607i 0 −3.00000 0 0 2.50000 + 0.866025i 2.23607i 0 0
269.1 2.23607i 0 −3.00000 0 0 2.50000 0.866025i 2.23607i 0 0
269.2 2.23607i 0 −3.00000 0 0 2.50000 0.866025i 2.23607i 0 0
\(n\): 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.i even 6 1 inner
63.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.i.c 4
3.b odd 2 1 inner 567.2.i.c 4
7.d odd 6 1 567.2.s.e 4
9.c even 3 1 189.2.p.c 4
9.c even 3 1 567.2.s.e 4
9.d odd 6 1 189.2.p.c 4
9.d odd 6 1 567.2.s.e 4
21.g even 6 1 567.2.s.e 4
63.h even 3 1 1323.2.c.b 4
63.i even 6 1 inner 567.2.i.c 4
63.i even 6 1 1323.2.c.b 4
63.j odd 6 1 1323.2.c.b 4
63.k odd 6 1 189.2.p.c 4
63.s even 6 1 189.2.p.c 4
63.t odd 6 1 inner 567.2.i.c 4
63.t odd 6 1 1323.2.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.c 4 9.c even 3 1
189.2.p.c 4 9.d odd 6 1
189.2.p.c 4 63.k odd 6 1
189.2.p.c 4 63.s even 6 1
567.2.i.c 4 1.a even 1 1 trivial
567.2.i.c 4 3.b odd 2 1 inner
567.2.i.c 4 63.i even 6 1 inner
567.2.i.c 4 63.t odd 6 1 inner
567.2.s.e 4 7.d odd 6 1
567.2.s.e 4 9.c even 3 1
567.2.s.e 4 9.d odd 6 1
567.2.s.e 4 21.g even 6 1
1323.2.c.b 4 63.h even 3 1
1323.2.c.b 4 63.i even 6 1
1323.2.c.b 4 63.j odd 6 1
1323.2.c.b 4 63.t odd 6 1

Hecke kernels

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

\( T_{2}^{2} + 5 \)
\( T_{11}^{4} - 20 T_{11}^{2} + 400 \)
\( T_{13}^{2} + 3 T_{13} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 5 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 7 - 5 T + T^{2} )^{2} \)
$11$ \( 400 - 20 T^{2} + T^{4} \)
$13$ \( ( 3 + 3 T + T^{2} )^{2} \)
$17$ \( 3600 + 60 T^{2} + T^{4} \)
$19$ \( ( 12 + 6 T + T^{2} )^{2} \)
$23$ \( 400 - 20 T^{2} + T^{4} \)
$29$ \( 400 - 20 T^{2} + T^{4} \)
$31$ \( ( 3 + T^{2} )^{2} \)
$37$ \( ( 25 + 5 T + T^{2} )^{2} \)
$41$ \( 3600 + 60 T^{2} + T^{4} \)
$43$ \( ( 49 - 7 T + T^{2} )^{2} \)
$47$ \( ( -60 + T^{2} )^{2} \)
$53$ \( 400 - 20 T^{2} + T^{4} \)
$59$ \( ( -60 + T^{2} )^{2} \)
$61$ \( ( 75 + T^{2} )^{2} \)
$67$ \( ( 1 + T )^{4} \)
$71$ \( ( 80 + T^{2} )^{2} \)
$73$ \( ( 48 - 12 T + T^{2} )^{2} \)
$79$ \( ( -11 + T )^{4} \)
$83$ \( 3600 + 60 T^{2} + T^{4} \)
$89$ \( 57600 + 240 T^{2} + T^{4} \)
$97$ \( ( 3 + 3 T + T^{2} )^{2} \)
show more
show less