# Properties

 Label 1155.2.a.d.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)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} +1.00000 q^{22} -3.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -5.00000 q^{32} +1.00000 q^{33} -6.00000 q^{34} +1.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -3.00000 q^{40} -10.0000 q^{41} -1.00000 q^{42} +4.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +1.00000 q^{55} -3.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} -8.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -1.00000 q^{63} +7.00000 q^{64} +2.00000 q^{65} -1.00000 q^{66} -8.00000 q^{67} -6.00000 q^{68} -1.00000 q^{70} -8.00000 q^{71} +3.00000 q^{72} -14.0000 q^{73} -6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +1.00000 q^{77} -2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -16.0000 q^{83} -1.00000 q^{84} -6.00000 q^{85} -4.00000 q^{86} +6.00000 q^{87} -3.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} +2.00000 q^{91} -8.00000 q^{94} -4.00000 q^{95} +5.00000 q^{96} -2.00000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})$$

## 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 $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ −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
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ −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
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −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
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ 1.00000 0.174078
$$34$$ −6.00000 −1.02899
$$35$$ 1.00000 0.169031
$$36$$ −1.00000 −0.166667
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 2.00000 0.320256
$$40$$ −3.00000 −0.474342
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ −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
$$46$$ 0 0
$$47$$ 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
$$51$$ −6.00000 −0.840168
$$52$$ 2.00000 0.277350
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 1.00000 0.134840
$$56$$ −3.00000 −0.400892
$$57$$ −4.00000 −0.529813
$$58$$ 6.00000 0.787839
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 7.00000 0.875000
$$65$$ 2.00000 0.248069
$$66$$ −1.00000 −0.123091
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 0 0
$$70$$ −1.00000 −0.119523
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 3.00000 0.353553
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ −1.00000 −0.115470
$$76$$ −4.00000 −0.458831
$$77$$ 1.00000 0.113961
$$78$$ −2.00000 −0.226455
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 10.0000 1.10432
$$83$$ −16.0000 −1.75623 −0.878114 0.478451i $$-0.841198\pi$$
−0.878114 + 0.478451i $$0.841198\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ −6.00000 −0.650791
$$86$$ −4.00000 −0.431331
$$87$$ 6.00000 0.643268
$$88$$ −3.00000 −0.319801
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ −4.00000 −0.410391
$$96$$ 5.00000 0.510310
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ −1.00000 −0.100504
$$100$$ −1.00000 −0.100000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 6.00000 0.594089
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ −1.00000 −0.0975900
$$106$$ 6.00000 0.582772
$$107$$ 20.0000 1.93347 0.966736 0.255774i $$-0.0823304\pi$$
0.966736 + 0.255774i $$0.0823304\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ −1.00000 −0.0953463
$$111$$ −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 $$0.529988\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ −2.00000 −0.184900
$$118$$ 8.00000 0.736460
$$119$$ −6.00000 −0.550019
$$120$$ 3.00000 0.273861
$$121$$ 1.00000 0.0909091
$$122$$ −2.00000 −0.181071
$$123$$ 10.0000 0.901670
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 1.00000 0.0890871
