# Properties

 Label 49.6.a.a.1.1 Level $49$ Weight $6$ Character 49.1 Self dual yes Analytic conductor $7.859$ 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] = [49,6,Mod(1,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("49.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 49.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: $$7.85880717084$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 49.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-10.0000 q^{2} +14.0000 q^{3} +68.0000 q^{4} +56.0000 q^{5} -140.000 q^{6} -360.000 q^{8} -47.0000 q^{9} +O(q^{10})$$ $$q-10.0000 q^{2} +14.0000 q^{3} +68.0000 q^{4} +56.0000 q^{5} -140.000 q^{6} -360.000 q^{8} -47.0000 q^{9} -560.000 q^{10} +232.000 q^{11} +952.000 q^{12} +140.000 q^{13} +784.000 q^{15} +1424.00 q^{16} +1722.00 q^{17} +470.000 q^{18} +98.0000 q^{19} +3808.00 q^{20} -2320.00 q^{22} +1824.00 q^{23} -5040.00 q^{24} +11.0000 q^{25} -1400.00 q^{26} -4060.00 q^{27} +3418.00 q^{29} -7840.00 q^{30} +7644.00 q^{31} -2720.00 q^{32} +3248.00 q^{33} -17220.0 q^{34} -3196.00 q^{36} -10398.0 q^{37} -980.000 q^{38} +1960.00 q^{39} -20160.0 q^{40} +17962.0 q^{41} +10880.0 q^{43} +15776.0 q^{44} -2632.00 q^{45} -18240.0 q^{46} -9324.00 q^{47} +19936.0 q^{48} -110.000 q^{50} +24108.0 q^{51} +9520.00 q^{52} +2262.00 q^{53} +40600.0 q^{54} +12992.0 q^{55} +1372.00 q^{57} -34180.0 q^{58} +2730.00 q^{59} +53312.0 q^{60} -25648.0 q^{61} -76440.0 q^{62} -18368.0 q^{64} +7840.00 q^{65} -32480.0 q^{66} -48404.0 q^{67} +117096. q^{68} +25536.0 q^{69} -58560.0 q^{71} +16920.0 q^{72} -68082.0 q^{73} +103980. q^{74} +154.000 q^{75} +6664.00 q^{76} -19600.0 q^{78} +31784.0 q^{79} +79744.0 q^{80} -45419.0 q^{81} -179620. q^{82} +20538.0 q^{83} +96432.0 q^{85} -108800. q^{86} +47852.0 q^{87} -83520.0 q^{88} +50582.0 q^{89} +26320.0 q^{90} +124032. q^{92} +107016. q^{93} +93240.0 q^{94} +5488.00 q^{95} -38080.0 q^{96} +58506.0 q^{97} -10904.0 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −10.0000 −1.76777 −0.883883 0.467707i $$-0.845080\pi$$
−0.883883 + 0.467707i $$0.845080\pi$$
$$3$$ 14.0000 0.898100 0.449050 0.893507i $$-0.351762\pi$$
0.449050 + 0.893507i $$0.351762\pi$$
$$4$$ 68.0000 2.12500
$$5$$ 56.0000 1.00176 0.500879 0.865517i $$-0.333010\pi$$
0.500879 + 0.865517i $$0.333010\pi$$
$$6$$ −140.000 −1.58763
$$7$$ 0 0
$$8$$ −360.000 −1.98874
$$9$$ −47.0000 −0.193416
$$10$$ −560.000 −1.77088
$$11$$ 232.000 0.578104 0.289052 0.957313i $$-0.406660\pi$$
0.289052 + 0.957313i $$0.406660\pi$$
$$12$$ 952.000 1.90846
$$13$$ 140.000 0.229757 0.114879 0.993380i $$-0.463352\pi$$
0.114879 + 0.993380i $$0.463352\pi$$
$$14$$ 0 0
$$15$$ 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
$$21$$ 0 0
$$22$$ −2320.00 −1.02195
$$23$$ 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
$$28$$ 0 0
$$29$$ 3418.00 0.754705 0.377352 0.926070i $$-0.376835\pi$$
0.377352 + 0.926070i $$0.376835\pi$$
$$30$$ −7840.00 −1.59042
$$31$$ 7644.00 1.42862 0.714310 0.699830i $$-0.246741\pi$$
0.714310 + 0.699830i $$0.246741\pi$$
$$32$$ −2720.00 −0.469563
$$33$$ 3248.00 0.519196
$$34$$ −17220.0 −2.55468
$$35$$ 0 0
$$36$$ −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 $$0.185825\pi$$
$$42$$ 0 0
$$43$$ 10880.0 0.897342 0.448671 0.893697i $$-0.351898\pi$$
0.448671 + 0.893697i $$0.351898\pi$$
$$44$$ 15776.0 1.22847
$$45$$ −2632.00 −0.193756
$$46$$ −18240.0 −1.27096
$$47$$ −9324.00 −0.615684 −0.307842 0.951438i $$-0.599607\pi$$
−0.307842 + 0.951438i $$0.599607\pi$$
$$48$$ 19936.0 1.24892
$$49$$ 0 0
$$50$$ −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
$$55$$ 12992.0 0.579121
