# Properties

 Label 950.2.r.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$200q + 25q^{2} + 25q^{4} + q^{5} + 28q^{7} - 50q^{8} + 13q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$200q + 25q^{2} + 25q^{4} + q^{5} + 28q^{7} - 50q^{8} + 13q^{9} - 4q^{10} + 14q^{11} + 8q^{13} + q^{14} - 4q^{15} + 25q^{16} - 5q^{17} + 184q^{18} + 24q^{19} - 2q^{20} + 32q^{21} + 8q^{22} + 11q^{23} - 23q^{25} + 64q^{26} - 36q^{27} + 6q^{28} + 12q^{29} + 8q^{30} - 12q^{31} - 100q^{32} + 12q^{33} + 2q^{35} + 13q^{36} + 8q^{37} + 2q^{38} - 52q^{39} + q^{40} + 14q^{41} + 32q^{42} - 10q^{43} + 8q^{44} - 52q^{45} + 8q^{46} - 22q^{47} + 204q^{49} - 34q^{50} - 10q^{51} + 8q^{52} - 18q^{53} - 12q^{54} + 10q^{55} - 12q^{56} + 34q^{57} - 24q^{58} - 2q^{59} - 14q^{60} + 16q^{61} - 4q^{62} - 21q^{63} - 50q^{64} + 96q^{65} - 18q^{66} - 54q^{67} - 20q^{68} - 20q^{69} - 3q^{70} - 5q^{71} + 33q^{72} - 16q^{73} - 4q^{74} + 56q^{75} + 14q^{76} - 116q^{77} - 34q^{78} + q^{80} + 57q^{81} - 36q^{82} - 72q^{83} + 76q^{84} - 5q^{85} - 112q^{87} + 14q^{88} - 4q^{89} - 49q^{90} - 64q^{91} - 4q^{92} - 124q^{93} + 44q^{94} + 41q^{95} + 13q^{97} + 43q^{98} - 28q^{99} + O(q^{100})$$

## 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 −1.98778 2.20765i −0.978148 + 0.207912i −1.48151 + 1.67485i −1.98778 + 2.20765i 5.05644 0.309017 + 0.951057i −0.608880 + 5.79310i 1.82053 + 1.29833i
11.2 −0.104528 0.994522i −1.96943 2.18727i −0.978148 + 0.207912i 2.20352 + 0.380149i −1.96943 + 2.18727i 1.45173 0.309017 + 0.951057i −0.591926 + 5.63180i 0.147736 2.23118i
11.3 −0.104528 0.994522i −1.82439 2.02619i −0.978148 + 0.207912i −2.20812 0.352454i −1.82439 + 2.02619i −1.52448 0.309017 + 0.951057i −0.463466 + 4.40959i −0.119712 + 2.23286i
11.4 −0.104528 0.994522i −1.56704 1.74038i −0.978148 + 0.207912i −1.01249 + 1.99370i −1.56704 + 1.74038i −2.70259 0.309017 + 0.951057i −0.259705 + 2.47093i 2.08862 + 0.798548i
11.5 −0.104528 0.994522i −1.54095 1.71140i −0.978148 + 0.207912i 1.52065 1.63939i −1.54095 + 1.71140i 0.0891276 0.309017 + 0.951057i −0.240774 + 2.29081i −1.78936 1.34096i
11.6 −0.104528 0.994522i −1.20234 1.33534i −0.978148 + 0.207912i −0.901641 2.04623i −1.20234 + 1.33534i 1.04913 0.309017 + 0.951057i −0.0239117 + 0.227505i −1.94077 + 1.11059i
