# Properties

 Label 21.6.a.b.1.1 Level $21$ Weight $6$ Character 21.1 Self dual yes Analytic conductor $3.368$ 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] = [21,6,Mod(1,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, 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("21.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 21.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: $$3.36806021607$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 21.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -9.00000 q^{3} -31.0000 q^{4} -34.0000 q^{5} -9.00000 q^{6} -49.0000 q^{7} -63.0000 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -9.00000 q^{3} -31.0000 q^{4} -34.0000 q^{5} -9.00000 q^{6} -49.0000 q^{7} -63.0000 q^{8} +81.0000 q^{9} -34.0000 q^{10} -340.000 q^{11} +279.000 q^{12} +454.000 q^{13} -49.0000 q^{14} +306.000 q^{15} +929.000 q^{16} -798.000 q^{17} +81.0000 q^{18} +892.000 q^{19} +1054.00 q^{20} +441.000 q^{21} -340.000 q^{22} -3192.00 q^{23} +567.000 q^{24} -1969.00 q^{25} +454.000 q^{26} -729.000 q^{27} +1519.00 q^{28} -8242.00 q^{29} +306.000 q^{30} -2496.00 q^{31} +2945.00 q^{32} +3060.00 q^{33} -798.000 q^{34} +1666.00 q^{35} -2511.00 q^{36} +9798.00 q^{37} +892.000 q^{38} -4086.00 q^{39} +2142.00 q^{40} +19834.0 q^{41} +441.000 q^{42} -17236.0 q^{43} +10540.0 q^{44} -2754.00 q^{45} -3192.00 q^{46} +8928.00 q^{47} -8361.00 q^{48} +2401.00 q^{49} -1969.00 q^{50} +7182.00 q^{51} -14074.0 q^{52} +150.000 q^{53} -729.000 q^{54} +11560.0 q^{55} +3087.00 q^{56} -8028.00 q^{57} -8242.00 q^{58} -42396.0 q^{59} -9486.00 q^{60} +14758.0 q^{61} -2496.00 q^{62} -3969.00 q^{63} -26783.0 q^{64} -15436.0 q^{65} +3060.00 q^{66} -1676.00 q^{67} +24738.0 q^{68} +28728.0 q^{69} +1666.00 q^{70} +14568.0 q^{71} -5103.00 q^{72} +78378.0 q^{73} +9798.00 q^{74} +17721.0 q^{75} -27652.0 q^{76} +16660.0 q^{77} -4086.00 q^{78} -2272.00 q^{79} -31586.0 q^{80} +6561.00 q^{81} +19834.0 q^{82} -37764.0 q^{83} -13671.0 q^{84} +27132.0 q^{85} -17236.0 q^{86} +74178.0 q^{87} +21420.0 q^{88} -117286. q^{89} -2754.00 q^{90} -22246.0 q^{91} +98952.0 q^{92} +22464.0 q^{93} +8928.00 q^{94} -30328.0 q^{95} -26505.0 q^{96} +10002.0 q^{97} +2401.00 q^{98} -27540.0 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.176777 0.0883883 0.996086i $$-0.471828\pi$$
0.0883883 + 0.996086i $$0.471828\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ −31.0000 −0.968750
$$5$$ −34.0000 −0.608210 −0.304105 0.952638i $$-0.598357\pi$$
−0.304105 + 0.952638i $$0.598357\pi$$
$$6$$ −9.00000 −0.102062
$$7$$ −49.0000 −0.377964
$$8$$ −63.0000 −0.348029
$$9$$ 81.0000 0.333333
$$10$$ −34.0000 −0.107517
$$11$$ −340.000 −0.847222 −0.423611 0.905844i $$-0.639238\pi$$
−0.423611 + 0.905844i $$0.639238\pi$$
$$12$$ 279.000 0.559308
$$13$$ 454.000 0.745071 0.372535 0.928018i $$-0.378489\pi$$
0.372535 + 0.928018i $$0.378489\pi$$
$$14$$ −49.0000 −0.0668153
$$15$$ 306.000 0.351150
$$16$$ 929.000 0.907227
$$17$$ −798.000 −0.669700 −0.334850 0.942271i $$-0.608686\pi$$
−0.334850 + 0.942271i $$0.608686\pi$$
$$18$$ 81.0000 0.0589256
$$19$$ 892.000 0.566867 0.283433 0.958992i $$-0.408527\pi$$
0.283433 + 0.958992i $$0.408527\pi$$
$$20$$ 1054.00 0.589204
$$21$$ 441.000 0.218218
$$22$$ −340.000 −0.149769
$$23$$ −3192.00 −1.25818 −0.629091 0.777332i $$-0.716573\pi$$
−0.629091 + 0.777332i $$0.716573\pi$$
$$24$$ 567.000 0.200935
$$25$$ −1969.00 −0.630080
$$26$$ 454.000 0.131711
$$27$$ −729.000 −0.192450
$$28$$ 1519.00 0.366153
$$29$$ −8242.00 −1.81986 −0.909929 0.414764i $$-0.863864\pi$$
−0.909929 + 0.414764i $$0.863864\pi$$
$$30$$ 306.000 0.0620752
$$31$$ −2496.00 −0.466488 −0.233244 0.972418i $$-0.574934\pi$$
−0.233244 + 0.972418i $$0.574934\pi$$
$$32$$ 2945.00 0.508406
$$33$$ 3060.00 0.489144
$$34$$ −798.000 −0.118387
$$35$$ 1666.00 0.229882
$$36$$ −2511.00 −0.322917
$$37$$ 9798.00 1.17661 0.588306 0.808639i $$-0.299795\pi$$
0.588306 + 0.808639i $$0.299795\pi$$
$$38$$ 892.000 0.100209
$$39$$ −4086.00 −0.430167
$$40$$ 2142.00 0.211675
$$41$$ 19834.0 1.84268 0.921342 0.388754i $$-0.127094\pi$$
0.921342 + 0.388754i $$0.127094\pi$$
$$42$$ 441.000 0.0385758
$$43$$ −17236.0 −1.42156 −0.710780 0.703414i $$-0.751658\pi$$
−0.710780 + 0.703414i $$0.751658\pi$$
$$44$$ 10540.0 0.820746
$$45$$ −2754.00 −0.202737
$$46$$ −3192.00 −0.222417
$$47$$ 8928.00 0.589535 0.294767 0.955569i $$-0.404758\pi$$
0.294767 + 0.955569i $$0.404758\pi$$
$$48$$ −8361.00 −0.523788
$$49$$ 2401.00 0.142857
$$50$$ −1969.00 −0.111383
$$51$$ 7182.00 0.386652
$$52$$ −14074.0 −0.721787
$$53$$ 150.000 0.00733502 0.00366751 0.999993i $$-0.498833\pi$$
0.00366751 + 0.999993i $$0.498833\pi$$
$$54$$ −729.000 −0.0340207
$$55$$ 11560.0 0.515289
$$56$$ 3087.00 0.131543
$$57$$ −8028.00 −0.327281
$$58$$ −8242.00 −0.321709
$$59$$ −42396.0 −1.58560 −0.792802 0.609479i $$-0.791379\pi$$
−0.792802 + 0.609479i $$0.791379\pi$$
$$60$$ −9486.00 −0.340177
$$61$$ 14758.0 0.507812 0.253906 0.967229i $$-0.418285\pi$$
0.253906 + 0.967229i $$0.418285\pi$$
$$62$$ −2496.00 −0.0824642
$$63$$ −3969.00 −0.125988
$$64$$ −26783.0 −0.817352
$$65$$ −15436.0 −0.453160
$$66$$ 3060.00 0.0864692
$$67$$ −1676.00 −0.0456128 −0.0228064 0.999740i $$-0.507260\pi$$
−0.0228064 + 0.999740i $$0.507260\pi$$
$$68$$ 24738.0 0.648772
$$69$$ 28728.0 0.726411
$$70$$ 1666.00 0.0406378
$$71$$ 14568.0 0.342968 0.171484 0.985187i $$-0.445144\pi$$
