# Properties

 Label 36.5.f.a Level 36 Weight 5 Character orbit 36.f Analytic conductor 3.721 Analytic rank 0 Dimension 44 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 36.f (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.72131867102$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44q - q^{2} - q^{4} - 2q^{5} + 15q^{6} + 122q^{8} - 60q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q - q^{2} - q^{4} - 2q^{5} + 15q^{6} + 122q^{8} - 60q^{9} + 28q^{10} - 228q^{12} - 2q^{13} + 252q^{14} - q^{16} - 56q^{17} + 72q^{18} - 140q^{20} + 138q^{21} - 33q^{22} - 951q^{24} - 1752q^{25} + 1096q^{26} - 516q^{28} + 526q^{29} - 1980q^{30} - 121q^{32} + 2994q^{33} + 385q^{34} - 1005q^{36} - 8q^{37} + 1395q^{38} - 2276q^{40} - 2762q^{41} + 3330q^{42} + 6714q^{44} + 4110q^{45} + 3576q^{46} + 2163q^{48} + 3428q^{49} + 6375q^{50} + 1438q^{52} - 10088q^{53} - 4983q^{54} - 7506q^{56} + 1752q^{57} - 4064q^{58} - 16392q^{60} - 2q^{61} - 18324q^{62} + 9026q^{64} - 2014q^{65} - 17358q^{66} - 11405q^{68} + 3354q^{69} + 3666q^{70} + 4083q^{72} - 3416q^{73} + 14620q^{74} + 1581q^{76} - 3942q^{77} + 34566q^{78} + 45520q^{80} - 1164q^{81} - 8486q^{82} + 51078q^{84} - 1252q^{85} + 22113q^{86} + 1995q^{88} + 13048q^{89} - 4692q^{90} - 30294q^{92} + 12090q^{93} + 7524q^{94} - 76164q^{96} + 5638q^{97} - 92938q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −3.96529 + 0.525827i −4.76702 + 7.63384i 15.4470 4.17011i 11.0746 + 19.1817i 14.8885 32.7770i −82.7885 47.7980i −59.0591 + 24.6582i −35.5511 72.7813i −54.0000 70.2376i
7.2 −3.93519 0.717143i 2.22096 8.72166i 14.9714 + 5.64418i −5.51579 9.55363i −14.9946 + 32.7286i −10.3188 5.95759i −54.8676 32.9476i −71.1347 38.7409i 14.8544 + 41.5509i
7.3 −3.86509 + 1.03009i 7.79916 + 4.49144i 13.8778 7.96277i −5.89438 10.2094i −34.7711 9.32599i 50.5548 + 29.1878i −45.4367 + 45.0722i 40.6539 + 70.0590i 33.2988 + 33.3884i
7.4 −3.28826 2.27757i −8.99820 + 0.179896i 5.62536 + 14.9785i −2.83091 4.90328i 29.9982 + 19.9025i 45.1595 + 26.0728i 15.6169 62.0654i 80.9353 3.23749i −1.85878 + 22.5709i
7.5 −3.05951 + 2.57671i −5.87916 6.81436i 2.72116 15.7669i 14.3046 + 24.7763i 35.5459 + 5.69969i 22.2124 + 12.8243i 32.3013 + 55.2506i −11.8710 + 80.1254i −107.606 38.9445i
7.6 −2.73046 2.92312i 8.98168 0.573987i −1.08921 + 15.9629i 19.5394 + 33.8433i −26.2019 24.6872i −10.5700 6.10260i 49.6354 40.4021i 80.3411 10.3107i 45.5763 149.524i
7.7 −2.23562 3.31693i 3.18543 + 8.41742i −6.00404 + 14.8308i −23.3466 40.4374i 20.7986 29.3840i −52.4363 30.2741i 62.6153 13.2409i −60.7060 + 53.6263i −81.9341 + 167.841i
7.8 −2.14060 + 3.37903i −6.53699 + 6.18609i −6.83568 14.4663i −14.8847 25.7811i −6.90992 35.3306i 51.8739 + 29.9494i 63.5145 + 7.86857i 4.46449 80.8769i 118.977 + 4.89106i
