# Properties

 Label 1155.2.a.m.1.1 Level $1155$ Weight $2$ Character 1155.1 Self dual yes Analytic conductor $9.223$ 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] = [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: $$0$$ 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 $$\chi$$ $$=$$ 1155.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +1.00000 q^{30} +4.00000 q^{31} +5.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{35} -1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +6.00000 q^{39} -3.00000 q^{40} +6.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} +1.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} -1.00000 q^{55} -3.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} -8.00000 q^{59} -1.00000 q^{60} +10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} +6.00000 q^{65} -1.00000 q^{66} -8.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} +1.00000 q^{70} -3.00000 q^{72} -14.0000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -1.00000 q^{77} +6.00000 q^{78} -16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} -1.00000 q^{84} +2.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} +3.00000 q^{88} -6.00000 q^{89} +1.00000 q^{90} +6.00000 q^{91} +4.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} +4.00000 q^{95} +5.00000 q^{96} +18.0000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})$$

## 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
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 1.00000 0.258199
$$16$$ −1.00000 −0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ −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
$$26$$ 6.00000 1.17670
$$27$$ 1.00000 0.192450
$$28$$ −1.00000 −0.188982
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 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
$$39$$ 6.00000 0.960769
$$40$$ −3.00000 −0.474342
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 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
$$52$$ −6.00000 −0.832050
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −1.00000 −0.134840
$$56$$ −3.00000 −0.400892
$$57$$ 4.00000 0.529813
$$58$$ −2.00000 −0.262613
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 1.00000 0.125988
$$64$$ 7.00000 0.875000
$$65$$ 6.00000 0.744208
$$66$$ −1.00000 −0.123091
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ −4.00000 −0.481543
$$70$$ 1.00000 0.119523
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 1.00000 0.115470
$$76$$ −4.00000 −0.458831
$$77$$ −1.00000 −0.113961
$$78$$ 6.00000 0.679366
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 2.00000 0.216930
$$86$$ 4.00000 0.431331
$$87$$ −2.00000 −0.214423
$$88$$ 3.00000 0.319801
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 6.00000 0.628971
$$92$$ 4.00000 0.417029
$$93$$ 4.00000 0.414781
$$94$$ 8.00000 0.825137
$$95$$ 4.00000 0.410391
$$96$$ 5.00000 0.510310
$$97$$ 18.0000 1.82762 0.913812 0.406138i $$-0.133125\pi$$
0.913812 + 0.406138i $$0.133125\pi$$
$$98$$ 1.00000 0.101015
$$99$$ −1.00000 −0.100504
$$100$$ −1.00000 −0.100000
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 2.00000 0.198030
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ −18.0000 −1.76505
$$105$$ 1.00000 0.0975900
$$106$$ −2.00000 −0.194257
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ −1.00000 −0.0953463
$$111$$ −2.00000 −0.189832
$$112$$ −1.00000 −0.0944911
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 4.00000 0.374634
$$115$$ −4.00000 −0.373002
$$116$$ 2.00000 0.185695
$$117$$ 6.00000 0.554700
$$118$$ −8.00000 −0.736460
$$119$$ 2.00000 0.183340
$$120$$ −3.00000 −0.273861
$$121$$ 1.00000 0.0909091
$$122$$ 10.0000 0.905357
$$123$$ 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
$$130$$ 6.00000 0.526235
$$131$$ 20.0000 1.74741 0.873704 0.486458i $$-0.161711\pi$$
