# Properties

 Label 1850.2.a.e.1.1 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$14.7723243739$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} -2.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +3.00000 q^{18} -6.00000 q^{19} -4.00000 q^{23} +2.00000 q^{26} +2.00000 q^{28} -4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} -3.00000 q^{36} +1.00000 q^{37} +6.00000 q^{38} -10.0000 q^{41} +4.00000 q^{43} +4.00000 q^{46} +2.00000 q^{47} -3.00000 q^{49} -2.00000 q^{52} -2.00000 q^{53} -2.00000 q^{56} -6.00000 q^{59} +4.00000 q^{62} -6.00000 q^{63} +1.00000 q^{64} -8.00000 q^{67} +6.00000 q^{68} +3.00000 q^{72} -8.00000 q^{73} -1.00000 q^{74} -6.00000 q^{76} +4.00000 q^{79} +9.00000 q^{81} +10.0000 q^{82} +12.0000 q^{83} -4.00000 q^{86} +6.00000 q^{89} -4.00000 q^{91} -4.00000 q^{92} -2.00000 q^{94} +10.0000 q^{97} +3.00000 q^{98} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 3.00000 0.707107
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ 2.00000 0.377964
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ −3.00000 −0.500000
$$37$$ 1.00000 0.164399
$$38$$ 6.00000 0.973329
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 4.00000 0.508001
$$63$$ −6.00000 −0.755929
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 3.00000 0.353553
$$73$$ −8.00000 −0.936329 −0.468165 0.883641i $$-0.655085\pi$$
−0.468165 + 0.883641i $$0.655085\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ −6.00000 −0.688247
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 10.0000 1.10432
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ −2.00000 −0.206284
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ 0 0
$$109$$ −20.0000 −1.91565 −0.957826 0.287348i $$-0.907226\pi$$
−0.957826 + 0.287348i $$0.907226\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 2.00000 0.188982
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 6.00000 0.554700
$$118$$ 6.00000 0.552345
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 6.00000 0.534522
$$127$$ −22.0000 −1.95218 −0.976092 0.217357i $$-0.930256\pi$$
−0.976092 + 0.217357i $$0.930256\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 0 0
$$133$$ −12.0000 −1.04053
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ 8.00000 0.662085
$$147$$ 0 0
$$148$$ 1.00000 0.0821995
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 6.00000 0.486664
$$153$$ −18.0000 −1.45521
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −22.0000 −1.75579 −0.877896 0.478852i $$-0.841053\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ −9.00000 −0.707107
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 18.0000 1.37649
$$172$$ 4.00000 0.304997
$$173$$ −2.00000 −0.152057 −0.0760286 0.997106i $$-0.524224\pi$$
−0.0760286 + 0.997106i $$0.524224\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ −14.0000 −1.04641 −0.523205 0.852207i $$-0.675264\pi$$
−0.523205 + 0.852207i $$0.675264\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 4.00000 0.296500
$$183$$ 0 0
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 2.00000 0.145865
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 0 0
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 10.0000 0.712470 0.356235 0.934396i $$-0.384060\pi$$
0.356235 + 0.934396i $$0.384060\pi$$
$$198$$ 0 0
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 10.0000 0.703598
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 12.0000 0.834058
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ 0 0
$$214$$ 8.00000 0.546869
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −8.00000 −0.543075
$$218$$ 20.0000 1.35457
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ −14.0000 −0.937509 −0.468755 0.883328i $$-0.655297\pi$$
−0.468755 + 0.883328i $$0.655297\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 24.0000 1.57229 0.786146 0.618041i $$-0.212073\pi$$
0.786146 + 0.618041i $$0.212073\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ 0 0
