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Properties

Label 172.2.f.a
Level $172$
Weight $2$
Character orbit 172.f
Analytic conductor $1.373$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [172,2,Mod(7,172)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(172, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("172.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 172 = 2^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 172.f (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: \(1.37342691477\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.157044178944.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 153x^{4} + 602x^{2} + 1849 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{2}) q^{2} + \beta_1 q^{3} - 2 q^{4} + ( - \beta_{6} - \beta_{4} + 1) q^{5} + ( - \beta_{7} + \beta_{3}) q^{6} + (\beta_{5} + \beta_1) q^{7} + (2 \beta_{6} - 2 \beta_{2}) q^{8} + (2 \beta_{6} + 4 \beta_{4} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{2}) q^{2} + \beta_1 q^{3} - 2 q^{4} + ( - \beta_{6} - \beta_{4} + 1) q^{5} + ( - \beta_{7} + \beta_{3}) q^{6} + (\beta_{5} + \beta_1) q^{7} + (2 \beta_{6} - 2 \beta_{2}) q^{8} + (2 \beta_{6} + 4 \beta_{4} - \beta_{2}) q^{9} + ( - \beta_{6} - 2 \beta_{4} + 2 \beta_{2} - 2) q^{10} + \beta_{3} q^{11} - 2 \beta_1 q^{12} + (2 \beta_{6} - \beta_{4} - \beta_{2}) q^{13} - \beta_{7} q^{14} + ( - \beta_{7} - \beta_{5} + \beta_1) q^{15} + 4 q^{16} + ( - 2 \beta_{6} - 3 \beta_{4} + \beta_{2}) q^{17} + (2 \beta_{4} - 4 \beta_{2} + 4) q^{18} - \beta_1 q^{19} + (2 \beta_{6} + 2 \beta_{4} - 2) q^{20} + (\beta_{6} + \beta_{2} - 7) q^{21} + 2 \beta_{5} q^{22} + (\beta_{7} - 2 \beta_{5} + \cdots - \beta_1) q^{23}+ \cdots + ( - 4 \beta_{7} - 2 \beta_{5} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 12 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 12 q^{5} - 16 q^{9} - 8 q^{10} + 4 q^{13} + 32 q^{16} + 12 q^{17} + 24 q^{18} - 24 q^{20} - 56 q^{21} + 24 q^{26} - 12 q^{29} + 24 q^{33} - 24 q^{34} + 32 q^{36} + 12 q^{37} + 16 q^{40} - 48 q^{50} - 8 q^{52} - 12 q^{53} + 28 q^{57} - 8 q^{58} + 12 q^{61} - 64 q^{64} - 56 q^{66} - 24 q^{68} + 60 q^{69} - 48 q^{72} + 12 q^{73} + 24 q^{74} + 24 q^{77} + 48 q^{80} - 4 q^{81} + 112 q^{84} + 36 q^{89} + 112 q^{90} - 84 q^{93} - 32 q^{97} - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14x^{6} + 153x^{4} + 602x^{2} + 1849 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{6} + 357\nu^{4} + 2805\nu^{2} + 17759 ) / 6579 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{7} + 357\nu^{5} + 2805\nu^{3} + 17759\nu ) / 6579 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\nu^{6} + 153\nu^{4} + 2142\nu^{2} + 1849 ) / 6579 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\nu^{7} + 153\nu^{5} + 2142\nu^{3} + 1849\nu ) / 6579 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -47\nu^{6} - 357\nu^{4} - 2805\nu^{2} + 2408 ) / 6579 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -47\nu^{7} - 357\nu^{5} - 2805\nu^{3} + 2408\nu ) / 6579 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{6} + 7\beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 7\beta_{5} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -14\beta_{6} - 55\beta_{4} + 28\beta_{2} - 55 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{7} - 55\beta_{5} + 28\beta_{3} - 55\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -153\beta_{6} - 153\beta_{2} + 469 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -153\beta_{7} - 153\beta_{3} + 469\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/172\mathbb{Z}\right)^\times\).

\(n\) \(87\) \(89\)
\(\chi(n)\) \(-1\) \(-\beta_{4}\)

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.

