# Properties

 Label 14.6.a.a.1.1 Level $14$ Weight $6$ Character 14.1 Self dual yes Analytic conductor $2.245$ 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] = [14,6,Mod(1,14)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(14, 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("14.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$14 = 2 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 14.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: $$2.24537347738$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 14.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.00000 q^{2} +10.0000 q^{3} +16.0000 q^{4} +84.0000 q^{5} -40.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} -143.000 q^{9} +O(q^{10})$$ $$q-4.00000 q^{2} +10.0000 q^{3} +16.0000 q^{4} +84.0000 q^{5} -40.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} -143.000 q^{9} -336.000 q^{10} -336.000 q^{11} +160.000 q^{12} +584.000 q^{13} -196.000 q^{14} +840.000 q^{15} +256.000 q^{16} -1458.00 q^{17} +572.000 q^{18} +470.000 q^{19} +1344.00 q^{20} +490.000 q^{21} +1344.00 q^{22} -4200.00 q^{23} -640.000 q^{24} +3931.00 q^{25} -2336.00 q^{26} -3860.00 q^{27} +784.000 q^{28} +4866.00 q^{29} -3360.00 q^{30} -7372.00 q^{31} -1024.00 q^{32} -3360.00 q^{33} +5832.00 q^{34} +4116.00 q^{35} -2288.00 q^{36} +14330.0 q^{37} -1880.00 q^{38} +5840.00 q^{39} -5376.00 q^{40} +6222.00 q^{41} -1960.00 q^{42} +3704.00 q^{43} -5376.00 q^{44} -12012.0 q^{45} +16800.0 q^{46} -1812.00 q^{47} +2560.00 q^{48} +2401.00 q^{49} -15724.0 q^{50} -14580.0 q^{51} +9344.00 q^{52} -37242.0 q^{53} +15440.0 q^{54} -28224.0 q^{55} -3136.00 q^{56} +4700.00 q^{57} -19464.0 q^{58} +34302.0 q^{59} +13440.0 q^{60} +24476.0 q^{61} +29488.0 q^{62} -7007.00 q^{63} +4096.00 q^{64} +49056.0 q^{65} +13440.0 q^{66} -17452.0 q^{67} -23328.0 q^{68} -42000.0 q^{69} -16464.0 q^{70} +28224.0 q^{71} +9152.00 q^{72} +3602.00 q^{73} -57320.0 q^{74} +39310.0 q^{75} +7520.00 q^{76} -16464.0 q^{77} -23360.0 q^{78} +42872.0 q^{79} +21504.0 q^{80} -3851.00 q^{81} -24888.0 q^{82} -35202.0 q^{83} +7840.00 q^{84} -122472. q^{85} -14816.0 q^{86} +48660.0 q^{87} +21504.0 q^{88} +26730.0 q^{89} +48048.0 q^{90} +28616.0 q^{91} -67200.0 q^{92} -73720.0 q^{93} +7248.00 q^{94} +39480.0 q^{95} -10240.0 q^{96} -16978.0 q^{97} -9604.00 q^{98} +48048.0 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.00000 −0.707107
$$3$$ 10.0000 0.641500 0.320750 0.947164i $$-0.396065\pi$$
0.320750 + 0.947164i $$0.396065\pi$$
$$4$$ 16.0000 0.500000
$$5$$ 84.0000 1.50264 0.751319 0.659939i $$-0.229418\pi$$
0.751319 + 0.659939i $$0.229418\pi$$
$$6$$ −40.0000 −0.453609
$$7$$ 49.0000 0.377964
$$8$$ −64.0000 −0.353553
$$9$$ −143.000 −0.588477
$$10$$ −336.000 −1.06253
$$11$$ −336.000 −0.837255 −0.418627 0.908158i $$-0.637489\pi$$
−0.418627 + 0.908158i $$0.637489\pi$$
$$12$$ 160.000 0.320750
$$13$$ 584.000 0.958417 0.479208 0.877701i $$-0.340924\pi$$
0.479208 + 0.877701i $$0.340924\pi$$
$$14$$ −196.000 −0.267261
$$15$$ 840.000 0.963943
$$16$$ 256.000 0.250000
$$17$$ −1458.00 −1.22359 −0.611794 0.791017i $$-0.709552\pi$$
−0.611794 + 0.791017i $$0.709552\pi$$
$$18$$ 572.000 0.416116
$$19$$ 470.000 0.298685 0.149343 0.988786i $$-0.452284\pi$$
0.149343 + 0.988786i $$0.452284\pi$$
$$20$$ 1344.00 0.751319
$$21$$ 490.000 0.242464
$$22$$ 1344.00 0.592028
$$23$$ −4200.00 −1.65550 −0.827751 0.561096i $$-0.810380\pi$$
−0.827751 + 0.561096i $$0.810380\pi$$
$$24$$ −640.000 −0.226805
$$25$$ 3931.00 1.25792
$$26$$ −2336.00 −0.677703
$$27$$ −3860.00 −1.01901
$$28$$ 784.000 0.188982
$$29$$ 4866.00 1.07443 0.537214 0.843446i $$-0.319477\pi$$
0.537214 + 0.843446i $$0.319477\pi$$
$$30$$ −3360.00 −0.681610
$$31$$ −7372.00 −1.37778 −0.688892 0.724864i $$-0.741903\pi$$
−0.688892 + 0.724864i $$0.741903\pi$$
$$32$$ −1024.00 −0.176777
$$33$$ −3360.00 −0.537099
$$34$$ 5832.00 0.865207
$$35$$ 4116.00 0.567944
$$36$$ −2288.00 −0.294239
$$37$$ 14330.0 1.72085 0.860423 0.509581i $$-0.170200\pi$$
0.860423 + 0.509581i $$0.170200\pi$$
$$38$$ −1880.00 −0.211202
$$39$$ 5840.00 0.614825
$$40$$ −5376.00 −0.531263
$$41$$ 6222.00 0.578057 0.289028 0.957321i $$-0.406668\pi$$
0.289028 + 0.957321i $$0.406668\pi$$
$$42$$ −1960.00 −0.171448
$$43$$ 3704.00 0.305492 0.152746 0.988265i $$-0.451188\pi$$
0.152746 + 0.988265i $$0.451188\pi$$
$$44$$ −5376.00 −0.418627
$$45$$ −12012.0 −0.884268
$$46$$ 16800.0 1.17062
$$47$$ −1812.00 −0.119650 −0.0598251 0.998209i $$-0.519054\pi$$
−0.0598251 + 0.998209i $$0.519054\pi$$
$$48$$ 2560.00 0.160375
$$49$$ 2401.00 0.142857
$$50$$ −15724.0 −0.889484
$$51$$ −14580.0 −0.784932
$$52$$ 9344.00 0.479208
$$53$$ −37242.0 −1.82114 −0.910570 0.413355i $$-0.864357\pi$$
−0.910570 + 0.413355i $$0.864357\pi$$
$$54$$ 15440.0 0.720548
$$55$$ −28224.0 −1.25809
$$56$$ −3136.00 −0.133631
$$57$$ 4700.00 0.191607
$$58$$ −19464.0 −0.759735
$$59$$ 34302.0 1.28289 0.641445 0.767169i $$-0.278335\pi$$
0.641445 + 0.767169i $$0.278335\pi$$
$$60$$ 13440.0 0.481971
$$61$$ 24476.0 0.842201 0.421101 0.907014i $$-0.361644\pi$$
0.421101 + 0.907014i $$0.361644\pi$$
$$62$$ 29488.0 0.974240
$$63$$ −7007.00 −0.222424
$$64$$ 4096.00 0.125000
$$65$$ 49056.0 1.44015
$$66$$ 13440.0 0.379786
$$67$$ −17452.0 −0.474961 −0.237481 0.971392i $$-0.576322\pi$$
−0.237481 + 0.971392i $$0.576322\pi$$
$$68$$ −23328.0 −0.611794
$$69$$ −42000.0 −1.06201
$$70$$ −16464.0 −0.401597
