# Properties

 Label 950.2.bc.b Level $950$ Weight $2$ Character orbit 950.bc Analytic conductor $7.586$ Analytic rank $0$ Dimension $600$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.bc (of order $$45$$, degree $$24$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$600$$ Relative dimension: $$25$$ over $$\Q(\zeta_{45})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{45}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$600q + 54q^{7} - 75q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$600q + 54q^{7} - 75q^{8} - 9q^{11} - 30q^{15} + 18q^{17} - 576q^{18} - 36q^{19} - 6q^{20} - 18q^{22} - 6q^{23} - 48q^{26} - 18q^{29} - 18q^{33} - 18q^{34} + 57q^{35} - 12q^{38} - 36q^{39} - 36q^{41} + 108q^{43} + 18q^{44} + 60q^{45} + 24q^{46} + 30q^{47} - 258q^{49} - 15q^{50} - 72q^{51} + 18q^{53} + 54q^{54} - 21q^{55} - 12q^{56} - 90q^{57} + 36q^{58} - 6q^{59} - 30q^{61} + 12q^{62} + 18q^{63} + 75q^{64} + 66q^{65} - 78q^{66} - 18q^{67} + 54q^{68} + 24q^{69} - 57q^{70} + 72q^{71} + 18q^{73} - 12q^{74} + 120q^{75} + 24q^{77} - 12q^{78} - 12q^{79} + 156q^{81} - 24q^{82} - 57q^{83} + 3q^{84} - 78q^{85} + 12q^{86} - 66q^{87} + 9q^{88} + 18q^{89} + 30q^{91} - 42q^{92} + 18q^{93} + 54q^{94} + 102q^{95} + 114q^{97} + 108q^{98} - 162q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1 −0.961262 + 0.275637i −2.93242 0.412125i 0.848048 0.529919i 2.12467 0.696979i 2.93242 0.412125i −0.00563586 + 0.00976159i −0.669131 + 0.743145i 5.54546 + 1.59014i −1.85025 + 1.25562i
61.2 −0.961262 + 0.275637i −2.91518 0.409702i 0.848048 0.529919i −2.13354 + 0.669334i 2.91518 0.409702i −1.27384 + 2.20635i −0.669131 + 0.743145i 5.44664 + 1.56180i 1.86640 1.23149i
61.3 −0.961262 + 0.275637i −2.76167 0.388127i 0.848048 0.529919i −0.487804 2.18221i 2.76167 0.388127i −0.573557 + 0.993430i −0.669131 + 0.743145i 4.59240 + 1.31685i 1.07041 + 1.96322i
61.4 −0.961262 + 0.275637i −2.67119 0.375412i 0.848048 0.529919i −2.08175 0.816285i 2.67119 0.375412i 2.22642 3.85627i −0.669131 + 0.743145i 4.11056 + 1.17868i 2.22610 + 0.210855i
61.5 −0.961262 + 0.275637i −2.27525 0.319766i 0.848048 0.529919i −0.523473 + 2.17393i 2.27525 0.319766i 0.513432 0.889290i −0.669131 + 0.743145i 2.19073 + 0.628182i −0.0960220 2.23401i
61.6 −0.961262 + 0.275637i −2.13154 0.299568i 0.848048 0.529919i 1.98945 + 1.02082i 2.13154 0.299568i 2.50721 4.34262i −0.669131 + 0.743145i 1.56993 + 0.450170i −2.19376 0.432908i
61.7 −0.961262 + 0.275637i −1.08231 0.152109i 0.848048 0.529919i 1.47691 + 1.67891i 1.08231 0.152109i 0.0170328 0.0295017i −0.669131 + 0.743145i −1.73552 0.497652i −1.88247 1.20678i
61.8 −0.961262 + 0.275637i −0.987842 0.138832i 0.848048 0.529919i −2.07952 + 0.821944i 0.987842 0.138832i −2.07405 + 3.59236i −0.669131 + 0.743145i −1.92723 0.552624i 1.77241 1.36330i
61.9 −0.961262 + 0.275637i −0.963527 0.135415i 0.848048 0.529919i −2.13219 0.673618i 0.963527 0.135415i 0.212802 0.368583i −0.669131 + 0.743145i −1.97374 0.565960i 2.23527 + 0.0598117i
61.10 −0.961262 + 0.275637i −0.902497 0.126838i 0.848048 0.529919i 1.77071 1.36550i 0.902497 0.126838i −2.28164 + 3.95191i −0.669131 + 0.743145i −2.08537 0.597971i −1.32573 + 1.80068i
