# Properties

 Label 1155.2.a.l.1.1 Level $1155$ Weight $2$ Character 1155.1 Self dual yes Analytic conductor $9.223$ 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] = [1155,2,Mod(1,1155)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1155, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1155.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1155 = 3 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1155.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: $$9.22272143346$$ 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 $$\chi$$ $$=$$ 1155.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} +5.00000 q^{32} +1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{35} -1.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +3.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} -4.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} +3.00000 q^{56} -4.00000 q^{57} -6.00000 q^{58} -12.0000 q^{59} +1.00000 q^{60} -2.00000 q^{61} -4.00000 q^{62} -1.00000 q^{63} +7.00000 q^{64} +2.00000 q^{65} +1.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} -4.00000 q^{69} +1.00000 q^{70} +16.0000 q^{71} -3.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -1.00000 q^{77} -2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +1.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} -3.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} +2.00000 q^{91} +4.00000 q^{92} -4.00000 q^{93} +8.00000 q^{94} +4.00000 q^{95} +5.00000 q^{96} +14.0000 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})$$

## 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 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 1.00000 0.408248
$$7$$ −1.00000 −0.377964
$$8$$ −3.00000 −1.06066
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 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
$$15$$ −1.00000 −0.258199
$$16$$ −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
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −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 $$0.636929\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ 1.00000 0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000 0.192450
$$28$$ 1.00000 0.188982
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 1.00000 0.174078
$$34$$ −2.00000 −0.342997
$$35$$ 1.00000 0.169031
$$36$$ −1.00000 −0.166667
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ −2.00000 −0.320256
$$40$$ 3.00000 0.474342
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ −1.00000 −0.149071
$$46$$ −4.00000 −0.589768
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ −2.00000 −0.280056
$$52$$ 2.00000 0.277350
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −1.00000 −0.134840
$$56$$ 3.00000 0.400892
$$57$$ −4.00000 −0.529813
$$58$$ −6.00000 −0.787839
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 1.00000 0.129099
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ −1.00000 −0.125988
$$64$$ 7.00000 0.875000
$$65$$ 2.00000 0.248069
$$66$$ 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
$$69$$ −4.00000 −0.481543
$$70$$ 1.00000 0.119523
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 1.00000 0.115470
$$76$$ 4.00000 0.458831
$$77$$ −1.00000 −0.113961
$$78$$ −2.00000 −0.226455
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 2.00000 0.216930
$$86$$ −4.00000 −0.431331
$$87$$ −6.00000 −0.643268
$$88$$ −3.00000 −0.319801
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 2.00000 0.209657
$$92$$ 4.00000 0.417029
$$93$$ −4.00000 −0.414781
$$94$$ 8.00000 0.825137
$$95$$ 4.00000 0.410391
$$96$$ 5.00000 0.510310
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 1.00000 0.100504
$$100$$ −1.00000 −0.100000
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 1.00000 0.0975900
$$106$$ 2.00000 0.194257
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ −1.00000 −0.0953463
$$111$$ −2.00000 −0.189832
$$112$$ 1.00000 0.0944911
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 4.00000 0.373002
$$116$$ 6.00000 0.557086
