# Properties

 Label 4650.2.a.bu.1.1 Level $4650$ Weight $2$ Character 4650.1 Self dual yes Analytic conductor $37.130$ 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] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.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: $$37.1304369399$$ 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 $$\chi$$ $$=$$ 4650.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +3.00000 q^{21} -3.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} +3.00000 q^{28} +10.0000 q^{29} +1.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +3.00000 q^{37} -1.00000 q^{39} +7.00000 q^{41} +3.00000 q^{42} -1.00000 q^{43} -3.00000 q^{44} +4.00000 q^{46} -7.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -2.00000 q^{51} -1.00000 q^{52} +9.00000 q^{53} +1.00000 q^{54} +3.00000 q^{56} +10.0000 q^{58} +7.00000 q^{61} +1.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} -2.00000 q^{67} -2.00000 q^{68} +4.00000 q^{69} +7.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} +3.00000 q^{74} -9.00000 q^{77} -1.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} +7.00000 q^{82} +9.00000 q^{83} +3.00000 q^{84} -1.00000 q^{86} +10.0000 q^{87} -3.00000 q^{88} +10.0000 q^{89} -3.00000 q^{91} +4.00000 q^{92} +1.00000 q^{93} -7.00000 q^{94} +1.00000 q^{96} -2.00000 q^{97} +2.00000 q^{98} -3.00000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 3.00000 0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ −3.00000 −0.639602
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 1.00000 0.192450
$$28$$ 3.00000 0.566947
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605
$$32$$ 1.00000 0.176777
$$33$$ −3.00000 −0.522233
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ 7.00000 1.09322 0.546608 0.837389i $$-0.315919\pi$$
0.546608 + 0.837389i $$0.315919\pi$$
$$42$$ 3.00000 0.462910
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ −3.00000 −0.452267
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ −7.00000 −1.02105 −0.510527 0.859861i $$-0.670550\pi$$
−0.510527 + 0.859861i $$0.670550\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ −1.00000 −0.138675
$$53$$ 9.00000 1.23625 0.618123 0.786082i $$-0.287894\pi$$
0.618123 + 0.786082i $$0.287894\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ 0 0
$$58$$ 10.0000 1.31306
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 1.00000 0.127000
$$63$$ 3.00000 0.377964
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −3.00000 −0.369274
$$67$$ −2.00000 −0.244339 −0.122169 0.992509i $$-0.538985\pi$$
−0.122169 + 0.992509i $$0.538985\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 7.00000 0.830747 0.415374 0.909651i $$-0.363651\pi$$
0.415374 + 0.909651i $$0.363651\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ 3.00000 0.348743
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −9.00000 −1.02565
$$78$$ −1.00000 −0.113228
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 7.00000 0.773021
$$83$$ 9.00000 0.987878 0.493939 0.869496i $$-0.335557\pi$$
0.493939 + 0.869496i $$0.335557\pi$$
$$84$$ 3.00000 0.327327
$$85$$ 0 0
$$86$$ −1.00000 −0.107833
$$87$$ 10.0000 1.07211
$$88$$ −3.00000 −0.319801
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ 4.00000 0.417029
$$93$$ 1.00000 0.103695
$$94$$ −7.00000 −0.721995
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 2.00000 0.202031
$$99$$ −3.00000 −0.301511
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ −1.00000 −0.0985329 −0.0492665 0.998786i $$-0.515688\pi$$
−0.0492665 + 0.998786i $$0.515688\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 9.00000 0.874157
$$107$$ 18.0000 1.74013 0.870063 0.492941i $$-0.164078\pi$$
0.870063 + 0.492941i $$0.164078\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ 3.00000 0.283473
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 10.0000 0.928477
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 7.00000 0.633750
$$123$$ 7.00000 0.631169
$$124$$ 1.00000 0.0898027
