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Properties

Label 726.2.i.d
Level $726$
Weight $2$
Character orbit 726.i
Analytic conductor $5.797$
Analytic rank $0$
Dimension $60$
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] = [726,2,Mod(67,726)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(726, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("726.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 726.i (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: \(5.79713918674\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 6 q^{2} + 60 q^{3} - 6 q^{4} + 6 q^{6} - q^{7} + 6 q^{8} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 6 q^{2} + 60 q^{3} - 6 q^{4} + 6 q^{6} - q^{7} + 6 q^{8} + 60 q^{9} + 11 q^{10} - 10 q^{11} - 6 q^{12} - 17 q^{13} - 10 q^{14} - 6 q^{16} + 20 q^{17} + 6 q^{18} + 6 q^{19} - q^{21} - 12 q^{22} + 14 q^{23} + 6 q^{24} - 2 q^{25} - 5 q^{26} + 60 q^{27} - q^{28} - 2 q^{29} + 11 q^{30} + 20 q^{31} + 6 q^{32} - 10 q^{33} - 20 q^{34} + 15 q^{35} - 6 q^{36} - 17 q^{37} - 28 q^{38} - 17 q^{39} - 22 q^{40} - 23 q^{41} - 10 q^{42} + 38 q^{43} + q^{44} + 19 q^{46} + 2 q^{47} - 6 q^{48} - 33 q^{49} + 2 q^{50} + 20 q^{51} + 5 q^{52} - 4 q^{53} + 6 q^{54} + 11 q^{55} + 12 q^{56} + 6 q^{57} - 20 q^{58} + 61 q^{59} + 34 q^{61} + 24 q^{62} - q^{63} - 6 q^{64} + 34 q^{65} - 12 q^{66} - 50 q^{67} + 9 q^{68} + 14 q^{69} - 4 q^{70} - 10 q^{71} + 6 q^{72} + 41 q^{73} + 6 q^{74} - 2 q^{75} - 16 q^{76} + 123 q^{77} - 5 q^{78} - 40 q^{79} + 60 q^{81} - 10 q^{82} + 30 q^{83} - q^{84} - 63 q^{85} + 6 q^{86} - 2 q^{87} + 10 q^{88} - 10 q^{89} + 11 q^{90} - 4 q^{91} - 19 q^{92} + 20 q^{93} - 90 q^{94} + 65 q^{95} + 6 q^{96} - 37 q^{97} + 55 q^{98} - 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 0.959493 − 0.281733i 1.00000 0.841254 − 0.540641i −2.31759 − 2.67464i 0.959493 − 0.281733i −1.34578 − 2.94685i 0.654861 − 0.755750i 1.00000 −2.97724 − 1.91336i
67.2 0.959493 − 0.281733i 1.00000 0.841254 − 0.540641i −2.13469 − 2.46356i 0.959493 − 0.281733i 1.34707 + 2.94968i 0.654861 − 0.755750i 1.00000 −2.74228 − 1.76236i
67.3 0.959493 − 0.281733i 1.00000 0.841254 − 0.540641i 0.0588228 + 0.0678851i 0.959493 − 0.281733i −1.31620 − 2.88208i 0.654861 − 0.755750i 1.00000 0.0755655 + 0.0485630i
67.4 0.959493 − 0.281733i 1.00000 0.841254 − 0.540641i 0.0930238 + 0.107355i 0.959493 − 0.281733i 1.48917 + 3.26083i 0.654861 − 0.755750i 1.00000 0.119501 + 0.0767987i
67.5 0.959493 − 0.281733i 1.00000 0.841254 − 0.540641i 2.01871 + 2.32971i 0.959493 − 0.281733i −1.42331 − 3.11662i 0.654861 − 0.755750i 1.00000 2.59329 + 1.66661i
67.6 0.959493 − 0.281733i 1.00000 0.841254 − 0.540641i 2.07381 + 2.39330i 0.959493 − 0.281733i 0.968499 + 2.12072i 0.654861 − 0.755750i 1.00000 2.66408 + 1.71210i