0.873704 + 0.486458i $$0.161711\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ 4.00000 0.346844
$$134$$ −8.00000 −0.691095
$$135$$ 1.00000 0.0860663
$$136$$ −6.00000 −0.514496
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ 8.00000 0.673722
$$142$$ 0 0
$$143$$ −6.00000 −0.501745
$$144$$ −1.00000 −0.0833333
$$145$$ −2.00000 −0.166091
$$146$$ −14.0000 −1.15865
$$147$$ 1.00000 0.0824786
$$148$$ 2.00000 0.164399
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ −12.0000 −0.973329
$$153$$ 2.00000 0.161690
$$154$$ −1.00000 −0.0805823
$$155$$ 4.00000 0.321288
$$156$$ −6.00000 −0.480384
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ −16.0000 −1.27289
$$159$$ −2.00000 −0.158610
$$160$$ 5.00000 0.395285
$$161$$ −4.00000 −0.315244
$$162$$ 1.00000 0.0785674
$$163$$ 24.0000 1.87983 0.939913 0.341415i $$-0.110906\pi$$
0.939913 + 0.341415i $$0.110906\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ −1.00000 −0.0778499
$$166$$ −4.00000 −0.310460
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ −3.00000 −0.231455
$$169$$ 23.0000 1.76923
$$170$$ 2.00000 0.153393
$$171$$ 4.00000 0.305888
$$172$$ −4.00000 −0.304997
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 1.00000 0.0755929
$$176$$ 1.00000 0.0753778
$$177$$ −8.00000 −0.601317
$$178$$ −6.00000 −0.449719
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 6.00000 0.444750
$$183$$ 10.0000 0.739221
$$184$$ 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
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 7.00000 0.505181
$$193$$ −18.0000 −1.29567 −0.647834 0.761781i $$-0.724325\pi$$
−0.647834 + 0.761781i $$0.724325\pi$$
$$194$$ 18.0000 1.29232
$$195$$ 6.00000 0.429669
$$196$$ −1.00000 −0.0714286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ −8.00000 −0.564276
$$202$$ 2.00000 0.140720
$$203$$ −2.00000 −0.140372
$$204$$ −2.00000 −0.140028
$$205$$ 6.00000 0.419058
$$206$$ −16.0000 −1.11477
$$207$$ −4.00000 −0.278019
$$208$$ −6.00000 −0.416025
$$209$$ −4.00000 −0.276686
$$210$$ 1.00000 0.0690066
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 4.00000 0.272798
$$216$$ −3.00000 −0.204124
$$217$$ 4.00000 0.271538
$$218$$ 6.00000 0.406371
$$219$$ −14.0000 −0.946032
$$220$$ 1.00000 0.0674200
$$221$$ 12.0000 0.807207
$$222$$ −2.00000 −0.134231
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 5.00000 0.334077
$$225$$ 1.00000 0.0666667
$$226$$ −6.00000 −0.399114
$$227$$ −28.0000 −1.85843 −0.929213 0.369546i $$-0.879513\pi$$
−0.929213 + 0.369546i $$0.879513\pi$$
$$228$$ −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
$$232$$ 6.00000 0.393919
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 8.00000 0.521862
$$236$$ 8.00000 0.520756
$$237$$ −16.0000 −1.03931
$$238$$ 2.00000 0.129641
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 1.00000 0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ 1.00000 0.0638877
$$246$$ 6.00000 0.382546
$$247$$ 24.0000 1.52708
$$248$$ −12.0000 −0.762001
$$249$$ −4.00000 −0.253490
$$250$$ 1.00000 0.0632456
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ 4.00000 0.251478
$$254$$ 8.00000 0.501965
$$255$$ 2.00000 0.125245
$$256$$ −17.0000 −1.06250
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 4.00000 0.249029
$$259$$ −2.00000 −0.124274
$$260$$ −6.00000 −0.372104
$$261$$ −2.00000 −0.123797
$$262$$ 20.0000 1.23560
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 3.00000 0.184637
$$265$$ −2.00000 −0.122859
$$266$$ 4.00000 0.245256
$$267$$ −6.00000 −0.367194
$$268$$ 8.00000 0.488678
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 24.0000 1.45790 0.728948 0.684569i $$-0.240010\pi$$
0.728948 + 0.684569i $$0.240010\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 6.00000 0.363137
$$274$$ 2.00000 0.120824
$$275$$ −1.00000 −0.0603023
$$276$$ 4.00000 0.240772
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ −12.0000 −0.719712
