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 disc. -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 \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0359326809096\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.648.1
Artin image size \(18\)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes
9.c Even 1 yes
72.p Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(72, [\chi])\).