# Properties

 Label 63.6.a.c.1.1 Level $63$ Weight $6$ Character 63.1 Self dual yes Analytic conductor $10.104$ 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] = [63,6,Mod(1,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("63.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 63.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: $$10.1041806482$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 63.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -31.0000 q^{4} +34.0000 q^{5} -49.0000 q^{7} +63.0000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} -31.0000 q^{4} +34.0000 q^{5} -49.0000 q^{7} +63.0000 q^{8} -34.0000 q^{10} +340.000 q^{11} +454.000 q^{13} +49.0000 q^{14} +929.000 q^{16} +798.000 q^{17} +892.000 q^{19} -1054.00 q^{20} -340.000 q^{22} +3192.00 q^{23} -1969.00 q^{25} -454.000 q^{26} +1519.00 q^{28} +8242.00 q^{29} -2496.00 q^{31} -2945.00 q^{32} -798.000 q^{34} -1666.00 q^{35} +9798.00 q^{37} -892.000 q^{38} +2142.00 q^{40} -19834.0 q^{41} -17236.0 q^{43} -10540.0 q^{44} -3192.00 q^{46} -8928.00 q^{47} +2401.00 q^{49} +1969.00 q^{50} -14074.0 q^{52} -150.000 q^{53} +11560.0 q^{55} -3087.00 q^{56} -8242.00 q^{58} +42396.0 q^{59} +14758.0 q^{61} +2496.00 q^{62} -26783.0 q^{64} +15436.0 q^{65} -1676.00 q^{67} -24738.0 q^{68} +1666.00 q^{70} -14568.0 q^{71} +78378.0 q^{73} -9798.00 q^{74} -27652.0 q^{76} -16660.0 q^{77} -2272.00 q^{79} +31586.0 q^{80} +19834.0 q^{82} +37764.0 q^{83} +27132.0 q^{85} +17236.0 q^{86} +21420.0 q^{88} +117286. q^{89} -22246.0 q^{91} -98952.0 q^{92} +8928.00 q^{94} +30328.0 q^{95} +10002.0 q^{97} -2401.00 q^{98} +O(q^{100})$$

## 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 −0.176777 −0.0883883 0.996086i $$-0.528172\pi$$
−0.0883883 + 0.996086i $$0.528172\pi$$
$$3$$ 0 0
$$4$$ −31.0000 −0.968750
$$5$$ 34.0000 0.608210 0.304105 0.952638i $$-0.401643\pi$$
0.304105 + 0.952638i $$0.401643\pi$$
$$6$$ 0 0
$$7$$ −49.0000 −0.377964
$$8$$ 63.0000 0.348029
$$9$$ 0 0
$$10$$ −34.0000 −0.107517
$$11$$ 340.000 0.847222 0.423611 0.905844i $$-0.360762\pi$$
0.423611 + 0.905844i $$0.360762\pi$$
$$12$$ 0 0
$$13$$ 454.000 0.745071 0.372535 0.928018i $$-0.378489\pi$$
0.372535 + 0.928018i $$0.378489\pi$$
$$14$$ 49.0000 0.0668153
$$15$$ 0 0
$$16$$ 929.000 0.907227
$$17$$ 798.000 0.669700 0.334850 0.942271i $$-0.391314\pi$$
0.334850 + 0.942271i $$0.391314\pi$$
$$18$$ 0 0
$$19$$ 892.000 0.566867 0.283433 0.958992i $$-0.408527\pi$$
0.283433 + 0.958992i $$0.408527\pi$$
$$20$$ −1054.00 −0.589204
$$21$$ 0 0
$$22$$ −340.000 −0.149769
$$23$$ 3192.00 1.25818 0.629091 0.777332i $$-0.283427\pi$$
0.629091 + 0.777332i $$0.283427\pi$$
$$24$$ 0 0
$$25$$ −1969.00 −0.630080
$$26$$ −454.000 −0.131711
$$27$$ 0 0
$$28$$ 1519.00 0.366153
$$29$$ 8242.00 1.81986 0.909929 0.414764i $$-0.136136\pi$$
0.909929 + 0.414764i $$0.136136\pi$$
$$30$$ 0 0
$$31$$ −2496.00 −0.466488 −0.233244 0.972418i $$-0.574934\pi$$
−0.233244 + 0.972418i $$0.574934\pi$$
$$32$$ −2945.00 −0.508406
$$33$$ 0 0
$$34$$ −798.000 −0.118387
$$35$$ −1666.00 −0.229882
$$36$$ 0 0
$$37$$ 9798.00 1.17661 0.588306 0.808639i $$-0.299795\pi$$
0.588306 + 0.808639i $$0.299795\pi$$
$$38$$ −892.000 −0.100209
$$39$$ 0 0
$$40$$ 2142.00 0.211675
$$41$$ −19834.0 −1.84268 −0.921342 0.388754i $$-0.872906\pi$$
−0.921342 + 0.388754i $$0.872906\pi$$
$$42$$ 0 0
$$43$$ −17236.0 −1.42156 −0.710780 0.703414i $$-0.751658\pi$$
−0.710780 + 0.703414i $$0.751658\pi$$
$$44$$ −10540.0 −0.820746
$$45$$ 0 0
$$46$$ −3192.00 −0.222417
$$47$$ −8928.00 −0.589535 −0.294767 0.955569i $$-0.595242\pi$$
−0.294767 + 0.955569i $$0.595242\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 1969.00 0.111383
$$51$$ 0 0
$$52$$ −14074.0 −0.721787
$$53$$ −150.000 −0.00733502 −0.00366751 0.999993i $$-0.501167\pi$$
−0.00366751 + 0.999993i $$0.501167\pi$$
$$54$$ 0 0
$$55$$ 11560.0 0.515289
$$56$$ −3087.00 −0.131543
$$57$$ 0 0
$$58$$ −8242.00 −0.321709
$$59$$ 42396.0 1.58560 0.792802 0.609479i $$-0.208621\pi$$
0.792802 + 0.609479i $$0.208621\pi$$
$$60$$ 0 0
$$61$$ 14758.0 0.507812 0.253906 0.967229i $$-0.418285\pi$$
0.253906 + 0.967229i $$0.418285\pi$$
$$62$$ 2496.00 0.0824642
$$63$$ 0 0
$$64$$ −26783.0 −0.817352
$$65$$ 15436.0 0.453160
$$66$$ 0 0
$$67$$ −1676.00 −0.0456128 −0.0228064 0.999740i $$-0.507260\pi$$
−0.0228064 + 0.999740i $$0.507260\pi$$
$$68$$ −24738.0 −0.648772
$$69$$ 0 0
$$70$$ 1666.00 0.0406378
$$71$$ −14568.0 −0.342968 −0.171484 0.985187i $$-0.554856\pi$$
−0.171484 + 0.985187i $$0.554856\pi$$
$$72$$ 0 0
$$73$$ 78378.0 1.72142 0.860710 0.509095i $$-0.170020\pi$$
0.860710 + 0.509095i $$0.170020\pi$$
$$74$$ −9798.00 −0.207998
$$75$$ 0 0
$$76$$ −27652.0 −0.549152