$$56$$ 0 0
$$57$$ 1372.00 0.0559329
$$58$$ −34180.0 −1.33414
$$59$$ 2730.00 0.102102 0.0510508 0.998696i $$-0.483743\pi$$
0.0510508 + 0.998696i $$0.483743\pi$$
$$60$$ 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
$$63$$ 0 0
$$64$$ −18368.0 −0.560547
$$65$$ 7840.00 0.230161
$$66$$ −32480.0 −0.917817
$$67$$ −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
$$70$$ 0 0
$$71$$ −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 $$-0.407511\pi$$
0.286491 + 0.958083i $$0.407511\pi$$
$$80$$ 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 $$0.447683\pi$$
$$84$$ 0 0
$$85$$ 96432.0 1.44768
$$86$$ −108800. −1.58629
$$87$$ 47852.0 0.677801
$$88$$ −83520.0 −1.14970
$$89$$ 50582.0 0.676894 0.338447 0.940985i $$-0.390098\pi$$
0.338447 + 0.940985i $$0.390098\pi$$
$$90$$ 26320.0 0.342515
$$91$$ 0 0
$$92$$ 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 $$-0.397769\pi$$
0.315676 + 0.948867i $$0.397769\pi$$
$$98$$ 0 0
$$99$$ −10904.0 −0.111814
$$100$$ 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 $$-0.579248\pi$$
−0.246402 + 0.969168i $$0.579248\pi$$
$$104$$ −50400.0 −0.456927
$$105$$ 0 0
$$106$$ −22620.0 −0.195537
$$107$$ −146324. −1.23554 −0.617769 0.786360i $$-0.711963\pi$$
−0.617769 + 0.786360i $$0.711963\pi$$
$$108$$ −276080. −2.27759
$$109$$ 92898.0 0.748928 0.374464 0.927241i $$-0.377827\pi$$
0.374464 + 0.927241i $$0.377827\pi$$
$$110$$ −129920. −1.02375
$$111$$ −145572. −1.12143
$$112$$ 0 0
$$113$$ −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
$$119$$ 0 0
$$120$$ −282240. −1.78923
$$121$$ −107227. −0.665795
$$122$$ 256480. 1.56011
$$123$$ 251468. 1.49872
$$124$$ 519792. 3.03582
$$125$$ −174384. −0.998232
$$126$$ 0 0
$$127$$ 60384.0 0.332210 0.166105 0.986108i $$-0.446881\pi$$
0.166105 + 0.986108i $$0.446881\pi$$
$$128$$ 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 $$0.449891\pi$$
$$132$$ 220864. 1.10329
$$133$$ 0 0
$$134$$ 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 $$0.475237\pi$$
$$140$$ 0 0
$$141$$ −130536. −0.552946
$$142$$ 585600. 2.43714
$$143$$ 32480.0 0.132824
$$144$$ −66928.0 −0.268969
$$145$$ 191408. 0.756032
$$146$$ 680820. 2.64332
$$147$$ 0 0
$$148$$ −707064. −2.65341
$$149$$ −20226.0 −0.0746353 −0.0373177 0.999303i $$-0.511881\pi$$
−0.0373177 + 0.999303i $$0.511881\pi$$
$$150$$ −1540.00 −0.00558847
$$151$$ 70904.0 0.253063 0.126531 0.991963i $$-0.459616\pi$$
0.126531 + 0.991963i $$0.459616\pi$$
$$152$$ −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 $$-0.657620\pi$$
−0.475187 + 0.879885i $$0.657620\pi$$
$$158$$ −317840. −1.01290
$$159$$ 31668.0 0.0993408
$$160$$ −152320. −0.470389
$$161$$ 0 0
$$162$$ 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
$$165$$ 181888. 0.520109
$$166$$ −205380. −0.578479
$$167$$ −493612. −1.36960 −0.684801 0.728730i $$-0.740111\pi$$
−0.684801 + 0.728730i $$0.740111\pi$$
$$168$$ 0 0
$$169$$ −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
$$175$$ 0 0
$$176$$ 330368. 0.803926
$$177$$ 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 $$-0.375140\pi$$
0.382277 + 0.924048i $$0.375140\pi$$
$$182$$ 0 0
$$183$$ −359072. −0.792600
$$184$$ −656640. −1.42982
$$185$$ −582288. −1.25086
$$186$$ −1.07016e6 −2.26812
$$187$$ 399504. 0.835444
$$188$$ −634032. −1.30833
$$189$$ 0 0
$$190$$ −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 $$-0.905364\pi$$
−0.956128 + 0.292948i $$0.905364\pi$$
$$194$$ −585060. −1.11608
$$195$$ 109760. 0.206708
$$196$$ 0 0
$$197$$ −990050. −1.81757 −0.908786 0.417263i $$-0.862989\pi$$
−0.908786 + 0.417263i $$0.862989\pi$$
$$198$$ 109040. 0.197662
$$199$$ 840756. 1.50500 0.752501 0.658591i $$-0.228847\pi$$