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
−1.53700 − 2.66216i
1.53700 + 2.66216i
−1.06660 − 1.84740i
1.06660 + 1.84740i
−1.06660 + 1.84740i
1.06660 − 1.84740i
−1.53700 + 2.66216i
1.53700 − 2.66216i
− 1.41421i −1.53700 − 2.66216i −2.00000 2.72474 − 1.57313i −3.76487 + 2.17365i 1.53700 − 2.66216i 2.82843i −3.22474 + 5.58542i −2.22474 − 3.85337i
7.2 − 1.41421i 1.53700 + 2.66216i −2.00000 2.72474 − 1.57313i 3.76487 − 2.17365i −1.53700 + 2.66216i 2.82843i −3.22474 + 5.58542i −2.22474 − 3.85337i
7.3 1.41421i −1.06660 − 1.84740i −2.00000 0.275255 − 0.158919i 2.61262 − 1.50839i 1.06660 − 1.84740i − 2.82843i −0.775255 + 1.34278i 0.224745 + 0.389270i
7.4 1.41421i 1.06660 + 1.84740i −2.00000 0.275255 − 0.158919i −2.61262 + 1.50839i −1.06660 + 1.84740i − 2.82843i −0.775255 + 1.34278i 0.224745 + 0.389270i
123.1 − 1.41421i −1.06660 + 1.84740i −2.00000 0.275255 + 0.158919i 2.61262 + 1.50839i 1.06660 + 1.84740i 2.82843i −0.775255 − 1.34278i 0.224745 − 0.389270i
123.2 − 1.41421i 1.06660 − 1.84740i −2.00000 0.275255 + 0.158919i −2.61262 − 1.50839i −1.06660 − 1.84740i 2.82843i −0.775255 − 1.34278i 0.224745 − 0.389270i
123.3 1.41421i −1.53700 + 2.66216i −2.00000 2.72474 + 1.57313i −3.76487 − 2.17365i 1.53700 + 2.66216i − 2.82843i −3.22474 − 5.58542i −2.22474 + 3.85337i
123.4 1.41421i 1.53700 − 2.66216i −2.00000 2.72474 + 1.57313i 3.76487 + 2.17365i −1.53700 − 2.66216i − 2.82843i −3.22474 − 5.58542i −2.22474 + 3.85337i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
43.d odd 6 1 inner
172.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 172.2.f.a ✓ 8
4.b odd 2 1 inner 172.2.f.a ✓ 8
43.d odd 6 1 inner 172.2.f.a ✓ 8
172.f even 6 1 inner 172.2.f.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
172.2.f.a ✓ 8 1.a even 1 1 trivial
172.2.f.a ✓ 8 4.b odd 2 1 inner
172.2.f.a ✓ 8 43.d odd 6 1 inner
172.2.f.a ✓ 8 172.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 14T_{3}^{6} + 153T_{3}^{4} + 602T_{3}^{2} + 1849 \) acting on \(S_{2}^{\mathrm{new}}(172, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 14 T^{6} + \cdots + 1849 \) Copy content Toggle raw display
$5$ \( (T^{4} - 6 T^{3} + 13 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 14 T^{6} + \cdots + 1849 \) Copy content Toggle raw display
$11$ \( (T^{4} + 28 T^{2} + 172)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 6 T^{3} + 33 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 14 T^{6} + \cdots + 1849 \) Copy content Toggle raw display
$23$ \( T^{8} - 46 T^{6} + \cdots + 1849 \) Copy content Toggle raw display
$29$ \( (T^{4} + 6 T^{3} + 13 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 42 T^{6} + \cdots + 149769 \) Copy content Toggle raw display
$37$ \( (T^{4} - 6 T^{3} + \cdots + 225)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 54)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + 80 T^{6} + \cdots + 3418801 \) Copy content Toggle raw display
$47$ \( (T^{4} + 112 T^{2} + 2752)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 6 T^{3} + \cdots + 225)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 184 T^{2} + 688)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 6 T^{3} + \cdots + 225)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 222 T^{6} + \cdots + 93605625 \) Copy content Toggle raw display
$71$ \( T^{8} + 138 T^{6} + \cdots + 149769 \) Copy content Toggle raw display
$73$ \( (T^{4} - 6 T^{3} + \cdots + 225)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 42 T^{6} + \cdots + 149769 \) Copy content Toggle raw display
$83$ \( T^{8} - 346 T^{6} + \cdots + 240963529 \) Copy content Toggle raw display
$89$ \( (T^{4} - 18 T^{3} + \cdots + 361)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T + 10)^{4} \) Copy content Toggle raw display
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