61.11 −0.961262 + 0.275637i −0.890453 0.125145i 0.848048 0.529919i 0.325758 + 2.21221i 0.890453 0.125145i −0.413882 + 0.716865i −0.669131 + 0.743145i −2.10654 0.604041i −0.922907 2.03672i
61.12 −0.961262 + 0.275637i −0.592065 0.0832092i 0.848048 0.529919i −0.453451 2.18961i 0.592065 0.0832092i 0.286355 0.495982i −0.669131 + 0.743145i −2.54017 0.728382i 1.03942 + 1.97980i
61.13 −0.961262 + 0.275637i −0.483134 0.0679001i 0.848048 0.529919i 0.667227 2.13420i 0.483134 0.0679001i 2.02846 3.51339i −0.669131 + 0.743145i −2.65498 0.761302i −0.0531142 + 2.23544i
61.14 −0.961262 + 0.275637i 0.797908 + 0.112139i 0.848048 0.529919i −2.22905 + 0.177042i −0.797908 + 0.112139i 1.52948 2.64913i −0.669131 + 0.743145i −2.25970 0.647959i 2.09390 0.784592i
61.15 −0.961262 + 0.275637i 0.959876 + 0.134902i 0.848048 0.529919i 2.19119 + 0.445725i −0.959876 + 0.134902i 1.91206 3.31179i −0.669131 + 0.743145i −1.98062 0.567934i −2.22917 + 0.175517i
61.16 −0.961262 + 0.275637i 1.00717 + 0.141549i 0.848048 0.529919i −1.45407 1.69873i −1.00717 + 0.141549i −1.19727 + 2.07373i −0.669131 + 0.743145i −1.88942 0.541783i 1.86598 + 1.23213i
61.17 −0.961262 + 0.275637i 1.01884 + 0.143188i 0.848048 0.529919i 1.62816 + 1.53268i −1.01884 + 0.143188i −0.935339 + 1.62005i −0.669131 + 0.743145i −1.86626 0.535142i −1.98755 1.02452i
61.18 −0.961262 + 0.275637i 1.19617 + 0.168111i 0.848048 0.529919i −1.05511 + 1.97148i −1.19617 + 0.168111i −2.18297 + 3.78102i −0.669131 + 0.743145i −1.48123 0.424734i 0.470822 2.18594i
61.19 −0.961262 + 0.275637i 1.24823 + 0.175428i 0.848048 0.529919i 2.17900 0.501953i −1.24823 + 0.175428i 0.830929 1.43921i −0.669131 + 0.743145i −1.35647 0.388962i −1.95623 + 1.08312i
61.20 −0.961262 + 0.275637i 1.26132 + 0.177267i 0.848048 0.529919i −1.27114 + 1.83962i −1.26132 + 0.177267i 0.704966 1.22104i −0.669131 + 0.743145i −1.32428 0.379731i 0.714832 2.11873i
See next 80 embeddings (of 600 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 921.25 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner
25.d even 5 1 inner
475.bc even 45 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.bc.b 600
19.e even 9 1 inner 950.2.bc.b 600
25.d even 5 1 inner 950.2.bc.b 600
475.bc even 45 1 inner 950.2.bc.b 600

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.bc.b 600 1.a even 1 1 trivial
950.2.bc.b 600 19.e even 9 1 inner
950.2.bc.b 600 25.d even 5 1 inner
950.2.bc.b 600 475.bc even 45 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$13\!\cdots\!62$$$$T_{3}^{577} -$$$$22\!\cdots\!28$$$$T_{3}^{576} + 549257851497 T_{3}^{575} +$$$$14\!\cdots\!85$$$$T_{3}^{574} +$$$$10\!\cdots\!54$$$$T_{3}^{573} +$$$$47\!\cdots\!22$$$$T_{3}^{572} -$$$$12\!\cdots\!70$$$$T_{3}^{571} +$$$$87\!\cdots\!54$$$$T_{3}^{570} -$$$$48\!\cdots\!84$$$$T_{3}^{569} -$$$$27\!\cdots\!70$$$$T_{3}^{568} -$$$$14\!\cdots\!94$$$$T_{3}^{567} -$$$$28\!\cdots\!47$$$$T_{3}^{566} +$$$$38\!\cdots\!12$$$$T_{3}^{565} -$$$$11\!\cdots\!83$$$$T_{3}^{564} +$$$$10\!\cdots\!95$$$$T_{3}^{563} +$$$$20\!\cdots\!37$$$$T_{3}^{562} -$$$$29\!\cdots\!19$$$$T_{3}^{561} +$$$$48\!\cdots\!60$$$$T_{3}^{560} +$$$$77\!\cdots\!89$$$$T_{3}^{559} +$$$$51\!\cdots\!04$$$$T_{3}^{558} -$$$$59\!\cdots\!74$$$$T_{3}^{557} -$$$$12\!\cdots\!95$$$$T_{3}^{556} +$$$$63\!\cdots\!76$$$$T_{3}^{555} -$$$$47\!\cdots\!35$$$$T_{3}^{554} -$$$$16\!