# Properties

 Label 310.2.a.b.1.1 Level $310$ Weight $2$ Character 310.1 Self dual yes Analytic conductor $2.475$ 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] = [310,2,Mod(1,310)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(310, 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("310.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$310 = 2 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 310.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: $$2.47536246266$$ 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 $$\chi$$ $$=$$ 310.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} +2.00000 q^{12} -2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} -4.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} -4.00000 q^{27} -4.00000 q^{29} -2.00000 q^{30} -1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} -4.00000 q^{38} -1.00000 q^{40} +6.00000 q^{41} +2.00000 q^{43} +2.00000 q^{44} -1.00000 q^{45} -4.00000 q^{46} +2.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} +8.00000 q^{53} -4.00000 q^{54} -2.00000 q^{55} -8.00000 q^{57} -4.00000 q^{58} +8.00000 q^{59} -2.00000 q^{60} -1.00000 q^{62} +1.00000 q^{64} +4.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} +1.00000 q^{72} +6.00000 q^{73} -8.00000 q^{74} +2.00000 q^{75} -4.00000 q^{76} -4.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} +6.00000 q^{83} -2.00000 q^{85} +2.00000 q^{86} -8.00000 q^{87} +2.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} -4.00000 q^{92} -2.00000 q^{93} +4.00000 q^{95} +2.00000 q^{96} -2.00000 q^{97} -7.00000 q^{98} +2.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 2.00000 0.816497
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 2.00000 0.577350
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 2.00000 0.426401
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 2.00000 0.408248
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −4.00000 −0.769800
$$28$$ 0 0
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ −1.00000 −0.179605
$$32$$ 1.00000 0.176777
$$33$$ 4.00000 0.696311
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ 2.00000 0.301511
$$45$$ −1.00000 −0.149071
$$46$$ −4.00000 −0.589768
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 2.00000 0.288675
$$49$$ −7.00000 −1.00000
$$50$$ 1.00000 0.141421
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ 8.00000 1.09888 0.549442 0.835532i $$-0.314840\pi$$
0.549442 + 0.835532i $$0.314840\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ −2.00000 −0.269680
$$56$$ 0 0
$$57$$ −8.00000 −1.05963
$$58$$ −4.00000 −0.525226
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ −2.00000 −0.258199
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ −1.00000 −0.127000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 2.00000 0.242536
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ 2.00000 0.230940
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ −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 $$0.393190\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ 2.00000 0.215666
$$87$$ −8.00000 −0.857690
$$88$$ 2.00000 0.213201
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −4.00000 −0.417029
$$93$$ −2.00000 −0.207390
$$94$$ 0 0
$$95$$ 4.00000 0.410391
$$96$$ 2.00000 0.204124
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ 2.00000 0.201008
$$100$$ 1.00000 0.100000
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 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
$$106$$ 8.00000 0.777029
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ −4.00000 −0.384900
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ −16.0000 −1.51865
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ −8.00000 −0.749269
$$115$$ 4.00000 0.373002
$$116$$ −4.00000 −0.371391
