# Properties

 Label 1850.2.a.k.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 $1$

# Learn more

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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -3.00000 q^{9} -4.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} -3.00000 q^{18} -4.00000 q^{19} -4.00000 q^{22} -2.00000 q^{26} -6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} -3.00000 q^{36} +1.00000 q^{37} -4.00000 q^{38} -6.00000 q^{41} -4.00000 q^{43} -4.00000 q^{44} +8.00000 q^{47} -7.00000 q^{49} -2.00000 q^{52} -10.0000 q^{53} -6.00000 q^{58} +4.00000 q^{59} +10.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +8.00000 q^{67} +2.00000 q^{68} -3.00000 q^{72} -10.0000 q^{73} +1.00000 q^{74} -4.00000 q^{76} -4.00000 q^{79} +9.00000 q^{81} -6.00000 q^{82} -4.00000 q^{86} -4.00000 q^{88} +2.00000 q^{89} +8.00000 q^{94} -6.00000 q^{97} -7.00000 q^{98} +12.0000 q^{99} +O(q^{100})$$

## 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
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ −3.00000 −0.707107
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 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
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ −3.00000 −0.500000
$$37$$ 1.00000 0.164399
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ −6.00000 −0.662589
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ −4.00000 −0.426401
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ 12.0000 1.20605
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 0 0
$$109$$ 18.0000 1.72409 0.862044 0.506834i $$-0.169184\pi$$
0.862044 + 0.506834i $$0.169184\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 6.00000 0.554700
$$118$$ 4.00000 0.368230
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000 0.905357
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 8.00000 0.668994
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ −10.0000 −0.827606
$$147$$ 0 0
$$148$$ 1.00000 0.0821995
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ −24.0000 −1.95309 −0.976546 0.215308i $$-0.930924\pi$$
−0.976546 + 0.215308i $$0.930924\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 6.00000 0.478852 0.239426 0.970915i $$-0.423041\pi$$
0.239426 + 0.970915i $$0.423041\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 9.00000 0.707107
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 12.0000 0.917663
$$172$$ −4.00000 −0.304997
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ 2.00000 0.149906
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −8.00000 −0.585018
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 12.0000 0.852803
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 6.00000 0.422159
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ 28.0000 1.92760 0.963800 0.266627i $$-0.0859092\pi$$
0.963800 + 0.266627i $$0.0859092\pi$$
$$212$$ −10.0000 −0.686803
$$213$$ 0 0
$$214$$ 8.00000 0.546869
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 18.0000 1.21911
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 0 0
$$229$$ −18.0000 −1.18947 −0.594737 0.803921i $$-0.702744\pi$$
−0.594737 + 0.803921i $$0.702744\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 0 0
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ −4.00000 −0.254000
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 18.0000 1.11417
$$262$$ −12.0000 −0.741362
$$263$$ 32.0000 1.97320 0.986602 0.163144i $$-0.0521635\pi$$
0.986602 + 0.163144i $$0.0521635\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 8.00000 0.488678
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 12.0000 0.718421
$$280$$ 0 0
$$281$$ 2.00000 0.119310 0.0596550 0.998219i $$-0.481000\pi$$
0.0596550 + 0.998219i $$0.481000\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ −3.00000 −0.176777
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −10.0000 −0.585206
$$293$$ −18.0000 −1.05157 −0.525786 0.850617i $$-0.676229\pi$$
−0.525786 + 0.850617i $$0.676229\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −24.0000 −1.38104
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ 32.0000 1.82634 0.913168 0.407583i $$-0.133628\pi$$
0.913168 + 0.407583i $$0.133628\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 6.00000 0.338600
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −8.00000 −0.445132
$$324$$ 9.00000 0.500000
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ −6.00000 −0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 0 0
$$333$$ −3.00000 −0.164399
$$334$$ 16.0000 0.875481