0.752501 + 0.658591i $$0.228847\pi$$
$$200$$ −3960.00 −0.00700036
$$201$$ −677656. −1.18309
$$202$$ 386960. 0.667249
$$203$$ 0 0
$$204$$ 1.63934e6 2.75800
$$205$$ 1.00587e6 1.67170
$$206$$ 530600. 0.871163
$$207$$ −85728.0 −0.139058
$$208$$ 199360. 0.319506
$$209$$ 22736.0 0.0360038
$$210$$ 0 0
$$211$$ 1.15073e6 1.77938 0.889689 0.456568i $$-0.150921\pi$$
0.889689 + 0.456568i $$0.150921\pi$$
$$212$$ 153816. 0.235051
$$213$$ −819840. −1.23817
$$214$$ 1.46324e6 2.18414
$$215$$ 609280. 0.898919
$$216$$ 1.46160e6 2.13154
$$217$$ 0 0
$$218$$ −928980. −1.32393
$$219$$ −953148. −1.34292
$$220$$ 883456. 1.23063
$$221$$ 241080. 0.332032
$$222$$ 1.45572e6 1.98242
$$223$$ 824264. 1.10995 0.554976 0.831866i $$-0.312727\pi$$
0.554976 + 0.831866i $$0.312727\pi$$
$$224$$ 0 0
$$225$$ −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 $$-0.752755\pi$$
−0.713199 + 0.700961i $$0.752755\pi$$
$$230$$ −1.02144e6 −1.27319
$$231$$ 0 0
$$232$$ −1.23048e6 −1.50091
$$233$$ −198726. −0.239809 −0.119904 0.992785i $$-0.538259\pi$$
−0.119904 + 0.992785i $$0.538259\pi$$
$$234$$ 65800.0 0.0785572
$$235$$ −522144. −0.616766
$$236$$ 185640. 0.216966
$$237$$ 444976. 0.514595
$$238$$ 0 0
$$239$$ 482904. 0.546847 0.273424 0.961894i $$-0.411844\pi$$
0.273424 + 0.961894i $$0.411844\pi$$
$$240$$ 1.11642e6 1.25112
$$241$$ −805910. −0.893807 −0.446904 0.894582i $$-0.647473\pi$$
−0.446904 + 0.894582i $$0.647473\pi$$
$$242$$ 1.07227e6 1.17697
$$243$$ 350714. 0.381011
$$244$$ −1.74406e6 −1.87537
$$245$$ 0 0
$$246$$ −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 $$-0.569227\pi$$
−0.215774 + 0.976443i $$0.569227\pi$$
$$252$$ 0 0
$$253$$ 423168. 0.415634
$$254$$ −603840. −0.587270
$$255$$ 1.35005e6 1.30017
$$256$$ −2.11942e6 −2.02124
$$257$$ 1.17691e6 1.11150 0.555751 0.831349i $$-0.312431\pi$$
0.555751 + 0.831349i $$0.312431\pi$$
$$258$$ −1.52320e6 −1.42465
$$259$$ 0 0
$$260$$ 533120. 0.489093
$$261$$ −160646. −0.145972
$$262$$ −615860. −0.554279
$$263$$ 1.29098e6 1.15088 0.575438 0.817845i $$-0.304831\pi$$
0.575438 + 0.817845i $$0.304831\pi$$
$$264$$ −1.16928e6 −1.03254
$$265$$ 126672. 0.110807
$$266$$ 0 0
$$267$$ 708148. 0.607919
$$268$$ −3.29147e6 −2.79932
$$269$$ 1.27756e6 1.07646 0.538232 0.842797i $$-0.319093\pi$$
0.538232 + 0.842797i $$0.319093\pi$$
$$270$$ 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
$$273$$ 0 0
$$274$$ 2.04462e6 1.64527
$$275$$ 2552.00 0.00203493
$$276$$ 1.73645e6 1.37211
$$277$$ −1.06409e6 −0.833257 −0.416628 0.909077i $$-0.636788\pi$$
−0.416628 + 0.909077i $$0.636788\pi$$
$$278$$ −354060. −0.274767
$$279$$ −359268. −0.276317
$$280$$ 0 0
$$281$$ −22342.0 −0.0168794 −0.00843969 0.999964i $$-0.502686\pi$$
−0.00843969 + 0.999964i $$0.502686\pi$$
$$282$$ 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
$$286$$ −324800. −0.234802
$$287$$ 0 0
$$288$$ 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 $$0.271702\pi$$
$$294$$ 0 0
$$295$$ 152880. 0.102281
$$296$$ 3.74328e6 2.48326
$$297$$ −941920. −0.619616
$$298$$ 202260. 0.131938
$$299$$ 255360. 0.165187
$$300$$ 10472.0 0.00671779
$$301$$ 0 0
$$302$$ −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 $$-0.562652\pi$$
−0.195559 + 0.980692i $$0.562652\pi$$
$$312$$ −705600. −0.410367
$$313$$ 111034. 0.0640612 0.0320306 0.999487i $$-0.489803\pi$$
0.0320306 + 0.999487i $$0.489803\pi$$
$$314$$ 2.93524e6 1.68004
$$315$$ 0 0
$$316$$ 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
$$319$$ 792976. 0.436298
$$320$$ −1.02861e6 −0.561533
$$321$$ −2.04854e6 −1.10964
$$322$$ 0 0
$$323$$ 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
$$329$$ 0 0
