# Properties

 Label 33.2.a.a.1.1 Level $33$ Weight $2$ Character 33.1 Self dual yes Analytic conductor $0.264$ 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] = [33,2,Mod(1,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, 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("33.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 33.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: $$0.263506326670$$ 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 $$\chi$$ $$=$$ 33.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} -4.00000 q^{21} +1.00000 q^{22} +8.00000 q^{23} +3.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -6.00000 q^{29} +2.00000 q^{30} -8.00000 q^{31} +5.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} -8.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} +2.00000 q^{39} +6.00000 q^{40} -2.00000 q^{41} -4.00000 q^{42} -1.00000 q^{44} -2.00000 q^{45} +8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -2.00000 q^{55} -12.0000 q^{56} -6.00000 q^{58} -4.00000 q^{59} -2.00000 q^{60} +6.00000 q^{61} -8.00000 q^{62} +4.00000 q^{63} +7.00000 q^{64} +4.00000 q^{65} -1.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} -8.00000 q^{70} -3.00000 q^{72} -14.0000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{77} +2.00000 q^{78} -4.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +12.0000 q^{83} +4.00000 q^{84} +4.00000 q^{85} +6.00000 q^{87} -3.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} -8.00000 q^{91} -8.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} -5.00000 q^{96} +2.00000 q^{97} +9.00000 q^{98} +1.00000 q^{99} +O(q^{100})$$

## 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 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.00000 −0.500000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 1.00000 0.333333
$$10$$ −2.00000 −0.632456
$$11$$ 1.00000 0.301511
$$12$$ 1.00000 0.288675
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 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
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 2.00000 0.447214
$$21$$ −4.00000 −0.872872
$$22$$ 1.00000 0.213201
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 3.00000 0.612372
$$25$$ −1.00000 −0.200000
$$26$$ −2.00000 −0.392232
$$27$$ −1.00000 −0.192450
$$28$$ −4.00000 −0.755929
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 2.00000 0.365148
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 5.00000 0.883883
$$33$$ −1.00000 −0.174078
$$34$$ −2.00000 −0.342997
$$35$$ −8.00000 −1.35225
$$36$$ −1.00000 −0.166667
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 6.00000 0.948683
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ −4.00000 −0.617213
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ −2.00000 −0.298142
$$46$$ 8.00000 1.17954
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 9.00000 1.28571
$$50$$ −1.00000 −0.141421
$$51$$ 2.00000 0.280056
$$52$$ 2.00000 0.277350
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −2.00000 −0.269680
$$56$$ −12.0000 −1.60357
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ −2.00000 −0.258199
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ 4.00000 0.503953
$$64$$ 7.00000 0.875000
$$65$$ 4.00000 0.496139
$$66$$ −1.00000 −0.123091
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 2.00000 0.242536
$$69$$ −8.00000 −0.963087
$$70$$ −8.00000 −0.956183
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 4.00000 0.455842
$$78$$ 2.00000 0.226455
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 1.00000 0.111111
$$82$$ −2.00000 −0.220863
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 4.00000 0.433861
$$86$$ 0 0
$$87$$ 6.00000 0.643268
$$88$$ −3.00000 −0.319801
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ −2.00000 −0.210819
$$91$$ −8.00000 −0.838628
$$92$$ −8.00000 −0.834058
$$93$$ 8.00000 0.829561
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ −5.00000 −0.510310
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 9.00000 0.909137
$$99$$ 1.00000 0.100504
$$100$$ 1.00000 0.100000
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 2.00000 0.198030
