Properties

Label 966.2.i.k
Level $966$
Weight $2$
Character orbit 966.i
Analytic conductor $7.714$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.29428272.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{4} - 4x^{3} - 42x^{2} + 343 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} - \beta_{3} q^{3} + \beta_{3} q^{4} + ( - \beta_{3} - 1) q^{5} + q^{6} + ( - \beta_{5} - 1) q^{7} - q^{8} + ( - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{2} - \beta_{3} q^{3} + \beta_{3} q^{4} + ( - \beta_{3} - 1) q^{5} + q^{6} + ( - \beta_{5} - 1) q^{7} - q^{8} + ( - \beta_{3} - 1) q^{9} - \beta_{3} q^{10} + (\beta_{5} - \beta_{2} + 1) q^{11} + (\beta_{3} + 1) q^{12} + ( - \beta_{5} - \beta_{3} - \beta_{2} + \cdots - 1) q^{13}+ \cdots + (\beta_{4} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} + 3 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} - 3 q^{7} - 6 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} + 3 q^{3} - 3 q^{4} - 3 q^{5} + 6 q^{6} - 3 q^{7} - 6 q^{8} - 3 q^{9} + 3 q^{10} + 3 q^{12} - 6 q^{13} - 3 q^{14} - 6 q^{15} - 3 q^{16} - 3 q^{17} + 3 q^{18} - 6 q^{19} + 6 q^{20} + 3 q^{23} - 3 q^{24} + 12 q^{25} - 3 q^{26} - 6 q^{27} + 24 q^{29} - 3 q^{30} + 3 q^{32} - 6 q^{34} + 3 q^{35} + 6 q^{36} + 12 q^{37} + 6 q^{38} - 3 q^{39} + 3 q^{40} - 36 q^{41} - 3 q^{42} - 3 q^{45} - 3 q^{46} + 3 q^{47} - 6 q^{48} - 3 q^{49} + 24 q^{50} + 3 q^{51} + 3 q^{52} + 3 q^{53} - 3 q^{54} + 3 q^{56} - 12 q^{57} + 12 q^{58} + 12 q^{59} + 3 q^{60} - 30 q^{61} + 3 q^{63} + 6 q^{64} + 3 q^{65} + 21 q^{67} - 3 q^{68} + 6 q^{69} - 6 q^{71} + 3 q^{72} - 15 q^{73} - 12 q^{74} - 12 q^{75} + 12 q^{76} + 24 q^{77} - 6 q^{78} + 12 q^{79} - 3 q^{80} - 3 q^{81} - 18 q^{82} - 24 q^{83} - 3 q^{84} + 6 q^{85} + 12 q^{87} - 12 q^{89} - 6 q^{90} + 33 q^{91} - 6 q^{92} - 3 q^{94} - 6 q^{95} - 3 q^{96} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 6x^{4} - 4x^{3} - 42x^{2} + 343 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{5} + 35\nu^{4} + 31\nu^{3} + 121\nu^{2} - 217\nu - 1519 ) / 490 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{5} + 7\nu^{4} + 19\nu^{3} - 13\nu^{2} + 203\nu - 147 ) / 490 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} - 7\nu^{4} - 19\nu^{3} + 13\nu^{2} + 287\nu + 147 ) / 490 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 7\nu^{4} - \nu^{3} + 11\nu^{2} + 91\nu + 245 ) / 70 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{5} - 35\nu^{4} + 29\nu^{3} - 81\nu^{2} + 637\nu + 1519 ) / 490 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + 2\beta_{4} - \beta_{2} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{5} + \beta_{4} - 11\beta_{3} + \beta_{2} + 4\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - 9\beta_{4} + 18\beta_{3} + 5\beta_{2} + 5\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -23\beta_{5} + 14\beta_{4} - 24\beta_{3} + 44\beta_{2} - 17\beta _1 - 10 \) Copy content Toggle raw display

Character values

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

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(-1 - \beta_{3}\) \(1\)

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
−0.0741344 + 2.64471i
2.63435 0.245357i
−2.56022 0.667305i
−0.0741344 2.64471i
2.63435 + 0.245357i
−2.56022 + 0.667305i
0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −2.25332 + 1.38656i −1.00000 −0.500000 0.866025i 0.500000 0.866025i
277.2 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −1.10469 2.40409i −1.00000 −0.500000 0.866025i 0.500000 0.866025i
277.3 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000 1.85801 + 1.88356i −1.00000 −0.500000 0.866025i 0.500000 0.866025i
415.1 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000 −2.25332 1.38656i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
415.2 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000 −1.10469 + 2.40409i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
415.3 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000 1.85801 1.88356i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.3
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.k 6
7.c even 3 1 inner 966.2.i.k 6
7.c even 3 1 6762.2.a.ce 3
7.d odd 6 1 6762.2.a.cf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.k 6 1.a even 1 1 trivial
966.2.i.k 6 7.c even 3 1 inner
6762.2.a.ce 3 7.c even 3 1
6762.2.a.cf 3 7.d odd 6 1

Hecke kernels

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

\( T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display
\( T_{11}^{6} + 15T_{11}^{4} - 12T_{11}^{3} + 225T_{11}^{2} - 90T_{11} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} + 15 T^{4} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( (T^{3} + 3 T^{2} - 24 T - 22)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + \cdots + 15876 \) Copy content Toggle raw display
$19$ \( T^{6} + 6 T^{5} + \cdots + 2304 \) Copy content Toggle raw display
$23$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$29$ \( (T^{3} - 12 T^{2} + \cdots + 48)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 75 T^{4} + \cdots + 45796 \) Copy content Toggle raw display
$37$ \( T^{6} - 12 T^{5} + \cdots + 345744 \) Copy content Toggle raw display
$41$ \( (T^{3} + 18 T^{2} + \cdots + 128)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 60 T + 48)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 3 T^{5} + \cdots + 19600 \) Copy content Toggle raw display
$53$ \( T^{6} - 3 T^{5} + \cdots + 121 \) Copy content Toggle raw display
$59$ \( T^{6} - 12 T^{5} + \cdots + 32400 \) Copy content Toggle raw display
$61$ \( (T^{2} + 10 T + 100)^{3} \) Copy content Toggle raw display
$67$ \( T^{6} - 21 T^{5} + \cdots + 156816 \) Copy content Toggle raw display
$71$ \( (T^{3} + 3 T^{2} + \cdots - 726)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 15 T^{5} + \cdots + 3136 \) Copy content Toggle raw display
$79$ \( T^{6} - 12 T^{5} + \cdots + 1140624 \) Copy content Toggle raw display
$83$ \( (T^{3} + 12 T^{2} + 33 T - 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 12 T^{5} + \cdots + 345744 \) Copy content Toggle raw display
$97$ \( (T^{3} - 243 T + 108)^{2} \) Copy content Toggle raw display
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