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 3.00000 0.265165
$$129$$ −4.00000 −0.352180
$$130$$ −2.00000 −0.175412
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ −4.00000 −0.346844
$$134$$ 8.00000 0.691095
$$135$$ 1.00000 0.0860663
$$136$$ 18.0000 1.54349
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ 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
$$142$$ 8.00000 0.671345
$$143$$ 2.00000 0.167248
$$144$$ −1.00000 −0.0833333
$$145$$ 6.00000 0.498273
$$146$$ 14.0000 1.15865
$$147$$ −1.00000 −0.0824786
$$148$$ −6.00000 −0.493197
$$149$$ −14.0000 −1.14692 −0.573462 0.819232i $$-0.694400\pi$$
−0.573462 + 0.819232i $$0.694400\pi$$
$$150$$ 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
$$153$$ 6.00000 0.485071
$$154$$ −1.00000 −0.0805823
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ −6.00000 −0.478852 −0.239426 0.970915i $$-0.576959\pi$$
−0.239426 + 0.970915i $$0.576959\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ 6.00000 0.475831
$$160$$ 5.00000 0.395285
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ 10.0000 0.780869
$$165$$ −1.00000 −0.0778499
$$166$$ 16.0000 1.24184
$$167$$ −4.00000 −0.309529 −0.154765 0.987951i $$-0.549462\pi$$
−0.154765 + 0.987951i $$0.549462\pi$$
$$168$$ 3.00000 0.231455
$$169$$ −9.00000 −0.692308
$$170$$ 6.00000 0.460179
$$171$$ 4.00000 0.305888
$$172$$ −4.00000 −0.304997
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ −1.00000 −0.0755929
$$176$$ 1.00000 0.0753778
$$177$$ 8.00000 0.601317
$$178$$ 10.0000 0.749532
$$179$$ −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 $$0.523682\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ −2.00000 −0.147844
$$184$$ 0 0
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ −6.00000 −0.438763
$$188$$ −8.00000 −0.583460
$$189$$ 1.00000 0.0727393
$$190$$ 4.00000 0.290191
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −7.00000 −0.505181
$$193$$ 18.0000 1.29567 0.647834 0.761781i $$-0.275675\pi$$
0.647834 + 0.761781i $$0.275675\pi$$
$$194$$ 2.00000 0.143592
$$195$$ −2.00000 −0.143223
$$196$$ −1.00000 −0.0714286
$$197$$ −26.0000 −1.85242 −0.926212 0.377004i $$-0.876954\pi$$
−0.926212 + 0.377004i $$0.876954\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 3.00000 0.212132
$$201$$ 8.00000 0.564276
$$202$$ 6.00000 0.422159
$$203$$ 6.00000 0.421117
$$204$$ 6.00000 0.420084
$$205$$ 10.0000 0.698430
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ −4.00000 −0.276686
$$210$$ 1.00000 0.0690066
$$211$$ 16.0000 1.10149 0.550743 0.834675i $$-0.314345\pi$$
0.550743 + 0.834675i $$0.314345\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 8.00000 0.548151
$$214$$ −20.0000 −1.36717
$$215$$ −4.00000 −0.272798
$$216$$ −3.00000 −0.204124
$$217$$ 0 0
$$218$$ 10.0000 0.677285
$$219$$ 14.0000 0.946032
$$220$$ −1.00000 −0.0674200
$$221$$ −12.0000 −0.807207
$$222$$ 6.00000 0.402694
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 5.00000 0.334077
$$225$$ 1.00000 0.0666667
$$226$$ 2.00000 0.133038
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ −1.00000 −0.0657952
$$232$$ −18.0000 −1.18176
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 2.00000 0.130744
$$235$$ −8.00000 −0.521862
$$236$$ 8.00000 0.520756
$$237$$ −4.00000 −0.259828
$$238$$ 6.00000 0.388922
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ −1.00000 −0.0638877
$$246$$ −10.0000 −0.637577
$$247$$ −8.00000 −0.509028
$$248$$ 0 0
$$249$$ 16.0000 1.01396
$$250$$ 1.00000 0.0632456
$$251$$ −16.0000 −1.00991 −0.504956 0.863145i $$-0.668491\pi$$
−0.504956 + 0.863145i $$0.668491\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 0 0
$$254$$ −16.0000 −1.00393
$$255$$ 6.00000 0.375735
$$256$$ −17.0000 −1.06250
$$257$$ 30.0000 1.87135 0.935674 0.352865i $$-0.114792\pi$$