$$279$$ 4.00000 0.239474
$$280$$ −3.00000 −0.179284
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 8.00000 0.476393
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ 4.00000 0.236940
$$286$$ −6.00000 −0.354787
$$287$$ 6.00000 0.354169
$$288$$ 5.00000 0.294628
$$289$$ −13.0000 −0.764706
$$290$$ −2.00000 −0.117444
$$291$$ 18.0000 1.05518
$$292$$ 14.0000 0.819288
$$293$$ −10.0000 −0.584206 −0.292103 0.956387i $$-0.594355\pi$$
−0.292103 + 0.956387i $$0.594355\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ −8.00000 −0.465778
$$296$$ 6.00000 0.348743
$$297$$ −1.00000 −0.0580259
$$298$$ −10.0000 −0.579284
$$299$$ −24.0000 −1.38796
$$300$$ −1.00000 −0.0577350
$$301$$ 4.00000 0.230556
$$302$$ −16.0000 −0.920697
$$303$$ 2.00000 0.114897
$$304$$ −4.00000 −0.229416
$$305$$ 10.0000 0.572598
$$306$$ 2.00000 0.114332
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 1.00000 0.0569803
$$309$$ −16.0000 −0.910208
$$310$$ 4.00000 0.227185
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ −18.0000 −1.01905
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 1.00000 0.0563436
$$316$$ 16.0000 0.900070
$$317$$ 14.0000 0.786318 0.393159 0.919470i $$-0.371382\pi$$
0.393159 + 0.919470i $$0.371382\pi$$
$$318$$ −2.00000 −0.112154
$$319$$ 2.00000 0.111979
$$320$$ 7.00000 0.391312
$$321$$ −12.0000 −0.669775
$$322$$ −4.00000 −0.222911
$$323$$ 8.00000 0.445132
$$324$$ −1.00000 −0.0555556
$$325$$ 6.00000 0.332820
$$326$$ 24.0000 1.32924
$$327$$ 6.00000 0.331801
$$328$$ −18.0000 −0.993884
$$329$$ 8.00000 0.441054
$$330$$ −1.00000 −0.0550482
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 4.00000 0.219529
$$333$$ −2.00000 −0.109599
$$334$$ −16.0000 −0.875481
$$335$$ −8.00000 −0.437087
$$336$$ −1.00000 −0.0545545
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ 23.0000 1.25104
$$339$$ −6.00000 −0.325875
$$340$$ −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
$$346$$ −18.0000 −0.967686
$$347$$ 28.0000 1.50312 0.751559 0.659665i $$-0.229302\pi$$
0.751559 + 0.659665i $$0.229302\pi$$
$$348$$ 2.00000 0.107211
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ 6.00000 0.320256
$$352$$ −5.00000 −0.266501
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ −8.00000 −0.425195
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 2.00000 0.105851
$$358$$ −12.0000 −0.634220
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ −3.00000 −0.158114
$$361$$ −3.00000 −0.157895
$$362$$ −2.00000 −0.105118
$$363$$ 1.00000 0.0524864
$$364$$ −6.00000 −0.314485
$$365$$ −14.0000 −0.732793
$$366$$ 10.0000 0.522708
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 6.00000 0.312348
$$370$$ −2.00000 −0.103975
$$371$$ −2.00000 −0.103835
$$372$$ −4.00000 −0.207390
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ −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 $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ 8.00000 0.409852
$$382$$ −8.00000 −0.409316
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ −1.00000 −0.0509647
$$386$$ −18.0000 −0.916176
$$387$$ 4.00000 0.203331
$$388$$ −18.0000 −0.913812
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 6.00000 0.303822
$$391$$ −8.00000 −0.404577
$$392$$ −3.00000 −0.151523
$$393$$ 20.0000 1.00887
$$394$$ −6.00000 −0.302276
$$395$$ −16.0000 −0.805047
$$396$$ 1.00000 0.0502519
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ 20.0000 1.00251
$$399$$ 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 $$0.351624\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ 24.0000 1.19553
$$404$$ −2.00000 −0.0995037
$$405$$ 1.00000 0.0496904
$$406$$ −2.00000 −0.0992583
$$407$$ 2.00000 0.0991363
$$408$$ −6.00000 −0.297044
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 6.00000 0.296319
$$411$$ 2.00000 0.0986527
$$412$$ 16.0000 0.788263
$$413$$ −8.00000 −0.393654