$$117$$ 0 0
$$118$$ 8.00000 0.736460
$$119$$ 0 0
$$120$$ −2.00000 −0.182574
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 12.0000 1.08200
$$124$$ −1.00000 −0.0898027
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 20.0000 1.74741 0.873704 0.486458i $$-0.161711\pi$$
0.873704 + 0.486458i $$0.161711\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 4.00000 0.344265
$$136$$ 2.00000 0.171499
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ −8.00000 −0.681005
$$139$$ 2.00000 0.169638 0.0848189 0.996396i $$-0.472969\pi$$
0.0848189 + 0.996396i $$0.472969\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 4.00000 0.332182
$$146$$ 6.00000 0.496564
$$147$$ −14.0000 −1.15470
$$148$$ −8.00000 −0.657596
$$149$$ −14.0000 −1.14692 −0.573462 0.819232i $$-0.694400\pi$$
−0.573462 + 0.819232i $$0.694400\pi$$
$$150$$ 2.00000 0.163299
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 1.00000 0.0803219
$$156$$ 0 0
$$157$$ 6.00000 0.478852 0.239426 0.970915i $$-0.423041\pi$$
0.239426 + 0.970915i $$0.423041\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ 16.0000 1.26888
$$160$$ −1.00000 −0.0790569
$$161$$ 0 0
$$162$$ −11.0000 −0.864242
$$163$$ 24.0000 1.87983 0.939913 0.341415i $$-0.110906\pi$$
0.939913 + 0.341415i $$0.110906\pi$$
$$164$$ 6.00000 0.468521
$$165$$ −4.00000 −0.311400
$$166$$ 6.00000 0.465690
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ −2.00000 −0.153393
$$171$$ −4.00000 −0.305888
$$172$$ 2.00000 0.152499
$$173$$ −2.00000 −0.152057 −0.0760286 0.997106i $$-0.524224\pi$$
−0.0760286 + 0.997106i $$0.524224\pi$$
$$174$$ −8.00000 −0.606478
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 16.0000 1.20263
$$178$$ −6.00000 −0.449719
$$179$$ 14.0000 1.04641 0.523205 0.852207i $$-0.324736\pi$$
0.523205 + 0.852207i $$0.324736\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −24.0000 −1.78391 −0.891953 0.452128i $$-0.850665\pi$$
−0.891953 + 0.452128i $$0.850665\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ 8.00000 0.588172
$$186$$ −2.00000 −0.146647
$$187$$ 4.00000 0.292509
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 4.00000 0.290191
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 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
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 8.00000 0.569976 0.284988 0.958531i $$-0.408010\pi$$
0.284988 + 0.958531i $$0.408010\pi$$
$$198$$ 2.00000 0.142134
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 8.00000 0.564276
$$202$$ 2.00000 0.140720
$$203$$ 0 0
$$204$$ 4.00000 0.280056
$$205$$ −6.00000 −0.419058
$$206$$ 8.00000 0.557386
$$207$$ −4.00000 −0.278019
$$208$$ 0 0
$$209$$ −8.00000 −0.553372
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ 8.00000 0.549442
$$213$$ 0 0
$$214$$ −8.00000 −0.546869
$$215$$ −2.00000 −0.136399
$$216$$ −4.00000 −0.272166
$$217$$ 0 0
$$218$$ −18.0000 −1.21911
$$219$$ 12.0000 0.810885
$$220$$ −2.00000 −0.134840
$$221$$ 0 0
$$222$$ −16.0000 −1.07385
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 14.0000 0.931266
$$227$$ 8.00000 0.530979 0.265489 0.964114i $$-0.414466\pi$$
0.265489 + 0.964114i $$0.414466\pi$$
$$228$$ −8.00000 −0.529813
$$229$$ 4.00000 0.264327 0.132164 0.991228i $$-0.457808\pi$$
0.132164 + 0.991228i $$0.457808\pi$$
$$230$$ 4.00000 0.263752
$$231$$ 0 0
$$232$$ −4.00000 −0.262613
$$233$$ −26.0000 −1.70332 −0.851658 0.524097i $$-0.824403\pi$$
−0.851658 + 0.524097i $$0.824403\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ −2.00000 −0.129099
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ −7.00000 −0.449977
$$243$$ −10.0000 −0.641500
$$244$$ 0 0
$$245$$ 7.00000 0.447214
$$246$$ 12.0000 0.765092
$$247$$ 0 0
$$248$$ −1.00000 −0.0635001
$$249$$ 12.0000 0.760469
$$250$$ −1.00000 −0.0632456
$$251$$ 14.0000 0.883672 0.441836 0.897096i $$-0.354327\pi$$
