# Properties

 Label 224.2.u.c Level $224$ Weight $2$ Character orbit 224.u Analytic conductor $1.789$ Analytic rank $0$ Dimension $52$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,2,Mod(29,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

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

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

chi := DirichletCharacter("224.29");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 224.u (of order $$8$$, 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: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$52$$ Relative dimension: $$13$$ over $$\Q(\zeta_{8})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$52 q - 20 q^{6}+O(q^{10})$$ 52 * q - 20 * q^6 $$\operatorname{Tr}(f)(q) =$$ $$52 q - 20 q^{6} - 8 q^{10} + 12 q^{12} - 12 q^{16} - 20 q^{18} + 20 q^{22} - 20 q^{23} - 8 q^{24} + 20 q^{26} - 24 q^{27} - 8 q^{28} + 20 q^{30} + 60 q^{32} - 48 q^{33} + 48 q^{34} + 8 q^{36} - 60 q^{38} - 24 q^{39} + 20 q^{40} - 44 q^{43} + 32 q^{44} + 40 q^{45} - 32 q^{46} - 84 q^{48} - 124 q^{50} + 16 q^{51} - 32 q^{52} - 36 q^{53} + 96 q^{54} + 32 q^{55} + 16 q^{56} + 4 q^{58} - 92 q^{60} - 32 q^{61} + 12 q^{62} + 68 q^{63} + 48 q^{64} + 80 q^{65} + 16 q^{66} + 28 q^{67} - 4 q^{68} - 32 q^{69} + 8 q^{70} - 88 q^{72} + 36 q^{74} + 32 q^{75} + 96 q^{76} - 12 q^{77} + 12 q^{78} - 108 q^{80} - 96 q^{82} + 64 q^{85} + 76 q^{86} - 56 q^{87} + 104 q^{88} - 132 q^{90} + 32 q^{92} - 4 q^{94} - 64 q^{95} + 8 q^{96} - 72 q^{97} - 64 q^{99}+O(q^{100})$$ 52 * q - 20 * q^6 - 8 * q^10 + 12 * q^12 - 12 * q^16 - 20 * q^18 + 20 * q^22 - 20 * q^23 - 8 * q^24 + 20 * q^26 - 24 * q^27 - 8 * q^28 + 20 * q^30 + 60 * q^32 - 48 * q^33 + 48 * q^34 + 8 * q^36 - 60 * q^38 - 24 * q^39 + 20 * q^40 - 44 * q^43 + 32 * q^44 + 40 * q^45 - 32 * q^46 - 84 * q^48 - 124 * q^50 + 16 * q^51 - 32 * q^52 - 36 * q^53 + 96 * q^54 + 32 * q^55 + 16 * q^56 + 4 * q^58 - 92 * q^60 - 32 * q^61 + 12 * q^62 + 68 * q^63 + 48 * q^64 + 80 * q^65 + 16 * q^66 + 28 * q^67 - 4 * q^68 - 32 * q^69 + 8 * q^70 - 88 * q^72 + 36 * q^74 + 32 * q^75 + 96 * q^76 - 12 * q^77 + 12 * q^78 - 108 * q^80 - 96 * q^82 + 64 * q^85 + 76 * q^86 - 56 * q^87 + 104 * q^88 - 132 * q^90 + 32 * q^92 - 4 * q^94 - 64 * q^95 + 8 * q^96 - 72 * q^97 - 64 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −1.33683 + 0.461403i −0.825255 0.341832i 1.57421 1.23363i −0.175588 0.423906i 1.26095 + 0.0761946i 0.707107 0.707107i −1.53525 + 2.37550i −1.55712 1.55712i 0.430322 + 0.485673i
29.2 −1.24749 0.666168i 2.18713 + 0.905938i 1.11244 + 1.66207i 0.797931 + 1.92638i −2.12490 2.58714i 0.707107 0.707107i −0.280536 2.81448i 1.84149 + 1.84149i 0.287882 2.93468i
29.3 −1.03545 + 0.963248i 2.47594 + 1.02557i 0.144306 1.99479i −1.23251 2.97553i −3.55158 + 1.32302i 0.707107 0.707107i 1.77205 + 2.20450i 2.95715 + 2.95715i 4.14237 + 1.89380i
29.4 −0.926451 1.06850i −1.34443 0.556881i −0.283377 + 1.97982i 0.598728 + 1.44546i 0.650522 + 1.95244i 0.707107 0.707107i 2.37797 1.53142i −0.623944 0.623944i 0.989776 1.97888i
29.5 −0.742328 + 1.20372i −2.57348 1.06597i −0.897897 1.78712i 0.224493 + 0.541974i 3.19350 2.30646i 0.707107 0.707107i 2.81773 + 0.245807i 3.36518 + 3.36518i −0.819034 0.132095i