$$330$$ −1.81888e6 −0.919431
$$331$$ −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
$$336$$ 0 0
$$337$$ 2.07729e6 0.996376 0.498188 0.867069i $$-0.333999\pi$$
0.498188 + 0.867069i $$0.333999\pi$$
$$338$$ 3.51693e6 1.67445
$$339$$ −1.16696e6 −0.551512
$$340$$ 6.55738e6 3.07633
$$341$$ 1.77341e6 0.825891
$$342$$ 46060.0 0.0212941
$$343$$ 0 0
$$344$$ −3.91680e6 −1.78458
$$345$$ 1.43002e6 0.646834
$$346$$ 2.40716e6 1.08097
$$347$$ −53248.0 −0.0237399 −0.0118700 0.999930i $$-0.503778\pi$$
−0.0118700 + 0.999930i $$0.503778\pi$$
$$348$$ 3.25394e6 1.44033
$$349$$ 2.27200e6 0.998494 0.499247 0.866460i $$-0.333610\pi$$
0.499247 + 0.866460i $$0.333610\pi$$
$$350$$ 0 0
$$351$$ −568400. −0.246256
$$352$$ −631040. −0.271456
$$353$$ −4.00645e6 −1.71129 −0.855644 0.517565i $$-0.826838\pi$$
−0.855644 + 0.517565i $$0.826838\pi$$
$$354$$ −382200. −0.162100
$$355$$ −3.27936e6 −1.38108
$$356$$ 3.43958e6 1.43840
$$357$$ 0 0
$$358$$ −2.94932e6 −1.21623
$$359$$ 73784.0 0.0302152 0.0151076 0.999886i $$-0.495191\pi$$
0.0151076 + 0.999886i $$0.495191\pi$$
$$360$$ 947520. 0.385329
$$361$$ −2.46650e6 −0.996121
$$362$$ −3.36980e6 −1.35155
$$363$$ −1.50118e6 −0.597951
$$364$$ 0 0
$$365$$ −3.81259e6 −1.49792
$$366$$ 3.59072e6 1.40113
$$367$$ −1.40431e6 −0.544250 −0.272125 0.962262i $$-0.587726\pi$$
−0.272125 + 0.962262i $$0.587726\pi$$
$$368$$ 2.59738e6 0.999805
$$369$$ −844214. −0.322765
$$370$$ 5.82288e6 2.21123
$$371$$ 0 0
$$372$$ 7.27709e6 2.72647
$$373$$ −1.60323e6 −0.596657 −0.298329 0.954463i $$-0.596429\pi$$
−0.298329 + 0.954463i $$0.596429\pi$$
$$374$$ −3.99504e6 −1.47687
$$375$$ −2.44138e6 −0.896513
$$376$$ 3.35664e6 1.22443
$$377$$ 478520. 0.173399
$$378$$ 0 0
$$379$$ −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 $$-0.372981\pi$$
0.388536 + 0.921434i $$0.372981\pi$$
$$384$$ 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
$$392$$ 0 0
$$393$$ 862204. 0.281597
$$394$$ 9.90050e6 3.21304
$$395$$ 1.77990e6 0.573989
$$396$$ −741472. −0.237606
$$397$$ −995820. −0.317106 −0.158553 0.987350i $$-0.550683\pi$$
−0.158553 + 0.987350i $$0.550683\pi$$
$$398$$ −8.40756e6 −2.66049
$$399$$ 0 0
$$400$$ 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
$$406$$ 0 0
$$407$$ −2.41234e6 −0.721858
$$408$$ −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
$$413$$ 0 0
$$414$$ 857280. 0.245823
$$415$$ 1.15013e6 0.327813
$$416$$ −380800. −0.107886
$$417$$ 495684. 0.139593
$$418$$ −227360. −0.0636463
$$419$$ −2.81438e6 −0.783154 −0.391577 0.920145i $$-0.628070\pi$$
−0.391577 + 0.920145i $$0.628070\pi$$
$$420$$ 0 0
$$421$$ 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
$$427$$ 0 0
$$428$$ −9.95003e6 −2.62552
$$429$$ 454720. 0.119289
$$430$$ −6.09280e6 −1.58908
$$431$$ 1.93750e6 0.502398 0.251199 0.967936i $$-0.419175\pi$$
0.251199 + 0.967936i $$0.419175\pi$$
$$432$$ −5.78144e6 −1.49048
$$433$$ −3.94790e6 −1.01192 −0.505961 0.862557i $$-0.668862\pi$$
−0.505961 + 0.862557i $$0.668862\pi$$
$$434$$ 0 0
$$435$$ 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 $$0.129403\pi$$
$$440$$ −4.67712e6 −1.15172
$$441$$ 0 0
$$442$$ −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
$$448$$ 0 0
$$449$$ −590574. −0.138248 −0.0691239 0.997608i $$-0.522020\pi$$
−0.0691239 + 0.997608i $$0.522020\pi$$
$$450$$ 5170.00 0.00120354
$$451$$ 4.16718e6 0.964720
$$452$$ −5.66807e6 −1.30494
$$453$$ 992656. 0.227276
$$454$$ 743820. 0.169367
$$455$$ 0 0
$$456$$ −493920. −0.111236
$$457$$ −2.90484e6 −0.650627 −0.325313 0.945606i $$-0.605470\pi$$
−0.325313 + 0.945606i $$0.605470\pi$$
$$458$$ 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 $$-0.479299\pi$$