0.441836 + 0.897096i $$0.354327\pi$$
$$252$$ 0 0
$$253$$ −8.00000 −0.502956
$$254$$ 16.0000 1.00393
$$255$$ −4.00000 −0.250490
$$256$$ 1.00000 0.0625000
$$257$$ 10.0000 0.623783 0.311891 0.950118i $$-0.399037\pi$$
0.311891 + 0.950118i $$0.399037\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 20.0000 1.23560
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 4.00000 0.246183
$$265$$ −8.00000 −0.491436
$$266$$ 0 0
$$267$$ −12.0000 −0.734388
$$268$$ 4.00000 0.244339
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 4.00000 0.243432
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ 2.00000 0.120605
$$276$$ −8.00000 −0.481543
$$277$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$278$$ 2.00000 0.119952
$$279$$ −1.00000 −0.0598684
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ −24.0000 −1.42665 −0.713326 0.700832i $$-0.752812\pi$$
−0.713326 + 0.700832i $$0.752812\pi$$
$$284$$ 0 0
$$285$$ 8.00000 0.473879
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 4.00000 0.234888
$$291$$ −4.00000 −0.234484
$$292$$ 6.00000 0.351123
$$293$$ −18.0000 −1.05157 −0.525786 0.850617i $$-0.676229\pi$$
−0.525786 + 0.850617i $$0.676229\pi$$
$$294$$ −14.0000 −0.816497
$$295$$ −8.00000 −0.465778
$$296$$ −8.00000 −0.464991
$$297$$ −8.00000 −0.464207
$$298$$ −14.0000 −0.810998
$$299$$ 0 0
$$300$$ 2.00000 0.115470
$$301$$ 0 0
$$302$$ 4.00000 0.230174
$$303$$ 4.00000 0.229794
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 0 0
$$309$$ 16.0000 0.910208
$$310$$ 1.00000 0.0567962
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ 6.00000 0.338600
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ 2.00000 0.112331 0.0561656 0.998421i $$-0.482113\pi$$
0.0561656 + 0.998421i $$0.482113\pi$$
$$318$$ 16.0000 0.897235
$$319$$ −8.00000 −0.447914
$$320$$ −1.00000 −0.0559017
$$321$$ −16.0000 −0.893033
$$322$$ 0 0
$$323$$ −8.00000 −0.445132
$$324$$ −11.0000 −0.611111
$$325$$ 0 0
$$326$$ 24.0000 1.32924
$$327$$ −36.0000 −1.99080
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ −4.00000 −0.220193
$$331$$ −30.0000 −1.64895 −0.824475 0.565899i $$-0.808529\pi$$
−0.824475 + 0.565899i $$0.808529\pi$$
$$332$$ 6.00000 0.329293
$$333$$ −8.00000 −0.438397
$$334$$ 12.0000 0.656611
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ −13.0000 −0.707107
$$339$$ 28.0000 1.52075
$$340$$ −2.00000 −0.108465
$$341$$ −2.00000 −0.108306
$$342$$ −4.00000 −0.216295
$$343$$ 0 0
$$344$$ 2.00000 0.107833
$$345$$ 8.00000 0.430706
$$346$$ −2.00000 −0.107521
$$347$$ −22.0000 −1.18102 −0.590511 0.807030i $$-0.701074\pi$$
−0.590511 + 0.807030i $$0.701074\pi$$
$$348$$ −8.00000 −0.428845
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.00000 0.106600
$$353$$ 30.0000 1.59674 0.798369 0.602168i $$-0.205696\pi$$
0.798369 + 0.602168i $$0.205696\pi$$
$$354$$ 16.0000 0.850390
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 14.0000 0.739923
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −3.00000 −0.157895
$$362$$ −24.0000 −1.26141
$$363$$ −14.0000 −0.734809
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ 4.00000 0.208798 0.104399 0.994535i $$-0.466708\pi$$
0.104399 + 0.994535i $$0.466708\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 6.00000 0.312348
$$370$$ 8.00000 0.415900
$$371$$ 0 0
$$372$$ −2.00000 −0.103695
$$373$$ −26.0000 −1.34623 −0.673114 0.739538i $$-0.735044\pi$$
−0.673114 + 0.739538i $$0.735044\pi$$
$$374$$ 4.00000 0.206835
$$375$$ −2.00000 −0.103280
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 32.0000 1.63941
$$382$$ −8.00000 −0.409316
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ 2.00000 0.102062
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ 2.00000 0.101666
$$388$$ −2.00000 −0.101535
$$389$$ −36.0000 −1.82527 −0.912636 0.408773i $$-0.865957\pi$$
