# Properties

 Label 66.2.a.a.1.1 Level $66$ Weight $2$ Character 66.1 Self dual yes Analytic conductor $0.527$ 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] = [66,2,Mod(1,66)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(66, 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("66.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$66 = 2 \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 66.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: $$0.527012653340$$ 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 $$\chi$$ $$=$$ 66.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{21} +1.00000 q^{22} +6.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} +6.00000 q^{29} +8.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -10.0000 q^{37} +4.00000 q^{38} -4.00000 q^{39} +6.00000 q^{41} -2.00000 q^{42} +8.00000 q^{43} -1.00000 q^{44} -6.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +5.00000 q^{50} -6.00000 q^{51} -4.00000 q^{52} -1.00000 q^{54} -2.00000 q^{56} -4.00000 q^{57} -6.00000 q^{58} +8.00000 q^{61} -8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} -4.00000 q^{67} -6.00000 q^{68} +6.00000 q^{69} +6.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} +10.0000 q^{74} -5.00000 q^{75} -4.00000 q^{76} -2.00000 q^{77} +4.00000 q^{78} +14.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} +2.00000 q^{84} -8.00000 q^{86} +6.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} -8.00000 q^{91} +6.00000 q^{92} +8.00000 q^{93} +6.00000 q^{94} -1.00000 q^{96} +14.0000 q^{97} +3.00000 q^{98} -1.00000 q^{99} +O(q^{100})$$

## 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 1.00000 0.288675
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 1.00000 0.213201
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −5.00000 −1.00000
$$26$$ 4.00000 0.784465
$$27$$ 1.00000 0.192450
$$28$$ 2.00000 0.377964
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −1.00000 −0.174078
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ 4.00000 0.648886
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 5.00000 0.707107
$$51$$ −6.00000 −0.840168
$$52$$ −4.00000 −0.554700
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ −4.00000 −0.529813
$$58$$ −6.00000 −0.787839
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 1.00000 0.123091
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 10.0000 1.16248
$$75$$ −5.00000 −0.577350
$$76$$ −4.00000 −0.458831
$$77$$ −2.00000 −0.227921
$$78$$ 4.00000 0.452911
$$79$$ 14.0000 1.57512 0.787562 0.616236i $$-0.211343\pi$$
0.787562 + 0.616236i $$0.211343\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 6.00000 0.643268
$$88$$ 1.00000 0.106600
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ 6.00000 0.625543
$$93$$ 8.00000 0.829561
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 3.00000 0.303046
$$99$$ −1.00000 −0.100504
$$100$$ −5.00000 −0.500000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 6.00000 0.594089
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ 2.00000 0.188982
$$113$$ 18.0000 1.69330 0.846649 0.532152i $$-0.178617\pi$$
0.846649 + 0.532152i $$0.178617\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ −4.00000 −0.369800
$$118$$ 0 0
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −8.00000 −0.724286
$$123$$ 6.00000 0.541002
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ 14.0000 1.24230 0.621150 0.783692i $$-0.286666\pi$$
0.621150 + 0.783692i $$0.286666\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ −8.00000 −0.693688
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ −6.00000 −0.510754
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ −6.00000 −0.503509
$$143$$ 4.00000 0.334497
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ −3.00000 −0.247436
$$148$$ −10.0000 −0.821995
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 5.00000 0.408248
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 4.00000 0.324443
