# Properties

 Label 1638.2.a.f.1.1 Level $1638$ Weight $2$ Character 1638.1 Self dual yes Analytic conductor $13.079$ 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] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.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: $$13.0794958511$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 182) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1638.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{19} -3.00000 q^{22} +3.00000 q^{23} -5.00000 q^{25} -1.00000 q^{26} +1.00000 q^{28} +5.00000 q^{31} -1.00000 q^{32} -7.00000 q^{37} -2.00000 q^{38} -3.00000 q^{41} +8.00000 q^{43} +3.00000 q^{44} -3.00000 q^{46} +3.00000 q^{47} +1.00000 q^{49} +5.00000 q^{50} +1.00000 q^{52} +12.0000 q^{53} -1.00000 q^{56} -6.00000 q^{59} -1.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} +5.00000 q^{67} -12.0000 q^{71} +11.0000 q^{73} +7.00000 q^{74} +2.00000 q^{76} +3.00000 q^{77} -1.00000 q^{79} +3.00000 q^{82} -12.0000 q^{83} -8.00000 q^{86} -3.00000 q^{88} +18.0000 q^{89} +1.00000 q^{91} +3.00000 q^{92} -3.00000 q^{94} +17.0000 q^{97} -1.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −3.00000 −0.639602
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −3.00000 −0.468521 −0.234261 0.972174i $$-0.575267\pi$$
−0.234261 + 0.972174i $$0.575267\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 0 0
$$46$$ −3.00000 −0.442326
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 5.00000 0.707107
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ 12.0000 1.64833 0.824163 0.566352i $$-0.191646\pi$$
0.824163 + 0.566352i $$0.191646\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ −1.00000 −0.128037 −0.0640184 0.997949i $$-0.520392\pi$$
−0.0640184 + 0.997949i $$0.520392\pi$$
$$62$$ −5.00000 −0.635001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 5.00000 0.610847 0.305424 0.952217i $$-0.401202\pi$$
0.305424 + 0.952217i $$0.401202\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ 7.00000 0.813733
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 3.00000 0.341882
$$78$$ 0 0
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 3.00000 0.331295
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ 18.0000 1.90800 0.953998 0.299813i $$-0.0969242\pi$$
0.953998 + 0.299813i $$0.0969242\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 3.00000 0.312772
$$93$$ 0 0
$$94$$ −3.00000 −0.309426
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 17.0000 1.72609 0.863044 0.505128i $$-0.168555\pi$$
0.863044 + 0.505128i $$0.168555\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ −5.00000 −0.500000
$$101$$ 3.00000 0.298511 0.149256 0.988799i $$-0.452312\pi$$
0.149256 + 0.988799i $$0.452312\pi$$
$$102$$ 0 0
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −12.0000 −1.16554
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ −9.00000 −0.846649 −0.423324 0.905978i $$-0.639137\pi$$
−0.423324 + 0.905978i $$0.639137\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 6.00000 0.552345
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 1.00000 0.0905357
$$123$$ 0 0
$$124$$ 5.00000 0.449013
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −7.00000 −0.621150 −0.310575 0.950549i $$-0.600522\pi$$
−0.310575 + 0.950549i $$0.600522\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 2.00000 0.173422
$$134$$ −5.00000 −0.431934
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 12.0000 1.00702
$$143$$ 3.00000 0.250873
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −11.0000 −0.910366
$$147$$ 0 0
$$148$$ −7.00000 −0.575396
$$149$$ 15.0000 1.22885 0.614424 0.788976i $$-0.289388\pi$$
0.614424 + 0.788976i $$0.289388\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 0 0
$$154$$ −3.00000 −0.241747
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −13.0000 −1.03751 −0.518756 0.854922i $$-0.673605\pi$$
