# Properties

 Label 363.2.a.a.1.1 Level $363$ Weight $2$ Character 363.1 Self dual yes Analytic conductor $2.899$ 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] = [363,2,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, 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("363.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 363.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$2.89856959337$$ 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 $$\chi$$ $$=$$ 363.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +4.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +4.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} -8.00000 q^{10} -2.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} -4.00000 q^{15} -4.00000 q^{16} +4.00000 q^{17} -2.00000 q^{18} -3.00000 q^{19} +8.00000 q^{20} -1.00000 q^{21} +2.00000 q^{23} +11.0000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +6.00000 q^{29} +8.00000 q^{30} -5.00000 q^{31} +8.00000 q^{32} -8.00000 q^{34} +4.00000 q^{35} +2.00000 q^{36} +3.00000 q^{37} +6.00000 q^{38} +2.00000 q^{39} -2.00000 q^{41} +2.00000 q^{42} +12.0000 q^{43} +4.00000 q^{45} -4.00000 q^{46} +2.00000 q^{47} +4.00000 q^{48} -6.00000 q^{49} -22.0000 q^{50} -4.00000 q^{51} -4.00000 q^{52} +6.00000 q^{53} +2.00000 q^{54} +3.00000 q^{57} -12.0000 q^{58} -10.0000 q^{59} -8.00000 q^{60} +3.00000 q^{61} +10.0000 q^{62} +1.00000 q^{63} -8.00000 q^{64} -8.00000 q^{65} -1.00000 q^{67} +8.00000 q^{68} -2.00000 q^{69} -8.00000 q^{70} -11.0000 q^{73} -6.00000 q^{74} -11.0000 q^{75} -6.00000 q^{76} -4.00000 q^{78} +11.0000 q^{79} -16.0000 q^{80} +1.00000 q^{81} +4.00000 q^{82} +6.00000 q^{83} -2.00000 q^{84} +16.0000 q^{85} -24.0000 q^{86} -6.00000 q^{87} +12.0000 q^{89} -8.00000 q^{90} -2.00000 q^{91} +4.00000 q^{92} +5.00000 q^{93} -4.00000 q^{94} -12.0000 q^{95} -8.00000 q^{96} +5.00000 q^{97} +12.0000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 2.00000 1.00000
$$5$$ 4.00000 1.78885 0.894427 0.447214i $$-0.147584\pi$$
0.894427 + 0.447214i $$0.147584\pi$$
$$6$$ 2.00000 0.816497
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ −8.00000 −2.52982
$$11$$ 0 0
$$12$$ −2.00000 −0.577350
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ −4.00000 −1.03280
$$16$$ −4.00000 −1.00000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ −2.00000 −0.471405
$$19$$ −3.00000 −0.688247 −0.344124 0.938924i $$-0.611824\pi$$
−0.344124 + 0.938924i $$0.611824\pi$$
$$20$$ 8.00000 1.78885
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 2.00000 0.417029 0.208514 0.978019i $$-0.433137\pi$$
0.208514 + 0.978019i $$0.433137\pi$$
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ 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 $$0.311919\pi$$
$$30$$ 8.00000 1.46059
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 8.00000 1.41421
$$33$$ 0 0
$$34$$ −8.00000 −1.37199
$$35$$ 4.00000 0.676123
$$36$$ 2.00000 0.333333
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 2.00000 0.308607
$$43$$ 12.0000 1.82998 0.914991 0.403473i $$-0.132197\pi$$
0.914991 + 0.403473i $$0.132197\pi$$
$$44$$ 0 0
$$45$$ 4.00000 0.596285
$$46$$ −4.00000 −0.589768
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 4.00000 0.577350
$$49$$ −6.00000 −0.857143
$$50$$ −22.0000 −3.11127
$$51$$ −4.00000 −0.560112
$$52$$ −4.00000 −0.554700
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 2.00000 0.272166
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 3.00000 0.397360
$$58$$ −12.0000 −1.57568
$$59$$ −10.0000 −1.30189 −0.650945 0.759125i $$-0.725627\pi$$
−0.650945 + 0.759125i $$0.725627\pi$$
$$60$$ −8.00000 −1.03280
$$61$$ 3.00000 0.384111 0.192055 0.981384i $$-0.438485\pi$$
0.192055 + 0.981384i $$0.438485\pi$$
$$62$$ 10.0000 1.27000
$$63$$ 1.00000 0.125988
$$64$$ −8.00000 −1.00000
$$65$$ −8.00000 −0.992278
$$66$$ 0 0
$$67$$ −1.00000 −0.122169 −0.0610847 0.998133i $$-0.519456\pi$$
−0.0610847 + 0.998133i $$0.519456\pi$$
$$68$$ 8.00000 0.970143
$$69$$ −2.00000 −0.240772
$$70$$ −8.00000 −0.956183
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −11.0000 −1.28745 −0.643726 0.765256i $$-0.722612\pi$$
−0.643726 + 0.765256i $$0.722612\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ −11.0000 −1.27017
$$76$$ −6.00000 −0.688247
$$77$$ 0 0
$$78$$ −4.00000 −0.452911
$$79$$ 11.0000 1.23760 0.618798 0.785550i $$-0.287620\pi$$
0.618798 + 0.785550i $$0.287620\pi$$
$$80$$ −16.0000 −1.78885
$$81$$ 1.00000 0.111111
$$82$$ 4.00000 0.441726
