Properties

Label 966.2.i.c
Level $966$
Weight $2$
Character orbit 966.i
Analytic conductor $7.714$
Analytic rank $0$
Dimension $2$
CM no
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} - q^{6} + ( - 3 \zeta_{6} + 1) q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} - q^{6} + ( - 3 \zeta_{6} + 1) q^{7} + q^{8} - \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{10} + ( - 3 \zeta_{6} + 3) q^{11} + \zeta_{6} q^{12} + 2 q^{13} + (2 \zeta_{6} - 3) q^{14} - q^{15} - \zeta_{6} q^{16} + ( - 2 \zeta_{6} + 2) q^{17} + (\zeta_{6} - 1) q^{18} + 2 \zeta_{6} q^{19} + q^{20} + ( - \zeta_{6} - 2) q^{21} - 3 q^{22} - \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{24} + ( - 4 \zeta_{6} + 4) q^{25} - 2 \zeta_{6} q^{26} - q^{27} + (\zeta_{6} + 2) q^{28} - 7 q^{29} + \zeta_{6} q^{30} + (7 \zeta_{6} - 7) q^{31} + (\zeta_{6} - 1) q^{32} - 3 \zeta_{6} q^{33} - 2 q^{34} + (2 \zeta_{6} - 3) q^{35} + q^{36} + 2 \zeta_{6} q^{37} + ( - 2 \zeta_{6} + 2) q^{38} + ( - 2 \zeta_{6} + 2) q^{39} - \zeta_{6} q^{40} - 2 q^{41} + (3 \zeta_{6} - 1) q^{42} - 6 q^{43} + 3 \zeta_{6} q^{44} + (\zeta_{6} - 1) q^{45} + (\zeta_{6} - 1) q^{46} - 6 \zeta_{6} q^{47} - q^{48} + (3 \zeta_{6} - 8) q^{49} - 4 q^{50} - 2 \zeta_{6} q^{51} + (2 \zeta_{6} - 2) q^{52} + ( - 9 \zeta_{6} + 9) q^{53} + \zeta_{6} q^{54} - 3 q^{55} + ( - 3 \zeta_{6} + 1) q^{56} + 2 q^{57} + 7 \zeta_{6} q^{58} + ( - 9 \zeta_{6} + 9) q^{59} + ( - \zeta_{6} + 1) q^{60} - 6 \zeta_{6} q^{61} + 7 q^{62} + (2 \zeta_{6} - 3) q^{63} + q^{64} - 2 \zeta_{6} q^{65} + (3 \zeta_{6} - 3) q^{66} + (10 \zeta_{6} - 10) q^{67} + 2 \zeta_{6} q^{68} - q^{69} + (\zeta_{6} + 2) q^{70} - 8 q^{71} - \zeta_{6} q^{72} + ( - 10 \zeta_{6} + 10) q^{73} + ( - 2 \zeta_{6} + 2) q^{74} - 4 \zeta_{6} q^{75} - 2 q^{76} + ( - 3 \zeta_{6} - 6) q^{77} - 2 q^{78} + 15 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{80} + (\zeta_{6} - 1) q^{81} + 2 \zeta_{6} q^{82} + q^{83} + ( - 2 \zeta_{6} + 3) q^{84} - 2 q^{85} + 6 \zeta_{6} q^{86} + (7 \zeta_{6} - 7) q^{87} + ( - 3 \zeta_{6} + 3) q^{88} + q^{90} + ( - 6 \zeta_{6} + 2) q^{91} + q^{92} + 7 \zeta_{6} q^{93} + (6 \zeta_{6} - 6) q^{94} + ( - 2 \zeta_{6} + 2) q^{95} + \zeta_{6} q^{96} + 5 q^{97} + (5 \zeta_{6} + 3) q^{98} - 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} - q^{5} - 2 q^{6} - q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{3} - q^{4} - q^{5} - 2 q^{6} - q^{7} + 2 q^{8} - q^{9} - q^{10} + 3 q^{11} + q^{12} + 4 q^{13} - 4 q^{14} - 2 q^{15} - q^{16} + 2 q^{17} - q^{18} + 2 q^{19} + 2 q^{20} - 5 q^{21} - 6 q^{22} - q^{23} + q^{24} + 4 q^{25} - 2 q^{26} - 2 q^{27} + 5 q^{28} - 14 q^{29} + q^{30} - 7 q^{31} - q^{32} - 3 q^{33} - 4 q^{34} - 4 q^{35} + 2 q^{36} + 2 q^{37} + 2 q^{38} + 2 q^{39} - q^{40} - 4 q^{41} + q^{42} - 12 q^{43} + 3 q^{44} - q^{45} - q^{46} - 6 q^{47} - 2 q^{48} - 13 q^{49} - 8 q^{50} - 2 q^{51} - 2 q^{52} + 9 q^{53} + q^{54} - 6 q^{55} - q^{56} + 4 q^{57} + 7 q^{58} + 9 q^{59} + q^{60} - 6 q^{61} + 14 q^{62} - 4 q^{63} + 2 q^{64} - 2 q^{65} - 3 q^{66} - 10 q^{67} + 2 q^{68} - 2 q^{69} + 5 q^{70} - 16 q^{71} - q^{72} + 10 q^{73} + 2 q^{74} - 4 q^{75} - 4 q^{76} - 15 q^{77} - 4 q^{78} + 15 q^{79} - q^{80} - q^{81} + 2 q^{82} + 2 q^{83} + 4 q^{84} - 4 q^{85} + 6 q^{86} - 7 q^{87} + 3 q^{88} + 2 q^{90} - 2 q^{91} + 2 q^{92} + 7 q^{93} - 6 q^{94} + 2 q^{95} + q^{96} + 10 q^{97} + 11 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.00000 −0.500000 2.59808i 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 −0.500000 + 2.59808i 1.00000 −0.500000 + 0.866025i −0.500000 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
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.c 2
7.c even 3 1 inner 966.2.i.c 2
7.c even 3 1 6762.2.a.bb 1
7.d odd 6 1 6762.2.a.bh 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.c 2 1.a even 1 1 trivial
966.2.i.c 2 7.c even 3 1 inner
6762.2.a.bb 1 7.c even 3 1
6762.2.a.bh 1 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
\( T_{11}^{2} - 3T_{11} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$29$ \( (T + 7)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T + 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$67$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$79$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$83$ \( (T - 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 5)^{2} \) Copy content Toggle raw display
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