Properties

 Label 1520.2.a.g.1.1 Level $1520$ Weight $2$ Character 1520.1 Self dual yes Analytic conductor $12.137$ 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] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.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: $$12.1372611072$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} -3.00000 q^{13} -1.00000 q^{15} -7.00000 q^{17} +1.00000 q^{19} +1.00000 q^{21} +5.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} -5.00000 q^{29} -10.0000 q^{31} -1.00000 q^{35} +2.00000 q^{37} -3.00000 q^{39} +2.00000 q^{41} -6.00000 q^{43} +2.00000 q^{45} -6.00000 q^{49} -7.00000 q^{51} +9.00000 q^{53} +1.00000 q^{57} +7.00000 q^{59} -4.00000 q^{61} -2.00000 q^{63} +3.00000 q^{65} -7.00000 q^{67} +5.00000 q^{69} -9.00000 q^{73} +1.00000 q^{75} +10.0000 q^{79} +1.00000 q^{81} +2.00000 q^{83} +7.00000 q^{85} -5.00000 q^{87} -10.0000 q^{89} -3.00000 q^{91} -10.0000 q^{93} -1.00000 q^{95} -18.0000 q^{97} +O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ −3.00000 −0.832050 −0.416025 0.909353i $$-0.636577\pi$$
−0.416025 + 0.909353i $$0.636577\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ −7.00000 −1.69775 −0.848875 0.528594i $$-0.822719\pi$$
−0.848875 + 0.528594i $$0.822719\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ 5.00000 1.04257 0.521286 0.853382i $$-0.325452\pi$$
0.521286 + 0.853382i $$0.325452\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ −10.0000 −1.79605 −0.898027 0.439941i $$-0.854999\pi$$
−0.898027 + 0.439941i $$0.854999\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ −3.00000 −0.480384
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ −6.00000 −0.914991 −0.457496 0.889212i $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ −7.00000 −0.980196
$$52$$ 0 0
$$53$$ 9.00000 1.23625 0.618123 0.786082i $$-0.287894\pi$$
0.618123 + 0.786082i $$0.287894\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ 7.00000 0.911322 0.455661 0.890153i $$-0.349403\pi$$
0.455661 + 0.890153i $$0.349403\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 0 0
$$63$$ −2.00000 −0.251976
$$64$$ 0 0
$$65$$ 3.00000 0.372104
$$66$$ 0 0
$$67$$ −7.00000 −0.855186 −0.427593 0.903971i $$-0.640638\pi$$
−0.427593 + 0.903971i $$0.640638\pi$$
$$68$$ 0 0
$$69$$ 5.00000 0.601929
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −9.00000 −1.05337 −0.526685 0.850060i $$-0.676565\pi$$
−0.526685 + 0.850060i $$0.676565\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 2.00000 0.219529 0.109764 0.993958i $$-0.464990\pi$$
0.109764 + 0.993958i $$0.464990\pi$$
$$84$$ 0 0
$$85$$ 7.00000 0.759257
$$86$$ 0 0
$$87$$ −5.00000 −0.536056
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ 0 0
$$93$$ −10.0000 −1.03695
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −18.0000 −1.82762 −0.913812 0.406138i $$-0.866875\pi$$
−0.913812 + 0.406138i $$0.866875\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −4.00000 −0.398015 −0.199007 0.979998i $$-0.563772\pi$$
−0.199007 + 0.979998i $$0.563772\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ −1.00000 −0.0975900
$$106$$ 0 0
$$107$$ −9.00000 −0.870063 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$108$$ 0 0
$$109$$ 13.0000 1.24517 0.622587 0.782551i $$-0.286082\pi$$
0.622587 + 0.782551i $$0.286082\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ 8.00000 0.752577 0.376288 0.926503i $$-0.377200\pi$$