0.171484 + 0.985187i $$0.445144\pi$$
$$72$$ −5103.00 −0.116010
$$73$$ 78378.0 1.72142 0.860710 0.509095i $$-0.170020\pi$$
0.860710 + 0.509095i $$0.170020\pi$$
$$74$$ 9798.00 0.207998
$$75$$ 17721.0 0.363777
$$76$$ −27652.0 −0.549152
$$77$$ 16660.0 0.320220
$$78$$ −4086.00 −0.0760435
$$79$$ −2272.00 −0.0409582 −0.0204791 0.999790i $$-0.506519\pi$$
−0.0204791 + 0.999790i $$0.506519\pi$$
$$80$$ −31586.0 −0.551785
$$81$$ 6561.00 0.111111
$$82$$ 19834.0 0.325743
$$83$$ −37764.0 −0.601704 −0.300852 0.953671i $$-0.597271\pi$$
−0.300852 + 0.953671i $$0.597271\pi$$
$$84$$ −13671.0 −0.211399
$$85$$ 27132.0 0.407319
$$86$$ −17236.0 −0.251299
$$87$$ 74178.0 1.05070
$$88$$ 21420.0 0.294858
$$89$$ −117286. −1.56954 −0.784768 0.619790i $$-0.787218\pi$$
−0.784768 + 0.619790i $$0.787218\pi$$
$$90$$ −2754.00 −0.0358391
$$91$$ −22246.0 −0.281610
$$92$$ 98952.0 1.21886
$$93$$ 22464.0 0.269327
$$94$$ 8928.00 0.104216
$$95$$ −30328.0 −0.344774
$$96$$ −26505.0 −0.293528
$$97$$ 10002.0 0.107934 0.0539669 0.998543i $$-0.482813\pi$$
0.0539669 + 0.998543i $$0.482813\pi$$
$$98$$ 2401.00 0.0252538
$$99$$ −27540.0 −0.282407
$$100$$ 61039.0 0.610390
$$101$$ −108770. −1.06098 −0.530488 0.847692i $$-0.677991\pi$$
−0.530488 + 0.847692i $$0.677991\pi$$
$$102$$ 7182.00 0.0683510
$$103$$ −199192. −1.85003 −0.925015 0.379930i $$-0.875948\pi$$
−0.925015 + 0.379930i $$0.875948\pi$$
$$104$$ −28602.0 −0.259306
$$105$$ −14994.0 −0.132722
$$106$$ 150.000 0.00129666
$$107$$ −79972.0 −0.675272 −0.337636 0.941277i $$-0.609627\pi$$
−0.337636 + 0.941277i $$0.609627\pi$$
$$108$$ 22599.0 0.186436
$$109$$ −46098.0 −0.371634 −0.185817 0.982584i $$-0.559493\pi$$
−0.185817 + 0.982584i $$0.559493\pi$$
$$110$$ 11560.0 0.0910911
$$111$$ −88182.0 −0.679317
$$112$$ −45521.0 −0.342899
$$113$$ 262706. 1.93541 0.967707 0.252078i $$-0.0811138\pi$$
0.967707 + 0.252078i $$0.0811138\pi$$
$$114$$ −8028.00 −0.0578556
$$115$$ 108528. 0.765239
$$116$$ 255502. 1.76299
$$117$$ 36774.0 0.248357
$$118$$ −42396.0 −0.280298
$$119$$ 39102.0 0.253123
$$120$$ −19278.0 −0.122211
$$121$$ −45451.0 −0.282215
$$122$$ 14758.0 0.0897693
$$123$$ −178506. −1.06387
$$124$$ 77376.0 0.451910
$$125$$ 173196. 0.991432
$$126$$ −3969.00 −0.0222718
$$127$$ 196608. 1.08166 0.540831 0.841131i $$-0.318110\pi$$
0.540831 + 0.841131i $$0.318110\pi$$
$$128$$ −121023. −0.652894
$$129$$ 155124. 0.820738
$$130$$ −15436.0 −0.0801081
$$131$$ −77140.0 −0.392737 −0.196368 0.980530i $$-0.562915\pi$$
−0.196368 + 0.980530i $$0.562915\pi$$
$$132$$ −94860.0 −0.473858
$$133$$ −43708.0 −0.214255
$$134$$ −1676.00 −0.00806329
$$135$$ 24786.0 0.117050
$$136$$ 50274.0 0.233075
$$137$$ 208170. 0.947582 0.473791 0.880637i $$-0.342885\pi$$
0.473791 + 0.880637i $$0.342885\pi$$
$$138$$ 28728.0 0.128413
$$139$$ −275580. −1.20979 −0.604896 0.796304i $$-0.706785\pi$$
−0.604896 + 0.796304i $$0.706785\pi$$
$$140$$ −51646.0 −0.222698
$$141$$ −80352.0 −0.340368
$$142$$ 14568.0 0.0606288
$$143$$ −154360. −0.631240
$$144$$ 75249.0 0.302409
$$145$$ 280228. 1.10686
$$146$$ 78378.0 0.304307
$$147$$ −21609.0 −0.0824786
$$148$$ −303738. −1.13984
$$149$$ −296106. −1.09265 −0.546326 0.837573i $$-0.683974\pi$$
−0.546326 + 0.837573i $$0.683974\pi$$
$$150$$ 17721.0 0.0643073
$$151$$ −426472. −1.52212 −0.761059 0.648683i $$-0.775320\pi$$
−0.761059 + 0.648683i $$0.775320\pi$$
$$152$$ −56196.0 −0.197286
$$153$$ −64638.0 −0.223233
$$154$$ 16660.0 0.0566074
$$155$$ 84864.0 0.283723
$$156$$ 126666. 0.416724
$$157$$ 178486. 0.577903 0.288952 0.957344i $$-0.406693\pi$$
0.288952 + 0.957344i $$0.406693\pi$$
$$158$$ −2272.00 −0.00724045
$$159$$ −1350.00 −0.00423488
$$160$$ −100130. −0.309218
$$161$$ 156408. 0.475548
$$162$$ 6561.00 0.0196419
$$163$$ 252772. 0.745178 0.372589 0.927996i $$-0.378470\pi$$
0.372589 + 0.927996i $$0.378470\pi$$
$$164$$ −614854. −1.78510
$$165$$ −104040. −0.297502
$$166$$ −37764.0 −0.106367
$$167$$ 508088. 1.40977 0.704884 0.709322i $$-0.250999\pi$$
0.704884 + 0.709322i $$0.250999\pi$$
$$168$$ −27783.0 −0.0759462
$$169$$ −165177. −0.444870
$$170$$ 27132.0 0.0720045
$$171$$ 72252.0 0.188956
$$172$$ 534316. 1.37714
$$173$$ −221834. −0.563525 −0.281762 0.959484i $$-0.590919\pi$$
−0.281762 + 0.959484i $$0.590919\pi$$
$$174$$ 74178.0 0.185739
$$175$$ 96481.0 0.238148
$$176$$ −315860. −0.768622
$$177$$ 381564. 0.915449
$$178$$ −117286. −0.277457
$$179$$ −113564. −0.264916 −0.132458 0.991189i $$-0.542287\pi$$
−0.132458 + 0.991189i $$0.542287\pi$$
$$180$$ 85374.0 0.196401
$$181$$ 663118. 1.50451 0.752254 0.658873i $$-0.228967\pi$$
0.752254 + 0.658873i $$0.228967\pi$$
$$182$$ −22246.0 −0.0497821
$$183$$ −132822. −0.293185
$$184$$ 201096. 0.437884
$$185$$ −333132. −0.715628
$$186$$ 22464.0 0.0476107
$$187$$ 271320. 0.567385
$$188$$ −276768. −0.571112
$$189$$ 35721.0 0.0727393
$$190$$ −30328.0 −0.0609480
$$191$$ 505664. 1.00295 0.501474 0.865173i $$-0.332791\pi$$
0.501474 + 0.865173i $$0.332791\pi$$
$$192$$ 241047. 0.471899
$$193$$ −432382. −0.835554 −0.417777 0.908550i $$-0.637191\pi$$
−0.417777 + 0.908550i $$0.637191\pi$$
$$194$$ 10002.0 0.0190802
$$195$$ 138924. 0.261632
$$196$$ −74431.0 −0.138393
$$197$$ −131962. −0.242261 −0.121130 0.992637i $$-0.538652\pi$$
−0.121130 + 0.992637i $$0.538652\pi$$
$$198$$ −27540.0 −0.0499230
$$199$$ 298536. 0.534397 0.267199 0.963642i $$-0.413902\pi$$
0.267199 + 0.963642i $$0.413902\pi$$
