# Properties

 Label 4650.2.a.bi.1.1 Level $4650$ Weight $2$ Character 4650.1 Self dual yes Analytic conductor $37.130$ 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] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.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: $$37.1304369399$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4650.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -5.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -5.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} -5.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +3.00000 q^{19} -5.00000 q^{21} -1.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{27} -5.00000 q^{28} +6.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} +3.00000 q^{38} +2.00000 q^{41} -5.00000 q^{42} +1.00000 q^{43} -1.00000 q^{44} +1.00000 q^{46} -4.00000 q^{47} +1.00000 q^{48} +18.0000 q^{49} +4.00000 q^{51} -3.00000 q^{53} +1.00000 q^{54} -5.00000 q^{56} +3.00000 q^{57} +6.00000 q^{58} -14.0000 q^{59} +14.0000 q^{61} -1.00000 q^{62} -5.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -10.0000 q^{67} +4.00000 q^{68} +1.00000 q^{69} +9.00000 q^{71} +1.00000 q^{72} +7.00000 q^{73} +4.00000 q^{74} +3.00000 q^{76} +5.00000 q^{77} +15.0000 q^{79} +1.00000 q^{81} +2.00000 q^{82} +10.0000 q^{83} -5.00000 q^{84} +1.00000 q^{86} +6.00000 q^{87} -1.00000 q^{88} -1.00000 q^{89} +1.00000 q^{92} -1.00000 q^{93} -4.00000 q^{94} +1.00000 q^{96} -10.0000 q^{97} +18.0000 q^{98} -1.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ −5.00000 −1.88982 −0.944911 0.327327i $$-0.893852\pi$$
−0.944911 + 0.327327i $$0.893852\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511 −0.150756 0.988571i $$-0.548171\pi$$
−0.150756 + 0.988571i $$0.548171\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ −5.00000 −1.33631
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 3.00000 0.688247 0.344124 0.938924i $$-0.388176\pi$$
0.344124 + 0.938924i $$0.388176\pi$$
$$20$$ 0 0
$$21$$ −5.00000 −1.09109
$$22$$ −1.00000 −0.213201
$$23$$ 1.00000 0.208514 0.104257 0.994550i $$-0.466753\pi$$
0.104257 + 0.994550i $$0.466753\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ −5.00000 −0.944911
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −1.00000 −0.179605
$$32$$ 1.00000 0.176777
$$33$$ −1.00000 −0.174078
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 3.00000 0.486664
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ −5.00000 −0.771517
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 1.00000 0.147442
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 18.0000 2.57143
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −5.00000 −0.668153
$$57$$ 3.00000 0.397360
$$58$$ 6.00000 0.787839
$$59$$ −14.0000 −1.82264 −0.911322 0.411693i $$-0.864937\pi$$
−0.911322 + 0.411693i $$0.864937\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ −1.00000 −0.127000
$$63$$ −5.00000 −0.629941
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −1.00000 −0.123091
$$67$$ −10.0000 −1.22169 −0.610847 0.791748i $$-0.709171\pi$$
−0.610847 + 0.791748i $$0.709171\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ 9.00000 1.06810 0.534052 0.845452i $$-0.320669\pi$$
0.534052 + 0.845452i $$0.320669\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 7.00000 0.819288 0.409644 0.912245i $$-0.365653\pi$$
0.409644 + 0.912245i $$0.365653\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ 3.00000 0.344124
$$77$$ 5.00000 0.569803
$$78$$ 0 0
$$79$$ 15.0000 1.68763 0.843816 0.536633i $$-0.180304\pi$$
0.843816 + 0.536633i $$0.180304\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.00000 0.220863
$$83$$ 10.0000 1.09764 0.548821 0.835940i $$-0.315077\pi$$
0.548821 + 0.835940i $$0.315077\pi$$
$$84$$ −5.00000 −0.545545
$$85$$ 0 0
$$86$$ 1.00000 0.107833
$$87$$ 6.00000 0.643268
$$88$$ −1.00000 −0.106600
$$89$$ −1.00000 −0.106000 −0.0529999 0.998595i $$-0.516878\pi$$
−0.0529999 + 0.998595i $$0.516878\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 1.00000 0.104257
$$93$$ −1.00000 −0.103695
$$94$$ −4.00000 −0.412568