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −34.0000 −1.85210 −0.926049 0.377403i $$-0.876817\pi$$
−0.926049 + 0.377403i $$0.876817\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ 12.0000 0.648886
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ −4.00000 −0.214731 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$348$$ 0 0
$$349$$ −18.0000 −0.963518 −0.481759 0.876304i $$-0.660002\pi$$
−0.481759 + 0.876304i $$0.660002\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −4.00000 −0.213201
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 2.00000 0.106000
$$357$$ 0 0
$$358$$ 4.00000 0.211407
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −2.00000 −0.105118
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ 18.0000 0.937043
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −12.0000 −0.613973
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 12.0000 0.609994
$$388$$ −6.00000 −0.304604
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −7.00000 −0.353553
$$393$$ 0 0
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ 12.0000 0.603023
$$397$$ −34.0000 −1.70641 −0.853206 0.521575i $$-0.825345\pi$$
−0.853206 + 0.521575i $$0.825345\pi$$
$$398$$ −20.0000 −1.00251
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −22.0000 −1.09863 −0.549314 0.835616i $$-0.685111\pi$$
−0.549314 + 0.835616i $$0.685111\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −4.00000 −0.198273
$$408$$ 0 0
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 0 0
$$418$$ 16.0000 0.782586
$$419$$ 28.0000 1.36789 0.683945 0.729534i $$-0.260263\pi$$
0.683945 + 0.729534i $$0.260263\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 28.0000 1.36302
$$423$$ −24.0000 −1.16692
$$424$$ −10.0000 −0.485643
$$425$$ 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 $$-0.659974\pi$$
−0.481683 + 0.876346i $$0.659974\pi$$
$$432$$ 0 0
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 18.0000 0.862044
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 0 0
$$441$$ 21.0000 1.00000
$$442$$ −4.00000 −0.190261
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 16.0000 0.757622
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ −14.0000 −0.658505
$$453$$ 0 0
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ −18.0000 −0.841085
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 10.0000 0.465746 0.232873 0.972507i $$-0.425187\pi$$
0.232873 + 0.972507i $$0.425187\pi$$
$$462$$ 0 0
$$463$$ 40.0000 1.85896 0.929479 0.368875i $$-0.120257\pi$$
0.929479 + 0.368875i $$0.120257\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 22.0000 1.01913
$$467$$ −28.0000 −1.29569 −0.647843 0.761774i $$-0.724329\pi$$
−0.647843 + 0.761774i $$0.724329\pi$$
$$468$$ 6.00000 0.277350
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 4.00000 0.184115
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 30.0000 1.37361
$$478$$ 12.0000 0.548867
$$479$$ 28.0000 1.27935 0.639676 0.768644i $$-0.279068\pi$$
0.639676 + 0.768644i $$0.279068\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 2.00000 0.0910975
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 16.0000 0.725029 0.362515 0.931978i $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 0 0
$$493$$ −12.0000 −0.540453
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 20.0000 0.892644
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −14.0000 −0.617514
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −32.0000 −1.40736
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −38.0000 −1.66481 −0.832405 0.554168i $$-0.813037\pi$$
−0.832405 + 0.554168i $$0.813037\pi$$
$$522$$ 18.0000 0.787839
$$523$$ −12.0000 −0.524723 −0.262362 0.964970i $$-0.584501\pi$$
−0.262362 + 0.964970i $$0.584501\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ 12.0000 0.519778
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ 0 0
$$538$$ 14.0000 0.603583
$$539$$ 28.0000 1.20605
$$540$$ 0 0
$$541$$ −6.00000 −0.257960 −0.128980 0.991647i $$-0.541170\pi$$
−0.128980 + 0.991647i $$0.541170\pi$$
$$542$$ −8.00000 −0.343629
$$543$$ 0 0
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −4.00000 −0.171028 −0.0855138 0.996337i $$-0.527253\pi$$
−0.0855138 + 0.996337i $$0.527253\pi$$
$$548$$ 6.00000 0.256307
$$549$$ −30.0000 −1.28037
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ 12.0000 0.508001
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 2.00000 0.0843649
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 8.00000 0.334497
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −3.00000 −0.125000
