Properties

 Label 370.2.a.d.1.1 Level $370$ Weight $2$ Character 370.1 Self dual yes Analytic conductor $2.954$ Analytic rank $0$ 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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{12} +2.00000 q^{13} +2.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} +1.00000 q^{20} -4.00000 q^{21} -2.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{27} +2.00000 q^{28} +6.00000 q^{29} -2.00000 q^{30} -10.0000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} +2.00000 q^{38} -4.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -4.00000 q^{42} -4.00000 q^{43} +1.00000 q^{45} -6.00000 q^{47} -2.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -12.0000 q^{51} +2.00000 q^{52} +6.00000 q^{53} +4.00000 q^{54} +2.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} -6.00000 q^{59} -2.00000 q^{60} -10.0000 q^{61} -10.0000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +2.00000 q^{67} +6.00000 q^{68} +2.00000 q^{70} +1.00000 q^{72} +2.00000 q^{73} +1.00000 q^{74} -2.00000 q^{75} +2.00000 q^{76} -4.00000 q^{78} -10.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -6.00000 q^{82} -6.00000 q^{83} -4.00000 q^{84} +6.00000 q^{85} -4.00000 q^{86} -12.0000 q^{87} -6.00000 q^{89} +1.00000 q^{90} +4.00000 q^{91} +20.0000 q^{93} -6.00000 q^{94} +2.00000 q^{95} -2.00000 q^{96} +2.00000 q^{97} -3.00000 q^{98} +O(q^{100})$$

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 $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −2.00000 −0.816497
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 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 $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 2.00000 0.534522
$$15$$ −2.00000 −0.516398
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −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
$$25$$ 1.00000 0.200000
$$26$$ 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 $$0.311919\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ −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
$$35$$ 2.00000 0.338062
$$36$$ 1.00000 0.166667
$$37$$ 1.00000 0.164399
$$38$$ 2.00000 0.324443
$$39$$ −4.00000 −0.640513
$$40$$ 1.00000 0.158114
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ −4.00000 −0.617213
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ −2.00000 −0.288675
$$49$$ −3.00000 −0.428571
$$50$$ 1.00000 0.141421
$$51$$ −12.0000 −1.68034
$$52$$ 2.00000 0.277350
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 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 $$0.627721\pi$$
$$60$$ −2.00000 −0.258199
$$61$$ −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
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 2.00000 0.239046
$$71$$ 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 $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 1.00000 0.116248
$$75$$ −2.00000 −0.230940
$$76$$ 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 $$0.690177\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −11.0000 −1.22222
$$82$$ −6.00000 −0.662589
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 6.00000 0.650791
$$86$$ −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 $$0.603011\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 4.00000 0.419314
$$92$$ 0 0
$$93$$ 20.0000 2.07390
$$94$$ −6.00000 −0.618853
$$95$$ 2.00000 0.205196
$$96$$ −2.00000 −0.204124
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 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 $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 2.00000 0.196116