$$71$$ 28224.0 0.664466 0.332233 0.943197i $$-0.392198\pi$$
0.332233 + 0.943197i $$0.392198\pi$$
$$72$$ 9152.00 0.208058
$$73$$ 3602.00 0.0791109 0.0395555 0.999217i $$-0.487406\pi$$
0.0395555 + 0.999217i $$0.487406\pi$$
$$74$$ −57320.0 −1.21682
$$75$$ 39310.0 0.806956
$$76$$ 7520.00 0.149343
$$77$$ −16464.0 −0.316453
$$78$$ −23360.0 −0.434747
$$79$$ 42872.0 0.772869 0.386435 0.922317i $$-0.373706\pi$$
0.386435 + 0.922317i $$0.373706\pi$$
$$80$$ 21504.0 0.375659
$$81$$ −3851.00 −0.0652170
$$82$$ −24888.0 −0.408748
$$83$$ −35202.0 −0.560883 −0.280441 0.959871i $$-0.590481\pi$$
−0.280441 + 0.959871i $$0.590481\pi$$
$$84$$ 7840.00 0.121232
$$85$$ −122472. −1.83861
$$86$$ −14816.0 −0.216015
$$87$$ 48660.0 0.689246
$$88$$ 21504.0 0.296014
$$89$$ 26730.0 0.357704 0.178852 0.983876i $$-0.442762\pi$$
0.178852 + 0.983876i $$0.442762\pi$$
$$90$$ 48048.0 0.625272
$$91$$ 28616.0 0.362248
$$92$$ −67200.0 −0.827751
$$93$$ −73720.0 −0.883849
$$94$$ 7248.00 0.0846055
$$95$$ 39480.0 0.448816
$$96$$ −10240.0 −0.113402
$$97$$ −16978.0 −0.183213 −0.0916067 0.995795i $$-0.529200\pi$$
−0.0916067 + 0.995795i $$0.529200\pi$$
$$98$$ −9604.00 −0.101015
$$99$$ 48048.0 0.492705
$$100$$ 62896.0 0.628960
$$101$$ 99204.0 0.967667 0.483833 0.875160i $$-0.339244\pi$$
0.483833 + 0.875160i $$0.339244\pi$$
$$102$$ 58320.0 0.555031
$$103$$ −131644. −1.22267 −0.611333 0.791373i $$-0.709366\pi$$
−0.611333 + 0.791373i $$0.709366\pi$$
$$104$$ −37376.0 −0.338852
$$105$$ 41160.0 0.364336
$$106$$ 148968. 1.28774
$$107$$ 48852.0 0.412499 0.206250 0.978499i $$-0.433874\pi$$
0.206250 + 0.978499i $$0.433874\pi$$
$$108$$ −61760.0 −0.509504
$$109$$ −56374.0 −0.454478 −0.227239 0.973839i $$-0.572970\pi$$
−0.227239 + 0.973839i $$0.572970\pi$$
$$110$$ 112896. 0.889604
$$111$$ 143300. 1.10392
$$112$$ 12544.0 0.0944911
$$113$$ 8742.00 0.0644043 0.0322021 0.999481i $$-0.489748\pi$$
0.0322021 + 0.999481i $$0.489748\pi$$
$$114$$ −18800.0 −0.135486
$$115$$ −352800. −2.48762
$$116$$ 77856.0 0.537214
$$117$$ −83512.0 −0.564007
$$118$$ −137208. −0.907140
$$119$$ −71442.0 −0.462473
$$120$$ −53760.0 −0.340805
$$121$$ −48155.0 −0.299005
$$122$$ −97904.0 −0.595526
$$123$$ 62220.0 0.370823
$$124$$ −117952. −0.688892
$$125$$ 67704.0 0.387560
$$126$$ 28028.0 0.157277
$$127$$ 315992. 1.73847 0.869234 0.494401i $$-0.164612\pi$$
0.869234 + 0.494401i $$0.164612\pi$$
$$128$$ −16384.0 −0.0883883
$$129$$ 37040.0 0.195973
$$130$$ −196224. −1.01834
$$131$$ −24666.0 −0.125580 −0.0627900 0.998027i $$-0.520000\pi$$
−0.0627900 + 0.998027i $$0.520000\pi$$
$$132$$ −53760.0 −0.268550
$$133$$ 23030.0 0.112892
$$134$$ 69808.0 0.335848
$$135$$ −324240. −1.53120
$$136$$ 93312.0 0.432604
$$137$$ 303234. 1.38031 0.690155 0.723662i $$-0.257542\pi$$
0.690155 + 0.723662i $$0.257542\pi$$
$$138$$ 168000. 0.750951
$$139$$ 250586. 1.10007 0.550034 0.835142i $$-0.314615\pi$$
0.550034 + 0.835142i $$0.314615\pi$$
$$140$$ 65856.0 0.283972
$$141$$ −18120.0 −0.0767557
$$142$$ −112896. −0.469848
$$143$$ −196224. −0.802439
$$144$$ −36608.0 −0.147119
$$145$$ 408744. 1.61448
$$146$$ −14408.0 −0.0559399
$$147$$ 24010.0 0.0916429
$$148$$ 229280. 0.860423
$$149$$ −60594.0 −0.223596 −0.111798 0.993731i $$-0.535661\pi$$
−0.111798 + 0.993731i $$0.535661\pi$$
$$150$$ −157240. −0.570604
$$151$$ 124448. 0.444166 0.222083 0.975028i $$-0.428714\pi$$
0.222083 + 0.975028i $$0.428714\pi$$
$$152$$ −30080.0 −0.105601
$$153$$ 208494. 0.720054
$$154$$ 65856.0 0.223766
$$155$$ −619248. −2.07031
$$156$$ 93440.0 0.307412
$$157$$ 76040.0 0.246203 0.123101 0.992394i $$-0.460716\pi$$
0.123101 + 0.992394i $$0.460716\pi$$
$$158$$ −171488. −0.546501
$$159$$ −372420. −1.16826
$$160$$ −86016.0 −0.265631
$$161$$ −205800. −0.625721
$$162$$ 15404.0 0.0461154
$$163$$ 124256. 0.366310 0.183155 0.983084i $$-0.441369\pi$$
0.183155 + 0.983084i $$0.441369\pi$$
$$164$$ 99552.0 0.289028
$$165$$ −282240. −0.807065
$$166$$ 140808. 0.396604
$$167$$ −72420.0 −0.200940 −0.100470 0.994940i $$-0.532035\pi$$
−0.100470 + 0.994940i $$0.532035\pi$$
$$168$$ −31360.0 −0.0857241
$$169$$ −30237.0 −0.0814370
$$170$$ 489888. 1.30009
$$171$$ −67210.0 −0.175770
$$172$$ 59264.0 0.152746
$$173$$ −441552. −1.12167 −0.560837 0.827926i $$-0.689521\pi$$
−0.560837 + 0.827926i $$0.689521\pi$$
$$174$$ −194640. −0.487370
$$175$$ 192619. 0.475449
$$176$$ −86016.0 −0.209314
$$177$$ 343020. 0.822974
$$178$$ −106920. −0.252935
$$179$$ −10692.0 −0.0249417 −0.0124709 0.999922i $$-0.503970\pi$$
−0.0124709 + 0.999922i $$0.503970\pi$$
$$180$$ −192192. −0.442134
$$181$$ −546064. −1.23893 −0.619465 0.785024i $$-0.712651\pi$$
−0.619465 + 0.785024i $$0.712651\pi$$
$$182$$ −114464. −0.256148
$$183$$ 244760. 0.540272
$$184$$ 268800. 0.585308
$$185$$ 1.20372e6 2.58581
$$186$$ 294880. 0.624975
$$187$$ 489888. 1.02445
$$188$$ −28992.0 −0.0598251
$$189$$ −189140. −0.385149
$$190$$ −157920. −0.317361
$$191$$ −575976. −1.14241 −0.571204 0.820808i $$-0.693523\pi$$
−0.571204 + 0.820808i $$0.693523\pi$$
$$192$$ 40960.0 0.0801875
$$193$$ −413938. −0.799912 −0.399956 0.916534i $$-0.630975\pi$$
−0.399956 + 0.916534i $$0.630975\pi$$
$$194$$ 67912.0 0.129551
$$195$$ 490560. 0.923859
$$196$$ 38416.0 0.0714286
$$197$$ −494946. −0.908641 −0.454320 0.890838i $$-0.650118\pi$$
−0.454320 + 0.890838i $$0.650118\pi$$
$$198$$ −192192. −0.348395
$$199$$ 520364. 0.931482 0.465741 0.884921i $$-0.345788\pi$$
0.465741 + 0.884921i $$0.345788\pi$$
