Properties

Label 966.2.i.i
Level $966$
Weight $2$
Character orbit 966.i
Analytic conductor $7.714$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 5 \nu^{2} - 5 \nu + 16 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4 \beta_{2} + \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(5 \beta_{3} - 4\)

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
−0.780776 1.35234i
1.28078 + 2.21837i
−0.780776 + 1.35234i
1.28078 2.21837i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.780776 1.35234i −1.00000 0.500000 + 2.59808i −1.00000 −0.500000 0.866025i 0.780776 1.35234i
277.2 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.28078 + 2.21837i −1.00000 0.500000 + 2.59808i −1.00000 −0.500000 0.866025i −1.28078 + 2.21837i
415.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.780776 + 1.35234i −1.00000 0.500000 2.59808i −1.00000 −0.500000 + 0.866025i 0.780776 + 1.35234i
415.2 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.28078 2.21837i −1.00000 0.500000 2.59808i −1.00000 −0.500000 + 0.866025i −1.28078 2.21837i
\(n\): 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.i 4
7.c even 3 1 inner 966.2.i.i 4
7.c even 3 1 6762.2.a.bx 2
7.d odd 6 1 6762.2.a.br 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.i 4 1.a even 1 1 trivial
966.2.i.i 4 7.c even 3 1 inner
6762.2.a.br 2 7.d odd 6 1
6762.2.a.bx 2 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 \(S_{2}^{\mathrm{new}}(966, [\chi])\):

\( T_{5}^{4} - T_{5}^{3} + 5 T_{5}^{2} + 4 T_{5} + 16 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( 16 + 4 T + 5 T^{2} - T^{3} + T^{4} \)
$7$ \( ( 7 - T + T^{2} )^{2} \)
$11$ \( ( 9 + 3 T + T^{2} )^{2} \)
$13$ \( ( -16 - 2 T + T^{2} )^{2} \)
$17$ \( 1296 - 108 T + 45 T^{2} + 3 T^{3} + T^{4} \)
$19$ \( 256 - 32 T + 20 T^{2} + 2 T^{3} + T^{4} \)
$23$ \( ( 1 - T + T^{2} )^{2} \)
$29$ \( ( -1 + 8 T + T^{2} )^{2} \)
$31$ \( 324 - 162 T + 99 T^{2} + 9 T^{3} + T^{4} \)
$37$ \( 64 + 80 T + 92 T^{2} + 10 T^{3} + T^{4} \)
$41$ \( ( -8 + 6 T + T^{2} )^{2} \)
$43$ \( ( -52 - 8 T + T^{2} )^{2} \)
$47$ \( 4 - 10 T + 23 T^{2} - 5 T^{3} + T^{4} \)
$53$ \( 4 - 6 T + 11 T^{2} + 3 T^{3} + T^{4} \)
$59$ \( 64 + 56 T + 41 T^{2} + 7 T^{3} + T^{4} \)
$61$ \( ( 4 - 2 T + T^{2} )^{2} \)
$67$ \( ( 4 - 2 T + T^{2} )^{2} \)
$71$ \( ( 2 - 5 T + T^{2} )^{2} \)
$73$ \( 40804 - 1010 T + 227 T^{2} + 5 T^{3} + T^{4} \)
$79$ \( 23409 + 153 T^{2} + T^{4} \)
$83$ \( ( 16 + 9 T + T^{2} )^{2} \)
$89$ \( 4624 + 68 T^{2} + T^{4} \)
$97$ \( ( -8 - 11 T + T^{2} )^{2} \)
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