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 16.0000 1.73544
$$86$$ −24.0000 −2.58799
$$87$$ −6.00000 −0.643268
$$88$$ 0 0
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ −8.00000 −0.843274
$$91$$ −2.00000 −0.209657
$$92$$ 4.00000 0.417029
$$93$$ 5.00000 0.518476
$$94$$ −4.00000 −0.412568
$$95$$ −12.0000 −1.23117
$$96$$ −8.00000 −0.816497
$$97$$ 5.00000 0.507673 0.253837 0.967247i $$-0.418307\pi$$
0.253837 + 0.967247i $$0.418307\pi$$
$$98$$ 12.0000 1.21218
$$99$$ 0 0
$$100$$ 22.0000 2.20000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 8.00000 0.792118
$$103$$ −7.00000 −0.689730 −0.344865 0.938652i $$-0.612075\pi$$
−0.344865 + 0.938652i $$0.612075\pi$$
$$104$$ 0 0
$$105$$ −4.00000 −0.390360
$$106$$ −12.0000 −1.16554
$$107$$ −18.0000 −1.74013 −0.870063 0.492941i $$-0.835922\pi$$
−0.870063 + 0.492941i $$0.835922\pi$$
$$108$$ −2.00000 −0.192450
$$109$$ 1.00000 0.0957826 0.0478913 0.998853i $$-0.484750\pi$$
0.0478913 + 0.998853i $$0.484750\pi$$
$$110$$ 0 0
$$111$$ −3.00000 −0.284747
$$112$$ −4.00000 −0.377964
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −6.00000 −0.561951
$$115$$ 8.00000 0.746004
$$116$$ 12.0000 1.11417
$$117$$ −2.00000 −0.184900
$$118$$ 20.0000 1.84115
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 0 0
$$122$$ −6.00000 −0.543214
$$123$$ 2.00000 0.180334
$$124$$ −10.0000 −0.898027
$$125$$ 24.0000 2.14663
$$126$$ −2.00000 −0.178174
$$127$$ −13.0000 −1.15356 −0.576782 0.816898i $$-0.695692\pi$$
−0.576782 + 0.816898i $$0.695692\pi$$
$$128$$ 0 0
$$129$$ −12.0000 −1.05654
$$130$$ 16.0000 1.40329
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 0 0
$$133$$ −3.00000 −0.260133
$$134$$ 2.00000 0.172774
$$135$$ −4.00000 −0.344265
$$136$$ 0 0
$$137$$ 8.00000 0.683486 0.341743 0.939793i $$-0.388983\pi$$
0.341743 + 0.939793i $$0.388983\pi$$
$$138$$ 4.00000 0.340503
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 8.00000 0.676123
$$141$$ −2.00000 −0.168430
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −4.00000 −0.333333
$$145$$ 24.0000 1.99309
$$146$$ 22.0000 1.82073
$$147$$ 6.00000 0.494872
$$148$$ 6.00000 0.493197
$$149$$ −16.0000 −1.31077 −0.655386 0.755295i $$-0.727494\pi$$
−0.655386 + 0.755295i $$0.727494\pi$$
$$150$$ 22.0000 1.79629
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ 4.00000 0.323381
$$154$$ 0 0
$$155$$ −20.0000 −1.60644
$$156$$ 4.00000 0.320256
$$157$$ −1.00000 −0.0798087 −0.0399043 0.999204i $$-0.512705\pi$$
−0.0399043 + 0.999204i $$0.512705\pi$$
$$158$$ −22.0000 −1.75023
$$159$$ −6.00000 −0.475831
$$160$$ 32.0000 2.52982
$$161$$ 2.00000 0.157622
$$162$$ −2.00000 −0.157135
$$163$$ 25.0000 1.95815 0.979076 0.203497i $$-0.0652307\pi$$
0.979076 + 0.203497i $$0.0652307\pi$$
$$164$$ −4.00000 −0.312348
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ −18.0000 −1.39288 −0.696441 0.717614i $$-0.745234\pi$$
−0.696441 + 0.717614i $$0.745234\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −32.0000 −2.45429
$$171$$ −3.00000 −0.229416
$$172$$ 24.0000 1.82998
$$173$$ 24.0000 1.82469 0.912343 0.409426i $$-0.134271\pi$$
0.912343 + 0.409426i $$0.134271\pi$$
$$174$$ 12.0000 0.909718
$$175$$ 11.0000 0.831522
$$176$$ 0 0
$$177$$ 10.0000 0.751646
$$178$$ −24.0000 −1.79888
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 8.00000 0.596285
$$181$$ −23.0000 −1.70958 −0.854788 0.518977i $$-0.826313\pi$$
−0.854788 + 0.518977i $$0.826313\pi$$
$$182$$ 4.00000 0.296500
$$183$$ −3.00000 −0.221766
$$184$$ 0 0
$$185$$ 12.0000 0.882258
$$186$$ −10.0000 −0.733236
$$187$$ 0 0
$$188$$ 4.00000 0.291730
$$189$$ −1.00000 −0.0727393
$$190$$ 24.0000 1.74114
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 8.00000 0.577350
$$193$$ −5.00000 −0.359908 −0.179954 0.983675i $$-0.557595\pi$$
−0.179954 + 0.983675i $$0.557595\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 8.00000 0.572892
$$196$$ −12.0000 −0.857143
$$197$$ −8.00000 −0.569976 −0.284988 0.958531i $$-0.591990\pi$$
−0.284988 + 0.958531i $$0.591990\pi$$
$$198$$ 0 0
$$199$$ −21.0000 −1.48865 −0.744325 0.667817i $$-0.767229\pi$$
−0.744325 + 0.667817i $$0.767229\pi$$
$$200$$ 0 0
$$201$$ 1.00000 0.0705346
$$202$$ 20.0000 1.40720
$$203$$ 6.00000 0.421117
$$204$$ −8.00000 −0.560112
$$205$$ −8.00000 −0.558744
$$206$$ 14.0000 0.975426
$$207$$ 2.00000 0.139010
$$208$$ 8.00000 0.554700
$$209$$ 0 0
$$210$$ 8.00000 0.552052
$$211$$ −21.0000 −1.44570 −0.722850 0.691005i $$-0.757168\pi$$
−0.722850 + 0.691005i $$0.757168\pi$$
