# Properties

 Label 27.3.f.a Level 27 Weight 3 Character orbit 27.f Analytic conductor 0.736 Analytic rank 0 Dimension 30 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$27 = 3^{3}$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 27.f (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.735696713773$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$5$$ over $$\Q(\zeta_{18})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q - 6q^{2} - 6q^{3} - 6q^{4} - 15q^{5} - 18q^{6} - 6q^{7} - 9q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q - 6q^{2} - 6q^{3} - 6q^{4} - 15q^{5} - 18q^{6} - 6q^{7} - 9q^{8} - 3q^{10} - 6q^{11} - 15q^{12} - 6q^{13} - 15q^{14} - 9q^{15} - 18q^{16} - 9q^{17} + 63q^{18} - 3q^{19} + 213q^{20} + 132q^{21} - 42q^{22} + 120q^{23} + 144q^{24} - 15q^{25} - 90q^{27} - 12q^{28} - 168q^{29} - 243q^{30} + 39q^{31} - 360q^{32} - 207q^{33} + 54q^{34} - 252q^{35} - 360q^{36} - 3q^{37} - 84q^{38} + 15q^{39} - 33q^{40} + 228q^{41} + 486q^{42} - 96q^{43} + 639q^{44} + 477q^{45} - 3q^{46} + 399q^{47} + 453q^{48} - 78q^{49} + 303q^{50} + 36q^{51} - 9q^{52} - 54q^{54} - 12q^{55} - 393q^{56} - 192q^{57} + 129q^{58} - 474q^{59} - 846q^{60} + 138q^{61} - 900q^{62} - 585q^{63} - 51q^{64} - 411q^{65} - 423q^{66} + 354q^{67} + 99q^{68} + 99q^{69} + 489q^{70} + 315q^{71} + 720q^{72} - 66q^{73} + 321q^{74} + 255q^{75} + 258q^{76} + 201q^{77} + 180q^{78} + 30q^{79} + 36q^{81} - 12q^{82} - 33q^{83} - 588q^{84} - 261q^{85} - 258q^{86} - 279q^{87} - 642q^{88} + 72q^{89} + 288q^{90} + 96q^{91} - 3q^{92} + 591q^{93} - 861q^{94} + 681q^{95} + 270q^{96} - 582q^{97} + 882q^{98} + 513q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −3.46291 0.610605i 1.47633 + 2.61160i 7.86014 + 2.86086i 3.16671 + 3.77394i −3.51774 9.94519i −3.18911 + 1.16074i −13.2912 7.67367i −4.64090 + 7.71116i −8.66165 15.0024i
2.2 −2.31604 0.408381i −1.62484 2.52189i 1.43851 + 0.523575i −3.71692 4.42965i 2.73330 + 6.50435i 4.57693 1.66587i 5.02894 + 2.90346i −3.71981 + 8.19530i 6.79956 + 11.7772i
2.3 0.115908 + 0.0204377i 2.32380 1.89736i −3.74575 1.36334i 3.98394 + 4.74788i 0.308125 0.172426i −7.49258 + 2.72708i −0.814011 0.469969i 1.80008 8.81815i 0.364735 + 0.631740i
2.4 1.14332 + 0.201599i 1.10490 + 2.78912i −2.49222 0.907094i −3.46013 4.12362i 0.700975 + 3.41162i 9.89907 3.60297i −6.68824 3.86146i −6.55839 + 6.16340i −3.12473 5.41219i
2.5 2.58003 + 0.454929i −2.92175 0.680712i 2.69082 + 0.979377i 0.519123 + 0.618667i −7.22853 3.08544i −5.56035 + 2.02380i −2.57852 1.48871i 8.07326 + 3.97774i 1.05790 + 1.83234i
5.1 −2.17948 2.59740i −2.99958 0.0504108i −1.30178 + 7.38274i −1.19547 3.28452i 6.40658 + 7.90098i −1.88718 10.7027i 10.2675 5.92795i 8.99492 + 0.302422i −5.92572 + 10.2637i
5.2 −1.24712 1.48626i 2.80515 + 1.06355i 0.0409354 0.232156i −1.47839 4.06185i −1.91764 5.49555i 1.54363 + 8.75434i −7.11704 + 4.10903i 6.73771 + 5.96685i −4.19322 + 7.26288i
5.3 −0.374063 0.445791i −0.428198 2.96928i 0.635786 3.60572i 2.62195 + 7.20376i −1.16351 + 1.30159i 0.231638 + 1.31369i −3.86112 + 2.22922i −8.63329 + 2.54288i 2.23059 3.86350i