0.0649881 + 0.997886i $$0.479299\pi$$
$$468$$ −447440. −0.0944322
$$469$$ 0 0
$$470$$ 5.22144e6 1.09030
$$471$$ −4.10934e6 −0.853531
$$472$$ −982800. −0.203053
$$473$$ 2.52416e6 0.518757
$$474$$ −4.44976e6 −0.909684
$$475$$ 1078.00 0.000219222 0
$$476$$ 0 0
$$477$$ −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
$$483$$ 0 0
$$484$$ −7.29144e6 −1.41482
$$485$$ 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
$$490$$ 0 0
$$491$$ 1.64090e6 0.307170 0.153585 0.988135i $$-0.450918\pi$$
0.153585 + 0.988135i $$0.450918\pi$$
$$492$$ 1.70998e7 3.18478
$$493$$ 5.88580e6 1.09066
$$494$$ −137200. −0.0252951
$$495$$ −610624. −0.112011
$$496$$ 1.08851e7 1.98667
$$497$$ 0 0
$$498$$ −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 $$0.292185\pi$$
$$504$$ 0 0
$$505$$ −2.16698e6 −0.378117
$$506$$ −4.23168e6 −0.734745
$$507$$ −4.92370e6 −0.850691
$$508$$ 4.10611e6 0.705946
$$509$$ −2.30476e6 −0.394305 −0.197152 0.980373i $$-0.563169\pi$$
−0.197152 + 0.980373i $$0.563169\pi$$
$$510$$ −1.35005e7 −2.29839
$$511$$ 0 0
$$512$$ 1.25312e7 2.11260
$$513$$ −397880. −0.0667511
$$514$$ −1.17691e7 −1.96488
$$515$$ −2.97136e6 −0.493671
$$516$$ 1.03578e7 1.71254
$$517$$ −2.16317e6 −0.355929
$$518$$ 0 0
$$519$$ −3.37002e6 −0.549180
$$520$$ −2.82240e6 −0.457731
$$521$$ 1.20960e7 1.95231 0.976155 0.217073i $$-0.0696509\pi$$
0.976155 + 0.217073i $$0.0696509\pi$$
$$522$$ 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
$$525$$ 0 0
$$526$$ −1.29098e7 −2.03448
$$527$$ 1.31630e7 2.06456
$$528$$ 4.62515e6 0.722007
$$529$$ −3.10937e6 −0.483095
$$530$$ −1.26672e6 −0.195880
$$531$$ −128310. −0.0197480
$$532$$ 0 0
$$533$$ 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 $$-0.664253\pi$$
−0.493420 + 0.869791i $$0.664253\pi$$
$$542$$ 1.65054e7 2.41340
$$543$$ 4.71772e6 0.686646
$$544$$ −4.68384e6 −0.678586
$$545$$ 5.20229e6 0.750245
$$546$$ 0 0
$$547$$ −5.00235e6 −0.714835 −0.357418 0.933945i $$-0.616343\pi$$
−0.357418 + 0.933945i $$0.616343\pi$$
$$548$$ −1.39034e7 −1.97774
$$549$$ 1.20546e6 0.170695
$$550$$ −25520.0 −0.00359728
$$551$$ 334964. 0.0470023
$$552$$ −9.19296e6 −1.28413
$$553$$ 0 0
$$554$$ 1.06409e7 1.47300
$$555$$ −8.15203e6 −1.12340
$$556$$ 2.40761e6 0.330293
$$557$$ 9.01961e6 1.23183 0.615913 0.787814i $$-0.288787\pi$$
0.615913 + 0.787814i $$0.288787\pi$$
$$558$$ 3.59268e6 0.488465
$$559$$ 1.52320e6 0.206171
$$560$$ 0 0
$$561$$ 5.59306e6 0.750312
$$562$$ 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
$$567$$ 0 0
$$568$$ 2.10816e7 2.74178
$$569$$ 6.48804e6 0.840103 0.420052 0.907500i $$-0.362012\pi$$
0.420052 + 0.907500i $$0.362012\pi$$
$$570$$ −768320. −0.0990501
$$571$$ −1.02285e7 −1.31287 −0.656435 0.754382i $$-0.727936\pi$$
−0.656435 + 0.754382i $$0.727936\pi$$
$$572$$ 2.20864e6 0.282251
$$573$$ 5.01570e6 0.638182
$$574$$ 0 0
$$575$$ 20064.0 0.00253074
$$576$$ 863296. 0.108419
$$577$$ −2.65338e6 −0.331787 −0.165894 0.986144i $$-0.553051\pi$$
−0.165894 + 0.986144i $$0.553051\pi$$
$$578$$ −1.54543e7 −1.92411
$$579$$ −1.38538e7 −1.71740
$$580$$ 1.30157e7 1.60657
$$581$$ 0 0
$$582$$ −8.19084e6 −1.00235
$$583$$ 524784. 0.0639454
$$584$$ 2.45095e7 2.97374
$$585$$ −368480. −0.0445168
$$586$$ −1.93178e7 −2.32387
$$587$$ 1.43044e7 1.71346 0.856729 0.515766i $$-0.172493\pi$$
0.856729 + 0.515766i $$0.172493\pi$$
$$588$$ 0 0
$$589$$ 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 $$0.300921\pi$$
$$594$$ 9.41920e6 1.09534
$$595$$ 0 0
$$596$$ −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 $$0.561240\pi$$
$$602$$ 0 0
$$603$$ 2.27499e6 0.254792
$$604$$ 4.82147e6 0.537759
$$605$$ −6.00471e6 −0.666966
