# Properties

 Label 108.3.j.a Level 108 Weight 3 Character orbit 108.j Analytic conductor 2.943 Analytic rank 0 Dimension 204 CM no Inner twists 4

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.94278685509$$ Analytic rank: $$0$$ Dimension: $$204$$ Relative dimension: $$34$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$204q - 6q^{2} - 6q^{4} - 12q^{5} - 6q^{6} - 3q^{8} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$204q - 6q^{2} - 6q^{4} - 12q^{5} - 6q^{6} - 3q^{8} - 12q^{9} - 3q^{10} + 39q^{12} - 12q^{13} + 39q^{14} - 6q^{16} - 6q^{17} - 27q^{18} - 69q^{20} - 12q^{21} - 6q^{22} - 138q^{24} - 12q^{25} - 174q^{26} - 12q^{28} + 60q^{29} - 153q^{30} - 96q^{32} + 48q^{33} + 6q^{34} + 24q^{36} - 6q^{37} + 72q^{38} + 69q^{40} - 192q^{41} - 126q^{42} - 219q^{44} - 132q^{45} - 3q^{46} - 219q^{48} - 12q^{49} - 165q^{50} + 21q^{52} - 24q^{53} + 78q^{54} + 99q^{56} - 150q^{57} - 141q^{58} + 210q^{60} - 12q^{61} + 294q^{62} - 3q^{64} - 156q^{65} + 393q^{66} + 375q^{68} - 60q^{69} - 165q^{70} + 228q^{72} - 6q^{73} + 447q^{74} - 54q^{76} + 132q^{77} + 750q^{78} + 798q^{80} + 228q^{81} - 12q^{82} + 762q^{84} + 138q^{85} + 606q^{86} - 198q^{88} - 114q^{89} + 894q^{90} + 723q^{92} - 1020q^{93} - 357q^{94} + 474q^{96} + 168q^{97} + 510q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −1.95448 + 0.424266i 0.0189322 + 2.99994i 3.64000 1.65844i −1.07944 6.12179i −1.30977 5.85530i 6.40210 7.62973i −6.41069 + 4.78571i −8.99928 + 0.113591i 4.70701 + 11.5070i
7.2 −1.95123 0.438997i −1.49386 2.60161i 3.61456 + 1.71316i 0.0372686 + 0.211361i 1.77277 + 5.73213i 7.43930 8.86581i −6.30076 4.92955i −4.53674 + 7.77290i 0.0200672 0.428773i
7.3 −1.91270 0.584441i 2.24560 + 1.98929i 3.31686 + 2.23572i 0.641948 + 3.64067i −3.13254 5.11734i −2.33709 + 2.78523i −5.03752 6.21478i 1.08543 + 8.93431i 0.899899 7.33869i
7.4 −1.90734 0.601703i −2.99807 + 0.107643i 3.27591 + 2.29531i −0.755508 4.28470i 5.78311 + 1.59863i −6.44162 + 7.67682i −4.86718 6.34906i 8.97683 0.645444i −1.13710 + 8.62697i
7.5 −1.89957 + 0.625818i 2.48687 1.67794i 3.21670 2.37757i 1.04974 + 5.95339i −3.67389 + 4.74368i 1.18627 1.41374i −4.62241 + 6.52942i 3.36905 8.34563i −5.71979 10.6519i
7.6 −1.85742 + 0.741613i −2.48687 + 1.67794i 2.90002 2.75497i 1.04974 + 5.95339i 3.37478 4.96093i −1.18627 + 1.41374i −3.34343 + 7.26784i 3.36905 8.34563i −6.36492 10.2794i
7.7 −1.76993 + 0.931310i −0.0189322 2.99994i 2.26532 3.29671i −1.07944 6.12179i 2.82738 + 5.29206i −6.40210 + 7.62973i −0.939205 + 7.94468i −8.99928 + 0.113591i 7.61182 + 9.82987i
7.8 −1.59321 1.20900i 2.60238 1.49253i 1.07665 + 3.85238i −0.890579 5.05073i −5.95060 0.768354i 0.0265129 0.0315969i 2.94218 7.43932i 4.54473 7.76823i −4.68743 + 9.12359i
7.9 −1.21254 + 1.59051i 1.49386 + 2.60161i −1.05947 3.85714i 0.0372686 + 0.211361i −5.94927 0.778554i −7.43930 + 8.86581i 7.41949 + 2.99184i −4.53674 + 7.77290i −0.381362 0.197008i