−0.518756 + 0.854922i $$0.673605\pi$$
$$158$$ 1.00000 0.0795557
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3.00000 0.236433
$$162$$ 0 0
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ −3.00000 −0.234261
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 24.0000 1.85718 0.928588 0.371113i $$-0.121024\pi$$
0.928588 + 0.371113i $$0.121024\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 8.00000 0.609994
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 0 0
$$175$$ −5.00000 −0.377964
$$176$$ 3.00000 0.226134
$$177$$ 0 0
$$178$$ −18.0000 −1.34916
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 0 0
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ −1.00000 −0.0741249
$$183$$ 0 0
$$184$$ −3.00000 −0.221163
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 3.00000 0.218797
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ −17.0000 −1.22053
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 3.00000 0.213741 0.106871 0.994273i $$-0.465917\pi$$
0.106871 + 0.994273i $$0.465917\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 5.00000 0.353553
$$201$$ 0 0
$$202$$ −3.00000 −0.211079
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ −22.0000 −1.51454 −0.757271 0.653101i $$-0.773468\pi$$
−0.757271 + 0.653101i $$0.773468\pi$$
$$212$$ 12.0000 0.824163
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 5.00000 0.339422
$$218$$ −2.00000 −0.135457
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −1.00000 −0.0669650 −0.0334825 0.999439i $$-0.510660\pi$$
−0.0334825 + 0.999439i $$0.510660\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 9.00000 0.598671
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ 0 0
$$229$$ −4.00000 −0.264327 −0.132164 0.991228i $$-0.542192\pi$$
−0.132164 + 0.991228i $$0.542192\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −3.00000 −0.196537 −0.0982683 0.995160i $$-0.531330\pi$$
−0.0982683 + 0.995160i $$0.531330\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 2.00000 0.128565
$$243$$ 0 0
$$244$$ −1.00000 −0.0640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ −5.00000 −0.317500
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 15.0000 0.946792 0.473396 0.880850i $$-0.343028\pi$$
0.473396 + 0.880850i $$0.343028\pi$$
$$252$$ 0 0
$$253$$ 9.00000 0.565825
$$254$$ 7.00000 0.439219
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −24.0000 −1.49708 −0.748539 0.663090i $$-0.769245\pi$$
−0.748539 + 0.663090i $$0.769245\pi$$
$$258$$ 0 0
$$259$$ −7.00000 −0.434959
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.00000 −0.122628
$$267$$ 0 0
$$268$$ 5.00000 0.305424
$$269$$ 9.00000 0.548740 0.274370 0.961624i $$-0.411531\pi$$
0.274370 + 0.961624i $$0.411531\pi$$
$$270$$ 0 0
$$271$$ 11.0000 0.668202 0.334101 0.942537i $$-0.391567\pi$$
0.334101 + 0.942537i $$0.391567\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ −15.0000 −0.904534
$$276$$ 0 0
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ 0 0
$$283$$ −31.0000 −1.84276 −0.921379 0.388664i $$-0.872937\pi$$
−0.921379 + 0.388664i $$0.872937\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ −3.00000 −0.177394
$$287$$ −3.00000 −0.177084
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 11.0000 0.643726
$$293$$ −18.0000 −1.05157 −0.525786 0.850617i $$-0.676229\pi$$
−0.525786 + 0.850617i $$0.676229\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 7.00000 0.406867
$$297$$ 0 0
$$298$$ −15.0000 −0.868927
$$299$$ 3.00000 0.173494
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ −8.00000 −0.460348
$$303$$ 0 0
$$304$$ 2.00000 0.114708
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 3.00000 0.170941
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −30.0000 −1.70114 −0.850572 0.525859i $$-0.823744\pi$$
−0.850572 + 0.525859i $$0.823744\pi$$