$$77$$ −16660.0 −0.320220
$$78$$ 0 0
$$79$$ −2272.00 −0.0409582 −0.0204791 0.999790i $$-0.506519\pi$$
−0.0204791 + 0.999790i $$0.506519\pi$$
$$80$$ 31586.0 0.551785
$$81$$ 0 0
$$82$$ 19834.0 0.325743
$$83$$ 37764.0 0.601704 0.300852 0.953671i $$-0.402729\pi$$
0.300852 + 0.953671i $$0.402729\pi$$
$$84$$ 0 0
$$85$$ 27132.0 0.407319
$$86$$ 17236.0 0.251299
$$87$$ 0 0
$$88$$ 21420.0 0.294858
$$89$$ 117286. 1.56954 0.784768 0.619790i $$-0.212782\pi$$
0.784768 + 0.619790i $$0.212782\pi$$
$$90$$ 0 0
$$91$$ −22246.0 −0.281610
$$92$$ −98952.0 −1.21886
$$93$$ 0 0
$$94$$ 8928.00 0.104216
$$95$$ 30328.0 0.344774
$$96$$ 0 0
$$97$$ 10002.0 0.107934 0.0539669 0.998543i $$-0.482813\pi$$
0.0539669 + 0.998543i $$0.482813\pi$$
$$98$$ −2401.00 −0.0252538
$$99$$ 0 0
$$100$$ 61039.0 0.610390
$$101$$ 108770. 1.06098 0.530488 0.847692i $$-0.322009\pi$$
0.530488 + 0.847692i $$0.322009\pi$$
$$102$$ 0 0
$$103$$ −199192. −1.85003 −0.925015 0.379930i $$-0.875948\pi$$
−0.925015 + 0.379930i $$0.875948\pi$$
$$104$$ 28602.0 0.259306
$$105$$ 0 0
$$106$$ 150.000 0.00129666
$$107$$ 79972.0 0.675272 0.337636 0.941277i $$-0.390373\pi$$
0.337636 + 0.941277i $$0.390373\pi$$
$$108$$ 0 0
$$109$$ −46098.0 −0.371634 −0.185817 0.982584i $$-0.559493\pi$$
−0.185817 + 0.982584i $$0.559493\pi$$
$$110$$ −11560.0 −0.0910911
$$111$$ 0 0
$$112$$ −45521.0 −0.342899
$$113$$ −262706. −1.93541 −0.967707 0.252078i $$-0.918886\pi$$
−0.967707 + 0.252078i $$0.918886\pi$$
$$114$$ 0 0
$$115$$ 108528. 0.765239
$$116$$ −255502. −1.76299
$$117$$ 0 0
$$118$$ −42396.0 −0.280298
$$119$$ −39102.0 −0.253123
$$120$$ 0 0
$$121$$ −45451.0 −0.282215
$$122$$ −14758.0 −0.0897693
$$123$$ 0 0
$$124$$ 77376.0 0.451910
$$125$$ −173196. −0.991432
$$126$$ 0 0
$$127$$ 196608. 1.08166 0.540831 0.841131i $$-0.318110\pi$$
0.540831 + 0.841131i $$0.318110\pi$$
$$128$$ 121023. 0.652894
$$129$$ 0 0
$$130$$ −15436.0 −0.0801081
$$131$$ 77140.0 0.392737 0.196368 0.980530i $$-0.437085\pi$$
0.196368 + 0.980530i $$0.437085\pi$$
$$132$$ 0 0
$$133$$ −43708.0 −0.214255
$$134$$ 1676.00 0.00806329
$$135$$ 0 0
$$136$$ 50274.0 0.233075
$$137$$ −208170. −0.947582 −0.473791 0.880637i $$-0.657115\pi$$
−0.473791 + 0.880637i $$0.657115\pi$$
$$138$$ 0 0
$$139$$ −275580. −1.20979 −0.604896 0.796304i $$-0.706785\pi$$
−0.604896 + 0.796304i $$0.706785\pi$$
$$140$$ 51646.0 0.222698
$$141$$ 0 0
$$142$$ 14568.0 0.0606288
$$143$$ 154360. 0.631240
$$144$$ 0 0
$$145$$ 280228. 1.10686
$$146$$ −78378.0 −0.304307
$$147$$ 0 0
$$148$$ −303738. −1.13984
$$149$$ 296106. 1.09265 0.546326 0.837573i $$-0.316026\pi$$
0.546326 + 0.837573i $$0.316026\pi$$
$$150$$ 0 0
$$151$$ −426472. −1.52212 −0.761059 0.648683i $$-0.775320\pi$$
−0.761059 + 0.648683i $$0.775320\pi$$
$$152$$ 56196.0 0.197286
$$153$$ 0 0
$$154$$ 16660.0 0.0566074
$$155$$ −84864.0 −0.283723
$$156$$ 0 0
$$157$$ 178486. 0.577903 0.288952 0.957344i $$-0.406693\pi$$
0.288952 + 0.957344i $$0.406693\pi$$
$$158$$ 2272.00 0.00724045
$$159$$ 0 0
$$160$$ −100130. −0.309218
$$161$$ −156408. −0.475548
$$162$$ 0 0
$$163$$ 252772. 0.745178 0.372589 0.927996i $$-0.378470\pi$$
0.372589 + 0.927996i $$0.378470\pi$$
$$164$$ 614854. 1.78510
$$165$$ 0 0
$$166$$ −37764.0 −0.106367
$$167$$ −508088. −1.40977 −0.704884 0.709322i $$-0.749001\pi$$
−0.704884 + 0.709322i $$0.749001\pi$$
$$168$$ 0 0
$$169$$ −165177. −0.444870
$$170$$ −27132.0 −0.0720045
$$171$$ 0 0
$$172$$ 534316. 1.37714
$$173$$ 221834. 0.563525 0.281762 0.959484i $$-0.409081\pi$$
0.281762 + 0.959484i $$0.409081\pi$$
$$174$$ 0 0
$$175$$ 96481.0 0.238148
$$176$$ 315860. 0.768622
$$177$$ 0 0
$$178$$ −117286. −0.277457
$$179$$ 113564. 0.264916 0.132458 0.991189i $$-0.457713\pi$$
0.132458 + 0.991189i $$0.457713\pi$$
$$180$$ 0 0
$$181$$ 663118. 1.50451 0.752254 0.658873i $$-0.228967\pi$$
0.752254 + 0.658873i $$0.228967\pi$$
$$182$$ 22246.0 0.0497821
$$183$$ 0 0
$$184$$ 201096. 0.437884
$$185$$ 333132. 0.715628
$$186$$ 0 0
$$187$$ 271320. 0.567385
$$188$$ 276768. 0.571112
$$189$$ 0 0
$$190$$ −30328.0 −0.0609480
$$191$$ −505664. −1.00295 −0.501474 0.865173i $$-0.667209\pi$$
−0.501474 + 0.865173i $$0.667209\pi$$
$$192$$ 0 0
$$193$$ −432382. −0.835554 −0.417777 0.908550i $$-0.637191\pi$$
−0.417777 + 0.908550i $$0.637191\pi$$
$$194$$ −10002.0 −0.0190802
$$195$$ 0 0
$$196$$ −74431.0 −0.138393
$$197$$ 131962. 0.242261 0.121130 0.992637i $$-0.461348\pi$$
0.121130 + 0.992637i $$0.461348\pi$$
$$198$$ 0 0
$$199$$ 298536. 0.534397 0.267199 0.963642i $$-0.413902\pi$$
0.267199 + 0.963642i $$0.413902\pi$$
$$200$$ −124047. −0.219286
$$201$$ 0 0
$$202$$ −108770. −0.187556
$$203$$ −403858. −0.687842
$$204$$ 0 0
$$205$$ −674356. −1.12074
$$206$$ 199192. 0.327042