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 8.00000 0.780720
$$106$$ 6.00000 0.582772
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ −6.00000 −0.569495
$$112$$ −4.00000 −0.377964
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ −16.0000 −1.49201
$$116$$ 6.00000 0.557086
$$117$$ −2.00000 −0.184900
$$118$$ −4.00000 −0.368230
$$119$$ −8.00000 −0.733359
$$120$$ −6.00000 −0.547723
$$121$$ 1.00000 0.0909091
$$122$$ 6.00000 0.543214
$$123$$ 2.00000 0.180334
$$124$$ 8.00000 0.718421
$$125$$ 12.0000 1.07331
$$126$$ 4.00000 0.356348
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ 4.00000 0.350823
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 2.00000 0.172133
$$136$$ 6.00000 0.514496
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ −8.00000 −0.681005
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 8.00000 0.676123
$$141$$ −8.00000 −0.673722
$$142$$ 0 0
$$143$$ −2.00000 −0.167248
$$144$$ −1.00000 −0.0833333
$$145$$ 12.0000 0.996546
$$146$$ −14.0000 −1.15865
$$147$$ −9.00000 −0.742307
$$148$$ −6.00000 −0.493197
$$149$$ −22.0000 −1.80231 −0.901155 0.433497i $$-0.857280\pi$$
−0.901155 + 0.433497i $$0.857280\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 4.00000 0.322329
$$155$$ 16.0000 1.28515
$$156$$ −2.00000 −0.160128
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ −6.00000 −0.475831
$$160$$ −10.0000 −0.790569
$$161$$ 32.0000 2.52195
$$162$$ 1.00000 0.0785674
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 2.00000 0.155700
$$166$$ 12.0000 0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 12.0000 0.925820
$$169$$ −9.00000 −0.692308
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 6.00000 0.454859
$$175$$ −4.00000 −0.302372
$$176$$ −1.00000 −0.0753778
$$177$$ 4.00000 0.300658
$$178$$ −6.00000 −0.449719
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ −8.00000 −0.592999
$$183$$ −6.00000 −0.443533
$$184$$ −24.0000 −1.76930
$$185$$ −12.0000 −0.882258
$$186$$ 8.00000 0.586588
$$187$$ −2.00000 −0.146254
$$188$$ −8.00000 −0.583460
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ −7.00000 −0.505181
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 2.00000 0.143592
$$195$$ −4.00000 −0.286446
$$196$$ −9.00000 −0.642857
$$197$$ −14.0000 −0.997459 −0.498729 0.866758i $$-0.666200\pi$$
−0.498729 + 0.866758i $$0.666200\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 3.00000 0.212132
$$201$$ 4.00000 0.282138
$$202$$ 2.00000 0.140720
$$203$$ −24.0000 −1.68447
$$204$$ −2.00000 −0.140028
$$205$$ 4.00000 0.279372
$$206$$ 8.00000 0.557386
$$207$$ 8.00000 0.556038
$$208$$ 2.00000 0.138675
$$209$$ 0 0
$$210$$ 8.00000 0.552052
$$211$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 3.00000 0.204124
$$217$$ −32.0000 −2.17230
$$218$$ −2.00000 −0.135457
$$219$$ 14.0000 0.946032
$$220$$ 2.00000 0.134840
$$221$$ 4.00000 0.269069
$$222$$ −6.00000 −0.402694
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 20.0000 1.33631
$$225$$ −1.00000 −0.0666667
$$226$$ −6.00000 −0.399114
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ −16.0000 −1.05501
$$231$$ −4.00000 −0.263181
$$232$$ 18.0000 1.18176
$$233$$ 30.0000 1.96537 0.982683 0.185296i $$-0.0593245\pi$$
0.982683 + 0.185296i $$0.0593245\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ −16.0000 −1.04372
$$236$$ 4.00000 0.260378
$$237$$ 4.00000 0.259828
$$238$$ −8.00000 −0.518563
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ −2.00000 −0.129099
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ −6.00000 −0.384111
$$245$$ −18.0000 −1.14998
$$246$$ 2.00000 0.127515
$$247$$ 0 0
$$248$$ 24.0000 1.52400
$$249$$ −12.0000 −0.760469
$$250$$ 12.0000 0.758947
$$251$$ 4.00000 0.252478 0.126239 0.992000i $$-0.459709\pi$$
0.126239 + 0.992000i $$0.459709\pi$$
$$252$$ −4.00000 −0.251976
$$253$$ 8.00000 0.502956
$$254$$ −4.00000 −0.250982
$$255$$ −4.00000 −0.250490
$$256$$ −17.0000 −1.06250
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 0 0
$$259$$ 24.0000 1.49129
$$260$$ −4.00000 −0.248069
$$261$$ −6.00000 −0.371391
$$262$$ −12.0000 −0.741362
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 3.00000 0.184637
$$265$$ −12.0000 −0.737154
