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Properties

Label 273.2.cd.e
Level $273$
Weight $2$
Character orbit 273.cd
Analytic conductor $2.180$
Analytic rank $0$
Dimension $112$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [273,2,Mod(44,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 4, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.44");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.cd (of order \(12\), degree \(4\), 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: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(28\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 12 q^{3} - 8 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 12 q^{3} - 8 q^{6} - 4 q^{7} + 8 q^{9} - 48 q^{13} - 12 q^{15} + 40 q^{16} - 26 q^{18} + 40 q^{19} - 10 q^{21} + 16 q^{22} + 32 q^{24} - 24 q^{27} - 52 q^{28} - 12 q^{31} - 44 q^{33} + 16 q^{34} - 8 q^{37} - 42 q^{39} - 160 q^{40} - 80 q^{42} + 6 q^{45} + 32 q^{46} + 72 q^{48} - 12 q^{52} + 34 q^{54} - 48 q^{55} - 24 q^{57} - 28 q^{58} + 44 q^{60} + 78 q^{63} + 4 q^{66} + 24 q^{67} - 12 q^{70} - 26 q^{72} - 40 q^{73} + 112 q^{76} + 32 q^{78} + 48 q^{79} + 128 q^{81} - 150 q^{84} + 160 q^{85} - 48 q^{87} + 24 q^{91} + 10 q^{93} - 8 q^{94} - 106 q^{96} + 56 q^{97} - 36 q^{99}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 −2.69539 + 0.722228i 1.57685 + 0.716628i 5.01147 − 2.89338i 1.51996 − 0.407273i −4.76779 − 0.792751i −0.780863 + 2.52790i −7.47188 + 7.47188i 1.97289 + 2.26003i −3.80275 + 2.19552i
44.2 −2.42391 + 0.649484i −1.28483 − 1.16155i 3.72144 − 2.14858i 3.58252 − 0.959933i 3.86873 + 1.98102i 1.55291 − 2.14207i −4.07612 + 4.07612i 0.301593 + 2.98480i −8.06023 + 4.65358i
44.3 −2.41247 + 0.646418i 0.872737 − 1.49611i 3.67009 − 2.11892i −1.51647 + 0.406338i −1.13834 + 4.17346i −1.79039 − 1.94795i −3.95215 + 3.95215i −1.47666 − 2.61141i 3.39578 − 1.96055i
44.4 −2.19229 + 0.587422i −1.51343 + 0.842335i 2.72902 − 1.57560i 0.959762 − 0.257167i 2.82307 − 2.73567i −2.15459 + 1.53549i −1.84753 + 1.84753i 1.58094 − 2.54963i −1.95301 + 1.12757i
44.5 −2.10280 + 0.563443i −0.755319 − 1.55868i 2.37223 − 1.36961i −2.37528 + 0.636456i 2.46651 + 2.85201i 1.72404 + 2.00691i −1.13792 + 1.13792i −1.85899 + 2.35461i 4.63613 − 2.67667i
44.6 −2.00566 + 0.537415i −0.123517 + 1.72764i 2.00181 − 1.15574i 0.878767 − 0.235465i −0.680727 − 3.53144i −1.67646 − 2.04682i −0.457340 + 0.457340i −2.96949 − 0.426786i −1.63596 + 0.944525i
44.7 −1.87036 + 0.501163i 1.45895 + 0.933521i 1.51505 − 0.874714i −1.33892 + 0.358762i −3.19662 − 1.01485i 1.90555 − 1.83545i 0.343083 − 0.343083i 1.25708 + 2.72392i 2.32447 − 1.34203i
44.8 −1.64181 + 0.439921i 1.02109 − 1.39906i 0.769949 − 0.444530i 3.49677 − 0.936956i −1.06096 + 2.74619i 0.612738 + 2.57382i 1.33522 − 1.33522i −0.914735 − 2.85714i −5.32883 + 3.07660i
44.9 −1.55291 + 0.416100i −1.66835 + 0.465425i 0.506326 − 0.292327i 0.0914679 − 0.0245087i 2.39712 − 1.41696i 2.58235 + 0.575745i 1.60897 − 1.60897i 2.56676 − 1.55298i −0.131843 + 0.0761195i
44.10 −1.27346 + 0.341222i 1.73165 − 0.0374536i −0.226792 + 0.130939i −3.02659 + 0.810973i −2.19240 + 0.638570i −0.631669 + 2.56924i 2.10860 − 2.10860i 2.99719 − 0.129713i 3.57751 − 2.06548i
44.11 −1.25001 + 0.334940i −1.60871 − 0.641920i −0.281700 + 0.162639i −3.70264 + 0.992121i 2.22591 + 0.263589i −1.90960 − 1.83124i 2.12780 − 2.12780i 2.17588 + 2.06532i 4.29606 − 2.48033i
