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Properties

Label 168.3.z.a
Level $168$
Weight $3$
Character orbit 168.z
Analytic conductor $4.578$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,3,Mod(73,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.73");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 168.z (of order \(6\), degree \(2\), 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: \(4.57766844125\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.126303473664.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 9x^{6} + 2x^{5} + 92x^{4} + 14x^{3} - 441x^{2} - 686x + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - 2) q^{3} + (\beta_{6} + 1) q^{5} + ( - \beta_{7} + \beta_{5} - 3 \beta_{4} + 1) q^{7} + ( - 3 \beta_{4} + 3) q^{9} + ( - \beta_{7} + 2 \beta_{5} + 3 \beta_{4} + \cdots + 1) q^{11} + ( - \beta_{7} + 4 \beta_{6} + \cdots - 3 \beta_1) q^{13}+ \cdots + ( - 3 \beta_{7} + 6 \beta_{5} + \cdots + 15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} + 6 q^{5} - 4 q^{7} + 12 q^{9} + 14 q^{11} - 12 q^{15} + 12 q^{17} + 78 q^{19} + 18 q^{21} - 6 q^{25} - 4 q^{29} - 24 q^{31} - 42 q^{33} - 156 q^{35} + 50 q^{37} - 6 q^{39} - 20 q^{43} + 18 q^{45}+ \cdots + 84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 9x^{6} + 2x^{5} + 92x^{4} + 14x^{3} - 441x^{2} - 686x + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -13\nu^{7} + 355\nu^{6} + 96\nu^{5} + 198\nu^{4} - 5242\nu^{3} - 1568\nu^{2} + 1029\nu + 28469 ) / 21952 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -25\nu^{7} + 15\nu^{6} + 736\nu^{5} - 274\nu^{4} - 3938\nu^{3} - 2688\nu^{2} + 18865\nu + 21609 ) / 10976 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -73\nu^{7} + 55\nu^{6} - 288\nu^{5} + 526\nu^{4} - 5330\nu^{3} - 672\nu^{2} + 21217\nu + 57281 ) / 21952 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 99\nu^{7} + 75\nu^{6} - 800\nu^{5} - 1818\nu^{4} + 4950\nu^{3} + 16800\nu^{2} - 10731\nu - 89523 ) / 21952 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -67\nu^{7} - 27\nu^{6} + 288\nu^{5} + 874\nu^{4} - 1782\nu^{3} - 6048\nu^{2} + 9163\nu + 40131 ) / 5488 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 181\nu^{7} + 177\nu^{6} - 992\nu^{5} - 2774\nu^{4} + 6754\nu^{3} + 22624\nu^{2} - 2989\nu - 153321 ) / 10976 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -349\nu^{7} - 177\nu^{6} + 2784\nu^{5} + 6246\nu^{4} - 16610\nu^{3} - 45696\nu^{2} + 36309\nu + 306985 ) / 10976 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{6} + \beta_{5} - 4\beta_{4} + \beta_{3} - \beta _1 + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - \beta_{5} + 12\beta_{4} + \beta_{3} - \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 4\beta_{6} + 13\beta_{5} - 4\beta_{4} - 13\beta_{3} - 2\beta_{2} + \beta _1 + 37 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{7} + 8\beta_{6} - 5\beta_{5} + 12\beta_{4} + 4\beta_{3} - 8\beta_{2} - 4\beta _1 - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 24\beta_{7} + 3\beta_{5} + 142\beta_{4} - 69\beta_{3} + 30\beta_{2} + 3\beta _1 + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -14\beta_{7} + 28\beta_{6} + 83\beta_{5} - 28\beta_{4} - 83\beta_{3} - 14\beta_{2} + 127\beta _1 + 141 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 224\beta_{7} + 562\beta_{6} - 671\beta_{5} - 1348\beta_{4} + 281\beta_{3} - 224\beta_{2} - 281\beta _1 + 1067 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(1 - \beta_{4}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
2.18070 − 1.49818i
−1.43536 + 2.22255i
