Properties

 Label 847.6.a.b.1.1 Level $847$ Weight $6$ Character 847.1 Self dual yes Analytic conductor $135.845$ 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] = [847,6,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, 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("847.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 847.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: $$135.845095382$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+10.0000 q^{2} -14.0000 q^{3} +68.0000 q^{4} -56.0000 q^{5} -140.000 q^{6} +49.0000 q^{7} +360.000 q^{8} -47.0000 q^{9} +O(q^{10})$$ $$q+10.0000 q^{2} -14.0000 q^{3} +68.0000 q^{4} -56.0000 q^{5} -140.000 q^{6} +49.0000 q^{7} +360.000 q^{8} -47.0000 q^{9} -560.000 q^{10} -952.000 q^{12} +140.000 q^{13} +490.000 q^{14} +784.000 q^{15} +1424.00 q^{16} +1722.00 q^{17} -470.000 q^{18} +98.0000 q^{19} -3808.00 q^{20} -686.000 q^{21} +1824.00 q^{23} -5040.00 q^{24} +11.0000 q^{25} +1400.00 q^{26} +4060.00 q^{27} +3332.00 q^{28} -3418.00 q^{29} +7840.00 q^{30} -7644.00 q^{31} +2720.00 q^{32} +17220.0 q^{34} -2744.00 q^{35} -3196.00 q^{36} -10398.0 q^{37} +980.000 q^{38} -1960.00 q^{39} -20160.0 q^{40} +17962.0 q^{41} -6860.00 q^{42} -10880.0 q^{43} +2632.00 q^{45} +18240.0 q^{46} +9324.00 q^{47} -19936.0 q^{48} +2401.00 q^{49} +110.000 q^{50} -24108.0 q^{51} +9520.00 q^{52} +2262.00 q^{53} +40600.0 q^{54} +17640.0 q^{56} -1372.00 q^{57} -34180.0 q^{58} -2730.00 q^{59} +53312.0 q^{60} -25648.0 q^{61} -76440.0 q^{62} -2303.00 q^{63} -18368.0 q^{64} -7840.00 q^{65} -48404.0 q^{67} +117096. q^{68} -25536.0 q^{69} -27440.0 q^{70} -58560.0 q^{71} -16920.0 q^{72} -68082.0 q^{73} -103980. q^{74} -154.000 q^{75} +6664.00 q^{76} -19600.0 q^{78} -31784.0 q^{79} -79744.0 q^{80} -45419.0 q^{81} +179620. q^{82} +20538.0 q^{83} -46648.0 q^{84} -96432.0 q^{85} -108800. q^{86} +47852.0 q^{87} -50582.0 q^{89} +26320.0 q^{90} +6860.00 q^{91} +124032. q^{92} +107016. q^{93} +93240.0 q^{94} -5488.00 q^{95} -38080.0 q^{96} -58506.0 q^{97} +24010.0 q^{98} +O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 10.0000 1.76777 0.883883 0.467707i $$-0.154920\pi$$
0.883883 + 0.467707i $$0.154920\pi$$
$$3$$ −14.0000 −0.898100 −0.449050 0.893507i $$-0.648238\pi$$
−0.449050 + 0.893507i $$0.648238\pi$$
$$4$$ 68.0000 2.12500
$$5$$ −56.0000 −1.00176 −0.500879 0.865517i $$-0.666990\pi$$
−0.500879 + 0.865517i $$0.666990\pi$$
$$6$$ −140.000 −1.58763
$$7$$ 49.0000 0.377964
$$8$$ 360.000 1.98874
$$9$$ −47.0000 −0.193416
$$10$$ −560.000 −1.77088
$$11$$ 0 0
$$12$$ −952.000 −1.90846
$$13$$ 140.000 0.229757 0.114879 0.993380i $$-0.463352\pi$$
0.114879 + 0.993380i $$0.463352\pi$$
$$14$$ 490.000 0.668153
$$15$$ 784.000 0.899680
$$16$$ 1424.00 1.39062
$$17$$ 1722.00 1.44514 0.722572 0.691296i $$-0.242960\pi$$
0.722572 + 0.691296i $$0.242960\pi$$
$$18$$ −470.000 −0.341914
$$19$$ 98.0000 0.0622791 0.0311395 0.999515i $$-0.490086\pi$$
0.0311395 + 0.999515i $$0.490086\pi$$
$$20$$ −3808.00 −2.12874
$$21$$ −686.000 −0.339450
$$22$$ 0 0
$$23$$ 1824.00 0.718961 0.359480 0.933153i $$-0.382954\pi$$
0.359480 + 0.933153i $$0.382954\pi$$
$$24$$ −5040.00 −1.78609
$$25$$ 11.0000 0.00352000
$$26$$ 1400.00 0.406158
$$27$$ 4060.00 1.07181
$$28$$ 3332.00 0.803175
$$29$$ −3418.00 −0.754705 −0.377352 0.926070i $$-0.623165\pi$$
−0.377352 + 0.926070i $$0.623165\pi$$
$$30$$ 7840.00 1.59042
$$31$$ −7644.00 −1.42862 −0.714310 0.699830i $$-0.753259\pi$$
−0.714310 + 0.699830i $$0.753259\pi$$
$$32$$ 2720.00 0.469563
$$33$$ 0 0
$$34$$ 17220.0 2.55468
$$35$$ −2744.00 −0.378629
$$36$$ −3196.00 −0.411008
$$37$$ −10398.0 −1.24866 −0.624332 0.781159i $$-0.714629\pi$$
−0.624332 + 0.781159i $$0.714629\pi$$
$$38$$ 980.000 0.110095
$$39$$ −1960.00 −0.206345
$$40$$ −20160.0 −1.99223
$$41$$ 17962.0 1.66876 0.834382 0.551186i $$-0.185825\pi$$
0.834382 + 0.551186i $$0.185825\pi$$
$$42$$ −6860.00 −0.600069
$$43$$ −10880.0 −0.897342 −0.448671 0.893697i $$-0.648102\pi$$
−0.448671 + 0.893697i $$0.648102\pi$$
$$44$$ 0 0
$$45$$ 2632.00 0.193756
$$46$$ 18240.0 1.27096
$$47$$ 9324.00 0.615684 0.307842 0.951438i $$-0.400393\pi$$
0.307842 + 0.951438i $$0.400393\pi$$
$$48$$ −19936.0 −1.24892
$$49$$ 2401.00 0.142857
$$50$$ 110.000 0.00622254
$$51$$ −24108.0 −1.29788
$$52$$ 9520.00 0.488235
$$53$$ 2262.00 0.110612 0.0553061 0.998469i $$-0.482387\pi$$
0.0553061 + 0.998469i $$0.482387\pi$$
$$54$$ 40600.0 1.89471
$$55$$ 0 0
$$56$$ 17640.0 0.751672
$$57$$ −1372.00 −0.0559329
$$58$$ −34180.0 −1.33414
$$59$$ −2730.00 −0.102102 −0.0510508 0.998696i $$-0.516257\pi$$
−0.0510508 + 0.998696i $$0.516257\pi$$
$$60$$ 53312.0 1.91182
$$61$$ −25648.0 −0.882529 −0.441264 0.897377i $$-0.645470\pi$$
−0.441264 + 0.897377i $$0.645470\pi$$
$$62$$ −76440.0 −2.52547
$$63$$ −2303.00 −0.0731042
$$64$$ −18368.0 −0.560547
$$65$$ −7840.00 −0.230161
$$66$$ 0 0
$$67$$ −48404.0 −1.31733 −0.658664 0.752437i $$-0.728878\pi$$
−0.658664 + 0.752437i $$0.728878\pi$$
$$68$$ 117096. 3.07093
$$69$$ −25536.0 −0.645699
$$70$$ −27440.0 −0.669328
$$71$$ −58560.0 −1.37865 −0.689327 0.724450i $$-0.742094\pi$$
−0.689327 + 0.724450i $$0.742094\pi$$
$$72$$ −16920.0 −0.384653
$$73$$ −68082.0 −1.49529 −0.747645 0.664099i $$-0.768815\pi$$
