↑

Properties

Label 726.2.i.b
Level $726$
Weight $2$
Character orbit 726.i
Analytic conductor $5.797$
Analytic rank $0$
Dimension $50$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(50\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 50 q + 5 q^{2} - 50 q^{3} - 5 q^{4} - 5 q^{6} - q^{7} + 5 q^{8} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q + 5 q^{2} - 50 q^{3} - 5 q^{4} - 5 q^{6} - q^{7} + 5 q^{8} + 50 q^{9} - 11 q^{11} + 5 q^{12} - 21 q^{13} - 10 q^{14} - 5 q^{16} - 12 q^{17} + 5 q^{18} + q^{21} - 11 q^{22} + 22 q^{23} - 5 q^{24} - 3 q^{25} - q^{26} - 50 q^{27} - q^{28} + 36 q^{31} + 5 q^{32} + 11 q^{33} + 12 q^{34} - 11 q^{35} - 5 q^{36} - 19 q^{37} + 22 q^{38} + 21 q^{39} + 23 q^{41} + 10 q^{42} - 44 q^{43} + 11 q^{44} + 11 q^{46} + 20 q^{47} + 5 q^{48} - 24 q^{49} + 3 q^{50} + 12 q^{51} + q^{52} - 11 q^{53} - 5 q^{54} - 33 q^{55} + 12 q^{56} + q^{59} + 38 q^{61} - 14 q^{62} - q^{63} - 5 q^{64} + 26 q^{65} + 11 q^{66} - 64 q^{67} - q^{68} - 22 q^{69} + 24 q^{71} + 5 q^{72} + 35 q^{73} - 14 q^{74} + 3 q^{75} + 22 q^{76} - 77 q^{77} + q^{78} - 32 q^{79} + 50 q^{81} - 12 q^{82} - 8 q^{83} + q^{84} - 5 q^{85} - 11 q^{88} - 12 q^{89} - 11 q^{92} - 36 q^{93} + 46 q^{94} + 57 q^{95} - 5 q^{96} + 33 q^{97} - 42 q^{98} - 11 q^{99}+O(q^{100}) \) 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 −2.41363 − 2.78548i −0.959493 + 0.281733i −0.0473710 − 0.103728i 0.654861 − 0.755750i 1.00000 −3.10062 − 1.99265i
67.2 0.959493 − 0.281733i −1.00000 0.841254 − 0.540641i −1.13110 − 1.30536i −0.959493 + 0.281733i 0.148877 + 0.325995i 0.654861 − 0.755750i 1.00000 −1.45304 − 0.933815i
67.3 0.959493 − 0.281733i −1.00000 0.841254 − 0.540641i 0.218702 + 0.252395i −0.959493 + 0.281733i 1.43275 + 3.13729i 0.654861 − 0.755750i 1.00000 0.280951 + 0.180556i
67.4 0.959493 − 0.281733i −1.00000 0.841254 − 0.540641i 0.564730 + 0.651733i −0.959493 + 0.281733i −1.20194 − 2.63189i 0.654861 − 0.755750i 1.00000 0.725468 + 0.466230i
67.5 0.959493 − 0.281733i −1.00000 0.841254 − 0.540641i 2.76130 + 3.18671i −0.959493 + 0.281733i −0.612872 − 1.34200i 0.654861 − 0.755750i 1.00000 3.54725 + 2.27968i
133.1 −0.841254 + 0.540641i −1.00000 0.415415 − 0.909632i −0.511211 + 3.55555i 0.841254 − 0.540641i −0.0418117 + 0.0482533i 0.142315 + 0.989821i 1.00000 −1.49222 − 3.26750i
133.2 −0.841254 + 0.540641i −1.00000 0.415415 − 0.909632i −0.171316 + 1.19153i 0.841254 − 0.540641i 2.77790 − 3.20587i 0.142315 + 0.989821i 1.00000 −0.500068 − 1.09500i
133.3 −0.841254 + 0.540641i −1.00000 0.415415 − 0.909632i 0.00194716 − 0.0135428i 0.841254 − 0.540641i −1.27711 + 1.47387i 0.142315 + 0.989821i 1.00000 0.00568373 + 0.0124456i