$$200$$ 124047. 0.219286
$$201$$ 15084.0 0.0263346
$$202$$ −108770. −0.187556
$$203$$ 403858. 0.687842
$$204$$ −222642. −0.374569
$$205$$ −674356. −1.12074
$$206$$ −199192. −0.327042
$$207$$ −258552. −0.419394
$$208$$ 421766. 0.675948
$$209$$ −303280. −0.480262
$$210$$ −14994.0 −0.0234622
$$211$$ −1.17062e6 −1.81013 −0.905065 0.425273i $$-0.860178\pi$$
−0.905065 + 0.425273i $$0.860178\pi$$
$$212$$ −4650.00 −0.00710581
$$213$$ −131112. −0.198013
$$214$$ −79972.0 −0.119372
$$215$$ 586024. 0.864608
$$216$$ 45927.0 0.0669782
$$217$$ 122304. 0.176316
$$218$$ −46098.0 −0.0656963
$$219$$ −705402. −0.993863
$$220$$ −358360. −0.499186
$$221$$ −362292. −0.498974
$$222$$ −88182.0 −0.120087
$$223$$ 399376. 0.537799 0.268899 0.963168i $$-0.413340\pi$$
0.268899 + 0.963168i $$0.413340\pi$$
$$224$$ −144305. −0.192159
$$225$$ −159489. −0.210027
$$226$$ 262706. 0.342136
$$227$$ 707916. 0.911837 0.455918 0.890022i $$-0.349311\pi$$
0.455918 + 0.890022i $$0.349311\pi$$
$$228$$ 248868. 0.317053
$$229$$ −735778. −0.927167 −0.463584 0.886053i $$-0.653437\pi$$
−0.463584 + 0.886053i $$0.653437\pi$$
$$230$$ 108528. 0.135276
$$231$$ −149940. −0.184879
$$232$$ 519246. 0.633364
$$233$$ −208758. −0.251915 −0.125957 0.992036i $$-0.540200\pi$$
−0.125957 + 0.992036i $$0.540200\pi$$
$$234$$ 36774.0 0.0439037
$$235$$ −303552. −0.358561
$$236$$ 1.31428e6 1.53605
$$237$$ 20448.0 0.0236472
$$238$$ 39102.0 0.0447462
$$239$$ 713376. 0.807837 0.403919 0.914795i $$-0.367648\pi$$
0.403919 + 0.914795i $$0.367648\pi$$
$$240$$ 284274. 0.318573
$$241$$ −505246. −0.560351 −0.280176 0.959949i $$-0.590393\pi$$
−0.280176 + 0.959949i $$0.590393\pi$$
$$242$$ −45451.0 −0.0498890
$$243$$ −59049.0 −0.0641500
$$244$$ −457498. −0.491943
$$245$$ −81634.0 −0.0868872
$$246$$ −178506. −0.188068
$$247$$ 404968. 0.422356
$$248$$ 157248. 0.162351
$$249$$ 339876. 0.347394
$$250$$ 173196. 0.175262
$$251$$ 317108. 0.317704 0.158852 0.987302i $$-0.449221\pi$$
0.158852 + 0.987302i $$0.449221\pi$$
$$252$$ 123039. 0.122051
$$253$$ 1.08528e6 1.06596
$$254$$ 196608. 0.191213
$$255$$ −244188. −0.235166
$$256$$ 736033. 0.701936
$$257$$ −1.44285e6 −1.36266 −0.681329 0.731977i $$-0.738598\pi$$
−0.681329 + 0.731977i $$0.738598\pi$$
$$258$$ 155124. 0.145087
$$259$$ −480102. −0.444717
$$260$$ 478516. 0.438999
$$261$$ −667602. −0.606619
$$262$$ −77140.0 −0.0694267
$$263$$ 271496. 0.242033 0.121016 0.992651i $$-0.461385\pi$$
0.121016 + 0.992651i $$0.461385\pi$$
$$264$$ −192780. −0.170236
$$265$$ −5100.00 −0.00446124
$$266$$ −43708.0 −0.0378754
$$267$$ 1.05557e6 0.906172
$$268$$ 51956.0 0.0441874
$$269$$ 850614. 0.716724 0.358362 0.933583i $$-0.383335\pi$$
0.358362 + 0.933583i $$0.383335\pi$$
$$270$$ 24786.0 0.0206917
$$271$$ −540128. −0.446759 −0.223380 0.974732i $$-0.571709\pi$$
−0.223380 + 0.974732i $$0.571709\pi$$
$$272$$ −741342. −0.607570
$$273$$ 200214. 0.162588
$$274$$ 208170. 0.167510
$$275$$ 669460. 0.533818
$$276$$ −890568. −0.703711
$$277$$ 513574. 0.402164 0.201082 0.979574i $$-0.435554\pi$$
0.201082 + 0.979574i $$0.435554\pi$$
$$278$$ −275580. −0.213863
$$279$$ −202176. −0.155496
$$280$$ −104958. −0.0800056
$$281$$ −1.35642e6 −1.02478 −0.512388 0.858754i $$-0.671239\pi$$
−0.512388 + 0.858754i $$0.671239\pi$$
$$282$$ −80352.0 −0.0601692
$$283$$ 286756. 0.212837 0.106418 0.994321i $$-0.466062\pi$$
0.106418 + 0.994321i $$0.466062\pi$$
$$284$$ −451608. −0.332251
$$285$$ 272952. 0.199055
$$286$$ −154360. −0.111589
$$287$$ −971866. −0.696469
$$288$$ 238545. 0.169469
$$289$$ −783053. −0.551501
$$290$$ 280228. 0.195667
$$291$$ −90018.0 −0.0623156
$$292$$ −2.42972e6 −1.66763
$$293$$ −1.70727e6 −1.16180 −0.580901 0.813974i $$-0.697300\pi$$
−0.580901 + 0.813974i $$0.697300\pi$$
$$294$$ −21609.0 −0.0145803
$$295$$ 1.44146e6 0.964381
$$296$$ −617274. −0.409495
$$297$$ 247860. 0.163048
$$298$$ −296106. −0.193155
$$299$$ −1.44917e6 −0.937434
$$300$$ −549351. −0.352409
$$301$$ 844564. 0.537299
$$302$$ −426472. −0.269075
$$303$$ 978930. 0.612555
$$304$$ 828668. 0.514276
$$305$$ −501772. −0.308857
$$306$$ −64638.0 −0.0394625
$$307$$ −546788. −0.331111 −0.165555 0.986201i $$-0.552942\pi$$
−0.165555 + 0.986201i $$0.552942\pi$$
$$308$$ −516460. −0.310213
$$309$$ 1.79273e6 1.06812
$$310$$ 84864.0 0.0501556
$$311$$ 3.23426e6 1.89616 0.948079 0.318035i $$-0.103023\pi$$
0.948079 + 0.318035i $$0.103023\pi$$
$$312$$ 257418. 0.149711
$$313$$ 1.81313e6 1.04609 0.523044 0.852306i $$-0.324796\pi$$
0.523044 + 0.852306i $$0.324796\pi$$
$$314$$ 178486. 0.102160
$$315$$ 134946. 0.0766273
$$316$$ 70432.0 0.0396782
$$317$$ −1.27658e6 −0.713509 −0.356754 0.934198i $$-0.616117\pi$$
−0.356754 + 0.934198i $$0.616117\pi$$
$$318$$ −1350.00 −0.000748628 0
$$319$$ 2.80228e6 1.54182
$$320$$ 910622. 0.497122
$$321$$ 719748. 0.389868
$$322$$ 156408. 0.0840658
$$323$$ −711816. −0.379631
$$324$$ −203391. −0.107639
$$325$$ −893926. −0.469454
$$326$$ 252772. 0.131730
$$327$$ 414882. 0.214563
$$328$$ −1.24954e6 −0.641307
$$329$$ −437472. −0.222823
$$330$$ −104040. −0.0525915
$$331$$ −1.73621e6 −0.871029 −0.435515 0.900182i $$-0.643434\pi$$
−0.435515 + 0.900182i $$0.643434\pi$$
$$332$$ 1.17068e6 0.582901
$$333$$ 793638. 0.392204
$$334$$ 508088. 0.249214
$$335$$ 56984.0 0.0277422
$$336$$ 409689. 0.197973
$$337$$ 2.07215e6 0.993907 0.496953 0.867777i $$-0.334452\pi$$
0.496953 + 0.867777i $$0.334452\pi$$