−0.747645 + 0.664099i $$0.768815\pi$$
$$74$$ −103980. −2.20735
$$75$$ −154.000 −0.00316131
$$76$$ 6664.00 0.132343
$$77$$ 0 0
$$78$$ −19600.0 −0.364770
$$79$$ −31784.0 −0.572982 −0.286491 0.958083i $$-0.592489\pi$$
−0.286491 + 0.958083i $$0.592489\pi$$
$$80$$ −79744.0 −1.39307
$$81$$ −45419.0 −0.769175
$$82$$ 179620. 2.94999
$$83$$ 20538.0 0.327237 0.163619 0.986524i $$-0.447683\pi$$
0.163619 + 0.986524i $$0.447683\pi$$
$$84$$ −46648.0 −0.721331
$$85$$ −96432.0 −1.44768
$$86$$ −108800. −1.58629
$$87$$ 47852.0 0.677801
$$88$$ 0 0
$$89$$ −50582.0 −0.676894 −0.338447 0.940985i $$-0.609902\pi$$
−0.338447 + 0.940985i $$0.609902\pi$$
$$90$$ 26320.0 0.342515
$$91$$ 6860.00 0.0868402
$$92$$ 124032. 1.52779
$$93$$ 107016. 1.28304
$$94$$ 93240.0 1.08839
$$95$$ −5488.00 −0.0623886
$$96$$ −38080.0 −0.421715
$$97$$ −58506.0 −0.631351 −0.315676 0.948867i $$-0.602231\pi$$
−0.315676 + 0.948867i $$0.602231\pi$$
$$98$$ 24010.0 0.252538
$$99$$ 0 0
$$100$$ 748.000 0.00748000
$$101$$ −38696.0 −0.377453 −0.188726 0.982030i $$-0.560436\pi$$
−0.188726 + 0.982030i $$0.560436\pi$$
$$102$$ −241080. −2.29436
$$103$$ 53060.0 0.492804 0.246402 0.969168i $$-0.420752\pi$$
0.246402 + 0.969168i $$0.420752\pi$$
$$104$$ 50400.0 0.456927
$$105$$ 38416.0 0.340047
$$106$$ 22620.0 0.195537
$$107$$ 146324. 1.23554 0.617769 0.786360i $$-0.288037\pi$$
0.617769 + 0.786360i $$0.288037\pi$$
$$108$$ 276080. 2.27759
$$109$$ −92898.0 −0.748928 −0.374464 0.927241i $$-0.622173\pi$$
−0.374464 + 0.927241i $$0.622173\pi$$
$$110$$ 0 0
$$111$$ 145572. 1.12143
$$112$$ 69776.0 0.525607
$$113$$ −83354.0 −0.614088 −0.307044 0.951695i $$-0.599340\pi$$
−0.307044 + 0.951695i $$0.599340\pi$$
$$114$$ −13720.0 −0.0988762
$$115$$ −102144. −0.720225
$$116$$ −232424. −1.60375
$$117$$ −6580.00 −0.0444387
$$118$$ −27300.0 −0.180492
$$119$$ 84378.0 0.546213
$$120$$ 282240. 1.78923
$$121$$ 0 0
$$122$$ −256480. −1.56011
$$123$$ −251468. −1.49872
$$124$$ −519792. −3.03582
$$125$$ 174384. 0.998232
$$126$$ −23030.0 −0.129231
$$127$$ −60384.0 −0.332210 −0.166105 0.986108i $$-0.553119\pi$$
−0.166105 + 0.986108i $$0.553119\pi$$
$$128$$ −270720. −1.46048
$$129$$ 152320. 0.805903
$$130$$ −78400.0 −0.406872
$$131$$ 61586.0 0.313548 0.156774 0.987635i $$-0.449891\pi$$
0.156774 + 0.987635i $$0.449891\pi$$
$$132$$ 0 0
$$133$$ 4802.00 0.0235393
$$134$$ −484040. −2.32873
$$135$$ −227360. −1.07369
$$136$$ 619920. 2.87401
$$137$$ −204462. −0.930703 −0.465352 0.885126i $$-0.654072\pi$$
−0.465352 + 0.885126i $$0.654072\pi$$
$$138$$ −255360. −1.14145
$$139$$ 35406.0 0.155432 0.0777159 0.996976i $$-0.475237\pi$$
0.0777159 + 0.996976i $$0.475237\pi$$
$$140$$ −186592. −0.804587
$$141$$ −130536. −0.552946
$$142$$ −585600. −2.43714
$$143$$ 0 0
$$144$$ −66928.0 −0.268969
$$145$$ 191408. 0.756032
$$146$$ −680820. −2.64332
$$147$$ −33614.0 −0.128300
$$148$$ −707064. −2.65341
$$149$$ 20226.0 0.0746353 0.0373177 0.999303i $$-0.488119\pi$$
0.0373177 + 0.999303i $$0.488119\pi$$
$$150$$ −1540.00 −0.00558847
$$151$$ −70904.0 −0.253063 −0.126531 0.991963i $$-0.540384\pi$$
−0.126531 + 0.991963i $$0.540384\pi$$
$$152$$ 35280.0 0.123857
$$153$$ −80934.0 −0.279513
$$154$$ 0 0
$$155$$ 428064. 1.43113
$$156$$ −133280. −0.438484
$$157$$ 293524. 0.950374 0.475187 0.879885i $$-0.342380\pi$$
0.475187 + 0.879885i $$0.342380\pi$$
$$158$$ −317840. −1.01290
$$159$$ −31668.0 −0.0993408
$$160$$ −152320. −0.470389
$$161$$ 89376.0 0.271742
$$162$$ −454190. −1.35972
$$163$$ 13192.0 0.0388903 0.0194452 0.999811i $$-0.493810\pi$$
0.0194452 + 0.999811i $$0.493810\pi$$
$$164$$ 1.22142e6 3.54612
$$165$$ 0 0
$$166$$ 205380. 0.578479
$$167$$ −493612. −1.36960 −0.684801 0.728730i $$-0.740111\pi$$
−0.684801 + 0.728730i $$0.740111\pi$$
$$168$$ −246960. −0.675077
$$169$$ −351693. −0.947212
$$170$$ −964320. −2.55917
$$171$$ −4606.00 −0.0120457
$$172$$ −739840. −1.90685
$$173$$ −240716. −0.611490 −0.305745 0.952113i $$-0.598906\pi$$
−0.305745 + 0.952113i $$0.598906\pi$$
$$174$$ 478520. 1.19819
$$175$$ 539.000 0.00133043
$$176$$ 0 0
$$177$$ 38220.0 0.0916975
$$178$$ −505820. −1.19659
$$179$$ 294932. 0.688001 0.344001 0.938969i $$-0.388218\pi$$
0.344001 + 0.938969i $$0.388218\pi$$
$$180$$ 178976. 0.411731
$$181$$ −336980. −0.764553 −0.382277 0.924048i $$-0.624860\pi$$
−0.382277 + 0.924048i $$0.624860\pi$$
$$182$$ 68600.0 0.153513
$$183$$ 359072. 0.792600
$$184$$ 656640. 1.42982
$$185$$ 582288. 1.25086
$$186$$ 1.07016e6 2.26812
$$187$$ 0 0
$$188$$ 634032. 1.30833
$$189$$ 198940. 0.405105
$$190$$ −54880.0 −0.110288
$$191$$ 358264. 0.710591 0.355296 0.934754i $$-0.384380\pi$$
0.355296 + 0.934754i $$0.384380\pi$$
$$192$$ 257152. 0.503427
$$193$$ 989554. 1.91226 0.956128 0.292948i $$-0.0946362\pi$$
0.956128 + 0.292948i $$0.0946362\pi$$
$$194$$ −585060. −1.11608
$$195$$ 109760. 0.206708
$$196$$ 163268. 0.303571
$$197$$ 990050. 1.81757 0.908786 0.417263i $$-0.137011\pi$$
0.908786 + 0.417263i $$0.137011\pi$$
$$198$$ 0 0
$$199$$ −840756. −1.50500 −0.752501 0.658591i $$-0.771153\pi$$
−0.752501 + 0.658591i $$0.771153\pi$$
$$200$$ 3960.00 0.00700036
$$201$$ 677656. 1.18309
$$202$$ −386960. −0.667249
$$203$$ −167482. −0.285252
$$204$$ −1.63934e6 −2.75800
$$205$$ −1.00587e6 −1.67170