0.376288 + 0.926503i $$0.377200\pi$$
$$114$$ 0 0
$$115$$ −5.00000 −0.466252
$$116$$ 0 0
$$117$$ 6.00000 0.554700
$$118$$ 0 0
$$119$$ −7.00000 −0.641689
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 2.00000 0.180334
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 6.00000 0.532414 0.266207 0.963916i $$-0.414230\pi$$
0.266207 + 0.963916i $$0.414230\pi$$
$$128$$ 0 0
$$129$$ −6.00000 −0.528271
$$130$$ 0 0
$$131$$ 20.0000 1.74741 0.873704 0.486458i $$-0.161711\pi$$
0.873704 + 0.486458i $$0.161711\pi$$
$$132$$ 0 0
$$133$$ 1.00000 0.0867110
$$134$$ 0 0
$$135$$ 5.00000 0.430331
$$136$$ 0 0
$$137$$ −3.00000 −0.256307 −0.128154 0.991754i $$-0.540905\pi$$
−0.128154 + 0.991754i $$0.540905\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 5.00000 0.415227
$$146$$ 0 0
$$147$$ −6.00000 −0.494872
$$148$$ 0 0
$$149$$ 4.00000 0.327693 0.163846 0.986486i $$-0.447610\pi$$
0.163846 + 0.986486i $$0.447610\pi$$
$$150$$ 0 0
$$151$$ 6.00000 0.488273 0.244137 0.969741i $$-0.421495\pi$$
0.244137 + 0.969741i $$0.421495\pi$$
$$152$$ 0 0
$$153$$ 14.0000 1.13183
$$154$$ 0 0
$$155$$ 10.0000 0.803219
$$156$$ 0 0
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 0 0
$$159$$ 9.00000 0.713746
$$160$$ 0 0
$$161$$ 5.00000 0.394055
$$162$$ 0 0
$$163$$ 18.0000 1.40987 0.704934 0.709273i $$-0.250976\pi$$
0.704934 + 0.709273i $$0.250976\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −14.0000 −1.08335 −0.541676 0.840587i $$-0.682210\pi$$
−0.541676 + 0.840587i $$0.682210\pi$$
$$168$$ 0 0
$$169$$ −4.00000 −0.307692
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 7.00000 0.526152
$$178$$ 0 0
$$179$$ −24.0000 −1.79384 −0.896922 0.442189i $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ −4.00000 −0.295689
$$184$$ 0 0
$$185$$ −2.00000 −0.147043
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −5.00000 −0.363696
$$190$$ 0 0
$$191$$ 7.00000 0.506502 0.253251 0.967401i $$-0.418500\pi$$
0.253251 + 0.967401i $$0.418500\pi$$
$$192$$ 0 0
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 0 0
$$195$$ 3.00000 0.214834
$$196$$ 0 0
$$197$$ −10.0000 −0.712470 −0.356235 0.934396i $$-0.615940\pi$$
−0.356235 + 0.934396i $$0.615940\pi$$
$$198$$ 0 0
$$199$$ −17.0000 −1.20510 −0.602549 0.798082i $$-0.705848\pi$$
−0.602549 + 0.798082i $$0.705848\pi$$
$$200$$ 0 0
$$201$$ −7.00000 −0.493742
$$202$$ 0 0
$$203$$ −5.00000 −0.350931
$$204$$ 0 0
$$205$$ −2.00000 −0.139686
$$206$$ 0 0
$$207$$ −10.0000 −0.695048
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −5.00000 −0.344214 −0.172107 0.985078i $$-0.555058\pi$$
−0.172107 + 0.985078i $$0.555058\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 6.00000 0.409197
$$216$$ 0 0
$$217$$ −10.0000 −0.678844
$$218$$ 0 0
$$219$$ −9.00000 −0.608164
$$220$$ 0 0
$$221$$ 21.0000 1.41261
$$222$$ 0 0
$$223$$ 22.0000 1.47323 0.736614 0.676313i $$-0.236423\pi$$
0.736614 + 0.676313i $$0.236423\pi$$
$$224$$ 0 0
$$225$$ −2.00000 −0.133333
$$226$$ 0 0
$$227$$ 25.0000 1.65931 0.829654 0.558278i $$-0.188538\pi$$
0.829654 + 0.558278i $$0.188538\pi$$
$$228$$ 0 0
$$229$$ −18.0000 −1.18947 −0.594737 0.803921i $$-0.702744\pi$$
−0.594737 + 0.803921i $$0.702744\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 10.0000 0.649570
$$238$$ 0 0
$$239$$ 27.0000 1.74648 0.873242 0.487286i $$-0.162013\pi$$
0.873242 + 0.487286i $$0.162013\pi$$
$$240$$ 0 0
$$241$$ −28.0000 −1.80364 −0.901819 0.432113i $$-0.857768\pi$$
