# Properties

 Label 370.2.a.b.1.1 Level $370$ Weight $2$ Character 370.1 Self dual yes Analytic conductor $2.954$ 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] = [370,2,Mod(1,370)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(370, 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("370.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 370.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: $$2.95446487479$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 370.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} -3.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +2.00000 q^{13} +1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} +1.00000 q^{25} -2.00000 q^{26} -6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} -3.00000 q^{36} -1.00000 q^{37} +4.00000 q^{38} +1.00000 q^{40} -6.00000 q^{41} +4.00000 q^{43} -4.00000 q^{44} +3.00000 q^{45} -8.00000 q^{47} -7.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} +10.0000 q^{53} +4.00000 q^{55} +6.00000 q^{58} +4.00000 q^{59} +10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -8.00000 q^{67} -2.00000 q^{68} +3.00000 q^{72} +10.0000 q^{73} +1.00000 q^{74} -4.00000 q^{76} -4.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} +6.00000 q^{82} +2.00000 q^{85} -4.00000 q^{86} +4.00000 q^{88} +2.00000 q^{89} -3.00000 q^{90} +8.00000 q^{94} +4.00000 q^{95} +6.00000 q^{97} +7.00000 q^{98} +12.0000 q^{99} +O(q^{100})$$

## 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
$$5$$ −1.00000 −0.447214
$$6$$ 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
$$10$$ 1.00000 0.316228
$$11$$ −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 $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 3.00000 0.707107
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −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
$$40$$ 1.00000 0.158114
$$41$$ −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 $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 3.00000 0.447214
$$46$$ 0 0
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 0 0
$$55$$ 4.00000 0.539360
$$56$$ 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
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ −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 $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 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 $$0.572244\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 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
$$85$$ 2.00000 0.216930
$$86$$ −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 $$0.466196\pi$$
$$90$$ −3.00000 −0.316228
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 4.00000 0.410391
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 7.00000 0.707107
$$99$$ 12.0000 1.20605
$$100$$ 1.00000 0.100000
$$101$$ 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 $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ 0 0
$$109$$ 18.0000 1.72409 0.862044 0.506834i $$-0.169184\pi$$
0.862044 + 0.506834i $$0.169184\pi$$
$$110$$ −4.00000 −0.381385
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 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
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 2.00000 0.175412
$$131$$ −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 $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 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
$$145$$ 6.00000 0.498273
$$146$$ −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
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ −6.00000 −0.478852 −0.239426 0.970915i $$-0.576959\pi$$
−0.239426 + 0.970915i $$0.576959\pi$$
$$158$$ 4.00000 0.318223
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ −9.00000 −0.707107
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −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
$$170$$ −2.00000 −0.153393
$$171$$ 12.0000 0.917663
$$172$$ 4.00000 0.304997
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ 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 $$0.452238\pi$$
$$180$$ 3.00000 0.223607
$$181$$ −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
$$185$$ 1.00000 0.0735215
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ −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 $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ −12.0000 −0.852803
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ −6.00000 −0.422159
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 6.00000 0.419058
$$206$$ 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
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −18.0000 −1.21911
$$219$$ 0 0