$$206$$ 530600. 0.871163
$$207$$ −85728.0 −0.139058
$$208$$ 199360. 0.319506
$$209$$ 0 0
$$210$$ 384160. 0.601124
$$211$$ −1.15073e6 −1.77938 −0.889689 0.456568i $$-0.849079\pi$$
−0.889689 + 0.456568i $$0.849079\pi$$
$$212$$ 153816. 0.235051
$$213$$ 819840. 1.23817
$$214$$ 1.46324e6 2.18414
$$215$$ 609280. 0.898919
$$216$$ 1.46160e6 2.13154
$$217$$ −374556. −0.539967
$$218$$ −928980. −1.32393
$$219$$ 953148. 1.34292
$$220$$ 0 0
$$221$$ 241080. 0.332032
$$222$$ 1.45572e6 1.98242
$$223$$ −824264. −1.10995 −0.554976 0.831866i $$-0.687273\pi$$
−0.554976 + 0.831866i $$0.687273\pi$$
$$224$$ 133280. 0.177478
$$225$$ −517.000 −0.000680823 0
$$226$$ −833540. −1.08556
$$227$$ −74382.0 −0.0958083 −0.0479042 0.998852i $$-0.515254\pi$$
−0.0479042 + 0.998852i $$0.515254\pi$$
$$228$$ −93296.0 −0.118857
$$229$$ 1.13196e6 1.42640 0.713199 0.700961i $$-0.247245\pi$$
0.713199 + 0.700961i $$0.247245\pi$$
$$230$$ −1.02144e6 −1.27319
$$231$$ 0 0
$$232$$ −1.23048e6 −1.50091
$$233$$ 198726. 0.239809 0.119904 0.992785i $$-0.461741\pi$$
0.119904 + 0.992785i $$0.461741\pi$$
$$234$$ −65800.0 −0.0785572
$$235$$ −522144. −0.616766
$$236$$ −185640. −0.216966
$$237$$ 444976. 0.514595
$$238$$ 843780. 0.965577
$$239$$ −482904. −0.546847 −0.273424 0.961894i $$-0.588156\pi$$
−0.273424 + 0.961894i $$0.588156\pi$$
$$240$$ 1.11642e6 1.25112
$$241$$ −805910. −0.893807 −0.446904 0.894582i $$-0.647473\pi$$
−0.446904 + 0.894582i $$0.647473\pi$$
$$242$$ 0 0
$$243$$ −350714. −0.381011
$$244$$ −1.74406e6 −1.87537
$$245$$ −134456. −0.143108
$$246$$ −2.51468e6 −2.64938
$$247$$ 13720.0 0.0143091
$$248$$ −2.75184e6 −2.84115
$$249$$ −287532. −0.293892
$$250$$ 1.74384e6 1.76464
$$251$$ 430738. 0.431548 0.215774 0.976443i $$-0.430773\pi$$
0.215774 + 0.976443i $$0.430773\pi$$
$$252$$ −156604. −0.155347
$$253$$ 0 0
$$254$$ −603840. −0.587270
$$255$$ 1.35005e6 1.30017
$$256$$ −2.11942e6 −2.02124
$$257$$ −1.17691e6 −1.11150 −0.555751 0.831349i $$-0.687569\pi$$
−0.555751 + 0.831349i $$0.687569\pi$$
$$258$$ 1.52320e6 1.42465
$$259$$ −509502. −0.471951
$$260$$ −533120. −0.489093
$$261$$ 160646. 0.145972
$$262$$ 615860. 0.554279
$$263$$ −1.29098e6 −1.15088 −0.575438 0.817845i $$-0.695169\pi$$
−0.575438 + 0.817845i $$0.695169\pi$$
$$264$$ 0 0
$$265$$ −126672. −0.110807
$$266$$ 48020.0 0.0416119
$$267$$ 708148. 0.607919
$$268$$ −3.29147e6 −2.79932
$$269$$ −1.27756e6 −1.07646 −0.538232 0.842797i $$-0.680907\pi$$
−0.538232 + 0.842797i $$0.680907\pi$$
$$270$$ −2.27360e6 −1.89804
$$271$$ −1.65054e6 −1.36522 −0.682612 0.730781i $$-0.739156\pi$$
−0.682612 + 0.730781i $$0.739156\pi$$
$$272$$ 2.45213e6 2.00965
$$273$$ −96040.0 −0.0779912
$$274$$ −2.04462e6 −1.64527
$$275$$ 0 0
$$276$$ −1.73645e6 −1.37211
$$277$$ 1.06409e6 0.833257 0.416628 0.909077i $$-0.363212\pi$$
0.416628 + 0.909077i $$0.363212\pi$$
$$278$$ 354060. 0.274767
$$279$$ 359268. 0.276317
$$280$$ −987840. −0.752994
$$281$$ 22342.0 0.0168794 0.00843969 0.999964i $$-0.497314\pi$$
0.00843969 + 0.999964i $$0.497314\pi$$
$$282$$ −1.30536e6 −0.977479
$$283$$ 2.49574e6 1.85239 0.926196 0.377042i $$-0.123059\pi$$
0.926196 + 0.377042i $$0.123059\pi$$
$$284$$ −3.98208e6 −2.92964
$$285$$ 76832.0 0.0560312
$$286$$ 0 0
$$287$$ 880138. 0.630734
$$288$$ −127840. −0.0908208
$$289$$ 1.54543e6 1.08844
$$290$$ 1.91408e6 1.33649
$$291$$ 819084. 0.567017
$$292$$ −4.62958e6 −3.17749
$$293$$ 1.93178e6 1.31458 0.657291 0.753637i $$-0.271702\pi$$
0.657291 + 0.753637i $$0.271702\pi$$
$$294$$ −336140. −0.226805
$$295$$ 152880. 0.102281
$$296$$ −3.74328e6 −2.48326
$$297$$ 0 0
$$298$$ 202260. 0.131938
$$299$$ 255360. 0.165187
$$300$$ −10472.0 −0.00671779
$$301$$ −533120. −0.339163
$$302$$ −709040. −0.447356
$$303$$ 541744. 0.338991
$$304$$ 139552. 0.0866068
$$305$$ 1.43629e6 0.884081
$$306$$ −809340. −0.494114
$$307$$ 459074. 0.277995 0.138997 0.990293i $$-0.455612\pi$$
0.138997 + 0.990293i $$0.455612\pi$$
$$308$$ 0 0
$$309$$ −742840. −0.442587
$$310$$ 4.28064e6 2.52991
$$311$$ 667128. 0.391118 0.195559 0.980692i $$-0.437348\pi$$
0.195559 + 0.980692i $$0.437348\pi$$
$$312$$ −705600. −0.410367
$$313$$ −111034. −0.0640612 −0.0320306 0.999487i $$-0.510197\pi$$
−0.0320306 + 0.999487i $$0.510197\pi$$
$$314$$ 2.93524e6 1.68004
$$315$$ 128968. 0.0732328
$$316$$ −2.16131e6 −1.21759
$$317$$ −68778.0 −0.0384416 −0.0192208 0.999815i $$-0.506119\pi$$
−0.0192208 + 0.999815i $$0.506119\pi$$
$$318$$ −316680. −0.175611
$$319$$ 0 0
$$320$$ 1.02861e6 0.561533
$$321$$ −2.04854e6 −1.10964
$$322$$ 893760. 0.480376
$$323$$ 168756. 0.0900022
$$324$$ −3.08849e6 −1.63450
$$325$$ 1540.00 0.000808746 0
$$326$$ 131920. 0.0687490
$$327$$ 1.30057e6 0.672613
$$328$$ 6.46632e6 3.31874
$$329$$ 456876. 0.232707
$$330$$ 0 0
$$331$$ −564448. −0.283174 −0.141587 0.989926i $$-0.545221\pi$$
−0.141587 + 0.989926i $$0.545221\pi$$
$$332$$ 1.39658e6 0.695379
$$333$$ 488706. 0.241511
$$334$$ −4.93612e6 −2.42114
$$335$$ 2.71062e6 1.31965
$$336$$ −976864. −0.472048
$$337$$ −2.07729e6 −0.996376 −0.498188 0.867069i $$-0.666001\pi$$
−0.498188 + 0.867069i $$0.666001\pi$$
$$338$$ −3.51693e6 −1.67445
$$339$$ 1.16696e6 0.551512
$$340$$ −6.55738e6 −3.07633
$$341$$ 0 0
$$342$$ −46060.0 −0.0212941
$$343$$ 117649. 0.0539949