$$125$$ 0 0
$$126$$ 3.00000 0.267261
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −1.00000 −0.0880451
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ −3.00000 −0.261116
$$133$$ 0 0
$$134$$ −2.00000 −0.172774
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 4.00000 0.340503
$$139$$ −5.00000 −0.424094 −0.212047 0.977259i $$-0.568013\pi$$
−0.212047 + 0.977259i $$0.568013\pi$$
$$140$$ 0 0
$$141$$ −7.00000 −0.589506
$$142$$ 7.00000 0.587427
$$143$$ 3.00000 0.250873
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ 2.00000 0.164957
$$148$$ 3.00000 0.246598
$$149$$ −20.0000 −1.63846 −0.819232 0.573462i $$-0.805600\pi$$
−0.819232 + 0.573462i $$0.805600\pi$$
$$150$$ 0 0
$$151$$ −18.0000 −1.46482 −0.732410 0.680864i $$-0.761604\pi$$
−0.732410 + 0.680864i $$0.761604\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ −9.00000 −0.725241
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ −22.0000 −1.75579 −0.877896 0.478852i $$-0.841053\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ 9.00000 0.713746
$$160$$ 0 0
$$161$$ 12.0000 0.945732
$$162$$ 1.00000 0.0785674
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 7.00000 0.546608
$$165$$ 0 0
$$166$$ 9.00000 0.698535
$$167$$ −2.00000 −0.154765 −0.0773823 0.997001i $$-0.524656\pi$$
−0.0773823 + 0.997001i $$0.524656\pi$$
$$168$$ 3.00000 0.231455
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −1.00000 −0.0762493
$$173$$ 4.00000 0.304114 0.152057 0.988372i $$-0.451410\pi$$
0.152057 + 0.988372i $$0.451410\pi$$
$$174$$ 10.0000 0.758098
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ −5.00000 −0.373718 −0.186859 0.982387i $$-0.559831\pi$$
−0.186859 + 0.982387i $$0.559831\pi$$
$$180$$ 0 0
$$181$$ −13.0000 −0.966282 −0.483141 0.875542i $$-0.660504\pi$$
−0.483141 + 0.875542i $$0.660504\pi$$
$$182$$ −3.00000 −0.222375
$$183$$ 7.00000 0.517455
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ 1.00000 0.0733236
$$187$$ 6.00000 0.438763
$$188$$ −7.00000 −0.510527
$$189$$ 3.00000 0.218218
$$190$$ 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
$$193$$ 9.00000 0.647834 0.323917 0.946085i $$-0.395000\pi$$
0.323917 + 0.946085i $$0.395000\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ −27.0000 −1.92367 −0.961835 0.273629i $$-0.911776\pi$$
−0.961835 + 0.273629i $$0.911776\pi$$
$$198$$ −3.00000 −0.213201
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ −2.00000 −0.141069
$$202$$ 2.00000 0.140720
$$203$$ 30.0000 2.10559
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ −1.00000 −0.0696733
$$207$$ 4.00000 0.278019
$$208$$ −1.00000 −0.0693375
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 2.00000 0.137686 0.0688428 0.997628i $$-0.478069\pi$$
0.0688428 + 0.997628i $$0.478069\pi$$
$$212$$ 9.00000 0.618123
$$213$$ 7.00000 0.479632
$$214$$ 18.0000 1.23045
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 3.00000 0.203653
$$218$$ −10.0000 −0.677285
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ 2.00000 0.134535
$$222$$ 3.00000 0.201347
$$223$$ −6.00000 −0.401790 −0.200895 0.979613i $$-0.564385\pi$$
−0.200895 + 0.979613i $$0.564385\pi$$
$$224$$ 3.00000 0.200446
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ −9.00000 −0.592157
$$232$$ 10.0000 0.656532
$$233$$ 9.00000 0.589610 0.294805 0.955557i $$-0.404745\pi$$
0.294805 + 0.955557i $$0.404745\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −10.0000 −0.649570
$$238$$ −6.00000 −0.388922
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 12.0000 0.772988 0.386494 0.922292i $$-0.373686\pi$$
0.386494 + 0.922292i $$0.373686\pi$$
$$242$$ −2.00000 −0.128565
$$243$$ 1.00000 0.0641500
$$244$$ 7.00000 0.448129
$$245$$ 0 0
$$246$$ 7.00000 0.446304
$$247$$ 0 0
$$248$$ 1.00000 0.0635001
$$249$$ 9.00000 0.570352
$$250$$ 0 0
$$251$$ −3.00000 −0.189358 −0.0946792 0.995508i $$-0.530183\pi$$
−0.0946792 + 0.995508i $$0.530183\pi$$
$$252$$ 3.00000 0.188982
$$253$$ −12.0000 −0.754434
