Properties

Label 666.2.f.j
Level $666$
Weight $2$
Character orbit 666.f
Analytic conductor $5.318$
Analytic rank $0$
Dimension $6$
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(343,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("666.343");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.f (of order \(3\), 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: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.4406832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} + \beta_{3} q^{4} - \beta_{5} q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{2} + \beta_{3} q^{4} - \beta_{5} q^{5} + ( - \beta_{4} - \beta_{2}) q^{7} - q^{8} + \beta_1 q^{10} + ( - \beta_{2} - \beta_1 - 1) q^{11} + 2 \beta_{3} q^{13} - \beta_{2} q^{14} + ( - \beta_{3} - 1) q^{16} + ( - 2 \beta_{4} - \beta_{3} - 1) q^{17} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_{2}) q^{19}+ \cdots + ( - \beta_{5} + \beta_{4} + \cdots + \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 3 q^{4} + q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} - 3 q^{4} + q^{5} - 6 q^{8} + 2 q^{10} - 8 q^{11} - 6 q^{13} - 3 q^{16} - 3 q^{17} - 8 q^{19} + q^{20} - 4 q^{22} - 16 q^{23} - 20 q^{25} - 12 q^{26} + 6 q^{29} + 8 q^{31} + 3 q^{32} + 3 q^{34} - 4 q^{35} - 11 q^{37} - 16 q^{38} - q^{40} - 7 q^{41} + 16 q^{43} + 4 q^{44} - 8 q^{46} + 32 q^{47} + 5 q^{49} + 20 q^{50} - 6 q^{52} + 22 q^{53} - 32 q^{55} + 3 q^{58} + 4 q^{59} - 9 q^{61} + 4 q^{62} + 6 q^{64} + 2 q^{65} + 16 q^{67} + 6 q^{68} + 4 q^{70} - 12 q^{71} + 4 q^{73} + 2 q^{74} - 8 q^{76} + 12 q^{77} + 4 q^{79} - 2 q^{80} - 14 q^{82} + 4 q^{83} - 18 q^{85} + 8 q^{86} + 8 q^{88} + 9 q^{89} + 8 q^{92} + 16 q^{94} - 36 q^{95} + 26 q^{97} - 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{5} + 30\nu^{4} - 31\nu^{3} + 120\nu^{2} + 25\nu + 595 ) / 149 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{5} - 42\nu^{4} + 103\nu^{3} - 168\nu^{2} - 35\nu - 386 ) / 149 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30\nu^{5} - 31\nu^{4} + 186\nu^{3} + 174\nu^{2} + 744\nu + 6 ) / 149 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 88\nu^{5} - 81\nu^{4} + 486\nu^{3} + 719\nu^{2} + 1944\nu + 405 ) / 149 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 125\nu^{5} - 154\nu^{4} + 775\nu^{3} + 725\nu^{2} + 2951\nu + 25 ) / 149 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 4\beta_{3} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{2} + 7\beta _1 - 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{5} + 3\beta_{4} + 28\beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -55\beta_{5} + 31\beta_{4} + 139\beta_{3} - 55\beta _1 + 139 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
343.1
1.43310 + 2.48220i
−0.827721 1.43366i
−0.105378 0.182520i
1.43310 2.48220i
−0.827721 + 1.43366i
−0.105378 + 0.182520i
0.500000 0.866025i 0 −0.500000 0.866025i −2.10755 3.65038i 0 0.258652 + 0.447998i −1.00000 0 −4.21509
343.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.629755 + 1.09077i 0 −1.52569 2.64257i −1.00000 0 1.25951
343.3 0.500000 0.866025i 0 −0.500000 0.866025i 1.97779 + 3.42563i 0 1.26704 + 2.19457i −1.00000 0 3.95558
433.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.10755 + 3.65038i 0 0.258652 0.447998i −1.00000 0 −4.21509
433.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.629755 1.09077i 0 −1.52569 + 2.64257i −1.00000 0 1.25951
433.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.97779 3.42563i 0 1.26704 2.19457i −1.00000 0 3.95558
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 343.3
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 666.2.f.j 6
3.b odd 2 1 74.2.c.c 6
12.b even 2 1 592.2.i.e 6
37.c even 3 1 inner 666.2.f.j 6
111.h odd 6 1 2738.2.a.n 3
111.i odd 6 1 74.2.c.c 6
111.i odd 6 1 2738.2.a.o 3
444.t even 6 1 592.2.i.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.c.c 6 3.b odd 2 1
74.2.c.c 6 111.i odd 6 1
592.2.i.e 6 12.b even 2 1
592.2.i.e 6 444.t even 6 1
666.2.f.j 6 1.a even 1 1 trivial
666.2.f.j 6 37.c even 3 1 inner
2738.2.a.n 3 111.h odd 6 1
2738.2.a.o 3 111.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\):

\( T_{5}^{6} - T_{5}^{5} + 18T_{5}^{4} - 25T_{5}^{3} + 310T_{5}^{2} - 357T_{5} + 441 \) Copy content Toggle raw display
\( T_{7}^{6} + 8T_{7}^{4} - 8T_{7}^{3} + 64T_{7}^{2} - 32T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} + \cdots + 441 \) Copy content Toggle raw display
$7$ \( T^{6} + 8 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{3} + 4 T^{2} - 16 T - 48)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 4)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + \cdots + 3969 \) Copy content Toggle raw display
$19$ \( T^{6} + 8 T^{5} + \cdots + 12544 \) Copy content Toggle raw display
$23$ \( (T^{3} + 8 T^{2} + 4 T - 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 3 T^{2} - 5 T + 3)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 4 T^{2} - 24 T - 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 11 T^{5} + \cdots + 50653 \) Copy content Toggle raw display
$41$ \( T^{6} + 7 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$43$ \( (T^{3} - 8 T^{2} + \cdots + 148)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} - 16 T^{2} + \cdots - 48)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} - 22 T^{5} + \cdots + 46656 \) Copy content Toggle raw display
$59$ \( T^{6} - 4 T^{5} + \cdots + 451584 \) Copy content Toggle raw display
$61$ \( T^{6} + 9 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$67$ \( T^{6} - 16 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( T^{6} + 12 T^{5} + \cdots + 1296 \) Copy content Toggle raw display
$73$ \( (T^{3} - 2 T^{2} - 20 T - 8)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 4 T^{5} + \cdots + 20736 \) Copy content Toggle raw display
$83$ \( T^{6} - 4 T^{5} + \cdots + 17424 \) Copy content Toggle raw display
$89$ \( T^{6} - 9 T^{5} + \cdots + 301401 \) Copy content Toggle raw display
$97$ \( (T^{3} - 13 T^{2} + 27 T - 7)^{2} \) Copy content Toggle raw display
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