$$344$$ −3.91680e6 −1.78458
$$345$$ 1.43002e6 0.646834
$$346$$ −2.40716e6 −1.08097
$$347$$ 53248.0 0.0237399 0.0118700 0.999930i $$-0.496222\pi$$
0.0118700 + 0.999930i $$0.496222\pi$$
$$348$$ 3.25394e6 1.44033
$$349$$ 2.27200e6 0.998494 0.499247 0.866460i $$-0.333610\pi$$
0.499247 + 0.866460i $$0.333610\pi$$
$$350$$ 5390.00 0.00235190
$$351$$ 568400. 0.246256
$$352$$ 0 0
$$353$$ 4.00645e6 1.71129 0.855644 0.517565i $$-0.173162\pi$$
0.855644 + 0.517565i $$0.173162\pi$$
$$354$$ 382200. 0.162100
$$355$$ 3.27936e6 1.38108
$$356$$ −3.43958e6 −1.43840
$$357$$ −1.18129e6 −0.490554
$$358$$ 2.94932e6 1.21623
$$359$$ −73784.0 −0.0302152 −0.0151076 0.999886i $$-0.504809\pi$$
−0.0151076 + 0.999886i $$0.504809\pi$$
$$360$$ 947520. 0.385329
$$361$$ −2.46650e6 −0.996121
$$362$$ −3.36980e6 −1.35155
$$363$$ 0 0
$$364$$ 466480. 0.184535
$$365$$ 3.81259e6 1.49792
$$366$$ 3.59072e6 1.40113
$$367$$ 1.40431e6 0.544250 0.272125 0.962262i $$-0.412274\pi$$
0.272125 + 0.962262i $$0.412274\pi$$
$$368$$ 2.59738e6 0.999805
$$369$$ −844214. −0.322765
$$370$$ 5.82288e6 2.21123
$$371$$ 110838. 0.0418075
$$372$$ 7.27709e6 2.72647
$$373$$ 1.60323e6 0.596657 0.298329 0.954463i $$-0.403571\pi$$
0.298329 + 0.954463i $$0.403571\pi$$
$$374$$ 0 0
$$375$$ −2.44138e6 −0.896513
$$376$$ 3.35664e6 1.22443
$$377$$ −478520. −0.173399
$$378$$ 1.98940e6 0.716131
$$379$$ −4.77012e6 −1.70581 −0.852906 0.522064i $$-0.825162\pi$$
−0.852906 + 0.522064i $$0.825162\pi$$
$$380$$ −373184. −0.132576
$$381$$ 845376. 0.298358
$$382$$ 3.58264e6 1.25616
$$383$$ −2.23079e6 −0.777072 −0.388536 0.921434i $$-0.627019\pi$$
−0.388536 + 0.921434i $$0.627019\pi$$
$$384$$ 3.79008e6 1.31166
$$385$$ 0 0
$$386$$ 9.89554e6 3.38042
$$387$$ 511360. 0.173560
$$388$$ −3.97841e6 −1.34162
$$389$$ 4.84024e6 1.62178 0.810892 0.585196i $$-0.198982\pi$$
0.810892 + 0.585196i $$0.198982\pi$$
$$390$$ 1.09760e6 0.365412
$$391$$ 3.14093e6 1.03900
$$392$$ 864360. 0.284105
$$393$$ −862204. −0.281597
$$394$$ 9.90050e6 3.21304
$$395$$ 1.77990e6 0.573989
$$396$$ 0 0
$$397$$ 995820. 0.317106 0.158553 0.987350i $$-0.449317\pi$$
0.158553 + 0.987350i $$0.449317\pi$$
$$398$$ −8.40756e6 −2.66049
$$399$$ −67228.0 −0.0211406
$$400$$ 15664.0 0.00489500
$$401$$ −3.31605e6 −1.02982 −0.514909 0.857245i $$-0.672174\pi$$
−0.514909 + 0.857245i $$0.672174\pi$$
$$402$$ 6.77656e6 2.09143
$$403$$ −1.07016e6 −0.328236
$$404$$ −2.63133e6 −0.802087
$$405$$ 2.54346e6 0.770527
$$406$$ −1.67482e6 −0.504258
$$407$$ 0 0
$$408$$ −8.67888e6 −2.58115
$$409$$ −3.07273e6 −0.908274 −0.454137 0.890932i $$-0.650052\pi$$
−0.454137 + 0.890932i $$0.650052\pi$$
$$410$$ −1.00587e7 −2.95517
$$411$$ 2.86247e6 0.835865
$$412$$ 3.60808e6 1.04721
$$413$$ −133770. −0.0385908
$$414$$ −857280. −0.245823
$$415$$ −1.15013e6 −0.327813
$$416$$ 380800. 0.107886
$$417$$ −495684. −0.139593
$$418$$ 0 0
$$419$$ 2.81438e6 0.783154 0.391577 0.920145i $$-0.371930\pi$$
0.391577 + 0.920145i $$0.371930\pi$$
$$420$$ 2.61229e6 0.722600
$$421$$ 3.05802e6 0.840883 0.420441 0.907320i $$-0.361875\pi$$
0.420441 + 0.907320i $$0.361875\pi$$
$$422$$ −1.15073e7 −3.14552
$$423$$ −438228. −0.119083
$$424$$ 814320. 0.219979
$$425$$ 18942.0 0.00508690
$$426$$ 8.19840e6 2.18880
$$427$$ −1.25675e6 −0.333565
$$428$$ 9.95003e6 2.62552
$$429$$ 0 0
$$430$$ 6.09280e6 1.58908
$$431$$ −1.93750e6 −0.502398 −0.251199 0.967936i $$-0.580825\pi$$
−0.251199 + 0.967936i $$0.580825\pi$$
$$432$$ 5.78144e6 1.49048
$$433$$ 3.94790e6 1.01192 0.505961 0.862557i $$-0.331138\pi$$
0.505961 + 0.862557i $$0.331138\pi$$
$$434$$ −3.74556e6 −0.954536
$$435$$ −2.67971e6 −0.678993
$$436$$ −6.31706e6 −1.59147
$$437$$ 178752. 0.0447762
$$438$$ 9.53148e6 2.37397
$$439$$ 7.41770e6 1.83700 0.918498 0.395426i $$-0.129403\pi$$
0.918498 + 0.395426i $$0.129403\pi$$
$$440$$ 0 0
$$441$$ −112847. −0.0276308
$$442$$ 2.41080e6 0.586956
$$443$$ 1.40269e6 0.339589 0.169794 0.985480i $$-0.445690\pi$$
0.169794 + 0.985480i $$0.445690\pi$$
$$444$$ 9.89890e6 2.38303
$$445$$ 2.83259e6 0.678085
$$446$$ −8.24264e6 −1.96214
$$447$$ −283164. −0.0670300
$$448$$ −900032. −0.211867
$$449$$ −590574. −0.138248 −0.0691239 0.997608i $$-0.522020\pi$$
−0.0691239 + 0.997608i $$0.522020\pi$$
$$450$$ −5170.00 −0.00120354
$$451$$ 0 0
$$452$$ −5.66807e6 −1.30494
$$453$$ 992656. 0.227276
$$454$$ −743820. −0.169367
$$455$$ −384160. −0.0869929
$$456$$ −493920. −0.111236
$$457$$ 2.90484e6 0.650627 0.325313 0.945606i $$-0.394530\pi$$
0.325313 + 0.945606i $$0.394530\pi$$
$$458$$ 1.13196e7 2.52154
$$459$$ 6.99132e6 1.54891
$$460$$ −6.94579e6 −1.53048
$$461$$ 922684. 0.202209 0.101105 0.994876i $$-0.467762\pi$$
0.101105 + 0.994876i $$0.467762\pi$$
$$462$$ 0 0
$$463$$ 7.18235e6 1.55709 0.778546 0.627588i $$-0.215958\pi$$
0.778546 + 0.627588i $$0.215958\pi$$
$$464$$ −4.86723e6 −1.04951
$$465$$ −5.99290e6 −1.28530
$$466$$ 1.98726e6 0.423926
$$467$$ −612570. −0.129976 −0.0649881 0.997886i $$-0.520701\pi$$
−0.0649881 + 0.997886i $$0.520701\pi$$
$$468$$ −447440. −0.0944322
$$469$$ −2.37180e6 −0.497904
$$470$$ −5.22144e6 −1.09030
$$471$$ −4.10934e6 −0.853531
$$472$$ −982800. −0.203053
$$473$$ 0 0
$$474$$ 4.44976e6 0.909684
$$475$$ 1078.00 0.000219222 0
$$476$$ 5.73770e6 1.16070
