Properties

 Label 1850.2.a.f.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)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -4.00000 q^{21} -2.00000 q^{24} +2.00000 q^{26} -4.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} -10.0000 q^{31} -1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} -2.00000 q^{38} -4.00000 q^{39} -6.00000 q^{41} +4.00000 q^{42} +4.00000 q^{43} +6.00000 q^{47} +2.00000 q^{48} -3.00000 q^{49} -12.0000 q^{51} -2.00000 q^{52} -6.00000 q^{53} +4.00000 q^{54} +2.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} -6.00000 q^{59} -10.0000 q^{61} +10.0000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -2.00000 q^{67} -6.00000 q^{68} -1.00000 q^{72} -2.00000 q^{73} +1.00000 q^{74} +2.00000 q^{76} +4.00000 q^{78} -10.0000 q^{79} -11.0000 q^{81} +6.00000 q^{82} +6.00000 q^{83} -4.00000 q^{84} -4.00000 q^{86} +12.0000 q^{87} -6.00000 q^{89} +4.00000 q^{91} -20.0000 q^{93} -6.00000 q^{94} -2.00000 q^{96} -2.00000 q^{97} +3.00000 q^{98} +O(q^{100})$$

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
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −2.00000 −0.816497
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 2.00000 0.577350
$$13$$ −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 $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −2.00000 −0.408248
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ −4.00000 −0.769800
$$28$$ −2.00000 −0.377964
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −10.0000 −1.79605 −0.898027 0.439941i $$-0.854999\pi$$
−0.898027 + 0.439941i $$0.854999\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −1.00000 −0.164399
$$38$$ −2.00000 −0.324443
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 4.00000 0.617213
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 2.00000 0.288675
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ −12.0000 −1.68034
$$52$$ −2.00000 −0.277350
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 4.00000 0.544331
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ 4.00000 0.529813
$$58$$ −6.00000 −0.787839
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 10.0000 1.27000
$$63$$ −2.00000 −0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −2.00000 −0.244339 −0.122169 0.992509i $$-0.538985\pi$$
−0.122169 + 0.992509i $$0.538985\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 0 0
$$78$$ 4.00000 0.452911
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 6.00000 0.662589
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 12.0000 1.28654
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 0 0
$$93$$ −20.0000 −2.07390
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ −2.00000 −0.204124
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 12.0000 1.18818
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ −4.00000 −0.384900
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ −2.00000 −0.188982
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ −2.00000 −0.184900
$$118$$ 6.00000 0.552345
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 10.0000 0.905357
$$123$$ −12.0000 −1.08200
$$124$$ −10.0000 −0.898027
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 0 0
$$133$$ −4.00000 −0.346844
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 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
$$141$$ 12.0000 1.01058
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ −6.00000 −0.494872
$$148$$ −1.00000 −0.0821995
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 10.0000 0.795557
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 11.0000 0.864242
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 4.00000 0.308607
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ 4.00000 0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ −12.0000 −0.909718
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −12.0000 −0.901975
$$178$$ 6.00000 0.449719
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ −4.00000 −0.296500
$$183$$ −20.0000 −1.47844
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 20.0000 1.46647
