# Properties

 Label 1183.2.e.l Level $1183$ Weight $2$ Character orbit 1183.e Analytic conductor $9.446$ Analytic rank $0$ Dimension $48$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1183,2,Mod(170,1183)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1183, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1183.170");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.e (of order $$3$$, degree $$2$$, 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: $$9.44630255912$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q + q^{2} - 23 q^{4} - 13 q^{5} + 28 q^{6} + 3 q^{7} - 26 q^{9}+O(q^{10})$$ 48 * q + q^2 - 23 * q^4 - 13 * q^5 + 28 * q^6 + 3 * q^7 - 26 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$48 q + q^{2} - 23 q^{4} - 13 q^{5} + 28 q^{6} + 3 q^{7} - 26 q^{9} - 5 q^{10} + q^{11} - 5 q^{12} - 2 q^{14} + 10 q^{15} - 17 q^{16} + 5 q^{17} - 24 q^{19} + 68 q^{20} - q^{21} - 28 q^{22} - 11 q^{23} - 32 q^{24} - 33 q^{25} - 42 q^{27} - 15 q^{28} + 8 q^{29} + 22 q^{30} - 40 q^{31} + 6 q^{32} - 24 q^{33} + 72 q^{34} + 44 q^{35} - 30 q^{36} + 4 q^{37} + 29 q^{38} + 4 q^{40} + 98 q^{41} - 9 q^{42} + 26 q^{43} - 10 q^{44} - 58 q^{45} + 10 q^{46} - 62 q^{47} + 178 q^{48} + 31 q^{49} - 46 q^{50} + 21 q^{51} + 18 q^{53} - 12 q^{54} - 28 q^{55} - 56 q^{56} - 26 q^{57} - 56 q^{58} - 79 q^{59} - 22 q^{60} - 13 q^{61} + 24 q^{62} + 22 q^{63} + 36 q^{64} + 38 q^{66} + 2 q^{67} + 12 q^{68} - 56 q^{69} + 85 q^{70} - 38 q^{71} - 81 q^{72} - 17 q^{73} - 17 q^{74} - 24 q^{75} + 116 q^{76} - 30 q^{77} + 9 q^{79} - 63 q^{80} - 16 q^{81} + 22 q^{82} + 162 q^{83} + 203 q^{84} - 68 q^{85} - 22 q^{86} - 70 q^{87} + 33 q^{88} - 72 q^{89} + 2 q^{90} - 8 q^{92} - 19 q^{93} + 30 q^{94} - 13 q^{95} - 11 q^{96} + 90 q^{97} + 81 q^{98} - 78 q^{99}+O(q^{100})$$ 48 * q + q^2 - 23 * q^4 - 13 * q^5 + 28 * q^6 + 3 * q^7 - 26 * q^9 - 5 * q^10 + q^11 - 5 * q^12 - 2 * q^14 + 10 * q^15 - 17 * q^16 + 5 * q^17 - 24 * q^19 + 68 * q^20 - q^21 - 28 * q^22 - 11 * q^23 - 32 * q^24 - 33 * q^25 - 42 * q^27 - 15 * q^28 + 8 * q^29 + 22 * q^30 - 40 * q^31 + 6 * q^32 - 24 * q^33 + 72 * q^34 + 44 * q^35 - 30 * q^36 + 4 * q^37 + 29 * q^38 + 4 * q^40 + 98 * q^41 - 9 * q^42 + 26 * q^43 - 10 * q^44 - 58 * q^45 + 10 * q^46 - 62 * q^47 + 178 * q^48 + 31 * q^49 - 46 * q^50 + 21 * q^51 + 18 * q^53 - 12 * q^54 - 28 * q^55 - 56 * q^56 - 26 * q^57 - 56 * q^58 - 79 * q^59 - 22 * q^60 - 13 * q^61 + 24 * q^62 + 22 * q^63 + 36 * q^64 + 38 * q^66 + 2 * q^67 + 12 * q^68 - 56 * q^69 + 85 * q^70 - 38 * q^71 - 81 * q^72 - 17 * q^73 - 17 * q^74 - 24 * q^75 + 116 * q^76 - 30 * q^77 + 9 * q^79 - 63 * q^80 - 16 * q^81 + 22 * q^82 + 162 * q^83 + 203 * q^84 - 68 * q^85 - 22 * q^86 - 70 * q^87 + 33 * q^88 - 72 * q^89 + 2 * q^90 - 8 * q^92 - 19 * q^93 + 30 * q^94 - 13 * q^95 - 11 * q^96 + 90 * q^97 + 81 * q^98 - 78 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
