# Properties

 Label 105.4.s.a Level $105$ Weight $4$ Character orbit 105.s Analytic conductor $6.195$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 105.s (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.19520055060$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 2q^{3} + 64q^{4} - 80q^{5} - 28q^{6} + 46q^{7} + 100q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 2q^{3} + 64q^{4} - 80q^{5} - 28q^{6} + 46q^{7} + 100q^{9} + 36q^{11} + 246q^{12} + 18q^{14} + 20q^{15} - 376q^{16} - 72q^{17} - 442q^{18} - 198q^{19} - 640q^{20} - 218q^{21} + 204q^{22} + 72q^{23} - 50q^{24} - 400q^{25} - 312q^{26} + 508q^{27} + 350q^{28} + 40q^{30} + 510q^{31} + 810q^{32} + 290q^{33} - 70q^{35} - 612q^{36} - 658q^{37} - 192q^{38} - 648q^{39} - 1404q^{41} + 1892q^{42} + 332q^{43} + 2034q^{44} - 490q^{45} - 468q^{46} + 408q^{47} + 2810q^{48} + 980q^{49} - 888q^{51} + 3378q^{52} + 1152q^{53} + 2714q^{54} - 3354q^{56} - 816q^{57} - 1080q^{58} - 48q^{59} - 420q^{60} - 1662q^{61} - 2076q^{62} + 874q^{63} - 1952q^{64} + 870q^{65} - 1892q^{66} - 1298q^{67} + 1182q^{68} + 2450q^{69} - 450q^{70} - 2708q^{72} + 378q^{73} + 2898q^{74} - 50q^{75} - 3528q^{77} - 1896q^{78} - 326q^{79} - 1880q^{80} - 3308q^{81} - 2916q^{82} - 1536q^{83} + 1380q^{84} + 720q^{85} + 5202q^{86} - 1090q^{87} + 1668q^{88} - 1590q^{89} + 910q^{90} + 2082q^{91} - 4950q^{93} - 1152q^{94} + 990q^{95} + 7416q^{96} - 7830q^{98} + 3128q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1 −4.57212 2.63971i 4.83065 1.91438i 9.93617 + 17.2100i −2.50000 + 4.33013i −27.1397 3.99877i −17.5731 + 5.84690i 62.6791i 19.6703 18.4954i 22.8606 13.1986i
26.2 −4.56345 2.63471i −4.97118 1.51242i 9.88336 + 17.1185i −2.50000 + 4.33013i 18.7009 + 19.9994i 12.9395 13.2503i 62.0037i 22.4252 + 15.0370i 22.8172 13.1735i
26.3 −3.31734 1.91526i 4.42611 + 2.72205i 3.33648 + 5.77895i −2.50000 + 4.33013i −9.46943 17.5071i 8.94735 16.2156i 5.08329i 12.1808 + 24.0962i 16.5867 9.57632i
26.4 −3.06378 1.76887i −0.433556 5.17803i 2.25781 + 3.91065i −2.50000 + 4.33013i −7.83096 + 16.6312i 5.77785 + 17.5959i 12.3268i −26.6241 + 4.48993i 15.3189 8.84436i
26.5 −2.51429 1.45163i −2.26232 + 4.67781i 0.214449 + 0.371437i −2.50000 + 4.33013i 12.4786 8.47733i 17.9959 + 4.37591i 21.9809i −16.7638 21.1654i 12.5715 7.25814i
26.6 −1.35336 0.781362i −5.18197 + 0.383668i −2.77895 4.81328i −2.50000 + 4.33013i 7.31285 + 3.52975i −17.7664 + 5.23007i 21.1872i 26.7056 3.97631i 6.76679 3.90681i
26.7 −1.20225 0.694119i 3.08234 4.18320i −3.03640 5.25919i −2.50000 + 4.33013i −6.60938 + 2.88975i −11.1211 14.8095i 19.5364i −7.99841 25.7881i 6.01125 3.47060i
26.8 −0.133140 0.0768685i 2.47104 + 4.57099i −3.98818 6.90773i −2.50000 + 4.33013i 0.0223714 0.798528i −15.0457 + 10.7995i 2.45616i −14.7880 + 22.5902i 0.665701 0.384343i