$$477$$ −106314. −0.0213941
$$478$$ −4.82904e6 −0.966699
$$479$$ −2.60330e6 −0.518424 −0.259212 0.965820i $$-0.583463\pi$$
−0.259212 + 0.965820i $$0.583463\pi$$
$$480$$ 2.13248e6 0.422456
$$481$$ −1.45572e6 −0.286890
$$482$$ −8.05910e6 −1.58004
$$483$$ −1.25126e6 −0.244051
$$484$$ 0 0
$$485$$ 3.27634e6 0.632461
$$486$$ −3.50714e6 −0.673539
$$487$$ 5.46309e6 1.04380 0.521898 0.853008i $$-0.325224\pi$$
0.521898 + 0.853008i $$0.325224\pi$$
$$488$$ −9.23328e6 −1.75512
$$489$$ −184688. −0.0349274
$$490$$ −1.34456e6 −0.252982
$$491$$ −1.64090e6 −0.307170 −0.153585 0.988135i $$-0.549082\pi$$
−0.153585 + 0.988135i $$0.549082\pi$$
$$492$$ −1.70998e7 −3.18478
$$493$$ −5.88580e6 −1.09066
$$494$$ 137200. 0.0252951
$$495$$ 0 0
$$496$$ −1.08851e7 −1.98667
$$497$$ −2.86944e6 −0.521082
$$498$$ −2.87532e6 −0.519533
$$499$$ 2.99796e6 0.538983 0.269491 0.963003i $$-0.413144\pi$$
0.269491 + 0.963003i $$0.413144\pi$$
$$500$$ 1.18581e7 2.12124
$$501$$ 6.91057e6 1.23004
$$502$$ 4.30738e6 0.762876
$$503$$ 6.89405e6 1.21494 0.607469 0.794343i $$-0.292185\pi$$
0.607469 + 0.794343i $$0.292185\pi$$
$$504$$ −829080. −0.145385
$$505$$ 2.16698e6 0.378117
$$506$$ 0 0
$$507$$ 4.92370e6 0.850691
$$508$$ −4.10611e6 −0.705946
$$509$$ 2.30476e6 0.394305 0.197152 0.980373i $$-0.436831\pi$$
0.197152 + 0.980373i $$0.436831\pi$$
$$510$$ 1.35005e7 2.29839
$$511$$ −3.33602e6 −0.565166
$$512$$ −1.25312e7 −2.11260
$$513$$ 397880. 0.0667511
$$514$$ −1.17691e7 −1.96488
$$515$$ −2.97136e6 −0.493671
$$516$$ 1.03578e7 1.71254
$$517$$ 0 0
$$518$$ −5.09502e6 −0.834299
$$519$$ 3.37002e6 0.549180
$$520$$ −2.82240e6 −0.457731
$$521$$ −1.20960e7 −1.95231 −0.976155 0.217073i $$-0.930349\pi$$
−0.976155 + 0.217073i $$0.930349\pi$$
$$522$$ 1.60646e6 0.258044
$$523$$ −5.48443e6 −0.876753 −0.438377 0.898791i $$-0.644446\pi$$
−0.438377 + 0.898791i $$0.644446\pi$$
$$524$$ 4.18785e6 0.666289
$$525$$ −7546.00 −0.00119486
$$526$$ −1.29098e7 −2.03448
$$527$$ −1.31630e7 −2.06456
$$528$$ 0 0
$$529$$ −3.10937e6 −0.483095
$$530$$ −1.26672e6 −0.195880
$$531$$ 128310. 0.0197480
$$532$$ 326536. 0.0500210
$$533$$ 2.51468e6 0.383411
$$534$$ 7.08148e6 1.07466
$$535$$ −8.19414e6 −1.23771
$$536$$ −1.74254e7 −2.61982
$$537$$ −4.12905e6 −0.617894
$$538$$ −1.27756e7 −1.90294
$$539$$ 0 0
$$540$$ −1.54605e7 −2.28160
$$541$$ 6.71799e6 0.986839 0.493420 0.869791i $$-0.335747\pi$$
0.493420 + 0.869791i $$0.335747\pi$$
$$542$$ −1.65054e7 −2.41340
$$543$$ 4.71772e6 0.686646
$$544$$ 4.68384e6 0.678586
$$545$$ 5.20229e6 0.750245
$$546$$ −960400. −0.137870
$$547$$ 5.00235e6 0.714835 0.357418 0.933945i $$-0.383657\pi$$
0.357418 + 0.933945i $$0.383657\pi$$
$$548$$ −1.39034e7 −1.97774
$$549$$ 1.20546e6 0.170695
$$550$$ 0 0
$$551$$ −334964. −0.0470023
$$552$$ −9.19296e6 −1.28413
$$553$$ −1.55742e6 −0.216567
$$554$$ 1.06409e7 1.47300
$$555$$ −8.15203e6 −1.12340
$$556$$ 2.40761e6 0.330293
$$557$$ −9.01961e6 −1.23183 −0.615913 0.787814i $$-0.711213\pi$$
−0.615913 + 0.787814i $$0.711213\pi$$
$$558$$ 3.59268e6 0.488465
$$559$$ −1.52320e6 −0.206171
$$560$$ −3.90746e6 −0.526531
$$561$$ 0 0
$$562$$ 223420. 0.0298388
$$563$$ −1.24051e7 −1.64941 −0.824707 0.565561i $$-0.808660\pi$$
−0.824707 + 0.565561i $$0.808660\pi$$
$$564$$ −8.87645e6 −1.17501
$$565$$ 4.66782e6 0.615167
$$566$$ 2.49574e7 3.27460
$$567$$ −2.22553e6 −0.290721
$$568$$ −2.10816e7 −2.74178
$$569$$ −6.48804e6 −0.840103 −0.420052 0.907500i $$-0.637988\pi$$
−0.420052 + 0.907500i $$0.637988\pi$$
$$570$$ 768320. 0.0990501
$$571$$ 1.02285e7 1.31287 0.656435 0.754382i $$-0.272064\pi$$
0.656435 + 0.754382i $$0.272064\pi$$
$$572$$ 0 0
$$573$$ −5.01570e6 −0.638182
$$574$$ 8.80138e6 1.11499
$$575$$ 20064.0 0.00253074
$$576$$ 863296. 0.108419
$$577$$ 2.65338e6 0.331787 0.165894 0.986144i $$-0.446949\pi$$
0.165894 + 0.986144i $$0.446949\pi$$
$$578$$ 1.54543e7 1.92411
$$579$$ −1.38538e7 −1.71740
$$580$$ 1.30157e7 1.60657
$$581$$ 1.00636e6 0.123684
$$582$$ 8.19084e6 1.00235
$$583$$ 0 0
$$584$$ −2.45095e7 −2.97374
$$585$$ 368480. 0.0445168
$$586$$ 1.93178e7 2.32387
$$587$$ −1.43044e7 −1.71346 −0.856729 0.515766i $$-0.827507\pi$$
−0.856729 + 0.515766i $$0.827507\pi$$
$$588$$ −2.28575e6 −0.272638
$$589$$ −749112. −0.0889731
$$590$$ 1.52880e6 0.180809
$$591$$ −1.38607e7 −1.63236
$$592$$ −1.48068e7 −1.73642
$$593$$ 1.00265e7 1.17088 0.585442 0.810714i $$-0.300921\pi$$
0.585442 + 0.810714i $$0.300921\pi$$
$$594$$ 0 0
$$595$$ −4.72517e6 −0.547173
$$596$$ 1.37537e6 0.158600
$$597$$ 1.17706e7 1.35164
$$598$$ 2.55360e6 0.292011
$$599$$ −7.52292e6 −0.856681 −0.428341 0.903617i $$-0.640902\pi$$
−0.428341 + 0.903617i $$0.640902\pi$$
$$600$$ −55440.0 −0.00628702
$$601$$ −3.38625e6 −0.382413 −0.191207 0.981550i $$-0.561240\pi$$
−0.191207 + 0.981550i $$0.561240\pi$$
$$602$$ −5.33120e6 −0.599562
$$603$$ 2.27499e6 0.254792
$$604$$ −4.82147e6 −0.537759
$$605$$ 0 0
$$606$$ 5.41744e6 0.599256
$$607$$ 6.90861e6 0.761060 0.380530 0.924769i $$-0.375742\pi$$
0.380530 + 0.924769i $$0.375742\pi$$
$$608$$ 266560. 0.0292439
$$609$$ 2.34475e6 0.256185
$$610$$ 1.43629e7 1.56285
$$611$$ 1.30536e6 0.141458
$$612$$ −5.50351e6 −0.593966
$$613$$ 9.68896e6 1.04142 0.520710 0.853734i $$-0.325667\pi$$
0.520710 + 0.853734i $$0.325667\pi$$
