# Properties

 Label 81.3.h.a Level $81$ Weight $3$ Character orbit 81.h Analytic conductor $2.207$ Analytic rank $0$ Dimension $306$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 81.h (of order $$54$$, degree $$18$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.20709014132$$ Analytic rank: $$0$$ Dimension: $$306$$ Relative dimension: $$17$$ over $$\Q(\zeta_{54})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{54}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$306 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10})$$ 306 * q - 18 * q^2 - 18 * q^3 - 18 * q^4 - 18 * q^5 - 18 * q^6 - 18 * q^7 - 18 * q^8 - 18 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$306 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9} - 18 q^{10} - 18 q^{11} - 18 q^{12} - 18 q^{13} - 18 q^{14} - 18 q^{15} - 18 q^{16} - 18 q^{17} - 90 q^{18} - 18 q^{19} - 234 q^{20} - 153 q^{21} - 18 q^{22} - 99 q^{23} - 126 q^{24} - 18 q^{25} - 27 q^{26} + 9 q^{27} - 9 q^{28} + 63 q^{29} + 198 q^{30} - 18 q^{31} + 306 q^{32} + 171 q^{33} - 18 q^{34} + 225 q^{35} + 342 q^{36} - 18 q^{37} + 90 q^{38} - 18 q^{39} - 18 q^{40} - 234 q^{41} - 513 q^{42} - 18 q^{43} - 666 q^{44} - 450 q^{45} - 18 q^{46} - 342 q^{47} - 513 q^{48} - 18 q^{49} - 369 q^{50} - 144 q^{51} - 54 q^{52} - 27 q^{53} + 108 q^{54} - 9 q^{55} + 396 q^{56} + 198 q^{57} - 18 q^{58} + 360 q^{59} + 801 q^{60} - 18 q^{61} + 873 q^{62} + 522 q^{63} - 18 q^{64} + 1170 q^{65} + 1926 q^{66} - 369 q^{67} + 2169 q^{68} + 1062 q^{69} - 558 q^{70} + 630 q^{71} + 1710 q^{72} - 18 q^{73} + 846 q^{74} + 432 q^{75} - 342 q^{76} + 414 q^{77} + 189 q^{78} - 72 q^{79} - 90 q^{81} - 36 q^{82} - 234 q^{83} - 945 q^{84} + 252 q^{85} - 882 q^{86} - 1026 q^{87} + 630 q^{88} - 1314 q^{89} - 2529 q^{90} - 18 q^{91} - 3960 q^{92} - 2214 q^{93} + 738 q^{94} - 2394 q^{95} - 3321 q^{96} + 441 q^{97} - 2853 q^{98} - 1566 q^{99}+O(q^{100})$$ 306 * q - 18 * q^2 - 18 * q^3 - 18 * q^4 - 18 * q^5 - 18 * q^6 - 18 * q^7 - 18 * q^8 - 18 * q^9 - 18 * q^10 - 18 * q^11 - 18 * q^12 - 18 * q^13 - 18 * q^14 - 18 * q^15 - 18 * q^16 - 18 * q^17 - 90 * q^18 - 18 * q^19 - 234 * q^20 - 153 * q^21 - 18 * q^22 - 99 * q^23 - 126 * q^24 - 18 * q^25 - 27 * q^26 + 9 * q^27 - 9 * q^28 + 63 * q^29 + 198 * q^30 - 18 * q^31 + 306 * q^32 + 171 * q^33 - 18 * q^34 + 225 * q^35 + 342 * q^36 - 18 * q^37 + 90 * q^38 - 18 * q^39 - 18 * q^40 - 234 * q^41 - 513 * q^42 - 18 * q^43 - 666 * q^44 - 450 * q^45 - 18 * q^46 - 342 * q^47 - 513 * q^48 - 18 * q^49 - 369 * q^50 - 144 * q^51 - 54 * q^52 - 27 * q^53 + 108 * q^54 - 9 * q^55 + 396 * q^56 + 198 * q^57 - 18 * q^58 + 360 * q^59 + 801 * q^60 - 18 * q^61 + 873 * q^62 + 522 * q^63 - 18 * q^64 + 1170 * q^65 + 1926 * q^66 - 369 * q^67 + 2169 * q^68 + 1062 * q^69 - 558 * q^70 + 630 * q^71 + 1710 * q^72 - 18 * q^73 + 846 * q^74 + 432 * q^75 - 342 * q^76 + 414 * q^77 + 189 * q^78 - 72 * q^79 - 90 * q^81 - 36 * q^82 - 234 * q^83 - 945 * q^84 + 252 * q^85 - 882 * q^86 - 1026 * q^87 + 630 * q^88 - 1314 * q^89 - 2529 * q^90 - 18 * q^91 - 3960 * q^92 - 2214 * q^93 + 738 * q^94 - 2394 * q^95 - 3321 * q^96 + 441 * q^97 - 2853 * q^98 - 1566 * q^99