$$238$$ −12.0000 −0.777844
$$239$$ 28.0000 1.81117 0.905585 0.424165i $$-0.139432\pi$$
0.905585 + 0.424165i $$0.139432\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 11.0000 0.707107
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 12.0000 0.763542
$$248$$ 4.00000 0.254000
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ −6.00000 −0.377964
$$253$$ 0 0
$$254$$ 22.0000 1.38040
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 30.0000 1.87135 0.935674 0.352865i $$-0.114792\pi$$
0.935674 + 0.352865i $$0.114792\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −14.0000 −0.864923
$$263$$ 2.00000 0.123325 0.0616626 0.998097i $$-0.480360\pi$$
0.0616626 + 0.998097i $$0.480360\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 12.0000 0.735767
$$267$$ 0 0
$$268$$ −8.00000 −0.488678
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 18.0000 1.08152 0.540758 0.841178i $$-0.318138\pi$$
0.540758 + 0.841178i $$0.318138\pi$$
$$278$$ 20.0000 1.19952
$$279$$ 12.0000 0.718421
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −28.0000 −1.66443 −0.832214 0.554455i $$-0.812927\pi$$
−0.832214 + 0.554455i $$0.812927\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −20.0000 −1.18056
$$288$$ 3.00000 0.176777
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −8.00000 −0.468165
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ 0 0
$$298$$ −14.0000 −0.810998
$$299$$ 8.00000 0.462652
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ −16.0000 −0.920697
$$303$$ 0 0
$$304$$ −6.00000 −0.344124
$$305$$ 0 0
$$306$$ 18.0000 1.02899
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ 22.0000 1.24153
$$315$$ 0 0
$$316$$ 4.00000 0.225018
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 8.00000 0.445823
$$323$$ −36.0000 −2.00309
$$324$$ 9.00000 0.500000
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ 0 0
$$328$$ 10.0000 0.552158
$$329$$ 4.00000 0.220527
$$330$$ 0 0
$$331$$ −34.0000 −1.86881 −0.934405 0.356214i $$-0.884068\pi$$
−0.934405 + 0.356214i $$0.884068\pi$$
$$332$$ 12.0000 0.658586
$$333$$ −3.00000 −0.164399
$$334$$ 16.0000 0.875481
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −8.00000 −0.435788 −0.217894 0.975972i $$-0.569919\pi$$
−0.217894 + 0.975972i $$0.569919\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −18.0000 −0.973329
$$343$$ −20.0000 −1.07990
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 2.00000 0.107521
$$347$$ −36.0000 −1.93258 −0.966291 0.257454i $$-0.917117\pi$$
−0.966291 + 0.257454i $$0.917117\pi$$
$$348$$ 0 0
$$349$$ 30.0000 1.60586 0.802932 0.596071i $$-0.203272\pi$$
0.802932 + 0.596071i $$0.203272\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 14.0000 0.739923
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ −2.00000 −0.105118
$$363$$ 0 0
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 26.0000 1.35719 0.678594 0.734513i $$-0.262589\pi$$
0.678594 + 0.734513i $$0.262589\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 30.0000 1.56174
$$370$$ 0 0
$$371$$ −4.00000 −0.207670
$$372$$ 0 0
$$373$$ 34.0000 1.76045 0.880227 0.474554i $$-0.157390\pi$$
0.880227 + 0.474554i $$0.157390\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −2.00000 −0.103142
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 12.0000 0.616399 0.308199 0.951322i $$-0.400274\pi$$
0.308199 + 0.951322i $$0.400274\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −4.00000 −0.204658
$$383$$ 28.0000 1.43073 0.715367 0.698749i $$-0.246260\pi$$
0.715367 + 0.698749i $$0.246260\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ −12.0000 −0.609994
$$388$$ 10.0000 0.507673
$$389$$ −16.0000 −0.811232 −0.405616 0.914044i $$-0.632943\pi$$
−0.405616 + 0.914044i $$0.632943\pi$$
$$390$$ 0 0
$$391$$ −24.0000 −1.21373
$$392$$ 3.00000 0.151523
$$393$$ 0 0
$$394$$ −10.0000 −0.503793
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 20.0000 1.00251
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2.00000 −0.0998752 −0.0499376 0.998752i $$-0.515902\pi$$