$$200$$ −251584. −0.444742
$$201$$ −174520. −0.304688
$$202$$ −396816. −0.684244
$$203$$ 238434. 0.406095
$$204$$ −233280. −0.392466
$$205$$ 522648. 0.868610
$$206$$ 526576. 0.864556
$$207$$ 600600. 0.974225
$$208$$ 149504. 0.239604
$$209$$ −157920. −0.250076
$$210$$ −164640. −0.257624
$$211$$ 183284. 0.283412 0.141706 0.989909i $$-0.454741\pi$$
0.141706 + 0.989909i $$0.454741\pi$$
$$212$$ −595872. −0.910570
$$213$$ 282240. 0.426255
$$214$$ −195408. −0.291681
$$215$$ 311136. 0.459044
$$216$$ 247040. 0.360274
$$217$$ −361228. −0.520753
$$218$$ 225496. 0.321364
$$219$$ 36020.0 0.0507497
$$220$$ −451584. −0.629045
$$221$$ −851472. −1.17271
$$222$$ −573200. −0.780591
$$223$$ −1.27746e6 −1.72023 −0.860115 0.510100i $$-0.829608\pi$$
−0.860115 + 0.510100i $$0.829608\pi$$
$$224$$ −50176.0 −0.0668153
$$225$$ −562133. −0.740257
$$226$$ −34968.0 −0.0455407
$$227$$ −1.28764e6 −1.65856 −0.829279 0.558835i $$-0.811248\pi$$
−0.829279 + 0.558835i $$0.811248\pi$$
$$228$$ 75200.0 0.0958034
$$229$$ 350936. 0.442221 0.221110 0.975249i $$-0.429032\pi$$
0.221110 + 0.975249i $$0.429032\pi$$
$$230$$ 1.41120e6 1.75901
$$231$$ −164640. −0.203004
$$232$$ −311424. −0.379867
$$233$$ 836154. 1.00901 0.504506 0.863408i $$-0.331675\pi$$
0.504506 + 0.863408i $$0.331675\pi$$
$$234$$ 334048. 0.398813
$$235$$ −152208. −0.179791
$$236$$ 548832. 0.641445
$$237$$ 428720. 0.495796
$$238$$ 285768. 0.327018
$$239$$ 774336. 0.876869 0.438434 0.898763i $$-0.355533\pi$$
0.438434 + 0.898763i $$0.355533\pi$$
$$240$$ 215040. 0.240986
$$241$$ −1.15285e6 −1.27859 −0.639293 0.768963i $$-0.720773\pi$$
−0.639293 + 0.768963i $$0.720773\pi$$
$$242$$ 192620. 0.211428
$$243$$ 899470. 0.977172
$$244$$ 391616. 0.421101
$$245$$ 201684. 0.214663
$$246$$ −248880. −0.262212
$$247$$ 274480. 0.286265
$$248$$ 471808. 0.487120
$$249$$ −352020. −0.359806
$$250$$ −270816. −0.274047
$$251$$ 1.35801e6 1.36056 0.680282 0.732951i $$-0.261858\pi$$
0.680282 + 0.732951i $$0.261858\pi$$
$$252$$ −112112. −0.111212
$$253$$ 1.41120e6 1.38608
$$254$$ −1.26397e6 −1.22928
$$255$$ −1.22472e6 −1.17947
$$256$$ 65536.0 0.0625000
$$257$$ −317742. −0.300083 −0.150042 0.988680i $$-0.547941\pi$$
−0.150042 + 0.988680i $$0.547941\pi$$
$$258$$ −148160. −0.138574
$$259$$ 702170. 0.650418
$$260$$ 784896. 0.720077
$$261$$ −695838. −0.632276
$$262$$ 98664.0 0.0887985
$$263$$ 1.05101e6 0.936951 0.468475 0.883477i $$-0.344804\pi$$
0.468475 + 0.883477i $$0.344804\pi$$
$$264$$ 215040. 0.189893
$$265$$ −3.12833e6 −2.73651
$$266$$ −92120.0 −0.0798270
$$267$$ 267300. 0.229467
$$268$$ −279232. −0.237481
$$269$$ 1.18958e6 1.00234 0.501169 0.865349i $$-0.332903\pi$$
0.501169 + 0.865349i $$0.332903\pi$$
$$270$$ 1.29696e6 1.08272
$$271$$ −1.43008e6 −1.18287 −0.591435 0.806353i $$-0.701438\pi$$
−0.591435 + 0.806353i $$0.701438\pi$$
$$272$$ −373248. −0.305897
$$273$$ 286160. 0.232382
$$274$$ −1.21294e6 −0.976026
$$275$$ −1.32082e6 −1.05320
$$276$$ −672000. −0.531003
$$277$$ 63302.0 0.0495699 0.0247849 0.999693i $$-0.492110\pi$$
0.0247849 + 0.999693i $$0.492110\pi$$
$$278$$ −1.00234e6 −0.777866
$$279$$ 1.05420e6 0.810795
$$280$$ −263424. −0.200798
$$281$$ −496614. −0.375192 −0.187596 0.982246i $$-0.560070\pi$$
−0.187596 + 0.982246i $$0.560070\pi$$
$$282$$ 72480.0 0.0542744
$$283$$ −1.15842e6 −0.859803 −0.429902 0.902876i $$-0.641452\pi$$
−0.429902 + 0.902876i $$0.641452\pi$$
$$284$$ 451584. 0.332233
$$285$$ 394800. 0.287915
$$286$$ 784896. 0.567410
$$287$$ 304878. 0.218485
$$288$$ 146432. 0.104029
$$289$$ 705907. 0.497168
$$290$$ −1.63498e6 −1.14161
$$291$$ −169780. −0.117531
$$292$$ 57632.0 0.0395555
$$293$$ 1.43886e6 0.979151 0.489575 0.871961i $$-0.337152\pi$$
0.489575 + 0.871961i $$0.337152\pi$$
$$294$$ −96040.0 −0.0648013
$$295$$ 2.88137e6 1.92772
$$296$$ −917120. −0.608411
$$297$$ 1.29696e6 0.853170
$$298$$ 242376. 0.158106
$$299$$ −2.45280e6 −1.58666
$$300$$ 628960. 0.403478
$$301$$ 181496. 0.115465
$$302$$ −497792. −0.314073
$$303$$ 992040. 0.620758
$$304$$ 120320. 0.0746713
$$305$$ 2.05598e6 1.26552
$$306$$ −833976. −0.509155
$$307$$ −989098. −0.598954 −0.299477 0.954104i $$-0.596812\pi$$
−0.299477 + 0.954104i $$0.596812\pi$$
$$308$$ −263424. −0.158226
$$309$$ −1.31644e6 −0.784341
$$310$$ 2.47699e6 1.46393
$$311$$ −2.22050e6 −1.30182 −0.650909 0.759155i $$-0.725612\pi$$
−0.650909 + 0.759155i $$0.725612\pi$$
$$312$$ −373760. −0.217373
$$313$$ 2.33008e6 1.34434 0.672171 0.740396i $$-0.265362\pi$$
0.672171 + 0.740396i $$0.265362\pi$$
$$314$$ −304160. −0.174092
$$315$$ −588588. −0.334222
$$316$$ 685952. 0.386435
$$317$$ 427542. 0.238963 0.119481 0.992836i $$-0.461877\pi$$
0.119481 + 0.992836i $$0.461877\pi$$
$$318$$ 1.48968e6 0.826086
$$319$$ −1.63498e6 −0.899569
$$320$$ 344064. 0.187830
$$321$$ 488520. 0.264618
$$322$$ 823200. 0.442452
$$323$$ −685260. −0.365468
$$324$$ −61616.0 −0.0326085
$$325$$ 2.29570e6 1.20561
$$326$$ −497024. −0.259020
$$327$$ −563740. −0.291548
$$328$$ −398208. −0.204374
$$329$$ −88788.0 −0.0452235
$$330$$ 1.12896e6 0.570681
$$331$$ −396616. −0.198976 −0.0994879 0.995039i $$-0.531720\pi$$
−0.0994879 + 0.995039i $$0.531720\pi$$
$$332$$ −563232. −0.280441
$$333$$ −2.04919e6 −1.01268
$$334$$ 289680. 0.142086
$$335$$ −1.46597e6 −0.713695
$$336$$ 125440. 0.0606161
$$337$$ −3.21819e6 −1.54361 −0.771805 0.635860i $$-0.780646\pi$$
−0.771805 + 0.635860i $$0.780646\pi$$
$$338$$ 120948. 0.0575847
$$339$$ 87420.0 0.0413154