−0.901819 + 0.432113i $$0.857768\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 0 0
$$245$$ 6.00000 0.383326
$$246$$ 0 0
$$247$$ −3.00000 −0.190885
$$248$$ 0 0
$$249$$ 2.00000 0.126745
$$250$$ 0 0
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 7.00000 0.438357
$$256$$ 0 0
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ 0 0
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ 0 0
$$265$$ −9.00000 −0.552866
$$266$$ 0 0
$$267$$ −10.0000 −0.611990
$$268$$ 0 0
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −5.00000 −0.303728 −0.151864 0.988401i $$-0.548528\pi$$
−0.151864 + 0.988401i $$0.548528\pi$$
$$272$$ 0 0
$$273$$ −3.00000 −0.181568
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 28.0000 1.68236 0.841178 0.540758i $$-0.181862\pi$$
0.841178 + 0.540758i $$0.181862\pi$$
$$278$$ 0 0
$$279$$ 20.0000 1.19737
$$280$$ 0 0
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 0 0
$$283$$ 26.0000 1.54554 0.772770 0.634686i $$-0.218871\pi$$
0.772770 + 0.634686i $$0.218871\pi$$
$$284$$ 0 0
$$285$$ −1.00000 −0.0592349
$$286$$ 0 0
$$287$$ 2.00000 0.118056
$$288$$ 0 0
$$289$$ 32.0000 1.88235
$$290$$ 0 0
$$291$$ −18.0000 −1.05518
$$292$$ 0 0
$$293$$ 31.0000 1.81104 0.905520 0.424304i $$-0.139481\pi$$
0.905520 + 0.424304i $$0.139481\pi$$
$$294$$ 0 0
$$295$$ −7.00000 −0.407556
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −15.0000 −0.867472
$$300$$ 0 0
$$301$$ −6.00000 −0.345834
$$302$$ 0 0
$$303$$ −4.00000 −0.229794
$$304$$ 0 0
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −33.0000 −1.87126 −0.935629 0.352985i $$-0.885167\pi$$
−0.935629 + 0.352985i $$0.885167\pi$$
$$312$$ 0 0
$$313$$ −29.0000 −1.63918 −0.819588 0.572953i $$-0.805798\pi$$
−0.819588 + 0.572953i $$0.805798\pi$$
$$314$$ 0 0
$$315$$ 2.00000 0.112687
$$316$$ 0 0
$$317$$ 3.00000 0.168497 0.0842484 0.996445i $$-0.473151\pi$$
0.0842484 + 0.996445i $$0.473151\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −9.00000 −0.502331
$$322$$ 0 0
$$323$$ −7.00000 −0.389490
$$324$$ 0 0
$$325$$ −3.00000 −0.166410
$$326$$ 0 0
$$327$$ 13.0000 0.718902
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 17.0000 0.934405 0.467202 0.884150i $$-0.345262\pi$$
0.467202 + 0.884150i $$0.345262\pi$$
$$332$$ 0 0
$$333$$ −4.00000 −0.219199
$$334$$ 0 0
$$335$$ 7.00000 0.382451
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ 8.00000 0.434500
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ 0 0
$$345$$ −5.00000 −0.269191
$$346$$ 0 0
$$347$$ 18.0000 0.966291 0.483145 0.875540i $$-0.339494\pi$$
0.483145 + 0.875540i $$0.339494\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 15.0000 0.800641
$$352$$ 0 0
$$353$$ 27.0000 1.43706 0.718532 0.695493i $$-0.244814\pi$$
0.718532 + 0.695493i $$0.244814\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −7.00000 −0.370479
$$358$$ 0 0
$$359$$ −27.0000 −1.42501 −0.712503 0.701669i $$-0.752438\pi$$
−0.712503 + 0.701669i $$0.752438\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −11.0000 −0.577350
$$364$$ 0 0
$$365$$ 9.00000 0.471082
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ −4.00000 −0.208232
$$370$$ 0 0
$$371$$ 9.00000 0.467257
$$372$$ 0 0
$$373$$ 3.00000 0.155334 0.0776671 0.996979i $$-0.475253\pi$$
0.0776671 + 0.996979i $$0.475253\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 15.0000 0.772539
$$378$$ 0 0
$$379$$ 7.00000 0.359566 0.179783 0.983706i $$-0.442460\pi$$
0.179783 + 0.983706i $$0.442460\pi$$
$$380$$ 0 0