$$606$$ 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
$$609$$ 0 0
$$610$$ 1.43629e7 1.56285
$$611$$ −1.30536e6 −0.141458
$$612$$ −5.50351e6 −0.593966
$$613$$ −9.68896e6 −1.04142 −0.520710 0.853734i $$-0.674333\pi$$
−0.520710 + 0.853734i $$0.674333\pi$$
$$614$$ −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 $$-0.320372\pi$$
0.534840 + 0.844953i $$0.320372\pi$$
$$620$$ 2.91084e7 3.04115
$$621$$ −7.40544e6 −0.770587
$$622$$ 6.67128e6 0.691406
$$623$$ 0 0
$$624$$ 2.79104e6 0.286949
$$625$$ −9.79988e6 −1.00351
$$626$$ −1.11034e6 −0.113245
$$627$$ 318304. 0.0323350
$$628$$ −1.99596e7 −2.01954
$$629$$ −1.79054e7 −1.80450
$$630$$ 0 0
$$631$$ −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
$$637$$ 0 0
$$638$$ −7.92976e6 −0.771273
$$639$$ 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 $$-0.804484\pi$$
−0.817217 + 0.576330i $$0.804484\pi$$
$$644$$ 0 0
$$645$$ 8.52992e6 0.807320
$$646$$ −1.68756e6 −0.159103
$$647$$ 54964.0 0.00516200 0.00258100 0.999997i $$-0.499178\pi$$
0.00258100 + 0.999997i $$0.499178\pi$$
$$648$$ 1.63508e7 1.52969
$$649$$ 633360. 0.0590254
$$650$$ −15400.0 −0.00142968
$$651$$ 0 0
$$652$$ 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
$$658$$ 0 0
$$659$$ −2.72136e6 −0.244103 −0.122051 0.992524i $$-0.538947\pi$$
−0.122051 + 0.992524i $$0.538947\pi$$
$$660$$ 1.23684e7 1.10523
$$661$$ 2.14525e6 0.190974 0.0954869 0.995431i $$-0.469559\pi$$
0.0954869 + 0.995431i $$0.469559\pi$$
$$662$$ 5.64448e6 0.500586
$$663$$ 3.37512e6 0.298198
$$664$$ −7.39368e6 −0.650789
$$665$$ 0 0
$$666$$ −4.88706e6 −0.426935
$$667$$ 6.23443e6 0.542603
$$668$$ −3.35656e7 −2.91041
$$669$$ 1.15397e7 0.996848
$$670$$ 2.71062e7 2.33283
$$671$$ −5.95034e6 −0.510194
$$672$$ 0 0
$$673$$ 2.92796e6 0.249188 0.124594 0.992208i $$-0.460237\pi$$
0.124594 + 0.992208i $$0.460237\pi$$
$$674$$ −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
$$679$$ 0 0
$$680$$ −3.47155e7 −2.87906
$$681$$ −1.04135e6 −0.0860455
$$682$$ −1.77341e7 −1.45998
$$683$$ −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
$$686$$ 0 0
$$687$$ −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 $$-0.811490\pi$$
−0.829702 + 0.558207i $$0.811490\pi$$
$$692$$ −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
$$700$$ 0 0
$$701$$ 2.35141e7 1.80731 0.903655 0.428261i $$-0.140874\pi$$
0.903655 + 0.428261i $$0.140874\pi$$
$$702$$ 5.68400e6 0.435323
$$703$$ −1.01900e6 −0.0777656
$$704$$ −4.26138e6 −0.324055
$$705$$ −7.31002e6 −0.553918
$$706$$ 4.00645e7 3.02516
$$707$$ 0 0
$$708$$ 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
$$714$$ 0 0
$$715$$ 1.81888e6 0.133057
$$716$$ 2.00554e7 1.46200
$$717$$ 6.76066e6 0.491124
$$718$$ −737840. −0.0534135
$$719$$ 2.61152e7 1.88396 0.941978 0.335674i $$-0.108964\pi$$
0.941978 + 0.335674i $$0.108964\pi$$
$$720$$ −3.74797e6 −0.269442
$$721$$ 0 0
$$722$$ 2.46650e7 1.76091
$$723$$ −1.12827e7 −0.802729
$$724$$ 2.29146e7 1.62468
$$725$$ 37598.0 0.00265656
$$726$$ 1.50118e7 1.05704
$$727$$ −1.54126e7 −1.08154 −0.540768 0.841172i $$-0.681866\pi$$
−0.540768 + 0.841172i $$0.681866\pi$$
$$728$$ 0 0
$$729$$ 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
$$735$$ 0 0
$$736$$ −4.96128e6 −0.337597
$$737$$ −1.12297e7 −0.761554
$$738$$ 8.44214e6 0.570574
$$739$$ 2.01511e6 0.135734 0.0678669 0.997694i $$-0.478381\pi$$
0.0678669 + 0.997694i $$0.478381\pi$$
$$740$$ −3.95956e7 −2.65808
$$741$$ 192080. 0.0128510
$$742$$ 0 0
$$743$$ −1.51381e7 −1.00600 −0.503001 0.864286i $$-0.667771\pi$$
−0.503001 + 0.864286i $$0.667771\pi$$
$$744$$ −3.85258e7 −2.55164
$$745$$ −1.13266e6 −0.0747666
$$746$$ 1.60323e7 1.05475