\cdots\!60$$$$T_{3}^{553} +$$$$63\!\cdots\!55$$$$T_{3}^{552} -$$$$36\!\cdots\!57$$$$T_{3}^{551} -$$$$15\!\cdots\!88$$$$T_{3}^{550} -$$$$15\!\cdots\!35$$$$T_{3}^{549} +$$$$36\!\cdots\!75$$$$T_{3}^{548} +$$$$14\!\cdots\!84$$$$T_{3}^{547} -$$$$13\!\cdots\!39$$$$T_{3}^{546} +$$$$69\!\cdots\!49$$$$T_{3}^{545} +$$$$31\!\cdots\!76$$$$T_{3}^{544} +$$$$76\!\cdots\!60$$$$T_{3}^{543} -$$$$15\!\cdots\!56$$$$T_{3}^{542} -$$$$13\!\cdots\!90$$$$T_{3}^{541} +$$$$10\!\cdots\!01$$$$T_{3}^{540} -$$$$17\!\cdots\!64$$$$T_{3}^{539} -$$$$30\!\cdots\!86$$$$T_{3}^{538} +$$$$12\!\cdots\!54$$$$T_{3}^{537} -$$$$21\!\cdots\!88$$$$T_{3}^{536} -$$$$15\!\cdots\!65$$$$T_{3}^{535} -$$$$43\!\cdots\!30$$$$T_{3}^{534} +$$$$19\!\cdots\!23$$$$T_{3}^{533} -$$$$14\!\cdots\!80$$$$T_{3}^{532} -$$$$20\!\cdots\!49$$$$T_{3}^{531} +$$$$62\!\cdots\!93$$$$T_{3}^{530} +$$$$11\!\cdots\!38$$$$T_{3}^{529} -$$$$75\!\cdots\!39$$$$T_{3}^{528} -$$$$34\!\cdots\!18$$$$T_{3}^{527} +$$$$46\!\cdots\!61$$$$T_{3}^{526} +$$$$48\!\cdots\!67$$$$T_{3}^{525} -$$$$54\!\cdots\!94$$$$T_{3}^{524} +$$$$37\!\cdots\!83$$$$T_{3}^{523} +$$$$69\!\cdots\!93$$$$T_{3}^{522} -$$$$13\!\cdots\!65$$$$T_{3}^{521} -$$$$21\!\cdots\!18$$$$T_{3}^{520} +$$$$91\!\cdots\!63$$$$T_{3}^{519} +$$$$44\!\cdots\!41$$$$T_{3}^{518} -$$$$10\!\cdots\!47$$$$T_{3}^{517} -$$$$10\!\cdots\!26$$$$T_{3}^{516} +$$$$12\!\cdots\!34$$$$T_{3}^{515} -$$$$14\!\cdots\!38$$$$T_{3}^{514} -$$$$78\!\cdots\!43$$$$T_{3}^{513} +$$$$29\!\cdots\!21$$$$T_{3}^{512} +$$$$35\!\cdots\!37$$$$T_{3}^{511} -$$$$22\!\cdots\!92$$$$T_{3}^{510} -$$$$12\!\cdots\!74$$$$T_{3}^{509} +$$$$18\!\cdots\!03$$$$T_{3}^{508} -$$$$53\!\cdots\!42$$$$T_{3}^{507} -$$$$20\!\cdots\!22$$$$T_{3}^{506} +$$$$25\!\cdots\!51$$$$T_{3}^{505} +$$$$16\!\cdots\!34$$$$T_{3}^{504} -$$$$41\!\cdots\!50$$$$T_{3}^{503} -$$$$32\!\cdots\!18$$$$T_{3}^{502} +$$$$26\!\cdots\!94$$$$T_{3}^{501} +$$$$11\!\cdots\!78$$$$T_{3}^{500} -$$$$24\!\cdots\!46$$$$T_{3}^{499} -$$$$16\!\cdots\!36$$$$T_{3}^{498} +$$$$24\!\cdots\!51$$$$T_{3}^{497} -$$$$44\!\cdots\!48$$$$T_{3}^{496} -$$$$10\!\cdots\!03$$$$T_{3}^{495} +$$$$54\!\cdots\!60$$$$T_{3}^{494} +$$$$27\!\cdots\!22$$$$T_{3}^{493} -$$$$25\!\cdots\!75$$$$T_{3}^{492} -$$$$76\!\cdots\!21$$$$T_{3}^{491} +$$$$27\!\cdots\!80$$$$T_{3}^{490} -$$$$19\!\cdots\!71$$$$T_{3}^{489} -$$$$27\!\cdots\!81$$$$T_{3}^{488} +$$$$43\!\cdots\!27$$$$T_{3}^{487} +$$$$12\!\cdots\!64$$$$T_{3}^{486} -$$$$48\!\cdots\!92$$$$T_{3}^{485} -$$$$19\!\cdots\!98$$$$T_{3}^{484} +$$$$25\!\cdots\!97$$$$T_{3}^{483} +$$$$35\!\cdots\!79$$$$T_{3}^{482} -$$$$21\!\cdots\!06$$$$T_{3}^{481} -$$$$24\!\cdots\!71$$$$T_{3}^{480} +$$$$23\!\cdots\!45$$$$T_{3}^{479} -$$$$40\!\cdots\!16$$$$T_{3}^{478} -$$$$71\!\cdots\!04$$$$T_{3}^{477} +$$$$24\!\cdots\!03$$$$T_{3}^{476} +$$$$17\!\cdots\!97$$$$T_{3}^{475} -$$$$39\!\cdots\!51$$$$T_{3}^{474} -$$$$40\!\cdots\!84$$$$T_{3}^{473} +$$$$21\!\cdots\!83$$$$T_{3}^{472} -$$$$84\!\cdots\!58$$$$T_{3}^{471} -$$$$17\!\cdots\!78$$$$T_{3}^{470} +$$$$22\!\cdots\!82$$$$T_{3}^{469} +$$$$37\!\cdots\!62$$$$T_{3}^{468} -$$$$18\!\cdots\!46$$$$T_{3}^{467} -$$$$36\!\cdots\!73$$$$T_{3}^{466} +$$$$77\!\cdots\!84$$$$T_{3}^{465} +$$$$55\!\cdots\!42$$$$T_{3}^{464} -$$$$12\!\cdots\!87$$$$T_{3}^{463} -$$$$25\!\cdots\!56$$$$T_{3}^{462} +$$$$16\!