# Properties

 Label 550.2.a.e.1.1 Level $550$ Weight $2$ Character 550.1 Self dual yes Analytic conductor $4.392$ 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] = [550,2,Mod(1,550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$550 = 2 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 550.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: $$4.39177211117$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 550.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{18} -5.00000 q^{19} -3.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +4.00000 q^{26} -5.00000 q^{27} -3.00000 q^{28} +5.00000 q^{29} +7.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +3.00000 q^{34} -2.00000 q^{36} +7.00000 q^{37} +5.00000 q^{38} -4.00000 q^{39} -8.00000 q^{41} +3.00000 q^{42} +6.00000 q^{43} +1.00000 q^{44} +4.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -3.00000 q^{51} -4.00000 q^{52} -9.00000 q^{53} +5.00000 q^{54} +3.00000 q^{56} -5.00000 q^{57} -5.00000 q^{58} -13.0000 q^{61} -7.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} +12.0000 q^{67} -3.00000 q^{68} -4.00000 q^{69} -3.00000 q^{71} +2.00000 q^{72} +6.00000 q^{73} -7.00000 q^{74} -5.00000 q^{76} -3.00000 q^{77} +4.00000 q^{78} +1.00000 q^{81} +8.00000 q^{82} -4.00000 q^{83} -3.00000 q^{84} -6.00000 q^{86} +5.00000 q^{87} -1.00000 q^{88} -15.0000 q^{89} +12.0000 q^{91} -4.00000 q^{92} +7.00000 q^{93} +8.00000 q^{94} -1.00000 q^{96} +12.0000 q^{97} -2.00000 q^{98} -2.00000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 1.00000 0.288675
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 3.00000 0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 2.00000 0.471405
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ −1.00000 −0.213201
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ −5.00000 −0.962250
$$28$$ −3.00000 −0.566947
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ 7.00000 1.25724 0.628619 0.777714i $$-0.283621\pi$$
0.628619 + 0.777714i $$0.283621\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 1.00000 0.174078
$$34$$ 3.00000 0.514496
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ 7.00000 1.15079 0.575396 0.817875i $$-0.304848\pi$$
0.575396 + 0.817875i $$0.304848\pi$$
$$38$$ 5.00000 0.811107
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ −8.00000 −1.24939 −0.624695 0.780869i $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ 3.00000 0.462910
$$43$$ 6.00000 0.914991 0.457496 0.889212i $$-0.348747\pi$$
0.457496 + 0.889212i $$0.348747\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −3.00000 −0.420084
$$52$$ −4.00000 −0.554700
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 5.00000 0.680414
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ −5.00000 −0.662266
$$58$$ −5.00000 −0.656532
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −13.0000 −1.66448 −0.832240 0.554416i $$-0.812942\pi$$
−0.832240 + 0.554416i $$0.812942\pi$$
$$62$$ −7.00000 −0.889001
$$63$$ 6.00000 0.755929
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −1.00000 −0.123091
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ −3.00000 −0.356034 −0.178017 0.984027i $$-0.556968\pi$$
−0.178017 + 0.984027i $$0.556968\pi$$
$$72$$ 2.00000 0.235702
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ −7.00000 −0.813733
$$75$$ 0 0
$$76$$ −5.00000 −0.573539
$$77$$ −3.00000 −0.341882
$$78$$ 4.00000 0.452911
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 8.00000 0.883452
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ −3.00000 −0.327327
$$85$$ 0 0
$$86$$ −6.00000 −0.646997
$$87$$ 5.00000 0.536056
$$88$$ −1.00000 −0.106600
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 0 0
$$91$$ 12.0000 1.25794
$$92$$ −4.00000 −0.417029
$$93$$ 7.00000 0.725866
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 12.0000 1.21842 0.609208 0.793011i $$-0.291488\pi$$
0.609208 + 0.793011i $$0.291488\pi$$
$$98$$ −2.00000 −0.202031
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 3.00000 0.297044
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ 9.00000 0.874157
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ −5.00000 −0.481125