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 18.0000 1.81827
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ 7.00000 0.696526 0.348263 0.937397i $$-0.386772\pi$$
0.348263 + 0.937397i $$0.386772\pi$$
$$102$$ 4.00000 0.396059
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −3.00000 −0.291386
$$107$$ −9.00000 −0.870063 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ −5.00000 −0.472456
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ 3.00000 0.280976
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ −14.0000 −1.28880
$$119$$ −20.0000 −1.83340
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 14.0000 1.26750
$$123$$ 2.00000 0.180334
$$124$$ −1.00000 −0.0898027
$$125$$ 0 0
$$126$$ −5.00000 −0.445435
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ 18.0000 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ −15.0000 −1.30066
$$134$$ −10.0000 −0.863868
$$135$$ 0 0
$$136$$ 4.00000 0.342997
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 1.00000 0.0851257
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ 9.00000 0.755263
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 7.00000 0.579324
$$147$$ 18.0000 1.48461
$$148$$ 4.00000 0.328798
$$149$$ 1.00000 0.0819232 0.0409616 0.999161i $$-0.486958\pi$$
0.0409616 + 0.999161i $$0.486958\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 3.00000 0.243332
$$153$$ 4.00000 0.323381
$$154$$ 5.00000 0.402911
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −1.00000 −0.0798087 −0.0399043 0.999204i $$-0.512705\pi$$
−0.0399043 + 0.999204i $$0.512705\pi$$
$$158$$ 15.0000 1.19334
$$159$$ −3.00000 −0.237915
$$160$$ 0 0
$$161$$ −5.00000 −0.394055
$$162$$ 1.00000 0.0785674
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 10.0000 0.776151
$$167$$ 9.00000 0.696441 0.348220 0.937413i $$-0.386786\pi$$
0.348220 + 0.937413i $$0.386786\pi$$
$$168$$ −5.00000 −0.385758
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 3.00000 0.229416
$$172$$ 1.00000 0.0762493
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ −14.0000 −1.05230
$$178$$ −1.00000 −0.0749532
$$179$$ −8.00000 −0.597948 −0.298974 0.954261i $$-0.596644\pi$$
−0.298974 + 0.954261i $$0.596644\pi$$
$$180$$ 0 0
$$181$$ 11.0000 0.817624 0.408812 0.912619i $$-0.365943\pi$$
0.408812 + 0.912619i $$0.365943\pi$$
$$182$$ 0 0
$$183$$ 14.0000 1.03491
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ −1.00000 −0.0733236
$$187$$ −4.00000 −0.292509
$$188$$ −4.00000 −0.291730
$$189$$ −5.00000 −0.363696
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 10.0000 0.719816 0.359908 0.932988i $$-0.382808\pi$$
0.359908 + 0.932988i $$0.382808\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 18.0000 1.28571
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ −9.00000 −0.637993 −0.318997 0.947756i $$-0.603346\pi$$
−0.318997 + 0.947756i $$0.603346\pi$$
$$200$$ 0 0
$$201$$ −10.0000 −0.705346
$$202$$ 7.00000 0.492518
$$203$$ −30.0000 −2.10559
$$204$$ 4.00000 0.280056
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ −21.0000 −1.44570 −0.722850 0.691005i $$-0.757168\pi$$
−0.722850 + 0.691005i $$0.757168\pi$$
$$212$$ −3.00000 −0.206041
$$213$$ 9.00000 0.616670
$$214$$ −9.00000 −0.615227
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 5.00000 0.339422
$$218$$ 10.0000 0.677285
$$219$$ 7.00000 0.473016
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 4.00000 0.268462
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ −5.00000 −0.334077
$$225$$ 0 0
$$226$$ 9.00000 0.598671
$$227$$ −29.0000 −1.92480 −0.962399 0.271640i $$-0.912434\pi$$
−0.962399 + 0.271640i $$0.912434\pi$$
$$228$$ 3.00000 0.198680
$$229$$ 23.0000 1.51988 0.759941 0.649992i $$-0.225228\pi$$
0.759941 + 0.649992i $$0.225228\pi$$
$$230$$ 0 0
$$231$$ 5.00000 0.328976
$$232$$ 6.00000 0.393919
$$233$$ −13.0000 −0.851658 −0.425829 0.904804i $$-0.640018\pi$$
−0.425829 + 0.904804i $$0.640018\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −14.0000 −0.911322
$$237$$ 15.0000 0.974355
$$238$$ −20.0000 −1.29641
$$239$$ −22.0000 −1.42306 −0.711531 0.702655i $$-0.751998\pi$$
