# Properties

 Label 20.10.e.b Level 20 Weight 10 Character orbit 20.e Analytic conductor 10.301 Analytic rank 0 Dimension 48 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 20.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3007167233$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 30q^{2} - 1440q^{5} + 6152q^{6} - 17100q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 30q^{2} - 1440q^{5} + 6152q^{6} - 17100q^{8} + 72790q^{10} - 155360q^{12} + 198960q^{13} + 527288q^{16} + 452880q^{17} - 49230q^{18} - 1562100q^{20} - 634176q^{21} + 3176920q^{22} + 6148560q^{25} - 20346636q^{26} + 12731840q^{28} + 15498680q^{30} - 37481400q^{32} - 2488000q^{33} + 24617980q^{36} + 23039440q^{37} - 26115120q^{38} - 48114900q^{40} - 18757824q^{41} + 82830200q^{42} + 78566320q^{45} - 92534488q^{46} + 46448320q^{48} - 28839030q^{50} + 234908620q^{52} - 171595440q^{53} - 177356448q^{56} + 62365440q^{57} + 150783080q^{58} + 166923520q^{60} - 252113984q^{61} - 292810200q^{62} - 72509520q^{65} + 614341200q^{66} - 782859900q^{68} - 203227600q^{70} + 1252703940q^{72} - 20961680q^{73} - 1061841600q^{76} - 277316160q^{77} + 49362600q^{78} + 586589520q^{80} - 933107568q^{81} - 349461720q^{82} + 3229742960q^{85} - 874588728q^{86} + 865939360q^{88} + 1476868470q^{90} - 1106673600q^{92} - 2554477120q^{93} + 1870685312q^{96} + 5873785520q^{97} - 5176905450q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −22.6111 + 0.858964i −163.414 + 163.414i 510.524 38.8443i 1390.97 + 135.329i 3554.60 3835.33i 7883.41 + 7883.41i −11510.2 + 1316.83i 33725.0i −31567.7 1865.14i
3.2 −22.1492 4.62751i 107.646 107.646i 469.172 + 204.991i −103.512 + 1393.70i −2882.39 + 1886.13i −1033.09 1033.09i −9443.19 6711.48i 3492.17i 8742.08 30390.4i
3.3 −21.5582 + 6.87337i 110.359 110.359i 417.514 296.355i 404.317 1337.78i −1620.61 + 3137.69i 1215.70 + 1215.70i −6963.90 + 9258.61i 4675.33i 478.682 + 31619.2i
3.4 −20.8440 + 8.80488i −99.4230 + 99.4230i 356.948 367.059i −1395.00 84.3181i 1196.97 2947.78i −5882.34 5882.34i −4208.33 + 10793.9i 86.8475i 29819.8 10525.2i
3.5 −19.1048 12.1247i −48.3412 + 48.3412i 217.985 + 463.278i −1393.61 104.823i 1509.67 337.427i 5614.42 + 5614.42i 1452.53 11493.8i 15009.3i 25353.6 + 18899.6i
3.6 −18.4240 13.1360i −30.0552 + 30.0552i 166.891 + 484.037i 965.172 1010.73i 948.542 158.933i −6801.74 6801.74i 3283.50 11110.2i 17876.4i −31059.3 + 5943.18i
3.7 −9.96719 20.3139i 181.251 181.251i −313.310 + 404.945i −997.734 978.597i −5488.49 1875.36i −1096.38 1096.38i 11348.8 + 2328.39i 46021.0i −9934.52 + 30021.7i
3.8 −8.80488 + 20.8440i 99.4230 99.4230i −356.948 367.059i −1395.00 84.3181i 1196.97 + 2947.78i 5882.34 + 5882.34i 10793.9 4208.33i 86.8475i 14040.3 28335.0i
3.9 −7.98663 21.1711i −175.077 + 175.077i −384.428 + 338.171i −373.256 + 1346.78i 5104.83 + 2308.29i −4888.23 4888.23i 10229.7 + 5437.90i 41620.6i 31493.7 2853.97i
3.10 −6.87337 + 21.5582i −110.359 + 110.359i −417.514 296.355i 404.317 1337.78i −1620.61 3137.69i −1215.70 1215.70i 9258.61 6963.90i 4675.33i 26061.1 + 17911.4i
3.11 −6.66069 21.6249i 40.3330 40.3330i −423.270 + 288.073i 1192.73 + 728.376i −1140.84 603.551i 3888.44 + 3888.44i 9048.82 + 7234.41i 16429.5i 7806.66 30644.0i
3.12 −0.858964 + 22.6111i 163.414 163.414i −510.524 38.8443i 1390.97 + 135.329i 3554.60 + 3835.33i −7883.41 7883.41i 1316.83 11510.2i 33725.0i −4254.74 + 31335.2i
3.13 3.74232 22.3158i −68.1019 + 68.1019i −483.990 167.026i −497.509 1305.99i 1264.89 + 1774.61i 2745.50 + 2745.50i −5538.56 + 10175.6i 10407.3i −31006.1 + 6214.87i
3.14 4.62751 + 22.1492i −107.646 + 107.646i −469.172 + 204.991i −103.512 + 1393.70i −2882.39 1886.13i 1033.09 + 1033.09i −6711.48 9443.19i 3492.17i −31348.4 + 4156.66i
3.15 9.06759 20.7311i 67.5601 67.5601i −347.558 375.962i −928.159 + 1044.82i −787.988 2013.20i −5705.62 5705.62i −10945.6 + 3796.18i 10554.3i 13244.1 + 28715.8i
3.16 12.1247 + 19.1048i 48.3412 48.3412i −217.985 + 463.278i −1393.61 104.823i 1509.67 + 337.427i −5614.42 5614.42i −11493.8 + 1452.53i 15009.3i −14894.4 27895.5i
3.17 13.1360 + 18.4240i 30.0552 30.0552i −166.891 + 484.037i 965.172 1010.73i 948.542 + 158.933i 6801.74 + 6801.74i −11110.2 + 3283.50i 17876.4i 31300.2 + 4505.46i
3.18 15.5607 16.4276i 139.769 139.769i −27.7316 511.248i 1375.58 246.768i −121.170 4470.96i 3052.81 + 3052.81i −8830.10 7499.80i 19387.7i 17351.2 26437.4i
3.19 16.4276 15.5607i −139.769 + 139.769i 27.7316 511.248i 1375.58 246.768i −121.170 + 4470.96i −3052.81 3052.81i −7499.80 8830.10i 19387.7i 18757.6 25458.8i
3.20 20.3139 + 9.96719i −181.251 + 181.251i 313.310 + 404.945i −997.734 978.597i −5488.49 + 1875.36i 1096.38 + 1096.38i 2328.39 + 11348.8i 46021.0i −10514.0 29823.7i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.24 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
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.10.e.b 48
4.b odd 2 1 inner 20.10.e.b 48
5.b even 2 1 100.10.e.e 48
5.c odd 4 1 inner 20.10.e.b 48
5.c odd 4 1 100.10.e.e 48
20.d odd 2 1 100.10.e.e 48
20.e even 4 1 inner 20.10.e.b 48
20.e even 4 1 100.10.e.e 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.10.e.b 48 1.a even 1 1 trivial
20.10.e.b 48 4.b odd 2 1 inner
20.10.e.b 48 5.c odd 4 1 inner
20.10.e.b 48 20.e even 4 1 inner
100.10.e.e 48 5.b even 2 1
100.10.e.e 48 5.c odd 4 1
100.10.e.e 48 20.d odd 2 1
100.10.e.e 48 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{48} + \cdots$$ acting on $$S_{10}^{\mathrm{new}}(20, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database