# Properties

 Label 462.2.a.b.1.1 Level $462$ Weight $2$ Character 462.1 Self dual yes Analytic conductor $3.689$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [462,2,Mod(1,462)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("462.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.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: $$3.68908857338$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 462.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} +1.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -10.0000 q^{29} +6.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} -6.00000 q^{38} +2.00000 q^{39} -12.0000 q^{41} -1.00000 q^{42} -8.00000 q^{43} -1.00000 q^{44} +4.00000 q^{46} +2.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +5.00000 q^{50} +4.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +1.00000 q^{56} -6.00000 q^{57} +10.0000 q^{58} -8.00000 q^{59} +6.00000 q^{61} -6.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -4.00000 q^{67} -4.00000 q^{68} +4.00000 q^{69} -1.00000 q^{72} -12.0000 q^{73} +6.00000 q^{74} +5.00000 q^{75} +6.00000 q^{76} +1.00000 q^{77} -2.00000 q^{78} +1.00000 q^{81} +12.0000 q^{82} +14.0000 q^{83} +1.00000 q^{84} +8.00000 q^{86} +10.0000 q^{87} +1.00000 q^{88} +10.0000 q^{89} +2.00000 q^{91} -4.00000 q^{92} -6.00000 q^{93} -2.00000 q^{94} +1.00000 q^{96} +10.0000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 1.00000 0.408248
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −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 $$0.589456\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 1.00000 0.213201
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −5.00000 −1.00000
$$26$$ 2.00000 0.392232
$$27$$ −1.00000 −0.192450
$$28$$ −1.00000 −0.188982
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 1.00000 0.174078
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ −6.00000 −0.973329
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ −12.0000 −1.87409 −0.937043 0.349215i $$-0.886448\pi$$
−0.937043 + 0.349215i $$0.886448\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ 5.00000 0.707107
$$51$$ 4.00000 0.560112
$$52$$ −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
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ −6.00000 −0.794719
$$58$$ 10.0000 1.31306
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ −6.00000 −0.762001
$$63$$ −1.00000 −0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −1.00000 −0.123091
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −12.0000 −1.40449 −0.702247 0.711934i $$-0.747820\pi$$
−0.702247 + 0.711934i $$0.747820\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 5.00000 0.577350
$$76$$ 6.00000 0.688247
$$77$$ 1.00000 0.113961
$$78$$ −2.00000 −0.226455
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 12.0000 1.32518
$$83$$ 14.0000 1.53670 0.768350 0.640030i $$-0.221078\pi$$
0.768350 + 0.640030i $$0.221078\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 10.0000 1.07211
$$88$$ 1.00000 0.106600
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ −4.00000 −0.417029
$$93$$ −6.00000 −0.622171
$$94$$ −2.00000 −0.206284
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ −1.00000 −0.100504
$$100$$ −5.00000 −0.500000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ −4.00000 −0.396059
$$103$$ 10.0000 0.985329 0.492665 0.870219i $$-0.336023\pi$$
0.492665 + 0.870219i $$0.336023\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 18.0000 1.72409 0.862044 0.506834i $$-0.169184\pi$$
0.862044 + 0.506834i $$0.169184\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ −1.00000 −0.0944911
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 6.00000 0.561951
$$115$$ 0 0
$$116$$ −10.0000 −0.928477
