# Properties

 Label 201.3.o.b Level 201 Weight 3 Character orbit 201.o Analytic conductor 5.477 Analytic rank 0 Dimension 840 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 201.o (of order $$66$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.47685331364$$ Analytic rank: $$0$$ Dimension: $$840$$ Relative dimension: $$42$$ over $$\Q(\zeta_{66})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$840q - 16q^{3} - 126q^{4} - 25q^{6} - 34q^{7} - 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$840q - 16q^{3} - 126q^{4} - 25q^{6} - 34q^{7} - 24q^{9} - 50q^{10} + 168q^{12} - 38q^{13} - 100q^{15} + 86q^{16} - 33q^{18} - 6q^{19} - 118q^{21} + 256q^{22} + 170q^{24} + 384q^{25} - 160q^{27} - 652q^{28} - 40q^{30} + 72q^{31} - 113q^{33} + 10q^{34} - 127q^{36} + 2q^{37} - 51q^{39} - 172q^{40} - 274q^{42} + 50q^{43} - 518q^{45} + 1070q^{46} + 281q^{48} + 132q^{49} - 37q^{51} - 2024q^{52} - 809q^{54} - 1810q^{55} + 546q^{57} - 716q^{58} - 2q^{60} + 410q^{61} + 1371q^{63} - 144q^{64} - 814q^{66} + 460q^{67} - 123q^{69} - 1296q^{70} + 1196q^{72} + 1324q^{73} + 208q^{75} + 1588q^{76} - 118q^{78} + 66q^{79} + 220q^{81} + 2412q^{82} - 2123q^{84} + 50q^{85} - 954q^{87} - 14q^{88} - 504q^{90} - 36q^{91} - 1271q^{93} - 1328q^{94} + 1335q^{96} - 90q^{97} - 16q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 −0.372771 3.90383i −0.867999 2.87169i −11.1732 + 2.15346i 2.04976 + 0.294710i −10.8870 + 4.45900i −4.95140 6.95327i 8.15244 + 27.7647i −7.49316 + 4.98524i 0.386411 8.11176i
17.2 −0.365428 3.82693i 2.09890 + 2.14350i −10.5841 + 2.03993i −7.82250 1.12471i 7.43604 8.81565i 5.16070 + 7.24719i 7.34209 + 25.0049i −0.189206 + 8.99801i −1.44561 + 30.3472i
17.3 −0.339674 3.55723i −2.79427 + 1.09181i −8.61080 + 1.65960i 4.72450 + 0.679280i 4.83296 + 9.56901i 4.33674 + 6.09010i 4.80145 + 16.3522i 6.61591 6.10162i 0.811566 17.0369i
17.4 −0.316674 3.31636i 2.98394 + 0.310028i −6.97027 + 1.34341i 4.38305 + 0.630187i 0.0832287 9.99400i −3.34678 4.69989i 2.90823 + 9.90451i 8.80777 + 1.85021i 0.701931 14.7353i
17.5 −0.315296 3.30193i −0.380346 + 2.97579i −6.87561 + 1.32517i 1.00410 + 0.144367i 9.94578 + 0.317620i −4.51680 6.34296i 2.80549 + 9.55461i −8.71067 2.26366i 0.160103 3.36097i
17.6 −0.300568 3.14769i −2.02846 2.21028i −5.88990 + 1.13519i −3.30027 0.474508i −6.34760 + 7.04931i 7.22926 + 10.1521i 1.78017 + 6.06270i −0.770704 + 8.96694i −0.501647 + 10.5309i
17.7 −0.287750 3.01346i 2.06232 2.17873i −5.07042 + 0.977244i −5.16381 0.742444i −7.15894 5.58777i −0.342776 0.481361i 0.992495 + 3.38013i −0.493707 8.98645i −0.751435 + 15.7746i
17.8 −0.258098 2.70292i 1.18857 2.75451i −3.31147 + 0.638234i 8.17030 + 1.17471i −7.75199 2.50167i 5.28772 + 7.42556i −0.480079 1.63500i −6.17462 6.54783i 1.06642 22.3869i