0.935674 + 0.352865i $$0.114792\pi$$
$$258$$ 4.00000 0.249029
$$259$$ −6.00000 −0.372822
$$260$$ −2.00000 −0.124035
$$261$$ −6.00000 −0.371391
$$262$$ 20.0000 1.23560
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 3.00000 0.184637
$$265$$ 6.00000 0.368577
$$266$$ 4.00000 0.245256
$$267$$ 10.0000 0.611990
$$268$$ 8.00000 0.488678
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ −2.00000 −0.121046
$$274$$ −6.00000 −0.362473
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 0 0
$$280$$ 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 $$0.462067\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 4.00000 0.236940
$$286$$ −2.00000 −0.118262
$$287$$ 10.0000 0.590281
$$288$$ −5.00000 −0.294628
$$289$$ 19.0000 1.11765
$$290$$ −6.00000 −0.352332
$$291$$ 2.00000 0.117242
$$292$$ 14.0000 0.819288
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ 8.00000 0.465778
$$296$$ 18.0000 1.04623
$$297$$ 1.00000 0.0580259
$$298$$ 14.0000 0.810998
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ −4.00000 −0.230556
$$302$$ 12.0000 0.690522
$$303$$ 6.00000 0.344691
$$304$$ −4.00000 −0.229416
$$305$$ −2.00000 −0.114520
$$306$$ −6.00000 −0.342997
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ −1.00000 −0.0569803
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ 6.00000 0.339683
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ 6.00000 0.338600
$$315$$ 1.00000 0.0563436
$$316$$ −4.00000 −0.225018
$$317$$ −22.0000 −1.23564 −0.617822 0.786318i $$-0.711985\pi$$
−0.617822 + 0.786318i $$0.711985\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ 6.00000 0.335936
$$320$$ −7.00000 −0.391312
$$321$$ −20.0000 −1.11629
$$322$$ 0 0
$$323$$ 24.0000 1.33540
$$324$$ −1.00000 −0.0555556
$$325$$ −2.00000 −0.110940
$$326$$ −16.0000 −0.886158
$$327$$ 10.0000 0.553001
$$328$$ −30.0000 −1.65647
$$329$$ −8.00000 −0.441054
$$330$$ 1.00000 0.0550482
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 16.0000 0.878114
$$333$$ 6.00000 0.328798
$$334$$ 4.00000 0.218870
$$335$$ 8.00000 0.437087
$$336$$ −1.00000 −0.0545545
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 2.00000 0.108625
$$340$$ 6.00000 0.325396
$$341$$ 0 0
$$342$$ −4.00000 −0.216295
$$343$$ −1.00000 −0.0539949
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ 2.00000 0.106752
$$352$$ 5.00000 0.266501
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ −8.00000 −0.425195
$$355$$ 8.00000 0.424596
$$356$$ 10.0000 0.529999
$$357$$ 6.00000 0.317554
$$358$$ 12.0000 0.634220
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ −3.00000 −0.158114
$$361$$ −3.00000 −0.157895
$$362$$ 2.00000 0.105118
$$363$$ −1.00000 −0.0524864
$$364$$ −2.00000 −0.104828
$$365$$ 14.0000 0.732793
$$366$$ 2.00000 0.104542
$$367$$ −32.0000 −1.67039 −0.835193 0.549957i $$-0.814644\pi$$
−0.835193 + 0.549957i $$0.814644\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ 6.00000 0.311925
$$371$$ 6.00000 0.311504
$$372$$ 0 0
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ 6.00000 0.310253
$$375$$ 1.00000 0.0516398
$$376$$ 24.0000 1.23771
$$377$$ 12.0000 0.618031
$$378$$ −1.00000 −0.0514344
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 4.00000 0.205196
$$381$$ −16.0000 −0.819705
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ −1.00000 −0.0509647
$$386$$ −18.0000 −0.916176
$$387$$ 4.00000 0.203331
$$388$$ 2.00000 0.101535
$$389$$ −10.0000 −0.507020 −0.253510 0.967333i $$-0.581585\pi$$
−0.253510 + 0.967333i $$0.581585\pi$$
$$390$$ 2.00000 0.101274
$$391$$ 0 0
$$392$$ 3.00000 0.151523
$$393$$ 20.0000 1.00887
$$394$$ 26.0000 1.30986
$$395$$ −4.00000 −0.201262
$$396$$ 1.00000 0.0502519
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 4.00000 0.200250
$$400$$ −1.00000 −0.0500000