$$266$$ 0 0
$$267$$ 6.00000 0.367194
$$268$$ 4.00000 0.244339
$$269$$ −2.00000 −0.121942 −0.0609711 0.998140i $$-0.519420\pi$$
−0.0609711 + 0.998140i $$0.519420\pi$$
$$270$$ 2.00000 0.121716
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 8.00000 0.484182
$$274$$ 2.00000 0.120824
$$275$$ −1.00000 −0.0603023
$$276$$ 8.00000 0.481543
$$277$$ −26.0000 −1.56219 −0.781094 0.624413i $$-0.785338\pi$$
−0.781094 + 0.624413i $$0.785338\pi$$
$$278$$ −8.00000 −0.479808
$$279$$ −8.00000 −0.478947
$$280$$ 24.0000 1.43427
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ 16.0000 0.951101 0.475551 0.879688i $$-0.342249\pi$$
0.475551 + 0.879688i $$0.342249\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ −8.00000 −0.472225
$$288$$ 5.00000 0.294628
$$289$$ −13.0000 −0.764706
$$290$$ 12.0000 0.704664
$$291$$ −2.00000 −0.117242
$$292$$ 14.0000 0.819288
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ −9.00000 −0.524891
$$295$$ 8.00000 0.465778
$$296$$ −18.0000 −1.04623
$$297$$ −1.00000 −0.0580259
$$298$$ −22.0000 −1.27443
$$299$$ −16.0000 −0.925304
$$300$$ −1.00000 −0.0577350
$$301$$ 0 0
$$302$$ 20.0000 1.15087
$$303$$ −2.00000 −0.114897
$$304$$ 0 0
$$305$$ −12.0000 −0.687118
$$306$$ −2.00000 −0.114332
$$307$$ 32.0000 1.82634 0.913168 0.407583i $$-0.133628\pi$$
0.913168 + 0.407583i $$0.133628\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ −8.00000 −0.455104
$$310$$ 16.0000 0.908739
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ −6.00000 −0.339683
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ 14.0000 0.790066
$$315$$ −8.00000 −0.450749
$$316$$ 4.00000 0.225018
$$317$$ 22.0000 1.23564 0.617822 0.786318i $$-0.288015\pi$$
0.617822 + 0.786318i $$0.288015\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ −6.00000 −0.335936
$$320$$ −14.0000 −0.782624
$$321$$ 12.0000 0.669775
$$322$$ 32.0000 1.78329
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 2.00000 0.110940
$$326$$ 4.00000 0.221540
$$327$$ 2.00000 0.110600
$$328$$ 6.00000 0.331295
$$329$$ 32.0000 1.76422
$$330$$ 2.00000 0.110096
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 6.00000 0.328798
$$334$$ 0 0
$$335$$ 8.00000 0.437087
$$336$$ 4.00000 0.218218
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 6.00000 0.325875
$$340$$ −4.00000 −0.216930
$$341$$ −8.00000 −0.433224
$$342$$ 0 0
$$343$$ 8.00000 0.431959
$$344$$ 0 0
$$345$$ 16.0000 0.861411
$$346$$ −6.00000 −0.322562
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ 6.00000 0.321173 0.160586 0.987022i $$-0.448662\pi$$
0.160586 + 0.987022i $$0.448662\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 2.00000 0.106752
$$352$$ 5.00000 0.266501
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 8.00000 0.423405
$$358$$ 12.0000 0.634220
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 6.00000 0.316228
$$361$$ −19.0000 −1.00000
$$362$$ 22.0000 1.15629
$$363$$ −1.00000 −0.0524864
$$364$$ 8.00000 0.419314
$$365$$ 28.0000 1.46559
$$366$$ −6.00000 −0.313625
$$367$$ −32.0000 −1.67039 −0.835193 0.549957i $$-0.814644\pi$$
−0.835193 + 0.549957i $$0.814644\pi$$
$$368$$ −8.00000 −0.417029
$$369$$ −2.00000 −0.104116
$$370$$ −12.0000 −0.623850
$$371$$ 24.0000 1.24602
$$372$$ −8.00000 −0.414781
$$373$$ −2.00000 −0.103556 −0.0517780 0.998659i $$-0.516489\pi$$
−0.0517780 + 0.998659i $$0.516489\pi$$
$$374$$ −2.00000 −0.103418
$$375$$ −12.0000 −0.619677
$$376$$ −24.0000 −1.23771
$$377$$ 12.0000 0.618031
$$378$$ −4.00000 −0.205738
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 0 0
$$381$$ 4.00000 0.204926
$$382$$ 8.00000 0.409316
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 3.00000 0.153093
$$385$$ −8.00000 −0.407718
$$386$$ −14.0000 −0.712581
$$387$$ 0 0
$$388$$ −2.00000 −0.101535
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ −4.00000 −0.202548
$$391$$ −16.0000 −0.809155
$$392$$ −27.0000 −1.36371
$$393$$ 12.0000 0.605320
$$394$$ −14.0000 −0.705310
$$395$$ 8.00000 0.402524
$$396$$ −1.00000 −0.0502519
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 26.0000 1.29838 0.649189 0.760627i $$-0.275108\pi$$
0.649189 + 0.760627i $$0.275108\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 16.0000 0.797017
