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Properties

Label 726.2.l.b
Level $726$
Weight $2$
Character orbit 726.l
Analytic conductor $5.797$
Analytic rank $0$
Dimension $220$
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(65,726)]
 
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([11, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("726.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 726.l (of order \(22\), 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: \(220\)
Relative dimension: \(22\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 220 q + 22 q^{2} + q^{3} - 22 q^{4} - q^{6} + 22 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 220 q + 22 q^{2} + q^{3} - 22 q^{4} - q^{6} + 22 q^{8} + 3 q^{9} + 11 q^{10} - 6 q^{11} - 10 q^{12} + 22 q^{13} - 18 q^{15} - 22 q^{16} - 2 q^{17} - 3 q^{18} + 12 q^{21} - 16 q^{22} - 22 q^{23} - q^{24} + 20 q^{25} - 11 q^{27} - 10 q^{29} - 4 q^{30} + 24 q^{31} + 22 q^{32} + 2 q^{33} + 2 q^{34} + 8 q^{35} + 3 q^{36} - 8 q^{37} + 3 q^{39} + 10 q^{41} + 32 q^{42} + 5 q^{44} - 52 q^{45} + q^{48} - 20 q^{50} - 26 q^{51} - 11 q^{53} + 11 q^{54} - 26 q^{55} - 27 q^{57} + 32 q^{58} + 37 q^{60} - 2 q^{62} + 33 q^{63} - 22 q^{64} - 38 q^{65} - 57 q^{66} + 6 q^{67} - 2 q^{68} - 30 q^{70} - 3 q^{72} + 22 q^{73} + 8 q^{74} + 9 q^{75} - 33 q^{76} + 32 q^{77} + 8 q^{78} - 44 q^{79} + 15 q^{81} + 12 q^{82} + 24 q^{83} - 32 q^{84} + 132 q^{85} - 59 q^{87} - 27 q^{88} - 154 q^{89} - 36 q^{90} + 100 q^{91} - 43 q^{93} + 4 q^{95} - q^{96} - 16 q^{97} - 44 q^{98} + 91 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0.959493 + 0.281733i −1.71801 + 0.220064i 0.841254 + 0.540641i 1.05435 + 0.913596i −1.71042 − 0.272871i −4.03160 − 1.84117i 0.654861 + 0.755750i 2.90314 − 0.756145i 0.754248 + 1.17363i
65.2 0.959493 + 0.281733i −1.69264 + 0.367369i 0.841254 + 0.540641i 1.92939 + 1.67183i −1.72758 − 0.124384i 1.36848 + 0.624963i 0.654861 + 0.755750i 2.73008 − 1.24365i 1.38023 + 2.14768i
65.3 0.959493 + 0.281733i −1.62222 − 0.606954i 0.841254 + 0.540641i −3.23305 − 2.80146i −1.38551 − 1.03940i −2.01015 − 0.918003i 0.654861 + 0.755750i 2.26321 + 1.96923i −2.31283 − 3.59884i
65.4 0.959493 + 0.281733i −1.49114 − 0.881191i 0.841254 + 0.540641i −0.211809 − 0.183533i −1.18248 − 1.26560i −0.168696 − 0.0770407i 0.654861 + 0.755750i 1.44700 + 2.62796i −0.151522 − 0.235772i
65.5 0.959493 + 0.281733i −1.44684 + 0.952189i 0.841254 + 0.540641i −2.30855 − 2.00037i −1.65649 + 0.505998i 2.03461 + 0.929176i 0.654861 + 0.755750i 1.18667 − 2.75532i −1.65147 − 2.56974i
65.6 0.959493 + 0.281733i −1.37226 + 1.05683i 0.841254 + 0.540641i 1.82244 + 1.57916i −1.61442 + 0.627411i 2.50381 + 1.14345i 0.654861 + 0.755750i 0.766216 − 2.90050i 1.30372 + 2.02863i
65.7 0.959493 + 0.281733i −1.23264 + 1.21680i 0.841254 + 0.540641i −0.822648 − 0.712829i −1.52552 + 0.820236i −4.17450 − 1.90643i 0.654861 + 0.755750i 0.0387970 − 2.99975i −0.588498 − 0.915721i
65.8 0.959493 + 0.281733i −0.801144 − 1.53563i 0.841254 + 0.540641i 1.07146 + 0.928424i −0.336054 − 1.69914i 3.37213 + 1.54000i 0.654861 + 0.755750i −1.71634 + 2.46053i 0.766490 + 1.19268i