$$254$$ −2.00000 −0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −7.00000 −0.436648 −0.218324 0.975876i $$-0.570059\pi$$
−0.218324 + 0.975876i $$0.570059\pi$$
$$258$$ −1.00000 −0.0622573
$$259$$ 9.00000 0.559233
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ 12.0000 0.741362
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 10.0000 0.611990
$$268$$ −2.00000 −0.122169
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ −3.00000 −0.181568
$$274$$ −2.00000 −0.120824
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ 18.0000 1.08152 0.540758 0.841178i $$-0.318138\pi$$
0.540758 + 0.841178i $$0.318138\pi$$
$$278$$ −5.00000 −0.299880
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ 7.00000 0.417585 0.208792 0.977960i $$-0.433047\pi$$
0.208792 + 0.977960i $$0.433047\pi$$
$$282$$ −7.00000 −0.416844
$$283$$ 24.0000 1.42665 0.713326 0.700832i $$-0.247188\pi$$
0.713326 + 0.700832i $$0.247188\pi$$
$$284$$ 7.00000 0.415374
$$285$$ 0 0
$$286$$ 3.00000 0.177394
$$287$$ 21.0000 1.23959
$$288$$ 1.00000 0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 4.00000 0.234082
$$293$$ 24.0000 1.40209 0.701047 0.713115i $$-0.252716\pi$$
0.701047 + 0.713115i $$0.252716\pi$$
$$294$$ 2.00000 0.116642
$$295$$ 0 0
$$296$$ 3.00000 0.174371
$$297$$ −3.00000 −0.174078
$$298$$ −20.0000 −1.15857
$$299$$ −4.00000 −0.231326
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ −18.0000 −1.03578
$$303$$ 2.00000 0.114897
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ −22.0000 −1.25561 −0.627803 0.778372i $$-0.716046\pi$$
−0.627803 + 0.778372i $$0.716046\pi$$
$$308$$ −9.00000 −0.512823
$$309$$ −1.00000 −0.0568880
$$310$$ 0 0
$$311$$ 27.0000 1.53103 0.765515 0.643418i $$-0.222484\pi$$
0.765515 + 0.643418i $$0.222484\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ 24.0000 1.35656 0.678280 0.734803i $$-0.262726\pi$$
0.678280 + 0.734803i $$0.262726\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ −22.0000 −1.23564 −0.617822 0.786318i $$-0.711985\pi$$
−0.617822 + 0.786318i $$0.711985\pi$$
$$318$$ 9.00000 0.504695
$$319$$ −30.0000 −1.67968
$$320$$ 0 0
$$321$$ 18.0000 1.00466
$$322$$ 12.0000 0.668734
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ −10.0000 −0.553001
$$328$$ 7.00000 0.386510
$$329$$ −21.0000 −1.15777
$$330$$ 0 0
$$331$$ −3.00000 −0.164895 −0.0824475 0.996595i $$-0.526274\pi$$
−0.0824475 + 0.996595i $$0.526274\pi$$
$$332$$ 9.00000 0.493939
$$333$$ 3.00000 0.164399
$$334$$ −2.00000 −0.109435
$$335$$ 0 0
$$336$$ 3.00000 0.163663
$$337$$ 8.00000 0.435788 0.217894 0.975972i $$-0.430081\pi$$
0.217894 + 0.975972i $$0.430081\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −3.00000 −0.162459
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ −1.00000 −0.0539164
$$345$$ 0 0
$$346$$ 4.00000 0.215041
$$347$$ −7.00000 −0.375780 −0.187890 0.982190i $$-0.560165\pi$$
−0.187890 + 0.982190i $$0.560165\pi$$
$$348$$ 10.0000 0.536056
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ −3.00000 −0.159901
$$353$$ 24.0000 1.27739 0.638696 0.769460i $$-0.279474\pi$$
0.638696 + 0.769460i $$0.279474\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ −6.00000 −0.317554
$$358$$ −5.00000 −0.264258
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ −13.0000 −0.683265
$$363$$ −2.00000 −0.104973
$$364$$ −3.00000 −0.157243
$$365$$ 0 0
$$366$$ 7.00000 0.365896
$$367$$ 28.0000 1.46159 0.730794 0.682598i $$-0.239150\pi$$
0.730794 + 0.682598i $$0.239150\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 7.00000 0.364405
$$370$$ 0 0
$$371$$ 27.0000 1.40177
$$372$$ 1.00000 0.0518476
$$373$$ 24.0000 1.24267 0.621336 0.783544i $$-0.286590\pi$$
0.621336 + 0.783544i $$0.286590\pi$$
$$374$$ 6.00000 0.310253
$$375$$ 0 0
$$376$$ −7.00000 −0.360997
$$377$$ −10.0000 −0.515026
$$378$$ 3.00000 0.154303
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ −2.00000 −0.102463
$$382$$ −8.00000 −0.409316
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 9.00000 0.458088
$$387$$ −1.00000 −0.0508329
$$388$$ −2.00000 −0.101535