\cdots\!79$$$$T_{3}^{461} -$$$$12\!\cdots\!14$$$$T_{3}^{460} -$$$$41\!\cdots\!23$$$$T_{3}^{459} -$$$$20\!\cdots\!86$$$$T_{3}^{458} +$$$$21\!\cdots\!70$$$$T_{3}^{457} +$$$$81\!\cdots\!09$$$$T_{3}^{456} -$$$$54\!\cdots\!33$$$$T_{3}^{455} +$$$$87\!\cdots\!06$$$$T_{3}^{454} +$$$$12\!\cdots\!24$$$$T_{3}^{453} -$$$$56\!\cdots\!65$$$$T_{3}^{452} +$$$$56\!\cdots\!75$$$$T_{3}^{451} -$$$$60\!\cdots\!11$$$$T_{3}^{450} +$$$$58\!\cdots\!74$$$$T_{3}^{449} -$$$$16\!\cdots\!07$$$$T_{3}^{448} +$$$$14\!\cdots\!62$$$$T_{3}^{447} +$$$$19\!\cdots\!91$$$$T_{3}^{446} -$$$$35\!\cdots\!96$$$$T_{3}^{445} -$$$$10\!\cdots\!15$$$$T_{3}^{444} +$$$$59\!\cdots\!37$$$$T_{3}^{443} -$$$$61\!\cdots\!52$$$$T_{3}^{442} -$$$$20\!\cdots\!72$$$$T_{3}^{441} -$$$$74\!\cdots\!09$$$$T_{3}^{440} +$$$$77\!\cdots\!25$$$$T_{3}^{439} +$$$$43\!\cdots\!19$$$$T_{3}^{438} -$$$$23\!\cdots\!08$$$$T_{3}^{437} +$$$$78\!\cdots\!27$$$$T_{3}^{436} +$$$$78\!\cdots\!70$$$$T_{3}^{435} -$$$$48\!\cdots\!31$$$$T_{3}^{434} -$$$$22\!\cdots\!27$$$$T_{3}^{433} -$$$$78\!\cdots\!58$$$$T_{3}^{432} +$$$$43\!\cdots\!24$$$$T_{3}^{431} -$$$$75\!\cdots\!40$$$$T_{3}^{430} -$$$$88\!\cdots\!83$$$$T_{3}^{429} -$$$$18\!\cdots\!98$$$$T_{3}^{428} +$$$$96\!\cdots\!01$$$$T_{3}^{427} +$$$$19\!\cdots\!28$$$$T_{3}^{426} -$$$$13\!\cdots\!85$$$$T_{3}^{425} -$$$$93\!\cdots\!66$$$$T_{3}^{424} -$$$$15\!\cdots\!52$$$$T_{3}^{423} +$$$$13\!\cdots\!04$$$$T_{3}^{422} -$$$$21\!\cdots\!67$$$$T_{3}^{421} +$$$$25\!\cdots\!31$$$$T_{3}^{420} -$$$$31\!\cdots\!16$$$$T_{3}^{419} +$$$$73\!\cdots\!70$$$$T_{3}^{418} +$$$$37\!\cdots\!96$$$$T_{3}^{417} -$$$$24\!\cdots\!35$$$$T_{3}^{416} -$$$$12\!\cdots\!22$$$$T_{3}^{415} -$$$$50\!\cdots\!03$$$$T_{3}^{414} +$$$$11\!\cdots\!26$$$$T_{3}^{413} -$$$$19\!\cdots\!67$$$$T_{3}^{412} -$$$$10\!\cdots\!08$$$$T_{3}^{411} +$$$$98\!\cdots\!20$$$$T_{3}^{410} +$$$$24\!\cdots\!35$$$$T_{3}^{409} -$$$$20\!\cdots\!52$$$$T_{3}^{408} -$$$$14\!\cdots\!05$$$$T_{3}^{407} +$$$$94\!\cdots\!34$$$$T_{3}^{406} +$$$$54\!\cdots\!59$$$$T_{3}^{405} +$$$$90\!\cdots\!16$$$$T_{3}^{404} -$$$$87\!\cdots\!60$$$$T_{3}^{403} +$$$$10\!\cdots\!72$$$$T_{3}^{402} -$$$$10\!\cdots\!20$$$$T_{3}^{401} +$$$$59\!\cdots\!82$$$$T_{3}^{400} -$$$$14\!\cdots\!43$$$$T_{3}^{399} -$$$$29\!\cdots\!30$$$$T_{3}^{398} +$$$$14\!\cdots\!90$$$$T_{3}^{397} -$$$$10\!\cdots\!53$$$$T_{3}^{396} +$$$$26\!\cdots\!90$$$$T_{3}^{395} -$$$$16\!\cdots\!96$$$$T_{3}^{394} +$$$$22\!\cdots\!80$$$$T_{3}^{393} +$$$$36\!\cdots\!91$$$$T_{3}^{392} -$$$$45\!\cdots\!24$$$$T_{3}^{391} -$$$$19\!\cdots\!92$$$$T_{3}^{390} -$$$$39\!\cdots\!29$$$$T_{3}^{389} +$$$$23\!\cdots\!40$$$$T_{3}^{388} -$$$$18\!\cdots\!87$$$$T_{3}^{387} -$$$$55\!\cdots\!91$$$$T_{3}^{386} -$$$$25\!\cdots\!39$$$$T_{3}^{385} +$$$$68\!\cdots\!94$$$$T_{3}^{384} +$$$$31\!\cdots\!34$$$$T_{3}^{383} -$$$$14\!\cdots\!35$$$$T_{3}^{382} -$$$$97\!\cdots\!13$$$$T_{3}^{381} -$$$$38\!\cdots\!57$$$$T_{3}^{380} +$$$$58\!\cdots\!67$$$$T_{3}^{379} -$$$$91\!\cdots\!83$$$$T_{3}^{378} +$$$$12\!\cdots\!52$$$$T_{3}^{377} -$$$$10\!\cdots\!65$$$$T_{3}^{376} +$$$$52\!\cdots\!71$$$$T_{3}^{375} +$$$$75\!\cdots\!00$$$$T_{3}^{374} -$$$$68\!\cdots\!66$$$$T_{3}^{373} +$$$$65\!\cdots\!21$$$$T_{3}^{372} -$$$$93\!\cdots\!63$$$$T_{3}^{371} +$$$$27\!\cdots\!74$$$$T_{3}^{370} -$$$$88\!