$$207$$ 0 0
$$208$$ 421766. 0.675948
$$209$$ 303280. 0.480262
$$210$$ 0 0
$$211$$ −1.17062e6 −1.81013 −0.905065 0.425273i $$-0.860178\pi$$
−0.905065 + 0.425273i $$0.860178\pi$$
$$212$$ 4650.00 0.00710581
$$213$$ 0 0
$$214$$ −79972.0 −0.119372
$$215$$ −586024. −0.864608
$$216$$ 0 0
$$217$$ 122304. 0.176316
$$218$$ 46098.0 0.0656963
$$219$$ 0 0
$$220$$ −358360. −0.499186
$$221$$ 362292. 0.498974
$$222$$ 0 0
$$223$$ 399376. 0.537799 0.268899 0.963168i $$-0.413340\pi$$
0.268899 + 0.963168i $$0.413340\pi$$
$$224$$ 144305. 0.192159
$$225$$ 0 0
$$226$$ 262706. 0.342136
$$227$$ −707916. −0.911837 −0.455918 0.890022i $$-0.650689\pi$$
−0.455918 + 0.890022i $$0.650689\pi$$
$$228$$ 0 0
$$229$$ −735778. −0.927167 −0.463584 0.886053i $$-0.653437\pi$$
−0.463584 + 0.886053i $$0.653437\pi$$
$$230$$ −108528. −0.135276
$$231$$ 0 0
$$232$$ 519246. 0.633364
$$233$$ 208758. 0.251915 0.125957 0.992036i $$-0.459800\pi$$
0.125957 + 0.992036i $$0.459800\pi$$
$$234$$ 0 0
$$235$$ −303552. −0.358561
$$236$$ −1.31428e6 −1.53605
$$237$$ 0 0
$$238$$ 39102.0 0.0447462
$$239$$ −713376. −0.807837 −0.403919 0.914795i $$-0.632352\pi$$
−0.403919 + 0.914795i $$0.632352\pi$$
$$240$$ 0 0
$$241$$ −505246. −0.560351 −0.280176 0.959949i $$-0.590393\pi$$
−0.280176 + 0.959949i $$0.590393\pi$$
$$242$$ 45451.0 0.0498890
$$243$$ 0 0
$$244$$ −457498. −0.491943
$$245$$ 81634.0 0.0868872
$$246$$ 0 0
$$247$$ 404968. 0.422356
$$248$$ −157248. −0.162351
$$249$$ 0 0
$$250$$ 173196. 0.175262
$$251$$ −317108. −0.317704 −0.158852 0.987302i $$-0.550779\pi$$
−0.158852 + 0.987302i $$0.550779\pi$$
$$252$$ 0 0
$$253$$ 1.08528e6 1.06596
$$254$$ −196608. −0.191213
$$255$$ 0 0
$$256$$ 736033. 0.701936
$$257$$ 1.44285e6 1.36266 0.681329 0.731977i $$-0.261402\pi$$
0.681329 + 0.731977i $$0.261402\pi$$
$$258$$ 0 0
$$259$$ −480102. −0.444717
$$260$$ −478516. −0.438999
$$261$$ 0 0
$$262$$ −77140.0 −0.0694267
$$263$$ −271496. −0.242033 −0.121016 0.992651i $$-0.538615\pi$$
−0.121016 + 0.992651i $$0.538615\pi$$
$$264$$ 0 0
$$265$$ −5100.00 −0.00446124
$$266$$ 43708.0 0.0378754
$$267$$ 0 0
$$268$$ 51956.0 0.0441874
$$269$$ −850614. −0.716724 −0.358362 0.933583i $$-0.616665\pi$$
−0.358362 + 0.933583i $$0.616665\pi$$
$$270$$ 0 0
$$271$$ −540128. −0.446759 −0.223380 0.974732i $$-0.571709\pi$$
−0.223380 + 0.974732i $$0.571709\pi$$
$$272$$ 741342. 0.607570
$$273$$ 0 0
$$274$$ 208170. 0.167510
$$275$$ −669460. −0.533818
$$276$$ 0 0
$$277$$ 513574. 0.402164 0.201082 0.979574i $$-0.435554\pi$$
0.201082 + 0.979574i $$0.435554\pi$$
$$278$$ 275580. 0.213863
$$279$$ 0 0
$$280$$ −104958. −0.0800056
$$281$$ 1.35642e6 1.02478 0.512388 0.858754i $$-0.328761\pi$$
0.512388 + 0.858754i $$0.328761\pi$$
$$282$$ 0 0
$$283$$ 286756. 0.212837 0.106418 0.994321i $$-0.466062\pi$$
0.106418 + 0.994321i $$0.466062\pi$$
$$284$$ 451608. 0.332251
$$285$$ 0 0
$$286$$ −154360. −0.111589
$$287$$ 971866. 0.696469
$$288$$ 0 0
$$289$$ −783053. −0.551501
$$290$$ −280228. −0.195667
$$291$$ 0 0
$$292$$ −2.42972e6 −1.66763
$$293$$ 1.70727e6 1.16180 0.580901 0.813974i $$-0.302700\pi$$
0.580901 + 0.813974i $$0.302700\pi$$
$$294$$ 0 0
$$295$$ 1.44146e6 0.964381
$$296$$ 617274. 0.409495
$$297$$ 0 0
$$298$$ −296106. −0.193155
$$299$$ 1.44917e6 0.937434
$$300$$ 0 0
$$301$$ 844564. 0.537299
$$302$$ 426472. 0.269075
$$303$$ 0 0
$$304$$ 828668. 0.514276
$$305$$ 501772. 0.308857
$$306$$ 0 0
$$307$$ −546788. −0.331111 −0.165555 0.986201i $$-0.552942\pi$$
−0.165555 + 0.986201i $$0.552942\pi$$
$$308$$ 516460. 0.310213
$$309$$ 0 0
$$310$$ 84864.0 0.0501556
$$311$$ −3.23426e6 −1.89616 −0.948079 0.318035i $$-0.896977\pi$$
−0.948079 + 0.318035i $$0.896977\pi$$
$$312$$ 0 0
$$313$$ 1.81313e6 1.04609 0.523044 0.852306i $$-0.324796\pi$$
0.523044 + 0.852306i $$0.324796\pi$$
$$314$$ −178486. −0.102160
$$315$$ 0 0
$$316$$ 70432.0 0.0396782
$$317$$ 1.27658e6 0.713509 0.356754 0.934198i $$-0.383883\pi$$
0.356754 + 0.934198i $$0.383883\pi$$
$$318$$ 0 0
$$319$$ 2.80228e6 1.54182
$$320$$ −910622. −0.497122
$$321$$ 0 0
$$322$$ 156408. 0.0840658
$$323$$ 711816. 0.379631
$$324$$ 0 0
$$325$$ −893926. −0.469454
$$326$$ −252772. −0.131730
$$327$$ 0 0
$$328$$ −1.24954e6 −0.641307
$$329$$ 437472. 0.222823
$$330$$ 0 0
$$331$$ −1.73621e6 −0.871029 −0.435515 0.900182i $$-0.643434\pi$$
−0.435515 + 0.900182i $$0.643434\pi$$
$$332$$ −1.17068e6 −0.582901
$$333$$ 0 0
$$334$$ 508088. 0.249214
$$335$$ −56984.0 −0.0277422
$$336$$ 0 0
$$337$$ 2.07215e6 0.993907 0.496953 0.867777i $$-0.334452\pi$$
0.496953 + 0.867777i $$0.334452\pi$$
$$338$$ 165177. 0.0786426
$$339$$ 0 0
$$340$$ −841092. −0.394590
$$341$$ −848640. −0.395219
$$342$$ 0 0
$$343$$ −117649. −0.0539949