$$212$$ 12.0000 0.824163
$$213$$ 0 0
$$214$$ 36.0000 2.46091
$$215$$ 48.0000 3.27357
$$216$$ 0 0
$$217$$ −5.00000 −0.339422
$$218$$ −2.00000 −0.135457
$$219$$ 11.0000 0.743311
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ 6.00000 0.402694
$$223$$ −17.0000 −1.13840 −0.569202 0.822198i $$-0.692748\pi$$
−0.569202 + 0.822198i $$0.692748\pi$$
$$224$$ 8.00000 0.534522
$$225$$ 11.0000 0.733333
$$226$$ −12.0000 −0.798228
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 6.00000 0.397360
$$229$$ −18.0000 −1.18947 −0.594737 0.803921i $$-0.702744\pi$$
−0.594737 + 0.803921i $$0.702744\pi$$
$$230$$ −16.0000 −1.05501
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 8.00000 0.521862
$$236$$ −20.0000 −1.30189
$$237$$ −11.0000 −0.714527
$$238$$ −8.00000 −0.518563
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 16.0000 1.03280
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 6.00000 0.384111
$$245$$ −24.0000 −1.53330
$$246$$ −4.00000 −0.255031
$$247$$ 6.00000 0.381771
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ −48.0000 −3.03579
$$251$$ −2.00000 −0.126239 −0.0631194 0.998006i $$-0.520105\pi$$
−0.0631194 + 0.998006i $$0.520105\pi$$
$$252$$ 2.00000 0.125988
$$253$$ 0 0
$$254$$ 26.0000 1.63139
$$255$$ −16.0000 −1.00196
$$256$$ 16.0000 1.00000
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 24.0000 1.49417
$$259$$ 3.00000 0.186411
$$260$$ −16.0000 −0.992278
$$261$$ 6.00000 0.371391
$$262$$ 12.0000 0.741362
$$263$$ −10.0000 −0.616626 −0.308313 0.951285i $$-0.599764\pi$$
−0.308313 + 0.951285i $$0.599764\pi$$
$$264$$ 0 0
$$265$$ 24.0000 1.47431
$$266$$ 6.00000 0.367884
$$267$$ −12.0000 −0.734388
$$268$$ −2.00000 −0.122169
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 8.00000 0.486864
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ −16.0000 −0.970143
$$273$$ 2.00000 0.121046
$$274$$ −16.0000 −0.966595
$$275$$ 0 0
$$276$$ −4.00000 −0.240772
$$277$$ −11.0000 −0.660926 −0.330463 0.943819i $$-0.607205\pi$$
−0.330463 + 0.943819i $$0.607205\pi$$
$$278$$ −32.0000 −1.91923
$$279$$ −5.00000 −0.299342
$$280$$ 0 0
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ 4.00000 0.238197
$$283$$ −11.0000 −0.653882 −0.326941 0.945045i $$-0.606018\pi$$
−0.326941 + 0.945045i $$0.606018\pi$$
$$284$$ 0 0
$$285$$ 12.0000 0.710819
$$286$$ 0 0
$$287$$ −2.00000 −0.118056
$$288$$ 8.00000 0.471405
$$289$$ −1.00000 −0.0588235
$$290$$ −48.0000 −2.81866
$$291$$ −5.00000 −0.293105
$$292$$ −22.0000 −1.28745
$$293$$ 12.0000 0.701047 0.350524 0.936554i $$-0.386004\pi$$
0.350524 + 0.936554i $$0.386004\pi$$
$$294$$ −12.0000 −0.699854
$$295$$ −40.0000 −2.32889
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 32.0000 1.85371
$$299$$ −4.00000 −0.231326
$$300$$ −22.0000 −1.27017
$$301$$ 12.0000 0.691669
$$302$$ 32.0000 1.84139
$$303$$ 10.0000 0.574485
$$304$$ 12.0000 0.688247
$$305$$ 12.0000 0.687118
$$306$$ −8.00000 −0.457330
$$307$$ −19.0000 −1.08439 −0.542194 0.840254i $$-0.682406\pi$$
−0.542194 + 0.840254i $$0.682406\pi$$
$$308$$ 0 0
$$309$$ 7.00000 0.398216
$$310$$ 40.0000 2.27185
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 4.00000 0.225374
$$316$$ 22.0000 1.23760
$$317$$ −20.0000 −1.12331 −0.561656 0.827371i $$-0.689836\pi$$
−0.561656 + 0.827371i $$0.689836\pi$$
$$318$$ 12.0000 0.672927
$$319$$ 0 0
$$320$$ −32.0000 −1.78885
$$321$$ 18.0000 1.00466
$$322$$ −4.00000 −0.222911
$$323$$ −12.0000 −0.667698
$$324$$ 2.00000 0.111111
$$325$$ −22.0000 −1.22034
$$326$$ −50.0000 −2.76924
$$327$$ −1.00000 −0.0553001
$$328$$ 0 0
$$329$$ 2.00000 0.110264
$$330$$ 0 0
$$331$$ −11.0000 −0.604615 −0.302307 0.953211i $$-0.597757\pi$$
−0.302307 + 0.953211i $$0.597757\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 3.00000 0.164399
$$334$$ 36.0000 1.96983
$$335$$ −4.00000 −0.218543
$$336$$ 4.00000 0.218218
$$337$$ 5.00000 0.272367 0.136184 0.990684i $$-0.456516\pi$$
0.136184 + 0.990684i $$0.456516\pi$$
$$338$$ 18.0000 0.979071
$$339$$ −6.00000 −0.325875
$$340$$ 32.0000 1.73544
$$341$$ 0 0
$$342$$ 6.00000 0.324443
$$343$$ −13.0000 −0.701934
$$344$$ 0 0
$$345$$ −8.00000 −0.430706
$$346$$ −48.0000 −2.58050
$$347$$ −2.00000 −0.107366 −0.0536828 0.998558i $$-0.517096\pi$$
−0.0536828 + 0.998558i $$0.517096\pi$$
$$348$$ −12.0000 −0.643268
$$349$$ 15.0000 0.802932 0.401466 0.915874i $$-0.368501\pi$$
0.401466 + 0.915874i $$0.368501\pi$$
$$350$$ −22.0000 −1.17595
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ −12.0000 −0.638696 −0.319348 0.947638i $$-0.603464\pi$$