5.4 0.837612 + 0.998227i −0.987122 + 2.83295i 0.399729 2.26698i 0.149473 + 0.410673i −3.65475 + 1.38754i −1.05651 5.99176i 7.11182 4.10601i −7.05118 5.59293i −0.284745 + 0.493192i
5.5 2.13670 + 2.54642i −2.03568 2.20363i −1.22417 + 6.94260i −2.35247 6.46335i 1.26173 9.89219i 1.10811 + 6.28443i −8.77937 + 5.06877i −0.712002 + 8.97179i 11.4319 19.8006i
11.1 −2.17948 + 2.59740i −2.99958 + 0.0504108i −1.30178 7.38274i −1.19547 + 3.28452i 6.40658 7.90098i −1.88718 + 10.7027i 10.2675 + 5.92795i 8.99492 0.302422i −5.92572 10.2637i
11.2 −1.24712 + 1.48626i 2.80515 1.06355i 0.0409354 + 0.232156i −1.47839 + 4.06185i −1.91764 + 5.49555i 1.54363 8.75434i −7.11704 4.10903i 6.73771 5.96685i −4.19322 7.26288i
11.3 −0.374063 + 0.445791i −0.428198 + 2.96928i 0.635786 + 3.60572i 2.62195 7.20376i −1.16351 1.30159i 0.231638 1.31369i −3.86112 2.22922i −8.63329 2.54288i 2.23059 + 3.86350i
11.4 0.837612 0.998227i −0.987122 2.83295i 0.399729 + 2.26698i 0.149473 0.410673i −3.65475 1.38754i −1.05651 + 5.99176i 7.11182 + 4.10601i −7.05118 + 5.59293i −0.284745 0.493192i
11.5 2.13670 2.54642i −2.03568 + 2.20363i −1.22417 6.94260i −2.35247 + 6.46335i 1.26173 + 9.89219i 1.10811 6.28443i −8.77937 5.06877i −0.712002 8.97179i 11.4319 + 19.8006i
14.1 −3.46291 + 0.610605i 1.47633 2.61160i 7.86014 2.86086i 3.16671 3.77394i −3.51774 + 9.94519i −3.18911 1.16074i −13.2912 + 7.67367i −4.64090 7.71116i −8.66165 + 15.0024i
14.2 −2.31604 + 0.408381i −1.62484 + 2.52189i 1.43851 0.523575i −3.71692 + 4.42965i 2.73330 6.50435i 4.57693 + 1.66587i 5.02894 2.90346i −3.71981 8.19530i 6.79956 11.7772i
14.3 0.115908 0.0204377i 2.32380 + 1.89736i −3.74575 + 1.36334i 3.98394 4.74788i 0.308125 + 0.172426i −7.49258 2.72708i −0.814011 + 0.469969i 1.80008 + 8.81815i 0.364735 0.631740i
14.4 1.14332 0.201599i 1.10490 2.78912i −2.49222 + 0.907094i −3.46013 + 4.12362i 0.700975 3.41162i 9.89907 + 3.60297i −6.68824 + 3.86146i −6.55839 6.16340i −3.12473 + 5.41219i
14.5 2.58003 0.454929i −2.92175 + 0.680712i 2.69082 0.979377i 0.519123 0.618667i −7.22853 + 3.08544i −5.56035 2.02380i −2.57852 + 1.48871i 8.07326 3.97774i 1.05790 1.83234i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.5 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 27.3.f.a 30
3.b odd 2 1 81.3.f.a 30
4.b odd 2 1 432.3.bc.a 30
9.c even 3 1 243.3.f.c 30
9.c even 3 1 243.3.f.d 30
9.d odd 6 1 243.3.f.a 30
9.d odd 6 1 243.3.f.b 30
27.e even 9 1 81.3.f.a 30
27.e even 9 1 243.3.f.a 30
27.e even 9 1 243.3.f.b 30
27.e even 9 1 729.3.b.a 30
27.f odd 18 1 inner 27.3.f.a 30
27.f odd 18 1 243.3.f.c 30
27.f odd 18 1 243.3.f.d 30
27.f odd 18 1 729.3.b.a 30
108.l even 18 1 432.3.bc.a 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.3.f.a 30 1.a even 1 1 trivial
27.3.f.a 30 27.f odd 18 1 inner
81.3.f.a 30 3.b odd 2 1
81.3.f.a 30 27.e even 9 1
243.3.f.a 30 9.d odd 6 1
243.3.f.a 30 27.e even 9 1
243.3.f.b 30 9.d odd 6 1
243.3.f.b 30 27.e even 9 1
243.3.f.c 30 9.c even 3 1
243.3.f.c 30 27.f odd 18 1
243.3.f.d 30 9.c even 3 1
243.3.f.d 30 27.f odd 18 1
432.3.bc.a 30 4.b odd 2 1
432.3.bc.a 30 108.l even 18 1
729.3.b.a 30 27.e even 9 1
729.3.b.a 30 27.f odd 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(27, [\chi])$$.