$$338$$ −165177. −0.0786426
$$339$$ −2.36435e6 −1.11741
$$340$$ −841092. −0.394590
$$341$$ 848640. 0.395219
$$342$$ 72252.0 0.0334029
$$343$$ −117649. −0.0539949
$$344$$ 1.08587e6 0.494744
$$345$$ −976752. −0.441811
$$346$$ −221834. −0.0996180
$$347$$ −1.65146e6 −0.736282 −0.368141 0.929770i $$-0.620006\pi$$
−0.368141 + 0.929770i $$0.620006\pi$$
$$348$$ −2.29952e6 −1.01786
$$349$$ 1.26645e6 0.556578 0.278289 0.960497i $$-0.410233\pi$$
0.278289 + 0.960497i $$0.410233\pi$$
$$350$$ 96481.0 0.0420990
$$351$$ −330966. −0.143389
$$352$$ −1.00130e6 −0.430732
$$353$$ 573218. 0.244840 0.122420 0.992478i $$-0.460934\pi$$
0.122420 + 0.992478i $$0.460934\pi$$
$$354$$ 381564. 0.161830
$$355$$ −495312. −0.208597
$$356$$ 3.63587e6 1.52049
$$357$$ −351918. −0.146141
$$358$$ −113564. −0.0468310
$$359$$ 4.46322e6 1.82773 0.913866 0.406016i $$-0.133082\pi$$
0.913866 + 0.406016i $$0.133082\pi$$
$$360$$ 173502. 0.0705583
$$361$$ −1.68044e6 −0.678662
$$362$$ 663118. 0.265962
$$363$$ 409059. 0.162937
$$364$$ 689626. 0.272810
$$365$$ −2.66485e6 −1.04699
$$366$$ −132822. −0.0518283
$$367$$ −4.50797e6 −1.74709 −0.873546 0.486742i $$-0.838185\pi$$
−0.873546 + 0.486742i $$0.838185\pi$$
$$368$$ −2.96537e6 −1.14146
$$369$$ 1.60655e6 0.614228
$$370$$ −333132. −0.126506
$$371$$ −7350.00 −0.00277238
$$372$$ −696384. −0.260910
$$373$$ 1.66535e6 0.619774 0.309887 0.950773i $$-0.399709\pi$$
0.309887 + 0.950773i $$0.399709\pi$$
$$374$$ 271320. 0.100300
$$375$$ −1.55876e6 −0.572403
$$376$$ −562464. −0.205175
$$377$$ −3.74187e6 −1.35592
$$378$$ 35721.0 0.0128586
$$379$$ −2.53232e6 −0.905568 −0.452784 0.891620i $$-0.649569\pi$$
−0.452784 + 0.891620i $$0.649569\pi$$
$$380$$ 940168. 0.334000
$$381$$ −1.76947e6 −0.624498
$$382$$ 505664. 0.177298
$$383$$ 796368. 0.277407 0.138703 0.990334i $$-0.455707\pi$$
0.138703 + 0.990334i $$0.455707\pi$$
$$384$$ 1.08921e6 0.376949
$$385$$ −566440. −0.194761
$$386$$ −432382. −0.147706
$$387$$ −1.39612e6 −0.473853
$$388$$ −310062. −0.104561
$$389$$ 1.94799e6 0.652699 0.326349 0.945249i $$-0.394181\pi$$
0.326349 + 0.945249i $$0.394181\pi$$
$$390$$ 138924. 0.0462504
$$391$$ 2.54722e6 0.842605
$$392$$ −151263. −0.0497184
$$393$$ 694260. 0.226747
$$394$$ −131962. −0.0428261
$$395$$ 77248.0 0.0249112
$$396$$ 853740. 0.273582
$$397$$ 1.08116e6 0.344281 0.172140 0.985072i $$-0.444932\pi$$
0.172140 + 0.985072i $$0.444932\pi$$
$$398$$ 298536. 0.0944689
$$399$$ 393372. 0.123700
$$400$$ −1.82920e6 −0.571625
$$401$$ 2.76770e6 0.859524 0.429762 0.902942i $$-0.358598\pi$$
0.429762 + 0.902942i $$0.358598\pi$$
$$402$$ 15084.0 0.00465534
$$403$$ −1.13318e6 −0.347566
$$404$$ 3.37187e6 1.02782
$$405$$ −223074. −0.0675789
$$406$$ 403858. 0.121594
$$407$$ −3.33132e6 −0.996851
$$408$$ −452466. −0.134566
$$409$$ 2.36350e6 0.698630 0.349315 0.937005i $$-0.386414\pi$$
0.349315 + 0.937005i $$0.386414\pi$$
$$410$$ −674356. −0.198121
$$411$$ −1.87353e6 −0.547087
$$412$$ 6.17495e6 1.79222
$$413$$ 2.07740e6 0.599302
$$414$$ −258552. −0.0741391
$$415$$ 1.28398e6 0.365963
$$416$$ 1.33703e6 0.378798
$$417$$ 2.48022e6 0.698474
$$418$$ −303280. −0.0848991
$$419$$ −2.98669e6 −0.831104 −0.415552 0.909569i $$-0.636412\pi$$
−0.415552 + 0.909569i $$0.636412\pi$$
$$420$$ 464814. 0.128575
$$421$$ −3.46331e6 −0.952326 −0.476163 0.879357i $$-0.657973\pi$$
−0.476163 + 0.879357i $$0.657973\pi$$
$$422$$ −1.17062e6 −0.319989
$$423$$ 723168. 0.196512
$$424$$ −9450.00 −0.00255280
$$425$$ 1.57126e6 0.421965
$$426$$ −131112. −0.0350041
$$427$$ −723142. −0.191935
$$428$$ 2.47913e6 0.654169
$$429$$ 1.38924e6 0.364447
$$430$$ 586024. 0.152843
$$431$$ 2.33693e6 0.605971 0.302986 0.952995i $$-0.402017\pi$$
0.302986 + 0.952995i $$0.402017\pi$$
$$432$$ −677241. −0.174596
$$433$$ −3.50838e6 −0.899264 −0.449632 0.893214i $$-0.648445\pi$$
−0.449632 + 0.893214i $$0.648445\pi$$
$$434$$ 122304. 0.0311685
$$435$$ −2.52205e6 −0.639044
$$436$$ 1.42904e6 0.360021
$$437$$ −2.84726e6 −0.713221
$$438$$ −705402. −0.175692
$$439$$ 3.54833e6 0.878744 0.439372 0.898305i $$-0.355201\pi$$
0.439372 + 0.898305i $$0.355201\pi$$
$$440$$ −728280. −0.179336
$$441$$ 194481. 0.0476190
$$442$$ −362292. −0.0882070
$$443$$ 1.76833e6 0.428109 0.214055 0.976822i $$-0.431333\pi$$
0.214055 + 0.976822i $$0.431333\pi$$
$$444$$ 2.73364e6 0.658088
$$445$$ 3.98772e6 0.954608
$$446$$ 399376. 0.0950703
$$447$$ 2.66495e6 0.630842
$$448$$ 1.31237e6 0.308930
$$449$$ −5.52579e6 −1.29354 −0.646768 0.762687i $$-0.723880\pi$$
−0.646768 + 0.762687i $$0.723880\pi$$
$$450$$ −159489. −0.0371278
$$451$$ −6.74356e6 −1.56116
$$452$$ −8.14389e6 −1.87493
$$453$$ 3.83825e6 0.878795
$$454$$ 707916. 0.161191
$$455$$ 756364. 0.171278
$$456$$ 505764. 0.113903
$$457$$ −2.96226e6 −0.663488 −0.331744 0.943369i $$-0.607637\pi$$
−0.331744 + 0.943369i $$0.607637\pi$$
$$458$$ −735778. −0.163902
$$459$$ 581742. 0.128884
$$460$$ −3.36437e6 −0.741325
$$461$$ 2.11884e6 0.464350 0.232175 0.972674i $$-0.425416\pi$$
0.232175 + 0.972674i $$0.425416\pi$$
$$462$$ −149940. −0.0326823
$$463$$ 3.19226e6 0.692062 0.346031 0.938223i $$-0.387529\pi$$
0.346031 + 0.938223i $$0.387529\pi$$
$$464$$ −7.65682e6 −1.65102
$$465$$ −763776. −0.163807
$$466$$ −208758. −0.0445326
$$467$$ −7.42621e6 −1.57571 −0.787853 0.615863i $$-0.788807\pi$$
−0.787853 + 0.615863i $$0.788807\pi$$
$$468$$ −1.13999e6 −0.240596
$$469$$ 82124.0 0.0172400
$$470$$ −303552. −0.0633853