170.1 −1.37124 + 2.37506i −1.20676 2.09017i −2.76062 4.78154i −0.312210 + 0.540764i 6.61904 1.27237 2.31971i 9.65697 −1.41253 + 2.44657i −0.856233 1.48304i
170.2 −1.21736 + 2.10853i −0.979025 1.69572i −1.96394 3.40165i −2.07488 + 3.59380i 4.76731 −2.28564 1.33262i 4.69388 −0.416980 + 0.722230i −5.05176 8.74991i
170.3 −1.19245 + 2.06538i −0.352721 0.610931i −1.84386 3.19366i 0.384905 0.666675i 1.68241 −2.49409 + 0.882891i 4.02505 1.25118 2.16710i 0.917958 + 1.58995i
170.4 −1.03405 + 1.79102i 1.27200 + 2.20318i −1.13851 1.97195i −0.427895 + 0.741135i −5.26125 0.562037 + 2.58537i 0.572885 −1.73599 + 3.00682i −0.884926 1.53274i
170.5 −0.883662 + 1.53055i 1.16401 + 2.01613i −0.561718 0.972924i −2.03380 + 3.52265i −4.11437 1.79914 + 1.93987i −1.54917 −1.20985 + 2.09552i −3.59438 6.22566i
170.6 −0.787207 + 1.36348i −1.10111 1.90717i −0.239388 0.414633i 1.88030 3.25678i 3.46719 −0.609781 + 2.57452i −2.39503 −0.924870 + 1.60192i 2.96037 + 5.12751i
170.7 −0.771913 + 1.33699i 0.676597 + 1.17190i −0.191698 0.332031i −0.170982 + 0.296150i −2.08910 2.12749 1.57282i −2.49575 0.584432 1.01227i −0.263967 0.457204i
170.8 −0.603060 + 1.04453i −0.516035 0.893798i 0.272638 + 0.472223i −1.51464 + 2.62344i 1.24480 −1.04407 + 2.43103i −3.06991 0.967416 1.67561i −1.82684 3.16418i
170.9 −0.545983 + 0.945670i −0.697705 1.20846i 0.403806 + 0.699412i 0.813252 1.40859i 1.52374 2.51623 0.817685i −3.06581 0.526415 0.911778i 0.888042 + 1.53813i
170.10 −0.438601 + 0.759680i 0.377512 + 0.653870i 0.615258 + 1.06566i 0.132920 0.230223i −0.662309 0.588870 2.57939i −2.83382 1.21497 2.10439i 0.116597 + 0.201953i
170.11 −0.00118363 + 0.00205011i 1.54914 + 2.68320i 0.999997 + 1.73205i 0.565709 0.979836i −0.00733446 −2.64547 + 0.0383280i −0.00946903 −3.29970 + 5.71524i 0.00133918 + 0.00231953i
170.12 0.0682269 0.118172i −1.48341 2.56934i 0.990690 + 1.71593i −1.64515 + 2.84948i −0.404833 −1.56259 + 2.13502i 0.543274 −2.90099 + 5.02467i 0.224487 + 0.388822i
170.13 0.127376 0.220623i 0.787620 + 1.36420i 0.967550 + 1.67585i −0.259002 + 0.448605i 0.401297 −1.92441 1.81567i 1.00248 0.259311 0.449140i 0.0659816 + 0.114283i
170.14 0.294236 0.509631i −0.386839 0.670024i 0.826851 + 1.43215i 1.34801 2.33482i −0.455287 2.52472 + 0.791073i 2.15010 1.20071 2.07969i −0.793264 1.37397i
170.15 0.308759 0.534786i −1.53664 2.66155i 0.809336 + 1.40181i −0.698024 + 1.20901i −1.89781 2.36540 1.18527i 2.23459 −3.22255 + 5.58162i 0.431043 + 0.746588i
170.16 0.514418 0.890998i −1.33163 2.30645i 0.470749 + 0.815361i −0.745563 + 1.29135i −2.74006 −1.93668 1.80257i 3.02632 −2.04647 + 3.54460i 0.767061 + 1.32859i
170.17 0.593734 1.02838i 1.46725 + 2.54136i 0.294961 + 0.510887i −1.70195 + 2.94787i 3.48463 −2.56390 0.653016i 3.07545 −2.80567 + 4.85955i 2.02101 + 3.50050i
170.18 0.680712 1.17903i 0.656465 + 1.13703i 0.0732621 + 0.126894i 1.52933 2.64889i 1.78745 2.33513 + 1.24384i 2.92233 0.638108 1.10524i −2.08207 3.60626i
