# Properties

 Label 105.4.s.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 2q^{3} + 64q^{4} + 80q^{5} + 28q^{6} + 46q^{7} - 98q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 2q^{3} + 64q^{4} + 80q^{5} + 28q^{6} + 46q^{7} - 98q^{9} - 36q^{11} + 84q^{12} - 18q^{14} + 20q^{15} - 376q^{16} + 72q^{17} + 260q^{18} - 198q^{19} + 640q^{20} - 256q^{21} + 204q^{22} - 72q^{23} - 94q^{24} - 400q^{25} + 312q^{26} - 508q^{27} + 350q^{28} + 100q^{30} + 510q^{31} - 810q^{32} + 454q^{33} + 70q^{35} - 612q^{36} - 658q^{37} + 192q^{38} + 576q^{39} + 1404q^{41} - 1790q^{42} + 332q^{43} - 2034q^{44} - 500q^{45} - 468q^{46} - 408q^{47} - 2810q^{48} + 980q^{49} + 2748q^{51} + 3378q^{52} - 1152q^{53} + 3322q^{54} + 3354q^{56} - 816q^{57} - 1080q^{58} + 48q^{59} + 1230q^{60} - 1662q^{61} + 2076q^{62} - 2306q^{63} - 1952q^{64} - 870q^{65} - 3808q^{66} - 1298q^{67} - 1182q^{68} - 2450q^{69} - 450q^{70} + 7678q^{72} + 378q^{73} - 2898q^{74} + 50q^{75} + 3528q^{77} - 1896q^{78} - 326q^{79} + 1880q^{80} + 1774q^{81} - 2916q^{82} + 1536q^{83} - 10680q^{84} + 720q^{85} - 5202q^{86} - 5666q^{87} + 1668q^{88} + 1590q^{89} - 910q^{90} + 2082q^{91} + 4086q^{93} - 1152q^{94} - 990q^{95} + 3996q^{96} + 7830q^{98} + 3128q^{99} + O(q^{100})$$

## 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 −4.69077 2.70822i 2.33527 + 4.64182i 10.6689 + 18.4790i 2.50000 4.33013i 1.61684 28.0981i 14.5222 + 11.4937i 72.2429i −16.0930 + 21.6798i −23.4538 + 13.5411i
26.2 −3.95258 2.28202i −1.04959 5.08904i 6.41528 + 11.1116i 2.50000 4.33013i −7.46472 + 22.5101i −15.3726 10.3287i 22.0469i −24.7967 + 10.6828i −19.7629 + 11.4101i
26.3 −3.64936 2.10696i 4.12131 3.16462i 4.87855 + 8.44990i 2.50000 4.33013i −21.7079 + 2.86543i 16.7999 7.79504i 7.40430i 6.97031 26.0848i −18.2468 + 10.5348i
26.4 −3.53626 2.04166i −5.19608 0.0272697i 4.33676 + 7.51148i 2.50000 4.33013i 18.3190 + 10.7051i 0.627588 + 18.5096i 2.75017i 26.9985 + 0.283391i −17.6813 + 10.2083i
26.5 −2.43022 1.40309i 2.69612 + 4.44195i −0.0626967 0.108594i 2.50000 4.33013i −0.319698 14.5778i −12.1846 13.9476i 22.8013i −12.4619 + 23.9520i −12.1511 + 7.01543i
26.6 −1.48940 0.859905i −4.80226 + 1.98451i −2.52113 4.36672i 2.50000 4.33013i 8.85898 + 1.17376i 2.28179 18.3792i 22.4302i 19.1235 19.0603i −7.44700 + 4.29953i
26.7 −0.815639 0.470909i −0.600797 + 5.16130i −3.55649 6.16002i 2.50000 4.33013i 2.92054 3.92684i 13.9941 + 12.1312i 14.2337i −26.2781 6.20179i −4.07820 + 2.35455i
26.8 −0.155491 0.0897728i 2.14006 4.73499i −3.98388 6.90029i 2.50000 4.33013i −0.757834 + 0.544130i 18.1774 + 3.54688i 2.86694i −17.8403 20.2663i −0.777456 + 0.448864i
26.9 0.133140 + 0.0768685i −2.72308 4.42548i −3.98818 6.90773i 2.50000 4.33013i −0.0223714 0.798528i −15.0457 + 10.7995i 2.45616i −12.1697 + 24.1018i 0.665701 0.384343i
26.10 1.20225 + 0.694119i 5.16393 0.577779i −3.03640 5.25919i 2.50000 4.33013i 6.60938 + 2.88975i −11.1211 14.8095i 19.5364i 26.3323 5.96722i 6.01125 3.47060i
26.11 1.35336 + 0.781362i −2.92325 + 4.29588i −2.77895 4.81328i 2.50000 4.33013i −7.31285 + 3.52975i −17.7664 + 5.23007i 21.1872i −9.90921 25.1159i 6.76679 3.90681i
26.12 2.51429 + 1.45163i −5.18226 0.379674i 0.214449 + 0.371437i 2.50000 4.33013i −12.4786 8.47733i 17.9959 + 4.37591i 21.9809i 26.7117 + 3.93514i 12.5715 7.25814i
26.13 3.06378 + 1.76887i 4.26753 + 2.96449i 2.25781 + 3.91065i 2.50000 4.33013i 7.83096 + 16.6312i 5.77785 + 17.5959i 12.3268i 9.42363 + 25.3021i 15.3189 8.84436i
26.14 3.31734 + 1.91526i −0.144315 5.19415i 3.33648 + 5.77895i 2.50000 4.33013i 9.46943 17.5071i 8.94735 16.2156i 5.08329i −26.9583 + 1.49918i 16.5867 9.57632i
26.15 4.56345 + 2.63471i −1.17580 + 5.06137i 9.88336 + 17.1185i 2.50000 4.33013i −18.7009 + 19.9994i 12.9395 13.2503i 62.0037i −24.2350 11.9023i 22.8172 13.1735i
26.16 4.57212 + 2.63971i 4.07322 3.22628i 9.93617 + 17.2100i 2.50000 4.33013i 27.1397 3.99877i −17.5731 + 5.84690i 62.6791i 6.18230 26.2827i 22.8606 13.1986i
101.1 −4.69077 + 2.70822i 2.33527 4.64182i 10.6689 18.4790i 2.50000 + 4.33013i 1.61684 + 28.0981i 14.5222 11.4937i 72.2429i −16.0930 21.6798i −23.4538 13.5411i
101.2 −3.95258 + 2.28202i −1.04959 + 5.08904i 6.41528 11.1116i 2.50000 + 4.33013i −7.46472 22.5101i −15.3726 + 10.3287i 22.0469i −24.7967 10.6828i −19.7629 11.4101i
101.3 −3.64936 + 2.10696i 4.12131 + 3.16462i 4.87855 8.44990i 2.50000 + 4.33013i −21.7079 2.86543i 16.7999 + 7.79504i 7.40430i 6.97031 + 26.0848i −18.2468 10.5348i
101.4 −3.53626 + 2.04166i −5.19608 + 0.0272697i 4.33676 7.51148i 2.50000 + 4.33013i 18.3190 10.7051i 0.627588 18.5096i 2.75017i 26.9985 0.283391i −17.6813 10.2083i
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.b yes 32
3.b odd 2 1 105.4.s.a 32
7.d odd 6 1 105.4.s.a 32
21.g even 6 1 inner 105.4.s.b yes 32

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

## 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 $$S_{4}^{\mathrm{new}}(105, [\chi])$$.