$$414$$ −4.00000 −0.196589
$$415$$ −4.00000 −0.196352
$$416$$ 30.0000 1.47087
$$417$$ −12.0000 −0.587643
$$418$$ −4.00000 −0.195646
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ −1.00000 −0.0487950
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 8.00000 0.388973
$$424$$ 6.00000 0.291386
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 10.0000 0.483934
$$428$$ 12.0000 0.580042
$$429$$ −6.00000 −0.289683
$$430$$ 4.00000 0.192897
$$431$$ −32.0000 −1.54139 −0.770693 0.637207i $$-0.780090\pi$$
−0.770693 + 0.637207i $$0.780090\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ 4.00000 0.192006
$$435$$ −2.00000 −0.0958927
$$436$$ −6.00000 −0.287348
$$437$$ −16.0000 −0.765384
$$438$$ −14.0000 −0.668946
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 3.00000 0.143019
$$441$$ 1.00000 0.0476190
$$442$$ 12.0000 0.570782
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ −6.00000 −0.284427
$$446$$ 0 0
$$447$$ −10.0000 −0.472984
$$448$$ 7.00000 0.330719
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ −6.00000 −0.282529
$$452$$ 6.00000 0.282216
$$453$$ −16.0000 −0.751746
$$454$$ −28.0000 −1.31411
$$455$$ 6.00000 0.281284
$$456$$ −12.0000 −0.561951
$$457$$ 30.0000 1.40334 0.701670 0.712502i $$-0.252438\pi$$
0.701670 + 0.712502i $$0.252438\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ 2.00000 0.0933520
$$460$$ 4.00000 0.186501
$$461$$ 34.0000 1.58354 0.791769 0.610821i $$-0.209160\pi$$
0.791769 + 0.610821i $$0.209160\pi$$
$$462$$ −1.00000 −0.0465242
$$463$$ 36.0000 1.67306 0.836531 0.547920i $$-0.184580\pi$$
0.836531 + 0.547920i $$0.184580\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 4.00000 0.185496
$$466$$ 6.00000 0.277945
$$467$$ 28.0000 1.29569 0.647843 0.761774i $$-0.275671\pi$$
0.647843 + 0.761774i $$0.275671\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ −8.00000 −0.369406
$$470$$ 8.00000 0.369012
$$471$$ −2.00000 −0.0921551
$$472$$ 24.0000 1.10469
$$473$$ −4.00000 −0.183920
$$474$$ −16.0000 −0.734904
$$475$$ 4.00000 0.183533
$$476$$ −2.00000 −0.0916698
$$477$$ −2.00000 −0.0915737
$$478$$ 8.00000 0.365911
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 5.00000 0.228218
$$481$$ −12.0000 −0.547153
$$482$$ −18.0000 −0.819878
$$483$$ −4.00000 −0.182006
$$484$$ −1.00000 −0.0454545
$$485$$ 18.0000 0.817338
$$486$$ 1.00000 0.0453609
$$487$$ −28.0000 −1.26880 −0.634401 0.773004i $$-0.718753\pi$$
−0.634401 + 0.773004i $$0.718753\pi$$
$$488$$ −30.0000 −1.35804
$$489$$ 24.0000 1.08532
$$490$$ 1.00000 0.0451754
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ −4.00000 −0.180151
$$494$$ 24.0000 1.07981
$$495$$ −1.00000 −0.0449467
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ −4.00000 −0.179244
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −16.0000 −0.714827
$$502$$ −24.0000 −1.07117
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 2.00000 0.0889988
$$506$$ 4.00000 0.177822
$$507$$ 23.0000 1.02147
$$508$$ −8.00000 −0.354943
$$509$$ 22.0000 0.975133 0.487566 0.873086i $$-0.337885\pi$$
0.487566 + 0.873086i $$0.337885\pi$$
$$510$$ 2.00000 0.0885615
$$511$$ −14.0000 −0.619324
$$512$$ −11.0000 −0.486136
$$513$$ 4.00000 0.176604
$$514$$ 18.0000 0.793946
$$515$$ −16.0000 −0.705044
$$516$$ −4.00000 −0.176090
$$517$$ −8.00000 −0.351840
$$518$$ −2.00000 −0.0878750
$$519$$ −18.0000 −0.790112
$$520$$ −18.0000 −0.789352
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ −20.0000 −0.873704
$$525$$ 1.00000 0.0436436
$$526$$ 16.0000 0.697633
$$527$$ 8.00000 0.348485
$$528$$ 1.00000 0.0435194
$$529$$ −7.00000 −0.304348
$$530$$ −2.00000 −0.0868744
$$531$$ −8.00000 −0.347170
$$532$$ −4.00000 −0.173422
$$533$$ 36.0000 1.55933
$$534$$ −6.00000 −0.259645
$$535$$ −12.0000 −0.518805
$$536$$ 24.0000 1.03664
$$537$$ −12.0000 −0.517838
$$538$$ −10.0000 −0.431131
$$539$$ −1.00000 −0.0430730