−0.0499376 + 0.998752i $$0.515902\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −12.0000 −0.590481
$$414$$ −12.0000 −0.589768
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ 36.0000 1.75453 0.877266 0.480004i $$-0.159365\pi$$
0.877266 + 0.480004i $$0.159365\pi$$
$$422$$ 16.0000 0.778868
$$423$$ −6.00000 −0.291730
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −8.00000 −0.386695
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 20.0000 0.963366 0.481683 0.876346i $$-0.340026\pi$$
0.481683 + 0.876346i $$0.340026\pi$$
$$432$$ 0 0
$$433$$ 16.0000 0.768911 0.384455 0.923144i $$-0.374389\pi$$
0.384455 + 0.923144i $$0.374389\pi$$
$$434$$ 8.00000 0.384012
$$435$$ 0 0
$$436$$ −20.0000 −0.957826
$$437$$ 24.0000 1.14808
$$438$$ 0 0
$$439$$ 4.00000 0.190910 0.0954548 0.995434i $$-0.469569\pi$$
0.0954548 + 0.995434i $$0.469569\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 12.0000 0.570782
$$443$$ −16.0000 −0.760183 −0.380091 0.924949i $$-0.624107\pi$$
−0.380091 + 0.924949i $$0.624107\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 14.0000 0.662919
$$447$$ 0 0
$$448$$ 2.00000 0.0944911
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −2.00000 −0.0940721
$$453$$ 0 0
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −24.0000 −1.11178
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 6.00000 0.277350
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 6.00000 0.276172
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 12.0000 0.550019
$$477$$ 6.00000 0.274721
$$478$$ −28.0000 −1.28069
$$479$$ 12.0000 0.548294 0.274147 0.961688i $$-0.411605\pi$$
0.274147 + 0.961688i $$0.411605\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ −10.0000 −0.455488
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −24.0000 −1.08754 −0.543772 0.839233i $$-0.683004\pi$$
−0.543772 + 0.839233i $$0.683004\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −12.0000 −0.539906
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 30.0000 1.34298 0.671492 0.741012i $$-0.265654\pi$$
0.671492 + 0.741012i $$0.265654\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −2.00000 −0.0892644
$$503$$ −40.0000 −1.78351 −0.891756 0.452517i $$-0.850526\pi$$
−0.891756 + 0.452517i $$0.850526\pi$$
$$504$$ 6.00000 0.267261
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −22.0000 −0.976092
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ −16.0000 −0.707798
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −30.0000 −1.32324
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −2.00000 −0.0878750
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 26.0000 1.13908 0.569540 0.821963i $$-0.307121\pi$$
0.569540 + 0.821963i $$0.307121\pi$$
$$522$$ 0 0
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 14.0000 0.611593
$$525$$ 0 0
$$526$$ −2.00000 −0.0872041
$$527$$ −24.0000 −1.04546
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 18.0000 0.781133
$$532$$ −12.0000 −0.520266
$$533$$ 20.0000 0.866296
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ 0 0
$$538$$ 10.0000 0.431131
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −16.0000 −0.687894 −0.343947 0.938989i $$-0.611764\pi$$
−0.343947 + 0.938989i $$0.611764\pi$$
$$542$$ 8.00000 0.343629
$$543$$ 0 0
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ −18.0000 −0.764747
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ −12.0000 −0.508001
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −6.00000 −0.253095
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 28.0000 1.17693
$$567$$ 18.0000 0.755929
$$568$$ 0 0
$$569$$ −34.0000 −1.42535 −0.712677 0.701492i $$-0.752517\pi$$
−0.712677 + 0.701492i $$0.752517\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 20.0000 0.834784
$$575$$ 0 0
$$576$$ −3.00000 −0.125000
$$577$$ 38.0000 1.58196 0.790980 0.611842i $$-0.209571\pi$$
0.790980 + 0.611842i $$0.209571\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 8.00000 0.331042