$$220$$ 4.00000 0.269680
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 0 0
$$225$$ −3.00000 −0.200000
$$226$$ −14.0000 −0.931266
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ 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 $$-0.756149\pi$$
−0.720634 + 0.693316i $$0.756149\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 8.00000 0.521862
$$236$$ 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
$$245$$ 7.00000 0.447214
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ 4.00000 0.254000
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ 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 $$-0.356166\pi$$
0.436648 + 0.899632i $$0.356166\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −2.00000 −0.124035
$$261$$ 18.0000 1.11417
$$262$$ 12.0000 0.741362
$$263$$ −32.0000 −1.97320 −0.986602 0.163144i $$-0.947836\pi$$
−0.986602 + 0.163144i $$0.947836\pi$$
$$264$$ 0 0
$$265$$ −10.0000 −0.614295
$$266$$ 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
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ 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 $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ 3.00000 0.176777
$$289$$ −13.0000 −0.764706
$$290$$ −6.00000 −0.352332
$$291$$ 0 0
$$292$$ 10.0000 0.585206
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 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
$$305$$ −10.0000 −0.572598
$$306$$ −6.00000 −0.342997
$$307$$ −32.0000 −1.82634 −0.913168 0.407583i $$-0.866372\pi$$
−0.913168 + 0.407583i $$0.866372\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −4.00000 −0.227185
$$311$$ 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 $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 6.00000 0.338600
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ 2.00000 0.112331 0.0561656 0.998421i $$-0.482113\pi$$
0.0561656 + 0.998421i $$0.482113\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 8.00000 0.445132
$$324$$ 9.00000 0.500000
$$325$$ 2.00000 0.110940
$$326$$ 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
$$335$$ 8.00000 0.437087
$$336$$ 0 0
$$337$$ 34.0000 1.85210 0.926049 0.377403i $$-0.123183\pi$$
0.926049 + 0.377403i $$0.123183\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 0 0
$$340$$ 2.00000 0.108465
$$341$$ 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 $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ 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 $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 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 $$0.5\pi$$
$$360$$ −3.00000 −0.158114
$$361$$ −3.00000 −0.157895
$$362$$ 2.00000 0.105118
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −10.0000 −0.523424
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 0 0
$$369$$ 18.0000 0.937043
$$370$$ −1.00000 −0.0519875
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −14.0000 −0.724893 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ −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 $$0.755460\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 0 0
$$382$$ 12.0000 0.613973
$$383$$ −24.0000 −1.22634 −0.613171 0.789950i $$-0.710106\pi$$
−0.613171 + 0.789950i $$0.710106\pi$$
$$384$$ 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
$$395$$ 4.00000 0.201262
$$396$$ 12.0000 0.603023
$$397$$ 34.0000 1.70641 0.853206 0.521575i $$-0.174655\pi$$
0.853206 + 0.521575i $$0.174655\pi$$
$$398$$ 20.0000 1.00251
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −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
$$405$$ −9.00000 −0.447214
$$406$$ 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 $$0.484254\pi$$
$$410$$ −6.00000 −0.296319
$$411$$ 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
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −8.00000 −0.386695
$$429$$ 0 0
$$430$$ 4.00000 0.192897
$$431$$ −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 $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 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 $$0.658375\pi$$
$$440$$ −4.00000 −0.190693
$$441$$ 21.0000 1.00000
$$442$$ 4.00000 0.190261
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ 0 0
$$445$$ −2.00000 −0.0948091
$$446$$ 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 $$0.545218\pi$$
$$450$$ 3.00000 0.141421
$$451$$ 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 $$-0.638322\pi$$
−0.421002 + 0.907060i $$0.638322\pi$$
$$458$$ 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 $$-0.879743\pi$$
−0.929479 + 0.368875i $$0.879743\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 22.0000 1.01913
$$467$$ 28.0000 1.29569 0.647843 0.761774i $$-0.275671\pi$$