$$404$$ −2.00000 −0.0995037
$$405$$ −2.00000 −0.0993808
$$406$$ −24.0000 −1.19110
$$407$$ 6.00000 0.297409
$$408$$ −6.00000 −0.297044
$$409$$ 18.0000 0.890043 0.445021 0.895520i $$-0.353196\pi$$
0.445021 + 0.895520i $$0.353196\pi$$
$$410$$ 4.00000 0.197546
$$411$$ −2.00000 −0.0986527
$$412$$ −8.00000 −0.394132
$$413$$ −16.0000 −0.787309
$$414$$ 8.00000 0.393179
$$415$$ −24.0000 −1.17811
$$416$$ −10.0000 −0.490290
$$417$$ 8.00000 0.391762
$$418$$ 0 0
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ −8.00000 −0.390360
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 0 0
$$423$$ 8.00000 0.388973
$$424$$ −18.0000 −0.874157
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 24.0000 1.16144
$$428$$ 12.0000 0.580042
$$429$$ 2.00000 0.0965609
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ −32.0000 −1.53605
$$435$$ −12.0000 −0.575356
$$436$$ 2.00000 0.0957826
$$437$$ 0 0
$$438$$ 14.0000 0.668946
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 6.00000 0.286039
$$441$$ 9.00000 0.428571
$$442$$ 4.00000 0.190261
$$443$$ 28.0000 1.33032 0.665160 0.746701i $$-0.268363\pi$$
0.665160 + 0.746701i $$0.268363\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 12.0000 0.568855
$$446$$ 16.0000 0.757622
$$447$$ 22.0000 1.04056
$$448$$ 28.0000 1.32288
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ −2.00000 −0.0941763
$$452$$ 6.00000 0.282216
$$453$$ −20.0000 −0.939682
$$454$$ 12.0000 0.563188
$$455$$ 16.0000 0.750092
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ 6.00000 0.280362
$$459$$ 2.00000 0.0933520
$$460$$ 16.0000 0.746004
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ −4.00000 −0.186097
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 6.00000 0.278543
$$465$$ −16.0000 −0.741982
$$466$$ 30.0000 1.38972
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ −16.0000 −0.738811
$$470$$ −16.0000 −0.738025
$$471$$ −14.0000 −0.645086
$$472$$ 12.0000 0.552345
$$473$$ 0 0
$$474$$ 4.00000 0.183726
$$475$$ 0 0
$$476$$ 8.00000 0.366679
$$477$$ 6.00000 0.274721
$$478$$ 24.0000 1.09773
$$479$$ 8.00000 0.365529 0.182765 0.983157i $$-0.441495\pi$$
0.182765 + 0.983157i $$0.441495\pi$$
$$480$$ 10.0000 0.456435
$$481$$ −12.0000 −0.547153
$$482$$ 10.0000 0.455488
$$483$$ −32.0000 −1.45605
$$484$$ −1.00000 −0.0454545
$$485$$ −4.00000 −0.181631
$$486$$ −1.00000 −0.0453609
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ −18.0000 −0.814822
$$489$$ −4.00000 −0.180886
$$490$$ −18.0000 −0.813157
$$491$$ 4.00000 0.180517 0.0902587 0.995918i $$-0.471231\pi$$
0.0902587 + 0.995918i $$0.471231\pi$$
$$492$$ −2.00000 −0.0901670
$$493$$ 12.0000 0.540453
$$494$$ 0 0
$$495$$ −2.00000 −0.0898933
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ −12.0000 −0.537733
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 0 0
$$502$$ 4.00000 0.178529
$$503$$ −32.0000 −1.42681 −0.713405 0.700752i $$-0.752848\pi$$
−0.713405 + 0.700752i $$0.752848\pi$$
$$504$$ −12.0000 −0.534522
$$505$$ −4.00000 −0.177998
$$506$$ 8.00000 0.355643
$$507$$ 9.00000 0.399704
$$508$$ 4.00000 0.177471
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ −4.00000 −0.177123
$$511$$ −56.0000 −2.47729
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ −14.0000 −0.617514
$$515$$ −16.0000 −0.705044
$$516$$ 0 0
$$517$$ 8.00000 0.351840
$$518$$ 24.0000 1.05450
$$519$$ 6.00000 0.263371
$$520$$ −12.0000 −0.526235
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ −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
$$525$$ 4.00000 0.174574
$$526$$ −16.0000 −0.697633
$$527$$ 16.0000 0.696971
$$528$$ 1.00000 0.0435194
$$529$$ 41.0000 1.78261
$$530$$ −12.0000 −0.521247
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ 4.00000 0.173259
$$534$$ 6.00000 0.259645
$$535$$ 24.0000 1.03761
$$536$$ 12.0000 0.518321
$$537$$ −12.0000 −0.517838
$$538$$ −2.00000 −0.0862261
$$539$$ 9.00000 0.387657
$$540$$ −2.00000 −0.0860663
$$541$$ 46.0000 1.97769 0.988847 0.148933i $$-0.0475840\pi$$
0.988847 + 0.148933i $$0.0475840\pi$$
$$542$$ 20.0000 0.859074
$$543$$ −22.0000 −0.944110
$$544$$ −10.0000 −0.428746
$$545$$ 4.00000 0.171341