44.12 −0.623254 + 0.167000i −1.63214 − 0.579750i −1.37149 + 0.791833i 2.98988 − 0.801137i 1.11406 + 0.0887629i −2.60516 + 0.461652i 1.63506 − 1.63506i 2.32778 + 1.89247i −1.72966 + 0.998622i
44.13 −0.578797 + 0.155088i 0.590016 − 1.62846i −1.42110 + 0.820471i −0.609213 + 0.163238i −0.0889444 + 1.03405i 2.12540 − 1.57565i 1.54270 − 1.54270i −2.30376 − 1.92163i 0.327294 − 0.188963i
44.14 −0.246180 + 0.0659636i −0.753906 + 1.55937i −1.67580 + 0.967522i −2.71744 + 0.728135i 0.0827348 − 0.433615i 1.41179 − 2.23760i 0.709158 − 0.709158i −1.86325 − 2.35123i 0.620947 − 0.358504i
44.15 0.246180 − 0.0659636i 1.72740 + 0.126782i −1.67580 + 0.967522i 2.71744 − 0.728135i 0.433615 − 0.0827348i 1.41179 − 2.23760i −0.709158 + 0.709158i 2.96785 + 0.438007i 0.620947 − 0.358504i
44.16 0.578797 − 0.155088i −1.70530 − 0.303261i −1.42110 + 0.820471i 0.609213 − 0.163238i −1.03405 + 0.0889444i 2.12540 − 1.57565i −1.54270 + 1.54270i 2.81607 + 1.03430i 0.327294 − 0.188963i
44.17 0.623254 − 0.167000i 0.313993 − 1.70335i −1.37149 + 0.791833i −2.98988 + 0.801137i −0.0887629 − 1.11406i −2.60516 + 0.461652i −1.63506 + 1.63506i −2.80282 − 1.06968i −1.72966 + 0.998622i
44.18 1.25001 − 0.334940i 0.248434 − 1.71414i −0.281700 + 0.162639i 3.70264 − 0.992121i −0.263589 − 2.22591i −1.90960 − 1.83124i −2.12780 + 2.12780i −2.87656 − 0.851703i 4.29606 − 2.48033i
44.19 1.27346 − 0.341222i −0.898259 + 1.48092i −0.226792 + 0.130939i 3.02659 − 0.810973i −0.638570 + 2.19240i −0.631669 + 2.56924i −2.10860 + 2.10860i −1.38626 − 2.66050i 3.57751 − 2.06548i
44.20 1.55291 − 0.416100i 1.23724 − 1.21212i 0.506326 − 0.292327i −0.0914679 + 0.0245087i 1.41696 − 2.39712i 2.58235 + 0.575745i −1.60897 + 1.60897i 0.0615393 − 2.99937i −0.131843 + 0.0761195i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
13.d odd 4 1 inner
21.h odd 6 1 inner
39.f even 4 1 inner
91.z odd 12 1 inner
273.cd even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.cd.e ✓ 112
3.b odd 2 1 inner 273.2.cd.e ✓ 112
7.c even 3 1 inner 273.2.cd.e ✓ 112
13.d odd 4 1 inner 273.2.cd.e ✓ 112
21.h odd 6 1 inner 273.2.cd.e ✓ 112
39.f even 4 1 inner 273.2.cd.e ✓ 112
91.z odd 12 1 inner 273.2.cd.e ✓ 112
273.cd even 12 1 inner 273.2.cd.e ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.cd.e ✓ 112 1.a even 1 1 trivial
273.2.cd.e ✓ 112 3.b odd 2 1 inner
273.2.cd.e ✓ 112 7.c even 3 1 inner
273.2.cd.e ✓ 112 13.d odd 4 1 inner
273.2.cd.e ✓ 112 21.h odd 6 1 inner
273.2.cd.e ✓ 112 39.f even 4 1 inner
273.2.cd.e ✓ 112 91.z odd 12 1 inner
273.2.cd.e ✓ 112 273.cd even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\):

\( T_{2}^{112} - 242 T_{2}^{108} + 33605 T_{2}^{104} - 3152030 T_{2}^{100} + 221630758 T_{2}^{96} - 12146123884 T_{2}^{92} + 535268243529 T_{2}^{88} - 19244548977438 T_{2}^{84} + \cdots + 42\!\cdots\!41 \) Copy content Toggle raw display
\( T_{5}^{112} - 882 T_{5}^{108} + 456457 T_{5}^{104} - 157971354 T_{5}^{100} + 40834781064 T_{5}^{96} - 8078915475000 T_{5}^{92} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
\( T_{19}^{56} - 20 T_{19}^{55} + 200 T_{19}^{54} - 1956 T_{19}^{53} + 15541 T_{19}^{52} - 71276 T_{19}^{51} + 230288 T_{19}^{50} + 87122 T_{19}^{49} - 13087904 T_{19}^{48} + 87101224 T_{19}^{47} + \cdots + 88\!\cdots\!25 \) Copy content Toggle raw display