$$312$$ 0 0
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ 13.0000 0.733632
$$315$$ 0 0
$$316$$ −1.00000 −0.0562544
$$317$$ 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
$$321$$ 0 0
$$322$$ −3.00000 −0.167183
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −5.00000 −0.277350
$$326$$ −20.0000 −1.10770
$$327$$ 0 0
$$328$$ 3.00000 0.165647
$$329$$ 3.00000 0.165395
$$330$$ 0 0
$$331$$ −19.0000 −1.04433 −0.522167 0.852843i $$-0.674876\pi$$
−0.522167 + 0.852843i $$0.674876\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 0 0
$$334$$ −24.0000 −1.31322
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −31.0000 −1.68868 −0.844339 0.535810i $$-0.820006\pi$$
−0.844339 + 0.535810i $$0.820006\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 15.0000 0.812296
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 5.00000 0.267261
$$351$$ 0 0
$$352$$ −3.00000 −0.159901
$$353$$ −15.0000 −0.798369 −0.399185 0.916871i $$-0.630707\pi$$
−0.399185 + 0.916871i $$0.630707\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 18.0000 0.953998
$$357$$ 0 0
$$358$$ −6.00000 −0.317110
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 7.00000 0.367912
$$363$$ 0 0
$$364$$ 1.00000 0.0524142
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 3.00000 0.156386
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 0 0
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −3.00000 −0.154713
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −3.00000 −0.153293 −0.0766464 0.997058i $$-0.524421\pi$$
−0.0766464 + 0.997058i $$0.524421\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ 0 0
$$388$$ 17.0000 0.863044
$$389$$ 36.0000 1.82527 0.912636 0.408773i $$-0.134043\pi$$
0.912636 + 0.408773i $$0.134043\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −1.00000 −0.0505076
$$393$$ 0 0
$$394$$ −3.00000 −0.151138
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 20.0000 1.00377 0.501886 0.864934i $$-0.332640\pi$$
0.501886 + 0.864934i $$0.332640\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 0 0
$$400$$ −5.00000 −0.250000
$$401$$ −24.0000 −1.19850 −0.599251 0.800561i $$-0.704535\pi$$
−0.599251 + 0.800561i $$0.704535\pi$$
$$402$$ 0 0
$$403$$ 5.00000 0.249068
$$404$$ 3.00000 0.149256
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −21.0000 −1.04093
$$408$$ 0 0
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 14.0000 0.689730
$$413$$ −6.00000 −0.295241
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ −6.00000 −0.293470
$$419$$ −21.0000 −1.02592 −0.512959 0.858413i $$-0.671451\pi$$
−0.512959 + 0.858413i $$0.671451\pi$$
$$420$$ 0 0
$$421$$ −37.0000 −1.80327 −0.901635 0.432498i $$-0.857632\pi$$
−0.901635 + 0.432498i $$0.857632\pi$$
$$422$$ 22.0000 1.07094
$$423$$ 0 0
$$424$$ −12.0000 −0.582772
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −1.00000 −0.0483934
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6.00000 0.289010 0.144505 0.989504i $$-0.453841\pi$$
0.144505 + 0.989504i $$0.453841\pi$$
$$432$$ 0 0
$$433$$ 38.0000 1.82616 0.913082 0.407777i $$-0.133696\pi$$
0.913082 + 0.407777i $$0.133696\pi$$
$$434$$ −5.00000 −0.240008
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 6.00000 0.287019
$$438$$ 0 0
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 1.00000 0.0473514
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ −9.00000 −0.423793
$$452$$ −9.00000 −0.423324
$$453$$ 0 0
$$454$$ −24.0000 −1.12638
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 26.0000 1.21623 0.608114 0.793849i $$-0.291926\pi$$
0.608114 + 0.793849i $$0.291926\pi$$
$$458$$ 4.00000 0.186908
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −24.0000 −1.11779 −0.558896 0.829238i $$-0.688775\pi$$
−0.558896 + 0.829238i $$0.688775\pi$$
$$462$$ 0 0
$$463$$ −40.0000 −1.85896 −0.929479 0.368875i $$-0.879743\pi$$
