# Properties

 Label 9610.2.a.a.1.1 Level $9610$ Weight $2$ Character 9610.1 Self dual yes Analytic conductor $76.736$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9610,2,Mod(1,9610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9610, 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("9610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9610 = 2 \cdot 5 \cdot 31^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9610.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: $$76.7362363425$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 310) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 9610.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

## 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 $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −2.00000 −0.816497
$$7$$ 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 $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ −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 $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 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 $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ −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 $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 2.00000 0.365148
$$31$$ 0 0
$$32$$ 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 $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ −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 $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ −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 $$-0.685160\pi$$
−0.549442 + 0.835532i $$0.685160\pi$$
$$54$$ 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 $$0.5\pi$$
$$62$$ 0 0
$$63$$ 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 $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ 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 $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ −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 $$0.606810\pi$$
$$84$$ 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 $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ 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
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 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 $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ −8.00000 −0.681005
$$139$$ −2.00000 −0.169638 −0.0848189 0.996396i $$-0.527031\pi$$
−0.0848189 + 0.996396i $$0.527031\pi$$
$$140$$ 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 $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 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 $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 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 $$-0.675264\pi$$
−0.523205 + 0.852207i $$0.675264\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 24.0000 1.78391 0.891953 0.452128i $$-0.149335\pi$$
0.891953 + 0.452128i $$0.149335\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 4.00000 0.294884
$$185$$ −8.00000 −0.588172
$$186$$ 0 0
$$187$$ 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 $$-0.591990\pi$$
−0.284988 + 0.958531i $$0.591990\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 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 $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 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 $$-0.542192\pi$$
−0.132164 + 0.991228i $$0.542192\pi$$
$$230$$ −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 $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 2.00000 0.129099
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ −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
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ −1.00000 −0.0632456
$$251$$ −14.0000 −0.883672 −0.441836 0.897096i $$-0.645673\pi$$
−0.441836 + 0.897096i $$0.645673\pi$$
$$252$$ 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 $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 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 $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ −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
$$279$$ 0 0
$$280$$ 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
$$310$$ 0 0
$$311$$ 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 $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ 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 $$-0.191471\pi$$
0.824475 + 0.565899i $$0.191471\pi$$
$$332$$ −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 $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ −13.0000 −0.707107
$$339$$ −28.0000 −1.52075
$$340$$ 2.00000 0.108465
$$341$$ 0 0
$$342$$ −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 $$-0.298926\pi$$
0.590511 + 0.807030i $$0.298926\pi$$
$$348$$ −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 $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ −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 $$-0.533292\pi$$
−0.104399 + 0.994535i $$0.533292\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 6.00000 0.312348
$$370$$ −8.00000 −0.415900
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −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 $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ −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 $$-0.134043\pi$$
0.912636 + 0.408773i $$0.134043\pi$$
$$390$$ 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 $$-0.897716\pi$$
−0.948815 + 0.315833i $$0.897716\pi$$
$$402$$ −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 $$-0.547393\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ −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 $$-0.714795\pi$$
−0.624740 + 0.780833i $$0.714795\pi$$
$$434$$ 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 $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 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 $$-0.514895\pi$$
−0.0467780 + 0.998905i $$0.514895\pi$$
$$458$$ −4.00000 −0.186908
$$459$$ −8.00000 −0.373408
$$460$$ −4.00000 −0.186501
$$461$$ 8.00000 0.372597 0.186299 0.982493i $$-0.440351\pi$$
0.186299 + 0.982493i $$0.440351\pi$$
$$462$$ 0 0
$$463$$ −36.0000 −1.67306 −0.836531 0.547920i $$-0.815420\pi$$
−0.836531 + 0.547920i $$0.815420\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ −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 $$-0.758174\pi$$