$$117$$ −2.00000 −0.184900
$$118$$ 8.00000 0.736460
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −6.00000 −0.543214
$$123$$ 12.0000 1.08200
$$124$$ 6.00000 0.538816
$$125$$ 0 0
$$126$$ 1.00000 0.0890871
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 10.0000 0.873704 0.436852 0.899533i $$-0.356093\pi$$
0.436852 + 0.899533i $$0.356093\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ −6.00000 −0.520266
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 4.00000 0.342997
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 0 0
$$141$$ −2.00000 −0.168430
$$142$$ 0 0
$$143$$ 2.00000 0.167248
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 12.0000 0.993127
$$147$$ −1.00000 −0.0824786
$$148$$ −6.00000 −0.493197
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ −5.00000 −0.408248
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ −6.00000 −0.486664
$$153$$ −4.00000 −0.323381
$$154$$ −1.00000 −0.0805823
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 12.0000 0.957704 0.478852 0.877896i $$-0.341053\pi$$
0.478852 + 0.877896i $$0.341053\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ −1.00000 −0.0785674
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ −12.0000 −0.937043
$$165$$ 0 0
$$166$$ −14.0000 −1.08661
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 6.00000 0.458831
$$172$$ −8.00000 −0.609994
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ −10.0000 −0.758098
$$175$$ 5.00000 0.377964
$$176$$ −1.00000 −0.0753778
$$177$$ 8.00000 0.601317
$$178$$ −10.0000 −0.749532
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ −8.00000 −0.594635 −0.297318 0.954779i $$-0.596092\pi$$
−0.297318 + 0.954779i $$0.596092\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ −6.00000 −0.443533
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ 6.00000 0.439941
$$187$$ 4.00000 0.292509
$$188$$ 2.00000 0.145865
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −22.0000 −1.58359 −0.791797 0.610784i $$-0.790854\pi$$
−0.791797 + 0.610784i $$0.790854\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ 5.00000 0.353553
$$201$$ 4.00000 0.282138
$$202$$ 6.00000 0.422159
$$203$$ 10.0000 0.701862
$$204$$ 4.00000 0.280056
$$205$$ 0 0
$$206$$ −10.0000 −0.696733
$$207$$ −4.00000 −0.278019
$$208$$ −2.00000 −0.138675
$$209$$ −6.00000 −0.415029
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ −8.00000 −0.546869
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ −6.00000 −0.407307
$$218$$ −18.0000 −1.21911
$$219$$ 12.0000 0.810885
$$220$$ 0 0
$$221$$ 8.00000 0.538138
$$222$$ −6.00000 −0.402694
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −5.00000 −0.333333
$$226$$ 2.00000 0.133038
$$227$$ −14.0000 −0.929213 −0.464606 0.885517i $$-0.653804\pi$$
−0.464606 + 0.885517i $$0.653804\pi$$
$$228$$ −6.00000 −0.397360
$$229$$ 24.0000 1.58596 0.792982 0.609245i $$-0.208527\pi$$
0.792982 + 0.609245i $$0.208527\pi$$
$$230$$ 0 0
$$231$$ −1.00000 −0.0657952
$$232$$ 10.0000 0.656532
$$233$$ −26.0000 −1.70332 −0.851658 0.524097i $$-0.824403\pi$$
−0.851658 + 0.524097i $$0.824403\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ −8.00000 −0.520756
$$237$$ 0 0
$$238$$ −4.00000 −0.259281
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 0 0
$$241$$ −28.0000 −1.80364 −0.901819 0.432113i $$-0.857768\pi$$
−0.901819 + 0.432113i $$0.857768\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ 6.00000 0.384111
$$245$$ 0 0
$$246$$ −12.0000 −0.765092
$$247$$ −12.0000 −0.763542
$$248$$ −6.00000 −0.381000
$$249$$ −14.0000 −0.887214
$$250$$ 0 0
$$251$$ 8.00000 0.504956 0.252478 0.967603i $$-0.418755\pi$$
0.252478 + 0.967603i $$0.418755\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ 4.00000 0.251478
