Properties

Label 966.2.g.d
Level $966$
Weight $2$
Character orbit 966.g
Analytic conductor $7.714$
Analytic rank $0$
Dimension $4$
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] = [966,2,Mod(643,966)]
 
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([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.643");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.g (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: \(\Q(i, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 3x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + \beta_{2} q^{5} - \beta_1 q^{6} + \beta_{2} q^{7} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + \beta_{2} q^{5} - \beta_1 q^{6} + \beta_{2} q^{7} - q^{8} - q^{9} - \beta_{2} q^{10} + \beta_1 q^{12} + 5 \beta_1 q^{13} - \beta_{2} q^{14} + \beta_{3} q^{15} + q^{16} + 2 \beta_{2} q^{17} + q^{18} - 2 \beta_{2} q^{19} + \beta_{2} q^{20} + \beta_{3} q^{21} + (\beta_{3} + 4) q^{23} - \beta_1 q^{24} + 2 q^{25} - 5 \beta_1 q^{26} - \beta_1 q^{27} + \beta_{2} q^{28} - q^{29} - \beta_{3} q^{30} - 4 \beta_1 q^{31} - q^{32} - 2 \beta_{2} q^{34} + 7 q^{35} - q^{36} + 3 \beta_{3} q^{37} + 2 \beta_{2} q^{38} - 5 q^{39} - \beta_{2} q^{40} - 9 \beta_1 q^{41} - \beta_{3} q^{42} - \beta_{3} q^{43} - \beta_{2} q^{45} + ( - \beta_{3} - 4) q^{46} - 13 \beta_1 q^{47} + \beta_1 q^{48} + 7 q^{49} - 2 q^{50} + 2 \beta_{3} q^{51} + 5 \beta_1 q^{52} + 2 \beta_{3} q^{53} + \beta_1 q^{54} - \beta_{2} q^{56} - 2 \beta_{3} q^{57} + q^{58} + 14 \beta_1 q^{59} + \beta_{3} q^{60} - 4 \beta_{2} q^{61} + 4 \beta_1 q^{62} - \beta_{2} q^{63} + q^{64} + 5 \beta_{3} q^{65} - 2 \beta_{3} q^{67} + 2 \beta_{2} q^{68} + ( - \beta_{2} + 4 \beta_1) q^{69} - 7 q^{70} + 6 q^{71} + q^{72} + 4 \beta_1 q^{73} - 3 \beta_{3} q^{74} + 2 \beta_1 q^{75} - 2 \beta_{2} q^{76} + 5 q^{78} + 2 \beta_{3} q^{79} + \beta_{2} q^{80} + q^{81} + 9 \beta_1 q^{82} + 6 \beta_{2} q^{83} + \beta_{3} q^{84} + 14 q^{85} + \beta_{3} q^{86} - \beta_1 q^{87} + \beta_{2} q^{90} + 5 \beta_{3} q^{91} + (\beta_{3} + 4) q^{92} + 4 q^{93} + 13 \beta_1 q^{94} - 14 q^{95} - \beta_1 q^{96} - \beta_{2} q^{97} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 4 q^{9} + 4 q^{16} + 4 q^{18} + 16 q^{23} + 8 q^{25} - 4 q^{29} - 4 q^{32} + 28 q^{35} - 4 q^{36} - 20 q^{39} - 16 q^{46} + 28 q^{49} - 8 q^{50} + 4 q^{58} + 4 q^{64} - 28 q^{70} + 24 q^{71} + 4 q^{72} + 20 q^{78} + 4 q^{81} + 56 q^{85} + 16 q^{92} + 16 q^{93} - 56 q^{95} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 3x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} + 5\beta_1 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
643.1
−1.32288 0.500000i
1.32288 0.500000i
−1.32288 + 0.500000i
1.32288 + 0.500000i
−1.00000 1.00000i 1.00000 −2.64575 1.00000i −2.64575 −1.00000 −1.00000 2.64575
643.2 −1.00000 1.00000i 1.00000 2.64575 1.00000i 2.64575 −1.00000 −1.00000 −2.64575
643.3 −1.00000 1.00000i 1.00000 −2.64575 1.00000i −2.64575 −1.00000 −1.00000 2.64575
643.4 −1.00000 1.00000i 1.00000 2.64575 1.00000i 2.64575 −1.00000 −1.00000 −2.64575
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.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.g.d 4
3.b odd 2 1 2898.2.g.d 4
7.b odd 2 1 inner 966.2.g.d 4
21.c even 2 1 2898.2.g.d 4
23.b odd 2 1 inner 966.2.g.d 4
69.c even 2 1 2898.2.g.d 4
161.c even 2 1 inner 966.2.g.d 4
483.c odd 2 1 2898.2.g.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.g.d 4 1.a even 1 1 trivial
966.2.g.d 4 7.b odd 2 1 inner
966.2.g.d 4 23.b odd 2 1 inner
966.2.g.d 4 161.c even 2 1 inner
2898.2.g.d 4 3.b odd 2 1
2898.2.g.d 4 21.c even 2 1
2898.2.g.d 4 69.c even 2 1
2898.2.g.d 4 483.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\):

\( T_{5}^{2} - 7 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 8 T + 23)^{2} \) Copy content Toggle raw display
$29$ \( (T + 1)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 63)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 7)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 169)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 112)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$71$ \( (T - 6)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 252)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
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