# Properties

 Label 950.2.r.a Level $950$ Weight $2$ Character orbit 950.r Analytic conductor $7.586$ Analytic rank $0$ Dimension $200$ 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.r (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$200$$ Relative dimension: $$25$$ over $$\Q(\zeta_{15})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$200q - 25q^{2} + 25q^{4} + q^{5} - 36q^{7} + 50q^{8} + 37q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$200q - 25q^{2} + 25q^{4} + q^{5} - 36q^{7} + 50q^{8} + 37q^{9} + 4q^{10} - 2q^{11} + 8q^{13} + 7q^{14} + 10q^{15} + 25q^{16} - 13q^{17} - 216q^{18} - 36q^{19} - 2q^{20} + 4q^{22} - 13q^{23} + 33q^{25} - 64q^{26} - 2q^{28} + 12q^{29} - 20q^{30} + 12q^{31} + 100q^{32} - 12q^{33} + 8q^{34} - 22q^{35} + 37q^{36} + 8q^{37} + 18q^{38} + 28q^{39} - q^{40} + 2q^{41} + 70q^{43} - 4q^{44} + 60q^{45} + 24q^{46} - 22q^{47} + 156q^{49} + 26q^{50} - 42q^{51} + 8q^{52} - 18q^{53} - 10q^{55} - 4q^{56} + 74q^{57} + 24q^{58} + 50q^{59} + 12q^{61} - 4q^{62} - 45q^{63} - 50q^{64} + 120q^{65} + 2q^{66} - 18q^{67} - 84q^{68} + 4q^{69} - 13q^{70} + 19q^{71} - 17q^{72} - 24q^{73} + 4q^{74} - 136q^{75} - 6q^{76} - 12q^{77} - 26q^{78} - 8q^{79} + q^{80} - 63q^{81} + 28q^{82} + 16q^{83} - 20q^{84} - 5q^{85} + 20q^{86} - 64q^{87} + 2q^{88} + 16q^{89} - 65q^{90} + 72q^{91} + 12q^{92} - 8q^{93} - 44q^{94} - 73q^{95} - 11q^{97} - 27q^{98} + 16q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 0.104528 + 0.994522i −2.23670 2.48410i −0.978148 + 0.207912i 2.20697 0.359579i 2.23670 2.48410i −4.60253 −0.309017 0.951057i −0.854373 + 8.12881i 0.588300 + 2.15729i
11.2 0.104528 + 0.994522i −2.02768 2.25196i −0.978148 + 0.207912i −0.209287 2.22625i 2.02768 2.25196i 1.71696 −0.309017 0.951057i −0.646281 + 6.14895i 2.19218 0.440848i
11.3 0.104528 + 0.994522i −1.82987 2.03228i −0.978148 + 0.207912i −1.78813 + 1.34260i 1.82987 2.03228i 1.23976 −0.309017 0.951057i −0.468141 + 4.45407i −1.52216 1.63800i
11.4 0.104528 + 0.994522i −1.76134 1.95617i −0.978148 + 0.207912i −1.69383 + 1.45977i 1.76134 1.95617i −4.49296 −0.309017 0.951057i −0.410684 + 3.90739i −1.62883 1.53197i
11.5 0.104528 + 0.994522i −1.74362 1.93649i −0.978148 + 0.207912i −1.31558 1.80811i 1.74362 1.93649i −0.103099 −0.309017 0.951057i −0.396182 + 3.76942i 1.66069 1.49737i