$$340$$ −1.95955e6 −0.919305
$$341$$ 2.47699e6 1.15356
$$342$$ 268840. 0.124288
$$343$$ 117649. 0.0539949
$$344$$ −237056. −0.108008
$$345$$ −3.52800e6 −1.59581
$$346$$ 1.76621e6 0.793143
$$347$$ 2.78018e6 1.23951 0.619755 0.784796i $$-0.287232\pi$$
0.619755 + 0.784796i $$0.287232\pi$$
$$348$$ 778560. 0.344623
$$349$$ −338800. −0.148895 −0.0744475 0.997225i $$-0.523719\pi$$
−0.0744475 + 0.997225i $$0.523719\pi$$
$$350$$ −770476. −0.336193
$$351$$ −2.25424e6 −0.976635
$$352$$ 344064. 0.148007
$$353$$ −362046. −0.154642 −0.0773209 0.997006i $$-0.524637\pi$$
−0.0773209 + 0.997006i $$0.524637\pi$$
$$354$$ −1.37208e6 −0.581931
$$355$$ 2.37082e6 0.998451
$$356$$ 427680. 0.178852
$$357$$ −714420. −0.296676
$$358$$ 42768.0 0.0176365
$$359$$ 876528. 0.358946 0.179473 0.983763i $$-0.442561\pi$$
0.179473 + 0.983763i $$0.442561\pi$$
$$360$$ 768768. 0.312636
$$361$$ −2.25520e6 −0.910787
$$362$$ 2.18426e6 0.876056
$$363$$ −481550. −0.191812
$$364$$ 457856. 0.181124
$$365$$ 302568. 0.118875
$$366$$ −979040. −0.382030
$$367$$ 2.98062e6 1.15516 0.577578 0.816335i $$-0.303998\pi$$
0.577578 + 0.816335i $$0.303998\pi$$
$$368$$ −1.07520e6 −0.413875
$$369$$ −889746. −0.340173
$$370$$ −4.81488e6 −1.82844
$$371$$ −1.82486e6 −0.688326
$$372$$ −1.17952e6 −0.441924
$$373$$ 3.91441e6 1.45678 0.728391 0.685162i $$-0.240268\pi$$
0.728391 + 0.685162i $$0.240268\pi$$
$$374$$ −1.95955e6 −0.724399
$$375$$ 677040. 0.248620
$$376$$ 115968. 0.0423027
$$377$$ 2.84174e6 1.02975
$$378$$ 756560. 0.272342
$$379$$ 3.60661e6 1.28974 0.644868 0.764294i $$-0.276912\pi$$
0.644868 + 0.764294i $$0.276912\pi$$
$$380$$ 631680. 0.224408
$$381$$ 3.15992e6 1.11523
$$382$$ 2.30390e6 0.807804
$$383$$ −2.66644e6 −0.928826 −0.464413 0.885619i $$-0.653735\pi$$
−0.464413 + 0.885619i $$0.653735\pi$$
$$384$$ −163840. −0.0567012
$$385$$ −1.38298e6 −0.475513
$$386$$ 1.65575e6 0.565623
$$387$$ −529672. −0.179775
$$388$$ −271648. −0.0916067
$$389$$ −213366. −0.0714910 −0.0357455 0.999361i $$-0.511381\pi$$
−0.0357455 + 0.999361i $$0.511381\pi$$
$$390$$ −1.96224e6 −0.653267
$$391$$ 6.12360e6 2.02565
$$392$$ −153664. −0.0505076
$$393$$ −246660. −0.0805596
$$394$$ 1.97978e6 0.642506
$$395$$ 3.60125e6 1.16134
$$396$$ 768768. 0.246353
$$397$$ −4.09408e6 −1.30371 −0.651854 0.758345i $$-0.726008\pi$$
−0.651854 + 0.758345i $$0.726008\pi$$
$$398$$ −2.08146e6 −0.658657
$$399$$ 230300. 0.0724205
$$400$$ 1.00634e6 0.314480
$$401$$ 942366. 0.292657 0.146328 0.989236i $$-0.453254\pi$$
0.146328 + 0.989236i $$0.453254\pi$$
$$402$$ 698080. 0.215447
$$403$$ −4.30525e6 −1.32049
$$404$$ 1.58726e6 0.483833
$$405$$ −323484. −0.0979976
$$406$$ −953736. −0.287153
$$407$$ −4.81488e6 −1.44079
$$408$$ 933120. 0.277515
$$409$$ −4.84561e6 −1.43232 −0.716160 0.697936i $$-0.754102\pi$$
−0.716160 + 0.697936i $$0.754102\pi$$
$$410$$ −2.09059e6 −0.614200
$$411$$ 3.03234e6 0.885469
$$412$$ −2.10630e6 −0.611333
$$413$$ 1.68080e6 0.484887
$$414$$ −2.40240e6 −0.688881
$$415$$ −2.95697e6 −0.842804
$$416$$ −598016. −0.169426
$$417$$ 2.50586e6 0.705694
$$418$$ 631680. 0.176830
$$419$$ −1.73485e6 −0.482754 −0.241377 0.970431i $$-0.577599\pi$$
−0.241377 + 0.970431i $$0.577599\pi$$
$$420$$ 658560. 0.182168
$$421$$ −1.65145e6 −0.454109 −0.227055 0.973882i $$-0.572910\pi$$
−0.227055 + 0.973882i $$0.572910\pi$$
$$422$$ −733136. −0.200403
$$423$$ 259116. 0.0704115
$$424$$ 2.38349e6 0.643870
$$425$$ −5.73140e6 −1.53918
$$426$$ −1.12896e6 −0.301408
$$427$$ 1.19932e6 0.318322
$$428$$ 781632. 0.206250
$$429$$ −1.96224e6 −0.514765
$$430$$ −1.24454e6 −0.324593
$$431$$ 4.14360e6 1.07445 0.537223 0.843440i $$-0.319473\pi$$
0.537223 + 0.843440i $$0.319473\pi$$
$$432$$ −988160. −0.254752
$$433$$ −3.03966e6 −0.779121 −0.389561 0.921001i $$-0.627373\pi$$
−0.389561 + 0.921001i $$0.627373\pi$$
$$434$$ 1.44491e6 0.368228
$$435$$ 4.08744e6 1.03569
$$436$$ −901984. −0.227239
$$437$$ −1.97400e6 −0.494474
$$438$$ −144080. −0.0358855
$$439$$ 2.54271e6 0.629703 0.314852 0.949141i $$-0.398045\pi$$
0.314852 + 0.949141i $$0.398045\pi$$
$$440$$ 1.80634e6 0.444802
$$441$$ −343343. −0.0840682
$$442$$ 3.40589e6 0.829229
$$443$$ −2.43210e6 −0.588806 −0.294403 0.955681i $$-0.595121\pi$$
−0.294403 + 0.955681i $$0.595121\pi$$
$$444$$ 2.29280e6 0.551961
$$445$$ 2.24532e6 0.537500
$$446$$ 5.10986e6 1.21639
$$447$$ −605940. −0.143437
$$448$$ 200704. 0.0472456
$$449$$ 1.82853e6 0.428042 0.214021 0.976829i $$-0.431344\pi$$
0.214021 + 0.976829i $$0.431344\pi$$
$$450$$ 2.24853e6 0.523441
$$451$$ −2.09059e6 −0.483981
$$452$$ 139872. 0.0322021
$$453$$ 1.24448e6 0.284933
$$454$$ 5.15057e6 1.17278
$$455$$ 2.40374e6 0.544327
$$456$$ −300800. −0.0677432
$$457$$ 1.58063e6 0.354030 0.177015 0.984208i $$-0.443356\pi$$
0.177015 + 0.984208i $$0.443356\pi$$
$$458$$ −1.40374e6 −0.312697
$$459$$ 5.62788e6 1.24685
$$460$$ −5.64480e6 −1.24381
$$461$$ 5.09604e6 1.11681 0.558407 0.829567i $$-0.311413\pi$$
0.558407 + 0.829567i $$0.311413\pi$$
$$462$$ 658560. 0.143546
$$463$$ −7.02338e6 −1.52263 −0.761313 0.648384i $$-0.775445\pi$$
−0.761313 + 0.648384i $$0.775445\pi$$
$$464$$ 1.24570e6 0.268607
$$465$$ −6.19248e6 −1.32810
$$466$$ −3.34462e6 −0.713479
$$467$$ −4.24845e6 −0.901443 −0.450722 0.892665i $$-0.648833\pi$$
−0.450722 + 0.892665i $$0.648833\pi$$
$$468$$ −1.33619e6 −0.282003
$$469$$ −855148. −0.179518
$$470$$ 608832. 0.127131
$$471$$ 760400. 0.157939
$$472$$ −2.19533e6 −0.453570
