Properties

Label 966.2.i.n
Level $966$
Weight $2$
Character orbit 966.i
Analytic conductor $7.714$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.173309020416.2
Defining polynomial: \(x^{8} - 5 x^{6} - 28 x^{5} - 4 x^{4} + 70 x^{3} + 51 x^{2} + 406 x + 841\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 5 x^{6} - 28 x^{5} - 4 x^{4} + 70 x^{3} + 51 x^{2} + 406 x + 841\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 859 \nu^{7} - 10005 \nu^{6} + 23922 \nu^{5} + 26031 \nu^{4} + 252808 \nu^{3} - 875758 \nu^{2} - 654047 \nu + 463246 \)\()/1219827\)
\(\beta_{2}\)\(=\)\((\)\( 1294 \nu^{7} - 31436 \nu^{6} - 33817 \nu^{5} - 96233 \nu^{4} + 609189 \nu^{3} + 599182 \nu^{2} + 841135 \nu + 4558220 \)\()/1219827\)
\(\beta_{3}\)\(=\)\((\)\( -4618 \nu^{7} + 27347 \nu^{6} + 66010 \nu^{5} + 220886 \nu^{4} - 712908 \nu^{3} - 1033528 \nu^{2} - 3780478 \nu - 3965315 \)\()/2439654\)
\(\beta_{4}\)\(=\)\((\)\( 55 \nu^{7} - 39 \nu^{6} + 44 \nu^{5} - 770 \nu^{4} - 3680 \nu^{3} - 4466 \nu^{2} - 9251 \nu + 57733 \)\()/28042\)
\(\beta_{5}\)\(=\)\((\)\( -875 \nu^{7} - 290 \nu^{6} - 700 \nu^{5} + 24268 \nu^{4} + 7560 \nu^{3} + 10960 \nu^{2} - 21077 \nu - 306472 \)\()/348522\)
\(\beta_{6}\)\(=\)\((\)\( 1108 \nu^{7} + 1363 \nu^{6} - 10731 \nu^{5} - 41551 \nu^{4} + 34573 \nu^{3} + 144782 \nu^{2} - 240046 \nu - 197635 \)\()/406609\)
\(\beta_{7}\)\(=\)\((\)\( -6167 \nu^{7} + 7250 \nu^{6} + 41536 \nu^{5} + 120389 \nu^{4} - 193006 \nu^{3} - 900939 \nu^{2} + 607045 \nu - 2154903 \)\()/1219827\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{6} - 2 \beta_{3} - \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{7} - 4 \beta_{6} - 6 \beta_{5} - 6 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 3 \beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{7} + \beta_{6} - 3 \beta_{5} - 15 \beta_{4} - \beta_{3} - 2 \beta_{2} + 6 \beta_{1} + 27\)\()/3\)
\(\nu^{4}\)\(=\)\(-10 \beta_{7} - 8 \beta_{6} + 14 \beta_{5} - 6 \beta_{3} - 8 \beta_{2} + 5 \beta_{1} + 9\)
\(\nu^{5}\)\(=\)\((\)\(-27 \beta_{7} - 104 \beta_{6} - 249 \beta_{5} - 159 \beta_{4} + 107 \beta_{3} + 52 \beta_{2} - 27 \beta_{1}\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-30 \beta_{7} + 76 \beta_{6} - 30 \beta_{5} - 234 \beta_{4} - 76 \beta_{3} - 152 \beta_{2} + 60 \beta_{1} + 861\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-864 \beta_{7} - 625 \beta_{6} + 78 \beta_{5} - 470 \beta_{3} - 625 \beta_{2} + 432 \beta_{1} - 354\)\()/3\)

Character values

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

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(1\) \(-1 - \beta_{5}\) \(1\)

