Properties

Label 864.1.t.a
Level $864$
Weight $1$
Character orbit 864.t
Analytic conductor $0.431$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -8
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 864.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.431192170915\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.648.1
Artin image $C_6\times S_3$
Artin field Galois closure of 12.0.8916100448256.2

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q -\zeta_{6}^{2} q^{11} + q^{17} + q^{19} + \zeta_{6}^{2} q^{25} -\zeta_{6} q^{41} + \zeta_{6}^{2} q^{43} -\zeta_{6} q^{49} + \zeta_{6} q^{59} -\zeta_{6} q^{67} - q^{73} + 2 \zeta_{6}^{2} q^{83} -2 q^{89} -\zeta_{6}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + q^{11} + 2q^{17} + 2q^{19} - q^{25} - q^{41} - q^{43} - q^{49} + q^{59} - q^{67} - 2q^{73} - 2q^{83} - 4q^{89} + q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 0 0 0 0
847.1 0 0 0 0 0 0 0 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
9.c even 3 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.1.t.a 2
3.b odd 2 1 288.1.t.a 2
4.b odd 2 1 216.1.p.a 2
8.b even 2 1 216.1.p.a 2
8.d odd 2 1 CM 864.1.t.a 2
9.c even 3 1 inner 864.1.t.a 2
9.c even 3 1 2592.1.b.a 1
9.d odd 6 1 288.1.t.a 2
9.d odd 6 1 2592.1.b.b 1
12.b even 2 1 72.1.p.a 2
24.f even 2 1 288.1.t.a 2
24.h odd 2 1 72.1.p.a 2
36.f odd 6 1 216.1.p.a 2
36.f odd 6 1 648.1.b.a 1
36.h even 6 1 72.1.p.a 2
36.h even 6 1 648.1.b.b 1
48.i odd 4 2 2304.1.o.c 4
48.k even 4 2 2304.1.o.c 4
60.h even 2 1 1800.1.bk.d 2
60.l odd 4 2 1800.1.ba.b 4
72.j odd 6 1 72.1.p.a 2
72.j odd 6 1 648.1.b.b 1
72.l even 6 1 288.1.t.a 2
72.l even 6 1 2592.1.b.b 1
72.n even 6 1 216.1.p.a 2
72.n even 6 1 648.1.b.a 1
72.p odd 6 1 inner 864.1.t.a 2
72.p odd 6 1 2592.1.b.a 1
84.h odd 2 1 3528.1.cg.a 2
84.j odd 6 1 3528.1.ba.a 2
84.j odd 6 1 3528.1.ce.b 2
84.n even 6 1 3528.1.ba.b 2
84.n even 6 1 3528.1.ce.a 2
120.i odd 2 1 1800.1.bk.d 2
120.w even 4 2 1800.1.ba.b 4
144.u even 12 2 2304.1.o.c 4
144.w odd 12 2 2304.1.o.c 4
168.i even 2 1 3528.1.cg.a 2
168.s odd 6 1 3528.1.ba.b 2
168.s odd 6 1 3528.1.ce.a 2
168.ba even 6 1 3528.1.ba.a 2
168.ba even 6 1 3528.1.ce.b 2
180.n even 6 1 1800.1.bk.d 2
180.v odd 12 2 1800.1.ba.b 4
252.o even 6 1 3528.1.ce.a 2
252.r odd 6 1 3528.1.ba.a 2
252.s odd 6 1 3528.1.cg.a 2
252.bb even 6 1 3528.1.ba.b 2
252.bn odd 6 1 3528.1.ce.b 2
360.bh odd 6 1 1800.1.bk.d 2
360.br even 12 2 1800.1.ba.b 4
504.y even 6 1 3528.1.ce.b 2
504.bi odd 6 1 3528.1.ba.b 2
504.ca even 6 1 3528.1.ba.a 2
504.cc even 6 1 3528.1.cg.a 2
504.db odd 6 1 3528.1.ce.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 12.b even 2 1
72.1.p.a 2 24.h odd 2 1
72.1.p.a 2 36.h even 6 1
72.1.p.a 2 72.j odd 6 1
216.1.p.a 2 4.b odd 2 1
216.1.p.a 2 8.b even 2 1
216.1.p.a 2 36.f odd 6 1
216.1.p.a 2 72.n even 6 1
288.1.t.a 2 3.b odd 2 1
288.1.t.a 2 9.d odd 6 1
288.1.t.a 2 24.f even 2 1
288.1.t.a 2 72.l even 6 1
648.1.b.a 1 36.f odd 6 1
648.1.b.a 1 72.n even 6 1
648.1.b.b 1 36.h even 6 1
648.1.b.b 1 72.j odd 6 1
864.1.t.a 2 1.a even 1 1 trivial
864.1.t.a 2 8.d odd 2 1 CM
864.1.t.a 2 9.c even 3 1 inner
864.1.t.a 2 72.p odd 6 1 inner
1800.1.ba.b 4 60.l odd 4 2
1800.1.ba.b 4 120.w even 4 2
1800.1.ba.b 4 180.v odd 12 2
1800.1.ba.b 4 360.br even 12 2
1800.1.bk.d 2 60.h even 2 1
1800.1.bk.d 2 120.i odd 2 1
1800.1.bk.d 2 180.n even 6 1
1800.1.bk.d 2 360.bh odd 6 1
2304.1.o.c 4 48.i odd 4 2
2304.1.o.c 4 48.k even 4 2
2304.1.o.c 4 144.u even 12 2
2304.1.o.c 4 144.w odd 12 2
2592.1.b.a 1 9.c even 3 1
2592.1.b.a 1 72.p odd 6 1
2592.1.b.b 1 9.d odd 6 1
2592.1.b.b 1 72.l even 6 1
3528.1.ba.a 2 84.j odd 6 1
3528.1.ba.a 2 168.ba even 6 1
3528.1.ba.a 2 252.r odd 6 1
3528.1.ba.a 2 504.ca even 6 1
3528.1.ba.b 2 84.n even 6 1
3528.1.ba.b 2 168.s odd 6 1
3528.1.ba.b 2 252.bb even 6 1
3528.1.ba.b 2 504.bi odd 6 1
3528.1.ce.a 2 84.n even 6 1
3528.1.ce.a 2 168.s odd 6 1
3528.1.ce.a 2 252.o even 6 1
3528.1.ce.a 2 504.db odd 6 1
3528.1.ce.b 2 84.j odd 6 1
3528.1.ce.b 2 168.ba even 6 1
3528.1.ce.b 2 252.bn odd 6 1
3528.1.ce.b 2 504.y even 6 1
3528.1.cg.a 2 84.h odd 2 1
3528.1.cg.a 2 168.i even 2 1
3528.1.cg.a 2 252.s odd 6 1
3528.1.cg.a 2 504.cc even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(864, [\chi])\).

Hecke characteristic polynomials

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