$$546$$ 8.00000 0.342368
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ −2.00000 −0.0854358
$$549$$ 6.00000 0.256074
$$550$$ −1.00000 −0.0426401
$$551$$ 0 0
$$552$$ 24.0000 1.02151
$$553$$ −16.0000 −0.680389
$$554$$ −26.0000 −1.10463
$$555$$ 12.0000 0.509372
$$556$$ 8.00000 0.339276
$$557$$ −14.0000 −0.593199 −0.296600 0.955002i $$-0.595853\pi$$
−0.296600 + 0.955002i $$0.595853\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ 0 0
$$560$$ 8.00000 0.338062
$$561$$ 2.00000 0.0844401
$$562$$ −18.0000 −0.759284
$$563$$ −44.0000 −1.85438 −0.927189 0.374593i $$-0.877783\pi$$
−0.927189 + 0.374593i $$0.877783\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 12.0000 0.504844
$$566$$ 16.0000 0.672530
$$567$$ 4.00000 0.167984
$$568$$ 0 0
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 0 0
$$571$$ −16.0000 −0.669579 −0.334790 0.942293i $$-0.608665\pi$$
−0.334790 + 0.942293i $$0.608665\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ −8.00000 −0.334205
$$574$$ −8.00000 −0.333914
$$575$$ −8.00000 −0.333623
$$576$$ 7.00000 0.291667
$$577$$ −30.0000 −1.24892 −0.624458 0.781058i $$-0.714680\pi$$
−0.624458 + 0.781058i $$0.714680\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 14.0000 0.581820
$$580$$ −12.0000 −0.498273
$$581$$ 48.0000 1.99138
$$582$$ −2.00000 −0.0829027
$$583$$ 6.00000 0.248495
$$584$$ 42.0000 1.73797
$$585$$ 4.00000 0.165380
$$586$$ −6.00000 −0.247858
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ 9.00000 0.371154
$$589$$ 0 0
$$590$$ 8.00000 0.329355
$$591$$ 14.0000 0.575883
$$592$$ −6.00000 −0.246598
$$593$$ 38.0000 1.56047 0.780236 0.625485i $$-0.215099\pi$$
0.780236 + 0.625485i $$0.215099\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ 16.0000 0.655936
$$596$$ 22.0000 0.901155
$$597$$ 0 0
$$598$$ −16.0000 −0.654289
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ −3.00000 −0.122474
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ −20.0000 −0.813788
$$605$$ −2.00000 −0.0813116
$$606$$ −2.00000 −0.0812444
$$607$$ −4.00000 −0.162355 −0.0811775 0.996700i $$-0.525868\pi$$
−0.0811775 + 0.996700i $$0.525868\pi$$
$$608$$ 0 0
$$609$$ 24.0000 0.972529
$$610$$ −12.0000 −0.485866
$$611$$ −16.0000 −0.647291
$$612$$ 2.00000 0.0808452
$$613$$ 14.0000 0.565455 0.282727 0.959200i $$-0.408761\pi$$
0.282727 + 0.959200i $$0.408761\pi$$
$$614$$ 32.0000 1.29141
$$615$$ −4.00000 −0.161296
$$616$$ −12.0000 −0.483494
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ 44.0000 1.76851 0.884255 0.467005i $$-0.154667\pi$$
0.884255 + 0.467005i $$0.154667\pi$$
$$620$$ −16.0000 −0.642575
$$621$$ −8.00000 −0.321029
$$622$$ −24.0000 −0.962312
$$623$$ −24.0000 −0.961540
$$624$$ −2.00000 −0.0800641
$$625$$ −19.0000 −0.760000
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ −14.0000 −0.558661
$$629$$ −12.0000 −0.478471
$$630$$ −8.00000 −0.318728
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 12.0000 0.477334
$$633$$ 0 0
$$634$$ 22.0000 0.873732
$$635$$ 8.00000 0.317470
$$636$$ 6.00000 0.237915
$$637$$ −18.0000 −0.713186
$$638$$ −6.00000 −0.237542
$$639$$ 0 0
$$640$$ 6.00000 0.237171
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 12.0000 0.473602
$$643$$ 20.0000 0.788723 0.394362 0.918955i $$-0.370966\pi$$
0.394362 + 0.918955i $$0.370966\pi$$
$$644$$ −32.0000 −1.26098
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 8.00000 0.314512 0.157256 0.987558i $$-0.449735\pi$$
0.157256 + 0.987558i $$0.449735\pi$$
$$648$$ −3.00000 −0.117851
$$649$$ −4.00000 −0.157014
$$650$$ 2.00000 0.0784465
$$651$$ 32.0000 1.25418
$$652$$ −4.00000 −0.156652
$$653$$ −2.00000 −0.0782660 −0.0391330 0.999234i $$-0.512460\pi$$
−0.0391330 + 0.999234i $$0.512460\pi$$
$$654$$ 2.00000 0.0782062
$$655$$ 24.0000 0.937758
$$656$$ 2.00000 0.0780869
$$657$$ −14.0000 −0.546192
$$658$$ 32.0000 1.24749
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ −2.00000 −0.0778499
$$661$$ −26.0000 −1.01128 −0.505641 0.862744i $$-0.668744\pi$$
−0.505641 + 0.862744i $$0.668744\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ −4.00000 −0.155347
$$664$$ −36.0000 −1.39707
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ −48.0000 −1.85857
$$668$$ 0 0