−0.929479 + 0.368875i $$0.879743\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 3.00000 0.138972
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 0 0
$$469$$ 5.00000 0.230879
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 6.00000 0.276172
$$473$$ 24.0000 1.10352
$$474$$ 0 0
$$475$$ −10.0000 −0.458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 6.00000 0.274434
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ −7.00000 −0.319173
$$482$$ −26.0000 −1.18427
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 2.00000 0.0906287 0.0453143 0.998973i $$-0.485571\pi$$
0.0453143 + 0.998973i $$0.485571\pi$$
$$488$$ 1.00000 0.0452679
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −2.00000 −0.0899843
$$495$$ 0 0
$$496$$ 5.00000 0.224507
$$497$$ −12.0000 −0.538274
$$498$$ 0 0
$$499$$ 23.0000 1.02962 0.514811 0.857304i $$-0.327862\pi$$
0.514811 + 0.857304i $$0.327862\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −15.0000 −0.669483
$$503$$ −12.0000 −0.535054 −0.267527 0.963550i $$-0.586206\pi$$
−0.267527 + 0.963550i $$0.586206\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −9.00000 −0.400099
$$507$$ 0 0
$$508$$ −7.00000 −0.310575
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ 11.0000 0.486611
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 24.0000 1.05859
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 9.00000 0.395820
$$518$$ 7.00000 0.307562
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ −7.00000 −0.306089 −0.153044 0.988219i $$-0.548908\pi$$
−0.153044 + 0.988219i $$0.548908\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.00000 0.0867110
$$533$$ −3.00000 −0.129944
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −5.00000 −0.215967
$$537$$ 0 0
$$538$$ −9.00000 −0.388018
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ −11.0000 −0.472490
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 0 0
$$550$$ 15.0000 0.639602
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −1.00000 −0.0425243
$$554$$ −26.0000 −1.10463
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 15.0000 0.635570 0.317785 0.948163i $$-0.397061\pi$$
0.317785 + 0.948163i $$0.397061\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 12.0000 0.506189
$$563$$ 39.0000 1.64365 0.821827 0.569737i $$-0.192955\pi$$
0.821827 + 0.569737i $$0.192955\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 31.0000 1.30303
$$567$$ 0 0
$$568$$ 12.0000 0.503509
$$569$$ −21.0000 −0.880366 −0.440183 0.897908i $$-0.645086\pi$$
−0.440183 + 0.897908i $$0.645086\pi$$
$$570$$ 0 0
$$571$$ −22.0000 −0.920671 −0.460336 0.887745i $$-0.652271\pi$$
−0.460336 + 0.887745i $$0.652271\pi$$
$$572$$ 3.00000 0.125436
$$573$$ 0 0
$$574$$ 3.00000 0.125218
$$575$$ −15.0000 −0.625543
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 17.0000 0.707107
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ 36.0000 1.49097
$$584$$ −11.0000 −0.455183
$$585$$ 0 0
$$586$$ 18.0000 0.743573
$$587$$ −18.0000 −0.742940 −0.371470 0.928445i $$-0.621146\pi$$
−0.371470 + 0.928445i $$0.621146\pi$$
$$588$$ 0 0
$$589$$ 10.0000 0.412043
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −7.00000 −0.287698
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 15.0000 0.614424
$$597$$ 0 0
$$598$$ −3.00000 −0.122679
$$599$$ −39.0000 −1.59350 −0.796748 0.604311i $$-0.793448\pi$$
−0.796748 + 0.604311i $$0.793448\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ 0 0
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 3.00000 0.121367
$$612$$ 0 0
$$613$$ −25.0000 −1.00974 −0.504870 0.863195i $$-0.668460\pi$$
−0.504870 + 0.863195i $$0.668460\pi$$
$$614$$ 16.0000 0.645707
$$615$$ 0 0
$$616$$ −3.00000 −0.120873
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ −10.0000 −0.401934 −0.200967 0.979598i $$-0.564408\pi$$