−0.725029 + 0.688718i $$0.758174\pi$$
$$488$$ 0 0
$$489$$ −48.0000 −2.17064
$$490$$ 7.00000 0.316228
$$491$$ −2.00000 −0.0902587 −0.0451294 0.998981i $$-0.514370\pi$$
−0.0451294 + 0.998981i $$0.514370\pi$$
$$492$$ −12.0000 −0.541002
$$493$$ −8.00000 −0.360302
$$494$$ 0 0
$$495$$ 2.00000 0.0898933
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ −10.0000 −0.447661 −0.223831 0.974628i $$-0.571856\pi$$
−0.223831 + 0.974628i $$0.571856\pi$$
$$500$$ −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 $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ −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 $$-0.812124\pi$$
−0.830812 + 0.556553i $$0.812124\pi$$
$$524$$ 20.0000 0.873704
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 0 0
$$528$$ 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 $$-0.669784\pi$$
−0.508456 + 0.861088i $$0.669784\pi$$
$$558$$ 0 0
$$559$$ 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 $$-0.157297\pi$$
0.880366 + 0.474295i $$0.157297\pi$$
$$570$$ −8.00000 −0.335083
$$571$$ 26.0000 1.08807 0.544033 0.839064i $$-0.316897\pi$$
0.544033 + 0.839064i $$0.316897\pi$$
$$572$$ 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 $$-0.486858\pi$$
0.0412744 + 0.999148i $$0.486858\pi$$
$$588$$ 14.0000 0.577350
$$589$$ 0 0
$$590$$ −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 $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 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 $$-0.151698\pi$$
0.888572 + 0.458738i $$0.151698\pi$$
$$614$$ −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 $$-0.260549\pi$$
0.683288 + 0.730149i $$0.260549\pi$$
$$620$$ 0 0
$$621$$ 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 $$-0.311825\pi$$
0.557331 + 0.830290i $$0.311825\pi$$
$$632$$ 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 $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 16.0000 0.631470
$$643$$ 2.00000 0.0788723 0.0394362 0.999222i $$-0.487444\pi$$
0.0394362 + 0.999222i $$0.487444\pi$$
$$644$$ 0 0
$$645$$ −4.00000 −0.157500
$$646$$ 8.00000 0.314756
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ −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 $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 4.00000 0.153960
$$676$$ −13.0000 −0.500000
$$677$$ 48.0000 1.84479 0.922395 0.386248i $$-0.126229\pi$$
0.922395 + 0.386248i $$0.126229\pi$$
$$678$$ −28.0000 −1.07533
$$679$$ 0 0
$$680$$ 2.00000 0.0766965
$$681$$ −16.0000 −0.613121
$$682$$ 0 0
$$683$$ −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 $$-0.402864\pi$$
0.300446 + 0.953799i $$0.402864\pi$$
$$710$$ 0 0
$$711$$ 4.00000 0.150012
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ 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 $$-0.658715\pi$$
−0.478213 + 0.878244i $$0.658715\pi$$
$$740$$ −8.00000 −0.294086
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ 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 $$-0.569977\pi$$
−0.218074 + 0.975932i $$0.569977\pi$$
$$758$$ 16.0000 0.581146
$$759$$ 16.0000 0.580763
$$760$$ 4.00000 0.145095
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ 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 $$-0.332029\pi$$
0.503545 + 0.863969i $$0.332029\pi$$
$$774$$ −2.00000 −0.0718885
$$775$$ 0 0
$$776$$ −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 $$-0.628254\pi$$
−0.392108 + 0.919919i $$0.628254\pi$$
$$788$$ −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 $$-0.639748\pi$$
−0.425062 + 0.905164i $$0.639748\pi$$
$$798$$ 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 $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 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 $$-0.613479\pi$$
−0.349002 + 0.937122i $$0.613479\pi$$
$$822$$ 36.0000 1.25564
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 4.00000 0.139262
$$826$$ 0 0
$$827$$ 14.0000 0.486828 0.243414 0.969923i $$-0.421733\pi$$
0.243414 + 0.969923i $$0.421733\pi$$
$$828$$ 4.00000 0.139010
$$829$$ −32.0000 −1.11141 −0.555703 0.831381i $$-0.687551\pi$$
−0.555703 + 0.831381i $$0.687551\pi$$
$$830$$ 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
$$837$$ 0 0
$$838$$ −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 $$-0.754262\pi$$
−0.716511 + 0.697576i $$0.754262\pi$$
$$860$$ 2.00000 0.0681994
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 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 $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ −7.00000 −0.235702
$$883$$ 26.0000 0.874970 0.437485 0.899226i $$-0.355869\pi$$
0.437485 + 0.899226i $$0.355869\pi$$
$$884$$ 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
$$899$$ 0 0
$$900$$ 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 $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 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 $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 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 $$-0.0669349\pi$$
0.977972 + 0.208736i $$0.0669349\pi$$
$$942$$ −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 $$-0.426955\pi$$
0.227469 + 0.973785i $$0.426955\pi$$
$$948$$ −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 $$-0.594167\pi$$
−0.291539 + 0.956559i $$0.594167\pi$$
$$954$$ −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
$$961$$ 0 0
$$962$$ 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 $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ −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 $$-0.540721\pi$$
−0.127580 + 0.991828i $$0.540721\pi$$
$$984$$ −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 $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 0 0
$$993$$ −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 9610.2.a.a.1.1 1
31.30 odd 2 310.2.a.b.1.1 1
93.92 even 2 2790.2.a.h.1.1 1
124.123 even 2 2480.2.a.c.1.1 1
155.92 even 4 1550.2.b.e.249.2 2
155.123 even 4 1550.2.b.e.249.1 2
155.154 odd 2 1550.2.a.a.1.1 1
248.61 odd 2 9920.2.a.d.1.1 1
248.123 even 2 9920.2.a.bg.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
310.2.a.b.1.1 1 31.30 odd 2
1550.2.a.a.1.1 1 155.154 odd 2
1550.2.b.e.249.1 2 155.123 even 4
1550.2.b.e.249.2 2 155.92 even 4
2480.2.a.c.1.1 1 124.123 even 2
2790.2.a.h.1.1 1 93.92 even 2
9610.2.a.a.1.1 1 1.1 even 1 trivial
9920.2.a.d.1.1 1 248.61 odd 2
9920.2.a.bg.1.1 1 248.123 even 2