$$540$$ −1.00000 −0.0430331
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 24.0000 1.03089
$$543$$ −2.00000 −0.0858282
$$544$$ 10.0000 0.428746
$$545$$ 6.00000 0.257012
$$546$$ 6.00000 0.256776
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ −2.00000 −0.0854358
$$549$$ 10.0000 0.426790
$$550$$ −1.00000 −0.0426401
$$551$$ −8.00000 −0.340811
$$552$$ 12.0000 0.510754
$$553$$ −16.0000 −0.680389
$$554$$ 2.00000 0.0849719
$$555$$ −2.00000 −0.0848953
$$556$$ 12.0000 0.508913
$$557$$ 2.00000 0.0847427 0.0423714 0.999102i $$-0.486509\pi$$
0.0423714 + 0.999102i $$0.486509\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 24.0000 1.01509
$$560$$ −1.00000 −0.0422577
$$561$$ −2.00000 −0.0844401
$$562$$ −6.00000 −0.253095
$$563$$ −44.0000 −1.85438 −0.927189 0.374593i $$-0.877783\pi$$
−0.927189 + 0.374593i $$0.877783\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ −6.00000 −0.252422
$$566$$ −4.00000 −0.168133
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 4.00000 0.167542
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ 6.00000 0.250873
$$573$$ −8.00000 −0.334205
$$574$$ 6.00000 0.250435
$$575$$ −4.00000 −0.166812
$$576$$ 7.00000 0.291667
$$577$$ 34.0000 1.41544 0.707719 0.706494i $$-0.249724\pi$$
0.707719 + 0.706494i $$0.249724\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ −18.0000 −0.748054
$$580$$ 2.00000 0.0830455
$$581$$ −4.00000 −0.165948
$$582$$ 18.0000 0.746124
$$583$$ 2.00000 0.0828315
$$584$$ 42.0000 1.73797
$$585$$ 6.00000 0.248069
$$586$$ −10.0000 −0.413096
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ 16.0000 0.659269
$$590$$ −8.00000 −0.329355
$$591$$ −6.00000 −0.246807
$$592$$ 2.00000 0.0821995
$$593$$ 42.0000 1.72473 0.862367 0.506284i $$-0.168981\pi$$
0.862367 + 0.506284i $$0.168981\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ 2.00000 0.0819920
$$596$$ 10.0000 0.409616
$$597$$ 20.0000 0.818546
$$598$$ −24.0000 −0.981433
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ −3.00000 −0.122474
$$601$$ 6.00000 0.244745 0.122373 0.992484i $$-0.460950\pi$$
0.122373 + 0.992484i $$0.460950\pi$$
$$602$$ 4.00000 0.163028
$$603$$ −8.00000 −0.325785
$$604$$ 16.0000 0.651031
$$605$$ 1.00000 0.0406558
$$606$$ 2.00000 0.0812444
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ 20.0000 0.811107
$$609$$ −2.00000 −0.0810441
$$610$$ 10.0000 0.404888
$$611$$ 48.0000 1.94187
$$612$$ −2.00000 −0.0808452
$$613$$ 18.0000 0.727013 0.363507 0.931592i $$-0.381579\pi$$
0.363507 + 0.931592i $$0.381579\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 6.00000 0.241943
$$616$$ 3.00000 0.120873
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ −16.0000 −0.643614
$$619$$ 24.0000 0.964641 0.482321 0.875995i $$-0.339794\pi$$
0.482321 + 0.875995i $$0.339794\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ −4.00000 −0.160514
$$622$$ 12.0000 0.481156
$$623$$ −6.00000 −0.240385
$$624$$ −6.00000 −0.240192
$$625$$ 1.00000 0.0400000
$$626$$ 26.0000 1.03917
$$627$$ −4.00000 −0.159745
$$628$$ 2.00000 0.0798087
$$629$$ −4.00000 −0.159490
$$630$$ 1.00000 0.0398410
$$631$$ −24.0000 −0.955425 −0.477712 0.878516i $$-0.658534\pi$$
−0.477712 + 0.878516i $$0.658534\pi$$
$$632$$ 48.0000 1.90934
$$633$$ 4.00000 0.158986
$$634$$ 14.0000 0.556011
$$635$$ 8.00000 0.317470
$$636$$ 2.00000 0.0793052
$$637$$ 6.00000 0.237729
$$638$$ 2.00000 0.0791808
$$639$$ 0 0
$$640$$ −3.00000 −0.118585
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ −20.0000 −0.788723 −0.394362 0.918955i $$-0.629034\pi$$
−0.394362 + 0.918955i $$0.629034\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 4.00000 0.157500
$$646$$ 8.00000 0.314756
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ −3.00000 −0.117851
$$649$$ 8.00000 0.314027
$$650$$ 6.00000 0.235339
$$651$$ 4.00000 0.156772
$$652$$ −24.0000 −0.939913