## 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 −3.74703 0.218240i −2.98307 0.318284i 10.0197 + 1.17113i 1.51414 + 6.38868i 11.1082 + 1.84364i −3.22727 4.33498i −22.5029 3.96787i 8.79739 + 1.89893i −4.27928 24.2690i
2.2 −3.51578 0.204771i 1.87859 + 2.33900i 8.34586 + 0.975491i −0.560130 2.36337i −6.12575 8.60809i 0.853586 + 1.14656i −15.2695 2.69243i −1.94182 + 8.78802i 1.48535 + 8.42381i
2.3 −3.19159 0.185889i 0.779433 2.89698i 6.17872 + 0.722189i −1.14257 4.82088i −3.02614 + 9.10107i 2.64358 + 3.55094i −6.99198 1.23288i −7.78497 4.51600i 2.75046 + 15.5986i
2.4 −2.38296 0.138791i 2.64624 1.41330i 1.68627 + 0.197097i 1.89333 + 7.98858i −6.50203 + 3.00056i −1.92617 2.58729i 5.41197 + 0.954277i 5.00517 7.47986i −3.40297 19.2992i
2.5 −2.16212 0.125929i −2.73727 + 1.22773i 0.685955 + 0.0801767i −1.17077 4.93986i 6.07293 2.30981i 5.31774 + 7.14296i 7.05851 + 1.24461i 5.98534 6.72129i 1.90927 + 10.8280i
2.6 −1.71539 0.0999103i −0.611041 + 2.93711i −1.04036 0.121601i 0.258193 + 1.08940i 1.34162 4.97725i −6.70340 9.00422i 8.54126 + 1.50605i −8.25326 3.58939i −0.334060 1.89455i
2.7 −1.04279 0.0607357i −2.04391 2.19600i −2.88923 0.337702i 1.21177 + 5.11287i 1.99800 + 2.41411i 4.42126 + 5.93878i 7.10711 + 1.25318i −0.644854 + 8.97687i −0.953092 5.40525i
2.8 −0.614022 0.0357627i 2.99545 0.165205i −3.59721 0.420453i −2.20305 9.29539i −1.84518 0.00568553i −3.03111 4.07148i 4.61661 + 0.814032i 8.94541 0.989729i 1.02029 + 5.78637i
2.9 −0.272030 0.0158440i 2.02197 + 2.21622i −3.89920 0.455752i 0.613475 + 2.58845i −0.514925 0.634914i 5.77313 + 7.75466i 2.12689 + 0.375028i −0.823240 + 8.96227i −0.125872 0.713857i
2.10 −0.218910 0.0127500i 0.445650 2.96671i −3.92519 0.458789i −0.196368 0.828540i −0.135383 + 0.643760i −2.81710 3.78402i 1.71721 + 0.302791i −8.60279 2.64424i 0.0324228 + 0.183879i
2.11 1.00396 + 0.0584742i −2.94698 0.561535i −2.96843 0.346960i −0.829724 3.50088i −2.92582 0.736083i −3.90171 5.24091i −6.92145 1.22044i 8.36936 + 3.30966i −0.628301 3.56327i
2.12 1.44133 + 0.0839482i −1.99809 + 2.23778i −1.90256 0.222377i 1.78114 + 7.51523i −3.06777 + 3.05765i 0.363641 + 0.488455i −8.41092 1.48307i −1.01527 8.94255i 1.93633 + 10.9815i
2.13 2.03769 + 0.118682i 2.77384 1.14271i 0.165158 + 0.0193041i 0.612887 + 2.58597i 5.78787 1.99929i 1.33468 + 1.79278i −7.70630 1.35883i 6.38843 6.33940i 0.941967 + 5.34216i
2.14 2.62917 + 0.153131i −0.579386 2.94352i 2.91611 + 0.340844i −1.61217 6.80228i −1.07256 7.82773i 6.82619 + 9.16917i −2.75970 0.486609i −8.32862 + 3.41087i −3.19702 18.1312i
2.15 2.79319 + 0.162685i 1.52144 + 2.58558i 3.80251 + 0.444450i −0.618939 2.61151i 3.82905 + 7.46953i −3.52360 4.73302i −0.472829 0.0833725i −4.37041 + 7.86762i −1.30396 7.39515i
2.16 3.40512 + 0.198326i −1.03823 2.81462i 7.58258 + 0.886276i 1.79915 + 7.59120i −2.97709 9.79003i −7.23042 9.71214i 12.2075 + 2.15252i −6.84415 + 5.84445i 4.62078 + 26.2058i
2.17 3.55891 + 0.207283i −2.59188 + 1.51068i 8.64995 + 1.01103i −0.346846 1.46346i −9.53743 + 4.83912i 4.11378 + 5.52576i 16.5317 + 2.91499i 4.43571 7.83100i −0.931045 5.28022i
5.1 −0.860368 3.63018i 0.315927 + 2.98332i −8.86342 + 4.45138i −2.70864 + 2.01651i 10.5582 3.71362i −9.77832 + 6.43130i 14.1928 + 16.9143i −8.80038 + 1.88502i 9.65071 + 8.09791i
5.2 −0.769839 3.24821i 2.36730 1.84279i −6.38367 + 3.20600i 5.00097 3.72308i −7.80821 6.27084i −2.54834 + 1.67607i 6.74517 + 8.03858i 2.20825 8.72489i −15.9433 13.3780i
5.3 −0.723056 3.05081i −0.981828 2.83479i −5.21012 + 2.61662i −4.98049 + 3.70784i −7.93849 + 5.04508i 4.70676 3.09568i 3.68862 + 4.39593i −7.07203 + 5.56655i 14.9131 + 12.5136i
See next 80 embeddings (of 306 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 77.17 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
81.h odd 54 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.3.h.a 306
3.b odd 2 1 243.3.h.a 306
81.g even 27 1 243.3.h.a 306
81.h odd 54 1 inner 81.3.h.a 306

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.3.h.a 306 1.a even 1 1 trivial
81.3.h.a 306 81.h odd 54 1 inner
243.3.h.a 306 3.b odd 2 1
243.3.h.a 306 81.g even 27 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(81, [\chi])$$.