\cdots\!83$$$$T_{3}^{369} -$$$$11\!\cdots\!82$$$$T_{3}^{368} +$$$$37\!\cdots\!12$$$$T_{3}^{367} -$$$$41\!\cdots\!54$$$$T_{3}^{366} +$$$$97\!\cdots\!15$$$$T_{3}^{365} -$$$$36\!\cdots\!82$$$$T_{3}^{364} +$$$$77\!\cdots\!38$$$$T_{3}^{363} +$$$$83\!\cdots\!20$$$$T_{3}^{362} +$$$$79\!\cdots\!50$$$$T_{3}^{361} +$$$$42\!\cdots\!13$$$$T_{3}^{360} +$$$$25\!\cdots\!99$$$$T_{3}^{359} +$$$$51\!\cdots\!17$$$$T_{3}^{358} +$$$$15\!\cdots\!92$$$$T_{3}^{357} +$$$$18\!\cdots\!59$$$$T_{3}^{356} +$$$$13\!\cdots\!96$$$$T_{3}^{355} +$$$$83\!\cdots\!98$$$$T_{3}^{354} +$$$$16\!\cdots\!44$$$$T_{3}^{353} +$$$$22\!\cdots\!58$$$$T_{3}^{352} +$$$$10\!\cdots\!27$$$$T_{3}^{351} +$$$$73\!\cdots\!15$$$$T_{3}^{350} +$$$$56\!\cdots\!10$$$$T_{3}^{349} +$$$$53\!\cdots\!09$$$$T_{3}^{348} +$$$$19\!\cdots\!24$$$$T_{3}^{347} +$$$$35\!\cdots\!76$$$$T_{3}^{346} +$$$$56\!\cdots\!04$$$$T_{3}^{345} +$$$$21\!\cdots\!99$$$$T_{3}^{344} +$$$$13\!\cdots\!66$$$$T_{3}^{343} +$$$$76\!\cdots\!95$$$$T_{3}^{342} +$$$$55\!\cdots\!58$$$$T_{3}^{341} +$$$$18\!\cdots\!23$$$$T_{3}^{340} +$$$$30\!\cdots\!29$$$$T_{3}^{339} -$$$$99\!\cdots\!95$$$$T_{3}^{338} +$$$$96\!\cdots\!98$$$$T_{3}^{337} -$$$$19\!\cdots\!77$$$$T_{3}^{336} +$$$$35\!\cdots\!75$$$$T_{3}^{335} -$$$$84\!\cdots\!99$$$$T_{3}^{334} -$$$$18\!\cdots\!28$$$$T_{3}^{333} -$$$$19\!\cdots\!65$$$$T_{3}^{332} -$$$$10\!\cdots\!11$$$$T_{3}^{331} -$$$$49\!\cdots\!19$$$$T_{3}^{330} -$$$$31\!\cdots\!53$$$$T_{3}^{329} -$$$$19\!\cdots\!81$$$$T_{3}^{328} -$$$$33\!\cdots\!03$$$$T_{3}^{327} -$$$$68\!\cdots\!07$$$$T_{3}^{326} +$$$$13\!\cdots\!94$$$$T_{3}^{325} -$$$$55\!\cdots\!47$$$$T_{3}^{324} +$$$$87\!\cdots\!07$$$$T_{3}^{323} +$$$$78\!\cdots\!87$$$$T_{3}^{322} +$$$$26\!\cdots\!54$$$$T_{3}^{321} +$$$$48\!\cdots\!85$$$$T_{3}^{320} +$$$$51\!\cdots\!28$$$$T_{3}^{319} +$$$$14\!\cdots\!91$$$$T_{3}^{318} +$$$$18\!\cdots\!02$$$$T_{3}^{317} +$$$$12\!\cdots\!09$$$$T_{3}^{316} -$$$$38\!\cdots\!34$$$$T_{3}^{315} -$$$$62\!\cdots\!24$$$$T_{3}^{314} -$$$$18\!\cdots\!27$$$$T_{3}^{313} -$$$$32\!\cdots\!07$$$$T_{3}^{312} -$$$$40\!\cdots\!47$$$$T_{3}^{311} -$$$$74\!\cdots\!19$$$$T_{3}^{310} -$$$$15\!\cdots\!10$$$$T_{3}^{309} -$$$$87\!\cdots\!68$$$$T_{3}^{308} +$$$$18\!\cdots\!87$$$$T_{3}^{307} +$$$$11\!\cdots\!21$$$$T_{3}^{306} +$$$$61\!\cdots\!65$$$$T_{3}^{305} +$$$$97\!\cdots\!21$$$$T_{3}^{304} +$$$$87\!\cdots\!11$$$$T_{3}^{303} +$$$$34\!\cdots\!84$$$$T_{3}^{302} +$$$$30\!\cdots\!06$$$$T_{3}^{301} +$$$$62\!\cdots\!98$$$$T_{3}^{300} -$$$$14\!\cdots\!44$$$$T_{3}^{299} -$$$$13\!\cdots\!79$$$$T_{3}^{298} -$$$$51\!\cdots\!22$$$$T_{3}^{297} -$$$$46\!\cdots\!11$$$$T_{3}^{296} -$$$$12\!\cdots\!80$$$$T_{3}^{295} -$$$$13\!\cdots\!91$$$$T_{3}^{294} -$$$$13\!\cdots\!88$$$$T_{3}^{293} -$$$$14\!\cdots\!99$$$$T_{3}^{292} +$$$$14\!\cdots\!08$$$$T_{3}^{291} +$$$$21\!\cdots\!88$$$$T_{3}^{290} -$$$$87\!\cdots\!91$$$$T_{3}^{289} +$$$$12\!\cdots\!01$$$$T_{3}^{288} -$$$$46\!\cdots\!02$$$$T_{3}^{287} +$$$$30\!\cdots\!63$$$$T_{3}^{286} -$$$$22\!\cdots\!38$$$$T_{3}^{285} +$$$$51\!\cdots\!08$$$$T_{3}^{284} -$$$$47\!\cdots\!35$$$$T_{3}^{283} +$$$$32\!\cdots\!34$$$$T_{3}^{282} -$$$$28\!\cdots\!97$$$$T_{3}^{281} -$$$$15\!\cdots\!74$$$$T_{3}^{280} +$$$$16\!\cdots\!28$$$$T_{3}^{279} -$$$$68\!\cdots\!56$$$$T_{3}^{278} +$$$$57\!