−0.711531 + 0.702655i $$0.751998\pi$$
$$240$$ 0 0
$$241$$ 4.00000 0.257663 0.128831 0.991667i $$-0.458877\pi$$
0.128831 + 0.991667i $$0.458877\pi$$
$$242$$ −10.0000 −0.642824
$$243$$ 1.00000 0.0641500
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ 2.00000 0.127515
$$247$$ 0 0
$$248$$ −1.00000 −0.0635001
$$249$$ 10.0000 0.633724
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ −5.00000 −0.314970
$$253$$ −1.00000 −0.0628695
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −3.00000 −0.187135 −0.0935674 0.995613i $$-0.529827\pi$$
−0.0935674 + 0.995613i $$0.529827\pi$$
$$258$$ 1.00000 0.0622573
$$259$$ −20.0000 −1.24274
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 18.0000 1.11204
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ −1.00000 −0.0615457
$$265$$ 0 0
$$266$$ −15.0000 −0.919709
$$267$$ −1.00000 −0.0611990
$$268$$ −10.0000 −0.610847
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ 13.0000 0.789694 0.394847 0.918747i $$-0.370798\pi$$
0.394847 + 0.918747i $$0.370798\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ −26.0000 −1.56219 −0.781094 0.624413i $$-0.785338\pi$$
−0.781094 + 0.624413i $$0.785338\pi$$
$$278$$ 4.00000 0.239904
$$279$$ −1.00000 −0.0598684
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ −4.00000 −0.238197
$$283$$ −22.0000 −1.30776 −0.653882 0.756596i $$-0.726861\pi$$
−0.653882 + 0.756596i $$0.726861\pi$$
$$284$$ 9.00000 0.534052
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −10.0000 −0.590281
$$288$$ 1.00000 0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 7.00000 0.409644
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 18.0000 1.04978
$$295$$ 0 0
$$296$$ 4.00000 0.232495
$$297$$ −1.00000 −0.0580259
$$298$$ 1.00000 0.0579284
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −5.00000 −0.288195
$$302$$ −8.00000 −0.460348
$$303$$ 7.00000 0.402139
$$304$$ 3.00000 0.172062
$$305$$ 0 0
$$306$$ 4.00000 0.228665
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 5.00000 0.284901
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ −1.00000 −0.0564333
$$315$$ 0 0
$$316$$ 15.0000 0.843816
$$317$$ 22.0000 1.23564 0.617822 0.786318i $$-0.288015\pi$$
0.617822 + 0.786318i $$0.288015\pi$$
$$318$$ −3.00000 −0.168232
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ −9.00000 −0.502331
$$322$$ −5.00000 −0.278639
$$323$$ 12.0000 0.667698
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 8.00000 0.443079
$$327$$ 10.0000 0.553001
$$328$$ 2.00000 0.110432
$$329$$ 20.0000 1.10264
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 10.0000 0.548821
$$333$$ 4.00000 0.219199
$$334$$ 9.00000 0.492458
$$335$$ 0 0
$$336$$ −5.00000 −0.272772
$$337$$ 10.0000 0.544735 0.272367 0.962193i $$-0.412193\pi$$
0.272367 + 0.962193i $$0.412193\pi$$
$$338$$ −13.0000 −0.707107
$$339$$ 9.00000 0.488813
$$340$$ 0 0
$$341$$ 1.00000 0.0541530
$$342$$ 3.00000 0.162221
$$343$$ −55.0000 −2.96972
$$344$$ 1.00000 0.0539164
$$345$$ 0 0
$$346$$ 2.00000 0.107521
$$347$$ −30.0000 −1.61048 −0.805242 0.592946i $$-0.797965\pi$$
−0.805242 + 0.592946i $$0.797965\pi$$
$$348$$ 6.00000 0.321634
$$349$$ −32.0000 −1.71292 −0.856460 0.516213i $$-0.827341\pi$$
−0.856460 + 0.516213i $$0.827341\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.00000 −0.0533002
$$353$$ 8.00000 0.425797 0.212899 0.977074i $$-0.431710\pi$$
0.212899 + 0.977074i $$0.431710\pi$$
$$354$$ −14.0000 −0.744092
$$355$$ 0 0
$$356$$ −1.00000 −0.0529999
$$357$$ −20.0000 −1.05851
$$358$$ −8.00000 −0.422813
$$359$$ −15.0000 −0.791670 −0.395835 0.918322i $$-0.629545\pi$$
−0.395835 + 0.918322i $$0.629545\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ 11.0000 0.578147
$$363$$ −10.0000 −0.524864
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 14.0000 0.731792
$$367$$ −26.0000 −1.35719 −0.678594 0.734513i $$-0.737411\pi$$
−0.678594 + 0.734513i $$0.737411\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ 15.0000 0.778761
$$372$$ −1.00000 −0.0518476
$$373$$ 3.00000 0.155334 0.0776671 0.996979i $$-0.475253\pi$$