−0.912636 + 0.408773i $$0.865957\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ −7.00000 −0.353553
$$393$$ 40.0000 2.01773
$$394$$ 8.00000 0.403034
$$395$$ 4.00000 0.201262
$$396$$ 2.00000 0.100504
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 38.0000 1.89763 0.948815 0.315833i $$-0.102284\pi$$
0.948815 + 0.315833i $$0.102284\pi$$
$$402$$ 8.00000 0.399004
$$403$$ 0 0
$$404$$ 2.00000 0.0995037
$$405$$ 11.0000 0.546594
$$406$$ 0 0
$$407$$ −16.0000 −0.793091
$$408$$ 4.00000 0.198030
$$409$$ 6.00000 0.296681 0.148340 0.988936i $$-0.452607\pi$$
0.148340 + 0.988936i $$0.452607\pi$$
$$410$$ −6.00000 −0.296319
$$411$$ 36.0000 1.77575
$$412$$ 8.00000 0.394132
$$413$$ 0 0
$$414$$ −4.00000 −0.196589
$$415$$ −6.00000 −0.294528
$$416$$ 0 0
$$417$$ 4.00000 0.195881
$$418$$ −8.00000 −0.391293
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ −16.0000 −0.778868
$$423$$ 0 0
$$424$$ 8.00000 0.388514
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −8.00000 −0.386695
$$429$$ 0 0
$$430$$ −2.00000 −0.0964486
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 0 0
$$435$$ 8.00000 0.383571
$$436$$ −18.0000 −0.862044
$$437$$ 16.0000 0.765384
$$438$$ 12.0000 0.573382
$$439$$ 40.0000 1.90910 0.954548 0.298057i $$-0.0963387\pi$$
0.954548 + 0.298057i $$0.0963387\pi$$
$$440$$ −2.00000 −0.0953463
$$441$$ −7.00000 −0.333333
$$442$$ 0 0
$$443$$ −28.0000 −1.33032 −0.665160 0.746701i $$-0.731637\pi$$
−0.665160 + 0.746701i $$0.731637\pi$$
$$444$$ −16.0000 −0.759326
$$445$$ 6.00000 0.284427
$$446$$ −8.00000 −0.378811
$$447$$ −28.0000 −1.32435
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 12.0000 0.565058
$$452$$ 14.0000 0.658505
$$453$$ 8.00000 0.375873
$$454$$ 8.00000 0.375459
$$455$$ 0 0
$$456$$ −8.00000 −0.374634
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ 4.00000 0.186908
$$459$$ −8.00000 −0.373408
$$460$$ 4.00000 0.186501
$$461$$ −8.00000 −0.372597 −0.186299 0.982493i $$-0.559649\pi$$
−0.186299 + 0.982493i $$0.559649\pi$$
$$462$$ 0 0
$$463$$ 36.0000 1.67306 0.836531 0.547920i $$-0.184580\pi$$
0.836531 + 0.547920i $$0.184580\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 2.00000 0.0927478
$$466$$ −26.0000 −1.20443
$$467$$ 8.00000 0.370196 0.185098 0.982720i $$-0.440740\pi$$
0.185098 + 0.982720i $$0.440740\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 12.0000 0.552931
$$472$$ 8.00000 0.368230
$$473$$ 4.00000 0.183920
$$474$$ −8.00000 −0.367452
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ 8.00000 0.366295
$$478$$ 12.0000 0.548867
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ −2.00000 −0.0912871
$$481$$ 0 0
$$482$$ −2.00000 −0.0910975
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ 2.00000 0.0908153
$$486$$ −10.0000 −0.453609
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 0 0
$$489$$ 48.0000 2.17064
$$490$$ 7.00000 0.316228
$$491$$ 2.00000 0.0902587 0.0451294 0.998981i $$-0.485630\pi$$
0.0451294 + 0.998981i $$0.485630\pi$$
$$492$$ 12.0000 0.541002
$$493$$ −8.00000 −0.360302
$$494$$ 0 0
$$495$$ −2.00000 −0.0898933
$$496$$ −1.00000 −0.0449013
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ 10.0000 0.447661 0.223831 0.974628i $$-0.428144\pi$$
0.223831 + 0.974628i $$0.428144\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 24.0000 1.07224
$$502$$ 14.0000 0.624851
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ −2.00000 −0.0889988
$$506$$ −8.00000 −0.355643
$$507$$ −26.0000 −1.15470
$$508$$ 16.0000 0.709885
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ −4.00000 −0.177123
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 16.0000 0.706417
$$514$$ 10.0000 0.441081
$$515$$ −8.00000 −0.352522
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −4.00000 −0.175581
$$520$$ 0 0
$$521$$ −38.0000 −1.66481 −0.832405 0.554168i $$-0.813037\pi$$
−0.832405 + 0.554168i $$0.813037\pi$$
$$522$$ −4.00000 −0.175075