$$117$$ −2.00000 −0.184900
$$118$$ −12.0000 −1.10469
$$119$$ 2.00000 0.183340
$$120$$ 3.00000 0.273861
$$121$$ 1.00000 0.0909091
$$122$$ −2.00000 −0.181071
$$123$$ −6.00000 −0.541002
$$124$$ 4.00000 0.359211
$$125$$ −1.00000 −0.0894427
$$126$$ −1.00000 −0.0890871
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ −4.00000 −0.352180
$$130$$ 2.00000 0.175412
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ 4.00000 0.346844
$$134$$ −4.00000 −0.345547
$$135$$ −1.00000 −0.0860663
$$136$$ 6.00000 0.514496
$$137$$ 14.0000 1.19610 0.598050 0.801459i $$-0.295942\pi$$
0.598050 + 0.801459i $$0.295942\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ 8.00000 0.673722
$$142$$ 16.0000 1.34269
$$143$$ −2.00000 −0.167248
$$144$$ −1.00000 −0.0833333
$$145$$ 6.00000 0.498273
$$146$$ 2.00000 0.165521
$$147$$ 1.00000 0.0824786
$$148$$ 2.00000 0.164399
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 12.0000 0.973329
$$153$$ −2.00000 −0.161690
$$154$$ −1.00000 −0.0805823
$$155$$ 4.00000 0.321288
$$156$$ 2.00000 0.160128
$$157$$ −22.0000 −1.75579 −0.877896 0.478852i $$-0.841053\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ 2.00000 0.158610
$$160$$ −5.00000 −0.395285
$$161$$ 4.00000 0.315244
$$162$$ 1.00000 0.0785674
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 6.00000 0.468521
$$165$$ −1.00000 −0.0778499
$$166$$ 0 0
$$167$$ −20.0000 −1.54765 −0.773823 0.633402i $$-0.781658\pi$$
−0.773823 + 0.633402i $$0.781658\pi$$
$$168$$ 3.00000 0.231455
$$169$$ −9.00000 −0.692308
$$170$$ 2.00000 0.153393
$$171$$ −4.00000 −0.305888
$$172$$ 4.00000 0.304997
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ −1.00000 −0.0755929
$$176$$ −1.00000 −0.0753778
$$177$$ −12.0000 −0.901975
$$178$$ 6.00000 0.449719
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 2.00000 0.148250
$$183$$ −2.00000 −0.147844
$$184$$ 12.0000 0.884652
$$185$$ 2.00000 0.147043
$$186$$ −4.00000 −0.293294
$$187$$ −2.00000 −0.146254
$$188$$ −8.00000 −0.583460
$$189$$ −1.00000 −0.0727393
$$190$$ 4.00000 0.290191
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 7.00000 0.505181
$$193$$ 6.00000 0.431889 0.215945 0.976406i $$-0.430717\pi$$
0.215945 + 0.976406i $$0.430717\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 2.00000 0.143223
$$196$$ −1.00000 −0.0714286
$$197$$ 26.0000 1.85242 0.926212 0.377004i $$-0.123046\pi$$
0.926212 + 0.377004i $$0.123046\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ −28.0000 −1.98487 −0.992434 0.122782i $$-0.960818\pi$$
−0.992434 + 0.122782i $$0.960818\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ −4.00000 −0.282138
$$202$$ −2.00000 −0.140720
$$203$$ 6.00000 0.421117
$$204$$ 2.00000 0.140028
$$205$$ 6.00000 0.419058
$$206$$ 0 0
$$207$$ −4.00000 −0.278019
$$208$$ 2.00000 0.138675
$$209$$ −4.00000 −0.276686
$$210$$ 1.00000 0.0690066
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ 16.0000 1.09630
$$214$$ 4.00000 0.273434
$$215$$ 4.00000 0.272798
$$216$$ −3.00000 −0.204124
$$217$$ 4.00000 0.271538
$$218$$ −2.00000 −0.135457
$$219$$ 2.00000 0.135147
$$220$$ 1.00000 0.0674200
$$221$$ 4.00000 0.269069
$$222$$ −2.00000 −0.134231
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ −5.00000 −0.334077
$$225$$ 1.00000 0.0666667
$$226$$ −10.0000 −0.665190
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 4.00000 0.263752
$$231$$ −1.00000 −0.0657952
$$232$$ 18.0000 1.18176
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ −8.00000 −0.521862
$$236$$ 12.0000 0.781133
$$237$$ −4.00000 −0.259828
$$238$$ 2.00000 0.129641
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 1.00000 0.0641500
$$244$$ 2.00000 0.128037
$$245$$ −1.00000 −0.0638877
$$246$$ −6.00000 −0.382546
$$247$$ 8.00000 0.509028
$$248$$ 12.0000 0.762001
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ −4.00000 −0.251478
$$254$$ 8.00000 0.501965
$$255$$ 2.00000 0.125245
$$256$$ −17.0000 −1.06250
$$257$$ 14.0000 0.873296 0.436648 0.899632i $$-0.356166\pi$$
0.436648 + 0.899632i $$0.356166\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ 2.00000 0.124274
