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Properties

Label 666.2.x.f
Level $666$
Weight $2$
Character orbit 666.x
Analytic conductor $5.318$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [666,2,Mod(127,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("666.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.x (of order \(9\), 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: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: 12.0.686339028913329.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 8x^{9} + 37x^{6} - 216x^{3} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 222)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{2}) q^{2} + \beta_{4} q^{4} + ( - \beta_{11} + \beta_{8} + \beta_{7} + \cdots + 1) q^{5}+ \cdots + (\beta_{3} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{2}) q^{2} + \beta_{4} q^{4} + ( - \beta_{11} + \beta_{8} + \beta_{7} + \cdots + 1) q^{5}+ \cdots + (2 \beta_{11} - 2 \beta_{10} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{5} - 9 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{5} - 9 q^{7} - 6 q^{8} - 12 q^{13} - 9 q^{17} - 21 q^{19} - 3 q^{20} + 6 q^{22} - 30 q^{25} + 6 q^{26} - 9 q^{29} + 6 q^{31} + 9 q^{34} - 21 q^{35} - 18 q^{37} - 24 q^{38} - 3 q^{40} + 39 q^{41} - 6 q^{43} + 6 q^{44} - 18 q^{46} + 15 q^{47} + 21 q^{49} + 15 q^{50} - 3 q^{52} - 9 q^{53} + 3 q^{55} + 9 q^{56} - 63 q^{59} + 3 q^{61} + 12 q^{62} - 6 q^{64} - 36 q^{65} + 6 q^{67} - 18 q^{68} + 33 q^{70} + 6 q^{71} - 54 q^{73} + 18 q^{74} - 21 q^{76} + 45 q^{77} - 12 q^{79} + 12 q^{82} + 39 q^{83} + 33 q^{85} + 27 q^{86} - 57 q^{89} - 3 q^{91} + 9 q^{92} + 12 q^{94} - 39 q^{95} + 45 q^{97} + 21 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 8x^{9} + 37x^{6} - 216x^{3} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8\nu^{11} - 37\nu^{8} + 1295\nu^{5} - 729\nu^{2} ) / 17982 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\nu^{10} - 185\nu^{7} + 481\nu^{4} - 2808\nu ) / 5994 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -8\nu^{9} + 37\nu^{6} - 296\nu^{3} + 1728 ) / 999 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 253\nu^{2} ) / 666 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{10} - \nu^{9} - 37\nu^{6} - 37\nu^{3} + 759\nu + 216 ) / 666 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -13\nu^{11} + 9\nu^{9} + 185\nu^{8} + 333\nu^{6} - 481\nu^{5} + 333\nu^{3} + 2808\nu^{2} - 1944 ) / 5994 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 40\nu^{10} - 185\nu^{7} + 481\nu^{4} - 3645\nu ) / 5994 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -16\nu^{10} - 3\nu^{9} + 74\nu^{7} - 111\nu^{6} - 592\nu^{4} - 111\nu^{3} + 1458\nu + 648 ) / 1998 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16\nu^{10} - 3\nu^{9} - 74\nu^{7} - 111\nu^{6} + 592\nu^{4} - 111\nu^{3} - 3456\nu + 648 ) / 1998 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2\nu^{11} - \nu^{9} - 37\nu^{6} - 37\nu^{3} + 160\nu^{2} + 216 ) / 666 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -35\nu^{10} + 120\nu^{9} + 37\nu^{7} - 555\nu^{6} - 1295\nu^{4} + 1443\nu^{3} + 7560\nu - 10935 ) / 5994 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} - \beta_{8} + 2\beta_{7} + 2\beta_{5} - 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - 6\beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{11} - 4\beta_{9} + 2\beta_{8} + 2\beta_{7} + 2\beta_{5} - 15\beta_{3} + 4\beta_{2} + 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{9} - 10\beta_{8} - 13\beta_{7} + 5\beta_{5} - 5\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{10} + 2\beta_{9} + 2\beta_{8} + 2\beta_{7} + 3\beta_{6} + 2\beta_{5} - 3\beta_{4} - 2\beta_{2} + 45\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -16\beta_{9} - 16\beta_{8} - 16\beta_{7} - 16\beta_{5} + 9\beta_{3} + 16\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 26\beta_{9} - 13\beta_{8} - 13\beta_{7} - 13\beta_{5} - 131\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 35\beta_{9} + 35\beta_{8} + 35\beta_{7} + 105\beta_{6} + 35\beta_{5} + 117\beta_{4} - 35\beta_{2} + 117\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 222\beta_{11} + 74\beta_{9} - 148\beta_{8} - 148\beta_{7} - 148\beta_{5} + 222\beta_{3} - 74\beta_{2} + 93 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -31\beta_{9} - 31\beta_{8} + 728\beta_{7} + 62\beta_{5} - 728\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 759\beta_{10} - 253\beta_{9} - 253\beta_{8} - 253\beta_{7} - 253\beta_{5} + 480\beta_{4} + 253\beta_{2} ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.0973299 − 1.72931i