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 7.00000 0.664411
$$112$$ −3.00000 −0.283473
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 5.00000 0.468293
$$115$$ 0 0
$$116$$ 5.00000 0.464238
$$117$$ 8.00000 0.739600
$$118$$ 0 0
$$119$$ 9.00000 0.825029
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 13.0000 1.17696
$$123$$ −8.00000 −0.721336
$$124$$ 7.00000 0.628619
$$125$$ 0 0
$$126$$ −6.00000 −0.534522
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ 7.00000 0.611593 0.305796 0.952097i $$-0.401077\pi$$
0.305796 + 0.952097i $$0.401077\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ 15.0000 1.30066
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 3.00000 0.257248
$$137$$ 12.0000 1.02523 0.512615 0.858619i $$-0.328677\pi$$
0.512615 + 0.858619i $$0.328677\pi$$
$$138$$ 4.00000 0.340503
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 3.00000 0.251754
$$143$$ −4.00000 −0.334497
$$144$$ −2.00000 −0.166667
$$145$$ 0 0
$$146$$ −6.00000 −0.496564
$$147$$ 2.00000 0.164957
$$148$$ 7.00000 0.575396
$$149$$ −5.00000 −0.409616 −0.204808 0.978802i $$-0.565657\pi$$
−0.204808 + 0.978802i $$0.565657\pi$$
$$150$$ 0 0
$$151$$ 2.00000 0.162758 0.0813788 0.996683i $$-0.474068\pi$$
0.0813788 + 0.996683i $$0.474068\pi$$
$$152$$ 5.00000 0.405554
$$153$$ 6.00000 0.485071
$$154$$ 3.00000 0.241747
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ −13.0000 −1.03751 −0.518756 0.854922i $$-0.673605\pi$$
−0.518756 + 0.854922i $$0.673605\pi$$
$$158$$ 0 0
$$159$$ −9.00000 −0.713746
$$160$$ 0 0
$$161$$ 12.0000 0.945732
$$162$$ −1.00000 −0.0785674
$$163$$ 21.0000 1.64485 0.822423 0.568876i $$-0.192621\pi$$
0.822423 + 0.568876i $$0.192621\pi$$
$$164$$ −8.00000 −0.624695
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ −23.0000 −1.77979 −0.889897 0.456162i $$-0.849224\pi$$
−0.889897 + 0.456162i $$0.849224\pi$$
$$168$$ 3.00000 0.231455
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 10.0000 0.764719
$$172$$ 6.00000 0.457496
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ −5.00000 −0.379049
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 0 0
$$178$$ 15.0000 1.12430
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ −12.0000 −0.889499
$$183$$ −13.0000 −0.960988
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ −7.00000 −0.513265
$$187$$ −3.00000 −0.219382
$$188$$ −8.00000 −0.583460
$$189$$ 15.0000 1.09109
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 1.00000 0.0719816 0.0359908 0.999352i $$-0.488541\pi$$
0.0359908 + 0.999352i $$0.488541\pi$$
$$194$$ −12.0000 −0.861550
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ 22.0000 1.56744 0.783718 0.621117i $$-0.213321\pi$$
0.783718 + 0.621117i $$0.213321\pi$$
$$198$$ 2.00000 0.142134
$$199$$ 25.0000 1.77220 0.886102 0.463491i $$-0.153403\pi$$
0.886102 + 0.463491i $$0.153403\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ −2.00000 −0.140720
$$203$$ −15.0000 −1.05279
$$204$$ −3.00000 −0.210042
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ 8.00000 0.556038
$$208$$ −4.00000 −0.277350
$$209$$ −5.00000 −0.345857
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ −9.00000 −0.618123
$$213$$ −3.00000 −0.205557
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 5.00000 0.340207
$$217$$ −21.0000 −1.42557
$$218$$ 10.0000 0.677285
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ −7.00000 −0.469809
$$223$$ −14.0000 −0.937509 −0.468755 0.883328i $$-0.655297\pi$$
−0.468755 + 0.883328i $$0.655297\pi$$
$$224$$ 3.00000 0.200446
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ −5.00000 −0.331133
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 0 0
$$231$$ −3.00000 −0.197386
$$232$$ −5.00000 −0.328266
$$233$$ −19.0000 −1.24473 −0.622366 0.782727i $$-0.713828\pi$$
−0.622366 + 0.782727i $$0.713828\pi$$
$$234$$ −8.00000 −0.522976
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ −9.00000 −0.583383
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −8.00000 −0.515325 −0.257663 0.966235i $$-0.582952\pi$$
−0.257663 + 0.966235i $$0.582952\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 16.0000 1.02640
