Properties

Label 148.1.i.a
Level 148
Weight 1
Character orbit 148.i
Analytic conductor 0.074
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM disc. -4
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 148 = 2^{2} \cdot 37 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 148.i (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0738616218697\)
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 3.1.5476.1
Artin image size \(18\)
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.87616.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \zeta_{6}^{2} q^{2} \) \( -\zeta_{6} q^{4} \) \( + \zeta_{6} q^{5} \) \(+ q^{8}\) \( + \zeta_{6}^{2} q^{9} \) \(+O(q^{10})\) \( q\) \( + \zeta_{6}^{2} q^{2} \) \( -\zeta_{6} q^{4} \) \( + \zeta_{6} q^{5} \) \(+ q^{8}\) \( + \zeta_{6}^{2} q^{9} \) \(- q^{10}\) \( -2 \zeta_{6} q^{13} \) \( + \zeta_{6}^{2} q^{16} \) \( -\zeta_{6}^{2} q^{17} \) \( -\zeta_{6} q^{18} \) \( -\zeta_{6}^{2} q^{20} \) \( + 2 q^{26} \) \(- q^{29}\) \( -\zeta_{6} q^{32} \) \( + \zeta_{6} q^{34} \) \(+ q^{36}\) \( -\zeta_{6} q^{37} \) \( + \zeta_{6} q^{40} \) \( + \zeta_{6} q^{41} \) \(- q^{45}\) \( + \zeta_{6}^{2} q^{49} \) \( + 2 \zeta_{6}^{2} q^{52} \) \( + 2 \zeta_{6}^{2} q^{53} \) \( -\zeta_{6}^{2} q^{58} \) \( + \zeta_{6} q^{61} \) \(+ q^{64}\) \( -2 \zeta_{6}^{2} q^{65} \) \(- q^{68}\) \( + \zeta_{6}^{2} q^{72} \) \( + 2 q^{73} \) \(+ q^{74}\) \(- q^{80}\) \( -\zeta_{6} q^{81} \) \(- q^{82}\) \(+ q^{85}\) \( -\zeta_{6}^{2} q^{89} \) \( -\zeta_{6}^{2} q^{90} \) \(- q^{97}\) \( -\zeta_{6} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut -\mathstrut 2q^{10} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut +\mathstrut q^{17} \) \(\mathstrut -\mathstrut q^{18} \) \(\mathstrut +\mathstrut q^{20} \) \(\mathstrut +\mathstrut 4q^{26} \) \(\mathstrut -\mathstrut 2q^{29} \) \(\mathstrut -\mathstrut q^{32} \) \(\mathstrut +\mathstrut q^{34} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut q^{37} \) \(\mathstrut +\mathstrut q^{40} \) \(\mathstrut +\mathstrut q^{41} \) \(\mathstrut -\mathstrut 2q^{45} \) \(\mathstrut -\mathstrut q^{49} \) \(\mathstrut -\mathstrut 2q^{52} \) \(\mathstrut -\mathstrut 2q^{53} \) \(\mathstrut +\mathstrut q^{58} \) \(\mathstrut +\mathstrut q^{61} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut -\mathstrut 2q^{68} \) \(\mathstrut -\mathstrut q^{72} \) \(\mathstrut +\mathstrut 4q^{73} \) \(\mathstrut +\mathstrut 2q^{74} \) \(\mathstrut -\mathstrut 2q^{80} \) \(\mathstrut -\mathstrut q^{81} \) \(\mathstrut -\mathstrut 2q^{82} \) \(\mathstrut +\mathstrut 2q^{85} \) \(\mathstrut +\mathstrut q^{89} \) \(\mathstrut +\mathstrut q^{90} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut -\mathstrut q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/148\mathbb{Z}\right)^\times\).

\(n\) \(75\) \(113\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 + 0.866025i 0 0 1.00000 −0.500000 + 0.866025i −1.00000
63.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 0.866025i 0 0 1.00000 −0.500000 0.866025i −1.00000
\(n\): 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
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes
37.c Even 1 yes
148.i Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(148, [\chi])\).