Properties

Label 966.2.i.h
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: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

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
0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
−0.707107 1.22474i
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 −2.62132 0.358719i 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 1.62132 + 2.09077i 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 −2.62132 + 0.358719i 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 1.62132 2.09077i 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
\(n\): 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.h 4
7.c even 3 1 inner 966.2.i.h 4
7.c even 3 1 6762.2.a.cc 2
7.d odd 6 1 6762.2.a.ca 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.h 4 1.a even 1 1 trivial
966.2.i.h 4 7.c even 3 1 inner
6762.2.a.ca 2 7.d odd 6 1
6762.2.a.cc 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 \(S_{2}^{\mathrm{new}}(966, [\chi])\):

\( T_{5}^{2} - T_{5} + 1 \)
\( T_{11}^{2} - 3 T_{11} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( 49 + 14 T - 3 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( ( 9 - 3 T + T^{2} )^{2} \)
$13$ \( ( -2 + T^{2} )^{2} \)
$17$ \( 4 + 2 T^{2} + T^{4} \)
$19$ \( 4 - 8 T + 14 T^{2} - 4 T^{3} + T^{4} \)
$23$ \( ( 1 - T + T^{2} )^{2} \)
$29$ \( ( 7 + 10 T + T^{2} )^{2} \)
$31$ \( 961 + 62 T + 35 T^{2} - 2 T^{3} + T^{4} \)
$37$ \( 2116 + 184 T + 62 T^{2} - 4 T^{3} + T^{4} \)
$41$ \( ( 18 - 12 T + T^{2} )^{2} \)
$43$ \( ( -28 + 4 T + T^{2} )^{2} \)
$47$ \( 15376 - 496 T + 140 T^{2} + 4 T^{3} + T^{4} \)
$53$ \( 289 - 170 T + 83 T^{2} - 10 T^{3} + T^{4} \)
$59$ \( 49 - 70 T + 93 T^{2} - 10 T^{3} + T^{4} \)
$61$ \( ( 36 - 6 T + T^{2} )^{2} \)
$67$ \( 4624 - 1360 T + 332 T^{2} - 20 T^{3} + T^{4} \)
$71$ \( ( -82 + 8 T + T^{2} )^{2} \)
$73$ \( 1024 + 32 T^{2} + T^{4} \)
$79$ \( 1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4} \)
$83$ \( ( -127 - 2 T + T^{2} )^{2} \)
$89$ \( 24964 - 632 T + 174 T^{2} + 4 T^{3} + T^{4} \)
$97$ \( ( 23 + 10 T + T^{2} )^{2} \)
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