$$244$$ −13.0000 −0.832240
$$245$$ 0 0
$$246$$ 8.00000 0.510061
$$247$$ 20.0000 1.27257
$$248$$ −7.00000 −0.444500
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ 6.00000 0.377964
$$253$$ −4.00000 −0.251478
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −28.0000 −1.74659 −0.873296 0.487190i $$-0.838022\pi$$
−0.873296 + 0.487190i $$0.838022\pi$$
$$258$$ −6.00000 −0.373544
$$259$$ −21.0000 −1.30488
$$260$$ 0 0
$$261$$ −10.0000 −0.618984
$$262$$ −7.00000 −0.432461
$$263$$ −9.00000 −0.554964 −0.277482 0.960731i $$-0.589500\pi$$
−0.277482 + 0.960731i $$0.589500\pi$$
$$264$$ −1.00000 −0.0615457
$$265$$ 0 0
$$266$$ −15.0000 −0.919709
$$267$$ −15.0000 −0.917985
$$268$$ 12.0000 0.733017
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 22.0000 1.33640 0.668202 0.743980i $$-0.267064\pi$$
0.668202 + 0.743980i $$0.267064\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 12.0000 0.726273
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ −4.00000 −0.240772
$$277$$ −28.0000 −1.68236 −0.841178 0.540758i $$-0.818138\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ −14.0000 −0.838158
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 8.00000 0.476393
$$283$$ 16.0000 0.951101 0.475551 0.879688i $$-0.342249\pi$$
0.475551 + 0.879688i $$0.342249\pi$$
$$284$$ −3.00000 −0.178017
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 24.0000 1.41668
$$288$$ 2.00000 0.117851
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 12.0000 0.703452
$$292$$ 6.00000 0.351123
$$293$$ 16.0000 0.934730 0.467365 0.884064i $$-0.345203\pi$$
0.467365 + 0.884064i $$0.345203\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 0 0
$$296$$ −7.00000 −0.406867
$$297$$ −5.00000 −0.290129
$$298$$ 5.00000 0.289642
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ −18.0000 −1.03750
$$302$$ −2.00000 −0.115087
$$303$$ 2.00000 0.114897
$$304$$ −5.00000 −0.286770
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ −8.00000 −0.456584 −0.228292 0.973593i $$-0.573314\pi$$
−0.228292 + 0.973593i $$0.573314\pi$$
$$308$$ −3.00000 −0.170941
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −13.0000 −0.737162 −0.368581 0.929596i $$-0.620156\pi$$
−0.368581 + 0.929596i $$0.620156\pi$$
$$312$$ 4.00000 0.226455
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 13.0000 0.733632
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −23.0000 −1.29181 −0.645904 0.763418i $$-0.723520\pi$$
−0.645904 + 0.763418i $$0.723520\pi$$
$$318$$ 9.00000 0.504695
$$319$$ 5.00000 0.279946
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ −12.0000 −0.668734
$$323$$ 15.0000 0.834622
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −21.0000 −1.16308
$$327$$ −10.0000 −0.553001
$$328$$ 8.00000 0.441726
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ −18.0000 −0.989369 −0.494685 0.869072i $$-0.664716\pi$$
−0.494685 + 0.869072i $$0.664716\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ −14.0000 −0.767195
$$334$$ 23.0000 1.25850
$$335$$ 0 0
$$336$$ −3.00000 −0.163663
$$337$$ 7.00000 0.381314 0.190657 0.981657i $$-0.438938\pi$$
0.190657 + 0.981657i $$0.438938\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 7.00000 0.379071
$$342$$ −10.0000 −0.540738
$$343$$ 15.0000 0.809924
$$344$$ −6.00000 −0.323498
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ −18.0000 −0.966291 −0.483145 0.875540i $$-0.660506\pi$$
−0.483145 + 0.875540i $$0.660506\pi$$
$$348$$ 5.00000 0.268028
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 20.0000 1.06752
$$352$$ −1.00000 −0.0533002
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −15.0000 −0.794998
$$357$$ 9.00000 0.476331
$$358$$ −10.0000 −0.528516
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 18.0000 0.946059
$$363$$ 1.00000 0.0524864
$$364$$ 12.0000 0.628971
$$365$$ 0 0
$$366$$ 13.0000 0.679521
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 16.0000 0.832927
$$370$$ 0 0
$$371$$ 27.0000 1.40177
$$372$$ 7.00000 0.362933
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 3.00000 0.155126
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ −20.0000 −1.03005
$$378$$ −15.0000 −0.771517