26.9 0.155491 + 0.0897728i 5.17065 + 0.514148i −3.98388 6.90029i −2.50000 + 4.33013i 0.757834 + 0.544130i 18.1774 + 3.54688i 2.86694i 26.4713 + 5.31696i −0.777456 + 0.448864i
26.10 0.815639 + 0.470909i −4.77022 2.06035i −3.55649 6.16002i −2.50000 + 4.33013i −2.92054 3.92684i 13.9941 + 12.1312i 14.2337i 18.5099 + 19.6566i −4.07820 + 2.35455i
26.11 1.48940 + 0.859905i −4.11977 + 3.16663i −2.52113 4.36672i −2.50000 + 4.33013i −8.85898 + 1.17376i 2.28179 18.3792i 22.4302i 6.94494 26.0915i −7.44700 + 4.29953i
26.12 2.43022 + 1.40309i −2.49879 4.55588i −0.0626967 0.108594i −2.50000 + 4.33013i 0.319698 14.5778i −12.1846 13.9476i 22.8013i −14.5121 + 22.7684i −12.1511 + 7.01543i
26.13 3.53626 + 2.04166i −2.57442 + 4.51357i 4.33676 + 7.51148i −2.50000 + 4.33013i −18.3190 + 10.7051i 0.627588 + 18.5096i 2.75017i −13.7447 23.2397i −17.6813 + 10.2083i
26.14 3.64936 + 2.10696i 4.80130 1.98684i 4.87855 + 8.44990i −2.50000 + 4.33013i 21.7079 + 2.86543i 16.7999 7.79504i 7.40430i 19.1049 19.0788i −18.2468 + 10.5348i
26.15 3.95258 + 2.28202i 3.88244 + 3.45350i 6.41528 + 11.1116i −2.50000 + 4.33013i 7.46472 + 22.5101i −15.3726 10.3287i 22.0469i 3.14674 + 26.8160i −19.7629 + 11.4101i
26.16 4.69077 + 2.70822i −2.85230 4.34332i 10.6689 + 18.4790i −2.50000 + 4.33013i −1.61684 28.0981i 14.5222 + 11.4937i 72.2429i −10.7288 + 24.7769i −23.4538 + 13.5411i
101.1 −4.57212 + 2.63971i 4.83065 + 1.91438i 9.93617 17.2100i −2.50000 4.33013i −27.1397 + 3.99877i −17.5731 5.84690i 62.6791i 19.6703 + 18.4954i 22.8606 + 13.1986i
101.2 −4.56345 + 2.63471i −4.97118 + 1.51242i 9.88336 17.1185i −2.50000 4.33013i 18.7009 19.9994i 12.9395 + 13.2503i 62.0037i 22.4252 15.0370i 22.8172 + 13.1735i
101.3 −3.31734 + 1.91526i 4.42611 2.72205i 3.33648 5.77895i −2.50000 4.33013i −9.46943 + 17.5071i 8.94735 + 16.2156i 5.08329i 12.1808 24.0962i 16.5867 + 9.57632i
101.4 −3.06378 + 1.76887i −0.433556 + 5.17803i 2.25781 3.91065i −2.50000 4.33013i −7.83096 16.6312i 5.77785 17.5959i 12.3268i −26.6241 4.48993i 15.3189 + 8.84436i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.s.a 32
3.b odd 2 1 105.4.s.b yes 32
7.d odd 6 1 105.4.s.b yes 32
21.g even 6 1 inner 105.4.s.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.s.a 32 1.a even 1 1 trivial
105.4.s.a 32 21.g even 6 1 inner
105.4.s.b yes 32 3.b odd 2 1
105.4.s.b yes 32 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$12\!\cdots\!44$$$$T_{2}^{14} - 335728806336 T_{2}^{13} +$$$$58\!\cdots\!84$$$$T_{2}^{12} +$$$$19\!\cdots\!28$$$$T_{2}^{11} -$$$$16\!\cdots\!24$$$$T_{2}^{10} -$$$$61\!\cdots\!80$$$$T_{2}^{9} +$$$$32\!\cdots\!68$$$$T_{2}^{8} +$$$$12\!\cdots\!88$$$$T_{2}^{7} -$$$$31\!\cdots\!72$$$$T_{2}^{6} -$$$$84\!\cdots\!40$$$$T_{2}^{5} +$$$$22\!\cdots\!40$$$$T_{2}^{4} - 714941199360 T_{2}^{3} - 602278336512 T_{2}^{2} + 18831605760 T_{2} + 16057958400$$">$$T_{2}^{32} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(105, [\chi])$$.