# Properties

 Label 588.2.x.a Level $588$ Weight $2$ Character orbit 588.x Analytic conductor $4.695$ Analytic rank $0$ Dimension $168$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,2,Mod(55,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([7, 0, 9]))

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

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

chi := DirichletCharacter("588.55");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 588.x (of order $$14$$, degree $$6$$, 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: $$4.69520363885$$ Analytic rank: $$0$$ Dimension: $$168$$ Relative dimension: $$28$$ over $$\Q(\zeta_{14})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$168 q - 28 q^{3} + 2 q^{7} + 6 q^{8} - 28 q^{9}+O(q^{10})$$ 168 * q - 28 * q^3 + 2 * q^7 + 6 * q^8 - 28 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$168 q - 28 q^{3} + 2 q^{7} + 6 q^{8} - 28 q^{9} + 20 q^{10} - 12 q^{14} + 36 q^{16} + 12 q^{19} - 25 q^{20} + 2 q^{21} - 6 q^{22} - 15 q^{24} + 32 q^{25} + 6 q^{26} - 28 q^{27} - 66 q^{28} - 8 q^{30} - 4 q^{31} + 25 q^{32} - 68 q^{34} - 12 q^{35} - 10 q^{37} + 35 q^{38} + 14 q^{39} + 16 q^{40} + 9 q^{42} + 20 q^{44} - 28 q^{46} - 8 q^{47} + 8 q^{48} - 8 q^{49} + 114 q^{50} + 20 q^{52} - 8 q^{53} - q^{56} + 12 q^{57} - 6 q^{58} + 20 q^{59} + 10 q^{60} - 14 q^{61} - 16 q^{62} - 12 q^{63} + 42 q^{64} - 8 q^{65} - 6 q^{66} - 16 q^{68} + 59 q^{70} + 28 q^{71} - 15 q^{72} + 22 q^{74} + 18 q^{75} + 7 q^{76} + 8 q^{77} + 6 q^{78} + 26 q^{80} - 28 q^{81} + 12 q^{82} + 10 q^{83} + 11 q^{84} - 24 q^{85} - 6 q^{86} - 242 q^{88} + 20 q^{90} - 16 q^{91} + 7 q^{92} - 4 q^{93} - 53 q^{94} - 10 q^{96} - 118 q^{98}+O(q^{100})$$ 168 * q - 28 * q^3 + 2 * q^7 + 6 * q^8 - 28 * q^9 + 20 * q^10 - 12 * q^14 + 36 * q^16 + 12 * q^19 - 25 * q^20 + 2 * q^21 - 6 * q^22 - 15 * q^24 + 32 * q^25 + 6 * q^26 - 28 * q^27 - 66 * q^28 - 8 * q^30 - 4 * q^31 + 25 * q^32 - 68 * q^34 - 12 * q^35 - 10 * q^37 + 35 * q^38 + 14 * q^39 + 16 * q^40 + 9 * q^42 + 20 * q^44 - 28 * q^46 - 8 * q^47 + 8 * q^48 - 8 * q^49 + 114 * q^50 + 20 * q^52 - 8 * q^53 - q^56 + 12 * q^57 - 6 * q^58 + 20 * q^59 + 10 * q^60 - 14 * q^61 - 16 * q^62 - 12 * q^63 + 42 * q^64 - 8 * q^65 - 6 * q^66 - 16 * q^68 + 59 * q^70 + 28 * q^71 - 15 * q^72 + 22 * q^74 + 18 * q^75 + 7 * q^76 + 8 * q^77 + 6 * q^78 + 26 * q^80 - 28 * q^81 + 12 * q^82 + 10 * q^83 + 11 * q^84 - 24 * q^85 - 6 * q^86 - 242 * q^88 + 20 * q^90 - 16 * q^91 + 7 * q^92 - 4 * q^93 - 53 * q^94 - 10 * q^96 - 118 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1 −1.40988 0.110588i −0.900969 0.433884i 1.97554 + 0.311833i 1.52760 3.17209i 1.22228 + 0.711362i −1.99561 + 1.73711i −2.75080 0.658120i 0.623490 + 0.781831i −2.50453 + 4.30333i
55.2 −1.39924 0.205230i −0.900969 0.433884i 1.91576 + 0.574332i −1.72652 + 3.58517i 1.17163 + 0.792014i 1.74761 + 1.98642i −2.56275 1.19680i 0.623490 + 0.781831i 3.15161 4.66218i
55.3 −1.35491 0.405223i −0.900969 0.433884i 1.67159 + 1.09809i −0.997385 + 2.07109i 1.04492 + 0.952969i −0.739943 2.54017i −1.81989 2.16518i 0.623490 + 0.781831i 2.19063 2.40199i
55.4 −1.27947 + 0.602464i −0.900969 0.433884i 1.27407 1.54167i 0.0432081 0.0897226i 1.41416 + 0.0123389i 0.103018 + 2.64374i −0.701339 + 2.74010i 0.623490 + 0.781831i −0.00122877 + 0.140828i
55.5 −1.20909 + 0.733551i −0.900969 0.433884i 0.923805 1.77386i −0.0202251 + 0.0419978i 1.40763 0.136302i 2.50508 0.851222i 0.184253 + 2.82242i 0.623490 + 0.781831i −0.00635357 0.0656153i