$$105$$ −4.00000 −0.390360
$$106$$ 6.00000 0.582772
$$107$$ 6.00000 0.580042 0.290021 0.957020i $$-0.406338\pi$$
0.290021 + 0.957020i $$0.406338\pi$$
$$108$$ 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 $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −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
$$120$$ −2.00000 −0.182574
$$121$$ −11.0000 −1.00000
$$122$$ −10.0000 −0.905357
$$123$$ 12.0000 1.08200
$$124$$ −10.0000 −0.898027
$$125$$ 1.00000 0.0894427
$$126$$ 2.00000 0.178174
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 2.00000 0.175412
$$131$$ −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
$$135$$ 4.00000 0.344265
$$136$$ 6.00000 0.514496
$$137$$ −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 $$0.554260\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 12.0000 1.01058
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 6.00000 0.498273
$$146$$ 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 $$0.236096\pi$$
$$150$$ −2.00000 −0.163299
$$151$$ 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
$$155$$ −10.0000 −0.803219
$$156$$ −4.00000 −0.320256
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ −12.0000 −0.951662
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ −11.0000 −0.864242
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ −4.00000 −0.308607
$$169$$ −9.00000 −0.692308
$$170$$ 6.00000 0.460179
$$171$$ 2.00000 0.152944
$$172$$ −4.00000 −0.304997
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ −12.0000 −0.909718
$$175$$ 2.00000 0.151186
$$176$$ 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 $$0.428013\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −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
$$185$$ 1.00000 0.0735215
$$186$$ 20.0000 1.46647
$$187$$ 0 0
$$188$$ −6.00000 −0.437595
$$189$$ 8.00000 0.581914
$$190$$ 2.00000 0.145095
$$191$$ −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 $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 2.00000 0.143592
$$195$$ −4.00000 −0.286446
$$196$$ −3.00000 −0.214286
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ −22.0000 −1.55954 −0.779769 0.626067i $$-0.784664\pi$$
−0.779769 + 0.626067i $$0.784664\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −4.00000 −0.282138
$$202$$ 18.0000 1.26648
$$203$$ 12.0000 0.842235
$$204$$ −12.0000 −0.840168
$$205$$ −6.00000 −0.419058
$$206$$ −4.00000 −0.278693
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ 0 0
$$210$$ −4.00000 −0.276026
$$211$$ 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
$$215$$ −4.00000 −0.272798
$$216$$ 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 $$-0.344703\pi$$
0.468755 + 0.883328i $$0.344703\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 1.00000 0.0666667
$$226$$ 6.00000 0.399114
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ −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 $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 2.00000 0.130744
$$235$$ −6.00000 −0.391397
$$236$$ −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 $$0.437836\pi$$
$$240$$ −2.00000 −0.129099
$$241$$ −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
$$245$$ −3.00000 −0.191663
$$246$$ 12.0000 0.765092
$$247$$ 4.00000 0.254514
$$248$$ −10.0000 −0.635001
$$249$$ 12.0000 0.760469
$$250$$ 1.00000 0.0632456
$$251$$ 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
$$255$$ −12.0000 −0.751469
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 8.00000 0.498058
$$259$$ 2.00000 0.124274
$$260$$ 2.00000 0.124035
$$261$$ 6.00000 0.371391
$$262$$ −6.00000 −0.370681