$$260$$ −2.00000 −0.124035
$$261$$ −6.00000 −0.371391
$$262$$ 12.0000 0.741362
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ −2.00000 −0.122859
$$266$$ 4.00000 0.245256
$$267$$ 6.00000 0.367194
$$268$$ 4.00000 0.244339
$$269$$ −22.0000 −1.34136 −0.670682 0.741745i $$-0.733998\pi$$
−0.670682 + 0.741745i $$0.733998\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ −32.0000 −1.94386 −0.971931 0.235267i $$-0.924404\pi$$
−0.971931 + 0.235267i $$0.924404\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 2.00000 0.121046
$$274$$ 14.0000 0.845771
$$275$$ 1.00000 0.0603023
$$276$$ 4.00000 0.240772
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ 4.00000 0.239904
$$279$$ −4.00000 −0.239474
$$280$$ −3.00000 −0.179284
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 8.00000 0.476393
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ −16.0000 −0.949425
$$285$$ 4.00000 0.236940
$$286$$ −2.00000 −0.118262
$$287$$ 6.00000 0.354169
$$288$$ 5.00000 0.294628
$$289$$ −13.0000 −0.764706
$$290$$ 6.00000 0.352332
$$291$$ 14.0000 0.820695
$$292$$ −2.00000 −0.117041
$$293$$ −22.0000 −1.28525 −0.642627 0.766179i $$-0.722155\pi$$
−0.642627 + 0.766179i $$0.722155\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ 12.0000 0.698667
$$296$$ 6.00000 0.348743
$$297$$ 1.00000 0.0580259
$$298$$ 2.00000 0.115857
$$299$$ 8.00000 0.462652
$$300$$ −1.00000 −0.0577350
$$301$$ 4.00000 0.230556
$$302$$ −12.0000 −0.690522
$$303$$ −2.00000 −0.114897
$$304$$ 4.00000 0.229416
$$305$$ 2.00000 0.114520
$$306$$ −2.00000 −0.114332
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 1.00000 0.0569803
$$309$$ 0 0
$$310$$ 4.00000 0.227185
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 6.00000 0.339683
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ 1.00000 0.0563436
$$316$$ 4.00000 0.225018
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 2.00000 0.112154
$$319$$ −6.00000 −0.335936
$$320$$ −7.00000 −0.391312
$$321$$ 4.00000 0.223258
$$322$$ 4.00000 0.222911
$$323$$ 8.00000 0.445132
$$324$$ −1.00000 −0.0555556
$$325$$ −2.00000 −0.110940
$$326$$ 4.00000 0.221540
$$327$$ −2.00000 −0.110600
$$328$$ 18.0000 0.993884
$$329$$ −8.00000 −0.441054
$$330$$ −1.00000 −0.0550482
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 0 0
$$333$$ −2.00000 −0.109599
$$334$$ −20.0000 −1.09435
$$335$$ 4.00000 0.218543
$$336$$ 1.00000 0.0545545
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ −10.0000 −0.543125
$$340$$ −2.00000 −0.108465
$$341$$ −4.00000 −0.216612
$$342$$ −4.00000 −0.216295
$$343$$ −1.00000 −0.0539949
$$344$$ 12.0000 0.646997
$$345$$ 4.00000 0.215353
$$346$$ 2.00000 0.107521
$$347$$ 20.0000 1.07366 0.536828 0.843692i $$-0.319622\pi$$
0.536828 + 0.843692i $$0.319622\pi$$
$$348$$ 6.00000 0.321634
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ −2.00000 −0.106752
$$352$$ 5.00000 0.266501
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ −16.0000 −0.849192
$$356$$ −6.00000 −0.317999
$$357$$ 2.00000 0.105851
$$358$$ 4.00000 0.211407
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 3.00000 0.158114
$$361$$ −3.00000 −0.157895
$$362$$ −10.0000 −0.525588
$$363$$ 1.00000 0.0524864
$$364$$ −2.00000 −0.104828
$$365$$ −2.00000 −0.104685
$$366$$ −2.00000 −0.104542
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ 4.00000 0.208514
$$369$$ −6.00000 −0.312348
$$370$$ 2.00000 0.103975
$$371$$ −2.00000 −0.103835
$$372$$ 4.00000 0.207390
$$373$$ 2.00000 0.103556 0.0517780 0.998659i $$-0.483511\pi$$
0.0517780 + 0.998659i $$0.483511\pi$$
$$374$$ −2.00000 −0.103418
$$375$$ −1.00000 −0.0516398
$$376$$ −24.0000 −1.23771
$$377$$ 12.0000 0.618031
$$378$$ −1.00000 −0.0514344
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ 8.00000 0.409852
$$382$$ 24.0000 1.22795
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 1.00000 0.0509647
$$386$$ 6.00000 0.305392
$$387$$ −4.00000 −0.203331
$$388$$ −14.0000 −0.710742
$$389$$ −26.0000 −1.31825 −0.659126 0.752032i $$-0.729074\pi$$