$$153$$ −6.00000 −0.485071
$$154$$ 2.00000 0.161165
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ −14.0000 −1.11378
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 12.0000 0.945732
$$162$$ −1.00000 −0.0785674
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 8.00000 0.609994
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ −10.0000 −0.755929
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 8.00000 0.592999
$$183$$ 8.00000 0.591377
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 6.00000 0.438763
$$188$$ −6.00000 −0.437595
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 5.00000 0.353553
$$201$$ −4.00000 −0.282138
$$202$$ −6.00000 −0.422159
$$203$$ 12.0000 0.842235
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ 6.00000 0.417029
$$208$$ −4.00000 −0.277350
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ 0 0
$$213$$ 6.00000 0.411113
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 16.0000 1.08615
$$218$$ 4.00000 0.270914
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 24.0000 1.61441
$$222$$ 10.0000 0.671156
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ −5.00000 −0.333333
$$226$$ −18.0000 −1.19734
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ −2.00000 −0.131590
$$232$$ −6.00000 −0.393919
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 14.0000 0.909398
$$238$$ 12.0000 0.777844
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 1.00000 0.0641500
$$244$$ 8.00000 0.512148
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 16.0000 1.01806
$$248$$ −8.00000 −0.508001
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 2.00000 0.125988
$$253$$ −6.00000 −0.377217
$$254$$ −14.0000 −0.878438
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −30.0000 −1.87135 −0.935674 0.352865i $$-0.885208\pi$$
−0.935674 + 0.352865i $$0.885208\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ −20.0000 −1.24274
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 12.0000 0.741362
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ 0 0
$$266$$ 8.00000 0.490511
$$267$$ −6.00000 −0.367194
$$268$$ −4.00000 −0.244339
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 0 0
$$271$$ 2.00000 0.121491 0.0607457 0.998153i $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ −8.00000 −0.484182
$$274$$ 18.0000 1.08742
$$275$$ 5.00000 0.301511
$$276$$ 6.00000 0.361158
$$277$$ −16.0000 −0.961347 −0.480673 0.876900i $$-0.659608\pi$$
−0.480673 + 0.876900i $$0.659608\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 6.00000 0.357295
$$283$$ 8.00000 0.475551 0.237775 0.971320i $$-0.423582\pi$$
0.237775 + 0.971320i $$0.423582\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ 12.0000 0.708338
$$288$$ −1.00000 −0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 2.00000 0.117041
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ 10.0000 0.581238
$$297$$ −1.00000 −0.0580259
$$298$$ 6.00000 0.347571
$$299$$ −24.0000 −1.38796
$$300$$ −5.00000 −0.288675
$$301$$ 16.0000 0.922225
$$302$$ 10.0000 0.575435
$$303$$ 6.00000 0.344691
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ −2.00000 −0.113961
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 4.00000 0.226455
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ 14.0000 0.787562
$$317$$ 12.0000 0.673987 0.336994 0.941507i $$-0.390590\pi$$
0.336994 + 0.941507i $$0.390590\pi$$
$$318$$ 0 0
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ −12.0000 −0.668734
$$323$$ 24.0000 1.33540
$$324$$ 1.00000 0.0555556
$$325$$ 20.0000 1.10940
$$326$$ 4.00000 0.221540
$$327$$ −4.00000 −0.221201
$$328$$ −6.00000 −0.331295
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ −10.0000 −0.547997
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ 18.0000 0.977626
$$340$$ 0 0
$$341$$ −8.00000 −0.433224
$$342$$ 4.00000 0.216295
$$343$$ −20.0000 −1.07990
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 36.0000 1.93258 0.966291 0.257454i $$-0.0828835\pi$$