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ 24.0000 0.988903
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1.00000 0.0410997
$$593$$ −24.0000 −0.985562 −0.492781 0.870153i $$-0.664020\pi$$
−0.492781 + 0.870153i $$0.664020\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 14.0000 0.573462
$$597$$ 0 0
$$598$$ −8.00000 −0.327144
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ 24.0000 0.977356
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 36.0000 1.46119 0.730597 0.682808i $$-0.239242\pi$$
0.730597 + 0.682808i $$0.239242\pi$$
$$608$$ 6.00000 0.243332
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −4.00000 −0.161823
$$612$$ −18.0000 −0.727607
$$613$$ 30.0000 1.21169 0.605844 0.795583i $$-0.292835\pi$$
0.605844 + 0.795583i $$0.292835\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12.0000 −0.483102 −0.241551 0.970388i $$-0.577656\pi$$
−0.241551 + 0.970388i $$0.577656\pi$$
$$618$$ 0 0
$$619$$ −32.0000 −1.28619 −0.643094 0.765787i $$-0.722350\pi$$
−0.643094 + 0.765787i $$0.722350\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 12.0000 0.480770
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −14.0000 −0.559553
$$627$$ 0 0
$$628$$ −22.0000 −0.877896
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ −4.00000 −0.159111
$$633$$ 0 0
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6.00000 0.237729
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 14.0000 0.552967 0.276483 0.961019i $$-0.410831\pi$$
0.276483 + 0.961019i $$0.410831\pi$$
$$642$$ 0 0
$$643$$ 44.0000 1.73519 0.867595 0.497271i $$-0.165665\pi$$
0.867595 + 0.497271i $$0.165665\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ 36.0000 1.41640
$$647$$ 16.0000 0.629025 0.314512 0.949253i $$-0.398159\pi$$
0.314512 + 0.949253i $$0.398159\pi$$
$$648$$ −9.00000 −0.353553
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 12.0000 0.469956
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −10.0000 −0.390434
$$657$$ 24.0000 0.936329
$$658$$ −4.00000 −0.155936
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ −28.0000 −1.08907 −0.544537 0.838737i $$-0.683295\pi$$
−0.544537 + 0.838737i $$0.683295\pi$$
$$662$$ 34.0000 1.32145
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ 3.00000 0.116248
$$667$$ 0 0
$$668$$ −16.0000 −0.619059
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 36.0000 1.38770 0.693849 0.720121i $$-0.255914\pi$$
0.693849 + 0.720121i $$0.255914\pi$$
$$674$$ 8.00000 0.308148
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −30.0000 −1.15299 −0.576497 0.817099i $$-0.695581\pi$$
−0.576497 + 0.817099i $$0.695581\pi$$
$$678$$ 0 0
$$679$$ 20.0000 0.767530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 20.0000 0.765279 0.382639 0.923898i $$-0.375015\pi$$
0.382639 + 0.923898i $$0.375015\pi$$
$$684$$ 18.0000 0.688247
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ 4.00000 0.152388
$$690$$ 0 0
$$691$$ 4.00000 0.152167 0.0760836 0.997101i $$-0.475758\pi$$
0.0760836 + 0.997101i $$0.475758\pi$$
$$692$$ −2.00000 −0.0760286
$$693$$ 0 0
$$694$$ 36.0000 1.36654
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −60.0000 −2.27266
$$698$$ −30.0000 −1.13552
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −4.00000 −0.151078 −0.0755390 0.997143i $$-0.524068\pi$$
−0.0755390 + 0.997143i $$0.524068\pi$$
$$702$$ 0 0
$$703$$ −6.00000 −0.226294
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ −20.0000 −0.752177
$$708$$ 0 0
$$709$$ −40.0000 −1.50223 −0.751116 0.660171i $$-0.770484\pi$$
−0.751116 + 0.660171i $$0.770484\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ −6.00000 −0.224860
$$713$$ 16.0000 0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −14.0000 −0.523205
$$717$$ 0 0
$$718$$ 8.00000 0.298557
$$719$$ −16.0000 −0.596699 −0.298350 0.954457i $$-0.596436\pi$$
−0.298350 + 0.954457i $$0.596436\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −17.0000 −0.632674
$$723$$ 0 0
