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Properties

Label 666.2.s.a
Level $666$
Weight $2$
Character orbit 666.s
Analytic conductor $5.318$
Analytic rank $0$
Dimension $4$
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(307,666)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("666.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.s (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: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + (\zeta_{12}^{2} - 2) q^{5} + 2 \zeta_{12}^{2} q^{7} + \zeta_{12}^{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + (\zeta_{12}^{2} - 2) q^{5} + 2 \zeta_{12}^{2} q^{7} + \zeta_{12}^{3} q^{8} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{10} + (\zeta_{12}^{3} - 2 \zeta_{12} + 3) q^{11} + (2 \zeta_{12}^{2} - 4) q^{13} + 2 \zeta_{12}^{3} q^{14} + (\zeta_{12}^{2} - 1) q^{16} + ( - \zeta_{12}^{2} + 6 \zeta_{12} - 1) q^{17} + (3 \zeta_{12}^{3} + \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{19} + ( - \zeta_{12}^{2} - 1) q^{20} + ( - \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{22} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{23} + (2 \zeta_{12}^{2} - 2) q^{25} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{26} + (2 \zeta_{12}^{2} - 2) q^{28} + (10 \zeta_{12}^{2} - 5) q^{29} + ( - 3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + ( - \zeta_{12}^{3} + 6 \zeta_{12}^{2} - \zeta_{12}) q^{34} + ( - 2 \zeta_{12}^{2} - 2) q^{35} + ( - 6 \zeta_{12}^{3} - \zeta_{12}^{2} + 6 \zeta_{12}) q^{37} + (\zeta_{12}^{3} - 2 \zeta_{12} - 3) q^{38} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{40} + ( - 4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 4 \zeta_{12}) q^{41} + (6 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{43} + ( - \zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12}) q^{44} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - \zeta_{12} + 3) q^{46} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 3) q^{47} + ( - 3 \zeta_{12}^{2} + 3) q^{49} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{50} + ( - 2 \zeta_{12}^{2} - 2) q^{52} + ( - 4 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 2 \zeta_{12} + 6) q^{53} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12} - 6) q^{55} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{56} + (10 \zeta_{12}^{3} - 5 \zeta_{12}) q^{58} + (2 \zeta_{12}^{2} + 6 \zeta_{12} + 2) q^{59} + ( - \zeta_{12}^{2} + 2) q^{61} + ( - 2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12} + 3) q^{62} - q^{64} + ( - 6 \zeta_{12}^{2} + 6) q^{65} + ( - 3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 3 \zeta_{12}) q^{67} + (6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{68} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{70} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{71} + 4 q^{73} + ( - \zeta_{12}^{3} + 6) q^{74} + ( - \zeta_{12}^{2} - 3 \zeta_{12} - 1) q^{76} + ( - 2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 2 \zeta_{12}) q^{77} + ( - 15 \zeta_{12}^{3} - \zeta_{12}^{2} + 15 \zeta_{12} + 2) q^{79} + ( - 2 \zeta_{12}^{2} + 1) q^{80} + (3 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{82} + ( - 10 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 5 \zeta_{12} + 3) q^{83} + (6 \zeta_{12}^{3} - 12 \zeta_{12} + 3) q^{85} + (8 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 4 \zeta_{12} - 6) q^{86} + (3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{88} + ( - 3 \zeta_{12}^{2} - 12 \zeta_{12} - 3) q^{89} + ( - 4 \zeta_{12}^{2} - 4) q^{91} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{92} + (\zeta_{12}^{2} - 3 \zeta_{12} + 1) q^{94} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{95} + (6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{97} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 6 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{11} - 12 q^{13} - 2 q^{16} - 6 q^{17} - 6 q^{19} - 6 q^{20} - 6 q^{22} - 4 q^{25} - 4 q^{28} + 12 q^{34} - 12 q^{35} - 2 q^{37} - 12 q^{38} + 6 q^{41} + 6 q^{44} + 6 q^{46} - 12 q^{47} + 6 q^{49} - 12 q^{52} + 12 q^{53} - 18 q^{55} + 12 q^{59} + 6 q^{61} + 6 q^{62} - 4 q^{64} + 12 q^{65} + 10 q^{67} + 16 q^{73} + 24 q^{74} - 6 q^{76} + 12 q^{77} + 6 q^{79} + 6 q^{83} + 12 q^{85} - 12 q^{86} - 18 q^{89} - 24 q^{91} - 6 q^{92} + 6 q^{94} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.50000 − 0.866025i 0 1.00000 − 1.73205i 1.00000i 0 1.73205
307.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.50000 − 0.866025i 0 1.00000 − 1.73205i − 1.00000i 0 −1.73205
397.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.50000 + 0.866025i 0 1.00000 + 1.73205i − 1.00000i 0 1.73205
397.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.50000 + 0.866025i 0 1.00000 + 1.73205i 1.00000i 0 −1.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.s.a 4
3.b odd 2 1 74.2.e.b ✓ 4
12.b even 2 1 592.2.w.e 4
37.e even 6 1 inner 666.2.s.a 4
111.h odd 6 1 74.2.e.b ✓ 4
111.m even 12 1 2738.2.a.e 2
111.m even 12 1 2738.2.a.i 2
444.p even 6 1 592.2.w.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.e.b ✓ 4 3.b odd 2 1
74.2.e.b ✓ 4 111.h odd 6 1
592.2.w.e 4 12.b even 2 1
592.2.w.e 4 444.p even 6 1
666.2.s.a 4 1.a even 1 1 trivial
666.2.s.a 4 37.e even 6 1 inner
2738.2.a.e 2 111.m even 12 1
2738.2.a.i 2 111.m even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} - 21 T^{2} + \cdots + 1089 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$23$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} - 33 T^{2} + \cdots + 1369 \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} + 75 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$43$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + 24 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$61$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 10 T^{3} + 102 T^{2} + 20 T + 4 \) Copy content Toggle raw display
$71$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$73$ \( (T - 4)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} - 6 T^{3} - 210 T^{2} + \cdots + 49284 \) Copy content Toggle raw display
$83$ \( T^{4} - 6 T^{3} + 102 T^{2} + \cdots + 4356 \) Copy content Toggle raw display
$89$ \( T^{4} + 18 T^{3} - 9 T^{2} + \cdots + 13689 \) Copy content Toggle raw display
$97$ \( T^{4} + 78T^{2} + 1089 \) Copy content Toggle raw display
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