$$381$$ 6.00000 0.307389
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 12.0000 0.609994
$$388$$ 0 0
$$389$$ 24.0000 1.21685 0.608424 0.793612i $$-0.291802\pi$$
0.608424 + 0.793612i $$0.291802\pi$$
$$390$$ 0 0
$$391$$ −35.0000 −1.77003
$$392$$ 0 0
$$393$$ 20.0000 1.00887
$$394$$ 0 0
$$395$$ −10.0000 −0.503155
$$396$$ 0 0
$$397$$ −20.0000 −1.00377 −0.501886 0.864934i $$-0.667360\pi$$
−0.501886 + 0.864934i $$0.667360\pi$$
$$398$$ 0 0
$$399$$ 1.00000 0.0500626
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 30.0000 1.49441
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ −3.00000 −0.147979
$$412$$ 0 0
$$413$$ 7.00000 0.344447
$$414$$ 0 0
$$415$$ −2.00000 −0.0981761
$$416$$ 0 0
$$417$$ −12.0000 −0.587643
$$418$$ 0 0
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ 31.0000 1.51085 0.755424 0.655237i $$-0.227431\pi$$
0.755424 + 0.655237i $$0.227431\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −7.00000 −0.339550
$$426$$ 0 0
$$427$$ −4.00000 −0.193574
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ 12.0000 0.576683 0.288342 0.957528i $$-0.406896\pi$$
0.288342 + 0.957528i $$0.406896\pi$$
$$434$$ 0 0
$$435$$ 5.00000 0.239732
$$436$$ 0 0
$$437$$ 5.00000 0.239182
$$438$$ 0 0
$$439$$ −10.0000 −0.477274 −0.238637 0.971109i $$-0.576701\pi$$
−0.238637 + 0.971109i $$0.576701\pi$$
$$440$$ 0 0
$$441$$ 12.0000 0.571429
$$442$$ 0 0
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ 0 0
$$445$$ 10.0000 0.474045
$$446$$ 0 0
$$447$$ 4.00000 0.189194
$$448$$ 0 0
$$449$$ 14.0000 0.660701 0.330350 0.943858i $$-0.392833\pi$$
0.330350 + 0.943858i $$0.392833\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 6.00000 0.281905
$$454$$ 0 0
$$455$$ 3.00000 0.140642
$$456$$ 0 0
$$457$$ −41.0000 −1.91790 −0.958950 0.283577i $$-0.908479\pi$$
−0.958950 + 0.283577i $$0.908479\pi$$
$$458$$ 0 0
$$459$$ 35.0000 1.63366
$$460$$ 0 0
$$461$$ −10.0000 −0.465746 −0.232873 0.972507i $$-0.574813\pi$$
−0.232873 + 0.972507i $$0.574813\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ 0 0
$$465$$ 10.0000 0.463739
$$466$$ 0 0
$$467$$ 8.00000 0.370196 0.185098 0.982720i $$-0.440740\pi$$
0.185098 + 0.982720i $$0.440740\pi$$
$$468$$ 0 0
$$469$$ −7.00000 −0.323230
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −18.0000 −0.824163
$$478$$ 0 0
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ 0 0
$$483$$ 5.00000 0.227508
$$484$$ 0 0
$$485$$ 18.0000 0.817338
$$486$$ 0 0
$$487$$ 2.00000 0.0906287 0.0453143 0.998973i $$-0.485571\pi$$
0.0453143 + 0.998973i $$0.485571\pi$$
$$488$$ 0 0
$$489$$ 18.0000 0.813988
$$490$$ 0 0
$$491$$ −14.0000 −0.631811 −0.315906 0.948791i $$-0.602308\pi$$
−0.315906 + 0.948791i $$0.602308\pi$$
$$492$$ 0 0
$$493$$ 35.0000 1.57632
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −42.0000 −1.88018 −0.940089 0.340929i $$-0.889258\pi$$
−0.940089 + 0.340929i $$0.889258\pi$$
$$500$$ 0 0
$$501$$ −14.0000 −0.625474
$$502$$ 0 0
$$503$$ 17.0000 0.757993 0.378996 0.925398i $$-0.376269\pi$$
0.378996 + 0.925398i $$0.376269\pi$$
$$504$$ 0 0
$$505$$ 4.00000 0.177998
$$506$$ 0 0
$$507$$ −4.00000 −0.177646
$$508$$ 0 0
$$509$$ −42.0000 −1.86162 −0.930809 0.365507i $$-0.880896\pi$$
−0.930809 + 0.365507i $$0.880896\pi$$
$$510$$ 0 0
$$511$$ −9.00000 −0.398137
$$512$$ 0 0
$$513$$ −5.00000 −0.220755
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 19.0000 0.830812 0.415406 0.909636i $$-0.363640\pi$$
0.415406 + 0.909636i $$0.363640\pi$$