$$523$$ 38.0000 1.66162 0.830812 0.556553i $$-0.187876\pi$$
0.830812 + 0.556553i $$0.187876\pi$$
$$524$$ 20.0000 0.873704
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ −2.00000 −0.0871214
$$528$$ 4.00000 0.174078
$$529$$ −7.00000 −0.304348
$$530$$ −8.00000 −0.347498
$$531$$ 8.00000 0.347170
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −12.0000 −0.519291
$$535$$ 8.00000 0.345870
$$536$$ 4.00000 0.172774
$$537$$ 28.0000 1.20829
$$538$$ 0 0
$$539$$ −14.0000 −0.603023
$$540$$ 4.00000 0.172133
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ −48.0000 −2.05988
$$544$$ 2.00000 0.0857493
$$545$$ 18.0000 0.771035
$$546$$ 0 0
$$547$$ −32.0000 −1.36822 −0.684111 0.729378i $$-0.739809\pi$$
−0.684111 + 0.729378i $$0.739809\pi$$
$$548$$ 18.0000 0.768922
$$549$$ 0 0
$$550$$ 2.00000 0.0852803
$$551$$ 16.0000 0.681623
$$552$$ −8.00000 −0.340503
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 16.0000 0.679162
$$556$$ 2.00000 0.0848189
$$557$$ 24.0000 1.01691 0.508456 0.861088i $$-0.330216\pi$$
0.508456 + 0.861088i $$0.330216\pi$$
$$558$$ −1.00000 −0.0423334
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 10.0000 0.421825
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ 0 0
$$565$$ −14.0000 −0.588984
$$566$$ −24.0000 −1.00880
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 8.00000 0.335083
$$571$$ −26.0000 −1.08807 −0.544033 0.839064i $$-0.683103\pi$$
−0.544033 + 0.839064i $$0.683103\pi$$
$$572$$ 0 0
$$573$$ −16.0000 −0.668410
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ 1.00000 0.0416667
$$577$$ 30.0000 1.24892 0.624458 0.781058i $$-0.285320\pi$$
0.624458 + 0.781058i $$0.285320\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 4.00000 0.166234
$$580$$ 4.00000 0.166091
$$581$$ 0 0
$$582$$ −4.00000 −0.165805
$$583$$ 16.0000 0.662652
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ −2.00000 −0.0825488 −0.0412744 0.999148i $$-0.513142\pi$$
−0.0412744 + 0.999148i $$0.513142\pi$$
$$588$$ −14.0000 −0.577350
$$589$$ 4.00000 0.164817
$$590$$ −8.00000 −0.329355
$$591$$ 16.0000 0.658152
$$592$$ −8.00000 −0.328798
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ −8.00000 −0.328244
$$595$$ 0 0
$$596$$ −14.0000 −0.573462
$$597$$ 32.0000 1.30967
$$598$$ 0 0
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ 2.00000 0.0816497
$$601$$ −14.0000 −0.571072 −0.285536 0.958368i $$-0.592172\pi$$
−0.285536 + 0.958368i $$0.592172\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 4.00000 0.162758
$$605$$ 7.00000 0.284590
$$606$$ 4.00000 0.162489
$$607$$ −24.0000 −0.974130 −0.487065 0.873366i $$-0.661933\pi$$
−0.487065 + 0.873366i $$0.661933\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 2.00000 0.0808452
$$613$$ −44.0000 −1.77714 −0.888572 0.458738i $$-0.848302\pi$$
−0.888572 + 0.458738i $$0.848302\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ −12.0000 −0.483887
$$616$$ 0 0
$$617$$ −38.0000 −1.52982 −0.764911 0.644136i $$-0.777217\pi$$
−0.764911 + 0.644136i $$0.777217\pi$$
$$618$$ 16.0000 0.643614
$$619$$ −34.0000 −1.36658 −0.683288 0.730149i $$-0.739451\pi$$
−0.683288 + 0.730149i $$0.739451\pi$$
$$620$$ 1.00000 0.0401610
$$621$$ 16.0000 0.642058
$$622$$ 8.00000 0.320771
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −22.0000 −0.879297
$$627$$ −16.0000 −0.638978
$$628$$ 6.00000 0.239426
$$629$$ −16.0000 −0.637962
$$630$$ 0 0
$$631$$ −28.0000 −1.11466 −0.557331 0.830290i $$-0.688175\pi$$
−0.557331 + 0.830290i $$0.688175\pi$$
$$632$$ −4.00000 −0.159111
$$633$$ −32.0000 −1.27189
$$634$$ 2.00000 0.0794301
$$635$$ −16.0000 −0.634941
$$636$$ 16.0000 0.634441
$$637$$ 0 0
$$638$$ −8.00000 −0.316723
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ −16.0000 −0.631470
$$643$$ −2.00000 −0.0788723 −0.0394362 0.999222i $$-0.512556\pi$$
−0.0394362 + 0.999222i $$0.512556\pi$$