$$379$$ −10.0000 −0.513665 −0.256833 0.966456i $$-0.582679\pi$$
−0.256833 + 0.966456i $$0.582679\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 8.00000 0.409316
$$383$$ 6.00000 0.306586 0.153293 0.988181i $$-0.451012\pi$$
0.153293 + 0.988181i $$0.451012\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −1.00000 −0.0508987
$$387$$ −12.0000 −0.609994
$$388$$ 12.0000 0.609208
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ −2.00000 −0.101015
$$393$$ 7.00000 0.353103
$$394$$ −22.0000 −1.10834
$$395$$ 0 0
$$396$$ −2.00000 −0.100504
$$397$$ 22.0000 1.10415 0.552074 0.833795i $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ −25.0000 −1.25314
$$399$$ 15.0000 0.750939
$$400$$ 0 0
$$401$$ 27.0000 1.34832 0.674158 0.738587i $$-0.264507\pi$$
0.674158 + 0.738587i $$0.264507\pi$$
$$402$$ −12.0000 −0.598506
$$403$$ −28.0000 −1.39478
$$404$$ 2.00000 0.0995037
$$405$$ 0 0
$$406$$ 15.0000 0.744438
$$407$$ 7.00000 0.346977
$$408$$ 3.00000 0.148522
$$409$$ 20.0000 0.988936 0.494468 0.869196i $$-0.335363\pi$$
0.494468 + 0.869196i $$0.335363\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ −4.00000 −0.197066
$$413$$ 0 0
$$414$$ −8.00000 −0.393179
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ 20.0000 0.979404
$$418$$ 5.00000 0.244558
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ −28.0000 −1.36464 −0.682318 0.731055i $$-0.739028\pi$$
−0.682318 + 0.731055i $$0.739028\pi$$
$$422$$ 13.0000 0.632830
$$423$$ 16.0000 0.777947
$$424$$ 9.00000 0.437079
$$425$$ 0 0
$$426$$ 3.00000 0.145350
$$427$$ 39.0000 1.88734
$$428$$ 12.0000 0.580042
$$429$$ −4.00000 −0.193122
$$430$$ 0 0
$$431$$ 32.0000 1.54139 0.770693 0.637207i $$-0.219910\pi$$
0.770693 + 0.637207i $$0.219910\pi$$
$$432$$ −5.00000 −0.240563
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 21.0000 1.00803
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ 20.0000 0.956730
$$438$$ −6.00000 −0.286691
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ −4.00000 −0.190476
$$442$$ −12.0000 −0.570782
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ 7.00000 0.332205
$$445$$ 0 0
$$446$$ 14.0000 0.662919
$$447$$ −5.00000 −0.236492
$$448$$ −3.00000 −0.141737
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ 6.00000 0.282216
$$453$$ 2.00000 0.0939682
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 5.00000 0.234146
$$457$$ 27.0000 1.26301 0.631503 0.775373i $$-0.282438\pi$$
0.631503 + 0.775373i $$0.282438\pi$$
$$458$$ 0 0
$$459$$ 15.0000 0.700140
$$460$$ 0 0
$$461$$ −13.0000 −0.605470 −0.302735 0.953075i $$-0.597900\pi$$
−0.302735 + 0.953075i $$0.597900\pi$$
$$462$$ 3.00000 0.139573
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 5.00000 0.232119
$$465$$ 0 0
$$466$$ 19.0000 0.880158
$$467$$ −3.00000 −0.138823 −0.0694117 0.997588i $$-0.522112\pi$$
−0.0694117 + 0.997588i $$0.522112\pi$$
$$468$$ 8.00000 0.369800
$$469$$ −36.0000 −1.66233
$$470$$ 0 0
$$471$$ −13.0000 −0.599008
$$472$$ 0 0
$$473$$ 6.00000 0.275880
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 9.00000 0.412514
$$477$$ 18.0000 0.824163
$$478$$ 0 0
$$479$$ −10.0000 −0.456912 −0.228456 0.973554i $$-0.573368\pi$$
−0.228456 + 0.973554i $$0.573368\pi$$
$$480$$ 0 0
$$481$$ −28.0000 −1.27669
$$482$$ 8.00000 0.364390
$$483$$ 12.0000 0.546019
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ −16.0000 −0.725775
$$487$$ 22.0000 0.996915 0.498458 0.866914i $$-0.333900\pi$$
0.498458 + 0.866914i $$0.333900\pi$$
$$488$$ 13.0000 0.588482
$$489$$ 21.0000 0.949653
$$490$$ 0 0
$$491$$ −33.0000 −1.48927 −0.744635 0.667472i $$-0.767376\pi$$
−0.744635 + 0.667472i $$0.767376\pi$$
$$492$$ −8.00000 −0.360668
$$493$$ −15.0000 −0.675566
$$494$$ −20.0000 −0.899843
$$495$$ 0 0
$$496$$ 7.00000 0.314309
$$497$$ 9.00000 0.403705
$$498$$ 4.00000 0.179244
$$499$$ −40.0000 −1.79065 −0.895323 0.445418i $$-0.853055\pi$$
−0.895323 + 0.445418i $$0.853055\pi$$
$$500$$ 0 0
$$501$$ −23.0000 −1.02756
$$502$$ −2.00000 −0.0892644
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ −6.00000 −0.267261
$$505$$ 0 0
$$506$$ 4.00000 0.177822
$$507$$ 3.00000 0.133235
$$508$$ −8.00000 −0.354943