11.6 0.104528 + 0.994522i −1.36974 1.52125i −0.978148 + 0.207912i 2.23379 0.100934i 1.36974 1.52125i 5.15541 −0.309017 0.951057i −0.124432 + 1.18389i 0.333876 + 2.21100i
11.7 0.104528 + 0.994522i −1.32901 1.47601i −0.978148 + 0.207912i 1.09596 + 1.94907i 1.32901 1.47601i −0.0211674 −0.309017 0.951057i −0.0987671 + 0.939706i −1.82383 + 1.29369i
11.8 0.104528 + 0.994522i −0.955473 1.06116i −0.978148 + 0.207912i 0.663784 2.13527i 0.955473 1.06116i −1.22437 −0.309017 0.951057i 0.100453 0.955746i 2.19296 + 0.436951i
11.9 0.104528 + 0.994522i −0.588897 0.654036i −0.978148 + 0.207912i −2.19070 0.448160i 0.588897 0.654036i −3.02472 −0.309017 0.951057i 0.232622 2.21325i 0.216715 2.22554i
11.10 0.104528 + 0.994522i −0.286801 0.318525i −0.978148 + 0.207912i −1.03010 + 1.98466i 0.286801 0.318525i 3.29125 −0.309017 0.951057i 0.294382 2.80086i −2.08147 0.817006i
11.11 0.104528 + 0.994522i −0.278343 0.309131i −0.978148 + 0.207912i −1.78815 1.34258i 0.278343 0.309131i 2.38873 −0.309017 0.951057i 0.295498 2.81148i 1.14831 1.91869i
11.12 0.104528 + 0.994522i −0.261407 0.290322i −0.978148 + 0.207912i 1.28550 + 1.82962i 0.261407 0.290322i −0.691099 −0.309017 0.951057i 0.297632 2.83178i −1.68522 + 1.46970i
11.13 0.104528 + 0.994522i −0.175744 0.195183i −0.978148 + 0.207912i 1.83434 1.27875i 0.175744 0.195183i −2.84995 −0.309017 0.951057i 0.306375 2.91496i 1.46349 + 1.69062i
11.14 0.104528 + 0.994522i −0.0896612 0.0995788i −0.978148 + 0.207912i −1.17911 + 1.89992i 0.0896612 0.0995788i 2.27794 −0.309017 0.951057i 0.311709 2.96571i −2.01277 0.974050i
11.15 0.104528 + 0.994522i 0.388315 + 0.431268i −0.978148 + 0.207912i 1.69879 + 1.45400i −0.388315 + 0.431268i −3.61865 −0.309017 0.951057i 0.278382 2.64863i −1.26846 + 1.84147i
11.16 0.104528 + 0.994522i 0.831705 + 0.923702i −0.978148 + 0.207912i 2.14903 + 0.617800i −0.831705 + 0.923702i 0.754763 −0.309017 0.951057i 0.152093 1.44707i −0.389781 + 2.20183i
11.17 0.104528 + 0.994522i 0.856524 + 0.951266i −0.978148 + 0.207912i −1.58933 1.57290i −0.856524 + 0.951266i 1.37782 −0.309017 0.951057i 0.142311 1.35400i 1.39816 1.74504i
11.18 0.104528 + 0.994522i 0.895592 + 0.994656i −0.978148 + 0.207912i −2.22420 + 0.230050i −0.895592 + 0.994656i −2.57007 −0.309017 0.951057i 0.126330 1.20195i −0.461283 2.18797i
11.19 0.104528 + 0.994522i 1.14496 + 1.27160i −0.978148 + 0.207912i 1.79074 1.33913i −1.14496 + 1.27160i 1.05033 −0.309017 0.951057i 0.00753716 0.0717113i 1.51897 + 1.64095i