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ −10.0000 −0.618984
$$262$$ −10.0000 −0.617802
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ −1.00000 −0.0615457
$$265$$ 0 0
$$266$$ 6.00000 0.367884
$$267$$ −10.0000 −0.611990
$$268$$ −4.00000 −0.244339
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ −2.00000 −0.121046
$$274$$ 10.0000 0.604122
$$275$$ 5.00000 0.301511
$$276$$ 4.00000 0.240772
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ 14.0000 0.839664
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 2.00000 0.119098
$$283$$ 2.00000 0.118888 0.0594438 0.998232i $$-0.481067\pi$$
0.0594438 + 0.998232i $$0.481067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ 12.0000 0.708338
$$288$$ −1.00000 −0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ −12.0000 −0.702247
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 1.00000 0.0580259
$$298$$ −10.0000 −0.579284
$$299$$ 8.00000 0.462652
$$300$$ 5.00000 0.288675
$$301$$ 8.00000 0.461112
$$302$$ 16.0000 0.920697
$$303$$ 6.00000 0.344691
$$304$$ 6.00000 0.344124
$$305$$ 0 0
$$306$$ 4.00000 0.228665
$$307$$ −14.0000 −0.799022 −0.399511 0.916728i $$-0.630820\pi$$
−0.399511 + 0.916728i $$0.630820\pi$$
$$308$$ 1.00000 0.0569803
$$309$$ −10.0000 −0.568880
$$310$$ 0 0
$$311$$ −10.0000 −0.567048 −0.283524 0.958965i $$-0.591504\pi$$
−0.283524 + 0.958965i $$0.591504\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ −12.0000 −0.677199
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 10.0000 0.559893
$$320$$ 0 0
$$321$$ −8.00000 −0.446516
$$322$$ −4.00000 −0.222911
$$323$$ −24.0000 −1.33540
$$324$$ 1.00000 0.0555556
$$325$$ 10.0000 0.554700
$$326$$ −12.0000 −0.664619
$$327$$ −18.0000 −0.995402
$$328$$ 12.0000 0.662589
$$329$$ −2.00000 −0.110264
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 14.0000 0.768350
$$333$$ −6.00000 −0.328798
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 1.00000 0.0545545
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ −6.00000 −0.324443
$$343$$ −1.00000 −0.0539949
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 16.0000 0.858925 0.429463 0.903085i $$-0.358703\pi$$
0.429463 + 0.903085i $$0.358703\pi$$
$$348$$ 10.0000 0.536056
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ −5.00000 −0.267261
$$351$$ 2.00000 0.106752
$$352$$ 1.00000 0.0533002
$$353$$ 10.0000 0.532246 0.266123 0.963939i $$-0.414257\pi$$
0.266123 + 0.963939i $$0.414257\pi$$
$$354$$ −8.00000 −0.425195
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ −4.00000 −0.211702
$$358$$ −20.0000 −1.05703
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 8.00000 0.420471
$$363$$ −1.00000 −0.0524864
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 6.00000 0.313625
$$367$$ −10.0000 −0.521996 −0.260998 0.965339i $$-0.584052\pi$$
−0.260998 + 0.965339i $$0.584052\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ −12.0000 −0.624695
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ −6.00000 −0.311086
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ −2.00000 −0.103142
$$377$$ 20.0000 1.03005
$$378$$ −1.00000 −0.0514344
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ 12.0000 0.613973
$$383$$ −10.0000 −0.510976 −0.255488 0.966812i $$-0.582236\pi$$
−0.255488 + 0.966812i $$0.582236\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 22.0000 1.11977
$$387$$ −8.00000 −0.406663
$$388$$ 10.0000 0.507673
$$389$$ 2.00000 0.101404 0.0507020 0.998714i $$-0.483854\pi$$
0.0507020 + 0.998714i $$0.483854\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ −1.00000 −0.0505076
$$393$$ −10.0000 −0.504433
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ −1.00000 −0.0502519
$$397$$ −4.00000 −0.200754 −0.100377 0.994949i $$-0.532005\pi$$
−0.100377 + 0.994949i $$0.532005\pi$$