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −28.0000 −1.03846 −0.519231 0.854634i $$-0.673782\pi$$
−0.519231 + 0.854634i $$0.673782\pi$$
$$728$$ 4.00000 0.148250
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ 38.0000 1.40356 0.701781 0.712393i $$-0.252388\pi$$
0.701781 + 0.712393i $$0.252388\pi$$
$$734$$ −26.0000 −0.959678
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 0 0
$$738$$ −30.0000 −1.10432
$$739$$ −36.0000 −1.32428 −0.662141 0.749380i $$-0.730352\pi$$
−0.662141 + 0.749380i $$0.730352\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 4.00000 0.146845
$$743$$ −26.0000 −0.953847 −0.476924 0.878945i $$-0.658248\pi$$
−0.476924 + 0.878945i $$0.658248\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −34.0000 −1.24483
$$747$$ −36.0000 −1.31717
$$748$$ 0 0
$$749$$ −16.0000 −0.584627
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 2.00000 0.0729325
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −14.0000 −0.508839 −0.254419 0.967094i $$-0.581884\pi$$
−0.254419 + 0.967094i $$0.581884\pi$$
$$758$$ −12.0000 −0.435860
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 34.0000 1.23250 0.616250 0.787551i $$-0.288651\pi$$
0.616250 + 0.787551i $$0.288651\pi$$
$$762$$ 0 0
$$763$$ −40.0000 −1.44810
$$764$$ 4.00000 0.144715
$$765$$ 0 0
$$766$$ −28.0000 −1.01168
$$767$$ 12.0000 0.433295
$$768$$ 0 0
$$769$$ 10.0000 0.360609 0.180305 0.983611i $$-0.442292\pi$$
0.180305 + 0.983611i $$0.442292\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −10.0000 −0.359908
$$773$$ 54.0000 1.94225 0.971123 0.238581i $$-0.0766824\pi$$
0.971123 + 0.238581i $$0.0766824\pi$$
$$774$$ 12.0000 0.431331
$$775$$ 0 0
$$776$$ −10.0000 −0.358979
$$777$$ 0 0
$$778$$ 16.0000 0.573628
$$779$$ 60.0000 2.14972
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 24.0000 0.858238
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −40.0000 −1.42585 −0.712923 0.701242i $$-0.752629\pi$$
−0.712923 + 0.701242i $$0.752629\pi$$
$$788$$ 10.0000 0.356235
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −4.00000 −0.142224
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ −20.0000 −0.708881
$$797$$ −6.00000 −0.212531 −0.106265 0.994338i $$-0.533889\pi$$
−0.106265 + 0.994338i $$0.533889\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ 0 0
$$801$$ −18.0000 −0.635999
$$802$$ 2.00000 0.0706225
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 0 0
$$808$$ 10.0000 0.351799
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ −36.0000 −1.26413 −0.632065 0.774915i $$-0.717793\pi$$
−0.632065 + 0.774915i $$0.717793\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −24.0000 −0.839654
$$818$$ −14.0000 −0.489499
$$819$$ 12.0000 0.419314
$$820$$ 0 0
$$821$$ −50.0000 −1.74501 −0.872506 0.488603i $$-0.837507\pi$$
−0.872506 + 0.488603i $$0.837507\pi$$
$$822$$ 0 0
$$823$$ 26.0000 0.906303 0.453152 0.891434i $$-0.350300\pi$$
0.453152 + 0.891434i $$0.350300\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 12.0000 0.417029
$$829$$ −20.0000 −0.694629 −0.347314 0.937749i $$-0.612906\pi$$
−0.347314 + 0.937749i $$0.612906\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −2.00000 −0.0693375
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −20.0000 −0.690889
$$839$$ −48.0000 −1.65714 −0.828572 0.559883i $$-0.810846\pi$$
−0.828572 + 0.559883i $$0.810846\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ −36.0000 −1.24064
$$843$$ 0 0
$$844$$ −16.0000 −0.550743
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ −22.0000 −0.755929
$$848$$ −2.00000 −0.0686803
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −4.00000 −0.137118
$$852$$ 0 0
$$853$$ −30.0000 −1.02718 −0.513590 0.858036i $$-0.671685\pi$$
−0.513590 + 0.858036i $$0.671685\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 8.00000 0.273434
$$857$$ 2.00000 0.0683187 0.0341593 0.999416i $$-0.489125\pi$$
0.0341593 + 0.999416i $$0.489125\pi$$
$$858$$ 0 0
$$859$$ 10.0000 0.341196 0.170598 0.985341i $$-0.445430\pi$$
