Properties

Label 148.1.f.a
Level 148
Weight 1
Character orbit 148.f
Analytic conductor 0.074
Analytic rank 0
Dimension 2
Projective image \(S_{4}\)
CM/RM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 148 = 2^{2} \cdot 37 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 148.f (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.0738616218697\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(S_{4}\)
Projective field Galois closure of 4.0.202612.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + i q^{3} - q^{7} +O(q^{10})\) \( q + i q^{3} - q^{7} -i q^{11} + ( -1 - i ) q^{17} + ( 1 + i ) q^{19} -i q^{21} + ( -1 - i ) q^{23} -i q^{25} + i q^{27} + ( -1 + i ) q^{29} + q^{33} + i q^{37} + i q^{41} + q^{47} + ( 1 - i ) q^{51} + q^{53} + ( -1 + i ) q^{57} + ( 1 - i ) q^{69} + q^{71} + i q^{73} + q^{75} + i q^{77} + ( -1 - i ) q^{79} - q^{81} - q^{83} + ( -1 - i ) q^{87} + ( -1 + i ) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} + O(q^{10}) \) \( 2q - 2q^{7} - 2q^{17} + 2q^{19} - 2q^{23} - 2q^{29} + 2q^{33} + 2q^{47} + 2q^{51} + 2q^{53} - 2q^{57} + 2q^{69} + 2q^{71} + 2q^{75} - 2q^{79} - 2q^{81} - 2q^{83} - 2q^{87} - 2q^{89} + O(q^{100}) \)

Character Values

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

\(n\) \(75\) \(113\)
\(\chi(n)\) \(1\) \(-i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
105.1
1.00000i
1.00000i
0 1.00000i 0 0 0 −1.00000 0 0 0
117.1 0 1.00000i 0 0 0 −1.00000 0 0 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
37.d Odd 1 yes

Hecke kernels

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