# Properties

 Label 115.3.i.a Level $115$ Weight $3$ Character orbit 115.i Analytic conductor $3.134$ Analytic rank $0$ Dimension $220$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,3,Mod(14,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 21]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("115.14");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 115.i (of order $$22$$, degree $$10$$, 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: $$3.13352304014$$ Analytic rank: $$0$$ Dimension: $$220$$ Relative dimension: $$22$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$220 q + 26 q^{4} - 11 q^{5} - 14 q^{6} + 32 q^{9}+O(q^{10})$$ 220 * q + 26 * q^4 - 11 * q^5 - 14 * q^6 + 32 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$220 q + 26 q^{4} - 11 q^{5} - 14 q^{6} + 32 q^{9} - 11 q^{10} - 22 q^{11} - 22 q^{14} - 88 q^{15} - 142 q^{16} - 22 q^{19} - 99 q^{20} - 22 q^{21} - 88 q^{24} + 17 q^{25} + 34 q^{26} + 92 q^{29} + 341 q^{30} - 152 q^{31} - 264 q^{34} - 13 q^{35} - 62 q^{36} - 118 q^{39} - 11 q^{40} - 80 q^{41} - 242 q^{44} + 226 q^{46} + 90 q^{49} - 211 q^{50} - 22 q^{51} + 658 q^{54} - 565 q^{55} + 770 q^{56} - 172 q^{59} - 891 q^{60} + 286 q^{61} - 474 q^{64} - 242 q^{65} - 44 q^{66} - 288 q^{69} + 790 q^{70} - 210 q^{71} + 506 q^{74} + 804 q^{75} - 2376 q^{76} + 462 q^{79} + 2398 q^{80} - 2408 q^{81} + 1034 q^{84} + 1197 q^{85} - 1518 q^{86} - 22 q^{89} + 154 q^{90} - 210 q^{94} - 338 q^{95} + 2772 q^{96} + 1628 q^{99}+O(q^{100})$$ 220 * q + 26 * q^4 - 11 * q^5 - 14 * q^6 + 32 * q^9 - 11 * q^10 - 22 * q^11 - 22 * q^14 - 88 * q^15 - 142 * q^16 - 22 * q^19 - 99 * q^20 - 22 * q^21 - 88 * q^24 + 17 * q^25 + 34 * q^26 + 92 * q^29 + 341 * q^30 - 152 * q^31 - 264 * q^34 - 13 * q^35 - 62 * q^36 - 118 * q^39 - 11 * q^40 - 80 * q^41 - 242 * q^44 + 226 * q^46 + 90 * q^49 - 211 * q^50 - 22 * q^51 + 658 * q^54 - 565 * q^55 + 770 * q^56 - 172 * q^59 - 891 * q^60 + 286 * q^61 - 474 * q^64 - 242 * q^65 - 44 * q^66 - 288 * q^69 + 790 * q^70 - 210 * q^71 + 506 * q^74 + 804 * q^75 - 2376 * q^76 + 462 * q^79 + 2398 * q^80 - 2408 * q^81 + 1034 * q^84 + 1197 * q^85 - 1518 * q^86 - 22 * q^89 + 154 * q^90 - 210 * q^94 - 338 * q^95 + 2772 * q^96 + 1628 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1 −1.06616 3.63102i −3.98118 + 3.44971i −8.68261 + 5.57997i 4.33958 2.48356i 16.7706 + 10.7778i −0.608324 + 1.33204i 18.0781 + 15.6648i 2.66845 18.5594i −13.6446 13.1092i
14.2 −1.01059 3.44174i −0.190798 + 0.165327i −7.45930 + 4.79380i −4.45644 + 2.26720i 0.761832 + 0.489600i 0.896990 1.96413i 13.1937 + 11.4324i −1.27176 + 8.84530i 12.3067 + 13.0467i
14.3 −0.909272 3.09670i 2.61799 2.26850i −5.39774 + 3.46891i 0.569021 4.96752i −9.40531 6.04443i −1.76617 + 3.86736i 5.89367 + 5.10690i 0.426936 2.96941i −15.9003 + 2.75474i
14.4 −0.761060 2.59193i 4.31740 3.74105i −2.77389 + 1.78267i −0.640865 + 4.95876i −12.9823 8.34325i 1.74741 3.82629i −1.43453 1.24303i 3.36366 23.3948i 13.3405 2.11284i
14.5 −0.723107 2.46268i −0.136015 + 0.117858i −2.17688 + 1.39899i 4.86226 + 1.16552i 0.388600 + 0.249738i 4.52870 9.91648i −2.73958 2.37386i −1.27622 + 8.87633i −0.645632 12.8170i