$$473$$ −1.24454e6 −0.255775
$$474$$ −1.71488e6 −0.350581
$$475$$ 1.84757e6 0.375722
$$476$$ −1.14307e6 −0.231236
$$477$$ 5.32561e6 1.07170
$$478$$ −3.09734e6 −0.620040
$$479$$ 559284. 0.111377 0.0556883 0.998448i $$-0.482265\pi$$
0.0556883 + 0.998448i $$0.482265\pi$$
$$480$$ −860160. −0.170403
$$481$$ 8.36872e6 1.64929
$$482$$ 4.61140e6 0.904097
$$483$$ −2.05800e6 −0.401400
$$484$$ −770480. −0.149502
$$485$$ −1.42615e6 −0.275303
$$486$$ −3.59788e6 −0.690965
$$487$$ −1.32057e6 −0.252312 −0.126156 0.992010i $$-0.540264\pi$$
−0.126156 + 0.992010i $$0.540264\pi$$
$$488$$ −1.56646e6 −0.297763
$$489$$ 1.24256e6 0.234988
$$490$$ −806736. −0.151789
$$491$$ 6.27193e6 1.17408 0.587040 0.809558i $$-0.300293\pi$$
0.587040 + 0.809558i $$0.300293\pi$$
$$492$$ 995520. 0.185412
$$493$$ −7.09463e6 −1.31466
$$494$$ −1.09792e6 −0.202420
$$495$$ 4.03603e6 0.740358
$$496$$ −1.88723e6 −0.344446
$$497$$ 1.38298e6 0.251144
$$498$$ 1.40808e6 0.254422
$$499$$ −3.93785e6 −0.707959 −0.353979 0.935253i $$-0.615172\pi$$
−0.353979 + 0.935253i $$0.615172\pi$$
$$500$$ 1.08326e6 0.193780
$$501$$ −724200. −0.128903
$$502$$ −5.43204e6 −0.962063
$$503$$ −7.59830e6 −1.33905 −0.669525 0.742790i $$-0.733502\pi$$
−0.669525 + 0.742790i $$0.733502\pi$$
$$504$$ 448448. 0.0786386
$$505$$ 8.33314e6 1.45405
$$506$$ −5.64480e6 −0.980104
$$507$$ −302370. −0.0522419
$$508$$ 5.05587e6 0.869234
$$509$$ −7.82664e6 −1.33900 −0.669501 0.742812i $$-0.733492\pi$$
−0.669501 + 0.742812i $$0.733492\pi$$
$$510$$ 4.89888e6 0.834010
$$511$$ 176498. 0.0299011
$$512$$ −262144. −0.0441942
$$513$$ −1.81420e6 −0.304363
$$514$$ 1.27097e6 0.212191
$$515$$ −1.10581e7 −1.83722
$$516$$ 592640. 0.0979866
$$517$$ 608832. 0.100178
$$518$$ −2.80868e6 −0.459915
$$519$$ −4.41552e6 −0.719554
$$520$$ −3.13958e6 −0.509171
$$521$$ 8.94454e6 1.44366 0.721828 0.692072i $$-0.243302\pi$$
0.721828 + 0.692072i $$0.243302\pi$$
$$522$$ 2.78335e6 0.447087
$$523$$ 4.07481e6 0.651407 0.325704 0.945472i $$-0.394399\pi$$
0.325704 + 0.945472i $$0.394399\pi$$
$$524$$ −394656. −0.0627900
$$525$$ 1.92619e6 0.305001
$$526$$ −4.20403e6 −0.662524
$$527$$ 1.07484e7 1.68584
$$528$$ −860160. −0.134275
$$529$$ 1.12037e7 1.74069
$$530$$ 1.25133e7 1.93501
$$531$$ −4.90519e6 −0.754952
$$532$$ 368480. 0.0564462
$$533$$ 3.63365e6 0.554019
$$534$$ −1.06920e6 −0.162258
$$535$$ 4.10357e6 0.619837
$$536$$ 1.11693e6 0.167924
$$537$$ −106920. −0.0160001
$$538$$ −4.75834e6 −0.708760
$$539$$ −806736. −0.119608
$$540$$ −5.18784e6 −0.765600
$$541$$ −1.18676e7 −1.74329 −0.871644 0.490140i $$-0.836946\pi$$
−0.871644 + 0.490140i $$0.836946\pi$$
$$542$$ 5.72032e6 0.836416
$$543$$ −5.46064e6 −0.794775
$$544$$ 1.49299e6 0.216302
$$545$$ −4.73542e6 −0.682915
$$546$$ −1.14464e6 −0.164319
$$547$$ −5.37801e6 −0.768516 −0.384258 0.923226i $$-0.625543\pi$$
−0.384258 + 0.923226i $$0.625543\pi$$
$$548$$ 4.85174e6 0.690155
$$549$$ −3.50007e6 −0.495616
$$550$$ 5.28326e6 0.744724
$$551$$ 2.28702e6 0.320916
$$552$$ 2.68800e6 0.375475
$$553$$ 2.10073e6 0.292117
$$554$$ −253208. −0.0350512
$$555$$ 1.20372e7 1.65880
$$556$$ 4.00938e6 0.550034
$$557$$ −5.64878e6 −0.771466 −0.385733 0.922611i $$-0.626051\pi$$
−0.385733 + 0.922611i $$0.626051\pi$$
$$558$$ −4.21678e6 −0.573318
$$559$$ 2.16314e6 0.292789
$$560$$ 1.05370e6 0.141986
$$561$$ 4.89888e6 0.657188
$$562$$ 1.98646e6 0.265301
$$563$$ 4.56407e6 0.606850 0.303425 0.952855i $$-0.401870\pi$$
0.303425 + 0.952855i $$0.401870\pi$$
$$564$$ −289920. −0.0383778
$$565$$ 734328. 0.0967763
$$566$$ 4.63367e6 0.607973
$$567$$ −188699. −0.0246497
$$568$$ −1.80634e6 −0.234924
$$569$$ 8.00165e6 1.03609 0.518047 0.855352i $$-0.326659\pi$$
0.518047 + 0.855352i $$0.326659\pi$$
$$570$$ −1.57920e6 −0.203587
$$571$$ −1.37164e7 −1.76055 −0.880275 0.474464i $$-0.842642\pi$$
−0.880275 + 0.474464i $$0.842642\pi$$
$$572$$ −3.13958e6 −0.401220
$$573$$ −5.75976e6 −0.732855
$$574$$ −1.21951e6 −0.154492
$$575$$ −1.65102e7 −2.08249
$$576$$ −585728. −0.0735597
$$577$$ 6.09797e6 0.762510 0.381255 0.924470i $$-0.375492\pi$$
0.381255 + 0.924470i $$0.375492\pi$$
$$578$$ −2.82363e6 −0.351551
$$579$$ −4.13938e6 −0.513144
$$580$$ 6.53990e6 0.807238
$$581$$ −1.72490e6 −0.211994
$$582$$ 679120. 0.0831073
$$583$$ 1.25133e7 1.52476
$$584$$ −230528. −0.0279699
$$585$$ −7.01501e6 −0.847498
$$586$$ −5.75544e6 −0.692364
$$587$$ −8.08462e6 −0.968422 −0.484211 0.874951i $$-0.660893\pi$$
−0.484211 + 0.874951i $$0.660893\pi$$
$$588$$ 384160. 0.0458214
$$589$$ −3.46484e6 −0.411524
$$590$$ −1.15255e7 −1.36310
$$591$$ −4.94946e6 −0.582893
$$592$$ 3.66848e6 0.430211
$$593$$ 1.41575e6 0.165330 0.0826649 0.996577i $$-0.473657\pi$$
0.0826649 + 0.996577i $$0.473657\pi$$
$$594$$ −5.18784e6 −0.603282
$$595$$ −6.00113e6 −0.694929
$$596$$ −969504. −0.111798
$$597$$ 5.20364e6 0.597546
$$598$$ 9.81120e6 1.12194
$$599$$ 8.75460e6 0.996941 0.498470 0.866907i $$-0.333895\pi$$
0.498470 + 0.866907i $$0.333895\pi$$
$$600$$ −2.51584e6 −0.285302
$$601$$ 8.70276e6 0.982813 0.491407 0.870930i $$-0.336483\pi$$
0.491407 + 0.870930i $$0.336483\pi$$
$$602$$ −725984. −0.0816462
$$603$$ 2.49564e6 0.279504
$$604$$ 1.99117e6 0.222083
$$605$$ −4.04502e6 −0.449296
$$606$$ −3.96816e6 −0.438942
$$607$$ −1.69578e7 −1.86809 −0.934045 0.357157i $$-0.883746\pi$$
−0.934045 + 0.357157i $$0.883746\pi$$
$$608$$ −481280. −0.0528006
$$609$$ 2.38434e6 0.260510
$$610$$ −8.22394e6 −0.894860
$$611$$ −1.05821e6 −0.114675