$$509$$ −20.0000 −0.886484 −0.443242 0.896402i $$-0.646172\pi$$
−0.443242 + 0.896402i $$0.646172\pi$$
$$510$$ 0 0
$$511$$ −18.0000 −0.796273
$$512$$ −1.00000 −0.0441942
$$513$$ 25.0000 1.10378
$$514$$ 28.0000 1.23503
$$515$$ 0 0
$$516$$ 6.00000 0.264135
$$517$$ −8.00000 −0.351840
$$518$$ 21.0000 0.922687
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 10.0000 0.437688
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ 7.00000 0.305796
$$525$$ 0 0
$$526$$ 9.00000 0.392419
$$527$$ −21.0000 −0.914774
$$528$$ 1.00000 0.0435194
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 15.0000 0.650332
$$533$$ 32.0000 1.38607
$$534$$ 15.0000 0.649113
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 10.0000 0.431532
$$538$$ 0 0
$$539$$ 2.00000 0.0861461
$$540$$ 0 0
$$541$$ 17.0000 0.730887 0.365444 0.930834i $$-0.380917\pi$$
0.365444 + 0.930834i $$0.380917\pi$$
$$542$$ −22.0000 −0.944981
$$543$$ −18.0000 −0.772454
$$544$$ 3.00000 0.128624
$$545$$ 0 0
$$546$$ −12.0000 −0.513553
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 12.0000 0.512615
$$549$$ 26.0000 1.10965
$$550$$ 0 0
$$551$$ −25.0000 −1.06504
$$552$$ 4.00000 0.170251
$$553$$ 0 0
$$554$$ 28.0000 1.18961
$$555$$ 0 0
$$556$$ 20.0000 0.848189
$$557$$ −28.0000 −1.18640 −0.593199 0.805056i $$-0.702135\pi$$
−0.593199 + 0.805056i $$0.702135\pi$$
$$558$$ 14.0000 0.592667
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ −3.00000 −0.126660
$$562$$ 18.0000 0.759284
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ −16.0000 −0.672530
$$567$$ −3.00000 −0.125988
$$568$$ 3.00000 0.125877
$$569$$ 20.0000 0.838444 0.419222 0.907884i $$-0.362303\pi$$
0.419222 + 0.907884i $$0.362303\pi$$
$$570$$ 0 0
$$571$$ −3.00000 −0.125546 −0.0627730 0.998028i $$-0.519994\pi$$
−0.0627730 + 0.998028i $$0.519994\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ −8.00000 −0.334205
$$574$$ −24.0000 −1.00174
$$575$$ 0 0
$$576$$ −2.00000 −0.0833333
$$577$$ −8.00000 −0.333044 −0.166522 0.986038i $$-0.553254\pi$$
−0.166522 + 0.986038i $$0.553254\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 1.00000 0.0415586
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ −12.0000 −0.497416
$$583$$ −9.00000 −0.372742
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ −16.0000 −0.660954
$$587$$ 7.00000 0.288921 0.144460 0.989511i $$-0.453855\pi$$
0.144460 + 0.989511i $$0.453855\pi$$
$$588$$ 2.00000 0.0824786
$$589$$ −35.0000 −1.44215
$$590$$ 0 0
$$591$$ 22.0000 0.904959
$$592$$ 7.00000 0.287698
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 5.00000 0.205152
$$595$$ 0 0
$$596$$ −5.00000 −0.204808
$$597$$ 25.0000 1.02318
$$598$$ −16.0000 −0.654289
$$599$$ −5.00000 −0.204294 −0.102147 0.994769i $$-0.532571\pi$$
−0.102147 + 0.994769i $$0.532571\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 18.0000 0.733625
$$603$$ −24.0000 −0.977356
$$604$$ 2.00000 0.0813788
$$605$$ 0 0
$$606$$ −2.00000 −0.0812444
$$607$$ 7.00000 0.284121 0.142061 0.989858i $$-0.454627\pi$$
0.142061 + 0.989858i $$0.454627\pi$$
$$608$$ 5.00000 0.202777
$$609$$ −15.0000 −0.607831
$$610$$ 0 0
$$611$$ 32.0000 1.29458
$$612$$ 6.00000 0.242536
$$613$$ 6.00000 0.242338 0.121169 0.992632i $$-0.461336\pi$$
0.121169 + 0.992632i $$0.461336\pi$$
$$614$$ 8.00000 0.322854
$$615$$ 0 0
$$616$$ 3.00000 0.120873
$$617$$ 12.0000 0.483102 0.241551 0.970388i $$-0.422344\pi$$
0.241551 + 0.970388i $$0.422344\pi$$
$$618$$ 4.00000 0.160904
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 20.0000 0.802572
$$622$$ 13.0000 0.521253
$$623$$ 45.0000 1.80289
$$624$$ −4.00000 −0.160128
$$625$$ 0 0
$$626$$ 14.0000 0.559553
$$627$$ −5.00000 −0.199681
$$628$$ −13.0000 −0.518756
$$629$$ −21.0000 −0.837325
$$630$$ 0 0
$$631$$ −43.0000 −1.71180 −0.855901 0.517139i $$-0.826997\pi$$
−0.855901 + 0.517139i $$0.826997\pi$$
$$632$$ 0 0
$$633$$ −13.0000 −0.516704
$$634$$ 23.0000 0.913447
$$635$$ 0 0
$$636$$ −9.00000 −0.356873
$$637$$ −8.00000 −0.316972
$$638$$ −5.00000 −0.197952
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ −33.0000 −1.30342 −0.651711 0.758468i $$-0.725948\pi$$
−0.651711 + 0.758468i $$0.725948\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ 1.00000 0.0394362 0.0197181 0.999806i $$-0.493723\pi$$
