# Properties

 Label 115.4.e.a Level $115$ Weight $4$ Character orbit 115.e Analytic conductor $6.785$ Analytic rank $0$ Dimension $68$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,4,Mod(22,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("115.22");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 115.e (of order $$4$$, 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: $$6.78521965066$$ Analytic rank: $$0$$ Dimension: $$68$$ Relative dimension: $$34$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$68 q - 4 q^{2} + 4 q^{3} + 32 q^{6} + 28 q^{8}+O(q^{10})$$ 68 * q - 4 * q^2 + 4 * q^3 + 32 * q^6 + 28 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$68 q - 4 q^{2} + 4 q^{3} + 32 q^{6} + 28 q^{8} + 200 q^{12} - 100 q^{13} - 832 q^{16} - 112 q^{18} - 46 q^{23} - 372 q^{25} - 312 q^{26} - 704 q^{27} - 1056 q^{31} + 560 q^{32} - 348 q^{35} + 2424 q^{36} + 2248 q^{41} - 96 q^{46} - 1412 q^{47} + 1532 q^{48} - 620 q^{50} + 112 q^{52} + 2688 q^{55} + 2044 q^{58} + 3948 q^{62} - 1836 q^{70} - 2272 q^{71} + 1128 q^{72} + 1100 q^{73} + 2828 q^{75} + 4216 q^{77} - 9180 q^{78} - 3580 q^{81} - 6516 q^{82} - 2684 q^{85} - 6304 q^{87} - 688 q^{92} - 1608 q^{93} - 8616 q^{95} - 544 q^{96} - 3612 q^{98}+O(q^{100})$$ 68 * q - 4 * q^2 + 4 * q^3 + 32 * q^6 + 28 * q^8 + 200 * q^12 - 100 * q^13 - 832 * q^16 - 112 * q^18 - 46 * q^23 - 372 * q^25 - 312 * q^26 - 704 * q^27 - 1056 * q^31 + 560 * q^32 - 348 * q^35 + 2424 * q^36 + 2248 * q^41 - 96 * q^46 - 1412 * q^47 + 1532 * q^48 - 620 * q^50 + 112 * q^52 + 2688 * q^55 + 2044 * q^58 + 3948 * q^62 - 1836 * q^70 - 2272 * q^71 + 1128 * q^72 + 1100 * q^73 + 2828 * q^75 + 4216 * q^77 - 9180 * q^78 - 3580 * q^81 - 6516 * q^82 - 2684 * q^85 - 6304 * q^87 - 688 * q^92 - 1608 * q^93 - 8616 * q^95 - 544 * q^96 - 3612 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1 −3.71645 3.71645i 3.49413 3.49413i 19.6239i −10.7999 + 2.89187i −25.9715 2.69551 2.69551i 43.1997 43.1997i 2.58205i 50.8846 + 29.3896i
22.2 −3.71645 3.71645i 3.49413 3.49413i 19.6239i 10.7999 2.89187i −25.9715 −2.69551 + 2.69551i 43.1997 43.1997i 2.58205i −50.8846 29.3896i
22.3 −3.67169 3.67169i −4.81114 + 4.81114i 18.9626i −0.398194 11.1732i 35.3300 20.1910 20.1910i 40.2512 40.2512i 19.2941i −39.5626 + 42.4867i
22.4 −3.67169 3.67169i −4.81114 + 4.81114i 18.9626i 0.398194 + 11.1732i 35.3300 −20.1910 + 20.1910i 40.2512 40.2512i 19.2941i 39.5626 42.4867i
22.5 −2.78118 2.78118i 1.69288 1.69288i 7.46996i −9.46680 5.94809i −9.41642 1.06728 1.06728i −1.47414 + 1.47414i 21.2683i 9.78619 + 42.8716i
22.6 −2.78118 2.78118i 1.69288 1.69288i 7.46996i 9.46680 + 5.94809i −9.41642 −1.06728 + 1.06728i −1.47414 + 1.47414i 21.2683i −9.78619 42.8716i