0.0776671 + 0.996979i $$0.475253\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ 0 0
$$378$$ −5.00000 −0.257172
$$379$$ −19.0000 −0.975964 −0.487982 0.872854i $$-0.662267\pi$$
−0.487982 + 0.872854i $$0.662267\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ 24.0000 1.22795
$$383$$ 20.0000 1.02195 0.510976 0.859595i $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ 1.00000 0.0508329
$$388$$ −10.0000 −0.507673
$$389$$ −16.0000 −0.811232 −0.405616 0.914044i $$-0.632943\pi$$
−0.405616 + 0.914044i $$0.632943\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 18.0000 0.909137
$$393$$ 18.0000 0.907980
$$394$$ 2.00000 0.100759
$$395$$ 0 0
$$396$$ −1.00000 −0.0502519
$$397$$ −17.0000 −0.853206 −0.426603 0.904439i $$-0.640290\pi$$
−0.426603 + 0.904439i $$0.640290\pi$$
$$398$$ −9.00000 −0.451129
$$399$$ −15.0000 −0.750939
$$400$$ 0 0
$$401$$ −29.0000 −1.44819 −0.724095 0.689700i $$-0.757743\pi$$
−0.724095 + 0.689700i $$0.757743\pi$$
$$402$$ −10.0000 −0.498755
$$403$$ 0 0
$$404$$ 7.00000 0.348263
$$405$$ 0 0
$$406$$ −30.0000 −1.48888
$$407$$ −4.00000 −0.198273
$$408$$ 4.00000 0.198030
$$409$$ −16.0000 −0.791149 −0.395575 0.918434i $$-0.629455\pi$$
−0.395575 + 0.918434i $$0.629455\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ 8.00000 0.394132
$$413$$ 70.0000 3.44447
$$414$$ 1.00000 0.0491473
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.00000 0.195881
$$418$$ −3.00000 −0.146735
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ −21.0000 −1.02226
$$423$$ −4.00000 −0.194487
$$424$$ −3.00000 −0.145693
$$425$$ 0 0
$$426$$ 9.00000 0.436051
$$427$$ −70.0000 −3.38754
$$428$$ −9.00000 −0.435031
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −11.0000 −0.528626 −0.264313 0.964437i $$-0.585145\pi$$
−0.264313 + 0.964437i $$0.585145\pi$$
$$434$$ 5.00000 0.240008
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ 3.00000 0.143509
$$438$$ 7.00000 0.334473
$$439$$ 36.0000 1.71819 0.859093 0.511819i $$-0.171028\pi$$
0.859093 + 0.511819i $$0.171028\pi$$
$$440$$ 0 0
$$441$$ 18.0000 0.857143
$$442$$ 0 0
$$443$$ −15.0000 −0.712672 −0.356336 0.934358i $$-0.615974\pi$$
−0.356336 + 0.934358i $$0.615974\pi$$
$$444$$ 4.00000 0.189832
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ 1.00000 0.0472984
$$448$$ −5.00000 −0.236228
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ −2.00000 −0.0941763
$$452$$ 9.00000 0.423324
$$453$$ −8.00000 −0.375873
$$454$$ −29.0000 −1.36104
$$455$$ 0 0
$$456$$ 3.00000 0.140488
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 23.0000 1.07472
$$459$$ 4.00000 0.186704
$$460$$ 0 0
$$461$$ 24.0000 1.11779 0.558896 0.829238i $$-0.311225\pi$$
0.558896 + 0.829238i $$0.311225\pi$$
$$462$$ 5.00000 0.232621
$$463$$ 6.00000 0.278844 0.139422 0.990233i $$-0.455476\pi$$
0.139422 + 0.990233i $$0.455476\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −13.0000 −0.602213
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ 50.0000 2.30879
$$470$$ 0 0
$$471$$ −1.00000 −0.0460776
$$472$$ −14.0000 −0.644402
$$473$$ −1.00000 −0.0459800
$$474$$ 15.0000 0.688973
$$475$$ 0 0
$$476$$ −20.0000 −0.916698
$$477$$ −3.00000 −0.137361
$$478$$ −22.0000 −1.00626
$$479$$ 29.0000 1.32504 0.662522 0.749043i $$-0.269486\pi$$
0.662522 + 0.749043i $$0.269486\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 4.00000 0.182195
$$483$$ −5.00000 −0.227508
$$484$$ −10.0000 −0.454545
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −36.0000 −1.63132 −0.815658 0.578535i $$-0.803625\pi$$
−0.815658 + 0.578535i $$0.803625\pi$$
$$488$$ 14.0000 0.633750
$$489$$ 8.00000 0.361773
$$490$$ 0 0
$$491$$ 27.0000 1.21849 0.609246 0.792981i $$-0.291472\pi$$
0.609246 + 0.792981i $$0.291472\pi$$
$$492$$ 2.00000 0.0901670
$$493$$ 24.0000 1.08091
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −1.00000 −0.0449013
$$497$$ −45.0000 −2.01853
$$498$$ 10.0000 0.448111
$$499$$ −42.0000 −1.88018 −0.940089 0.340929i $$-0.889258\pi$$
−0.940089 + 0.340929i $$0.889258\pi$$
$$500$$ 0 0
$$501$$ 9.00000 0.402090
$$502$$ −12.0000 −0.535586
$$503$$ −42.0000 −1.87269 −0.936344 0.351085i $$-0.885813\pi$$