$$344$$ −1.08587e6 −0.494744
$$345$$ 0 0
$$346$$ −221834. −0.0996180
$$347$$ 1.65146e6 0.736282 0.368141 0.929770i $$-0.379994\pi$$
0.368141 + 0.929770i $$0.379994\pi$$
$$348$$ 0 0
$$349$$ 1.26645e6 0.556578 0.278289 0.960497i $$-0.410233\pi$$
0.278289 + 0.960497i $$0.410233\pi$$
$$350$$ −96481.0 −0.0420990
$$351$$ 0 0
$$352$$ −1.00130e6 −0.430732
$$353$$ −573218. −0.244840 −0.122420 0.992478i $$-0.539066\pi$$
−0.122420 + 0.992478i $$0.539066\pi$$
$$354$$ 0 0
$$355$$ −495312. −0.208597
$$356$$ −3.63587e6 −1.52049
$$357$$ 0 0
$$358$$ −113564. −0.0468310
$$359$$ −4.46322e6 −1.82773 −0.913866 0.406016i $$-0.866918\pi$$
−0.913866 + 0.406016i $$0.866918\pi$$
$$360$$ 0 0
$$361$$ −1.68044e6 −0.678662
$$362$$ −663118. −0.265962
$$363$$ 0 0
$$364$$ 689626. 0.272810
$$365$$ 2.66485e6 1.04699
$$366$$ 0 0
$$367$$ −4.50797e6 −1.74709 −0.873546 0.486742i $$-0.838185\pi$$
−0.873546 + 0.486742i $$0.838185\pi$$
$$368$$ 2.96537e6 1.14146
$$369$$ 0 0
$$370$$ −333132. −0.126506
$$371$$ 7350.00 0.00277238
$$372$$ 0 0
$$373$$ 1.66535e6 0.619774 0.309887 0.950773i $$-0.399709\pi$$
0.309887 + 0.950773i $$0.399709\pi$$
$$374$$ −271320. −0.100300
$$375$$ 0 0
$$376$$ −562464. −0.205175
$$377$$ 3.74187e6 1.35592
$$378$$ 0 0
$$379$$ −2.53232e6 −0.905568 −0.452784 0.891620i $$-0.649569\pi$$
−0.452784 + 0.891620i $$0.649569\pi$$
$$380$$ −940168. −0.334000
$$381$$ 0 0
$$382$$ 505664. 0.177298
$$383$$ −796368. −0.277407 −0.138703 0.990334i $$-0.544293\pi$$
−0.138703 + 0.990334i $$0.544293\pi$$
$$384$$ 0 0
$$385$$ −566440. −0.194761
$$386$$ 432382. 0.147706
$$387$$ 0 0
$$388$$ −310062. −0.104561
$$389$$ −1.94799e6 −0.652699 −0.326349 0.945249i $$-0.605819\pi$$
−0.326349 + 0.945249i $$0.605819\pi$$
$$390$$ 0 0
$$391$$ 2.54722e6 0.842605
$$392$$ 151263. 0.0497184
$$393$$ 0 0
$$394$$ −131962. −0.0428261
$$395$$ −77248.0 −0.0249112
$$396$$ 0 0
$$397$$ 1.08116e6 0.344281 0.172140 0.985072i $$-0.444932\pi$$
0.172140 + 0.985072i $$0.444932\pi$$
$$398$$ −298536. −0.0944689
$$399$$ 0 0
$$400$$ −1.82920e6 −0.571625
$$401$$ −2.76770e6 −0.859524 −0.429762 0.902942i $$-0.641402\pi$$
−0.429762 + 0.902942i $$0.641402\pi$$
$$402$$ 0 0
$$403$$ −1.13318e6 −0.347566
$$404$$ −3.37187e6 −1.02782
$$405$$ 0 0
$$406$$ 403858. 0.121594
$$407$$ 3.33132e6 0.996851
$$408$$ 0 0
$$409$$ 2.36350e6 0.698630 0.349315 0.937005i $$-0.386414\pi$$
0.349315 + 0.937005i $$0.386414\pi$$
$$410$$ 674356. 0.198121
$$411$$ 0 0
$$412$$ 6.17495e6 1.79222
$$413$$ −2.07740e6 −0.599302
$$414$$ 0 0
$$415$$ 1.28398e6 0.365963
$$416$$ −1.33703e6 −0.378798
$$417$$ 0 0
$$418$$ −303280. −0.0848991
$$419$$ 2.98669e6 0.831104 0.415552 0.909569i $$-0.363588\pi$$
0.415552 + 0.909569i $$0.363588\pi$$
$$420$$ 0 0
$$421$$ −3.46331e6 −0.952326 −0.476163 0.879357i $$-0.657973\pi$$
−0.476163 + 0.879357i $$0.657973\pi$$
$$422$$ 1.17062e6 0.319989
$$423$$ 0 0
$$424$$ −9450.00 −0.00255280
$$425$$ −1.57126e6 −0.421965
$$426$$ 0 0
$$427$$ −723142. −0.191935
$$428$$ −2.47913e6 −0.654169
$$429$$ 0 0
$$430$$ 586024. 0.152843
$$431$$ −2.33693e6 −0.605971 −0.302986 0.952995i $$-0.597983\pi$$
−0.302986 + 0.952995i $$0.597983\pi$$
$$432$$ 0 0
$$433$$ −3.50838e6 −0.899264 −0.449632 0.893214i $$-0.648445\pi$$
−0.449632 + 0.893214i $$0.648445\pi$$
$$434$$ −122304. −0.0311685
$$435$$ 0 0
$$436$$ 1.42904e6 0.360021
$$437$$ 2.84726e6 0.713221
$$438$$ 0 0
$$439$$ 3.54833e6 0.878744 0.439372 0.898305i $$-0.355201\pi$$
0.439372 + 0.898305i $$0.355201\pi$$
$$440$$ 728280. 0.179336
$$441$$ 0 0
$$442$$ −362292. −0.0882070
$$443$$ −1.76833e6 −0.428109 −0.214055 0.976822i $$-0.568667\pi$$
−0.214055 + 0.976822i $$0.568667\pi$$
$$444$$ 0 0
$$445$$ 3.98772e6 0.954608
$$446$$ −399376. −0.0950703
$$447$$ 0 0
$$448$$ 1.31237e6 0.308930
$$449$$ 5.52579e6 1.29354 0.646768 0.762687i $$-0.276120\pi$$
0.646768 + 0.762687i $$0.276120\pi$$
$$450$$ 0 0
$$451$$ −6.74356e6 −1.56116
$$452$$ 8.14389e6 1.87493
$$453$$ 0 0
$$454$$ 707916. 0.161191
$$455$$ −756364. −0.171278
$$456$$ 0 0
$$457$$ −2.96226e6 −0.663488 −0.331744 0.943369i $$-0.607637\pi$$
−0.331744 + 0.943369i $$0.607637\pi$$
$$458$$ 735778. 0.163902
$$459$$ 0 0
$$460$$ −3.36437e6 −0.741325
$$461$$ −2.11884e6 −0.464350 −0.232175 0.972674i $$-0.574584\pi$$
−0.232175 + 0.972674i $$0.574584\pi$$
$$462$$ 0 0
$$463$$ 3.19226e6 0.692062 0.346031 0.938223i $$-0.387529\pi$$
0.346031 + 0.938223i $$0.387529\pi$$
$$464$$ 7.65682e6 1.65102
$$465$$ 0 0
$$466$$ −208758. −0.0445326
$$467$$ 7.42621e6 1.57571 0.787853 0.615863i $$-0.211193\pi$$
0.787853 + 0.615863i $$0.211193\pi$$
$$468$$ 0 0
$$469$$ 82124.0 0.0172400
$$470$$ 303552. 0.0633853
$$471$$ 0 0
$$472$$ 2.67095e6 0.551837
$$473$$ −5.86024e6 −1.20438
