Properties

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

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [966,2,Mod(461,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.461");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.f (of order \(2\), degree \(1\), 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: \(4\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{3} - q^{4} - \beta_{3} q^{5} + \beta_{3} q^{6} + ( - \beta_{2} + 2) q^{7} + \beta_1 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{3} - q^{4} - \beta_{3} q^{5} + \beta_{3} q^{6} + ( - \beta_{2} + 2) q^{7} + \beta_1 q^{8} - 3 q^{9} + \beta_{2} q^{10} - \beta_{2} q^{12} + \beta_{2} q^{13} + ( - \beta_{3} - 2 \beta_1) q^{14} - 3 \beta_1 q^{15} + q^{16} - 2 \beta_{3} q^{17} + 3 \beta_1 q^{18} + 4 \beta_{2} q^{19} + \beta_{3} q^{20} + (2 \beta_{2} + 3) q^{21} + \beta_1 q^{23} - \beta_{3} q^{24} - 2 q^{25} + \beta_{3} q^{26} - 3 \beta_{2} q^{27} + (\beta_{2} - 2) q^{28} - 9 \beta_1 q^{29} - 3 q^{30} + 6 \beta_{2} q^{31} - \beta_1 q^{32} + 2 \beta_{2} q^{34} + ( - 2 \beta_{3} + 3 \beta_1) q^{35} + 3 q^{36} - 5 q^{37} + 4 \beta_{3} q^{38} - 3 q^{39} - \beta_{2} q^{40} - 7 \beta_{3} q^{41} + (2 \beta_{3} - 3 \beta_1) q^{42} + q^{43} + 3 \beta_{3} q^{45} + q^{46} - 5 \beta_{3} q^{47} + \beta_{2} q^{48} + ( - 4 \beta_{2} + 1) q^{49} + 2 \beta_1 q^{50} - 6 \beta_1 q^{51} - \beta_{2} q^{52} + 12 \beta_1 q^{53} - 3 \beta_{3} q^{54} + (\beta_{3} + 2 \beta_1) q^{56} - 12 q^{57} - 9 q^{58} + 2 \beta_{3} q^{59} + 3 \beta_1 q^{60} + 6 \beta_{3} q^{62} + (3 \beta_{2} - 6) q^{63} - q^{64} - 3 \beta_1 q^{65} - 4 q^{67} + 2 \beta_{3} q^{68} - \beta_{3} q^{69} + (2 \beta_{2} + 3) q^{70} + 12 \beta_1 q^{71} - 3 \beta_1 q^{72} + 4 \beta_{2} q^{73} + 5 \beta_1 q^{74} - 2 \beta_{2} q^{75} - 4 \beta_{2} q^{76} + 3 \beta_1 q^{78} - 10 q^{79} - \beta_{3} q^{80} + 9 q^{81} + 7 \beta_{2} q^{82} - 2 \beta_{3} q^{83} + ( - 2 \beta_{2} - 3) q^{84} + 6 q^{85} - \beta_1 q^{86} + 9 \beta_{3} q^{87} + 2 \beta_{3} q^{89} - 3 \beta_{2} q^{90} + (2 \beta_{2} + 3) q^{91} - \beta_1 q^{92} - 18 q^{93} + 5 \beta_{2} q^{94} - 12 \beta_1 q^{95} + \beta_{3} q^{96} - 7 \beta_{2} q^{97} + ( - 4 \beta_{3} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 8 q^{7} - 12 q^{9} + 4 q^{16} + 12 q^{21} - 8 q^{25} - 8 q^{28} - 12 q^{30} + 12 q^{36} - 20 q^{37} - 12 q^{39} + 4 q^{43} + 4 q^{46} + 4 q^{49} - 48 q^{57} - 36 q^{58} - 24 q^{63} - 4 q^{64} - 16 q^{67} + 12 q^{70} - 40 q^{79} + 36 q^{81} - 12 q^{84} + 24 q^{85} + 12 q^{91} - 72 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
461.1
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
1.00000i 1.73205i −1.00000 1.73205 −1.73205 2.00000 + 1.73205i 1.00000i −3.00000 1.73205i
461.2 1.00000i 1.73205i −1.00000 −1.73205 1.73205 2.00000 1.73205i 1.00000i −3.00000 1.73205i
461.3 1.00000i 1.73205i −1.00000 −1.73205 1.73205 2.00000 + 1.73205i 1.00000i −3.00000 1.73205i
461.4 1.00000i 1.73205i −1.00000 1.73205 −1.73205 2.00000 1.73205i 1.00000i −3.00000 1.73205i
\(n\): 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.b odd 2 1 inner
21.c even 2 1 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$37$ \( (T + 5)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 147)^{2} \) Copy content Toggle raw display
$43$ \( (T - 1)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T + 4)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$79$ \( (T + 10)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 147)^{2} \) Copy content Toggle raw display
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