# Properties

 Label 108.4.h.b Level $108$ Weight $4$ Character orbit 108.h Analytic conductor $6.372$ Analytic rank $0$ Dimension $24$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,4,Mod(35,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1]))

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

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

chi := DirichletCharacter("108.35");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 108.h (of order $$6$$, degree $$2$$, not 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: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 12 q^{4} + 72 q^{5}+O(q^{10})$$ 24 * q - 12 * q^4 + 72 * q^5 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 12 q^{4} + 72 q^{5} + 96 q^{10} - 216 q^{13} + 36 q^{14} - 72 q^{16} + 540 q^{20} - 192 q^{22} + 252 q^{25} - 672 q^{28} - 576 q^{29} - 360 q^{32} - 660 q^{34} + 1248 q^{37} + 144 q^{38} + 636 q^{40} - 1116 q^{41} + 960 q^{46} + 348 q^{49} + 648 q^{50} + 132 q^{52} + 1692 q^{56} + 516 q^{58} - 264 q^{61} + 960 q^{64} + 2592 q^{65} - 5688 q^{68} + 564 q^{70} - 4776 q^{73} - 5652 q^{74} - 600 q^{76} - 648 q^{77} - 4104 q^{82} + 720 q^{85} + 9540 q^{86} + 1956 q^{88} + 7416 q^{92} - 1188 q^{94} + 588 q^{97}+O(q^{100})$$ 24 * q - 12 * q^4 + 72 * q^5 + 96 * q^10 - 216 * q^13 + 36 * q^14 - 72 * q^16 + 540 * q^20 - 192 * q^22 + 252 * q^25 - 672 * q^28 - 576 * q^29 - 360 * q^32 - 660 * q^34 + 1248 * q^37 + 144 * q^38 + 636 * q^40 - 1116 * q^41 + 960 * q^46 + 348 * q^49 + 648 * q^50 + 132 * q^52 + 1692 * q^56 + 516 * q^58 - 264 * q^61 + 960 * q^64 + 2592 * q^65 - 5688 * q^68 + 564 * q^70 - 4776 * q^73 - 5652 * q^74 - 600 * q^76 - 648 * q^77 - 4104 * q^82 + 720 * q^85 + 9540 * q^86 + 1956 * q^88 + 7416 * q^92 - 1188 * q^94 + 588 * q^97