11.7 −0.104528 0.994522i −1.08024 1.19973i −0.978148 + 0.207912i 0.757710 + 2.10378i −1.08024 + 1.19973i 2.77680 0.309017 + 0.951057i 0.0411561 0.391575i 2.01305 0.973464i
11.8 −0.104528 0.994522i −1.00310 1.11405i −0.978148 + 0.207912i 2.10151 + 0.763969i −1.00310 + 1.11405i −3.65812 0.309017 + 0.951057i 0.0786767 0.748558i 0.540117 2.16986i
11.9 −0.104528 0.994522i −0.809690 0.899251i −0.978148 + 0.207912i −1.75482 1.38587i −0.809690 + 0.899251i 2.11907 0.309017 + 0.951057i 0.160530 1.52734i −1.19485 + 1.89006i
11.10 −0.104528 0.994522i −0.475190 0.527752i −0.978148 + 0.207912i 1.37456 1.76369i −0.475190 + 0.527752i −4.63378 0.309017 + 0.951057i 0.260869 2.48200i −1.89771 1.18267i
11.11 −0.104528 0.994522i −0.358798 0.398486i −0.978148 + 0.207912i −0.0794850 + 2.23465i −0.358798 + 0.398486i −0.651916 0.309017 + 0.951057i 0.283531 2.69761i 2.23072 0.154535i
11.12 −0.104528 0.994522i −0.302553 0.336019i −0.978148 + 0.207912i 0.565124 2.16348i −0.302553 + 0.336019i 3.20863 0.309017 + 0.951057i 0.292215 2.78024i −2.21070 0.335883i
11.13 −0.104528 0.994522i 0.220840 + 0.245267i −0.978148 + 0.207912i 2.16400 0.563111i 0.220840 0.245267i 4.03900 0.309017 + 0.951057i 0.302200 2.87524i −0.786226 2.09329i
11.14 −0.104528 0.994522i 0.343520 + 0.381518i −0.978148 + 0.207912i 1.76429 + 1.37378i 0.343520 0.381518i −3.26090 0.309017 + 0.951057i 0.286036 2.72145i 1.18184 1.89822i
11.15 −0.104528 0.994522i 0.370227 + 0.411179i −0.978148 + 0.207912i −1.78679 + 1.34439i 0.370227 0.411179i −1.46528 0.309017 + 0.951057i 0.281585 2.67911i 1.52379 + 1.63648i
11.16 −0.104528 0.994522i 0.459328 + 0.510136i −0.978148 + 0.207912i −2.23563 0.0442828i 0.459328 0.510136i 4.99570 0.309017 + 0.951057i 0.264329 2.51493i 0.189647 + 2.22801i
11.17 −0.104528 0.994522i 0.525231 + 0.583328i −0.978148 + 0.207912i −1.47087 1.68420i 0.525231 0.583328i −3.06829 0.309017 + 0.951057i 0.249181 2.37080i −1.52123 + 1.63886i
11.18 −0.104528 0.994522i 0.918911 + 1.02055i −0.978148 + 0.207912i −1.88261 + 1.20655i 0.918911 1.02055i −0.500112 0.309017 + 0.951057i 0.116452 1.10797i 1.39673 + 1.74618i
11.19 −0.104528 0.994522i 0.942721 + 1.04700i −0.978148 + 0.207912i −0.192905 2.22773i 0.942721 1.04700i −0.469500 0.309017 + 0.951057i 0.106104 1.00951i −2.19536 + 0.424709i