−0.319348 + 0.947638i $$0.603464\pi$$
$$354$$ −20.0000 −1.06299
$$355$$ 0 0
$$356$$ 24.0000 1.27200
$$357$$ −4.00000 −0.211702
$$358$$ −12.0000 −0.634220
$$359$$ 4.00000 0.211112 0.105556 0.994413i $$-0.466338\pi$$
0.105556 + 0.994413i $$0.466338\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ 46.0000 2.41771
$$363$$ 0 0
$$364$$ −4.00000 −0.209657
$$365$$ −44.0000 −2.30307
$$366$$ 6.00000 0.313625
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ −8.00000 −0.417029
$$369$$ −2.00000 −0.104116
$$370$$ −24.0000 −1.24770
$$371$$ 6.00000 0.311504
$$372$$ 10.0000 0.518476
$$373$$ 7.00000 0.362446 0.181223 0.983442i $$-0.441994\pi$$
0.181223 + 0.983442i $$0.441994\pi$$
$$374$$ 0 0
$$375$$ −24.0000 −1.23935
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 2.00000 0.102869
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ −24.0000 −1.23117
$$381$$ 13.0000 0.666010
$$382$$ −16.0000 −0.818631
$$383$$ 26.0000 1.32854 0.664269 0.747494i $$-0.268743\pi$$
0.664269 + 0.747494i $$0.268743\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ 12.0000 0.609994
$$388$$ 10.0000 0.507673
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ −16.0000 −0.810191
$$391$$ 8.00000 0.404577
$$392$$ 0 0
$$393$$ 6.00000 0.302660
$$394$$ 16.0000 0.806068
$$395$$ 44.0000 2.21388
$$396$$ 0 0
$$397$$ 31.0000 1.55585 0.777923 0.628360i $$-0.216273\pi$$
0.777923 + 0.628360i $$0.216273\pi$$
$$398$$ 42.0000 2.10527
$$399$$ 3.00000 0.150188
$$400$$ −44.0000 −2.20000
$$401$$ −28.0000 −1.39825 −0.699127 0.714998i $$-0.746428\pi$$
−0.699127 + 0.714998i $$0.746428\pi$$
$$402$$ −2.00000 −0.0997509
$$403$$ 10.0000 0.498135
$$404$$ −20.0000 −0.995037
$$405$$ 4.00000 0.198762
$$406$$ −12.0000 −0.595550
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 21.0000 1.03838 0.519192 0.854658i $$-0.326233\pi$$
0.519192 + 0.854658i $$0.326233\pi$$
$$410$$ 16.0000 0.790184
$$411$$ −8.00000 −0.394611
$$412$$ −14.0000 −0.689730
$$413$$ −10.0000 −0.492068
$$414$$ −4.00000 −0.196589
$$415$$ 24.0000 1.17811
$$416$$ −16.0000 −0.784465
$$417$$ −16.0000 −0.783523
$$418$$ 0 0
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ −8.00000 −0.390360
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 42.0000 2.04453
$$423$$ 2.00000 0.0972433
$$424$$ 0 0
$$425$$ 44.0000 2.13431
$$426$$ 0 0
$$427$$ 3.00000 0.145180
$$428$$ −36.0000 −1.74013
$$429$$ 0 0
$$430$$ −96.0000 −4.62953
$$431$$ 18.0000 0.867029 0.433515 0.901146i $$-0.357273\pi$$
0.433515 + 0.901146i $$0.357273\pi$$
$$432$$ 4.00000 0.192450
$$433$$ −17.0000 −0.816968 −0.408484 0.912766i $$-0.633942\pi$$
−0.408484 + 0.912766i $$0.633942\pi$$
$$434$$ 10.0000 0.480015
$$435$$ −24.0000 −1.15071
$$436$$ 2.00000 0.0957826
$$437$$ −6.00000 −0.287019
$$438$$ −22.0000 −1.05120
$$439$$ 37.0000 1.76591 0.882957 0.469454i $$-0.155549\pi$$
0.882957 + 0.469454i $$0.155549\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 16.0000 0.761042
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ 48.0000 2.27542
$$446$$ 34.0000 1.60995
$$447$$ 16.0000 0.756774
$$448$$ −8.00000 −0.377964
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ −22.0000 −1.03709
$$451$$ 0 0
$$452$$ 12.0000 0.564433
$$453$$ 16.0000 0.751746
$$454$$ 0 0
$$455$$ −8.00000 −0.375046
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ 36.0000 1.68217
$$459$$ −4.00000 −0.186704
$$460$$ 16.0000 0.746004
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ −24.0000 −1.11417
$$465$$ 20.0000 0.927478
$$466$$ −36.0000 −1.66767
$$467$$ 24.0000 1.11059 0.555294 0.831654i $$-0.312606\pi$$
0.555294 + 0.831654i $$0.312606\pi$$
$$468$$ −4.00000 −0.184900
$$469$$ −1.00000 −0.0461757
$$470$$ −16.0000 −0.738025
$$471$$ 1.00000 0.0460776
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 22.0000 1.01049
$$475$$ −33.0000 −1.51414
$$476$$ 8.00000 0.366679
$$477$$ 6.00000 0.274721
$$478$$ −12.0000 −0.548867
$$479$$ −22.0000 −1.00521 −0.502603 0.864517i $$-0.667624\pi$$
−0.502603 + 0.864517i $$0.667624\pi$$
$$480$$ −32.0000 −1.46059
$$481$$ −6.00000 −0.273576
$$482$$ 28.0000 1.27537
$$483$$ −2.00000 −0.0910032
$$484$$ 0 0
$$485$$ 20.0000 0.908153
$$486$$ 2.00000 0.0907218
$$487$$ −40.0000 −1.81257 −0.906287 0.422664i $$-0.861095\pi$$
−0.906287 + 0.422664i $$0.861095\pi$$
$$488$$ 0 0
$$489$$ −25.0000 −1.13054
$$490$$ 48.0000 2.16842
$$491$$ −14.0000 −0.631811 −0.315906 0.948791i $$-0.602308\pi$$
−0.315906 + 0.948791i $$0.602308\pi$$
$$492$$ 4.00000 0.180334
$$493$$ 24.0000 1.08091