$$471$$ −1.60637e6 −0.333653
$$472$$ 2.67095e6 0.551837
$$473$$ 5.86024e6 1.20438
$$474$$ 20448.0 0.00418028
$$475$$ −1.75635e6 −0.357171
$$476$$ −1.21216e6 −0.245213
$$477$$ 12150.0 0.00244501
$$478$$ 713376. 0.142807
$$479$$ −3.39685e6 −0.676453 −0.338226 0.941065i $$-0.609827\pi$$
−0.338226 + 0.941065i $$0.609827\pi$$
$$480$$ 901170. 0.178527
$$481$$ 4.44829e6 0.876659
$$482$$ −505246. −0.0990570
$$483$$ −1.40767e6 −0.274558
$$484$$ 1.40898e6 0.273396
$$485$$ −340068. −0.0656465
$$486$$ −59049.0 −0.0113402
$$487$$ −3.71382e6 −0.709574 −0.354787 0.934947i $$-0.615447\pi$$
−0.354787 + 0.934947i $$0.615447\pi$$
$$488$$ −929754. −0.176733
$$489$$ −2.27495e6 −0.430229
$$490$$ −81634.0 −0.0153596
$$491$$ 5.57494e6 1.04361 0.521803 0.853066i $$-0.325260\pi$$
0.521803 + 0.853066i $$0.325260\pi$$
$$492$$ 5.53369e6 1.03063
$$493$$ 6.57712e6 1.21876
$$494$$ 404968. 0.0746626
$$495$$ 936360. 0.171763
$$496$$ −2.31878e6 −0.423210
$$497$$ −713832. −0.129630
$$498$$ 339876. 0.0614111
$$499$$ 3.92698e6 0.706004 0.353002 0.935623i $$-0.385161\pi$$
0.353002 + 0.935623i $$0.385161\pi$$
$$500$$ −5.36908e6 −0.960450
$$501$$ −4.57279e6 −0.813930
$$502$$ 317108. 0.0561627
$$503$$ 6.42079e6 1.13154 0.565768 0.824564i $$-0.308580\pi$$
0.565768 + 0.824564i $$0.308580\pi$$
$$504$$ 250047. 0.0438475
$$505$$ 3.69818e6 0.645297
$$506$$ 1.08528e6 0.188437
$$507$$ 1.48659e6 0.256846
$$508$$ −6.09485e6 −1.04786
$$509$$ 146278. 0.0250256 0.0125128 0.999922i $$-0.496017\pi$$
0.0125128 + 0.999922i $$0.496017\pi$$
$$510$$ −244188. −0.0415718
$$511$$ −3.84052e6 −0.650636
$$512$$ 4.60877e6 0.776980
$$513$$ −650268. −0.109094
$$514$$ −1.44285e6 −0.240886
$$515$$ 6.77253e6 1.12521
$$516$$ −4.80884e6 −0.795090
$$517$$ −3.03552e6 −0.499467
$$518$$ −480102. −0.0786157
$$519$$ 1.99651e6 0.325351
$$520$$ 972468. 0.157713
$$521$$ 7.70937e6 1.24430 0.622149 0.782899i $$-0.286260\pi$$
0.622149 + 0.782899i $$0.286260\pi$$
$$522$$ −667602. −0.107236
$$523$$ −569420. −0.0910287 −0.0455144 0.998964i $$-0.514493\pi$$
−0.0455144 + 0.998964i $$0.514493\pi$$
$$524$$ 2.39134e6 0.380464
$$525$$ −868329. −0.137495
$$526$$ 271496. 0.0427857
$$527$$ 1.99181e6 0.312407
$$528$$ 2.84274e6 0.443764
$$529$$ 3.75252e6 0.583021
$$530$$ −5100.00 −0.000788643 0
$$531$$ −3.43408e6 −0.528535
$$532$$ 1.35495e6 0.207560
$$533$$ 9.00464e6 1.37293
$$534$$ 1.05557e6 0.160190
$$535$$ 2.71905e6 0.410707
$$536$$ 105588. 0.0158746
$$537$$ 1.02208e6 0.152949
$$538$$ 850614. 0.126700
$$539$$ −816340. −0.121032
$$540$$ −768366. −0.113392
$$541$$ −9.44802e6 −1.38787 −0.693933 0.720040i $$-0.744124\pi$$
−0.693933 + 0.720040i $$0.744124\pi$$
$$542$$ −540128. −0.0789766
$$543$$ −5.96806e6 −0.868628
$$544$$ −2.35011e6 −0.340479
$$545$$ 1.56733e6 0.226032
$$546$$ 200214. 0.0287417
$$547$$ −1.35321e6 −0.193374 −0.0966869 0.995315i $$-0.530825\pi$$
−0.0966869 + 0.995315i $$0.530825\pi$$
$$548$$ −6.45327e6 −0.917970
$$549$$ 1.19540e6 0.169271
$$550$$ 669460. 0.0943665
$$551$$ −7.35186e6 −1.03162
$$552$$ −1.80986e6 −0.252812
$$553$$ 111328. 0.0154807
$$554$$ 513574. 0.0710933
$$555$$ 2.99819e6 0.413168
$$556$$ 8.54298e6 1.17199
$$557$$ 8.19390e6 1.11906 0.559529 0.828811i $$-0.310982\pi$$
0.559529 + 0.828811i $$0.310982\pi$$
$$558$$ −202176. −0.0274881
$$559$$ −7.82514e6 −1.05916
$$560$$ 1.54771e6 0.208555
$$561$$ −2.44188e6 −0.327580
$$562$$ −1.35642e6 −0.181157
$$563$$ −1.05796e7 −1.40669 −0.703347 0.710847i $$-0.748312\pi$$
−0.703347 + 0.710847i $$0.748312\pi$$
$$564$$ 2.49091e6 0.329732
$$565$$ −8.93200e6 −1.17714
$$566$$ 286756. 0.0376246
$$567$$ −321489. −0.0419961
$$568$$ −917784. −0.119363
$$569$$ −1.20205e7 −1.55648 −0.778238 0.627969i $$-0.783886\pi$$
−0.778238 + 0.627969i $$0.783886\pi$$
$$570$$ 272952. 0.0351884
$$571$$ −2.48948e6 −0.319534 −0.159767 0.987155i $$-0.551074\pi$$
−0.159767 + 0.987155i $$0.551074\pi$$
$$572$$ 4.78516e6 0.611514
$$573$$ −4.55098e6 −0.579053
$$574$$ −971866. −0.123119
$$575$$ 6.28505e6 0.792755
$$576$$ −2.16942e6 −0.272451
$$577$$ 8.21322e6 1.02701 0.513504 0.858087i $$-0.328347\pi$$
0.513504 + 0.858087i $$0.328347\pi$$
$$578$$ −783053. −0.0974926
$$579$$ 3.89144e6 0.482407
$$580$$ −8.68707e6 −1.07227
$$581$$ 1.85044e6 0.227423
$$582$$ −90018.0 −0.0110159
$$583$$ −51000.0 −0.00621439
$$584$$ −4.93781e6 −0.599105
$$585$$ −1.25032e6 −0.151053
$$586$$ −1.70727e6 −0.205380
$$587$$ −1.21827e6 −0.145931 −0.0729655 0.997334i $$-0.523246\pi$$
−0.0729655 + 0.997334i $$0.523246\pi$$
$$588$$ 669879. 0.0799012
$$589$$ −2.22643e6 −0.264436
$$590$$ 1.44146e6 0.170480
$$591$$ 1.18766e6 0.139869
$$592$$ 9.10234e6 1.06745
$$593$$ −8.42379e6 −0.983718 −0.491859 0.870675i $$-0.663683\pi$$
−0.491859 + 0.870675i $$0.663683\pi$$
$$594$$ 247860. 0.0288231
$$595$$ −1.32947e6 −0.153952
$$596$$ 9.17929e6 1.05851
$$597$$ −2.68682e6 −0.308534
$$598$$ −1.44917e6 −0.165717
$$599$$ 8.21254e6 0.935212 0.467606 0.883937i $$-0.345117\pi$$
0.467606 + 0.883937i $$0.345117\pi$$
$$600$$ −1.11642e6 −0.126605
$$601$$ 3.25478e6 0.367566 0.183783 0.982967i $$-0.441166\pi$$
0.183783 + 0.982967i $$0.441166\pi$$
$$602$$ 844564. 0.0949820
$$603$$ −135756. −0.0152043
$$604$$ 1.32206e7 1.47455
$$605$$ 1.54533e6 0.171646
$$606$$ 978930. 0.108285
$$607$$ 7.82101e6 0.861571 0.430785 0.902454i $$-0.358237\pi$$
0.430785 + 0.902454i $$0.358237\pi$$
$$608$$ 2.62694e6 0.288198
$$609$$ −3.63472e6 −0.397126
