Properties

Label 966.2.i.k
Level $966$
Weight $2$
Character orbit 966.i
Analytic conductor $7.714$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

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

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
Defining polynomial: \(x^{6} - 6 x^{4} - 4 x^{3} - 42 x^{2} + 343\)
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

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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{3} ) q^{2} -\beta_{3} q^{3} + \beta_{3} q^{4} + ( -1 - \beta_{3} ) q^{5} + q^{6} + ( -1 - \beta_{5} ) q^{7} - q^{8} + ( -1 - \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{3} ) q^{2} -\beta_{3} q^{3} + \beta_{3} q^{4} + ( -1 - \beta_{3} ) q^{5} + q^{6} + ( -1 - \beta_{5} ) q^{7} - q^{8} + ( -1 - \beta_{3} ) q^{9} -\beta_{3} q^{10} + ( 1 - \beta_{2} + \beta_{5} ) q^{11} + ( 1 + \beta_{3} ) q^{12} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{13} + ( -1 + \beta_{1} ) q^{14} - q^{15} + ( -1 - \beta_{3} ) q^{16} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{17} -\beta_{3} q^{18} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{19} + q^{20} + ( -\beta_{1} - \beta_{5} ) q^{21} + ( -\beta_{1} - \beta_{4} ) q^{22} + ( 1 + \beta_{3} ) q^{23} + \beta_{3} q^{24} -4 \beta_{3} q^{25} + ( -2 - \beta_{3} - \beta_{4} - \beta_{5} ) q^{26} - q^{27} + ( \beta_{1} + \beta_{5} ) q^{28} + ( 4 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{29} + ( -1 - \beta_{3} ) q^{30} + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{31} -\beta_{3} q^{32} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{33} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{34} + ( 1 - \beta_{1} ) q^{35} + q^{36} + ( 6 + 3 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{37} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{38} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{39} + ( 1 + \beta_{3} ) q^{40} + ( -6 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{41} + ( -1 - \beta_{5} ) q^{42} + ( -2 \beta_{1} - 2 \beta_{4} ) q^{43} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{44} + \beta_{3} q^{45} + \beta_{3} q^{46} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{47} - q^{48} + ( 2 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{49} + 4 q^{50} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{51} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{52} + ( -2 + 2 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{53} + ( -1 - \beta_{3} ) q^{54} + ( \beta_{1} + \beta_{4} ) q^{55} + ( 1 + \beta_{5} ) q^{56} + ( -2 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{57} + ( 3 + 4 \beta_{3} - \beta_{4} - \beta_{5} ) q^{58} + ( -1 - \beta_{1} + \beta_{2} - 5 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{59} -\beta_{3} q^{60} + ( -10 - 10 \beta_{3} ) q^{61} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{62} + ( 1 - \beta_{1} ) q^{63} + q^{64} + ( 2 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{65} + ( 1 - \beta_{2} + \beta_{5} ) q^{66} + ( -1 + 2 \beta_{1} + \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{67} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{68} + q^{69} + ( -\beta_{1} - \beta_{5} ) q^{70} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{71} + ( 1 + \beta_{3} ) q^{72} + ( 1 - \beta_{2} + 5 \beta_{3} + \beta_{5} ) q^{73} + ( 2 + \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{74} + ( -4 - 4 \beta_{3} ) q^{75} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{76} + ( 5 + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{77} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{78} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{79} + \beta_{3} q^{80} + \beta_{3} q^{81} + ( -6 - \beta_{1} + \beta_{2} - 6 \beta_{3} ) q^{82} + ( -4 + \beta_{1} + \beta_{4} ) q^{83} + ( -1 + \beta_{1} ) q^{84} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{85} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{86} + ( 1 - \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} ) q^{87} + ( -1 + \beta_{2} - \beta_{5} ) q^{88} + ( -6 - 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{89} - q^{90} + ( 12 + 2 \beta_{1} - 3 \beta_{2} + 8 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{91} - q^{92} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{93} + ( -1 + 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{94} + ( -\beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{95} + ( -1 - \beta_{3} ) q^{96} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{5} ) q^{97} + ( -2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{98} + ( \beta_{1} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{2} + 3q^{3} - 3q^{4} - 3q^{5} + 6q^{6} - 3q^{7} - 6q^{8} - 3q^{9} + O(q^{10}) \) \( 6q + 3q^{2} + 3q^{3} - 3q^{4} - 3q^{5} + 6q^{6} - 3q^{7} - 6q^{8} - 3q^{9} + 3q^{10} + 3q^{12} - 6q^{13} - 3q^{14} - 6q^{15} - 3q^{16} - 3q^{17} + 3q^{18} - 6q^{19} + 6q^{20} + 3q^{23} - 3q^{24} + 12q^{25} - 3q^{26} - 6q^{27} + 24q^{29} - 3q^{30} + 3q^{32} - 6q^{34} + 3q^{35} + 6q^{36} + 12q^{37} + 6q^{38} - 3q^{39} + 3q^{40} - 36q^{41} - 3q^{42} - 3q^{45} - 3q^{46} + 3q^{47} - 6q^{48} - 3q^{49} + 24q^{50} + 3q^{51} + 3q^{52} + 3q^{53} - 3q^{54} + 3q^{56} - 12q^{57} + 12q^{58} + 12q^{59} + 3q^{60} - 30q^{61} + 3q^{63} + 6q^{64} + 3q^{65} + 21q^{67} - 3q^{68} + 6q^{69} - 6q^{71} + 3q^{72} - 15q^{73} - 12q^{74} - 12q^{75} + 12q^{76} + 24q^{77} - 6q^{78} + 12q^{79} - 3q^{80} - 3q^{81} - 18q^{82} - 24q^{83} - 3q^{84} + 6q^{85} + 12q^{87} - 12q^{89} - 6q^{90} + 33q^{91} - 6q^{92} - 3q^{94} - 6q^{95} - 3q^{96} - 12q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 3 \nu^{5} + 35 \nu^{4} + 31 \nu^{3} + 121 \nu^{2} - 217 \nu - 1519 \)\()/490\)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{5} + 7 \nu^{4} + 19 \nu^{3} - 13 \nu^{2} + 203 \nu - 147 \)\()/490\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{5} - 7 \nu^{4} - 19 \nu^{3} + 13 \nu^{2} + 287 \nu + 147 \)\()/490\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} - 7 \nu^{4} - \nu^{3} + 11 \nu^{2} + 91 \nu + 245 \)\()/70\)
\(\beta_{5}\)\(=\)\((\)\( -13 \nu^{5} - 35 \nu^{4} + 29 \nu^{3} - 81 \nu^{2} + 637 \nu + 1519 \)\()/490\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{3} + \beta_{2}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + 2 \beta_{4} - \beta_{2} + 2 \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(5 \beta_{5} + \beta_{4} - 11 \beta_{3} + \beta_{2} + 4 \beta_{1} - 3\)
\(\nu^{4}\)\(=\)\(\beta_{5} - 9 \beta_{4} + 18 \beta_{3} + 5 \beta_{2} + 5 \beta_{1} + 40\)
\(\nu^{5}\)\(=\)\(-23 \beta_{5} + 14 \beta_{4} - 24 \beta_{3} + 44 \beta_{2} - 17 \beta_{1} - 10\)