$$669$$ −16.0000 −0.618596
$$670$$ 8.00000 0.309067
$$671$$ 6.00000 0.231627
$$672$$ −20.0000 −0.771517
$$673$$ −46.0000 −1.77317 −0.886585 0.462566i $$-0.846929\pi$$
−0.886585 + 0.462566i $$0.846929\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 1.00000 0.0384900
$$676$$ 9.00000 0.346154
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 8.00000 0.307012
$$680$$ −12.0000 −0.460179
$$681$$ −12.0000 −0.459841
$$682$$ −8.00000 −0.306336
$$683$$ 20.0000 0.765279 0.382639 0.923898i $$-0.375015\pi$$
0.382639 + 0.923898i $$0.375015\pi$$
$$684$$ 0 0
$$685$$ −4.00000 −0.152832
$$686$$ 8.00000 0.305441
$$687$$ −6.00000 −0.228914
$$688$$ 0 0
$$689$$ −12.0000 −0.457164
$$690$$ 16.0000 0.609110
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 4.00000 0.151947
$$694$$ 4.00000 0.151838
$$695$$ 16.0000 0.606915
$$696$$ −18.0000 −0.682288
$$697$$ 4.00000 0.151511
$$698$$ 6.00000 0.227103
$$699$$ −30.0000 −1.13470
$$700$$ 4.00000 0.151186
$$701$$ 50.0000 1.88847 0.944237 0.329267i $$-0.106802\pi$$
0.944237 + 0.329267i $$0.106802\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ 0 0
$$704$$ 7.00000 0.263822
$$705$$ 16.0000 0.602595
$$706$$ 18.0000 0.677439
$$707$$ 8.00000 0.300871
$$708$$ −4.00000 −0.150329
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 18.0000 0.674579
$$713$$ −64.0000 −2.39682
$$714$$ 8.00000 0.299392
$$715$$ 4.00000 0.149592
$$716$$ −12.0000 −0.448461
$$717$$ −24.0000 −0.896296
$$718$$ −8.00000 −0.298557
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ 32.0000 1.19174
$$722$$ −19.0000 −0.707107
$$723$$ −10.0000 −0.371904
$$724$$ −22.0000 −0.817624
$$725$$ 6.00000 0.222834
$$726$$ −1.00000 −0.0371135
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 24.0000 0.889499
$$729$$ 1.00000 0.0370370
$$730$$ 28.0000 1.03633
$$731$$ 0 0
$$732$$ 6.00000 0.221766
$$733$$ 30.0000 1.10808 0.554038 0.832492i $$-0.313086\pi$$
0.554038 + 0.832492i $$0.313086\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 18.0000 0.663940
$$736$$ 40.0000 1.47442
$$737$$ −4.00000 −0.147342
$$738$$ −2.00000 −0.0736210
$$739$$ 8.00000 0.294285 0.147142 0.989115i $$-0.452992\pi$$
0.147142 + 0.989115i $$0.452992\pi$$
$$740$$ 12.0000 0.441129
$$741$$ 0 0
$$742$$ 24.0000 0.881068
$$743$$ 40.0000 1.46746 0.733729 0.679442i $$-0.237778\pi$$
0.733729 + 0.679442i $$0.237778\pi$$
$$744$$ −24.0000 −0.879883
$$745$$ 44.0000 1.61204
$$746$$ −2.00000 −0.0732252
$$747$$ 12.0000 0.439057
$$748$$ 2.00000 0.0731272
$$749$$ −48.0000 −1.75388
$$750$$ −12.0000 −0.438178
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ −4.00000 −0.145768
$$754$$ 12.0000 0.437014
$$755$$ −40.0000 −1.45575
$$756$$ 4.00000 0.145479
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ 28.0000 1.01701
$$759$$ −8.00000 −0.290382
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 4.00000 0.144905
$$763$$ −8.00000 −0.289619
$$764$$ −8.00000 −0.289430
$$765$$ 4.00000 0.144620
$$766$$ −16.0000 −0.578103
$$767$$ 8.00000 0.288863
$$768$$ 17.0000 0.613435
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ −8.00000 −0.288300
$$771$$ 14.0000 0.504198
$$772$$ 14.0000 0.503871
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ 0 0
$$775$$ 8.00000 0.287368
$$776$$ −6.00000 −0.215387
$$777$$ −24.0000 −0.860995
$$778$$ −18.0000 −0.645331
$$779$$ 0 0
$$780$$ 4.00000 0.143223
$$781$$ 0 0
$$782$$ −16.0000 −0.572159
$$783$$ 6.00000 0.214423
$$784$$ −9.00000 −0.321429
$$785$$ −28.0000 −0.999363
$$786$$ 12.0000 0.428026
$$787$$ −8.00000 −0.285169 −0.142585 0.989783i $$-0.545541\pi$$
−0.142585 + 0.989783i $$0.545541\pi$$
$$788$$ 14.0000 0.498729
$$789$$ 16.0000 0.569615
$$790$$ 8.00000 0.284627
$$791$$ −24.0000 −0.853342
$$792$$ −3.00000 −0.106600
$$793$$ −12.0000 −0.426132
$$794$$ −2.00000 −0.0709773
$$795$$ 12.0000 0.425596
$$796$$ 0 0
$$797$$ −10.0000 −0.354218 −0.177109 0.984191i $$-0.556675\pi$$
−0.177109 + 0.984191i $$0.556675\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ −5.00000 −0.176777
$$801$$ −6.00000 −0.212000
$$802$$ 26.0000 0.918092
$$803$$ −14.0000 −0.494049
$$804$$ −4.00000 −0.141069
$$805$$ −64.0000 −2.25570
$$806$$ 16.0000 0.563576
$$807$$ 2.00000 0.0704033