$$610$$ −501772. −0.0545986
$$611$$ 4.05331e6 0.439245
$$612$$ 2.00378e6 0.216257
$$613$$ −9.51670e6 −1.02290 −0.511452 0.859312i $$-0.670892\pi$$
−0.511452 + 0.859312i $$0.670892\pi$$
$$614$$ −546788. −0.0585326
$$615$$ 6.06920e6 0.647059
$$616$$ −1.04958e6 −0.111446
$$617$$ −7.04895e6 −0.745438 −0.372719 0.927944i $$-0.621574\pi$$
−0.372719 + 0.927944i $$0.621574\pi$$
$$618$$ 1.79273e6 0.188818
$$619$$ −6.32174e6 −0.663147 −0.331574 0.943429i $$-0.607580\pi$$
−0.331574 + 0.943429i $$0.607580\pi$$
$$620$$ −2.63078e6 −0.274856
$$621$$ 2.32697e6 0.242137
$$622$$ 3.23426e6 0.335197
$$623$$ 5.74701e6 0.593229
$$624$$ −3.79589e6 −0.390259
$$625$$ 264461. 0.0270808
$$626$$ 1.81313e6 0.184924
$$627$$ 2.72952e6 0.277279
$$628$$ −5.53307e6 −0.559844
$$629$$ −7.81880e6 −0.787977
$$630$$ 134946. 0.0135459
$$631$$ 8.61236e6 0.861090 0.430545 0.902569i $$-0.358321\pi$$
0.430545 + 0.902569i $$0.358321\pi$$
$$632$$ 143136. 0.0142546
$$633$$ 1.05356e7 1.04508
$$634$$ −1.27658e6 −0.126132
$$635$$ −6.68467e6 −0.657879
$$636$$ 41850.0 0.00410254
$$637$$ 1.09005e6 0.106439
$$638$$ 2.80228e6 0.272559
$$639$$ 1.18001e6 0.114323
$$640$$ 4.11478e6 0.397097
$$641$$ −5.22829e6 −0.502590 −0.251295 0.967910i $$-0.580857\pi$$
−0.251295 + 0.967910i $$0.580857\pi$$
$$642$$ 719748. 0.0689196
$$643$$ 1.61373e7 1.53923 0.769615 0.638508i $$-0.220448\pi$$
0.769615 + 0.638508i $$0.220448\pi$$
$$644$$ −4.84865e6 −0.460687
$$645$$ −5.27422e6 −0.499182
$$646$$ −711816. −0.0671099
$$647$$ −1.58749e7 −1.49090 −0.745451 0.666560i $$-0.767766\pi$$
−0.745451 + 0.666560i $$0.767766\pi$$
$$648$$ −413343. −0.0386699
$$649$$ 1.44146e7 1.34336
$$650$$ −893926. −0.0829886
$$651$$ −1.10074e6 −0.101796
$$652$$ −7.83593e6 −0.721891
$$653$$ −5.94112e6 −0.545237 −0.272619 0.962122i $$-0.587890\pi$$
−0.272619 + 0.962122i $$0.587890\pi$$
$$654$$ 414882. 0.0379298
$$655$$ 2.62276e6 0.238867
$$656$$ 1.84258e7 1.67173
$$657$$ 6.34862e6 0.573807
$$658$$ −437472. −0.0393900
$$659$$ −7.64430e6 −0.685684 −0.342842 0.939393i $$-0.611390\pi$$
−0.342842 + 0.939393i $$0.611390\pi$$
$$660$$ 3.22524e6 0.288205
$$661$$ −7.58688e6 −0.675398 −0.337699 0.941254i $$-0.609649\pi$$
−0.337699 + 0.941254i $$0.609649\pi$$
$$662$$ −1.73621e6 −0.153978
$$663$$ 3.26063e6 0.288083
$$664$$ 2.37913e6 0.209410
$$665$$ 1.48607e6 0.130312
$$666$$ 793638. 0.0693325
$$667$$ 2.63085e7 2.28971
$$668$$ −1.57507e7 −1.36571
$$669$$ −3.59438e6 −0.310498
$$670$$ 56984.0 0.00490417
$$671$$ −5.01772e6 −0.430229
$$672$$ 1.29874e6 0.110943
$$673$$ −2.06681e7 −1.75899 −0.879494 0.475910i $$-0.842119\pi$$
−0.879494 + 0.475910i $$0.842119\pi$$
$$674$$ 2.07215e6 0.175700
$$675$$ 1.43540e6 0.121259
$$676$$ 5.12049e6 0.430968
$$677$$ 7.89541e6 0.662068 0.331034 0.943619i $$-0.392602\pi$$
0.331034 + 0.943619i $$0.392602\pi$$
$$678$$ −2.36435e6 −0.197532
$$679$$ −490098. −0.0407951
$$680$$ −1.70932e6 −0.141759
$$681$$ −6.37124e6 −0.526449
$$682$$ 848640. 0.0698655
$$683$$ −1.96015e7 −1.60782 −0.803911 0.594750i $$-0.797251\pi$$
−0.803911 + 0.594750i $$0.797251\pi$$
$$684$$ −2.23981e6 −0.183051
$$685$$ −7.07778e6 −0.576329
$$686$$ −117649. −0.00954504
$$687$$ 6.62200e6 0.535300
$$688$$ −1.60122e7 −1.28968
$$689$$ 68100.0 0.00546511
$$690$$ −976752. −0.0781019
$$691$$ −1.72710e7 −1.37601 −0.688005 0.725706i $$-0.741513\pi$$
−0.688005 + 0.725706i $$0.741513\pi$$
$$692$$ 6.87685e6 0.545914
$$693$$ 1.34946e6 0.106740
$$694$$ −1.65146e6 −0.130158
$$695$$ 9.36972e6 0.735808
$$696$$ −4.67321e6 −0.365673
$$697$$ −1.58275e7 −1.23405
$$698$$ 1.26645e6 0.0983900
$$699$$ 1.87882e6 0.145443
$$700$$ −2.99091e6 −0.230706
$$701$$ −5.36344e6 −0.412238 −0.206119 0.978527i $$-0.566083\pi$$
−0.206119 + 0.978527i $$0.566083\pi$$
$$702$$ −330966. −0.0253478
$$703$$ 8.73982e6 0.666982
$$704$$ 9.10622e6 0.692479
$$705$$ 2.73197e6 0.207015
$$706$$ 573218. 0.0432821
$$707$$ 5.32973e6 0.401011
$$708$$ −1.18285e7 −0.886841
$$709$$ −1.73733e7 −1.29798 −0.648988 0.760798i $$-0.724808\pi$$
−0.648988 + 0.760798i $$0.724808\pi$$
$$710$$ −495312. −0.0368751
$$711$$ −184032. −0.0136527
$$712$$ 7.38902e6 0.546244
$$713$$ 7.96723e6 0.586926
$$714$$ −351918. −0.0258343
$$715$$ 5.24824e6 0.383927
$$716$$ 3.52048e6 0.256637
$$717$$ −6.42038e6 −0.466405
$$718$$ 4.46322e6 0.323100
$$719$$ 424608. 0.0306313 0.0153157 0.999883i $$-0.495125\pi$$
0.0153157 + 0.999883i $$0.495125\pi$$
$$720$$ −2.55847e6 −0.183928
$$721$$ 9.76041e6 0.699246
$$722$$ −1.68044e6 −0.119972
$$723$$ 4.54721e6 0.323519
$$724$$ −2.05567e7 −1.45749
$$725$$ 1.62285e7 1.14666
$$726$$ 409059. 0.0288034
$$727$$ 2.18290e7 1.53179 0.765893 0.642968i $$-0.222297\pi$$
0.765893 + 0.642968i $$0.222297\pi$$
$$728$$ 1.40150e6 0.0980086
$$729$$ 531441. 0.0370370
$$730$$ −2.66485e6 −0.185083
$$731$$ 1.37543e7 0.952020
$$732$$ 4.11748e6 0.284023
$$733$$ 2.17675e7 1.49640 0.748202 0.663470i $$-0.230917\pi$$
0.748202 + 0.663470i $$0.230917\pi$$
$$734$$ −4.50797e6 −0.308845
$$735$$ 734706. 0.0501644
$$736$$ −9.40044e6 −0.639667
$$737$$ 569840. 0.0386442
$$738$$ 1.60655e6 0.108581
$$739$$ 6.21786e6 0.418822 0.209411 0.977828i $$-0.432845\pi$$
0.209411 + 0.977828i $$0.432845\pi$$
$$740$$ 1.03271e7 0.693264
$$741$$ −3.64471e6 −0.243847
$$742$$ −7350.00 −0.000490092 0
$$743$$ 3.77647e6 0.250966 0.125483 0.992096i $$-0.459952\pi$$
0.125483 + 0.992096i $$0.459952\pi$$
$$744$$ −1.41523e6 −0.0937336