$$494$$ −12.0000 −0.539906
$$495$$ 0 0
$$496$$ 20.0000 0.898027
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ 23.0000 1.02962 0.514811 0.857304i $$-0.327862\pi$$
0.514811 + 0.857304i $$0.327862\pi$$
$$500$$ 48.0000 2.14663
$$501$$ 18.0000 0.804181
$$502$$ 4.00000 0.178529
$$503$$ −32.0000 −1.42681 −0.713405 0.700752i $$-0.752848\pi$$
−0.713405 + 0.700752i $$0.752848\pi$$
$$504$$ 0 0
$$505$$ −40.0000 −1.77998
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ −26.0000 −1.15356
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 32.0000 1.41698
$$511$$ −11.0000 −0.486611
$$512$$ −32.0000 −1.41421
$$513$$ 3.00000 0.132453
$$514$$ 28.0000 1.23503
$$515$$ −28.0000 −1.23383
$$516$$ −24.0000 −1.05654
$$517$$ 0 0
$$518$$ −6.00000 −0.263625
$$519$$ −24.0000 −1.05348
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ −12.0000 −0.525226
$$523$$ 29.0000 1.26808 0.634041 0.773300i $$-0.281395\pi$$
0.634041 + 0.773300i $$0.281395\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ −11.0000 −0.480079
$$526$$ 20.0000 0.872041
$$527$$ −20.0000 −0.871214
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ −48.0000 −2.08499
$$531$$ −10.0000 −0.433963
$$532$$ −6.00000 −0.260133
$$533$$ 4.00000 0.173259
$$534$$ 24.0000 1.03858
$$535$$ −72.0000 −3.11283
$$536$$ 0 0
$$537$$ −6.00000 −0.258919
$$538$$ 28.0000 1.20717
$$539$$ 0 0
$$540$$ −8.00000 −0.344265
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 23.0000 0.987024
$$544$$ 32.0000 1.37199
$$545$$ 4.00000 0.171341
$$546$$ −4.00000 −0.171184
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 16.0000 0.683486
$$549$$ 3.00000 0.128037
$$550$$ 0 0
$$551$$ −18.0000 −0.766826
$$552$$ 0 0
$$553$$ 11.0000 0.467768
$$554$$ 22.0000 0.934690
$$555$$ −12.0000 −0.509372
$$556$$ 32.0000 1.35710
$$557$$ −8.00000 −0.338971 −0.169485 0.985533i $$-0.554211\pi$$
−0.169485 + 0.985533i $$0.554211\pi$$
$$558$$ 10.0000 0.423334
$$559$$ −24.0000 −1.01509
$$560$$ −16.0000 −0.676123
$$561$$ 0 0
$$562$$ 24.0000 1.01238
$$563$$ 28.0000 1.18006 0.590030 0.807382i $$-0.299116\pi$$
0.590030 + 0.807382i $$0.299116\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 24.0000 1.00969
$$566$$ 22.0000 0.924729
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ 12.0000 0.503066 0.251533 0.967849i $$-0.419065\pi$$
0.251533 + 0.967849i $$0.419065\pi$$
$$570$$ −24.0000 −1.00525
$$571$$ −25.0000 −1.04622 −0.523109 0.852266i $$-0.675228\pi$$
−0.523109 + 0.852266i $$0.675228\pi$$
$$572$$ 0 0
$$573$$ −8.00000 −0.334205
$$574$$ 4.00000 0.166957
$$575$$ 22.0000 0.917463
$$576$$ −8.00000 −0.333333
$$577$$ 15.0000 0.624458 0.312229 0.950007i $$-0.398924\pi$$
0.312229 + 0.950007i $$0.398924\pi$$
$$578$$ 2.00000 0.0831890
$$579$$ 5.00000 0.207793
$$580$$ 48.0000 1.99309
$$581$$ 6.00000 0.248922
$$582$$ 10.0000 0.414513
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −8.00000 −0.330759
$$586$$ −24.0000 −0.991431
$$587$$ 4.00000 0.165098 0.0825488 0.996587i $$-0.473694\pi$$
0.0825488 + 0.996587i $$0.473694\pi$$
$$588$$ 12.0000 0.494872
$$589$$ 15.0000 0.618064
$$590$$ 80.0000 3.29355
$$591$$ 8.00000 0.329076
$$592$$ −12.0000 −0.493197
$$593$$ −46.0000 −1.88899 −0.944497 0.328521i $$-0.893450\pi$$
−0.944497 + 0.328521i $$0.893450\pi$$
$$594$$ 0 0
$$595$$ 16.0000 0.655936
$$596$$ −32.0000 −1.31077
$$597$$ 21.0000 0.859473
$$598$$ 8.00000 0.327144
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ 0 0
$$601$$ −1.00000 −0.0407909 −0.0203954 0.999792i $$-0.506493\pi$$
−0.0203954 + 0.999792i $$0.506493\pi$$
$$602$$ −24.0000 −0.978167
$$603$$ −1.00000 −0.0407231
$$604$$ −32.0000 −1.30206
$$605$$ 0 0
$$606$$ −20.0000 −0.812444
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ −24.0000 −0.973329
$$609$$ −6.00000 −0.243132
$$610$$ −24.0000 −0.971732
$$611$$ −4.00000 −0.161823
$$612$$ 8.00000 0.323381
$$613$$ −13.0000 −0.525065 −0.262533 0.964923i $$-0.584558\pi$$
−0.262533 + 0.964923i $$0.584558\pi$$
$$614$$ 38.0000 1.53356
$$615$$ 8.00000 0.322591
$$616$$ 0 0
$$617$$ −24.0000 −0.966204 −0.483102 0.875564i $$-0.660490\pi$$
−0.483102 + 0.875564i $$0.660490\pi$$
$$618$$ −14.0000 −0.563163
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ −40.0000 −1.60644
$$621$$ −2.00000 −0.0802572
$$622$$ −48.0000 −1.92462
$$623$$ 12.0000 0.480770
$$624$$ −8.00000 −0.320256
$$625$$ 41.0000 1.64000
$$626$$ 20.0000 0.799361
$$627$$ 0 0
$$628$$ −2.00000 −0.0798087
$$629$$ 12.0000 0.478471
$$630$$ −8.00000 −0.318728
