# Properties

 Label 726.2.a.i.1.1 Level $726$ Weight $2$ Character 726.1 Self dual yes Analytic conductor $5.797$ 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] = [726,2,Mod(1,726)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(726, 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("726.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$726 = 2 \cdot 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 726.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: $$5.79713918674$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 66) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 726.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

## 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 $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 1.00000 0.288675
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ 0 0
$$23$$ 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 $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 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
$$33$$ 0 0
$$34$$ 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 $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 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 $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 8.00000 1.01600
$$63$$ −2.00000 −0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −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 $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ −5.00000 −0.577350
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 4.00000 0.452911
$$79$$ −14.0000 −1.57512 −0.787562 0.616236i $$-0.788657\pi$$
−0.787562 + 0.616236i $$0.788657\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ −6.00000 −0.643268
$$88$$ 0 0
$$89$$ −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
$$99$$ 0 0
$$100$$ −5.00000 −0.500000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 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 $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 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
$$121$$ 0 0
$$122$$ −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 $$-0.713334\pi$$
−0.621150 + 0.783692i $$0.713334\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ −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 $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 6.00000 0.503509
$$143$$ 0 0
$$144$$ 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 $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ −5.00000 −0.408248
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 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 $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ −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 $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 10.0000 0.755929
$$176$$ 0 0
$$177$$ 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
$$187$$ 0 0
$$188$$ −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 $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −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
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 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 $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 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
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 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 $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 0 0
$$243$$ 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
$$253$$ 0 0
$$254$$ −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 $$0.5\pi$$
$$264$$ 0 0
$$265$$ 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 $$-0.519348\pi$$
−0.0607457 + 0.998153i $$0.519348\pi$$
$$272$$ 6.00000 0.363803
$$273$$ −8.00000 −0.484182
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ 6.00000 0.361158
$$277$$ 16.0000 0.961347 0.480673 0.876900i $$-0.340392\pi$$
0.480673 + 0.876900i $$0.340392\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ −6.00000 −0.357295
$$283$$ −8.00000 −0.475551 −0.237775 0.971320i $$-0.576418\pi$$
−0.237775 + 0.971320i $$0.576418\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 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 $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ −10.0000 −0.581238
$$297$$ 0 0
$$298$$ 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 $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 0 0
$$309$$ −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
$$319$$ 0 0
$$320$$ 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 $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ 3.00000 0.163178
$$339$$ 18.0000 0.977626
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 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 $$-0.917117\pi$$
−0.966291 + 0.257454i $$0.917117\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ 4.00000 0.214115 0.107058 0.994253i $$-0.465857\pi$$
0.107058 + 0.994253i $$0.465857\pi$$