$$398$$ −10.0000 −0.501255
$$399$$ 6.00000 0.300376
$$400$$ −5.00000 −0.250000
$$401$$ −34.0000 −1.69788 −0.848939 0.528490i $$-0.822758\pi$$
−0.848939 + 0.528490i $$0.822758\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ −12.0000 −0.597763
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ −10.0000 −0.496292
$$407$$ 6.00000 0.297409
$$408$$ −4.00000 −0.198030
$$409$$ 24.0000 1.18672 0.593362 0.804936i $$-0.297800\pi$$
0.593362 + 0.804936i $$0.297800\pi$$
$$410$$ 0 0
$$411$$ 10.0000 0.493264
$$412$$ 10.0000 0.492665
$$413$$ 8.00000 0.393654
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 14.0000 0.685583
$$418$$ 6.00000 0.293470
$$419$$ 28.0000 1.36789 0.683945 0.729534i $$-0.260263\pi$$
0.683945 + 0.729534i $$0.260263\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 8.00000 0.389434
$$423$$ 2.00000 0.0972433
$$424$$ −6.00000 −0.291386
$$425$$ 20.0000 0.970143
$$426$$ 0 0
$$427$$ −6.00000 −0.290360
$$428$$ 8.00000 0.386695
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −22.0000 −1.05725 −0.528626 0.848855i $$-0.677293\pi$$
−0.528626 + 0.848855i $$0.677293\pi$$
$$434$$ 6.00000 0.288009
$$435$$ 0 0
$$436$$ 18.0000 0.862044
$$437$$ −24.0000 −1.14808
$$438$$ −12.0000 −0.573382
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ −8.00000 −0.380521
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 0 0
$$446$$ 2.00000 0.0947027
$$447$$ −10.0000 −0.472984
$$448$$ −1.00000 −0.0472456
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 5.00000 0.235702
$$451$$ 12.0000 0.565058
$$452$$ −2.00000 −0.0940721
$$453$$ 16.0000 0.751746
$$454$$ 14.0000 0.657053
$$455$$ 0 0
$$456$$ 6.00000 0.280976
$$457$$ −6.00000 −0.280668 −0.140334 0.990104i $$-0.544818\pi$$
−0.140334 + 0.990104i $$0.544818\pi$$
$$458$$ −24.0000 −1.12145
$$459$$ 4.00000 0.186704
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 1.00000 0.0465242
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ −10.0000 −0.464238
$$465$$ 0 0
$$466$$ 26.0000 1.20443
$$467$$ 4.00000 0.185098 0.0925490 0.995708i $$-0.470499\pi$$
0.0925490 + 0.995708i $$0.470499\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ −12.0000 −0.552931
$$472$$ 8.00000 0.368230
$$473$$ 8.00000 0.367840
$$474$$ 0 0
$$475$$ −30.0000 −1.37649
$$476$$ 4.00000 0.183340
$$477$$ 6.00000 0.274721
$$478$$ 8.00000 0.365911
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ 12.0000 0.547153
$$482$$ 28.0000 1.27537
$$483$$ −4.00000 −0.182006
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −28.0000 −1.26880 −0.634401 0.773004i $$-0.718753\pi$$
−0.634401 + 0.773004i $$0.718753\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 12.0000 0.541002
$$493$$ 40.0000 1.80151
$$494$$ 12.0000 0.539906
$$495$$ 0 0
$$496$$ 6.00000 0.269408
$$497$$ 0 0
$$498$$ 14.0000 0.627355
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ −8.00000 −0.357057
$$503$$ −4.00000 −0.178351 −0.0891756 0.996016i $$-0.528423\pi$$
−0.0891756 + 0.996016i $$0.528423\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 0 0
$$506$$ −4.00000 −0.177822
$$507$$ 9.00000 0.399704
$$508$$ −16.0000 −0.709885
$$509$$ −32.0000 −1.41838 −0.709188 0.705020i $$-0.750938\pi$$
−0.709188 + 0.705020i $$0.750938\pi$$
$$510$$ 0 0
$$511$$ 12.0000 0.530849
$$512$$ −1.00000 −0.0441942
$$513$$ −6.00000 −0.264906
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ −2.00000 −0.0879599
$$518$$ −6.00000 −0.263625
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 10.0000 0.437688
$$523$$ −22.0000 −0.961993 −0.480996 0.876723i $$-0.659725\pi$$
−0.480996 + 0.876723i $$0.659725\pi$$
$$524$$ 10.0000 0.436852
$$525$$ −5.00000 −0.218218
$$526$$ −16.0000 −0.697633
$$527$$ −24.0000 −1.04546
$$528$$ 1.00000 0.0435194
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ −6.00000 −0.260133
