Properties

Label 148.1.p.a
Level 148
Weight 1
Character orbit 148.p
Analytic conductor 0.074
Analytic rank 0
Dimension 6
Projective image \(D_{9}\)
CM disc. -4
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 148 = 2^{2} \cdot 37 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 148.p (of order \(18\) and degree \(6\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0738616218697\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{9}\)
Projective field Galois closure of 9.1.899194740203776.1

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\zeta_{18} q^{2} \) \( + \zeta_{18}^{2} q^{4} \) \( + ( \zeta_{18}^{4} + \zeta_{18}^{6} ) q^{5} \) \( -\zeta_{18}^{3} q^{8} \) \( -\zeta_{18}^{7} q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{18} q^{2} \) \( + \zeta_{18}^{2} q^{4} \) \( + ( \zeta_{18}^{4} + \zeta_{18}^{6} ) q^{5} \) \( -\zeta_{18}^{3} q^{8} \) \( -\zeta_{18}^{7} q^{9} \) \( + ( -\zeta_{18}^{5} - \zeta_{18}^{7} ) q^{10} \) \( -\zeta_{18}^{2} q^{13} \) \( + \zeta_{18}^{4} q^{16} \) \( + ( \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{17} \) \( + \zeta_{18}^{8} q^{18} \) \( + ( \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{20} \) \( + ( -\zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{8} ) q^{25} \) \( + \zeta_{18}^{3} q^{26} \) \( + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{29} \) \( -\zeta_{18}^{5} q^{32} \) \( + ( -\zeta_{18}^{3} + \zeta_{18}^{4} ) q^{34} \) \(+ q^{36}\) \( + \zeta_{18}^{2} q^{37} \) \( + ( 1 - \zeta_{18}^{7} ) q^{40} \) \( + ( \zeta_{18}^{6} - \zeta_{18}^{7} ) q^{41} \) \( + ( \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{45} \) \( -\zeta_{18} q^{49} \) \( + ( 1 + \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{50} \) \( -\zeta_{18}^{4} q^{52} \) \( -\zeta_{18}^{4} q^{53} \) \( + ( \zeta_{18}^{2} + \zeta_{18}^{6} ) q^{58} \) \( + ( 1 + \zeta_{18}^{4} ) q^{61} \) \( + \zeta_{18}^{6} q^{64} \) \( + ( -\zeta_{18}^{6} - \zeta_{18}^{8} ) q^{65} \) \( + ( \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{68} \) \( -\zeta_{18} q^{72} \) \(- q^{73}\) \( -\zeta_{18}^{3} q^{74} \) \( + ( -\zeta_{18} + \zeta_{18}^{8} ) q^{80} \) \( -\zeta_{18}^{5} q^{81} \) \( + ( -\zeta_{18}^{7} + \zeta_{18}^{8} ) q^{82} \) \( + ( 1 + \zeta_{18}^{6} - \zeta_{18}^{7} + \zeta_{18}^{8} ) q^{85} \) \( + ( 1 + \zeta_{18}^{8} ) q^{89} \) \( + ( -\zeta_{18}^{3} - \zeta_{18}^{5} ) q^{90} \) \( + ( \zeta_{18}^{4} + \zeta_{18}^{8} ) q^{97} \) \( + \zeta_{18}^{2} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut -\mathstrut 3q^{17} \) \(\mathstrut -\mathstrut 3q^{20} \) \(\mathstrut -\mathstrut 3q^{25} \) \(\mathstrut +\mathstrut 3q^{26} \) \(\mathstrut -\mathstrut 3q^{34} \) \(\mathstrut +\mathstrut 6q^{36} \) \(\mathstrut +\mathstrut 6q^{40} \) \(\mathstrut -\mathstrut 3q^{41} \) \(\mathstrut +\mathstrut 6q^{50} \) \(\mathstrut -\mathstrut 3q^{58} \) \(\mathstrut +\mathstrut 6q^{61} \) \(\mathstrut -\mathstrut 3q^{64} \) \(\mathstrut +\mathstrut 3q^{65} \) \(\mathstrut -\mathstrut 6q^{73} \) \(\mathstrut -\mathstrut 3q^{74} \) \(\mathstrut +\mathstrut 3q^{85} \) \(\mathstrut +\mathstrut 6q^{89} \) \(\mathstrut -\mathstrut 3q^{90} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(75\) \(113\)
\(\chi(n)\) \(-1\) \(\zeta_{18}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.939693 0.342020i
−0.766044 0.642788i
−0.173648 0.984808i
−0.173648 + 0.984808i
−0.766044 + 0.642788i
0.939693 + 0.342020i
−0.939693 + 0.342020i 0 0.766044 0.642788i −0.326352 1.85083i 0 0 −0.500000 + 0.866025i 0.766044 + 0.642788i 0.939693 + 1.62760i
71.1 0.766044 + 0.642788i 0 0.173648 + 0.984808i −1.43969 0.524005i 0 0 −0.500000 + 0.866025i 0.173648 0.984808i −0.766044 1.32683i
83.1 0.173648 + 0.984808i 0 −0.939693 + 0.342020i 0.266044 + 0.223238i 0 0 −0.500000 0.866025i −0.939693 0.342020i −0.173648 + 0.300767i
107.1 0.173648 0.984808i 0 −0.939693 0.342020i 0.266044 0.223238i 0 0 −0.500000 + 0.866025i −0.939693 + 0.342020i −0.173648 0.300767i
123.1 0.766044 0.642788i 0 0.173648 0.984808i −1.43969 + 0.524005i 0 0 −0.500000 0.866025i 0.173648 + 0.984808i −0.766044 + 1.32683i
127.1 −0.939693 0.342020i 0 0.766044 + 0.642788i −0.326352 + 1.85083i 0 0 −0.500000 0.866025i 0.766044 0.642788i 0.939693 1.62760i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.1
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.f Even 1 yes
148.p Odd 1 yes

Hecke kernels

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