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Properties

Label 966.2.q.g
Level $966$
Weight $2$
Character orbit 966.q
Analytic conductor $7.714$
Analytic rank $0$
Dimension $30$
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: \(30\)
Relative dimension: \(3\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{2} + 3 q^{3} - 3 q^{4} - 10 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 3 q^{2} + 3 q^{3} - 3 q^{4} - 10 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9} + 10 q^{10} + 9 q^{11} + 3 q^{12} - 12 q^{13} - 3 q^{14} - q^{15} - 3 q^{16} + 5 q^{17} + 3 q^{18} + 18 q^{19} + q^{20} - 3 q^{21} + 2 q^{22} + 21 q^{23} + 30 q^{24} + 13 q^{25} - 10 q^{26} + 3 q^{27} + 3 q^{28} + 17 q^{29} - 10 q^{30} + 12 q^{31} + 3 q^{32} + 13 q^{33} + 17 q^{34} - q^{35} - 3 q^{36} + 16 q^{37} + 15 q^{38} + 12 q^{39} - 12 q^{40} + 10 q^{41} + 3 q^{42} - 35 q^{43} + 9 q^{44} + 12 q^{45} + q^{46} - 8 q^{47} + 3 q^{48} - 3 q^{49} - 2 q^{50} + 6 q^{51} - q^{52} + 42 q^{53} - 3 q^{54} + 49 q^{55} - 3 q^{56} + 15 q^{57} + 5 q^{58} - 6 q^{59} + 10 q^{60} - 18 q^{61} - 34 q^{62} + 3 q^{63} - 3 q^{64} + 34 q^{65} - 2 q^{66} + 72 q^{67} - 6 q^{68} - 10 q^{69} + 12 q^{70} + 17 q^{71} + 3 q^{72} + 9 q^{73} - 16 q^{74} - 2 q^{75} + 18 q^{76} + 2 q^{77} + 10 q^{78} - 56 q^{79} + q^{80} - 3 q^{81} + 12 q^{82} + 52 q^{83} - 3 q^{84} - 53 q^{85} - 31 q^{86} + 5 q^{87} + 13 q^{88} - 104 q^{89} - q^{90} + 34 q^{91} - 12 q^{92} + 32 q^{93} - 14 q^{94} - 92 q^{95} - 3 q^{96} - 82 q^{97} + 3 q^{98} - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

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 −0.635376 − 1.39128i −0.142315 + 0.989821i 0.959493 − 0.281733i −0.841254 − 0.540641i 0.415415 − 0.909632i −1.46754 − 0.430909i
85.2 0.654861 − 0.755750i −0.841254 + 0.540641i −0.142315 − 0.989821i 0.477791 + 1.04622i −0.142315 + 0.989821i 0.959493 − 0.281733i −0.841254 − 0.540641i 0.415415 − 0.909632i 1.10356 + 0.324036i
85.3 0.654861 − 0.755750i −0.841254 + 0.540641i −0.142315 − 0.989821i 0.906796 + 1.98561i −0.142315 + 0.989821i 0.959493 − 0.281733i −0.841254 − 0.540641i 0.415415 − 0.909632i 2.09445 + 0.614985i
127.1 −0.841254 + 0.540641i 0.142315 − 0.989821i 0.415415 − 0.909632i −4.09857 + 1.20345i 0.415415 + 0.909632i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 − 0.281733i 2.79730 − 3.22826i
127.2 −0.841254 + 0.540641i 0.142315 − 0.989821i 0.415415 − 0.909632i −1.03525 + 0.303978i 0.415415 + 0.909632i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 − 0.281733i 0.706568 − 0.815423i
127.3 −0.841254 + 0.540641i 0.142315 − 0.989821i 0.415415 − 0.909632i 0.279230 − 0.0819895i 0.415415 + 0.909632i 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 − 0.281733i −0.190577 + 0.219937i
169.1 0.142315 − 0.989821i −0.415415 − 0.909632i −0.959493 − 0.281733i −1.94500 − 2.24464i −0.959493 + 0.281733i −0.841254 − 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i −2.49860 + 1.60575i