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ 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 $$0.558553\pi$$
$$270$$ 4.00000 0.243432
$$271$$ −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 $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ −10.0000 −0.598684
$$280$$ 2.00000 0.119523
$$281$$ 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 $$-0.657751\pi$$
−0.475551 + 0.879688i $$0.657751\pi$$
$$284$$ 0 0
$$285$$ −4.00000 −0.236940
$$286$$ 0 0
$$287$$ −12.0000 −0.708338
$$288$$ 1.00000 0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 6.00000 0.352332
$$291$$ −4.00000 −0.234484
$$292$$ 2.00000 0.117041
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 6.00000 0.349927
$$295$$ −6.00000 −0.349334
$$296$$ 1.00000 0.0581238
$$297$$ 0 0
$$298$$ 18.0000 1.04271
$$299$$ 0 0
$$300$$ −2.00000 −0.115470
$$301$$ −8.00000 −0.461112
$$302$$ 20.0000 1.15087
$$303$$ −36.0000 −2.06815
$$304$$ 2.00000 0.114708
$$305$$ −10.0000 −0.572598
$$306$$ 6.00000 0.342997
$$307$$ 26.0000 1.48390 0.741949 0.670456i $$-0.233902\pi$$
0.741949 + 0.670456i $$0.233902\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ −10.0000 −0.567962
$$311$$ −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 $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 2.00000 0.112687
$$316$$ −10.0000 −0.562544
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ −12.0000 −0.672927
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 12.0000 0.667698
$$324$$ −11.0000 −0.611111
$$325$$ 2.00000 0.110940
$$326$$ −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
$$335$$ 2.00000 0.109272
$$336$$ −4.00000 −0.218218
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ −12.0000 −0.651751
$$340$$ 6.00000 0.325396
$$341$$ 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 $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ −12.0000 −0.643268
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 8.00000 0.427008
$$352$$ 0 0
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ 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 $$0.101079\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −15.0000 −0.789474
$$362$$ −22.0000 −1.15629
$$363$$ 22.0000 1.15470
$$364$$ 4.00000 0.209657
$$365$$ 2.00000 0.104685
$$366$$ 20.0000 1.04542
$$367$$ 2.00000 0.104399 0.0521996 0.998637i $$-0.483377\pi$$
0.0521996 + 0.998637i $$0.483377\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 1.00000 0.0519875
$$371$$ 12.0000 0.623009
$$372$$ 20.0000 1.03695
$$373$$ −34.0000 −1.76045 −0.880227 0.474554i $$-0.842610\pi$$
−0.880227 + 0.474554i $$0.842610\pi$$
$$374$$ 0 0
$$375$$ −2.00000 −0.103280
$$376$$ −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 $$0.634797\pi$$
$$380$$ 2.00000 0.102598
$$381$$ −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 $$0.451394\pi$$
$$390$$ −4.00000 −0.202548
$$391$$ 0 0
$$392$$ −3.00000 −0.151523
$$393$$ 12.0000 0.605320
$$394$$ −18.0000 −0.906827
$$395$$ −10.0000 −0.503155
$$396$$ 0 0
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ −22.0000 −1.10276
$$399$$ −8.00000 −0.400501
$$400$$ 1.00000 0.0500000
$$401$$ −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
$$405$$ −11.0000 −0.546594
$$406$$ 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 $$0.484254\pi$$
$$410$$ −6.00000 −0.296319
$$411$$ 12.0000 0.591916
$$412$$ −4.00000 −0.197066
$$413$$ −12.0000 −0.590481
$$414$$ 0 0
$$415$$ −6.00000 −0.294528
$$416$$ 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 $$0.300608\pi$$
$$420$$ −4.00000 −0.195180
$$421$$ −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