$$745$$ 1.00676e7 0.664562
$$746$$ 1.66535e6 0.109562
$$747$$ −3.05888e6 −0.200568
$$748$$ −8.41092e6 −0.549654
$$749$$ 3.91863e6 0.255229
$$750$$ −1.55876e6 −0.101188
$$751$$ −2.88795e6 −0.186849 −0.0934244 0.995626i $$-0.529781\pi$$
−0.0934244 + 0.995626i $$0.529781\pi$$
$$752$$ 8.29411e6 0.534842
$$753$$ −2.85397e6 −0.183427
$$754$$ −3.74187e6 −0.239696
$$755$$ 1.45000e7 0.925768
$$756$$ −1.10735e6 −0.0704662
$$757$$ 1.25519e6 0.0796104 0.0398052 0.999207i $$-0.487326\pi$$
0.0398052 + 0.999207i $$0.487326\pi$$
$$758$$ −2.53232e6 −0.160083
$$759$$ −9.76752e6 −0.615432
$$760$$ 1.91066e6 0.119991
$$761$$ −1.42623e7 −0.892746 −0.446373 0.894847i $$-0.647284\pi$$
−0.446373 + 0.894847i $$0.647284\pi$$
$$762$$ −1.76947e6 −0.110397
$$763$$ 2.25880e6 0.140465
$$764$$ −1.56756e7 −0.971606
$$765$$ 2.19769e6 0.135773
$$766$$ 796368. 0.0490390
$$767$$ −1.92478e7 −1.18139
$$768$$ −6.62430e6 −0.405263
$$769$$ −2.02261e7 −1.23338 −0.616689 0.787207i $$-0.711526\pi$$
−0.616689 + 0.787207i $$0.711526\pi$$
$$770$$ −566440. −0.0344292
$$771$$ 1.29856e7 0.786732
$$772$$ 1.34038e7 0.809443
$$773$$ 2.62288e7 1.57881 0.789406 0.613872i $$-0.210389\pi$$
0.789406 + 0.613872i $$0.210389\pi$$
$$774$$ −1.39612e6 −0.0837663
$$775$$ 4.91462e6 0.293925
$$776$$ −630126. −0.0375641
$$777$$ 4.32092e6 0.256758
$$778$$ 1.94799e6 0.115382
$$779$$ 1.76919e7 1.04456
$$780$$ −4.30664e6 −0.253456
$$781$$ −4.95312e6 −0.290570
$$782$$ 2.54722e6 0.148953
$$783$$ 6.00842e6 0.350232
$$784$$ 2.23053e6 0.129604
$$785$$ −6.06852e6 −0.351487
$$786$$ 694260. 0.0400835
$$787$$ −9.92829e6 −0.571397 −0.285698 0.958320i $$-0.592226\pi$$
−0.285698 + 0.958320i $$0.592226\pi$$
$$788$$ 4.09082e6 0.234690
$$789$$ −2.44346e6 −0.139738
$$790$$ 77248.0 0.00440372
$$791$$ −1.28726e7 −0.731518
$$792$$ 1.73502e6 0.0982860
$$793$$ 6.70013e6 0.378356
$$794$$ 1.08116e6 0.0608608
$$795$$ 45900.0 0.00257570
$$796$$ −9.25462e6 −0.517697
$$797$$ 1.09033e7 0.608014 0.304007 0.952670i $$-0.401675\pi$$
0.304007 + 0.952670i $$0.401675\pi$$
$$798$$ 393372. 0.0218674
$$799$$ −7.12454e6 −0.394812
$$800$$ −5.79871e6 −0.320336
$$801$$ −9.50017e6 −0.523179
$$802$$ 2.76770e6 0.151944
$$803$$ −2.66485e7 −1.45843
$$804$$ −467604. −0.0255116
$$805$$ −5.31787e6 −0.289233
$$806$$ −1.13318e6 −0.0614416
$$807$$ −7.65553e6 −0.413801
$$808$$ 6.85251e6 0.369251
$$809$$ 6.06398e6 0.325751 0.162876 0.986647i $$-0.447923\pi$$
0.162876 + 0.986647i $$0.447923\pi$$
$$810$$ −223074. −0.0119464
$$811$$ −8.59438e6 −0.458841 −0.229421 0.973327i $$-0.573683\pi$$
−0.229421 + 0.973327i $$0.573683\pi$$
$$812$$ −1.25196e7 −0.666347
$$813$$ 4.86115e6 0.257937
$$814$$ −3.33132e6 −0.176220
$$815$$ −8.59425e6 −0.453225
$$816$$ 6.67208e6 0.350781
$$817$$ −1.53745e7 −0.805835
$$818$$ 2.36350e6 0.123501
$$819$$ −1.80193e6 −0.0938701
$$820$$ 2.09050e7 1.08572
$$821$$ −2.01396e6 −0.104278 −0.0521391 0.998640i $$-0.516604\pi$$
−0.0521391 + 0.998640i $$0.516604\pi$$
$$822$$ −1.87353e6 −0.0967122
$$823$$ −2.64679e7 −1.36213 −0.681067 0.732221i $$-0.738484\pi$$
−0.681067 + 0.732221i $$0.738484\pi$$
$$824$$ 1.25491e7 0.643864
$$825$$ −6.02514e6 −0.308200
$$826$$ 2.07740e6 0.105943
$$827$$ −3.90229e6 −0.198407 −0.0992033 0.995067i $$-0.531629\pi$$
−0.0992033 + 0.995067i $$0.531629\pi$$
$$828$$ 8.01511e6 0.406288
$$829$$ −1.95595e7 −0.988487 −0.494244 0.869323i $$-0.664555\pi$$
−0.494244 + 0.869323i $$0.664555\pi$$
$$830$$ 1.28398e6 0.0646937
$$831$$ −4.62217e6 −0.232190
$$832$$ −1.21595e7 −0.608985
$$833$$ −1.91600e6 −0.0956715
$$834$$ 2.48022e6 0.123474
$$835$$ −1.72750e7 −0.857436
$$836$$ 9.40168e6 0.465254
$$837$$ 1.81958e6 0.0897756
$$838$$ −2.98669e6 −0.146920
$$839$$ −2.45448e7 −1.20380 −0.601901 0.798570i $$-0.705590\pi$$
−0.601901 + 0.798570i $$0.705590\pi$$
$$840$$ 944622. 0.0461913
$$841$$ 4.74194e7 2.31188
$$842$$ −3.46331e6 −0.168349
$$843$$ 1.22078e7 0.591655
$$844$$ 3.62892e7 1.75356
$$845$$ 5.61602e6 0.270574
$$846$$ 723168. 0.0347387
$$847$$ 2.22710e6 0.106667
$$848$$ 139350. 0.00665453
$$849$$ −2.58080e6 −0.122881
$$850$$ 1.57126e6 0.0745936
$$851$$ −3.12752e7 −1.48039
$$852$$ 4.06447e6 0.191825
$$853$$ 3.38305e7 1.59197 0.795987 0.605314i $$-0.206952\pi$$
0.795987 + 0.605314i $$0.206952\pi$$
$$854$$ −723142. −0.0339296
$$855$$ −2.45657e6 −0.114925
$$856$$ 5.03824e6 0.235014
$$857$$ 3.18009e7 1.47907 0.739534 0.673120i $$-0.235046\pi$$
0.739534 + 0.673120i $$0.235046\pi$$
$$858$$ 1.38924e6 0.0644257
$$859$$ 638420. 0.0295205 0.0147602 0.999891i $$-0.495301\pi$$
0.0147602 + 0.999891i $$0.495301\pi$$
$$860$$ −1.81667e7 −0.837589
$$861$$ 8.74679e6 0.402106
$$862$$ 2.33693e6 0.107122
$$863$$ −4.22256e6 −0.192996 −0.0964981 0.995333i $$-0.530764\pi$$
−0.0964981 + 0.995333i $$0.530764\pi$$
$$864$$ −2.14690e6 −0.0978427
$$865$$ 7.54236e6 0.342742
$$866$$ −3.50838e6 −0.158969
$$867$$ 7.04748e6 0.318409
$$868$$ −3.79142e6 −0.170806
$$869$$ 772480. 0.0347007
$$870$$ −2.52205e6 −0.112968
$$871$$ −760904. −0.0339848
$$872$$ 2.90417e6 0.129340
$$873$$ 810162. 0.0359779
$$874$$ −2.84726e6 −0.126081
$$875$$ −8.48660e6 −0.374726
$$876$$ 2.18675e7 0.962805
$$877$$ −2.45043e7 −1.07583 −0.537915 0.842999i $$-0.680788\pi$$
−0.537915 + 0.842999i $$0.680788\pi$$
$$878$$ 3.54833e6 0.155341
$$879$$ 1.53654e7 0.670767
$$880$$ 1.07392e7 0.467484
$$881$$ −2.77630e7 −1.20511 −0.602555 0.798078i $$-0.705850\pi$$
−0.602555 + 0.798078i $$0.705850\pi$$