−0.200967 + 0.979598i $$0.564408\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 30.0000 1.20289
$$623$$ 18.0000 0.721155
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ −26.0000 −1.03917
$$627$$ 0 0
$$628$$ −13.0000 −0.518756
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −34.0000 −1.35352 −0.676759 0.736204i $$-0.736616\pi$$
−0.676759 + 0.736204i $$0.736616\pi$$
$$632$$ 1.00000 0.0397779
$$633$$ 0 0
$$634$$ −3.00000 −0.119145
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 33.0000 1.30342 0.651711 0.758468i $$-0.274052\pi$$
0.651711 + 0.758468i $$0.274052\pi$$
$$642$$ 0 0
$$643$$ 32.0000 1.26196 0.630978 0.775800i $$-0.282654\pi$$
0.630978 + 0.775800i $$0.282654\pi$$
$$644$$ 3.00000 0.118217
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −12.0000 −0.471769 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$648$$ 0 0
$$649$$ −18.0000 −0.706562
$$650$$ 5.00000 0.196116
$$651$$ 0 0
$$652$$ 20.0000 0.783260
$$653$$ −24.0000 −0.939193 −0.469596 0.882881i $$-0.655601\pi$$
−0.469596 + 0.882881i $$0.655601\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −3.00000 −0.117130
$$657$$ 0 0
$$658$$ −3.00000 −0.116952
$$659$$ −42.0000 −1.63609 −0.818044 0.575156i $$-0.804941\pi$$
−0.818044 + 0.575156i $$0.804941\pi$$
$$660$$ 0 0
$$661$$ 32.0000 1.24466 0.622328 0.782757i $$-0.286187\pi$$
0.622328 + 0.782757i $$0.286187\pi$$
$$662$$ 19.0000 0.738456
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 24.0000 0.928588
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −3.00000 −0.115814
$$672$$ 0 0
$$673$$ −1.00000 −0.0385472 −0.0192736 0.999814i $$-0.506135\pi$$
−0.0192736 + 0.999814i $$0.506135\pi$$
$$674$$ 31.0000 1.19408
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 39.0000 1.49889 0.749446 0.662066i $$-0.230320\pi$$
0.749446 + 0.662066i $$0.230320\pi$$
$$678$$ 0 0
$$679$$ 17.0000 0.652400
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −15.0000 −0.574380
$$683$$ 27.0000 1.03313 0.516563 0.856249i $$-0.327211\pi$$
0.516563 + 0.856249i $$0.327211\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ 8.00000 0.304997
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 0 0
$$694$$ 24.0000 0.911028
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 10.0000 0.378506
$$699$$ 0 0
$$700$$ −5.00000 −0.188982
$$701$$ 36.0000 1.35970 0.679851 0.733351i $$-0.262045\pi$$
0.679851 + 0.733351i $$0.262045\pi$$
$$702$$ 0 0
$$703$$ −14.0000 −0.528020
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ 15.0000 0.564532
$$707$$ 3.00000 0.112827
$$708$$ 0 0
$$709$$ 35.0000 1.31445 0.657226 0.753693i $$-0.271730\pi$$
0.657226 + 0.753693i $$0.271730\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −18.0000 −0.674579
$$713$$ 15.0000 0.561754
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 6.00000 0.224231
$$717$$ 0 0
$$718$$ −24.0000 −0.895672
$$719$$ −6.00000 −0.223762 −0.111881 0.993722i $$-0.535688\pi$$
−0.111881 + 0.993722i $$0.535688\pi$$
$$720$$ 0 0
$$721$$ 14.0000 0.521387
$$722$$ 15.0000 0.558242
$$723$$ 0 0
$$724$$ −7.00000 −0.260153
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 26.0000 0.964287 0.482143 0.876092i $$-0.339858\pi$$
0.482143 + 0.876092i $$0.339858\pi$$
$$728$$ −1.00000 −0.0370625
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −4.00000 −0.147743 −0.0738717 0.997268i $$-0.523536\pi$$
−0.0738717 + 0.997268i $$0.523536\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 0 0
$$736$$ −3.00000 −0.110581
$$737$$ 15.0000 0.552532
$$738$$ 0 0
$$739$$ −16.0000 −0.588570 −0.294285 0.955718i $$-0.595081\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −12.0000 −0.440534
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 4.00000 0.146450
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −31.0000 −1.13121 −0.565603 0.824678i $$-0.691357\pi$$
−0.565603 + 0.824678i $$0.691357\pi$$