11.20 0.104528 + 0.994522i 1.32430 + 1.47079i −0.978148 + 0.207912i 0.202278 2.22690i −1.32430 + 1.47079i −4.37854 −0.309017 0.951057i −0.0958531 + 0.911982i 2.23584 0.0316046i
See next 80 embeddings (of 200 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 881.25 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.c even 3 1 inner
25.d even 5 1 inner
475.r even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.r.a 200
19.c even 3 1 inner 950.2.r.a 200
25.d even 5 1 inner 950.2.r.a 200
475.r even 15 1 inner 950.2.r.a 200

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.r.a 200 1.a even 1 1 trivial
950.2.r.a 200 19.c even 3 1 inner
950.2.r.a 200 25.d even 5 1 inner
950.2.r.a 200 475.r even 15 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$11\!\cdots\!72$$$$T_{3}^{178} -$$$$12\!\cdots\!85$$$$T_{3}^{177} +$$$$58\!\cdots\!71$$$$T_{3}^{176} +$$$$26\!\cdots\!45$$$$T_{3}^{175} +$$$$33\!\cdots\!23$$$$T_{3}^{174} -$$$$16\!\cdots\!00$$$$T_{3}^{173} -$$$$10\!\cdots\!09$$$$T_{3}^{172} -$$$$94\!\cdots\!30$$$$T_{3}^{171} +$$$$10\!\cdots\!99$$$$T_{3}^{170} +$$$$26\!\cdots\!35$$$$T_{3}^{169} -$$$$40\!\cdots\!22$$$$T_{3}^{168} -$$$$16\!\cdots\!80$$$$T_{3}^{167} -$$$$29\!\cdots\!62$$$$T_{3}^{166} -$$$$12\!\cdots\!20$$$$T_{3}^{165} +$$$$63\!\cdots\!95$$$$T_{3}^{164} +$$$$31\!\cdots\!45$$$$T_{3}^{163} -$$$$51\!\cdots\!43$$$$T_{3}^{162} -$$$$25\!\cdots\!25$$$$T_{3}^{161} +$$$$15\!\cdots\!18$$$$T_{3}^{160} +$$$$37\!\cdots\!95$$$$T_{3}^{159} +$$$$14\!\cdots\!04$$$$T_{3}^{158} +$$$$15\!\cdots\!95$$$$T_{3}^{157} -$$$$23\!\cdots\!99$$$$T_{3}^{156} -$$$$20\!\cdots\!70$$$$T_{3}^{155} +$$$$14\!\cdots\!55$$$$T_{3}^{154} +$$$$11\!\cdots\!20$$$$T_{3}^{153} -$$$$24\!\cdots\!46$$$$T_{3}^{152} +$$$$60\!\cdots\!25$$$$T_{3}^{151} -$$$$45\!\cdots\!47$$$$T_{3}^{150} -$$$$63\!\cdots\!30$$$$T_{3}^{149} +$$$$50\!\cdots\!63$$$$T_{3}^{148} +$$$$54\!\cdots\!60$$$$T_{3}^{147} -$$$$24\!\cdots\!45$$$$T_{3}^{146} -$$$$18\!\cdots\!55$$$$T_{3}^{145} +$$$$57\!\cdots\!31$$$$T_{3}^{144} -$$$$69\!\cdots\!80$$$$T_{3}^{143} +$$$$86\!\cdots\!69$$$$T_{3}^{142} +$$$$12\!\cdots\!80$$$$T_{3}^{141} -$$$$69\!\cdots\!81$$$$T_{3}^{140} -$$$$73\!\cdots\!60$$$$T_{3}^{139} +$$$$23\!\cdots\!43$$$$T_{3}^{138} +$$$$11\!\cdots\!10$$$$T_{3}^{137} +$$$$43\!\cdots\!02$$$$T_{3}^{136} +$$$$14\!