# Properties

 Label 201.4.p.a Level 201 Weight 4 Character orbit 201.p Analytic conductor 11.859 Analytic rank 0 Dimension 1320 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 201.p (of order $$66$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.8593839112$$ Analytic rank: $$0$$ Dimension: $$1320$$ Relative dimension: $$66$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$1320q - 22q^{3} + 214q^{4} + q^{6} + 22q^{7} + 48q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$1320q - 22q^{3} + 214q^{4} + q^{6} + 22q^{7} + 48q^{9} - 26q^{10} - 4q^{12} + 136q^{13} + 166q^{15} + 694q^{16} - 181q^{18} + 32q^{19} + 1004q^{21} + 544q^{22} - 230q^{24} - 2552q^{25} - 22q^{27} + 100q^{28} + 810q^{30} + 532q^{31} + 800q^{33} + 718q^{34} - 243q^{36} + 216q^{37} - 1938q^{39} + 820q^{40} - 22q^{42} + 1672q^{43} + 4488q^{45} - 3182q^{46} + 2547q^{48} - 2360q^{49} + 287q^{51} + 2156q^{52} - 3793q^{54} + 11272q^{55} + 1091q^{57} + 308q^{58} - 56q^{60} - 4544q^{61} + 512q^{63} - 22064q^{64} - 1734q^{67} + 350q^{69} - 5588q^{70} + 10648q^{72} - 7992q^{73} - 8459q^{75} + 4540q^{76} + 4664q^{78} + 1178q^{79} - 2448q^{81} + 21556q^{82} - 1183q^{84} + 1864q^{85} - 7051q^{87} - 13694q^{88} + 1138q^{90} - 6308q^{91} + 9792q^{93} - 7172q^{94} - 5417q^{96} - 1140q^{97} - 3678q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −0.263188 + 5.52500i 2.55934 + 4.52214i −22.4926 2.14778i 12.6391 14.5863i −25.6584 + 12.9502i −13.6688 26.5138i 11.4888 79.9067i −13.8995 + 23.1474i 77.2627 + 73.6698i
2.2 −0.257205 + 5.39940i 0.0986149 + 5.19522i −21.1236 2.01706i −9.22534 + 10.6466i −28.0764 0.803775i 10.8621 + 21.0696i 10.1697 70.7317i −26.9806 + 1.02465i −55.1125 52.5496i
2.3 −0.243816 + 5.11833i 0.0384771 5.19601i −18.1741 1.73542i 1.53871 1.77577i 26.5855 + 1.46381i −8.55363 16.5917i 7.47964 52.0221i −26.9970 0.399854i 8.71382 + 8.30861i
2.4 −0.241447 + 5.06860i −4.87773 + 1.79102i −17.6686 1.68715i −4.20480 + 4.85260i −7.90026 25.1557i −5.52731 10.7215i 7.04027 48.9662i 20.5845 17.4722i −23.5806 22.4841i
2.5 −0.240289 + 5.04428i 5.03306 1.29161i −17.4232 1.66372i −12.5554 + 14.4897i 5.30585 + 25.6985i −10.2423 19.8673i 6.82934 47.4991i 23.6635 13.0015i −70.0733 66.8147i
2.6 −0.239166 + 5.02071i 2.74669 4.41086i −17.1866 1.64112i 3.46556 3.99947i 21.4887 + 14.8453i 6.74721 + 13.0878i 6.62737 46.0944i −11.9113 24.2306i 19.2513 + 18.3561i
2.7 −0.235330 + 4.94018i 5.19611 0.0202274i −16.3863 1.56470i 2.61014 3.01227i −1.12287 + 25.6745i 8.63758 + 16.7546i 5.95520 41.4193i 26.9992 0.210208i 14.2669 + 13.6035i
2.8 −0.234299 + 4.91853i −4.94170 1.60611i −16.1733 1.54436i 9.16856 10.5811i 9.05754 23.9296i 6.12309 + 11.8771i 5.77918 40.1951i 21.8408 + 15.8738i 49.8952 + 47.5750i