11.20 −0.104528 0.994522i 1.04186 + 1.15710i −0.978148 + 0.207912i 0.550620 + 2.16721i 1.04186 1.15710i 3.43419 0.309017 + 0.951057i 0.0601717 0.572496i 2.09779 0.774140i
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.b 200
19.c even 3 1 inner 950.2.r.b 200
25.d even 5 1 inner 950.2.r.b 200
475.r even 15 1 inner 950.2.r.b 200

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$20\!\cdots\!60$$$$T_{3}^{178} -$$$$10\!\cdots\!35$$$$T_{3}^{177} -$$$$59\!\cdots\!97$$$$T_{3}^{176} +$$$$42\!\cdots\!19$$$$T_{3}^{175} +$$$$58\!\cdots\!97$$$$T_{3}^{174} -$$$$41\!\cdots\!52$$$$T_{3}^{173} -$$$$18\!\cdots\!97$$$$T_{3}^{172} +$$$$11\!\cdots\!64$$$$T_{3}^{171} -$$$$26\!\cdots\!65$$$$T_{3}^{170} +$$$$17\!\cdots\!07$$$$T_{3}^{169} +$$$$47\!\cdots\!26$$$$T_{3}^{168} -$$$$25\!\cdots\!02$$$$T_{3}^{167} -$$$$34\!\cdots\!16$$$$T_{3}^{166} +$$$$13\!\cdots\!02$$$$T_{3}^{165} +$$$$43\!\cdots\!67$$$$T_{3}^{164} +$$$$22\!\cdots\!53$$$$T_{3}^{163} +$$$$18\!\cdots\!67$$$$T_{3}^{162} -$$$$89\!\cdots\!77$$$$T_{3}^{161} -$$$$22\!\cdots\!56$$$$T_{3}^{160} +$$$$56\!\cdots\!71$$$$T_{3}^{159} +$$$$11\!\cdots\!54$$$$T_{3}^{158} +$$$$75\!\cdots\!85$$$$T_{3}^{157} +$$$$12\!\cdots\!43$$$$T_{3}^{156} -$$$$34\!\cdots\!72$$$$T_{3}^{155} -$$$$75\!\cdots\!63$$$$T_{3}^{154} +$$$$16\!\cdots\!42$$$$T_{3}^{153} +$$$$63\!\cdots\!56$$$$T_{3}^{152} +$$$$91\!\cdots\!17$$$$T_{3}^{151} -$$$$21\!\cdots\!95$$$$T_{3}^{150} -$$$$17\!\cdots\!40$$$$T_{3}^{149} -$$$$10\!\cdots\!01$$$$T_{3}^{148} +$$$$88\!\cdots\!96$$$$T_{3}^{147} +$$$$18\!\cdots\!41$$$$T_{3}^{146} +$$$$22\!\cdots\!33$$$$T_{3}^{145} -$$$$10\!\cdots\!79$$$$T_{3}^{144} -$$$$67\!\cdots\!56$$$$T_{3}^{143} +$$$$14\!\cdots\!57$$$$T_{3}^{142} +$$$$46\!\cdots\!30$$$$T_{3}^{141} +$$$$27\!\cdots\!55$$$$T_{3}^{140} -$$$$77\!\cdots\!16$$$$T_{3}^{139} -$$$$25\!\cdots\!29$$$$T_{3}^{138} -$$$$13\!\cdots\!98$$$$T_{3}^{137} +$$$$96\!\cdots\!04$$$$T_{3}^{136} +$$$$13\!\cdots\!15$$$$T_{3}^{135} +$$$$11\!\cdots\!40$$$$T_{3}^{134} -$$$$51\!\cdots\!62$$$$T_{3}^{133} -$$$$35\!\cdots\!88$$$$T_{3}^{132} -$$$$64\!\cdots\!73$$$$T_{3}^{131} +$$$$20\!\cdots\!90$$$$T_{3}^{130} +$$$$20\!\cdots\!42$$$$T_{3}^{129} -$$$$39\!