−0.659126 + 0.752032i $$0.729074\pi$$
$$390$$ 2.00000 0.101274
$$391$$ 8.00000 0.404577
$$392$$ −3.00000 −0.151523
$$393$$ 12.0000 0.605320
$$394$$ 26.0000 1.30986
$$395$$ 4.00000 0.201262
$$396$$ −1.00000 −0.0502519
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ −28.0000 −1.40351
$$399$$ 4.00000 0.200250
$$400$$ −1.00000 −0.0500000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ 8.00000 0.398508
$$404$$ 2.00000 0.0995037
$$405$$ −1.00000 −0.0496904
$$406$$ 6.00000 0.297775
$$407$$ −2.00000 −0.0991363
$$408$$ 6.00000 0.297044
$$409$$ −38.0000 −1.87898 −0.939490 0.342578i $$-0.888700\pi$$
−0.939490 + 0.342578i $$0.888700\pi$$
$$410$$ 6.00000 0.296319
$$411$$ 14.0000 0.690569
$$412$$ 0 0
$$413$$ 12.0000 0.590481
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ −10.0000 −0.490290
$$417$$ 4.00000 0.195881
$$418$$ −4.00000 −0.195646
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ −1.00000 −0.0487950
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 8.00000 0.389434
$$423$$ 8.00000 0.388973
$$424$$ −6.00000 −0.291386
$$425$$ −2.00000 −0.0970143
$$426$$ 16.0000 0.775203
$$427$$ 2.00000 0.0967868
$$428$$ −4.00000 −0.193347
$$429$$ −2.00000 −0.0965609
$$430$$ 4.00000 0.192897
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 22.0000 1.05725 0.528626 0.848855i $$-0.322707\pi$$
0.528626 + 0.848855i $$0.322707\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 6.00000 0.287678
$$436$$ 2.00000 0.0957826
$$437$$ 16.0000 0.765384
$$438$$ 2.00000 0.0955637
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 3.00000 0.143019
$$441$$ 1.00000 0.0476190
$$442$$ 4.00000 0.190261
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ −6.00000 −0.284427
$$446$$ −16.0000 −0.757622
$$447$$ 2.00000 0.0945968
$$448$$ −7.00000 −0.330719
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ −6.00000 −0.282529
$$452$$ 10.0000 0.470360
$$453$$ −12.0000 −0.563809
$$454$$ 0 0
$$455$$ −2.00000 −0.0937614
$$456$$ 12.0000 0.561951
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ −2.00000 −0.0933520
$$460$$ −4.00000 −0.186501
$$461$$ −2.00000 −0.0931493 −0.0465746 0.998915i $$-0.514831\pi$$
−0.0465746 + 0.998915i $$0.514831\pi$$
$$462$$ −1.00000 −0.0465242
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 4.00000 0.185496
$$466$$ −10.0000 −0.463241
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 4.00000 0.184703
$$470$$ −8.00000 −0.369012
$$471$$ −22.0000 −1.01371
$$472$$ 36.0000 1.65703
$$473$$ −4.00000 −0.183920
$$474$$ −4.00000 −0.183726
$$475$$ −4.00000 −0.183533
$$476$$ −2.00000 −0.0916698
$$477$$ 2.00000 0.0915737
$$478$$ −16.0000 −0.731823
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ −5.00000 −0.228218
$$481$$ 4.00000 0.182384
$$482$$ −14.0000 −0.637683
$$483$$ 4.00000 0.182006
$$484$$ −1.00000 −0.0454545
$$485$$ −14.0000 −0.635707
$$486$$ 1.00000 0.0453609
$$487$$ 16.0000 0.725029 0.362515 0.931978i $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ 6.00000 0.271607
$$489$$ 4.00000 0.180886
$$490$$ −1.00000 −0.0451754
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 6.00000 0.270501
$$493$$ 12.0000 0.540453
$$494$$ 8.00000 0.359937
$$495$$ −1.00000 −0.0449467
$$496$$ 4.00000 0.179605
$$497$$ −16.0000 −0.717698
$$498$$ 0 0
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −20.0000 −0.893534
$$502$$ 12.0000 0.535586
$$503$$ −28.0000 −1.24846 −0.624229 0.781241i $$-0.714587\pi$$
−0.624229 + 0.781241i $$0.714587\pi$$
$$504$$ 3.00000 0.133631
$$505$$ 2.00000 0.0889988
$$506$$ −4.00000 −0.177822
$$507$$ −9.00000 −0.399704
$$508$$ −8.00000 −0.354943
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 2.00000 0.0885615
$$511$$ −2.00000 −0.0884748
$$512$$ −11.0000 −0.486136
$$513$$ −4.00000 −0.176604
$$514$$ 14.0000 0.617514
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 8.00000 0.351840
$$518$$ 2.00000 0.0878750
$$519$$ 2.00000 0.0877903
$$520$$ −6.00000 −0.263117
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ 12.0000 0.524723 0.262362 0.964970i $$-0.415499\pi$$