$$644$$ 0 0
$$645$$ −4.00000 −0.157500
$$646$$ −8.00000 −0.314756
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ −11.0000 −0.432121
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 24.0000 0.939913
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ −36.0000 −1.40771
$$655$$ −20.0000 −0.781465
$$656$$ 6.00000 0.234261
$$657$$ 6.00000 0.234082
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ −4.00000 −0.155700
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ −30.0000 −1.16598
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ −8.00000 −0.309994
$$667$$ 16.0000 0.619522
$$668$$ 12.0000 0.464294
$$669$$ −16.0000 −0.618596
$$670$$ −4.00000 −0.154533
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ 14.0000 0.539260
$$675$$ −4.00000 −0.153960
$$676$$ −13.0000 −0.500000
$$677$$ −48.0000 −1.84479 −0.922395 0.386248i $$-0.873771\pi$$
−0.922395 + 0.386248i $$0.873771\pi$$
$$678$$ 28.0000 1.07533
$$679$$ 0 0
$$680$$ −2.00000 −0.0766965
$$681$$ 16.0000 0.613121
$$682$$ −2.00000 −0.0765840
$$683$$ −48.0000 −1.83667 −0.918334 0.395805i $$-0.870466\pi$$
−0.918334 + 0.395805i $$0.870466\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ −18.0000 −0.687745
$$686$$ 0 0
$$687$$ 8.00000 0.305219
$$688$$ 2.00000 0.0762493
$$689$$ 0 0
$$690$$ 8.00000 0.304555
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ −2.00000 −0.0760286
$$693$$ 0 0
$$694$$ −22.0000 −0.835109
$$695$$ −2.00000 −0.0758643
$$696$$ −8.00000 −0.303239
$$697$$ 12.0000 0.454532
$$698$$ 10.0000 0.378506
$$699$$ −52.0000 −1.96682
$$700$$ 0 0
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 0 0
$$703$$ 32.0000 1.20690
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ 30.0000 1.12906
$$707$$ 0 0
$$708$$ 16.0000 0.601317
$$709$$ −16.0000 −0.600893 −0.300446 0.953799i $$-0.597136\pi$$
−0.300446 + 0.953799i $$0.597136\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ −6.00000 −0.224860
$$713$$ 4.00000 0.149801
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 14.0000 0.523205
$$717$$ 24.0000 0.896296
$$718$$ −8.00000 −0.298557
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 0 0
$$722$$ −3.00000 −0.111648
$$723$$ −4.00000 −0.148762
$$724$$ −24.0000 −0.891953
$$725$$ −4.00000 −0.148556
$$726$$ −14.0000 −0.519589
$$727$$ 40.0000 1.48352 0.741759 0.670667i $$-0.233992\pi$$
0.741759 + 0.670667i $$0.233992\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ −6.00000 −0.222070
$$731$$ 4.00000 0.147945
$$732$$ 0 0
$$733$$ 2.00000 0.0738717 0.0369358 0.999318i $$-0.488240\pi$$
0.0369358 + 0.999318i $$0.488240\pi$$
$$734$$ 4.00000 0.147643
$$735$$ 14.0000 0.516398
$$736$$ −4.00000 −0.147442
$$737$$ 8.00000 0.294684
$$738$$ 6.00000 0.220863
$$739$$ 26.0000 0.956425 0.478213 0.878244i $$-0.341285\pi$$
0.478213 + 0.878244i $$0.341285\pi$$
$$740$$ 8.00000 0.294086
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ −2.00000 −0.0733236
$$745$$ 14.0000 0.512920
$$746$$ −26.0000 −0.951928
$$747$$ 6.00000 0.219529
$$748$$ 4.00000 0.146254
$$749$$ 0 0
$$750$$ −2.00000 −0.0730297
$$751$$ 48.0000 1.75154 0.875772 0.482724i $$-0.160353\pi$$
0.875772 + 0.482724i $$0.160353\pi$$
$$752$$ 0 0
$$753$$ 28.0000 1.02038
$$754$$ 0 0
$$755$$ −4.00000 −0.145575
$$756$$ 0 0
$$757$$ 12.0000 0.436147 0.218074 0.975932i $$-0.430023\pi$$
0.218074 + 0.975932i $$0.430023\pi$$
$$758$$ 16.0000 0.581146
$$759$$ −16.0000 −0.580763
$$760$$ 4.00000 0.145095
$$761$$ 22.0000 0.797499 0.398750 0.917060i $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ 32.0000 1.15924
$$763$$ 0 0
$$764$$ −8.00000 −0.289430
$$765$$ −2.00000 −0.0723102
$$766$$ 16.0000 0.578103
$$767$$ 0 0
$$768$$ 2.00000 0.0721688
$$769$$ −2.00000 −0.0721218 −0.0360609 0.999350i $$-0.511481\pi$$
−0.0360609 + 0.999350i $$0.511481\pi$$
$$770$$ 0 0
$$771$$ 20.0000 0.720282
$$772$$ 2.00000 0.0719816