$$389$$ 10.0000 0.507020 0.253510 0.967333i $$-0.418415\pi$$
0.253510 + 0.967333i $$0.418415\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ 2.00000 0.101015
$$393$$ 12.0000 0.605320
$$394$$ −27.0000 −1.36024
$$395$$ 0 0
$$396$$ −3.00000 −0.150756
$$397$$ 28.0000 1.40528 0.702640 0.711546i $$-0.252005\pi$$
0.702640 + 0.711546i $$0.252005\pi$$
$$398$$ −20.0000 −1.00251
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −8.00000 −0.399501 −0.199750 0.979847i $$-0.564013\pi$$
−0.199750 + 0.979847i $$0.564013\pi$$
$$402$$ −2.00000 −0.0997509
$$403$$ −1.00000 −0.0498135
$$404$$ 2.00000 0.0995037
$$405$$ 0 0
$$406$$ 30.0000 1.48888
$$407$$ −9.00000 −0.446113
$$408$$ −2.00000 −0.0990148
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ −2.00000 −0.0986527
$$412$$ −1.00000 −0.0492665
$$413$$ 0 0
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ −5.00000 −0.244851
$$418$$ 0 0
$$419$$ −30.0000 −1.46560 −0.732798 0.680446i $$-0.761786\pi$$
−0.732798 + 0.680446i $$0.761786\pi$$
$$420$$ 0 0
$$421$$ −28.0000 −1.36464 −0.682318 0.731055i $$-0.739028\pi$$
−0.682318 + 0.731055i $$0.739028\pi$$
$$422$$ 2.00000 0.0973585
$$423$$ −7.00000 −0.340352
$$424$$ 9.00000 0.437079
$$425$$ 0 0
$$426$$ 7.00000 0.339151
$$427$$ 21.0000 1.01626
$$428$$ 18.0000 0.870063
$$429$$ 3.00000 0.144841
$$430$$ 0 0
$$431$$ 7.00000 0.337178 0.168589 0.985686i $$-0.446079\pi$$
0.168589 + 0.985686i $$0.446079\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 4.00000 0.192228 0.0961139 0.995370i $$-0.469359\pi$$
0.0961139 + 0.995370i $$0.469359\pi$$
$$434$$ 3.00000 0.144005
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ 0 0
$$438$$ 4.00000 0.191127
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 2.00000 0.0951303
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ 3.00000 0.142374
$$445$$ 0 0
$$446$$ −6.00000 −0.284108
$$447$$ −20.0000 −0.945968
$$448$$ 3.00000 0.141737
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ 0 0
$$451$$ −21.0000 −0.988851
$$452$$ −6.00000 −0.282216
$$453$$ −18.0000 −0.845714
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 8.00000 0.374224 0.187112 0.982339i $$-0.440087\pi$$
0.187112 + 0.982339i $$0.440087\pi$$
$$458$$ 10.0000 0.467269
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ −3.00000 −0.139724 −0.0698620 0.997557i $$-0.522256\pi$$
−0.0698620 + 0.997557i $$0.522256\pi$$
$$462$$ −9.00000 −0.418718
$$463$$ 14.0000 0.650635 0.325318 0.945605i $$-0.394529\pi$$
0.325318 + 0.945605i $$0.394529\pi$$
$$464$$ 10.0000 0.464238
$$465$$ 0 0
$$466$$ 9.00000 0.416917
$$467$$ 18.0000 0.832941 0.416470 0.909149i $$-0.363267\pi$$
0.416470 + 0.909149i $$0.363267\pi$$
$$468$$ −1.00000 −0.0462250
$$469$$ −6.00000 −0.277054
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ 0 0
$$473$$ 3.00000 0.137940
$$474$$ −10.0000 −0.459315
$$475$$ 0 0
$$476$$ −6.00000 −0.275010
$$477$$ 9.00000 0.412082
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −3.00000 −0.136788
$$482$$ 12.0000 0.546585
$$483$$ 12.0000 0.546019
$$484$$ −2.00000 −0.0909091
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 28.0000 1.26880 0.634401 0.773004i $$-0.281247\pi$$
0.634401 + 0.773004i $$0.281247\pi$$
$$488$$ 7.00000 0.316875
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 7.00000 0.315584
$$493$$ −20.0000 −0.900755
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 1.00000 0.0449013
$$497$$ 21.0000 0.941979
$$498$$ 9.00000 0.403300
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 0 0
$$501$$ −2.00000 −0.0893534
$$502$$ −3.00000 −0.133897
$$503$$ −11.0000 −0.490466 −0.245233 0.969464i $$-0.578864\pi$$
−0.245233 + 0.969464i $$0.578864\pi$$
$$504$$ 3.00000 0.133631
$$505$$ 0 0
$$506$$ −12.0000 −0.533465
$$507$$ −12.0000 −0.532939
$$508$$ −2.00000 −0.0887357
$$509$$ −15.0000 −0.664863 −0.332432 0.943127i $$-0.607869\pi$$
−0.332432 + 0.943127i $$0.607869\pi$$
$$510$$ 0 0
$$511$$ 12.0000 0.530849
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −7.00000 −0.308757