0.0197181 + 0.999806i $$0.493723\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ −15.0000 −0.590167
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −21.0000 −0.823055
$$652$$ 21.0000 0.822423
$$653$$ 21.0000 0.821794 0.410897 0.911682i $$-0.365216\pi$$
0.410897 + 0.911682i $$0.365216\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ −8.00000 −0.312348
$$657$$ −12.0000 −0.468165
$$658$$ −24.0000 −0.935617
$$659$$ −35.0000 −1.36341 −0.681703 0.731629i $$-0.738760\pi$$
−0.681703 + 0.731629i $$0.738760\pi$$
$$660$$ 0 0
$$661$$ 12.0000 0.466746 0.233373 0.972387i $$-0.425024\pi$$
0.233373 + 0.972387i $$0.425024\pi$$
$$662$$ 18.0000 0.699590
$$663$$ 12.0000 0.466041
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ 14.0000 0.542489
$$667$$ −20.0000 −0.774403
$$668$$ −23.0000 −0.889897
$$669$$ −14.0000 −0.541271
$$670$$ 0 0
$$671$$ −13.0000 −0.501859
$$672$$ 3.00000 0.115728
$$673$$ 1.00000 0.0385472 0.0192736 0.999814i $$-0.493865\pi$$
0.0192736 + 0.999814i $$0.493865\pi$$
$$674$$ −7.00000 −0.269630
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ −8.00000 −0.307465 −0.153732 0.988113i $$-0.549129\pi$$
−0.153732 + 0.988113i $$0.549129\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ −36.0000 −1.38155
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ −7.00000 −0.268044
$$683$$ −49.0000 −1.87493 −0.937466 0.348076i $$-0.886835\pi$$
−0.937466 + 0.348076i $$0.886835\pi$$
$$684$$ 10.0000 0.382360
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ 0 0
$$688$$ 6.00000 0.228748
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 6.00000 0.227921
$$694$$ 18.0000 0.683271
$$695$$ 0 0
$$696$$ −5.00000 −0.189525
$$697$$ 24.0000 0.909065
$$698$$ 10.0000 0.378506
$$699$$ −19.0000 −0.718646
$$700$$ 0 0
$$701$$ 27.0000 1.01978 0.509888 0.860241i $$-0.329687\pi$$
0.509888 + 0.860241i $$0.329687\pi$$
$$702$$ −20.0000 −0.754851
$$703$$ −35.0000 −1.32005
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ 14.0000 0.526897
$$707$$ −6.00000 −0.225653
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 15.0000 0.562149
$$713$$ −28.0000 −1.04861
$$714$$ −9.00000 −0.336817
$$715$$ 0 0
$$716$$ 10.0000 0.373718
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 15.0000 0.559406 0.279703 0.960087i $$-0.409764\pi$$
0.279703 + 0.960087i $$0.409764\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ −6.00000 −0.223297
$$723$$ −8.00000 −0.297523
$$724$$ −18.0000 −0.668965
$$725$$ 0 0
$$726$$ −1.00000 −0.0371135
$$727$$ −18.0000 −0.667583 −0.333792 0.942647i $$-0.608328\pi$$
−0.333792 + 0.942647i $$0.608328\pi$$
$$728$$ −12.0000 −0.444750
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −18.0000 −0.665754
$$732$$ −13.0000 −0.480494
$$733$$ −14.0000 −0.517102 −0.258551 0.965998i $$-0.583245\pi$$
−0.258551 + 0.965998i $$0.583245\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 12.0000 0.442026
$$738$$ −16.0000 −0.588968
$$739$$ 40.0000 1.47142 0.735712 0.677295i $$-0.236848\pi$$
0.735712 + 0.677295i $$0.236848\pi$$
$$740$$ 0 0
$$741$$ 20.0000 0.734718
$$742$$ −27.0000 −0.991201
$$743$$ 1.00000 0.0366864 0.0183432 0.999832i $$-0.494161\pi$$
0.0183432 + 0.999832i $$0.494161\pi$$
$$744$$ −7.00000 −0.256632
$$745$$ 0 0
$$746$$ 4.00000 0.146450
$$747$$ 8.00000 0.292705
$$748$$ −3.00000 −0.109691
$$749$$ −36.0000 −1.31541
$$750$$ 0 0
$$751$$ 27.0000 0.985244 0.492622 0.870243i $$-0.336039\pi$$
0.492622 + 0.870243i $$0.336039\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 2.00000 0.0728841
$$754$$ 20.0000 0.728357
$$755$$ 0 0
$$756$$ 15.0000 0.545545
$$757$$ −18.0000 −0.654221 −0.327111 0.944986i $$-0.606075\pi$$
−0.327111 + 0.944986i $$0.606075\pi$$
$$758$$ 10.0000 0.363216
$$759$$ −4.00000 −0.145191
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ 8.00000 0.289809
$$763$$ 30.0000 1.08607
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ −6.00000 −0.216789
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 30.0000 1.08183 0.540914 0.841078i $$-0.318079\pi$$
0.540914 + 0.841078i $$0.318079\pi$$
$$770$$ 0 0
$$771$$ −28.0000 −1.00840