0.262362 + 0.964970i $$0.415499\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ −1.00000 −0.0436436
$$526$$ 8.00000 0.348817
$$527$$ 8.00000 0.348485
$$528$$ −1.00000 −0.0435194
$$529$$ −7.00000 −0.304348
$$530$$ −2.00000 −0.0868744
$$531$$ −12.0000 −0.520756
$$532$$ −4.00000 −0.173422
$$533$$ 12.0000 0.519778
$$534$$ 6.00000 0.259645
$$535$$ −4.00000 −0.172935
$$536$$ 12.0000 0.518321
$$537$$ 4.00000 0.172613
$$538$$ −22.0000 −0.948487
$$539$$ 1.00000 0.0430730
$$540$$ 1.00000 0.0430331
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ −32.0000 −1.37452
$$543$$ −10.0000 −0.429141
$$544$$ −10.0000 −0.428746
$$545$$ 2.00000 0.0856706
$$546$$ 2.00000 0.0855921
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ −14.0000 −0.598050
$$549$$ −2.00000 −0.0853579
$$550$$ 1.00000 0.0426401
$$551$$ 24.0000 1.02243
$$552$$ 12.0000 0.510754
$$553$$ 4.00000 0.170097
$$554$$ −22.0000 −0.934690
$$555$$ 2.00000 0.0848953
$$556$$ −4.00000 −0.169638
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ 8.00000 0.338364
$$560$$ −1.00000 −0.0422577
$$561$$ −2.00000 −0.0844401
$$562$$ 30.0000 1.26547
$$563$$ −32.0000 −1.34864 −0.674320 0.738440i $$-0.735563\pi$$
−0.674320 + 0.738440i $$0.735563\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 10.0000 0.420703
$$566$$ 4.00000 0.168133
$$567$$ −1.00000 −0.0419961
$$568$$ −48.0000 −2.01404
$$569$$ −26.0000 −1.08998 −0.544988 0.838444i $$-0.683466\pi$$
−0.544988 + 0.838444i $$0.683466\pi$$
$$570$$ 4.00000 0.167542
$$571$$ −32.0000 −1.33916 −0.669579 0.742741i $$-0.733526\pi$$
−0.669579 + 0.742741i $$0.733526\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ 24.0000 1.00261
$$574$$ 6.00000 0.250435
$$575$$ −4.00000 −0.166812
$$576$$ 7.00000 0.291667
$$577$$ −42.0000 −1.74848 −0.874241 0.485491i $$-0.838641\pi$$
−0.874241 + 0.485491i $$0.838641\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 6.00000 0.249351
$$580$$ −6.00000 −0.249136
$$581$$ 0 0
$$582$$ 14.0000 0.580319
$$583$$ 2.00000 0.0828315
$$584$$ −6.00000 −0.248282
$$585$$ 2.00000 0.0826898
$$586$$ −22.0000 −0.908812
$$587$$ −28.0000 −1.15568 −0.577842 0.816149i $$-0.696105\pi$$
−0.577842 + 0.816149i $$0.696105\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ 16.0000 0.659269
$$590$$ 12.0000 0.494032
$$591$$ 26.0000 1.06950
$$592$$ 2.00000 0.0821995
$$593$$ −10.0000 −0.410651 −0.205325 0.978694i $$-0.565825\pi$$
−0.205325 + 0.978694i $$0.565825\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ −2.00000 −0.0819920
$$596$$ −2.00000 −0.0819232
$$597$$ −28.0000 −1.14596
$$598$$ 8.00000 0.327144
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ −3.00000 −0.122474
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 4.00000 0.163028
$$603$$ −4.00000 −0.162893
$$604$$ 12.0000 0.488273
$$605$$ −1.00000 −0.0406558
$$606$$ −2.00000 −0.0812444
$$607$$ 16.0000 0.649420 0.324710 0.945814i $$-0.394733\pi$$
0.324710 + 0.945814i $$0.394733\pi$$
$$608$$ −20.0000 −0.811107
$$609$$ 6.00000 0.243132
$$610$$ 2.00000 0.0809776
$$611$$ −16.0000 −0.647291
$$612$$ 2.00000 0.0808452
$$613$$ −38.0000 −1.53481 −0.767403 0.641165i $$-0.778451\pi$$
−0.767403 + 0.641165i $$0.778451\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 6.00000 0.241943
$$616$$ 3.00000 0.120873
$$617$$ −10.0000 −0.402585 −0.201292 0.979531i $$-0.564514\pi$$
−0.201292 + 0.979531i $$0.564514\pi$$
$$618$$ 0 0
$$619$$ −32.0000 −1.28619 −0.643094 0.765787i $$-0.722350\pi$$
−0.643094 + 0.765787i $$0.722350\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ −4.00000 −0.160514
$$622$$ −24.0000 −0.962312
$$623$$ −6.00000 −0.240385
$$624$$ 2.00000 0.0800641
$$625$$ 1.00000 0.0400000
$$626$$ 6.00000 0.239808
$$627$$ −4.00000 −0.159745
$$628$$ 22.0000 0.877896
$$629$$ 4.00000 0.159490
$$630$$ 1.00000 0.0398410
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 12.0000 0.477334
$$633$$ 8.00000 0.317971
$$634$$ −6.00000 −0.238290
$$635$$ −8.00000 −0.317470
$$636$$ −2.00000 −0.0793052
$$637$$ −2.00000 −0.0792429
$$638$$ −6.00000 −0.237542