17.9 −0.253296 2.65263i −2.66207 1.38324i −3.04458 + 0.586794i −6.58359 0.946578i −2.99494 + 7.41187i −3.47491 4.87983i −0.675202 2.29953i 5.17328 + 7.36459i −0.843327 + 17.7036i
17.10 −0.238781 2.50063i 2.20075 + 2.03880i −2.26841 + 0.437200i 4.46322 + 0.641713i 4.57279 5.99007i 5.28893 + 7.42726i −1.19593 4.07295i 0.686561 + 8.97377i 0.538955 11.3141i
17.11 −0.223275 2.33824i −1.10838 + 2.78774i −1.48980 + 0.287135i −3.05677 0.439497i 6.76587 + 1.96923i 2.18219 + 3.06445i −1.64299 5.59553i −6.54298 6.17976i −0.345149 + 7.24558i
17.12 −0.217915 2.28210i −2.97742 0.367411i −1.23280 + 0.237603i 5.68528 + 0.817420i −0.189647 + 6.87484i −3.86064 5.42151i −1.77259 6.03689i 8.73002 + 2.18787i 0.626532 13.1525i
17.13 −0.155968 1.63338i 2.15774 + 2.08427i 1.28412 0.247495i −9.24803 1.32967i 3.06785 3.84948i −6.62008 9.29660i −2.45361 8.35622i 0.311677 + 8.99460i −0.729443 + 15.3129i
17.14 −0.152393 1.59594i 0.135358 2.99694i 1.40393 0.270585i 0.199274 + 0.0286513i −4.80356 + 0.240692i −2.27333 3.19244i −2.45247 8.35236i −8.96336 0.811320i 0.0153576 0.322395i
17.15 −0.132486 1.38746i −2.67554 + 1.35701i 2.02022 0.389366i −4.99218 0.717767i 2.23728 + 3.53242i 1.84316 + 2.58836i −2.37856 8.10065i 5.31703 7.26149i −0.334478 + 7.02155i
17.16 −0.127499 1.33522i 1.42593 + 2.63945i 2.16114 0.416527i 5.12044 + 0.736208i 3.34246 2.24047i −4.38585 6.15907i −2.34325 7.98037i −4.93343 + 7.52737i 0.330155 6.93081i
17.17 −0.105003 1.09964i 2.99034 0.240568i 2.72952 0.526072i −2.36755 0.340403i −0.578535 3.26305i 3.93452 + 5.52527i −2.10996 7.18587i 8.88425 1.43876i −0.125721 + 2.63921i
17.18 −0.0816192 0.854754i 2.61588 1.46873i 3.20377 0.617476i 6.65997 + 0.957559i −1.46891 2.11606i −5.73553 8.05442i −1.75691 5.98349i 4.68569 7.68403i 0.274897 5.77079i
17.19 −0.0720449 0.754488i −0.695849 2.91818i 3.36365 0.648291i −3.62054 0.520555i −2.15160 + 0.735250i −1.07696 1.51238i −1.58559 5.40001i −8.03159 + 4.06123i −0.131911 + 2.76916i
17.20 −0.0661166 0.692405i −0.991594 + 2.83139i 3.45266 0.665446i 8.44666 + 1.21445i 2.02602 + 0.499383i 3.65289 + 5.12977i −1.47288 5.01616i −7.03348 5.61517i 0.282424 5.92880i
See next 80 embeddings (of 840 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 194.42 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
67.g even 33 1 inner
201.o odd 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.3.o.b 840
3.b odd 2 1 inner 201.3.o.b 840
67.g even 33 1 inner 201.3.o.b 840
201.o odd 66 1 inner 201.3.o.b 840

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.3.o.b 840 1.a even 1 1 trivial
201.3.o.b 840 3.b odd 2 1 inner
201.3.o.b 840 67.g even 33 1 inner
201.3.o.b 840 201.o odd 66 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{840} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database