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ 0 0
$$404$$ 6.00000 0.298511
$$405$$ −1.00000 −0.0496904
$$406$$ −6.00000 −0.297775
$$407$$ −6.00000 −0.297409
$$408$$ −18.0000 −0.891133
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ −10.0000 −0.493865
$$411$$ −6.00000 −0.295958
$$412$$ 0 0
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ 16.0000 0.785409
$$416$$ 10.0000 0.490290
$$417$$ −4.00000 −0.195881
$$418$$ 4.00000 0.195646
$$419$$ −16.0000 −0.781651 −0.390826 0.920465i $$-0.627810\pi$$
−0.390826 + 0.920465i $$0.627810\pi$$
$$420$$ 1.00000 0.0487950
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ −16.0000 −0.778868
$$423$$ 8.00000 0.388973
$$424$$ −18.0000 −0.874157
$$425$$ 6.00000 0.291043
$$426$$ −8.00000 −0.387601
$$427$$ −2.00000 −0.0967868
$$428$$ −20.0000 −0.966736
$$429$$ −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
$$433$$ −10.0000 −0.480569 −0.240285 0.970702i $$-0.577241\pi$$
−0.240285 + 0.970702i $$0.577241\pi$$
$$434$$ 0 0
$$435$$ −6.00000 −0.287678
$$436$$ 10.0000 0.478913
$$437$$ 0 0
$$438$$ −14.0000 −0.668946
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 3.00000 0.143019
$$441$$ 1.00000 0.0476190
$$442$$ 12.0000 0.570782
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 10.0000 0.474045
$$446$$ −8.00000 −0.378811
$$447$$ 14.0000 0.662177
$$448$$ −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
$$451$$ 10.0000 0.470882
$$452$$ 2.00000 0.0940721
$$453$$ 12.0000 0.563809
$$454$$ −24.0000 −1.12638
$$455$$ −2.00000 −0.0937614
$$456$$ −12.0000 −0.561951
$$457$$ 26.0000 1.21623 0.608114 0.793849i $$-0.291926\pi$$
0.608114 + 0.793849i $$0.291926\pi$$
$$458$$ 10.0000 0.467269
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 1.00000 0.0465242
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ −28.0000 −1.29569 −0.647843 0.761774i $$-0.724329\pi$$
−0.647843 + 0.761774i $$0.724329\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 8.00000 0.369406
$$470$$ 8.00000 0.369012
$$471$$ 6.00000 0.276465
$$472$$ −24.0000 −1.10469
$$473$$ −4.00000 −0.183920
$$474$$ 4.00000 0.183726
$$475$$ 4.00000 0.183533
$$476$$ 6.00000 0.275010
$$477$$ −6.00000 −0.274721
$$478$$ −24.0000 −1.09773
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ −5.00000 −0.228218
$$481$$ −12.0000 −0.547153
$$482$$ −22.0000 −1.00207
$$483$$ 0 0
$$484$$ −1.00000 −0.0454545
$$485$$ 2.00000 0.0908153
$$486$$ 1.00000 0.0453609
$$487$$ 4.00000 0.181257 0.0906287 0.995885i $$-0.471112\pi$$
0.0906287 + 0.995885i $$0.471112\pi$$
$$488$$ 6.00000 0.271607
$$489$$ −16.0000 −0.723545
$$490$$ 1.00000 0.0451754
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ −10.0000 −0.450835
$$493$$ −36.0000 −1.62136
$$494$$ 8.00000 0.359937
$$495$$ 1.00000 0.0449467
$$496$$ 0 0
$$497$$ 8.00000 0.358849
$$498$$ −16.0000 −0.716977
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 4.00000 0.178707
$$502$$ 16.0000 0.714115
$$503$$ 20.0000 0.891756 0.445878 0.895094i $$-0.352892\pi$$
0.445878 + 0.895094i $$0.352892\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 6.00000 0.266996
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ −16.0000 −0.709885
$$509$$ −14.0000 −0.620539 −0.310270 0.950649i $$-0.600419\pi$$
−0.310270 + 0.950649i $$0.600419\pi$$
$$510$$ −6.00000 −0.265684
$$511$$ 14.0000 0.619324
$$512$$ 11.0000 0.486136
$$513$$ −4.00000 −0.176604
$$514$$ −30.0000 −1.32324
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ −8.00000 −0.351840
$$518$$ 6.00000 0.263625
$$519$$ 6.00000 0.263371
$$520$$ 6.00000 0.263117
$$521$$ 14.0000 0.613351 0.306676 0.951814i $$-0.400783\pi$$
0.306676 + 0.951814i $$0.400783\pi$$
$$522$$ 6.00000 0.262613
$$523$$ 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
$$526$$ 24.0000 1.04645
$$527$$ 0 0