0.647843 + 0.761774i $$0.275671\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ 0 0
$$470$$ −8.00000 −0.369012
$$471$$ 0 0
$$472$$ −4.00000 −0.184115
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 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
$$485$$ −6.00000 −0.272446
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ 0 0
$$490$$ −7.00000 −0.316228
$$491$$ 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
$$495$$ −12.0000 −0.539360
$$496$$ −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 $$0.647743\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ −20.0000 −0.892644
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ −6.00000 −0.266996
$$506$$ 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
$$520$$ 2.00000 0.0877058
$$521$$ −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 $$-0.415499\pi$$
0.262362 + 0.964970i $$0.415499\pi$$
$$524$$ −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
$$530$$ 10.0000 0.434372
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ −12.0000 −0.519778
$$534$$ 0 0
$$535$$ 8.00000 0.345870
$$536$$ 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
$$545$$ −18.0000 −0.771035
$$546$$ 0 0
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ −30.0000 −1.28037
$$550$$ 4.00000 0.170561
$$551$$ 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 $$-0.486509\pi$$
0.0423714 + 0.999102i $$0.486509\pi$$
$$558$$ −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 $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ 0 0
$$565$$ −14.0000 −0.588984
$$566$$ −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 $$-0.0931316\pi$$
0.957503 + 0.288425i $$0.0931316\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 0 0
$$580$$ 6.00000 0.249136
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −40.0000 −1.65663
$$584$$ −10.0000 −0.413803
$$585$$ 6.00000 0.248069
$$586$$ −18.0000 −0.743573
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ 16.0000 0.659269
$$590$$ 4.00000 0.164677
$$591$$ 0 0
$$592$$ −1.00000 −0.0410997
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ 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
$$605$$ −5.00000 −0.203279
$$606$$ 0 0
$$607$$ 24.0000 0.974130 0.487065 0.873366i $$-0.338067\pi$$
0.487065 + 0.873366i $$0.338067\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ 10.0000 0.404888
$$611$$ −16.0000 −0.647291
$$612$$ 6.00000 0.242536
$$613$$ −6.00000 −0.242338 −0.121169 0.992632i $$-0.538664\pi$$
−0.121169 + 0.992632i $$0.538664\pi$$
$$614$$ 32.0000 1.29141
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 0 0
$$622$$ −12.0000 −0.481156
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 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
$$635$$ −8.00000 −0.317470
$$636$$ 0 0
$$637$$ −14.0000 −0.554700
$$638$$ −24.0000 −0.950169
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ 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 $$-0.834335\pi$$
−0.867595 + 0.497271i $$0.834335\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ 8.00000 0.314512 0.157256 0.987558i $$-0.449735\pi$$
0.157256 + 0.987558i $$0.449735\pi$$
$$648$$ −9.00000 −0.353553
$$649$$ −16.0000 −0.628055
$$650$$ −2.00000 −0.0784465
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 34.0000 1.33052 0.665261 0.746611i $$-0.268320\pi$$
0.665261 + 0.746611i $$0.268320\pi$$
$$654$$ 0 0
$$655$$ 12.0000 0.468879
$$656$$ −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
$$670$$ −8.00000 −0.309067
$$671$$ −40.0000 −1.54418
$$672$$ 0 0
$$673$$ 10.0000 0.385472 0.192736 0.981251i $$-0.438264\pi$$
0.192736 + 0.981251i $$0.438264\pi$$
$$674$$ −34.0000 −1.30963
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 2.00000 0.0768662 0.0384331 0.999261i $$-0.487763\pi$$
0.0384331 + 0.999261i $$0.487763\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −2.00000 −0.0766965
$$681$$ 0 0
$$682$$ −16.0000 −0.612672
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 12.0000 0.458831
$$685$$ 6.00000 0.229248
$$686$$ 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
$$695$$ 4.00000 0.151729
$$696$$ 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
$$715$$ 8.00000 0.299183
$$716$$ 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 $$0.296479\pi$$
$$720$$ 3.00000 0.111803
$$721$$ 0 0
$$722$$ 3.00000 0.111648
$$723$$ 0 0
$$724$$ −2.00000 −0.0743294
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 10.0000 0.370117
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ −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 $$0.269648\pi$$
$$740$$ 1.00000 0.0367607
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ 10.0000 0.366372