$$187$$ 0 0
$$188$$ 6.00000 0.437595
$$189$$ 8.00000 0.581914
$$190$$ 0 0
$$191$$ −6.00000 −0.434145 −0.217072 0.976156i $$-0.569651\pi$$
−0.217072 + 0.976156i $$0.569651\pi$$
$$192$$ 2.00000 0.144338
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 0 0
$$199$$ −22.0000 −1.55954 −0.779769 0.626067i $$-0.784664\pi$$
−0.779769 + 0.626067i $$0.784664\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ −18.0000 −1.26648
$$203$$ −12.0000 −0.842235
$$204$$ −12.0000 −0.840168
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ 6.00000 0.410152
$$215$$ 0 0
$$216$$ 4.00000 0.272166
$$217$$ 20.0000 1.35769
$$218$$ −14.0000 −0.948200
$$219$$ −4.00000 −0.270295
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 2.00000 0.134231
$$223$$ −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
$$226$$ 6.00000 0.399114
$$227$$ −24.0000 −1.59294 −0.796468 0.604681i $$-0.793301\pi$$
−0.796468 + 0.604681i $$0.793301\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ −20.0000 −1.29914
$$238$$ −12.0000 −0.777844
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ 11.0000 0.707107
$$243$$ −10.0000 −0.641500
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ 12.0000 0.765092
$$247$$ −4.00000 −0.254514
$$248$$ 10.0000 0.635001
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 0 0
$$254$$ 2.00000 0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 6.00000 0.370681
$$263$$ 6.00000 0.369976 0.184988 0.982741i $$-0.440775\pi$$
0.184988 + 0.982741i $$0.440775\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 4.00000 0.245256
$$267$$ −12.0000 −0.734388
$$268$$ −2.00000 −0.122169
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 8.00000 0.484182
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −26.0000 −1.56219 −0.781094 0.624413i $$-0.785338\pi$$
−0.781094 + 0.624413i $$0.785338\pi$$
$$278$$ 4.00000 0.239904
$$279$$ −10.0000 −0.598684
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ −12.0000 −0.714590
$$283$$ 16.0000 0.951101 0.475551 0.879688i $$-0.342249\pi$$
0.475551 + 0.879688i $$0.342249\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 12.0000 0.708338
$$288$$ −1.00000 −0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ −4.00000 −0.234484
$$292$$ −2.00000 −0.117041
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 6.00000 0.349927
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ −20.0000 −1.15087
$$303$$ 36.0000 2.06815
$$304$$ 2.00000 0.114708
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ −26.0000 −1.48390 −0.741949 0.670456i $$-0.766098\pi$$
−0.741949 + 0.670456i $$0.766098\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 4.00000 0.226455
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 12.0000 0.672927
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ −12.0000 −0.667698
$$324$$ −11.0000 −0.611111
$$325$$ 0 0
$$326$$ −16.0000 −0.886158
$$327$$ 28.0000 1.54840
$$328$$ 6.00000 0.331295
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ −10.0000 −0.549650 −0.274825 0.961494i $$-0.588620\pi$$
−0.274825 + 0.961494i $$0.588620\pi$$
$$332$$ 6.00000 0.329293
$$333$$ −1.00000 −0.0547997
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ 9.00000 0.489535
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ 0 0
$$342$$ −2.00000 −0.108148
$$343$$ 20.0000 1.07990
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 12.0000 0.643268
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ 8.00000 0.427008
$$352$$ 0 0
$$353$$ 30.0000 1.59674 0.798369 0.602168i $$-0.205696\pi$$
0.798369 + 0.602168i $$0.205696\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 24.0000 1.27021
$$358$$ −6.00000 −0.317110
$$359$$ 36.0000 1.90001 0.950004 0.312239i $$-0.101079\pi$$
0.950004 + 0.312239i $$0.101079\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 22.0000 1.15629
$$363$$ −22.0000 −1.15470
$$364$$ 4.00000 0.209657
$$365$$ 0 0
$$366$$ 20.0000 1.04542
$$367$$ −2.00000 −0.104399 −0.0521996 0.998637i $$-0.516623\pi$$
−0.0521996 + 0.998637i $$0.516623\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ −20.0000 −1.03695
$$373$$ 34.0000 1.76045 0.880227 0.474554i $$-0.157390\pi$$