0.966291 + 0.257454i $$0.0828835\pi$$
$$348$$ 6.00000 0.321634
$$349$$ −4.00000 −0.214115 −0.107058 0.994253i $$-0.534143\pi$$
−0.107058 + 0.994253i $$0.534143\pi$$
$$350$$ 10.0000 0.534522
$$351$$ −4.00000 −0.213504
$$352$$ 1.00000 0.0533002
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ −12.0000 −0.635107
$$358$$ −24.0000 −1.26844
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 22.0000 1.15629
$$363$$ 1.00000 0.0524864
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ −8.00000 −0.418167
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 6.00000 0.312772
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 8.00000 0.414781
$$373$$ 20.0000 1.03556 0.517780 0.855514i $$-0.326758\pi$$
0.517780 + 0.855514i $$0.326758\pi$$
$$374$$ −6.00000 −0.310253
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ −24.0000 −1.23606
$$378$$ −2.00000 −0.102869
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 14.0000 0.717242
$$382$$ −18.0000 −0.920960
$$383$$ 6.00000 0.306586 0.153293 0.988181i $$-0.451012\pi$$
0.153293 + 0.988181i $$0.451012\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 8.00000 0.406663
$$388$$ 14.0000 0.710742
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ −36.0000 −1.82060
$$392$$ 3.00000 0.151523
$$393$$ −12.0000 −0.605320
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ −1.00000 −0.0502519
$$397$$ 26.0000 1.30490 0.652451 0.757831i $$-0.273741\pi$$
0.652451 + 0.757831i $$0.273741\pi$$
$$398$$ 4.00000 0.200502
$$399$$ −8.00000 −0.400501
$$400$$ −5.00000 −0.250000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 4.00000 0.199502
$$403$$ −32.0000 −1.59403
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ −12.0000 −0.595550
$$407$$ 10.0000 0.495682
$$408$$ 6.00000 0.297044
$$409$$ −34.0000 −1.68119 −0.840596 0.541663i $$-0.817795\pi$$
−0.840596 + 0.541663i $$0.817795\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ −4.00000 −0.197066
$$413$$ 0 0
$$414$$ −6.00000 −0.294884
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ −4.00000 −0.195881
$$418$$ −4.00000 −0.195646
$$419$$ 24.0000 1.17248 0.586238 0.810139i $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ −8.00000 −0.389434
$$423$$ −6.00000 −0.291730
$$424$$ 0 0
$$425$$ 30.0000 1.45521
$$426$$ −6.00000 −0.290701
$$427$$ 16.0000 0.774294
$$428$$ −12.0000 −0.580042
$$429$$ 4.00000 0.193122
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ −24.0000 −1.14808
$$438$$ −2.00000 −0.0955637
$$439$$ −10.0000 −0.477274 −0.238637 0.971109i $$-0.576701\pi$$
−0.238637 + 0.971109i $$0.576701\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ −24.0000 −1.14156
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ −10.0000 −0.474579
$$445$$ 0 0
$$446$$ 16.0000 0.757622
$$447$$ −6.00000 −0.283790
$$448$$ 2.00000 0.0944911
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 5.00000 0.235702
$$451$$ −6.00000 −0.282529
$$452$$ 18.0000 0.846649
$$453$$ −10.0000 −0.469841
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 22.0000 1.02799
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ 42.0000 1.95614 0.978068 0.208288i $$-0.0667892\pi$$
0.978068 + 0.208288i $$0.0667892\pi$$
$$462$$ 2.00000 0.0930484
$$463$$ −4.00000 −0.185896 −0.0929479 0.995671i $$-0.529629\pi$$
−0.0929479 + 0.995671i $$0.529629\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ −4.00000 −0.184900
$$469$$ −8.00000 −0.369406
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ −8.00000 −0.367840
$$474$$ −14.0000 −0.643041
$$475$$ 20.0000 0.917663
$$476$$ −12.0000 −0.550019
$$477$$ 0 0
$$478$$ 12.0000 0.548867
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 40.0000 1.82384
$$482$$ 10.0000 0.455488
$$483$$ 12.0000 0.546019
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ −8.00000 −0.362143
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 6.00000 0.270501
$$493$$ −36.0000 −1.62136
$$494$$ −16.0000 −0.719874
