# 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 $8$

# Related objects

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.

 $$\operatorname{Tr}(f)(q) =$$ $$112 q - 12 q^{3} - 8 q^{6} - 4 q^{7} + 8 q^{9}+O(q^{10})$$ 112 * q - 12 * q^3 - 8 * q^6 - 4 * q^7 + 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$112 q - 12 q^{3} - 8 q^{6} - 4 q^{7} + 8 q^{9} - 48 q^{13} - 12 q^{15} + 40 q^{16} - 26 q^{18} + 40 q^{19} - 10 q^{21} + 16 q^{22} + 32 q^{24} - 24 q^{27} - 52 q^{28} - 12 q^{31} - 44 q^{33} + 16 q^{34} - 8 q^{37} - 42 q^{39} - 160 q^{40} - 80 q^{42} + 6 q^{45} + 32 q^{46} + 72 q^{48} - 12 q^{52} + 34 q^{54} - 48 q^{55} - 24 q^{57} - 28 q^{58} + 44 q^{60} + 78 q^{63} + 4 q^{66} + 24 q^{67} - 12 q^{70} - 26 q^{72} - 40 q^{73} + 112 q^{76} + 32 q^{78} + 48 q^{79} + 128 q^{81} - 150 q^{84} + 160 q^{85} - 48 q^{87} + 24 q^{91} + 10 q^{93} - 8 q^{94} - 106 q^{96} + 56 q^{97} - 36 q^{99}+O(q^{100})$$ 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(273, [\chi])$$:

 $$T_{2}^{112} - 242 T_{2}^{108} + 33605 T_{2}^{104} - 3152030 T_{2}^{100} + 221630758 T_{2}^{96} - 12146123884 T_{2}^{92} + 535268243529 T_{2}^{88} - 19244548977438 T_{2}^{84} + \cdots + 42\!\cdots\!41$$ T2^112 - 242*T2^108 + 33605*T2^104 - 3152030*T2^100 + 221630758*T2^96 - 12146123884*T2^92 + 535268243529*T2^88 - 19244548977438*T2^84 + 571531364156364*T2^80 - 14087419199051836*T2^76 + 289371402969369848*T2^72 - 4947544815600179222*T2^68 + 70325135619674275312*T2^64 - 825891083362765260718*T2^60 + 7972128152964277438287*T2^56 - 62534656970261644424566*T2^52 + 395373865993299788105839*T2^48 - 1976538819243115954313132*T2^44 + 7714878230752995812498137*T2^40 - 22668136225995317214192338*T2^36 + 49480223013185711954874138*T2^32 - 72944950171055391523562452*T2^28 + 68431162019210861568858530*T2^24 - 18939997212760164158126778*T2^20 + 3766090228782095742127941*T2^16 - 365135200078911904684368*T2^12 + 25205300558855823695534*T2^8 - 106070068600492137124*T2^4 + 421301638652745841 $$T_{5}^{112} - 882 T_{5}^{108} + 456457 T_{5}^{104} - 157971354 T_{5}^{100} + 40834781064 T_{5}^{96} - 8078915475000 T_{5}^{92} + \cdots + 27\!\cdots\!76$$ T5^112 - 882*T5^108 + 456457*T5^104 - 157971354*T5^100 + 40834781064*T5^96 - 8078915475000*T5^92 + 1262487309973485*T5^88 - 156189528458475184*T5^84 + 15465815506348258083*T5^80 - 1214839847978749680632*T5^76 + 75653861075346976192461*T5^72 - 3645804993148278500727088*T5^68 + 134499721421576454676980201*T5^64 - 3592822498821587566062713838*T5^60 + 68636649603551965699235079246*T5^56 - 812830903838947194267037274706*T5^52 + 6926247291760602252170316988006*T5^48 - 41064835212451037482734516707676*T5^44 + 180142781840508447944564146539396*T5^40 - 542833553794273143048880012433348*T5^36 + 1169523632706045629273775828776161*T5^32 - 1499377974262428251194370832021680*T5^28 + 1371787011209022459881661626139424*T5^24 - 752608103208928875292550654425088*T5^20 + 284480609913662017750972578155264*T5^16 - 40525316003592617935197689757696*T5^12 + 4273718558751695773703621582848*T5^8 - 343640974709329952384417792*T5^4 + 27610424190746351435776 $$T_{19}^{56} - 20 T_{19}^{55} + 200 T_{19}^{54} - 1956 T_{19}^{53} + 15541 T_{19}^{52} - 71276 T_{19}^{51} + 230288 T_{19}^{50} + 87122 T_{19}^{49} - 13087904 T_{19}^{48} + 87101224 T_{19}^{47} + \cdots + 88\!\cdots\!25$$ T19^56 - 20*T19^55 + 200*T19^54 - 1956*T19^53 + 15541*T19^52 - 71276*T19^51 + 230288*T19^50 + 87122*T19^49 - 13087904*T19^48 + 87101224*T19^47 - 333626032*T19^46 + 614257792*T19^45 + 12192800159*T19^44 - 110068954864*T19^43 + 555768503610*T19^42 - 2734650750678*T19^41 + 4652402504145*T19^40 + 22640422699612*T19^39 - 153729637440626*T19^38 + 1065836747607532*T19^37 - 3127800855517798*T19^36 - 4910220432153312*T19^35 + 47936699927081150*T19^34 - 410959341215141580*T19^33 + 2040350933409896529*T19^32 - 4349647432786143224*T19^31 + 14513644087952283956*T19^30 - 28308976505383991184*T19^29 - 102314418941743731253*T19^28 + 69830763303226627912*T19^27 + 8744389502254459458*T19^26 + 2970126623315837821690*T19^25 + 4836637190825093663877*T19^24 - 13073894607098794005676*T19^23 - 31661651104481730287134*T19^22 - 50442116166856136274396*T19^21 - 87696424553585703426174*T19^20 + 100724292066962664043424*T19^19 + 746472094179434834779418*T19^18 + 1891144969124758835776388*T19^17 + 4054799083319981736456309*T19^16 + 7325991269713980225751968*T19^15 + 10791186901403444719770084*T19^14 + 14232534587557031083032408*T19^13 + 17035321736525587120984839*T19^12 + 17575026587861262396638448*T19^11 + 16108619748409848216692246*T19^10 + 13464377313745918712790394*T19^9 + 9540671759853458683245172*T19^8 + 5614747283113833478785512*T19^7 + 2925179734773123729764662*T19^6 + 1138817725156812837826904*T19^5 + 168867203697339052084449*T19^4 - 25609191340135807594060*T19^3 + 925657717104802976450*T19^2 - 127964895351481106750*T19 + 8845069910685750625