$$612$$ 3.33590e6 0.360027
$$613$$ 1.76743e7 1.89973 0.949866 0.312658i $$-0.101220\pi$$
0.949866 + 0.312658i $$0.101220\pi$$
$$614$$ 3.95639e6 0.423524
$$615$$ 5.22648e6 0.557213
$$616$$ 1.05370e6 0.111883
$$617$$ −9.70636e6 −1.02646 −0.513232 0.858250i $$-0.671552\pi$$
−0.513232 + 0.858250i $$0.671552\pi$$
$$618$$ 5.26576e6 0.554613
$$619$$ 1.48739e7 1.56027 0.780133 0.625613i $$-0.215151\pi$$
0.780133 + 0.625613i $$0.215151\pi$$
$$620$$ −9.90797e6 −1.03515
$$621$$ 1.62120e7 1.68697
$$622$$ 8.88202e6 0.920525
$$623$$ 1.30977e6 0.135199
$$624$$ 1.49504e6 0.153706
$$625$$ −6.59724e6 −0.675557
$$626$$ −9.32031e6 −0.950593
$$627$$ −1.57920e6 −0.160424
$$628$$ 1.21664e6 0.123101
$$629$$ −2.08931e7 −2.10561
$$630$$ 2.35435e6 0.236331
$$631$$ 1.26353e7 1.26331 0.631656 0.775248i $$-0.282375\pi$$
0.631656 + 0.775248i $$0.282375\pi$$
$$632$$ −2.74381e6 −0.273251
$$633$$ 1.83284e6 0.181809
$$634$$ −1.71017e6 −0.168972
$$635$$ 2.65433e7 2.61229
$$636$$ −5.95872e6 −0.584131
$$637$$ 1.40218e6 0.136917
$$638$$ 6.53990e6 0.636092
$$639$$ −4.03603e6 −0.391023
$$640$$ −1.37626e6 −0.132816
$$641$$ 6.23398e6 0.599267 0.299634 0.954054i $$-0.403136\pi$$
0.299634 + 0.954054i $$0.403136\pi$$
$$642$$ −1.95408e6 −0.187113
$$643$$ 1.06874e7 1.01940 0.509701 0.860352i $$-0.329756\pi$$
0.509701 + 0.860352i $$0.329756\pi$$
$$644$$ −3.29280e6 −0.312860
$$645$$ 3.11136e6 0.294477
$$646$$ 2.74104e6 0.258425
$$647$$ 1.83258e7 1.72109 0.860544 0.509376i $$-0.170124\pi$$
0.860544 + 0.509376i $$0.170124\pi$$
$$648$$ 246464. 0.0230577
$$649$$ −1.15255e7 −1.07411
$$650$$ −9.18282e6 −0.852496
$$651$$ −3.61228e6 −0.334063
$$652$$ 1.98810e6 0.183155
$$653$$ −7.28857e6 −0.668897 −0.334448 0.942414i $$-0.608550\pi$$
−0.334448 + 0.942414i $$0.608550\pi$$
$$654$$ 2.25496e6 0.206155
$$655$$ −2.07194e6 −0.188701
$$656$$ 1.59283e6 0.144514
$$657$$ −515086. −0.0465550
$$658$$ 355152. 0.0319779
$$659$$ 4.54337e6 0.407534 0.203767 0.979019i $$-0.434681\pi$$
0.203767 + 0.979019i $$0.434681\pi$$
$$660$$ −4.51584e6 −0.403533
$$661$$ −2.10021e7 −1.86964 −0.934821 0.355120i $$-0.884440\pi$$
−0.934821 + 0.355120i $$0.884440\pi$$
$$662$$ 1.58646e6 0.140697
$$663$$ −8.51472e6 −0.752292
$$664$$ 2.25293e6 0.198302
$$665$$ 1.93452e6 0.169636
$$666$$ 8.19676e6 0.716072
$$667$$ −2.04372e7 −1.77872
$$668$$ −1.15872e6 −0.100470
$$669$$ −1.27746e7 −1.10353
$$670$$ 5.86387e6 0.504658
$$671$$ −8.22394e6 −0.705137
$$672$$ −501760. −0.0428620
$$673$$ 3.46923e6 0.295253 0.147627 0.989043i $$-0.452837\pi$$
0.147627 + 0.989043i $$0.452837\pi$$
$$674$$ 1.28728e7 1.09150
$$675$$ −1.51737e7 −1.28183
$$676$$ −483792. −0.0407185
$$677$$ −1.80916e7 −1.51707 −0.758536 0.651631i $$-0.774085\pi$$
−0.758536 + 0.651631i $$0.774085\pi$$
$$678$$ −349680. −0.0292144
$$679$$ −831922. −0.0692481
$$680$$ 7.83821e6 0.650047
$$681$$ −1.28764e7 −1.06397
$$682$$ −9.90797e6 −0.815687
$$683$$ 4.67752e6 0.383675 0.191838 0.981427i $$-0.438555\pi$$
0.191838 + 0.981427i $$0.438555\pi$$
$$684$$ −1.07536e6 −0.0878848
$$685$$ 2.54717e7 2.07411
$$686$$ −470596. −0.0381802
$$687$$ 3.50936e6 0.283685
$$688$$ 948224. 0.0763730
$$689$$ −2.17493e7 −1.74541
$$690$$ 1.41120e7 1.12841
$$691$$ 1.68960e7 1.34614 0.673069 0.739579i $$-0.264976\pi$$
0.673069 + 0.739579i $$0.264976\pi$$
$$692$$ −7.06483e6 −0.560837
$$693$$ 2.35435e6 0.186225
$$694$$ −1.11207e7 −0.876466
$$695$$ 2.10492e7 1.65300
$$696$$ −3.11424e6 −0.243685
$$697$$ −9.07168e6 −0.707303
$$698$$ 1.35520e6 0.105285
$$699$$ 8.36154e6 0.647282
$$700$$ 3.08190e6 0.237725
$$701$$ 2.40964e6 0.185207 0.0926035 0.995703i $$-0.470481\pi$$
0.0926035 + 0.995703i $$0.470481\pi$$
$$702$$ 9.01696e6 0.690585
$$703$$ 6.73510e6 0.513991
$$704$$ −1.37626e6 −0.104657
$$705$$ −1.52208e6 −0.115336
$$706$$ 1.44818e6 0.109348
$$707$$ 4.86100e6 0.365744
$$708$$ 5.48832e6 0.411487
$$709$$ −5.77010e6 −0.431090 −0.215545 0.976494i $$-0.569153\pi$$
−0.215545 + 0.976494i $$0.569153\pi$$
$$710$$ −9.48326e6 −0.706012
$$711$$ −6.13070e6 −0.454816
$$712$$ −1.71072e6 −0.126468
$$713$$ 3.09624e7 2.28092
$$714$$ 2.85768e6 0.209782
$$715$$ −1.64828e7 −1.20578
$$716$$ −171072. −0.0124709
$$717$$ 7.74336e6 0.562512
$$718$$ −3.50611e6 −0.253813
$$719$$ −1.43716e7 −1.03677 −0.518385 0.855147i $$-0.673467\pi$$
−0.518385 + 0.855147i $$0.673467\pi$$
$$720$$ −3.07507e6 −0.221067
$$721$$ −6.45056e6 −0.462124
$$722$$ 9.02080e6 0.644024
$$723$$ −1.15285e7 −0.820214
$$724$$ −8.73702e6 −0.619465
$$725$$ 1.91282e7 1.35154
$$726$$ 1.92620e6 0.135631
$$727$$ −1.40705e7 −0.987353 −0.493676 0.869646i $$-0.664347\pi$$
−0.493676 + 0.869646i $$0.664347\pi$$
$$728$$ −1.83142e6 −0.128074
$$729$$ 9.93049e6 0.692073
$$730$$ −1.21027e6 −0.0840574
$$731$$ −5.40043e6 −0.373796
$$732$$ 3.91616e6 0.270136
$$733$$ −3.75000e6 −0.257793 −0.128897 0.991658i $$-0.541144\pi$$
−0.128897 + 0.991658i $$0.541144\pi$$
$$734$$ −1.19225e7 −0.816819
$$735$$ 2.01684e6 0.137706
$$736$$ 4.30080e6 0.292654
$$737$$ 5.86387e6 0.397664
$$738$$ 3.55898e6 0.240539
$$739$$ 2.61318e7 1.76019 0.880093 0.474802i $$-0.157480\pi$$
0.880093 + 0.474802i $$0.157480\pi$$
$$740$$ 1.92595e7 1.29290
$$741$$ 2.74480e6 0.183639
$$742$$ 7.29943e6 0.486720
$$743$$ −159072. −0.0105711 −0.00528557 0.999986i $$-0.501682\pi$$
−0.00528557 + 0.999986i $$0.501682\pi$$
$$744$$ 4.71808e6 0.312488
$$745$$ −5.08990e6 −0.335984
$$746$$ −1.56577e7 −1.03010
$$747$$ 5.03389e6 0.330067