$$533$$ 24.0000 1.03956
$$534$$ 10.0000 0.432742
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ −20.0000 −0.863064
$$538$$ −24.0000 −1.03471
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 4.00000 0.171815
$$543$$ 8.00000 0.343313
$$544$$ 4.00000 0.171499
$$545$$ 0 0
$$546$$ 2.00000 0.0855921
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ −10.0000 −0.427179
$$549$$ 6.00000 0.256074
$$550$$ −5.00000 −0.213201
$$551$$ −60.0000 −2.55609
$$552$$ −4.00000 −0.170251
$$553$$ 0 0
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ −14.0000 −0.593732
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ −6.00000 −0.254000
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ −30.0000 −1.26547
$$563$$ −6.00000 −0.252870 −0.126435 0.991975i $$-0.540353\pi$$
−0.126435 + 0.991975i $$0.540353\pi$$
$$564$$ −2.00000 −0.0842152
$$565$$ 0 0
$$566$$ −2.00000 −0.0840663
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 0 0
$$571$$ −44.0000 −1.84134 −0.920671 0.390339i $$-0.872358\pi$$
−0.920671 + 0.390339i $$0.872358\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ 12.0000 0.501307
$$574$$ −12.0000 −0.500870
$$575$$ 20.0000 0.834058
$$576$$ 1.00000 0.0416667
$$577$$ 14.0000 0.582828 0.291414 0.956597i $$-0.405874\pi$$
0.291414 + 0.956597i $$0.405874\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 22.0000 0.914289
$$580$$ 0 0
$$581$$ −14.0000 −0.580818
$$582$$ 10.0000 0.414513
$$583$$ −6.00000 −0.248495
$$584$$ 12.0000 0.496564
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ −32.0000 −1.32078 −0.660391 0.750922i $$-0.729609\pi$$
−0.660391 + 0.750922i $$0.729609\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ 36.0000 1.48335
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ −6.00000 −0.246598
$$593$$ −4.00000 −0.164260 −0.0821302 0.996622i $$-0.526172\pi$$
−0.0821302 + 0.996622i $$0.526172\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ −10.0000 −0.409273
$$598$$ −8.00000 −0.327144
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ −5.00000 −0.204124
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ −4.00000 −0.162893
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ −6.00000 −0.243733
$$607$$ 24.0000 0.974130 0.487065 0.873366i $$-0.338067\pi$$
0.487065 + 0.873366i $$0.338067\pi$$
$$608$$ −6.00000 −0.243332
$$609$$ −10.0000 −0.405220
$$610$$ 0 0
$$611$$ −4.00000 −0.161823
$$612$$ −4.00000 −0.161690
$$613$$ 22.0000 0.888572 0.444286 0.895885i $$-0.353457\pi$$
0.444286 + 0.895885i $$0.353457\pi$$
$$614$$ 14.0000 0.564994
$$615$$ 0 0
$$616$$ −1.00000 −0.0402911
$$617$$ 2.00000 0.0805170 0.0402585 0.999189i $$-0.487182\pi$$
0.0402585 + 0.999189i $$0.487182\pi$$
$$618$$ 10.0000 0.402259
$$619$$ 12.0000 0.482321 0.241160 0.970485i $$-0.422472\pi$$
0.241160 + 0.970485i $$0.422472\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 10.0000 0.400963
$$623$$ −10.0000 −0.400642
$$624$$ 2.00000 0.0800641
$$625$$ 25.0000 1.00000
$$626$$ −6.00000 −0.239808
$$627$$ 6.00000 0.239617
$$628$$ 12.0000 0.478852
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ −44.0000 −1.75161 −0.875806 0.482663i $$-0.839670\pi$$
−0.875806 + 0.482663i $$0.839670\pi$$
$$632$$ 0 0
$$633$$ 8.00000 0.317971
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ −2.00000 −0.0792429
$$638$$ −10.0000 −0.395904
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −38.0000 −1.50091 −0.750455 0.660922i $$-0.770166\pi$$
−0.750455 + 0.660922i $$0.770166\pi$$
$$642$$ 8.00000 0.315735
$$643$$ −16.0000 −0.630978 −0.315489 0.948929i $$-0.602169\pi$$
−0.315489 + 0.948929i $$0.602169\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ −10.0000 −0.393141 −0.196570 0.980490i $$-0.562980\pi$$
−0.196570 + 0.980490i $$0.562980\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 8.00000 0.314027