0.880227 + 0.474554i $$0.157390\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ −12.0000 −0.618031
$$378$$ −8.00000 −0.411476
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 6.00000 0.306987
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ −2.00000 −0.102062
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ 4.00000 0.203331
$$388$$ −2.00000 −0.101535
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 3.00000 0.151523
$$393$$ −12.0000 −0.605320
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 22.0000 1.10276
$$399$$ −8.00000 −0.400501
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 20.0000 0.996271
$$404$$ 18.0000 0.895533
$$405$$ 0 0
$$406$$ 12.0000 0.595550
$$407$$ 0 0
$$408$$ 12.0000 0.594089
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 4.00000 0.197066
$$413$$ 12.0000 0.590481
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ −8.00000 −0.391762
$$418$$ 0 0
$$419$$ 24.0000 1.17248 0.586238 0.810139i $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 6.00000 0.291730
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 20.0000 0.967868
$$428$$ −6.00000 −0.290021
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 30.0000 1.44505 0.722525 0.691345i $$-0.242982\pi$$
0.722525 + 0.691345i $$0.242982\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ −20.0000 −0.960031
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 0 0
$$438$$ 4.00000 0.191127
$$439$$ −22.0000 −1.05000 −0.525001 0.851101i $$-0.675935\pi$$
−0.525001 + 0.851101i $$0.675935\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ −12.0000 −0.570782
$$443$$ −6.00000 −0.285069 −0.142534 0.989790i $$-0.545525\pi$$
−0.142534 + 0.989790i $$0.545525\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ 14.0000 0.662919
$$447$$ 36.0000 1.70274
$$448$$ −2.00000 −0.0944911
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −6.00000 −0.282216
$$453$$ 40.0000 1.87936
$$454$$ 24.0000 1.12638
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ −26.0000 −1.21623 −0.608114 0.793849i $$-0.708074\pi$$
−0.608114 + 0.793849i $$0.708074\pi$$
$$458$$ 10.0000 0.467269
$$459$$ 24.0000 1.12022
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 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
$$466$$ 18.0000 0.833834
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 20.0000 0.921551
$$472$$ 6.00000 0.276172
$$473$$ 0 0
$$474$$ 20.0000 0.918630
$$475$$ 0 0
$$476$$ 12.0000 0.550019
$$477$$ −6.00000 −0.274721
$$478$$ −6.00000 −0.274434
$$479$$ −18.0000 −0.822441 −0.411220 0.911536i $$-0.634897\pi$$
−0.411220 + 0.911536i $$0.634897\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 22.0000 1.00207
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 0 0
$$486$$ 10.0000 0.453609
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 32.0000 1.44709
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ −12.0000 −0.541002
$$493$$ −36.0000 −1.62136
$$494$$ 4.00000 0.179969
$$495$$ 0 0
$$496$$ −10.0000 −0.449013
$$497$$ 0 0
$$498$$ −12.0000 −0.537733
$$499$$ 14.0000 0.626726 0.313363 0.949633i $$-0.398544\pi$$
0.313363 + 0.949633i $$0.398544\pi$$
$$500$$ 0 0
$$501$$ −24.0000 −1.07224
$$502$$ −18.0000 −0.803379
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −18.0000 −0.799408
$$508$$ −2.00000 −0.0887357
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ −1.00000 −0.0441942
$$513$$ −8.00000 −0.353209
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ −2.00000 −0.0878750
$$519$$ 36.0000 1.58022
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ −6.00000 −0.262111
$$525$$ 0 0
$$526$$ −6.00000 −0.261612
$$527$$ 60.0000 2.61364
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ −4.00000 −0.173422
$$533$$ 12.0000 0.519778
$$534$$ 12.0000 0.519291
$$535$$ 0 0
$$536$$ 2.00000 0.0863868
$$537$$ 12.0000 0.517838
$$538$$ 6.00000 0.258678
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ 16.0000 0.687259
$$543$$ −44.0000 −1.88822
$$544$$ 6.00000 0.257248
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ −44.0000 −1.88130 −0.940652 0.339372i $$-0.889785\pi$$
