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Properties

Label 966.2.i.n
Level $966$
Weight $2$
Character orbit 966.i
Analytic conductor $7.714$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.173309020416.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 5x^{6} - 28x^{5} - 4x^{4} + 70x^{3} + 51x^{2} + 406x + 841 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} + 1) q^{2} - \beta_{5} q^{3} + \beta_{5} q^{4} - \beta_1 q^{5} + q^{6} + ( - \beta_{4} + \beta_{3}) q^{7} - q^{8} + ( - \beta_{5} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + 1) q^{2} - \beta_{5} q^{3} + \beta_{5} q^{4} - \beta_1 q^{5} + q^{6} + ( - \beta_{4} + \beta_{3}) q^{7} - q^{8} + ( - \beta_{5} - 1) q^{9} + (\beta_{7} - \beta_1 + 1) q^{10} + ( - \beta_{7} + \beta_{5} - \beta_{2} + \cdots - 1) q^{11}+ \cdots + (\beta_{7} + \beta_{6} + \beta_{4} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 4 q^{3} - 4 q^{4} - 2 q^{5} + 8 q^{6} - 8 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 4 q^{3} - 4 q^{4} - 2 q^{5} + 8 q^{6} - 8 q^{8} - 4 q^{9} + 2 q^{10} - 6 q^{11} + 4 q^{12} - 4 q^{13} - 4 q^{15} - 4 q^{16} + 2 q^{17} + 4 q^{18} + 6 q^{19} + 4 q^{20} - 12 q^{22} - 4 q^{23} - 4 q^{24} - 6 q^{25} - 2 q^{26} - 8 q^{27} - 20 q^{29} - 2 q^{30} + 14 q^{31} + 4 q^{32} + 6 q^{33} + 4 q^{34} + 8 q^{36} - 6 q^{37} - 6 q^{38} - 2 q^{39} + 2 q^{40} + 20 q^{41} + 16 q^{43} - 6 q^{44} - 2 q^{45} + 4 q^{46} + 10 q^{47} - 8 q^{48} - 28 q^{49} - 12 q^{50} - 2 q^{51} + 2 q^{52} - 6 q^{53} - 4 q^{54} + 44 q^{55} + 12 q^{57} - 10 q^{58} - 24 q^{59} + 2 q^{60} + 28 q^{61} + 28 q^{62} + 8 q^{64} - 18 q^{65} - 6 q^{66} - 28 q^{67} + 2 q^{68} - 8 q^{69} + 32 q^{71} + 4 q^{72} - 6 q^{73} + 6 q^{74} + 6 q^{75} - 12 q^{76} - 14 q^{77} - 4 q^{78} + 4 q^{79} - 2 q^{80} - 4 q^{81} + 10 q^{82} + 28 q^{83} - 52 q^{85} + 8 q^{86} - 10 q^{87} + 6 q^{88} + 14 q^{89} - 4 q^{90} + 14 q^{91} + 8 q^{92} - 14 q^{93} - 10 q^{94} + 34 q^{95} - 4 q^{96} + 28 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 5x^{6} - 28x^{5} - 4x^{4} + 70x^{3} + 51x^{2} + 406x + 841 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 859 \nu^{7} - 10005 \nu^{6} + 23922 \nu^{5} + 26031 \nu^{4} + 252808 \nu^{3} - 875758 \nu^{2} + \cdots + 463246 ) / 1219827 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1294 \nu^{7} - 31436 \nu^{6} - 33817 \nu^{5} - 96233 \nu^{4} + 609189 \nu^{3} + 599182 \nu^{2} + \cdots + 4558220 ) / 1219827 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4618 \nu^{7} + 27347 \nu^{6} + 66010 \nu^{5} + 220886 \nu^{4} - 712908 \nu^{3} - 1033528 \nu^{2} + \cdots - 3965315 ) / 2439654 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 55\nu^{7} - 39\nu^{6} + 44\nu^{5} - 770\nu^{4} - 3680\nu^{3} - 4466\nu^{2} - 9251\nu + 57733 ) / 28042 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -875\nu^{7} - 290\nu^{6} - 700\nu^{5} + 24268\nu^{4} + 7560\nu^{3} + 10960\nu^{2} - 21077\nu - 306472 ) / 348522 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1108 \nu^{7} + 1363 \nu^{6} - 10731 \nu^{5} - 41551 \nu^{4} + 34573 \nu^{3} + 144782 \nu^{2} + \cdots - 197635 ) / 406609 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6167 \nu^{7} + 7250 \nu^{6} + 41536 \nu^{5} + 120389 \nu^{4} - 193006 \nu^{3} - 900939 \nu^{2} + \cdots - 2154903 ) / 1219827 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{6} - 2\beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} - 4\beta_{6} - 6\beta_{5} - 6\beta_{4} + 4\beta_{3} + 2\beta_{2} - 3\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} + \beta_{6} - 3\beta_{5} - 15\beta_{4} - \beta_{3} - 2\beta_{2} + 6\beta _1 + 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{7} - 8\beta_{6} + 14\beta_{5} - 6\beta_{3} - 8\beta_{2} + 5\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -27\beta_{7} - 104\beta_{6} - 249\beta_{5} - 159\beta_{4} + 107\beta_{3} + 52\beta_{2} - 27\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -30\beta_{7} + 76\beta_{6} - 30\beta_{5} - 234\beta_{4} - 76\beta_{3} - 152\beta_{2} + 60\beta _1 + 861 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -864\beta_{7} - 625\beta_{6} + 78\beta_{5} - 470\beta_{3} - 625\beta_{2} + 432\beta _1 - 354 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(-1 - \beta_{5}\) \(1\)