$$650$$ −10.0000 −0.392232
$$651$$ 6.00000 0.235159
$$652$$ 12.0000 0.469956
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ 18.0000 0.703856
$$655$$ 0 0
$$656$$ −12.0000 −0.468521
$$657$$ −12.0000 −0.468165
$$658$$ 2.00000 0.0779681
$$659$$ −16.0000 −0.623272 −0.311636 0.950202i $$-0.600877\pi$$
−0.311636 + 0.950202i $$0.600877\pi$$
$$660$$ 0 0
$$661$$ −32.0000 −1.24466 −0.622328 0.782757i $$-0.713813\pi$$
−0.622328 + 0.782757i $$0.713813\pi$$
$$662$$ 20.0000 0.777322
$$663$$ −8.00000 −0.310694
$$664$$ −14.0000 −0.543305
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ 40.0000 1.54881
$$668$$ 12.0000 0.464294
$$669$$ 2.00000 0.0773245
$$670$$ 0 0
$$671$$ −6.00000 −0.231627
$$672$$ −1.00000 −0.0385758
$$673$$ −10.0000 −0.385472 −0.192736 0.981251i $$-0.561736\pi$$
−0.192736 + 0.981251i $$0.561736\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 5.00000 0.192450
$$676$$ −9.00000 −0.346154
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ −2.00000 −0.0768095
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ 14.0000 0.536481
$$682$$ 6.00000 0.229752
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 6.00000 0.229416
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ −24.0000 −0.915657
$$688$$ −8.00000 −0.304997
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −40.0000 −1.52167 −0.760836 0.648944i $$-0.775211\pi$$
−0.760836 + 0.648944i $$0.775211\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 1.00000 0.0379869
$$694$$ −16.0000 −0.607352
$$695$$ 0 0
$$696$$ −10.0000 −0.379049
$$697$$ 48.0000 1.81813
$$698$$ −2.00000 −0.0757011
$$699$$ 26.0000 0.983410
$$700$$ 5.00000 0.188982
$$701$$ 50.0000 1.88847 0.944237 0.329267i $$-0.106802\pi$$
0.944237 + 0.329267i $$0.106802\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ −36.0000 −1.35777
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ −10.0000 −0.376355
$$707$$ 6.00000 0.225653
$$708$$ 8.00000 0.300658
$$709$$ −6.00000 −0.225335 −0.112667 0.993633i $$-0.535939\pi$$
−0.112667 + 0.993633i $$0.535939\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −10.0000 −0.374766
$$713$$ −24.0000 −0.898807
$$714$$ 4.00000 0.149696
$$715$$ 0 0
$$716$$ 20.0000 0.747435
$$717$$ 8.00000 0.298765
$$718$$ −8.00000 −0.298557
$$719$$ 38.0000 1.41716 0.708580 0.705630i $$-0.249336\pi$$
0.708580 + 0.705630i $$0.249336\pi$$
$$720$$ 0 0
$$721$$ −10.0000 −0.372419
$$722$$ −17.0000 −0.632674
$$723$$ 28.0000 1.04133
$$724$$ −8.00000 −0.297318
$$725$$ 50.0000 1.85695
$$726$$ 1.00000 0.0371135
$$727$$ −34.0000 −1.26099 −0.630495 0.776193i $$-0.717148\pi$$
−0.630495 + 0.776193i $$0.717148\pi$$
$$728$$ −2.00000 −0.0741249
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 32.0000 1.18356
$$732$$ −6.00000 −0.221766
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ 10.0000 0.369107
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 4.00000 0.147342
$$738$$ 12.0000 0.441726
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 0 0
$$741$$ 12.0000 0.440831
$$742$$ 6.00000 0.220267
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 6.00000 0.219971
$$745$$ 0 0
$$746$$ −22.0000 −0.805477
$$747$$ 14.0000 0.512233
$$748$$ 4.00000 0.146254
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ 44.0000 1.60558 0.802791 0.596260i $$-0.203347\pi$$
0.802791 + 0.596260i $$0.203347\pi$$
$$752$$ 2.00000 0.0729325
$$753$$ −8.00000 −0.291536
$$754$$ −20.0000 −0.728357
$$755$$ 0 0
$$756$$ 1.00000 0.0363696
$$757$$ 50.0000 1.81728 0.908640 0.417579i $$-0.137121\pi$$
0.908640 + 0.417579i $$0.137121\pi$$
$$758$$ 4.00000 0.145287
$$759$$ −4.00000 −0.145191
$$760$$ 0 0
$$761$$ −48.0000 −1.74000 −0.869999 0.493053i $$-0.835881\pi$$
−0.869999 + 0.493053i $$0.835881\pi$$
