# Properties

 Label 108.3.k.a Level 108 Weight 3 Character orbit 108.k Analytic conductor 2.943 Analytic rank 0 Dimension 36 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 108.k (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.94278685509$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$6$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 9q^{5} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 9q^{5} + 6q^{9} + 36q^{11} + 45q^{15} + 42q^{21} - 18q^{23} - 9q^{25} - 18q^{29} + 45q^{31} - 153q^{33} - 243q^{35} - 123q^{39} - 198q^{41} + 90q^{43} - 333q^{45} - 243q^{47} + 72q^{49} - 99q^{51} + 243q^{57} + 252q^{59} - 144q^{61} + 381q^{63} + 747q^{65} + 108q^{67} + 585q^{69} + 324q^{71} - 63q^{73} + 597q^{75} + 495q^{77} + 36q^{79} - 54q^{81} - 27q^{83} - 180q^{85} - 441q^{87} - 567q^{89} + 99q^{91} - 699q^{93} - 1044q^{95} - 216q^{97} - 945q^{99} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 0 −2.92720 0.656882i 0 0.740753 + 2.03520i 0 1.08248 + 6.13906i 0 8.13701 + 3.84565i 0
5.2 0 −1.95183 + 2.27824i 0 −3.25030 8.93012i 0 0.410040 + 2.32545i 0 −1.38071 8.89346i 0
5.3 0 −0.727525 2.91045i 0 0.0686711 + 0.188672i 0 −1.47862 8.38565i 0 −7.94141 + 4.23485i 0
5.4 0 1.02038 + 2.82114i 0 1.26650 + 3.47969i 0 −0.0728181 0.412972i 0 −6.91763 + 5.75729i 0
5.5 0 2.70588 1.29546i 0 1.65461 + 4.54600i 0 1.68621 + 9.56295i 0 5.64356 7.01072i 0
5.6 0 2.99363 0.195330i 0 −2.50118 6.87194i 0 −1.62729 9.22884i 0 8.92369 1.16949i 0
29.1 0 −2.78257 + 1.12129i 0 −2.69546 3.21232i 0 11.1367 4.05342i 0 6.48544 6.24012i 0
29.2 0 −2.59609 1.50344i 0 −0.298552 0.355800i 0 −10.1488 + 3.69384i 0 4.47934 + 7.80612i 0
29.3 0 −0.776129 + 2.89787i 0 2.68656 + 3.20172i 0 −4.88621 + 1.77844i 0 −7.79525 4.49824i 0
29.4 0 0.0634556 2.99933i 0 5.64904 + 6.73227i 0 4.05297 1.47516i 0 −8.99195 0.380648i 0
29.5 0 1.49775 2.59937i 0 −5.00278 5.96208i 0 −3.39388 + 1.23527i 0 −4.51349 7.78642i 0
29.6 0 2.88785 + 0.812609i 0 0.980262 + 1.16823i 0 3.23920 1.17897i 0 7.67933 + 4.69338i 0
41.1 0 −2.78257 1.12129i 0 −2.69546 + 3.21232i 0 11.1367 + 4.05342i 0 6.48544 + 6.24012i 0
41.2 0 −2.59609 + 1.50344i 0 −0.298552 + 0.355800i 0 −10.1488 3.69384i 0 4.47934 7.80612i 0
41.3 0 −0.776129 2.89787i 0 2.68656 3.20172i 0 −4.88621 1.77844i 0 −7.79525 + 4.49824i 0
41.4 0 0.0634556 + 2.99933i 0 5.64904 6.73227i 0 4.05297 + 1.47516i 0 −8.99195 + 0.380648i 0
41.5 0 1.49775 + 2.59937i 0 −5.00278 + 5.96208i 0 −3.39388 1.23527i 0 −4.51349 + 7.78642i 0
41.6 0 2.88785 0.812609i 0 0.980262 1.16823i 0 3.23920 + 1.17897i 0 7.67933 4.69338i 0
65.1 0 −2.92720 + 0.656882i 0 0.740753 2.03520i 0 1.08248 6.13906i 0 8.13701 3.84565i 0
65.2 0 −1.95183 2.27824i 0 −3.25030 + 8.93012i 0 0.410040 2.32545i 0 −1.38071 + 8.89346i 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.6 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.f odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.3.k.a 36
3.b odd 2 1 324.3.k.a 36
4.b odd 2 1 432.3.bc.b 36
27.e even 9 1 324.3.k.a 36
27.e even 9 1 2916.3.c.b 36
27.f odd 18 1 inner 108.3.k.a 36
27.f odd 18 1 2916.3.c.b 36
108.l even 18 1 432.3.bc.b 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.k.a 36 1.a even 1 1 trivial
108.3.k.a 36 27.f odd 18 1 inner
324.3.k.a 36 3.b odd 2 1
324.3.k.a 36 27.e even 9 1
432.3.bc.b 36 4.b odd 2 1
432.3.bc.b 36 108.l even 18 1
2916.3.c.b 36 27.e even 9 1
2916.3.c.b 36 27.f odd 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(108, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database