$$524$$ 0 0
$$525$$ 1.00000 0.0436436
$$526$$ 0 0
$$527$$ 70.0000 3.04925
$$528$$ 0 0
$$529$$ 2.00000 0.0869565
$$530$$ 0 0
$$531$$ −14.0000 −0.607548
$$532$$ 0 0
$$533$$ −6.00000 −0.259889
$$534$$ 0 0
$$535$$ 9.00000 0.389104
$$536$$ 0 0
$$537$$ −24.0000 −1.03568
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −28.0000 −1.20381 −0.601907 0.798566i $$-0.705592\pi$$
−0.601907 + 0.798566i $$0.705592\pi$$
$$542$$ 0 0
$$543$$ −10.0000 −0.429141
$$544$$ 0 0
$$545$$ −13.0000 −0.556859
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 0 0
$$549$$ 8.00000 0.341432
$$550$$ 0 0
$$551$$ −5.00000 −0.213007
$$552$$ 0 0
$$553$$ 10.0000 0.425243
$$554$$ 0 0
$$555$$ −2.00000 −0.0848953
$$556$$ 0 0
$$557$$ 8.00000 0.338971 0.169485 0.985533i $$-0.445789\pi$$
0.169485 + 0.985533i $$0.445789\pi$$
$$558$$ 0 0
$$559$$ 18.0000 0.761319
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 44.0000 1.85438 0.927189 0.374593i $$-0.122217\pi$$
0.927189 + 0.374593i $$0.122217\pi$$
$$564$$ 0 0
$$565$$ −8.00000 −0.336563
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −8.00000 −0.335377 −0.167689 0.985840i $$-0.553630\pi$$
−0.167689 + 0.985840i $$0.553630\pi$$
$$570$$ 0 0
$$571$$ −10.0000 −0.418487 −0.209243 0.977864i $$-0.567100\pi$$
−0.209243 + 0.977864i $$0.567100\pi$$
$$572$$ 0 0
$$573$$ 7.00000 0.292429
$$574$$ 0 0
$$575$$ 5.00000 0.208514
$$576$$ 0 0
$$577$$ −11.0000 −0.457936 −0.228968 0.973434i $$-0.573535\pi$$
−0.228968 + 0.973434i $$0.573535\pi$$
$$578$$ 0 0
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ 2.00000 0.0829740
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −6.00000 −0.248069
$$586$$ 0 0
$$587$$ −30.0000 −1.23823 −0.619116 0.785299i $$-0.712509\pi$$
−0.619116 + 0.785299i $$0.712509\pi$$
$$588$$ 0 0
$$589$$ −10.0000 −0.412043
$$590$$ 0 0
$$591$$ −10.0000 −0.411345
$$592$$ 0 0
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ 0 0
$$595$$ 7.00000 0.286972
$$596$$ 0 0
$$597$$ −17.0000 −0.695764
$$598$$ 0 0
$$599$$ −30.0000 −1.22577 −0.612883 0.790173i $$-0.709990\pi$$
−0.612883 + 0.790173i $$0.709990\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ 14.0000 0.570124
$$604$$ 0 0
$$605$$ 11.0000 0.447214
$$606$$ 0 0
$$607$$ −22.0000 −0.892952 −0.446476 0.894795i $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ 0 0
$$609$$ −5.00000 −0.202610
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ 0 0
$$615$$ −2.00000 −0.0806478
$$616$$ 0 0
$$617$$ −2.00000 −0.0805170 −0.0402585 0.999189i $$-0.512818\pi$$
−0.0402585 + 0.999189i $$0.512818\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ −25.0000 −1.00322
$$622$$ 0 0
$$623$$ −10.0000 −0.400642
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −14.0000 −0.558217
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ −5.00000 −0.198732
$$634$$ 0 0
$$635$$ −6.00000 −0.238103
$$636$$ 0 0
$$637$$ 18.0000 0.713186
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −12.0000 −0.473972 −0.236986 0.971513i $$-0.576159\pi$$
−0.236986 + 0.971513i $$0.576159\pi$$
$$642$$ 0 0
$$643$$ 2.00000 0.0788723 0.0394362 0.999222i $$-0.487444\pi$$
0.0394362 + 0.999222i $$0.487444\pi$$
$$644$$ 0 0
$$645$$ 6.00000 0.236250
$$646$$ 0 0
$$647$$ 9.00000 0.353827 0.176913 0.984226i $$-0.443389\pi$$
0.176913 + 0.984226i $$0.443389\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −10.0000 −0.391931
$$652$$ 0 0
$$653$$ −12.0000 −0.469596 −0.234798 0.972044i $$-0.575443\pi$$