−0.940652 + 0.339372i $$0.889785\pi$$
$$548$$ 6.00000 0.256307
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ 20.0000 0.850487
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 10.0000 0.423334
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −18.0000 −0.759284
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ 12.0000 0.505291
$$565$$ 0 0
$$566$$ −16.0000 −0.672530
$$567$$ 22.0000 0.923913
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ −12.0000 −0.501307
$$574$$ −12.0000 −0.500870
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −38.0000 −1.58196 −0.790980 0.611842i $$-0.790429\pi$$
−0.790980 + 0.611842i $$0.790429\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ −4.00000 −0.166234
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 4.00000 0.165805
$$583$$ 0 0
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ 36.0000 1.48588 0.742940 0.669359i $$-0.233431\pi$$
0.742940 + 0.669359i $$0.233431\pi$$
$$588$$ −6.00000 −0.247436
$$589$$ −20.0000 −0.824086
$$590$$ 0 0
$$591$$ 36.0000 1.48084
$$592$$ −1.00000 −0.0410997
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ −44.0000 −1.80080
$$598$$ 0 0
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ 8.00000 0.326056
$$603$$ −2.00000 −0.0814463
$$604$$ 20.0000 0.813788
$$605$$ 0 0
$$606$$ −36.0000 −1.46240
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ −24.0000 −0.972529
$$610$$ 0 0
$$611$$ −12.0000 −0.485468
$$612$$ −6.00000 −0.242536
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ 26.0000 1.04927
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 30.0000 1.20775 0.603877 0.797077i $$-0.293622\pi$$
0.603877 + 0.797077i $$0.293622\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 18.0000 0.721734
$$623$$ 12.0000 0.480770
$$624$$ −4.00000 −0.160128
$$625$$ 0 0
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ 10.0000 0.399043
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 14.0000 0.557331 0.278666 0.960388i $$-0.410108\pi$$
0.278666 + 0.960388i $$0.410108\pi$$
$$632$$ 10.0000 0.397779
$$633$$ 40.0000 1.58986
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ −12.0000 −0.475831
$$637$$ 6.00000 0.237729
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 12.0000 0.473602
$$643$$ −32.0000 −1.26196 −0.630978 0.775800i $$-0.717346\pi$$
−0.630978 + 0.775800i $$0.717346\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 12.0000 0.472134
$$647$$ −48.0000 −1.88707 −0.943537 0.331266i $$-0.892524\pi$$
−0.943537 + 0.331266i $$0.892524\pi$$
$$648$$ 11.0000 0.432121
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 40.0000 1.56772
$$652$$ 16.0000 0.626608
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ −28.0000 −1.09489
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ −2.00000 −0.0780274
$$658$$ 12.0000 0.467809
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ 10.0000 0.388661
$$663$$ 24.0000 0.932083
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ 1.00000 0.0387492
$$667$$ 0 0
$$668$$ −12.0000 −0.464294
$$669$$ −28.0000 −1.08254
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 4.00000 0.154303
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 42.0000 1.61419 0.807096 0.590421i $$-0.201038\pi$$
0.807096 + 0.590421i $$0.201038\pi$$
$$678$$ 12.0000 0.460857
$$679$$ 4.00000 0.153506
$$680$$ 0 0
$$681$$ −48.0000 −1.83936
$$682$$ 0 0
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ −20.0000 −0.763048
$$688$$ 4.00000 0.152499
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 0 0
$$694$$ −12.0000 −0.455514
$$695$$ 0 0
$$696$$ −12.0000 −0.454859
$$697$$ 36.0000 1.36360
$$698$$ −26.0000 −0.984115
$$699$$ −36.0000 −1.36165
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ −8.00000 −0.301941
$$703$$ −2.00000 −0.0754314
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ −36.0000 −1.35392
$$708$$ −12.0000 −0.450988
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ 0 0
$$711$$ −10.0000 −0.375029
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ −24.0000 −0.898177
$$715$$ 0 0
$$716$$ 6.00000 0.224231
$$717$$ 12.0000 0.448148
