Properties

Label 966.2.i.m
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: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 13x^{6} + 10x^{5} + 47x^{4} + 180x^{3} + 220x^{2} + 768x + 1164 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} - \beta_1 q^{3} + \beta_1 q^{4} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{5} - q^{6} + ( - \beta_{5} - \beta_1) q^{7} + q^{8} + ( - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} - \beta_1 q^{3} + \beta_1 q^{4} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{5} - q^{6} + ( - \beta_{5} - \beta_1) q^{7} + q^{8} + ( - \beta_1 - 1) q^{9} + ( - \beta_{5} - \beta_{4}) q^{10} + (\beta_{5} - \beta_{3} + \beta_{2} + 1) q^{11} + (\beta_1 + 1) q^{12} + (\beta_{6} + 1) q^{13} + (\beta_{5} - \beta_{3}) q^{14} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 1) q^{15}+ \cdots + ( - \beta_{4} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 4 q^{3} - 4 q^{4} - 2 q^{5} - 8 q^{6} + 6 q^{7} + 8 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 4 q^{3} - 4 q^{4} - 2 q^{5} - 8 q^{6} + 6 q^{7} + 8 q^{8} - 4 q^{9} - 2 q^{10} + 4 q^{12} + 12 q^{13} - 6 q^{14} - 4 q^{15} - 4 q^{16} - 2 q^{17} - 4 q^{18} + 8 q^{19} + 4 q^{20} + 4 q^{23} + 4 q^{24} - 10 q^{25} - 6 q^{26} - 8 q^{27} + 28 q^{29} + 2 q^{30} + 8 q^{31} - 4 q^{32} + 4 q^{34} + 26 q^{35} + 8 q^{36} - 20 q^{37} + 8 q^{38} + 6 q^{39} - 2 q^{40} + 8 q^{41} - 6 q^{42} + 16 q^{43} - 2 q^{45} + 4 q^{46} + 4 q^{47} - 8 q^{48} + 12 q^{49} + 20 q^{50} + 2 q^{51} - 6 q^{52} - 18 q^{53} + 4 q^{54} - 32 q^{55} + 6 q^{56} + 16 q^{57} - 14 q^{58} - 8 q^{59} + 2 q^{60} - 4 q^{61} - 16 q^{62} - 6 q^{63} + 8 q^{64} - 14 q^{65} + 2 q^{67} - 2 q^{68} + 8 q^{69} - 16 q^{70} + 8 q^{71} - 4 q^{72} - 8 q^{73} - 20 q^{74} + 10 q^{75} - 16 q^{76} + 62 q^{77} - 12 q^{78} - 32 q^{79} - 2 q^{80} - 4 q^{81} - 4 q^{82} - 8 q^{83} + 6 q^{84} + 28 q^{85} - 8 q^{86} + 14 q^{87} + 8 q^{89} + 4 q^{90} + 2 q^{91} - 8 q^{92} - 8 q^{93} + 4 q^{94} + 4 q^{95} + 4 q^{96} - 64 q^{97} + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 13x^{6} + 10x^{5} + 47x^{4} + 180x^{3} + 220x^{2} + 768x + 1164 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 19\nu^{5} + 8\nu^{4} - 145\nu^{3} - 142\nu^{2} - 254\nu - 1064 ) / 784 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 6\nu^{6} - 37\nu^{5} + 106\nu^{4} - 183\nu^{3} - 108\nu^{2} - 742\nu - 2228 ) / 784 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} + 33\nu^{6} - 11\nu^{5} + 275\nu^{4} + 215\nu^{3} + 2227\nu^{2} + 4228\nu + 6758 ) / 2744 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 3\nu^{6} - 9\nu^{5} + 49\nu^{4} - 19\nu^{3} + 17\nu^{2} + 148\nu - 582 ) / 392 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{7} + 41\nu^{6} - 107\nu^{5} + 99\nu^{4} - 337\nu^{3} + 235\nu^{2} + 476\nu - 3138 ) / 2744 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} - 9\nu^{6} + 24\nu^{5} - 40\nu^{4} - 100\nu^{3} + 120\nu^{2} - 840\nu - 529 ) / 343 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -39\nu^{7} + 32\nu^{6} - 517\nu^{5} - 648\nu^{4} - 3223\nu^{3} - 3474\nu^{2} - 16786\nu - 19592 ) / 5488 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{4} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 2\beta_{5} + 2\beta_{3} - \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{6} - 5\beta_{5} - 5\beta_{4} + 2\beta_{3} + 4\beta_{2} - 5\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{7} - \beta_{6} + 2\beta_{5} - 18\beta_{4} + 16\beta_{2} + 7\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -25\beta_{7} + 13\beta_{6} + 59\beta_{5} - 11\beta_{4} - 29\beta_{3} + 5\beta_{2} + 4\beta _1 + 73 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{7} + 31\beta_{6} + 156\beta_{5} + 132\beta_{4} - 78\beta_{3} - 74\beta_{2} - 114\beta _1 + 235 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 132\beta_{7} + 35\beta_{6} - 96\beta_{5} + 536\beta_{4} - 23\beta_{3} - 293\beta_{2} - 191\beta _1 + 176 \) Copy content Toggle raw display