\cdots\!37$$$$T_{3}^{277} -$$$$11\!\cdots\!08$$$$T_{3}^{276} +$$$$71\!\cdots\!35$$$$T_{3}^{275} +$$$$14\!\cdots\!31$$$$T_{3}^{274} -$$$$68\!\cdots\!73$$$$T_{3}^{273} +$$$$51\!\cdots\!28$$$$T_{3}^{272} -$$$$39\!\cdots\!42$$$$T_{3}^{271} +$$$$95\!\cdots\!93$$$$T_{3}^{270} -$$$$36\!\cdots\!10$$$$T_{3}^{269} -$$$$27\!\cdots\!37$$$$T_{3}^{268} +$$$$11\!\cdots\!30$$$$T_{3}^{267} -$$$$46\!\cdots\!05$$$$T_{3}^{266} +$$$$52\!\cdots\!02$$$$T_{3}^{265} -$$$$10\!\cdots\!31$$$$T_{3}^{264} +$$$$10\!\cdots\!07$$$$T_{3}^{263} -$$$$13\!\cdots\!78$$$$T_{3}^{262} +$$$$18\!\cdots\!23$$$$T_{3}^{261} -$$$$92\!\cdots\!71$$$$T_{3}^{260} +$$$$68\!\cdots\!93$$$$T_{3}^{259} +$$$$35\!\cdots\!84$$$$T_{3}^{258} -$$$$92\!\cdots\!47$$$$T_{3}^{257} +$$$$20\!\cdots\!05$$$$T_{3}^{256} -$$$$40\!\cdots\!30$$$$T_{3}^{255} +$$$$60\!\cdots\!56$$$$T_{3}^{254} -$$$$83\!\cdots\!24$$$$T_{3}^{253} +$$$$87\!\cdots\!86$$$$T_{3}^{252} -$$$$69\!\cdots\!44$$$$T_{3}^{251} +$$$$27\!\cdots\!03$$$$T_{3}^{250} +$$$$40\!\cdots\!23$$$$T_{3}^{249} -$$$$72\!\cdots\!97$$$$T_{3}^{248} +$$$$43\!\cdots\!88$$$$T_{3}^{247} +$$$$15\!\cdots\!14$$$$T_{3}^{246} -$$$$63\!\cdots\!38$$$$T_{3}^{245} +$$$$12\!\cdots\!20$$$$T_{3}^{244} -$$$$18\!\cdots\!42$$$$T_{3}^{243} +$$$$22\!\cdots\!71$$$$T_{3}^{242} -$$$$23\!\cdots\!32$$$$T_{3}^{241} +$$$$20\!\cdots\!77$$$$T_{3}^{240} -$$$$10\!\cdots\!16$$$$T_{3}^{239} +$$$$18\!\cdots\!24$$$$T_{3}^{238} -$$$$54\!\cdots\!63$$$$T_{3}^{237} +$$$$19\!\cdots\!19$$$$T_{3}^{236} -$$$$27\!\cdots\!59$$$$T_{3}^{235} +$$$$13\!\cdots\!66$$$$T_{3}^{234} +$$$$98\!\cdots\!30$$$$T_{3}^{233} -$$$$32\!\cdots\!17$$$$T_{3}^{232} +$$$$67\!\cdots\!46$$$$T_{3}^{231} -$$$$10\!\cdots\!92$$$$T_{3}^{230} +$$$$14\!\cdots\!86$$$$T_{3}^{229} -$$$$18\!\cdots\!46$$$$T_{3}^{228} +$$$$15\!\cdots\!30$$$$T_{3}^{227} -$$$$60\!\cdots\!84$$$$T_{3}^{226} -$$$$25\!\cdots\!36$$$$T_{3}^{225} +$$$$11\!\cdots\!11$$$$T_{3}^{224} -$$$$27\!\cdots\!81$$$$T_{3}^{223} +$$$$38\!\cdots\!02$$$$T_{3}^{222} -$$$$34\!\cdots\!85$$$$T_{3}^{221} +$$$$47\!\cdots\!30$$$$T_{3}^{220} -$$$$88\!\cdots\!71$$$$T_{3}^{219} +$$$$12\!\cdots\!55$$$$T_{3}^{218} -$$$$18\!\cdots\!83$$$$T_{3}^{217} +$$$$29\!\cdots\!95$$$$T_{3}^{216} -$$$$37\!\cdots\!94$$$$T_{3}^{215} +$$$$34\!\cdots\!07$$$$T_{3}^{214} -$$$$33\!\cdots\!12$$$$T_{3}^{213} +$$$$31\!\cdots\!97$$$$T_{3}^{212} -$$$$22\!\cdots\!45$$$$T_{3}^{211} -$$$$20\!\cdots\!34$$$$T_{3}^{210} -$$$$75\!\cdots\!39$$$$T_{3}^{209} +$$$$24\!\cdots\!62$$$$T_{3}^{208} +$$$$60\!\cdots\!87$$$$T_{3}^{207} +$$$$37\!\cdots\!33$$$$T_{3}^{206} -$$$$16\!\cdots\!65$$$$T_{3}^{205} +$$$$17\!\cdots\!04$$$$T_{3}^{204} -$$$$67\!\cdots\!92$$$$T_{3}^{203} +$$$$78\!\cdots\!73$$$$T_{3}^{202} -$$$$11\!\cdots\!17$$$$T_{3}^{201} -$$$$31\!\cdots\!13$$$$T_{3}^{200} +$$$$10\!\cdots\!45$$$$T_{3}^{199} +$$$$22\!\cdots\!87$$$$T_{3}^{198} +$$$$37\!\cdots\!26$$$$T_{3}^{197} +$$$$56\!\cdots\!99$$$$T_{3}^{196} -$$$$50\!\cdots\!53$$$$T_{3}^{195} +$$$$56\!\cdots\!24$$$$T_{3}^{194} +$$$$57\!\cdots\!12$$$$T_{3}^{193} -$$$$11\!\cdots\!49$$$$T_{3}^{192} +$$$$11\!\cdots\!33$$$$T_{3}^{191} -$$$$49\!\cdots\!24$$$$T_{3}^{190} +$$$$18\!\cdots\!47$$$$T_{3}^{189} -$$$$16\!\cdots\!54$$$$T_{3}^{188} -$$$$20\!\cdots\!81$$$$T_{3}^{187} +$$$$42\!\cdots\!72$$$$T_{3}^{186} -$$$$24\!