$$718$$ −36.0000 −1.34351
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 15.0000 0.558242
$$723$$ −44.0000 −1.63638
$$724$$ −22.0000 −0.817624
$$725$$ 0 0
$$726$$ 22.0000 0.816497
$$727$$ −32.0000 −1.18681 −0.593407 0.804902i $$-0.702218\pi$$
−0.593407 + 0.804902i $$0.702218\pi$$
$$728$$ −4.00000 −0.148250
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ −20.0000 −0.739221
$$733$$ −14.0000 −0.517102 −0.258551 0.965998i $$-0.583245\pi$$
−0.258551 + 0.965998i $$0.583245\pi$$
$$734$$ 2.00000 0.0738213
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 6.00000 0.220863
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ −12.0000 −0.440534
$$743$$ −18.0000 −0.660356 −0.330178 0.943919i $$-0.607109\pi$$
−0.330178 + 0.943919i $$0.607109\pi$$
$$744$$ 20.0000 0.733236
$$745$$ 0 0
$$746$$ −34.0000 −1.24483
$$747$$ 6.00000 0.219529
$$748$$ 0 0
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 36.0000 1.31191
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ 8.00000 0.290957
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 16.0000 0.581146
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 4.00000 0.144905
$$763$$ −28.0000 −1.01367
$$764$$ −6.00000 −0.217072
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 12.0000 0.433295
$$768$$ 2.00000 0.0721688
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ −12.0000 −0.432169
$$772$$ −2.00000 −0.0719816
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ 4.00000 0.143499
$$778$$ −6.00000 −0.215110
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −24.0000 −0.857690
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 10.0000 0.356462 0.178231 0.983989i $$-0.442963\pi$$
0.178231 + 0.983989i $$0.442963\pi$$
$$788$$ 18.0000 0.641223
$$789$$ 12.0000 0.427211
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ 0 0
$$793$$ 20.0000 0.710221
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ −22.0000 −0.779769
$$797$$ 30.0000 1.06265 0.531327 0.847167i $$-0.321693\pi$$
0.531327 + 0.847167i $$0.321693\pi$$
$$798$$ 8.00000 0.283197
$$799$$ −36.0000 −1.27359
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 6.00000 0.211867
$$803$$ 0 0
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ −20.0000 −0.704470
$$807$$ −12.0000 −0.422420
$$808$$ −18.0000 −0.633238
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ −12.0000 −0.421117
$$813$$ −32.0000 −1.12229
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −12.0000 −0.420084
$$817$$ 8.00000 0.279885
$$818$$ −2.00000 −0.0699284
$$819$$ 4.00000 0.139771
$$820$$ 0 0
$$821$$ −42.0000 −1.46581 −0.732905 0.680331i $$-0.761836\pi$$
−0.732905 + 0.680331i $$0.761836\pi$$
$$822$$ −12.0000 −0.418548
$$823$$ 34.0000 1.18517 0.592583 0.805510i $$-0.298108\pi$$
0.592583 + 0.805510i $$0.298108\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ −24.0000 −0.834562 −0.417281 0.908778i $$-0.637017\pi$$
−0.417281 + 0.908778i $$0.637017\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 0 0
$$831$$ −52.0000 −1.80386
$$832$$ −2.00000 −0.0693375
$$833$$ 18.0000 0.623663
$$834$$ 8.00000 0.277017
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 40.0000 1.38260
$$838$$ −24.0000 −0.829066
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 10.0000 0.344623
$$843$$ 36.0000 1.23991
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ −6.00000 −0.206284
$$847$$ 22.0000 0.755929
$$848$$ −6.00000 −0.206041
$$849$$ 32.0000 1.09824
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ −20.0000 −0.684386
$$855$$ 0 0
$$856$$ 6.00000 0.205076
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ 50.0000 1.70598 0.852989 0.521929i $$-0.174787\pi$$
0.852989 + 0.521929i $$0.174787\pi$$
$$860$$ 0 0
$$861$$ 24.0000 0.817918
$$862$$ −30.0000 −1.02180
$$863$$ −54.0000 −1.83818 −0.919091 0.394046i $$-0.871075\pi$$
−0.919091 + 0.394046i $$0.871075\pi$$
$$864$$ 4.00000 0.136083
$$865$$ 0 0
$$866$$ 2.00000 0.0679628
$$867$$ 38.0000 1.29055
$$868$$ 20.0000 0.678844
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ −14.0000 −0.474100
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −4.00000 −0.135147