133.4 −0.841254 + 0.540641i −1.00000 0.415415 − 0.909632i 0.268475 − 1.86729i 0.841254 − 0.540641i −0.505972 + 0.583923i 0.142315 + 0.989821i 1.00000 0.783675 + 1.71601i
133.5 −0.841254 + 0.540641i −1.00000 0.415415 − 0.909632i 0.412105 − 2.86625i 0.841254 − 0.540641i 2.24673 − 2.59287i 0.142315 + 0.989821i 1.00000 1.20293 + 2.63405i
199.1 0.654861 − 0.755750i −1.00000 −0.142315 − 0.989821i −2.50081 + 1.60717i −0.654861 + 0.755750i 1.17421 + 0.344779i −0.841254 − 0.540641i 1.00000 −0.423062 + 2.94246i
199.2 0.654861 − 0.755750i −1.00000 −0.142315 − 0.989821i −1.85223 + 1.19036i −0.654861 + 0.755750i −0.134204 − 0.0394059i −0.841254 − 0.540641i 1.00000 −0.313342 + 2.17934i
199.3 0.654861 − 0.755750i −1.00000 −0.142315 − 0.989821i −0.421646 + 0.270975i −0.654861 + 0.755750i −3.29192 − 0.966596i −0.841254 − 0.540641i 1.00000 −0.0713298 + 0.496110i
199.4 0.654861 − 0.755750i −1.00000 −0.142315 − 0.989821i 2.33561 − 1.50101i −0.654861 + 0.755750i −0.107206 − 0.0314786i −0.841254 − 0.540641i 1.00000 0.395116 − 2.74809i
199.5 0.654861 − 0.755750i −1.00000 −0.142315 − 0.989821i 2.43907 − 1.56750i −0.654861 + 0.755750i −0.349582 − 0.102646i −0.841254 − 0.540641i 1.00000 0.412618 − 2.86982i
265.1 −0.415415 + 0.909632i −1.00000 −0.654861 − 0.755750i −2.63232 − 0.772918i 0.415415 − 0.909632i 0.388005 + 2.69864i 0.959493 − 0.281733i 1.00000 1.79657 − 2.07336i
265.2 −0.415415 + 0.909632i −1.00000 −0.654861 − 0.755750i −2.04383 − 0.600123i 0.415415 − 0.909632i 0.0100077 + 0.0696052i 0.959493 − 0.281733i 1.00000 1.39493 − 1.60983i
265.3 −0.415415 + 0.909632i −1.00000 −0.654861 − 0.755750i 0.120444 + 0.0353657i 0.415415 − 0.909632i −0.477670 − 3.32227i 0.959493 − 0.281733i 1.00000 −0.0822041 + 0.0948686i
265.4 −0.415415 + 0.909632i −1.00000 −0.654861 − 0.755750i 1.86664 + 0.548096i 0.415415 − 0.909632i −0.239369 − 1.66485i 0.959493 − 0.281733i 1.00000 −1.27400 + 1.47027i
265.5 −0.415415 + 0.909632i −1.00000 −0.654861 − 0.755750i 2.68906 + 0.789580i 0.415415 − 0.909632i 0.613235 + 4.26514i 0.959493 − 0.281733i 1.00000 −1.83530 + 2.11805i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.5
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.b ✓ 50
121.e even 11 1 inner 726.2.i.b ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
726.2.i.b ✓ 50 1.a even 1 1 trivial
726.2.i.b ✓ 50 121.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{50} + 14 T_{5}^{48} + 2 T_{5}^{47} + 295 T_{5}^{46} + 320 T_{5}^{45} + 2836 T_{5}^{44} + \cdots + 2076481 \) acting on \(S_{2}^{\mathrm{new}}(726, [\chi])\). Copy content Toggle raw display