65.9 0.959493 + 0.281733i −0.736251 − 1.56778i 0.841254 + 0.540641i −1.63520 − 1.41691i −0.264733 − 1.71170i −0.639460 − 0.292032i 0.654861 + 0.755750i −1.91587 + 2.30856i −1.16977 − 1.82020i
65.10 0.959493 + 0.281733i −0.647553 + 1.60645i 0.841254 + 0.540641i −0.770713 − 0.667826i −1.07391 + 1.35894i −0.0890865 − 0.0406845i 0.654861 + 0.755750i −2.16135 − 2.08052i −0.551345 − 0.857909i
65.11 0.959493 + 0.281733i 0.0643254 − 1.73086i 0.841254 + 0.540641i 2.67928 + 2.32161i 0.549358 − 1.64262i −0.555302 − 0.253598i 0.654861 + 0.755750i −2.99172 − 0.222676i 1.91668 + 2.98241i
65.12 0.959493 + 0.281733i 0.200058 + 1.72046i 0.841254 + 0.540641i 1.47810 + 1.28078i −0.292755 + 1.70713i 1.22772 + 0.560682i 0.654861 + 0.755750i −2.91995 + 0.688384i 1.05739 + 1.64533i
65.13 0.959493 + 0.281733i 0.246545 + 1.71441i 0.841254 + 0.540641i 3.13973 + 2.72059i −0.246448 + 1.71443i −3.61725 − 1.65194i 0.654861 + 0.755750i −2.87843 + 0.845360i 2.24607 + 3.49495i
65.14 0.959493 + 0.281733i 0.477628 − 1.66489i 0.841254 + 0.540641i −0.181554 − 0.157318i 0.927335 − 1.46289i −4.37409 − 1.99758i 0.654861 + 0.755750i −2.54374 − 1.59040i −0.129878 − 0.202095i
65.15 0.959493 + 0.281733i 0.950849 + 1.44772i 0.841254 + 0.540641i −3.12938 − 2.71163i 0.504464 + 1.65696i 4.17347 + 1.90596i 0.654861 + 0.755750i −1.19177 + 2.75312i −2.23867 − 3.48344i
65.16 0.959493 + 0.281733i 1.09242 − 1.34410i 0.841254 + 0.540641i −0.944423 − 0.818347i 1.42685 − 0.981887i 1.94737 + 0.889332i 0.654861 + 0.755750i −0.613230 − 2.93666i −0.675612 − 1.05127i
65.17 0.959493 + 0.281733i 1.25365 − 1.19514i 0.841254 + 0.540641i −0.298131 − 0.258332i 1.53958 − 0.793540i 4.42797 + 2.02219i 0.654861 + 0.755750i 0.143260 − 2.99658i −0.213274 − 0.331861i
65.18 0.959493 + 0.281733i 1.38689 + 1.03756i 0.841254 + 0.540641i −0.493288 − 0.427437i 1.03840 + 1.38626i 1.22731 + 0.560492i 0.654861 + 0.755750i 0.846940 + 2.87797i −0.352884 − 0.549098i
65.19 0.959493 + 0.281733i 1.41433 − 0.999833i 0.841254 + 0.540641i 2.51420 + 2.17857i 1.63873 − 0.560869i −0.782416 − 0.357317i 0.654861 + 0.755750i 1.00067 − 2.82819i 1.79858 + 2.79865i
65.20 0.959493 + 0.281733i 1.52926 + 0.813234i 0.841254 + 0.540641i 1.58903 + 1.37690i 1.23820 + 1.21114i 2.05961 + 0.940594i 0.654861 + 0.755750i 1.67730 + 2.48730i 1.13674 + 1.76881i
See next 80 embeddings (of 220 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
363.j even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 726.2.l.b yes 220
3.b odd 2 1 726.2.l.a ✓ 220
121.f odd 22 1 726.2.l.a ✓ 220
363.j even 22 1 inner 726.2.l.b yes 220
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
726.2.l.a ✓ 220 3.b odd 2 1
726.2.l.a ✓ 220 121.f odd 22 1
726.2.l.b yes 220 1.a even 1 1 trivial
726.2.l.b yes 220 363.j even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{220} - 65 T_{5}^{218} + 55 T_{5}^{217} + 2470 T_{5}^{216} - 4037 T_{5}^{215} + \cdots + 14\!\cdots\!64 \) acting on \(S_{2}^{\mathrm{new}}(726, [\chi])\). Copy content Toggle raw display