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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This is the first weight $1$ newform with projective image $S_4$.

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.0738616218697\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 Parity Ord Mult Type
1.a even 1 1 trivial
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 148.1.f.a 2
3.b odd 2 1 1332.1.o.a 2
4.b odd 2 1 592.1.k.b 2
5.b even 2 1 3700.1.j.c 2
5.c odd 4 1 3700.1.t.a 2
5.c odd 4 1 3700.1.t.b 2
8.b even 2 1 2368.1.k.a 2
8.d odd 2 1 2368.1.k.b 2
37.d odd 4 1 inner 148.1.f.a 2
111.g even 4 1 1332.1.o.a 2
148.g even 4 1 592.1.k.b 2
185.f even 4 1 3700.1.t.a 2
185.j odd 4 1 3700.1.j.c 2
185.k even 4 1 3700.1.t.b 2
296.j even 4 1 2368.1.k.b 2
296.m odd 4 1 2368.1.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.1.f.a 2 1.a even 1 1 trivial
148.1.f.a 2 37.d odd 4 1 inner
592.1.k.b 2 4.b odd 2 1
592.1.k.b 2 148.g even 4 1
1332.1.o.a 2 3.b odd 2 1
1332.1.o.a 2 111.g even 4 1
2368.1.k.a 2 8.b even 2 1
2368.1.k.a 2 296.m odd 4 1
2368.1.k.b 2 8.d odd 2 1
2368.1.k.b 2 296.j even 4 1
3700.1.j.c 2 5.b even 2 1
3700.1.j.c 2 185.j odd 4 1
3700.1.t.a 2 5.c odd 4 1
3700.1.t.a 2 185.f even 4 1
3700.1.t.b 2 5.c odd 4 1
3700.1.t.b 2 185.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(148, [\chi])\).

Hecke characteristic polynomials

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