# Properties

 Label 950.2.bc.a 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.

 $$\operatorname{Tr}(f)(q) =$$ $$600q - 42q^{7} + 75q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$600q - 42q^{7} + 75q^{8} - 9q^{11} - 24q^{15} + 18q^{17} + 624q^{18} + 36q^{19} - 6q^{20} - 18q^{22} - 6q^{23} + 120q^{25} + 48q^{26} - 18q^{29} - 90q^{33} + 18q^{34} - 51q^{35} + 12q^{38} + 36q^{39} + 36q^{41} + 108q^{43} - 18q^{44} + 24q^{45} + 36q^{46} - 66q^{47} - 282q^{49} - 9q^{50} - 48q^{51} + 18q^{53} - 18q^{54} - 87q^{55} - 36q^{56} - 18q^{57} - 36q^{58} - 30q^{59} + 6q^{60} + 174q^{61} + 12q^{62} + 18q^{63} + 75q^{64} + 54q^{65} - 18q^{66} + 18q^{67} - 42q^{68} + 24q^{69} + 69q^{70} + 48q^{71} + 18q^{73} + 12q^{74} - 120q^{75} - 72q^{77} + 12q^{78} - 36q^{79} - 60q^{81} - 24q^{82} + 3q^{83} + 75q^{84} - 18q^{85} - 12q^{86} + 6q^{87} - 9q^{88} - 90q^{89} - 36q^{90} - 30q^{91} - 42q^{92} - 102q^{93} - 150q^{94} - 102q^{95} + 42q^{97} + 108q^{98} - 30q^{99} + O(q^{100})$$

## 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 −3.36425 0.472815i 0.848048 0.529919i −1.20385 1.88434i −3.36425 + 0.472815i 0.735407 1.27376i 0.669131 0.743145i 8.21086 + 2.35443i −1.67661 1.47952i
61.2 0.961262 0.275637i −3.10593 0.436510i 0.848048 0.529919i −0.194755 + 2.22757i −3.10593 + 0.436510i 1.10411 1.91237i 0.669131 0.743145i 6.57247 + 1.88463i 0.426791 + 2.19496i
61.3 0.961262 0.275637i −2.52861 0.355373i 0.848048 0.529919i 2.12329 + 0.701180i −2.52861 + 0.355373i 1.00675 1.74375i 0.669131 0.743145i 3.38381 + 0.970291i 2.23431 + 0.0887607i
61.4 0.961262 0.275637i −2.47903 0.348405i 0.848048 0.529919i 1.80399 1.32121i −2.47903 + 0.348405i −0.861925 + 1.49290i 0.669131 0.743145i 3.14043 + 0.900503i 1.36993 1.76728i
61.5 0.961262 0.275637i −2.26586 0.318446i 0.848048 0.529919i 0.868613 2.06046i −2.26586 + 0.318446i −1.77884 + 3.08104i 0.669131 0.743145i 2.14894 + 0.616198i 0.267024 2.22007i
61.6 0.961262 0.275637i −1.99159 0.279900i 0.848048 0.529919i −2.16545 0.557508i −1.99159 + 0.279900i 0.493923 0.855500i 0.669131 0.743145i 1.00432 + 0.287984i −2.23524 + 0.0609689i
61.7 0.961262 0.275637i −1.97569 0.277665i 0.848048 0.529919i −0.0284411 + 2.23589i −1.97569 + 0.277665i −1.68657 + 2.92123i 0.669131 0.743145i 0.942462 + 0.270247i 0.588955 + 2.15711i
61.8 0.961262 0.275637i −1.54503 0.217140i 0.848048 0.529919i 1.55455 1.60729i −1.54503 + 0.217140i 2.43412 4.21602i 0.669131 0.743145i −0.543813 0.155936i 1.05130 1.97352i
61.9 0.961262 0.275637i −1.18489 0.166526i 0.848048 0.529919i −2.08057 + 0.819293i −1.18489 + 0.166526i −0.801519 + 1.38827i 0.669131 0.743145i −1.50754 0.432281i −1.77414 + 1.36104i
61.10 0.961262 0.275637i −1.08040 0.151840i 0.848048 0.529919i 1.81971 + 1.29948i −1.08040 + 0.151840i −1.07705 + 1.86551i 0.669131 0.743145i −1.73958 0.498818i 2.10741 + 0.747557i
61.11 0.961262 0.275637i −0.363561 0.0510952i 0.848048 0.529919i −1.52899 + 1.63162i −0.363561 + 0.0510952i 1.89241 3.27775i 0.669131 0.743145i −2.75422 0.789760i −1.02003 + 1.98986i
61.12 0.961262 0.275637i 0.0711673 + 0.0100019i 0.848048 0.529919i 2.19008 + 0.451146i 0.0711673 0.0100019i 0.840373 1.45557i 0.669131 0.743145i −2.87882 0.825488i 2.22960 0.170000i
61.13 0.961262 0.275637i 0.181877 + 0.0255611i 0.848048 0.529919i −1.97721 1.04434i 0.181877 0.0255611i −1.51682 + 2.62720i 0.669131 0.743145i −2.85136 0.817614i −2.18847 0.458892i
61.14 0.961262 0.275637i 0.198824 + 0.0279429i 0.848048 0.529919i −0.662616 2.13564i 0.198824 0.0279429i 0.864339 1.49708i 0.669131 0.743145i −2.84503 0.815801i −1.22561 1.87026i
61.15 0.961262 0.275637i 0.287928 + 0.0404657i 0.848048 0.529919i −0.0993678 2.23386i 0.287928 0.0404657i −0.701041 + 1.21424i 0.669131 0.743145i −2.80252 0.803610i −0.711253 2.11993i
61.16 0.961262 0.275637i 0.897293 + 0.126106i 0.848048 0.529919i 0.788589 + 2.09240i 0.897293 0.126106i −2.61399 + 4.52757i 0.669131 0.743145i −2.09455 0.600603i 1.33478 + 1.79398i
61.17 0.961262 0.275637i 1.28122 + 0.180063i 0.848048 0.529919i 0.0954887 + 2.23403i 1.28122 0.180063i 0.778156 1.34781i 0.669131 0.743145i −1.27469 0.365513i 0.707571 + 2.12117i
61.18 0.961262 0.275637i 1.47109 + 0.206748i 0.848048 0.529919i −2.22682 + 0.203157i 1.47109 0.206748i 2.40262 4.16146i 0.669131 0.743145i −0.762436 0.218625i −2.08456 + 0.809082i
61.19 0.961262 0.275637i 2.08096 + 0.292460i 0.848048 0.529919i 1.28691 1.82862i 2.08096 0.292460i 0.875211 1.51591i 0.669131 0.743145i 1.36107 + 0.390281i 0.733018 2.11251i
61.20 0.961262 0.275637i 2.20161 + 0.309416i 0.848048 0.529919i 1.57433 + 1.58792i 2.20161 0.309416i 1.15550 2.00139i 0.669131 0.743145i 1.86756 + 0.535515i 1.95103 + 1.09246i
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.a 600
19.e even 9 1 inner 950.2.bc.a 600
25.d even 5 1 inner 950.2.bc.a 600
475.bc even 45 1 inner 950.2.bc.a 600

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

## Hecke kernels

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