−0.234798 + 0.972044i $$0.575443\pi$$
$$654$$ 0 0
$$655$$ −20.0000 −0.781465
$$656$$ 0 0
$$657$$ 18.0000 0.702247
$$658$$ 0 0
$$659$$ −15.0000 −0.584317 −0.292159 0.956370i $$-0.594373\pi$$
−0.292159 + 0.956370i $$0.594373\pi$$
$$660$$ 0 0
$$661$$ 17.0000 0.661223 0.330612 0.943767i $$-0.392745\pi$$
0.330612 + 0.943767i $$0.392745\pi$$
$$662$$ 0 0
$$663$$ 21.0000 0.815572
$$664$$ 0 0
$$665$$ −1.00000 −0.0387783
$$666$$ 0 0
$$667$$ −25.0000 −0.968004
$$668$$ 0 0
$$669$$ 22.0000 0.850569
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −28.0000 −1.07932 −0.539660 0.841883i $$-0.681447\pi$$
−0.539660 + 0.841883i $$0.681447\pi$$
$$674$$ 0 0
$$675$$ −5.00000 −0.192450
$$676$$ 0 0
$$677$$ 19.0000 0.730229 0.365115 0.930963i $$-0.381030\pi$$
0.365115 + 0.930963i $$0.381030\pi$$
$$678$$ 0 0
$$679$$ −18.0000 −0.690777
$$680$$ 0 0
$$681$$ 25.0000 0.958002
$$682$$ 0 0
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 0 0
$$685$$ 3.00000 0.114624
$$686$$ 0 0
$$687$$ −18.0000 −0.686743
$$688$$ 0 0
$$689$$ −27.0000 −1.02862
$$690$$ 0 0
$$691$$ −10.0000 −0.380418 −0.190209 0.981744i $$-0.560917\pi$$
−0.190209 + 0.981744i $$0.560917\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ −14.0000 −0.530288
$$698$$ 0 0
$$699$$ 26.0000 0.983410
$$700$$ 0 0
$$701$$ 24.0000 0.906467 0.453234 0.891392i $$-0.350270\pi$$
0.453234 + 0.891392i $$0.350270\pi$$
$$702$$ 0 0
$$703$$ 2.00000 0.0754314
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −4.00000 −0.150435
$$708$$ 0 0
$$709$$ −34.0000 −1.27690 −0.638448 0.769665i $$-0.720423\pi$$
−0.638448 + 0.769665i $$0.720423\pi$$
$$710$$ 0 0
$$711$$ −20.0000 −0.750059
$$712$$ 0 0
$$713$$ −50.0000 −1.87251
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 27.0000 1.00833
$$718$$ 0 0
$$719$$ −29.0000 −1.08152 −0.540759 0.841178i $$-0.681863\pi$$
−0.540759 + 0.841178i $$0.681863\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −28.0000 −1.04133
$$724$$ 0 0
$$725$$ −5.00000 −0.185695
$$726$$ 0 0
$$727$$ −11.0000 −0.407967 −0.203984 0.978974i $$-0.565389\pi$$
−0.203984 + 0.978974i $$0.565389\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 42.0000 1.55343
$$732$$ 0 0
$$733$$ 24.0000 0.886460 0.443230 0.896408i $$-0.353832\pi$$
0.443230 + 0.896408i $$0.353832\pi$$
$$734$$ 0 0
$$735$$ 6.00000 0.221313
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −50.0000 −1.83928 −0.919640 0.392763i $$-0.871519\pi$$
−0.919640 + 0.392763i $$0.871519\pi$$
$$740$$ 0 0
$$741$$ −3.00000 −0.110208
$$742$$ 0 0
$$743$$ −34.0000 −1.24734 −0.623670 0.781688i $$-0.714359\pi$$
−0.623670 + 0.781688i $$0.714359\pi$$
$$744$$ 0 0
$$745$$ −4.00000 −0.146549
$$746$$ 0 0
$$747$$ −4.00000 −0.146352
$$748$$ 0 0
$$749$$ −9.00000 −0.328853
$$750$$ 0 0
$$751$$ −46.0000 −1.67856 −0.839282 0.543696i $$-0.817024\pi$$
−0.839282 + 0.543696i $$0.817024\pi$$
$$752$$ 0 0
$$753$$ −4.00000 −0.145768
$$754$$ 0 0
$$755$$ −6.00000 −0.218362
$$756$$ 0 0
$$757$$ 22.0000 0.799604 0.399802 0.916602i $$-0.369079\pi$$
0.399802 + 0.916602i $$0.369079\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −45.0000 −1.63125 −0.815624 0.578582i $$-0.803606\pi$$
−0.815624 + 0.578582i $$0.803606\pi$$
$$762$$ 0 0
$$763$$ 13.0000 0.470632
$$764$$ 0 0
$$765$$ −14.0000 −0.506171
$$766$$ 0 0
$$767$$ −21.0000 −0.758266
$$768$$ 0 0
$$769$$ 49.0000 1.76699 0.883493 0.468445i $$-0.155186\pi$$
0.883493 + 0.468445i $$0.155186\pi$$
$$770$$ 0 0
$$771$$ −6.00000 −0.216085