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 12.0000 0.538274
$$498$$ 12.0000 0.537733
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ 12.0000 0.535054 0.267527 0.963550i $$-0.413794\pi$$
0.267527 + 0.963550i $$0.413794\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 6.00000 0.266733
$$507$$ 3.00000 0.133235
$$508$$ 14.0000 0.621150
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ −1.00000 −0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ 30.0000 1.32324
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 6.00000 0.263880
$$518$$ 20.0000 0.878750
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ −10.0000 −0.436436
$$526$$ 0 0
$$527$$ −48.0000 −2.09091
$$528$$ −1.00000 −0.0435194
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −8.00000 −0.346844
$$533$$ −24.0000 −1.03956
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ 24.0000 1.03568
$$538$$ 24.0000 1.03471
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 20.0000 0.859867 0.429934 0.902861i $$-0.358537\pi$$
0.429934 + 0.902861i $$0.358537\pi$$
$$542$$ −2.00000 −0.0859074
$$543$$ −22.0000 −0.944110
$$544$$ 6.00000 0.257248
$$545$$ 0 0
$$546$$ 8.00000 0.342368
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ 8.00000 0.341432
$$550$$ −5.00000 −0.213201
$$551$$ −24.0000 −1.02243
$$552$$ −6.00000 −0.255377
$$553$$ 28.0000 1.19068
$$554$$ 16.0000 0.679775
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ −32.0000 −1.35346
$$560$$ 0 0
$$561$$ 6.00000 0.253320
$$562$$ 6.00000 0.253095
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ −6.00000 −0.252646
$$565$$ 0 0
$$566$$ −8.00000 −0.336265
$$567$$ 2.00000 0.0839921
$$568$$ −6.00000 −0.251754
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 18.0000 0.751961
$$574$$ −12.0000 −0.500870
$$575$$ −30.0000 −1.25109
$$576$$ 1.00000 0.0416667
$$577$$ −34.0000 −1.41544 −0.707719 0.706494i $$-0.750276\pi$$
−0.707719 + 0.706494i $$0.750276\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ −24.0000 −0.995688
$$582$$ −14.0000 −0.580319
$$583$$ 0 0
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ 24.0000 0.990586 0.495293 0.868726i $$-0.335061\pi$$
0.495293 + 0.868726i $$0.335061\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ −10.0000 −0.410997
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ −4.00000 −0.163709
$$598$$ 24.0000 0.981433
$$599$$ 30.0000 1.22577 0.612883 0.790173i $$-0.290010\pi$$
0.612883 + 0.790173i $$0.290010\pi$$
$$600$$ 5.00000 0.204124
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ −16.0000 −0.652111
$$603$$ −4.00000 −0.162893
$$604$$ −10.0000 −0.406894
$$605$$ 0 0
$$606$$ −6.00000 −0.243733
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 12.0000 0.486265
$$610$$ 0 0
$$611$$ 24.0000 0.970936
$$612$$ −6.00000 −0.242536
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 2.00000 0.0805823
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ 4.00000 0.160904
$$619$$ 44.0000 1.76851 0.884255 0.467005i $$-0.154667\pi$$
0.884255 + 0.467005i $$0.154667\pi$$
$$620$$ 0 0
$$621$$ 6.00000 0.240772
$$622$$ 18.0000 0.721734
$$623$$ −12.0000 −0.480770
$$624$$ −4.00000 −0.160128
$$625$$ 25.0000 1.00000
$$626$$ −26.0000 −1.03917
$$627$$ 4.00000 0.159745
$$628$$ 2.00000 0.0798087
$$629$$ 60.0000 2.39236
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ −14.0000 −0.556890
$$633$$ 8.00000 0.317971
$$634$$ −12.0000 −0.476581
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 12.0000 0.475457
$$638$$ 6.00000 0.237542
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ 12.0000 0.473602
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ 6.00000 0.235884 0.117942 0.993020i $$-0.462370\pi$$
0.117942 + 0.993020i $$0.462370\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ −20.0000 −0.784465
$$651$$ 16.0000 0.627089
$$652$$ −4.00000 −0.156652
$$653$$ −36.0000 −1.40879 −0.704394 0.709809i $$-0.748781\pi$$
−0.704394 + 0.709809i $$0.748781\pi$$
$$654$$ 4.00000 0.156412