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
0.435461 − 1.77894i
−1.61272 + 2.45863i
2.93560 − 0.167344i
−1.75834 − 0.512349i
0.435461 + 1.77894i
−1.61272 − 2.45863i
2.93560 + 0.167344i
−1.75834 + 0.512349i
0.500000 + 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i −1.94864 − 3.37514i 1.00000 −1.32288 + 2.29129i −1.00000 −0.500000 − 0.866025i 1.94864 − 3.37514i
277.2 0.500000 + 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i −0.739514 − 1.28088i 1.00000 1.32288 − 2.29129i −1.00000 −0.500000 − 0.866025i 0.739514 − 1.28088i
277.3 0.500000 + 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i 0.239514 + 0.414851i 1.00000 1.32288 − 2.29129i −1.00000 −0.500000 − 0.866025i −0.239514 + 0.414851i
277.4 0.500000 + 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i 1.44864 + 2.50912i 1.00000 −1.32288 + 2.29129i −1.00000 −0.500000 − 0.866025i −1.44864 + 2.50912i
415.1 0.500000 − 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i −1.94864 + 3.37514i 1.00000 −1.32288 − 2.29129i −1.00000 −0.500000 + 0.866025i 1.94864 + 3.37514i
415.2 0.500000 − 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i −0.739514 + 1.28088i 1.00000 1.32288 + 2.29129i −1.00000 −0.500000 + 0.866025i 0.739514 + 1.28088i
415.3 0.500000 − 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i 0.239514 − 0.414851i 1.00000 1.32288 + 2.29129i −1.00000 −0.500000 + 0.866025i −0.239514 − 0.414851i
415.4 0.500000 − 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i 1.44864 − 2.50912i 1.00000 −1.32288 − 2.29129i −1.00000 −0.500000 + 0.866025i −1.44864 − 2.50912i
\(n\): 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.n ✓ 8
7.c even 3 1 inner 966.2.i.n ✓ 8
7.c even 3 1 6762.2.a.cg 4
7.d odd 6 1 6762.2.a.ch 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.n ✓ 8 1.a even 1 1 trivial
966.2.i.n ✓ 8 7.c even 3 1 inner
6762.2.a.cg 4 7.c even 3 1
6762.2.a.ch 4 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 \(S_{2}^{\mathrm{new}}(966, [\chi])\):

\( T_{5}^{8} + 2T_{5}^{7} + 15T_{5}^{6} + 2T_{5}^{5} + 137T_{5}^{4} + 100T_{5}^{3} + 232T_{5}^{2} - 96T_{5} + 64 \) Copy content Toggle raw display
\( T_{11}^{8} + 6T_{11}^{7} + 58T_{11}^{6} + 208T_{11}^{5} + 1703T_{11}^{4} + 6128T_{11}^{3} + 24522T_{11}^{2} + 33830T_{11} + 39601 \) 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
$5$ \( T^{8} + 2 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{4} + 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + \cdots + 39601 \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} - 34 T^{2} + \cdots - 56)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots + 3844 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + \cdots + 87616 \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 10 T^{3} + \cdots + 189)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 14 T^{7} + \cdots + 404496 \) Copy content Toggle raw display
$37$ \( T^{8} + 6 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( (T^{4} - 10 T^{3} + \cdots - 56)^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{8} \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} + \cdots + 6697744 \) Copy content Toggle raw display
$53$ \( T^{8} + 6 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$59$ \( T^{8} + 24 T^{7} + \cdots + 50466816 \) Copy content Toggle raw display
$61$ \( T^{8} - 28 T^{7} + \cdots + 16384 \) Copy content Toggle raw display
$67$ \( T^{8} + 28 T^{7} + \cdots + 81144064 \) Copy content Toggle raw display
$71$ \( (T^{4} - 16 T^{3} + \cdots + 18)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 6 T^{7} + \cdots + 2637376 \) Copy content Toggle raw display
$79$ \( T^{8} - 4 T^{7} + \cdots + 1471369 \) Copy content Toggle raw display
$83$ \( (T^{4} - 14 T^{3} + \cdots - 3708)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 14 T^{7} + \cdots + 9437184 \) Copy content Toggle raw display
$97$ \( (T^{4} - 319 T^{2} + \cdots + 23888)^{2} \) Copy content Toggle raw display
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