$$772$$ 0 0
$$773$$ 15.0000 0.539513 0.269756 0.962929i $$-0.413057\pi$$
0.269756 + 0.962929i $$0.413057\pi$$
$$774$$ 0 0
$$775$$ −10.0000 −0.359211
$$776$$ 0 0
$$777$$ 2.00000 0.0717496
$$778$$ 0 0
$$779$$ 2.00000 0.0716574
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 25.0000 0.893427
$$784$$ 0 0
$$785$$ −18.0000 −0.642448
$$786$$ 0 0
$$787$$ −11.0000 −0.392108 −0.196054 0.980593i $$-0.562813\pi$$
−0.196054 + 0.980593i $$0.562813\pi$$
$$788$$ 0 0
$$789$$ 8.00000 0.284808
$$790$$ 0 0
$$791$$ 8.00000 0.284447
$$792$$ 0 0
$$793$$ 12.0000 0.426132
$$794$$ 0 0
$$795$$ −9.00000 −0.319197
$$796$$ 0 0
$$797$$ 5.00000 0.177109 0.0885545 0.996071i $$-0.471775\pi$$
0.0885545 + 0.996071i $$0.471775\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 20.0000 0.706665
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −5.00000 −0.176227
$$806$$ 0 0
$$807$$ −10.0000 −0.352017
$$808$$ 0 0
$$809$$ −39.0000 −1.37117 −0.685583 0.727994i $$-0.740453\pi$$
−0.685583 + 0.727994i $$0.740453\pi$$
$$810$$ 0 0
$$811$$ −11.0000 −0.386262 −0.193131 0.981173i $$-0.561864\pi$$
−0.193131 + 0.981173i $$0.561864\pi$$
$$812$$ 0 0
$$813$$ −5.00000 −0.175358
$$814$$ 0 0
$$815$$ −18.0000 −0.630512
$$816$$ 0 0
$$817$$ −6.00000 −0.209913
$$818$$ 0 0
$$819$$ 6.00000 0.209657
$$820$$ 0 0
$$821$$ −56.0000 −1.95441 −0.977207 0.212290i $$-0.931908\pi$$
−0.977207 + 0.212290i $$0.931908\pi$$
$$822$$ 0 0
$$823$$ 31.0000 1.08059 0.540296 0.841475i $$-0.318312\pi$$
0.540296 + 0.841475i $$0.318312\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 9.00000 0.312961 0.156480 0.987681i $$-0.449985\pi$$
0.156480 + 0.987681i $$0.449985\pi$$
$$828$$ 0 0
$$829$$ −51.0000 −1.77130 −0.885652 0.464350i $$-0.846288\pi$$
−0.885652 + 0.464350i $$0.846288\pi$$
$$830$$ 0 0
$$831$$ 28.0000 0.971309
$$832$$ 0 0
$$833$$ 42.0000 1.45521
$$834$$ 0 0
$$835$$ 14.0000 0.484490
$$836$$ 0 0
$$837$$ 50.0000 1.72825
$$838$$ 0 0
$$839$$ 48.0000 1.65714 0.828572 0.559883i $$-0.189154\pi$$
0.828572 + 0.559883i $$0.189154\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 0 0
$$843$$ −26.0000 −0.895488
$$844$$ 0 0
$$845$$ 4.00000 0.137604
$$846$$ 0 0
$$847$$ −11.0000 −0.377964
$$848$$ 0 0
$$849$$ 26.0000 0.892318
$$850$$ 0 0
$$851$$ 10.0000 0.342796
$$852$$ 0 0
$$853$$ −46.0000 −1.57501 −0.787505 0.616308i $$-0.788628\pi$$
−0.787505 + 0.616308i $$0.788628\pi$$
$$854$$ 0 0
$$855$$ 2.00000 0.0683986
$$856$$ 0 0
$$857$$ 24.0000 0.819824 0.409912 0.912125i $$-0.365559\pi$$
0.409912 + 0.912125i $$0.365559\pi$$
$$858$$ 0 0
$$859$$ −56.0000 −1.91070 −0.955348 0.295484i $$-0.904519\pi$$
−0.955348 + 0.295484i $$0.904519\pi$$
$$860$$ 0 0
$$861$$ 2.00000 0.0681598
$$862$$ 0 0
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ −6.00000 −0.204006
$$866$$ 0 0
$$867$$ 32.0000 1.08678
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 21.0000 0.711558
$$872$$ 0 0
$$873$$ 36.0000 1.21842
$$874$$ 0 0
$$875$$ −1.00000 −0.0338062
$$876$$ 0 0
$$877$$ −5.00000 −0.168838 −0.0844190 0.996430i $$-0.526903\pi$$
−0.0844190 + 0.996430i $$0.526903\pi$$
$$878$$ 0 0
$$879$$ 31.0000 1.04560
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 0 0
$$883$$ −26.0000 −0.874970 −0.437485 0.899226i $$-0.644131\pi$$
−0.437485 + 0.899226i $$0.644131\pi$$
$$884$$ 0 0
$$885$$ −7.00000 −0.235302
$$886$$ 0 0
$$887$$ −52.0000 −1.74599 −0.872995 0.487730i $$-0.837825\pi$$
−0.872995 + 0.487730i $$0.837825\pi$$
$$888$$ 0 0