$$773$$ −28.0000 −1.00709 −0.503545 0.863969i $$-0.667971\pi$$
−0.503545 + 0.863969i $$0.667971\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ −1.00000 −0.0359211
$$776$$ −2.00000 −0.0717958
$$777$$ 0 0
$$778$$ −36.0000 −1.29066
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −8.00000 −0.286079
$$783$$ 16.0000 0.571793
$$784$$ −7.00000 −0.250000
$$785$$ −6.00000 −0.214149
$$786$$ 40.0000 1.42675
$$787$$ 22.0000 0.784215 0.392108 0.919919i $$-0.371746\pi$$
0.392108 + 0.919919i $$0.371746\pi$$
$$788$$ 8.00000 0.284988
$$789$$ −48.0000 −1.70885
$$790$$ 4.00000 0.142314
$$791$$ 0 0
$$792$$ 2.00000 0.0710669
$$793$$ 0 0
$$794$$ −14.0000 −0.496841
$$795$$ −16.0000 −0.567462
$$796$$ 16.0000 0.567105
$$797$$ 24.0000 0.850124 0.425062 0.905164i $$-0.360252\pi$$
0.425062 + 0.905164i $$0.360252\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 1.00000 0.0353553
$$801$$ −6.00000 −0.212000
$$802$$ 38.0000 1.34183
$$803$$ 12.0000 0.423471
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 2.00000 0.0703598
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 11.0000 0.386501
$$811$$ 24.0000 0.842754 0.421377 0.906886i $$-0.361547\pi$$
0.421377 + 0.906886i $$0.361547\pi$$
$$812$$ 0 0
$$813$$ −40.0000 −1.40286
$$814$$ −16.0000 −0.560800
$$815$$ −24.0000 −0.840683
$$816$$ 4.00000 0.140028
$$817$$ −8.00000 −0.279885
$$818$$ 6.00000 0.209785
$$819$$ 0 0
$$820$$ −6.00000 −0.209529
$$821$$ 20.0000 0.698005 0.349002 0.937122i $$-0.386521\pi$$
0.349002 + 0.937122i $$0.386521\pi$$
$$822$$ 36.0000 1.25564
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 4.00000 0.139262
$$826$$ 0 0
$$827$$ −14.0000 −0.486828 −0.243414 0.969923i $$-0.578267\pi$$
−0.243414 + 0.969923i $$0.578267\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ 32.0000 1.11141 0.555703 0.831381i $$-0.312449\pi$$
0.555703 + 0.831381i $$0.312449\pi$$
$$830$$ −6.00000 −0.208263
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −14.0000 −0.485071
$$834$$ 4.00000 0.138509
$$835$$ −12.0000 −0.415277
$$836$$ −8.00000 −0.276686
$$837$$ 4.00000 0.138260
$$838$$ −24.0000 −0.829066
$$839$$ 16.0000 0.552381 0.276191 0.961103i $$-0.410928\pi$$
0.276191 + 0.961103i $$0.410928\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ −22.0000 −0.758170
$$843$$ 20.0000 0.688837
$$844$$ −16.0000 −0.550743
$$845$$ 13.0000 0.447214
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 8.00000 0.274721
$$849$$ −48.0000 −1.64736
$$850$$ 2.00000 0.0685994
$$851$$ 32.0000 1.09695
$$852$$ 0 0
$$853$$ −14.0000 −0.479351 −0.239675 0.970853i $$-0.577041\pi$$
−0.239675 + 0.970853i $$0.577041\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ −8.00000 −0.273434
$$857$$ −6.00000 −0.204956 −0.102478 0.994735i $$-0.532677\pi$$
−0.102478 + 0.994735i $$0.532677\pi$$
$$858$$ 0 0
$$859$$ 42.0000 1.43302 0.716511 0.697576i $$-0.245738\pi$$
0.716511 + 0.697576i $$0.245738\pi$$
$$860$$ −2.00000 −0.0681994
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 2.00000 0.0680020
$$866$$ 26.0000 0.883516
$$867$$ −26.0000 −0.883006
$$868$$ 0 0
$$869$$ −8.00000 −0.271381
$$870$$ 8.00000 0.271225
$$871$$ 0 0
$$872$$ −18.0000 −0.609557
$$873$$ −2.00000 −0.0676897
$$874$$ 16.0000 0.541208
$$875$$ 0 0
$$876$$ 12.0000 0.405442
$$877$$ 34.0000 1.14810 0.574049 0.818821i $$-0.305372\pi$$
0.574049 + 0.818821i $$0.305372\pi$$
$$878$$ 40.0000 1.34993
$$879$$ −36.0000 −1.21425
$$880$$ −2.00000 −0.0674200
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ −7.00000 −0.235702
$$883$$ −26.0000 −0.874970 −0.437485 0.899226i $$-0.644131\pi$$
−0.437485 + 0.899226i $$0.644131\pi$$
$$884$$ 0 0
$$885$$ −16.0000 −0.537834
$$886$$ −28.0000 −0.940678
$$887$$ 48.0000 1.61168 0.805841 0.592132i $$-0.201714\pi$$
0.805841 + 0.592132i $$0.201714\pi$$
$$888$$ −16.0000 −0.536925
$$889$$ 0 0
$$890$$ 6.00000 0.201120