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.

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 415.3
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 \(S_{2}^{\mathrm{new}}(966, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{3} \)
$3$ \( ( 1 - T + T^{2} )^{3} \)
$5$ \( ( 1 + T + T^{2} )^{3} \)
$7$ \( 343 + 147 T + 42 T^{2} + 5 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} \)
$11$ \( 36 - 90 T + 225 T^{2} - 12 T^{3} + 15 T^{4} + T^{6} \)
$13$ \( ( -22 - 24 T + 3 T^{2} + T^{3} )^{2} \)
$17$ \( 15876 + 4536 T + 1674 T^{2} + 144 T^{3} + 45 T^{4} + 3 T^{5} + T^{6} \)
$19$ \( 2304 + 288 T + 324 T^{2} + 60 T^{3} + 42 T^{4} + 6 T^{5} + T^{6} \)
$23$ \( ( 1 - T + T^{2} )^{3} \)
$29$ \( ( 48 + 21 T - 12 T^{2} + T^{3} )^{2} \)
$31$ \( 45796 + 16050 T + 5625 T^{2} + 428 T^{3} + 75 T^{4} + T^{6} \)
$37$ \( 345744 - 24696 T + 8820 T^{2} - 672 T^{3} + 186 T^{4} - 12 T^{5} + T^{6} \)
$41$ \( ( 128 + 90 T + 18 T^{2} + T^{3} )^{2} \)
$43$ \( ( 48 - 60 T + T^{3} )^{2} \)
$47$ \( 19600 + 10080 T + 4764 T^{2} + 496 T^{3} + 81 T^{4} - 3 T^{5} + T^{6} \)
$53$ \( 121 - 627 T + 3282 T^{2} + 149 T^{3} + 66 T^{4} - 3 T^{5} + T^{6} \)
$59$ \( 32400 + 1620 T + 2241 T^{2} - 468 T^{3} + 135 T^{4} - 12 T^{5} + T^{6} \)
$61$ \( ( 100 + 10 T + T^{2} )^{3} \)
$67$ \( 156816 + 28512 T + 13500 T^{2} - 2304 T^{3} + 369 T^{4} - 21 T^{5} + T^{6} \)
$71$ \( ( -726 - 132 T + 3 T^{2} + T^{3} )^{2} \)
$73$ \( 3136 + 3360 T + 2760 T^{2} + 788 T^{3} + 165 T^{4} + 15 T^{5} + T^{6} \)
$79$ \( 1140624 - 92916 T + 20385 T^{2} - 1092 T^{3} + 231 T^{4} - 12 T^{5} + T^{6} \)
$83$ \( ( -2 + 33 T + 12 T^{2} + T^{3} )^{2} \)
$89$ \( 345744 + 24696 T + 8820 T^{2} + 672 T^{3} + 186 T^{4} + 12 T^{5} + T^{6} \)
$97$ \( ( 108 - 243 T + T^{3} )^{2} \)
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