\cdots\!86$$$$T_{3}^{185} +$$$$12\!\cdots\!87$$$$T_{3}^{184} -$$$$31\!\cdots\!63$$$$T_{3}^{183} +$$$$43\!\cdots\!05$$$$T_{3}^{182} -$$$$12\!\cdots\!30$$$$T_{3}^{181} -$$$$38\!\cdots\!95$$$$T_{3}^{180} +$$$$53\!\cdots\!13$$$$T_{3}^{179} -$$$$10\!\cdots\!64$$$$T_{3}^{178} -$$$$28\!\cdots\!96$$$$T_{3}^{177} -$$$$12\!\cdots\!00$$$$T_{3}^{176} +$$$$74\!\cdots\!76$$$$T_{3}^{175} -$$$$24\!\cdots\!50$$$$T_{3}^{174} -$$$$77\!\cdots\!14$$$$T_{3}^{173} +$$$$27\!\cdots\!71$$$$T_{3}^{172} +$$$$13\!\cdots\!76$$$$T_{3}^{171} -$$$$14\!\cdots\!55$$$$T_{3}^{170} -$$$$40\!\cdots\!44$$$$T_{3}^{169} +$$$$11\!\cdots\!09$$$$T_{3}^{168} +$$$$10\!\cdots\!47$$$$T_{3}^{167} -$$$$26\!\cdots\!58$$$$T_{3}^{166} -$$$$17\!\cdots\!36$$$$T_{3}^{165} +$$$$27\!\cdots\!91$$$$T_{3}^{164} -$$$$77\!\cdots\!40$$$$T_{3}^{163} -$$$$12\!\cdots\!79$$$$T_{3}^{162} +$$$$75\!\cdots\!41$$$$T_{3}^{161} +$$$$26\!\cdots\!12$$$$T_{3}^{160} +$$$$52\!\cdots\!13$$$$T_{3}^{159} -$$$$17\!\cdots\!23$$$$T_{3}^{158} +$$$$98\!\cdots\!84$$$$T_{3}^{157} +$$$$96\!\cdots\!08$$$$T_{3}^{156} -$$$$15\!\cdots\!64$$$$T_{3}^{155} +$$$$15\!\cdots\!66$$$$T_{3}^{154} +$$$$14\!\cdots\!62$$$$T_{3}^{153} -$$$$17\!\cdots\!24$$$$T_{3}^{152} +$$$$10\!\cdots\!55$$$$T_{3}^{151} -$$$$25\!\cdots\!98$$$$T_{3}^{150} -$$$$31\!\cdots\!61$$$$T_{3}^{149} +$$$$99\!\cdots\!80$$$$T_{3}^{148} -$$$$16\!\cdots\!68$$$$T_{3}^{147} +$$$$15\!\cdots\!33$$$$T_{3}^{146} -$$$$58\!\cdots\!93$$$$T_{3}^{145} -$$$$25\!\cdots\!80$$$$T_{3}^{144} +$$$$17\!\cdots\!49$$$$T_{3}^{143} +$$$$26\!\cdots\!79$$$$T_{3}^{142} -$$$$11\!\cdots\!98$$$$T_{3}^{141} -$$$$57\!\cdots\!45$$$$T_{3}^{140} +$$$$96\!\cdots\!01$$$$T_{3}^{139} -$$$$68\!\cdots\!22$$$$T_{3}^{138} +$$$$10\!\cdots\!70$$$$T_{3}^{137} +$$$$25\!\cdots\!70$$$$T_{3}^{136} -$$$$20\!\cdots\!22$$$$T_{3}^{135} -$$$$84\!\cdots\!09$$$$T_{3}^{134} +$$$$34\!\cdots\!16$$$$T_{3}^{133} -$$$$36\!\cdots\!40$$$$T_{3}^{132} +$$$$16\!\cdots\!56$$$$T_{3}^{131} +$$$$11\!\cdots\!17$$$$T_{3}^{130} -$$$$31\!\cdots\!10$$$$T_{3}^{129} +$$$$37\!\cdots\!43$$$$T_{3}^{128} -$$$$34\!\cdots\!40$$$$T_{3}^{127} +$$$$27\!\cdots\!33$$$$T_{3}^{126} -$$$$19\!\cdots\!41$$$$T_{3}^{125} +$$$$76\!\cdots\!80$$$$T_{3}^{124} +$$$$64\!\cdots\!58$$$$T_{3}^{123} -$$$$16\!\cdots\!61$$$$T_{3}^{122} +$$$$19\!\cdots\!95$$$$T_{3}^{121} -$$$$17\!\cdots\!63$$$$T_{3}^{120} +$$$$13\!\cdots\!63$$$$T_{3}^{119} -$$$$83\!\cdots\!99$$$$T_{3}^{118} +$$$$20\!\cdots\!65$$$$T_{3}^{117} +$$$$29\!\cdots\!03$$$$T_{3}^{116} -$$$$43\!\cdots\!02$$$$T_{3}^{115} +$$$$30\!\cdots\!01$$$$T_{3}^{114} -$$$$17\!\cdots\!57$$$$T_{3}^{113} +$$$$20\!\cdots\!17$$$$T_{3}^{112} -$$$$28\!\cdots\!18$$$$T_{3}^{111} +$$$$27\!\cdots\!01$$$$T_{3}^{110} -$$$$15\!\cdots\!33$$$$T_{3}^{109} +$$$$44\!\cdots\!83$$$$T_{3}^{108} -$$$$11\!\cdots\!49$$$$T_{3}^{107} +$$$$25\!\cdots\!49$$$$T_{3}^{106} -$$$$16\!\cdots\!69$$$$T_{3}^{105} -$$$$37\!\cdots\!17$$$$T_{3}^{104} +$$$$94\!\cdots\!02$$$$T_{3}^{103} -$$$$11\!\cdots\!55$$$$T_{3}^{102} +$$$$86\!\cdots\!17$$$$T_{3}^{101} -$$$$39\!\cdots\!60$$$$T_{3}^{100} +$$$$63\!\cdots\!68$$$$T_{3}^{99} +$$$$18\!\cdots\!28$$$$T_{3}^{98} -$$$$20\!\cdots\!66$$$$T_{3}^{97} +$$$$14\!\cdots\!33$$$$T_{3}^{96} -$$$$84\!\cdots\!57$$$$T_{3}^{95} +$$$$38\!\cdots\!28$$$$T_{3}^{94} -$$$$16\!