−1.03702 + 1.38729i
1.44896 − 0.948947i
−0.682920 + 1.59173i
1.71994 − 0.204441i
−1.54629 − 0.780367i
1.44896 + 0.948947i
−0.682920 − 1.59173i
1.71994 + 0.204441i
−1.54629 + 0.780367i
0.0973299 + 1.72931i
−1.03702 − 1.38729i
−0.939693 − 0.342020i 0 0.766044 + 0.642788i −0.537023 + 3.04561i 0 −0.563528 + 3.19592i −0.500000 − 0.866025i 0 1.54629 − 2.67826i
127.2 −0.939693 − 0.342020i 0 0.766044 + 0.642788i 0.597330 − 3.38763i 0 0.176868 − 1.00307i −0.500000 − 0.866025i 0 −1.71994 + 2.97903i
145.1 0.766044 + 0.642788i 0 0.173648 + 0.984808i −0.182920 − 0.0665776i 0 −4.67213 − 1.70052i −0.500000 + 0.866025i 0 −0.0973299 − 0.168580i
145.2 0.766044 + 0.642788i 0 0.173648 + 0.984808i 1.94896 + 0.709365i 0 1.46639 + 0.533723i −0.500000 + 0.866025i 0 1.03702 + 1.79618i
181.1 0.173648 − 0.984808i 0 −0.939693 − 0.342020i −1.04629 + 0.877946i 0 0.415163 − 0.348363i −0.500000 + 0.866025i 0 0.682920 + 1.18285i
181.2 0.173648 − 0.984808i 0 −0.939693 − 0.342020i 2.21994 − 1.86275i 0 −1.32277 + 1.10993i −0.500000 + 0.866025i 0 −1.44896 − 2.50968i
271.1 0.766044 − 0.642788i 0 0.173648 − 0.984808i −0.182920 + 0.0665776i 0 −4.67213 + 1.70052i −0.500000 − 0.866025i 0 −0.0973299 + 0.168580i
271.2 0.766044 − 0.642788i 0 0.173648 − 0.984808i 1.94896 − 0.709365i 0 1.46639 − 0.533723i −0.500000 − 0.866025i 0 1.03702 − 1.79618i
379.1 0.173648 + 0.984808i 0 −0.939693 + 0.342020i −1.04629 − 0.877946i 0 0.415163 + 0.348363i −0.500000 − 0.866025i 0 0.682920 − 1.18285i
379.2 0.173648 + 0.984808i 0 −0.939693 + 0.342020i 2.21994 + 1.86275i 0 −1.32277 − 1.10993i −0.500000 − 0.866025i 0 −1.44896 + 2.50968i
451.1 −0.939693 + 0.342020i 0 0.766044 − 0.642788i −0.537023 − 3.04561i 0 −0.563528 − 3.19592i −0.500000 + 0.866025i 0 1.54629 + 2.67826i
451.2 −0.939693 + 0.342020i 0 0.766044 − 0.642788i 0.597330 + 3.38763i 0 0.176868 + 1.00307i −0.500000 + 0.866025i 0 −1.71994 − 2.97903i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.x.f 12
3.b odd 2 1 222.2.k.e ✓ 12
37.f even 9 1 inner 666.2.x.f 12
111.n odd 18 1 8214.2.a.bc 6
111.p odd 18 1 222.2.k.e ✓ 12
111.p odd 18 1 8214.2.a.bb 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.k.e ✓ 12 3.b odd 2 1
222.2.k.e ✓ 12 111.p odd 18 1
666.2.x.f 12 1.a even 1 1 trivial
666.2.x.f 12 37.f even 9 1 inner
8214.2.a.bb 6 111.p odd 18 1
8214.2.a.bc 6 111.n odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 6 T_{5}^{11} + 33 T_{5}^{10} - 118 T_{5}^{9} + 333 T_{5}^{8} - 666 T_{5}^{7} + 1089 T_{5}^{6} + \cdots + 289 \) acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{11} + \cdots + 289 \) Copy content Toggle raw display
$7$ \( T^{12} + 9 T^{11} + \cdots + 576 \) Copy content Toggle raw display
$11$ \( T^{12} + 51 T^{10} + \cdots + 262144 \) Copy content Toggle raw display
$13$ \( T^{12} + 12 T^{11} + \cdots + 12321 \) Copy content Toggle raw display
$17$ \( T^{12} + 9 T^{11} + \cdots + 14722569 \) Copy content Toggle raw display
$19$ \( T^{12} + 21 T^{11} + \cdots + 3474496 \) Copy content Toggle raw display
$23$ \( T^{12} + 63 T^{10} + \cdots + 46656 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 172108161 \) Copy content Toggle raw display
$31$ \( (T^{6} - 3 T^{5} + \cdots + 136)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 7386371136 \) Copy content Toggle raw display
$43$ \( (T^{6} + 3 T^{5} - 78 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 5061468736 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 509359761 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 2109381184 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 16360712281 \) Copy content Toggle raw display
$67$ \( T^{12} - 6 T^{11} + \cdots + 7096896 \) Copy content Toggle raw display
$71$ \( T^{12} - 6 T^{11} + \cdots + 36864 \) Copy content Toggle raw display
$73$ \( (T^{6} + 27 T^{5} + \cdots - 90008)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 12 T^{11} + \cdots + 788544 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 14424970816 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 37603275886801 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 8606087361 \) Copy content Toggle raw display
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