Properties

Label 666.2.be.b
Level $666$
Weight $2$
Character orbit 666.be
Analytic conductor $5.318$
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] = [666,2,Mod(125,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("666.125");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.be (of order \(12\), degree \(4\), 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: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{24}^{7} q^{2} - \zeta_{24}^{2} q^{4} + (2 \zeta_{24}^{7} + 2 \zeta_{24}^{3}) q^{5} + ( - \zeta_{24}^{4} + 1) q^{7} + ( - \zeta_{24}^{5} + \zeta_{24}) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{7} q^{2} - \zeta_{24}^{2} q^{4} + (2 \zeta_{24}^{7} + 2 \zeta_{24}^{3}) q^{5} + ( - \zeta_{24}^{4} + 1) q^{7} + ( - \zeta_{24}^{5} + \zeta_{24}) q^{8} + (2 \zeta_{24}^{6} - 4 \zeta_{24}^{2}) q^{10} + ( - 2 \zeta_{24}^{7} + 2 \zeta_{24}^{3} + 2 \zeta_{24}) q^{11} + ( - 3 \zeta_{24}^{6} + 3 \zeta_{24}^{2} + 3) q^{13} + \zeta_{24}^{3} q^{14} + \zeta_{24}^{4} q^{16} + ( - \zeta_{24}^{7} + 2 \zeta_{24}^{3} + \zeta_{24}) q^{17} + (4 \zeta_{24}^{6} - \zeta_{24}^{4} - 3 \zeta_{24}^{2} - 3) q^{19} + ( - 4 \zeta_{24}^{5} + 2 \zeta_{24}) q^{20} + (2 \zeta_{24}^{6} + 2 \zeta_{24}^{4} - 2) q^{22} + (2 \zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} - \zeta_{24}) q^{23} + (7 \zeta_{24}^{6} - 7 \zeta_{24}^{2}) q^{25} + (3 \zeta_{24}^{7} + 3 \zeta_{24}^{5}) q^{26} + (\zeta_{24}^{6} - \zeta_{24}^{2}) q^{28} + (3 \zeta_{24}^{5} - 5 \zeta_{24}^{3} + 3 \zeta_{24}) q^{29} + (4 \zeta_{24}^{6} - \zeta_{24}^{4} + \zeta_{24}^{2} - 4) q^{31} + (\zeta_{24}^{7} - \zeta_{24}^{3}) q^{32} + (2 \zeta_{24}^{6} + \zeta_{24}^{4} - \zeta_{24}^{2} - 1) q^{34} + ( - 2 \zeta_{24}^{7} + 4 \zeta_{24}^{3}) q^{35} + (\zeta_{24}^{6} + 6) q^{37} + ( - 4 \zeta_{24}^{7} - 3 \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}) q^{38} + (2 \zeta_{24}^{4} + 2) q^{40} + ( - 4 \zeta_{24}^{3} + 4 \zeta_{24}) q^{41} + (2 \zeta_{24}^{6} - 7 \zeta_{24}^{4} - 7 \zeta_{24}^{2} + 2) q^{43} + ( - 2 \zeta_{24}^{3} - 2 \zeta_{24}) q^{44} + ( - \zeta_{24}^{6} - \zeta_{24}^{4} - \zeta_{24}^{2}) q^{46} + (6 \zeta_{24}^{7} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{3} + 2 \zeta_{24}) q^{47} + 6 \zeta_{24}^{4} q^{49} - 7 \zeta_{24}^{5} q^{50} + ( - 3 \zeta_{24}^{2} - 3) q^{52} + (2 \zeta_{24}^{7} - 2 \zeta_{24}) q^{53} + (4 \zeta_{24}^{6} + 8 \zeta_{24}^{4} + 4 \zeta_{24}^{2} - 4) q^{55} - \zeta_{24}^{5} q^{56} + ( - 5 \zeta_{24}^{6} + 3 \zeta_{24}^{4} + 5 \zeta_{24}^{2} - 6) q^{58} + ( - \zeta_{24}^{7} + 2 \zeta_{24}^{3} + 3 \zeta_{24}) q^{59} + ( - 2 \zeta_{24}^{6} - \zeta_{24}^{4} + \zeta_{24}^{2} - 1) q^{61} + ( - 5 \zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} - 5 \zeta_{24}) q^{62} - \zeta_{24}^{6} q^{64} + (6 \zeta_{24}^{7} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{3} + 6 \zeta_{24}) q^{65} - 3 \zeta_{24}^{2} q^{67} + ( - \zeta_{24}^{5} - \zeta_{24}^{3} - \zeta_{24}) q^{68} + (4 \zeta_{24}^{6} - 2 \zeta_{24}^{2}) q^{70} + ( - 2 \zeta_{24}^{7} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3}) q^{71} + ( - 5 \zeta_{24}^{6} - 12 \zeta_{24}^{4} + 6) q^{73} + (6 \zeta_{24}^{7} - \zeta_{24}) q^{74} + (\zeta_{24}^{6} - \zeta_{24}^{4} + 3 \zeta_{24}^{2} + 4) q^{76} + ( - 2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} + 2 \zeta_{24}) q^{77} + (3 \zeta_{24}^{6} - 4 \zeta_{24}^{4} + \zeta_{24}^{2} + 1) q^{79} + (4 \zeta_{24}^{7} - 2 \zeta_{24}^{3}) q^{80} + ( - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{2} - 4) q^{82} + (7 \zeta_{24}^{7} + 12 \zeta_{24}^{5} - 12 \zeta_{24}^{3} - 7 \zeta_{24}) q^{83} + (6 \zeta_{24}^{6} + 4 \zeta_{24}^{4} - 2) q^{85} + ( - 5 \zeta_{24}^{7} - 7 \zeta_{24}^{5} + 7 \zeta_{24}^{3} + 5 \zeta_{24}) q^{86} + ( - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{2} + 2) q^{88} + (2 \zeta_{24}^{7} - 12 \zeta_{24}^{5} + 6 \zeta_{24}) q^{89} + ( - 3 \zeta_{24}^{6} - 3 \zeta_{24}^{4} + 3) q^{91} + ( - \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2 \zeta_{24}) q^{92} + ( - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{4} - 4 \zeta_{24}^{2} - 6) q^{94} + ( - 10 \zeta_{24}^{7} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{3} - 10 \zeta_{24}) q^{95} + ( - 4 \zeta_{24}^{6} + 11 \zeta_{24}^{4} + 11 \zeta_{24}^{2} - 4) q^{97} + (6 \zeta_{24}^{7} - 6 \zeta_{24}^{3}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{7} + 24 q^{13} + 4 q^{16} - 28 q^{19} - 8 q^{22} - 36 q^{31} - 4 q^{34} + 48 q^{37} + 24 q^{40} - 12 q^{43} - 4 q^{46} + 24 q^{49} - 24 q^{52} - 36 q^{58} - 12 q^{61} + 28 q^{76} - 8 q^{79} - 16 q^{82} + 8 q^{88} + 12 q^{91} - 40 q^{94} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(371\) \(631\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1
0.965926 0.258819i
−0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
−0.258819 0.965926i 0 −0.866025 + 0.500000i 0.896575 3.34607i 0 0.500000 + 0.866025i 0.707107 + 0.707107i 0 −3.46410
125.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −0.896575 + 3.34607i 0 0.500000 + 0.866025i −0.707107 0.707107i 0 −3.46410
251.1 −0.965926 0.258819i 0 0.866025 + 0.500000i −3.34607 + 0.896575i 0 0.500000 0.866025i −0.707107 0.707107i 0 3.46410
251.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i 3.34607 0.896575i 0 0.500000 0.866025i 0.707107 + 0.707107i 0 3.46410
341.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0.896575 + 3.34607i 0 0.500000 0.866025i 0.707107 0.707107i 0 −3.46410
341.2 0.258819 0.965926i 0 −0.866025 0.500000i −0.896575 3.34607i 0 0.500000 0.866025i −0.707107 + 0.707107i 0 −3.46410
467.1 −0.965926 + 0.258819i 0 0.866025 0.500000i −3.34607 0.896575i 0 0.500000 + 0.866025i −0.707107 + 0.707107i 0 3.46410
467.2 0.965926 0.258819i 0 0.866025 0.500000i 3.34607 + 0.896575i 0 0.500000 + 0.866025i 0.707107 0.707107i 0 3.46410
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.g odd 12 1 inner
111.m even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.be.b 8
3.b odd 2 1 inner 666.2.be.b 8
37.g odd 12 1 inner 666.2.be.b 8
111.m even 12 1 inner 666.2.be.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.be.b 8 1.a even 1 1 trivial
666.2.be.b 8 3.b odd 2 1 inner
666.2.be.b 8 37.g odd 12 1 inner
666.2.be.b 8 111.m even 12 1 inner

