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Properties

Label 133.2.o.d
Level $133$
Weight $2$
Character orbit 133.o
Analytic conductor $1.062$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [133,2,Mod(75,133)] 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("133.75"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(133, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 133 = 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 133.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.06201034688\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.6967728.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{3} - \beta_{3} q^{4} + ( - \beta_{5} - 1) q^{5} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{6} + ( - \beta_{4} - \beta_{3} - \beta_1) q^{7}+ \cdots + ( - 3 \beta_{5} + 3 \beta_{4} + \cdots - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{5} + 5 q^{7} - 7 q^{9} + 3 q^{10} + 5 q^{11} - 2 q^{12} + 8 q^{13} - 3 q^{14} + 15 q^{16} - 12 q^{17} + 21 q^{18} - 7 q^{19} + 14 q^{21} - q^{23} - 6 q^{24} + 2 q^{25}+ \cdots - 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 8\nu^{4} + 64\nu^{3} - 50\nu^{2} + 7\nu - 56 ) / 393 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 56\nu^{5} - 55\nu^{4} + 440\nu^{3} + 344\nu^{2} + 2750\nu - 385 ) / 393 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 70\nu^{5} - 36\nu^{4} + 550\nu^{3} + 561\nu^{2} + 3634\nu + 534 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 77\nu^{5} - 92\nu^{4} + 605\nu^{3} + 211\nu^{2} + 3683\nu - 1430 ) / 393 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{5} - \beta_{4} + 4\beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + 7\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{5} + 16\beta_{4} - 31\beta_{3} - 6\beta _1 - 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 28\beta_{5} + 14\beta_{4} - 48\beta_{3} - 55\beta_{2} - 55\beta _1 + 28 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/133\mathbb{Z}\right)^\times\).

\(n\) \(78\) \(115\)
\(\chi(n)\) \(-1\) \(1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
75.1
1.56632 − 2.71294i
0.0702177 − 0.121621i
−1.13654 + 1.96854i
1.56632 + 2.71294i
0.0702177 + 0.121621i
−1.13654 − 1.96854i
−1.50000 − 0.866025i −1.06632 − 1.84692i 0.500000 + 0.866025i −2.90671 − 1.67819i 3.69384i 1.84039 + 1.90078i 1.73205i −0.774071 + 1.34073i 2.90671 + 5.03457i
75.2 −1.50000 − 0.866025i 0.429782 + 0.744405i 0.500000 + 0.866025i 1.99014 + 1.14901i − 1.48881i −1.56036 + 2.13665i 1.73205i 1.13057 − 1.95821i −1.99014 − 3.44702i
75.3 −1.50000 − 0.866025i 1.63654 + 2.83456i 0.500000 + 0.866025i −0.583430 − 0.336844i − 5.66913i 2.21997 − 1.43936i 1.73205i −3.85650 + 6.67966i 0.583430 + 1.01053i
94.1 −1.50000 + 0.866025i −1.06632 + 1.84692i 0.500000 − 0.866025i −2.90671 + 1.67819i − 3.69384i 1.84039 − 1.90078i − 1.73205i −0.774071 − 1.34073i 2.90671 − 5.03457i
94.2 −1.50000 + 0.866025i 0.429782 − 0.744405i 0.500000 − 0.866025i 1.99014 − 1.14901i 1.48881i −1.56036 − 2.13665i − 1.73205i 1.13057 + 1.95821i −1.99014 + 3.44702i
94.3 −1.50000 + 0.866025i 1.63654 − 2.83456i 0.500000 − 0.866025i −0.583430 + 0.336844i 5.66913i 2.21997 + 1.43936i − 1.73205i −3.85650 − 6.67966i 0.583430 − 1.01053i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 75.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 133.2.o.d ✓ 6
7.b odd 2 1 931.2.o.e 6
7.c even 3 1 931.2.c.d 6
7.c even 3 1 931.2.o.f 6
7.d odd 6 1 133.2.o.e yes 6
7.d odd 6 1 931.2.c.e 6
19.b odd 2 1 133.2.o.e yes 6
133.c even 2 1 931.2.o.f 6
133.o even 6 1 inner 133.2.o.d ✓ 6
133.o even 6 1 931.2.c.d 6
133.r odd 6 1 931.2.c.e 6
133.r odd 6 1 931.2.o.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.o.d ✓ 6 1.a even 1 1 trivial
133.2.o.d ✓ 6 133.o even 6 1 inner
133.2.o.e yes 6 7.d odd 6 1
133.2.o.e yes 6 19.b odd 2 1
931.2.c.d 6 7.c even 3 1
931.2.c.d 6 133.o even 6 1
931.2.c.e 6 7.d odd 6 1
931.2.c.e 6 133.r odd 6 1
931.2.o.e 6 7.b odd 2 1
931.2.o.e 6 133.r odd 6 1
931.2.o.f 6 7.c even 3 1
931.2.o.f 6 133.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(133, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 3 T + 3)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} - 2 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + \cdots + 27 \) Copy content Toggle raw display
$7$ \( T^{6} - 5 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} - 5 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T^{3} - 4 T^{2} - 12 T - 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 12 T^{5} + \cdots + 48 \) Copy content Toggle raw display
$19$ \( T^{6} + 7 T^{5} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + T^{5} + \cdots + 441 \) Copy content Toggle raw display
$29$ \( T^{6} + 108 T^{4} + \cdots + 38988 \) Copy content Toggle raw display
$31$ \( T^{6} + 16 T^{5} + \cdots + 12996 \) Copy content Toggle raw display
$37$ \( T^{6} + 12 T^{5} + \cdots + 972 \) Copy content Toggle raw display
$41$ \( (T^{3} - 6 T^{2} + \cdots + 216)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 5 T^{2} + \cdots + 167)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 27 T^{5} + \cdots + 6627 \) Copy content Toggle raw display
$53$ \( T^{6} - 6 T^{5} + \cdots + 428652 \) Copy content Toggle raw display
$59$ \( (T^{2} + 6 T + 36)^{3} \) Copy content Toggle raw display
$61$ \( T^{6} - 15 T^{5} + \cdots + 3 \) Copy content Toggle raw display
$67$ \( T^{6} - 6 T^{5} + \cdots + 1728 \) Copy content Toggle raw display
$71$ \( T^{6} + 108 T^{4} + \cdots + 108 \) Copy content Toggle raw display
$73$ \( T^{6} + 3 T^{5} + \cdots + 11907 \) Copy content Toggle raw display
$79$ \( T^{6} - 198 T^{4} + \cdots + 38988 \) Copy content Toggle raw display
$83$ \( T^{6} + 341 T^{4} + \cdots + 1232643 \) Copy content Toggle raw display
$89$ \( T^{6} + 48 T^{4} + \cdots + 15876 \) Copy content Toggle raw display
$97$ \( (T^{3} + 4 T^{2} + \cdots - 288)^{2} \) Copy content Toggle raw display
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