Properties

Label 72.1.p.a
Level 72
Weight 1
Character orbit 72.p
Analytic conductor 0.036
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM discriminant -8
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 72.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0359326809096\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.648.1
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.41472.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} + \zeta_{6}^{2} q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{6} + q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{2} + \zeta_{6}^{2} q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{6} + q^{8} -\zeta_{6} q^{9} -\zeta_{6}^{2} q^{11} + q^{12} + \zeta_{6}^{2} q^{16} - q^{17} + q^{18} - q^{19} + \zeta_{6} q^{22} + \zeta_{6}^{2} q^{24} + \zeta_{6}^{2} q^{25} + q^{27} -\zeta_{6} q^{32} + \zeta_{6} q^{33} -\zeta_{6}^{2} q^{34} + \zeta_{6}^{2} q^{36} -\zeta_{6}^{2} q^{38} + \zeta_{6} q^{41} -\zeta_{6}^{2} q^{43} - q^{44} -\zeta_{6} q^{48} -\zeta_{6} q^{49} -\zeta_{6} q^{50} -\zeta_{6}^{2} q^{51} + \zeta_{6}^{2} q^{54} -\zeta_{6}^{2} q^{57} + \zeta_{6} q^{59} + q^{64} - q^{66} + \zeta_{6} q^{67} + \zeta_{6} q^{68} -\zeta_{6} q^{72} - q^{73} -\zeta_{6} q^{75} + \zeta_{6} q^{76} + \zeta_{6}^{2} q^{81} - q^{82} + 2 \zeta_{6}^{2} q^{83} + \zeta_{6} q^{86} -\zeta_{6}^{2} q^{88} + 2 q^{89} + q^{96} -\zeta_{6}^{2} q^{97} + q^{98} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{3} - q^{4} - q^{6} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q - q^{2} - q^{3} - q^{4} - q^{6} + 2q^{8} - q^{9} + q^{11} + 2q^{12} - q^{16} - 2q^{17} + 2q^{18} - 2q^{19} + q^{22} - q^{24} - q^{25} + 2q^{27} - q^{32} + q^{33} + q^{34} - q^{36} + q^{38} + q^{41} + q^{43} - 2q^{44} - q^{48} - q^{49} - q^{50} + q^{51} - q^{54} + q^{57} + q^{59} + 2q^{64} - 2q^{66} + q^{67} + q^{68} - q^{72} - 2q^{73} - q^{75} + q^{76} - q^{81} - 2q^{82} - 2q^{83} + q^{86} + q^{88} + 4q^{89} + 2q^{96} + q^{97} + 2q^{98} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 −0.500000 0.866025i 0 1.00000 −0.500000 0.866025i 0
67.1 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 1.00000 −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 72.1.p.a 2
3.b odd 2 1 216.1.p.a 2
4.b odd 2 1 288.1.t.a 2
5.b even 2 1 1800.1.bk.d 2
5.c odd 4 2 1800.1.ba.b 4
7.b odd 2 1 3528.1.cg.a 2
7.c even 3 1 3528.1.ba.b 2
7.c even 3 1 3528.1.ce.a 2
7.d odd 6 1 3528.1.ba.a 2
7.d odd 6 1 3528.1.ce.b 2
8.b even 2 1 288.1.t.a 2
8.d odd 2 1 CM 72.1.p.a 2
9.c even 3 1 inner 72.1.p.a 2
9.c even 3 1 648.1.b.b 1
9.d odd 6 1 216.1.p.a 2
9.d odd 6 1 648.1.b.a 1
12.b even 2 1 864.1.t.a 2
16.e even 4 2 2304.1.o.c 4