−0.936344 + 0.351085i $$0.885813\pi$$
$$504$$ −5.00000 −0.222718
$$505$$ 0 0
$$506$$ −1.00000 −0.0444554
$$507$$ −13.0000 −0.577350
$$508$$ 16.0000 0.709885
$$509$$ −36.0000 −1.59567 −0.797836 0.602875i $$-0.794022\pi$$
−0.797836 + 0.602875i $$0.794022\pi$$
$$510$$ 0 0
$$511$$ −35.0000 −1.54831
$$512$$ 1.00000 0.0441942
$$513$$ 3.00000 0.132453
$$514$$ −3.00000 −0.132324
$$515$$ 0 0
$$516$$ 1.00000 0.0440225
$$517$$ 4.00000 0.175920
$$518$$ −20.0000 −0.878750
$$519$$ 2.00000 0.0877903
$$520$$ 0 0
$$521$$ −20.0000 −0.876216 −0.438108 0.898922i $$-0.644351\pi$$
−0.438108 + 0.898922i $$0.644351\pi$$
$$522$$ 6.00000 0.262613
$$523$$ 41.0000 1.79280 0.896402 0.443241i $$-0.146171\pi$$
0.896402 + 0.443241i $$0.146171\pi$$
$$524$$ 18.0000 0.786334
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ −4.00000 −0.174243
$$528$$ −1.00000 −0.0435194
$$529$$ −22.0000 −0.956522
$$530$$ 0 0
$$531$$ −14.0000 −0.607548
$$532$$ −15.0000 −0.650332
$$533$$ 0 0
$$534$$ −1.00000 −0.0432742
$$535$$ 0 0
$$536$$ −10.0000 −0.431934
$$537$$ −8.00000 −0.345225
$$538$$ 30.0000 1.29339
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ 4.00000 0.171973 0.0859867 0.996296i $$-0.472596\pi$$
0.0859867 + 0.996296i $$0.472596\pi$$
$$542$$ 13.0000 0.558398
$$543$$ 11.0000 0.472055
$$544$$ 4.00000 0.171499
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ 18.0000 0.766826
$$552$$ 1.00000 0.0425628
$$553$$ −75.0000 −3.18932
$$554$$ −26.0000 −1.10463
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ 33.0000 1.39825 0.699127 0.714997i $$-0.253572\pi$$
0.699127 + 0.714997i $$0.253572\pi$$
$$558$$ −1.00000 −0.0423334
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 0 0
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 0 0
$$566$$ −22.0000 −0.924729
$$567$$ −5.00000 −0.209980
$$568$$ 9.00000 0.377632
$$569$$ −1.00000 −0.0419222 −0.0209611 0.999780i $$-0.506673\pi$$
−0.0209611 + 0.999780i $$0.506673\pi$$
$$570$$ 0 0
$$571$$ 34.0000 1.42286 0.711428 0.702759i $$-0.248049\pi$$
0.711428 + 0.702759i $$0.248049\pi$$
$$572$$ 0 0
$$573$$ 24.0000 1.00261
$$574$$ −10.0000 −0.417392
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 10.0000 0.416305 0.208153 0.978096i $$-0.433255\pi$$
0.208153 + 0.978096i $$0.433255\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 10.0000 0.415586
$$580$$ 0 0
$$581$$ −50.0000 −2.07435
$$582$$ −10.0000 −0.414513
$$583$$ 3.00000 0.124247
$$584$$ 7.00000 0.289662
$$585$$ 0 0
$$586$$ −30.0000 −1.23929
$$587$$ 30.0000 1.23823 0.619116 0.785299i $$-0.287491\pi$$
0.619116 + 0.785299i $$0.287491\pi$$
$$588$$ 18.0000 0.742307
$$589$$ −3.00000 −0.123613
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ 4.00000 0.164399
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ 0 0
$$596$$ 1.00000 0.0409616
$$597$$ −9.00000 −0.368345
$$598$$ 0 0
$$599$$ 15.0000 0.612883 0.306442 0.951889i $$-0.400862\pi$$
0.306442 + 0.951889i $$0.400862\pi$$
$$600$$ 0 0
$$601$$ −28.0000 −1.14214 −0.571072 0.820900i $$-0.693472\pi$$
−0.571072 + 0.820900i $$0.693472\pi$$
$$602$$ −5.00000 −0.203785
$$603$$ −10.0000 −0.407231
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 7.00000 0.284356
$$607$$ 23.0000 0.933541 0.466771 0.884378i $$-0.345417\pi$$
0.466771 + 0.884378i $$0.345417\pi$$
$$608$$ 3.00000 0.121666
$$609$$ −30.0000 −1.21566
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 4.00000 0.161690
$$613$$ 16.0000 0.646234 0.323117 0.946359i $$-0.395269\pi$$
0.323117 + 0.946359i $$0.395269\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 5.00000 0.201456
$$617$$ 13.0000 0.523360 0.261680 0.965155i $$-0.415723\pi$$
0.261680 + 0.965155i $$0.415723\pi$$
$$618$$ 8.00000 0.321807
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ 5.00000 0.200321
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −26.0000 −1.03917
$$627$$ −3.00000 −0.119808
$$628$$ −1.00000 −0.0399043
$$629$$ 16.0000 0.637962
$$630$$ 0 0
$$631$$ 29.0000 1.15447 0.577236 0.816577i $$-0.304131\pi$$
0.577236 + 0.816577i $$0.304131\pi$$
$$632$$ 15.0000 0.596668
$$633$$ −21.0000 −0.834675
$$634$$ 22.0000 0.873732
$$635$$ 0 0