Hecke kernels

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

\( T_{5}^{8} - 144T_{5}^{4} + 20736 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 144 T^{4} + 20736 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 16 T^{2} + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 12 T^{3} + 45 T^{2} - 54 T + 81)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{6} + 44 T^{4} + 48 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{4} + 14 T^{3} + 74 T^{2} + 220 T + 484)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 56T^{4} + 16 \) Copy content Toggle raw display
$29$ \( T^{8} + 10808T^{4} + 16 \) Copy content Toggle raw display
$31$ \( (T^{4} + 18 T^{3} + 162 T^{2} + 702 T + 1521)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 12 T + 37)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + 64 T^{6} + 3840 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$43$ \( (T^{4} + 6 T^{3} + 18 T^{2} - 414 T + 4761)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 112 T^{2} + 2704)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 36 T^{6} + 396 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$61$ \( (T^{4} + 6 T^{3} + 18 T^{2} + 36 T + 36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 266 T^{2} + 6889)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + 53 T^{2} + 14 T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 436 T^{6} + \cdots + 2097273616 \) Copy content Toggle raw display
$89$ \( T^{8} - 144 T^{6} + \cdots + 116985856 \) Copy content Toggle raw display
$97$ \( (T^{4} - 6 T^{3} + 18 T^{2} + 1062 T + 31329)^{2} \) Copy content Toggle raw display
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