Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [966,2,Mod(277,966)]
 
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([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.277");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.i (of order \(3\), 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 \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{2} - 1) q^{4} + \beta_1 q^{5} - q^{6} + (3 \beta_{2} - 1) q^{7} - q^{8} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{2} - 1) q^{4} + \beta_1 q^{5} - q^{6} + (3 \beta_{2} - 1) q^{7} - q^{8} - \beta_{2} q^{9} + (\beta_{3} + \beta_1) q^{10} + (3 \beta_{2} - 3) q^{11} - \beta_{2} q^{12} + (2 \beta_{3} + 2) q^{13} + (2 \beta_{2} - 3) q^{14} + \beta_{3} q^{15} - \beta_{2} q^{16} + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots - 3) q^{17}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + q^{5} - 4 q^{6} + 2 q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + q^{5} - 4 q^{6} + 2 q^{7} - 4 q^{8} - 2 q^{9} - q^{10} - 6 q^{11} - 2 q^{12} + 4 q^{13} - 8 q^{14} - 2 q^{15} - 2 q^{16} - 3 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 10 q^{21} - 12 q^{22} + 2 q^{23} + 2 q^{24} + q^{25} + 2 q^{26} + 4 q^{27} - 10 q^{28} - 16 q^{29} - q^{30} - 9 q^{31} + 2 q^{32} - 6 q^{33} - 6 q^{34} - 4 q^{35} + 4 q^{36} - 10 q^{37} + 2 q^{38} - 2 q^{39} - q^{40} - 12 q^{41} - 2 q^{42} + 16 q^{43} - 6 q^{44} + q^{45} - 2 q^{46} + 5 q^{47} + 4 q^{48} - 26 q^{49} + 2 q^{50} - 3 q^{51} - 2 q^{52} - 3 q^{53} + 2 q^{54} - 6 q^{55} - 2 q^{56} + 4 q^{57} - 8 q^{58} - 7 q^{59} + q^{60} + 4 q^{61} - 18 q^{62} + 8 q^{63} + 4 q^{64} - 16 q^{65} + 6 q^{66} + 4 q^{67} - 3 q^{68} - 4 q^{69} - 5 q^{70} + 10 q^{71} + 2 q^{72} - 5 q^{73} + 10 q^{74} + q^{75} + 4 q^{76} - 30 q^{77} - 4 q^{78} + q^{80} - 2 q^{81} - 6 q^{82} - 18 q^{83} + 8 q^{84} + 48 q^{85} + 8 q^{86} + 8 q^{87} + 6 q^{88} + 2 q^{90} + 2 q^{91} - 4 q^{92} - 9 q^{93} - 5 q^{94} - 16 q^{95} + 2 q^{96} + 22 q^{97} - 22 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) Copy content Toggle raw display

Character values

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

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
−0.780776 1.35234i
1.28078 + 2.21837i
−0.780776 + 1.35234i
1.28078 2.21837i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.780776 1.35234i −1.00000 0.500000 + 2.59808i −1.00000 −0.500000 0.866025i 0.780776 1.35234i
277.2 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.28078 + 2.21837i −1.00000 0.500000 + 2.59808i −1.00000 −0.500000 0.866025i −1.28078 + 2.21837i
415.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.780776 + 1.35234i −1.00000 0.500000 2.59808i −1.00000 −0.500000 + 0.866025i 0.780776 + 1.35234i
415.2 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.28078 2.21837i −1.00000 0.500000 2.59808i −1.00000 −0.500000 + 0.866025i −1.28078 2.21837i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.i.i 4
7.c even 3 1 inner 966.2.i.i 4
7.c even 3 1 6762.2.a.bx 2
7.d odd 6 1 6762.2.a.br 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.i 4 1.a even 1 1 trivial
966.2.i.i 4 7.c even 3 1 inner
6762.2.a.br 2 7.d odd 6 1
6762.2.a.bx 2 7.c even 3 1

Hecke kernels

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

\( T_{5}^{4} - T_{5}^{3} + 5T_{5}^{2} + 4T_{5} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{3} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T - 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 3 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 9 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$37$ \( T^{4} + 10 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T - 52)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 5 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$53$ \( T^{4} + 3 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$59$ \( T^{4} + 7 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 5 T + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 5 T^{3} + \cdots + 40804 \) Copy content Toggle raw display
$79$ \( T^{4} + 153 T^{2} + 23409 \) Copy content Toggle raw display
$83$ \( (T^{2} + 9 T + 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 68T^{2} + 4624 \) Copy content Toggle raw display
$97$ \( (T^{2} - 11 T - 8)^{2} \) Copy content Toggle raw display
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