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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [148,1,Mod(7,148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(148, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 16]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("148.7");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 148 = 2^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 148.p (of order \(18\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.0738616218697\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 7.1
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

$p$ $F_p(T)$
$2$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{6} \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} - 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( (T + 1)^{6} \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} - 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
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