# 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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$0.735696713773$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$5$$ over $$\Q(\zeta_{18})$$ 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.

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

## 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

This newform does not have CM; other inner twists have not been computed.

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(27, [\chi])$$.