$$891$$ −22.0000 −0.737028
$$892$$ −8.00000 −0.267860
$$893$$ 0 0
$$894$$ −28.0000 −0.936460
$$895$$ −14.0000 −0.467968
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 6.00000 0.200223
$$899$$ 4.00000 0.133407
$$900$$ 1.00000 0.0333333
$$901$$ 16.0000 0.533037
$$902$$ 12.0000 0.399556
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ 24.0000 0.797787
$$906$$ 8.00000 0.265782
$$907$$ −32.0000 −1.06254 −0.531271 0.847202i $$-0.678286\pi$$
−0.531271 + 0.847202i $$0.678286\pi$$
$$908$$ 8.00000 0.265489
$$909$$ 2.00000 0.0663358
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ −8.00000 −0.264906
$$913$$ 12.0000 0.397142
$$914$$ 2.00000 0.0661541
$$915$$ 0 0
$$916$$ 4.00000 0.132164
$$917$$ 0 0
$$918$$ −8.00000 −0.264039
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 4.00000 0.131876
$$921$$ −32.0000 −1.05444
$$922$$ −8.00000 −0.263466
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −8.00000 −0.263038
$$926$$ 36.0000 1.18303
$$927$$ 8.00000 0.262754
$$928$$ −4.00000 −0.131306
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 2.00000 0.0655826
$$931$$ 28.0000 0.917663
$$932$$ −26.0000 −0.851658
$$933$$ 16.0000 0.523816
$$934$$ 8.00000 0.261768
$$935$$ −4.00000 −0.130814
$$936$$ 0 0
$$937$$ −14.0000 −0.457360 −0.228680 0.973502i $$-0.573441\pi$$
−0.228680 + 0.973502i $$0.573441\pi$$
$$938$$ 0 0
$$939$$ −44.0000 −1.43589
$$940$$ 0 0
$$941$$ −60.0000 −1.95594 −0.977972 0.208736i $$-0.933065\pi$$
−0.977972 + 0.208736i $$0.933065\pi$$
$$942$$ 12.0000 0.390981
$$943$$ −24.0000 −0.781548
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ −14.0000 −0.454939 −0.227469 0.973785i $$-0.573045\pi$$
−0.227469 + 0.973785i $$0.573045\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 0 0
$$950$$ −4.00000 −0.129777
$$951$$ 4.00000 0.129709
$$952$$ 0 0
$$953$$ 18.0000 0.583077 0.291539 0.956559i $$-0.405833\pi$$
0.291539 + 0.956559i $$0.405833\pi$$
$$954$$ 8.00000 0.259010
$$955$$ 8.00000 0.258874
$$956$$ 12.0000 0.388108
$$957$$ −16.0000 −0.517207
$$958$$ 24.0000 0.775405
$$959$$ 0 0
$$960$$ −2.00000 −0.0645497
$$961$$ 1.00000 0.0322581
$$962$$ 0 0
$$963$$ −8.00000 −0.257796
$$964$$ −2.00000 −0.0644157
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ −7.00000 −0.224989
$$969$$ −16.0000 −0.513994
$$970$$ 2.00000 0.0642161
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ −10.0000 −0.320750
$$973$$ 0 0
$$974$$ 32.0000 1.02535
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 34.0000 1.08776 0.543878 0.839164i $$-0.316955\pi$$
0.543878 + 0.839164i $$0.316955\pi$$
$$978$$ 48.0000 1.53487
$$979$$ −12.0000 −0.383522
$$980$$ 7.00000 0.223607
$$981$$ −18.0000 −0.574696
$$982$$ 2.00000 0.0638226
$$983$$ 8.00000 0.255160 0.127580 0.991828i $$-0.459279\pi$$
0.127580 + 0.991828i $$0.459279\pi$$
$$984$$ 12.0000 0.382546
$$985$$ −8.00000 −0.254901
$$986$$ −8.00000 −0.254772
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −8.00000 −0.254385
$$990$$ −2.00000 −0.0635642
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ −1.00000 −0.0317500
$$993$$ −60.0000 −1.90404
$$994$$ 0 0
$$995$$ −16.0000 −0.507234
$$996$$ 12.0000 0.380235
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ 10.0000 0.316544
$$999$$ 32.0000 1.01244
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 310.2.a.b.1.1 1
3.2 odd 2 2790.2.a.h.1.1 1
4.3 odd 2 2480.2.a.c.1.1 1
5.2 odd 4 1550.2.b.e.249.2 2
5.3 odd 4 1550.2.b.e.249.1 2
5.4 even 2 1550.2.a.a.1.1 1
8.3 odd 2 9920.2.a.bg.1.1 1
8.5 even 2 9920.2.a.d.1.1 1
31.30 odd 2 9610.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
310.2.a.b.1.1 1 1.1 even 1 trivial
1550.2.a.a.1.1 1 5.4 even 2
1550.2.b.e.249.1 2 5.3 odd 4
1550.2.b.e.249.2 2 5.2 odd 4
2480.2.a.c.1.1 1 4.3 odd 2
2790.2.a.h.1.1 1 3.2 odd 2
9610.2.a.a.1.1 1 31.30 odd 2
9920.2.a.d.1.1 1 8.5 even 2
9920.2.a.bg.1.1 1 8.3 odd 2