$$636$$ −3.00000 −0.118958
$$637$$ 0 0
$$638$$ −6.00000 −0.237542
$$639$$ 9.00000 0.356034
$$640$$ 0 0
$$641$$ 38.0000 1.50091 0.750455 0.660922i $$-0.229834\pi$$
0.750455 + 0.660922i $$0.229834\pi$$
$$642$$ −9.00000 −0.355202
$$643$$ 3.00000 0.118308 0.0591542 0.998249i $$-0.481160\pi$$
0.0591542 + 0.998249i $$0.481160\pi$$
$$644$$ −5.00000 −0.197028
$$645$$ 0 0
$$646$$ 12.0000 0.472134
$$647$$ 27.0000 1.06148 0.530740 0.847535i $$-0.321914\pi$$
0.530740 + 0.847535i $$0.321914\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 14.0000 0.549548
$$650$$ 0 0
$$651$$ 5.00000 0.195965
$$652$$ 8.00000 0.313304
$$653$$ −2.00000 −0.0782660 −0.0391330 0.999234i $$-0.512460\pi$$
−0.0391330 + 0.999234i $$0.512460\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ 7.00000 0.273096
$$658$$ 20.0000 0.779681
$$659$$ 10.0000 0.389545 0.194772 0.980848i $$-0.437603\pi$$
0.194772 + 0.980848i $$0.437603\pi$$
$$660$$ 0 0
$$661$$ −40.0000 −1.55582 −0.777910 0.628376i $$-0.783720\pi$$
−0.777910 + 0.628376i $$0.783720\pi$$
$$662$$ 28.0000 1.08825
$$663$$ 0 0
$$664$$ 10.0000 0.388075
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ 6.00000 0.232321
$$668$$ 9.00000 0.348220
$$669$$ −8.00000 −0.309298
$$670$$ 0 0
$$671$$ −14.0000 −0.540464
$$672$$ −5.00000 −0.192879
$$673$$ 6.00000 0.231283 0.115642 0.993291i $$-0.463108\pi$$
0.115642 + 0.993291i $$0.463108\pi$$
$$674$$ 10.0000 0.385186
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ −11.0000 −0.422764 −0.211382 0.977403i $$-0.567796\pi$$
−0.211382 + 0.977403i $$0.567796\pi$$
$$678$$ 9.00000 0.345643
$$679$$ 50.0000 1.91882
$$680$$ 0 0
$$681$$ −29.0000 −1.11128
$$682$$ 1.00000 0.0382920
$$683$$ 31.0000 1.18618 0.593091 0.805135i $$-0.297907\pi$$
0.593091 + 0.805135i $$0.297907\pi$$
$$684$$ 3.00000 0.114708
$$685$$ 0 0
$$686$$ −55.0000 −2.09991
$$687$$ 23.0000 0.877505
$$688$$ 1.00000 0.0381246
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −5.00000 −0.190209 −0.0951045 0.995467i $$-0.530319\pi$$
−0.0951045 + 0.995467i $$0.530319\pi$$
$$692$$ 2.00000 0.0760286
$$693$$ 5.00000 0.189934
$$694$$ −30.0000 −1.13878
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ 8.00000 0.303022
$$698$$ −32.0000 −1.21122
$$699$$ −13.0000 −0.491705
$$700$$ 0 0
$$701$$ 35.0000 1.32193 0.660966 0.750416i $$-0.270147\pi$$
0.660966 + 0.750416i $$0.270147\pi$$
$$702$$ 0 0
$$703$$ 12.0000 0.452589
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 8.00000 0.301084
$$707$$ −35.0000 −1.31631
$$708$$ −14.0000 −0.526152
$$709$$ 21.0000 0.788672 0.394336 0.918966i $$-0.370975\pi$$
0.394336 + 0.918966i $$0.370975\pi$$
$$710$$ 0 0
$$711$$ 15.0000 0.562544
$$712$$ −1.00000 −0.0374766
$$713$$ −1.00000 −0.0374503
$$714$$ −20.0000 −0.748481
$$715$$ 0 0
$$716$$ −8.00000 −0.298974
$$717$$ −22.0000 −0.821605
$$718$$ −15.0000 −0.559795
$$719$$ 50.0000 1.86469 0.932343 0.361576i $$-0.117761\pi$$
0.932343 + 0.361576i $$0.117761\pi$$
$$720$$ 0 0
$$721$$ −40.0000 −1.48968
$$722$$ −10.0000 −0.372161
$$723$$ 4.00000 0.148762
$$724$$ 11.0000 0.408812
$$725$$ 0 0
$$726$$ −10.0000 −0.371135
$$727$$ 27.0000 1.00137 0.500687 0.865628i $$-0.333081\pi$$
0.500687 + 0.865628i $$0.333081\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ 14.0000 0.517455
$$733$$ −34.0000 −1.25582 −0.627909 0.778287i $$-0.716089\pi$$
−0.627909 + 0.778287i $$0.716089\pi$$
$$734$$ −26.0000 −0.959678
$$735$$ 0 0
$$736$$ 1.00000 0.0368605
$$737$$ 10.0000 0.368355
$$738$$ 2.00000 0.0736210
$$739$$ 46.0000 1.69214 0.846069 0.533074i $$-0.178963\pi$$
0.846069 + 0.533074i $$0.178963\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 15.0000 0.550667
$$743$$ −21.0000 −0.770415 −0.385208 0.922830i $$-0.625870\pi$$
−0.385208 + 0.922830i $$0.625870\pi$$
$$744$$ −1.00000 −0.0366618
$$745$$ 0 0
$$746$$ 3.00000 0.109838
$$747$$ 10.0000 0.365881
$$748$$ −4.00000 −0.146254
$$749$$ 45.0000 1.64426
$$750$$ 0 0
$$751$$ 2.00000 0.0729810 0.0364905 0.999334i $$-0.488382\pi$$
0.0364905 + 0.999334i $$0.488382\pi$$
$$752$$ −4.00000 −0.145865
$$753$$ −12.0000 −0.437304
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −5.00000 −0.181848