$$747$$ −965286. −0.0632928
$$748$$ 2.71663e7 1.77532
$$749$$ 0 0
$$750$$ 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
$$756$$ 0 0
$$757$$ −1.09697e7 −0.695755 −0.347877 0.937540i $$-0.613097\pi$$
−0.347877 + 0.937540i $$0.613097\pi$$
$$758$$ 4.77012e7 3.01548
$$759$$ 5.92435e6 0.373281
$$760$$ −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
$$763$$ 0 0
$$764$$ 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 $$-0.689338\pi$$
−0.560363 + 0.828247i $$0.689338\pi$$
$$774$$ 5.11360e6 0.306813
$$775$$ 84084.0 0.00502874
$$776$$ −2.10622e7 −1.25559
$$777$$ 0 0
$$778$$ −4.84024e7 −2.86694
$$779$$ 1.76028e6 0.103929
$$780$$ 7.46368e6 0.439255
$$781$$ −1.35859e7 −0.797006
$$782$$ −3.14093e7 −1.83671
$$783$$ −1.38771e7 −0.808898
$$784$$ 0 0
$$785$$ −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
$$791$$ 0 0
$$792$$ 3.92544e6 0.222370
$$793$$ −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 $$-0.409712\pi$$
0.279860 + 0.960041i $$0.409712\pi$$
$$798$$ 0 0
$$799$$ −1.60559e7 −0.889751
$$800$$ −29920.0 −0.00165286
$$801$$ −2.37735e6 −0.130922
$$802$$ 3.31605e7 1.82048
$$803$$ −1.57950e7 −0.864433
$$804$$ −4.60806e7 −2.51407
$$805$$ 0 0
$$806$$ −1.07016e7 −0.580245
$$807$$ 1.78858e7 0.966772
$$808$$ 1.39306e7 0.750655
$$809$$ 1.40884e7 0.756816 0.378408 0.925639i $$-0.376472\pi$$
0.378408 + 0.925639i $$0.376472\pi$$
$$810$$ 2.54346e7 1.36211
$$811$$ −1.81433e7 −0.968646 −0.484323 0.874889i $$-0.660934\pi$$
−0.484323 + 0.874889i $$0.660934\pi$$
$$812$$ 0 0
$$813$$ −2.31076e7 −1.22611
$$814$$ 2.41234e7 1.27608
$$815$$ 738752. 0.0389587
$$816$$ 3.43298e7 1.80487
$$817$$ 1.06624e6 0.0558856
$$818$$ 3.07273e7 1.60562
$$819$$ 0 0
$$820$$ 6.83993e7 3.55236
$$821$$ −2.13669e7 −1.10633 −0.553164 0.833072i $$-0.686580\pi$$
−0.553164 + 0.833072i $$0.686580\pi$$
$$822$$ 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
$$825$$ 35728.0 0.00182757
$$826$$ 0 0
$$827$$ 1.62921e7 0.828350 0.414175 0.910197i $$-0.364070\pi$$
0.414175 + 0.910197i $$0.364070\pi$$
$$828$$ −5.82950e6 −0.295499
$$829$$ 2.08499e6 0.105370 0.0526851 0.998611i $$-0.483222\pi$$
0.0526851 + 0.998611i $$0.483222\pi$$
$$830$$ −1.15013e7 −0.579497
$$831$$ −1.48973e7 −0.748348
$$832$$ −2.57152e6 −0.128790
$$833$$ 0 0
$$834$$ −4.95684e6 −0.246769
$$835$$ −2.76423e7 −1.37201
$$836$$ 1.54605e6 0.0765081
$$837$$ −3.10346e7 −1.53120
$$838$$ 2.81438e7 1.38443
$$839$$ 2.27850e7 1.11749 0.558745 0.829340i $$-0.311283\pi$$
0.558745 + 0.829340i $$0.311283\pi$$
$$840$$ 0 0
$$841$$ −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 $$0.320672\pi$$
$$854$$ 0 0
$$855$$ −257936. −0.0120669
$$856$$ 5.26766e7 2.45716
$$857$$ −2.52900e7 −1.17624 −0.588120 0.808774i $$-0.700132\pi$$
−0.588120 + 0.808774i $$0.700132\pi$$
$$858$$ −4.54720e6 −0.210875
$$859$$ 1.03947e7 0.480652 0.240326 0.970692i $$-0.422746\pi$$
0.240326 + 0.970692i $$0.422746\pi$$
$$860$$ 4.14310e7 1.91020
$$861$$ 0 0
$$862$$ −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
$$868$$ 0 0
$$869$$ 7.37389e6 0.331243
$$870$$ −2.67971e7 −1.20030
$$871$$ −6.77656e6 −0.302666
$$872$$ −3.34433e7 −1.48942
$$873$$ −2.74978e6 −0.122113
$$874$$ −1.78752e6 −0.0791539
$$875$$ 0 0
$$876$$ −6.48141e7 −2.85370
$$877$$ 3.71659e7 1.63172 0.815861 0.578248i $$-0.196264\pi$$
0.815861 + 0.578248i $$0.196264\pi$$
$$878$$ −7.41770e7 −3.24738
$$879$$ 2.70449e7 1.18063
$$880$$ 1.85006e7 0.805340
$$881$$ −9.04785e6 −0.392740 −0.196370 0.980530i $$-0.562915\pi$$
−0.196370 + 0.980530i $$0.562915\pi$$
$$882$$ 0 0
$$883$$ 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
$$889$$ 0 0
$$890$$ −2.83259e7 −1.19870