$$577$$ −46.0000 −1.91501 −0.957503 0.288425i $$-0.906868\pi$$
−0.957503 + 0.288425i $$0.906868\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 40.0000 1.65663
$$584$$ −10.0000 −0.413803
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ 16.0000 0.659269
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1.00000 0.0410997
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ −24.0000 −0.977356
$$604$$ −24.0000 −0.976546
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −24.0000 −0.974130 −0.487065 0.873366i $$-0.661933\pi$$
−0.487065 + 0.873366i $$0.661933\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −16.0000 −0.647291
$$612$$ −6.00000 −0.242536
$$613$$ 6.00000 0.242338 0.121169 0.992632i $$-0.461336\pi$$
0.121169 + 0.992632i $$0.461336\pi$$
$$614$$ 32.0000 1.29141
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 30.0000 1.20775 0.603877 0.797077i $$-0.293622\pi$$
0.603877 + 0.797077i $$0.293622\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 12.0000 0.481156
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 0 0
$$628$$ 6.00000 0.239426
$$629$$ 2.00000 0.0797452
$$630$$ 0 0
$$631$$ −36.0000 −1.43314 −0.716569 0.697517i $$-0.754288\pi$$
−0.716569 + 0.697517i $$0.754288\pi$$
$$632$$ −4.00000 −0.159111
$$633$$ 0 0
$$634$$ −2.00000 −0.0794301
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 14.0000 0.554700
$$638$$ 24.0000 0.950169
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ 44.0000 1.73519 0.867595 0.497271i $$-0.165665\pi$$
0.867595 + 0.497271i $$0.165665\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ −8.00000 −0.314512 −0.157256 0.987558i $$-0.550265\pi$$
−0.157256 + 0.987558i $$0.550265\pi$$
$$648$$ 9.00000 0.353553
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ −34.0000 −1.33052 −0.665261 0.746611i $$-0.731680\pi$$
−0.665261 + 0.746611i $$0.731680\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 30.0000 1.17041
$$658$$ 0 0
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −3.00000 −0.116248
$$667$$ 0 0
$$668$$ 16.0000 0.619059
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ 0 0
$$673$$ −10.0000 −0.385472 −0.192736 0.981251i $$-0.561736\pi$$
−0.192736 + 0.981251i $$0.561736\pi$$
$$674$$ −34.0000 −1.30963
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −2.00000 −0.0768662 −0.0384331 0.999261i $$-0.512237\pi$$
−0.0384331 + 0.999261i $$0.512237\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 16.0000 0.612672
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 12.0000 0.458831
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 14.0000 0.532200
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ −18.0000 −0.681310
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 50.0000 1.88847 0.944237 0.329267i $$-0.106802\pi$$
0.944237 + 0.329267i $$0.106802\pi$$
$$702$$ 0 0
$$703$$ −4.00000 −0.150863
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −30.0000 −1.12667 −0.563337 0.826227i $$-0.690483\pi$$
−0.563337 + 0.826227i $$0.690483\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ 2.00000 0.0749532
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 32.0000 1.19340 0.596699 0.802465i $$-0.296479\pi$$
0.596699 + 0.802465i $$0.296479\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −3.00000 −0.111648
$$723$$ 0 0
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −16.0000 −0.593407 −0.296704 0.954970i $$-0.595887\pi$$
−0.296704 + 0.954970i $$0.595887\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ −34.0000 −1.25582 −0.627909 0.778287i $$-0.716089\pi$$
−0.627909 + 0.778287i $$0.716089\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −32.0000 −1.17874
$$738$$ 18.0000 0.662589
$$739$$ 36.0000 1.32428 0.662141 0.749380i $$-0.269648\pi$$
0.662141 + 0.749380i $$0.269648\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 14.0000 0.512576
$$747$$ 0 0
$$748$$ −8.00000 −0.292509
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 0 0
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ −28.0000 −1.01701
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −14.0000 −0.503871
$$773$$ 22.0000 0.791285 0.395643 0.918405i $$-0.370522\pi$$
0.395643 + 0.918405i $$0.370522\pi$$
$$774$$ 12.0000 0.431331
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ −14.0000 −0.501924
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −7.00000 −0.250000
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −48.0000 −1.71102 −0.855508 0.517790i $$-0.826755\pi$$
−0.855508 + 0.517790i $$0.826755\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 12.0000 0.426401