$$653$$ −10.0000 −0.391330 −0.195665 0.980671i $$-0.562687\pi$$
−0.195665 + 0.980671i $$0.562687\pi$$
$$654$$ 6.00000 0.234619
$$655$$ 20.0000 0.781465
$$656$$ −6.00000 −0.234261
$$657$$ −14.0000 −0.546192
$$658$$ 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 $$0.562301\pi$$
$$662$$ −12.0000 −0.466393
$$663$$ 12.0000 0.466041
$$664$$ 12.0000 0.465690
$$665$$ 4.00000 0.155113
$$666$$ −2.00000 −0.0774984
$$667$$ 8.00000 0.309761
$$668$$ 16.0000 0.619059
$$669$$ 0 0
$$670$$ −8.00000 −0.309067
$$671$$ −10.0000 −0.386046
$$672$$ 5.00000 0.192879
$$673$$ −10.0000 −0.385472 −0.192736 0.981251i $$-0.561736\pi$$
−0.192736 + 0.981251i $$0.561736\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 1.00000 0.0384900
$$676$$ −23.0000 −0.884615
$$677$$ −2.00000 −0.0768662 −0.0384331 0.999261i $$-0.512237\pi$$
−0.0384331 + 0.999261i $$0.512237\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ 18.0000 0.690777
$$680$$ −6.00000 −0.230089
$$681$$ −28.0000 −1.07296
$$682$$ −4.00000 −0.153168
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 2.00000 0.0764161
$$686$$ 1.00000 0.0381802
$$687$$ −2.00000 −0.0763048
$$688$$ −4.00000 −0.152499
$$689$$ −12.0000 −0.457164
$$690$$ −4.00000 −0.152277
$$691$$ −24.0000 −0.913003 −0.456502 0.889723i $$-0.650898\pi$$
−0.456502 + 0.889723i $$0.650898\pi$$
$$692$$ 18.0000 0.684257
$$693$$ −1.00000 −0.0379869
$$694$$ 28.0000 1.06287
$$695$$ −12.0000 −0.455186
$$696$$ 6.00000 0.227429
$$697$$ 12.0000 0.454532
$$698$$ −22.0000 −0.832712
$$699$$ 6.00000 0.226941
$$700$$ −1.00000 −0.0377964
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 6.00000 0.226455
$$703$$ −8.00000 −0.301726
$$704$$ −7.00000 −0.263822
$$705$$ 8.00000 0.301297
$$706$$ −30.0000 −1.12906
$$707$$ 2.00000 0.0752177
$$708$$ 8.00000 0.300658
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ 18.0000 0.674579
$$713$$ −16.0000 −0.599205
$$714$$ 2.00000 0.0748481
$$715$$ −6.00000 −0.224387
$$716$$ 12.0000 0.448461
$$717$$ 8.00000 0.298765
$$718$$ −24.0000 −0.895672
$$719$$ 44.0000 1.64092 0.820462 0.571702i $$-0.193717\pi$$
0.820462 + 0.571702i $$0.193717\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ −16.0000 −0.595871
$$722$$ −3.00000 −0.111648
$$723$$ −18.0000 −0.669427
$$724$$ 2.00000 0.0743294
$$725$$ −2.00000 −0.0742781
$$726$$ 1.00000 0.0371135
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ −18.0000 −0.667124
$$729$$ 1.00000 0.0370370
$$730$$ −14.0000 −0.518163
$$731$$ 8.00000 0.295891
$$732$$ −10.0000 −0.369611
$$733$$ −10.0000 −0.369358 −0.184679 0.982799i $$-0.559125\pi$$
−0.184679 + 0.982799i $$0.559125\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 1.00000 0.0368856
$$736$$ −20.0000 −0.737210
$$737$$ 8.00000 0.294684
$$738$$ 6.00000 0.220863
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 2.00000 0.0735215
$$741$$ 24.0000 0.881662
$$742$$ −2.00000 −0.0734223
$$743$$ −48.0000 −1.76095 −0.880475 0.474093i $$-0.842776\pi$$
−0.880475 + 0.474093i $$0.842776\pi$$
$$744$$ −12.0000 −0.439941
$$745$$ −10.0000 −0.366372
$$746$$ 26.0000 0.951928
$$747$$ −4.00000 −0.146352
$$748$$ 2.00000 0.0731272
$$749$$ −12.0000 −0.438470
$$750$$ 1.00000 0.0365148
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ −24.0000 −0.874609
$$754$$ −12.0000 −0.437014
$$755$$ −16.0000 −0.582300
$$756$$ −1.00000 −0.0363696
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ −4.00000 −0.145287
$$759$$ 4.00000 0.145191
$$760$$ −12.0000 −0.435286
$$761$$ −42.0000 −1.52250 −0.761249 0.648459i $$-0.775414\pi$$
−0.761249 + 0.648459i $$0.775414\pi$$
$$762$$ 8.00000 0.289809
$$763$$ 6.00000 0.217215
$$764$$ 8.00000 0.289430
$$765$$ 2.00000 0.0723102
$$766$$ 16.0000 0.578103
$$767$$ −48.0000 −1.73318
$$768$$ −17.0000 −0.613435
$$769$$ −10.0000 −0.360609 −0.180305 0.983611i $$-0.557708\pi$$
−0.180305 + 0.983611i $$0.557708\pi$$
$$770$$ −1.00000 −0.0360375
$$771$$ 18.0000 0.648254