$$528$$ −1.00000 −0.0435194
$$529$$ −23.0000 −1.00000
$$530$$ −6.00000 −0.260623
$$531$$ −8.00000 −0.347170
$$532$$ 4.00000 0.173422
$$533$$ 20.0000 0.866296
$$534$$ −10.0000 −0.432742
$$535$$ −20.0000 −0.864675
$$536$$ −24.0000 −1.03664
$$537$$ 12.0000 0.517838
$$538$$ −18.0000 −0.776035
$$539$$ −1.00000 −0.0430730
$$540$$ −1.00000 −0.0430331
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 2.00000 0.0858282
$$544$$ −30.0000 −1.28624
$$545$$ 10.0000 0.428353
$$546$$ 2.00000 0.0855921
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 2.00000 0.0853579
$$550$$ 1.00000 0.0426401
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ −4.00000 −0.170097
$$554$$ −22.0000 −0.934690
$$555$$ 6.00000 0.254686
$$556$$ −4.00000 −0.169638
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ −1.00000 −0.0422577
$$561$$ 6.00000 0.253320
$$562$$ −30.0000 −1.26547
$$563$$ −8.00000 −0.337160 −0.168580 0.985688i $$-0.553918\pi$$
−0.168580 + 0.985688i $$0.553918\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 2.00000 0.0841406
$$566$$ −4.00000 −0.168133
$$567$$ −1.00000 −0.0419961
$$568$$ −24.0000 −1.00702
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ −4.00000 −0.167542
$$571$$ 16.0000 0.669579 0.334790 0.942293i $$-0.391335\pi$$
0.334790 + 0.942293i $$0.391335\pi$$
$$572$$ −2.00000 −0.0836242
$$573$$ 0 0
$$574$$ −10.0000 −0.417392
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ −42.0000 −1.74848 −0.874241 0.485491i $$-0.838641\pi$$
−0.874241 + 0.485491i $$0.838641\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ −18.0000 −0.748054
$$580$$ −6.00000 −0.249136
$$581$$ 16.0000 0.663792
$$582$$ −2.00000 −0.0829027
$$583$$ 6.00000 0.248495
$$584$$ −42.0000 −1.73797
$$585$$ 2.00000 0.0826898
$$586$$ 30.0000 1.23929
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ 0 0
$$590$$ −8.00000 −0.329355
$$591$$ 26.0000 1.06950
$$592$$ −6.00000 −0.246598
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ −1.00000 −0.0410305
$$595$$ 6.00000 0.245976
$$596$$ 14.0000 0.573462
$$597$$ 16.0000 0.654836
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ −3.00000 −0.122474
$$601$$ −34.0000 −1.38689 −0.693444 0.720510i $$-0.743908\pi$$
−0.693444 + 0.720510i $$0.743908\pi$$
$$602$$ 4.00000 0.163028
$$603$$ −8.00000 −0.325785
$$604$$ 12.0000 0.488273
$$605$$ −1.00000 −0.0406558
$$606$$ −6.00000 −0.243733
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ −20.0000 −0.811107
$$609$$ −6.00000 −0.243132
$$610$$ 2.00000 0.0809776
$$611$$ −16.0000 −0.647291
$$612$$ −6.00000 −0.242536
$$613$$ 38.0000 1.53481 0.767403 0.641165i $$-0.221549\pi$$
0.767403 + 0.641165i $$0.221549\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ −10.0000 −0.403239
$$616$$ 3.00000 0.120873
$$617$$ 30.0000 1.20775 0.603877 0.797077i $$-0.293622\pi$$
0.603877 + 0.797077i $$0.293622\pi$$
$$618$$ 0 0
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −4.00000 −0.160385
$$623$$ 10.0000 0.400642
$$624$$ −2.00000 −0.0800641
$$625$$ 1.00000 0.0400000
$$626$$ 26.0000 1.03917
$$627$$ 4.00000 0.159745
$$628$$ 6.00000 0.239426
$$629$$ 36.0000 1.43541
$$630$$ −1.00000 −0.0398410
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 12.0000 0.477334
$$633$$ −16.0000 −0.635943
$$634$$ 22.0000 0.873732
$$635$$ −16.0000 −0.634941
$$636$$ −6.00000 −0.237915
$$637$$ −2.00000 −0.0792429
$$638$$ −6.00000 −0.237542
$$639$$ −8.00000 −0.316475
$$640$$ −3.00000 −0.118585
$$641$$ −14.0000 −0.552967 −0.276483 0.961019i $$-0.589169\pi$$
−0.276483 + 0.961019i $$0.589169\pi$$
$$642$$ 20.0000 0.789337
$$643$$ −36.0000 −1.41970 −0.709851 0.704352i $$-0.751238\pi$$
−0.709851 + 0.704352i $$0.751238\pi$$
$$644$$ 0 0
$$645$$ 4.00000 0.157500
$$646$$ −24.0000 −0.944267
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 3.00000 0.117851
