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 discriminant -4
Inner twists 4

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Newspace parameters

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

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: \( 1 \)
Twist minimal: yes
Projective image \(D_{9}\)
Projective field Galois closure of 9.1.899194740203776.1

$q$-expansion

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
37.f even 9 1 inner
148.p odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 148.1.p.a 6
3.b odd 2 1 1332.1.cz.a 6
4.b odd 2 1 CM 148.1.p.a 6
5.b even 2 1 3700.1.ce.a 6
5.c odd 4 2 3700.1.cb.a 12
8.b even 2 1 2368.1.ck.a 6
8.d odd 2 1 2368.1.ck.a 6
12.b even 2 1 1332.1.cz.a 6
20.d odd 2 1 3700.1.ce.a 6
20.e even 4 2 3700.1.cb.a 12
37.f even 9 1 inner 148.1.p.a 6
111.p odd 18 1 1332.1.cz.a 6
148.p odd 18 1 inner 148.1.p.a 6
185.x even 18 1 3700.1.ce.a 6
185.bd odd 36 2 3700.1.cb.a 12
296.bb odd 18 1 2368.1.ck.a 6
296.bc even 18 1 2368.1.ck.a 6
444.z even 18 1 1332.1.cz.a 6
740.bs odd 18 1 3700.1.ce.a 6
740.cf even 36 2 3700.1.cb.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.1.p.a 6 1.a even 1 1 trivial
148.1.p.a 6 4.b odd 2 1 CM
148.1.p.a 6 37.f even 9 1 inner
148.1.p.a 6 148.p odd 18 1 inner
1332.1.cz.a 6 3.b odd 2 1
1332.1.cz.a 6 12.b even 2 1
1332.1.cz.a 6 111.p odd 18 1
1332.1.cz.a 6 444.z even 18 1
2368.1.ck.a 6 8.b even 2 1
2368.1.ck.a 6 8.d odd 2 1
2368.1.ck.a 6 296.bb odd 18 1
2368.1.ck.a 6 296.bc even 18 1
3700.1.cb.a 12 5.c odd 4 2
3700.1.cb.a 12 20.e even 4 2
3700.1.cb.a 12 185.bd odd 36 2
3700.1.cb.a 12 740.cf even 36 2
3700.1.ce.a 6 5.b even 2 1
3700.1.ce.a 6 20.d odd 2 1
3700.1.ce.a 6 185.x even 18 1
3700.1.ce.a 6 740.bs odd 18 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{3} + T^{6} \)
$3$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$5$ \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
$7$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$11$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$13$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$17$ \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
$19$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$23$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$29$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$31$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$37$ \( 1 + T^{3} + T^{6} \)
$41$ \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \)
$43$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$47$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$53$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$59$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$61$ \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
$67$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$71$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$73$ \( ( 1 + T + T^{2} )^{6} \)
$79$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$83$ \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \)
$89$ \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \)
$97$ \( ( 1 + T^{3} + T^{6} )^{2} \)
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