$$474$$ 0 0
$$475$$ −1.75635e6 −0.357171
$$476$$ 1.21216e6 0.245213
$$477$$ 0 0
$$478$$ 713376. 0.142807
$$479$$ 3.39685e6 0.676453 0.338226 0.941065i $$-0.390173\pi$$
0.338226 + 0.941065i $$0.390173\pi$$
$$480$$ 0 0
$$481$$ 4.44829e6 0.876659
$$482$$ 505246. 0.0990570
$$483$$ 0 0
$$484$$ 1.40898e6 0.273396
$$485$$ 340068. 0.0656465
$$486$$ 0 0
$$487$$ −3.71382e6 −0.709574 −0.354787 0.934947i $$-0.615447\pi$$
−0.354787 + 0.934947i $$0.615447\pi$$
$$488$$ 929754. 0.176733
$$489$$ 0 0
$$490$$ −81634.0 −0.0153596
$$491$$ −5.57494e6 −1.04361 −0.521803 0.853066i $$-0.674740\pi$$
−0.521803 + 0.853066i $$0.674740\pi$$
$$492$$ 0 0
$$493$$ 6.57712e6 1.21876
$$494$$ −404968. −0.0746626
$$495$$ 0 0
$$496$$ −2.31878e6 −0.423210
$$497$$ 713832. 0.129630
$$498$$ 0 0
$$499$$ 3.92698e6 0.706004 0.353002 0.935623i $$-0.385161\pi$$
0.353002 + 0.935623i $$0.385161\pi$$
$$500$$ 5.36908e6 0.960450
$$501$$ 0 0
$$502$$ 317108. 0.0561627
$$503$$ −6.42079e6 −1.13154 −0.565768 0.824564i $$-0.691420\pi$$
−0.565768 + 0.824564i $$0.691420\pi$$
$$504$$ 0 0
$$505$$ 3.69818e6 0.645297
$$506$$ −1.08528e6 −0.188437
$$507$$ 0 0
$$508$$ −6.09485e6 −1.04786
$$509$$ −146278. −0.0250256 −0.0125128 0.999922i $$-0.503983\pi$$
−0.0125128 + 0.999922i $$0.503983\pi$$
$$510$$ 0 0
$$511$$ −3.84052e6 −0.650636
$$512$$ −4.60877e6 −0.776980
$$513$$ 0 0
$$514$$ −1.44285e6 −0.240886
$$515$$ −6.77253e6 −1.12521
$$516$$ 0 0
$$517$$ −3.03552e6 −0.499467
$$518$$ 480102. 0.0786157
$$519$$ 0 0
$$520$$ 972468. 0.157713
$$521$$ −7.70937e6 −1.24430 −0.622149 0.782899i $$-0.713740\pi$$
−0.622149 + 0.782899i $$0.713740\pi$$
$$522$$ 0 0
$$523$$ −569420. −0.0910287 −0.0455144 0.998964i $$-0.514493\pi$$
−0.0455144 + 0.998964i $$0.514493\pi$$
$$524$$ −2.39134e6 −0.380464
$$525$$ 0 0
$$526$$ 271496. 0.0427857
$$527$$ −1.99181e6 −0.312407
$$528$$ 0 0
$$529$$ 3.75252e6 0.583021
$$530$$ 5100.00 0.000788643 0
$$531$$ 0 0
$$532$$ 1.35495e6 0.207560
$$533$$ −9.00464e6 −1.37293
$$534$$ 0 0
$$535$$ 2.71905e6 0.410707
$$536$$ −105588. −0.0158746
$$537$$ 0 0
$$538$$ 850614. 0.126700
$$539$$ 816340. 0.121032
$$540$$ 0 0
$$541$$ −9.44802e6 −1.38787 −0.693933 0.720040i $$-0.744124\pi$$
−0.693933 + 0.720040i $$0.744124\pi$$
$$542$$ 540128. 0.0789766
$$543$$ 0 0
$$544$$ −2.35011e6 −0.340479
$$545$$ −1.56733e6 −0.226032
$$546$$ 0 0
$$547$$ −1.35321e6 −0.193374 −0.0966869 0.995315i $$-0.530825\pi$$
−0.0966869 + 0.995315i $$0.530825\pi$$
$$548$$ 6.45327e6 0.917970
$$549$$ 0 0
$$550$$ 669460. 0.0943665
$$551$$ 7.35186e6 1.03162
$$552$$ 0 0
$$553$$ 111328. 0.0154807
$$554$$ −513574. −0.0710933
$$555$$ 0 0
$$556$$ 8.54298e6 1.17199
$$557$$ −8.19390e6 −1.11906 −0.559529 0.828811i $$-0.689018\pi$$
−0.559529 + 0.828811i $$0.689018\pi$$
$$558$$ 0 0
$$559$$ −7.82514e6 −1.05916
$$560$$ −1.54771e6 −0.208555
$$561$$ 0 0
$$562$$ −1.35642e6 −0.181157
$$563$$ 1.05796e7 1.40669 0.703347 0.710847i $$-0.251688\pi$$
0.703347 + 0.710847i $$0.251688\pi$$
$$564$$ 0 0
$$565$$ −8.93200e6 −1.17714
$$566$$ −286756. −0.0376246
$$567$$ 0 0
$$568$$ −917784. −0.119363
$$569$$ 1.20205e7 1.55648 0.778238 0.627969i $$-0.216114\pi$$
0.778238 + 0.627969i $$0.216114\pi$$
$$570$$ 0 0
$$571$$ −2.48948e6 −0.319534 −0.159767 0.987155i $$-0.551074\pi$$
−0.159767 + 0.987155i $$0.551074\pi$$
$$572$$ −4.78516e6 −0.611514
$$573$$ 0 0
$$574$$ −971866. −0.123119
$$575$$ −6.28505e6 −0.792755
$$576$$ 0 0
$$577$$ 8.21322e6 1.02701 0.513504 0.858087i $$-0.328347\pi$$
0.513504 + 0.858087i $$0.328347\pi$$
$$578$$ 783053. 0.0974926
$$579$$ 0 0
$$580$$ −8.68707e6 −1.07227
$$581$$ −1.85044e6 −0.227423
$$582$$ 0 0
$$583$$ −51000.0 −0.00621439
$$584$$ 4.93781e6 0.599105
$$585$$ 0 0
$$586$$ −1.70727e6 −0.205380
$$587$$ 1.21827e6 0.145931 0.0729655 0.997334i $$-0.476754\pi$$
0.0729655 + 0.997334i $$0.476754\pi$$
$$588$$ 0 0
$$589$$ −2.22643e6 −0.264436
$$590$$ −1.44146e6 −0.170480
$$591$$ 0 0
$$592$$ 9.10234e6 1.06745
$$593$$ 8.42379e6 0.983718 0.491859 0.870675i $$-0.336317\pi$$
0.491859 + 0.870675i $$0.336317\pi$$
$$594$$ 0 0
$$595$$ −1.32947e6 −0.153952
$$596$$ −9.17929e6 −1.05851
$$597$$ 0 0
$$598$$ −1.44917e6 −0.165717
$$599$$ −8.21254e6 −0.935212 −0.467606 0.883937i $$-0.654883\pi$$
−0.467606 + 0.883937i $$0.654883\pi$$
$$600$$ 0 0
$$601$$ 3.25478e6 0.367566 0.183783 0.982967i $$-0.441166\pi$$
0.183783 + 0.982967i $$0.441166\pi$$
$$602$$ −844564. −0.0949820
$$603$$ 0 0
$$604$$ 1.32206e7 1.47455
$$605$$ −1.54533e6 −0.171646
$$606$$ 0 0
$$607$$ 7.82101e6 0.861571 0.430785 0.902454i $$-0.358237\pi$$
0.430785 + 0.902454i $$0.358237\pi$$
$$608$$ −2.62694e6 −0.288198
$$609$$ 0 0
$$610$$ −501772. −0.0545986
$$611$$ −4.05331e6 −0.439245
$$612$$ 0 0