$$649$$ 8.00000 0.314027
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ −16.0000 −0.626608
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ 20.0000 0.781465
$$656$$ 10.0000 0.390434
$$657$$ −14.0000 −0.546192
$$658$$ 8.00000 0.311872
$$659$$ 20.0000 0.779089 0.389545 0.921008i $$-0.372632\pi$$
0.389545 + 0.921008i $$0.372632\pi$$
$$660$$ 1.00000 0.0389249
$$661$$ −26.0000 −1.01128 −0.505641 0.862744i $$-0.668744\pi$$
−0.505641 + 0.862744i $$0.668744\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 12.0000 0.466041
$$664$$ −48.0000 −1.86276
$$665$$ 4.00000 0.155113
$$666$$ −6.00000 −0.232495
$$667$$ 0 0
$$668$$ 4.00000 0.154765
$$669$$ −8.00000 −0.309298
$$670$$ −8.00000 −0.309067
$$671$$ −2.00000 −0.0772091
$$672$$ −5.00000 −0.192879
$$673$$ 2.00000 0.0770943 0.0385472 0.999257i $$-0.487727\pi$$
0.0385472 + 0.999257i $$0.487727\pi$$
$$674$$ 14.0000 0.539260
$$675$$ −1.00000 −0.0384900
$$676$$ 9.00000 0.346154
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ −2.00000 −0.0768095
$$679$$ 2.00000 0.0767530
$$680$$ −18.0000 −0.690268
$$681$$ −24.0000 −0.919682
$$682$$ 0 0
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ −6.00000 −0.229248
$$686$$ 1.00000 0.0381802
$$687$$ 10.0000 0.381524
$$688$$ −4.00000 −0.152499
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 1.00000 0.0379869
$$694$$ −12.0000 −0.455514
$$695$$ −4.00000 −0.151729
$$696$$ 18.0000 0.682288
$$697$$ −60.0000 −2.27266
$$698$$ 14.0000 0.529908
$$699$$ 6.00000 0.226941
$$700$$ 1.00000 0.0377964
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ 24.0000 0.905177
$$704$$ −7.00000 −0.263822
$$705$$ 8.00000 0.301297
$$706$$ 18.0000 0.677439
$$707$$ 6.00000 0.225653
$$708$$ −8.00000 −0.300658
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ −8.00000 −0.300235
$$711$$ 4.00000 0.150012
$$712$$ −30.0000 −1.12430
$$713$$ 0 0
$$714$$ −6.00000 −0.224544
$$715$$ −2.00000 −0.0747958
$$716$$ 12.0000 0.448461
$$717$$ −24.0000 −0.896296
$$718$$ 0 0
$$719$$ −12.0000 −0.447524 −0.223762 0.974644i $$-0.571834\pi$$
−0.223762 + 0.974644i $$0.571834\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 0 0
$$722$$ 3.00000 0.111648
$$723$$ −22.0000 −0.818189
$$724$$ 2.00000 0.0743294
$$725$$ −6.00000 −0.222834
$$726$$ 1.00000 0.0371135
$$727$$ 32.0000 1.18681 0.593407 0.804902i $$-0.297782\pi$$
0.593407 + 0.804902i $$0.297782\pi$$
$$728$$ 6.00000 0.222375
$$729$$ 1.00000 0.0370370
$$730$$ −14.0000 −0.518163
$$731$$ 24.0000 0.887672
$$732$$ 2.00000 0.0739221
$$733$$ −26.0000 −0.960332 −0.480166 0.877178i $$-0.659424\pi$$
−0.480166 + 0.877178i $$0.659424\pi$$
$$734$$ 32.0000 1.18114
$$735$$ 1.00000 0.0368856
$$736$$ 0 0
$$737$$ 8.00000 0.294684
$$738$$ 10.0000 0.368105
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ 6.00000 0.220564
$$741$$ 8.00000 0.293887
$$742$$ −6.00000 −0.220267
$$743$$ 8.00000 0.293492 0.146746 0.989174i $$-0.453120\pi$$
0.146746 + 0.989174i $$0.453120\pi$$
$$744$$ 0 0
$$745$$ 14.0000 0.512920
$$746$$ 10.0000 0.366126
$$747$$ −16.0000 −0.585409
$$748$$ 6.00000 0.219382
$$749$$ −20.0000 −0.730784
$$750$$ −1.00000 −0.0365148
$$751$$ 48.0000 1.75154 0.875772 0.482724i $$-0.160353\pi$$
0.875772 + 0.482724i $$0.160353\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 16.0000 0.583072
$$754$$ −12.0000 −0.437014
$$755$$ 12.0000 0.436725
$$756$$ −1.00000 −0.0363696
$$757$$ −34.0000 −1.23575 −0.617876 0.786276i $$-0.712006\pi$$
−0.617876 + 0.786276i $$0.712006\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 0 0
$$760$$ −12.0000 −0.435286
$$761$$ 22.0000 0.797499 0.398750 0.917060i $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ 16.0000 0.579619
$$763$$ 10.0000 0.362024
$$764$$ 0 0
$$765$$ −6.00000 −0.216930
$$766$$ 0 0
$$767$$ 16.0000 0.577727