$$350$$ 10.0000 0.534522
$$351$$ 4.00000 0.213504
$$352$$ 0 0
$$353$$ −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 $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −22.0000 −1.15629
$$363$$ 0 0
$$364$$ −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 $$-0.673242\pi$$
−0.517780 + 0.855514i $$0.673242\pi$$
$$374$$ 0 0
$$375$$ 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
$$396$$ 0 0
$$397$$ 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
$$407$$ 0 0
$$408$$ 6.00000 0.297044
$$409$$ 34.0000 1.68119 0.840596 0.541663i $$-0.182205\pi$$
0.840596 + 0.541663i $$0.182205\pi$$
$$410$$ 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
$$418$$ 0 0
$$419$$ 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
$$429$$ 0 0
$$430$$ 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 $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 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
$$451$$ 0 0
$$452$$ 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 $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −22.0000 −1.02799
$$459$$ 6.00000 0.280056
$$460$$ 0 0
$$461$$ −42.0000 −1.95614 −0.978068 0.208288i $$-0.933211\pi$$
−0.978068 + 0.208288i $$0.933211\pi$$
$$462$$ 0 0
$$463$$ −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
$$473$$ 0 0
$$474$$ −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 $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −40.0000 −1.82384
$$482$$ 10.0000 0.455488
$$483$$ −12.0000 −0.546019
$$484$$ 0 0
$$485$$ 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 $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ −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 $$-0.586206\pi$$
−0.267527 + 0.963550i $$0.586206\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 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
$$517$$ 0 0
$$518$$ 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 $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 10.0000 0.436436
$$526$$ 0 0
$$527$$ 48.0000 2.09091
$$528$$ 0 0
$$529$$ 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
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −20.0000 −0.859867 −0.429934 0.902861i $$-0.641463\pi$$
−0.429934 + 0.902861i $$0.641463\pi$$
$$542$$ −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 $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ −8.00000 −0.341432
$$550$$ 0 0
$$551$$ −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 $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 8.00000 0.338667
$$559$$ −32.0000 −1.35346
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ −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 $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ 0 0
$$573$$ 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 $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ 0 0
$$595$$ 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 $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 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 $$-0.591702\pi$$
−0.284121 + 0.958788i $$0.591702\pi$$
$$608$$ 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 $$-0.395269\pi$$
0.323117 + 0.946359i $$0.395269\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −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
$$627$$ 0 0
$$628$$ 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
$$638$$ 0 0
$$639$$ 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 $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 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
$$671$$ 0 0
$$672$$ −2.00000 −0.0771517
$$673$$ −14.0000 −0.539660 −0.269830 0.962908i $$-0.586968\pi$$
−0.269830 + 0.962908i $$0.586968\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ −5.00000 −0.192450
$$676$$ 3.00000 0.115385
$$677$$ −30.0000 −1.15299 −0.576497 0.817099i $$-0.695581\pi$$
−0.576497 + 0.817099i $$0.695581\pi$$
$$678$$ 18.0000 0.691286
$$679$$ −28.0000 −1.07454
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ −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
$$693$$ 0 0
$$694$$ −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 $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 4.00000 0.150970
$$703$$ −40.0000 −1.50863
$$704$$ 0 0
$$705$$ 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
$$726$$ 0 0
$$727$$ −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 $$-0.476464\pi$$
0.0738717 + 0.997268i $$0.476464\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ 0 0
$$738$$ −6.00000 −0.220863
$$739$$ −8.00000 −0.294285 −0.147142 0.989115i $$-0.547008\pi$$
−0.147142 + 0.989115i $$0.547008\pi$$
$$740$$ 0 0
$$741$$ 16.0000 0.587775
$$742$$ 0 0
$$743$$ −36.0000 −1.32071 −0.660356 0.750953i $$-0.729595\pi$$
−0.660356 + 0.750953i $$0.729595\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ −20.0000 −0.732252
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ −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