$$515$$ 0 0
$$516$$ −1.00000 −0.0440225
$$517$$ 21.0000 0.923579
$$518$$ 9.00000 0.395437
$$519$$ 4.00000 0.175581
$$520$$ 0 0
$$521$$ −33.0000 −1.44576 −0.722878 0.690976i $$-0.757181\pi$$
−0.722878 + 0.690976i $$0.757181\pi$$
$$522$$ 10.0000 0.437688
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ −6.00000 −0.261612
$$527$$ −2.00000 −0.0871214
$$528$$ −3.00000 −0.130558
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −7.00000 −0.303204
$$534$$ 10.0000 0.432742
$$535$$ 0 0
$$536$$ −2.00000 −0.0863868
$$537$$ −5.00000 −0.215766
$$538$$ −10.0000 −0.431131
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ −8.00000 −0.343947 −0.171973 0.985102i $$-0.555014\pi$$
−0.171973 + 0.985102i $$0.555014\pi$$
$$542$$ −8.00000 −0.343629
$$543$$ −13.0000 −0.557883
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ −3.00000 −0.128388
$$547$$ −22.0000 −0.940652 −0.470326 0.882493i $$-0.655864\pi$$
−0.470326 + 0.882493i $$0.655864\pi$$
$$548$$ −2.00000 −0.0854358
$$549$$ 7.00000 0.298753
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 4.00000 0.170251
$$553$$ −30.0000 −1.27573
$$554$$ 18.0000 0.764747
$$555$$ 0 0
$$556$$ −5.00000 −0.212047
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ 1.00000 0.0423334
$$559$$ 1.00000 0.0422955
$$560$$ 0 0
$$561$$ 6.00000 0.253320
$$562$$ 7.00000 0.295277
$$563$$ 34.0000 1.43293 0.716465 0.697623i $$-0.245759\pi$$
0.716465 + 0.697623i $$0.245759\pi$$
$$564$$ −7.00000 −0.294753
$$565$$ 0 0
$$566$$ 24.0000 1.00880
$$567$$ 3.00000 0.125988
$$568$$ 7.00000 0.293713
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ −23.0000 −0.962520 −0.481260 0.876578i $$-0.659821\pi$$
−0.481260 + 0.876578i $$0.659821\pi$$
$$572$$ 3.00000 0.125436
$$573$$ −8.00000 −0.334205
$$574$$ 21.0000 0.876523
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −22.0000 −0.915872 −0.457936 0.888985i $$-0.651411\pi$$
−0.457936 + 0.888985i $$0.651411\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 9.00000 0.374027
$$580$$ 0 0
$$581$$ 27.0000 1.12015
$$582$$ −2.00000 −0.0829027
$$583$$ −27.0000 −1.11823
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ 24.0000 0.991431
$$587$$ −17.0000 −0.701665 −0.350833 0.936438i $$-0.614101\pi$$
−0.350833 + 0.936438i $$0.614101\pi$$
$$588$$ 2.00000 0.0824786
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −27.0000 −1.11063
$$592$$ 3.00000 0.123299
$$593$$ −1.00000 −0.0410651 −0.0205325 0.999789i $$-0.506536\pi$$
−0.0205325 + 0.999789i $$0.506536\pi$$
$$594$$ −3.00000 −0.123091
$$595$$ 0 0
$$596$$ −20.0000 −0.819232
$$597$$ −20.0000 −0.818546
$$598$$ −4.00000 −0.163572
$$599$$ −15.0000 −0.612883 −0.306442 0.951889i $$-0.599138\pi$$
−0.306442 + 0.951889i $$0.599138\pi$$
$$600$$ 0 0
$$601$$ 12.0000 0.489490 0.244745 0.969587i $$-0.421296\pi$$
0.244745 + 0.969587i $$0.421296\pi$$
$$602$$ −3.00000 −0.122271
$$603$$ −2.00000 −0.0814463
$$604$$ −18.0000 −0.732410
$$605$$ 0 0
$$606$$ 2.00000 0.0812444
$$607$$ 43.0000 1.74532 0.872658 0.488332i $$-0.162394\pi$$
0.872658 + 0.488332i $$0.162394\pi$$
$$608$$ 0 0
$$609$$ 30.0000 1.21566
$$610$$ 0 0
$$611$$ 7.00000 0.283190
$$612$$ −2.00000 −0.0808452
$$613$$ −26.0000 −1.05013 −0.525065 0.851062i $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ −22.0000 −0.887848
$$615$$ 0 0
$$616$$ −9.00000 −0.362620
$$617$$ 3.00000 0.120775 0.0603877 0.998175i $$-0.480766\pi$$
0.0603877 + 0.998175i $$0.480766\pi$$
$$618$$ −1.00000 −0.0402259
$$619$$ −5.00000 −0.200967 −0.100483 0.994939i $$-0.532039\pi$$
−0.100483 + 0.994939i $$0.532039\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 27.0000 1.08260
$$623$$ 30.0000 1.20192
$$624$$ −1.00000 −0.0400320
$$625$$ 0 0
$$626$$ 24.0000 0.959233
$$627$$ 0 0
$$628$$ −22.0000 −0.877896
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ 2.00000 0.0796187 0.0398094 0.999207i $$-0.487325\pi$$
0.0398094 + 0.999207i $$0.487325\pi$$
$$632$$ −10.0000 −0.397779
$$633$$ 2.00000 0.0794929
$$634$$ −22.0000 −0.873732
$$635$$ 0 0
$$636$$ 9.00000 0.356873