$$752$$ 3.00000 0.109399
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ 16.0000 0.581146
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −3.00000 −0.108750 −0.0543750 0.998521i $$-0.517317\pi$$
−0.0543750 + 0.998521i $$0.517317\pi$$
$$762$$ 0 0
$$763$$ 2.00000 0.0724049
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 3.00000 0.108394
$$767$$ −6.00000 −0.216647
$$768$$ 0 0
$$769$$ −13.0000 −0.468792 −0.234396 0.972141i $$-0.575311\pi$$
−0.234396 + 0.972141i $$0.575311\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −4.00000 −0.143963
$$773$$ 54.0000 1.94225 0.971123 0.238581i $$-0.0766824\pi$$
0.971123 + 0.238581i $$0.0766824\pi$$
$$774$$ 0 0
$$775$$ −25.0000 −0.898027
$$776$$ −17.0000 −0.610264
$$777$$ 0 0
$$778$$ −36.0000 −1.29066
$$779$$ −6.00000 −0.214972
$$780$$ 0 0
$$781$$ −36.0000 −1.28818
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 3.00000 0.106871
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −9.00000 −0.320003
$$792$$ 0 0
$$793$$ −1.00000 −0.0355110
$$794$$ −20.0000 −0.709773
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ −9.00000 −0.318796 −0.159398 0.987214i $$-0.550955\pi$$
−0.159398 + 0.987214i $$0.550955\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 5.00000 0.176777
$$801$$ 0 0
$$802$$ 24.0000 0.847469
$$803$$ 33.0000 1.16454
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −5.00000 −0.176117
$$807$$ 0 0
$$808$$ −3.00000 −0.105540
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 21.0000 0.736050
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 16.0000 0.559769
$$818$$ 22.0000 0.769212
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ 41.0000 1.42917 0.714585 0.699549i $$-0.246616\pi$$
0.714585 + 0.699549i $$0.246616\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ 0 0
$$826$$ 6.00000 0.208767
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 0 0
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 6.00000 0.207514
$$837$$ 0 0
$$838$$ 21.0000 0.725433
$$839$$ 21.0000 0.725001 0.362500 0.931984i $$-0.381923\pi$$
0.362500 + 0.931984i $$0.381923\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 37.0000 1.27510
$$843$$ 0 0
$$844$$ −22.0000 −0.757271
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −2.00000 −0.0687208
$$848$$ 12.0000 0.412082
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −21.0000 −0.719871
$$852$$ 0 0
$$853$$ 44.0000 1.50653 0.753266 0.657716i $$-0.228477\pi$$
0.753266 + 0.657716i $$0.228477\pi$$
$$854$$ 1.00000 0.0342193
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −54.0000 −1.84460 −0.922302 0.386469i $$-0.873695\pi$$
−0.922302 + 0.386469i $$0.873695\pi$$
$$858$$ 0 0
$$859$$ 23.0000 0.784750 0.392375 0.919805i $$-0.371654\pi$$
0.392375 + 0.919805i $$0.371654\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −6.00000 −0.204361
$$863$$ −12.0000 −0.408485 −0.204242 0.978920i $$-0.565473\pi$$
−0.204242 + 0.978920i $$0.565473\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −38.0000 −1.29129
$$867$$ 0 0
$$868$$ 5.00000 0.169711
$$869$$ −3.00000 −0.101768
$$870$$ 0 0
$$871$$ 5.00000 0.169419
$$872$$ −2.00000 −0.0677285
$$873$$ 0 0
$$874$$ −6.00000 −0.202953
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −13.0000 −0.438979 −0.219489 0.975615i $$-0.570439\pi$$
−0.219489 + 0.975615i $$0.570439\pi$$
$$878$$ −26.0000 −0.877457
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 0 0
$$883$$ 2.00000 0.0673054 0.0336527 0.999434i $$-0.489286\pi$$
0.0336527 + 0.999434i $$0.489286\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ −42.0000 −1.41022 −0.705111 0.709097i $$-0.749103\pi$$
−0.705111 + 0.709097i $$0.749103\pi$$
$$888$$ 0 0
$$889$$ −7.00000 −0.234772
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −1.00000 −0.0334825
$$893$$ 6.00000 0.200782
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ −6.00000 −0.200223