\cdots\!65$$$$T_{3}^{135} -$$$$10\!\cdots\!42$$$$T_{3}^{134} -$$$$13\!\cdots\!30$$$$T_{3}^{133} +$$$$58\!\cdots\!40$$$$T_{3}^{132} +$$$$48\!\cdots\!15$$$$T_{3}^{131} -$$$$12\!\cdots\!52$$$$T_{3}^{130} +$$$$34\!\cdots\!60$$$$T_{3}^{129} -$$$$70\!\cdots\!91$$$$T_{3}^{128} -$$$$13\!\cdots\!20$$$$T_{3}^{127} +$$$$73\!\cdots\!55$$$$T_{3}^{126} +$$$$71\!\cdots\!50$$$$T_{3}^{125} -$$$$28\!\cdots\!12$$$$T_{3}^{124} -$$$$13\!\cdots\!50$$$$T_{3}^{123} +$$$$12\!\cdots\!17$$$$T_{3}^{122} -$$$$65\!\cdots\!80$$$$T_{3}^{121} +$$$$49\!\cdots\!13$$$$T_{3}^{120} +$$$$57\!\cdots\!95$$$$T_{3}^{119} -$$$$29\!\cdots\!72$$$$T_{3}^{118} -$$$$17\!\cdots\!60$$$$T_{3}^{117} +$$$$73\!\cdots\!19$$$$T_{3}^{116} -$$$$56\!\cdots\!00$$$$T_{3}^{115} +$$$$11\!\cdots\!64$$$$T_{3}^{114} +$$$$27\!\cdots\!30$$$$T_{3}^{113} -$$$$17\!\cdots\!63$$$$T_{3}^{112} -$$$$10\!\cdots\!70$$$$T_{3}^{111} +$$$$69\!\cdots\!34$$$$T_{3}^{110} +$$$$67\!\cdots\!25$$$$T_{3}^{109} -$$$$80\!\cdots\!27$$$$T_{3}^{108} +$$$$95\!\cdots\!35$$$$T_{3}^{107} -$$$$53\!\cdots\!54$$$$T_{3}^{106} -$$$$38\!\cdots\!20$$$$T_{3}^{105} +$$$$33\!\cdots\!01$$$$T_{3}^{104} +$$$$70\!\cdots\!10$$$$T_{3}^{103} -$$$$80\!\cdots\!42$$$$T_{3}^{102} +$$$$46\!\cdots\!20$$$$T_{3}^{101} -$$$$24\!\cdots\!77$$$$T_{3}^{100} -$$$$16\!\cdots\!95$$$$T_{3}^{99} +$$$$89\!\cdots\!44$$$$T_{3}^{98} -$$$$50\!\cdots\!00$$$$T_{3}^{97} -$$$$30\!\cdots\!56$$$$T_{3}^{96} +$$$$22\!\cdots\!50$$$$T_{3}^{95} +$$$$28\!\cdots\!06$$$$T_{3}^{94} -$$$$71\!\cdots\!80$$$$T_{3}^{93} +$$$$14\!\cdots\!05$$$$T_{3}^{92} +$$$$17\!\cdots\!30$$$$T_{3}^{91} -$$$$64\!\cdots\!17$$$$T_{3}^{90} +$$$$54\!\cdots\!75$$$$T_{3}^{89} +$$$$88\!\cdots\!93$$$$T_{3}^{88} -$$$$17\!\cdots\!20$$$$T_{3}^{87} +$$$$14\!\cdots\!13$$$$T_{3}^{86} +$$$$15\!\cdots\!60$$$$T_{3}^{85} -$$$$85\!\cdots\!63$$$$T_{3}^{84} +$$$$60\!\cdots\!30$$$$T_{3}^{83} +$$$$14\!\cdots\!05$$$$T_{3}^{82} -$$$$24\!\cdots\!05$$$$T_{3}^{81} +$$$$23\!\cdots\!74$$$$T_{3}^{80} +$$$$37\!\cdots\!90$$$$T_{3}^{79} -$$$$70\!\cdots\!57$$$$T_{3}^{78} +$$$$15\!\cdots\!20$$$$T_{3}^{77} +$$$$16\!\cdots\!10$$$$T_{3}^{76} -$$$$20\!\cdots\!00$$$$T_{3}^{75} -$$$$11\!\cdots\!06$$$$T_{3}^{74} +$$$$43\!\cdots\!90$$$$T_{3}^{73} -$$$$33\!\cdots\!74$$$$T_{3}^{72} -$$$$29\!\cdots\!85$$$$T_{3}^{71} +$$$$11\!\cdots\!78$$$$T_{3}^{70} -$$$$72\!\cdots\!70$$$$T_{3}^{69} -$$$$13\!\cdots\!26$$$$T_{3}^{68} +$$$$23\!