−2.38781 + 1.13946i
2.64247 − 0.131782i
2.18070 + 1.49818i
−1.43536 − 2.22255i
−2.38781 − 1.13946i
2.64247 + 0.131782i
0 −1.50000 − 0.866025i 0 −2.31749 + 1.33800i 0 6.66796 − 2.13033i 0 1.50000 + 2.59808i 0
73.2 0 −1.50000 − 0.866025i 0 −2.05105 + 1.18418i 0 4.55981 + 5.31113i 0 1.50000 + 2.59808i 0
73.3 0 −1.50000 − 0.866025i 0 −0.425150 + 0.245461i 0 −6.25375 + 3.14494i 0 1.50000 + 2.59808i 0
73.4 0 −1.50000 − 0.866025i 0 7.79369 − 4.49969i 0 −6.97403 + 0.602461i 0 1.50000 + 2.59808i 0
145.1 0 −1.50000 + 0.866025i 0 −2.31749 − 1.33800i 0 6.66796 + 2.13033i 0 1.50000 − 2.59808i 0
145.2 0 −1.50000 + 0.866025i 0 −2.05105 − 1.18418i 0 4.55981 − 5.31113i 0 1.50000 − 2.59808i 0
145.3 0 −1.50000 + 0.866025i 0 −0.425150 − 0.245461i 0 −6.25375 − 3.14494i 0 1.50000 − 2.59808i 0
145.4 0 −1.50000 + 0.866025i 0 7.79369 + 4.49969i 0 −6.97403 − 0.602461i 0 1.50000 − 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.3.z.a ✓ 8
3.b odd 2 1 504.3.by.a 8
4.b odd 2 1 336.3.bh.h 8
7.b odd 2 1 1176.3.z.d 8
7.c even 3 1 1176.3.f.a 8
7.c even 3 1 1176.3.z.d 8
7.d odd 6 1 inner 168.3.z.a ✓ 8
7.d odd 6 1 1176.3.f.a 8
12.b even 2 1 1008.3.cg.n 8
21.g even 6 1 504.3.by.a 8
21.g even 6 1 3528.3.f.f 8
21.h odd 6 1 3528.3.f.f 8
28.f even 6 1 336.3.bh.h 8
28.f even 6 1 2352.3.f.k 8
28.g odd 6 1 2352.3.f.k 8
84.j odd 6 1 1008.3.cg.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.z.a ✓ 8 1.a even 1 1 trivial
168.3.z.a ✓ 8 7.d odd 6 1 inner
336.3.bh.h 8 4.b odd 2 1
336.3.bh.h 8 28.f even 6 1
504.3.by.a 8 3.b odd 2 1
504.3.by.a 8 21.g even 6 1
1008.3.cg.n 8 12.b even 2 1
1008.3.cg.n 8 84.j odd 6 1
1176.3.f.a 8 7.c even 3 1
1176.3.f.a 8 7.d odd 6 1
1176.3.z.d 8 7.b odd 2 1
1176.3.z.d 8 7.c even 3 1
2352.3.f.k 8 28.f even 6 1
2352.3.f.k 8 28.g odd 6 1
3528.3.f.f 8 21.g even 6 1
3528.3.f.f 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 6T_{5}^{7} - 29T_{5}^{6} + 246T_{5}^{5} + 1973T_{5}^{4} + 5412T_{5}^{3} + 6956T_{5}^{2} + 3696T_{5} + 784 \) acting on \(S_{3}^{\mathrm{new}}(168, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 6 T^{7} + \cdots + 784 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{8} - 14 T^{7} + \cdots + 20322064 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 4781999104 \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + \cdots + 2166784 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 123878657296 \) Copy content Toggle raw display
$23$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2 T^{3} + \cdots + 278272)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 4336090801 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 6736469776 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 22503866142976 \) Copy content Toggle raw display
$43$ \( (T^{4} + 10 T^{3} + \cdots + 994948)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 107206475776 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 618123019264 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 562746188841984 \) Copy content Toggle raw display
$61$ \( (T^{2} + 60 T + 1200)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 583524876544 \) Copy content Toggle raw display
$71$ \( (T^{4} - 212 T^{3} + \cdots - 11002304)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 97154396416 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 450927645793249 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 53944676847616 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 656906741956864 \) Copy content Toggle raw display
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