Properties

Label 966.2.l.a
Level $966$
Weight $2$
Character orbit 966.l
Analytic conductor $7.714$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [966,2,Mod(47,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.l (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: \(7.71354883526\)
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: yes
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{2} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{2} + 1) q^{6} + ( - 3 \zeta_{12}^{2} + 2) q^{7} - \zeta_{12}^{3} q^{8} - 3 \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{2} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{2} + 1) q^{6} + ( - 3 \zeta_{12}^{2} + 2) q^{7} - \zeta_{12}^{3} q^{8} - 3 \zeta_{12}^{2} q^{9} + ( - \zeta_{12}^{2} - 1) q^{10} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{12} + (2 \zeta_{12}^{2} - 1) q^{13} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{14} - 3 q^{15} - \zeta_{12}^{2} q^{16} + (6 \zeta_{12}^{3} - 3 \zeta_{12}) q^{17} - 3 \zeta_{12} q^{18} + ( - 2 \zeta_{12}^{2} + 4) q^{19} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{20} + ( - \zeta_{12}^{3} - 4 \zeta_{12}) q^{21} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{23} + ( - \zeta_{12}^{2} - 1) q^{24} + ( - 2 \zeta_{12}^{2} + 2) q^{25} + (\zeta_{12}^{3} + \zeta_{12}) q^{26} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + ( - 2 \zeta_{12}^{2} - 1) q^{28} + 6 \zeta_{12}^{3} q^{29} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{30} + (2 \zeta_{12}^{2} + 2) q^{31} - \zeta_{12} q^{32} + (6 \zeta_{12}^{2} - 3) q^{34} + (4 \zeta_{12}^{3} - 5 \zeta_{12}) q^{35} - 3 q^{36} - 4 \zeta_{12}^{2} q^{37} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{38} + 3 \zeta_{12} q^{39} + (\zeta_{12}^{2} - 2) q^{40} + ( - \zeta_{12}^{2} - 4) q^{42} + 4 q^{43} + (6 \zeta_{12}^{3} - 3 \zeta_{12}) q^{45} + ( - \zeta_{12}^{2} + 1) q^{46} + (3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{47} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{48} + ( - 3 \zeta_{12}^{2} - 5) q^{49} - 2 \zeta_{12}^{3} q^{50} + 9 \zeta_{12}^{2} q^{51} + (\zeta_{12}^{2} + 1) q^{52} - 9 \zeta_{12} q^{53} + (3 \zeta_{12}^{2} - 6) q^{54} + (\zeta_{12}^{3} - 3 \zeta_{12}) q^{56} - 6 \zeta_{12}^{3} q^{57} + 6 \zeta_{12}^{2} q^{58} + ( - 12 \zeta_{12}^{3} + 6 \zeta_{12}) q^{59} + (3 \zeta_{12}^{2} - 3) q^{60} + ( - 8 \zeta_{12}^{2} + 16) q^{61} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{62} + (3 \zeta_{12}^{2} - 9) q^{63} - q^{64} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{65} + (5 \zeta_{12}^{2} - 5) q^{67} + (3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{68} + ( - 2 \zeta_{12}^{2} + 1) q^{69} + (4 \zeta_{12}^{2} - 5) q^{70} - 15 \zeta_{12}^{3} q^{71} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{72} + (7 \zeta_{12}^{2} + 7) q^{73} - 4 \zeta_{12} q^{74} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{75} + ( - 4 \zeta_{12}^{2} + 2) q^{76} + 3 q^{78} - 8 \zeta_{12}^{2} q^{79} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{80} + (9 \zeta_{12}^{2} - 9) q^{81} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{83} + (4 \zeta_{12}^{3} - 5 \zeta_{12}) q^{84} + 9 q^{85} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{86} + (6 \zeta_{12}^{2} + 6) q^{87} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{89} + (6 \zeta_{12}^{2} - 3) q^{90} + (\zeta_{12}^{2} + 4) q^{91} - \zeta_{12}^{3} q^{92} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{93} + (3 \zeta_{12}^{2} + 3) q^{94} - 6 \zeta_{12} q^{95} + (\zeta_{12}^{2} - 2) q^{96} + (12 \zeta_{12}^{2} - 6) q^{97} + (5 \zeta_{12}^{3} - 8 \zeta_{12}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{7} - 6 q^{9} - 6 q^{10} - 12 q^{15} - 2 q^{16} + 12 q^{19} - 6 q^{24} + 4 q^{25} - 8 q^{28} + 12 q^{31} - 12 q^{36} - 8 q^{37} - 6 q^{40} - 18 q^{42} + 16 q^{43} + 2 q^{46} - 26 q^{49} + 18 q^{51} + 6 q^{52} - 18 q^{54} + 12 q^{58} - 6 q^{60} + 48 q^{61} - 30 q^{63} - 4 q^{64} - 10 q^{67} - 12 q^{70} + 42 q^{73} + 12 q^{78} - 16 q^{79} - 18 q^{81} + 36 q^{85} + 36 q^{87} + 18 q^{91} + 18 q^{94} - 6 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/966\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i −0.866025 1.50000i 0.500000 + 0.866025i 0.866025 1.50000i 1.73205i 0.500000 + 2.59808i 1.00000i −1.50000 + 2.59808i −1.50000 + 0.866025i
47.2 0.866025 + 0.500000i 0.866025 + 1.50000i 0.500000 + 0.866025i −0.866025 + 1.50000i 1.73205i 0.500000 + 2.59808i 1.00000i −1.50000 + 2.59808i −1.50000 + 0.866025i
185.1 −0.866025 + 0.500000i −0.866025 + 1.50000i 0.500000 0.866025i 0.866025 + 1.50000i 1.73205i 0.500000 2.59808i 1.00000i −1.50000 2.59808i −1.50000 0.866025i
185.2 0.866025 0.500000i 0.866025 1.50000i 0.500000 0.866025i −0.866025 1.50000i 1.73205i 0.500000 2.59808i 1.00000i −1.50000 2.59808i −1.50000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.l.a 4
3.b odd 2 1 inner 966.2.l.a 4
7.d odd 6 1 inner 966.2.l.a 4
21.g even 6 1 inner 966.2.l.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.l.a 4 1.a even 1 1 trivial
966.2.l.a 4 3.b odd 2 1 inner
966.2.l.a 4 7.d odd 6 1 inner
966.2.l.a 4 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 3T_{5}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T - 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$53$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$59$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$61$ \( (T^{2} - 24 T + 192)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 225)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 21 T + 147)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$97$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
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