# Properties

 Label 105.4.j.a Level $105$ Weight $4$ Character orbit 105.j Analytic conductor $6.195$ Analytic rank $0$ Dimension $72$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 105.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.19520055060$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$36$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72q + 8q^{3} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q + 8q^{3} + 144q^{10} - 128q^{12} - 144q^{13} - 16q^{15} - 1608q^{16} + 460q^{18} + 112q^{21} + 576q^{22} + 504q^{25} - 592q^{27} - 580q^{30} - 960q^{31} - 56q^{33} + 928q^{36} + 2088q^{37} + 144q^{40} - 140q^{42} + 240q^{43} - 880q^{45} + 528q^{46} + 3208q^{48} + 1960q^{51} + 240q^{52} + 1200q^{55} - 1112q^{57} + 840q^{58} - 1528q^{60} - 1824q^{61} - 1064q^{63} - 1408q^{66} - 2832q^{67} - 1008q^{70} - 296q^{72} + 1776q^{73} + 5280q^{75} + 7296q^{76} - 4500q^{78} - 4064q^{81} + 1680q^{82} - 10536q^{85} - 392q^{87} - 5352q^{88} - 5664q^{90} + 1008q^{91} - 5488q^{93} - 288q^{96} - 7872q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1 −3.89881 3.89881i −3.58895 3.75759i 22.4014i −10.6015 3.55092i −0.657498 + 28.6427i −4.94975 + 4.94975i 56.1481 56.1481i −1.23893 + 26.9716i 27.4887 + 55.1774i
8.2 −3.77908 3.77908i 3.12291 4.15300i 20.5629i 9.30912 + 6.19195i −27.4963 + 3.89279i 4.94975 4.94975i 47.4763 47.4763i −7.49484 25.9389i −11.7801 58.5798i
8.3 −3.62179 3.62179i −2.33204 + 4.64345i 18.2347i 5.19694 9.89908i 25.2637 8.37145i 4.94975 4.94975i 37.0681 37.0681i −16.1232 21.6574i −54.6746 + 17.0302i
8.4 −3.36864 3.36864i 0.501648 + 5.17188i 14.6955i −2.34376 + 10.9319i 15.7323 19.1121i −4.94975 + 4.94975i 22.5546 22.5546i −26.4967 + 5.18893i 44.7210 28.9304i
8.5 −3.31161 3.31161i 5.02457 + 1.32428i 13.9335i −11.0620 + 1.62271i −12.2539 21.0249i 4.94975 4.94975i 19.6493 19.6493i 23.4926 + 13.3078i 42.0066 + 31.2590i
8.6 −2.97398 2.97398i 4.37077 + 2.81005i 9.68906i 11.0280 1.83931i −4.64156 21.3556i −4.94975 + 4.94975i 5.02323 5.02323i 11.2073 + 24.5641i −38.2671 27.3270i
8.7 −2.82935 2.82935i −5.15708 + 0.636028i 8.01043i −3.73979 + 10.5363i 16.3907 + 12.7916i 4.94975 4.94975i 0.0295179 0.0295179i 26.1909 6.56009i 40.3921 19.2297i
8.8 −2.78584 2.78584i 4.19015 3.07289i 7.52176i −2.19464 10.9628i −20.2336 3.11248i −4.94975 + 4.94975i −1.33231 + 1.33231i 8.11466 25.7517i −24.4267 + 36.6545i
8.9 −2.42501 2.42501i −0.644450 5.15603i 3.76131i 4.31658 + 10.3134i −10.9406 + 14.0662i −4.94975 + 4.94975i −10.2789 + 10.2789i −26.1694 + 6.64561i 14.5424 35.4779i
8.10 −2.24956 2.24956i −3.89791 + 3.43603i 2.12102i −8.51062 7.25047i 16.4981 + 1.03901i −4.94975 + 4.94975i −13.2251 + 13.2251i 3.38735 26.7867i 2.83478 + 35.4555i
8.11 −1.92888 1.92888i 0.415036 5.17955i 0.558817i −11.1325 + 1.03353i −10.7913 + 9.19019i 4.94975 4.94975i −16.5090 + 16.5090i −26.6555 4.29940i 23.4668 + 19.4797i
8.12 −1.59773 1.59773i 3.12266 + 4.15319i 2.89452i −1.68350 11.0529i 1.64652 11.6248i 4.94975 4.94975i −17.4065 + 17.4065i −7.49804 + 25.9380i −14.9697 + 20.3493i
8.13 −1.39449 1.39449i −0.298666 + 5.18756i 4.11082i 8.15594 + 7.64726i 7.65046 6.81749i 4.94975 4.94975i −16.8884 + 16.8884i −26.8216 3.09870i −0.709344 22.0373i
8.14 −0.948593 0.948593i −4.45711 + 2.67099i 6.20034i 11.1478 + 0.851917i 6.76166 + 1.69430i −4.94975 + 4.94975i −13.4703 + 13.4703i 12.7317 23.8098i −9.76663 11.3829i
8.15 −0.777009 0.777009i 4.13454 3.14732i 6.79251i 10.0976 4.79984i −5.65807 0.767075i 4.94975 4.94975i −11.4939 + 11.4939i 7.18876 26.0254i −11.5754 4.11641i
8.16 −0.694283 0.694283i 2.36844 + 4.62499i 7.03594i −10.8783 + 2.58102i 1.56668 4.85541i −4.94975 + 4.94975i −10.4392 + 10.4392i −15.7810 + 21.9080i 9.34460 + 5.76069i
8.17 −0.554582 0.554582i −4.87809 1.79005i 7.38488i −7.42269 8.36084i 1.71257 + 3.69803i 4.94975 4.94975i −8.53218 + 8.53218i 20.5915 + 17.4640i −0.520282 + 8.75326i
8.18 −0.152180 0.152180i −1.39229 5.00615i 7.95368i 3.10522 10.7405i −0.549957 + 0.973712i −4.94975 + 4.94975i −2.42783 + 2.42783i −23.1231 + 13.9400i −2.10703 + 1.16193i
8.19 0.152180 + 0.152180i −5.00615 1.39229i 7.95368i −3.10522 + 10.7405i −0.549957 0.973712i −4.94975 + 4.94975i 2.42783 2.42783i 23.1231 + 13.9400i −2.10703 + 1.16193i
8.20 0.554582 + 0.554582i −1.79005 4.87809i 7.38488i 7.42269 + 8.36084i 1.71257 3.69803i 4.94975 4.94975i 8.53218 8.53218i −20.5915 + 17.4640i −0.520282 + 8.75326i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 92.36 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
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.j.a 72
3.b odd 2 1 inner 105.4.j.a 72
5.c odd 4 1 inner 105.4.j.a 72
15.e even 4 1 inner 105.4.j.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.j.a 72 1.a even 1 1 trivial
105.4.j.a 72 3.b odd 2 1 inner
105.4.j.a 72 5.c odd 4 1 inner
105.4.j.a 72 15.e even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(105, [\chi])$$.