# Properties

 Label 27.4.e.a Level 27 Weight 4 Character orbit 27.e Analytic conductor 1.593 Analytic rank 0 Dimension 48 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$27 = 3^{3}$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 27.e (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.59305157016$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$8$$ over $$\Q(\zeta_{9})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 6q^{2} - 6q^{3} - 6q^{4} + 6q^{5} - 18q^{6} - 6q^{7} - 75q^{8} - 54q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 6q^{2} - 6q^{3} - 6q^{4} + 6q^{5} - 18q^{6} - 6q^{7} - 75q^{8} - 54q^{9} - 3q^{10} + 57q^{11} + 147q^{12} - 6q^{13} + 51q^{14} - 36q^{15} + 18q^{16} - 207q^{17} - 639q^{18} - 3q^{19} - 597q^{20} - 138q^{21} - 60q^{22} + 402q^{23} + 1170q^{24} - 222q^{25} + 1914q^{26} + 1125q^{27} - 12q^{28} + 480q^{29} + 459q^{30} - 60q^{31} - 648q^{32} - 639q^{33} + 288q^{34} - 1257q^{35} - 2088q^{36} - 3q^{37} - 1524q^{38} - 768q^{39} + 561q^{40} - 1731q^{41} - 3078q^{42} + 507q^{43} - 2211q^{44} - 360q^{45} - 3q^{46} + 984q^{47} + 2289q^{48} - 600q^{49} + 4359q^{50} + 2655q^{51} - 1431q^{52} + 2736q^{53} + 5454q^{54} - 12q^{55} + 5907q^{56} + 3426q^{57} - 897q^{58} + 2238q^{59} + 1314q^{60} + 48q^{61} - 2118q^{62} - 2610q^{63} - 195q^{64} - 6990q^{65} - 11115q^{66} - 681q^{67} - 11169q^{68} - 6138q^{69} - 33q^{70} - 3105q^{71} - 36q^{72} - 219q^{73} + 3543q^{74} + 2604q^{75} + 3426q^{76} + 4722q^{77} + 6066q^{78} + 2802q^{79} + 9870q^{80} + 3438q^{81} - 12q^{82} + 3468q^{83} + 7674q^{84} + 2529q^{85} + 3624q^{86} + 2880q^{87} + 2850q^{88} - 5202q^{89} - 12510q^{90} + 267q^{91} - 18453q^{92} - 11802q^{93} - 1653q^{94} - 10113q^{95} - 14094q^{96} - 3381q^{97} - 4392q^{98} - 1242q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −4.97861 1.81207i −3.67671 3.67176i 15.3746 + 12.9008i −1.82513 + 10.3508i 11.6515 + 24.9427i −5.92244 + 4.96952i −31.9745 55.3815i 0.0364268 + 27.0000i 27.8430 48.2255i
4.2 −4.05233 1.47493i 4.37853 + 2.79794i 8.11761 + 6.81148i 3.22691 18.3007i −13.6165 17.7962i 14.4087 12.0903i −5.59920 9.69809i 11.3431 + 24.5017i −40.0687 + 69.4011i
4.3 −2.12171 0.772239i −0.789571 + 5.13581i −2.22306 1.86537i −3.33920 + 18.9376i 5.64131 10.2870i 0.500915 0.420318i 12.3077 + 21.3175i −25.7532 8.11018i 21.7091 37.6013i
4.4 −0.982993 0.357780i −5.19600 0.0395600i −5.29009 4.43891i 2.68017 15.2000i 5.09348 + 1.89791i −18.0349 + 15.1331i 7.79628 + 13.5036i 26.9969 + 0.411107i −8.07284 + 13.9826i
4.5 −0.932125 0.339266i 1.23404 5.04749i −5.37460 4.50982i 0.0250883 0.142283i −2.86272 + 4.28623i 19.4381 16.3105i 7.44756 + 12.8996i −23.9543 12.4576i −0.0716572 + 0.124114i
4.6 1.82340 + 0.663664i 5.12138 + 0.878355i −3.24401 2.72205i −0.470388 + 2.66770i 8.75540 + 5.00047i −9.12942 + 7.66050i −11.8703 20.5600i 25.4570 + 8.99678i −2.62817 + 4.55212i
4.7 3.53135 + 1.28531i −2.46998 + 4.57157i 4.69008 + 3.93544i 1.06026 6.01302i −14.5982 + 12.9691i 13.2194 11.0924i −3.52788 6.11047i −14.7984 22.5833i 11.4727 19.8713i
4.8 4.06758 + 1.48048i −1.65472 4.92564i 8.22505 + 6.90164i −0.745703 + 4.22909i 0.561612 22.4852i −15.6540 + 13.1353i 5.92381 + 10.2603i −21.5238 + 16.3011i −9.29428 + 16.0982i