$$762$$ −16.0000 −0.579619
$$763$$ −18.0000 −0.651644
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 10.0000 0.361315
$$767$$ 16.0000 0.577727
$$768$$ −1.00000 −0.0360844
$$769$$ −40.0000 −1.44244 −0.721218 0.692708i $$-0.756418\pi$$
−0.721218 + 0.692708i $$0.756418\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ −22.0000 −0.791797
$$773$$ −28.0000 −1.00709 −0.503545 0.863969i $$-0.667971\pi$$
−0.503545 + 0.863969i $$0.667971\pi$$
$$774$$ 8.00000 0.287554
$$775$$ −30.0000 −1.07763
$$776$$ −10.0000 −0.358979
$$777$$ −6.00000 −0.215249
$$778$$ −2.00000 −0.0717035
$$779$$ −72.0000 −2.57967
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −16.0000 −0.572159
$$783$$ 10.0000 0.357371
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 10.0000 0.356688
$$787$$ 22.0000 0.784215 0.392108 0.919919i $$-0.371746\pi$$
0.392108 + 0.919919i $$0.371746\pi$$
$$788$$ 6.00000 0.213741
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ 2.00000 0.0711118
$$792$$ 1.00000 0.0355335
$$793$$ −12.0000 −0.426132
$$794$$ 4.00000 0.141955
$$795$$ 0 0
$$796$$ 10.0000 0.354441
$$797$$ 12.0000 0.425062 0.212531 0.977154i $$-0.431829\pi$$
0.212531 + 0.977154i $$0.431829\pi$$
$$798$$ −6.00000 −0.212398
$$799$$ −8.00000 −0.283020
$$800$$ 5.00000 0.176777
$$801$$ 10.0000 0.353333
$$802$$ 34.0000 1.20058
$$803$$ 12.0000 0.423471
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ 12.0000 0.422682
$$807$$ −24.0000 −0.844840
$$808$$ 6.00000 0.211079
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 2.00000 0.0702295 0.0351147 0.999383i $$-0.488820\pi$$
0.0351147 + 0.999383i $$0.488820\pi$$
$$812$$ 10.0000 0.350931
$$813$$ 4.00000 0.140286
$$814$$ −6.00000 −0.210300
$$815$$ 0 0
$$816$$ 4.00000 0.140028
$$817$$ −48.0000 −1.67931
$$818$$ −24.0000 −0.839140
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ −10.0000 −0.348790
$$823$$ 44.0000 1.53374 0.766872 0.641800i $$-0.221812\pi$$
0.766872 + 0.641800i $$0.221812\pi$$
$$824$$ −10.0000 −0.348367
$$825$$ −5.00000 −0.174078
$$826$$ −8.00000 −0.278356
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ 56.0000 1.94496 0.972480 0.232986i $$-0.0748495\pi$$
0.972480 + 0.232986i $$0.0748495\pi$$
$$830$$ 0 0
$$831$$ 22.0000 0.763172
$$832$$ −2.00000 −0.0693375
$$833$$ −4.00000 −0.138592
$$834$$ −14.0000 −0.484780
$$835$$ 0 0
$$836$$ −6.00000 −0.207514
$$837$$ −6.00000 −0.207390
$$838$$ −28.0000 −0.967244
$$839$$ 14.0000 0.483334 0.241667 0.970359i $$-0.422306\pi$$
0.241667 + 0.970359i $$0.422306\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ −6.00000 −0.206774
$$843$$ −30.0000 −1.03325
$$844$$ −8.00000 −0.275371
$$845$$ 0 0
$$846$$ −2.00000 −0.0687614
$$847$$ −1.00000 −0.0343604
$$848$$ 6.00000 0.206041
$$849$$ −2.00000 −0.0686398
$$850$$ −20.0000 −0.685994
$$851$$ 24.0000 0.822709
$$852$$ 0 0
$$853$$ −18.0000 −0.616308 −0.308154 0.951336i $$-0.599711\pi$$
−0.308154 + 0.951336i $$0.599711\pi$$
$$854$$ 6.00000 0.205316
$$855$$ 0 0
$$856$$ −8.00000 −0.273434
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 2.00000 0.0682789
$$859$$ 8.00000 0.272956 0.136478 0.990643i $$-0.456422\pi$$
0.136478 + 0.990643i $$0.456422\pi$$
$$860$$ 0 0
$$861$$ −12.0000 −0.408959
$$862$$ −8.00000 −0.272481
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 22.0000 0.747590
$$867$$ 1.00000 0.0339618
$$868$$ −6.00000 −0.203653
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ −18.0000 −0.609557
$$873$$ 10.0000 0.338449
$$874$$ 24.0000 0.811812
$$875$$ 0 0
$$876$$ 12.0000 0.405442
$$877$$ 50.0000 1.68838 0.844190 0.536044i $$-0.180082\pi$$
0.844190 + 0.536044i $$0.180082\pi$$
$$878$$ −28.0000 −0.944954
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ −28.0000 −0.942275 −0.471138 0.882060i $$-0.656156\pi$$
