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Properties

Label 966.2.q.j
Level $966$
Weight $2$
Character orbit 966.q
Analytic conductor $7.714$
Analytic rank $0$
Dimension $40$
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.q (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(4\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

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

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 − 0.989821i −1.36169 − 2.98169i 0.142315 − 0.989821i 0.959493 − 0.281733i 0.841254 + 0.540641i 0.415415 − 0.909632i 3.14512 + 0.923492i
85.2 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 − 0.989821i −0.568699 − 1.24528i 0.142315 − 0.989821i 0.959493 − 0.281733i 0.841254 + 0.540641i 0.415415 − 0.909632i 1.31354 + 0.385689i
85.3 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 − 0.989821i 0.479771 + 1.05055i 0.142315 − 0.989821i 0.959493 − 0.281733i 0.841254 + 0.540641i 0.415415 − 0.909632i −1.10814 − 0.325379i
85.4 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 − 0.989821i 1.60548 + 3.51551i 0.142315 − 0.989821i 0.959493 − 0.281733i 0.841254 + 0.540641i 0.415415 − 0.909632i −3.70821 − 1.08883i
127.1 0.841254 − 0.540641i 0.142315 − 0.989821i 0.415415 − 0.909632i −3.54832 + 1.04188i −0.415415 − 0.909632i 0.654861 + 0.755750i −0.142315 − 0.989821i −0.959493 − 0.281733i −2.42175 + 2.79485i
127.2 0.841254 − 0.540641i 0.142315 − 0.989821i 0.415415 − 0.909632i −0.571809 + 0.167898i −0.415415 − 0.909632i 0.654861 + 0.755750i −0.142315 − 0.989821i −0.959493 − 0.281733i −0.390264 + 0.450389i
127.3 0.841254 − 0.540641i 0.142315 − 0.989821i 0.415415 − 0.909632i −0.309565 + 0.0908964i −0.415415 − 0.909632i 0.654861 + 0.755750i −0.142315 − 0.989821i −0.959493 − 0.281733i −0.211280 + 0.243830i
127.4 0.841254 − 0.540641i 0.142315 − 0.989821i 0.415415 − 0.909632i 3.08844 − 0.906847i −0.415415 − 0.909632i 0.654861 + 0.755750i −0.142315 − 0.989821i −0.959493 − 0.281733i 2.10788 − 2.43262i
169.1 −0.142315 + 0.989821i −0.415415 − 0.909632i −0.959493 − 0.281733i −2.74645 − 3.16958i 0.959493 − 0.281733i −0.841254 − 0.540641i 0.415415 − 0.909632i −0.654861 + 0.755750i 3.52818 − 2.26742i
169.2 −0.142315 + 0.989821i −0.415415 − 0.909632i −0.959493 − 0.281733i −0.463708 − 0.535148i 0.959493 − 0.281733i −0.841254 − 0.540641i 0.415415 − 0.909632i −0.654861 + 0.755750i 0.595693 − 0.382829i
169.3 −0.142315 + 0.989821i −0.415415 − 0.909632i −0.959493 − 0.281733i 0.0335087 + 0.0386712i 0.959493 − 0.281733i −0.841254 − 0.540641i 0.415415 − 0.909632i −0.654861 + 0.755750i −0.0430463 + 0.0276642i
169.4 −0.142315 + 0.989821i −0.415415 − 0.909632i −0.959493 − 0.281733i 2.81897 + 3.25326i 0.959493 − 0.281733i −0.841254 − 0.540641i 0.415415 − 0.909632i −0.654861 + 0.755750i −3.62133 + 2.32729i
211.1 0.415415 + 0.909632i 0.959493 − 0.281733i −0.654861 + 0.755750i −3.48521 − 2.23981i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 − 0.281733i 0.841254 − 0.540641i 0.589593 − 4.10071i
211.2 0.415415 + 0.909632i 0.959493 − 0.281733i −0.654861 + 0.755750i −2.25359 − 1.44829i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 − 0.281733i 0.841254 − 0.540641i 0.381240 − 2.65158i
211.3 0.415415 + 0.909632i 0.959493 − 0.281733i −0.654861 + 0.755750i 1.73540 + 1.11528i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 − 0.281733i 0.841254 − 0.540641i −0.293578 + 2.04188i
211.4 0.415415 + 0.909632i 0.959493 − 0.281733i −0.654861 + 0.755750i 3.08798 + 1.98452i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 − 0.281733i 0.841254 − 0.540641i −0.522394 + 3.63333i
463.1 −0.142315 − 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i −2.74645 + 3.16958i 0.959493 + 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 − 0.755750i 3.52818 + 2.26742i
463.2 −0.142315 − 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i −0.463708 + 0.535148i 0.959493 + 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 − 0.755750i 0.595693 + 0.382829i
463.3 −0.142315 − 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 0.0335087 − 0.0386712i 0.959493 + 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 − 0.755750i −0.0430463 − 0.0276642i
463.4 −0.142315 − 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 2.81897 − 3.25326i 0.959493 + 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 − 0.755750i −3.62133 − 2.32729i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.q.j ✓ 40
23.c even 11 1 inner 966.2.q.j ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.j ✓ 40 1.a even 1 1 trivial
966.2.q.j ✓ 40 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(18\!\cdots\!18\)\( T_{5}^{18} + \)\(21\!\cdots\!84\)\( T_{5}^{17} + \)\(66\!\cdots\!51\)\( T_{5}^{16} + \)\(80\!\cdots\!67\)\( T_{5}^{15} + \)\(29\!\cdots\!83\)\( T_{5}^{14} + \)\(56\!\cdots\!56\)\( T_{5}^{13} + \)\(12\!\cdots\!69\)\( T_{5}^{12} + \)\(19\!\cdots\!34\)\( T_{5}^{11} + \)\(26\!\cdots\!45\)\( T_{5}^{10} + \)\(31\!\cdots\!92\)\( T_{5}^{9} + \)\(32\!\cdots\!36\)\( T_{5}^{8} + \)\(28\!\cdots\!32\)\( T_{5}^{7} + \)\(20\!\cdots\!32\)\( T_{5}^{6} + \)\(11\!\cdots\!04\)\( T_{5}^{5} + \)\(38\!\cdots\!40\)\( T_{5}^{4} + \)\(72\!\cdots\!48\)\( T_{5}^{3} + 376549054208 T_{5}^{2} - 35888090112 T_{5} + 2431673344 \)">\(T_{5}^{40} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\).