Properties

Label 966.2.i.m
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: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} + 13 x^{6} + 10 x^{5} + 47 x^{4} + 180 x^{3} + 220 x^{2} + 768 x + 1164\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 13 x^{6} + 10 x^{5} + 47 x^{4} + 180 x^{3} + 220 x^{2} + 768 x + 1164\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} - 19 \nu^{5} + 8 \nu^{4} - 145 \nu^{3} - 142 \nu^{2} - 254 \nu - 1064 \)\()/784\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 6 \nu^{6} - 37 \nu^{5} + 106 \nu^{4} - 183 \nu^{3} - 108 \nu^{2} - 742 \nu - 2228 \)\()/784\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{7} + 33 \nu^{6} - 11 \nu^{5} + 275 \nu^{4} + 215 \nu^{3} + 2227 \nu^{2} + 4228 \nu + 6758 \)\()/2744\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 3 \nu^{6} - 9 \nu^{5} + 49 \nu^{4} - 19 \nu^{3} + 17 \nu^{2} + 148 \nu - 582 \)\()/392\)
\(\beta_{5}\)\(=\)\((\)\( -13 \nu^{7} + 41 \nu^{6} - 107 \nu^{5} + 99 \nu^{4} - 337 \nu^{3} + 235 \nu^{2} + 476 \nu - 3138 \)\()/2744\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{7} - 9 \nu^{6} + 24 \nu^{5} - 40 \nu^{4} - 100 \nu^{3} + 120 \nu^{2} - 840 \nu - 529 \)\()/343\)
\(\beta_{7}\)\(=\)\((\)\( -39 \nu^{7} + 32 \nu^{6} - 517 \nu^{5} - 648 \nu^{4} - 3223 \nu^{3} - 3474 \nu^{2} - 16786 \nu - 19592 \)\()/5488\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{4} - \beta_{2} + \beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - 2 \beta_{5} + 2 \beta_{3} - \beta_{1} - 5\)
\(\nu^{3}\)\(=\)\(\beta_{7} - 2 \beta_{6} - 5 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 5 \beta_{1} - 13\)
\(\nu^{4}\)\(=\)\(-7 \beta_{7} - \beta_{6} + 2 \beta_{5} - 18 \beta_{4} + 16 \beta_{2} + 7 \beta_{1} + 4\)
\(\nu^{5}\)\(=\)\(-25 \beta_{7} + 13 \beta_{6} + 59 \beta_{5} - 11 \beta_{4} - 29 \beta_{3} + 5 \beta_{2} + 4 \beta_{1} + 73\)
\(\nu^{6}\)\(=\)\(-4 \beta_{7} + 31 \beta_{6} + 156 \beta_{5} + 132 \beta_{4} - 78 \beta_{3} - 74 \beta_{2} - 114 \beta_{1} + 235\)
\(\nu^{7}\)\(=\)\(132 \beta_{7} + 35 \beta_{6} - 96 \beta_{5} + 536 \beta_{4} - 23 \beta_{3} - 293 \beta_{2} - 191 \beta_{1} + 176\)

Character values

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

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

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
1.76655 + 2.53227i
1.26426 2.63252i
−0.415888 + 2.23501i
−1.61492 0.402708i
1.76655 2.53227i
1.26426 + 2.63252i
−0.415888 2.23501i
−1.61492 + 0.402708i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.76655 3.05976i −1.00000 1.80974 + 1.92999i 1.00000 −0.500000 0.866025i −1.76655 + 3.05976i
277.2 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.26426 2.18976i −1.00000 −2.41196 1.08740i 1.00000 −0.500000 0.866025i −1.26426 + 2.18976i
277.3 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.415888 + 0.720339i −1.00000 2.64352 0.108691i 1.00000 −0.500000 0.866025i 0.415888 0.720339i
277.4 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.61492 + 2.79713i −1.00000 0.958707 2.46594i 1.00000 −0.500000 0.866025i 1.61492 2.79713i
415.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.76655 + 3.05976i −1.00000 1.80974 1.92999i 1.00000 −0.500000 + 0.866025i −1.76655 3.05976i
415.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.26426 + 2.18976i −1.00000 −2.41196 + 1.08740i 1.00000 −0.500000 + 0.866025i −1.26426 2.18976i
415.3 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.415888 0.720339i −1.00000 2.64352 + 0.108691i 1.00000 −0.500000 + 0.866025i 0.415888 + 0.720339i
415.4 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.61492 2.79713i −1.00000 0.958707 + 2.46594i 1.00000 −0.500000 + 0.866025i 1.61492 + 2.79713i
\(n\): 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.m 8
7.c even 3 1 inner 966.2.i.m 8
7.c even 3 1 6762.2.a.cl 4
7.d odd 6 1 6762.2.a.cr 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.m 8 1.a even 1 1 trivial
966.2.i.m 8 7.c even 3 1 inner
6762.2.a.cl 4 7.c even 3 1
6762.2.a.cr 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 \(S_{2}^{\mathrm{new}}(966, [\chi])\):