$$768$$ 17.0000 0.613435
$$769$$ −18.0000 −0.649097 −0.324548 0.945869i $$-0.605212\pi$$
−0.324548 + 0.945869i $$0.605212\pi$$
$$770$$ 1.00000 0.0360375
$$771$$ −30.0000 −1.08042
$$772$$ −18.0000 −0.647834
$$773$$ 26.0000 0.935155 0.467578 0.883952i $$-0.345127\pi$$
0.467578 + 0.883952i $$0.345127\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ 6.00000 0.215249
$$778$$ 10.0000 0.358517
$$779$$ −40.0000 −1.43315
$$780$$ 2.00000 0.0716115
$$781$$ 8.00000 0.286263
$$782$$ 0 0
$$783$$ 6.00000 0.214423
$$784$$ −1.00000 −0.0357143
$$785$$ 6.00000 0.214149
$$786$$ −20.0000 −0.713376
$$787$$ 12.0000 0.427754 0.213877 0.976861i $$-0.431391\pi$$
0.213877 + 0.976861i $$0.431391\pi$$
$$788$$ 26.0000 0.926212
$$789$$ 24.0000 0.854423
$$790$$ 4.00000 0.142314
$$791$$ 2.00000 0.0711118
$$792$$ −3.00000 −0.106600
$$793$$ −4.00000 −0.142044
$$794$$ 14.0000 0.496841
$$795$$ −6.00000 −0.212798
$$796$$ 16.0000 0.567105
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ −4.00000 −0.141598
$$799$$ 48.0000 1.69812
$$800$$ −5.00000 −0.176777
$$801$$ −10.0000 −0.353333
$$802$$ −2.00000 −0.0706225
$$803$$ 14.0000 0.494049
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −18.0000 −0.633630
$$808$$ −18.0000 −0.633238
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ −6.00000 −0.210559
$$813$$ 16.0000 0.561144
$$814$$ 6.00000 0.210300
$$815$$ −16.0000 −0.560456
$$816$$ 6.00000 0.210042
$$817$$ 16.0000 0.559769
$$818$$ 2.00000 0.0699284
$$819$$ 2.00000 0.0698857
$$820$$ −10.0000 −0.349215
$$821$$ −14.0000 −0.488603 −0.244302 0.969699i $$-0.578559\pi$$
−0.244302 + 0.969699i $$0.578559\pi$$
$$822$$ 6.00000 0.209274
$$823$$ 20.0000 0.697156 0.348578 0.937280i $$-0.386665\pi$$
0.348578 + 0.937280i $$0.386665\pi$$
$$824$$ 0 0
$$825$$ 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 $$0.715277\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ −16.0000 −0.555368
$$831$$ −22.0000 −0.763172
$$832$$ −14.0000 −0.485363
$$833$$ 6.00000 0.207888
$$834$$ 4.00000 0.138509
$$835$$ 4.00000 0.138426
$$836$$ 4.00000 0.138343
$$837$$ 0 0
$$838$$ 16.0000 0.552711
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ −3.00000 −0.103510
$$841$$ 7.00000 0.241379
$$842$$ 26.0000 0.896019
$$843$$ −30.0000 −1.03325
$$844$$ −16.0000 −0.550743
$$845$$ 9.00000 0.309609
$$846$$ −8.00000 −0.275046
$$847$$ −1.00000 −0.0343604
$$848$$ 6.00000 0.206041
$$849$$ −4.00000 −0.137280
$$850$$ −6.00000 −0.205798
$$851$$ 0 0
$$852$$ −8.00000 −0.274075
$$853$$ 22.0000 0.753266 0.376633 0.926363i $$-0.377082\pi$$
0.376633 + 0.926363i $$0.377082\pi$$
$$854$$ 2.00000 0.0684386
$$855$$ −4.00000 −0.136797
$$856$$ 60.0000 2.05076
$$857$$ 22.0000 0.751506 0.375753 0.926720i $$-0.377384\pi$$
0.375753 + 0.926720i $$0.377384\pi$$
$$858$$ 2.00000 0.0682789
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 4.00000 0.136399
$$861$$ −10.0000 −0.340799
$$862$$ 0 0
$$863$$ −48.0000 −1.63394 −0.816970 0.576681i $$-0.804348\pi$$
−0.816970 + 0.576681i $$0.804348\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 6.00000 0.204006
$$866$$ 10.0000 0.339814
$$867$$ −19.0000 −0.645274
$$868$$ 0 0
$$869$$ −4.00000 −0.135691
$$870$$ 6.00000 0.203419
$$871$$ 16.0000 0.542139
$$872$$ −30.0000 −1.01593
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ 1.00000 0.0338062
$$876$$ −14.0000 −0.473016
$$877$$ 14.0000 0.472746 0.236373 0.971662i $$-0.424041\pi$$
0.236373 + 0.971662i $$0.424041\pi$$
$$878$$ 0 0
$$879$$ 30.0000 1.01187
$$880$$ −1.00000 −0.0337100
$$881$$ 54.0000 1.81931 0.909653 0.415369i $$-0.136347\pi$$
0.909653 + 0.415369i $$0.136347\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ 32.0000 1.07689 0.538443 0.842662i $$-0.319013\pi$$
0.538443 + 0.842662i $$0.319013\pi$$
$$884$$ 12.0000 0.403604