$$639$$ 16.0000 0.632950
$$640$$ 3.00000 0.118585
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 4.00000 0.157867
$$643$$ 12.0000 0.473234 0.236617 0.971603i $$-0.423961\pi$$
0.236617 + 0.971603i $$0.423961\pi$$
$$644$$ −4.00000 −0.157622
$$645$$ 4.00000 0.157500
$$646$$ 8.00000 0.314756
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ −3.00000 −0.117851
$$649$$ −12.0000 −0.471041
$$650$$ −2.00000 −0.0784465
$$651$$ 4.00000 0.156772
$$652$$ −4.00000 −0.156652
$$653$$ 10.0000 0.391330 0.195665 0.980671i $$-0.437313\pi$$
0.195665 + 0.980671i $$0.437313\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ −12.0000 −0.468879
$$656$$ 6.00000 0.234261
$$657$$ 2.00000 0.0780274
$$658$$ −8.00000 −0.311872
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 1.00000 0.0389249
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 4.00000 0.155347
$$664$$ 0 0
$$665$$ −4.00000 −0.155113
$$666$$ −2.00000 −0.0774984
$$667$$ 24.0000 0.929284
$$668$$ 20.0000 0.773823
$$669$$ −16.0000 −0.618596
$$670$$ 4.00000 0.154533
$$671$$ −2.00000 −0.0772091
$$672$$ −5.00000 −0.192879
$$673$$ 14.0000 0.539660 0.269830 0.962908i $$-0.413032\pi$$
0.269830 + 0.962908i $$0.413032\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 1.00000 0.0384900
$$676$$ 9.00000 0.346154
$$677$$ −14.0000 −0.538064 −0.269032 0.963131i $$-0.586704\pi$$
−0.269032 + 0.963131i $$0.586704\pi$$
$$678$$ −10.0000 −0.384048
$$679$$ −14.0000 −0.537271
$$680$$ −6.00000 −0.230089
$$681$$ 0 0
$$682$$ −4.00000 −0.153168
$$683$$ 8.00000 0.306111 0.153056 0.988218i $$-0.451089\pi$$
0.153056 + 0.988218i $$0.451089\pi$$
$$684$$ 4.00000 0.152944
$$685$$ −14.0000 −0.534913
$$686$$ −1.00000 −0.0381802
$$687$$ −2.00000 −0.0763048
$$688$$ 4.00000 0.152499
$$689$$ −4.00000 −0.152388
$$690$$ 4.00000 0.152277
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ −2.00000 −0.0760286
$$693$$ −1.00000 −0.0379869
$$694$$ 20.0000 0.759190
$$695$$ −4.00000 −0.151729
$$696$$ 18.0000 0.682288
$$697$$ 12.0000 0.454532
$$698$$ −2.00000 −0.0757011
$$699$$ −10.0000 −0.378235
$$700$$ 1.00000 0.0377964
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ 8.00000 0.301726
$$704$$ 7.00000 0.263822
$$705$$ −8.00000 −0.301297
$$706$$ 14.0000 0.526897
$$707$$ 2.00000 0.0752177
$$708$$ 12.0000 0.450988
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ −16.0000 −0.600469
$$711$$ −4.00000 −0.150012
$$712$$ −18.0000 −0.674579
$$713$$ 16.0000 0.599205
$$714$$ 2.00000 0.0748481
$$715$$ 2.00000 0.0747958
$$716$$ −4.00000 −0.149487
$$717$$ −16.0000 −0.597531
$$718$$ 16.0000 0.597115
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 0 0
$$722$$ −3.00000 −0.111648
$$723$$ −14.0000 −0.520666
$$724$$ 10.0000 0.371647
$$725$$ −6.00000 −0.222834
$$726$$ 1.00000 0.0371135
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ −6.00000 −0.222375
$$729$$ 1.00000 0.0370370
$$730$$ −2.00000 −0.0740233
$$731$$ 8.00000 0.295891
$$732$$ 2.00000 0.0739221
$$733$$ 22.0000 0.812589 0.406294 0.913742i $$-0.366821\pi$$
0.406294 + 0.913742i $$0.366821\pi$$
$$734$$ 16.0000 0.590571
$$735$$ −1.00000 −0.0368856
$$736$$ −20.0000 −0.737210
$$737$$ −4.00000 −0.147342
$$738$$ −6.00000 −0.220863
$$739$$ −24.0000 −0.882854 −0.441427 0.897297i $$-0.645528\pi$$
−0.441427 + 0.897297i $$0.645528\pi$$
$$740$$ −2.00000 −0.0735215
$$741$$ 8.00000 0.293887
$$742$$ −2.00000 −0.0734223
$$743$$ −40.0000 −1.46746 −0.733729 0.679442i $$-0.762222\pi$$
−0.733729 + 0.679442i $$0.762222\pi$$
$$744$$ 12.0000 0.439941
$$745$$ −2.00000 −0.0732743
$$746$$ 2.00000 0.0732252
$$747$$ 0 0
$$748$$ 2.00000 0.0731272
$$749$$ −4.00000 −0.146157
$$750$$ −1.00000 −0.0365148
$$751$$ −24.0000 −0.875772 −0.437886 0.899030i $$-0.644273\pi$$
−0.437886 + 0.899030i $$0.644273\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 12.0000 0.437304
$$754$$ 12.0000 0.437014
$$755$$ 12.0000 0.436725
$$756$$ 1.00000 0.0363696
$$757$$ 46.0000 1.67190 0.835949 0.548807i $$-0.184918\pi$$
0.835949 + 0.548807i $$0.184918\pi$$
$$758$$ 4.00000 0.145287