$$772$$ 18.0000 0.647834
$$773$$ 54.0000 1.94225 0.971123 0.238581i $$-0.0766824\pi$$
0.971123 + 0.238581i $$0.0766824\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 4.00000 0.143684
$$776$$ −54.0000 −1.93849
$$777$$ −2.00000 −0.0717496
$$778$$ 6.00000 0.215110
$$779$$ 24.0000 0.859889
$$780$$ −6.00000 −0.214834
$$781$$ 0 0
$$782$$ −8.00000 −0.286079
$$783$$ −2.00000 −0.0714742
$$784$$ −1.00000 −0.0357143
$$785$$ −2.00000 −0.0713831
$$786$$ 20.0000 0.713376
$$787$$ −44.0000 −1.56843 −0.784215 0.620489i $$-0.786934\pi$$
−0.784215 + 0.620489i $$0.786934\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 16.0000 0.569615
$$790$$ −16.0000 −0.569254
$$791$$ −6.00000 −0.213335
$$792$$ 3.00000 0.106600
$$793$$ 60.0000 2.13066
$$794$$ 14.0000 0.496841
$$795$$ −2.00000 −0.0709327
$$796$$ −20.0000 −0.708881
$$797$$ 30.0000 1.06265 0.531327 0.847167i $$-0.321693\pi$$
0.531327 + 0.847167i $$0.321693\pi$$
$$798$$ 4.00000 0.141598
$$799$$ 16.0000 0.566039
$$800$$ 5.00000 0.176777
$$801$$ −6.00000 −0.212000
$$802$$ 18.0000 0.635602
$$803$$ 14.0000 0.494049
$$804$$ 8.00000 0.282138
$$805$$ −4.00000 −0.140981
$$806$$ 24.0000 0.845364
$$807$$ −10.0000 −0.352017
$$808$$ −6.00000 −0.211079
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −12.0000 −0.421377 −0.210688 0.977553i $$-0.567571\pi$$
−0.210688 + 0.977553i $$0.567571\pi$$
$$812$$ 2.00000 0.0701862
$$813$$ 24.0000 0.841717
$$814$$ 2.00000 0.0701000
$$815$$ 24.0000 0.840683
$$816$$ −2.00000 −0.0700140
$$817$$ 16.0000 0.559769
$$818$$ −10.0000 −0.349642
$$819$$ 6.00000 0.209657
$$820$$ −6.00000 −0.209529
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 2.00000 0.0697580
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 48.0000 1.67216
$$825$$ −1.00000 −0.0348155
$$826$$ −8.00000 −0.278356
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 4.00000 0.139010
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ −4.00000 −0.138842
$$831$$ 2.00000 0.0693792
$$832$$ 42.0000 1.45609
$$833$$ 2.00000 0.0692959
$$834$$ −12.0000 −0.415526
$$835$$ −16.0000 −0.553703
$$836$$ 4.00000 0.138343
$$837$$ 4.00000 0.138260
$$838$$ −24.0000 −0.829066
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ −3.00000 −0.103510
$$841$$ −25.0000 −0.862069
$$842$$ −26.0000 −0.896019
$$843$$ −6.00000 −0.206651
$$844$$ −4.00000 −0.137686
$$845$$ 23.0000 0.791224
$$846$$ 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
$$852$$ 0 0
$$853$$ 54.0000 1.84892 0.924462 0.381273i $$-0.124514\pi$$
0.924462 + 0.381273i $$0.124514\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 4.00000 0.136797
$$856$$ 36.0000 1.23045
$$857$$ −30.0000 −1.02478 −0.512390 0.858753i $$-0.671240\pi$$
−0.512390 + 0.858753i $$0.671240\pi$$
$$858$$ −6.00000 −0.204837
$$859$$ 8.00000 0.272956 0.136478 0.990643i $$-0.456422\pi$$
0.136478 + 0.990643i $$0.456422\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 6.00000 0.204479
$$862$$ −32.0000 −1.08992
$$863$$ −12.0000 −0.408485 −0.204242 0.978920i $$-0.565473\pi$$
−0.204242 + 0.978920i $$0.565473\pi$$
$$864$$ 5.00000 0.170103
$$865$$ −18.0000 −0.612018
$$866$$ 34.0000 1.15537
$$867$$ −13.0000 −0.441503
$$868$$ −4.00000 −0.135769
$$869$$ 16.0000 0.542763
$$870$$ −2.00000 −0.0678064
$$871$$ −48.0000 −1.62642
$$872$$ −18.0000 −0.609557
$$873$$ 18.0000 0.609208
$$874$$ −16.0000 −0.541208
$$875$$ 1.00000 0.0338062
$$876$$ 14.0000 0.473016
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ −32.0000 −1.07995
$$879$$ −10.0000 −0.337292
$$880$$ 1.00000 0.0337100
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ 16.0000 0.538443 0.269221 0.963078i $$-0.413234\pi$$
0.269221 + 0.963078i $$0.413234\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ −8.00000 −0.268917
$$886$$ −24.0000 −0.806296
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ 6.00000 0.201347
$$889$$ 8.00000 0.268311
