Properties

Label 966.2.i.l
Level $966$
Weight $2$
Character orbit 966.i
Analytic conductor $7.714$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1768034304.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - x^{6} - 6x^{5} + 14x^{4} + 18x^{3} - 31x^{2} - 14x + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - x^{6} - 6x^{5} + 14x^{4} + 18x^{3} - 31x^{2} - 14x + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{7} - 67\nu^{6} + 173\nu^{5} - 208\nu^{4} - 56\nu^{3} - 965\nu^{2} + 1303\nu + 3997 ) / 1659 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 34\nu^{7} - 33\nu^{6} + 176\nu^{5} - 218\nu^{4} + 154\nu^{3} - 1474\nu^{2} - 522\nu - 623 ) / 3318 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 128\nu^{7} - 417\nu^{6} - 1094\nu^{5} - 40\nu^{4} + 5264\nu^{3} + 7918\nu^{2} + 1548\nu - 22351 ) / 9954 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\nu^{7} - 117\nu^{6} - 8\nu^{5} - 184\nu^{4} + 704\nu^{3} + 304\nu^{2} - 1506\nu + 1655 ) / 1422 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -332\nu^{7} + 615\nu^{6} + 38\nu^{5} + 1348\nu^{4} - 6188\nu^{3} - 9028\nu^{2} + 21492\nu + 16135 ) / 9954 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 338\nu^{7} + 843\nu^{6} - 1178\nu^{5} - 1972\nu^{4} - 7252\nu^{3} + 11110\nu^{2} + 666\nu - 22393 ) / 9954 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -254\nu^{7} - 339\nu^{6} + 149\nu^{5} + 3190\nu^{4} + 3241\nu^{3} - 1870\nu^{2} - 8982\nu + 4459 ) / 4977 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - \beta_{4} - 2\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} - 4\beta_{6} - 2\beta_{5} - 3\beta_{4} + \beta_{3} - 3\beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} + 3\beta_{6} - 3\beta_{4} + 3\beta_{3} + 5\beta_{2} + 3\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{6} - 18\beta_{5} - 23\beta_{4} - 3\beta_{3} - 12\beta_{2} + 17\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{7} - 12\beta_{6} - 8\beta_{5} - 15\beta_{4} - 3\beta_{3} - 15\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 32\beta_{7} + 54\beta_{6} - 37\beta_{4} + 59\beta_{3} + 148\beta_{2} + 37\beta _1 + 207 ) / 2 \) Copy content Toggle raw display

Character values

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

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

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
−0.885685 + 1.90520i
2.09279 + 0.185573i
0.972165 0.800426i
−1.17927 + 0.441707i
−0.885685 1.90520i
2.09279 0.185573i
0.972165 + 0.800426i
−1.17927 0.441707i
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.914214 1.58346i 1.00000 −1.75255 1.98206i 1.00000 −0.500000 0.866025i −0.914214 + 1.58346i
277.2 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.914214 1.58346i 1.00000 2.45965 0.974732i 1.00000 −0.500000 0.866025i −0.914214 + 1.58346i
277.3 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.91421 + 3.31552i 1.00000 −1.87485 + 1.86680i 1.00000 −0.500000 0.866025i 1.91421 3.31552i
277.4 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.91421 + 3.31552i 1.00000 1.16774 2.37411i 1.00000 −0.500000 0.866025i 1.91421 3.31552i
415.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.914214 + 1.58346i 1.00000 −1.75255 + 1.98206i 1.00000 −0.500000 + 0.866025i −0.914214 1.58346i
415.2 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.914214 + 1.58346i 1.00000 2.45965 + 0.974732i 1.00000 −0.500000 + 0.866025i −0.914214 1.58346i
415.3 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.91421 3.31552i 1.00000 −1.87485 1.86680i 1.00000 −0.500000 + 0.866025i 1.91421 + 3.31552i
415.4 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.91421 3.31552i 1.00000 1.16774 + 2.37411i 1.00000 −0.500000 + 0.866025i 1.91421 + 3.31552i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.4
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.l 8
7.c even 3 1 inner 966.2.i.l 8
7.c even 3 1 6762.2.a.cp 4
7.d odd 6 1 6762.2.a.cm 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.l 8 1.a even 1 1 trivial
966.2.i.l 8 7.c even 3 1 inner
6762.2.a.cm 4 7.d odd 6 1
6762.2.a.cp 4 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 \(S_{2}^{\mathrm{new}}(966, [\chi])\):

\( T_{5}^{4} - 2T_{5}^{3} + 11T_{5}^{2} + 14T_{5} + 49 \) Copy content Toggle raw display
\( T_{11}^{8} + 6T_{11}^{7} + 55T_{11}^{6} + 198T_{11}^{5} + 1485T_{11}^{4} + 5220T_{11}^{3} + 20764T_{11}^{2} + 29328T_{11} + 35344 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{3} + 11 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 12 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + \cdots + 35344 \) Copy content Toggle raw display
$13$ \( (T^{4} - 6 T^{3} - T^{2} + \cdots - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 10 T^{7} + \cdots + 21316 \) Copy content Toggle raw display
$19$ \( T^{8} - 4 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 6 T^{3} - T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 2 T^{7} + \cdots + 35344 \) Copy content Toggle raw display
$37$ \( T^{8} + 58 T^{6} + \cdots + 18496 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} - 30 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 20 T^{3} + \cdots - 5696)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 10 T^{7} + \cdots + 1817104 \) Copy content Toggle raw display
$53$ \( T^{8} + 58 T^{6} + \cdots + 305809 \) Copy content Toggle raw display
$59$ \( T^{8} + 6 T^{7} + \cdots + 12544 \) Copy content Toggle raw display
$61$ \( T^{8} + 232 T^{6} + \cdots + 78287104 \) Copy content Toggle raw display
$67$ \( T^{8} + 18 T^{7} + \cdots + 125350416 \) Copy content Toggle raw display
$71$ \( (T^{4} - 42 T^{3} + \cdots + 10654)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 6 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$79$ \( T^{8} - 6 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( (T^{4} + 10 T^{3} + \cdots + 1252)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 106 T^{6} + \cdots + 399424 \) Copy content Toggle raw display
$97$ \( (T^{4} + 10 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
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