0.170598 + 0.985341i $$0.445430\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −20.0000 −0.681203
$$863$$ 6.00000 0.204242 0.102121 0.994772i $$-0.467437\pi$$
0.102121 + 0.994772i $$0.467437\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −16.0000 −0.543702
$$867$$ 0 0
$$868$$ −8.00000 −0.271538
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ 20.0000 0.677285
$$873$$ −30.0000 −1.01535
$$874$$ −24.0000 −0.811812
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 34.0000 1.14810 0.574049 0.818821i $$-0.305372\pi$$
0.574049 + 0.818821i $$0.305372\pi$$
$$878$$ −4.00000 −0.134993
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ −9.00000 −0.303046
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ 16.0000 0.537531
$$887$$ 50.0000 1.67884 0.839418 0.543487i $$-0.182896\pi$$
0.839418 + 0.543487i $$0.182896\pi$$
$$888$$ 0 0
$$889$$ −44.0000 −1.47571
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −14.0000 −0.468755
$$893$$ −12.0000 −0.401565
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ −30.0000 −1.00111
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 2.00000 0.0665190
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 20.0000 0.664089 0.332045 0.943264i $$-0.392262\pi$$
0.332045 + 0.943264i $$0.392262\pi$$
$$908$$ −20.0000 −0.663723
$$909$$ 30.0000 0.995037
$$910$$ 0 0
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ 2.00000 0.0660819
$$917$$ 28.0000 0.924641
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −4.00000 −0.131448
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ 24.0000 0.786146
$$933$$ 0 0
$$934$$ −36.0000 −1.17796
$$935$$ 0 0
$$936$$ −6.00000 −0.196116
$$937$$ −4.00000 −0.130674 −0.0653372 0.997863i $$-0.520812\pi$$
−0.0653372 + 0.997863i $$0.520812\pi$$
$$938$$ 16.0000 0.522419
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −22.0000 −0.717180 −0.358590 0.933495i $$-0.616742\pi$$
−0.358590 + 0.933495i $$0.616742\pi$$
$$942$$ 0 0
$$943$$ 40.0000 1.30258
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −28.0000 −0.909878 −0.454939 0.890523i $$-0.650339\pi$$
−0.454939 + 0.890523i $$0.650339\pi$$
$$948$$ 0 0
$$949$$ 16.0000 0.519382
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −12.0000 −0.388922
$$953$$ 24.0000 0.777436 0.388718 0.921357i $$-0.372918\pi$$
0.388718 + 0.921357i $$0.372918\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 28.0000 0.905585
$$957$$ 0 0
$$958$$ −12.0000 −0.387702
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 2.00000 0.0644826
$$963$$ 24.0000 0.773389
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 28.0000 0.900419 0.450210 0.892923i $$-0.351349\pi$$
0.450210 + 0.892923i $$0.351349\pi$$
$$968$$ 11.0000 0.353553
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 32.0000 1.02693 0.513464 0.858111i $$-0.328362\pi$$
0.513464 + 0.858111i $$0.328362\pi$$
$$972$$ 0 0
$$973$$ −40.0000 −1.28234
$$974$$ 24.0000 0.769010
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 60.0000 1.91565
$$982$$ −24.0000 −0.765871
$$983$$ 38.0000 1.21201 0.606006 0.795460i $$-0.292771\pi$$
0.606006 + 0.795460i $$0.292771\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 12.0000 0.381771
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ 48.0000 1.52477 0.762385 0.647124i $$-0.224028\pi$$
0.762385 + 0.647124i $$0.224028\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 26.0000 0.823428 0.411714 0.911313i $$-0.364930\pi$$
0.411714 + 0.911313i $$0.364930\pi$$
$$998$$ −30.0000 −0.949633
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1850.2.a.e.1.1 1
5.2 odd 4 370.2.b.a.149.1 2
5.3 odd 4 370.2.b.a.149.2 yes 2
5.4 even 2 1850.2.a.j.1.1 1
15.2 even 4 3330.2.d.f.1999.2 2
15.8 even 4 3330.2.d.f.1999.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.a.149.1 2 5.2 odd 4
370.2.b.a.149.2 yes 2 5.3 odd 4
1850.2.a.e.1.1 1 1.1 even 1 trivial
1850.2.a.j.1.1 1 5.4 even 2
3330.2.d.f.1999.1 2 15.8 even 4
3330.2.d.f.1999.2 2 15.2 even 4