$$808$$ −6.00000 −0.211079
$$809$$ 54.0000 1.89854 0.949269 0.314464i $$-0.101825\pi$$
0.949269 + 0.314464i $$0.101825\pi$$
$$810$$ −2.00000 −0.0702728
$$811$$ −56.0000 −1.96643 −0.983213 0.182462i $$-0.941593\pi$$
−0.983213 + 0.182462i $$0.941593\pi$$
$$812$$ 24.0000 0.842235
$$813$$ −20.0000 −0.701431
$$814$$ 6.00000 0.210300
$$815$$ −8.00000 −0.280228
$$816$$ −2.00000 −0.0700140
$$817$$ 0 0
$$818$$ 18.0000 0.629355
$$819$$ −8.00000 −0.279543
$$820$$ −4.00000 −0.139686
$$821$$ −14.0000 −0.488603 −0.244302 0.969699i $$-0.578559\pi$$
−0.244302 + 0.969699i $$0.578559\pi$$
$$822$$ −2.00000 −0.0697580
$$823$$ 24.0000 0.836587 0.418294 0.908312i $$-0.362628\pi$$
0.418294 + 0.908312i $$0.362628\pi$$
$$824$$ −24.0000 −0.836080
$$825$$ 1.00000 0.0348155
$$826$$ −16.0000 −0.556711
$$827$$ 20.0000 0.695468 0.347734 0.937593i $$-0.386951\pi$$
0.347734 + 0.937593i $$0.386951\pi$$
$$828$$ −8.00000 −0.278019
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ −24.0000 −0.833052
$$831$$ 26.0000 0.901930
$$832$$ −14.0000 −0.485363
$$833$$ −18.0000 −0.623663
$$834$$ 8.00000 0.277017
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 8.00000 0.276520
$$838$$ −4.00000 −0.138178
$$839$$ −56.0000 −1.93333 −0.966667 0.256036i $$-0.917584\pi$$
−0.966667 + 0.256036i $$0.917584\pi$$
$$840$$ −24.0000 −0.828079
$$841$$ 7.00000 0.241379
$$842$$ −26.0000 −0.896019
$$843$$ 18.0000 0.619953
$$844$$ 0 0
$$845$$ 18.0000 0.619219
$$846$$ 8.00000 0.275046
$$847$$ 4.00000 0.137442
$$848$$ −6.00000 −0.206041
$$849$$ −16.0000 −0.549119
$$850$$ 2.00000 0.0685994
$$851$$ 48.0000 1.64542
$$852$$ 0 0
$$853$$ −34.0000 −1.16414 −0.582069 0.813139i $$-0.697757\pi$$
−0.582069 + 0.813139i $$0.697757\pi$$
$$854$$ 24.0000 0.821263
$$855$$ 0 0
$$856$$ 36.0000 1.23045
$$857$$ −10.0000 −0.341593 −0.170797 0.985306i $$-0.554634\pi$$
−0.170797 + 0.985306i $$0.554634\pi$$
$$858$$ 2.00000 0.0682789
$$859$$ −36.0000 −1.22830 −0.614152 0.789188i $$-0.710502\pi$$
−0.614152 + 0.789188i $$0.710502\pi$$
$$860$$ 0 0
$$861$$ 8.00000 0.272639
$$862$$ −24.0000 −0.817443
$$863$$ 48.0000 1.63394 0.816970 0.576681i $$-0.195652\pi$$
0.816970 + 0.576681i $$0.195652\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 12.0000 0.408012
$$866$$ 34.0000 1.15537
$$867$$ 13.0000 0.441503
$$868$$ 32.0000 1.08615
$$869$$ −4.00000 −0.135691
$$870$$ −12.0000 −0.406838
$$871$$ 8.00000 0.271070
$$872$$ 6.00000 0.203186
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ 48.0000 1.62270
$$876$$ −14.0000 −0.473016
$$877$$ 6.00000 0.202606 0.101303 0.994856i $$-0.467699\pi$$
0.101303 + 0.994856i $$0.467699\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ 6.00000 0.202375
$$880$$ 2.00000 0.0674200
$$881$$ 26.0000 0.875962 0.437981 0.898984i $$-0.355694\pi$$
0.437981 + 0.898984i $$0.355694\pi$$
$$882$$ 9.00000 0.303046
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ −8.00000 −0.268917
$$886$$ 28.0000 0.940678
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ 18.0000 0.604040
$$889$$ −16.0000 −0.536623
$$890$$ 12.0000 0.402241
$$891$$ 1.00000 0.0335013
$$892$$ −16.0000 −0.535720
$$893$$ 0 0
$$894$$ 22.0000 0.735790
$$895$$ −24.0000 −0.802232
$$896$$ −12.0000 −0.400892
$$897$$ 16.0000 0.534224
$$898$$ 2.00000 0.0667409
$$899$$ 48.0000 1.60089
$$900$$ 1.00000 0.0333333
$$901$$ −12.0000 −0.399778
$$902$$ −2.00000 −0.0665927
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ −44.0000 −1.46261
$$906$$ −20.0000 −0.664455
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 2.00000 0.0663358
$$910$$ 16.0000 0.530395
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 18.0000 0.595387
$$915$$ 12.0000 0.396708
$$916$$ −6.00000 −0.198246
$$917$$ −48.0000 −1.58510
$$918$$ 2.00000 0.0660098
$$919$$ −20.0000 −0.659739 −0.329870 0.944027i $$-0.607005\pi$$
−0.329870 + 0.944027i $$0.607005\pi$$
$$920$$ 48.0000 1.58251
$$921$$ −32.0000 −1.05444
$$922$$ −30.0000 −0.987997
$$923$$ 0 0
$$924$$ 4.00000 0.131590
$$925$$ −6.00000 −0.197279
$$926$$ 16.0000 0.525793
$$927$$ 8.00000 0.262754
$$928$$ −30.0000 −0.984798
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ −16.0000 −0.524661
$$931$$ 0 0
$$932$$ −30.0000 −0.982683