$$746$$ 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
$$755$$ 24.0000 0.873449
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 28.0000 1.01701
$$759$$ 0 0
$$760$$ −4.00000 −0.145095
$$761$$ 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
$$765$$ −6.00000 −0.216930
$$766$$ 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 $$-0.629478\pi$$
−0.395643 + 0.918405i $$0.629478\pi$$
$$774$$ 12.0000 0.431331
$$775$$ −4.00000 −0.143684
$$776$$ −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
$$785$$ 6.00000 0.214149
$$786$$ 0 0
$$787$$ 48.0000 1.71102 0.855508 0.517790i $$-0.173245\pi$$
0.855508 + 0.517790i $$0.173245\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 0 0
$$790$$ −4.00000 −0.142314
$$791$$ 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 $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ −1.00000 −0.0353553
$$801$$ −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 $$0.348904\pi$$
$$810$$ 9.00000 0.316228
$$811$$ −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
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ −2.00000 −0.0699284
$$819$$ 0 0
$$820$$ 6.00000 0.209529
$$821$$ 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 $$-0.455473\pi$$
0.139431 + 0.990232i $$0.455473\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 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
$$835$$ 16.0000 0.553703
$$836$$ 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
$$845$$ 9.00000 0.309609
$$846$$ −24.0000 −0.825137
$$847$$ 0 0
$$848$$ 10.0000 0.343401
$$849$$ 0 0
$$850$$ 2.00000 0.0685994
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ 0 0
$$855$$ −12.0000 −0.410391
$$856$$ 8.00000 0.273434
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 0 0
$$862$$ 20.0000 0.681203
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ 14.0000 0.476014
$$866$$ 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 $$-0.249093\pi$$
0.709120 + 0.705088i $$0.249093\pi$$
$$878$$ 20.0000 0.674967
$$879$$ 0 0
$$880$$ 4.00000 0.134840
$$881$$ −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 $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ 56.0000 1.88030 0.940148 0.340766i $$-0.110687\pi$$
0.940148 + 0.340766i $$0.110687\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 2.00000 0.0670402
$$891$$ −36.0000 −1.20605
$$892$$ −16.0000 −0.535720
$$893$$ 32.0000 1.07084
$$894$$ 0 0
$$895$$ −4.00000 −0.133705
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 6.00000 0.200223
$$899$$ 24.0000 0.800445
$$900$$ −3.00000 −0.100000
$$901$$ −20.0000 −0.666297
$$902$$ −24.0000 −0.799113
$$903$$ 0 0
$$904$$ −14.0000 −0.465633
$$905$$ 2.00000 0.0664822
$$906$$ 0 0
$$907$$ −20.0000 −0.664089 −0.332045 0.943264i $$-0.607738\pi$$
−0.332045 + 0.943264i $$0.607738\pi$$
$$908$$ 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
$$925$$ −1.00000 −0.0328798
$$926$$ 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
$$935$$ −8.00000 −0.261628
$$936$$ 6.00000 0.196116
$$937$$ −14.0000 −0.457360 −0.228680 0.973502i $$-0.573441\pi$$
−0.228680 + 0.973502i $$0.573441\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 8.00000 0.260931
$$941$$ −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 $$-0.520702\pi$$
−0.0649913 + 0.997886i $$0.520702\pi$$
$$948$$ 0 0
$$949$$ 20.0000 0.649227
$$950$$ 4.00000 0.129777
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 30.0000 0.971286
$$955$$ 12.0000 0.388311
$$956$$ 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
$$965$$ −14.0000 −0.450676
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 0 0
$$970$$ 6.00000 0.192648
$$971$$ −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 $$-0.795072\pi$$
−0.799821 + 0.600239i $$0.795072\pi$$
$$978$$ 0 0
$$979$$ −8.00000 −0.255681
$$980$$ 7.00000 0.223607
$$981$$ −54.0000 −1.72409
$$982$$ −20.0000 −0.638226
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 0 0
$$985$$ −2.00000 −0.0637253
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ −8.00000 −0.254514
$$989$$ 0 0
$$990$$ 12.0000 0.381385
$$991$$ 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
$$995$$ 20.0000 0.634043
$$996$$ 0 0
$$997$$ 10.0000 0.316703 0.158352 0.987383i $$-0.449382\pi$$
0.158352 + 0.987383i $$0.449382\pi$$
$$998$$ 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 370.2.a.b.1.1 1
3.2 odd 2 3330.2.a.w.1.1 1
4.3 odd 2 2960.2.a.g.1.1 1
5.2 odd 4 1850.2.b.d.149.1 2
5.3 odd 4 1850.2.b.d.149.2 2
5.4 even 2 1850.2.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.b.1.1 1 1.1 even 1 trivial
1850.2.a.k.1.1 1 5.4 even 2
1850.2.b.d.149.1 2 5.2 odd 4
1850.2.b.d.149.2 2 5.3 odd 4
2960.2.a.g.1.1 1 4.3 odd 2
3330.2.a.w.1.1 1 3.2 odd 2