$$613$$ −9.51670e6 −1.02290 −0.511452 0.859312i $$-0.670892\pi$$
−0.511452 + 0.859312i $$0.670892\pi$$
$$614$$ 546788. 0.0585326
$$615$$ 0 0
$$616$$ −1.04958e6 −0.111446
$$617$$ 7.04895e6 0.745438 0.372719 0.927944i $$-0.378426\pi$$
0.372719 + 0.927944i $$0.378426\pi$$
$$618$$ 0 0
$$619$$ −6.32174e6 −0.663147 −0.331574 0.943429i $$-0.607580\pi$$
−0.331574 + 0.943429i $$0.607580\pi$$
$$620$$ 2.63078e6 0.274856
$$621$$ 0 0
$$622$$ 3.23426e6 0.335197
$$623$$ −5.74701e6 −0.593229
$$624$$ 0 0
$$625$$ 264461. 0.0270808
$$626$$ −1.81313e6 −0.184924
$$627$$ 0 0
$$628$$ −5.53307e6 −0.559844
$$629$$ 7.81880e6 0.787977
$$630$$ 0 0
$$631$$ 8.61236e6 0.861090 0.430545 0.902569i $$-0.358321\pi$$
0.430545 + 0.902569i $$0.358321\pi$$
$$632$$ −143136. −0.0142546
$$633$$ 0 0
$$634$$ −1.27658e6 −0.126132
$$635$$ 6.68467e6 0.657879
$$636$$ 0 0
$$637$$ 1.09005e6 0.106439
$$638$$ −2.80228e6 −0.272559
$$639$$ 0 0
$$640$$ 4.11478e6 0.397097
$$641$$ 5.22829e6 0.502590 0.251295 0.967910i $$-0.419143\pi$$
0.251295 + 0.967910i $$0.419143\pi$$
$$642$$ 0 0
$$643$$ 1.61373e7 1.53923 0.769615 0.638508i $$-0.220448\pi$$
0.769615 + 0.638508i $$0.220448\pi$$
$$644$$ 4.84865e6 0.460687
$$645$$ 0 0
$$646$$ −711816. −0.0671099
$$647$$ 1.58749e7 1.49090 0.745451 0.666560i $$-0.232234\pi$$
0.745451 + 0.666560i $$0.232234\pi$$
$$648$$ 0 0
$$649$$ 1.44146e7 1.34336
$$650$$ 893926. 0.0829886
$$651$$ 0 0
$$652$$ −7.83593e6 −0.721891
$$653$$ 5.94112e6 0.545237 0.272619 0.962122i $$-0.412110\pi$$
0.272619 + 0.962122i $$0.412110\pi$$
$$654$$ 0 0
$$655$$ 2.62276e6 0.238867
$$656$$ −1.84258e7 −1.67173
$$657$$ 0 0
$$658$$ −437472. −0.0393900
$$659$$ 7.64430e6 0.685684 0.342842 0.939393i $$-0.388610\pi$$
0.342842 + 0.939393i $$0.388610\pi$$
$$660$$ 0 0
$$661$$ −7.58688e6 −0.675398 −0.337699 0.941254i $$-0.609649\pi$$
−0.337699 + 0.941254i $$0.609649\pi$$
$$662$$ 1.73621e6 0.153978
$$663$$ 0 0
$$664$$ 2.37913e6 0.209410
$$665$$ −1.48607e6 −0.130312
$$666$$ 0 0
$$667$$ 2.63085e7 2.28971
$$668$$ 1.57507e7 1.36571
$$669$$ 0 0
$$670$$ 56984.0 0.00490417
$$671$$ 5.01772e6 0.430229
$$672$$ 0 0
$$673$$ −2.06681e7 −1.75899 −0.879494 0.475910i $$-0.842119\pi$$
−0.879494 + 0.475910i $$0.842119\pi$$
$$674$$ −2.07215e6 −0.175700
$$675$$ 0 0
$$676$$ 5.12049e6 0.430968
$$677$$ −7.89541e6 −0.662068 −0.331034 0.943619i $$-0.607398\pi$$
−0.331034 + 0.943619i $$0.607398\pi$$
$$678$$ 0 0
$$679$$ −490098. −0.0407951
$$680$$ 1.70932e6 0.141759
$$681$$ 0 0
$$682$$ 848640. 0.0698655
$$683$$ 1.96015e7 1.60782 0.803911 0.594750i $$-0.202749\pi$$
0.803911 + 0.594750i $$0.202749\pi$$
$$684$$ 0 0
$$685$$ −7.07778e6 −0.576329
$$686$$ 117649. 0.00954504
$$687$$ 0 0
$$688$$ −1.60122e7 −1.28968
$$689$$ −68100.0 −0.00546511
$$690$$ 0 0
$$691$$ −1.72710e7 −1.37601 −0.688005 0.725706i $$-0.741513\pi$$
−0.688005 + 0.725706i $$0.741513\pi$$
$$692$$ −6.87685e6 −0.545914
$$693$$ 0 0
$$694$$ −1.65146e6 −0.130158
$$695$$ −9.36972e6 −0.735808
$$696$$ 0 0
$$697$$ −1.58275e7 −1.23405
$$698$$ −1.26645e6 −0.0983900
$$699$$ 0 0
$$700$$ −2.99091e6 −0.230706
$$701$$ 5.36344e6 0.412238 0.206119 0.978527i $$-0.433917\pi$$
0.206119 + 0.978527i $$0.433917\pi$$
$$702$$ 0 0
$$703$$ 8.73982e6 0.666982
$$704$$ −9.10622e6 −0.692479
$$705$$ 0 0
$$706$$ 573218. 0.0432821
$$707$$ −5.32973e6 −0.401011
$$708$$ 0 0
$$709$$ −1.73733e7 −1.29798 −0.648988 0.760798i $$-0.724808\pi$$
−0.648988 + 0.760798i $$0.724808\pi$$
$$710$$ 495312. 0.0368751
$$711$$ 0 0
$$712$$ 7.38902e6 0.546244
$$713$$ −7.96723e6 −0.586926
$$714$$ 0 0
$$715$$ 5.24824e6 0.383927
$$716$$ −3.52048e6 −0.256637
$$717$$ 0 0
$$718$$ 4.46322e6 0.323100
$$719$$ −424608. −0.0306313 −0.0153157 0.999883i $$-0.504875\pi$$
−0.0153157 + 0.999883i $$0.504875\pi$$
$$720$$ 0 0
$$721$$ 9.76041e6 0.699246
$$722$$ 1.68044e6 0.119972
$$723$$ 0 0
$$724$$ −2.05567e7 −1.45749
$$725$$ −1.62285e7 −1.14666
$$726$$ 0 0
$$727$$ 2.18290e7 1.53179 0.765893 0.642968i $$-0.222297\pi$$
0.765893 + 0.642968i $$0.222297\pi$$
$$728$$ −1.40150e6 −0.0980086
$$729$$ 0 0
$$730$$ −2.66485e6 −0.185083
$$731$$ −1.37543e7 −0.952020
$$732$$ 0 0
$$733$$ 2.17675e7 1.49640 0.748202 0.663470i $$-0.230917\pi$$
0.748202 + 0.663470i $$0.230917\pi$$
$$734$$ 4.50797e6 0.308845
$$735$$ 0 0
$$736$$ −9.40044e6 −0.639667
$$737$$ −569840. −0.0386442
$$738$$ 0 0
$$739$$ 6.21786e6 0.418822 0.209411 0.977828i $$-0.432845\pi$$
0.209411 + 0.977828i $$0.432845\pi$$
$$740$$ −1.03271e7 −0.693264
$$741$$ 0 0
$$742$$ −7350.00 −0.000490092 0
$$743$$ −3.77647e6 −0.250966 −0.125483 0.992096i $$-0.540048\pi$$
−0.125483 + 0.992096i $$0.540048\pi$$
$$744$$ 0 0
$$745$$ 1.00676e7 0.664562
$$746$$ −1.66535e6 −0.109562
$$747$$ 0 0