$$748$$ 7.83821e6 0.512227
$$749$$ 2.39375e6 0.155910
$$750$$ −2.70816e6 −0.175801
$$751$$ −2.65311e7 −1.71654 −0.858272 0.513196i $$-0.828461\pi$$
−0.858272 + 0.513196i $$0.828461\pi$$
$$752$$ −463872. −0.0299126
$$753$$ 1.35801e7 0.872802
$$754$$ −1.13670e7 −0.728143
$$755$$ 1.04536e7 0.667421
$$756$$ −3.02624e6 −0.192575
$$757$$ −1.52032e7 −0.964260 −0.482130 0.876100i $$-0.660137\pi$$
−0.482130 + 0.876100i $$0.660137\pi$$
$$758$$ −1.44264e7 −0.911981
$$759$$ 1.41120e7 0.889169
$$760$$ −2.52672e6 −0.158680
$$761$$ 4.71380e6 0.295059 0.147530 0.989058i $$-0.452868\pi$$
0.147530 + 0.989058i $$0.452868\pi$$
$$762$$ −1.26397e7 −0.788585
$$763$$ −2.76233e6 −0.171776
$$764$$ −9.21562e6 −0.571204
$$765$$ 1.75135e7 1.08198
$$766$$ 1.06657e7 0.656779
$$767$$ 2.00324e7 1.22954
$$768$$ 655360. 0.0400938
$$769$$ −1.58977e6 −0.0969434 −0.0484717 0.998825i $$-0.515435\pi$$
−0.0484717 + 0.998825i $$0.515435\pi$$
$$770$$ 5.53190e6 0.336239
$$771$$ −3.17742e6 −0.192504
$$772$$ −6.62301e6 −0.399956
$$773$$ −9.69095e6 −0.583334 −0.291667 0.956520i $$-0.594210\pi$$
−0.291667 + 0.956520i $$0.594210\pi$$
$$774$$ 2.11869e6 0.127120
$$775$$ −2.89793e7 −1.73314
$$776$$ 1.08659e6 0.0647757
$$777$$ 7.02170e6 0.417244
$$778$$ 853464. 0.0505518
$$779$$ 2.92434e6 0.172657
$$780$$ 7.84896e6 0.461929
$$781$$ −9.48326e6 −0.556327
$$782$$ −2.44944e7 −1.43235
$$783$$ −1.87828e7 −1.09485
$$784$$ 614656. 0.0357143
$$785$$ 6.38736e6 0.369954
$$786$$ 986640. 0.0569642
$$787$$ −1.57170e6 −0.0904549 −0.0452275 0.998977i $$-0.514401\pi$$
−0.0452275 + 0.998977i $$0.514401\pi$$
$$788$$ −7.91914e6 −0.454320
$$789$$ 1.05101e7 0.601054
$$790$$ −1.44050e7 −0.821193
$$791$$ 428358. 0.0243425
$$792$$ −3.07507e6 −0.174198
$$793$$ 1.42940e7 0.807180
$$794$$ 1.63763e7 0.921860
$$795$$ −3.12833e7 −1.75547
$$796$$ 8.32582e6 0.465741
$$797$$ −2.25298e6 −0.125635 −0.0628175 0.998025i $$-0.520009\pi$$
−0.0628175 + 0.998025i $$0.520009\pi$$
$$798$$ −921200. −0.0512090
$$799$$ 2.64190e6 0.146403
$$800$$ −4.02534e6 −0.222371
$$801$$ −3.82239e6 −0.210501
$$802$$ −3.76946e6 −0.206940
$$803$$ −1.21027e6 −0.0662360
$$804$$ −2.79232e6 −0.152344
$$805$$ −1.72872e7 −0.940232
$$806$$ 1.72210e7 0.933728
$$807$$ 1.18958e7 0.643000
$$808$$ −6.34906e6 −0.342122
$$809$$ −2.37938e7 −1.27818 −0.639090 0.769132i $$-0.720689\pi$$
−0.639090 + 0.769132i $$0.720689\pi$$
$$810$$ 1.29394e6 0.0692947
$$811$$ 5.32300e6 0.284187 0.142093 0.989853i $$-0.454617\pi$$
0.142093 + 0.989853i $$0.454617\pi$$
$$812$$ 3.81494e6 0.203048
$$813$$ −1.43008e7 −0.758812
$$814$$ 1.92595e7 1.01879
$$815$$ 1.04375e7 0.550431
$$816$$ −3.73248e6 −0.196233
$$817$$ 1.74088e6 0.0912460
$$818$$ 1.93824e7 1.01280
$$819$$ −4.09209e6 −0.213174
$$820$$ 8.36237e6 0.434305
$$821$$ 1.48802e7 0.770464 0.385232 0.922820i $$-0.374121\pi$$
0.385232 + 0.922820i $$0.374121\pi$$
$$822$$ −1.21294e7 −0.626121
$$823$$ 2.00601e7 1.03236 0.516182 0.856479i $$-0.327353\pi$$
0.516182 + 0.856479i $$0.327353\pi$$
$$824$$ 8.42522e6 0.432278
$$825$$ −1.32082e7 −0.675628
$$826$$ −6.72319e6 −0.342867
$$827$$ 1.21539e7 0.617949 0.308975 0.951070i $$-0.400014\pi$$
0.308975 + 0.951070i $$0.400014\pi$$
$$828$$ 9.60960e6 0.487113
$$829$$ 3.21197e7 1.62325 0.811625 0.584179i $$-0.198583\pi$$
0.811625 + 0.584179i $$0.198583\pi$$
$$830$$ 1.18279e7 0.595952
$$831$$ 633020. 0.0317991
$$832$$ 2.39206e6 0.119802
$$833$$ −3.50066e6 −0.174798
$$834$$ −1.00234e7 −0.499001
$$835$$ −6.08328e6 −0.301941
$$836$$ −2.52672e6 −0.125038
$$837$$ 2.84559e7 1.40397
$$838$$ 6.93938e6 0.341359
$$839$$ −1.01320e6 −0.0496922 −0.0248461 0.999691i $$-0.507910\pi$$
−0.0248461 + 0.999691i $$0.507910\pi$$
$$840$$ −2.63424e6 −0.128812
$$841$$ 3.16681e6 0.154394
$$842$$ 6.60580e6 0.321104
$$843$$ −4.96614e6 −0.240686
$$844$$ 2.93254e6 0.141706
$$845$$ −2.53991e6 −0.122370
$$846$$ −1.03646e6 −0.0497884
$$847$$ −2.35960e6 −0.113013
$$848$$ −9.53395e6 −0.455285
$$849$$ −1.15842e7 −0.551564
$$850$$ 2.29256e7 1.08836
$$851$$ −6.01860e7 −2.84886
$$852$$ 4.51584e6 0.213128
$$853$$ 234824. 0.0110502 0.00552510 0.999985i $$-0.498241\pi$$
0.00552510 + 0.999985i $$0.498241\pi$$
$$854$$ −4.79730e6 −0.225088
$$855$$ −5.64564e6 −0.264118
$$856$$ −3.12653e6 −0.145840
$$857$$ 2.83802e7 1.31997 0.659985 0.751279i $$-0.270563\pi$$
0.659985 + 0.751279i $$0.270563\pi$$
$$858$$ 7.84896e6 0.363994
$$859$$ 4.00081e7 1.84997 0.924986 0.380001i $$-0.124076\pi$$
0.924986 + 0.380001i $$0.124076\pi$$
$$860$$ 4.97818e6 0.229522
$$861$$ 3.04878e6 0.140158
$$862$$ −1.65744e7 −0.759748
$$863$$ −2.08030e7 −0.950823 −0.475411 0.879764i $$-0.657701\pi$$
−0.475411 + 0.879764i $$0.657701\pi$$
$$864$$ 3.95264e6 0.180137
$$865$$ −3.70904e7 −1.68547
$$866$$ 1.21586e7 0.550922
$$867$$ 7.05907e6 0.318933
$$868$$ −5.77965e6 −0.260377
$$869$$ −1.44050e7 −0.647088
$$870$$ −1.63498e7 −0.732341
$$871$$ −1.01920e7 −0.455211
$$872$$ 3.60794e6 0.160682
$$873$$ 2.42785e6 0.107817
$$874$$ 7.89600e6 0.349646
$$875$$ 3.31750e6 0.146484
$$876$$ 576320. 0.0253748
$$877$$ 3.03559e7 1.33273 0.666367 0.745624i $$-0.267848\pi$$
0.666367 + 0.745624i $$0.267848\pi$$
$$878$$ −1.01708e7 −0.445267
$$879$$ 1.43886e7 0.628125
$$880$$ −7.22534e6 −0.314523
$$881$$ −2.58936e7 −1.12396 −0.561981 0.827150i $$-0.689961\pi$$
−0.561981 + 0.827150i $$0.689961\pi$$
$$882$$ 1.37337e6 0.0594452
$$883$$ −1.88813e7 −0.814950 −0.407475 0.913216i $$-0.633591\pi$$