$$890$$ −6.00000 −0.201120
$$891$$ −1.00000 −0.0335013
$$892$$ 0 0
$$893$$ 32.0000 1.07084
$$894$$ −10.0000 −0.334450
$$895$$ −12.0000 −0.401116
$$896$$ −3.00000 −0.100223
$$897$$ −24.0000 −0.801337
$$898$$ 18.0000 0.600668
$$899$$ −8.00000 −0.266815
$$900$$ −1.00000 −0.0333333
$$901$$ −4.00000 −0.133259
$$902$$ −6.00000 −0.199778
$$903$$ 4.00000 0.133112
$$904$$ 18.0000 0.598671
$$905$$ −2.00000 −0.0664822
$$906$$ −16.0000 −0.531564
$$907$$ −32.0000 −1.06254 −0.531271 0.847202i $$-0.678286\pi$$
−0.531271 + 0.847202i $$0.678286\pi$$
$$908$$ 28.0000 0.929213
$$909$$ 2.00000 0.0663358
$$910$$ 6.00000 0.198898
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ 4.00000 0.132381
$$914$$ 30.0000 0.992312
$$915$$ 10.0000 0.330590
$$916$$ 2.00000 0.0660819
$$917$$ 20.0000 0.660458
$$918$$ 2.00000 0.0660098
$$919$$ −24.0000 −0.791687 −0.395843 0.918318i $$-0.629548\pi$$
−0.395843 + 0.918318i $$0.629548\pi$$
$$920$$ 12.0000 0.395628
$$921$$ 12.0000 0.395413
$$922$$ 34.0000 1.11973
$$923$$ 0 0
$$924$$ 1.00000 0.0328976
$$925$$ −2.00000 −0.0657596
$$926$$ 36.0000 1.18303
$$927$$ −16.0000 −0.525509
$$928$$ −10.0000 −0.328266
$$929$$ 10.0000 0.328089 0.164045 0.986453i $$-0.447546\pi$$
0.164045 + 0.986453i $$0.447546\pi$$
$$930$$ 4.00000 0.131165
$$931$$ 4.00000 0.131095
$$932$$ −6.00000 −0.196537
$$933$$ 12.0000 0.392862
$$934$$ 28.0000 0.916188
$$935$$ −2.00000 −0.0654070
$$936$$ −18.0000 −0.588348
$$937$$ 10.0000 0.326686 0.163343 0.986569i $$-0.447772\pi$$
0.163343 + 0.986569i $$0.447772\pi$$
$$938$$ −8.00000 −0.261209
$$939$$ 26.0000 0.848478
$$940$$ −8.00000 −0.260931
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ −2.00000 −0.0651635
$$943$$ −24.0000 −0.781548
$$944$$ 8.00000 0.260378
$$945$$ 1.00000 0.0325300
$$946$$ −4.00000 −0.130051
$$947$$ 32.0000 1.03986 0.519930 0.854209i $$-0.325958\pi$$
0.519930 + 0.854209i $$0.325958\pi$$
$$948$$ 16.0000 0.519656
$$949$$ −84.0000 −2.72676
$$950$$ 4.00000 0.129777
$$951$$ 14.0000 0.453981
$$952$$ −6.00000 −0.194461
$$953$$ 46.0000 1.49009 0.745043 0.667016i $$-0.232429\pi$$
0.745043 + 0.667016i $$0.232429\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ −8.00000 −0.258874
$$956$$ −8.00000 −0.258738
$$957$$ 2.00000 0.0646508
$$958$$ −16.0000 −0.516937
$$959$$ 2.00000 0.0645834
$$960$$ 7.00000 0.225924
$$961$$ −15.0000 −0.483871
$$962$$ −12.0000 −0.386896
$$963$$ −12.0000 −0.386695
$$964$$ 18.0000 0.579741
$$965$$ −18.0000 −0.579441
$$966$$ −4.00000 −0.128698
$$967$$ −56.0000 −1.80084 −0.900419 0.435023i $$-0.856740\pi$$
−0.900419 + 0.435023i $$0.856740\pi$$
$$968$$ −3.00000 −0.0964237
$$969$$ 8.00000 0.256997
$$970$$ 18.0000 0.577945
$$971$$ 32.0000 1.02693 0.513464 0.858111i $$-0.328362\pi$$
0.513464 + 0.858111i $$0.328362\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −12.0000 −0.384702
$$974$$ −28.0000 −0.897178
$$975$$ 6.00000 0.192154
$$976$$ −10.0000 −0.320092
$$977$$ 26.0000 0.831814 0.415907 0.909407i $$-0.363464\pi$$
0.415907 + 0.909407i $$0.363464\pi$$
$$978$$ 24.0000 0.767435
$$979$$ 6.00000 0.191761
$$980$$ −1.00000 −0.0319438
$$981$$ 6.00000 0.191565
$$982$$ 12.0000 0.382935
$$983$$ −16.0000 −0.510321 −0.255160 0.966899i $$-0.582128\pi$$
−0.255160 + 0.966899i $$0.582128\pi$$
$$984$$ −18.0000 −0.573819
$$985$$ −6.00000 −0.191176
$$986$$ −4.00000 −0.127386
$$987$$ 8.00000 0.254643
$$988$$ −24.0000 −0.763542
$$989$$ −16.0000 −0.508770
$$990$$ −1.00000 −0.0317821
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 20.0000 0.635001
$$993$$ −12.0000 −0.380808
$$994$$ 0 0
$$995$$ 20.0000 0.634043
$$996$$ 4.00000 0.126745
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ 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.m.1.1 1
3.2 odd 2 3465.2.a.c.1.1 1
5.4 even 2 5775.2.a.d.1.1 1
7.6 odd 2 8085.2.a.r.1.1 1

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