Properties

Label 966.2.i.b
Level $966$
Weight $2$
Character orbit 966.i
Analytic conductor $7.714$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 1.00000 0.500000 + 2.59808i 1.00000 −0.500000 0.866025i 0
415.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 1.00000 0.500000 2.59808i 1.00000 −0.500000 + 0.866025i 0
\(n\): 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.b 2
7.c even 3 1 inner 966.2.i.b 2
7.c even 3 1 6762.2.a.bk 1
7.d odd 6 1 6762.2.a.ba 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.b 2 1.a even 1 1 trivial
966.2.i.b 2 7.c even 3 1 inner
6762.2.a.ba 1 7.d odd 6 1
6762.2.a.bk 1 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 \(S_{2}^{\mathrm{new}}(966, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( 1 + T + T^{2} \)
$13$ \( ( 6 + T )^{2} \)
$17$ \( 25 + 5 T + T^{2} \)
$19$ \( 36 + 6 T + T^{2} \)
$23$ \( 1 + T + T^{2} \)
$29$ \( ( -9 + T )^{2} \)
$31$ \( 36 - 6 T + T^{2} \)
$37$ \( 36 + 6 T + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( 6 + T )^{2} \)
$47$ \( 121 + 11 T + T^{2} \)
$53$ \( 196 + 14 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 4 - 2 T + T^{2} \)
$67$ \( 100 + 10 T + T^{2} \)
$71$ \( ( 3 + T )^{2} \)
$73$ \( 25 - 5 T + T^{2} \)
$79$ \( 25 + 5 T + T^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( 100 + 10 T + T^{2} \)
$97$ \( ( 4 + T )^{2} \)
show more
show less