$$748$$ −8.41092e6 −0.549654
$$749$$ −3.91863e6 −0.255229
$$750$$ 0 0
$$751$$ −2.88795e6 −0.186849 −0.0934244 0.995626i $$-0.529781\pi$$
−0.0934244 + 0.995626i $$0.529781\pi$$
$$752$$ −8.29411e6 −0.534842
$$753$$ 0 0
$$754$$ −3.74187e6 −0.239696
$$755$$ −1.45000e7 −0.925768
$$756$$ 0 0
$$757$$ 1.25519e6 0.0796104 0.0398052 0.999207i $$-0.487326\pi$$
0.0398052 + 0.999207i $$0.487326\pi$$
$$758$$ 2.53232e6 0.160083
$$759$$ 0 0
$$760$$ 1.91066e6 0.119991
$$761$$ 1.42623e7 0.892746 0.446373 0.894847i $$-0.352716\pi$$
0.446373 + 0.894847i $$0.352716\pi$$
$$762$$ 0 0
$$763$$ 2.25880e6 0.140465
$$764$$ 1.56756e7 0.971606
$$765$$ 0 0
$$766$$ 796368. 0.0490390
$$767$$ 1.92478e7 1.18139
$$768$$ 0 0
$$769$$ −2.02261e7 −1.23338 −0.616689 0.787207i $$-0.711526\pi$$
−0.616689 + 0.787207i $$0.711526\pi$$
$$770$$ 566440. 0.0344292
$$771$$ 0 0
$$772$$ 1.34038e7 0.809443
$$773$$ −2.62288e7 −1.57881 −0.789406 0.613872i $$-0.789611\pi$$
−0.789406 + 0.613872i $$0.789611\pi$$
$$774$$ 0 0
$$775$$ 4.91462e6 0.293925
$$776$$ 630126. 0.0375641
$$777$$ 0 0
$$778$$ 1.94799e6 0.115382
$$779$$ −1.76919e7 −1.04456
$$780$$ 0 0
$$781$$ −4.95312e6 −0.290570
$$782$$ −2.54722e6 −0.148953
$$783$$ 0 0
$$784$$ 2.23053e6 0.129604
$$785$$ 6.06852e6 0.351487
$$786$$ 0 0
$$787$$ −9.92829e6 −0.571397 −0.285698 0.958320i $$-0.592226\pi$$
−0.285698 + 0.958320i $$0.592226\pi$$
$$788$$ −4.09082e6 −0.234690
$$789$$ 0 0
$$790$$ 77248.0 0.00440372
$$791$$ 1.28726e7 0.731518
$$792$$ 0 0
$$793$$ 6.70013e6 0.378356
$$794$$ −1.08116e6 −0.0608608
$$795$$ 0 0
$$796$$ −9.25462e6 −0.517697
$$797$$ −1.09033e7 −0.608014 −0.304007 0.952670i $$-0.598325\pi$$
−0.304007 + 0.952670i $$0.598325\pi$$
$$798$$ 0 0
$$799$$ −7.12454e6 −0.394812
$$800$$ 5.79871e6 0.320336
$$801$$ 0 0
$$802$$ 2.76770e6 0.151944
$$803$$ 2.66485e7 1.45843
$$804$$ 0 0
$$805$$ −5.31787e6 −0.289233
$$806$$ 1.13318e6 0.0614416
$$807$$ 0 0
$$808$$ 6.85251e6 0.369251
$$809$$ −6.06398e6 −0.325751 −0.162876 0.986647i $$-0.552077\pi$$
−0.162876 + 0.986647i $$0.552077\pi$$
$$810$$ 0 0
$$811$$ −8.59438e6 −0.458841 −0.229421 0.973327i $$-0.573683\pi$$
−0.229421 + 0.973327i $$0.573683\pi$$
$$812$$ 1.25196e7 0.666347
$$813$$ 0 0
$$814$$ −3.33132e6 −0.176220
$$815$$ 8.59425e6 0.453225
$$816$$ 0 0
$$817$$ −1.53745e7 −0.805835
$$818$$ −2.36350e6 −0.123501
$$819$$ 0 0
$$820$$ 2.09050e7 1.08572
$$821$$ 2.01396e6 0.104278 0.0521391 0.998640i $$-0.483396\pi$$
0.0521391 + 0.998640i $$0.483396\pi$$
$$822$$ 0 0
$$823$$ −2.64679e7 −1.36213 −0.681067 0.732221i $$-0.738484\pi$$
−0.681067 + 0.732221i $$0.738484\pi$$
$$824$$ −1.25491e7 −0.643864
$$825$$ 0 0
$$826$$ 2.07740e6 0.105943
$$827$$ 3.90229e6 0.198407 0.0992033 0.995067i $$-0.468371\pi$$
0.0992033 + 0.995067i $$0.468371\pi$$
$$828$$ 0 0
$$829$$ −1.95595e7 −0.988487 −0.494244 0.869323i $$-0.664555\pi$$
−0.494244 + 0.869323i $$0.664555\pi$$
$$830$$ −1.28398e6 −0.0646937
$$831$$ 0 0
$$832$$ −1.21595e7 −0.608985
$$833$$ 1.91600e6 0.0956715
$$834$$ 0 0
$$835$$ −1.72750e7 −0.857436
$$836$$ −9.40168e6 −0.465254
$$837$$ 0 0
$$838$$ −2.98669e6 −0.146920
$$839$$ 2.45448e7 1.20380 0.601901 0.798570i $$-0.294410\pi$$
0.601901 + 0.798570i $$0.294410\pi$$
$$840$$ 0 0
$$841$$ 4.74194e7 2.31188
$$842$$ 3.46331e6 0.168349
$$843$$ 0 0
$$844$$ 3.62892e7 1.75356
$$845$$ −5.61602e6 −0.270574
$$846$$ 0 0
$$847$$ 2.22710e6 0.106667
$$848$$ −139350. −0.00665453
$$849$$ 0 0
$$850$$ 1.57126e6 0.0745936
$$851$$ 3.12752e7 1.48039
$$852$$ 0 0
$$853$$ 3.38305e7 1.59197 0.795987 0.605314i $$-0.206952\pi$$
0.795987 + 0.605314i $$0.206952\pi$$
$$854$$ 723142. 0.0339296
$$855$$ 0 0
$$856$$ 5.03824e6 0.235014
$$857$$ −3.18009e7 −1.47907 −0.739534 0.673120i $$-0.764954\pi$$
−0.739534 + 0.673120i $$0.764954\pi$$
$$858$$ 0 0
$$859$$ 638420. 0.0295205 0.0147602 0.999891i $$-0.495301\pi$$
0.0147602 + 0.999891i $$0.495301\pi$$
$$860$$ 1.81667e7 0.837589
$$861$$ 0 0
$$862$$ 2.33693e6 0.107122
$$863$$ 4.22256e6 0.192996 0.0964981 0.995333i $$-0.469236\pi$$
0.0964981 + 0.995333i $$0.469236\pi$$
$$864$$ 0 0
$$865$$ 7.54236e6 0.342742
$$866$$ 3.50838e6 0.158969
$$867$$ 0 0
$$868$$ −3.79142e6 −0.170806
$$869$$ −772480. −0.0347007
$$870$$ 0 0
$$871$$ −760904. −0.0339848
$$872$$ −2.90417e6 −0.129340
$$873$$ 0 0
$$874$$ −2.84726e6 −0.126081
$$875$$ 8.48660e6 0.374726
$$876$$ 0 0
$$877$$ −2.45043e7 −1.07583 −0.537915 0.842999i $$-0.680788\pi$$
−0.537915 + 0.842999i $$0.680788\pi$$
$$878$$ −3.54833e6 −0.155341
$$879$$ 0 0
$$880$$ 1.07392e7 0.467484
$$881$$ 2.77630e7 1.20511 0.602555 0.798078i $$-0.294150\pi$$
0.602555 + 0.798078i $$0.294150\pi$$
$$882$$ 0 0
$$883$$ 3.30170e7 1.42507 0.712534 0.701638i $$-0.247548\pi$$