\cdots\!80$$$$T_{3}^{67} -$$$$20\!\cdots\!17$$$$T_{3}^{66} -$$$$28\!\cdots\!95$$$$T_{3}^{65} +$$$$32\!\cdots\!19$$$$T_{3}^{64} +$$$$11\!\cdots\!55$$$$T_{3}^{63} -$$$$52\!\cdots\!82$$$$T_{3}^{62} +$$$$45\!\cdots\!80$$$$T_{3}^{61} +$$$$34\!\cdots\!18$$$$T_{3}^{60} -$$$$69\!\cdots\!65$$$$T_{3}^{59} +$$$$16\!\cdots\!63$$$$T_{3}^{58} +$$$$46\!\cdots\!55$$$$T_{3}^{57} -$$$$53\!\cdots\!83$$$$T_{3}^{56} +$$$$78\!\cdots\!75$$$$T_{3}^{55} +$$$$57\!\cdots\!95$$$$T_{3}^{54} -$$$$43\!\cdots\!05$$$$T_{3}^{53} -$$$$41\!\cdots\!02$$$$T_{3}^{52} +$$$$35\!\cdots\!25$$$$T_{3}^{51} +$$$$14\!\cdots\!23$$$$T_{3}^{50} -$$$$11\!\cdots\!15$$$$T_{3}^{49} +$$$$78\!\cdots\!44$$$$T_{3}^{48} -$$$$34\!\cdots\!10$$$$T_{3}^{47} -$$$$12\!\cdots\!79$$$$T_{3}^{46} +$$$$63\!\cdots\!95$$$$T_{3}^{45} +$$$$50\!\cdots\!32$$$$T_{3}^{44} -$$$$32\!\cdots\!10$$$$T_{3}^{43} +$$$$10\!\cdots\!87$$$$T_{3}^{42} -$$$$14\!\cdots\!45$$$$T_{3}^{41} -$$$$13\!\cdots\!44$$$$T_{3}^{40} +$$$$79\!\cdots\!30$$$$T_{3}^{39} +$$$$32\!\cdots\!05$$$$T_{3}^{38} -$$$$30\!\cdots\!35$$$$T_{3}^{37} +$$$$83\!\cdots\!65$$$$T_{3}^{36} -$$$$34\!\cdots\!45$$$$T_{3}^{35} -$$$$33\!\cdots\!85$$$$T_{3}^{34} +$$$$37\!\cdots\!00$$$$T_{3}^{33} +$$$$33\!\cdots\!90$$$$T_{3}^{32} -$$$$73\!\cdots\!75$$$$T_{3}^{31} -$$$$33\!\cdots\!50$$$$T_{3}^{30} -$$$$99\!\cdots\!00$$$$T_{3}^{29} +$$$$65\!\cdots\!25$$$$T_{3}^{28} +$$$$44\!\cdots\!25$$$$T_{3}^{27} +$$$$79\!\cdots\!25$$$$T_{3}^{26} -$$$$17\!\cdots\!00$$$$T_{3}^{25} -$$$$54\!\cdots\!00$$$$T_{3}^{24} +$$$$27\!\cdots\!00$$$$T_{3}^{23} +$$$$11\!\cdots\!75$$$$T_{3}^{22} +$$$$13\!\cdots\!25$$$$T_{3}^{21} -$$$$11\!\cdots\!50$$$$T_{3}^{20} -$$$$29\!\cdots\!00$$$$T_{3}^{19} +$$$$80\!\cdots\!75$$$$T_{3}^{18} -$$$$12\!\cdots\!00$$$$T_{3}^{17} +$$$$29\!\cdots\!00$$$$T_{3}^{16} -$$$$14\!\cdots\!75$$$$T_{3}^{15} +$$$$33\!\cdots\!25$$$$T_{3}^{14} -$$$$15\!\cdots\!25$$$$T_{3}^{13} +$$$$75\!\cdots\!25$$$$T_{3}^{12} -$$$$19\!\cdots\!50$$$$T_{3}^{11} +$$$$61\!\cdots\!00$$$$T_{3}^{10} -$$$$23\!\cdots\!00$$$$T_{3}^{9} +$$$$50\!\cdots\!25$$$$T_{3}^{8} -$$$$73\!\cdots\!25$$$$T_{3}^{7} +$$$$26\!\cdots\!75$$$$T_{3}^{6} -$$$$76\!\cdots\!75$$$$T_{3}^{5} +$$$$82\!\cdots\!75$$$$T_{3}^{4} -$$$$26\!\cdots\!75$$$$T_{3}^{3} +$$$$99\!\cdots\!00$$$$T_{3}^{2} -$$$$23\!\cdots\!75$$$$T_{3} +$$$$13\!\cdots\!25$$">$$T_{3}^{200} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.