$$772$$ 1.00000 0.0359908
$$773$$ 11.0000 0.395643 0.197821 0.980238i $$-0.436613\pi$$
0.197821 + 0.980238i $$0.436613\pi$$
$$774$$ 12.0000 0.431331
$$775$$ 0 0
$$776$$ −12.0000 −0.430775
$$777$$ −21.0000 −0.753371
$$778$$ 30.0000 1.07555
$$779$$ 40.0000 1.43315
$$780$$ 0 0
$$781$$ −3.00000 −0.107348
$$782$$ −12.0000 −0.429119
$$783$$ −25.0000 −0.893427
$$784$$ 2.00000 0.0714286
$$785$$ 0 0
$$786$$ −7.00000 −0.249682
$$787$$ −38.0000 −1.35455 −0.677277 0.735728i $$-0.736840\pi$$
−0.677277 + 0.735728i $$0.736840\pi$$
$$788$$ 22.0000 0.783718
$$789$$ −9.00000 −0.320408
$$790$$ 0 0
$$791$$ −18.0000 −0.640006
$$792$$ 2.00000 0.0710669
$$793$$ 52.0000 1.84657
$$794$$ −22.0000 −0.780751
$$795$$ 0 0
$$796$$ 25.0000 0.886102
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ −15.0000 −0.530994
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ 30.0000 1.06000
$$802$$ −27.0000 −0.953403
$$803$$ 6.00000 0.211735
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ 28.0000 0.986258
$$807$$ 0 0
$$808$$ −2.00000 −0.0703598
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 0 0
$$811$$ 37.0000 1.29925 0.649623 0.760257i $$-0.274927\pi$$
0.649623 + 0.760257i $$0.274927\pi$$
$$812$$ −15.0000 −0.526397
$$813$$ 22.0000 0.771574
$$814$$ −7.00000 −0.245350
$$815$$ 0 0
$$816$$ −3.00000 −0.105021
$$817$$ −30.0000 −1.04957
$$818$$ −20.0000 −0.699284
$$819$$ −24.0000 −0.838628
$$820$$ 0 0
$$821$$ 22.0000 0.767805 0.383903 0.923374i $$-0.374580\pi$$
0.383903 + 0.923374i $$0.374580\pi$$
$$822$$ −12.0000 −0.418548
$$823$$ 26.0000 0.906303 0.453152 0.891434i $$-0.350300\pi$$
0.453152 + 0.891434i $$0.350300\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 22.0000 0.765015 0.382507 0.923952i $$-0.375061\pi$$
0.382507 + 0.923952i $$0.375061\pi$$
$$828$$ 8.00000 0.278019
$$829$$ −20.0000 −0.694629 −0.347314 0.937749i $$-0.612906\pi$$
−0.347314 + 0.937749i $$0.612906\pi$$
$$830$$ 0 0
$$831$$ −28.0000 −0.971309
$$832$$ −4.00000 −0.138675
$$833$$ −6.00000 −0.207888
$$834$$ −20.0000 −0.692543
$$835$$ 0 0
$$836$$ −5.00000 −0.172929
$$837$$ −35.0000 −1.20978
$$838$$ 20.0000 0.690889
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 28.0000 0.964944
$$843$$ −18.0000 −0.619953
$$844$$ −13.0000 −0.447478
$$845$$ 0 0
$$846$$ −16.0000 −0.550091
$$847$$ −3.00000 −0.103081
$$848$$ −9.00000 −0.309061
$$849$$ 16.0000 0.549119
$$850$$ 0 0
$$851$$ −28.0000 −0.959828
$$852$$ −3.00000 −0.102778
$$853$$ −44.0000 −1.50653 −0.753266 0.657716i $$-0.771523\pi$$
−0.753266 + 0.657716i $$0.771523\pi$$
$$854$$ −39.0000 −1.33455
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 17.0000 0.580709 0.290354 0.956919i $$-0.406227\pi$$
0.290354 + 0.956919i $$0.406227\pi$$
$$858$$ 4.00000 0.136558
$$859$$ −10.0000 −0.341196 −0.170598 0.985341i $$-0.554570\pi$$
−0.170598 + 0.985341i $$0.554570\pi$$
$$860$$ 0 0
$$861$$ 24.0000 0.817918
$$862$$ −32.0000 −1.08992
$$863$$ 36.0000 1.22545 0.612727 0.790295i $$-0.290072\pi$$
0.612727 + 0.790295i $$0.290072\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ 14.0000 0.475739
$$867$$ −8.00000 −0.271694
$$868$$ −21.0000 −0.712786
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −48.0000 −1.62642
$$872$$ 10.0000 0.338643
$$873$$ −24.0000 −0.812277
$$874$$ −20.0000 −0.676510
$$875$$ 0 0
$$876$$ 6.00000 0.202721
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ −10.0000 −0.337484
$$879$$ 16.0000 0.539667
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 4.00000 0.134687
$$883$$ −9.00000 −0.302874 −0.151437 0.988467i $$-0.548390\pi$$
−0.151437 + 0.988467i $$0.548390\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ 32.0000 1.07445 0.537227 0.843437i $$-0.319472\pi$$
0.537227 + 0.843437i $$0.319472\pi$$
$$888$$ −7.00000 −0.234905
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ −14.0000 −0.468755
$$893$$ 40.0000 1.33855
$$894$$ 5.00000 0.167225
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ 16.0000 0.534224
$$898$$ −30.0000 −1.00111
$$899$$ 35.0000 1.16732
$$900$$ 0 0
$$901$$ 27.0000 0.899500
$$902$$ 8.00000 0.266371
$$903$$ −18.0000 −0.599002
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ −2.00000 −0.0664455