−0.407475 + 0.913216i $$0.633591\pi$$
$$884$$ −1.36236e7 −0.586354
$$885$$ 2.88137e7 1.23663
$$886$$ 9.72840e6 0.416349
$$887$$ −2.34431e7 −1.00048 −0.500238 0.865888i $$-0.666754\pi$$
−0.500238 + 0.865888i $$0.666754\pi$$
$$888$$ −9.17120e6 −0.390296
$$889$$ 1.54836e7 0.657079
$$890$$ −8.98128e6 −0.380070
$$891$$ 1.29394e6 0.0546033
$$892$$ −2.04394e7 −0.860115
$$893$$ −851640. −0.0357378
$$894$$ 2.42376e6 0.101425
$$895$$ −898128. −0.0374784
$$896$$ −802816. −0.0334077
$$897$$ −2.45280e7 −1.01784
$$898$$ −7.31412e6 −0.302671
$$899$$ −3.58722e7 −1.48033
$$900$$ −8.99413e6 −0.370129
$$901$$ 5.42988e7 2.22833
$$902$$ 8.36237e6 0.342226
$$903$$ 1.81496e6 0.0740709
$$904$$ −559488. −0.0227703
$$905$$ −4.58694e7 −1.86166
$$906$$ −4.97792e6 −0.201478
$$907$$ −5.60873e6 −0.226384 −0.113192 0.993573i $$-0.536108\pi$$
−0.113192 + 0.993573i $$0.536108\pi$$
$$908$$ −2.06023e7 −0.829279
$$909$$ −1.41862e7 −0.569450
$$910$$ −9.61498e6 −0.384897
$$911$$ 2.16215e7 0.863156 0.431578 0.902076i $$-0.357957\pi$$
0.431578 + 0.902076i $$0.357957\pi$$
$$912$$ 1.20320e6 0.0479017
$$913$$ 1.18279e7 0.469602
$$914$$ −6.32252e6 −0.250337
$$915$$ 2.05598e7 0.811834
$$916$$ 5.61498e6 0.221110
$$917$$ −1.20863e6 −0.0474648
$$918$$ −2.25115e7 −0.881654
$$919$$ 4.51695e7 1.76424 0.882119 0.471028i $$-0.156117\pi$$
0.882119 + 0.471028i $$0.156117\pi$$
$$920$$ 2.25792e7 0.879506
$$921$$ −9.89098e6 −0.384229
$$922$$ −2.03842e7 −0.789706
$$923$$ 1.64828e7 0.636835
$$924$$ −2.63424e6 −0.101502
$$925$$ 5.63312e7 2.16469
$$926$$ 2.80935e7 1.07666
$$927$$ 1.88251e7 0.719512
$$928$$ −4.98278e6 −0.189934
$$929$$ −2.28729e7 −0.869524 −0.434762 0.900545i $$-0.643168\pi$$
−0.434762 + 0.900545i $$0.643168\pi$$
$$930$$ 2.47699e7 0.939112
$$931$$ 1.12847e6 0.0426693
$$932$$ 1.33785e7 0.504506
$$933$$ −2.22050e7 −0.835117
$$934$$ 1.69938e7 0.637417
$$935$$ 4.11506e7 1.53938
$$936$$ 5.34477e6 0.199406
$$937$$ −1.79616e7 −0.668336 −0.334168 0.942514i $$-0.608455\pi$$
−0.334168 + 0.942514i $$0.608455\pi$$
$$938$$ 3.42059e6 0.126939
$$939$$ 2.33008e7 0.862395
$$940$$ −2.43533e6 −0.0898955
$$941$$ −1.79697e7 −0.661558 −0.330779 0.943708i $$-0.607311\pi$$
−0.330779 + 0.943708i $$0.607311\pi$$
$$942$$ −3.04160e6 −0.111680
$$943$$ −2.61324e7 −0.956974
$$944$$ 8.78131e6 0.320722
$$945$$ −1.58878e7 −0.578740
$$946$$ 4.97818e6 0.180860
$$947$$ 4.32115e7 1.56576 0.782879 0.622174i $$-0.213750\pi$$
0.782879 + 0.622174i $$0.213750\pi$$
$$948$$ 6.85952e6 0.247898
$$949$$ 2.10357e6 0.0758213
$$950$$ −7.39028e6 −0.265676
$$951$$ 4.27542e6 0.153295
$$952$$ 4.57229e6 0.163509
$$953$$ −7.50965e6 −0.267848 −0.133924 0.990992i $$-0.542758\pi$$
−0.133924 + 0.990992i $$0.542758\pi$$
$$954$$ −2.13024e7 −0.757806
$$955$$ −4.83820e7 −1.71662
$$956$$ 1.23894e7 0.438434
$$957$$ −1.63498e7 −0.577074
$$958$$ −2.23714e6 −0.0787551
$$959$$ 1.48585e7 0.521708
$$960$$ 3.44064e6 0.120493
$$961$$ 2.57172e7 0.898288
$$962$$ −3.34749e7 −1.16622
$$963$$ −6.98584e6 −0.242746
$$964$$ −1.84456e7 −0.639293
$$965$$ −3.47708e7 −1.20198
$$966$$ 8.23200e6 0.283833
$$967$$ −1.69305e7 −0.582242 −0.291121 0.956686i $$-0.594028\pi$$
−0.291121 + 0.956686i $$0.594028\pi$$
$$968$$ 3.08192e6 0.105714
$$969$$ −6.85260e6 −0.234448
$$970$$ 5.70461e6 0.194669
$$971$$ 2.86144e7 0.973949 0.486974 0.873416i $$-0.338101\pi$$
0.486974 + 0.873416i $$0.338101\pi$$
$$972$$ 1.43915e7 0.488586
$$973$$ 1.22787e7 0.415787
$$974$$ 5.28227e6 0.178412
$$975$$ 2.29570e7 0.773400
$$976$$ 6.26586e6 0.210550
$$977$$ 3.69445e7 1.23826 0.619132 0.785287i $$-0.287485\pi$$
0.619132 + 0.785287i $$0.287485\pi$$
$$978$$ −4.97024e6 −0.166161
$$979$$ −8.98128e6 −0.299489
$$980$$ 3.22694e6 0.107331
$$981$$ 8.06148e6 0.267450
$$982$$ −2.50877e7 −0.830200
$$983$$ −3.88787e7 −1.28330 −0.641650 0.766998i $$-0.721750\pi$$
−0.641650 + 0.766998i $$0.721750\pi$$
$$984$$ −3.98208e6 −0.131106
$$985$$ −4.15755e7 −1.36536
$$986$$ 2.83785e7 0.929603
$$987$$ −887880. −0.0290109
$$988$$ 4.39168e6 0.143133
$$989$$ −1.55568e7 −0.505743
$$990$$ −1.61441e7 −0.523512
$$991$$ 2.49212e7 0.806092 0.403046 0.915180i $$-0.367951\pi$$
0.403046 + 0.915180i $$0.367951\pi$$
$$992$$ 7.54893e6 0.243560
$$993$$ −3.96616e6 −0.127643
$$994$$ −5.53190e6 −0.177586
$$995$$ 4.37106e7 1.39968
$$996$$ −5.63232e6 −0.179903
$$997$$ 1.01956e7 0.324845 0.162422 0.986721i $$-0.448069\pi$$
0.162422 + 0.986721i $$0.448069\pi$$
$$998$$ 1.57514e7 0.500603
$$999$$ −5.53138e7 −1.75356
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 14.6.a.a.1.1 1
3.2 odd 2 126.6.a.f.1.1 1
4.3 odd 2 112.6.a.c.1.1 1
5.2 odd 4 350.6.c.d.99.1 2
5.3 odd 4 350.6.c.d.99.2 2
5.4 even 2 350.6.a.i.1.1 1
7.2 even 3 98.6.c.c.67.1 2
7.3 odd 6 98.6.c.d.79.1 2
7.4 even 3 98.6.c.c.79.1 2
7.5 odd 6 98.6.c.d.67.1 2
7.6 odd 2 98.6.a.a.1.1 1
8.3 odd 2 448.6.a.l.1.1 1
8.5 even 2 448.6.a.e.1.1 1
12.11 even 2 1008.6.a.b.1.1 1
21.20 even 2 882.6.a.x.1.1 1
28.27 even 2 784.6.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
14.6.a.a.1.1 1 1.1 even 1 trivial
98.6.a.a.1.1 1 7.6 odd 2
98.6.c.c.67.1 2 7.2 even 3
98.6.c.c.79.1 2 7.4 even 3
98.6.c.d.67.1 2 7.5 odd 6
98.6.c.d.79.1 2 7.3 odd 6
112.6.a.c.1.1 1 4.3 odd 2
126.6.a.f.1.1 1 3.2 odd 2
350.6.a.i.1.1 1 5.4 even 2
350.6.c.d.99.1 2 5.2 odd 4
350.6.c.d.99.2 2 5.3 odd 4
448.6.a.e.1.1 1 8.5 even 2
448.6.a.l.1.1 1 8.3 odd 2
784.6.a.i.1.1 1 28.27 even 2
882.6.a.x.1.1 1 21.20 even 2
1008.6.a.b.1.1 1 12.11 even 2