$$757$$ 4.00000 0.145382 0.0726912 0.997354i $$-0.476841\pi$$
0.0726912 + 0.997354i $$0.476841\pi$$
$$758$$ −19.0000 −0.690111
$$759$$ −1.00000 −0.0362977
$$760$$ 0 0
$$761$$ −21.0000 −0.761249 −0.380625 0.924730i $$-0.624291\pi$$
−0.380625 + 0.924730i $$0.624291\pi$$
$$762$$ 16.0000 0.579619
$$763$$ −50.0000 −1.81012
$$764$$ 24.0000 0.868290
$$765$$ 0 0
$$766$$ 20.0000 0.722629
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ −27.0000 −0.973645 −0.486822 0.873501i $$-0.661844\pi$$
−0.486822 + 0.873501i $$0.661844\pi$$
$$770$$ 0 0
$$771$$ −3.00000 −0.108042
$$772$$ 10.0000 0.359908
$$773$$ 51.0000 1.83434 0.917171 0.398493i $$-0.130467\pi$$
0.917171 + 0.398493i $$0.130467\pi$$
$$774$$ 1.00000 0.0359443
$$775$$ 0 0
$$776$$ −10.0000 −0.358979
$$777$$ −20.0000 −0.717496
$$778$$ −16.0000 −0.573628
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ −9.00000 −0.322045
$$782$$ 4.00000 0.143040
$$783$$ 6.00000 0.214423
$$784$$ 18.0000 0.642857
$$785$$ 0 0
$$786$$ 18.0000 0.642039
$$787$$ 25.0000 0.891154 0.445577 0.895244i $$-0.352999\pi$$
0.445577 + 0.895244i $$0.352999\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ −8.00000 −0.284808
$$790$$ 0 0
$$791$$ −45.0000 −1.60002
$$792$$ −1.00000 −0.0355335
$$793$$ 0 0
$$794$$ −17.0000 −0.603307
$$795$$ 0 0
$$796$$ −9.00000 −0.318997
$$797$$ 34.0000 1.20434 0.602171 0.798367i $$-0.294303\pi$$
0.602171 + 0.798367i $$0.294303\pi$$
$$798$$ −15.0000 −0.530994
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ −1.00000 −0.0353333
$$802$$ −29.0000 −1.02403
$$803$$ −7.00000 −0.247025
$$804$$ −10.0000 −0.352673
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 30.0000 1.05605
$$808$$ 7.00000 0.246259
$$809$$ 15.0000 0.527372 0.263686 0.964609i $$-0.415062\pi$$
0.263686 + 0.964609i $$0.415062\pi$$
$$810$$ 0 0
$$811$$ −17.0000 −0.596951 −0.298475 0.954417i $$-0.596478\pi$$
−0.298475 + 0.954417i $$0.596478\pi$$
$$812$$ −30.0000 −1.05279
$$813$$ 13.0000 0.455930
$$814$$ −4.00000 −0.140200
$$815$$ 0 0
$$816$$ 4.00000 0.140028
$$817$$ 3.00000 0.104957
$$818$$ −16.0000 −0.559427
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 6.00000 0.209274
$$823$$ −18.0000 −0.627441 −0.313720 0.949515i $$-0.601575\pi$$
−0.313720 + 0.949515i $$0.601575\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 70.0000 2.43561
$$827$$ 34.0000 1.18230 0.591148 0.806563i $$-0.298675\pi$$
0.591148 + 0.806563i $$0.298675\pi$$
$$828$$ 1.00000 0.0347524
$$829$$ 39.0000 1.35453 0.677263 0.735741i $$-0.263166\pi$$
0.677263 + 0.735741i $$0.263166\pi$$
$$830$$ 0 0
$$831$$ −26.0000 −0.901930
$$832$$ 0 0
$$833$$ 72.0000 2.49465
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ −3.00000 −0.103757
$$837$$ −1.00000 −0.0345651
$$838$$ 26.0000 0.898155
$$839$$ −35.0000 −1.20833 −0.604167 0.796858i $$-0.706494\pi$$
−0.604167 + 0.796858i $$0.706494\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 22.0000 0.758170
$$843$$ 0 0
$$844$$ −21.0000 −0.722850
$$845$$ 0 0
$$846$$ −4.00000 −0.137523
$$847$$ 50.0000 1.71802
$$848$$ −3.00000 −0.103020
$$849$$ −22.0000 −0.755038
$$850$$ 0 0
$$851$$ 4.00000 0.137118
$$852$$ 9.00000 0.308335
$$853$$ −49.0000 −1.67773 −0.838864 0.544341i $$-0.816780\pi$$
−0.838864 + 0.544341i $$0.816780\pi$$
$$854$$ −70.0000 −2.39535
$$855$$ 0 0
$$856$$ −9.00000 −0.307614
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ 0 0
$$861$$ −10.0000 −0.340799
$$862$$ 8.00000 0.272481
$$863$$ −57.0000 −1.94030 −0.970151 0.242500i $$-0.922032\pi$$
−0.970151 + 0.242500i $$0.922032\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −11.0000 −0.373795
$$867$$ −1.00000 −0.0339618
$$868$$ 5.00000 0.169711
$$869$$ −15.0000 −0.508840
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 10.0000 0.338643
$$873$$ −10.0000 −0.338449
$$874$$ 3.00000 0.101477
$$875$$ 0 0
$$876$$ 7.00000 0.236508
$$877$$ −34.0000 −1.14810 −0.574049 0.818821i $$-0.694628\pi$$
−0.574049 + 0.818821i $$0.694628\pi$$
$$878$$ 36.0000 1.21494
$$879$$ −30.0000 −1.01187
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ 18.0000 0.606092