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
0.435461 1.77894i
−1.61272 + 2.45863i
2.93560 0.167344i
−1.75834 0.512349i
0.435461 + 1.77894i
−1.61272 2.45863i
2.93560 + 0.167344i
−1.75834 + 0.512349i
0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.94864 3.37514i 1.00000 −1.32288 + 2.29129i −1.00000 −0.500000 0.866025i 1.94864 3.37514i
277.2 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.739514 1.28088i 1.00000 1.32288 2.29129i −1.00000 −0.500000 0.866025i 0.739514 1.28088i
277.3 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.239514 + 0.414851i 1.00000 1.32288 2.29129i −1.00000 −0.500000 0.866025i −0.239514 + 0.414851i
277.4 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.44864 + 2.50912i 1.00000 −1.32288 + 2.29129i −1.00000 −0.500000 0.866025i −1.44864 + 2.50912i
415.1 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.94864 + 3.37514i 1.00000 −1.32288 2.29129i −1.00000 −0.500000 + 0.866025i 1.94864 + 3.37514i
415.2 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.739514 + 1.28088i 1.00000 1.32288 + 2.29129i −1.00000 −0.500000 + 0.866025i 0.739514 + 1.28088i
415.3 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.239514 0.414851i 1.00000 1.32288 + 2.29129i −1.00000 −0.500000 + 0.866025i −0.239514 0.414851i
415.4 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.44864 2.50912i 1.00000 −1.32288 2.29129i −1.00000 −0.500000 + 0.866025i −1.44864 2.50912i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.4
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.n 8
7.c even 3 1 inner 966.2.i.n 8
7.c even 3 1 6762.2.a.cg 4
7.d odd 6 1 6762.2.a.ch 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.n 8 1.a even 1 1 trivial
966.2.i.n 8 7.c even 3 1 inner
6762.2.a.cg 4 7.c even 3 1
6762.2.a.ch 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 \(S_{2}^{\mathrm{new}}(966, [\chi])\):

\(T_{5}^{8} + \cdots\)
\(T_{11}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{4} \)
$3$ \( ( 1 - T + T^{2} )^{4} \)
$5$ \( 64 - 96 T + 232 T^{2} + 100 T^{3} + 137 T^{4} + 2 T^{5} + 15 T^{6} + 2 T^{7} + T^{8} \)
$7$ \( ( 49 + 7 T^{2} + T^{4} )^{2} \)
$11$ \( 39601 + 33830 T + 24522 T^{2} + 6128 T^{3} + 1703 T^{4} + 208 T^{5} + 58 T^{6} + 6 T^{7} + T^{8} \)
$13$ \( ( -56 - 112 T - 34 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$17$ \( 3844 - 5084 T + 5174 T^{2} - 2298 T^{3} + 851 T^{4} - 114 T^{5} + 29 T^{6} - 2 T^{7} + T^{8} \)
$19$ \( 87616 + 23680 T + 16464 T^{2} + 832 T^{3} + 1340 T^{4} + 44 T^{5} + 70 T^{6} - 6 T^{7} + T^{8} \)
$23$ \( ( 1 + T + T^{2} )^{4} \)
$29$ \( ( 189 - 126 T - 10 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$31$ \( 404496 - 267120 T + 179580 T^{2} - 15708 T^{3} + 6541 T^{4} - 910 T^{5} + 191 T^{6} - 14 T^{7} + T^{8} \)
$37$ \( 1024 - 512 T + 960 T^{2} - 32 T^{3} + 548 T^{4} - 100 T^{5} + 58 T^{6} + 6 T^{7} + T^{8} \)
$41$ \( ( -56 + 224 T - 10 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$43$ \( ( -2 + T )^{8} \)
$47$ \( 6697744 - 2722576 T + 902252 T^{2} - 134868 T^{3} + 19349 T^{4} - 1314 T^{5} + 179 T^{6} - 10 T^{7} + T^{8} \)
$53$ \( 144 + 288 T + 588 T^{2} + 120 T^{3} + 157 T^{4} + 54 T^{5} + 35 T^{6} + 6 T^{7} + T^{8} \)
$59$ \( 50466816 + 8183808 T + 2016192 T^{2} + 229248 T^{3} + 44161 T^{4} + 4632 T^{5} + 479 T^{6} + 24 T^{7} + T^{8} \)
$61$ \( 16384 - 86016 T + 420352 T^{2} - 156800 T^{3} + 40592 T^{4} - 5488 T^{5} + 540 T^{6} - 28 T^{7} + T^{8} \)
$67$ \( 81144064 + 11097856 T + 2887040 T^{2} + 317184 T^{3} + 66608 T^{4} + 6720 T^{5} + 632 T^{6} + 28 T^{7} + T^{8} \)
$71$ \( ( 18 - 66 T + 67 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$73$ \( 2637376 + 1409632 T + 586152 T^{2} + 108892 T^{3} + 17441 T^{4} + 1118 T^{5} + 139 T^{6} + 6 T^{7} + T^{8} \)
$79$ \( 1471369 + 1508972 T + 1821674 T^{2} - 271440 T^{3} + 54839 T^{4} - 1584 T^{5} + 242 T^{6} - 4 T^{7} + T^{8} \)
$83$ \( ( -3708 + 1176 T - 43 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$89$ \( 9437184 - 4128768 T + 1542144 T^{2} - 201600 T^{3} + 29284 T^{4} - 1484 T^{5} + 282 T^{6} - 14 T^{7} + T^{8} \)
$97$ \( ( 23888 + 112 T - 319 T^{2} + T^{4} )^{2} \)
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