$$882$$ 194481. 0.00841794
$$883$$ 3.30170e7 1.42507 0.712534 0.701638i $$-0.247548\pi$$
0.712534 + 0.701638i $$0.247548\pi$$
$$884$$ 1.12311e7 0.483381
$$885$$ −1.29732e7 −0.556786
$$886$$ 1.76833e6 0.0756797
$$887$$ 4.34462e6 0.185414 0.0927070 0.995693i $$-0.470448\pi$$
0.0927070 + 0.995693i $$0.470448\pi$$
$$888$$ 5.55547e6 0.236422
$$889$$ −9.63379e6 −0.408830
$$890$$ 3.98772e6 0.168752
$$891$$ −2.23074e6 −0.0941358
$$892$$ −1.23807e7 −0.520993
$$893$$ 7.96378e6 0.334188
$$894$$ 2.66495e6 0.111518
$$895$$ 3.86118e6 0.161125
$$896$$ 5.93013e6 0.246771
$$897$$ 1.30425e7 0.541228
$$898$$ −5.52579e6 −0.228667
$$899$$ 2.05720e7 0.848942
$$900$$ 4.94416e6 0.203463
$$901$$ −119700. −0.00491227
$$902$$ −6.74356e6 −0.275977
$$903$$ −7.60108e6 −0.310210
$$904$$ −1.65505e7 −0.673580
$$905$$ −2.25460e7 −0.915057
$$906$$ 3.83825e6 0.155350
$$907$$ 1.96499e7 0.793128 0.396564 0.918007i $$-0.370203\pi$$
0.396564 + 0.918007i $$0.370203\pi$$
$$908$$ −2.19454e7 −0.883342
$$909$$ −8.81037e6 −0.353659
$$910$$ 756364. 0.0302780
$$911$$ −7.26518e6 −0.290035 −0.145018 0.989429i $$-0.546324\pi$$
−0.145018 + 0.989429i $$0.546324\pi$$
$$912$$ −7.45801e6 −0.296918
$$913$$ 1.28398e7 0.509777
$$914$$ −2.96226e6 −0.117289
$$915$$ 4.51595e6 0.178318
$$916$$ 2.28091e7 0.898193
$$917$$ 3.77986e6 0.148440
$$918$$ 581742. 0.0227837
$$919$$ 9.82532e6 0.383758 0.191879 0.981419i $$-0.438542\pi$$
0.191879 + 0.981419i $$0.438542\pi$$
$$920$$ −6.83726e6 −0.266326
$$921$$ 4.92109e6 0.191167
$$922$$ 2.11884e6 0.0820863
$$923$$ 6.61387e6 0.255536
$$924$$ 4.64814e6 0.179102
$$925$$ −1.92923e7 −0.741359
$$926$$ 3.19226e6 0.122340
$$927$$ −1.61346e7 −0.616677
$$928$$ −2.42727e7 −0.925226
$$929$$ 2.71152e7 1.03080 0.515399 0.856951i $$-0.327644\pi$$
0.515399 + 0.856951i $$0.327644\pi$$
$$930$$ −763776. −0.0289573
$$931$$ 2.14169e6 0.0809809
$$932$$ 6.47150e6 0.244042
$$933$$ −2.91084e7 −1.09475
$$934$$ −7.42621e6 −0.278548
$$935$$ −9.22488e6 −0.345089
$$936$$ −2.31676e6 −0.0864354
$$937$$ −4.53522e7 −1.68752 −0.843761 0.536720i $$-0.819663\pi$$
−0.843761 + 0.536720i $$0.819663\pi$$
$$938$$ 82124.0 0.00304764
$$939$$ −1.63182e7 −0.603959
$$940$$ 9.41011e6 0.347356
$$941$$ 4.65780e7 1.71477 0.857387 0.514672i $$-0.172086\pi$$
0.857387 + 0.514672i $$0.172086\pi$$
$$942$$ −1.60637e6 −0.0589820
$$943$$ −6.33101e7 −2.31843
$$944$$ −3.93859e7 −1.43850
$$945$$ −1.21451e6 −0.0442408
$$946$$ 5.86024e6 0.212906
$$947$$ 2.53799e7 0.919632 0.459816 0.888014i $$-0.347915\pi$$
0.459816 + 0.888014i $$0.347915\pi$$
$$948$$ −633888. −0.0229082
$$949$$ 3.55836e7 1.28258
$$950$$ −1.75635e6 −0.0631396
$$951$$ 1.14892e7 0.411944
$$952$$ −2.46343e6 −0.0880942
$$953$$ 1.52948e7 0.545520 0.272760 0.962082i $$-0.412063\pi$$
0.272760 + 0.962082i $$0.412063\pi$$
$$954$$ 12150.0 0.000432220 0
$$955$$ −1.71926e7 −0.610004
$$956$$ −2.21147e7 −0.782592
$$957$$ −2.52205e7 −0.890173
$$958$$ −3.39685e6 −0.119581
$$959$$ −1.02003e7 −0.358152
$$960$$ −8.19560e6 −0.287014
$$961$$ −2.23991e7 −0.782389
$$962$$ 4.44829e6 0.154973
$$963$$ −6.47773e6 −0.225091
$$964$$ 1.56626e7 0.542840
$$965$$ 1.47010e7 0.508192
$$966$$ −1.40767e6 −0.0485354
$$967$$ −5.71465e6 −0.196527 −0.0982637 0.995160i $$-0.531329\pi$$
−0.0982637 + 0.995160i $$0.531329\pi$$
$$968$$ 2.86341e6 0.0982190
$$969$$ 6.40634e6 0.219180
$$970$$ −340068. −0.0116048
$$971$$ 1.30250e7 0.443332 0.221666 0.975123i $$-0.428851\pi$$
0.221666 + 0.975123i $$0.428851\pi$$
$$972$$ 1.83052e6 0.0621453
$$973$$ 1.35034e7 0.457258
$$974$$ −3.71382e6 −0.125436
$$975$$ 8.04533e6 0.271039
$$976$$ 1.37102e7 0.460700
$$977$$ −1.70360e7 −0.570992 −0.285496 0.958380i $$-0.592158\pi$$
−0.285496 + 0.958380i $$0.592158\pi$$
$$978$$ −2.27495e6 −0.0760544
$$979$$ 3.98772e7 1.32974
$$980$$ 2.53065e6 0.0841720
$$981$$ −3.73394e6 −0.123878
$$982$$ 5.57494e6 0.184485
$$983$$ −1.36985e7 −0.452156 −0.226078 0.974109i $$-0.572590\pi$$
−0.226078 + 0.974109i $$0.572590\pi$$
$$984$$ 1.12459e7 0.370259
$$985$$ 4.48671e6 0.147346
$$986$$ 6.57712e6 0.215448
$$987$$ 3.93725e6 0.128647
$$988$$ −1.25540e7 −0.409157
$$989$$ 5.50173e7 1.78858
$$990$$ 936360. 0.0303637
$$991$$ −3.49088e7 −1.12915 −0.564574 0.825383i $$-0.690959\pi$$
−0.564574 + 0.825383i $$0.690959\pi$$
$$992$$ −7.35072e6 −0.237165
$$993$$ 1.56259e7 0.502889
$$994$$ −713832. −0.0229155
$$995$$ −1.01502e7 −0.325026
$$996$$ −1.05362e7 −0.336538
$$997$$ 875662. 0.0278996 0.0139498 0.999903i $$-0.495559\pi$$
0.0139498 + 0.999903i $$0.495559\pi$$
$$998$$ 3.92698e6 0.124805
$$999$$ −7.14274e6 −0.226439
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 21.6.a.b.1.1 1
3.2 odd 2 63.6.a.c.1.1 1
4.3 odd 2 336.6.a.l.1.1 1
5.2 odd 4 525.6.d.d.274.2 2
5.3 odd 4 525.6.d.d.274.1 2
5.4 even 2 525.6.a.c.1.1 1
7.2 even 3 147.6.e.f.67.1 2
7.3 odd 6 147.6.e.e.79.1 2
7.4 even 3 147.6.e.f.79.1 2
7.5 odd 6 147.6.e.e.67.1 2
7.6 odd 2 147.6.a.e.1.1 1
12.11 even 2 1008.6.a.t.1.1 1
21.20 even 2 441.6.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.b.1.1 1 1.1 even 1 trivial
63.6.a.c.1.1 1 3.2 odd 2
147.6.a.e.1.1 1 7.6 odd 2
147.6.e.e.67.1 2 7.5 odd 6
147.6.e.e.79.1 2 7.3 odd 6
147.6.e.f.67.1 2 7.2 even 3
147.6.e.f.79.1 2 7.4 even 3
336.6.a.l.1.1 1 4.3 odd 2
441.6.a.d.1.1 1 21.20 even 2
525.6.a.c.1.1 1 5.4 even 2
525.6.d.d.274.1 2 5.3 odd 4
525.6.d.d.274.2 2 5.2 odd 4
1008.6.a.t.1.1 1 12.11 even 2