↑

Properties

Label 148.2.e.a
Level $148$
Weight $2$
Character orbit 148.e
Analytic conductor $1.182$
Analytic rank $0$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [148,2,Mod(121,148)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("148.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(148, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 148 = 2^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 148.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.18178594991\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.27870912.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{4} q^{5} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{7} + (\beta_{5} + \beta_{4} + \beta_{2} + \cdots - 1) q^{9} + \beta_{3} q^{11} + (\beta_{5} - 2 \beta_{4} + \cdots - \beta_1) q^{13}+ \cdots + (\beta_{5} - 8 \beta_{4} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{3} + 3 q^{5} - q^{7} - 4 q^{9} - 7 q^{13} + q^{15} + 3 q^{17} + 7 q^{19} - 9 q^{21} - 8 q^{23} + 12 q^{25} + 14 q^{27} - 4 q^{33} + q^{35} + 4 q^{37} - 19 q^{39} - 17 q^{41} - 16 q^{43} - 8 q^{45}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 7\nu^{4} + 49\nu^{3} - 38\nu^{2} + 12\nu - 84 ) / 254 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{5} + 49\nu^{4} - 89\nu^{3} + 266\nu^{2} - 84\nu + 1096 ) / 254 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\nu^{5} - 20\nu^{4} + 140\nu^{3} + 91\nu^{2} + 760\nu + 14 ) / 254 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -85\nu^{5} + 87\nu^{4} - 609\nu^{3} - 72\nu^{2} - 3306\nu + 1044 ) / 254 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 4\beta_{4} + \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 7\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{5} - 26\beta_{4} + 7\beta_{3} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{5} - 30\beta_{4} - 51\beta_{2} - 51\beta _1 + 30 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/148\mathbb{Z}\right)^\times\).

\(n\) \(75\) \(113\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
1.42789 + 2.47317i
0.160819 + 0.278546i
−1.08870 − 1.88569i
1.42789 − 2.47317i
0.160819 − 0.278546i
−1.08870 + 1.88569i
0 −1.42789 − 2.47317i 0 0.500000 + 0.866025i 0 −2.07772 − 3.59871i 0 −2.57772 + 4.46474i 0
121.2 0 −0.160819 − 0.278546i 0 0.500000 + 0.866025i 0 1.94827 + 3.37451i 0 1.44827 − 2.50849i 0
121.3 0 1.08870 + 1.88569i 0 0.500000 + 0.866025i 0 −0.370556 − 0.641823i 0 −0.870556 + 1.50785i 0
137.1 0 −1.42789 + 2.47317i 0 0.500000 − 0.866025i 0 −2.07772 + 3.59871i 0 −2.57772 − 4.46474i 0
137.2 0 −0.160819 + 0.278546i 0 0.500000 − 0.866025i 0 1.94827 − 3.37451i 0 1.44827 + 2.50849i 0
137.3 0 1.08870 − 1.88569i 0 0.500000 − 0.866025i 0 −0.370556 + 0.641823i 0 −0.870556 − 1.50785i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 121.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 148.2.e.a ✓ 6
3.b odd 2 1 1332.2.j.e 6
4.b odd 2 1 592.2.i.f 6
37.c even 3 1 inner 148.2.e.a ✓ 6
37.c even 3 1 5476.2.a.f 3
37.e even 6 1 5476.2.a.g 3
111.i odd 6 1 1332.2.j.e 6
148.i odd 6 1 592.2.i.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.2.e.a ✓ 6 1.a even 1 1 trivial
148.2.e.a ✓ 6 37.c even 3 1 inner
592.2.i.f 6 4.b odd 2 1
592.2.i.f 6 148.i odd 6 1
1332.2.j.e 6 3.b odd 2 1
1332.2.j.e 6 111.i odd 6 1
5476.2.a.f 3 37.c even 3 1
5476.2.a.g 3 37.e even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(148, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} + 7 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + T^{5} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( (T^{3} - 14 T - 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 7 T^{5} + \cdots + 5184 \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{5} + \cdots + 33489 \) Copy content Toggle raw display
$19$ \( T^{6} - 7 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$23$ \( (T^{3} + 4 T^{2} - 26 T - 96)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 59 T - 174)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 14 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 4 T^{5} + \cdots + 50653 \) Copy content Toggle raw display
$41$ \( T^{6} + 17 T^{5} + \cdots + 1521 \) Copy content Toggle raw display
$43$ \( (T^{3} + 8 T^{2} + \cdots - 208)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} + 8 T^{2} + \cdots - 592)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 19 T^{5} + \cdots + 20736 \) Copy content Toggle raw display
$59$ \( T^{6} - 9 T^{5} + \cdots + 246016 \) Copy content Toggle raw display
$61$ \( T^{6} - 7 T^{5} + \cdots + 2209 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} + \cdots + 887364 \) Copy content Toggle raw display
$71$ \( T^{6} - 15 T^{5} + \cdots + 318096 \) Copy content Toggle raw display
$73$ \( (T^{3} - 22 T^{2} + \cdots - 208)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 15 T^{5} + \cdots + 26244 \) Copy content Toggle raw display
$83$ \( T^{6} + 15 T^{5} + \cdots + 1249924 \) Copy content Toggle raw display
$89$ \( T^{6} + 17 T^{5} + \cdots + 157609 \) Copy content Toggle raw display
$97$ \( (T^{3} + 4 T^{2} + \cdots + 474)^{2} \) Copy content Toggle raw display
show more
show less