$$631$$ −32.0000 −1.27390 −0.636950 0.770905i $$-0.719804\pi$$
−0.636950 + 0.770905i $$0.719804\pi$$
$$632$$ 0 0
$$633$$ 21.0000 0.834675
$$634$$ 40.0000 1.58860
$$635$$ −52.0000 −2.06356
$$636$$ −12.0000 −0.475831
$$637$$ 12.0000 0.475457
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ −36.0000 −1.42081
$$643$$ −37.0000 −1.45914 −0.729569 0.683907i $$-0.760279\pi$$
−0.729569 + 0.683907i $$0.760279\pi$$
$$644$$ 4.00000 0.157622
$$645$$ −48.0000 −1.89000
$$646$$ 24.0000 0.944267
$$647$$ −4.00000 −0.157256 −0.0786281 0.996904i $$-0.525054\pi$$
−0.0786281 + 0.996904i $$0.525054\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 44.0000 1.72582
$$651$$ 5.00000 0.195965
$$652$$ 50.0000 1.95815
$$653$$ 10.0000 0.391330 0.195665 0.980671i $$-0.437313\pi$$
0.195665 + 0.980671i $$0.437313\pi$$
$$654$$ 2.00000 0.0782062
$$655$$ −24.0000 −0.937758
$$656$$ 8.00000 0.312348
$$657$$ −11.0000 −0.429151
$$658$$ −4.00000 −0.155936
$$659$$ 46.0000 1.79191 0.895953 0.444149i $$-0.146494\pi$$
0.895953 + 0.444149i $$0.146494\pi$$
$$660$$ 0 0
$$661$$ −5.00000 −0.194477 −0.0972387 0.995261i $$-0.531001\pi$$
−0.0972387 + 0.995261i $$0.531001\pi$$
$$662$$ 22.0000 0.855054
$$663$$ 8.00000 0.310694
$$664$$ 0 0
$$665$$ −12.0000 −0.465340
$$666$$ −6.00000 −0.232495
$$667$$ 12.0000 0.464642
$$668$$ −36.0000 −1.39288
$$669$$ 17.0000 0.657258
$$670$$ 8.00000 0.309067
$$671$$ 0 0
$$672$$ −8.00000 −0.308607
$$673$$ −13.0000 −0.501113 −0.250557 0.968102i $$-0.580614\pi$$
−0.250557 + 0.968102i $$0.580614\pi$$
$$674$$ −10.0000 −0.385186
$$675$$ −11.0000 −0.423390
$$676$$ −18.0000 −0.692308
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ 12.0000 0.460857
$$679$$ 5.00000 0.191882
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −34.0000 −1.30097 −0.650487 0.759517i $$-0.725435\pi$$
−0.650487 + 0.759517i $$0.725435\pi$$
$$684$$ −6.00000 −0.229416
$$685$$ 32.0000 1.22266
$$686$$ 26.0000 0.992685
$$687$$ 18.0000 0.686743
$$688$$ −48.0000 −1.82998
$$689$$ −12.0000 −0.457164
$$690$$ 16.0000 0.609110
$$691$$ 11.0000 0.418460 0.209230 0.977866i $$-0.432904\pi$$
0.209230 + 0.977866i $$0.432904\pi$$
$$692$$ 48.0000 1.82469
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ 64.0000 2.42766
$$696$$ 0 0
$$697$$ −8.00000 −0.303022
$$698$$ −30.0000 −1.13552
$$699$$ −18.0000 −0.680823
$$700$$ 22.0000 0.831522
$$701$$ 50.0000 1.88847 0.944237 0.329267i $$-0.106802\pi$$
0.944237 + 0.329267i $$0.106802\pi$$
$$702$$ −4.00000 −0.150970
$$703$$ −9.00000 −0.339441
$$704$$ 0 0
$$705$$ −8.00000 −0.301297
$$706$$ 24.0000 0.903252
$$707$$ −10.0000 −0.376089
$$708$$ 20.0000 0.751646
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ 11.0000 0.412532
$$712$$ 0 0
$$713$$ −10.0000 −0.374503
$$714$$ 8.00000 0.299392
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ −6.00000 −0.224074
$$718$$ −8.00000 −0.298557
$$719$$ −6.00000 −0.223762 −0.111881 0.993722i $$-0.535688\pi$$
−0.111881 + 0.993722i $$0.535688\pi$$
$$720$$ −16.0000 −0.596285
$$721$$ −7.00000 −0.260694
$$722$$ 20.0000 0.744323
$$723$$ 14.0000 0.520666
$$724$$ −46.0000 −1.70958
$$725$$ 66.0000 2.45118
$$726$$ 0 0
$$727$$ −12.0000 −0.445055 −0.222528 0.974926i $$-0.571431\pi$$
−0.222528 + 0.974926i $$0.571431\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 88.0000 3.25703
$$731$$ 48.0000 1.77534
$$732$$ −6.00000 −0.221766
$$733$$ −30.0000 −1.10808 −0.554038 0.832492i $$-0.686914\pi$$
−0.554038 + 0.832492i $$0.686914\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 24.0000 0.885253
$$736$$ 16.0000 0.589768
$$737$$ 0 0
$$738$$ 4.00000 0.147242
$$739$$ 41.0000 1.50821 0.754105 0.656754i $$-0.228071\pi$$
0.754105 + 0.656754i $$0.228071\pi$$
$$740$$ 24.0000 0.882258
$$741$$ −6.00000 −0.220416
$$742$$ −12.0000 −0.440534
$$743$$ −20.0000 −0.733729 −0.366864 0.930274i $$-0.619569\pi$$
−0.366864 + 0.930274i $$0.619569\pi$$
$$744$$ 0 0
$$745$$ −64.0000 −2.34478
$$746$$ −14.0000 −0.512576
$$747$$ 6.00000 0.219529
$$748$$ 0 0
$$749$$ −18.0000 −0.657706
$$750$$ 48.0000 1.75271
$$751$$ 19.0000 0.693320 0.346660 0.937991i $$-0.387316\pi$$
0.346660 + 0.937991i $$0.387316\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 2.00000 0.0728841
$$754$$ 24.0000 0.874028
$$755$$ −64.0000 −2.32920
$$756$$ −2.00000 −0.0727393
$$757$$ 5.00000 0.181728 0.0908640 0.995863i $$-0.471037\pi$$
0.0908640 + 0.995863i $$0.471037\pi$$
$$758$$ −32.0000 −1.16229
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −24.0000 −0.869999 −0.435000 0.900431i $$-0.643252\pi$$