−0.471138 + 0.882060i $$0.656156\pi$$
$$884$$ 8.00000 0.269069
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ −6.00000 −0.201347
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ −2.00000 −0.0669650
$$893$$ 12.0000 0.401565
$$894$$ 10.0000 0.334450
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ −8.00000 −0.267112
$$898$$ 2.00000 0.0667409
$$899$$ −60.0000 −2.00111
$$900$$ −5.00000 −0.166667
$$901$$ −24.0000 −0.799556
$$902$$ −12.0000 −0.399556
$$903$$ −8.00000 −0.266223
$$904$$ 2.00000 0.0665190
$$905$$ 0 0
$$906$$ −16.0000 −0.531564
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ −14.0000 −0.464606
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ −6.00000 −0.198680
$$913$$ −14.0000 −0.463332
$$914$$ 6.00000 0.198462
$$915$$ 0 0
$$916$$ 24.0000 0.792982
$$917$$ −10.0000 −0.330229
$$918$$ −4.00000 −0.132020
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 0 0
$$921$$ 14.0000 0.461316
$$922$$ 6.00000 0.197599
$$923$$ 0 0
$$924$$ −1.00000 −0.0328976
$$925$$ 30.0000 0.986394
$$926$$ −16.0000 −0.525793
$$927$$ 10.0000 0.328443
$$928$$ 10.0000 0.328266
$$929$$ 42.0000 1.37798 0.688988 0.724773i $$-0.258055\pi$$
0.688988 + 0.724773i $$0.258055\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ −26.0000 −0.851658
$$933$$ 10.0000 0.327385
$$934$$ −4.00000 −0.130884
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ −4.00000 −0.130674 −0.0653372 0.997863i $$-0.520812\pi$$
−0.0653372 + 0.997863i $$0.520812\pi$$
$$938$$ −4.00000 −0.130605
$$939$$ −6.00000 −0.195803
$$940$$ 0 0
$$941$$ 6.00000 0.195594 0.0977972 0.995206i $$-0.468820\pi$$
0.0977972 + 0.995206i $$0.468820\pi$$
$$942$$ 12.0000 0.390981
$$943$$ 48.0000 1.56310
$$944$$ −8.00000 −0.260378
$$945$$ 0 0
$$946$$ −8.00000 −0.260102
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 0 0
$$949$$ 24.0000 0.779073
$$950$$ 30.0000 0.973329
$$951$$ 18.0000 0.583690
$$952$$ −4.00000 −0.129641
$$953$$ 50.0000 1.61966 0.809829 0.586665i $$-0.199560\pi$$
0.809829 + 0.586665i $$0.199560\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ −8.00000 −0.258738
$$957$$ −10.0000 −0.323254
$$958$$ 8.00000 0.258468
$$959$$ 10.0000 0.322917
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ −12.0000 −0.386896
$$963$$ 8.00000 0.257796
$$964$$ −28.0000 −0.901819
$$965$$ 0 0
$$966$$ 4.00000 0.128698
$$967$$ −56.0000 −1.80084 −0.900419 0.435023i $$-0.856740\pi$$
−0.900419 + 0.435023i $$0.856740\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 14.0000 0.448819
$$974$$ 28.0000 0.897178
$$975$$ −10.0000 −0.320256
$$976$$ 6.00000 0.192055
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 12.0000 0.383718
$$979$$ −10.0000 −0.319601
$$980$$ 0 0
$$981$$ 18.0000 0.574696
$$982$$ −12.0000 −0.382935
$$983$$ −14.0000 −0.446531 −0.223265 0.974758i $$-0.571672\pi$$
−0.223265 + 0.974758i $$0.571672\pi$$
$$984$$ −12.0000 −0.382546
$$985$$ 0 0
$$986$$ −40.0000 −1.27386
$$987$$ 2.00000 0.0636607
$$988$$ −12.0000 −0.381771
$$989$$ 32.0000 1.01754
$$990$$ 0 0
$$991$$ −48.0000 −1.52477 −0.762385 0.647124i $$-0.775972\pi$$
−0.762385 + 0.647124i $$0.775972\pi$$
$$992$$ −6.00000 −0.190500
$$993$$ 20.0000 0.634681
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −14.0000 −0.443607
$$997$$ 10.0000 0.316703 0.158352 0.987383i $$-0.449382\pi$$
0.158352 + 0.987383i $$0.449382\pi$$
$$998$$ −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 462.2.a.b.1.1 1
3.2 odd 2 1386.2.a.i.1.1 1
4.3 odd 2 3696.2.a.y.1.1 1
7.6 odd 2 3234.2.a.k.1.1 1
11.10 odd 2 5082.2.a.s.1.1 1
21.20 even 2 9702.2.a.bt.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.b.1.1 1 1.1 even 1 trivial
1386.2.a.i.1.1 1 3.2 odd 2
3234.2.a.k.1.1 1 7.6 odd 2
3696.2.a.y.1.1 1 4.3 odd 2
5082.2.a.s.1.1 1 11.10 odd 2
9702.2.a.bt.1.1 1 21.20 even 2