$$883$$ 39.0000 1.31245 0.656227 0.754563i $$-0.272151\pi$$
0.656227 + 0.754563i $$0.272151\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −15.0000 −0.503935
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ 4.00000 0.134231
$$889$$ −80.0000 −2.68311
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ −8.00000 −0.267860
$$893$$ −12.0000 −0.401565
$$894$$ 1.00000 0.0334450
$$895$$ 0 0
$$896$$ −5.00000 −0.167038
$$897$$ 0 0
$$898$$ 6.00000 0.200223
$$899$$ −6.00000 −0.200111
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ −2.00000 −0.0665927
$$903$$ −5.00000 −0.166390
$$904$$ 9.00000 0.299336
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ −24.0000 −0.796907 −0.398453 0.917189i $$-0.630453\pi$$
−0.398453 + 0.917189i $$0.630453\pi$$
$$908$$ −29.0000 −0.962399
$$909$$ 7.00000 0.232175
$$910$$ 0 0
$$911$$ 18.0000 0.596367 0.298183 0.954509i $$-0.403619\pi$$
0.298183 + 0.954509i $$0.403619\pi$$
$$912$$ 3.00000 0.0993399
$$913$$ −10.0000 −0.330952
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ 23.0000 0.759941
$$917$$ −90.0000 −2.97206
$$918$$ 4.00000 0.132020
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ 24.0000 0.790398
$$923$$ 0 0
$$924$$ 5.00000 0.164488
$$925$$ 0 0
$$926$$ 6.00000 0.197172
$$927$$ 8.00000 0.262754
$$928$$ 6.00000 0.196960
$$929$$ 33.0000 1.08269 0.541347 0.840799i $$-0.317914\pi$$
0.541347 + 0.840799i $$0.317914\pi$$
$$930$$ 0 0
$$931$$ 54.0000 1.76978
$$932$$ −13.0000 −0.425829
$$933$$ 0 0
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 38.0000 1.24141 0.620703 0.784046i $$-0.286847\pi$$
0.620703 + 0.784046i $$0.286847\pi$$
$$938$$ 50.0000 1.63256
$$939$$ −26.0000 −0.848478
$$940$$ 0 0
$$941$$ −52.0000 −1.69515 −0.847576 0.530674i $$-0.821939\pi$$
−0.847576 + 0.530674i $$0.821939\pi$$
$$942$$ −1.00000 −0.0325818
$$943$$ 2.00000 0.0651290
$$944$$ −14.0000 −0.455661
$$945$$ 0 0
$$946$$ −1.00000 −0.0325128
$$947$$ −18.0000 −0.584921 −0.292461 0.956278i $$-0.594474\pi$$
−0.292461 + 0.956278i $$0.594474\pi$$
$$948$$ 15.0000 0.487177
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 22.0000 0.713399
$$952$$ −20.0000 −0.648204
$$953$$ −4.00000 −0.129573 −0.0647864 0.997899i $$-0.520637\pi$$
−0.0647864 + 0.997899i $$0.520637\pi$$
$$954$$ −3.00000 −0.0971286
$$955$$ 0 0
$$956$$ −22.0000 −0.711531
$$957$$ −6.00000 −0.193952
$$958$$ 29.0000 0.936947
$$959$$ −30.0000 −0.968751
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 0 0
$$963$$ −9.00000 −0.290021
$$964$$ 4.00000 0.128831
$$965$$ 0 0
$$966$$ −5.00000 −0.160872
$$967$$ −28.0000 −0.900419 −0.450210 0.892923i $$-0.648651\pi$$
−0.450210 + 0.892923i $$0.648651\pi$$
$$968$$ −10.0000 −0.321412
$$969$$ 12.0000 0.385496
$$970$$ 0 0
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −20.0000 −0.641171
$$974$$ −36.0000 −1.15351
$$975$$ 0 0
$$976$$ 14.0000 0.448129
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 8.00000 0.255812
$$979$$ 1.00000 0.0319601
$$980$$ 0 0
$$981$$ 10.0000 0.319275
$$982$$ 27.0000 0.861605
$$983$$ −28.0000 −0.893061 −0.446531 0.894768i $$-0.647341\pi$$
−0.446531 + 0.894768i $$0.647341\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ 0 0
$$986$$ 24.0000 0.764316
$$987$$ 20.0000 0.636607
$$988$$ 0 0
$$989$$ 1.00000 0.0317982
$$990$$ 0 0
$$991$$ −17.0000 −0.540023 −0.270011 0.962857i $$-0.587027\pi$$
−0.270011 + 0.962857i $$0.587027\pi$$
$$992$$ −1.00000 −0.0317500
$$993$$ 28.0000 0.888553
$$994$$ −45.0000 −1.42731
$$995$$ 0 0
$$996$$ 10.0000 0.316862
$$997$$ −6.00000 −0.190022 −0.0950110 0.995476i $$-0.530289\pi$$
−0.0950110 + 0.995476i $$0.530289\pi$$
$$998$$ −42.0000 −1.32949
$$999$$ 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 4650.2.a.bi.1.1 1
5.2 odd 4 930.2.d.c.559.2 yes 2
5.3 odd 4 930.2.d.c.559.1 2
5.4 even 2 4650.2.a.m.1.1 1
15.2 even 4 2790.2.d.e.559.1 2
15.8 even 4 2790.2.d.e.559.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.d.c.559.1 2 5.3 odd 4
930.2.d.c.559.2 yes 2 5.2 odd 4
2790.2.d.e.559.1 2 15.2 even 4
2790.2.d.e.559.2 2 15.8 even 4
4650.2.a.m.1.1 1 5.4 even 2
4650.2.a.bi.1.1 1 1.1 even 1 trivial