14.6 −0.636390 2.16735i −2.13320 + 1.84843i −0.927382 + 0.595992i −2.88883 4.08101i 5.36373 + 3.44706i −1.73429 + 3.79756i −4.94659 4.28624i −0.146981 + 1.02227i −7.00655 + 8.85821i
14.7 −0.618956 2.10797i −1.36297 + 1.18102i −0.695419 + 0.446919i 1.65130 + 4.71945i 3.33318 + 2.14211i −4.87531 + 10.6754i −5.26889 4.56552i −0.817952 + 5.68898i 8.92638 6.40203i
14.8 −0.435846 1.48436i −4.32985 + 3.75184i 1.35166 0.868661i −3.10020 + 3.92285i 7.45621 + 4.79182i 4.16584 9.12191i −6.55516 5.68008i 3.39049 23.5814i 7.17412 + 2.89204i
14.9 −0.254893 0.868087i 1.66868 1.44592i 2.67641 1.72002i −4.78864 1.43838i −1.68052 1.08001i 2.63365 5.76689i −4.91034 4.25483i −0.587022 + 4.08283i −0.0280477 + 4.52359i
14.10 −0.245567 0.836325i 2.37235 2.05565i 2.72588 1.75182i 4.33861 2.48524i −2.30176 1.47925i −2.30176 + 5.04015i −4.76942 4.13272i 0.121499 0.845043i −3.14389 3.01820i
14.11 −0.0253139 0.0862113i −2.10252 + 1.82185i 3.35822 2.15820i 0.240436 4.99422i 0.210287 + 0.135143i 0.329456 0.721408i −0.542690 0.470243i −0.179355 + 1.24744i −0.436644 + 0.105695i
14.12 0.0253139 + 0.0862113i 2.10252 1.82185i 3.35822 2.15820i 1.63773 + 4.72418i 0.210287 + 0.135143i −0.329456 + 0.721408i 0.542690 + 0.470243i −0.179355 + 1.24744i −0.365820 + 0.260778i
14.13 0.245567 + 0.836325i −2.37235 + 2.05565i 2.72588 1.75182i 4.86304 + 1.16224i −2.30176 1.47925i 2.30176 5.04015i 4.76942 + 4.13272i 0.121499 0.845043i 0.222190 + 4.35249i
14.14 0.254893 + 0.868087i −1.66868 + 1.44592i 2.67641 1.72002i −4.18943 + 2.72923i −1.68052 1.08001i −2.63365 + 5.76689i 4.91034 + 4.25483i −0.587022 + 4.08283i −3.43707 2.94112i
14.15 0.435846 + 1.48436i 4.32985 3.75184i 1.35166 0.868661i −4.07982 2.89052i 7.45621 + 4.79182i −4.16584 + 9.12191i 6.55516 + 5.68008i 3.39049 23.5814i 2.51239 7.31572i
14.16 0.618956 + 2.10797i 1.36297 1.18102i −0.695419 + 0.446919i 0.254791 4.99350i 3.33318 + 2.14211i 4.87531 10.6754i 5.26889 + 4.56552i −0.817952 + 5.68898i 10.6839 2.55367i
14.17 0.636390 + 2.16735i 2.13320 1.84843i −0.927382 + 0.595992i −1.62206 + 4.72958i 5.36373 + 3.44706i 1.73429 3.79756i 4.94659 + 4.28624i −0.146981 + 1.02227i −11.2829 0.505697i
14.18 0.723107 + 2.46268i 0.136015 0.117858i −2.17688 + 1.39899i 4.33694 2.48817i 0.388600 + 0.249738i −4.52870 + 9.91648i 2.73958 + 2.37386i −1.27622 + 8.87633i 9.26362 + 8.88126i
14.19 0.761060 + 2.59193i −4.31740 + 3.74105i −2.77389 + 1.78267i −2.01195 4.57734i −12.9823 8.34325i −1.74741 + 3.82629i 1.43453 + 1.24303i 3.36366 23.3948i 10.3330 8.69847i
14.20 0.909272 + 3.09670i −2.61799 + 2.26850i −5.39774 + 3.46891i 1.94548 + 4.60598i −9.40531 6.04443i 1.76617 3.86736i −5.89367 5.10690i 0.426936 2.96941i −12.4944 + 10.2127i
See next 80 embeddings (of 220 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 14.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.d odd 22 1 inner
115.i odd 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.3.i.a 220
5.b even 2 1 inner 115.3.i.a 220
23.d odd 22 1 inner 115.3.i.a 220
115.i odd 22 1 inner 115.3.i.a 220

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.3.i.a 220 1.a even 1 1 trivial
115.3.i.a 220 5.b even 2 1 inner
115.3.i.a 220 23.d odd 22 1 inner
115.3.i.a 220 115.i odd 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(115, [\chi])$$.