−0.435000 + 0.900431i $$0.643252\pi$$
$$762$$ −26.0000 −0.941881
$$763$$ 1.00000 0.0362024
$$764$$ 16.0000 0.578860
$$765$$ 16.0000 0.578481
$$766$$ −52.0000 −1.87884
$$767$$ 20.0000 0.722158
$$768$$ −16.0000 −0.577350
$$769$$ 11.0000 0.396670 0.198335 0.980134i $$-0.436447\pi$$
0.198335 + 0.980134i $$0.436447\pi$$
$$770$$ 0 0
$$771$$ 14.0000 0.504198
$$772$$ −10.0000 −0.359908
$$773$$ 36.0000 1.29483 0.647415 0.762138i $$-0.275850\pi$$
0.647415 + 0.762138i $$0.275850\pi$$
$$774$$ −24.0000 −0.862662
$$775$$ −55.0000 −1.97566
$$776$$ 0 0
$$777$$ −3.00000 −0.107624
$$778$$ −36.0000 −1.29066
$$779$$ 6.00000 0.214972
$$780$$ 16.0000 0.572892
$$781$$ 0 0
$$782$$ −16.0000 −0.572159
$$783$$ −6.00000 −0.214423
$$784$$ 24.0000 0.857143
$$785$$ −4.00000 −0.142766
$$786$$ −12.0000 −0.428026
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ −16.0000 −0.569976
$$789$$ 10.0000 0.356009
$$790$$ −88.0000 −3.13090
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ −6.00000 −0.213066
$$794$$ −62.0000 −2.20030
$$795$$ −24.0000 −0.851192
$$796$$ −42.0000 −1.48865
$$797$$ −10.0000 −0.354218 −0.177109 0.984191i $$-0.556675\pi$$
−0.177109 + 0.984191i $$0.556675\pi$$
$$798$$ −6.00000 −0.212398
$$799$$ 8.00000 0.283020
$$800$$ 88.0000 3.11127
$$801$$ 12.0000 0.423999
$$802$$ 56.0000 1.97743
$$803$$ 0 0
$$804$$ 2.00000 0.0705346
$$805$$ 8.00000 0.281963
$$806$$ −20.0000 −0.704470
$$807$$ 14.0000 0.492823
$$808$$ 0 0
$$809$$ −48.0000 −1.68759 −0.843795 0.536666i $$-0.819684\pi$$
−0.843795 + 0.536666i $$0.819684\pi$$
$$810$$ −8.00000 −0.281091
$$811$$ −17.0000 −0.596951 −0.298475 0.954417i $$-0.596478\pi$$
−0.298475 + 0.954417i $$0.596478\pi$$
$$812$$ 12.0000 0.421117
$$813$$ −8.00000 −0.280572
$$814$$ 0 0
$$815$$ 100.000 3.50285
$$816$$ 16.0000 0.560112
$$817$$ −36.0000 −1.25948
$$818$$ −42.0000 −1.46850
$$819$$ −2.00000 −0.0698857
$$820$$ −16.0000 −0.558744
$$821$$ −38.0000 −1.32621 −0.663105 0.748527i $$-0.730762\pi$$
−0.663105 + 0.748527i $$0.730762\pi$$
$$822$$ 16.0000 0.558064
$$823$$ −27.0000 −0.941161 −0.470580 0.882357i $$-0.655955\pi$$
−0.470580 + 0.882357i $$0.655955\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 20.0000 0.695889
$$827$$ −10.0000 −0.347734 −0.173867 0.984769i $$-0.555626\pi$$
−0.173867 + 0.984769i $$0.555626\pi$$
$$828$$ 4.00000 0.139010
$$829$$ −11.0000 −0.382046 −0.191023 0.981586i $$-0.561180\pi$$
−0.191023 + 0.981586i $$0.561180\pi$$
$$830$$ −48.0000 −1.66610
$$831$$ 11.0000 0.381586
$$832$$ 16.0000 0.554700
$$833$$ −24.0000 −0.831551
$$834$$ 32.0000 1.10807
$$835$$ −72.0000 −2.49166
$$836$$ 0 0
$$837$$ 5.00000 0.172825
$$838$$ −52.0000 −1.79631
$$839$$ 16.0000 0.552381 0.276191 0.961103i $$-0.410928\pi$$
0.276191 + 0.961103i $$0.410928\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 4.00000 0.137849
$$843$$ 12.0000 0.413302
$$844$$ −42.0000 −1.44570
$$845$$ −36.0000 −1.23844
$$846$$ −4.00000 −0.137523
$$847$$ 0 0
$$848$$ −24.0000 −0.824163
$$849$$ 11.0000 0.377519
$$850$$ −88.0000 −3.01838
$$851$$ 6.00000 0.205677
$$852$$ 0 0
$$853$$ 11.0000 0.376633 0.188316 0.982108i $$-0.439697\pi$$
0.188316 + 0.982108i $$0.439697\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ −12.0000 −0.410391
$$856$$ 0 0
$$857$$ −4.00000 −0.136637 −0.0683187 0.997664i $$-0.521763\pi$$
−0.0683187 + 0.997664i $$0.521763\pi$$
$$858$$ 0 0
$$859$$ −45.0000 −1.53538 −0.767690 0.640821i $$-0.778594\pi$$
−0.767690 + 0.640821i $$0.778594\pi$$
$$860$$ 96.0000 3.27357
$$861$$ 2.00000 0.0681598
$$862$$ −36.0000 −1.22616
$$863$$ 30.0000 1.02121 0.510606 0.859815i $$-0.329421\pi$$
0.510606 + 0.859815i $$0.329421\pi$$
$$864$$ −8.00000 −0.272166
$$865$$ 96.0000 3.26410
$$866$$ 34.0000 1.15537
$$867$$ 1.00000 0.0339618
$$868$$ −10.0000 −0.339422
$$869$$ 0 0
$$870$$ 48.0000 1.62735
$$871$$ 2.00000 0.0677674
$$872$$ 0 0
$$873$$ 5.00000 0.169224
$$874$$ 12.0000 0.405906
$$875$$ 24.0000 0.811348
$$876$$ 22.0000 0.743311
$$877$$ −45.0000 −1.51954 −0.759771 0.650191i $$-0.774689\pi$$
−0.759771 + 0.650191i $$0.774689\pi$$
$$878$$ −74.0000 −2.49738
$$879$$ −12.0000 −0.404750
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 12.0000 0.404061
$$883$$ 49.0000 1.64898 0.824491 0.565876i $$-0.191462\pi$$
0.824491 + 0.565876i $$0.191462\pi$$
$$884$$ −16.0000 −0.538138
$$885$$ 40.0000 1.34459
$$886$$ −8.00000 −0.268765
$$887$$ −22.0000 −0.738688 −0.369344 0.929293i $$-0.620418\pi$$
−0.369344 + 0.929293i $$0.620418\pi$$
$$888$$ 0 0
$$889$$ −13.0000 −0.436006