$$759$$ −4.00000 −0.145191
$$760$$ −12.0000 −0.435286
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 8.00000 0.289809
$$763$$ 2.00000 0.0724049
$$764$$ −24.0000 −0.868290
$$765$$ 2.00000 0.0723102
$$766$$ 24.0000 0.867155
$$767$$ 24.0000 0.866590
$$768$$ −17.0000 −0.613435
$$769$$ 18.0000 0.649097 0.324548 0.945869i $$-0.394788\pi$$
0.324548 + 0.945869i $$0.394788\pi$$
$$770$$ 1.00000 0.0360375
$$771$$ 14.0000 0.504198
$$772$$ −6.00000 −0.215945
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ −4.00000 −0.143684
$$776$$ −42.0000 −1.50771
$$777$$ 2.00000 0.0717496
$$778$$ −26.0000 −0.932145
$$779$$ 24.0000 0.859889
$$780$$ −2.00000 −0.0716115
$$781$$ 16.0000 0.572525
$$782$$ 8.00000 0.286079
$$783$$ −6.00000 −0.214423
$$784$$ −1.00000 −0.0357143
$$785$$ 22.0000 0.785214
$$786$$ 12.0000 0.428026
$$787$$ −4.00000 −0.142585 −0.0712923 0.997455i $$-0.522712\pi$$
−0.0712923 + 0.997455i $$0.522712\pi$$
$$788$$ −26.0000 −0.926212
$$789$$ 8.00000 0.284808
$$790$$ 4.00000 0.142314
$$791$$ 10.0000 0.355559
$$792$$ −3.00000 −0.106600
$$793$$ 4.00000 0.142044
$$794$$ 2.00000 0.0709773
$$795$$ −2.00000 −0.0709327
$$796$$ 28.0000 0.992434
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ 4.00000 0.141598
$$799$$ −16.0000 −0.566039
$$800$$ 5.00000 0.176777
$$801$$ 6.00000 0.212000
$$802$$ 18.0000 0.635602
$$803$$ 2.00000 0.0705785
$$804$$ 4.00000 0.141069
$$805$$ −4.00000 −0.140981
$$806$$ 8.00000 0.281788
$$807$$ −22.0000 −0.774437
$$808$$ 6.00000 0.211079
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ −6.00000 −0.210559
$$813$$ −32.0000 −1.12229
$$814$$ −2.00000 −0.0701000
$$815$$ −4.00000 −0.140114
$$816$$ 2.00000 0.0700140
$$817$$ 16.0000 0.559769
$$818$$ −38.0000 −1.32864
$$819$$ 2.00000 0.0698857
$$820$$ −6.00000 −0.209529
$$821$$ −38.0000 −1.32621 −0.663105 0.748527i $$-0.730762\pi$$
−0.663105 + 0.748527i $$0.730762\pi$$
$$822$$ 14.0000 0.488306
$$823$$ 8.00000 0.278862 0.139431 0.990232i $$-0.455473\pi$$
0.139431 + 0.990232i $$0.455473\pi$$
$$824$$ 0 0
$$825$$ 1.00000 0.0348155
$$826$$ 12.0000 0.417533
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ 4.00000 0.139010
$$829$$ −18.0000 −0.625166 −0.312583 0.949890i $$-0.601194\pi$$
−0.312583 + 0.949890i $$0.601194\pi$$
$$830$$ 0 0
$$831$$ −22.0000 −0.763172
$$832$$ −14.0000 −0.485363
$$833$$ −2.00000 −0.0692959
$$834$$ 4.00000 0.138509
$$835$$ 20.0000 0.692129
$$836$$ 4.00000 0.138343
$$837$$ −4.00000 −0.138260
$$838$$ 20.0000 0.690889
$$839$$ 48.0000 1.65714 0.828572 0.559883i $$-0.189154\pi$$
0.828572 + 0.559883i $$0.189154\pi$$
$$840$$ −3.00000 −0.103510
$$841$$ 7.00000 0.241379
$$842$$ −26.0000 −0.896019
$$843$$ 30.0000 1.03325
$$844$$ −8.00000 −0.275371
$$845$$ 9.00000 0.309609
$$846$$ 8.00000 0.275046
$$847$$ −1.00000 −0.0343604
$$848$$ −2.00000 −0.0686803
$$849$$ 4.00000 0.137280
$$850$$ −2.00000 −0.0685994
$$851$$ 8.00000 0.274236
$$852$$ −16.0000 −0.548151
$$853$$ 46.0000 1.57501 0.787505 0.616308i $$-0.211372\pi$$
0.787505 + 0.616308i $$0.211372\pi$$
$$854$$ 2.00000 0.0684386
$$855$$ 4.00000 0.136797
$$856$$ −12.0000 −0.410152
$$857$$ 22.0000 0.751506 0.375753 0.926720i $$-0.377384\pi$$
0.375753 + 0.926720i $$0.377384\pi$$
$$858$$ −2.00000 −0.0682789
$$859$$ 32.0000 1.09183 0.545913 0.837842i $$-0.316183\pi$$
0.545913 + 0.837842i $$0.316183\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 6.00000 0.204479
$$862$$ 0 0
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ 5.00000 0.170103
$$865$$ −2.00000 −0.0680020
$$866$$ 22.0000 0.747590
$$867$$ −13.0000 −0.441503
$$868$$ −4.00000 −0.135769
$$869$$ −4.00000 −0.135691
$$870$$ 6.00000 0.203419
$$871$$ 8.00000 0.271070
$$872$$ 6.00000 0.203186
$$873$$ 14.0000 0.473828
$$874$$ 16.0000 0.541208
$$875$$ 1.00000 0.0338062
$$876$$ −2.00000 −0.0675737
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ 0 0
$$879$$ −22.0000 −0.742042
$$880$$ 1.00000 0.0337100
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ 44.0000 1.48072 0.740359 0.672212i $$-0.234656\pi$$