\cdots\!63$$$$T_{3}^{128} -$$$$11\!\cdots\!98$$$$T_{3}^{127} -$$$$24\!\cdots\!17$$$$T_{3}^{126} +$$$$24\!\cdots\!98$$$$T_{3}^{125} +$$$$24\!\cdots\!86$$$$T_{3}^{124} +$$$$12\!\cdots\!18$$$$T_{3}^{123} -$$$$89\!\cdots\!47$$$$T_{3}^{122} -$$$$13\!\cdots\!54$$$$T_{3}^{121} +$$$$25\!\cdots\!89$$$$T_{3}^{120} +$$$$52\!\cdots\!45$$$$T_{3}^{119} +$$$$15\!\cdots\!16$$$$T_{3}^{118} -$$$$42\!\cdots\!34$$$$T_{3}^{117} -$$$$84\!\cdots\!19$$$$T_{3}^{116} -$$$$66\!\cdots\!92$$$$T_{3}^{115} +$$$$18\!\cdots\!26$$$$T_{3}^{114} +$$$$41\!\cdots\!14$$$$T_{3}^{113} +$$$$26\!\cdots\!37$$$$T_{3}^{112} -$$$$10\!\cdots\!40$$$$T_{3}^{111} -$$$$36\!\cdots\!36$$$$T_{3}^{110} +$$$$29\!\cdots\!49$$$$T_{3}^{109} +$$$$13\!\cdots\!75$$$$T_{3}^{108} +$$$$11\!\cdots\!85$$$$T_{3}^{107} -$$$$20\!\cdots\!98$$$$T_{3}^{106} -$$$$48\!\cdots\!70$$$$T_{3}^{105} -$$$$40\!\cdots\!67$$$$T_{3}^{104} +$$$$99\!\cdots\!86$$$$T_{3}^{103} +$$$$33\!\cdots\!70$$$$T_{3}^{102} -$$$$30\!\cdots\!26$$$$T_{3}^{101} -$$$$99\!\cdots\!57$$$$T_{3}^{100} -$$$$55\!\cdots\!09$$$$T_{3}^{99} +$$$$13\!\cdots\!30$$$$T_{3}^{98} +$$$$21\!\cdots\!96$$$$T_{3}^{97} +$$$$18\!\cdots\!14$$$$T_{3}^{96} -$$$$40\!\cdots\!74$$$$T_{3}^{95} -$$$$14\!\cdots\!52$$$$T_{3}^{94} +$$$$24\!\cdots\!26$$$$T_{3}^{93} +$$$$36\!\cdots\!13$$$$T_{3}^{92} +$$$$10\!\cdots\!82$$$$T_{3}^{91} -$$$$41\!\cdots\!91$$$$T_{3}^{90} -$$$$37\!\cdots\!31$$$$T_{3}^{89} -$$$$53\!\cdots\!11$$$$T_{3}^{88} +$$$$67\!\cdots\!26$$$$T_{3}^{87} +$$$$35\!\cdots\!93$$$$T_{3}^{86} -$$$$56\!\cdots\!42$$$$T_{3}^{85} -$$$$81\!\cdots\!71$$$$T_{3}^{84} -$$$$38\!\cdots\!30$$$$T_{3}^{83} +$$$$93\!\cdots\!07$$$$T_{3}^{82} +$$$$18\!\cdots\!39$$$$T_{3}^{81} +$$$$34\!\cdots\!38$$$$T_{3}^{80} -$$$$26\!\cdots\!80$$$$T_{3}^{79} -$$$$37\!\cdots\!69$$$$T_{3}^{78} +$$$$19\!\cdots\!28$$$$T_{3}^{77} +$$$$77\!\cdots\!78$$$$T_{3}^{76} -$$$$16\!\cdots\!44$$$$T_{3}^{75} -$$$$75\!\cdots\!02$$$$T_{3}^{74} +$$$$48\!\cdots\!84$$$$T_{3}^{73} -$$$$24\!\cdots\!90$$$$T_{3}^{72} -$$$$14\!\cdots\!11$$$$T_{3}^{71} +$$$$21\!\cdots\!64$$$$T_{3}^{70} +$$$$26\!\cdots\!86$$$$T_{3}^{69} -$$$$38\!\cdots\!34$$$$T_{3}^{68} -$$$$16\!\cdots\!12$$$$T_{3}^{67} +$$$$33\!\cdots\!91$$$$T_{3}^{66} -$$$$68\!\cdots\!59$$$$T_{3}^{65} -$$$$14\!\cdots\!05$$$$T_{3}^{64} +$$$$21\!