Properties

Label 966.2.q.i
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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [966,2,Mod(85,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.85");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 −1.70670 3.73716i −0.142315 + 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i 3.94201 + 1.15748i
85.2 −0.654861 + 0.755750i 0.841254 0.540641i −0.142315 0.989821i −0.446804 0.978364i −0.142315 + 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i 1.03199 + 0.303020i
85.3 −0.654861 + 0.755750i 0.841254 0.540641i −0.142315 0.989821i 0.431857 + 0.945634i −0.142315 + 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −0.997469 0.292883i
85.4 −0.654861 + 0.755750i 0.841254 0.540641i −0.142315 0.989821i 1.56679 + 3.43079i −0.142315 + 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −3.61885 1.06259i
127.1 0.841254 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i −2.99676 + 0.879928i 0.415415 + 0.909632i −0.654861 0.755750i −0.142315 0.989821i −0.959493 0.281733i −2.04531 + 2.36041i
127.2 0.841254 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i −1.32090 + 0.387851i 0.415415 + 0.909632i −0.654861 0.755750i −0.142315 0.989821i −0.959493 0.281733i −0.901523 + 1.04041i
127.3 0.841254 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i 2.16398 0.635401i 0.415415 + 0.909632i −0.654861 0.755750i −0.142315 0.989821i −0.959493 0.281733i 1.47693 1.70447i
127.4 0.841254 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i 3.49494 1.02621i 0.415415 + 0.909632i −0.654861 0.755750i −0.142315 0.989821i −0.959493 0.281733i 2.38532 2.75280i
169.1 −0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −1.80470 2.08274i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i 2.31838 1.48993i
169.2 −0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −0.987362 1.13948i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i 1.26839 0.815148i
169.3 −0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i 0.751750 + 0.867566i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i −0.965721 + 0.620631i
169.4 −0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i 2.39800 + 2.76744i −0.959493 + 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i −3.08054 + 1.97974i
211.1 0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −2.63117 1.69095i −0.654861 0.755750i −0.142315 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0.445115 3.09584i
211.2 0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i −0.627520 0.403283i −0.654861 0.755750i −0.142315 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0.106158 0.738342i
211.3 0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 1.41748 + 0.910960i −0.654861 0.755750i −0.142315 0.989821i −0.959493 0.281733i 0.841254 0.540641i −0.239795 + 1.66781i
211.4 0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 2.75663 + 1.77158i −0.654861 0.755750i −0.142315 0.989821i −0.959493 0.281733i 0.841254 0.540641i −0.466338 + 3.24345i
463.1 −0.142315 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −1.80470 + 2.08274i −0.959493 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i 2.31838 + 1.48993i
463.2 −0.142315 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −0.987362 + 1.13948i −0.959493 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i 1.26839 + 0.815148i
463.3 −0.142315 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i 0.751750 0.867566i −0.959493 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i −0.965721 0.620631i
463.4 −0.142315 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i 2.39800 2.76744i −0.959493 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i −3.08054 1.97974i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.4
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.i 40
23.c even 11 1 inner 966.2.q.i 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.i 40 1.a even 1 1 trivial
966.2.q.i 40 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} - 4 T_{5}^{39} + 29 T_{5}^{38} - 118 T_{5}^{37} + 490 T_{5}^{36} - 1276 T_{5}^{35} + 3416 T_{5}^{34} - 7890 T_{5}^{33} + 25074 T_{5}^{32} - 51327 T_{5}^{31} + 482723 T_{5}^{30} - 1600581 T_{5}^{29} + \cdots + 6847307293696 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\). Copy content Toggle raw display