29.6 −0.729813 + 1.21135i 1.74965 + 0.724727i −0.934745 1.76812i 1.44353 + 3.48498i −2.15481 + 1.59052i 0.707107 0.707107i 2.82401 + 0.158094i 0.414709 + 0.414709i −5.27504 0.794768i
29.7 −0.0564658 1.41309i −2.95511 1.22405i −1.99362 + 0.159582i −1.21151 2.92484i −1.56282 + 4.24494i 0.707107 0.707107i 0.338075 + 2.80815i 5.11306 + 5.11306i −4.06464 + 1.87712i
29.8 0.171231 + 1.40381i −0.380077 0.157433i −1.94136 + 0.480752i −1.58477 3.82598i 0.155925 0.560513i 0.707107 0.707107i −1.00731 2.64298i −2.00165 2.00165i 5.09958 2.87984i
29.9 0.631975 1.26515i 2.91760 + 1.20851i −1.20122 1.59909i −0.629460 1.51965i 3.37279 2.92745i 0.707107 0.707107i −2.78223 + 0.509137i 4.93055 + 4.93055i −2.32039 0.164019i
29.10 0.863963 1.11963i −0.301186 0.124755i −0.507135 1.93464i −0.107860 0.260396i −0.399893 + 0.229432i 0.707107 0.707107i −2.60422 1.10365i −2.04617 2.04617i −0.384733 0.104210i
29.11 0.875361 + 1.11074i 0.999166 + 0.413868i −0.467486 + 1.94460i 0.523077 + 1.26282i 0.414931 + 1.47210i 0.707107 0.707107i −2.56916 + 1.18297i −1.29427 1.29427i −0.944783 + 1.68643i
29.12 1.41113 0.0933551i −2.44686 1.01352i 1.98257 0.263472i 1.68815 + 4.07556i −3.54745 1.20178i 0.707107 0.707107i 2.77306 0.556876i 2.83858 + 2.83858i 2.76268 + 5.59355i
29.13 1.41405 0.0212810i 0.496926 + 0.205834i 1.99909 0.0601851i −0.334218 0.806875i 0.707060 + 0.280485i 0.707107 0.707107i 2.82555 0.127648i −1.91675 1.91675i −0.489774 1.13385i
85.1 −1.33683 0.461403i −0.825255 + 0.341832i 1.57421 + 1.23363i −0.175588 + 0.423906i 1.26095 0.0761946i 0.707107 + 0.707107i −1.53525 2.37550i −1.55712 + 1.55712i 0.430322 0.485673i
85.2 −1.24749 + 0.666168i 2.18713 0.905938i 1.11244 1.66207i 0.797931 1.92638i −2.12490 + 2.58714i 0.707107 + 0.707107i −0.280536 + 2.81448i 1.84149 1.84149i 0.287882 + 2.93468i
85.3 −1.03545 0.963248i 2.47594 1.02557i 0.144306 + 1.99479i −1.23251 + 2.97553i −3.55158 1.32302i 0.707107 + 0.707107i 1.77205 2.20450i 2.95715 2.95715i 4.14237 1.89380i
85.4 −0.926451 + 1.06850i −1.34443 + 0.556881i −0.283377 1.97982i 0.598728 1.44546i 0.650522 1.95244i 0.707107 + 0.707107i 2.37797 + 1.53142i −0.623944 + 0.623944i 0.989776 + 1.97888i
85.5 −0.742328 1.20372i −2.57348 + 1.06597i −0.897897 + 1.78712i 0.224493 0.541974i 3.19350 + 2.30646i 0.707107 + 0.707107i 2.81773 0.245807i 3.36518 3.36518i −0.819034 + 0.132095i
85.6 −0.729813 1.21135i 1.74965 0.724727i −0.934745 + 1.76812i 1.44353 3.48498i −2.15481 1.59052i 0.707107 + 0.707107i 2.82401 0.158094i 0.414709 0.414709i −5.27504 + 0.794768i
85.7 −0.0564658 + 1.41309i −2.95511 + 1.22405i −1.99362 0.159582i −1.21151 + 2.92484i −1.56282 4.24494i 0.707107 + 0.707107i 0.338075 2.80815i 5.11306 5.11306i −4.06464 1.87712i
See all 52 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.13 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.u.c 52
4.b odd 2 1 896.2.u.c 52
32.g even 8 1 inner 224.2.u.c 52
32.h odd 8 1 896.2.u.c 52

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.u.c 52 1.a even 1 1 trivial
224.2.u.c 52 32.g even 8 1 inner
896.2.u.c 52 4.b odd 2 1
896.2.u.c 52 32.h odd 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{52} + 8 T_{3}^{49} - 72 T_{3}^{47} - 40 T_{3}^{46} - 88 T_{3}^{45} + 16440 T_{3}^{44} - 1328 T_{3}^{43} - 12224 T_{3}^{42} + 108960 T_{3}^{41} + 96 T_{3}^{40} - 1101488 T_{3}^{39} + 796048 T_{3}^{38} + \cdots + 151519232$$ acting on $$S_{2}^{\mathrm{new}}(224, [\chi])$$.