$$890$$ −96.0000 −3.21793
$$891$$ 0 0
$$892$$ −34.0000 −1.13840
$$893$$ −6.00000 −0.200782
$$894$$ −32.0000 −1.07024
$$895$$ 24.0000 0.802232
$$896$$ 0 0
$$897$$ 4.00000 0.133556
$$898$$ −40.0000 −1.33482
$$899$$ −30.0000 −1.00056
$$900$$ 22.0000 0.733333
$$901$$ 24.0000 0.799556
$$902$$ 0 0
$$903$$ −12.0000 −0.399335
$$904$$ 0 0
$$905$$ −92.0000 −3.05818
$$906$$ −32.0000 −1.06313
$$907$$ 33.0000 1.09575 0.547874 0.836561i $$-0.315438\pi$$
0.547874 + 0.836561i $$0.315438\pi$$
$$908$$ 0 0
$$909$$ −10.0000 −0.331679
$$910$$ 16.0000 0.530395
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ −12.0000 −0.397360
$$913$$ 0 0
$$914$$ −36.0000 −1.19077
$$915$$ −12.0000 −0.396708
$$916$$ −36.0000 −1.18947
$$917$$ −6.00000 −0.198137
$$918$$ 8.00000 0.264039
$$919$$ −5.00000 −0.164935 −0.0824674 0.996594i $$-0.526280\pi$$
−0.0824674 + 0.996594i $$0.526280\pi$$
$$920$$ 0 0
$$921$$ 19.0000 0.626071
$$922$$ 12.0000 0.395199
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 33.0000 1.08503
$$926$$ −32.0000 −1.05159
$$927$$ −7.00000 −0.229910
$$928$$ 48.0000 1.57568
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ −40.0000 −1.31165
$$931$$ 18.0000 0.589926
$$932$$ 36.0000 1.17922
$$933$$ −24.0000 −0.785725
$$934$$ −48.0000 −1.57061
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 23.0000 0.751377 0.375689 0.926746i $$-0.377406\pi$$
0.375689 + 0.926746i $$0.377406\pi$$
$$938$$ 2.00000 0.0653023
$$939$$ 10.0000 0.326338
$$940$$ 16.0000 0.521862
$$941$$ 42.0000 1.36916 0.684580 0.728937i $$-0.259985\pi$$
0.684580 + 0.728937i $$0.259985\pi$$
$$942$$ −2.00000 −0.0651635
$$943$$ −4.00000 −0.130258
$$944$$ 40.0000 1.30189
$$945$$ −4.00000 −0.130120
$$946$$ 0 0
$$947$$ 54.0000 1.75476 0.877382 0.479792i $$-0.159288\pi$$
0.877382 + 0.479792i $$0.159288\pi$$
$$948$$ −22.0000 −0.714527
$$949$$ 22.0000 0.714150
$$950$$ 66.0000 2.14132
$$951$$ 20.0000 0.648544
$$952$$ 0 0
$$953$$ 34.0000 1.10137 0.550684 0.834714i $$-0.314367\pi$$
0.550684 + 0.834714i $$0.314367\pi$$
$$954$$ −12.0000 −0.388514
$$955$$ 32.0000 1.03550
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ 44.0000 1.42158
$$959$$ 8.00000 0.258333
$$960$$ 32.0000 1.03280
$$961$$ −6.00000 −0.193548
$$962$$ 12.0000 0.386896
$$963$$ −18.0000 −0.580042
$$964$$ −28.0000 −0.901819
$$965$$ −20.0000 −0.643823
$$966$$ 4.00000 0.128698
$$967$$ 13.0000 0.418052 0.209026 0.977910i $$-0.432971\pi$$
0.209026 + 0.977910i $$0.432971\pi$$
$$968$$ 0 0
$$969$$ 12.0000 0.385496
$$970$$ −40.0000 −1.28432
$$971$$ 2.00000 0.0641831 0.0320915 0.999485i $$-0.489783\pi$$
0.0320915 + 0.999485i $$0.489783\pi$$
$$972$$ −2.00000 −0.0641500
$$973$$ 16.0000 0.512936
$$974$$ 80.0000 2.56337
$$975$$ 22.0000 0.704564
$$976$$ −12.0000 −0.384111
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 50.0000 1.59882
$$979$$ 0 0
$$980$$ −48.0000 −1.53330
$$981$$ 1.00000 0.0319275
$$982$$ 28.0000 0.893516
$$983$$ 6.00000 0.191370 0.0956851 0.995412i $$-0.469496\pi$$
0.0956851 + 0.995412i $$0.469496\pi$$
$$984$$ 0 0
$$985$$ −32.0000 −1.01960
$$986$$ −48.0000 −1.52863
$$987$$ −2.00000 −0.0636607
$$988$$ 12.0000 0.381771
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ −40.0000 −1.27000
$$993$$ 11.0000 0.349074
$$994$$ 0 0
$$995$$ −84.0000 −2.66298
$$996$$ −12.0000 −0.380235
$$997$$ −49.0000 −1.55185 −0.775923 0.630828i $$-0.782715\pi$$
−0.775923 + 0.630828i $$0.782715\pi$$
$$998$$ −46.0000 −1.45610
$$999$$ −3.00000 −0.0949158
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 363.2.a.a.1.1 1
3.2 odd 2 1089.2.a.k.1.1 1
4.3 odd 2 5808.2.a.bh.1.1 1
5.4 even 2 9075.2.a.t.1.1 1
11.2 odd 10 363.2.e.d.202.1 4
11.3 even 5 363.2.e.i.130.1 4
11.4 even 5 363.2.e.i.148.1 4
11.5 even 5 363.2.e.i.124.1 4
11.6 odd 10 363.2.e.d.124.1 4
11.7 odd 10 363.2.e.d.148.1 4
11.8 odd 10 363.2.e.d.130.1 4
11.9 even 5 363.2.e.i.202.1 4
11.10 odd 2 363.2.a.c.1.1 yes 1
33.32 even 2 1089.2.a.a.1.1 1
44.43 even 2 5808.2.a.bi.1.1 1
55.54 odd 2 9075.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
363.2.a.a.1.1 1 1.1 even 1 trivial
363.2.a.c.1.1 yes 1 11.10 odd 2
363.2.e.d.124.1 4 11.6 odd 10
363.2.e.d.130.1 4 11.8 odd 10
363.2.e.d.148.1 4 11.7 odd 10
363.2.e.d.202.1 4 11.2 odd 10
363.2.e.i.124.1 4 11.5 even 5
363.2.e.i.130.1 4 11.3 even 5
363.2.e.i.148.1 4 11.4 even 5
363.2.e.i.202.1 4 11.9 even 5
1089.2.a.a.1.1 1 33.32 even 2
1089.2.a.k.1.1 1 3.2 odd 2
5808.2.a.bh.1.1 1 4.3 odd 2
5808.2.a.bi.1.1 1 44.43 even 2
9075.2.a.b.1.1 1 55.54 odd 2
9075.2.a.t.1.1 1 5.4 even 2