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ −20.0000 −0.967868
$$428$$ 6.00000 0.290021
$$429$$ 0 0
$$430$$ −4.00000 −0.192897
$$431$$ 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 $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ −20.0000 −0.960031
$$435$$ −12.0000 −0.575356
$$436$$ 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 $$-0.454475\pi$$
0.142534 + 0.989790i $$0.454475\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ −6.00000 −0.284427
$$446$$ 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 $$0.750354\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 0 0
$$452$$ 6.00000 0.282216
$$453$$ −40.0000 −1.87936
$$454$$ 24.0000 1.12638
$$455$$ 4.00000 0.187523
$$456$$ −4.00000 −0.187317
$$457$$ 26.0000 1.21623 0.608114 0.793849i $$-0.291926\pi$$
0.608114 + 0.793849i $$0.291926\pi$$
$$458$$ −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 $$-0.879743\pi$$
−0.929479 + 0.368875i $$0.879743\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 20.0000 0.927478
$$466$$ 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
$$470$$ −6.00000 −0.276759
$$471$$ 20.0000 0.921551
$$472$$ −6.00000 −0.276172
$$473$$ 0 0
$$474$$ 20.0000 0.918630
$$475$$ 2.00000 0.0917663
$$476$$ 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 $$0.634897\pi$$
$$480$$ −2.00000 −0.0912871
$$481$$ 2.00000 0.0911922
$$482$$ −22.0000 −1.00207
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 2.00000 0.0908153
$$486$$ 10.0000 0.453609
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ 32.0000 1.44709
$$490$$ −3.00000 −0.135526
$$491$$ 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 $$0.398544\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −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
$$505$$ 18.0000 0.800989
$$506$$ 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 $$0.268490\pi$$
$$510$$ −12.0000 −0.531369
$$511$$ 4.00000 0.176950
$$512$$ 1.00000 0.0441942
$$513$$ 8.00000 0.353209
$$514$$ 6.00000 0.264649
$$515$$ −4.00000 −0.176261
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ 2.00000 0.0878750
$$519$$ 36.0000 1.58022
$$520$$ 2.00000 0.0877058
$$521$$ 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 $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ −6.00000 −0.262111
$$525$$ −4.00000 −0.174574
$$526$$ −6.00000 −0.261612
$$527$$ −60.0000 −2.61364
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 6.00000 0.260623
$$531$$ −6.00000 −0.260378
$$532$$ 4.00000 0.173422
$$533$$ −12.0000 −0.519778
$$534$$ 12.0000 0.519291
$$535$$ 6.00000 0.259403
$$536$$ 2.00000 0.0863868
$$537$$ −12.0000 −0.517838
$$538$$ −6.00000 −0.258678
$$539$$ 0 0
$$540$$ 4.00000 0.172133
$$541$$ 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
$$545$$ 14.0000 0.599694
$$546$$ −8.00000 −0.342368
$$547$$ 44.0000 1.88130 0.940652 0.339372i $$-0.110215\pi$$
0.940652 + 0.339372i $$0.110215\pi$$
$$548$$ −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
$$555$$ −2.00000 −0.0848953
$$556$$ −4.00000 −0.169638
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ −10.0000 −0.423334
$$559$$ −8.00000 −0.338364
$$560$$ 2.00000 0.0845154
$$561$$ 0 0
$$562$$ 18.0000 0.759284
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ 12.0000 0.505291
$$565$$ 6.00000 0.252422
$$566$$ −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 $$0.540139\pi$$
$$570$$ −4.00000 −0.167542
$$571$$ −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 $$-0.209571\pi$$
0.790980 + 0.611842i $$0.209571\pi$$
$$578$$ 19.0000 0.790296
$$579$$ −4.00000 −0.166234
$$580$$ 6.00000 0.249136