$$637$$ −2.00000 −0.0792429
$$638$$ −30.0000 −1.18771
$$639$$ 7.00000 0.276916
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 18.0000 0.710403
$$643$$ −31.0000 −1.22252 −0.611260 0.791430i $$-0.709337\pi$$
−0.611260 + 0.791430i $$0.709337\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −42.0000 −1.65119 −0.825595 0.564263i $$-0.809160\pi$$
−0.825595 + 0.564263i $$0.809160\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 3.00000 0.117579
$$652$$ 4.00000 0.156652
$$653$$ −16.0000 −0.626128 −0.313064 0.949732i $$-0.601356\pi$$
−0.313064 + 0.949732i $$0.601356\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ 0 0
$$656$$ 7.00000 0.273304
$$657$$ 4.00000 0.156055
$$658$$ −21.0000 −0.818665
$$659$$ −40.0000 −1.55818 −0.779089 0.626913i $$-0.784318\pi$$
−0.779089 + 0.626913i $$0.784318\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ −3.00000 −0.116598
$$663$$ 2.00000 0.0776736
$$664$$ 9.00000 0.349268
$$665$$ 0 0
$$666$$ 3.00000 0.116248
$$667$$ 40.0000 1.54881
$$668$$ −2.00000 −0.0773823
$$669$$ −6.00000 −0.231973
$$670$$ 0 0
$$671$$ −21.0000 −0.810696
$$672$$ 3.00000 0.115728
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ 8.00000 0.308148
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ −27.0000 −1.03769 −0.518847 0.854867i $$-0.673639\pi$$
−0.518847 + 0.854867i $$0.673639\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ −6.00000 −0.230259
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ −3.00000 −0.114876
$$683$$ −26.0000 −0.994862 −0.497431 0.867503i $$-0.665723\pi$$
−0.497431 + 0.867503i $$0.665723\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ 10.0000 0.381524
$$688$$ −1.00000 −0.0381246
$$689$$ −9.00000 −0.342873
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 4.00000 0.152057
$$693$$ −9.00000 −0.341882
$$694$$ −7.00000 −0.265716
$$695$$ 0 0
$$696$$ 10.0000 0.379049
$$697$$ −14.0000 −0.530288
$$698$$ 0 0
$$699$$ 9.00000 0.340411
$$700$$ 0 0
$$701$$ −48.0000 −1.81293 −0.906467 0.422276i $$-0.861231\pi$$
−0.906467 + 0.422276i $$0.861231\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ 0 0
$$704$$ −3.00000 −0.113067
$$705$$ 0 0
$$706$$ 24.0000 0.903252
$$707$$ 6.00000 0.225653
$$708$$ 0 0
$$709$$ 50.0000 1.87779 0.938895 0.344204i $$-0.111851\pi$$
0.938895 + 0.344204i $$0.111851\pi$$
$$710$$ 0 0
$$711$$ −10.0000 −0.375029
$$712$$ 10.0000 0.374766
$$713$$ 4.00000 0.149801
$$714$$ −6.00000 −0.224544
$$715$$ 0 0
$$716$$ −5.00000 −0.186859
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 50.0000 1.86469 0.932343 0.361576i $$-0.117761\pi$$
0.932343 + 0.361576i $$0.117761\pi$$
$$720$$ 0 0
$$721$$ −3.00000 −0.111726
$$722$$ −19.0000 −0.707107
$$723$$ 12.0000 0.446285
$$724$$ −13.0000 −0.483141
$$725$$ 0 0
$$726$$ −2.00000 −0.0742270
$$727$$ −27.0000 −1.00137 −0.500687 0.865628i $$-0.666919\pi$$
−0.500687 + 0.865628i $$0.666919\pi$$
$$728$$ −3.00000 −0.111187
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 2.00000 0.0739727
$$732$$ 7.00000 0.258727
$$733$$ −36.0000 −1.32969 −0.664845 0.746981i $$-0.731502\pi$$
−0.664845 + 0.746981i $$0.731502\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 6.00000 0.221013
$$738$$ 7.00000 0.257674
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 27.0000 0.991201
$$743$$ 4.00000 0.146746 0.0733729 0.997305i $$-0.476624\pi$$
0.0733729 + 0.997305i $$0.476624\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 0 0
$$746$$ 24.0000 0.878702
$$747$$ 9.00000 0.329293
$$748$$ 6.00000 0.219382
$$749$$ 54.0000 1.97312
$$750$$ 0 0
$$751$$ 7.00000 0.255434 0.127717 0.991811i $$-0.459235\pi$$
0.127717 + 0.991811i $$0.459235\pi$$
$$752$$ −7.00000 −0.255264
$$753$$ −3.00000 −0.109326
$$754$$ −10.0000 −0.364179
$$755$$ 0 0
$$756$$ 3.00000 0.109109
$$757$$ 13.0000 0.472493 0.236247 0.971693i $$-0.424083\pi$$
0.236247 + 0.971693i $$0.424083\pi$$
$$758$$ 0 0
$$759$$ −12.0000 −0.435572
$$760$$ 0 0
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ −2.00000 −0.0724524