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 2.00000 0.0780274
$$658$$ 12.0000 0.467809
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 24.0000 0.932083
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 10.0000 0.387492
$$667$$ 36.0000 1.39393
$$668$$ 12.0000 0.464294
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ −2.00000 −0.0771517
$$673$$ 14.0000 0.539660 0.269830 0.962908i $$-0.413032\pi$$
0.269830 + 0.962908i $$0.413032\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ −5.00000 −0.192450
$$676$$ 3.00000 0.115385
$$677$$ 30.0000 1.15299 0.576497 0.817099i $$-0.304419\pi$$
0.576497 + 0.817099i $$0.304419\pi$$
$$678$$ −18.0000 −0.691286
$$679$$ 28.0000 1.07454
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 8.00000 0.306336
$$683$$ −24.0000 −0.918334 −0.459167 0.888350i $$-0.651852\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ −22.0000 −0.839352
$$688$$ 8.00000 0.304997
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 6.00000 0.228086
$$693$$ −2.00000 −0.0759737
$$694$$ −36.0000 −1.36654
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ −36.0000 −1.36360
$$698$$ 4.00000 0.151402
$$699$$ −18.0000 −0.680823
$$700$$ −10.0000 −0.377964
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 4.00000 0.150970
$$703$$ 40.0000 1.50863
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 12.0000 0.451306
$$708$$ 0 0
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ 14.0000 0.525041
$$712$$ 6.00000 0.224860
$$713$$ 48.0000 1.79761
$$714$$ 12.0000 0.449089
$$715$$ 0 0
$$716$$ 24.0000 0.896922
$$717$$ −12.0000 −0.448148
$$718$$ 12.0000 0.447836
$$719$$ 30.0000 1.11881 0.559406 0.828894i $$-0.311029\pi$$
0.559406 + 0.828894i $$0.311029\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 3.00000 0.111648
$$723$$ −10.0000 −0.371904
$$724$$ −22.0000 −0.817624
$$725$$ −30.0000 −1.11417
$$726$$ −1.00000 −0.0371135
$$727$$ −28.0000 −1.03846 −0.519231 0.854634i $$-0.673782\pi$$
−0.519231 + 0.854634i $$0.673782\pi$$
$$728$$ 8.00000 0.296500
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −48.0000 −1.77534
$$732$$ 8.00000 0.295689
$$733$$ −4.00000 −0.147743 −0.0738717 0.997268i $$-0.523536\pi$$
−0.0738717 + 0.997268i $$0.523536\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ 4.00000 0.147342
$$738$$ −6.00000 −0.220863
$$739$$ 8.00000 0.294285 0.147142 0.989115i $$-0.452992\pi$$
0.147142 + 0.989115i $$0.452992\pi$$
$$740$$ 0 0
$$741$$ 16.0000 0.587775
$$742$$ 0 0
$$743$$ 36.0000 1.32071 0.660356 0.750953i $$-0.270405\pi$$
0.660356 + 0.750953i $$0.270405\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ −20.0000 −0.732252
$$747$$ −12.0000 −0.439057
$$748$$ 6.00000 0.219382
$$749$$ −24.0000 −0.876941
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ −6.00000 −0.218797
$$753$$ 0 0
$$754$$ 24.0000 0.874028
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ −34.0000 −1.23575 −0.617876 0.786276i $$-0.712006\pi$$
−0.617876 + 0.786276i $$0.712006\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ −6.00000 −0.217786
$$760$$ 0 0
$$761$$ 18.0000 0.652499 0.326250 0.945284i $$-0.394215\pi$$
0.326250 + 0.945284i $$0.394215\pi$$
$$762$$ −14.0000 −0.507166
$$763$$ −8.00000 −0.289619
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ −6.00000 −0.216789
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ −34.0000 −1.22607 −0.613036 0.790055i $$-0.710052\pi$$
−0.613036 + 0.790055i $$0.710052\pi$$
$$770$$ 0 0
$$771$$ −30.0000 −1.08042
$$772$$ 14.0000 0.503871
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ −40.0000 −1.43684
$$776$$ −14.0000 −0.502571
$$777$$ −20.0000 −0.717496
$$778$$ 0 0
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ −6.00000 −0.214697
$$782$$ 36.0000 1.28736
$$783$$ 6.00000 0.214423
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 36.0000 1.28001
$$792$$ 1.00000 0.0355335
$$793$$ −32.0000 −1.13635
$$794$$ −26.0000 −0.922705
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 8.00000 0.283197
