Properties

Label 288.1.t.a
Level $288$
Weight $1$
Character orbit 288.t
Analytic conductor $0.144$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 288.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.143730723638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 12.0.110075314176.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{3} -\zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{3} -\zeta_{6} q^{9} + \zeta_{6}^{2} q^{11} - q^{17} + q^{19} + \zeta_{6}^{2} q^{25} - q^{27} + \zeta_{6} q^{33} + \zeta_{6} q^{41} + \zeta_{6}^{2} q^{43} -\zeta_{6} q^{49} + \zeta_{6}^{2} q^{51} -\zeta_{6}^{2} q^{57} -\zeta_{6} q^{59} -\zeta_{6} q^{67} - q^{73} + \zeta_{6} q^{75} + \zeta_{6}^{2} q^{81} -2 \zeta_{6}^{2} q^{83} + 2 q^{89} -\zeta_{6}^{2} q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - q^{9} - q^{11} - 2q^{17} + 2q^{19} - q^{25} - 2q^{27} + q^{33} + q^{41} - q^{43} - q^{49} - q^{51} + q^{57} - q^{59} - q^{67} - 2q^{73} + q^{75} - q^{81} + 2q^{83} + 4q^{89} + q^{97} + 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 0 0 0 0 −0.500000 0.866025i 0
175.1 0 0.500000 + 0.866025i 0 0 0 0 0 −0.500000 + 0.866025i 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 288.1.t.a 2
3.b odd 2 1 864.1.t.a 2
4.b odd 2 1 72.1.p.a 2
8.b even 2 1 72.1.p.a 2
8.d odd 2 1 CM 288.1.t.a 2
9.c even 3 1 inner 288.1.t.a 2
9.c even 3 1 2592.1.b.b 1
9.d odd 6 1 864.1.t.a 2
9.d odd 6 1 2592.1.b.a 1
12.b even 2 1 216.1.p.a 2
16.e even 4 2 2304.1.o.c 4
16.f odd 4 2 2304.1.o.c 4
20.d odd 2 1 1800.1.bk.d 2
20.e even 4 2 1800.1.ba.b 4
24.f even 2 1 864.1.t.a 2
24.h odd 2 1 216.1.p.a 2
28.d even 2 1 3528.1.cg.a 2
28.f even 6 1 3528.1.ba.a 2
28.f even 6 1 3528.1.ce.b 2
28.g odd 6 1 3528.1.ba.b 2
28.g odd 6 1 3528.1.ce.a 2
36.f odd 6 1 72.1.p.a 2
36.f odd 6 1 648.1.b.b 1
36.h even 6 1 216.1.p.a 2
36.h even 6 1 648.1.b.a 1
40.f even 2 1 1800.1.bk.d 2
40.i odd 4 2 1800.1.ba.b 4
56.h odd 2 1 3528.1.cg.a 2
56.j odd 6 1 3528.1.ba.a 2
56.j odd 6 1 3528.1.ce.b 2
56.p even 6 1 3528.1.ba.b 2
56.p even 6 1 3528.1.ce.a 2
72.j odd 6 1 216.1.p.a 2
72.j odd 6 1 648.1.b.a 1
72.l even 6 1 864.1.t.a 2
72.l even 6 1 2592.1.b.a 1
72.n even 6 1 72.1.p.a 2
72.n even 6 1 648.1.b.b 1
72.p odd 6 1 inner 288.1.t.a 2
72.p odd 6 1 2592.1.b.b 1
144.v odd 12 2 2304.1.o.c 4
144.x even 12 2 2304.1.o.c 4
180.p odd 6 1 1800.1.bk.d 2
180.x even 12 2 1800.1.ba.b 4
252.n even 6 1 3528.1.ce.b 2
252.u odd 6 1 3528.1.ba.b 2
252.bi even 6 1 3528.1.cg.a 2
252.bj even 6 1 3528.1.ba.a 2
252.bl odd 6 1 3528.1.ce.a 2
360.bk even 6 1 1800.1.bk.d 2
360.bu odd 12 2 1800.1.ba.b 4
504.w even 6 1 3528.1.ce.a 2
504.bn odd 6 1 3528.1.cg.a 2
504.bp odd 6 1 3528.1.ba.a 2
504.cq even 6 1 3528.1.ba.b 2
504.cw odd 6 1 3528.1.ce.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 4.b odd 2 1
72.1.p.a 2 8.b even 2 1
72.1.p.a 2 36.f odd 6 1
72.1.p.a 2 72.n even 6 1
216.1.p.a 2 12.b even 2 1
216.1.p.a 2 24.h odd 2 1
216.1.p.a 2 36.h even 6 1
216.1.p.a 2 72.j odd 6 1
288.1.t.a 2 1.a even 1 1 trivial
288.1.t.a 2 8.d odd 2 1 CM
288.1.t.a 2 9.c even 3 1 inner
288.1.t.a 2 72.p odd 6 1 inner
648.1.b.a 1 36.h even 6 1
648.1.b.a 1 72.j odd 6 1
648.1.b.b 1 36.f odd 6 1
648.1.b.b 1 72.n even 6 1
864.1.t.a 2 3.b odd 2 1
864.1.t.a 2 9.d odd 6 1
864.1.t.a 2 24.f even 2 1
864.1.t.a 2 72.l even 6 1
1800.1.ba.b 4 20.e even 4 2
1800.1.ba.b 4 40.i odd 4 2
1800.1.ba.b 4 180.x even 12 2
1800.1.ba.b 4 360.bu odd 12 2
1800.1.bk.d 2 20.d odd 2 1
1800.1.bk.d 2 40.f even 2 1
1800.1.bk.d 2 180.p odd 6 1
1800.1.bk.d 2 360.bk even 6 1
2304.1.o.c 4 16.e even 4 2
2304.1.o.c 4 16.f odd 4 2
2304.1.o.c 4 144.v odd 12 2
2304.1.o.c 4 144.x even 12 2
2592.1.b.a 1 9.d odd 6 1
2592.1.b.a 1 72.l even 6 1
2592.1.b.b 1 9.c even 3 1
2592.1.b.b 1 72.p odd 6 1
3528.1.ba.a 2 28.f even 6 1
3528.1.ba.a 2 56.j odd 6 1
3528.1.ba.a 2 252.bj even 6 1
3528.1.ba.a 2 504.bp odd 6 1
3528.1.ba.b 2 28.g odd 6 1
3528.1.ba.b 2 56.p even 6 1
3528.1.ba.b 2 252.u odd 6 1
3528.1.ba.b 2 504.cq even 6 1
3528.1.ce.a 2 28.g odd 6 1
3528.1.ce.a 2 56.p even 6 1
3528.1.ce.a 2 252.bl odd 6 1
3528.1.ce.a 2 504.w even 6 1
3528.1.ce.b 2 28.f even 6 1
3528.1.ce.b 2 56.j odd 6 1
3528.1.ce.b 2 252.n even 6 1
3528.1.ce.b 2 504.cw odd 6 1
3528.1.cg.a 2 28.d even 2 1
3528.1.cg.a 2 56.h odd 2 1
3528.1.cg.a 2 252.bi even 6 1
3528.1.cg.a 2 504.bn odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(288, [\chi])\).

Hecke characteristic polynomials

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