Character values

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

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

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
1.76655 + 2.53227i
1.26426 2.63252i
−0.415888 + 2.23501i
−1.61492 0.402708i
1.76655 2.53227i
1.26426 + 2.63252i
−0.415888 2.23501i
−1.61492 + 0.402708i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.76655 3.05976i −1.00000 1.80974 + 1.92999i 1.00000 −0.500000 0.866025i −1.76655 + 3.05976i
277.2 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.26426 2.18976i −1.00000 −2.41196 1.08740i 1.00000 −0.500000 0.866025i −1.26426 + 2.18976i
277.3 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.415888 + 0.720339i −1.00000 2.64352 0.108691i 1.00000 −0.500000 0.866025i 0.415888 0.720339i
277.4 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.61492 + 2.79713i −1.00000 0.958707 2.46594i 1.00000 −0.500000 0.866025i 1.61492 2.79713i
415.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.76655 + 3.05976i −1.00000 1.80974 1.92999i 1.00000 −0.500000 + 0.866025i −1.76655 3.05976i
415.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.26426 + 2.18976i −1.00000 −2.41196 + 1.08740i 1.00000 −0.500000 + 0.866025i −1.26426 2.18976i
415.3 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.415888 0.720339i −1.00000 2.64352 + 0.108691i 1.00000 −0.500000 + 0.866025i 0.415888 + 0.720339i
415.4 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.61492 2.79713i −1.00000 0.958707 + 2.46594i 1.00000 −0.500000 + 0.866025i 1.61492 + 2.79713i
\(n\): 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.m 8
7.c even 3 1 inner 966.2.i.m 8
7.c even 3 1 6762.2.a.cl 4
7.d odd 6 1 6762.2.a.cr 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.m 8 1.a even 1 1 trivial
966.2.i.m 8 7.c even 3 1 inner
6762.2.a.cl 4 7.c even 3 1
6762.2.a.cr 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 \(S_{2}^{\mathrm{new}}(966, [\chi])\):

\( T_{5}^{8} + 2T_{5}^{7} + 17T_{5}^{6} + 14T_{5}^{5} + 185T_{5}^{4} + 164T_{5}^{3} + 712T_{5}^{2} - 480T_{5} + 576 \) Copy content Toggle raw display
\( T_{11}^{8} + 31T_{11}^{6} + 24T_{11}^{5} + 781T_{11}^{4} + 372T_{11}^{3} + 5724T_{11}^{2} - 2160T_{11} + 32400 \) 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
$5$ \( T^{8} + 2 T^{7} + \cdots + 576 \) Copy content Toggle raw display
$7$ \( T^{8} - 6 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 31 T^{6} + \cdots + 32400 \) Copy content Toggle raw display
$13$ \( (T^{4} - 6 T^{3} - 27 T^{2} + \cdots - 60)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 14 T^{3} + \cdots - 40)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + 20 T^{7} + \cdots + 331776 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} + \cdots + 480)^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{8} \) Copy content Toggle raw display
$47$ \( T^{8} - 4 T^{7} + \cdots + 7080921 \) Copy content Toggle raw display
$53$ \( T^{8} + 18 T^{7} + \cdots + 104976 \) Copy content Toggle raw display
$59$ \( T^{8} + 8 T^{7} + \cdots + 37748736 \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{7} + \cdots + 4194304 \) Copy content Toggle raw display
$67$ \( T^{8} - 2 T^{7} + \cdots + 10942864 \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{3} + \cdots + 8181)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 8 T^{7} + \cdots + 4100625 \) Copy content Toggle raw display
$79$ \( T^{8} + 32 T^{7} + \cdots + 5702544 \) Copy content Toggle raw display
$83$ \( (T^{4} + 4 T^{3} - 112 T^{2} + \cdots + 48)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 8 T^{7} + \cdots + 5308416 \) Copy content Toggle raw display
$97$ \( (T + 8)^{8} \) Copy content Toggle raw display
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