2.9 −0.223576 + 4.69343i −2.83186 4.35667i −14.0145 1.33822i −11.6480 + 13.4425i 21.0808 12.3171i 15.7082 + 30.4697i 4.06453 28.2695i −10.9611 + 24.6750i −60.4871 57.6743i
2.10 −0.203341 + 4.26865i −2.80662 + 4.37297i −10.2162 0.975531i 1.38641 1.60001i −18.0960 12.8697i −2.67187 5.18271i 1.37612 9.57115i −11.2458 24.5465i 6.54795 + 6.24346i
2.11 −0.189440 + 3.97683i −0.298544 + 5.18757i −7.81552 0.746291i 7.60791 8.78000i −20.5735 2.16999i 14.2616 + 27.6636i −0.0843859 + 0.586917i −26.8217 3.09743i 33.4753 + 31.9187i
2.12 −0.181829 + 3.81706i 3.18386 + 4.10646i −6.57310 0.627655i −5.58017 + 6.43986i −16.2535 + 11.4063i −3.89368 7.55269i −0.759744 + 5.28414i −6.72605 + 26.1488i −23.5667 22.4708i
2.13 −0.178074 + 3.73823i 5.00732 + 1.38806i −5.97885 0.570910i 6.59041 7.60574i −6.08055 + 18.4713i 2.99850 + 5.81628i −1.06200 + 7.38637i 23.1466 + 13.9009i 27.2584 + 25.9908i
2.14 −0.176877 + 3.71311i −4.49634 2.60440i −5.79212 0.553080i −9.84104 + 11.3572i 10.4657 16.2348i −11.0761 21.4847i −1.15410 + 8.02692i 13.4342 + 23.4206i −40.4297 38.5497i
2.15 −0.168996 + 3.54767i −2.66059 4.46332i −4.59361 0.438637i 8.27468 9.54949i 16.2840 8.68460i −5.63397 10.9284i −1.71122 + 11.9018i −12.8425 + 23.7501i 32.4800 + 30.9696i
2.16 −0.160415 + 3.36752i 1.06419 5.08601i −3.35071 0.319954i −6.11753 + 7.06001i 16.9565 + 4.39956i −0.0381180 0.0739387i −2.22338 + 15.4640i −24.7350 10.8250i −22.7934 21.7335i
2.17 −0.147428 + 3.09489i −3.36537 + 3.95908i −1.59284 0.152098i 11.9344 13.7731i −11.7568 10.9991i −6.80969 13.2090i −2.82203 + 19.6277i −4.34859 26.6475i 40.8667 + 38.9663i
2.18 −0.146703 + 3.07967i 4.58334 2.44805i −1.49905 0.143142i 7.20079 8.31015i 6.86680 + 14.4743i −15.1390 29.3655i −2.84949 + 19.8186i 15.0141 22.4405i 24.5361 + 23.3951i
2.19 −0.134842 + 2.83067i −5.15314 + 0.667213i −0.0307503 0.00293630i −3.07901 + 3.55336i −1.19380 14.6768i 11.0994 + 21.5299i −3.21397 + 22.3537i 26.1097 6.87648i −9.64323 9.19480i
2.20 −0.119268 + 2.50375i 4.20022 3.05911i 1.70924 + 0.163213i −3.68755 + 4.25566i 7.15829 + 10.8812i 5.98781 + 11.6147i −3.46630 + 24.1086i 8.28370 25.6979i −10.2153 9.74027i
See next 80 embeddings (of 1320 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 197.66 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.h odd 66 1 inner
201.p even 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.p.a 1320
3.b odd 2 1 inner 201.4.p.a 1320
67.h odd 66 1 inner 201.4.p.a 1320
201.p even 66 1 inner 201.4.p.a 1320

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.p.a 1320 1.a even 1 1 trivial
201.4.p.a 1320 3.b odd 2 1 inner
201.4.p.a 1320 67.h odd 66 1 inner
201.4.p.a 1320 201.p even 66 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database