# Properties

 Label 588.2.ba.a Level $588$ Weight $2$ Character orbit 588.ba Analytic conductor $4.695$ Analytic rank $0$ Dimension $336$ 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(103,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([21, 0, 29]))

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

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

chi := DirichletCharacter("588.103");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 588.ba (of order $$42$$, degree $$12$$, 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: $$336$$ Relative dimension: $$28$$ over $$\Q(\zeta_{42})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10})$$ 336 * q - 28 * q^3 + 2 * q^7 - 6 * q^8 + 28 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$336 q - 28 q^{3} + 2 q^{7} - 6 q^{8} + 28 q^{9} + 41 q^{10} + 6 q^{11} + 52 q^{14} + 20 q^{16} - 6 q^{19} - 25 q^{20} + 4 q^{21} + 6 q^{22} + 9 q^{24} - 26 q^{25} - 18 q^{26} + 56 q^{27} + 5 q^{30} + 2 q^{31} + 15 q^{32} + 6 q^{33} + 44 q^{34} - 12 q^{35} + 16 q^{37} - 100 q^{38} + 8 q^{39} - 7 q^{40} - 7 q^{42} - 53 q^{44} - 10 q^{46} + 4 q^{47} + 8 q^{48} - 4 q^{49} - 114 q^{50} - 28 q^{52} - 4 q^{53} + q^{56} - 12 q^{57} + 27 q^{58} - 10 q^{59} - 7 q^{60} + 2 q^{61} - 16 q^{62} - 12 q^{63} - 84 q^{64} - 4 q^{65} + 21 q^{66} + 42 q^{67} + 26 q^{68} - 70 q^{70} + 28 q^{71} + 15 q^{72} + 18 q^{73} + 20 q^{74} + 54 q^{75} + 49 q^{76} - 8 q^{77} - 6 q^{78} - 6 q^{79} - 40 q^{80} + 28 q^{81} - 99 q^{82} + 10 q^{83} + 54 q^{84} + 24 q^{85} - 314 q^{86} - 40 q^{88} + 20 q^{90} - 34 q^{91} + 14 q^{92} - 2 q^{93} - 152 q^{94} - 24 q^{95} - 10 q^{96} - 156 q^{98}+O(q^{100})$$ 336 * q - 28 * q^3 + 2 * q^7 - 6 * q^8 + 28 * q^9 + 41 * q^10 + 6 * q^11 + 52 * q^14 + 20 * q^16 - 6 * q^19 - 25 * q^20 + 4 * q^21 + 6 * q^22 + 9 * q^24 - 26 * q^25 - 18 * q^26 + 56 * q^27 + 5 * q^30 + 2 * q^31 + 15 * q^32 + 6 * q^33 + 44 * q^34 - 12 * q^35 + 16 * q^37 - 100 * q^38 + 8 * q^39 - 7 * q^40 - 7 * q^42 - 53 * q^44 - 10 * q^46 + 4 * q^47 + 8 * q^48 - 4 * q^49 - 114 * q^50 - 28 * q^52 - 4 * q^53 + q^56 - 12 * q^57 + 27 * q^58 - 10 * q^59 - 7 * q^60 + 2 * q^61 - 16 * q^62 - 12 * q^63 - 84 * q^64 - 4 * q^65 + 21 * q^66 + 42 * q^67 + 26 * q^68 - 70 * q^70 + 28 * q^71 + 15 * q^72 + 18 * q^73 + 20 * q^74 + 54 * q^75 + 49 * q^76 - 8 * q^77 - 6 * q^78 - 6 * q^79 - 40 * q^80 + 28 * q^81 - 99 * q^82 + 10 * q^83 + 54 * q^84 + 24 * q^85 - 314 * q^86 - 40 * q^88 + 20 * q^90 - 34 * q^91 + 14 * q^92 - 2 * q^93 - 152 * q^94 - 24 * q^95 - 10 * q^96 - 156 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
103.1 −1.40964 0.113611i −0.826239 0.563320i 1.97419 + 0.320301i 3.32240 + 0.248980i 1.10070 + 0.887950i −2.08944 1.62303i −2.74651 0.675798i 0.365341 + 0.930874i −4.65512 0.728433i
103.2 −1.36468 + 0.371026i −0.826239 0.563320i 1.72468 1.01266i −0.793563 0.0594693i 1.33655 + 0.462193i 0.251406 + 2.63378i −1.97791 + 2.02185i 0.365341 + 0.930874i 1.10502 0.213276i
103.3 −1.33039 0.479649i −0.826239 0.563320i 1.53987 + 1.27624i −1.47341 0.110417i 0.829024 + 1.14574i 2.51658 + 0.816579i −1.43648 2.43650i 0.365341 + 0.930874i 1.90724 + 0.853615i
103.4 −1.24022 0.679596i −0.826239 0.563320i 1.07630 + 1.68570i −2.59884 0.194756i 0.641889 + 1.26015i −2.28001 + 1.34222i −0.189256 2.82209i 0.365341 + 0.930874i 3.09079 + 2.00770i
103.5 −1.23869 + 0.682383i −0.826239 0.563320i 1.06871 1.69052i 0.961592 + 0.0720614i 1.40785 + 0.133968i 2.50588 0.848875i −0.170216 + 2.82330i 0.365341 + 0.930874i −1.24029 + 0.566912i