$$763$$ −30.0000 −1.08607
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ −45.0000 −1.62274 −0.811371 0.584532i $$-0.801278\pi$$
−0.811371 + 0.584532i $$0.801278\pi$$
$$770$$ 0 0
$$771$$ −7.00000 −0.252099
$$772$$ 9.00000 0.323917
$$773$$ 34.0000 1.22290 0.611448 0.791285i $$-0.290588\pi$$
0.611448 + 0.791285i $$0.290588\pi$$
$$774$$ −1.00000 −0.0359443
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ 9.00000 0.322873
$$778$$ 10.0000 0.358517
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −21.0000 −0.751439
$$782$$ −8.00000 −0.286079
$$783$$ 10.0000 0.357371
$$784$$ 2.00000 0.0714286
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ −7.00000 −0.249523 −0.124762 0.992187i $$-0.539817\pi$$
−0.124762 + 0.992187i $$0.539817\pi$$
$$788$$ −27.0000 −0.961835
$$789$$ −6.00000 −0.213606
$$790$$ 0 0
$$791$$ −18.0000 −0.640006
$$792$$ −3.00000 −0.106600
$$793$$ −7.00000 −0.248577
$$794$$ 28.0000 0.993683
$$795$$ 0 0
$$796$$ −20.0000 −0.708881
$$797$$ −2.00000 −0.0708436 −0.0354218 0.999372i $$-0.511277\pi$$
−0.0354218 + 0.999372i $$0.511277\pi$$
$$798$$ 0 0
$$799$$ 14.0000 0.495284
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ −8.00000 −0.282490
$$803$$ −12.0000 −0.423471
$$804$$ −2.00000 −0.0705346
$$805$$ 0 0
$$806$$ −1.00000 −0.0352235
$$807$$ −10.0000 −0.352017
$$808$$ 2.00000 0.0703598
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 30.0000 1.05279
$$813$$ −8.00000 −0.280572
$$814$$ −9.00000 −0.315450
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ 0 0
$$818$$ −10.0000 −0.349642
$$819$$ −3.00000 −0.104828
$$820$$ 0 0
$$821$$ −53.0000 −1.84971 −0.924856 0.380317i $$-0.875815\pi$$
−0.924856 + 0.380317i $$0.875815\pi$$
$$822$$ −2.00000 −0.0697580
$$823$$ 24.0000 0.836587 0.418294 0.908312i $$-0.362628\pi$$
0.418294 + 0.908312i $$0.362628\pi$$
$$824$$ −1.00000 −0.0348367
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 28.0000 0.973655 0.486828 0.873498i $$-0.338154\pi$$
0.486828 + 0.873498i $$0.338154\pi$$
$$828$$ 4.00000 0.139010
$$829$$ 55.0000 1.91023 0.955114 0.296237i $$-0.0957318\pi$$
0.955114 + 0.296237i $$0.0957318\pi$$
$$830$$ 0 0
$$831$$ 18.0000 0.624413
$$832$$ −1.00000 −0.0346688
$$833$$ −4.00000 −0.138592
$$834$$ −5.00000 −0.173136
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 1.00000 0.0345651
$$838$$ −30.0000 −1.03633
$$839$$ 15.0000 0.517858 0.258929 0.965896i $$-0.416631\pi$$
0.258929 + 0.965896i $$0.416631\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ −28.0000 −0.964944
$$843$$ 7.00000 0.241093
$$844$$ 2.00000 0.0688428
$$845$$ 0 0
$$846$$ −7.00000 −0.240665
$$847$$ −6.00000 −0.206162
$$848$$ 9.00000 0.309061
$$849$$ 24.0000 0.823678
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ 7.00000 0.239816
$$853$$ −16.0000 −0.547830 −0.273915 0.961754i $$-0.588319\pi$$
−0.273915 + 0.961754i $$0.588319\pi$$
$$854$$ 21.0000 0.718605
$$855$$ 0 0
$$856$$ 18.0000 0.615227
$$857$$ 3.00000 0.102478 0.0512390 0.998686i $$-0.483683\pi$$
0.0512390 + 0.998686i $$0.483683\pi$$
$$858$$ 3.00000 0.102418
$$859$$ −25.0000 −0.852989 −0.426494 0.904490i $$-0.640252\pi$$
−0.426494 + 0.904490i $$0.640252\pi$$
$$860$$ 0 0
$$861$$ 21.0000 0.715678
$$862$$ 7.00000 0.238421
$$863$$ −46.0000 −1.56586 −0.782929 0.622111i $$-0.786275\pi$$
−0.782929 + 0.622111i $$0.786275\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 4.00000 0.135926
$$867$$ −13.0000 −0.441503
$$868$$ 3.00000 0.101827
$$869$$ 30.0000 1.01768
$$870$$ 0 0
$$871$$ 2.00000 0.0677674
$$872$$ −10.0000 −0.338643
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 4.00000 0.135147
$$877$$ 8.00000 0.270141 0.135070 0.990836i $$-0.456874\pi$$
0.135070 + 0.990836i $$0.456874\pi$$
$$878$$ 0 0
$$879$$ 24.0000 0.809500
$$880$$ 0 0
$$881$$ 32.0000 1.07811 0.539054 0.842271i $$-0.318782\pi$$
0.539054 + 0.842271i $$0.318782\pi$$
$$882$$ 2.00000 0.0673435
$$883$$ −11.0000 −0.370179 −0.185090 0.982722i $$-0.559258\pi$$