$$581$$ −12.0000 −0.497844
$$582$$ −4.00000 −0.165805
$$583$$ 0 0
$$584$$ 2.00000 0.0827606
$$585$$ 2.00000 0.0826898
$$586$$ 6.00000 0.247858
$$587$$ −36.0000 −1.48588 −0.742940 0.669359i $$-0.766569\pi$$
−0.742940 + 0.669359i $$0.766569\pi$$
$$588$$ 6.00000 0.247436
$$589$$ −20.0000 −0.824086
$$590$$ −6.00000 −0.247016
$$591$$ 36.0000 1.48084
$$592$$ 1.00000 0.0410997
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 12.0000 0.491952
$$596$$ 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 $$0.763034\pi$$
$$600$$ −2.00000 −0.0816497
$$601$$ 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
$$605$$ −11.0000 −0.447214
$$606$$ −36.0000 −1.46240
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ −24.0000 −0.972529
$$610$$ −10.0000 −0.404888
$$611$$ −12.0000 −0.485468
$$612$$ 6.00000 0.242536
$$613$$ −34.0000 −1.37325 −0.686624 0.727013i $$-0.740908\pi$$
−0.686624 + 0.727013i $$0.740908\pi$$
$$614$$ 26.0000 1.04927
$$615$$ 12.0000 0.483887
$$616$$ 0 0
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ 8.00000 0.321807
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ −10.0000 −0.401610
$$621$$ 0 0
$$622$$ −18.0000 −0.721734
$$623$$ −12.0000 −0.480770
$$624$$ −4.00000 −0.160128
$$625$$ 1.00000 0.0400000
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ −10.0000 −0.399043
$$629$$ 6.00000 0.239236
$$630$$ 2.00000 0.0796819
$$631$$ 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
$$635$$ 2.00000 0.0793676
$$636$$ −12.0000 −0.475831
$$637$$ −6.00000 −0.237729
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ 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 $$-0.282654\pi$$
0.630978 + 0.775800i $$0.282654\pi$$
$$644$$ 0 0
$$645$$ 8.00000 0.315000
$$646$$ 12.0000 0.472134
$$647$$ 48.0000 1.88707 0.943537 0.331266i $$-0.107476\pi$$
0.943537 + 0.331266i $$0.107476\pi$$
$$648$$ −11.0000 −0.432121
$$649$$ 0 0
$$650$$ 2.00000 0.0784465
$$651$$ 40.0000 1.56772
$$652$$ −16.0000 −0.626608
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ −28.0000 −1.09489
$$655$$ −6.00000 −0.234439
$$656$$ −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
$$665$$ 4.00000 0.155113
$$666$$ 1.00000 0.0387492
$$667$$ 0 0
$$668$$ 12.0000 0.464294
$$669$$ −28.0000 −1.08254
$$670$$ 2.00000 0.0772667
$$671$$ 0 0
$$672$$ −4.00000 −0.154303
$$673$$ 26.0000 1.00223 0.501113 0.865382i $$-0.332924\pi$$
0.501113 + 0.865382i $$0.332924\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ 4.00000 0.153960
$$676$$ −9.00000 −0.346154
$$677$$ −42.0000 −1.61419 −0.807096 0.590421i $$-0.798962\pi$$
−0.807096 + 0.590421i $$0.798962\pi$$
$$678$$ −12.0000 −0.460857
$$679$$ 4.00000 0.153506
$$680$$ 6.00000 0.230089
$$681$$ −48.0000 −1.83936
$$682$$ 0 0
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ −6.00000 −0.229248
$$686$$ −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
$$695$$ −4.00000 −0.151729
$$696$$ −12.0000 −0.454859
$$697$$ −36.0000 −1.36360
$$698$$ 26.0000 0.984115
$$699$$ −36.0000 −1.36165
$$700$$ 2.00000 0.0755929
$$701$$ 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
$$705$$ 12.0000 0.451946
$$706$$ −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 $$0.734258\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ −8.00000 −0.297936
$$722$$ −15.0000 −0.558242
$$723$$ 44.0000 1.63638
$$724$$ −22.0000 −0.817624
$$725$$ 6.00000 0.222834
$$726$$ 22.0000 0.816497
$$727$$ 32.0000 1.18681 0.593407 0.804902i $$-0.297782\pi$$
0.593407 + 0.804902i $$0.297782\pi$$
$$728$$ 4.00000 0.148250
$$729$$ 13.0000 0.481481
$$730$$ 2.00000 0.0740233
$$731$$ −24.0000 −0.887672
$$732$$ 20.0000 0.739221