$$877$$ 10.0000 0.337676 0.168838 0.985644i $$-0.445999\pi$$
0.168838 + 0.985644i $$0.445999\pi$$
$$878$$ 22.0000 0.742464
$$879$$ −12.0000 −0.404750
$$880$$ 0 0
$$881$$ −42.0000 −1.41502 −0.707508 0.706705i $$-0.750181\pi$$
−0.707508 + 0.706705i $$0.750181\pi$$
$$882$$ 3.00000 0.101015
$$883$$ −56.0000 −1.88455 −0.942275 0.334840i $$-0.891318\pi$$
−0.942275 + 0.334840i $$0.891318\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ 6.00000 0.201574
$$887$$ 42.0000 1.41022 0.705111 0.709097i $$-0.250897\pi$$
0.705111 + 0.709097i $$0.250897\pi$$
$$888$$ 2.00000 0.0671156
$$889$$ 4.00000 0.134156
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −14.0000 −0.468755
$$893$$ 12.0000 0.401565
$$894$$ −36.0000 −1.20402
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ 30.0000 1.00111
$$899$$ −60.0000 −2.00111
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ −16.0000 −0.532447
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ −40.0000 −1.32891
$$907$$ −8.00000 −0.265636 −0.132818 0.991140i $$-0.542403\pi$$
−0.132818 + 0.991140i $$0.542403\pi$$
$$908$$ −24.0000 −0.796468
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ −30.0000 −0.993944 −0.496972 0.867766i $$-0.665555\pi$$
−0.496972 + 0.867766i $$0.665555\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 0 0
$$914$$ 26.0000 0.860004
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 12.0000 0.396275
$$918$$ −24.0000 −0.792118
$$919$$ −46.0000 −1.51740 −0.758700 0.651440i $$-0.774165\pi$$
−0.758700 + 0.651440i $$0.774165\pi$$
$$920$$ 0 0
$$921$$ −52.0000 −1.71346
$$922$$ −6.00000 −0.197599
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −40.0000 −1.31448
$$927$$ 4.00000 0.131377
$$928$$ −6.00000 −0.196960
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ −18.0000 −0.589610
$$933$$ −36.0000 −1.17859
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ −26.0000 −0.849383 −0.424691 0.905338i $$-0.639617\pi$$
−0.424691 + 0.905338i $$0.639617\pi$$
$$938$$ −4.00000 −0.130605
$$939$$ 44.0000 1.43589
$$940$$ 0 0
$$941$$ −30.0000 −0.977972 −0.488986 0.872292i $$-0.662633\pi$$
−0.488986 + 0.872292i $$0.662633\pi$$
$$942$$ −20.0000 −0.651635
$$943$$ 0 0
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −48.0000 −1.55979 −0.779895 0.625910i $$-0.784728\pi$$
−0.779895 + 0.625910i $$0.784728\pi$$
$$948$$ −20.0000 −0.649570
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ 36.0000 1.16738
$$952$$ −12.0000 −0.388922
$$953$$ −42.0000 −1.36051 −0.680257 0.732974i $$-0.738132\pi$$
−0.680257 + 0.732974i $$0.738132\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 6.00000 0.194054
$$957$$ 0 0
$$958$$ 18.0000 0.581554
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ −2.00000 −0.0644826
$$963$$ −6.00000 −0.193347
$$964$$ −22.0000 −0.708572
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 4.00000 0.128631 0.0643157 0.997930i $$-0.479514\pi$$
0.0643157 + 0.997930i $$0.479514\pi$$
$$968$$ 11.0000 0.353553
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ −10.0000 −0.320750
$$973$$ 8.00000 0.256468
$$974$$ 20.0000 0.640841
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ −32.0000 −1.02325
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 14.0000 0.446986
$$982$$ −36.0000 −1.14881
$$983$$ −18.0000 −0.574111 −0.287055 0.957914i $$-0.592676\pi$$
−0.287055 + 0.957914i $$0.592676\pi$$
$$984$$ 12.0000 0.382546
$$985$$ 0 0
$$986$$ 36.0000 1.14647
$$987$$ −24.0000 −0.763928
$$988$$ −4.00000 −0.127257
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −34.0000 −1.08005 −0.540023 0.841650i $$-0.681584\pi$$
−0.540023 + 0.841650i $$0.681584\pi$$
$$992$$ 10.0000 0.317500
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ 10.0000 0.316703 0.158352 0.987383i $$-0.449382\pi$$
0.158352 + 0.987383i $$0.449382\pi$$
$$998$$ −14.0000 −0.443162
$$999$$ 4.00000 0.126554
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.f.1.1 1
5.2 odd 4 1850.2.b.b.149.1 2
5.3 odd 4 1850.2.b.b.149.2 2
5.4 even 2 370.2.a.d.1.1 1
15.14 odd 2 3330.2.a.d.1.1 1
20.19 odd 2 2960.2.a.m.1.1 1

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