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ −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 $$-0.289948\pi$$
0.613036 + 0.790055i $$0.289948\pi$$
$$770$$ 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
$$781$$ 0 0
$$782$$ 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 $$-0.693188\pi$$
−0.570338 + 0.821410i $$0.693188\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −36.0000 −1.28001
$$792$$ 0 0
$$793$$ −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
$$803$$ 0 0
$$804$$ −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 $$-0.235619\pi$$
0.738321 + 0.674450i $$0.235619\pi$$
$$810$$ 0 0
$$811$$ −8.00000 −0.280918 −0.140459 0.990086i $$-0.544858\pi$$
−0.140459 + 0.990086i $$0.544858\pi$$
$$812$$ 12.0000 0.421117
$$813$$ −2.00000 −0.0701431
$$814$$ 0 0
$$815$$ 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 $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ −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
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 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
$$836$$ 0 0
$$837$$ 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
$$847$$ 0 0
$$848$$ 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 $$-0.543732\pi$$
−0.136957 + 0.990577i $$0.543732\pi$$
$$854$$ 16.0000 0.547509
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ −54.0000 −1.84460 −0.922302 0.386469i $$-0.873695\pi$$
−0.922302 + 0.386469i $$0.873695\pi$$
$$858$$ 0 0
$$859$$ 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
$$869$$ 0 0
$$870$$ 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 $$-0.158906\pi$$
0.877958 + 0.478738i $$0.158906\pi$$
$$878$$ 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 $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ −10.0000 −0.335578
$$889$$ 28.0000 0.939090
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −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
$$902$$ 0 0
$$903$$ 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
$$913$$ 0 0
$$914$$ 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 $$-0.510502\pi$$
−0.0329870 + 0.999456i $$0.510502\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ −42.0000 −1.38320
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ 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 $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 8.00000 0.261209
$$939$$ 26.0000 0.848478
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 2.00000 0.0651635
$$943$$ −36.0000 −1.17232
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 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 $$-0.261868\pi$$
0.680257 + 0.732974i $$0.261868\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ 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 $$-0.572272\pi$$
−0.225105 + 0.974335i $$0.572272\pi$$
$$968$$ 0 0
$$969$$ 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
$$979$$ 0 0
$$980$$ 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 $$-0.745370\pi$$
−0.696747 + 0.717317i $$0.745370\pi$$
$$998$$ −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 726.2.a.i.1.1 1
3.2 odd 2 2178.2.a.b.1.1 1
4.3 odd 2 5808.2.a.l.1.1 1
11.2 odd 10 726.2.e.k.565.1 4
11.3 even 5 726.2.e.b.493.1 4
11.4 even 5 726.2.e.b.511.1 4
11.5 even 5 726.2.e.b.487.1 4
11.6 odd 10 726.2.e.k.487.1 4
11.7 odd 10 726.2.e.k.511.1 4
11.8 odd 10 726.2.e.k.493.1 4
11.9 even 5 726.2.e.b.565.1 4
11.10 odd 2 66.2.a.a.1.1 1
33.32 even 2 198.2.a.e.1.1 1
44.43 even 2 528.2.a.d.1.1 1
55.32 even 4 1650.2.c.d.199.1 2
55.43 even 4 1650.2.c.d.199.2 2
55.54 odd 2 1650.2.a.m.1.1 1
77.76 even 2 3234.2.a.d.1.1 1
88.21 odd 2 2112.2.a.i.1.1 1
88.43 even 2 2112.2.a.v.1.1 1
99.32 even 6 1782.2.e.f.1189.1 2
99.43 odd 6 1782.2.e.s.595.1 2
99.65 even 6 1782.2.e.f.595.1 2
99.76 odd 6 1782.2.e.s.1189.1 2
132.131 odd 2 1584.2.a.h.1.1 1
165.32 odd 4 4950.2.c.r.199.2 2
165.98 odd 4 4950.2.c.r.199.1 2
165.164 even 2 4950.2.a.g.1.1 1
231.230 odd 2 9702.2.a.bu.1.1 1
264.131 odd 2 6336.2.a.bf.1.1 1
264.197 even 2 6336.2.a.bj.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.a.1.1 1 11.10 odd 2
198.2.a.e.1.1 1 33.32 even 2
528.2.a.d.1.1 1 44.43 even 2
726.2.a.i.1.1 1 1.1 even 1 trivial
726.2.e.b.487.1 4 11.5 even 5
726.2.e.b.493.1 4 11.3 even 5
726.2.e.b.511.1 4 11.4 even 5
726.2.e.b.565.1 4 11.9 even 5
726.2.e.k.487.1 4 11.6 odd 10
726.2.e.k.493.1 4 11.8 odd 10
726.2.e.k.511.1 4 11.7 odd 10
726.2.e.k.565.1 4 11.2 odd 10
1584.2.a.h.1.1 1 132.131 odd 2
1650.2.a.m.1.1 1 55.54 odd 2
1650.2.c.d.199.1 2 55.32 even 4
1650.2.c.d.199.2 2 55.43 even 4
1782.2.e.f.595.1 2 99.65 even 6
1782.2.e.f.1189.1 2 99.32 even 6
1782.2.e.s.595.1 2 99.43 odd 6
1782.2.e.s.1189.1 2 99.76 odd 6
2112.2.a.i.1.1 1 88.21 odd 2
2112.2.a.v.1.1 1 88.43 even 2
2178.2.a.b.1.1 1 3.2 odd 2
3234.2.a.d.1.1 1 77.76 even 2
4950.2.a.g.1.1 1 165.164 even 2
4950.2.c.r.199.1 2 165.98 odd 4
4950.2.c.r.199.2 2 165.32 odd 4
5808.2.a.l.1.1 1 4.3 odd 2
6336.2.a.bf.1.1 1 264.131 odd 2
6336.2.a.bj.1.1 1 264.197 even 2
9702.2.a.bu.1.1 1 231.230 odd 2