# Properties

 Label 108.5.f.a Level 108 Weight 5 Character orbit 108.f Analytic conductor 11.164 Analytic rank 0 Dimension 44 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 108.f (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1639560131$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$44q + q^{2} - q^{4} + 2q^{5} - 122q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q + q^{2} - q^{4} + 2q^{5} - 122q^{8} + 28q^{10} - 2q^{13} - 252q^{14} - q^{16} + 56q^{17} + 140q^{20} - 33q^{22} - 1752q^{25} - 1096q^{26} - 516q^{28} - 526q^{29} + 121q^{32} + 385q^{34} - 8q^{37} - 1395q^{38} - 2276q^{40} + 2762q^{41} - 6714q^{44} + 3576q^{46} + 3428q^{49} - 6375q^{50} + 1438q^{52} + 10088q^{53} + 7506q^{56} - 4064q^{58} - 2q^{61} + 18324q^{62} + 9026q^{64} + 2014q^{65} + 11405q^{68} + 3666q^{70} - 3416q^{73} - 14620q^{74} + 1581q^{76} + 3942q^{77} - 45520q^{80} - 8486q^{82} - 1252q^{85} - 22113q^{86} + 1995q^{88} - 13048q^{89} + 30294q^{92} + 7524q^{94} + 5638q^{97} + 92938q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −3.99035 0.277636i 0 15.8458 + 2.21573i 23.3466 + 40.4374i 0 52.4363 + 30.2741i −62.6153 13.2409i 0 −81.9341 167.841i
19.2 −3.89672 0.903087i 0 14.3689 + 7.03816i −19.5394 33.8433i 0 10.5700 + 6.10260i −49.6354 40.4021i 0 45.5763 + 149.524i
19.3 −3.68138 + 1.56444i 0 11.1051 11.5186i −1.01545 1.75881i 0 20.0352 + 11.5673i −22.8618 + 59.7774i 0 6.48981 + 4.88624i
19.4 −3.61656 1.70894i 0 10.1591 + 12.3610i 2.83091 + 4.90328i 0 −45.1595 26.0728i −15.6169 62.0654i 0 −1.85878 22.5709i
19.5 −3.42968 + 2.05847i 0 7.52540 14.1198i −16.6139 28.7760i 0 −39.9759 23.0801i 3.25547 + 63.9171i 0 116.215 + 64.4935i
19.6 −2.63137 + 3.01262i 0 −2.15179 15.8546i 10.5756 + 18.3175i 0 −38.6407 22.3092i 53.4262 + 35.2369i 0 −83.0119 16.3397i
19.7 −2.58866 3.04940i 0 −2.59770 + 15.7877i 5.51579 + 9.55363i 0 10.3188 + 5.95759i 54.8676 32.9476i 0 14.8544 41.5509i
19.8 −1.52726 3.69695i 0 −11.3349 + 11.2924i −11.0746 19.1817i 0 82.7885 + 47.7980i 59.0591 + 24.6582i 0 −54.0000 + 70.2376i
19.9 −1.29332 + 3.78514i 0 −12.6546 9.79083i 10.5756 + 18.3175i 0 38.6407 + 22.3092i 53.4262 35.2369i 0 −83.0119 + 16.3397i
19.10 −1.04046 3.86231i 0 −13.8349 + 8.03717i 5.89438 + 10.2094i 0 −50.5548 29.1878i 45.4367 + 45.0722i 0 33.2988 33.3884i
19.11 −0.0678484 + 3.99942i 0 −15.9908 0.542709i −16.6139 28.7760i 0 39.9759 + 23.0801i 3.25547 63.9171i 0 116.215 64.4935i
19.12 0.485844 + 3.97038i 0 −15.5279 + 3.85798i −1.01545 1.75881i 0 −20.0352 11.5673i −22.8618 59.7774i 0 6.48981 4.88624i
19.13 0.701741 3.93796i 0 −15.0151 5.52686i −14.3046 24.7763i 0 −22.2124 12.8243i −32.3013 + 55.2506i 0 −107.606 + 38.9445i
19.14 1.85603 3.54333i 0 −9.11034 13.1530i 14.8847 + 25.7811i 0 −51.8739 29.9494i −63.5145 + 7.86857i 0 118.977 4.89106i
19.15 2.14060 3.37903i 0 −6.83568 14.4663i 14.8847 + 25.7811i 0 51.8739 + 29.9494i −63.5145 7.86857i 0 118.977 + 4.89106i
19.16 2.23562 + 3.31693i 0 −6.00404 + 14.8308i 23.3466 + 40.4374i 0 −52.4363 30.2741i −62.6153 + 13.2409i 0 −81.9341 + 167.841i
19.17 2.73046 + 2.92312i 0 −1.08921 + 15.9629i −19.5394 33.8433i 0 −10.5700 6.10260i −49.6354 + 40.4021i 0 45.5763 149.524i
19.18 3.05951 2.57671i 0 2.72116 15.7669i −14.3046 24.7763i 0 22.2124 + 12.8243i −32.3013 55.2506i 0 −107.606 38.9445i
19.19 3.28826 + 2.27757i 0 5.62536 + 14.9785i 2.83091 + 4.90328i 0 45.1595 + 26.0728i −15.6169 + 62.0654i 0 −1.85878 + 22.5709i
19.20 3.86509 1.03009i 0 13.8778 7.96277i 5.89438 + 10.2094i 0 50.5548 + 29.1878i 45.4367 45.0722i 0 33.2988 + 33.3884i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.22 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
9.c even 3 1 inner
36.f odd 6 1 inner

## Twists

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

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