# 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 $2$

# Related objects

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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