# Properties

 Label 324.4.i.a Level 324 Weight 4 Character orbit 324.i Analytic conductor 19.117 Analytic rank 0 Dimension 54 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 324.i (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.1166188419$$ Analytic rank: $$0$$ Dimension: $$54$$ Relative dimension: $$9$$ over $$\Q(\zeta_{9})$$ Twist minimal: no (minimal twist has level 108) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$54q - 12q^{5} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$54q - 12q^{5} + 87q^{11} - 204q^{17} - 96q^{23} - 216q^{25} - 318q^{29} - 54q^{31} - 6q^{35} - 867q^{41} - 513q^{43} + 1548q^{47} + 594q^{49} + 1068q^{53} + 1218q^{59} - 54q^{61} - 96q^{65} - 2997q^{67} + 120q^{71} - 216q^{73} - 3480q^{77} + 2808q^{79} - 4464q^{83} + 2160q^{85} - 4029q^{89} + 270q^{91} + 1650q^{95} - 3483q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 0 0 0 −19.7528 + 7.18944i 0 0.723940 + 4.10567i 0 0 0
37.2 0 0 0 −13.2490 + 4.82225i 0 3.17294 + 17.9946i 0 0 0
37.3 0 0 0 −11.1609 + 4.06223i 0 −4.28132 24.2806i 0 0 0
37.4 0 0 0 −5.21695 + 1.89881i 0 1.58868 + 9.00985i 0 0 0
37.5 0 0 0 4.27250 1.55506i 0 −4.69568 26.6305i 0 0 0
37.6 0 0 0 4.80495 1.74886i 0 5.74341 + 32.5725i 0 0 0
37.7 0 0 0 6.50317 2.36696i 0 1.34685 + 7.63837i 0 0 0
37.8 0 0 0 10.0092 3.64305i 0 −2.90933 16.4997i 0 0 0
37.9 0 0 0 13.0875 4.76347i 0 −0.689494 3.91032i 0 0 0
73.1 0 0 0 −15.7004 + 13.1742i 0 −8.81657 + 3.20897i 0 0 0
73.2 0 0 0 −8.18465 + 6.86774i 0 −24.4722 + 8.90714i 0 0 0
73.3 0 0 0 −7.52940 + 6.31792i 0 24.9076 9.06562i 0 0 0
73.4 0 0 0 −6.54095 + 5.48851i 0 11.9650 4.35489i 0 0 0
73.5 0 0 0 1.73722 1.45770i 0 −5.32680 + 1.93880i 0 0 0
73.6 0 0 0 5.18194 4.34817i 0 16.8240 6.12342i 0 0 0
73.7 0 0 0 5.19451 4.35871i 0 −17.8009 + 6.47901i 0 0 0
73.8 0 0 0 12.8040 10.7439i 0 −20.8033 + 7.57178i 0 0 0
73.9 0 0 0 14.9393 12.5356i 0 23.5233 8.56177i 0 0 0
145.1 0 0 0 −3.27839 + 18.5927i 0 4.82725 4.05055i 0 0 0
145.2 0 0 0 −2.26368 + 12.8380i 0 −3.04053 + 2.55131i 0 0 0
See all 54 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.9 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 324.4.i.a 54
3.b odd 2 1 108.4.i.a 54
27.e even 9 1 inner 324.4.i.a 54
27.f odd 18 1 108.4.i.a 54

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.i.a 54 3.b odd 2 1
108.4.i.a 54 27.f odd 18 1
324.4.i.a 54 1.a even 1 1 trivial
324.4.i.a 54 27.e even 9 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(324, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database