7.1 −4.97861 + 1.81207i −3.67671 + 3.67176i 15.3746 12.9008i −1.82513 10.3508i 11.6515 24.9427i −5.92244 4.96952i −31.9745 + 55.3815i 0.0364268 27.0000i 27.8430 + 48.2255i
7.2 −4.05233 + 1.47493i 4.37853 2.79794i 8.11761 6.81148i 3.22691 + 18.3007i −13.6165 + 17.7962i 14.4087 + 12.0903i −5.59920 + 9.69809i 11.3431 24.5017i −40.0687 69.4011i
7.3 −2.12171 + 0.772239i −0.789571 5.13581i −2.22306 + 1.86537i −3.33920 18.9376i 5.64131 + 10.2870i 0.500915 + 0.420318i 12.3077 21.3175i −25.7532 + 8.11018i 21.7091 + 37.6013i
7.4 −0.982993 + 0.357780i −5.19600 + 0.0395600i −5.29009 + 4.43891i 2.68017 + 15.2000i 5.09348 1.89791i −18.0349 15.1331i 7.79628 13.5036i 26.9969 0.411107i −8.07284 13.9826i
7.5 −0.932125 + 0.339266i 1.23404 + 5.04749i −5.37460 + 4.50982i 0.0250883 + 0.142283i −2.86272 4.28623i 19.4381 + 16.3105i 7.44756 12.8996i −23.9543 + 12.4576i −0.0716572 0.124114i
7.6 1.82340 0.663664i 5.12138 0.878355i −3.24401 + 2.72205i −0.470388 2.66770i 8.75540 5.00047i −9.12942 7.66050i −11.8703 + 20.5600i 25.4570 8.99678i −2.62817 4.55212i
7.7 3.53135 1.28531i −2.46998 4.57157i 4.69008 3.93544i 1.06026 + 6.01302i −14.5982 12.9691i 13.2194 + 11.0924i −3.52788 + 6.11047i −14.7984 + 22.5833i 11.4727 + 19.8713i
7.8 4.06758 1.48048i −1.65472 + 4.92564i 8.22505 6.90164i −0.745703 4.22909i 0.561612 + 22.4852i −15.6540 13.1353i 5.92381 10.2603i −21.5238 16.3011i −9.29428 16.0982i
13.1 −0.874714 4.96075i 2.71299 4.43167i −16.3263 + 5.94230i 11.9326 + 10.0126i −24.3575 9.58200i −5.29732 1.92807i 23.6100 + 40.8938i −12.2794 24.0461i 39.2325 67.9527i
13.2 −0.592608 3.36085i −5.15968 0.614572i −3.42657 + 1.24717i −8.41296 7.05931i 0.992184 + 17.7051i 4.14024 + 1.50692i −7.42861 12.8667i 26.2446 + 6.34199i −18.7397 + 32.4581i
13.3 −0.404735 2.29536i 4.52897 + 2.54724i 2.41266 0.878135i −4.78113 4.01185i 4.01380 11.4266i 2.53990 + 0.924448i −12.3152 21.3306i 14.0232 + 23.0727i −7.27356 + 12.5982i
13.4 −0.0518956 0.294314i −2.82128 + 4.36353i 7.43361 2.70561i 15.3604 + 12.8889i 1.43066 + 0.603897i −20.1945 7.35018i −2.37749 4.11794i −11.0807 24.6215i 2.99626 5.18968i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 25.8 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.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.4.e.a 48
3.b odd 2 1 81.4.e.a 48
9.c even 3 1 243.4.e.c 48
9.c even 3 1 243.4.e.d 48
9.d odd 6 1 243.4.e.a 48
9.d odd 6 1 243.4.e.b 48
27.e even 9 1 inner 27.4.e.a 48
27.e even 9 1 243.4.e.c 48
27.e even 9 1 243.4.e.d 48
27.e even 9 1 729.4.a.d 24
27.f odd 18 1 81.4.e.a 48
27.f odd 18 1 243.4.e.a 48
27.f odd 18 1 243.4.e.b 48
27.f odd 18 1 729.4.a.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.e.a 48 1.a even 1 1 trivial
27.4.e.a 48 27.e even 9 1 inner
81.4.e.a 48 3.b odd 2 1
81.4.e.a 48 27.f odd 18 1
243.4.e.a 48 9.d odd 6 1
243.4.e.a 48 27.f odd 18 1
243.4.e.b 48 9.d odd 6 1
243.4.e.b 48 27.f odd 18 1
243.4.e.c 48 9.c even 3 1
243.4.e.c 48 27.e even 9 1
243.4.e.d 48 9.c even 3 1
243.4.e.d 48 27.e even 9 1
729.4.a.c 24 27.f odd 18 1
729.4.a.d 24 27.e even 9 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(27, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database