7.9 −1.85603 + 3.54333i 6.53699 6.18609i −9.11034 13.1530i −14.8847 25.7811i 9.78653 + 34.6442i −51.8739 29.9494i 63.5145 7.86857i 4.46449 80.8769i 118.977 4.89106i
7.10 −0.701741 + 3.93796i 5.87916 + 6.81436i −15.0151 5.52686i 14.3046 + 24.7763i −30.9603 + 18.3700i −22.2124 12.8243i 32.3013 55.2506i −11.8710 + 80.1254i −107.606 + 38.9445i
7.11 −0.485844 3.97038i −4.25251 7.93197i −15.5279 + 3.85798i 1.01545 + 1.75881i −29.4269 + 20.7378i −20.0352 11.5673i 22.8618 + 59.7774i −44.8324 + 67.4615i 6.48981 4.88624i
7.12 0.0678484 3.99942i −2.72970 + 8.57606i −15.9908 0.542709i 16.6139 + 28.7760i 34.1141 + 11.4991i 39.9759 + 23.0801i −3.25547 + 63.9171i −66.0975 46.8201i 116.215 64.4935i
7.13 1.04046 + 3.86231i −7.79916 4.49144i −13.8349 + 8.03717i −5.89438 10.2094i 9.23260 34.7960i −50.5548 29.1878i −45.4367 45.0722i 40.6539 + 70.0590i 33.2988 33.3884i
7.14 1.29332 3.78514i 8.65442 2.47004i −12.6546 9.79083i −10.5756 18.3175i 1.84350 35.9528i 38.6407 + 22.3092i −53.4262 + 35.2369i 68.7978 42.7535i −83.0119 + 16.3397i
7.15 1.52726 + 3.69695i 4.76702 7.63384i −11.3349 + 11.2924i 11.0746 + 19.1817i 35.5025 + 5.96455i 82.7885 + 47.7980i −59.0591 24.6582i −35.5511 72.7813i −54.0000 + 70.2376i
7.16 2.58866 + 3.04940i −2.22096 + 8.72166i −2.59770 + 15.7877i −5.51579 9.55363i −32.3451 + 15.8048i 10.3188 + 5.95759i −54.8676 + 32.9476i −71.1347 38.7409i 14.8544 41.5509i
7.17 2.63137 3.01262i −8.65442 + 2.47004i −2.15179 15.8546i −10.5756 18.3175i −15.3317 + 32.5721i −38.6407 22.3092i −53.4262 35.2369i 68.7978 42.7535i −83.0119 16.3397i
7.18 3.42968 2.05847i 2.72970 8.57606i 7.52540 14.1198i 16.6139 + 28.7760i −8.29157 35.0321i −39.9759 23.0801i −3.25547 63.9171i −66.0975 46.8201i 116.215 + 64.4935i
7.19 3.61656 + 1.70894i 8.99820 0.179896i 10.1591 + 12.3610i −2.83091 4.90328i 32.8500 + 14.7267i −45.1595 26.0728i 15.6169 + 62.0654i 80.9353 3.23749i −1.85878 22.5709i
7.20 3.68138 1.56444i 4.25251 + 7.93197i 11.1051 11.5186i 1.01545 + 1.75881i 28.0642 + 22.5478i 20.0352 + 11.5673i 22.8618 59.7774i −44.8324 + 67.4615i 6.48981 + 4.88624i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.22 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 36.5.f.a 44
3.b odd 2 1 108.5.f.a 44
4.b odd 2 1 inner 36.5.f.a 44
9.c even 3 1 inner 36.5.f.a 44
9.c even 3 1 324.5.d.f 22
9.d odd 6 1 108.5.f.a 44
9.d odd 6 1 324.5.d.e 22
12.b even 2 1 108.5.f.a 44
36.f odd 6 1 inner 36.5.f.a 44
36.f odd 6 1 324.5.d.f 22
36.h even 6 1 108.5.f.a 44
36.h even 6 1 324.5.d.e 22

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(36, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database