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ 2.00000 0.0738213
$$735$$ 6.00000 0.221313
$$736$$ 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 $$0.380092\pi$$
$$740$$ 1.00000 0.0367607
$$741$$ −8.00000 −0.293887
$$742$$ 12.0000 0.440534
$$743$$ 18.0000 0.660356 0.330178 0.943919i $$-0.392891\pi$$
0.330178 + 0.943919i $$0.392891\pi$$
$$744$$ 20.0000 0.733236
$$745$$ 18.0000 0.659469
$$746$$ −34.0000 −1.24483
$$747$$ −6.00000 −0.219529
$$748$$ 0 0
$$749$$ 12.0000 0.438470
$$750$$ −2.00000 −0.0730297
$$751$$ −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
$$755$$ 20.0000 0.727875
$$756$$ 8.00000 0.290957
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ −16.0000 −0.581146
$$759$$ 0 0
$$760$$ 2.00000 0.0725476
$$761$$ −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
$$765$$ 6.00000 0.216930
$$766$$ 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 $$-0.604929\pi$$
−0.323708 + 0.946157i $$0.604929\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ −10.0000 −0.359211
$$776$$ 2.00000 0.0717958
$$777$$ −4.00000 −0.143499
$$778$$ 6.00000 0.215110
$$779$$ −12.0000 −0.429945
$$780$$ −4.00000 −0.143223
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 24.0000 0.857690
$$784$$ −3.00000 −0.107143
$$785$$ −10.0000 −0.356915
$$786$$ 12.0000 0.428026
$$787$$ −10.0000 −0.356462 −0.178231 0.983989i $$-0.557037\pi$$
−0.178231 + 0.983989i $$0.557037\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ 12.0000 0.427211
$$790$$ −10.0000 −0.355784
$$791$$ 12.0000 0.426671
$$792$$ 0 0
$$793$$ −20.0000 −0.710221
$$794$$ 14.0000 0.496841
$$795$$ −12.0000 −0.425596
$$796$$ −22.0000 −0.779769
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ −8.00000 −0.283197
$$799$$ −36.0000 −1.27359
$$800$$ 1.00000 0.0353553
$$801$$ −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 $$0.533636\pi$$
$$810$$ −11.0000 −0.386501
$$811$$ 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
$$815$$ −16.0000 −0.560456
$$816$$ −12.0000 −0.420084
$$817$$ −8.00000 −0.279885
$$818$$ 2.00000 0.0699284
$$819$$ 4.00000 0.139771
$$820$$ −6.00000 −0.209529
$$821$$ −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 $$-0.701892\pi$$
−0.592583 + 0.805510i $$0.701892\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ 24.0000 0.834562 0.417281 0.908778i $$-0.362983\pi$$
0.417281 + 0.908778i $$0.362983\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ −6.00000 −0.208263
$$831$$ −52.0000 −1.80386
$$832$$ 2.00000 0.0693375
$$833$$ −18.0000 −0.623663
$$834$$ 8.00000 0.277017
$$835$$ 12.0000 0.415277
$$836$$ 0 0
$$837$$ −40.0000 −1.38260
$$838$$ 24.0000 0.829066
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ −4.00000 −0.138013
$$841$$ 7.00000 0.241379
$$842$$ −10.0000 −0.344623
$$843$$ −36.0000 −1.23991
$$844$$ 20.0000 0.688428
$$845$$ −9.00000 −0.309609
$$846$$ −6.00000 −0.206284
$$847$$ −22.0000 −0.755929
$$848$$ 6.00000 0.206041
$$849$$ 32.0000 1.09824
$$850$$ 6.00000 0.205798
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −10.0000 −0.342393 −0.171197 0.985237i $$-0.554763\pi$$
−0.171197 + 0.985237i $$0.554763\pi$$
$$854$$ −20.0000 −0.684386
$$855$$ 2.00000 0.0683986
$$856$$ 6.00000 0.205076
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 0 0
$$859$$ 50.0000 1.70598 0.852989 0.521929i $$-0.174787\pi$$
0.852989 + 0.521929i $$0.174787\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 24.0000 0.817918
$$862$$ 30.0000 1.02180
$$863$$ 54.0000 1.83818 0.919091 0.394046i $$-0.128925\pi$$
0.919091 + 0.394046i $$0.128925\pi$$
$$864$$ 4.00000 0.136083
$$865$$ −18.0000 −0.612018
