# Properties

 Label 324.3.o.a Level $324$ Weight $3$ Character orbit 324.o Analytic conductor $8.828$ Analytic rank $0$ Dimension $324$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 324.o (of order $$54$$, degree $$18$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.82836056527$$ Analytic rank: $$0$$ Dimension: $$324$$ Relative dimension: $$18$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$324 q+O(q^{10})$$ 324 * q $$\operatorname{Tr}(f)(q) =$$ $$324 q - 135 q^{21} - 81 q^{23} + 27 q^{27} + 81 q^{29} + 189 q^{33} + 243 q^{35} + 216 q^{41} + 432 q^{45} + 324 q^{47} + 126 q^{51} - 216 q^{57} - 378 q^{59} - 540 q^{63} - 108 q^{65} - 351 q^{67} + 504 q^{69} + 648 q^{71} + 450 q^{75} + 432 q^{77} - 54 q^{79} - 72 q^{81} - 216 q^{83} + 270 q^{85} - 1008 q^{87} - 648 q^{89} - 684 q^{93} - 432 q^{95} + 459 q^{97} - 252 q^{99}+O(q^{100})$$ 324 * q - 135 * q^21 - 81 * q^23 + 27 * q^27 + 81 * q^29 + 189 * q^33 + 243 * q^35 + 216 * q^41 + 432 * q^45 + 324 * q^47 + 126 * q^51 - 216 * q^57 - 378 * q^59 - 540 * q^63 - 108 * q^65 - 351 * q^67 + 504 * q^69 + 648 * q^71 + 450 * q^75 + 432 * q^77 - 54 * q^79 - 72 * q^81 - 216 * q^83 + 270 * q^85 - 1008 * q^87 - 648 * q^89 - 684 * q^93 - 432 * q^95 + 459 * q^97 - 252 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 0 −2.85677 + 0.915910i 0 −4.55987 + 3.39470i 0 9.74231 6.40762i 0 7.32222 5.23308i 0
5.2 0 −2.80896 1.05344i 0 −7.15515 + 5.32681i 0 −7.59481 + 4.99518i 0 6.78053 + 5.91815i 0
5.3 0 −2.78702 + 1.11018i 0 2.99904 2.23270i 0 −3.97531 + 2.61460i 0 6.53498 6.18822i 0
5.4 0 −2.76434 1.16550i 0 −0.148228 + 0.110352i 0 5.09912 3.35374i 0 6.28320 + 6.44371i 0
5.5 0 −2.61407 1.47195i 0 7.35484 5.47548i 0 2.16180 1.42184i 0 4.66673 + 7.69556i 0
5.6 0 −1.59067 + 2.54358i 0 −2.52291 + 1.87823i 0 −4.60759 + 3.03046i 0 −3.93956 8.09196i 0
5.7 0 −1.40095 2.65280i 0 1.05662 0.786622i 0 −3.63828 + 2.39294i 0 −5.07467 + 7.43288i 0
5.8 0 −0.763445 + 2.90123i 0 3.55757 2.64851i 0 5.32637 3.50321i 0 −7.83430 4.42986i 0
5.9 0 −0.463613 2.96396i 0 −2.73041 + 2.03271i 0 −1.12154 + 0.737649i 0 −8.57013 + 2.74826i 0
5.10 0 0.876176 + 2.86920i 0 5.98178 4.45327i 0 −8.57475 + 5.63970i 0 −7.46463 + 5.02785i 0
5.11 0 0.897733 + 2.86253i 0 −1.94378 + 1.44709i 0 9.49800 6.24693i 0 −7.38815 + 5.13957i 0
5.12 0 1.44773 2.62756i 0 3.57760 2.66342i 0 6.85076 4.50582i 0 −4.80816 7.60799i 0
5.13 0 1.79639 2.40271i 0 3.71691 2.76713i 0 −11.2502 + 7.39935i 0 −2.54600 8.63237i 0
5.14 0 1.81604 + 2.38789i 0 −4.15318 + 3.09193i 0 −3.40769 + 2.24128i 0 −2.40402 + 8.67299i 0
5.15 0 1.87615 2.34095i 0 −7.80193 + 5.80832i 0 5.92166 3.89473i 0 −1.96012 8.78396i 0
5.16 0 2.91181 0.722057i 0 −2.43946 + 1.81611i 0 −7.41916 + 4.87966i 0 7.95727 4.20498i 0
5.17 0 2.94286 + 0.582717i 0 5.13522 3.82303i 0 3.31229 2.17853i 0 8.32088 + 3.42971i 0
5.18 0 2.97960 + 0.349247i 0 −2.56932 + 1.91279i 0 3.67698 2.41839i 0 8.75605 + 2.08123i 0
29.1 0 −2.99940 0.0601358i 0 2.76376 2.60747i 0 −5.64514 + 0.659822i 0 8.99277 + 0.360742i 0
29.2 0 −2.97121 + 0.414613i 0 −0.778741 + 0.734705i 0 1.20130 0.140411i 0 8.65619 2.46381i 0
See next 80 embeddings (of 324 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 317.18 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 324.3.o.a 324
81.h odd 54 1 inner 324.3.o.a 324

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.3.o.a 324 1.a even 1 1 trivial
324.3.o.a 324 81.h odd 54 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(324, [\chi])$$.