133.1 −0.841254 + 0.540641i 1.00000 0.415415 − 0.909632i −0.547646 + 3.80896i −0.841254 + 0.540641i 2.14910 − 2.48019i 0.142315 + 0.989821i 1.00000 −1.59857 − 3.50038i
133.2 −0.841254 + 0.540641i 1.00000 0.415415 − 0.909632i −0.246106 + 1.71170i −0.841254 + 0.540641i 2.20064 − 2.53967i 0.142315 + 0.989821i 1.00000 −0.718379 − 1.57303i
133.3 −0.841254 + 0.540641i 1.00000 0.415415 − 0.909632i −0.203162 + 1.41302i −0.841254 + 0.540641i −1.72809 + 1.99432i 0.142315 + 0.989821i 1.00000 −0.593027 − 1.29855i
133.4 −0.841254 + 0.540641i 1.00000 0.415415 − 0.909632i −0.0302575 + 0.210446i −0.841254 + 0.540641i 0.338748 − 0.390936i 0.142315 + 0.989821i 1.00000 −0.0883213 − 0.193397i
133.5 −0.841254 + 0.540641i 1.00000 0.415415 − 0.909632i 0.286225 − 1.99074i −0.841254 + 0.540641i −2.77964 + 3.20788i 0.142315 + 0.989821i 1.00000 0.835486 + 1.82946i
133.6 −0.841254 + 0.540641i 1.00000 0.415415 − 0.909632i 0.574556 − 3.99613i −0.841254 + 0.540641i 3.01898 − 3.48408i 0.142315 + 0.989821i 1.00000 1.67712 + 3.67238i
199.1 0.654861 − 0.755750i 1.00000 −0.142315 − 0.989821i −2.81568 + 1.80953i 0.654861 − 0.755750i −2.04582 − 0.600708i −0.841254 − 0.540641i 1.00000 −0.476328 + 3.31294i
199.2 0.654861 − 0.755750i 1.00000 −0.142315 − 0.989821i −2.17390 + 1.39708i 0.654861 − 0.755750i −2.66489 − 0.782483i −0.841254 − 0.540641i 1.00000 −0.367758 + 2.55781i
199.3 0.654861 − 0.755750i 1.00000 −0.142315 − 0.989821i −0.303083 + 0.194780i 0.654861 − 0.755750i 3.43038 + 1.00725i −0.841254 − 0.540641i 1.00000 −0.0512725 + 0.356608i
199.4 0.654861 − 0.755750i 1.00000 −0.142315 − 0.989821i 1.20970 − 0.777427i 0.654861 − 0.755750i 2.44803 + 0.718805i −0.841254 − 0.540641i 1.00000 0.204645 − 1.42334i
199.5 0.654861 − 0.755750i 1.00000 −0.142315 − 0.989821i 2.70290 − 1.73705i 0.654861 − 0.755750i −4.65498 − 1.36682i −0.841254 − 0.540641i 1.00000 0.457249 − 3.18024i
199.6 0.654861 − 0.755750i 1.00000 −0.142315 − 0.989821i 3.30201 − 2.12207i 0.654861 − 0.755750i 0.778580 + 0.228612i −0.841254 − 0.540641i 1.00000 0.558601 − 3.88516i
265.1 −0.415415 + 0.909632i 1.00000 −0.654861 − 0.755750i −3.66604 − 1.07645i −0.415415 + 0.909632i −0.135379 − 0.941584i 0.959493 − 0.281733i 1.00000 2.50210 − 2.88757i
265.2 −0.415415 + 0.909632i 1.00000 −0.654861 − 0.755750i −2.70671 − 0.794763i −0.415415 + 0.909632i 0.702736 + 4.88764i 0.959493 − 0.281733i 1.00000 1.84735 − 2.13196i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 726.2.i.d ✓ 60
121.e even 11 1 inner 726.2.i.d ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
726.2.i.d ✓ 60 1.a even 1 1 trivial
726.2.i.d ✓ 60 121.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{60} + 16 T_{5}^{58} + 61 T_{5}^{57} + 247 T_{5}^{56} + 808 T_{5}^{55} + 6407 T_{5}^{54} + \cdots + 2036987689 \) acting on \(S_{2}^{\mathrm{new}}(726, [\chi])\). Copy content Toggle raw display