$$614$$ 4.59074e6 0.491430
$$615$$ 1.40822e7 1.50135
$$616$$ 0 0
$$617$$ −7.84742e6 −0.829877 −0.414939 0.909849i $$-0.636197\pi$$
−0.414939 + 0.909849i $$0.636197\pi$$
$$618$$ −7.42840e6 −0.782391
$$619$$ −1.01972e7 −1.06968 −0.534840 0.844953i $$-0.679628\pi$$
−0.534840 + 0.844953i $$0.679628\pi$$
$$620$$ 2.91084e7 3.04115
$$621$$ 7.40544e6 0.770587
$$622$$ 6.67128e6 0.691406
$$623$$ −2.47852e6 −0.255842
$$624$$ −2.79104e6 −0.286949
$$625$$ −9.79988e6 −1.00351
$$626$$ −1.11034e6 −0.113245
$$627$$ 0 0
$$628$$ 1.99596e7 2.01954
$$629$$ −1.79054e7 −1.80450
$$630$$ 1.28968e6 0.129459
$$631$$ −8.36258e6 −0.836116 −0.418058 0.908420i $$-0.637289\pi$$
−0.418058 + 0.908420i $$0.637289\pi$$
$$632$$ −1.14422e7 −1.13951
$$633$$ 1.61102e7 1.59806
$$634$$ −687780. −0.0679558
$$635$$ 3.38150e6 0.332794
$$636$$ −2.15342e6 −0.211099
$$637$$ 336140. 0.0328225
$$638$$ 0 0
$$639$$ 2.75232e6 0.266653
$$640$$ 1.51603e7 1.46305
$$641$$ 1.10283e6 0.106014 0.0530070 0.998594i $$-0.483119\pi$$
0.0530070 + 0.998594i $$0.483119\pi$$
$$642$$ −2.04854e7 −1.96158
$$643$$ 1.71354e7 1.63443 0.817217 0.576330i $$-0.195516\pi$$
0.817217 + 0.576330i $$0.195516\pi$$
$$644$$ 6.07757e6 0.577451
$$645$$ −8.52992e6 −0.807320
$$646$$ 1.68756e6 0.159103
$$647$$ −54964.0 −0.00516200 −0.00258100 0.999997i $$-0.500822\pi$$
−0.00258100 + 0.999997i $$0.500822\pi$$
$$648$$ −1.63508e7 −1.52969
$$649$$ 0 0
$$650$$ 15400.0 0.00142968
$$651$$ 5.24378e6 0.484945
$$652$$ 897056. 0.0826420
$$653$$ −485166. −0.0445254 −0.0222627 0.999752i $$-0.507087\pi$$
−0.0222627 + 0.999752i $$0.507087\pi$$
$$654$$ 1.30057e7 1.18902
$$655$$ −3.44882e6 −0.314099
$$656$$ 2.55779e7 2.32063
$$657$$ 3.19985e6 0.289212
$$658$$ 4.56876e6 0.411371
$$659$$ 2.72136e6 0.244103 0.122051 0.992524i $$-0.461053\pi$$
0.122051 + 0.992524i $$0.461053\pi$$
$$660$$ 0 0
$$661$$ −2.14525e6 −0.190974 −0.0954869 0.995431i $$-0.530441\pi$$
−0.0954869 + 0.995431i $$0.530441\pi$$
$$662$$ −5.64448e6 −0.500586
$$663$$ −3.37512e6 −0.298198
$$664$$ 7.39368e6 0.650789
$$665$$ −268912. −0.0235807
$$666$$ 4.88706e6 0.426935
$$667$$ −6.23443e6 −0.542603
$$668$$ −3.35656e7 −2.91041
$$669$$ 1.15397e7 0.996848
$$670$$ 2.71062e7 2.33283
$$671$$ 0 0
$$672$$ −1.86592e6 −0.159393
$$673$$ −2.92796e6 −0.249188 −0.124594 0.992208i $$-0.539763\pi$$
−0.124594 + 0.992208i $$0.539763\pi$$
$$674$$ −2.07729e7 −1.76136
$$675$$ 44660.0 0.00377276
$$676$$ −2.39151e7 −2.01282
$$677$$ 1.34992e7 1.13198 0.565988 0.824414i $$-0.308495\pi$$
0.565988 + 0.824414i $$0.308495\pi$$
$$678$$ 1.16696e7 0.974945
$$679$$ −2.86679e6 −0.238628
$$680$$ −3.47155e7 −2.87906
$$681$$ 1.04135e6 0.0860455
$$682$$ 0 0
$$683$$ −5.42972e6 −0.445375 −0.222688 0.974890i $$-0.571483\pi$$
−0.222688 + 0.974890i $$0.571483\pi$$
$$684$$ −313208. −0.0255972
$$685$$ 1.14499e7 0.932340
$$686$$ 1.17649e6 0.0954504
$$687$$ −1.58474e7 −1.28105
$$688$$ −1.54931e7 −1.24787
$$689$$ 316680. 0.0254140
$$690$$ 1.43002e7 1.14345
$$691$$ 2.08280e7 1.65940 0.829702 0.558207i $$-0.188510\pi$$
0.829702 + 0.558207i $$0.188510\pi$$
$$692$$ −1.63687e7 −1.29942
$$693$$ 0 0
$$694$$ 532480. 0.0419667
$$695$$ −1.98274e6 −0.155705
$$696$$ 1.72267e7 1.34797
$$697$$ 3.09306e7 2.41160
$$698$$ 2.27200e7 1.76510
$$699$$ −2.78216e6 −0.215372
$$700$$ 36652.0 0.00282717
$$701$$ −2.35141e7 −1.80731 −0.903655 0.428261i $$-0.859126\pi$$
−0.903655 + 0.428261i $$0.859126\pi$$
$$702$$ 5.68400e6 0.435323
$$703$$ −1.01900e6 −0.0777656
$$704$$ 0 0
$$705$$ 7.31002e6 0.553918
$$706$$ 4.00645e7 3.02516
$$707$$ −1.89610e6 −0.142664
$$708$$ 2.59896e6 0.194857
$$709$$ −1.95747e7 −1.46244 −0.731221 0.682140i $$-0.761049\pi$$
−0.731221 + 0.682140i $$0.761049\pi$$
$$710$$ 3.27936e7 2.44142
$$711$$ 1.49385e6 0.110824
$$712$$ −1.82095e7 −1.34617
$$713$$ −1.39427e7 −1.02712
$$714$$ −1.18129e7 −0.867185
$$715$$ 0 0
$$716$$ 2.00554e7 1.46200
$$717$$ 6.76066e6 0.491124
$$718$$ −737840. −0.0534135
$$719$$ −2.61152e7 −1.88396 −0.941978 0.335674i $$-0.891036\pi$$
−0.941978 + 0.335674i $$0.891036\pi$$
$$720$$ 3.74797e6 0.269442
$$721$$ 2.59994e6 0.186262
$$722$$ −2.46650e7 −1.76091
$$723$$ 1.12827e7 0.802729
$$724$$ −2.29146e7 −1.62468
$$725$$ −37598.0 −0.00265656
$$726$$ 0 0
$$727$$ 1.54126e7 1.08154 0.540768 0.841172i $$-0.318134\pi$$
0.540768 + 0.841172i $$0.318134\pi$$
$$728$$ 2.46960e6 0.172702
$$729$$ 1.59468e7 1.11136
$$730$$ 3.81259e7 2.64797
$$731$$ −1.87354e7 −1.29679
$$732$$ 2.44169e7 1.68427
$$733$$ 1.69868e7 1.16776 0.583878 0.811841i $$-0.301535\pi$$
0.583878 + 0.811841i $$0.301535\pi$$
$$734$$ 1.40431e7 0.962107
$$735$$ 1.88238e6 0.128526
$$736$$ 4.96128e6 0.337597
$$737$$ 0 0
$$738$$ −8.44214e6 −0.570574
$$739$$ −2.01511e6 −0.135734 −0.0678669 0.997694i $$-0.521619\pi$$
−0.0678669 + 0.997694i $$0.521619\pi$$
$$740$$ 3.95956e7 2.65808
$$741$$ −192080. −0.0128510
$$742$$ 1.10838e6 0.0739059
$$743$$ 1.51381e7 1.00600 0.503001 0.864286i $$-0.332229\pi$$
0.503001 + 0.864286i $$0.332229\pi$$
$$744$$ 3.85258e7 2.55164
$$745$$ −1.13266e6 −0.0747666
$$746$$ 1.60323e7 1.05475
$$747$$ −965286. −0.0632928
$$748$$ 0 0
$$749$$ 7.16988e6 0.466989
$$750$$ −2.44138e7 −1.58483
$$751$$ 7.21401e6 0.466742 0.233371 0.972388i $$-0.425024\pi$$