55.6 −1.17933 0.780496i −0.900969 0.433884i 0.781653 + 1.84093i −0.352634 + 0.732252i 0.723898 + 1.21490i −1.67777 + 2.04575i 0.515009 2.78114i 0.623490 + 0.781831i 0.987392 0.588339i
55.7 −1.11034 0.875865i −0.900969 0.433884i 0.465720 + 1.94502i 0.698680 1.45082i 0.620360 + 1.27089i 1.90070 1.84047i 1.18647 2.56755i 0.623490 + 0.781831i −2.04650 + 0.998961i
55.8 −0.977782 + 1.02174i −0.900969 0.433884i −0.0878851 1.99807i −1.73990 + 3.61294i 1.32427 0.496308i −2.50782 0.843102i 2.12743 + 1.86388i 0.623490 + 0.781831i −1.99022 5.31038i
55.9 −0.782021 + 1.17832i −0.900969 0.433884i −0.776887 1.84295i 1.10901 2.30289i 1.21583 0.722326i −0.953739 2.46787i 2.77913 + 0.525799i 0.623490 + 0.781831i 1.84628 + 3.10769i
55.10 −0.712285 1.22174i −0.900969 0.433884i −0.985301 + 1.74045i 1.07827 2.23906i 0.111653 + 1.40980i −2.56980 0.629382i 2.82820 0.0359170i 0.623490 + 0.781831i −3.50359 + 0.277476i
55.11 −0.537130 1.30824i −0.900969 0.433884i −1.42298 + 1.40539i −1.35460 + 2.81286i −0.0836867 + 1.41174i 2.57260 0.617858i 2.60291 + 1.10673i 0.623490 + 0.781831i 4.40750 + 0.261274i
55.12 −0.365015 + 1.36630i −0.900969 0.433884i −1.73353 0.997438i −0.617993 + 1.28328i 0.921681 1.07262i 2.62009 0.367586i 1.99556 2.00443i 0.623490 + 0.781831i −1.52776 1.31278i
55.13 −0.0562843 1.41309i −0.900969 0.433884i −1.99366 + 0.159070i 0.302206 0.627538i −0.562408 + 1.29757i 1.01653 + 2.44268i 0.336993 + 2.80828i 0.623490 + 0.781831i −0.903779 0.391725i
55.14 −0.0312592 + 1.41387i −0.900969 0.433884i −1.99805 0.0883928i 1.63442 3.39390i 0.641618 1.26029i 0.364664 + 2.62050i 0.187433 2.82221i 0.623490 + 0.781831i 4.74743 + 2.41694i
55.15 0.0809236 1.41190i −0.900969 0.433884i −1.98690 0.228511i −0.881995 + 1.83148i −0.685509 + 1.23696i −1.70818 2.02043i −0.483422 + 2.78681i 0.623490 + 0.781831i 2.51449 + 1.39349i
55.16 0.131027 + 1.40813i −0.900969 0.433884i −1.96566 + 0.369007i −1.40445 + 2.91638i 0.492913 1.32553i −0.602515 + 2.57623i −0.777166 2.71956i 0.623490 + 0.781831i −4.29066 1.59553i
55.17 0.311114 + 1.37957i −0.900969 0.433884i −1.80642 + 0.858405i 0.477768 0.992096i 0.318268 1.37794i −2.27188 1.35593i −1.74623 2.22501i 0.623490 + 0.781831i 1.51730 + 0.350459i
55.18 0.505797 1.32067i −0.900969 0.433884i −1.48834 1.33598i 1.55098 3.22064i −1.02872 + 0.970425i 1.05984 2.42420i −2.51719 + 1.28987i 0.623490 + 0.781831i −3.46892 3.67732i
55.19 0.652439 + 1.25472i −0.900969 0.433884i −1.14865 + 1.63726i −0.0842420 + 0.174930i −0.0434247 1.41355i 2.61856 + 0.378316i −2.80372 0.373018i 0.623490 + 0.781831i −0.274451 + 0.00843125i
55.20 0.745941 1.20149i −0.900969 0.433884i −0.887143 1.79248i 0.0475607 0.0987608i −1.19338 + 0.758851i −1.69059 + 2.03517i −2.81540 0.271193i 0.623490 + 0.781831i −0.0831823 0.130813i
See next 80 embeddings (of 168 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 559.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
196.j even 14 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.x.a 168
4.b odd 2 1 588.2.x.b yes 168
49.f odd 14 1 588.2.x.b yes 168
196.j even 14 1 inner 588.2.x.a 168

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.x.a 168 1.a even 1 1 trivial
588.2.x.a 168 196.j even 14 1 inner
588.2.x.b yes 168 4.b odd 2 1
588.2.x.b yes 168 49.f odd 14 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{168} - 162 T_{11}^{166} - 168 T_{11}^{165} + 15627 T_{11}^{164} + 33768 T_{11}^{163} - 1126194 T_{11}^{162} - 3686760 T_{11}^{161} + 66139323 T_{11}^{160} + 288456056 T_{11}^{159} - 3324030176 T_{11}^{158} + \cdots + 89\!\cdots\!56$$ acting on $$S_{2}^{\mathrm{new}}(588, [\chi])$$.