# Properties

 Label 108.4.i.a Level $108$ Weight $4$ Character orbit 108.i Analytic conductor $6.372$ Analytic rank $0$ Dimension $54$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,4,Mod(13,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 8]))

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

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

chi := DirichletCharacter("108.13");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 108.i (of order $$9$$, 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: $$6.37220628062$$ Analytic rank: $$0$$ Dimension: $$54$$ Relative dimension: $$9$$ over $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$54 q + 12 q^{5} - 48 q^{9}+O(q^{10})$$ 54 * q + 12 * q^5 - 48 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$54 q + 12 q^{5} - 48 q^{9} - 87 q^{11} + 234 q^{15} + 204 q^{17} - 12 q^{21} + 96 q^{23} - 216 q^{25} + 27 q^{27} + 318 q^{29} - 54 q^{31} + 63 q^{33} + 6 q^{35} + 66 q^{39} + 867 q^{41} - 513 q^{43} - 306 q^{45} - 1548 q^{47} + 594 q^{49} - 1368 q^{51} - 1068 q^{53} - 1269 q^{57} - 1218 q^{59} - 54 q^{61} + 30 q^{63} + 96 q^{65} - 2997 q^{67} + 1476 q^{69} - 120 q^{71} - 216 q^{73} + 732 q^{75} + 3480 q^{77} + 2808 q^{79} + 3348 q^{81} + 4464 q^{83} + 2160 q^{85} + 4824 q^{87} + 4029 q^{89} + 270 q^{91} + 1164 q^{93} - 1650 q^{95} - 3483 q^{97} - 5076 q^{99}+O(q^{100})$$ 54 * q + 12 * q^5 - 48 * q^9 - 87 * q^11 + 234 * q^15 + 204 * q^17 - 12 * q^21 + 96 * q^23 - 216 * q^25 + 27 * q^27 + 318 * q^29 - 54 * q^31 + 63 * q^33 + 6 * q^35 + 66 * q^39 + 867 * q^41 - 513 * q^43 - 306 * q^45 - 1548 * q^47 + 594 * q^49 - 1368 * q^51 - 1068 * q^53 - 1269 * q^57 - 1218 * q^59 - 54 * q^61 + 30 * q^63 + 96 * q^65 - 2997 * q^67 + 1476 * q^69 - 120 * q^71 - 216 * q^73 + 732 * q^75 + 3480 * q^77 + 2808 * q^79 + 3348 * q^81 + 4464 * q^83 + 2160 * q^85 + 4824 * q^87 + 4029 * q^89 + 270 * q^91 + 1164 * q^93 - 1650 * q^95 - 3483 * q^97 - 5076 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 0 −4.52704 + 2.55067i 0 −1.73722 1.45770i 0 −5.32680 1.93880i 0 13.9881 23.0940i 0
13.2 0 −4.40571 2.75494i 0 8.18465 + 6.86774i 0 −24.4722 8.90714i 0 11.8206 + 24.2749i 0
13.3 0 −4.40211 2.76069i 0 −14.9393 12.5356i 0 23.5233 + 8.56177i 0 11.7572 + 24.3057i 0
13.4 0 −0.716497 5.14652i 0 7.52940 + 6.31792i 0 24.9076 + 9.06562i 0 −25.9733 + 7.37493i 0
13.5 0 −0.173032 + 5.19327i 0 6.54095 + 5.48851i 0 11.9650 + 4.35489i 0 −26.9401 1.79720i 0
13.6 0 2.34095 4.63896i 0 −5.19451 4.35871i 0 −17.8009 6.47901i 0 −16.0399 21.7191i 0
13.7 0 2.66172 + 4.46265i 0 −12.8040 10.7439i 0 −20.8033 7.57178i 0 −12.8305 + 23.7566i 0
13.8 0 5.14478 + 0.728858i 0 −5.18194 4.34817i 0 16.8240 + 6.12342i 0 25.9375 + 7.49962i 0
13.9 0 5.19029 0.246754i 0 15.7004 + 13.1742i 0 −8.81657 3.20897i 0 26.8782 2.56145i 0
25.1 0 −4.52704 2.55067i 0 −1.73722 + 1.45770i 0 −5.32680 + 1.93880i 0 13.9881 + 23.0940i 0
25.2 0 −4.40571 + 2.75494i 0 8.18465 6.86774i 0 −24.4722 + 8.90714i 0 11.8206 24.2749i 0
25.3 0 −4.40211 + 2.76069i 0 −14.9393 + 12.5356i 0 23.5233 8.56177i 0 11.7572 24.3057i 0
25.4 0 −0.716497 + 5.14652i 0 7.52940 6.31792i 0 24.9076 9.06562i 0 −25.9733 7.37493i 0
25.5 0 −0.173032 5.19327i 0 6.54095 5.48851i 0 11.9650 4.35489i 0 −26.9401 + 1.79720i 0
25.6 0 2.34095 + 4.63896i 0 −5.19451 + 4.35871i 0 −17.8009 + 6.47901i 0 −16.0399 + 21.7191i 0
25.7 0 2.66172 4.46265i 0 −12.8040 + 10.7439i 0 −20.8033 + 7.57178i 0 −12.8305 23.7566i 0
25.8 0 5.14478 0.728858i 0 −5.18194 + 4.34817i 0 16.8240 6.12342i 0 25.9375 7.49962i 0
25.9 0 5.19029 + 0.246754i 0 15.7004 13.1742i 0 −8.81657 + 3.20897i 0 26.8782 + 2.56145i 0
49.1 0 −5.16735 + 0.546343i 0 5.21695 1.89881i 0 1.58868 + 9.00985i 0 26.4030 5.64629i 0
49.2 0 −4.14053 + 3.13943i 0 −4.27250 + 1.55506i 0 −4.69568 26.6305i 0 7.28791 25.9978i 0
See all 54 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.i.a 54
3.b odd 2 1 324.4.i.a 54
27.e even 9 1 inner 108.4.i.a 54
27.f odd 18 1 324.4.i.a 54

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.i.a 54 1.a even 1 1 trivial
108.4.i.a 54 27.e even 9 1 inner
324.4.i.a 54 3.b odd 2 1
324.4.i.a 54 27.f odd 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(108, [\chi])$$.