0.233371 + 0.972388i $$0.425024\pi$$
$$752$$ 1.32774e7 0.856185
$$753$$ −6.03033e6 −0.387573
$$754$$ −4.78520e6 −0.306529
$$755$$ 3.97062e6 0.253508
$$756$$ 1.35279e7 0.860848
$$757$$ −1.09697e7 −0.695755 −0.347877 0.937540i $$-0.613097\pi$$
−0.347877 + 0.937540i $$0.613097\pi$$
$$758$$ −4.77012e7 −3.01548
$$759$$ 0 0
$$760$$ −1.97568e6 −0.124075
$$761$$ −1.92442e7 −1.20459 −0.602293 0.798275i $$-0.705746\pi$$
−0.602293 + 0.798275i $$0.705746\pi$$
$$762$$ 8.45376e6 0.527427
$$763$$ −4.55200e6 −0.283068
$$764$$ 2.43620e7 1.51001
$$765$$ 4.53230e6 0.280005
$$766$$ −2.23079e7 −1.37368
$$767$$ −382200. −0.0234586
$$768$$ 2.96719e7 1.81528
$$769$$ −8.21185e6 −0.500755 −0.250378 0.968148i $$-0.580555\pi$$
−0.250378 + 0.968148i $$0.580555\pi$$
$$770$$ 0 0
$$771$$ 1.64767e7 0.998241
$$772$$ 6.72897e7 4.06355
$$773$$ 1.86187e7 1.12073 0.560363 0.828247i $$-0.310662\pi$$
0.560363 + 0.828247i $$0.310662\pi$$
$$774$$ 5.11360e6 0.306813
$$775$$ −84084.0 −0.00502874
$$776$$ −2.10622e7 −1.25559
$$777$$ 7.13303e6 0.423859
$$778$$ 4.84024e7 2.86694
$$779$$ 1.76028e6 0.103929
$$780$$ 7.46368e6 0.439255
$$781$$ 0 0
$$782$$ 3.14093e7 1.83671
$$783$$ −1.38771e7 −0.808898
$$784$$ 3.41902e6 0.198661
$$785$$ −1.64373e7 −0.952045
$$786$$ −8.62204e6 −0.497799
$$787$$ −2.62501e7 −1.51075 −0.755377 0.655291i $$-0.772546\pi$$
−0.755377 + 0.655291i $$0.772546\pi$$
$$788$$ 6.73234e7 3.86234
$$789$$ 1.80737e7 1.03360
$$790$$ 1.77990e7 1.01468
$$791$$ −4.08435e6 −0.232103
$$792$$ 0 0
$$793$$ −3.59072e6 −0.202768
$$794$$ 9.95820e6 0.560570
$$795$$ 1.77341e6 0.0995155
$$796$$ −5.71714e7 −3.19813
$$797$$ −1.00373e7 −0.559720 −0.279860 0.960041i $$-0.590288\pi$$
−0.279860 + 0.960041i $$0.590288\pi$$
$$798$$ −672280. −0.0373717
$$799$$ 1.60559e7 0.889751
$$800$$ 29920.0 0.00165286
$$801$$ 2.37735e6 0.130922
$$802$$ −3.31605e7 −1.82048
$$803$$ 0 0
$$804$$ 4.60806e7 2.51407
$$805$$ −5.00506e6 −0.272220
$$806$$ −1.07016e7 −0.580245
$$807$$ 1.78858e7 0.966772
$$808$$ −1.39306e7 −0.750655
$$809$$ −1.40884e7 −0.756816 −0.378408 0.925639i $$-0.623528\pi$$
−0.378408 + 0.925639i $$0.623528\pi$$
$$810$$ 2.54346e7 1.36211
$$811$$ −1.81433e7 −0.968646 −0.484323 0.874889i $$-0.660934\pi$$
−0.484323 + 0.874889i $$0.660934\pi$$
$$812$$ −1.13888e7 −0.606160
$$813$$ 2.31076e7 1.22611
$$814$$ 0 0
$$815$$ −738752. −0.0389587
$$816$$ −3.43298e7 −1.80487
$$817$$ −1.06624e6 −0.0558856
$$818$$ −3.07273e7 −1.60562
$$819$$ −322420. −0.0167962
$$820$$ −6.83993e7 −3.55236
$$821$$ 2.13669e7 1.10633 0.553164 0.833072i $$-0.313420\pi$$
0.553164 + 0.833072i $$0.313420\pi$$
$$822$$ 2.86247e7 1.47761
$$823$$ 1.78017e7 0.916142 0.458071 0.888916i $$-0.348541\pi$$
0.458071 + 0.888916i $$0.348541\pi$$
$$824$$ 1.91016e7 0.980058
$$825$$ 0 0
$$826$$ −1.33770e6 −0.0682195
$$827$$ −1.62921e7 −0.828350 −0.414175 0.910197i $$-0.635930\pi$$
−0.414175 + 0.910197i $$0.635930\pi$$
$$828$$ −5.82950e6 −0.295499
$$829$$ −2.08499e6 −0.105370 −0.0526851 0.998611i $$-0.516778\pi$$
−0.0526851 + 0.998611i $$0.516778\pi$$
$$830$$ −1.15013e7 −0.579497
$$831$$ −1.48973e7 −0.748348
$$832$$ −2.57152e6 −0.128790
$$833$$ 4.13452e6 0.206449
$$834$$ −4.95684e6 −0.246769
$$835$$ 2.76423e7 1.37201
$$836$$ 0 0
$$837$$ −3.10346e7 −1.53120
$$838$$ 2.81438e7 1.38443
$$839$$ −2.27850e7 −1.11749 −0.558745 0.829340i $$-0.688717\pi$$
−0.558745 + 0.829340i $$0.688717\pi$$
$$840$$ 1.38298e7 0.676264
$$841$$ −8.82842e6 −0.430421
$$842$$ 3.05802e7 1.48648
$$843$$ −312788. −0.0151594
$$844$$ −7.82498e7 −3.78118
$$845$$ 1.96948e7 0.948877
$$846$$ −4.38228e6 −0.210511
$$847$$ 0 0
$$848$$ 3.22109e6 0.153820
$$849$$ −3.49403e7 −1.66363
$$850$$ 189420. 0.00899246
$$851$$ −1.89660e7 −0.897740
$$852$$ 5.57491e7 2.63111
$$853$$ 2.26975e7 1.06808 0.534042 0.845458i $$-0.320672\pi$$
0.534042 + 0.845458i $$0.320672\pi$$
$$854$$ −1.25675e7 −0.589664
$$855$$ 257936. 0.0120669
$$856$$ 5.26766e7 2.45716
$$857$$ −2.52900e7 −1.17624 −0.588120 0.808774i $$-0.700132\pi$$
−0.588120 + 0.808774i $$0.700132\pi$$
$$858$$ 0 0
$$859$$ −1.03947e7 −0.480652 −0.240326 0.970692i $$-0.577254\pi$$
−0.240326 + 0.970692i $$0.577254\pi$$
$$860$$ 4.14310e7 1.91020
$$861$$ −1.23219e7 −0.566462
$$862$$ −1.93750e7 −0.888122
$$863$$ 4.33399e7 1.98089 0.990447 0.137892i $$-0.0440327\pi$$
0.990447 + 0.137892i $$0.0440327\pi$$
$$864$$ 1.10432e7 0.503281
$$865$$ 1.34801e7 0.612566
$$866$$ 3.94790e7 1.78884
$$867$$ −2.16360e7 −0.977527
$$868$$ −2.54698e7 −1.14743
$$869$$ 0 0
$$870$$ −2.67971e7 −1.20030
$$871$$ −6.77656e6 −0.302666
$$872$$ −3.34433e7 −1.48942
$$873$$ 2.74978e6 0.122113
$$874$$ 1.78752e6 0.0791539
$$875$$ 8.54482e6 0.377296
$$876$$ 6.48141e7 2.85370
$$877$$ −3.71659e7 −1.63172 −0.815861 0.578248i $$-0.803736\pi$$
−0.815861 + 0.578248i $$0.803736\pi$$
$$878$$ 7.41770e7 3.24738
$$879$$ −2.70449e7 −1.18063
$$880$$ 0 0
$$881$$ 9.04785e6 0.392740 0.196370 0.980530i $$-0.437085\pi$$
0.196370 + 0.980530i $$0.437085\pi$$
$$882$$ −1.12847e6 −0.0488448
$$883$$ 3.29679e7 1.42295 0.711474 0.702712i $$-0.248028\pi$$
0.711474 + 0.702712i $$0.248028\pi$$
$$884$$ 1.63934e7 0.705569
$$885$$ −2.14032e6 −0.0918588
$$886$$ 1.40269e7 0.600313
$$887$$ 1.61099e7 0.687517 0.343758 0.939058i $$-0.388300\pi$$
