# Properties

 Label 108.5.k.a Level 108 Weight 5 Character orbit 108.k Analytic conductor 11.164 Analytic rank 0 Dimension 72 CM no Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 108.k (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1639560131$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$12$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$72q + 9q^{5} - 102q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q + 9q^{5} - 102q^{9} + 18q^{11} - 225q^{15} - 282q^{21} - 1278q^{23} + 441q^{25} + 54q^{27} + 1854q^{29} - 1665q^{31} - 45q^{33} - 2673q^{35} + 6951q^{39} - 5472q^{41} + 1260q^{43} + 5553q^{45} + 5103q^{47} - 5904q^{49} + 1899q^{51} + 1107q^{57} - 10944q^{59} + 8352q^{61} - 11985q^{63} + 8757q^{65} + 378q^{67} + 5607q^{69} - 19764q^{71} + 6111q^{73} - 3453q^{75} - 5679q^{77} - 5652q^{79} - 20466q^{81} - 20061q^{83} + 26100q^{85} + 40545q^{87} + 15633q^{89} - 6039q^{91} + 40179q^{93} + 48024q^{95} - 37530q^{97} + 12177q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 0 −8.89497 1.37096i 0 −1.59659 4.38660i 0 12.9775 + 73.5989i 0 77.2410 + 24.3892i 0
5.2 0 −8.09115 3.94122i 0 14.0198 + 38.5191i 0 −13.2730 75.2747i 0 49.9335 + 63.7781i 0
5.3 0 −7.74246 + 4.58850i 0 −2.39469 6.57935i 0 −4.62159 26.2104i 0 38.8914 71.0525i 0
5.4 0 −5.53499 7.09676i 0 −14.0519 38.6073i 0 −5.57324 31.6074i 0 −19.7279 + 78.5609i 0
5.5 0 −2.33607 + 8.69153i 0 15.1037 + 41.4971i 0 8.53787 + 48.4207i 0 −70.0855 40.6081i 0
5.6 0 −1.23486 8.91488i 0 5.87553 + 16.1429i 0 8.12577 + 46.0835i 0 −77.9503 + 22.0172i 0
5.7 0 −0.126357 + 8.99911i 0 −6.31102 17.3394i 0 2.68521 + 15.2286i 0 −80.9681 2.27420i 0
5.8 0 4.78712 + 7.62125i 0 −3.22217 8.85284i 0 −16.2288 92.0381i 0 −35.1670 + 72.9677i 0
5.9 0 5.44219 7.16816i 0 4.60350 + 12.6480i 0 −6.68463 37.9104i 0 −21.7651 78.0210i 0
5.10 0 6.98131 5.67991i 0 −8.75895 24.0650i 0 −0.632293 3.58591i 0 16.4774 79.3063i 0
5.11 0 8.08622 + 3.95134i 0 −10.7879 29.6395i 0 15.2291 + 86.3688i 0 49.7738 + 63.9028i 0
5.12 0 8.73547 + 2.16600i 0 9.54164 + 26.2154i 0 −0.541932 3.07345i 0 71.6168 + 37.8421i 0
29.1 0 −8.99629 0.258464i 0 −25.3131 30.1670i 0 −27.6070 + 10.0481i 0 80.8664 + 4.65043i 0
29.2 0 −8.23149 + 3.63904i 0 18.1168 + 21.5908i 0 22.9581 8.35608i 0 54.5147 59.9095i 0
29.3 0 −6.90497 5.77247i 0 11.0592 + 13.1798i 0 7.77764 2.83083i 0 14.3573 + 79.7174i 0
29.4 0 −3.84126 + 8.13909i 0 0.0201223 + 0.0239808i 0 −60.3934 + 21.9814i 0 −51.4895 62.5287i 0
29.5 0 −2.24301 + 8.71602i 0 −14.1660 16.8824i 0 35.7790 13.0225i 0 −70.9378 39.1002i 0
29.6 0 −1.21427 8.91771i 0 −22.3567 26.6437i 0 80.4961 29.2982i 0 −78.0511 + 21.6570i 0
29.7 0 1.21143 8.91810i 0 1.79646 + 2.14093i 0 −63.3056 + 23.0413i 0 −78.0649 21.6073i 0
29.8 0 4.39288 + 7.85510i 0 6.32074 + 7.53276i 0 66.1128 24.0631i 0 −42.4052 + 69.0131i 0
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.12 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.f odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.5.k.a 72
3.b odd 2 1 324.5.k.a 72
27.e even 9 1 324.5.k.a 72
27.f odd 18 1 inner 108.5.k.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.5.k.a 72 1.a even 1 1 trivial
108.5.k.a 72 27.f odd 18 1 inner
324.5.k.a 72 3.b odd 2 1
324.5.k.a 72 27.e even 9 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(108, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database