$$889$$ 6.00000 0.201234
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ 0 0
$$897$$ −15.0000 −0.500835
$$898$$ 0 0
$$899$$ 50.0000 1.66759
$$900$$ 0 0
$$901$$ −63.0000 −2.09883
$$902$$ 0 0
$$903$$ −6.00000 −0.199667
$$904$$ 0 0
$$905$$ 10.0000 0.332411
$$906$$ 0 0
$$907$$ −5.00000 −0.166022 −0.0830111 0.996549i $$-0.526454\pi$$
−0.0830111 + 0.996549i $$0.526454\pi$$
$$908$$ 0 0
$$909$$ 8.00000 0.265343
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 4.00000 0.132236
$$916$$ 0 0
$$917$$ 20.0000 0.660458
$$918$$ 0 0
$$919$$ −11.0000 −0.362857 −0.181428 0.983404i $$-0.558072\pi$$
−0.181428 + 0.983404i $$0.558072\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 21.0000 0.688988 0.344494 0.938789i $$-0.388051\pi$$
0.344494 + 0.938789i $$0.388051\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 0 0
$$933$$ −33.0000 −1.08037
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −13.0000 −0.424691 −0.212346 0.977195i $$-0.568110\pi$$
−0.212346 + 0.977195i $$0.568110\pi$$
$$938$$ 0 0
$$939$$ −29.0000 −0.946379
$$940$$ 0 0
$$941$$ −37.0000 −1.20617 −0.603083 0.797679i $$-0.706061\pi$$
−0.603083 + 0.797679i $$0.706061\pi$$
$$942$$ 0 0
$$943$$ 10.0000 0.325645
$$944$$ 0 0
$$945$$ 5.00000 0.162650
$$946$$ 0 0
$$947$$ 28.0000 0.909878 0.454939 0.890523i $$-0.349661\pi$$
0.454939 + 0.890523i $$0.349661\pi$$
$$948$$ 0 0
$$949$$ 27.0000 0.876457
$$950$$ 0 0
$$951$$ 3.00000 0.0972817
$$952$$ 0 0
$$953$$ 44.0000 1.42530 0.712650 0.701520i $$-0.247495\pi$$
0.712650 + 0.701520i $$0.247495\pi$$
$$954$$ 0 0
$$955$$ −7.00000 −0.226515
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −3.00000 −0.0968751
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ 0 0
$$963$$ 18.0000 0.580042
$$964$$ 0 0
$$965$$ 2.00000 0.0643823
$$966$$ 0 0
$$967$$ −12.0000 −0.385894 −0.192947 0.981209i $$-0.561805\pi$$
−0.192947 + 0.981209i $$0.561805\pi$$
$$968$$ 0 0
$$969$$ −7.00000 −0.224872
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ 0 0
$$973$$ −12.0000 −0.384702
$$974$$ 0 0
$$975$$ −3.00000 −0.0960769
$$976$$ 0 0
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −26.0000 −0.830116
$$982$$ 0 0
$$983$$ 18.0000 0.574111 0.287055 0.957914i $$-0.407324\pi$$
0.287055 + 0.957914i $$0.407324\pi$$
$$984$$ 0 0
$$985$$ 10.0000 0.318626
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −30.0000 −0.953945
$$990$$ 0 0
$$991$$ 46.0000 1.46124 0.730619 0.682785i $$-0.239232\pi$$
0.730619 + 0.682785i $$0.239232\pi$$
$$992$$ 0 0
$$993$$ 17.0000 0.539479
$$994$$ 0 0
$$995$$ 17.0000 0.538936
$$996$$ 0 0
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ 0 0
$$999$$ −10.0000 −0.316386
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 1520.2.a.g.1.1 1
4.3 odd 2 190.2.a.a.1.1 1
5.4 even 2 7600.2.a.g.1.1 1
8.3 odd 2 6080.2.a.r.1.1 1
8.5 even 2 6080.2.a.i.1.1 1
12.11 even 2 1710.2.a.r.1.1 1
20.3 even 4 950.2.b.d.799.2 2
20.7 even 4 950.2.b.d.799.1 2
20.19 odd 2 950.2.a.e.1.1 1
28.27 even 2 9310.2.a.i.1.1 1
60.59 even 2 8550.2.a.l.1.1 1
76.75 even 2 3610.2.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.a.a.1.1 1 4.3 odd 2
950.2.a.e.1.1 1 20.19 odd 2
950.2.b.d.799.1 2 20.7 even 4
950.2.b.d.799.2 2 20.3 even 4
1520.2.a.g.1.1 1 1.1 even 1 trivial
1710.2.a.r.1.1 1 12.11 even 2
3610.2.a.h.1.1 1 76.75 even 2
6080.2.a.i.1.1 1 8.5 even 2
6080.2.a.r.1.1 1 8.3 odd 2
7600.2.a.g.1.1 1 5.4 even 2
8550.2.a.l.1.1 1 60.59 even 2
9310.2.a.i.1.1 1 28.27 even 2