$$799$$ 36.0000 1.27359
$$800$$ 5.00000 0.176777
$$801$$ −6.00000 −0.212000
$$802$$ −30.0000 −1.05934
$$803$$ −2.00000 −0.0705785
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ 32.0000 1.12715
$$807$$ −24.0000 −0.844840
$$808$$ −6.00000 −0.211079
$$809$$ −42.0000 −1.47664 −0.738321 0.674450i $$-0.764381\pi$$
−0.738321 + 0.674450i $$0.764381\pi$$
$$810$$ 0 0
$$811$$ 8.00000 0.280918 0.140459 0.990086i $$-0.455142\pi$$
0.140459 + 0.990086i $$0.455142\pi$$
$$812$$ 12.0000 0.421117
$$813$$ 2.00000 0.0701431
$$814$$ −10.0000 −0.350500
$$815$$ 0 0
$$816$$ −6.00000 −0.210042
$$817$$ −32.0000 −1.11954
$$818$$ 34.0000 1.18878
$$819$$ −8.00000 −0.279543
$$820$$ 0 0
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 18.0000 0.627822
$$823$$ 8.00000 0.278862 0.139431 0.990232i $$-0.455473\pi$$
0.139431 + 0.990232i $$0.455473\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 5.00000 0.174078
$$826$$ 0 0
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ 6.00000 0.208514
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ −16.0000 −0.555034
$$832$$ −4.00000 −0.138675
$$833$$ 18.0000 0.623663
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ 8.00000 0.276520
$$838$$ −24.0000 −0.829066
$$839$$ 18.0000 0.621429 0.310715 0.950503i $$-0.399432\pi$$
0.310715 + 0.950503i $$0.399432\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 10.0000 0.344623
$$843$$ −6.00000 −0.206651
$$844$$ 8.00000 0.275371
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ 2.00000 0.0687208
$$848$$ 0 0
$$849$$ 8.00000 0.274559
$$850$$ −30.0000 −1.02899
$$851$$ −60.0000 −2.05677
$$852$$ 6.00000 0.205557
$$853$$ 8.00000 0.273915 0.136957 0.990577i $$-0.456268\pi$$
0.136957 + 0.990577i $$0.456268\pi$$
$$854$$ −16.0000 −0.547509
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 54.0000 1.84460 0.922302 0.386469i $$-0.126305\pi$$
0.922302 + 0.386469i $$0.126305\pi$$
$$858$$ −4.00000 −0.136558
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ 12.0000 0.408959
$$862$$ 0 0
$$863$$ 42.0000 1.42970 0.714848 0.699280i $$-0.246496\pi$$
0.714848 + 0.699280i $$0.246496\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −26.0000 −0.883516
$$867$$ 19.0000 0.645274
$$868$$ 16.0000 0.543075
$$869$$ −14.0000 −0.474917
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ 4.00000 0.135457
$$873$$ 14.0000 0.473828
$$874$$ 24.0000 0.811812
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ −52.0000 −1.75592 −0.877958 0.478738i $$-0.841094\pi$$
−0.877958 + 0.478738i $$0.841094\pi$$
$$878$$ 10.0000 0.337484
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 3.00000 0.101015
$$883$$ −28.0000 −0.942275 −0.471138 0.882060i $$-0.656156\pi$$
−0.471138 + 0.882060i $$0.656156\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 10.0000 0.335578
$$889$$ 28.0000 0.939090
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ −16.0000 −0.535720
$$893$$ 24.0000 0.803129
$$894$$ 6.00000 0.200670
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ −24.0000 −0.801337
$$898$$ −6.00000 −0.200223
$$899$$ 48.0000 1.60089
$$900$$ −5.00000 −0.166667
$$901$$ 0 0
$$902$$ 6.00000 0.199778
$$903$$ 16.0000 0.532447
$$904$$ −18.0000 −0.598671
$$905$$ 0 0
$$906$$ 10.0000 0.332228
$$907$$ 20.0000 0.664089 0.332045 0.943264i $$-0.392262\pi$$
0.332045 + 0.943264i $$0.392262\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ −42.0000 −1.39152 −0.695761 0.718273i $$-0.744933\pi$$
−0.695761 + 0.718273i $$0.744933\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ 12.0000 0.397142
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ −24.0000 −0.792550
$$918$$ 6.00000 0.198030
$$919$$ 2.00000 0.0659739 0.0329870 0.999456i $$-0.489498\pi$$
0.0329870 + 0.999456i $$0.489498\pi$$
$$920$$ 0 0
$$921$$ 20.0000 0.659022
$$922$$ −42.0000 −1.38320
$$923$$ −24.0000 −0.789970
$$924$$ −2.00000 −0.0657952
$$925$$ 50.0000 1.64399
$$926$$ 4.00000 0.131448
$$927$$ −4.00000 −0.131377