$$933$$ 24.0000 0.785725
$$934$$ −12.0000 −0.392652
$$935$$ 4.00000 0.130814
$$936$$ 6.00000 0.196116
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ −16.0000 −0.522419
$$939$$ 22.0000 0.717943
$$940$$ 16.0000 0.521862
$$941$$ −54.0000 −1.76035 −0.880175 0.474650i $$-0.842575\pi$$
−0.880175 + 0.474650i $$0.842575\pi$$
$$942$$ −14.0000 −0.456145
$$943$$ −16.0000 −0.521032
$$944$$ 4.00000 0.130189
$$945$$ 8.00000 0.260240
$$946$$ 0 0
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ −4.00000 −0.129914
$$949$$ 28.0000 0.908918
$$950$$ 0 0
$$951$$ −22.0000 −0.713399
$$952$$ 24.0000 0.777844
$$953$$ 22.0000 0.712650 0.356325 0.934362i $$-0.384030\pi$$
0.356325 + 0.934362i $$0.384030\pi$$
$$954$$ 6.00000 0.194257
$$955$$ −16.0000 −0.517748
$$956$$ −24.0000 −0.776215
$$957$$ 6.00000 0.193952
$$958$$ 8.00000 0.258468
$$959$$ 8.00000 0.258333
$$960$$ 14.0000 0.451848
$$961$$ 33.0000 1.06452
$$962$$ −12.0000 −0.386896
$$963$$ −12.0000 −0.386695
$$964$$ −10.0000 −0.322078
$$965$$ 28.0000 0.901352
$$966$$ −32.0000 −1.02958
$$967$$ 4.00000 0.128631 0.0643157 0.997930i $$-0.479514\pi$$
0.0643157 + 0.997930i $$0.479514\pi$$
$$968$$ −3.00000 −0.0964237
$$969$$ 0 0
$$970$$ −4.00000 −0.128432
$$971$$ −52.0000 −1.66876 −0.834380 0.551190i $$-0.814174\pi$$
−0.834380 + 0.551190i $$0.814174\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −32.0000 −1.02587
$$974$$ −16.0000 −0.512673
$$975$$ −2.00000 −0.0640513
$$976$$ −6.00000 −0.192055
$$977$$ −6.00000 −0.191957 −0.0959785 0.995383i $$-0.530598\pi$$
−0.0959785 + 0.995383i $$0.530598\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ −6.00000 −0.191761
$$980$$ 18.0000 0.574989
$$981$$ −2.00000 −0.0638551
$$982$$ 4.00000 0.127645
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 28.0000 0.892154
$$986$$ 12.0000 0.382158
$$987$$ −32.0000 −1.01857
$$988$$ 0 0
$$989$$ 0 0
$$990$$ −2.00000 −0.0635642
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ −40.0000 −1.27000
$$993$$ 20.0000 0.634681
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ −6.00000 −0.189832
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 33.2.a.a.1.1 1
3.2 odd 2 99.2.a.b.1.1 1
4.3 odd 2 528.2.a.g.1.1 1
5.2 odd 4 825.2.c.a.199.2 2
5.3 odd 4 825.2.c.a.199.1 2
5.4 even 2 825.2.a.a.1.1 1
7.6 odd 2 1617.2.a.j.1.1 1
8.3 odd 2 2112.2.a.j.1.1 1
8.5 even 2 2112.2.a.bb.1.1 1
9.2 odd 6 891.2.e.g.595.1 2
9.4 even 3 891.2.e.e.298.1 2
9.5 odd 6 891.2.e.g.298.1 2
9.7 even 3 891.2.e.e.595.1 2
11.2 odd 10 363.2.e.g.202.1 4
11.3 even 5 363.2.e.e.130.1 4
11.4 even 5 363.2.e.e.148.1 4
11.5 even 5 363.2.e.e.124.1 4
11.6 odd 10 363.2.e.g.124.1 4
11.7 odd 10 363.2.e.g.148.1 4
11.8 odd 10 363.2.e.g.130.1 4
11.9 even 5 363.2.e.e.202.1 4
11.10 odd 2 363.2.a.b.1.1 1
12.11 even 2 1584.2.a.o.1.1 1
13.12 even 2 5577.2.a.a.1.1 1
15.2 even 4 2475.2.c.d.199.1 2
15.8 even 4 2475.2.c.d.199.2 2
15.14 odd 2 2475.2.a.g.1.1 1
17.16 even 2 9537.2.a.m.1.1 1
21.20 even 2 4851.2.a.b.1.1 1
24.5 odd 2 6336.2.a.x.1.1 1
24.11 even 2 6336.2.a.n.1.1 1
33.32 even 2 1089.2.a.j.1.1 1
44.43 even 2 5808.2.a.t.1.1 1
55.54 odd 2 9075.2.a.q.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.a.a.1.1 1 1.1 even 1 trivial
99.2.a.b.1.1 1 3.2 odd 2
363.2.a.b.1.1 1 11.10 odd 2
363.2.e.e.124.1 4 11.5 even 5
363.2.e.e.130.1 4 11.3 even 5
363.2.e.e.148.1 4 11.4 even 5
363.2.e.e.202.1 4 11.9 even 5
363.2.e.g.124.1 4 11.6 odd 10
363.2.e.g.130.1 4 11.8 odd 10
363.2.e.g.148.1 4 11.7 odd 10
363.2.e.g.202.1 4 11.2 odd 10
528.2.a.g.1.1 1 4.3 odd 2
825.2.a.a.1.1 1 5.4 even 2
825.2.c.a.199.1 2 5.3 odd 4
825.2.c.a.199.2 2 5.2 odd 4
891.2.e.e.298.1 2 9.4 even 3
891.2.e.e.595.1 2 9.7 even 3
891.2.e.g.298.1 2 9.5 odd 6
891.2.e.g.595.1 2 9.2 odd 6
1089.2.a.j.1.1 1 33.32 even 2
1584.2.a.o.1.1 1 12.11 even 2
1617.2.a.j.1.1 1 7.6 odd 2
2112.2.a.j.1.1 1 8.3 odd 2
2112.2.a.bb.1.1 1 8.5 even 2
2475.2.a.g.1.1 1 15.14 odd 2
2475.2.c.d.199.1 2 15.2 even 4
2475.2.c.d.199.2 2 15.8 even 4
4851.2.a.b.1.1 1 21.20 even 2
5577.2.a.a.1.1 1 13.12 even 2
5808.2.a.t.1.1 1 44.43 even 2
6336.2.a.n.1.1 1 24.11 even 2
6336.2.a.x.1.1 1 24.5 odd 2
9075.2.a.q.1.1 1 55.54 odd 2
9537.2.a.m.1.1 1 17.16 even 2