0.712534 + 0.701638i $$0.247548\pi$$
$$884$$ −1.12311e7 −0.483381
$$885$$ 0 0
$$886$$ 1.76833e6 0.0756797
$$887$$ −4.34462e6 −0.185414 −0.0927070 0.995693i $$-0.529552\pi$$
−0.0927070 + 0.995693i $$0.529552\pi$$
$$888$$ 0 0
$$889$$ −9.63379e6 −0.408830
$$890$$ −3.98772e6 −0.168752
$$891$$ 0 0
$$892$$ −1.23807e7 −0.520993
$$893$$ −7.96378e6 −0.334188
$$894$$ 0 0
$$895$$ 3.86118e6 0.161125
$$896$$ −5.93013e6 −0.246771
$$897$$ 0 0
$$898$$ −5.52579e6 −0.228667
$$899$$ −2.05720e7 −0.848942
$$900$$ 0 0
$$901$$ −119700. −0.00491227
$$902$$ 6.74356e6 0.275977
$$903$$ 0 0
$$904$$ −1.65505e7 −0.673580
$$905$$ 2.25460e7 0.915057
$$906$$ 0 0
$$907$$ 1.96499e7 0.793128 0.396564 0.918007i $$-0.370203\pi$$
0.396564 + 0.918007i $$0.370203\pi$$
$$908$$ 2.19454e7 0.883342
$$909$$ 0 0
$$910$$ 756364. 0.0302780
$$911$$ 7.26518e6 0.290035 0.145018 0.989429i $$-0.453676\pi$$
0.145018 + 0.989429i $$0.453676\pi$$
$$912$$ 0 0
$$913$$ 1.28398e7 0.509777
$$914$$ 2.96226e6 0.117289
$$915$$ 0 0
$$916$$ 2.28091e7 0.898193
$$917$$ −3.77986e6 −0.148440
$$918$$ 0 0
$$919$$ 9.82532e6 0.383758 0.191879 0.981419i $$-0.438542\pi$$
0.191879 + 0.981419i $$0.438542\pi$$
$$920$$ 6.83726e6 0.266326
$$921$$ 0 0
$$922$$ 2.11884e6 0.0820863
$$923$$ −6.61387e6 −0.255536
$$924$$ 0 0
$$925$$ −1.92923e7 −0.741359
$$926$$ −3.19226e6 −0.122340
$$927$$ 0 0
$$928$$ −2.42727e7 −0.925226
$$929$$ −2.71152e7 −1.03080 −0.515399 0.856951i $$-0.672356\pi$$
−0.515399 + 0.856951i $$0.672356\pi$$
$$930$$ 0 0
$$931$$ 2.14169e6 0.0809809
$$932$$ −6.47150e6 −0.244042
$$933$$ 0 0
$$934$$ −7.42621e6 −0.278548
$$935$$ 9.22488e6 0.345089
$$936$$ 0 0
$$937$$ −4.53522e7 −1.68752 −0.843761 0.536720i $$-0.819663\pi$$
−0.843761 + 0.536720i $$0.819663\pi$$
$$938$$ −82124.0 −0.00304764
$$939$$ 0 0
$$940$$ 9.41011e6 0.347356
$$941$$ −4.65780e7 −1.71477 −0.857387 0.514672i $$-0.827914\pi$$
−0.857387 + 0.514672i $$0.827914\pi$$
$$942$$ 0 0
$$943$$ −6.33101e7 −2.31843
$$944$$ 3.93859e7 1.43850
$$945$$ 0 0
$$946$$ 5.86024e6 0.212906
$$947$$ −2.53799e7 −0.919632 −0.459816 0.888014i $$-0.652085\pi$$
−0.459816 + 0.888014i $$0.652085\pi$$
$$948$$ 0 0
$$949$$ 3.55836e7 1.28258
$$950$$ 1.75635e6 0.0631396
$$951$$ 0 0
$$952$$ −2.46343e6 −0.0880942
$$953$$ −1.52948e7 −0.545520 −0.272760 0.962082i $$-0.587937\pi$$
−0.272760 + 0.962082i $$0.587937\pi$$
$$954$$ 0 0
$$955$$ −1.71926e7 −0.610004
$$956$$ 2.21147e7 0.782592
$$957$$ 0 0
$$958$$ −3.39685e6 −0.119581
$$959$$ 1.02003e7 0.358152
$$960$$ 0 0
$$961$$ −2.23991e7 −0.782389
$$962$$ −4.44829e6 −0.154973
$$963$$ 0 0
$$964$$ 1.56626e7 0.542840
$$965$$ −1.47010e7 −0.508192
$$966$$ 0 0
$$967$$ −5.71465e6 −0.196527 −0.0982637 0.995160i $$-0.531329\pi$$
−0.0982637 + 0.995160i $$0.531329\pi$$
$$968$$ −2.86341e6 −0.0982190
$$969$$ 0 0
$$970$$ −340068. −0.0116048
$$971$$ −1.30250e7 −0.443332 −0.221666 0.975123i $$-0.571149\pi$$
−0.221666 + 0.975123i $$0.571149\pi$$
$$972$$ 0 0
$$973$$ 1.35034e7 0.457258
$$974$$ 3.71382e6 0.125436
$$975$$ 0 0
$$976$$ 1.37102e7 0.460700
$$977$$ 1.70360e7 0.570992 0.285496 0.958380i $$-0.407842\pi$$
0.285496 + 0.958380i $$0.407842\pi$$
$$978$$ 0 0
$$979$$ 3.98772e7 1.32974
$$980$$ −2.53065e6 −0.0841720
$$981$$ 0 0
$$982$$ 5.57494e6 0.184485
$$983$$ 1.36985e7 0.452156 0.226078 0.974109i $$-0.427410\pi$$
0.226078 + 0.974109i $$0.427410\pi$$
$$984$$ 0 0
$$985$$ 4.48671e6 0.147346
$$986$$ −6.57712e6 −0.215448
$$987$$ 0 0
$$988$$ −1.25540e7 −0.409157
$$989$$ −5.50173e7 −1.78858
$$990$$ 0 0
$$991$$ −3.49088e7 −1.12915 −0.564574 0.825383i $$-0.690959\pi$$
−0.564574 + 0.825383i $$0.690959\pi$$
$$992$$ 7.35072e6 0.237165
$$993$$ 0 0
$$994$$ −713832. −0.0229155
$$995$$ 1.01502e7 0.325026
$$996$$ 0 0
$$997$$ 875662. 0.0278996 0.0139498 0.999903i $$-0.495559\pi$$
0.0139498 + 0.999903i $$0.495559\pi$$
$$998$$ −3.92698e6 −0.124805
$$999$$ 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 63.6.a.c.1.1 1
3.2 odd 2 21.6.a.b.1.1 1
4.3 odd 2 1008.6.a.t.1.1 1
7.6 odd 2 441.6.a.d.1.1 1
12.11 even 2 336.6.a.l.1.1 1
15.2 even 4 525.6.d.d.274.2 2
15.8 even 4 525.6.d.d.274.1 2
15.14 odd 2 525.6.a.c.1.1 1
21.2 odd 6 147.6.e.f.67.1 2
21.5 even 6 147.6.e.e.67.1 2
21.11 odd 6 147.6.e.f.79.1 2
21.17 even 6 147.6.e.e.79.1 2
21.20 even 2 147.6.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.b.1.1 1 3.2 odd 2
63.6.a.c.1.1 1 1.1 even 1 trivial
147.6.a.e.1.1 1 21.20 even 2
147.6.e.e.67.1 2 21.5 even 6
147.6.e.e.79.1 2 21.17 even 6
147.6.e.f.67.1 2 21.2 odd 6
147.6.e.f.79.1 2 21.11 odd 6
336.6.a.l.1.1 1 12.11 even 2
441.6.a.d.1.1 1 7.6 odd 2
525.6.a.c.1.1 1 15.14 odd 2
525.6.d.d.274.1 2 15.8 even 4
525.6.d.d.274.2 2 15.2 even 4
1008.6.a.t.1.1 1 4.3 odd 2