\cdots\!03$$$$T_{3}^{63} -$$$$34\!\cdots\!90$$$$T_{3}^{62} -$$$$28\!\cdots\!50$$$$T_{3}^{61} +$$$$46\!\cdots\!26$$$$T_{3}^{60} +$$$$16\!\cdots\!55$$$$T_{3}^{59} -$$$$30\!\cdots\!09$$$$T_{3}^{58} +$$$$40\!\cdots\!67$$$$T_{3}^{57} +$$$$56\!\cdots\!37$$$$T_{3}^{56} -$$$$14\!\cdots\!73$$$$T_{3}^{55} +$$$$10\!\cdots\!07$$$$T_{3}^{54} +$$$$12\!\cdots\!69$$$$T_{3}^{53} -$$$$12\!\cdots\!24$$$$T_{3}^{52} -$$$$64\!\cdots\!73$$$$T_{3}^{51} +$$$$61\!\cdots\!87$$$$T_{3}^{50} +$$$$17\!\cdots\!85$$$$T_{3}^{49} -$$$$55\!\cdots\!54$$$$T_{3}^{48} +$$$$41\!\cdots\!70$$$$T_{3}^{47} -$$$$16\!\cdots\!55$$$$T_{3}^{46} -$$$$74\!\cdots\!25$$$$T_{3}^{45} +$$$$16\!\cdots\!40$$$$T_{3}^{44} +$$$$42\!\cdots\!80$$$$T_{3}^{43} -$$$$89\!\cdots\!15$$$$T_{3}^{42} -$$$$15\!\cdots\!55$$$$T_{3}^{41} +$$$$32\!\cdots\!60$$$$T_{3}^{40} +$$$$36\!\cdots\!00$$$$T_{3}^{39} -$$$$65\!\cdots\!75$$$$T_{3}^{38} -$$$$42\!\cdots\!75$$$$T_{3}^{37} -$$$$19\!\cdots\!75$$$$T_{3}^{36} +$$$$49\!\cdots\!75$$$$T_{3}^{35} +$$$$71\!\cdots\!25$$$$T_{3}^{34} -$$$$17\!\cdots\!50$$$$T_{3}^{33} -$$$$30\!\cdots\!50$$$$T_{3}^{32} +$$$$20\!\cdots\!25$$$$T_{3}^{31} +$$$$71\!\cdots\!00$$$$T_{3}^{30} -$$$$97\!\cdots\!00$$$$T_{3}^{29} -$$$$57\!\cdots\!25$$$$T_{3}^{28} +$$$$34\!\cdots\!25$$$$T_{3}^{27} -$$$$25\!\cdots\!25$$$$T_{3}^{26} -$$$$84\!\cdots\!50$$$$T_{3}^{25} +$$$$12\!\cdots\!50$$$$T_{3}^{24} +$$$$12\!\cdots\!50$$$$T_{3}^{23} -$$$$28\!\cdots\!75$$$$T_{3}^{22} -$$$$60\!\cdots\!25$$$$T_{3}^{21} +$$$$39\!\cdots\!00$$$$T_{3}^{20} -$$$$19\!\cdots\!00$$$$T_{3}^{19} -$$$$78\!\cdots\!75$$$$T_{3}^{18} +$$$$13\!\cdots\!50$$$$T_{3}^{17} -$$$$86\!\cdots\!50$$$$T_{3}^{16} -$$$$36\!\cdots\!75$$$$T_{3}^{15} +$$$$16\!\cdots\!25$$$$T_{3}^{14} +$$$$44\!\cdots\!75$$$$T_{3}^{13} -$$$$14\!\cdots\!25$$$$T_{3}^{12} -$$$$14\!\cdots\!00$$$$T_{3}^{11} +$$$$28\!\cdots\!00$$$$T_{3}^{10} +$$$$28\!\cdots\!00$$$$T_{3}^{9} +$$$$19\!\cdots\!25$$$$T_{3}^{8} +$$$$59\!\cdots\!75$$$$T_{3}^{7} -$$$$23\!\cdots\!75$$$$T_{3}^{6} +$$$$33\!\cdots\!25$$$$T_{3}^{5} +$$$$14\!\cdots\!25$$$$T_{3}^{4} +$$$$10\!\cdots\!75$$$$T_{3}^{3} +$$$$47\!\cdots\!50$$$$T_{3}^{2} +$$$$21\!\cdots\!25$$$$T_{3} +$$$$57\!\cdots\!25$$">$$T_{3}^{200} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.