−0.185090 + 0.982722i $$0.559258\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ 13.0000 0.436497 0.218249 0.975893i $$-0.429966\pi$$
0.218249 + 0.975893i $$0.429966\pi$$
$$888$$ 3.00000 0.100673
$$889$$ −6.00000 −0.201234
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ −6.00000 −0.200895
$$893$$ 0 0
$$894$$ −20.0000 −0.668900
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ −4.00000 −0.133556
$$898$$ 20.0000 0.667409
$$899$$ 10.0000 0.333519
$$900$$ 0 0
$$901$$ −18.0000 −0.599667
$$902$$ −21.0000 −0.699224
$$903$$ −3.00000 −0.0998337
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ −18.0000 −0.598010
$$907$$ −32.0000 −1.06254 −0.531271 0.847202i $$-0.678286\pi$$
−0.531271 + 0.847202i $$0.678286\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 2.00000 0.0663358
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ 0 0
$$913$$ −27.0000 −0.893570
$$914$$ 8.00000 0.264616
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 36.0000 1.18882
$$918$$ −2.00000 −0.0660098
$$919$$ 25.0000 0.824674 0.412337 0.911031i $$-0.364713\pi$$
0.412337 + 0.911031i $$0.364713\pi$$
$$920$$ 0 0
$$921$$ −22.0000 −0.724925
$$922$$ −3.00000 −0.0987997
$$923$$ −7.00000 −0.230408
$$924$$ −9.00000 −0.296078
$$925$$ 0 0
$$926$$ 14.0000 0.460069
$$927$$ −1.00000 −0.0328443
$$928$$ 10.0000 0.328266
$$929$$ 40.0000 1.31236 0.656179 0.754606i $$-0.272172\pi$$
0.656179 + 0.754606i $$0.272172\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 9.00000 0.294805
$$933$$ 27.0000 0.883940
$$934$$ 18.0000 0.588978
$$935$$ 0 0
$$936$$ −1.00000 −0.0326860
$$937$$ −2.00000 −0.0653372 −0.0326686 0.999466i $$-0.510401\pi$$
−0.0326686 + 0.999466i $$0.510401\pi$$
$$938$$ −6.00000 −0.195907
$$939$$ 24.0000 0.783210
$$940$$ 0 0
$$941$$ −38.0000 −1.23876 −0.619382 0.785090i $$-0.712617\pi$$
−0.619382 + 0.785090i $$0.712617\pi$$
$$942$$ −22.0000 −0.716799
$$943$$ 28.0000 0.911805
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 3.00000 0.0975384
$$947$$ 3.00000 0.0974869 0.0487435 0.998811i $$-0.484478\pi$$
0.0487435 + 0.998811i $$0.484478\pi$$
$$948$$ −10.0000 −0.324785
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ −22.0000 −0.713399
$$952$$ −6.00000 −0.194461
$$953$$ 34.0000 1.10137 0.550684 0.834714i $$-0.314367\pi$$
0.550684 + 0.834714i $$0.314367\pi$$
$$954$$ 9.00000 0.291386
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −30.0000 −0.969762
$$958$$ 0 0
$$959$$ −6.00000 −0.193750
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ −3.00000 −0.0967239
$$963$$ 18.0000 0.580042
$$964$$ 12.0000 0.386494
$$965$$ 0 0
$$966$$ 12.0000 0.386094
$$967$$ 18.0000 0.578841 0.289420 0.957202i $$-0.406537\pi$$
0.289420 + 0.957202i $$0.406537\pi$$
$$968$$ −2.00000 −0.0642824
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −15.0000 −0.480878
$$974$$ 28.0000 0.897178
$$975$$ 0 0
$$976$$ 7.00000 0.224065
$$977$$ 13.0000 0.415907 0.207953 0.978139i $$-0.433320\pi$$
0.207953 + 0.978139i $$0.433320\pi$$
$$978$$ 4.00000 0.127906
$$979$$ −30.0000 −0.958804
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ −28.0000 −0.893516
$$983$$ −16.0000 −0.510321 −0.255160 0.966899i $$-0.582128\pi$$
−0.255160 + 0.966899i $$0.582128\pi$$
$$984$$ 7.00000 0.223152
$$985$$ 0 0
$$986$$ −20.0000 −0.636930
$$987$$ −21.0000 −0.668437
$$988$$ 0 0
$$989$$ −4.00000 −0.127193
$$990$$ 0 0
$$991$$ 2.00000 0.0635321 0.0317660 0.999495i $$-0.489887\pi$$
0.0317660 + 0.999495i $$0.489887\pi$$
$$992$$ 1.00000 0.0317500
$$993$$ −3.00000 −0.0952021
$$994$$ 21.0000 0.666080
$$995$$ 0 0
$$996$$ 9.00000 0.285176
$$997$$ −42.0000 −1.33015 −0.665077 0.746775i $$-0.731601\pi$$
−0.665077 + 0.746775i $$0.731601\pi$$
$$998$$ −20.0000 −0.633089
$$999$$ 3.00000 0.0949158
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 4650.2.a.bu.1.1 yes 1
5.2 odd 4 4650.2.d.q.3349.2 2
5.3 odd 4 4650.2.d.q.3349.1 2
5.4 even 2 4650.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
4650.2.a.b.1.1 1 5.4 even 2
4650.2.a.bu.1.1 yes 1 1.1 even 1 trivial
4650.2.d.q.3349.1 2 5.3 odd 4
4650.2.d.q.3349.2 2 5.2 odd 4