103.6 −1.15847 + 0.811146i −0.826239 0.563320i 0.684086 1.87937i 1.33423 + 0.0999870i 1.41410 0.0176130i −2.38917 1.13661i 0.731952 + 2.73208i 0.365341 + 0.930874i −1.62677 + 0.966426i
103.7 −1.08950 0.901663i −0.826239 0.563320i 0.374008 + 1.96472i 2.88553 + 0.216241i 0.392260 + 1.35872i 0.232552 + 2.63551i 1.36403 2.47778i 0.365341 + 0.930874i −2.94880 2.83737i
103.8 −0.839211 + 1.13830i −0.826239 0.563320i −0.591451 1.91055i −4.04500 0.303131i 1.33462 0.467763i −2.59666 + 0.507311i 2.67113 + 0.930102i 0.365341 + 0.930874i 3.73966 4.35003i
103.9 −0.706021 1.22537i −0.826239 0.563320i −1.00307 + 1.73028i −0.481855 0.0361100i −0.106934 + 1.41016i −1.51904 2.16623i 2.82842 + 0.00752107i 0.365341 + 0.930874i 0.295951 + 0.615945i
103.10 −0.677412 1.24142i −0.826239 0.563320i −1.08223 + 1.68190i −2.31910 0.173792i −0.139610 + 1.40731i 2.40273 1.10766i 2.82105 + 0.204152i 0.365341 + 0.930874i 1.35524 + 2.99670i
103.11 −0.593055 + 1.28386i −0.826239 0.563320i −1.29657 1.52279i 4.20665 + 0.315245i 1.21323 0.726692i 2.58503 + 0.563570i 2.72399 0.761513i 0.365341 + 0.930874i −2.89950 + 5.21378i
103.12 −0.476542 + 1.33151i −0.826239 0.563320i −1.54582 1.26904i −0.786913 0.0589710i 1.14380 0.831696i 0.621496 2.57172i 2.42638 1.45351i 0.365341 + 0.930874i 0.453517 1.01968i
103.13 −0.194188 1.40082i −0.826239 0.563320i −1.92458 + 0.544043i 1.91992 + 0.143878i −0.628664 + 1.26680i −0.854457 + 2.50398i 1.13584 + 2.59034i 0.365341 + 0.930874i −0.171278 2.71740i
103.14 −0.0925486 + 1.41118i −0.826239 0.563320i −1.98287 0.261206i −2.91108 0.218156i 0.871414 1.11384i 2.63369 + 0.252375i 0.552121 2.77402i 0.365341 + 0.930874i 0.577274 4.08788i
103.15 −0.0532621 + 1.41321i −0.826239 0.563320i −1.99433 0.150541i 2.71904 + 0.203764i 0.840097 1.13765i −2.62441 + 0.335390i 0.318968 2.81038i 0.365341 + 0.930874i −0.432783 + 3.83173i
103.16 0.0278575 1.41394i −0.826239 0.563320i −1.99845 0.0787776i 4.26189 + 0.319384i −0.819517 + 1.15256i 0.230798 2.63567i −0.167058 + 2.82349i 0.365341 + 0.930874i 0.570316 6.01715i
103.17 0.201122 1.39984i −0.826239 0.563320i −1.91910 0.563077i −1.87993 0.140882i −0.954732 + 1.04331i 1.80315 + 1.93615i −1.17419 + 2.57318i 0.365341 + 0.930874i −0.575308 + 2.60327i
103.18 0.672250 1.24422i −0.826239 0.563320i −1.09616 1.67285i −0.188298 0.0141110i −1.25633 + 0.649330i 1.32214 2.29171i −2.81829 + 0.239290i 0.365341 + 0.930874i −0.144141 + 0.224798i
103.19 0.700855 + 1.22833i −0.826239 0.563320i −1.01760 + 1.72177i −2.57232 0.192768i 0.112871 1.40970i −0.135490 2.64228i −2.82810 0.0432451i 0.365341 + 0.930874i −1.56604 3.29476i
103.20 0.826670 1.14744i −0.826239 0.563320i −0.633232 1.89711i −1.15601 0.0866309i −1.32940 + 0.482379i −2.18569 + 1.49090i −2.70029 0.841687i 0.365341 + 0.930874i −1.05504 + 1.25483i
See next 80 embeddings (of 336 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 535.28 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.p even 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.ba.a 336
4.b odd 2 1 588.2.ba.b yes 336
49.h odd 42 1 588.2.ba.b yes 336
196.p even 42 1 inner 588.2.ba.a 336

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.ba.a 336 1.a even 1 1 trivial
588.2.ba.a 336 196.p even 42 1 inner
588.2.ba.b yes 336 4.b odd 2 1
588.2.ba.b yes 336 49.h odd 42 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{336} - 6 T_{11}^{335} + 189 T_{11}^{334} - 1230 T_{11}^{333} + 16023 T_{11}^{332} - 109080 T_{11}^{331} + 693180 T_{11}^{330} - 4574286 T_{11}^{329} + 4014444 T_{11}^{328} + 13601918 T_{11}^{327} + \cdots + 41\!\cdots\!16$$ acting on $$S_{2}^{\mathrm{new}}(588, [\chi])$$.