$$928$$ −6.00000 −0.196960
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ −18.0000 −0.589610
$$933$$ −18.0000 −0.589294
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ 4.00000 0.130744
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ 8.00000 0.261209
$$939$$ 26.0000 0.848478
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ −2.00000 −0.0651635
$$943$$ 36.0000 1.17232
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 14.0000 0.454699
$$949$$ −8.00000 −0.259691
$$950$$ −20.0000 −0.648886
$$951$$ 12.0000 0.389127
$$952$$ 12.0000 0.388922
$$953$$ −42.0000 −1.36051 −0.680257 0.732974i $$-0.738132\pi$$
−0.680257 + 0.732974i $$0.738132\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ −6.00000 −0.193952
$$958$$ 24.0000 0.775405
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −40.0000 −1.28965
$$963$$ −12.0000 −0.386695
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ −12.0000 −0.386094
$$967$$ 14.0000 0.450210 0.225105 0.974335i $$-0.427728\pi$$
0.225105 + 0.974335i $$0.427728\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −8.00000 −0.256468
$$974$$ −20.0000 −0.640841
$$975$$ 20.0000 0.640513
$$976$$ 8.00000 0.256074
$$977$$ 54.0000 1.72761 0.863807 0.503824i $$-0.168074\pi$$
0.863807 + 0.503824i $$0.168074\pi$$
$$978$$ 4.00000 0.127906
$$979$$ 6.00000 0.191761
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ −12.0000 −0.382935
$$983$$ 30.0000 0.956851 0.478426 0.878128i $$-0.341208\pi$$
0.478426 + 0.878128i $$0.341208\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 36.0000 1.14647
$$987$$ −12.0000 −0.381964
$$988$$ 16.0000 0.509028
$$989$$ 48.0000 1.52631
$$990$$ 0 0
$$991$$ 56.0000 1.77890 0.889449 0.457034i $$-0.151088\pi$$
0.889449 + 0.457034i $$0.151088\pi$$
$$992$$ −8.00000 −0.254000
$$993$$ −4.00000 −0.126936
$$994$$ −12.0000 −0.380617
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ 44.0000 1.39349 0.696747 0.717317i $$-0.254630\pi$$
0.696747 + 0.717317i $$0.254630\pi$$
$$998$$ 4.00000 0.126618
$$999$$ −10.0000 −0.316386
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 66.2.a.a.1.1 1
3.2 odd 2 198.2.a.e.1.1 1
4.3 odd 2 528.2.a.d.1.1 1
5.2 odd 4 1650.2.c.d.199.1 2
5.3 odd 4 1650.2.c.d.199.2 2
5.4 even 2 1650.2.a.m.1.1 1
7.6 odd 2 3234.2.a.d.1.1 1
8.3 odd 2 2112.2.a.v.1.1 1
8.5 even 2 2112.2.a.i.1.1 1
9.2 odd 6 1782.2.e.f.595.1 2
9.4 even 3 1782.2.e.s.1189.1 2
9.5 odd 6 1782.2.e.f.1189.1 2
9.7 even 3 1782.2.e.s.595.1 2
11.2 odd 10 726.2.e.b.565.1 4
11.3 even 5 726.2.e.k.493.1 4
11.4 even 5 726.2.e.k.511.1 4
11.5 even 5 726.2.e.k.487.1 4
11.6 odd 10 726.2.e.b.487.1 4
11.7 odd 10 726.2.e.b.511.1 4
11.8 odd 10 726.2.e.b.493.1 4
11.9 even 5 726.2.e.k.565.1 4
11.10 odd 2 726.2.a.i.1.1 1
12.11 even 2 1584.2.a.h.1.1 1
15.2 even 4 4950.2.c.r.199.2 2
15.8 even 4 4950.2.c.r.199.1 2
15.14 odd 2 4950.2.a.g.1.1 1
21.20 even 2 9702.2.a.bu.1.1 1
24.5 odd 2 6336.2.a.bj.1.1 1
24.11 even 2 6336.2.a.bf.1.1 1
33.32 even 2 2178.2.a.b.1.1 1
44.43 even 2 5808.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.a.1.1 1 1.1 even 1 trivial
198.2.a.e.1.1 1 3.2 odd 2
528.2.a.d.1.1 1 4.3 odd 2
726.2.a.i.1.1 1 11.10 odd 2
726.2.e.b.487.1 4 11.6 odd 10
726.2.e.b.493.1 4 11.8 odd 10
726.2.e.b.511.1 4 11.7 odd 10
726.2.e.b.565.1 4 11.2 odd 10
726.2.e.k.487.1 4 11.5 even 5
726.2.e.k.493.1 4 11.3 even 5
726.2.e.k.511.1 4 11.4 even 5
726.2.e.k.565.1 4 11.9 even 5
1584.2.a.h.1.1 1 12.11 even 2
1650.2.a.m.1.1 1 5.4 even 2
1650.2.c.d.199.1 2 5.2 odd 4
1650.2.c.d.199.2 2 5.3 odd 4
1782.2.e.f.595.1 2 9.2 odd 6
1782.2.e.f.1189.1 2 9.5 odd 6
1782.2.e.s.595.1 2 9.7 even 3
1782.2.e.s.1189.1 2 9.4 even 3
2112.2.a.i.1.1 1 8.5 even 2
2112.2.a.v.1.1 1 8.3 odd 2
2178.2.a.b.1.1 1 33.32 even 2
3234.2.a.d.1.1 1 7.6 odd 2
4950.2.a.g.1.1 1 15.14 odd 2
4950.2.c.r.199.1 2 15.8 even 4
4950.2.c.r.199.2 2 15.2 even 4
5808.2.a.l.1.1 1 44.43 even 2
6336.2.a.bf.1.1 1 24.11 even 2
6336.2.a.bj.1.1 1 24.5 odd 2
9702.2.a.bu.1.1 1 21.20 even 2