22.7 −2.66509 2.66509i −2.23283 + 2.23283i 6.20541i −7.41137 + 8.37088i 11.9014 10.9048 10.9048i −4.78275 + 4.78275i 17.0290i 42.0611 2.55718i
22.8 −2.66509 2.66509i −2.23283 + 2.23283i 6.20541i 7.41137 8.37088i 11.9014 −10.9048 + 10.9048i −4.78275 + 4.78275i 17.0290i −42.0611 + 2.55718i
22.9 −2.37561 2.37561i 6.86048 6.86048i 3.28704i −2.55138 10.8853i −32.5957 −19.7347 + 19.7347i −11.1962 + 11.1962i 67.1325i −19.7982 + 31.9204i
22.10 −2.37561 2.37561i 6.86048 6.86048i 3.28704i 2.55138 + 10.8853i −32.5957 19.7347 19.7347i −11.1962 + 11.1962i 67.1325i 19.7982 31.9204i
22.11 −1.84990 1.84990i −5.85544 + 5.85544i 1.15573i −10.6129 3.51653i 21.6640 −7.25929 + 7.25929i −16.9372 + 16.9372i 41.5723i 13.1276 + 26.1381i
22.12 −1.84990 1.84990i −5.85544 + 5.85544i 1.15573i 10.6129 + 3.51653i 21.6640 7.25929 7.25929i −16.9372 + 16.9372i 41.5723i −13.1276 26.1381i
22.13 −1.32617 1.32617i 2.49169 2.49169i 4.48253i −3.09011 + 10.7448i −6.60883 −19.9254 + 19.9254i −16.5540 + 16.5540i 14.5829i 18.3475 10.1515i
22.14 −1.32617 1.32617i 2.49169 2.49169i 4.48253i 3.09011 10.7448i −6.60883 19.9254 19.9254i −16.5540 + 16.5540i 14.5829i −18.3475 + 10.1515i
22.15 −0.386420 0.386420i −0.970983 + 0.970983i 7.70136i −8.97655 6.66495i 0.750415 −6.23903 + 6.23903i −6.06732 + 6.06732i 25.1144i 0.893251 + 6.04419i
22.16 −0.386420 0.386420i −0.970983 + 0.970983i 7.70136i 8.97655 + 6.66495i 0.750415 6.23903 6.23903i −6.06732 + 6.06732i 25.1144i −0.893251 6.04419i
22.17 −0.152517 0.152517i −2.82764 + 2.82764i 7.95348i −5.56672 + 9.69596i 0.862527 20.1728 20.1728i −2.43318 + 2.43318i 11.0089i 2.32782 0.629778i
22.18 −0.152517 0.152517i −2.82764 + 2.82764i 7.95348i 5.56672 9.69596i 0.862527 −20.1728 + 20.1728i −2.43318 + 2.43318i 11.0089i −2.32782 + 0.629778i
22.19 0.304528 + 0.304528i 5.13977 5.13977i 7.81453i −11.1201 + 1.15901i 3.13040 2.31330 2.31330i 4.81596 4.81596i 25.8345i −3.73933 3.03343i
22.20 0.304528 + 0.304528i 5.13977 5.13977i 7.81453i 11.1201 1.15901i 3.13040 −2.31330 + 2.31330i 4.81596 4.81596i 25.8345i 3.73933 + 3.03343i
See all 68 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 22.34 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.c odd 4 1 inner
23.b odd 2 1 inner
115.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.4.e.a 68
5.c odd 4 1 inner 115.4.e.a 68
23.b odd 2 1 inner 115.4.e.a 68
115.e even 4 1 inner 115.4.e.a 68

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.e.a 68 1.a even 1 1 trivial
115.4.e.a 68 5.c odd 4 1 inner
115.4.e.a 68 23.b odd 2 1 inner
115.4.e.a 68 115.e even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(115, [\chi])$$.