$$891$$ −1.05372e7 −0.444663
$$892$$ 5.60500e7 2.35865
$$893$$ −913752. −0.0383442
$$894$$ 2.83164e6 0.118493
$$895$$ 1.65162e7 0.689211
$$896$$ 0 0
$$897$$ 3.57504e6 0.148354
$$898$$ 5.90574e6 0.244390
$$899$$ 2.61272e7 1.07819
$$900$$ −35156.0 −0.00144675
$$901$$ 3.89516e6 0.159850
$$902$$ −4.16718e7 −1.70540
$$903$$ 0 0
$$904$$ 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
$$910$$ 0 0
$$911$$ −6.60518e6 −0.263687 −0.131844 0.991271i $$-0.542090\pi$$
−0.131844 + 0.991271i $$0.542090\pi$$
$$912$$ 1.95373e6 0.0777816
$$913$$ 4.76482e6 0.189177
$$914$$ 2.90484e7 1.15016
$$915$$ −2.01080e7 −0.793993
$$916$$ −7.69730e7 −3.03110
$$917$$ 0 0
$$918$$ 6.99132e7 2.73812
$$919$$ −3.08930e7 −1.20662 −0.603311 0.797506i $$-0.706152\pi$$
−0.603311 + 0.797506i $$0.706152\pi$$
$$920$$ −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 $$-0.470479\pi$$
0.0926087 + 0.995703i $$0.470479\pi$$
$$930$$ −5.99290e7 −2.27211
$$931$$ 0 0
$$932$$ −1.35134e7 −0.509593
$$933$$ −9.33979e6 −0.351264
$$934$$ −6.12570e6 −0.229767
$$935$$ 2.23722e7 0.836913
$$936$$ 2.36880e6 0.0883769
$$937$$ 3.25004e7 1.20932 0.604658 0.796485i $$-0.293310\pi$$
0.604658 + 0.796485i $$0.293310\pi$$
$$938$$ 0 0
$$939$$ 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
$$945$$ 0 0
$$946$$ −2.52416e7 −0.917042
$$947$$ −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
$$952$$ 0 0
$$953$$ −6.71983e6 −0.239677 −0.119838 0.992793i $$-0.538238\pi$$
−0.119838 + 0.992793i $$0.538238\pi$$
$$954$$ 1.06314e6 0.0378198
$$955$$ 2.00628e7 0.711841
$$956$$ 3.28375e7 1.16205
$$957$$ 1.11017e7 0.391840
$$958$$ 2.60330e7 0.916454
$$959$$ 0 0
$$960$$ −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
$$966$$ 0 0
$$967$$ −2.78979e6 −0.0959413 −0.0479707 0.998849i $$-0.515275\pi$$
−0.0479707 + 0.998849i $$0.515275\pi$$
$$968$$ 3.86017e7 1.32409
$$969$$ 2.36258e6 0.0808310
$$970$$ −3.27634e7 −1.11804
$$971$$ −3.33594e7 −1.13545 −0.567727 0.823217i $$-0.692177\pi$$
−0.567727 + 0.823217i $$0.692177\pi$$
$$972$$ 2.38486e7 0.809648
$$973$$ 0 0
$$974$$ −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
$$979$$ 1.17350e7 0.391316
$$980$$ 0 0
$$981$$ −4.36621e6 −0.144854
$$982$$ −1.64090e7 −0.543004
$$983$$ 5.79760e6 0.191366 0.0956829 0.995412i $$-0.469497\pi$$
0.0956829 + 0.995412i $$0.469497\pi$$
$$984$$ −9.05285e7 −2.98056
$$985$$ −5.54428e7 −1.82077
$$986$$ −5.88580e7 −1.92803
$$987$$ 0 0
$$988$$ 932960. 0.0304068
$$989$$ 1.98451e7 0.645153
$$990$$ 6.10624e6 0.198009
$$991$$ 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
$$994$$ 0 0
$$995$$ 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 49.6.a.a.1.1 1
3.2 odd 2 441.6.a.k.1.1 1
4.3 odd 2 784.6.a.c.1.1 1
7.2 even 3 49.6.c.b.18.1 2
7.3 odd 6 49.6.c.c.30.1 2
7.4 even 3 49.6.c.b.30.1 2
7.5 odd 6 49.6.c.c.18.1 2
7.6 odd 2 7.6.a.a.1.1 1
21.20 even 2 63.6.a.e.1.1 1
28.27 even 2 112.6.a.g.1.1 1
35.13 even 4 175.6.b.a.99.2 2
35.27 even 4 175.6.b.a.99.1 2
35.34 odd 2 175.6.a.b.1.1 1
56.13 odd 2 448.6.a.m.1.1 1
56.27 even 2 448.6.a.c.1.1 1
77.76 even 2 847.6.a.b.1.1 1
84.83 odd 2 1008.6.a.y.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.a.1.1 1 7.6 odd 2
49.6.a.a.1.1 1 1.1 even 1 trivial
49.6.c.b.18.1 2 7.2 even 3
49.6.c.b.30.1 2 7.4 even 3
49.6.c.c.18.1 2 7.5 odd 6
49.6.c.c.30.1 2 7.3 odd 6
63.6.a.e.1.1 1 21.20 even 2
112.6.a.g.1.1 1 28.27 even 2
175.6.a.b.1.1 1 35.34 odd 2
175.6.b.a.99.1 2 35.27 even 4
175.6.b.a.99.2 2 35.13 even 4
441.6.a.k.1.1 1 3.2 odd 2
448.6.a.c.1.1 1 56.27 even 2
448.6.a.m.1.1 1 56.13 odd 2
784.6.a.c.1.1 1 4.3 odd 2
847.6.a.b.1.1 1 77.76 even 2
1008.6.a.y.1.1 1 84.83 odd 2