$$866$$ 2.00000 0.0679628
$$867$$ −38.0000 −1.29055
$$868$$ −20.0000 −0.678844
$$869$$ 0 0
$$870$$ −12.0000 −0.406838
$$871$$ 4.00000 0.135535
$$872$$ 14.0000 0.474100
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ 2.00000 0.0676123
$$876$$ −4.00000 −0.135147
$$877$$ −10.0000 −0.337676 −0.168838 0.985644i $$-0.554001\pi$$
−0.168838 + 0.985644i $$0.554001\pi$$
$$878$$ −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 $$-0.108682\pi$$
0.942275 + 0.334840i $$0.108682\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 12.0000 0.403376
$$886$$ 6.00000 0.201574
$$887$$ −42.0000 −1.41022 −0.705111 0.709097i $$-0.749103\pi$$
−0.705111 + 0.709097i $$0.749103\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 4.00000 0.134156
$$890$$ −6.00000 −0.201120
$$891$$ 0 0
$$892$$ 14.0000 0.468755
$$893$$ −12.0000 −0.401565
$$894$$ −36.0000 −1.20402
$$895$$ 6.00000 0.200558
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ −30.0000 −1.00111
$$899$$ −60.0000 −2.00111
$$900$$ 1.00000 0.0333333
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 16.0000 0.532447
$$904$$ 6.00000 0.199557
$$905$$ −22.0000 −0.731305
$$906$$ −40.0000 −1.32891
$$907$$ 8.00000 0.265636 0.132818 0.991140i $$-0.457597\pi$$
0.132818 + 0.991140i $$0.457597\pi$$
$$908$$ 24.0000 0.796468
$$909$$ 18.0000 0.597022
$$910$$ 4.00000 0.132599
$$911$$ −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
$$915$$ 20.0000 0.661180
$$916$$ −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
$$925$$ 1.00000 0.0328798
$$926$$ −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 $$0.663783\pi$$
$$930$$ 20.0000 0.655826
$$931$$ −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 $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ 4.00000 0.130605
$$939$$ 44.0000 1.43589
$$940$$ −6.00000 −0.195698
$$941$$ −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
$$945$$ 8.00000 0.260240
$$946$$ 0 0
$$947$$ 48.0000 1.55979 0.779895 0.625910i $$-0.215272\pi$$
0.779895 + 0.625910i $$0.215272\pi$$
$$948$$ 20.0000 0.649570
$$949$$ 4.00000 0.129845
$$950$$ 2.00000 0.0648886
$$951$$ 36.0000 1.16738
$$952$$ 12.0000 0.388922
$$953$$ 42.0000 1.36051 0.680257 0.732974i $$-0.261868\pi$$
0.680257 + 0.732974i $$0.261868\pi$$
$$954$$ 6.00000 0.194257
$$955$$ −6.00000 −0.194155
$$956$$ 6.00000 0.194054
$$957$$ 0 0
$$958$$ −18.0000 −0.581554
$$959$$ −12.0000 −0.387500
$$960$$ −2.00000 −0.0645497
$$961$$ 69.0000 2.22581
$$962$$ 2.00000 0.0644826
$$963$$ 6.00000 0.193347
$$964$$ −22.0000 −0.708572
$$965$$ 2.00000 0.0643823
$$966$$ 0 0
$$967$$ −4.00000 −0.128631 −0.0643157 0.997930i $$-0.520486\pi$$
−0.0643157 + 0.997930i $$0.520486\pi$$
$$968$$ −11.0000 −0.353553
$$969$$ −24.0000 −0.770991
$$970$$ 2.00000 0.0642161
$$971$$ −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
$$975$$ −4.00000 −0.128103
$$976$$ −10.0000 −0.320092
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 32.0000 1.02325
$$979$$ 0 0
$$980$$ −3.00000 −0.0958315
$$981$$ 14.0000 0.446986
$$982$$ 36.0000 1.14881
$$983$$ 18.0000 0.574111 0.287055 0.957914i $$-0.407324\pi$$
0.287055 + 0.957914i $$0.407324\pi$$
$$984$$ 12.0000 0.382546
$$985$$ −18.0000 −0.573528
$$986$$ 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
$$995$$ −22.0000 −0.697447
$$996$$ 12.0000 0.380235
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ 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 370.2.a.d.1.1 1
3.2 odd 2 3330.2.a.d.1.1 1
4.3 odd 2 2960.2.a.m.1.1 1
5.2 odd 4 1850.2.b.b.149.2 2
5.3 odd 4 1850.2.b.b.149.1 2
5.4 even 2 1850.2.a.f.1.1 1

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