$$793$$ −20.0000 −0.710221
$$794$$ −34.0000 −1.20661
$$795$$ 0 0
$$796$$ −20.0000 −0.708881
$$797$$ 30.0000 1.06265 0.531327 0.847167i $$-0.321693\pi$$
0.531327 + 0.847167i $$0.321693\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ −22.0000 −0.776847
$$803$$ 40.0000 1.41157
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 0 0
$$808$$ 6.00000 0.211079
$$809$$ 26.0000 0.914111 0.457056 0.889438i $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ 0 0
$$811$$ −20.0000 −0.702295 −0.351147 0.936320i $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −4.00000 −0.140200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 16.0000 0.559769
$$818$$ 2.00000 0.0699284
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ −8.00000 −0.278862 −0.139431 0.990232i $$-0.544527\pi$$
−0.139431 + 0.990232i $$0.544527\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −2.00000 −0.0693375
$$833$$ −14.0000 −0.485071
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ 28.0000 0.967244
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −6.00000 −0.206774
$$843$$ 0 0
$$844$$ 28.0000 0.963800
$$845$$ 0 0
$$846$$ −24.0000 −0.825137
$$847$$ 0 0
$$848$$ −10.0000 −0.343401
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −10.0000 −0.342393 −0.171197 0.985237i $$-0.554763\pi$$
−0.171197 + 0.985237i $$0.554763\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 8.00000 0.273434
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −20.0000 −0.681203
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 14.0000 0.475739
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ −16.0000 −0.542139
$$872$$ 18.0000 0.609557
$$873$$ 18.0000 0.609208
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −42.0000 −1.41824 −0.709120 0.705088i $$-0.750907\pi$$
−0.709120 + 0.705088i $$0.750907\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 21.0000 0.707107
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ −56.0000 −1.88030 −0.940148 0.340766i $$-0.889313\pi$$
−0.940148 + 0.340766i $$0.889313\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −36.0000 −1.20605
$$892$$ 16.0000 0.535720
$$893$$ −32.0000 −1.07084
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −6.00000 −0.200223
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ −20.0000 −0.666297
$$902$$ 24.0000 0.799113
$$903$$ 0 0
$$904$$ −14.0000 −0.465633
$$905$$ 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
$$909$$ −18.0000 −0.597022
$$910$$ 0 0
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 18.0000 0.595387
$$915$$ 0 0
$$916$$ −18.0000 −0.594737
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 20.0000 0.659739 0.329870 0.944027i $$-0.392995\pi$$
0.329870 + 0.944027i $$0.392995\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 10.0000 0.329332
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 40.0000 1.31448
$$927$$ 0 0
$$928$$ −6.00000 −0.196960
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ 28.0000 0.917663
$$932$$ 22.0000 0.720634
$$933$$ 0 0
$$934$$ −28.0000 −0.916188
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ 14.0000 0.457360 0.228680 0.973502i $$-0.426559\pi$$
0.228680 + 0.973502i $$0.426559\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −42.0000 −1.36916 −0.684580 0.728937i $$-0.740015\pi$$
−0.684580 + 0.728937i $$0.740015\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ 4.00000 0.129983 0.0649913 0.997886i $$-0.479298\pi$$
0.0649913 + 0.997886i $$0.479298\pi$$
$$948$$ 0 0
$$949$$ 20.0000 0.649227
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 30.0000 0.971286
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ 28.0000 0.904639
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −2.00000 −0.0644826
$$963$$ −24.0000 −0.773389
$$964$$ 2.00000 0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ 50.0000 1.59964 0.799821 0.600239i $$-0.204928\pi$$
0.799821 + 0.600239i $$0.204928\pi$$
$$978$$ 0 0
$$979$$ −8.00000 −0.255681
$$980$$ 0 0
$$981$$ −54.0000 −1.72409
$$982$$ 20.0000 0.638226
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 20.0000 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ −20.0000 −0.633089
$$999$$ 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.k.1.1 1
5.2 odd 4 1850.2.b.d.149.2 2
5.3 odd 4 1850.2.b.d.149.1 2
5.4 even 2 370.2.a.b.1.1 1
15.14 odd 2 3330.2.a.w.1.1 1
20.19 odd 2 2960.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.b.1.1 1 5.4 even 2
1850.2.a.k.1.1 1 1.1 even 1 trivial
1850.2.b.d.149.1 2 5.3 odd 4
1850.2.b.d.149.2 2 5.2 odd 4
2960.2.a.g.1.1 1 20.19 odd 2
3330.2.a.w.1.1 1 15.14 odd 2