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 9.00000 0.299667
$$903$$ 0 0
$$904$$ 9.00000 0.299336
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 26.0000 0.863316 0.431658 0.902037i $$-0.357929\pi$$
0.431658 + 0.902037i $$0.357929\pi$$
$$908$$ 24.0000 0.796468
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ −36.0000 −1.19143
$$914$$ −26.0000 −0.860004
$$915$$ 0 0
$$916$$ −4.00000 −0.132164
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −25.0000 −0.824674 −0.412337 0.911031i $$-0.635287\pi$$
−0.412337 + 0.911031i $$0.635287\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 24.0000 0.790398
$$923$$ −12.0000 −0.394985
$$924$$ 0 0
$$925$$ 35.0000 1.15079
$$926$$ 40.0000 1.31448
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 9.00000 0.295280 0.147640 0.989041i $$-0.452832\pi$$
0.147640 + 0.989041i $$0.452832\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ −3.00000 −0.0982683
$$933$$ 0 0
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −52.0000 −1.69877 −0.849383 0.527777i $$-0.823026\pi$$
−0.849383 + 0.527777i $$0.823026\pi$$
$$938$$ −5.00000 −0.163256
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −30.0000 −0.977972 −0.488986 0.872292i $$-0.662633\pi$$
−0.488986 + 0.872292i $$0.662633\pi$$
$$942$$ 0 0
$$943$$ −9.00000 −0.293080
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ −24.0000 −0.780307
$$947$$ −48.0000 −1.55979 −0.779895 0.625910i $$-0.784728\pi$$
−0.779895 + 0.625910i $$0.784728\pi$$
$$948$$ 0 0
$$949$$ 11.0000 0.357075
$$950$$ 10.0000 0.324443
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 42.0000 1.36051 0.680257 0.732974i $$-0.261868\pi$$
0.680257 + 0.732974i $$0.261868\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −6.00000 −0.194054
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 7.00000 0.225689
$$963$$ 0 0
$$964$$ 26.0000 0.837404
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −22.0000 −0.707472 −0.353736 0.935345i $$-0.615089\pi$$
−0.353736 + 0.935345i $$0.615089\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 21.0000 0.673922 0.336961 0.941519i $$-0.390601\pi$$
0.336961 + 0.941519i $$0.390601\pi$$
$$972$$ 0 0
$$973$$ −4.00000 −0.128234
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ −1.00000 −0.0320092
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ 54.0000 1.72585
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 12.0000 0.382935
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 2.00000 0.0636285
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 11.0000 0.349427 0.174713 0.984619i $$-0.444100\pi$$
0.174713 + 0.984619i $$0.444100\pi$$
$$992$$ −5.00000 −0.158750
$$993$$ 0 0
$$994$$ 12.0000 0.380617
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −37.0000 −1.17180 −0.585901 0.810383i $$-0.699259\pi$$
−0.585901 + 0.810383i $$0.699259\pi$$
$$998$$ −23.0000 −0.728052
$$999$$ 0 0
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 1638.2.a.f.1.1 1
3.2 odd 2 182.2.a.d.1.1 1
12.11 even 2 1456.2.a.d.1.1 1
15.14 odd 2 4550.2.a.c.1.1 1
21.2 odd 6 1274.2.f.d.1145.1 2
21.5 even 6 1274.2.f.i.1145.1 2
21.11 odd 6 1274.2.f.d.79.1 2
21.17 even 6 1274.2.f.i.79.1 2
21.20 even 2 1274.2.a.j.1.1 1
24.5 odd 2 5824.2.a.k.1.1 1
24.11 even 2 5824.2.a.x.1.1 1
39.5 even 4 2366.2.d.e.337.1 2
39.8 even 4 2366.2.d.e.337.2 2
39.38 odd 2 2366.2.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.a.d.1.1 1 3.2 odd 2
1274.2.a.j.1.1 1 21.20 even 2
1274.2.f.d.79.1 2 21.11 odd 6
1274.2.f.d.1145.1 2 21.2 odd 6
1274.2.f.i.79.1 2 21.17 even 6
1274.2.f.i.1145.1 2 21.5 even 6
1456.2.a.d.1.1 1 12.11 even 2
1638.2.a.f.1.1 1 1.1 even 1 trivial
2366.2.a.e.1.1 1 39.38 odd 2
2366.2.d.e.337.1 2 39.5 even 4
2366.2.d.e.337.2 2 39.8 even 4
4550.2.a.c.1.1 1 15.14 odd 2
5824.2.a.k.1.1 1 24.5 odd 2
5824.2.a.x.1.1 1 24.11 even 2