16.f odd 4 2 2304.1.o.c 4
24.f even 2 1 216.1.p.a 2
24.h odd 2 1 864.1.t.a 2
36.f odd 6 1 288.1.t.a 2
36.f odd 6 1 2592.1.b.b 1
36.h even 6 1 864.1.t.a 2
36.h even 6 1 2592.1.b.a 1
40.e odd 2 1 1800.1.bk.d 2
40.k even 4 2 1800.1.ba.b 4
45.j even 6 1 1800.1.bk.d 2
45.k odd 12 2 1800.1.ba.b 4
56.e even 2 1 3528.1.cg.a 2
56.k odd 6 1 3528.1.ba.b 2
56.k odd 6 1 3528.1.ce.a 2
56.m even 6 1 3528.1.ba.a 2
56.m even 6 1 3528.1.ce.b 2
63.g even 3 1 3528.1.ce.a 2
63.h even 3 1 3528.1.ba.b 2
63.k odd 6 1 3528.1.ce.b 2
63.l odd 6 1 3528.1.cg.a 2
63.t odd 6 1 3528.1.ba.a 2
72.j odd 6 1 864.1.t.a 2
72.j odd 6 1 2592.1.b.a 1
72.l even 6 1 216.1.p.a 2
72.l even 6 1 648.1.b.a 1
72.n even 6 1 288.1.t.a 2
72.n even 6 1 2592.1.b.b 1
72.p odd 6 1 inner 72.1.p.a 2
72.p odd 6 1 648.1.b.b 1
144.v odd 12 2 2304.1.o.c 4
144.x even 12 2 2304.1.o.c 4
360.z odd 6 1 1800.1.bk.d 2
360.bo even 12 2 1800.1.ba.b 4
504.ba odd 6 1 3528.1.ce.a 2
504.be even 6 1 3528.1.cg.a 2
504.bf even 6 1 3528.1.ba.a 2
504.ce odd 6 1 3528.1.ba.b 2
504.cz even 6 1 3528.1.ce.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 1.a even 1 1 trivial
72.1.p.a 2 8.d odd 2 1 CM
72.1.p.a 2 9.c even 3 1 inner
72.1.p.a 2 72.p odd 6 1 inner
216.1.p.a 2 3.b odd 2 1
216.1.p.a 2 9.d odd 6 1
216.1.p.a 2 24.f even 2 1
216.1.p.a 2 72.l even 6 1
288.1.t.a 2 4.b odd 2 1
288.1.t.a 2 8.b even 2 1
288.1.t.a 2 36.f odd 6 1
288.1.t.a 2 72.n even 6 1
648.1.b.a 1 9.d odd 6 1
648.1.b.a 1 72.l even 6 1
648.1.b.b 1 9.c even 3 1
648.1.b.b 1 72.p odd 6 1
864.1.t.a 2 12.b even 2 1
864.1.t.a 2 24.h odd 2 1
864.1.t.a 2 36.h even 6 1
864.1.t.a 2 72.j odd 6 1
1800.1.ba.b 4 5.c odd 4 2
1800.1.ba.b 4 40.k even 4 2
1800.1.ba.b 4 45.k odd 12 2
1800.1.ba.b 4 360.bo even 12 2
1800.1.bk.d 2 5.b even 2 1
1800.1.bk.d 2 40.e odd 2 1
1800.1.bk.d 2 45.j even 6 1
1800.1.bk.d 2 360.z odd 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 36.h even 6 1
2592.1.b.a 1 72.j odd 6 1
2592.1.b.b 1 36.f odd 6 1
2592.1.b.b 1 72.n even 6 1
3528.1.ba.a 2 7.d odd 6 1
3528.1.ba.a 2 56.m even 6 1
3528.1.ba.a 2 63.t odd 6 1
3528.1.ba.a 2 504.bf even 6 1
3528.1.ba.b 2 7.c even 3 1
3528.1.ba.b 2 56.k odd 6 1
3528.1.ba.b 2 63.h even 3 1
3528.1.ba.b 2 504.ce odd 6 1
3528.1.ce.a 2 7.c even 3 1
3528.1.ce.a 2 56.k odd 6 1
3528.1.ce.a 2 63.g even 3 1
3528.1.ce.a 2 504.ba odd 6 1
3528.1.ce.b 2 7.d odd 6 1
3528.1.ce.b 2 56.m even 6 1
3528.1.ce.b 2 63.k odd 6 1
3528.1.ce.b 2 504.cz even 6 1
3528.1.cg.a 2 7.b odd 2 1
3528.1.cg.a 2 56.e even 2 1
3528.1.cg.a 2 63.l odd 6 1
3528.1.cg.a 2 504.be even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(72, [\chi])\).

Hecke characteristic polynomials

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