# Properties

 Label 324.5.d.f Level 324 Weight 5 Character orbit 324.d Analytic conductor 33.492 Analytic rank 0 Dimension 22 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 324.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.4918680392$$ Analytic rank: $$0$$ Dimension: $$22$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$22q + q^{2} + q^{4} + 2q^{5} + 61q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q + q^{2} + q^{4} + 2q^{5} + 61q^{8} + 14q^{10} + 2q^{13} - 252q^{14} + q^{16} - 28q^{17} + 140q^{20} + 33q^{22} + 1752q^{25} + 548q^{26} - 258q^{28} - 526q^{29} + 121q^{32} - 385q^{34} - 4q^{37} - 1395q^{38} + 2276q^{40} + 2762q^{41} + 3357q^{44} + 1788q^{46} - 3428q^{49} - 6375q^{50} - 1438q^{52} - 5044q^{53} + 7506q^{56} + 4064q^{58} + 2q^{61} - 9162q^{62} + 4513q^{64} + 2014q^{65} + 11405q^{68} - 3666q^{70} - 1708q^{73} - 14620q^{74} - 1581q^{76} + 3942q^{77} + 22760q^{80} - 4243q^{82} + 1252q^{85} - 22113q^{86} - 1995q^{88} + 6524q^{89} + 30294q^{92} - 7524q^{94} - 5638q^{97} - 46469q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1 −3.92469 0.772521i 0 14.8064 + 6.06382i 21.1512 0 44.6184i −53.4262 35.2369i 0 −83.0119 16.3397i
163.2 −3.92469 + 0.772521i 0 14.8064 6.06382i 21.1512 0 44.6184i −53.4262 + 35.2369i 0 −83.0119 + 16.3397i
163.3 −3.49753 1.94095i 0 8.46540 + 13.5771i −33.2277 0 46.1602i −3.25547 63.9171i 0 116.215 + 64.4935i
163.4 −3.49753 + 1.94095i 0 8.46540 13.5771i −33.2277 0 46.1602i −3.25547 + 63.9171i 0 116.215 64.4935i
163.5 −3.19553 2.40595i 0 4.42285 + 15.3766i −2.03090 0 23.1347i 22.8618 59.7774i 0 6.48981 + 4.88624i
163.6 −3.19553 + 2.40595i 0 4.42285 15.3766i −2.03090 0 23.1347i 22.8618 + 59.7774i 0 6.48981 4.88624i
163.7 −1.75474 3.59457i 0 −9.84180 + 12.6150i 46.6931 0 60.5483i 62.6153 + 13.2409i 0 −81.9341 167.841i
163.8 −1.75474 + 3.59457i 0 −9.84180 12.6150i 46.6931 0 60.5483i 62.6153 13.2409i 0 −81.9341 + 167.841i
163.9 −1.16626 3.82620i 0 −13.2797 + 8.92473i −39.0789 0 12.2052i 49.6354 + 40.4021i 0 45.5763 + 149.524i
163.10 −1.16626 + 3.82620i 0 −13.2797 8.92473i −39.0789 0 12.2052i 49.6354 40.4021i 0 45.5763 149.524i
163.11 −0.328300 3.98650i 0 −15.7844 + 2.61754i 5.66182 0 52.1456i 15.6169 + 62.0654i 0 −1.85878 22.5709i
163.12 −0.328300 + 3.98650i 0 −15.7844 2.61754i 5.66182 0 52.1456i 15.6169 62.0654i 0 −1.85878 + 22.5709i
163.13 1.34653 3.76654i 0 −12.3737 10.1435i 11.0316 0 11.9152i −54.8676 + 32.9476i 0 14.8544 41.5509i
163.14 1.34653 + 3.76654i 0 −12.3737 + 10.1435i 11.0316 0 11.9152i −54.8676 32.9476i 0 14.8544 + 41.5509i
163.15 2.43802 3.17113i 0 −4.11208 15.4626i −22.1491 0 95.5959i −59.0591 24.6582i 0 −54.0000 + 70.2376i
163.16 2.43802 + 3.17113i 0 −4.11208 + 15.4626i −22.1491 0 95.5959i −59.0591 + 24.6582i 0 −54.0000 70.2376i
163.17 2.82463 2.83222i 0 −0.0429591 15.9999i 11.7888 0 58.3756i −45.4367 45.0722i 0 33.2988 33.3884i
163.18 2.82463 + 2.83222i 0 −0.0429591 + 15.9999i 11.7888 0 58.3756i −45.4367 + 45.0722i 0 33.2988 + 33.3884i
163.19 3.76125 1.36126i 0 12.2940 10.2400i −28.6092 0 25.6487i 32.3013 55.2506i 0 −107.606 + 38.9445i
163.20 3.76125 + 1.36126i 0 12.2940 + 10.2400i −28.6092 0 25.6487i 32.3013 + 55.2506i 0 −107.606 38.9445i
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 163.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.5.d.f 22
3.b odd 2 1 324.5.d.e 22
4.b odd 2 1 inner 324.5.d.f 22
9.c even 3 2 36.5.f.a 44
9.d odd 6 2 108.5.f.a 44
12.b even 2 1 324.5.d.e 22
36.f odd 6 2 36.5.f.a 44
36.h even 6 2 108.5.f.a 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.5.f.a 44 9.c even 3 2
36.5.f.a 44 36.f odd 6 2
108.5.f.a 44 9.d odd 6 2
108.5.f.a 44 36.h even 6 2
324.5.d.e 22 3.b odd 2 1
324.5.d.e 22 12.b even 2 1
324.5.d.f 22 1.a even 1 1 trivial
324.5.d.f 22 4.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{11} - \cdots$$ acting on $$S_{5}^{\mathrm{new}}(324, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database