\(T_{5}^{8} + \cdots\)
\( T_{11}^{8} + 31 T_{11}^{6} + 24 T_{11}^{5} + 781 T_{11}^{4} + 372 T_{11}^{3} + 5724 T_{11}^{2} - 2160 T_{11} + 32400 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{4} \)
$3$ \( ( 1 - T + T^{2} )^{4} \)
$5$ \( 576 - 480 T + 712 T^{2} + 164 T^{3} + 185 T^{4} + 14 T^{5} + 17 T^{6} + 2 T^{7} + T^{8} \)
$7$ \( 2401 - 2058 T + 588 T^{2} + 84 T^{3} - 107 T^{4} + 12 T^{5} + 12 T^{6} - 6 T^{7} + T^{8} \)
$11$ \( 32400 - 2160 T + 5724 T^{2} + 372 T^{3} + 781 T^{4} + 24 T^{5} + 31 T^{6} + T^{8} \)
$13$ \( ( -60 + 164 T - 27 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$17$ \( 9 + 102 T + 1240 T^{2} - 964 T^{3} + 713 T^{4} - 124 T^{5} + 32 T^{6} + 2 T^{7} + T^{8} \)
$19$ \( ( 4 - 2 T + T^{2} )^{4} \)
$23$ \( ( 1 - T + T^{2} )^{4} \)
$29$ \( ( -40 + 56 T + 33 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$31$ \( ( 4 - 2 T + T^{2} )^{4} \)
$37$ \( 331776 + 73728 T + 69376 T^{2} + 11264 T^{3} + 11600 T^{4} + 2096 T^{5} + 308 T^{6} + 20 T^{7} + T^{8} \)
$41$ \( ( 480 + 64 T - 52 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$43$ \( ( -2 + T )^{8} \)
$47$ \( 7080921 + 904740 T + 429598 T^{2} - 18832 T^{3} + 12623 T^{4} - 208 T^{5} + 134 T^{6} - 4 T^{7} + T^{8} \)
$53$ \( 104976 + 409536 T + 1587004 T^{2} + 53376 T^{3} + 24165 T^{4} + 1934 T^{5} + 357 T^{6} + 18 T^{7} + T^{8} \)
$59$ \( 37748736 - 7864320 T + 2916352 T^{2} + 167936 T^{3} + 47360 T^{4} + 896 T^{5} + 272 T^{6} + 8 T^{7} + T^{8} \)
$61$ \( 4194304 + 262144 T + 335872 T^{2} - 36352 T^{3} + 21776 T^{4} - 880 T^{5} + 172 T^{6} + 4 T^{7} + T^{8} \)
$67$ \( 10942864 + 1296736 T + 659788 T^{2} - 46744 T^{3} + 20885 T^{4} - 478 T^{5} + 157 T^{6} - 2 T^{7} + T^{8} \)
$71$ \( ( 8181 + 612 T - 246 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$73$ \( 4100625 - 340200 T + 307674 T^{2} - 9216 T^{3} + 18363 T^{4} - 768 T^{5} + 202 T^{6} + 8 T^{7} + T^{8} \)
$79$ \( 5702544 + 3734832 T + 1603132 T^{2} + 399260 T^{3} + 72173 T^{4} + 8168 T^{5} + 671 T^{6} + 32 T^{7} + T^{8} \)
$83$ \( ( 48 + 272 T - 112 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$89$ \( 5308416 + 2211840 T + 1271808 T^{2} - 109056 T^{3} + 28480 T^{4} - 704 T^{5} + 216 T^{6} - 8 T^{7} + T^{8} \)
$97$ \( ( 8 + T )^{8} \)
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