169.2 0.142315 − 0.989821i −0.415415 − 0.909632i −0.959493 − 0.281733i 1.14899 + 1.32600i −0.959493 + 0.281733i −0.841254 − 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i 1.47603 − 0.948584i
169.3 0.142315 − 0.989821i −0.415415 − 0.909632i −0.959493 − 0.281733i 1.64194 + 1.89490i −0.959493 + 0.281733i −0.841254 − 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i 2.10928 − 1.35555i
211.1 −0.415415 − 0.909632i 0.959493 − 0.281733i −0.654861 + 0.755750i −2.54794 − 1.63746i −0.654861 − 0.755750i 0.142315 + 0.989821i 0.959493 + 0.281733i 0.841254 − 0.540641i −0.431035 + 2.99791i
211.2 −0.415415 − 0.909632i 0.959493 − 0.281733i −0.654861 + 0.755750i −0.642037 − 0.412612i −0.654861 − 0.755750i 0.142315 + 0.989821i 0.959493 + 0.281733i 0.841254 − 0.540641i −0.108613 + 0.755423i
211.3 −0.415415 − 0.909632i 0.959493 − 0.281733i −0.654861 + 0.755750i 1.75345 + 1.12688i −0.654861 − 0.755750i 0.142315 + 0.989821i 0.959493 + 0.281733i 0.841254 − 0.540641i 0.296631 − 2.06312i
463.1 0.142315 + 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i −1.94500 + 2.24464i −0.959493 − 0.281733i −0.841254 + 0.540641i −0.415415 − 0.909632i −0.654861 − 0.755750i −2.49860 − 1.60575i
463.2 0.142315 + 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.14899 − 1.32600i −0.959493 − 0.281733i −0.841254 + 0.540641i −0.415415 − 0.909632i −0.654861 − 0.755750i 1.47603 + 0.948584i
463.3 0.142315 + 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 1.64194 − 1.89490i −0.959493 − 0.281733i −0.841254 + 0.540641i −0.415415 − 0.909632i −0.654861 − 0.755750i 2.10928 + 1.35555i
547.1 0.959493 + 0.281733i 0.654861 − 0.755750i 0.841254 + 0.540641i −0.471267 + 3.27773i 0.841254 − 0.540641i −0.415415 − 0.909632i 0.654861 + 0.755750i −0.142315 − 0.989821i −1.37562 + 3.01219i
547.2 0.959493 + 0.281733i 0.654861 − 0.755750i 0.841254 + 0.540641i −0.121006 + 0.841612i 0.841254 − 0.540641i −0.415415 − 0.909632i 0.654861 + 0.755750i −0.142315 − 0.989821i −0.353213 + 0.773429i
547.3 0.959493 + 0.281733i 0.654861 − 0.755750i 0.841254 + 0.540641i 0.288244 − 2.00478i 0.841254 − 0.540641i −0.415415 − 0.909632i 0.654861 + 0.755750i −0.142315 − 0.989821i 0.841381 − 1.84237i
673.1 −0.415415 + 0.909632i 0.959493 + 0.281733i −0.654861 − 0.755750i −2.54794 + 1.63746i −0.654861 + 0.755750i 0.142315 − 0.989821i 0.959493 − 0.281733i 0.841254 + 0.540641i −0.431035 − 2.99791i
673.2 −0.415415 + 0.909632i 0.959493 + 0.281733i −0.654861 − 0.755750i −0.642037 + 0.412612i −0.654861 + 0.755750i 0.142315 − 0.989821i 0.959493 − 0.281733i 0.841254 + 0.540641i −0.108613 − 0.755423i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.3
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.g ✓ 30
23.c even 11 1 inner 966.2.q.g ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.q.g ✓ 30 1.a even 1 1 trivial
966.2.q.g ✓ 30 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{30} + 10 T_{5}^{29} + 51 T_{5}^{28} + 204 T_{5}^{27} + 710 T_{5}^{26} + 2154 T_{5}^{25} + \cdots + 3418801 \) acting on \(S_{2}^{\mathrm{new}}(966, [\chi])\). Copy content Toggle raw display