\cdots\!86$$$$T_{3}^{93} +$$$$14\!\cdots\!31$$$$T_{3}^{92} -$$$$22\!\cdots\!39$$$$T_{3}^{91} +$$$$29\!\cdots\!15$$$$T_{3}^{90} -$$$$28\!\cdots\!01$$$$T_{3}^{89} +$$$$19\!\cdots\!62$$$$T_{3}^{88} -$$$$51\!\cdots\!87$$$$T_{3}^{87} -$$$$71\!\cdots\!30$$$$T_{3}^{86} +$$$$13\!\cdots\!57$$$$T_{3}^{85} -$$$$13\!\cdots\!54$$$$T_{3}^{84} +$$$$10\!\cdots\!06$$$$T_{3}^{83} -$$$$49\!\cdots\!07$$$$T_{3}^{82} +$$$$77\!\cdots\!06$$$$T_{3}^{81} +$$$$15\!\cdots\!87$$$$T_{3}^{80} -$$$$21\!\cdots\!89$$$$T_{3}^{79} +$$$$17\!\cdots\!93$$$$T_{3}^{78} -$$$$98\!\cdots\!36$$$$T_{3}^{77} +$$$$35\!\cdots\!61$$$$T_{3}^{76} -$$$$95\!\cdots\!31$$$$T_{3}^{75} -$$$$12\!\cdots\!30$$$$T_{3}^{74} +$$$$12\!\cdots\!29$$$$T_{3}^{73} -$$$$84\!\cdots\!91$$$$T_{3}^{72} +$$$$42\!\cdots\!79$$$$T_{3}^{71} -$$$$14\!\cdots\!02$$$$T_{3}^{70} -$$$$55\!\cdots\!02$$$$T_{3}^{69} +$$$$48\!\cdots\!77$$$$T_{3}^{68} -$$$$46\!\cdots\!51$$$$T_{3}^{67} +$$$$29\!\cdots\!88$$$$T_{3}^{66} -$$$$14\!\cdots\!94$$$$T_{3}^{65} +$$$$47\!\cdots\!32$$$$T_{3}^{64} -$$$$53\!\cdots\!54$$$$T_{3}^{63} -$$$$86\!\cdots\!95$$$$T_{3}^{62} +$$$$10\!\cdots\!11$$$$T_{3}^{61} -$$$$72\!\cdots\!05$$$$T_{3}^{60} +$$$$38\!\cdots\!23$$$$T_{3}^{59} -$$$$14\!\cdots\!53$$$$T_{3}^{58} +$$$$26\!\cdots\!68$$$$T_{3}^{57} +$$$$15\!\cdots\!74$$$$T_{3}^{56} -$$$$21\!\cdots\!61$$$$T_{3}^{55} +$$$$15\!\cdots\!12$$$$T_{3}^{54} -$$$$88\!\cdots\!63$$$$T_{3}^{53} +$$$$41\!\cdots\!69$$$$T_{3}^{52} -$$$$15\!\cdots\!19$$$$T_{3}^{51} +$$$$41\!\cdots\!64$$$$T_{3}^{50} +$$$$81\!\cdots\!50$$$$T_{3}^{49} -$$$$99\!\cdots\!65$$$$T_{3}^{48} +$$$$81\!\cdots\!23$$$$T_{3}^{47} -$$$$46\!\cdots\!51$$$$T_{3}^{46} +$$$$21\!\cdots\!40$$$$T_{3}^{45} -$$$$81\!\cdots\!42$$$$T_{3}^{44} +$$$$22\!\cdots\!80$$$$T_{3}^{43} -$$$$19\!\cdots\!34$$$$T_{3}^{42} -$$$$29\!\cdots\!51$$$$T_{3}^{41} +$$$$27\!\cdots\!73$$$$T_{3}^{40} -$$$$16\!\cdots\!16$$$$T_{3}^{39} +$$$$82\!\cdots\!77$$$$T_{3}^{38} -$$$$36\!\cdots\!55$$$$T_{3}^{37} +$$$$14\!\cdots\!22$$$$T_{3}^{36} -$$$$49\!\cdots\!90$$$$T_{3}^{35} +$$$$16\!\cdots\!14$$$$T_{3}^{34} -$$$$49\!\cdots\!14$$$$T_{3}^{33} +$$$$13\!\cdots\!56$$$$T_{3}^{32} -$$$$36\!\cdots\!68$$$$T_{3}^{31} +$$$$92\!\cdots\!82$$$$T_{3}^{30} -$$$$22\!\cdots\!66$$$$T_{3}^{29} +$$$$53\!\cdots\!18$$$$T_{3}^{28} -$$$$12\!\cdots\!83$$$$T_{3}^{27} +$$$$27\!\cdots\!35$$$$T_{3}^{26} -$$$$58\!\cdots\!93$$$$T_{3}^{25} +$$$$11\!\cdots\!93$$$$T_{3}^{24} -$$$$22\!\cdots\!53$$$$T_{3}^{23} +$$$$40\!\cdots\!52$$$$T_{3}^{22} -$$$$69\!\cdots\!69$$$$T_{3}^{21} +$$$$11\!\cdots\!22$$$$T_{3}^{20} -$$$$17\!\cdots\!67$$$$T_{3}^{19} +$$$$24\!\cdots\!20$$$$T_{3}^{18} -$$$$33\!\cdots\!49$$$$T_{3}^{17} +$$$$43\!\cdots\!03$$$$T_{3}^{16} -$$$$55\!\cdots\!26$$$$T_{3}^{15} +$$$$70\!\cdots\!41$$$$T_{3}^{14} -$$$$87\!\cdots\!08$$$$T_{3}^{13} +$$$$10\!\cdots\!30$$$$T_{3}^{12} -$$$$10\!\cdots\!81$$$$T_{3}^{11} +$$$$90\!\cdots\!83$$$$T_{3}^{10} -$$$$64\!\cdots\!48$$$$T_{3}^{9} +$$$$33\!\cdots\!35$$$$T_{3}^{8} +$$$$24\!\cdots\!72$$$$T_{3}^{7} -$$$$20\!\cdots\!03$$$$T_{3}^{6} +$$$$24\!\cdots\!26$$$$T_{3}^{5} +$$$$85\!\cdots\!90$$$$T_{3}^{4} -$$$$19\!\cdots\!32$$$$T_{3}^{3} +$$$$26\!\cdots\!40$$$$T_{3}^{2} +$$$$12\!\cdots\!84$$$$T_{3} +$$$$28\!\cdots\!41$$">$$T_{3}^{600} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.