Properties

Label 666.2.j.d
Level $666$
Weight $2$
Character orbit 666.j
Analytic conductor $5.318$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [666,2,Mod(179,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("666.179");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.j (of order \(4\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{4} + (\beta_{7} - \beta_{2} + 1) q^{7} + \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{4} + (\beta_{7} - \beta_{2} + 1) q^{7} + \beta_{6} q^{8} + ( - \beta_{6} + \beta_{4} + \beta_1) q^{11} + (2 \beta_{3} + \beta_{2} + 2) q^{13} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{14} - q^{16} + (\beta_{6} + 2 \beta_{5} - 2 \beta_{4}) q^{17} + (3 \beta_{3} + \beta_{2} + 3) q^{19} + (\beta_{3} + \beta_{2} + 1) q^{22} + (\beta_{6} - \beta_{5} + \beta_{4}) q^{23} + 5 \beta_{3} q^{25} + (2 \beta_{6} + \beta_{5} + 2 \beta_1) q^{26} + ( - \beta_{7} + \beta_{3} - \beta_{2}) q^{28} + ( - \beta_{5} - \beta_{4} - 2 \beta_1) q^{29} + (2 \beta_{7} + \beta_{3} - 1) q^{31} - \beta_1 q^{32} + (2 \beta_{7} - 2 \beta_{2} - 1) q^{34} + ( - 4 \beta_{3} - \beta_{2} + 4) q^{37} + (3 \beta_{6} + \beta_{5} + 3 \beta_1) q^{38} + (4 \beta_{6} + 2 \beta_{4} - 4 \beta_1) q^{41} + (4 \beta_{3} + 2 \beta_{2} + 4) q^{43} + (\beta_{6} + \beta_{5} + \beta_1) q^{44} + ( - \beta_{7} + \beta_{2} - 1) q^{46} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_1) q^{47} + (2 \beta_{7} - 2 \beta_{2} + 4) q^{49} + 5 \beta_{6} q^{50} + (\beta_{7} + 2 \beta_{3} - 2) q^{52} + ( - 2 \beta_{6} - 5 \beta_{5} - 2 \beta_1) q^{53} + (\beta_{6} - \beta_{5} + \beta_{4}) q^{56} + ( - \beta_{7} - 2 \beta_{3} - \beta_{2}) q^{58} + ( - 4 \beta_{7} - 2 \beta_{3} + 2) q^{61} + (\beta_{6} - 2 \beta_{4} - \beta_1) q^{62} - \beta_{3} q^{64} + ( - 2 \beta_{7} - 2 \beta_{3} - 2 \beta_{2}) q^{67} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_1) q^{68} + (\beta_{6} + 4 \beta_{5} + \beta_1) q^{71} + (2 \beta_{7} - 9 \beta_{3} + 2 \beta_{2}) q^{73} + ( - 4 \beta_{6} - \beta_{5} + 4 \beta_1) q^{74} + (\beta_{7} + 3 \beta_{3} - 3) q^{76} + (4 \beta_{6} - \beta_{4} - 4 \beta_1) q^{77} + (2 \beta_{3} + 6 \beta_{2} + 2) q^{79} + ( - 4 \beta_{3} + 2 \beta_{2} - 4) q^{82} + ( - \beta_{6} + 5 \beta_{5} - \beta_1) q^{83} + (4 \beta_{6} + 2 \beta_{5} + 4 \beta_1) q^{86} + (\beta_{7} + \beta_{3} - 1) q^{88} + (2 \beta_{5} + 2 \beta_{4} - 5 \beta_1) q^{89} + ( - 3 \beta_{3} - 3 \beta_{2} - 3) q^{91} + (\beta_{5} + \beta_{4} - \beta_1) q^{92} + ( - 2 \beta_{7} - 2 \beta_{3} + 2) q^{94} + ( - 4 \beta_{3} - 2 \beta_{2} - 4) q^{97} + ( - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} + 16 q^{13} - 8 q^{16} + 24 q^{19} + 8 q^{22} - 8 q^{31} - 8 q^{34} + 32 q^{37} + 32 q^{43} - 8 q^{46} + 32 q^{49} - 16 q^{52} + 16 q^{61} - 24 q^{76} + 16 q^{79} - 32 q^{82} - 8 q^{88} - 24 q^{91} + 16 q^{94} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 8\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{4} + 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} + 6\nu^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} + 13\nu^{3} ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} + 29\nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 3\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{4} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{2} + 11\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{5} - 9\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} + 29\beta_{6} ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1
1.14412 1.14412i
−0.437016 + 0.437016i
0.437016 0.437016i
−1.14412 + 1.14412i
1.14412 + 1.14412i
−0.437016 0.437016i
0.437016 + 0.437016i
−1.14412 1.14412i
−0.707107 + 0.707107i 0 1.00000i 0 0 −2.16228 0.707107 + 0.707107i 0 0
179.2 −0.707107 + 0.707107i 0 1.00000i 0 0 4.16228 0.707107 + 0.707107i 0 0
179.3 0.707107 0.707107i 0 1.00000i 0 0 −2.16228 −0.707107 0.707107i 0 0
179.4 0.707107 0.707107i 0 1.00000i 0 0 4.16228 −0.707107 0.707107i 0 0
413.1 −0.707107 0.707107i 0 1.00000i 0 0 −2.16228 0.707107 0.707107i 0 0
413.2 −0.707107 0.707107i 0 1.00000i 0 0 4.16228 0.707107 0.707107i 0 0
413.3 0.707107 + 0.707107i 0 1.00000i 0 0 −2.16228 −0.707107 + 0.707107i 0 0
413.4 0.707107 + 0.707107i 0 1.00000i 0 0 4.16228 −0.707107 + 0.707107i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.d odd 4 1 inner
111.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.j.d 8
3.b odd 2 1 inner 666.2.j.d 8
37.d odd 4 1 inner 666.2.j.d 8
111.g even 4 1 inner 666.2.j.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.j.d 8 1.a even 1 1 trivial
666.2.j.d 8 3.b odd 2 1 inner
666.2.j.d 8 37.d odd 4 1 inner
666.2.j.d 8 111.g even 4 1 inner

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T - 9)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 14 T^{2} + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 8 T^{3} + 32 T^{2} - 24 T + 9)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 3682 T^{4} + \cdots + 2313441 \) Copy content Toggle raw display
$19$ \( (T^{4} - 12 T^{3} + 72 T^{2} - 156 T + 169)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 322T^{4} + 6561 \) Copy content Toggle raw display
$29$ \( T^{8} + 712T^{4} + 1296 \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} + 8 T^{2} - 72 T + 324)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 16 T^{3} + 128 T^{2} - 592 T + 1369)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 104 T^{2} + 144)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 16 T^{3} + 128 T^{2} - 192 T + 144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 56 T^{2} + 144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 266 T^{2} + 13689)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} + 32 T^{2} + 576 T + 5184)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 88 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 164 T^{2} + 6084)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 242 T^{2} + 1681)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + 32 T^{2} + 1376 T + 29584)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 254 T^{2} + 15129)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 16450 T^{4} + 50625 \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} + 128 T^{2} + 192 T + 144)^{2} \) Copy content Toggle raw display
show more
show less