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 discriminant -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.i (of order \(6\), degree \(2\), minimal)

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: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.5476.1
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.

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
37.c even 3 1 inner
148.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 148.1.i.a 2
3.b odd 2 1 1332.1.y.a 2
4.b odd 2 1 CM 148.1.i.a 2
5.b even 2 1 3700.1.z.a 2
5.c odd 4 2 3700.1.x.a 4
8.b even 2 1 2368.1.z.a 2
8.d odd 2 1 2368.1.z.a 2
12.b even 2 1 1332.1.y.a 2
20.d odd 2 1 3700.1.z.a 2
20.e even 4 2 3700.1.x.a 4
37.c even 3 1 inner 148.1.i.a 2
111.i odd 6 1 1332.1.y.a 2
148.i odd 6 1 inner 148.1.i.a 2
185.n even 6 1 3700.1.z.a 2
185.s odd 12 2 3700.1.x.a 4
296.p odd 6 1 2368.1.z.a 2
296.s even 6 1 2368.1.z.a 2
444.t even 6 1 1332.1.y.a 2
740.w odd 6 1 3700.1.z.a 2
740.bg even 12 2 3700.1.x.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.1.i.a 2 1.a even 1 1 trivial
148.1.i.a 2 4.b odd 2 1 CM
148.1.i.a 2 37.c even 3 1 inner
148.1.i.a 2 148.i odd 6 1 inner
1332.1.y.a 2 3.b odd 2 1
1332.1.y.a 2 12.b even 2 1
1332.1.y.a 2 111.i odd 6 1
1332.1.y.a 2 444.t even 6 1
2368.1.z.a 2 8.b even 2 1
2368.1.z.a 2 8.d odd 2 1
2368.1.z.a 2 296.p odd 6 1
2368.1.z.a 2 296.s even 6 1
3700.1.x.a 4 5.c odd 4 2
3700.1.x.a 4 20.e even 4 2
3700.1.x.a 4 185.s odd 12 2
3700.1.x.a 4 740.bg even 12 2
3700.1.z.a 2 5.b even 2 1
3700.1.z.a 2 20.d odd 2 1
3700.1.z.a 2 185.n even 6 1
3700.1.z.a 2 740.w odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
$5$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
$7$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
$11$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$13$ \( ( 1 + T + T^{2} )^{2} \)
$17$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
$19$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
$23$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$29$ \( ( 1 + T + T^{2} )^{2} \)
$31$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$37$ \( 1 + T + T^{2} \)
$41$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
$43$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$47$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$53$ \( ( 1 + T + T^{2} )^{2} \)
$59$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
$61$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
$67$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
$71$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
$73$ \( ( 1 - T )^{4} \)
$79$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
$83$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
$89$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
$97$ \( ( 1 + T + T^{2} )^{2} \)
show more
show less