$$885$$ −8.00000 −0.268917
$$886$$ 12.0000 0.403148
$$887$$ −12.0000 −0.402921 −0.201460 0.979497i $$-0.564569\pi$$
−0.201460 + 0.979497i $$0.564569\pi$$
$$888$$ −18.0000 −0.604040
$$889$$ −16.0000 −0.536623
$$890$$ −10.0000 −0.335201
$$891$$ −1.00000 −0.0335013
$$892$$ −8.00000 −0.267860
$$893$$ 32.0000 1.07084
$$894$$ −14.0000 −0.468230
$$895$$ 12.0000 0.401116
$$896$$ −3.00000 −0.100223
$$897$$ 0 0
$$898$$ 14.0000 0.467186
$$899$$ 0 0
$$900$$ −1.00000 −0.0333333
$$901$$ −36.0000 −1.19933
$$902$$ −10.0000 −0.332964
$$903$$ 4.00000 0.133112
$$904$$ −6.00000 −0.199557
$$905$$ 2.00000 0.0664822
$$906$$ −12.0000 −0.398673
$$907$$ 16.0000 0.531271 0.265636 0.964073i $$-0.414418\pi$$
0.265636 + 0.964073i $$0.414418\pi$$
$$908$$ −24.0000 −0.796468
$$909$$ −6.00000 −0.199007
$$910$$ 2.00000 0.0662994
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 16.0000 0.529523
$$914$$ −26.0000 −0.860004
$$915$$ 2.00000 0.0661180
$$916$$ 10.0000 0.330409
$$917$$ 20.0000 0.660458
$$918$$ 6.00000 0.198030
$$919$$ 52.0000 1.71532 0.857661 0.514216i $$-0.171917\pi$$
0.857661 + 0.514216i $$0.171917\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 30.0000 0.987997
$$923$$ 16.0000 0.526646
$$924$$ 1.00000 0.0328976
$$925$$ 6.00000 0.197279
$$926$$ −4.00000 −0.131448
$$927$$ 0 0
$$928$$ 30.0000 0.984798
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ 6.00000 0.196537
$$933$$ −4.00000 −0.130954
$$934$$ 28.0000 0.916188
$$935$$ 6.00000 0.196221
$$936$$ −6.00000 −0.196116
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ −8.00000 −0.261209
$$939$$ 26.0000 0.848478
$$940$$ 8.00000 0.260931
$$941$$ −46.0000 −1.49956 −0.749779 0.661689i $$-0.769840\pi$$
−0.749779 + 0.661689i $$0.769840\pi$$
$$942$$ −6.00000 −0.195491
$$943$$ 0 0
$$944$$ 8.00000 0.260378
$$945$$ −1.00000 −0.0325300
$$946$$ 4.00000 0.130051
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ 4.00000 0.129914
$$949$$ 28.0000 0.908918
$$950$$ −4.00000 −0.129777
$$951$$ 22.0000 0.713399
$$952$$ −18.0000 −0.583383
$$953$$ −46.0000 −1.49009 −0.745043 0.667016i $$-0.767571\pi$$
−0.745043 + 0.667016i $$0.767571\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ −24.0000 −0.776215
$$957$$ −6.00000 −0.193952
$$958$$ −24.0000 −0.775405
$$959$$ −6.00000 −0.193750
$$960$$ 7.00000 0.225924
$$961$$ −31.0000 −1.00000
$$962$$ 12.0000 0.386896
$$963$$ 20.0000 0.644491
$$964$$ −22.0000 −0.708572
$$965$$ −18.0000 −0.579441
$$966$$ 0 0
$$967$$ −56.0000 −1.80084 −0.900419 0.435023i $$-0.856740\pi$$
−0.900419 + 0.435023i $$0.856740\pi$$
$$968$$ 3.00000 0.0964237
$$969$$ −24.0000 −0.770991
$$970$$ −2.00000 −0.0642161
$$971$$ −16.0000 −0.513464 −0.256732 0.966483i $$-0.582646\pi$$
−0.256732 + 0.966483i $$0.582646\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −4.00000 −0.128234
$$974$$ −4.00000 −0.128168
$$975$$ 2.00000 0.0640513
$$976$$ −2.00000 −0.0640184
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 16.0000 0.511624
$$979$$ 10.0000 0.319601
$$980$$ 1.00000 0.0319438
$$981$$ −10.0000 −0.319275
$$982$$ −12.0000 −0.382935
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 30.0000 0.956365
$$985$$ 26.0000 0.828429
$$986$$ 36.0000 1.14647
$$987$$ 8.00000 0.254643
$$988$$ 8.00000 0.254514
$$989$$ 0 0
$$990$$ −1.00000 −0.0317821
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 0 0
$$993$$ 20.0000 0.634681
$$994$$ −8.00000 −0.253745
$$995$$ 16.0000 0.507234
$$996$$ −16.0000 −0.506979
$$997$$ −58.0000 −1.83688 −0.918439 0.395562i $$-0.870550\pi$$
−0.918439 + 0.395562i $$0.870550\pi$$
$$998$$ −20.0000 −0.633089
$$999$$ −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 1155.2.a.d.1.1 1
3.2 odd 2 3465.2.a.q.1.1 1
5.4 even 2 5775.2.a.u.1.1 1
7.6 odd 2 8085.2.a.h.1.1 1

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