0.740359 + 0.672212i $$0.234656\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 12.0000 0.403376
$$886$$ 24.0000 0.806296
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 6.00000 0.201347
$$889$$ −8.00000 −0.268311
$$890$$ −6.00000 −0.201120
$$891$$ 1.00000 0.0335013
$$892$$ 16.0000 0.535720
$$893$$ −32.0000 −1.07084
$$894$$ 2.00000 0.0668900
$$895$$ −4.00000 −0.133705
$$896$$ 3.00000 0.100223
$$897$$ 8.00000 0.267112
$$898$$ −14.0000 −0.467186
$$899$$ 24.0000 0.800445
$$900$$ −1.00000 −0.0333333
$$901$$ −4.00000 −0.133259
$$902$$ −6.00000 −0.199778
$$903$$ 4.00000 0.133112
$$904$$ 30.0000 0.997785
$$905$$ 10.0000 0.332411
$$906$$ −12.0000 −0.398673
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 0 0
$$909$$ −2.00000 −0.0663358
$$910$$ −2.00000 −0.0662994
$$911$$ 56.0000 1.85536 0.927681 0.373373i $$-0.121799\pi$$
0.927681 + 0.373373i $$0.121799\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 0 0
$$914$$ 6.00000 0.198462
$$915$$ 2.00000 0.0661180
$$916$$ 2.00000 0.0660819
$$917$$ −12.0000 −0.396275
$$918$$ −2.00000 −0.0660098
$$919$$ 44.0000 1.45143 0.725713 0.687998i $$-0.241510\pi$$
0.725713 + 0.687998i $$0.241510\pi$$
$$920$$ −12.0000 −0.395628
$$921$$ −20.0000 −0.659022
$$922$$ −2.00000 −0.0658665
$$923$$ −32.0000 −1.05329
$$924$$ 1.00000 0.0328976
$$925$$ −2.00000 −0.0657596
$$926$$ −8.00000 −0.262896
$$927$$ 0 0
$$928$$ −30.0000 −0.984798
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 4.00000 0.131165
$$931$$ −4.00000 −0.131095
$$932$$ 10.0000 0.327561
$$933$$ −24.0000 −0.785725
$$934$$ −12.0000 −0.392652
$$935$$ 2.00000 0.0654070
$$936$$ 6.00000 0.196116
$$937$$ −38.0000 −1.24141 −0.620703 0.784046i $$-0.713153\pi$$
−0.620703 + 0.784046i $$0.713153\pi$$
$$938$$ 4.00000 0.130605
$$939$$ 6.00000 0.195803
$$940$$ 8.00000 0.260931
$$941$$ −50.0000 −1.62995 −0.814977 0.579494i $$-0.803250\pi$$
−0.814977 + 0.579494i $$0.803250\pi$$
$$942$$ −22.0000 −0.716799
$$943$$ 24.0000 0.781548
$$944$$ 12.0000 0.390567
$$945$$ 1.00000 0.0325300
$$946$$ −4.00000 −0.130051
$$947$$ −8.00000 −0.259965 −0.129983 0.991516i $$-0.541492\pi$$
−0.129983 + 0.991516i $$0.541492\pi$$
$$948$$ 4.00000 0.129914
$$949$$ −4.00000 −0.129845
$$950$$ −4.00000 −0.129777
$$951$$ −6.00000 −0.194563
$$952$$ −6.00000 −0.194461
$$953$$ −42.0000 −1.36051 −0.680257 0.732974i $$-0.738132\pi$$
−0.680257 + 0.732974i $$0.738132\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ −24.0000 −0.776622
$$956$$ 16.0000 0.517477
$$957$$ −6.00000 −0.193952
$$958$$ −8.00000 −0.258468
$$959$$ −14.0000 −0.452084
$$960$$ −7.00000 −0.225924
$$961$$ −15.0000 −0.483871
$$962$$ 4.00000 0.128965
$$963$$ 4.00000 0.128898
$$964$$ 14.0000 0.450910
$$965$$ −6.00000 −0.193147
$$966$$ 4.00000 0.128698
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ −3.00000 −0.0964237
$$969$$ 8.00000 0.256997
$$970$$ −14.0000 −0.449513
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −4.00000 −0.128234
$$974$$ 16.0000 0.512673
$$975$$ −2.00000 −0.0640513
$$976$$ 2.00000 0.0640184
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 4.00000 0.127906
$$979$$ 6.00000 0.191761
$$980$$ 1.00000 0.0319438
$$981$$ −2.00000 −0.0638551
$$982$$ −20.0000 −0.638226
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 18.0000 0.573819
$$985$$ −26.0000 −0.828429
$$986$$ 12.0000 0.382158
$$987$$ −8.00000 −0.254643
$$988$$ −8.00000 −0.254514
$$989$$ 16.0000 0.508770
$$990$$ −1.00000 −0.0317821
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ −20.0000 −0.635001
$$993$$ 4.00000 0.126936
$$994$$ −16.0000 −0.507489
$$995$$ 28.0000 0.887660
$$996$$ 0 0
$$997$$ 46.0000 1.45683 0.728417 0.685134i $$-0.240256\pi$$
0.728417 + 0.685134i $$0.240256\pi$$
$$998$$ 20.0000 0.633089
$$999$$ −2.00000 −0.0632772
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 1155.2.a.l.1.1 1
3.2 odd 2 3465.2.a.d.1.1 1
5.4 even 2 5775.2.a.e.1.1 1
7.6 odd 2 8085.2.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.l.1.1 1 1.1 even 1 trivial
3465.2.a.d.1.1 1 3.2 odd 2
5775.2.a.e.1.1 1 5.4 even 2
8085.2.a.t.1.1 1 7.6 odd 2