0.343758 + 0.939058i $$0.388300\pi$$
$$888$$ 5.24059e7 2.23022
$$889$$ −2.95882e6 −0.125564
$$890$$ 2.83259e7 1.19870
$$891$$ 0 0
$$892$$ −5.60500e7 −2.35865
$$893$$ 913752. 0.0383442
$$894$$ −2.83164e6 −0.118493
$$895$$ −1.65162e7 −0.689211
$$896$$ −1.32653e7 −0.552009
$$897$$ −3.57504e6 −0.148354
$$898$$ −5.90574e6 −0.244390
$$899$$ 2.61272e7 1.07819
$$900$$ −35156.0 −0.00144675
$$901$$ 3.89516e6 0.159850
$$902$$ 0 0
$$903$$ 7.46368e6 0.304603
$$904$$ −3.00074e7 −1.22126
$$905$$ 1.88709e7 0.765898
$$906$$ 9.92656e6 0.401771
$$907$$ −4.47286e7 −1.80537 −0.902686 0.430300i $$-0.858408\pi$$
−0.902686 + 0.430300i $$0.858408\pi$$
$$908$$ −5.05798e6 −0.203593
$$909$$ 1.81871e6 0.0730053
$$910$$ −3.84160e6 −0.153783
$$911$$ −6.60518e6 −0.263687 −0.131844 0.991271i $$-0.542090\pi$$
−0.131844 + 0.991271i $$0.542090\pi$$
$$912$$ −1.95373e6 −0.0777816
$$913$$ 0 0
$$914$$ 2.90484e7 1.15016
$$915$$ −2.01080e7 −0.793993
$$916$$ 7.69730e7 3.03110
$$917$$ 3.01771e6 0.118510
$$918$$ 6.99132e7 2.73812
$$919$$ 3.08930e7 1.20662 0.603311 0.797506i $$-0.293848\pi$$
0.603311 + 0.797506i $$0.293848\pi$$
$$920$$ −3.67718e7 −1.43234
$$921$$ −6.42704e6 −0.249667
$$922$$ 9.22684e6 0.357459
$$923$$ −8.19840e6 −0.316756
$$924$$ 0 0
$$925$$ −114378. −0.00439530
$$926$$ 7.18235e7 2.75258
$$927$$ −2.49382e6 −0.0953160
$$928$$ −9.29696e6 −0.354381
$$929$$ −4.87215e6 −0.185217 −0.0926087 0.995703i $$-0.529521\pi$$
−0.0926087 + 0.995703i $$0.529521\pi$$
$$930$$ −5.99290e7 −2.27211
$$931$$ 235298. 0.00889701
$$932$$ 1.35134e7 0.509593
$$933$$ −9.33979e6 −0.351264
$$934$$ −6.12570e6 −0.229767
$$935$$ 0 0
$$936$$ −2.36880e6 −0.0883769
$$937$$ 3.25004e7 1.20932 0.604658 0.796485i $$-0.293310\pi$$
0.604658 + 0.796485i $$0.293310\pi$$
$$938$$ −2.37180e7 −0.880177
$$939$$ 1.55448e6 0.0575334
$$940$$ −3.55058e7 −1.31063
$$941$$ 2.64040e6 0.0972066 0.0486033 0.998818i $$-0.484523\pi$$
0.0486033 + 0.998818i $$0.484523\pi$$
$$942$$ −4.10934e7 −1.50884
$$943$$ 3.27627e7 1.19978
$$944$$ −3.88752e6 −0.141985
$$945$$ −1.11406e7 −0.405817
$$946$$ 0 0
$$947$$ −4.08179e7 −1.47903 −0.739513 0.673142i $$-0.764944\pi$$
−0.739513 + 0.673142i $$0.764944\pi$$
$$948$$ 3.02584e7 1.09351
$$949$$ −9.53148e6 −0.343554
$$950$$ 10780.0 0.000387534 0
$$951$$ 962892. 0.0345244
$$952$$ 3.03761e7 1.08627
$$953$$ 6.71983e6 0.239677 0.119838 0.992793i $$-0.461762\pi$$
0.119838 + 0.992793i $$0.461762\pi$$
$$954$$ −1.06314e6 −0.0378198
$$955$$ −2.00628e7 −0.711841
$$956$$ −3.28375e7 −1.16205
$$957$$ 0 0
$$958$$ −2.60330e7 −0.916454
$$959$$ −1.00186e7 −0.351773
$$960$$ −1.44005e7 −0.504313
$$961$$ 2.98016e7 1.04095
$$962$$ −1.45572e7 −0.507154
$$963$$ −6.87723e6 −0.238972
$$964$$ −5.48019e7 −1.89934
$$965$$ −5.54150e7 −1.91562
$$966$$ −1.25126e7 −0.431426
$$967$$ 2.78979e6 0.0959413 0.0479707 0.998849i $$-0.484725\pi$$
0.0479707 + 0.998849i $$0.484725\pi$$
$$968$$ 0 0
$$969$$ −2.36258e6 −0.0808310
$$970$$ 3.27634e7 1.11804
$$971$$ 3.33594e7 1.13545 0.567727 0.823217i $$-0.307823\pi$$
0.567727 + 0.823217i $$0.307823\pi$$
$$972$$ −2.38486e7 −0.809648
$$973$$ 1.73489e6 0.0587477
$$974$$ 5.46309e7 1.84519
$$975$$ −21560.0 −0.000726335 0
$$976$$ −3.65228e7 −1.22727
$$977$$ −7.60033e6 −0.254739 −0.127370 0.991855i $$-0.540653\pi$$
−0.127370 + 0.991855i $$0.540653\pi$$
$$978$$ −1.84688e6 −0.0617435
$$979$$ 0 0
$$980$$ −9.14301e6 −0.304105
$$981$$ 4.36621e6 0.144854
$$982$$ −1.64090e7 −0.543004
$$983$$ −5.79760e6 −0.191366 −0.0956829 0.995412i $$-0.530503\pi$$
−0.0956829 + 0.995412i $$0.530503\pi$$
$$984$$ −9.05285e7 −2.98056
$$985$$ −5.54428e7 −1.82077
$$986$$ −5.88580e7 −1.92803
$$987$$ −6.39626e6 −0.208994
$$988$$ 932960. 0.0304068
$$989$$ −1.98451e7 −0.645153
$$990$$ 0 0
$$991$$ 1.26825e7 0.410224 0.205112 0.978739i $$-0.434244\pi$$
0.205112 + 0.978739i $$0.434244\pi$$
$$992$$ −2.07917e7 −0.670827
$$993$$ 7.90227e6 0.254319
$$994$$ −2.86944e7 −0.921152
$$995$$ 4.70823e7 1.50765
$$996$$ −1.95522e7 −0.624521
$$997$$ −1.44400e7 −0.460077 −0.230039 0.973182i $$-0.573885\pi$$
−0.230039 + 0.973182i $$0.573885\pi$$
$$998$$ 2.99796e7 0.952796
$$999$$ −4.22159e7 −1.33833
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 847.6.a.b.1.1 1
11.10 odd 2 7.6.a.a.1.1 1
33.32 even 2 63.6.a.e.1.1 1
44.43 even 2 112.6.a.g.1.1 1
55.32 even 4 175.6.b.a.99.1 2
55.43 even 4 175.6.b.a.99.2 2
55.54 odd 2 175.6.a.b.1.1 1
77.10 even 6 49.6.c.b.30.1 2
77.32 odd 6 49.6.c.c.30.1 2
77.54 even 6 49.6.c.b.18.1 2
77.65 odd 6 49.6.c.c.18.1 2
77.76 even 2 49.6.a.a.1.1 1
88.21 odd 2 448.6.a.m.1.1 1
88.43 even 2 448.6.a.c.1.1 1
132.131 odd 2 1008.6.a.y.1.1 1
231.230 odd 2 441.6.a.k.1.1 1
308.307 odd 2 784.6.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 1 11.10 odd 2
49.6.a.a.1.1 1 77.76 even 2
49.6.c.b.18.1 2 77.54 even 6
49.6.c.b.30.1 2 77.10 even 6
49.6.c.c.18.1 2 77.65 odd 6
49.6.c.c.30.1 2 77.32 odd 6
63.6.a.e.1.1 1 33.32 even 2
112.6.a.g.1.1 1 44.43 even 2
175.6.a.b.1.1 1 55.54 odd 2
175.6.b.a.99.1 2 55.32 even 4
175.6.b.a.99.2 2 55.43 even 4
441.6.a.k.1.1 1 231.230 odd 2
448.6.a.c.1.1 1 88.43 even 2
448.6.a.m.1.1 1 88.21 odd 2
784.6.a.c.1.1 1 308.307 odd 2
847.6.a.b.1.1 1 1.1 even 1 trivial
1008.6.a.y.1.1 1 132.131 odd 2