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1 −2.71307 + 0.799546i 0 6.72145 4.33844i 14.2911 8.25096i 0 −19.2620 11.1209i −14.7670 + 17.1446i 0 −32.1756 + 33.8118i
35.2 −2.66543 0.946295i 0 6.20905 + 5.04457i −2.08666 + 1.20474i 0 2.30362 + 1.33000i −11.7761 19.3216i 0 6.70190 1.23654i
35.3 −2.15223 1.83518i 0 1.26420 + 7.89948i −2.08666 + 1.20474i 0 −2.30362 1.33000i 11.7761 19.3216i 0 6.70190 + 1.23654i
35.4 −1.44536 + 2.43124i 0 −3.82189 7.02803i −14.6499 + 8.45813i 0 3.08966 + 1.78382i 22.6108 + 0.866066i 0 0.610574 47.8425i
35.5 −0.823719 + 2.70582i 0 −6.64298 4.45768i 4.71466 2.72201i 0 20.9358 + 12.0873i 17.5336 14.3029i 0 3.48173 + 14.9992i
35.6 −0.664105 2.74936i 0 −7.11793 + 3.65173i 14.2911 8.25096i 0 19.2620 + 11.1209i 14.7670 + 17.1446i 0 −32.1756 33.8118i
35.7 0.157323 + 2.82405i 0 −7.95050 + 0.888573i 1.23846 0.715028i 0 −23.8818 13.7882i −3.76017 22.3128i 0 2.21411 + 3.38499i
35.8 1.38284 2.46734i 0 −4.17551 6.82387i −14.6499 + 8.45813i 0 −3.08966 1.78382i −22.6108 + 0.866066i 0 0.610574 + 47.8425i
35.9 1.65391 + 2.29447i 0 −2.52915 + 7.58969i 14.4924 8.36717i 0 16.7175 + 9.65186i −21.5973 + 6.74964i 0 43.1673 + 19.4137i
35.10 1.93145 2.06627i 0 −0.538974 7.98182i 4.71466 2.72201i 0 −20.9358 12.0873i −17.5336 14.3029i 0 3.48173 14.9992i
35.11 2.52436 1.27578i 0 4.74478 6.44105i 1.23846 0.715028i 0 23.8818 + 13.7882i 3.76017 22.3128i 0 2.21411 3.38499i
35.12 2.81402 + 0.285097i 0 7.83744 + 1.60454i 14.4924 8.36717i 0 −16.7175 9.65186i 21.5973 + 6.74964i 0 43.1673 19.4137i
71.1 −2.71307 0.799546i 0 6.72145 + 4.33844i 14.2911 + 8.25096i 0 −19.2620 + 11.1209i −14.7670 17.1446i 0 −32.1756 33.8118i
71.2 −2.66543 + 0.946295i 0 6.20905 5.04457i −2.08666 1.20474i 0 2.30362 1.33000i −11.7761 + 19.3216i 0 6.70190 + 1.23654i
71.3 −2.15223 + 1.83518i 0 1.26420 7.89948i −2.08666 1.20474i 0 −2.30362 + 1.33000i 11.7761 + 19.3216i 0 6.70190 1.23654i
71.4 −1.44536 2.43124i 0 −3.82189 + 7.02803i −14.6499 8.45813i 0 3.08966 1.78382i 22.6108 0.866066i 0 0.610574 + 47.8425i
71.5 −0.823719 2.70582i 0 −6.64298 + 4.45768i 4.71466 + 2.72201i 0 20.9358 12.0873i 17.5336 + 14.3029i 0 3.48173 14.9992i
71.6 −0.664105 + 2.74936i 0 −7.11793 3.65173i 14.2911 + 8.25096i 0 19.2620 11.1209i 14.7670 17.1446i 0 −32.1756 + 33.8118i
71.7 0.157323 2.82405i 0 −7.95050 0.888573i 1.23846 + 0.715028i 0 −23.8818 + 13.7882i −3.76017 + 22.3128i 0 2.21411 3.38499i
71.8 1.38284 + 2.46734i 0 −4.17551 + 6.82387i −14.6499 8.45813i 0 −3.08966 + 1.78382i −22.6108 0.866066i 0 0.610574 47.8425i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 35.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.h.b 24
3.b odd 2 1 36.4.h.b 24
4.b odd 2 1 inner 108.4.h.b 24
9.c even 3 1 36.4.h.b 24
9.c even 3 1 324.4.b.c 24
9.d odd 6 1 inner 108.4.h.b 24
9.d odd 6 1 324.4.b.c 24
12.b even 2 1 36.4.h.b 24
36.f odd 6 1 36.4.h.b 24
36.f odd 6 1 324.4.b.c 24
36.h even 6 1 inner 108.4.h.b 24
36.h even 6 1 324.4.b.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.h.b 24 3.b odd 2 1
36.4.h.b 24 9.c even 3 1
36.4.h.b 24 12.b even 2 1
36.4.h.b 24 36.f odd 6 1
108.4.h.b 24 1.a even 1 1 trivial
108.4.h.b 24 4.b odd 2 1 inner
108.4.h.b 24 9.d odd 6 1 inner
108.4.h.b 24 36.h even 6 1 inner
324.4.b.c 24 9.c even 3 1
324.4.b.c 24 9.d odd 6 1
324.4.b.c 24 36.f odd 6 1
324.4.b.c 24 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} - 36 T_{5}^{11} + 210 T_{5}^{10} + 7992 T_{5}^{9} - 59073 T_{5}^{8} - 2269332 T_{5}^{7} + \cdots + 7678666384$$ acting on $$S_{4}^{\mathrm{new}}(108, [\chi])$$.