170.19 0.875911 1.51712i −0.335909 0.581811i −0.534440 0.925678i 1.33875 2.31878i −1.17690 −1.17520 + 2.37042i 1.63116 1.27433 2.20721i −2.34525 4.06209i
170.20 0.973551 1.68624i 0.867274 + 1.50216i −0.895604 1.55123i −1.85966 + 3.22103i 3.37734 1.95410 1.78368i 0.406541 −0.00432987 + 0.00749956i 3.62095 + 6.27168i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 170.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.e.l yes 48
7.c even 3 1 inner 1183.2.e.l yes 48
7.c even 3 1 8281.2.a.cu 24
7.d odd 6 1 8281.2.a.ct 24
13.b even 2 1 1183.2.e.k 48
91.r even 6 1 1183.2.e.k 48
91.r even 6 1 8281.2.a.cv 24
91.s odd 6 1 8281.2.a.cw 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.2.e.k 48 13.b even 2 1
1183.2.e.k 48 91.r even 6 1
1183.2.e.l yes 48 1.a even 1 1 trivial
1183.2.e.l yes 48 7.c even 3 1 inner
8281.2.a.ct 24 7.d odd 6 1
8281.2.a.cu 24 7.c even 3 1
8281.2.a.cv 24 91.r even 6 1
8281.2.a.cw 24 91.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1183, [\chi])$$:

 $$T_{2}^{48} - T_{2}^{47} + 36 T_{2}^{46} - 33 T_{2}^{45} + 732 T_{2}^{44} - 632 T_{2}^{43} + 10151 T_{2}^{42} + \cdots + 1$$ T2^48 - T2^47 + 36*T2^46 - 33*T2^45 + 732*T2^44 - 632*T2^43 + 10151*T2^42 - 8243*T2^41 + 106087*T2^40 - 81539*T2^39 + 870994*T2^38 - 635067*T2^37 + 5784748*T2^36 - 4027379*T2^35 + 31544454*T2^34 - 21098051*T2^33 + 142824395*T2^32 - 92649281*T2^31 + 539578858*T2^30 - 342740897*T2^29 + 1707005233*T2^28 - 1073774747*T2^27 + 4519907017*T2^26 - 2842327653*T2^25 + 10008413532*T2^24 - 6341979220*T2^23 + 18445450356*T2^22 - 11816033318*T2^21 + 28167291869*T2^20 - 18217427312*T2^19 + 35274496336*T2^18 - 22870877696*T2^17 + 35867842038*T2^16 - 23004178701*T2^15 + 28974567039*T2^14 - 18000910641*T2^13 + 18187182197*T2^12 - 10587246410*T2^11 + 8389502964*T2^10 - 4340674208*T2^9 + 2674229836*T2^8 - 1117654469*T2^7 + 460013699*T2^6 - 115146543*T2^5 + 23201747*T2^4 - 2341432*T2^3 + 172905*T2^2 + 409*T2 + 1 $$T_{3}^{48} + 49 T_{3}^{46} + 14 T_{3}^{45} + 1371 T_{3}^{44} + 625 T_{3}^{43} + 26183 T_{3}^{42} + \cdots + 810483961$$ T3^48 + 49*T3^46 + 14*T3^45 + 1371*T3^44 + 625*T3^43 + 26183*T3^42 + 15656*T3^41 + 377639*T3^40 + 265248*T3^39 + 4276082*T3^38 + 3352995*T3^37 + 39137350*T3^36 + 32936152*T3^35 + 293341345*T3^34 + 258463976*T3^33 + 1820021470*T3^32 + 1644030309*T3^31 + 9380286332*T3^30 + 8561820000*T3^29 + 40297420793*T3^28 + 36705670001*T3^27 + 144137621881*T3^26 + 129928923099*T3^25 + 429283628038*T3^24 + 380148112164*T3^23 + 1060445105592*T3^22 + 917156320984*T3^21 + 2166761284917*T3^20 + 1816468407179*T3^19 + 3632221493931*T3^18 + 2926411244206*T3^17 + 4958186844061*T3^16 + 3789915207554*T3^15 + 5423102564747*T3^14 + 3870961606576*T3^13 + 4684342042750*T3^12 + 3054624770578*T3^11 + 3103562854573*T3^10 + 1805907682167*T3^9 + 1550393131196*T3^8 + 790250544491*T3^7 + 561525929980*T3^6 + 238793889789*T3^5 + 138029745763*T3^4 + 46857186096*T3^3 + 20032294513*T3^2 + 3978485812*T3 + 810483961