7.10 −1.20427 1.59678i −0.204519 2.99302i −1.09944 + 3.84594i 1.50670 + 8.54492i −4.53291 + 3.93099i −7.53806 + 8.98351i 7.46516 2.87599i −8.91634 + 1.22426i 11.8299 12.6963i
7.11 −1.08954 + 1.67717i −2.24560 1.98929i −1.62579 3.65470i 0.641948 + 3.64067i 5.78306 1.59883i 2.33709 2.78523i 7.90091 + 1.25523i 1.08543 + 8.93431i −6.80544 2.89001i
7.12 −1.07434 + 1.68695i 2.99807 0.107643i −1.69158 3.62471i −0.755508 4.28470i −3.03936 + 5.17323i 6.44162 7.67682i 7.93204 + 1.04057i 8.97683 0.645444i 8.03973 + 3.32873i
7.13 −1.00307 1.73027i −2.96394 + 0.463733i −1.98770 + 3.47117i 0.608002 + 3.44815i 3.77543 + 4.66328i 5.71739 6.81372i 7.99989 0.0425704i 8.56990 2.74895i 5.35638 4.51075i
7.14 −0.974385 1.74659i −0.586935 + 2.94202i −2.10115 + 3.40370i −1.26933 7.19876i 5.71041 1.84153i −2.67622 + 3.18940i 7.99219 + 0.353324i −8.31101 3.45355i −11.3364 + 9.23137i
7.15 −0.619770 1.90155i 2.41946 + 1.77376i −3.23177 + 2.35705i 0.463207 + 2.62698i 1.87339 5.70004i 4.34885 5.18275i 6.48499 + 4.68454i 2.70753 + 8.58308i 4.70825 2.50894i
7.16 −0.443343 + 1.95024i −2.60238 + 1.49253i −3.60689 1.72925i −0.890579 5.05073i −1.75705 5.73697i −0.0265129 + 0.0315969i 4.97155 6.26767i 4.54473 7.76823i 10.2450 + 0.502357i
7.17 0.103865 + 1.99730i 0.204519 + 2.99302i −3.97842 + 0.414901i 1.50670 + 8.54492i −5.95672 + 0.719357i 7.53806 8.98351i −1.24190 7.90302i −8.91634 + 1.22426i −16.9103 + 3.89685i
7.18 0.210380 1.98890i 1.64282 2.51021i −3.91148 0.836851i −0.133306 0.756014i −4.64695 3.79550i 3.08121 3.67204i −2.48731 + 7.60350i −3.60231 8.24763i −1.53168 + 0.106082i
7.19 0.277531 1.98065i −2.64062 1.42377i −3.84595 1.09938i −0.542224 3.07510i −3.55285 + 4.83500i −2.98812 + 3.56110i −3.24486 + 7.31238i 4.94574 + 7.51929i −6.24119 + 0.220520i
7.20 0.343802 + 1.97023i 2.96394 0.463733i −3.76360 + 1.35474i 0.608002 + 3.44815i 1.93267 + 5.68021i −5.71739 + 6.81372i −3.96308 6.94939i 8.56990 2.74895i −6.58462 + 2.38339i
See next 80 embeddings (of 204 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 103.34 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
27.e even 9 1 inner
108.j 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.j.a 204
3.b odd 2 1 324.3.j.a 204
4.b odd 2 1 inner 108.3.j.a 204
12.b even 2 1 324.3.j.a 204
27.e even 9 1 inner 108.3.j.a 204
27.f odd 18 1 324.3.j.a 204
108.j odd 18 1 inner 108.3.j.a 204
108.l even 18 1 324.3.j.a 204

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.j.a 204 1.a even 1 1 trivial
108.3.j.a 204 4.b odd 2 1 inner
108.3.j.a 204 27.e even 9 1 inner
108.3.j.a 204 108.j odd 18 1 inner
324.3.j.a 204 3.b odd 2 1
324.3.j.a 204 12.b even 2 1
324.3.j.a 204 27.f odd 18 1
324.3.j.a 204 108.l even 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