$$907$$ −23.0000 −0.763702 −0.381851 0.924224i $$-0.624713\pi$$
−0.381851 + 0.924224i $$0.624713\pi$$
$$908$$ 12.0000 0.398234
$$909$$ −4.00000 −0.132672
$$910$$ 0 0
$$911$$ 37.0000 1.22586 0.612932 0.790135i $$-0.289990\pi$$
0.612932 + 0.790135i $$0.289990\pi$$
$$912$$ −5.00000 −0.165567
$$913$$ −4.00000 −0.132381
$$914$$ −27.0000 −0.893081
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −21.0000 −0.693481
$$918$$ −15.0000 −0.495074
$$919$$ 50.0000 1.64935 0.824674 0.565608i $$-0.191359\pi$$
0.824674 + 0.565608i $$0.191359\pi$$
$$920$$ 0 0
$$921$$ −8.00000 −0.263609
$$922$$ 13.0000 0.428132
$$923$$ 12.0000 0.394985
$$924$$ −3.00000 −0.0986928
$$925$$ 0 0
$$926$$ −16.0000 −0.525793
$$927$$ 8.00000 0.262754
$$928$$ −5.00000 −0.164133
$$929$$ 15.0000 0.492134 0.246067 0.969253i $$-0.420862\pi$$
0.246067 + 0.969253i $$0.420862\pi$$
$$930$$ 0 0
$$931$$ −10.0000 −0.327737
$$932$$ −19.0000 −0.622366
$$933$$ −13.0000 −0.425601
$$934$$ 3.00000 0.0981630
$$935$$ 0 0
$$936$$ −8.00000 −0.261488
$$937$$ 42.0000 1.37208 0.686040 0.727564i $$-0.259347\pi$$
0.686040 + 0.727564i $$0.259347\pi$$
$$938$$ 36.0000 1.17544
$$939$$ −14.0000 −0.456873
$$940$$ 0 0
$$941$$ 17.0000 0.554184 0.277092 0.960843i $$-0.410629\pi$$
0.277092 + 0.960843i $$0.410629\pi$$
$$942$$ 13.0000 0.423563
$$943$$ 32.0000 1.04206
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −6.00000 −0.195077
$$947$$ −53.0000 −1.72227 −0.861134 0.508378i $$-0.830245\pi$$
−0.861134 + 0.508378i $$0.830245\pi$$
$$948$$ 0 0
$$949$$ −24.0000 −0.779073
$$950$$ 0 0
$$951$$ −23.0000 −0.745826
$$952$$ −9.00000 −0.291692
$$953$$ −29.0000 −0.939402 −0.469701 0.882826i $$-0.655638\pi$$
−0.469701 + 0.882826i $$0.655638\pi$$
$$954$$ −18.0000 −0.582772
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 5.00000 0.161627
$$958$$ 10.0000 0.323085
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 28.0000 0.902756
$$963$$ −24.0000 −0.773389
$$964$$ −8.00000 −0.257663
$$965$$ 0 0
$$966$$ −12.0000 −0.386094
$$967$$ −13.0000 −0.418052 −0.209026 0.977910i $$-0.567029\pi$$
−0.209026 + 0.977910i $$0.567029\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 15.0000 0.481869
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 16.0000 0.513200
$$973$$ −60.0000 −1.92351
$$974$$ −22.0000 −0.704925
$$975$$ 0 0
$$976$$ −13.0000 −0.416120
$$977$$ −28.0000 −0.895799 −0.447900 0.894084i $$-0.647828\pi$$
−0.447900 + 0.894084i $$0.647828\pi$$
$$978$$ −21.0000 −0.671506
$$979$$ −15.0000 −0.479402
$$980$$ 0 0
$$981$$ 20.0000 0.638551
$$982$$ 33.0000 1.05307
$$983$$ −14.0000 −0.446531 −0.223265 0.974758i $$-0.571672\pi$$
−0.223265 + 0.974758i $$0.571672\pi$$
$$984$$ 8.00000 0.255031
$$985$$ 0 0
$$986$$ 15.0000 0.477697
$$987$$ 24.0000 0.763928
$$988$$ 20.0000 0.636285
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ −7.00000 −0.222250
$$993$$ −18.0000 −0.571213
$$994$$ −9.00000 −0.285463
$$995$$ 0 0
$$996$$ −4.00000 −0.126745
$$997$$ 2.00000 0.0633406 0.0316703 0.999498i $$-0.489917\pi$$
0.0316703 + 0.999498i $$0.489917\pi$$
$$998$$ 40.0000 1.26618
$$999$$ −35.0000 −1.10735
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 550.2.a.e.1.1 1
3.2 odd 2 4950.2.a.ba.1.1 1
4.3 odd 2 4400.2.a.k.1.1 1
5.2 odd 4 110.2.b.a.89.1 2
5.3 odd 4 110.2.b.a.89.2 yes 2
5.4 even 2 550.2.a.j.1.1 1
11.10 odd 2 6050.2.a.bk.1.1 1
15.2 even 4 990.2.c.d.199.2 2
15.8 even 4 990.2.c.d.199.1 2
15.14 odd 2 4950.2.a.q.1.1 1
20.3 even 4 880.2.b.a.529.1 2
20.7 even 4 880.2.b.a.529.2 2
20.19 odd 2 4400.2.a.s.1.1 1
55.32 even 4 1210.2.b.a.969.2 2
55.43 even 4 1210.2.b.a.969.1 2
55.54 odd 2 6050.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.b.a.89.1 2 5.2 odd 4
110.2.b.a.89.2 yes 2 5.3 odd 4
550.2.a.e.1.1 1 1.1 even 1 trivial
550.2.a.j.1.1 1 5.4 even 2
880.2.b.a.529.1 2 20.3 even 4
880.2.b.a.529.2 2 20.7 even 4
990.2.c.d.199.1 2 15.8 even 4
990.2.c.d.199.2 2 15.2 even 4
1210.2.b.a.969.1 2 55.43 even 4
1210.2.b.a.969.2 2 55.32 even 4
4400.2.a.k.1.1 1 4.3 odd 2
4400.2.a.s.1.1 1 20.19 odd 2
4950.2.a.q.1.1 1 15.14 odd 2
4950.2.a.ba.1.1 1 3.2 odd 2
6050.2.a.f.1.1 1 55.54 odd 2
6050.2.a.bk.1.1 1 11.10 odd 2